parametric multicriteria integer programming

P. Hansen. ed., Studies on Graphs and Discrete Prograrnrning@ North-HoIland Publishing Cornpany (1981) 371-379

PARAMETRIC MULTICRITERIA INTEGERPROGRAMMING*

Bernardo VILLARREALInstituto Tecnologico y de Estudios Superiores de Monterrey, Monterrey, Mexico

Mark H. KARWANState University of New York at Buffalo, Buffalo, NY 14260, USA

The pararnetric prograrnrning problern on the right hand side for rnulticriteria integerlinear prograrnrning problerns is treated under a (hybrid) dynarnic prograrnrning approach.Cornputational results are reported.

1. Introduction

This paper presents an extension of the dynamic programming scheme formulticriteria discrete programming problems suggested in [7,8], to solving theparametric problem on the right hand side. The multicriteria parametric integerlinear programming problem of concern is formulated as follows.

v-maxN

L en . Xn,n=1

(MPILP) s.t.N

L a n . Xn ~ {b + ed},n=l

integer, (n = 1, . . . , N),

where en = (Cln, . . . , cpn)t; an = (aln, . . . , aMn)t denotes a set of integers; kn isa positive integer constant; d = (di> . . . , dM)t denotes a direction vector; e E

[O, 1] is a positive parameter; and v-max (vector maximization) is used todifferentiate the problem from the common case of single-objective maximiza-tion. The solution to (MPILP) corresponds to obtaining the sets of efficient orPareto optimal solutions for the family of multicriteria integer programmingproblems whose right hand side vectors líe along the ray defined by (b + ed),e E [O, 1]. This problem has not yet been approached under a multicriteriaframework. Previous work on parametric integer programming on the righthand side has concentrated on the single objective case [5,6,4,3].

* This research was supported in part by the Consejo Nacional de Ciencia y Technologia, Mexico.

371

372 B. Villarreal, M.H. Karwan

The mathematical model of concern is of practical interest when manage-ment would like to perform a sensitivity analysis of the Pareto optimal set forseveral values of the vector of resources. This situation may occur when precisepoint estimates or values of the amount of resources available are not known.Thus, obtaining the solution set for the family of problems generated byvarying the resources over a range of values of interest, will allow managementto determine how sensitive this set is to changes in the amount of resources. Inmany investment and location problems containing 0-1 decision variables, a setof alternatives must be judged under different criteria. Some essential con-straints are of the budgetary type and the effects of varying budgets should beone of the main concerns of the decision makers.

The plan of this paper is as follows. The following section deals with thedevelopment of a scheme for solving (MPILP). This procedure is modified inSection 3 by the inclusion of bounds to obtain a more efficient procedure.Section 4 offers some computational results as well as conc1usions and com-ments.

2. Development01 aIgorithm

The parametric extension of the earlier works of Villarreal and Karwan [7, 8]can be handled by modifying the dyn~mic recursive procedure which theydevelop. This is done by taking advantage of the characteristics of the recur-sions. Specifically, one can obtain the set of efficient solutions for any nonnega-tive vector of resources, y(~b). This implies that one could solve (MPILP) bysolving a multicriteria integer linear programming problem with a right handside vector with components bi = max{b;,bi+ dJ, i = 1, . . . , M. Then at the finalstage, instead of requiring the set of efficient points for only one vector ofresources, one would compute the set of efficient solutions for each vector,b + ed; e E [O, 1]. In order to c1arify these ideas, let us outline the imbeddedstate dynamic programming procedure suggested in [7,8] without consideringthe bounding feature reported there.

Step O. Set m = 1 and let cf>o be an empty set.Step 1. Obtain the set of m-dimensional integer points (x¡,. . . , xm) such

that O ~ Xm ~ km and (X¡, . . . , Xm-l) E cf>m-l.

Step 2. Eliminate all those m -dimensional points that are infeasible, Le.,those for which some component, i, of their resource consumption vectorsatisfies the following expression.

m N

L ain . Xn > bi + Ln=l n=m+l

ieHn

--

Parametric multicriteria integer programming 373

Step 3. Obtaín the set of resource efficíent solutions, <1> m, vía pairwisecomparisons.

Step 4. If m = N go to Step 5. Otherwise, set m = m + 1 and go to Step 2.Step 5. For any given vector of resources, YN (~b), obtain the set of efficient

solutions by solving the following problem via pairwise comparisons.

v-maxttl en . x.. I (Xl' . . . , XN)E <l>N and ntl an . Xn ~ YN).

