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""3 UBRARY
2%” Michigan State
University

 

This is to certify that the
dissertation entitled

PROCESS OPTIMIZATION FOR MOIST AIR IMPINGEMENT
COOKING OF MEAT PATTIES

presented by

SANGHYUP JEONG

has been accepted towards fulﬁllment
of the requirements for the

PhD. degree in Biosystems Engineering

 

 

WM;

Cirtléjot‘lPrﬁfessor’s Signature

December 14, 2005

 

Date

MSU is an Affirmative Action/Equal Opportunity Institution

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DATE DUE DATE DUE DATE DUE
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2/05 p:/CIRC/DaleDue.indd-p.1

PROCESS OPTIMIZATION FOR MOIST AIR IMPINGEMENT COOKING OF
MEAT PATTIES

By

Sanghyup Jeong

A DISSERTATION

Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Biosystems and Agricultural Engineering

2005

ABSTRACT

PROCESS OPTIMIZATION FOR MOIST-AIR IMPINGEMENT COOKING OF
MEAT PATTIES

By
Sanghyup Jeong

Process conditions of a moist air impingement cooking system were optimized to
achieve maximum yield and to satisfy safety and quality constraints, simultaneously. To
accomplish this goal, various strategies were tested by combining different modeling
approaches, global optimization algorithms, and parameterization of control proﬁles.

In this study, a ﬁnite element model (FEM) predicting yield, Salmonella
inactivation, internal color change, and surface color change was considered as an actual
experiment with which all the results were compared. Static neural network models
(SNNM) and a dynamic neural network model (DNNM) were utilized as potential, faster
alternatives to the ﬁnite element model. For the global optimization algorithms, genetic
algorithms (GA), simulated annealing (SA), and integrated controlled random search in
dynamic system (ICRS/DS) algorithms were tested along with the ﬁnite element model
and alternative models. In addition, piecewise linear interpolation (PLI) and Fourier
series (FS) were used for the control proﬁle parameterization.

This study was conducted in two different ways. In the ﬁrst part, overall aspects
of this optimization problem and the effectiveness of the various strategies were
investigated to identify the best strategy for ideal dynamic control proﬁles. Secondly,
based on prior knowledge, the optimization strategies were applied to several

industrially-relevant case studies.

The performance of the alternative models (DNNM and SNNM) was fast, general,
and robust, with a few exceptions. Even though the accuracy and the power of
classiﬁcation were not as high as the ﬁnite element model results, the neural network
models showed potential as reliable alternative models. The highest goal (yield) was 73%,
which was obtained by using the ICRS algorithm, FEM, and PLI. However, the
optimization strategies with alternative models could not ﬁnd such a high yield; rather,
they committed critical classiﬁcation errors at the later stages of the optimization process.
Generally, all the global optimization algorithms showed convergence to an optimal
solution, albeit with different convergence speed and goal achievement. Although
comprehensive evaluation was impossible, ICRS was observed as the most
recommendable algorithm.

Single-stage, double-stage, and multi-zone processes were studied by using three
different models (FEM, DNNM, and SNNM) and the ICRS algorithm. The maximum
yield (67%) was achieved in the double-stage process. The case studies showed that a
simple and minor design change of the single-stage oven might improve the performance.

In addition, the objective ﬁanction (yield) for the single-stage oven was replaced
with a cost function, and the operating conditions for maximum proﬁt were determined,
which were different from the results when the objective function was yield. Finally,
Monte Carlo simulation showed that all the optimal proﬁles were highly sensitive to
small perturbations, which implied difﬁculties in the actual application of the optimal

solutions, due to unavoidable control errors of a cooking system.

To my family

iv

ACKNOWLEDGEMENTS

I would like to express my sincere appreciation to Dr. Bradley P. Marks, a truly
inspiring teacher and counselor; for his guidance and friendship during the course of this
investigation. Grateful appreciation is also extended to Dr. Alden M. Booren, Dr. Kirk
Dolan and Dr. James Steffe for serving on my guidance committee.

The author is indebted to the United States Department of Agriculture (USDA)
Cooperative State Research, Education, and Extension Service for supporting project
2003-51110-02081 of which this study is a part.

Thanks go to our research team members and colleagues of graduate ofﬁce 5.
Special thanks also goes to college group of the New Hope Baptist Church for their
prayer and good memories in God. Dr. Sukjung Chang, MD and his wife deserve to have
my sincere thanks for their spiritual encouragement all the way through this journey. Of
course, my family and parents have been the essence of refreshment for every new

challenge in my life.

TABLE OF CONTENTS

 

 

 

 

 

 

 

LIST OF TABLES - _ ix
LIST OF FIGURES ...... - _ xi
NOMENCLATURE- - -- - - -- - - -- - xvi
1 INTRODUCTION - - - - - - - -- -- l
1.1 Background ......................................................................................................... 1
1.1.1 Food Quality and Safety ............................................................................. 1
1.1.2 The Drive for Food Quality and Safety Innovation .................................... 3
1.1.3 Economic Signiﬁcance of the Meat Industry .............................................. 5
1.1.4 Recent Innovation in Meat Patty Processing .............................................. 6

1 .2 Objectives ........................................................................................................... 8

2 LITERATURE REVIEW - - -- - - - - 9
2.1 Achieving Extrema in Food Processing .............................................................. 9
2.2 Characteristics of Optimization in Food Processing ........................................... 9
2.3 The State of the Art of Optimization Techniques ............................................. 12
2.4 Applications in Food Process Engineering ....................................................... 14
2.4.1 Sterilization of Canned Product ................................................................ 14
2.4.2 Food Dehydration ..................................................................................... 17
2.4.3 Meat Patty Cooking .................................................................................. 19
2.4.4 Various Processing Areas ......................................................................... 20

3 THEORIES- - - - - 23
3.1 Overview of Optimization Theory .................................................................... 23

3. 1 .1 Introduction ............................................................................................... 23
3.1.2 Theory and Methods ................................................................................. 26

3.2 Global Optimization Techniques ...................................................................... 40
3.2.1 Genetic Algorithm (GA) ........................................................................... 40
3.2.2 Simulated Annealing (SA) ........................................................................ 44
3.2.3 Integrated Controlled Random Search for Dynamic Systems (ICRS/D8) 48

3.3 Artiﬁcial Neural Network ................................................................................. 53

3 .3. 1 Fundamentals ............................................................................................ 53
3.3.2 Back-Propagation F eed-Forward Neural Network (FF NN) ..................... 55
3.3.3 Generalized Regression Neural Network (GRNN) ................................... 61

4 METHODS AND PROCEDURES _- - 65
4.1 Overall Methods and Procedures ...................................................................... 65
4.2 Impingement Cooking Technology .................................................................. 65
4.3 Integrated Process Model Development ........................................................... 68

vi

4.4
4.5

4.6

4.7

4.8

5.1
5.2

5.3

5.4
5.5

5.6

 

4.3.1 Basic Finite Element Model ...................................................................... 69
4.3.2 Kinetic Parameters for Microbial Inactivation in a Meat Patty ................ 73
4.3.3 Kinetic Parameters for Internal Color Change of a Meat Patty ................ 74
4.3.4 Kinetic Parameters for Surface Color Change of a Meat Patty ................ 76
4.3.5 Practical Considerations for Integration of Models .................................. 82
Formulating the Optimization Problem ............................................................ 84
Alternative Modeling by Using Artiﬁcial Neural Networks ............................ 86
4.5.1 Parameterization of the Control Function ................................................. 86
4.5.2 Training and Validation Data Groups ....................................................... 88
4.5.3 Static Training ........................................................................................... 90
4.5.4 Dynamic Training ..................................................................................... 92
4.5.5 Neural Network Validation ....................................................................... 96
4.5.6 Neural Network Parameter Optimization ................................................. 96
Process Optimization Strategies ....................................................................... 97
4.6.1 Combinations of Techniques .................................................................... 97
4.6.2 GA Based Strategy .................................................................................... 98
4.6.3 SA Based Strategy .................................................................................... 99
4.6.4 ICRS/DSBased Strategy ........................................................................ 101
4.6.5 Benchmark Test for Optimization Algorithms ....................................... 102
Case Studies .................................................................................................... 103
4.7.1 Single-Stage Oven .................................................................................. 104
4.7.2 Double-Stage Oven ................................................................................. 104
4.7.3 Multi—Zone Oven .................................................................................... 105
4.7.4 An Example of Economic-Based Optimization ...................................... 105
Programming and Computation Tools ............................................................ 109
RESULTS AND DISCUSSION - 111
Integrated Process Model Performance .......................................................... 111
Training and Optimizing Neural Networks .................................................... 114
5.2.1 Dynamic Neural Network Model (DNNM) ............................................ 114
5.2.2 Static Neural Network Models (SNNM) ................................................ 116
Validation of the Trained Neural Networks .................................................... 119
5.3.1 The Performance of DNNM ................................................................... 119
5.3.2 The Performance of SNNM .................................................................... 130
Benchmark Test for Optimization Algorithms ............................................... 135
Optimal Conditions from the Various Strategies ............................................ 137
5.5.1 Summary of the Various Optimization Strategies .................................. 137
5.5.2 Optimization Performance of the Alternative Models Coupled with
Algorithms .............................................................................................. 142
5.5.3 Efﬁciencies of Various Strategies ........................................................... 143
5.5.4 Constraints Satisfaction Problem ............................................................ 146
5.5.5 Sensitivity of the Optimal Control Proﬁles ............................................ 149
Case Studies .................................................................................................... 152
5.6.1 Single-Stage Process ............................................................................... 152
5.6.2 Double-Stage Process ............................................................................. 157
5.6.3 Multi-Zone Process ................................................................................. 160

vii

5.6.4 Economic-Based Optimization ............................................................... 164

 

 

6 CONCLUSIONS - 167
6. 1 General Conclusions ....................................................................................... 167
6.2 Suggestions for Future Research .................................................................... 169

APPENDICES - 172

APPENDIX A Tables and Figures of Optimization Process .................................. 173
APPENDIX B Computer Codes ............................................................................. 198
REFERENCES . -- - -- ..... 257

 

viii

LIST OF TABLES

Table 2.1 Analysis of basic elements used to optimize sterilization processes of
prepackaged conduction-heated foods. ..................................................................... 16

Table 3.1 Unconstrained multivariable optimization methods categorized by how they
generate the search direction (Edgar et al., 2001 ;Venkataraman, 2002). ................. 31

Table 4.1 Comparison chart of thermal inactivation parameters of Salmonella cocktail for
ground beef (1 :(Murphy et al., 2002); 2:(Juneja et al., 2001); 3:(Murphy et al.,
2004)). ....................................................................................................................... 73

Table 4.2 Reference decimal reduction time (Dr) and z-value of internal color change in
the various muscles at the reference temperature of 60 °C (Geileskey et al., 1998). 75

Table 4.3 Digitized data of surface time-temperature plot (recipe A) and time-color
change plot (recipe C) of Dagerskog and Bengtsson (1974). ................................... 78

Table 4.4 The names and contents of data groups for training and testing neural networks.
................................................................................................................................... 89

Table 4.5 Conditions of each process type to produce training data groups; D1, single-
stage (D3), double-stage (D4), and multi-zone (D5) process. .................................. 90

Table 4.6 Optimization strategies (by strategy number), according to their process model,
the method of control proﬁle parameterization, and optimization algorithm. .......... 97

Table 5.1 The performance of DNNM for a random data group (D2 of Table 4.4), in

terms of RMSE. ...................................................................................................... 119
Table 5.2 Four possible classiﬁcation categories of the constraints satisfaction ............ 123
Table 5.3 Classiﬁcation rate of DNNM for random processes ....................................... 125

Table 5.4 The performance of DNNM for single-stage, double-stage, and multi-zone
processes in terms of RMSE. .................................................................................. 126

Table 5.5 Classiﬁcation rate of DNNM for single-stage, double-stage, and multi-zone
processes. ................................................................................................................ 127

Table 5.6 The performance (RMSE) of SNNM for random, single-stage, double-stage,
and multi-zone processes. ....................................................................................... 131

ix

Table 5.7 Classiﬁcation rate of SNNM for random, single-stage, double-stage, and multi-
zone process. ........................................................................................................... 134

Table 5.8 Comparison of the convergence of three different optimization algorithms in
terms of RMSE for the Bezier parametric curve ﬁtting problem. .......................... 135

Table 5.9 Brief results of the various optimization strategies. ....................................... 138

Table 5.10 Predicted goal and constraint values of DNNM and SNNM, compared with
the result of FEM. ................................................................................................... 142

Table 5.11 Total execution time and the achieved maximum yield for various
optimization strategies. ........................................................................................... 144

Table 5.12 Sensitivity of the optimal solutions for F EM_ICRS_PLI and FEM_SA_FS
was obtained by using Monte Carlo simulation ...................................................... 150

Table 5.13 The results of the single-stage optimization process by using ICRS-FEM.. 152

Table 5.14 The accuracy of ICRS-DNNM (strategy #26) and ICRS-SNNM (strategy #27)
were validated by using FEM results. ..................................................................... 155

Table 5.15 Maximum yield and optimal control variables were found by using ICRS-
FEM for double-stage process. ............................................................................... 158

Table 5.16 Maximum yield and optimal control variables were found by using ICRS-
FEM and ICRS-DNNM for multi-zone process. .................................................... 162

Table 5.17 Sensitivity of the optimal solutions was obtained by using Monte Carlo
simulation for economic-based optimization problem. .......................................... 166

LIST OF FIGURES

Figure 1.1 Schematic diagram of a multi-staged, moist-air impingement cooking system.

..................................................................................................................................... 7
Figure 3.1 Locating minimum point of a quadratic function with ﬁrst derivative
information ................................................................................................................ 27
Figure 3.2 Illustration of the operation of captain “A” and captain “B”. ......................... 30
Figure 3.3 Concept of linear programming problem in a two-dimensional case .............. 33
Figure 3.4 Global and local optimum in 2-dimensional case (The picture was adopted
from the help manual of MATLAB®) ....................................................................... 35
Figure 3.5 General procedure of a genetic algorithm. ...................................................... 42
Figure 3.6 An example of simple crossover operation in a genetic algorithm. ................ 43
Figure 3.7 Flowchart of modiﬁed ICRS algorithm. .......................................................... 52
Figure 3.8 Physical structure of a neuron. ........................................................................ 54
Figure 3.9 Mathematical model of biological neuron ....................................................... 55
Figure 3.10 Topology of back-propagation of feed-forward neural network. .................. 57
Figure 3.11 Architecture of generalized regression neural network (GRNN). ................. 63
Figure 3.12 GRNN response depending on the size of receptive ﬁeld (a) in two-
dimensional case. ...................................................................................................... 64
Figure 4.1 A schematic diagram of overall procedures and methods. .............................. 66
Figure 4.2 An actual and a cross-sectional image of the ISO-IV Jet Stream® Oven of
Stein-DSI (FMC FoodTech). .................................................................................... 68
Figure 4.3 Finite element mesh utilized for one quarter of the 2-D cylindrical model
product. ..................................................................................................................... 72
Figure 4.4 A sigmoid function (logistic function) of surface temperature (Ts) showing
gradual change in a certain factor around a critical value, Tdcw in this case. ............ 81
Figure 4.5 Examples of random time-varying conditions by using Fourier series control
proﬁle parameterization. ........................................................................................... 91

xi

Figure 4.6 Conceptual diagram of dynamic training paradigm. ....................................... 93

Figure 4.7 The structure of an input/output data pair for dynamic training to identify
dynamic characteristics of meat patty cooking. ........................................................ 94

Figure 4.8 Acceptance probability (Boltzmann probability) along with cooling schedule
(T 0:25, T N=O.001, N=150, and average increment of accepted objective value=0.3)
for SA. ..................................................................................................................... 100

Figure 5.1 An example of the integrated process model performance under constant
process condition: (a) Process conditions, yield, center, and surface temperature; (b)
Salmonella inactivation; (0) Internal color change; ((1) Surface color change ........ 112

Figure 5.2 An example of the integrated process model performance under dynamic
process condition: (a) Process conditions, yield, center, and surface temperature; (b)
Salmonella inactivation; (c) Internal color change; ((1) Surface color change ........ 113

Figure 5.3 Optimal “spread” values at the lowest RSME were determined by trying
“spread” for each GRNN in DNNM: (a) Yield prediction; (b) Salmonella
inactivation, internal color change, and surface color change. ............................... 115

Figure 5.4 Average normalized RMSE of each combination for the number of neurons in
each hidden layer of SNNM_R. .............................................................................. 117

Figure 5.5 RMSE (yield, Salmonella inactivation, internal color change, and surface color
change) of SNNM_D for different “spread” values. .............................................. 118

Figure 5.6 RMSE of SNNM_M for different “spread” values: (a) Yield prediction; (b)
Constraints prediction. ............................................................................................ 1 18

Figure 5.7 Example qualitative performance of the DNNM, with respect to: (a) Yield; (b)
Salmonella inactivation; (c) Internal color change; ((1) Surface color change ........ 120

Figure 5.8 The performance of DNNM for random process as a predictor and classiﬁer.
(a) Yield; (b) Salmonella inactivation; (c) Internal color change; ((1) Surface color
change (some data in NE region was not plotted). ................................................. 121

Figure 5.9 Accepted patties (170 patties) were superimposed on the 1,000 random data.
................................................................................................................................. 124

Figure 5.10 Goal and constraints prediction performance of the DNNM for single-stage
process: (a) Yield; (b) Salmonella inactivation; (c) Internal color change; ((1) Surface
color change. ........................................................................................................... 128

Figure 5.11 Goal and constraints prediction performance of the SNNM for random
Fourier process condition: (a) Yield; (b) Salmonella inactivation; (c) Internal color
change; (d) Surface color change ............................................................................ 132

xii

Figure 5.12 Examples of objective ﬁrnction value (RMSE) for every generation of three
different optimization algorithms in the Bezier parametric curve ﬁtting problem. (a)
GA; (b) SA; (c) ICRS. ............................................................................................ 136

Figure 5.13 ICRS-F EM optimization process and convergence (strategy #3): (a) PLI
parameterization; (b) F S parameterization .............................................................. 139

Figure 5.14 Optimal control proﬁle of temperature, humidity, impingement velocity, and
cooking duration found by ICRS-FEM optimization strategy (strategy #3): (a) PLI
parameterization; (b) F S parameterization .............................................................. 140

Figure 5.15 Optimal control proﬁle of temperature, humidity, impingement velocity, and
cooking duration found by SA-FEM optimization strategy (strategy #6): (a) PLI
parameterization; (b) FS parameterization .............................................................. 141

Figure 5.16 Convergence history of each control parameters shows exploration and
exploitation features of the optimization algorithms: (a) GA-FEM-PLI (strategy #1);
(b) SA-FEM-FS (strategy #6); (c) ICRS-FEM-PLI (strategy #3). ......................... 145

Figure 5.17 A typical search path committing false-pass classiﬁcation errors (ICRS-
DNNM-F S): (3) Yield; (b) Salmonella reduction; (c) Internal color change; ((1)
Surface color change. .............................................................................................. 147

Figure 5.18 Directional constraint satisfaction (DCS) algorithm prevents search from
drifting toward false-pass region (ICRS-DNNM-FS): (a) Yield; (b) Salmonella
reduction; (c) Internal color change; ((1) Surface color change. ............................. 148

Figure 5.19 Histograms of objectives and constraints were plotted from the population of
Monte Carlo simulation for the strategy #6: (a) Yield distribution of passed
simulations; (b) Salmonella inactivation distribution; (c) Internal color change
distribution; ((1) Surface color change distribution. ................................................ 151

Figure 5.20 Optimization processes for single-stage oven: (a) Process for low temperature
and high humidity optimal proﬁle; (b) Process for high temperature and low
humidity proﬁle. ..................................................................................................... 153

Figure 5.21 Optimization processes and optimal control proﬁles for double-stage oven
obtained by ICRS-FEM strategy: (a) Optimization process; (b) Optimal control
proﬁle of trial #1; (c) Optimal control proﬁle of trial #2. ....................................... 159

Figure 5.22 Optimization process and optimal control proﬁle were obtained by using
ICRS-FEM and ICRS-DNNM for multi-zone process: (a) Optimization process of
strategy #33; (b) Optimal control proﬁles of strategy #33; (c) Optimization process
of strategy #34; (d) Optimal control proﬁles of strategy #34. ................................ 163

Figure 5.23 Optimization processes and optimal control variables for maximum proﬁt of
single-stage oven were found by using ICRS-F EM strategy: (a) Convergence
history; (b) Comparison of net proﬁt and patty yield; (c) Various cost history. 165

xiii

Figure A.1 The performance of DNNM for double-stage process: (a) Yield; (b)
Salmonella inactivation; (c) Internal color change; (d) Surface color change ........ 174

Figure A.2 The performance of the DNNM for multi-zone process: (a) Yield; (b)
Salmonella inactivation; (c) Internal color change; ((1) Surface color change ........ 176

Figure A.3 The performance of the SNNM for single-stage process: (a) Yield; (b)
Salmonella inactivation; (c) Internal color change; ((1) Surface color change ........ 178

Figure A.4 The performance of the SNNM for double-stage process: (a) Yield; (b)
Salmonella inactivation; (0) Internal color change; ((1) Surface color change ........ 180

Figure A5 The performance of the SNNM for multi-zone process: (a) Yield; (b)
Salmonella inactivation; (c) Internal color change; ((1) Surface color change ........ 182

Figure A.6 GA-F EM optimization process and convergence: (a) PLI parameterization;
(b) F S parameterization ........................................................................................... 184

Figure A.7 Optimal control proﬁle of temperature, humidity, impingement velocity, and
cooking duration found by GA-FEM optimization strategy: (a) PLI
parameterization; (b) FS parameterization .............................................................. 185

Figure A.8 SA-FEM optimization process and convergence: (a) PLI parameterization; (b)
F S parameterization. ............................................................................................... 186

Figure A.9 GA-DNNM optimization process and convergence: (a) PLI parameterization;
(b) FS parameterization ........................................................................................... 187

Figure A.10 Optimal control proﬁle of temperature, humidity, impingement velocity, and
cooking duration found by GA-DNNM optimization strategy: (a) PLI
parameterization; (b) F S parameterization .............................................................. 188

Figure A.11 SA-DNNM optimization process and convergence: (3) PLI parameterization;
(b) FS parameterization ........................................................................................... 189

Figure A. 12 Optimal control proﬁle of temperature, humidity, impingement velocity, and
cooking duration found by SA-DNNM optimization strategy: (a) PLI
parameterization; (b) FS parameterization .............................................................. 190

Figure A.13 ICRS-DNNM optimization process and convergence: (a) PLI
parameterization; (b) FS parameterization .............................................................. 191

Figure A. 14 Optimal control proﬁle of temperature, humidity, impingement velocity, and
cooking duration found by ICRS-DNNM optimization strategy: (a) PLI
parameterization; (b) F S parameterization .............................................................. 192

xiv

Figure A. 15 Optimization process and optimal control proﬁles found by GA-SNNM_R
optimization strategy and Fourier series parameterization: (a) Convergence history;
(b) Optimal control proﬁle of temperature, humidity, impingement velocity, and
cooking duration. .................................................................................................... 193

Figure A.16 Optimization process and optimal control proﬁles found by SA-SNNM_R
optimization strategy and Fourier series parameterization: (a) Convergence history;
(b) Optimal control proﬁle of temperature, humidity, impingement velocity, and
cooking duration. .................................................................................................... 194

Figure A.17 Optimization process and optimal control proﬁles found by ICRS-SNNM_R
optimization strategy and Fourier series parameterization: (a) Convergence history;
(b) Optimal control proﬁle of temperature, humidity, impingement velocity, and
cooking duration. .................................................................................................... 195

Figure A.18 Two different optimization processes and optimal control proﬁles found by
single-stage ICRS-FEM optimization strategy: (a) Convergence history of case I
(LTHH); (b) Convergence history of case 11 (HTLH); (c) Optimal control proﬁle of
temperature, humidity, impingement velocity, and cooking duration of case I; ((1)
Optimal control proﬁle of temperature, humidity, impingement velocity, and
cooking duration of case 11. .................................................................................... 196

XV

NOMENCLATURE

Roman Letters:

§rm\a5m

laseesreeckos

5
as

a parameter of a sigmoid activation function

color value of redness

coefﬁcient of Fourier series

kth coefﬁcients of Fourier series

color value of yellowness

concentration, [kg/m3]

surface color change

cooling rate

mass capacity, [kg/kg]

speciﬁc heat capacity, [J/(kg °C)]

decimal reduction time, [s]

capillary diffusivity, [mz/s]

fat capillary diffusivity, [mz/s]

expected value

activation energy, [J/gmol]

failure counter in ICRS/D8

nonlinear activation ﬁmction, or probability density function
humidity (% moisture by volume), [%MV]

output of hidden layer neuron, or enthalpy, [kJ/kg]
convective mass transfer coefﬁcient, [m/s]

convective heat transfer coefﬁcient, [W/(m2 °C)]
internal color change at center

objective function, or performance index

Boltzmann constant, or inactivation rate constant, [l/s]
heuristic parameter of ICRS/DS

heuristic parameter of ICRS/DS

moisture conductivity, [kgmoism/(m 5)]

thermal conductivity, [W/(m °C)]

color value of lightness

log cycles of microbial destruction

a multiplier, or moisture or fat content-decimal dry basis
number of control variables

ﬁnal averaged fat content based on non-fat solids, [kg/kg]

initial fat content based on non-fat solids, [kg/kg]
ﬁnal average water content based on non-fat solids, [kg/kg]

initial water content based on non-fat solids, [kg/kg]

neighborhood function, number of discretization, total number of cooling cycle or
number of microorganism, [CFU/g]

normal to surface

heuristic parameter in ICRS/DS

xvi

SN~<‘<§<V~1§‘“~JM‘>‘UV
D1 gm

parameters of control functions, or probability
ideal gas constant, [J/(kmol K)]

random number, or r-direction in radial coordinates
step size

temperature, [°C]

time, [3]

total execution time, [s]

dew point temperature, [°C]

impingement exit velocity, [m/s]

sum of neuron

weight

output of neuron

patty yield, [%]

vertical direction in cylindrical coordinates, or microbial z-value, [°C]
total color change

Greek Letters:

geompmﬁgm—IQ

search direction

vector of random numbers (Gaussian distribution)

error, or small number

efﬁciency of algorithm [% yield/s]

learning rate constant

time coordinates of control proﬁle parameterization grid
minimum interval, or latent heat [kJ/kg]

decision variables vector

density, [kg/m3]

standard deviation vector, or constant for the size of receptive ﬁeld
control coordinates of control proﬁle parameterization grid

Bold style (vector or matrix):

x€:~’~:

output vector of individual neuron
target vector

control vector, or training vector
weight matrix

input column vector, or control vector

xvii

Superscript:

HL hidden layer

i ith iteration , point, or initial state
j jth neuron
k kth iteration, or time step, or neuron

OL output layer

Subscript:

0 initial state

c critical value

f ﬁnal state

L lower limit

11 number of step

r reference state

3 surface

U upper limit

Abbreviations:

ANN artiﬁcial neural network

BB branch and bound

FEM ﬁnite element model

FFNN feed-forward neural network

GA genetic algorithm

GO global optimization

GRG generalized reduced gradient

GRNN generalized regression neural network
ICRS/DS integrated controlled random search for dynamic system
LP linear programming

MILP mixed integer linear programming
MINLP mixed integer nonlinear programming
MIP mixed integer programming

NLP nonlinear programming

OMb oxymyoglobin

PLI piecewise linear interpolation

QP quadratic programming

RBF radial basis-function, or radial basis-function neural network
RMSE root mean squared error

SA simulated annealing

SLP successive linear programming

SSE sum of squared error

xviii

1 INTRODUCTION
1.1 Background

1.1.] Food Quality and Safety

Food processing technology has been advanced along with human history.
However, the fundamental concept of food processing has not changed considerably,
even though many innovative processes and products are being developed. Basically,
manufacturing of foods encompasses two types of conversions, physical and chemical.
The main purposes of these conversions are preserving and improving the quality of
processed food products. One of the predominant unit operations, inducing both physical
and chemical conversions, is heat or thermal processing.

Thermal processing is transferring heat to a food material to induce desirable
results, but this also can cause concurrent undesirable results. Thermal processing
increases digestibility (e.g., protein denaturation), reduces enzyme and microorganism
activity, and enhances food characteristics (e.g., carbohydrate gelatinization, color
development, texture, and ﬂavor changes). However, thermal processing also produces
undesirable results, such as loss of heat sensitive nutrients and undesirable color and
ﬂavor changes due to over-cooking. Therefore, a compromise must be found between
intensity of thermal processing and its various effects on the product (Trystram, 2004).

According to the results of a recent survey published in “Trends In The United
States - Consumer Attitudes and The Supermarket 1999”(FMI, 1999), the top food
selection concerns and the percentages of the shopping public that consider these factors
“very important” in their food selection were as follows: (1) Taste, (92% of those

interviewed), (2) Nutrition, (70%), (3) Product Safety (70%), (4) Price, (63%), (5)
1

Storability, (42%); (6) Ease of Preparation, (35%); (7) Food Preparation Time (35%) and
(8) Product packaging that can be recycled, (29%). Therefore, food manufacturing must
address not just production volume, but also a variety of other competing criteria.

For the consumer’s food safety concern, the USDA Economic Research Service
(ERS) reports that the percent of consumers “completely conﬁdent” in the safety of the
food supply increased from a low of 72% in 1992-93 to a high of 83% in 1996, with
levels declining to 74% in 2000 (ERS, 2002).

From the economic viewpoint, food safety is the most critical and nonnegotiable
quality factor. In the United States, foodbome diseases have been estimated to cause 6
million to 81 million illnesses and up to 9,000 deaths each year (Mead et al., 1999). For
six speciﬁc bacterial pathogens, the costs of human illness are estimated to be $9.3-12.9
billion annually (Buzby et al., 1996). In 2000, ERS estimated the annual costsl due to
selected foodbome pathogens2 as $6.9 billion (ERS, 2000). These estimated costs are an
enormous burden for society and also for food manufacturers.

Therefore, it is clear that the major sectors (i.e., consumers, industry, and

regulatory agencies) in the food market must collaborate to improve the current situation.

 

' includes medical costs, productivity losses, and costs of premature deaths

2 Campylobacter (all serotypes), Salmonella (nontyphoidal), E. coli 0157, E. coli non-0157 STEC, and
Listeria monocytogenes

1.1.2 The Drive for Food Quality and Safety Innovation

Driving forces can be passive or active. Since the 1993 outbreak of E. coli
0157:H7, consumer awareness and demand for food safety has increased (Golan et al.,
2004). The passive driving forces often come from foodbome illness outbreaks and
recalls3, and they trigger consumer awareness, regulatory changes, or industrial
innovation.

“The number and size of recalls have increased dramatically over the last decade.
During 1993-96, the number of meat and poultry Class I4 recalls averaged about 24 per
year and amounted to 1.5 million pounds annually; during 1997-2000, Class I recalls
averaged 41 per year and reached 24 million pounds annually (Ollinger and Ballenger,
2003).” These increasing recall cases are not because of loose control, but because the
ability to detect pathogens on products has increased dramatically, which can generate
more recalls (AMI, 2002). In addition, the ability to track foodbome disease and tie it to a
speciﬁc food product has evolved into a practical technology (AMI, 2002). Recalls result
in bad reputation and catastrophic ﬁnancial damage to a manufacturer.

Therefore, efﬁcient quality assurance has become a critical issue for consumers,
manufacturers, and related government organizations. Instrumentation, food safety
practices, and lethality criteria are of central importance, with particular emphasis on very

high sanitary and hygienic operating standards. Evolving federal regulations, such as

 

3 A food recall is a voluntary action by a manufacturer or distributor to protect the public from products that
may cause health problems or possible death. The purpose of a recall is to remove meat or poultry from
commerce when there is reason to believe it may be adulterated (injurious to health or unﬁt for human
consumption) or misbranded (false or misleading labeling and/or packaging) (F SIS, “F SIS Recalls”, USDA,
http://www.fsis.usda.gov/Fsis_Recalls/index.asp, March 22, 2005.).

4 Recalls that involve meat or poultry products that could, especially without cooking to safe temperature,
cause serious illness or death.

9CFR318.17 (F SIS, 1999 & 2001), change safety regulations from passive to active
compliance required of the food industry. Traditionally, regulations have provided a
speciﬁc endpoint temperature and holding time to achieve target lethality in a meat and
poultry product. Thus, the traditional approach discourages the food industry from
adopting new technology and voluntary compliance to the regulations. However, the
evolving regulations require manufacturers to prove, via scientiﬁcally supportable means,
that their process or Operating policy achieves a target lethality performance standard.
The transition from command-and-control to performance standards allows more
freedom of choosing process design and operation policy, but also moves more
responsibility to the industry.

Contrary to the above passive driving forces, there might be an active driving
force that originates from industry. Food manufacturers invest in the development of new
methodologies to improve safety and quality of their food product. When industry
successfully innovates to produce safe foods, a win-win situation arises, with the
innovating ﬁrm, consumers, and government all beneﬁting from improved food safety
(Golan et al., 2004).

Many attempts have been made to maximize desirable quality and simultaneously
minimize undesirable effects by adding ingredients, developing innovative process
equipment, improving process conditions, and so forth. Among those approaches,
improving process conditions is advantageous, in that it uses existing systems. Therefore,
additional capital investment is not necessary to resolve these contradicting factors to
achieve both safety requirements and maximize quality and proﬁt. Finding the best

operating condition is critical from the perspective of industry, because the need to

improve efﬁciency, reduce energy consumption, increase productivity, and comply with
regulations pushes industry to adopt improved safety practices if they can see

simultaneous beneﬁts in yield, which translate to proﬁt.

1.1.3 Economic Significance of the Meat Industry

In spite of the increased safety concerns, the US. meat and poultry industry
contributes signiﬁcantly to the US. agricultural economy. Total meat and poultry
production in 2000 exceeded 80 billion pounds, a 31 percent increase since 1987. The
meat and poultry industry is the largest segment of US. agriculture, contributing over
$100 billion in annual sales to the GNP (AMI, 2001).

The products affected by regulatory changes related to ready-to-eat products
account for over $28 billion in annual sales (F SIS, 2001), and consumer trends for ready-
to-eat products also suggest continued rapid growth in this category. The size of this
market is important as a spur to greater proﬁt in this industry. Accordingly, the industry
aims to develop novel products and to increase the efﬁciency of its production lines. For
an example, even a modest 0.5% improvement of cooking yields based on the $28 billion
annual sales in this category would give an impact of approximately $140 million
increase in annual revenue for ready-to-eat products in the US.

Therefore, given the regulatory changes and the economic importance of ready-
to-eat (RTE) meat products, there are compelling needs for integrated simulation tools
that will allow industry to design and operate processes that meet the lethality
performance standards and simultaneously increase quality and proﬁt. Optimization
techniques that ﬁnd the conditions for the best result from a given situation are needed to

meet these demands. Process optimization is the most economic approach for industry to

maximize yields while ensuring safety and quality factors with existing facilities.
Therefore, the information and tools that will enable the industry to design and operate

the optimal processes are essential.

I. 1.4 Recent Innovation in Meat Patty Processing

Impingement cooking technology has been popular in certain segments of the
food processing industry, because of its efﬁciency. Speciﬁcally, moist air impingement
cooking systems are widely used in the ready-to-eat meat product industry, because the
system (Figure 1.1) results in short cooking time and relatively high cooking yield.

The moist air impingement cooking systems jets a steam-air mixture through
arrays of nozzles onto products, yielding a high heat transfer rate by reducing the
thickness of the boundary layer at the surface of the product. Also, at the initial stage of
cooking, steam is condensed on the surface of the product, which results in effective
transfer of latent heat into the product at low temperature. Therefore, moist air
impingement cooking systems are characterized by fast cooking and suppression of
moisture loss. Moist air impingement cooking systems involve many control variables,
such as cooking duration, air temperature, air moisture content, impingement exit

velocity, and impingement geometry (e.g., jet width, spacing, and height).

Impingement unit \

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Figure 1.1 Schematic diagram of a multi-staged, moist-air impingement cooking
system.

Currently, most oven operators select the oven operating conditions ‘(i.e., the
control variables) based on their experience or simple rules of thumb, which are not
necessarily proven scientiﬁcally. For a single-stage oven, one might operate the oven
within sub-optimal conditions. However, if multiple ovens (Figure 1.1) are connected to
increase the rate of production with various cooking zones, then the complexity and size
of the problem is too large to select optimal conditions with experience. Also, meat
cooking under moist air impingement cooking environments involves multiple control
Variables, mass transfer coupled with heat transfer, phase transitions, and complex

cOndensing boundary conditions, which results in complex combinations of control

PrOﬁles.

Therefore, optimizing the process conditions of moist air impingement cooking
systems is a signiﬁcant challenge and a worthwhile endeavor from the perspective of

researchers, food processors, oven manufacturers, and government agencies.

1.2 Objectives

The overall goal of this study was to develop an efﬁcient process optimization
method, in terms of speed and objective-achievement, and to evaluate the method for
maximizing cooking yield, while ensuring the microbial safety and quality of ready-to-eat
meat products (ground and formed meat and poultry products) cooked in commercial
moist air impingement cooking systems. To achieve the overall goal, the speciﬁc

objectives were:

1. To add color prediction capabilities to an existing ﬁnite element model.

2. To develop an alternative process model for moist air impingement cooking of
meat patties by using artiﬁcial neural network (ANN) to replace an existing
ﬁnite element model.

3. To identify the best strategy for process optimization among many
combinations of optimization algorithms, process models, and
parameterization of control functions.

4. To apply the developed optimization strategies to three case studies: single-
stage, double-stage, and multi-zone oven systems.

5. To examine the performance of the optimization strategy for a single-stage

oven system, given an economic-based objective function.

2 LITERATURE REVIEW

2.1 Achieving Extrema in Food Processing

An ultimate goal of most production activity is to increase proﬁtability at given
conditions, and food processing is no exception. Typically, improving the efﬁciency,
reducing process time, and increasing yield and quality are concerns for most food
processors and equipment designers. Generally, a typical industry consists of
management, process design and equipment speciﬁcation, and plant operations (Edgar et
al., 2001). Because these components are inter-connected, achieving those improvements
are not simple tasks. Therefore, depending on the level of complexity and difﬁculty, the
scope of a problem can be the entire enterprise, 3 plant, a process, a single unit operation,
a single piece of equipment in that operation, or any intermediate stage between these
(Beveridge and Schechter, 1970).

Generally, these improving activities are maximizing the capabilities of existing
facilities or equipments by changing their conventional operating policies, conditions,
numbers, and so on. All these attempts can be described in a single word: “optimization.”
A more formal deﬁnition would be “the collective process of ﬁnding the set of conditions
required to achieve the best result from a given situation (Beveridge and Schechter,

1970).”

2-2 Characteristics of Optimization in Food Processing

Food processing is unique in that the process involves materials having irregular
Shapes, non-homogeneous compositions, and individual variance even in the same

material. In addition to the materials, quality factors for product evaluation can be

subjective, such as taste, aroma, and ﬂavor. Also, food processing encompasses various
techniques, such as frying, baking, boiling, blanching, fermenting, drying, and so on.
Therefore, modeling food processing phenomena is very challenging.

