THE EFFICIENCY OF COMPUTER ALGORITHMS FOR PLANT LAYOUT Thesis for the Degree of D. B. A. MICHIGAN STATE UNIVERSITY LARRY P. RITZMAN 1968 1.333.324 R. Y Michigan State University This is to certify that the thesis entitled THE EFFICIENCY OF COMPUTER ALGORITHMS FOR PLANT LAYOUT presented by LARRY P . RITZMAN has been accepted towards fulfillment of the requirements for D.B.A. degree mm” MW 0 7 Major professor Date JULY 18, 1968 0-169 _‘\/—-—»_“ : «Indiana av IIOAB & SONS' BOOK BRIBERY INC. LIBRARY BINDERS SPIIHOPDIT, MICHIGAN \v I=~ .1; A4 ABSTRACT THE EFFICIENCY OF COMPUTER ALGORITHMS FOR PLANT LAYOUT by Larry Paul Ritzman This thesis provides a comparative appraisal of suboptimal computer algorithms for the plant layout problem. The research objective is determining which ones output the best solutions and whether their performances are dependent upon the specific problem. The layout problem is viewed as assigning centers to locations to minimize a cost function, subject to certain constraints. Its mathematical formula- tion recognizes three types of costs: linear, special quadratic, and general quadratic. This formulation accommo- dates the objectives usually attributed to a layout. Al- though most algorithms deal explicitly with only the special quadratic costs, the other two cost components can be accounted for with prohibited assignment constraints and transformations to the cost data. The existing algorithms which are amenable to this formulation, or can be revised to do so, are examined. Particular attention is paid to their theory, strengths, weaknesses, and omissions. Four algorithms are selected for Larry Paul Ritzman further study: CRAFT, Hillier's algorithm, Wimmert's proce— dure, and a random selection algorithm. Computer programs had not been available for the last two algorithms, and so they are written specifically for this thesis. Since several concepts of unknown merit are added to Wimmert's original formulation, thirteen versions of it are developed for evaluation. The algorithms are then applied to twenty-six realis- tic test problems using the CDC 3600 computer. The resulting output provides data for comparing the algorithms' computa- tional time, abilities to satisfy constraints, and the cost- liness of their solutions. The test results support the conclusion that the better algorithms are consistently good, regardless of the problem characteristics. CRAFT performs better than any other algorithm in terms of solution feasi- bility, solution cost, computer time, and the ability to produce many good solutions to the same problem. Hillier's algorithm is competitive with CRAFT; the differences are not significant. The total performance of the random selection algorithm is inferior, in spite of the small amount of time it consumes per solution. The results for most of Wimmert's versions are not encouraging. However, two versions do provide satisfactory solutions. In terms of average solution costs, the differ- ences between them and CRAFT are not statistically signif- icant. Although they require much more computer time than Larry Paul Ritzman CRAFT, several modifications are suggested which may signif- icantly improve their total performances. The findings of this thesis offer several insights tangential to the main research objective. Even the better algorithms intermittently generate poor solutions to the same problem. This conclusion underscores the need for find- ing several suboptimal solutions to a problem, which in turn requires some type of stopping rule. Preliminary evidence suggests a satisfactory rule could be constructed from information on the lower and upper bounds as well as by monitoring output information during the actual solution process. Another finding of interest is that alternative criteria for computing distances between locations provide comparable results. Several areas for future research are described, including: revisions to existing algorithms, satisfying unequal area requirements, and developing new algorithms. THE EFFICIENCY OF COMPUTER ALGORITHMS FOR PLANT LAYOUT BY Larry PL Ritzman A THESIS Submitted to ,Michigan.State University in partial fulfillment of the requirements for the degree of DOCTOR OF BUSINESS ADMINISTRATION Department of Management 1968 .2 ’1 z , ‘ .' > 'f/ *[y _,.I L/‘t' " //’3 /r; 7 I I L9 Copyright by LARRY PAUL RITZMAN 1969 ACKNOWLEDGMENTS As this thesis progressed from its conception to completion I received the aid and advice of several persons. I sincerely appreciate the generous assistance given by: Professor John Muth, chairman; Professor Stanley Bryan; and Professor Richard Gonzalez. I owe a special debt to Dr. Muth for his invaluable insights and guidance. The author also wishes to thank the Ford Foundation for the financial support during the period of research. My constant gratitude goes to my wife, Barbara, for her aid, sacrifice, and encouragement. This thesis is dedicated to her. iii Chapter I. II. III. IV. TABLE OF CONTENTS THESIS OBJECTIVE.AND PROBLEM DESCRIPTION Introduction . . . . . . . . . . . . . Research Objective . . . . . . . . . .Sequential Steps to Plant Layout . . A Formal Statement of the Layout Problem . Layout Objectives and the Problem Formulation . . . . . . . . . . . . Unequal Area Requirements . . . . . . Obstacles to Solution . . . . . . Summary and Organization of Thesis . . REVIEW AND APPRAISAL OF EXISTING DECISIONS MODELS . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . Visual Aids to Design . . . . . . . . Evaluation of Alternatives . . . . . . .Algorithms for the Layout Problem ALGORITHMS FOR.WIMMERT'S METHOD Introduction . . . . . . . . . . . Selecting Quadruplets . . . . . . . Entering Tallies . . . . . . . . . . . Updating CRITERIA . . . . . . . . . . Infeasibility Testing . . . . . . . . Revising SOLUTION . ... . . . . . . . Making Diad Assignments . . . . . . . Detecting Post-Assignment Infeasibility Recycling . . . . . . . . . . . . . . Deducting the Last Diad Assignment . . Rule Combinations for Each Version . . >Summary . . . . . . . . . . . . . . . ANALYSIS OF FINDINGS . . . . . . . . . . Additional Algorithms Analyzed . . . . Test Problems . . . . . . . . . . . iv 26 26 28 33 33 80 80 81 83 84 86 100 100 102 103 104 104 108 109 109 111 Chapter Desired Information on Variables . . . .Solution Quality . . . . . . . . . . . AConstraint Satisfaction . . . . . . . .Computer Time Requirements . . . . . . Combining Cost and Time Considerations Findings Tangential to the Research Objective . . . . . . . . . . . . . V. SUMMARY'AND FUTURE RESEARCH . . . . . . Summary and Conclusions . . . . . . . Future Research Needs . . . . . . . . APPENDICES . . . . . . . . . . . . . . . . . . . BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . Page 112 114 124 125 130 135 144 144 149 155 253 Table 1-1. 3-9. 4-1. 4-2. LIST OF TABLES Relation of layout objective to problem formulation . . . . . . . . . . . Computing time for the branch-and-bound method . . . . . . . . . . . . . . TALLY matrix . . . . . . . . . . . Time to generate, store and read combina- tions (in seconds) . . . . . . . . .COST matrix for example A . . . . TALLY matrix for example A . . . . COST matrix for example B . . . TALLY matrix for example B . . . . .COST matrix for example C . . . . TALLY matrix for example C . . . . Rule sets for each version . . . . Solution cost rankings for problem Solution cost rankings for problem Cost effects of decision rules . . .Solution cost rankings for problem Solution cost rankings for problem Phase II time effects of decision rules Probabilities of RDM solutions having costs set set set set III IV less than or equal to the average cost of selected versions after one and t trials . vi Page 16 49 87 92 93 94 95 95 96 97 105 116 117 119 121 123 128 134 Table 4-8. 4-9. Page Occurrences of conflict as percentages . . . 136 Penetration of Wimmert's versions expressed as percentages . . . . . . . . . . . . . . . 138 Iterations before convergence for H-R, H—S, C—R, and C-S . . . . . . . . . . . . . . . . 139 vii LIST OF FIGURES Descriptive flow chart of Steinberg's algorithm . . . . . . . . . . . . . . . . . Descriptive flow chart of CRAFT . . . . . . Descriptive flow chart of Hillier's algorithm . . . . . . . . . . . . . . . . . Simplified descriptive flow chart of Wimmert's method . . . . . . . . . . . . Descriptive flow chart for all of Wimmert's vers ions 0 O O O I O O O O O O O O O O O O Two-way chart for problem set II . . . . . Two-way chart for problem set IV . . . . . viii Page 54 67 70 75 107 131 132 LIST OF APPENDICES Appendix Page I. GLOSSARY OF TERMS FOR WIMMERT VERSIONS . 155 II. LISTING OF COMBINATION GENERATOR . . . . 161 III. LISTING OF 3-C . . . . . . . . . . . . . 162 IV. LISTING OF 49A . . . . . . . . . . . . 171 V. LISTING OF 4-B . . . . . . . . . . . . 177 VI. LISTING OF 4-C . . . . . . . . . . . . . 184 VII. LISTING OF 5-B . . . . . . . . . . . . 192 VIII. LISTING OF RDM . . . . . . . . . . . . . 201 IX. PROBLEM DESCRIPTIONS . . . . . . . . . . . 203 X. RANDOM STARTING SOLUTIONS . . . . . . . . . . 232 XI. PROBLEM STATISTICS . . . . . . . . . . . . . 235 XII. ,SUMMARY OF COST AND TIME OUTPUT FOR TEST PROBLEMS . . . . . . . . . . . . . . . . . . 236 XIII. AVERAGE PHASE I TIME AS A FUNCTION OF N . . . 247 XIV. COEFFICIENTS AND STATISTICS OF REGRESSION EQUATIONS FOR PHASE I TIME . . . . . . . . . 248 XV. AVERAGE PHASE II TIME.AS A FUNCTION OF N . . 249 XVI. COEFFICIENTS AND STATISTICS OF REGRESSION EQUATIONS FOR PHASE II TIME . . . . . . . . . 250 XVII. SOLUTION COST STATISTICS OF RDM FOR ALL UNCONSTRAINED PROBLEMS . . . . . . . . . . . 251 XVIII. LEAST COST SOLUTIONS AND RANDOM MEAN INCREMENTS . . . . . . . . . . . . . . . . . 252 ix CHAPTER I THESIS OBJECTIVE AND PROBLEM DESCRIPTION Introduction Plant layout is defined as the arrangement of cen- ters to meet a firm's production requirements economically. Economic centers can be machines, groups of machines, departments, storage areas, material handling systems, or other types of supporting service systems. Optimal location‘ of centers has been of theoretical and practical concern for several centuries. The ancient Greeks, for example, consid- ered the problem of how a person could most quickly travel between two points, both of which are on the same side of a river, given that he must get water at some third point at the river bank during his trip.1 The problem is to find that point on the bank which minimizes the total length of two lines drawn from it to the other two points. The solu- tion was found by drawing ellipses using the origin and destination of the trip as foci, finding the smallest .lWilliam-Miehle, "Link-Length Minimization in Net- works," The Journal of the Operations Research Society_of America, VI, No. 2 (MarcheApril, 1958), 235. ellipse which touches but does not cross the river, and selecting the point where it touches as the location of the third point. The whole area of Spatial location theory, of which plant layout is but one facet, has advanced significantly since this early age. This growth of knowledge has been uneven, however, depending on the particular problem formu- lation of interest. The plant layout problem has defied optimal solution by a computationally feasible procedure. "Plant layout is largely an art today. . . . There is no overall theory that makes it possible to relate the magni- tude of influencing factors into a composite design."1 Fortunately, several suboptimal algorithms2 of considerable promise have been made available in the last decade. These contributions to layout theory come from diverse sources--mathematics, physics, industrial engineer- ing, computer design, management science, operations re- search and business administration. They can be found under such seemingly unrelated headings as the backboard wiring problem, kitchen layout, ergonomics, parking lot design, lElwood S. Buffa, Modern Production Management (2d ed.; New York: John Wiley and Sons, Inc., 1965), p. 400. 280me authors prefer the term "heuristic" or “heuris- tic algorithm," which Feigenbaum and Feldman define as "a rule of thumb, strategy, trick, simplification or any other kind of device which drastically limits search for solutions in large problem spaces." Edward A. Feigenbaum and Julian Feldman, Computers and Thought (New YOrk: McGraw-Hill Book Co., 1963), p. 6. storeroom design, quadratic assignment problem, assignment problem, plant design, network minimization problem, link- length minimization in networks, location analysis, and equipment location analysis. Centers of economic activity may not be those normally associated with the layout problem. Centers may be knobs to be attached to a control panel or computer components to be wired together. In a more trivial example, centers can be people who must be seated relative to each other to minimize some measure of conflict. All of these problems are compatible in terms of the objective func- tion and constraints. Research Objective The objective of this thesis is to compare the per- formance characteristics of the most promising suboptimal algorithms now in existence. This is to be achieved by first obtaining computer programs for each of the procedures selected for detailed study and by then applying them to several realistic test problems. The resulting output pro- vides data for comparing the algorithms' computational time, abilities to satisfy constraints, and the costliness of their solutions. The most obvious reason for research along the sug- gested lines is the important part of a good layout design plays in efficient production operations. Material handling costs are very much dependent on the layout design. Although their exact magnitude is unknown, one author estimated that they range between 10 and 40 per cent of total production costs.1 Layout decisions often possess an irrecoverable quality, due to the cost of relocating centers. They are long-run commitments affecting the very design of the pro- duction system. It is true that in some production situa- tions mounting machines on pallet bases or using air cushion systems can be an inexpensive means to relocate machines and to gain the advantages of line production for large produc- tion runs. In such a case, the layout problem is of minor concern, giving way to a scheduling problem.2 However, this type of production system is definitely not as prevalent as the one of selecting a layout design acceptable over a long time horizon. Reaching the research objective is beneficial due to the paucity of data on the comparative performances of exist- ing algorithms. The main thrust of recent research is to develop still another algorithm. Although significant con- tributions, these efforts do little to apprise the practi— tioner as to-which one performs best under varying conditions. 1Philip R. Reimert, "An Investigation of the Feasi- bility and Cost of Flexible Plant Layout Using Movable Pro- duction Machinery and a Computerized Scheduling Program" (unpublished M.S.I.E. dissertation, Central Library, Arizona State University, 1963), p. 3. 2Philip Reimert addresses himself mainly to this technological solution in his thesis. Philip R. Reimert, ibid. This is not to say that no comparisons have been made. The most rigorous comparative analysis1 now available involves the Hillier algorithm, the Hillier—Connors algorithm, and a Gilmore algorithm.2 However, the cost data of the sixteen test problems were derived from random two—digit numbers—— an unrealistic assumption. Other comparisons have been based on one or two test problems at best. At any rate, comparative research is not available in the quality and quantity as is true with a production problem such as line balancing.3 The interrelated nature of computational time and the costliness of layout design solutions is unexplored. Sequential Stepsfito Layout Design It is desirable at the outset to put into perspec— tive the problem formulated for this study. ,Solving this problem is actually only one of three steps necessary to reach a satisfactory layout design. lF. Hillier and M. Connors, "Quadratic Assignment Problem Algorithms and the Location of Indivisible Facil— ities," Technical Report No. 6, Program in Operations Re- search, Stanford University (Stanford: By the authors, 1965), pp. 29-34. ZAnother study, one by C. Nugent, T. Vollman, and J. Ruml, has just been reported in "An Experimental Compari— son of Techniques for the Assignment of Facilities to Loca- tions," Operations Research, XVI, No. l (January—February, 1968), 150-173. 3E. Ignall offers a comprehensive analysis in "A Re— view of Assembly Line Balancing," The Journal of Industrial Engineering, XVI (July-August, 1965), 244-254. A comparable study is presented by M. Kilbridge and L. Wester in "A Review of Analytic Systems of Line Balancing,“ Operations Research,.X, No. 5 (September—October, 1962), 626. The first step is to gather information on: product demand; sequences of production; alternative handling sys- tems; relocation costs (or location costs in the case of a new plant); economic advantages of adjacency and line produc- tion; the flow of materials between each pair of centers; the number, type, capacity and physical size of centers; total area available; and the way any other of the multiple plant layout objectives are affected by the design. With this information on production requirements and economic relationships, the analyst turns to the second step--assigning centers to locations. Locations are usually, but not necessarily, represented as equal discrete areas in a Cartesian plane having the axes intersecting at one corner of the layout. The distances between pairs of locations can be calculated from the configuration; they are assumed to be constant regardless of how centers are arranged in the final solution.1 .A decision must then be made on how to assign the N centers to the N discrete locations to meet production requirements and yet keep the resulting costs at a satisfac- torily low level. In this thesis, this decision will be made using algorithms. (With the second step completed, the analyst has an idealized plan or block diagram to which can be added the detailed planning necessary to reach the final design. The 1A three-dimensional location system can be accommo- dated as long as this assumption is valid. precise locations and configuration of all centers and sub- systems within the centers must be carefully weighed. .We shall be concerned in this thesis with only the second step--assigning N centers to N locations to minimize some sort of total cost function subject to a number of constraints. This is what we consider to be "the layout problem.” A Formal Statement of the Layout Problem Since the hypothesis to be tested is that existing algorithms perform differently in solving the layout problem and that these differences can be determined by trying them out on realistic test problems using the computer, a more precise formulation of the problem is in order. The advan— tage is not that all algorithms use the formulation directly, but that it helps conceptualize the relevant independent variables and their linkages to the layout criteria. There are at least three relevant costs: special quadratic, linear, and general quadratic. They may appear either in the objective function or as constraints. The objective function and constraints are first to be stated mathemati- cally. This is followed by a description of their relation- ship with the layout objectives. Special_Quadratic.Costs There are layout costs dependent only on the rela- tive locations of center pairs. Considering centers i and k, this cost (c. ) is a function of the distance between 1jk£ locations j and E to which the two centers are respectively assigned. If fi measures the desirability of locating k centers 1 and k close together and djfl is the distance between locations j and 2, then Cijkfi is calculated as: Cijk£= fik djfl (1) If xij is equal to one when center i is assigned to location j and is zero otherwise, then special quadratic costs are expressed as: N 1/2 (2) C.. X..X i,j,k,£=1 13kg 13 k3 Three constraints must be imposed to assure a fea- siblei solution. Condition 3 is an indivisibility require- ment to prohibit the assignment of fractional centers. _ 0 - -_ xij "' l (llj_ll2I'°°lN) (3) Constraint 4 requires that each center is assigned to a location. Similarly, condition 5 assures that all locations are assigned. Taken together, they make it impos— sible to assign a center to more than one location or to have more than one center assigned to the same location. N 2x.. = 1 (i=l,2,...,N) (4) j=l 13 (j=l.2,..-,N) (5) II MZ K II H Linear Costs There can be a certain linear cost, call it bij' incurred by assigning center i to location j. This cost (positive or negative) is unaffected by the center's relative location to the other (N-l) centers. Subject to constraints 3, 4, and 5, linear costs are formulated as: N 2 b.. x.. (6) 3 Conditions 4, 5, and 6 are to be recognized as the ordinary assignment problem, a special case of the transpor- tation problem with unity rim conditions. There are N sources to supply N sinks; all or none of the capacity of source i can be assigned to sink j. The integrality prop- erty of the transporation problem assures that all positive Xi' values will be equal to one, thereby satisfying condi- tion 3 automatically. .A feasible solution to this transpor— tation problem is degenerate, since less than (ZN-l) of the xij values are positive. This necessitates the use of a perturbation method. Other algorithms for solving this problem are presented in Chapter II. It should be noted that relative location costs which are a function of distance become linear rather than quadratic when only one center is to be added2to an existing 10 layout. In this restricted case, there are at least three alternative formulations (other than conditions 2 through 5) of relative location costs. They are based on the assump— tion that the new center can be "squeezed in" at any point on the Cartesian plane, rather than requiring a discrete area. In each formulation, x and y are the abscissa and ordinate of the new center, whereas fi measures the desir- ability of locating the new center close to fixed center i. The centroids of the M existing centers are fixed at points (Xi’yi)’ where i=l,2,...,M. Formulation 7 presumes that distances between loca- tions are best computed with the Pythagorean theorem. For ease of solution, formulation 8 approximates costs by squar- ing the terms in the function. Finally, formulation 9 computes distances using the rectilinear criterion. M 2 2 1/2 1: M .21 £12 [(X-Xi)2 + (y—yi) 2:] (8) 1: fi [Ix'xil + IY‘YiI] (9) ’TIM z 1 11 General Quadratic Costs A certain cost (call it Cijkz tive location of centers, but is not strictly a linear func- ) depends on the rela- tion of the distance between them. Let Cijkfi take on values in the following manner: ( aO 1f djfl g_ 0 al 1f 0 < djz g, l Cijkfi = < . (10) L a if —1 < d. < p (p ) 32 p Let the general quadratic cost term Cijkfl be com- puted as follows: I l = l Cijkz Cijkfi + Cijkz (11) Then the general quadratic cost function, subject to conditions 3, 4, and 5, becomes: (12) N 1/2 E Cijkfi xij sz i,j 2:1 Two Formulations of the Layout Problem The layout problem which recognizes all three types of costs can be formulated in two different ways. The most direct formulation is given in condition 13, subject to equations 3, 4, and 5. A St 4 "1 12 Minimize: (13) N 1 I C I I + — 'I '0 I O i b X 2 Z cijkfl X13 xkfi i,'=l 13 13 i,j,k,£=l This formulation, although the most complete, has several shortcomings. The foremost deficiency is that exist- ing algorithms seldom recognize bij and never recognize I Cijkz . . l . object1ves represented by Cijkfi ficult to quantify. The analyst may prefer to rule out cer- in the objective function. Secondly, the layout terms are particularly dif- tain assignments involving large terms without formally CijkE including all of them in the objective function. Finally, the objectives represented by bij do not generally represent a stream of costs over a time horizon, as opposed to Cijkfi and c! This necessitates present value calculations and 1jk£° the additional complication to the solution process may not be worthwhile. In light of these objections to the first formula- tion, the second one appears to be more satisfactory. This alternative formulation, which is the one used in our study, considers bij and c! terms only in an approximate way. 1jk£ Linear costs can be recognized by adding new constraints requiring some xij variables to be one and others to be zero. Consider the assignment of center 1 to location 2. There are problem situations where b12 is so much less than other 1These are discussed in the next section. 13 blj values that a satisfactory solution obviously must have x12 equal to l, regardless of the effect on Special qua- dratic costs. It is also conceivable that b12 is so much higher than other blj values that x12 must obviously be constrained to zero. Let Pi be the set of prohibited loca— tion assignments for center i. The constraints1 acting in the place of a linear cost function become: xij = 0 1f jePi (1=1,2,...,N) (14) The alternative way of including_cijk£ terms in the solution process is to transform selected fik values into arbitrarily large values. If certain Cijkfi terms make it desirable to cluster centers one, three, four, and eight, then f f f f f 13' 14’ 18’ 34’ 38’ 48 large. .An effective algorithm would automatically bring them and f can be made arbitrarily together. If it is desired to separate centers, the appro— priate fi values can be arbitrarily small. k The second problem formulation, which recognizes bij and Cijkfl terms only implicitly, is therefore subject to transformations in fi as well as conditions 3, 4, 5, and 14. k It can be expressed as: 1Required constraints are also implied by condition 14, simply by including in Pi all locations except the one to which center i must be assigned. However, this may not be the most efficient procedure for an algorithm to use. ‘. L b 11' l4 Minimize: N 1/2 2 i,j,k,£=1 Cijke Xij XkE (15) Although it is an imperfect substitute for including b.. and c! 13 1jk£ d1rectly 1n the object1ve funct1on, it would seem to be acceptable if the layout analyst obtains solu— tions with and without the various constraints and transfor- mations. The differences in the objective function values measure the additional costs caused by the constraints and transformations; a final selection can be reached on an incremental cost basis. Layout Objectives and the Problem Formulation Traditional layout theory specifies a bewildering assortment of objectives, the attainment of which are sup- posed to be dependent on the solution selected to the layout problem. This large assortment of objectives has led some authors to suggest that their algorithms are apprOpriate only for a pure "process" layout.l It is the purpose of this section to show that this is an unnecessary restriction; solving the layout problem as formulated in the previous section can provide satisfactory designs in many types of 1For example, Elwood S. Buffa, Gordon C. Armour and Thomas E. Vollman suggest CRAFT is applicable mainly to job or machine shops in "Allocating Facilities with CRAFT," Harvard Business Review, XLII, No. 2 (March—April, 1964). o!) (J. (I Q.) 15 "mixed" situations. Conceiving of layout problems as either "pure line" or "pure process" layouts is unrealistic and not very useful; it does nothing in terms of assigning centers to locations. In addition, there are few instances of a pure type in a real world application. To demonstrate the generality of our problem formu- lation, consider the objectives commonly cited as being 1 These objectives dependent on the layout objectives (xij). are classified in Table 1-1 as to the appropriate costs, constraints and transformations. These relationships are admittedly conjectural. Little research has been done in the area of cost functions and it would seem that many of the objectives are not always related to Xij' When they are, the exact linkages are situational; the decision-maker must be cognizant of them and tailor the problem with them in mind. Consider in turn each of the objectives listed in Table l-l. Material handliaq is usually represented by lTypical statements of layout objectives are found in: Richard C. Wilson, "Evaluation of Spatial Relations and Empirical Plant Layout Criteria by Digital Computer" (unpub- lished Ph.D. dissertation, University of Michigan, 1961); James M. Moore, Plant Layout and Desiga (New York: The Macmillan Co., 1962), p. 93; James M. Apple,_ Plant Layout and Materials Handling (2d ed.; New York: The Ronald Press Company, 1950), pp. 7-11; and Roy D. Harris and Roland K. Smith, A Cost-Effectiveness Approach to Facilities Layout, Working Paper 67-22, Graduate School of Business, The University of Texas at.Austin (Austin: By the authors, August, 1967), p. 15. 16 x x x MDHHHQmpcmmxo paw muflHHQmeHm x doflumwflaflus mommm Hooam x x mmoa mmuom paw muHHMSO DUSGOHm x x x OUGOHQO>QOU Hoxuo3 cam muommm x coauommmflumm now can coaum>fluoz x x Amcflnoummmflp paw mcHHSUOLUm MHHHmEflHmv Honma uomuflde x mmoco>wuommmm muomfl>ummsm x ucmEumm>GH udmfimflsvm x usm£m50H£u cam xuouco>cfl mmoooumICH x x Amhmamp Ho COHDMNHHmaoomm mo monmopv Honma DUOHHQ x doHumaamumdfl unmemflsvm. x x mcflapcm: HMHHOPEE. saw on ea axflau m>auowflno mzowumauommcmne ucflmuumcoo no wxnwo no flan doaumasanom EmHQOHm mo cofluuom ucm>mamm coflumasauom EOHQOHQ on mo>fluomnno usowma mo coHumamm .HIH magma 17 c.. terms in the objective function.1 The f. term is a ijkfl 1k measure of the flow between centers 1 and k. It should reflect the true cost of moving the required number of loads between the centers for each unit of distance. It can be estimated from data on the numbercflfloads per unit time, the characteristics of the load, time standards and labor rates. The value of fi is zero when i and k are equal. When k appearing in a total cost function with linear costs, fik is correctly expressed as the present value of the unit cost. The djg term is a measure of the distance between locations j and E. The distance along the actual path traveled can be adjusted to account for modes of transportation and "impedi- ments."2 Equipment installation is a linear cost. It includes not only the actual movement of the center to its assigned location, but also constructing foundations and providing access to water, compressed air, gas, and electricity. The objectives of direct labor, in—process inventory, and eqaipment investment can all be related to positive terms (or transformations in fi Each objective can cijkfi k)° l . . . It 18 a l1near cost if one or more new, unrelated centers are to be added to an existing layout. 2The calculation of d. is described by Robert J. Wimmert in "A Quantitative Appéoach to Equipment Location in Intermittent Manufacturing" (unpublished Ph.D. dissertation, Purdue University), pp. 40-80. 18 be affected by whether or not a series of centers are located adjacently; i.e., in the form of a production line. Depending on such exogenous variables as product demand, line production may increase the efficiency of labor (due to specialization), increase the throughput of materials, and increase the utilization of equipment (thereby decreas- ing the number of machines which must be purchased). Since these savings are not realized if two of the sequentially linked centers of the line are separated by intervening centers, positive cijkfi terms or fik transformations are relevant. Cases where material handling delays increase direct labor costs or where direct laborers are used for material handling purposes can be accommodated by the Spe- cial quadratic cost function. Similarly the objectives of supervisory_effective- nessL indirect labopL motivation, job satisfactionL and direct labor are related in the sense that all may be affected by clustering certain centers. Traditional layout theory suggests that, under certain conditions, grouping centers together into a "process" layout increases the effectiveness of a supervisor, due to his familiarity with the workers and the technology. It may also decrease in— direct labor costs as a result of the concomitant simplicity in scheduling and diSpatching. Small group theory suggests that motivation and job satisfaction can sometimes be in- creased by clustering (or separating) certain groups of 19 employees (centers). Finally, clustering centers may permit a worker to operate more than one machine simultaneously. .Safety and worker convenience can be represented by all three cost components. The danger of separating centers between which many cumbersome parts must travel is recog- nized as a general quadratic cost. It may also be dangerous to make two centers adjacent, such as locating a painting area emitting noxious fumes next to a center having a high concentration of workers. This situation is accommodated by fixing the painting center in an isolated location (con- straint 14) or making the flow between them arbitrarily small. ,Worker convenience is adversely affected by assign— ing large concentrations of workers to areas remote from service facilities. This is to be recognized as a special quadratic cost. Most worker convenience considerations, however, would seem to be reserved for the detailed planning stage. Product qaalipy_and scrap loss objectives are treated in a similar fashion. If expensive, easily damaged parts must move between two centers, the centers should be located close together. If one center creates conditions adversely affecting the quality of products in another center, such as a foundry located adjacent to a "clean" room, they should be separated. .Another possibility is to assign the foundry to an isolated location. 20 The best use of floor space seems primarily a prob- lem for the third step in layout design--detailed planning. How components within a center are dovetailed together can materially affect area utilization. There is one way, how- ever, that xij has direct bearing on area utilization. Making centers i and k adjacent for line production purposes eliminates the need of an area to store materials exchanged between them. Flexibilipy and expandabilipy are nebulous terms in the context of the layout problem. The connotation seems to be that the layout design should accommodate all future foreseeable and unforeseeable changes in production require— ments. In the case of special quadratic costs, a determinis— tic model could be built after forecasting changes in fik terms during the planning period; present value calculations could then be made. In the case of risk, expected value calculations would be appropriate. There is little that can be done if the situation is one of uncertainty. In summary, our problem formulations are quite flexible in accommodating relevant objectives if the analyst carefully considers how they are related to xij and revises the problem statement accordingly. The algorithms tested in this thesis apply to the second formulation or else can be easily revised to this end. It must be acknowledged that it is rare when a center or group of centers must be assigned in a certain manner. The statement that "if they don't, 21 certain costs will be incurred" is more valid than the all- k' Unfortunately, little experience has been reported on how several of the or-nothing modifications made to xij and fi objectives can be quantified in terms of one dependent vari- able1 (dollars) and how they are related to Xij' If it is found that not all of the objectives can be translated into dollar values, it still is possible to handle several depen- dent variables simultaneously.2 The inescapable conclusion is that a satisfactory algorithm addressed to the quadratic cost component and amenable to the suggested constraints and transformations is a workable, but certainly imperfect tool for the layout analyst. Judgment and nonquantitative factors are still essential ingredients. It is necessary to "weigh and decide, balancing quantitative and nonquantitative factors . . . since usually, measures of effectiveness are not capable of reflecting all aspects of performance. . . . One of the lRoy Harris and Roland Smith, using a "system" and "cost-effectiveness" approach, show how a dollar figure can be derived for each objective. Harris and Smith, pp. 10-20. They do not take the additional step of tying each objec- tive to the design variables of this thesis, i.e., xi-. Their proposal serves only to evaluate a design after it is generated. 2One possible approach along this line is offered by Lawrence E. Briskin; his methodology is applied to the shipping problem where there are two objectives--minimum cost and minimum time. "A Method of Unifying Multiple Objec- tive Functions," Management Science, XII, No. 10 (June, 1966), 406-416. .5. 22 great traps in quantitative analysis is the siren song of the optimal solutions."1 Unequal Area Requirements The algorithms of this thesis can be distinguished by whether or not they explicitly accommodate centers with unequal area requirements. For those algorithms possessing such a provision, the shape of a center is pap determined by cost considerations; rather the center takes on any configur- ation which meets very limited qualifications. The result can be clearly unacceptable center shapes which are not justifiable on the basis of costs. These algorithms do have an important advantage in reduced computational time, since N must not be increased to handle unequal center areas. The algorithms not having an eXplicit provision can still accommodate problems having unequal center areas. Several techniques for doing so are as follows: 1. Combine small centers with large fik values into one center before applying the algorithm. 2. Ignore small centers and "squeeze them in" after the algorithm provides a solution. Any one of the methods for adding centers to a layout having fixed centers could be used. 3. Divide the larger centers into two or more subcen- ters possessing equally shared flows. Make all flows between subcenters arbitrarily large. 4. .Add one or more "dummy" centers and set all of the fik values equal to zero. When the block diagram lBuffa,Modern Production Management, pp. 55-56. 