A SYSTEMS APPROACH TO A PRODUCTION
EHEDUUNG PROBLEM SNVOLVING ASSEMBLY

Thesis ior the 993m ofPh. D.

MICHiGAN STATE UMVERsm

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1967

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This is to certify that the

thesis entitled
A SYSTEMS APPROACH TO A PRODUCTION SCHEDULING
PROBLEM INVOLVING ASSEMBLY

presented by

Rodolfo Camanzo Yaptenco

has been accepted towards fulfillment
of the requirements for

Ph. D. degree in Wood Technology

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ABSTRACT

A SYSTEMS APPROACH TO A PRODUCTION SCHEDULING
PROBLEM INVOLVING ASSEMBLY

by Rodolfo Camanzo Yaptenco

A scheduling problem involving assembly of the type
that generally characterizes plywood manufacturing is
identified and modelled mathematically in terms of alge—
braic and difference equations. The facility is viewed
as a system made up of a discrete number of components,
viz., production centers, that interact with each other
only at a discrete number of points, viz., points where
inputs are received and outputs removed. The mathematical
formulation is shown in considerable detail to illustrate
the logic of the approach taken.

Following hypothetical considerations, an actual
scheduling problem is subsequently described and modelled.
firoduction centers in the facility are identified and de-
fined, modelled independently of each other and the com—
ponent models combined (using the inter-connection pattern
of the system) to form the system model.

Practical implications of the dynamic case are dis-
cussed, with some emphasis given to the economic feasi-
bility of implementing such a model in the real world.

The static case (where only one interval is considered

Rodolfo Camanzo Yaptenco

and taken equal to the planning horizon) is also dis-
cussed. Results from a computer run of the static model
using actual data, e.g., order file, initial inventories,
machine capacities available, space limitations, etc.,
are presented in the appendix.

Certain aspects of the model where improvements can

be effected are mentioned.

A SYSTEMS APPROACH TO A PRODUCTION SCHEDULING
PROBLEM INVOLVING ASSEMBLY

BY

Rodolfo Camanzo Yaptenco

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Forest Products

1967

ACKNOWLEDGMENTS

I thank the College of Forestry, University of the
Philippines, the National Economic Council, Philippine
Government, and the Agency for International Development,
United States Department of State which collectively made
it possible for me to pursue a doctoral program at Michigan
State University. I also thank the Department of Forest
Products, Michigan State University, through its chairman,
Dr. Alexis J. Panshin, for the pleasure of doing graduate
work with them.

Special thanks are due to Boise Cascade Corporation,
especially the Department of Operations Research and the
Lumber and Plywood Division at Yakima, Washington, for
allowing me to conduct the study in one of their plywood
mills and for giving me access to actual production data.
Acknowledgment is also given for their financial assistance,
especially that for computer time.

I am indebted to my major professor, Dr. Aubrey E.
Wylie, for his guidance during the course of my program,
to the members of my committee, Drs. Richard F. Gonzales,
David N. Milstein and Otto Suchsland, for their invaluable
help in shaping my program, and to Dr. Herman E. Koenig

for reviewing the manuscript and for his suggestions.

ii

Finally, many thanks are due to a wonderful family,
my wife Tita and daughters Jig and Jinkle, for their
understanding and patience during those trying days of

graduate work.

111

TABLE OF CONTENTS

ACKNOWLEDGMENTS

LIST OF TABLES . . . . . . . . . . .

LIST OF FIGURES .

LIST OF APPENDICES . . . .
Chapter
I. INTRODUCTION. . . . . .
II. THE JOB-SHOP SCHEDULING PROBLEM
III. A SYSTEMS APPROACH TO SCHEDULING .
IV. THE PLYWOOD PRODUCTION SCHEDULING PROBLEM
V. FORMULATION OF A SCHEDULING MODEL.
VI. PRACTICAL IMPLICATIONS
VII. CONCLUSIONS . . . . . . . . .
APPENDICES. . . . .
REFERENCES.

iv

Page
ii

vi

viii

26

35
75
80
85
137

LIST OF TABLES

Table Page
V-2.l. Green End Recoveries in Percent . . . A3
V-A.l—a. Dryer Grade Yields (Face Stock) . . . U9
V-A.l-b. Dryer Grade Yields (Core Stock) . . . 50
V-9.l-a. Panel Falldown Summary (Exterior

Sheathing) . . . . . . . . . 68
V-9.l-b. Panel Falldown Summary (Sanded

Products) . . . . . . . . . 69
V—lO.l. 8-Foot Lathe Production Rates. . . . 72
V-lO.2. Drying Productivity . . . . . . . 73
V-10.3. Patching Productivity . . . . . . 7A

Figure

III-2.1.

III-2.2.

III-2.3.

IV-l.l.

V-5.l.

V-6.l.

V-7.l.

LIST OF FIGURES

Schematic Diagram of a Hypothetical
Production System .

Equivalent System Graph of Production
System Shown in Figure III-2.1.

Region of Feasibility.

Typical Construction of a Plywood
Panel . . . .

Relationship Between the Magnitudes
of Lead Time and Period . .

Schematic Diagram of Processor No. l
Representing the 8-Foot Lathe,
Two 8-Foot Clippers and Associ-
ated Equipment . . . . . .

Schematic Diagram of Processor No. 2
Representing the Core Lathe,
Clippers and Associated Equip—
ment . . . . . .

Fish Tail.

Schematic Diagram of Processor No. 3
Representing Three Veneer Dryers

Schematic Diagram of Processor No. A
Representing a Veneer Saw . .

Schematic Diagram of Processor No. 5
Representing Veneer Jointer and
Edge Gluer . . . . . .

Schematic Diagram of Processor No. 7
Representing Four Veneer Patchers.

vi

Page

l5

15
2A

28

38

Al

Al
A8

A8

A8

53

53

Figure Page

V—8.l. Schematic Diagram of Green and Dry End
Sections of Plywood Mill . . . . 55

vii

LIST OF APPENDICES

Appendix Page
A. List of Vectors and Variables . . . . 86
B. Lay-Up Schedule. . . . . . . . . 95
C. Veneer Requirements . . . . . . . 98
D. Production Schedule for Week . . . . 100

E. Yakima Plywood Mill Scheduling Model,
January 1967.. . . . . . 103

viii

I. INTRODUCTION

A scheduling problem is said to exist if in the pro-
duction of goods there are (or there become available) a
number of alternatives for sequencing a number of jobs or
performing a series of operations on a number of machines
and it is desired to choose from these alternatives those
that would optimize some chosen objective.

In general, the number of alternatives is very large;
consequently, the Job of a production scheduler is expected
to be complex. It is a wonder, therefore, why schedulers
do not seem to find scheduling a problem at all. Pounds
(13) provides us with some explanation. He reports that
in most cases there are no scheduling problems to start
with because ". . . the organization which surrounds the
schedulers reacts to protect them from strongly inter—
dependent sequencing problems." Mellor (l2) concurs, "In
effect," he observes, "industrial schedulers are being
asked to get a pint out of a quart pot and are experiencing
no difficulty in doing so. The scheduler's protection, of
course, comes from extravagant provision of shop capacity
or poor commercial performance."

We conclude, therefore, that multiplicity of alter-

natives per se does not make scheduling difficult; rather,

it is how restrictive our objective is or how good a
schedule is desired and how well the capacity of the
facility is utilized. Indeed, if only mediocre schedules
are made or if plenty of excess capacity is available,
the Job of a production scheduler would be trivial. On
the other hand, choosing an optimal schedule for a plant
operating at or near capacity is a formidable Job. Ob-
taining optimal schedules is of course the object of to-
day's factories, especially with the specter of increas-
ing costs and stiffening competition looming in the future.
Thus, the need for efficient methods of selecting Optimal
or near optimal schedules is very real.

i The purpose of the research which forms the basis
of this dissertation was to find a method of solution to
a certain class of scheduling problems involving assembly,
to which plywood manufacturing belongs. The work herein
described is an attempt to provide the scheduler assis—
tance by reducing the scale of the problem to a point
where it would require only Judgment to select a near
optimal or Optimal schedule. It is expected to free the
scheduler from having to contend with normally multi-

farious alternatives.

II. THE JOB-SHOP SCHEDULING PROBLEM

II—l. Background

 

Much of the literature about scheduling is concen-
trated on "job-shop scheduling," a consequence of the fact
that it is regarded as the most complex. We define "job—
shop scheduling" as the sequencing of a number of jobs
that require varied sets of operations and follow diverse
routings through a facility of several production centers
and compete with each other for productive capacity on
common machines. It becomes a problem when sequencing is
to be done with some objective in mind, e.g., minimizing
total processing time, minimizing total production cost,
minimizing lateness of deliveries, minimizing downtime,
or some combinations thereof.

The simplest approach to the problem is without any
doubt an exhaustive enumeration of all feasible schedules
and selecting from the set the schedule that optimizes the
objective. The method, however, is not practical since
the number of feasible schedules is very large even for
small problems. For example, if we have j jobs and m
machines and each job needs to be processed by each

machine, then the number of feasible schedules is (j!)m.

For five jobs and four machines this number is 207,360,000
--a number which even the modern digital computer cannot

search through economically for the optimal schedule.

 

Intuitively, though, one gets the impression that surely
the optimal schedule must come only from a much smaller
subset of the whole set of feasible schedules. What is
obviously needed are some criteria for discarding those
schedules that have no possibilities of becoming optimal
and searching only those that do, for the optimal schedule.
Giffler and Thompson (8) narrowed the number of
feasible schedules to be included in the enumeration by
considering only "active" schedules. (They define an
"active schedule" as a feasible schedule such that no
machine may be idle for a period of time greater than or
equal to the processing time required by a job that is
available for processing and that processing of a job is
started as soon as both machine and job are free.) For
moderate-sized problems the method has distinct advantages.
However, when considered in terms of the size of real
problems it likewise suffers from the combinatorial character
of job-shop problems, since the number of active schedules
also increases very rapidly with an increase in the number
of jobs or the number of operations required on each job.
To get around the immensity of the number of alter-
natives, a Monte Carlo sampling technique was devised (9)
for drawing samples from the set of active schedules, and

searching only the samples for the best schedule. Since

only samples are considered, the method does not guarantee
an optimal schedule, although the probability of obtaining
one can be increased by taking more samples. The impli-
cation of course is that as more samples are taken to in—
crease the probability of obtaining an optimal schedule,
the amount of computation required will correspondingly in-
crease.

A method which has attracted a great deal of attention,
and rightfully so, is the heuristics approach (2, 7, 18).
In this method simple but effective rules of thumb are
used for discriminating between alternative ways of sequenc—
ing jobs. These rules may be "borrowed" from rules of
thumb used by capable schedulers, or they may be the result
of simulated studies of the problem, or from some other
appropriate sources. Unfortunately, it cannot show, ex-
cept perhaps intuitively, that the schedules chosen on the
basis of these rules are consistently good. The only justifi-
cation for their use is that they either represent the sound
judgment of skillful schedulers and/or were found effective
inder simulated conditions. Heuristics, however, provides
the opportunity to narrow, rather drastically, the number
of alternatives down to manageable proportions, permitting
the selection of a schedule which is at least as good (if
the rules are chosen properly) as a human scheduler could
make. With more experience in the selection of rules, it
may be the most practical approach to certain types of

scheduling problems.

There are of course other methods (I, 10, 12, 15,
16, 19) that have been proposed for certain idealizations
of the scheduling problem. However, there exists a gap
between the problems assumed in these formulations and
those of the real world. Moreover, the amount of compu-
tation seems to limit the size of the problem that can be
practically handled. Consequently, applicability is still
somewhat limited.

In each of the methods of solution reviewed above,
the object was always to find some optimal solution to the
problem. Analytical methods tend to require too much
computation (at least for now) and empirical methods tend
to oversimplify the problem. A combination of analytical
and empirical methods might provide the right combination.
Indeed, it is a distinct possibility. In other words,
instead of trying to obtain a complete solution to the
problem, what is suggested here is cutting down the size
of the problem by analytical methods to a point where the
human scheduler can use his judgment effectively and con-
sistently.

The subject of this dissertation is the formulation
of the scheduling problem in a format that is amenable to
already available optimizing techniques. It approaches the
problem with the systems concept by viewing the production
facility as a "system" and describing it in terms of pro-
duct flows and inprocess inventories in the form of alge-

braic and difference equations. Its objective is to

narrow down the number of alternatives a scheduler must
choose from and the number of decisions he has to make.

It hopes to present to the human scheduler the problem in
a much simplified form and leave to him the details of
bringing the problem to complete solution. In effect what
is proposed here is some form of sub-optimization.

II-2. Scheduling Problems
Involving Assembly

 

We also note from the literature that attention given
to scheduling problems is generally directed at job-shop
problems of the type that does not involve intermediate or
final assembly. Consequently, the solutions proposed ex-
clude assembly—type problems where jobs (and the parts
they require for assembly) compete with each other for
productive capacity. In this variation of the job-shop
problem, jobs generally require certain parts that are
also required by other jobs. Since parts are processed
in batches, we can conceive of the problem as made up of
jobs which in later stages of manufacture break up and re-
combine with others to form new jobs. Thus, the problem
basically is sequencing the right quantities of material
or semi-finished products at each processing center so
that assembly of jobs at a later stage in time can proceed
with as little interruption as possible without building

up in-process inventories.

The development following concerns an assembly—type
of scheduling although the method here proposed is not

necessarily limited to such types of problems.

III. A SYSTEMS APPROACH TO SCHEDULING

A system is a collection of objects or entities
which are related to and interact with one another in some
fashion such that each object or entity performs a function
that contributes to the objectives of the group. A formal-
ized awareness of the interactions between parts of a

system is what is popularly known as systems engineering

 

(6). The key to the whole concept of systems engineering
is the simultaneous consideration of the relationships be-
tween components of the system and the emphasis given to
the effectiveness or the objectives of the whole system
rather than that of the parts taken separately.

A production facility, in the context of the definition
given above, is a system. It is a collection of processing
centers functioning together as a group with a unified
objective-—the production of goods. It should, therefore,
be amenable to the same techniques of systems analysis
used in engineering to describe physical systems, provided
certain conditions are met. Indeed, if these conditions
can in fact be met, then we would have made significant
progress because system theory provides a rigorous and
consistent analytical framework for describing the behavior

of systems.

IO

III-l. System Theory

 

System theory has its origins in the analysis of
electrical networks. It has been extended to include dis—
crete physical systems, and very recently efforts have
been made to extend its usefulness to socio-economic
systems.

A rigorous treatment of system theory as applied to
physical systems may be found in a recent book by Koenig,
Tokad, and Kesavan (11). To attempt will be made here to
discuss the fundamentals of the theory, except very briefly

those that have pertinence to the discussion.

III-1.1 Discreteness

One fundamental requirement of system theory is that
a system must be identifiable into a finite number of com-
ponents that are interconnected and interact only at a dis—
crete number of points. Consequently, each component can
be isolated and modelled independently from other compon-
ents and, through the topology of their interconnections,
combined into a system model. System theory, therefore,
assumes that the model of a component characterizes that
component as an entity without regard to how the component
may be interconnected with other components of the system.
This assumption is not true of course in cases where the
presence of a component has induced effects on other com-
ponents. In such cases, we may either neglect these in-

duced effects or isolate the components only conceptually.

11

We note that a production system is identifiable
into a discrete number of components, e.g., machines or
groups of machines, that constrain or interact with each
other only at the points where inputs are received and
outputs removed. Hence, the above requirement of dis-

creteness can be satisfied.