Models are essential components of modern process systems engineering (i. e.,
simulation, optimization, and control), and they are usually classiﬁed into three
categories, which are ﬁrst-principle models (or white-box), data-driven models (or black-
box), and hybrid models (or gray-box) (Banga et al., 2003). Without adequate models, it
is impossible to carry out optimization. From the view point of the application of
optimization technique, the ﬁrst-principle models are highly desirable, because the
response time of the models is short, and it is convenient to apply various mathematical
operations, such as differentiation. However, because ﬁrst-principle models are difﬁcult
to obtain, data-driven models and hybrid models are popular in food processing.
Generally, mathematical modeling of a food process requires knowledge of transport
phenomena and reaction kinetics. Transport phenomena involve heat, mass, and
momentum transfer into a food, and reaction kinetics cover degradation or inactivation of
nutritional and organoleptic factors or microbial and enzymatic activity (Oliveira and
Oliveira, 1999). Usually, these multi-physical phenomena are expressed as sets of
algebraic, partial, and ordinary differential equations in the mathematical model. Due to
the lack of theoretical methods for solving those highly complex, nonlinear systems of
equations, most of the problems are solved by using numerical techniques, such as the
ﬁnite difference method and the ﬁnite element method. Considering that most

0Immization techniques require numerous iterations of a process model, the numerical,

10

computational requirements of the model are often signiﬁcant impediments to
optimization.

One of the popular methods to overcome the above difﬁculties of modeling is
alternative modeling, which can be fast, simple, and robust, with reliable accuracy. Even
though alternative models sacriﬁce some degree of accuracy, compared to numerical
models, the overall beneﬁts of the alternative modeling techniques must be considered
when evaluating performance in an actual optimization problem.

Artiﬁcial neural networks (ANN) have emerged as a potential alternative to
physical-based models for food process engineering, because of their simple structure,
robustness, no requirement of prior knowledge, and adaptive performance (Torrecilla et
al., 2004). ANN have been successfully applied to the modeling of food processes (Mittal
and Zhang, 2000; Sablani and Shayya, 2001; Chen and Ramaswamy, 2002; Chen and
Ramaswamy, 2003; Horiuchi et al., 2004; Torrecilla et al., 2004), property and quality
prediction (Berg et al., 1997; Xie and Xiong, 1999; Raptis et al., 2000; Albert et al.,

2001; Therdthai and Zhou, 2001; Tominaga et al., 2001; Hussain et al., 2002; Boillereaux
et al., 2003; Ganjyal et al., 2003), machine vision and image analysis (Chao et al., 2002;
Marique et al., 2003; Diaz et al., 2004), extrusion (Ganjyal and Hanna, 2002), microbial
growth and inactivation (Geeraerd et al., 1998; Garcia-Gimeno et al., 2003), high-
pressure processes (Torrecilla et al., 2005), and ﬂuid ﬂow (Adhikari and Jindal, 2000;
Sablani and Shayya, 2003; Singh and Jindal, 2003). Once an ANN is established, the
Computation time of the network is very small with reliable accuracy. ANN is ideal for

SyStem identiﬁcation and replacement of an existing ﬁrst-principle model. For example, it

11

was shown that ANN were very effective for replacing a ﬁnite difference computer
simulation of retort processes (Chen and Ramaswamy, 2002).

Another difﬁculty in food processing optimization arises at the characteristics of
the response space of a problem. If the response space has just a single unique maximum
(or minimum), ﬁnding the extremum can be guaranteed. However, food processing
models are generally characterized as a system of nonlinear partial differential algebraic
equations, which usually exhibit a multimodal nature (Banga et al., 2003). In this
situation, an effective and systematic procedure (or algorithm) is essential for

optimization.

2.3 The State of the Art of Optimization Techniques

In the previous section, characteristics of food processing were discussed by
focusing on model related issues. However, to solve practical optimization problems,
effective techniques that are capable of consistently ﬁnding the best solution to the
problems must be available.

A popular optimization technique is nonlinear programming (NLP, Section 3.1.2),
which is usually using gradient information to decide the search direction (e.g., steepest
ascent path). If NLP is applied to highly nonlinear, constrained, and multimodal problems,
it usually converges to the “nearest” local solutions, because its search direction and size

are determined from its starting point (Edgar et al., 2001). Therefore, NLP cannot

guarantee a global solution”, if the starting point is not close enough to the global

x

5
l"3f€=l‘s to local maxima or mrnrma in a section of the entire solution space

6
reft'tl's to the highest or the lowest point among other local maxima or minima.

12

optimum. However, the local nature of NLP algorithms can be overcome by global
optimization (GO, Section 3.1.2.5) techniques designed to ﬁnd a global solution.

GO algorithms are designed to escape from local solutions and explore promising
regions where a global solution may exist. Also, GO can be utilized with ANN in parallel,
without any modiﬁcations to the process model. In food process engineering, G0 has
been used with numerical models, such as ﬁnite difference models (F DM), ﬁnite element
models (FEM), and ANN (Chen and Ramaswamy, 2002). Banga et al. (2001) and
Zorrilla et al. (2003) coupled a FDM model of double-sided cooking of meat patties
directly with the Integrated Controlled Random Search for Dynamic Systems (ICRS/D8),
which is an adaptive stochastic GO algorithm.

Response surface methodology (RSM7) is currently the most popular optimization
technique in food science, because of its comprehensive method, reasonably high
efﬁciency, visualization, and simplicity, even though RSM is inefﬁcient and cannot be
automated in ﬁnding the overall optimum (Arteaga et al., 1994). Also, Banga et al.
(2003) pointed out some important drawbacks of RSM methods, such as the empirical,
local, and stationary nature of these statistical techniques. RSM has uncertainty of model
equations, which means the model might not represent a real physical model, because
RSM is a statistically designed experimental optimization method. Generally, the method

is not considered as a formal optimization technique. However, RSM has been

\

7 RSM uses quantitative data from an appropriate experimental design to determine and simultaneously
solve multivariate problems. The equations describe the effect of the test variables on the response,
determine interrelationships among test variables, and represent the combined effect of all test variables in
the I‘CSponse. This approach enables an experimenter to make efﬁcient exploration of a process or system
(PORCIano S. Madamba, “The response surface methodology: an application to optimize dehydration
operations of selected agricultural crops”, Lebensm.-Wiss. u.-Technol., v. 35, p. 584)

13

successfully applied to product development and static process identiﬁcation when the

system behavior or the process is unknown, complicated, and static.
2.4 Applications in Food Process Engineering

2. 4.1 Sterilization of Canned Product

Optimization techniques have been rigorously applied to sterilization of
prepackaged conduction-heated food, such as retorting of canned foods. The basic form
of this application is to ﬁnd the best combination of retort temperature and process time
for a constant heating process or the best retort time-temperature history for optimal
control, which can simultaneously achieve the required lethality of the target

microorganism and maximize quality factors.

The ﬁrst attempt by Teixeria et al. (1969) found the best combination of retort
temperature and process time for a constant retort process of conduction-heated foods.
The best set of time and temperature was found by plotting thiamine retention against
equivalent process conditions producing the same level of lethality. Even though the
attempt introduced optimization concepts to food process engineering, the study did not
apply formal optimization methods to the problem.

Saguy and Karel (1979) applied Pontryagin’s maximum principle (PMP) to
maximize thiamine retention in retort process and found a single optimal variable retort
temperature proﬁle. The signiﬁcance of this study was the ﬁrst application of formal
0Inimization theory to food process engineering. However, because the application of
PMP requires quite a modiﬁcation of the original problem, it might not be suitable for a

ComPlex problem.

14

Numerous attempts have been made since the work of Teixeira et al. (1969) to
optimize retort operation. However, the essential features of those optimization problems
have many similar aspects (Table 2.1). The popular objectives of sterilizing prepackaged
conduction-heated foods were maximizing retention of a single nutrient, minimizing
quality degradation, and minimizing energy consumption. Most of the prior studies dealt
with microbial lethality as an inequality constraint and a two-dimensional numerical

model (heat conduction only) as an equality constraint.

Although there was similarity in the formulation of the problem among these
many studies, the major difference among them was the method used to ﬁnd the optimal
condition or proﬁle. Application of formal optimization methods has been increasing, so
that several studies (Saguy and Karel, 1979; Nadkami and Hatton, 1985; Banga et al.,
1991; Chalabi et al., 1999; Kleis and Sachs, 2000; Erdogdu, 2002; Erdogdu and Balaban,
2003) found optimal retort solutions, with respect to speciﬁc assumptions. The type of
Optimal solutions can be categorized into: (a) combinations of process time and constant
temperature, (b) piecewise continuous temperature proﬁles, and (c) on-off type of control
proﬁles. Even though the piecewise continuous temperature proﬁle is the true optimal
solution, the solution cannot always be considered as the best, because it is not practically

possible to implement a continuous proﬁle in many conventional processes.

The biggest advantage in applying process optimization to retorts is the relatively
simple process model, as compared with unpackaged food processes. Usually, the retort
optimization encompasses a single control variable, temperature, which limits the burden
of computation. Also, the process model does not involve mass transfer, which is a very

Complex phenomenon in meat cooking process.

15

Table 2.1 Analysis of basic elements used to optimize sterilization processes of
prepackaged conduction-heated foods.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

References 0b 'ective function a Constraints Optimization Optimal
’ IC 1’ EC " method Solution ‘
Teixeira et al. Max. (thiamine
0 969) retention) VAL 2D FDM GS (T, t)opt.
Teixeira et al. Max. (thiamine VAL
(1975) retention) TL / TU 2D FDM GS “Tim"
Saguy and Karel Max. (thiamine VAL 2D FDM PMP 5:133:33
(1979) retention) TL / TU proﬁle
(T, t)opt w/
Min. (surface & volume Comparison linear
Ohlsson (1980) averaged cook value) Fc 2D FDM (diagrams) come-up-
time
Chart
. . (Unsteady- Comparison Constant
aggro et al. :glﬁnzg) Fc state heat (time- time &
p conduction temperature) temperature
)
2D
Nadkrani and . . VAL . On-off
Ha tton (1985) Max. (nutrient retention) Tl. / Tu numerical PMDP control
solution
Ban a e t a 1 Max. (nutrient and VAL
(199%) ' surface quality retention) Fc 2D FDM ICRS/DS VRT
Min. (process time)
F Step
. c Davis, Swann function w/
Silva et al. (1992) M“ .(surface ““31”” Tend 1D FDM and Campey linear
retentlon)
tend method come-up-
time
. 2D Open-loop
815913;” et al. Max. (nutrient retention) Fc Theoretical optimal Eggglrang
2D F DM control
. Max. (vitamins FC
22:03:)?“ Sachs retention) Min. ( energy Tl. / Tu 1D FEM SQP f(T,t)
consumption) 1‘93!
Fc Equidistant
Erdogdu (2002) Max. (thiamine) Tl/ Tu mm) “mm“ ramp
FDM method .
Tend function
Max. (thiamine of two FC Complex Piecewise
Erdogdu and . . method .
Balaban (2003) different objects) — TL/ Tu FDM Wei ghtin continuous
multi-objective Tend method g proﬁle

 

 

IC: inequality constraints; EC: equality constraints; TL: lower temperature limit; TU: upper temperature
limit.
2 Max., maximizing; Min., minimizing

VAL, volume averaged lethality; Fc, critical lethality

C

d FDM, ﬁnite difference method; FEM, ﬁnite element method
GS, graphical search; PMP, Pontryagin’s maximum principle; PMDP, Pontryagin’s minimum

distributed principle; ICRS/DS; Integrated Controlled Random Search for Dynamic Search; SQP,
Sequential quadratic programming
e . .
VRT, variable retort temperature; opt., optimum

16

Chen and Ramaswamy (2002) developed an ANN model with training and testing
data produced by a ﬁnite element model, and coupled the ANN model with GA. In that
study, the control function (i. e., retort temperature) was parameterized with sine and
exponential functions to replace the traditional constant retort process. The coupled
ANN-GA model was able to identify the relationship between the operating variables and
control function parameters. Even though the ANN -GA found the optimal processing
condition of a retort process, the optimal solution could be improved if additional
parameters for the control function were used to increased ﬂexibility. Also, the work of
Chen and Ramaswamy (2002) was for prepackaged food retorting, a simpler process

model than convection cooking of meat patties.

2. 4.2 Food Dehydration

Dehydration is a common method to extend the shelf life of foods. This operation
is normally removing water in a foodstuff- via evaporation or sublimation (Brennan,
1990). Among many techniques, such as freeze drying, spray drying, super-heated drying,
inﬁared drying, microwave drying, heated air drying of a solid food block is the focus of
this review, because of the physical similarities with meat patty cooking under moist air
impingement cooking. Using heated air is the typical method of dehydration, in which a
food product is placed in contact with a moving stream of heated air. Therefore, drying
can be an optimization problem, in which the optimal conditions are sought to maximize
retention of nutrients, such as ascorbic acid, while minimizing enzyme or microbial
activities (Banga and Singh, 1994).

Drying processes are generally driven by evaporation at the surface, which causes

Water transport within a foodstuff. F ick’s equation of diffusion is often used to describe

l7

water transport in drying. For hot air drying, the mass transfer model must be coupled
with a heat transfer model, often assuming a one-dimensional thin slab and an
evaporation term to account for the energy balance. These partial differential equations
are usually solved by the ﬁnite difference method or rarely by existing analytical
solutions with empirical equations for model parameters, such as diffusion coefﬁcient,
convective heat transfer coefﬁcient, and ﬁrst-order rate constant for ascorbic acid
degradation (Mishkin et al., 1982). Food drying quality parameters are often modeled by
using ﬁrst-order reaction kinetics (Mishkin et al., 1983; Banga and Singh, 1994).

Mishkin et al. (1982) used Pontryagin’s maximum principle (PMP) and the
complex method to ﬁnd the optimal air temperature proﬁle maximizing ascorbic acid
retention in a model system (a slab composed of water, cellulose, and ascorbic acid) with
fixed relative humidity. The complex method was selected, because the method was
convenient to use along with any type of process model and constraints without
modiﬁcation. Mishkin et al. (1983) extended their research to a multi-stage drying
process. They found optimal stepwise temperature and humidity proﬁles for three stages
by using the complex method.

Banga and Singh (1994) set up four different optimization problems for drying of
a thin slab of cellulose: (a) maximizing ascorbic acid retention with a constraint on the
ﬁnal moisture content, using air dry bulb temperature as the control variable; (b)
minimizing process time with ﬁnal retention of ascorbic acid, using air dry bulb
temperature control; (c) maximizing ascorbic acid retention with ﬁnal retention of
enzyme, using dry bulb temperature and relative humidity control; (d) maximizing energy

efﬁciency with ﬁnal ascorbic acid retention, with dry bulb temperature control. These

18

problems were solved by using an ICRS/DS algorithm, and they found optimal piecewise
linear control proﬁles in all cases.

In spite of many similarities with meat cooking processes under moist air
condition, drying processes are different from meat cooking, in that cooking involves
more complex mass transfer phenomena (water-fat mixing and fat dripping), microbial

inactivation, phase change, and air humidity sufﬁciently high to cause condensation.

2. 4.3 Meat Patty Cooking

The abundance of prior research in retort operation is due to the availability of the
process model, which encompasses relatively simple phenomena, such as depletion of
certain nutrients coupled with heat transfer. However, the nature of meat patty cooking is
an unpackaged process that generally involves various mass transfer phenomena, such as
evaporation and dripping of fat and water. In addition, geometry change and phase
transition is typical for this process. These phenomena must be coupled with heat transfer
and solved. Other difﬁculties in the modeling of meat patty cooking arise in the heating
medium. The heating medium in retort processes is water or steam, which does not
interact with the food material in the package and has simple thermal properties.
However, in an unpackaged food product, the heating medium interacts with the surface
of the food material, which therefore leads to more elaborate boundary conditions.

Banga et al. (2001) and Zorrilla et al. (2003) applied the dynamic optimization
technique (Section 3.1.2.6) to contact cooking of meat patties, which is considered as the
first attempt to optimize a meat patty cooking process. The objective of those studies was
to minimize cooking loss of patties, while ensuring inactivation of E. coli 0157:H7 and

ﬁnal product center temperature. One-dimensional coupled heat and mass transfer solved

l9

with ﬁnite difference method (Pan, 1998) was used as a process model. The above
optimization problem was solved by a global optimization algorithm, ICRS/DS
(Integrated Controlled Random Search for Dynamic Systems), which found the optimal
piecewise step grill surface temperature proﬁle (Banga et al., 2001; Zorrilla et al., 2003).
The process model is an important part of an optimization problem, because the
model is used to predict the objective value and constraints. The process model
developed by Pan, (1998) and used by Banga et al. (2001) and Zorrilla et al. (2003),
assumed the patty as a one dimensional inﬁnite slab, which is less accurate than two-
dimensional modeling. Even though the model could predict water and fat transfer and
microbial log reduction, other important quality factors, such as internal color change and
surface color change, were not considered. Also, the nature of the cooking method was
contact cooking, which was modeled using simpliﬁed, effective boundary conditions,
compared with the condensing-convective boundary conditions during meat patty
cooking under a moist air environment. In their optimization problem, grill surface
temperature was the single decision variable. However, moist-air impingement meat patty
cooking involves the additional control variables of air humidity, impingement velocity,
and impingement geometry. Even though the performance of ICRS/DS was good enough
to locate the global optimum for the contact cooking problem, comparison with other GO

methods was not conducted.

2. 4.4 Various Processing Areas

In addition to the application for canning, cooking, and drying, optimization
techniques have been applied to the other processes, such as ultra-ﬁltration, baking,

extrusion, cheese manufacturing, mixing, and so forth. Each application provides some

20

lesson about the implementation of formal optimization techniques to the unique nature
of various food processes.

Usually, the objectives of wine ﬁltration are to minimize colloid content,
maximize color intensity, and maximize ﬂux by varying pore size and the recycle rate of
membranes (Gergely et al., 2003). Gergely et al. (2003) expressed the objective ﬁrnction
as a second-order form of regression functions of the membrane pore size and recycle
ﬂow rate, which was a response surface model (RSM). Even though the RSM can solve
some Optimization problems, generally the method is not recognized as a formal
optimization method (Section 2.4).

In optimizing commercial bread baking, the biggest challenge is getting a reliable
process model. Therdthai et al. (2002) used four different temperature zones of a
commercial oven and baking time to ﬁnd optimal condition for minimizing loss while
controlling the top crust color, side crust color, and average crust color within acceptable
ranges. Statistical methods were used to construct a model equation, which was a RSM.
However, some researchers have used neural networks coupled with a Genetic Algorithm
(GA) for leavening process optimization in a bread-making industrial plant (Fravolini et
al., 2003). Fravolini et al. (2003) used a nonlinear system identiﬁcation method, called
NARMA (Nonlinear Autoregressive-Moving Average), to model the leavening process.

Extrusion is very complex process. The most common objectives are expansion
ratio, shearing strength, and sensory texture, which are functions of temperature, feed
moisture, and process variables, such as screw speed, screw compression ratio, feed
speed, and die diameter. RSM is the most popular method to model the process and to

apply optimization methods, because the results of RSM are analytically differentiable

21

mathematical expressions (Chévez-Jéuregui et al., 2000); Frazier et al., 1983; Iwe et al.,
1998; Karwe and Godavarti, 1997; Olkku et al., 1983; Quintero-Ramos et al., 1998;
Vainionpaa, 1991). Therefore, optimal condition could be found by applying theoretical
optimization techniques directly to the model.

In addition to applications for food processes, well-organized dynamic
optimization problems also exist in biochemical processes, such as fermentation
(Tartakovsky et al., 1995; Banga et al., 1997; Berber et al., 1998; Tsoneva et al., 1998;
Faqir, 1998);Lee et al. 1999; Radhakrishnan et al., 1999; Halsall-Whitney et al., 2003;
Levisauskas et al., 2003). The nature of biochemical processes is similar to food
processes, in that their dynamic behavior is inherently nonlinear. However, most
biochemical systems can be modeled by a set of ordinary differential equations, which
makes many optimization theories applicable, because the model equations are
differentiable. Generally, biochemical processes have been treated as problems of optimal

control, which use a special form of the performance function.

22

3 THEORIES

3.1 Overview of Optimization Theory
3.1.] Introduction

3. 1 .1.1 What is Optimization?

Our daily life is always full of choices in various activities, such as traveling,
shopping, or eating. Decision—making is the cognitive process of selecting a course of
action from among multiple alternatives. The purpose of decision-making is to choose the
best alternatives ﬁtting with our goals. Even though the purpose of decision-making is
clear, making decisions involves many considerations and uncertainties. In the case of a
simple problem, we can guess the results of some trials or test the effect of each possible
alternative. However, as the number of variables and the interactions between variables
increase, these attempts lose the ability of identifying the best solution among the many
possible good solutions. For example, we can increase the thickness of insulation to
decrease energy loss, but increased insulation thickness increases cost, which is the trade-
off. In that case, the problem is to ﬁnd the thickness of insulation minimizing the total
cost, which cannot be solved easily.

Therefore, systematic methods and procedures are necessary to solve those
problems containing many variables, restrictions, and factors, which compete with each
other for the best solution. Optimization can be deﬁned as “the collective process of
ﬁnding the set of conditions required to achieve the best result from a given situation”
(Beveridge and Schechter, 1970). By the help of optimization techniques, we can explore

more complex decision-making situations with more certainty and effectiveness.

23

3.1.1.2 Essential Features of Optimization Problems

By observing optimization problems, some common features can be found. For

example, if a driver wants to travel from city “A” to city “B” in the least time, then the

minimum traveling time will be the primary objective for the driver. To accomplish this

goal, the driver has to consider the speed limit of each state, weather conditions, physical

limits of driver and car, paths, and so on.

The basic components of an optimization problem are the objective function,

decision variables, constraints, and mathematical model. These components are well

explained by Evans (1982).

1.

Objective function (performance function, or cost function): This is the quantity
to be maximized (or minimized). It is often referred to as the cost or performance
function. Whether measured in dollars, efﬁciency, or other terms, the performance
function evaluates alternative solutions to the problem to determine which one is
the best.

Decision variables: These are the parameters in the process or system that can be
adjusted to improve the objective. They are the free or independent variables that
must be speciﬁed in the traditional case-study approach to problem-solving.
Constraints: All optimization problem have constraints on the allowed solutions.
These constraints may limit values of the decision variables or of other dependent
variables that describe the behavior of the system.

Mathematical Model: If the problem is to be solved other than by trial-and-error
physical experimentation, we must have a model of the system. The model is the

mathematical representation of the system that determines the objective function

24

 

in terms of the decision variables. It also determines other dependent variables

that may be subject to constraints.

Now, we can deﬁne optimization by using the terms described above. Therefore,
optimization is ﬁnding a set of conditions maximizing or minimizing the objective

function while satisfying the imposed constraints of the problem.

3.1.1.3 Solving Optimization Problems

To solve an optimization problem, one must analyze the essential features of
problem. This means that the objective and constraints of the problem should be
identiﬁed ﬁrst, but not necessarily in the form of mathematical expressions. Then the
objective function must be expressed in terms of the system variables by using
mathematical expressions. Interrelationships, internal restrictions, and system models
must be expressed in mathematical form. Now, the problem is reconstructed according to
the essential components of a formal optimization problem. Upon this mathematically
redeﬁned optimization problem, an appropriate optimization method and algorithm can

be applied to obtain the optimal conditions to achieve the objective.

3 . 1.1.4 Hierarchy of Optimization Problems

Optimization can be employed at any level in a plant, ranging from a small piece of
equipment to management of a whole company. Someone may need more workers to
maximize the production rate. However, from the management perspective, this
0Dtirnization result might not be favorable, because of labor cost. Therefore, the scope of

Optimization is important, because one level of optimization does not guarantee the

25

 

optimality of another level of a system. Typical industrial company optimization
encompasses three levels: management, process design and equipment speciﬁcation, and
plant operation (Edgar et al., 2001). The highest level of optimization of a company is the
management to maximize net proﬁt. To accomplish the objective, someone would need
to model the whole company, including every piece of equipment, which is practically
impossible. Therefore, the scope of an optimization problem must be reviewed before

proceeding to the next step.
3.1.2 Theory and Methods

3.1.2.1 Basic Concepts of Optimization
The basic concepts of optimization can be clearly illustrated by observing an
unconstrained one-dimensional case. “Unconstrained” means that there are no equality or
inequality constraints, so that the independent or dependent variables can be simulated
without any limitations. Let’s take a simple arithmetic example. If we deﬁne a quadratic
ﬁrnction y with an independent variable x and coefﬁcients a, b, and c, then:
y=f(x)=ax2+bx+c [3,1]
The objective is to ﬁnd x minimizing or maximizing the function value. One might plot
the function to see the actual shape of the curve and ﬁnd a maximum or minimum point
graphically (Figure 3.1). However, there is a mathematical tool to see the feature of the
Curve, which is the ﬁrst derivative 0») information obtained by differentiating the above

equation with respect to the variable x. This is shown in Equation [3.2].

y'=§%r—)=2ax+b [3.21

26

f(X)

‘—y=f(x)=ax2+bx+c

 

 

df(x)/dx “ ‘

 

0 -b/2a

 

 

Figure 3.1 Locating minimum point of a quadratic function with first derivative
information.

27

 

This ﬁrst derivative information implies that there is an extreme point between the
sign changes of the gradient (Equation [3.2]), which is an inﬂection point. The inﬂection
point (stationary point) of zero gradient can be calculated by setting the right hand side of
Equation [3.2] equal to zero. Thus, the stationary point is x= —b/2a. If the x value is
inserted to the Equation [3.1], then we can have an extreme value

of y = (—b2 + 4ac) /(4a). However, we still do not know whether the value is maximum or

minimum if we do not have graphical information. To determine this, the second

derivative information of the Equation [3.1] is necessary. The second derivative (y") is:
y" = 2a [3.3]

If the second derivative is negative, then the extreme value is a maximum or vice versa.
Hence, a necessary condition for a minimum or maximum of f(x) is that the ﬁrst
derivative of f(x) becomes zero at x. However, the necessary condition does not tell
whether the extremum is the minimum or maximum. So, we need the second derivative
information as a sufﬁciency condition. Now, we can identify a minimum or maximum of
f(x) mathematically with the necessary and sufﬁcient conditions.

The above case is very simple, but the basic concepts can be extended to the
multivariable cases with some help of mathematical techniques. Let f(xl, x2,...,x,,) be an n-
dimensional function. To meet the necessary condition, the ﬁrst derivative of each

Variable x1, x2...x,. must be zero as follow.

 

af(x)=6f(x)=,,,=i(x_)=0 [34]
6x1 6x2 a'xn .

28

A set of variables satisfying the set of Equations [3.4] represents an extreme
point. The sufﬁcient condition for the extreme point can be checked with the nature of the
matrix of second partial derivatives (Hessian matrix) of f(xl, x2,...,x,.) at the point, which
is the extension of Equation [3.3] to the multivariable case. If all the objective functions
of our problems were quadratic and differentiable, then life would be simple. However,
this analytical method has many limitations in real life applications. Therefore, we need a
more general approach to solve optimization problems.

Before addressing this issue, imagining a movie scene will give us insight for the
generalized optimization method. Let’s imagine a situation that two special soldiers
(captain “A” and captain “B”) are dropped from an airplane into an enemy region in a
night having no moon light (Figure 3.2). Their mission is to ﬁnd the highest point from
sea level and communicate with the nearby resistance. Their landing location is different
from each other, so that they have to complete the mission individually. Captain A has
the map of the region, compass, and GPS (global positioning system), but captain B lost
his equipment because of a tough landing. Captain A saw the map and located the highest
point and also his current location with GPS. Then he set the direction toward the highest
point and ﬁnally got to the point. However, captain B does not have any information or

equipment, except a small ﬂashlight in the complete darkness, which means he only can
get terrain features a few yards around him. How could he get to the highest point? So, he
marked his current location on the ground and looked around with the ﬂashlight. He
found that the steepness of terrain was increasing in the northeast direction. So, he kept
moving a few yards from his current location to the next location in this direction if there

Was an increase of elevation. He repeated this strategy until he could observe no more

29

increase of elevation. Fortunately, he reached the same location where captain A arrived,
because there was only one peak in the region. However, if there were several peaks, it

could take more time or he might be stranded at a sub-peak.

 

 

 

 

/: I : : 1: i \
' f I ' j 3 ' 7 ' Sea Level

 

Sea Level

Figure 3.2 Illustration of the operation of captain “A” and captain “B”.

In the above story, our focus is captain B’s approach, because his situation is very

Slmilar to ours, like ﬁnding a maximum or minimum within a region that we cannot

30

 

ﬁgure out, which means no analytically differentiable mathematical expressions. Captain
B’s approach gives us a clue for setting up a general rule for ﬁnding such an extreme
point in an unknown region. There are three factors to determine the next point from the
current point. Current point (15"), search direction (of), and step size (5") are needed to
determine the next point (xw). Also, the relationship can be expressed mathematically as

following:

xk“ =xk +a" -S" [3.5]

The above iterative search procedure stops based on some criteria. If there is no
signiﬁcant improvement, the search stops. There are many methods to perform the
search; however, the differences between methods are mainly in how they generate the
search direction. A popular method is using the ﬁrst derivative information. However,
function value, and ﬁnite difference approximation are also used in lieu of derivatives

(Edgar et al., 2001). Table 3.1 shows some methods according to their approaches.

Table 3.1 Unconstrained multivariable optimization methods categorized by how
they generate the search direction (Edgar et al., 2001;Venkataraman, 2002).

 

Function values only Gradient based

 

@irect Method) (Indirect Method)
Random Search Steepest Descent
Grid Search Conjugate Gradient
Univariate Search Davidon-Fletcher—Powell Method (DF P)
Simplex Search Broydon-Fletcher-Goldfarb-Shanno Method (BFGS)

Conjugate Search

Until now, we discussed handling unconstrained function optimization. However,
the real life optimization problem always has equality or inequality constraints. The
Constraints have to be handled so that the constrained optimization problem can be

Converted into an unconstrained optimization problem. The basic idea is to set the -

31

objective function free from constraints by using mathematical manipulations of the
constraints.

Let us ﬁrst look at the handling of equality constraints. Three methods are mainly
used for the solution of such problems: direct substitution, constrained variation, and
Lagrange multipliers.

Direct substitution is to substitute equality constraints to the objective function
when the equality constraints are all linear equations. Then, the constraints vanish, and
the problem can be handled as an unconstrained case.

Constrained optimization subjected to inequality constraints is treated as in the
case of equality constrained problem after transformation of inequality constraints to
equality ones by introducing a slack variable, which is a buffer between the original
inequality constraint and the transformed equality constraint.

Therefore, a general approach to solve a constrained optimization problem is to
convert the constrained problem into an unconstrained problem. Then, unconstrained

multivariable optimization techniques (Table 3.1) can be used to ﬁnd optimal solution.

3.1.2.2 Linear Programming

A linear programming8 (LP) problem is one in which the objective and all of the
constraints are linear functions of the decision variables, so that the linear constraints
(lines in 2-dimensional case) form boundaries. Therefore, an objective function,
represented as a line, is moving through the bounded region to ﬁnd an optimal set of

variables to get an extreme value in a two-dimensional case (Figure 3.3).

 

8 “The word programming here does not refer to computer programming, but means optimization.”
Edgar, T. F ., D. M. Himmelblau and L. S. Lasdon (2001). Optimization of chemical processes. New York,
McGraw-Hill.

32

X2 (Constraint)

if Maximum
\

’ Constraints

   
 
 
   
   
       

Feasible region

  
 

 

   

'

“\\\
\\ f (Objective Function)
§\\\ \\ ‘ £7 X1 (Constraint)

Constraints

Figure 3.3 Concept of linear programming problem in a two-dimensional case.

One of the important characteristics of LP is that the extremum of a linear
program always occurs at a vertex or corner of the system boundaries, which is the
intersection of constraints in the feasible region (Beveridge and Schechter, 1970). The
basic idea is a systematic examination of these boundaries, which is converting a set of
constraints into a set of equality equations and applying linear algebra and matrix
manipulation (Venkataraman, 2002). For instance, the simplex algorithm is designed to
explore one intersection after another to the direction of improving the objective function,
according to a set of rules, until the best objective function attainable is found (Beveridge

and Schechter, 1970).

33

3.1.2.3 Nonlinear Programming

Nonlinear programming (NLP) problems contain at least one nonlinear equation
of the objective function or constraints (Venkataraman, 2002). No single optimization
algorithm can possibly be efﬁcient or even successful in all cases of interest. However,
unique techniques are well developed for each speciﬁc case. For instance, a nonlinear
objective function with linear equality constraints can be handled with the direct
substitution method, which solves the objective function explicitly for one variable and
eliminates that variable from the problem formulation (Edgar et al., 2001). A nonlinear
objective function with linear inequality constraints can be solved by using Kuhn-Tucker
conditions9 and Lagrange multipliers"). If an optimization problem has a quadratic
objective function and linear inequality or equality constraints, quadratic programming
(QP) can be used (Edgar et al., 2001). Another strategy to solve nonlinear optimization
problems is to replace all nonlinear functions in the problem with their Taylor series
approximations and apply linear programming, which is called successive linear
programming (SLP) (Edgar et al., 2001).

The most robust and generally accepted nonlinear optimization technique is the
generalized reduced gradient (GRG) method, which is also implemented as the “Solver”
in the spreadsheet program Microsoft Excel (Edgar et al., 2001). The concept of GRG is

to reduce the dimension of the ﬁrst derivative of the objective ﬁrnction by using the ﬁrst

 

9 At any local constrained optimum, no (small) allowable change in the problem variables can improve the
value of the objective function (Edgar, T. F., D. M. Himmelblau and L. S. Lasdon (2001). thimization of
chemical proces_s_e§. New York, McGraw-Hill.). This is a generalization of the Lagrange multiplier method.

'0 By introducing an unknown scalar variable to the constraints, a linear combination is formed, which

reduces a constrained problem into an unconstrained problem (Venkataraman, P. (2002). Applied
optimization with matlab proggamming. New York, John Wiley & Sons).

34

derivative of constraints. All major NLP algorithms are based on estimation of ﬁrst
derivatives of the problem to obtain a solution and to evaluate the optimality conditions
(necessary and sufﬁcient conditions) (Edgar et al., 2001). Because NLP is mainly based
on the gradient information, the solution has a local nature, which means the solution is a
local maximum or minimum, but not necessarily the global one in general cases of

nonlinear optimization problems (Figure 3.4).

Global maximum

Local maximum

, ﬂ"
l’ I,
’0’:

g:::

 

Figure 3.4 Global and local optimum in 2-dimensiona1 case (The picture was
adopted from the help manual of MATLAB®).

35

3.1.2.4 Mixed-Integer Programming

Another special type of optimization problem is mixed integer programming
(MIP). As the name implies, this type of problem includes integer variables that are not
continuous. The integer variable is common in plant operation, design, location, and
scheduling (Edgar et al., 2001). For example, number of equipment, yes-no decisions,
and number of stages are integer variables. If the objective function and constraints are
linear in a MIP, then it is called mixed-integer linear programming (MILP). And if the
MIP involves nonlinear objective and constraints, then it is called mixed-integer
nonlinear programming (MINLP) (Edgar et al., 2001). If the number of integer variables
is small, then exhaustive or complete enumeration is possible. However, the effort grows
exponentially to examine all possible solutions (V enkataraman, 2002). Therefore,
systematic ways are necessary to get to the optimal solution with less effort and time and
also to handle large scale problems.

One popular solution for this kind of the problem is branch and bound (BB). The
basic concept of BB is systematically ﬁnding the closest set of discrete variables from the
optimum set of continuous variables. To do this, a relaxation technique is used to convert
the discrete variables into continuous variables bounded by their maximum and minimum
value. BB uses two strategies to ﬁnd the best set of integer variables. Branching is the
efﬁcient way of covering the feasible region by several smaller feasible sub-regions, and
bounding is comparing and selecting the sub-region having its upper bound that is less

than the lower bound of any other sub-region.

36

3.1.2.5 Global Optimization

We have discussed unconstrained, constrained multi-dimensional cases, and some
special cases, such as LP, NLP, MILP, and MINLP. One of the underlying assumptions
for optimality is that the problem must be convex, which means there is only one
minimum or maximum. Thus, if the convex condition is not met, then the above methods
can be stranded at a local optimum, not at a global optimum (Figure 3.4). This situation
frequently happens when the model has nonlinear equality constraints, such as a
nonlinear material balance, nonlinear physical property relations, nonlinear process
models, and so on (Edgar et al., 2001).

In addition, if a gradient-based method is involved, then the search method can
become stranded around the vicinity of local minima or maxima, without guarantee of
optimality. However, in spite of this fundamental limitation, there are several effective
and practical optimization techniques that explore inﬁnite solution spaces systematically
with minimum effort and high possibility of locating the global optimum. This type of
optimization is called global optimization (GO).

In a G0 problem, a local solution can be considered as a global optimum if the
local solution is the best among many local solutions or if multiple search trials from
different starting points reach to the same local solution. Widely used GO methods can be
classiﬁed as deterministic (exact) or stochastic strategies (heuristic) (Banga et al., 2003).

Deterministic global optimization methods are based on the systematic gradient-
based local search methods following the systematically divided sub-regions of attraction

until they meet their termination criteria. This type of method can guarantee global

37

optimum in certain problems, but not in a general GO problem. Branch-and-boundll
methods, methods based on interval arithmetic (Kearfott, 1996), and multi-start methods
are classiﬁed as deterministic GO techniques (Edgar et al., 2001). Even though
deterministic GO methods have sound theoretical convergence properties, the associated
computational effort increases very rapidly (often exponentially) with the problem size
(Banga et al., 2003).

Stochastic GO methods do not follow systematically divided regions of attraction.
Instead, they set their search direction based on a logic found in natural processes, such
as cooling of metals and genetic evolution processes. Many stochastic GO methods can
locate the vicinity of global solutions with relatively good efﬁciency, but the downside is
that global optimality cannot be guaranteed (Banga et al., 2003). Scatter search, tabu
search, simulated annealing, and genetic and evolutionary methods can be classiﬁed as
stochastic GO methods (Edgar et al., 2001). Stochastic GO methods are applicable to
almost any problem, without modiﬁcation of the original process model (Edgar et al.,
2001). In the case of Genetic and evolutionary GO methods, the method produces a
population that is a set of solutions and keeps updating the population with improved
solutions according to the rule of biological processes of crossover and mutation.
Although stochastic GO cannot guarantee a global optimum, it is widely adapted to solve

real life problems, because of its relative simplicity and robustness.

 

" Branch and bound algorithms are a variety of adaptive partition strategies that have been proposed to
solve global optimization models. These are based upon partition, sampling, and subsequent lower and
upper bounding procedures: these operations are applied iteratively to the collection of active ('candidate')
subsets within the feasible set D. Their exhaustive search feature is guaranteed in similar spirit to the
analogous integer linear programming methodology (Eric W. Weisstein et al. "Branch and Bound
Algorithm." From Math World--A Wolfram Web Resource.
http://mathworld.wolfram.com/BranchandBoundAlgorithm.html, March 22, 2005).

38

3.1.2.6 Static and Dynamic Optimization

In the previous sections, we have discussed how to solve optimization problems
in each special case, especially when all the constraints are algebraic equations. In other
words, the problem is not changing with time. For instance, ﬁnding the optimum
diameter of a pressure vessel does not involve a time factor. However, what if the
constraints have time varying equations, such as ordinary differential equations in which
a variable is changing with respect to time? This kind of optimization problem can be
called “dynamic” optimization, as opposed to the “static” optimization problem. To be
precise, this type of problem is generally referred to as an optimal-control problem,
which is the area of control. However, some optimization techniques can be employed to
solve such problems.

There are two general approaches. The ﬁrst approach is discretization of the
control function, which can be understood as dividing the control function into pieces.
This method is replacing a differential term by the ﬁrst order Eulerian difference

expression, which is:

g z 36(0) - x(tl—l)
dt At

 

[3.6]

Now the ordinary differential equation can be expressed as a set of algebraic
equality constraints, which is the standard form of a constrained nonlinear programming
technique.

The second method is parameterization of the control function. We need an
inﬁnite number of points to express a varying control function, which is not practical.

However, if the control ﬁlnction is expressed with Fourier series or a simple polynomial

39

form, the inﬁnite solution space can be explored with a ﬁnite number of parameters. By
applying NLP, the optimal set of parameters can be obtained as the optimal control

function, even though there is no guarantee of optimality.