23 provided by the algorithm is translated into the final floor plan, expand actual centers into the areas assigned to the dummies. Which approach provides the best solutions in an equivalent amount of computer time is as yet unknown. This question is beyond the scope of this thesis. The third technique given above is arbitrarily chosen so that the effective portions of each algorithm can be tested on an equal footing. Explicit provisions for unequal areas are simply ignored. Obstacles to Solution There are several obstacles defying an easy solution to plant layout problems. It is important to keep these obstacles in mind during the evaluation of existing algo- rithms. The obstacles are the multiplicity of objectives, the current vagueness of these objectives, changes in system parameters over time, the cost of data collection, simplify— ing model assumptions,l indivisibility of centers, and the very rapid increase in computational difficulty as the prob- lem size increases. The last obstacle is undoubtedly the most critical one. 1An example of simplified model assumptions is the discontinuity of material handling costs, due to the imper- fect ability to hire a fraction of a material handler. A thorough treatment of model assumptions is offered by Thomas E. Vollman in “An Investigation of Bases for the Relative Location of Facilities" (unpublished Ph.D. dissertation, University of California, Los Angeles, 1964). l) 01 r) () (fl 24 Summary and Organization of Thesis The main research objective is the comparative appraisal of several promising algorithms relating to the layout problem. The plant layout problem is defined as assigning N centers to N discrete locations in such a way as to satisfy all constraints and attain a reasonably low value in the objective function. Solving the layout problem is really the second step of a larger decision process. It comes after information gathering and is followed by de- tailed planning. A complete statement of the layout problem j’ Cijkfi’ and CijkE cost terms, as well as constraints assuring that each center is assigned, each recognizes bi center is indivisible, and each location is assigned. A less complete, but more tractable statement explicitly recognizes only special quadratic costs. The other two cost components are built into the problem formu— lation with additional constraints on xij and transforma- tions of fi The computer algorithms of this thesis are k' addressed to this latter formulation and therefore must be recognized as valuable, but admittedly imperfect decision- making tools. The purpose of the remaining chapters is to select and evaluate algorithms which fit--or can be made to fit—- this problem formulation. Chapter II contains a summary and appraisal of existing decision models in the area of plant layout. These tools are divided into three groups: (1) 77‘ 1'. m .y. UV 4.. ll: nu a.“ ‘v r1 25 visual aids to design, (2) evaluation techniques, and (3) algorithms for the layout problem. Four algorithms are selected for additional analysis: CRAFT, Hillier's algo— rithm, Wimmert's algorithm and ALDEP. Since computer programs had been available only for the first two algorithms, they must be written for the last two. Wimmert's procedure is the most complex and incomplete of these two, and so Chapter III is devoted to it. Since Wimmert's algorithm can be translated into many different sets of decision rules, thirteen unique versions are devel- oped for evaluation. Chapter IV provides an analysis of the results of applying the selected algorithms to twenty-six test problems. This analysis places particular emphasis on computational time, objective function values, and satisfaction of con- straints. Other properties of the algorithms, heretofore unknown, are offered. It is also a concern in Chapter IV to develop more insights as to when a "satisfactory" solu— tion is reached. This will aid in constructing a "stopping rule." Chapter V is set aside for the conclusions of this thesis. Necessary revisions in algorithms and questions for future research are presented. II!-A h.» "in“. «cm, ‘ CHAPTER II REVIEW AND APPRAISAL OF EXISTING DECISION MODELS Introduction There is a surprisingly large number of plant layout algorithms, or concepts which could be transformed into algo- righms, which have recently been put at the disposal of the practitioner. Most of them have merit as long as their assumptions describe reasonably well the problem being solved. In searching the list for the most flexible, effi- cient algorithms which solve the layout problem we have for- mulated, several procedures can be deleted. These deletions are made mainly on the basis of their assumptions or re- ported experience with their computational effectiveness. Of the remaining algorithms, four of them appear to be particularly promising. The reported experience with CRAFT and Hillier's algorithms is especially favorable. Wimmert's procedure and a random—generating procedure are also inter- esting, but for a different reason. Both appear to be plausible tools and are rather strongly advocated in some quarters on a priori grounds. This chapter is devoted to an evaluation of relevant layout models from all contributing disciplines. Particular 26 .~ 5 \II T I y‘ H 27 attention is paid to their theory, strengths, weaknesses and omissions. These models are grouped into three categories: (1) visual aids to design, (2) evaluation techniques, and (3) algorithms for the layout problem. Reports on plant visits1 and a review of trade journals suggest that the time-honored models of the first two categories are the most commonly used tools in business today. Unfortunately, none offer a basis for an algorithm of any promise. They do not provide rigorously defined decision rules for minimizing a cost function. The assign— ment of centers is left to the "judgment" of the analyst using the trial-and-error method. For this reason, none of these models are selected for additional study. They would appear to be most useful for small uncomplicated problems. The value of N need not be very large, for example, before a flow diagram becomes a mass of directed lines which are completely unintelligible. Perhaps the best conclusion is that models of the first two categories have an important role to play, partic- ularly during the first and third steps of the layout process. In regard to the second step, they must be supplemented by algorithms explicitly directed to the layout assignment problem. lHubert F. Lund, "Plant Planning Tools," Factory, CXXI, No. 9 (September, 1963), 86-91. 28 Visual Aids to Design These techniques have been available in varying forms of refinement for several decades. They emphasize the data collection and detailed planning steps, although the implication is that the analyst somehow uses them to gener- ate satisfactory solutions. The multiproduct_process chart, operation process chart, and flow process chartl are convenient means of deter- mining the sequence of operations for each product. Com- bined with product demand information, they facilitate the computation of the "flow" between any two centers (fik)' The MAGnitude chart2 accounts for differences in material handling difficulty and therefore is of value in pointing out critical handling problems and adjusting fik terms. Layout drawinga, machine data cards, the layout_planning chart,3 3-D models, scale templates and isometric drawings4 are useful tools for data collection or detailed planning. lTwo references are: John A. Shubin and H. Madeheim, Plant Layout (Englewood Cliffs, N.J.: Prentice-Hall, Inc., 1951), and Roy E. Elicker, "Operating Procedures Used in Plant Layout" (unpublished M.S. dissertation, Michigan State University, 1951). 2Robert S. Rice, "Three New Tools for Better Plant Layout," Factory, XVIII, No. 6 (June, 1960), 101-103. 3Ruddell Reed, Jr., Plant Layout: Factors, Princi- ples and Techniques (Homewood, Illinois: Richard D. Irwin, Inc., 1961). 4Lund, Factory, pp. 86-91. u. up. n- 5C uh: .1 ~\.V\ . \~u. \qs\ NIU 29 The product-quantity chart1 is intended to help distinguish centers to be clustered in a process layout from those to be located in a production line. Demand, in units per time period, is plotted on the y—axis of the graph. On the x—axis are listed the product lines, ranked in monotoni- cally nonincreasing order with respect to demand. When a curve so plotted is convex, a production line is to be used for products in the upper region of the curve, a "mixed" design for the central region, and a process layout for the lower region. This tool provides a primitive notion as to what centers should be adjacent and clues as to what type of cost data should be collected. Notable deficiencies are the failure to specify region boundaries and the lack of an interpretation for a line, particularly a horizontal one. In its simplest form, the travel (cross) chart is a matrix showing for a given product the volume of material flow exchanged between centers in both directions.2 Modi- fied versions display the combined material flows for all products, use rating scales, use fi djfi rather than fi k k' 1This chart was proposed by Richard Muther in Systematic Layout Planning, Industrial Education Institute (Boston: By the author, 1962). Another reference is Robert S. Rice's "Three New Tools for Better Plant Layout," Factory, CVIII, No. 6 (June, 1960), 101-103. 2Travel charts as originally proposed are found in: D. C. Cameron, "Travel Charts," Modern Materials Handling, VII, No. 1 (January, 1952), 37-40 and M. L. Levy, "Let Travel Charting Simplify Your Material Movement Problems," Mill and Factopy, XLVIII, No. 5 (May, 1951), 100-101. 30 or transform fik and djfl to better reflect costs.1 The usual suggestion is to use the charts as a "guide," with alternative block diagrams being drawn by trial and error to reduce the special quadratic costs or the sum of nonadjacent material flows.2 One author,3 however, offers two rather concrete decision rules for assigning centers to locations. After a center, such as the receiving area, is fixed at the end of the layout grid, assignment decisions are made for locations one grid square away from it. The next stage is to assign centers to locations two grid squares removed, and so on. The decision rules are to make centers adjacent if the fi value associated with them is large or if they are k connected by one-directional flows involving only one product. lRepresentative sources on such modifications are: Wayland P. Smith, "Travel Charting," Journal of Industrial Engineering, VI, No. 1 (January, 1955); James L. Lundy, ”A Reply to Wayland P. Smith's Article," Journal of Industrial Engineering, VI, No. 3 (May-June, 1955), 29; and Glenn E. Anderson and Irvin L. Reis, "Relative Importance Factors in Layout Analysis," Journal of Industrial Engineering, II, No. 4 (July-August, 1960), 312-316. 2Elwood S. Buffa, "Sequence Analysis for Functional Layout," Journal of Industrial Engineeriag, VI, No. 2 (March— April, 1955), 12-25. 3Marshall Schneider, "Cross Charting Technique as a Basis for Plant Layout," Journal of Industrial Engineeriag, II, No. 6 (November—December, 1960), 478-483. 31 The R§p_chartl is similar to the cross chart, except that an ordinal rating scale is used to assign values to the matrix elements above the main diagonal. Usually five or six levels are used to describe the desirability of locating adjacently each possible pair of centers. Centers with a high "closeness" rating (rik) are given priority when assign- ing centers to locations. Although the use of rik provides the analyst considerable flexibility in formulating the problem, the REL chart leaves unanswered the following ques- tion. Let the REL values be such that: Is a solution placing center 1 adjacent to center 2 but non— adjacent to centers 3 and 4 a better solution than the con— 9 verse one. Perhaps the use of fik djfi or even rik djE would better reflect true costs. A flow diagram2 is a scaled drawing of the layout area for a given alternative; the location of each center is specified on it. Directed lines are drawn between centers 1RichardMuther and John D. Wheeler, "Simplified Systematic Layout Planning," Factory, CXX, Nos. 8-10 (August- September, 1962). 2John R. Immer, Layout Planning_Technigues (New York: McGraw-Hill Book Company, Inc., 1950). >r u'v rfli .U h a: .n u fiuv «pa ‘5 32 to represent material flows. The stringdiagraml is identi- cal to the flow diagram, except that an unbroken string represents material flows. A pin is placed at each center centroid and a string is run between each pin. The total str1ng length measures fik djfi' The apiral method2 is a visual technique whereby each center is represented by a node; directed lines are added to show flow from preceding to following centers. The branches are labelled with appropriate fik terms. The nodes are then joined by trial and error so as to reduce the sum of nonadjacent flows. Both the location and shape of cen- ters are considered to be design variables. The straight-line method3 is a visual aid which, since it provides a rather specific solution procedure, possesses some of the properties of an algorithm. However, it is relevant only if the sequences of operations are nearly identical for all products. Products are listed along the y-axis in decreasing order of importance from top to bottom. A possible sequence of centers is listed along the x-axis. A bar chart is then constructed showing "split 1R. T. Ronan, "String Diagrams Cut Handling Bottle- necks," Modern Materials Handliag, VIII, No. 8 (August, 1953), 67-71. 2Ruddell Reed, Jr., Plant Location, Layout and Main- tenance, Vol. V of the Irwin Series in Operations Managament, ed. by H. L. Timms (Homewood, Illinois: Richard D. Irwin, Inc., 1967), pp. 85-87. 3Ibid., pp. 87-91. 5: ¢.- 33 centers," which are to be eliminated when possible by chang- ing the sequence of centers. The final bar chart is a guide for drawing center boundaries. Evaluation of Alternatives Measures of effectiveness can be obtained from travel and REL charts, which are discussed in the previous section. Other simple devices for ranking alternatives in terms of several criteria are: tally of gains and losses, pros and cons, ranking, and value ratinga.l Alternatives are ranked for each criteria and the alternative with the highest score is deemed the best. An ordinal number system is used as if it were cardinal when computing the total measure of effectiveness. .Alqorithms for the Layout Problem There are several computational procedures purport— ing to solve a center assignment problem similar to one of those stated in the first chapter. Analytic method Numeric method Level curves Physical analogies "Candidate Area" methods LACH Demand-position method Enumeration of alternatives Integer programming Loooqowmbwwt—a lRichard Muther, Practical Plant Layout, (lst ed.; New York: McGraw—Hill Book Company, Inc., 1955), pp. 239- 250. 34 10. Branch and bound ll. Steinberg's algorithm 12. Kodres' algorithm 13. Gilmore's N4 algorithm 14. Gilmore's N5 algorithm 15. Hillier-Connors' algorithm 16. CORELAP l7. CRAFT l8. Hillier's algorithm 19. ,ALDEP 20. Wimmert's method. The first six techniques are addressed to the spe- cial problem of adding new unrelated centers to an existing layout. Some of these techniques can be embedded in more comprehensive algorithms for the generalized layout prob— lem.1 However, algorithms for the general problem (the last thirteen listed) are more flexible. It is from their ranks that four algorithms are selected to be applied to the test problems of this thesis. 1The algorithms of Steinberg, Kodres, Gilmore and Hillier-Connors use one or more of these techniques. 35 Analytic Methodl Consider problem formulations 1.7, 1.8, and 1.9,2 which treat material handling as a linear cost function. Taking the partial derivatives of expression 1.7 yields two extremal equations to be solved simultaneously: llhdz fi (x-xi)/ [(x-xi)2 + (y-yi)2:]l/2 = O (1) i l f. (y-yiv [(X‘Xi’2 + (ix-3192]”2 = o (2) 1 “MK 1 l Rationalizing these equations is a fundamental difficulty. If M exceeds three, the number of cross terms created by squaring both sides of the equations exceeds the number of lMany authors address themselves to this approach, including: R. T. Eddison, K. Pennycuick, and B. H. P. Rivett, Qperational Research in Management (London: English Union Press, 1962), pp. 183—185; Richard L. Francis, "A Note on the Optimal Location of New Machines in Existing Plant Layouts," The Journal of Industrial Engineering, XIV, No. l (January-February, 1963), 33-40; Andre H. McHose, "A Quad- ratic Formulation of the Activity Location Problem," Journal of Industrial Engineering, XII, No. 5 (September-October, 1961), 334-337; James M. Moore, "Mathematical Models for Optimizing Plant Layouts" (unpublished Ph.D. dissertation, Department of Industrial Engineering, Stanford University, 1965); F. P. Palermo, V, No. 4 (October, 1961), 335-337; and Roger C. Vergin and Jack D. Rogers, "An Algorithm and Compu- tational Procedure for Locating Economic Facilities," Man- agement Science, XIII, No. 6 (February, 1967), 240-254. 2The expressions 1.7, 1.8, and 1.9 reference condi- tions 7, 8, and 9 respectively found in Chapter I. (h ()1 r (3‘ 36 terms from which the radical sign has been removed. If more than one new center is to be added (N >1), the extremal equations become even more complex. Formulation 1.8 lends itself to easy solution. The optimal x and y values are found to be: M 2 M 2 x = 2 f. x./ [2 f. ] (3) =1 l 1 i=1 1 M 2 2 fi Yi/ [5 £11 (4) 1 1 If N is greater than one, it is possible to intro- duce the first new center and locate it ”optimally” in rela- tion to the M fixed centers. The second new center is then located "optimally" with respect to M+l fixed centers, and so on. It is yet to be determined in what order the N centers are to be introduced and to what degree the result— ing solution approaches optimality. An optimal solution to 1.9 is also possible. It can be proved that the best x value is obtained by arranging the xi terms in monotonically increasing order and repeating each xi by a number equal to £1“ The optimal x value is equal to the median of the resulting sequence. The optimal y value is obtained in the same fashion. If (1 + 2 + ..._+ M) is an even number, the solution can be a line or area rather than a discrete point. 37 Numeric Method Since the extremal equations to formulation 1.7 becomes difficult to solve as N and M become larger, a numeric solution process must be considered. It has been proved that the cost function for the straight-line movement case is convex.1 Therefore the iterative, "one—direction-at- a-timefijprocedure can be used without fear of finding a local minimum, If N is one, this method fixes x at some arbitrary value and varies y until the partial derivative in respect to y is zero. The computed y is then taken as given, with x being varied until the partial derivative in respect to x is equal to zero. This constitutes a full iteration. The procedure is repeated until neither x nor y change-dur- ing a full iteration. This procedure lends itself to com- puter solution.2 The only real difficulty is the initial placement of-x and the size of the incremental steps. If N is greater than one, a suboptimal solution could be generated with the following procedure. Temporarily fix all but one of the centers and "optimally" locate the variable center. Select and locate successively all of the other centers in turn. This constitutes one full iteration. Begin a new 1K. B. Haley, "The Siting of Depots," International Journal of Production Research, II, No. 1 (March, 1963), 41-45. ZAn optimal solution to a problem having M equal to 100 was solved in seven seconds on the IBM 7094. Vergin and Rogers, Management Science, p. 242. 38 iteration unless no assignment changes were made during the last full iteration. Two problems are associated with this suboptimal procedure. Suboptimization is possible if fik is partic- ularly large and center k is introduced for relocation immediately after center i. The second problem is that centers could cluster together. Clustering is avoidable if constraints are imposed on the distance between center pairs using the Lagrange multiplier. If a.. is a given distance 1] parameter, the constraint is as follows: (Xi'xj’2 + (yi-yj)2 - aij = o (5) Level Curves This method1 provides a graphic solution to the one center addition problem by computing the total cost of locating the machine at several points on the layout grid. All points having equal values are connected to form isocost curves. The center is then assigned to a feasible location on the lowest possible isocost curve; in this way, center area requirements can be considered in the analysis. 1James M. Moore, "Level Curve Approximation for Lay- out Analysis" (unpublished paper, Department of Industrial Engineering, Stanford University, 1960); Andre E. Bindsched- 1er and James M. Moore, "Optimal Location of New Machines in Existing Plant Layouts," Journal of Industrial Engineering, XII, No. l (January—February, 1961), 41—48. 39 Formulas were developed to calculate the slope of the curves in various segments of the graph. This made it possible to program the procedure for the IBM 1620 computer.1 This procedure is reportedly being programmed for the IBM 7090 so that problem size (M) can be increased. Level curves are most attractive if only one center is to be located. If N is greater than one, it is still possible to use them by successively introducing one center at a time and locating it "optimally" in relation to the centers already assigned. Physical Analogies There are at least three physical analogies relating to the addition prOblem: the soap-film solution, mechanical link-length minimizer, and the analogue computer method. The first method2 exploits that propensity of soap film to take a shape minimizing its potential energy (and therefore its area). .A scale model of plexiglass sheets (representing the grid) and brass posts (representing the M centers) is constructed. After being submerged in a soap solution, the optimal network is observed as a film on the 1James M. Moore and Martin R. Mariner, "Layout ,Planning: New Role for Computers," Modern Materials Han- dling, XVIII, No. 3 (March, 1963), 38—42. 2Miehle, The Journal of the Operations Research Society of America, p. 239. 40 plexiglass. Unfortunately, N cannot be specified in advance and the links cannot be weighted with fik values. The second method1 consists of a scale layout, pegs, pulleys and strings. One unbroken string is looped around all movable and fixed pegs (representing centers) in such a way as to represent total material flow. Applying tension to the string locates the N centers Optimally if friction does not become an insurmountable complication. The analogue computer method2 allows the analyst to locate one center optimally in relation to the other (N+M-1) centers. The analyst varies x and y values while the ana— logue computer is operating. When N is greater than one, the new centers can be located sequentially in the same manner suggested for level curves. "Candidate Area" Methods If N new centers are to be assigned to N discrete "candidate areas" and if the material flows between these new centers are negligible, the layout problem takes the form of the ordinary assignment problem. There are several 1Ibid., p. 240. 2Edward L. Brink and John S. deCani, "An Analogue Solution of the Generalized Transportation Problem with Specific Application to Marketing Location, Proceedings of the First International Conference on Operational Research (Baltimore: Operations Research Society of America, 1957), pp. 123-137; and Samuel Eilon and D. P. Dexiel, "Siting a Distribution Center, An Analogue Computer Application," Management Science, XII, No. 6 (February, 1966), 245-254. 41 optimization algorithms available, including the tranSporta— tion method, Kuhn's Hungarian Method, Munkres' algorithms, Flood's technique, and the Ford-Fulkerson model.1 If the assignment of fraction centers is permitted as a rough approximation, the problem can be reformulated as a zero-sum two-person game or as a linear programming prob- lem. Considering the latter approach, let bij be the cost of assigning center i to location j. The linear programming formulation in N2 unknowns is as follows: Minimize: (6) N z x.. = 1 (j=l,2,...,N) (7) N Z x.. = 1 (i=l,2,...,N) (8) lFour references to these techniques are: Richard E. Beckwith and Ram Vaswani, "The Assignment Problem--A Special Case of Linear Programming," Journal of Industrial Engineer- ipg, VIII, No. 3 (May-June, 1957), 167-172; H. W. Kuhn, "The Hungarian Method for the Assignment Problem," Naval Research Logistics Qaarterly, II, Nos. 1 and 2 (March-June, 1955), 83-97; James Munkres, "Algorithms for the Assignment and TranSportation Problems," Journal of the Society for Indus— trial and Applied Mathematics, V, No. 1 (March, 1957), 32-38; and A. Yaspan, "On Finding a Maximal Assignment," Operations Research, XIV, No. 4 (July-August, 1966), 646-651. 42 xij 2.0 (1,3=l,2,...,N) (9) An integer linear programming problem is obtained by replacing 2.9 with 1.3. 919E Location.assignment by Qost of Handling is a dynamic programming approach1 for assigning N new centers to k candi— date areas in a layout already having M fixed centers. Each of the new centers are introduced sequentially for analysis, with the centers having the highest kgl fik values intro— duced first. The first center ignores the interaction with the other (N-l) new centers, the second one ignores the other (N-2) new centers, and so on. After all new centers (stages) are considered, the assignment is selected which has the minimum operating costs and does not violate the budget con- straint on the purchase of material handling equipment. LACH is unique in its attempt to integrate decisions on layout assignment with decisions on handling equipment purchases. It has not been programmed and data are not available on its computational efficiency. Some of its assumptions and information requirements are somewhat unreal- istic. Hewever, it may be a useful tool when additional lDavid W._Willoughby, "A Technique for Integrating Facility Location and Materials Handling Equipment Selection,‘ (unpublished M.S. dissertation, Department of Industrial Engineering, Purdue University, 1967). 43 materials handling equipment must be purchased for the new centers and there are several alternative handling systems. When any of the other algorithms in this chapter are used, it is assumed that equipment purchasing decisions have either already been made or are to be finalized when a feasible layout solution is obtained.1 Demand-Position Method This unprogrammed procedure is the most primitive and probably the least satisfactory one of those listed.2 Assume a unidirectional grid is given, with N discrete loca— tions numbered consecutively from left to right. Determine the production sequence for each product or part, with the sequence being a permutation of centers. Compute the total flow passing through each center. For center i, the total flow is equal to: N 2: f . k=l 1k The next step is construct a matrix agof the N x N order, with element sij being the sum of flows for all products having center i as the j Ilstep in their production sequences. The "demand-position" is then calculated for 1 . . . For example, buy1ng a conveyor to link centers 1 and k would require a solution having them located adjacently. 2Peter C. Noy, "Make the Right Plant Layout--Mathe- matically," American Machinist, CI, No. 6 (March, 1957), 76-78. a: A... u -r is. ‘5 -1 44 each center using the moment analogy of mechanics. If pi is the demand-position of center i, then: s.. (10) pi= 1 13 IIMZ IIDGZ 1 (sinjvj j The final step is to take each center in order of its total flow and assign it to the unassigned integer loca- tion closest to its demand-position. This process is con— tinued without duplication until all N centers are assigned. The limitations to this method are four. It is limited to a one-dimensional grid——a very restrictive assump- tion. It cannot accommodate centers of unequal size. In addition, if the number of centers in product sequences vary, a solution is more difficult to obtain. Finally, the solu- tion is not generated or evaluated by a cost function. For these reasons, this method is not selected for further study. Enumeration of Alternatives This method1 guarantees an optimal solution; all permutations of centers are considered and the one with the least cost is selected. There are certain steps common to any enumerative procedure: 1There are at least two programs producing an exhaus- tive search. Listed in the order of program generality, the references are as follows: George Conrade, "Computers: Impartial Judge of.Kitchen Layout," Institutions Magazine, September, 1967, pp. 119-122; and P. Giles et al., Facilipy Allocation Project, Department of Industrial Engineering and .Administration, Cornell University Library (May 22, 1962), p. 8. ' 45 1. Read in f. and d. matrix elements. 1k jE 2. Generate a permutation of centers (a solution). 3. Determine if the cost of the permutation is lower than that of any solution generated previously. If it is, put the solution and its cost in temporary storage. 4. Return to step two until all N! permutations have been considered. 5. Print out the best solution found. Conrade's procedure, which was programmed in Fortran for the CDC 3600 computer, also includes an option for computing internally the distances between locations. It also prints out a permutation each time a solution cost is found to be lower than any previously generated. The fundamental problem with these programs, one that rules them out as workable tools if N exceeds 10 or 11, is the rapid growth of computational time as N is increased. There are 20! or 2.43x1018 permutations of centers to be generated when N is equal to twenty. It is true that if the layout grid is rectangular, the number of really different permutations is one-fourth as great; if the grid is square, there only one-eighth as many solutions. This characteris- tic derives from the ability to flip a permutation matrix about its diagonals, to convert to its mirror image, or to rotate it 180 degrees. If constraints are imposed on some xi. values, the number of permutations is reduced by an unknown amount. Unfortunately, an enumeration algorithm-- even one taking into account identical solutions to limit 46 the search of the solution space--is computationally in— feasible for larger values of N. If N is set equal to twenty and a computer evaluates one combination each micro— second, working eight hours per day and each day of the year, it would take one-quarter of a million years to reach the final solution.1 Experience with Conrade's program indi- cates that computer time when N is 10, 11, and 12 would be approximately 4, 44, and 528 hours of computer time respectively.2 Integer Programming_ Integer programming procedures for qaadratic cost functions and linear constraints are of only theoretical interest in solving the layout problem. Even for small problems, "a significant increase in computing efficiencies is required for integer programming to become as practical a tool as linear programming."3 Even suboptimal algorithms, which combine intelligently directed and random search pro- cedures, "require computer time per iteration which increases at an increasing rate as the number of variables increases. . . . The class of problem is also a major determinant of tO IWimmert, "A Quantitative Approach to Equipment Location in Intermittent Manufacturing," p. 95. 2Conversation with George Conrade. 3Donald Blessing Rice, "Discrete Optimizing Solutions to Linear and Nonlinear Integer Programming Problems" (unpub- lished Ph.D. dissertation, Purdue University, 1965), p. 2. 47 [time] since adding quadratic forms increases geometrically the number of calculations required at any stage of the procedure."1 Since the layout problem has a quadratic cost func- tion and an enormous number of variables and constraints, integer quadratic programming must currently be discarded as a practical solution to the layout problem. Another possibility is to convert the layout problem into an integer linear program by defining Yijkfi to be xij xk£°2 The following integer linear program is then appropriate: Minimize: N Z c. . y.. (11) 1,j,k,£=1 1.ka :ijfi Subject to constraints 1.3, 1.4, 1.5 and: N 2 2 y.. = N (12) ijkE 13kg .. + - .. ' ' = ,..., xlj ka Zyijkfi Z_0 (1,j,k,£ 1,2 N) (13) y. = O (i j k 2:1 2 N) (14) ljkfi l I I I I I'°°I Ibid., p. 49. 2Eugene L. Lawler, "The Quadratic Assignment Problem, Management Science, IX, No. 4 (July, 1963), 586-599. 48 Unfortunately, and xij involve N4 and N2 vari- Yijkz ables respectively. This condition and the erratic nature of integer programming mean that it is now an impractical tool for solving the layout problem. Branch and Bound There are several versions of branch and bound which guarantee an Optimal solution.l Most of them are amenable to both linear and Special quadratic cost components. The versions possess an identical philosophy which can be described in the following manner. Let the assignment prob— lem be viewed as a tree containing all possible permutations. Move out along the branches in stages, eliminating those branches which need not be investigated further. This is ascertained by computing the lower bound of the branches at each stage and eliminating all those having lower bound values higher than the cost of a known solution. Continue lThese versions are discussed in: Paul C. Gilmore, "Optimal and Suboptimal Algorithms for the Quadratic Assign— ment Problem," Journal of the Society_for Industrial and Applied Mathematics, X, No. 2 (June, 1962), 305-313; Eugene L. Lawler, "The Quadratic Assignment Problem," Management Science, IX, No. 4 (July, 1963), 586-599; Eugene L. Lawler and D. E. Wood, "Branch-and-Bound Methods: A Survey," Operations Research, XIX, No. 4 (July-August, 1966), 699-719; A. H. Land, "A Problem of Assignment with Interrelated Costs," Operational Research Quarterly, II, No. 2 (June, 1963), 185- 199; J. W. Gavett and Norman V. Plyter, "The Optimal Assign— ment of Facilities to Locations by Branch and Bound," Operations Research, XIX, No. 2 (March-April, 1966), 210-232; and J. D. C. Little et al., "An Algorithm for the Traveling Salesman Problem," Operations Research, II, No. 6 (November- December, 1963), 972-989. 49 in this manner until all but.one of the branches have been "pruned." This is the optimal solution. The branch-and- bound technique systematically searches only a portion of the solution space to find the optimal solution. The several branch-and-bound versions differ on two counts: (1) the order in which partial permutations are introduced for consideration and (2) how the lower bounds are calculated. The first source of difference is not to be considered here, since branch-and-bound methods have the same deficiency as the other Optimization techniques; i.e., they are computationally infeasible for larger problems. The version of Gavett and Plyter has been programmed in Fortran II for the IBM 7074. The computing times for five different values of N are given in Table 2—1. Table 2-1. Computing times for the branch- and-bound methoda N_ Computing Time 4 . . . . . . . . . 3 sec° 5 . . . . . . . . . 15 sec. 6 . . . . . . . . . 45 sec. 7 . . . . . . . . . 14 min. 8 . . . . . . . . . 42 min. aGavett and Plyter, Operations Research, p. 228. 50 The rate of increase in time is too great to select it as an algorithm for further study and there is no reason to believe the other versions will significantly reduce com- putational time. This is particularly true when the costs of various permutations are nearly equal. Perhaps the greatest promise for this technique would be when an excel— lent subOptimal solution is already known, but it is desir- able to find the optimal solution. Using the suboptimal solution as a starting permutation may eliminate so many branches in early stages as to make it computationally feasible. The second source of difference in the versions-- computing lower bounds—-does justify additional comment. One of the methods will be applied to the test problems in conjunction with selected suboptimal algorithms. There are at least five methods of computing the lower bound. The first method1 uses the matrix g:of the N(N-l)/2 x N(N-l)/2 order, which is computed by multiplying vector f_by the transpose of vector a. The lower bound is equal to the sum of column (or row) minima which are still allowed (not as yet constrained to zero in the solution process). If vectors f and a_were monotonically ranked in opposite order before computing O, lower bound would be the sum of the elements in the first column and last row of O. lLand, Operational Research Quarterly, p. 186. II and =\ 51 The second method1 is take the sum of the elements in the main diagonal of the ranked 9 matrix. This is equiv— alent to adding the product of the largest fik and smallest d. values with the product of the second largest fik and 33 second smallest djfi' and so on. The permuted dot product so obtained can be added to the lower bound of linear costs (which is found by solving the ordinary assignment problem) to get a lower bound accounting for both linear and quad- ratic costs.2 This technique is also appropriate for the case of a partial permutation having some, but not all of the N centers are already assigned. It is superior to the first method since its lower bound can never be less than the one produced by the first method. The third method, purported to give an even more realistic lower bound value,3 computes the lower bound using a matrix g:of the N x N order. The matrix 2 is the sum of matrix 2 and matrix 2; The elements ofig are the familiar bij values. Calculating the g matrix is more complicated. To compute the element c12, rank all elements of vector p which reference center one; rank all elements of vector d which reference location two in the reverse order. Set c12 equal to the permuted dot product of these two vectors. lGavett and Plyter, Operations Research, p. 217. 2Gilmore, Journal of the Sociepy for Industrial and Applied Mathematics, p. 307. 3Lawler, Management Science, p. 590. 52 The element t obtained by adding b12 to C12’ is the lower 12’ bound of the cost contributed by assigning center one to location two. After all N2 elements of g are computed in a like manner, the total lower bound is found by treating 2 as an ordinary assignment problem. The fourth method1 is to solve the integer linear program of the preceding section by eliminating the integer requirement; i.e., substitute xij Z_0 for xij = <2. The fifth method2 is to enter the Yijkz variables of equation 2.12 into a matrix of the N2 x N2 order and partitioning it into N2 minors in such a way that each minor becomes a linear assignment problem. Since little experience has been reported as to which procedure provides the best lower bound (the one with the highest value), the second method is chosen for use in this thesis. The reason for this choice is mainly the ease in computing it. Ste inberg ' 3 Algorithm This is a many—staged algorithm;3 at each stage the layout problem is viewed as an ordinary assignment problem having a linear cost function. lIbid., p. 591. 2Ibid. 3Leon Steinberg, "The Backboard Wiring Problem: A Placement Algorithm," Society for Industrial and Applied Mathematics Review, III, No. 1 (January, 1961), 37-50° On: \.AV a}; "LIL NV .3 I o: ‘1' ‘nd .1~.4‘ 53 There are two unique concepts in the algorithm. The first is to generate a family of unconnected sets of centers; i.e., centers having no materials flowing between them. It is preferable to have all centers mentioned at least once in the family. One set is introduced at each iteration for assignment. The second concept is to treat the assignment of such a set as the ordinary assignment problem, using any of the existing algorithms which give an optimal solution.1 If the resulting solution is feasible, it becomes the starting solution for the next iteration. A solution is feasible when none of the centers in the set occupy the locations already assigned to the (N-M) centers not in the set. The main steps in the algorithm are specified by the flow chart of Figure 2-1. This suboptimal procedure was originally programmed on the Univac I system, but is cur- rently being rewritten in Fortran.2 The final assignments are dependent on the initial solution, the unconnected sets considered, and the order in which they are listed. 