an interactive method for bicriteria integer programming

An Interactive Method for Bicriteria Integer Programming

R. RAMESH, MARK H. KARWAN, A N D STANLEY ZIONTS

Abstract -The bicriteria problem is an important problem in multioh- jective optimization that has been studied extensively in the literature. Practical applications and theoretical investigations of this problem are several. In the research an efficient interactive solution framework for bicriteria integer programming is developed. The proposed methodology follows the implicit utility maximization approach. The decisionmaker's underlying utility function is assumed to be pseudoconcave and nondecreasing, and the problem is solved using an interactive branch-and- bound methodology. Several new concepts on bicriteria integer programming that offer great efficiency in the solution process are developed. The framework has been tested extensively, and results with problems having up to 80 variables and 40 constraints are presented. The results show that the proposed methodology is an effective approach to solving practical bicriteria problems.

I . INTRODUCTION

HIS PAPER presents an efficient interactive solution T framework for solving bicriteria integer programming problems. A decisionmaker is assumed to have only an implicit utility function of the two objectives he wishes to maximize. The problem is solved by an interactive branch- and-bound method in which the decisionmaker's preference structure is explored and determined using painvise comparision questions. The proposed framework exploits the structure and special properties of the bicriteria problem. A number of new concepts on bicriteria integer programming are introduced in this paper, which offer great efficiency in the solution process.

The motivation for the present research arises from several real-world managerial applications of multicriteria integer programming reported in the literature. Multicri- teria integer programming has been applied in manpower planning [ 141, resource allocation [ 151, production planning [ 131 and public investment decisionmaking [ 161 among others. Bicriteria integer programming inodeIs for the previous applications can be derived using the principle of satisficing [22] by considering only the two most important objectives for optimization, and treating the other objectives as problem constraints having certain minimum levels of attainment. In this approach, the special structure of the bicriteria problem can be exploited to obtain good

Manuscript received April 23. 1YXX; revised November 1 I , 1YXX. R. Ramesh and S. Zionts are with the School of Management, State

M. H. Kanvan is with the Department of Industrial Engineering, State

IEEE Log Number XY33033.

University of New York at Buffalo, Buffalo, NY 14260.

University of New York at Buffalo, Buffalo, NY 14260.

acceptable solutions relatively easily, with minimal computational and cognitive effort. Furthermore, specific applications of bicriteria integer programming are several. These include university examination scheduling [ 11, production scheduling [24], network optimization [8] and cluster analysis of large data sets [91 among others.

Literature on the theoretical development of multicriteria mathematical programming problems is also extensive. The multicriteria programming problems have been addressed using three approaches: vector optimization, maximization of an explicit utility function derived using the multiattribute utility theory, and the maximization of an implicit utility function using interactive methods. A comprehensive survey of the multicriteria, mathematical programming methods is provided in Evans [5]. In particular, Teghem and Kunsch [23] survey the interactive methods for multicriteria integer programming. The methods of Gabbani and Magazine [6], Marcotte and Soland [I71 and Ramesh et al. [20] belong to this category.

A theoretical characterization of bicriteria mathematical programs is given in Geoffrion [7]. Pasternak and Passy [18], Benson [2] and Chalmet et al. [3] consider the bicriteria problem using the vector optimization approach. These methods obtain all the efficient solutions to the bicriteria problem. However, the proposed methodology follows the implicit utility maximization approach, and obtains the most preferred solution to the problem through an interaction with the decisionmaker. In this regard, the proposed method is an extension of the literature on the generation of all the efficient solutions to obtaining a workable, most preferred solution to the bicriteria problem. The significant contributions arising out of this work are threefold. First, a branch-and-bound search strategy that effectively utilizes the special structure of the objective function space in bicriteria problems is developed. The search is conducted in the decision variable space by taking into account the behavior of the objective function space during the search process. Sec- ond, a scheme of ordered representation of the decisionmaker's preference structure in the objective function space is developed. The preference structure is assessed interactively during the search process. Third, the interaction with the decisionmaker is integrated with the search process to solve the problem with minimal cognitive effort. The proposed methodology is specifically tailored to

0018-9472/90/0300-0395$01.00 01990 IEEE

the interactive solution of bicriteria integer programming problems and employs certain special convex properties of the preference structure. In this respect, this methodology is different from other more general methods for multicriteria integer programming, and is a new contribu- tion to the literature on bicriteria programming.

The problem of concern is formulated as follows.

BCIP: max g(Cx) s.t. Ax G b

x 2 0, integer

where C and A are 2 x n and m x n matrices of criteria and constraint coefficients respectively, b E R”’ is the vector of resources available, x E R” denotes the vector of decision variables and g ( . ) is the implicit utility function of the decisionmaker that is assumed to be pseudoconcave and nondecreasing. The linear relaxation of problem (BCIP) is denoted as (BCLP).