3.2 Global Optimization Techniques

3. 2. 1 Genetic Algorithm (GA)

Genetic algorithms (GA) are well-known stochastic global optimization
algorithms based on biological evolution theory. Holland (1962) was the ﬁrst to use the
technique, but its use as an optimization tool began in earnest in the late 19805,
developed momentum in the mid-19908, and continues to attract serious interest today
(Venkataraman, 2002). Genetic algorithms have been successfully applied to a wide
range of problems, such as engineering design, scheduling, signal processing, optimal
control, transportation, and so on.

The biological evolution theory is the combination of Darwin's theory of natural
selection and Mendel's theory of genetics. According to the theory, a chromosome in a
gene pool is modiﬁed by simple rules of genetics, such as crossover and mutation. The
biological system having the gene that is the ﬁttest among others to the environment
survives and produces the next generation, having individuals with more desirable
characteristics. These biological concepts are implemented in genetic algorithms via
numerical operations. For example, a vector of design variables is considered as a
chromosome in genetic algorithms. The following is a description of genetic algorithm
datails; the explanations are largely adopted from the work of Venkataraman (2002).

Figure 3.5 shows the general procedures of a genetic algorithm. The initial

population is constructed with a certain number of design vectors that take higher rank in

40

terms of a performance index or an objective function value, in the case of an
unconstrained problem among randomly produced individuals”. For the constrained
problem, these individuals must satisfy the speciﬁc constraints. The initial population
size usually remains the same throughout the whole generation. However, there are no
speciﬁc criteria to determine the optimal initial population size (Venkataraman, 2002).
Next generation candidates are now produced by using selected parents and
genetic operators. A crossover genetic operator exchanges a piece of the chromosome of
each parent. A simple crossover exchanges a piece of chromosome in the same location
of each parent, and the ratio of the piece is generated randomly at each operation (Figure
3.6). Arithmetic crossover uses a linear combination of parent chromosomes to produce

two children.

 

'2 This refers to a vector of design variables. A piece of the chromosome or a portion of a design vector is
called as allele.

41

 

 

Set up an initial
population

Evaluation & Parents selection

t

Perform Genetic Operations

 

 

 

 

 

 

 

 

Crossover Mutation Immigration

 

 

 

 

 

 

 

 

 

 

 

 

Reevaluation of the population
(Fitness)

 

 

 

 

 

Perform competitive selection

 

 

 

 

 

< New generation. >

  
 
     
 
    

  

Check if generation No
reached to a target # of

generations

 

Yes

C Terminate D

 

Figure 3.5 General procedure of a genetic algorithm.

42

-------

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

* i 3" =iii¥iliifiii :
g x5 i i 3’5 i i X5 l l 3’5 g
gysi in:
gyvé in:
E xs i E Y8 g i X3 3 i YB 3
Parents Children

Figure 3.6 An example of simple crossover operation in a genetic algorithm.

The children are deﬁned as:

C1 = XIX-I- le

[3.7]
C2 = 12X + lilY

where, Ii] +12 =1; 11,12 > 0

X, Y: parents

C1, C2: children
The parameter A, and A; are generated randomly. Mutation selects a design variable
randomly and replaces it with a randomly generated value. Generally, these genetic
operators, crossover and mutation, narrow a search to a promising region that might have

a local extremum (Venkataraman, 2002). However, it is hard for a global optimization

algorithm to ﬁnd a global optimum if it is stranded around a local solution. Another

43

feature of GA is immigration, in which a set of randomly generated, unbiased population
is added to the existing population before the selection is made for the next generation.
By using this feature, GA can reduce the possibility of being trapped in a local solution
and effectively explore an entire solution space.

The newly generated population now must be evaluated to identify suitable
parents for the next generation. A simple method of selecting parents is ranking or sorting
the individuals of the population according to their objective function value. A fraction of
the best individuals can be used for reproduction of the next generation. In a strategy
called tournament selection, the remaining portion of the population, excluding the
fraction of the best individuals, is fed back to the next generation as new immigrants.

The above genetic operation and ﬁtness evaluation are performed until the
generation number reaches a certain number or the solution does not show any
improvement. GA is very useful for handling ill-behaved, discontinuous, and
nondifferentiable problems, because the algorithm generates a possible solution group at
each iteration, instead of a search direction, and is effective for handling continuous

problem (Venkataraman, 2002).

3.2.2 Simulated Annealing (SA)

Annealing is a heat treatment technique to alter material characteristic by
removing internal stresses and crystal defects. When a material is in molten state, atoms
can move freely before they form crystal structure. However, if a molten material is
cooled down rapidly, atoms lose chances to form crystal structure, which creates internal

stresses and crystal defects. Therefore, the rapidly cooled material is in a higher energy

44

state than the material cooled slowly. The cooling schedule is also called the annealing
schedule, which reduces the surrounding temperature from the critical temperature to a
sufﬁciently lower temperature stepoby-step. In each cooling step, the material being
treated is allowed to cool in the ﬁrmace until it reaches thermal equilibrium with its
surrounding temperature. By using the technique, internal energy of a material can be
minimized, which means stable crystalline order. Usually, annealing is used to soften a
material and make it more ductile, to relieve residual stresses, and to reﬁne the crystal
structure (Shigley and Mischke, 1989).

The above ideas can be applied to ﬁnding the global optimum in an optimization
problem, which has become popular for combinatorial optimization” problems. The
molten state of the material can be an initial design space in which every combination of
variables has equal opportunity to be searched out to ﬁnd the best combination. The
objective is to reduce the internal energy level to the lowest state, which is crystalline
order. As the annealing temperature goes down, the mobility of atoms decreases, and the
total internal energy level goes down at the same time, which means that the search
direction is being ﬁxed to a direction without considering other opportunities. When the
temperature reaches the ﬁnal scheduled temperature, atoms form crystalline structure,
which represents the lowest energy level, a global minimum.

The above analogy is realized by the basic SA procedure (Floquet et al., 1994)
summarized by Edgar et al. (2001) as follows:

0 Choose an initial solution x , an initial temperature T, a lower limit of temperature

T Low, and an inner iteration limit L.

 

'3 A method to search for the best possible solution out of very large number of discrete feasible solutions.

45

0 While (DTLow), ('10

O

O

O

Fork= 1, 2,..., L, do

Make a random choice of an element x' e N (56) , where N( x) = x + a - S
(or: search direction; S: stepsize)

Move_value= f (x') - f (x)

If move_value S 0 (downhill move), set x = x'

If move_value > 0 (uphill move), set x = x' with probability p = exp(-
move_value / T) > r (uniformly distributed random number in [0, 1])
(Kirkpatrick et al. , 1983)

End inner loop

0 Reduce temperature according to an annealing schedule. An example is new

T =C’T 0, where 0 < c < 1 (c is the rate of annealing schedule).

0 End temperature loop

Basically, the algorithm is more supportive of a solution that improves the

objective function, while permitting adverse solutions to give potential for the algorithm

to discover the global optimum and to escape a local optimum (Venkataraman, 2002).

Some of the critical parameters are the perturbation method of neighbor selection,

transitional probability,14 and the rate of annealing schedule (c).

To obtain a neighbor state of X0, a search direction and a stepsize must be

determined. Given a neighborhood structure, simulated annealing can be viewed as an

algorithm that continuously attempts to transform the current conﬁguration into one of its

 

" Probability to accept a worse solution.

46

neighbors. Practically, the calculation of the search direction and stepsize could be
determined by using the traditional one-dimensional case (Venkataraman, 2002).

For the transitional probability ﬁrnction, Metropolis algorithm15 and Glauber
algorithm16 are related to Boltzman probability distribution17 (Edgar et al., 2001).

The way in which the temperature is decreased is known as the cooling schedule.
The rate of annealing schedule must be appropriate for its application so as not to get
trapped in a local minimum due to fast cooling. The annealing rate can be ﬁxed to a value
or can vary in each step of cooling (Laarhoven and Aarts, 1987). For the algorithm to be
effective, it is recommended that the probability be in the range of 0.5 S p S 0.9
(Venkataraman, 2002).

This mechanism is mathematically best described by means of a Markov chain”,
a sequence of trial, where the outcome of each trial depends only on the outcome of the
previous one (Laarhoven and Aarts, 1987). An example of a Markov chain is a random
walk").

A Simulated Annealing program consists of a pair of nested DO-loops. The outer-
most loop sets the temperature, and the inner-most loop runs a Metropolis Monte Carlo

simulation at that temperature.

 

'5 p = exp (-move_value / T)
MP = exp I-move_value / T) / (l + exp (-move_value / T)

‘7 p = -k / T (k: Boltzmann constant; T: annealing temperature)

‘8 A collection of random variables {X,} (where the index t runs through 0, 1,...) having the property that,
given the present, the ﬁrture is conditionally independent of the past. In other words, P( X, = j l X0 = to, X]
= i1, Xr) = i)- 1) = P( X, = jl X,- I = i,-1). (Eric W. Weisstein. "Markov Chain." From Math World--A
Wolfram Web Resource. http://mathworld.wolfram.com/MarkovChain.html, May 18, 2005)

19 . . . . .
Idea of taking successrve steps in a random direction.

47

Like other discrete optimization techniques, such as Branch-and-Bound and Tabu
Search, SA is a popular method to solve combinatorial optimization problems, which
seek the best combination out of many possible combinations. Typically, these
techniques are being used in production system planning, scheduling, transportation, and

logistics.

3.2.3 Integrated Controlled Random Search for Dynamic Systems (I CRS/DS)

ICRS/DS is a modiﬁcation of the ICRS algorithm (Banga and Casares Long,
1987; Casares and Rodriguez, 1989), which was a generalization of the method proposed
by Goulcher and Casares Long (1987) for steady-state optimization problems (Banga et
al., 1997). As a generic algorithm, ICRS/DS is not as popular as GA or SA, but it has
been applied to various problems, such as bioprocesses (Banga 'et al., 1997), wastewater
treatment (Banga and Casares Long, 1987), retorting (Banga et al., 1991), drying (Banga
and Singh, 1994), and meat patty cooking (Banga et al., 2001), and proved its ability to
solve dynamic Optimization problems.

Basically, ICRS/D8 uses two strategies: control vector parameterization and a
stochastic direct search procedure. The original constrained dynamic optimization
problem is transformed into a constrained nonlinear programming (NLP) problem by
using a ﬂexible parameterization of the control function. The constrained NLP problem is
solved using the stochastic direct search procedure (Banga et al., 1997). Like GA and SA,
the search uses search direction and step size to determine the next feasible move. This
search mechanism starts with a user-speciﬁed feasible control vector and perturbs the
vector randomly with a normal probability distribution, which has the control vector as

the average and a standard deviation vector. The standard deviation vector is a set of

48

smaller distances between the control values and their upper and lower bounds. If the
algorithm cannot ﬁnd any improvement within a limited number of trials, the standard
deviation vector is reduced by a heuristic parameter, which results in smaller search step.
The algorithm can be terminated by checking user-speciﬁed tolerances for the decision
vector and/or the performance index at the end of each iteration (Banga et al., 1997).
The detailed procedures of ICRS/DS can be found in the work of Banga et al.
(1997). These procedures consist of control proﬁle parameterization and the modiﬁed

ICRS algorithm.

Control proﬁle parameterization:

The control function u(t) over t e [to ,tf] is parameterized using N
points, (6,- , (0)) (i = 1...N). The value of u(t) at iteration k can be calculated using
variable-length piecewise linear interpolation (Equation [3.8]) within the optimization

procedure and Q-k _<. t 3 61/11;

k k
(t)- - 0)-
u" (t) = 3124—“: - 91‘) + of [3.8]

By using the above parameterization technique, the original optimal control
problem is transformed into a 2N (or 2NM if there are M control variables) dimensional
nonlinear programming (NLP) problem. For implementation purpose, the parameterized

time and control values (0,- , 60,-) are replaced with the decision variable vector 4‘ for any

iteration k by using the following rule:

49

5126,", i=1....N

k k [3.9]
g, =ro,_N, i=N+l...2N

Also, the upper and lower bounds for the decision variables are expressed in a

 

 

simple way:
it” = 0r”, A.3-L = 6}; i=1...N
U U L L [3.10]
.- = 0,-_N. e“.- = 0,-_N; i: N+ l...2N
In addition, the parameterization must satisfy following conditions.
For time-intervals:
ref 3 l9," s a,” ‘
k k .
l9,- s cm r1 =1...N, Vk [3.11]
L U L
92' s e,- s 9.41 I
For control vector limits:
(0,-1’ 3 ref s (in , i=1...N, Vk [3.12]
For initial and ﬁnal time-interval limit:
L U U
61 =61 =t0, 6N =tf [3.13]

The above parameterization method is designed for variable-length intervals. However, if

a ﬁxed constant discretization ratio of time is desired, it sufﬁces to take

 

a,” = of + e =[ tf )(i—1)+ e [3.14]

where e is a small number.

50

The Modiﬁed ICRS Algorithm (Figure 3.7):

The algorithm starts with a feasible decision vector, which can be chosen
randomly. Subsequently, a standard deviation vector is calculated by multiplying
minimum intervals (smaller interval between the decision vector and upper and lower
bound vector) by a heuristic parameter (k,). Based on the initial feasible vector, the
algorithm generates a next decision vector by using the standard deviation vector
(stepsize) and Gaussian distribution (random direction). Once the next decision vector is
evaluated as feasible (satisfying all the constraints) and improved (increasing or
decreasing objective function value), the above iteration is repeated until a convergence
criteria is satisﬁed. However, if the next generation decision vector is determined as an
infeasible or retrogressive (or remains the same), the failure counter (F) is increased by
one up to a pre-set value (recommending ne >< total dimension of a problem). Every
failure counter increment, another new decision vector is generated with the same
standard deviation vector and tested. If the algorithm fails to get an improved decision
vector within a pre-set failure counter value, the standard deviation vector (stepsize) is
decreased by multiplying another heuristic parameter (k2). This stepsize reduction is
continued until the algorithm encounters a feasible and improved decision vector with the
frequency of the maximum failure counter value. Once the feasible and improved
decision vector is found, the failure counter is set to zero.

In this algorithm, the three heuristic parameters (k1, kg, and n.) are critical to be
successful in optimization process. Pan (1998) recommended 1/3, 1/2, and 4 as default

value for k), kg, and ne, respectively.

51

 

Initial feasible decision
vector 60

Evaluate objective function:

104a")

 

 

 

 

 

 

Calculate minimum intervals of decision vector 6*:

if? = mink,” — riﬂe," - g} )j i=1...2N

 

 

 

 

 

Calculate standard deviation vector:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

     
     

 

 

 

 

 

of = rel/if, i=1...2N
Generate new decision vector f H:
."+‘ =g," + 0,-1",, i=1...2N
Increase failure counter: Evaluate objective ﬁinction:
F=F+1 Me)
No
@
Yes Yes
Reset F =0
of e—kzoi", i=1...2N
Yes

 

 

 

 

 

 

Convergence criterion?
igk+1—g."l k-i-l _ Jki
———'<tol(§-)and/or (t [J No
U L I 0( )
5i ‘5: IJ I
‘Yes

 

C Termination D

Figure 3.7 Flowchart of modiﬁed ICRS algorithm.

52

3.3 Artiﬁcial Neural Network

3.3.1 Fundamentals

As mentioned before, availability of a model is essential to solving optimization
problems in various applications. Even when a model is available, the computational
demand and the compatibility with optimization algorithms are critical to run an
optimization procedure effectively. Taking into account the above facts, simpler and
faster models often must be considered. Therefore, artiﬁcial neural network (ANN)
appears to be a good candidate for fast nonlinear dynamic models (Trelea et al., 1997).

In cognitive neuroscience, a neural network (also known as a neuronal network or
biological neural network to distinguish from artiﬁcial neural networks) is a population of
interconnected neurons. As the primary cells in the nervous system, neurons are structural
constituents of the brain. Generally, neurons are ﬁve to six orders of magnitude slower
than a silicon chip; events in a silicon chip happen in the nanosecond (10'9 8) range,
whereas neural events happen in the millisecond (10'3 3) range. However, the brain
processes tremendous amounts of information by using massive interconnections
between approximately 10 billion neurons and 60 trillion synapses (Haykin, 1999).
Neurons react to electrochemical impulses. Generally, neurons consists of several
dendrites (receptive zone) and one axon (transmission line to output). If the sum of the
input signals from dendrites surpasses a certain threshhold, the neurons encodes their
outputs (action potentials or spikes), which originate at the cell body (soma) of neurons

and transmit the electrical signal along the axon (Figure 3.8).

53

 

     
   

/,-~~——~— I Cell body (som-aﬂ

 

 

/* [teed
\

L [ Dendrites—j

 

 

 

Synaptic —\
terminals \

 

 

 

Figure 3.8 Physical structure of a neuron.

An artiﬁcial neural network is deﬁned as “a massively parallel distributed processor
made up of simple processing units, which has a natural propensity for storing
experiential knowledge and making it available for use” (Haykin, 1999). The
mathematical model of a neuron is comprised of three basic elements that are a set of
synapses20 (connecting links), an adder (summing junction), and an activation function

(squashing ﬁrnction) in the Figure 3.9.

 

20 Elementary structural and functional units that mediate the interactions between neurons (Simon Hayakin,
Neural Networks: A Comprehensive Foundation, p. 6).

54

 

 

   

 

 

 

 

 

 

Inputs < f —’ Output
, 5 Summing Activation
'unction function
K . O ’
Synaptic
weights

Figure 3.9 Mathematical model of biological neuron.

An input signal is multiplied by the synaptic weight while passing through the
synapse. All the weighted input signals are summed up at the adder. Finally, the
activation function regulates the amplitude of the output of a neuron. Basically, training a
neural network or learning process is the process of adjusting the synaptic weights to
minimize the error between outputs and targets. Depending on the methods of learning

and the architectures of network, a large variety of neural networks exists.

3. 3.2 Back-Propagation F eed-Forward Neural Network (F FNN)

The feed-forward neural network trained by back-propagation is reckoned as a
major advance in the history of artiﬁcial neural network, because it provides a
theoretically sound technique for training multilayer, feed-forward networks with

nonlinear neurons when there is no efﬁcient, theoretically sound method for training

55

 

(Wasserman, 1993). As a training method, back-propagation became highly popular in
neural network training (Haykin, 1999).

In the feed-forward neural network, the input data ﬂows only in one direction
from input layer to output layer. Basically, training a neural network means to ﬁnd the
best set of weights of neurons that minimizes the error between neural network prediction
and measured data. Thus, the back-propagation method starts with initial weights to
produce prediction and uses its error to adjust the weights in the direction of reducing the
error by using a learning rule or weight updating rule. Because adjusting weights starts
from outmost layer to inner hidden layer, the method goes by the name of “back-
propagation.” The following is the fundamental principles of the back-propagation
training method. The notations of Figure 3.10 will be used for further mathematical
representation. Details can be found in the work of Wasserman (1993).’

The ﬁrst step for back-propagation is to calculate outputs with given inputs and
initial weights of each layer. The input column vector (x) is multiplied by hidden layer
weight matrix (Wm‘), then the result is transferred to the nonlinear activation function (f).

Thus,
y j = f(wjfoi) [3.15]

Finally, at the output layer, the vector y is multiplied by output layer weight matrix

(WOL). Therefore,

yk = WELL- = wiLfIWﬁLxrl [3.161

56

Hidden Layer Output Layer

 

 

    
   

 

 

      
 

‘ A
I
I
Input : Target
Vector i Vector
i" \ ‘x‘ g%//'
“ ./I \
x’ i
nggj'
. . t2

 

 

  

    

 

 

 

tk

 

 

‘ f () : Activation Function 2 ( ti- yi) 2

 

Figure 3.10 Topology of back-propagation of feed-forward neural network.

After obtaining the network output, error between output vector and target vector is

computed. Sum of squared error (SSE) is used for error measurement. Hence,

e=SSE=Zek =Zkrk —y,,)2] [3.17]
k k

Now, the error information is propagated from the output layer to the hidden layer

in the form of a gradient vector (V), which is the set of derivatives for all weights with

57

respect to the output error. Thus, a positive gradient of a weight means that overall error
is increased, which requires the weight to be decreased to reduce the overall error. In case
of a negative gradient, the opposite is true. The gradient vector of output layer can be

evaluated as follows:

 

 

 

as 66
V32: z 6w01‘ {67] 332 [3.18]
jk k jk
where,
65
ay—k = —2(tr — yr) [3.19]
$32 = y,- [3.20]
jk

V1015” : the gradient vector component associated with the weight from neuron j in

the hidden layer to neuron k in the output layer
kaL : the weight connecting neuron j in the hidden layer to neuron k in the output

layer
y j : output of neuron j in the hidden layer
y k : output of neuron k in the output layer
tk : the target value for neuron k in the output layer

By using the calculated gradient of each weight, the following weight update rule or

learning rule can be deﬁned.

58

0L 0L 0L
wjk (n+1)=wjk (n)—77 gvjk [3.21]
where
ij (n): the value of the weight at time n

77: a learning rate constant, typically < 1.0

According to the weight update rule, a weight of a negative gradient gains more weight
and a weight of a positive gradient loses its weight.
The next step is to ﬁnd the gradient vector of the hidden layer. According to the

chain rule of calculus, this gradient can be expressed as:

as- [as] a. . aw
J awyf-IL Z6” 5Yj avj dwgll‘ [ I

By substituting the following relations to the Equation 3.27

 

 

 

 

68
—=‘2(tk 'yk)55k [3.231
aYk
5311‘— : w’f [3.24]
6y,- 1
5y;
— = ' - 3.25
avj f (V,) l 1
83L = x, [3.26]
5W
and deﬁning

59

on} = 26,, wjjfftvj) [3.27]

where f ' is the derivative of the nonlinear function. Thus, Equation 3.22 becomes
V5?“ = or]. -x,. [3.28]

Now, the weights of the hidden layer can be updated by using the same updating rule of

the output layer.
W5.” (n + 1) = wijO-L (n) — n - V ,3“ [3.29]

The above two-layer example can be generalized for multi-layered feed-forward neural
network. By calculating 6 for each neuron using 6 from the previous layer, V for each
weight can be calculated, and the weight can be adjusted.

The training can be done by using sequential method and batch method.
Sequential method updates weight at the time each input vector is applied. In batch mode,
however, changes of weights are done after presenting all input vectors to the network.
Because the batch training averages the derivatives over a pass through the training set,
the batch training gives a more accurate estimate of the overall gradient.

The back-propagation method is generally slow to train a network. However,
some methods are available to speed the process. These methods are using second
derivative information for fast convergence. Due to the second-order information,

training time may be reduced by up to a factor of 100. Conjugate gradient descent and
quasi-Newton method are examples of these fast convergence methods (Wasserman,

1993)

60

3.3.3 Generalized Regression Neural Network (GRNN)

GRNN is a neural network architecture that has many similarities with the radial
basis-function21 neural network (RBF), in which each hidden neuron uses a radial basis-
function as the activation function, and the output neurons implement linear combinations
of these radial basis—function. Among many neural networks, nonlinear regression theory
based GRNN can approximate any arbitrary function between input and output vectors
(Wasserman, 1993). Therefore, if the GRNN is used for predicting the future value of
observed variables that are dependent variables related to input variables in a process,
plan, or system, the GRNN can be used to model the process, plant, or system
(Christodoulou and Georgiopoulos, 2001).

GRNN is based upon well-established nonlinear regression theory, which is the

following formula (Wasserman, 1993):

Iy-f(x,y)dy

Emir]: ‘2) [330]

If (x, y) dy

 

where
y = output of the estimator

x = the estimator input vector

E ( ylx) = the expected value of out, given the input vector x

f (x, y) = the joint probability density function (pdf) of x and y

 

2' An activation function which is centered at a point speciﬁed by the connection weight vector and whose
position and width are adjusted by learning. The most popular radial basis-function is the Gaussian ﬁrnction.

61

GRNN is, in essence, a method for estimating f (x, y) , given only a training set.

Detail architecture is shown in Figure 3.11. Specht (1991) shows that y, (function value)

is estimated optimally as follows:

n n
yi = Zhiwlj Zhi [3.31]
i=1 i=1

where

wij = the target (desired) output corresponding to input training vector x, and
output j

2

_ D-
hi = exp': 3 :l, the output of a hidden layer neuron
20

 

D} = (x — ii ,- )T (x - u,) (the squared distance between the input vector x and the
training vector u
x = the input vector (a column vector)

u,- = training vector i, the center of neuron i (a column vector)

0' = a constant controlling the size of the receptive region

62

T . . Training
raining —>[xn,X2l,...xml]= u] 0 Target vector

input vector @

\
XlO._- . -,®

\

X2 0

Km 0' I I , a",

. ‘ ~. yk ' '
[Kim X2n,...,xmn] “fl 0 r, ,I k
" . “-31. ’ 'I I
" “ix ‘ ‘ it I
a \ \ ‘ I
V Wnk \‘ r‘i' ’
s \\ U ’
\‘ “ . I,
‘ \ “ ' I

  

IICIIIIIIII
s
Q
.IUIIUIIII.
COCO-CI...-

Figure 3.11 Architecture of generalized regression neural network (GRNN).

The GRNN copies the training cases into the network to be used to estimate the

response on new points. In GRNN, one simply assigns to wij the target value directly

from the training set associated with input training vector i and component j of its
corresponding output vector (Wasserman, 1993). Because of the above fact, a GRNN
trains almost instantly. Thus, as the number of pairs of inputs/output increases, the more

computation time is required, because of the corresponding increase of the number of

hidden neurons. The a is the standard deviation of the response curve of the neuron.

Therefore, points nearby contribute most heavily to the estimate. If a is small, the neuron

responds only to the input vector close to the weight of the neuron (Figure 3.12). Again,

63

if 0 is large, the neuron responds to the wide range of input vector. GRNN can only be
used for regression problems, because of the nature of the fundamental equation. Like a
radial basis-function (RBF) network, a GRNN does not extrapolate. Thus, it is important
to construct the training data set carefully, so that the data set covers the region of

interest.

 

 

 

small a

  

Figure 3.12 GRNN response depending on the size of receptive ﬁeld (a) in two-
dimensional case.

64

4 METHODS AND PROCEDURES

4.1 Overall Methods and Procedures

Figure 4.1 is a schematic diagram to show the overall procedures and methods of
this project. In this study, the model outputs from an existing, validated ﬁnite element
model (FEM) of meat patty cooking (Watkins, 2004) were considered as actual ﬁeld
experiments. For this study, the model was expanded to include quality prediction
capability, such as internal color change (Section 4.3.3) and surface color change
(Section 4.3.4). By using the integrated FEM, various data groups (train and test or
validation) were generated to develop various artiﬁcial neural networks (ANN) as
alternatives to the FEM (Section 4.5). Then, various optimization strategies were applied
to both models (FEM and ANN’s) to ﬁnd a theoretically possible optimal condition and
to evaluate the various strategies (Section 4.6). Optimization strategies were then applied
to industrially-relevant case studies (single-stage, double-stage, multi-zone, and an

economic-based problem) to illustrate potential utility of these technologies (Section 4.7).

4.2 Impingement Cooking Technology

As the term “impingement” represents, it is a sharp collision produced by striking
or dashing a heating medium against a product. Because of the “sweeping away” effect
around the product, the boundary layer thickness is reduced greatly, which increases the
heat transfer rate. Therefore, impingement technology can be characterized as a fast and

efﬁcient means of processing.

65

 

Quality model
+
Basic FEM
I|

Integrated
FEM
(Sec.4.3)

 

 

 

 

 

 

 

 

 

   

 

 

 

Optimization
(Sec. 4.6)

 

 

 

 

      

Optimal
Solutions

 

 

 

Train Data Test Data
Sets Sets

Train §§ Validation

 

Artiﬁcial Neural
Network
(Sec. 4.5)

 

----------------------

 

Optimization

 

 

(Sec. 4.6)

 

 

Validation

 

Optimal

 

 

Solutions

    

 

I

/

 

Integrated
€§ FEM

(Sec.4.3)

 

 

 

 

Applications

 

 

 

 

Conclusions

     
 

I

 

 

1

Case Studies
(Sec. 4.7)

 
   
 

 

 

 

Figure 4.1 A schematic diagram of overall procedures and methods.

66

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_--_.._---_-----_----------J

This technology is widely adopted in the industrial applications, including:
annealing of non-ferrous sheet metals, tempering of glass, drying of paper and textile,
cooling of electrical components and turbine blades, and processing of food products
(Saad et al., 1980). Especially, if the impingement technology is used with moist air
instead of dry air, some unique advantages can be obtained

When the moist air meets a surface cooler than the dew point temperature of the
moist air, water condenses on the product surface. Due to condensation at the initial stage
of cooking, the condensed steam gives off its latent energy to the product. This efﬁcient
heat transfer happens until the product surface temperature increases beyond the dew
point temperature of the moist air.

In this research, a moist air impingement oven (model J SO-IV, Figure 4.2), which
is a commercial product of Stein-DSI (a business of FMC FoodTech, Sandusky, OH),
was the example oven used for process modeling. The major process variables of the
oven are impingement exit velocity, process duration, moisture content of the impinging
gas (volumetric basis), and impinging gas temperature. These four process variables were
used for the process modeling and optimization. Even though there are some physical
system conﬁguration parameters, such as open slot ratio, jet spacing, and gap between jet
exit and surface, these parameters were held constant for this study and not included in

the control variable group.

67

 

 

 

 

 

Figure 4.2 An actual and a cross-sectional image of the JSO-IV Jet Stream® Oven
of Stein-DSI (FMC FoodTech).

4.3 Integrated Process Model Development

As a complex, multi-physical phenomenon, meat cooking involves many factors,
such as yield, microbial inactivation, and quality changes. A ﬁnite element model
developed by Watkins (2004) could predict yield (via fat and moisture prediction) and
microbial inactivation within a meat patty subjected to moist air impingement cooking.
However, the model could not predict other quality factors.

Internal cooked appearance of ground beef patties is used to evaluate doneness by
many consumers (Hunt et al., 1999). In addition, ﬂavor is a very important component,
which is affected by the Maillard reaction on the surface of meat product (Mottram,
1998). Therefore, internal color change and ﬂavor (with surface color change) models
were incorporated into the ﬁnite element model (Section 4.3.3 and 4.3.4).

Many of the reactions in food processing follow ﬁrst-order kinetics, which can be

expressed by the familiar D-value and z—value (Toledo, 1991). Those kinetic parameters

68

for the quality indices in this project were calculated based on literature data (Section

4.3.3 & 4.3.4).

4. 3. I Basic Finite Element Model

The process model is a central component in optimization problems, because it is
used as an equality constraint. Thus, moist air impingement cooking of meat patties needs
to be modeled before the cooking system can be optimized. Watkins (2004) developed
this model by using a coupled heat and mass (moisture) transfer model. The model also
encompassed fat transport and Salmonella inactivation. The model utilized the ﬁnite
element method to solve separate equations for heat, moisture, and fat transport. These
equations were coupled through boundary conditions and interdependent thermo-physical
property relationships. An enthalpy formulation for heat transfer was utilized to avoid
discontinuities related to solid-to-liquid phase changes of water and fat within the
product. A solution for modeling condensing-convective boundary conditions during
moist air impingement cooking, developed by Millsap (2002), was incorporated into the
model. These boundary conditions accounted for the additional heating effects of surface
condensation that can occur within moist air impingement systems (Watkins, 2004).

A ground and formed meat patty (2-D cylindrical object) was used as a model
product. Input temperatures were converted to enthalpy using an equation based on the
work of Voller and Cross (1981). Heat transfer was modeled via following equation.

@411 ,. kT a_” +3: kT @ [4.1]
at rar cT-par dz cT-paz

 

 

Moisture transfer within the product was modeled using a two-dimensional equation for

diffusion in radial coordinates.

69

 

 

 

 

at r 6r cm,water 'p ar 52 cm,water 'p 62

M-[li[r km,water amwater]+_a_[ km,water 6mm,” ]]

[4.2]

For fat transfer modeling, a two—dimensional formulation of Darcy’s law for diffusion of

liquids through porous media was used (Datta, 2002).

am fat 1 6 am fat 5 am fat
—at_ = [-— r ' Dcap,fat —a—r—_ + E Dcap,fat '3— [4'3]

rar

The heat transfer boundary condition was formulated with a convection term and a

moisture transport (evaporation/condensation) term:

6T
kT a: = hT (T air — T surface) + hm, water ‘ ’l'vaporization ' (Cair T Csurface) [4'4]

Mass transfer at the product surface was modeled by using a convective boundary

condition.

6m
k m, water 3:!” = m,water (Coir '— Csurface) [4'5]

 

The fat content at the surface of the patty was modeled using an equation derived
from experimental data (Equation [4.6]) (Watkins, 2004). The equation [4.6] was utilized
to set the values of the fat content at each boundary node as a function of temperature and

product composition.

m fa, = 0.7062 — 0.0193 . T + 0.0001 - T2 + 0.0069 . mi, fa, + 0.0002 - T . mi, f,,

[4.6]

70

 

In addition to these boundary conditions, heat and mass transfer at the radial and
vertical centerlines of the patty were assumed to be zero due to product symmetry. These
coupled partial differential and ordinary differential equations were solved by using the
ﬁnite element method (Watkins, 2004). The ﬁnite element solution of this time-
dependent ﬁeld problem was solved with a ﬁnite difference approximation in the time
domain to generate the time—step solution, and the time step was 1 second. The time step
was smaller than the allowable time step (AF2.16 s), which was calculated to prevent the
ﬁnite element model from oscillating and deviating from physical limits for a right

triangle element, by using Equation [4.7] (Segerlind, 1984).

At < L [4.7]
l— 6

where

a: det([c(°)]-a[k(°)])=0; det: determinant; c & k: element matrices

0:1/2 for central difference method

However, in some cases of actually running this FEM model, instability of the
prediction of yield and surface temperature were observed at the very early stage and
rarely at the later stage. Therefore, the time step was decreased from 1 to 0.5 second to

reduce the instability of the model.

Watkins (2004) validated the ﬁnite element model using data generated in
industrial cooking tests with beef patties in a J SO-IV (Stein~DSI, a business of FMC
FoodTech, Sandusky, OH) and additional published data for ground chicken breast

patties cooked in a pilot-scale impingement oven (Stein, FMC FoodTech). Predictions of

71

cooking yield had errors ranging from 0.1 to 15.4%, with an average deviation of 5.9%
(27 sets of process conditions) in the industrial cooking tests. Comparisons with
published data were also favorable (standard errors of prediction for yield ranged from

1.1 to 15.7%).

4.3.1.1 Model Product

The dimension of the patty was 120 mm in diameter and 10 mm in thickness,
which was modeled as a two-dimensional, axisymmetric body. Figure 4.3 shows the

element mesh conﬁguration, which consisted of 36 nodes and 50 elements.

The model patty was assumed to initially be 60% water, 20% fat, and 20% protein.

The initial temperature was set to 5 °C.

 

 

 

 

 

 

 

 

 

 

 

 

 

Center

Patty

 

 

Figure 4.3 Finite element mesh utilized for one quarter of the 2-D cylindrical model
product.

72

4. 3.2 Kinetic Parameters for Microbial Inactivation in a Meat Patty

Salmonella is the most heat-resistant organism among other food related
microorganisms, such as E. coli 0157:H7 and Listeria monocytogenes (Murphy et al.,
2004). A few works (Juneja et al., 2001; Murphy et al., 2002; Murphy et al., 2004) of
thermal inactivation kinetics for Salmonella cocktails in ground beef were found and
compared to get a conservative reference D and z-value.

In Table 4.1, D values at 62.5 °C, which is the center between 55 and 70 °C, were
compared, and the D value of 2.62 min was selected as a conservative value.

For z-value, the highest value (9.14 °C) was chosen among other values, because

the value reﬂects the least sensitivity to temperature change.

Table 4.1 Comparison chart of thermal inactivation parameters of Salmonella
cocktail for ground beef (1:(Murphy et al., 2002); 2:(Juneja et al., 2001); 3:(Murphy et al., 2004)).

 

 

 

 

 

 

 

 

 

 

 

 

l 2 3
Fat [%] 18.56 12.45 34.4
Water [%] 51.30 65.5 49.7
Method 0.7 mm 1-2 mm Thin bag
thin metal thin bag
container
Culture Type Cocktail Cocktail Cocktail
Tr [°C] [min] [min] [min]
55 9.09 37.05
57.5 7.70 18.35
58 8.65
60 4.80 5.48 6.90
D-value [ 6:55 I 2.40 1.50 | 2.62 l
65 0.97 0.67 1.03
67.5 0.57 0.30
68
70 0.25 0.066
z-value 9.14 [°C] 6.01 [°C] 5.74 [°C]

 

73

4. 3.3 Kinetic Parameters for Internal Color Change of a Meat Patty

Kinetic modeling is a very efﬁcient tool to model food quality, which otherwise
can be very subjective. Once the rate and temperature dependence of a reaction is known,
the reaction can be controlled and predicted (Martins et al., 2001).

The color of the center region of a meat patty is an important quality index related
to acceptability. As cooking proceeds, the center color gradually changes from red-pink
to brown (1'. e., “well done”) due to the heat denaturation of various types of myoglobin.
The rate of cooked meat hemoprotein formation (via the rate of loss of myoglobin
solubility) was found to obey ﬁrst-order kinetics in aqueous muscle extracts and mixtures
of myglobin and bovine serum albumin (Geileskey et al., 1998). They measured ﬁrst-
order rate constants for the loss of myoglobin solubility in various muscles at 60, 65, 70,
and 80 °C.

To implement ﬁrst-order kinetics in the basic ﬁnite element model, a reference
decimal reduction time (D,) and z-value were calculated based on the rate constants at the
various temperatures (Toledo, 1991). The reference temperature was 60 °C. D, was
calculated by using Equation [4.8] and the ﬁrst-order rate constant (Geileskey et al.,

1998) at the reference temperature.

_ ln(10)
- k

 

Dr [4.8]

To calculate the z-value in the temperature ranging from 60 to 80 °C, activation energy
(E3) was obtained by using temperature and rate constant data from (Geileskey et al.,
1998), assuming an Arrhenius relationship (Toledo, 1991). The z-value was then

computed as follows:

74

_ ln(10)
z —- (Ea /R) Tsz [4.9]

 

Table 4.2 shows the derived D, and z-values for different muscles. Considering that most
ground beef comes from the chuck, sirloin, or round, the D, and z-value of beef chuck
were used for estimating acceptability (via center color change) of model product.

Even though the color for acceptability can be predicted with the above parameters, an
adequate log reduction value of internal color change must be speciﬁed to use the quality

as a constraint in this optimization problem.

Table 4.2 Reference decimal reduction time (D,) and z-value of internal color change
in the various muscles at the reference temperature of 60 °C (Geileskey et al., 1998).

 

 

Muscle D, [min] 2 [°C]
Beef shin 85.28 9.63
Beef chuck 65.79 9.79
Beef m. I. dorsi 35.42 10.41

 

Hunt et al. (1999) observed that 95.9% denaturation of oxymyoglobin (OMb),
which is contained in signiﬁcant amount in most ground beef, was visually scored 5 (no
evidence of pink color or “well done”). Also, the ground beef used in that research
contained 20% fat, and the pigment was predominantly OMb (Hunt et al., 1999), which
implies a good match with the model product in this study. Thus, the target level of
denaturation (95.9%) was converted to the equivalent value of 1.387 log reductions,
which was used as a minimum criterion for internal color change of meat patty. Even
though there is some variation in the concentration of myoglobin in beef, the differences
in concentration do not appear to be a factor in the dependence of denaturation on muscle

type (Geileskey et al., 1998).

75

4. 3.4 Kinetic Parameters for Surface Color Change of a Meat Patty

Non-enzymatic browning in food processing is the major cause of surface
browning, aroma, and taste (Martins, 2003). Non-enzymatic browning of baked products
is a very complex chemical reaction, which encompasses two major reactions, the
Maillard reaction and caramelization (Zanoni et al., 1995).