1These algorithms are mentioned in the section on "candidate area" methods. Steinberg uses James Munkres' algorithm, which is found in ”Algorithms for the Assignment and TranSportation Problems," Journal of the Society for Industrial and Applied Mathematics, V, No. 1 (March, 1957), 32-38. ZA Univac representative reports that the algorithm was originally written in machine language for the Univac I computer. It was then reprogrammed in assembly language (SALT) for the Univac II. It is now being written in Fortran for the Univac 1107 and is somewhat of a proprietary nature. 54 Generate a family of p unconnected center sets. Mention each center at least once, if possible. I List the sets in any arbitrary order. I Generate or input the starting solution. I Select the next set to be checked. I Fix the (N-M) centers not in the set at the locations specified by the current starting solution. I Find the least—cost placement of the M centers in the set. I Is the solution fea- sible; i.e., is each No center assigned to only one location? 1 Make the new assignment Figure 2-1. the current starting solution. 1L Have p consecutive No sets been considered without producing a new starting solution? Yes Print out the current starting solution as the final assignment. Descriptive flow chart of Steinberg's algorithm. (Sets are chosen in order from the top to the bottom of the list. After the p set is checked, the algorithm returns again to the first set in the list.) 55 This dependency is more of an advantage than a disadvantage, Since it provides three ways of "perturbing" a final solu- tion in the search for an even better solution. Computa- tional time has never been reported, although computational feasibility is a safe assumption. There are three weaknesses in applying this algo- rithm to the layout problem. The first one is that con- straint 1.14 is not recognized, nor is the linear cost com- ponent included in the problem formulation. This problem is of a minor nature. Linear costs could be added to the data matrices prior to finding the least-cost solution at each stage. Constraint 1.14 could be imposed with an additional .infeasibility step. The second problem is also minor; some problems may have few fik values which are zero. This could drastically reduce the number of possible unconnected sets. The chances of this occurrence are rather remote for realis- tic problems of sufficient size. If it does occur, it is possible to include centers 1 and k in a set if fik is non— zero but negligible, such as two standard deviations below the mean. A third point, also of a minor nature, is letting the M centers of a set to be mapped into any of the N loca- tions, rather than restricting them to the M locations not already occupied. There is no apparent justification for such a provision. The most important problem, however, is the apparent failure of the algorithm to produce a solution close enough 56 to optimality. Reported eXperience with this algorithm is limited to one test problem (problem number five of Appendix IX). CRAFT and the algorithms of Gilmore and Hillier pro- duce answers of substantially lower cost.1 Although one test problem provides only preliminary evidence, it is deemed sufficient to select other algorithms for more detailed study. Kodres' Alggrithm Kodres2 considers a subclass of the layout problem by squaring the distance between location pairs in the objective function. Let xi be the abscissa of the location assigned to center i. Let yi be the ordinate of its center location. The layout problem can then be formulated as follows: Minimize: if (-)2+(-)2 (15) i k=l ik Xi Xk Y1 Yk Subject to: (x-x)2+(y-y)2>o (173k) (16) i k i k lSee Appendix XII. 2U. R. Kodres, "Geometrical Positioning of Circuit Elements in a Computer," Conference Paper 1172, AIEE Fall General Meeting, October, 1959. 57 xi an integer and (17a) 0 g_xi g_P yi an integer and (17b) 0512,30 Kodres demonstrates how several types of constraints can be built into the objective function to assure, for example, that a center is located on a particular line in the layout grid or that two centers are made adjacent and located along the same line. To obtain an intermediate solution, drop the con— straints stated in his original formulation. Minimize the positive definite quadratic function by setting all partial derivatives equal to zero and solve the resulting set of Simultaneous equations with the Gauss-Seidel interation technique.1 The final solution is constructed manually by using the relative location of centers in the intermediate solu- tion as a guide. "The assumption on which the construction is based is that in the complete solution the points will retain approximately the same relative positions of the . . . . 2 1ntermed1ate, non-integer solution." lKodres reports that "a representative lOO-variable system.was solved in less than five minutes on the IBM 704." Ibid., p. 6. 2Ibid., pp. 7-8. lvlchfi =.\ ..11 (N F‘ 58 The deficiencies of this algorithm are obvious. The squared distance criterion is not the one normally used.1 More importantly, the intermediate solution can leave much to be desired. Locations are points rather than discrete areas. ,Several centers can be assigned to the same point. If no constraints are built into the objective function to fix centers in certain grid areas, "bunching" can occur. It is also noteworthy that the flow matrix of his test problem is extremely simple. All but a few of the fi values are k zero, simplifying the last phase of the solution process. Gilmore's N4 and N5 Algorithms Gilmore offers two versions of a suboptimal proce— dure embodying an (N-l)-stage decision process.2 At each stage an unassigned center is matched with one of the remain- ing locations. The solution procedure involves 3! a matrix used in the third method of computing lower bounds as described in the previous section on branch-and—bound tech— niques. Let k be the number of centers already assigned and T be of the (N-k) x (N-k) order when the (k+1) center is to 2k be assigned. This assignment choice is made by picking one 1This thesis provides substantial evidence that such changes in d.£ values do not materially affect the qual- ity of solutions generated by a satisfactory algorithm. See the discussion in Chapter IV on alternative distance criteria. 2P..C. Gilmore, Journal of the Sociepy for Indus- trial ang_Applied.Mathematics, pp. 310-313. 59 of the elements in 2k' be assigned to location j. If it is tij’ then center i is to Gilmore suggests two decision rules for making this choice. The first decision rule (for the N4 algorithm) is to make a max-min choice, i.e., pick the maximum element from the minima of all lines. The second decision rule (for the NS algorithm) requires more computational effort. It calls for the maximum element to be chosen from that set of cells obtained by treating 2k as an ordinary assignment problem. These two versions were programmed on the IBM 7090 and applied to a test problem, with the N4 algorithm sur- prisingly providing a better answer than the N5 algorithm. Gilmore explains that this was due to the arbitrary way of breaking ties when implementing the decision rules. The N4 algorithm took less than one minute of computer time and the N5 algorithm took less than three minutes when N was 36.1 Both versions are very efficient and explicitly recognize linear costs. However, recent comparative tests based on random test data indicate that a slightly different algorithm, although somewhat slower, provides better solutions than 1Ibid., p. 312. Gi 60 . 4 . . . Gilmore's N ver31on.l For this reason, G1lmore's algo— rithms are not among those selected for further analysis. The Hillier-Connors Algorithm This algorithm3 retains the philosophy of Gilmore's algorithms, differing only in how to choose an element from Ti. tion Method for the transportation problem in linear pro- The new criterion was suggested by Vogel's Approxima— gramming.4 Determine the arithmetic difference between the smallest and next smallest elements in each line of ER“ "This quantity provides a measure of the proper priorities for making allocations to respective rows and columns since it indicates the minimum cost penalty incurred by failing to make the assignment to the smallest cost cell in that row or column."5 Select that element of g for assignment which is the smallest cost cell in the line having the largest differ— ence. This algorithm, which is programmed in Fortran for the IBM 7090, seemingly performs better than Gilmore's N4 lHillier and Connors, pp. 27—34. 2Another reason is that it was learned that Gilmore's computer programs "are no longer available in any form." Letter from P. C. Gilmore, January, 1968. 3Hillier and Connors, pp. 15-20. 4Nyles V. Reinfeld and William R. Vogel, Mathematical Programming (Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1958). 5 Hillier and Connors, p. 17. 61 version. Computational times should be comparable if both are programmed equally well. However, a comparison of it to Hillier's earlier algorithm using random number flow data with no linear costs indicates that Hillier's version is somewhat superior in terms of solution costs. The Hillier- Connors algorithm also has the disadvantage of producing only one suboptimal solution; this is also true with Gilmore's versions. CORELAP This computer algorithm1 translates a REL chart into an assignment solution. Inputs to the program are the REL chart,2 the area requirements of each center (expressed in terms of the total number of units squares), the total num- ber of unit squares for all centers, N, and the maximum length to width ratio allowable for the layout grid boundary. The algorithm begins by assigning the center with the highest closeness rating to a location in the center of the grid. This center is called a "winner." The "total closeness rating" for center i is equal to: 1This procedure was initially made available in Robert C. Lee's thesis. "QOmputerized RElationship Egyout Planning" (unpublished M.S. dissertation, Northeastern University, 1966). A more recent reference is: Robert C. Lee and James M. Moore, "CORELAP-OOmputerized RElationship Egyout Planning," Journal of Industrial Engineering, XVIII, No. 3 (March, 1967), 195-200. 2The REL chart is discussed in an earlier section of this chapter. 62 (i # k) (18) IIMZ r1 ik The list of unassigned centers, presumably ranked by total closeness ratings, is then searched for a victor. A "victor" is an unassigned center having the maximum possible REL value (call it "A") with the winner. If the victor also has an "A" rating with another center located near the win- ner, the victor is immediately assigned to a location as close as possible to the winner. An attempt is made to make the overall shape of the center as square as possible. The search then begins for another victor. If one is found, the same procedure is followed. If one is not found, all victors are checked for an "A" rating with an unassigned center. If such a victor is found, it becomes the new winner and the center having an "A" rating with it is made the victor. If no victors qualify as winners, the victors are searched again for a relationship with an unassigned center having a next to the highest rating. This process continues until all centers are assigned. This [solution process] results in the apparent "growth" of several crystals in a loosely con- nected fashion. The center of each crystal is a Winner with it's Victors around it, but several Winners may share a single Victor which provides the logical tie between the growing "crystals." 1Lee and Moore, Journal of Industrial Engineering, p. 196. 63 This algorithm is computationally efficient. A Fortran program for the IBM 7090 computer took 2.46 minutes for a problem having an N of forty-five. It generates a common-sense sequence of events which would seem to resemble the sort of thought process a layout analyst would have if relying solely on his judgment and a REL chart. There are several a priori judgments as to the relative worth of CORELAP which seem relevant. Firstly, the assignment of a center is made by considering the relation- ships of it with only two other centers. If a center has more than two high REL values, they would be ignored in making the assignment. A second consideration is the use of a REL chart rather than the objective function provided in the first chapter. Lee and Moore consider this an advantage, since otherwise it is not possible to "take into consideration the problems involved in placing service facilities, such as washrooms, cafeterias, maintenance shops, and the like."1 The validity of this statement can be questioned. It can be concluded, however, that the use of a REL chart prohibits the calculation of the usual total measure of efficiency unless the following two assumptions are made: 1. A nonzero closeness rating between two centers implies that, ceteris paribus, the closer they are located, the better is the solution. Likewise, the 1Ibid., p. 195. 64 further away they are located, the worse is the solution. 2. The analyst can construct a scale that cardinally reflects the importance of each relationship. If these two assumptions are met, then the algorithms of this section are compatible with CORELAP and the total measure of effectiveness would be: 1/2 . . z _ rik djg (19) Other considerations relevant to CORELAP are: (1) its current inability to generate more than one solution, (2) its ability to handle centers of unequal size, (3) its inability to handle constraints necessitating or ruling out the assignment of a center to a particular area, and (4) the individual and collective shapes of centers which are output. Since any suboptimal procedure is capable of produc- ing a relatively poor solution, the first consideration is important. It could be remedied rather easily, however, by listing all potential victors (or winners) which meet all other qualifying requirements and selecting from this list randomly. This would reduce the possibility of generating only one answer. Provisions for constraint satisfaction are also possible, although this is complicated by the way cen- ters of unequal size are handled. 65 The authors cite the last consideration as an advan- tage, since the grid boundaries provide a basis for design- ing the building which will house the centers.1 Hewever, CORELAP can produce very unconventional center and building shapes. It is even possible to have holes in the solution grid. Although the analyst can adjust the shapes to obtain a realistic solution, he may unwittingly eliminate some of the relationships which make the unadjusted solution a low- cost one. If production requirements are perceived to be ag constant over time that a less conventional (and flexible) building shape is desired, it has been pointed out that CRAFT2 (as well as all of the other algorithms of this sec- tion) is also amenable to building design simply by adding dummy centers and an equal number of locations. The disadvantages of CORELAP seem to outweigh its advantages when compared with several other algorithms. In addition, research3 is now underway to compare CRAFT with CORELAP. Since CRAFT is tested in this thesis, a common yardstick is available to compare the research of this thesis with CORELAP at a future date. lIbid., p. 200. 2Elwood S. Buffa, "Reader Comment," Journal of Industrial Engineering, XVIII, No. 8 (August, 1967), 502. 3James M. Moore, "Author's Comments," Journal of Industrial Engineering, XVIII, No. 8 (August, 1967), 502. 66 CRAFT This algorithm, apparently an outgrowth of Glaser's concepts,l has been developed into a comprehensive computer program.2 The bulk of it is written for the IBM 7094 in Fortran.3 A flow chart of CRAFT is found in Figure 2-2. Given an initial starting solution, the algorithm executes successive exchanges of center locations which reduce the value of the objective function. For each itera- tion, the exchange leading to the greatest apparent cost reduction is selected. Recyling continues until no exchange reducing costs can be found. Only those centers having equal areas or centers of unequal area sharing a common border in the previous solution are considered for exchange. lMurphy suggests that Armour was aided by Robert H. Glaser, whose research dealt with "airborne digital computers where the purpose was to avoid high frequency signal loss due to excessive inductive reactance and stray capacitance." Daniel J. Murphy, "Machine Location Patterns for Facility Analysis" (unpubliShed M.S.I.E. dissertation, University of Pittsburgh, Pittsburgh, 1957), p. 87. 2Buffa, Vollman, and Armour appear to be the major authors of CRAFT: Gordon C. Armour, "A Heuristic Algorithm and Simulation Approach to Relative Location of Facilities" (unpublished Ph.D. dissertation, University of California, Los Angeles, 1961); Thomas E. Vollman, "An Investigation of Bases for the Relative Location of Facilities" (unpublished Ph.D. dissertation, University of California, Los Angeles, 1964); and Elwood S. Buffa, Gordon C. Armour, and Thomas E. Vollman, "Allocating Facilities with CRAFT," Harvard Business Review, XLII, No. 2 (March-April, 1964). Another program for the algorithm is found in P. Giles et al., FacilityyAlloca- tion Project. 3This program is in the SHARE Library of computer programs (SDA 3391). 67 Start 1 Input parameters and starting solution. I :y> Output cost and other desired information. Analyze all feasible exchanges. Select one with largest potential cost reduction. I - (Has one been found? I Yes Make the exchange. I Recompute affected matrices. No g I: Stop. Figure 2-2. Descriptive flow chart of CRAFT. 68 If all centers require equal areas and the search is re- stricted to pair Switches, the total number of exchanges investigated per iteration is: N! 2 2,(N_2) = (N -N)/2 (20) The evaluation of an exchange requires computation of two vector dot products, switching the two vectors, recomputing the dot products, and computing the difference between the two products. If center area requirements are unequal, actual cost reduction after an exchange need not be what is expected. This is because new center centroids are computed after the algorithm is committed to the exchange. The program is amenable to three types of alternatives; two-center exchanges, three-center exchanges, or the best of the first two alternatives. The effective portion of the CRAFT program is a small part of its total length. Most of the program is concerned with allocating unit squares after an exchange is selected. Since the authors1 of CRAFT feel it has considerable promise, CRAFT is selected for additional study. It would seem that the CRAFT model should yield results superior to those obtained by Hillier's model since the 1It is reported that "Gordon Armour is currently working on the inclusion of relayout costs to generate changes based on a return on investment capital run-off criterion." Vollman, "An Investigation of Bases for the Relative Location of Facilities," p. 28. 69 former considers more possibilities and is not restricted to adjacent single step moves. Hillier's Algorithm This computer algorithm2 is also an iterative scheme; a flow chart for it is given in Figure 2-3. At each itera- tion a pair exchange is selected which, according to an a priori indication, will reduce the objective function the most. Only rectilinear and diagonal pair exchanges which are k units separated are considered. The value of k can be initialized to any integer number equal to or less than the maximum distance between any pair of locations in the layout grid. AS soon as no more exchanges are found to reduce the objective function, k is reduced by one and the algorithm recycles. When k is one and no cost-reducing pair exchanges are thought to exist, the algorithm terminates. The criterion providing an a priori indication of a favorable interchange is called a "k-step move desirability 1Thomas E. Vollman, "An Investigation of Bases for the Relative Location of Facilities" (unpublished Ph.D. dissertation, University of California, Los Angeles, 1964), p. 37. 2A listing and description are found in Hillier and Connors, "Quadratic Assignment Problem Algorithms and the Location of Indivisible Facilities," pp. 20-73. Other sources are: Frederick S. Hillier, "Quantitative Tools for Plant Layout Analysis," The Journal of Industrial Engineer- ipg, XIV, No. l (January-February, 1963), 33-40; and Frederick S. Hillier and Michael N. Connors, "Quadratic Assignment PrOblem Algorithms and the Location of Indivisible Facilities," Management Science, XIII, No. 1 (September, 1966), 42-57. 7O Start J. Input parameters and starting solution. I Print out current solu- tion and its actual cost. Compute the k-step move desirability table. Find candidate, i.e., center having largest element in table. Will the suggested horizontal (vertical) Yes exchange be favorable? L... Will one of the dia- gonal interchanges be favorable? lYes Make the switch. E Give the cell just con- sidered an arbitrary negative number, so it will not be considered again. Have at least 4N cells been considered in the current table? NO k>0? \ Yes Make "last pass.“ Stop. Figure 2-3. Descriptive flow chart of Hillier's algorithm. (A'Tast pass" sets k back to its original value; therefore the whole solution process is repeated once more.) 71 table." This matrix is of the N x 4 order. The rows repre- sent centers and the columns represent the four possible rectilinear directions--up, down, left, and right. An ele- ment in the first column, for example, is equal to the cost reduction of moving the center under consideration k units of distance up on the grid without changing any other center assignments. It is therefore a measure of the center's "desire" to move up. Diagonal move values are equated to the sum of horizontal and vertical move values. At each iteration, the matrix is searched for the largest cell not yet considered. Each of the three alterna- tives associated with this cell are investigated. If the element represents a center to be moved to the right, exchanges are considered with centers k steps to the right, k steps up and k step to the right, and k step down and to the right. If cost approximations indicate the exchange is favorable, it is made. Otherwise, the next largest positive matrix element not yet considered is evaluated for inter- change. Since it is suggested that "applying the Hillier algorithm once yields a better solution on the average than 1 any other available suboptimal algorithm," Hillier's approach justifies further study. lHillier and Connors, pp. 27-34. 72 ALDEP The Automated payout QESign‘grograml is an IBM 7090 Fortran program generating random solutions and evaluating them with the REL chart. All solutions having a certain minimum score are stored on magnetic tape to be later output in the form of line drawings using a CALCOMP plotter. The score of any solution is the sum of the REL values for all adjacent center pairs. It is therefore a measure of savings rather than cost, with maximization being the objective. The authors suggest two variations on this theme: A variation . . . allows the first layout to be developed randomly and proceeds to permute pairs of units randomly and retains the new layout only if the score improves. . . . This method . . . converges much more quickly to a maximum score. The better technique would be several randpmly generated layouts used as starting points. The second variation would be a modified random— selection technique. Initially, any available department is randomly selected. (After the selected department is processed, the preference table REL chart for that department is searched to find any depart- ment with a demand preference, that is, the preference of highest priority. If an avail- able department is found with demand preference, this department is processed next. If no avail— able department is found, a department is selected randomly. This procedure is repeated until all departments are processed. - lJerrold M. Seehof et al., "Automated Facilities Lay- out Program," Proceedings-A.C.M. National Meeting, 1966, pp. 191-199. 2Ibid., p. 1920 3Ibid., p. 194. 73 This computer program handles unequal center areas, but not linear costs or constraints on Xij' The probability that a completely random selection process would generate a good solution at any one iteration is small. However, a solution is generated so quickly that it could conceivably reach a satisfactory solution in less time than other algo- rithms. It is also of interest in providing a measure of the average solution costs which act as a sort of upper bound. For these reasons, a random selection procedure is tested in this thesis. Since ALDEP is "not generally available outside IBM,"l a very simple version is written expressly for this thesis. It uses the more conventional measure of costs based on f. d. values. 1k jfi Wimmert's Method The manual computational scheme2 originated by Wimmert, when made operational and extended, provides the basis for several computer algorithms. However, the essence of any version is as follows: select successive elements in lIbid., p. 191. 2Relevant references are: Robert J. Wimmert, "A Quantitative Approach to Equipment Location in Intermittent Manufacturing" (unpublished Ph.D. dissertation, Purdue University, 1957); Robert J. Wimmert, "A Mathematical Model of Equipment Location," Journal of Industrial Engineering, IX, No. 6 (November-December, 1958), 498-505; and David N. _Willoughby, "A Technique for Integrating Facility Location and Materials Handling Equipment Selection" (unpublished M.S. dissertation, Department of Industrial Engineering, Purdue University, 1967). 74 matrix 23 eliminate the quadraplets implied by these ele- ments with the use of the tally matrix T_until only one feasible diad remains for either a center or location; enter this diad into the final solution by using the g matrix; and continue this process until N diad assignments have been made. The versions can be thought of as ruling out solu- tions involving the largest Cijkfl terms until this cannot be done any more without eliminating the last possible fea- sible solution. A flow chart of Wimmert's method is given in Figure 2-4. Definitions for the new terms used in this paragraph are as follows: O_or COST.--This ranked matrix is of the (NZ-N)/2 x (NZ—N)/2 order; its elements are c. as defined in equation 1jk£ 1.1. Each row of the matrix refers to a different set of centers [i,k}, where i # k. Each column refers to a differ— ent set of locations {j,£}, where j # B. The matrix is unique in the way c. elements are arranged. Let: 1jk£ = a' f (21) H0 The column vector Q} has (NZ-N)/2 d. elements, 32 where j # 2, which are ranked in non-increasing monotone order from top to bottom. The row vector p consists of (NZ-N)/2 f. terms, where i # k, which are ranked in non- 1 k decreasing monotone order from left to right. Our computer algorithms never store O_in memory, since a and §_can be used directly; however, the matrix is of value for expository purposes. Figure 2-4. 75 I Start I Select the next c.. ijkE term according to some rule. L Enter tallies to eliminate quadruplet (iojtklae) I Has (i,j)o (i,£), (klj)l or (111.2) now become infeasible? 3LY’es Make appropriate entries in the §_matrix to refledf the new infeasible diad(s). Search the lines in to see if any cente (location) has been disqualified from all but one location (center). .5. f' Has such a line been found? I Yes Revise §_to Show the new centér assignment. Have (N-l) assignments now been made? I L——————-— iYe S Deduct final assignment and output solution. NO I _4 Stop Simplified descriptive flow chart of Wimmert's method. 76 Quadraplet (i,j,k,£).--This term refers to the four labels (i,j,k,£) implied by each element of g” When we Speak of eliminating (i,j,k,£) from the final solution, this means that all of the following conditional assignments are to be disallowed: xij = 1 if Xkfi = l Xifi = 1 if xkj = 1 Xkfi = 1 if xij = l xkj = 1 if X12 = 1 However, eliminating quadruplet (i,j,k,£) need not rule out any of the following conditional assignments: xij = 1 if Xkfl = 0 xifl = 1 if xkj = 0 Xkfl = 1 if xij = 0 xkj = 1 if Xifi = 0 Diad (i,j).-—This term refers to the assignment of center i to location j. .It is used synonymously with the xij variable defined in Chapter I. Four diads derive from quadruplet (i,j,k,£): diads (i,j), (k,£). (1,2), and (k,j). For this quadruplet, let the "complement" of diad (i,j) be taken to mean (k,£). Similarly, the complement of (1,2) is (k,j). There are two "diad sets" for this quadruplet: [(i,j), (k,£)} and ((1,3), (k,j)}. Diad (i,j) is said to be 77 "infeasible" if it has been equated to zero; it is "assigned" after being equated to one.1 Egor Tally.--A quadruplet is eliminated by incre- menting up to four elements in this matrix. The incremented elements are the derivatives of the eliminated quadruplet. Since each element in 2 references a diad, the element's value is the number of the diad's complements which have been tallied. There are limits to the number of complements which can be tallied without ruling out all feasible solu- tions not involving the quadruplets already eliminated. Since these limits are specified by XOUT, g:is used with XOUT to determine when diads are to be made infeasible. This system therefore is intended to prevent tallied qua- druplets from entering the final solution. The size oft; varies, depending on the algorithm used. XOO£.--This is either a parameter or a vector of (NZ-N) elements, depending on the version used. The purpose of XOUT is to judge when a TALLY element (tij) increases to the critical value where diad (i,j) is deemed infeasible. fiLcu'SOLUTION.--This (N+1) x (N+1) matrix provides the current feasibility status of each diad. Let the follow- ing scheme be used in giving Sij a numeric value. At the start of the solution process, initialize §Iat zero. After 1This refers to the problem formulation of Chapter I. Different conventions are adopted for the SOLUTION matrix in our computer programs. 78 a sufficient number of quadruplets have been eliminated to make a diad infeasible, the element in g corresponding to the infeasible diad is equated to one. If either of the two lines passing through this element now have (N-l) elements which are nonzero, the zero element must be equated to two. This means the "open" diad has been assigned; i.e., entered into the final solution. All other (ZN-2) elements in the lines passing through the assigned diad are equated to three to show that they can no longer be assigned. The (N+1) col— umn stores the total number of nonzero elements in each row and the (N+1) row stores the number of nonzero elements in each column. The order in which elements in g are selected is prescribed by Wimmert to begin with the northeast corner of the matrix, then the elements in the minor diagonal next to the northeast corner, then the elements in the minor diago- nal two steps from the northeast corner, and so on. This particular order greatly simplifies infeasibility testing, k 1 but does not guarantee that fi djz values are selected in the order of their magnitude. This possibility, which was recognized by Wimmert, led some authors to conclude that the procedures have been "disproved."2 Conway and Maxwell 1R. W. Conway and W. L. Maxwell, "A Note on the .Assignment of Facility Location," Journal of Industrial Engineering, XII, No. 1 (January-February, 1961), 7-13. 2Armour and Buffa, Management Science, p. 294. 79 suggested an alternative approach would be better, that is, searching systematically in the neighborhood of the main diagonal for feasible solutions. This alternative has been attempted, but with very poor results.1 However, mathemati- cal criticisms of Wimmert's original procedure, which is admittedly suboptimal, are not appropriate. There is no assurance that optimal solutions will be generated even if Cijkfi values are selected in the order of their magnitude. Wimmert's method had never been programmed for the computer and never compared with other approaches. Propo- nents describe it in rather glowing terms; it is selected for further analysis in this thesis. In general, Wimmert's method will result in an optimum solution for n machines in n possible locations. However, the optimality of the solu— tion using successive diagonals . . . does not necessarily hold. An optimal or near optimal solution may be obtained by eliminating the high value cell and its dependents before the lower diagonal value. lP. Giles et al., Facility Allocation Project. 2Reed, Plant Location, Layput and Maintenance, p. 100. a.. ¢~s Tu 9X A FL 8 CHAPTER III ALGORITHMS FOR WIMMERT'S METHOD Introduction In this chapter, the computer algorithms derived from Wimmert's original scheme are described. There are . 1 nine stages *1. *2. *3. *4. 5. 6. *7. *8. 9. to each algorithm: Selecting quadruplets Entering tallies Updating CRITERIA Infeasibility testing Revising SOLUTION Making diad assignments Detecting post—assignment infeasibility Recycling Deducting the last diad assignment. These stages provide the organization of this chapter; each stage is taken in order as a separate section. For the stages marked by an asterisk, several alternative decision rules of unknown merit are used. All decision rules are explained and matched with the versions which use them. It is convenient to adopt a set of terminology for describing the algorithms. The more frequently used terms are defined in the last section of Chapter II. Additional 1As Shown in Figure 2-4, Wimmert's original scheme recognizes only six stages and therefore is not completely specified. 80 81 terms are defined as needed in this chapter. A more compre- hensive list of definitions is given in Appendix I. Selecting Quadrupiets Quadruplets are selected for elimination indirectly. The choice is made by first selecting a c.. element and ijkfi then eliminating the quadruplet implied by it, that is, quadruplet (i,j,k.£). There are three decision rules for selecting g elements. Rule 1—1. The first element chosen is the one in the north- east corner of the matrix. The next two elements belong to the diagonal next to the northeast corner element. The next three elements belong to the second diagonal closest to the northeast corner element, and so on. Elements in a diagonal are introduced one at a time, beginning with the one directly south of the northeast corner and proceed- ing up along the diagonal until reaching the element in the same row as the northeast corner element. Rule 1-2. Select diagonals in the order specified by rule 1-1, but eliminate all diagonal elements before searching SOLUTION for diad assignment demands. Rule 1-3. Select the largest element in g not previously chosen. Break ties with a random choice from a1; tied elements. Rule 1-1 is suggested by Wimmert to simplify infeasi- bility testing. Rule 1-2 is a modification of it. It should be noted that neither rule guarantees the selection of cijkE elements in the order of their magnitude. Rule l-l implies that all terms in a minor diagonal are greater than any element in diagonals to the west. It also assumes all CijkE values in a diagonal are ranked in nonincreasing size moving 82 northwest from the southeast corner. Rule 1-2 does not make the latter assumption, but instead assumes that all diagonal elements are of equal value. All of these assumptions can be violated. This is particularly true when the variance of (fi-fi-l) is large or when (di-di-l) var1es cons1derably as 1 increases from 1 to (NZ-N/Z). There ia one property of O (not fully SXploited by any version) which is always true. Element Cijkfi can never be less than that portion of the matrix partitioned with two lines drawn from it. The first line is drawn from Cijkfl directly south and the second directly east. The computer routine for implementing rule 1—1 or 1-2 is simple. KOUNT7 is the index of the minor diagonal being considered, where KOUNT7 is one for the first diagonal (northeast corner element). The row and column indices (KOUNTlO and KOUNT9) are incremented to consider each ele- ment in the KOUNT7 diagonal. On the other hand, the routine for implementing rule 1-3 can be described as follows; 1. If KOUNT 27 is greater than zero, go to step 3. KOUNT27 indicates the number of the quadruplets stored in matrix CELLTIE. 2. Of the elements in the first row of O which have not yet been selected (their SCORE vglues are zero), identify the element(s) with the largest value. Store its (their) labels in CELLTIE. Discontinue the search if all elements have already been eliminated or as soon as an element is found having a value less than one to its east. Search the succeeding rows the same way, discard— ing all previous CELLTIE elements when an element is found with an even larger value. 83 3. Choose a quadruplet randomly from the KOUNT27 rows of CELLTIE. Strike out the row selected in CELLTIE, move up all following rows by one notch, reduce KOUNT27 by one, and set the corresponding SCORE element to one. Entering Tallies The way tallies are recorded depends on the size of the 3 matrix as well as the particular quadruplet being eliminated. The Tlmatrix takes on one of three sizes in this thesis: N x N, (NZ—N) x N, or N2 x N2. The rows of the first matrix refer to centers; the columns reference locations. There is only one tij element for diad (i,j). The columns of the (NZ-N) x N matrix also refer to locations. However, the row pairs {1,2}, {3,4}, ..., {(NZ-N-l), (NZ-N)} reference the center labels of fik values which have been ranked in nonincreasing monotone order from top to bottom. This arrangement simplifies infeasibility testing. There are (N-l) tij elements for each diad. If quadruplet (i,j, k,£) is selected for elimination and corresponds to the p I row of C, the relevant TALLY elements are t ., t . = ZED-1,] 2p!) t , and t . Each element in the N2 x N2 TALLY 2p-ll£ 2p1£ matrix references a complement of a diad, with every comple- ment of each possible diad being allotted an element in the matrix. There are (N2) tij elements for each diad. When this matrix is used, a supplementary tally matrix (MINTALLY) of the N x N order is used to make infeasibility testing more efficient. QT vi A: ans“.— u “A; OI 84 There are two alternative decision rules for tally entry. Rule 2-1 is implied by Wimmert, whereas rule 2-2 seems superior on a priori grounds. Rule 2-1. Always add one tally to each possible TALLY element (t. ,t. ,t . and t ). 