Here, <l>mdenotes the set of re so urce efficient solutions for the m-stageproblem. These solutions have the property that if XOE <l>mthere is no otherfeasible point, X, such that the following expressions are satisfied with at leastone strict inequality in the first expression.

m m

L en . x.. ~ L en . x~,n=l n=l

m m

L an . Xn ~ L an . X~.n=l n=l

The set Hn required in the procedure is the set {i I ain < O}. For the case ofmulticriteria multidimensional knapsack problems, Hn is empty.

We can immediately notice that changing the right hand side vector ofresources will only affect the feasibility test of Step 2, and Step 5 whenobtaining the set of efficient solutions for the particular vectors of resources ofinterest. This follows from observing that the scheme computes, as an ínitialgeneral step, the set of resource efficient points for the relaxed problem:

N

"" en . X~ n>n=l(P')

v-max

s.t. integer, (n = 1, . . . , N).

Then, the constraints imposed on the resources, YN, are used in the feasibilitytesting step during the recursion and at the final step of the scheme. (This isone of the main differences of the dynamic programming approach to thatsuggested by Bitran [1] for the case in which 1<..= 1, n = 1, . . . , N. He first findsthe set of efficient (not resource efficient) points for the relaxed problem (P')before imposing the feasibility restrictions. Then, he proceeds to find otherpoints which were not efficient for (P') but which are efficient to the morerestricted problem.) This observation leads one to further conc1ude tliat regard-les s of the right hand vector of resources, the procedure would remain almostthe same with the exception of the feasibility test, until the final step, when aspecific vector of resources must be determined. Hence, since the parametricinteger linear programming problem with multiple criteria (PMILP) differsfrom the original multiple criteria integer linear programming problem by thestructure of their right hand vectors of resources, one could use the dynamicrecursions, with slight modifications, to solve (PMILP).

374 B. Villa"eal, M.H. Karwan

Let us define B and 1] as

and

where

b. = max ( b. b. + d. )1 ~, I I and Q¡ = min(b¡, b¡ + dJ.

Given these definitions, one can modify the recursive scheme as follows:

(a) Replace Step 2 byStep 2'. Eliminate all those m-dimensional points for which some compo-

nent, i, of their resource consumption vector satisfies the following condition.m N

L ain. xn>b¡+ Ln=l n=m+l

¡eHn

\ll¡n . Xn l.

(b) Replace Step 5 byStep 5'. For each vector of resources, YN, such that 1]::::;;YN::::;;B, obtain .the

set of efficient solutions by solving the following problem via pairwise compari-so ns..

v-maxttl en . Xn I(Xi>...' xN)e cbNand ntl an . xn::::;;YN}.

Even though the formulation of this problem suggests the direct variation ofthe vector of resources, YN,in order to solve the family of problems associatedwith all the values of B, one actually works only with those vectors of resourceswhich make possible new alternate feasible solutions. In this procedure, westart at YN = 1](B = O), and consider only those solutions that are feasible, todetermine the efficient set of solutions. The set of infeasible solutions is storedto be considered for obtaining the set of efficient solutions for subsequentvectors of resources. The process is continued considering next feasible solu-tions one by one until the set becomes empty. O~e determines if each newfe asible solution is efficient by simple comparison with those currently efficient.Each time a solution is dominated, it is discarded from further consideration.The next value of B (~O) is obtained by using the following relationship.

where Yiq is the ith component of the resource consumption vector for the qth. . resource efficientpoint which is infeasible for the current value of B. Obvi-

ously, the next feasible solution would be that associated with BO.As was the case for multicriteria integer linear programming problems, one

could also adapt the prior scheme to allow for a bounding procedure. Develop-

ing such a procedure is the aim of the following section.


3. A hybrid approach

The hybrid dynamic scheme developed in [8] can also be extended to allowfor the parametric case on the right hand side. Assume that one has partialIysolved the original problem by the dynamic programming recursion until them th stage, considering as the maximum vector of resources the vector 13, andhaving at hand the set of resource efficient points for that stage. The residualproblem for a particular vector of resources, b + (9d; (9 E [O, 1], and a specificresource efficient point, say q, can be expressed as follows:

v-maxN

L en. x..n=m+l

(RP(9)S.t.

N

L an. x.. :S:;,(b+(9d-yq),n=m+l

(n = m + 1, . . . , N)

where Yq denotes the resource consumption vector for the corresponding qthresource efficient solution. Let us denote by UBm+l (Yq, (9) a set of upperbounds which for given (9 is a set of points which satisfies the followingconditions:

(1) Each element is either efficient or dominates at least one of the efficientsolutions of the problem.