The Maillard reaction starts with an initial reaction of a reducing sugar with an
amino compound, followed by consecutive and parallel reactions to form a variety of
colored and colorless products (Martins and Van Boekel, 2005). In addition to the
desirable effects, the reaction generates undesirable results, such as discoloration, off-
ﬂavor, and mutagenic and carcinogenic components. Therefore, it is necessary to
optimize the reaction by ﬁnding the best balance between the favorable and unfavorable
effects of the reaction in a given process (Lingnert, 1990). The Maillard reaction is very
complex reaction and very difficult to control (Martins et al., 2001). For instance, over
1,000 volatile compounds are formed during cooking (Mottram, 1998). Many factors
inﬂuence the reaction, such as temperature, time, pH, water activity, type of reactants,
and availability of reactants (Lingnert, 1990).

The Maillard reaction kinetics has been studied in various applications, such as
frying and drying. However, detailed observations for the Maillard reaction during
cooking of meat patties, especially in the moist air impingement convection cooking
environment, were not found in the literature.

Another non-enzymatic browning is caramelization. Caramelization is deﬁned as

the thermal degradation of sugars leading to the formation of volatiles (caramel aroma)

76

and brown-colored products (caramel colors). Caramelization usually occurs at high
temperature (>120 °C), compared to the Maillard reaction temperature (>50 °C).
Non-enzymatic browning is very complex and encompasses not just a single
reaction pathway, but a whole network of various chemical reactions (Martins et al.,
2001). Although complex reactions are involved, general perceptions for non-enzymatic
browning reactions are color and aroma. The resultant effect of the various reactions can
be lumped into crust color development during a meat cooking process. Dagerskog and
Bengtsson (1974) studied the relationship among crust color formation, yield,
composition, and processing conditions for double-sided pan frying of meat patties. They
found kinetic parameters for crust color changes, in terms of reaction rate and activation
energy. Ateba and Mittal (1994) obtained kinetic parameters for crust total color change
and ﬁrmness. The above studies for crust color change of meat product used the total
crust color change (AE) to describe the overall quality changes due to non-enzymatic
browning. The total color change was calculated with the following equation (Dagerskog

and Bengtsson, 1974):

 

AE = \/(AL)2 + (M)2 + (Ab)2 [4.10]

where,
AL: lightness
Aa: redness

Ab: yellowness

Color development is a surface phenomena, so that the plot of surface temperature
(recipe A) and color change (recipe C) (Dagerskog and Bengtsson, 1974) were digitized

77

 

to obtain a D, and z-value (Table 4.3). Even though the recipes were different, both data
were assumed for one recipe in this study, because only two graphical data sets were
available. Recipe “A” consisted of 59.6% water, 15.6% fat, and 9% breadcrumb, and
recipe “C” consisted of 66.6% water, 14.1% fat, and no breadcrumb. Therefore, recipe

“C” is closer to the model patty in this study, in terms of composition.

Table 4.3 Digitized data of surface time-temperature plot (recipe A) and time-color
change plot (recipe C) of Dagerskog and Bengtsson (1974).

 

Pan Temperatures

 

 

 

140[°C] 160£C1 180 [°C] 200 [°C]
Time [min] T,[°C] AB Ts[°C] AE T,[°C] AE T,[°C] AB
0 10.00 0 10 0 10 o 10 0
1 101.40 5.51 107.60 3.95 121.60 6.14 130.30 7.70
1.5 103.40 3.48 110.20 6.76 123.50 7.80 132.60 9.99
2 103.70 6.02 111.20 9.46 122.20 9.88 129.40 12.17
2.5 104.70 8.62 112.50 9.04 122.90 12.27 130.70 14.82
3 105.60 8.57 115.10 12.21 126.40 15.03 135.90 15.60
4 109.50 12.36 122.20 15.17 134.90 16.37 148.90 18.40
5 113.40 14.90 127.70 17.51 142.70 17.87 158.60 20.37
6 116.70 16.04 132.00 19.01 149.80 19.48 166.10 20.21

 

Ts: product surface temperature

The AE was replaced with the difference with the maximum color change
(AEM.=22). Then, a log reduction of color change was calculated. By using the Microsoft
Excel SOLVER (Microsoft Excel Version 2000: Redmond, WA) and Equation [4.11], the
D-value and z-values for this kinetics were estimated based on minimization of the

RMSE of color change.

78

’f
C 1
1 ——=— —dt 4.11
ogicoj ID I I
0

where,

 

19:1), .10 z [4.12]

The estimated D-value and z-value for the surface color kinetics were 7.75 min
and 90.55 °C, respectively (average RMSE=1.33 [AE]).

Now, dynamic surface color change can be predicted by integrating Equation
[4.11] over time with the given D and z—values. However, there are several hurdles for the
non-enzymatic browning to take place on the surface of a meat patty in the moist air
cooking environment. The ﬁrst barrier is water ﬁlm formation due to condensation on the
meat patty surface when the surface temperature is lower than the dew point temperature
of the moist air. The water ﬁlm keeps the surface temperature under 100 °C, which is far
less than the critical caramelization temperature of 120 °C. For the Maillard reaction,
some water is needed, because water participates in the later stage of the reaction.
However, excessive water suppresses the reaction and also dilutes surface concentration
of amino acids and sugars, which result in retarded or no reactions. Therefore, under the
presence of condensed water on the surface, it is hard to expect non-enzymatic browning
mainly due to the Maillard reaction. Even after overcoming the water ﬁlm formation,
evaporation occurs, which decreases the surface temperature until the ﬁlm disappears.
When the product surface temperature reaches 100 °C, the water evaporation zone
recedes toward the center. The surface continues to lose water, the temperature rises, and

79

structural and chemical changes in the protein result in crust formation (Ateba and Mittal,
1994). Finally, non-enzymatic browning of the surface takes place rapidly when the
surface temperature passes the critical temperature of the caramelization reaction. After
this point, caramelization is accelerated signiﬁcantly. The above limitation to non-
enzymatic browning on the surface of meat patty under moist air cooking conditions need
to be implemented in a simple non-enzymatic browning model to predict surface quality
of the meat patty.

To account for the adverse effects, a function cueing the non-enzymatic browning
on the product surface is necessary. A sigmoid function, which is the common form of an
activation function, was adopted for this purpose via the logistic function (Equation

[4.13]).

l

f = 1+e—a-(Ts—Tdew)

[4.13]

 

The sigmoid function is ideal, because it is convenient to account for an abrupt change
around a critical point and also it is continuous (Figure 4.4). The logistic function
parameter a was determined at the value where 99.5% change is achieved in the range of

+/- 5 °C around the critical value. Therefore,

a = _]n(0%5_) e 1.06 [4.14]

80

fit

 

 
  

Increasing a

 

 

O Tdew

._]V

Figure 4.4 A sigmoid function (logistic function) of surface temperature (T‘s)
showing gradual change in a certain factor around a critical value, Tm, in this case.

The sigmoid function incorporating surface temperature and dew point
temperature was inserted into Equation [4.11] to provide a status of surface browning for
integration over time. Thus, the ﬁnal equation of non-enzymatic browning, accounting

for condensation effects, was expressed as follow:

 

 

’f

log[—EC—] = — B; dt [4.15]
O
’0
where,

T )- 1 4 16
f( S — l+e—a(Ts—Tdew) I ° ]

Tr—T
13:1), .10 z [4.17]

81

In Equation [4.14], the surface temperature was obtained from the ﬁnite element solution,
and the dew point temperature was interpolated by using the tabulated data22 of dew point
temperature and moisture content by volume (which is a model input).

For non-enzymatic browning, both minimum and maximum critical values are
necessary for the cooking process to control the desirable color development on the
surface of the product, so that the product is neither too pale nor over-cooked. Generally,
the surface non-enzymatic browning is measured by total color change (Dagerskog and
Bengtsson, 1974; Ateba and Mittal, 1994). Dagerskog and Bengtsson (1974) observed
that the maximum total color change was 22, and suggested 10 as a desirable color
change. In this optimization problem, 10 and 15 were chosen as lower and upper bounds
for surface color change values, respectively. Unlike with microbial inactivation, the
intensity of color is increasing instead of decreasing. Thus, relative color change, which
is deﬁned by subtracting a color change value from the maximum observable color
change value (AEmax=22), was used to calculate the log reduction as follow:

AE — AE
1og[—"&—] [4.18]
AEmax

Therefore, 0.263 and 0.497 were used as a minimum and a maximum log reduction of

relative color change, respectively.

4.3.5 Practical Considerations for Integration of Models

Log reduction value is a good tool for describing the amount of reduction from

initial concentration of a material. However, if the reference D-value is small, which

 

22 In Humidity/Moisture Handbook by Machine Application Corporation (Sandusky, OH)

82

means fast reaction, the log reduction value may be too large for computer programming
to handle such a large number. For computer programming, the double data type,
according to IEEE Standard 754 for double precision, allows +/-308 for maximum and
minimum exponent to 10. Thus, ﬂoating-point overﬂow problems occur when the
reference decimal reduction time is too small.

Final log reduction of a non-isothermal process can be integrated as:

n

 

log All =log—1Yi + log& + + log
N0 N0 N1 Nn—l

_ At/D, At/D, At/D, l4-19l

_ + +...+——————
(Tr’Tl] [Tr—T2] (Tr'Tn)
10 z 10 z 10 2

Even though the log(N,,/N,,-,) at early stages of cooking has no overﬂowing problem,

 

log(N,./N,,-,) of a later stage can become very large, which results in the abortion of the
computer program. To resolve this problem, a multiplier (m) was adopted to slow the
process, so that the log reduction number stays within the computational limit. After

ﬁnishing the process, the original log reduction number was calculated back by using the

multiplier. For this, an increased D-value (D, ) was deﬁned as follow:

A

D, = m x D, [4.20]

Then the Equation [4.19] can be re-written by using Equation [4.20]. Thus,

[Gigi—n] :logivl—+log]—V;+..+log——N" = At/D, At/Dr At/D,

+ +..+-————
0 N0 N1 Nn—l [Tr-Tl) [Tr-T2) (Tr-Tn)
10 z 10 z 10 z

where, * means resultant log reduction using the increased Dr-value.

 

[4.21]

83

Now, dividing Equation [4.19] with Equation [4.21] produces the following result.

 

 

 

 

 

At/D, + At/D, +...+ At/D,
agile] [2:1] [2:2] [7:14] All ..... 1 .
No*=10 2“ 10 Z. 10 Z“ :13. =_Qr_=m’Dr=m
.451] A”? + Mr ....._____A”Dr 4’4 ..... 1 r Dr
No [Tr—41] [112] [ii] Dr
10 Z 10 Z 10 z
[4.22]

Therefore, the original log reduction value can be restored by using the multiplier m.
Even though the size of m can vary depending on the size of the maximum exponent, the

value of 106 was used for microbial inactivation calculation in this research.

4.4 Formulating the Optimization Problem

The objective of the problem is to maximize patty yield by ﬁnding optimal
process temperature, humidity, impingement air velocity, and cooking duration, while
ensuring the target microbial lethality, internal color change, and surface color change.
This optimization problem can be formulated with mathematical expressions to apply
formal optimization techniques. The formulation of the problem contains three essential
components: the objective function, inequality constraints, and equality constraints.
Objective Function:

The objective is to maximize performance index J, which is the ﬁnal patty yield
(%) by ﬁnding the optimal process temperature proﬁle T (t), humidity H(t), impingement
air velocity V(t), and cooking time tf over t e [0, LI]:

_ +_ +1
J= f(i,)=m"’f m” x100 [4.23]
mw0+mfo+l

 

84

The ﬁnal yield, f(tf), was calculated by dividing the ﬁnal patty weight by the initial patty

weight. The weights were calculated with water and fat content based on non-fat solids.

Inequality Constraints:

Final log reduction of microbial inactivation and internal color change (Cc) of the
center of patty should be greater or equal than 6.5 (beef) and 1.3 87, respectively. Also,
the acceptability via surface color change (Cs) log reduction should be between 0.263 and

0.497. These limits are:

L0,) 2 Lc(= 6.5) [4.24]
Cc (t f) 2 Cc,c(= 1.387) [4.25]
Cs, 1. (= 0.263) s C, (tf) 3 CW (= 0.497) , [4.26]

The control variables (process temperature, humidity, impingement air velocity,
and cooking time) are also bounded by upper and lower limits. For temperature, the JSO-
IV oven can achieve 260°C and is not normally operated below 100°C. Also, the
maximum humidity level is 90%MV, and the maximum impingement velocity is 30.6
m/s. Based on the above actual capacities and reasonable range of cooking duration, the

following upper and lower bounds were set:
TL (= 100) s T(t) s TU (= 250) [°C] [4.27]
HL(= 0) s H(t) _<_ HU(100) [%MV] [4.28]
(Instead of 90% MV, 100%MV was used to test maximum theoretical limit)

VL (= 0) s V s VU (= 30) [m /s] [4.29]

85

t f, L (= 60) s t f s t ,1, (= 600) [s] [430]

Equality Constraints (Process Model):

The ﬁnite element process model (Section 4.3) developed for moist air
impingement cooking of ground formed meat patties was the equality constraint for this

problem.

4.5 Alternative Modeling by Using Artificial Neural Networks

Artiﬁcial neural networks were used to replace the ﬁnite element process model,
in order to speed simulation time. The networks were trained and tested to determine
whether the networks were representing the ﬁnite element process model sufﬁciently
well. Two different training strategies were applied to two different types of networks.
Static training (Section 4.5.3) was used to develop static neural network model (SNNM),
and dynamic training (Section 4.5.4) was used to develop dynamic neural network model
(DNNM). The ﬁrst step for training and optimization was to parameterize the control

proﬁles.

4. 5.1 Parameterization of the Control Function

To explore the inﬁnite solution domain, an efﬁcient method of representing the
dynamic control vectors must be available. In this research, control vectors were
parameterized in two ways: by using piecewise linear interpolation, which was used by
Banga et al. (1997), and by Fourier series.

Piecewise linear interpolation (PLI):

The piecewise linear interpolation method was used to represent a control

function with a ﬁxed number of points and ﬁnal process time. The interval between the

86

points can vary, but in this application, the control functions were evenly discretized into
20 pieces. Thus, by using those points, any other points can be calculated by using
Equation [3.8] (piecewise linear interpolation). Therefore, a total of 21 points, including a
ﬂexible endpoint time, were used to parameterize each control function (process
temperature, humidity, and impingement velocity). Although the size of interval varied
depending on the endpoint time, due to the ﬁxed 21 control points, the effect of the
interval size was not considered in this research. Therefore, the maximum interval was 30
[s] for the duration of 600 [s], and the minimum interval was 3 [s] for the duration of 60
[s]. Thus, the total number of parameters to represent three control functions and process
duration was 64 (=3 X21+1). Each designated point followed the conditions of Section
3.2.3.1 (Equation [3.11-14]).

In the case of linear interpolation, the total 64 discrete values comprise the
following structure.

A set of discrete values= {p1, p2, p64}, where

p1~p2/: discrete values of temperature

p22~p421 discrete values of humidity

p43~p63z discrete values of impingement velocity

p64: cooking duration
Fourier series (FS):

The Fourier series is a robust parameterization tool. Fourier synthesis can
generate all possible continuous functions with a sum of sine and cosine functions called
a Fourier series. The function is uniquely deﬁned by constants known as Fourier

coefﬁcients, which are shown in Equation [4.31]:

87

T(t) = 329 + Z (a k cos(kt) +6, sin(kt)) [4.31]
k=l

In this application, the inﬁnite Fourier series was limited to ten terms, which is sufﬁcient
to closely approximate even step functions. Also, the period (2n) of cosine and sine
functions needs to be adjusted to the cooking duration tf. Thus, Equation [4.31] was

rearranged on the interval {-0, tf] as follow:
10
T(t) = 194.202,, cos(k§—'1)+o,, sin(k 11)) [4.32]
2 k 1 t f l f

Therefore, according to the Equation [4.32], each control proﬁle was parameterized by
using 21 Fourier series coefﬁcients for a given process duration.

In the case of Fourier parameterization, the 64 parameters comprise the following
structure.

A set of parameters={p1, p2, p64}, where

p1~pzo (i.e. ak, bk): Fourier coefﬁcients for temperature; p2, (i.e. a0): shift

pzrp4 1: Fourier coefﬁcients for humidity; p42: shift

p43~p6zz Fourier coefﬁcients for impingement velocity; p63: shift

p64: cooking duration

4. 5.2 Training and Validation Data Groups

As previously mentioned, training and validation data groups for neural networks
were produced by using the integrated FEM. In this study, a total of ﬁve neural networks
(one DNNM and four SNNM) were trained and validated with various groups of data.

Each group of data had a different number of input-target data sets and contents

88

depending on process type, which creates complexity. Thus, each group of data sets was
named for further reference (Table 4.4). In addition, the detailed conditions for each data
group are provided in Table 4.5. For example, the alternative model SNNM_S was
trained by using D3 and validated with D7 (Table 4.4), and the conditions for those data
sets are reported in Table 4.5.

Data group D2 was generated with 63 Fourier coefﬁcients and a process time.
Fourier coefﬁcients (Equation [4.32]), a], and bk, were selected randomly between ~100
and 100, and a shift value (up) was also chosen between 5 and 200. Among these random
combinations, only the proﬁle satisfying the upper and lower bounds of the control
variables (temperature, humidity, and velocity) were accepted and applied to the FEM to

generate results.

Table 4.4 The names and contents of data groups for training and testing neural
networks.

 

 

 

 

 

 

 

 

 

 

Neural Networks Training Data Group Testing Data Group
Type Name Name Contents Name Contents
1,000 processes were generated with 64
910 constant D2 random Fourier coefﬁcients including
processes coogg time.
(Table 4.5) 1,000 processes (randomly generated
D7 constant processes with random cooking
DNNM DNNM D1 (54,600 duration)
conditions/states D4 2187 processes
sampled out for (double-sgge, Table 4.5)
training) D5 2187 processes
multi-zone, Table 4.5)
1,000 processes were generated with 64
SNNM_R D2 D6 random Fourier coefﬁcients including
cooking time. (#D2)
1,296 processes 1,000 processes (randomly generated
SNNM SNNM_S D3 (single-stage, D7 constant processes with random cooking
Table 4.5) duration)
SNNM_D D4 D8 1,000 processes (randomly generated

double-stage conditions)

1,000 processes (randomly generated
D9 . . .
multi-zone conditions)

 

 

 

 

 

 

 

SNNM_M D5

 

89

Table 4.5 Conditions of each process type to produce training data groups; D1,
single-stage (D3), double-stage (D4), and multi-zone (D5) process.

 

Control Variables

 

 

 

 

 

 

 

Process Total
T H V tf nu r
”is 1°C] 1% MV1 im/si [s] (1.2:):an
100,112.5, 1, 15,30, 1,5, 10, 15, 20, 25, 30
125, 137.5, 40, 50, 60,
Constant 150, 162.5, 70, 80, 90,
Process“ 175, 187.5, 100 600 910
(1)1) 200, 212.5,
225, 237.5,
250
Single- 100, I30, 0, 20, 0.5, 6.4, 12.3, 18.2, 24, 3O 60, 180
Stage 160, 190, 40, 60, 300, 420 1,296
(D3) 220, 250 80, 100 540, 660
Double- T1 T2 H1 H2 V1 V2 tr
stage 100 100 0 0 0.5 0 60
(D4) 175 175 50 50 15.25 15.25 330 2,187
250 250 100 100 30 30 600
Multi- T H V1 V2 V3 V4 tf
”ﬁgs?“ 100 0 0.5 0.5 0.5 0.5 60
(D5) 175 50 15.25 15.25 15.25 15.25 330 2,187
250 100 30 30 30 30 600

 

* Process conditions remain constant during cooking

4.5.3 Static Training

Depending on the characteristics of the training data sets and the architecture of
neural network, the mapping relationship between input and output vectors may be static,
where each application of a given input vector always produces the same output vector
(Wasserman, 1993). Therefore, a neural network trained by this method cannot account
for process history; rather, it merely identiﬁes a relationship between initial conditions
and results. This kind of neural network has to be trained whenever different process
types are engaged.

For SNNM_R (Table 4.4), 1,000 random time-varying conditions were used to
generate data sets by using the ﬁnite element process model. Some examples of random

time-varying conditions are illustrated in Figure 4.5.

90

    

 

 

 

 

S‘ 250 — 5- 250 .

2 is

32 ._.

E 200 - z. 200 «

’9 "7 Tern erature E 7

E E, 150 ~ p = E 150 .

E a 5 a?

88 100 . . §§ 100 _ Humidity

“‘0 ‘ o-Jlumidity as; 3“...

g > I ~ 5 \ '~ I s

u \ Q - ’ ‘

8 50 r ' E 50 I“ ‘ ,’ Velocity ‘

8- ~ ’ ﬂ . Velocity E ‘ ’ g _ a “ h‘.

g :- -’- - \ n ’ 0 2.. g

.- 0 k 1 ﬂ —*1 F o - r r f i r '—
0 100 200 300 0 100 200 300 400 500

Process time [5] Process time [s]

Figure 4.5 Examples of random time-varying conditions by using Fourier series
control proﬁle parameterization.

Each training data set consisted of 64 parameters, which consisted of 21 randomly
generated Fourier coefﬁcients for 3 different control vectors (temperature, humidity, and
impingement velocity) plus process duration. Those 64 parameters were used as the
system input variables. By using the 64 parameters and Fourier series (Equation [4.32]),
three continuous random proﬁles were generated and used for the ﬁnite element process
model input to predict the yield, microbial lethality, internal color change, and surface
color. The outputs of the ﬁnite element model were used as target data sets. The data
group (D2) of combined input and target data sets was used to train the SNNM_R
(Section 3.3.2). Those data groups were also used for validation of GRNN_R.

For the training of SNNM_S, SNNM_D, and SNNM_M, training data groups (D3,

D4, and D5) were generated according to their conditions in Table 4.5.

91

4. 5.4 Dynamic Training

Compared to the static training method, the dynamic training method teaches
system behavior so that the neural network can predict output depending upon previous,
as well as current, input and/or output (Wasserman, 1993). Therefore, the dynamic neural
network reacts like a physical model, not just an input-output vector mapping.

If a full factorial design is planned with 64 parameters and just two extreme
values for each parameter, it results in 264 combinations, which is an impractical number
of trials. However, dynamic training can capacitate the neural network to understand
system behavior with a limited amount of information or training data set. Morimoto et
al. (1997) used historical input and output data to describe the dynamic characteristics of
fruit color changing behavior with time and temperature. The only disadvantage of
dynamic training is that it takes a little more time to produce outputs, because the neural
network has to predict the next point based on the previous point until it gets to the ﬁnal
point.

Depending on the interval size, the accuracy of the prediction also changes. The
smaller the interval is, a more accurate result is possible. In this research, process
temperature, humidity, and impingement velocity were used as system control vectors,
and yield, microbial lethality, internal color change, and surface color were included in
states (Figure 4.6). In addition, past time was included in the inputs as a factor accounting
for history of the system until it became past state. Morimoto et al. (1997) added linear
data to the input of neural network and observed that the identiﬁcation accuracy for any
cumulative responses was signiﬁcantly improved. However, the linear data were replaced

with actual current time of the system.

92

To obtain the training data group D1, 910 constant process conditions (Table 4.5)
were produced, and the results were calculated with the ﬁnite element model. The
combination has a maximum possible duration (600 [8]) by which other durations can be

covered, because the process is constant.

t(n-l) t(n)

 

 

 

Past time >Current time

////////// ///// /////////////
I I ' I %’ast/ St//a//t//% éurrent State W? I I I I
//////////////////// ////////////

[ Process direction >

Figure 4.6 Conceptual diagram of dynamic training paradigm.

\s

 

 

 

Out of the 910 resultant processes, 54,600 data sets were sampled with 10 seconds
interval. By using the sampled data, input and output pairs for neural network training
were reconstructed. Figure 4.7 shows the structure of input/output pairs.

The control force vectors, which have direction and magnitude, are implicitly
dissolved in the past and current controls. Although the four current outputs can be
predicted with a single GRNN, an individual GRNN for each output was trained to
increase accuracy for each prediction (Trelea et al., 1997). The network parameters were

also optimized independently.

93

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

rT(k) \
Current
Controls < H(k)
(Scalar)
\VUQ
Current Outputs
,
To“) GRNNl
Past
Controlsi H(k-l) E ::> AY(k)
(Scalar)
‘ Wk”) GRNN2
g :1) Auk)
P t
Tiattsle {‘(k‘l) j ‘ GRNN3
g 12> 40.00
r Y(1"1) ' GRNN4
Past L(k-l) 0 g :1) AC,(k)
States i l
(2‘0“) ° Y(k)=Y(k-1)+AY(k)
\ C,(k-1) L(k)=L(k-1)+AL(k)

 

Cc(k)=Cc(k-l)+ACc(k)
C.(k)=Cs(k-l)+AC.(k)

 

 

 

 

I

Figure 4.7 The structure of an input/output data pair for dynamic training to
identify dynamic characteristics of meat patty cooking.

 

Time delay

 

 

 

 

94

How well the input data are represented to the network is a critical issue to a
successful application of artiﬁcial neural networks (Wasserman, 1993). There are two
important points to be considered in the training of the GRNN. First, the target vectors
(yield and the other outputs) are monotonic, which are always increasing or decreasing in
one direction. If the GRNN is trained with actual values, then there is a possibility for the
GRNN to forecast a value that does not coincide with the monotonic characteristics of
output target vectors. This abnormality of forecasting can be minimized by training the
network not with actual values but with the difference. At the end of prediction, a
cumulative summation for each time was calculated to get the actual value for each time,
which renders all the training data to be always positive or negative.

Secondly, thermal inactivation kinetics of a small z-value generate large log
reduction values, because of high sensitivity to temperature change. The z-values for
thermal inactivation of Salmonella and the depletion of Myoglobin (internal color
change) are 5.9 °C and 10.41 0C, respectively, which are relatively smaller than 90.55 °C
for kinetics of surface color change. Therefore, there are huge variations in the predicted
results. The large variations in the magnitude of the components of a vector may not
convey meaningful information, but can confuse the network (Wasserman, 1993). In the
above case, raw data range from zero to several orders of magnitude, while the critical or
valuable information is located in a small range. Wasserman (1993) suggests that “taking
the logarithm of the data will adjust the range so that large values can be ‘squashed’ more
than small values, thereby allocating a constant range to a given percentage deviation”,

which is called nonlinear normalization. In this research, nonlinear normalization using

95

base-10 logarithm was used for the training GRNN with the above two reaction kinetic

data (Salmonella inactivation and internal color change).

4. 5.5 Neural Network Validation

All the neural network based models were validated with the testing data groups
that were prescribed in Table 4.4. For all the validations, root mean squared errors for
yield, microbial lethality, internal color change, and surface color were calculated to

quantify the performance of the neural networks.

4. 5. 6 Neural Network Parameter Optimization

F FNN based models consist of layers and neurons embedded in the hidden layer.
For the best performance of the network, the number of layers and neurons must be
optimized. Generally, there is no speciﬁc rule to set these numbers except by trial-and-
error. In this speciﬁc application, the number of layers was two, and nine combinations of
the numbers of neurons for each layer were examined in terms of root mean squared error
to ﬁnd the optimal number of neurons in each layer. For each layer, 5, 10, and 15 neurons
were selected for the combinations. For each case, 500 data sets were used to test each
combination.

DNNM has a ﬁxed number of layers and neurons according to the principles of
network architecture (Section 3.3.3). Therefore, the size of the receptive ﬁeld that
determines the robustness of the network needs to be optimized by using trial-and-error.
The parameter is represented as “spread” in the actual computer code. With varying

“spread”, the results in terms of RMSE were compared, and the value shows the

96

minimum RMSE was selected as an optimal “spread.” For this trial-and-error procedure,

100 data sets from D2 were used to test each case.

4.6 Process Optimization Strategies

4. 6.1 Combinations of Techniques

The optimization techniques utilized in this research are global optimization
algorithms, which can be coupled with any type of process model. Thus, various
combinations of optimization strategies are possible, depending on the optimization
algorithm, type of process model, and method of control function parameterization. Table
4.6 shows the optimization strategies that were evaluated in this study. The numbers in

Table 4.6 were used throughout this study to refer to a speciﬁc optimization strategy.

Table 4.6 Optimization strategies (by strategy number), according to their process
model, the method of control profile parameterization, and optimization algorithm.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

\\ Optimization Algorithms

Model “mm" PM.” GA SA ICRS/DS
parameterization

PLI 1 2 3

FEM FS 5 6 7

111 ll [ PLI 9 10 11

D FS 13 14 15

Ill 11 l PLI N/A N/A N/A

S ’R FS 21 22 23

Single-stage 25

FEM Double-stage 29

Multi-zone 33

Single-stage 26

DNNM Double-stage N/A 30

Multi-zone 34

SNNM;S Single-stage 27

SNNM D Double-stage 31

SNNM_M Multi-zone 35

 

 

 

 

 

PLI: piecewise linear interpolation; F S: Fourier series

97

SNNM_ R was not trained with actual control values, but with Fourier
coefﬁcients. Thus, control function parameterization with PLI could not be applied,
because SNNM was speciﬁc for trained data type. All of these combinations (Table 4.6)

were designed to ﬁnd ideal and continuous optimal control proﬁles.

4. 6.2 GA Based Strategy

Genetic algorithms (Section 3.2.1) have some parameters that must be set before
running the algorithm properly. For genetic algorithms, setting a convergence criterion
using the objective function value is difﬁcult, because sometimes the same objective
function value wins the competition for several or more generations. Therefore, the total
number of generation cycles was set to 400 as a default value, because most of the
signiﬁcant convergences tended to occur around 200 generations, according to prior
observations. The algorithm was terminated when it reached the target generation number.
However, if convergence was not achieved at the ﬁnal number of generations, the
optimization process was continued until no improvement was observed. The size of
initial population and the population of each generation was set to 10 in this research,
because the number is not too large to demand too much computation, nor too small to
cause slow convergence. In addition, there is no speciﬁc rule for setting the number in the
initial population (Venkataraman, 2002). Other parameters are also problem-speciﬁc. If
large numbers are selected for the other parameters, it will demand huge computation
time, depending on the process model. Thus, rate of simple crossover, arithmetic
crossover, mutation, and immigrants of each generation were set to produce 6, 6, 4, and 2
Offspring, respectively. Mutation rate was 0.016 (=1/64), which replaced one parameter

with a random value among 64 parameters. At the end of each generation, all the

98

population was tested to identify qualiﬁed individuals that satisﬁed all the constraints,

and they were moved to the next generation.

4. 6.3 SA Based Strategy

As indicated in Section 3.2.2, there are several critical parameters for a simulated
annealing algorithm, such as cooling schedule, number of Markov chains at a temperature,
transitional probability ﬁinction, and convergence criteria.

The following cooling schedule was used:

A
T =—+B 4.33
k k+1 [ I

where,

k=1...N
(To-TNXNH)

A =
N
B = To - A
The above cooling schedule generates more moderate acceptance probability
(between 0.5~0.8) than high acceptance probability (above 0.9), which reduces
exhaustive exploration at the initial search. The initial temperature, ﬁnal temperature, and
the number of cooling steps were set to 25, 0.001, and 150, respectively. These

optimization parameters generate a probability curve (Figure 4.8) showing less

exhaustive exploration in the early stage, which is desirable in this speciﬁc application.

99

 

 

 

 

 

 

 

 

30] - 1.2
l Cooling Schedule I [
- - - -Acceptance Probability;
25 _ —— '— +1
20 "l ‘ s -. 08 g
s D
g 15 ., ~‘ -. 0.6 3
E ~. =
I- . a
10 -- r ‘ -- 0.4 §
‘ <
\
8
5 ‘ . 1» 0.2
S
\
0 i 4 i . i t T . * I: ' ‘r . . i 1 0

0 10 20 30 40 50 60 70 80 90 100110120130140150
Number of temperature reduction

Figure 4.8 Acceptance probability (Boltzmann probability) along with cooling
schedule (T ,=25, TN=0.001, N=150, and average increment of accepted objective
value=0.3) for SA.

In this research, a Boltzmann probability distribution function was used as a
transitional probability function (acceptance probability), which is classic and simple.
The algorithm was coded to terminate its search when there is no acceptance and no
improvement of the objective ﬁrnction. The number of iterations at each cooling cycle,
which is the length of Markov chain, was set to 20 from prior observations.

The algorithm generates the next move by using random search direction and
stepsize for each parameter or discrete value. Variable stepsizes were used, depending on
the control vectors (temperature, humidity, and impingement velocity) and type of

control function representation (discretization or parameterization), because the

100

sensitivity of each parameter or value is different. Depending on the structure of the
control vector (Section 4.4.1), different step-sizes were used as follows:
Linear interpolation parameterization:
o Stepsize[p1~p21]=5; temperature proﬁle
0 Stepsize[pzz~p42]=5; humidity proﬁle
- Stepsize[p43~p63]=5; impingement velocity proﬁle
0 Stepsize[p64]=10; cooking duration
For Fourier parameterization:
o Stepsize[p,~p20]=2; stepsize[p2,]=10; temperature proﬁle
0 Stepsize[p27~p41]=2; stepsize[p42]=5; humidity proﬁle
0 Stepsize[p43~p62]=2; stepsize[p63]=2; impingement velocity proﬁle
0 Stepsize[p64]=10; cooking duration
The search direction was produced by using uniform random distribution of interval

between -O.5 and 0.5.

4. 6.4 I CRS/DS Based Strategy

The ICRS algorithm has three important heuristic parameters, kl, kg, and ne. The
k, parameter controls the size of the search step by changing the magnitude of the
standard deviation vector. However, if the search fails to ﬁnd improvement before the
number of trial reaches the failure counter (F), the standard deviation vector is
subsequently reduced by using kg. The ne integer parameter controls the rate of
convergence by changing the failure counter value. Pan (1998) recommended default
values of kl=1/3, k2=1/2, and ne=4. However, the stepsize of search depends on the
method of control proﬁle parameterization. If the control parameters are highly sensitive,

101

then k1 needs to be reduced. A k, value of 0.05 was used for the case of Fourier
parameterization, because the Fourier coefﬁcients are more sensitive than the discretized
values in PLI parameterization. For the case of linear interpolation parameterization (PLI),
the default value k,=1/3 was used.

According to the original ICRS/DS algorithm, the failure counter (F) is ac times
twice the problem dimension. The problem dimension of this research is 64. Thus, the
failure counter must be 512, which demands too much computation time. By using some
trial runs, the failure counter was set to 100, because it was rare to reach 100 or more. For

convergence criterion, Equation 3.19 was used with the tolerance value of 1.5X10'5.

4. 6.5 Benchmark T est for Optimization Algorithms

Even though the principles of GA, SA, and ICRS optimization algorithms are well
established and proven, the developed computer codes must be validated for their
effectiveness. Especially, in this application, a multi-variable problem is desirable,
because the optimization problem of this study is multivariate. The Bezier23 curve is a
good tool to observe the behavior of all the variables in two-dimensional space
(Venkataraman, 2002). A ﬁﬁh-order Bezier curve-ﬁtting problem was presented to each

computer code. The ﬁfth-order Bezier curve is:

B(t)=Z[:]ﬂ(l—t)"_iti, t6 [0,1] [4.34]
i=0

where,

 

23 Bézier curves were widely publicized in 1962 by the French engineer Pierre Bezier, who used them to
design automobile bodies. The curves were developed in 1959 by Paul de Casteljau using de Casteljau‘s
algorithm. Bézier curves are widely used in computer graphics to model smooth curves.
(http://en.wikipedia.org/wiki/Bezier_curve)

102

n=degree
[qti(1—t)"—i, i = 0,---,n Bernstein basis polynomials
2

For n=5, the ﬁfth order Bezier function becomes:
B(t) = P0 (1 —t)5 + 5Plt(l -z)4 +10P2t2(1—t)3 +10P3t3(1-t)4 + 5P4t4(1—t) + 105:5
[4.35]

Equation [4.35] was used to minimize root mean squared error with the following curve:

f (x) = 1 + 0.25x + 2e_x cos 3x [4.36]

A total of eight design variables were used to ﬁt the above curve. All the computer codes

associated for this benchmark test were adopted from the works of Venkataraman (2002).

4.7 Case Studies

Global optimization techniques developed and tested in the above sections were
applied to practical situations to validate their potential application to real, industrially-
relevant processes. Even though the optimal control proﬁles are the “best” in terms of
mathematical application of the theories, it is practically challenging to apply the
theoretically “best” proﬁle in the real life situation, given constraints of existing
equipment and knowledge. Therefore, a series of case studies (single-stage, double-stage,
and multi-zone) that are currently available, or at least practical with minimal changes of

cooking system conﬁguration, were solved and compared to each other.

103

4. 7. 1 Single-Stage Oven

The single-stage oven case represents an oven (J SO-IV) using constant control
proﬁles during its process. The control variables were constant temperature, humidity,
impingement velocity, and cooking duration. Because the oven uses a constant proﬁle,
the actual number of control variables can be reduced down to four instead of 64, which
was used in the previous applications.

Three different models were used in this case study, FEM, DNNM, and SNNM_S.
The previously developed FEM and DNNM models were used without alterations.
However, the SNNM_S, was trained with the D3 training data group, because the static
neural network model was speciﬁc for training data set. The FEM, GRNN, and SNNM_S

were coupled with SA, GA, and ICRS/DS, respectively.

4. 7.2 Double-Stage Oven

The double-stage oven case is considered as the second possible option that
occurs in real commercial applications, by connecting two single-stage ovens in series.
This is a relatively simple solution for the food processing industry. The double-stage
system generates an equidistant, two-step constant control proﬁle, which has more
dynamics in control than does the single-stage oven. Although more capital cost is
necessary to purchase an additional unit, increased yield and production rate may justify
cost. In this case study, only the total yield of the system was considered and optimized.

The total number of control variables was seven, given two steps for each control
proﬁle and a cooking duration. FEM and DNNM were not altered, but the SNNM_D was
trained with 2,187 data sets of data group D4, because the SNNM_D was a static model.

The 2,187 training data sets were combinations of three different levels and the seven

104

control variables (Table 4.5). The FEM, DNNM, and SNNM_D were coupled with GA,

SA, and ICRS/DS, respectively.

4. 7.3 Multi-Zone Oven

This conﬁguration may be a suggestion for the oven manufacturing industry.
Among the three major control variables (temperature, humidity, and impingement
velocity), varying impingement velocity is the easiest design change to implement in the
existing system, because the local jet exit velocity can be controlled by the size of the jet
exit or other ﬂow control mechanisms. By observing this case study, the effect of a multi-
zone concept was tested.

Control temperature and humidity remained constant throughout the process, and
four equidistance zones were considered for different impingement velocities. Thus, the
total number of control variables was seven, including the process duration. SNNM_M
was trained with 2,187 data sets of data group D5, which were the combinations of three

different levels for each variable according to Table 4.5.

4. 7.4 An Example of Economic-Based Optimization

As the level of optimization is expanded from a single unit operation to an entire
plant, the best control proﬁle of the single unit may be reevaluated to improve the overall
objective of the plant. Compared to unit operation optimization, economic-based
optimization requires much more information related to management, such as product
value, energy cost, labor cost, warehouse management, raw material prices, and so on. In
addition, the objective can vary, such as minimizing energy consumption, maximizing

Processing rates, maximizing proﬁt, or maximizing a practical quality attribute.

