Two-Dimensional Cutting Stock Problems

A Search-Based Approach for Solving Regular

Ahmed W. El-Bouri

A thesis presentecl to the Uliversity of Manitoba in fulfilment of the thesis lequirement for the cleglee of Mastel of Science in 14eclianical Engineeling

Winnipeg, Manitoba, Canada 1993

@Ahmecl W. trl-Bouli 1993 Bibliothèque nationale !*l )'t:îå'o'Jo'"" du Canada Acquisitions and D¡rection des acquisitions et Bibliographic Serv¡ces Branch dês serv¡ces bibliograph¡ques

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TI.IO- DIMENS IONAL CUTTING STOCK PROBLEMS

Alt¡fED W. EL-BOURI

A Thesis subtrritted to the Faculty of G¡aduate Studies of the University of Manitoba in ¡rartial fulñllment of the tequí¡enents for the degree of

M¡.STER OF SCÍENCE

@ 1993

Pe¡stission has been granted to the LIBRåRY OF TI{E IJNTI¡ERSITY OF ¡víÄNITOBA to lend or sell copies of this thesis, to the NÂTIONAL UBRÂRY OF CANADA to rrisoËlm this thesis ¿nd to lend or sell copies of the 6!n, and TINTYERSITY MICROETLMS to publish an abstract of this thesis. The author rese¡ves Oth€r publicãtion lighis, a¡d neiihe¡ the thesis nor exte$ive exEacts Éom it may be printed or othe¡wise reproduced without the authofs permission I hereby cleclale that I arl the sole author. of this thesis.

I authotize the Univelsity of N4anitoba to lend this thesis to othel institutions ol individuals for the ¡rurpose of scliolarly r.eseaLch.

I fulthel authorize tlie University of Manitoba to i.eprocluce this thesis by pho- tocopying or by other rneans, in total ol in paÌt, at tlìe l.equest of othel institutions ol individuals for the purpose of scholally research. This tl¿esis is tled,i,cated to mg uife, Hanø, and. rng d,aughter, May Abstract

The plesent stucly proposes au algorithm fol two-dimensional cutting stock pr.oblems. The problem of cutting small rectangular. pieces fi.om lar.ger. uniform stock plates is consiclered lvith a view to minimize the unusable poltions of tlle stock material resulting fiom the cutting. The srnall pieces lange in size fr.om 0.4% to

100% of the size of the stock plate. Existilg heuristic methocls for solving cutting stock ploblems are ploblem-dependent, and the solutiols fol a given problern using cliffelent heuristics ca' be highly valiable. A critical analysis o{ existilg heu.istics resulted in the developrnent of a new algorithm rvhere the objective is a r¡ore col- sistent efficiency arcl a rvicler applicability to diferent cìasses o{ the problern. The proposecl algorithrn is designecl fol al IBM conrpatible rniclocomputer., It coml¡ines a restrictecl tlee search procedure with priority-rule assignments for the demandecl pieces in a lranner such that the solution pr.oceclure is guided by the instantaneous latio of the avelage alea of the clernandecl pieces relative to the alea of the stock plate. The algorithrn is testecl ancl compalerl lvith several existing heulistics using lhe sarne test data. The results show the proposed algorithrn to be highly competitive and well-suitecl fol serving as a tool for an expert system application to the cutting stock problern, Acknowledgements

I arn cleeply glateful to prof, S. Balaklishnan for.his kind assistance, guiclance and supervision of this ivork, and to plof. N. Poppleivell for' his generous advice, encoulagernent, and many valuable suggestions. I rvoulcl like to equally acknorvledge plof. A. Alpha fol his help and nuch-appreciatecl involvemeut, particular.ly in the

Operations Research aspects of this rvolk.

My glatitucle also goes to prof. Pechycz fol his revierv and comments of this work. I also wisli to acknowleclge the coopelation of pr.of. S. Yakorvitz of the Dept. of Systerns and Industrial Ðngileeling at the University of Arizona, and R. Dietrich of Bell Cornmurications Resealch, il ploviding higlily useful data from their.wor.k on the topic considered in this thesis. Finally, I rvish to extencl my thanks to the

National Oil Company (Gas Projects) and Sirte Oil Cornpany, both of Libya, fol their financial support. TABLE OF CONTENTS

CHAPTER PÀGE

Äbstlact

A ckn owl ed ge ments vt

List of Figures x¡

List of Tables xlll

Nomenclature . . XIV

1. fntroduction 1

1.1 Background 1

7.2 Classification of Cutting Stock Problerns 2

1.2,L Dirnensional Chalacteristic 2

L2.2 Numbel of Available Stock Sizes 3

1.2.3 Orclel Piece Geornetly 5

1.3 Ploblem Colstraints 5

1.4 Probler¡ FormuÌation 7 1.5 Methods of Solution o

1.6 Factors Affecting Trim Loss 10

L7 Measure of Perfolmance 11 1.8 Statement of the Ploblem I2

1.9 Ovel vierv 13

2. Literature Review . 15

2.7 Introduction 15

2.2 One-Dimensional Ploblerns 15

2.3 Trvo-Dir¡ensional Problems 18

2.3.1 Optirnizing Algolithms 18

2.3.2 Heulistic Algorithrns 24

vll CHAPTER PAGE

2.4 L'r'egular Til'o- Dimensional Prol¡lerns 31 2.5 Thlee-Dimensional Prol¡ìenrs

3. Compalison of Three Heuristic Methods .1.) 3.1 Intlocluction .tK

3.2 The First-Fit Heuristic of Islani ancl Sandels 36 3.2.I Description of the Heuristic 3.3 A Combinecl Heuristic-Ðxact Proceclule 40

3.3.1 Details of the Algolithm 40 3.3.2 Contlol of Strip Lengths 46 3.4 A Priolity Rule Heuristic 47 3.4.1 Description of the Algolithm +t

3.4.2 Expansion of Ulfilled Subsheets 51 3.4.3 Moclifiecl Velsior of Dietrich ancl Yakowitz's Algolithn 3.4.4 Description of the Modifiecl Algolithm 54

3.5 Tests ancl Analysis 57 3.5.1 Experirnental Test Data 57

3.5.2 Test Results Fol Modified Dietrich-Yakorvitz Algorithrn . 57

3.6 Courpalison of Heulist ic Results . 59

3.7 Conclusions 59

4, Search-Based Heuristic 63 4.1 The Plogramrning Language o4

4.2 The Plolog Algorithm bf)

4.2.1 Input Data 66

4.2.2 The SEARCH routine 67

4.2.3 Sheet Subdivision - the STRIPS routine 71 4.2.4 Managernent of Waste Areas - the EXPAND t.outine

4.3 Descliption of the Algorithmic Iteratior 77 4.4 Ðvaluation of the Search-Basecl Algorithrn 70

+.4.7 Test Results fol the Sealch-Basecl Algolitlim 81

4.4.2 Analysis of Results 83

vlll CHAPTER PAGE

4.4.3 Conclusion ... 87

5. Analysis of the Search-Based Algorithm 89

5.1 Solution by Compouncl Heuristics 90

5.2 Implernentation of Thresholds 91

5.3 Priolity-Basecl Contlol of the Algorithrn 98 5.3.1 Assignmelt of Priolities oo

5.3.2 Test Results 102

5.4 F inal Design of the Nelv Algorithrn - A Pliority-Rule Methocl 104

5.5 Testing of priority-basecl algorithrn 104

5.5.1 Comparison rvith Dietrich-Yakorvitz (1991) 106

5.5.2 Cornpa'-ison with Bengtsson (i982) . 108

5.5.3 Cornpalison rvith islani-Sanclers (1982) 109

5.5.4 Compa.-ison rvith Albano-Orsili (1980) 110 5.6 Analysis of Results 172 5.6.1 Computational time fol the priority-r'ule algor.ithm rt2 5.6.2 Restrictions r12

5.6.3 Relationship l¡etween APSA r.atio ancl Sheet Utilization. 115

6. Conclusions and Recommendations . . l2L 6,1 Irnplementatiou of an Expelt System ..,....122 6.2 Conclusions ....,,.123 6.3 Recornmenclations fol' Futule Research ...... I24 REFERENCES...... I25 ,A.PPENDICES...... 129

A. Results of Experimental Test Runs using Dietrich-Yakowitz Data setfl . 190 4.1 Results for Categoly #I...... 181 4.2 Results fol Category #2...... 18s .{.3 Results for Category #3...... 139

IX 4.4 Results for Category # 4. , ...... I43 4.5 Results fol Category #5...... 147 4.6 Results for Category #6...... 151 4.7 Results fol Category #7 ...... 155 LIST OF FIGURÐS

FIGURE PAGE

1.1 Cutting Papel Roll Orclers fror¡ Stock Rolls 3 7.2 Ðxample of Two-Dimensional Multi-plate Problem ...... 4

1.3 Exarnples of Guillotine and Non-Guillotine Cutting Patterns 6

1.4 Examples of Multistage Cutting Pattelns ...... 6

3.1 Layout Obtainecl by Filst-fit Heulistic 39 3.2 Holizontal ancl veltical hornogeneous strips 3.3 Genelation of Strips ...... 42 3.4 Exarnple of a Strip Layout Generatecl by Dynamic Programmilg . . . 45 3.5 Layout Obtainecl by Combined Exact-Heuristic ...... 48 3.6 Sheet Subdivisions rvith a Zelo-clegree Fit . . . . 49 3.7 Example of "Stand-Still" State ...... 52 3.8 Expansion of Unfillecl Subsheets using Two Rules . . . . . 53 3.9 Layout Obtained by Modified Dietrich-Yakorvitz Heur.istic ...... 56 4.7 Layout Composecl o{ One-clegree Fits Only . . . . 68 4.2 Layout producecl by StrARCH-A ...... 70 4.3 Layout proclucecl by StrARCH-Z. . . , ...... 7i 4.4 Tlee Search Enumelatiol with SEARCH-A and SEARCH-Z . . . . . 72 4.5 Partitionirg Possibilities of Space Remaining on Sheet aftel Alloca-

tion ofa Piece...... 73 4.6 Adjacent Waste Areas .... 76 4.7 Exarnple of Horizontal Integlation ,..., 76 4.8 Flowchart for the Search-Based Algorithm . . . . . 78 4.9 Cornparison of Results : Search-Basecl Algorithm vs. Refelence . . . . 82 4,10 Cornparisol of Standarcl Deviations l¡etrveen Results fot.T*'o Variants

of the Sealch-Basecl Algorithrn ...... 84 4.1 1 Cornparison of Computational Tir:res for Trvo Valiants of the Search- Basecl AlgorithÍr ..... 85 FIGURE PÀGE

5.1 Cornparison betrveen Compouncl Heulistics Appr.oach ancl Trvo Vat.i-

ants of tlie Search-Basecl Algorithm 92

5.2 Layouts attainal¡le lvith cliffelent thleshold pararneters 93

5.3 Layout of Sample B.O.M. # 2 . . . 94 5.4 Ðfect ofThlesholds on Sealch-Based Algorithm (Categories 1, 2, ancl

3).... 96

5.5 Effect of Thlesholcls ol Sealch-Basecl Algorithrn (Categor.ies 4, 5, 6,

ancl 7) 97 5.6 Result Compalison : Priority-Rule Algor.ithm vs. Original Sear.ch- Based Algolithn ,.....103 5.7 Results of Pliolity-Rule Algolithm fol Dietrich-Yakor'ç'itz Test Data . i05 5.8 Serni-logPIot fol Dietrich-YakowitzDataSet fI Results ...... 118 5.9 Serai-log Plot fol Dietricli-Yakorvitz Dala Set $2 Results ...... 119 5.10 Semi-log Plot for Dietrich-Yakorvitz Dala Set f3 Results ...... I20

xll LIST OF TABLES

TABLE PAGE 3.1 Sample Bill-of-Materials f 1...... 36 3.2 Test Results for Moclified Dietricìr-Yakowitz Algor.ithrn . . . 58 3.3 Cornpa.-ison of Test Results front tlie Investigatecl Heur.istics . . . . . 60 4.1 Sample Bill-of-Matelials f2...... 69 4.2 Test Results fol Four Valiants of the Sear.ch-Basecl Algolithm . . . . 80 5.1 Results fol the Conpouncl Heuristics Run...... g0 5.2 Sarrple Bill-of-Matelials f 3.,...... 91 5.3 Results for Runs Unclel Diffelent Threshold Settings . . . . 95 5.4 Resuìts of Pliority-Rule Algorithn Dietr.ich-Yakorvitz Data Set S 1 . . 102 5.5 Result Compalisol with Dietrich-Yakorvitz Data Set fl ...... 106 5.6 Result Cornpalison rvith Dietrich-Yakowilz Data Sef ff2 ...... 106 5.7 Result Cornparison with Dietricli-Yakoivitz Data Set f3 ...... 107 5.8 Average CPU Times using Dietlich-Yakorvitz Test Data...... 107 5.9 Result Comparison with Bengtsson (sheet size 25 x 10) . . . 109 5.10 Result Com¡ralison with Bengtsson (sheet size 40 x 25) . 110 5.11 Result Compalison rvith lslani-Sancler.s . . . . . 111 5.12 Result Compalison rvith All¡ano-Or.sini . . . . ll1 5.13 Percentage utilization results fol Bengtsson's test ploblern 1 rvitli diferent lule combinatiolrs ....114 4.1 Results for Category $1 of Dietrich-Yakorvitz Data Set 1 ...... 131 4.2 Results fol Categoly f2 of Dietr.ich-Yakorvitz Data Set I ...... 135 4.3 Results for Categoly f3 of Dietr.ich-Yakoivitz Data Set, 1 ...... 139 4.4 Results for Category f4 of Dietr.ich-Yakorvitz Data Set 1 ...... 145 4.5 Results fol Categoly f5 of Dietr.ich-Yakorvitz Data Set i ...... 147 4,6 Results for Categoly S6 of Dietrich-Yakorvitz Data Set 1 ...... 1bl 4.7 Results for Category f7 of Dietrich-Yakorvitz Data Set 1 ...... 155

xlll Nomenclature

Latin Letters

L length of stock sheet w ividth of stock sheet

I lelgth of demancled piece u ividth of clem andecl piece n nurnbel of clernarded pieces of a specific type a nurnber of unique piece sizes demanclecl in a B.O.M.

?72 nur¡ì¡el of stock sheets utilised for filling a B.O.M.

T average pelceltage utilization of stoch for'filling a B.O.M.

Greek Letters o stanclald deviation

Subscripts

stock sheet

der¡andecl piece type Miscellaneous

B.O.M. Bill-of- Mateliaìs CHAFTER. I.

Introduction

1,L Background

Whenever "stock" pieces, whether they be sheets of metal, wood, or glass; or

¡olls of textile or paper, are to be cut into smalle¡ "demandedt' pieces, it is important to minimize trim wastage. Fo¡ reasons of economy, stock pieces are normally supplied in standardized units of size. In secondary manufacturing processes the stock is cut to satisfy customer orders, The orders are usually placed in the form of bills-of-materials (B.O.Ms) specifying the demanded sizes and quantities of the smaller pieces, also known as the order pieces.

The cutting stock problem deals specifically rvith the cutting of the pieces iisted on the B.O.M. from the available stock pieces in a manner such that the total scrap

(or trim-loss) generated in satisfying the demand is minimized, Alternatively, the problem can be described as that of allocating the B.O.M. to a selection of stock sheets such that the sum of unusable portions of the sheets is a minimum.

It is evident that an eficient cutting strategy rvhich satisfies B.O.Ms as well as utilizes a minimum number of stock sheets will reduce manufacturing costs, These reduced costs take the form of less material rvasted and, to a lesser extent, lower material handling and inventory costs associated with lower levels of scrap and stock piece handling and storage. 1.2 Classification of Cutting Stock Froblems

Cutting stock problems may be classified based upon the following characteristics:

1. number of significant dimensions;

2, number of different stock piece sizes available; and

3. the geometry of the order pieces.

L,2.1 Dimensional Characteristic

A cutting stock problem can be one-dimensional, two-dimensional or three- dimensional. In a one-dimensional problem, only one dimension on the stock and order pieces is valiable, with the other dimensions being constant. An example is the slitting of paper rolls of given stock lengths into smaller demanded lengths as shown in Figure 1.1, whe¡e the demanded amounts of each length are indicated in parentheses. The rolls have a constant diameter, and the slitting is done onl3l along the length of the rolls. Hence, only the longitudinal dimension is of significance in the problem.

In a two-dimensional problem, trvo dimensions on the stock an

The last categorization is the three-dimensional problem, where the length, width and height of the orcler pieces and stock sheets are ¡elevant to the solution of the problem. An example of ¿ three-dimensional problem is the loading of containers l_-rq' I

(10). (8) (%\ f-r-r'---r æ æ Bill-0f-Materials .--E 115) -- f8)

Xj- cuting pattem no.j

0.5' residuat ' Dêmanded qlenlilíêsol €ach size

Figure 1.1: Cutting Paper Roll Orders from Stock Rolls

'with boxes of varying volumes. The containers represent tlie stock units, and the loaded boxes are the pieces demanded in the B.O.M. t,2,2 Number of Available Stock Sizes

Cutting stock problems can be of the single-plate or multi-plate varieties. A single plate problem involves stock pieces that are identical in shape, size and dimensions. In multi-plate problems, more than one stock size is available for meeting the requirements specifred in the B.O.M. The combinatorial possibilities for a multi-plate problem increase almost exponentially as the number of diferent available stock sizes increases. These problems are much harder', therefole, to solve than the single plate variant of the problem. EE s3

(7) (3) (3) (4) 12) BílÌ-o f-l'laLerial s E L1] l"l f-;-l E(s) E

B E E feasible layout B F IB c N B DID

Figure 1.2: Example of Two-Dimensional Multi-Plate Problem (numbers in paren-

theses indicate amounts demanded) I.2.3 Order Piece Geometry

A cutting stock problem is described as regular when the B.O.M. is made up of purely rectangular pieces. Whenever other shapes are to be cut from the stock, the problem is classified as irregular. The irregular probÌem is common in the textile, shoe and aerospace industries. The regular problem is more common in the glass, furniture and metal cutting industries.

1.3 ProblemConstraints

Cutting stock ploblem are also frequently characterised by the constraints imposed upon the problern. A basic constraint for all types of problems is that all patterns allocated to the stock piece must not overlap and must 1ie entirely within the stock piece. Most regular two-dimensional problems are also subject to an orthogonal cut constraint, where all cuts must be parallel to one of the sheet edges.

Other constraints found in cutting stock problems include :

1 . Guillotine Cuts

This constraint in two-dimensional problems requires each and every cut made

on the stock plate to extend fi'or¡ one edge of the sheet (or subsheets produced

by previous cuts) across to the opposite edge, as shown in Figure 1.3. Many

real-rvorld applications, particularly in the glass and furniture industries, have

cutting processes and machinery that permit only guillotine type cuts. This

constraint is necessitated, therefore, by the nature of the cutting operations.

Cutting stock problems constrained to guillotine cuts are easier to solve than

non-guillotine cut problems. (a) (b)

Figure 1.3: Examples of Guillotine and Non-Guillotine Cutting Patterns

(a) (b)

Figure 1.4: Examples of Multistage Cutting Patterns

Multi-stage cuts occur wherever pieces produced as a result of a guilÌotine cut are further subject to guillotine cuts to obtain the pieces with the dernanded dimensions. Figure 1.4 shows multi-stage guillotine cut patterns.

Orientation Constraints

This restriction requires soÌre or all of ihe order pieces to adopt specific ori- entatio¡rs in the cutting pattern. A Problem solution is usually facilitated by these const¡aints l¡ecause of the reduction in combinatori¿l alternatives. 3. Location Constr¿ints

There may be requirements for some of the pieces to be placed or cut from

certain areas of the stock sheet. This location restriction also reduces combinatorial possibilities and, hence, the problem difficulty.

4. Number of Pieces of each Type

This const¡aint limits the number of pieces of each type cut to the exact num-

ber demanded in the B.O.M. Oversupply is not accepted and is considered

waste. Problems subject to this restriction are generally termed t'constrained"

problems. Constrained problems are mole difficult to solve than their uncon-

strained counterparts, t.4 Froblem Formulation

The generalized form of the problem is best formulated as in Hinxman [1] : minimize DT=t"¡ subject to 2 Lr;¡ d,¿ Ío, i, :7..'n (1.1) ;=t - lvhere

cj trim loss in stock piece j.

!¡L = number of different piece sizes (types) specifled in B.O.M.

i:ij number of pieces of type i cut from stock piece j.

d,¿ = number of pieces of type i demanded in B.O.M.

rn number of stock sheets utilised for satisfying B.O.M. The above formulation describes the constrained problem. In the unconstrained version, the equality in the constraint equation is changed to a gleater-than or equal- to comparison. A feasible cutting pattern on stock item j can thus be described by lhe se| l:r1¡, r.2¡,' . . ,r.il.

Cutting problems are closely related to the bin-packing problem, rvhere a B.O.M. is to be packed onto a minimal number of bins. The diference between the two problems lies in the number of demanded sizes and their quantities. In bin-packing,

B.O.Ms are usually composed of a few pieces of many different sizes (i.e large z and low d¡). In cutting stock ptoblems the case is the reverse, whete many pieces of relatively few different sizes (low ri and high d;) are demanded by the B.O.M.

The cutting stock problem is also related, in awa¡ to the knapsack problem. In the knapsack problem, the objective is to load a stock item with pieces of different values, such that the total value of the load is maximized. The solution to a knapsack problem is a selection of pieces from the B.O.M. If the value of each B.O.M. piece is taken as proportional to its surface area, then the knapsack solution minimizes trim loss on the loaded stock piece. Attempting to allocate all the items on the B.O.M. using the knapsack problem approach for each iridividual sheet does not, in general, produce good results. The paradox of the problem is that the sum of local minima for the individual sheets does not necessarily add up to a global minimum for the problem as a whole. This is attributable to the dynamism involved in the B.O.M.'s composition as pieces are taken off it, 1,5 Methods of Solution

Approaches for solving cutting stock problems can be classified as either exact or heuristic. Exact solution techniques are more commonly applied in one-dimensional ploblems, less commonly in two-dimensional problems, and very rarely in three- dimensional problem, Three-dimensional problems are almost always solved rvith heuristics because of the computational burdens involved in such problems. This also holds true, to a lesser extent, in two-dimensional problems and certain ore- dimensional problems,

1. Exact Solution Techniques

Exact methods seek optimal solutions to the problem. They use mainly linear

proglamming, dynamic programming and state-space search methods. In the

state-space search lnethods, the problem is solved by searching nodes in a

graph model to find a path to the optimal. Integer programming (branch and

bound algorithms) are frequently implemented in the search techniques.

2. Heuristic Solution Techni

Heuristics are used for solving cutting stock problems rvhen no exact algorithm

is known to exist, or where the use of available exact methods is computational-

ly proliibitive. A heuristic method does not attempt to find optimal solutions.

Rather, it aims for a solution judged to be rvithin a tolerable range of the

optimum in an acceptable computation time. Many heuristic techniques are

structured upon exact m¿thematical algorithms. In such cases the heuristic's

role is generally to confine the search by bounding partial solutions {ound by

the search algorithms. Another flequently used method in heu¡istics is that

of problem reduction, where the initial problem is decomposed into smalle¡

sub-problems, which may in turn be decomposed further. The basis for the decomposition is that the smaller sub-problems rnay be more manageabìe than

the initial problem. The sub-problems are solved separately, and the union of

the solutions to all the sub-problems constitutes the solution to the or.iginal problem.

1.6 Factors Äffecting Tlim Loss

There are four important factors affecting the trim-loss in cutting stock problerns. These are :

1. Total number of demanded in B.O.M.

Trim loss decreases as the total number of pieces specifred in the B.O.M.

increases. This is observed [2] because the greater availability of pieces permits more combinations on the individual stock pieces.

2. Average piece-to-stock area (APSA) ratio

This ratio defines the average a¡ea of the B.O.M. pieces to the area of the

stock piece. Larger B.O.M. pieces are more difficult to accommodate, and

contribute greater trim loss. As the average area of the pieces to be allocated

decreases in relation to the stock sheet, better utilization can be expected

because of the obviously better likelihood of pieces fitting in sr¡aller available

spaces on the sheet.

3. Number of unique piece types (Q) demanded in B.O.M.

The availability o{ a number of pieces with a wider variety of dimensions

prornotes the chalces of finding pieces to fit in unused spaces on the stock

sheet. Thus, the greater is the variety of the sizes specified in the B.O.M.,

the lorver is the trim losses.

10 4. Aspect ratio of pieces in B.O.M.

The aspect ratio of a rectangle is the height (sraaller dimension) of the rectan-

gle divided by its length (larger dimension). Pieces with aspect ratios ap-

proaching 1.0 (i.e. near squares) reduce the combinatorial alternatives in

pattern layout because they have little or no efect if laid out in different orientations.

-1.,7 Measure of Performance

Throughout this thesis, the average percentage utilization of the stock sheets consumed in filling a B,O.M, is used to represent the measure by which results are compared. If z¿ is the total number of sheets consumed in filling the B.O.M., then the average percentage utilization is calculated by summing the total area of the demanded pieces allocated to the first rn - 1 sheets, and dividing it by the combined surface area of those stock sheets. The last sheet is not included in tlie calculations because it is considered as partially filled ancl not completed. Expressed as an equation, the average percentage utilization, ø, is calculated as follows :

D Area of B.O.M. pieces on first (r¿ 1) sheets *.t : - (1.2) 100 x (zz - 1) x Area of Stock Sheet

1t 1.8 Statement of the Problem

The application of interest in this work is the heuristic solution of regular, trvo- dimensional, siugle-plate cutting stock problems. The problem under study is a constrained one, with a requirement that the B.O.M. be satisfied exactly and with no restriction to guillotine-type cuts. The objective is to maximize the average percentage utilization of stock sheets consumed in meeting the B.O.M. demand and, thereby, minimize trim losses. Generally, problems in which the order pieces have low areas in relation to the stock sheets produce layouts with high stock utilization.

Research into further improvements will very likely ploduce only marginal results.

The large B.O.Ms invaliably associated with such problems will likely requile higher computation tir¡es to achieve better results, and this wil] ofset any gains attained in area utilization. It is also a common occu¡rence that most heuristics employed in such cases will produce layouts rvith na¡rorv differences between one and another in the overall stock sheet utilization.

The diferences in the performances of different heuristics are more higlily illustrated in B.O.Ms composed of medium to large pieces in relation to the stock sheet.

The larger the B.O.Ms in such cases, the more pronounced will be the difference.

One major shortcoming in many existing heuristics is their applicability to problems having limitations on the total number of pieces demanded, and the B.O.M. mix, which is represented by the valiation between demanded piece sizes. For example, a heuristic may be eficient for problems with a large number of relatively small pieces, but mediocre for one with a small number of large pieces having few different sizes. Another difficulty rvith some heuristics is the degree to which they are

12 data-dependent. Such heuristics are inconsistent, having excellent performance in some cases, and poor performance in other, simiiar cases.

The research in this thesis will cover two areas :

1. An analysis of the existing heuristic approaches with the aim of identifying

their limitations will be undertaken first. The conclusions reached will serve

as the basis fol the design of a new algorithm. Only methods adaptable to

implementation on microcomputers will l¡e considered.

2. Development of an algorithm that will perform competitively over a wider

range of problem mixes. The nerv algorithm is expected to produce, in par-

ticular, good results for problems of B.O.Ms composed rnainly of medium to

larger-sized pieces in relation to the stock sheet, while maintaining an accept-

able performance for problems with smaller B.O.M. pieces, or a near-even mix

between large and small pieces.

The objective of extending the applicability of the proposed algorithm over a wider range of cutting stock problems entails the inclusion of a rule-based system.

Specific parameters based on the B.O.M.'s composition are identified and used in rules designed to control the solution path follorved by the algorithm. Such a system enhances flexibility and allorvs a wider applicability.

1.9 Overview

The organization of this thesis will be as follows :

Chapter 2 reviews publications in the past 30 years on topics related to cutting stock problems, with palticular emphasis on the two-dimensional probiem that is

13 the subject of this thesis.

Chapter 3 details three different heuristic algorithms developed by other in- vestigators. The first heuristic is a pure one using first-fit principles; the second algorithm combines a heuristic technique for generating an input to an exact method; and the third heuristic employs priority rules in decicling layout formations.

Chapter 4 presents a fuil description of the algorithm that is proposed in this thesis.

Chapter 5 describes and evaluates techniques for improving the proposed algorithm's performance. The fina1 form of the algorithm is tested by comparing its results with those of previously published algorithms, using the same test problems.

Finally, conclusions are presented and recommendations for future research are suggested in Chapter 6.

L4 CHAPTER,2

titerature Review

2,I Introduction

A good numl¡er of publications on the subject of cutting and packing problems for the one, two and three dimensional cases are available. It is the inten- tion of this survey to focus primarily on publications related to rectangular, two- dimensional cutting stock problems. Very brief reviews for the one-dimensional and three-dimensional problems will also be presented, the emphasis being on those publications which are relevant or rvhich can be extended to ttvo-dimensional cases.

The irregular trvo-dimensional problem rvill also be included rvith a brief overview of the research done in th¿t area. The survey is divided into four sections. The fir'st section covers cutting stock problems in one-dimension. This is followeci by a review of trvo-dimensional problems in the second section. The thild section briefly surveys two-dimensional problems with irregular older pieces, and finally three-dimensional packing problems are reviewed in the fourth section.

