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dissertation entitled

A Comparative Study of Manpower Scheduling Algorithm:
A Multi-Objective Heuristic Procedure, Tabu Search,
and Simulated Annealing

presented by
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has been accepted towards fulfillment
of the requirements for

Ph.D. degree in Operations Management

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Date 7/10/ 93
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summons-9.1

 

 

 

 

A COMPARATIVE STUDY OF MANPOWER SCHEDULING ALGORITHMS: A
MULTI-OBJECI‘IVE HEURISTIC PROCEDURE, TABU SEARCH, AND SIMULATED
ANNEALING

By
Aaron Paul Blossom

I A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Management
1 993

ABSTRACT

A COMPARATIVE STUDY OF MANPOWER SCHEDULING ALGORITHMS: A
MULTI-OBJECTIVE HEURISTIC PROCEDURE, TABU SEARCH, AND SIMULATED
ANNEALING

By A. P. Blossom

Achieving minimum staffing costs, meeting worker's requests for hours worked
and time off while maintaining required staffing levels are all important but conflicting
Objectives when scheduling service employees. Current employee scheduling methods
address only one or two of these Obj ectives.

This research implements two algorithms new to the manpower scheduling field,
simulated annealing and tabu search. The algorithms were then compared against each
other and a multi-obj ective heuristic developed by Loucks on several measures of
solution quality. This research was conducted in the domain of the tour scheduling
problem with non-interchangeable workers and using the full set of possible shifts. All
previous comparative research has used a working subset approach.

Tests were conducted with problems constructed with data from service
businesses and with synthetic problems. The solution methods were tested for the effect
of several independent variables measuring solution quality, cost and time-to-completion.
Simulated Annealing was found to generally outperform Loucks' heuristic and tabu

search on the measures of solution quality and cost, while Loucks' heuristic was clearly

better in time-tO-completion.

Accepted by the Graduate Faculty, Michigan State University, in partial
fulfillment of the requirements of the degree of Doctor of Philosophy

RamNarasimhan, BET). ‘ - '

Jam. m

/Soumen Ghosh, PhD.

 

 

 

TAIL
L. ga by.

July 20, 1993

Coryright by
Aaron Paul Blossom
1993

Dedicated to my boys,
Aaron M. Blossom,
Andrew P. Blossom, and
Joseph C. Blossom
for their patience and help.

iv

ACKNOWLEDGEMENTS

I wish to thank my dissertation committee, Dr. Ram Narasimhan, Dr. Soumen
Ghosh, and Dr. Gary Ragatz for their assistance, guidance and support in the preparation
of this research. This dissertation could not have been completed without their
contributions.

The Chairman of the Dissertation Committee, Dr. Ram Narasimhan, has my
profound gratitude for the mentoring, counsel and support throughout my doctoral
student career. His confidence and continuing interest in this student and the research
made the dissertation possible.

Dr. Gary Ragatz and Dr. Soumen Ghosh generously took time to discuss this
research with me whenever necessary. They Offered important suggestions throughout
the dissertation process and contributed greatly to my ability to complete the work.

I also wish to thank my boys, Aaron, Andrew and Joseph, for their sacrifices of
time and attention with their father. Without their help, this dissertation would have been

impossible to complete.

Table Of Contents

 

 

 

 

List of Tables- - - - - -- - -- - x
List of Figures - - -xii
Chapter 1. Introduction.--.---_ - __ . . .......... - ........ l
1.1.Introduction ................................................................................................... 1
1.2.Special Characteristics Of Service Businesses ................................................ l
1.3.0bjectives for Scheduling in Service Systems ................................................ 3
1.4.Problem setting .............................................................................................. 3
1.5.Problem Statement ......................................................................................... 4
1.5.1. Three Step Manpower Scheduling Process 5
1.5.2. Restrictions on Work Tours For This Research Setting 7
1.6.Research Focus and Objectives ...................................................................... 9
1.6.1. Critical aspects of research setting 10
1.6.2. New Methods Used For Solution Of This Problem 11
1.7.Algorithms Previously Applied in this Setting ............................................. 11
1.8.0rganization of the Dissertation ................................................................... 12
Chapter 2. Review of Related Literature- - _ -- -- - - - - - - - l3
2.1.0rganization of Literature Review ............................................................... 13
2.2.No Work-Time Scheduling Decision ........................................................... 16
2.2.1. No Work Time Decision - Unlimited Worker
Interchangeability 16
2.2.2. No Work Time Decision - Limited Worker
Interchangeability 17
2.3.Days-Ofi‘ Scheduling Decision ..................................................................... 17
2.3.1. Days-Ofi‘ Scheduling Decision - Unlimited Worker
Interchangeability 17
2.3.2. Days-Off Scheduling Decision - Limited Worker
Interchangeability l9
2.4.Shift Scheduling Decision ............................................................................ 19
2.4.1. Shifi Scheduling Decision - Unlimited Worker
Interchangeability 19
2.4.2. Shift Scheduling Decision - Limited Worker
Interchangeability 21
2.5.Tour Scheduling Decision ............................................................................ 22

vi

2.5.1. Tour Scheduling Decision - Unlimited Worker

 

 

Interchangeability 22
2.5.2. Tour Scheduling Decision - Limited Worker
Interchangeability 24
2.6.Summary of the Review Of Labor Scheduling Literature ............................. 25
2.7 .Opportunities for Research .......................................................................... 25
2.7.1. Application of new algorithms 25
2.7.2. Comparison of Previously Applied Algorithms 26
Chapter 3, Mathematical Formulation ........ - -- - - 27
3.1.Choice of Formulation ................................................................................. 27
3.2.Problem Formulation ................................................................................... 28
3.3.Assumptions of Formulation ........................................................................ 31
Chapter 4, Generation of Initial Feasible Solution .................................................... 33
4. 1.1ntroduction ................................................................................................. 33
4.2.Strategy of initial feasible solution generator ............................................... 35
4.2.1. "Fat" solution, vs. "Lean" solution 35
4.2.2. Margin on task hours 36
4.2.3. Choice of worker for task hour 36
4.2.4. Elimination of Constraint Violations 39
4.3.Step by Step Explanation Of IFS Generator ................................................. 43
4.3.1. Explanation of Steps in Flow Chart 45
4.3.2. Pseudo Pascal Description of [F S Generator Algorithm 49
4.3.3. Conclusion 50
Chapter 5 - Tabu Search: Theory and Implementation ............................................ 51
5.1.Introduction ................................................................................................. 51
5.1.1. Overview of Tabu Search 51
5.1.2. Description of Tabu Search Algorithm In Pseudo-Pascal 56
5.1.3. Example of Tabu Search Algorithm 57
5.2.Tabu Search Implementation Issues ............................................................. 58
5.2.1. Tabu List Types 58
5.2.2. Aspiration Criteria 59
5.2.3. Selection of Parameter Values 60
5.2.4. Comparison of Results for Different Tabu List Lengths. 61
5.2.6. Generation of Moves or New Solutions 61
5.2.7. Move Generators 62
5.2.8. Formulation of constraints. 80
5.2.9. Objective Function Estimator 80
5.3.Conclusion ................................................................................................... 81
Chapter 6 - Simulated Annealing; Theory and Implementation.-_- -- - -- 82
6.1 .Introduction ................................................................................................. 82
6.1.1. Statistical Mechanics 82

vii

6.1.2. The Simulation of Particles at Thermal Equilibrium 83

 

6.1.3. The Relationship to Combinatorial Optimization 84
6.1.4. Requirements for Simulated Annealing Optimization 85
6.1.5. Description of Simulated Annealing Algorithm in Pseudo-
Pascal 86
6.1.6. Flow Chart of Simulated Annealing 86
6.1.7. Example of The Simulated Annealing Algorithm 88
6.2.Simulated Annealing Implementation issues ................................................ 89
6.2.1. The Annealing Schedule 89
6.2.2. Setting the parameter values 90
6.2.3. Stopping Rules 91
6.2.4. Perturbation Using Configuration Generators 92
6.2.5. Inclusion of Both Improvement And Random
Configuration Generators 116
6.2.6. Probabilities used for selection of configuration generators 119
6.2.7. Improvement Path for Simulated Annealing 120
6.3.Conclusion ................................................................................................. 120
Chapter 7 - Research Methodology .......................................................................... 122
7. 1 .Introduction ............................................................................................... 123
7.2.Choice of Algorithm for Comparison ......................................................... 124
7.3.Hypotheses ................................................................................................ 124
7.3.1. Time - Speed of Algorithm 124
7.3.2. Cost - Total Manpower Cost of the Schedule. 125
7.3.3. Quality Of Solution 125
7.3.4. TSD - Sum of squared deviations between scheduled and
targeted work hours. 126
7.3.5. OBJ - Objective Function Value 126
7.3.6. Interactions among variables 127
7 .4.Selection of Experimental Design .............................................................. 127
7.4.1. Factor Levels 127
7.4.2. Choice of Design 128
7.4.3. Sample size of experiment 128
7.4.4. Test data 129
7.4.5. Analysis 130
7.4.6. Choice of sample size 130
7.4.7. Replications for Simulated Annealing 131
7.5.Generation of Test Problems ...................................................................... 133
7 .6.Data Gathering .......................................................................................... 135
Chapter 8 - Results of Experimentation - - - - -- 136
8. 1.1ntroduction ............................................................................................... 136
8.2.Tests for Assumptions of ANOVA ............................................................ 136
8.2.1. Normality of error residuals 137
8.2.2. Normal Probability Plots of Residuals 138

viii

8.2.3. Independent distribution of error residuals 140
8.2.4. Homogeneity of variance 141
8.2.5. Bartlett's Test 144
8.3.Box and Cox Analytical Method for Determining Transformation ............. 144

8.3.1. Explanation of Analytical Determination of Transformation 145
8.3.2. Results of the Analytical Determination of Transformation 146

 

 

 

 

 

 

8.3.3. Normal probability plots of residuals 147
8.3.4. Independent distribution of error residuals 150
8.3.5. Homogeneity of Variance 151
8.3.6. Bartlett's test after transformations 154
8.4.Justification for use of Analysis of Variance .............................................. 154
8.5.Results of ANOVA .................................................................................... 155
8.5.1. Detailed Results 156 .
8.6.Differential Difficulty of Problem Types ................................................... 177
Chapter 9 - Summary, Conclusions and Recommendations..-. -- -- -- - 179
9.1.Summary ................................................................................................... 179
9.2.Contributions ............................................................................................. 180
9.3 .Opportunities for Further Research ............................................................ 182
Appendix 1. ..... - ,_ __ -- -l84
Calculation of Theoretrcal Number Of WOrk Toms adapted from Loucks [19871184
Notation .......................................................................................................... 184
Assumptions .................................................................................................... 184
Equation .......................................................................................................... 185
Example .......................................................................................................... 185
Appendix 2 ....... - -- -- - _______ 186
Transformation Of PrOblem ForIDrIlatiOn Into Single F unctron Form ............... 186
Appendix 3 - - - - - .................. . .......... 188
Synthetic Problem GeneratOr ........................................................................... 188
Appendix 4 ..... -- - - - .......... - 190
Raw Data ......................................................................................................... 190
Appendix 5 - -- -- ---- 217
Transformed Treatment Averages .................................................................... 217
Bibliography--- - - -- - - -- - 220

 

ix

 

LIST OF TABLES

Table 1-1, Table for Translation of Sales into Required Manpower ................................ 6

Table 5-1, Comparison of Tabu List Length and Objective Function Value ................. 61
Table 6-1, ANOVA for Objective Function ................................................................ 117

Table 6-2, TUKEY HSD Multiple Comparison Test, Objective Function Value ......... 118

Table 7-1, Factors and Factor Levels .......................................................................... 128
Table 7-2 Results of the analytic procedure for determination of sample size ............. 130
Table 8-1, Autocorrelation Values For Dependent Variables ...................................... 141
Table 8-2, Bartlett‘s Test For Homogeneity of Variance ............................................. 144
Table 8-3, Results of Procedure for Analytic Determination of Transformation ......... 147
Table 8-4, Autocorrelation of transformed residuals .................................................. 150
Table 8-5, Bartlett's Test For Homogeneity of Variance ............................................. 154
Table 8-6 Analysis Of Variance Results .................................................................... 156
Table 8-7. ANOVA on TITVIE, Transformed ............................................................. 159
Table 8-8, Table of the best TIME performance on problem type by algorithm .......... 161

Table 8-9, Table of the worst TIME performance on problem type by algorithm ...... 161
Table 8-10. ANOVA on COST, Transformed ........................................................... 163
Table 8-11, Correlation coefficients on COST performance between Algorithms ....... 164
Table 8-12, Best COST performance across problem types by algorithm ................... 165

Table 8-13, Worst COST performance across problem type by algorithm .............. 166

Table 8-14. ANOVA on TOS, Transformed .............................................................. 168

Table 8-15, Best TOS performance across problem type by algorithm ...................... 170
Table 8-16, Worst TOS performance across problem type by algorithm .................... 170
Table 8-17. ANOVA on TSD, Transformed .............................................................. 172
Table 8-18 Best TSD performance across problem type by algorithm ...................... 174
Table 8-19, Worst TSD performance across problem type by algorithm .................... 175
Table 8-20, Best OBJ performance across problem type by algorithm ....................... 177
Table 8-21, Worst OBJ performance across problem type by algorithm .................... 177
Table A4-1. Raw Data ................................................................................................ 189
Table A5-1, Treatment Averages Of Transformed Data ............................................. 215

xi

LIST OF FIGURES

Figure 1-1,Typical Demand Pattern For a Fast Food Restaurant ..................................... 4
Figure 2-1, Taxonomy of Literature Review ................................................................ 14
Figure 2-2, A Classified Sample of Labor Scheduling Research ................................... 15
Figure 4-1, Selecting a Worker for Assignment ........................................................... 37
Figure 4-2, Eliminate Constraint VIOlations ................................................................ 40
Figure 4-3, Procedure to Generate Initial Feasible Solution .......................................... 44
Figure 4—4, Create or Add to Shift ............................................................................... 48
Figure 5-1, Tabu Search Algorithm .............................................................................. 53
Figure 5-2, Tabu Search Improvement Path for Different Tabu List Lengths ............... 61
Figure 5-3, Improvement Path for Tabu List Length = 7 .............................................. 62
Figure 5-4, Reduce Over Staffmg ................................................................................. 65
Figure 5-5, Increase Over Stafi‘ing ............................................................................... 67
Figure 5-6, Give An Hour to Another Worker ............................................................... 69
Figure 5-7, Give A Day to Another Worker ................................................................. 71
Figure 5-8, Trade Shifis With Another Worker ........................................................... 73
Figure 5-9, Cross Trade Shifts With Another Worker ................................................. 75
Figure 5-10, Give Away of a Worker’s Shift ................................................................ 77
Figure 5-11, Give Away A Short Shifi, Hour By Hour ................................................ 79
Figure 6-1, Simulated Annealing Flowchart ................................................................ 87

xii

Figure 6-2, Reduce Over Staffing ................................................................................ 93

Figure 6-3, Increase Over Staffing .............................................................................. 95
Figure 6-4, Give An Hour to Another Randomly ......................................................... 97
Figure 6-5, Give A Day To Another Randomly ........................................................... 99
Figure 6-6, Trade Tours Randomly ........................................................................... 101
Figure 6-7, Cross Trade Tours Randomly .................................................................. 103
Figure 6-8, Give An Hour to Another ......................................................................... 105
Figure 69, Give A Day to Another ............................................................................ 107
Figure 6-10, Trade Tours Between Workers ............................................................... 109
Figure 6-11, Cross Trade Tours Between Workers ..................................................... 111
Figure 6-12, Give Away Part Of Shift To Other Workers ........................................... 113
Figure 6-13, Give Away a Short Shift ........................................................................ 115
Figure 6-14, Simulated Annealing Improvement Path ............................................... 120
Figure 8-1, Normal probability plot of TOS residuals ............................................... 138
Figure 8-2, Normal probability plot of TSD residuals ................................................ 138
Figure 8-3, Normal probability plot of OBJ residuals ................................................. 139
Figure 8-4, Normal probability plot of COST residuals ............................................... 139
Figure 8-5, Normal probability plot of TIME residuals ............................................. 140
Figure 8-6, Plot of Residual vs..Estimate - TOS ......................................................... 141
Figure 8-7, Plot of Residual vs..Estimate - TSD .......................................................... 142
Figure 8-8 Plot of Residual vs..Estimate - OBJ .......................................................... 142
Figure 8-9, Plot of Residual vs..Estimate - COST ....................................................... 143
Figure 8-10, Plot of Residual vs..Estimate - TIME ..................................................... 143
Figure 8-11, Normal probability plot of TOS residuals After Transformation ............ 148
Figure 8-12, Normal probability plot of TSD residuals Afier Transformation ............ 148

xiii

Figure 8-13, Normal probability plot of OBJ residuals After Transformation ............. 149

Figure 8-14, Normal probability plot of COST residuals After Transformation .......... 149
Figure 8-15, Normal probability plot of TIME residuals After Transformation .......... 150
Figure 8-16, Plot of Residual vs..Estimate - TOS After Transformation ..................... 151
Figure 8-17, Plot of Residual vs..Estimate - TSD After Transformation ..................... 152
Figure 8-18, Plot of Residual vs..Estimate - OBJ After Transformation ..................... 152
Figure 8-19, Plot of Residual vs..Estimate - COST After Transformation .................. 153
Figure 8-20, Plot of Residual vs..Estimate - TIME After Transformation ................... 153
Figure 8-21, Plot of TIME by Algorithm vs..Problem Type ........................................ 160
Figure 8—22, Plot of COST by Algorithm vs..Problem Type ........................................ 164
Figure 8-22, Plot of TOS by Algorithm vs..Problem Type .......................................... 169
Figure 8-22, Plot of TSD by Algorithm vs..Problem Type .......................................... 173
Figure 8-23, Plot of OBJ by Algorithm vs..Problem Type ........................................... 176

xiv

CHAPTER 1. INTRODUCTION

1. I. INTRODUCUON

Labor scheduling is a required managerial task in any firm which employs
workers in the production of goods and services. Labor is typically a limited and
indispensable resource which must be managed effectively in order for the firm to
achieve its objectives. The effective scheduling of manpower is a major component of
the management of labor. As such, manpower scheduling has a great deal of impact on
profitability, customer service and employee morale. The specific manpower scheduling
decisions that a manager must consider depend upon the environment, business and
labor, that the scheduling decisions are made in. This research focuses on manpower
scheduling in the service sector, which has characteristics distinct from other sectors of

the economy.

1.2. SPECIAL CHARACTERISTICS OF SERVICE BUSHVES'SES'

The special characteristics of the service environment relevant to manpower

scheduling are [Aggarwal, 1982]:

. The output of service systems (a service system is a "factory" that
provides services), generally cannot be placed into inventory. The
exceptions are products, such as hamburgers, that have a very short shelf
life (10 minutes). Consequently in service systems it is not possible to
level output to meet demand.

. Service systems require equipment, labor and space capacity to handle
peak or near peak demand conditions. In other words, the capacity
decision in service systems is to buy capacity for peak, rather than
average, demand levels.

. The demand for output of service systems varies greatly from month-to-
month, week-to—week and day-to-day. Extreme variations from hour-to-
hour exist in some service systems. The demand may also have seasonal
variations.

. The demand for output of service systems cannot be easily backlogged.
Usually unsatisfied demand is lost business. In cases where the demand
can be backlogged, the indirect costs may be significant.

. Since in most service systems, the customer receives the service directly
from the server, they are labor intensive.

Since demand must be satisfied or lost and material costs are fixed for a particular

product, the most significant opportunity for reducing costs is through better scheduling
of manpower. In addition, the opportunities for improving customer service and

employee morale lie primarily in improved manpower scheduling.

1. 3. OBJECTIVES FOR SCHEDULING IN SERVICE SYSTM

Service systems also differ from manufacturing systems in that they have
different objectives. Indisputably, cost minimization is an objective of service systems,
but their primary objectives are often different because of the direct interaction with the

customer. The objectives of a service firm might include the following:

. Minimization of average response time to start of service delivery.

. Minimization of the average customer waiting time.

o Minimization of average customer service time.

. Minimization of the average number of facilities or work crews.

. Improvement and maintenance of employee morale through scheduling

employees at preferred times (of the employee).
Additionally, the manager of the service system must operate within a large
number of legal, regulatory, union, policy, and budgetary constraints.
In summary, labor scheduling is a major concern in service businesses, with the
conflicting objectives of maintaining desired levels of customer service and minimizing

costs.

I. 4. PROBLWSETHNG

The research is concerned with scheduling problems faced by four environments,
food service, retail, banking and a series of synthetic problems. The research will
consider scheduling of work tours with limited worker interchangeability in the four

service environments mentioned above. Limited worker interchangeability refers to

restrictions placed on the assignments of workers. Not all workers can perform all tasks,
and must not be assigned to tasks for which they are not qualified.

A typical demand pattern from a fast food restaurant is shown in figure 1-1.

 

 

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Figure 1-1, Typical Demand Pattern For A Fast Food Restaurant

Note that the demand varies significantly over short time periods. In order to
minimize costs, scheduling is typically done in relatively small time increments, one hour
or less, creating a large set of possible shifts. A review and taxonomy of labor scheduling
problems is presented in chapter 2. The particular problem studied here of work tour

scheduling with limited worker interchangeability is detailed in section 2.5.2.

1. 5. PROBLEM S TAIWT

The manpower scheduling problem presented here is one of deciding who works

when and at what task. Stated more precisely, the manager must decide an employees

work shifts and task assignment over the time span chosen. In this research a one week
time horizon is chosen for the scheduling activity. Further, each day is broken into 1 hour
increments. The manager, in order to provide adequate customer service, will desire to
have on duty, during every hour of each day scheduled, enough employees to satisfy the
demand for services forecasted during that hour. Because of restrictions in the problem
setting, such as minimum shift length, a manager may not be able to schedule the
optimum number of workers during a particular hour. The manager may then decide to

over staff a particular hour.

1.5.1. Three Step Manpower Scheduling Process

The manpower scheduling process consists of three steps, Swart and Donno
[1981]. First, a weekly sales/demand forecast is developed. Second, the weekly
sales/demand forecast is converted into hourly staffmg requirements. And third, the
manpower schedule is developed, which specifies for each worker, the hours on duty, i.e.
the "tour", and the task to be performed during each on-duty hour.

Sales / Volume F orecasting. Short-term, i.e. one week, sales/demand forecasting
is the first step in the manpower scheduling process. This forecast is typically made using
three sources of data, historical sales/demand data, current trends of sales/demand and the
schedule of special events. The manager will gather past sales/demand data (for example,
last week's sales/demand data, or same week last year sales/demand data). This data may
be used as is or the manager may average several past periods to provide an initial
sales/demand forecast.

Second, the manager will adjust the forecast using the two remaining sources of

information, current sales/demand trends and the schedule of special events. Trends in

the sales/demand patterns of the firm may cause the manager to adjust the forecast of
individual hours or the entire week up or down. Special events, such as a sale, athletic
event, or parties may cause increased demand for the services of the organization. In this
case, the manager would estimate the effect of the special event and adjust the forecast
accordingly.

Determination of Stafiing Requirements. Once the sales/demand forecast is
complete, the forecast is converted into staffing requirements for each hour. These
requirements would be the number of workers required to staff each task for each hour
forecasted.

Often, a table is used to convert hourly sales forecasts into staffing requirements.
Such a table is shown in table 1.1. This particular table divides the work load into eight
tasks, each of which may or may not be staffed at a particular demand level. Note that
this table specifies tasks and staffmg levels associated with meeting customer demand
and does not include activities where the timing is not critical, such as planned
maintenance. Tables like these may be developed in a variety of ways, such as expert

Opinion and simulation, Swart and Donno [1981].

Table 1-1, Table for Translation of Sales into Required Manpower.

Task 1 Task Task 3 Task Task 5 Task Total

1
1 1
1 1
2 1
2 2
2 2
3 2
3 2
3 3
3 3

 

Two additional points about staffing requirements may be made. First, the
staffing requirements shown on the table are minimum staffing requirements. And
second, typically in fast food and retail establishments, some amount of staffing is
required before opening and after closing. The staffing requirements for these hours
cannot be determined by using a table based on hourly sales/demand volumes.

Task and Time Scheduling of Workers. The final stage of the manpower
scheduling process is the development of a manpower schedule that specifies the workers
on duty and the task assignments. A schedule usually consists Of at least two reports, one
for the manager that lists the workers for each hour and task, and the second for each
worker that lists the days scheduled, the hours scheduled during those days, and the tasks
assigned during those hours.

The objectives for the manpower schedule development include minimizing
overstaff'mg of task hours, minimize under staffmg of task hours, and meeting the
worker's preferences for the number of hours scheduled, while not violating work rules
such as minimum shift length, maximum shift length, task qualifications, etc.

It might seem that once a schedule for a week is developed, it would only need
minor modification for use in subsequent weeks. There are several factors that cause that
supposition to be true only occasionally. These factors include, high worker turnover (at
least in some industries/firms), variations in sales/demand patterns, special events,
changes in worker availability, and etc.

This final stage of the manpower scheduling process is the focus of this research.

8

I. 5. 2. Restrictions on Work Tours For This Research Setting

To facilitate discussion of the problem, related literature, mathematical
formulation and results, a common set of terms will de defined. After these definitions
common to the scheduling literature are established, three aspects of the problem, limited
worker interchangeability, limited worker availability, and worker task qualifications will
be discussed.

Definitions/Terminology. The first terms to be defined are "shift" and "work
shift". These two terms may be used interchangeably. A shifi is a period of work during a
day. A work shift has a start and a finish time. The period of time between the start time
and finish time is sometimes called a "work stretc ". A work stretch may contain rest
breaks or lunch breaks. In some scheduling environments "split" shifts exist. A split shifi
is a daily work shift that has a break of more than one half hour. A split shift would be
defined by two sets of start and finish times for a single day. These two sets would not,
of course, overlap.

The term "work tour" is used to describe the set of shifis scheduled for a worker
over the scheduling horizon. A work tour would consist of the shifts scheduled as well as
the days off, or "relief days". The "scheduling horizon" is the length of time for which a
schedule is produced. For our research, the scheduling horizon is one week. In this
research the scheduling process produces schedules where the shifts are generally not the
same from workday to workday. Such a tour is said to contain "mixed shifts."

Limited Worker Interchangeability. The term limited worker interchangeability
refers to existence of characteristic differences between workers that must be considered
during the third step of the scheduling process. Ifthere are no such differences, the
workforce is called homogenous. A homogenous workforce has workers that are

completely interchangeable. On the other hand, a heterogeneous workforce has workers

whose differences must be taken into account when the schedule is constructed. These
differences might be due to differing times available during the day and/or week, and
different tasks the worker is qualified to perform.

Limited Worker Availability. A worker's availability is defined as the specific
hours during the week that the worker has agreed to be available for work. Because the
variety of workers in service fums ranges from full-time professionals to part-time high
school students, it can be reasonably inferred that the availability of workers would vary
from one to another, being limited for some or all employees. School, family,
transportation, and curfews are examples of conditions that can restrict a worker‘s
availability.

A part-time employee will accept a schedule with mixed shifts and non-
consecutive workdays in order to restrict his availability and to have the option of
changing that availability to accommodate needed (or desired) changes in schedule. The
manager, however, would like to have unrestricted availability to ease the scheduling
process and to maintain the greatest degree of flexibility for use in the scheduling
process. Some companies require unrestricted availability of their full time employees, in
exchange for the benefits of full time employment, such as health insurance,
unemployment insurance, guaranteed number of hours, etc.

Worker Task Qualifications. The particular tasks (or functions) that a worker is
qualified to perform are referred to as the worker‘s task qualifications. Because of the
cost of training, and the rate of employee turnover (sometimes high), the manager may
not train every new employee on every task as quickly as possible. The manager may
train the employee on those tasks which the manager deems most necessary, probably
based on an assessment of task coverage needs. Task coverage is the degree to which the
task required to be performed by the forecast can be covered by the employees available

to be Scheduled.

10

1. 6. RESEARCH Foc US AND OBJECIIVEs

1.6.1. Critical aspects of research setting

The characteristic of service systems that output cannot be inventoried requires
scheduling enough workers for adequate customer service or, alternatively, minimization
of staffing shortage hours (too few workers scheduled can cause unacceptable service
delivery times). When delivery of service is critical, such as in an health care
environment, stafi'mg shortage hours cannot be tolerated. On the other hand, high volume
and low margins in food service require minimization of staffing surplus hours
(scheduling too many workers does not improve service delivery times sufficiently to
offset increases in costs). Changes in the relative importance of surplus and shortage
hours can be accommodated in the problem formulation to be used in this research
(chapter 3) by adjusting parameters associated with the decision variables.

Task specific, non-interchangeable workers exist in the problem settings ,
considered in this research. Workers can be qualified in more than one task, however.
This means that workers must be scheduled for specific tasks at specific times.

As noted before, in the food service environment demand patterns vary from
hour-to-hour, day-to-day, week-to-week, and etc. This requires the scheduling to be
performed with short time intervals in order to minimize surplus hours, and hence cost.

In order to accommodate part-time workers, to increase worker satisfaction levels
and to reduce worker turnover, worker preferences must be taken into account. The
worker preferences would include: minimum and maximum work hours per day and per
week, preferable work times during the day (mornings, afternoons or evenings) as well as

days off.

11

These characteristics suggest that the number of potential solutions to be
considered to identify the "best" solution is large. As an example, consider a fast food
restaurant with these characteristics: it is open 18 hours per day, 7 days per week, the
worker shifts are from 3 to 8 hours in length and the number of days worked per week
can vary from 1 to 5. In this example there are 74,747,846,439 possible combinations of
shifts or work tours. For a calculation method for the theoretical number of work tours,
please see appendix 1.

In addition, a mathematical characterization of the work tour scheduling problem
leads to a combinatorial optimization problem, since many of the variables are integer
valued. Because of the combinatorial nature of the problem and its large size, integer
programming methods are currently not useful. There are two primary reasons integer
programming solution methods are not useful [Glover, 1990]: First, solution times are
unacceptably long, given the size of the problem for even small businesses, and second,
most computer codes available for small computers will not accommodate the large

number of variables necessary to model the problem.

1.6.2. New Methods Used For Solution Of This Problem

This research applied two new methods to this research problem, simulated
annealing and tabu search. Their implementation in this research demonstrates that more
general procedures can be applied to this labor scheduling problem and provides

additional information about the relative merits of tabu search and simulated annealing.

12

1. 7. ALGORITHMS PREVIOUSLY APPLIED IN ms SETTING

There have been three algorithms discussed in the literature to solve the problem
of work tour scheduling with limited worker interchangeability; Heuristic Programming
[Glover, 1988], Multi-Objective Heuristic [Loucks, 1987], and Network Solution
Methods [Love and Hoey, 1990]. A distinction might be made between the research
problems used by Glover and Love and Hoey, and Loucks. In the Glover and Love and
Hoey research the length and position of the shifts during the working day were fixed. In
Loucks, neither the shift length nor the position of a shift is fixed within the day. The
research describing the development and application of these three techniques are

detailed in the literature review section 2.5.2.

1.8. ORGANIZATION or IHEDISS‘ERTAHON

The remainder of the dissertation consists of: a literature review comprises
chapter 2, and the mathematical formulation of the research problem is given in chapter
3. The initial feasible solution generator is presented in chapter 4, and the
implementation of tabu search is discussed in chapter 5. The subject of chapter 6 is the
theory and implementation of simulated annealing. Chapter 7 is devoted to the
experimental design. Results of the experiment are discussed in chapter 8, with the

summary and conclusions presented in chapter 9.

13

CHAPTER 2. REVIEW OR RELATED LITERATURE

2.1. ORGANIZATION OFLITERATURE REVIEW

The organization of this review is an adaptation of Loucks' (1987) scheme for the
categorization of the labor scheduling problem. He proposed two criteria for
classification of labor scheduling problems, work tour set composition and worker
interchangeability. A work tour set is the set of different work shifts (tours) from which a
worker’s schedule is selected. A work tour set can be multiple shifts or a single shift per
day. The work tour set can be further characterized by its "days-off" patterns, whether
there is a single days-off pattern or multiple days-Off patterns.

Worker interchangeability is defined as the degree to which the workers being
scheduled differ in their qualifications to perform the various tasks in the work
environment, times available to work, or other relevant properties. Worker
interchangeability, for the purposes of this literature review , can be split into two
categories, limited and unlimited. Figure 2-1 shows a taxonomy which can be used to
classify the existing body of literature. Figure 2-2 shows a representative sample of the

literature classified into the taxonomy presented in figure 2-1.

14

2.1.1. Taxonomy of Literature Review

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Work-Time Degree Of
Scheduling Worker
Decision Interchangeability
Unlimited
None
Limited
Unlimited
Days-Off
Limited
Labor
Scheduling
Problem
unlimited
Shift
Limited
unlimited
Tour
Limited

 

 

 

Figure 2-1, Taxonomy of Literature Review

15

2.1.2. A Classification of Labor Scheduling Research

Labor Degree of Worker Interchangeability
Scheduling
Problem Unlimited Limited

 

None" Holstein/Berry [1972] Chames/COOper [1961]
Job Shop Workers Job Shop Workers

Church [1973]
Telephone Service Reps.

Miller/Berry [1974]
Job Shop Workers

 

Days-Off Baker [1974a]
Generalized

BrownelllLowerre [1976]
Generalized

Monroe [1970]
Baggage Handlers

 

Shift Baker/er a1. [1973] Pappas [1967]
Baggage Handlers Railroad Personnel

Hmderson/Berry [1977]
Telephone Operators

Keith [1979]
Telephone Operators

 

Tour Showalter/et a1. [1977] Warner/Prawda [1972]
Mail Sorters Hospital Nrnses

Morris/Showalter [1983] Ritzman/et a1. [1976]
Generalized Mail Sorters

Mabert/Watts [1982] Loucks [1987]
Check Encoders Fast Food

 

* This category is characterized by an absence of any work time decision, as described in the text. The
decision is primarily one of job assignment.

Figure 2-2. A Classified Sample of Labor Scheduling Research

l6

2. 2. NO WORK-7M SCHEDULING DECISION

This category represents the single shift - single days-off pattern. In the absence
of any work-time scheduling decision, the decision is primarily job assignment. This
scheduling decision is primarily used in an operation that operates one shift per day, five

days per week.

2.2.1. No Work Time Decision - Unlimited Worker Interchangeability

Given a single work tour and manageable work loads, the scheduling problem
becomes one of job assignment. Miller and Berry [1974] examine the problem of
assigning workers to several semi-automatic machines. Two heuristics were developed
for determining labor assignments: the "man-loading" and "labor-saved" heuristics. Their
heuristics minimize the combined costs of idle labor and machine time. They compare
their heuristics against the optimal solutions found by a branch and bound algorithm.

The scheduling of staff to answer phones and process paperwork was studied by
Church [1973]. The problem he studied was to assign workers to one of two tasks over
half hour time intervals. His heuristic assigned workers to tasks with the objective of
equalizing work load among workers and keeping a worker's task the same through

several consecutive periods and then switching tasks.

l7

2. 2. 2. No Work Time Decision - Limited Worker Interchangeability

The scheduling problem with a single work tour and limited worker
interchangeability is a restricted assignment problem. This type of problem has been
dealt with in the linear programming literature and is efficiently solved by such methods

[Charnes and Cooper, 1961].

2. 3. DAYS-OFF SCHEDULING DECISION

This category represents the single shift - multiple days-off pattern. This
scheduling decision is typically found in a business that functions 8 hours per day and
more than 5 days per week. Thus the decision is categorized as a days-off problem or a
days-off and task assignment problem, for unlimited and limited worker

interchangeability, respectively [Loucks, 1987].

2.3.1. Days-0,5r Scheduling Decision - Unlimited Worker Interchangeability

The most common form of this problem is scheduling five day work tours with a
seven day operation. The objective is to find the minimum labor force size while meeting
the demand for the worker's services, and permitting two consecutive days off. If excess
staffmg is necessary to assure feasibility, the algorithms by Baker [1974a] and Tibrewala
[1972] are optimal and simple to use.

Several variations of this problem can be found in the literature. Bartholdi and

Ratliff [197 8] examine the problem with several different days-off policies. Monroe

18

maximized consecutive days-off without employing part-time workers. Baker [1974b]
relaxes the five day work tour constraint to minimize the number of part-time man-days
to exactly meet the staffing requirements.

Brownell and Lowerre [1976] compare days off policies, with regard to the
manpower requirements for each different policy.

Miller, Pierskalla and Rath [1976] use a cyclic coordinate descent algorithm to
find near optimal solutions. They compared their algorithm to an optimal branch and
bound algorithm, and found that their solutions were near-optimal and that the algorithm
was much faster than the branch and bound. Their algorithm was also found to be better
than the manual methods in use at the time.