105

In this case study, capacity and geometric information for the oven was obtained
from speciﬁcations of the J SO-IV (model 4044). For simplicity, raw material feeding rate,
energy cost, and ﬁnal product yield were considered as factors inﬂuencing the net proﬁt
of an entire system. Even though many of the values for these factors were “best guess”,
it was considered adequate to illustrate the concept of economic-based optimization. To
make the problem as a single objective optimization, all the factors were converted into
price ($) with the following equations. The economic-based optimization problem was

deﬁned as follow:

Objective function (f): maximizing proﬁt [S]
f = product value - feed cost - energy cost :

. Energy cost ($/hr) = (steam cost) + (electricity cost) + (thermal cost)
= a 'fH(H)+ﬂ-(fV(V)+fB(B))+r°K'fr(T)
' Product value ($/hr)= ,u - Y ,where Y = fy (H ,V,T ,t) - F
I Feed cost (raw material: frozen beef patties)($/hr)=¢ - F ,where F = 3600 - A - B
Variables:
- fy (T, H, V,t) [%]: process model
- f H (H ) = 0 ~ H [kg/h]: steam consumption rate as a function of H

' fV (V) = u - f ' A - V [kW]: electrical demand as a ﬁanction of V

2
' f B (B) = 3%6—3) - 10'3 [kW]: electrical demand for transporting patties

(kinetic energy required to transport patties on conveyor belt was divided

by residence time of a patty(L/B))

106

Parameters:

500,000
1 50

 

fT (T) = [ -(1.8T + 32) + 1,170,000] - (1055.04) [J/h] (Fulton®

thermal ﬂuid boiler (capacity: 2.2 MMBtu/h) was modeled. A linear

relationship between operating temperature and power was assumed)

H: humidity set point [%MV]

V: impingement velocity set point [m/s]

T: temperature set point [°C]

fH: amount of steam as a ﬁmction of humidity set point [kg]

fv: electricity'as a function of impingement velocity set point [kW]
fT: amount of natural gas as a function of temperature set point [J/h]
fy: cooking model output or yield [%]

f3: electricity as a function of belt speed [kW]

Y: production rate [kg/h]

1F: feed rate [kg/h] as a function of process duration

B: belt speed [m/s] (from (24 ﬁ)/(60 s) to (24 ﬁ)/(600 s)) as a function of

process duration

o=0.0092: steam cost [$/kg]

B=O.30: electricity cost [$/kWh]

7:0.18: natural gas cost [$/kg]

u=7.7l: product value [$fkg] (3.50 S/lbs)
(p=4.19: raw patty price [$/kg] (1.90 $/lbs)

107

N34: loading capacity of patties per unit length of oven [kg/m] (258 patty

(120X10mm) on cooking area assumed)

I 0:5: amount of steam per hour proportional to the process humidity (linear
relationship was assumed from 90%-1,000 lbs/hr required) [kg/(h'%MV)]

I 0:0.33: electricity demand unit ﬂow rate of air [kW/(m3/h)]

I K=2E-82 kg of natural gas to produce 1 J [kg/J]

I f: fraction of nozzle open area (0.0683): total slot open area per total belt
area or cooking zone area

I L=7.32: cooking zone length (24 ft) [m]

I w=1.85: width of conveyor belt [m]

I A=L><w=13.6: cooking zone area (146 ftz) [m2]

I m=64.6: total product weight on cooking area (43 X6=258 patties><113.5

g=29.3 [kg]

Control variables:
I T: Process temperature [°C]
I H: Process humidity [%MV]
I V: Impingement exit velocity [m/s]
I t: Process duration [5]
Explicit constraints:
I Upper and lower bound of T, H, and V (Equation [4.26—28])
I Cooking duration can be expressed via belt speed:

Bf,L(= 0.012) s B s Bf,U(= 0.12) [m/s]

108

Implicit constraints:
I The constraints, such as microbial lethality, internal color change, and surface
color, are imbedded in the cooking model implicitly.
Assumptions:
I J SO-IV was assumed to have no transitional (inlet or outlet) cooking/equilibration
zone (e.g., steam tunnel), and was therefore assumed in this entire project to be
just a impingement zone.

I Only steady-state operation was considered.

The above objective function seems like a linear function, but it is nonlinear,
because of the nonlinear cooking model. Thus, global optimization techniques were
applied to the problem. Different cooking models (FEM and SNNM_S) were used for
comparison. Even though the parameters of the above equations were estimated by guess,

the effect of parameters was assessed through sensitivity analysis.

4.8 Programming and Computation Tools

A high level computing language, MATLAB® (The MathWorks, Inc., Natick,
MA), was used for all the computer programming of the research, because of its
convenience of handling matrix and vector formulations, which are the major operations
of artiﬁcial neural networks, optimization algorithms, and ﬁnite element modeling. Also,
neural network toolbox 4.0.1 in the MATLAB 7.0.0 provides neural networks, such as
GRNN and FFNN, which were used in the computer codes.

High performance PC’s (3.2 GHz, Intel Pentium 4 hyper threaded) were used to

run the optimization algorithms using GRNN and FFNN. To run optimization algorithm

109

coupled with FEM, server computers (Sun Fire v880, 4 UltraSPARC-III 750MHz 64 bits
CPU) were used.

By using those computers, approximately >20,000 FEM runs, > 20,000 ANN runs,
and >100 optimization runs were calculated. In total, all of these runs required an

estimate of >250 h of CPU time.

110

5 RESULTS AND DISCUSSION

5.1 Integrated Process Model Performance

The integrated process model was developed by adding quality prediction kinetics
(Section 4.3.3 and 4.3.4) to the existing ﬁnite element model (Watkins, 2004) that was
previously validated (Section 4.3.1). The performance of the integrated model was tested
qualitatively with various control proﬁles, but not quantitatively, because of the lack of
actual experimental data encompassing the complex output from the model. Even though
the accuracy of the model was not measured with experimental values, a reasonable
working model is sufﬁcient to test the effectiveness of various Optimization techniques
and alternative models. Figure 5.1 and Figure 5.2 are example model predictions under
constant control proﬁles and dynamic control proﬁles, respectively.

Although there were huge log reductions in some cases, the predicted log
reduction of Salmonella and internal color change were used without truncation at the
target log reductions (6.5 and 1.4, respectively) to increase the performance of neural
networks. In some cases, instability of the prediction of yield and surface temperature
were observed at the very early stage. However, the instability was minimized by
decreasing the time step of model calculation from 1 to 0.5 second.

The surface color prediction in the moist process condition (Sec. 4.2.4) was well
represented in Figure 5.1 and Figure 5.2, which represent high and low humidity process
conditions, respectively. The surface color change in Figure 5.1 showed a very slow rate
throughout the process, because the surface temperature could not reach the process dew
point temperature (2197.2 °C dew point, 90%MV). Under the ambient dew point

temperature, condensation of moisture is predominant on the surface of a patty, which

111

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113

suppresses the browning reaction. However, the surface temperature in Figure 5.2 was
much higher than the highest dew point temperature (281.7 °C dew point, 50%MV),
which accelerated the simulated browning reaction on the surface. In Figure 5.2(d),
surface color change remained within upper (0.497) and lower (0.263) bounds for about

200 seconds in the later stage.

5.2 Training and Optimizing Neural Networks

5. 2.1 Dynamic Neural Network Model (DNNM)

The DNNM was developed by using four GRNN trained with a training data set
(in data group D1) consisting of 53,442 states that were sampled out of the 910 results of
the ﬁnite element model, according to the dynamic training methodology of the Section
4.4.3. The only network parameter of the GRNN was “spread”, which determines the size
of receptive ﬁeld of the network. In this study, each “spread” was determined at a point
minimizing RMSE of each GRNN (yield, Salmonella inactivation, internal color change,
and surface color change) in DNNM. Firstly, the networks were trained with data group
D1, and their prediction RMSE for each trial “spread” was computed by using 100
random data sets from data group D2 and plotted with respect to each trial “spread”
(Figure 5.3).

The DNNM consisted of four different GRNN. For the yield prediction GRNN,
0.25 was determined as the optimal “spread” around the lowest RMSE. For the other
constraint GRNN, 0.2 was to be a commonly optimal “spread.” Thus, those optimal

“spreads” were used to train each GRNN that was speciﬁc for each of the four outputs.

114

RMSE of Yield Prediction [%]

 

0 0.1 0.2 0.3 0.4 0.5 0.6

     
 

  

 

SPREAD
(a)
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as V ‘°
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0.49 ”33—" +—+—+—+- 9—1—1—9- + ——1 0.100
0.05 0.10 0.20 0.30 0.40 0.50
SPREAD
(b)

Figure 5.3 Optimal “spread” values at the lowest RSME were determined by trying
“spread” for each GRNN in DNNM: (a) Yield prediction; (b) Salmonella
inactivation, internal color change, and surface color change.

115

5. 2.2 Static Neural Network Models (SNNM)

5.2.2.1 SNNM for random and single-stage process

The optimal number of neurons for each hidden layer needs to be determined for
SNNM_R (SNNM for random process), because the model uses a feed-forward neural
network. The optimal number of neurons of the two hidden layers was determined by
simulating 9 different combinations. The combinations were designed with full factorial
(5, 20, and 40 neurons). The RMSE for yield, Salmonella reduction, internal color change,
and surface color change were calculated for each combination. However, the RMSE of
each category could not be compared with each other, due to the scale of each category.
Thus, to determine the optimal combination of neurons, the RMSE of each category was
normalized with its possible maximum value (100, 10, 10, and 2) and averaged. Then the
averages of the normalized RMSE were compared with each other.

The lowest average of the normalized RMSE (0.029) was observed with the
combination of 40 and 20 neurons in the ﬁrst and second hidden layer, respectively
(Figure 5.4). Therefore, two hidden layers containing 40 and 20 neurons were used for
the architecture of SNNM for the random process (SNNM_R). This architecture was also
used for the training of SNNM for a single-stage process (SNNM_S), without validation,
because the effect of the variation of the number of neurons was insigniﬁcant by

observation.

116

Neurons of
the Second
Hidden Layer

 

Neurons of the First
Hidden Layer

Figure 5.4 Average normalized RMSE of each combination for the number of
neurons in each hidden layer of SNNM_R.

5.2.2.2 SNNM for double-stage and multi-zone process

The SNNM for double-stage (SNNM_D) and multi-zone processes (SNNM_M)
was developed by using GRNN instead of FFNN, because of the poor performance of
F FNN by trial and observations. Various values for “spread” were tested in terms of
RMSE. Because the SNNM had a neural network to predict yield, Salmonella reduction,
internal color change, and surface color all at the same time, the “spread” should be
chosen by negotiating among RMSE of the outputs. The “spread” for SNNM_D was
negotiated around 0.6, which gives more accuracy to constraints prediction than goal
prediction (Figure 5.5). Also, 0.55 was chosen for the “spread” of GRNN_M by
negotiating accuracies among outputs (Figure 5.6). Other values can be used as long as

those values are not offset too much from a reasonable “spread.”

117

 

1 —<>— RMSE(% Yield)
12 " -I- RMSE(Sa|monella Inactivation)
+ RMSE (lntemal Color Change)

~— 0.35

.a
O

+ RMSE (Surface Color Change)

 

~ 0.3

 

>- 0.25

 

RMSE (Yield, Salmonella
lnactviation, lntemal Color Change)
RMSE (Surface Color Change)

 

 

 

 

 

Figure 5.5 RMSE (yield, Salmonella inactivation, internal color change, and surface
color change) of SNNM_D for different “spread” values.

 

 

 

 

   

 

 

—<>— RMSE (% Yield)
10 +Salmonella Inactivation 0’28
’g 9 +lntemal Color Change 10.25
c + Surface Color Chan e 0’
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Figure 5.6 RMSE of SNNM_M for different “spread” values: (a) Yield prediction;
(b) Constraints prediction.

118

5.3 Validation of the Trained Neural Networks

5. 3.1 The Performance of DNNM

5.3.1.1 DNNM for Random Fourier Process

To increase the accuracy for yield, Salmonella inactivation, internal color change,
and surface color change prediction, four DNNM’s were trained by using GRNN. For
validation of the networks, 1,000 conditions were generated randomly by varying the 20
coefﬁcients of the Fourier series for each control vector (temperature, humidity, and
impingement velocity) and cooking duration. Thereafter, the 1,000 conditions were
applied to the ﬁnite element model, and 1,000 resultant data were obtained. The
conditions at every 10 seconds and the initial state values were cast to the four
individually trained DNNM, and the output vectors were compared with the result of the
ﬁnite element model. The accuracy of the trained DNNM was calculated in terms of

RMSE (Table 5.1).

Table 5.1 The performance of DNNM for a random data group (D2 of Table 4.4), in
terms of RMSE.

 

 

 

Salmonella Internal Color Surface Color
Yield Inactivation Change Change
[%] (>O.813)** (>O.146)** (0.26~0.497)
[log(log(N/No))] [10g(logLC/Co))] [10g(C/Co)]
RMSE 2.05 0.63 0.55 0.12
Number Of 1,000 1,000 1,000 999*

Samples
RMSE: Root mean squared error.
* One of the data in the 1,000 samples was eliminated due to overﬂow.
** log of critical limit for each factor (O.813=log(6.5), 0.146=log(1.4))
Figure 5.7 shows several example of the network prediction qualitatively. In

Figure 5.8, the performance of the individual DNNM was represented graphically.

119

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120

The slope (0.98) of the regression line of yield (Figure 5.8 (a)) prediction was
close to l, and the intercept (1.83) was also small, which means that the model represents
the characteristics of the ﬁnite element model. Even though some points deviated more

than 10% from the 1:1 line, the overall accuracy was fairly good (RMSE = 2.05).

.A

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Patty Yield by DNNM [%]

 

(ﬂ

    
 
 

   

_..-.. 0813 /
, ’///.-'////// 1 Tru e -p a S 5
Fa | s e-p

./..

  
 
 

byDNNM

True-fail

Salmonella Inactivation (log(log(N/No)))

Salmonella Inactivation (log(log(N/No))) by
FEM

(b)

Figure 5.8 The performance of DNNM for random process as a predictor and
classiﬁer. (a) Yield; (b) Salmonella inactivation; (c) Internal color change; (d)
Surface color change (some data in NE region was not plotted).

121

Surface Color Change (log(C/Co)) by DNNM
O

lntemal Color Change (log(log(C/Co))) by

0.8

9
a)

 
   
     
  

  

True-pass

 

// l /’

   

.. A "

    

lntemal Color Change (Iog(log(C/Co))) by FEM

(C)

  

   

01263 0.497

E E True-fail
True-fail EFalse-fail E ,- '

: an... .3 . .

' '1‘ 32727-74577'2212 0-497

         

2 False-pass }

 
 

   

 

0.2 0.4 0.6 0.8
Surface Color Change (log(C/Co)) by FEM

((1)

Figure 5.8 (cont’d).

122

The Salmonella inactivation results were divided into four regions, with a
crosshairs at the value 0.813 (=log 6.5) in Figure 5.8 (b). Salmonella inactivation works
as a constraint in the actual optimization problem. There are two critical aspects of the
model prediction: constraints satisfaction (i.e., classiﬁcation) and goal prediction
accuracy. If there is an order of priority between those two aspects, the constraints
satisfaction must be placed ﬁrst, because it determines whether a process “passes”.
Therefore, the prediction results were analyzed according to classiﬁcation categories
(Table 5.2).

The samples in the true-fail region (Figure 5.8 b) are under-cooked sample, in
terms of microorganism inactivation. Again, the samples in the true-pass region passed
the critical standard log reduction value. However, some points in the false-fail region
were under-estimated samples, which mean safe estimation while sacriﬁcing chance to be
explored. In the false-pass region, there were about 1.7% of over-estimated samples out
of the 1,000 points, which was an small number of false-passes. However, the false-pass
region is the most dangerous one that must be avoided among the classiﬁcation

categories.

Table 5.2 Four possible classiﬁcation categories of the constraints satisfaction.

 

Neural Network Prediction

 

 

FEM Prediction Pass Fail
Pass True-pass False-fail
Fail False-pass* True-fail

 

* Most dangerous prediction

Plot (c) in Figure 5.8 shows the result of internal color change prediction. With

the same rationale as the Salmonella inactivation case, samples in the true-pass, true-fail,

123

and false-fail region are considered as safe classiﬁcation. Only a small portion (53 points)
of the 1,000 data points is located in the false-pass region, which is not desirable.

Data points in Figure 5.8 (d), were divided into 9 regions, because the surface
color change constraint has lower (0.26) and upper critical limits of (0.497). The points in
the true-fail, true-pass, and true-fail region were classiﬁed correctly and safely. However,
about 11% of the 1000 points in the two false-pass regions were classiﬁed as a dangerous
category.

Generally, the trained DNNM showed good prediction and classiﬁcation
performance for the random process, even though the DNNM was trained by using the
results of constant processes. The false-pass region is inevitable for a model, but the
region must be avoided when the optimization algorithms explore solution space with the
model.

Among the 1,000 random data, 170 points that satisﬁed the safety and quality

constraints were sorted out and superimposed on the yield plot (Figure 5.9).

 

10° “1 h-1000randofn data

 

 

      

E 90 1 L oAccepted Patties ('6‘; '
2 80 1 . "
Z . . -
E 70 _. 360/0 51% . f ' .
> I ,oo 3...
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E 40 ~ _'
30 ﬁb—L‘T—TJ _ T' ' "#ﬁ’ _ "‘T’VW*_1

 

30 40 50 60 70 80 90 100
Patty Yield by FEM [%]

Figure 5.9 Accepted patties (170 patties) were superimposed on the 1,000 random
data.

124

The accepted patties were located between 36% and 51% yield. This fact does not
mean that the optimal solution may exist in this range. Rather, the optimal point may be
in some region beyond 51%, but with a very low probability. However, GRNN based
dynamic neural network model can search the region beyond 51%, because of the nature
of the model, which is explained in detail in the Section 4.5.4. Also, the plot implies that
optimization algorithms will progress fast in this range (36 - 51%), because of the high
density of training data.

The performance of a neural network can be measured by two different criteria,
depending on the characteristics of the network. Goal network predicting scalar, such as
patty yield, can be estimated via the RMSE. However, constraint networks that determine
“Yes” or “No”, “Passed” or “Failed”, or “True” or “False” (such as safety and quality)
can be estimated by using classiﬁcation rate. In Table 5.3, the classiﬁcation rates of
DNNM as a constraint network were divided in four categories. A critical value is the
error of classiﬁcation for false-pass category. The rate was 0.15 for DNNM network

applied to random processes.

Table 5.3 Classiﬁcation rate of DNNM for random processes.

 

Classiﬁcation Categories

 

 

 

 

 

Correct Incorrect
True-pass* True-fail F alse-fail False-pass Total
NN prediction 170 647 71 112 1,000
JFEM results) (241) (759) (241) (759) (1,000)
R,“ 0.71 0.85 0.29 0.15

 

* Sample passed all constraints (Salmonella inactivation, internal color change,
and surface color change)
** Rate of classiﬁcation = DNNM prediction / FEM results

125

5.3.1.2 DNNM for Single-Stage, Double-Stage, and Multi-Zone Process

DNNM is a dynamic neural network model, which can be applied to any type of
input patterns. Therefore, the performance of the DNNM must be general and robust. To
measure the robustness, the performance of DNNM was measured for single-stage,
double-stage, and multi-zone processes.

Randomly produced single-stage, double-stage, and multi-zone data were applied
to the DNNM, and RMSE’s for yield, Salmonella inactivation, internal color, and surface
color prediction were calculated for each case (Table 5.4). The RMSE of the single-stage
case showed good accuracy for goal prediction (yield, 1.67%) and constraint predictions
(0.49 for Salmonella inactivation, 0.42 for internal color change, and 0.76 for surface
color change). Such accuracy was possible, because the DNNM was trained with single-
stage data. It is not strange that the accuracy for the multi-zone case was not much
different from the accuracy of the single-stage case, because the multi-zone process has a
similar input data pattern to the single-stage, except for a variable velocity proﬁle.
Generally, the accuracies of double-stage and multi-zone cases were in the reasonable

range, which supports the robustness of the dynamic neural network model.

Table 5.4 The performance of DNNM for single-stage, double-stage, and multi-zone
processes in terms of RMSE.

 

 

 

 

 

Salmonella lntemal Color Surface Color
Yield Inactivation Change Change

[%] (>0.813)** (>0.146)** (0.26~0.497)"
Process Type Performance [log(log(N/No))] [log(1og(C/Co))] [long/Coﬂ
Single-stage 1333:) 1.67 (998*) 0.49 (998*) 0.42 (998*) 0.76 (998*)
Double-stage ($1335) 3.22 (994*) 1.20 (994*) 1.04 (994*) 0.90 (990*)

Multi-zone RMSE

Process (# data) 1.91 (1,000) 0.57 (1,000) 0.50 (1,000) 2.09 (999*)

 

RMSE: Root mean squared error
* Some data in the 1,000 samples was eliminated due to computation overﬂow.
** Critical limits

126

Figure 5.10 shows validation plots of DNNM for the single—stage oven. All the
predictions were very close to the 1:1 line. Other validation plots for double-stage
(Appendix A: Figure A.l) and multi-zone process (Appendix A: Figure A.2) can be
found in the appendix.

The performance of DNNM for various processes was also measured in terms of
the rate of classiﬁcation (Table 5.5). Classiﬁcation rate for false-pass category for the
single-stage case was very low (0.02) as expected. However, false-pass rates for double-
stage process were high compared with the other processes. DNNM is a robust model but
is not perfectly free from the characteristics of training data sets, which is a single-stage
process. The double-stage process deviates more from a single-stage process than does
the multi-zone process, and therefore shows the highest false-pass error for DNNM

among other process types.

Table 5.5 Classification rate of DNNM for single-stage, double-stage, and multi-zone
processes.

 

Classiﬁcation Categories

 

 

 

 

 

 

 

 

P Correct Incorrect
TOCCSS
Type . ' False-
True-pass“ True-fall False-fall pass Total
. NN prediction 162 785 38 15
3:281“ (FEM results) (200) (800) (200) (800) (1,000)
g" R," 0.81 0.98 0.19 0.02
NN prediction 80 682 116 122
”$2“ (FEM results) (196) (804) (196) (804) (1,000)
g R, 0.41 0.85 0.59 0.15
Multi- NN prediction 129 772 71 27
zone (FEM results) (200) (799) (200) (799) (9991)
Process Rc 0.65 0.97 0.36 0.03

 

* Sample passed all constraints (Salmonella inactivation, internal color change,
and surface color change)

" Rate of classiﬁcation = DNNM prediction / FEM results

1' Some data was eliminated due to computation overﬂow.

127

Patty Yield by DNNM [%]

 

 

 

o 7 I f 7 l V ‘ﬁ
0 20 4o 60 80 100
Patty Yield by FEM [%]

(a)

   

   
      
  
  

    
 

 

Salmonella Inactivation (log(log(N/N0))) by

 

- --------------- 0.813
4 6 8 10
-1O
Salmonella Inactiwltion (Iog(log(N/N0))) by FEM

(b)

Figure 5.10 Goal and constraints prediction performance of the DNNM for single-
stage process: (a) Yield; (b) Salmonella inactivation; (c) Internal color change; (d)
Surface color change.

128

     

     
 

  

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129

5.3.2 The Performance of SNNM

The performance of a static neural network model is speciﬁc to the training data,
which is in contrast to the performance of the DNNM trained by a single data group D1.
Four different SNNM were trained by using random, single-stage, double-stage, and
multi-zone process data. The training data for single, double, and multi-zone process
were generated according to Section 4.4.2. Then the feed-forward neural network was
used for all predictions (yield, Salmonella inactivation, internal color change, and surface
color change). However, the feed-forward neural network was not trained well for double
and multi-zone processes. Thus, a generalized regression neural network was used for
double and multi-zone processes, and improved results were obtained.

The performances of the trained SNNM were validated by using test data groups
in terms of RMSE (Table 5.6). The RMSE of yield for random and single-stage
processes were 1.78 and 2.52, which were much smaller than the RMSE for double-stage
and multi-zone process. For safety and quality constraints, the same was true. In addition,
training result of SNNM for the double-stage and multi-zone process, in terms of RMSE
and correlation coefﬁcient, were poor (Figure A.4 and Figure A5 in appendix A).

The performance of SNNM for 1,000 random processes was graphically
represented in Figure 5.11. Compared to the performance of the DNNM (Table 5.1) for
random processes, the SNNM were more accurate for yield, Salmonella reduction, and

internal color change. Surface color change prediction accuracy was almost identical.

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Salmonella Inactivation (Iog(log(N/N0))) by
FEM

(b)

Salmonella Inactivation (log(log(N/N0))) by

 

Figure 5.11 Goal and constraints prediction performance of the SNNM for random
Fourier process condition: (a) Yield; (b) Salmonella inactivation; (c) Internal color
change; (d) Surface color change.

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133

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Suface Color Change (log(C/CO)) by FEM

0.2

The performance of the SNNM as a classiﬁer was also measured (Table 5.7).

Although false-pass rates were similar to each other, false-fail rates of the double-stage

and multi-zone process were double those for random and single-stage processes. Based

on RMSE and classiﬁcation rate, SNNM’s for double-stage and multi-zone processes

were a poor predictor and classiﬁer, and are therefore not appropriate for further

utilization. Therefore, those two SNNM were not used with optimization algorithms. The

reason of such a poor performance could be some problem with the training data group or

inappropriate neural network type.

Table 5.7 Classification rate of SNNM for random, single-stage, double-stage, and
multi-zone process.

 

Classiﬁcation Categories

 

 

 

 

 

 

 

 

 

 

Process Correct Incorrect
Type True- F alse- False-
jass“ True-fail fail pass Total
Random NN prediction 190 702 53 55
Process (FEM results) (243) (757) (243) (757) 41,000)
Rc 0.78 0.93 0.22 0.07
Single- NN prediction 146 739 53 62
stage (FEM results) (199) (801) (199) (801) (1,000)
Rc 0.73 0.92 0.27 0.08
Double- NN prediction 90 729 107 70
stage (FEM results) (197) (797) (197) (797) (9961')
Rc 0.46 0.91 0.54 0.09
Multi- NN prediction 99 733 104 63
zone (FEM results) (203) (796) (203) (796) (9991‘)
Process Rc 0.49 0.92 0.51 0.08

 

"‘ Sample passed all constraints (Salmonella inactivation, internal color change,
and surface color change)
RC: Classiﬁcation rate = NN prediction / FEM results.

1' Some data was eliminated due to computation overﬂow.

5.4 Benchmark Test for Optimization Algorithms

A Bezier curve ﬁtting problem (Section 4.6.5) was solved with three Optimization
algorithms (GA, SA, and ICRS), and the results were compared (Table 5.8). Ten
simulations were executed for each algorithm, and the average and variance were
calculated. The lowest average RMSE was achieved by the ICRS algorithm. The three
average RMSE were statistically compared, and the averages were not signiﬁcantly
different from each other (F -value=1 .42; p-value=0.26). In other words, their

performances were almost identical.

Table 5.8 Comparison of the convergence of three different optimization algorithms
in terms of RMSE for the Bezier parametric curve fitting problem.

 

 

 

RMSE
Algorithms Simulations Average Variance
GA 10 0.4804 0.1334
SA 10 0.2871 0.0586
ICRS 10 0.2580 0.1167

 

GA showed a gradual and moderate rate of convergence through out the total 200
generations (Figure 5.12 (a)). SA showed lots of exploration at the early stages but not
many in the later iterations (Figure 5.12 (b)). The ICRS algorithm converged before the
20th iteration (Figure 5.12 (c)). However, this does not mean that the ICRS is the fastest
algorithm, because the execution time for one generation is heuristic, which is different

from the other algorithms.

135

 

    

 

 

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0 7 rl' l l l l 7

0 50 100 150 40 60 80 100
Generation Generation
(b) (c)

Figure 5.12 Examples of objective function value (RMSE) for every generation of
three different optimization algorithms in the Bezier parametric curve ﬁtting

problem. (a) GA; (b) SA; (c) ICRS.

136

Generally, all computer codes for the optimization algorithms successfully
optimized the Bezier curve-ﬁtting problem. However, the speed of the algorithms could
not be compared, because the total execution time depends on the values of the algorithm

parameters.

5.5 Optimal Conditions from the Various Strategies

5. 5.1 Summary of the Various Optimization Strategies

To understand comprehensive characteristics of speciﬁc optimization strategies
for meat patty cooking under moist air impingement conditions, three models, two
control proﬁle parameterization methods, and three optimization algorithms were
combined (Table 5.9) and tested. However, the SNNM-PLI combinations (strategy #17,
#18, and #19) were not tested, because of problem previously described (Section 4.6.1)

The highest maximum yield (73%) was achieved with the ICRS-FEM-PLI
strategy (#3), and the second highest yield (70%) was achieved with SA-FEM-FS
(strategy #6). However, neural network model based optimization strategies showed
constraints satisfaction problems, meeting safety and quality constraints. This is an
expected classiﬁcation error, because those alternative models were trained using FEM
results, which was considered as an experiment in this research. However, even though
the FEM-based optimization strategies satisﬁed all the constraints, those might actually
not be free from the constraint satisfaction problem, if the results were validated with

actual ﬁeld experiments.

137

Table 5.9 Brief results of the various optimization strategies.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Max Convergence
1 . . ’ Constraints and Optimal
Model CP Algorithm Strategy Y[(l;l]d Satisfaction Proﬁles
0 (Appendix A)
Figure A.6
GA 1 56 S Figure A.7
Figure 5.15
PLI SA 2 64 S Figure A.8
Figure 5.13
ICRS 3 73 S Figure 5.14
FEM Figure A 6
GA 5 60 S Figure A.7
Figure 5.15
FS SA 6 70 S Figure A.8
Figure 5.13
ICRS 7 65 S Figure 5.14
Figure A.9
GA 9 57 US Figure A. 10
Figure A.11
PLI SA 10 52 US Figure A.12
Figure A. 13
ICRS 11 60 US Figure A. 14
DNNM Figure A 9
GA 13 69 US Figure A. 10
Figure A.11
FS SA 14 49 S Figure A. 12
Figure A. 13
ICRS 15 55 US Figure A.14
GA 17 N/A N/A N/A
PLI SA 18 N/A N/A N/A
SNNM_ ICRS 19 N/A N/A N/A
R GA 21 51 S Figure A. 15
PS SA 22 53 US Figure A. 16
ICRS 23 54 S Figure A.17

 

l

S: Satisﬁed; US: Unsatisﬁed

Control proﬁle parameterization method

138

 

Generally speaking, FEM based approaches were the slowest approaches, and DNNM
and SNNM-based approaches were much faster, while sacriﬁcing accuracy. All the
different strategies converged at the end of the optimization process (Figure 5.13, strategy
#3). However, all the converged optimal control proﬁles were not consistent with each
other, which implies that the solution domain might be highly multi-modal and
constrained. In addition, the total number of control parameters was 64 in this study,
which made it somewhat difﬁcult to achieve uniform and consistent optimal yields and
proﬁles. Even so, some strategies showed meaningful optimal control proﬁles,
suggesting stepwise control policies in Figure 5.14 and Figure 5.15 . Polynomial
regression curves were added to the PLI proﬁles to illustrate general trends in process
conditions. Convergence and optimal control proﬁles of the other strategies can be found

in Appendix A.

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(a)
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(a) (b)

Figure 5.13 ICRS-FEM optimization process and convergence (strategy #3): (a) PLI
parameterization; (b) FS parameterization.

139

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Process time [s]

 

 

 

 

 

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0 50 100 150 200 250

Process time [s]

(b)

Figure 5.14 Optimal control proﬁle of temperature, humidity, impingement velocity,
and cooking duration found by ICRS-FEM optimization strategy (strategy #3): (a)
PLI parameterization; (b) FS parameterization.

140

 

I 49— Temperature 0 l
+ Humidity i
+ Velocity |
— Poly. (Temperature) ’
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Temperature [C] I Humidity [%MV] /
Velocity [mls]

 

 

 

 

 
      

 

 

 

 

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Process time [s]
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and cooking duration found by SA-FEM optimization strategy (strategy #6): (a) PLI
parameterization; (b) FS parameterization.

141

5. 5.2 Optimization Performance of the Alternative Models Coupled with Algorithms

In this research, FEM was used as a replacement for actual experiments.
Therefore, the results of DNNM and SNNM were compared with FEM results to test the
performance of alternative models when those models were coupled with various
optimization algorithms.

To measure the model performance via speed, the execution time of the model
itself was measured by using the MATLAB proﬁler function and constant control proﬁles
(600 seconds process duration). FEM, DNNM, and SNNM took 140, 16, and 0.22
seconds, respectively. Compared with FEM, therefore, DNNM and SNNM showed huge
beneﬁt in total execution time of algorithms, due to the iterative nature of the algorithms.

In addition to the execution speed, the model performance was measured via
prediction accuracy, in terms of constraints satisfaction. Table 5.10 showed that most of
the cases committed false-pass classiﬁcation errors, except three cases (strategy #14, #21,

and #23).

Table 5.10 Predicted goal and constraint values of DNNM and SNNM, compared
with the result of FEM.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Objective Constraints
S Salmonella log lntemal color Surface color
trate . reduction change change
gy Y‘e'd [%] (>0.813) (>0.146) (0.26~0.497) 5
(Table [log(log(N/No))] [logﬂgm/COD] [log(C/Co)]
Model 5.9) A F E [%] A F A F A F
9 57 44 29 1.44 4.01 0.15 2.19 0.27 >100*
10 52 48 9 1.48 1.76 0.15 0.29 0.26 0. l 7*
DNNM 1 1 60 57 5 1.47 3.09 0.15 1.43 0.45 0.22“
13 69 65 6 1.71 -0.93* 0.15 -2.14* 0.39 0.15“
14 49 44 13 1.61 3.33 0.15 1.66 0.27 0.37 P
15 55 50 10 1.7 1.41 0.19 -0.04"' 0.30 0.27
21 51 47 10 1.64 2.37 0.15 0.80 0.26 0.32 P
SNNM 22 53 51 4 1.64 1.87 0.15 0.37 0.26 0.21“
23 54 48 13 1.65 2.78 0.15 1.14 0.26 0.41 P

 

A: alternative model prediction; F: ﬁnite element model result; S: constraints satisfaction status; P: passed
all the three constraints; E (Prediction error) =100>< lA-F] / F [%]; *: Unsatisﬁed constraint

142

Also, the prediction error for yield of those successful cases were 10 to 13%.
Another interesting point was that there were more false-pass classiﬁcation errors in
surface color change prediction than the other two constraints, indicating that the surface

color change is reacting sensitively to the control proﬁles.

5. 5.3 Efﬁciencies of Various Strategies

Efﬁciency of an optimization can be measured by dividing total goal achievement

by total execution time (Equation [5.1]).
5A =lYmax “Yol/texe [5°11

However, comparing efﬁciencies of multiple strategies based on absolute fair conditions
seemed impossible, because of the various parameters in the algorithms and a heuristic
search algorithm. Although absolute comparison was difﬁcult, total execution time and
achieved maximum yield are provided in Table 5.11.

To measure combined performance of each strategy, averaged execution time per
unit iteration was deﬁned, which was calculated at the point where a strategy converged
to 99% of the ﬁnal converged value. Generally, the efﬁciencies of DNNM and SNNM
were higher than the efﬁciency of FEM, due to fast convergence. However, the efﬁciency
does not account for accuracy. The efﬁciencies in Table 5.11 do not provide an absolute
fair comparison, but provide a rough reference. Even though a fair comparison was
impossible, the performance of the optimization algorithms thereby can be viewed

qualitatively.

143

Table 5.11 Total execution time and the achieved maximum yield for various
optimization strategies.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Algori Yo Ymax D 8A
Model CP thm Strategy NG,99% tG ten [%] [%] [%] [% yEId/s]
GA 1 86 1.33 114.38 48 56 8 0.07

PLI SA 2 95 10.94 1039.30 40 64 23 0.02

FEM ICRS 3 46 6.72 309.12 38 73 34 0.1 1
GA 5 165 0.58 95.70 53 60 6 0.07

FS SA 6 127 4.9 622.30 40 70 29 0.05

ICRS 7 96 6.03 578.88 53 65 11 0.02

GA 9 554 0.08 44.32 52 57 5 0.1 1

PLI SA 10 130 0.48 62.40 43 52 9 0.14

ICRS 11 231 0.23 53.13 48 60 12 0.22

DNNM GA 13 215 0.05 10.75 51 69 17 1.60
FS SA 14 147 0.12 17.64 38 49 11 0.62

ICRS 15 7 0.08 0.56 38 55 16 28.91

GA 21 23 0.0039 0.09 50 51 l l 1.59

SNNM FS SA 22 N/A 0.00044 N/A N/A N/A N/A N/A
ICRS 23 26 0.0034 0.09 48 54 5 59.39

 

CP: Control proﬁle parameterization; NOW/p Iteration number achieving 99% of the ﬁnal objective value;
to: Averaged execution time per iteration; texe: total execution time; Yo: Initial objective value; Ymax:
Final objective value; D: |Yo-Ymax|; 8A: Optimization efﬁciency [% yield/s]

Strategy #1~7 were run on Sun Fire V880 (4 UltraSPARC-III 750MHz 64 bits CPU), and 9~23 were run on
PC (3.2 GHz, Intel Pentium 4 hyper threaded). Thus, to compare with FEM results, time must be multiplied
by about 4.5.

Global optimization has exploration and exploitation features in its algorithm.
Figure 5.16 shows how the control variables converge throughout the whole optimization
process in GA, SA, and ICRS. In GA (Figure 5.16, (a)), there is no distinct point dividing
exploration and exploitation mode, because the algorithm has the two features in every
generation. However, in SA, there is a distinct section of exploration, which is over 50%
of the entire optimization process, which indicates the algorithm was effective for
searching many possibilities at the early stages of optimization. Following exploration,
the exploitation process gradually starts as the process goes to the later stage of
optimization. However, the exhaustive exploration at the early stage could be negative to

the efﬁciency of SA algorithm, especially when the process model is slow. Compared

144

with SA, ICRS showed no distinctive section of exploration and exploitation; rather, it
was gradual. Therefore, ICRS could avoid exhaustive searching at the early stage by

transitioning smoothly from exploration to exploitation within a small number of iteration.

 

 

 

     

 

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Figure 5.16 Convergence history of each control parameters shows exploration and
exploitation features of the optimization algorithms: (a) GA-FEM-PLI (strategy #1);
(b) SA-FEM-FS (strategy #6); (c) ICRS-FEM-PLI (strategy #3).

145

5. 5.4 Constraints Satisfaction Problem

To analyze the search pattern of a DNNM or SNNM based optimization strategy
that is committing false-pass classiﬁcation errors frequently, the search pathways were
validated with FEM results. A typical example is presented in the Figure 5.17. The
performance index or goal (yield) moved along the 1:1 prediction line, even though the
accuracy was not high (Figure 5.17 (a)). However, the search algorithm traveled to the
false-pass region at the later stage of optimization for all three constraints. This strange
behavior was likely due to erroneous moves that had small slope in the validation plot.
However, if the pathways of the constraints are guided to follow the 1:1 line, the
possibility of the erroneous move can be reduced.

Therefore, a directional constraints satisfaction (DCS) algorithm was developed
and implemented in the ICRS algorithm. The DCS algorithm guides the progress of
constraints in one direction, which is determined at the starting point, until no progress is
observed, and then turns the direction into a direction of more progress. This alternate
process is kept until the goal is converged. Figure 5.18 shows an example of a DCS
implemented ICRS algorithm search behavior. In the ﬁgures, all three constraints moved
along the 1:1 line without deviatory moves, even though a constraint fell into a false-pass
region. Another positive side effect is that the pathway of the goal was also straight,
which means fast convergence. However, the goal achieved was relatively lower than the
algorithm without DCS. This drawback might be possible, because the search algorithm

was initially constrained in one direction while sacriﬁcing explorations chances.

146

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Another strategy to avoid false-pass classiﬁcation errors is using under-estimated
conditions as a starting point of the optimization algorithm. For strategy #23, a highly
under-estimated point was intentionally used as a starting point, which was successful in

avoiding false-pass classiﬁcation error throughout the several simulations.

