In presenting the dissertation as a partial fulfillment of the requirements for an advanced degree from the Georgia Institute of Technology, I agree

Transcription

1 In preenting the diertation a a partial fulfillment of the requirement for an advanced degree from the Georgia Intitute of Technology, I agree that the Library of the Intitute hall make it available for inpection and circulation in accordance with it regulation governing material of thi type. I agree that permiion to copy from, or to publih from, thi diertation may be granted by the profeor under whoe direction it wa written, or, in hi abence, by the Dean of the Graduate Diviion when uch copying or publication i olely for cholarly purpoe and doe not involve potential financial gain. It i undertood that any copying from, or publication of, thi diertation which involve potential financial gain will not be allowed without written permiion.

2 AN IMPLICIT ENUMERATION ALGORITHM FOR INTEGER PROGRAMMING A THESIS Preented to The Faculty of the Graduate Diviion by Lelie Earl Trotter, Jr. In Partial Fulfillment of the Requirement for the Degree Mater of Science in the School of Indutrial Engineering Georgia Intitute of Technology September, 1970

3 IMPLICIT ENUMERATION ALGORITHM FOR INTEGER PROGRAMMING Approved: Chairman f-l 72. * r\kw>\ V ( f t / Date approved by Chairman: Jfc<c /'/Y^

4 ACKNOWLEDGMENTS The author expree hi deepet appreciation and thank to Dr. C. M. Shetty whoe perfect combination of intruction and inpiration made thi work poible; hi keen inight erve a a continuing inpiration to the author. In addition, I would like to thank Dr. D. E. Fyffe and Dr. J. J. Jarvi for their helpful comment and uggetion concerning all apect of the reearch preented herein. Special thank are alo due to G. E. Moore of Control Data Corporation who granted the CDC 6600 computer time required for the computational invetigation purued in thi thei. Such an in-depth computational tudy would certainly have been impoible without thi aitance. I would alo like to expre my gratitude to my wife, Jomi, for her continued initence upon the excellence of thi work depite the many long hour of my attention which it required. Latly, I wih to thank Betty Sim for her excellent typing job.

5 iii TABLE OF CONTENTS Page AC KNOWLEDGMENTS LIST OF TABLES LIST OF ILLUSTRATIONS SUMMARY i i v vi vii Chapter I. INTRODUCTION 1 Cutting-Plane Algorithm Branch and Bound Method Geoffrion' Algorithm Scope of thi Study II. BASIC ALGORITHM 8 Problem Statement Definition Statement of the Algorithm Non-Redundance and Exhautivene III. MATHEMATICAL IMPLEMENTATION 22 Initial Solution Simple Tet for Fathoming Augmentation Fathoming Several Partial Solution Simultaneouly IV. SURROGATE CONSTRAINTS 34 Introduction Stronget Surrogate Contraint Sequential Solution to the Imbedded Linear Program Fathoming by Optimality Conideration Relative Bound Redefinition Near Optimal Solution An Alternative View of the Imbedded Linear Program Geometric Interpretation of Surrogate Contraint

6 iv Chapter Page V. COMPUTATIONAL RESULTS 57 Baic Conideration Reult VI. CONCLUSIONS AND RECOMMENDATIONS 74 APPENDIX A 81 B 87 C 105 BIBLIOGRAPHY 118

7 V LIST OF TABLES Table Page 1. Problem Characteritic Reult Uing Minimum-Branch Augmentation and No Bound Redefinition Reult Uing Bala' Augmentation Rule and No Bound Redefinition Reult Uing Minimum-Branch Augmentation and Bound Redefinition Reult Uing Bala' Augmentation Rule and Bound Redefinition, A Comparion Between Binary and Direct Implicit Enumeration 73

8 vi LIST OF ILLUSTRATIONS Figure Page 1. Flow Diagram of Implicit Enumeration Algorithm Flow Diagram of Imbedded Linear Program Fathoming Proce... 59

9 vii SUMMARY Thi tudy deal mainly with the bounded variable pure integer programming problem of the form Min cx ubject to linear contraint where the variable mut aume bounded integer value. The algorithm propoed for olving uch a problem i an implicit enumeration procedure cloely related to that advanced by Geoffrion for binary programming problem. However, the propoed algorithm deal with the bound on program variable directly. Following Geoffrion, we develop variou tet for implicitly enumerating "completion" to partial olution; thee tet are then extended to permit implicit enumeration of "completion" to everal partial olution imultaneouly. The ue of urrogate contraint in uch implicit enumeration i examined in depth and two new interpretation of uch urrogate contraint are given, including a geometric interpretation formulated in term of determining a eparating hyperplane between two ubet of the pace of lattice point being enumerated. Alo, mean of obtaining near optimum olution through a light modification of the algorithm are dicued. The theoretical conideration are upplemented by a computational tudy which deal not only generally with the efficiency of the algorithm developed a a whole, but alo pecifically with an evaluation of ome of the more important deciion tep of the algorithm. The reult of thi computational tudy indicate not only that the propoed algorithm i far more effective than exiting method for olving

10 viii integer programming problem but alo that it ability to dynamically redefine bound on program variable relative to pecific partial olution yield ignificant improvement in the efficiency of the enumeration proce. The computational invetigation alo indicate that the propoed algorithm will allow the efficient olution of integer programming problem ubject only to core memory retriction of preent-day computer.