Khan [197 9] uses a capacitated network flow method for producing manpower
schedules. Khan found that this method was substantially more efficient than the Simplex
method in this application.

Baker and Magazine [1977] present a lower bound for workforce size and
feasible schedule construction algorithms for 4 different days-off policies.

Baker, Burns and Carter [197 9] Develop lower a bound on workforce size and
manual methods for constructing schedules to meet these lower bounds.

Shepardson and Marston [1980] reformulate the two duty period scheduling
problem as a one duty period problem with side constraints. Since the one duty period
problem can be solved using a minimal cost network flow model, dualization is used
with respect to the side constraints, forming a Lagrangean relaxation that is easily solved.

Bartholdi [1981] solves the cyclic staff scheduling problem using linear
programming with a round-off procedure. He shows that his round off algorithm is
better than those presented earlier.

Burns and Carter [1985] developed a simple, one pass algorithm to generate

schedules and to compute lower bounds on the workforce size.

19

2. 3. 2. Days-Ofl Scheduling Decision - Limited Worker Interchangeability

There are undoubtedly scheduling environments with single work tours and

limited worker interchangeability. There seems to be no published work in this area.

2. 4. SIHFT SCHEDULING DECISION

This category represents the multiple shifts - single days-off pattern. The shifts
assigned to different workers might overlap or be contiguous. Overlapping shifts are
typical of a service environment. Labor costs can be reduced while maintaining customer

service levels with overlapping shifts.

2.4.1. Shifi Scheduling Decision - Unlimited Worker Interchangeability

Baker [1973] formulates the problem of scheduling 8 hour shifts with 4 hour
overlaps as an integer program. He develops results that show that a linear program
solution to his formulation will always produce an integer solution provided that the
period requirements are integer valued.

As requirement periods are shortened and the number of shifts are increased,
problem size grows. Problem size in business environments easily exceeds the limits of
existing, commonly available integer programming codes. As an additional complication
to the solution of the labor scheduling problem, LP solutions do not typically provide
integer solutions. In that sense, Baker's [1973] LP solution is uncommon in that it

provides integer solutions.

20

As a result of the limitations of integer programming codes, researchers have
developed rounding heuristics for LP solutions. Keith [1979] developed a heuristic to
find an integer solution close to the optimal solution. First he rounds the solution to the
nearest integer, then he attempts to add or remove workers from the rounded solution.

For some large problems, such as those encountered in labor scheduling, even
linear programming can prove to be inefficient. Segal [1974] developed a network-flow
formulation which guarantees integer solutions for very large problems. The limitation of
Segal's formulation is that lunch and breaks cannot be explicitly modeled, or scheduled.

Henderson and Berry [1976] develop heuristics to select "working subse " of
shift schedules from the "master set" of all possible shifts, without appreciably reducing
solution quality.

Buffa [1976] outlines an integrated work shift scheduling system used to
schedule telephone operators. A heuristic is used to schedule the operators. An outline of
requirements for heuristics to solve manpower scheduling systems is given.

Mabert and Raedels [1977] compares two heuristic methods with an integer
programming formulation for scheduling part-time workers to meet varying daily work
loads. Computation time was the criterion for ranking methods.

Henderson and Berry [1977] have developed a branch and bound algorithm to
efficiently solve problems having as many as 100 shifts per day. The large number of
shifts allows breaks and lunch to be explicitly scheduled, assigning workers to both work
and break shifts.

Mabert [1979] develops a non-deterministic model for scheduling workers in a
bank. He also develops the idea of safety capacity, which is analogous to safety stock in
inventory theory, to meet varying volume demands when forecast errors are present.

Wilson and Willis [1983] developed a network flow and a LP model for this
problem, constrained by forecasted demand, shift sizes, number of supervisors and space

limitations for workers. Although the network flow model produced optimal solutions,

21

the LP model was implemented because Of the excessive input requirements for the
network flow model.

Sinuany-Stem and T eomi [1986] develop both an optimal and heuristic algorithm
to schedule security guards. The Optimal algorithm was found to be inefficient for the
problem, which was rather large. The heuristic produced nearly optimal solutions, as well
as almost 50% cost savings over the manual system.

Burns and Koop [1987] present a way to develop optimal cyclic schedules (what
they call master schedules). Their approach allows differing (deterministic) demand

patterns for each shift.

2. 4. 2. Shift Scheduling Decision - Limited Worker Interchangeability

Pappas [1967] is one of few authors who have looked at limited worker
interchangeability. In his study several scheduling restrictions exist: no employee should
be scheduled to work on his preset rest days, and no morning shifts should be assigned to
a worker that worked an evening shift the day before.

Warner [1976] poses the scheduling decision as a large multiple choice
programming problem whose objective function quantifies preferences of individual
nurses. The problem is solved using a modified version of Balintfy and Blackburn's
algorithm for multiple choice programming problems.

Franz, et a1 [1989] develop a multi-Objective integer linear program for
scheduling an staffing multiple clinics. As we might expect optimal procedures for

solving this problem were not found to be computationally efficient. The problem was

22

reformulated to resemble a generalized network flow model with substantially reduced

computation requirements.

2. 5. TOUR SCHEDULING DECISION

This is the labor scheduling problem that the proposed research focuses on. This
category represents the multiple shifts - multiple days-off pattern. A business that would
have this type of labor scheduling decision to make would typically be functioning more
than 8 hours per day and more than 5 days per week. Shifts may or may not overlap,

depending on demand patterns for workers and may change from day-to-day.

2.5.1. Tour Scheduling Decision - Unlimited Worker Interchangeability

The research referenced above has studied either shift scheduling or days-off
scheduling, not both simultaneously. There are many problem environments where both
decisions must be made such as in restaurants, banking, and retail operations .

Morris and Showalter [1983] treat this issue of integration, using two approaches,
bottom-up and top-down. The bottom-up approach solves the two problems sequentially,
while the top-down approach solves the two problems simultaneously. Their solution
method for the top-down approach used a LP model with a heuristic to round the solution
down. They found that the top-down approach was superior.

Mabert and Watts [1982] follow On the Henderson and Berry [1976] study of tour
scheduling problem. They develop and compare six techniques for selecting a working

subset of tours from the master set. They found that the techniques based on historical

23

demand patterns produced the lowest cost schedules. The other techniques tried were
based on random selection and employee convenience.

Bechtold [1988] develops implicit optimal and heuristic methods for solving the
problem in a multi-objective, multi-location environment. He explicitly models the
tradeoffs between idle time, the number of employees required to work at multiple
locations and the size of the total labor pool.

Li, Robinson and Mabert [1991] compare 3 different heuristics for the tour
scheduling problem with unlimited worker interchangeability. They minimized total
labor costs, rather than minimization of the number of hours scheduled or the
minimization of the number of employees.

Bechtold, Brusco and Showalter [1991] compare 8 different heuristics with
optimal solutions generated by solution of an integer linear program. The primary
criterion for comparison was labor hours scheduled above optimum. Secondary criteria
evaluated for each of the 8 heuristics were percentage of employees with two consecutive
days off, number of active tours, mean number of daily shifts, and computational times.

Easton and Rossin [1991] develop a method for efficiently generating equivalent
alternative solutions for the tour scheduling problem. They show that their method works

best when it has a limited number of tours with which to work.

24

2. 5. 2. Tour Scheduling Decision - Limited Worker Interchangeability

Ritzman, Krajewski and Showalter [1976] developed a combined heuristic and
simulation algorithm to construct tour assignments in a post office where workers differ
by the work centers in which they are qualified to work. Their algorithm has several
objectives; maximize service, minimize administrative costs and minimize night and
overtime wage premiums.

A nurse scheduling problem with different skill levels was investigated by
Warner and Prawda [1972]. The scheduling horizon used in the research was 3 or 4 days.
The problem is formulated as a mixed integer quadratic programming problem and is
decomposed into a linear 0,1 programming master problem with small quadratic
programming subproblems.

Loucks [1987] and Loucks and Jacobs [1991] developed a heuristic to solve a
goal programming formulation of this problem. His contention was that the problem was
so large that it precluded solution by optimal methods. His research indicated that the
heuristic produced high quality schedules but no comparison was made with other
solution methods.

A general heuristic for solution of this problem was outlined by Glover et. a1.
[1984, 1988]. This procedure was implemented and tested in a fast food restaurant, and is
unavailable for testing. The tests showed that the solutions were of high quality but the
only comparison (informal) was against manual methods.

Love and Hoey [1990] presented an integer formulation for this scheduling
problem, deveIOped from a fast food setting. The integer problem formulation is
amenable to a minimum cost network flow solution, after decomposition. The tests
showed that the solutions were of high quality but the only comparison (informal) was

against manual methods.

25

2. 6. SUMMARY OF THE REVIEW OF LABOR SCHEDULING LITERATURE

As can be seen from reviewing the literature, real world labor scheduling can be a
complex problem. A researcher or practitioner has to accommodate multiple objectives,
multiple constraints, and many variables. Prior investigations have shown that the
solution difficulty is dramatically affected by limited worker interchangeability. Reliance
on specially developed heuristics is common because of the solution difficulty and
problem size, causing a large number of heuristics to be developed, but no general
methods. In addition, precedent exists for construction/improvement approaches, as will

be used in this research.

2. 7. OPPORTUNITIES FOR RESEARCH

2. 7.1. Application of new algorithms

Two recently developed optimization techniques, tabu search [Glover, 1989] and
simulated annealing [Kirkpatrick 1983, van Laarhoven and Aarts, 1987], show promise
in the area of combinatorial optimization. These algorithms have not been applied in any
of the labor scheduling problems discussed above. Previous research [Knox, 1989 and
Skorin-Kapov, 1990] has shown tabu search and simulated annealing to be superior to
existing techniques in the problem areas tested (the traveling salesman problem and the
quadratic assignment problem, respectively). Adaptation of these algorithms to the
research problem considered in this proposal may yield a superior technique in terms of
solution quality and solution speed. In addition, incorporation of heuristic information

into these optimization techniques may provide some extensions to the theory of tabu

26

search and simulated annealing. These algorithms are explained in further detail in

chapters 5 and 6.

2. 7. 2. Comparison of Previously Applied Algorithms

As noted in the review of the solution techniques above, no comparative study of
different solution techniques for the work tour scheduling problem with limited worker
interchangeability has been performed. A comparison of available techniques will be
useful to guide future research in several ways.

Since the research problem is large and combinatorial, any research that compares
algorithms may have general applicability in guiding further research into areas that
appear to be more fruitful in terms of solution quality and solution speed.

Several algorithms have been suggested, apparently without knowledge of the
others, for the solution of the work tour scheduling problem with limited worker
interchangeability. A comparative study of the algorithms can provide guidance for
improvements in solution quality and speed. In addition to the computation aspects, the
new algorithms might enable the researcher to address the need for a different set of

objectives for each firm and provide a tool that can be used for more than one problem

type-

27

CHAPTER 3, MATHEMATICAL FORMULATION

3.1. CHOICE OF FORMULATION

There are several ways to formulate the problem of scheduling work tours with
limited worker interchangeability [Love and Hoey, 1990], [Loucks, 1987], [Glover and
McMillan, 1986], [Warner, 1976], [Warner and Prawda, 1972]. The formulation selected
and described below is adapted from one presented by Loucks [1987].

The primary differentiator between formulations is that of fixed shifts or variable
shifts. In the fixed shift formulations [Love and Hoey, 1990] , [Glover and McMillan,
1986], [Warner, 1976], [Warner and Prawda, 1972] the manager defines, beforehand, the
shifts to be used in scheduling the workforce, so the problem becomes assignment of
workers to the predefined shifts. In the variable shift models [Loucks, 1986] , the authors
define their models in such a way that the Shifts are generated simultaneously with the
worker assignments to those shifis. The fixed shift model, because of the fixed shifts, is
more constrained than the variable shift model. The more highly constrained fixed shift
model has as smaller solution space and takes less time to solve, (compare, for example,
the results of [Glover and McMillan, 1986] to the results of the tabu search

implementation here).

28

The advantage of using a variable shift model is that the labor costs can be lower
than those of a fixed shift model because that algorithm can more closely match the
demand for workers with the assignment of workers. The primary disadvantage of these
formulation is that they are more complex and time consuming to solve.

Loucks presents his formulation as a O-l/integer goal programming problem. His
formulation consists of ranked goals and constraints, and for solution, uses a two phase
procedure; Phase 1) the construction of an initial feasible solution and Phase 2)
improvement of the solution. Without loss of generality, and with the goal of reducing
implementation complexity, Loucks' formulation will be modified to be one phase.

There are several reasons for using Loucks' formulation. First, his formulation has
a large number of 0,] variables, easing application of simulated annealing.

Second, his formulation uses variable rather than fixed shifts. This has the
advantage of potentially lower costs than a formulation using fixed shifts, but may
require more computation time.

Third, Loucks' formulation explicitly considers worker preferences, an advantage

when workers are scarce.

3. 2. PROBLEM FORMULATION

This particular formulation of the work tour scheduling problem with limited
worker interchangeability was adapted from one presented by Loucks [1987]. His
applications were in a fast food restaurant, a banking environment and synthetic

problems. The mathematical formulation uses the following notation:

29

Definition of Notation:

A capital letter with one or more subscripts represents a finite set.

The number of elements of a finite set, say A, is denoted by |A|.

I = number of workers being scheduled/assigned in labor pool,

J = number of tasks in operation,

K = number of operation hours per day,

T = number of operating days in scheduling horizon,

ci = wage rate for employee i,

Dht = Demand maximum for workers on day t, by task hour,

Dlt = Demand minimum for workers on day t, by task hour,

ejkt = number of workers scheduled in excess of the number required for task j in

hour k of day t,

bi = number of tour hours targeted for worker i,

Qj = workers qualified for task j,

rjkt = number of workers required for task j in hour k of day t,

8min = allowed minimum number of hours in any shift,

Smax = allowed maximum number of hours in any shift,

Ti = tasks for which worker i is qualified,

Uit = worker i's available hours in day t,

“it = worker i's first available hour in day t,

Vi = worker i's available days (partial or whole),

Vit = worker i's last available hour in day t,

Wmax = maximum number of days worker can be assigned in scheduling period,

xijkt = 1 if worker i is scheduled to perform task j in hour k of day t,
‘ = 0 otherwise, and

Yit = 1 if worker i is scheduled to work any hours in day t,

= 0 otherwise.

30

The objective of the work tour scheduling problem with limited worker
interchangeability is to minimize the total man-hours of overstaffmg and to match as

closely as possible the workers desired hours and time off. So the objective function is

minimize:
w1* Zejkt +W2*(injkt-hi).
jkt ijkt

In Loucks' formulation, the two terms in the objective function were stated as
goals in his goal programming formulation. The weights W1 and W2 will be chosen to
ensure that this integer program behaves as his goal program does. The first term has first
priority and as such will have a large W1 value, W2 will have a relatively smaller value.

Subject to:

Z xijkt - ejkt = rjkt for all j,tEVit, kEUit,(1)
iEQj
Constraint set (1) states that the labor schedule must meet, but preferably not

exceed, the staffmg requirement for each task in each operating hour of the week.

2 Z xijk, =<1 for alli(where |Ti| > 1, tEVi and KEUit, (2)
lETi kEUit

Constraint set (2) restricts a worker assignment to not more than one task per

hour.
. 2 2 xijkt - 5min(Yit) >= 0 for all i and tEVi, (3)
JEUitkEUit
Z Z xijkt - lUitlYik =< 0 for all i and tEVi, (4)
lETi kEUit

Constraint sets (3) and (4) confines a worker's shift in any workday to be greater
than or equal to the minimum allowable duration, 3min. Constraint set (4) controls the

value (0 or 1) of Yit-

31
2: Z xijkt =< Smax for all i and tEVi, (5)
jETi kEUit
Constraint set (5) restricts the worker's shift length to be less than or equal to the

maximum allowable shift length, Smax-

Z Yit =< Wmax for all i where NM > Wmax: (6)
tEVi

Constraint set (6) restricts the number of days worked per week for any worker to

less or equal to than the allowable maximum, Wmax-

E (Xijkt - xij(k+1)t + xij(k+2)t) =< 1 for all i
1 Ti

where lUitl>3mim k=(uit,..,vit-smin+l), and tEVi. (7)

The last constraint set (7) ensures that a worker’s shift will be contiguous.

3. 3. ASSUll/fl’IYONS OF FORMULATION

There are several assumptions inherent in this formulation. The first assumption
is that the forecast of worker requirements for the scheduling period is given; there are no
provisions in the algorithm or formulation to develop a forecast. The implications of this
assumption are two-fold; 1) performance differences between algorithms will be due to
the suitability of the algorithm to the problem and not superior performance of
forecasting methods and 2) the staffmg requirements are deterministic, that is, the
number of workers needed for each task in each hour is assumed to be known.

Second, the solution is constrained by limited worker interchangeability, and
specific task assignments. This assumption is derived from application to service

businesses. It makes the problem larger and more difficult to solve.

32

Third, the schedule encompasses less than 24 hours, and no accommodation has
been made for shifts to overlap days. Reforrnulation of the problem would be necessary
to accommodate 24 hour schedules.

Fourth, the time period length has been set at 1 hour. There are two reasons for
this length. 1) A 1 hour scheduling period will allow the manager enough flexibility to
schedule lunch and breaks during lulls in customer demand without having to explicitly
model such break times. Reducing the time period would allow us to model the lunch and
rest breaks, but since demand is variable, even though it is assumed to be fixed, modeling
with reduced time periods may cause reductions in customer service. 2) Managers may
not care to adjust the staffing levels more often than once per hour, more frequent
changes might reduce productivity.

Fifth, this formulation assumes that the operating days are short enough to ensure
that there is sufficient rest time between shifts. So a worker may "close" the facility one
day and "open" the next day.

Last, the model allows customer or client demand patterns that are independent
from day-to-day. This assumption or feature of the model is a necessity for 7 day

operations.

33

CHAPTER 4, GENERATION OF INITIAL FEASIBLE SOLUTION

4. I. INTRODUCTION

There are at least three approaches to the design and implementation of iterative
improvement algorithms. The first method is to generate an initial solution, possibly
feasible, using random methods, which is then acted upon by the improvement algorithm.
The second method is to generate an initial solution that is known to be feasible, which is
then acted upon by the improvement algorithm. The third method is to construct and
improve the solution simultaneously, this is the method used by Loucks [1987]. An
examination of the research literature of both simulated annealing and tabu search
revealed that the prevailing approach is to generate an initial solution using random
methods, which is then acted upon by the respective algorithm. The two predominant
reasons for previous researchers using this method are these. First, according to van
Laarhoven and Aarts [1989], simulated annealing is not sensitive to initial solution or
configuration. No such results are available for tabu search. Second, the problems
considered by previous research of both methods, simulated annealing and tabu search,
are such that it is a relatively simple matter to use random methods to generate an initial,

feasible configuration, because the problems previously considered have few constraints.

34

The use of a random approach was deemed inappropriate in this research for
several reasons. First, no known, or easily conceived, methods are known for generating
an initial, feasible configuration randomly, for this highly constrained problem. Second,
after some investigation, it was determined that the algorithms, as implemented here,
could not always (less that 50% of the time) remove the infeasibilities present in an
infeasible initial configuration. Simulated annealing, given enough run time, tries all
possible configurations and so would remove all infeasibilities. Tabu search, being a
deterministic algorithm does not try all possible configurations and may never remove an
infeasibility. Last, it was found that estimating the change in the Objective function was
an excellent way to improve the speed of the algorithms, as explained in the following
chapters. It is relatively simple to estimate the effect of a feasible transition or move on
the objective function, but it is not easy to estimate the effect on the objective function of
causing an additional infeasibility, or partial removal of an infeasibility.

Previous authors have used the output from a previously developed algorithm as
the input to simulated annealing and tabu search. On reason this approach was not taken
is because Loucks' heuristic was shown to produce optimal results for a significant
proportion of the problems tested by Loucks [1987]. It seemed foolish to apply an
improvement algorithm to a heuristic that produces optimal solutions Often. Additionally,
Loucks presents evidence that his algorithm will not solve all problems produced by the
problem generator he used. So, the possibility existed that improvements could be made
in the number of problems that could be solved by developing a new method for
generating initial, feasible configurations. These considerations prompted development
of an initial configuration generator that, upon successful completion, produces feasible

configurations or solutions.

35

4. 2. STRATEGY OF INITIAL FEASIBLE SOLUTION GENERATOR

Several objectives and strategies for the development of an IFS generator were
identified as a result of the considerations above. The objectives for the IFS generator
were to have the algorithm work reliably and to produce a feasible solution quickly. This
algorithm works more reliably than the only other method known for this problem,
Loucks' heuristic. The method is also reasonably fast, it normally takes less than 10
seconds on a personal computer (33 MHz, 80386 processor) to produce the feasible

solution. The strategies are detailed below.

4.2.1. ”F at " solution, vs. "Lean" solution

One consideration that guides the development of the IFS generator is that of the
type of solution to be produced. In this case the strategy was to develop the IFS generator
in such a way that a "fat" solution is produced. This means that there is not much attempt
to keep from over scheduling task hours. This is direct contrast to Loucks' heuristic,
which attempts to minimize over scheduling task hours at every opportunity. The result
of this strategy is that the improvement algorithms, simulated annealing and tabu search,
have more possible configurations to generate and choose from, hopefully improving the
prospects of a good final solution. This strategy is also responsible for the improved

ability of the IFS generator to produce feasible configurations.

36

4. 2. 2. Margin on task hours

Some method must be used to determine the priority in which to schedule task
hours. While there are several methods to assign priority, such as starting at the
beginning of the week and proceeding in a linear fashion to the end, the priorities could
be assigned randomly, and etc., the method used in this research was to calculate the
margin for each task hour and use that margin as the basis for assigning priority. The
margin of a taskhour is defined as the difference between the number of workers required
for the taskhour and the number of workers available and qualified for that taskhour.
These margins are sorted from lowest to highest and the taskhours are assigned the
highest priority from lowest to highest margin.

This method of assigning priority based on margin has the advantage of ensuring
that when the scheduling assignments are made, those workers that were identified as
available at the beginning of the algorithm are available when they are needed for

assignment to the taskhour. The idea of margin is attributable to Loucks [1987].

4. 2. 3. Choice of worker for task hour

Another strategic issue is the order in which workers are chosen for scheduling in
a particular taskhour. When a pool of workers for a particular taskhour is identified,
some method must be used to pick among the workers in that pool for assignment to that
taskhour. The method used in this research is to create a worker priority score. This

score, then, is used to differentiate among the workers. See figure 4-1.

37

 

Begin Subroutine

 

 

 

 

l

 

 

 

Road florkar "Huber Prom Pool Array

 

 

l

 

 

Calculate Pill-in Hours

 

 

V

 

 

 

Calculate NUmbar o: Off-Target Hours

 

 

 

 

Calculata Horkars Scorn

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

      
 

score - 0
Score - Score +Workar'a Fill In Hour-
Scora - Scoro - Wbrkar'a of: Targat Houra “ 2
Scora - Scorn + unabar 0t Tasks Known by Worker
Score - score + number Hours Workar Available on Day
scora . Score - Workar NUmbor
Scora - acora + number of Daya lbrkar 1a Schedulad

I

Put Scoro Into Pool Array
Yea

Any More
Horkara?

 

 

Sort Pool Array By Scorn, Ascending

 

1

8nd Subroutina I

 

Figure 4-1, Selecting a Worker for Assignment

 

 

 

 

38

The first consideration in determining a worker score is the worker's scheduling
flexibility. This flexibility is determined by the number of tasks known by the worker,
the workers availability, and the number of days already scheduled. A worker is more
flexible when that worker knows more tasks, when that worker is more available, and
when that worker is scheduled for fewer days. Since any worker that is available and
qualified may be chosen for assignment to the taskhour at hand, it makes intuitive sense
to select the least flexible workers for assignment first, saving the more flexible workers
for later in the assignment process when greater flexibility is needed.

The other consideration in choosing a worker for assignment is that of hours
previously assigned to the worker. The method for calculating the score attempts to
minimize the number of fill-in hours and the number of off-target hours.

The score considers the number of fill-in hours between the hour to be scheduled
and the hours already scheduled, the number of hours a worker is under or over targeted
hours, the number of tasks known by the worker, the number of hours the worker is
available for the day, and the number of days the worker is already scheduled. The
equation for calculating the score is:

Score = (Number of F ill-In Hours) - (Number of Hours Off Target) A 2 +
(Number of Tasks Known) + (Number of Hours Available on Day) + (Number of Days
Already Scheduled) + (Worker Number)

The reader will note that the equation for worker score puts preference on lower
worker flexibility and endeavors to minimize the number of fill-in hours and off target

hours. The worker number is added to the score to alleviate potential ties among workers.

39

4. 2. 4. Elimination of Constraint Violations

The methods used to produce an initial schedule typically produce a number of
constraint violations. In order to produce a feasible solution, these constraint violations
must be eliminated. As elucidated in chapter 3,there are seven types of constraints in the
mathematical formulation of this problem. Seven separate methods were developed to fix

the various constraint violations. See figure 4-2.

 

Determine Type
or Constraint

 

 

 

4O

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Violation
Constraint
Violatlon IYpe Action
Uhdar- Scheduled Add Workers
“'"‘ Task Hour To Task flour
Worker Assigned Remove Highest Margin Task Proa
_4.. More Than 1 Schedule In Hour And Giva To
Task per Hour Another Worker
Worker Is Add Low Margin
__g.. Scheduled for Less Hour(s) To
than Win. Shift Worker's Shift
Worker Scheduled Give Non-Available
t... During Non- Hours or Shift

 

Available Hours

To Another Worker

 

 

 

 

 

 

Worker is
Scheduled for Moro
Than Wax. Shift

 

Give Hours To
Another Worker,
Based on Highest

Margin

 

 

 

 

 

 

Worker Is
Scheduled For Wore
Than Maximum Days

 

 

Give Avay Day
With Highest
Average Margin To
Another Worker

 

 

 

 

 

 

Worker's Shift
Must Be
Contiguous

 

 

Give Away Part of Shift With
Highest Margin To Another
Worker Or Fill In Hours
Between Scheduled Hours

 

 

 

 

 

 

 

 

8nd Subroutine

 

 

 

 

 

Figure 4—2, Eliminate Constraint Violations

41

Constraint Type 1: The labor schedule must meet or exceed demand
requirements expressed in man-hours needed per taskhour. For constraint violations of
this type the correction is to add a worker(s) to the taskhour in violation. If the Average
Worker Utilization is low enough, this is an easy correction to make. If the AWU is too
high, the procedure fails, and the algorithm fails to find a feasible solution.

Constraint T we 2: A worker must not be assigned to more than one task in an
hour. Given the data structure of the programs representing the implemented algorithm, it
is impossible to have a constraint violation of this type.

Constraint T we 3: A worker must not be scheduled for less than the minimum
shift in any day. A rectification of this type of constraint violation requires that an
hour(s) be added to one end of the shift in violation. The method used examines the
taskhours on either end of the shift that the worker is available and qualified to perform,
chooses the taskhour(s) with the lowest margin, and assigns the worker to that
taskhour(s).

Constraint T we 4: The labor schedule must not exceed worker's available hours.
In this case, the worker has been inadvertently assigned to hours that the worker is not
available to work. If the shift length of the worker for the day, less the number of hours
scheduled in error, leaves a shift equal to or greater than the minimum shift then the
hours in error are given to another worker, available and qualified, of course. If no
worker can take the add on hours, the another worker is found who is available and
qualified and not scheduled for the day. This worker is then given the hours in error and
a shift of minimum length is created around the hours given to the second worker. In the
case that after subtracting the hours in error, the remaining shift is shorter in length than
the minimum shifi, the entire shift is given to another worker, who is available and
qualified.

Constraint Type 5: A worker must not be scheduled for more than the maximum

shift in any day. In this case, the worker has been inadvertently assigned to hours that

42

extend the shift beyond the maximum length. If the shift length of the worker for the day,
less the number of hours scheduled in error, leaves a shift equal to or greater than the
minimum shifi, then the hours in error are given to another worker, that is available and
qualified. If no worker can take the add on hours, the another worker is found who is
available and qualified and not scheduled for the day. This worker is then given the hours
in error and a shift of minimum length is created around the hours given to the second
worker. In the case that after subtracting the hours in error, the remaining shifi is shorter
in length than the minimum shifi, the entire shift is given to another worker, who is
available and qualified.

Constraint Type 6: A worker must not be scheduled to work more than the
maximum number of days per week. In case of a constraint violation of this type, the
shifi(s) with the highest average margin is given to another worker(s). The reason the
highest average margin shift is given away is that it is easier to find a worker to take the
shift for higher margin task hours than for lower margin task hours.

Constraint T we 7: The worker's shift must consist of contiguous hours. This
type of constraint violation has the worker working several hours in a day, but with a gap
of a non-scheduled taskhour(s) somewhere in the shift. If the total shifi length (including
the non-scheduled taskhours), is less than the maximum shift, then the gap is filled in by
adding taskhours with the smallest margin. If the aforementioned shift length is longer
than the maximum shift, then the part of the shift with the highest average margin is
given to another worker.

Experience with the constraint violation correction routine showed that for some
constraint violations, especially violation of constraint 7, that not only was the intended
violation corrected, but that another violation(s) was corrected at the same time. For
example, let us suppose that the constraint violation was of constraint 7. If the shift
length was over the maximum shift length, then there would be a violation of constraint 5

also associated with the same scheduling error. Correction of the violation of constraint 7

43

would fix the violation of constraint 5, but not vice versa. After some experimentation,
the order that the constraints were corrected, 6-1-5-4-3 -7 (by number of constraint type),
was found to be the most effective in correcting violations. It was also found that often,
two iterations through the violations in the above order would completely correct the
constraint violations where one iteration would not. This behavior may be attributable to

the interactions between methods for fixing the violations.

4. 3. STEP BY STEP EHLANAHON 0F IFS GENERATOR

A brief description of the steps that comprise the initial feasible solution
generator is given below. A flow chart, figure 4-3, is presented as illustration of the
algorithm.

44

<:: Read Problem Pile <::

Calculate Margins
For All Taskflours

L

Sort Margin Array
In Acending Order

i

Pick Remaining Taskflour
" With Lowest Margin

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

  
 
    

Is WUmber
Needed in Task
Hour > 0?

 

 

Determine Pool of
Workers Available
For TaskHour

i

Sort Worker Pool
According to
Assignment Criteria

L

Select Worker
Prom Pool

1

Create or
Add To Shift

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

  
   
   
  

Is Wuuber
Scheduled >- unmber
Required?

 

  
   
   

Y0“ Are There

More Taskfloure
To Fill?

 

 

Eliminate Constraint
Violations

<::::j— Write Feasible Solution To Disk (::::

Figure 4-3, Procedure to Generate Initial Feasible Solution

 

 

 

 

 

45

4.3.1. Explanation of Steps in Flow Chart

The description of steps below follow the flow chart in figure 4-3. The titles of
the sections below match the text in the boxes in the flow chart.

Step 1. Read Problem File. This step reads a problem from a disk (generated
previously and stored on disk) and puts the problem data into the appropriate data
structures.

Step 2. Calculate Margin For All Task Hours. Step 2 loops through all taskhours
for the scheduling period, tallying the total number of workers available and qualified for
each taskhour and calculating the margin for each taskhour according to the following
equation:

Margin(Hour,Task,Day) =NumberOfiNorkersAvailable(Hour,Task,Day) -
NumberOflNorkersRequiredO-Iour,Task,Day).

These margins are stored in an array for future reference.

Step 3. Sort Margin Array in Ascending Order. The purpose of Step 3 is to sort
the array in which the margins are stored in ascending order so that the taskhours may be
selected from lowest margin to greatest.

Step 4. Pick Remaining Task Hour With Lowest Margin. This step begins a loop
that, when finished, assigns workers to all taskhours. This loop steps through the sorted
margin array from beginning to end, picking the taskhour with the lowest margin that has
not yet been scheduled.

Step 5. Check T 0 See if This Task Hour Needs to Be Scheduled. During the
assignment of hours, performed in step 9, workers usually are assigned more to than one
taskhour. As these assignments occur, some or all of the taskhour requirements in
neighboring hours to the taskhour currently being scheduled are being filled.
Occasionally early, and especially late in the algorithm, a taskhour may not need any

further worker assignments. If a particular task hour's requirements are already met by

46

previous worker assignment, there is no need to execute the loop, and this step skips the
remainder of the loop, returning processing to step 4.

Step 6. Create POOl Of A vailable Workers for Task Hour. Once the taskhour is
chosen for scheduling, workers must be found that are available and qualified for the
taskhour. This step creates a pool of these workers by looking at all worker's availability
and task qualifications, and putting the worker numbers of all workers available and
qualified for that taskhour into an array for further processing.

Step 7. Sort Worker POOl According to Assignment Criteria, Figure 4-1. After
the all available and qualified workers are identified, a method for choosing a worker
from that pool must be used. While such methods as random selection might be used, the
method used, shown in figure 4-2 and detailed above, selects workers based upon their
flexibility and schedule thus far in the algorithm. This method considers the number of
fill-in hours between the hour to be scheduled and the hours already scheduled, the
number of hours a worker is under or over targeted hours, the number of tasks known by
the worker, the number of hours the worker is available for the day, and the number of
days the worker is already scheduled. As stated above the workers are assigned a score
based on the aforementioned criteria, the scores set into the worker pool array and then
the worker pool array is sorted into ascending order.

Step 8. Select Worker From Pool. Workers are selected from the pool, starting at
the beginning of the array. If more than one worker is needed, the workers are taken in
the order in which they were sorted in step 7.

Step 9. Create New Shift or Add to Existing Shifi, Figure 4-4. This step assigns
the worker chosen in step 8 to the taskhour. If the worker is already scheduled for to
work on the day of the taskhour, the hour(s) between the taskhour and the previously
scheduled shift are filled in with task assignments. If the worker was not scheduled to

work on the day of the taskhour, then a shift is created around the taskhour being

47

scheduled. A more complete discussion is given above and the reader may also refer to
figure 4-4.

Step 10. Check tO See if Enough Workers have Been Scheduled for Task Hour.
Frequently more than one worker is required for a particular taskhour. This step
determines if the taskhour requirements have been met. If not, this step redirects the
processing to step 8.

Step 11. Check tO See if there are more Task Hours to Schedule. The algorithm
schedules workers into taskhours according to the order in the margin array. This step
checks whether any taskhours remain to be scheduled. If there are more taskhours to be
scheduled, this step directs the processing to step 4.

Step 12. Eliminate Constraint Violations, Figure 4-2. After all the taskhours have
been scheduled,. a schedule has been created for the problem read in step 1. Typically the
schedule created by the preceding steps will violate the constraints detailed in chapter 3.
This step rectifies constraint violations. A more complete description is given above, and
a graphical description of the procedure to eliminate constraint violations is given in
figure 4-2.

Step 13. Write Feasible Solution to Disk This step saves the initial feasible
solution in a disk file for improvement by the simulated annealing and Tabu search

algorithms.

Yes

 

  
   

 

Determine Task(s)

For Fill-in Hours

Based on Samllest
Margin(s)

 

 

 

 

 

Assign Worker
To Hour, Fill-in
Hours and Tasks

 

 

 

 

48

 

Start

 

 

 

   

Determine if Worker
is Already Scheduled

 

I

 

 

Determine TaskHours, Before
and/or After Hour Passed ,
To Which To Assign Worker,
Based on Smallest Margins

 

 

 

Assign Worker
To New Shift

 

 

 

 

 

 

 

End Subroutine

 

 

 

Figure 4-4, Create or Add to Shift

 

49

4. 3. 2. Pseudo Pascal Description of IFS Generator Algorithm

The following is a description of the algorithm in pseudo pascal.

Procedure Initial Feasible Solution
Begin
Initialize;
Read Problem File;
Repeat
Calculate Worker Margin For Task Hour;
Until All Task Hours Are Done;
Sort Margins in Ascending Order;
Repeat
Pick Remaining Task Hour with Lowest Margin;
IfNumber Needed > 0 then
Repeat
Determine Worker Pool Available For Task Hour;
Sort Worker Pool According to Assignment Criteria;
Select Worker From Pool;
If Worker Is Already Scheduled Then
Add To Current Shift;
ElseIf
Create Shift Around Task Hour;
Endif;
Endif;
Until Number Scheduled >= Number Needed For Task Hour;
Until All Task Hours Have Been Scheduled;
Eliminate Constraint Violations;
Write Feasible Solution To Disk;
End:

50

4. 3. 3. Conclusion

An algorithm for generating feasible solutions to the work tour scheduling
problem with non-interchangeable workers, as formulated in chapter 3, has been
presented. As discussed before, the performance of this algorithm exceeds previous
algorithms in terms of the number of problems for which it is able to produce a feasible
solution. This initial configuration generator produces initial, feasible configurations on
problems for which Loucks' heuristic produces an abnormal stop. The initial, feasible
configuration (IFS) generator produces feasible configurations on all problems that
Loucks' heuristic would run to completion, but Loucks' heuristic would not run to
completion on all the problems that the IFS generator would successfully run to

completion.