5. 5.5 Sensitivity of the Optimal Control Proﬁles

Monte Carlo simulation was applied to the optimal solutions to check the

sensitivity of the answer by using Equation [5.2]:
13'.- =pi+0-1‘ [5.2]

By using control parameter (p,) as an average, a normal distribution (F), and a practical

standard deviation (0), a perturbed control parameter (5,- ) was generated for 64 control

parameters. The standard deviations were set as follows: 1 for PLI parameters, which
means actual perturbation of 2H (°C, %MV, m/s, and s) and 0.01 for PS parameters,
which perturbs control proﬁles with almost the same magnitude of the PLI case, except
process duration (=1). A total of 100 simulations were executed for the optimal results of

strategy #3 and #6, because these were the ﬁrst and the second best results, in terms of
patty yield.

For strategy #3 (FEM-PLl-ICRS), 37 simulations showed computational overﬂow,
and the other 63 simulations gave proper predictions (Table 5.12). However, all the 64
simulations were rejected due to the constraints satisfaction test. This is a reasonable
result, because the search algorithm locates a ﬁnal solution with very small stepsize
moves in all directions, which result in a solution very close to the constraint boundaries.

If this high sensitivity of the optimal solution is also true to actual experiments, the

149

adoption of the optimal solution must be careful, because of the variance of the actual
control proﬁles. Therefore, a sub-optimal solution of strategy #3 was checked for
sensitivity. The sub-optimal solution was arbitrarily chosen at 45th iteration (48 iteration
total). Even though the simulation showed almost the same number of computational
overﬂows, the risk of failure was lowered from 100% to 85% (Table 5.12), and the
expected average yield was 64.51i2.02 (95% conﬁdence) among 62 non-failed
simulations, compared with 71% for the optimum. The Monte Carlo simulation was also
applied to strategy #6, with standard deviation 0.01 for Fourier coefﬁcients and 1 for
process duration. This case showed less computational overﬂows, much lower risk of
failure (49%), and higher expected average yield of 68.36i0.62 (95% conﬁdence) than
the case of strategy #3. Therefore, Fourier parameterization can be considered to have

more advantages in real application of the optimal solution in terms of risk of failure.

Table 5.12 Sensitivity of the optimal solutions for FEM_ICRS_PLI and
FEM_SA_FS was obtained by using Monte Carlo simulation.

Number of Total Number Risk of Failure Yield Conﬁdence

 

 

Maximum Failures 0f Samples (Nf/ N1) Interval for N,
Strategy Yield [%] N] N,“ * [%] (95% conﬁdence)
#3 73 63 63 100 N/ A
71 53 62 85 64.51i2.02*
#6 7O 39 79 49 68.36zt0.62

 

* Among nine true-pass cases, an outlier (20% yield) was excluded
** Among 100 simulations, some simulations failed to predict results due to computational over ﬂow.

The simulation results of strategy #6 were represented by using histograms
(Figure 5.19) to see more aspects of the simulations. Figure 5.19 (a) showed that a point
of 10% deviation from the optimal solution was rare but possible. In contrast, there was
also a possibility to have a little more improved result. The constraints for Salmonella
reduction and internal color change had a small possibility of rejection. However, the

surface color change constraint showed more vulnerability to rejection.

150

 

 

 
 
 
 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

    

 

 

 

 

 

 

 

 

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population of Monte Carlo simulation for the strategy #6: (a) Yield distribution of
passed simulations; (b) Salmonella inactivation distribution; (c) Internal color
change distribution; ((1) Surface color change distribution.

151

5.6 Case Studies

5. 6. 1 Single-Stage Process

The ICRS algorithm was applied to ﬁnd optimal constant proﬁles for a single-

stage oven process. Two independent trials, having different starting conditions, were

made for strategy #25, and the results were summarized in Table 5.13.

Table 5.13 The results of the single-stage optimization process by using ICRS-FEM.

 

 

 

 

 

 

Objective Constraints Control Variables
Surface Color
Salmonella lntemal Color Change

Maximum Inactivation Change (0-26~0-497)
Yield (>65) (>14) (0.1 1~0.87)" T H v Pt
Trial [%] [Iog(N/N0)] [log(C/Co)] [Iog(C/Co)] [°C] [%MV] [m/s] [5]
1 68* 53.711- i.54* 039* 102 97 28 212
1M 57 959.01 21.08 0.27 104 95 28 306
2 57 49.14 1.40 0.31 248 10* 29 222
3" 57 49.14 1.40 0.31” 248 10 29 222
4" 67 47.58 1.40 0.14“ 100 89 29 216
5‘ 67 47.71 1.41 0.26 101 87 10 230

 

* The values are meaningless due to computational failure.

" 17th iteration of trial 1 (21 iteration total).

1' Lower bound was 10% MV that is achievable in practical application.
# Surface color change constraint was relaxed. (0.11~O.87).

I A reduced D-value (=233 s which is the V2 of the original D-value) was used for surface color change

kinetics.

At the ﬁrst trial, 68% yield was achieved with low temperature (close to 100 °C)

and high humidity (close to 100 %MV) (LTHH) condition. However, the algorithm took

a wrong path at the 17th iteration and showed abrupt improvement of the yield, because of

failure of the predictive model at certain conditions (Figure 5.20). Even so, the

optimization process already showed a converging trend at the 13th iteration, and the

result of the 17th iteration was considered as the ﬁnal converged answer. Therefore, the

maximum yield was 57% at the 17th iteration, with low temperature and high humidity

152

condition, which could be a generally accepted operating policy in commercial
application. In addition, the optimal proﬁles at thel7th and the 21St iteration were very

close to each other, except for total process duration.

  
   

 

 

 

 

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Figure 5.20 Optimization processes for single-stage oven: (a) Process for low
temperature and high humidity optimal profile; (b) Process for high temperature
and low humidity proﬁle.

The second trial (Table 5.13) showed 57% yield with high temperature (close to
250 °C) and low humidity (#10 %MV), which is opposite to the conditions of the 1St trial.
The HTLH (high temperature and low humidity) condition processed the patty minimally
in terms of Salmonella reduction and internal color change, which was the advantage,
compared with the case of the LTHH. In addition, the total process time is also shorter

than the duration of the LTHH.

153

Therefore, those results suggest that the HTLH condition is more favorable for
single-stage cooking, according to the facts found in this optimization. However, this is
opposite to the conventional operating concept in industry, which favors LTHH (or
HTHH) to achieve high yield. The surface color change factor incorporated into process
model might be the cause of this opposite result, because HTLH condition is
advantageous to surface browning.

In addition, the effects of a relaxed constraint and a kinetic parameter change
were studied via trial 3, 4, and 5 in Table 5.13. Trial 3 and 4 were conducted with relaxed
surface color constraints (0.11~0.87), which showed two extreme cases, HTLH and
LTHH. These patterns were similar with the patterns of trial 1 and 2 in Table 5.13, except
improved yield (66%) and decreased process time (216 s), which were possible because
the surface color change could go beyond the previous lower bound (0.26) and resulted in
O. 14. However, the results of trial 3 were identical with trial 2, which showed that the
increased range of surface color constraint had no effect on this HTLH pattern. For trial 5,
the D-value of surface color change was decreased by half of the original value (7.75
min), which increased reaction rate. LTHH proﬁle was found as an optimal solution,
because the decreased D-value was close to the D-value of Salmonella inactivation.

Neural network model based optimization strategies were also tested, and the
results were summarized in Table 5.14. The yield prediction accuracies of SNNM were
better the DNNM, even though the model performance of DNNM was better than SNNM.
The third trial (#27-3 in Table 5.14) was very accurate (2% error) and passed all the

constraints.

154

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The optimal control proﬁles found by DNNM and SNNM showed similar
tendency with FEM based optimization. Temperature proﬁle patterns were polarized into
upper and lower limits, while humidity proﬁle patterns were in the mid range between the
upper and lower limits. To avoid frequent false-pass classiﬁcation errors, the DCS
algorithm was applied to the two trials of SNNM based optimization (#27-1 and #27-2).
Even though one of the trials failed to satisfy surface color constraints, the predicted
value (0.26) was close to the result (0.24) of FEM.

The optimal solutions for the single-stage oven were compared with the previous
actual experiment (Watkins, 2004). The actual experiment was for the single stage moist-
air impingement oven, and the conditions consisted of temperature (121~232 °C),
humidity (50~88 %MV), impingement velocity (11.4~21.8 m/s), and cooking duration
(l80~660 3). Among 27 different conditions, 19 conditions were veriﬁed as fully-
cooked24 cases, and the yield was ranged from 58% to 73%. Considering the differences
in the constitutions of the sample patties (65.7 % water and 10% fat), patty dimensions,
the process conditions, and the number of quality constraints, the optimal solutions of this

study can be thought to be in a reasonable range.

 

2‘ According to F SIS, required lethalities (6.5-log10 or 7—log10 reduction of Salmonella) are achieved
instantly when the internal temperature of a cooked meat product reaches 71 . 1°C or above.

156

5. 6.2 Double-Stage Process

The double-stage process is a simple and practical solution to give dynamics to
the process, via serial arrangement of two single-stage ovens. Two trials with ICRS-FEM
strategy (strategy #29) were conducted, and the results are summarized in the Table 5.15.
The two trials showed consistent results in every aspect. The yields achieved by double-
stage process were 67%, which were about 10% point higher than the yields of single-
stage process. The advantage is not just improved yield, but also the shorter process
duration, which is almost half that of the single-stage process. This is a very encouraging
result for both food processors and oven manufacturers. For food processors, high yield
and short process time increase proﬁts and productivity, which can justify the capital cost
of two ovens. In addition, the optimal process satisﬁes multiple safety and quality
constraints within a single process.

In the previous section, the results from strategies #3 and #7 already suggested
that a 2- or 3-step process might be optimal. Thus, that indication was further supported
by the results of the double-stage process optimization. In Figure 5.21, temperature and
velocity proﬁles were not signiﬁcantly different in each step. However, humidity showed
two signiﬁcantly different conditions, dry process (20 %MV) and wet process
(2100 %MV). This implies that humidity plays major role in the double-stage process.
Also, the pattern of humidity condition (low to high humidity) was opposite to industrial
convention (high to low humidity). This interesting result might be caused by the surface

color change factor in the integrated FEM.

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Patty yield [%]
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Figure 5.21 Optimization processes and optimal control proﬁles for double-stage
oven obtained by ICRS-FEM strategy: (a) Optimization process; (b) Optimal
control proﬁle of trial #1; (c) Optimal control profile of trial #2.

159

The effects of relaxed constraint were also demonstrated via trial 3 and 4 in Table
5.15. Trial 3 showed almost identical results with the previous trials (trial 1 and 2).
However, trial 4 suggested LTHH proﬁle because of the relaxed surface color change
constraint. The yield was almost same with the other trials but the process time was
longer than the other trials. Thus, the high temperature and low-high humidity proﬁle
seems more desirable in case of double-stage oven process.

Neural network model based optimizations were also conducted using DNNM.
Although the ICRS-DNNM found optimal solutions, they fell into false-pass
classiﬁcation errors. Also, SNNM application was not considered, due to low model

accuracy.

5. 6.3 Multi-Zone Process

A multi-zone process was investigated by using the ICRS algorithm coupled with
FEM (strategy #33) and DNNM (strategy #34). In Table 5.16, maximum yield achieved
by the ICRS-FEM strategy was 65%, which is about 8% point higher than the maximum
yield of the single-stage process.

The process time of strategy #33 was also about half the process time of the
single-stage process (trial 1 in Table 5.13) even in HTLH conditions. This result is also
opposite to industrial convention, which would suggest that LTHH achieves high yield.
In LT HH proﬁle, internal temperature of the patty can be easily increased to inactivate
Salmonella and to achieve internal color change, but the high humidity also hampers
nonenzymatic surface browning. However, in HTLH proﬁle, the low impingement
velocity in the % portion of the entire process was minimizing yield loss until the last

portion of the process, in which the impingement velocity was increased to meet the

160

surface color constraint. These internal reactions (Salmonella inactivation and internal
color change) and surface reaction (surface color change) took place one aﬁer the other in
the LTHH proﬁle, which resulted in longer cooking duration. However, these internal
and surface reactions took place simultaneously in the HTLH proﬁle, which seemed to
shorten the process.

In this comparison, only ICRS-DNNM strategy was tested, because the model
performance of SNNM for multi-zone process was poor (Figure A.5). Although the
maximum value achieved by ICRS-DNNM was 61%, the value was over-estimated by
about 5% points, compared with the FEM validation result. Figure 5.22 shows stepwise
velocity proﬁles for strategy #33 and #34. ICRS-DNNM found LTHH proﬁle as an
optimal control proﬁle, while ICRS-FEM suggests HTLH control proﬁles.

When the surface color change constraint was relaxed, which ranged from 0.11 to
0.87, two trials (33-1 and 33-2 in Table 5.16) showed improved yields and HTLH type
proﬁles. Trial 33-2 was quite impressive, in that the process achieved better results in
yield and process time than the case of the double-stage oven. The improved yield (69%)
was achieved just by varying impingement velocity. However, the ultimate dry
conditions (~0%MV) of all the trials of strategy 33 must be considered in the practical

sense. A practical lower bound for humidity was considered necessary.

161

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using ICRS-FEM and ICRS-DNNM for multi—zone process: (a) Optimization

process of strategy #33; (b) Optimal control proﬁles of strategy #33; (c)

Optimization process of strategy #34; ((1) Optimal control proﬁles of strategy #34.

163

5. 6.4 Economic-Based Optimization

Based on the single-stage process model, a proﬁt model was developed according
to the equations in Section 4.7.4. Then, the ICRS algorithm was used to search for a
maximum proﬁt, rather than simply a maximum yield. All the control variables, such as
temperature, humidity, velocity, process duration, and yield, converged to 243 °C,
2 %MV, 30 m/s, 154 s, and 64%, respectively (Figure 5.23 (a)). The optimal proﬁle
suggested HTLH policy, which is also consistent with the second trial of the single-stage
optimization. The maximum yield reached 64%, which is much higher than the yield of
the single-stage process, even though the optimization was based on the same predictive
model. This is possible, because the objective function value is not yield, but proﬁt,
which might affect the solution space feature. Therefore, the same algorithm could reach
a much higher objective value. The proﬁt started from negative value and turned to gain
aﬁer passing around 55% yield (Figure 5.23 (b)). Plot (c) of Figure 5.23 shows that the
energy cost is ﬂuctuating as the control variables changes. However, the impact of the
energy cost was much smaller than the feed cost or product value. The product value was
proportional to the yield, and the feed cost was inversely proportional to the process

duration.

164

 

 

 

 

 

 

 

 

 

 

 

 

  

 

  
 
  

 

 

 

 

 

 

 

 

 

 

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0 -+—l—l-+~H—+—+—l—_i—+—l—+-+—l++ -300 11.0 a .. lemma -1000 0'
1 3 5 7 911131517192123 1 4 7 1013161922
Iteration Iteration
(b) (C)

Figure 5.23 Optimization processes and optimal control variables for maximum
proﬁt of single-stage oven were found by using ICRS-FEM strategy: (a)
Convergence history; (b) Comparison of net proﬁt and patty yield; (c) Various cost
history.

165

In addition, the sensitivity of the optimal solution was investigated by using
Monte Carlo simulation (Equation [52]). As in the previous cases, the optimal solution
was also highly sensitive to the constraints. All the 100 simulations with standard
deviation of 2 °C, 2 %MV, 0.1 m/s, and 1 s (for temperature, humidity, velocity, and
process duration) failed to satisfy the constraints (Table 5.17). However, in Table 5.17, a
sub-optimal solution (241°C, 3 %MV, 29 m/s, and 175 s at 10th iteration) showed 47%
risk of failure, which was much lower than the risk of failure of the optimal solution.
Even though the sub-optimal solution was more practical for application, there was 200
S/h difference with the optimal solution, which was estimated $720,000 difference

annually (based on 12 h operation/day for 300 day/yr.)

Table 5.17 Sensitivity of the optimal solutions was obtained by using Monte Carlo
simulation for economic-based optimization problem.

 

Conﬁdence Interval

 

 

(95% conﬁdence)
Optimal Yield @ Number Total RiSk Of Failure
Strate Proﬁt Optimal Proﬁt of Fails Simulations (Ni/Ni) Yield Proﬁt
gy [$/h] [%] N. N, [%] [%] [MI]
#3 7 518 64 100 100 100% N/A N/A
318 61 47 100 47% 61 .6zl:O. 12 324.98i5.49

 

166

6.1

6 CONCLUSIONS

General Conclusions

Global optimization algorithms, integrated process models, and control proﬁle

parameterization methods were combined and utilized to ﬁnd optimal control proﬁles

maximizing patty yield while satisfying microbial safety and quality constraints. Based

on the results of the previous chapter, the following conclusions were drawn:

1.

Artiﬁcial neural networks were developed and substituted as alternative process
models for ﬁnite element model. Dynamic neural network models (DNNM)
showed robust performance for random, single-stage, double-stage, and multi-
zone processes. However, static neural network models (SNNM) demonstrated
good performance just for random and single-stage processes. The DNNM and
SNNM were much faster than the FEM (about 9X and 636X, respectively).
Although the accuracy of DNNM and SNNM were lower than FEM, those neural
network models were viable alternatives to FEM, due to improved execution time.
Various optimization strategies were designed with combinations of models,
algorithms, and parameterization methods. The highest yield (73%) was obtained
by strategy #3 (ICRS-FEM-PLI), which converged to control proﬁles with
apparent step changes in the control variables. The strategies using neural network
models found optimal solutions 10~1,000 times faster than did FEM. However,
most of them committed false-pass classiﬁcation errors or showed low accuracy
for yield prediction, even though the DCS algorithm helped the model not commit
false-pass classiﬁcation errors. Compared to the other algorithms, ICRS was the

most recommendable algorithm, because it was easy to set the algorithm

167

parameters, and ICRS showed well-balanced exploration and exploitation features.
In addition, ICRS-related strategies achieved signiﬁcant improvement of the
objective value within relatively few iterations. Even though the highest yield was
achieved by a strategy using PLI, the solutions obtained by PLI were much more
sensitive than the solutions by F S. Therefore, F S is recommended for practical
usefulness of the obtained solution. Even though the ideal and optimal proﬁles are
impossible to generate practically, those ideal proﬁles suggest what type of
general process might be a desirable strategy.

. Single-stage, double-stage, and multi-zone processes were studied by using three
different models and the ICRS algorithm. The maximum yield (67%) was
achieved in the double-stage process, and the control proﬁles showed similar
control patterns with the results from strategy #3 (FEM-PLI-ICRS) applied to a
continuously varying process. Case studies showed many possible variations of
the single-stage process to achieve improved solutions. In addition to the
optimization for yield, a simple economic-based example was illustrated from the
viewpoint of net proﬁt for a single-stage process. Maximum proﬁt was achieved
at 64% yield, which was different from the single-stage case with a yield
objective.

. Monte Carlo simulation showed that the sensitivity of optimal solutions was very
critical for the usefulness of the solution. Small perturbations of the optimal
control proﬁles could result in failure to satisfy safety and quality constraints.

Instead of using the optimal solution, however, slightly sub-optimal solutions

168

were less sensitive to the perturbations and therefore less susceptible to fail (for

constraints).

This study was designed to build a foundation for multivariate nonlinear dynamic
process optimization, such as meat patty cooking under moist air impingement
environments. This type of problem is very difﬁcult and complex from the viewpoint of
product and process. In this study, various factors, such as product yield, microbial safety,
and colors, were investigated to identify the best optimization strategy for this type of
process. Finally, many valuable aspects related to developing the model, selecting
optimization algorithms, interpreting optimal results, and handling various pitfalls were
carefully studied, and knowledge was established. However, a direct application of the
resultant numbers of this study to a problem requires discretion by users, because the
resultant numbers, such as maximum yield and the status of constraint satisfaction, can be
signiﬁcantly affected by the properties related to various calculations, such as yield,
safety, and quality predictions. Even so, the knowledge of process optimization strategy
in this study remains sound for the applications to various complex food processing

operations, such as ﬂying, drying, extrusion, baking, and retorting.

6.2 Suggestions for Future Research
Although the results of this project are instructive and potentially valuable, further
research and improvements are still necessary. The following suggestions are made for
ﬁiture research.
1. The process model plays a very important role in optimization. The FEM, DNNM,

and SNNM models must be validated with actual experiments. For this purpose,

169

the integrated FEM needs to be ﬁne-tuned with validated predictive model
parameters of quality.

. Optimal solutions found throughout this research need to be validated
experimentally. Even though some optimal proﬁles are impractical to generate,
single or double-stage results can be validated by experiment.

. Methods or algorithms need to be developed to avoid false-pass classiﬁcation
errors when alternative models are used. A neural network trained with the
relationship between goal and constraints might help optimization algorithms
maneuver the solution space without committing false-pass classiﬁcation error.

. Even though the alternative models did not show satisfactory performance
locating optimal solutions, they can be used to acquire prior knowledge for the
target model. For example, sensitivity analysis can be used along with alternative
models to reduce control variables, test the effect of parameters, determine the
approximate location of optimal solution, etc.

. Effective control proﬁle parameterization methods need to be devised to address
the sensitivity issues for optimal solutions. Conﬁdence interval embedded control
proﬁles can reduce the risk of failure involved in the high sensitivity of the
optimal control proﬁle. Also, such a high sensitivity problem needs to be
validated by experiment.

. The ﬁnite element model used in this study had some degree of error with
physical experiment. In this study, how the source (ﬁnite element model) error
affects the optimal solutions was not studied. However, as long as the ﬁnite

element model represents the general characteristics of the solution space, the

170

general conclusions seem to remain sound, even though some numeric value
might change. Even so, the effect of the source error on the optimal solution
should be studied.

. A new efﬁcient global optimization algorithm having advantages from GA, SA,
and ICRS can be developed and applied to this problem.

. Typical installations of a single-stage oven have an inlet and outlet tunnel (often
with saturated condition) in addition to the major cooking zone. These actual
process conditions need to be implemented in the control proﬁles to reﬂect these
systems and to obtain results that are directly practical and interesting to the

industry.

171

APPENDICES

172

APPENDIX A

Tables and Figures of Optimization Processes

Polynomial regression lines were added to smooth piecewise control proﬁles so that
general trend could be observed in the following plots:

Figure A.7 (a), Figure A.10 (a), Figure A.12 (a), and Figure AM (a)

173

Patty Yield by GRNN [%]

 

 

o ,, ﬂ; .
o 10 20 30 4o 50 60 7o 80 90 100
Patty Yield by FEM [%]

7T) ﬁ—Vni'i ﬁ—V iﬁ‘l

   
   
  
 
 
     

  
       

   

(a)
////////////{///;///////// /
0 /////¢//
g” p ss

   

Salmonella Inactivation (Iog(log(N/M») by

Salmonella Inactivation (log(log(N/No») by FEM

(b)

Figure A.l The performance of DNNM for double-stage process: (a) Yield; (b)
Salmonella inactivation; (c) Internal color change; (d) Surface color change.

174

Surface Color Change (log(C/Co)) by DNNM

7
-//

7
l

-8

lntemal Color Change (log(log(C/Co))) by DNNM

0.8 «

      

f—i'Y—iﬁi

 

     

 
 

 

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, ,/ / ,///,V/ , .

  
   
   

,//g False-pass 3/3 . '3'

  
 

        
     
 
 

//

  

-6

False-fail

   

‘3': TnJe-fall

-8
lntemal Color Change (log(log(C/Co») by FEM

(C)

. O)

True-fail

' False-fail

-3.-----_.'-...----- iv

 

 

 

   
 

 

0.2 0.4 0.6 0.8
Surface Color Change (log(C/Co» by FEM

(d)

Figure A.l (cont’d)

175

Patty Yield by DNNM [%]

 

 

0 j 7' "T. *T ”_T#" ﬁr—l’ ’_—_T "— T‘ﬁ—T— _—_IV_—“l
0 10 2030405060 70 80 90100
Patty Yield by FEM [%]

(a)

False-pass

/////////// ”3

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DNNM

. False-fail
True-fail

l
------------

Salmonella Inactivation (Iog(log(N/Nl)» by

 

Salmonella Inactivation (Iog(log(N/No)» by FEM

(b)

Figure A.2 The performance of the DNNM for multi-zone process: (a) Yield; (b)
Salmonella inactivation; (c) Internal color change; (d) Surface color change.

176

    

False-fail

5.220
3 $6660.50; $85 E28 .325

lntemal Color Change (log(log(C/Co)» by FEM

(C)

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%

 

 

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Surface Color Change (log(C/Co)) by FEM

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Figure A.2 (cont’d)

177

 

 

a a < < a 2 Jll 11
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4|

3.. muzz.._.._ E 22> and

 

10 20 30 40 50 60 70 80 90 100

0

Patty Yield by FEM [%]

(a)

              

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True-fail

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....r m... ................
%M. . . e

Salmonella Inactivation (log(log(N/No)» by FEM

(b)

Figure A.3 The performance of the SNNM for single-stage process: (a) Yield; (b)

Salmonella inactivation; (c) Internal color change; ((1) Surface color change.

178

lntemal Color Change (log(log(N/Ml») by

Surface Color Change (log(C/Co)) by SNNM_S

 
 

/

    
   

 

 

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0.8 -

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lntemal Color Change (Iog(log(C/Co») by FEM

  
    
 

(C)
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Tme-fail

...—..----LLI.-.‘ .'

 

.0 0.2 0.4 0.6 0.8 1.0 1.2

Surface Color Change (log(C/Co» by FEM

((1)

Figure A.3 (cont’d)
l 79

100 —
90 — RMSE=6.32 __=
so . R2=0.93 -. - ", . -
7o -
60< ',;.,:.'.. '.
50 - "

40 —
30 ~
20

ll
ii:
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Yield by SNNM_D [%]

 

0 ‘1' I I l l l l T l l
010 20 30 40 50 60 70 80 90100
Yield by FEM [%]

 

(3)
221252256 W08”
////2

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)m....'i..,.-'232:: W

Salmonella Inactivation (log(log(N/No)» by FEM

   
    
      
 
 

Salmonella Inactivation (log(log(N/Nl») by

(b)

Figure A.4 The performance of the SNNM for double-stage process: (a) Yield; (b)
Salmonella inactivation; (c) Internal color change; ((1) Surface color change.

180

lntemal Color Change (log(log(C/Co») by

Surface Color Change (log(C/Co)) by SNNM_D

 

40.146

 

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lntemal Color Change (Iog(log(C/Co») by FEM

 

   

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(C)

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True-fail: False-fail:
0 0.2 0.4 0.6 0.8 1

 

Surface Color Change (Iog(C/Co» by FEM

(d)

Figure A.4 (cont’d)

181

90 - RMSE=5.48
R2=0.96

Patty Yield by SNNM_M [%]

 

 

o 1 Hi I l l I . l I l l
0 10 20 30 40 50 60 70 80 90 100
Patty Yield by FEM [%]

 

(a)

/ ....

// /

' /
- False-fail %
/ 40 ,. //////////////////////

Salmonella Inactivation (Iog(log(N/No») by FEM

   
  
 
   

  
 
  

  

Salmonella lnactivation(log(log(N/M))) by

(b)

Figure A.5 The performance of the SNNM for multi-zone process: (a) Yield; (b)
Salmonella inactivation; (c) Internal color change; ((1) Surface color change.

182

lntemal Color Change (log(log(C/Co») by

Surface Color Change (Iog(C/Co)) by SNNM_M

/

%

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Interanl Color Change (Iog(log(C/Co») by FEM
(C)

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R2=0.59 0.497

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True-fail I ,-."'":.'-.~ _-

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Tale-fail

 

 

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Surface Color Change (Iog(C/Co)) by FEM
(d)
Figure A.5 (cont’d)

183

Strategy #1 & #5

 

 

 

 

  

 

58 .. 62 —
56 . 60 4
g 54 . E 58 4
'0 2
E 52 1'1; 56 'i
>s
> 50 J 2: 54
g 43 E
46 .' 52
44 "'i——T———'T——_T—_ T "—1 50 ' 7—I— '—_’“l—_""—"T_ ‘—l
0 50 100 150 200 250 O 100 200 300 400
Generation Generation
(21) (b)

Figure A.6 GA-FEM optimization process and convergence: (a) PLI
parameterization; (b) FS parameterization.

184

Temperature [C] / Humidity [%MV] /
Velocity [mls]

Temperature [C] / Humidity [%MV]
I Velocrty [mls]

 

 

—a— Temperature
—a— Humidity

-o— Velocity

—— Poly. (Humidity)
— Poly. (Velocity) I
— Poly. (Temperature) J

 

 

 

 

 

 

200

 

 

 

 

 

 

0 100 300 400
Process time [s]
(a)
250 1 —Temperature [C]
- - - -Humidity [%MV]
— - -Velocity [mls]
200 .
150 «
100 - .. . ..
50 - ,. '
I
‘5. .5 q I - ‘- - - -
~ 5 ~ . In... : J—I' ‘I-u ‘ H. ‘ __ ' ﬂ
0 i -r- . i r T ~ ' , 1
0 50 100 150 200 250 300

Process time [s]

(b)

Figure A.7 Optimal control proﬁle of temperature, humidity, impingement velocity,
and cooking duration found by GA-FEM optimization strategy: (a) PLI
parameterization; (b) FS parameterization.

185

Strategy #2 & #6

70 n
60 d
55 ~
50 ~
45 ~
40 —
35 ~
30 TH— —’7 "T—-"—_‘_— __—1

0 50 100 0 50 100 150

Generation Generation

Patty yield [%]
Patty yield [%]

 

 

 

 

(a) (b)

Figure A.8 SA-FEM optimization process and convergence: (a) PLI
parameterization; (b) FS parameterization.

186

Strategy #9 & #13

 

 

 

58 .

574 64,
LE 56 _ 62 _
"‘ e\°
% 55 i 1; 60 i
g 54 g “as" 58 r
E 53 — 5' 56 i

if 54 .
52 l 52 i
51 ‘T—T‘ —" ' "'v _I_—_"' "'— "_ "_“l 50 ‘1“ I _ ‘~T_‘ _‘
0 500 1000 0 50 100
Generation Generation
(a) (b)

Figure A.9 GA-DNNM optimization process and convergence: (a) PLI
parameterization; (b) FS parameterization.

187

 

 

—9— Temperature ‘1
250 + Humidity t
_ —o—— Velocity |
200 "l — Poly. (Temperature) 1
l —Poly. (Humidity)
150 — Poly. (Velocity) '

 

Temperature [C] / Humidity [%MV] I
Velocity [mls]

 

 

 

0 100 200 300 400
Process time [s]

 

 

 

 

 

 

(a)

250 _____ Temperature [C]
.. l I — — Humidity [%MV]
E i 1 — . -VeIocity [mls]
2*: 200 l _
a I
3‘3 E
:5, £150 «
I 2:
5'8
9 £100 .
a \
e
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'— . ‘5. I. N ' — - a— . _. w":

0 r“ . "' - :- - - -.- —' "" . —.
0 50 100 150 200 250

Process time [s]

(b)
Figure A.10 Optimal control proﬁle of temperature, humidity, impingement velocity,

and cooking duration found by GA-DNNM optimization strategy: (a) PLI
parameterization; (b) FS parameterization.

188

Strategy #10 & #14

##U'IUI
NCO-FOO

45 .
43
41 .
39 .
37 4

O 50 100 150 0 50 100 150

Generation Generation

  
 

 

Patty yield [%]
Patty yield [%]

 

 

 

 

 

(a) (b)

Figure A.11 SA-DNNM optimization process and convergence: (a) PLI
parameterization; (b) FS parameterization.

189

'lm—er—atﬁre j
. —ei— Humidity 1
, ‘ . —o— Velocity i
’. ‘ — Poly. (Temperature)‘
' 1 y — Poly. (Humidity) .

1 L— Poly. (Velocity) j

 

 

Temperature [C] / Humidity [%MV]l
Velocity [mls]

 

0 1 00 200 300 400
Process time [s]

 

 

 

(a)
{—- Temperature [C] i
- . - .Humidity [%MV]
_ . -Velocit m/s
200 1 Y I l

N

01

O
bl -__-__ _.——L—___—‘ - .————J

Y [mls]
8

Temperature [C] / Humidity [%MV]
l V_eJocit
8

 

 

Process time [s]

(b)
Figure A.12 Optimal control proﬁle of temperature, humidity, impingement velocity,

and cooking duration found by SA-DNNM optimization strategy: (a) PLI
parameterization; (b) FS parameterization.

190

Strategy #11 & #15

 

 

 

 

 

 

61 ~ 60 -
59 ~ 55
:3 57 4 S j
9.. 2.. 50 4
2 55 T 2
a:
3 53 ~ .1; 45 -
>‘ >.
g 51 g 40 ~
0- 49 n.
47 35 .
45 i . - . - . 30 I- . . __ _.,
0 50 100 150 0 5 10 15
Generation Generation
(a) (b)

Figure A.13 ICRS-DNNM optimization process and convergence: (a) PLI
parameterization; (b) FS parameterization.

191

 

250 . :9: Temperature 7‘
—¢— Humidity 1
200 + Velocity l
, .’ 1 “ — Poly. (Temperature).
A V! — Poly (Humidity) .

I

150 '
\ ’ (} “\V — Poly. (Velocity)

 

 

Velocity [mls]

01
O

100.
" V 'l

Temperature [C] I Humidity [%MV]l

 

 

0 i
0 50 100 150 200 250 300 350

Process time [s]

 

 

 

 

 

(a)
250 >
§
5.
E. 200 ~
3
g £150 Temperature [C]
E .é‘ . - - - -Humidity [%MV]
9: %100 i! — - -Velocity [mls]
2 > . —
3 ‘ l
E - q " a ' " I. g
8 50 j ‘ a a " p T V v a '-
E ’ “.._’£:_
(D . d- " 'bu. -. Q
I‘- f ‘ ﬁ _ d.” a
0 r" ' I , T _
O 100 200 300

Process time [s]

(b)

Figure A.14 Optimal control profile of temperature, humidity, impingement velocity,
and cooking duration found by ICRS-DNNM optimization strategy: (a) PLI
parameterization; (b) FS parameterization.

192

Strategy #21

 

 

 

 

 

 

   

55 «
54 .4
53 .
,3. 52 -
a}. 51 .
E, 50 f
3" 49 l
a 4,.
47 ~
46 1
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0 100 200 300 400
Generation
(a)
250 .. Temperature [C]
. . . .Humidity [%MV]
200 J _ . -Velocity [mls]

 

  

 

y [93/8]
OI
O

 

Temperature [C] / Humidity [%MV]
/ Velocit
O
O

 

 

0 50 1 00 1 50 200 250 300 350

Process time [s]

(b)

Figure A.15 Optimization process and optimal control proﬁles found by GA-
SNNM_R optimization strategy and Fourier series parameterization: (a)
Convergence history; (b) Optimal control proﬁle of temperature, humidity,
impingement velocity, and cooking duration.

193

Strategy #22

 

 

 

 

 

  

 

 

 

 

541
52 -
32°: 50—
'O
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ti“
441
42
40 l— T‘ - _— T—T‘ ﬂ_ —— 'T—_- _‘_'——_'l
0 50 100 150
Generation
(a)
é ““ '1
Tem rature C
250 1 ”9 I l
- - - -Humidity [%MV]
3 — - -Velocity [mls]
OE 200 — J
b
5, £150 4
I a
g3
9 £100 _ . . _ _ ~ ~ .
3 \ ‘ I. . . o F o '5‘
E ' ~ ‘
g 50 — x
5 »-~ ,-”'—‘4-::S:—
0 L—i— T. ' e e
0 100 200 300

Process time [s]

(m

Figure A.16 Optimization process and optimal control proﬁles found by SA-
SNNM_R optimization strategy and Fourier series parameterization: (a)
Convergence history; (b) Optimal control proﬁle of temperature, humidity,
impingement velocity, and cooking duration.

194

 

Temperature [C] l Humidity [%MV]

 

 

 

 

Strategy #23
54 -.
_ 52 .
e: 50 ~
'0
E2 48 1
>
3 46 1
S 44 -
42 .
40 «t—W—WW WWW-W
0 20 40
Generation
(a)
250 1 Temperature [C]
— — Humidity [%MV]
200 L_ Velocny [mls]

I Velocit

100 ~

 

y [511/8]
8

 

Process time [s]

(b)

Figure A.17 Optimization process and optimal control proﬁles found by ICRS-
SNNM_R optimization strategy and Fourier series parameterization: (a)
Convergence history; (b) Optimal control proﬁle of temperature, humidity,
impingement velocity, and cooking duration.

195

 

 

 

 

 

 

  
 

 

 

 

 

 

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[.—
0 ——++44 l—i—i—l—F+—+—H—+H~—l—F—+ +e— 0
1 5 9 13 17 21
Iteration
(a)
300 ~ 58
(D
33
§§ 250 ~ —~ 57
.‘Q
E CL
3 %' 200 ~~ ~~ 56
53.5 —o—T as.
H>~<D150-~- —o—H +552
2 F .E .9
3 § ~ +V >-
gg’ 100 -o—tf 4» 54
E \ -——Yield
a: S.
l— 2 50 + ~~ 53
i H—ﬁ—A—ﬁ—ﬁ—d—ﬁ—b—O—ﬁ—ﬁ—A
o—o—o—o—o—o—o—o—o—o—o—o—o
0 i i i + i i + i i i i 52

 

 

Iteration

(b)

Figure A.18 Two different optimization processes and optimal control proﬁles found
by single-stage ICRS-FEM optimization strategy: (a) Convergence history of case I
(LTHH); (b) Convergence history of case 11 (HTLH); (c) Optimal control proﬁle of
temperature, humidity, impingement velocity, and cooking duration of case I; (d)
Optimal control proﬁle of temperature, humidity, impingement velocity, and
cooking duration of case 11.

196

250

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Process time [s]

((1)

Figure A.18 (cont’d)

197

g 'r Temperature [C] ﬂ
é. 200 J 1 - . . .Humidity [%MV]
.3 Velocity [mls] J
:2 E L _ i _ _._

:53 £150 4 211.94
I 2:
9- %100 . ..................... .

9 >
a \

E

a 50 4

E
I.—

O ‘17—— T—‘—‘ '—'__7— ‘ l l 1' i
O 50 100 150 200 250
Process time [s]
(C)
300 4

2" 221.98

°\° 250

3‘

3% g 200 A f Temperature [C]

:E’ ‘; 150 - - - -Humidity [%MV]

(2 g t Velocity [mls]

g 2 100 1

2“»;

E 50

*9 L 1

0 ' - - - -1 ..... I ----- i - - - - .1 - i ﬂ
0 50 100 150 200 250

APPENDIX B

Computer Codes

198

‘ Iii—MW __a.' :- . ____». . _

 

User Guide

In this study, various optimization strategies and case studies were coded via
Matlab. Even though there are many computer codes for each cases, some typical codes
were represented in the appendix. Many other codes were simple variations of those
typical codes. Also, some sub-programs were commonly used in the other applications
with slight variations.

The appendix has codes related to two neural network model development
(DNNM and SNNM), three GO algorithms (GA, SA, and ICRS), and two
parameterization methods (PLI and FS). However, the entire FEM code was not included,
except some quality prediction sub-programs, because the large portion of the work was
done by Watkins (2004).

MATLAB codes for case studies have the same frameworks with other strategies.
The only difference is the sub-program generating random proﬁles, which has speciﬁc
number of control variables for each applications, l-stage (4 control variables), 2-stage (7
control variables), and multi-zone process (7 control variables). Therefore, those codes

were not included in this appendix.

199

Dynamic Neural Network Model (DNNM)
T rain_DNNM.m has following two sub programs.

0 Generate_Squata_ Train. m
0 Generate_Squata_ Test. m

200

T rain_DNNM. m

% Train__DNNM . m

% This program trains a dynamic neural network model as an alternative
to

% a finite element model by using a nonlinear system identification
method.