III-1.2. Generalized Kirchoff Postulates

Another requirement of system theory is that the per-
formance characteristics of each component should be de-
scribed in terms of two complementary variables that satisfy
the two generalized Kirchoff postulates. This pair of
variables, which we shall denote as Y and Y, should have
analogous connotations, respectively, as voltage and cur-
rent in electrical systems or pressure and fluid flow in

hydraulic systems.

III-1.2.1. Flow Variable, Yi
Let Yi be the flow of materials or products in the
system. If the flows are expressed in the same units,

then the flow variable Y satisfies the first Kirchoff

i
postulate. That is, "The algebraic sum of all directed
flows at any point in the system must vanish." In symbols,

for any cut-set m_and any time t, we write,

ZYmi(t) = 0 (III-1.2.1)

12

Therefore, it only needs to be established that all flows
in the system are expressable in the same units. It will
be shown later, in the discussion of an actual problem,

that in fact this can be done.

III-1.2.2. Propensity Variable, Xi

The propensity variable, X is most difficult to

1’
identify at the present time because little is known about
the interrelationships between factors that influence the
flow of materials in a production system. We know intui-
tively and from experience that the flows of materials be-
tween any two points is a function of (i) demand for such
materials elsewhere in the system, (ii) the levels of in-
process inventories, (iii) the costs associated with such
flows, (iv) difficulty associated with holding materials
in inventory (such as due to space limitations), (v) the
capacities of machines in which these materials need to be
processed, and (vi) still probably some other factors. Be—
cause the relationships between these factors are not known,
it is not now possible to identify a propensity variable
that could, in addition to the flow variable, be used to
describe the performance characteristics of a production
system. More research and experience are undoubtedly re—
quired before an appropriate propensity variable can be
identified.

We can, however, proceed to model a production system

in terms of flows and the factors suggested above without

l3

necessarily knowing the interrelationships of these factors,
for we know in general how these factors independently af-
fect the selection of a schedule. This is best shown by
considering a hypothetical example in the next section.
There we will model a system in terms of material flows

and in-process inventories and use such factors as demnad,
costs, space limitations and machine capacities to con—
strain the selection of a schedule, at the same time

making the most use of the resources of the system.

III-2. A Scheduling Problem
Involving Assembly

 

 

In the classical1 job-shop problem, the usual assumption
is that a succeeding operation on a job cannot be started un-
less a preceding operation has been completed on the whole
job. In other words, no two machines may work on the same
job simultaneously. This assumption is obviously suited
only for jobs that cannot be split physically (such as jobs
consisting only of one unit each) or for jobs that require
small processing times on each machine. 0n the other hand,
if the processing times involved are in the order of, say,
an hour or more and it is possible and practical to split
jobs (such as batch types), the assumption introduces con-

siderable error in the form of excessive and unnecessary

 

1Defined as Q_jobs, m machines, the processing time
of each job on each machine being known and the object
is to find the sequence at each machine center that re-
quires the least total processing time.

lA

waiting time. In assembly-type production, the parts

that go into assembly are generally processed in batches.

At earlier stages of production, i.e., before assembly,
these batches are themselves jobs which can be worked on
concurrently by two or more machines after allowing suffi-
cient lead time in the preceding operations. In such a
case, it is more desirable to use a time interval as a
criterion for moving completed portions of jobs to the

next operations. For example, if the interval chosen were
an hour, then we can for instance make the assumption that
the portion of a job completed during a given hour may be
moved to and worked on at the next operation during the

next hour, but not before then. We also make the restriction
that the movement of material between two processing centers
occurs in "spurts" and is possible only at the beginning of
each period.

Obviously, the choice of an appropriate interval will
depend on how discrete the jobs are, i.e., how many units
make up a job, and the time it takes to process a unit of
a job.

Consider now a facility of several processing centers
manufacturing products that require assembly in the last
operation, shown schematically in Figure III-2.1. The
solid circles represent processing centers and the dotted
circles in-process inventories. If we let Z1(t) repre-
sent inventories and Yij(t) represent material flows during

any time 2, then we can write

15

 

Figure III-2.l--Schematic diagram of a hypothetical produc-
tion system. Solid circles represent processing centers and
dotted circles represent in—process inventories.

 

Figure III—2.2--Equivalent system graph of production system
shown in Figure III—2.1.

l6

Zi(t+l) = pi zi(t) + zyji(t) - 2yki(t) (III-2.1)

where yki(t) represents flow from inventory i to machine
5, yji(t) represents flow from machine j|to inventory i

and p1 is some constant. In the example shown in Figure

III-2.1, i 1,2...8, j 1,2,3,u, and k = 1,2,3,u. If

we write Equation III-2.1 for all i, we have

zl(t+l) = pl zl(t) + yll(t) — y2l(t) (III-2.2)
22(t+l) = p2 22(t) + yl2(t) - y22(t) (III—2.3)
z3(t+l) = p3 23(t) + yl3(t) - y36(t) - y33(t) (III-2.A)
zu(t+l) = Pu zu(t) + y1u(t) - yuu(t) (III—2.5)
25(t+l) = p5 25(t) + y25(t) - y35(t) (III-2.6)
z6(t+l) = p6 26(t) + y26(t) + y36(t) - yu6(t) (III-2.7)

27(t+l) = p7 27(t) + y27(t) + y37(t) - yu7(t) (III-2.8)

+

28(t+l) p8 28(t) y38(t) - yA8(t) (III-2.9)

Equations III-2.2 through III—2.9 can all be written together

simply as

Z(t+1) = P z(t) + Cl Ll(t) - T2 L2(t) (III-2.10)

17

where Z(t) is a vector of in—process inventories, L1(t)
is a vector of outputs (yji(t))’ J = 1,2,3, and L2(t)
a vector of inputs (yki(t)), k = 2,3,A. P, G1 and G2 are
matrices. Let us assume for the moment that we have valid

models of machines 1, 2 and 3 of the form

in(t+l) = R in(t) + S Yki(t) (III-2.11a)

or in(t) = D Yki(t) (III—2.11b)

Then we can combine Equations III-2.10 and III—2.1, elimi-
nating outputs (yji)’ j = 1,2,3, and write the result in

the following matrix form:
Z(t+1) = P Z(t) + Q E(t) (III-2.12)

where E(t) is a vector of inputs, (Yki)’ k = 1,2,3,A and
Q is a new matrix resulting from the operations implied
above.

Equation III-2.12 can also be obtained by representing
Figure III-2.1 by an equivalent system graph shown in
Figure III-2.2 and selecting a tree, T (shown in heavy
lines). Writing the outset equations for the tree T, we

obtain a mathematical model of the interconnection pattern

of the system:

l8

 

 

 

 

 

 

 

 

r ‘ F ‘ F ‘
l 1 A yl(t) -1 i; (t)
y (t) -1 ll(
1 y2<t> -1 1 $12<E§
1 y3<t> + -1 yl3<t>
l y5(t) -1 ylu(t)
1 y6(t) -1 -1 y22(t)
1 y7<t> -1 -1 y§7<t>
_ :L—J _y8(t)_ — -l_i $36EE§
37
y (t)
1.38 _
- r _
Al y (t)
l y§%(t)
l y33(t)
1 y (t) = o (III-2.13)
1 35(t)
1 y“(w
1 §16(t)
1 ”7(t)
_, J .FAB .1

 

 

 

 

The general equation for in-process inventory is given by

Zi(t+l) = pi Zi(t) + Y1(t) (III—2.1A)

Combining Equation III-2.1A with Equations III-2.11 and
III-2.13, Equation III-2.12 can be obtained.

Let the vector E(t) be rearranged such that we can
partition it into (1) inputs to machines 1,2, and 3 and
(ii) inputs to machine A. Then we can write

El<t)
E(t) = E (III-2.15)
2

where El(t) represents inputs to machines 1, 2, and 3 and
E2(t) represents inputs to machine A (an assembly process).

We can therefore rewrite Equation III-2.12 as

19

E (t)
_ l
Z(t+l) - P Z(t) + [O1 Qé] EETFI (III-2.16)
or Z(t+l) = P Z(t) + Ql El(t) + Q2 E2(t) (III-2.17)
Jl(t)
Let J(t) = J2(t) represent the assembly schedule for

 

 

‘O3(t)_
machine A during time t. The quantities of E2(t) demanded

by the assembly process during the same period is given by

E2(t) = W J(t) (III-2.18)

where W is a matrix whose entries represent the quantities
of each part in E2(t) that is required in the assembly of
each unit of product in J(t). Substituting III-2.18 in

III-2.17, we obtain

Z(t+l) P Z(t) + Ql El(t) + Q2 W J(t) (III-2.19)

or Z(t+l) P Z(t) + Ql El(t) + R J(t) (III-2.20)

where R = Q2 W.

For any given period, Z(t) represents initial inven-
tories and Z(t+l) represent ending inventories. J(t) is
the schedule of assembly at machine A and El(t) represents
the schedules at machines 1, 2, and 3 during the same

period. In general J(t), El(t) and Z(t+l) are unknowns.

20

However, Z(t) is known so that if we specify J(t) and re-
quire that Z(t+l) be non-negative, then we can choose
El(t). It is clear that J(t) and El(t) are both arbitrary;
therefore, they can be selected so that some objective is
realized.

Equation III-2.20 represents a mathematical model of
the system shown in Figure III—2.1. If we assume for
simplicity a planning horizon of three periods, then

Equation III-2.20 can be solved for t = 0, l, 2 as follows:

2(1) = P 2(0) + Ql 21(0) + R J(O) (III-2.2la)
2(2) = P 2(1) + Ql El(l) + R J(l) (III—2.2lb)
2(3) = P 2(2) + Ql E1(2) + R J(2) (III-2.21c)

Equations III-2.2la, III-2.2lb and III-2.2lc can be written

in simplified form as

Ql R -U .7 El(0)
= I 1—2.22
Q R P -U El(l) 0 ( I )

 

 

 

 

Ql R P -u‘ El(2) __0
J(O)
J(l)
J(2)
Z(l)

Z(2)

 

 

2(3)

21

Equation III-2.20 can also be solved recursively for three

periods to obtain a much more compact form

2(1) = P 2(0) + Q1 El(O) + R J(O)
Z(2) = P Z(l) + Ql El(l) + R J(l)
= P2Z(O) + P Ql El(O) + P R J(O)
+ Q1 21(1) + R J(l)
2(3) = P 2(2) + Ql El(2) + R J(2)

2

P32(o) + P2Ql El(O) + P R J(O) + P Q1E1(1)

+ P R J(l) + Ql El(2) + R J(2) (III-2.23)

Equation III-2.23 can also be written in matrix form.

2 2 _ A 1 = - 3 III-2.2A)
[P Ql PQl Q1 P R PR R u] El(0) ['P 2(0)] (
31(1)
El(2)
J(o)
J(l)
J(2)
2(3) J

 

 

The form given in Equation III-2.22 is preferable, however,
since it is important that we be able to impose certain
restrictions on the magnitudes that 2(1) and Z(2) can

attain. For example, we must require that the value of

22

Z(t) during any time t should be non-negative and not
exceed the maximum that can be tolerated in the system.
In Equation III-2.2A, there is no way of doing this so
that we have no assurance that the quantities of in-pro-
cess inventories during any period will always be within
the range that has meaning in an actual scheduling problem.
It is also necessary to impose restrictions on the
magnitudes of El(t) since the quantities that can be pro-
cessed during any period should not exceed the capacities
available from the machines during the same period. In
addition, there may be other restrictions that are peculiar
to the problem. Finally, if we add an objective function,
such as a processing cost function, then Equation III-2.22

can be rewritten together with the restrictions and ob-

jective function, as follows:

 

 

 

 

 

 

r -1 r- r '1
01 R —U 21(0)T = -P 2(0)
Q1 R P —U El(l) = 0
01 R P -U El(2) =
A1 J(O) : K
A2 J(l) f K (III-2.25)
A3 J(2) f K
D D D U 2(1) = OF
Bl Z(2) f V
B2 Z(3) 3
B3 J'(3) : V
_ _. T“‘_“‘
= Mi
£31 Ca C3 Cu 05 Co C7 C8 C9 ClO_ __ n ..

 

23

The vectors K and V represent, respectively, machine and

inventory space capacities and the row vectors C d =

d’
l,2,....,10, are costs associated with each activity.
J'(3) represents orders that cannot be satisfied during
the planning horizon. Thus, C10 is a penalty cost. In
Equation III-2.25,

Al, A2, A3 = matrices with entries representing
productivity coefficients of each
production center for each input

D = unit matrix
U = unit matrix
81’ B2, B3 = row vectors with entries of 1's

K = column vector of machine capacities

order file

0
’13
II

V = in-process inventory space capacity

It is now evident from Equation III—2.25 that the
scheduling problem has now been formulated in a form that
can be solved with linear programming (A). The number of
feasible schedules is still very large. However, we do
nave in the linear programming algorithm an efficient
technique for arriving at an optimal solution since the
algorithm considers only those combinations that potenti-
ally can become optimal. For example, in Figure III-2.3
the range of feasibilities due to the indicated linear
constraints is represented by the enclosed area A. If the

variables x and y were continuous, the area 5 represents

2A

 

 

Figure III-2.3--Region of feasibility.

25

an infinite number of feasible schedules. Linear pro-
gramming, however, considers only the feasible schedules
represented by points a, b, c, d and e; surely a small
number compared to the whole set.

We note that originally the problem was dynamic in
nature in the sense that the element of time is involved.
But through the use of systems concepts we have reduced
the problem to an equivalent static problem that is amen-
able to linear programming. It should be clear, however,
that the solution to Equation III—2.20 need not be obtained
with linear programming. Possibly other methods, e.g.,
:ynamic programming, would be just as effective. Linear
programming was selected, however, because of its simpli-
city and because of the value of other information obtain-
able from the solution, e.g., the dual.

The above developments form the basis of the formu-
;ation of a mathematical model of an actual scheduling

roblem.

 

IV. THE PLYWOOD PRODUCTION

SCHEDULING PROBLEM

Assembly-type problems occur very frequently in
industry; in fact, it is more the rule than the exception.
One such problem occurs in plywood manufacturing, where
various veneers that go into the lay-up of plywood panels
have to be processed on several machines with sufficient
lead time so assembly can later proceed with as little
interruption as possible. Since veneers are generally
processed on common machines, sequencing the right quanti-
ties of material through these machines is of prime impor-

tance.1

The following description of a scheduling problem
is that of a mill that utilizes relatively poorer quality

logs than generally are used.

IV-l. The Process

 

Plywood manufacturing (5) includes three major pro-

cesses: conversion of logs to veneer (also called the

 

lSequencing is even more important in many of today's
mills, where poorer quality logs are increasingly utilized
due to increasing log costs and scarcity of supply, thus
requiring more veneer preparation and more competition for
machine time. Although this situation is not yet typical
of the whole industry at the present time, it is expected
to be so in the near future when further increases in log
costs, scarcity of supplies and more competition from other
materials are to be expected.

26

27

green end), preparation of veneers to usable form (the

dry end), and assembly of correct mixtures of veneers into
plywood panels. Green end equipment includes one barker,
two cut-off saws, eight steam vats, one high—speed 8-foot
lathe, one A-foot lathe, and three clippers. Veneer prepar-
ation is accomplished with three dryers, one edge gluer,
one veneer saw, and four veneer patchers. Lay-up and
finishing machinery includes four glue Spreaders, two pre-
presses, two automatic 30-opening hot presses, one panel
saw, one high speed wide belt sander and one panel sorter.
In addition, there are other equipment associated with
special finishing and packaging of products.