51

CHAPTER 5 - TABU SEARCH: THEORY AND IMPLEMENTATION

5. I. INTRODUCTION

Tabu search is a strategy to solve combinatorial optimization problems
that uses an adaptive mechanism to guide other algorithms, such as linear programming
or other specialized heuristics, to avoid the limitations of local optimality. Tabu search
was initially developed by Glover [1986] and has since been the subject of considerable

research.

5.1.1. Overview Of T abu Search

Tabu search may be considered a supervisory heuristic imposed on another
heuristic in order to more rapidly solve combinatorial optimization problems. In general,
tabu search uses another heuristic(s) to generate a number of potential intermediate
solutions (moves) from which it picks the "best". This leads to rapid arrival at a local or
glObal optima. The approach endeavors to overcome local optima by imposing a strategy
of forbidding certain moves and forcing moves away from the local optima. Reversal of

these moves or directions are forbidden or tabu for a number of iterations so that cycling

52

might be avoided. The flow chart and related discussion following will make the
operation of the tabu search algorithm clear. The tabu search described below is an

adaptation of that presented by Glover [1989].

53

 

Procedure Tabu Search

L

Read Initial
Feasible
Solution

 

 

 

 

 

l

Generate Set or
Potential Moves

T

Rank Potential Moves
Based on Estimated
Objective Function Value

. Select Potential Move L.,
————e+ J“

 

 

 

 

 

 

 

 

 

 
 

Test For Test

Tabu Status . .
Criteria

 

Accept Potential Move
as New Solution

 

 

 

 

 

 

 

 

[ Implement New Solution I
[ Update Tabu Lists I

 

l

[Update Aspiration Criteria List I

f

Update Current Best Solution,
If Best

 

 

 

 

 

   

 

   

Check Status Against
Stopping Criteria

  

Figure 5-1, Tabu Search Algorithm

  
 
  
 
  

 
  
  
 

Against Aspiration

  
 

54

As shown in the flow chart, the first step is to generate an initial feasible solution. This
initial solution is generated using the procedure explained in chapter 4. The current best
solution is set to the initial solution. The next step is to generate a set of potential moves.
As mentioned before, these moves can be generated by other heuristic methods, by
optimization methods used on partitions of the problem, etc. The move generators used
in this investigation are detailed below. The potential moves are then sorted or ranked.
Ranking of potential moves might be done in a number of different ways, such as, on
objective function value, estimates of objective function value, change in objective
function value, or some other criterion. The ranking of potential moves for this research
is based on an estimate of the change in objective function value.

Testing the solution for tabu status is the next step. If the solution is not tabu, the
solution is accepted as the new solution. The tabu list, aspiration criteria list (to be
described shortly) and current best solution would be updated as necessary. If the prior
solution was a global or local optima, it is very likely that all of the potential solutions in
the current iteration would be worse than the current best solution. In this situation, there
are two items of interest; First, this is an example of the algorithm moving away from an
(local) optima in search of a better (local or global) optima and Second, in this situation
the current best solution as well as the aspiration criteria would not be updated.

If the solution is tabu, then the solution is tested against the aspiration criteria. An
aspiration criteria is a change in objective function value of sufficient magnitude to
override the tabu status. The aspiration criteria is set in such a manner so as to prevent
cycling of the algorithm. For example: if the algorithm was moving away from an
optima, the aspiration criteria must be set in such a way that it would not override the
tabu status and cycle back to the previously found optima. The aspiration criteria used in
this implementation of tabu search is to require a move to produce a solution better than
the current best solution, in order to override tabu status. Glover [1989] covers aSpiration

criteria in considerable detail. If the solution meets or exceeds the aspiration criteria, tabu

55

status is overridden and the tabu list, aspiration criteria and current best solution would
be updated as necessary.

At each iteration, the algorithm checks against the stopping criteria. The stopping
criteria used in this research was based on two rules. First, a limit of one thousand
iterations was set, based on trial runs of the algorithm. Glover [1989] recommends that
such an overall iteration limit be set. This limit seems to be somewhat past the "point of
diminishing returns". The second rule used was to stop the algorithm if there had been no
improvement in the current best solution in 300 iterations. Again this rule is based on

Glover [1989] and trial runs of the algorithm.

56

5.1.2. Description Of T abu Search Algorithm In Pseudo-Pascal

Procedure Tabu Search
Begin
Initialize;
M := 0;
repeat
Generate {Potential Solutions}
Sort {Potential Solutions} ...... (based on Objective Function Value,
Best First)
repeat
N := O;
TabuTest (Solution N, Tabu Status);
If (Solution N) = Tabu Then
AspirationTest (Solution N, Aspiration Criteria);
If (Solution N) > Aspiration Criteria Then
Accept (Solution N);
Else
Accept (Solution N) ;
Update (Tabu List, Aspiration Criteria, Current
Best Solution)
Accept Flag := l;
Endif;
Until Accept Flag = 1;
Until Stop Criterion = True;
End.

57

5.1.3. Example Of T abu Search Algorithm

Consider maximizing the function f(x) = x2 : where x is an integer and permitted
to vary from 0 to 31. For this example the move generator will be a random number
generator that generates integers from O to 32. Furthermore, we will arbitrarily set the
number of moves to be generated at 4 and the tabu list length at 3.

Initialize: We select an initial solution at random, say x=3. The objective
function value is 9. We set the current best solution to x=3. The tabu list is empty as is
the aspiration criteria list.

Generate: We pick 4 potential solutions from our random number generator, 1,
8, l9, and 6.

Sort: We sort the values from high to low, 19, 8, 6, 1.

TabuTest: We test the lst solution (19) for tabu status. Since the tabu list is
empty the solution is not tabu.

Accept: We accept the solution (19) as the new solution.

Update: We update the tabu list, (it includes 19 now), the aspiration criteria list,
(a solution must beat 19), and the current best solution (set to 19).

Generate: We pick 4 potential solutions from our random number generator, 12,
7, 19, and 5.

Sort: We sort the values from high to low, 19, 12, 7, 5.

TabuTest: We test the lst solution (19) for tabu status. Since the tabu list
contains 19 the solution is tabu.

AspirationTest: The solution does not meet the aspiration criteria that the
solution must beat 19, so we go the next potential solution (12).

TabuTest: We test the 2nd solution for tabu status. The tabu list does not contain

12 so the solution is not tabu.

58

Accept: We accept the 2nd solution as our new solution.

Update: We update the tabu list (it includes 12 and 19 now). The aspiration
criteria list and the current best solution remain unchanged.

The algorithm would continue until a stopping criteria is met.

Because the example is simple for clarity's sake, the aspiration criteria is almost
trivial in its simplicity. Normally, the aspiration criteria would allow a tabu variable back
into the solution if two conditions are met; I) the magnitude of the change in objective
function value is sufficiently large, and 2) there is some change in the solution, i.e. the
solution has a different variable set than it had when the tabu variable was in the solution

previously or equivalently, the solution is not identical to any previous solution.

5. 2. TABU SEARCH IWLMNTAHON ISSUES

5. 2. 1. Tabu List T mes

The basic idea of a tabu list is to restrict previous solutions (or some aspect of a
previous solution) from being used for a specified number of iterations. Each move that
is selected as a potential move is checked against the tabu list. Ifthat move is found on
the tabu list, that potential move is eliminated from consideration, and another potential
move is chosen. In prior implementations tabu search, Glover [1989], Knox [1990], used
tabu lists that prohibit reentry of a solution based on move type. Using the traveling
salesman problem as an example (that is what Knox researched), if a move had changed
the connection between two edges, the tabu list would not accept a move that
reconnected the two edges in the original order, for a certain period of time (based on the

length of the tabu list). In the implementation of tabu search for the work tour scheduling

59

problem with non-interchangeable workers, the move types used precluded the use of
such conceptually simple tabu lists.

The tabu lists that are kept for this implementation of tabu search are based on
the types of move generators used. There are three types of move generators used; ones
based on hours (i.e. GiveAnHour, IncreaseOverScheduling, GiveAwayPartShift and
ReduceOverScheduling) , ones based on shifts (i.e. GiveADay, TradeTours, and
CrossTradeTours) and hybrids of the two (i.e. GiveAwayMinShifi and
GiveAwayPartShift). As mentioned before, in prior research, tabu lists were used for
each different type of move. In this research, however, separate lists for each type of
move would be very cumbersome to maintain and check for tabu status. For these
reasons, four lists were created that would preserve the idea of tabu lists and also allow
for easy maintenance and checking. These four lists track hours deleted from a workers
schedule on a particular day, hours added to a worker's schedule on a particular day,
shifts deleted from a workers schedule on a particular day and shifts added to a worker’s
tour for a particular day. As a particular move is implemented, the appropriate list is
updated with the relevant information from the move.

These four tabu lists were found to be adequate to prevent cycling, and to force

the algorithm out of local optima, which are the two objectives of using tabu lists.

5. 2. 2. Aspiration Criteria

The tabu lists, when properly specified, prevent the tabu search algorithm from
cycling. As was mentioned before, this is accomplished by prohibiting a previous
solution or some aspect of a previous solution from reentering the solution, for some

number of iterations. Aspiration criteria allow parts of a previous solution back into the

 

60

solution, providing that the reentry of this part of the solution does not cause cycling.
Glover [1987] suggests that aspiration criteria allow a tabu variable, or solution, back
into the solution if two conditions are met; 1) the magnitude of the change in objective
function value is sufficiently large, and 2) there is some change in the solution, i.e. the
solution has a different variable set than it had when the tabu variable was in the solution
previously or equivalently, the solution is not identical to any previous solution. These

are the aspiration criteria used in this investigation.

5. 2. 3. Selection Of Parameter Values

The only parameter value to be determined is the length of the tabu list. Although
there is some guidance given in the literature [Glover, 1989], the tabu list size is usually
determined by experimentation. Experimentation was used in this research to determine
the tabu list length. Results of the experimentation is shown in Figure 5-2, and in Table

5-1.

5. 2. 4. Comparison of Results for Difi'erent T abu List
Lengths.

61

 

 

 

 

 

 

 

[Tabu Search Improvement Peth'
336i
315 .,
me)
«A
w "M ’W *V w”
265T W ’ d . «M»
2x0 ‘1 1| R"
215.L
195,. f
175
O O N V N D O Q N V 0 G N
3$§§2§§N §a§ssaita§§rfi s§§as§
lhraienfl mber
—--— TabuListLengt =16 ----- TabuListLengh =17 — — - stuListhngh =18

 

 

 

Figure 5-2, Tabu Search Improvement Path for Different Tabu List Lengths

 

As can be seen from Figure 5-2, the tabu list length of 17 seems to outperform
either 16 or 18. Table 5-1 shows that for other lengths of the tabu list, the best objective
function value is always higher than when the length is 17.

Table 5-1, Comparison of Tabu List Length and Objective Function Value
[:abu List Length 7 11 13 15 16 17 18 19
[Minimum r 221 225 225 209 203 185 233 215

 

30]
219|

 

 

 

 

 

 

 

 

 

 

 

As a demonstration that the tabu list length choice is important in eliminating the
threat of cycling, Figure 5-3 is offered. Cycling is clearly evident after about 600

iterations.

62

 

 

LObjecive chion Value for Tabu ListLengh = '

 

 

 

.340l
:320
l'
2§3°°-
5;280..
5.9260
09240
J
“220
200
°38%.8'3§§E§§§§SESEESSSE§§§§§§§§
Imsraion Number

 

 

 

Figure 5-3, Improvement Path for Tabu List Length = 7

5. 2. 6. Generation of Moves or New Solutions

The generation of moves in the implementation of tabu search has been
approached in a problem dependent manner in the past [Glover, 1989]. Moves or
potential solutions have been generated by mechanisms based on salient aspects of the
problems to be solved or by utilizing portions of heuristics previously applied to the
problem [Knox, 1987]. As a result, there are no "generic" move generators, so for the
labor scheduling problem in this research, the methods below are used for generating
moves. In the computer program written to implement tabu search for this research, each
of the move generators was contained in a separate subroutine and each of those
subroutines are explained below. i

There are several strategies that guided the development of these move

generators. First, each of the move generators was designed to produce a move without

63

creating constraint violations. The reasons behind this strategy are; 1) There is no good
way to estimate the change in objective function without assurance that the move is
feasible, and 2) If moves that produce constraint violations are allowed, there is no
assurance that a constraint violation will be rectified without a routine with which to fix
that violation. Therefore, move generators were designed so that no constraint violations
were generated along with the move.

The second strategy utilized was to design move generators that gave the
algorithms the greatest opportunity to arrive at a good solution. While adherence to such
a strategy, with a given set of move generators, probably cannot be proven, it is posited
that the set of move generators utilized in this research do, in substantial measure, adhere
to this strategy.

The last strategy utilized for designing move generators for this research was to
use moves that had the same underlying concepts for both tabu search and simulated
annealing. This strategy was rigidly adhered to because of the overriding policy of giving

no advantage to one algorithm over another.

5. 2. 7. Move Generators

Those subroutines that implement the move generators are explained in the
following sections. As each of the move generator subroutines are executed, all possible
potential moves of the respective type are generated. All these potential moves are then

evaluated and one is chosen for implementation.

 

64

Reduce OverStafiing. This subroutine produces potential moves or transitions that
eliminate an hour from one end of a worker's (workerl) shifi. A flow chart of this
subroutine is given in figure 54. First, a list of all taskhours that are overstaffed is
generated. Then, for each overstafl'ed taskhour, a list of workers (workerz) is generated
that meet the following criteria: 1) workerz is scheduled for that taskhour, 2) the
taskhour is either the beginning or end of workerz's shift on that day, and 3) workerz is
scheduled for more than the minimum shift on that day. Finally, this list is combined
with the lists from all other overstaffed taskhours and added to the potential move list for
further processing.

The motivation behind this move is to reduce the cost of the schedule, and
secondarily to increase the availability, and hence flexibility of the worker, so that other

moves might be more easily made.

 

 

 

Start
Subroutine
ReduceOverStaffing

 

 

 

 

 

Get List of All'
OverStaffed Hours

 

 

 

 

 

Select First Hour

 

 

 

l

 

Select First Worker

 

 

 

 

  
 

Is Worker
Scheduled
TaskHour Day?

  
 

Is Worker's
First or Last
TaskHour=Selected
Taskfiour

   
    
  
 
     

 

Add Worker and
TaskHour to
Potential Mbve List

 

 

 

l

 

Select Next Worker,
Until All Workers
Have Been Processed

 

 

 

65

No

 

 

 

l

 

Select Next OverStaffed
___ TaskHour, Uhtil A11
Have Been Processed

 

 

 

Figure 5-4, Reduce Over Staffing

 

 

 

66

Increase Over Stafiing. This subroutine produces potential moves that add an
hour to one end of a worker's shift. A flow chart for this subroutine is given in figure 5-5.
For each day, potential moves are generated by adding an hour to either end of the shift
for workers that are scheduled. The particular task the to which the worker would be
assigned is determined at the time of assignment, based on the taskhour margin, as
discussed in chapter 4. All p0tential moves generated are combined with the potential
move list for additional processing.

This move type is motivated by the idea that adding to a worker’s shifi might
decrease his deviation from targeted hours, and may provide additional opportunities for

the other move generators.

 

Start
Subroutine
IncreaseOverStaffing

 

 

 

 

l
Select First Worker

l

-— Select First Day

 

 

 

 

 

 

 

 

 

  
    
   

  

Is Worker
Scheduled
For Day?

 

Add Hour to
Beginning of
Original Shift

l

Set Worker
And Hour Into
Potential Move List

L

Add Hour to
'End of
Original Shift

i

Set Worker
And Hour Into
Potential Move List

l

Select Next Day,
Until All Days
Have Been Processed

1

Select Next WOrker,
Until All Are Processed

i

End Subroutine

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5-5, Increase Over Scheduling

68

Give An Hour T 0 Another Worker. In this subroutine, potential moves are
generated by moving a taskhour assignment from one worker to another worker. The
flow char for this move is given in figure 5-6. First, the shifts for each worker are
considered one-by-one and for each shift that is longer than the minimum shifi, a second
worker (workerz) is sought for the first or last hour of the shift. A candidate for workerz
will be considered only if the taskhour to be given away is either the hour immediately
prior to workerzs existing shift, or the hour immediately following workerzs existing
shift, and if workerz is available and qualified for the taskhour. All workerzs and
taskhours so identified are added to the potential move list for further processing.

This type of move is meant to simultaneously decrease two worker's deviation
from targeted hours. It may also decrease the cost of the schedule, if the two workers

have different wage rates.

 

69

 

Start
Subroutine
GiveAnHour

 

 

 

 

l
[ASelect First Workerl

 

 

 

Select First Day

 

 

 

 

V
M

 

 

Find All WorkerZS
For Which Workerl's
First or Last TaskHour
Is Just Before or After
Worker2's Shift

 

 

 

 

l

 

Set All Workerz,
TaskHour Pairs Into
the Potential Move List

 

 

 

 

 

Select Next Day,
Until All Days
Are Processed

 

 

 

 

 

 

 

Select Next Workerl,
Until All Workerls
Have Been Processed

 

 

 

 

 

 

 

End Subroutine

 

 

 

Figure 5-6, Give An Hour to Another Worker

70

Give A Day T 0 Another Worker. The subroutine that generates these potential
moves gives away one workers (workerl) shift to another (workerz). The flow chart for
this subroutine is given in figure 5-7. First, the shifts for each worker] are considered
one-by-one and for each shift, a workerz is sought for the entire shift. A candidate for
workerz will be considered for the potential move only if workerz is available and
qualified for the taskhour. All workers and taskhours so identified are added to the
potential move list for further processing.

This type of move is meant to simultaneously decrease two worker’s deviation
from targeted hours. It may also decrease the cost of the schedule, if the two workers

have different wage rates.

71

 

Start
Subroutine
GiveADay

 

 

 

 

l
Select First Workerl

 

 

 

 

I
Select First Day

 

 

 

 

 

 

l

 

Find All WorkerZS,
Available and Qualified,
That Can Take
Workerl's Shift

 

 

 

 

 

Set All Workerz,
Day Pairs Into
the Potential Move List

 

 

 

 

l

 

Select Next Day,
until All Days
Are Processed

 

 

 

 

 

 

 

Select Next Workerl,
Until All Workerls
Have Been Processed

 

 

 

 

 

 

 

End Subroutine

 

 

 

Figure 5-7, Give A Day to Another Worker

72

Trade Shifts With Another Worker. In this subroutine, potential moves are
generated by trading two worker's shifts on the same day. A flow chart for this move is
given in figure 5-8. For each worker and each shift for that worker, all other workers are
tested to see whether they are eligible to trade shifts. Both workers must be scheduled on
the same day, available for the hours of the other worker's shift and be qualified to
perform the task on the other worker's shift. If all these conditions hold, then a potential
move has been identified and is added to the list of potential moves.

Moves of this type are useful for reducing both worker‘s deviation from targeted
hour, and may also reduce the total schedule cost if the wage rates of the two workers are

different.

73

 

Start
Subroutine
TradeTours

i

[Select First Workerl

l

Select First Day
Workerl is Scheduled

 

 

 

 

 

 

 

 

 

 

 

Find All Worker28,
Available and Qualified,
That Can Trade Shifts
With Workerl

 

 

 

l

 

Set All Workerl, Workerz,
Day Triples Into
the Potential Move List

 

 

 

l

 

Select Next Day Scheduled,
Until All Days
Are Processed

 

 

 

 

 

 

 

Select Next Workerl,
Until All Workerls
Have Been Processed

 

 

 

 

 

 

 

End Subroutine

 

 

 

Figure 5-8, Trade Shifts With Another Worker

74

Cross Trade Shifis With Another Worker. In this subroutine, potential moves are
generated by trading two worker's shifts on different days. This move is illustrated with
flow chart 5-9. For each worker] and each shift for that worker], all other workers
(workerz) are tested to see whether they are eligible to trade shifts. Both worker] and
workerz must be scheduled on the different days, available for the hours of the other
worker’s shift and be qualified to perform the task on the other worker's shift. If all these
conditions hold, then a potential move has been identified and is added to the list of
potential moves.

Moves of this type are useful for reducing both worker's deviation from targeted
hour, and may also reduce the total schedule cost if the wage rates of the two workers are

different.

75

 

Start
Subroutine
CrossTradeTours

 

 

 

 

Select First Workerl}

l

Select First Dayl
Workerl is Scheduled

i

Select First Day2

 

 

 

 

 

 

 

 

 

 

Find All WorkerZs, (on Day2)
Available and Qualified,
That Can Trade Shifts
With Workerl (on Dayl)

 

 

 

l

 

Set All Workerl, WOrkerZ,
Day1, Dayz Quadruples Into
the Potential Mbve List

 

 

 

 

Select Next Day2 Scheduled,
Until All Dast
Are Processed

 

 

 

 

 

Select Next Dayl Scheduled,
Until A11 Dayle
Are Processed

 

 

 

 

 

i

 

Select Next Workerl,
until All Workerls
Have Been Processed

 

 

 

 

 

l

End Subroutine

 

 

 

 

Figure 5-9, Cross Trade Shifts With Another Worker

76

Give Away Part Of A Worker's Shift. This subroutine gives away part of a
worker’s (workerl) shift to other workers (workerz) on an hour-by-hour basis. This move
is depicted in figure 5-10. First, the shifts for each worker] are considered one-by-one
and for each shift that is longer than the minimum shift, workerzs are sought for the first
and last hours of the shift. The number of hours to be given away is either the worker's
shift length less the minimum shift or the number of hours off target for the worker,
which ever is less. A candidate for workerz will be considered only if the taskhour to be
given away is either the hour immediately prior to workerz's existing shift, or the hour
immediately following workerz's existing shift, and if workerz's is available and
qualified for the taskhour. Once workerz's have been identified for all taskhours to be
given away, all possible combinations of those workerz's and taskhours are generated for
the appropriate end of the shift of worker]. All workerzs and taskhours combinations so

identified are added to the potential move list for further processing.

77

 

Start
Subroutine
GiveAwayPartShift

 

 

 

 

LSelect First Worker1_fi]

Select First Day
Workerl is Scheduled

 

 

 

 

 

 

l

Select First TaskHour

 

 

 

 

 

 

 

Find All WorkerZs,
Available and Qualified,
That Can Take The TaskHour
From Workerl

 

 

 

 

 

Select Next TaskHour
Until "Number“ Are Processed

L

 

 

 

 

 

Create All Possible, Feasible
Comnbinations of WorkerZ-TaskHours,
Set All Workerl, Workerz,
TaskHours Triples Into
the Potential Move List

 

 

 

l

Select Next Day Scheduled,
Until All Days

 

 

 

Are Processed

 

 

 

 

7
Select Next Workerl,

Until All Workerls
Have Been Processed

l

End Subroutine

 

 

 

 

 

 

 

 

 

 

Figure 5-10, Give Away Part of a Worker's Shift

78

Give Away A Short Shift, Hour By Hour. This subroutine gives away a worker's
(workerl) short shift to other workers (workerz) on an hour-by-hour basis. A flow chart
depiction of this move is given in figure 5-11. First, the shifts for each worker] are
considered one-by-one and for each shift that is equal to or less than the minimum shift
plus one hour, workerzs are sought for the hours of the shift. The workerz will be
considered only if the taskhour to be given away is either the hour immediately prior to
workerz's existing shift, or the hour immediately following workerz's existing shift, and
if workerz's is available and qualified for the taskhour. Once workerz's have been
identified for all taskhours to be given away, all possible combinations of those workerz's
and taskhours are generated for the appropriate end of the shift of worker]. All workerzs
and taskhours combinations so identified are added to the potential move list for further

processing.

79

 

Start
Subroutine
GiveAwayMinShift

t

Select First Worker1

i

Select First Day
Worker1 is Scheduled
For Less Than Minimum

Shift + 1 Hours

 

 

 

 

 

 

 

V

 

 

7

Select First TaskHour

 

 

 

 

 

 

l

Find All WorkerZs,
Available and Qualified,
That Can Take The TaskHour
From.W0rker1

 

 

 

 

 

l

Select Next TaskHour
Uhtil Entire Shift is Processed

L

Create All Possible, Feasible
Comnbinations of WorkerZ-TaskHours,
Set All Worker1, Workerz,
TaskHours Triples Into
the Potential Move List

i

Select Next Day Scheduled For
More than Minimum Shift +1 Hours

Until All Days Are Processed

 

 

 

p—d

 

 

 

 

 

 

 

 

 

 

 

 

 

7

Select Next Worker1,
Until All WOrkers
Have Been Processed

l

End Subroutine

 

 

 

 

 

 

 

 

 

 

Figure 5-11, Give Away A Short Shift, Hour By Hour

80

5. 2. 8. Formulation Of constraints.

Formulation of the problem for solution with tabu search is shown in appendix 2.
Although a complete exposition of the single function for the objective function and
constraints of the problem is given in appendix 2, the implementation of tabu search
utilized in this research does not require the problem to be in single function form. In
fact, the because the constraints are not violated when a potential move is implemented,
they are not required to be checked or calculated with the objective function. The routine
that calculates the objective function does, however, check for constraint violations every

time the objective function is calculated.

5. 2. 9. Objective Function Estimator

A routine to estimate the change in objective function value was developed and
implemented in the tabu search algorithm used in this research. An estimator was used
for several reasons. First, calculating the entire objective function is a time consuming
procedure. Second, computer memory constraints allowed only one copy of the schedule
in memory at a time (calculating the objective function requires a complete schedule), so
calculating the objective function rather than estimating would cause a great deal of disk
access, also very time consuming. Last, the change in objective function can be estimated
very accurately. The estimator that was developed determines the changes to be made to
the schedule, given the move type and information about the workers, and calculates the
changes to the objective function value based on differences in targeted hours, before and

after the move, and differences in overstaffmg, before and after the move. These

81

estimates are then used to rank the potential moves, best first. The estimator also allows

the potential moves to be evaluated without any changes to the master schedule.

5. 3. CONCLUSTON

This chapter has described the implementation of tabu search used in this
investigation. The overall flow of the algorithm was shown and explained as were the
various move generators. The methods used to set parameters was explained and results
of those experiments were shown. A demonstration of the fact that the algorithm can
cycle with incorrect parameter settings for the tabu list length was demonstrated.
Contributions based on the implementation of tabu search include new types of tabu lists,

ranking the potential moves, and estimation of objective function value.

82

CHAPTER 6 - SIMULATED ANNEALING; THEORY AND IMPLEMENTATION

6. I. INHODUCHON

Simulated annealing is a stochastic optimization technique derived from statistical
mechanics. It is used for finding (near) globally minimum cost solutions to wide variety
of large, combinatorial optimization problems. Kirkpatrick et a1 [1983] were the first to
propose and demonstrate the application of simulation techniques from statistical
mechanics to problems of combinatorial optimization. For a complete discussion of

simulated annealing the reader is referred to van Laarhoven and Aarts, [1987].

6.1.]. Statistical Mechanics

An understanding of statistical mechanics is necessary to comprehend the
relationship between the techniques of statistical physics and the solution of large
combinatorial optimization problems. Statistical mechanics is a body of methods for
analyzing aggregate properties of the large numbers of atoms to be found in samples of
liquids or solids, in thermal equilibrium at finite temperatures. Because the number of

atoms is on the order of 1023 per cubic centimeter, only the most probable behavior of

83

the system in thermal equilibrium is observed. This behavior can be characterized by the
average and small fluctuations about the average behavior of the system, when the
average is taken over the set of such systems defined by the configurations of the
systems. Suppose that the configuration (configi) of the system is analogous to the set of
spatial positions of the components (atoms or molecules). If the system is in thermal
equilibrium at a given temperature c, then the probability p(config.i) that the system is in
a given configuration (configi) depends on the energy E(config.i) of the configuration.
This probability follows the Boltzmann distribution: p(config.i) = exp(-E(config.i)/ch),
where E(config.i) is the energy of the configuration, c is the temperature, and k3 is

Boltzmann's constant (kB=l .38 x 10'”).

6.1.2. 77w Simulation of Particles at Iliermal Equilibrium

The behavior of a system of particles or components in thermal equilibrium can
be simulated using a stochastic relaxation technique developed be Metropolis et a1
[1953]. Suppose that at time t, the system is in configuration (configa). A candidate
(configb) for the configuration at time t+1 is generated using a random process. The test
for selecting or rejecting configuration (configb) as the configuration at time t+1 is
based on the difference between the energy levels of configurations (configa) and
(configb). If the energy level of (config.b) < (configa) then (configb) is accepted as the
new configuration. If the energy of (config.b) 2 (config.a) then (configb) is accepted
with probability p = exp(-(E(config.b) - E(config.a))/ch). It must be noted that higher
energy states can be attained (with probability p), but that the general trend of
configuration energy levels will be toward the lower energy states. This trend is also

confirmed by noting that the temperature, c, is decreasing periodically (as described in

84

the definition of annealing, below). This decreases the probability that a configuration

with a higher energy level would be accepted.

6.1.3. The Relationship to Combinatorial Optimization

A primary question in statistical mechanics is the nature of the system at low
temperatures, for example, whether the atoms remain fluid or solidify, and if they
solidify, whether they form a crystalline solid or a glass. Very low energy states
predominate at low temperatures, because of the nature of the Boltzmann distribution. To
achieve low-energy configurations, simply lowering the temperature is not sufficient,
because of the possibility of the production of glasses (localized higher energy states). To
avoid these glasses or higher energy states, an annealing process must be used. In an
annealing process the temperature is raised, and then slowly lowered, spending enough
time at each intermediate temperature level to achieve thermal equilibrium.

The problem of finding the low-temperature state of a system when a formula for
its energy level calculation is given is similar to the optimization of a combinatorial
problem. This similarity comes from the fact that annealing attempts to transform the
entire solid into its lowest energy state, avoiding the formation of glasses in m part of
the solid. Such a solid is, of course, composed of very many molecules, that individually,
must be cooled evenly to avoid the formation of glasses.

For the analogy combinatorial optimization, the variables may be thought of as
molecules. When we simulate annealing we are attempting to reduce the objective
function value (energy level) by manipulating the variables (molecules and temperature).
The objective function value must be lowered evenly and slowly to avoid local optima

(glasses) as described above.

85

6.1.4. Requirements for Simulated Annealing Optimization

Simulated annealing as applied to optimization problems involves several steps:

Identification of the analogs of the physical process in the optimization

process itself:

- the energy function becomes the objective function,
- the configurations of atoms becomes the configuration of parameters,
- finding a low energy configuration becomes finding a near-optimal

solution
- temperature becomes a control parameter for the simulation.

Formation of an annealing schedule comprised of the following parts:

- a set of steadily decreasing temperatures or control parameters designated c
in the description of the algorithm. The 0 parameters control the probability of
a higher temperature configuration being accepted as the new configuration at
each iteration.
- the amount of time (number of iterations) spent at each temperature. The
objective is to spend enough time at each value of c to minimize the formation
of glasses or locally optimal solutions.

The development of a method(s) of generating new configurations.

86

6.1.5. Description of Simulated Annealing Algorithm in Pseudo-Pascal

Procedure Simulated Annealing
Begin
Initialize;
M = 0;
repeat
repeat
Perturb(Config.i -> Config. j , ACij)
if ACij =< 0 then accept else
if exp(-ACij/c)>random[0, 1] then accept;
if accept then Update(configuration j);
until equilibrium is approached sufficiently closely;
cM+1 := f (CM);
M := M+1;
until stop criterion = true;
end.

Where Ci is the objective function value for configuration i and A Cij := Cj - Ci.

6.1.6. F low Chart of Simulated Annealing

A flow chart for the simulated annealing algorithm is given in figure 6-1.

87

 

Procedure Simulated Annealing

Read Initial
Solution

Initialize
Temperature (c)and
Cooling Schedule

l

Generate Potential
Configuration j

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Estimate ACij

 

 

 

 
     
 
   

if exp(-ACij/c)>
random[0,1]

 

Implement
Configuration j,
Set to i

 

 

 

 

   
 
 
     
   

Enough
Searches?

 

Set New Temperature
Based on Cooling Schedule,
Reset Search Counter

 

 

 

Stopping
Criteria
Met?

 

Figure 6-1, Simulated Annealing Flowchart

88

6.1. 7. Example of The Simulated Annealing Algorithm

Consider maximizing the function f(x) = x2, where x is an integer and permitted
to vary from 0 to 31. To maximize the function using simulated annealing, we must first
code the decision variable as a finite length string. For this problem we will code the
variable as an unsigned binary integer string of length 5 (25:32). We also note that the
algorithm is inherently a minimization algorithm. For the purposes of this example we
will define AC3 to be Ci - Cj (rather than Cj-Ci), and we will arbitrarily set c at 2 (c is
the control parameter or temperature in the annealing schedule).

Initialize: We select an initial configuration at random, say 00101. The Ci value
for this configuration is 5.

Perturb: We select a bit to change at random, in this case the bit at position 3.
The Cj value is then 1.

ACij = 4 (5-1). This configuration fails the first if-then test. exp(-ACij/c)= 0.135,
a random U[0,1] pick (re-picked at each generation) gives .729, so we do not update and
loop back to perturb.

Perturb: We select a bit to change at random, bit 2, giving 00111, with Cj = 7.

ACij = -1 so we accept the new configuration.

Following the pseudo-PASCAL description above, the algorithm would continue
the inner loop until equilibrium is reached (determined by some stopping rule), when the
next c value is used and the process would repeat itself until the algorithm loops with the
final value of c. At that point, an optimal or near optimal solution would have been

reached.

89

6. 2. SHl/IULATED ANNEALING MLMNTATTON ISSUES

6. 2. 1. The Annealing Schedule

The tradeoff in developing a cooling schedule is between solution speed and
solution quality. A faster annealing schedule will usually give a poorer solution. Several
types of cooling schedules are classified in van Laarhoven and Aarts [1987]. They
distinguish between two general classes of cooling schedules:

Class A: A variable number of searches, the number of configurations generated
and tried at a given temperature and a fixed decrement of the control parameter, or
temperature, and

Class B: A fixed number of searches and a variable decrement of the control
parameter.

Both class A and class B cooling schedules are theoretically sound, although most
prior research has used class A cooling schedules with good success. Prior research on
scheduling problems (V akharia and Chang [1990] and Brusco and Jacobs [1993]), has
also shown good results with class A cooling schedules. Given the prior research and
some preliminary experimentation, a class A cooling schedule was selected and
implemented for this research. The method for setting the number of searches per value
of the control parameter (iteration) was determined in accordance with theoretical
concerns. Those concerns are: 1) the algorithm must have a finite probability of
accepting all possible configurations, and 2) the probability of accepting a configuration
decreases as the value of the control parameter decreases. One method for increasing the
likelihood of acceptance for new configurations is to increase the number of
configurations tried as the control parameter decreases. A rather elaborate method due to

Romeo [1985] was tried first and then compared to a simple method that multiplies the

90

initial number of searches by the iteration number. The comparison of final objective
fiinction values indicated that the simpler method was preferred. The simpler method was
used in the remainder of this investigation, to multiply the initial number of searches by
the iteration number. After selecting the type of cooling schedule, experimentation was
performed to determine good cooling schedule parameters, as explained in the next

section.

6. 2. 2. Setting the parameter values

The cooling schedule used for this implementation of simulated annealing has
three parameters that require setting; the initial value of the control parameter, the
decrement of the control parameter, and the initial number of searches per iteration.

The initial value of the control parameter, Co, was set using the method of White
[1984]. In this method, a number of configurations are generated and the mean and
standard deviation of those configurations are calculated. The mean was calculated to be
1.38 and the standard deviation, 11.94. White proposes that CO 2 a, so Co was set at 12.
Setting the parameter in this manner allows most configurations to be accepted early in
the execution of the algorithm, which, theoretically, improves the performance of the
algorithm.

The rule for the decrement of the control parameter was first proposed by
Kirkpatrick [1982], and used in most of the applications published since then. This rule
is: ck+1= a ~ ck, k = 1, 2, ,n. a is usually set in the range of .5 to .99. After some
experimentation it was found that at = .8 produced satisfactory results. A more elaborate
method for decrementing the control parameter, c, due to Huang, et al [1986] was tried,

but the results were not as good as the simpler decrement rule.

91

The initial number of searches per iteration was set using experimentation.
Several values for the initial number of searches were tried ranging from 75 to 300.
Analysis of the final objective function values from the range of values indicated that
there was no significant difference among the values. The implication is that this
implementation of simulated annealing is insensitive to the setting of the initial number
of searches per iteration. The initial number of searches per iteration was set at 75 for the

remainder of the research.