%
% TRAINING PHASE
%

 

 

clear all
fprintf('Network Training Started...\n')
fprintf('>Training Phase:\n')

counter=0;

=1; % Global training matrix head number
Nt=0; % Global training matrix tail number

m=2; % length of past time series
mg=3*m+1+1; % total number of rows of training matrix
mk=3*m+l; % row # of target output value at global training matrix

% Constructing global trainging data set
for j=1z1
start =[l 209]; % start file #
finish=[910 328]; % finish file #
fprint£('Total number of training data = %d\n',finish(j))
for instart(j):finish(j)

% clear interim variables
clear a dPl dP2 dP3 dP4 dTl dT2 dT3 dT4 P1 P2 P3 P4 T1 T2 T3
T4
path=2i :
switch path
case 1

fid3=£open(['C:\MATLAB?\work\Dissertation\Data_Pool\Train\FE Solution
Piles\Constant\FEsolution_',num2str(i),'.dat']);
case 2

fid3=fopen(['C:\MATLAB?\work\Dissertation\Data_Pool\Test\FE Solution
Piles\FEsolution_',num2str(i),'.dat']);
end

% Reading individual data files

a=fscanf(fid3,'%f %f %f %f %f %f %e %e %e %e %f %f',[12,
infl); % [time temp steam.velocity yield vavqu SamenellaLR D1 034 DEavg
Tc Ts]

a=a';

fclose(fid3);

% Data Correction: Abnormal data trend will be truncated

201

% The total row# of data file is 1201. If the file has abnormal
data, fscanf terminates its reading right at the beginning of the
abnomality. In this case the total number of row of the file will be
less than 1200

[rn cn]=size(a);

if rn < 1200

a=a(1:rn-20,:);
end

% Converting concurrent data to sequential data format

[P1, T1]=Generate_Squata_Train(a,5); % 5 Yield

[P2, T2]=Generate_Squata_Train(a,7); % 7 Salmonella Reduction

[P3, T3]=Generate_Squata‘Train(a,8); % 8 D1 (Doneness via
internal color change

[P4, T4]=Generate_Squata_Train(a,10);% 10 DEavg (Averaged
Doneness via surface color change

[rnl,cn1]=size(P1(mg,:
[rn2,cn2]=size(P2(mg,:
[rn3,cn3]=size(P3(mg,:
[rn4,cn4]=size(P4(mg,:

V V v v
V V V v
s. \g \o ‘0

% Log10(P2 E P3, T2 8 T3)for scaling
P2=prelog('p',P2,m); T2=prelog('t',T2,m);
P3=prelog('p',P3,m); T3=prelog('t',T3,m);

% Calculating difference of target values in an interval
for h=1zcn3
if h==1
dTl(1,h)=P1(mk,h)-T1(1,h);
d'r2 (1,h)=P2 (mk,h) -'1'2 (1,11);
dT3(1,h)=P3(mk,h)-T3(1ph)i
dT4(1,h)=T4(1,h)-0;
else
dTl(1,h)=T1(1,h-1)-T1(l,h);
dT2(l,h)=T2(1,h-1)-T2(1,h);
dT3(1,h)=T3(1,h-1)-T3(1,h);
dT4(1,h)=T4(1,h)-T4(1,h-1):
end
end

% Replacing the target value with the difference
T1(1,1:cn3)=dT1;
T2(1,1:cn3)=dT2;
T3(1,1:cn3)=dT3;
T4(1,1:cn3)=dT4;

% Creating a global training file from individual file
Nt=Nh+cn1;

SP1(1:mg,Nh:Nt-1)=P1(1:mg,:
SP2(l:mg,Nh:Nt-1)=P2(1:mg,:
Salmonella LR
SP3(1:mg,Nh:Nt-1)=P3(1:mg,:
SP4(1:mg,Nh:Nt-1)=P4(1:mg,:

° STl(1,Nh:Nt-1)=T1(l,:); % Yield
° ST2(1,Nh:Nt-1)-T2(l,:); %

vv
‘ ‘

ST3(1,Nh:Nt-1)=T3(1,:); % DI
ST4(1,Nh:Nt-1)-T4(1,:); % DEavg

vv

‘. ‘g

202

NhaNt;
end
end % End of constructing a global training data set
fprintf('End of constructing a global training data set\n')

% Normalization

[SP1n,minSP1,maxSP1] = premnmx(SP1);
[SP2n,minSP2,maxSP2] = premnmx(SP2);
[SP3n,minSP3,maxSP3] 8 premnmx(SP3);
[SP4n,minSP4,maxSP4] = premnmx(SP4);
[ST1n,minST1,maxST1] = premnmx(ST1);
[ST2n,minST2,maxST2] 8 premnmx(ST2);
[ST3n,minST3,maxST3] a premnmx(ST3);
[ST4n,minST4,maxST4] 8 premnmx(ST4);

% Training networks for each output by using Generalized Regression NN
net1=newgrnn(SP1n, STln, 0.5); % Yield

net2=newgrnn(SP2n, ST2n, 0.22); % Salmonella

net3=newgrnn(SP3n, ST3n, 0.22); % DI

net4=newgrnn(SP4n, STdn, 0.20); % DE

fprintf('\n>End of Training\n')

% Saving trained networks

save('C:\MATLAB?\work\Dissertation\NN\TrainedNetworks\netl','net1');
save('C:\MmTLAB7\work\Dissertation\NN\TrainedNethrks\net2','net2');
save('C:\MATLAB?\work\Dissertation\NN\TrainedNetworks\net3',‘net3');
save('C:\MATLAB?\work\Dissertation\NN\TrainedNetworks\net4',‘net4');

% Saving normalization parameters

% SP1 SP2 SP3 SP4 8T1 ST2 ST3 8T4

% min max min max

% row 1~8

% row 9

Outputs(1:8,1)=minSP1; Outputs(1:8,3)=minSP2; Outputs(1:8,5):minSP3;
Outputs(1:8,7)=minSP4;

Outputs(1:8,2)=maxSP1; Outputs(1:8,4)=maxSP2; Outputs(1:8,6):maxSP3;
Outputs(1:8,8)=maxSP4;

Outputs(9,1)=minSTl; Outputs(9,3)=minST2; Outputs(9,5):minST3;
Outputs(9,7)=minST4;

Outputs(9,2)=maxST1; Outputs(9,4)=maxST2; Outputs(9,6)=maxST3;
Outputs(9,8)=maxST4;

fida
fopen('C:\MATLAB7\work\Dissertation\NN\TrainedNetworks\normal_para.dat'
.‘wt'H

£printf(fid,'%10.8f %10.8f %10.8f %10.8£ %10.8f %10.8f %10.8f %10.8f
\n',0utputs');

fclose(fid);

% Retrieving stored networks and parameters

clear net
load('C:\MATLAB?\work\Dissertation\NN\TrainedNetworks\netl',‘net1');
loadt'C:\MATLAB?\work\Dissertation\NN\TrainedNethrks\net2','net2');

203

load('C:\MATLAB7\work\Dissertation\NN\TrainedNetworks\net3',‘net3');
load('C:\MATLAB?\work\Dissertation\NN\TrainedNetworks\net4',‘net4');

fid=fopen('C:\MATLAB?\work\Dissertation\NN\TrainedNetworks\normal_para.
dat'); ,
a=fscanf(fid,'%f %f %f %f %f %f %f %f',[8, infll; % [time temp steam
velocity yield vavqu SamonellaLR DI D24 DEavg Tc Ts]

a=a';

fclose(fid);

minSP1=a(1:8,1); minSP2=a(1:8,3); minSP3=a(1:8,5); mdnSP4=a(1:8,7);
maxSP1=a(l:8,2); maxSP2=a(1:8,4); maxSP3=a(1:8,6); maxSP4=a(l:8,8);
minST1=a(9,l); minST2=a(9,3); minST3=a(9,5); minST4=a(9,7);
maxSTl=a(9,2); maxST2=a(9,4); maxST3=a(9,6); maxST4=a(9,8);

 

 

m=2; % length of past time series

mg=3*m+1+l; % total number of rows of training matrix
mk=3*m+1; % row # of target output value at training matrix
% SIMULATION PHASE

fprintf('>Simu1ation Phase\n')

clear Outputs
clear Y;pred b_pred DI_pred D3_pred t
clear X_meas L;meas DI_meas DE_meas

for counter=10:10 % 13
clear a b Cond ittcl ittc2 ittc3 ittcd
clear ittl itt2 itt3 itt4
clear dPPl dPP2 dPP3 dPP4 dTTl dTT2 dTT3 dTT4
clear PP1 PP2 PP3 PP4 TTl TT2 TT3 TT4

% Reading result files

fid:fopen(['C:\MATLAB?\work\Dissertation\Data_Pool\Test\FE Solution
Piles\FEsolution_',num23tr(counter),'.dat']);

a=fscanf(fid,'%f %f %f %f %f %f %e %e %e %e %f %f',[12, inﬁll; %
[time temp steam velocity yield vav M SamonellaLR DI D34 DEavg Tc Ts]

asa';

fclose(fid);

% Reading condition files

fid=fopen(['C:\MATLAB?\work\Dissertation\Data;Pool\Test\Condition
Files\Condition_',num2str(counter),'.dat']);

b=fscanf(fid,'%f %f %f %f',[4, infl); % [time temp steam.velocity
yield vangiSamonellaLR DI DE4 DEavg Tc Ts]

bsb';

fclose(fid);

% Reconstructing testing matrix

[PP1, TTl]=Generate_Squata_Test(a,5); % Yield

[PP2, TT2]=Generate_Squata_Test(a,7); % Salmonella LR
[PP3, TT3]=Generate_Squata_Test(a,8); % DI

[PP4, TT4]=Generate_Squata_Test(a,10); % DEavg

204

% log10(ipp2, ipp3)scaling
PP2=prelog('p',PP2,m);
PP3=prelog('p',PP3,m);

% Recording measured data for comparison
measuredTTlsPP1(mk,:);
measuredTT2=PP2(mk,:);
measuredTT3=PP3(mk,:);
measuredTT4=PP4(mk,:);

% Sampling contol profile from condition file
CondsGenerate_Squata_Cond(b);
[rt1,ct1]ssize(PP1);

% Assigning initial values

% Yield

ippl(1:mg,1)=Cond(1:mg,1); % copy
ippl(mk,l)=100; % 100 %

itt1(l:mrl,1)=0; % initial difference is zero

% Salmonella reduction
ipp2 (1:mg,1)=Cond(1:mg,1);
ipp2(mk,1)=-10; % ~0
itt2(1:m-1,1)=0;

% Internal doneness via internal color change
ipp3(1:mg,1)=Cond(1:mg,l);

ipp3(mk,l)8-10; % ~0

itt3(l:m-1,l)=0;

% External doneness via surface color chane
ipp4 (l:mg,1)=Cond(1 :mg,1);

ipp4(mk,l)=0; % 0

itt4(l:m-l,1)=0;

% Initial simulation

% Scaling input data by using Minna: value at the trining phase
ippln = tramnmx(ipp1,minSP1,maxSP1);
ipp2n = tramnmx(ipp2,minSP2,maxSP2);
ipp3n a tramnmx(ipp3,minSP3,maxSP3);
ipp4n = tramnmx(ipp4,minSP4,maxSPd);

% Simulation

itlssim(netl,ipp1n);
it2=sim(net2,ipp2n);
it3=sim(net3,ipp3n);
it4=sim(net4,ipp4n);

% Denormalization

it1 a postmnmx(it1,minST1,maxST1);
it2 a postmnmx(it2,minST2,maxST2);
it3 a postmnmx(it3,minST3,maxST3);
it4 8 postmnmx(it4,min8T4,maxST4);

205

ittcl(1,1)=itl(1,1);
ittc2(1,1)=it2(1,1);
ittc3(1,l)=it3(l,l);
ittc4(1,1)=it4(1,1);

% ...............................................
% Beginning of time stepping simulation
% ...............................................
for j=2:ctl

i-j;

% Constructing input data set
ippl(l:mk-1,1)=Cond(1:mk-1,i); % copying conditions
ipp1(mg,1)=Cond(mg,i); % copying time sequence
ittl(1,1)=it1(l,1);

ipp2(1:mk-1,1)8Cond(1:mk-1,i);

ipp2(mg,1)=Cond(mg,i);
itt2(1,1)=it2(1,1);

ipp3(l:mk-1,1)=Cond(1:mk-1,i);
iPP3(m9.1)=Condlmq.i):
itt3(l,l)-it3(l,1);

ipp4(1:mk-1,1)-Cond(1:mk-1,i);
ipp4(mg,1)=Cond(mg,i):
itt4(1,1)=it4(l,1);

% replacing past time series of output

ipp1(mk,1)=ipp1(mk,1)-itt1(l,1); % value(t=n)-value(t=n-1)-
difference

ipp2 (mk.1)=ipp2 (mk.1) -itt2 (1,1);

ipp3(mk,l)=ipp3(mk,1)-itt3(l,1);

ipp4(mk,1)=ipp4(mk,1)-itt4(1,1);

% Normalization

ippln s tramnmx(ipp1,minSP1,maxSP1);
ipp2n = tramnmx(ipp2,minSP2,maxSP2);
ipp3n I tramnmx(ipp3,minSP3,maxSP3);
ipp4n a tramnmx(ipp4,minSP4,maxSP4);

% Simulation

itl=siminet1,ipp1n);
it2=simlnet2,ipp2n);
it3-sim(net3,ipp3n):
it4=sim(net4,ipp4n);

% Denormalization

itl a postmnmx(it1,minST1,maxST1);
it2 = postmnmx(it2,minST2,maxST2);
it3 = postmnmx(it3,minST3,maxST3);
itd - postmnmx(it4,minST4,maxST4);

% record simulation result; difference at each time
ittc1(1,i)8it1(1,1);
ittc2(1,i)=it2(1,1);
ittc3(1,i)=it3(1,1);

206

ittc4(1,i)=it4(l,1)i

% shift down past time series of output
ittl(2:m~1,1)=itt1(1:m-2,1);
itt2(2:m~1,1)=itt2(1:m-2,1);
itt3(2:mrl,1)=itt3(l:m-2,1);
itt4(2:mr1,l)=itt4(1:m-2,1);

end % End of time stepping simulation

% copying time series to t
tsPP1(mg,l:i);
t=t';

% Calculating cumulative results

[b c]=size(ittc1);
ittc1m41,1)2100;
ittc1m11,2:c+1)-100-cumsum(ittcl);
TT1=measuredTT1;

[b c]=size(ittc2);
ittc2mil,1)=-10;
ittc2m41,2:c+l)--10-cumsum(ittc2);
TTZImeasuredTTZ;

[b c]=size(ittc3);
ittc3mil,1)--10;
ittc3m(1,2:c+1)--10-cmmmum(ittc3);
TT3=measuredTT3;

[b c]=size(ittc4);
ittc4m41,1)80;
ittc4mil,2:c+1)=0+cumsum(ittc4);
TT4=measuredTT4;

% Transposing data for graphical representation
TplsTTl'; ittclsittclm}:
Tp2=TT2'; ittc2=ittc2mﬂ;
Tp3=TT3'; ittc3-ittc3mﬂ;
Tp4=TT4'; ittc4-ittc4mf;

% Storing result of each simulation
Y;pred(counter,1)-ittc1(ct1,1);
Y;meas(counter,1)=Tp1(ct1,1);

D_pred(counter,l)=ittc2(ctl,1);
L_meas(counter,1)-Tp2(ct1,1);

DI_pred(counter,1)=ittc3(ctl,1);
DI_meas(counter,1)=Tp3(ct1,1);

DE_pred(counter,1)=ittc4(ct1,1):
DR_meas(counter,1)=Tp4(ct1,1);

207

% Graphical Validation

% szrnu’prediction ittc:Network prediction
figure
haplot(t,Tp1(1:i,1),'k+',t,ittc1(1:i,1),‘ko');
%title('Yield');

xlabel('Process Time [s]');

ylabel('Yield [%1');

legend('FEM','GRNN',4);
set(h,'MarkerSize',5,'LineWidth',l.5);

grid on

figure
h=plot(t,Tp2(1:i,1),'k+',t,ittc2(1:i,l),'ko');
%title('Sa1monella LR');

xlabel('Process Time [s]');
ylabe1('Log(Log Reduction of Salmonella)');
legend('FEM','GRNN',4);
set(h,'MarkerSize',5,'LineWidth',l.5);
grid on

figure
h:plot(t,Tp3(1:i,1),'k+',t,ittc3(1:i,1),'ko'):
%title('DI');

xlabel('Process Time [s]');

ylabel('Log(Log Reduction of Internal color change)');
1egend('FEM','GRNN',4);
set(h,'MarkerSize',5,'LineWidth',l.5);

grid on

figure
hgplot(t,Tp4(1:i,l),'k+',t,ittc4(1:i,1),'ko');
%title('DEavg');

xlabel('Process Time [s]');

ylabe1('Log(Log Reduction of Crust Color Change)');
legend('FEM','GRNN',4);
set(h,'MarkerSize',5,'Linewidth',1.5);

grid on

fprintf('%d\n',counter)
end

% Exporting final results

Outputs(:,1)=Y_meas(:,1);
Outputs(:,2)-Y_pred(:,1);
Outputs(:,3)=b_meas(:,1);
Outputs(:,4)=L_pred(:.1):
Outputs(:,5)-DI_meas(:,1);
Outputs(:,6)=DI_pred(:,1);
Outputs(:,7)-DE_meas(:,1);
Outputs(:,8)=DE_pred(:,1):

208

fid=
fopen(['C:\MATLAB7\work\Dissertation\NN\Validation\result_final_opt.dat
ll'lwtl)’.

fprintf(fid,'%10.4f %10.4f %10.4f %10.4f %10.4f %10.4f %10.4f %10.4f
\n',Outputs');

fclose(fid);

209

Gen erate_Squata_ Train. m

% Generate_Squata_Train.m
% Reconstruct P(input) & Tg(output) with original data to train DNNM

function [rv1, rv2]=Generate_Squata_Train(a,resu1t)

% Input 8 output matrix structure

% [input output]

% inputsle Sd Vd / Tm Sm vm./ Y L Tc Ts / Yp Lp Tcp Tsp / t ]
% output=[Yf Lf ch Tsf]

[r c]=size(a); % [#rows #columns]

m=2; % number of past time series
i=0;
dt=10; % [sec]

for mi=(dt*2+1):(2*dt):(r-O);% (2*dt) due to the time interval of data
set is 0.5 s
i=i+1;

%Conditions @ t(n+1)
%Process Temperature
for n=1:m
if (mi-2*dt*n)<1
T(n)=25; % Ambient Temp. 25C
else
T(n)-a(mi-2*dt*n,2);
end
end

%Process Humidity
for nalzm
if (mi-2*dt*n)<1
H(n)-5; % Ambient Humidity. 30%
else
H(n)=a(mi-2*dt*n,3);
end
end

%Jet Velocity
for n=1zm
if (mi-2*dt*n)<1
V(n)=0; % Ambient Jet. 0 m/s
else
V(n)=a(mi-2*dt*n,4);
end
end

% Time data

if (mi-2*dt*l)<= 0
t=((mi-2*dt*1)+1)/2-1;

else

210

t=a(mi-2*dt*1,1); %a(mi-2*n,1);
end

%State @ t(n)

%Output
for n=1:m-1
if (mi-2*dt*n)< 1 % initial state
switch result

case 5
Yp(n)=100; %initial yield
case 7
Yp(n)=0; %initial Salmonella LR
case 8
Yp(n)=0; %initial DI
case 10
Yp(n)=0; %initial DEavg
end
else

dumy=a(mi-2*dt*n,result);
Yp(n)=dumy; % 5: row# contains output variable

end

end

dumy=a(mi,resu1t); % 5: row# contains output variable
Yf=dumy;

% Constructing input and output set
k=i; %k=i-m4 %(m-l);

% Input data set

P(l:m,k)=T(1:m); % Temperature series
P(m+1:2*m,k)=H(l:m); % Humidity series
P(2*m+l:3*m,k)¢V(1:m); % velocity series
P(3*m+1:4*m-1,k)=Yp(1:mrl); % Output series
P(4*m,k)=t; % Linear Data series

% P(4*m:5*mrl,k)=t(1:m);

% T matrix
Tg(1,k)=Yf; % State @ t(n+1)

end
% Return variables

rvlsP;
rv2=Tg;

211

Gen erate_Squata_ Test. m

% Generate_Squata_Test.m
% Reconstruct G(input)with test data to simulate DNNM

function rv1=Generate_Squata_Test(a)
[r c]=size(a); % [#rows #columns]

m=2; % number of past time series
i=0;
dt=10; % [sec]

% Tailoring original condition data set
T1=a(1:dt*2:r,2); [i j]=size(T1); Tl(i+1,1)=a(r,2);
H1=a(l:dt*2:r,3); [i j]=size(H1); H1(i+1,1)8a(r,3);
Vl=a(l:dt*2:r,4); [i j]=size(V1); V1(i+l,1)=a(r,4);
t=a(1:dt*2:r,1); [i j]=size(t); t(i+l,1)=a(r,1);

% Transposing
T1=T1'; H1=H1'; VISVI’; =t';

[nr nc]=size(T1);

% Assigning intial values

T2(1,1)=25; T2(1,2:nc)=T1(l,1:nc-1);
H2(1,1)=25; H2(1,2:nc)=H1(1,1:nc-1);
V2(1,1)=25; V2(1,2:nc)=V1(1,1:nc-1);

% Merging into a single matrix
G(1,:)=T1;
G(2,:)=T2;
G(3,:)=H1;
G(4,:)=H2;
G(5,:)=V1;
G(6,:)=V2;
G(8,:)=t;

% Return value
rvl=G;

212

Static Neural Network Model (SNNM)

213

T rain_SNNM. m

 

% Train_SNNM. m
% This program is to train SNNM with random.profile

clear all

 

TRAINING

 

:5 1P ¢IP

% Reading files
fprintf('**************\n');
fprintf('Training Phase\n')

ct=0 ; % counter
for i=1:1000

% Reading Fourier parameters

fid=fopen(['C:\MATLAB?\work\Dissertation\Data_Pool\D2\Condition
Para\Parameter_' ,num2str(i) , ' .dat' J) ;

=fscanf(fid,'%f',[1, infll; % [temp(1-21) steamt22-42)
velocity(43-63) time(64) yield.SamonellaLR DI DEavg RMSE(T) RMSE(H)
RMSE (V) ]

h=h';

fclose(fid);

% Reading results

fidsfopen(['C:\MATLAB7\work\Dissertation\Data_Pool\D2\PB Solution
Files\FEsolution_',num23tr(i),'.dat']);

g=fscanf(fid,'%f %f %f %f %f %f %e %e %e %e %f %f',[12, inf]); %
[temp(l-21) steam(22-42) velocity(43-63) time(64) yield SamonellaLR DI
DEavg RMSE (T) RMSE (H) RMSE (V) ]

9=9' :

fclose(fid);

[rn cn]=size(g);

% Rearranging data structure
ct=ct+1;
P(1:64,ct)=h(1:64,1); % input training

% Yield
T(1,ct)=g(rn,5); % output training

% lethality
if g(rn,7)<0.0000000001
T(2,ct)=-10;
else
T(2,ct)-log10(g(rn,7)); % output training
end

% DI (interanal color change)

if g(rn,8)<0.000001
T(3,ct)=-6;

214

else
T(3,ct)=log10(g(rn,8)); % output training
end

% DE (surface color change)
T(4,ct)=g(rn.10);

end

% Normalization
[Pn,minP,maxP,Tn,minT,maxT] = premnmx(P,T);

% Network training: feed-forward neural network

net 3 newff([minP maxP],[40 20 4],('tansig' 'tansig'
'purelin'},'traincgp'); %40 20
net.trainParam~epochs 8 1500;
net.trainParam~show=10;

net 8 train(net,Pn,Tn);

fprintf('\n>8nd of Training\n')

% Saving trained network 8 parameters
save('C:\MATLAB?\work\Dissertation\NN\SNNM_R\net',’net');

Outputs(1:64,1)=minP;
Outputs(1:64,2)=maxP;
Outputs(65:68,1)=minT;
Outputs(65:68,2)=maxT;

fid: fopen('C:\MATLAB?\work\Dissertation\NN\SNNM_R\net;para.dat','wt');
fprintf(fid,'%10.8f %10.8£\n',0utputs');
fclose(fid);

 
 

  
    

-4- -_,._+-, o . —.- e . +++4-—-4.- ro—r -..—...._-,,.__,_ o . o-o- . o o. ... .- o4o-4- - o—.——o4-+ . ,- o 04—. 4. o 7.— - 9 H4
o o o

- -_ . ._._.- o o . .__- H—.. o . -__.—.e—-o—-—.+---——..vo4-+ .—- +04 . - .- 4—0 —- -.—.—-+—. o—v— . --.—. o—o—+~- 4. . ..

 
 

 
     

% Testing

ct=0;

clear h g Pt Tt Outputs
nst;

fprintf(' ---------------- \n')
fprintf('Simulation Phase\n')

% Retrieving stored network and parameters
clear net
load('C:\MATLAB?\work\Dissertation\NN\SNNM;R\net','net');

fidafopen('C:\MATLAB?\work\Dissertation\NN\SNNMLR\net‘para.dat');
asfscanf(fid,'%f %f',[2, inf]); % [time temp steam velocity yield.vang
SamonellaLR DI D34 DEavg Tc Ts]

asa';

fclose(fid);

215

minP=a(1:64,1);
maxP=a(l:64,2);
minT=a(65:68,1);
maxT=a(65:68,2);

for i8121000
nf=nf+1;

% Reading Fourier parameters

fidnfopen(['C:\MATLAB7\work\Dissertation\Data_Pool\D6\Para\Parameter_',
num2str(i) , ‘ .dat']);

hsfscanf(fid,'%f',[1, infl); % [temp(1-21) steam(22-42)
velocity(43-63) time(64) yield.SamonellaLR DI DEavg RMSB(T) RMSE(H)
RMSBW) ]

h=h';

fclose(fid);

% Reading results

fidsfopen(['C:\MATLAB?\work\Dissertation\Data_Pool\D6\Test_Pourier',num
2str(i) , ' .dat']);

g=fscanf(fid,'%f %f %f %f %f %f %e %e %e %e %f %f',[12, infl); %
[temp(1-21) steam(22-42) velocity(43-63) time(64) yield SamonellaLR DI
DEavg RMSE ('1') mass (3) RMSE (V) 1

939';

fclose(fid);

[rn cn]=size(g);

% Rearranging data structure
ct=ct+l;
Pt(1:64,ct)=h(1:64,1); % input training

% Yield
Tt(1,ct)=g(rn,5); % output training

% lethality
if g(rn,7)<0.0000000001
Tt(2,ct)--10;
else
Tt(2,ct)=loglO(g(rn,7)); % output training
end

% DI
if g(rn,8)<0.000001
Tt(3,ct)=-6;
else
Tt(3,ct)=loglO(g(rn,8)); % output training
end

% DE
Tt(4,ct)=g(rn,10);
end

% Nermalization

216

[Ptn] = tramnmx(Pt,minP,maxP);

% simulation
Tsimnssim(net,Ptn);
Tsimspostmnmx(Tsimn,minT,maxT);

 

 

Graphical validation

*—-O-o-o——o—<v—o -*—.—- o *9 -797-9— o-o-o o --.—--- o—o o~-.-—-.o~+a HM+_._4_._._.

##1"

     

figure

plot(Tt(1,:),Tsim(l,:),'.r');
title('Yield')

grid on
RMSE_Yield:RMSE(Tt(1,:)',Tsim(1,:)',0);

figure
plot(Tt(2,:),Tsim(2,:),'.r');
title('Salmonella LR')

grid on

RMSE_LR=RMSE(Tt(2, :) ' ,Tsim(2, :) ' ,0);

figure
plot(Tt(3,:),Tsim(3,:),'.r');
title('DI')

grid on

RMSE_DI=RMSE (Tt(3, :) ' ,Tsim(3, :) ' ,0);

figure

plot(Tt(4,:),Tsim(4,:),'.r');
title('DEavg')

grid on

RMSE_DEavg:-RMSE (Tt(4, :) ' ,Tsim(d, :) ' ,0);

fprintf('%6.4f\n%6.4f\n%6.4f\n%6.4f\n',RMSE_Yie1d,RMBE_LR,RMSE_DI,RMSE_
DEavg);

 

%sm «44 ~- 44 II

% Exporting

 

 

% Changing variable name

F1(:,1)=Tt(1,:)'; P1(:,2)-Tsim(1,:
F2(:,1)=Tt(2,:)'; F2(:,2)=Tsim(2,:
P3(:,1)=Tt(3,:)'; FB(:,2)=Tsim(3,:
F4(:,1)=Tt(4,:)'; F4(:,2)=Tsim(4,:

savvy
- c o c

‘0 ‘g ‘0 ‘.

% Exporting final results
Outputs(:,1)=F1(:,1);
Outputs(:,2)-Fl(:,2):
Outputs(:,3)=F2(:,1);
Outputs(:,4)=F2(:,2);
Outputs(:,5)=F3(:,l):
Outputs(:,6)-F3(:,2);
Outputs(:,7)=F4(:,1);

217

Outputs(: ,8)=F4(: ,2);
Outputs=0utputs';

fids fopen(['C:\MATLAB?\work\Dissertation\NN\SNNM_R\RMSE.dat'],'wt');
fprintf(fid,'RMSE(Y)8 %10.4f\n',RMSE_Yield);

fprintf(fid, 'RMSE(L)= %10. 4f\n' ,RMBE _LR);

fprintf(fid,,'RMSE(DI)= %10. 4f\n' ,RMSE _DI);

fprintf(fid.,'RMSE(DE)= %10. 4f\n' ,RMSE _DEavg);

fprintf(fid, '%10. 4f %10. 4f %10. 4f %10. 4f %10. 4f %10. 4f %10. 4f %10. 4f
\n' ,Outputs);

fclose(fid);

clear Outputs

 

Classification 5 Exporting

«Dd’c’

 

% Initial values
c1=0;c2=0;c3=0;c4=0;
c5=0;c6=0;c7=0;c8=0;
Outputs_aa=[0; 0; 0; 0; ; , ;
Output8_rr=[0; OI. O; o; I I I
Outputs_typeI=[0; 0; 0; ; , ; ;
Outputs_typeII=[0; 0; 0; 0; ; 0; 0; 0;];
index_aa=0; index_rr=0; index typeI=0; index _typeII=0;

for i=1:1000

% Rate of the accepted for the accepted
if
(P2(i,1)>=0.813)£&(F3(i,1)>=0.146)8&(F4(i,1)>=0.26)5&(F4(i,1)<=0.497)%
accpeted patties among the measured
c1=c1+1;
if
(F2(i,2)>=0.813)8&(F3(i,2)>=0.146)8&(F4(i,2)>=0.26)5&(F4(i,2)<=0.497)
c2=c2+1;
Outputs_aa(c2,l)=Fl(i,1);
Outputs_aa(c2,2)=Fl(i,2);
Outputs_aa(c2,3)=F2(i,l);
Outputs_aa(c2,4)=F2(i,2);
Outputs_aa(c2,5)=F3(i,1);
Outputs_aa(c2,6)=F3(i,2);
Outputs_aa(c2,7)=F4(i,1);
Outputs_aa(c2,8)=F4(i,2);
index_aa(c2,1)=i;
end
end

% Rate of the rejected for the rejected
if
(F2(i,1)<0.813)||(F3(i,1)<0.146)|[(P4(i,1)<0.26)||(F4(i,1)>0.497)%
rejected patties among the measured
c3=c3+1;

218

if

(F2(i,2)<0.813)||(F3(i,2)<0.146)||(F4(i,2)<0.26)ll(F4(i,2)>O.497)

end

end

c4=c4+1;
Outputs_rr(c4,1)=F1(i,1);
Outputs_rr(c4,2)=F1(i,2);
Outputs_rr(c4,3)=F2(i,l);
Outputs_rr(c4,4)=P2(i,2);
Outputs_rr(c4,5)=F3(i,1);
Outputs_rr(c4,6)=F3(i,2);
Outputs_rr(c4,7)=F4(i,1);
Outputs_rr(c4,8)=F4(i,2);
index_rr(c4,1)-i;

% Rate of Type I error

if

(F2(i,1)>=0.813)5&(F3(i,1)>-0.146)&&(F4(i,1)>=0.26)8&(F4(i,1)<=0.497)%
accpeted patties among the measured
c5=c5+1;

if

(F2(i,2)<0.813)||(F3(i,2)<0.146)||(F4(i,2)<0.26)||(F4(i,2)>0.497)

end

end

c6=c6+1;
Outputs_typeI(c6,1)=F1(i,1):
Outputs_typeI(c6,2)-F1(i,2):
Outputs_typeI(c6,3)=P2(i,1);
Outputs_typeI(c6,4)=F2(i,2);
Outputs_typeI(c6,5)=F3(i,1);
Outputs_typeI(c6,6)-F3(i,2);
Outputs_typeI(c6,7)=F4(i,1);
Outputs_typeI(c6,8)=F4(i,2);
index_typeI(c6,1)=i;

% Rate of False-pass classification error

if

(P2(i,1)<0.813)||(F3(i,1)<0.146)ll(P4(i,1)<0.26)||(P4(i,1)>0.497)%
rejected patties among the measured
c7=c7+l;

if

(F2(i,2)>=0.813)8&(P3(i,2)>=0.146)5&(F4(i,2)>80.26)8&(F4(i,2)<=0.497)

end

end

c8=c8+1;
Outputs_typeII(c8,1)=F1(i,1);
Outputs_typeII(c8,2)=F1(i,2);
Outputs_typeII(c8,3)=F2(i,1);
Outputs_typeII(c8,4)-F2(i,2):
Outputs_typeII(c8,5)=F3(i,1);
Outputs_typeII(c8,6)=F3(i,2);
Outputs_typeII(c8,7)=F4(i,1);
Outputs_typeII(c8,8)=F4(i,2);
index_typeII(c8,1)=i;

219

end

R_aa=c2/c1;
R;rr=c4/c3;
R_typeI=c6/c5;
R_typeII=c8/c7;

% Exporting Classification Results

fids fopen(['C: \MATLAB7\work\Dissertation\NN\SNNM;R\DA_aa.dat'],'wt');
fprintf(fid, 'R _aa= %10. 4f=%d/%d\n' ,R _aa,c2, cl);

fprintf(fid, '%10. 4f %10. 4f %10. 4f %10. 4f %10. 4f %10. 4f %10. 4f %10. 4f
\n' ,Outputs_aa');

fprintf(fid,'%d\n',index_aa');

fclose(fid);

fids fopen(['C: \MATLAB7\work\Dissertation\NN\SNNM;R\DA_rr.dat'],‘wt');
fprintf(fid, 'R_ rr= %10. 4f=%d/%d\n' ,R_ rr, c4,c3);

fprintf(fid, '%10. 4f %10. 4f %10. 4f %10. 4f %10. 4f %10. 4f %10. 4f %10. 4f
\n' ,Outputs_rr' );

fprintf(fid,'%d\n',index_rr');

fclose(fid);

fids
fopen(['C:\MATLAB7\work\Dissertation\NN\SNNMngDA;typeI.dat'],'wt');
fprintf (fid, 'R_typeI= %10 . 4f=%d/%d\n' , R_typeI , c6 , c5) ,-
fprintf(fid,'%10.4f %10.4f %10.4£ %10.4f %10.4f %10.4£ %10.4f %10.4f
\n',Outputs_typeI');

fprintf(fid,'%d\n',index_typeI');

fclose(fid);

fida

fopen([' C: \MATLAB7\work\Dissertation\NN\SNNMiR\DA _typeII.dat'],'wt');
fprintf(fid, 'R _typeII= %10. 4f=%d/%d\n' ,R _typeII, c8 ,c7);

fprintf(fid, '%10. 4f %10. 4f %10. 4f %10. 4f %10. 4f %10. 4f %10. 4f %10. 4f
\n' ,Outputs_typeII' );

fprintf(fid,'%d\n',index_typeII');

fclose(fid);

220

Genetic Algorithm (GA)

This case was coupled with FEM and PLI (piecewise linear interpolation)
parameterizaiton method.

GA_FEM_PLI.m has following sub-programs. Some sub-programs for genetic operations
(crossover and mutation) were adopted from the work of Venkataraman (2002) and
modiﬁed.
0 SimpleCrossover.m
ArithmeticCrossoverm
Mutation_PLI.m
Populator_PLI.m
CheckRange_PLI.m
GenProﬁle_PLI.m
fun_FEM_PLI.m
o FunGenFEM_PLI.m
o Model_F EM (Finite element model was not included. Refer the model
developed by Watkins (2004).

/ fun_DNNM_FS.m :
This sub-program can be included when the GA is coupled with DNNM and
Fourier series parameterization method while maintaing framworks of
GA_FEM_PLI.m.
o FourierFunGen.m (included DNNM package)
0 Generate_Squata_Test.m (included in DNNM package)

221

GA_ FEM_PLLm

% GA_FEM_PLI.m
% Genetic algorithm coupled with FEM and piecewise linear interpolation
% parameterization (PLI) were used.

clear all
global Fbest_trace Xbest_traceIMu_count

% GA parameters
fprintf('GA Optimization Started!\n')

nDes=64; % i of design parameters [duration temp moisture velocity]
nG=200; % total number of generation

nPi=10; % i of initial population

nP=4; % i of population for each generation

nSC=3; % number of simple crossover - each yields 2 children
nAC=3; % number of arithmetic crossover - each yields 2 children
nIM=4; % number of imigrants

nMU=2; % number of mutation

% Generating initial population
fprintf('Generating initial population\n')

% Getting initial population information from a file
for i=1:nPi

fid=fopen(['lhome/jeongsal/Matlab/1/IP/IDsmooth_',num2str(i),'.dat']);

g=fscanf(fid,'%g',[1, infl); % [temp(1-21) steam(22-42)
velocity(43-63) time(64) yield SamonellaLR DI DEavg RMSE(T) RMSE(H)
RMSE (V) ]

939';

fclose(fid);

vector=g(l:64,1);
status=1; % lacontraints satisfied; Oanot satisfied
results=g(65:68,1); % predicted yield, Salmonella reduction, DI, DE

ind(i,:)=vector';
ind_Y(i,1)=results(l,1); % yield

ind_status(i,1)=status;
ind_resu1ts(i,:)=results'; % Evaluation of individual: Yield

% ind_L(i,1)=Lf; %fun_nn_1stg_nlec(vector); % Evaluation of
individual: Lethality

fprintf('%d ',i)

end

fprintf('\n')

% If you want to generate an initial pop., unmark following marked code

and
% mark the above code.