The green end process consists of cutting logs to
length, steaming, peeling to veneer, clipping veneer to
standard widths, and sorting. As much as possible, veneers
are clipped as wide as defects would allow to 5A inches,
27 inches and narrower random widths. The mill utilizes
six species of logs for use as faces, backs and centers.1
Three of these are being peeled to two thicknesses (l/lO
and 1/6) and the rest to three thicknesses (1/10, 1/6 and
7/32) or a total of 15 different inputs at the 8-foot lathe
distinguished by species and thickness of peel. At the
A-foot lathe (or core lathe), four basic species are used.
Three are peeled to 1/6, 7/32, and 5/16 and the fourth to

only one thickness of 1/6 or a total of 10 possible inputs

 

1See Figure IV-l.l for description of a plywood
panel.

28

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on oofin one

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ooozmam ofimmm Song oopooo< .Hocma ooozzflo a no cofiuosnpmcoo aonQSBIIH.HI>H

 

 

Han LoccH u mmmezmo
: Amocoooaoaov mmmoo

u Mo<m

u mo<m

ohswfim

 

29

at the core lathe. Clipping practice at the core lathe
is similar to that at the 8—foot lathe. However, because
of poorer quality of logs being peeled specifically for
core, clipping is generally in random fashion.

The dry end process consists of drying veneers to
appropriate moisture contents, sorting dryer output by
widths and grades, and further processing on certain types
of veneers as the situation requires. Veneers are graded
into four grades after drying. Eight-foot stock is sorted
to Ang, g, D and NC grades.l Some of the SA-inch stock
is directly available after drying for faces and backs (g
and D grade) or centers (9, D, and NC grades) while ngp
grade needs further processing at the veneer patchers
where they eventually split up to grades A, B, Op, g, and
D. The C, D. and NC grades of 27-inch veneer may be used
as centers without further processing or edge glued (ex-
cept N9 grade) for faces and backs. Alternatively, they
may be sawn in half for core. The ABQR grade of 27-inch
stock is generally edge glued and patched for faces and
backs, although some of it is used for patching material.

Core veneer is sorted into four grades: Csolid,

Q, Q and NC and needs no further processing.
Assembly of veneers into plywood starts at the glue

spreaders where certain quantities, grades and thicknesses

 

lABCp is a veneer grade from which grades A, B and
Cp can be recovered. Grade 5 is the highest, B second
highest and NC is the lowest grade.

30

of veneer are assembled in certain combinations to obtain
required panel dimensions and grades. Then assemblies in
multiples of 30's or 60's (depending on the thickness) are
pre—pressed in preparation for automatic loading and curing
in hot presses. Thence, cured panels are trimmed to proper
dimensions, sanded and graded for defects. Panels that

are defective and cannot be accepted as "on—grade" are re-
classified to appropriate lower grades. Thus, only a cer-
tain percentage of the original number laid-up at the
spreaders can be applied to customer orders so that allow—
ances have to be made to compensate for "falldowns." For
sheathing products, the process would have ended after sort-
ing for grades; however, for sanded products and other
specialty items additional work on the panels is required,
such as repairing minor defects, additional sanding, groov-
ing, coating and other special work the customer may re—

quire.

IV—2. The Scheduling Problem

The processes after the spreaders do not pose much of
a scheduling problem since the operations follow one an—
other in a fixed sequence and so long as no breakdowns
occur in the line, the rest of the operations proceed with-
out requiring too much attention. Indeed, the heart of the
scheduling problem lies in the green and dry ends where
choices have to be made from among a large number of alter—

natives. In general, the choice of one alternative over

31

another is influenced by the composition of the order file
and of the veneer and product inventories on the floor and
by the available machine and storage capacities.

Customer orders for the coming week's production (an
order file) are received at the plant superintendent's
office on the Friday of the preceding week and may consist
of about 15 to 25 customer orders representing approxi-
mately 75 percent of plant capacity. Each customer order
may consist of any number of items in the some 152 basic
products that the mill manufactures. Usually, though, an
order ranges from one to ten items, altogether usually form-
ing a carload, sometimes two. Because of limited space in
the mill it is not possible to inventory dry veneers or
finished products in large quantities. Railroad cars are
brought in five at a time and must be loaded and ready
within three days at the latest. Because no additional
cars can be brought in until previous cars have been cleared
away, it is desirable to complete loading whatever cars are
on the docks within 2A hours. This means that orders have
to be completed and loaded at the rate of approximately
four to five orders a day. The problem has been basically
that of properly timing veneer preparation so that the
right types and quantities of veneer are available at the
time they are needed at the spreaders. Moreover, these
should be in quantities sufficient to maintain production
with a minimum of change—overs but small enough so as not

to build prohibitively large in-process inventories.

32

The availability of the right types and quantities
of veneer is important, too, to the proper use of material.
Experience has shown that misuse of veneer is extensive
when the proper grades are not available in sufficient
quantities. In an effort to maintain production, down-
grading1 is sometimes resorted to. Although downgrading
is not a policy of the mill, it is common knowledge that
it is practiced. At the present time the mill has no mea—
sure of how much loss due to improper use of material is
being sustained, mainly because it is very difficult (if
not impossible) to get factual data about it. However, to
those knowledgeable with losses in revenue due to mis-
application of material, the feeling is that the practice
might be reaching alarming proportions.

In general, if the schedule were to lay-up by customer
orders, finished products inventory would be minimized but
at the expense of probably too many changes in machine set-
ups and a probable build-up of dry veneer inventory,
especially if customer orders lean heavily toward one or
two product grades. This is a consequence of the fact that
the production of certain grades of veneer is always ac-
companied by the generation of other grades as well which
may not be needed immediately, if at all, or which may be

in excess of what is or will be required.

 

lUsing better quality veneer than what is required.

33

On the other hand, laying—up products that comple-
ment each other, i.e., products that use veneers in the
proportion they are generated, minimizes veneer inventory.
However, these products may belong to various orders making
it necessary to hold them in inventory until the other
items in the orders are likewise completed.

Efforts are being made by the company to seek orders
in the right combinations (through an allocation model) and
using the information as a guide for sales efforts. Un-
fortunately discrepancies between what is desired and what
is obtained always exist. Thus, failing to influence the
composition of demand for its products, it then becomes
necessary for the mill to control production effectively
so that whatever discrepancies may exist between the order
file and the mixture of veneers obtainable from the raw
material could be worked into the plant and still operate
within the constraints of machine and storage capacities
and delivery dates without incurring unnecessary increases
in production costs.

The presence of a number of alternatives for getting
required veneers to the spreaders gives the mill flexi-
bility for working around whatever limitations the order
file may impose on the mill. It also makes scheduling
that much more difficult since compatible decisions have

to be made consistently at each production center.

 

3A

Because of the formidable number of alternatives
that a production scheduler has to choose from, it is very
unlikely that the choices he makes every day or every week
are consistently good because it is simply impossible to
consider or to be aware of all interacting factors that
should enter into his decisions during the short time
available for decision making. Indeed, not even the best
and most experienced scheduler can or would be willing to
go through even a small number of, say, 200 alternatives
every week to determine which combination of alternatives
will give him his best (according to some criterion)
schedule.

Thus it is desirable to quantify the problem so that,
for any given week, order file and initial inventories, a
schedule that takes into account all important factors and

limitations can be determined.

V. FORMULATION OF A SCHEDULING MODEL

The formulations that will be developed in this
chapter concern only the green and dry ends of the ply—
wood mill where the scheduling problem resides. Because
of the great number of variables involved, the flows are
handled as vectors to simplify the algebra. A listing
of the variables included in each vector can be found in

Appendix A.

V-l. Choice of Interval

 

The choice of an appropriate interval of time to be
used in the formulation of the problem is an important
consideration since it determines the quality of infor—
mation that can be derived from the solution to the pro-
blem and the corresponding cost of computation required
to obtain the solution. Ideally the interval would be
the smallest lead time being experienced in the system.
However, a practical interval, during which it is useful
to have information on the level of activities, should
probably be something much larger. Indeed, it may be
only necessary or helpful to know the state of the system
once every two hours, or once a day, or once a month, even
though there exists in the system lead times in the order

Of, Say, 10 minutes.

35

36

In the problem studied, it is difficult to make a
definite statement on what might be the most practical
interval of time to use. Practicability has to be mea-
sured in terms of the difference between the cost to be
incurred and the benefits to be derived. Moreover, the
benefits to be derived have to be measured relative to
some datum, generally current performance. The cost of
computation can be reasonably determined; unfortunately
a measure of current performance is not readily available.
However, an evaluation of current performance should yield
some useful measure of effectiveness. Only then can
practicability or economic feasibility be truly determined.

In the following developments, an 8-hour interval is
used because it seems the most ideal from the plant superin—
tendent's point of view. Certainly his work would be much
simpler if for any given week the schedule of activities
of all processing centers has already been determined for
all shifts. As a consequence, decision making would be
narrowed down and localized to 8-hour periods since there
would be no more need to consider the relationship between
activities of different shifts, these relationships having

been considered already in the solution to the problem.

V-2. Model for Processor No. l

 

Processor no. I basically consists of the 8-foot
lathe, belt conveyors, two clippers, a sorting table,
sorting carts, and the men associated with each activity.

This is shown schematically in Figure V-2.2.

37

It takes approximately only an average of 10 minutes
to process veneer on the lathe and clippers, starting at
a point in time when a log is charged to the lathe to a
point when it becomes clipped veneer and available for
drying. If the interval chosen were 10 minutes, then the
lead time required for processing is one period, which
means that inputs to the lathe during any 10-minute period
will not be available as output from the clippers until
the next 10-minute period. If the interval chosen were
20 minutes instead, the lead time will still be 10 minutes;
however, inputs to the lathe during any 20-minute period
will all be available as clipped veneer after the first half
of the next period. The relationship between lead time and
period is shown schematically in Figure V-2.l. In the
figures, 5 represents the first period during which input
is being taken in and B represents the time (also equivalent
to one period) during which output is available. The first
input occurs at al and first output becomes available at
time bl‘ The last input (during the interval under consider-
ation) is taken in at time a2 which subsequently becomes
available as output at time b2. The lead time, represented
by alb1 or a2b2, is 10 minutes regardless of the magnitude
of the period chosen. However, if lead time is considered
in terms of a period, in (a) it is equivalent to one period,
in (b) half of a period and in (c) l/A8th of a period.
Thus, as the interval is chosen to be larger the lead time

becomes relatively smaller. At the chosen interval of

 

 

 

 

 

B
lead time
b
. .g 2
’I

 

 

 

 

let period 2nd period

(a) Lead time and period of 10 minutes each.

 

 

 

 

 

 

 

 

 

 

B
lead time
1
b1 _ ibz
a
a1 2
A
let period 2nd period

(b) Lead time of IO minutes and period of 20 minutes.

 

 

 

 

lead time
B
“i1 b2
a
a1 2
A
' let period 7 2nd period

 

(c) Lead time of 10 minutes and period of 8 houre(not

drawn to scale).

Figure V-2.1. Relationship between the magnitudes of lead
time and period.

39

8 hours the relative importance of a 10-minute delay
becomes very small and insignificant. Moreover, the lathe
and clippers have excess capacity relative to the dryers
so that the production of the lathe during any period can
keep the dryers busy during the same period. We thus
assume for practical purposes no delay in processor no. 1,
that is, no lead time is necessary for peeling logs to
veneer before drying can be initiated, provided that the
period in question is large enough. We therefore write

for the 8-foot lathe:

Y121(t)T Ril
Y122(t) R21
Yl23(t) = Rél Yllo(t) (v—2.1)
Y12u(t) RA
LY_125(t) _Ré1_

 

 

 

 

We note from Appendix A that YllO’ Yl2l’ Y122, Y123 and

Yl2A are vector flows. Consequently, Rll’ R21’ R31, R51
and R' are matrices whose entries represent the quantities

51
of veneer of each type that is obtainable from every unit

of input of each variable in YIIO' Y125 represents the
flow of material that is not usable for veneer. There-
fore, as far as the scheduling problem is concerned, it

is best excluded from further consideration. If we let

_ — R.‘i
Y122(t) 21
Yl2o(t) = Y123(t) and R11 ’ R31

 

 

 

 

%u(t)_ LRul

A0

then we can write V-2.l as

Yl21(t) a Ei}. Y (t) (V-2 2)

Equation V-2.2 is a mathematical model of processor no.
1 shown schematically in Figure V—2.l. To illustrate the
structure of Equation V-2.2, we write it in detail as
Equation V-2.3 using data from Table V-2.1. In Equation
' V-2.3, log inputs are expressed in thousand board feet1
(MBF) and veneer outputs are in thousand surface feet 3/8
basis2 (M3/8). The first column in the recovery matrix
indicates that for every MBF of Douglas fir logs peeled
to l/lOth veneer, 1.13 M3/8 of 5A-inch veneers, 0.82 M3/8
of 27-inch veneers, O.A6 M3/8 of strips and 0.12 M3/8 of
fish tails3 can be recovered.

We also note from Equation V-2.3 that during a given
period any number of inputs can be greater than zero. In
other words, two or more inputs can be processed concur-

rently during any 8-hour period. In practice, of course,

inputs are processed sequentially. This is perfectly

 

1A board foot is the volume of a rectangular piece
of wood one-inch thick, 12 inches wide and 12 inches long.

2Volume of a panel 3/8-inch thick, lO—feet wide and
100—feet long.

3Veneer generated at the 8-foot lathe before the

log gets perfectly round so that only one end may be
utilized, and only for core. See Figure V-5.l.

121

122
110

 

V

~—4r—e 1L_l23

 

12A

125

Figure V-2.2--Schematic diagram of processor no. 1 repre-
senting the 8-foot lathe, two 8—foot clippers and associated
equipment.

221

210 see

 

222

Figure V‘3.l~—Schematic diagram of processor no. 2 repre—
senting the core lathe, clippers and associated equipment.

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valid since, as will be seen later, the total demand for
machine time will always be equal to or less than what

is available.

V—3. Model for Processor No. 2

 

Processor no. 2 is made up of a cut-off saw, a core
lathe, a drum clipper, a guillotine clipper and veneer
handling accessories. Schematically it is shown in
Figure V—3.l as a single processor receiving logs at one
end and producing veneer at the other.

Following the discussion of Section V-2, we likewise
write for processor no. 2,

*
Y221(t) _ R12 Y

- (t) (V-3.l)
Y222(t) .252 210

where Y (t) represents veneer waste that cannot be

222
used for plywood production. We therefore drop it from

further consideration and write simply

Y (t) = R Y t) (V—3.2)

221 20 210(

where R20 is equivalent to R§2 and represents veneer re-
covery at the core lathe. Because the logs generally
used for core are of poorer quality, core veneer can
rarely be clipped into standard widths. Thus veneers
from the core lathe are not sorted by widths unlike those
from the 8-foot lathe. The entries of R20, therefore,

represent total veneer recoveries in M3/8 units from each

A5

MBF of log input in Y210. Equation V-3.2 or V-3.3 repre-

sents a mathematical model of processor no. 2.

V-A. Model for Processor No. 3

 

Three veneer dryers make up processor no. 3. Veneers
are brought to the dryers from green in-process inventories
and fed manually as fast as is operationally possible. On
the other side, veneers are sorted by grades and moved to
dry inventory.