(2) Each efficient solution of the problem is dominated by at least onemember of the set or is contained in the seto

Let us also denote by LB( (9) a set of lower bounds for the original problem(given (9) with the characteristic that each of its elements is either efficient ordominated by at least one efficient solution of the problem.

We shall describe now the extension in the following resulto

Theorem 1. Let a resource efficient point for the m-stage problem, say x,with resource consumption vector, Yq, and p dimensional multicriteria value Hmx,available. If for each value of (9 E [O, 1], and every element1

1 EB means that vector addition is performed with Hmx and each member of the set

UB",+1(Yn' (9). If

then

B. Villarreal, M. H. Karwan

gkEHntxEBUBm+l(yme) there exists an element LBj(k)ELB(e) such thatgk ~ LBj(k) with at least one strict inequality, then no completion of x would leadto an efficient solution for any e E [O, 1].

Proof. The proof follows the one given in [8] for each fixed value of e. Thus, itis omitted.

With this result one can modify the previous procedure to allow for the useof bounds for fathoming purposes. The changes to be made are the following:

(a) Step O must inc1ude the computation of the set of lower bounds for eachmember of the family of original problems generated by the range of vector ofresources, b + ed; e E [O, 1]. One must also initialize a counter of stages thatwill serve as an indicator to know when the bounding scheme will be used.

(b) After Step 3 one must check if the bounding scheme is to be used. If not,continue on Step 4. Otherwise, set(s) of upper bounds, UBm+l(Yq, e); e E [O, 1]for each resource efficient solution (q), are computed. AIso note that the set(s)of lower bounds, LB( e), can be improved using heuristics (if required) tocompute feasible solutions for the associated residual problems correspondingto the qth resource efficient point.

Now, it is a matter of generating the family of sets of lower and upperbounds for all the values of e. All the sets suggested in [8] can be extended tobe used for these purposes. Unfortunately, most of them depend upon thevalue of the vector of resources that in turn depends on the value of e. Thiscircumstance has as a result that the degree of difficulty to obtain them increases.As a consequence, it would be desirable to compute a fixed set of boundswhich can be used for fathoming purposes for any member of the family ofproblems described by the vector of resources b + ed, e E [O, 1].

A weaker set of lower bounds for the original problem can be constructed bysimply considering as the vector of resources the vector 1], and generatingeither efficient solutions by maximizing composite objective functions from theoriginal set of objective functions or computing good fe asible solutions usingheuristics. SimilarIy, weaker sets of upper bounds can be determined by usingas vector of resources the vector B which does not vary with e. Even thoughthese sets are expected to be less efficient, it is also true that less computationaleffort will be devoted to obtain them. In order to illustrate the effect of usingthese type of bounds in a hybrid dynamic procedure various problems gener-ated at random were solved using the heuristic suggested by Loulou andMichaelides [3] for zero-one problems to obtain sets of lower bounds, and thefollowing concepts were used to obtain sets of upper bounds. The set of upperbounds is composed of only one member that is obtained from the use ofduality theory and each and every objective function of the problem. Its


development is as follows. Let us firstrecallthe residual problem (RP8):

v-maxN

L Cn. x..,n=m+l

(RP8)s.t.

N

L an. Xn ~ (b + ed - yq), .n= t+1

integer, (n = m + 1, . . . , N).

An upper bound for the solution of thisproblem isobtained by maximizingeach and every objective function of the problem, subject to the same con-straint set (relaxing integrality). Then each solution is used to form a p-dimensional vector which obviously will dominate each efficient solution for(RP8). Letusdenote by Ci the objective function to be maximized in each ofthese problems. The dual problem of the resulting problem is given as

{ktw + (b + 8d - yq)tu},

T={

WI+UA~C;,w, u ~ O,

s.t.

where w and u represent the dual variables. Let us denote by ni the set oíextreme points for the constraint set T. The dual solution, UBi.m+1(Yq, 8), canbe obtained and equivalentIy expressed as follows.

UBi.m+1 (Yq,8) = min {ktw + (b + 8d)tu - y~u}.(w.u)e{};

As can be seen, any íeasible solution (w, u) E ni is also an upperbound forany resource efficient point q. That is

ktw + (b + 8d)tu - YqU ~ UBi.m+l(Yq, e).

Obviously, if one uses the vector B (~(b + 8d)inequality), one has that since u ~ O

with at least one strict

ktw + Btu - Y~u ~ ktw + (b + 8d)tu - Y~u ~ UBi.m+1(Yq, e).

This is a less tight upper bound butitdoes not depend on thevalueoí 8. Thisresult is used to solve several problems whose solution times are given inSection 4.