2,2 One-Dimensional Problems

The mathematical programming techniques for solving one-dimensional cutting stock problems are led by the column generation method used by Gilmore and

Gomory [3] for a linear programming formulation of the problem. Solving even

15 moderately-sized problems with linear programming, using the simplex method, requires the enumeration of millions of columns in the simplex tableau, rvherein each column represents a feasible cutting pattern. The computational difficulty involved in such circumstances can be reduced to the solution of a single knapsack problem at each pivot in the simplex procedure. The best entering solution can then be determined without having to perform the computationally burdensome tasl< of enumerating all the columns. Thus, the problem is t¡ansformed into one of efrcientiy solving an auxiliary knapsack problem. Gilmore and Gomory suggested in ref. [3] and [4] a method for reducing the size of the knapsack problem. In addition, a more eficient technique for its solution is also presented. Horvever, the column generation method still requires considerable computational time for large problems. One approach to overcome this diflûculty has been to terminate the procedure whenever the improvement in the value of the objective function recorded ovel a certain number of iterations does not exceed a certain level, Gilmore and GomoÌy used a criterion of less than 0.1 % reduction in waste over ten consecutive iterations. The strategy is a reasonable one because the rnagnitude of the improvements in the objective function lessens rvith successive iterations, so that only a matginal imptovement can be expected at the pivot steps in advanced iterations.

Dyckhof [5] also used a linear programming approach for the problem. However, he formulated the problern such that, in general, there rvere fewer variables but more constraints than in Gilmore and Gomory's model. The method guarantees an optimal solution, and is probably more efficient for problems with a greatet number of diferent stock lengths, and/or a larger number of order lengths, in comparison to the column generation technique.

16 Pierce 16l presented, in 1966, a branch and bound solution to an integer proglamming formulation of the single-plate variant. Computational times were extremely high for large problems and, as a result, the method's usefulness is confined to small B.O.Ms.

Heulistic solution procedures for the one-dimensional problem are abundant.

Most are custom-made for real-world industrial case studies. Many others take into consideration other constraints and factors that relegate the trimloss problem to a sub-objective in optimizating a system process as a whole. General heuristic solution methods found in the literature frecluently appear as combined heuristic-exact techniques. Roodman [7], for example, irnplemented a set of heuristic procedures to improve a feasible starting solution derived from the column generation procedure of

Gilmore and Gomory. Aspilation levels are often set to speed up the mathematical programming solutions. Pierce [8] used repeated exhaustion reduction to generate cutting patterns. Those patterns meeting an aspiration level were introduced into a linear programming model. Non-promising patterns rvere excluded heuristically from the analysis by the exact procedures. In a related work, Coverclale and Whar- ton [9] developed a repeated exhaustion reduction heuristic designed to improve the seÌection of aspiration levels and restrict the generated patterns to only suitable ones.

Haessler [10] addressed the problem where a limit is desired on the numbel of pattern changes made in a solution. The method of solution fixes a charge to each cutting pattern prior to a sequential search for the final solution. Haessler [11] had also previously studied the non-linear cutting stock probiem founcl in the paper industry. The probiem was characterised by the inclusion of finishing costs, which depended on the choice of cutting patterns, as a joint objective to be considered

T7 with trim-losses in minimizing the controllable costs.

A few authors have studied an on-line solution for cutting stock problems. Tilanus and Gerhardt [12], for instance, utilized heuristics in assigning values to steel slabs based on the due date, slab quality ¿nd the state of completion of the order.

These values were formulated as a knapsack problem and solved by mathematical programming methods. Tokuyama and Ueno [13] used a heuristic combinatorial technique to solve a specifrc on-line problem in a Japanese steel mill. The problem is characterised by a variety of objectives, trim-loss minimization being one, all subject to a set of constraints imposed by the physical nature of the production ptocess,

2.3 Two-Dimensional Problems

The survey in this section will be divided into two parts. The first part will discuss the literature that has dealt analytically with the problem with the aim of achieving optimality, while the second part will review the heuristic techniques developed to solve the problem. It is noted that the literature contains substantial examples of heuristics developed specifrcally for particular real-life applications in industry. These customized heuristics rvill be touched upon only briefly, primarily where important concepts or approaches warrant mention.

2.3,I Optimizing Algorithms

The generalized cutting stock problem is thought to be intractable [14] and it is unlikely that any polynomially bounded solution can be found. However, con-

18 str¿ined varieties of the two-dimensional problem have been solved optimally by using mathematical prograraming techniques, primarily linear and integer programming, as well as recursive procedures.

Linear Prograrnming

Gilmore and Gomory extended their techniques for solving the one-dimensional problem, [3] and [ ], to the two-dimensional case. In the one-dimensional problem, the main computational difficulty of enumerating the large number of columns in the simplex tableau rvas resolved by soiving an auxiliary knapsack problem. The entering pattern that best improves the current value of the objective at each pivot step of the simplex procedure is determined by solving the auxiliary knapsack problem. Theoreticall¡ this column generation method could be extended to two- dimensional problems. Each column of the simplex tableau would represent a feasible two-dimensional cutting pattern. However, the procedure would require the solution of a two-dimensional knapsack problem at each of the simplex algorithm steps. This form of the knapsack problem is very dificult to solve, and no quick algorithms are known to exist. Nevertheless, Gilmore and Gomory [15] went on io show that two-dimensional problems restricted to multistage guillotine cutting (see section 1.3) can be solved by linear programming, with the aid of dynamic proglamming. The dynamic programming procedules are used at every simplex pivot step to solve trvo one-dimensional knapsack problems fol determining the next entering column. One knapsack problem optimizes the packing of strips of all possible rvidths, and the second knapsack pÌoblem optimizes the seiection from those strips for layout on the stock sheet,

19 There are several cases in industry rvhere cutting is of the multi-stage guillotine type for which Gilmore and Gomory's methods [3] are applicable. Much research has been carried out since the publication of [15] to find more efficient means of solving the trvo-dimensionai knapsack problem. In a later paper, Gilmore and Gomory

[16] suggested two algorithms designed to solve more quickly one and two dimensional knapsack problems. They presented a characterisation of knapsack problems, lvhich they then used in a modifled dynamic programming procedure. The procedure, though based on dynamic programming, is essentially iterative. The second algorithm contained restrictions designed to improve performance over the first algorithm. The tlvo algorithms, though they may be faster computationally than the standard dynamic programming procedures, still required substantial computer memory.

Recursive Procedures

Helz [17] used a recursive technique in his approach to the same problem dealt with by Gilmore and Gomory, namely tlie unconstrained problem restÌicted to guillotine cuts. If the problern were to be constrained, Herz's procedure rvould not guarantee optimality. The algorithm proposed by Herz optimally divides the stock sheet into subsheets such that each subsheet will accommodate a piece from the

B.O.M. Every possible cut (dissection) of the sheet (or subsheet) under consideration is tried ancl its value calculated. The dissection having the maximum value is retained at each recursion. The optimal solution rvill eventually have the recursive property wherein either the sub-sheet matches a member of the B.O.M. or else it is dissected optimally. In speeding up computations, the author considered upper bounds for the dissection values of every rectangle. Bounds could also be set to permit waste up to levels deemed acceptable, in which case the computation can be

20 speeded up further, but at the expense of sacrificing optimality. Herz demonstrated, using three example problems, a 20To increase in computational efficiency of the recursive procedure over Gilmo¡e and Gomory's iterative algorithms provided in ref,

[16]. The improvement is due mainly to the decreased memory requirements of the recursive procedure.

Haims and Freeman [18] used a dynamic programming algorithm to find a near optimal packing of the two-dimensional knapsack problem. At each stage, one order piece is allocated using a hypothetical index rectangle that assumes all the possible length-rvidth combinations that can be contained within the stock sheet. For each possible allocation rvithin the index rectangle, the remaining unused areas can be divided into sub-sheets in three diferent ways. The return function for each stage t¿kes on the maximum value obtained by summing the values of the allocated rectangle and its best sub-sheet combination. Computational efficiency of the algorithm is not very cle¿r'. The authors quoted a run time with a variation of less than one and upto fifteen ninutes for a smali sample problem on different IBM machines. It is not believed tliat the algorithm rvill be very quick for problems involving large

B.O.Ms, particularly those with 1ow piece/stock area ratios.

In regards to unconstrained guillotine cut problems, the most powerful algorithmic soiution would appear to be that of Gilmore and Gomory's linear programming approach, using Herz's procedure for solving the auxiliary knapsack problem at each iteration. There ale, however, no references in the iiterature as to what performance such a procedure may produce.

21 Constrained Problems

Upto this point all the solution methods described have dealt with the unconstrained case and assume that no limits exist on the number of pieces of any size that may be cut from the B.O.M. In most applications of cutting stock problems, this assumption is not valid, and B.O.Ms have to be satisfied exactl¡ with no overproduction. This is the constrained version of the problem, and methods to obtain a solution are described in the rest of this sectiotr.

Christofldes and Whitlock [i9] employed tree search techniques in an algorithm for solving the constrained problem. The permissible cutting patterns are restricted to normalised guillotine cuts. In normalised patterns, each rectangle is allocated left-justifled at the lowest possible position, and adjacent to previously allocated rectangles. The computational dificulty in tree search methods lies in the size of the tree, which generally grows almost exponentially as the problem size increases.

Except for small sized problems, tree search techniques are computationally pro- hibitive, uniess bounds are placed to limit the search, Christofides and Whitlock employed a procedure to eliminate cluplications resulting fi'om symmetry, as well as similar pattelns arising from different cutting sequences. In the tree, each node represented a list of lectangles generated as a consequence of the cut rnade at the node on the ir¡nediately preceecling level. Each branch emanating from a node represented a possible cut for one of the rectangles included on that node, The algorithm calculates a value for the upper bound to the node under consideration.

If this value is greater than the current best solution, then further branching from that node may lead to an improvernent in the solution. Otherwise branching is fu- ti1e, and the algorithm backtracks to try othel branches. The upper bound at each node is found by solving a transportation problem for a subset of the rectangles

22 for that node, and a dynamic programming routine for solving the unconst¡ained two-dimensional cutting problem for the remaining rectangles on that node.

Computational efrciency appeats acceptable for medium-sized problems, in the range of 40 to 80 pieces per B.O.M. An example ivith a total o{ 62 pieces involving

20 diferent sizes was solved in 66 seconds on a CDC 7600. Computational times, however, can be expected to rise near-exponentially as problems containing more pieces with lower piece/stock area ratios ale lun. This is due to the greater number of nodes that would be generated il the trees for these problems.

Non-guillotine Cut Solution

Beasley [20] presented an algorithm for problems in which non-guillotine cutting is permitted. The objective is to maximize the total value of the pieces cut from a single stock sheet.The problem is formulated as a 0-1 integer problem, and its solution is performed in a 3-stage procedure. In the first stage, a Lagrangean rel¿xation of the formulation is used to determine an upper bound fo¡ the optimal objective. In the following stage, a subgradient optimization method minimizes this upper value and, in the last stage of the solution, the problem is resoived in a tree search procedure. If, at the end of the subgradient procedure, the minimum upper bound and the minimum lorvel bound corresponding to a feasible solution coincide, then the optimal solution to the original 2-dimensional cutting problem will have been found. Otherwise the tree search has to be performed to solve the problem. A number of reduction tests derived from both the original problem and the Lagrangean relax¿tion may be applied to reduce the size of the tree that must be searched. The performance of the search and reduction tests depends upon the quality of the value obtained for the iorver bound of the optirnal objective value, A

23 heuristic procedure is also incorporated to find a good feasible solution from any solution to the Lagrangean prograrn. A feasible solution having the maximum value is used as the initial value for the lower bound of the optimal solution. The heuristic procedure is used, at each subgradient iteration, to update the iower bound.

The algorithm was shorvn to be capable of solving rnoderately-sized problems within reasonable computational times, about 60 seconds on a CDC 7600 for a B.O.M. containing between 15 and 20 pieces.

The optimization algorithms, whether they are iterative, recursive, or tree search procedures, all have ¿ common shortcoming : computational ineficiency for large problems. Most cutting stock problems in real-world applications are often large and complex, so that rrany researchers have turned to heuristic methods.

2,3.2 Ileuristic .A.lgorithms

The heuristics employed vary from pure heuristic methods to rnethods that rely heavily on analytical optimization of portions of the problem.

Adamowicz and Albano [21] used a heuristic to generate strips out of the B.O.M. pieces. A dynamic programming routine was used to optimize the layout of these strips on the stock sheet. Thus the method uses a heuristic to convert the trvo- dimensional problem into a one-dimensional knapsack problem that can be solved by dynamic programming. The quality of the solution, which is necessarily a guillotine cut pattern, is related to the eficiency of the strip generation procedure.

Homogeneous strips, rvhich contaitr only a single type of rectangle, are the quickest and easiest to generate, while non-homogeneous strips require more computation.

24 As a result, a solution allorving non-homogeneous strips rvill require a longer computation time. Howevet, its quality will likely be better because of the greater number of strip possibilities aforded at the knapsack stage of the problem. Tlie algorithm is discussed in greater detail in the following chapter.

The heuristic of ref. [21] was upgraded by Albano and Orsini [22] to cover the generation of non-homogeneous, uniform and quasi-uniform strips. A uniform strip is composed of pieces of different types, but having a common width. Quasi-uniform strips carry pieces of diferent heights. Threshold values are introduced to control waste levels during strip generation, and in selecting how the stock sheet may be subdivided. The heuristics of [21] and [22] are computationally very efficient, but best results ale obtained when problems have low average piece to stock area ratios, rvhich permit more strip configurations and may allow upto a hundred or more pieces per sheet.

Bengtsson [23] proposed an algorithm suitable for packing relatively larger pieces on stock sheets, rvitli an average piece to stock area ratio of upto 0.15. The algorithm selects pieces to serve as bases for sections, Each section is created by placing pieces on top of the base in decreasing horizontal dimensions, such that the rightmost edge of any piece does not extend beyond that of the piece or configuration of pieces lying immediately l¡elorv. Each time a section is built, a comparison is carried out to see whether continuation may produce an implovement. If not, the section is discarded and another one built using a diferent base piece. Sections are built starting at the left edge of the stock sheet and across to the opposite edge. When the right edge is reached and no furthe¡ sections can be accommodated, backtracking is performed and othe¡ sections are tested in the search for the minimal waste layout. In problems rvith large B.O.Ms wliich require more than one stock piece, an approximate initiai solution is generated, and the least favourable layouts are discarded. The pieces from the discarded layouts are collected in a "pool". The packing algorithm just described is then run fo¡ the non-discarded sheets using the pooled pieces to try and implove their initial layout formations. The time spent by the algorithm in trying to improve the layout in each sheet can be fixed, such that computational time is kept to within desired levels. Obviousl¡ the smaller the time limits set, the greater is the compromise of the solution quality in favour of the computational speed.

Wang 124] approaclìed the constrained two-dimensional problem in a diferent manner. Instead of enumerating the patterns on the stock sheet, Wang proposed that the order pieces be combined iteratively into larger ones, Rectangles could be added to t'build" larger enclosing rectangles. A vertical build occurs when two rectangles are joined vertically with one on top of the other. A horizontal build is the product of joining two rectangles side by side.

The proposed algorithm generates al1 possible builds of the B.O.M. pieces subject to the following constraints :

1. A "built" rectanglets dimensions must not exceed those of the stock sheet.

2. A "built" rectangle shall not contain more pieces of any type than tlie quantity

specified in the B.O.M. . This condition is necessary to ensure that the solution

is a constlained one.

3. The waste area within a built rectangle must be rvithin an allowable amount.

When no more possible combinations of order pieces can be built, the algorithm reiterates using the set of newly-built rectangles âs a nelv B,O.M, In this fashion, larger pieces are continually being built, ancl the process continues until no further builds can take place. Then the algorithm terminates and the patterns on the final

26 B,O,M. represent the solution to ihe problem, with each pattern constituting a layout fol one stock sheet. Trvo algorithms, based on different methods of defining allorvable rvaste, could be run by using Wang's concept. The first algorithm defines waste carried by a built combination in terrns of a percentage of the total stock sheet area, The second algoriihm considers the waste to be that unused area in a build expressed as a percentage of the area of the rectangle enclosing the build.

The efficiency of the algorithms depends on the settings for acceptable waste levels.

Wang used error bounds to measure horv much the waste in the patterns deviated from the optimal value. The end result is that the algorithms are fast and efficient for problems with small B.O.Ms and small order quantities for each type. For larger problems, a solutionts quality can be traded off in favour of gleater computational speed, depending upon how much the user is willing to deviate from the optimal solution.

Dietrich and Yakorvitz 125] used priority rules in a problem reduction technique for the single-plate problem. B.O.M. pieces are classified as two-degree, one-degree or zero-degree flts in relation to the current stock sheet or subsheet depending how many dimensions a piece has in common with the sheet. Two-degree fits have overall priority. One-degree fits have priority over zero-degree fits. Ties between pieces of equal degree fits are settled by heuristic rules based on the areas and lengths of the pieces and sheet/sub-sheet under consideration. The solution method also adopts ¡ules for combining the waste areas created as a result of each piece allocation, as well as rules for selecting which of the available subsheets is to be processed next. Six different rules were used in deciding which piece from a set of equal priority candidates is to be selected for allocation onto rvhich available stock sheets/subsheets. These six rules were tested on randomly generated sample problems. The results shorved that area-based rules generally gave better solutions

27 in terr¡s of higher average utilizations. Computer tirne for the alogrithm run on an

IBM PS/2 was reported in the range of 0.009 and 0.025 seconds per piece.

Israni and Sanders [26] put forth a recursive heuristic characterised by simplicity and special adaptabiÌity to personal computers. The patterns generated can be guillotine or non-guillotine types, aud provision is made to allorv for use¡ intervention to aid in obtaining better solutions. The lieuristic pre-orders the B.O.M. into a list of decreasing lengths. The first piece on the list is placed at the lower left corner of the stock sheet. The other pieces are allocated sequentially in an alternating fashion along the length and width of the stock sheet. A piece that will not fit in the available space is retained, ¿t its same position, on the pre-ordered list.

When no further pieces from the list can be allocated, that stage is considered complete and the procedure is repeated for the residual sub-sheet. Gaps in the layout can always be filled with unallocated pieces, where possible, by means of user intervention. Computational time for randomly generated B.O.Ms ranged from 5 to

15 minutes, with most of that time consumed in user intervention.

Zinober, Yanasse, and Harris [27] examined the constrained multiplate guillotine-cut problem, whe¡e stock sheets of seve¡al different sizes and numbers were available. The problem rvas to assign the B.O.M. pieces to a mix of the available stock sheets such that the total unused area on the stock sheets is minirnized.

The solution rnethod involved a procedure that combined pattern genelating techniques and enumeration. Patterns rvere created by sorting the B.O.M, according to priorities. The priorities sorted pieces according to their area, length, rvidth and weighted area. Weighted area is defined as the alea of a piece multiplied by ilie demand for its type. Pieces ale selected secluentially from the prioritized lists and laid at the lowest possible leftliand space on the sheet. A scoring system is irnplemented

28 to assist in deciding how unused spaces, created each time a piece is allocated, can be partitioned into subsheets. The enumerative part of the algorithrn generates the sequence in which the available stock sheets are processed. This is done in a tree search for diferent sequence combinations, The sequence having the least percentage of waste areas is selected as the solution. The algorithm is designed for solving relativeiy small problems where the B.O.M. cannot contain more than gg pieces in all, regardless of size. The problems handled could have upto nine diferent stock sheet sizes and sixty diffelent sizes for the B.O.M. pieces. Computational times are very highly data-dependent, and may range from a few minutes to severai hours.

Hodgson [28] dealt rvith the pallet loading problem in which a B.O.M. composed of pieces of varying dimensions is to be loaded onto a rectangular pallet. The approach was ¿ combined heuristic-dynamic programming method, in which the dynamic programming maximizes the area covered on the pallet. Heuristics are implemented to limit the computational effort.

Steudel [29] also used dynamic programming in pallet-loading problems where only uniforrnly-sized order rectangles are considered in the B.O.M, Four optimum sets rvere placed, by dynamic programming, along the inside of each edge of the stock sheet in an efort to maximize utilization of the sheet perimeter. The unused central area served as a ¡educed stock sheet for the next algorithmic iteration.

Other authors have investigated the cutting stock problem for specific constraints imposed by the nature of real world cutting processes. Hahn [30] studied the problem where the stock sheet contained defective areas. These areas had to be avoided in cutting the B.O.M. Hahn developed an iterative rnethod for a three-stage cutting process, Pieces rvere assigned values, and dynamic programming was used to obtain

to patterns in which the sum of the heuristic values was maximized.

Dyson and Gregory [31] considered the problem where constraints are imposed on the sequence of the cutting in order to avoid breaks in the production, Breaks arise when partially completed orders are not included in the immediate patterns follorving. This situation is undesirable because it leads to increased costs of machine set-up, storage, handling and tracking. Dyson and Gregory outlined two approaches to this problem. The first is linear programming based and uses Gilmore and

Gomory's algorithm [15] to generate cutting patterns, which are sequenced subse- quently with a branch and bound technique. The second approach is a repeated exhaustion heuristic in which higher values are assigned to pieces that must be included in the pattern if a break in production is to be avoided. The pattern having the best total value is selected at each iteration of the heuristic.

Hinxman [32] also used a repeated exhaustion reduction technique to solve sequentially constrained problems. He formulated the problem in a manner such that identical pieces ordered by different customers are treated separately. At each iteration a problem reduction is carried out in an attempt to inciude those pieces whose allocation is critical if a production discontinuity is to be avoided. Heuristic values are given to pieces to help the solution process run more eficiently.

In a more recent publication, Madsen 133] used a 2-st,age solution method for sequence-constrained problems. In the first stage, a cutting stock ptoblem is solved by the methods of Gilmore and Gomory [15] and, in the second stage, the sequencing constraint is incorporated to folrnulate a travelling salesman model. The model is solved optimally or near optimally using a 3-optimal method [34].

30 2,4 frregularTwo-Dimensional Problems

Researchers studying the problem of allocating irregularly-shaped pieces on rectangular stock sheets have grouped approaches to the problem into three general categories:

1. Manual, rvherein an operator may manipulate shapes on a computer screen

and arrange them into patterns judged to be good. The computer then caicu-

lates wastages for generated patterns to assist the operator in evaluating the

trial soiutions.

2. Automatic, in rvirich algorithmic procedures lay out the B.O.M. on the stock

sheet by either (a) working directly on the order pieces in search procedures for

optimal arrangements or (b) enclosing (nesting) pieces in rectangular modules

which are then allocated by known rectangular two-dimensional layout meth-

ods,

3. Semi-,{utomatic where the system automatically generates tentative solu-

tions which may be improved manually by means of operator interaction, An

example of such a system is described by Albano [35].

Many efñcient nesting programs are known to have been developed commercially but have not been published for proprietary leasons. This section will survey briefly the most important nesting algorithms and techniques that have been published.

Adamorvicz and All¡ano [36] developed an algorithm for clustering irregular pieces, defind as polygons, inside rectangular modules. The procedure allows the user to select preferred types of clustering, the clusters being obtained from pairs of similar shapes or va¡ious clusters of both irregular and rectangular shapes. The

3l algorithm is capable o{ rotating shapes in search of the orientation that produces a minimum area for a rectangular enclosure of the piece. This enclosure is known as a minimum enclosing rectangle (MER). The algorithm is also able to pair polygons in minimum MERs by testing various positions and orientations of the pieces in relation to each other. The clustering of several pieces into a single integrated unit is achieved by treating the problem as a multi-stage problem where pieces are combined into similar composite shapes that are clustered pairwise,

Bronsoiler and Sanders [37] proposed an algorithm that pre-sorts the B.O.M. to give priority to those pieces that are largest and most rectangular in shape.

The pieces are allocated on the stock sheet in order of priorit¡ and the program examines diferent positions and orientations for the selected part. The placement contributing best to ¿ denser packing is chosen.

All¡ano and Sappupo [38] used an automatic approach based on a state-space search procedure, combined rvith heuristics designed to shorten the search. A piece selected for allocation is always placed in the position and orientation contributing least to the extension of the lower left-most profile created by the right-most ex- tremities of the previously allocated pieces. A state of the solution is composed of two ordered lists, one containing the position and orientation of the allocated pieces, and the second containing the current/instantaneous status of the B.O.M.

A nesting algorithm that uses composites of rectangles to approximate irregular shapes rvas proposed by Qu and Sande¡s [2]. The greater is the number of rectangies in the representation of the shape, the finer is the approximation. Minimun enclosing rectangles (MER"s) for each piece are created and sorted in ordel of decreasing height. A problem-reduction technirlue, based on a first-fit process, is initiated to

32 lay out the MERs sequentially from left to right ac¡oss the stock sheet.

Dagli and Tatoglu [39] suggested a two-stage hierarchical approach to the problem. In the first stage an initial ailocation is obtained through mathematical programming methods. The allocation is refined in the second stage by means of heuristic procedures that allocate pieces in a sequence determined by value-based priority rules.

2,5 Three-Dimensional Problems

Not many papers deal with the three-dimensional problem. Those that have been published generally deal with highly restricted forms of the problem.

George and Robinson [40], for example, used priority lists for packing a given B.O.M. into a container whose volume is known to be larger than the sum total of the volumes of the individual pieces on the B.O.M. The algorithm's objective is the successful packing of the B.O.M. into a single container.

Many rnethods used in solving the three-dirnensional problem are extensions of those developed for two-dimensional problems. A case in point is the method proposed by Hodgson [28]. He illustrated two ways in which his previously published algorithm for the trvo-dimensionai pallet loading problem can be extended to the loading of three-dimensional pieces onto rectangulat pallets which are restticted to maxir¡um allowable heights. Other authors, whose algorithms for solving the two-dimensional problem can be rnodified or extended to the th¡ee-dimensional case, include Hairns [41], Steudel

129),1421, Gilmore and Gomory [15] as well as Herz [17] rvhose recursive procedure can be extended providing cuts remain restricted to the guillotine type.

Han et als. [43] proposed a heuristic procedure based on a dynamic programming approach for solving problems where a given number of identically sized boxes are to be loaded into the minimum numbe¡ of containers. The algorithm is a problem reduction technique in which the base and one of the vertical faces of the container are loaded with boxes. The objective is to maximize utilization of the surface area of the two faces. Dynamic programming is used in this step, which results in the loaded boxes assuming an L-shaped prism within the container. The area left unused by this prism serves as a sub-space to be filled in a manner similar to that in the previous iteration.

34 CF{AFT'ER, 3

Comparison of Three Heuristic Methods

3.1 fntroduction

In this chapter the heuristic approaches of Israni and Sanders [26], Adamowicz and Albano [21] as well as Dietrich and Yakowitz [25] to the cutting stock prob-

1em are described in greater detail. The performances of these three heuristics are evaluated approximately on the l¡asis of computer programs written to mimic the algorithms. The main goal is to study the foilowing thtee characteristics for each heuristic:

Quality of Solution, as measured by stock sheet utilization;

Computational Speed; and

Range of Applicability.

The third characteristic is meant to examine how efective a heuristic is for B.O.Ms of varying sizes and composition.

The main reason rvhy these particular three heuristics rvere selected is the ease rvith rvhicli they can be proglammed on a microcolnputer. Each one is also repre- sentative of a particular approach to the cutting stock problem, as explained later in this chapter'. The goai here is basically to uncover the weaknesses in the perfo¡mance o{ these heuristics. The information generated will l¡e utilised to develop

35 an improved method that avoids the observed shortcomings. The methodology of each algorithm is illustrated, using as an example, the arbitrarily compiled Bill-of-

Materials presented in Table 3.1.

Table 3.1: Sample Bill-of-Materials f 1.

Piece f Length width Quantiiy i I h 2 8 8

3 7 n 15

4 5 4 11 5 J 3 t2

3.2 The First-Fit Heuristic of fsrani and Sanders

A heuristic algorithm developed by Israni and Sanders [26] is an example of a first-fit heuristic. First-fit heuristics perform by pre-ordering the B.O.M. according to pre-deterrnined rules. The pieces are allocated sequentially from the ordered list sequentially to the stock sheet in a manner conforming to a set of rules that define the heuristic. The Israni-Sanders heuristic uses the flrst-frt method in a problem reduction procedure that solves the problem by allocating ihe B.O.l\{. to stock sheets one sheet at a time. The heuristic was developed basically for solving very quickly large problems on a microcomputer. It allows user interaction to improve solutions.

The user may manually allocate pieces from the B.O.M. and place them at positions on the current layout by specifying the coordinates of the location.

36 A program in Turbo-C language has been rvritten to implement this heuristic.

No provision is made, hotvever, to allow user intervention. Solutions enhanced by user interaction are bound, of course to be better than those that are not. However, the rnain purpose here is to assess the behaviour of a first-fit heuristic with B.O.Ms of varying sizes.

3.2.L Description of the Heuristic

STEPi-Datalnput

The following is all the input data recluired by the heuristic :

1. stock sheet length (.L,),

2. stock sheet width (I4l"),

3. number of diferent types of order rectangle (ri,),

4. lengtlr of rectangle type i (/¿, i,:1.,.'. ,n),

5. width of rectangle type i (to;, i:1,..',n), and

6. number of type i rectangles demanded (d,;, i - 1.,... ,n).

If, in any rectangle, the input width exceeds its input length, the algorithm automatically exchanges the two values so that the length always assumes the greater dimension,

STEP 2 - Sorting

The B.O.M. is sorted in order of decreøsdng length of the rectangle. STEP 3 - Layout

Starting at the bottom left-hand corner of the stock sheet, rectangles are alternatively allocated along the length and ividth of the stock sheet, such that the sheet is loaded along its half-perimeter. Whenever the remaining space will not accommodate the next piece on the ordered list, the list is traversed down until the first piece that can fit is found. That piece is allocated and immediately removed from ihe B.O.M. so that the B.O.M. is updated instantaneously rvith every allocation.