6. 2. 3. Stopping Rules

As is shown in the general description of simulated annealing given above,
stopping rules are necessary for termination of the algorithm. Two types of stopping
rules were implemented in this research. The first rule is based on a final value of the
control parameter, c. The final value of the control parameter can be set in a number of
different ways. The method chosen for this research is due to Lundy and Mees [1986].
The calculation yields a final value close to .05, so that was the value used for the final
value of the control parameter. The other stOpping rule used was to stop execution of the
algorithm if there had been no improvement in the objective fimction value after some
number of iterations. The rule developed after some experimentation was to stop the
algorithm when there had been no improvement for two iterations if at least 10 iterations
had been done (this would correspond to at least 1575 configurations tried with no

improvement).

92

6. 2. 4. Perturbation Using Configuration Generators

There are several methods for generating new configurations explained in van
Laarhoven and Aarts [1987]. These methods were deemed unsuitable because there was
no protection from the generation of constraint violations along with the new
configuration. In order to ensure that each new configuration was feasible, and to keep
the comparison between simulated annealing and tabu search as fair as possible, the same
move generating methods used in tabu search were used in generating perturbations, or
new configurations (or moves), for simulated annealing.

The configuration generators used in the implementation of simulated annealing
use the same concepts as those used in tabu search. The primary difference is that the
move generators for tabu search generate all possible moves of the particular type each
time the generator is used, while the move generators used in simulated annealing
generate only one move of the particular type each time the generator is used.
Descriptions of each of the move generators follows.

Reduce Over Stafiing. The subroutine used to produce a transition to reduce
overstaffmg is shown in figure 6-2. This subroutine finds the task hour that is most over
staffed. Next, a pool of workers scheduled to work during that task hour and whose first
or last hour in the shift is the hour most over staffed are identified. From this pool of
workers, the worker most over targeted hours is selected. If that worker is not scheduled
in the over staffed task for that hour, an attempt to swap the workers task for the over
staffed one is made. Ifthe swap is not successful, then the next most over target worker
is selected. When a worker from the pool is found that is scheduled for the over staffed
task hour (whether the worker swapped for the task or not) that worker, and task hour is

returned to be tested for acceptance by the main program.

93

 

Subroutine
ReduceOverStaffing

 

 

 

l

 

Find the TaskHour
That is Most
OverStaffed

 

 

 

 

 

l

 

Find All Workers
Scheduled For Hour
on First or Last Hour

 

 

 

 

 

l

 

Pick Remaining Worker
Most Over Target

 

 

 

 

l

 

If Qualified, and Necessary
Swap With Another Worker
for TaskHour

 

 

 

  
  

 

  

Swap
Successful?

  

 

Return Worker, TaskHour Pair
As Potential Configuration

 

 

 

Figure 6-2, Reduce Over Scheduling

94

Increase Over Stafiing. The flow chart for the transition generator is given in
figure 6-3. The first step is to identify all workers under targeted hours, sort them
according to the number of hours under target and select the worker most under target.
Next that worker's shortest shift is found. Once the shortest shift is known an hour either
just before the shift or just after the shift is picked randomly. The task is set to be the
same as for assignment of the hour immediately before or after the hour picked, as the
case may be. This task hour and worker combination is returned to the main program for

acceptance or rejection according to the metropolis equation.

 

Subroutine
IncreaseOverStaffing

1

Find Worker Most
Under Target

l

Find Worker's
Shortest Shift

l

Pick Hour at
Beginning-1 or End+1
of Shift Randomly

l -

Set Task Same as
Preceding or Following
Hour

l

Return Worker,
TaskHour Pair as
Potential Configuration

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 6-3, Increase Over Scheduling

95

96

Give An Hour To Another Worker Randomly. Figure 64 gives the flowchart of
the subroutine that creates a potential configuration that gives an hour to another worker
randomly. The first step is to pick a day in the scheduling period randomly, from a
uniform distribution. Next the pool of all workers scheduled for the day is created. From
the pool, a worker is picked randomly, and the rest of the pool is searched for all other
workers that can take for the first worker's first or last hour, considering availabilities and
task qualifications. From the pool of all workers that can trade, a second worker is
chosen, based on the shortest shift length. Based on the availability of the second worker,
the task hour is determined, and the first and second worker and the task hour are

returned to the main program for fiirther processing.

97

 

Subroutine
GiveAnHourToAnotherRandomly

 

 

 

 

 

Pick a Day
Randomly

 

 

 

 

 

Find Pool of All Workers
Scheduled on Day

 

 

 

 

 

Pick A Worker1
From Pool Randomly

 

 

 

 

l

Find All Workers in Pool
Available For Workerl's
First or Last Hour in Day

 

 

 

 

 

 

Pick Worker2
Based On Shortest
Shift Length

Based on Availability of

Worker2, Pick Workerl's
First or Last TaskHour

l

Return the Worker1,
Worker2, TaskHour Triple
As Potential Configuration

 

 

 

 

 

 

 

 

 

 

 

Figure 6-4, Give An Hour to Another Randomly

98

Give A Day To Another Worker Randomly. The flowchart of the subroutine that
creates a potential configuration that gives a day to another worker randomly is given in
Figure 6-5. The first step is to pick a day in the scheduling period randomly, from a
uniform distribution. Next the pool of all workers scheduled for the day is created. From
the pool, a worker is picked randomly. A second pool of worker is formed that can trade
for the first worker's shift , considering availabilities and task qualifications. From the
pool of all workers that can trade, a second worker is chosen, based on the most hours
under target. Finally, the first and second worker and the shift are returned to the main

program for further processing.

99

 

Subroutine
GiveADayToAnotherRandomly

l

Pick a Day
Randomly

l

Find Pool of All Workers
Scheduled on Day

 

 

 

 

 

 

 

 

 

 

 

 

 

Pick A Worker1
From Pool Randomly

l

Find All Workers in Pool
Available For Workerl's
Shift on Day

l

Pick Worker2
Based On Most Hours
under Target

 

 

 

 

 

 

 

 

 

 

 

 

 

Return the Worker1,
Worker2, Shift Triple
As Potential Configuration

 

 

 

Figure 6-5, Give A Day To Another Randomly

100

Trade Tours Between Workers Randomly. The subroutine that generates a new
configuration that trades tours between two workers randomly is given in figure 6-6. The
first step is to pick a day in the scheduling period randomly, from a uniform distribution.
Next the pool of all workers scheduled for the day is created. From the pool, a worker is
picked randomly. A second pool of worker is formed that can trade for the first worker's
shift (and that the first worker can trade with), considering availabilities and task
qualifications. From the pool of all workers that can trade, a second worker is chosen,
based on the most hours under target. Finally, the first and second worker and the shift

are returned to the main program for further processing.

101

 

Subroutine
TradeToursRandomly

 

 

 

 

l

Pick a Day
Randomly

 

 

 

 

 

l

Find Pool of All Workers
Scheduled on Day

 

 

 

 

 

l

Pick A Worker1
From Pool Randomly

 

 

 

 

 

 

Find All Workers in Pool
That Can Trade Shifts on Day.
Both Workers Must be Available
and Qualified For the Other Shift

 

 

 

 

7

Pick Worker2
Based On Most Hours
Under Target

 

 

 

 

 

 

Return the Worker1,
Worker2, Shift Triple
As Potential Configuration

 

 

 

Figure 6-6, Trade Tours Randomly

102

Cross Trade Tours Between Workers Randomly. Figure 6-7 gives the flowchart of
the subroutine that creates a potential configuration that cross trades tours randomly. The
first step is to pick a day in the scheduling period randomly, from a uniform distribution.
Next the pool of all workers scheduled for that day is created. From the pool, a worker is
picked randomly. Another pool of workers is created with workers that are under target
and are available and qualified to take the first worker's shift. The worker most under
target is picked as the second worker. All the days the second worker is scheduled to
work are checked to determine whether the first worker is available and qualified to take
the second workers shift on that day. If so, a match has been found and the first worker,
the second worker and the two shifts are retumed to the main program for further

processing. If not, another second worker is chosen and the process repeats.

103

 

Subroutine
CrossTradeToursRandomly

l

Pick a Day
Randomly

 

 

 

 

 

 

 

 

1

Find Pool of All Workers
Scheduled on Day

 

 

 

 

1

Pick A Worker1
From Pool Randomly

 

 

 

 

 

 

1

Pick a (The Next) Worker2
r... Based On Most Hours
Under Target

 

 

 

 

 

 

Loop Through Other Days

To Find Shift To Trade,

Based on Avalibilities
and Qualifications

 

 

 

      
 

Was Trade
Found?

 

 

Return the Worker1,
Worker2, Shift Triple
As Potential Configuration

 

 

 

Figure 6-7, Cross Trade Tours Randomly

104

Give An Hour To Another Worker. The reader may refer to the flowchart shown
figure 6-8 for a depiction of the subroutine that generates a configuration based on the
method of give an hour to another worker. This subroutine first creates a pool of workers
that are over targeted hours. The worker with the most hours over target is picked from
the pool. From the worker's tour, the longest shift is selected. A second pool is created of
workers that are available and qualified for either the first or last hour of the first
worker's longest shift. This second pool's workers are required to have the shift they are
scheduled for on the day in question adjacent to the first worker's shift, so the second
worker might take either the first or last hour of the first worker's shifi, without fill-in
hours. Once a second worker is found, the subroutine returns the first and second workers

and the task hour to the main program for further processing.

105

 

Subroutine
GiveAnHourToAnother

J

Find Pool of All Workers
Over Targeted Hours

l

Pick A Worker1
From Pool Based On Most
Over Targeted Hours

I

Find Workerl's
Longest Shift

J

Find All Workers in Pool
Available For Workerl's
First or Last Hour Of Shift

l

Pick Worker2
Based On Most Under
Target Hours

1

Based on Availability of
Worker2, Pick Workerl's
First or Last TaskHour

l

Return the Worker1,
Worker2, TaskHour Triple
As Potential Configuration

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 6-8, Give An Hour To Another

106

Give A Day To Another Worker. Figure 6-9 illustrates the subroutine that
generates a new configuration based on the method of give a day to another. This
subroutine first creates a pool of workers that are over targeted hours. The worker with
the most hours over target is picked from the pool. Next, the worker's tour is searched for
the shift whose length most closely matches the number of hours over target. Then a pool
of workers, available and qualified for the first worker's shift, is created. A worker with
the most hours under target is picked from this second pool. The first worker, the second

worker and the shift are returned to the main program for fiirther processing.

107

 

Subroutine
GiveADayToAnother

 

 

 

 

 

Find Pool of All Workers
Over Targeted Hours

 

 

 

1

Pick A Worker1
From Pool Based on Most
Hours Over Target

 

 

 

 

 

 

l

 

Pick The Shift Whose Length
Is Closest To The Number Of
Hours Over Target

 

 

 

 

 

Find All Workers
Available For Workerl's
Shift on Day

 

 

 

 

 

Pick Worker2
Based On Most Hours
Under Target

 

 

 

 

1

Return the Worker1,
Worker2, Shift Triple
As Potential Configuration

 

 

 

 

Figure 6-9, Give A Day To Another

108

Trade Tours Between Workers. Figure 6—10 illustrates the subroutine that
generates a new configuration based on the method of trade tours. This subroutine first
creates a pool of workers that are over targeted hours. The worker with the most hours
over target is picked from the pool. Next, the worker's tour is searched for the shift
whose length most closely matches the number of hours over target. Then a pool of
workers, available and qualified for the first worker's shift, is created. A worker with the
most hours under target is picked from this second pool. The second worker's shift is
tested to determine whether the first worker is available and qualified to take it. If so, a
match has been found and the first worker, the second worker and the shift are returned
to the main program for further processing. If not, another second worker is picked and

the process repeats.

109

 

Subroutine
TradeTours

 

 

 

 

 

Find Pool of All Workers
Over Targeted Hours

 

 

 

 

 

Pick A Worker1
From Pool Based On Most
Hours Over Target

 

 

 

 

 

Pick the Shift Whose Length
Most Closely Matches The
Number Of Hours Over Target

 

 

 

 

 

Find All Workers
That Can Trade Shifts on Day.
Both Workers Must be Available
and Qualified For the Other Shift

 

 

 

 

1

Pick Worker2
Based On Most Hours
Under Target

 

 

 

 

 

7

Return the Worker1,
Worker2, Shift Triple
As Potential Configuration

 

 

 

 

Figure 6-10, Trade Tours Between Workers

110

Cross Trade Tours Between Workers. The flowchart in figure 6-11 shows the
method for generating configurations by cross trading tours. This subroutine first creates
a pool of workers that are over targeted hours. The worker with the most hours over
target is picked from the pool. Next, the worker's tour is searched for the shift whose
length most closely matches the number of hours over target. Then a pool of workers,
available and qualified for the first worker's shift, is created. A worker with the most
hours under target is picked from this second pool. All the days the second worker is
scheduled to work are checked to determine whether the first worker is available and
qualified to take the second workers shift on that day. If so, a match has been found and
the first worker, the second worker and the two shifts (on different days) are returned to
the main program for further processing. If not, another second worker is chosen and the

process repeats .

111

 

Subroutine
CrossTradeTours

 

 

 

 

 

Find Pool of All Workers
Over Targeted Hours

 

 

 

 

 

Pick A Worker1
From Pool Based On The
Number Of Hours Over Target

 

 

 

l

Pick the Day Whose Shift Length
Most Closely Matches the Number —r__
Of Hours Over Target

 

 

 

 

 

 

 

Pick The Next Worker2
_. Based On Most Hours
Under Target

 

 

 

 

l

 

Loop Through Other Days

To Find Shift To Trade,

Based on Availabilities
and Qualifications

 

 

 

      
 

NO

Was Trade
Found?

 

 

 

 

Return the Worker1,
Worker2, Shift Triple
As Potential Configuration

 

 

 

Figure 6-11, Cross Trade Tours Between Workers

112

Give Away Part Of Shift T o Other Workers. The subroutine used to produce a
transition to give away part of a worker's shift is shown in figure 6-2. This subroutine
first finds the task hour that is most over staffed. Next, a pool of workers scheduled to
work during that task hour is created and the worker that is most over targeted hours is
chosen as the first worker. The task hour chosen in the first step lies somewhere in the
first worker's shift. The portion and length of the shift to be given away are determined
by the location of the task hour in the shift and the number of hours the worker is over
target. Once the portion of the shift to be given away has been identified, a second
worker, available and qualified, and most under targeted hours is found. The first worker,
second worker, and portion of the shifi are returned to the main program for further

processing.

113

 

Subroutine
GiveAwayPartShift

 

 

 

 

 

Find Most Overstaffed
TaskHour

 

 

 

 

 

 

1

Find Most Over Target
Worker(1) That Is Scheduled
For TaskHour

 

 

 

 

 

 

 

 

Determine Which Part
Of Shift Is To Be Given
Away Based on Number Of
Workerl's OverTarget Hours

 

 

 

 

 

Find Worker(2) Most
UnderTarget, Available and
Qualified to Take Part Shift

 

 

 

 

 

 

 

Return Worker1, Worker2,
PartShift Triple As
Potential Configuration

 

 

 

Figure 6-12, Give Away Part Of Shift To Other Workers

114

Give Away A Short Shift T o Other Workers. An illustration of the subroutine to
give away a short shift is given in figure 6-13. This subroutine gives away a worker's
shortest shift (either minimum hours or minimum hours + 1), to other workers, one hour
at a time. First the worker with the most, over targeted hours is identified. Next that
workers shortest shift is identified. Other workers are identified that can take the hours,
one hour at a time. Then lists of the hours, and workers are returned to the main program

for further processing.

115

 

Subroutine
GiveAwayMinShift

 

 

 

 

Find Worker(1)
Most Over
Targeted Hours

 

 

 

 

 

 

I

 

Find Workerl's
Shortest Shift

1

Select First or Next
Hour Of
Workerl's Shift

1

 

 

 

 

 

 

 

 

 

Find Worker2 To
Take Hour, Must Be
Adjacent to First or Last
Hour Already Scheduled

 

 

 

 

 

  

 

 
 

Are All Hours
Given Away?

  

 

Return Worker1,Worker2,
Shift Data as Potential
Configuration

 

 

 

Figure 6-13, Give Away A Short Shift

116

6.2.5. Inclusion of Both Improvement And Random Configuration Generators

The type of configuration generators described in the simulated annealing
literature have been what might be termed random configuration generators. These
random configuration generators produce a perturbation in a random manner.
Configuration generators of this type have been used for several reasons, the first of
which is that annealing, in a physical sense, is a random process driven by the brownian
motion of the molecules in the solid. Additionally, the problems to which simulated
annealing has been applied lend themselves to random perturbations. As the
configurations generators for this problem were being developed, it became evident that
both random and "improvement" configuration generators could be developed. An
improvement configuration generator produces potential configurations that are chosen to
improve the objective function (solution), as apposed to the random configuration
generators that may or may not improve the objective function. Theoretically, there was
no reason why improvement configuration generators should not be utilized. Practically,
it was not known, however, whether immovement configuration generators would be
useful. As the reader will have noted from the descriptions of the configuration
generators, both random and improvement configuration generators were used in this
implementation of simulated annealing. Further, the reader will have noted that several
of the configuration generators had both a good and a random version. Statistical
justification for use of both types follows.

Separate runs on 15 test problems were made for the simulated annealing
algorithm with the set of both random and improvement configuration generators
(designated SA in tables 6-1 and 6-2), the set of configuration generators without the
random versions (SANR), and the set of configuration generators without the
improvement versions (SARO). An ANOVA was calculated as the statistical test for the

hypothesis that all sets of configuration generators perform similarly, vs. the alternative

 

117

that there are significant differences among the sets of configuration generators. Results

of the ANOVA are given in table 6-1.

Table 6-1. AN OVA for Objective Function

 

ANOVA for Objective Function Value
VAR$: SA, SANR, SARO
DEP VAR: Objective Function Value
MULTIPLE R: 0.759
SQUARED MULTIPLE R: 0.577
ANALYSIS OF VARIANCE
SOURCE SUM-OF-SQUARES DF MEAN-SQUARE F-RATIO P

VAR$ 1922542.178 2 961271.089 28.596 0.000
ERROR 1411841.467 42 33615.273

 

 

 

118

Table 6-2, TUKEY HSD Multiple Comparison Test For Objective Function Value

 

TUKEY HSD Multiple Comparison Test For
Objective Function Value

VAR$
SA, SANR, SARO

USING LEAST SQUARES MEANS.

USING MODEL MSE OF 33615.273 WITH 42. DF.
MATRIX OF PAIRWISE MEAN DIFFERENCES:

SA SANR SARO
SA 0.000
SANR 479.000 0.000
SARO 381.533 -97.467 0.000

TUKEY HSD MULTIPLE COMPARISONS.
MATRIX 0F PAIRWISE COMPARISON PROBABILITIES:

SA SANR SARO
SA 1.000
SANR 0.000 1.000
SARO 0.000 0.322 1.000

 

If we take the objective function value as our measure of solution " goodness", we
find that there are significant differences between the three sets of configuration
generators. So we reject the null hypothesis that there are no significant differences
among the three sets of configuration generators. Table 6-2 shows the Tukey HSD
multiple comparison test for differences between the means of objective function values
for the three different sets of configuration generators. The ANOVA results in table 6-1
show that there are significant differences among means of the three sets of configuration
generators at p < .001. The Tukey HSD multiple comparison test shown in table 6—2

indicates that there are significant differences between SA and SANR and SARO but not

 

119

between SANR and SARO. Further, the mean for SA is smaller than both SANR's and
SARO's. The implication is that the combination of both random and improvement
configuration generators is better than either alone, at least for this implementation of
simulated annealing. Full justification for the use of the Tukey HSD multiple comparison

test is given in chapter 8.

6.2. 6. Probabilities used for selection of configuration generators

As is mentioned before, there are 12 different configuration generators. With 12
generators, a question naturally arises: "Would the algorithm perform better if some
configuration generators were used more frequently than others, or would choice of
configuration generator by equal probability random choice be better? ". To answer this
question some experimentation was done. Several schemes were tried, equal
probabilities, adaptive probabilities based on percentage of configurations accepted,
unequal probabilities produced randomly, and probabilities based on expectations of
what configuration generators would work best at the beginning and end of the run. The
results of the experimentation indicated the probabilities based on expectations of what
configurations would work best at the beginning of the run and what configuration
generators would work best at the end of the run produced better solutions, in terms of
objective function values. All configuration generators had positive probabilities at every
stage of the optimization procedure. Those configuration generators that were thought to
perform better at the beginning of the run had relatively higher probabilities of being
used at the beginning of the run, with decreasing probabilities each iteration. Those

configuration generators that were thought to perform better at the end of the run had

120

relatively lower probabilities of being used at the beginning of the run, with increasing

probabilities each iteration.

6.2. 7. Improvement Path for Simulated Annealing

The improvement path of the objective function value is shown in figure 6-14.
This path has marked similarities to those shown in other investigations, see Knox

[1991], and Brusco and Jacobs [1993].

 

 

’ Simulated Annealing Improvement Pathl

 

 

 

g 1400..
33 12001.
g 1000..
us 800..
0—
3; 600..
‘g 400..
25" 200$
0 0 fi%4r4r%:+e seifil t—tk %:L+:: +f
v-I-v-r-v-PPPPPPFv-PFPPPV-x-Fv-v-FFFP
IDOIOOIDOIDOtOOIDOIDOIDOIDOIOOIOOIDOIOO
FMVONODONMIDCDQODV-avt-DNOOFCOXONO’
Fv-Fv-v-v-r-N NNNNMVOCO c0006)
SearchNumber

 

 

 

Figure 6-14, Simulated Annealing Improvement Path

6. 3. CONCLUSION

The theory and implementation details of simulated annealing have been given.

Among the contributions of this investigation are the introduction of "improvement"

121

configuration generators, and the use of weights to determine how often a configuration

generator is used at a particular stage of the algorithm.

122

CHAPTER 7 - RESEARCH METHODOLOGY

7.1. ImoovcnoN

The objectives of this research were to adapt and implement two new algorithms,
simulated annealing and tabu search, and to compare the performance of these two
algorithms with an existing heuristic, Loucks', for solving large scale labor scheduling
problems. The comparison was to be made on three areas of performance; speed to
solution (measured in seconds), manpower cost of final schedule (measured in dollars),
and quality of solution, measured by the number of hours overstaffed, by the sum of
squared deviations from workers targeted hours and by the Objective firnction value (a
linear combination of the number Of hours overstaffed and the sum of squared deviations
from workers targeted hours).

The algorithms were tested across a spectrum of problems varying in difficulty as
measured by three factors; Average Worker Utilization, Average Percentage of Number
Of Tasks Known, and a measure of peak-tO-trough taskhour demand variation,
designated Max—Min Ratio.

Average worker utilization is the ratio, averaged across all workers, of the
number of hours scheduled to number of hours targeted, expressed as a percentage. This

factor is intended to measure the flexibility that a scheduler would have in assigning

123

workers to particular task hours. As average worker utilization increases, the problem
becomes more difficult to solve, because there are fewer workers available to schedule
for each task hour.

The measure, average percentage of number of tasks known, is the ratio, averaged
across all workers, of the number of tasks known by a worker to the total number of
tasks, expressed as a percentage. This factor also is intended to measure the flexibility
that a scheduler would have in assigning a worker to a particular taskhour. As the
measure, average percentage of tasks known by worker, is increased the scheduler has
more flexibility to schedule workers, making the problem easier to solve.

The min-max ratio is the maximum demand in task hours for the day (summed
across all tasks) divided by the minimum demand for task hours (summed across all
tasks) within a four hour period. This factor was included in order to capture the
difficulty a scheduler might encounter in minimizing the total number of task hours
scheduled in the presence of peakedness in customer demand for task hours and a
constraint on the minimum number of hours scheduled in a shift for a worker (in this
case, 3). A higher level of this ratio indicates a more difficult problem.

This investigation was conducted in three phases. First a set of hypotheses were
developed, based on the Obj ectives of the research as stated above. Second, an
experimental design was chosen to accomplish the objectives of the research. And third,
the new methods as well as Loucks' heuristic were used to solve the labor tour scheduling
problems associated with the worker flexibility and labor requirements characteristics

specified in this research.

124

7. 2. CHOICE OF ALGORITWFOR COMPARISON

The heuristic solution method of Loucks [1987] was chosen for comparison for
several reasons; the method has found optimal solutions in research settings [Loucks
1987], software is available for comparisons between algorithms, and the underlying
formulation used in the sofiware is similar enough to the one chosen for the research

problem that it can be used for comparative purposes.

7. 3. HYPOTHES’ES'

As stated earlier, an Objective of the research was to compare the three
algorithms, Loucks' heuristic, tabu search, and simulated annealing. TO accomplish that
objective several hypotheses were developed relating to the speed, cost and quality of
solution. In the following discussion Of the hypotheses, the subscript attached to the
dependent variable of interest refers to the solution method. For example, Time] refers to
the Time for Loucks' heuristic, Timez refers to the time for simulated annealing, and

Time; refers to the Time for tabu search.

7. 3. 1. Time - Speed Of Algorithm

Ho : Timel = Timez = Time3
H1 :Time1¢ Timez ¢ Time3
Speculation at the outset of this research was that tabu search and simulated

annealing will outperform the Loucks' method on this measure [Knox, 1989]. Further,

125

related evidence [Knox, 1989, Skorin-Kapov, 1990], shows that tabu search outperforms
simulated annealing in solution time for small problems. However, since the problems to
be used in this research are large, the a priori expectation was that simulated annealing

will outperform tabu search in Total CPU Time. Accordingly, the alternative hypothesis

states that solution speeds are unequal.

7. 3.2. Cost - Total Manpower Cost of the Schedule.

H0 : Costl = Costz = Cost3
H1 : Cost] at Costz ¢ Cost3
There was no evidence at the outset of this research to suggest that there is a cost

difference between schedules developed by the different algorithms.

7. 3. 3. Quality Of Solution

TOS - Total man-hours of overstaffmg.

H0 2 TOS] = T082 = T083

H1 : TOS] #3 T082 at T033

There is no evidence to suggest any a priori ranking on this measure. This does
not diminish its importance, however, because as overstaff'mg increases so do costs. An

algorithm with a higher TOS will produce higher cost schedules.

126

7. 3. 4. T SD - Sum of squared deviations between scheduled and targeted work hours.

H0 : TSDl = TSD2 = TSD3

H1 : TSD] at TSD2 at TSD3

The measure TSD is an important indicator of a worker's perception of schedule
quality. There is no evidence to suggest any rankings, however. Since simulated
annealing can be shown to converge to an optimal solution, given sufficient time [van

Laarhoven and Aarts, 1987], it may outperform the other algorithms on this performance

measure.

7. 3.5. OBJ - Objective Function Value

H0 : OBJ] = OBJZ = 03.13

H1 : 03.11 at 0312 at 0BJ3

The measure OBJ is a linear combination of TOS and TSD. This measure is
important because it is a measure of the overall level of quality of the schedule produced
by the algorithm, from the perspectives of both management and labor. Again there is no

i

evidence to suggest a priori rankings.

127

7. 3. 6. Interactions among variables

Because there were several factors intended to measure problem difficulty, and
because the set of problems reflected those factors, it was deemed necessary that the data
be tested for the presence of significant interactions. At the time of the research, there

was no prior evidence to suggest which interactions might be significant.

7. 4. SELECTION OF EHERMNTAL DESIGN

7. 4. 1. Factor Levels

As discussed above, the problem difficulty factors are Average Worker
Utilization (AWU), Average Percentage of Tasks Known (PCTNTN) and the Max-Min
Ratio (MMR). Eighteen managers of firms in the service industry were interviewed to
determine the number of levels and the value of levels to be used in the experiment and
to substantiate assumptions basic to this research. These firms were chosen to be
representative of the different types of food service and retail establishments in the
locality. Based on the interviews, three levels were chosen for each factor, with the
values of the levels based on the average of the responses. The factor "Algorithm" has
three levels, Loucks, simulated annealing and tabu search. Those values are shown in

table 7-1.

128

Table 7-1 - Factors and Factor Levels

Factor 2
Average . > .3 and S .5

verage . >35 and ..<. .625

OfTasksKnown
>23nd$5

 

7. 4. 2. Choice of Design

Because of the interest in investigating the interactions among factors and because
there were no prior expectations regarding interactions, other than that some would be
significant, a full factorial design was selected, in order to investigate all possible
interactions. Given three levels for each of the factors, the design is 3x3x3x3, for a total

of 81 treatments.

7. 4. 3. Sample size of experiment

As is widely known, sample size determination is a tradeoff between power of the
test, that is the probability of a type H error, and the amount of resources necessary to
carry out the experiment. For this investigation, the probability of a type two error was
desired to be no more than .05.

A number of methods exist for determining sample size for an experiment, some
quite general or "guesstimates", some quite analytical, see Kirk [1982], Larsen and Marx

[1981] and others. Montgomery [1984] outlines several analytical methods for

129

determining sample size using Operating characteristic curves. The method of

determining sample size chosen for this research relies on estimates of treatment means

and overall variance.

7. 4. 4. Test data

Because no estimates of means or variances were available from past experiment
or experience, problems generated for use in the experiment were used to estimate the
means and variances required for the calculation of <1), a variable used in the analytical
method for determining sample size using operating characteristic curves. Four problem
types (a total of 12 treatments) were chosen at random to provide estimates of the
variance and treatment means. Each of the algorithms was run against 15 problems of
each type. Means and variances were calculated for each treatment. Further, treatment
effects were calculated for each treatment by subtracting from the treatment mean the
grand mean. Using the results of these calculations values Of (I) were calculated for each
of the five performance measures.

For the purposes of this analysis, the variable chosen was TSD - Sum of squared
deviations between workers scheduled and targeted work hours. This variable was chosen
because, based on the sample data used, the value of <1) had the largest magnitude of all
the performance measures, except for the performance measure time-to-solution (or
time). Since the performance measures associated with solution quality were deemed
more important that of time-tO-solution, TSD was used for the subsequent analysis. The
use of the performance measure with the largest value of (1) provides the most

conservative results from the analytical procedure, i. e. the largest sample size.

130

7. 4. 5. Analysis

After analysis, the data calculated and given in table 6-2 indicated that a sample
size of 6 would be adequate to ensure that the probability of committing a type 11 error

would be less than .05.

Table 7-2 Results of the analytic procedure for determination of sample size.

 

 

 

 

 

 

 

n <I>"2 (I) a(n-1) B Power(1-B)
4 2.09882 1.44873 36 0.10 .90
5 2.623 525 1.61973 48 0.06 .94
6 3.14823 1.774325 60 0.03 .97

 

 

 

 

 

7. 4. 6. Choice Of sample size

Although 12 treatments out of 81 represents a relatively high percentage of total
treatments for determination of sample size, some uncertainty remained about how
representative were the observed values of both the treatment means and the variances.
Because of that uncertainty, and because of additional uncertainty regarding the ability of
the algorithms to come to solution on the various problem types, it was decided to
increase the sample size to 15. In this investigation, increasing the sample size merely
increases the amount of personal computer time spent on solutions, so increasing the
sample size was relatively cost free. It was felt that the tradeoff of increasing the total run
time was well worth the effort since a larger sample size would increase the power of the
experiment.

The total number of computer runs was set at, sample size * number of cells, or

15 * 81 =1215.

131

7. 4. 7. Replications for Simulated Annealing

As discussed in chapter 6, simulated annealing is a randomized search process,
that requires the use of a probability distribution associated with the "transition" and
therefore produces a different solution each time the algorithm stops (if it is stopped
before reaching optimality). Since Optimality for the simulated annealing algorithm can
only be guaranteed with an infinite number of transitions, the implementation used in this
research cannot be guaranteed to find the globally optimal solution. Since global
optimality is not assured, and since the algorithm produces a different solution, with a
different Obj ective function value, for each run on the same problem, some method to
characterize the performance of the algorithm was sought. Both Loucks' heuristic and
tabu search are deterministic, that is, the solution generated is the best one for each
problem solved, so they do not require repetition.

There are two ways to evaluate simulated annealing's performance - with no
repetitions and with repetitions. Using one run, the first solution and associated
performance measurement values, is an attractive suggestion. This has the benefit of
reduced computer time for the research, but leaves the question of "Is this the best
solution that this algorithm will generate?" open. It also rejects the probability that a
manager would run the algorithm at least twice to find the better solution. After some
discussion, the decision was made to run two or more replications of simulated annealing
on each problem for two reasons. First, a single run would not capture indications Of
simulated annealing's best performance, and second, it is likely that in a firm the
scheduler would run the algorithm at least twice, picking the best solution.

The number of replications for simulated annealing for each problem was set at 5,
based on two considerations. First, the total time required for 5 replications of simulated

annealing is just less than or approximately equal to the time that the tabu search

132

algorithm takes for completion, that is, five is the largest integer multiple of simulated
annealing's run time that does not exceed tabu search's run time, on average. Second, the
analytical procedure for determining sample size indicated that for simulated annealing,
five replications would give a power of about .95, which was deemed sufficient for this
research.

The data used for the statistical analysis for simulated annealing is the best
solution of the five replications. The best of the five was chosen on the basis of the
lowest objective function value. Ties were broken based on the fastest run time. All the
associated values of performance measures of the best solution were used in the statistical
analysis.

Some might argue that the average of the five solutions be used in the subsequent
analysis, rather than the best solution. This argument was considered and rejected
because; 1) scheduling managers were very unlikely to run the algorithm five times and
then choose the average solution, and 2) while it is possible to average performance
indicators, it is impractical to average two (or more) schedules together. One of the
schedules, of the five, will have to be implemented. The obvious choice is the schedule
corresponding to the best objective function value. So the idea of an average schedule

represented by averages of performances is unworkable.

133

7. 5. GENERATION 0F TES‘TPROBLEMS‘

The generation of test problems followed the procedure of Loucks. For a full
discussion of his procedure, see Loucks [1987], but in general terms, his procedure is to
generate a work tour for each worker, to generate a set of available hours for each worker
that are inclusive of the work tour, to sum across workers to find the task hours required
for each hour of each day, and to use those hour by hour, task by task sums to construct a
scheduling problem. As you will note, since the schedule is constructed from an actual
schedule, Loucks' procedure provides a problem for which we know the optimal answer.
It was felt, however, that Loucks' procedure should be modified for the following
reasons.

First, actual demand patterns for workers, in terms of task hours, probably would
not have the implicit characteristic of a minimum shift of 3 hours. When the Loucks
procedure creates a scheduling problem, it sums the workers scheduled by task hour, with
the restriction that all workers shifts will be at least the minimum shift (3 hours). This
abnormally restricts the range of the Max-Min Ratio (MMR), or alternatively, reduces
the peakedness of the demand pattern. A review of demand patterns published in the
literature indicates that frequently the MMR will exceed 10. Measurement of the MMR
from the unmodified Loucks procedure revealed a maximum of seven. After the
restriction on minimum shift was lifled (by setting the minimum shifl = one hour) the
modified procedure showed MMR values up to 12. Unfortunately, the modification
causes the procedure to produce problems for which the optimal solution is not known.

Last, modification of Loucks' procedure, by removing the minimum shift
constraint, would also eliminate any favorable bias toward one algorithm or another
because no algorithm would possess prior knowledge about the characteristics of the

problem to be solved.

134

As specified, Loucks' procedure produces problems in 6 of the 27 problem types.
To generate the other 21 problem types, the cumulative probability distributions for
number Of hours scheduled, number of tasks known and window of availability were
adjusted, simultaneously and singly. Adjusting the cumulative probability distribution of
the number of hours scheduled to increase the average of the number of hours scheduled
produces problems with increased average worker utilization. Adjusting the cumulative
probability distribution of the number of tasks known to decrease the mean of the
number of tasks known produces problems with a lower average number of tasks known.
Adjusting the cumulative probability distribution of the window of availability to reduce
the average size of the availability window produces problems that have higher average
worker utilization. These adjustments were performed to produce the problems for all 27
problem types.

This use of adjusting cumulative probability distributions as a method of
producing problems, coupled with the relaxation of the minimum shift constraint(as
explained above), causes the problem generator to produce a significant proportion of
infeasible problems. Depending on the problem type, as many as 60 problems were
generated to find 15 that were feasible. The problems were tested for feasibility by using
the program used to generate initial feasible solutions for the simulated annealing and

tabu search algorithms as described in chapter 4.

135

7. 6. DATA GATHERING

Once the problems were generated, running the algorithm on the problems
commenced. After initial calculation of approximate run times for the algorithms and in
view of the total number of problems, additional computing power was arranged for this
research. A total of 17 computers were used to collect the data, at three locations. All but
five of these computers ran at different speeds. Several test problems were run on all
computers to establish the factors by which to adjust the run time of the algorithm on
each problem by computer on which it was run.