11 1 CHAPTER I INTRODUCTION In recent year the field of Operation Reearch ha witneed a izable increae in the methodology concerned with dicrete variable programming problem; a formal tatement of uch a problem i given by Equation (l)-(4) in Chapter II. The literature now reflect many varied approache for olving the general integer programming problem. The computational ucce, however, of virtually all exiting algorithm i limited, thu leaving the iue of an economical mean for determining the olution to a given integer program largely unettled. Both the ize of the current reearch effort in the field of integer programming A [3,4] and the large number of problem which can be tated in dicrete variable attet to the importance of thi area. In fact, given computationally efficient mean of olving integer program, many indutrial problem currently tated a linear programming model, would be more realitically phraed in the context of integer programming. Thi thei deal mainly with an extenion of a particular integer programming algorithm [10]; conequently only a broad view of exiting method of integer programming will be dicued here. In thi regard, the work of Balinki [3,4] provide an excellent urvey of all apect of integer programming. Bracketed number refer to reference cited in the Bibliography.

12 2 Cutting-Plane Algorithm Depite the variety of available algorithm for integer programming, the majority of thoe propoed to date belong to either of two broad categorie: cutting-plane method and branch and bound method. Hitorically, cutting-plane technique were developed firt, with the bulk of the initial contribution in thi area due to R. E. Gomory [15,16,17]. The baic cutting-plane approach require that one firt remove the integrality retriction on all program variable and olve the reulting linear program; then by alternating the introduction of additional contraint (cut) with re-optimization, one continually retrict the feaible region of olution until an optimum olution thu obtained i integer-valued. Of coure, the character of an algorithm of thi type require that one take appropriate tep to inure that no feaible integral-valued olution are paed over during the proce; thi demontrate the dual nature of mot cutting-plane algorithm in the ene that intermediate reult are infeaible until one imultaneouly achieve optimality and feaibility of the original problem. Primal cutting-plane algorithm have alo been developed [13,27] which attain optimality by paing through a continually improving equence of feaible olution. More recently, promiing theoretical reult have been produced [18] for cutting-plane algorithm by uing a group theoretic approach to develop cut aociated with the face of the convex A more detailed categorization i, of coure, poible. For example, [3,4] conider additional method uch a dynamic programming and heuritic technique.

13 3 hull of feaible integer olution to the original integer programming problem. To a large extent, computational experience with cutting-plane algorithm ha proven diappointing. Some problem are olved quite eaily while other, differing only lightly, yield reult only after an inordinate amount of computation. A comprehenive explanation of the extreme enitivity of cutting-plane algorithm to problem tructure ha not yet been propoed. Branch and Bound Method Recent development in integer linear programming how an increaing amount of attention being focued on branch and bound method of olution. The idea central to branch and bound technique i that of conducting a highly pecialized earch of the entire olution pace in uch a manner that a large number of poible olution will be eliminated from conideration by excluion tet baed on the character of the algorithm. Thee method may be conceptualized a tree-earch algorithm which implicitly or explicitly conider each poible olution in it turn in a ytematic effort to determine an optimal olution. Intuitively, an effective algorithm of thi variety mut poe an efficient mean of remembering which olution have been conidered; further, the trength of a branch and bound algorithm will lie in it ability to implicitly enumerate a large proportion of the poible olution. Thee two ingredient, an efficient hitory-remembering cheme and a high rate of implicit excluion, are eential in evaluating the computational effectivene of a branch and bound algorithm.

14 4 Initial branch and bound approache were advanced by Eatman [6], Land and Doig [20], Little, et al. [22], and Bala [1]. The urvey article by Lawler and Wood [21] give an excellent account of branch and bound hitory and methodology. In addition, the work of Mitten [24] provide a good dicuion on the tructure of thi methodology. Bala ha extended hi work in [1] to produce hi Filter Algorithm [2] for binary (0-1) integer linear program. Thi algorithm, dual in nature, ue the olution to a linear programming problem to develop a "filter contraint" which peed the convergence of the algorithm to an optimal olution. Thi concept of uing a urrogate contraint (o-called becaue of it direct derivation from the "parent" contraint of the original problem) to accelerate convergence wa firt propoed by Glover [12] and later modified and extended by other [2,8,10,14]. Geoffrion' Algorithm The reearch of Geoffrion [7,8,9,10,11] in thi area appear to be epecially promiing in term of computational reult. Geoffrion' original work [7] wa a refinement of Bala' additive algorithm [1] which wa very amenable to eae of computation. Hi later extenion [8,9,10,11] include an imbedded linear program which i olved at ucceive iteration of the algorithm in order to produce urrogate contraint which are highly effective in relation to the excluion tet of the algorithm. The computational reult preented by Geoffrion in [10] illutrating the improvement gained by uing thi type of urrogate contraint are very convincing; thi evidence eem to indicate that, for hi tet problem, the increae in olution time a a function of

15 5 the number of binary variable in the original problem i linear or loworder monomial. Scope of thi Study Geoffrion' algorithm i tated within the context of the binary programming problem; ince any integer linear program whoe variable are bounded can be repreented in binary form, hi algorithm i alo applicable to the general bounded variable pure integer programming problem. We note that any bounded integer variable o < x ^ n may be repreented a the um of binary variable; i.e., x = 2 x o + + k k x^ where k i the mallet integer uch that n < 2-1 and x., for j = 0,l,**',k are binary variable. we are preented with 2 k+1 Clearly, in thi formulation poible value for x (one value for each of k+1 the 2 combination of the binary variable x^,x^,,x^). Since o < x < n and we have defined k to be the mallet integer uch that k+1 k+1 n < 2-1, it i clear that 2 > n + 1. Hence one might upect that an enumeration algorithm dealing only with the n+1 permiible value of x directly hould prove more efficient than an approach allowk+1 ing 2 > n + 1 permiible value of x. In a computational context, an additional practical advantage of an algorithm which avoid binary expanion of variable would be it.ability to handle problem with large bound on the variable more effectively ince an increae in the number of variable of a problem alo implie an increae in the amount of core torage required for handling the problem. The extenion of Geoffrion' baic algorithm [7,10] to allow direct treatment of bounded integer variable i traightforward; in the following chapter we turn