13 1E kj k2 3219 2-2a. If any of the four derivative diads of the quadruplet are already assigned, enter no tallies. Make the complement of the assigned diad infeasible in SOLUTION, if this has not already been done. The reason for this rule follows from the definition of quadruplet elimination. Rule 2-2b. If rule 2-2a is not evoked and at least one diad in a set is already infeasible (equal to one in SOLUTION), enter no tallies for the set. The justi- fication for this rule is that if a diad is already infeasible, its complement should not be "penalized." The diad and its complemen cannot both enter into the final solution anyway. This rule is applied separately to both diad sets: ((i,j), (k,£)} and {(i1£)1 (klj)}° _Rule 2-2c. For each diad set not evoking rules 2-2a or 2—2b, enter two tallies. If tallies are to be entered for both sets, for example, add 1 to tij’ tiZ’ tkfl' and tkj' gpdatinquRITERIA When searching SOLUTION, it is possible to find one OI? more lines containing no assigned diad, but at the same tiine having no zero element. Another possibility is that Serveral lines have only one nonzero element. One cause of 1This need not be true if the infeasible diad bfiElongs to a SOLUTION line having no assigned diad and has all its elements "closed" (nonzero). 85 this "conflict" is that, even if only one quadruplet is eliminated at a time (rule 1-2 is not used), up to four diads are tallied. This may make more than one of them infeasible simultaneously. Another cause is that when a diad is assigned, all of the other (ZN-2) elements in its two orthogonal lines are simultaneously made infeasible. The possibility of conflict is not recognized in Wimmert's original procedure and therefore some technique must be developed to resolve conflicting diad assignment demands. One possible alternative is to select the diad with the smallest lower bound. This approach is like Gilmore's algorithm but is not eXplored in this thesis. The approach we do use involves a matrix called CRITERIA. There is one element in this matrix for each possible diad. The value of this element provides an approximate measure of the costliness of the corresponding diad assignment. The mea- sure is necessarily approximate, because the cost of an assigned diad depends on which of the other diads also enter into the final solution. This is known only after a solu- tion is reached, not during the solution process. Four decision rules seem promising on an a priori lDasis. All four of them record the number (or cost) of Eireviously eliminated quadruplets which will be disqualified Iif the diad is made infeasible. 86 Rule 3-1. Add one to CRITERIA values corresponding to the four derivative diads of the quadruplet eliminated. Rule 3-2. .Add one to the CRITERIA values of each diad in a set which has not evoked rule 2-2a when the quadru- plet was tallied. Rule 3-3. Add a given fraction of f- - to CRITERIA values of the four diads derivingg from the quadru— plet being eliminated. Rule 3-4. Add a given fraction of fm dag to CRITERIA values of each diad in a set lwhi has not evoked rule 2- 2a when quadruplet (i,j,k,Z) was tallied. Infeasibility Testing A diad should be made infeasible for either of three reasons. The first reason is that one of the diads in a set belonging to an eliminated quadruplet has already been assigned. This type of infeasibility is accounted for in rule 2-2a. The second reason is that its center has already been assigned to another location (or vice versa). The third reason is that quadruplets involving the right "mix" of diad complements have been eliminated so as to make it impossible for all complements to satisfy condi- tions 1.3, 1.4, and 1.5 if the diad is assigned. The logic 3behind this third type of infeasibility is developed with “the examples of the following sections and the TALLY matrix c>f Table 3-1. This matrix is identical to the N2 x N2 TALLY Inatrix described in the section on tally entries except that ciiad sets having i=k or j=£ are excluded. There are 3 (N4-2N +N2) matrix elements, one for each diad complement. Since there are four derivative diads to each quadruplet, 87 Table 3-1. TALLY matriX* j l 2 3 4 N 1k1234...N 134...N 124...u 123...»: 1234... :2 It'll!» a a a a c c 3 b b b b 1 4 d b d b b b g, l a c a c ace 3 c c c c 2 4 d c c d c c £1 1 b b b b 2 c c c c 3 4 d d N b b 2000 (”NH @ n U" Z eeeubuNI-l I L *Circled groups of tallies indicate an infeasible died for the various examples. C: ‘PL 88 four tally entries are appropriate per quadruplet. No qua- druplet is tallied more than once, making it impossible for a matrix element's value to exceed one. The question to be answered is how many of a diad's complements can be tallied before a diad becomes infeasible. Example A jN=4).--If N is equal to 4 and three com- plements of a diad have been talliedl which all reference the same center, the diad must be made infeasible to assure that none of the quadruplets enter into the final solution. Consider the following quadruplets involving diad (1,1): (l,l,2,2), (l,l,2,3) and (l,l,2,4). Their elimination means that if center 1 is assigned to location 1, center 2 cannot be assigned to any of the remaining locations. Example B (N=4).--If four complements of a diad have been tallied which reference the same two centers and two locations, the diad is infeasible. Consider quadruplets (1,3,3,2), (l,3,3,4), (l,3,4,2) and (l,3,4,4). If they are to be eliminated, diad (1,3) must be made infeasible. Other- wise, only location 1 is left for centers 3 and 4. This violates conditions 1.3, 1.4, and 1.5. Example C (N=4).--If the following six quadruplets are eliminated, diad (2,4) is still feasible: (2,4,3,3), (2,4,3,1), (2,4,4,2), (2,4,4,1), (2,4,1,2), and (2,4,1,3). 1The tallies for example A are indicated by the letter "a" in Table 3-1. .0 l r—J a! 89 For example, one feasible solution which includes none of the quadruplets in the final solution when diad (2,4) is a551gned would be: xll=l; x24=1, x32=1; and x43=1. This example demonstrates that the number and type of quadru- plets both bear on infeasibility. Example D jN=4).--If three complements of a diad are tallied which reference the same location, the diad is infea- sible. Let quadruplets (4,3,3,l), (4,3,2,1) and (4,3,1,1) be eliminated. Since it is not possible to assign centers 1, 2, and 3 to only two locations (2 and 4), diad (4,3) is infea- sible. Generalized Case.—-Let the number of centers (N) take on any positive integer value and O take on integer values from 1 through (N-l). Then the rule for the gener- alized case is: Rule 4—1. A diad is infeasible when at least (¢N_¢2) of its complements have been tallied which reference the same o centers and (N-¢) locations. The number of tallied complements required to make a diad infeasible is therefore variable. It depends on the center and location labels of each tallied complement. The minimum number of tallies is (N-2). When the number increases to (N-l), infeasibility becomes possible; i.e., the case where 4 equals 1 or (N-l). The maximum number (NZ/4) occurs when 4 equals N/2, with fractional values truncated when N is an odd number. 90 6 = eN-¢2 Siti— _ carp—NM ¢* = u/z 2 It is impossible to eliminate (N /4+1) diad complements without also making the diad itself infeasible. ProgrammianRule 4-1 Only one version implements rule 4-1 when rule 1-3 is also used. The version's two limitations make it only of academic interest. First of all, there is an enormous com- puter storage Space problem for the 2 matrix. Even if.l is a logical array and the matrix is reduced to have only 3 one element for each quadruplet rather than four, (N4-2N +N2) /128 words are required. Secondly, the storage space and computational time required to make an infeasibility test becomes forbidding. This is evident when analyzing how rule 4-1 can be programmed. Let (N-A) be the number of centers yet to be assigned before reaching the final solution and diad (1,1) be tested for infeasibility. Take each possible combination of ¢ centers not involving center 1, where O = 1,2,..., (N-A-l). Each combination provides the center labels to a set of comple- ments to be checked. Determine if at least one combination references the same (N-O) locations, other than location 1. If one is found, diad (1,1) must be made infeasible. In our version, each center label of a combination is stored in a 91 separate word. The number of combinations grows very rapidly as N increases. If N is 40, 1.1 x 1012 combinations must be generated and the number of computer words required is 2.2 x 1013. The number of combinations is the sum of coefficients of the binomial distribution; it is equal to: N-l z N: = N_ ¢=1 e: (N-¢)1 2 2 Not only are storage space requirements excessive, but the time spent generating, storing and reading them is also significant. A routine has been written to generate combinations, store them on the drum, and then read them. This routine is given in Appendix II. The computer time for varying levels of N is given in Table 3-2. Even though the time can be reduced by buffering and blocking the data, it is excessive. Each quadruplet requires four tests2 and the number of quadruplets to be eliminated ranges from 15 to 35 per cent of the total number of elements in g; 1Approximately 100,000 unblocked words can be stored on the drum and 280,000 words on a tape. If blocked, the number which can be stored on tape is increased to only 1.6 million‘words. 2Version 5-B uses MINTALLY to cut down on the number of tests. If a diad's value in MINTALLY is less than (N-l), a test is not made. AM urn r. ‘c 92 Table 3-2. Time to generate, store, and read combinations (in seconds) N Generation and Storage Reading 2 .10 .04 3 .12 .07 4 .16 .13 5 .30 .21 6 .44 .43 7 .84 .81 8 1.66 1.69 9 3.22 3.26 10 6.45 6.67 11 12.96 13.31 12 26.05 26.51 13 50.51 50.70 Wimmert's Test If quadruplets are introduced by the size of their Ciij values, the order of which differ for each problem, rule 4-1 must be discarded as impractical. Fortunately, another testing procedure is available; it is suggested by Wimmert. Quadruplets are selected for elimination with rule 1-1 and TALLY is the one of the (N2xN) x N order. XOUT is a column vector with (NZ-N) elements, there being one element correSponding to all elements in each TALLY row. XOUT is computed with rule 4-2a and rule 4-2b governs the actual test. mi. II. VI. 93 Rule 4-2a. Equate each XOUT element to N minus A minus ¢. ,A is the number of diads already assigned. (¢-1) is the number of times ti' appears in previous TALLY rows. ¢ takes on values of 1,2,...(N-1). Rule 4-2b. If a tij is equal to or greater than the value of its correSponding XOUT element, diad (i,j) is infeasible. Rule 4-2 yields the same result as rule 4-1, glyan the specific way quadruplets are introduced. Since this is not obvious, three examples are given to demonstrate this equivalency; a proof for the generalized case is then given. Example A (N=4).—-Let the problem be such that the first quadruplets eliminated are (l,2,2,3), (l,2,3,3), (l,3,2,4), (3,2,4,3) and (l,3,3,4). The relevant COST and TALLY matrices are given in Tables 3-3 and 3-4, where num- bers in parentheses depict the order of their elimination. Table 3-3. COST matrix for example A Location Pairs Center Pairs . . . 3 - 4 2 - 3 1 - 2 (3) (l) 1 - 3 (5) (2) 3 — 4 (4) W M 1The discussion of the generalize case shows that this 9 is identical to the one of rule 4—1. 94 Table 3-4. TALLY matrix for example A ations l 2 3 4 XOUT ¢ Centers 1 (l) (1) (3) (3) 3 1 2 (1) (l) (3) (3) 3 1 l (2) (5) 2 2 . . (2) GER-P? (5) 3 1 3 (4) (4) 2 2 4 (4) (4) 3 1 . W WWW“ “W Num The elimination of quadruplets (l,2,2,3), (l,2,3,3), (l,3,2,4), and (l,3,3,4) evokes rule 4-1, making diad (1,3) infeasible (the case where ¢=2). Infeasibility is also detected using rule 4-2, since t13 in the fourth row is equal to two and this is the value of the corresponding XOUT element. Example B jN=5).--Using rule 1-1, the order of quadruplet elimination for a hypothetical problem is shown in the COST matrix of Table 3—5. The corresponding TALLY matrix is given in Table 3-6. The elimination of quadruplets (l,l,2,2), (1,1,3,2), (l,l,2,3), (l,l,3,3), (l,l,2,4), and (l,l,3,4) evokes rule 4-1 when 9 equals 2, making diad (1,1) infeasible. The in- feasibility is also detected in the third row of the TALLY matr1x where tll 95 equals the value of the corresponding XOUT element. Table 3-5. COST matrix for example B ocations g i . 4-5 1-4 1-3 1-2 Centers 1-2 (10) (6) (3) (1) 1-3 (9) (5) (2) 4-5 (8) (4) 3-5 (7) N’NM WW Table 3-6. TALLY matrix for example B ocations l 2 3 4 5 XOUT ¢ enters l (l) . (3) . (6) (l) (3) (6) , (10) (10) 4 l 2 (l) . (3) . (6) (l) (3) (6) , (10) (10) 4 l l (2) . (5) . (9) (2) (5) (9) 3 2 3 2 , 5 , 9 (2) (5) (9) 4 l 4 (4). (8) (4) (8) 4 1 5 (4).(8) (4) (8) 4 1 3 (7) (7) 3 2 5 (7) (7) 3 2 W'Wv—JNWNMWN 96 Example C (N=§).-—If quadruplets (l,l,2,2), (l,l,3,2), (l,l,2,3), (l,l,4,2), (l,l,3,3), and (l,l,4,3) are eliminated, diad (1,1) is infeasible. Rule 4-1 is evoked when ¢ equals 3° Tables 3-7 and 3-8 show that rule 4-2 also detects this infeasibility. Table 3-7. COST matrix for example C Locations . . . 3-5 4-5 1—3 1-2 Centers 1-2 (10) (6) (3) (1) 1-3 (9) (5) (2) 1—4 (8) (4) 4-5 (7) KW “AM-aw 97 Table 3-8. TALLY matrix for example C ocations l 2 3 4 5 XOUT ¢ Centers 1 (1).(3) (l) (3).(10) (6) (6).(10) 4 l 2 (l).(3) (l) (3).(10) (6) (6),(10) 4 l l (2).(5) (2) (5) (9) (9) 3 2 3 (2).(5) (2) (5) (9) (9) 4 1 1 _ (4) (8) 2 3 4 (4),(8 (4) (8) 4 1 4 (7) (7) 3 2 5 (7) (7) 4 1 Generalized case.--Let the value of a tij element be (N-¢). If A is zero, this makes diad (i,j) infeasible if rule 442 is used.li It must be proved that rule 4-l would also make xij infeasible. Consider the provision of rule 4-l which stipulates that ¢(N-¢) of a diad's complements must be tallied which reference the same (N-¢) locations. Let: fik > fiu’ where i¢k and i#u. 1There is no loss of generality by equating A to zero. If A is not zero and no tallies are entered for a quadruplet if one of its derivative diads is already assigned, N can be simply redefined as the number of unassigned centers. 98 Due to the arrangement of elements in g! Cijkfi is always in a diagonal closer to the northeast corner than is cijufi' This means quadruplet (i,j,k,£) is always tallied before (i,j,u,£). Furthermore, since center pairs are associated with TALLY rows in the order of the flow between them, tallies for quadruplet (i,j,k,£) are always entered in higher TALLY rows than the rows in which quadruplet (i,j,u,£) is tallied. Therefore, for each tally entered in a tij ele- ment, there must also be at least one tally in each tij ele- ment of the ¢ previous rows. The complements of this diad must also reference the same location in each such row. This must be true regardless of whether an element has one tally or (N—¢) tallies. The reason is that a diad and its complement are tallied simultaneously; therefore, a diad with (N-¢) tallies must have at least ¢(N-¢) of its comple— ments tallied. Furthermore, these complements must refer- ence the same (N-¢) locations. The other condition of rule 4—1 is that all of these ¢(N-¢) tallied complements reference the same ¢ centers. Each pair of center labels in a diad set are associated with a different pair of TALLY rows. Since one of the center labels in the ¢(N-¢) quadruplets producing these tallies is always the same (center i), all of the complement center labels must be different. Since there are only ¢ rows of complements being considered, there must be exactly ¢ dif- ferent labels. 99 It is therefore concluded that rule 4-1 makes diad (i,j) infeasible in the same instances as rule 4-2, provid- ing Wimmert's method of quadruplet selection is followed (rule l-l). XOUT as a Parameter Rule 4—2 is not appropriate if quadruplets are selected with rule 1-3. One alternative pursued in this thesis is to combine rules 1-3 and 4-2 anyway as an approx— imate testing approach. The assumption is that rule l-l does not lead to an order of c. values that is signif— ijkfl icantly different from the order produced by rule l-3. If this assumption is true, rule 4—2 should provide a good testing criterion. A second approach is to make XOUT some arbitrary con- stant and use the N x N TALLY matrix in the following manner: Rule 4—3. Diad (i,j) is to be made infeasible when ti' is equal to or greater than (N-l). 3 Rule 4-4. _Diad Xi' is to be made infeasible when tij is‘1 equal to or greater than (N+N2/4)/2.l Infeasibility testing for versions using rule 4—3 or 4-4 takes on a new meaning. After tallying ”enough" undesir- able quadruplets, all of which have diad (i,j) as one of l(N-l) is the minimum number of tallies which can make a diad infeasible and (NZ/4+1) is the number which always will make it infeasible. The value of XOUT in rule 4-4 is therefore an average of these two numbers. 0 c~ lOO their derivatives, there is good reason to believe that the diad will result in a costly solution and does not justify further consideration. It is therefore made infeasible. Revising SOLUTION When it is decided to make a diad infeasible, its new status is recorded in the SOLUTION matrix. The manner of making this change is the same for all versions. If diad (i,j) is to be made infeasible, the value of sij is checked. If it is not zero, no change in SOLUTION is needed. If it is zero, sij is set equal to one. Furthermore, and SN+1.j si N+l are incremented by one, since the (N+1) element of a I line indicates the number of infeasible diads in it. Making Diad Assignments After revisions have been made to the SOLUTION matrix, it must be searched for conditions dictating diad assignment, that is, unassigned lines having either zero or one remaining feasible diads. The routine governing this search, handling conflicting demands, and reaching an assignment decision is the same for all versions: STEP A. Make a random choice as to whether rows or columns are~searched first. Let ROW be zero to indicate that rows are to be searched first. A value of one calls for a column search. (n lOl STEP B. Search SOLUTION for closed or Open lines.1 A search for open lines cannot take place unless unsuccessful searches for both closed rows and columns have just been completed. STEP C. If the (N+l) element of one or more lines is equal to N (or (N-l) in the case of an open search), store all unassigned center and location labels implied by each line in CONFLICT. STEP D. If the search is unsuccessful, go to STEP I. STEP E. Transfer selected CRITERIA element values into the appropriate CONFLICT elements, as determined by the labels already stored in CRITERIA. STEP F. Find among the set of elements which received new values in STEP E the ones with the minimum value. Store the center and location label of each such diad in a SELECT row. STEP G. Choose a diad from SELECT at random. STEP H. Update SOLUTION to reflect the new diad assignment by equating the appropriate elements to either 2, 3, or 1,ooo,ooo.3 Go to STEP I if less than (N-l) diads have been assigned. STEP I. Determine where to go next with the following rules: 1. If a closed search by rows was successful, go to STEP B for another closed search by rows° 2. If a closed by rows was unsuccessful: a. Go to STEP B to search for Open rows if the next to the last search was for closed columns and was unsuccessful. 1A closed line has N infeasible diads; an Open line has (N-l) infeasible diads and no assigned diads. 2The adjective “unsuccessful" means that the search did not result in a diad assignment. 3The (N+1) element of a line having an assigned diad is equated to 1,000,000. The assigned diad element is set equal to 2, while all other elements are equated to 3. 102 b. Otherwise, go to STEP B for a closed search of columns. 3. If an open search by rows was successful, go to STEP B for a closed search by rows. 4. If a search for open rows was unsuccessful, then: a. If the next to the last search was for open columns and unsuccessful, return to the por— tion of the program where the next quadru— plet is selected for elimination. b. Otherwise, return to STEP B for an open search by columns. 5-8. These rules are identical to rules 1, 2, 3, and 4, respectively, with the exception that the words "rows" and "columns" are substituted for each other. Detecting Post-Assignment Infeasibility Due to the possibility of closed lines, even infeasi- bility testing with rule 4-1 need not prevent all previously eliminated quadruplets from actually entering into the final solution. This problem would seem particularly critical for versions using rules 4-3 or 4-4. This possibility can be reduced by making another infeasibility check after a diad is assigned. This test, as described in rule 7-1, applies only to those previously tallied quadruplets which have the newly assigned diad as a derivative. Since Wimmert's orig- inal scheme did not recognize the possibility of conflict, some of our versions do not include rule 7—1, that is, they use rule 7-2. 103 Rule 7-1. _When a diad is assigned, make all of its tallied complements infeasible if this has not been done previously. Rule 7-2. ‘Continue the solution process without evoking rule 7-l. Recycling Whenever a diad is made infeasible (equated to one or three in SOLUTION), all of the tallies recorded for its complements should be erased, since all of the quadruplets that they derive from can not enter the final solution any- way. This also means that all sij elements equal to one should be reconfirmed and CRITERIA should be adjusted after a diad is made infeasible. These provisions are not included in any versions of this thesis and are not recognized by proponents of Wimmert's method. However, a recycling scheme is employed for two versions to reduce the possibility of obsolete sij and tij terms. This is done by clearing the matrices after each diad assignment. The assignment problem therefore is broken up into (N-l) separate stages. It should be noted that rule 8-1 is appropriate for versions having XOUT as an array, whereas rule 8-2 is relevant when XOUT is a parameter. Furthermore, any versions combining either rule 8-l or 8-2 with rule 2-2a will indirectly satisfy rule 7-l. 104 Rule 8-1. Reduce XOUT by one. Set all elements in SCORE, CRITERIA and TALLY equal to zero. Set all Si' values now equal to one back to zero. Revise the (N+1) line elements as required. Rule 8-2. Let XOUTLOG be a logical (NZ-N) x l array, with one element corresponding to each row in XOUT. When no diad has yet been assigned, all XOUTLOG elements are zero. When xi' is assigned, examine each pair of rows in TALLY to see if either reference center i. If one does, set the corresponding two rows of XOUTLOG equal to one. XOUTLOG is then used to reset selected XOUT elements. Other necessary changes are setting all CRITERIA and TALLY elements back to zero and adjusting sij values as specified in rule 8-1. Rule 8-3. Continue the solution process without recycling. Deducting the Last Diad Assignment As soon as (N-l) diads have been assigned, the solu- tion is fully specified. The center label of the last diad is that of the SOLUTION row having no assigned diad. The location label belongs to the SOLUTION column having no assigned diad. The same routine for deducting this last assignment is used for all versions. Rule Combinations for Each Version Thirteen unique versions of Wimmert's procedure have been programmed: 14A, l-B, 2-A, 2-B, 3-T, 3-A(L), 3-A(H), 3-B, 3-C, 4-A, 4-B, 4—C, and 5-B. Each version incorporates a different set of rules as specified in Table 3-9. The most important differences between versions are few. Version 4-A is patterned after Wimmert's concepts, except that it accommodates conflicting assignment demands .mlm no Hum.masn Honuflm mm HHmB mm mum masu moumasmflum DH mocflm manomnflpca.aln many mmflmmwumm aonHm> manan .mcfiaomomn cam .muHHHQHmmmmGH ucmficmflmmMIumom mafiuomump .mcflpmmp MuHHflQHmmmmGH .«HmmBHmU mcflumpms .mmwaamu mcfl Inmucm .mumamsnpmsv mcfluomamm no» aam>fluommmmn oumaou m 6cm .5 .¢ .m .N .H mmsouwm N N N X N N N N um um an Ml 105 m: mum one mu¢ ale onm. mum me “MM sum mum «rm mna «ra masm one A COHmHm> coamum> sumo Mom wumm masm .mlm magma 106 in SOLUTION. Version 4—A differs from l-A only in that the latter version eliminates all quadruplets in a diagonal before searching SOLUTION for possible assignments. Ver- sions l—B and 4-B also differ only in this respect. All “B" versions, as opposed to "A" versions, use rule 7—1 for post-assignment infeasibility checks. All "C" versions recycle after each diad assignment. All "B" and "C" ver- sions have provisions to assign some diads in advance. .All "1" and "4" versions use the diagonal elimination approach of Wimmert‘s, whereas "2" and "3" versions select the largest cijk ence between 3-T and 3-A(L) is that SCORE is stored on the 2 element not yet considered. The only differ- drum for the former version. Versions 3—A(L) and 3-A(H) differ only in the given value of XOUT. Figure 3-1 is a descriptive flow chart which is generalized for all Wimmert versions. The branches taken for each procedure are shown. Program listings for 3-C, 4-A, 4-B, 4—C, and S-B are found in Appendices III, IV, V, VI, and VII, respectively. Listings of all other versions are found in a supplementary volume. 107 Initialize random number generaton Input number of problems to be sol ‘ Input and FLODA:A DSTDA Output solut 101: BALI)! f Initillizo matric's. Enter tallies rules 2—l Must any of their complements be made infeasible? Update CRITERIA per rules 3—l, 3-2, 3—3, Test for infea514 bility per rules 4—l,4~2, 3, or 4—4. S ReVise _ infeasible ciads. Are all quadru~ plets in the diag— onal now tallied? KC‘UEJTL‘: e l I l l Equate ROW to zero or one randomly. Has no died just been assignej with KOUNT14 = O? Yes Was the next to the last search made for closed rows w/o resulting n an assignment? Has a diad just been assigned? Deduct last diad assi nment; compute cost; output final solution and other da Have enough solu- tions been out r this probl put em? Have all problems been solved? Figure 3-l. Descriptive flow chart for all of Wimmert's versions. (Capital letters and arabic numerals depict differing paths taken by versions.) all the 11:1 IQC H (D USE if”. dr: 108 Summary Wimmert's original concepts have been augmented to develOp thirteen unique algorithms for the layout problems. Each algorithm contains a different assortment of decision rules. The eighteen different rules pertain either to Cijkfi selection, tally entry, CRITERIA revision, infeasibility testing, or recycling. Version 3-T is designed to store the SCORE matrix on the drum (or magnetic tape), whereas all others use core storage. Generally, the versions use almost all of the core storage space in the CDC 3600. Even then, the maximum value of N ranges from only 27 to 52. This limitation can be relaxed in several ways. One way is to reduce the problem size by combining several highly inter- related centers into one module. Another alternative is to use overlays. A third possibility is to pack several values in one computer word. A fourth alternative is to use the drum or magnetic tape to store the larger matrices with appropriate buffering and blocking provisions. None of these alternatives are pursued, since the versions handle problems of sufficient size to reach the objective of this thesis. CHAPTER IV ANALYSIS OF FINDINGS This chapter contains additional remarks on the five algorithms not described in Chapter III which are also to be included in this study. The main intent, however, is to analyze the computer output and judge the relative perfor- mances of all eighteen algorithms. After examining the test problems used as well as the types of variables included in the analysis, the following topics are considered: solution quality, constraint satisfaction, computer time requirements, and findings tangential to the main research objective. Additional Algorithms Analyzed Five algorithms, in addition to the thirteen ver- sions of the previous chapter, are to be compared. A com- puter program, referred to as RDM, is listed in Appendix VIII. This computer algorithm generates random solutions, thereby approximating the ALDEP algorithm of Chapter II. Centers one through N are assigned sequentially. Center k is assigned a location by randomly selecting from a list of the k unassigned location labels. The newly assigned loca- tion is then removed from the list so that only (k-l) un- are assigned locations remain. The values of fik and djfl 109 110 read in and the solution cost computed in the same way as is done for Wimmert's versions. RDM has no provision for con- strained diad assignments. Computer programs for Hillier's algorithms are derived from his original program listing.l Two versions of this algorithm, as discussed in Chapter II, are compared. The first one implements the rectangular distance criterion in computing the distance between centers. This version, referred to as H—R, is identical to Hillier's listing with two exceptions. It is modified for compatibility with the CDC 3600 computer. Secondly, provisions are added to output the time spent on a solution and the number of trials (iter- ations) that were required. The second version, referred to as H—S, calculates the straight-line distance between locations. In addition to the modifications in H-R, it also is changed to assure a cost reduction after each trial. Hillier's versions contain three options. The alternatives chosen for our purposes are as follows: 1. Always make a "last pass." 2. The order of the first "Move Desirability Table" is the maximum grid dimension (length or width) minus one. 3. The minimum allowable decrease in the objective function is zero. lHillier and Connors, pp. 65-73. lll Versions C-R and C-S of CRAFT use the rectangular and straight-line distance criteria respectively. These versions differ from the original program1 on only three counts: changes for compatibility, calculating the computer time per solution, and recording the number of iterations. CRAFT has options as to how many centers can be involved in an exchange. The option selected for this thesis is to choose the best of two or three center moves at each iter— ation. Test Problems Test problems can be secured either from field research or from data existing in the literature. Since plant visits would be necessary for field research, time constraints would drastically limit our sample size. For this reason, the twenty-six test problems used for our analysis are derived from the literature. Several of them appear to be "hypothetical" in the sense that they do not have an empirical basis. However, the implication is that the problems are similar to those actually found in industry. The authors of several problems provide only fik values. In such cases, a reasonable lattice configuration is arbitrarily defined to compute djfi' Appendix IX provides problem infor- mation relevant to the source, fik terms, constraints, and 1The program is in the SHARE Library (SDA 3391). 112 lattice configurations. For layout configurations with unequally spaced locations, djfi values are also provided. Random starting solutions used for versions H-R, H-S, C-R, and C-8 are provided in Appendix X. Desired Information on Variable§_ Solution quality (cost eXpressed as a per cent of the lower bound), constraint satisfaction, and computer time represent output information bearing directly on the thesis objective. A summary of these output variables is found in Appendix XII. Computer time is divided into two components. "Phase I" for Wimmert's versions is the time taken to input data, initialize matrices, compute XOUT and calculate the lower bound. In the case of version S—B, it also includes the generation and storage of center combinations. Phase I for versions RDM, H-R, H-S, C-R, and C-S is the time taken to input problem data and initialize matrices. Program segments relating to Phase I time never need be repeated for a problem, no matter how many solutions are generated. Phase II time which is all execution time not included in Phase I, must be repeated for each solution. Several other output variables are also of interest: solutions diversity, number of iterations, penetration and instances of conflict. Solution diversity is defined as the ability of an algorithm to produce many satisfactory solu- tions. If an algorithm produces the same suboptimal solution, 113 it is of much less value than one producing many solutions, even if the average solution cost is unchanged. "Penetra- tion" is defined as the per cent of quadruplets eliminated by Wimmert's versions before reaching a solution. In regard to the recycling versions (3-C and 4-C), information is obtained as to the average per cent eliminated for all (N—l) cycles as well as the maximum per cent eliminated in any one cycle. Whether the penetration remains relatively constant for all types of problems is of interest as well as whether the degree of penetration affects the cost of the solution. 2 3+N ), where k is the num- Penetration is equal to 4k/(N4—2N ber of quadruplets eliminated. Three types of conflict are output for Wimmert's versions, primarily to determine if the decision rules deal- ing with it adversely affect solution costs. The three types of conflict are as follows: 1. More than one line (open or closed) is encountered dictating an assignment. 2. One or more closed lines are found. 3. Conflict must be resolved with a random choice among diads with the same minimum.CRITERIA value. These several output variables can be tested on how they are related to certain independent variables. The independent variables are either the algorithm or problem- related characteristics. The most obvious problem char— acteristic is N and powers of it. Other easily computed 114 statistics, which would seem to be related to the dependent variables, are: The coefficient of variation (Vf) of fik values. The coefficient of variation (Vd) of d. values. 33 The per cent (Z) of fi values that are equal to zero. k These statistics are given for each problem in Appendix XI. Solution Quality The quality of solutions (cost expressed as a per cent of the lower bound) produced by the algorithms is analyzed in one or more of the following five ways: 1. 2. A ranking is given for the average cost to a set of problems, where there are four solutions per problem. Solution diversity is measured by ranking the algo- rithms in relation to the least-cost solutions to the problems. The difference between the average and least-cost values measures solution diversity. Another measure of it is given by the standard deviation of costs generated to the same problem. The hypothesis of equal cost means is tested for each pair of versions with a two—tailed ”t" test. This is done for both the average and least cost rankings. The level of significance is taken to be .05. This test is based on the assumptions that the standard deviations are unknown and not necessarily equal for both cost distributions. Since it is also assumed that both distributions are normally dis- tributed, the significant differences should be considered with caution. Another reason is that enumerating and testing all possible pairs in this manner overstates the differences which are signif- icant. The large number of pairs can cause some significant differences to be fictitious. 115 The hypothesis that mean values produced by any two versions are equal is also tested using analysis of variance for a one-way classification. More pre- cisely, the null hypothesis is that the separate category (version) means do not account for any of the sum of squared deviations from the overall mean. This hypothesis is rejected only if the significance probability of the F statistic is less than or equal to .05. Since this form of testing provides, in our case, results identical to those of the "t" test, the statements on significant differences in this section apply equally well to either test. The regression model is also used to measure the net effects of the decision rules used in Wimmert's ver- sions. Since our concern was more to develop a good algorithm rather than to test the effects of each rule, our experimental design is not very efficient. The best design would be a factorial experiment. Since our design (see Table 3-9) is not orthogonal and since the factors are not independent, the regression data must be viewed with caution. How- ever, the critical rules are fairly obvious and several direct comparisons are possible, that is, some versions differ only in respect to one rule. Although these five forms of comparisons are imper- fect in themselves, they all support the same conclusions; their use seems well justified. One other preliminary remark is in order. Several versions were not applied to every test problem. The reasons are storage problems, excessive computer time requirements, or the inability of H-R, H-S, C-R, C-S to accept d. values not specified by a 32 rectangular lattice. It is therefore necessary to use several problem sets in comparing versions. Wimmert's Versions and RDM The rankings of these versions are given in Tables 4—1 and 4-2. Table 4-2 restricts the number of problems to include 3—C and 3-A(H) in the comparison. The approximate 116 SDH3 UoumHUOmmm on on one mosam> onB onu mscfle ofluouflno umoo ommuo>m onu co comma sooE on» On Hmsvo ma .EoHQOHm oEmm one How mumoo COAuSHOm .mcoflmno> mo mcflumfla .Cofluoufluo umoo ESEHCHE one ocooom onu moans cmoE osHo> mane C an sunaflnm IHHm> ogu cone Honumu mEoHQOHm soonoQ coaumflum> oflu ou moflammm UHDmemum mHSBo .ma.ha.¢H.NH.HH.m.h.©.¢.m.N.H .pcson Hoon on» mo ucoo Mom m mm poomoumxo oum mumoo .om pom Q .mm.mm.-.om.ma "mEoHQoum coounmflo wo mumflocoo pom annoum maneo H.ma m.~v m.oma Sam m.em m.¢sa sum OH q.o v.mv m.mmH ans m.a¢ ¢.mma mum m m.H m.Hv H.mea «us m.mv «.mmH sue m H.m m.am m.oqa mua o.mm m.ama mus e o.m a.om ¢.mva mum o.a¢ o.oma «Ia o o.~ ¢.mm e.HvH mus 4.4m m.o¢a arm m m.m H.mm m.omH mum 0.4m n.mva mue a m.oa 0.0m o.mma «rm o.m~ m.mvH Aqv COHUNH>OQ Gmmz COHmHm> COHumH>mQ Cmmz COHmHm> vemm Cmmz CH UUHMUCmum UUHMUGmum COHHUDU mm Qt coHHouHHO umou.EsEHcH2 QsOHHouHHU umoo omouo>¢ MH uom annonm How mosaxcmu umoo sofludaom .HI¢ oHQmB .muflmuo>flp COHDSHOm mo ouSmooE pcooom o mcH©H>onm anouonu .mEoHQOHm cooBqu cmcu Hogumn GH£UH3 mumoo COHUSHOm one on monoaon coflumfl>o© oumpcmum one .mcoflmuo> mo mcflumfla ocooom osu cuHB woumaoomoo on on one mosHm> omoneQ .mm 000 .0m.0H.0H.