STEP 4 - Processing Sul¡-sheets

When no rnore lectangles can be allocated along either dimeusion of the stock sheet, the program repeats step 3 for the reduced subsheet that presently exists in the area between the allocated edges of the stock sheet. The dimensions of the reduced subsheet are determined by the longest and the highest (widest) pieces allocated to the previous subsheet. If piece type k is the longest piece allocated, and piece type j is the highest, the dimensions of the new (reduced) subsheet will be (2, - /¡) and (W" - ú¡).

STEP 5 - Starting New Sheets

If no allocations are possible, at all, on a curLent sub-sheet and the B.O.M. is not empty, the proglam starts a new stock sheet and proceeds fi.om step 3. The heuristic terminates once the B.O.M. is empty.

A run of ihe Israni-Sanders heuristic for ihe B.O.M. presented in Table 3.1 and

40 x 30 stock plates gives the layout shown in Figure 3.1

3B 5N 3 3 3 2 3 2 3 3 3

1 3

3 3 L 2

2 I 3 2

1 I 1 I 1 14

.É) ( Àvefâge She€t Util¡zål¡o = 86;l

' Numbers ¡nside the ajlocated rectangles identify rectangle type

Figure 3.1: Layout given by First-fit Heuristic for B.O.M. of Table 3.1 3.3 A Combined Heuristic-Exact Frocedure

The algorithm selected as an example of a method tliat coml¡ine heuristics lvith exact techniques is that of Adamowicz and Albano [21]. It rvas chosen because of its computational speed and ability to solve problems with large Bills-of-Materials, as well as its adaptability for implementation on a microcomputer. A program was rvritten in Turbo C to implement, as closely as possible, the algorithm published in [21]. The program incorporates the rnain features of the Adamorvicz-Albano algorithm, namely :

c Pariitioning of sheets into subsheets,

c Generation of homogenous strips using B.O.M. pieces,

o Use of a dynamic programming routine to select the optimal combination of

strips for maximising utilization of the sheet (or subsheet) under consideration,

c Decision made whether the solution is acceptable o¡ can l¡e improved. If the latter holds, the algoritirm allows a reduction (lengthwise or widthwise) of

the sheet/sub-sheet to permit alternative layouts.

Tlre solutions found by this algorithm are single or multi-stage guillotine øú patterns.

3.3.L Details of the Algorithm

Stepl-Datalnput

The algorithm takes the input data in the same form as that in the previously described fi rst-fit heuristic.

40 Figure 3.2: Horizontal and vertical homogeneous strips

Ste 2 - Sheet Subdivision

At this point, a decision is made on how, if at all, to divicÌe the available stock sheet into subsheets. The strategy employed gives priority to the allocaiion of the largest-sized piece on the B.O.M. This procedure permits better strip generation combinations for the later stages by maintaining availability of the smaller sizes towards the final stages of the solution.

Let -L¡ be the length of tÌre longest strip generated from type k rectangles, where a type k rectangle has the longest length amongst the rernaining rectangles capable of fitting in the space available on the stock sheet. If the subsheet (L" - L¡) x W" can accommodate at least one of the rectangles remaining in the B.O.M, the stock sheet will be subdivided into two subrectangles with dimensions (.L¡ x I7, ) and (L" - L*) x I7,. Otherwise, the stock sheet remains undivided and retains the dimensions (L" x W"). 2 x dfìPtt?a B

+2+

Figure 3.3: Strips generated for 20, type i pieces each having dimension 3 x 2

Step 3 - Strip generation

Only uniform homogenous strips are used as candidate strips. A uniform strip is one in which all the rectangles making up the strip have a uniform orientation, either a1l veltical or all horizontal. A homogeneous strip is one where all the component rectangles are identical. Figure 3.2 shows two uniform homogeneous strips generated using rectangular pieces o1 size a x ô (ø > å). Type A is the strip formed with a horizontal orientation of pieces, and type B with a vertical one.

The length of a strip generated by the algorithm is the longest that can fit within the length of the subsheet under consideration. For instance, assume that 20 units of rectangle type i with dimensions I¿ : 3, u¿ : 2, ate demanded. Assuming a length (I,) of 14 for the cu¡rent subsheet, the strips that rvill be generated, and the quantity of each strip, are as shown in Figure 3.3.

Strip type B is chosen over strip type A as a candidate for allocation because it is longer. The 20 demanded pieces of rectangle type i are sufficient for generating two type B strips which are put on the list of candidate strips for layout on the subsheet. The remaining six unused pieces are insuficient in number to form a third strip. These pieces are left on the B.O.M. and remain available for subsequent use.

42 Step 4 - Strip Selection and Layout

Once all the rectangles remaining in the B.O.M. have l¡een grouped into the longest possible hornogeneous strips, the set of these strips and their quantities is analysed to determine which and how many of each type will be assigned for layout on the sub-sheet. This is done by dynamic programming.

The problem of assigning a coml¡ination of the candidate strips to the subsheet under consideration is equivalent to the following knapsack problem.

Maximize

rtkt*rzkz*...*rnkn (3.1)

subject to

(.rkt + uzkz I '..* w"k" <- I4l"). (3.2)

Here,

k¡ number of units of strip type i (¿ : 1, '.. ,n) . uì - width of strip type i. r"i : return from allocating one strip type i (: (I;lL") x w;). l; = length of strip type i. L" = Iength of sheet (or sub-sheet).

W" width of sheet (or sub-sheet).

In the dynamic prograrnming formulation, each strip type is represented by a stage (i), and the states are variables of the sul¡sheet's width. The width variables can assume values from 0 to lWlw;1, the largest integer contained in Wf u¡, lot

4ó stage i. The knapsack value of each strip is determined by the ratio of the strip's length to the length of the subsheet, rnultiplied by the strip widtli.

The genelal recursive equation for this problem is given by :

/t(tt) : man {k1r1} (3.3)

år = 0, 1, "' ,LW"l.,)

l;@¡) - max {k¿r¡ * f¡-t(x¡ - k;W¡,)} (3.4)

å; : 0, 1, .'. ,LWl-¡), i :2,... ,n

and 0 1: n¡ 1: W" for all i.

The output of the dynarnic programming routine is a set

,9 : (sr,,.,,si,..,,s",),

where s; is the number of strips of type i selected for the layout. The strips are sorted in order o{ decreasing length and laid out in order on the subsheet, starting with the longest strip at the bottom of the sheet. A layout will typically take the form shown in Figure 3.4. The unfilled areas between each strip and the edge of the subsheet are called "holes". These holes are indicated bv Hi through H5 in Figure.

3.4.

The percentage utilization (U) of the sheet is calculated as

Y - (\r;k;)1w". (3.5)

44 Figure 3.4: Example of a Strip Layout Generated by Dynamic Programming

STEP 5 - Improvement of the Solution

At this stage, the user is allowed to judge rvhether the layout is acceptable in terms of the wastage generated. If the solution is acceptable, the B.O.M. is updated to account for the allocated pieces, and the algorithm returns to STÐP 2 to fill the next subsheet. If the layout is judged inadequate, the procedure calls for a

¡eduction in the subsheet length. STEPS 2 through 5 are repeated for the revised sheet sul¡divisions.

The new subsheet length is set by using the formula

rk:L(U xm¡1100 Ill (3.6) where

U percentage utilization of the rejected solution

mk -- number of rectangles in the strip composed of the longest rectangle

appearing in the unacceptable solution L,J - largest, integer in x

45 If r¡ < rn¡, the length of the new subsheet will be tr/ = rtl¡, where /¡ is the length of rectangle k, the longest rectangle appearing in the rejected layout. If rk : Ính or if the area (.L" - Lt) x W" is not lalge enough to accommodate any of the pieces left on the B.O.M., consideration is given to dividing the sheet along its width. To divide the sheet rvidthrvise, a condition that must l¡e satisfied is that, if Wf represents the sum of the strip rvidths appearing in the unacceptable solution, then (l,T/ - W') x,L" must be large enough to accommodate at least one of the unallocated rectangles. If the attempt to obtain smalier subsheets fails, the solution obtained at the end of STEP 4 has to be accepted. The solution procedure continues by returning to STEP 2 and starting on a new sheet ol subsheet, if any still remain available.

STEP 6 - Termination of the Algo¡ithm

The algorithm terminates when no further order rectangles remain on the B.O.M,

Solutions are presented in the form of a graphical display of the layout for each sheet, together with data on the average utilization and total number of sheets used in satisfying the B.O.M. The algorithm does not attempt to filI in any unused holes resulting from STEP 4.

3.3,2 Control of Stlip Lengths

The algorithm provides a rnechanism for controlling the lengths of the strips generated in STEP 3. These are :

o Ç- The maxirnum nurnber of rectangles pennitted in any strip. This thresh-

old. affecl.,s the solutions by restricting the lengths of the subsheets. A lorv 7"

causes the generation of shorter strips, which results in the algorithm rvork-

46 ing with smaller subsheets of the original stock plate. It also plevents undue

consumption of the smalier pieces early in the solution.

c [- Waste contribution by any strip. This is defined to be the minimum acceptable strip length expressed as a percentage of the sheet's length. The

?, specifies the range of acceptable strips, and any strip contributing more

than the specified percentage is prevented frorn becoming a candidate for a

solution. A lower ?" results in more strips qualifying for allocation in the

knapsack stage (STEP 4) of the solution.

A computer run for the Adamowicz-Albano heuristic rvith [, : 9999 (to practically represent infinite) and ?" : 0.95 for the B.O.M. of Table 3.1 gives the layout illustrated in Figure 3.5.

3.4 A Priority Rule Heuristic

The third heuristic is that presented by Dietrich and Yakowitz [25]. This heuristic uses rules for directing the selection of pieces from the B.O.M. for their allocation.

Rules also control the order in lvhich available subsheets are processed, and unfilled subsheets are cornbined into larger ones,

3.4,L Descliption of the .A,lgorithm

The Dieirich-Yakowitz algorithm is built around the concept of degree-of-fit.

There are three possible clegrees-of-flt that describe the relation between an o¡- der piece and a stock sheet. A two-degree fit occurs when both the order. piece dimensions equal the two dimensions of the stock sheet. In this case the order piece

47 4 4 4 414t 4

2 3 3 3 3 3

4 3

4 3 3 3 3 3 ; sls 5 515 5 5 515 51515

4 1, 1 I I

( A.verage S hee.l Ulilizâti oî = 96.8 )

' Numbers inslde the allocaled rectangl€s identity rectangle typê

Figure 3.5: Layout given by the combined exact-heuristic for the B,O.M. of Table

48 subsheet I sùbsheet 2

subsheet 3

Figure 3.6: Sheet sul¡divisions upon allocation of zero-degree fit, piece P completely fills the stock sheet. A one-degree fit describes the case rvhere only one dimension of an o¡der piece equals either one of the stock sheet dimensions. A one- degree allocation results in the creation of a single reduced sul¡-sheet. A zero-degree fit is one where neither dimension of the order piece equals either stock piece di mension. A zero-degree allocation at the lower left-hand corner of the stock sheet will produce an L-shaped are¿ of unused space. Dietrich and Yakowitz divided this area into three subsheets as shown in Figure 3.6.

The program sea¡ches the B.O.M. for preferred degrees of fit. The two-degree frt receives iop priority, followed by one-degree fits and, lastly, by zero-degree fits.

In the event that more than one piece qualifies for allocation on the basis of having identical degrees of fit, Dietrich and Yakowitz used one of six diferent heuristics to settle the conflict. These rules are listed as follows :

¡ Rule L1

Allocate the longest piece to the subsheet lvith the longest dimension.

e Rule L2

Allocate the longest piece to the subsheet rvith the shortest dimension,

49 o Rule L3

Allocate tlie longest piece to the subsheet possessing the preferred degree-of-fit

for that piece.

e Rule A4

Allocate largest qualifying piece to the ¿vailable subsheet with the largest

surface area.

c Rule A5

Allocate the largest piece to the smallest available subsheet.

¡ Rule A6 Allocate the largest piece to the subsheet possessing the preferred

degree-of-flt.

In any run of the algorithm, only one of these heuristics is chosen and applied consistently throughout the solution process. For two-degree and one-degree fits, the orientation of the allocated piece is determined by the common dimension between the piece and the sheet, In zero-degree fits, the piece may assume either vertical or horizontal orientations. The algorithm heuristically selects the orientation which aligns tlre longer of the allocated piece's dimensions with the longer of the sheet's dimensio¡s.

Areas remaining unfllled after an allocation of a piece aÌe stored on a list of available subsheets. One and three nelv subsheets (holes) are added, respectively, every time a one-degree or a zero-degree frt is allocated. The next sheet to be picked

{rom the list of available sheets is selected according to the rule implernented from rules 1 through 6. When none of the availal¡le subsheets is large enough to hold any of the pieces remaining on the B.O.M., the list of sub-sheets is processed rvith the aim of "expanding" them, An expansion rvill occur tvhen two adjacent sub-sheets

50 sharing a border of common length are integrated into one larger sub-sheet.

The Dietrich-Yakorvitz algorithm is a problem reduction type of algorithm. Ev- ery time a piece is allocated, new subproblems in the form of subsheets are created and solved separately. The B.O.M. is updated instantaneosly after every allocation.

When no further pieces can be allocated on the available subsheets, a new stock sheet is started, The unused subsheets on the previous stock sheet constitute the waste area. The program terminates rvhen no more pieces remain on the B.O.M.

3,4,2 Expansion of Unfflled Subsheets

Dietrich and Yakowitz defined a method for combining acljacent, unused aleas into larger ones that may have a better chance of accommodating further pieces.

The main dificulties in the algorithm occur when zero-degree fits are selected. The number of "holes" on the current stock sheet increase by three each time a zeÌo- degree fit is allocated. Tliis results in an increased number of possibÌe ways to expand these holes. Fig. 3.7 shows a situation described by Dietrich ¿nd Yakowitz as a "stand-still" state. In this state, none of tlie available holes is large enough to accommodate any one of the remaining pieces. If expanded, however, there is a possibility of fitting other pieces. As can be seen from Figure 3.7, there exists a number of possible combinations under which these holes can be expande

51 Figure 3.7: Example of "stand-still" state with frve holes

3.4,3 Modiffed Version of Dietrich and Yakowitzts Algorithm

Dietrich and Yakowitz's rules for expanding adjacent holes are not adequately ex-

plained in their paper, particularly in cases where a hole is adjacent to mo¡e than one

hole. The "stand-still" states are difficult to resolve, and they introduce a further complexity to the problem. Therefore, it was decided to moclify the Dietrich-

Yakowitz algorithm to avoid the so-called "stand-still" state âs much as possible.

This was accomplished by the imrne

every time a zero-degree fit is allocated. Tlvo rules are used to effect this expansion.

These are :

e Rule 1

Expand the largest hole first. This means that the largest unfilled subsheet is

expancled to include the next largest subsheet adj acent to it.

¡ Rule 2

Expand the smallest hole first. This rule is the same as rule 1, with the

exception that the smallest subsheets receive priority.

The algorithm pre-sorts the holes, depending on which rule is used, in either decreasing (for rule 1) or (for ruie 2) increasing areas. The first hole on the sorted list

52 ;ri:.11i:.r.:-subsneet l.r::::

:i:Subsheet 2i.

rirri::r:;::::;rir::i* P i:t:11....-;:::Lir.:.:.:.:.1;¡

(a) Rule 1 (b) Rule 2

Figure 3.8: Expansion of unfilled subsheets using (a) Rule 1 and (b) Rule 2

is expanded by integration with the next highest ad,jacent hole on the list. Figure

3.8(a) illustrates how an unused area is partitioned by rule 1 after a zero-degree-fit P

is laid out. Figure 3.8(b) shows the subsheet partitioning created by implementing

rule 2.

The modified version of the Diet¡ich-Yakowitz algorithm was coded in Turbo-C

and run for four diferent variations. In the first variant, rule 1 for hole expansion

was used, and in the second variant rule 2 was employed. In both cases, a zero-

degree fit rvas always laid out rvith its longer dimension along the stock sheet's

longer dimension. In variants 3 and 4, rules 1 and 2 were used, respectively, but no

restriction was enforced on the orientation of the zero-degree flt piece. Input data

was generated randomly, so that the length dimension did not have to be the larger

of the piece's trvo dimensions, Thus, allocation of the zero-degree fits using variants

3 and 4 resulted in a random orientation of the selected pieces in the layouts.

Abbreviating the modified algorithm as MDY, using the notation R (restricted orientation) to denote variants 1 and 2, and U (unrestricted orientation) for variants

3 and 4, the follorving notation is used to identify the four variants of the modified

53 heuristic.

e MDY-RI - variant i

o MDY-R2 - variant 2 .

e MDY-UI - variant 3 .

e MDY-U2 - variant 4 .

3.4.4 Description of the Modiffed Algorithm

STEPi-InputData

The data required as input for the algorithm consists of the stock sheet length and width, as well as the B.O.M. specifiying the length, width and quantity orde¡ed for each size of demanded rectangle. In the "restricted orientation" variants of the algorithm, the stock sheet and order piece dimensions are arranged automatically such that the longer dimensions are represented as the "length" and the shorter dimension as tlie width. In the "unrestricted orientationt' variants, there is no such rest¡iction and the widths may assume values greater than the lengths, depending on how the input data is presented to the algorithm.

STEP 2 - Selection of Heuristic Rules

The rule which determines which piece from a set of equally qualifying pieces is selected for allocation, and to rvhich subsheet it is allocated, is defined at this point. The chosen rule remains in effect until the termination of the algorithm. The rule governing the expansion of unfilled subsheets is also selected here.

STEP 3 - Categorization of Degrees-of-fit

The algorithm categorizes ihe B.O.M. pieces according to their degree-of-frt on the

54 culrent subsheet, determined from STEP 2, Pieces tvill fall into one of three categories depending on tireir qualifrcation as a two-degree, one-degree or zero-degree fir.

STEP 4 - Allocation of Piece

The heuristic rule chosen in STEP 2 is implemented for seiecting the next piece allocated on the current subsheet, and the B.O.M. is updated immediately. Once allocatecl, a piece cannot be reconsidered.

STEP 5 - Expansion of Unfilled Subsheets

Adjacent unused areas are integrated into larger areas using the expansion rule selected in STEP 2. The expanded regions seÌve as new subsheets available for layout.

STEP 6 - Iteration

If pieces still remain on the B.O.M., the procedure is reiterated fi'om STEP 3 for a new subsheet. If no subsheets remain, a nerv stock sheet is started. Otherwise, the average utilization of the stock sheets is calculated, the result is presented, and the algorithm is terminated.

Figure 3.9 displays the layout produced by tlie modified Dietrich-Yakowitz algorithm, using the heuristic area rule that allocates the largest piece to the largest available subsheet, for the example B.O.M. shorvn in Table 3.1. The stock plates in the example are 40 x 30 units. 4 0 ------..-.

2 2 2 2 2 2 2 3

3 3 3 3 4

3 3 3 3 3 4

L L L 4 4 4

I 1 L 1 4

( AverageSheet Ulilizalion = 94,3 %)

'Numbers insided allocated rectangles identìfy the rectanOle type

Figure 3.9: Layout given by MDY-U1 for.B.O.M. of Table 3.1 3.5 Tests and Analysis

3.5.1 Experimental Test Data

The sample set of test problems used in this chapter for evaluating and comparing the three different heuristics is taken from the first data set of Dietrich and Yakowitz given ref. [25]. The test data rvas generated randomly in eight diferent categories.

Each category is defined by the maximum size a B.O.M. piece can assume, expressed as a percentage of the stock sheet size. This parameter is shown in parenthesis next to the category numbers given in column 1 of Table 3.2.

There are 100 test problems in each category, The number of different sizes demanded in the B.O.Ms of the test data ranges from 25 to 50. The stock sheet length and width range from 25 to i00 units; an order piece length from 1 unit to the stock sheet length; an order piece width from 1 unit to the stock sheet width; and the quantity for each demanded size from 1 to 25 pieces. The test problems of the data set are generated with the above parameters and fi'om a distribution that is weighted linearly in favour of the larger pieces by a two-to-one margin. Therefore, the problems in the test data considered here have a high proportion of larger-sized pieces in the B.O.M. mix.

3.5.2 Test Results For Modified Dietrich-Yakowitz Algorithm

Table 3.2 shows the average percentage utilizations and standard deviations obtained for each of the four variants of the modifled Dietrich-Yakowitz algorithm in each of the eight test categories of the test data. In all cases tested, the heuristic rule that allocates the largest piece to the available subsheet with ihe largest surface area (rule Al) was used throughout.

57 Table 3.2: Test Results fo¡ Modified Dietrich-Yakorvitz Algorithm

MDY - R1 MDY - R2 MDY - U1 MDY - U2

J ategor¡ Average 7 it andard Average I tandard Average % tandard Avenge 7t tand ard Ntmber utilization utiliz¿tion utilization (õ) (") (ã) (") (ø) (o\ (d) (")

1(1.000) 74.5 4.97 72.6 5.30 76.2 4.72 74.4 5.07 2 (o.5oo) 78.6 5.53 75.1 6.97 80.5 5.47 78.0 6.45 3 (0.250) 88.6 2.31 88.3 2.78 89.0 2.44 88.6 3.06 4 (0.100) 94.L 1.80 94.4 7.79 94.2 7.73 94.2 1.87 5 (o.o4o) s8.2 L.64 s5.4 2.94 95.6 1.97 95.0 3.88 6 (0.025) s7 .5 1.78 96.5 4.46 96.3 2.61 94.8 6.29 7 (0.010) 98.6 1.77 98.0 2.41. 98.0 1.84 97.4 3.27 8 (o.oo4) 99.9 0.16 100.0 0.04 99.A 0.36 99.9 0.64

The results shown in Table 3.2 highlight trvo important observations :

1. Rule 1 for subsheet expansion generally outperforms rule 2. This holds true

particularly for those categories where the average piece-to-stock ratios are

large. As this ratio decreases, the distinction between the the two rules

becomes less significant. This result is expected because expanding the largest

of the holes provides a gleater opportunity to accomodate larger pieces.

2. The results for the cases in rvhich there is no definition of tire length as the

larger of the allocated piece's two dirnensions are better for categories having

high average piece-to-stock area (APSA) ratios. This de¡nonstrates that the

selected orientation of zero-degree flts is an important factor in the finai result.

Again, this importance diminishes as the APSA ¡atio decreases.

The conclusion to be drarvn is that, as the APSA ratio decreases below a value around 0,10, the area of a piece and its orientation become less significant factors,

58 and warrant less weight in heurisiic decisions. Priority in selecting candidate pieces in such cases need not be weighted very strongly torvards the size of the piece. The preferred policy for expanding holes also changes in favour of expanding smaller holes (ru1e 2) rvith decreasing APSA ratios.

3,6 Comparison of Heuristic Results

Table 3.3 shows the results from all three heuristic methods described in this chapter using the test data set f 1 of ref. [25]. For the present version of the Dietrich-Yakowitz algorithm, only the best variant (MDY-U1) is used in the comparison. The results for the original Dietrich-Yakowitz algorithm (using heuristic rule A4), as published in ref. [25], are shown in the last column of Table 3.3. The results compare the heu¡istics with respect to average percentage utilizations of the stock sheets and to the computation times on a 386, IBM compatible, personal computer.

3.7 Conclusions

1. The first-frt heuristic is consistently poorer lor all categories tested, par.tic-

ularly for those with very small APSA ratios. The heuristic may give more

satisfactory results in cases where there are high order quantities for each

piece type, and where the aspect ratio of the demanded piece types is close

to uniform. Results for the B.O.Ms with low APSA ratios can be improved

considerably by filling in the holes in existing layouts prior to commencing a

nerv sheet. Filst-fit heuristics are very highly problem-dependent. The vari-

ability in the aspect ¡atio between the pieces on the B.O.M. is a major factor

in the outcome. Patterns produced by first-fit methods tend to have small

Israni- Adamolvicz- MDY - U1 Dietrich- Sanders Albano Yakorvitz

Category Average % Comp.* Average % Comp. average Vo Comp. average To Comp. Number utilization time time utilization time time (r) (sec.) (r) (sec.) (¿) (sec.) (¿) (sec.) i 69.0 1.58 68.4 58.69 76.2 4.72 75.5 NIA 2 66.5 1.56 64.1 36.07 80.5 5.47 80.4 NIA 78.0 1.62 76.9 27.1.3 89.0 2.44 91.1 NIA 4 79.3 1.24 ott i5.85 94.2 95.8 NIA 5 56.9 0.92 95.1 9.82 95.6 1.97 97.4 N/A t) 37.9 0.76 94.4 7.76 96.3 2.6r 98.1 NIA 40.0 0.56 90.2 4.18 98.0 1.84 98.8 NIA 8 57.2 0.39 89.5 7.34 99.8 0.36 100.0 N/A

* Quoted computational times are average per individual test problern

unfilled gaps between the allocated pieces. These gaps are often too small

to accommodate remaining pieces, and too dispersed over the stocl< sheet to

allow integration into larger areas that may support further pieces. The chief

advantage of a first-fit heuristic is the very low demand on computer time.

2. The result for the combined exact-heuristic technique of Adamowicz and Al-

bano shows poor results for the first three categories in Table 3.3. The method

is fairly competitive in the fourth, fifth and sixth categories, but not quite so

in the last two categories. The algorithrnts poor performance in problems in

which there are many pieces rvith high APSA ratios can be attributed to two

reasons. First, by generating strips, the algorithm makes large pieces even

larger and, therefore, more dificult to allocate rvith minimal trim-loss. Sec-

ond, small pieces, rvith the exception of the reference piece, are formed into

60 lalgel strips ancl allocated on an equal footilg rvith sttips corr posed of lar.g- el pieces. Thelefore the solution tnay include a la::ge pr.opottion of iclentical, small-sizecl pieces allocated together. on the sarne plate. Thus the algor.ithm cloes not give aclequate priority to the largel pieces rernaining on the B.O.M.

Anothel difÊculty irr the Aclanorvicz ancl Albano algorithm is the choice of tlre sti-ip length control paLameters, 7,,, alcl T". These pararnetets aÌe nìore significant in problerns rvith low APSA latios, rvher.e generatecl stlips rr. ay have lengths that fall well sholt of the sheet length. In these cases, a high valiability in the dernands for the differ.ent sizes can procluce high ivastage if the sheet is not clivicled pr.eviously irlto smallel subsheets ì:y setting appropti- ate 7l,, arcl ?l or, alternately, by running the ¡rroglam so that it fills in the holes l¡etrveen the strip encls ancl the sheet's leftmost eclge. Setting good valr:es for 7l,, and 7" can laclically improve the solutions. This, horvever., ...equiles a goocl amount of experimentation ancl/ol user. experience. A ful.ther rveah- less obselvecl il the inplernentation of these thresholcls is their neglect of the clynarnic change in tlie cornposition of the B.O.M. as the solution progresses.

Ideally, 2,, and ?, should also change during the course of the solution to reflect the instantateous state of the B.O.M. In conclusion, the Aclamorvicz-

Albano algolithn is generally satisfactor.y for B.O.Ms rvith APSA r.atios belorv about 0.10, but failly u'eak for B.O.Ms rvith larger APSA ratios. The methocl can also be eficient, both in terms of solution and computational time, fol problerrs lvith very lorv APSA ratios if either (1) non-hornogeneous and quasi- unifor.m [22] pieces ar-e pelmittecl in the solution or' (2) the algorithm leviet,s e\¡eì')¡ completed sheet or subsheet to organize unusecl aleas in a malner. that alloivs other pieces renaining on the B.O.M. to be allocatecl to them. This

D1 second alteÌnative, horvever, incleases cornputational tirne ancl r.aises decisio¡l problems ivith respect to the orcler. in rvhich existing holes ¿le processecl.

The Dietlich-Yakowitz algorithr¡'s shortcomings rest il tu'o main aLeas associated with zelo-deglee fits; namely, subsheet (hole) ualipulation ancl orientatiou of allocatecl pieces. Many holes cau be cleated on an indiviclual sheet in the course of the solution, and the rnanagernelt of these holes cal plesent a sizeal¡le ploblern. If mole zero-degtee fits ar.e allocated, the gr-eater. ivill be the nutrber- of holes generatecl and, hence, tnore nuûterous cornl¡inatorial possibilities for hole expansion rvill r.esult. The secord disadi,antage of the

Dietrich-Yakorvitz algorithm r.esults fi.oru heuristically flxing the orientations of the allocated pieces on the layouts. The modified algorithrn (MDY-UI), lvhich dicl tìot ltave this I'estliction ancl left a piece's orientation to be cleter- rnined by the random input clata, showecl that this factor is an important one in B.O.Ms with high APSA ratios. A fulthel disaclvaltage of the Dietrich-

Yakolvitz algolithrn is the selection, at the outset, of one fixecl rule to settle conflicts involving $'hich piece is to be allocatecl to rvhich of the available subsheets. These rules increase the pr.oblen-clependency of the algor.itlun alcl do not plovicle the flexibility neeclecl for. dealing rvith a B.O.M, that is changing clynamically in cornposition with the progr.ess of the algor.ithm.

62 CHAPTER 4

Search-Based Heuristic

The heul'istic ploceclure pr.oposecl in this chaptel to genet'ate cutting layouts

on stock sheets has been clesignecl after careful study of the'esults cliscussed in

the previous chapter'. The corlesponcling algorithrn rvas lv'itten to ernphasize the follorving three principles.

1. The paltitioning of a sheet irto subsheets must be contlollecl so that icleally

(a) ouly a minimal number of subsheets ale createcl, ancl (b) zero tr.in loss

Ìayouts exist for the cre¿tecl subsheets, given the current B.O.M. status.