It takes approximately 10 to 30 minutes for veneer
to dry while traversing the length of the dryers, depending
on the thickness and species. The delay due to processing
time alone seems small enough to be neglected as argued
for processors 1 and 2; however, there are other factors
that must be taken into consideration. Firstly, veneers
proliferate into several grades after drying so that specific
grades do not accumulate fast enough, making it imperative
to accumulate them in sufficient quantities before they
can be moved to the next operation. For example, ABCp
grade veneers which come in small quantities relative to
the total output of the dryers cannot operationally be
transferred to the patchers as soon as a few sheets are
available. Rather, they are accumulated until there is
enough to keep a patcher busy for a sufficiently long
period.

Secondly, there are technical requirements that must

be satisfied. For instance, in the case of g and 2 grade

A6

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veneers that are 5A inches wide, the next operation would
be at the spreaders. However, these should be allowed to
cool down to room temperature before using due to techni-
cal requirements at the spreaders.

It is therefore, necessary that sufficient lead time
be allowed for the drying process so that the requirements
of succeeding operations are satisfied. A lead time equi—
valent to one shift was found the most appropriate so that

we write as a mathematical model of the drying process,

"' - F'- '1 r"

Y330(t+1) R12 1A Y310(tfl
Y340(t+1) _ R21 Y320(t) (v—u.1)
Y350(t+l) - R31 Y37O(t)

Y36O(t+l) R31 R33 LY380(t)

 

 

 

 

 

 

which indicates that inputs to the dryers druing any 8-hour
period will not be available for further processing until
the next period. Because of the size of the recovery matrix
for the dryers, Equation V-A.l is not shown here in detail.
However, it can be constructed from data given in Table

V-A.l and the listing of vectors in Appendix A.

V-5. Model for Processor No. A

Processor no. A consists only of one veneer saw. It
provides the opportunity to convert 8—foot veneer to core,
hence also the opportunity to peel core material at the
8-foot lathe. Since very little time is required to saw

veneer, no lead time is required and we write,

 

 

 

A“ 0 52."

Figure V-3.2-—Fish Tail. Core material may be recovered from
it by cutting at dotted line. Unusable portion goes to chipper.

 

 

 

310

330
370 3UO
320 350
380 360

Figure V-A.l-—Schematic diagram of processor no 3. represent-
ing three veneer dryers.

A15
A10 A30
A2 “35
AAO

Figure V—S.1—-Schematic diagram of processor no. A represent-
ing a veneer saw.

TABLE V—A.l-a.

A9

Dryer grade yieldsl (percent).

 

Face

 

 

Stock Width ABCp _2 2 39
1/10 5A's 18.89 30.2A A9.55 1.32
Doug. Fir 27's 5.A0 21.50 60.03 7.07
Strips 1.A0 20.A0 68.A7 9.63
1/10 5A's A.81 A3.58 A6.l3 5.A8
White Fir 27's 0.71 36.30 51.07 11.92
Strips 6.17 77.59 16.2A
1/10 5A's 3.08 52.AA Al.15 3.33
Spruce 27's 0.06 A5.8l A7.69 6.AA
Strips 22.28 68.75 8.97
1/10 SA'S 22.10 3A.8A A0.11 2.95
Larch 27's 8.69 30.31 5A.65 6.35
Strips 3.A6 21.33 6A.72 lO.A9
1/10 5A's A.57 23.18 68.05 A.20
Pond. 27's 13.82 78.05 8.13
Pine Strips 2.68 77.66 19.66
1/10 5A's 11.5A 3A.A5 50.09 3.92
Hemlock 27's 2.21 30.55 58.88 8.36
Strips 9.76 77.65 12.59
1/6 5A's 10.6A A2.65 A3.58 3.13
White Fir 27's 3.50 A5.58 A0.00 10.92
Strips AO.A3 “9.72 9.85
1/6 SA'S 12.00 A5.00 39.00 A.00
3pruce3 27's 2.00 A2.00 A6.00 10.00
Strips 26.00 62.00 ‘l2.00
1/6 5A's 5.50 28.50 62.50 3.50
Pond. 27's 1.50 18.50 73.00 7.00
Pine3 Strips 15.00 75.00 10.00
1/6 5A's 18.80 38.9A A0.63 1.63
Hemlock 27's A.A5 A3.3A A6.29 5.92
Strips 1.21 39.92 53.17 12.70
1Based on May, June and August dryer production
reports.

2

Includes 9 E9110:

3Estimated figures.

50

TABLE V-A.l-b.-—Dryer grade yields.

 

 

 

 

Core Stock C 1
Type Solid C D NC

1/6 Pine Random 9.00 32.98 A9.51 8.51
1/6 Hemlock " 11.00 31.83 A6.60 10.57
1/6 Mix " 10.00 2A.61 56.16 9.23
1/6 Redry " 7.00 12.12 67.63 13.25
7/32 D. Fir " 12.00 A6.92 33.AA 7.6A
7/32 W. Fir " 10.00 56.A2 27.58 6.00
7/32 Spruce " 10.00 56.59 26.53 6.87
7/32 Larch " 15.00 58.69 22.10 A.21
7/32 Pine " 11.00 50.08 32.80 6.11
7/32 Hemlock " 13.00 37.00 A0.A7 9.53
7/32 Mix " 12.00 A6.30 3A.07 7.63
5/16 P. Pine " 12.00 1A.90 68.08 5.02
5/16 Mix " 13.00 25.73 5A.A1 6.86

 

lEstimated figures. It is difficult to determine
precisely the percentage of C solid that can be recovered
from core stock because core is only sorted for C solid
when there is a need for it. Otherwise, it is sorted as
C grade. However, the sum of C solid and C grade is

accurate.

 

51

 

 

 

 

YA30(t) F51 S2 S3 — F1A10(t;_
Y03u(t) = S“ Yu20(t) (v—5.1)
31435832, - SSj YMOOJ)

-Y415(t)J

 

 

where Y415(t) represents a vector of fish tailsl originat-
ing from the 8-foot lathe. YA10(t)’ Yu20(t) and YAA0(t)
represent, respectively, dry strips, 27—inch, and 5A-inch
veneers that are to be converted to core, the entries in
81’ S2 and 83 are all 1's indicating 100 percent recovery.
Y03u(t) is the quantity of core recovered from fish tails

and YA35(t) is the waste material that goes to the chippers.

V—6. Model for Processor No. 5

Processor no. 5 is made up of a jointer and an edge
gluer that is used for joining narrower veneer, i.e., 27
inches wide and narrower strips, to obtain full-sized
veneer sheets for use as faces, backs or centers. Edge
gluing is a slow process and it is necessary that some lead
time be allowed for the operation to accumulate enough out—
put sufficient to be considered for input to the next
operation. The lead time required may not be quite 8
hours. However, for simplicity we also assume a lead time
equivalent to one period. This has the effect of a con-

servative schedule since it implies that processing of

 

1See Figure V—5.l.

52

needed veneers has to be done one full shift ahead, al—
though in certain cases (such as when only small quanti—
ties are required) it might be possible to process veneers
during the same shift it is needed. For the edge gluing

process we, therefore, write

 

F1531(t+l)1 r811 G121 Y510(t)

Y532(t+1) = 021 G22 Y520(t) (v-6.1)
Y533(t+1) G31 G32

B53A<t+l)_ £111 GA;

 

 

 

where as before, G11’ G12’ G21, G22, G31, G32, GAl and
GA2 are matrices with entries that represent the percent—
ages of each output that can be recovered from each unit

of input. Processor no. 5 is shown schematically in

Figure V-6.l.

V—7. Model for Processor No. 6

Four veneer patchers make up processor no. 6. It
takes in for input full—sized veneer sheets of AB R grade
originating directly from the dryers, from purchased dry
veneer, or from the edge gluer. The process involves
patching knotholes and similar defects that can be
patched and segregating processed veneers into grades
5’ E: 92) 9.3nd 23

Like edge gluing, patching is also a slow process.

Moreover, it takes a while to accumulate the usual

531
510
532
533
520
' 5314

Figure V-6.1--Schematic diagram of processor no. 5 represent-
ing veneer jointer and edge gluer.

 

 

621
531
622
612 + at: 623
62A
630
625

7.l--Schematic diagram of processor no. 6 represent-

Figure V-
ing four veneer patchers.

5A

quantities required of A, B and Cp_grades, due to poorer
quality of logs available to the mill. Consequently, it
is necessary to allow processor no. 6 a lead time of one
period and we write the mathematical model of the process

as follows:

 

 

 

 

 

 

 

_ 1 _ l l 1
Y621(t+1) T11 T12 T13 Y611(t)

Y622(t+l) T21 T22 T23 Y612(t)

R2393”) = T31 T32 T33 LY630(t)_ (V-7.1)
Y62A(t+l) TA1 TA2 TA3

LY625(t+l)_ LT51 T52 T53_

 

Y625 represents waste generated by the process and may be

dropped from further consideration. Schematically the
patching process is represented by Figure V-7.l. Data
such as that shown in Table V—7.l for Douglas Fir required

by Equation V-7.l can be obtained from production reports.

V—8. In-Process Inventories
Figure V—8.l is a schematic diagram of the green and

dry end sections of the plywood mill showing the manner

component processors previously discussed are interconnected

together. The symbols and connotations parallel those in
Figure III—2.1 except that the flows indicated are vector

flows, that is, each arrow in the diagram represents, in

general, more than one flow. The solid circles also

represent one or more machines. For example, Y110(t)

55

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56

represents 15 different flows and processor no. 6 repre—
sents four veneer patchers.
With reference to Figure V-8.l, we now write the

following equations for in-process inventories:

Green Core Originating from Fish Tails

 

 

 

 

 

 

 

 

Z800(t+l) = U012800(t) + U02Y03u(t) — U03Y380(t) (v-8,1)
Green Core from A-foot Lathe
2810(t+l) = U112810(t) + U12Y221(t) - U13Y32O(t) (V—8.2)
Green Face Stock from 8-foot Lathe
2820(t+l) = U212820(t) + U22Y120(t) ' U23Y31o(t) (V‘8°3)
Y122(t) Y3ll(t)
where Y120(t) = Yl23(t) and Y310(t) - Y312(t)
t
Y12A(t) Lf3l3( )
Green Face Stock from Purchased Veneer
(V—8.A)

2830(t+1) = U31Z830(t) + U32YO37(t) - U33Y37O(t)

57

Dry Core

28A0(t+l) = UA128A0(t) + UA2Y330(t) ‘ YA3Y100(t)

 

 

 

 

 

 

Y331(t)
Y (t)
where Y 3O(t) = 332 and Y
3 Y (t)
333
Y (t)
A
L33 _.
Dry Strips
2850(t+1) = U512850(t) + U52Y3uo(t)
U5AY510(t)
Purchased Dry Veneer
2880(t+l) = v3l2880(t) + V32YO38(t)
‘ 1
Y3A1(t)
Y (t)
where Y3A0<t) = 3A2 , YAlO(t) =
Y3u3(t)
LY3uu(t)_
’ 1
Y511(t)
and Y510(t) = Y512(t)
LY513(t)_

 

100

<t>1

Y101
Y

102(t)
Y103(t)

(t) =

 

 

L?10A(t)

' U53YA10(t) ’

’ V33Y63o(t)

— 1
YA11(t)
YA12(t)

YA13(t)

 

 

LFA1A(t)

(V—8.5)

(V-8.6)

(V-8.7)

Dry 27's

58

2860(t+1) = U612860(t) + U62Y350(t)

where Yu20(t) =

Dry 5A's

2870(t+l) =

where Y36O(t) =

U6AY520(t) ' U65Y012(t)

U67Y015(t)

P.

YA21(t;7
Yu22(t)

Yu23(t)

 

 

LFA2A(t{_

U712870
U7AY532
U77Y011
T71Y017

1
Y361(t)

Y362(t)
Y363(t)

(t)

(t)

(t)

(t)

 

 

LF36A(tZ_

and Y52O(t)

+

U72Y360(t)
’ U75YAAO(t)
' U78Y01A(t)

‘ T72Y021

(t)

+

U63YA20(t) ‘

U66Y013(t) ‘

Y521
Y522(t)

 

U73

U76Y612(t) ‘

U79Y016(t) -

— .7
YAA1(t)
YAA2(t)
Yuu3(t)

 

 

LYAAA(P)

<t)1

 

Ly523(t)_

Y62A(t) +

(V-8.8)

(V—8.9)

59

Processed Core

 

2890(t+l) = U91Z890(t) + U92Y100(t) - U93Y200(t) (v—8.10)
Y201(t)1
Y202(t)
where Y200(t) = Y (t) and YlOO(t) = Y33O(t).
203
L?20A<t)l

 

 

Processed Centers

 

2900(t+1) = V112900(t) + V12Y013(t) + V13Y012(t) +
V1AY533(t) + V15Y01l(t) + V16Y01A(t) +
V17Y3O(t) - V18Y300(t) (V-8.11)
" T
Y3Ol(t)
where Y300(t) = Y302(t) and Y3O(t) = Y015(t) + Y016(t)
L3303(t)_

 

Faces and Backg

 

(t+1) = V212910(t) + V22Y017(t) + V23Y021(t) +

 

 

Z910

V2AY620(t) ‘ V25YA00(t) (V-8.12)
1’ 1

Yu01(t) _ fl_

YA02(t) Y621(t)

where YAOO(t) = YA03(t) and Y620(t) = Y622(t)
YA0A(t) Li623(t)J
Lyu05(t)_

 

 

60
From Equations V-2.1 and V-5.l,
Y03A(t) = SA YA15(t)
YA15(t) = Y121(t) = R11Y110(t)
Substituting in Equation V—8.l, we obtain

2800(t+l) = U012800(t) + U02SAR11Y110(t) ‘ U03Y380(t)

(V-8.13)
We also have from Equation V—3.2,
Y221(t) = R20Y210(t)
Substituting into V-8.2, we have
2810(t+l) = U11Z810(t) + U12R2OY210 - U13Y32O(t) (V—8.lA)

V.9. System Model
Equations V-A.l, V—6.l, V—7.l, V-8.3, V-8.A, V-8.5,
V-8.6, V—8.7, V—8.8, V—8.9, V-8.10, V-8.ll, V—8.l2,

V-8.13, and V—8.lA can all be written in matrix form as

Equation V—9.l, where

 

 

 

A11 = U02 SA R11 ’ A12 U12 R20 ’ A13 _ U22 R11 ’
—7 ’1 5111
T11 T12 13

— '—

Til = T21 ’ T12 ’ T22 ’ and T13 T23

T
[F3r [F31 L31

 

 

 

If we let,

S(t)

2800(t)
Z810(t)
2820(t)
2830(t)
2850(t)
2860(t)
2870(t)
2880(t)

' 2890(t)

Z900(t)
(t)
Y33O(t)
Y3Ao(t)
Y350(t)
Y360(t)
(t)
(t)
(t)

2910

Y531
Y532
533

Y620(t)

 

 

L?62A<t)

, F(t)

61

i- 1
Y011(t)
012(t)

Y013(t)

Y01A(t)

015(t)
Y016(t)

YOl7(t)

021(t)
Y037(t)

YO38(t)

110(t)

210(t)

310(t)

F<+<r<

320(t)
Y37O(t)
Y38O(t)
YA10(t)
YA20(t)
YAA0(t)
Y510(t)
Y520(t)

Y612(t)

 

 

3630‘W

, E(t) =

I’m K a?)