~H.HH.0.s.0.0.m.m.a "mamanoum campuses no mumamcoo umm smanoum macaw 117 m.0H 0.0a p.00 0.00H 20m 0.00 n.00H 20m NA 0.0 0.0 m.e0 H.00H «:0 0.00 0.00s mum as 0.H e.a 0.m0 n.0ma «Ia m.e0 n.m0a ¢r0 0H 0.0 0.0 0.0m 5.5ma mum 0.00 e.m0a Anselm 0 m.0 0.m 0.0m 0.0ma mus 0.0m , «.m0a mnH m 0.0 s.m 0.0m 0.0ma 0-0 0.m0 0.00s era n 0.0 m.HH m.0m 0.Hma Anselm 0.0m m.sma 0:0 0 0.0 m.0 H.em 0.0NH mum 0.0m 0.0ma Aqvarm m 0.HH m.HH 0.5m 0.0NH «um 0.0m n.0ma 0:0 0 0.0a 5.0H 0.0m 0.mma Anselm 0.0m “.mma mum m 0.0 m.0 n.5H 0.m~H 0.0 0.5a m.mma 010 N 0.0 H.m «.ma 0.0NH oum 0.0a 0.00s oum H mEoHQOHm mosam> COHHMH>oQ coo: soamuo> soauofl>on smoz GOHmHo> xcmm Q GHSuHB. coo: Ca Unopcmum Cumwcmum >oQ Usmum sofluosoom soauoufluo umou EDEHGHS COHHouHHU umoo ommuo>< mHH pom EoHQOHm How mmcflxcmu umoo soflusaom .va oanma 118 effects of each rule used in Wimmert's versions are shown in Table 4-3. On the basis of these three tables, several con— clusions are evident. Firstly, the only pairs of means found statistically significant always involve the RDM version. In reference to Table 4-1, versions 4—C, 3-B, and 3-A(L) are statistically superior to it in terms of average costs, whereas versions 4—C and 3—A(L) are superior to it in terms of least-cost solutions. In regard to Table 4—2, only 3—C and 4—C are superior to RDM when considering the average cost criterion; in terms of the second criterion, only 3-C is different in the statistical sense. All of these versions use rule l-3 except version 4-C, although 4-C has the additional advan- tage of recycling. A second conclusion is that RDM produces the greatest solution diversity. The best of Wimmert's versions in this respect employ rule l-3. This becomes apparent after observ- ing the "reduction in mean values" of Tables 4-1 and 4-2 as well as the standard deviations within problems shown in Table 4-2. The reason is that for the typical test problem, the random selection provision of rule 1-3 for breaking ties is evoked about 90 per cent of the time. This tends to generate a different order of quadruplets for elimination each time the test problem is solved. If rule 8-1 is used, however, the diversity is reduced substantially. Table 4-3. .Cost effects of decision rules 119 a Significance Rule Group Group Description Rule Cost Effectsb Rule Groups 1 0.0 .... l Quadruplet 2 7.4 .641 .636 selection 3 2.8 .356 Tally l 0.0 .... 2 entry 2 -6.2 .205 “205 l 0.0 .... CRITERIA 2 —8.7 .297 3 revisions 3 -3.1 .698 '658 4 -7.6 .360 . . . 2 0.0 .... 4 Infigzéggllty 3 -7.3 .6ll .328 g 4 4.3 .204 7 Post-assignment l -6.3 .205 205 infeasibility 2 0.0 .... ' l -l6.9 .007 8 Recycling 2 —l6.6 .008 .004 3 0.0 .... aThese data are based on problem set II of Table 4-2. The mean cost is 136.8. bThe values in this column are the regression coefficients when each rule is introduced as a dummy variable equal to either zero or one. The values relate to average rather than least-cost data. Rules can be compared by computing the net difference between their respective cost effects. It should be emphasized, however, that the effects of all other rules are confounded in such comparisons, due to the eXperimental design. cWhen a rule group is introduced in the least squares equa— tion, these values provide the approximate significance probabilities that the coefficients are simultaneously zero for the independent variables. It is computed using the F test that they account for none of the squared deviations from the mean cost. 120 Thirdly, the group of rules most affecting solution costs involves recycling. The significance probability of this group in explaining the deviations from the mean is .004, as shown in Table 4-3. If a version recycles, its costs are reduced significantly. Of all Wimmert's versions, 3-C and 4-C show the most promise. It can also be concluded that the better the version, the smaller the standard deviation between problems. The conclusion must be that the better versions are more consis— tent in their performance over the whole range of test problems. A fifth conclusion is that combining rule l~3 with Wimmert's method of infeasibility testing (rule 4—2) is not particularly fruitful, such as is done with the "2" versions. Another important conclusion is that the value of XOUT has an important bearing on solution quality. Table 4-3 as well as a direct comparison between 30A(L) and 3-A(H) show rule 4—3 to be definitely superior to rule 4-4. Since both values of XOUT are arbitrary, there is reason to believe that values even lower than (N-l) would produce better answers in less time for 3-A, 3-B, and 3-C. The last conclusion is that rules involving tally entry and post—assignment infeasibility are not particularly important. On the surface, it appears that rules 2-2 and 7-1 are much better than their counterparts. It should be noted, however, that both recycling versions use rule 2—2. The 121 least squares equations for Table 4-3 consider that these two versions also use 7-1, even though the rule is implicit. If the opposite assumption were made, the cost effect for rule 7-1 would be (4.1) rather than (-6.3). This suggests that effects attributed to rule groups 2 and 7 stem mainly from the confounding of the recycling rules. Hillier's Versions and CRAFT A comparison of Hillier's and CRAFT versions is found in Table 4-4. Table 4-4. _Solution cost rankings for problem set IIIa Ave. Cost Criterion Min. Cost Criterion Reduction Stand. Stand. in Mean Rank Version Mean Dev. Version Mean Dev. Values 1 C-S 123.2 14.0 C—S 119.3 10.9 3.9 2 H—S 125.5 12.1 C—R 119.4 12.4 6.6 3 C-R 126.6 14.6 H-S 120.0 10.6 5.5 4 H-R 130.4 14.7 H-R 123.3 ‘12.1 7.1 aThis problem set consists of 17 problems: 2,4,5,6,7,9,10, 12,l3,14,15,18,21,22,23,25,27. The CRAFT versions perform slightly better than Hillier's versions. However, the differences in means are not statistically significant. Since most test problems use the straight-line distance criterion, it is interesting that 122 the algorithms incorporating this criterion perform only slightly better than their counterparts computing rectangular distances. The differences are particularly small in terms of the second criterion. One must conclude that any of these versions provide comparable answers regardless of which dis- tance criterion is chosen by the analyst to describe a layout problem. This conclusion is also supported by the slight change in solution costs if a squared rather than linear distance criterion is used.1 All four versions possess the ability to output many suboptimal solutions, particularly for the larger problems where the optimal solution is more diffi- cult to attain. This diversity in solutions is apparent from the computer output as well as the reductions in mean values given in Table 4-4. Wimmert'sL_Hillier's and CRAFT Versions The last problem set, given in Table 4-5, serves to compare Hillier's versions, CRAFT and the best two Wimmert versions. The null hypothesis for each pair of versions must be accepted in regard to the average cost criterion. How— ever, the results are different for the second criterion, owing to the tendency of 4-C to produce the same solution to a problem indefinitely. The following versions are lSee problem five in Appendix XII. Steinberg's and Gilmore's versions provide solutions of equivalent cost for either criterion. 123 .mm.mm.~m.0a.ma.0.e.0.0.m ”mEoHQOHQ Cop mo mumflmcou pom annoum onem 0.0 m.o o.ma h.mma 010 0.0a m.mma 010 o 0.0 H.v H.0H 0.0NH Ulm N.©H H.Hma Ulm m m.h m.© o.m ¢.mHH mum m.ma m.mma mum v m.m N.m m.m 0.0HH mum N.NH m.ama mum m 0.0 m.m N.OH m.mHH mIU m.ma m.HNH mIU N 0.0 0.0 m.HH H.mHH mIU 0.0H H.mHH mIU H mEoHQOHm modam> cOHuoH>oQ Coos coflmuo> COHumH>oD coo: soflmuo> xcmm CHSuHB .>oQ Goo: an Onmpcmum pumocmum pumpcmum cofluozpom GOHHouHHU umoo EOEHCHZ sofluoufluo umoo omouo>¢ m>H pom EoHQOHQ MOM mmCHmeu uooo COHesHOm .mI¢ oHQmB 124 significantly different from 4-C: C-R, C—S, H—R, and H-S. Hillier's versions and CRAFT provide more diversity than 3-C, but the differences in their means are not significant. The relative worth of the eighteen versions in terms of solution quality for our test problems can now be sum- marized. CRAFT and Hillier versions are superior, with CRAFT having a slight advantage over Hillier's versions. Of Wimmert's versions, only 3-C provides sufficiently good answers to not reject the hypothesis that its mean is equal to those of H-R, H-S, C—R, and C-S. Of the other Wimmert versions, 4—C is the best. Ignoring computer time, all of Wimmert's versions are superior to RDM in regard to the typical solutions produced. Constraint Satisfaction Of the twenty-six test problems, only seven of them involve constraints (problems 8, 10, 13, 15, 16, 21, and 24). Wimmert's versions are applied to all Of these problems, whereas Hillier's versions and CRAFT are not tested with prOblems 8 and 16.1 Almost all of the constraints are imposed by making selected fi values arbitrarily large. k None of Wimmert's versions consistently satisfies constrained problems, particularly when the number of con- straints is large. The best Wimmert version in this respect lOnly problems with a lattice configuration are accepted by Hillier's versions and CRAFT. 125 is 3—C, which satisfies the constraints of three problems. Versions 2-B, 3-A(L), 3-B, and 4-C do this for only two problems. All other versions are even less successful. The failure to satisfy constraints is understandable for all versions not using rule 1-3. Since no version penetrates to the main diagonal of g! many of the arbitrarily large Cijkfi terms are never selected for elimination. For ver- sions using rule 1-3, this failure can be excused only in the case of problems 8 and 16. As is explained in Appendix IX, the distance between locations defined by the authors as "adjacent" is not the minimum djz terms. Failure with the other problems, however, suggests the need for addi— tional routines to guard against constraint violation. Although H-S and C-S produce at least one solution satisfying all problems, versions H-R and C-R fail in the case of problem 24. It must be concluded that CRAFT and Hillier's versions are superior in terms of constrained problems, but even they do not always place centers 1 and k adjacently when fi is arbitrarily large. k Computer Time Requirements Phase I Time Phase I time observations are reported in Appendix XIII. With the exception of 5-B, which combines rules 1—3 and 4-1, the time expenditure is nominal. Projecting Phase I time for higher levels of N makes it clear that version 126 5-B is computationally infeasible. For the other Wimmert versions, Phase I time can be expected to average about 115 seconds when N equals forty. This is considerable higher than the time required for RDM, H-R, H-S, C-R, and C-S. The main reason for this divergence is that Wimmert's versions must rank fik and djz values as well as compute the lower bound; other versions do not perform these functions. It must also be true that the versions programmed for this thesis are less efficient, since RDM is more time consuming than Hillier's versions. Appendix XIV shows that Phase I time is very pre- dictable when N is known. The coefficient of determination of each regression equation is very satisfactory. Since the number of fik and djfl terms is a function of N2, the rejec- tion of the hypothesis that the nonlinear coefficients explain none of the squared deviations from the mean comes as no surprise. However, Phase I time is still acceptable if a forty-center problem is considered to be one of the larger ones encountered in industry. Phase II Time Phase II time data are reported in Appendices XV and XVI. Phase II time's functional relationship with powers of N is statistically significant for each version. The non- linear coefficients are also significant, except for ver- sions 2-A, 2-B, and 3-B. For these versions, the fit of the 127 regression equations is particularly poor. The importance Of the nonlinear coefficients derives from the size of the matrices manipulated. The order of the largest matrix manipulated by all versions is at least related to N2. For versions using rule 1-3, g must also be searched. The order of this matrix is a function of N4. The time requirements are nominal for RDM, H-R, H—S, C-R, and C-S, as well as for Wimmert's versions not using rules 1-3, 8-1, or 8-2. RDM is the most efficient routine, as is to be expected. If N is forty, RDM would require less than one minute of Phase II time. Versions H—R, C-R, and C—S require slightly more time, consuming about 2.2 minutes for a forty-center problem. Versions 1-A, l-B, 4-A, 4-B, and H-S require approximately five minutes for such a prob- lem.1 All of the remaining versions (Wimmert versions using rules 1-3, 8-1, or 8-2) are much more time consuming. The adverse effect of these rules on computer time is lVersion H-S consumes more time than H-R due to the added provision assuring a cost reduction at each iteration. This provision is desirable since, as can be detected in Appendix XII, H-R rather frequently gets into an infinite loop producing the same set of solutions indefinitely. The reason for this loop is that the "move desirability table” is used to approximate the cost reduction stemming from a center exchange rather than the true cost function. Al- though the probability is small for any one iteration that a cost change deemed negative by the move desirability table is actually positive, the large number of iterations neces— sary to reach a solution makes the problem a serious one. 128 demonstrated in Table 4—6, which is derived from the regres- sion model in the same manner as Table 4—3. The version which uses rule 1-3 and also recycles (3-C) is particularly time consuming. As they are 29w programmed, the use of recycling versions (3-C and 4-C) must be restricted to smaller problems unless they provide excellent solutions not duplicated by other algorithms (which is not the case). This is made apparent in Appendix XV. Versions 3-T and 5-B are clearly out of the question. Storing and updating a matrix which is so Often used as TALLY on a sequential access medium (such as is done with 3-T) is too costly. Table 406. Phase II time effects of decision rulesa Significance Rule Group Group Description Rule Time Effects Rule Group 1 0.0 .... 1 Eiiiifiiiit 2 -0.9 .828 .243 3 4.4 .185 Tally l 0.0 .... 2 entry 2 2.4 .410 '410 l 0.0 .... 3 CRITERIA 2 0.9 .812 222 revisions 3 3.0 .442 ' 4 7.6 .081 . . . 2 0.0 .... 4 Infiastplllty 3 5.8 .081 .197 es ing 4 2.3 .615 7 Post-assignment l 2.4 .410 410 infeasibility 2 0.0 .... ' 1 14.5 <.0005 8 Recycling 2 0.5 .771 <.0005 3 0.0 .... aThe mean time is 4.17 seconds. See the footnotes to Table 4-3 for additional information on the column headings. n1. 129 The versions requiring the least amount of Phase II time (RDM, H—R, H-S, C-R, C-S, l-A, and 4-A) also have the most predictable relationship to powers of N. The coeffi— cients of determination (R2) are at least .99 for their computed regression equations. The fit of the regression equations is much less satisfactory for the other versions. In the case of B versions, R2 can be increased considerably by recognizing the number of problem constraints as another explanatory variable. As the number of constraints in— creases, the penetration decreases which in turn reduces Phase II time. Recognizing this additional variable in- creases R2 for versions l-B and 4—B to .95 and .98 respec- tively. Similarly, the value of R2 increases by .18 for versions 2—B and 3-B. However, versions based on rule 1-3 still have an additional variability in Phase II time which is unexplained by powers of N and the number of constraints.l For example, less time is required for the test problems with an N of 23 or 24 than is consumed for twenty-center problems. Intro- ducing other problem statistics as explanatory variables, particularly V improves somewhat the fit of the regression dl equations. Even then, the value of R2 is relatively low, 1The coefficient of determination is high for 3-A(H), 3—T and 5-B. However, the main reason is the smaller sample of test problems to which they were applied. 130 supporting the conclusion that at least part of the routine implementing rule 1-3 is particularly inefficient. In summary, Phase II time is nominal and predictable for RDM, H-R, H-S, C—R, C-S, 1-A, l-B, 4-A, and 4—B. Ver- sions 2—A, 2-B, 3-A(L), 3-A(H), and 3-B require considerably more time as they are now programmed and the time expendi- ture is much less predictable. Versions 3-C and 4-C do not seem to justify consideration for large problems unless they can be revised to provide better solutions in less time. Versions 3-T and 5-B are computationally infeasible for moderately sized problems. Combining_Cost and Time Considerations Since Computer time requirements vary considerably between versions, the most relevant question is how a ver- sion compares with the others in terms Of the solutions it outputs for an equal amount of computer time. One way of analyzing this total performance is provided in Figures 4-1 and 4—2. These two—way charts show the average quality produced and computer time required for problem sets II and IV. .A second way of answering the question of total performance is to make two assumptions: 1. RDM produces solutions with costs normally distrib- uted about its mean. 2. The sample statistics of Appendix XVII adequately represent the population mean and standard deviation of the RDM solution costs. 155‘ 150" 145* 140“ 135“ 130’ 125” ’ 4-A . 1-B 'l-A ’4-B 131 ‘ 2-B ' 3-A (H) 'Z-A '3-A(IJ O 3_B 120 Figure 4-1. 1.0 2.0 3 .‘0 4.0 Input (Phase II time in seconds) Two-way chart for problem set II. (Output is the average cost expressed as a per cent of the lower bound.) 132 Output 180" \ 1 140‘ 1350 o 4-C r 130‘ 125‘L ' H‘R ' H-S 120+ 115“ 110 230 430 6,6 8.0 1020 12 "57.6 Input (Phase II time in seconds) Figure 4-2. Two-way chart for problem set IV. (Output is the average cost expressed as a per cent of the lower bound.) 133 The question then can be answered probabilistically by using RDM as a yardstick to compare other versions of interest. For each problem, let t be the Phase II time of a version being considered divided by the correSponding Phase II time of RDM. This ratio eXpresses the number of random solutions which can be generated in the time required to get one solution from the version of interest. Let Pl be the probability that RDM will produce a solution with a cost less than or equal to the average cost of the version being evaluated. Then P the probability that at least one of ti the t RDM solutions will be of equal or less cost, can be calculated as: _ t Pt — 1-(1-Pl) (1) The probability statements for the test problems are given in Table 4-7 for H-R, H-S, C—S, 3-C, and 4—C. The probability that RDM will produce a solution as good as C-S in the time C-S requires for one solution is extremely small. On the other hand, the large amount of computer time consumed by 3-C makes it almost probable that RDM will produce better results in the time 3-C requires for one solution. The conclusion therefore is that C-S performs much better than 3-C when both solution costs and computer time are considered. The second line of averages in the appendix shows that the versions ranked by their total per- formance are: C-S, H-R, H-S, 4—C and 3-C. 11341 oumflumoummm on» ma .chHmuo> OoumHH osu mo HHm Ou mcoHu:HOm mcfl>mz mEoHnOHm omonu Ou >Hco wouMHou momouo>o mo 3cm mazes .mconuo> OCHquEOO new 3ou A .oHQMHHm>m one oGOHum>uomno o: oDMOHOcH muOQ .GESHOU um onu CH OOHgo. Sofia mooH on 0» UOCHMOG mH DH .OOHHHO. can» mmoH on Ca GESHOO Hm onu CH oocHNOQ mH :Hszm Nome. thNo. ¢.HN mHhe. oHHo. 0.00H mmvNo. mmHNo. h.N Ohho. ommmo. m.hH NfiHo. mmNNo. H.o nommuo>< 000s. smso. 0.- mamm. N040. n.0n 000m0. 00HN0. 0.0 0000. 0mmmo. 0.0a mono. momN0. 0.5 mmmum>< HHz HHz m.~h ... ... ... ... ... ... ... ... ... ... ... ... o~ Hommo. ommmo. H.mN ommh. mMHo. ¢.mOH HHz bmmvo. m.m omvmo. omHmo. m.m~ mflmNO. Chemo. N.m mm HHZ OQHQO. 0.00 HH2 HHZ $.0NN HH2 HH2 h.H HH2 HH2 h.m HHZ. HH2 H.¢ mm hvfio. NHNNO. 0.HN ooo.H meo. o.mm mmomo. NHNNO. m.N Ohmv. Nomo. h.mm howo. «NHQi. 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H0mo. h.¢H ... ... ... ... ... ... ... ... ... m mHHO. mMHNo. v.m mNHNo. Otho. H.mm HH2 ObHvo. H.m ommmo. OhH¢o. h.HH omH O. ONHVO. h.m N HHz HHz o.m HHz HH2 m.vH ... ... ... ... ... ... ... ... ... H N& Hm U (mm Hm u um Hm u um Hm u um H& u HoQEDZ p EoHQOum Ulv Ulm mIU mum mlm :onuo> anamsuu u 0:0 oco nouum m:0«muo> pouuoHom mo umou omouo>o onu Ou Hmsgo no cmnu mmoH mumoo oca>mc mcoHusHom Sam mo moHuHHHQMQOHE .510 oHnme 135 This analysis allows RDM to be compared with other versions only if we assume that the analyst is constrained as to computer time usage. If computer time is allowed to take on an arbitrarily large value, the probability that RDM will equal or surpass any version approaches 1.00. However, there are three reasons why C-S (and to a lesser extent H-R, and H-S) seem superior to RDM for a reasonable amount of 4 computer time. The value of Pt is usually less than .0 100 for most of the test problems. To the extent Pt is related to N, the relationship is inverse.l Finally, C-S is not I“frozen" to any one solution. It can not be concluded that RDM is inferior to 3-C and this is less likely to be valid for 4—C, owing to the latter versions' tendency to output only a few different solutions to a problem. Findings Tangential to the Research Objective There are several interesting findings which relate less directly to the research Objective. They are the topics Of the remaining four sections of this chapter. Incidents of Conflict It is interesting to know the number of times ”con- flict" occurs in the SOLUTION matrix of Wimmert's versions lFor H-R, H-S, C-S, and 4-C, the simple correlations between P and N are negative, but not statistically signif— icant. The correlation for 3-C is positive, but not signif- icant. 136 and the effect this has on solution costs. Data on three types of conflict were output. The first type occurs when more than one line (open or closed) is found which dictates an assignment. The second type occurs when one or more closed lines are found. The third type of conflict results when more than one of the diads qualifying for assignment have the same minimum value in CRITERIA, thereby necessi- tating a random choice. These three types of conflict are reported as percentages in Table 4-8. Since only (N-l) decisions determine N assignments, the percentages are set equal to the number of occurrences divided by (N-1)/100. The averages apply to the same mix of test problems. Table 4-8. Occurrences of conflict as percentages TYPgoggliCt Type 1 Type 2 Type 3 Stand. Stand. Stand. Version Mean Dev. Mean Dev. Mean Dev. 1-A 13.7 16.8 25.2 18.8 13.6 20.1 1-B 40.6 30.2 59.7 23.1 10.9 12.2 2-A 12.2 11.4 30.9 20.7 15.1 17.3 2—B 8.7 10.2 20.0 17.8 10.6 13.0 3-A(L) 4.1 7.7 19.2 16.5 15.9 15.5 3-B 3.2 9.4 9.0 13.7 4.5 6.8 3-C 0.5 2.1 1.8 4.0 1.4 3.4 4-A 5.1 7.4 8.4 9.8 3.2 7.5 4-B 8.7 13.7 13.8 16.0 3.8 9.5 4-C 0.0 0.0 0.7 2.8 0.0 0.0 137 Incidents of conflict are surprisingly numerous for some versions. For example, over one-half of the decisions made with l-B involve closed lines and over 40 per cent involved more than one line (closed or open). However con- flict does not appear to affect solution costs. It is true that versions 3-B, 3-C, and 4-C produce the best solution and also encounter the least number of conflicting diad assignments. However, this does not mean that they are better because conflicting demands are fewer. The question of interest is whether a version outputs better solutions as an accompaniment to reductions in conflict. The answer is Obtained by examining the simple correlations between aver- age solution costs and the three types of conflict. With the exception of l-B, none of these correlations are statis- tically significant.l Some of them are even negative. Con- flict and the routines available to handle conflict do not seem to change solution costs in either direction. Even for l-B, the fit of a regression equation having average cost as the dependent variable improves only slightly with the addi- tion of conflict types as explanatory variables. 1The significance test used in this chapter for a correlation coefficient is equivalent to testing whether the regression coefficient of the independent variable is equal to zero. _A "t" test is used with the level of significance set at .05. 138 Penetration and Iterations Data on the penetration Of Wimmert's versions, defined as the per cent of quadruplets tallied before reach- ing a final solution, are supplied in Table 4-9. Table 4—9. Penetration of Wimmert's versions eXpressed as percentages Version Average Penetration Standard Deviation (%) 1-A 39.6 5.2 l-B 30.9 7.3 2-A 29.6 8.4 2-B 27.8 9.7 3-A(L) 22.8 10.5 3-A(H) 31.3 7.6 3-B 22.2 8.8 3-Ca 17.8 6.1 4—A 39.1 4.6 4-B 30.3 6.8 4-cb 28.5 4.2 aThe reported statistics are for the cycle (out of (N-l) cycles) which had the maximum penetration. The average cycle penetration and standard deviation are 12.3 and 4.7 reSpectively. b . . . . These statistics also apply to max1mum penetration. The average penetration and standard deviation are 17.7 and 4.3 respectively. The relatively small standard deviations suggest that a versions' penetration varies little between the test problems. The penetration does tend to be correlated with 139 N and V but the reason is that penetration is most f. affected by the number of constraints, which in turn are more characteristic of larger test problems. It is interesting that,of all the versions, penetra- tion is significantly related to solution costs only for 3-C and 4—C. The correlation is negative. Penetration's rela- tionship to computer time is described in a previous section. The number of iterations preceding convergence to a final solution for CRAFT and Hillier's versions is provided in Table 4-10. Table 4-10. Iterations before convergence for H-R, H—S, C-R, and C—S Version Mean Standard Deviation H-R 22.3 20.8 H-S 19.6 17.6 C-R 7.2 7.3 C-S 7.5 7.2 The number of iterations possesses a very signif- icant positive functional relationship with N. The simple correlations with N are approximately .96 for all four ver— sions. For this reason, variables related to the number of 1In regard to V , the simple correlation is negative and significant for all versions except 1-A and 4-B. A sta- tistically significant negative correlation also exists for 1-B, 3—C, 4—A, 4-B, and 4-C. 140 iterations, such as solution costs and computer time, can also be predicted with knowledge of N. Relating Problem Statistics to Solution Costs The simple correlation between average solution costs and either N, Vf, or Vd is statistically significant and positive for every version.1 The simple correlations range from .53 to .85, with .70 being the average. For Wimmert's versions, the value of R2 averages about .75 for regression equations having average costs as the dependent variable and the problem statistics as eXplanatory variables. However, the hypothesis we are most interested in is whether an algorithm's relative performance depends on the type of problem. This hypothesis receives little support, at least f’ Vd’ and Z. The better algorithms provide consistently good solutions when the "type of problem" is measured by N, V for the whole spectrum of test problems. Average solution costs are related to the problem statistics in the same way for all versions. Recognizing a Satisfactory Solution The findings Of this thesis show that it would be unwise to base a final layout decision on only one solution generated by an algorithm. Even CRAFT or Hillier's versions 1A significant simple correlation between average solution cost and Vf exists only for unconstrained problems. When several fik values are made arbitrarily large, the cor- relation is not significant. 141 often provide both good and poor solutions to the same prob- lem, depending on the starting solutions. For example, pairs of centers having arbitrarily large flows between them were not located adjacently in several instances. Generat- ing several solutions to a problem would therefore be pru- dent. This reduces the danger of accepting a costly assign- ment. This multi-solution approach introduces the question Of knowing when to stop the solution process. Of course, whether a solution is "good" is a question having a defini- tive answer only when the optimal solution is known. Fortu- nately, the data generated for this thesis suggest several ways of recognizing a good solution without knowing the optimal one. A workable "stopping rule" can be based on at least three types of information. The first source of information is the solution cost, expressed as a percentage Of the lower bound. Appendix XVIII, which provides the least-cost solu- tions for each problem, shows that the cost percentages of "good" solutions1 do not differ widely, particularly when problems of equivalent size are compared. The cost percent— ages are significantly and positively correlated with the 1We assume that the least-cost solutions are reason— ably close to Optimality. This is not proved except for the five problems having a known Optimal solution. In these cases, the least-cost solution in the appendix is also Optimal. 142 1 . . . . . d' The coeffiCient of determination is .74 for a regression equation having N and V problem size and V d as eXplanatory variables. The fit is not as good when constrained problems are included in the analysis, perhaps because the lower bound was computed without recognizing constraints. At any rate, our preliminary findings suggest a cost which is 150 per cent of the lower bound is suspect, even if N is as large as forty. Similarly if it is only 105 per cent, additional computer time will probably be of little value. A second indicator of a solutionsi acceptability is its comparison to the average RDM solution. RDM acts as a type of upper bound; it gives every assignment, good or poor, an equal chance of selection. Cost considerations play no part in the solution process. The difference between the cost of a solution and the average cost of random solutions, when each is expressed as a per cent Of the lower bound, is defined as the "random mean increment" in Appendix XVIII. The data in this appendix suggest that, for unconstrained problems, a solution is not attractive if its random mean increment is less than 20 per cent. The random increments of the appendix are significantly correlated with both N and Vd' _As the value of these problem statistics increase, the 1It is not known whether this correlation exists because the difference between the Optimal solution and the lower bound widens as N increases or because the quality of our best solutions is less at higher values of N. 143 random mean increments of good solutions also tend to in- crease. Since the increments of a good solution are reason- ably predictable and since they can be easily derived from the starting solutions of H-R, H-S, C-R, and C—S, the random mean increments could be of value in constructing a stopping rule. A stopping rule could also involve an analysis of the solutions during the solution process itself. The num- ber of solutions so far generated, the average cost, and the standard deviation could be particularly enlightening. For example, if CRAFT uses random starting solutions and the best solution so far generated has a cost more than a spec- ified proportion of the standard deviation below the mean, the solution process could be terminated. A similar stop- ping rule could determine after specified time intervals the difference in cost between the best known solutions after interval k and interval k—l. If this difference is less than a specified value, the solution process could be terminated. CHAPTER V SUMMARY AND FUTURE RESEARCH The first part Of this chapter is devoted to a brief synopsis of this study, including its overall conclusions. The second portion contains a description of the future research needs which seem most important. Summary and Conclusions The problem of assigning centers to locations to minimize a specified cost function has been the subject of several suboptimal algorithms. Optimization techniques are not now possible due to excessive memory requirements and computational time, as is true with several other combina— torial problems. Plant layout is often termed the "quadratic assignment problem,“ which is usually taken to refer only to material handling costs. Recognizing only material handling costs is an unnecessarily restricted view of the problem, particularly when one recalls the many objectives tradition— ally cited as being affected by center assignments. .Although the precise relationship between several of these objectives and center assignments apparently has never been clearly ascertained, the costs accompanying the more predictable 144 145 relationships seem to fall into at least one of three cate- gories: linear, special quadratic, or general quadratic costs. Even though computationally feasible algorithms are addressed only to Special quadratic costs, the other two cost components can be taken into account with prohibited and required assignment constraints as well as modifications to the appropriate fik values. These represent rather triv- ial changes to existing algorithms. An algorithm which solves the special quadratic cost problem is a valuable tool for the layout analyst. In light Of this conclusion, some of the most press- ing research questions seem to be: (1) which of the recent alternative algorithms best solves the quadratic cost func- tion and (2) whether the best one provides consistently good answers regardless of the type of problem. An examination of the total array of decision models indicates four algo- rithms to be of particular interest: random selection, Hillier's version, CRAFT, and Wimmert's tally system. These models are tested in this study for one Of two reasons. Pre- liminary reports suggest two of them (CRAFT and Hillier's version) have considerable promise. In regard to random selection and Wimmert's system, comparative studies are not available. However, both possess a certain logic which is attractive and there is no a priori reason why they should not perform well. 146 Each of these four algorithms are translated into computer programs compatible with the CDC 3600 computer. In the cases of Wimmert's system and random selection, pro— grams are developed Specifically for this thesis. Since several concepts of unknown merit are added to Wimmert's original formulation, thirteen variations on his theme are tested. The test results indicate the better algorithms are consistently good, regardless of the problem type. CRAFT is superior to any Of the algorithms in terms of solution feasi- bility, solution cost, computer time, and the ability to produce many good solutions to the same problem. Hillier's version is competitive with CRAFT; the differences between their performances is not significant. In comparison, the random selection algorithm seems inferior in terms of its total performance. This last conclusion must be qualified with the assumption that a layout analyst is constrained by the amount of computer time he can economically justify. Although the amount of justifiable time is situational, this study's findings indicate it is very improbable that a ran- dom search will provide better answers than CRAFT in a typical industrial setting. The results for Wimmert's versions are not partic— ularly encouraging. The incongruity between this study's findings and the Optimistic statements by proponents of the algorithm can be explained. All of Wimmert's versions, 147 including the one most resembling Wimmert's manual tally system (4-A), perform very well for small problems. Since Wimmert's system was manual, it was tested only for problems having N equal to four or five. Unfortunately, as the size increases to the point where manual solution is out Of the question, a marked decline in performance occurs with most of Wimmert's versions. This could only be detected with a computer algorithm. This is not to say that Wimmert's framework is without merit. Versions 3-C and 4—C return very satisfactory answers. In terms of average solution costs, the difference between them and CRAFT is not statis- tically significant. Furthermore, Wimmert's versions are amenable to layout configurations not in the shape Of a lattice, whereas CRAFT and Hillier's version are not able to do this. The comparison of CRAFT with Hillier's algorithm has recently been made in two other studies. Hillier and Connors, on the basis of one test problem, found the Hillier algorithm superior to CRAFT in terms of solution quality, although it required considerably more computer time.1 On the other hand, Nugent, Vollman and Ruml found, on the basis of eight problems, that "CRAFT seems to produce solutions of somewhat higher quality . . . , but the experimental results lHillier and Connors, "Quadratic Assignment Problem Algorithms and the Location of Indivisible Facilities,” Technical Report NO. 6, p. 26. 148 have not firmly established that fact."1 For some reason, the authors found Hillier's algorithm to be much less time- consuming than CRAFT. In our study, we found CRAFT to be slightly better in respect to both solution quality and computational time. The findings of this study Offer several insights tangential to the main research objective. One important conclusion is that even the best algorithms can generate intermittently poor solutions to the same problem. A second finding is that, except for the Wimmert versions using rule 1-3 for quadruplet selection, computer time is very predict- able. It bears a strong functional relationship with powers of N. The nonlinear coefficients of a least-squares poly- nomial equation are statistically significant, indicating a limit to problem sizes for even the most efficient routines. However, this limit is well beyond a problem with forty centers, which is by no means a small problem. A third insight indicates that for version 3-C, solu— tion costs are inversely related to the size Of the XOUT parameter as well as to the per cent of penetration. Pene- tration, in turn, is directly related to XOUT. Another find- ing relating to all of Wimmert's versions is that incidents of conflict do not adversely affect solution costs, even though they are surprisingly numerous for some versions. lNugent, Vollman, and Ruml, Operations Research, p. 164. 149 It has also been found that the per cent of pene- tration is reasonably constant over a whole range of test problems for each Wimmert version. In general, "C" versions require less penetration than their "B" counterparts; the "A" versions require the most penetration. In respect to CRAFT and Hillier's versions, the number of iterations gen- erated before convergence to a final solution is definitely related to N. Two other tangential findings are interesting. First of all, the choice of a distance criterion is rela- tively immaterial, be it straight-line or rectangular. Secondly, preliminary evidence suggests a very satisfactory stopping rule can be constructed from information on the lower bound,1 from data on the upper bound (mean cost of random solutions), and by monitoring solution costs during the solution process itself. Future Research Needs The findings Of this study point to several areas which are particularly in need of additional research: revising existing algorithms, unequal area requirements, and new algorithms. lNugent, Vollman, and Ruml also found a significant correlation between N and the best solution cost expressed as a per cent of the lower bound. Operations Research, p. 164. 