2. The proceclur-e must be guided, at any ilstant, by the sizes of the pieces culrently rernailing unallocatecl on tlie B.O.M.

3. The of heu'istic clecision branch points the algorith'ric 'umbel' i' solution nust be rnini¡ral. The more the solutio'is shaped by heuristic decisions, the greater is the data-clependency of the algorithrn.

The fir'st ancl thi.d poirts are conside.ed inplicitly i. the algorithn by rnea's of a heru'istic tree search that is constrainecl to a pre-defi.ed suL¡set of the B.o,M. pieces.

This subset depencls on the dimensions of the sheet under culrent consideration. The second point implies that the procedure must be a clynamic one, adjusting itself to suit the instantaleous composition of the B.O.M. This clynamism is achievecl l:y a set of mles that give pieces varyilg priorities basecl on theil climensional cha¡acteristics.

63 The nerv algorithrn is telrned a "search-basecl" algorithrn because of its clepen- clence o' tlee sealch techniq.es to solve sub-proble'rs. A tlee sear.ch basically enumeLates alì the possible combinations that give feasible solutions to the pr.oblem at hand, alcl selects the one that best satisfies the objective. In cutting stock ploblems, this is equivale't to sea,..ching a1l possible pattelns that can be cr.eated on the stock sheet with the given B.O.M. pieces. For large B.O.Ms, such a procedule is viltually irnpossible clue to the enormous numbel of possible combinatiols, rvhich i¡- creases llear-exponentially rvith every additional piece incluclecl iu the enutlelatiol. situations in ivhich the nulnl¡e. of feasible patte''s is so la.ge that the co'rputa- tional time neerled to enumeLate all the alterlatives is atr.onomic¿l ar.e descr.ibed as " coubinatorially ex¡:losive". The algo.ithrn attempts to exploit tr.ee sear.ching as a tool fol obtainilg goocl solutions, rvhile r¡ilimizing the cornputational l¡urclen inherertly associatecl with search techniques.

The algoi'ithrn lias l¡een coded in Tu'bo-P'olog, a la'guage ideal fo. ar.tificial i.- telligence progranming. It takes a Bill-of-Mate.ials as inp't, and outputs a g.aphic iustruction of horv to lay the pieces on each stock sheet, together with the per.cent- age utilization for each sheet. A final, ovelall avelage utilization value cover.ing the total nurnber of sheets needed to fill the Bill-of-Materials is also proviclecl.

4.1 The Programming Language

Prolog was selected as the plograr:lning language in lieu of other. procedural languages, such as C or Pascal, for the follorving thr.ee reasons.

1. The algorithm makes extensive use of tr.ee sealch procedur.es and pr.olog, as

a language built for Ai'tificial Lrtellige'ce applications, facilitates tlie ivr.iting

a.d executio. of these search proceclu.es. P'olog has a l¡uilt-in backtr.acking

64 capability that allorvs database facts to be searchecl exhaustively for. nratches

in a path leacling to the objective. This bactracking capability is the main

charactelistic in tree search techniques.

2. The conputel pÌogram for tlie algor.ithm uses l¡oth cleclar.ative ancl pr.oceclu-

ral sul¡routines. The ploceclural sul¡r'outines ar.e those that irnplement the

heuristic rules, sort the clata, gener.ate the layouts ancì calculate t,he tr.irn

losses a.c[ other data required by the algorith'r. The clecla'ative sub.outi'es

query a clynamic database to select pieces fi'om candidates for allocation. The

database keeps track of available slieets ancl subsheets, as well as waste ateas.

The B.O.M. is also ¡ralt of the database. Ðach B.O.M. piece is stor.ed with its

rlimensious ancl curÌent clualtity as a clatabase pleclicate (fact). Prolog, rvith

its facilities for both cleclar.ative ând procedural pr.ograrnrning, is rvell-adapted fol the search-basecl algorithrn.

3. The prograrn is open-endecl, which mealis it can accept new {acts ol con-

straints, ol have old ones cleleted or. modified. The constraints rnay be of

the form such as no trvo pieces may be allorved togethel on the same sheet,

or certain pieces have to be placed at specific locations on the stock plates.

These constraints can be proglammed very easily in the {orm of rules in the

datal¡ase files. Other lules that may be used are priority r.ules. priority rules

ale clesigned to assign each B.O.M. piece a level of priority based on the irn-

por:tance of allocating that piece before other pieces. These priority levels help

to steer the solution along a desir.ecl path. Rules for. assigning the pr.iot.ities

cal be acldecl or deleted fi.orn the

Plolog provides excellent support for a dylamic database, rvhere r.ules rnay be

added ol removed tluring program execution.

65 4,2 The Prolog Algorithrn

The cleveloped ploglarr utilizes the deglee-of-fit concept of Dietr.ich ancl

Yakowitz (cliscussed il the p.evious chapter) as palt of an algo'ithrnic sea'ch for satisfactoly layouts. By constraining the search to those pieces that fit the clefinition of trvo-degree ol one-deglee fits, it is possible to contr.ol the scope of the search and avoid the combinatorial ex¡rlosion that rvould otherwise occul if ever.y available B.O.M. piece rvas included. The mechanics of the algorithn involve, if necessaly) the subclivision of the stock-sheet into smallel sul¡-sheets, to rvhiclr the search p.ocedure is applied. The sealch is more efficient when implernentecl on srnaller areas.

The algorithm can be classified as a pr.oblern reduction techlique. The search ailns to regulate the reduction pÌocess as rnuch as possible so that only one r.e- clucecl sul¡sheet is createcl at evely iteration. This sirnplifies the solutiol process ancl reduces the n.rnbel of alternate paths (or clecision branches) cleated at each reduction.

The basis of the search ploceclure is carriecl out by a loutine narned SEARCH.

StrARCH strives to achieve full utilization of a sheet rvith one-deglee and/ol trvo- degree fit pieces.

4.2.L fnput Data

The ploglam olganizes the input data as a set of facts rvithit the clatabase.

A pledicate clefined by current-piece(Id,L,.W,Q) is used to specify the B.O.M. pieces in the database. The variables L, W, ancl Q repr.esent, r'espectively, the length, rviclth and qualtity demancled of piece type " Id" . The stock sheet dirnensiols

t) f) X ancl Y ale irput as the predicate stock_size(X,Y ,CL,CZ). Hele, C1 ancl C2 ale

the cool'dinates of the upper left cornet. of the sheet. They ar.e used as a refeÌerce for the location of the stock sheet rvhen p.esenting g.aphic output. The p.eclicates

can l¡e moclified ol deletecl fi'om the clatabase rvhile the pÌogÌam is running. For exaurple, the preclicate current-piece(Id,L,W,Q) can be updatecl so that the variaÌ¡le "Q" is recluced by ole, upon allocation of ole piece of tvpe ',Icl,,. Both current-piece and stock-size preclicates are user'-definecl. Details of the relationships ancl liieralchies of the pleclicates used in the plogram, as rvell as the plogram cocle, ale inclucled in a separate clocument [44].

4.2.2 The SEARCH routine

The SEARCH routine is a tr.ee sear.ch confinecl entir.ely to those pieces of the

B.O.M. that can be described as either ole- or two-dimensional fits with lespect to the sheet under consicleration. A trvo-dirnensional fit occurs rvhen an order piece fits exactly into the available space on the stock-sheet ol subsheet. A one-climensional fit is rvhere only ole dirnension of an ol'der piece matches either one of the sheet ol subsheet's climensions.

The SEARCH routine explores the cor¡biratio's of one- ancl trvo-clir.ensional fits on the stock sheet and sealches for the best, namely the one ploclucing the least trin'r loss. An occurerìce of a two-dimensional fit will successfully finish the sealch ancl plocluce a uo-waste layor:t. Listances rvhele mole than one piece qualify equally as one-climensional fits serve as blanch points in the tree. In this case, the ploglall backtlacl

~~67 Figure 4,1: Example of Layout Composed purely of Ole_Degr.ee Fits~~

~~the exhaustive sea,-ch does .ot yield a layout havi'g a r00 To úilizatio., the best~~

~~solutiou achievecl in the corrrse of the tlee sear.ch is letur.necl to the main algorithrn by the call to the StrARCH subroutire.~~

~~Figure 4.1 shorvs a layout procluced by the SÐARCH routine. The layout rvas~~

~~constructecl in the alphabetical older of the lettels shorvn in each of the allocatecl~~

rectangles. Layouts generated by this routine are str.ictly cut patterns. 'rultistage The patter' of Figure 4.1 shows that, ivhen a full utilizatio'of a sheet does 'ot exist with only one a.cl/o. two-clegree fits, the layout will necessarily car.r.y an unfillecl

~~a.ea o'the sheet. This unusecl area alivays appears as a'ectangle at the uppe. right halcl colnel of the layout.~~

The SEARCH '-outine is'ot computationally bur.clensome because it stops once a zero trim-loss solution is founcl. If there are rnany qualifying pieces available fo. the sea.ch, the' the cha'ce to obtain a quicker. sorutio. is increasecl. I' fact, the gr'eater the numbe. of pieces available for a sea,-.ch, the quicke. a'cl rnor.e likely

~~SEARCH will achieve its goal. This behaviour is i'crirect cont.ast with that of othe. exhaustive p.oceclures, rvhere the scope of the sear.ch i'creases ,ear'-exponentially as~~

~~68 more alternatives are availabie at each branch point. On the other hand, if there~~

~~are fewer qualifying pieces available for the search, then SEARCH can perforrn a~~

~~cornplete enurneration with reasonable cornputer tirnes. The likelihoocl of StrARCH~~

~~encountering a combinatorially explosive situation is rninirnal in most cases.~~

~~SEARCH achieves its goal once a zero trirn-loss pattern is discovered, ancl~~

~~cloes not proceed to enurnerate alternative zero trirn-loss patterns. As a result, the~~

~~manner in which it carries out a search influences the composition of the selectecl~~

~~pattern. Two variations have been considered in the SEARCH subroutine. These are called SEARCH-A and SEARCH-Z.~~

~~In SEARCH-A, the first piece qualifying as a one-degree fit always has prece-~~

~~dence' This means that the subroutine will attempt to obtain a pattern containing as~~

~~many pieces as possible of the iniiially selected one-degree fit. Figure 4.2 shows the~~

~~layout resulting when SEARCH-A is implemented on the sample Bill-of-Materials~~

~~of Table 4.1 for a 10 x 10 unit stock sheet. The layout is characterised by rnany~~

~~sirnilar pieces of a few different types.~~

~~Table 4.1: Sample Bill-of-Materials ffZ.~~

~~Piece f Length Wrdth Quantity 1 10 1 5 .) 2 9 L) 2 o 3 7 ,f 2 4 6 1 ù 5 5 ù 2~~

In SEARCH-Z, a piece selected at one point is given the last prior-ity in selecting the one-degree fit candidates at the next point. Patterns composed by StrARCH-Z have a mole mixed make-up of pieces than those produced by SEARCH-A. This can

~~69 IU X 1~~

~~l0 x 1~~

~~10 x 1~~

~~L0 x 1~~

~~7x3~~

~~7 x3~~

~~Figure 4.2: Layout proclucecl by SEARCH-A l¡e seen in the patteur shorvl in Figure 4.3, rvhich was obtained by irnplementir.rg~~

~~SEARCH-Z on a 10 x 10 unit stock sheet for the sanple B.O.M. #2 clescribecl in~~

~~Tal¡le 4.1. By pre-solting the qualifying one-degree fits in order of decreasing area, the largel pieces receive priolity for inclusion in pattelns composecl by SÐARCH-~~

~~Z. Figure 4.4 illustrates the tlee seaLch enurneration and shows the solution paths tracecl by the SEARCH-A ancl SEARCH-Z sul¡r.outines.~~

The aclvantage of the SEARCH routine is that it allows a forward look to see the consequences of decisions rnade at certain points. Most other cutting stock heuristics arbitra.ily ernploy fixed .ules to resolve corfilicts betrvee' equally qualifyi'g candidates. For exarnple, if mol'e tlian one piece has a cornnon clirnension rvith the stock sheet, then o'e and only one rule, such as priority to the piece rvith the largest area, is usecl. This is the case in the Dietrich-Yakowitz algorithrn. Once a lule for lesolving couflicts is selected, that lule is irnplernented rigiclly thloughout the solution plocess. The SEARCH routi'es, in contrast, can backtrack a'cl leverse a clecision if it is founcl that the decision leacls ultimat,ely to a less efficient solutiol,

~~70 X~~

~~7 x3~~

~~6x1 3x9~~

~~x 3x5~~

~~Figure 4.3: Layout pr.oduced by SEARCH-Z~~

~~hr orcler to be effective, SEARCH rnust have at its clisposal pieces that cau provirle o'e-degree fits. SEARCH is less likely to fincl full fits for subsheets in cases rvhele:~~

~~o the number of diffelent available pieces decr.eases;~~

~~¡ the total number of unallocated pieces clecreases;~~

~~r the avelage size of pieces in relation to the stock sheet incleases.~~

~~4.2,3 Sheet Subdivision - the STRIPS r.outine~~

Wlien the SEARCH subroutine fails to find a 100 % utilization of the initial stocl< sheet, then that stock sheet is sub-divicled in a mannel such that conditions nore favourable to the successful functioning of SEARCH ar.e created. The STRIpS routine controls the nannel in rvhich the sheet is subcliviclecl. It is callecl STRIpS l¡ecause its output, actilg in conjunction rvith the SEARCH loutine, is a layout of a rectangular- segment along the eltire length of one of the sheet,s dimensions.

~~77 <- rssÌdual shssl d¡mÊnsions <- allocal€d pi€cê dimensions~~

~~SEARCH . A~~

~~. SEAIìCH, Z~~

~~Figure 4.4: Tlee Sealch Enumeratiol rvith SEARCH-A ancl SEARCH-Z sd s/ Sa Sc~~

S¿

~~(a)~~

~~Figu'e 4.5: Subdivision Possibilities foL (a) P horizo'tal, a'cl (b) p ver.tical~~

~~whe. callecl, the STRIPS .outine i'itiates the sub-division by selecti.g a sta't-~~

~~ing piece P for the lorver left colner of the stock-sheet. Figure 4.5 clemonstrates the~~

~~rÌarner in which subsheets aÌe generatecl as a result of allocating piece p at the~~

~~lower left-hand cor'er of the stock sheet. The subclivision possibilities rvhen p is~~

~~oriented ho.izontally are shown in Figu.e 4,5(a) rvhereas Figure 4.5(b) illustlates the~~

~~subdivisions rvlren P is oliented vertically. The 5ø ancl só ar.e the associated, sr t-~~

~~sheets of P in a holizontal oriertatior, ancl ,gd ancl ,ge are the associated, subsheets of P il a veltical or.ientatiol.~~

~~STRIPS uses the SEARCH routire in an attempt to obtain a zero tr.i¡r-loss for.~~

either one of tlie associated subsheets of piece P. Beginning rvith the subdivision preseltecl i' Figure 4.5(a), SEARCH is appliecl to sð. If SEARCH retur.rs a zero t.i'r-loss, then P ancl the layout for ,gl¡ are accepted a'd ,g¿ a'cl ,gc a'e combi'ed to fo'l a la.ger subsheet. The combined subsheet corstitutes the problem fo' 'eclucecl the follorving iteration. If SEARCH cloes not retur.n a zero trim-loss, the. the value that is returnecl is stored and sÐARCH is executecl similarly o' ,gø. If strARCH still cloes not return a zero tlirn-loss, then the proceclure is repeated for. tlie s.bsheets

~~Sd a.c1 ,9e associated rvith a orientation of p as illustratecl in Figur.e 'ertical 4.5(b). If SEARCH still cannot fincl a zer.o rvaste pattei.n a.fter pr.ocessing all the possible~~

~~sub-sheets, then the associated subsheet of P having the ìeast *'aste (expressecl as~~

~~percentage utilization of the subsheet's ar.ea) is selecte~~

~~selected layout has a zelo trim-loss ol not, the lernaining two subsheets are corll-~~

~~l¡ine~~

Thele ale trvo policies fol the selection of stalting piece P, The first policy employs a heulistic rule that gives priority to the available piece rvith the lalgerst surface area. This policy rvill be identified as the "Largest-Filst" (LG) policy. Tlie seconcl policy iuvolves searching through the cliffelent available piece sizes until a piece is fould for P that rvill procluce a zero-waste r.etur.n fron SEARCH fol any one of the four respective subsheets associatecl with tlie curr.ent piece P, If no zero- rvaste is achievable, the algorithm rvill have searched unsuccessfully through all the available cancliclate pieces for P. The selection the' goes to that piece rvhich gives the lowest percentage trim-loss for the testecl subsheets. Before sealching for. the stalting piece P, the available pieces ale sor.tecl in ol.clet of area, Thus the first chance to satisfy the STRIPS routine is alivays given to the larger pieces. with the second policy, identified as the "Extensive Search" (EXS) policy, the stalting piece does not necessalily have to be the largest available qualifying piece.

~~'Waste 4.2.4 Management of Areas - the EXPAND routine~~

Ulfillecl aleas occuling in accepted pattelns generatecl by the SEARCH routines ale put at the bottont of tlie list of available sul¡sheets. Whenevet'an itelation of the algorithm plocluces no allocation of pieces on a subsheet, that sub-sheet is r.emovecl frorn the list of available subsheets ancl loa

D are the x ancl y coolclilates of the area's upper left-hand comer'. The infol'mation coltained in the pledicate is sufÊcient to clefine the area occuppied by the ivaste alea ol the stock sheet. Though a waste area cannot accommodate, by clefinition, any available olcler pieces, it rray clo so if it is colnÌrined with othel pr.e-existing lvaste areas. Thele are trvo conclitions uncler rvhich such an integratiott can tal

The nervly integrated area (W3) constitutes a new subsheet and it is adcled to the list of available subsheets. Tlie rvaste ar.ea W4 shorvn in Figure 4.7 Lemains oll the rvaste-list as a permanent rvaste region and it is not considerecl in subsequent expansion trials.

Each tir¡e ân aÌea is acldecl to the Waste-List, a routine callecl trXPAND is invokecl to search the Waste-List fol areas rvith an expansion potential. Beginning rvith the first rvaste area on the list, ÐXPAND sea,-ches the list for.a seconcl area that has characteristics matching the two conclitions requilecl for a ho-,-izontal integlation. If no such a'ea is founcl, the proceclure is repeatecl for the second membe' of the 'rVaste-List, and so on. Once an integlation is accornplished, the r.outine terminates. If the entile waste-Lisi is trave'secl lvithout a successful irtegr.ation, trXPAND repeats the plocedure, but this time in sealch o{ ar.eas that can be in-

~~75 (a) (b)~~

~~Figule 4.6: (a) horizontally adjacent u,aste areas (b) veltically acljacent lvaste areas~~

~~(a) (b)~~

~~Figure 4.7: Exarrrple of Horizontal Integration~~

tegratecl vertically. If no aleas can be integratecl in any rnanner, EXpAND is terminated and the waste-List lem¿i's ulìaltel-ed. The policy of preferring horizou- tal i.tegrations ove. vertical o'es is a heuristic policy implernentecl to siuplify the functions and scope of the EXPAND routine.

Ð¿ch time a u'aste aLea is added to the database, the EXPAND routine atter'pts an expansion. The EXPAND loutine rvill not exPancl arìy areas that retain a clirnension less than the slnallest climension on the remaiuing B.o.M. pieces. Tliis plevents futile expansions.

~~76 4.3 Description of the Algorithrnic Itelation~~

~~Havilg iutrocluced the tlu'ee plincipal subroutines SEARCH, STRIPS and ÐX-~~

PAND, the sealch-based algolitlirn can now be geleralll' descr.il¡ecl rvith the aid of the florvcha.t shorvn i' Figure 4.8. The B.o.M. is pre-so.ted first in orcler. of cle- creasing piece a.ea. The list of availal¡le sheets, refellerl to as ssLIST, is loacled initially rvith oue fresh stock sheet.

~~The ite'ation corn'erces by rer'ovi'g the first sheet o' the ssLIST. This sheet seÌves âs the subproblem to be solved in the current iter.ation of the algorithrn.~~

The selectecl StrARCH uoutine is ru. for the sheet as a rvhole a.cl, if a full fit of pieces rvith no trin.l losses can be alranged on the sheet, the sheet is consiclered processecl ancl a new iteration begins using the next sheet heading the ssllsr. If

SEARCH fails to a.rive at a full utilizatio., the algorithrn tu'rs to STRIpS to select a sta.ting piece. The outcone of the STRIPS routine is a layout for a str.ip- like segme't exte'cli'g thefull length ofone of the sheet's sides. srRIpS, i' effect, clecicles the orientation of the stalting piece ancl the rnanner in rvhich the sheet is partitione

unfilled areas c.eated during the run of STRIPS are classifiecl as subsrreets and irnmediately put at the bottorn of the SSLIST. The SSLIST per.forms o' the l¡asis of a "first-corne first-serve" queue of availal¡le subsheets at any instant. Cases in rvhich STRIPS fails to allocate any piece result in the tlansfer of the subsheet uncle¡ consideration from the SSLIST to tlie waste-List. Neivly c'eatecl rvaste aleas ar.e put at the top of the waste-List, and the trxPAND routine is r.un irnmediately to check rvhether the nerv u'aste alea cal be integlatecl into pleviously existilg waste aLeas to c'eate nerv, pote'tially usable subsheets. An itelation ends rvhen the subsheet is

~~(( START~~

~~SEARCH~~

~~STRIPS~~

Figure 4.8: Florvchart for the Search-Based Algorithrn processed into either (1) a full layout rvith no trim losses; (2) a partial layout rvith unfillerl aleas usal¡le in the forr¡ of r.educecl subsheets, or (B) a rvaste area. Subse- quent itel'ations follorv in a similar' rnanneÌ, ahvays starting tvith the slieet heaclilg the SSLIST, ancl continuing as long as B.O.M. pieces r.emain unallocatecl. Wllen- evel the SSLIST is empty, a nerv stocl< sheet is started. The Bill-of-Materials is updatecl itstaltaneously with every allocation of an olclel piece, ancl the algor.ithrn ter¡linates irarnediately upon allocation of the final B.O.M. piece.

~~4,4 Evaluation of the Search-Based Algorithm~~

The algorithrn just desc.il¡ed rvas tested i'itially usi'g the flr'st seve' categor.ies of Data set S 1 of ref. [25]. The eighth category, in rvhich the largest or.cler. piece does 'ot exceecl 11250t'h, of the stock sheet size, was omitted fi'om the tests. (The RAM lequired for some cases il this categor.y exceedecl the available computer capacity. In any case, the category does not lequir.e much attention because a rtear-optirnal 99.99% stock sheet utiliz¿tior has been achievecl [25].)

The search-based algorithrn was ÌurÌ four separ.ate times for. each category, each run irnplementing one of the two variations fo' the SEARCH routine, togethel rvith one of the two policies usecl for selecting the lorvel left cornel piece. (see sectio. 4.2.3.)

~~For pur'¡roses of clarifrcation, the foul clifferent heulistic policies are identifiecl as follorvs :~~

~~1' Heuristic LG-A is one in *'hich the La.gest-Fi'st (LG) policy is used in conjunction with the SEARCH-A version of the SEARCH routile.~~

~~79 2. Heulistic LG-Z also implements policy LG but, in this case, SEARCH-Z is~~

~~usecl.~~

~~Heuristic EXS-A uses the extenclecl search policy together. rvith SEARCH-A~~

~~Heulistic EXS-Z implements the extenclecl search policy rvith SEARCH-Z.~~

~~The test results for each of the four clifer.ent heuristics are given in Table 4.2,~~

The values giverl aÌe avelage pelcentage utilizations of stock sheets in each categor.y, together with the stancla'-cl cleviations rvithin inclividual categolies. The categor.ies ale no lolgel identified in Table 4.2 by the maximurn per.missible size in a B.O.M. but, rather, by the average piece-to-stock ar.ea (APSA) ratio for the B.O.Ms comprising the category. This appeals to be a better categor.ization because it gives a clearer indication of the dimensional chalactelistic actually defiling the category.

~~Table 4.2: Test Results for Four Valiants of the Search-Based Algoritlun~~

~~Category Avelage LG-Z LG-A EXS-Z EXS-A~~

~~Number Piece/Stock Average % std Average % std Average % srd Average % srd Area Ratio utilization dev. utiüzation lev utilization dev. utilization dev. (APSA) (r) (") (ã) (") (r) (") (r) (o)~~

~~I 0.446 77 .5 4.98 77 .5 ,l oç 74.3 +. D.l 74.',ì 4 65~~

~~2 0.284 82.6 5.70 82.6 5.70 6.90 6.91 3 0.154 01 t 2.54 91.3 tAa 97.2 91.1 3.34 4 0.067 96.2 i.35 96.2 96.5 1.46 97.6 1.10~~

~~5 0.026 97.5 2.10 97 .7 2.05 98.1 1.74 98.6 1.51 6 0.014 98.0 2.64 oa t 2.56 07 (]¡ 2.98 98.5 2.72~~

~~7 0.007 9 9.3 1.05 oo ¿ 1.10 99.0 2.77 99.3~~

~~80 4.4,L Test Results for the Search-Based Algorithm~~

~~Test results indicate that two of the four tested variants, namely EXS-A and~~

~~LG-A, are superior generally in terms of sheet utilizations. LG-A is more efective~~

~~for the categories with larger average order piece sizes, while EXS-A produces less~~

~~waste in categories rvhere the average size of the orders on the B.O.M. is less than~~

~~10 % of the stock sheet size.~~

~~A comparison of these two dominating heuristics, using as refelence the results~~

~~f¡om Dietrich-Yakowitz [25], is shown in Figure 4.9. The reference curve is compiled from the best values ol¡tained in each category in six independent runs. Each run~~

~~uses one of the six heuristic ruies defined in Dietrich-Yakowitz l25l and discussed in section 3.4. Figure 4.9 reveals a supeliolity of the LG-A policy in the first two~~

~~categories, and of the EXS-A policy in the last four categories. The LG-A and~~

~~EXS-A policies, together rvith the teference, give nearly identical results in the~~

~~third category. EXS-A is noticeably the worst of the heuristics for categories 1 and~~

~~2. The LG-A results closely ap¡rroximate those of the reference in categories 4~~

~~through 7.~~

The standard deviations betrveen the results of individual problems in each category is a relatively useful parameter. to compare diferent heuristic rules. In general, a lower average standard deviation indicates greater consistency and less

data-dependency, Too much emphasis on the significance of this parameter should be avoided, holever, because it is conceivable that oear-optimal lesults may have lower average standald deviation in many instances than the optimal. The comparison of the ÐXS-A and LG-A heuristic policies with the Dietrich-Yakowitz reference given in Figure 4.10 shows lower average standard deviations in the reference heuristic for the majority of tested categories. The EXS-A and LG-A heu¡istics have

~~B1 Fig.4.9 Comparison of LG-A and EXS-A Policies with the Reference~~

~~òe c .9 (ú .N oo '..E f\, = o) Q) .E ct)~~

~~Average piece/stock area ratio @ higher average standard deviations in the categories with the smallest APSA ratios.~~

EXS-A, on the other hand, has the highest average standard deviation in fou¡ of the seven categories. An average standard deviation that is signiflcantly higher for one heuristic, as in the EXS-A policy for category 2, indicates the existence of two possibilities. These are either (1) the heuristic performs exceptionally well for some particular problems in the tested set, or (2) the heuristic gives poorer than average results in some of the test problems. In any event, an unusually high average standard deviation in test problems of a similar class warrants further investigation to uncover the causes of the higher variability.

~~The average computational time per problem on a 486, IBM compatible microcomputer for the two dominant heuristic policies in each test category is shown in~~

Figure 4.11. The EXS-A policy generally demands more cornputer time. This demand is particularly notervorthy for high APSA ratios, where the algorithm spends time searching more numerous stock sheets for zero-rvaste subsheet partitions. The two heuristics require nearly equal computer time for problems with low B.O.M.

~~APSA ratios. Tliis is attributable primarily to the fewer number of stock sheets needed to fill the B.O.Ms making up these categories. Consequently less searching is demantled.~~

~~4,4.2 Analysis of Results~~

~~The results in Figure 4.9 show a weaker performance by the Extended Search variants of the algorithm for the first trvo categories that have a relatively high APSA~~

¡atio, This can be explained by the Extended Search's main feature, namely the rigorous pursuit of obtaining minimal trim-loss on the current sheet or subsheet, The consequences of this approach, for category 1 and 2 problems, is that the smaller pieces in the B,O,M,, rvhich should serve idealiy as "f.ller" pieces, are favoured

~~83 Fig. 4.10 Comparison of Standard Deviation of LG-A and EXS-A Results~~

~~oc '=(ú q) -õo @ .It(ú (úc ct)~~

~~o.os o-io 0.1s 0.20 0. Average Piece/Stock Area Ratio $g. a.11 Comparison of Computation Time for LG-A and EXS-A Hêuristics 500-~~

~~450-~~

~~400-~~

~~350-~~

~~300- 6'(J (¡) co I zso- ür c) E F 200-~~

~~150-~~

~~100-~~

~~50'-~~

0r 4 Category number ear-lier in the process because of their greatel combinatorial capacity. The net result is that the Extended Search heuristics will end up producing good layouts composed of small pieces for the first stock sheets processed, rvith poorer utilization of the later stock sheets due to the larger pieces predominating among the remaining pieces on the B.O.M. On the other hand, results for categories 1 and 2 using the Largest-Filst policy are very satisfactory, mainly because the largest pieces are allocated earlier in the process, with the smaller pieces serving principally as fillers for the areas left unfilled by the larger pieces. The net result is generally a fewer nurnber of stock sheets needed to fill the B.O.M., and consequently a lower trim-loss.