16 6 to the development, refinement, and computational examination of uch an algorithm. Since the approach taken i cloely related to that advanced by Geoffrion for binary programming problem, the aumption i made that the reader i familiar with reference [7] and [10] although the preentation here i reaonably elf-contained. In the chapter which follow we will preent the tructure of a baic algorithm deigned for olving bounded variable pure integer program; bounded variable will be treated directly rather than by converion to um of binary variable a decribed above. In addition we will how that the algorithm i both non-redundant and exhautive. Chapter III will give the mathematical mean for implementing the baic algorithm preented in Chapter II; thi implementation will include the development of method of implicit enumeration capable of accelerating the enumeration proce by the imultaneou conideration of everal olution. Chapter IV extend thee mean of implementation through examination of an imbedded linear program imilar to that of Geoffrion in [10]. Alo given in Chapter IV are two alternative interpretation of the type of urrogate contraint generated by thi imbedded linear program; Chapter IV alo preent convenient mean for the development of an algorithm which i deigned to find near optimal (rather than exactly optimal) olution. Many of the technique indicated in Chapter III and IV take advantage of the pecific tructure of an algorithm which treat bounded integer variable directly. Thee improvement are imply unavailable to an algorithm which relie on the binary converion of uch variable. Thee conideration are then implemented in Chapter

17 7 V in the form of a computational tudy which deal with a general evaluation of the efficiency of the algorithm a well a with a pecific invetigation of two important apect of the algorithm. The computer timing reult preented here indicate that the algorithm developed in Chapter II, III, and IV i certainly among the mot efficient, if not the mot efficient, mean available for olving integer programming problem. Furthermore, thee reult alo indicate that uch a direct algorithm a that preented in Chapter II, III, and IV i far uperior in virtually all cae conidered to algorithm which rely on binary converion of bounded integer variable. Finally we indicate in Chapter VI the concluion implied by thi thei and thoe avenue open to further invetigation.

18 8 CHAPTER II BASIC ALGORITHM Problem Statement The bounded variable pure integer programming problem may be tated in the following form. Determine x uch that: Min cx (1) ubject to b t Ax > 0 (2) 0 < x < d (3) x i integer-valued (4) where c, x, and d are n-vector, b i an m-vector, and A i an m*n matrix. Note that "a priori" knowledge of an upper bound d. on the value of each element x. of the olution vector x i aumed. In addition, complete flexibility on bounding a given variable from below i provided by (3) ince if one require that 1_. < x_. < d_. then the imple change of variable x! = x provide the equivalent contraint 0 < x! ^ d. - : : : : : 1.. In mot practical problem uch bound a 1. and d. are readily : : : available; if not, a lower bound 1_. for the variable x_. can be obtained a the olution to the linear program Min x. ubject to b t Ax > 0 and

19 9 x > 0 while an upper bound d_. can be obtained by olving Max x_. ubject to b + Ax > 0 and x ^ 0. Further aumption about (l)-(m-), without lo of generality, are that d i an integer-valued vector and that each element of c i non-negative. For each x., 3 thoe integer value of x. uch that 0 < x. ^ d. will be called the admiible value for x.. Each x-vector which atifie contraint 3 (3) and O) will be called an admiible olution; if (2) i alo atified, the olution will be termed feaible; any olution which verifie (l)-(m-) will be deignated a an optimal olution. Inadmiible value, inadmiible olution, and infeaible olution are all defined by reference to the obviou negation of the condition for definition given above. Since each element of the olution vector i bounded, it i apparent that the number of admiible olution to (3) and (4) i n finite and, m fact, equal to IT (d. + l). The particular technique to j=l 3 be developed for olving (l)-(4) will proceed through a ytematic enumeration of all admiible olution to (3) and (4) while retaining candidate for the optimal olution to the problem a they are dicovered. Since very few practical problem are amenable to total enumeration of admiible olution, effective mean of implicitly conidering portion If ome dj i not integral then we can replace dj by [dj], where the quare bracket denote the greatet integer le than or equal to dj, to obtain an equivalent tatement of (l)-(m-) in which d i an integer-valued vector. A For each j uch that C J < 0, the tranformation x-j = dj - Xj will reult in an equivalent formulation of (l)-(m-) in which c-; 0.

20 10 of the admiible olution pace mut be conidered; thee technique for implicit enumeration will be baed on thoe conideration which characterize the feaible olution to (2)-(4) and the optimal olution to (l)-(4). Definition The approach for enumerating admiible olution which follow i very cloely related to that given by Geoffrion in [7] and [10]; although the definition tated in the above tudy will be reviewed in our context for the ake of completene, familiarity with thee reference i eential. One quite natural approach for examining admiible olution i to attempt to reduce the number of variable involved by conidering the olution which reult when certain variable are held contant at admiible value. We refer to uch variable which have been aigned one of their admiible value a fixed and, imilarly, we denote the remaining variable a free. Thi dichotomy of the problem variable implie that we will be dealing with admiible olution which reult from aigning admiible value to thoe variable which are free. Hence we define a collection of fixed variable a a "partialolution; any aignment of admiible value to thoe variable not in a given partial olution will be called a completion to that partial olution. Note the implication that there may be many different completion to a given partial olution. A completion to a partial olution reulting in a feaible (infeaible) olution to the problem under conideration will be called a feaible (infeaible) completion. It i clear that a given partial olution may poe completion of either variety.