Loucks' heuristic would not run to completion on 20 of the feasible problems.
Upon examination of Loucks' computer code, it was found that the method by which the
workers were assigned to hours in Phase 1 of his algorithm, caused unresolvable
constraint violations, which in turn caused the algorithm to terminate prematurely. The
specific cause of the failures is thought to be related to the emphasis Loucks' algorithm
places on minimizing unnecessary over scheduling of workers during the early stages of
the algorithm. This caused a reduction of sample size in some cells to 13. To ensure a
balanced design, data points were eliminated randomly from cells to equalize the sample
size at 13 for all cells. As discussed previously above, the original sample size was set at
15 so that adequate power could be preserved in such a case as this.

The total time for computing on problems spanned 1014 hours of (adjusted to

33MHz, 386 standard time) computer time over the period of about 10 days:

136

CHAPTER 8 - RESULTS OF EXPERIMENTATION

8. I. INTRODUCTTON

Before discussing the results of experimentation based on ANOVA, tests of the
underlying assumptions of analysis of variance that were carried out are explained, so
that there might be confidence in the results and the ensuing discussion. After the validity
of using analysis of variance is established, results of significance tests will be presented
and discussed. The level of significance used throughout the remainder of this discussion
of statistical tests is a = .05. All statistical tests were carried out using SYSTAT for

Windows.

8. 2. TFSTSFOR ASSUMPTIONS OFANO VA

To have confidence in the results obtained from an analysis of variance,
verification that the data conform to the assumptions underlying the analysis of variance
is necessary. The three assumptions of analysis of variance to be tested are; 1) that the

error residuals are normally distributed, 2) that the error residuals are independently

137

distributed, and that all treatments have constant variance or that all treatments display
homogeneity of variance.

First, normality of error residuals was tested by using normal probability plots of
the error residuals. Second, first order autocorrelations are given to test for independence
of the distribution of the residuals, in tabular form. Last, homogeneity of variance was
tested by plots of residuals against estimates and by Bartlett's test for Homogeneity of

Variance.

8. 2. 1. Normalin of error residuals

Montgomery [1984] suggests that a useful procedure for checking the normality
assumption is to construct a normal probability plot of the residuals. If the plot resembles
a straight line, then the underlying error distribution is normal. Figures 8-1 through 8-5
show normal probability plots of residuals corresponding to each of the performance

measures.

138

8. 2. 2. Normal Probability Plots of Residuals

 

 

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2-

EXPECTED VALUE
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-200 -100 O 100 200 300

RESIDUAL

 

 

 

Figure 8-1, Normal probability plot of TOS residuals

 

 

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Figure 8-2, Normal probability plot of TSD residuals

139

 

 

 

 

 

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Figure 8-3, Normal probability plot of OBJ residuals

 

 

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-3 t o’ a

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_ r .4 r l 1 r
-4000-3000-2000-1000 0 1000 2000 3000

RESIDUAL

 

 

 

Figure 8-4, Normal probability plot of COST residuals

140

 

 

 

 

 

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RESIDUAL

 

 

 

Figure 8-5, Normal probability plot of TIME residuals

As can be seen from figures 8-1 and 8-4, the residuals plot seems to approximate
a straight line. However, figures 8-2, 8-3 and 8-5 indicate that the residuals may not be
normally distributed since the plots differ appreciably from a straight line. The non-
norrnal distributions for TSD, OBJ, and TIME may indicate that a transformation is

desirable.

8. 2. 3. Independent distribution of error residuals

The independent distribution of error residuals can be assessed by measuring the
autocorrelation of the data. This is done automatically by SYSTAT. The values of

autocorrelation are listed in the following table. There seems to be no significant

141

autocorrelation, so it may be concluded that the error residuals were independently

distributed.

Table 8-1, Autocorrelation Values For Dependent Variables

 

I TOS TSD OBJ | COST | TIME

 

 

 

 

I Autocorrelation -0.057 -0.002 -0.03O | -0.068 I 0.041

 

8. 2. 4. Homogeneity of variance

The plots of residuals vs. estimate for all dependent variables are shown in figures
8-6 through 8-10. Homogeneity of variance is indicated by a random dispersal of

residuals around a mean value of zero. There should be no apparent patterns in the data.

 

 

 

 

 

 

300 T T
200 r- . A
.00 . l

a

a.

a o L .
-1oo - -
-200 1 .

0 100 200 300

ESTIMATE

 

 

 

Figure 8-6, Plot of Residual vs. Estimate - TOS

142

 

 

 

 

 

 

 

300

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150 200 250
ESTRAATE

100

50

 

 

Figure 8-7, Plot of Residual vs. Estimate - TSD

 

 

 

 

 

 

 

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200 300 400

ESTWAATE

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Figure 8-8, Plot of Residual vs. Estimate - OBJ

143

 

 

 

 

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I
2000 3000 4000 5000 6000 7000 8000

 

 

3000

”Pg No.0.
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m m o m m m
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Figure 8-9, Plot of Residual vs. Estimate - COST

 

 

 

 

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10000

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6000

ESTIMATE

 

 

Figure 8-10, Plot of Residual vs. Estimate - TIME

144

An examination of the figures 8-6 through 8-10 reveals that the plots of residual
versus estimate for all dependent variables exhibit a funnel shaped distribution. The
presence of firnnel shaped distributions is indication that a transformation of the data is

desirable.

8. 2. 5. Bartlett's Test

Bartlett's test for homogeneity of variance tests the hypothesis that the variances
of all treatments are equal for each performance measure. This hypothesis is rejected if
the test statistic exceeds the appropriate chi-square value.

The test statistics for each of the dependent variables is shown in Table 8-2.

Table 8-2, Bartlett's Test For Homogeneity of Variance

 

 

 

 

[ |TOS TSD | OBJ | cosr TIME |
LChi-Square, soar | 60.39 60.39 | 60.39 | 60.39 60.39 |
lTest Statistic, 80df |600.381 1334.183 [445.353 |242.852 3650.298 |

 

 

As can be seen from the table, each of the test statistic values exceeds the chi-
square value, so we reject the hypothesis that all variances are equal for each dependent

variable.

8. 3. B OX AND C OX ANALYTICAL WHOD FOR DETWNING TRANSFORMATTON

145

The evidence presented indicates that the data violate the assumptions of
normality of residuals and homogeneity of variance. The usual approach for dealing with
non-homogenous variance is to apply a variance stabilizing transformation and then to
run the analysis of variance on the transformed data. In this approach, any conclusions
postulated apply only to the transformed populations.

A substantial amount of research has been devoted to the subject of the selection
of an appropriate transformation. Since there was no theoretical distribution for the
observations, there were two options for selection of the appropriate transformation:
empirical selection of a transformation or analytical selection of a transformation. A
researcher using the empirical selection method would try the various transformations
mentioned in the literature, one at a time, looking for a transformation that modified the
data in such a manner so as to remove heterogeneity of variance and non-normality of
error residuals (see Kirk, [1982] for example).

The use of the analytical method for selection of a transformation performs a
series of predefined transforms on the data, measuring the error sum of squares for each.
The intent is to generate a set of points from which the minimum error sum of squares
can be estimated. The selection of the transform is based on the minimum error sum of
squares. For this research, the analytical approach for selection of a transformation was

chosen.

8. 3. 1. Explanation of Analytical Determination of Transformation

Box and Cox [1964] have demonstrated how the transformation parameter it, in
y* = y1 may be estimated using the method of maximum likelihood, where y represents

the data to be transformed, y* is the transformed data, and A. is the exponent of the

146

transform. The procedure involves of performing a standard analysis Of variance, for a

range of values for it, on the equation(s):

y?» =(yK-1)/(T.g(I-1)) for T. 00, and

y In y for 2. =0, Equations 8-1

where y = 1n"1 [(1/n) 2 1n y)] is the geometric mean of the (n) observations y.
The maximum likelihood estimate of it is the value for which the error sum of squares is
a minimum. This is designated as SSe(L) in table 8-2. We cannot select the value of A by
directly comparing the error sum of squares from analyses of variance on y)“, because for
each value of it the error sum of squares is measured on a different scale. For a more

complete discussion of this procedure, see Montgomery [1984].

8. 3. 2. Results of the Analytical Determination of Transformation

For this inquiry, nine values of A were chosen as shown in table 8-3. The SSe(it)
values are shown next to the 1. values under the appropriate column heading for
dependent variable. Each SSe(Tt) value represents the error sum or squares from
performing an analysis of variance on the dependent variable as transformed by equation
8-1. In table 83, the row SSe(L) represents the minimum value found. If the minimum
SSe(T.) is near the value it = 1, this implies that the data do not support the need for
transformation. It can be seen from table 8-3 that none of the minimum values of SSe(Tt)
are close to the value of it = 1. This implies that transformation is desirable for each
dependent variable and that the value of 1. is close to the 3. associated with the minimum

SSe(A). At the bottom of the table the 3. associated with the minimum SSe(l.) is listed

147

along with the transformation determined by using the Box and Cox procedure. The

remainder of the data analysis will use the dependent variables transformed as indicated.

Table 8-3, Results of Procedure for Analytic Determination of Transformation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

p. TSD TOS OBJ cosr TIME
l1.00 2038104 1799965 4255743 6.01E+08 2.02E+09
“175 1255173 1500566 3657613 5.98E+08 6.48E+08
[0.50 876023 1345510 3277028 6.05E+08 2.24E+08
E25 73 8865 1298441 3076010 6.24E+08 8.87E+07
10.00 820865 1346459 3039912 6.57E+08 5.14E+07
-0.25 1289105 1495409 3176398 7.07E+08 7.34E+07
-0.50 2. 88E+06 1770814 3519015 7.79E+08 2.22E+08
_0.75 8.62E+06 2224573 4136779 8. 80E+08 8.5 1E+08
-1.00 3. 19E+07 2949697 5153033 1.02E+09 3 .48E+09
SSe(L) 738865 1298441 3039912 5.98E+08 5.14E+07
T. at SSE(L) 0.25 0.25 0.00 [0.75 0.00
Transformation yU-23 M15 ln(y) yU-73 ln(y)

 

 

 

 

 

8. 3. 3. Normal probability plots of residuals

 

After transformation, the variables are expected to adhere more closely to the
assumptions of the analysis of variance, those are, normality of error residuals,
independently distributed error residuals and homogeneity of variance of the residuals.
The analysis that follows tests the transformed variables against the assumptions of the
analysis of variance model. The tests follow the pattern set earlier, tests for normality of

residuals, results of autocorrelation analysis, and tests for homogeneity of variance.

148

 

 

EXPECTED VALUE
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Figure 8-11, Normal probability plot of TOS residuals After Transformation

 

 

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149

 

 

 

 

 

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RESIDUAL

 

 

 

Figure 8-14, Normal probability plot Of COST residuals After Transformation

150

 

 

EXPECTED VALUE
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RESIDUAL

 

 

 

Figure 8-15, Normal probability plot of TIME residuals After Transformation
All normal probability plots now indicate that the residuals Of the transformed
dependent variables are approximately normally distributed (i. e. no major departures

from normality are indicated by the plots).

8. 3. 4. Independent distribution of error residuals

The independent distribution of error residuals can be assessed by measuring the
autocorrelation of the data. The values of autocorrelation, after transformation, are listed
in the following table. There seems to be no autocorrelation, and it can be concluded that
the error residuals are independently distributed.

Table 8-4, Autocorrelation of transformed residuals

 

I [TOS |TSD OBJ COST TIME |
|Autocorrelation |-0.007 |0.047 0.003 -0.052 -0.047 |

 

 

 

 

 

151

8. 3.5. Homogeneity of Variance

After transformation of the data, verification of homogeneity of variance is
necessary to establish credibility of results. The tests for homogeneity of variance are the
same as used before; plots of residuals versus estimates and Bartlett's test for
homogeneity of variance. Figures 8-16 through 8-20 present the plots of residual versus

estimate for the dependent variables.

 

 

 

 

 

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152

 

 

 

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40

Figure 8-18, Plot of Residual vs. Estimate - OBJ After Transformation

 

 

153

 

 

 

 

 

 

200 r 7’
- I '.
100 '- :o 5 . .' "
:, fig 2
o. 0 .{
l2 :5: .
O - :0 .: _
1’ 5:: l
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E -100 — E. 9‘. z ..
-200 ._ ..
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-300 I 1 l ‘ I I
300 400 500 600 700 800 900
ESTIMATE

 

 

 

Figure 8-19, Plot of Residual vs. Estimate - COST After Transformation

 

 

 

 

 

2 T r T 17 T fir
. I
1 r- “ , . . _
:1 " .: ',
8" s vf¥s” :Ifl'”
i .
a o _ a : ’ z - a .
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8 O. .t
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-1.— . —
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ESTIMATE

 

 

 

Figure 8-20, Plot of Residual vs. Estimate - TIME After Transformation

154

The plots of estimate vs. residual for the transformed dependent variables exhibit

no obvious patterns and hence no obvious heterogeneity of variance.

8. 3. 6. Bartlett's test after transformations

As stated earlier, Bartlett's test for homogeneity of variance tests the
hypothesis that the variances of all the treatments are equal for each performance
measure. We reject this hypothesis if the test statistic exceeds the appropriate chi-
square value.

The statistics for each of the transformed dependent variables is shown in

Table 8-5.

Table 8-5, Bartlett's Test For Homogeneity of Variance

 

 

 

I | TOS [TSD [OBJ | COST | TIME |
| Chi-Square, 80df | 60.39 | 60.39 [60.39 | 60.39 | 60.39 |
[Test Statistic, 80df |343.413 |481.981 |201.125 [236.360 |288.969J

 

As can be seen from the table each of the test statistic values exceeds the chi-
square value, so we reject the hypothesis that all variances are equal for all transformed

dependent variables.

155

8. 4. JUSHHCATYON FOR USE OF ANALYSIS OF VARIANCE

At this juncture, it has been shown that the transformed dependent variables have
normally and independently distributed error residuals, and that the variance among cells
is heterogeneous. Therefore, the data still violate the assumptions of the analysis of
variance model. Both Montgomery [1984] and Kirk [1982] state that for a balanced,
fixed effects design, the F test is only slightly affected by heterogeneous variances if the
error residuals are normally distributed. Since the design is balanced, at a sample size of
13 (after random deletion of data points in treatment cells to balance the design), and
since the error residuals are normally and independently distributed, this research will
rely on the robustness of the F test to heterogeneity of variance as the justification for

using analysis of variance.

8. 5. RESULTS OFANO VA

The data, collected as described in chapter 7, were analyzed using the multi-factor
ANOVA routines in SYSTAT for Windows. A summary of significant effects is given in
Table 8-6.

156

Table 8-6 Analysis Of Variance Results
indicates significance at p < .01, ”y" indicates significance at p < .05.

 

I Source TOS TSD OBJ C
[AWU
[WITNTN

t-l

 

 

 

 

MHMWM
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As can be seen from table 8-6, the all main effects and the majority of interaction
effects are significant at the p < .01 level. All interaction effects were significant with

one or another dependent variable(s).

8. 5. I . Detailed Results

In this section, each of the hypotheses proposed will be considered. As was noted
before, a hypothesis was constructed for each of the dependent variables. In all cases
direct statements cannot be offered to reject or accept any hypothesis because of the
presence of significant interactions, Kirk [1982]. The data do offer supporting evidence

for hypotheses, as discussed below.

157

In order to compare the performance of the solution methods by problem type, a
Tukey HSD multiple comparison test was performed on the data representing each of the
five performance measures. This test compares each of the 81 treatments against all the
other treatments, testing for significant differences among the treatment means. There is
very little agreement among statisticians about the relative merits of the various multiple
comparison procedures. Tukey's HSD multiple comparison test was chosen for several
reasons. First, the data meet the assumptions of the test. Second, the Tukey test is more
powerful than other tests for large number of comparisons. Third, the Tukey HSD
multiple comparison test controls the error at a for the entire collection of tests.

Following the presentation of relevant statistics for each dependent variable,
interactions of interest to this research will be examined and discussed. Since this
research is primarily interested in a comparison of algorithms, emphasis will be placed
on discerning the main effect of ALGORITHM in the presence of interactions.

The reader is reminded that when considering a hypothesis, the subscript attached
to the dependent variable represents an algorithm. For example, TIME] refers to the
TIME for Loucks heuristic, TIME2 refers to the TIME for simulated annealing and
TIME3 refers to the TIME for tabu search.

In the charts and discussion that follows an abbreviation for problem types will be
used consisting of a three letter code. The first letter in the code stands for the level of
Average Worker Utilization, H for high, M for medium and L for low. The second letter
in the abbreviation stands for the level of average percentage of tasks known per worker,
using the same letters as the abbreviation for AWU. The last letter stands for the level of
max-to-min ratio, again using the same letters, H, M, and L. As an example, the problem
with the label, HLM, would be a problem type having a high level (H) of average worker
utilization, a low level (L) of average percentage of tasks known per worker, and a

medium (M) level of max-to-min ratio.

158

8.5.1.1. TIME - Speed of Algorithm

The hypotheses relating to this aspect of algorithm performance is:

Ho : TIME] = TIME2 = TIME3

H1 : TIME] 1: TIME; ¢ TIME3

This hypothesis tests whether there are significant speed differences among the
algorithms. Since there are significant interactions among the factors the hypothesis
cannot be tested directly.
Results of ANO VA on M, Transformed. The analysis of variance for the dependent
variable TIME as presented in table 8-7 indicates that all main effects and interaction
effects are significant at p < .001, with two exceptions. First, the main effect PCTNTN,
or average percentage of tasks known per worker, was not significant, and second, the
interaction of PCTNTN*MMR, or the interaction of average percentage of tasks known
per worker with the max-min ratio, was significant at the p < .05 level. The multiple r
correlation coefficient was 0.990 (squared value = 0.980), indicating that the factors and
interactions accounted for most of the variation in the dependent variable TIME.
Examination of the mean-square for each of the effects indicated that the mean-square
for the main effect ALGORITHM is at least 2 orders of magnitude larger than any other
effect and about 3 orders of magnitude larger than the other main effects. Mean-squares
for all effects are an order of magnitude larger than the mean-square for error for the

variable TIME.

 

159

Table 8-7. ANOVA on TIME, Transformed

 

ANOVA on TIME, Transformed

LEVELS ENCOUNTERED DURING PROCESSING ARE:

AWU$: High Low Med

PCTNTN$: High Low Med

MMRS: High Low Med

ALGORTHM$: Loucks Sim_Ann Tabu_Srch

DEP VAR: TIME N: 1053 MULTIPLE R: 0.990 SQUARED MULTIPLE R: 0.980

ANALYSIS OF VARIANCE

SOURCE SUM-OF—SQUARES DF MEAN-SQUARE F-RATIO P
AWU$ 4.199 2 2.100 21.515 0.000
PCTNTN$ 0.305 2 0.153 1.564 0.210
MMR$ 4.607 2 2.303 23.603 0.000
ALGORTHM$ 4456.638 2 2228.319 22833.320 0.000
AWU$*PCTNTN$ 4.645 4 1.161 11.898 0.000
AWU$*MMR$ 2.178 4 0.545 5.580 0.000
AWU$

*ALGORTHM$ 17.080 4 4.270 43.753 0.000
PCTNTN$*MMR$ 1.012 4 0.253 2.592 0.035
PCTNTN$

*ALGORTHM$ 19.868 4 4.967 50.897 0.000
MMRS

*ALGORTHM$ 12.223 4 3.056 31.312 0.000
AWU$*PCTNTN$

*MMR$ 4.800 8 0.600 6.148 0.000
AWU$*PCTNTN$

*ALGORTHM$ 16.451 8 2.056 21.071 0.000
AWU$*MMR$

*ALGORTHMS 5.094 8 0.637 6.524 0.000
PCTNTN$*MMR$

*ALGORTHM$ 5.183 8 0.648 6.639 0.000
AWU$*PCTNTN$

*MMRS

*ALGORTHMS 12.414 16 0.776 7.950 0.000
ERROR 94.858 972 0.098

DURBIN-WATSON D STATISTIC 2.090
FIRST ORDER AUTOCORRELATION -.047

 

 

160

TIME by Algorithm vs. Problem Type. As can be seen from figure 8-21, the time to
completion for Loucks' algorithm is always less than simulated annealing which is
always less than tabu search (indicating the presence of an ordinal interaction). The
Tukey HSD multiple comparison test (3x3x3x3, all factors, all factor levels) verifies that
these visual results are, with two exceptions, statistically significant (for problems LLH
and MLL, the differences are not significant). The comparison was carried out by
looking for significant differences between solution methods on a pairwise basis for each
of the problem types. Such pairwise differences were found to be significant for all pairs

except between tabu search and simulated annealing on problem types LLH and MLL.

 

 

lTime, Transformed'

 

 

Loucks

" ' " "' Sim_Ann

4 W — - Tabu_SId't

“(Thai
0o

 

 

 

 

 

 

 

 

Figure 8-21, Plot of TIME by Algorithm vs. Problem Type

While nothing statistical can properly be said about other differences in the
performance of the algorithms on TIME-to-completion, examination of the charts leads

to several speculations. The performance of tabu search seems to exhibit a pattern of

161

better performance when the problem type has a high level of MMR. Similarly,

simulated annealing seems to perform better (than others in the problem type group, i.e.

PM vs. HMH and HMM) with problems with a low level of MMR, whereas Loucks'

heuristic performs worse with a low level of MMR.

Due to the ordinal interaction effects (an ordinal interaction is the case when the

rank order of the treatment effects remains constant, Pedhazur [1982]), we cannot

directly test the hypothesis. The data does provide evidence in support of the alternate

hypothesis H1, that is, that the TIME-to-completion for each of the algorithms is not

equal.

Table 8-8, Table of the best TIME performance on problem type by algorithm .

 

 

 

 

 

 

 

 

 

TIME - Best
AWU AWU AWU
Low Low Low Med Med Med High High High

PCTNTN PCTNTN PCTNTN
MMR Low Med High Low Med High Low Med High
Low Loucks Loucks Loucks Loucks Loucks Loucks Loucks Loucks Loucks
Med Loucks Loucks Loucks Loucks Loucks Loucks Loucks Loucks Loucks
High Loucks Loucks Loucks Loucks Loucks Loucks Loucks Loucks Loucks

 

 

 

 

 

 

 

 

 

 

 

Table 8-9, Table of the worst TIME performance on problem type by algorithm.

 

 

 

 

 

 

 

 

 

TIME -Worst
AWU AWU AWU
Low Low Low Med Med Med High High High

PCTNTN PCTNTN PCTNTN
MMR Low Med High Low Med High Low Med High
Low Tabu_S Tabu_S Tabu_S Tabu_S Tabu_S Tabu_S Tabu_S Tabu_S Tabu_S
Med Tabu_S Tabu_S Tabu_S Tabu_S Tabu_S Tabu_S Tabu_S Tabu_S Tabu_S
High Tie Tabu_S Tabu_S Tabu_S Tabu_S Tabu_S Tabu_S Tabu_S Tabu_S

 

 

 

 

 

 

 

 

 

 

As can be seen from tables 8-8 and 8-9, Loucks' heuristic always had the best

TIME-to-completion and tabu search always had the worst TlME-to-completion.

 

162

8.5.1.2. COST - Total Manpower Cost of the Schedule.

To test for differences in COST, the following hypotheses were proposed:

H0 1 COST] = COST2 = COST3

H1 : COST] at COST2 at COST3
Results of ANOVA on COST, Transformed. The results of analysis of variance for the
dependent variable COST show that all main effects were significant at p < .001. The
interaction effects AWU*PCTNTN, AWU*MMR PCTNTN*MMR, and
AWU*PCTNTN*MMR are significant at p < .001 and AWU*ALGORITHM is
significant at p < .01. Most of the interaction effects were not significant at p < .05. An
examination of the mean-square for the each of the effects show that the mean-square for
PCTNTN is twice as large as for AWU and more that an order of magnitude larger than
any other effect. The multiple r correlation is 0.813 (squared value = 0.660), indicating
that the factors and their interactions accounted for most of the variation in the dependent
variable COST.

No evidence can be found for rejection of the null hypothesis,
H20 : COST] = COSTZ = COST3. Statistically, we cannot support the idea that one or
more algorithms is different in COST performance than any other.

163

Table 8-10. ANOVA on COST, Transformed

 

 

ANOVA on COST, Transformed

LEVELS ENCOUNTERED DURING PROCESSING ARE:

AWUs: High Low Med
PCTNTN$: High Low Med
MMR$= High Low Med
ALGORTHMs: Loucks Sim_Ann Tabu_Srch

DEP VAR: COST N: 1053 MULTIPLE R: 0.813 SQUARED MULTIPLE R: 0.660

ANALYSIS OF VARIANCE

SOURCE SUM-OF-SQUARES DF MEAN-SQUARE F-RATIO P
AWUS 2428046.223 2 1214023.111 235.282 0.000
PCTNTN$ 5234622.308 2 2617311.154 507.245 0.000
MMRS 246073.570 2 123036.785 23.845 0 000
ALGORTHM$ 182134.677 2 91067.339 17.649 0 000
AWU$*PCTNTN$ 146193.949 4 36548.487 7.083 0.000
AWU$*MMR$ 445239.567 4 111309.892 21.572 0 000
AWU$

*ALGORTHM$ 68764.822 4 17191.205 3.332 0.010
PCTNTN$*MMR$ 494109.619 4 123527.405 23.940 0.000
PCTNTN$

*ALGORTHMS 41125.660 4 10281.415 1.993 0.094
MMRS

*ALGORTHMS 43090.215 4 10772.554 2.088 0.080
AWU$*PCTNTN$

*MMR$ 252429.220 8 31553.653 6.115 0 000
AWU$*PCTNTN$

*ALGORTHMS 18298.392 8 2287.299 0.443 0.895
AWU$*MMR$

*ALGORTHMs 67488.884 8 8436.111 1.635 0.111
PCTNTN$*MMR$

*ALGORTHMs 32790.962 8 4098.870 0.794 0.608
AWU$*PCTNTN$

*MMR$

*ALGORTHMS 44171.530 16 2760.721 0.535 0.930
ERROR 5015379.444 972 5159.855

DURBIN-WATSON D STATISTIC 2.102
FIRST ORDER AUTOCORRELATION -.052

 

 

164

COST by Algorithm vs. Problem Type .Figure 8-22 plots the COST of the schedules
produced by the algorithms, by problem type.

 

 

ICost, Transformed'

 

 

 

 

 

 

 

 

300..
700..
smy
a ""“Loucku
-5oo..
E: "’ """" Sim_Ann
§4004~
~— Tabu_Srdm
3001
zmdb
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0+fi4—e 14.5::::+ tefifif+4fi+4fi
.4 1421-1213 5421-1233 1:421:42:
-’ .4 I I .4 I
333533355=§,§;§2§§=§=§33:35

 

 

 

Figure 8-22, Plot of COST by Algorithm vs. Problem Type
Upon examination of the chart, the first pattern that might be noticed is that the
COST performance of the three algorithms seems to be highly correlated across problem

type. This is, indeed, the case. The correlation coefficients are shown in table 8-11.

Table 8-11, Correlation coefficients on COST performance between Algorithms

Coefficients
T abu
- COST 1.0

im Ann - COST 0.96 l.
abu Srch — COST 0.93 O. l.

 

As can be seen from table 8-10, all correlations are very high, over .93. This

indicates that COST performance is more influenced by problem type than by algorithm
type.

165

It is also apparent, from the chart, that the COST performance varies by problem
type. If we group the problem types, first by AWU (the first letter) and then by PCTNTN
(the second letter), we note that when PCTNTN is low, the COST performance is lowest,
in the AWU group. This speculation is borne out statistically. A Tukey HSD multiple
comparison test (3x3x3x3, all factors, all factor levels) was performed on the COST data.
The Tukey test indicates that the COST performance for all the algorithms on problems
LLL, LLM, and LLH is lower than problem types LML, LMM, LMH, LHL, LHM, and
LHH at p < .001.

Another pattern in the data is that it seems that the COSTs are highest when
problem level PCTNTN is highest, and medium when the problem type has a level of
PCTNTN of medium. This speculation has only partial statistical support across problem

types.
Table 8-12, Best COST performance across problem types by algorithm.

COST - Best
AWU AWU
Med Med High High High
PCTNTN PCTNTN

Med High Low Med High

 

Inspection of table 8-12 reveals that across all problem types there were ties for

best performance.

166

Table 8-13, Worst COST performance across problem type by algorithm

COST - Worst
AWU AWU
Med Med High High High
PCTNTN PCTNTN

Med High Low Med High
abu S

 

As can be seen from table 8-13, tabu search has the only worst performances at

problem type MHL. All others are tied for both best and worst performance.

8.5.1.3. TOS - Total man-hours of overstaffing.

The hypotheses to test total man-hours of overstaffmg were formulated as
follows:

H0 : T081 = TOS2 = TOS3

H1 : TOSl ¢ TOS2 ¢ TOS3

As with all other hypotheses proposed in this research, no direct test can be made
nor can general, statistically valid statements be made about main or interaction effects
because of the presence of significant disordinal interaction effects (a disordinal
interaction is present when an interaction changes the rank order of the treatment effects,
Pedhazur [1982]). As will be shown below, evidence can be offered in support of the
alternate hypothesis.

Results of ANO VA on TOS, Transformed. The analysis of variance for the

dependent variable TOS indicates that all main effects are significant at p < .001. The

167

interaction effects, AWU*ALGORITHM, PCTNTN*ALGORITHM, PCTNTN*MMR,
MMR*ALGORITHM, AWU*PCTNTN*ALGORITHM and
AWU*MMR“ALGORITHM are significant at p < .001. The interaction effects,
AWU*PCTNTN, AWU*MMR, and AWU*PCTNTN“MMR*ALGORITHM are
significant at p < .05. Examination of the mean-square of all effects indicates that the
factor ALGORITHM has a mean square nearly an order of magnitude larger than any
other main effect. Only the interaction effect PCTNTN*MMR*ALGORITHM has a
lower mean-square than error. The multiple r correlation of 0.734 (squared value =
0.539) indicated that the factors and interactions accounted for most of the variation in

the variable TOS. Complete ANOVA results for TOS are given in table 8-14.

Table 8-14. ANOVA on TOS, Transformed.

168

 

ANOVA on TOS,

Transformed.

LEVELS ENCOUNTERED DURING PROCESSING ARE:

 

AWU$: High Low Med

PCTNTN$: High Low Med

MMRS: High Low Med

ALGORTHMs: Loucks Sim_Ann Tabu_Srch

DEP VAR: TOS N: 1053 MULTIPLE R: 0.734 SQUARED MULTIPLE R: 0.539

ANALYSIS OF VARIANCE

SOURCE SUM-OF-SQUARES DF MEAN-SQUARE F-RATIO P
AWU$ 3.024 2 1.512 16.237 0 000
PCTNTN$ 4.368 2 2.184 23.456 0.000
MMR$ 2.269 2 1.135 12.186 0.000
ALGORTHMS 26.243 2 13.121 140.918 0.000
AWU$*PCTNTN$ 1.170 4 0.292 3.141 0.014
AWU$*MMR$ 1.082 4 0.271 2.905 0.021
Awus

*ALGORTHM$ 13.547 4 3.387 36.373 0.000
PCTNTN$*MMR$ 1.805 4 0.451 4.846 0.001
PCTNTN$

*ALGORTHM$ 36.714 4 9.179 98.575 0.000
MMRs

*ALGORTHM$ 4.218 4 1.054 11.324 0.000
AWU$*PCTNTN$

*MMRS 1.129 8 0.141 1.516 0.147
AWU$*PCTNTN$

*ALGORTHMS 2.574 8 0.322 3.455 0.001
AWU$*MMR$

*ALGORTHM$ 3.490 8 0.436 4.685 0.000
PCTNTN$*MMR$

*ALGORTHM$ 0.625 8 0.078 0.838 0.569
AWU$*PCTNTN$

*MMR$

*ALGORTHM$ 3.370 16 0.211 2.262 0.003
ERROR 90.506 972 0.093

DURBIN-WATSON n STATISTIC 2.013

FIRST ORDER AUTOCORRELATION -.007

 

 

169

T OS by Algorithm vs. Problem Type. Figure 8-22 plots the total number of hours
overstaffed that each algorithm generated in the schedules produced, by problem type.

 

ITOS, Transformed'

 

— — Loucks

---- Sim_Ann

 

Tabu_Srch

 

 

 

 

 

 

LLIIIIPI l Jl J L]
2 IIIITIM 1 If If ‘fi
gasgsgs=5355955955955535935
-J-' 3 4542222252221:I:::I:1

 

 

 

Figure 8-22, Plot of TOS by Algorithm vs. Problem Type

Inspection of the chart indicated that there may be differences between Loucks'
algorithm and the other two algorithms when problem type has a low level of the factor
PCTNTN. A 3x3x3x3 (all factor, all factor level) Tukey HSD multiple comparison test
was performed on the TOS data. The results indicate that differences in total man-hours
Of overstaffing for problem types LLL, LLM, LLH, MLL, MLM, MLH, I-ILL, and HLH
are significant between Loucks' heuristic and the pair, simulated annealing and tabu
search. The exceptions to the "rule" that tabu search and simulated annealing outperform
Loucks' heuristic on problems with a low level of the factor PCTNTN are that for
problem type I-ILM there are no significant differences among the algorithms and for
problem type MLL the difference between Loucks' heuristic and tabu search is not

significant. There are no significant differences between simulated annealing and tabu

170

search on these problems. These results provide evidence for the alternate hypothesis,

H1

: TOS] i T082 i T033.

Table 8-15, Best TOS performance across problem type by algorithm

 

 

 

 

 

 

 

 

 

TOS - Best
AWU AWU AWU
Low Low Low Med Med Med High High High

PCTNTN PCTNTN PCTNTN
MMR Low Med High Low Med High Low Med High
Low Tie Tie Tie Tie Loucks Tie Tie Tie Loucks
Med Tie Sim_Ann Sim_A Tie Tie Tie Tie Tie Loucks
High Tie Sim_Ann Sim_A Tie Tie Tie Tie Tie Sim_A

 

 

 

 

 

 

 

 

 

 

The best performances across problem types by algorithm is presented in table 8-

15. These results may be summarized as follows, simulated annealing performed best on

5 of 27 problem types, Loucks' heuristic on 3 of 27 and ties for 19 of 27. There seems to

be no pattern to the best performances.

Table 8-16, Worst TOS performance across problem type by algorithm

 

 

 

 

 

 

 

 

 

TOS - Worst
AWU AWU AWU
Low Low Low Med Med Med High High High
PCTNTN PCTNTN I PCTNTN
MMR Low Med High Low Med High Low Med High
Low Loucks Tie Tie Loucks Tie Tie Loucks Tie TieL
Med Loucks Tie Tie Loucks Tie Tabu_S Loucks Tie Tie
High Loucks Tie Tie Loucks Tie Tie Loucks Tie Tie

 

 

 

 

 

 

 

 

 

 

 

As shown in table 8-16, worst performances on TOS are dominated by Louck's

heuristic, with 9 "worsts" of 27 problem types, while tabu search had 1 "worst" of 27

problem types. There were 17 statistical ties.

 

171

8.5.1.4. TSD - Sum of squared deviations between workers scheduled and targeted
work hours.

The hypothesis proposed to test for differences among the algorithms for the sum
of squared deviations between workers scheduled and targeted work hours is:

H0 : TSD] = TSD2 = TSD3

H1 : TSD] ¢ TSD2 :4 TSD3

As noted before, no direct tests on main effects are statistically permissible when
significant disordinal interactions are present. Nevertheless, indirect evidence will be
offered in support of the alternate hypothesis, H1 : TSDl ¢ TSD2 at TSD3.
Results of ANOVA on T SD, T ransformed. The results of the analysis of variance for the
dependent variable TSD show that all main and interaction effects are significant at p <
.001 except the two interaction effects, AWU*PCTNTN and
AWU*PCTNTN*ALGORITHM. Examination of the mean-square for each of the effects
show that ALGORITHM and PCTNTN and the interaction ALGORITIIM*PCTNTN are
the dominant sources of variation. All mean-squares for main and interaction effects are
larger that the mean-square for error. The multiple r correlation coefficient for this
analysis of variance is 0.846 (squared value = 0.715), indicating that the main and

interaction effects account for most of the variation of the dependent variable TSD.