In category 3, where the largest size in the B,O,M. does not exceed 25% of the stock sheet size, the results for the Largest-First and Extended Search policies are nearly identical. This is partly due to the fact that, as the sizes of the pieces are

¡educed in relation to the stock sheet, the distinction between pieces that warrant priority on account of their size and those that would serve better as "filler" pieces becomes less clear. The main diference between the results of the two policies for this category involves the standard deviations. The standard deviation in the

~~EXS policy is approximately 35% gleater than that for the LG policy. This is an indication that this particular category still poses difficulties for the EXS policy.~~

~~The dificulties are related to the priority requirements of the largest pieces on the B.O.M.~~

In categories 4, 5 and 6, the area of the order piece practically ceases to be a signifrcant factor in the solution process. Consequently, tlie EXS policy emerges as superior because of its higher efficiency in searching for localised minimal rvaste combinations on each individual stock sheet or processed subsheet.

~~86 The superiority of the EXS policy over the LG policy begins to fade in category~~

~~7 and, to a lesser extent, in category 6. Though the difference in the utilization~~

~~results for the two policies is only slight, ihe EXS results have a standard deviation~~

~~that is more than twice that of the LG results. The rveakening of the EXS policy~~

~~can be traced to problems due to pieces on the B.O.M. having very lorv aspect ratios~~

~~(i.e. pieces that are very long and thin). Such pieces are very likely to be rejected~~

~~by the EXS policy in the early stages of the solution because their lengths limit the~~

~~chances of their satisfying the STRIPS routine. These long pieces are also more~~

~~difficult to allocate on the subsequent smaller sul¡sheets of the original stock piece.~~

~~The result is tliat these pieces remain to be alloated until the end, when a nelv stock~~

~~sheet is usually required to accommodate them. This produces substantially larger~~

~~trim-loss. With the LG policy, the problem is not serious because the offending~~

~~pieces generally receive priority on the strength of the size of their surface areas.~~

~~4.4,3 Conclusion~~

~~The search-based algorithm appears capable of finding good solutions to cutting stock problems with B.O.Ms containing pieces that have APSA ratios ranging from~~

0.005 to 0.450. Tests have revealed that the trvo heuristic policies for controlling the execution of the algorithm difer in their efectiveness depending on the APSA ratio. The poiicy that recognizes priority based on the area size of the order pieces is stronger for B.O.Ms in the higher APSA ratio range and rveaker for those with lorv APSA ratios. The second policy, rvhich strives for localized minimal rvastes on subsheets of the stock sheet, demonst¡ates the opposite behaviour. The trvo policies, howeveL, overlap in their efectiveness for B.O.Ms ivith APSA ratios that are near the median of the range, betrveen apploximately 0.05 and 0.20.

87 The above conclusion set the stage for a rule-based system to direct the execution of the algorithm in the dilection of the policy that best serves the current B.O.M., using the B.O.M's APSA latio as the guiding parameter. Ideally, the rule-driven system rvould be expected to behave closer to the LG policy for problems rvith

B.O.Ms of higher APSA ratios, and more like the EXS policy for B.O.Ms rvith lower APSA ratios. The effect of rules in controlling the algorithm to satisfactorily cover a rvide range of size and piece mixes in the B.O.Ms are discussed in the following chapter.

~~88 CHAPTER, s~~

~~Analysis of the Search-Based Algorithm~~

This chapter clescribes and evaluates methods for improving the search-based algoritlim described in the previous chapter. The objective is for improvement in two areas. First, it is desired to achieve consistently good performance regardless of tlie B.O.M's composition. Second, it is sought to further enhance stock sheet utilization. Three different methods for running valiations of tlie algorithm are investigated rvith a view to achieving these objectives.

The first approach involves compound heuristics, whereby all four heuristic rules defined in section 4.4 are run separately, The best result from the four runs for each individual problem serves as the solution to that problem, The second method introduces threshold values for controlling the ryaste levels that can be tolerated in the solutions produced by the SEARCH subroutines. Finally, the ihird method suggests priority rules that aim to control the algorithmic execution so that the most appropriate procedure, based on the predominant shapes and sizes making up the

~~B.O.M,, is irnplemented.~~

The results derived from the three methods are to be used in designing the final form of the search-based algorithm. Data from this final algorithm are then compared with that {rom previously publishecl heuristics.

~~89 5,1 Solution by Compound }leuristics~~

~~This approach calls for the separate running of all four heuristic variants (LG-A,~~

~~LG-Z, EXS-A and EXS-Z) of the algorithm desclibed in section 4.4, and selecting the~~

~~best outcome as the solution for the given tesü problern. The major disadvantage of~~

~~this technique is obviously the escalation in computational time which can increase,~~

~~for example, upto an avarage of 20 minutes lor problems defined in category 1 o{~~

~~test data set f1 published in Dietrich-Yakowitz [25]. The results that rvould be obtained if the compound heulistics approach is implemented for data set f 1 of~~

~~ref. [25] are given in Table 5.1. Appendix A details the four heuristic results for the individual problems, where the best result for each test problem is listed under~~

~~the column headed "Compound Heuristics.". The average utilization percentage~~

~~and standard deviation from the compound heuristics appÌoach for the seven tested~~

~~categories are shown in Table 5.1,~~

~~Table 5.i: Results for the Compound Heu¡istics Run.~~

~~Cat. 1 Cat.2 Cat. 3 Cat. 4 Cat. 5 Cat.6 Cat.7~~

~~Ave. Utilization 77.6 83.0 92.3 97.9 99.0 oot 99.8~~

~~Std. deviation 4.99 5.67 2.52 0.76 1.24 r.47 0.49~~

Fig. 5.1 shows a graphical comparison between the results of the compound heuristics approach with those obtained by the EXS-A and LG-A variants of the search-based algorithm. The compound heuristics' superiority over both LG-A and

~~EXS-A is most noticeable for problems with APSA ratios of between 0.10 and 0.20.~~

~~Belorv 0.10, the compound heuristics produce results that are only slightly better~~

~~90 than EXS-A. The LG-A results approach those ofthe compound heuristics for APSA ratios above 0,20.~~

~~5,2 Implementation of Thresholds~~

The SEARCH subroutine is limited in the choice of pieces that can be allocated by the constraint requiring that only one- or two-degree fits be considered. This restriction may cause SEARCH to form layouts that are inferior to other alternatives.

This weakness is more pronounced when the choice between different pieces remaining to be allocated is limited to only a few sizes. An example of the efects of the restrictive nature of SEARCH is illustrated using the sample B.O.M. f3 presented in Table 5.2. Table 5.2: Sample Bill-of-Matelials f 3.

~~Piec< Length widrh Quantity~~

~~A t2 4 2~~

~~B l1 4 2~~

~~C 10 I I~~

Figure 5.2 (a) shorvs the layout produced by the SEARCH-A routine for sample B.O.M. # 3. The layout carries a trim-loss of 44 units. An alternative layout with a trim-loss of 18 units is shown in Figule 5.2 (b). The SEARCH routines, on the other hand, cannot produce the layout of Figure 5.2 (b) because piece C constitutes neither a one-degree nor a trvo-degree fit on the 11 x 8 square unit subsheet depicted as rectangle abcd. Consequentl¡ SEARCH-A favours the seconc{ type B piece for allocation because it satisfles the one-degree fit restriction.

~~91 Fig.5.l Comparison of Compound Heuristics wirh LG-A and EXS-A~~

~~\oo\ .o (ú è (.o t9 = (l) (¡)~~

~~-cU)~~

~~Average piece/stock area ratio B~~

~~(a) ûìreshold = 1.0 (b) threshold = 0.85~~

~~Figure 5.2: Layouts attainable rvith diferent threshold parameters~~

~~The SEARCH routines can be rnade to include patterns such as that of Figure~~

5.2 (b) in their tree searches by simply relaxing the definition of what constitutes a two-degree flt. This relaxation is effected by the use of lower threshold,limits on the dimensions of the pieces qualifying as a two-degree fit. The threshold defines the minimum percentage of the lengtli of the subsheetts dimensions which must be matched by an order piece for that piece to qualify as a two-degree fit. This concept is illustrated in Figure 5.2 (b), rvhere piece C, under a threshold setting of 0.85, can be allocated within the prescribed region abcd because its length is greater than

~~0.85 x the dimension a-d., and. its wiclth is also greater than 0.85 x the dimension a-b. If the threshold was set, instead, at 0.90, piece C woulcl not be allocated.~~

The threshold value of 0.85 permits layouts of pieces that may introduce localised trim-losses of upto 19 % (I00 - 0.92) for the subsheet in the case illustrated. Piece C is considered a tivo-degree fit in fig. 5,2(b), and the processed subsheet is assumed by SEARCH to have met the condition for zero trim-loss, even though there is, in fact, a wastage associated rvith the allocation of piece C, However, despite a de facto two-deglee fit, the SEARCH routine does not terminate at this point but, rather, continues searching for zero trim-loss layouts. If no such layouts

93 Figure 5.3: Layout of Sample B.O.M. f 2 are possible, the layout rvith the least wastage recorded during the search with the existing threshold limit is selectecl. The overiding priority in the SEARCH routines for a Ìayout involving purely one or two-degree fits can be illustrated further by considering a slight modification to the sample B.O.M. #3. If the B.O.M. demanded three units of the type B rectangle iristead of the current two, the output of SEARCH-A rvould be that shown in Figure 5.3 regardless of ihe tlireshold limit.

~~This is because a solution like the one depicted in Figure 5.2 (b) is accepted only on a tentative basis during the process, pending confirmation that no zero trim-loss layout can be formed by SEARCH.~~

Thresholds improve the output of SEARCH by allowing a larger number of patterns to be considered in the tree search, albeit at a cost of increased computational tìme. The threshold limits rnay range from 0 to 1. A zero threshold allows no allocations whatsoever, and the subroutine will return a 100 % trim-loss. A threshold of one corresponds to no effect on the original algorithm, where a trvo-degree fit must have both dimensions exactly matching the two dimensions of the subsheet.

~~94 Results obtaiued fi'om running the LG-A variant with various threshold values for the test problems in the seven categories of ref. [25] are presented in Table 5.3.~~

~~They are also depicted graphically in Figures 5.4 and 5.5.~~

~~Table 5.3: Results for Runs Under Diferent Threshold Settings~~

~~THRESHOLD SÐTTING 0.80 0.82 0.85 0.87 0.90 0.92 0.93 0.95 0.97 1.00~~

~~Cat. Ave. Util. 77.4 I t .:'t I I -al 77.5 77.5~~

~~1 std" dev" 5.01 5.02 5.03 5.02 5.02 4.98 Cat. Ave. uril. 82.6 83.0 83.2 83.0 82.7 82.6~~

~~2 std. dev. 5.86 5.94 5.87 5.83 5.75 5.70 Cat. Ave. uril 91-l) 91.6 91.8 91.8 91.7 91.3 ð std. dev 2.24 2.15 2.L2 2.19 2.2s 2.49 Cat. Ave. util. 96.0 96.3 96.3 96.4 96.5 96.5 s6.2 96.2~~

~~4 std. dev. 1.34 i.35 1.38 r.39 1.43 1.46 1.53 1.33~~

~~Cat. Ave. uril. 97.6 97.8 97.9 98.0 s7.9 97.8 97.7 5 std. dev. 1.85 1.63 1.96 1..67 t-u/ 1.86 2.05~~

~~Cat. Ave. uti1, 98.3 98.5 98.3 98.4 98.4 98.1 oat 98.3 ôat tl std. dev" 2.03 2.11 2.34 2.07 1.84 2.56 2.37 2.38 2.56~~

~~UAT. Ave. IIril. 98.7 99.0 99.1 99.3 99.3 99.3 99.4 99.4~~

~~7 std. dev. 1.51 1.40 1.31 1.11 i.10 1.13 1.10~~

~~The results of Table 5.3 indicate no significant improvement in the overall average percentage utilization in categories 1 and 7, despite lorver threshold lirnits.~~

The improvement is slightly more discernible in categories 2 through 6, tvith a peak improvement of about 0.5% in categories 2 and 3 for tliresholds around 0.87. The results also point to a relation betrveen the threshold lir¡it and the ApSA ¡atio of the B.O.M. B.O.Ms with medium APSA ratios (in the range of approximately 0.10 Fig. 5.4 Effects of Differenr Thresholds on LG-A Results 92-

~~90-~~

~~88- s c 86- .9 (ú .F.N s¿- l q)"-a) R?- U)~~

~~ï 0.78 0.86 0.90 0.98 0.80 0-84 0.88 0.92 1.00 Threshold Limit Fig. 5.5 Effects of Different Thresholds on I-G-A Results~~

~~òe .9 (ú .N (o *J :l q) (I) U)~~

0.88 Threshold Limit to 0.30) are served betier by threshold limits between 0.85 ancl 0.90. Thresholds above 0.90 are more favourable for B.O.Ms with higher or lower APSA ratios. These conclusions, of course, relate only to the test dat¿ used,

In conclusion, the implementation of threshold limits àppears to have only a marginal effect in lowering trim iosses. A threshold of between 0.9 and 0.92 seems to be optimal for the set of problems tested here. Threshold values below 0.9 would have to be implemented rvith more care because of the greater inherent waste levels introduced by the pieces entering the layout fo¡mations.

~~5.3 Priority-Based Control of the Algorithm~~

The final attempt at improving the basic search-based algorithm stemmed from the need to develop a single heuristic that will produce results similar to those obtained by both the trXS-A and LG-A heuristics over the rvhole range of categories tested. EXS-A was observed to be more efficient for categories 4 to 7 than for the first trvo categories, whereas LG-A's performance rvas opposite. Therefore it lvas decided that the new heuristic policy for controliing the algorithm be designed to take this fact into account. This meant that LG-A should be emulated in cases whe¡e the APSA ratio is high. Likewise, the algolithm's l¡ehaviour should approach more closely that of the trXS-A heuristic in problems with lolver APSA ratios.

Prol¡lems rvith APSA ratios that l¡order between the category extrernes pose rìo major difficulties. The final utilization results from l¡oth EXS-A and LG-A in this area are approximately equivalent,

~~The effect of giving priorities to specifrc piece types is to divide the B.O.M. between what can be termed "critical" pieces and "filler" pieces. The critical pieces~~

98 are those rvith priority for early allocation in the solution, while filler pieces serve to fi1l the unused spaces remaining after the critical pieces have been assigned. If no priorities are in effect, the algorithm's tendency will be to seek a partitioning of the current sheet into two distinct rectangular regions; one frlled completely by as large and complete a layout of pieces as possible, and the other remaining unfilled.

This could result in layouts favouring fi1ler pieces in the initial allocations. The critical pieces rvill likely remain on the B.O.l\4. for later allocation because they can provide only a small number of combinations rvith othel pieces on the same stocl< sheet or subsheet. This situation, which almost always results in an increase in the total number of stock sheets consumed to satisfy the B.O.M., is avoided through prioritization of specific piece sizes. This prioritization forces the early allocation of the critical pieces. The net result is that higher trim losses will be tolelated in the initiai sheets, in the interest of a lower overall trim loss after the entire B.O.M. is allocated.

The rnost practical method for combining the procedural elements of the trXS-A and LG-A heuristics into a single algorithm is to employ one or more priority rules that control the solution for a given subsheet. Four rules are suggested here for achieving this. Each piece of the Bill-of-Materials is given a priority rating ranging from first to fourth priorit¡ or no priority at all, depending on the dimensional characteristics of the piece in relation to the original stock sheet, These four rules are explained in more detail in the following subsection.

~~5,3.1 Assignment of Priorities~~

Priorities are assigned to inclividual pieces on the basis of four selected rules. The priorities ai-e ranked from 1 to 4 it decreasing order. of importance. A piece rnay be assigned more tlìan one priolity if it possesses the qualifying characteristics.

~~99 The allocation priorites are assigned according to the follorving rules~~

~~o Rule 1~~

~~If a piece has both dimensions greater than 50% of both stock sheet dimensions,~~

~~then that type of piece receives first priority.~~

~~a Rule 2~~

~~If the area of a piece is greater than ol equai to 10% of the size of the original~~

~~stock sheet, that piece is granted second priority status.~~

~~e Rule 3~~

~~If a piece has one dimension that is greater than the lesser of the tlvo stock sheet dimensions, that piece is given third priority.~~

~~o Rule 4~~

~~If the longer dimension of a piece is greater than 75% of the length of the~~

~~longer dimension of the stock sheet, then that piece is assigned fourth priority.~~

~~The reasons for the selection of these particular rules are as follows :~~

~~1. Rule 1 gives top priority to the type of piece which cannot be laid together~~

~~rvith anothe¡ of its type on the same stock sheet. The number of pieces of this~~

~~type in the B.O.M. sets the lower bound on the minimum number of stock sheets lequired to satisfy the B.O.M.~~

~~2. Rule 2 is based on a judgment that pieces of area greater than 10% of the stock~~

~~sheetts area have diminished chances of allocation in subsequent iterations on~~

~~the same stock sheet.~~

~~100 3. Rule 3 is desiglecl to allocate ear.ly, those pieces that car be laid out onÌy in~~

~~one orientation. Such pieces limit cornì¡inatorial possibilities ancl ale þetter.~~

~~ricl of quickly il'hen the B.o.M. has mo.e pieces of diferent sizes that can act~~

~~as fillels for the single-oriertation pieces.~~

~~4. R'le 4 is basecl on the observatio¡r that relatively long pieces can be difficult to~~

~~allocate in subsheets ancl rnust be dealt rvith preferably ir the fir'st iter.ation(s)~~

~~oll a llelv stock sheet. Pieces coverecl by this priority are usually long aucl thil and, iu the al¡se'ce of this rule, ivould probably fail to qualify u'cler. a'y of~~

~~the first tht'ee rules because of their small areas.~~

The algorithrn's l¡ehaviou' under these p.iority rules is straightfor*'ard. If, at any tirne, a piece carlying aly pliolity status is available ancl can fit into the space on a current sheet ol subsheet, then the algorithm rvill allocate that piece, ancl the layout on the sheet lvill be forrnecl in a l¡anner similar to that follorvecl by the

LG-A heuristic. If.one of the pieces qualifying for allocatior caÌr,y arìy priority, ther the EXS-A heul'istic policy clete'rni'es the layout. I' case of r¡ore than one qualifying piece having a priority, the piece rvith the hige. r'arkecl p'iority is given precedence. If trvo pieces calrying the same plio.ity ratki.g qualify for. allocation, the piece rvith the large. surface a'ea is favourecl. Therefo.e, the prior.ity-basecl version of the algorithm uses a heuristic tech.ique that is a hybrid of both the LG-

A ancl EXS-A proceclures. The result is that the p'inciples behind both the LG-A and EXS-A heuristic policies can be utilized i' fo.mi'g layouts on the s¿me stock slieet. The deglee to which each policy influences the layout clepencls on the priority status of the pieces in the B.o.M ard the size of the subsheet under consideratiol.

Fo. exa'rple, Rules 1 a'cl 2 ensu.e that the eÌer¡e'ts of the LG-A pr.oceclure have a gleater influence rvhen the B.o.N4. contains pieces that ar.e la.ge in relation to the stock sheet. In this rìa¡111s¡, the algorithrn is al¡le to tu',e the proceclrrre i.

~~t01 order to.elate ili'ectly to the cly'arlically chalging cornpositio' of the B.o.M. as the solution proglesses.~~

~~5,3,2 Test Results~~

Tlie p.io.ity-rule version of the algo'ithrn rvas testecl usi'g, once again, the same set of sample p.oblerns (clata set f i of ref. [25]). The .esults are given i' Table 5.4. They a'e co'rpa'-ecl rvith the LG-A ancl EXS-A heu'isitc r.esults obtainecr in section 4.4.

~~Table 5.4: Results of Priority-Rule Algolithrn Dietr.ich-yakorvitz Data Set g1~~

~~Category Average Pliolity-Rule LG-A ÐXS-A~~

~~Number' Piece/Stock Average % std Average 76 srd Avelage 76 Stcl APSA utilization clev. utilizat,ion dev. utilization dev. Ratio (') (") (¿) (") (r) (o)~~

~~1 0.446 77.9 4.95 77.5 4.98 4.65~~

~~2 0.284 83.2 82.6 5.70 ¡ i.,) 6.91~~

~~3 0,154 91.3 2.50 91,3 2.49 91 .1~~

~~4 0.067 97.5 1.09 orî t s7.6 1 .10~~

~~i) 0.026 ool 0.75 97.7 2.05 98.6 1.51 6 0.014 O0 zl 0.92 98.2 2.56 98.5 ,7.) 7 0.007 99.5 7.14 99.4 1.10 99.3~~

The lesults in Tal¡le 5.4 ale illustrated glaphically il Fig. 5.6. The pliolity- basecl of t'he algorithrn is clearly superior to 'e'sion both LG-A a.cl EXS-A, con- fir'rning the advantage of the hybrid rule-basecl approach.

~~r02 Fig. 5.6 Comparison for Priority-Rule Algorithm~~

~~àS c .9 (ú .N = (l) (I) -c U)~~

~~Average piece/stock area ratio @ 5.4 Final Design of the New Algorithrn - A priority-Rule Method~~

The'esults from the p'evious section i'clicate that the irnpleme'tatio' of pr.ì- oritJ' rules to the trXS-A algolithrn are sufficient to achieve a control in rvhich the solution path traced by the algo.ithm can l¡e lelated to the illstantaneous composition of the B.o.M. The four rules for assig'i'g priorities helpecl achieve the objective of combining the favour.able char.acteristics of tlie EXS-A and LG-A policies. consequertly, the sa're fou' rules rvill l¡e usecl to test the hyb'id algor.ith'r, called the "p'iority-r'ule algo.ithrn". Testing will be pe.founecl using the same sets of test problerns employecl to obtain previously publisliecl solutions. The r.esults ar.e cornpared, in each case, rvith previously publishecl clata.

The priority-r'le algorithrrr is modified slightly so that the SEARCH routire is uot pelforurecl on newly stalted stock sheets. Thefirst algoritlunic iter.ation o¡ anerv stock sheet bypasses the SEARCH routine ¿ncl proceecls, i'steacl, to trre srRlps routine. The justification for this proceclure is that, the SEARCH loutine, with its stringent sealch fol a zeLo-waste, will likely cleate patte'rs cornposed of rnany srnall and sirnilal pieces on the same stock sheet. Pieces are consurnecl, ther.efore, that rvould selve bette. as "fillers". Also, the requirer:re.t il the STRIpS loutine for a

~~100% utilization fol an associatecl sul¡sheet is clorvngr.aclecl slightly to gg.b%.~~

~~5,5 Testing of priority-based algorithrn~~

~~The priority-based algo'ithrn, usi.g the fou' selected .ules, is e'aluatecr in trre follo*'ing sectio's by compa.i'g its pe.founa'ce rvith four available proceclures. The~~

~~104 Fig.5,7 Results from Priority Rule Algorithm for Three Test Daia Sets '.-.R_~~

~~95-~~

~~àe .õ e0- (tf È O f 8s- õq) -c U)~~

~~8G.~~

~~75- 0 .0 0.1 0.2 0.3 0.5 Average Piece/Stock Ratio coÌ¡parisorì is l¡asecì upon the results achieved by the pr.iolity-rule algor.itlun fol the~~

~~sarne test clata utilizecl i' the respective publicatio.s for the other. pr.oceclur.es.~~

~~5.5.1 Compalison with Dietrich-Yakowitz (1S01)~~

~~The priority-basecl algo'ithrn is ru. fo. all th.ee data sets usecl by Diet.ich ancl~~

~~Yakorvitz in their publication [25]. The'esults are cornparecr in Tables 5.5 through~~

Table 5.7 with those published by Dietrich ancl Yakorvitz. The format for.presenting the results is the same as that used in ref. [2b]. Figure b.z gives a g'aphic prese.- tation of the utilization results fo'the three testecl rlata sets as calculated by the priority-r'ule algo'ithrn. The ave'age cornputatio'al time per problem in eacli of the testecl categories in tlie test data is provided in Table 5.8.

~~Tal¡le 5.5: Result Cornpalison ivith Dietrich-Yakowitz Data Set fJ~~

CATEGORY ALGORITHM 1.000 0.500 0.250 0.100 0.040 0.025 0.010 q x: o t o q o t o Dietrich-Yakowitz 7 5.5 5.02 80.4 5.45 91.l 2.16 l5. f 1.62 97 .7 1.40 98.6 0.87 100.0 0.03 Priority-rule AJ gorithm 77.9 4.95 83.2 91 .3 2.50 97.5 1.09 99.1 0.75 99.4 0.9 2 99.5 1.14 2.4 2.8 0.2 i.6 I.4 0.8 -0.5

~~Table 5.6: Result Comparison rvith Dietrich-Yakorv itz Data Se!, fi2~~

~~CATEGORY ALGORITIIM 1.000 0.5 00 0.250 0.100 0.040 0.025 0.010~~

a o o a o x) o xl o x: o o Dietrich-Yakorvitz 83.8 5.05 87 .7 4.7 A 94.4 2.45 97 .7 1.39 98,9 t.4I 99.4 0.71 99.9 0.25 Priority-rule Algorithur 85.7 4.35 19. f 4.14 94.5 2.35 98.8 0.88 90.4 t.1 I9.7 0.65 99.7 1.9 1.9 0.1 1,1 0.5 0.3 -0.2

~~106 Table 5.7: Result Cornparison rvith Dietr.ich-Yakorvitz Data Set f3~~

~~CATEGORY ALGORITHM 1.000 0.500 0.2ö0 0.100 0.040 0.0 25 0.010~~

lx o E o x: î, a o a o xi Dietrich-Yakorvitz 92.2 1l 91 .8 4.83 97 .7 2.67 98.7 1.21 99,5 0.97 99. t 0.67 100.t t.0! Priority-rule Aleo¡ithm 92.5 3.99 4.58 I6.0 2.1ß 99.1 0.9 0 99.7 0.6 5 99. t 0,66 100. r 0. 17 0.3 0.8 - I.7 0.4 0.2 0.0 0.0

Table 5.8: Avelage CPU Tir¡esx for Test Data of ,-e1, l24l CATÐGORY Data Set 1.000 1.500 0.250 0.100 0.040 0.025 0.010 #1 140.2 130.3 90.9 61.8 27.8 25.1 27.7 #2 112.0 95.2 59.7 32.0 27.5 23.0 25.1 #3 t.t.t o f.ì-) 49.3 31.1 20.1 20.1 18.9

~~+ in seconds per individual problem on a 486 IBM compatible~~

The pei'formance of the priority-rule algorithm is seen to cotrpare very u,ell, giving supeliol results fol percentage sheet utilizatiols and standarcl deviations in the majority of tlie testecl categories. The improvements brought about in the solutions by the priority-rule algorithm are especially r¡anifest in the categories rvith B.O.Ms of higlì APSA ratios and, u'ith the exceptio' of categories 3 a'd Z, also prorninelt in the other categories. The improvemert is more significant rvhen consiclerecl in the context of optimality. Fol exarn¡rle, assuming the optirnal solution il the foultli category (0.100) of Dataset l tobe 100%, the 1.64% diference betrveen the results of the proposed algolithrn and that of Dietrich-yakowitz tr.anslates to a 39,8% (1.64/(100 - 95.88)) improvement, at the aery le¿sl torvar.cls the optirnal solution. The thilcl and seventh categories are the only ones to show ver.y little, if

r07 an)¡, iùlploveùlent. This cleflciencS, coulcl be attlibuted to the r.ules selectecì for. the algorithrn. Expelimentation, as l'ell as trial ancl erÌor, nìay iclentify more effective lules for ¡rroblems similal to those falling under these trvo categories.

~~5.5.2 Comparison with Bengtsson (fSSZ)~~

~~Tal¡les 5.9 ancl 5.10 cornpare the prio.ity-r'ule algorithrn results rvith those of~~

Dietrich-Yakorvitz [25] and Bengtsson [23] fo' the trvo test cases usecl ir Bengtsson's exPel'iments. Table 5.9 compares all three algolithrns fol the five lalclorlly ge¡elatecl problerns tested by Bengtsson (1982) on a 25 x 10 unit stock sheet. Tal¡le 5.10 plesents a similar comparison, also for the five problerns testecl by Beugtsson (1g82) on a 40 x 25 ulit stock sheet. The test pr.oblerns wer.e cornposed of rectangular pieces generated ratdornly fi'orn a uriforr¡ clistril¡ution with a lange of 1 to 12 ulits for the length clirneusions ald 1 to 8 units fol the ividth. The folmat for preselting the results il Tables 5.9 and 5.10 is that used in Bengtsson,s publication. The pool represents those pieces that are left in the last ancl uncornpleted sheet of the solution.

The compa.ison shows that the pr.iority-r'ule algorith.r gives better. r'esults rvith large'B.o.Ms a'd rvith decreasi'g APSA.atios. This is de'ronstrated clear.ly in the secotd test case, for rvhich the algor.ithm achieves zero waste. Bengtsson,s algorithrn, holever, appeaÌs superior fol small B.O.Ms containiug less than 60 pieces anil having high APSA ratios.

~~108 Table 5.9: Result Cornpar.ison rvith Bengtsson (sheet size 25 x 10)~~

Test Problem: 2 4 5 ALG ORITHM 1\ UmDer ol rleces: 20 40 ôU öU rUU A¡ea of Pieces: 741 1420 2090 2673 Numbe¡ of Sheets 3 5 8 10 13 'Waste Bengtssou 34 7 29 Waste rate 8.570 2.7% r.6% 0.3% 0.\Vo A.rea left in pool 55 204 123 180 109 Dietrich- N/A N/A N/A N/À N/A Yakorvitz

Number of Sheets J 5 8 10 13 Priority-rule Waste tt2 61 66 L9 Algorithm Waste rate t4.9% 4.9Vo 3.3% 0.1% 0.6To Area left in pool 103 23t 156 I to 99 uomputatronal ¡1me t.ÐJ t.El 6 59 (486 IBM Compatible) secs. secs, secs secs, secs.