21 11 Any enumeration procedure which i computationally efficient mut poe the following propertie: The procedure mut be exhautive, in that it mut conider (whether explicitly or implicitly) all admiible olution to the problem being conidered. Secondly, the proce mut be non-redundant; i.e., it hould never return (whether explicitly or implicitly) to a olution previouly enumerated. Conequently, any effective enumeration proce mut be equipped with mean for "remembering" at each tage which olution have previouly been examined. In thi regard, we will ue the following notation: The ordered index et S = {j^,j2»«-«} will contain the indice of thoe variable which are currently fixed in the order in which they became fixed; the partial olution vector X = (x.,x.,...) will be ued to record the value aumed by thoe variable in the current partial olution. The correpondence between the element of X and thoe of S will be that the order of appearance of the element of S (from left to right) determine the ame ordering of the element of X c. For example, if S = {5,4,1,3} and Xg = (3,8,9,0) then the partial olution currently under conideration i: Xj- = 3, x^ = 8, x^ = 9, x g = 0; the remaining variable are currently free. To inure the implicit character of the algorithm, mean of determining information pertinent to optimality and feaibility from a pecific partial olution mut be conidered. Since we are olving a minimization problem, the following quetion i of interet: Doe a More precie definition of exhautive and non-redundant are given in a later ection of thi chapter.

22 12 given partial olution admit to a feaible completion with objective value lower than the bet feaible olution dicovered thu far? Thi quetion may be anwered in one of three way: (1) By demontrating the non-exitence of a completion with the required propertie. (2) By exhibiting the bet feaible completion available. (3) By reorting to additional explicit enumeration of the completion to the partial olution under conideration. In any event, if, at any given tage of the enumeration procedure either (1) or (2) can be hown, we hall ay that the partial olution being conidered ha been fathomed. The fathoming proce indicate that all relevant information ha been extracted from a partial olution; hence, after fathoming a particular partial olution, the enumeration proce hould proceed to conider a different partial olution. Thi proce of directing the enumeration to another partial olution after one ha been fathomed i called backtracking. A previouly indicated, effective mean mut be provided which will not only guide the enumeration proce to the next partial olution for conideration after fathoming but will alo maintain record of which partial olution have been previouly invetigated. At each tage of the algorithm we attempt to fathom the current partial olution X. If thi attempt i ucceful, we backtrack to a new partial olution a follow: we imply change the value of the rightmot element of X, requiring only that thi new value ha not been previouly aigned to thi variable (i.e., the variable whoe index i the rightmot element o

23 13 of S) ince it index lat entered S. Furthermore, if the value aumed by a fixed variable under uch an operation i it lat admiible value, we underline the rightmot element of S to indicate that the current partial olution ha been fathomed for all value, except the current value, of that variable. If, on the other hand, an attempt to fathom X i unucceful, we append the ubcript of ome free variable a the new rightmot element of S and enter one of the admiible value of thi variable a the new rightmot element of X. Thi augmentation proce i then followed by an attempt to fathom the new partial olution, and o the proce continue. Thee idea are illutrated in the following example. Suppoe S = {1,4,2} and Xg = (3,7,2), indicating that the current partial olution i x^ = 3, x^ = 7 and x^ - 2. Firt we attempt to fathom X^; if ucceful, we then proceed to a new partial olution by altering the value of x 2> Suppoe then that 0 < x^ < 5, of which value x^ ha aumed 5,4, 3 and 2 ince the index 2 lat entered S. Aume ucce on thi attempt to f athom X c ; we might now chooe to et x = 1, proceeding to the partial olution with S = {1,4,2} and X = (3,7,1). Continuing, we now attempt to fathom thi new partial olution; again aume thi attempt i ucceful. The next partial olution to be conidered mut be S = {l,4,2_} and X^ = (3,7,0). Note the ignificance of the underline; we have now fathomed X g for each admiible value of x^ except for it current value of 0. Once more an attempt i made to fathom the new partial olution. If again ucceful, then all poible completion of X will have been enumerated, indicating that the partial

24 14 olution given by S = {1,4}, X = (3,7) ha been fathomed. Hence, allowing the proce to telecope, the next partial olution would be obtained by dropping the index 2 from S and altering x^. On the other hand, uppoe that the attempt to fathom the partial olution with S = {1,4,2} and X = (3,7,0) i unucceful. In thi event, ome free variable would then be fixed at one of it admiible value, x^ = 4 for intance, and the next partial olution conidered would be given by S = {1,4,2_,7} and X g = (3,7,0,4). The proce would then continue a above. Thu we have characterized the implet form of the enumeration algorithm. Each iteration of the proce i begun with an attempt to fathom the current partial olution. If thi i ucceful the enumeration proce backtrack to a different partial olution; fathoming i attempted for the new partial olution, etc. If unucceful, however, the enumeration proceed to a new partial olution obtained by fixing ome free varible at one of it admiible value; fathoming i attempted for the new partial olution, etc. Thi proce i continued until the partial olution with S = <f> (or, e qui vale nt ly, a partial olution in which all element of S are underlined) ha been fathomed, indicating that all admiible olution to the original problem have been examined, and the proce terminate. Statement of the Algorithm The following tep demontrate the equence of operation followed by the enumeration proce: (1) Chooe an initial partial olution. Proceed to tep (2).