 

200(t)

300(t)

A00<t14

 

 

 

 

 

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then we can write Equation V-9.l simply as
S(t+l) = P S(t) + Q F(t) — G E(t) (V-9.2)

where P, Q and G are matrices. For any given period,
E(t) represents veneers demanded by the spreaders due to

a lay—up schedule J'(t), that is
E(t) = C J'(t) (V-9.3)

where C is a construction matrix with entries representing
the quantities of each type of veneer required by each pro-
duct in J'(t). The matrix C is shown in detail in Equation
V-9.3-b for certain selected products.

The lay-up schedule at the spreaders, represented by
J'(t), includes allowances for falldown.l The relation—
ship between the quantities required by the order file and

the quantities to be laid-up is given by

D J'(t) (V—9.A)

J(t)

or J'(t) D-lJ(t) (v-9.5)

where J(t) represents the quantities required by the order
file. The falldown matrix D is always square and lower
triangular (if product grades are arranged in decreasing

order) whose rows are independent of each other. Therefore,

 

lProducts not acceptable in the grade they were
originally laid-up due to defects that developed during

manufacture.

65

D—1 exists (17). To illustrate the construction of D,

we show it for the products used for an example in Equation
V-9.A-b, using data from Table V-9.l. The complete fall-
down matrix can be constructed from data given in Tables
V-9.l-a and V—9.l-b.

Substituting Equation V—9.A in V-9.2, we obtain

P S(t) + Q F(t) - G c J'(t) (V—9.6)

S(t+l)

or S(t+l) P S(t) + Q F(t) — H J'(t) (V—9.7)

where H = G C. Equation V-9.7 is a mathematical model of

the production process. For any period t_we have the

following:
S(t) = Veneer inventories at the beginning of the
period.
S(t+l) = Veneer inventories at the end of the period.
F(t) = Schedule of activities at all processing
centers preceding the spreaders.
J'(t) = Lay—up schedule at the spreaders.

We are now in a position to solve Equation V-9.7 for t = 0,1,

2.,,,1A,15, as follows.

P 8(0) + Q F(O) - H J'(O) (V-9.8)

S(l)

P 8(1) + Q F(l) — H J'(l) (V-9.9)

U)
A
R.)
v
ll

S(15) = P S(lA) + Q F(lA) - H J'(lA) (V—9.22)

66

 

 

 

 

 

 

 

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N\H
0\m
0\m
N\H
0\m
0\m
N\H
0\m
0\m
m\a

0\m

 

 

 

 

68

TABLE V-9.l-a.--Pane1 falldown summaryl (exterior sheathing).

 

 

On- Non- Sample
Product Grade Sub-Grade Cert Shop Blows Size2
3/8 CC 80.68 3/8 00—8.51 9.A3 0.63 0.75 1599
1/2 00 8A.17 1/2 CD-A.28 10.A9 0.82 0.2A A955
5/8 00 72.89 5/8 00—2A.67 2.AA A50
5/16 CD 85.10 12.37 0.7A 1.79 258A5
5/16 CD #2 77.89 17.17 1.83 3.11 1A17
5/16 CD #3 76.01 18.17 3.A9 2.33 2063
3/8 CD 83.65 1A.29 1.09 0.97 A5828
3/8 CD #2 82.68 13.53 1.91 1.88 3829
3/8 CD #3 83.89 12.A2 2.88 0.81 3576
1/2 CD 8A.59 13.53 1.18 0.70 150382
1/2 CD #2 82.30 15.28 1.72 0.70 76A32
1/2 CD #3 85.11 12.61 1.5A 0.7A 15579
5/8 CD 75.19 15.09 1.82 7.90 169A6
5/8 CD #2 8A.07 13.78 1.37 0.78 A969
5/8 CD #3 85.52 11.82 1.57 1.09 _-3
3/A CD 85.6A 11.37 1.9A 1.05 11578
3/A CD #2 86.08 lO.A6 2.78 0.68 1185
3/A CD #3 86.38 11.56 1.6A 0.A2 7A19
5/16 TRU PLY 96.81 3.19 660
1/2 TRU PLY 9A.01 2.80 3.19 20187
5/8 TRU PLY 91.A5 5.96 2.59 772

 

lData taken from July, August, September, and up to
October 18 (1966) production reports.
2Panels.

3No data available. Interpolated from 1/2 CD #3 and
3/A CD #3 figures.

69

.oaanfim>m poc nonvona an menswfim czopaamma

 

 

o.ooa 0.0m 0.00 00
0.00H 0.0a o.m 0.0a om
o.ooa o.ma o.ma o.ma mm
o.ooa o.mm 0.0 0.00 0a
o.ooa o.am o.ma 0.30 ea
0.00H 0.0a o.m 0.0a 0.0m ma
0.00a o.ma 0.0 o.mm o.mm as
asses 0000 00 00 00 ea 00 me ea oosso
Heefiwaso

030000090 000nm

 

H

.pcoonmm CH mpozpong UmUQdm pom assessm czooaamm Hocmm||.nna.0|> 00009

7O

Transposing terms and rearranging, we can write Equations

V-9.8 through V-9.22 as Equation V—9.23. In Equation

V-9.23,

A1

0;
F(J)
J'(J)
8(0)

3(15)

Matrix whose entries are productivity coeffici-
ents, that is, processing time for each unit
of input at each machine center,

Matrix with entries of 1's,

Falldown matrix,

Products that cannot be satisfied and must be
backlogged,

Order file,

Column vector of machine capacities,

Column vector of inventory space capacities,
Penalty cost for backlogged orders,

Cost associated with each activity,

Cost of holding veneers on inventory,
Production schedule during the (J+l)th period,
Lay—up schedule during the (j+l)th period,
Beginning inventories, and

Ending inventories.

The submatrices, A1, are constructed from data such as

shown in Tables V-lO.l, V—lO.2, and V-lO.3.

71

 

Amm.0u>0

ESEHCHE
mHH

:HH

 

1 0

 

VII

VII

VII

VII

VII

VII

VII

 

Aav.h
on.h

Azavm

Aavm

onm

 

 

 

3'

ma

ml!

2...

H00

 

 

72

.COHposuomm 00 mason 000 :03» pmummpw .mmazpmzpo

 

.pzacfi mo Ameom xooanv pm 00 0000 000 003000

.psapso mo pmmm mommpsm 0000 Log mpsom

0 .AmHmom 000000 050: 000 pmmm 000000

.Am\mv mwoa mo up 00 000 0mg pm mommpsm :0 mpo>oomm

n

.00\mv 0:00 000 0020

.coapozvopm mo mpson 00 cmsp mmm0**
.cofiposuona mo mason om can» mmmq

a.

 

 

 

*x0000.0 **Hm00.0 mm00.0 0000.0 0000.0 0000.0
**m0m.0 *xm00 *xomm.00 000.0 0mm 2:0.0m 002.0 0mm 000.00 @000
0000.0 m000.0
000.0 0mm :zm.:0 £0000
*00ma.0 *mm00.0 ##00ma.0 **m:mo.0 2000.0 0000.0
*00m.0 *mom *m00.00 **0mm.0 **0mm **mmm.00 000.0 :mm 00m.00 005000
*0mma.0 *0000.0 0mm0.0 0000.0 0000.0 0000.0
*m00.m *0mm *zmm.m0 002.0 0mm 000.00 000.0 :mm 000.00 xooHEmm
mmm0.0 mm00.0 0000.0 0000.0 00m
mm:.0 0mm 000.00 000.0 20m m0:.:0 mp0£3
00m0.0 0000.0 $0000.0 *0mm0.0 mm0m0.0 0:mmo.0 00m
000.0 000 000.00 #00:.0 0000 0000.00 0000.0 0000 0000.00 0000000
pSQCH oomm psauzo psasH comm unmade uSQCH oomm psapso mmaomam
mm\0 0\0 00\0

mmmcxofige 000m

.mmpmp coaposuopa mzpma poomlwll.0.00|> mqmqe

73

TABLE V-lO.2.--Drying productivity.l

 

 

Veneer

Thickness Species Hrs/MSF3/8 Hrs/MBF
l/lO Douglas Fir .OMM9 .1175
1/10 White Fir .0815 .1737
1/10 Spruce .0590 .13U5
1/10 Larch .O6u8 .15U3
l/lO Ponderosa Pine .0602 .1U21
1/10 Hemlock .0633 .1419
1/6 Douglas Fir .0783 .1980
1/6 White Fir .1282 .2731
1/6 Spruce .1011 .2304
1/6 Larch .1008 .2900
1/6 Ponderosa Fine .1005 .2333
l/6 Hemlock .0924 .2200
7/32 Spruce .1078 .238u
7/32 Ponderosa Pine .11uu .2u13
7/32 Hemlock .1019 .2282
5/16 Spruce .1110 .2u53
5/16 Ponderosa Pine .1177 .2483
5/16 Hemlock .1130 .2U3O

 

lAverage for all widths.

74

 

 

TABLE V-10.3.--Patching productivity.l
Veneer Original
Thickness Species Grade Width Hrs/MSF3/8
1/10 Douglas Fir AB 54 1.43
1/10 Douglas Fir ABCp 54 1.47
1/10 White Fir ABCp 54 1.70
1/10 Larch " " 1.33
1/10 Hemlock " " 1.77
1/6 Douglas Fir " " 0.94
1/6 White Sir " " 1.02
1/6 Larch " " 0.80
1/6 Hemlock " " 1.07
1/10 Douglas Fir " 27 1.80
1/10 White Fir " " 1.97
1/10 Larch " " 1.53
1/10 Hemlock " " 2.04
1/6 Douglas Fir " " 1.08
1/6 White Fir " " 1.18
1/6 Larch " " 0.99
1/6 Hemlock " " 1-2”
1/10 Douglas Fir " Strips 1.97
1/10 White Fir " " 2.13
1/10 Larch " " 1.67
1/10 Hemlock " " 2-2“
1/6 Douglas Fir " " 1.17
1/6 White Fir " " 1.29
1/6 Larch " " 1.01
1/6 Hemlock " " 1°35

 

lEstimated figures or based on small sample.

VI. PRACTICAL IMPLICATIONS

The practical usefulness of the preceding formu-
lation (or any other formulation) depends on two impor-
tant considerations which must be taken into account
before any attempt can be made to implement the proposed
method. The first consideration is the improvement that
can be expected over the present method of scheduling and
second, the cost of implementation, in particular, the
cost of computing time required to obtain a solution to
the problem.

The first consideration is not by any means easy to
evaluate because means for measuring scheduling effective-
ness are not generally available. Moreover, the proposed
method has to be tried and tested for a long enough period
before a comparison can be made. Such parameters as (1)
unit processing cost, (ii) weekly production "through-put,"
or (iii) lateness in deliveries, can possible be used to
measure scheduling effectiveness. However, it should be
borne in mind that the magnitudes of these parameters
could change, not as a result of a change in scheduling
techniques, but due to some other factors.

There is no question that current methods can be

improved upon. This conclusion comes from knowledge that

75

76

materials are every now and then misapplied, from observ-
ing frequent slowdown in production because sufficient
lead time was not allowed in preceding operations, from
observing the pile-up of in-process inventories because
activities at various processing centers were not coordi-
nated properly, and from similar omissions that every now
and then hinders the productivity of the mill. But whether
or not the improvement can be effected at a relatively
smaller cost to make implementation economically feasible
is another matter. Indeed, in any industrial innovation,
it is the "pay-off" that is the deciding factor. Conse—
quently, it is possible for an improvement to be practical
in one application and impractical in another if the dollar
savings in one is more than the other, although the size
and complexity of the problem in both cases are comparable.

The cost of computation (which for practical purposes
represents the cost of implementation) can be easily esti-
mated because the running time for any given size of a
problem does not vary very significantly from one run to
another. Thus, given the expected cost of computation, the
question that remains to be answered is Whether it is less
than the savings it is anticipated to make.

The scheduling problem of the mill as formulated in
Chapter V would result in a linear programming matrix of
dimensions in the order of 3000 rows by 5000 columns. The

matrix, however, is sparse with a density of approximately

77

0.53 percent. From past experience on the UNIVAC 1107,

it is estimated that the model would entail a computation
time of approximately two hours for each weekly schedule
and, depending on where it is run, may cost anywhere from
$600 to $1000 a run. The question then becomes: Would
the expected improvement be worth more than $1000 a week?
Initially, of course, this question can be answered only
in somewhat vague terms. However, a production manager
who has been keeping an eye on inefficiencies or deficien-
cies in the mill should be able to answer the question

emperically with reasonable accuracy.

V—l. The Static Case

 

The company for some time had been trying to formulate
a computer model of a "veneer scheduler" which in effect
would determine the quantities of materials (logs and
veneers) that would be required by a given order file. In
effect, the information to be obtained from the model would
be a summary of material requirements for the whole week.
Undoubtedly, there would be questions raised as to the
usefulness of such information. Admittedly, a lot still
remains for the production scheduler to do as far as mak-
ing specific machine time allocations is concerned. But
the information to be derived from such a veneer scheduler
will serve a very useful purpose of a guide for quantity

requirements of each type of material.

78

The same type of information can be obtained from
the model outlined in Chapter V with very little modifi—
cation. This is easily done by taking the period equal
to the planning horizon, i.e., a week, and making adjust—
ments in the lead times required by each processing
center. This is what might be referred to as the static
case and was considered an excellent starting point for
evaluating the value of the model. The static model as
formulated for the mill can be found in Appendix E. It
is intended to be run at the beginning of each week with
the order file and initial inventories as inputs. Its

solution consists of two stages.

VI-1.1. Phase I

The first stage is the determination of the lay-up
schedule, i.e., the quantities of each product that are
required to satisfy the order file, and of the correspond—
ing quantities of veneers that will be required. The
following are the factors considered in the calculation
of the lay-up schedule:

1. Products already on inventory at the beginning

of the week

2. Allowances to be made for product falldown

3. The order file
Because pressing is done by the press loads, viz., in
multiples of 60's for 1/4- and 3/8-inch plywood or 30’s

for l/2—inch or thicker plywood, the initial lay-up

79

schedule together with other pertinent data is shown in
Appendix B. Corresponding veneer requirements are shown

in Appendix C.

VI-l.2. Phase 11

Given the veneer requirements shown in Appendix C,
the next step is to determine how best these veneers can
be made available at the spreaders. This is accomplished
with the model shown in Appendix E. The matrix has 491
rows and 927 columns and takes about 11 minutes of com-
puter time on the UNIVAC 1107. The information obtained
from the solution includes (1) production schedules for
each processing center, (ii) outside veneer required to
balance what is available from the raw material against
the requirements of the order file, and (iv) veneer re-
quirements that cannot be met during the week and must be
backlogged. Valuable information can also be obtained
from the dual solution of the problem. The most important
of these is the information on machine capacities that are
expected to be inadequate (as far as the current order file
is concerned), thereby giving the scheduler an opportunity
to make necessary arrangements for overtime and other
similar adjustments. The production schedule correspond—
ing to veneer requirements indicated in Appendix C is

shown in Appnedix D.