~~5.5.3 Comparison with Israni-Sanders (1932)~~

Tal¡le 5.11 cornpares the results of the priority-rule algorithrn for the test problems used by Israni-Sanders [26]. The test pr.oblerns were generated by a ranclorn lurnber generator using a method desc'ibed by christofides and whitlocÌr [1g]. The results obtainecl by Dietrich ancl Yakowitz are also incluclecl in Table 5.11. Both the priority-r'ule algorithrn and that of Dietrich-Yakorvitz achieve zero-waste on trvo stocl< sheets for each of the trvo test cases. The prio.ity-rule algor.ithrn requiles cpu tirnes of 5.9 seco'cls for test proble'r no. 1 a'<1 3.g seconds for test problem .o. 2 on a 486 IBM cornpatible,

~~109 Table 5.10: Result Cornparison rvith Bengtsson (sheet size 40 x 25)~~

~~Test Problem: I 2 4 5~~

~~A LGORITHM Number of Pieces: 40 80 120 180 200~~

~~A¡ea of Pieces: r420 4027 50 08 6217~~

~~Number of Sheets I 2 4 5 0 Bengtsson Waste !2L 85 113 100 s7 Waste rate 12.1% 4.\Vo 2.8Vo 2.0% 1.6% Area left in pool 541 758 140 108 314~~

~~Numl:e¡ of Sheets I 2 4 5 6 'Waste Dietrich- 0 0 1 0 0 Yakorvitz Waste rate 0,0% 0.\Vo 0.0% 0.0% 0.\Vo Area left in pool 420 otó 28 8 2r7~~

~~Number of Sheets Ì 2 4 5 6 Priority-rule Waste 0 0 0 0 0 Algoritìrm Waste rate 0.0% 0.0% 0.0% 0.0% 0.0% Area left in pooÌ 420 D/ó 8 217~~

~~Computational time 2.85 o. Ðó 17.68 21.25 (486 IBN,I Compatible) secs. secs, secs. secs. secs.~~

~~5.5.4 Compalison with Albano-Orsini (fSSO)~~

The priority-rule algo.ithrn perfo'rr-'s quite r¡/ell in terms of sheet utilization in comparison rvith the results of Albano ancl orsini (See Table 5,12.) fol the three test pÌoblems f'-om ref. 122). rt is,, however, a diffe'ent matter altogether as regarcls the computational time. The priority-rule algorithm is very expensive corlputationalll, fo' the types of B,o.Ms used in Alba'o a.d o.si'i's expe.irne'ts, r.egiste'ing 6212,

~~1737, a'd 3027 seconcls, respectively, for test pr.oblerns 1, 2 a'cl 3 on a 4g6 IBM~~

Cornpatible. The test problems run weÌe collected frorn inclustrial case stuclies and involve non-integel climensiols. These climensions are conveltecl by rrultiplication to integels {or processing in both heuristic algo.ithrns comparecl here. This highel scale has the effect of increasing precisiolì, rvhich r¡akes it more unlikely tliat the

~~r10 Tal¡le 5.11: Result Compalison rvith Istani-Saldets~~

~~Test Plobìern: No. 1 No.2 ALGORITHM Number of Pieces: tLa~~

~~Israli- Sanclel's % utilization oo ,4 98.6 Pliority-rule Algolithrr % utilization 100 100 Dietrich-Yakorvitz Ta utihzatiott 100 100~~

StrARCH sub.outines ca' find exact dimensional matches between orcle. pieces a'cl the space available on the sheet being p.ocessed. For exarnple, a piece rvith a le'gth specifiecl as 4.05 metres on the B.O.M. will not be considerecl by the SEARCH loutine as a fit unless thele is space on the slieet with a clin.rension equal to exactly

~~4.05 rnetres. with these high tolera'ces, co'rputer ti're is expenclecl by the 'ruch SÐARCH loutines in futile searches. Hence, the excessive computatio.al tirne.~~

~~Table 5.12: Result Compar.ison rvith Albano-Or.sini~~

~~Test Problem: No. 1 No. 2 no.3~~

~~ALGORITHM Stock size : 12030 x 2550 6015 x 2550 12030 x 2550 Number of Pieces: 398 398 :tt),f~~

~~AIbano-Orsini % utilization 98.5 98.1 96.6 Ptiority-rule Algorithm % utilization 98.0 98.6 98.8~~

~~11i 5.6 Analysis of Results~~

~~5.6.1 Cornputational time fol the priority-rule algorithrn~~

~~The courputatioual time fol the pliolity-r'ule algolithm is highly clata-clepenclent,~~

~~cl.e to the enumeLative natule of tlie SEARCH and EXPAND subrouti.es. B.o.Ms~~

~~ivith lalge APSA ratios requi.e more computatio'al tirne than B.o.Ms of sirnilal~~

~~size but lorver APSA .atios. Ave.age computational ti'res, usi.g Data set f 1 of r.ef.~~

~~[25] as a' illustrative exampÌe, ralged fi'om 2g seconcls for catego.y z ivith ave'age~~

~~B.O.M. size of 485 pieces, to 140 secolcls for category 1, rvhere the avelage B.O.M.~~

~~size is 494 pieces. The conputational ti're ca' be quite excessive for B.o.Ms having~~

~~APSA ratios belorv 0.01. A pleclominance of relatively very small pieces may pose a~~

~~problern of coml¡inatorial ex¡rlosions fol the SEARCH subroutine. such clificulties~~

~~can be avoidecl by splitting the stock sheet into smaller sul¡sheets to boost the ApsA~~

~~latio or', altelnately, moclifying the priolity assignment r.ules to give more pieces~~

~~priolity status.~~

~~5.6.2 Restrictions~~

The p.io.ity-rule algoritlirl presented i'this chapter âppears to be highly co'r- petitive based on comparisous il'ith existing heuristic methocls. It cloes, horvever', have trvo identifiable shortcomings. These ale (1) poor.er sheet utilization for B.o.Ms composecl of a small total cler¡ancl for pieces, and (2) an unacceptably high cornputational tir¡e for B,o.Ms cau'ying pieces that are veÌy srlall il i'elation to the stock sheet size.

~~112 The first rveak point rvas demonstr.atecl in the fir.st test case of Bengtsson (See~~

~~Table 5.9.) A fulther investigation of the algolithrn's behavioul rvith respect to this~~

~~test case is carliecl out by testing cliferent cor.. binatious and fir.ing o.cler.s fo. the~~

~~foul rules usecl in the algotithm. Table 5.13 clisplays the lesults of this investigation.~~

~~The numbels in the fir'st column shorv rvhich rules ale activatecl ancl in rvhat olcler~~

~~they a.e fir'ecl to give the p.iority assignr'e'ts. As a' exarlple, the seque'ce 2-4-~~

~~3-1 neans that .ule 2 is usecl for assigning fir'st priority, rule 4 for second p.ioritv,~~

~~ancl so on. TabÌe 5.13 inclicates only a minor valiation in the results obtainecl with~~

~~different lule olclelings. A genelal trend is that rvhere rule 3 prececles rule 2, the~~

~~lesults irnprove for test p'oblems 1 ancl 2, but cleteliolate by an almost equal~~

~~margin in the othel tluee test ploblems. The opposite effect is observecl il,herr rule~~

~~2 plecedes rule 3 i, the firi'g orcler. The most inte.esti'g co'clusio', horve'er,~~

~~is that the best results occur whe. no priority lules ar.e implementecl. This is not~~

~~unusual because, rvith srnall B.o.Ms, only a ferv sheets are used in the solutions. hl~~

~~such cilcur¡stances, urethods that seek to locally minimize tr.im losses on inclividual~~

~~sheets ale likely to give better results than the plio.ity-driven methocls, rvhich tencl to distlil¡ute the tlirn-loss rnore evenly amongst the sheets.~~

The second obse.r'ecl weak'ess in the priority-based algor.ithni is the high co'rpu- tational tilne rvhich a.ises in problems having a high clernancl fo. ver.y small pieces in a B.o.M. of ferv clifelent piece sizes. In these ploblems, the co'rputatiolral l¡ur.clen occu'-s nainly in the STRIPS loutine cluling the search for a starting piece for.the sheet's lorver left-hand courer. cornputer times can l¡e controllecl easily in such cases by using less restrictive palarneter.s for rules 2 and 4.

~~113 Tal¡ìe 5.13: Pelcentage utilization lesults fol Bengtsson's test pr.oblen.r 1 rvith clifferent lule colnl¡inations~~

~~Firing Olcler TEST PROBLEMS~~

~~of Rules 2 3 4 5 none e0 t 96. 96.7 99.9 aol~~

1 89.2 96. 96.7 99.9 99.4 2 85.1 o¡< 96.7 ôoô 99.4 3 89.2 96. 90.7 100.0 98.9 4 89.2 96. 96.7 99.9 99.4 1-2 85.1 ô( 96.7 99.9 99.4 2-1 85.1 95. 96.7 oo o 99.4 85.1 o< 96.7 99.9 99.4 1ao 90.0 97. olí o 96.3 99.I 2-1-3 85. i 95. 96,7 99.9 99.4 85.1 o( 96.7 99.9 99.4 90.0 o7 95.9 96.3 oo1 3-2-1 90.0 97. 95.9 96,3 99.1 7-2-3-4 85.1 95. 96.7 99.9 99.4

~~1-2-4-3 85,1 95. 96.7 oo o oo ,1 2-3-1-4 85.1 o< 96.7 ooo 0o ¿ 1-3-4-2 90.0 97. OK Ô 96.3 oo1 1-4-3-2 90.0 97. 95.9 96.3 99.1 I-4-2-3 85.1 o¡i 96.7 99.9 q0¿~~

~~1-3 89.2 96.f 90.7 r 00.0 98.9 1-3-4 89.2 96.f 90.7 100,0 98.9~~

~~4-2-3-1 85.1 95. 96.7 99.9 oo ,.r 4-I-3-2 90.0 97., 95.9 96.3 ôo1~~

~~4-3-2-1 90.0 o7 1 95.9 96.3 99.1 4-3-r-2 90.0 97.1 95.9 96.3 99.1 3-1-4-2 90.0 97.1 95.9 96.3 99.1~~

~~3-2-4-I 90.0 ô7 1 o< o 96.3 99.1 3-4-2-r 90.0 97.1 95.9 96.3 oo1 5.6.3 Relationship between APSA ratio and Sheet Utilization~~

~~Figu.e 5.7 p.ese*ts an i'teresti'g patte'n for the achievecl rvith the 'esults p'iorit)'-rule algorithm fo. the th.ee testecl data sets of ref. [25]. The results in~~

~~each set follorv an elongatecl s-shaped culve, suggestirg a lelationship betiveen the~~

~~APSA latio ancl the ave.age stock sheet utilizatio' (or. aver.age tlirn ìoss). Fur.ther~~

~~i'r'estigation usi,g semi-logarithmic plots (shorvn in Figu.es 5.8 th'ough b.10) r.e-~~

~~veals an almost linear lelation in each set. The implication is tllat there exists a~~

~~possibility that the betrvee' the APSA ratio of a 'elatio'ship B.o.N4. ancl the e¿- ltected, ftin-loss it gelelates can be quantified ¡nathernatically, taking into account,~~

~~of cou'se, the valiation (stanclalcl deviation) l¡etween the orrle. sizes demanclecl~~

~~in the i.itial B.o.M. using a very app'oxi'rative for fitting st.aight li'es 'rethod to the plotted points in Figures 5.8 through 5.10, the relationsliip betwee. sheet~~

~~utilization alcl APSA ratio is quantifiecl as follows :~~

~~For clata set S 1, y : 100 * er,¡t-o.56 " (APSA Ratio)~~

~~For clata set S 2, y : 100 + e:r\fo 336 a (ÀPsA Ratio)~~

~~For data set S 3, g : 100 * e¿p-otaz,. (APSA Ra|io\~~

The clifel'ences in the expo'entials l¡etrveen the three clata sets highlights the ploblern-clepen dency of the algo.ithrn. The differences in tl:e exponentials car be attributed to the diferent palameters usecl in genelating the landorn clata for. each set. This leads to the logical conclusion that the stanclar.d deviation between the

~~115 sizes of the piece types clemanclecl ìn the B.o.M. afects the slopes of lhe lines plotted~~

~~in Figures 5.8 th.ough 5.10. A highel standa.d cleviatior incLeases the exponential,~~

~~ancl theleby the sheet utilizatio', because the B.o.Nil. mix ivould be rnore distinctly~~

~~clelineatecl betrvee. "clitical" ancl "flller." pieces. The corverse is tr-ue for B.O.Nis~~

~~lvith a lorv standalcl deviation betrveen demancled sizes.~~

~~Anothel factor influencing the natule of the relationship betrveen ApsA ratio~~

~~ancl sheet utilization is Ç, the total number of different piece types clemandecl. In~~

~~the data usecl to obtai' the results of Figu.es 5.8 th.ough 5.10, Q ra.gecl betwee'~~

~~25 a'd 50. For Ç less tha. 25, the slopes of the li'es increase ancl 'egati'ely, the p.oblem-clepencle.cy lises shalply. For very lorv Ç, the suggestecl r.elatio'ship~~

~~betryeen APSA latio and sheet utilization cannot holcl. Taking Ç : 1 ancl al ApSA~~

~~ratio of 1.0, fo. exa'rple, the sheet utilization can o.ly l¡e either. 100 % or. 0 %. This~~

~~clemonstrates that the plotted fonns in Figu.es 5.8 th'ougli s.l0 are data-clepencle't~~

~~and therefore not applicable i' general. Thus, as Ç is reduced, the r.elative aspect~~

~~ratios of the clemanclecl ¡riece types and the stock sheet become more cr.itical factors~~

~~in clet elil inilg slreet utilizatiorrs.~~

~~In conclusion, it rvoulcl appear. that for B.O.Ms rvitli a lar.ge Ç, say gr.eater.than~~

~~15, the APSA ratio is the clotninant factol affectilg sheet utilizatious. It is st¡essecl~~

that the suggested relationship betrveen the APSA ratio antl sheet utilizatiol as shorvr i'Figu.es 5.8 th.ougli 5.i0 is based on aveÌages fro'r 100 trial B.o.x4s i' each catego'-y. An irdiviclual B.o.M. may have a sheet utilization that ca'be sig- nificantly lorver or higher than that pleclicted by the APSA ratio - sheet utilization relationship. what Figules 5.8 thlough 5.10 clemonstrate, basecl on the limitecl

~~116 data used, is a genelal fo.nr relating ave.age sheet utilization to a cha'actel.istic of tlie B.O.À4,, namely the APSA ratio.~~

~~117 Fig. 5.8 Semi-logarithmic Plot for DataSet#lResults I ru~~

~~s .9 (ú .N co 5 (l) (1) -cU'~~

~~00 0.10 0.15 0.20 0.25 0.30 0.40 0.45 Average piece/stock area ratio F'ig. 5.9 Semi-logarithmic Plor for DataSet#2Results 100-~~

~~\o o (ú .N~~

~~(I, (l) U'~~

~~I 0+- 0.00 0.10 0.r5 0.20 i0 Average piece/stock area ratio Fig. 5.10 Semi-logarithmic Plot for DataSet#3Results~~

~~\oo\ E .o (ú aN) .N f (I) (l) U' CHAPTER 6~~

~~Conclusions and Recommendations~~

An algolithm courbinilg a constlailed enumelative sealch ploceclule ancl pr.ior.i- ty assig'ments rvas developecl fol solving the si'gle-plate, trvo-dirnensional cutting stock problem. observ¿tions f'orn test .uns of cliferent heuristics letl to the conclusion that the effect of area-l¡ased priority rules di'.inishes rvith smaller. ApsA ratios, and vice-versa. This il'as the main consideratiol in the design of the rules used in the algorithm for assigning allocatio' priorities to the pieces on the given

B.O.M. Trvo of the lules were wr.itten so that a lalge APSA r.atio translates into mole B.O.M. pieces qualifying fol priolities ancl, consequently, a solution that is prirnarily rule-clriven. On the othel hand, lvith low APSA ratios only a ferv, if any, of the pieces gain priority. As a result, the algolithmic execution is orientecl rnor.e torvarcls sealches for layouts that minimize trir¡ losses at the local level, r.ather than globally. Thus, the algorithm, in conjunctiorr rvith the four pi'iolitv rules chosen for. it, is equipped rvith a rnechanism that allorvs it to dynarnically tune its pr.oceclure i. accorda'ce with the current make-up of the B.o.M. This fe¿tur.e enhances the aìgoritlun's suitability for B.o.Ms having a t'ide.ange and mix of piece sizes ancl quantities.

~~The prio.ity-rule algorith'r was co'rpa.ecl rvith four p'eviously publishecl heulistics. It pro'idecl the best sìreet utilizations compared rvith the algorithn of Be.gts-~~

~~121 sorì [23] fo. Bengtsso''s seconcl test probleu. It also outPe.formecl the Dietr.ich-~~

Yakou'itz [25] rnethod in the rnajor.ity of the test pr.oblems used in their paper published in 1991. consiclerìng that the compalison rvas basecl o' the l¡est utilization results obtainecl by Dietlich-Yal

[26]. Fi'ally, a cornpa'iso' with the i980 heuristic of All¡a'o ancl orsi.i clernor- stlatecl the competitiveness of the algorithrn rvith lespect to stock sheet utilization, but not computational time.

~~6.1 Implementation of an Expert System~~

The algorithrn clevelopecl in the p'e'ious chapter is an icleal one around rvhich an exPert system can be buili fol solving cutting stock ¡rroblerns. The lulebase i'colporatecl in the algolithrn provicles a flexibility that allows r.ules to be adclecl, moclifiecl, cleletecl ol le-orclered. Theoretically, the lole of an expert system in this application rvo.ld be to ch au' upo' a knorvledge base to classify the gi'en

B.O.M. into an iclertifiable class of problems. Once the B.O.M is classifiecl, the system rvould again consult the knorvledge base to pick the appropriate pr.io,-'ity assignment lules fol the algorithm ancl anange thetr in the corl.ect firing olcler..

Thelefole, the ¡tain lole for.the expelt system is that of classifying given B.O.Ms accorcling to their cha.acteristic composition. of coulse, before such a' expe.t systern can be implemented, the knorvledge of ivhat char.acteristics clefine *'hat

~~122 classes ancl l'hich lules u'olk best in ivhich orclel rnust be gailed througli experience fiom experirnentation.~~

~~The rnain chalacteristics tliat ale useful in classifying B.O.Ms rvoulcl be:~~

~~1. APSA ratio~~

~~2. Total nurnbel of pieces in the B.O.l\{.~~

~~3. Nurnl¡er of cliferent piece sizes clemanclecl by the B.O.M,~~

~~4. Average qualtity clemandecl of each piece type~~

~~5. Standard cleviatiol between ar.ea sizes of the B.O.M. pieces~~

~~This approach to the implernentation of an expert system is souncl because of the ivell-clocurnented capabilities of expelt systems in classificatio' applications.~~

~~6,2 Conclusions~~

~~The rnain conclusions drarvn from this research are :~~

~~l. The so-callecl APSA ratio is a major facto'that should be consicler.ed in ir'-~~

~~plementilg heulistics fol trvo-dimensional, cuttilg stock ploblems.~~

~~2. Pliority allocations are essential in plocessing B.O.Ms having a size that r.e-~~

~~quites more than a ferv, say five, stock sheets. The gleatel is the nurnber of~~

~~incliviclual pieces in the B.O.M., the gleater.is the aclvantage of plioritizing~~

~~allocations.~~

~~3. The scope of the heulistic method is rviclenecl to include rnor.e B.O.Ms of~~

~~var-ying sizes aud corlpositions by using r¡ror.e than one rule for plioritizing~~

~~123 allocatious.~~

~~B.O.Ms having similar char.acteristics ar.e usually ser.vecl best by the sarne rule~~

~~or set of lules.~~

~~6.3 Recommendations for F\rtur.e Research~~

~~The author wishes to suggest the follorving areas for further resealch :~~

~~1. Developi'g a set of sta'da'clized test p'oblerns ivhich ca' be usecl to assess,~~

~~fairly, the performance of diferent heur.istic alcl optirnizing algor.ithms.~~

~~2. Collecting data flon the pr.ior.ity-rule algolithm for a tvider.variety of B.O.Ms~~

~~havi'g cliffere't rnixes of ilema'cled Pieces. Differe't rnay l¡e tested to 'ules build up a knowleclge base.~~

~~3. usi'g a.tiflcial neural to solve the In this approach, rvhich 'etwo-,-ks 1>r'oblern. presumably can be based on the Hopfield- Tank rnodel, the tr.im-loss is repre-~~

~~sented as an enelgy function. The cut,ting stock ploblem must be for.mulatecl~~

~~such that in rni'imizi.g the e'ergy function, the artificial neural netrvor.k is~~

~~al¡le to find a solution to the give'B.O.NI. Depencling o'the for.'r'lation,~~

~~the.esults give'in section 5.6.3 of this thesis'ray help in setti'g the i'put pa-~~

~~lalneters and connection weights to the neural netrvork. The ¡rlimar.y lesear.ch,~~

~~nonetheless, Lemains in the area of problem-fotmulatiol.~~

~~4. Exte'ding the'erv algo.ithrnic technique to th.ee-climensio'al paching prob-~~

~~lems and to two-climensional, ilregular, cutting stock problems.~~

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~~128 APPENDICES APPENDIX A~~

~~Results of Experimental Test Runs using~~

~~Dietrich-Yakowitz Data set f 1 ,A'.1 Results for Category f I~~

~~Table 4.1: Avelage StocÌ< Sheet Utilizations (r) il Categor.y $ 1~~

~~B.O.M Numbel Total A PSA ÐXS-Z Ðxs- LG-Z LG-A (ro nìpounc Prioritl # of Types Pieces Ratio Policy Polìcy Policl PoIic¡ Heuristic Rule Dernanded Demanderl Based~~

1 2S 365 10.438 t4.-tt) 74.16 80.94 80.94 80.94 80.94 2 413 In.n, 69.36 69.36 69.50 69.50 69.50 69.50 3 40 556 I o.nro 69.60 69.60 72.63 72.63 72.63 72.63 4 41 571 lo.nrn 78.05 78.05 81.65 81.65 81.65 81.65 I 5 35 424 0.386 72.59 79.20 79.20 79.20 79.20 6 496 0.4i6 76.35 76.35 82.88 82.88 82.88 82.1.2

7 28 360 0.437 67.70 67.70 71 .47 7 r.47 7r.47 71.15 8 45 611 0.425 71.35 71.35 75.92 75.92 75.92 75.92 I 40 530 0.456 tð.oJ 75.63 81.39 81.39 81.39 80.93 10 49 0.516 73.58 73.58 74.98 74.98 7 4.98 74.98 11 30 441 0.460 74.23 74.23 77 .78 77 .78 77.78 78.39 12 49 611 0.467 83.79 83.79 85.73 85.73 85.73 85.93 13 44 559 0.404 76.60 76.60 76.42 76.60 76.42 14 48 658 0.490 76.81 76.81 80.21 80.21 80.21 80.21

15 49 602 0.410 75.32 75.32 76.75 7ß.7 5 76.7 5 78.22 16 34 516 0.354 80.31 80.31 82.15 82.15 82.15 82.07 77 Jt) 462 0.448 68.79 68.79 75.24 75.24 75.24 75.24 18 40 528 0.412 69.75 69.75 69.99 69.99 69.99 69.99 19 49 ooz 0.403 74.80 74.80 79.26 79.26 79.26 79.26

~~21 36 481 0.498 77.62 77.62 76.13 76. i3 I I ,IJ¿ 78.18 22 44 40ll 0.413 82.55 82.55 83.29 83.29 83.29 84.31 l 45 496 0.40 0 7l.47 I +.4ó | 74.43 74.43 77.59~~

24 627 0.428 ] 72.60 72.60 77.561 77 .56 77.56 77.56 l 25 31 423 0.5071 81.00 8i.00 80.s41 so.e4 81.00 80.94 | 26 I 38 491 0.475 73.17 78.13 78.13 | 78.13 78.13 | 47 524 0.4551 77.79 77 .79 79.27 I 7s.27ll 79.27 Table A1 (cont.)

~~B,O.M Number Tot al APSA EXS-2 EXS-A LG-Z LG-A Jompound Priolity JJ- of Types Pieces Ratio Policy Policy loli cl ?oLic¡ Heuristic Rule~~

~~Demandei B ased~~

~~28 306 0.477 82.58 82.5 8 85.64 85.64 85.64 85.60 29 44 562 0.413 / o.,J / 76.63 82.46 82.4ß 82.46 82.46 30 25 0.428 ß7 .44 67 .44 70.55 70.55 70.55 76.01~~

26 0.454 82.89 82.89 85.47 85.47 85.47 85.3 5 451 0.430 80.00 80.00 77.45 77 .45 80.00 77.45 33 28 360 0.534 76.94 76.94 I f ..t t) 77.36 I l.Jt) I t.Jt) 34 44 555 0.495 77.72 71.r2 73.60 73.60 73.60 73.91

~~J l-) 28 325 0.439 68.75 68.75 68.75 68.75 68,75 68.54 36 40 554 0.446 68.91 68.91 72.14 72.74 72.14 74.75~~

39 474 0.495 76.29 76.29 77 .07 77 .07 77 .07 38 26 393 0.515 65.61 65.61 65.62 65.62 65,62 65.61 to 50 734 0.462 78.59 78.59 83.22 83.22 83.22 83.22 40 31 0.403 70.66 70.66 76.34 76.34 76.34 76.34 41 to 4L2 0.349 77.1.1 77.77 77.17 78.36 42 32 434 0.489 79.17 79.17 81.10 81.10 81.10 81.16 43 42 525 0.428 65.84 65.84 69.30 69.30 69.30 70.77 44 39 504 0.366 79.22 83.93 84.32 84.32 86.09

45 39 435 0.494 76.84 76.84 78.02 78.02 78.02 f / .o4 46 39 4Ð4 0.423 72.56 72.5ß 77.15 77.15 77 .15 79.4ß to 422 0.426 76.23 76.23 76.23 77.74 48 34 394 0.415 I ¿t.o I 78.67 83.26 83.26 I3.26 83.26 49 39 436 0.478 77.22 71.22 72.50 72.50 72,50 72.50

~~50 30 3i6 0.437 70.49 70.49 73.75 73.75 73.7 5 51 48 607 0.453 74.84 74.84 81.02 81.02 81.02 82.73 38 0.413 82.40 82.40 87 .37 87.37 87.37 88.60 26 0.420 83.93 83.93 86.25 86.25 86.25 86.52~~

54 40 431 0.458 77.57 77.51 8 3.99 83.99 83.99 84.22 l 55 31 0.557 69.85 69.85 71.30 i 71.30 71.30 71.30 56 46 660 0.470 69.63 69.63 70.771 70.7 7 70.77 72.09 AA t¿.¿ 57 642 0.401 I r 76.21 80.20 I 80.20 80.20 80.20

~~r32 Table A1 (cont. )~~

~~B.O.M Nurnber Total APSA EXS-2 EXS-A LG-Z LG-A Sonr poun( Priolit¡ of Types Pieces R.a.tio Poücy Policv Pol.icy Policy [Ieuristic Rule Dernanded Demander Based~~

58 42 502 0.470 76.28 76.28 | 80.51 80.51 80.51 80.51 59 39 418 0.444 74.37 I ',0.oo 79.54 79.54 79.54 I 60 49 0.415 75.34 180.,'r3 80.53 80.53 81.79 61 49 581 0.442 67.65 67.65 Ioo.r, 69.37 69.3 7 70.13 ¿o 62 628 0.428 68.30 68.30 l rr.rn 7r.34 7L.34 71.91 63 40 573 0.459 76.67 76.67 Lo.u" 75.62 79.62 79.62 64 47 702 0.476 77.53 77.53 79.90 79.90 79.90 79.90 65 50 602 0.457 74.02 74.02 77.37 78.03 t)t) 419 0.423 66.19 66.19 69.30 69.30 69.30 70.13 67 401 0.392 72.31 75.31 75.31 75.31 76.04 68 40 488 0.488 73.88 73.88 77.24 77.24 77.24 77.49 69 45 669 0.456 ( ó,+:) 73.45 74.12 to 70 370 0.446 81.80 81.80 85.36 ðÐ. JO 85.36 85.36 71. 50 615 0.432 74.38 74.38 79.46 79.46 79.46 79,69 466 0.399 77.54 77 .54 81.65 81.65 81.65 83.11 to 378 0.445 74.89 74.89 80.59 80.59 80.59 80.59

74 429 0.428 70.34 70.34 74.44 74.44 7 4.42 75 44 ll t).J 0.502 67.82 67.82 70.12 70.12 70.72 70.29 76 456 0.418 72.55 72.55 81.23 81.23 81.23 a1 0D 469 0.41.5 77 .77 77.71 80.76 80.76 80.76 82.50 78 29 425 0.4i3 77.54 71-.54 74.1.4 74.L4 74.14 /D.UI