172

Table 8-17. ANOVA on TSD, Transformed

 

 

ANOVA on TSD, Transformed

LEVELS ENCOUNTERED DURING PROCESSING ARE:

AWU$ : High Low Med

PCTNTN$ : High Low Med

MMRS: High Low Med

ALGOR'I‘HM$ : Loucks Sim_Ann Tabu_Srch

DEP VAR: TSD N: 1053 MULTIPLE R: 0.846 SQUARED MULTIPLE R: 0.715

ANALYSIS OF VARIANCE

SOURCE SUM-OF-SQUARES DF MEAN-SQUARE F-RATIO P
AWUS 3.353 2 1.676 12.497 0.000
PCTNTN$ 70.965 2 35.482 264.518 0.000
MMRS 2.526 2 1.263 9.416 0.000
ALGORTHM$ 105.198 2 52.599 392.123 0.000
AWU$*PCTNTN$ 0.612 4 0.153 1.141 0.336
AWU$*MMR$ 4.113 4 1.028 7.666 0.000
AWU$

*ALGORTHM$ 20.303 4 5.076 37.840 0.000
PCTNTN$*MMR$ 2.485 4 0.621 4.632 0.001
PCTNTN$

*ALGORTHM$ 64.567 4 16.142 120.336 0.000
MMRS

*ALGORTHM$ 20.264 4 5.066 37.766 0.000
AWU$*PCTNTN$

*MMR$ 4.550 8 0.569 4.240 0.000
AWU$*PCTNTN$

*ALGORTHM$ 1.199 8 0.150 1.117 0.349
AWU$*MMR$

*ALGORTHM$ 10.064 8 1.258 9.379 0.000
PCTNTN$*MMR$

*ALGORTHM$ 8.206 8 1.026 7.647 0.000
AWU$*PCTNTN$

*MMRS

*ALGORTHM$ 8.843 16 0.553 4.120 0.000
ERROR 130.384 972 0.134

DURBIN-WATSON D STATISTIC 1.905
FIRST ORDER AUTOCORRELATION .047

 

 

173

T SD by Algorithm vs. Problem Type. Figure 8-22 plots that total squared deviations
between all worker's desired number Of target hours and the number of hours scheduled

for the workers by the algorithms, by problem type.

 

 

‘TSD, Transformed'

 

— —" Loucks

- - - - Sim.Ann

 

Tabu_Srd'I

 

 

 

 

1.5 «-

 

 

1 l 11 LLLI LLL LL 1
WT FIT r UHMITIM I T r TM!
.1 .1 3.1 3.1 1.1 3.1 1.1 1.1 :4 3:
33§3§35§:4542§3:§:gfiaa§zz§x
.1 4422122222: Itxxzixz:

 

 

Figure 8-22, Plot of TSD by Algorithm vs. Problem Type

Review of figure 8-22 indicates that there may be a significant difference between
the performance of the pair, simulated annealing and tabu search, and Loucks' heuristic,
when the level of PCTNTN is low. A Tukey HSD test was performed on the TSD data (a
3x3x3x3, all factor, all factor level test). Results of the test show that differences are
significant for problem types LLL, LLM, MLL, MLM, MLH, HLL, HLM, and HLH.
For these problems Loucks' heuristic outperformed both simulated annealing and tabu
search.

There are other significant differences between algorithms, among them being

significant differences between simulated annealing and Loucks on problem types LHH,

 

174

LHL, LHM, LML, MHH, and MMH, where simulated annealing outperforms Loucks'
heuristic.

Further examination of the plots show that Loucks' heuristic "spikes up" and tabu
search "spikes down" when the level Of MMR is low. In addition, simulated annealing is
highest when the level of PCTNTN is low and lowest when PCTNTN is high.

All these significant differences provide evidence for the alternate hypothesis,

H1 : TSD] :6 TSD2 ¢ TSD3. That is, there are significant differences between the
performance of the algorithms on the squared differences between a worker's scheduled

and targeted hours.

Table 8-18 Best TSD performance across problem type by algorithm

 

 

 

 

 

 

 

 

 

 

TSD - Best
AWU AWU AWU
Low Low Low Med Med Med High High High

PCTNTN PCTNTN I PCTNTN
MMR Low Med High Low Med High Low Med High
Low Loucks Sim_Ann Tie Tie Tie Tie Loucks Tie Tie
Med Loucks Tie Tie Loucks Sim_A Sim_A Loucks Tue Tie
High Tie Tie Sim_A Loucks Sim_A Sim_A Loucks Tie Tie

 

 

 

 

 

 

 

 

 

 

For TSD, an examination of table 8-18 shows thatLoucks' heuristic performed
best on 7 of 27 problem types. Simulated annealing had the best performance on TSD for
6 of 27 problem types and tabu search for none of 27 problem types.

175

Table 8-19, Worst TSD performance across problem type by algorithm

 

 

 

 

 

TSD - Worst
AWU AWU AWU
Low Low Low Med Med Med High High High

PCTNTN PCTNTN PCTNTN
MMR Low Med High Low Med High Low Med High
Low Tie Tie Loucks Tie Loucks Loucks Tie Tabu_S Tabu_S
Med Tie Tabu_S Tabu_S Tie Tabu_S Tie Tie Tabu_S Tabu_S
High Tie Tabu_S Tabu_S Tie Tabu_S Tie Tie Tabu_S Tabu_S

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 8-19 shows which algorithm had the worst performance by problem type.
Tabu search had the largest number of worst performances for 12 of 27 problem types,
Loucks' heuristic was the worst for 3 of 27, and there were ties for the worst for 12 of 27

problem types.

8.5.1.5. OBJ - Objective Function Value

The hypothesis as proposed to test for differences among the algorithms with
respect to the objective function value is:

H0 : OBJ1 = OBJz = OBJg

H1 : OBJ] at OBJ2 at OBJ3

As before no direct confirmation or rejection of hypotheses can be made in the
presence of significant interactions. The data do provide evidence in support of the
alternate hypothesis, H1 : OBJl at OBJZ at OBJ3.
Results of ANO VA on OBJ, Transformed. The analysis of variance results for the
dependent variable OBJ show that the main effects AWU, PCTNTN and ALGORITHM

are significant at p < .001, and MMR is significant at p < .002. All interaction effects are

176

significant at p < .001 except for AWU*PCTNTN, AWU*PCTNTN*ALGORITHM and

PCTNTN*MMR*ALGORITHM which are significant at p < .05 and PCTNTN*MMR

which is not significant. An examination of the mean-squares for all effects indicates that

ALGORITHM is 4 times larger than any other effect, and that all main and interaction

effects are larger than the error mean-square. The multiple r correlation for this analysis

of variance is 0.755 (squared value = .570) indicating, along with the small error mean-

square that the main and interaction effects account for most of the variation in OBJ.

OBJ by Algorithm vs. Problem Type. This figure, 8-23, shows the objective firnction

performance of the algorithms by problem type.

 

LIIIOIJI

 

 

 

 

OBJ, Transformedl

 

— "' Loucks

" " ‘ ‘ Sim_Ann

 

 

 

Tabu_Srch

 

I
r

 

 

.1 .1 .1 .1 .1 .1
jgsg=§453955435=55355.§§.55
4 3 44452225252: 1111:“:11

 

Figure 8-23, Plot Of OBJ by Algorithm vs. Problem Type

An examination of figure 8-23 reveals that the only discernible pattern is that the

performance of all three algorithms performance is most similar at LLL, MLL, and HLL.

Simulated annealing seems to perform worse than other problems when the level of

 

177

MMR is low, this occurs 7 of 9 times. Another notable feature is that there appear to be
significant differences between the performance of algorithms at several points. A Tukey
HSD multiple comparison test (3x3x3x3, all factor, all factor level) was performed on
the OBJ data. Simulate annealing outperformed Loucks' heuristic and tabu search on
problem types LHH, LHM, LMH, and LMM. Simulated annealing and Loucks' heuristic
outperformed tabu search on problem types HMH, I-IML, and HMM.

These significant differences provide evidence in support of the alternate
hypothesis, H1 : OBJ 1 at OBJ2 1t OBJ3. That is, there is significant difference among the

algorithms for the dependent performance measure OBJ on some problem types.

Table 8-20, Best OBJ performance across problem type by algorithm

 

 

 

 

0 BJ - Best
A WU A WU A WU
Low Low Low Med Med Med High High High
PCTNTN PCTNTN PCTNTN

 

 

MMR Low Med High Low Med High Low Med High
Low Tie Sim_A Sim_A 'fie Tie Tie Loucks Tie Tie
Med Sim_A Sim_A Sim_A Tie Sim_A Tie Loucks Tie Tie
High Sim_A Sim_A Sim_A Tie Sim_A Tie Loucks Tie Tie

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 8-20 shows that simulated annealing has the best performance of all three
algorithms on OBJ for 10 of 27 problem types, Loucks' heuristic for 3 of 27, and ties for
14 of 27 .

178

Table 8-21, Worst OBJ performance across problem type by algorithm

 

 

 

 

 

 

 

 

 

 

OBJ - Worst
AWU AWU AWU
Low Low Low Med Med Med High High High

PCTNTN PCTNTN I PCTNTN
MMR Low Med High Low Med High Low Med High
Low Tie Tie Tie Tie Tie Tie Tie Tabu_S Tabu_S
Med Tie Tie Tie Tie Tabu_S Tabu_S Tie Tabu_S Tabu_S
High Tie Tie Tabu_S Tie Tabu_S Tabu_S Tie Tabu_S Tabu_S

 

 

 

 

 

 

 

 

 

 

In table 8-21 it can be seen that tabu search has the worst performance for 11 of

27 problem types, the rest were ties. No patterns are apparent in the worst performances.

8. 6. DIFFERENTIAL DIFFIC ULTY 0F PROBLEM TYPES

The presence of significant interactions in the data presented, restricts what might
be said, on a statistically sound basis, about the differential difficulty of the problems,
and problem types used in this research. Further, these interactions have obscured any
indications that one or another problem type might be more difficult to solve than
another. Evidence for this statement is that the measures of problem difficulty, AWU,
PCTNTN, and MMR, all have significant disordinal interactions with ALGORITHM.
The nature of these disordinal interactions have been explored above and will not be
repeated here. This research provides no evidence in support of general conclusions
about problem difficulty, with the problems and algOrithms used here. Other research,
see Jacobs and Bechtol [1993] for example, have found that some problems are more
difficult to solve than others, for a single algorithm. This research would support the
conclusion that some problems are more difficult to solve for one algorithm or another,

but not when all three algorithms are considered jointly.

179

CHAPTER 9 - SUMMARY, CONCLUSIONS AND RECOMMENDATIONS

9.1. SUMMARY

Determining the work tours for workers with limited interchangeability is a task
faced regularly, usually weekly, by managers in thousands of service businesses. Prior
research of this problem is sparse. Only Loucks has investigated the problem considering
both dissimilarity of workers' available hours and task qualifications. This heterogeneity
of workforce characteristics is the key aspect of Loucks' research, and of the research
presented here.

The Objectives of this study were to apply and implement two new algorithms,
simulated annealing and tabu search, to the work tour scheduling problem with limited
worker interchangeability, and to compare these two new algorithms to Loucks' heuristic.
Any of these methods might be used, on a computer, by a manager in the execution of
the tour of duty scheduling task, on a regular basis. All would be an improvement on the
manual, trial and error methods used in most service firms.

The data presented provide evidence in support of alternate hypotheses for all
performance measures. Stated another way, evidence was presented to support the
statement that there are significant differences among all three algorithms on all five

performance measures. The data further indicate that the algorithms perform very

180

similarly with regard to cost, and very differently with regard to time-to-completion. For
the measures of solution quality, TOS, TDS and OBJ, the differences weren't clear cut,
over all problem types each algorithm was best at least once, and worst at least once.

On time-to-completion, the performance of the algorithms suggest that Loucks'
heuristic is better than simulated annealing and simulated annealing is better than tabu
search. On the other measures, simulated annealing might be preferred, having 21 of 34
"bests" and no "worsts". On the same measures, Loucks' heuristic had 13 of 34 "bests"
and 12 of 36 "worsts" and tabu search had no "bests" and 24 of 36 "worsts".

From a practical standpoint, a manager might prefer simulated annealing over the
other solution methods, based on the performance on the quality of solution performance
measures. Although simulated annealing was not the fastest solution method in time-to-
solution, its solution time was acceptably short, about 22 minutes on average.

Prior research concluded that tabu search outperformed simulated annealing, for
example see Knox [1990]. This research does not support that conclusion. Not only was
simulated annealing faster in time-to-eompletion, but, as shown above, outperformed

tabu search in measures of solution quality and cost.

9. 2. Commons

There are several contributions that this research offers the field of employee

scheduling.

. Comparative results of three algorithms on five performance measures for

the work tour scheduling problem, as detailed above.

181

. Results contrary to previous research for the relative performance of
simulated annealing vs. tabu search, i.e. that simulated annealing generally outperformed
tabu search both on time-to-completion, cost, and quality of solution.

. Development of a method for generating initial, feasible solutions for the
work tour scheduling problem.

. Conception and development of transition generators for simulated
annealing and tabu search.

e Origination and development of the idea of differentiating between "good"
and "random" transition generators for simulated annealing.

. Evidence to show that the combination of "good" and "random" transition
generators for simulated annealing is more effective than either "good" or "random"
transition generators alone.

. The first use of the construction - improvement paradigm for simulated
annealing and tabu search. Published research, to date, has applied tabu search and
simulated annealing in an environment where an initial solution can easily be generated
with the implemented algorithm or where the generation of an initial can be done
randomly.

. The first use of simulated annealing and tabu search on a highly
constrained problem. Published research, to date, has applied tabu search and simulated
annealing in an environment where feasibility is easily maintained or guaranteed.
Designing transition generators that don't create constraint violations for the work tour

scheduling problem is a non-trivial process.

182

9. 3. OPPORTUNITIES FOR FURmER RESEARCH

Recently, a number of comparative studies of existing algorithms for employee
scheduling have been performed, see Bechtold, Brusco and Showalter [1991] and Li,
Robinson and Mabert [1991]. Each study used test problems generated with unique
problem generators, including this study. While these methods for generation of test
problems have merit, use of a standard technique for generation of test problems, or a
standardized set of test problems would make results comparable across studies, and save
researchers time and expense, while guaranteeing that the algorithm under investigation
gets a fair and vigorous (as the case may be) test. The use of a standardized problem set,
or a standard method for generating such problems, would also give the consumers of
such research more confidence in the results of research into new methods of problem
solution. Several other areas, such as the traveling salesman problem, have standard test
problems, with varying degrees of difficulty, that researchers use to test their algorithms.

The performance of both simulated annealing and Tabu search rely in large part
on the quality of the transition generators. In this research, effort was expended to ensure
as much as possible that neither algorithm, simulated annealing and tabu search, had an
advantage based on the set of transition generators used in the research. This was done by
using the same transition generators for each algorithm, with algorithm specific
modifications made during implementation. Further investigation into transition
generators probably would yield useful improvements in the performance of both
algorithms.

During each iteration of tabu search, every possible transition is generated using
the transition generators applied in the particular implementation. The time-to-
completion performance of tabu search could be improved by developing a method of

reusing previously generated solutions that would not have been affected by use of a

183

transition to improve the current solution. If such a method were found and used, a
substantial proportion of the previously generated potential transitions could be reused,
saving computation time.

Prior research into simulated annealing has not established a set of guidelines on
when a particular type of cooling schedule might be useful. Most research has used what
van Laarhoven and Aarts [1987] call "conceptually simple" cooling schedules. This
research compared such conceptually simple cooling schedules with more complex
cooling schedules, and found that at least one of the more complex schedules yielded
better results, and, therefore, was used in the remainder of the research. While several
cooling schedules were investigated during the course of this research, there are many
types and permutations of cooling schedules that have not been investigated. Further
research into cooling schedules for simulated annealing probably would find a cooling
schedule that would converge to a better solution more quickly.

Practical testing of these algorithms in service firms is a reasonable next step in
continuing this research. Since the Obj ective firnction in tabu search and simulated
annealing is easily changed to accommodate the needs of different firms, they would

seem to be prime candidates for such real-world testing.

APPENDICES

184

APPENDIX 1.

CALCULATION OF THEORETICAL NUMBER OF WORK TOURS ADAPTED FROM LOUCKS
[1987].

NOTATION

D = Number of Operation Days per Week.

Dmin = Allowed Minimum Number of Workdays per Week

Dmax = Allowed Maximum Number of Workdays per Week
H = Number of Operating Hours per Day

Hmin = Allowed Minimum Number of Work Hours per Day

Hmax = Allowed Maximum number of Work Hours per Day

ASSUMPTIONS

A worker's work hours within a day must be contiguous, a worker's work days
within a week need not be contiguous, and the worker's number of hours per day need

not be constant.

185

EQ UAHON

D H
28x11 230mm 1N Bum! (1)-N)!»
N=Dmin M=Hmin

EXAWLE
D=7Days H=18Hours
Dmin = 1 Day Hmin = 3 Hours

Dmax = 5 Days Hmax = 8 Hours

 

Numb er Of Number Of
Workdays (N) Tours In Subset
l 567
2 137,781
3 18,600,435
4 1,506,635,235
5 73,222,472,421

 

Total Tours 74,747,846,439

186

APPENDIX 2

TRANSFORMATION OF PROBLEM FORMULATION INTO SINGLE FUNCTION FORM

Transformation of the integer program formulation into single function form is
necessary for the application of the tabu search and simulated annealing. In this
appendix, that transformation will be accomplished.

In a tabu search and simulated annealing there is no way to accommodate
constraints directly. Since there are constraints in the problem formulation, they need to
be incorporated into the objective function. This is done by using the penalty method . In
a penalty method, a constrained problem in optimization is transformed to an
unconstrained problem by associating a cost or penalty with each violation of a
constraint.

Suppose for example we have a constrained problem such as :

minimize g(x)

subject to hi(x >= 0, i = 1,2,...,n

where x is an m vector.

We transform this to the unconstrained form:

minimize g(x) + 6*}; (I>[hi(x)]
i

where (I) = penalty function

187

p = penalty coefficient.

Several alternatives exist for the penalty function (I). A common penalty function
is to square the violation of the constraint, this is the firnction that we will use. p values
are generally proportioned so that moderate violations of the constraint yield a penalty
that is a significant percentage of the nominal operating cost. The p values will be
assigned after field experience.

When we apply the method outlined above, the following objective firnction is

generated:

Fr= W1*Zejk1 + Wz" (ZXijkm - hi)

jkl ijkl
+ 91*(rjk1+ eikl - Z xijkl)2

ijkl

+ 92*(1 - ZXijsz

ijkl
+ 93 *(Smin - Exam?

jk
+ 94*(IUill * 912743102
jk
‘1' 95*(2Xijkl ' smax)2
jk

+ 96*(23'11 - wmmoz
1

+ 97*(ZXijk1 - Xij(k+l)l + xij(k+2)l)2
1

It will be noted that the p], p2, etc. subscripts correspond to and indicate the
constraints as listed in the formulation in section 3. Since the most of the constraints are
inequality constraints, the value of the term to be squared determines whether it gets
included as a penalty, i.e. if the summation takes on any positive value then the constraint
is violated, so the term gets squared and included as a penalty, otherwise (the term's

value less than or equal to 0) it is ignored.

188

APPENDIX 3

SYNTHETIC PROBLEM GENERATOR

A computer program, due in large part to Loucks [1987], will be used to generate
test problems in which experimental factors can be controlled and for which Optimal
solutions are known.

The first step in the program is to read in several general problem parameters;
Number of Workers (I), Number of Tasks (I), Number of Operating Hours (K), Number
of Days (L), Allowed Minimum (3min) and Maximum (Smax) Shift Length, and the
Allowed Maximum Number of WorkDays (Wmax)-

The next step in the procedure is to decide for each worker i, how many tasks ”ii
and which specific tasks Ti the worker is qualified to perform. |Ti| is chosen by sampling
from a discrete probability distribution and Ti is chosen by sampling from a uniform
distribution.

Then for each worker i, IWiI, the number of work days, is chosen from a discrete
probability distribution. The workers specific workdays, Wi, are chosen by sampling
from a uniform distribution.

The worker's first assigned hour bi] is chosen by sampling from a discrete
probability distribution. Then the length of worker i's assigned shift for workday 1, Sil, is

decided by sampling from another discrete probability distribution.

189

Using the same worker i and workday l task j is chosen from Ti for each hour k
from bi] to Ci] by sampling from a uniform distribution. The steps outlined above are
then repeated for each day and each worker to constitute a schedule.

At this point, decisions regarding a specific worker's availability must be
addressed. These three decisions are what days , how many hours on those days, and the
first hour the worker is available. These decisions are made by looking at the schedule
already developed and setting availabilities equivalent to the schedule extant. Additional
availability is determined by sampling from a uniform distribution for days not already
scheduled, hours not already scheduled and hours earlier than those scheduled.

Finally, calculation of the staffing requirements, rijkl, for each task j in each hour
k of each day l, and of the workers targeted hour hi is made. The staffing requirements
are set equal to the number of workers scheduled to the task-hour. The number of
targeted tour hours for each worker is set to the number of hours scheduled.

The three solution methods would receive as input parameters, the rijkl: hi, and

the several parameters input to the problem generator.

 

Table A4-1. Raw Data, Continued.

190

APPENDIX 4

RAW DATA

The raw data from runs of the algorithms is as follows in table A4-l.

Table A4-1. Raw Data

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Awus PCTNTN$ MMRs Afdorrmms TOS TSD OBJ COST TIME

High High High Loucks 91 25 116 6237 52.66992
High High High Loucks 58 12 70 4264.25 27.73828
High High High Loucks 88 29 117 6575.25 54.15039
High High High Loucks 52 26 78 6179 42.07031
High High High Loucks 75 23 98 4736 33.55859
High High High Loucks 57 24 81 5781.25 36.62891
High High High Loucks 77 13 90 3506.25 20.04883
High High High Loucks 142 10 152 3422.5 24.92969
High High High Loucks 73 12 85 3996 24.21875
High High High Loucks 91 17 108 4495.5 27.58008
High High High Loucks 115 13 128 4930.25 35.15039
High High High Loucks 82 12 94 3646.5 27.52148
High High High Loucks 111 18 129 6354.75 49.88086
High High High Loucks 72 10 82 3795 26.4707
High High High Loucks 28 18 46 4141.5 29.21875
High High High Sim_Ann 80 41 121 5808 870.9895
High High High Sim_Ann 67 30 97 4042 1136.027
High High High Sim_Ann 78 37 115 6130 1194.244
High High High Simjnn 76 18 94 5809 952.1747
High High High Simjnn 39 11 50 4422 1080.146
High High High Sim_Ann 67 17 84 5421 942.2617
High High High Sim_Ann 41 10 51 3399 1074.136
High High High Sim_LAnn 34 21 55 3376 1271.589

 

 

 

 

 

 

 

 

 

 

Table A4-l. Raw Data, Continued.

191

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

High High High Sim_Ann 47 40 87 3867 1 141 .091
High High High Sim_Ann 36 14 50 4246 1657.1 1 5
High High High Sim_Ann 62 13 75 4690 1 222.279
High High High Sim_Ann 72 18 90 3531 1497.032
High High High Sim_Ann 75 42 1 17 6022 1080.31 2
High High High Sim_Ann 84 44 128 3762 91 8.1 513
High High High Sim_Ann 64 17 81 4051 1491 .598
High High High Tabu_Srch 126 31 1 57 6204 6585.908
High High High Tabu_Srch 1 13 1 34 247 4541 .75 4460.722
High High High Tabu_Srch 1 16 19 135 6484.5 8563.681
High High High Tabu_Srch 1 15 35 150 6225.25 6221 .88
High High High Tabu_Srch 88 74 162 4948.75 571 1 .454
High High High Tabu_Srch 103 33 136 581 8.25 8010.408
High High High Tabu_Srch 85 104 189 3852.75 5596.472
High High High Tabu_Srch 85 208 293 3894.25 2776.129
High High High Tabu_Srch 86 101 1 87 4347. 5 3048.144
High High High TahISrch 90 94 184 4800.75 3567.977
High High High Tabu_Srch 103 1 16 21 9 5050. 5 7707.466
High High High Tabu_Srch 122 1 36 258 3960 2735.551
High High High Tabu_Srch 1 14 47 161 6428.75 6650.193
High High High Tabu_Srch 107 71 178 3984.75 5354.23
High High High Tabu_Srch 93 88 181 4331 .25 551 1 .541
High High Low Loucks 128 17 145 6327 54. 54004
High High Low Loucks 23 39 62 7353.75 51 .95996
High High Low Loucks 33 20 53 6662.5 45.86035
High High Low Loucks 84 32 1 16 7187.25 61 .18945
High High Low Loucks 87 37 1 24 5948.25 45.3701 2
High High Low Loucks 90 22 1 12 7390.75 50.42969
High High Low Loucks 62 35 97 7045.5 58.22949
High High Low Loucks 32 28 60 4669.5 32.02051
High High Low Loucks 102 25 127 8722.75 75. 52051
High High Low Loucks 70 34 104 7078.5 51 .58008
High High Low Loucks 151 23 174 5975.5 63.26953
High High Low Loucks 59 30 89 6410.25 41 .46973
High High Low Loucks 107 36 143 6195.75 54.93066
High High Low Loucks 77 20 97 7210.5 54.27051
High High Low Loucks 108 29 1 37 6839.25 52.73047
High High Low Sim_Ann 131 37 168 601 3 851 .1406
High High Low Sim_Ann 90 38 128 6762 1051 .93
High High Low Sim_Ann 80 4 84 6468 1 297
High High Low Sim_Ann 89 58 147 6651 1437.078
High High Low Sim_Ann 103 25 128 5602 952.4062
High High Low Sim_Ann 138 33 171 7104 81 1 .4726
High High Low Sim_Ann 95 46 141 6575 1043.68
High High Low Sim_Ann 59 10 69 4389 1089.508
High High Low Sim_Ann 1 60 40 200 8408 832.7734

 

 

 

 

 

 

 

 

 

 

Table A4-1. Raw Data, Continued.

192

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

High High Low Sim_Ann 98 35 133 6551 1085.328
High High Low Sim_Ann 93 18 1 1 1 5661 917.461
High High Low Sim_Ann 75 1 1 86 6105 684.4594
High High Low Sim_Ann 106 15 121 5841 1060.281
High High Low Sim_Ann 137 21 1 58 6938 1268.766
High High Low Sim_Ann 1 10 45 1 55 631 1 1038.438
High High Low Tabu_Srch 163 59 222 6280.75 5279.299
High High Low Tabu_Srch 137 1 1 1 248 7215 9323.22
High High Low Tabu_Srch 140 186 326 7154.5 4141 .21 1
High High Low TabeSrch 146 53 199 7224.25 5043.595
High High Low Tabu_Srch 141 83 224 5907 8455.643
High High Low Tabu_Srch 178 51 229 7474 5767.549
High High Low Tabu_Srch 139 1 16 255 6963 9455.264
High High Low Tabu_Srch 93 76 169 4686 7685.332
High High Low Tabu_Srch 194 94 288 8704.25 7929.292
High High Low Tabu_Srch 156 257 41 3 7012.5 5741 .82
High High Low Tabu_Srch 129 76 205 6012.5 8303.015
High High Low Tabu_Srch 127 89 21 6 6660 41 49.88
High High Low Tabu_Srch 137 38 175 6105 8883.984
High High Low Tabu_Srch 165 29 194 7177.5 6987.894
High High Low Tabu_Srch 161 22 183 6748. 5 9041 .403
High High Med Loucks 105 22 127 6391 .75 48.56055
High High Med Loucks 43 40 83 7390.75 61 .50977
High High Med Loucks 27 21 48 6703.5 41 .29883
High High Med Loucks 48 32 80 7298.25 79.41992
High High Med Loucks 118 26 144 5923. 5 51.52148
High High Med Loucks 62 26 88 7501 .75 57.60938
High High Med Loucks 78 31 109 7062 58.05078
High High Med Loucks 18 28 46 4686 31 .201 17
High High Med Loucks 66 28 94 8833.75 79.9707
High High Med Loucks 60 33 93 7078. 5 55.46875
High High Med Loucks 1 19 28 147 6031 51.95898
High High Med Loucks 63 32 95 6484.25 44.43945
High High Med Loucks 45 36 81 6245.25 49.44141
High High Med Loucks 67 32 99 7359 55.31055
High High Med Loucks 96 27 123 6839.25 55.08984
High High Med Sim_Ann 139 39 178 6050 1368.738
High High Med Sim_Ann 101 21 122 6845 1666. 99
High High Med Sim_Ann 74 1 75 6478 1094.17
High High Med Sim_Ann 95 16 11 1 6854 1 190.841
High High Med Sim_Ann 89 22 11 1 5561 1237.52
High High Med Sim_Ann 141 15 156 7160 1463.881
High High Med Sim_Ann 86 37 123 6584 1461 .34
High High Med Sim_Ann 99 30 129 4422 1783.32
High High Med Sim_Ann 124 46 170 81 86 1350.07
High High Med Simhn 124 19 143 6749 1 518.141

 

 

 

 

 

 

 

 

 

 

 

Table A4-1. Raw Data, Continued.

193

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

High High Med Sim_Ann 88 21 109 5633 1 540.77
High High Med Sim_Ann 80 4 84 61 51 1 1 52.434
High High Med Sim_Ann 93 16 109 5816 1433.445
High High Med Sim_Ann 133 33 166 6971 1767.952
High High Med Sim_Ann 128 22 1 50 6485 1471 .392
High High Med Tabu_Srch 163 59 222 6271 .5 8252.854
High High Med Tabu_Srch 146 94 240 7316.75 5862.253
High High Med Tabu_Srch 139 196 335 7246.75 5933.308
High High Med Tabu_Srch 127 30 1 57 7205.75 7637.562
High High Med Tabu_Srch 140 85 225 6014.25 8065.934
High High Med Tabu_Srch 190 132 322 7612.75 8357.039
High High Med Tabu_Srch 129 58 187 6921 .75 6007.556
High High Med Tabu_Srch 126 73 1 99 4669. 5 4475.663
High High Med Tabu_Srch 184 70 254 8713.5 7101 .482
High High Med Tabu_Srch 165 202 367 7086.75 6491 .224
High High Med Tabu_Srch 131 96 227 6068 3626.68
High High Med Tabu_Srch 122 64 1 86 6576.75 5468.371
High High Med Tabu_Srch 142 89 231 6270 4127.216
High High Med Tabu_Srch 153 1 1 164 7169.25 6120.851
High High Med Tabu_Srch 157 25 182 6765 6385.818
High Low High Loucks 184 29 213 4142.25 41 .13965
High Low High Loucks 104 25 1 29 3874. 5 36.85059
High Low High Loucks 91 23 1 14 3433. 5 29.77051
High Low High Loucks 103 41 144 4812.5 38.1 709
High Low High Loucks 65 24 89 4281 .25 35.04004
High Low High Loucks 1 54 24 178 3769.5 29.21 973
High Low High Loucks 1 10 23 133 3979. 5 38.87988
High Low High Loucks 104 21 125 3549 39.65039
High Low High Loucks 109 25 134 3916.5 41 .46973
High Low High Loucks 145 30 175 4163.25 43.71973
High Low High Loucks 197 22 219 3412.5 54.26953
High Low High Loucks 52 36 88 3869.25 21 .86035
High Low High Loucks 126 21 147 3501 .75 28.71973
High Low High Loucks 142 1 8 1 60 3402 25.0498
High Low High Sim_Ann 56 1 67 223 4279 1 679.924
High Low High Sim_Ann 45 1 59 204 3948 1 852.921
High Low High Sim_Ann 49 1 73 222 3570 1 876.672
High Low High Sim_Ann 84 32 1 16 4594 1353.788
High Low High Sim_Ann 67 51 1 18 4275 1 604.355
High Low High Sim_Ann 42 1 30 1 72 3612 1724.24
High Low High Sim:Ann 49 168 217 3864 1728.107
High Low High Sim_Ann 45 147 1 92 4095 2591 .735
High Low High Sim_Ann 63 1 44 207 3659 1 605.01 8
High Low High Sim_Ann 78 203 281 4053 2301 .054
High Low High Sim_Ann 77 433 510 4389 3225.4
High Low High Simjflm 57 242 299 3539 1850.497

 

 

Table A4-1. Raw Data, Continued.

194

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

High Low High Sim_Ann 35 1 50 185 3848 1 605.744
High Low High Sim_Ann 44 87 131 3575 1 303.976
High Low High Sim_Ann 45 81 126 3439 1 704.242
High Low High Tabu_Srch 48 1 39 187 4247.25 7049.514
High Low High Tabu_Srch 45 1 59 204 3942.75 7550.317
High Low High Tabu_Srch 35 1 15 150 3512.25 9792.619
High Low High Tabu_Srch 1 19 1 19 238 4850 7469.008
High Low High Tabu_Srch 1 15 247 362 4625 6355.903
High Low High Tabu_Srch 37 89 126 3591 4862.726
High Low High Tabu:Srch 33 142 175 3780 7528. 504
High Low High Tabu_Srch 25 65 90 4032 9206.476
High Low High Tabu_Srch 49 130 179 3601 .5 9594.32
High Low High Tabu_Srch 50 97 147 3906 7845.326
High Low High Tabu_Srch 46 196 242 4231 .5 8563.857
High Low High Tabu_Srch 60 179 239 3554.25 5538.024
High Low High Tabu:Srch 30 109 139 3827.25 10025.25
High Low High TahT__§rch 49 136 185 3601 .5 10067.6
High Low High Tabu_Srch 40 88 128 3412.5 9181 .166
High Low Low Loucks 132 19 1 51 3979.5 76.00977
High Low Low Loucks 162 44 206 3787.5 63.49023
High Low Low Loucks 107 36 143 41 50 46.1 2988
High Low . Low Loucks 143 27 170 3585.75 31.95996
High Low Low Loucks 234 31 265 4882.5 54.87012
High Low Low Loucks 64 29 93 3701 .25 37.57031
High Low Low Loucks 1 52 35 187 4341 .75 58.92969
High Low Low Loucks 1 37 45 1 82 4475 47.73047
High Low Low Loucks 78 20| 98 3360 24.5
High Low Low Loucks 1 13 25I 138 3942.75 37.24023
High Low Low Loucks 165 37I 202 4550 42.73047
High Low Low Loucks 167 30I 1 97 3638.25 41 .63086
High Low Low Loucks 1 28 43 171 4368 44.80957
High Low Low Loucks 140 22 162 3906 36.41992
High Low Low Sim_Ann 76 308 384 4195 2359.82
High Low Low Sim_Ann 96 75 171 3744 1 543.281
High Low Low Sim_Ann 74 59 1 33 4088 1 149.1 64
High Low Low Sim_Ann 55 90 145 3686 1677.21 1
High Low Low Sim_Ann 78 247 325 5009 1 636.944
High Low Low Sim_Ann 49 167 216 3796 1 447.719
High Low Low Sim_Ann 51 1 51 202 4426 1 61 3. 501
High Low Low Sim_Ann 801 88 168 4413 1777.16
High Low Low Sim_Ann 53 214 267 3491 1 503.414
High Low Low Sim_Ann 44 87 1 31 4043 2500.98
High Low Low Sim_Ann 100 147 247 4481 1488.922
High Low Low Sim_Ann 55 189 244 3770 2268.296
High Low Low Sim_Ann 65 268 333 4484 1 579.536
High Low Low Sim_Ann 48 125 1 73 4037 1 764.32

 

 

 

 

 

 

 

 

 

 

 

Table A4-1. Raw Data, Continued.

195

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

High Low Low Sim_Ann 83 1 1 94 5763 201 1 .916
High Low Low Tabu_Srch 61 199 260 4105.5 10421 .29
High Low Low Tabu_Srch 1 24 213 337 3931 .25 7760.36
High Low Low Tabu_Srch 99 146 245 4262.5 10343.34
High Low Low Tabu_Srch 39 66 105 3617.25 6817.057
High Low Low Tabu_Srch 63 200 263 4940.25 8790.045
High Low Low Tabu_Srch 39 1 13 1 52 3780 7635.036
High Low Low Tabu_Srch 43 1 1 5 158 4399.5 1 1071 .67
High Low Low Tabu_Srch 1 14 262 376 461 8.75 9586.769
High Low Low Tabu_Srch 29 68 97 3375.75 8850.957
High Low Low Tabu_Srch 37 86 123 4000. 5 10687.5
High Low Low Tabu_Srch 139 338 477 4743.75 1 1 169.62
High Low Low Tabu_Srch 36 76 1 1 2 3706. 5 10281 .32
High Low Low Tabu_Srch 45 144 189 4389 1 1087.76
High Low Low Tabu_Srch 37 108 145 3979. 5 10342.25
High Low Low Tabu_Srch 135 345 480 61 31.25 10499.45
High Low Med Loucks 85 22 107 4047.75 46.79004
High Low Med Loucks 83 33 1 16 3881 .25 37.1 8945
High Low Med Loucks 134 36 170 4193.75 41 .25
High Low Med Loucks 147 19 166 3596.25 35.96973
High Low Med Loucks 186 31 217 4924.5 52.56055
High Low Med Loucks 70 31 101 3732.75 31.25977
High Low Med Loucks 170 27 197 4352.25 59.10059
High Low Med Loucks 127 48 175 4550 46.95996
High Low Med Loucks 70 19 89 3370.5 24.66016
High Low Med Loucks 1 17 22 139 3953.25 34.7207
High Low Med Loucks 140 37 177 4581.25 41 .74023
High Low Med Loucks 1 30 25 1 55 3654 34
High Low Med Loucks 1 54 40 194 4378. 5 50.37012
High Low Med Loucks 80 26 106 3948 30.20996
High Low Med Sim_Ann 73 345 41 8 4221 4899.347
High Low Med Sim_Ann 94 1 10 204 3888 4409.528
High Low Med Sim_Ann 60 60 120 4081 2993.818
High Low Med Sim_Ann 49 1 27 176 371 2 4306.466
High Low Med Sim_Ann 76 259 335 5061 3958.614
High Low Med Sim_Ann 48 1 82 230 3812 1645.01 5
High Low Med Sim_Ann 49 1 81 230 4468 1 695.454
High Low Med Sim_Ann 86 44 130 4363 1 556.766
High Low Med Sim_Ann 51 209 260 3497 1687.77
High Low Med Sim_Ann 46 1 12 158 4069 1 541 .541
High Low Med Sim_Ann 96 1 56 252 4513 1765.36
High Low Med Sim_Ann 51 1 93 244 3791 1403.4
High Low Med Sim_Ann 57 1 56 213 4463 1446.859
High Low Med Sim_Ann 46 1 1 5 161 4048 1388.879
High Low Med Sim_Ann 74 12 86 5756 1322.522
High Low Med Tabu_Srch 63 285 348 4163.25 1 1078.23

 

 

Table A4-l. Raw Data, Continued.