VII. CONCLUSIONS

The discussion in Chapters III, IV and V outlines a
procedure by which a scheduling problem involving assembly
may be modelled mathematically so that a realistic sche-
dule can be derived from its solution. The procedure is
built around the premise that alternatives are selected
over other alternatives on the basis of cost, within the
limitations of machine and storage capacities. To illus-
trate in part the behavior of the model, we assume a situ-
ation in which there is a demand for g centers at the
spreaders. Specifications allow the use of Douglas fir,
larch, white fir, spruce, ponderosa pine, or hemlock for
centers. Normally, the model would choose the most
economical source of g centers, since the objective is to
minimize total processing cost. Assuming that there are
sufficient machine capacities available, it would probably
call for a peel of such less expensive material as ponderosa
pine or spruce even if there is already available on inven-
tory 0 grade of Douglas fir. Douglas fir, of course, has
more valuable applications. Suppose, however, that there
is a shortage in dryer capacity. Then the model would be
constrained to use Douglas fir veneer (sustaining a higher

material cost because of capacity limitations) or backlog

80

81

the demand for g centers and suffer a corresponding penalty
cost, whichever is the cheaper. 0n the other hand, if 9
grade of hemlock were also available from inventory then
hemlock would be selected since it is a less valuable
material than Douglas fir.

It is clear that the behavior of the model is highly
dependent on the relative "costs" associated with each
activity or input; therefore, it is important that these
"costs" be accurately determined or appropriately chosen.
It is highly debatable whether what is referred to here as
"cost" is truly cost or a combination of cost and value.
In certain cases it can be actual direct processing and
material cost, in other instances, a combination of pro-
cessing cost and value (of material) is more appropriate.
For example, in applications where more than one species
are equally acceptable (such as the previous example),
actual cost or a combination of processing cost and value
can be used. However, if the choice were between materials
of the same species but of different grades, the picture
changes somewhat. Accounting practices in the mill con—
siders actual cost of Q and D veneer of the same species
to be the same, the processing cost being the same for
both grades and there being no accurate method of allocat-
ing the cost of material to various grades recoverable from
the log.

The model as formulated in the dynamic case (Chapter

V) or the static case (Appendix E) is rather formidable in

82

size. Questions might, therefore, be raised regarding the
efficiency of the method here proposed for describing sche-
duling problems. It is difficult at this point to assess
the efficiency since there is no basis for comparison.
However, it can be pointed out that the problem modelled
is not an idealization of reality but an actual problem
that involves 152 basic products that are assembled from
43 basic veneer types, 210 dry veneer in-process inven-
tories, 50 green veneer in-process inventories, and 6 pro-
cessing centers. A typical week might involve 20 customer
orders (Jobs) each requiring one to ten of the 152 pro-
ducts that the mill manufactures. Surely by any standard,
the problem is not trivial.

The model, however, can be simplified by excluding
certain activities or inputs that can never or very remotely
will ever be chosen over other alternatives; for example,
sawing 27-inch 9 grade of Douglas fir for core (Appendix
E, 881, column 1234) or edge gluing 2 grade of 7/32 spruce
strips (Appendix E, EGI, column 1209). Another possibility
is consolidating certain veneer in-process inventories that
can be combined; for example, combining all inventories
that are 54 inches wide (Appendix E, V154 and VRFX—X,
VRC8—X) and combining all core inventories (VIC4 and VRC4-X).
It is estimated that the number of variables can be reduced
by about 20 to 30 percent if the above possibilities are

undertaken. It will, however, require careful study and

83

time to make sure the removal of these variables does
not impair the effectiveness of the model.

Any mathematical model is only as good as the data
fed to it. The static case presented in Appendix E was
constructed with some estimated figures. While these
figures were chosen with care and are believed to be
realistic, they still remain estimates of certain pro-
duction statistics that were not readily available at
the time the model was formulated. The following are the
data in the model that need upgrading:

l. Recoveries at the edge gluer.

2. Productivity of the edge gluer for each species

and each thickness.

3. Productivity of the Riemann veneer patchers.

4. Falldown figures for sanded products.

5. Processing cost at each production center.

The production schedule presented in Appendices B,
C and D was obtained from a computer run using actual data.
These results were discussed with production people to
determine whether or not they are reasonable in the light
of what might be expected from usual methods of scheduling.
The major difference was in the tendency of the model to
bring in outside veneer (through purchase) even before
machine capacities are exhausted as contrasted with the
usual tendency of peeling all veneers that can be peeled

until machine capacities run out. This difference was

84

traced to the fact that it is the policy of the mill to
utilize its resources (logs and machines) whenever it can
without too much regard to the resulting in-process in-
ventories on the floor. The model, on the other hand

sees to it that levels of in-process inventories on the
floor do not exceed the prescribed maximum. We point

out that bringing in outside veneer is a good way to
balance the veneer grades obtainable from the logs against
the requirements of the order file, thereby minimizing
extraneous veneers not needed for production.

The model also has a greater tendency to up-grade
veneers (through edge gluing) than what is normally done.
No comments, however, can be made regarding the logic of
this tendency until more accurate data from the edge glu-
ing operation is available, viz., productivity of the
edge gluer for each species and grade, recoveries for each

grade and edge gluing costs.

APPENDICES

85

it

 

 

APPENDIX A

LIST OF VECTORS AND VARIABLES

86

 

YllO Vector--Log inputs to

87

8-foot lathe

 

 

 

1. Douglas Fir peel to l/lOth

2. White Fir " " n

3. Spruce " n n

4. Larch " H n

5. Ponderosa Pine " " "

6. Hemlock " " "

7. Douglas Fir peel to l/6th

8. White Fir " " "

9. Spruce " " "
10. Larch n n n
11. Ponderosa Pine " " "
l2. Hemlock " " n
13. Spruce peel to 7/32nd
l4. Ponderosa Pine " " "
l5. Hemlock " " "
Y210 Vector--Log inputs to core lathe

1. White Fir peel to l/6th

2. Spruce n n n

3. Ponderosa Pine " " "

4. Hemlock " " "

5. Spruce peel to 7/32nd

6. Ponderosa Pine " " "

7. Hemlock " " "

8. Spruce peel to 5/16th

9. Ponderosa Pine " " "
10. Hemlock " " "

-- 8-f ot veneer
Y120’ Y310, and 2820 Vector Green o

1. 1/10 Douglas Fir 54s

2. H " 27s

3. " " strips

4. " White Fir 54s

5. H H 278

6. " " strips

7. " Spruce 548

8. n n 273

9. " " strips
10. " Larch 548
ll. " " 27s
12. " " strips
13. " Ponderosa Pine 54s
1”. n n 27S
15. " " strips
16. " Hemlock 548
17. H H 275
18. " " strips

(Continued)

 

88

Y120’ Y310, and 2820 Vectors (continued)

 

19. 146 Douglas Fir 54s
1 "

 

20. 27s
21. " " strips
22. " White Fir 54s
23. n n 278
24. " " strips
25. " Spruce 54s
26. " " 27s
27. " " strips
28. " Larch 54s
29. n n 278
30. " " strips
31. " Ponderosa Pine 54s
32. n n 278
33. " " strips
34. " Hemlock 54s
35. n n 278
36. " " strips
37. 7/32 Spruce 54s
38. n n 278
39. " " strips
Y121 and Y415 Vectors-~Fish tails
1 1/10 Douglas Fir

2 " White Fir

3 " Spruce

4. " Larch

5. " Ponderosa Pine

6 " Hemlock

7 1/6 Douglas Fir

8. " White Fir

9. " Spruce

10. " Larch

ll. " Ponderosa Pine

12. " Hemlock

13. 7/32 Spruce
" Ponderosa Pine
15. " Hemlock

-- r v
Y037’ Y370’ and 2830 Vectors Purchased g een eneer

l. l/lOth Douglas Fir AB
2 . H I! H CD

89

Y22l’ Y320, and Z810 Vectors--Green core

 

l/6th White Fir random width

n Spruce n u

" Pond. Pine " "

" Hemlock " "
7/32 Spruce " "

" Pond. Pine " "

" Hemlock " "
5/16 Spruce " "

" Pond. Pine " "

" Hemlock " "

Y and 2800 Vectors--Green core from fishtails

380

l l/lOth Douglas Fir random width
2 " White Fir " "
3 n Spruce n n
4. " Larch " "
5. " Pond. Pine " "
6 " Hemlock " "
7 1/6th Douglas Fir " "
8 " White Fir " "
9. " Spruce " "
10. " Larch " "
ll. " Pond. Pine " "
l2. " Hemlock "
13. 7/32nd Spruce " "
" Pond. Pine " "

15: " Hemlock "

Y 30’ YlOO’ Y200’ 2840’ 2890 and Yu3O—-Dry core

3

l 1/10th C solid random Width
2 " C H n

3 " D H n
4. " NC n n
5. l/6th C solid :: a

n

7 " g n n
8

9

H NC
7/32nd 0 solid " "
H C

11: " D
12. " NC
13. 5/l6th C solid "

. " c
15. n D
16. " NC

Y34O’ Y410 and 2850 Vectors-—Dry strips

 

1.
2.
3.
4.
5.
6
7
8

9.
10.
ll.
12.
13.
14.
15.
l6.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.

l/lOth
H

Douglas Fir
" H

H H
H H

White Fir

H I!
I! H
H H

Spruce
H

Hemlock
H

H
H

Douglas Fir
n H

H "
H H

White Fir
H H

(continued)

ABCp
C

D

NC
ABCp
C

D

NC
ABCp
C

D

NC
ABCp
C

D

NC
ABCp
C

D

NC
ABCp
C

D

NC
ABCp
C

D

NC
ABCp
C

D

NC
ABCp
C

D

NC
ABCp
C

D

NC
ABCp
C

D

n
V

ABCp
C

D

NC

 

Y340’ Y410

and 2850 Vectors

(continued)

:3. 7/32nd Spruce ABCp

. C

5 l . I! H

5 2 . I! I! EC

53. " Pond. Pine ABCp

5 LI . H n n C

5 5 . H H H D

5 6 . n n :1 NC

2%. 3 Hemlock ABCp
H

5 9 . H H g

60. " " NC

Y510 Vector-—Strip input to edge gluer

 

Same as Y340’ escept without NC grades.

 

 

Y350, Y420’ and Z860--Dry 27—inch veneer
Identical with Y340 vector

Y520 Vector--27-inch veneer input to edge gluer
Identical with Y510 vector

Y360’ Y440’ and Z870 Vector--Dry 54-inch veneer

 

Identical with Y34O vector

Y and Z

Vectors--Centers

 

 

300 900
l. l/lOth 0 Any species
2 o H D H H
3 O H NC " fl
4. l/6th C " "
5 . H D H H
6 . n NC I! n
7. 7/32nd C " "
8 . u D n u
9 . I: NC n n

Y400 and 2910 Vector-—Faces and Backs
l 1/10th Doug. Fir/Larch A
2 . H H B
3 . H H Cp
Ll . n n C
5 H II D

(continued)

92

Y400 and Z910 Vector--Faces and Backs (continued)

 

6.

 

l/lOth Wh. Fir/Hemlock Cp
7 . H H C
8. " " D
9. " P. Pine/Spruce C
10 . n n D
11. l/6th Wh. Fir/Hemlock Op
12 . N H C
13 . N H D
YOll--54-inch C center input to spreaders, and
Y013--27-inch C center input to spreaders
1. l/lOth Doug. Fir
2. " White Fir
3. " Spruce
4. " Larch
5. " Ponderosa Pine
6. " Hemlock
7. 1/6th Douglas Fir
8. " White Fir
9. " Spruce
10. " Larch
ll. " Ponderosa Pine
12. " Hemlock
13. 7/32nd Spruce
l4. " Ponderosa Pine
15. " Hemlock
Y012--27-inch D center input to spreaders, and
YOlu—-54-inch D center input to spreaders

 

Identical with Y011 and Y013, except it is D grade

Y

Y015

016

Identical with Y011 and Y013,

Y

--27-inch NC center input
—-54-inch NC center input

--54-inch D back input to
021

Identical with Y014

Y017

Identical with YOll

to spreaders, and
to spreaders

except it is NC grade

spreaders

-—54—inch C back/face input to spreaders

 

93

Y6l2--54-inch ABCp input to patchers

 

1. 1/10th Douglas Fir ABCp
2. " White Fir "
3. " Hemlock "
4. " Larch "
5. l/6th Douglas Fir "
6. " White Fir "
7 " Larch "
8. " Hemlock "

Y62u-—edge glued C grade, 54-inch veneer

 

l l/lOth Douglas Fir C
2 " White Fir "
3 " Spruce
4. " Larch "
5. " Pond. Pine "
6. " Hemlock "
7 l/6th Douglas Fir "
8 " White Fir "
9. " Spruce
10. " Larch "
ll. " Pond. Pine "
12. " Hemlock "

Y038’ Y630 and 2880—-Purchased dry veneer

 

l. l/lOth Douglas Fir AB
1' H H

 

2. CD
Y53l--Edge glued 54-inch ABCp
l. l/lOth Douglas Fir edge glued 27's
2. " White Fir " " "
3 o H LarCh N n N
4. n Hemlock n n n
5. l/6th Douglas Fir " " "
6. " White Fir " " "
7 . H LaI‘Ch H H H
8. n Hemlock " n n
9. l/lOth Douglas Fir edge glued strips
10- " White Fir " " "
l l . H LarCh I! I! H
12. " Hemlock " " "
13. 1/6th Douglas Fir " " "
l4. " White Fir " " "
15 . H LaI‘Ch H H H
16. " Hemlock " " "

 

94

Y125, Y222 and Yu35-—Green veneer waste, goes to chippers

 

Y534 and Y625--Dry veneer waste, goes to hogs

 

Y533--D centers originating from edge gluing process

1. l/lOth D grade Mixed species
2 . l/6th n n n n

 

APPENDIX B

LAY—UP SCHEDULE

95

96

 

L

 

 

mad m
cehmq\eflm .mson mu :\m mam mam
sopmq\pfim .wsom no w\m mma mma
nehmq\ham .wson no mxa Ha ooom mmmm Noam 3am
nenqueam .wzon no m\m BAH maa
neequham .msoa no mH\m men men
nosmq\sfim
.o zmomsme\msm :\m mom mom
nepmq\eam .wsoo om s\m m cam mom mam
sohquham .wsoa om m\H ma 0am smm mma
eehmn\sam .msoa om mxm ea ozm mmm mad
cesqunam .wsoo om :\H mm osm mom mam
sad m sea .wsoo mm :\m em: mm:
ham .msom om mxm mm mm
has .wson am m\H mmH mma
ham .msoa om mxm : mm mm
has .wzoo mm :\H mm me
sag m has .wsom om :\m mmma assa
ham .wsoa om m\m mm mm
has .wsom om m\H Hana Hana
ham .wsom om w\m mam mzm
ham .msoo om :\H mm me
sad m has .wson mm s\m AH ooem msem comm aama
sag m ham .wson ma :\m cm cam osm Baa
has .mzom ma mxm ma om m: mm am
has .msoo ma :\H om oem cam gem AHA
sag m ham .wsoo o< :\m w om mm gem Hmm
has .msoa ea m\m mam mam
has .wsoo o< m\H em 03m mam 0mm
hem .wsom ca m\m m osm amm sow NHH
hem .msoo ea :\H :m: mam 0mm
the .mzoo << mxfi om om
has .wzom << m\m mHH mHH
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podUOAm wcfiocm msnmmq Hmcmm pomxm mafim Amuse Hmcmm

mcaccfimmm

 

£0LWJ\LH&
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3m NBA 00 m\m

 

 

eonm ewes: m\H-H

e :\m

2 m\m

= m\H

= m\m

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= zxm

.— ®\m

: N\H

aonm emecmm m\m

seesaw aonm :\H

emm mooqmsme w\H-H

emm Hueum mxana

emm memsmafie

= 9290 :\m

= .ozoo m\m

e ozao N\H

= 0290 mxm

meseemxmcam ozoo mH\m

7. = co :\m

o, moseem\ecam .m no maxm

= e no: mam no mxm
sooHEem\ham

.3 an: mam oo N\H

: Q0 :\m

xeoasmmxeam than; no mxm

= e an: mam oo m\H
xeoasmmxpam

.3 an: mam oo m\m

xeoasmm\sam means no m\m

xooasmmxnam\mpanz no mxa

xooasmmxpflm means 00 mxm
nopm4\uam

.wsoa gas mam mu :\m

= e an: mam no N\H

: = AQD mBm 00 :\m
nome\aam

.a an: mam oo mxm
nonmq\nam

.a gnaw mam no m\m

mas

mma
NNHH
mu
wmm
mmmm
mm:
mmw
mmHH

:NHH

omm

0mm
omoa

ONH
0mm

mmm

mmm
mooa

mod
mmm

mwm

mm»

mam

CNN
wax

mm
mam

map
:a
OHH
maoa
mu
mNN
MHHN
mm:
Nam
mmoa
mm:
:NHH
Nmm
ms
:3:
paw
ommm
QOH
0:»
Dad
msma

::

mza
mma

ONH
pom

mmm
NHH
ONOH
umma

mm

APPENDIX C

VENEER REQUIREMENTS

98

99

 

 

 

Quantity* Thickness Species Grade Use/Type
11.8932 1/10 D.Fir/Larch BCp Face/Back
37.9894 1/10 D.Fir/Larch C "
24.4638 1/10 " D "

3.2118 1/10 W.Fir/Hem. BCp " E
14.4718 " " c "
17.6836 " " D "
10.0750 " Doug. Fir A "
38.3776 " . BCp n

3.6401 ” " C "

3.0903 " " D "
39.8792 " Mixed Species D Center
49.4737 1/6 " C "

3.6300 " " D "
16.8483 l/lO " C Core
56.8064 " " D "

4.3742 1/6 " C solid "
35.4826 " " D "

114.8121 7/32 " C Core

9.5260 " " D "

 

*M3/8

 

APPENDIX D

PRODUCTION SCHEDULE FOR WEEK

100

101

BeginningVeneer Inventories

41.470
46.613
48.560
194.320
179.840
37.870
148.900

697.