79 349 0.463 74.97 7 4.97 82.06 82_06 82.06 82.06 80 39 522 0.536 72.71 72.7r 72.85 72.A5 72.85 72.91 l 81 355 0.446 73.15 73.15 79.80 79.80 | 79.80 79.50 ¿o 82 600 0.470 80.71 80.71 82.54 s2.541 82.54 82.54 83 48 611 0.405 77 .16 77 .16 82.26 82.261 82.26 83.52 84 38 509 0.425 80.08 80.08 81.94 81.e41 81.94 83.00 85 41 545 0.407 78.80 78.80 83.18 83.18 83.18 83.18 | 86 529 0.497 62.t4 62.r4 oì1. Jð 65.38 t 65.38 OD.JD 87 38 46L 0.442 73.54 73.94 77.34 77 341 78.24 Table A1 (cont. )

~~B.O.M Numbel Total AP SA IJ'\ò - I EXS.A LG-Z LG-A Jompouni Priority # of Types Pieces Ratio Policy Polìcy )olicy ?olic¡ Heuristic Rule Jema,nded Demanded Based~~

~~88 35 442 0.444 66.46 66.46 69.17 69.17 69.17 69.17 89 45 572 0.459 72.23 72.23 76.47 í tJ.4 í 76.47 to,4(~~

90 34 400 0.470 71.85 71.85 75.57 75.57 75.57 7 5.57 91 431 0.480 tv.o/ 79.67 82.92 82.92 82.92 82.92 92 602 0.425 64.85 64.85 69.36 69.36 69.36 69.36 93 30 389 0.516 75.92 75.92 77.29 iT ro 77 .29 77.38 g4 37 478 0.470 78.60 78.60 82.76 82.76 82.76 82.76 95 30 389 0.396 77.59 77.59 83.81 83.81 83.81 83.50 96 362 0.426 69.01 69.01 72.90 72.90 72.90 72.90 o'7 37 495 0.440 78.01 78.01 78.29 78.29 78.29 78.25 98 438 0.433 72.57 76.92 76.92 76.92 76.92 99 47 468 0.442 72.65 72.65

~~100 45 680 0.439 77.75 77 .75 77.75 78.56~~

~~134 Results for Category S 2 ^.2~~

~~Tal:le 4.2: Avelage Stock Sheet Utilizations (e) in Categor.y $ 2~~

B.O.M Ntrlber Total APSA Ðxs-2 S- A LG-Z LG-A Jornpound Priority JI of Types Pieces Ratio Policy Policy Policy Policy Heuristic Rule Dernalderl Dernanded Based to 1 449 I 0.296 81.92 81.92 89.37 189.37 89.37 89.48 I Ð/J Io,r, 7 4.63 82.46 83.42 83.42 81.82 42 542 0.237 81.08 80.57 89.84 89.84 89.84 90.13 4 34 499 0.2 80 63.93 63.93 73.62 73.62 73.62 74.02 a1 5 369 0.23 0 7 4.78 74.78 7 6.47 78.87 78.87 84.25 6 48 660 0.256 86.67 86.67 88.65 88.65 88.65 89.02 7 34 512 0.315 78.82 78.82 82,64 82.64 82.64 83.46 8 28 3t4 0.239 89.10 89.10 89.99 89.66 89.99 89.11 I 44 608 0.256 81.86 81.86 85.94 85.94 85.94 84.90 10 50 624 0.303 /o.+J 76.43 87.24 87.24 87.27 11 28 0.257 82.78 82.78 88.10 88.10 88.10 87.82 12 36 486 0.250 89.63 89.63 sl.1.2 97.22 91 .22 91.02

13 44 t) 14 0.279 71.27 78.67 78.67 / ¡J.t) I 80.29 l4 31 390 0.272 82.34 82.34 87.93 87.93 87.93 87.87 to n 15 328 too 72.82 72.82 79.18 79.18 79.18 79.17 16 50 ß23 0.296 85.95 85.95 88.40 88.93 88.93 87.14 17 31 391 0.242 83.36 83.36 86.78 86.63 86.78 87.81

18 40 631 0.283 77.46 7 t.46 84.96 87.08 87.08 86.61 19 442 0.296 75.68 75.68 80.60 80.60 80.60 80.60 20 4i) 663 0.252 78.47 78.47 89.56 89.56 89.56 89.32 2I 44 s57 0.305 89.59 89.59 89.59 88.66 89.59 89.19 22 422 0.299 71.45 70.65 82.09 82.0I 82.09 82.09 ¿¡t 40 506 0.293 74.50 74.12 77 .11 79.18 79.18 79.30 24 36 45t 0.275 80.31 80.31 87 .82 88.25 88.25 87.21. 25 42 611 0.279 78.37 78.37 85.90l 85.90 85.90 86.67 26 34 506 0.353 58.15 58.15 61.781 61.78 61.78 62.87 27 25 340 0.283 70.95 70.95 77 .341 77.34 77.34 Table A2 (cont.)

~~B,O,M Nurnl¡ er Total APSA E xs- z Ðxs-l Llr- L LG-A Jompounr Priority # of Types Pieces R.atio Policy Policy Polic¡ Polic¡ Heuristic Rule J ern a,r d erl Dernanded Based~~

~~28 35 41-3 0.265 76.18 76.18 85.19 185.08 85.19 84.63 to I 45 594 0.260 72.43 76.6I 177.41 77 .47 82.38~~

~~30 30 0.254 74.03 7 4.03 8t.24 87.24 81.24 82.19 31 663 0.274 84.09 84.09 84.37 84.37 84.37 84.09 47 585 0.269 88.19 88.19 88.84 89.05 89.05 89.15~~

ð,1 289 0.298 81.38 81.38 85.52 85.52 8 5.52 85.52 34 38 509 0.315 73.10 81.33 81.33 81.33 81.33 35 40 509 0.372 66.6 7 66.67 72.6r 73.29 73.29 ól) 48 601 0.278 85.22 85.22 84.r7 ó4,r( 85.22 84.52 37 36 0.323 68.72 68.72 80.98 80.98 80.98 80.98 40 502 0.310 8i.40 81.40 82.09 82.09 82.09 81.90 39 42 597 0.268 81.06 81,06 88.62 91.09 91.09 88.49 40 390 0.284 59.67 Ð9.b / 63.15 63.15 63.15 63.08 41 44 530 0.278 89.54 89.54 88.09 88.09 89.54 87.83 42 48 701 0.261 75.89 75.89 87.44 87 .44 87.44 87.61 43 30 439 0.281 65.37 65.37 74.93 74.93 74.93 75.25 44 409 0.289 80.79 80.79 80.79 80.79 45 34 463 0.307 77.11 77.1r 79.67 79.67 79.67 79.80 46 26 348 0.306 72.1.1 72,11 73.68 73.68 73.68 76.88 50 673 0.266 82.27 82.27 86.73 87.40 87.40 88.19 48 26 tol 0.314 83.37 83.37 87.83 87.83 87.83 88.03 49 31 369 0.267 70.72 IU.tZ 80.68 80.68 80.68 80.68 50 39 500 0.277 73.89 74.69 82.74 84.26 84.26 83.24

51 39 449 0.290 73.46 73.46 78.56 78.56 78.56 / ð.J I 52 637 0.291 ly.ÐJ 79.53 81.77 82.50 82.50 82.87 53 30 374 0.320 85.82 85.82 81.41 82.84 85.82 òl).4 / 54 43 533 0.30 0 75.39 76.11 85.04 85.24 85.24 85.82

~~55 47 644 0.281 68.10 68.10 81.26 I 8 0.89 81.26 82.47 Ðl) 38 507 0.357 I r Ð.úó 7e.541 7e.54l 81.37 57 46 472 0.3001 84.07 84.07 s.221ü.221 84.07 81.22~~

~~136 Table A2 (cont.)~~

~~B.O.M Nurnber Total APSA ÐXS-2 ÐXS-A LG.Z LG-A !ompounr Priorit¡ of Types Pieces R.a.tio Policy Policy Policy Policy Heuristic Rule Dernanded Dernandec Based~~

58 26 367 0.257 80.35 80.35 I I0.01 90.01 90.01 89.69 59 35 480 0.270 68.47 ß8.47 Lo.oo 76.99 76.99 76.91 60 49 578 0.301 66.28 66.28 l* uu 75.92 75.92 /o./t) 61 39 488 0.298 ttJ,ó¿ 76.32 79.94 80.05 80.05 79.83 62 35 420 0.230 84.67 84.67 88.64 88.64 88.64 89.13 63 27 0.232 88.42 88.42 87.87 88.01 88.42 87 .43 ß4 to 475 0.293 82.37 82.37 84.28 83.70 84.28 84.28

~~65 42 570 0.291 87.1 1 87.11 84.71 84.71 87.11 87.89 66 48 614 0.308 79.94 79.94 86.43 86.83 86.83 87.16~~

tll 48 671 0.272 71.87 7 r.87 80.60 82.25 82.25 81.97 68 50 690 0.281 73,83 73.83 78.08 78.08 78.08 78.08 69 442 0.324 68.99 68.99 73.98 73.98 73.98 75.04 70 26 283 0.248 78.57 78.57 80.12 80.12 80.12 80.12 7l 4I o t4 0.30i 77.07 77 .07 72.67 77 .07 79.50 477 0.291 o t.4t) 67.46 7r.93 71.93 71.93 76.72 39 530 0.302 79.78 79.78 83.56 83.56 83.56 84.35 74 36 493 0.302 82.62 82.62 88.21 87.95 88.21 87.95 75 39 595 0.284 77.00 76.3 0 82.38 81.98 82.38 82.30 76 35 452 0.282 66.57 66.57 77.64 77.64 78.60 77 45 679 0.285 81.10 81.10 88.00 87.60 88.00 87.60 78 36 427 0.314 69.41 69.41 79.88 79.88 79.88 79.88 70 39 ,l4f) 0.278 82.86 82.86 85.55 85.43 85.55 86.15 80 36 507 0.288 73.10 73.10 80.43 81.10 81.10 81.21 81 31 387 0.287 68.34 68.34 73.69 73.6I 73.69 73.69 82 48 612 0.285 82.23 82.23 84.27 84.21 84.21 84.06 83 294 0.265 81.67 81.67 85.61 85.61 85.61 85.61 l 84 47 525 0.281 80.8i 80.96 83.12 83.i 2 t 83.12 82.81 9K 85 514 0.285 78.17 78.17 80.34 80.7e 1 80.79 81.23 86 óo 387 0.248 85.48 85.48 84.38 84.16 85.48 84.26 | 87 38 500 0.277 77 .26 77 .26 85.00 s5.001 85.00 85.74

~~t,1t Table A2 (cont. )~~

~~B.O.N4 Nurnber ?ot al APSA EXS.2 EXS-A LG-Z LG-A Oorr pou Priorit¡ # of Types Pieces Ratio Policy PoJìcy Policy Heuristic Rule Demanded Dem a-n d erl Based~~

88 25 JJI) 0.312 67.33 o ¡..1ó 78.74 78.74 78.7 4 82.61 89 38 457 0.258 78.98 84.98 84.65 84.98 85,98 90 ¿Ð 355 0.249 88.75 88.75 92.4t 93.10 93.10 at ¿0 91 31 387 0.259 80.48 80.48 90.03 90.03 90.03 89.23 92 49 626 0.276 73.27 73.27 85.73 85.73 85.73 85.73 93 39 541 0.288 77 .78 77.78 85.74 85.74 85.74 85.97 aL 47 606 0.288 83.18 83.18 87.89 87 .77 87.89 87.67

95 34 411 0.317 64,95 64.95 72.70 74.12 74.12 7 4.71 96 to 0.267 84.01 84.01 89.35 89.35 89.35 89,35 g7 518 0.266 76.80 76.80 82.98 82.48 82.98 82.86 98 49 749 0.319 It). / t, 76.70 81.32 81.88 81.88 80.36 99 365 0.310 77.85 77.85 84.44 84.44 84.44 84.30 100 0.315 78.50 78.50 80.76 80.76 80.76 I0.76

~~138 A'.3 Results for Category f 3~~

~~Tal¡le 4.3: Avelage Stock Sheet Utilizations (ã) in Categor.y f B~~

~~B,O.M Number Total APSA EXS- 2 EXS-A LG-Z LG-A Compound Priolit¡ tl of Types Pieces Ratio Policy Policy Policy Policy Heuristic Rule D ern a,n rì er )ern an ded Based~~

1 30 349 0. r56 93.75 93.03 92.28 92.28 93.75 90.74 2 28 395 0. r30 95.55 95.86 96.08 95.2t 96.08 95.40 3 47 705 0. 139 93.68 94.00 94.09 94.71. 94.7r 94.98 4 49 668 0. t54 90.01 89.90 90.70 91.57 91.57 90.32 5 29 381 0. t49 88.84 88.84 90.29 90.40 90.40 90.19 t) 497 0. t61 88.62 88.67 90.45 89.96 90.45 89.52 7 43 488 0. r54 92.76 90.52 ot to 92.91 92.97 92.73 8 39 564 0. 89.85 88.82 88.76 89.57 89.85 90.39 (¡ 30 444 0. t49 94.i8 94.18 94.12 92.80 94.18 93.42 10 324 0. t70 85.72 85.72 84.37 86.83 86.83 86.83 11 534 0. i50 91.01 91.13 89.94 89.47 91.13 89.59 T2 468 0. 89.93 89.93 88.92 89.00 89.93 88.92

13 39 521 0. .56 92.38 91.35 90.84 91.51 92.38 91 .40 14 44 550 0. o1 90.93 90,93 92.10 92.23 92.23 91.79 15 34 462 0. 39 89.84 89.84 90.98 91.15 91.i5 90.98 16 39 574 0. 51 94.64 95.49 92.96 91.89 95.49 92.68 17 357 0. 67 87.08 85.95 88.30 88.35 88.35 87,26 ()? 18 44 546 0. 52 92.97 09 9i.08 91.29 92.97 91.00 19 46 557 0. 58 93.87 92.63 92.89 93.05 93.87 93.07 20 41 677 0. 34 95.05 92.95 I3.79 94.08 95.05 95.04 21 39 543 0, 77 84.14 85.48 88.0 0 89.62 89.62 88.76 22 43 538 0. 55 91.20 90.99 e2.s1l ot (l1 92.91 93.20 a^ 17 38 536 0. 59 94.47 13.31 I 92.83 94.47 93.17 24 47 505 0. 41 93.95 94.73 s2.42192.00 94.73 92.04 31 384 0. 69 90.91 90.91 86.701 86.70 90.91 86.70 I 26 48 632 0. 60 89.85 90.76 s0.60 91 _36 91.36 91.55 | ,).1 0. 47 91.36 91.36 91.54 | 91.54 91.54 91 .44 Table A3 (colt. )

~~B. O.lvl Nurnl¡ er Total A PSA DXS-Z EXS-A LG-Z LG-A -'otL pou n( Pr iolit¡ # of Types Pieces Ratio PoIicy Policy Policy Polic¡ Heuristic Rule~~

~~Deliandei Demander B aserl~~

28 34 380 0.752 87 .11 87.11 89.50 90.37 90.37 leo.37 29 38 448 0.144 95.83 95.83 t 94.49 93.90 95.83 94.59 30 35 454 0.160 89.63 91.89 lrn.rn 92.05 92.05 92.05 31 39 507 0.157 92.81 93.37 lnr.r, 91.79 93.78 92.78 26 316 0.158 90.05 90.05 88.77 90.05 88.45 laa.zz 39 48r 0.151 8 i.19 81.19 t89.37 89.37 89.3 7 89.37 34 510 0.156 92.32 93.40 lnr.n, 93.02 93.40 92.86 35 35 455 0.140 92.06 92.06 93.46 93.46 93.46 93.29 36 365 0.165 91.80 91.80 90.81 90.07 91.80 90.81 49 548 0.137 86.16 85.23 91..52 93.35 93.35 92.54 38 25 0.145 87.26 85.31 93.64 93.99 93.99 93.99 39 44 536 0.160 93.37 93.78 91.70 91.88 93.78 91.83 40 31 384 0.155 89.18 89.18 90.54 90.54 90.54 90.54 41 43 619 0.164 93.10 92.25 92.68 92.36 93.10 92.68 (? 42 47 534 0.158 09 92.38 93.29 92.56 93.29 92.28 43 4ll 574 0.159 86.01 85.21 86.43 86.71 86.71 87.26 49 725 0.165 92.70 91.98 90.45 90.06 92.70 89.38 45 433 0.159 89.46 89.46 91.23 91.79 91.79 46 35 442 0.145 94.57 94.95 on r'7 91.64 94.95 92.11 47 0.148 93.25 93.25 92.82 92.20 93.25 9r.72 48 28 0.157 90.21 89.40 9L.23 92.07 92.07 90.45 ,10 50 7

~~140 Table A3 (cont. )~~

~~B.O.M. Nurnber Total APSA EXS-2 EXS-I LG-Z LG-A 3ompoun Pliorit¡ of Types Pieces Ratio Policy Policy Policy Policy Heu ristic Rule~~

~~Dernanded Deuranded B ased~~

58 35 412 0. 142 92.73 93.23 92.52 92.66 93.23 93.40 59 46 699 0. t55 91.01 90.86 91.10 91.91 91.91 91.45 60 29 390 0. r43 91.23 97.23 90.02 91.19 91.23 01 Áo 61 30 390 0. t58 87 .43 87.43 89.84 89.65 89.84 89,84 520 0. r78 90.43 90.43 90.07 90.07 90.43 90.07 63 31 418 0. t43 95.84 95.5 7 95.87 93.99 95.87 94.85 45 582 0. t48 95,41 95.16 94.04 94.15 95.41 94.15 65 46 707 0. 95.36 95.32 91.18 91.87 95.36 92.47 66 ót) 442 0. t56 91.67 91.67 89.55 89.55 91.67 89.44 67 43 564 0. :60 93.70 93.70 91..52 90.85 93.70 91.52 68 44 585 0. .oll 90.75 91.51 93.01 91.58 93.01 92.93 69 41 529 0. .44 96.18 96.76 95.49 95.99 96.76 o|< (¡7 70 424 0. 41 93.46 94.07 I i.53 92.24 94.07 92.24 71 40 558 0. .64 86.20 86.20 89.80 90.62 90.62 89.80 72 41 588 0. 89.23 89.23 91.00 90.68 91.00 89.78 49 637 0. 28 94.44 92.55 94.78 94.80 94.80 94.92 49 ol t 0. 92.98 01 ()9 91.93 93.64 93.64 ot o( 75 49 647 0. 46 95.30 95.16 94.2r 93.78 95.30 94.06 76 29 385 0. 40 89.88 89.88 84.70 84.70 89.88 84.70 77 39 484 0. 80 80.29 80.08 81.43 81.43 81.43 81.43 78 38 484 0. 55 94,78 93.72 92.09 91.06 94.78 92.32 70 28 0. 84 84.40 84.40 88.86 87 .ß2 88.86 87.81 80 2ß 348 0. 49 85.85 85.85 88.93 88.47 88.93 88.48 81 46 660 0. 43 94.71 94.16 94.25 94.i31 94.71 94.26 82 35 497 0. 65 94.61 94.52 93.18 s2.821 94.61 93.38

~~83 44 674 0. 64 90.77 90.58 90.09 s0.0e l 90.77 90.01~~

~~84 +l) 686 0. 68 93.95 93.46 91.46 s 0.31 I 93.95 90.75~~

~~I 85 36 459 0. 58 93.86 92.93 92.06 e3.zrj 93.86 93.11 I 86 44 545 0. 40 ot oa 92.98 94.14 s4.141 94.r4 94.14 87 406 0. 39 89.87 88.44 90.72 e0.351 90.72 91.64~~

~~141 Table A3 (cont.)~~

~~B.O.I4. Nunber Total APSA EX s- 7 ÐXS-A LG-Z l,tt--1l. Cornpounc Priority of Tvpes Pieces Ratio Policy Policy Policy Þ^li^,, Heul istic Rule Demalded Dernanded Based~~

88 27 0.182 o( 17 94.65 96.18 93.40 96.18 94.16 89 4t 572 0.146 88.83 86.98 90.18 89.28 90.18 89.52 90 48 656 0.i56 91.91 o, '71 90.74 89.54 92.7r 91.45 91 36 446 0.156 91.67 91.67 92.02 92.00 92.02 90.80 92 417 0.156 92.23 93.64 90.17 90.89 93.64 91.04 93 27 294 0.140 89.58 88.23 92.38 92.99 92.99 93.52 ol 453 0.772 88.02 88.02 87.30 87.30 88.02 87.30 or- 49 599 0.168 92.13 92.30 90.69 91.32 92.30 91.01 96 34 381 0.169 86.89 86.89 87.76 87.02 87.76 87.87 o7 46 521 0.142 93.82 94.38 93.6 i 93.93 94.38 93.58 98 3t7 0.727 92.36 92.36 on ()1 90.91 92.36 90.88 99 34 526 0.175 90.52 90.52 91.31 91.31 I i.31 90.69 100 544 0.134 95.44 94.87 93.36 92.20 s5.44 92.74

~~142 4,4 Results for Category f 4~~

~~Tal¡le 4.4: Avelage Stock Sheet Utilizations (c) in Categor.y f 4~~

~~B.O.M Number Total APSA ÐXS-2 EXS-A LG-Z LG-A Compound Priorit¡ JI of Types Pieces R.atio Policy Policy Policy Policl Heuristic Rule Derrander Based~~

~~1 29 0.058 96.71 93.59 94.11 95.40 96.7 i 98.17~~

2 529 0.066 98.63 99.01 98.37 9 7.31 99.01 98.80 3 30 332 0.076 98.34 98.05 95.50 94.55 98.34 95.31 4 30 409 0.075 97.85 97.17 94.21 oÀ.)1 97.85 (l7 <ô 5 47 529 0.065 98.20 97.37 96.74 95.80 98.20 96.97 6 330 0,077 96.15 93.40 93.82 93.01 96.i5 96.71 7 41 484 0.067 s7.40 s7.25 96.85 96.68 97.40 98.76 8 3i 402 0.079 94.96 97.59 95.22 95.38 97.59 97.4r I 26 335 0.060 97.55 97.48 96.31 95.07 97.55 97.30 10 46 517 0.069 97.66 07 ¿o 95.86 96.80 97.66 97.88 11 43 598 0,059 98.44 97.33 96.67 96.93 98.44 98.08 12 391 0.069 96.78 97.65 92.77 93.52 97.65 96.19 13 42 474 0.061 95.78 96.70 95.68 96.57 96.70 98.07 \4 587 0.072 98.63 97.69 oa 9rl 96.65 98.63 97.55 15 40 512 0.075 98.00 97.21 95.67 96.15 98,00 97.74 1t) 28 418 0.070 96.58 96.97 95.26 96.01 96.97 97.21 17 29 364 0.067 97 .1.1 96.99 95.38 95.01 97.11 97.13 18 27 0.066 97.06 97.28 oÃ t1 96.05 97.28 97.22 19 35 382 0.066 97.92 96.77 95.73 96.10 97.92 97.82 20 42 581 0.065 98.58 98.22 96.34 97.93 98.58 98.42 21 44 602 0.063 98.08 96.76 96.59 96.33 98.08 97.80 22 40 512 0.061 97.00 98.08 96.76 97.24 98.08 98.12 zt) 3\4 0.066 97 .47 96.21 92.24 91.63 97.47 96.69 24 42 596 0.068 97.97 98.99 95.83 97.95 98.99 99,45 44 621 0.063 97.69 98.49 97.10 96.79 98.49 98.29 26 38 495 0.072 97 .49 98.68 96.39 97.73 98.68 98.80 27 50 684 0.070 98.98 98.24 97.7L 96.63 98,98 97.91

~~r43 Table A4 (colt.)~~

~~B.O.M. Number Total \PSA E 2 EXS-Ä LG-Z LG-A Jorrpoun Priority # of Types Pieces Ratio Policy PoIicy Policy Policy Heuristic Rule Dernanded Demandec Based~~

28 446 0.068 97.64 97.64 94.79 94.79 97.64 96.16 2S 458 0.066 96.57 97.57 96.68 96.51 97.57 96.47 30 30 JtÐ 0.065 96.94 97.50 96.58 95.93 97.50 96.38 31 31 462 0.070 98.30 99.05 97.86 97.29 99.05 98.i1

35 510 0.071 97.45 97.57 94.67 95.60 97 .57 a7 ^7 44 597 0.059 98.18 98.04 96.84 s7.42 98.18 97.79 34 39 464 0.070 97.08 95.81 96,78 95.28 97.08 o,/,70 (o7 35 48 0.066 97.42 96.98 95.38 94.66 97.42 96.86 36 42 545 0.069 96.75 98.55 95.37 95.61 98.55 96.74 37 47 589 0.068 99.25 98.22 98.85 98.36 99.25 97.78 38 30 0.065 97.00 96.91 95.46 95.77 97.00 96.61 39 31 368 0.064 97.18 oP,,1) 94.80 94.63 97.18 07 1<

40 40 498 0.065 97.36 97.41 96.79 vÐ. J t) 97.41 96.34 41 46 ß26 0.064 oa tÃ, 97.03 98.00 97.32 98.25 97.61 42 43 508 0.069 98.60 97.92 97.87 97.24 98.60 98.49 43 45 572 0.070 98.84 98.69 98.24 97.03 98.84 98.76 44 39 436 0.064 98.31 97.71 97.07 97.72 98.31 98.71 45 44 Ð/ó 0.064 97.48 96.81 95.44 97.48 07 0t 46 28 403 0.072 97.55 96.07 95.21 95.04 97.55 96.07 28 348 0,068 97.46 97.74 96.79 96.50 97.74 97.47 48 50 653 0.071 97.81 95.77 96.85 97.81 96.96 ¿o 38 0.074 96.75 96.75 94.48 o,t ot vD. tìl 96.31 50 26 350 0.068 97.23 96.63 93.77 93.77 97.23 97.26 51 ¿t) 263 0,076 94.42 96.19 93.84 92.90 96.19 92.89 52 40 437 0.070 96.81 98.38 07 7À 95.81 98.38 98.31 53 28 0.069 s8.22 96.85 96.39 96.61 98,22 97.58 54 40 5r4 0.067 98.29 98.27 96.70 97.79 98.29 98.09 55 38 õ4t 0.074 95.99 95.91 95.32 95.04 95.99 96.98 56 467 0.075 07 11 o'7 1t 94.91 94.44) 97.21 95.77

~~28 4TI 0.075 95.86 97.85 94.84 e6.0s l 97.85 98.20~~

~~744 Tabìe A4 (cont.)~~

~~B.O.M Nulr''ber Total APSA EXS-2 EXS-A LG-Z LG-A 0orrpou n( Priorit¡ # of Types Pieces Ratio Policy Policy ?olicy Policl Heuristic Rule Demanded Detrander Based~~

~~58 44 581 0.065 98.38 98.28 96.66 96.09 98.38 96.61~~

~~65 28 37t 0.072 97.57 s7.07 95.05 95.72 s7.57 97 .20~~

tr l) .J DI) 0.065 96.42 94.70 94.51 94.97 96.42 96.45 ß7 44 507 0.063 98.62 98.64 96.68 96.81 98.64 98.02 68 280 0.069 97.11 94.95 94.38 95.21 97.11 94.62 69 50 666 0.069 97.86 96.54 95.78 96.09 97.86 98.19 70 28 430 0.074 96.65 97.29 92.56 94.90 97.29 97.96 7t 36 478 0.070 96.81 98.69 96.99 96.39 98.69 99.4i 72 47 636 0.066 97.62 97.63 97.51 96.71 97.63 97.70 31 474 0.062 97.27 97.69 97 .04 96.84 97.69 98.18 40 475 0.066 98.78 98.73 97 .47 97.19 98.78 s8.72 75 42 530 0.065 98.44 98.61 97.62 97.91 98.61 98.67 76 38 457 0.057 96.70 96.73 96.00 96.47 vD.ló 96.95

77 29 369 0.057 98.52 98.15 96.83 96.93 98.52 97 ,17 78 50 651 0.066 94.80 98.37 96.1.2 97.16 98.37 98.41 70 41. 565 0.070 9D, ( I 95.57 96.76 95.69 96.76 98.11 80 505 0.068 98.18 98.16 98.00 98.28 98.28 97.60 81 28 430 0.061 99.31 99.21 97.70 97.16 99.31 99.15 82 to 380 0.070 97.80 99.03 97.09 98.32 99.03 99.03 83 4l) 674 0.062 97,25 97.74 07.1 I 96.97 07 to 98.53 84 Jt) 507 0.064 98.75 98.77 97.19 97.18 98.77 98.45

~~85 42 539 0.063 97.64 97.97 97.16 9ti _ 63 97.97 97.81 l 86 30 329 0.074 96.62 97.73 95.66 l I5.89 97 .73 96.04~~

~~87 28 0.064 97,18 97.90 e4.s4l 95.37 97.90 9 7.01-r~~

~~145 Table A4 (cont. )~~

~~B.O.M Number Total AP SA EXS-2 DXS-A LG-Z LG-A -ìompound Priority JT of Types Pieces Ratio PoIicy Policy Policy Policy Heuristic Rule~~

~~Dernanded B ased~~

88 28 0.05 6 98.56 98.38 97 .64 95,17 98.56 97.60 89 40 564 0.060 96.35 97.78 97 .27 97 .49 97.78 97.70 90 28 387 0.067 98.04 oo t1 97.63 98.65 s9.21. 99.43 91 42 550 0.062 97.09 97.57 95.96 94.72 97 .57 96.83 ot 34 525 0.066 96.31 98.00 96.32 96.41 98.00 98.16 93 39 526 0.066 96.13 98.48 97.34 97.98 98.48 96.12 94 40 544 0.06 i 97.56 99.33 97.12 98,70 99.33 99.01 95 4t 567 0.059 97.84 99. i8 98.77 98.38 99.18 oo 9Í¡ 96 50 649 0.061 97 ,17 96.75 96.13 97.54 97.54 98.17 o,7 48 622 0.067 98.28 97.74 96.30 96.20 98.28 97.93 98 26 JòI) 0.061 97.03 97.39 95.88 96.88 97.39 96.52 oo 29 364 0.066 97.70 97.38 95.34 95.03 97.70 95.41 100 4ß 619 0.068 96.81 97.14 94.96 95.02 97.74 I6.56