4. Some comments and computationaI experience

Table 1 illustrates a comparison oí the solution times spent by three differentprocedures in solving a sample of 0-1 bicriterion multidimensional knapsack

378 B. Vil/arreal, M.H. Karwan

a Current time at stage 10.

problems. The objective and constraint matrix coefficients were randomlygenerated within the range [0,99]. The constraint matrix has a 90% densityand the right hand side vector, b, corresponds to 0.25 and 0.45 times the sumof the coefficients of the associated row matrix respectively. It is assumed thatd = 0.5b and d = 0.75b respectively. The coding was done in Fortran on aCDC 6400 computer system. The hybrid dynamic programming approach usesthe heuristic developed by Loulou and Michaelides [3] to generate sets oflower bound vectors after forming various composite objective functions. Tbeupper bound vectors computed at each stage (after stage 3) are obtained bysetting the remaining variables to their upper bound, and at stage 6, using thesolutions to the dual problems associated with the linear relaxations for theremaining problems. The branch-and-bound scheme is the same proceduredescribed without the use of dominance (Le., the resource efficient concept).

One can observe from Table 1 that the branch-and-bound scheme appears tobe a better approach to solve the type of problems described. Notice that forproblems with a relatively small vector of resources (b = 0.25 and d = 0.50b),the dynamic programming recursion is computationally advantageous over thehybrid procedure. However, as these vectors increase, the use of a boundingscheme, in addition to the normal recursions, is justified by its ability toeliminate points not leading to efficient solutions. This is illustrated by the last

Table 1Comparison of solution times for various solution methodologies (N =10, DP A-Dynamic Programming Approach, HDP A-HybridDynamic Programming Approach, BBA-Branch and Bound Ap-proach)

Problemnumber M b d DPA HDPA BBA

1 10 0.25 0.50b 1.25 1.51 1.062 1.01 1.18 0.783 1.73 2.02 1.084 1.56 1.60 0.975 1.12 1.12 0.726 1.30 1.61 1.037 1.72 2.05 1.298 1.66 1.91 1.189 1.56 1.77 1.03

10 1.52 1.81 1.1111 4 0.45 0.75b 52.72 47.23 43.4812 111.28 83.55 45.3313 98.06 84.88 55.4414 67.23 63.07 51. 0415 143.45a 131.51 51.75

five sample problems of Table 1. Obviously, further computational experienceis necessary to decide on which of these approaches is best.

Table 2 illustrates the number of difIerent solutions obtained for the prob-lems of Table 1, for several specific values of €J, when we allow for thevariation of the values of €J from zero to 0.50. This sample corresponds to thefamily of problems generated for all possible values of b ~ 6 ~ b + €Jd; O~ €J ~

1. One can notice that the same set of efficient solutions may be shared byseveral problems with difIerent vectors of resources.

References

[1] G.R. Bitran, Linear muItiple objective programs with zero-one variables, Math. Programming13 (1977).

[2] R. Loulou and E. Michaelides, New greedy-like heuristics for the muItidimensional 0-1knapsack problem, Working paper, McGill University (1977).

[3] R.E. Marsten and T.L. Morin, Parametric integer programming: The right-hand side case,Ann. Discrete Math. 1 (1977).

[4] c.J. Piper and A.A. Zoltners, Implicit enumeration based algorithms for postoptimizingzero-one progams, Management Science Research Report No. 313, Graduate School ofIndustrial Administration, Carnegie Mellon University, Pittsburgh, PA (1973).

[5] G.M. Roodman, Postoptimality analysis in zero-one programming by implicit enumeration,Naval Res. Logist. Quart. 19 (1972).

[6] G.M. Roodman, Postoptimality analysis in integer programmed by implicit enumeration: Themixed integer case, The Amos Tuck School of Business Administration, Dartmouth College(1973).

[7] B. Villarreal and M.H. Karwan, Dynamic programming approaches for multicriteria integerprogramming, Research Report No. 78-3, State University of New York at Bufialo, NY(1978).

[8] B. Villarreal and M.H. Karwan, Multicriteria integer. programming: A (hybrid) dynamicprogramming recursive approach, Math. Programming 21 (1981) 204-223.


Table 2Number of difierent solutions3 for RHS less than b + ed(M=10, N=lO, P=2, b=0.25, d=0.5b)

0.0 0.1 0.2 0.3 0.4 0.5Problem no.

1 1 2 4 6 9 112 1 2 3 3 5 53 1 3 3 3 3 34 1 1 3 4 5 55 1 1 2 2 2 36 1 1 3 4 4 47 1 2 4 6 8 88 1 1 3 3 7 79 1 3 3 4 5 5

10 1 1 3 4 7 8

3 Sets of efficient solutions.

parametric multicriteria integer programming

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