573 M3/8
8-ft Lathe

 

56.770 MBF

37.280
101.630
74.560

4-ft Lathe
11.160 MBF
Purchased Veneer

16.450 M3/8

5-59

Band Saw
38.010 M3/8

35.490
1.41
21.15
5.69
32.33
30.65
7.06
21.143
1.61
12.143
3.15
9.66
9.53

H
N
H

H

H
H
H
H
H
H
H
H
fl
1'
‘fl
1'
H

1/10 Larch 548 D
1/6 Spruce 54s C
1/6 Spruce 543 D

1/10

1/6 Pine

7/32

Pine

Pine

Strips
Strips
Strips

7/32 Mixed D Core

Larch peel to 1/10
Hemlock peel to 1/10
White Fir peel to 1/6
Spruce peel to 7/32

White Fir, peel to 1/6

Douglas Fir,
Douglas Fir,

7/32
7/32
1/10
1/10
l/lO
1/6

7/32
1/10
1/10
1/10
1/10

Spruce
Spruce
Larch
Larch
Hemlock

green,
green,

54s
54s
27s
27s
27s

OUOUO

1/10 AB
1/10 CD

White Fir 278 D

Spruce

2750

Larch Strips C
Larch Strips D
Hemlock Strips C
Hemlock Strips D
1/6 White Fir Strips D
7/32 Spruce Strips C
7/32 Spruce Strips D

Edge Gluer

 

3.26
10.30
.74
-55
.80
.80
.15
.63
.47

20.27
179.84

OOl—‘Wl’U-D'O

Raimann Patchers

M3/

H

 

12.75
2.62

5.64'

8.37
28.86

3.50

M3/8

1/10
1/10
1/10
1/10
1/10

102

Larch 27s ABCp
Larch C
Hemlock ABCp
Hemlock 27s 0
Hemlock 273 D

1/6 White Fir 27s ABCp

1/10
1/10

Larch strips ABCp
Hemlock Strips D

1/6 White Fir strips ABCp
1/6 White Fir strips D
1/6 Pine strips D

1/10
1/10
1/10
1/10
1/10
1/10

Larch 54s ABCp

Larch 543 ABCp (from edge glued 275)
Douglas Fir 54s AB

Douglas Fir ABCp 543

Douglas Fir 54s ABCp (edge glued 27s)
Hemlock 54s ABCp

 

APPENDIX E

YAKIMA PLYWOOD MILL
SCHEDULING MODEL

January 1967

103

NOTES:

104

CODE FOR READING THE LP1107 OUTPUT

(Next 31 pages)

All veneers are given in 1000 SF 3/8 (MSF3/8)
and all logs are in 1000 bd. ft. (MBF).

Productivity coefficients are in Hrs. per
MSF3/8 (dryers, patchers, edge gluer, etc.)
and Hrs. per MBF (lathes).

Products are in number of panels.

To use MATRIX GENERATOR II

a.

Beginning panel inventories and order file (in
panels) are punched within the first 10 columns.
The product number (row number) is punched in
the next 5 columns (11 to 15) right Justified.

A 1 must be punched in column 20 to indicate that
it is an element of a vector.

Example: (Format is F10.4,2I5)

264.0 25 1 to indicate that 264
panels 0f 3/8 BB is on the order file or
beginning inventory, as the case may be.

The generator outputs cards in a format
compatible with the format required by the
LP1107 routine. The first card is a FIRST
B card and the last card is an EOF card.
Since a FIRST B and EOF cards are already
in the LP routine from previous runs, make
sure they are not duplicated.

 

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para N N ~3.3 :

 

 

106
YAKIMA PLYWOOD MILL MODEL

ROW DESCRIPTION

VRFB (D. Fir/Larch face veneer requirements)

 

Row No. Description
1 Objective function
2 1/10 Doug.Fir/Larch 5M5 BCp
3 II II " C
)4 II H H D
5 " Wh. Fir/Hemlock " BCp
6 H H N c
7 II II If D
8 " P. Pine/Spruce " C
9 H H H D
10 1/6 Wh. Fir/Hemlock " BCp
11 H H H C
12 H H I! D
13 " P. Pine/Spruce " C
114 II II II D
VRFX (Doug. Fir face/back veneer requirements)
15 1/10 Doug. Fir 5M8 A
l 6 II II II BCp
17 H H I? C
18 II II II D
19 " P. Pine " C
VRC8 (Center veneer requirements)
20 1/10 Mix
21 .. .. Mix 9
22 II II II \IC
2 3 1/6 II II C
214 II II II D
2 5 H H II NC
26 7/32 " " c
27 II II II D
28 H II II NC
VRC“ (Core veneer requirements)
29 l/lO Mix Random C solid
3 O H II n C
31 II II n D
32 II II I" NC
33 1/6 " " C solid

32"; H If H C

 

Row No.

107
Description

VRCH (Core veneer requirements continued)

35
36
37
38
39
HO
Ml
42
”3
an

V154 (Bus Veneer

1/6

7<32

H
H

5/l6

l/lO

H

Mix
II

inventory)

Doug. Fir

Wh. Fir

Doug. Fir
II

Wh. Fir

Random
II

D

C solid

NC
C solid

NC

ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

Row No.

V15“ (54s Veneer inventory continued)

80
81
82
83
8M
85
86
87
'88
89
90
91
92
93
9M
95
96
97
98
99
100
101
102
103
10“
105
106
107
108
109
110
111
112
113

114
115
116
117
118
119
120
121

122
123
124
125
126
127
128

1/6

108

Description

Spruce

Larch
H

Hemlock
H

H

H

Doug. Fir
H

Doug. Fir/Larch
Wh. Fir/Hemlock
Wh. Fir/Hemlock

Mix
H

H

Doug. Fir/Larch

Wh. Fir
Larch
Hemlock
Doug. Fir
Wh. Fir
Larch
Hemlock
Doug. Fir

Wh. Fir
Larch
Hemlock
Doug. Fir
Wh. Fir
Larch
Hemlock

NC
ABCp
C

D

NC
ABCp
C

D

NC
ABCp
C

D

NC
ABCp
C

D

NC
ABCp
C

D

NC
ABCp
C

D

NC
AB

A
BCp
BCp
BCp

D Centers
H

H

ABCp (edge
glued 278)

n H

n H

" (edged glued
strips)
H

 

Row No.

VI27 (27s veneer

129
130
131
132
133
138
135
136
137
138
139
1&0
11:1
142
143
1m:
1115
1246
1M
1148
1119
150
151
. 152
153
151.
155
156
157
158
159
160
161
162
163
1614
165
166
167
168
169
170
171
172
173
1714
175
176

1/10
I!

109
Description

 

inventory)

Doug. Fir

Wh.

Hemlock
H

H

H

Doug. Fir
H

Wh.

H

Hemlock
H

H

H

ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC

 

110

Row No. Description

VI27 (27s veneer inventory continued)

177
178
179
180
181
182
183
184
185
186
187
188

VIRD

189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220

(8 ft

7‘32

1/10
M

1/6

H
H
H
H
H

Spruce
H

Hemlock
H

H
H

Doug. Fir
H

H

H

Wh. Fir

Hemlock
H

H
H

Doug. Fir
H

n
u

Wh. Fir
H
n
17

random veneer inventory)

ABCp

NC
ABCp

NC
ABCp

NC

ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC

 

111
Row No. Description

 

VIRD (8 ft random veneer inventory continued)

221 " Spruce Strips ABCp
222 " n n C
223 " H n D
2211 " " " Egg
225 " Larch " ABCp
226 n H n C
227 H H n D
228 H n n {J
229 " Pine " ABCp
2 O H H H C
231 H H H D
232 H H H NC
233 ” Hemlock ” ABCp
2314 H n H C
235 H H H D
236 n n n NC
237 7/32 Spruce " ABCp
2 8 n u n C
239 H H H D

214 O n H H NC
241 " Pine " ABCp
214 2 " " " C

214 3 " " " D
288 .. " " NC
245 " Hemlock " ABCp
2L: 6 " ” " C

24 " " " D
248 " ” ” NC

VIC4 (core veneer inventory)

249 1/10 Mix Random E 50111
250 n n ' V

H H V D
3;; n .. n NC _ ‘
253 1/6 4 n C SOlid
25“ H V I J

H H 1 D
322 n n H NC

' " id

257 7/32 I n g sol
258 " 1 n D

H H
260 " " " NC

H

261 5/16 " H g solid
262 " " H D

H H
263 H NC

264 " "

 

Row No.

CAPACITY Constraints

265
266
267
268
269
270
271

Hours
Hours
Hours
Space
Hours
Hours
Hours

available
available
available
available
available
available
available

112

Description

 

from
from
from

8 ft lathe
core lathe
dryers

for veneer storage

from

edge gluer

from band saw

from

BEGINNING VENEER INVENTORIES

272
273
270
275
276
277
278
279
280
281
282
283
280
285
286
287
288
289
290
291
292
293
290
295
296
297
298
299
300
301
302
303
300
305
306
307
308
309

1/10
M

Doug.
H

Hemlo

H

Fir

Fir

ck

Fir

Fir

Raimann patchers

548
H

ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

D
NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

 

Row No.

310
311
312
313
314
315
316
317
318
319
320
321
322
323
32“
325
326
327
328
329
330
331
332
333
3314
335
336
337
338
339
3ND

3M1
3&2
343
3uu
345
3M6
3147
3M8

3149
350
351
352
353
35“
355

1/6

113

Descrigtion

 

Larch

N

Pine
1!

H
H

Hemlock
I!

H

Hemlock
H

H

Doug. Fir
H

Doug. Fir/Larch
White Fir/Hemlock
H

Mixed
H

H

Doug. Fir

White Fir
Larch
Hemlock
Doug. Fir
White Fir
Larch
Hemlock
Doug. Fir

White Fir
Larch
Hemlock
Doug. Fir
White Fir
Larch
Hemlock

543

D

NC
ABCp
C

D

NC
ABCp
C

D

NC
ABCp
C

D

NC
ABCp
C

D

NC
ABCp
C

D

NC
AB

A
BCp
H

D centers

H

H

ABCp (edge
glued 273)
H

ABCp (edge
glued strips)
H

H H
H H
H H
H H
H H

H H

 

Row No.
(278 Veneer)

356 1/10
357 "
358 n
359 "
360 H
361 u
362 n
363 u
36“ n
365 n
366 n
367 H
368 n
369 n
370 H
371 H
372 H
373 "
37“ H
375 "
376 H
377 "
378 H
379 "
380 1/6
381 H
382 H
383 H
38“ H
385 n
386 H
387 H
388 H
389 H
390 H
391 H
392 H
393 ”
39“ H
395 "
396 H
397 "
398 H
399 "
MOO "
U01 "
1:02 "
“03 H

11a

Descrigtion

 

Douglas Fir
H

White Fir
n

Hemlock
H

H
"

Douglas Fir
H

H
H

White Fir
H

H

Hemlock
H

H
H

ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp
C

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC

Row No.

(27s Veneer'continued)

HOH
H05
H06
H07
H08
H09
H10
H11
H13
HlH
H15

(Strips)

H16
H17
H18
H19
H20
H22
H23
HBH
H25
H26
H27
H2-
H29
H30

1:32
433
H3H
H35
1:33
1437
H38
H39
HHO
uul
H42
HH3
HHH
HHS
HH6

HH7

7532

1/10

7!

Spruce
H

Hemlock

H
H
H

Douglas Fir
H

H
H

White Fin
1?
H
"'

Spruce
H

H

Hemlock

H
H
H

Douglas Fir
H

Fir

115

Description

0 f)?»
to
O
'U

NC
ABCp

ZUOb‘I‘itJO
CO?)
C)
'6

r3

7‘
—q

[T] (‘3
(3
'U

A’dem>2tjC)bLi
mci
0
'0

 

Row No.

(Strips continued)

'.HH8
HH9
H50
1451
H52
H53
H5H
H55
H56
H57
H58
H59
H60
H61
H62
H63
H6H
H65
H66
H67
H68
H69
H70
H71
H72
H73
H7H
H75

(cores)

H76
H77
H78
H79
H80
H81
H82
H83
H8H
H85
H86
H87
H88
H89
H90
H91

146

H
H
H
H
H
H
H
H
H
H
H
H
H

Hemlock
H
H
H
Spruce
H

H
N

Pine
H

H
H

Hemlock
H

H
H

116

Description

Strips
H

ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC

C solid
C

D

NC

C solid
C

D

NC

C solid
C

D

NC

C solid
C

D

NC

COLUMN DESCRIPTION

117

 

VTF8 (veneers transferable directly from veneer to
inventory to spreader w/out further processing, 5H3)
--combination of species

Column No.

 

1001
1002
1003
100H
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
102H
1025
1026
1027

VTFX (Doug. Fir veneers directly transferable from inventory

1/10

M

1/6

Doug. Fir
H

Wh. Fir
H

Spruce

H

Larch

H

Pine
H

Hemlock

n
Doug. Fir
Wh."Fir
Spruce
Larch
Pine
n

Hemlock
H

Doug. Fir/Larch
Wh. Fir/Hemlock
H

Description

 

5H8

UOUOUOUOUOUOUOUOUOUOUOUO

-‘

C.
-.
-—

to spreaders without further processing)-—for orders
specifying Doug. Fir/pine

1028
1029
1030
1031

1/10

M
H
H

Doug. Fir
H

N

Pine

5H8

H

OUOID

H

mm
00
'U'U

nat.