150 Revising Existing Algorithms Of Wimmert's versions, 3-C most justifies additional research. .Several modifications to it are imperative. The first change is adding a routine to satisfy prohibited and required assignment constraints. This change involves the input Of the SOLUTION matrix to reflect all assignment constraints. Later, if a closed line is encountered, all locations or centers in it which violate the original con- straints are disqualified. Lastly, when a diad xij is assigned, center i is checked for arbitrarily large flows with other unassigned centers. The diads placing such cen- ters at locations not adjacent to location j can be disqual- ified with appropriate entries in SOLUTION. This routine should reduce significantly the chance of a final solution violating the constraints. A second mandatory revision is to find ways of reduc- ing Phase II time requirements. One way is to write a more efficient routine for nominating quadruplets for elimination. Exploiting the unique properties of the ranked matrix 2 and breaking ties arbitrarily rather than randomly should net a substantial time reduction. Another avenue leading to time savings is to store the ordered list of nominated quadru- plets on a sequential storage medium (either on drum or magnetic tape) rather than generating a new list for each cycle. In this way,.g must be searched only once, no matter how many solutions are desired, rather than (N-l) times for 151 each solution. The disastrous results using the drum.with 3-T and 5—B will not occur, since the records will always be read sequentially. .A third revision to 3-C is to eXperiment with vary- ing levels of XOUT. There is good reason to believe that the algorithm's performance will be improved (both in terms of solution cost and computer time) if XOUT values are made less than (N-l). There is another reason for experimenting with XOUT levels. If the second revision already cited is implemented, the main source of solution variability in 3-C is lost. .An average of 90 per cent of the quadruplet selec- tion decisions involves ties and therefore random choices. The result is a different quadruplet ordering for each cycle and a variety of final assignments. Fortunately, it seems likely that initializing XOUT at different values will have the same beneficial effect. A final revision in 3-C which may have value is to use a different rule to resolve conflicting assignment demands. Picking the diad with the smallest lower bound is one alternative. It certainly would reduce storage and computational requirements. Due to the relatively few incidents of conflict for 3-C and the fact that existing rules seem to perform well, this revision has a lower pri- ority than the previous ones. Several revisions to CRAFT and Hillier's versions are suggested by this study. The first change is to make 152 them amenable to problem constraints. CRAFT accepts only required assignments, whereas Hillier's versions accept neither required nor prohibited assignment constraints. These changes are not only easy to make, but both algo- rithms can be adapted to explicitly recognize linear costs. A second change, also rather trivial, applies only to Hillier's version. As was demonstrated by H—R, approximat— igg_cost reductions can cause indefinite looping. This should not be allowed. A third change is to add to both algorithms a random selection routine which is modified to satisfy constraints. In this way, starting solutions could be generated internally. The final change suggested by this study is to add a stopping rule. Our findings suggest that a satisfactory rule can be developed. Unequal Area Requirements As is discussed in Chapter I, there are two distinct alternatives for accommodating unequal area requirements. The first alternative, which is embodied in CRAFT and CORE- LAP, is to assign unit location blocks according to a few specified rules. This alternative has several disadvantages. The shape of a center is constructed without considering its effect on the Objective function. Secondly, in the case of CRAFT, the number of exchanges per iteration is limited to centers of equal area or centers sharing a common border. Finally, unreasonable and unconventional shapes are a dis- tinct possibility. It can be said that what constitutes a 153 reasonable shape is situational and this is a decision best left to the analyst when he enters the detailed planning stage. The other alternative accommodating unequal area requirements also has a disadvantage, as it usually involves an increase in N. Even if area requirements are levelled only approximately, increases in N and the concomitant computer time may make the first alternative more desirable. Which of the alternatives is best provides an excellent topic for future research. New Algorithms Although CRAFT possesses demonstrated effectiveness, it certainly does not rule out the possibility that new algorithms can surpass it in total performance. Three algorithms would seem, a priori, to provide solid bases for additional research. One possible algorithm is analogous to Tonge's heuristic program for the assembly line balancing problem.1 The key idea of his approach is to simplify the combinatorial problem until a problem is obtained which can be solved through direct means. Detail is reintroduced later in the solution process. In terms of the layout prob— lem, an algorithm could group highly related centers together 1Fred M. Tonge, A Heuristic Program for Assembly Line Balancing_(Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1961). 154 into a "module" and locate the modules with one of the optimization techniques. The centers within each module could then be arranged in a similar fashion. This signif— icantly reduces the problem size and makes optimization techniques feasible. A second approach has also been applied to the line- balancing problem.l Decisions are made by selecting ran- domly from a group of competing rules, by "learning" which rules provide the best answers, and then by increasing the probability that they will be chosen for future decisions. A third algorithm on which no information has been reported, would be to assigned centers by chance, but in- crease the probability that certain selections are made with the use of a few decision rules. This synthesizes little computational effort with logically derived decisions. Answers to these research questions would enhance the theory as well as the practice of plant layout. Hope— fully, they will help close the apparent gap between theo— retical formulation and actual industrial application. 1Fred M. Tonge, "Assembly Line Balancing Using Probabilistic Combinations of Heruistics,‘I Management Science, XI, NO. 7 (May, 1965), 727-735. ASSIGNC ASSIGNL BOUND — APPENDIX I GLOSSARY OF TERMS FOR WIMMERT VERSIONS - The center label of a diad just selected for assignment. - The location label of a diad just selected for assignment. The lower bound of a problem. If f- refers to the ranked values of FLODATA, d- refers to the ranked values in DSTDATA, k equals (NZ-N)/2, and j equals (k-i+l), the lower bound is calculated as: k 2 f. d. i=1lJ All arbitrarily large (9999) and arbitrarily small (-9999) values are assumed to be zero in this calculation. CELLTIE(300,2) - This matrix stores the indices (for the FLODATA and DSTDATA arrays) of the quadruplets nominated for elimination. If more than one pair Of indices are stored in it, a selection is made randomly. Complement - Considering the quadruplet (i,j,k,£), diads (i,j) and (k,£) are complements of each other. Taken together, they are a "diad set." CONFLICT(M1+1,M1+1) — This matrix is used to resolve con- COST(or flicting diad assignment demands. Up to M1 such diads can be handled at one time. The first row and first column store location and center labels. The elements of conflicting diads are assigned values from.CRITERIA. Q) - This ranked matrix is of the (NZ-N)/2 x (NZ-N)/2 Order. Its elements are the special quadratic cost terms. The matrix is never stored in the computer, as the algorithms work directly with FLODATA and DSTDATA arrays. 155 156 CRITERIA(N,N) - An element of this matrix stores the number or cost of all eliminated quadruplets having the diad implied by the element as a derivative. Derivative - Derivatives of the quadruplet (i,j,k,£) are diads (i,j). (k.£). (1.3) and (k,j)- Diad(i,j) - This term refers to the assignment of center i to location j. Diad Set - See "complement." DSTDATA((N2—N)/2) — The array containing ranked d- values. There is no value greater than that of DSTDATA(1) and none less than that of DSTDATA((N2—N)/2). DSTL1((N2-N)/2) - The array storing the smaller location label for the corresponding element in DSTDATA. DSTL2((N2-N)/2) — The array storing the larger location label for the corresponding element in DSTDATA. FINAL(N+1,N+1) - This matrix shares storage space with SOLUTION. After (N-l) diads are assigned, the last column is used to store the permutation Of locations specified by the final solution. FLOWCI((N2-N)/2) - The array storing the smaller center label for its FLODATA counterpart. FLOWC2((N2—N)/2) - The array storing the larger center label for its FLODATA counterpart. FLODATA((N2-N)/2) - The array containing fik values ranked in the same fashion as DSTDATA elements. I,J,K - Multipurpose indices. Infeasibility - This term applies to a diad which has been disqualified from entering into the final solution. Diad (i,j) can be infeasible due to any of four reasons: (1) center i has already been assigned to another location, (2) location j has already been assigned, (3) quadruplet (i,j,k.E) has been elimi- nated and complement (k,£) is already assigned, and (4) a sufficient number of quadruplets having (i,j) as a derivative have been eliminated. KOUNT - This variable name takes on different meanings, depending on the number appended to it. Some of the most important KOUNT variables for understanding the computer programs are: 157 KOUNTl: The name number of the problem being solved. It is used later by the programs as a general purpose index. KOUNT2: The number Of solutions requested to be gener- ated for a problem. KOUNT6: The name number of the solution being generated to a problem. KOUNT7, KOUNT9, KOUNT10: If rules 1—1 or 1—2 of Chapter III are used, KOUNT7 stores the number of elements to be eliminated in a g'diagonal. KOUNT9 and KOUNTlO store the indices of the diagonal element being con- sidered. If rule 1-3 is used, KOUNT9 and KOUNT10 store the indices of a quadruplet selected for elimination. KOUNT8: This variable stores the number of diads deemed infeasible by the tally system since the last time SOLUTION was searched. KOUNT14: A variable equal to zero if SOLUTION is being searched for closed lines. It is equal to one if the search is for open lines. Unassigned lines hav- ing one or none nonzero elements are referred to as "Open" and "closed" respectively. KOUNT15: This variable takes on the value of one if a diad assignment has just resulted from a row search of SOLUTION. Otherwise, it is zero. .KOUNT16: A variable equal to one if a diad assignment has just resulted from a column search of SOLUTION. Otherwise, it is equal to zero. KOUNT19: The number Of diads assigned so far is stored in this memory location. KOUNT20: The variable detects whether two consecutive searches of SOLUTION (one by rows and the other by columns) have failed to produce a diad assignment. KOUNT27: The number of quadruplets currently stored in CELLTIE for future elimination. .M - A parameter Specifying the number Of rows of SELECT which can be used. MINTALLY(or M) - This N x N matrix is used for infeasibility testIng in the case of version 5-B. Element mi- gives the number of times (i,j) has been a derivative 158 of tallied quadruplets. An infeasibility test is not continued for (i,j) if the value of mij is less than (N-l). Ml - A parameter Specifying the portion of CONFLICT to be used. N — The number of centers (and locations) for the problem being solved. NUMBER - The number Of problems to be solved for the current run. PHI - If rules 1—1 or 1-2 are used, this variable stores the number of previous rows in TALLY which repeat the center label of the TALLY row being considered. It is used to calculate XOUT. For 5—B, PHI is given meaning by rule 4-1 of Chapter III. Quadruplet (i,j,k.£) - This term refers to the four labels (i,j,k,£) corresponding to an element in 9, Labels 1 and k refer to centers, whereas j and Z-refer to locations. Quadruplet Elimination - When we Speak of eliminating qua— druplet (i,j,k.£) from the final solution, this means that all of the following conditional assign— ments are to be disallowed: Xij = 1 if XkE = 1 Xifi = 1 if xkj = l xkz = 1 if xij = 1 xkj = 1 if Xifi = 1 However, eliminating (i,j,k,£) need not rule out any of the following conditional assignments: xij = 1 if Xkfi = 0 xil = 1 if xkj = 0 Xkfi = 1 if xij = 0 xkj = 1 if X12 = 0 ROW - If this variable is equal to zero, the rows in SOLUTION are searched prior to columns for conditions requir- ing a diad assignment. If it is equal to one, the columns are searched first. Each time a new search Of SOLUTION is initiated, the value of ROW is deter- mined randomly. 159 SELECT(M,2) - This matrix stores for a random selection the diads having the lowest values in CONFLICT. The first column stores the appropriate index to the FLODATA array and the second column stores the index needed to enter the DSTDATA array. SCORE((N2-N)/2, (NZ-N)/2) - This logical array is used to record which quadruplets have already been elimi- nated. An element equal to one means the corre- Sponding quadruplet has been tallied. Otherwise, it is equal to zero. SCORE is used only for ver- sions incorporating rule 1-3. SOLUTION(Or S) - This (N+1)x(N+1) matrix provides the cur— rent—Status of each diad in terms of whether it is assigned or infeasible. The following scheme is used to give sij a numeric value. At the start of the solution process, initialize §_at zero. After a sufficient number of quadrupletS has been elimi- nated to make a diad infeasible, the corresponding §_element is equated to one. If either of the two Iines passing through this element now have (N-l) nonzero elements, the zero element is to be equated to two. This means the "open" diad has been assigned. All other (ZN-2) elements in the lines passing through the assigned diad are equated to three to Show that they can no longer be entered into the final solution. Therefore, the numbers zero, one, two, and three are used to reflect the current status of each diad. For convenience, (N+1) column stores the total number of nonzero elements in each row and the (N+1) row stores the number of nonzero elements in each column. TALLY(or 2) - A quadruplet is eliminated by incrementing up to four elements in this matrix. The incremented elements represent the derivatives of the quadruplet eliminated. The T matrix is then used in conjunc- tion with XOUT to-prevent the tallied quadruplets from entering the final solution. There are three possible sizes of T, depending on the version used. One T matrix is Of-the N x N order, where rows refefence centers and columns represent locations. Only one ti' element is allotted to a diad. The second matrix is of the (NZ—N) x N order. Each column again references one location. However, a pair Of rows corresponds to a pair of centers. The center pairs are ordered according to the Size of their fik values, as given in FLODATA. There are (N-l) tij elements for each diad in this matrix. 160 The third matrix has N2 rows and N2 columns. Each row corresponds to a pair of centers and each column corresponds to a pair of locations. There are (N2) tij elements for each diad. XOUT - If the N x N TALLY matrix is used, XOUT is a param- eter. For our purposes, it is arbitrarily set at either (N-l) or (N+N2/4)/2, depending on whether rule 4-3 or 4-4 is used. Diad (i,j) is made infea- sible when ti' is egual to or greater than the value of XOUT. If the (N -N)xN matrix is applicable, XOUT is a one-dimensional array. Each element corresponds to a row in TALLY. XOUT is computed on the basis of rule 4-2a and implemented with rule 4-2b. If the N2xN2 matrix is used, rule 4-1 is evoked and XOUT is not relevant. APPENDIX II LISTING OF COMBINATION GENERATOR PROGRA” Leona DIMENSION COMBIN(20) INTEGE4 PHI.PIVOT.c0H8!N RE‘DioVUqBER 1 FORMAT<12) REHIND 20 ' ’ ‘ " ” “” ”“”" TOLD:T1MEF(4) o“ - .. _- DOZIHEBP‘laNUMBER READSIV ‘0UNT19 PRINT 149 N, KOUN719 9"” "1‘ 'M ”‘w"”‘”"‘_m”"“”"”'"‘”""”" 149 FORMAT(1H .4 IN THIS PROBLEM N _15 v 124* nnp Kpu~119 !§ _3-123_ 'KO'N-l'KDuN119 ‘“ 3 FORMAT(12,!2) DOSOiUPHI%1.Kb s FIVUTsPRI 0030111=1.91VOT 3011 COMBIN(ISSI“ " ”""" “ “’ 7 "' ” ”“"”””’"’m’"”""'“”"” GO T0‘3012 ‘ 3013 COMBINIPHI)sCOHBIN(PHI)41 301? CONTINJE ““‘WTTE‘T‘AW C‘WI’WJTTJEE PHI Y IF(COM51N(PH!) .Lr. N)Go 10 3013 5 K-O IF(PHI .LE. PIVOT>GO To 3014 3n15 K= K+1 5 IF(COMBIN(PHI' K) .66. N'K)GO TO 3016 5 KPBPHI-K " INDEx180 S JTEHP:COHBIN(KP) 0030171=KP. PHI 3 [NDEX181NUEXI*1 ““‘3017 COMBINfrJaJTEMP+TNDEX1 GO To 3012 . ’3016"CONTINUE“E‘IF(PHI-K .GYz'PIVUTIGO’To’3015“ ‘“’ "”“”'"”M” “W‘ 3014 CONTINJE S IF(COHBIN(P!vor) 9LT. N-PHIQP1807360 To 3918 '"‘ PIVoTaPIVoT- -1 S IFprVoT .LE. U560 10‘3010 “' “ ""““‘”"““"’ 3018 INDEx1=0 S JTEHPaCOMBIN(Plv0T) 1‘1 301° COMBIN(I):JTEMP¢INDEX1 . GO To 3012 1 iii ..““1 _.._.__ .11_il_-r-_-,wiim-_ 3010 CoNTINJE REHIND 20 TNEH:T1MEF<4) S T: (TNEH TOLD)/1000. S PRINT 1472? '_—_I—I7—TUWNTTT"W—T?*TTEE—SPENT“UENERITTNUHCUHUTNITTUNSIWTl—_TTFTTTT7fiflflflflT-”" 1DS:*) TOLD-TVEH 003021PHI=1,K0 s IOTALisi s TOTAL201 0030221=1aPHI S TOTAL1ITOTAL131'5 Y'N'l*1 3n2? TOTAL2'TOTAL20Y NCUHBI°=TUTILZITUIAL1 003023181,NCOMBIN ' 3023 READ TAPE 20:(COHBINIJY.J=1.PHI) 3021 CONTINJE . H 1 'HM’ REVIND 20 "“' I ‘4’" ' _" "”""" ' ‘ ‘ "'” ' TNEH-TIMEF(4) S Y: (TNEH- -TOLD)/1000. S PRINT 140:? 148 FORMATT1H .0 TIME SPENT—RETDTNU‘TREH‘WIS“TTFIFT3TTIEUUNDS.t) TOLD-TVEN "2’ CONTINUE ’ ---,---1"--. " " I ---_ " ..-“-.- ' -.. ' " 11“-...” "mm." b ' END 1C1 APPENDIX III LISTING OF 3-C --—— . fl -—_.. 2-.., ,___.- . PROGRAW PRocsc DIMENSION FLODATA(595I.FL0N01(595I.FLORczts95IIOthNEICBIaORIGN02( 18IIORIGND(8IITALLV(35.35IIUSIDATA(595IIDSIL1Ib95).DSIL2CS95IoSOLUT 20N(36.36I.CRITERAI35535IIC0NFLCT(36536IISELECT(SSIQIIEINILT35.36I.’ 3UNRANKU(595.3)ICELLTIE(300!2)ISCORE1(500595)'SCORE2(501595)OSCOHE3 4(50 595). scuR54(su.¢3oI'SCURE5(50.S9SI SCONEEISOIS9SI;SCORE7I50’59 '55)ISC0dE8(500595), SCORE9(50,595I 5C0RE19(59.595I SCORE11(59.595I S 6CORE12‘50 595IIIFIXC(33I.IFIXC(33I EQUIVA ENCE (FINAL SOLUTONI INTEGE'I ASSIGNL.ASSIGNC, ORIGN01.0RIGN02 LoGICA. Sr0RE1.F‘CuRE2.Sc0Rt3oSCQREA SCORE5. SCUREbo SCORE7oSCORE8 SC 1ORE9. S OQEIUOSCORhillSCORElz TOLD=T1MEF(4) XsTIMEE(5) CALL RANFsevcxI RANDN0=RANF(-1I READ 600.NUNB&R 600 FORMAT/2 $ 1:1 3 J=L $ KUNT= o s BOUanfi. ”””“"”" 990 CONTINJE 9 IFcFLOOATAct) .bo. 99999. .OR. FLODATA(1) .50. «9999. . 1AND. 0510ATA(J>‘.t91”99999.‘.0H.'DSTDATABU To 990 PQINT OUT MATRIX SthS AND PROBLEM NUMBER. J: M1+1 3 PRINT 21 KCUNT1.N M1J180UN0 21 FoRMAT(66X.. PRUBLEM NUMBtR * 13///. THE SIZE OF IHIS PROBLEM IS N 1 EQUALS 1,13,..1/1 THERE ARE 1,13,. RUNS ALLOITED TO THE SELECT MA 2TRIX.t/* THE CCN*LItT MATRIX IS UP THE *aisa*“SOUARED ORDEkgt/t 3THE LONE4 RQUNU FOR.[H}S PHOupgn [$19.517.2)'H PRINTZU.XOUT pn FORMAT(1H ,1XOUT IS ‘9F800) READ 1001.NHRFIX 1 I=o I .1. .- m. u-v- u . 1 - A "an IQ : 1 s .‘vu--,-."-'1 » 1601 FORMAT (12) IFtNBRFix .EO. n)90 T0 1004 1noa =l+1 » QEAD 1093.1r1x0(1>.1+1x1(1) in”! FORMAT(212) IF(I .;T. NNRFIX)GO T0400z lfiod CONTINJE TNEN=T1MEF(4) T=(TNEN'TULD)/130N, PRINT 14o,T _~_ _ 146 FORMAT(i TIME SPENT 0N PHtLlMINARY NORA WAS io?16.3.¢ SECONDS.*) TOLDETVEJ GCNERATE A TOTAL OF KOUNTZ- SOLUTIONS FOR THE PHoaLtM; U032KOJNTO=11KOUNI?A‘ 2 ‘1 xoUTarwpxour 9 LAsTDEPzg S MAXUEP='99999o PANDN0=RANr<-1> $ PHINT 999.NAN0N0 ‘ _ 999 FORMAT(1H ,. THE FIRST RANDOM NUMBER GENERA?ED“NAS'.IF18.151“ KOUNT24= 0 S PRINT SSaKOUNTb 3 KOUNT19=O 5 KOUNT25= a KUUNT30= 6's KOUbT51:d $ KOUNT27=O INITIALIZ: aAThICES nT ZERO. 31 FORMAT(/I/1H 55x. 9 SOLUTION NUMBER 9.13) KOUN72°= o $ RELUP= 6 KoUNT1=N+1 D038181 ‘OUNTl n039J=1.KuUNT1 39 SOLUTUV u-vwv- - ZnUR CONTINJE D034121}N ' t "m‘ Do 35 J=1,N 55 TALLY(1,J)=0. 54 CONTINJE DO 36 1:1:N C (70000 1005 I=I91 b IFC21FIXCII) $ IFL= IIIXL(1) 996 CELLTIEII. 2)=CELL IEII‘192 603 CONTINJE 604 CONTINJE 698 CONTINUE 60‘ CONTINJE 807 CONTINUE 164 P0 37 J=1,M 37 CRITERAII,J)=0. IA CONTINUE , KOUNTl‘INo929N)/2' 00124131050 00125J= 1 KouNTl SCOREl‘T J>=o $ SCORE2II. JI=O S SCORESII sto 8 SCOREQII J390 SCORESII. J):0 $ SCOREOII. J): 0 s SCORE7I1141=O s SCOREBII 41:0 SCURF9IIoJ)=0 $ SCOR§10(1:4)3O S SCORE11II.J)§QY . 195 SCOREIZII.J)=U 124 CONTINJE RFVISE THE SOLUTION MATRIX TO RELECT CWXED CENTERS. IFIRELUP .EQ. 1>Gu T0.90 IFINBRFIX .eo 61uo To 40 1 1:0 ‘0'- ‘ "- D010 06J=1,M 1 IFISOLUTON(JoIFL) .NE. 35’50 TO 10066“ SOLUT0V(J N‘l):50LUTON(J.N*1)+; 1006 SOLUTONIJ, IPL)=3 DOloo7J=1.N $ lFIsOLUTONIIICIJ) .NE. 3160 TO 10B7 SOLUTOVIN+1. J)=SOLUT5N(N+1aJ)+1 1on7 SOLUTOVIIIC, J):3. SOLUTOVIIFC.IFL)=2. S SOLUIUNIIFCaN¢1I31000000.*S EOLUTONIN¥1.IFL) 1=1oooouo. S KOUNT19=K0UNT19+1 $ IFIKOUNT19I.GE. N-1IGO To 32 IFII .LT. NBRFIX)b0 To 1005 s 60 TO 836 SELECT NE“ QUADRUpLEi FOR ELIMINATION FROM THE CELLTIE* MATRIX IF ANY ARE SIORED IN IT. I? THERE ARE NONE. FIND ALL nuADRUPLETE NoT YET ELIMINATED HAVINu THE LARUEST FLOR-DISTANCE PRODUCT. STORE THE LABEL5 OF THESE QUADIUPLETS IN THE CELLTIEv MATRIX AND MAKE A RANDOM CHOICE. ~ 4n KOUNT8=U KOUNTZIS‘UUNH727 1 $ IFIKOUNT27 ._LE. OIGO To 997 00996I=IVDICAT KOUNT27 S CtLLTIE(Ipl)=CELLTIEII*1 1) IFIKOUVT27 E0. 1IGO T0 995 .uwmv T _ ”.mw.wuuwr.1 GO TO 135 997 CONTINUE BIGN0=-99999. I KUUNTl'IvaZ'N)/2 5 I90 3 KOUNT27=Q 12R 131615 J: -1 S L: 0 129 CONT1NJE IFII .LE. 50960 TO 603' IFII .LE. 100160 IO_604 IFII .LE. 150)GO TO 305 IFII .LE. 200180 To 606 IFII .NE. 250160 10 607 IFII .EE. SUOIGO *0 608. IFII .CE. 350)GO ID 609 IFII .;E. 4ODIGO .o 610 IFII .LE. 450160 10 611 IFII .LE. 500160 I0 612 IFII .LE. 550160 [0 613 IFII .LE. 600160 614 IFISCOHEiII JIIGO TU 130 s 60 To 139 1F¢Sc0HE2(l-509J))GU T0 131 5 60 TU 139 1F¢Sc0RE3II-1ooaJ)IUO To 130 5 GO 15 139 [FISCORE4II-150oJIIGO TO 130 5 60 IO 139 '1F¢S(;0NE5II-200.J))Go’TO 136 5 Go TO 139" fitfiéfiéfiifl Iv '--- ~ . r a ' I v - n C Aug 609 ,‘19 611 619 613 614 13° 131 13‘ 130 131 995 135 136 616 517 61R 619 620 621 622 623 524 623 62* 627 14? ENTER TALLIES RHERE APPROPRIAIE. 1026 165 . ..-—-..“, -————_.-—-—~..—-.~ “--.... IFISc0NE6(I-250aJIIbU TO 130 3 GO '0 139 CONTINJE 1 CONTINJE 1 IFISCOKE7(I-3000J))GU T0 :30 I GO «0"139 " CONTINJE I IFISCOMEB‘IYZSOOJ’IEU.J0.130 *nGO I0 139. 1. , V CoNTINJE 1 IFISc0NE9(I-4oopJIIGO To 130 s so 50 13b ' ’"" CONTINJE $_IF(SC0HE10(I~45DJJ)IGO To_1§g S GO TO 1 9 CONTINJE s IFISCOHE11II~500.J)IGO To 130 I GO TU"1 ”" CONTINJE s IFISCOEE1ZII-SSOIJIIGO To 130 3 GO TO 139 CONTINJE 1 1F¢FL0uATAII)iDSTUATAIUI'.LI. BIGNOTCd"?U”131' IFIFLOUATAII)*DSTUATA(J)I.EQ. UIGNO)GQ r0 132 H KOUNT27§1 1 816N0=FL00ATAII)*usIDATAGo TO 128 IF(KOUVT27 .NE. 1)GO TO 135 1:1 I 50 T0 136 _ RANDN0=RANFI'1) s X=KOUNT27 s x=xiRANoN0 S I'ij K0UNT29=IUUNT26+LW 1111. . 1.,11 , ,.L 1 A... KOUNT10=CELLTIEII.1) S KOUNT9=CELLTIEII.2I s INDICATSI IF'.NE. 0160 T0'1051 “ ‘ TALLY(MMC1.TMT2) IALLYCHMCInHMLZI+1. TALLY(WMC 2.MML1):TALLY .N 15. 0.)JO TO 1011i SOLUTON(MH02:NHL2 )31 SOLUTOV(MM02.N+1)= SOLUTON= SOLUI0N(N+1:HNL2)¢1 s KOUNTB-KOUNT8+1 1011 TARYIMUCZ $ MMC2=MM01 1nln MMClaTARY TARY=NWL2 $ MML2=MML1 1009 MML1:TARV KOUNT24=KOUNT24+1 s IFTKOUNTB .eo. CIGU To 40 ‘ SEARCH TH: SOLUTION MATRIX FOR CONDITIONS DICTATING ONE OR KORE UIAI AQSIGNFENTS. non KOUNT14=o $ KOUNT15=1 s KOUNT16=1 s KOUNTzozo MAKE A RAVDOM CHOICE ASWTO WHETHEH'RONS'OR COLUMNS'IRE'SEARCHED" FIRST. LOOK FOR AND RECONclLE DEMANDS MADE av CLOSED} LTNES PRIOR TO SEARCHING FOR LINES HITN ONLY ONE REMAINING FEASIBLE DIAD." RANuNosnAvrt-1) IranNDNO .GT. .49160 To 47 s Nou=o. s GO TO 43“ .47 R0‘“ 1- _ w .. . a. .1 _ . ,;1“E”M- 48 1:0 " ' 1 0049K0JNT1=1.M Do5oJ= 1 2 5n SELECTKKOUNT1.U)=U. 49 CONTINJE ' ‘ "‘ "““ .~ n-u- - \ ll .. _ v' A- A ~ 0'. --« -. ~ ‘_ » «‘1 «n: n. “o l— c ' -V- ' n. ”'1. »- n‘ - so a ,..v ~ .5 .' «.5‘ " *V _ ... ...,. .... og-A 000 DC) CO 167 L=H1+1 DO51KOUNT1=1,L 0052J312L _ 52 CONFLCT(‘OUNT10J)=D. 51 CONTINUE . STORE CENTEQ AND LOCATION LABELS UF OIAOS REQUIRING"ZSSIGNMENT'IN THE CONF.ICT MAIHIx. TRANSFEN INTO IT Tug'nlAOs1flyALues As sroaen IN THE ‘CilTERIA' NAIRIX. "" Jaofls LABELH=0 s LABELC=O $ IFFRQNW.NEe 09269'1Q153. 54 181*1 3 IF(SULUTON(I:N*1) 0E0. 10000000)GO TD 55 IrcN-SJLOTON¢;.N+1> ,NE. KOONTL4190 TO 55 s LABLLBpLABELR*1. IF(LAB:LR ,NE. 1 .AND. KOUNT14 .NEu'1)GO To 56 5 Lin 3 LABELC=U IF‘KOUVT14 .Ne. OIGO TO 57 I. 1. M _ .1 -.WW_W-2 -.H 5“ L=L+1 3 IF(SOLUT0N(IaL) .Nh. 1’60 T0 59 $ LABELCILABELC+1 CONFLCI‘thABELc+1??L.$ IF‘LABFLC yGE-.H1>90 IO 26 w” 50 CONTINJE s IFcL .LT.‘N)eo TO 53-: so re So “ 57 L=L+1 5 IF(SOLUT0N(IOL"QEUO ooIGO TO 60 IFCL ..T. N)Go TU 57 3 GO ID 56 6n CONFLCVLLABELR92)=L $_coNFLCIGo To 62 m KOUNT1580 3 GO To 63 53 J=J+1 t IF<$OLUTON60.TQ,642, ,.NH-.1 IF(N.SJLUTON(N+1:U) .NE. KUUNI14)GO TU 64 3 LABELCILABELC¢1 IFILABELC ,NE, 1 .AND. KOUNT14 ;Ne,-1gso To 65 3.L'0 s LABELR;0 Ireo T0 66 b7 L'L*1 5 IF(SOLU10N(L2J? [NF. 1300,10 63.3 LABFLBELABELR*11. CONFchILABELR*1a1)=L S IF60 TO 6713.60 TO 65. as L=L+1 A TFtSOLUTON(L.U> .EU. o.)GO To 69 IF‘L QFTOLN)GO T0 66 5,60 '0 65 _ “1 H . . q._.1..__ H,.. 6O CONFchcLABELc.1)=L s CONFLCT1LABELC.2)=J 3 GO TO 112 as CONFLcr<1,LABELc+11=U M "U 112 IFGO TO 63 3 IF(CUNFLcT(1.2) .NE, 1 O.)GO TO 62 $ KOUNT16'O S GO TO 63 62 1:1 s IF $ J'1 . 71 J=J¢1'$ KOUNY18:CUNFECT(1A‘) E'EONFLCTlIoJVzCRITERXKKOUNT17$KOJN11 8) ‘ 1IFCJ {LT} LABELC+1YGU”T0‘73 S‘[F11'.LT§'LABELR31IGO’TO'72 KOUNT31t‘0UNT31+1 s IF(I .LE. ZIGO T0 114 s KOUN1303KOUNT30¢1 .60 To 114 m _,hqfl,fiufiwm.--u Hnw__~,, ”._1 u,.flwfluflnu.qufl L 71 CONTINUE s IF(ROH .60. 1.)60 I0 116 s K=LAaeLN 5 GO TO 117 11~K=LARELC” 1“ - .-Wmngrmmu. .w,"u ,.“M.-mfl-m,hw-wwwww-au, 1 1 7 com my; -§~.12011,5.1_.=12L§_ towniscszmcu 14.1’,-£.-§.9.UNI.1.§!C.9.NEEFT5 1.:2 1) 115 CONFLCT(I.33=CRITEPA(K0UNT17oKUUNT18) IF(I“;LE§'1)GO'TO'114 S'KOUNTSOEKOUN730+1" J‘C" .. a..-p. . - ‘-I .~- .I— .‘ - ~- - . .vw .-o. ”3" ., -.- a..." 9 ;-Q\~"1 ~ .. . t. . . ..~~.-~ . .4 - 7“" "v C C C C 168 H (:umcll: C3NFLICIINU DIAD ASSIGNMENTS BY CHOOSING _THAT DIAD NITH THE I_“NEST CilTERlA VALUE, BREAK IIES RANDOMLY. ‘ “M” ”' " " 114 SMALLNJz 99999999999. 5 It; 8 L=Q IFIKOUVT14 .Eo. 1,00 T0"118 7Q I:I+1 i J: 1 I . 79 J2J+1 s IFICONFLCIII J) .GT. SMALLNOIGU TU 76"! IVYCONFLCTIIIJ) 3L 1T. SMA- LVO)GO TU 17 S IFIL .GE. HIGO TO 76 S LIL¢1 SELECI‘L01)=CONFLUTIID1I S SELECTILazItcONFLC|I%oJI 8 60 IO 76 77 L- 1 m DMALLNOBCONP LCT(I J) S SELECTI1 1I'CNNFLC IIL1I SELECTI1. 2):CONFLUT(1 J) 76 CONTINJE_$ 1Fuo To 75 s lrcl .LT. LABeLR+1IGo r0 174 S IF(L .EO. 0’60 T0 197 S IFIL .GT. 1IGn TU 79 ASSIGNJ:SELECTI1 1) s ASSIGNL= -SELECT(1.2I 1 EU T9 106 119 CONTINJE S IF(R0H ,E0.1)uu ID 119 S KcLABELR 3 Go To 120 119 K‘LAREECM. , I ... _. . ... 12h CONTINJE $ u01211= 1, K s IFICUNFLCTIlaSI ,Gf; SNAELNOIEO To 12' 11 $ IFICDNFLCTIIas) .LT. SHALLNOIGO T0 122 IFIL .SE. 8:60 T0 121 s L= L+1 S SELECT(LI1)=c0NFLCY(1,1) SELECTILo 2):CONFLCT(I 2) 8 GU T0 121 129 L31 $‘sNALLN03coNILcTiI;3) s SELECTIIo1I=€oNFCCTTIIiIWS"SELECTIIo2 ‘1):CONFEQIII,2) 121 CONTINJE IFtL .20, 0)60 To 197 IFIL .ST. 1)Go T0 79 3 ASSIGNC=SELECTI1 13 ASSIGN.=SELECT(1.2) s GO to 106 7Q RANDNO= NAur(-1) $ x=L s x: XwHANDNO s 1-x+; s ASSIGNCISELECTII 1) ASSIGNL= SELECT(1.2) s KOUNI2beoUN12§+1. ‘ _ 106 CONTINJE REVISE TH: SOLUTION MATRIX T0 REFLECT THE NEH UIAD ASSIGNMENT. 00801:1. N ' " '“' ' ' ‘ ‘” -IFISOLJTON(I: ASSIGNLI .NE. ooIUO TO‘EU S bOLJloNII N+1)850LUION(IA 1N+1)¢1. $ IFIASSIHNL .NE, IIGU T0 80 SOLUT0‘(N+1,I)= SOLUTQN(N*1AII+1I 8n SOLUTOVII ASSIGNL)=3o ' D081J= 1. N $ IF‘SOLUIONIASSIGNCOJ) .NE. g.)co lo 81 S SOLUT0N(N+1.4 1)=SOLUT0N(N¢1,J)+1. "“ ' '"‘”‘” ‘”' ‘ ” 81 SOLUTOVIASSIGNC J)=3. SOLUTOVIASSIGNC. ASSIGNL)= 2. $ bOLUIONIASSIGNCoN+1Ilf6 _SOLDTOV(“+1'ASSIGUL)=JDDOUUO' 3 K0UNT193KUUNTl9f1 3 I SUBTRACT JNE FROM x007. PREPARE SOLUTION MATRIX FOR RECYCLING. XOUT=X3UT. -1 $ ReLUP=1 S KOUNT2730 """ D0200 OI: -1,N $ 002 001J= -1,N 5 IFISOLUTONIIOJI oNEo 1IGO To 2001 SOLUTOVII: JI‘D S SOLUTONIIAN*1I= SOLUTONIIpN+1Icl SOLUTov-; ._l " A‘ ' " "".—r .o--.. ~' In. on - -\_ - \ ,-.’ -,-.~.1 9.. o g. u ""9.“ ‘U ""':v v - ‘1 I 00000- E‘KOQQI19 :55 2mg] CONTINJE 2000 CONTINJE DEPTH=«0JIT24. LASIDEP {’LASTDEP=KOUN72A IFIDEPIH .LE. MAXHFPIGU To 2002 s MAXDEP‘DEPTH 2n0? CONTINJE Go To £008 63 CONTINUE s IF(KOUHT14 . 0. .OR RON NE ogIGO TO 83 ' IFIKOUVT15 .EO. 1)GO T6 48 s KOUNTzoaKOUNT2001 s IFIKOUNTZO oGE. 2 1)GO T0 84 S Rod=1.$ GO TO 48 84 KOUNT14E1 $ KOUNT20: o S GU TU 48 81 CONTINJE $ IF(K0UNT14 .NE- 0 .UR. RON .NE. 1IGOVT0 85 I...- -~ as 8% an 87 169 IFIKOUVTIO .Eo. GO TO 48 $ KUUNTZDIKOUNIQDPl S [FIKOUNT20 06E. 2 V l)GU TU 86 $ ROW: 0. 3 GO TU 48 KOUNTl4=1 $ KOUNT20: o 9 GO To 48 coNTINJE s ITINOUNTix“;NE. 1 “.OR; Row 1N5; c;930“Yb'§7“"“ “ IFIKOUVT 9 NE. )GO To as s KUUNT;QIC s Go To 45 . KOUNT20=NOUN120+1 s IFIKOUNTZD. 2590 To In I Rbfli1. s*co To 46 CONTTNJE 9 IFIKOUNT16 .NE. 116a To 39 : KOUNI14co 3 GO To as MIKE THE EAST DIAD ASSIGNMENT BY DEDUCTION AND PRINT OUT ALL RESULTS R: LEVANT TO THIS SOLUTION. 59 82 123 91 9? 9P 94 93 93 97 9: 10“ 101 99 10? 103 Dana; 2004 Znofi KOUN720=KOUNT20+1 1 IFIKOUNTZD .GE. 2>GO TH 40 S Rowan. S 60 To 48 CONTINJE 5 DO 193 I=1aN FINAL(1 V91)=-0 D0 90 1:1. N no91J=1;N s'IFISOLUTONIIQJ) ,Eu CONTINJE KOUNTzléI 9 GO T0 96 FINAL‘I N+1)3J CONTINJE 1.093121 V D094J319V $ !F(FIHAL(JON*1) 0E“. 1’50 T0 93 CONTINUE ‘ GO T0 95 CONTINJE FINALIKOUNT21,N+1)=1" pRINT 96 _ . _ ,. 9-. a FORMATI/I1H .SSXIECENTERS LOCATION ASSIGNED') D0971=1, N PRINT 98 I I1:uAL(1.N.1) FORHAT<14 .57x912 10X F3.0) COSTSUo S KOUNT1=(Nw*2-N)/2 00991=11K0UNT1 K0UNT2¢=FL0N01II> KOUNT2§;FL0NC2(1) DO100J=1.N0DN71 ITIDSTL1 ~. ' of. . -- ~Ir 1T26.N+1) .OH. DSTL2(J) .Eu. FINALIKOUN1229N+15 .AND. DSTthJ) .60. 2 FINALIKDUNT239N+1))GO To 101 CONTINJE so To ?9. CONTINJE IFIFLOQATAII)..EQ. 99999, .OH. FLODATAIII .EO; .9999, )00 To 99 IFIDSTUATAIJ) .Eu. 99999. .oR. DSTDATAIJ) .ED. -9999.)ao To 99 COST:CUST¢FLUDATAII)tDSTDAIA(JI CONTINJE '”""”' PRINT }D2.CD§T _ ‘9, E .. T . F0RMATI1H ,/9 THE TOTAL COST OF THIS SOLUTION TS 53F23I1) PRINT 103 K0UNT24 ‘ -‘« x.--aw. x-KouN124 s YEN 1 s 1: _X/Y RRINT2003.Z FoRmATI1H ,tTHE AqERAGE EklMlNATED IS'OF20,1_)_ PRINT2UO4. MAXDEP FORMATI1H ..TNE MAXIMuM PENETRATION AFTER N- -2 ASSIGNMENTS ISP I16) nePTN=«ouNT24 LASIDEP ' PRINT 2005 DEPTH FoRMATI1H ..PENETRATION FOR THE NEXT To LAST ASSIGNMENT ISN.F15PO) u“..- ~.~-~~ ., -. ......w—7 1.- . .. ‘1, .I. -... 1 . m9"- ,-v-.- o' .- - » . , v -..;.-..- -. . ..,. ,.. ~ .. ..‘. -. 0‘ . .n .- '9‘ 170 ...—.__———-.——_._- . -7 PRINT ’39.KUUNT$0 739 FORMATI1H o'THE CRITERIA MATRIX HAS USED To HAKE A PARTIAL ASSIGNM 1ENT A IOTAE.0F PIIEE!HTIN§§o1)H PRINT 105.K0UNT25 505 FoRMATI1H ..A TOTAL DE #91200* TIEs_IN TNQ SEEQQTHNATRIX ARE RESOL 1VED BY RANDOM CHOICE.9) ' ” ” PRINT_974{KOUNT31 v-“ ”-4 . ,V.‘.,_ , 974 FoRMATI1H ,.cLoseo LINES NERE ENCOUNTERED I TUTALsoF 9.155511NESPI PRINI.L45:KOUNT26 N“. - ,1 , . I , . .-I.L”.D.W.I.I.. . , 146 FORMATI1H ,, A TOTAL OF 9.19.9 TIES HERE BROKEN RANDOMLY NHEN SELE 1CTING pAIR ASSIGNMENTS FUR ELIMINATION.*) x=KOUN126 9 Z:X/Y “ * ""’” ' PRINT 2006.2 N anos FoRNATI1N ,9THE ATERAGE NUMBER 15 95F22.19 TNEngIMEF(4)_NB .Iw.mc TEITNEN—TOLc>/1coo.' PRINT 147.T . . 147 FORMNTIPTIME SPENT oN'TRIS SOEUTION‘NASNIP1e;s;iSE60NDs;PI TOLDgTVEN 3? CONTINJE Go T0W901 197 PRINT 198 199 FORMAT(1H ,.CRITENII VNLUES ARE TDO LARGE To IEST..I 601 CONTINJE "' '" '"” *"" ”"'“’"'“”' END agw -" nw -‘~'0- ‘ ~I' v ’ cv- ~ vsv'w - - rs ,...,-“-pI-pgu ar ' I .—L-I-.-m1..i v'V' Ix~m .. an -0 9".» v ,v , - ad‘Q‘ APPENDIX IV LISTING OF 4-A PRUIIFIA‘I PROCAA DIMENSION‘FLODATAISSI){FLDNCIISSIIIFLUDCZISSITIURIENCIIB).DRTGNCEI" 18IAORIBND(8).TALLYI702.27).DSTUATAISSII.DSTL1I351I.DSTL2I351>.SOLU '2T0NI28P28){XOUTI7fi2)SCRITERAIZVC2713CUVFCCYIQBAQH){SELECTI5712).FI' “On 0' ITERATE FOR A TOTAL OF BNUNBERA PROBLEMS C' READ AND 2ANK FLow AND DISTANCE DATA 1 3 4. 1D . 11 12 13 14 16 15 17 1A ? ’C' COMPUTE LUNER BOUND 09” SNALI?8:28).UNRANKDI702,27) EQUIVALEVCE (UNRAquiTALLYW,IFIN[L“50LUTQN5"“”F ~~w~a~rw-~www 1 INTEGE-I ASSIG IL.ASSIGNC ORIGNC1 URIGNCZ TOLD: TIMEFIA) XzTIMEFISI CALL'RANFSETIXI RANDNU=RANF(.1) READ 600.NUN85R FORMATIIZI D0°01 ‘0UNT99=1:NU"BER I“. I 1,mm.mmw,.,u READ 1 M. N. N1.K0UNT11K0UNI2 FORMATIBIBT DO 2 KJUNTS = 1. 2 $ 1: 0 READ 4:(ORIGNC1IJ).0RIGNC2IJ).URIDNDIJIAJ91.8o1) PORNAT58<12.12.T6.1)) .“., DO 5 J=1l8 5 131*1 IFIORIJNC1IJ) .LT. DRIGNCZIJIIHO TO 6 S UNRANKDII.1)IORIGN02IJ) UNRANKUII.2)= 0916 ”c (J) 6 Go TO 7' UNRANKDII.1)=ORIGNC1IJI s UNHANKDII 2I¢0RIGNCgIJI UNRANNDII.3)=ORIGND(JS IFII-IYP*2 MN)/2)5 8,9 CONTINJE GO TO 6 1:0 9 «OJNT93(NwPa-N)/2 J:0 9 519Noz-99999999. J: J+1 n IF(UNRANNJI4.3)IBIEN0112{12I11"' 816N0= JNRANrucd 3> $HKOUNT5=QW$.K119836NKDI9'I! 5.szNNBANNDIJ:2>.. CONTINJE IFIJ ..T. KOUNT4IGO T0 10 1:111 1 IFIKOUNT3 .EO 1)GO TO 13”!"DSTDATAIIIiBIGNO S DSIL1III=x1' DSTnglszz $ 60 T0 14" " - . N“ . . - . FLODATAIII=BIGNU A FLOHC1II)'X1 $ FLOH02III=X2 CONTINJE $ IFIKOUNTS .EO. KOUNT4)GO TO 15 CM‘ .‘u‘fl I'I’ “t ‘A'\‘ " 0‘1. / ‘u.’ 9 VIA‘P'O‘ I. ".1 .I‘H ."9"lOL T UNRANKD<GO To 969 IFIFLODATAII) .NE.“99999. .ANDIFLDDATAII) .NE.'-9999 )GU TO 988_. 171 ' .._—___.—__-..--.-__.._. . ... m_—_____-___-—__ _L- I..- - __ ....-‘—_.___—_...‘ -- . -m—n ... _ .——o .-x , r- . - ..1... m-0 A'I'N “nan-v4 ...... -»...—-. 4 no»- - . ~¢< ' u -A ——--- - ._»- _—-. .- »--» (\- u—\ «n- ~ ~I\r ~~4>¢ -‘- .-¢——' A>~ . 172 I:I+1 t GO TO 957 . 969 CONTINJE s’IFIDsTuATAIJI"}NE- 99999; (AND. OSIOAYXIJI'.Ne;“39999.I 160 To 986 S J=J-1 S GO TO 987 H , ,. 956 BOUNO=30JND+D3TOATAIJIarLODATAIII "”" '” ' ”'““"”"““ W”' " 989 I=I+1 3 J=J-1 ' 937 KONTzKJNfil S IF(KONT .LT.'LIGU TO 990””"" J:Ni*2‘N DOZOI=1,J 2n XOUTII)=0 . fl . _h c PQINT OUT MATRIX SIZES AND PROBLEM NUMBER. J=M1+1 S PRINT 21, KOUNT1.N MIJOBOUND 21 FORMAT(60X.t PROBLEM NUMBER *.13///a THE 5125 OF THI§ PROELEK 15 N ' 1 EQUAL: t,13;*.*/t THERE ARE *91399 R023 ALLOITED 70 THE SELECT MA 2TRIX. */. THE CONFLICT MATRIX 13 OF THE 9.13.- SOUARED ORDER ./. 3THE LONE? BOUND FIIR THIS PROBLEN 18 99717.2) c 'cnmpuTE x3uT VECTOR. ' KOUNTl‘IN..2-N)/2 ‘ O022L=1.---. n’ V~9-'-<§r0 40., a— no COO ()0 COO (10(1 907 K0UNT24= KOUNT24 + 1 S IF(KUUNT8 ONE. 0360 TO UOé.yr 173 SELEcT NEXT DIAGQNAL 0T OUADHUPLETS FOR ELIHINATION. 4n KOUNT7=K3UNT7+1'T KOUNT6: o S KUUNT9=1'$‘KGUNTI63KOUN77MTJ ENTER FOU< TALLIES FDR EACH UUAURUPLET ELIMINATEU. 41 J:0 $ KOUNT11= DsTE1(KDUNT9) ’ " ” "‘“ 49 TALLY(gtKDUNTlo 1 KOUNT11)=TALLY(2tKOUNTV11oKOUNT 1)+1. S TALLY(Z 1*K0UNT10 KOUNT11):TALLY(2*KOUN710 K0UNT11T .1. I KOUNT11-DETL2(KOUN 2T9) T J= J+1 N . -1 1 . IFTJ .50. 1:50 To 42 1 Jan 3 NUUN112=rLowci(KUUNi1bT'i‘KOUNT1gusLo 1NCZ(KOJeroT uonATE 'CTTTERTA TATRIX AND CHECK'FDR'INFEASIBLE“DTXDS DUE To TALLY SPDPES. 43 CRTTFRA(K0UNT12, KUUNT11)BURITEHA(KOUNT120KDUVT11)+1. S IF(TALLY(2v IKOUNT1U-1..K0UNT11) .LT. XOUT(2*KOUNT1O'1’)GO '0 44 5 IF(SOLUTON(KU 2UNT12.‘OUNT11).NE. n. )6” T0 44 S SOLUTON"OJerzfiKOUNT11’.1Q ‘ SOLUTOV(KOUNT12,N+1)=SOLUIUNTKUUNT12,N*1T+1. S SOLUTONTN+1TK0UNT11 1)=SOLUTOV(N+1.KOUNT111) +1. T KUUNTaa KOUNTdti 44 CRITFRA(KDUNT13, KUUN )=CHITEHA(KOUNT131K0UNI11)+1.s IF(TALLY(2t 1KOUNT1U.KOUNT11) .LT. XOUT(2*KUUNT1Q ))GO TO 4: s 1F(SOLUTON(K0UNT1 23.K0UN111) .NE. 0. )GO To 45 SOLUTO:(KOUNT13 KOUNT11)=1. 3 bOLUTON(KOUNT13-N+1TISOLUTON(KOUNT13 1 N*1)* SOLUTOV(N*1. KOUNT11)=SOLUTUN(N+1;KOUNT11)¢{, $ KOUNTBtKOUNT8+1 AS J=J+1 IF(J .VE. 1)60 10 807 $ kOUNT11=DSTL1TKDUN79T s 60 T0 43 If NO CHAVGE IN SOLUTION HAS OCCURRED BY ELIMINATING THIS QUADRUP- LFTo ELIMINATE THE NEXT ONE IN THE DIAGONALg"DTHERdlsE. BEGIN IMFEASIBILITY TESTINN. a“ , ‘1 IFTKOUVTIU - 1 .LE, 0)GO TU 40 s KOUNT10=K0UNI10 7’1 _ ”,1 , KOUNTV=NOUNT9*1 T 60 T0 41 ' "' SEARCH TH: SOLUTION MATRIX FOR CONDITIONS DICTATING ONE OR MORE DTAD AQSIGNMENTS. ’ ‘ ' ' ' ' ' ‘ 909 KOUNT14: 0 T KOUNT15: 1 $ KUUNle=1 S KOUNT2030 MAKE A RANDOM cHoIcE AS TO NHETHER RUNS 0R COLUMNS ARE SEARCHED FIRST. LOOK FOR AND RECONCILE DEMANDS MADE BY CLOSED LINES PRIOR In SEARCHING FDR LINES NITH ONLY ONE REMAINING FEASTBLE DIAD. RANDN03RANF(-1) IFTRANUND .GT. .49130 To 4/ A Huw=o. s so To 45 47 ROW=1o 48 T30 0049K0JNT1=11M 0050J=152 5n SELECTTKDUN11.J)=U. 49 CONTINUE ‘ L8M161 D051KOJNT1:1,L 0052J=1JL _ 1‘ 59 coNFLcTTKOUN71.1)=n;' 51 CONTINJE STORE CENTER AND LOCATION LABELS OF DIADS‘REDUIRING’KSSIGNMENT IN“ Tue CONF.ICT MATRIX. TRANSFER INTO 1T THE DIAUS- VALUES AS STORED I“ THE ”CQITERIA" hAiRIX Wm WM E .W, waflflmwpmnm“w-WHWLEL... Jao $ LABELRBO s LABELC: o s IF(RON .NE. 0.)GO To 53 54 Isl¢1 f TFTSOLUTONTITN61) .EO.'1000OOU.)GO TO 55 ””"”" IFTNoSJLUTONTITN¢13 .NE. KUUNT14)OO TO 55 S LABELRILABELH*1. IF(LABELR .NE. 1 .AND.fNOUNT14 .NE. 1)Go Th 56 $"qu s LABELCid -- - ..."...--~.~ rm .. .- . _F-v‘lrn-J . n - .. .--. . . ---s . .. . fl! , \_ .r 11-..!1 ,. .4 ... 1 1 -I- max '- -'§. ‘0. . . -‘fiu . - . r- q. u--\ a . 1.»: - 90 ...- ..— ~ 174 IFCAOUVT14 .NE. OTGD TO 5/ 5q L=L+1 5 TF(SOLUT0N(IIL) .NE. 1>Go To 59FS’LABELCianeLc11 CONFLCT(1.LABELC+1)=L s TFTLAUELC .GE. H11G0 To 56 59 CONTINJE s TFTL .LT.'NTGO'T0 58‘s GO TO 56 ‘”'*”“ 57 L=L+1 s IFTSOLUTOTTI L) .EU. 0.)Go TO 61 IFTL ..T. N)Go TO 57 3 GO T0 56 ' 61 CONFLCTTLABELR.2)= L 3 coNFLCTTLABELR.1)=T 9 GD To 110 56 CONFLCTTLABELR+1.1T=T 11m CONTINJE T IFTLABELR .GE. M1TGU TO 61 59 CONTINJE s 1F(T .ET.'N) GO TO 54 61 CONTINJE T IFcKouNr 4 .EO.1)GU TO A IFTCUNFLCTT172) .NE. o.>u 10 TO 62 T IFTCONFECTT2T1) .NE. 9.)G$ To 62 G KOUNT1580 3 Go To 63 111 CONTINJE i [FTCONFLCT(1 1) NE. 90. )GO TO 62 5 TFCCONFLCT(1.2) NE, 1 0 )GO TO 62 s KOTNT1530 3 GO TO 63 ‘ 53 JsJ+1 t TFTSOLUTONTN+1au> .EU 000060.)Go TO 64 '7 IF‘N'SJLJTON(N+10J) .NE. KOUNT1TTGO TO 64 9 LABELCILABELC+£“” IFtLAchC NE 1 AND KOUNT14 ,NE1 1TGO To 69 s LIO S LABELR=U IF(KOUVT14 .Né. 0360 To 66 1" ' """ “ ” '”‘ b7 L=L+1 * IFTSOLUTON(Lo J) .NE, 1)G0 T0 68 S LAEELR‘LABELRTI CONFLCTTLABEL9+1 1)=L s TFTLABELP .GE. H1300 To 65 69 CONTTNJE s IFTL .-T. NTGO To 67 $ Go TO 65 . 6‘ L:L+1 i IT(SOLUTO (LaJ) .EU. 0. )60 To 69 IFTL ..r. N)GD T0 66 5 GO TO 65 go CONFLCTTLABELCa1): L's CONFLCTTLABELC.2T:U 1 GO To 112“’ 66 CONFLCTTT LABELC+1T=J _ 119 TFTLAB=LC .GE. M1)G0 T0 70 64 CONTINJE s IF(J .TT NTGO T0 56 w. 7nC0NT1NJE S IF(KOUNT14 .EO. 1)GU TO 113 S TFTCUNFLCTT1T2) .NE, o.>fi 10 To 6: s IFTCONFL CT<2.1) .NE. o.)GO T6 62 s Kou~116uo 5 Go To 63 113 CONTINJE 1 IrccoN LcT(1 1) .NE. 0.350 To 67 s TFTcoNrLcTc1.2) .N5. 1 0.)GU TO 62_$ KOUNT16‘O s GU To 63 1““ , .. 69 1:1 3 TFTROUNT14 .Eo. 1>GO To 71 ‘ 7? I 1+1 A KOUNT17- CUNFLCT11 1) 3 J: 1 73 JaJ+1 S KOUNT18: CUNFLCT(1:J) 3 CONFLCT!I.4)=CRITEKA‘KOUN717AKOUNT1 ” 8) 11FTJ .-T. LABELc+1TGO T0 73 s IFtI .LT. LAGELR11TGG'TD'72 KOUNT31= KOUNT31+1 $_TET1 .LEo2TGOT0 114 s KUUNtsoéNDUNang1 GO TO 114 71 CONTTNJE 3 IFTRDN .EQ- 1TTGU TU 116 S K=LARELR 5 GD To 117 116 KsLABELC ” 117 CONTINJE 5 001151: 1 K s K0UNT17sco~FLCTxCONrLCI(1.53'S GO 10'76’ 77 L31 5 5MALLN0= CONTLCT(T J) 3 SELECT(1T1TECONFLCT(191) SELECT‘1,2)8CONTLTT(1 J) 76 coNTTNJE $ IF¢J .ET. LABELC+1)GO To 75 s IFTI .LT. LABELR91TGO To 174 s IFcL .EO. O)UO T0 197 3 IFTL .GT.‘1;G6.TO 79 ‘ ” rA-c 1 ‘r T , . ,h ... -. ,...,AL .1 l C C 175 A5319 NU=SELECTTJTET T ASSTUNLz SELECT(1TZT 4 EU TO 106 119 CUNTINJE 1 IF(ROH ,EU.1TGU TU 119 s KnLAUELR s 60 To 120 119 KELAREL C . 1.. , ‘ 120 CONTINJE s u0121l 1. K T TTTCUNTLCT.ORIGN62‘J’:URIGNDIJ’5J31'3‘I" F09N4T160 --.--w' -.-—-_'.