25 15 (2) Attempt to fathom the current partial olution. If unucceful, go to tep (6). If ucceful, proceed to tep (3). (3) If the current partial olution admit to a feaible completion with objective value lower than that of the current bet feaible olution (inoumbent), replace the incumbent by thi improved olution. Proceed to tep (4). If all element of S are underlined, top; the current incumbent i optimal. Otherwie, proceed to tep (5). (5) Alter the value of the fixed variable whoe index i the rightmot non-underlined element of S to a new (not yet conidered) admiible value and drop all element of S to the right of thi index. Underline the new rightmot element of S if it correponding variable ha now aumed each of it admiible value ince it index lat entered S. Proceed to tep (2). (6) Fix ome free variable to one of it admiible value, update S and X accordingly, and go to tep (2). We next focu attention on mean by which the enumeration proce decribed above may be proven non-redundant and exhautive, after which ome mathematical tet ufficient for fathoming partial olution are preented in Chapter III. Non-Redundance and Exhautivene The two eentialitie which are central to the ucce of an enumerative algorithm are non-redundance and exhautivene. It eem reaonable that one hould require that the nature of an enumerative olution proce be uch that neither doe it repeat itelf nor doe it

26 16 omit any admiible candidate for olution. Thee heuritic concept are pecified more preciely when tated in term of the equence of partial olution to (l)-(4) examined by the enumeration algorithm. A equence of partial olution which hare the following property i called non-redundant: No partial olution admit to a completion which duplicate a completion of a partial olution previouly fathomed. Further, if an enumerative algorithm ceae only after the implicit or explicit generation of all admiible olution to a problem, it will be called exhautive. We hall refer to any one of the (d_.+l)! poible complete ordering of the admiible value of the variable x_. a an arrangement of thoe value. A pecific arrangement for x_. will be ued to determine the order in which ucceive admiible value of the fixed variable x_. are conidered. It i natural, for reaon relating to computational implicity, when implementing the algorithm preented in the Earlier ection of thi chapter, to retrict attention to the following two arrangement for the value of x_.: 0,.l,...,d_. andd_.,d_.-1,...,1,0 (i.e., thoe arrangement which define a monotone equence of the admiible value of x_.). It will become clear that the following proof for non-redundance and exhautivene apply in complete generality with repect to any particular arrangement choen for the value of a given fixed variable. However, for convenience we hall aume that the arrangement for x_. i given by 0,1,...,d_. in the proof which follow. The proof of non-redundance preented here i very imilar to that given by Geoffrion in [7]; everal lemma are firt provided in

27 17 order to facilitate the final proof of non-redundance. The following lemma etablihe a mean of perpetuating the non-redundant property throughout a equence of partial olution. For notational convenience we uppre the reference to the aociated index et and denote a 1 k equence of partial olution a X,,X where it i undertood that 1 k the index et S,«.«,S which precribe the variable in thee partial olution are not necearily identical (and alo not necearily different ). 1 k Lemma 1: Given a non-redundant equence of partial olution, X,,X k+1 and a partial olution X which contain at leat one element differ- 1 k ent from each fathomed partial olution in the equence X,,X, the 1 k+1 equence X,,X i alo non-redundant. k+1 Proof. Since X contain at leat one element different from 1 k each fathomed partial olution of the equence X,,X, it i clear k+1 that no completion of X can duplicate a completion of any fathomed 1 k+1 partial olution in that equence. Hence X,,X i non-redundant. Next uppoe j i the rightmot element of S at ome tage of the enumeration proce. We can inure that the property of nonredundance i maintained throughout a equence of partial olution obtained by allowing x.. to aume each of it admiible value in k-1 turn, according to it arrangement. To thi end, let X be a partial olution with repect to a pecific index et S of fixed variable. Further, let S' : Su{j} for ome j uch that x. i free relative to X c with admiible value 0,l,»»',d_.. In the following lemma we conider the

28 18 k k+d-" d_. + 1 partial olution X,,X ^ (recall notational implification) with repect to the index et S T, where x. aume the value 0,1,...d. ] 3 k k+1 k +dn m the partial olution X,X,,X, repectively. 1 k-1 Lemma 2: Aume the equence of partial olution X,,X i nonk k+1 k+d-j J redundant, ana that the partial olution X,X,,X are formed k+i in the following manner: X i obtained by including x. at the value k-1 k-1 i in X. Further aume that the partial olution X ha not been 1 fathomed. Then the entire equence X,...,X k + d i J k 2 i non-redundant. Proof. Aume, on the contrary, that X ha a completion which k k l l duplicate a completion of X where X wa fathomed, 1 < k^ < k^ < 1 k-1 k + d_.. If k^ ^ k-1 we violate the non-redundancy of X,,X ince k^ < k^. Hence k^ ^ k and we conider the following mutually excluive and collectively exhautive cae: (1) k^ ^ k and k^ > k-1. In thi cae it i impoible for a k k 2. l. completion of X to duplicate one of X ince x. aume different ki k 2 value in X and X (2) k 2 ^ k and k = k-1. Thi contradict the aumption that k l k-1 X ha been fathomed wherea X ha not been fathomed. (3) k^ ^ k and k^ < k-1. We have aumed that X ha a comk l pletion which duplicate a completion of X. But, every completion of k 2 X k-1 k-1 i alo a completion of X. Hence X ha a completion which duplicate a completion of X k l, which contradict the non-redundancy of 1 k-1 the equence X,,X. The deired concluion follow. The following theorem etablihe the property of non-redundance for the enumerative algorithm preented in the previou ection of thi chapter. k 2