196

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

High Low Med Tabu_Srch 1 16 266 382 4012.5 6866.375
High Low Med Tabu_Srch 105 191 296 4412.5 7480.438
High Low Med Tabu_Srch 31 81 1 12 3627.75 8283.397
High Low Med Tabu_Srch 56 241 297 4956 1 1325.34
High Low Med Tabu_Srch 39 93 1 32 3801 101 34.56
High Low Med Tabu_Srch 40 176 216 4410 10501 .86
High Low Med Tabu_Srch 1 1 7 285 402 4587.5 10724.9
High Low Med Tabu_Srch 24 72 96 3365.25 9092.756
High Low Med Tabu_Srch 35 91 1 26 4042.5 10700.86
High Low Med Tabu_Srch 1 19 243 362 4656.25 8503.333
High Low Med Tabu_Srch 33 63 96 3738 101 95.12
High Low Med Tabu_Srch 43 1 16 1 59 4394.25 9061 .921
High Low Med Tabu_Srch 35 90 125 4005.75 1031 5.41
High Low Med Tabu_Srch 127 163 290 6125 1 1819.57
High Med High Loucks 1 19 26 145 5038.75 44.33008
High Med High Loucks 74 22 96 3045 22.95996
High Med High Loucks 82 23 105 3443.75 30.65039
High Med High Loucks 127 35 162 561 1 .5 53.87988
High Med High Loucks 171 40 21 1 5169.25 48.71973
High Med High Loucks 69 39 108 5821.75 48.06055
High Med High Loucks 131 35 166 4516.75 36. 58008
High Med High Loucks 85 29 1 14 4828.5 35.25977
High Med High Loucks 55 41 96 4799.5 32.79004
High Med High Loucks 23 42 65 5858 39.33008
High Med High Loucks 82 20 102 4183.25 30.20996
High Med High Loucks 60! 19 79 2834.75 20.65039
High Med High Loucks 52 14 66 321 1.75 21.3701 2
High Med High Loucks 194 42 236 5183.75 47.07031
High Med High Sim_Ann 69 40 109 4865 2276.568
High Med High Sim_Ann 601 25 85 2980 1 370.1 17
High Med High Sim_Ann 64 43 107 3386 1 81 8.139
High Med High Sim_Ann 82 25 107 5365 1433.609
High Med High Sim_Ann 62 1 5 77 4814 1259.098
High Med High Sim_Ann 98 14 1 12 5525 1 519.795
High Med High Sim_Ann 67 59 126 4343 1 869.921
High Med High Sim_Ann 64 22 86 4575 1686.413
High Med High Sim_Ann 68 17 85 4524 1 303.401
High Med High Sim_Ann 94 9 103 5575 1446.48
High Med High Sim_Ann 65 44 109 4053 2151 .914
High Med High Sim_Ann 51 21 72 2748 2397.154
High Med High Sim_Ann 73 38 1 1 1 3125 1846.21 1
High Med High Sim_Ann 66 1 5 81 3857 1 620.617
High Med High Sim_Ann 76 34 1 10 4916 1081 .051
High Med High Tabu_Srch 125 246 371 5256.25 8622.104
High Med High Tabu_Srch 92 141 233 3342.25 2306. 329
High Med High Tabu_Srch 108 1 97 305 3741 3987.81 9

 

 

 

 

 

 

 

 

 

 

Table A4-1. Raw Data, Continued.

197

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

High Med High Tabu_Srch 1 1 9 78 197 5647.75 891 5.841
High Med High Tabu_Srch 104 95 199 5154.75 6051 .842
High Med High Tabu_Srch 121 35 156 5713 8225. 55
High Med High Tabu_Srch 1 14 212 326 4734.25 7728.248
High Med High Tabu_Srch 98 84 182 4857. 5 8344.492
High Med High Tabu_Srch 108 1 35 243 4850.25 7839.282
High Med High Tabu_Srch 144 1 1 5 259 5981 .25 4402.274
High Med High Tabu_Srch 103 1 80 283 4371 .75 6198.898
High Med High Tabu_Srch 73 85 158 2921 .75 3127.562
High Med High Tabu_Srch 100I 101 201 3342.25 4538.172
High Med High Tabu_Srch 96 1 19 215 4103.5 7776.352
High Med High Tabu_Srch 128 1 86 314 5292.5 3691 .746
High Med Low Loucks 81 31 1 12 5459.25 44.37988
High Med Low Loucks 124 39 163 5604.25 54.65039
High Med Low Loucks 140 35 175 5879.75 48.00977
High Med Low Loucks 137 34 171 4908.25 48.71973
High Med Low Loucks 102 28 130 5278 47.24023
High Med Low Loucks 120 27 147 4676.25 45.37012
High Med Low Loucks 58 41 99 5539 43.73047
High Med Low Loucks 89 31 120 4792.25 40.04004
High Med Low Loucks 87 41 128 5655 42.2998
High Med Low Loucks 69 31 100 5082.25 49.60059
High Med Low Loucks 55 37 92 4886.5 38.40039
High Med Low Loucks 11 1 34 145 51 18.5 43.12012
High Med Low Loucks 74 35 109 5814.5 44.92969
High Med Low Loucks 65 47 1 12 6169.75 44.31934
High Med Low Sim_Ann 96 12 108 5271 1379.929
High Med Low Sim_Ann 102 28 130 5278 2564.102
High Med Low Sim_Ann 109 33 142 5612 1066.021
High Med Low Sim_Ann 55 41 96 4568 2029.533
High Med Low Sim_Ann 80 17 97 4995 1442.489
High Med Low Sim_Ann 81 31 1 12 4568 1887.909
High Med Low Sim_Ann 83 35 1 18 5307 1561 .881
High Med Low Sim_Ann 74 29 103 4553 1430.566
High Med Low Sim_Ann 82 46 128 5271 1763.677
High Med Low Sim_Ann 79 1 1 90 4821 1243.98
High Med Low Sim_Ann 97 25 122 6010 1 600.959
High Med Low Simjnn 48 20 68 4604 1404.496
High Med Low Sim_Ann 72 16 88 4887 1832.316
High Med Low Sim_Ann 1 10 40 1 50 5496 1465.438
High Med Low Sim_Ann 103 9 1 12 5800 1410.733
High Med Low Tabu_Srch 123 95 218 5488.25 9620.726
High Med Low Tabu_Srch 147 167 314 5589.75 9669.143
High Med Low Tabu_‘Srch 145 93 238 5901 .5 91 1 1 .43
High Med Low Tabu‘Srch 125 283 408 5089.5 6948.048
High Med Low Tabu_Srch 128 149 277 5350. 5 9769.343

 

 

 

 

 

 

 

 

 

 

Table A4-1. Raw Data, Continued.

198

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

High Med Low Tabu_Srch 105 1 19 224 4799.5 9298.901
High Med Low Tabu_Srch 129 271 400 5655 9337.635
High Med Low Tabu:Srch 1 12 21 9 331 4843 5824.058
High Med Low Tabu_Srch 140 214 354 5676.75 8790.442
High Med Low Tabu_Srch 133 125 258 5285.25 6662.191
High Med Low Tabu_Srch 146 66 21 2 6351 6346.297
High Med Low Tabu_Srch 93 97 1 90 4951 .75 7956.849
High Med Low Tabu_Srch 103 67 170 5125.75 91 51 .667
High Med Low TahufSrch 137 69 206 5698.5 10176.05
High Med Low Tabu_Srch 141 1 67 308 6075.5 5976.499
High Med Med Loucks 41 32 73 5502.75 43
High Med Med Loucks 131 34 165 5611.5 48.05957
High Med Med Loucks 84 40 1 24 5952.25 45.4209
High Med Med Loucks 82 39 1 21 4988 43.94043
High Med Med Loucks 102 28 1 30 5278 47.23047
High Med Med Loucks 94 24 1 18 4676.25 38.73047
High Med Med Loucks 71 38 109 5560.75 60.68945
High Med Med Loucks 69 30 99 4806.75 37.23926
High Med Med Loucks 85 35 120 5640.5 45.30957
High Med Med Loucks 92 32 124 51 1 8.5 49.70996
High Med Med Loucks 97 54 1 51 6568.5 52.33984
High Med Med Loucks 60 40 100 4922.75 36.4707
High Med Med Loucks 105] 30 135 5147.5 41.7998
High Med Med Loucks 68] 38 106 5858 46.09082
High Med Med Loucks 62] 45 107 61 77 53.94043
High Med Med Sim_Ann 100 9 109 5293 2783.278
High Med Med Sim_Ann 85 55 140 5213 3167.208
High Med Med Sim_Ann 1 14 31 145 5612 2058.607
High Med Med Sim_Ann 52 26 78 4589.25 2089.365
High Med Med Sim_Ann 68 27 95 4908 2984.844
High Med Med Sim_Ann 81 27 108 4524 3102.753
High Med Med Sim_Ann 83 29 1 12 5263.5 2814.46
High Med Med Sim_Ann 65 38 103 4487.75 1054.1 18
High Med Med Sim_Ann 83 44 127 5357.75 1613.1 14
High Med Med Sim_Ann 94 2 96 4901 2587.141
High Med Med Sim:Ann 107 12 1 19 6104.5 2830.647
High Med Med Sim_Ann 53 19 72 4654.5 2327.558
High Med Med Sim_Ann 73 23 96 4908.25 1996.978
High Med Med Sim:Ann 123 1 1 134 5604.25 1 101 .294
High Med Med Sim:Ann 102 10 1 12 5821.75 3132.777
High Med Med Tabu_Srch 1 29 108 237 5502.75 8048.395
High Med Med Tabu_Srch 158 298 456 5756.5 1021 1 .44
High Med Med Tabu_Srch 1 55 136 291 5974 6743.694
High Med Med Tabu_Srch 1 16 220 336 5075 7556.386
High Med Med Tabu_Srch 128 149 277 5350.5 9769.5
High Med Med TabuiSrch 1 26 170 296 4843 71 98.274

 

 

 

Table A4-l. Raw Data, Continued.

199

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

High Med Med Tabu_Srch 1 15 1 17 232 5553.5 9259.342
High Med Med Tabu_Srch 125 248 373 4995.25 9666.027
High Med Med Tabu_Srch 1 32 21 5 347 5763.75 10024.34
High Med Med Tabu_Srch 129 77 206 5183.75 9218.443
High Med Med Tabu_Srch 1 36 67 203 6314.75 10302.79
High Med Med Tabu_Srch 94 98 192 5009.75 7389.456
High Med Med Tabu_Srch 107 85 192 5147.5 9775.879
High Med Med Tabu_Srch 141 67 208 5756. 5 10023.01
High Med Med Tabu_Srch 136 1 16 252 6068.25 7241 .36
Low High High Loucks 219 36 255 3778. 5 49.1001
Low High High Loucks 73 34 107 5296. 5 43.01025
Low High High Loucks 53 47 100 5984.75 46.56982
Low High High Loucks 141 47 188 3316.5 34. 5498
Low High High Loucks 72 32 104 4125 25.42969
Low High High Loucks 58 38 96 5280 102.6104
Low High High Loucks 137 51 188 5032. 5 42.24023
Low High High Loucks 85 47 132 5124.5 36.41992
Low High High Loucks 86 39 125 3481 .5 30.3701 2
Low High High Loucks 292 47 339 5057.25 39.9301 8
Low High High Loucks 43 35 78 41 74.5 33.00977
Low High High Loucks 163 48 21 1 4207. 5 31 .25977
Low High High Loucks 103 54 1 57 4083.75 29.77002
Low High High Loucks 107 38 145 3456.75 29.72021
Low High High Loucks 91 51 142 6080.25 54.47998
Low High High Sim_Ann 28 24 52 3514. 5 1262.175
Low High High Sim_Ann 62 6 68 4983 618.2425
Low High High Sim_Ann 61 1 62 5540.75 739.0205
Low High High Sim_Ann 38 17 55 301 9. 5 1025.09
Low High High Sim_Ann 42 3 45 3836.25 895.12
Low High High Sim_Ann 84 12 96 4884 761 .5492
Low High High Sim_Ann 59 24 83 4578.75 690.6294
Low High High Sim_Ann 58 2 60 4671 .25 604.34
Low High High Sim_Ann 36 23 59 3267 588.0055
Low High High Sim_Ann 63 1 64 4661 .25 805.6214
Low High High Sim_Ann 51 3 54 3877.5 743.4317
Low High High Sim_Ann '49 26 75 3861 632.9564
Low High High Sim_Ann 47 1 1 58 3696 716.1358
Low High High Sim_Ann 41 36 77 3242.25 1045.64
Low High High Sim_Ann 91 1 8 109 5527. 5 676.9068
Low High High Tabu_Srch 102 190 292 4191 6806. 579
Low High High Tabu_Srch 106 58 164 5403.75 1 1225.35
Low High High Tabu_Srch 120 102 222 6086. 5 121 14.73
Low High High Tabu_Srch 1 1 1 210 321 3638.25 4286.863
Low High High Tabu_Srch 131 194 325 4603.5 9085.792
Low High High Tabu_Srch 141 73 214 5403.75 9445.887
Low High High Tabu_Srch 1 37 174 31 1 5247 9061 .55

 

 

 

 

 

 

 

 

 

 

Table A4-1. Raw Data, Continued.

200

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Low High High Tabu_Srch 1 16 1 22 238 5244.75 10897.02
Low High High Tabu_Srch 107 206 313 391 8.75 6260.292
Low High High Tabu_Srch 148 174 322 5387.25 12266.45
Low High High Tabu_Srch 106 1 38 244 4356 5831 .373
Low High High Tabu_Srch 140 243 383 4644.75 71 58.472
Low High High Tabu_Srch 101 135 236 4207.5 10363.71
Low High High Tabu_Srch 97 188 285 3770.25 9280.053
Low High High Tabu_Srch 140 39 1 79 5956.5 12449.53
Low High Low Loucks 73 60 1 33 4941 .75 42. 57007
Low High Low Loucks 195 66 261 5254 48.6101 1
Low High Low Loucks 185 67 252 7380 1 17.27
Low High Low Loucks 174 75 249 6270 56.07007
Low High Low Loucks 133 51 1 84 3802.75 40.20996
Low High Low Loucks 85 63 148 6252. 5 56.67993
Low High Low Loucks 85 42 1 27 3330 29.27002
Low High Low Loucks 121 77 1 98 6456.5 86.5
Low High Low Loucks 93 39 1 32 3034 27.67993
Low High Low Loucks 251 65 316 5254 49.55005
Low High Low Loucks 81 85 166 6971 .25 96.16992
Low High Low Loucks 108 68 176 5123.25 52.72998
Low High Low Loucks 177 64 241 4892.25 41 .03003
Low High Low Loucks 207 94 301 7647.75 107.4299
Low High Low Loucks 91 70 161 6954.75 105.73
Low High Low Sim_Ann 101 10 1 1 1 4364.25 776.708
Low High Low Sim_Ann 92 1 2 104 4551 568.9242
Low High Low Sim_Ann 164 25 1 89 6437 649.8895
Low High Low Sim_Ann 136 201 1 56 5502.75 645.7544 .
Low High Low Sim_Ann 35 6 41 3239 51 8.4334
Low High Low Sim_Ann 1 11 25 136 5412 551 .6926
Low High Low Sim_Ann 34 9 43 2876.75 732.98
Low High Low Sim_Ann 148 19 167 5605.5 549.7141
Low High Low Sim_Ann 44 9 53 2682.5 818.6563
Low High Low Sim_Ann 83 16 99 4541 .75 565.993
Low High Low Sim_Ann 161 1 5 176 6146.25 768.1727
Low High Low Sim_Ann 1 1 1 5 1 16 4537.5 644.2188
Low High Low Sim_Ann 78 21 99 4191 821 .1 563
Low High Low Sim_Ann 265 32 297 6608.25 669. 5452
Low High Low Sim_Ann 200 26 226 6179.25 628.918
Low High Low Tabu_Srch 148 63 21 1 481 8 9574.344
Low High Low Tabu_Srch 134 44 178 4976.5 141 96.18
Low High Low Tabu_Srch 194 5 1 99 6734.25 10025.82
Low High Low Tabu_Srch 160 18 178 5717.25 10530.81
Low High Low Tabu_Srch 91 106 197 3843.75 1 1088.67
Low High Low Tabu_Srch 143 23 166 5729.75 10381 .87
Low High Low Tabu_Srch 98 1 1 1 209 3468.75 10269.04
Low High Low Tabu_Srch 167 14 1 81 5836.75 1 5019.46

 

 

 

 

 

 

 

 

 

 

Table A4-l. Raw Data, Continued.

201

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Low High Low Tabu_Srch 93 94 1 87 31 82 7797.804
Low High Low Tabu_Srch 108 17 1 25 4837.75 1 1 825.61
Low High Low Tabu_Srch 172 8 1 80 6303 1 5021 .73
Low High Low Tabu_Srch 130 1 8 148 4752 1 2400.55
Low High Low Tabu_Srch 125 36 1 61 4603. 5 8240.825
Low High Low Tabu_Srch 279 1 8 297 6707.25 20470.95
Low High Low Tabu:Srch 231 9 240 6435 20707.08
Low High Med Loucks 187 58 245 5222.25 47.01001
Low High Med Loucks 233 52 285 471 9 44.92993
Low High Med Loucks 194 44 238 5618.25 49.20996
Low High Med Loucks 1 63 46 209 5428. 5 52.83984
Low High Med Loucks 66 48 1 14 5255.25 40.42993
Low High Med Loucks 1 93 33 226 2945.25 36.58008
Low High Med Loucks 139 22 161 1716 25.76001
Low High Med Loucks 260 41 301 3638.25 41 .03003
Low High Med Loucks 69 54 123 4661 .25 33.38989
Low High Med Loucks 134 41 175 2813.25 37.9001 5
Low High Med Loucks 318 32 350 4290 54.70996
Low High Med Loucks 63 37 100 3902.25 31 .68994
Low High Med Loucks 281 45 326 3968.25 47.1 8018
Low High Med Loucks 51 40 91 3588.75 26.74976
Low High Med Loucks 108 41 149 4504. 5 40.47998
Low High Med Sim_Ann 67 14 81 4760.25 720.7294
Low High Med Simjnn 44 3 47 4281 .75 953.06
Low High Med Sim_Ann 89 9 98 5197.5 1332.813
Low High Med Sim:Ann 72 15 87 4925.25 942.5352
Low High Med Sim_Ann 50 12 62 4793.25 812.043
Low High Med Sim:Ann 39 41 80 2813.25 1145.645
Low High Med Sim_Ann 25 83 108 1707.75 728.7656
Low High Med Sim_Ann 37 19 56 3357.75 945. 81 26
Low High Med Sim_Ann 51 3 54 4191 742.7032
Low High Med Sim_Ann 33 50 83 2 590. 5 686. 5468
Low High Med Sim_Ann 58 51 109 41 16.75 1027.223
Low High Med Sim_Ann 39 7 46 3638.25 734.4688
Low High Med Sim_Ann 52 97 149 3770.25 2481 .734
Low High Med Sim_Ann 30 5 35 3300 1407.609
Low High Med Sim_Ann 51 3 54 4191 1 1 80.266
Low High Med Tabu_Srch 1 18 161 279 5197.5 9737.797
Low High Med Tabu_Srch 134 213 347 5057.25 7058.071
Low High Med Tabu_Srch 144 84 228 5676 9737.406
Low High Med Tabu_Srch 129 74 203 5445 9508.459
Low High Med Tabu_Srch 129 147 276 5461 .5 10870.24
Low High Med Tabu_Srch 89 1 59 248 3242.25 9921 .771
Low High Med Tabu_Srch 45 107 1 52 1947 7619.292
Low High Med Tabu_Srch 105 1 97 302 3960 3957.21 5
Low High Med Tabu_Srch 1 10 1 14 224 4702.5 1 1 338.54

 

 

 

 

 

 

 

 

 

 

Table A4-l. Raw Data, Continued.

202

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Low High Med Tabu_Srch 82 1 55 237 3102 6129.08
Low High Med Tabu_Srch 102 147 249 4529.25 9745.893
Low High Med Tabu_Srch 1 1 1 169 280 4273.5 6335.663
Low High Med Tabu_Srch 93 1 64 257 4207.5 8612.497
Low High Med Tabu_Srch 89 142 231 3828 10255.3
Low High Med Tabu_Srch 106 1 24 230 4686 7409.092
Low Low High Loucks 165 41 206 3001 .5 48.5
Low Low High Loucks 148 42 1 90 4581 .5 33.73
Low Low High Loucks 125 50 175 3687. 5 43.78
Low Low High Loucks 75 59 134 3512.5 33.17
Low Low High Loucks 295 52 347 2468.75 33.61
Low Low High Loucks 198 58 256 3175 37.01999
Low Low High Loucks 161 57 218 3968.75 44.16
Low Low High Loucks 483 40 523 2793 77.28
Low Low High Loucks 175 58 233 3056.25 37.46002
Low Low High Loucks 191 75 266 4425 40.63998
Low Low High Loucks 519 69 588 4131.25 57.99997
Low Low High Loucks 303 70 373 3687.5 46.35004
Low Low High Loucks 179 59 238 2831 .25 32.24005
Low Low High Sim_Ann 27 62 89 2849.25 1 189.801
Low Low High Sim_Ann 31 20 51 4307 1 668.34
Low Low High Sim_Ann 55 29 84 3493.75 2542.29
Low Low High Sim_Ann 52 36 88 3268.75 2662.776
Low Low High Sim_Ann 33 80 1 13 2331 .25 1 1 72.422
Low Low High Sim_Ann 46 54 100 2950 2347.958
Low Low High Sim_Ann 59 64 123 3775 1 234.07
Low Low High Sim_Ann 22 94 1 16 2000.25 1 557.25
Low Low High Sim_Ann 70 242 312 3318 994.3298
Low Low High Sim_Ann 33 159 192 2756.25 1082.375
Low Low High Sim_Ann 65 125 190 3012.5 3080.866
Low Low High Sim_Ann 79 107 186 4062.5 3002.002
Low Low High Sim_Ann 62 60 122 3887.5 1 1 45.521
Low Low High Sim_Ann 64 84 148 3450 3424.082
Low Low High Sim_Ann 45 63 108 2631 .25 1009.422
Low Low High Tabu_Srch 40 71 1 1 1 3008.75 2412.863
Low Low High Tabu_Srch 49 48 97 4502 2762.699
Low Low High Tabu_Srch 93 125 218 3756.25 2355.824
Low Low High Tabu_Srch 87 1 1 1 198 3506.25 2630.1 17
Low Low High Tabu_Srch 71 172 243 2643.75 1612.938
Low Low High Tabu_Srch 83 135 21 8 3231 .25 2737.1 56
Low Low High Tabu_Srch 121 258 379 4212.5 2564.738
Low Low High Tabu_Srch 24 66 90 2037 1 267.51 6
Low Low High Tabu_Srch 58 128 1 86 3265.5 2863.637
Low Low High Tabu_Srch 27 91 118 2735.25 1954.31
Low Low High Tabu_Srch 103 229 332 3306.25 1667.656
Low Low High TahiZSrch 109 141 250 4312.5 1636.696

 

 

 

Table A4-1. Raw Data, Continued.

203

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Low Low High Tabu_Srch 1 35 379 514 4400 1 531 .27
Low Low High Tabu_Srch 96 1 64 260 3668.75 2345 .96
Low Low High Tabu_Srch 92 224 316 2962.5 999.8594
Low Low Low Loucks 194 39 233 2604 89.14844
Low Low Low Loucks 1 54 25 179 1438.5 59.03906
Low Low Low Loucks 178 52 230 2766.75 77.32813
Low Low Low Loucks 141 58 199 2950 34.71094
Low Low Low Loucks 92 65 1 57 3486 69.26563
Low Low Low Loucks 331 51 382 3055.5 1 83.5625
Low Low Low Loucks 283 56 339 251 8.75 51 .57031
Low Low Low Loucks 181 32 213 1727.25 83.32031
Low Low Low Loucks 164 63 227 2803. 5 74. 58594
Low Low Low Loucks 137 49 1 86 321 8.25 83.1 6406
Low Low Low Loucks 250 67 317 2850 51 .46094
Low Low Low Loucks 209 40 249 2609.25 102.6016
Low Low Low Loucks 1 26 27 1 53 1449 58.9921 9
Low Low Low Loucks 290 53 343 2772 95. 52344
Low Low Low Sim_Ann 67 207 274 2751 728
Low Low Low Sim_Ann 51 179 230 1 575 638.2579
Low Low Low Sim_Ann 83 399 482 2929. 5 1 197.246
Low Low Low Sim_Ann 66 106 172 2862.5 909.4943
Low Low Low Sim_Ann 41 101 142 1601.25 814.3293
Low Low Low Sim_Ann 95 297 392 3643.5 2076.354
Low Low Low Sim_Ann 105 480 585 3333.75 1368.514
Low Low Low Sim_Ann 81 161 242 2537.5 1467.724
Low Low Low Sim_Ann 66 186 252 1905.75 658.981 1
Low Low Low Sim_Ann 76 206 282 2871 .75 757.8456
Low Low Low Sim_Ann 73 313 386 3344.25 777.4648
Low Low Low Sim_Ann 62 76 1 38 2643.75 1 795.322
Low Low Low Sim_Ann 70 1 56 226 2766.75 952.5607
Low Low Low Sim_Ann 51 1 39 1 90 1 575 839.6924
Low Low Low Sim_Ann 86 357 443 2945.25 891 .9806
Low Low Low Tabu_Srch 43 1 57 200 2661 .75 5751 .403
Low Low Low Tabu_Srch 31 351 382 1485.75 2129.258
Low Low Low Tabufich 60 224 284 2845.5 4205.222
Low Low Low Tabu_Srch 102 208 310 31 37.5 5414.227
Low Low Low Tabu_Srch 31 71 102 1 596 2846.293
Low Low Low Tabu_Srch 82 224 306 361 2 7750.327
Low Low Low Tabu_Srch 68 251 319 3144.75 4476.091
Low Low Low Tabu_Srch 102 220 322 2706.25 6259.887
Low Low Low Tabu_Srch 57 1 59 216 191 6.25 3349.801
Low Low Low Tabu_Srch 67 1 51 218 2856 441 1 .91 5
Low Low Low Tabu_Srch 55 141 1 96 3270.75 6864.893
Low Low Low Tabu_Srch 1 13 223 336 2993.75 4822.892
Low Low Low Tabu_Srch 50 1 84 234 2682.75 7514.857
Low Low Low Tabu_Srch 31 297 328 1 51 2 2860.348

 

 

 

 

 

 

 

 

 

 

Table A4-1. Raw Data, Continued.

204

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Low Low Low Tabu_Srch 67 1 34 201 2856 4226.961
Low Low Med Loucks 199 56 255 3031 .25 38.66406
Low Low Med Loucks 299 57 356 3531.25 52.00781
Low Low Med Loucks 128 49 177 2800 35.14844
Low Low Med Loucks 404 57 461 3087. 5 51 .1 9531
Low Low Med Loucks 237 56 293 3575 43.39063
Low Low Med Loucks 179 53 232 3037. 5 44.42969
Low Low Med Loucks 122 37 1 59 3024 36.96094
Low Low Med Loucks 330I 68 398 3400 68. 5
Low Low Med Loucks 239 85 324 4318.75 40.101 56
Low Low Med Loucks 181 20 201 2344.25 41 .02344
Low Low Med Loucks 144 34 178 3385 .7 5 41 .91406
Low Low Med Loucks 225 64 289 4500 57.781 25
Low Low Med Loucks 167 50 217 3837. 5 49.601 56
Low Low Med Loucks 191 63 254 4306.25 42.57
Low Low Med Sim_Ann 33 47 80 2825 803.16
Low Low Med Sim_Ann 78 162 240 3512.5 1939.897
Low Low Med Sim_Ann 48 88 1 36 271 8.75 934.2608
Low Low Med Sim_Ann 66 1 37 203 3050 891 .964
Low Low Med Sim_Ann 65 1 33 198 3481 .25 1 641 .486
Low Low Med Sim_Ann 60 151 211 2975 912.8812
Low Low Med Sim_Ann 59 182 241 2968.75 1071 .983
Low Low Med Sim_Ann 75 192 267 3325 1 566.278
Low Low Med Sim_Ann 801 86 166 3987. 5 1 314.506
Low Low Med Sim_Ann 18 13 31 2276 1071 .214
Low Low Med Sim_Ann 25 16 41 3226.25 834.933
Low Low Med Sim_Ann 65 83I 148 4293.75 919.4609
Low Low Med Sim_Ann 64 93 1 57 3087. 5 1 629.605
Low Low Med Sim_Ann 51 56 107 3675 141 1 .74
Low Low Med Sim_Ann 100 21 6 316 4200 2004.764
Low Low Med Tabu_Srch 88 190 278 3187.5 17213.29
Low Low Med Tabu_Srch 107 231 338 3743.75 20874.25
Low Low Med Tabu_Srch 81 221 302 3006.25 1 1595.97
Low Low Med Tabu_Srch 105 286 391 3356.25 13071.19
Low Low Med Tabu_Srch 103 193 296 3762.5 7646.517
Low Low Med Tabu_Srch 104 269 373 3300 5459.535
Low Low Med Tabu_Srch 32 125 1 57 301 3.5 7682.857
Low Low Med Tabu_Srch 105 294 399 3550 7842.982
Low Low Med Tabu_Srch 129 231 360 4337. 5 7160.318
Low Low Med Tabu_Srch 36 61 97 2469 251 1 .988
Low Low Med Tabu_Srch 40 43 83 3385.75 7146.1 39
Low Low Med Tabu_Srch 1 1 1 201 312 4625 9174.21 1
Low Low Med Tabu_Srch 102 1 91 293 3375 4743.5 54
Low Low Med Tabu_Srch 100 1 63 263 4037.5 8462.1 67
Low Low Med Tabu_Srch 131 303 434 4437.5 6777.957
Low Med High Loucks 170 40 210 3414.75 32 .34998

 

 

 

 

 

 

 

 

 

 

 

 

 

Table A4-l. Raw Data, Continued.

205

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Low Med High Loucks 1 71 59 230 4451 .5 38.78003
Low Med High Loucks 71 47 1 18 3378.5 31.08997
Low Med High Loucks 294 45 339 2363.5 41 .58008
Low Med High Loucks 138 59 197 4567. 5 46.03003
Low Med High Loucks 1 66 59 225 3886 34.77002
Low Med High Loucks 105 47 1 52 401 6. 5 34
Low Med High Loucks 99 51 150 321 1 .75 33.22998
Low Med High Loucks 94 41 135 3849.75 37.17993
Low Med High Loucks 167 64 231 471 2. 5 41 .46997
Low Med High Loucks 207 48 255 4226.75 46.91016
Low Med High Loucks 259 43 302 3356.75 43.93994
Low Med High Loucks 1 83 41 224 4328.25 48.43994
Low Med High Loucks 102 32 134 3530.75 34.98999
Low Med High Loucks 47 51 98 4502.25 35.37012
Low Med High Sim_Ann 50] 37 87 3262. 5 1 610.704
Low Med High Sim_Ann 58I 20 78 4067.25 1 372. 542
Low Med High Sim_Ann 39I 27 66 3175.5 734.5739
Low Med High SimjAhh 30[ 64 94 221 1 .25 3063.1 97
Low Med High Sim_Ann 70 9 79 4205 1 857.685
Low Med High Sim_Ann 46 36 82 3545.25 1 180.669
Low Med High Sim_Thn 71 57 128 3842. 5 1259.727
Low Med High Simjnn 38 4 42 2842 3335.497
Low Med High Sim_Ann 50 1 9 69 3646.75 1 349.329
Low Med High Sim_Ann 69 16 85 4190.5 959.6131
Low Med High SimjAhn 59 26 85 3922.25 752.0529
Low Med High Sim_Ann 41 15 56 3110.25 1795.531
Low Med High Sim_Ann 63 44 107 4161 .5 1 524.585
Low Med High Sim_LAnn 51 61 1 12 3451 1 126.704
Law Med High Sim:Ann 57 14 71 4176 2012.282
Low Med High Tabu_Srch 92 133 225 3588.75 9622.247
Low Med High Tabu_Srch 120 146 266 4560.25 6893.072
Low Med High Tabu_Srch 1 1OT 224 334 3726. 5 7197.274
Low Med High Tabu_Srch 78 190 268 2617.25 5367.087
Low Med High Tabu_Srch 141 190 331 4727 10583.81
Low Med High Tabu_Srch 1 13 1 89 302 4089 5653.604
Low Med High Tabu_Srch 123 1 81 304 4263 4796.558
Low Med High Tabu_Srch 89 1 1 1 200 3277 7127.227
Low Med High Tabu_Srch 1 12 165 277 4147 1071 1 .26
Low Med High Tabu_Srch 122 139 261 4574.75 101 16.65
Low Med High Tabu_Srch 126 171 297 4458.75 7375.626
Low Med High Tabu_Srch 76 98 1 74 3422 8735.873
Low Med High Tabu_Srch 127 226 353 4698 10293.9
Low Med High Tabu_Srch 1 10I 206 316 3944 7339.302
Low Med High Tabu_Srch 114 123 237 4618.25 10709.55
Low Med Low Loucks 279 66 345 371 9.25 55.48004
Low Med Low Loucks 262 68 330 3132 42.45001

 

 

 

Table A4-1. Raw Data, Continued.

206

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Low Med Low Loucks 1 16 55 171 3596 52.45996
Low Med Low Loucks 1 98 46 244 2660.75 34.54999
Low Med Low Loucks 234 83 317 4748.75 58.01001
Low Med Low Loucks 140 74 214 4879.25 48.98999
Low Med Low Loucks 234 69 303 3820. 75 43.1 7999
Low Med Low Loucks 164 77 241 4132.5 42.29004
Low Med Low Loucks 192 58 250 3182.75 39.38
Low Med Low Loucks 73 64 1 37 4698 44.05005
Low Med Low Loucks 338 79 417 4219.5 55.1 5002
Low Med Low Loucks 96 83 179 5995.75 65.25
Low Med Low Loucks 1 17 81 198 5763.75 70.41003
Low Med Low Loucks 63 93 156 6111.75 56.95996
Low Med Low Loucks 28 92 120 5843.5 49.42993
Low Med Low Sim_Ann 89 32 121 3400.25 646. 5704
Low Med Low Sim_Ann 58 33 91 2776.75 1 543.07
Low Med Low Sim_Ann 57 17 74 3146.5 636.9218
Low Med Low Sim_Ann 33 12 45 2356.25 550.8906
Low Med Low Sim_Ann 87 15 102 41 54.25 728.3282
Low Med Low Sim_Ann 95 3 98 4335.5 633. 5938
Low Med Low Sim_Ann 87 1O 97 3364 610.1094
Low Med Low Sim_Ann 93 1 5 108 3567 635.7968
Low Med Low Sim_Ann 46 16 62 2733.25 670.09 38
Low Med Low Sim_Ann 95 16 1 1 1 41 18 644.8126
Low Med Low Sim_Ann 66 9 75 3668.5 686.4688
Low Med Low Sim_Ann 149 1 1 160 5372.25 592.125
Low Med Low Sim_Ann 134 5 1 39 51 54.75 586.3046
Low Med Low Sim_Ann 153 18 171 5459.25 591.9766
Low Med Low Sim_Ann 155 15 170 5299.75 702.711
Low Med Low Tabu_Srch 1 13 74 187 3581.5 10758.3
Low LMed Low Tabu_Srch 101 1 26 227 3103 10010.37
Low Med Low Tabu_Srch 107 75 1 82 3523. 5 10530.1 1
Low Med Low Tabu_Srch 80 1 19 199 2755 4277.104
Low Med Low Tabu_Srch 133 83 216 4495 7287.255
Low Med Low Tabu_Srch 128 44 172 4589.25 1 1978.08
Low Med Low Tabu_Srch 1 33 1 20 253 3741 7298.421
Low Med Low Tabu_Srch 124 50 174 3828 10637
Low Med Low Tabu_Srch 96 86 182 3146.5 51 94.014
Low Med Low Tabu_Srch 121 12 133 4357.25 10088. 52
Low Med Low Tabu_Srch 105 74 179 3958.5 10457.64
Low Med Low Tabu_Srch 184 36 220 5633.25 1 1 821 .77
Low Med Low Tabu_Srch 166 35 201 541 5.75 13043.45
Low Med Low Tabu_Srch 175 6 181 5640.5 1 1681 .94
Low Med Low Tabu_Srch 175 5 180 5459.25 17760.98
Low Med Med Loucks 215 40 255 3349. 5 36.91003
Low Med Med Loucks 306 48 354 4255.75 56.95996
Low Med Med Loucks 94 56 150 3581.5 30.31995

 

 

 

Table A4-1. Raw Data, Continued.