- 1 - , « —- ,7. ...1- v - < . . -..rvr...-,,_, ...,, w v-7 .. c GcNERATF « YUTAL 0r KOUN12- SULuTIONS FOR THE PROBLEM. DUSZVOJNT6:1,kUUNr2 » ~- ‘ KOUNT24=0‘$‘PRINT”33{KUDNT5 Y KOUNT19=D'$’ROUNT25=0 33 C IvITIALI£= “ATFICES AT_2ERO. 39 34 KoUhT3u=0 $ KOUNT33=C FoRpATK/l/ih’,53x,w‘$0[UTIUN NUMBERm95I3’ KOUFT1=N*w?.N‘ -»~~-- ._ -m 00641=1,KOUII1 ‘DO 35 J=T;MHWV.W wwvnnw"r . TALLY(loJ)=Oo CONTINJE 7' no 36 1:1,N DO 37 J:1,M 179 $7 CRIIFRA(I,J)=C. 5A CONTINJE‘ KQUMT1=N¢1 DOSEI=IL‘OUFT1‘ - D069J81.‘0Uw11 $0 SOLUTOV(1.J)=0. 3" CONTINJE KOUkT7=0 ; HCUNT102'1 t DETECT=O $ KOUNT9:-1 C REVISE SJLJTOH MA THIX IO REFLECT FIXED CE~1gRs. IF(wFHFIK .EC n)u0 T0 an $ 1:0 “ '”“'“‘ "““‘f" “ 1mm; Izl+1 r IFC:IF1XC(I) i IFL= 1*1XL(I) D01006J¢1,N $ IF(SOLLTON(J ILL) .NE.”0.)GO TO‘1005”” SoLLTOV(J. N+J)-QULU10N(J,N+1)+1 1mg SOLUTUV(J. IFL)=5 DOlun7J=1 ~ $ IF(SOLUION(IFCaJ) .NE. G)Gu TO 1007 SnLLTUV(V31;J)£SOLUTCNFN+1aJ)¥I ' "“ '”' ""’"j””“”'””" 1QU7 SOLUTUV(IFC:J)=3O ‘ SOLLTOV(IFC;IFL):2. s SOLUTONIIFC;N;1)biooooonj"s“soLuvoN¢n;1 {FL} 1=10000U0. $ KOUmT19=KOUL119+1 i IFCKOUNI19 .Gt. N- ~1)Go To 52 ‘ IF‘I ;LT. NBPFIX)uO 10'1005 i 60 To 808 "‘“”““““ “ C SFLECT NEKT DIAGONAL 0F QUAUHUPLETS FOR ELIMINATION. ' 4r KUUhT7=KOUNT7+1 $‘KUUNT8=U S'KUUNTqii'S‘KOUNleEKUUNT7““ “‘ "" lflza MMcleLONC1Gu TO 1051 TALLVLZLKouuTl“LignnL21:TALth2¢KUUN716‘12MML27¥1'" TALLY\-oO--_.— . ...-- . .- 4? CRITFRKIKDUNT12.KOUITII)‘CRITEKA(KDUNT123KUUVTf1711. 3 IVVTILLYK2*' 1KOUNT1U'1:KOUNT11) .LT. XUUI(2*K0UNT1Cr1))co lo 44 s lF‘SOLUTON(KU 2UNT12.&UUNT11) .NE. U. 760 TO 44 s SOLUTONKKOUNleiK 0UN711)t1, ' 80LUYUV(60 TD 62 3 ROUNT16=G 3 GU T0 63 '* 62 I: 1 V 1F(KOUNT14 .EQ, 1)GU T0 /1 79 1: 1+1 N (0UN717z CONF[CT(I 1) 3 J: .1 ‘ ’ " 73 J: 4+1 3 ""““ IF(J .aT. LAHELC+1)GU T0 76 N IFCI .LT. LARELR+1TGO To 72 KOUNT31=30UNT31+1 $"TFII ;LE. 2160 T0 114 s KUJNTJO‘KOUNT30+1' GO Tn 114 71 CONTfNJE’3‘IEiR0N .EU.Mf})GU TUW116'3"K=LAEELR 3 GO Td'iil” 116 KaLAHELC 117 CONTINJE 3 D01151=1;K“$ KOUNT17=CONFLCT(I.1) 3 KOUNT18-CONFLCT(I.Z 1) . 11% CONFLCT=r:0NFLCT(1 J) 76 CONTINUE 3 IF(J .LT; LABELC+1)GU'TO 75 S IF(I 3LT;”LABELR+1)GO TO 174 3 IP(L .EN. Q)Gn T0 19! 3 [N(L oGT. 1’60 TU 79 ASSIGNCESELECT(1 1Y S’ASSTGNt:SELECT(1:2) 3 SU T0 106" “‘ 119 CONTINJE 5 1F(R0N ,Eu. 1)GU To 119 3 KaLAeELR 3 Go To 120 11c KaLARE C ’”"‘””' ' "' ”' ‘" 12" CONTTNJE 3 U()]211:1,K S IECCUNELCTHN3) gGT. SWALLNO)GO TD 12 11 3 Ir1caurLr27c1. 3) .LT:‘SMALLNU)GO'T0 22 - * * IF‘L .aE. M)GO T0 123 3 L¢L+1 N SELECTfio T0 197’ 182 IF‘L ,JT. 1)H0 TO 79 3 ASSINNu=SELECI(1.1) ASSIf4-=SELE(‘T(1.2) N GU TU 106 7a RaNIWMJ RANF(-1) $ x=L.1 X= XthANDNO 5 18x11 S ASSlGNCISELECY(l 1) ' ASSIGN -=SELECT(I}27 $‘KUUNT25YKUUNT§S+f‘””WW “*W*"’””'”‘ 10* CONT'NJE C"R‘VISE Tfic‘ SOLUTIUN ‘HATRIXWTO REFLECT“THE‘NER“DIAD“ISSIGNnEn11~ 00501=1,M " ' IF‘SOLJT3N(I.ASSTGNL7‘INE.‘0i)GU”TU‘EU'S”SOLUTDNtliNéifaSOKUTUVI1. 1N¢1)*1o 1 IFfASSIGNL .NE. Ijhu T0 ac SOLUTOV(N+1 W)'50LUTHNIN+1 IT+I.' ,2 " _”1 21,,“ , ,-_uapwwmw"2.h 8n SOLUTUV(I. ASSIGNL):3. 1 0081J=1,V s,IF(SOLUTDN(ASSIGNCaU)'}NE;'0;)60‘13‘51 S SUEUTUNTN*11J 1)=SQLUFOV(N+1,J)*1. a1 SOLUTQV(ASSIG c J)- 3.1“..2- -W-'~K ~ -—»- ~w~~»m-uv..www_fl-n-m._,~ SOLUT'~ s 60 TO 1015 1n14 CONTthE s IFcAsSIGNt' .Nt. rLuwcz(1T);go 7D 1015 S IC'FLDWCItIT) ”2. 1016 CONT} JE $ IF(ASSIGNL .NE. 05 [L1(KC))GO TO 1017 3 IL'DSTL2(KC) ' no Tn 1515 ‘~ ~-» . .2" 1N17 CONTINJE S 1FuU TU 69 1 KOUNT14=0 $ GO TO 48 1 89‘K0U1T20=K0ULT20;1 W“IF(KOUNT20 {GE5'?)GO TO 46 S RHKKU. S 80 TD 18" C MAKF THF LAsr DIAD ASSluNNEN1 bY DEDUcTIBN AND PRINF OUT ALL C RcsuLTs RtLEVANT To was SOLUfIUN. ' 89 CONTIINJE 3 b0 123 1:1.N 123 FINAL(1,V+1)=C - ww-v » ~ ~-‘-L A. . .,_1222 “0 Th bob 4* CONTIIJE 3 IFIUFTECT ;EQ§‘1)GU‘TU'1r21“3'KOUVTlU‘KnUNTIU' 1" KOUNT9: KOUNT9+1 3 KULNTgso 3 1*(KUUN710: 067. 0)GO To 1026 G0 TN 40 ---_-, «~— RO-f‘ CONT111JC 7 1 DD 90 1:1;N ’ ‘ ' ‘ 7 ~ .1 ,1 , 1-22..Mb ... -.-. .. .. .« . ..- .. 1....-1.-." .- .-~... . ..A ._.'_. fl , V __, ., __1 P _ .1 . _,,__. _ ‘1fi1__ ..._ _ *1 . 1,-A_.-,1, V”. '1‘ 183 91 CONTINJE‘ ‘”"‘””---- - ...A._ _ _ KOU1T2l=l R (0 To 901 9? F I NAL ( I V'T'iw ) . .J W... .- rm ...T .. ...”. - .1. . “.....-“ ..- “L...“ --...AW... -, , , N 9n CONTINJE ' DO9SI=I}V H .H. me_hwu .4 "Th__qn- _flh .UMWTML-VA "094JEIAV $ IFIFINALIJANvl) AEU, 1150 10 93 94 (HUNTIIJJEMW“ V'Mfi" ‘”" ””“' ' ~~--w~»- wu--w «mwmiquLw.wfl- _ -W.“A 7,.“ 9? CoNT1qJEMWV. ML.N.AMLH.MW,. .,..WWN.NWW.,A,HW.NMLE.1“,-JA,U-LAH 9s FINALI‘OJNT¢J.N+1A=I ‘PRIAY 96 -U-,-u..ww,nw.p -..1Lw-.L“,.AL--H. A , 7,- 9A FoRAATI/llH A55AAAC5ATER5 LOCATION ASSIGNED.) 00971811 “7* - ~ ~ .. -,.. - ~ A -- - ... - - . -~ -—~r-r~—. - --- -> -.~... ~~-1. 9----..........A.-. T1. 97 PRINT 98A1AF11A1IIAN611 99 FoRHATIIR”;S7xA12.157”F3;g)‘ ““'"“”‘ “ ’"'““ ”""“*"*“““" coSng. $ K0UNT1=1N¢*2-N)/2 no991=1, KOUAT1 ”— K0U1122=FL0AC1111 KOUN1253FL0LC2(1) -~W~-~ - ~ ~A-. -~ - ~~ --.d-m-.~~ DUiUOJ’laKOLNTl ’ ”“IFIDETL113Y'iEO.”FINALIRUUNT223N¥1) 1ARDA nSTLé'.EU. 99999. Auk; FLODAYAII) .EUo -9999.>co To 99' " IFCDSTUATA(J) .[0. 99999. A0h. DSTDAVAIJ} .EQo -9999A)GO 10 99 COST:CJSTZFLODATA(1)¢DSTUATA(J)”“‘"" ""””*M””M"m'""”""””"" 99 CONTINJE " PRINT 102 CuST 109 FORHATI1H ,/¢ THE TOTAL COST or THIS SOLUI10N IS w,F23A1) ”’ PRINY 1033KuhnT24 '” "' " " 103 F0R1A111H ,A A TOTAL OF *AI9At PAIR ASSIGNMENIS HERE ELIMINATEdoi) ' PRINT 105 KOUNT25 ”“' " 105 FOHAATIIH ,9 A TOTAL OF «A19AtT1ES 1N THE SELEcT HAIRIX HERE RESQL 1ve0 RY RAVDUN CHOICEAA) ” ” PRIIT 139, KoUAT30 739 FORIATIlq AiTHE'cRITERTA MATRIX‘HAS"USBD To “A‘E A PARTIAL ASSIGNH lFNI A TOTAL 0? oAISA‘ TIMESA') PRIAY 974 KUINT31 ”“"'" "‘ 974 FoR1A111R ,ACLUSED LINES NERE ENCOUNTERED A TUTAL of wAISAaTINES-A TNfivsTIMEFIA) '"' "”" T=(TNt~-TOLU)/1f00o PRINT 147,T ,WWML”.. _1H_TW.H.H._N_ L. -w_.‘mw -fl,wh- 147 F0R1AT1tTIME SPENT 0A 1A1$ SHLUTION HASVIFjb AA-becowus.*1 ‘ TOLDETVEJ‘" " 39 CONTINJE 1 G0 11y SDI _“ .w .. ”LL "WWW“, ,A ”A M,m_w- _ 1"",,.-wnm-m 197 PRINT 198 ' 19R FORIATT1H”59CRITERIA VALUES ARE TOU LARGE 70 TEST. 9) '“‘w 601 CONTINJE Go To 108m“-1 m..._flv.wmmw ”9 Am - - - - .- . ”A .- . 107 PRILT 109 100 F0R1AT1A AN ERROR WAS MADE READING 1N FLUN DATA. *1 10R CONTIHJE """ . . F [\J D . . --. ,. - .-.._ ..- -..,... ...“ ___ .. _ .. -_ .7 . ‘ . . —- ---«cwo-m. -—-~._.._ . 4— - - _ 9--.. -..--—— ...-A _ . . ,, “71». . ,. . --- _ ...,, .. . __ __,- hrwr--v-r~- “aw-n A 1’ - APPENDIX VI LISTING OF 4-C PRUGPA‘ 3R0C4c _ nlnenSIOV FLODATA(351).FLUHC1(6S1).FLOHCZ(351).OQIGNC1(8)aORIGMcz(, 18)o0RIJND(8)aTALLY(7n2027)onTUATA‘SSL’o057L1‘351’aDSTL2(35 ).SOLU 2T0N(?8o28).XOUT(702)aCRlTERA(21327)iCONFLCT¢28p28).SELECT‘2;12);FI smaL(28.2a),UNRAhKuc7o2.27)aIfIxc(25).IEIXL¢25)aXOUTLOG(702) EQUIVAgEVCE (uNHANKD.TALLY).(F1NAL.SOLUTON) ”‘ INTEGE* ASSlGNL,ASSIGNC.0HIGN01,ORIGNCZ LOGICAs XOUTLUG . TOLD=T1MEF(4) X=TINEE(5) CALL RANFSET(X) RANDN03RANF(-1) PEAD 6U0.NUanH son FoRmaT‘IZ) C [TERATE FJR A TOTAL OF NUMBER PROBLEMS. nobg1 <0JNT99s1,NUMbER C RFAD AND *AVK FLUw AND DISTANCE DATA. READ 1nMaN.H1,KOUNT1;KOUNT2 FORMAT(515) no 2 Kduvrs's 1.2 9 1:n PEAD 4:(ORIBNC1(J).OHIGNC2(J):URIGNUCJ).J=1.8o1) FORMAT{8(I?.12aF6.1)) DO 5 J3108 m I=I*1 _ Irconxawcch) .LT. ORIGNC2tJ))uu To 6 s UNRAVKO(I:1)BORIGNCZ‘J’ UNHANKU(I,2):QRIGN01(J) $ 50 IU 7 . UNRANKU=BIG~0 s FLowc1(1)=x1 s FLowczcx)=xzf 14 CONTINUE $ IF(K0UNTS .EO. KOUNI4)GO TO 15 ' 1A UNRAMKU(<0UnTS.1)=UNRANKD(K0UNI591.1) $ UNRAVKDCKOUNTS.2’IUNRANKD( 1K0UNT5¢ ,2) S UNRANKD(K0UNTSo5):UNRANKD(KOUNY5¢1,3) ' KOUNT5= OUNT5+1 IFfKnUVT5«1-KOUNT4)16:16:15 1S KOUNT4=KDUNT4.1 IF/1ooo, W PRINT 146,T FORMAT‘t TIME SPENT QNHERngmlfiARY HORK was '.§1¢.3qrnsgpqnu§,-1 TOLDITVEH LASTDE’sfl $ MAXDEP0299999,’ 7 4* KOUN724IU s PRIHT33,KOUN76 s KUUN72530 FURNATSIKI1H asaxaf‘SQEUTIUN Npflfiéfinf'16’ KOUNTSUIO $ K0UMT31:0 .. ... ”..— 186 c {VITIALIZB NATNIcEs AT ZERO. 30 2009 35 34 37 36 C REVISE TH; SOLUTION MATRIX T0 HELECT FIXED celeRs. 1n05 1n06 1007 C SFLECT NEXT DIAGONAL OF QUA 4n 1n26 KOUNT1'N+1 D038121,KOUNT1 D069J=1,KUUNT1 SOLUTOVIISJT=CT CONTINJE CONTINJE KOUNT13Nit2-N 0034191o<0UNT1 DO 35 J'loN TALLY(T JT'Qa CONTINJE D0 36 1:1, N no 37 JI1,N CRITERA‘I; J)=09' coNTINJE K0UNT730 S KOUNTloa-i $ KUUN|9=91 IFTRELUp .EQ. 1)GU T0 40 ...-.. IrtNRRle .to. oTGO T0 40 S 1:0 121.1 s 1rc=1r1xc<1) S IFL8IF1XL(IT 001006J=1,N {S iFuU T0 111 S 1F(CUNFLCT(1o2) .NE9 0o’G 10 Tu 64 ‘ IF‘CUNFLCT‘le) .Nt. o.)Go T0 62 s KOUNTlsfo S Go 10 63 111 CONTINJE $ IF(CUNFLCT(1.1) .Ntu 0.)GO T0 69 5 IF‘CONFLCT‘1azJ .NE. 1 00’60 T3 62 $ KOUNT1530 S GU '0 63 51 J=J+1 t IFGo T0 66 67 L=L.1 t F‘SOLbroNtL'J) .Nto 1’b0 To 68 3 LABt R. ‘8ELR‘1 CONFLC'(LAHELR+1:1)3L S lF(LABtLR .GE. M1)no I. 6 69 CONTINJE 3 IF(L .LT. N)GO I0 6/ 5 GO TO 65 66 L=L+1 h IF(SOLUTON(L.J) .EU. U.)GU T0 69 IrtL ,-T. N)Gu TU 66 3 GO '0 69 6° CONFLC?(LABELC:1)=L $ CONFLCI(LABhLC02):J t GU T3 112 65 CONFLC'(1aLABELC*1)=J 51? IFuo To 97 s Irun To 79 ASS GNu=SELECTt1o1) $ A S UNL=5ELECT(102’ ; GU 10 106 11g CON INJE $ 1F(Ruw .F0. 1) U IU 119 s KxLAbELR 5 GO To 120 11° KzLARE-C 12n CONTINJE $ p01731:1,x s IF(cUNrLCT(1.3) ,GT. SWALLNO)GO T0 12 11 $ 1F(CDMFLCT(I:5) .LT. SMALLNU)GO T0 122 IrcL ,se. M)GO To 121 s L2L+1 3 ShLECT¢La1)=CUV£LCT(lo1) SELECT‘L.2)=CCNFLCT(I.2) s GU Iu 121 12? L=; % DMALLh0=CLNPLCTCIa3) $ StLECT(1o1)=CONFLCT(l.1) s ShLECTtloz 1)=L0NFLCT(I.2) 189 121 (HNVTINJt IF‘L .=0. 0)GU T0 197 IF‘L .JT. 1)Gg TO 79 S AsSIGNc=SELEcY(1.1) 79 RANDNO=RANF(- G X: -L $ XSXtKANUNU S I¥x*1 6 ASSIG~C=SELECT(!.1) ASSIGNL=SELEC%(I.2) $ K0UNI25= nou~725.1 104 CONTINJE C REVISE TH: SOLUTION MATRIX l0 REFLECT IHE NEH DIAD ASSIGNMENT. P05 131,\1 lF‘gOLJTON(IpASSIUNL) .NE. 0.)uo T0 80 s SOLJIDNCI;N¢1)ISOLUTON([p 1N+1)+1. i I$(ASSIGNL .NE. 1)b0 to 60 SOLUTUV(V.1 I)=30LUT0N(N¢1 I>+1 80 SOLUTUVtt. ASSIGHL):3, DOU1J=1,V $ IF(SOLUTON(ASSIGNCoJ) .NEo oo)GO '3 81 S SOLUTON(N*1oJ 1)=SOLUIOVno T6 88 s KUUNT14=0 5 GO 010 4a - an KoUN7203<0ULT, 0»1 $ IF(KOUNT .GE. 2)GU in 46 S R0H81. $ 00 T0 48 37 CONTINJE s 1F. 601C0NTINJE GO TQ 108 1 ' v0 u 'vvvm» .--r' v - 1 —~ v-v‘wiavnv- Inn-- =er—‘vv'riw ' - - Jam '- ‘m r”. ‘0” -v\. w.~--vr w ~- w—u—q ...,... .¢-. ‘ u 107 PRINT 109 109 FORMATIA AN ERROR WAS MADE READING IN FLOR DAIA.') NTINJE 108 co END - ... a- ‘ o ‘.« 4" N-V‘ 1. “D 0' “mi " L. ~— - ... ow - '- .. .. ' h .. .. _ Q .... .— ..- ... - ...— - A "lv’ cw" .nvv :va wwn- H "vb. l-n-wv-II~Onr‘-fl' -‘ ' r ' VFW-V9 v‘n'mflWHQO-vn’ ‘13-.me .It-1 'fl.‘<"‘"fin finooovzv‘fir'“: n'~ A- l' . n _v ...- ’ “ w "I ... -~— “5. W ' .. - 4 - ... ' _.._ ..- - - ...... C. ... ._ ~.._ _ _, - m .. _1 .. , -. .r'u— - a ,5- ... tux: - "l-‘p‘t‘.A - . my... -. g- . -—.-~u -—-- v1, 1.. 1-‘1'.\1r~—~«' up r04. - Iv “ ‘pr-vvw-Huw-‘ 'EC'A'CV'.‘ vr -—:—~ I C ‘v' "Htic'. .a- ct -.u!I-v - -v - q ., . .—..._. \ - .. - .. - .. a u ur- p an - m n v- wrnl‘ “ o ‘P H n. r' —- ‘ no w~ «...,... ,. _.__, _ .. .... .. _ Q -7 ,. ,.-.,-.1 m ~10 - u-uv- o m y"\ In ‘ -4I"-~« *‘C” a w ‘0 “I Q; “0“ <\ II.— a. -... -1 _. ; .... ’7 .... I 7.-“ . . 1...- ..-... .... .....- _-. I ....~ .--—1.»..- V ...... MW.-~....~ V. -v~~,--‘ ”...-...... ...-.---.... v. .— ~— ”...... “...-..-...- _v o - " vr‘ our '- w an . In“! ' raw m. '9'! a..-n ct " 3' 4.71"“ ill v- n a" ‘4' h - a -\-1 a _ ... ...... ... .... - .. .. ...,... ...... .. .. .... .... ...f V —- — - n. _. n. s. ...-...,. nan: ~m" -.-—“1.1, r~ ,. . 41- o ' ,.».'.-.“ - no , a... A L . -7 - .. ,v.. «...---1 .u...‘—..- ...‘.-.“.~ ... - . .11. 1‘. . .-.«M--..I .. v‘u—w. —. .1 .1.— “wt-WW '0- MW "-'-fi~"4 - -1~ n. ...—...r - , “HI" [‘5‘- APPENDIX VII LISTING OF 5-B PROGRA* pROCSB ”DTfiEfi§7fifififofiifITKEYTFCOWCIK6673FC6WC§T63$I3RIGNC1(8).OWIGN02(8)o‘" 10RIGND(8).TALLY(144,144).DSTUATA(66)aDSTL2(66’oSOLUT0N(13a13’oDSYL 21(66).CRITERA(12.12).CONFLCTc13.13).SELECTc12.2).FINAL(13.13).UNRA 3NK0(66o3).CELLTIEtaoo.2)oItIXC(1a)aIFIXL‘10)aHlNTALY(12a12)oCOHBIN 4‘11).SCORE(66.66)oSTORE(3a300)oINDX(4012’ EQUIVALEVCE (FlNAL.SOLUTON) 'INTEGE?"ISS[GNC;ASSIGNE;ORIGNCi.ORIGch.PHI}PIVUT)cUH§YNWWWW LOGICA; TALLY.5C0HE TOLD:T1MEF(4) X:TIME€(5) CALL RANFSET(X) RANDN0=RANF<-1) w‘WETD'"SUD—{NU}?!E3E17W W ‘ ’ " 60¢ FORMAT‘IZ) c IYERATE Fun A TOTAL or NUMBER PROBLEMS. 00601 *0UNT9931.NuMaER C READ AND RAVK FLOW AND DISTANCE DATA. RE‘D 1aM. N. M1. KOUNT1.KOUN|2 ”iTOWHITTm)”“““”” " W” DO 2 KJUVTS = 1.2 $ [=0 3 READ 4a(ORIGNC1tJ).ORIGNCZ(J).URIGND(J).J31.B.1) 4 FORMAT‘8(12.12:F6.1)) DO 5 J31.8 S I‘i‘l IF¢0R16N01(J) .LT. ORIGNC2(J))b0 To 6 s UNPAVKD(lp1)IORIGN02(J) UNRANKD(T72):0RIGNC1(JT I GO TO 7'“ '”‘ a UNRANKU(I.1)=0RIGN61(J) s UNHANKD(!.2)IOR|GNCZ(J) 7 UNRANKU(I 3)=0RIGND(J) IF¢I. s KOUNT5=J S x1IUNRANKntJo1) S xzaUNRANKD(J02) 1; CoerNJE ' H IF‘J onT. KOUNT4)DO TO 10 “'“‘”‘ TiT?I'$“TFTKUUNT3“'EUT'ITGO TO TT 5 DSTDITA(IIiBTBflU—UmvsTEITT7€XI 057L2(1)=X2 3 GO To 14 1x FLODATA(I)=BIGN0 s FLOHC1(I)= x1 5 FLOH62(I)=XZ 14 CONTINJE S 1F¢K0UN75 .Eo. KOUNT4)GO 10 15 . 16 UNRANKU(KOUNTS:1)=UNRANKD(KOUN15¢1.1) s uuaANKo(KouN15.2’suNRANKDc _- 1KOUNTS*1.2> S UNRANKD(K0UNYSo5’8UNRANKD(K0UNY561a3) WWWW'WUUNTSVKUUNT591 “”“”””“ IFtKOUVT5¢1.KoUNT4)16.16.15 15 K0UNT4=K00N14.1 IFCKOUVT4 .GT. 1)60 70 9 18101 17 CONTINJE s IF(K0UN73 .Eo. 1)GU TO 13 s DSIDATA(I)8UNRANKD(1:3) ‘”“ ’“DSTEIITTEUNRINKDri;fT“$“DSTL21IIiUNRINKDK;;2T"I"RU”TU"§"“'“”““““” 1“ FLODATA(I):uNRANKU(1.3) S FLUHC1(I)IUNRANKD(1a1) FLOKC2(I)=UNRANKD(1.2) a CONTINJE C CfiHPUTE LDHER BOUND. L=(N**Z- V)/2 S I=1 s JIL S KUNT=O S BQUNDso. W99n CGNTINJE I‘( FLUDKTIco TO 3010 shin INDExlf JTEHP= COMBIN(PIVOT) IMP+ITyDEYiWW‘ ""” -1. " -— -—---~--—-«-~-~.——-.....-~.. 3n19 COMBIN(I)=JTEMP+INDEX1 GO TO 3012 3010 CONTINJE TNEH:T1MEF(4) T:(TNEd-TOLD)/1QOU, --- ~~mm-~..—.P_.T——Wg;—T-c ..— m- ~ -———_.. . --.- 7 ..,- --.—... ---_—. ~—- -.....___--._.. n.— 146 FORMATtt TIME SPENT ON PRhLIHINHQY HORK HAS 9.!16. 3.9 SECONDS. 9) TOLDBTVEJ GENERATF h YOTAL 0F KOUNT2 SOLUI;0NS FOR THE PROBLEM. 0032K0JNT6=1,K0uN12 _ KOUN724=0 s PRINT 33¢KOUNT6 $ Kcuuv13a9-s KQJNT2530 _KOUNT3080 S KOUNT_;=OW S KOUN72/30m 194 _w 33 FORMAT‘///1H .58X,9 SOLUTION NUMBER 0953) C” INITIALtzt MATRTCES AT ZERO KOUNTza-o “KdUNTiiN31”" 0038131 KOUNT1 00394.1.K00NT1 39 SOLUTOV(I JTIQ. 1‘. . —_.__.__.—_.._.__..AH ._..._._ _, ___,._ ..-...___-- -..— . H. .-1.._.‘ _-__.... ...-“W— .--.V. 3! CONTINUE KOUNT1'N992 00341-1.K6UNT1 _00354-1, KQUNT1 W39 TALLY(I JTIo 34 CONTINUE D0 36 181 N 1 DO 37 Jul, N H1 MINTALV(I J)'O 37 CRITER‘(!. J)_'Oo ’” 35 chNTTNUE ““ KOUNT1'(N9-2-N)/2 00124x=1 KOUNT1 DOiZSJ'iaKOUNTlufi ' 125 SCORE(I*J)-o’ W124 CONTINUE n~UETECTiU S KOUNT7T= TFTNBREIK .20. 0100 T0 40 8 Tag C REVI§E THE‘ SOLUTION Hilllx '0 REEECT FIXED CEH'ER‘A 1005 III¢1 9 IFCIIFIXC(! 5 IFLPIFIXLTI ”w""w*m”UUIOOBJiITN“S“TFTSO UTUNIUTIFLT". (”UTTGDWTUW1UIS _ SOLUTOVTJ.~-1T-spLUT0NTJ.N9LT+_L 10 06 SOLUTOVKJ YFL)'3 0010074-1 N s !F(SOLUTON(IFCaJ) .NE. 0100 TO 1997 SOLUTUV(N91.J)'SULUT0N(N31{JT*1 - 1007 SOLUTOV(IFC J)-3. “”“SOLUTOV(TFC IFL>-2. I SOLUTON(lFUiN91D31000000- 3 80LUT°NTV*t'!FL° _1I1pngpo. s KOUNT19aK0UNT19+1 s TrcKougT19 6E. N-1)go T0 02 V IFTI .ET. NBRFIXTGOTTO1005 5 Go To a c SELEcT NEN OUAuRuPLET FOR ELIMINATION Pngn THE CELLTIE MATRIX IF AVY A " i N I . "- "N a D c yer ELIMIVATED HAVING THE LARGEST FLOW-DISTANCE Paooucr. ”F11BELY‘OF'THESEW—oUHJ‘R‘UFUETFTN‘THEW‘_CU'LI’TTE c RANDOM CHOICE. "“‘“**”36‘K0UNT020“3“K00NTTE1 L"UA"'PL N07 0. H _ u-_1 W «STQEEMI!§_-11 MATRIX AND flfiKE A ».__._.-.._..—.._—.—w.—.__........_.._.- 1, . 24 ._..- . KOUNTzluKOUNT27-1 s TTTKOUNT27 .LE. 00 To 997 009961‘TVDICAT9KOUNT27 s CELLTIE‘I013; iELLTlETTK191) 996 CELLTIETI.2)-CELLTIETI .2! “ IFTK6UVT27 .Eo. 1)66’ 995 00 TO 135 ---—RE-2,_ -__-._,__.~.—..._.._.9,___,_.,___.._ ._ ,..._.1- _ _. ..1 . -.._- “M"“VWCUNTWUE BIGNoI-99999. S KOUNT19(N992-N)/2 8 Tue S KOUNT27sg 125—T'I‘1 T’J814I:L= -0 129 IF‘SCOKE(10J))GO T013 m 139 CUNTINUE S IF(FLUDATI(97‘D8TDATA-IJ’W .L'o BTG~0)GU 7°“151 IF(FLUUATA(I)tDSTDATl(J) oEQI 510N0380 T0 M'HMFKOUN7278ffls BIGNO:FLODKTI‘I)*DSTUAY‘(J,'s 30210-133 11.- 132 CONTINJE S IF(KOUNTZ7 .659 300,00 T0 133 K0UNT27:K0UNT27.1 -. H.___~-_ ... ..._. ___m--—_H - .‘_..._..._._._. -2-.._ —-—-—_.-.--. .... _ --- - - -92-... ...-1. - -22- . .. 1... . .. ._ .... .1. ..-2- o-o . .- ...—..., 195 3333 CELLTI=CK9UN727 1)=1 s CELLTIE(KOUNT2732)tJ s 1:131 C 1(128 HMCIIF.O¢JC1(KOUNY1") s ""62- FLUHCZ‘KOUNTID) ”,1,“ C C C” C IF‘L .VE. 1)GO T0 134 S X53FLUUATA(I)'DSTDATA(J) 5 GO TO 130 134 CONTINJE s IF(FLOUATA(I)*DSIDATA(J) .LI. x3>au Ia 131_s GO TO 130 130 Jgd¢1 t IF(J .Le. KOUNT1)GU to 129 131 CONTINJE 5 IF(1 .LT. (Nch-N)/Z)GO TO 128 IF‘KOUVT27 .NE. 1)GO T0 135 995 1:1 3 no T0 136 {3a RKNDNoaRANF(.1T 3 XEROUNTET S XixcRANfiRO S Tix.1 KOUNYzb- KOUN726+1 136 KOUNT1086ELLTIE(I 1) S KOUNT9: CELLTIE(I 2) S INDICATSI SCORE(‘0UNT10.K0UNT9)=1 KOUNT248KOUN724¢1 ENTER TALLIES RHERE APPROPRIATE HMLlnDSTL1(KOUNT9) S HHL2= USYL2(K0UNT9) IF(SOLJTON(MMC1,N+‘)‘.EG. 1000000.)GO V0 1020 IFCSOLJT0N(MH02.N+1) .Eo. 1000000.)GO 10 1020 IF(SOLJTON(N¢1,MML1) .EQ. 1000000. )60 To 1020 IFGO_ To 5926 CONTiNJE ”“ "”""*”””m”"'”“ HIT:0 DOSOQB‘L'loN IF!SOLJTON(N¢1»KL) .EO. 1000000.)GO To 3023 IF‘KL .EO. MML1)GU TO 3028 0050291631,9H1 "TCOMEC?EEH§TN(KEY- ...- v v . . , , . . . . « .. ,~ . ., '-~--wv—-’ ...--- --- 196 JC‘ISTARTC+ICONPL-1 JL'I§TKRTT¥RL41""M IFITALEYIJC.JL))GU TO 3029 IFISOLJTONIICOHPL.KL) {NE.,0)GU YO 3029 Go To 3028 3029 CONTINJE HITzHII¢1 “‘"”“TfrFTT_TGET’TIRGETSGU"T6”3036”””""’“'"”” *' 302R CONTINUE 3023 CONTINUE 3021 CONTINJE GO TO 3037 3035 KOUNTBIKOUNTBSl S SOLUIONIMH011NHL1I'1 “”""“'SUEUTUVTVHU“.N3fi'7 SUCUTONTHHC13N91T31 ““ “”“” SOLUTOV(V¢1,MML1)= SOLUTONIN+1:MHL1)¢1 3037 TARVEchz 3 MMC2= NN01 ‘ 3033 MNClaTARY TARYaMWLZ S MMLZ'MMLl 3034 HMLlarfiRY ' 'C""TFWNUWCWTVCE’IN SOLUYIUN "HKSWUCCURREDFBY'EEiHINTTTNEETNTgfiaUKUWUFS*”m C LET. ELIMINATE THE NEXT ONE IN THE DIAGONAL; IF‘KOUVTB .Eo, 0)GO To 40 S 80 To 808 C CHECK FOR UIAD INFEASIBILITY DUE [0 THE ELIMINATION OF A OUADRUPLEI C IVvoLVING A PREV10usLY AssIGNED DIAD. 1020 CONTINJE s 0010091-1.2 5 001010J-1.2 ""””IFTSU“UTDNTNHc1 HMLET- .NET 2IGO“YU’1011 3 1FI$OCUTONIHNC2INHE23 1E. o.)50 TO 1011 $ SOLUIONIHM02;HHL2)=1 SOLUTOV(VM62.N31)= SOLUTONIHH620N01)01 SOLUTOVIN¢1,MmL2)= SOLUTONIN+1:HHL2)¢1 S KOUNTBIKOUNTBO1 1n11 TARYBH‘CZ S MMC2=MMC1 101nMMHC18TARY ' ’"ffim {$4311.23 MELT ” """’ ’"‘" 1n09 MMLlcT‘RV IFIKOUVTB .EO. 0)u0 T0 40 C SEARCH TH: SOLUTION MATRIX FOR CONDITIONS DICIATING ONE OR MORE DIAD c ASSIGNMENTS. _ Rn? KoUNTl430 $ KOUNT15=1 S KOUNT16=1 S KOUNT20=0 ”U"“FZKF"I'RIVUUNMEFBTEE“T§“TU‘RHETHER‘RUW§—UR"CUEUWN§"TRE'SEARCHEfi C FIRSY. LJOK FOR AND RecONcILE DEMANDS MADE BY CLOSED LINES PRIOR C Tn SEARCHING FOR LINbS NIYH ONLY ONE REMAINING FEASIBLE DIAD# RANDN0=RANF(-1) . IF(RANUNO .GT. .49)GO Yo 47 S House. I GO TO 40 47 R0W=1. ”‘75”51” .....- """""”""" ""“” ' W" PRINT I78,Row 77R FoRMATI1H .F5.0) 0049K0JNT121.M DO5OJ31. 2 Sn SELECTIKOUNT1 J’gfl' “IB’CENTTNUE”””"’ "“"“""” ' ”‘ ’“‘“”""'“‘”“‘”"” L:"1¢1 DO51k0JNTl=1oL 0052J310L 52 CONFLCTIKOUNT13J):0. 51 CONTINUE ...- -- _— 4.. -1 .- . . -. .7- . A771. -... .. -1.1. - V I .. “av—.... -—-———_A——‘1--.- ~nw¢—.-_—-.—.—¢“ A..- .-A ...- . —. -——-——o—- -_ “*7 -.. --...--.—q—....—. .. ..- “a..-‘q .2-.....-. - H - - . +,.....q.,..m.. no “‘{15”I?YLA8:LC .GE. MI100—TOfl70mm"‘"' 197 T4E CQNE&!CI"_N518159fnlfififl§f§§mlfli9wllwlfi§.DIQEEMLYALUESH*5 S'OREDW_ Iv THE C602192922$.K99NT1£!0 5 09 F0 63” .+ _. . r 6? [:1 i lFGO F0 116 s K=LA9ELN 5 GO To 117 116 KaLABEBC ‘ 117 CONTINJE s 001151=1.K_s K0UNT12=CONFLCT(I.11 s KOUNT18-CONFLCT(I;2 1 . 1131C0NKLCV(Io3)IcRthRA(KOUNT17aKUUNT18) ... .7.-, , _ .....- -.-—.-. Irtx .55. 1160 TO 114 s‘ROUNTSUEkounfioo1 " RECONCILE_CONFLIcTING DIAD ASSIGNMENTS BY CHOOSING_IHAT DIAD HITH THE LouEST Co-ocro~acrowmow>o~ocrow>owoo~ocrummunmxnuururmh* x ’ l '0 “(I Problem 5e (continued) «yo . Vom~4owmc1ocn\401mtso p A D I» I l I unna-5615abafith “u.- - - ...—.... . ' 3"“ ‘36" '3‘" ' a. 6' 35 3? '0 .1. 15. fik 5 3 H“ '01“ 5 9 7395 7‘57 10 7333 W6 7 '090 6 , ]_‘ ‘090 6 9 1330 “"6 10 190 7 8 490 7 9 .690 7 ..-_. 1°___'-'_Q-1.9 3 ”000 5. --.1° 1..-:9'0 9 16 U 490 L fik _ .BHM___IE_ 9 2 10 ,1 8 g “1‘. 13 .' f : t . canawuxflqO%>O~Q P. AQMJuAJMIoKanaunvnamrvn>utdrswwApaH+4+4994»&H94+3Htavtptdvsw+a k? E. fik 7 99999.00 9 9.30 10 o0,00 11 -0.00 12 -n.00 3 9.30 w— £2.12 2 99999.00 3 -fl.00 4 -U.00 5 -n.00 6 .0.00 7 -0,00 8 -0.00 9 -0.00 10 -0.00 11 -0.00 12 -U.00 13 .0000 14 .n,on 15 "“000 16 -U.00 17 .u.no 18 .0.00 19 00.00 20 .n,on 21 -n.00 2 -fl.00 23 “0000 24 .fl,00 3 98.30 4 98.30 5 98,30 6 'U.no 7 44.00 8 44.00 9 44.00 10 «4,00 11 -0.00 12 -Uon0 13 -0.00 1‘ ~n.00 15 .0000 15 ~0.00 17 -n.nn Problem 15° (continued) SM '0007 9999900 999990U 99999.0 9999900 9999900 99999.0 99999.0 99999.0 99999.0 9999900 99999.0 99999.0 9999900 99999.0 99999.0 99999.0 99999.0 99999.0 9999900 9999900 99999.0 99999.0 16.5 19.0 21.5 2400 2605 29.0 51.5 34.0 36.5 39.0 3990 36.5 3400 5105 79.0 213 l. L. fik 7 9 9.30 7 10 «0.00 7 11 ”0000 7 12 -U.OO 8 9 99999.00 8 1” 13.02 8 11 13.02 Problem 16p P. Jhblhahblbb.5lbA(HOWUCROHROHOHNOHOKdOWU(d000H309GHhRONHUhJNHvha k: i. E. fik 8 12 4.15 9 10 13.02 9 11 13.02 9 12 4.15 910 11 99999.00 10 12 40.20 11 12 40.20 £95,319 -n,oo 26.5 '".00 2400 ‘0.00 21.5 .0000 1900 -".00 1605 '".00 1400 -U.00 9999900 99999.00 16.5 -".00 19.0 56.70 21.5 .0000 24.0 ““000 2605 .0000 29in .u.00 31.9 29.30 34.0 29.30 36.5 00.00 36.5 .0000 34.0 10040 31.5 10.40 29.0 10.40 26.5 10.40 2400 29.30 21.5 29.30 19.0 -0.00 16.5 '0.00 1‘00 .0000 16.5 00.00 99999.0 99999.00 16.5 56.70 19.0 .0000 21.5 .0000 2400 -0.00 26.5 .0.00 29.0 29.30 31.5 29.30 34.0 on.00 34.0 .u.no 31.5 [E \JVVNNOOO‘O*O‘O‘OO‘O~O~O~OO‘O‘O‘OO‘O‘WWU‘UWU‘U‘U’U‘WU'IUYU‘U‘UIU‘U'Ivmbbbbbbbbbb 52$ fik 15 10.40 16 10.40 17 10.40 18 10.40 19 29.30 20 29.30 21 -0.00 22 -0.00 23 .0.00 24 -0.00 6 56.70 7 .0.00 8 .0000 9 .0.00 10 .0.00 11 29.30 12 29.30 13 “0.00 14 -n.00 15 10.40 16 10.40 17 10.40 18 10.40 19 29.30 20 29.30 21 .n.00 22 .0.00 25 .0.00 24 -0.00 a 44.00 9 44.00 10 44.00 11 88.00 12 88.00 13 85.00 12 85.00 19 -0.00 1. .n,oo 17 ‘0000 18 .0000 19 .0.00 20 .0.00 21 -0.00 22 0.0.00 23 00.00 24 .0.00 8 99999.00 9 .0.00 10 .0,00 11 -0.00 12 -n.no 214 Problem 16p (continued) Er 29.0 26.5 24.0 21.5 19.0 16.5 14.0 16.5 19.0 90999.0 16.5 19.0 21.5 24.0 26.5 29.0 31.5 31.5 29.0 26.5 26.0 71.5 19.0 16.5 14.0 16.5 19.0 21.5 99999.0 16.5 19.0 21.5 24.0 26.5 29.0 29.0 2605 24.0 21.5 19.0 16.5 14.0 16.5 19.0 21.5 24.0 99999.0 16.5 19.0 21.5 74.0 26.5 l’éi cooooocooooooeomwmmocmaammmozamonmmvuvvuvvquuuu Hr+9+sw+tuw4bn 000000000 322. fik 13 .0000 14 -0.oo 15 -0.00 16 ~0.00 17 -0.00 18 .0.00 19 .0.00 20 n0.00 21 .0000 22 “0000 23 ~0.00 24 .0.00 9 99999.00 10 ~0.00 11 00.00 12 .0.00 13 .0.00 14 «0.00 15 p0.00 16 .0000 17 .0.00 18 .0.00 19 «0.00 20 «0.00 21 «0.00 22 .0000 23 -0.00 24 .0.00 10 99999.00 11 w0.00‘ 12 ’0000 13 .0.00 14 .0.00 15 .n,oo 16 a0.00 17 -o.oo 18 .0000 19 ”"000 20 .0000 21 «0.00 22 c0.00 23 -0.00 24 «0.00 11 ~0.0n 12 '0000 13 .0.00 14 -0.00 15 _0.00 16 .n.oo 17 ~0.00 13 -0.00 19 .0.00 djfl 26.5 24.0 21.5 1900 16.5 14.0 16.5 19.0 21.5 24.0 26.5 99999.0 15.5 19.0 21.5 24.0 24.0 21.5 19.0 15.5 14.0 1..5 19.0 21.5 24.0 26.5 29.0 99999.0 16.5 19.0 21.5 21.5 19.0 16.5 14.0 16.5 19.0 21.5 24.0 26.5 29.0 31.5 99999.0 16.5 19.0 19.0 1605 1490 16.5 19.0 21.5 24.0 Lil; -0.00 .0.00 .n,oo .0.00 .0,00 99999.00 .n,00 .0.00 p0,00 .0.00 .0.00 -0.00 .0.00 .0,00 -a.oo -0.00 .0.00 '0.00 .0.0n ~0.00 .0.00 -0.00 90.00 .0.00 -0.nn -0.00 -n,oo .0.00 .0.00 .n,oo 99999.00 15.60 15.60 15.60 15.60 .n,no .0.00 85.00 31.25 31.25 -0.00 15.60 15.60 15.60 15.60 .0,00 “0.00 35.00 31.25 215 Problem 16p (continued) djE 26.5 ?9.0 31.5 34.0 99999.0 16.5 16.5 14.0 16.5 19.0 21.5 24.0 26.5 29.0 31.5 34.0 36.5 99999.0 14.0 16.5 19.0 21.5 94.0 26.5 29.0 31.5 34.0 36.5 39.0 99999.0 16.5 19.0 21.5 24.0 26.5 29.0 31.5 34.0 36.5 39.0 99999.0 16.5 19.0 21.5 24.0 96.5 29.0 31.5 34.0 .161 £811 £13 14 23 31.25 14 24 .0.00 15 16 99999.00 15 17 ””000 15 18 .0.00 15 19 .0.00 15 20 .0.00 15 21 .n.oo 15 22 .0.00 15 23 .0.00 15 24 .0.00 16 17 99999.00 16 13 P0.00 16 19 «0.00 16 20 .0.00 16 21 ~0.00 16 22 .0.00 16 23 .0.00 16 24 '0.00 17 18 99999.00 17 19 '0.00 17 20 .0.00 17 21 -U.00 17 22 .0000 17 23 .0.00 17 24 --0.00 18 19 .0.00 18 20 .0000 18 22 «0.00 18 23 .o.oo 18 24 «0.00 19 20 99999.00 19 22 44.00 19 23 44.00 19 24 v0.00 20 21 -n.00 20 22 44.00 20 23 44.00 20 24 o0.00 21 22 35.00 21 23 85.00 21 24 -0.00 22 23 99999.00 22 24 -0.00 23 24 99999.00 93.1; 36.5 99999.0 16.5 19.0 21.5 '24.0. 26.5 29.0 31.5 34.0 99999.0 16.5 19.0 21.5 24.0 26.5 29.0 31.5 99999.0 16.5 19.0 21.5 24.0 26.5 29.0 99999.0 16.5 119.0 21.5 24.0 26.5 99999.0 16.5 19.0 21.5 24.0 99999.0 16.5 19.0 21.5 99999.0 16.5 19.0 99999.0 16.5 99999.0 0.0 IH° ‘JPHHPH (if; QQCAQUO' CI 1,450.0 ZA'QN'N'NNN'YN‘R'N'K"\"‘\“\"\‘f\.'\"\"\uk‘u‘-“-‘F“"“I-‘HF‘“HF“ fill h N \3 'X’ C! 3‘- \J‘Y‘ Jquy—J hidfi‘l-‘C; Nr»r-H-4Lo.. 3 010V rd. in' 4 .f‘ \I ’9‘ 'J' b 11 12 13 14 15 lb 17 18 19 23 r~4-w 11 1? 13 14 15 1h 17 1‘ ('3 C) (J D 1 0.3 C) O C) C.) (‘3 C) 20.00 50.f0 216 problem 17q ‘1. E. fik 5 1:} 13.515 3 29 $0.00 4 5 1.08 4 . “.70 9 7 7,50 4 F‘ 30.00 4 Q 2.34 4 10 ;O.CC 4 1.1 30.00 4 l? 1.40 4 13 30.00 4 1.6 'n000 4 15 ‘C.00 ‘1 1“ 50.00 '1 1.” “0.00 4 19 1.F,C 4 l9 1‘.7b "1 237 .50. PU 5 9 60.00 5 7 9.?5 t) ‘f 1.15 5 9' “1.00 5 10 1.90 ‘7 11 630.00 3. 1? 30.00 i 13 -0.03 5 1‘ 60.00 5 13 1.35 5 16 00.00 5 17 ac.n0 5 IF 50.00 5 19 30.03 5 a) 50.00 5 7 6.15 b R a0.00 6 0 ~0.00 O '0 30.90 6 l? 0.02 6 16 oij,93 5 14 30.00 5 19 80.03 5 1" $0.93 6 17 6.6.30 6 I“ 1.09 5 19 £0.03 5 24 30.03 7 9 24.C0 7 9 -o,gj 7 1; 1.») 7 11 53.90 7 1? .0.“J cooccocccocmanocmc‘mmasmmmookuvuwvu IF. fik 60.90 0.95 80.00 30.33 30.00 1.55 60.03 3.75 30.03 30.00 .0000. 30.00 0.50 50.00 30.00 60.00 30.00 90.00 7.00 33.45 80.00 I0.00 90.00 00.00 7.90 90.C0 00.90 7,q0 90.? n0.00 n0,90 0.X6 17.00 .0.00 13.60 1.02 .IU.'JO 80.00 90.00 5.95 50.90 ?.75 80.00 3.30 0.96 2?.‘0 90.00 60.00 30.00 30.90 30.90 80.00 12 12 12 12 12 12 13 13 13 13 13 13 W0h0<~0JNO~GPOhJNfUhJNfUhHVhJAHVRJHE‘F‘HP‘k’Hf‘P‘Hf“FJHFJIF. h. 15 16 17 18 19 20 14 15 16 17 18 19 2.1.65— .C.00 15.00 00.00 8.40 30.00 8.00 1.04 6.00 00.00 30.00 80.00 217 Problem 17q (cOntinued) 1 .1 E. fik 13 20 00.00 14 .15..“9.75 14 16 00.00 14 18 0.90 14' 19. 50.00 :15. 16. 30.00 15 17 5.25 '15 18 00.00 15 19 00.00 15 20 00.00 r Problem 18 .1 h. fik 3 12. 93" 3 13 '0 3 14 no 3 15 8 _45 5“. _.a 4 6 -0 4 7. -0 4 8 -0 4 9 n0 4 10 -0 4 11. .a. 4 12 92 4 13 '0 4 14 '0 4 15 , 884 5 6 .0 ,5 7 .30 5 8 12 5 9-_ 12 5 10 -0 5 11 -0 5 12 8 5 13 P 170 5 1 14 .175 5 15 .0 6 7 376 O 8 -D 6 9 -0 6 13 ‘0 .b u 11 .30 6 12 80 6 13 no “6.“ 14 -0 6 15 348 7 8 910 1. L. fik 15-.wLqu12r00. 16 18. 00.00 15;.-19 .I0.00 16 20 90.00' 12 - 18 .- 60.00 17 19 7.50. 11 “2.0 -.---.D, 00.4 18 19 4.65 lauhw20.m00.00 ‘19 20 90.00 1. .15. . 7.MWMQ.1 .50 1 7 10 '0 I 11.- ,90 7 12 '0 7. .13.. 12. 7 14 -0 . 7 .. -15...---.. 1.110 3 9 12 8 ”-10 .90 8 11 -0 8. 12.1- 90, 8- 13 '0 .Bum114 H _886 8 15 '0 9 10.. 90 9 11 7.0 9 12 _ .8 9 13 '0 .91 ¢¢wwmn.2zn 9 15 10 10.9 11 -L-0 10 12 '0 10 137- .:!0. 10 14 10 .10.WH151mems0m 11 12 -0 11-" 13 -1 .-0 11 1‘ '0 11» 15 552 12 13 .o 12-n-14wmww180 12 15 '0‘ 13 14 156 13 15 '0 14 15 1320 an Ej H‘ “wk. . H' NHPHP lg, IF. I... HHHHH_ MMMNUHHHHHHH _k 15 f ik 2 76 n 3 305 4 12p 12.51. f ik ? 1n .3 IS 4 1 5 15 5 a E f ik 2 99999 3 3 4 3 5 4 6 1 1‘. f ik 2 -n 3 18 4 m S 7 6 7 7 13 8 3 9 -n 3 13 4 w 5 -n ‘6 2 44 100 142 _1_. 23 33 2? 2n fl Abauuuumummm Problem 218 Problem 20t Problem Zlu iv- (AMMNEM WV IP- If fik 1e 2 1 6 24 59 '0 14 13 '0' 28 15 i&j k& 2 3 2 4 3 4 in_ k&£ 2 4 2 5 3 4 3 5 4 5 _i. 3 3 4 4 5 i 4 4 5 5 5 5 6 a 6 7 7 8 OOQOWVOQDNO~OD H1 H- 3' ' ' H I acnassm If 0001001 If _J_. 108 140 130 9.12. 4n ,25 so 13 IP- OWNOUUAMN PPHPHH Aumbumwo uuuuuaauNNNMNNNNNNNNNPPFPHHPHHHPHHP IH oomvomou~Wr HHPHHHPPH ...; 219 Problem 23w on 3 12 '0 7 9 .0 7000 3 13 '0 7 10 -n '0 3 1‘ '0 q 7 11 '0 16000 3 15 '0 7 12 12000 4000 2 Z '2 7 13 90 o - 4 " .3 4 7 2000 g is ,3 'n ‘ 1° '0 a 11 -o '0 ‘ 11 '0 a 12 -o ’0 ‘ 12 '0 8 13 -o -o 1 15 '0 9 10 -o 3000 5 6 '0 9 11 ca 4000 5 3 6000 9 13 on '0 g 3 .'g 9 14 to u 1 . 5 .0 4003 5 11 '0 13 :1 -o '0 5 12 '0 1o 12 -n '0 5 1‘ '“ 1o 14 -o ,0 5 15 '0 1° 15 .0 In 6 8 .0 11 13 '0 '0 ° ° '° 11 14 -o '0 6 13 '° T12 15 -n '0 6 14 '° 13 14 -n '0 6 15 ’0 13' 15 -0 Problem 24x .££E -i 5- £35_ 2. 5. fig. 4 1 11 4 1 20 3 4 1 12 4 1 21 2 3 1 13 2 1 22 2 3 1 14 2 1 23 9 2 1 15 2 3 2 3 4 2 1 16 2 2 4 2 2 1 17 2 2 5 3 3 1 18 2 2 6 2 4 1 19 2 2 7 2 Iv- 55555 $5515bbb55AUUUWUUUMUUUWUQ'OIUUGWUNNNNNNNNMNNNNNK)“, 03.x- m H. W MM’U‘OMMMAbeMMMONflNmNMMMNMMbc-wavvwflvH3bMMMM$MMALNMV 220 Problem 24x (continued) [H [3‘ H1 1'“ 7s" unuuvuvuwwvuoooooo ooooooooooomwwmwmwmwwmwmwmwmu’ubao ... co ooaaooanmnwownvmuuumuwuoamnbonmwuvmvvvvmo)anmmmumml N335bébbbbNVVVI‘JV\)\)MNM.\J\3\3-\7MVMUEA\IVVMVM‘UO‘OVMUMMRMI; 9999 ”NMMHOMN 4 ...—M 221 Problem 24X (continued) 3— 1‘. f1}: .1. 11 f ik .1. .15.. f 1k 11 17 2 13 21 2 16 21 2 11 13 x 13 22 2 16 22 9 11 19 7 13 23 p 16 23 p 11 20 7 14 1§ 99999 17 18 99999 11 21 x 14 16 -n 17 19 '0 11 2? ; 14 17 -n 17 20 9 11 23 z 14 14 -n 17 21 2 12 13 2 14 19 -u 17 22 7 12 14 2 14 20 7 17 23 ? 12 15 2 14 21 2 18 19 99999 12 15 P 14 22 9 16 20 2 12 17 ? 14 23 9 10 21 2 12 1R 2 15 16 99999 18 22 2 1; 19 2 15 17 -g 18 23 2 12 20 7 15 18 -n 19 20 2 12 21 7 15 19 .n 19 21 2 12 22 9 15 an 9 19 22 2 12 23 9 15 21 2 19 23 2 16 14 99999 15 22 9 20 21 3 13 15 -n 15 23 2 20 2? 2 15 16 -r 1b 17 99999 ?0 23 2 15 17 -, ie 14 .9 21 22 2 1s 18 -c 16 19 -n 21 23 2 13 19 -n 16 20 9 ?2 23 2 13 20 7 Problem 25y A. E. fik i. 5- £15. 3- 3' £592 1 2 5 2 11 -n 5 b 1n 1 3 y 2 1?. "fl 5 7 "n 1 4 4 3 4 'a 5 8 ”B 1 5 1 3 5 '9 5 9 “0 1 6 -c 3 6 '0 S 10 5 1 7 -c 3 7 ’0 5 11 1 1 R b 3 8 5 5 12 1 1 9 2 3 9 5 6 7 b 1 10 1 3 10 2 6 8 1 1 11 1 3 11 2 6 9 1 1 12 1 3 12 2 6 10 5 2 3 5 4 5 5 6 11 4 2 4 -c 4 6 7 6 12' ’0 2 5 9 4 7 ? 7 8 10 2 6 ? 4 8 1n 7 9 5 2 7 2 4 9 -n 7 1o 2 2 8 ‘0 4 10 ’n 7 11 3 2 9 4 4 11 5 7 12 3 2 1o 5 4 1’ 5 a 9 -n —— IP- xzna. IP- “H IX 10 11 12 [W b’d\ oar.» P‘P ULQOWVF‘U\013V to! Hon u'r 13 ‘00500 -0.000 ”00300 ‘0.000 '00300 -OIOOO -0.000 -0.000 '3.303 -0.000 .00800 ”00000 '00300 '00000 .OICPO '00300 '0.000 '00000 '0.000 1.831 0.660 0.103 0.390 '00990 0.578 '00090 0.270 0.663 0.135 0.3L54 .00330 0.243 0.530 -0.7?o 0.351 0.482 2.211 0.363 0.561 0.697 IP- 10130 U1U\’U‘ 1- ‘ U\HU\J\LUWIUTA35beJib.bJ>bwbJifilbbmbbJAOJ“CHOMNONMLNOHWUHMI P0 222 11 10 11 12 Problem'ZSy (continued) fik -n 10 1o problem 26z 0.561 1.683_ 0.054 0.091 1.419 0.027 0.479 “00000 0.270 0.650 6.831 1.155 1.039 0.330 0.216 0.697 0.001 0.606 0.0?7 0.330 0.660 0.5?5 0.054 '00000 0.001 0.135 0.650 0.528 0,693 0.108 0.0?7 0.132 0.866 “00000 0.630 0,495 .0.000 .0.000 .00000 .00000 -0.300 .1 1o 11 (’ c, . .1, H. k 11 12 E. 18 19 20 7 8 9 10 11 12 13 14 15 16 17 19 19 20 10 11 1? 14 15 16 17 18 19 20 10 11 12 13 14 15 16 18 19 fik 0.366 0.0?7 0.162 0.001 -0.000 0.162 0.054 0.270 0.081 .00000 '00000 .0.000 -0.000. 0.091 0.216 0.054 0.5?8 .0,000 0.108 -0.000 -0.000 0.135 -0.000 “00000 -0.000 -0.000 -0.003 0.363 0.3?7 0.330 0.216 .00000 0.270 0.660 '00000 0.0?7 ’00000 .00003 “00900 0.297 '00000 IP- ‘C(§L~C(30\O\3CIJQ'L' 12 13 1. 15 1e 17 1s 19 20 11 1? 13 16 15 16 17 1Q 10 23 12 £11: 0.1“) .0090: 0.333 0,269 0.397 0.207 -0.300 .0.900 -0.300 0.4‘2 0.660 0.396 .00903 '0.3?0 .00003 -0.000 '00000 .00003 .00000 '0.300 .39300 0.243 0.270 Prgblem 262 (continued) '4-J\J;-‘H ...... “...,.dflf‘H—OHfi-‘HHfl'JH cuouuuwc~o~unvncmamuunun2H44r.Hwar-ku Ir. 223 fik 0.330 0.381 -3.003 -0.000 0.199 0.135 0.330 -0.000 0.270 0.495 '0.000 “00300 0.333 0.598 0.132 0.391 .00000 '00000 '00090 .00000 0.094 0.097 13 1,. 14 14 1. 1. .14; 15 15 15 15 15 16 16 16 16 17 S7 17 18 18 19 E. 17 19 in: 0.0?7 '0.000 '00000 “00300 0.001 “00300 1.186 '0090 ’0000 '00300 ’00300 '00000 '00000 1 00.000 .00300 0.330 0.363 0.0?7 0.270 ‘00‘79 3.597 2.112 224 Footnotes to Appendix IX aRelevant references to problem 1 are: Robert J. Wimmert, "A Quantitative Approach to Equipment Location in Intermittent Manufacturing," p. 95; Robert J. Wimmert, "A Mathematical Model of Equipment Location," Journal of Indus- trial Enqineerinq, IX, No. 6 (November-December, 1958) 500- 501; and James M. Moore, Plant Layout and Design (New York: The Macmillan Company, 1962), p. 180. The hypothetical problem is not based on a lattice layout grid. All four centers require equal areas; no constraints are specified° bThis hypothetical problem is provided by Peter C. Noy, American Machinist, p. 58. There are no constraints; either distance criterion is appropriate, since the follow- ing one-dimensional lattice is specified: The numbers in the grid specify the location labels. CProblem 3 is offered by Gavett and Plyter, Opera- tions Research, pp. 212-213. It is hypothetical, imposes no constraints and is based on the assumption that all center areas are equal. dGeorge Conrade supplies problem 4, Institutions, pp. 12-121. This hypothetical kitchen layout problem specifies equal center areas and no constraints. The author computes d. terms using the straight—line criterion and the following iéttice: l 3 5 7 2 4 6 8 eProblem 5 is Steinberg's backboard wiring problem, Society for Industrial and Applied Mathematics Review, pp. 43-44. .A 4 x 9 lattice is used, with centers 35 and 36 added as dummies to make the lattice rectangular. Distances are calculated using the straight—line criterion. 225 10 ll 12 l3 14 15 l6 17 18 19 20 21 22 23 24 25 26 27 28 29 3O 31 32 33 34 35 36 fTwo sources of problem 6 are: Smith, Journal of Industrial Engineering, p. 14 and Gerald Nadler, Work Design (Homewood, Illinois: Richard D. Irwin, Inc., l963), p. 224. No constraints are imposed and centers are assumed to be of equal area. The fik values are empirically derived. The author's assumption that backtracking is twice as costly as forward movement is discarded for our purposes. The rectan- gular and straight-line distance criteria provide the same results, as the author specifies the following one-dimen- sional lattice: l 2 3 4 5 6 7 8 9 10 9Problem 7 is also based on an actual case study, Smith, Journal of Industrial Engineering, p. 26. All other information in the preceding, footnote also applies to this problem, including the lattice. hThe source of problem 8 is: Wimmert, "A Quantita- tive Approach to Equipment Location in Intermittent Manu- facturing," p. 152. This problem is derived from an actual case, with distances Specified without a lattice configura- tion. The author specifies four constraints: 1. Center l5 must be assigned to location 15. This constraint is imposed for "A" procedures by creating dummy center 16 (and location 16). The term flS,l6 is made arbitrarily large. The term d15,l6 is equated to zero, whereas all other dj,l6 values are made arbitrarily large. 2, Centers 12 and 13 must be adjacent. 3. Centers 12 and 14 must be adjacent. 4. Centers 13 and 14 must be adjacent. Since the first constraint can be handled without adding a dummy center, the problem statistics vary. It should be noted that the given layout configuration will not permit the last three constraints to be satisfied simultaneously. The author intended centers 12, 13, and 14 to be assigned 226 ‘to locations 5, 6, and 7 or else 12, 13, and 14. However, the distances between these "adjacent" locations are signif— icantly larger than several other d'g values. As a result, the algorithms do not produce a feasible solution. This dilemma is best solved by equating to one all diads involv- ing centers 12, 13, and 14 which reference locations other than 5, 6, 7, 12, 13, or 14 at the beginning of the solution process. The layout configuration is as follows: Q -. 7 \\\\\ 2 l \ //// 3 \\\i 5 8 10 \\ \ 12 6 9 11 \ 13 \§ 1 l. 7 R \ 5 1Problem 9 is taken from two sources: Reed, Plant Layout: Factors, Principles and Techniques, p. 210 and Daniel J. Murphy, "Machine Location Patterns for Facility Analysis" (unpublished M.S° thesis, Department of Industrial Engineering, Engineering Library, University of Pittsburg, 1957), p. 24. The problem is hypothetical and only fik values are provided. For our purposes, center areas are assumed to be equal and a 3 x 3 lattice is specified. Center 9 is a dummy added to make the grid square and dis- tance is computed with the straight—line criteria. l 2 3 4 5 6 7 8 9 JThis hypothetical problem was also taken from Reed, Plant Layout: FactorerPrinciples and Techniques, p. 210. Center 16 is a dummy and the following pairs of centers must be adjacent: {1,2}, {3,4}, {7,8}, {9,10}, {11,12}, and [13,14]. The following lattice is constructed, with distances computed on a straight-line basis: 227 5 6 7 8 9 10 ll 12 kThis hypothetical problem of Lawler, Management Science, p. 593, imposes no constraints; distances are specified without a lattice configuration. The author's linear costs are ignored for our purposes. Centers are assumed to require equal areas. EProblem 12, Reed, Plant Layout . . . , p. 391) is hypothetical and no constraints are imposed. The following lattice is used, with distances computed with the straight— line criteria: l 2 3 4 5 6 mThis problem is equivalent to problem 12, except that dummy centers are added to level the area requirements. Two sets of centers must be adjacent: {1,2} and {5,6}. The lattice is as follows: l 2 3 4 5 6 7 8 9 nProblem l4, Reed, Plant Location, Layout and Maintenance, p. 93, is hypothetical and has no constraints. A 3 x 3 lattice is arbitrarily chosen for our purposes, with distances computed using the straight-line criterion. Center 9 is added to complete the grid. 228 OProblem 15~is equivalent to the previous problem, except that three dummy centers are added to level area requirements. The following pairs of centers are con- strained to be adjacent: {3,4}, {6,7}, {8,9}, and {10,11}. l 2 3 4 5 6 7 8 9 10 ll 12 pKase and Nishiyama provide this hypothetical prob- lem in "An Industrial Engineering Game Model for Factory Layout," The Journal of Industrial Engineering, XV, No. 3 (May-June, 1964), 149-150. The fik values are based upon discounted present value calculations with an annual rate of return of 7 per cent. Linear and backtracking costs are ignored. A modified rectangular distance criteria is used to compute djg. Fourteen dummy centers are added to equate center area requirements. The following constraints are imposed: 1. x11 must be assigned. 2. x24 must be assigned. 3. The following pairs of centers must be adjacent: {1,2}, {3,4}. {4,5}. (7.8}. {8.91, {9,10}. {11,12}. {13,14} {15,16}, {16,17}, {17,18}, {19,20}, {22,23}, and {23,24}. Adjacency is defined to occur when a distance value is 16.5. Since some distance values are less than 16.5, the algo- rithms need not necessarily return feasible answers. The layout configuration is as follows: l3 14 15 l6 17 18 19 2O 21 22 23 24 \ \ \\\\\Aisle \\\\\\\\ 12 ll 10 9 8 7 6 5 4 3 2 l qProblem 17 is derived from an actual case study, Armour and Buffa, Management Science, p. 297. For our purposes,area requirements are assumed to be equal. The following 5 x 4 lattice is arbitrarily chosen, with the straight-line criterion used to compute distances. 