29 19 Theorem 1: The equence of partial olution examined by the baic enumeration proce decribed previouly (under the ection "Statement of the Algorithm") i non-redundant. Proof: Proceeding by induction, it i clear that the initial 1 k partial olution alone i non-redundant; hence we aume that X,,X i a non-redundant equence of partial olution and non-redundance 1 k k+1 k+1 will now be hown for the equence X,,X,X where X i the partial olution generated by the algorithm after attempting to fathom X. In the context of the algorithm decribed there are only three k+1 k mean by which X can be generated from X : (1) Step (6) of the baic algorithm: After an unucceful k k+1 attempt to fathom X,X the partial olution X i formed by augmenting a free variable into at value 0 (recall our retriction on the arrangement of the value of x. : 0,1,...,d.). 3 3 (2) Step (5) of the baic algorithm: After uccefully k k+1 fathoming X, X i formed by a unit increae in the rightmot element of X k. (3) Step (5) of the baic algorithm: After uccefully fathomk k+1 ing X, X i formed by dropping a tring of underlined element from the right of X, and then increaing the new rightmot element of the partial olution thu obtained by 1. In cae (1) and (2) the deired reult follow from Lemma 2. In cae k mot recent introduction (at the value of 0 0) of the variable which wa (3) uppoe that the partial olution X, k^ < k correpond to the k+1 k increaed by unity to obtain X from X. Then it i clear that the

30 20 k+1 variable whoe value i the lat element of X now aume a different ^0 k value than it aumed in the equence of partial olution X,,X. k+1 By Lemma 1 we conclude that X It remain to verify that the equence of partial olution examined by the enumeration proce i exhautive. i non-redundant with repect to the 0 k equence X,«««S X. A in cae (2) it i alo clear from Lemma 2 that X i non-redundant with repect to the equence X,,X. Cone- 1 k+1 quently the entire equence X.. 9 X i non-redundant, a required. Theorem 2: The equence of partial olution examined by the baic enumeration proce decribed previouly (under the ection "Statement of the Algorithm") i exhautive. Proof: The proof relie heavily on the manner in which the algorithm accomplihe it "hitory remembering." Termination of the algorithm occur only if a partial olution with an index et S of all underlined element i fathomed. After Geoffrion ([7], page 190), it i clear that a ufficient condition for exhautivene i that the underlining of a certain element of S connote the following: that we have uccefully fathomed (implicitly or explicitly) all poible completion up to and including all other admiible value for the variable indicated by the underlined index. Thi property i evident from the decription of the algorithm, ince we underline an element of S if and only if it correponding variable i currently aumed fixed at the final admiible value in it arrangement. Hence, when a partial olution coniting of all underlined element i fathomed, the fathoming

31 21 proce telecope, indicating the complete examination of all admiible olution. Hence the enumeration algorithm under invetigation doe indeed poe the requiite for computability: non-redundance and exhautivene. Note that for the mot part the decription of the algorithm thu far ha dealt in generalitie, thu permitting a maximum of flexibility in it implementation. Simultaneouly, however, we have precluded the poibility of making quantitative tatement concerning the efficiency of the algorithm. Thi characteritic will depend on the ability of the algorithm to function implicitly rather than explicitly a feature which can only be evaluated within the framework of a pecific implementation of the algorithm. Thu we next conider the development of uch mathematical tet a will be ued in implementing the algorithm.

32 22 CHAPTER III MATHEMATICAL IMPLEMENTATION The implicit enumeration algorithm decribed in Chapter II ha everal deciion tep for which mathematical rule mut be provided. Three pecific area of the algorithm will be conidered in detail: (1) mean for chooing a partial olution for initializing computation, (2) tet for fathoming a partial olution, and (3) criteria for augmentation which pecify not only which variable hould be fixed, but alo at which value it hould be fixed. We will alo invetigate how certain fathoming tet can be eaily extended in order to fathom everal partial olution imultaneouly. Initial Solution The inductive nature of the proof for non-redundance aure u of the preervation of thi property, regardle of the initial partial olution. However, the depepdence of the exhautive property on the method of remembering which olution have been conidered retrict the initial et S to be of the following form: Any ubet of the problem variable may be fixed, but each fixed variable mut be initialized at the firt value of it repective arrangement of admiible value. Thi i intuitively clear; it i cloely related to the connotation

33 23 given to an underlined element of S. Within thee conideration, ome permiible tarting convention would be: (1) Let S Q denote the initial index et of fixed variable. Then = <J> repreent a partial olution with which the algorithm can be initialized. (2) Computation may begin with every variable in the initial partial olution. To be conitent with the convention tated in Chapter II of conidering only the arrangement 0,l,«««,d_. and d^.,d_.-l,,1,0 for the variable x_., any uch initial partial olution (i.e., with S = {l,2,«..,n}) mut have that element of X correpond- U b 0 ing to x. fixed at either d. or 0. Two poible X vector correponding to S N = (1,2,-.,n} are X c = (0,0,-..,0) and X_ = (d n,d.,...,d ). 0 b Q S Q 1 I n (3) Sq could be initialized to contain only thoe indice correponding to variable which are integer-valued in the continuou correpondent of the integer program under conideration. In thi cae a uitable arrangement hould be choen for each x., j e S o that x. ] 0 3 would be initially fixed at ome value near that taken by x_. in the optimal olution to the continuou problem. (4) If a feaible olution to the initial problem i known, it can be ued a a ource for everal different initial partial olution. For intance, one might have = {j x_. > 0 in the given feaible olution} with X choen a decribed in (3). An alternative would be to b 0 have S = {j x. = 0 in the given feaible olution} with X = (0,»««,0). 0 1 S 0 Heuritic cheme for initialization which reflect the tructure of certain clae of problem could alo be ued. In fact,