222

% % Generating initial population

% ctsO;

% for i=1:nPi

% status=0; % 0:constaints are not satisfied, lzsatisfied

% while statues-0

% vectoraPopulator_PLI(nDes); % Generating individuals
having target lethality

% [status, results]=fun_PEM_PLI(vector); % Calculating
result with GRNN and vector

%

% end

% ind(i,:)=vector'; % transposed

% ind_Y(i,1)=results(l,1); %100-fun_nn_1stg(vector); %
Evaluation of individual: Yield

%

% ind_status(i,1)=status;

% ind_results(i,:)=resu1ts'; % Evaluation of individual: Yield
53

% passedCond(1:nDes,1)=vector;

% passedCond(nDes+1:nDes+4,1)=results;

% ctsct+1;

%
fidnfopen(['C:\MATLAB?\work\Dissertation\Optimizer\InitialPopulation\ID

',num2str(ct),'.dat'],'wt');
fprintf(fid,'%f\n',passedCond');
fclose(fid);

'5

%

%

% fprintf('%d ',i)
% end
% fprintf('\n')
% Sorting to select parents

[value, index]=sort(ind;Y,'descend');

% Selecting parents
for i-lznP
Xgen(i,:)=ind(index(i),:);

Xgen_status(i,1)=ind_status(index(i),:);
Xgen_resu1ts(i,:)=ind_results(index(i),:);

E_Origin(i,1)=ind;results(index(i),1);
x_Origin(i,:)=ind(index(i),:);

end

clear index

***********************************************************

% Generating offsprings and immigrants
for iG=1:nG

fprintf('***\n')
fprintf('Generation %d\n',iG)

% Simple crossover
[XChild] s SimpleCrossover(nDes, nP,nSC,Xgen);
for i=1:2*nSC

223

Xgen(nP+i,:) = XChild(i,:);
end
fprintf('\nFinished: Simple Crossover')

% Arithmetic crossover
[XAChild] a ArithmeticCrossover(nDes, nP,nAC,Xgen);
for i=l:2*nAC
Xgen(nP+2*nSC+i,:) - XAChild(i,:);
end
fprintf('\nFinished: Arithmetic Crossover')

% Mutation
[n1 m1] = size(Xgen);
[XMu]=.Mutation_PLI(nDes,Xgen);% mutate all members

for i=1:nMU % nMU mutated.member selected among all muatated
population

Xgen(nP+2*nSC+2*nAC+i,:) 8 XMu(i,:);

end

fprintf('\nPinished; Mutation')

% Immigration
[n2 m2] I size(Xgen);

for j = 1:nIM
vect = Populator_PLI(nDes);
Xgen(n2+j,:) 8 vect';

end

fprintf('\nFinished: Immigration\n')

% Checking the validity of parameters: testing if the actual control
% profiles are in the proper ranges
i=0:
clear temegen temegen_status temegen_results
[n4 m4] 8 size(Xgen);
for i=1:n4
s(i,1)=CheckRange_PLI(Xgen(i,:)');
if s(i,1)=-=1
j=j+1;
temegen(j,:)=Xgen(i,:);
if (i<=nP)
temegen_status(j,:)=Xgen_status(i,1);
temegen_results(j,:)=Xgen_resu1ts(i,:);
end
end
end

clear Xgen

Xgenstemegen;

Xgen_status=temegen_status;
Xgen_results=temegen_results;

fprintf('Finished: Range check: ')

[n3 m3] - size(Xgen);

fprintf('Rejection rate(range)=%d/%d\n',(n4-n3),n4)

224

% Evaluation
fprintf('Evaluation process\n')
for i = (nP+1):n3
vector 8 Xgen(i,:);
[status, results]=fun_PEM_PLI(vector');

Xgen_status(i,1)=status;
Xgen_results(i,:)-resu1ts';
end

% Inspecting constraints satisfaction
kk=0;
for i=1:n3
if (Xgen_status(i,1)-1)&(isnan(Xgen_results(1,1))==0)
kkakk+1;
passengen(kk,:)=Xgen(i,:);
passengen_results(kk,:)=Xgen_resu1ts(i,:);
passengen_status(kk,:)=Xgen_status(i,1);
end
end

if kk<2
fprintf('\nSpieces terminated\n')
break;

end

% Sorting
[value, indexlssort(passengen_results(:,1),'descend');

clear temp temp_status temp_results

% Temporary storage

tempspassengen;
temp_status-passengen_status;
temp_results-passengen_resu1ts;

fprintf('Number of passed generations %d',kk)
passengen_results(index,:);
fprintf('Pass ratio8 %d/%d\n',kk,n3)

Xbest=passengen(index(1),:); % best condition at each generation
Fbest=passengen_resu1ts(index(1),:); % best yield at each

generation
Xbest_trace(iG,:)=Xbest; % progressive history
Ebest_trace(iG,:)-Fbest; % progressive history

clear Xgen Xgen_status Xgen_resu1ts XChild XAChild xuu passengen
passengen_results passengen_status

% Selecting parents: Same vector will be skipped for diversity

for i=1:nP
Xgen(i,:)-temp(index(i),:);
Xgen_status(i,1)=temp_status(index(i),1);
Xgen_results(i,:)=temp_results(index(i),:);

end

225

clear index
Xgen_resu1ts

% Saving workspace at every 50 iteration
if(mod(iG,50)==0)

save(['C:\MATLAB7\work\Dissertation\Packages\1\1\NerkSpace_',num2str(iG
)I"Mt']);
end

end % End of cycle of a generation

% ............................................................
% Plotting 8 Exporting

% ............................................................
fprintf('Plotting 5 Exporting\n')

[n m] I size(Xbest_trace);

% Plotting the best profiles
p=FunGenFEM_PLI(Xbest_trace(n,:)');
time=p(:,1);

tM(: I2);

hum-Pt: .3);

vel=p(:,4);

figure

h=plot(time,temp,'r—',time,hum,'b--',time,ve1,'mr.');
x1abe1('Process time [s]');

ylabel('Temperature [deg. C] / Humidity [%MV] / velocity [m/sl');
1egend('Temperature','Humidity','velocity',4);
set(h,'LineWidth',1.5);

totznP+2*nSC+2*nAC+Mu_count+nIM

SimpleCrossover_rate=(2*nSC)ltot % each crossover produce two children
ArithmaticCrossover_rate=(2*nAC)ltot

MMtation_ratedMu_count/tot

Immigration_rate=nIM/tot

% Exporting result
% History of generations

fid=fopen(['C:\MATLAB7\work\Dissertation\Packages\1\1\Result_generation
s.dat'],'wt');
for i=1:nG

for j=1:m

fprintf(fid,'%10.4f',Xbest_trace(i,j));

end

fprintf(fid,'%10.4f %10.4f %10.4f %10.4f\n',Ebest_trace(i,:));
end

fclose(fid);

% Reporting GA summary

fid3=
fopen(['C:\MATLAB7\work\Dissertation\Packages\1\1\Result;summary.dat'],
'vt');

226

fprintf(fid3,‘ ---------------------------------------------------------
\n');

fprintf(fid3,’ SUMMARY OF GA-FEMZOPTIMIZATION\n');

fprintf(fid3,‘ >Discretized profile for each control vector\n');
fprintf(fid3,‘ >GA-FEM found the best combination of 64
coefficients\n');

fprintf(fid3,‘ ---------------------------------------------------------
\n');

fprintf(fid3,'Yield;max= %10.4f [%%]

\nlog(LRamicrob)= %10.4f\nlog(DI)- %10.4f\nlog(DE)8 %10.4f\n\n',Fbest);
fprintf(fid3,‘[Optimization Parameters]\n');

fprintf(fid3,'1. Design variables: %d\n',nDes);
fprintf(fid3,'2. Initial population: %d\n',nPi);
fprintf(fid3,'3. Parents: %d\n',nP);

fprintf(fid3,'4. Generation cycle: %d\n',nG);

fprintf(fid3,'5. Simple crossover: %d\n',nSC);
fprintf(fid3,'6. Arithmatic crossover: %d\n',nAC);
fprintf(fid3,'7. Immigration: %d\n',nIM);

fprintf(fid3,'8. Population/genertion: %d\n\n',tot);
fprintf(fid3,'9. Simple crossover

rate: %6.2f\n',SimpleCrossover_rate);
fprintf(fid3,'10. Arithmetic crossover

rate: %6.2f\n',ArithmaticCrossover_rate);

fprintf (fid3, ' 11 . Mutation rate: %6. 2f\n' ,Mutation_rate) ;
fprintf(fid3,'12. Immigaration

rate: %6.2f\n',Immigration_rate);

fprintf(fid3,‘ --------------------------------------------------------
\n') ;

fclose(fid3);

227

SimpleCrossover.m

 

% creating children by simple crossover
function ret = SimpleCrossover(nDes,nP,nSC,X)

XPl = X(l,:);
XP2 a X(2,:);

for k=1:nSC
r - rand(1,1);

if r > 0.5

r1 = floor(nDes*r);
else

r1 = cei1(nDes*r);
end

for i = l:1:length(XP1)
if i < r1
XC1(i) = XP1 (i);
XC2 (i) a XP2 (i);
else
XCl(i)= XP2(i);
XC2(i)= XP1(i);
end
end

temp=2*(k-1)+1;
C(temp,:)=XC1;
C(temp+1,:)=xc2;

end

ret 3C;

228

ArithmeticCrossover.m

function ret = ArithmeticCrossover(nDes,nP,nAC,X)

XP1=X(1,=);
XP2=X(2,:);

for k=l:nAC
r1 = rand(1,1);
r2 = 1 - r1;

for i = l:nDes
XC1(1,i) = r1*XPl(l,i) + r2*XP2(1,i);
XC2(l,i) = r2*XP1(1,i) + r1*XP2(l,i);
end
temp=2*(k—1)+1;
C(temp,:)=XC1;
C(temp+l,:)=XC2;

end

ret =C;

229

Mutation_PLI.m

% Mutating all the vectors by changing a single element

function ret I Mutation_PLI(nDes,X)
global xvl yvl xv2 yv2 Mu_count

Mu_countIO; % mutation counter

xx I X;
[n m] =size(X);

for

end

ret

i I 1:n
r I rand(1,l);
if r > 0.5

r1 I floor(nDes*r);
else

:1 I ceil(nDes*r);
end

for j = 1:m
if(j II r1)
x=Populator_PLI(nDes);
xxu,j) - xij):
Mu_countIMu_count+1;
end
end

= XX;

230

Populator_PLI.m

% Generating control parameters within feasible ranges

function rv2=Populator_PLI(nDes)

% generates a random control vector

n=(nDes-1)/3; % w/o duration; 21 points for each control vector
t_L=60; t_D=600; % time bounds [s]; L=lower UIupper

T;L=100; T_D=250; % Temperature bounds [C]
H_LIO; H_D=100; % Steam content bounds [% by volume]

v;L=0; V;D=30; % Impingement velocity bounds [mls]

% generating random profiles
x(1:n,1)IT_L+(T_D-T_L)*rand(n,1); % Temperature
x(n+1:2*n,1)=H_L+(H_D-H_L)*rand(n,1); % Humidity
x(2*n+1:3*n,l)=v;L+(v;D-V;L)*rand(n,1); % velocity

x(3*n+1,1)Iround(t_L+(t_D-t_L)*rand(1,1)); % duration

rv2Ix;

231

CWeothngquthn

% Check if the control profiles are within the upper and lower bounds.
function status=CheckRange_PLI(X)

% Temperature, humidity, and velocity bounds
Bounds=[100 250; 0 100; 0 30]; %[TL TU; HL HU; VL VU]

[r c]=size(X);

duration=X(r,1);
PIE‘unGenFEM_PLI (X) ;
timestep=0.5;

T(111)=P(3:2):
313111=P(1:3);
v(:11)=P(:I4);

% Check violations

if ((max(T) <= Bounds(l,2)) && (min(T) >= Bounds(1,1)))
Sl=l; .

else
81:0;

end

if ((max(H) <I Bounds(2,2)) as (min(H) >I Bounds(2,1)))
S2=1;
else

end

if ((max(V) <= Bounds(3,2)) && (min(V) >= Bounds(3,1)))
S3=1;

else
33:0;

end

status = 81 * $2 * S3; % 1=passed, 0=not passed

232

GenProﬁIe_PLI.m

 

% Calculating actual time-condition data with parameters
function rvl = GenProfi1e_PLI(v) % v=vector

% Spliting parameters for each control vector
T(1:21,1)=v(l:21,1); % Temperature
M(1:21,1)=v(22:42,1); % Humidity
V(1:21,l)=v(43:63,1); % velocity
duration=v(64,1);

tIO:(duration/(21-1)):duration;

% Generating 0.5 sec time intervals
ti=0:0.5:duration;

t=t';
ti=ti';

% Interpolation with parameters
Ti = interpl(t,T,ti);
Mi interp1(t,M,ti) ;
Vi interp1(t,v,ti);

% Merging time-condition data into a matrix
Cond(:,1)=ti;
Cond(:,2)=Ti;
Cond(:,3)=Mi;
Cond(:,4)IVi;

% return value
rv1=Cond;

233

 

fun_FEM_PLI.m

% Process evaluation with FEM and PLI

function [status, results]=fun_PEM_PLI(v) % nlc: nonlinear constraints

% Getting function value at each time
CIFunGenFEM_PLI (v) ;

duration=v(64,1);

TeIC(:,2);

Se=C(:,3);

ve=C(:,4);

dt=0.5;

% FEM Solution
[rvl rv2 rv3 rv4 rv5 rv6 rv7 rv8]IModle_PEM(duration,Te,Se,ve,dt);

[nr nc]=size(rv1);

results(1,1)=rv1(nr,1); % Yield
results(2,1)=rv3(nr,1); % Lethality
results(3,1)=rv4(nr,1); % DI
results(4,1)=rv6(nr,1); % DE

Lfsresults(2,l);
DIfIresults(3,1);
DEf=results(4,1);

if (Lf >- 6.5)
Det_Lf=1; % passed
else
Det_Lf=0; % failed
end

if (DIf >I 1.4)
DetgprIl; % passed
else
Det_DIf=0; % failed
end

if (DEf >I 0.263)&& (DEf <= 0.497)
Det_DEf=1; % passed

else
Det_DEfI0; % failed

and

% 1: feasible; 2: infeasible
status=Det_Lf*Det_DIf*Det;DEf;

234

Fun GenFEM_PLI.m

 

% Calculating actual time-condition data with parameters
function rvl = FunGenFEM_PLI (v) % v=vector

% Spliting parameters for each control vector
T(1:21,1)=v(1:21,1); % Temperature
M(1:21,1)=v(22:42,1); % Humidity
V(1:21,1)Iv(43:63,1); % velocity
duration=v(64,1);

t=0:(duration/(21-1)):duration;

% Generating 0.5 sec time intervals
ti=0:0.5:duration;

tIt';
tiIti';

% Interpolation with parameters
Ti I interp1(t,T,ti);
Mi I interp1(t,M,ti);
Vi I interp1(t,V,ti);

% Merging time-condition data into a matrix
Cond(:,1)Iti;
Cond(:,2)=Ti;
Cond(:,3)IMi;
Cond(:,4)IVi;

% return value
rleCond;

235

 

fun_DNNM_FS.m

% Dynamic neural network model and Fourier parameter inputs
function [status, results]=fun_DNNM_FS(v)

% Retrieving stored networks and parameters
clear net

load( 'M: \Matlab\NNs\net1' , 'netl ') ;

load( 'M: \Matlab\NNs\net2 ' , ' net2 ' ) ;

load( 'M:\Mat1ab\NNs\net3 ' , 'net3') ;
load('M:\Met1ab\NNs\net4','net4');

% Open scaling parameters

fid=fopen('M: \Matlab\NNs\normal_para.dat') ;

aIfscanf(fid,'%f %f %f %f %f %f %f %f',[8, infll; % [time temp steam
velocity yield vev M SamonellaLR DI DEG DEavg Tc Ts]

a=a';

fclose(fid);

% Assigning scaling parameters

minSP1=a(l:8,1); minSP2=a(1:8,3); minSP3Ia(1:8,5); minSPAIa(1:8,7);
maxSPl=a(1:8,2); maxSP2=a(1:8,4); maxSP3=a(1:8,6); maxSP4Ia(1:8,8);
minSTlIa(9,1); minST2=a(9,3); minST3=a(9,5); minST4Ia(9,7);
maxST1=a(9,2); maxST2=a(9,4); maxST3Ia(9,6); maxST4=a(9,8);

m=2; % number of system parameter (length of past time series)
mg=3*m+1+1; % total row number of training matrix
mk=3*m+1; % row location of target output value at training matrix

 

SIMULATION PHASE

cl’ & «ID

 

clear Outputs

clear Y_pred L_pred DI_pred DE_pred t
clear b Cond ittcl ittc2 ittc3 ittc4
clear ittl itt2 itt3 itt4

% Getting profiles
C=FourierFunGen(v);

% Rearranging input data for DNNM
CondIGenerate_Squata_Test(C);

[rt1,ct1]=size(Cond);

% Setting initial conditions
ippl(l:mg,1)=Cond(1:mg,1); % copy
ipp1(mk,1)=100; % 100

ittl(1:m-1,1)=0; % initial difference is zero

ipp2 (1:mg,1)ICond(1:mg,1) ;
ipp2(mk,1)=-10; % which is almost zero

236

itt2(1:m-1,1)IO;

ipp3(1:mg,1)ICond(1:mg,l);
ipp3(mk,l)I-10; % 0
itt3(1:m-1,1)=0;

ipp4(l:ng,1)ICond(1:mg,1);
ipp4(mk,1)=0;
itt4(1:m-1,1)I0;

% Initial simulation

% Transform data using a precalculated.minimum.and.maximum.value
ippln I tramnmx(ipp1,minSP1,maxSPl);
ipp2n I tramnmx(ipp2,minSP2,maxSP2);
ipp3n I tramnmx(ipp3,minSP3,maxSP3);
ipp4n I tramnmx(ipp4,minSP4,maxSP4);

% Simulation

itlIsim(net1,ippln); %ittc1(l,1)Iit1(1,1);
it2Isim(net2,ipp2n); %ittc2(l,1)=it2(1,1);
it3Isim(net3,ipp3n); %ittc3(1,1)Iit3(l,l);
it4=sim(net4,ipp4n); %ittc4(1,1)=it4(1,1);

% Denormalization

itl I postmnmx(it1,minST1,maxST1);
it2 I postmnmx(it2,minST2,maxST2);
it3 I postmnmx(it3,minST3,maxST3);
it4 I postmnmx(it4,minST4,maxST4);

ittc1(1,1)=it1(1,1);
ittc2(1,1)Iit2(l,1);
ittc3(1,1)=it3(1,1);
ittc4(1,1)Iit4(1,1);

g ...............................................
% Beginning of time stepping simulation

% ...............................................
for jI2:ct1

in;

% Constructing input data set
ippl(1:mk-1,1)=Cond(1:mk-l,i); % copying conditions
ipp1(mg,1)ICond(mg,i); % copying time
itt1(1,1)=it1(1,1);

ipp2(l:mk-1,1)=Cond(l:mk-1,i);
1PP2(m9.1)=Cond(m9.i);
itt2(1,1)Iit2(1,1);

ipp3(1:mk-l,1)ICond(1:mk-1,i);
ipp3(m9.1)-Cond(mg.i);
itt3(1,1)=it3(1,1);

ippd(1:mk-1,1)ICond(1:mk-1,i);
ipp4(mg,l)ICond(mg,i);

237

itt4(1,l)=it4(1,l);

% replacing past time series of output
ipp1(mk,1)Iippl(mk,1)-itt1(1,1); % value(tIn)Ivalue(tIn-1)-difference
ipp2 (mk,1)=ipp2 (11091) -itt2 (1.1):

ipp3(mk,l)Iipp3(mk,1)-itt3(1,1);

ipp4(mk,1)Iipp4(mk,1)-itt4(1,l);

% Normalization

ippln I tramnmx(ippl,minSP1,maxSPl);
ipp2n I tramnmx(ipp2,minSP2,maxSP2);
ipp3n I tramnmx(ipp3,minSP3,maxSP3);
ippdn I tramnmx(ipp4,minSP4,maxSP4);

% Simulation

itlIsim(net1,ipp1n); % yield
it2Isim(net2,ipp2n); % Salmonella inactivation
it3Isim(net3,ipp3n); % Internal color change
it4Isim(net4,ipp4n); % Surface color change

% Denormalization

itl I postmnmx(it1,minST1,maxSTl);
it2 I postmnmx(it2,minST2,maxST2);
it3 I postmnmx(it3,minST3,maxST3);
it4 I postmnmx(it4,minST4,maxST4);

% record simulation result; difference at each time
ittc1(1,i)Iit1(1,l);
ittc2(l,i)Iit2(1,1);
ittc3(1,i)=it3(l,1);
ittc4(l,i)Iit4(1,l);

% shift down past time series of output
ittl(2:m-1,1)Iittl(1:m-2,1);
itt2(2:m-l,l)Iitt2(1:m-2,l);
itt3(2:m-1,1)Iitt3(1:m-2,l);
itt4(2:m-l,1)Iitt4(1:m-2,1);

end % End of time stepping simulation

% copying time series to t
tICond(mg,l:ctl);

% Calculate cumulative results

% Yield

[b c]=size(ittc1);
ittc1m(1,1)I100;
ittc1m(1,2:c+l)I100-cumsum(ittc1);

% Salmonella inactivation

[b c]=size(ittc2);
ittc2m(1,1)I-10;
ittc2m(1,2:c+1)I-10-cumsum(ittc2);

% Internal color change
[b c]=size(ittc3);

238

 

ittc3m(1,l)I-10;
ittc3m(1,2:c+1)I-10-cumsum(ittc3);

% Surface color change

[b c]=size(ittc4);
ittc4m(1,l)I0;
ittc4m(1,2:c+1)I0+cumswm(ittc4);

% Transposing data
ittclIittch';
ittc2Iittc2m';
ittc3=ittc3m';
ittc4Iittc4m';

% Testing feasibility

results(1,1)Iittc1(ct1,1); % final yield
results(2,l)=ittc2(ct1,1); % final log(Salmone11a.LR)
results(3,1)Iittc3(ct1,1); % final log(DI.LR)
results(4,1)=ittc4(ct1,l); % final DE.LR

LfIresults(2,1);
DIfIresults(3,1);
DEfIresults(4,1);

if (Lf >I 0.813) % LfcI0.813, 2.5I316.2 log reduction
Det_LfI1; % passed

else
Det_LfI0; % failed

end

if (DIf >I 0.146) % DIfcI0.146
Det_DIfI1; % passed

else
DetgprIO; % failed

end

if (DEf >I 0.26)&& (DEf <I 0.497) % [0.26 0.497]
Det_DEfI1; % passed

else
Det_DEfI0; % failed

and

% 1: feasible, 2: infeasible
statusIDet_Lf*Det_DIf*Det_DEf;

239

Simulated Annealing (SA)

File association:

SA_FEM_FS.m has following sub-programs:

GenFourierParaSeLm
fun_FEM_FS.m
CheckRangeJn

F ourierF un Gen.m

240

SA_FEM_ FS.m

% SA;FEM_FS.m
% Simulated annealing algorithm coupled with FEM and Fourier series
% parameterization method.

clear all
sprintf('Simulated Annealing Algorithm Started...')

% Parameters
NiI20; % number of inner iteration

aI-0.5; % lower bound of random direction
bI0.5; % upper bound of random direction

I21; % number of design variables of a control vector

iGIO ;
nGIlSO; % total number of iteration

T0I100; % initial temp
TnI0.001; % final temp

% Stepsize for each parameters

stepsize(1:20,l)I2; stepsize(21,1)=10; % Temperature
stepsize(22:41,l)I2; stepsize(42,1)I5; % Humidity
stepsize(43:62,1)I2; stepsize(63,1)I2; % velocity
stepsize(64,l)=10; % duration

% Finding initial feasible control vector

% By using FEM
statusIO;
while statusIIO
XIGenFourierParaSet(N);
% Evaluate performance index
[status results]=fun_FEM_FS(X); % JI100-Yie1d
end

% % By reading from file

% fidIfopen(['/home/jeongsal/Matlab/G/IP/IP_',num2str(4),'.dat']);

% ngscanf(fid,'%f',[l, infl); % [temp(1-21) steam(22-42) velocity(43-
63) time(64) yield SamonellaLR DI DEavg RMSE(T) RMSE(H) RMSE(V) ]
9:9'F

fclose(fid);

XIg(1:64,1);

statusIl;

results=g(65:68,1);

d’ﬂﬂcpw

% Assinging initial objective value
JIresults(1,1)
iIl;

241

JinitIJ;
XinitIX;

[rn cn]=size(X);
TITo;

% Starting iteration

for iGI1:(nG+1)
kIO;
C_acceptI0; % Counter for acceptance
C_improvI0; % Counter for improvement

fprintf('%dth Generation\n',iG)

while k < Ni % Inner loop begins
statusIO;
while statusIIO
S£I0;

while Sf II 0

% generating random direction [a, b] @ n-dim.
hypersphere
s = a + (b - a) * rand(rn,1);

% Calculating new control vector
Knew I X + stepsize .* S;

% checking feasiblility of Xnew
SfICheckRange(Xnew);
end

% Obtaining prediction results by using process model
[status resultslIfun;FEM_FS(Knew);
end

Jneeresults(1,1);
delJIJnew-J;

% Checking if the new result is improved
if delJ > 0 % improved
P31;
CLimprov=C_ﬂmprov+1;
else
pIexp(-abs(delJ)/T);% Boltzmann probability distribution
end

rIrand(l); %0.7; %rand(1)

% New objective value is accepted and the design vector is
updated ‘
if p>r
XIXnew;
JIJnew;
ka+1;

242

 

iIi+l;
q_accepted(i,1)IJ;
xgaccepted(i,:)IX;
C_acceptIC_accept+1;

else
ka+1;
and
fprintf('kI%d pI%f r=%f JI%f\n',k,p,r,J)
end

% Cooling Schedule

NIiG;

AI(To-Tn)*(nG+1)/nG; % meta variable
BITo-A; % meta variable
TIA/(N+1)+B

% Recording convergence history

q_history(iG,l)IJ; % last value at every temperature
XLhistory(iG,:)Ix;

Rgaccept(iG,1)IC_accept; % number of acceptance
R;improv(iG,l)IC_improv; % number of improvement

% Saving workspace at every 5 iteration
if(mod(iG,5)II0)

save(['lhome/jeongsal/Matlab/6/6/WbrkSpace_',num2str(iG),'.mat']);
end
end

% Plotting optimal control profiles
pIFourierFunGen(x,10);

time=p(:,1);

1533135433);

lump(: '3);

vel=p(:,4);

figure

hIplot(time,temp,'r-',time,hum,'b--',time,ve1,'mr.');
x1abe1('Process time [s]');

ylabel('Temperature [deg. C] / Humidity [%Mv] / velocity [m/sl');
legend('Temperature','Humidity','velocity',4);
set(h,'LineWidth',1.5);

243

GenFourierParaSeLm

% Generating Fourier series parameter set

function rv2IGenFourierParaSet(N)

% Upper and lower limit of each control variables

t_ LI60; t _UI600; % time bounds [s]; LIlower UIupper
T :LI100; T bUIZSO % Temperature bounds [C]

H_LI0; H_DI100; % Steam content bounds [% by volume]
V;LI0; V;DI30; % Impingement velocity bounds [m/s]

% generating a random cooking duration
durationIround(t_L+rand(l,1)*(t_D-t_L));

remain=mod(duration,10); % time should be integer and multiple of 10
durationIduration-remain;

% calculating Fourier coefficients

x(1: N, 1)ICalculateFourierCoeffs(T_ U, T_ L, duration, 1); % Temperature
x(N+1:2*N,1)ICalculateFourierCoeffs(H_U, H_ L, duration, 1); % Humidity
x(2*N+l: 3*N, 1)ICalculateFourierCoeffs(V_ D, ‘V_ L, duration, 1);

velocity

x(3*N+1,1)Iduration; % Cooking duration

rv2Ix;

244

fun_FEM_Para.m

% Calculating prediction results by using FEM and Fourier parametes
function [status, results]=fun_£EM_Para(v) % nlc: nonlinear constraints

% Getting profiles with 0.5s interval
C=FourierFunGen(v, 0.5);
duration=v(64,1);

Te=C(:,2);

Se=C(:,3);

vs=C(:,4);

dt=0.5:

% FEM Solution
[rvl rv2 rv3 rv4 rv5 rv6 rv7
rv8]=FH_pataGen_half(duration,Te,Se,Ve,dt):

[nr nc]=size(rvl);

% Re-arranging results
results(l,l)=rv1(nr,l); % Yield
results(2,l)=rv3(nr,1); % Lethality
results(3,1)=rv4(nr,l); % DI
results(4,l)-rv6(nr,l); % DH

Lfaresults(2,1);
DIf=results(3,1);
DHfsresults(4,l);

% Checking feasibility

% Salmonella reduction
if (Lf >3 6.5)
Det_Lf=l; % passed
else
Det_Lf=O; % failed
end

% Internal color change
if (DIf >2 1.4)
Det_pr=l; % passed
else
Det_pr=O; % failed
end

% Surface color change

if (DEf >8 0.263)&& (DEf <2 0.497)
Det;pEf=1; % passed

else
Det_pEf=0; % failed

end

% lsfeasible; 0=infeasible
sumDet=Det_Lf*Det_pIf*Det_pEf;

245

% Check if the data points generated by Fourier coeffs. are in the

upper
% and lower bounds.

function status=CheckRange(X)

% Limits of variables
Bounds=[100 250; O 100; O 30]; %[TL TU; HL HU; VL VU]

[r c]=size(X);

duration=X(r,1);

timestep=l;

kmax=10; % maximum # of series
s=2*pi/((duration/2)*kmax); % period

for k=l:3
base-1+21*(k-l);
a=X(base:base+(kmax-l),1);
b=X(base+kmax:base+2*kmax-l,1);
f0=X(base+2*kmax,1);
i=0;
for ti=0:timestep:duration
i=i+1;
susz;

for m=lzkmax
sum:sum*a(m,1)*sin(s*mﬁti)+b(m,l)*cos(s*mﬁti);
end
fs(i,1)=f0+swm;
end

if ((max(fs) <= Bounds(k,2)) && (min(fs) >= Bounds(k,1)))
S(k.1)=l;
else
S(k,l)=0;
end
end

% l=feasible, O=infeasible
status=S(1,l)*S(2,l)*S(3,l);

246

 

F ourierFun Gen.m

% Generating control profile with 0.5s interval by using Fourier
coeffs.

function rvl = FourierFunGen(v,timestep) % v=vector

% Assigning Fourier coefficients

aT(l:lO,l)=v(1:10,l); bT(1:lO,l)=v(11:20,1); foT=v(21,1); % Temperature
aMKl:10,l)=v(22:31,l); bM(1:10,l)=v(32:41,1); foM-v(42,l); % Humidity
aV(1:10,1)=v(43:52,l); bV(1:lO,1)=v(53:62,1); fthv(63,1); % velocity
duration=v(64,1);

kmaxslo; % maximum.# of series
s=2*pi/((duration/2)*kmax); % period

i=0;

% Calculating
for t=0:timestep:duration %j=0:(quotient+1)

i=i+1;

sumT=O:

sumMsO;

sumvbo;

for kalzkmax
sustsumT+aT(k,l)*sin(s*k*t)+ bT(k,1)*cos(s*k*t);
sumM=sumM+aMXk,l)*sin(s*k*t)+ bM(k,l)*cos(s*k*t);
sumvesumv+aV(k,l)*sin(s*k*t)+ bV(k,l)*cos(s*k*t);

end

fT=foT+sumT;

fusfouwsumu;

thfoV+sumV;

Cond(i,l)=t;

Cond(i,2)=fT;
Cond(i,3)=fM;
Cond(i,4)=fv;

end

% returning profile data
rvl=Cond;

247

 

ICRS Algorithm

File association:

ICRS_FEM_PLI.m has following sub-programs:

- GetMinIntervaLm
o fun_FEM_PLI.m **
o CheckRange_PLI.m * *

** These ﬁles are included in the genetic algorithm in this study.

248

 

 

 

ICRS_ FEM_PLLm

 

% ICRS Algorithm

clear all
fprintf('ICRS+FHM.Algorithm.started\n')

% Parameters and input data

kal; % iteration counter
tol_criterion=0.0001; % convergence criterion
iG=400; % max. iteration

% Heuristic parameters

k1=l/3;

k2=l/2;

neleO; % Failure counter limit

N=21; % 21 control points for each control vector
M34; % # of control vectors
TotDim=N*(M-l)+1; % Total # of dimensions

% Ranges of parameters

XiU( lzN, 1)=250; % temperature
XiU( N+1:2*N,1)=100; % steam
XiU(2*N+l:3*N,l)=30; % velocity
XiU(3*N+1,l)=600; % time

XiL( 1:N, l)=100;
XiL( N+1z2*N,l)=0;
XiL(2*N+1:3*N,l)=O.5;
XiL(3*N+1,1)=60;

% Getting initial contol vectors by reading a file
fidsfopen(['/home/jeongsal/Matlab/3/IP/IDsmooth_',num2str(5),'.dat']);
g=fscanf(fid,'%f',[1, inf]);

9‘9 r
fclose(fid);

Xik=g(l:64,l);
status=l;
results=g(65:68,1);

% Assigning initial objective value
Josresults(1,l);

% Storing initial control vector
Xik_History(k,:)=Xik';
J;History(k,l:4)=resu1ts';

tolal;

% Starting iteration
while (k < i6)

249

% Calculate the minimum.intervals of the decision variables
Lik=GetMinInterval(XiU, XiL, Xik);

% Calculate the vector of standard deviations, using kl
Sik=k1*Lik;

F:0;
Jns-l; % just for safety

while (status ~= l)||(Jn < Jo)

[r c]=size(Xik);

P;

if F > (no)
Sik=k2*Sik;

F=0;
end

ParaStatussO;

% Generate a new decision vector using Gaussian distribution
while ParaStatus==0
XikNew I Xik + Sik .* randn(r,1);
Parastatus=CheckRange_PLI(XikNew);
end

% Evaluating performance index
[status results]=fun_FEM_PLI(XikNew);
Jn=results(1,1);

if (status == O)||(Jn <= Jo)
F=F+1; % Increase failure counter by 1
end
end

% Check the convergence

if isnan(Jn)-=O
tol=abs(Jn-Jo)/abs(Jo);
k=k+l;
Xik=XikNew;
Joan;

Xik_History(k,:)=Xik';
J_History(k,1:4)=results';
tol_History(k,1)=tol;
else
fprintf('Skipped because of NaN\n');
end

fprintf('k=%d F=%d J=%f\n',k,F,Jn)

% Saving workspace at every 2 iteration
if(mod(k,2)8=0)

save(['/home/jeongsal/Matlab/3/IP/Wbrk8pace_',num2str(k),'.mat']);
end

250

end
fprintf('End of ICRS+FEM.Algorithm\n');

% Exporting result
% History of generations

fid=fopen(['[home/jeongsal[Matlab/3/3/Result_history_Fourier.dat'],‘wt'
);

for i=1:k
for j=l:TotDim
fprintf(fid,'%10.4f',Xik_History(i,j));
end
fprintf(fid,'%10.4f %10.4f %10.4f %10.4f',q_History(i,:));
fprintf(fid,'%10.4f\n',tol_History(i,l));
end
fclose(fid);

% Reporting ICRS summary
fid3=
fopen(['/home/jeongsa1[Matlab/3/3/Result_summary_Fourier.dat'],'wt');

fprintf(fid3,' ---------------------------------------------------------
\n');

fprintf(fid3,' SUMMARY OF ICRS-GRNN OPTIMIZATION\n');

fprintf(fid3,' >Control vector parameterization with Fourier
series\n');

fprintf(fid3,' >ICRS-GRNN found the best combination of 64 descision
parameters\n');

fprintf(fid3,' ---------------------------------------------------------
\n');

fprintf(fid3,'YieldLmax= %10.4f [%%] \nlog(LR~microb)= %lO.4f\nlog(DI)=
%10.4f\nlog(D£)= %10.4f\n',q4History(k,l:4));

fprintf(fid3,'Cooking duration: %10.4f [s] \n',Xik_History(k,64));
fprintf(fid3,'tolerance= %10.4f \n\n',tol_History(k,l));
fprintf(fid3,'[Optimization Parameters]\n');

fprintf(fid3,'1. # of Decision parameters: %d\n',N);
fprintf(fid3,'2. # of Control variables: %d\n'pM);
fprintf(fid3,'3. Total dimension of problem: %d\n',TotDim);
fprintf(fid3,'4. kl heuristic parameter: %6.4f\n',k1);
fprintf(fid3,'5. k2 heuristic parameter: %6.4f\n',k2);
fprintf(fid3,'6. ne heuristic parameter: %6.4f\n',ne);
fprintf(fid3,'7. Convergence criterion: %6.4d\n',tol_criterion);
fprintf(fid3,'8. # of Fourier series: %d\n',10);
fprintf(fid3,'9. Total # of iteration: %d\n',k);

fprintf(fid3,' --------------------------------------------------------
\n');

fclose(fid3);

251

GetMinIntervaLm

% Calculating minimum intervals of the decision variables
function rv=GetMinInterval(U, L, V)
[r c]=size(V);
for i=1:r
dL¢V(i,l)-L(i,l);
dU=U(i,1) -V(i,l);
if dL <8 dU
r(i,1)=dL;
else
r(i,l)=dU;

end

rv=r;

252

Quality Prediction Sub-Programs

These sub-programs were added in the FEM of Watkins (2004).

o Salmonella inactivation: CalculateSurvivors.m
0 Internal color change: CalculateCenterColorChange.m
0 Surface color change: CalculateA veragedSurfaceCalorChange.m

253

CalculateSurvivors.m

% Calculate the number of surviving microorganisms for eanch node at
each time step

function []=CalculateSurvivors()

global NumNodes timestep temperature Ninitial No NW Nnew Nneww
logreduction logreductionw Dvalue Tref z TimeToLimit

m31000000; % factor to reduce Dr
% Log—linear inactivation equation

for X s 1:NumNodes
d = m*Dvalue * 10 “ ((Tref - temperature(X)) / Z);
Nnew(X) = No(X) / (10 “ (timestep / d));
No(X) 8 Nnew(X);
logreduction(X) a -log(No(X) / Ninitial) / log(lO);
logreduction(X)=mﬁlogreduction(X); % restoring original Dr

% weibull inactivation equation

if logreduction" < 9
b = 0.000000000011047 * exp(0.4l758 * temperature(l)); %‘0.03 *
(temperature(l)) * 2 - (2.7 * temperature(l)) + 72.19
n s 1.12;
NnewW(X) 8 NWO!) * (10 0 (-b * ((timestep / 60) " n)));
if Nnew(X) < l
Nnew(X) 8 1;
end
NWO!) a NnewW(X);
logreductionw a -log(NW(X) / Ninitial) / log(lO);
else
logreductionw 8 9;
end

end

254

CalculateCenterColorChange.m

% Calculate log reduction OMB(Oxy-myoglobin) concentration at center
function [rv]=CalculateCenterColorChange(t, Tc,delT)

% Geileskey A. et al. (1998)

% Target log reduction = 1.387 which is 95.9% denaturation of OMB which
is

% no evidence of pink.

% The worst case parameters:
Dr=5116.86; % [s] beef shin

2810.41; % [C] beef m. l. dorsi

Tr=60; % [C]

m=1000; % mDr=m*Dr Overflow Preventing Multiplier

mDr=m*Dr;

t=t';

Dt=mDr*10.“((Tr—Tc)/z); % Calculating D-value at time t

LRt=(-1 ./ Dt)*delT; % Calculating Log Reduction at time t for delT
mCumulative_LR=cumsum(LRt'); % Cumulative Log Reduction

Cumulative_LR:mﬁmCumulative_LR; % Restoring cumulative log reduction

rv=(-Cumulative_LR');

255

CalculateA veragedSmfaceColorChangem

% Calculate color change at surface

function [rv]=CalculateSurfaceColorChange(t, Td, Ts, delT)

% Dagerskog and Bengtsson (1974)

% The crust color kinetic parameters were obtained from the graphical

data

Dr=465.30; %[s]
2:90.55; %[C]

Tr=110; %[C]

a=1.06; % Activation function parameter(99.5% activation level
within +/- 5C)

m=1000; % mDr=m*Dr Overflow Preventing Multiflier

mDr=m*Dr;

[rn cn]=size(Ts);

t=t';

for i=1:rn
f=1/(1+exp(-a*(Ts(i,1)-Td(i,1)))); % Activation function
Dt=mDr*10“((Tr-Ts(i,1))lz); % Calculating D-value at time t
LRt(i,1)=f*(-1 / Dt)*de1T; % Calculating Log Reduction at time t

for delT

end

mCumulative_LR=cumsum(LRt'); % Cumulative Log Reduction

Cumulative_LR=mﬁmCumulative_LR; % Restoring cumulative log reduction

rv=(-Cumulative_LR');

256

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257

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