229 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 rThe fik values of problem 18 were derived empiri- cally by Muther in Systematic Layout Planning, pp. 4-18. The following lattice is chosen for our purposes, with the straight—line criteria used to compute distances. Center 16 must be added as a dummy for versions C-R, C-S, H-R, and H-S. 1 2 3 4 5 6 7 8 9 10 ll 12 13 14 15 16 5Problem 19 is supplied by Armour, "A Heuristic Algorithm and Simulation Approach . . . ," p. 32. The f- and d'fi values of problem 1 were changed to demonstrate the inadequacy of Wimmert's quadruplet selection procedure. tThis hypothetical problem of Land, Qperational Researchpggarterly, pp. 181-198, specifies no constraints and center areas are assumed to be equal. The author pro- vides djg values without a lattice grid. uProblem 21 is hypothetical and Reed specifies a 2 x 3 lattice in Plant Location, Layout and Maintenance, p. 106. Constraints are imposed to make two pairs of centers adjacent: {1,2} and {3,4}. The straight-line distance criterion is used, with the distance between adjacent centers being 30 (rather than 1). 230 vThis problem is adapted from a case study found in: Arch R. Dooley et al., Operation Planning and Control (New York: John Wiley and Sons, Inc., 1964), pp. 212-213. There are no constraints and all centers are assumed to require equal areas. The following lattice is chosen for our pur— poses, as is the straight-line distance criterion: 1 2 3 4 5 6 7 8 9 wProblem 23 is hypothetical. No constraints are imposed and the centers require equal areas. The layout grid is suggested by the configuration of Schneider's recom- mended solution, Journal of Industrial Engineering, p. 480. Straight-line distances are computed. l 2 3 4 5 6 7 8 9 10 11 12 l3 14 15 xThe problem was presented by Charles G. Haskins to the Computer Aided Plant Layout Institute, Milwaukee, Wiscon- sin, 1967. The problem is empirically derived and originally has eighteen centers. The f- terms are REL values. Two small centers are dropped from the analysis, since they can be added later with an "addition" algorithm. One very large center has been broken into seven centers (13 through 19) to level area requirements. The fik values are adjusted to link these seven centers sequentially in the final solution. A 4 x 6 lattice is chosen, with center 24 being a dummy. This is needed for H-R, H-S, C-R, and C-S to complete the grid. Straight-line distances are computed. Seven pairs of centers are constrained to be adjacent: {11,12}, {13,14}, {14,15}, {15,16}, {16,17}, {17,18}, and {18,19}. The lattice is as follows: 231 l 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 yThere are two sources of problem 25: Frederick S. Hillier, "Quantitative Tools for Plant Layout Analysis," The Journal of Industrial Engineering, XIV, No. 1 (January- February, 1963), 35; and Vollman, "An Investigation of Bases for the Relative Location of Facilities," pp. 28-35. The authors specify for this hypothetical problem a 3 x 4 lat- tice, no constraints, equal center areas, and the rectangu- lar distance criterion. The lattice is as follows: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 2This problem was empirically derived by Vollman, "An Investigation of Bases for the Relative Location of Facilities," p. 85. It is modified for the equal center area assumption, although no constraints are imposed. Distances are computed with the straight-line criterion. The grid used for our purposes is as follows: l 2 3 4 5 6 7 8 9 10 ll 12 13 14 15 16 17 18 19 20 APPENDIX X RANDOM STARTING SOLUTIONSa Prob- Solution lem Numbers Permutation of Locations 2 02,03,01,05,04,06 05,01,03,02,06,04 02,04,06,05,03,01 03,06,04,05,01,02 05, 08,03, 06, 07, 01,04, 02 01,04,08,06,03,02,07,05 01,07,03,06,04,05,08,02 03,05,02,01,04,08,06,07 08,34,05,l7,01,32,16,31,26,18,25,12,21,13,11, 33,22,36,23,04,07,27,09,24,06,20,10,03,14,35, 28, 19,30, 15, 02,29 2 25,19,16,26,11,12,35,34,18,02,07,33,05,17,22, 03,30,09,23,29,06,20,28,04,01,32,10,15,14,31, l3,36,08,21,27,24 3 12,22,20,l7,04,10,30,25,11,24,07,34,33,28,0l, 03,05,36,09,16,14,02,26,31,32,06,08,29,18,21, l9,26,23,35,l3,15 4 36,03,06,29,32,05,34,01,23,19,09,22,15,l3,35, 10,14,04,08,12,26,24,16,28,25,31,02,20,07,18, 33,21,17,27,30,11 07,03,0l,08,06,02,09,10,05,04 02,07,05,01,09,06,08,10,04,03 02,04,01,09,05,10,06,08,03,07 06,04,08,10,05,07,02,03,09,01 06,04,08,09,05,07,01,10,02,03 04,05,07,01,08,03,09,02,06,10 04,09,03,10,02,08,05,07,06,0l 03,09,10,02,01,04,07,06,08,05 04,01,07,09,08,05,02,06,03 01,03,04,05,08,09,06,07,02 06, 05, 02,01,03, 08,09, 07, 04 07,06,02,03,08,01,04,09,05 04,02,03, 10, 05, 16, O6, 13, 14,01,08, 10, 09, 15,07, 11 15,01, 10, 12, 06, 14, 08,07, 16, 03, 02, 13,09,05, ll, 04 b H-bwmr—abwwp—a waNwaNwawl—a 10 N aThese are the random starting solutions used for H-R, H-S, C-R, and C-S. A permutation of locations specifies the sequence of locations assigned to centers l,2,...,N. 232 233 Prob- Solution lem Numbers Permutation of Locations 3 01, 16, 15, 08, 14, 13, 07, 12, ll, 09, 05, 04, 03, 10, 06, 02 07, 08, 04, 14, 12, 11, 15, 09, 02, 06, 03, 10, 16, 13, 01, 05 04,03,06,01,05,02 06, 01, 05, 02, 03, 04 01,03,05,06,02,04 05,04,02,06,03,01 05,04,01,06,03,02,09,07,08 09,04,03,06,02,07,01,05,08 04,08,02,01,09,06,07,03,05 06,04,09,02,07,08,03,01,05 09,02,05,03,08,01,07,04,06 08,02,04,05,06,03,09,07,0l 09,05,04,03,01,06,02,07,08 02,04,06,07,03,08,01,05,09 05,11,07,04,03,08,01,06,09,12,02,10 11,09,04,02,12,05,01,08,06,03,10,07 12,01,11,08,10,09,07,04,03,06,05,02 03, 04, 10, 08, 05, 09, 07, 06,02, 12, 01, 11 17, 14, 05, 01, 18, 15, 07, 08, 20, 02, 13, 03, 04, 12, 06, 16,09,19,1l,10 04, 06, 01, 11, 13, 03, 10, 20, 08, 17, 15, 07, 19, 12, 18, 09,16,14,02,05 3 13, 06, 12, 16, 15, 07, 08, 05, 14, 19, 10, 11, 09, 01,20, 04,02,17,03,18 4 02, 11, 17,05, 13, 20, 10, 12, 08, 03, 01, 09, 19, 14, 18, 15,07,04,06,16 18 1 01,10,11,08,06,05,07,03,14,04,09,13,02,12,15, 16 2 05,04,10,12,08,15,02,09,03,11,13,14,01,06,07, 16 3 08,07,06,02,05,14,10,12,03,04,11,15,01,09,13, 16 05,10,12,11,03,01,08,14,06,02,07,15,09,13,04, l6 05,02,04,06,01,03 02,04,01,03,06,05 03,02,01,06,04,05 05,06,04,0l,02,03 01,09,07,05,04,02,03,06,08 03,07,05,02,06,04,01,09,08 07,05,03,08,04,06,02,09,01 09,06,02,07,03,01,05,08,04 .b 12 13 14 15 HIbWNHthNH-PWNl-‘nwal-J 17 N 4s 21 22 uwaI—IowaH 234 Prob- Solution lem Numbers Permutation of Locations 23 13, 04, 09, 05, 12, 11, 14, 07, 10, 02, 03, 01, 08, 15, 06 12,15,03,07,06,11,13,08,10,02,04,01,09,14,05 13,04,05,07,08,1l,12,14,01,09,02,10,15,03,06 10,15,04,01,08,09,05,l3,1l,07,03,12,06,14,02 12,06,13,04,10,16,17,08,0l,03,09,19,23,11,05, 22,20,07,18,15,14,21,02,24 22,09, 19, 15, 16, 04, 05, 01,08, 13, 02, 12,06, 17, 10, 18,21,14,20,07,03,23,1l,24 3 l4,11,10,03,09,12,04,22,l9,13,17,20,02,16,06, 18,05,21,23,01,07,15,08,24 20,08,23,02,l4,10,11,06,22,07,05,18,15,03,17, 13, 04, 09, 01, 21, 12, 16, 19,24 07,12f10,09,08,06,11,01,02,04,05,03 11,04,07,0l,08,05,03,09,02,10,12,06 12,04,09,03,10,02,07,01,08,1l,05,06 11,03,12,07,08,01,06,09,04,05,02,10 kdhcnkJH 24 N 4:. 25 waH APPENDIX XI PROBLEM STATISTICS a. b c Number of Problem N Z Vf Vd Lower Bound Constraints 1 4 0.0 52.3 44.8 86,540.0 0 2 4 33.3 90.4 53.5 2,250.0 0 3 4 0.0 45.6 49.3 389.0 0 4 8 46.4 165.5 42.3 6,661.2 0 5 36 72.4 394.2 52.6 3,002.9 0 6 10 44.4 215.1 60.3 455.5 0 7 10 4.4 124.1 60.3 3,049.0 0 sd 15 66.7 564.2 58.9 1,023,132.0 4 9 9 38.9 108.0 35.0 690.0 0 10 16 42.5 435.4 39.9 799,2 6 ll 7 14.3 75.8 57.4 454.0 0 12 6 33.3 82.3 33.0 481.0 0 13 9 50.0 411.4 35.1 485.2 2 14 9 75.0 166.4 33.9 552.4 0 15 12 50.0 393.1 39.7 644.8 4 16 24 15.9 431.8 232.4 40,529.6 6 17 20 66.0 246.9 42.8 396.6 0 18e 15 60.0 248.8 40.1 13,769.0 0 19 4 0.0 72.6 30.4 155,830.0 0 20 5 0.0 74.5 27.5 1,246.0 0 21 6 13.3 254.9 33.5 1,212.0 2 22 9 27.8 154.9 35.0 502.2 0 23 15 86.7 300.8 44.3 92,000.0 0 24 23 6.0 592.3 45.1 1,288.8 7 25 12 32.0 107.8 46.2 243.0 0 26 20 44.0 235.8 47.2 66.4 0 aZ is the number of zero fi terms divided by (NZ—N)/200. It is expressed as a per cent. bV is the coefficient of variation for fik terms and is expressed as a per cent. ch is the coefficient of variation expressed as a per cent for dj}; terms. dFor "A" versions: N=l6, Z=70.0, Vf=525.3, and Vd = 266.7. eFor C—R, C-S, H-R, and H-S versions: N=l6 and Z=65.0. 235 APPENDIX XII SUMMARY OF COST AND TIME OUTPUT FOR TEST PROBLEMSa Average Average CostC Phase I Phase II Problem b Time in Time in Number Procedure Average Minimum Seconds Seconds 1 Wimmert's ... 100.8** ... ... RDM 130.0 125.7 0.061 0.104 1-A 103.9 100.8** 0.169 0.209 1-B 102.4 100.8** 0.232 0.235 2—A 100.8 100.8** 0.159 0.505 2-B 100.8 100.8** 0.176 0.504 3-T 100.8 100.8** 0.210 2.758 32A(L) 100.8 100.8 0.161 0.653 3eA(H) 100.8 100.8** 0.182 0.726 3-B 100.8 100.8** 0.183 0.596 3-C 100.8 100.8** 0.192 1.513 4-A 100.8 100.8** 0.172 0.236 4-B 100.8 100.8** 0.168 0.251 4-C 100.8 100.8** 0.179 0.314 5—B 100.8 100.8** 0.278 1.280 2 Noy's ... 133.3 ... ... RDM 161.7 151.1 0.113 0.143 l-A 131.1 131.1 0.223 0.445 1-B 137.8 137.8 0.223 0.474 2-A 151.1 128.9 0.225 1.190 2-B 156.7 131.1 0.232 1.438 3-T 148.3 126.7 0.200 17.251 3-A(L) 154.4 126.7 0.214 1.555 3—A(H) 174.4 155.6 0.223 1.714 3-B 142.2 126.7 0.226 1.348 3-C 113.2 113.2* 0.224 5.996 4-A 131.1 131.1 0.220 0.531 4-B 126.7 126.7 0.234 0.536 4-C 126.7 126.7 0.238 0.944 5-B 153.3 144.5 0.606 16.600 H-R 113.2 113.2* 0.162 0.760 H-S 113.2 113.2* 0.121 1.325 C-R 113.2 113.2 0.498 0.367 C-S 113.2 113.2 0.504 0.351 3 Gavett's 103.6 103.6** ... ... RDM 129.4 118.3 0.061 0.102 1-A 104.4 103.6** 0.121 0.210 l-B 122.9 122.9 0.132 0.235 22A 103.6 103.6** 0.124 0.470 2-B 113.2 103.6** 0.136 0.492 3-T 106.3 103.6** 0.128 2.704 236 237 c Average Average Cost Phase I Phase II Problem b Time in Time in Number Procedure Average Minimum Seconds Seconds 3-A(L) 103.6 103.6** 0.135 0.656 3-A(H) 103.6 103.6** 0.140 0.728 3—B 103.6 103.6** 0.142 0.564 3-C 103.6 103.6** 0.138 1.502 4-A 114.4 114.4 0.140 0.232 4-B 110.0 103.6** 0.129 0.245 4-C 116.2 116.2 0.139 0.308 5-B 113.2 103.6** 0.217 1.414 4 Conrade's ... 113.0** ... ... RDM 154.0 141.7 0.192 0.221 l-A 126.8 126.8 0.423 1.036 l-B 121.6 116.9 0.442 1.078 2-A 134.1 118.9 0.428 3.852 2-B 134.9 125.0 0.442 3.461 3eA(L) 131.8 122.8 0.397 4.043 3-A(H) 136.7 135.5 0.437 6.447 3-B 140.0 130.9 0.412 4.284 3-C 126.2 114.7 0.404 17.377 42A 129.2 126.7 0.422 1.204 4-B 136.5 136.5 0.441 1.294 4-C 135.8 135.8 0.448 2.739 5-B 137.1 133.8 1.954 36.200 H-R 118.2 114.8 0.123 1.500 H-S 115.0 113.0** 0.131 4.074 C-R 116.3 113.7 0.669 0.529 C-S 116.0 113.0** 0.663 0.580 5 Steinberg-l ... 165.5 . . ... Steinberg-2 ... 162.8 ... ... Steinberg-3d ... 163.5 . . Steinberg-4 .. 162.9 . .. Gilmore's-N ... 151.1 .. Gilmore's-N ... 155.8 ... . Gilmore's-N d ... 154.6 ... . Gilmore's-NSd ... 164.9 ... ... RDM 303.1 278.9 3.392 35.894 H-R (3-R) 153.8 148.7 1.312 155.927 H-S 150.1 145.2 1.263 1324.784 C-R 156.8 150.0 8.890 86.532 C-S 151.1 144.5* 8.894 84.453 6 Smith'se ... 165.1 ... ... RDM 306.7 236.5 0.269 0.362 l-A 255.9 255.9 0.744 2.108 1-B 209.5 193.6 0.771 2.561 2-A 223.5 201.5 0.760 11.648 2-B 275.5 240.3 0.797 13.448 238 c Average Average Cost Phase I Phase II Problem b Time in Time in Number Procedure Average Minimum Seconds Seconds 3-A(L) 203.1 145.4 0.683 10.436 3-A(H) 266.0 192.0 0.660 13.323 3-B 196.5 185.8 0.706 9.848 3-C 163.8 163.8 0.700 45.986 4-A 281.9 281.9 0.738 2.228 4-B 220.3 200.4 0.781 3.110 4—C 159.2 159.2 0.798 9.702 H—R 142.1 121.0* 0.194 3.282 H-S 142.1 121.0* 0.155 8.276 C—R 146.0 124.9 0.870 0.841 C-S 146.0 124.9 0.863 1.002 7 Smith'se ... 201.8 ... ... RDM 252.9 228.6 0.277 0.364 l—A 182.1 182.1 0.759 2.274 l-B 200.1 196.2 0.788 2.456 2-A 181.7 158.1 0.783 18.250 2-B 198.9 188.6 0.812 16.272 3-A(L) 172.1 166.3 0.690 9.556 3-A(H) 178.5 173.4 0.672 13.496 3-B 179.2 179.2 0.714 12.163 3-C 150.4 150.4 0.705 53.995 4-A 183.0 183.0 0.763 2.600 4-B 183.0 183.0 0.783 3.216 4-C 157.8 157.8 0.813 8.748 H-R 143.0 141.0 0.167 3.318 H-S 143.0 141.0 0.157 8.435 C-R 143.1 136.6* 0.884 1.102 C—8 143.1 136.6* 0.886 1.126 8 Wimmert'sf 224.3 224.3 ... ... RDM U U 0.667 1.595 l-A U U 3.201 14.263 l—B U U 2.770 14.636 22A U(l) U(l) 3.476 476.232 2-B U U 2.854 419.602 3-A(L) U U 2.981 560.641 3—B U U 2.460 323.446 3-C U(l) U(l) 2.509 1196.874 4-A U U 3.279 15.796 4—B U U 2.814 14.939 4-C U U 2.826 66.508 9 RDM 132.9 126.6 0.235 0.282 1-A 153.0 153.0 0.570 1.358 1—B 148.6 131.3 0.595 1.918 2-A 140.8 133.3 0.576 3.700 2-B 147.3 141.0 0.603 6.337 239 c Average Average Cost Phase I Phase II Problem b Time in Time in Number Procedure .Average Minimum Seconds Seconds 3-T 143.8 141.8 0.483 186.978 3-A(L) 132.8 127.5 0.517 4.210 3-A(H) 132.1 124.0 0.549 4.685 3-B 140.7 137.0 0.566 4.553 3-C 123.7 123.7 0.537 24.165 4-A 153.0 153.0 0.567 1.708 4-B 120.6 120.6 0.604 2.321 4-C 117.5 117.5 0.611 5.849 S-B 151.2 144.1 3.720 597.252 H-R 115.4 112.1 0.146 1.760 H-S 116.3 115.8 0.139 5.070 C—R 113.1 100.2* 0.755 0.782 C-8 101.3 100.2* 0.759 0.802 10 RDM U U 0.686 1.581 1-A U U 3.246 12.770 l-B U U 3.515 19.147 2-A U U 3.432 26.703 2—B U U 3.578 19.114 3-A(L) U U 2.911 17.862 3—B U U 3.031 20.600 4-A U U 3.566 15.273 4-B U U 3.768 21.692 4-C U U 3.532 83.387 H—R(2—R) 138.2 131.7 0.326 21.751 H-S 135.0 124.2 0.312 36.682 C-R 143.0 120.5* 2.070 3.262 C-S 132.2 127.5 2.057 3.551 11 Lawler's ... 122.0 ... ... RDM 173.8 159.9 0.144 0.174 1-A 120.3 120.3 0.305 0.677 l-B 149.3 144.9 0.325 0.802 2-A 143.1 132.2 0.311 1.897 2-B 142.5 133.9 0.327 2.095 3-T 149.2 135.7 0.280 37.453 3-A(L) 147.1 145.8 0.288 2.176 3-A(H) 137.9 127.3 0.282 2.521 3—B 129.8 129.8 0.303 2.202 3-C 141.5 135.4 0.300 10.355 4-A 120.3 120.3 0.304 0.749 4-B 135.7 135.7 0.320 0.890 4-C 117.0 117.0* 0.330 1.614 5-B 152.8 137.8 1.050 45.837 240 E — L ! c Average Average Cost Phase I Phase II Problem b Time in Time in Number Procedure Average Minimum Seconds Seconds 12 RDM 133.1 129.7 0.109 0.149 leA 125.7 117.5 0.217 0.421 l-B 129.4 124.0 0.235 0.518 2-A 116.2 107.0* 0.222 1.144 2-B 137.1 132.2 0.235 1.310 3-T 128.7 126.1 0.201 21.346 3-A(L) 123.2 114.6 0.213 1.653 3-A(H) 126.4 112.4 0.215 1.908 3-B 131.4 121.0 0.225 1.543 3-C 113.2 113.2 0.223 6.123 42A 116.8 116.8 0.217 0.496 4-B 120.8 112.4 0.238 0.613 4—C 114.2 112.0 0.240 0.998 5—B 134.5 126.5 0.600 15.471 H—R 114.2 111.1 0.112 0.523 H-S 109.5 107.0* 0.091 1.379 C-R 109.5 107.0* 0.504 0.304 C—S 108.3 107.0* 0.491 0.339 13 RDM U U 0.230 1.125 l-A U U 0.567 1.318 l-B U U 0.597 2.124 2eA U U 0.582 3.890 2-B 159.0 159.0 0.602 4.537 3-A(L) 143.5 143.5 0.517 4.424 3-A(H) U U 0.511 5.236 3-B 153.4 153.4 0.544 4.838 3—C 143.0 138.8 0.533 21.818 4-A U U 0.567 1.741 4-B U U 0.597 2.470 4-C U U 0.607 5.379 H-R(1—R) 122.8 120.2 0.148 2.377 H-S 120.0 117.8* 0.136 3.765 C—R 121.2 117.8* 0.757 0.584 C-S 120.2 117.8* 0.762 0.626 14 RDM 158.4 143.7 0.233 0.280 12A 164.7 161.9 0.569 1.376 l-B 164.2 155.9 0.595 1.840 2-A 145.1 124.0 0.580 4.547 2-B 158.1 153.9 0.605 7.882 3eA(L) 156.0 141.6 0.522 4.529 32A(H) 149.0 142.5 0.512 7.263 3-B 142.7 140.8 0.540 6.066 3-C 123.9 117.5 0.530 27.235 4eA 164.7 161.9 0.569 1.612 4-B 145.6 145.6 0.738 2.368 241 c Average Average Cost Phase I Phase II Problem b Time in Time in Number Procedure Average Minimum Seconds Seconds 4—C 117.3 117.3 0.609 5.174 H-R 118.2 112.5 0.144 1.384 H-S 114.8 109.2* 0.133 4.054 C-R 115.0 112.3 0.765 0.687 C-S 113.2 112.3 0.762 0.822 15 RDM U U 0.409 0.603 1-A U U 1.311 4.215 1-B U U 1.384 6.042 2-A U U 1.368 11.988 2-B U U 1.402 10.040 3-A(L) U U 1.179 9.430 34A(H) U U 1.151 17.992 3-B U U 1.198 12.607 3-C 156.7 145.7 1.209 63.866 42A U U 1.314 5.026 4-B U U 1.412 6.830 4-C 122.0 122.0 1.396 23.318 H-R(4—R) 153.8 146.1 0.239 ... H-S 133.2 119.2* 0.211 12.198 C-R 137.5 124.6 1.405 1.361 C-S 126.2 124.6 1.401 1.639 16 RDM U U 1.509 7.214 1-A U(2) U(2) 13.946 65.056 1-B U(2) U(2) 18.027 25.208 2-A U(2) U(2) 15.194 516.901 2-B U(2) U(2) 15.879 440.166 3—A(L) U(2) U(2) 13.248 507.734 3-B U(2) U(2) 14.282 399.499 4-A U(2) U(2) 14.162 70.236 4-B U(2) U(2) 14.951 31.730 17 RDM 247.9 237.0 1.196 4.275 1-A 221.8(2) 221.8(2) 7.232 30.792 1-B 206.0(2) 193.0(2) 7.613 46.964 22A 196.5(2) 195.2(2) 7.600 538.387 2-B 203.5(2) 194.7(2) 8.132 500.046 3-A(L) 188.0(2) 179.8(2) 6.572 271.851 3-B 196.2(2) 195.7(2) 6.940 255.168 4—A 223.2(2) 223.2(2) 8.525 40.702 4—B 214.4(2) 214.4(2) 8.987 53.335 4-C 168.5(2) 168.5(2) 7.741 213.732 H-R(2-R) 144.0 129.0 0.482 26.195 H-S 133.8 130.9 0.437 92.549 C-R 137.5 130.0 2.822 9.865 C-S 135.0 126.2* 2.820 9.189 m 242 c Average Average Cost Phase I Phase II Problem b Time in Time in Number Procedure Average Minimum Seconds Seconds 18 RDM 197.7 169.9 0.616 1.274 leA 180.5 180.5 2.617 9.572 1-B 176.5 175.2 2.789 14.136 2-A 154.5 144.3 2.797 125.173 2-B 157.9 153.6 2.893 115.052 3-A(L) 157.7 148.6 2.439 111.160 3-B 141.2 135.0 2.536 104.174 44A 180.5 180.5 3.106 11.304 4-B 159.8 159.8 2.836 15.178 4-C 147.1 147.1 2.843 59.391 H-R(2-R) 137.0 122.5 0.498 10.201 H-S 125.0 122.5 0.302 39.031 C-R 125.7 122.1 1.992 3.350 C-S 123.0 120.6* 2.040 3.244 19 Armour's ... 100.2 ... ... RDM 130.5 116.4 0.066 0.102 14A 100.2 100.1* 0.127 0.217 1-B 117.0 116.4 0.130 0.235 2-A 100.2 100.2 0.127 0.475 2-B 100.2 100.2 0.129 0.510 3-T 100.2 100.2 0.123 2.493 3-A(L) 100.2 100.2 0.130 0.654 3-A(H) 100.2 100.2 0.139 0.714 3-B 100.2 100.2 0.139 0.558 3-C 100.2 100.2 0.136 1.458 4-A 116.5 116.5 0.128 0.232 4-B 116.5 116.5 0.137 0.252 4-C 116.5 116.5 0.135 0.312 S-B 100.1 100.1* 0.243 1.336 20 Land's ... 108.1** ... ... RDM 122.3 114.4 0.105 0.120 1-A 115.5 114.3 0.179 0.283 1-B 111.5 111.5 0.188 0.327 2-A 113.1 109.5 0.183 0.800 2-B 116.9 114.2 0.187 0.855 3-T 111.0 109.2 0.169 9.166 3-A(L) 118.0 109.2 0.183 1.077 3-A(H) 120.2 115.9 0.188 1.110 3-B 120.2 115.9 0.192 0.972 3-C 110.0 108.1** 0.186 3.180 42A 114.3 114.3 0.184 0.316 4-B 114.3 114.3 0.191 0.343 4—C 111.5 111.5 0.192 0.431 S-B 113.2 108.1** 0.358 2.147 243 c Average Average Cost Phase I Phase II Problem b Time in Time in Number Procedure Average Minimum Seconds Seconds 21 Willoughby's ... 118.3 ... ... RDM 137.0 137.0 0.109 0.142 1-A U U 0.219 0.409 1-B 113.9 113.9* 0.237 0.520 2—A U U 0.252 0.987 2-B 123.1 123.1 0.232 1.056 3-A(L) 123.1 123.1 0.215 1.488 3-A(H) U U 0.212 1.578 3-B 123.2 123.2 0.220 1.296 3-C 121.5 121.4 0.224 5.482 4-A U U 0.219 0.464 4-B U U 0.232 0.541 4-C 113.8 113.8* 0.242 0.904 5—B U U 0.593 4.122 H—R(1-R) 114.0 113.8* 0.118 0.999 H-S 119.0 113.8* 0.102 1.356 C-R 116.2 114.0 0.494 0.334 C-S 116.2 114.0 0.490 0.351 22 RDM 134.7 115.9 0.224 0.282 1eA 142.2 136.2 0.566 1.590 1-B 134.3 125.6 0.588 1.901 2~A 134.3 120.7 0.582 6.064 2-B 127.9 125.6 0.605 5.440 3-A(L) 127.8 119.4 0.514 6.065 3wA(H) 127.3 124.9 0.512 7.282 3-B 110.3 110.0* 0.538 6.560 3-C 126.0 125.2 0.534 26.239 4-A 142.2 139.2 0.567 1.828 4-B 120.1 120.1 0.599 2.096 4-C 111.3 111.3 0.611 6.092 H-R(1-R) 118.1 114.8 0.146 2.211 H-S 113.5 110.0* 0.138 6.690 C-R 115.0 111.2 0.771 0.732 C—S 111.2 110.0* 0.763 0.820 23 Schneider's ... 140.8 ... ... RDM 220.3 204.7 0.613 1.268 1-A 150.1 150.1 2.611 11.551 1-B 140.7 136.9 2.785 9.843 2-A 188.4 163.4 2.753 41.226 2-B 163.7 158.2 2.870 47.149 3-A(L) 148.3 132.5 2.452 46.605 3-B 160.5 143.3 2.530 47.281 3-C 128.9 118.0 2.546 287.719 44A 150.1 150.1 2.866 12.256 4-B 158.1 158.1 2.830 12.522 244 c Average Average Cost Phase I Phase II Problem b Time in Time in Number Procedure Average Minimum Seconds Seconds 4—C 159.5 159.5 2.836 59.078 H-R 132.2 119.0 0.298 5.200 H-S 122.0 113.9 0.269 12.360 C-R 118.0 104.3* 1.872 2.259 C-S 114.8 113.4 1.856 2.214 24 RDM U U 1.437 6.202 1-A U(2) U(2) 12.304 59.144 1-B U(2) U(2) 13.466 67.822 2-A U(2) U(2) 12.777 208.992 2—B U(2) U(2) 13.595 178.283 3-A(L) U(2) U(2) 11.446 118.504 3-B U(2) U(2) 11.568 149.672 4—A U(2) U(2) 12.691 67.109 4-B U(2) U(2) 12.840 71.075 4-C U(2) U(2) 13.009 476.156 H-R(3-R) U U 0.728 ... H-S 111.3 111.3 0.670 297.057 C-R U U 4.459 14.823 C—S 111.9 111.1* 4.480 17.738 25 RDM 163.8 154.0 0.438 0.668 1-A 172.6 160.0 1.306 3.956 1-B 180.6 178.2 1.409 6.008 2-A 158.2 151.5 1.340 11.020 2-B 165.2 161.0 1.423 13.653 3-A(L) 156.1 145.1 1.143 8.624 3-B 160.0 151.1 1.265 9.530 3-C 142.0 130.0 1.262 73.096 4-A 184.8 184.8 1.553 5.732 4-B 150.1 149.5 1.629 7.972 4-C 130.0 130.0 1.392 18.784 H-R(1-R) 147.2 124.0 0.238 5.488 H-S 128.8 121.9* 0.204 15.549 C-R 128.8 127.5 1.423 1.572 C-S 124.0 122.8 1.400 1.906 26 RDM 215.0 206.0 1.169 3.998 l-A 150.0 150.0 7.123 32.902 1-B 181.0 180.5 7.774 52.858 2-A 151.8 148.1 7.515 971.584 2-B 161.3 160.2 7.975 1029.603 32A(L) 153.5 151.5 6.567 943.000 3-B 151.8 150.0 6.837 979.177 4-A 151.0 151.0 8.435 44.070 4-B 153.9 152.7 9.159 59.688 4-C 137.8 137.8* 7.724 290.968 1n ' \. .. .I‘llllllllll I‘ll. ...l'llllil' 245 Footnotes to Appendix XII aDots indicate that the data is not appropriate or not available. bHillier's original algorithm (H-R) does not guaran- tee that solution costs are reduced after each iteration, since cost reductions are only approximated. For this rea- son,the solution produced by the final trial can have a cost higher than that of a previous one. Furthermore, it is pos- sible for a set of solution to be generated repeatedly with— out over reaching the final trial. If two such recycling incidents occurred for a problem, this information is noted by "2—R" in the second column. Version H-S is modified to assure that solution costs are reduced at each successive trial, thereby avoiding the recycling problem but increasing Phase II time. CUnless otherwise noted by a number in parentheses, each cost value is derived from four observations per prob- lem. The cost is expressed as a per cent of the lower bound. When computing the lower bound, all arbitrarily large fik terms are equated to zero. The letter "U" indicates that no feasible solution was generated. The minimum cost column applies to the solution having the least cost. Cost values are given only for solutions satisfying all constraints. One asterisk singles out the best solution for a problem, whereas two indicate a known optimal solution. dThis solution was generated on the basis of a cost function having distance squared between each pair of loca- tions. eThe author's solution procedure was based on the assumption that backtracking is twice as costly as forward movement. .Although this assumption has been severely crit— icized in the literature, the important consideration for this analysis is that Smith's solution is not strictly comparable to the other solutions generated. fIt is not clear how Wimmert arrived at this solu- tion if his manual solution process was followed without deviation. First of all, many of the quadruplets with arbitrarily large costs would never be selected for elimina- tion. Secondly, several types of conflict would be encoun- tered in the SOLUTION matrix. 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Moo. 0 o awoammao.fl aammmmoe.au Ho¢.amu AqO<-m Hoo.v nano.a maaa. aaaa. mooo.v mammaoam. aooaaomm.man maaamaom.aa oooa¢aaa.mmmu ~v¢.a~¢ sum oo¢ «mom.oma amaa. came. Hoo. o o «Haemomo.a anamooao.au maa.a¢: mum 004 aam~.maa meow. maoa. aooo.v o o moaamamo.a okaowomo.m ~aa.¢au <-m amo. mama.m mama. Hams. mooo.v o o mmaaamma. mmooqaoa.n mam. mga Hoo.v ammm. aaaa. aaaa. mooo.v o omammoao. adamqoma.- acmmmoam.a mma.~u ¢-H Na.aam a mm mm Hm.awm an an an an an musomooua UmUHumHumum AmucmHUkumoo :onmmummm Amficoomm GHV mmEHB HH mmdmm mom mZOHfidbom ZOHmmmmUmm m0 mUHBmHBX XHQmemfl APPENDIX XVII SOLUTION COST STATISTICS OF RDM FOR ALL UNCONSTRAINED PROBLEMS Problem Number Average Cost Standard Deviation 1 130.0 3.2 2 161.7 11.7 3 129.4 13.7 4 154.0 10.1 5 303.1 19.6 6 306.7 72.0 7 252.9 32.5 9 132.9 5.6 11 173.8 4.4 12 133.1 2.9 14 158.4 10.9 17 247.9 11.3 18 197.7 27.7 19 130.5 9.9 20 122.3 8.0 22 134.7 8.2 23 220.3 11.9 25 163.8 9.9 26 215.0 10.0 MEAN 182.5 14.9 251 APPENDIX XVIII LEAST-COST SOLUTIONS AND RANDOM MEAN INCREMENTS Problem Least-Cost Random Mean Numbera Solution IncrementC Procedure 1 100.8* 29.2 1-A,1-B,2-A,2-B,3-T,3-A(L), 3-A(H),3-B,3-C,4-A,4-B, 4-C,5-B 2 113.2 48.5 3-C,H-R,H-S,C—R,C-S 3 103.6* 25.8 1-A,2-A,2-B,3-T,3-A(L), 3-A(H),3-B,3-C,4-B,5-B 4 113.0 141.0 H-S,C-S 5 144.5 158.6 C-S 6 121.0 185.7 H-R,H-S 7 136.6 116.3 C-R,C-S 8 U U ... 9 100.2 32.7 C-R,C-S 10(C) 120.5 24.2 C-R 11 117.0* 56.8 4-C 12 107.0 26.1 2-A,H-S,C-R,C-S 13(C) 107.8 22.5 H-S,C-R,C-S 14 109.2 49.2 H-S 15(C) 119.2 32.6 H-S 16 U U ... 17 126.2 121.7 C-S 18 120.6 77.1 C-S 19 100.1* 30.4 1-A,5-B 20 108.1* 14.2 3-C,5-B 21(C) 113.8 10.3 4-C,H-R,H-S 22 110.0 24.6 3—B,H—S,C-S 23 104.3 116.0 C-R 24(C) 111.1 9.7 C-S 25 121.9 41.9 H-S 26 137.8* 4.8 4-C aThe letter "C" designates constrained problems. bThe cost of the best solution produced in this study is expressed as a per cent of the lower bound in this column. A "U" tions. indicates the failure to generate any feasible solu- An asterisk designates those problems for which no solutions are available from Procedures H-R, H-S, C—R, and C-S. CThis value is equal to the mean cost of RDM solutions minus the corresponding value in the second column of this appendix. dIn this column are listed all procedures producing the least-cost solution at least once. 252 BIBLIOGRAPHY Books Apple, James M. Plant Layout and Materials Handling. New York: The Ronald Press Company, 1950. Buffa, Elwood S. Modern Production Management. 2d ed. New York: John Wiley and Sons, Inc., 1965. Dooley, Arch R., et a1. Operation Planning and Control. New York: John Wiley and Sons, Inc., 1964. Eddison, R. T., Pennycuick, K., and Rivett, B. H. P. Operational Research in Management. London: English Union Press, 1962. Feigenbaum, Edward A., and Feldman, Julian. Computers and Thought. New York: McGraw-Hill Book Co., 1963. General Electric Corporation. Technique of Makinqulant Layouts. Schenectady, New York: General Electric Corporation, 1952. Immer, John R. Layout Planning Techniques. New York: McGraw-Hill Book Co., 1950. Ireson, William.G. Factornylanningyand Plant Layout. New York: Prentice-Hall, Inc., 1952. Moore, James M. Plant Layout and Design. New York: MacMillan Co., 1962. Muther, Richard. Practical Plant Layout. New YOrk: McGraw— Hill Book Co., 1955. . Systematic Layout Planning. Boston: Industrial Education Institute, 1961. Nadler, Gerald. Work Design. Homewood, Illinois: Richard D. Irwin, Inc., 1963. Reed, Ruddell, Jr. Plant Laygut: Factorgy Principles and Techniques. Homewood, Illinois: Richard D. Irwin, Inc., 1961. 253 254 Reed, Ruddell, Jr. Plant Location, Layout and Maintenance. Vol. V of The Irwin Series in Operations Management. Edited by H. L. Timms. Homewood, Illinois: Richard D. Irwin, Inc., 1967. Reinfeld, Nyles V., and Vogel, William R. Mathematical Programming. Englewood Cliffs, New Jersey: Prentice- Hall, Inc., 1958. Shubin, John A., and Madeheim, H. Plant Layout. Englewood Cliffs, New Jersey: Prentice—Hall, Inc., 1951. Tonge, Fred M. A Heuristic Program for Assembly Line Balancing, Englewood Cliffs, New Jersey: Prentice—Hall, Inc., 1961. Articles and Reports Armour, Gordon C., and Buffa, Elwood S. "A Heuristic Algorithm and Simulation Approach to Relative Location of Facilities," Management Science, IX, No. 2 (January, 1963), 294-309. Beckwith, Richard E., and Vaswani, Ram. "The Assignment Problem--A Special Case of Linear Programming," Journal of Industrial Engineering, XII, No. 1 (January-February, 1961), 41-48. Bindschedler, Andre E., and Moore, James M. "Optimal Location of New Machines in Existing Plant Layouts," Journal of Industrial Engineering, XII, No. 1 (January- February, 1961), 41-48. Brink, Edward L., and deCani, John S. ”An Analogue Solution ‘ of the Generalized Transportation Problem with Specific Application to Marketing Location," Proceedings of the First International Conference on Operational Research. Baltimore: Operations Research Society of America, 1957, pp. 123-137. Briskin, Lawrence E. "A Method of Unifying Multiple Objec- tive Functions," Management Science, XII, No. 10 (June, 1966), 406-416. Brooks, Samuel H. "A Discussion of Random Methods for Seek- ing Maxima," Operations Research, VI, No. 2 (March—April, 1958), 244-251. 255 Buffa, Elwood S. "Reader Comment," Journal of Industrial Engineering, XVIII, No. 8 (August, 1967), 502. . "Sequence Analysis for Functional Layout," Journal of Industrial Engineering, VI, No. 2 (March- April, 1955), 12-25. Buffa, Elwood S., Armour, Gordon C., and Vollman, Thomas E. "Allocating Facilities with CRAFT," Harvard Business Review, XLII, No. 2 (March-April, 1964), 70-78. Cameron, D. C. "Travel Charts," Modern Materials Handling, VII, No. 1 (January, 1952), 37-40. Carroll, Charles W. "The Created Response Surface Technique for Optimizing Nonlinear, Restrained Systems," Opera- tional Research, IX, No. 2 (March-April, 1961), 169-183. Coleman, J. R., Jr., Smidt, S., and York, R. "Optimum Plant Design for Seasonal Production," Management Science, X, No. 4 (July, 1964), 778-785. "Computers: Impartial Judge of Kitchen Layout," Institutions, September, 1967, pp. 119-122. Conway, R. W., and Maxwell, W. L. "A Note on the Assignment of Facility Location," Journal of Industrial Engineering, XII, No. 1 (January-February, 1961), 7-13. Eilon, Samuel, and Deziel, D. P. "Siting a Distribution Center, An Analogue Computer Application," Management Science, XII, No. 6 (February, 1966), 245-254. Francis, Richard L. "A Note on the Optimal Layout of New Machines in Existing Plant Layouts," Journal of Indus- trial Engineering, XIV, No. 1 (January-February, 1963), 33-40. . "Sufficient Conditions for Some Optimum-Property Facility Designs," Operations Research, XV, No. 3 (May— June, 1967), 448-466. Frank, H. "Optimum Locations on a Graph with Probabilistic Demands," _perations Research, XIV, No. 3 (May-June, 1966), 409-421. Gavett, J. W., and Plyter, Norman V. "The Optimal Assign- ment of Facilities to Locations by Branch and Bound," Operations Research, XIV, No. 2 (March-April, 1966), 210-232. 256 Gilmore, P. C. "Optimal and Suboptimal Algorithms for the Quadratic Assignment Problem," Journal of the Society for Industrial andgApplied Mathematics, X, No. 2 (June, 1962), 305-313. Hakimi, S. L. "Optimum Locations of Switching Centers and the Absolute Centers and Medians of a Graph," Operations Research, XII, No. 3 (May-June, 1964), 450—459. Haley, K. B. "The Siting of Depots," International Journal of Production Research, II, No. 1 (March, 1963), 41-45. Hillier, Frederick S. "Quantitative Tools for Plant Layout Analysis," Journal of Industrial Engineering, XIV, No. 1 (January-February, 1963), 33-40. Hillier, Frederick S., and Connors, Michael N. “Quadratic Assignment Problem Algorithms and the Location of Indivisible Facilities," Manggement Science, XIII, No. 1 (September, 1966), 42-57. Ignall, Edward J. ”A Review of Assembly Line Balancing," Journal of Industrial Engineering, XVI, No. 4 (July- August, 1965), 244-254. Kase, Shigeo, and Nishiyama, Noriyuki. "An Industrial Engineering Game Model for Factory Layout," Journal of Industrial Engineering, XV, No. 3 (May-June, 1964), 148-150. Kilbridge, Maurice D., and Wester, Leon. "A Review of Analytic Systems of Line Balancing," Operations Research, X, No. 5 (September-October, 1962), 626. Kodres, U. R. "Geometrical Positioning of Circuit Elements in a Computer," Conference Paper 1172, AIEE Fall General Meeting, October, 1959. Koopmans, Tjalling C., and Beckman, Martin. "Assignment Problems and the Location of Economic Activities," Econometrica, XXV, No. 1 (January, 1957), 53-76. Kruskal, J. B., Jr. "On the Shortest Spanning Subtree of a Graph and the Travelling Salesman Problem," Proceedings of the American Mathematical Society, VII (1956), 48-50. Kuhn, H. W. "The Hungarian Method for the Assignment Problem," Naval Research Logistics Quarterly, II (March- June, 1955), 83-97. 257 Land, A. H. "A Problem of Assignment with Interrelated Costs," Operational Research Ogarterly, XIV, No. 2 (June, 1963), 185-199. Lawler, Eugene L. "The Quadratic Assignment Problem," Management Science, IX, No. 4 (July, 1963), 586-599. Lawler, Eugene L., and Wood, D. E. "Branch-and-Bound Methods: A Survey," Operations Research, XIV, No. 4 (July-August, 1966), 699-719. Lee, Robert C., and Moore, James M. "CORELAP-OOmputerized RElationship LAyout Planning," Journal of Industrial Engineering, XVIII, No. 3 (March, 1967), 195-200. Levy, M. L. "Let Travel Charting Simplify Your Material Movement Problems," Mill and Factory, XLVIII, No. 5 (May, 1951), 100-101. Little, J. D. C., Murty, K. G., Sweeney, D. W., and Karel, C. "An Algorithm for the Traveling Salesman Problem," Operations Research, XI, No. 6 (November-December, 1963), 972-989. Llewellyn, Robert W. "Travel Charting with Realistic Criteria," Journal of Industrial Engineering, IX, No. 3 (May-June, 1958), 217-220. Loberman, H., and Weinberger, A. "Formal Procedures for Connecting Terminals with a Minimum Total Wire Length," Journal of the Association for Computing Machinery, IV (1957), 428-33. Lund, Herbert F. "Plant Planning Tools," Factory, CXXI, Part 2, No. 9 (September, 1963), 86-91. Lundy, James L. "A Reply to Wayland P. Smith's Article," Journal of Industrial Engineering, XI, No. 3 (May—June, 1955), 29. McHose, Andre H. "A Quadratic Formulation of the Activity Location Problem," Journal of Industrial Engineering, XII, No. 5 (September-October, 1961), 334-37. Miehle, William. "Link-Length Minimization in Networks," The Journal of the Operations Research Society of America, VI, No. 2 (MarchHApril, 1958), 232-240. Miller, Robert F. "Quantitative Approaches to Facilities Planning and the Planning of Manufacturing Processes," Journal of Industrial Engineering, XVIII, No. 1 (January, 1967), 10-13. 258 Moore, James M. "Author's Comments," Journal of Industrial Engineering, XVIII, No. 8 (August, 1967), 502. "Optimal Locations for Multiple Machines," Journal of Industrial Engineering, XII, No. 5 (September- October, 1961), 307-13. Moore, James M., and Mariner, Martin R. "Layout Planning: New Role for Computers," Modern Materials Handling, XVIII, No. 3 (March, 1963), 38-42. Moore, James M., and Whinston, A. B. "Experimental Methods in Quadratic Programming," Management Science, XIII, No. 1 (September, 1966), 58-76. Munkres, James. "Algorithms for the Assignment and Trans— portation Problems," Journal of the Society for Indus- trial and Applied Mathematics, V, No. 1 (March, 1957), 32—38. Muther, Richard, and Wheeler, John D. "Simplified Systematic Layout Planning," Factory, CXX, Part 2, Nos. 8, 9, 10 (1962). Newman, D. J. "A Parking Lot Design," Society for Industrial and Applied Mathematics Review, XI, No. 1 (January, 62-66. 1964), Noy, Peter C. "Make the Right Plant Layout—Mathematically," American Machinist, CI, No. 6 (March, 1957), 76—78. Nugent, Christopher E., Vollman, Thomas E., Ruml, John. "An Experimental Comparison of Techniques for the Assignment of Facilities to Locations," Operations Research, XVI, No. 1 (January—February, Palermo, F. P. 1968), 150-173. "A Network Minimization Problem," IBM Journal of Research and Development, V, No. 4 (October, 1961), 335-337. Prim, R. C. "Shortest Connection Networks and Some Generalizations," The Bell Sy§tem Technical Journal, XXXVI, No. 6 (November, 1957), 1389-1401. Reis, and Anderson, Glenn E. "Relative Importance Factors in Layout Analysis," Journal of Industrial Engineering, XI, No. 4 (July-August, Rice, Robert S. Irvin L., 1960), 312-316. "Three New Tools for Better Plant Layout," Factory, CXVIII, Part 1, No. 6 (June, 1960), 101-103. 259 Richman, Eugene. "Trends in Plant Layout and Design," Journal of Industrial Engineering, VII, No. 1 (January- February, 1956), 29—30. Ronan, R. T. (ed.). "String Diagrams Cut Handling Bottle- neck," Modern Materials Handling, VIII, No. 8 (August, 1953), 67-71. Schneider, Marshall. "Cross Charting Technique as a Basis for Plant Layout," Journal of Industrial Engineering, XI, No. 6 (November-December, 1960), 478-483. Smith, Wayland P. "Travel Charting," Journal of Industrial Engineering, VI, No. 1 (January, 1955), 26-29. Tonge, F. M. "Assembly Line Balancing Using Probabilistic Combinations of Heuristics," Management Science, XI, No. 7 (May, 1965), 727—735. Steinberg, L. "The Backboard Wiring Problem; A Placement Algorithm," Society for Industrial and Applied Mathe- matics Review, III, No. 1 (January, 1961), 37-50. Tideman, M. "Comment on 'A Network Minimization Problem,'" IBM Journal of Research and Development, VI, No. 2 (April, 1962), 259. Vergin, Roger C., and Rogers, Jack D. "An Algorithm and Computational Procedure for Locating Economic Facilities,I Management Science, XIII, No. 6 (February, 1967), 240-254. Vollman, Thomas E., and Buffa, Elwood S. "The Facilities Layout Problem in Perspective," Managpment Science, XII, No. 10 (June, 1966), 167-176. Wilson, Richard C. "A Review of Facility Design Models," The Journal of Industrial Engineering, XV, No. 3 (May- June, 1964), 115-121. Wimmert, Robert J. "A Mathematical Model of Equipment Location," Journal of Industrial Engineering, IX, No. 6 (November—December, 1958), 498-505. Yaspan, A. "On Finding a Maximal Assignment," Operations Research, XIV, No. 4 (JulyHAugust, 1966), 646—651. 260 Unpublished Material Armour, Gordon C. "A Heuristic Algorithm and Simulation Approach to Relative Location of Facilities." Unpub- lished Ph.D. dissertation, UCLA, Los Angeles, California, 1961. Elicker, Roy E. "Operating Procedures Used in Plant Layout." Unpublished Master's dissertation, Michigan State University, East Lansing, 1951. Giles, P., Kimmelman, E., Hirschfeld, H., and Keet, E. Facility Allocation Prnject, Department of Industrial Engineering and Administration, Cornell University Library, May 22, 1962. Harris, Roy D., and Smith, Roland K. A Cost-Effectiveness Approach to Facilities Layout, Working Paper 67-22, Graduate School of Business, The University of Texas at Austin, August, 1967. Hillier, Frederick S., and Connors, Michael M. "Quadratic Assignment Problem.Algorithms and the Location of Indi- visible Facilities," Technical Report No. 6, Program in Operations Research, Stanford University, 1965. Lee, Robert C. "COmputerized RElationship LAyout Planning," Unpublished MS dissertation, Northeastern University, Boston, Massachusetts, 1966. Moore, James M. "Level Curve Approximation for Location Analysis." Unpublished paper, Department of Industrial Engineering, Stanford University. . "Mathematical Models for Optimizing Plant Layouts. Unpublished Ph.D. dissertation, Department of Industrial Engineering, Stanford University, 1965. Murphy, Daniel J. "Machine Location Patterns for Facility Analysis." Unpublished M.S.I.E. dissertation, Engineer- ing Library, University of Pittsburgh, Pittsburgh 13, Pennsylvania, April, 1957. Reimert, Philip R. "An Investigation of the Feasibility and Cost of Flexible Plant Layout Using Movable Production Machinery and a Computerized Scheduling Program." Central Library, Arizona State University, Tempe, Arizona, June, 1963. 261 Rice, Donald Blessing. "Discrete Optimizing Solutions to Linear and Nonlinear Integer Programming Problems." Unpublished Ph.D. dissertation, Purdue University, 1965. Seehof, J. M., et a1. "Status of Automated Plant Layout Program at Rochester (IBM).” Unpublished memorandum, 1964. Vollman, Thomas E. "An Investigation of Bases for the Relative Location of Facilities." Unpublished Ph.D. dissertation, University of California, Los Angeles, 1964. Willoughby, David W. "A Technique for Integrating Facility Location and Materials Handling Equipment Selection." Unpublished M.S.I.E. dissertation, Purdue University, 1967. Wilson, Richard Christian. "Evaluation of Spatial Relations and Empirical Plant Layout Criteria by Digital Computer.‘ Unpublished Ph.D. dissertation, University of Michigan, 1961. Wimmert, Robert J. "A Quantitative Approach to Equipment Location in Intermittent Manufacturing." Unpublished Ph.D. dissertation, Purdue University, 1957.