34 24 initialization can be a very important factor in peed of olution; e.g., a good initial tarting point may lead quickly to a good feaible olution, thu allowing optimality conideration (ee the following ection on fathoming) to exclude large number of olution from explicit conideration. Simple Tet for Fathoming To a large extent, the trength of any implicit enumeration algorithm lie in it fathoming proce, ince it i by fathoming that olution are evaluated implicitly rather than explicitly. To fathom a given partial olution it i ufficient either to demontrate a bet feaible completion to that partial olution or to determine that there are no feaible completion to that partial olution or finally to how that no feaible completion to the partial olution improve on the objective value of the incumbent. We hall conider fathoming tet baed on each of thee three conideration. Fathoming by Optimality Recalling that the problem formulation (refer to Chapter II) require that c_, > 0, j = l,...,n, a trivial tet for fathoming by optimality i available. Let z be the current incumbent objective v value and z = i ex. be the objective value of the current partial jes 1 : - olution. Then, clearly, if z < z the partial olution X ha been fathomed by optimality. A another poibility, note that at any pecific iteration of the algorithm, the problem of determining a bet feaible completion to the current partial olution i again an integer programming problem. In certain intance the optimal olution to thi

35 25 ubproblem i eaily obtained. The following imple tet provide ufficient mean for etablihing a bet feaible completion to the partial olution being examined: Form the completion correponding to an aignment of 0 value to all free variable. If thi particular completion i feaible, it i certainly a bet feaible completion ince to aign a poitive value to any free variable would increae the value of the objective function. Thi particular completion i alo a bet poible completion from the current partial olution in the ene that it determine a lower bound on the value of the objective function which can be obtained by any completion of the current partial olution. We are thu provided with ufficient mean for fathoming a partial olution via determination of it bet feaible completion; if thi tet prove ucceful, the new incumbent olution i recorded and the enumeration proce backtrack to a new partial olution (a dicued in Chapter II). If thi tet i unucceful, however, we next attempt to fathom the current partial olution by demontrating either that it can poe no feaible completion or that no feaible completion to the current partial olution ha objective value improving on that of the incumbent. Fathoming by Infeaibility Further mean of fathoming a partial olution can be developed baed on feaibility conideration alone. Note that at any pecific iteration, the problem under conideration relative to the current partial olution X Q may be expreed a: Min z S + c.x. (5) j4 1 3

36 1 26 v ubject to b. + 2, a - - x -» i=l,"-,ni j4s d. ^ x. ^ 0 and x. integer, j^s where we define z S = V ex. and b S. = b. + 7 a..x c 3 3 c 3 3 J J J J jes jes A pecific contraint of (5) will be termed Unfeaible if no aignment of admiible value to the free variable exit which atifie that contraint. More preciely, a neceary and ufficient condition that the ith contraint of (5) be infeaible i that b. + Max T a..x. < 0. Note that the maximization in thi expreion i accomplih by aigning a value of 0 to thoe x_. for which a^_. < 0 and l a value of d. to thoe x. for which a S Y 1 Thu if any contraint of (5) atifie the inequality b. + ) Max(0,a..d.) < 0, then the partial olution given by Xg ha no feaible completion and hence ha been fathomed. We next conider a mean for fathoming a partial olution by howing that any feaible completion to that partial olution cannot improve on the objective value of the incumbent olution. Fathoming by Condition Related to Both Optimality and Feaibility Conider again the problem (5) which i encountered at any tage of the enumerative procedure. If we denote the incumbent objective value a z, then it i clear that no x., j^s for which z S + c. ^ z can 3 3 be non-zero in any completion to Xg which improve (trictly) the objective value of the incumbent olution. Similarly, in the event that augmentation i neceary we would like to aign poitive value only to thoe free variable which can lead to a reduction in the infeaibility

37 27 of (5) (i.e., to thoe x.. j 4 S for which a.. > 0 for ome i uch that 1 i: < 0). In thi context we define the et T a follow: T S = {j j< S; z + c. < z and a.. > 0 for ome i uch that b. < 0}. I T thu contain indice of thoe free variable which are candidate for becoming fixed; i.e., augmentation of one or everal of thoe variable indicated by the element of T into the et S may lead to an improved feaible olution (i.e., a new incumbent). Note in particular that if T = <{> then the partial olution given by Xg ha been fathomed by demontration that no feaible completion of X^ reult in an improved incumbent objective value. Thee variou imple tet for fathoming are eaily implemented computationally to provide everal ufficient mean for fathoming a given partial olution. Note, in particular, that the tet for contraint infeaibility i omewhat limited in that in it preent form it i only applicable to individual contraint. Many powerful mean of fathoming, baed on conidering non-negative linear combination of the contraint of (5) will be developed in Chapter IV. We now turn to conideration of the augmentation tep of the algorithm. Augmentation The augmentation proce i the explicit enumeration portion of the algorithm. When attempt to enumerate implicitly (i.e., via fathoming) fail, we reort to explicit enumeration until fathoming ucce

38 28 occur. Given that there i coniderable latitude in determining the pecific of augmentation, it eem reaonable to conider an augmentation proce which might increae the efficiency of the algorithm (ee reult in Chapter V) by precribing the augmentation of variable which lead to partial olution which are amenable to the fathoming tet ued. Conequently everal poible augmentation dicipline will be decribed; each i baed on heuritic conideration of the enumerative procedure. (1) In an attempt to proceed quickly to a feaible olution, Bala' augmentation rule i available: Fix at it upper bound that m variable x. which maximize the expreion J min(0,b.+a..d.) over all 1 1 : i=l D D j e T. In a ene the noted expreion give a meaure of the total ytem infeaibility remaining after x_. i fixed at value d_., o that maximization of thi non-poitive expreion minimize ytem infeaibility for the next iteration. (2) In direct contrat to (1), we might fix at it upper bound that variable which maximize total ytem infeaibility. Thi approach might prove ueful (during for intance, the phae in which the algorithm i verifying optimality of an incumbent olution) by making the partial olution obtained at the next iteration more uceptible to fathoming tet baed on infeaibility. (3) Another approach would be to fix that variable (at either it upper or lower bound) which minimize d_. for j«s ince thi would minimize the number of additional branche added into the explicit olution tree on the ubequent iteration. The computational comparion