VTC8 (Centers directly transferable from inventory to
spreaders w/out processing)

1032
1033
103H
1035

1/10
M

H
H

Doug. Fir
H

Wh. Fir

H

5H8 D
H NC
H D
H NC

 

1036
1037
1038
1039
10H0
10H1
10H2
10H3
10HH
10H5
10H6
10H7
10H8
10H9
1050
1051
1052
1053
105H
1055
1056
1057
1058
1059
1060
1061
1062
1063
106H
1065
1066
1067
1068
1069
1070
1071
1072
1073
107H
1075
1076
1077
1078
1079
1080
1081
1082
1083
108H
1085

118
Spruce
H

Larch

N

Pine
H
Hemlock
H
Doug. Fir
H
Wh. Fir
H
Spruce
H

Larch

N

Pine

H

Hemlock
H

Spruce

H

Hemlock

H
H

Doug. Fir
H

H

Wh. Fir

H

H
Spruce

H

H

Larch
H

N

Pine
H

H

Hemlock
H

H

D
NC

NC
NC
NC
NC
NC
‘C
NC

NC

 

NC
ABCp

NC
ABCp

NC
ABCp

1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118

H

1/10
1/6
7/32

inventory)

1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134

1/10
H

H
N

1/6

119

Doug. Fir
H

H

Wh. Fir

H
H

Spruce
ll

Larch

H
N

Pine
H

H

Hemlock
H

H
Spruce
H

H
N

Pine
H

H
H

Hemlock
H

H
H

Mix
H

H

Mix
H

545

H

Random

H

H

ABCp

NC
ABCp

NC
ABCp

NC
D Cn

H

VTC4 (Core veneer directly transferable to spreaders from

C solid
C

D

NC

C solid
C

D

NC

C solid
C

D

NC

C solid
C

D

NC

 

1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
115H
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185

1/10
H

120

EC: (inputs to edge gluer)

Doug. Fir
H

H
Wh. Fir
H
H
Spruce
H
H
Larch
H
H
Pine
H
H
Hemlock
H

H
Doug. Fir
H
H
Wh. Fir
H
H
Spruce
H
H

Larch
H
H
Pine
H

H

Hemlock
H

H
Doug. Fir
H
H
Wh. Fir
H
H
Spruce

H

H
Larch

H

H
Pine

H

H

ABCp
C
D
ABCp
C
D
ABCp
C
D
ABCp
C
D
ABCp
C
D
ABCp
C
D
ABCp
C
D
ABCp

UZJ U3 ED
0 C) O
'0 'U ’U

(D
O
'U

(D 00 EU
0 O O
’U *0 ‘U

O O
*C *0

 

1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215

1216
1217
1218
1219
1220
1221

1222
1223
1224
1225
1226
1227
1228
1229

Doug. Fir)

1/10
M

7<32

121

Hemlock
H

H
Doug. Fir
H

H

Wh. Fir

H
H

Spruce
H

H

Larch

H

H

Pine

H

H

Hemlock
H

H

Spruce
H

H

Pine

H

H

Hemlock
H

H

Doug. Fir
H

BSI (inputs to bandsaw for core)

Spruce
H

EGIX (inputs to edge gluer for products requiring

275
H
H

Strips
H

H

ABCp

NC
ABCp

NC

 

1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281

Hemlock
H
H
H
Doug. Fir
H
H

Wh. Fir
l

V
H

Spruce
H

Hemlock
H

H
Doug. Fir
H

H

Wh. Fir

H
H

Spruce
H

Hemlock
H

H

Spruce
H

ABCp

ABCp

ABCp

ABCp

NC
ABCp'

ABCpi

NC
ABCp

NC

 

1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333

Hemlock
H

H
H

Doug. Fir
H

H

Wh. Fir

H
H

Spruce
H

H

Hemlock
H

H
Doug. Fir
H

H

Wh. Fir

H
H

Spruce
H

Hemlock
H

H

Spruce
H

ABCp

ABCp

ABCp
T)
0

NC
ABCp

NC
ABCp

NC

 

1334
1335
1336
1337

1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362

1363
1364
1365
1366

7/32

H
H
H

1/10

H
H

1/10

H
H
H

Hemlock
H

H
H

Doug. Fir
H

Wh. Fir
Larch
Hemlock
Doug. Fir
Wh. Fir
Larch
Hemlock
Doug. Fir
Wh. Fir
Larch
Hemlock
Doug. Fir
Wh. Fir
Larch
Hemlock
Doug. Fir
Wh. Fir
Larch
Hemlock
Doug. Fir
Wh. Fir
Larch
Hemlock

Doug. Fir
H

H

H

124

RPI (inputs to Raimann patchers)

RPIX (Raimann patcher inputs for
Doug. Fir)

Strips ABCp
H

543 AB-54 (originally 548)
" ABCp-54 "

products Specifying all

543 AB—54 (originally 545)
" ABCp—54 "
" ABCp—27(edge glued 27s)
" ABCp—R (edge glued strips)

PDV (Purchase Dry Veneer—-Doug. Fir)

1367
1368

1/10

H

Doug. Fir
H

H AB
H CD

PGV (Purchase Green Veneer--Doug. Fir)

1369
1370

1/10

H

Doug. Fir
H

543 AB
H CD

 

1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385

1386
1387
1388
1389
1390
1391
1392
1393
139H
1395

VRF8-B

1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408

L18 (8 ft lathe log inputs)

Doug. Fir
Wh. Fir
Spruce
Larch
Pine
Hemlock
Doug. Fir
Wh. Fir
Spruce
Larch
Pine
Hemlock
Spruce
Pine
Hemlock

L14 (core lathe inputs)

White Fir
Spruce
Pine
Hemlock
Spruce
Pine
Hemlock
Spruce
Pine
Hemlock

125

Peel to
H H

H H
H H
H H
H H

Peel to

H H

1/10 veneer
H H

7/32 veneer
H H

H H

1/6 core
H H

H H

H H

7/32 core
H H

H H

5/16 core

H H

(Face/back veneer requirements that cannot be met—-

backlogged)

1/10 Doug. Fir/Larch
H H

1/6 Wh. Fir/Hemlock
H H

H

" Wh. Fir/Hemlock
H

H

Pine/Spruce
H

H

Pine/Spruce
H

I
O
'U

l
O
'U

UOUOCIDUOUOCUUOW
O
'0

 

VRFX-B

1409
1410
1411
1412
1413

VRCB-B

1414
1415
1416
1417
1418
1419
1420
1421
1422

VRC4—B

1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438

VIS4-E

1439
1440
1441
1442
1443
1444
1445
1446

126

(All Pon. Pine Doug. Fir face/back veneer that
cannot be met--backlogged)

1/10
H

H
H
H

Doug. Fir
H

H
H

Pine

A
B-Cp
C
D
C Nat. (for
knotty pine)

(Veneer requirements for centers that cannot be

met--back1ogged)

1/10
H
H
1/6
H
H

7432

H

(Veneer requirements

-—backlogged)

1/10
H

H
H

1/6

H
H

7‘32

H
H

5/16

H
H

(Ending inventory of

1/10
H

Mix
H

Mix
n

H

Doug. Fir

Wh. Fir

for core that cannot

Random
“H

54 veneers)

543

NC
C
D
NC
C
D
NC

be met
solid
D
NC
solid
D

C
solid

Q—u

D
C

C
C
C
C
V
C
C
N
C solid
C
D
N

C

ABCp

NC
ABCp

NC

 

1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498

Doug. Fir
H

ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NCth
ABCp

NC
ABCp

NC
ABCp

NC
ABCp
0

NC
ABCp

NC

 

1499
1500
1501
1502
1503
1504
1505
1506
1507

1508
1509
1510
1511
1512
1513
1514
1515

1516
1517
1518
1519
1520
1521
1522

VI27—E

1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546

128
Doug. Fir
H

Doug. Fir/Larch
White Fir/Hemlock
H

Mix
H

H

Doug. Fir

White Fir
Larch
Hemlock
Doug. Fir
Wh. Fir
Larch
Hemlock
Doug. Fir

Wh. Fir
Larch
Hemlock
Doug. Fir
Wh. Fir
Larch
Hemlock

(Ending inventory of 27 veneers)

1/10

H

Doug. Fir

Wh. Fir

275
H

AB

A
BCp
BCp
H

D ce
H

H

ABCp

H

ABCp

H

H

ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC

nters

-27 (edge
glued 278)
H

-R (edge
glued strips)
H

 

1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1559
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582

VIRD-E

1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594

1/6

7/32

(Ending inventory of

1/10

H

Wh. Fir

Hemlock
H

H

H

Doug. Fir
H

Wh. Fir

strips)

ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC

ABCp

NC
ABCr

NC
AFC?

we

 

1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1535
1636
1537
1638
1639
1640
1641
1642

Doug. Fir

Wh. Fir

Hemlock
H

H

H

ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp
0

NC
ABCp

NC

 

131

 

VIC4-E (Ending inventory for core veneers)
1243 1(10 Mix Random g solid
1645 " " n D
1336 n n n NC
1 7 1/6 " n i
1648 " n n 8 SOl”d
l6u9 " " v1 D
1650 " " I! NC
1651 7/32 " " C solid
1652 H n n C
1653 " " n D
165“ n n n NC
1655 5/16 " " C solid
1656 n n v: C '
1657 " " n D
1658 n n n NC
VRF8-X (Excess face/back veneer—-returned to inventory)
1659 1/10 Doug. Fir/Larch 54s BCp
1660 n H n C
1661 n n n D
1662 " Wh. Fir/Hemlock " BCp
1663 H H n C
1664 " " " D
1665 " Pine/Spruce " C
1666 " " " D
1667 1/6 Wh. Fir/Hemlock " BCp
1668 " " " C
1669 " " " D
1670 " Pine/Spruce " C
1671 " " " D

VRFX-X (Excess Doug. Fir/Pine face/back veneers—~returned to
inventory)

1672 1/10 Doug. Fir 54s A
1673 n n n BCp
167“ n n v: C

1675 I! n n D
1676 " Pine " C Nat.

VRC8—X (Excess center veneer—returned to inventory)

1677 1/10 Mix Mix C
1678 n u 11 D
1679 " " " NC
1680 1/6 " " C
1681 n " " 0
1682 " " " NC
1683 7/32 M n C
168“ n n n D
" NC

1685 n n

1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701

1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733

VRC4-X (Excess core

1/10
H

H
H

1/6

BEGINNING VENEER

1/10

132

Mix
1!

INVENTORIES (545)

Douglas Fir
H

White Fir
H

H

Hemlock

H
H
H

Douglas Fir
H

White Fir
H

Random
H

veneer--returned to inventory)

C solid
C

D

NC

C solid

(3

solid

0

solid

ZUOOZUOOZUO

O

ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC
ABCp

NC

 

1734
1735
1736
1737
1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
1759
1760
1761
1762
1763
1764
1765
1766
1767
1768
1769
1770

1771
1772
1773
1774
1775
1776
1777
1778

1779
1780
1781
1782
1783
1784
1785

Hemlock
H

H
H

Douglas Fir
H

Douglas Fir/Larch
White Fir/Hemlock
H

Mixed
H

H

Douglas Fir

White Fir
Larch
Hemlock
Douglas Fir
White Fir
Larch
Hemlock
Douglas Fir

White Fir
Larch
Hemlock
Douglas Fir
White Fir
Larch
Hemlock

Mixed

H

ABCp (edge

glued

H

u 2

m
C?

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1786
1787
1788
1789
1790
1791
1792
1793
1794
1795
1796
1797
1798
1799
1800
1801
1802
1803
1804
1805
1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837

(27s veneers)

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1839
1840
1841
1842
1843
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1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889

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1891
1892
1893
1894
1895
1896
1897
1898
1899
1900
1901
1902
1903
1904
1905

(cores)

1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921

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1922
1923

1924
1925
1926
1927

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REFERENCES

137

 

10.

REFERENCES

Brooks, G. H. and White, C. R. "An Algorithm for
Finding Optimal or Near Optimal Solution to
Production Scheduling Problems," Journal of
Industrial Engineering, Vol. 16, 1965, pp. 34-40.

 

 

Burstall, R. M. "A Heuristic Method for a Job-
Scheduling Problem," Operational Research
Quarterly, Vol. 17, No. 3, September 1966.

 

Charnes, A. and Cooper. W. W. Management Models and
Industrial Applications of Linear Programming.
New York: John Wiley & Sons, Inc., 1961.

Dantzig, G. B. Linear Programming and Extensions.
Princeton, New Jersey: Princeton University
Press, 1963.

Demas, Ted. Basic Plywood Processing. American Ply-
wood Association, 1965.

Forrester, J. W. Industrial Dynamics. Cambridge,
Massachusetts: The M. I. T. Press, 1961.

Gere, William 8., Jr. "Heuristics in Job ShOp Sche-
duling," Management Science, Vol. 13, No. 3,
November 1966, pp. 167-190.

Giffler, B. and Thompson, G. L. "Algorithms for
Solving Production Scheduling Problems,"
Operations Research, Vol. 8, No. 4, July-Aug.,

1960, pp. 487—503.

Giffler, B., Thompson, G. L. and Van Ness, V. "Numeri-
cal Experience with the Linear and Monte Carlo
Algoithms for Solving Production Scheduling
Problems," in J. F. Muth and G. L. Thompson (eds.),
Industrial Scheduling, Englewood Cliffs, New
Jersey: Prentice-Hall, Inc., 1963.

Ignall, E. and Schrage, L. "Application of the Branch
and Bound Technique to Some Flow Shop Sequencing
Problems," Operations Researqh, Vol. 13, 1965,

138

 

11.

12.

13.

14.

15.

16.

17.

18.

19.

139

Koenig, H. E., Tokad, Y. and Kesavan H. K. Analysis
of Discrete Physical Systems. Néw York: McGraw
Hill Book Co., 1967.

Manne, A. S. "On the Job Shop Sequencing Problem,"
Operations Research, Vol. 8, 1960, pp. 219-223.

Mellor, P. "A Review of Job Shop Scheduling,"
Operational Research Quarterly, Vol. 17, No. 2,
pp. 161—171,

Pounds, W. F. "The Scheduling Environment," in
J. F. Muth and G. L. Thompson (eds.), Industrial
Scheduling. Englewood Cliffs, New Jersey:
Prentice-Hall, Inc., 1963.

 

Rowe, A. J. "Toward a Theory of Scheduling," Journal
’ of Industrial Engineering, Vol. 11, No. 2, March-
April, 1960, pp. 125—136.

Smith, R. D. and Dudek, R. A. "A General Algorithm
for Solution of the n-Job, M—Machine Sequencing
Problem of the Flow Shop," Operations Research,
Vol. 15, No. l, January—February, 1967, pp. 71-82.

Shields, P. C. Linear Algebra. Reading, Massachusetts:

Addison-Wesley Publishing Co., Inc., 1964

Van Woerner, T. "The Trimmer: A HeuriStic SOIution
to the Trim Problem in the Corrugated Container
Industry." Unpublished Ph.D. Dissertation,
Carnegie Institute of Technology, 1963.

Zangwill, W. I. "A Deterministic Multi-Period
Scheduling Model," Management Sciengg, Vol. 13,
No. 1, September 1966} pp. 105—119.

l—L’" '! rim

 

 

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