207

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Low Med Med Loucks 318 51 369 2892.75 43.98999
Low Med Med Loucks 21 1 47 258 4589.25 49.54004
Low Med Med Loucks 227 50 277 4850.25 38.60999
Low Med Med Loucks 109 64 173 4618.25 42.13
Low Med Med Loucks 231 57 288 3849.75 41 .46997
Low Med Med Loucks 801 47 127 2994.25 25.42993
Low Med Med Loucks 62 61 123 4980.75 39.92993
Low Med Med Loucks 98 44 142 3298.75 34.1 5991
Low Med Med Loucks 254 45 299 2552 42.72998
Low Med Med Loucks 145 53 1 98 4089 31 .91003
Low Med Med Loucks 238 63 301 3936.75 42.12
Low Med Med Loucks 277 54 331 3458.25 31.91992
Low Med Med Sim_Ann 58 66 1 24 3204.5 985.2095
Low Med Med Sim_Ann 63 79 142 41 18 1066.227
Low Med Med Sim_Ann 39 34 73 3262.5 973.1096
Low Med Med Sim_Ann 65 127 192 2878.25 806.3563
Low Med Med Sim_Ann SCI 78 1 68 4408 1 81 6.872
Low Med Med Sim_Ann 591 1 60 4495 971 .3322
Low Med Med Sim_Ann 66 1 8 84 4226.75 931 .1801
Low Med Med Sim_Ann 54 26 80 3523.5 1018.833
Low Med Med Sim_Ann 28 18 46 271 1 .5 1 517.404
Low Med Med Sim_Ann 54 4 58 4524 728.7225
Low Med Med Sim_Ann 37 13 50 3016 1019.898
Low Med Med Sim_Ann 35 65 100 2392.5 1496.205
Low Med Med Sim_Ann 501 12 62 3733.75 1231 .175
Low Med Med Sim_Ann 63 38 101 3610.5 885.9002
Low Med Med Sim_Ann 41 50 91 3226.25 1478.394
Low Med Med Tabu_Srch 99 169 268 3523.5 8479.467
Low Med Med Tabu_Srch 1 18 208 326 4574.75 9061.945
Low Med Med Tabu_Srch 94 1 53 247 3719.25 7776.426
Low Med Med Tabu_Srch 125 317 442 3371 .25 7401 .425
Low Med Med Tabu_Srch 135 1 55 290 4770.5 7829.894
Low Med Med Tabu_Srch 101 83 184 4843 1 1087.57
Low Med Med Tabu_Srch 122 128 250 4640 1 171 1 .88
Low Med Med Tabu_Srch 105 157 262 3951 .25 5642.389
Low Med Med Tabu_Srch 76 1 20 1 96 3103 4886.335
Low Med Med Tabu_Srch 131 143 274 5118.5 3117.3
Low Med Med Tabu_Srch 92 144 236 3458.25 8230. 509
Low Med Med Tabu_Srch 92 216 308 2820.25 1 187.308
Low Med Med Tabu_Srch 107 145 252 4197.75 5736.227
Low Med Med Tabu_Srch 95 96 191 391 5 2773.574
Low Med Med Tabu_Srch 1 17 280 397 3842.5 2989.664
Med High High Loucks 104 34 1 38 5387.25 43.66992
Med High High Loucks 79 51 1 30 7260 73.81934
Med High High Loucks 81 58 139 6154.5 44.31934
Med High High Loucks 901 60 1 50 6245.25 50.75

 

 

 

 

 

 

 

 

 

 

 

 

 

Table A4-l. Raw Data, Continued.

208

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Med High High Loucks 100 50 1 50 5544 50.31055
Med High High Loucks 31 59 90 6575.25 48.61035
Med High High Loucks 93 41 134 5799.75 45.80957
Med High High Loucks 140 55 195 7242.75 59.26953
Med High High Loucks 100 23 123 5445 83
Med High High Loucks 97 66 163 8084. 5 63.92969
Med High High Loucks 99 45 144 5197.5 40.08984
Med High High Loucks 35 51 86 7501.75 59.25977
Med High High Loucks 185 38 223 4677.75 41 .2002
Med High High Loucks 1 12 36 148 4372. 5 42.73047
Med High High Loucks 283 46 329 5098.5 56.68945
Med High High Sim_Ann 87 29 1 16 51 15 1 195.172
Med High High Sim_Ann 132 12 144 6740.25 913.1876
Med High High Sim_Ann 90 10' 100 5593. 5 904.6094
Med High High Sim_Ann 118 17I 135 5610 915.4804
Med High High Sim_Ann 69 10l 79 5082 1212.875
Med High High 516311111 91 1 92 6096.75 1072.234
Med High High Sim_Ann 73 7 80 5403.75 1303.594
Med High High Sim_Ann 109 8 1 17 6660 1084.355
Med High High Sim_Ann 83 1 84 5247 754.1338
Med High High Sim_Ann 150 19 169 7298.25 891.9688
Med High High Sim_Ann 50 12 62 4727.25 1 279.75
Med High High Sim_Ann 102 18 120 6863.5 866.1856
Med High High Sim_Ann 51 201 71 4397.25 1 123.555
Med High High Sim_Ann 49 19 68 41 16.75 869.4766
Med High High SE Ann 57 5 62 4677.75 91 7.8984
Med High High Tabu_Srch 146 144 290 5610 7438.066
Med High High Tabu_Srch 196 86 282 7284.75 5000.086
Med High High Tabu_Srch 150 86 236 6096.75 6859.852
Med High High Tabu_Srch 166 45 21 1 6055.5 9909.55
Med High High Tabu_Srch 138 129 267 5742 4147.257
Med High High Tabu_Srch 123 39 162 6402 10546.68
Med High High Tabu_Srch 129 69 198 5890.5 7472.1 19
Med High High Tabu:Srch 153 46 199 7094.75 1 1427.97
Med High High Tabu_Srch 152 128 280 5832.75 8102.617
Med High High Tabu_Srch 194 31 225 7751.5 9681 .277
Med High High Tabu_Srch 129 139 268 5436.75 6968. 51
Med High High Tabu_Srch 171 99 270 7548 5792.108
Med High High Tabu_Srch 138 221 359 5148 891 1 .778
Med High High Tabu_Srch 1 15 159 274 4686 5416.699
Med High High Tabu:S_rch 135 171 306 5370.75 5305.424
Med High Low Loucks 100 58 158 8029 1 1 5.4502
Med High Low Loucks 236 90 326 7210.5 84. 25
Med High Low Loucks 22 69 91 8047.5 285.3398
Med High Low Loucks 44 66 1 10 6756.75 94.2002
Med High Low Loucks 1 54 81 235 8352.75 147.5303

 

 

 

Table A4-1. Raw Data, Continued.

209

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Med High Low Loucks 141 84 225 7334.25 103.9697
Med High Low Loucks 129 84 213 7359 126.9307
Med High Low Loucks 107 71 178 7326 131 .8701
Med High Low Loucks 184 83 267 71 52.75 72.83008
Med High Low Sim_Ann 187 35 222 7224.25 818.6016
Med High Low Sim_Ann 178 45 223 4560.25 1 197.063
Med High Low Sim_Ann 32 8 40 2557.5 961 .9688
Med High Low Sim_Ann 178 16 1 94 6352.5 734.8046
Med High Low Sim_Ann 178 8 186 7335.25 1064.1 33
Med High Low Sim_Ann 84 14 98 4551 667.4532
Med High Low Sim_Ann 182 35 217 6789.75 944.375
Med High Low Sim_Ann 46 18 64 2886 731 .9532
Med High Low Sim_Ann 144 29 173 6435 983.836
Med High Low Sim_Ann 171 13 1 84 6558.75 744.5782
Med High Low Sim_Ann 1 52 8 1 60 6493. 5 936.9394
Med High Low Sim_Ann 171 10 1 81 6660 763.145
Med High Low Sim_Ann 139 22 161 6591 .75 892.2002
Med High Low Sim_Ann 204 14 218 6459.75 755.7578
Med High Low Sim_Ann 37 12 49 2598.75 1058.305
Med High Low Tabu_Srch 210 1 2 222 7409.25 8462.034
Med High Low Tabu_Srch 1 99 24 223 4726.75 51 55.003
Med High Low Tabu_Srch 55 53 108 2763.75 3679.271
Med High Low Tabu_Srch 190 g 200 6484.5 8295.267
Med High Low Tabu_Srch 180 10' 190 7390.75 1 1 326.3
Med High Low Tabu_Srch 101 5 106 4699 6079.922
Med High Low Tabu_Srch 186 31 217 6864 5497.299
Med High Low Tabu_Srch 59 45 104 3061 .75 9085.802
Med High Low Tabu_Srch 164 1 1 175 6624.75 1 1838.1
Med High Low Tabu_Srch 191 1 1 202 6756.75 10701.26
Med High Low Tabu_Srch 199 43 242 6900.5 141 1 3.67
Med High Low Tabu_Srch 186 7 193 6798.75 15656.73
Med High Low Tabu_Srch 162 1 1 173 6748.5 1 1756.63
Med High Low Tabu_Srch 205 1 3 21 8 6484.5 1471 0.02
Med High Low Tabu_Srch 60 41 101 2796.75 5923.634
Med High Med Loucks 141 33 174 5832.75 71 .9502
Med High Med Loucks 81 35 1 16 5387.25 41 .95996
Med High Med Loucks 801 32 1 12 4851 44.97949
Med High Med Loucks 53 59 1 12 6501 52.39941
Med High Med Loucks 26 50 76 5883 53.2793
Med High Med Loucks 61 44 105 6550. 5 66.24023
Med High Med Loucks 135 42 177 5395. 5 45.75977
Med High Med Loucks 17 66 83 6740.25 50.2002
Med High Med Loucks 46 45 91 6575.25 52.78027
Med High Med Loucks 131 49 1 80 5824.5 46.84961
Med High Med Loucks 77 39 1 16 5469.75 41 .41992
Med High Med Loucks 105 46 1 51 5214 37.12988

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

210
Table A4-1. Raw Data, Continued.
Med High Med Loucks 53 39 92 5717.25 40.47949
Med High Med Loucks 109 39 148 6278.25 56.08008
Med High Med Loucks 76 53 129 5758.5 42.18066
Med High Med Sim_Ann 95 10 105 5494.5 959.5468
Med High Med Sim_Ann 59 7 66 5040.75 1096.398
Med High Med Sim_Ann 59 16 75 4653 1 664.34
Med High Med Sim_Ann 91 9 100 5940 1078.641
Med High Med Sim_Ann 56 9 65 5337.25 1488.914
Med High Med Sim_Ann 137 15 152 6162.75 1019.625
Med High Med Sim_Ann 90I 5 95 5007.75 712.0626
Med High Med Sim_Ann 1 12 16 128 6063.75 820.3594
Med High Med Sim_Ann 97 17 1 14 6063.75 856.2968
Med High Med Sim_Ann 81 7 88 5412 823.8798
Med High Med Sim_Ann 69 21 90 4974.75 847.1718
Med High Med Sim_Ann 65 19 84 4776.75 819.9238
Med High Med Sim_Ann 801 8 88 5346 897.0468
Med High Med Sim_Ann 121 5 126 591 5.25 877.6894
Med High Med Sim_Ann 83 5 88 5346 702.8282
Med High Med Tabu_Srch 130 51 181 5865.75 9813.695
Med High Med Tabu_Srch 134 1 32 266 5700.75 6041 .461
Med High Med Tabu_Srch 104 1 1 1 215 5073.75 6935.43
Med High Med Tabu_Srch 164 1 18 282 6550.5 10330.87
Med High Med Tabu_Srch 141 1 68 309 6169.75 9889.275
Med High Med Tabu_Srch 171 59 230 6451 .5 10876.08
Med High Med Tabu_Srch 156 121 277 5618.25 8148.905
Med High Med Tabu_Srch 145 23 168 6402 1 1330.81
Med High Med Tabu_Srch 171 93 264 6690.75 5700.898
Med High Med Tabu_Srch 131 83 214 5857.5 9263.625
Med High Med Tabu_Srch 139 75 214 5593.5 6712.223
Med High Med Tabu_Srch 125 93 218 5304.75 5198.697
Med High Med Tabu_Srch 130 98 228 5832.75 3562.57
Med High Med Tabu_Srch 190 1 26 316 6492.75 6949.164
Med High Med Tabu_Srch 1 61 1 71 332 6030.75 4281 .1 76
Med Low High Loucks 229 57 286 4331 .25 43.39014
Med Low High Loucks 180 39 219 3365.25 48.93994
Med Low High Loucks 125 65 190 4868.75 41 .46973
Med Low High Loucks 234 71 305 4487. 5 42.24023
Med Low High Loucks 158 69 227 4831 .25 52.67969
Med Low High Loucks 21 1 72 283 4650 43.83008
Med Low High Loucks 269 78 347 4437. 5 56. 52002
Med Low High Loucks 96 43 1 39 3937.5 47.07031
Med Low High Loucks 243 37 280 3097.5 45.64014
Med Low High Loucks 237 34 271 3265.5 60.68994
Med Low High Loucks 132 38 170 3108 41 .8501
Med Low High Sim_Ann 96 21 1 307 3906 1 360.652
Med Low High Sim_Ann 57 281 338 3654 1030.231

 

 

I

 

 

Table A4-1. Raw Data, Continued.

211

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Med Low High Sim_Ann 79 147 226 4181 .25 1 197.615
Med Low High Sim_Ann 42 103 145 3375.75 1 394.682
Med Low High Sim_Ann 64 72 136 3825 1533.71 1

Med Low High Sim_Ann 55 14 69 4475 1 670.368
Med Low High Sim_Ann 91 130' 221 4331.25 1714.152
Med Low High Sim_Ann 1 12 148 260 4675 1436.969
Med Low High Sim_Ann 68 47 1 15 4293.75 1434.713
Med Low High Sim_Ann 76 82 1 58 4100 1393.759
Med Low High Sim_Ann 71 222. 293 3953.25 1264.553
Med Low High Sim_Ann 45 127 172 3696 1076.865
Med Low High Sim_Ann 41 194 235 31 13.25 1029.591

Med Low High Sim_Ann 36 136 172 3276 1072.928
Med Low High Sim_Ann 42 125 167 3123.75 1 104.441

Med Low High Tabu_Srch 78 199 277 3801 4229.696
Med Low High Tabu_Srch 36 98 1 34 3570 4198
Med Low High Tabu_Srch 108 1 64 272 4431 .25 8448.304
Med Low High Tabu_Srch 42 1 19 161 3417.75 4226.261

Med Low High Tabu_Srch 120I 272 392 4206.25 8657.672
Med Low High Tabu_Srch 104 99 203 4843.75 6208.404
Med Low High Tabu_Srch 145 298 443 4700 9253.6
Med Low High Tabu_Srch 148 278 426 4931 .25 6265.438
Med Low High Tabu_Srch 124 1 51 275 4712.5 8576.398
Med Low High Tabu_Srch 126 286 412 4456.25 5879.25
Med Low High Tabu_Srch 63 1 54 217 391 1 .25 4883.992
Med Low High Tabu_Srch 44 1 54 198 3701 .25 6220.18
Med Low High Tabu_Srch 29 98 127 3071 .25 5299.454
Med Low High Tabu_Srch 28 1 02 1 30 3244. 5 5769.06
Med Low High Tabu_Srch 34 87 1 21 3102.75 3732.306
Med Low Low Loucks 65 39 104 241 5 28.5
Med Low Low Loucks 142 35 177 3506.25 39.48975
Med Low Low Loucks 146 44 1 90 2893.75 24.99023
Med Low Low Loucks 114 38 152 2982 43.17969
Med Low Low Loucks 101 33 1 34 2350 24.10986
Med Low Low Loucks 88 60 148 4331 .25 35.81006
Med Low Low Loucks 134 55 1 89 4425.75 89.08984
Med Low Low Loucks 1 56 90 246 5500 67.72998
Med Low Low Loucks 89 69 1 58 4814.25 67.33008
Med Low Low Sim_Ann 43 81 124 2425.5 1261 .41 1

Med Low Low Sim_Ann 70 1 1 81 3306.25 1 747.551

Med Low Low Sim_Ann 51 26 77 2706.25 2784.96
Med Low Low Sim_Ann 41 27 68 2662. 5 1721 .535
Med Low Low Sim_Ann 44 1 52 1 96 3013.5 1666.869
Med Low Low Sim_Ann 45 49 94 2275 2821 .674
Med Low Low Sim_Ann 30 89 1 19 2252.25 2642.253
Med Low Low Sim_Ann 33 54 87 2404. 5 2276.967
Med Low Low Sim_Ann 102 39 141 4037. 5 1 507.436

 

 

 

 

 

 

 

 

 

 

 

 

Table A4-1. Raw Data, Continued.

212

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Med Low Low Sim_Ann 45 107 152 4373.25 1527.623
Med Low Low Sim_Ann 67 66 1 33 3300 1 650.5
Med Low Low Sim_Ann 38 82 120 3302.25 21 21 .262
Med Low Low Sim_Ann 62 146 208 4788 1 71 9.307
Med Low Low Sim_Ann 115 1 116 4931.25 967.1354
Med Low Low Sim_Ann 60 1 14 174 4756. 5 2174.806
Med Low Low Tabu_Srch 37 107 144 2425.5 2503.1 56
Med Low Low Tabu_Srch 99 80 179 3537.5 3329.594
Med Low Low Tabu_Srch 61 40' 101 2800 5886.1 53
Med Low Low Tabu_Srch 73 69 142 2893.75 3573.966
Med Low Low Tabu_Srch 40 98 138 3050.25 2735.606
Med Low Low Tabu_Srch 60 96 1 56 2387.5 2122.75
Med Low Low Tabu_Srch 30 93 123 2294.25 2442.207
Med Low Low Tabu_Srch 27 126 153 2399.25 21 54.414
Med Low Low Tabu_Srch 120 53 173 4187.5 6332.785
Med Low Low Tabu_Srch 51 99 150 4425.75 401 9.943
Med Low Low Tabu_Srch 102 143 245 3568.75 3961 .482
Med Low Low Tabu_Srch 41 105 146 3328. 5 2407.207
Med Low Low Tabu_Srch 73 171 244 4872 441 5.373
Med Low Low Tabu_Srch 151 39 190 5187.5 1 1 124.87
Med Low Low Tabu_Srch 64 1 12 176 4798. 5 7548.639
Med Low Med Loucks 203 67 270 4375 47.61963
Med Low Med Loucks 248 27 275 3291 .75 58.71973
Med Low Med Loucks 126 69 195 4456.25 45.58984
Med Low Med Loucks 193 33 226 31 18.5 39.33008
Med Low Med Loucks 184 38 222 3454. 5 40.75
Med Low Med Loucks 212 51 263 4212.5 58
Med Low Med Loucks 253 57 310 4493.75 52.02002
Med Low Med Loucks 185 85 270 4531.25 45.75977
Med Low Med Loucks 233 59 292 4500 56.07959
Med Low Med Loucks 213 52 265 4743.75 74.1 5039
Med Low Med Loucks 317 45 362 3685.5 106.1201
Med Low Med Loucks 120 59 179 4162.5 38.56006
Med Low Med Loucks 1 19 66 185 4825 38.78027
Med Low Med Loucks 214 55 269 3868.75 41.35986
Med Low Med Sim_Ann 28 182 210 3302.25 1 176.266
Med Low Med Sim_Ann 50 165 21 5 3407.25 1289.344
Med Low Med Sim_Ann 34 1 12 146 3155.25 1044.578
Med Low Med Sim_Ann 38 1 18 156 3491 .25 1 128.602
Med Low Med Sim_Ann 24 330 354 3071 .25 1202.633
Med Low Med Sim_Ann 38 182 220 3433.5 1576.258
Med Low Med Sim_Ann 39 105 144 3895.5 1 101 .039
Med Low Med Sim_Ann 70 45 1 1 5 4375 1 148.465
Med Low Med Sim_Ann 75 83 1 58 4306.25 1 384.24
Med Low Med Sim_Ann 101 89 190 4937.5 2208.998
Med Low Med Sim_Ann 94 196 290 4731 .25 1 138.279

 

 

 

 

 

 

 

 

 

 

 

 

Table A4-1. Raw Data, Continued.

213

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Med Low Med Sim_Ann 42 127 169 3643.5 1 143.998
Med Low Med Sim_Ann 38 140 178 3664. 5 1047.859
Med Low Med Sim_Ann 64 183 247 3958.5 1768.383
Med Low Med Sim_Ann 73 39 1 12 3668.75 935.7226
Med Low Med Tabu_Srch 30 52 82 3354.75 3638.032
Med Low Med Tabu_Srch 32 77 109 3323.25 5996.126
Med Low Med Tabu_Srch 24 104 1 28 31 50 5800.89
Med Low Med Tabu_Srch 32 86 1 18 3501 .75 2217.226
Med Low Med Tabu_Srch 26 90' 1 16 3123.75 5171 .868
Med Low Med Tabu_Srch 39 123 1 62 3459.75 3668.602
Med Low Med Tabu_Srch 37 83 1 20 3942.75 9106.852
Med Low Med Tabu_Srch 125 192 317 4737.5 6393.968
Med Low Med Tabu_Srch 1 29 179 308 4681 .25 8052.21
Med Low Med Tabu_Srch 165 233 398 5412.5 9754.562
Med Low Med Tabu_Srch 1 15 201 316 4906.25 6961 .242
Med Low Med Tabu_Srch 29 58 87 3638.25 3230.71
Med Low Med Tabu_Srch 44 1 56 200 3690.75 4212.336
Med Low Med Tabu_Srch 55 146 201 3900.75 4812.554
Med Low Med Tabu_Srch 1 17 167 284 3975 2940.602
Med Med High Loucks 1 81 63 244 4872 1 17.3701
Med Med High Loucks 1 19 61 1 80 6409 49.97949
Med Med High Loucks 85 56 141 5183.75 46.2998
Med Med High Loucks 19 57 76 5089. 5 33.29004
Med Med High Loucks 66 49 1 15 5133 39.06055
Med Med High Loucks 73 59 132 491 5.5 42.01953
Med Med High Loucks 85 56 141 5829 45.25977
Med Med High Loucks 1 36 44 180 4640 46.95996
Med Med High Loucks 275 63 338 4002 33.71973
Med Med High Loucks 86 49 135 5227.25 43.5
Med Med High Loucks 1 1O 55 1 65 4930 43.83008
Med Med High Loucks 1 19 53 172 5046 43.33984
Med Med High Loucks 1 44 49 1 93 5829 58.33008
Med Med High Loucks 1 33 62 1 95 5234. 5 41 .4707
Med Med High Sim_Ann 69 14 83 441 5.25 978.2188
Med Med High Sim_Ann 104 16 120 5850.75 1 1 1 1.57
Med Med High Sim_Ann 95 0 95 4777.75 860.6718
Med Med High Sim_Ann 63 2 65 4690.75 1 108.484
Med Med High Sim_Ann 66 7 73 4756 997.9922
Med Med High Sim_Ann 64 19 83 4364. 5 1 121 .1 33
Med Med High Sim_Ann 105 9 1 14 5372.25 1007.43
Med Med High Sim_Ann 74 38 1 12 4277. 5 1062.391
Med Med High Sim_Ann 44 61 105 3639.5 1 356.375
Med Med High Sim_Ann 81 2 83 4857.5 1 123.564
Med Med High Sim_Ann 73 33 106 4625.5 1 503.859
Med Med High Sim_Ann 80 1 1 91 4712.5 783.127
Med Med High Sim_Ann 87 28 1 15 5328.75 1042.172

 

 

 

 

 

 

 

 

 

 

Table A4-1. Raw Data, Continued.

214

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Med Med High Sim_Ann 86 56 142 4886. 5 818.7188
Med Med High Sim_Ann 74 41 1 1 5 4451 .5 965.4766
Med Med High Tabu_Srch 90' 51 141 4603.75 10170.55
Med Med High Tabu_Srch 1 38 32 170 6126.25 12442.79
Med Med High Tabu_Srch 137 82 219 5133 1 1 192.97
Med Med High Tabu_Srch ‘ 1 57 238 395 5408. 5 5442. 532
Med Med High Tabu_Srch 1 14 71 185 5162 1 1338.63
Med Med High Tabu_Srch 126 85 21 1 4879.25 8157.196
Med Med High Tabu_Srch 182 126 308 5952.25 1 1762.29
Med Med High Tabu_Srch 138 136 274 4785 8059.454
Med Med High Tabu_Srch 121 390 511 4263 1316.773
Med Med High Tabu_Srch 1 54 139 293 5444.75 9688.102
Med Med High Tabu_Srch 1 27 149 276 5053.25 3079.884
Med Med High Tabu_Srch 141 156 297 5220 6573.16
Med Med High Tabu_Srch 149 60 209 5814.5 2695.346
Med Med High Tabu_Srch 1 76 300' 476 5568 2780.887
Med Med High Tabu_Srch 148 177 325 5038.75 8395.376
Med Med Low Loucks 64 57 1 21 4161 .5 38.06006
Med Med Low Loucks 32 52 84 6496 92.06006
Med Med Low Loucks 84 89 1 73 7068.75 109.6299
Med Med Low Loucks 1 29 77 206 5887 59.38037
Med Med Low Loucks 75 73 148 6510.5 69.08984
Med Med Low Loucks 162 74 236 6387.25 235.1899
Med Med Low Loucks 101 74 175 6235 68.99023
Med Med Low Loucks 1 1 1 74 185 6285.75 72.06006
Med Med Low Loucks 94 89 1 83 6256.75 60.7002
Med Med Low Loucks 47 81 128 5923.25 52.11963
Med Med Low Loucks 29 94 1 23 6293 70.4101 6
Med Med Low Loucks 47 80 1 27 6278. 5 81 .02002
Med Med Low Loucks 50' 80 1 30 6278.5 81 .33984
Med Med Low Loucks 46 76 122 6300.25 74.85986
Med Med Low Sim_Ann 82 29 1 1 1 3770 859.4949
Med Med Low Sim_Ann 1 59 3 1 62 5720.25 1002.1 32
Med Med Low Sim_Ann 1 77 10' 1 87 6061 963.931 1
Med Med Low Sim_Ann 133 19 1 52 5321 .5 794.2955
Med Med Low Sim_Ann 107 24 131 5183.75 682.6358
Med Med Low Sim_Ann 324 53 377 5626 1 122.554
Med Med Low Sim_Ann 1 66 10 176 5807.25 732.8471
Med Med Low Sim_Ann 1 51 15 166 5604.25 801 .4178
Med Med Low Sim_Ann 1 57 12 169 5676.75 745.9403
Med Med Low Sim_Ann 131 23 154 5473.75 963.7977
Med Med Low Sim_Ann 1 35 13 148 5444.75 745.7781
Med Med Low Sim_Ann 149 30 179 5597 789.9418
Med Med Low Sim_Ann 172 26 198 5756.5 798.5589
Med Med Low Sim_Ann 165 34 199 5705.75 774.1353
Med Med Low Sim_Ann 1 57 31 188 5676.75 777.951 1

 

 

 

 

 

 

 

 

 

 

Table A4-l. Raw Data, Continued.

215

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Med Med Low Tabu_Srch 1 21 72 193 4089 4214.455
Med Med Low Tabu_Srch 1 55 17 172 5749.25 9894.763
Med Med Low Tabu_Srch 1 78 9 1 87 6068.25 9910.408
Med Med Low Tabu_Srch 1 56 20 176 5495.5 10657.65
Med Med Low Tabu_Srch 130 17 147 5336 1 1946.71
Med Med Low Tabu_Srch 344 33 377 5785.5 5604.286
Med Med Low Tabu_Srch 190' 22 212 6024.75 1 1 574. 52
Med Med Low Tabu_Srch 1 63 9 172 5684 10457.44
Med Med Low Tabu_Srch 155 16 171 5655 12828.04
Med Med Low TabugSrch 145 1 5 160 561 1 .5 12542.54
Med Med Low Tabu_Srch 150 12 162 5589.75 12582.35
Med Med Low Tabu_Srch 164 1 9 183 5676.75 6026.188
Med Med Low Tabu_Srch 190' 16 206 5879.75 6502.94
Med Med Low Tabu_Srch 1 86 1 9 205 5821 .75 10400.03
Med Med Low Tabu_Srch 168 20 188 5756. 5 9122.642
Med Med Med Loucks 81 50 1 31 4886. 5 40.59033
Med Med Med Loucks 122 68 1 90 5604.25 174.4399
Med Med Med Loucks 77 43 120 4277.5 35.41992
Med Med Med Loucks 32 58 90 6691 .75 44.49023
Med Med Med Loucks 39 56 95 5959. 5 42.79004
Med Med Med Loucks 169 42 21 1 5633.25 49.98975
Med Med Med Loucks 68 53 1 21 5046 109.46
Med Med Med Loucks 128 42 170 4930 55.81055
Med Med Med Loucks 58 55 1 13 5270.75 41.62988
Med Med Med Loucks 212 43 255 5974 56.9091 8
Med Med Med Loucks 102 37 139 5270.75 55.96973
Med Med Med Loucks 144 54 198 4748.75 40. 59082
Med Med Med Loucks 69 50 1 19 5017 40.91992
Med Med Med Loucks 87 41 128 5140.25 38.4502
Med Med Med Loucks 1 50] 63 213 5959.5 72.4502
Med Med Med Sim_Ann 80r 4 84 4538.5 1471 .757 '
Med Med Med Sim_Ann 69 19 88 4988 1488.24
Med Med Med Sim_Ann 51 14 65 4038.25 869.3614
Med Med Med Sim_Ann 1 19 8 127 6213.25 970.9021
Med Med Med Sim_Ann 99 6 105 5539 1285.601
Med Med Med Sim_Ann 85 15 100 5234.5 1037.638
Med Med Med Sim_Ann 73 2 75 4676.25 1533.612
Med Med Med Sim_Ann 72 33 105 4632.75 968.059
Med Med Med Sim_Ann 59 5 64 4864.75 949.3903
Med Med Med Sim_Ann 69 21 90 4937.25 841 .5786
Med Med Med Sim_Ann 84 26 1 10 4959 1 292.436
Med Med Med Sim_Ann 60 14 74 441 5.25 1038.638
Med Med Med Sim_Ann 79 24 103 4669 1 228.901
Med Med Med Sim_Ann 62 5 67 4821 .25 1029.248
Med Med Med Sim_Ann 101 3 104 5481 861 .9123
Med Med Med Tabu_Srch 1 1 3 81 1 94 4843 10782.52

 

 

 

 

 

 

 

 

 

 

 

Table A4-l. Raw Data, Continued.

216

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Med Med Med Tabu_Srch 138 104 242 5524.5 7365.099
Med Med Med Tabu_Srch 95 1 12 207 4437 10046.44
Med Med Med Tabu_Srch 187 1 10 297 6749.75 6023.649
Med Med Med Tabu_Srch 145 62 207 5894.25 10660. 56
Med Med Med Tabu_Srch 161 1 33 294 5836.25 1 1688.02
Med Med Med Tabu_Srch 136 1 31 267 5169.25 6490.522
Med Med Med Tabu_Srch 148 169 317 5198.25 6636.888
Med Med Med Tabu_Srch 154 222 376 5568 791 1 .051
Med Med Med Tabu_Srch 171 107 278 5966.75 2069.955
Med Med Med Tabu_Srch 140 1 56 296 5379.5 6363.073
Med Med Med Tabu_Srch 1 14 124 238 4835.75 8740.931
Med Med Med Tabu_Srch 162 223 385 5307 5679.787
Med Med Med Tabu_Srch 156 223 379 5560.75 10999.57
Med Med Med Tabu=Srch 139 45 184 5778.25 8055.93

 

 

 

 

 

 

 

 

 

 

217

APPENDIX 5

TRANSFORMED TREATMENT AVERAGES

The following table contains the averages for all treatments, transformed.

Table A5-l, Treatment Averages Of Transformed Data.

3.68756981
3.8091
3.8047441
3.
3. 1
.413541
3.294643
3.
3.2091 1
3.2876537
3. 1
3.64453096
2.91 130408
3213831325
3.195047
3.1051
2.901241
3.1684964
3.39205746
3.2609843
3.28419721
3.1042442
2. 714
3.00204943
2.9123058
2.8091

2.
2.
2.

2.91631
2.
2.641
2.81
2.
2.
2.
2.6821
2. 1
2.946
2.6422
2.720401
2.

2.5821
2.

337
2.321

2.4 3

2.

2.31
2.309074
2.32481

.21

.1
5.305293
5.
5.12701
5.
5.441
5.

5.08181
5.1
4.
5.
5.086168
4.954461
4.94931
4.
4. 1
. 1
4. 46
4

499.1
629.01
534.1 921
564.

481 .
51 1
507.3

696.
612.

600.
806.8586
677.1428

673.64
503.
508.3235
482.9435
6 9.

.6

. 76
738.3506
743.9041

 

.01571
3.
4.
3.90451

3.
2.91
2.8091
2.9768455

2.

2.
2.640491
3.102083
2.67403013
2.

3.13553
2.
2.71

2.643633
2.

2.8236
3.

2.

2.
3.49085141
3.00671

3. 9421
2.85191

2.

2. 1

3.0147
3.0311
2.88237
3
3.216
2. 49413

2.
3.
2.94361
3.3716023
3.196581
3
3.51
3.2431
3.
2.

2. 1
2.8407323
3.

3.
3.426403
3.44933091

2.31

2.
2.32481
2.01

TSD
3.811668
3.1
2.91

1.91
2.
2.217111
1.941

2.071
2.169311
TSD
3.6721

3.
3.

2.
3.5301
3.580901
2.259808
3.44
3.38104
3.

3.3211

3.
2.102858
3.41

3.31

2.

218

4.
4.
4.561
OBJ
5.
4.9901
4.
4.679741
4.502941
4.36417
4.6
4.35251
4
4.8151
.1944

.17
. 3

.11

1
4.68061
4.692431
4.

4.
4.
4.3
OBJ
.48897
. 1
5.341951

5.561
5.611
5.
5.086821
.1
241
5.
5.111

43:9041
.5501

COST

363.01
.77
1

1
468.
563.7331
485.4361

.386
.813
4 8.901
494.41
634.
58 .841
9.897
.501
625.
.57
14

499.
609
613.611
530.
728.
716.

0051‘

372.1 1

461.

446.

525.

496.

506.
623.

583.151

448.

49 .9061
492. 1

61261
642.1

 

Table AS-l, Treatment Averages Of Transformed Data, Continued.

6.
.102331
.38961

6.

.01
451

6.

6.87511
6.

.1
.1

6.

.0131

6.9141

6. 4

6. 1

6.94831

7. 11
7.506961
.3284
.41
6.91 1

7

8.
9
8.62763
8.
9.4051
9.

9.
8.221

219

Table AS-l, Treatment Averages Of Transformed Data, Continued.

3.4 3.11 5. 683. 1 8.91

3.48 11 3.181 5.5541 693.311 8.8586
2.82931291 3.50541 . 523. 9.1664
2.8267 3. 5. 537. 9.1717
2.6691 3. 5.21 493 8.962131
3.353 3.401 . 638 8.9847

3.368651 3. .603 635.4891
3.2287 3.33051 5.

3. 2.921 5.411 744

3. 2.919401 5. 1

3.1743237 2.95141 .21 .1031

 

BIBLIOGRAPHY

220

BIBLIOGRAPHY

Andrews, B. H. and Parsons, H. L., 1989, "L. L. Bean Chooses a Telephone Agent
Scheduling System," Interfaces Vol. 19, Nov-Dec 1989, pp. 1-9.

 

Aggarwal, S. C., 1982, "A Focused Review of Scheduling in Services," European Journal
pf Operational Research, Vol. 9, Iss. 2, pp. 114-121.

Bechtold, S. E. and Showalter, M. J., "Methodology for Labor Scheduling in a Service
Operating System," Decisipn Sciences, Vol. 18, Iss. 1, pp. 89-107.

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