39 29 of thi rule and (1) above given in Chapter V eem to indicate that thi i a worthwhile heuritic. (4) One might alternatively conider a minimum cot augmentation dicipline, baed on the heuritic that an algorithm uing thi rule would tend to explore more quickly thoe olution with lower objective value and hence obtain a good feaible olution more quickly. Such a rule would augment at it lower bound that variable which minimized c. for all j<is. Fathoming Several Partial Solution Simultaneouly The neceary mean for a imple implementation of the enumerative algorithm are now at hand; tet for fathoming partial olution and rule for augmentation can be choen from thoe preented above with coniderable flexibility. Before conidering more powerful method of fathoming baed on infeaibility of ytem of contraint, we firt conider ome traightforward mean of extending the above tet in order to fathom everal partial olution imultaneouly, taking advantage of an appropriate arrangement of the admiible value of the variable whoe index i rightmot in S. We note that uch tet are, of coure, applicable only to an algorithm which deal with bounded integer variable directly rather than in piecemeal fahion a in the binary converion of a problem with bounded integer variable. Succe in fathoming i alway followed by backtracking, and conequently one might upect that fathoming everal olution imultaneouly hould tend to accelerate the enumeration proce.

40 30 Optimality Tet Fathoming by optimality occur when the bet poible olution i feaible (a decribed earlier). Suppoe that a partial olution X g i fathomed by optimality, and let j be the rightmot index of S. In thi cae, if the fixed variable x.. i being equenced through increaing value (arrangement 0,1,...,d_.), we may immediately eliminate j from S and backtrack accordingly, ince c.. ^ 0 implie that an increae in x_. will lead to partial olution with a non-optimal objective value. Alternatively, uppoe thi ame ituation i encountered and x.. i currently being decreaed (arrangement d..,d_.-l,...,1,0). Then conideration of feaibility imply that x_. may be reduced to 0, x. = Max <, 3 x. - 3 l a.. 13 for i uch that a.. < , for i uch that a.. > 0 13 before violating feaibility while improving the incumbent olution. Infeaibility Tet Another mean of fathoming developed for ingle partial olution wa by contraint infeaibility. The extenion in thi cae to fathoming multiple partial olution i quite natural. Suppoe that the ith contraint i infeaible relative to the partial olution X g ; i.e., 3 = b S. + J Max(0,a..d.) < 0. Let j be the current rightmot index in S. If x. i being increaed (arrangement 0,1,...,d.) and a.. < 0 then 1 -'o "'o all admiible value of x. have been fathomed, ince to increae x. 3 ] o o

41 31 under uch condition cannot diminih the infeaibility of the ith contraint (i.e., cannot increae 3). A imilar reult i obtained if x. -'o i being decreaed (arrangement d.,d.-1,...,1,0) and a.. > 0. On the 1 1 il other hand, uppoe that x_. i increaing and a > 0. In thi cae o we can only increae x. by -'o a.. o, at which point the ith equation become feaible. Again, a imilar reult hold if x. i being -'o decreaed and a.. < 0. It i ueful to note the applicability of thee 1 -'o reult to urrogate contraint, which are deigned to be pecifically uceptible to uch tet (ee Chapter IV). Tet for Relative Bound Redefinition Another form of fathoming everal partial olution imultaneouly i by recomputing the bound on the free variable relative to a given partial olution. At any tage of the enumeration proce the quetion of interet in attempting to redefine bound on the free variable i: What are the upper and lower bound on, j^s for all feaible completion to the current partial olution which improve on the incumbent value of the objective? It i clear that ufficient tet for anwering thi quetion will provide mean for imultaneouly fathoming everal partial olution ince they will limit the range of a variable to be augmented. Certainly no variable hould be fixed at a value which implie a non-optimality (multiple fathoming via optimality); i.e., no variable hould be fixed at one of it admiible value uch that z + ex. ^ z. 3 1 Similarly, no variable hould be fixed at a value which force one of the contraint of (5) to be infeaible (multiple fathoming via g _

42 32 infeaibility); i.e., no x_. hould be fixed at a value uch that b S. + a..x. + Max(0,a.,d, ) < 0. Thee conideration lead to the 1 ± J 3 k*j,k4s ^ k poibility of defining a new (and poibly trengthened) upper bound on each free variable; uch bound will be valid in any improving feaible olution to the current partial olution. Thi upper bound redefinition may be accomplihed by computing a temporary upper bound d\ for x_.,j^s by conidering the following contraint: x_. > 0 and integer (6) x. < d. : 3. z - z X. < 3 c. x. < b^ + J Max(0,a. k d k ) k*i kis tll 13, for all i uch that a.. < 0. In addition, it i eaily een that a new lower bound redefinition may be accomplihed by computing a temporary lower bound i\ for x_.,j< S by conidering the following contraint: x_. ^ 0 and integer (7) x. < d. 3 3 b. + I Max(0,a. k d k ) k x. > ^ ' k( * S, for all i uch that a.. > a.. 13 J 13