~~146 Ä',5 Results for Category f 5~~

~~Table 4.5: Avelage Stock Sheet Utilizations (e) in Categor.y f 5~~

~~B.O.M Nulrber Total APSA ÐXS-2 EXS-A LG-Z LG.A lo tn po11n¿ Priorit¡ JI of Types Pieces Ratio PoIicy PoLicy Policl PoIic¡ Heuristic Rule Demander Based~~

1 423 0.025 97.67 97.54 95.15 97.22 97.67 99.50 2 36 566 0.028 96.60 99.61 98.73 97.12 99.61 oo a/î 3 44 589 0.024 99.84 99.88 98.91 95.77 99.88 99.66 4 27 0.028 98.5 i 98.83 92.98 92.98 98.83 99.17 5 50 6i1 0.025 98.39 99.91 97.96 99.8 i oo o1 99.89 6 48 660 0.025 99.69 99.72 97.32 99.84 99.84 99.41 7 35 462 0.026 98.78 99.18 99.05 96.51 99.18 99.26 8 47 614 0.025 95.51 99.50 95.29 98.70 99.50 99.58 I 44 555 0.023 99.90 oo o,1 99.62 97.02 99.94 oo 70 10 tq 404 0.025 98.66 98.30 97.48 98.53 98.66 99.09 11 48 667 0.026 99.52 99.50 97.74 99.28 99.52 99.74 72 49 625 0.027 98.10 97.39 96.88 97.99 98.10 98.79 13 43 590 0.030 99.44 99.15 98.52 98.66 99.44 99.25 1.4 46 663 0.024 98.90 97.88 98.96 98.86 98.96 99.2i 15 38 4ðO 0.025 98.87 99.73 98.77 98.88 99.73 99.57 tb 38 545 0.025 99.90 99.92 99.87 99.69 oo ot 99.69 17 389 0.030 95.36 97.08 96.97 96.70 97.08 99.23 18 391 0.027 95.01 96.06 98.44 96.06 98.44 98.81 1() 49 632 0.024 95.31 99.23 96.48 99.06 99.23 99.53 20 25 318 0.025 95.8 7 97.98 96.34 95.89 97.98 97.80 21 29 375 0.027 98.92 98.76 96.53 92.09 98.92 98.53 22 447 0.024 98.42 99.84 99.48 98.78 99.84 98.76 40 585 0.025 98.93 98.38 98.37 99.04 99.04 98.97 24 36 427 0.027 v,5. òJ 94.75 99.09 93,96 99.09 99.65 25 36 4i3 0.023 99.93 99.88 oo oa 99.91 oo oa 99.82 2ß 425 0.029 94.86 99.67 98.54 99.13 99.67 99.45 27 +ò 646 0.027 99.55 99,82 99.53 99.56 oo er) 99.86

~~747 Table A5 (cont. )~~

~~B.O.M. Number' Total APSA EXS.2 EXS-A LG-Z LG.A Sorn Pou n( P rroÌrtJ # of Types Pieces Ratio Policy Policv Policy Policy Heuristic Rule Demanded Dern a¡ r'ler Based~~

28 29 3r4 0.026 98.67 98.95 94.81 94.81 98.95 98.50 29 34 547 0.026 98.99 98.68 98.62 97.97 98.99 99.23 30 436 0.026 99,37 99.54 94.49 98,67 99.54 99.54 31 28 353 0.026 99.38 91.64 I6.57 99.20 99.38 99.38 25 336 0.024 95.44 96.91 90.67 98.75 98.75 98.78 406 0.025 98.92 99.37 99.08 98.52 99.37 99.37 34 30 429 0.023 98.51 oo 9? 97.46 97.31 99.27 98.84 lti 420 0.023 99.45 oo ¡í,1 oo tt 97.65 99.54 99.14 36 50 614 0.026 99.15 98.46 98.70 98.43 99.15 99.41 31 428 0.023 07,10 96.93 93.42 97.38 s7.49 98.18 38 44 624 0.026 99.00 99.19 98.74 96.22 99.19 99.13 .)a 39 ,7¿ 416 0.023 99.38 97.72 oo 98.96 99.38 99.56 40 384 0.024 99.80 97.99 91.25 99.34 99.80 98.55 41 31 +t) ( 0.027 99.28 99.37 95.26 97.83 99.37 98.61 42 48 675 0.025 99.46 99.46 99.21 99.26 99.46 99.48 43 40 485 0.028 98.71 99.00 99.46 99.20 99.46 oo 90 44 26 360 0.022 97.86 99.67 98.11 96.00 99.67 97.93 45 38 507 0.026 99.37 99.05 99.66 99.16 99.66 99.79 46 536 0.026 96.25 98.38 96.77 98.38 98.38 99.05 47 0.023 92.61 92.19 90,25 90.27 92.61 99.48 48 30 410 0.028 97.13 97.02 96.80 98.62 98.62 99.30 49 43 608 0.028 99.09 oo 11 99.06 98.11 99.41 99.30 50 49 730 0.029 98.08 96.07 98.19 95.27 98.19 98.82 51 437 0.027 97.69 97.19 96.96 94.73 97.69 97.33 52 36 465 0.027 99.88 96.40 98.95 99.45 99.88 99.61 40 505 0.024 98.33 98.59 98.36 98.73 98.73 oo t1 54 34 513 0.023 97.20 99.52 97.25 96.65 99.52 98.30 55 47 572 0,026 07 90 97.41. 99.36 g9.44 99.44 99.63 56 42 586 0.026 99.91 oo ot 99.61 99.30 99.92 99.85 57 42 471 0.030 oo 10 99.54 98.50 98.54 99.54 99.64

~~148 Table A5 (cont.)~~

~~B.O.M Nunber' Total AP SA EXS-Z EXS-A LG-Z LG.A Jornpound Priolity of Types Pieces Ratio Policy Policy Policy Polic¡ Heuristic Rule ) ernan rled Detranded Based~~

~~58 36 430 0.026 98.10 07 0t 97.89 98.65 98.65 98.85 59 30 369 0.025 96.12 98.04 96.98 89.57 98.04 98.74~~

60 27 380 0.025 98.5 7 98.67 97 .84 97 .32 98.67 9 8.54 61 455 0.026 96.86 97.36 97.51 97.95 97.95 99.53 62 42 534 0.024 96.80 98.25 98.58 98.57 98.58 98.60 63 4t4 0.028 99.84 99.41 98.37 96.05 99.84 99.80 64 44 556 0.026 97.03 97.40 97.76 97.65 97.76 99.84 65 40 479 0.028 99.73 99.97 99.73 99.79 oo (ì7 99.97 66 45 570 0.023 98.68 98.71 97.86 98.35 98.71 99.83 67 31 364 0.025 94.16 96.44 96.46 98.68 98.68 98.66 68 26 311 0.022 99.58 99.41 95.18 98.90 99.58 99.50 69 411 0.026 95.77 oo ¿K 99.66 99.18 99.66 oo at 70 30 388 0.024 91.31 98.77 98.68 98.96 91.31 99.51 71 ¿I 334 0.027 97.85 98.62 97.40 o7 t1 98.62 98.89 72 45 586 0.024 96.90 98.67 98.61 93.24 98.67 99.42 73 30 394 0.027 93.24 93.3 7 93.62 97.23 99.19 74 407 0.024 95.93 t¡o ,11 94.97 07 tIi 99.41 99.62 75 39 498 0.028 97.93 98.87 91.67 96.92 98.87 98.23 76 518 0.026 98.95 99.29 95.89 98.73 99.29 99.21 77 50 636 0.026 99.59 99.85 97.53 99.36 99.85 99.66 78 to 344 0.031 97.10 97.40 97.76 98.01 98.01 98.83 79 30 392 0.029 98.67 99.10 97.10 97.75 99.10 99.01 80 30 0.028 99.94 99.88 99.65 99.86 99.88 99.88 81 49 605 0.024 99.86 99.96 99.83 99.68 99.96 99,73 82 368 0.023 99.06 99.55 93.88 99.35 99.55 99.07 83 40 517 0.029 98.92 oo t,4 98.92 98.54 99.24 99.50 84 46 590 0.024 99.88 99.78 99.35 99.59 99.88 99.41 85 48 647 0.025 99.60 99.36 98.58 98,83 99.60 99.32 86 49 624 0.023 99.02 99.40 98.93 98.89 99,40 99.79

~~87 404 0.023 98.57 oo /e 98.i 1 98.80 99.48 99.36~~

~~149 Table A5 (cont.)~~

~~B.O.M Number Total A,PSA EXS-7 EXS-A LG-Z LG-A Compounr Priorit¡ # of Types Pieces R.¿,tio Policy Policy PoIicy Policy Heulistic Rule~~

~~Dernan d ed Demander B ased~~

88 34 J/¿t 0.029 98.79 99.82 99.26 99.12 oo at 99.92 (o 89 Ji) 452 0.026 98.65 oo s9.22 98.92 99.59 99.i6 90 42 523 0.024 95.73 99.59 99.06 98.65 99.59 99.54 91 28 303 0.026 96.85 94.95 95.14 93.05 96.85 94.84 ot 46 580 0.024 98.35 98.41 97.74 97.79 98.41 98.98 93 50 594 0.027 99.37 99.09 98.38 99.14 99.37 99.03 94 45 0.027 99.08 99.29 99.03 98.61 99.29 99.73 349 0,025 99. i4 99.10 96,74 96.95 99.14 97.25

96 J l-) 572 0.026 98.05 95.86 95.11 95.71 98.05 98.60 97 28 382 0.024 99.00 99.13 97.15 97.09 99.13 98.82 98 30 378 0.024 98.14 97 .70 95.01 95.01 98.14 96.51 oo 30 383 0.029 99.86 98.61 98.70 99.86 99.97 t00 30 to7 0.027 99.12 oo t1 97.85 97.96 99.21 99.21

~~150 ,4.,6 Results for Category f 6~~

~~Table 4.6: Average Stock Sheet Utilizations (ã) in Categor-y f 6~~

~~B.O.M. Nurnber Total APSA EXS-2 EXS-A LG-Z LG-A Cornpound Priority # of Types Pieces Ratio Policy Policy Polcy Policy Heuristic Rule Dernanded Demanded Based~~

l 40 563 0.014 97.57 98.45 98.29 98.63 98.63 99.20 2 35 407 0,014 99.95 oo oo 99.67 99.60 99.99 100.00 3 30 443 0.016 99.72 98. i3 97.64 98.90 99.72 99.50 4 41 554 0.014 99.99 99.93 oo oo oo ot 99.99 99.93 5 304 0.014 99.13 98.98 98.79 98.44 99.13 98.04 6 485 0.014 91.37 92.58 94.01 95.50 95.50 99.61 7 39 0.013 95.55 98.67 98.04 96.31 98.67 98.67 8 563 0.013 99.84 99.73 99.53 98.94 99.84 99.79 9 48 644 0.014 97.89 99.11 99.19 oo ta oo ta 98.96 10 26 338 0.014 94.35 95.90 96.28 99.08 98.08 96.72 11 45 570 0.015 86.32 96.28 86.54 93.83 96.28 99.25 t2 392 0.015 94.36 93.66 94.36 93.89 94.36 99.55 13 40 450 0.013 99.97 oo at 99.37 95.77 99.97 99.83 14 45 663 0.015 100.00 100.00 99.91 99.91 100.00 100.00 15 43 583 0.012 98.05 98.35 98.49 96,77 98.49 99.54 lb 501 0.013 99.97 99.68 98.82 s9.22 oo (ì7 99.68 17 28 J1Ð 0.013 98.96 99.86 98.34 98.68 99.86 99.86 18 49 588 0.014 99.26 98.38 99.55 99.18 99.55 98.71 19 45 640 0.013 oo no 99.82 99.36 99.51 oo at 99.82 20 456 0.015 99.69 100.00 99.45 99.60 100.00 100.00 21 40 i)J I 0.012 99.90 100.00 99.61 99.75 100.00 oo ?o 22 389 0.014 100.00 99.94 99.61 99.88 100.00 99.94 34 4ó l) 0.015 99.84 99.96 98.72 99.73 99.96 99.76 24 30 369 0.012 oo oo 99.83 99.91 99.83 99.99 99.71 ¿Ð 27 339 0.014 90.43 99.92 94.8 0 97.98 oo ot 97.50 l g9.74 26 26 301 0.014 99.75 90.83 | 98.55 99.75 99.46 27 522 0.014 99.64 99.75 ee.07l 99.i0 99.75 99.75

~~151 Table A6 (cont.)~~

~~B.O.M Nurnber Total APSA EXS-Z EXS.A LG-Z LG-A Cornpound Pliolity # of Types Pieces Ratio Policy PoIicy Policv Policy Heulistic Rule Derranded Derrander Baserì~~

28 ÐJ I 0.014 99.35 99.66 99.43 98.97 99.66 99.62 t(¡ 30 424 0.015 96.17 97.03 97.21 95.48 97.21 99.52 30 44 622 0.014 99.83 99.75 99.92 99.51 oo ot 99.89 31 35 499 0.013 82.86 91.65 83.73 87.77 91.65 97.01 479 0.013 98.31 99.81 98.86 98.76 99.81 99.81 47 610 0.013 99.89 99.86 94.91 99.75 99.89 99.80 34 49 616 0.014 96.55 99.71 95.99 99.58 99.7i 98.01 26 328 0.013 99.83 99.52 97.30 84.35 99.83 99.52 Jt) 40 490 0.013 99.75 oo 7a 96.42 97.19 99.78 99.78 45 596 0.013 95.23 99.66 99.37 94.80 99.66 99.71 38 35 480 0.015 94.54 89.55 95.47 95.95 95.95 oo 7l 39 44 629 0.012 100.00 100.00 99.60 99.95 100.00 99.95 40 299 0.013 94.95 96.94 9ß.72 96.54 96.94 98.79 4I ta 471 0.012 98.32 99.58 98.45 97.03 99.58 99.55 42 45 631 0.013 99.74 99.67 93. i5 99.27 oo 71 99.77 43 39 540 0.0 i5 oo to 99.68 98.87 99.68 99.68 99.68 44 0.014 99.16 99.53 98.00 99.26 99.53 99.36 45 46 61i 0.015 98.21 100.00 98.95 99.59 100.00 100.00 46 29 346 0.014 100.00 99.97 100.00 99.86 100.00 99.97 9ô 0.013 97.61 99.36 98.56 99.36 99.36 99.36 48 26 361 0.015 99.83 99,73 99.20 99.64 99.83 99.93 49 44 621 0.012 95.39 95.39 99.41 99.47 99.47 99.83 50 29 340 0.014 92.06 94.22 ot ta 94.34 94.34 99.62 51 386 0.012 99.79 100.00 99.79 97.94 100.00 i 00.00 443 0.014 98.21 99.70 98.47 99.15 99.70 99.14 50 683 0.015 99.73 99.99 99.24 oo o,1 99.99 99.80 54 46 594 0.015 94.89 95.95 99.55 I9.02 99.55 99.58 55 28 414 0.014 93.76 95.47 93.69 I9.46 99.46 99.79 56 616' 0.013 99.50 99.39 99.48 99.87l 99.87 99.39 50 681 0.014 99.78 99.99 99.76 ee.60l oo oo 99.93

~~r52 Table A6 (cont.)~~

~~B.O.M Nurnber Total APSA EXS-2 EXS.A LG-Z LG-A Jompound Priority JI of Types Pieces Ratio Policy PoIicy Policy PoLicy Heuristic Rule~~

~~Demanded Demander B ased~~

58 48 685 0.0 L4 99.50 99.88 99.06 99.37 99.88 99.88 rio 389 0.0 95.20 96.55 98.26 98.09 98.26 96.70 60 39 520 0.0 12 99.60 99.67 99.20 99.56 99.67 99.67 61 41. 596 0.0 t5 95.09 98.77 96.43 99.04 oo n/ 98.98 62 350 0.0 t4 99.58 99.63 93.91 93.72 99.63 99.63 oó 43 501 0.0 t5 99.95 100.00 99.98 100.0t 100.00 100.00 o4 41. 455 0.0 l4 99.87 99.71 99.90 91.69 99.90 oo 7(l 65 43 538 0.0 95.24 99.94 95.39 99.68 99.94 99.47 bl) 44 0.0 12 99.23 89.94 99.18 98.03 99.23 99.82 67 271 0.0 .6 98.77 99.77 oo t¿ 99.57 99.77 99.77 68 39 510 0.0 .3 99.65 99.88 99.74 98.88 99.88 100.00 69 30 415 0.0 .4 oo 70 99.9 i o0 Ã9 92.82 99.91 99.55 70 34 383 0.0: 4 99.92 100.00 oo rî¿ 100.0c 100.00 100.00 71 27 292 0.0 5 96.46 84.17 94.41 94.25 96.46 97.91 72 28 310 0.0 3 95.35 95.83 95.51 97.68 97.68 98.03 49 696 0.0. 5 99.91 99.95 99.90 99.87 99.95 99.91 74 25 326 0.0. 5 97.39 99.60 98.82 97.31 99.60 98.68 75 27 340 0.0 i 5 100.00 100.00 100.00 100.00 100.00 100.00 76 47 569 0.01 95.20 99.89 95.86 99.40 99.89 99.86 77 49 660 0.01 3 98.09 99.98 99.57 99.88 99.98 99.98 78 49 691 0,01 3 99.43 99.70 99.55 99.59 99.70 99.70 79 622 0.0l 99.76 93.32 99.06 99.62 99.76 99.78 80 30 368 0.01 96.11 99.76 98.49 98.64 99.76 99.76 81 43 602 0.0l 5 95.79 99.70 99.01 98.00 99.70 99.24 82 652 0.0l 98.85 98.88 99.79 98.85 99.79 98.88 83 J¿) 427 0.01 99.77 99.83 99.44 99.59 99.77 99.98 84 26 308 0.01 99.78 93.82 99.18 98.06 99.78 99.66 85 50 652 0.0l 91.62 99.39 97.35 98.38 99.39 99.69 86 27 3i9 0.0 r 99.82 95.2r 99.23 99.04 99.82 99.43 87 38 531 0.01 99.93 100.00 98.15 99.88 100.00 99.96

~~1l<9 Table A6 (cont.)~~

~~B.O.M. Nurnber Tot al APSA EXS-Z EXS-A LG-Z LG.A Jornpotld Priorit¡ # of Types Pieces Ratio Policy Policy Policy PoIicy Heuristic Rule Dernandei Dernanded Based~~

88 37 471 0.0 t3 99.29 99.75 98.87 98.66 oo 7ri 97.85 89 542 0.0 t5 96.25 98.30 95.19 98.54 98.54 99.84 90 50 704 0.0 l2 s4.42 94.39 93.81 94.39 94.42 99.95 91 394 0.0 t5 99.50 o< t,i 99.59 99.96 oo o/î 95.24 0, 38 583 0.0 t4 99.76 99.50 99.00 98.94 99.76 99,51 93 45 649 0.0 l4 100.0i i00.00 99.98 100.0c 100.00 100.00 0,,t 40 ì)t+ 0.0 t5 99.88 100.00 98.90 99.95 i00,00 100.00 o( 27 290 0.0 t6 99.31 100.00 99.17 99.93 100.00 100.00 96 49 610 0.0 .4 99.44 99,84 98.20 99.46 99.84 99.33 o7 436 0.0 .6 99.64 99.49 99.28 99.70 99.70 99.11 98 28 352 0.0 99.84 99.87 99.63 99.40 99.87 95.37 o0 45 ß25 0.0 J 97.78 100.00 100.0( 99.98 100.00 100.00

~~100 44 660 0.0 99.81 9 9.86 99.82 oo 7( 99.86 99.86~~

~~154 L.7 Results fol Category f 7~~

~~Table 4.7: Average Stock Sheet Utilizations (n) in Category S 7~~

~~B.O.M Ntrnber Total APSA E \¡e t 0,xs-A LG-Z LG-A Jonlpound Priority of Types Pieces Ratio Policy PoIicy PoLicy PoIicy Heulistic Rule Detnander Demauded Based~~

1 404 0.008 99.45 99.95 99.72 99.45 99.95 99.95 2 45 6L2 0.007 100.0c 100.00 oo (l7 100.0( 100.00 100.00 .J 47 635 0.007 100.0c 99.91 100.0( 99.87 100.00 oo (l1 4 35 462 0.007 99.83 99.83 99.60 99.41 99.83 99.83 5 34 516 0.006 98.74 99.67 98.55 99.94 oo o,t 99.67 6 40 516 0.007 100.00 100.00 100.0( 100.0( 100.00 100.00 7 38 468 0.008 99.80 99.80 99.80 99.85 99.85 99.80 8 48 636 0.008 99.82 96.13 oa to 99.32 99.82 96.13 I ¿o 634 0.007 99.94 100.00 99.98 100.0t 100.00 100.00 10 31 405 0.008 99.73 99.86 99.04 99.68 99.86 99.86 11 415 0.007 94.37 94.29 99.92 oo e7 oo ot 99.92 12 30 394 0.008 95.11 98.28 95.18 I /.Ðl) 98.28 98.63 13 48 t)tl4 0.007 100.00 100.00 99.98 100.0c 100.00 100.00 1.4 25 0.008 99,92 99.75 99,39 99.70 99.92 99.75 15 to 393 0.008 99.63 99.76 99.07 98.63 99.76 99.76 16 31 352 0.007 100.00 98.79 99.68 98.11 100.00 98.79 17 25 329 0.007 99.79 79.80 99.48 99.85 99.85 99.73 18 316 0.008 100.00 99.67 100.00 99.67 100.00 99.67 19 43 0.008 100.00 99.97 99.81 99.78 100.00 99.97 20 27 296 0.006 86.51 i00.00 oo (t 98.76 100.00 100.00 21 454 0.007 100.00 10 0.0 0 99.84 oo t1 100.00 100.00 22 47 594 0.008 99.87 100.0 0 99.87 100.00 i 00.00 100.00 to 354 0.007 99.77 99_40 96.76 98.02 99.77 99.49 24 578 0.006 100.00 100.001 100.00 oo 07 100.00 100.00 ¿i) ,J+ 44i) 0.007 99.82 100.001 99.85 99.99 100.00 100.00 26 41. 560 0.008 99.61 99.?6 97.27 99.33 99.76 99.76 | 27 493 0.00 7 100.00 r00.00 | 100.00 r 00.00 100.00 10 0.00

~~155 Table A7 (cont. )~~

~~B.O.À4. Nurnber Total A PSA ÐXS-Z ÐXS-A LG-Z LG-A Co t'll poun(] Priorit # of Types Pieces R.a.tio Policy PoLicy Policy Policy Heuristic Rule Dern an dec Dernanded Based~~

28 39 572 0.008 100.0t i 00.00 100.0t | 10 0.0t 100.00 100.00 I 29 42 500 0.007 oo ol< oo oo 99.93 lqe.qr 99.99 99.99 30 42 536 0.007 98.61 99.83 99.24 I nn.r, 99.83 99.83 31 40 489 0.007 100.0( 100.00 100.0t hnn.n, 100.00 100.00

I 48 583 0.007 97.38 99.58 98.92 99.61 99.6 i 99.58 357 0.006 97.06 99.25 99.25 I9.99 99.99 99.25 34 428 0.007 95.20 95.80 97.76 97.11 97.76 95.80 300 0.007 99.97 99.39 98.59 99.66 oo o7 99.39 36 38 501 0.006 99.06 98.05 98.32 97.06 99.06 98.05 48 590 0.007 99.84 100.00 99.18 100.00 100.00 100.00 38 50 591 0.007 98.83 99.52 99.36 99.37 99.52 99.52 39 43 595 0.007 100.00 99.92 99.97 99.91 100.00 oo 09 40 0.008 98.82 99.76 99.43 99.00 99.76 99.76 47 42 500 0.007 100.00 100.00 100.00 100.00 100.00 100.00 42 30 464 0.007 99.90 99.79 98.01 99.87 99.90 oo 70 39 464 0.006 100.00 100.00 99.98 100.00 100.00 100.00 44 41 580 0,009 99.91 99.94 99.91 oo o,1 99.94 oo o¿ 45 38 540 0.007 100.00 99.94 100.00 99.54 100.00 oo o¿ 4t) 45 553 0.007 100.00 100.00 r00.00 99.98 100.00 100.00 47 AO 609 0.007 100.00 i 00.00 99.80 100.00 100.00 100.00 48 ß41 0.007 100.00 100,00 98.33 99.87 100.00 100.00 49 48 587 0.007 97.94 99.93 99.01 99.13 99.93 99.93 50 38 465 0.007 100.00 100.00 100.00 100.00 100.00 100.00 51 31 376 0.007 99,25 99.96 99.66 99,96 99.96 99.96 52 ó¿ 428 0.007 95.20 95.80 97.76 97.11 97.76 95.80 òJ 'K 300 0.007 99.97 99.39 98.59 99.66 99.97 99.39 54 38 501 0.006 99.06 98,05 98.32 97.06 99.06 98.05 55 48 590 0.007 99.84 100.00 99.18 100.00 100.00 100.00 56 50 591 0.007 98.83 oo

~~156 Table A7 (cont. )~~

~~B.O.M. Ntrnber Tot al A PSA ÐXS-Z EXS-A LG-Z LG-A -oltì Poù nc Priorit¡ of Types Pieces Ratio Policy Policy Policy Policy Heuristic Rule Dernanderi Demanded Based~~

58 31 0.008 98.82 99.76 199.43 lee.00 99.76 99.76 lío I 42 500 0.007 i 00.00 100.00 100.0( Iroo.or 100.00 100.00 I 60 30 464 0.007 99.90 99,79 98.01 I9.87 99.90 99.79 61 39 464 0.006 100.00 100.00 99.98 10 0.0c 100.00 100.00 62 41 580 0.009 99.91 99.94 99.91 99.94 oo o,1 99.94 63 38 540 0.007 100.00 99.94 100,0( I9.54 100.00 99.94 45 553 0.007 100.00 100.00 100.0( 99.98 100.00 100.00 65 49 609 0.007 100.00 100.00 99.80 100.00 100.00 100.00 DD 47 64t 0.007 100.00 100.00 98.33 99.87 100.00 100.00 67 48 587 0.007 07 0^ 99.93 99.01 99.13 99.93 99.93 68 38 4DÐ 0.007 100.00 100.00 100.0t 100.00 100.00 100.00 69 31 376 0.007 99.25 99.96 99.66 99.96 99.96 99.96 70 38 491 0.007 98.14 99.78 98.18 98.63 99.78 99.94 7l 38 451 0.007 99,94 99.94 99.44 99.94 99.94 99.77 72 288 0.007 99.86 99.77 99.58 98.58 99.86 99.53 73 432 0.007 99.66 99.53 97.43 99.7 4 99.74 99.76 74 29 351 0.008 95.29 99.76 99.76 99.76 99.76 98.23 75 30 391 0.008 98.12 98.23 99.98 99.82 99.98 99.92 It) 36 446 0.008 99.87 oo ()9 99.73 99.83 99.92 100.00 77 31 455 0.006 100.00 100.00 100.00 100.00 i 00.00 100.00 78 28 365 0.008 90.76 100.00 99.93 100.00 100.00 100.00 70 44 596 0.008 100.00 100.00 100.00 100.00 100.00 100.00 80 50 591 0.008 10 0.0 0 100.00 100.00 100.00 100.00 99.83 81 31 0.007 100.00 99.83 99.66 99.83 100.00 100.00 82 40 468 0.008 10 0.00 100.00 99.62 99.98 100.00 98.85 83 388 0.008 99.84 98.85 99.84 98.85 99.84 100.00 84 549 0.007 100.001 100.00 100.00 99.9 7 i 00.00 96.73 85 28 338 0.006 e7.17 96.73 98.03 93.71 98.03 96.76 | 86 28 355 0.007 95.61 I yl).ll) 97.57 93.98 97.57 100.00 I l t),-t 87 50 / 0.0 07 99.95 I 100.00l 99.65 99.85 | i 00.00 I6.32

~~157 Table A7 (con t. )~~

~~B.O.M. Ntrrber Total A PS ,A EXS-2 Ðxs-A LG-Z LG-A Compound Pr ioÌitl II of Types Pieces R.a.tio Policy Policy Policy Policy Heu ris t ic Rule~~

~~Demander Demandei B ased~~

88 48 0.007 94.25 99.81 98.73 99.63 99.81 93.14 89 29 319 0.008 94.25 93.14 93.65 98.75 98.75 100.00 90 38 51i 0.007 100.00 100.00 100.0( 100.0( 100.00 99.87 91 36 450 0.007 99.90 99.87 99.91 99.74 99.91 100.00 92 42 519 0.007 100.00 100.00 100.0( 99.97 100.00 99.85 93 48 638 0.007 99.93 99.85 98.i7 99.91 99.93 100.00 94 31 402 0.008 100.00 100.00 99.91 99.31 100.00 100.00 95 J/¿t 0.007 100.00 100.00 100.0( 100.0t 100.00 100.00 96 46 573 0.008 100.00 100.00 99.47 99.56 100.00 100.00 97 36 461 0.008 100.00 100.00 100.0t 100.0i 100.00 99.81 98 46 ÐÐJ 0.007 97.53 99.81 98.39 96.76 99.81 100.00 99 42 518 0.007 100.00 100.00 100.0c 100.0i i 00.00 99.89

~~158~~