The organization of this paper is as follows. Section I1 presents the decisionmaking framework of the proposed methodology and the underlying implications. Section 111 develops the proposed algorithm. The computational experience using computer simulations of the decisionmaking strategy appropriate to this methodology is discussed in Section IV, and the conclusions are presented in Sec- tion V.

11. THE DECISIONMAKING FRAMEWORK

Decisionmaking has been the principal focus of research in several disciplines over the years. Simon [22] first presented a structural framework of the principles of decisionmaking from a behavioral perspective. Alongside this development, theories of decisionmaking have evolved from the field of economics, and subsequently, the field of decision analysis has been developed through a successful integration of the behavioral and economic principles in decisionmaking. Simon [22] studied decisionmaking in an administrative environment, and identified the critical determinants of choice behavior. The selection of a course of action from a set of decision alternatives in the pres- ence of multiple conflicting objectives is affected by the decisionmaker’s perception of the facts on the alternatives, the values ascribed by him to the alternatives, the element of rationality in judging these values, and the need for efficiency in decisionmaking. The interaction of these determinants leads to limits on the extent of rationality in the decisionmaking process. These limits result in a decisionmaker restricting the choice problem by first eliminating certain alternatives from consideration, and then eliminating certain attributes of the alternatives as well, in evaluating the remaining set. Based on this obser- vation, Simon [22] developed the concept of bounded rationality and identified the limits in terms of the decisionmaker’s skills, habits, reflexes, values, conceptions of purpose and the knowledge of the decision problem. Subsequently, behavioral scientists have developed the psychological foundations of bounded rationality by relat-

ing it to the cognitive strain in decisionmaking. Accord- ingly, multicriteria decisionmaking methodologies have focussed on minimizing the cognitive burden by assessing the decisionmaker’s preference structure, and using this assessment to identify a good acceptable decision with minimal cognitive effort from the decisionmaker. How- ever, this assessment requires certain assumptions about the decisionmaker’s underlying preference structure and his decisionmaking strategy. While these assumptions may be considered to be restrictive in the scope of these methodologies, they also provide useful mechanisms to model different classes of decisionmaker behavior, and develop convergent solution strategies.

The underlying assumption in the proposed methodology is the existence of an implicit utility function. Given the bounds on the decisionmaker’s rationality, the pri- mary objective of the proposed methodology is to mini- mize the cognitive strain by effectively utilizing an assessment of his preference structure in the search for a good acceptable solution. The decisionmaker’s preference structure is assessed using pairwise comparisions of at- tainable decision alternatives. The methodology employs a modified version of the method Zionts and Wallenius [25] within a branch-and-bound framework. An experimental investigation of the method of Zionts and Walle- nius conducted by De Samblanckx et al. [4] shows that the use of the method can improve the quality decisions, and that the method is easy to use and strongly convergent. Another experimental study comparing the performance of a paired comparison method with that of an explicit utility assessment method is presented in Klein er al. [lo]. They report that the paired comparison method has been perceived to be easier to use by decisionmakers, requires less cognitive effort, and yields better solutions in general than the explicit assessment approach. On the other hand, the explicit method provides greater insights on the problem to the decisionmaker than the pairwise comparison method. This is probably because, the cognitive complex- ity of the questions asked in explicit utility assessment is greater than that of simple painvise comparisons, thus requiring the decisionmaker to think deeper about the problem. Larichev and Nikiforov [ 121 discuss the cognitive and computational limitations of several interactive methods for multicriteria decisionmaking, and indicate that these limitations may not be significant for problems involving just two or three objectives. In summary, these studies characterize utility assessment methodologies, and the empirical evidence suggests the viability of the interactive paired comparison methods for practical problems.

111. THE ALGORITHMIC FRAMEWORK

The problem (BCIP) is solved by employing an interactive bicriteria linear programming algorithm within a branch-and-bound framework. The bicriteria linear programming algorithm takes into account the special structure of the bicriteria problem, and is based on the method Zionts and Wallenius [251. The decisionmaker’s prefer-

ence structure is assessed using pairwise comparison questions during the course of the solution process. In the method of Zionts and Wallenius [25], constraints on the set of feasible weights on the objectives are derived from the decisionmaker's responses, and are used as a representation of his preference structure. The difficulty with this representation is that inconsistency arises in the set of constraints on the weights when the underlying utility function is nonlinear. Therefore, some of the constraints will have to be dropped, resulting in a loss of information on the preference structure. In the proposed methodology, this problem is avoided by representing the preference structure using certain convex cones in the objective function space as well as the constraints on the weights. The convex cones accurately represent any quasiconcave (and hence pseudoconcave) utility function. This representation scheme is used to deduce the decisionmaker's responses wherever possible, so that the number of questions asked of the decisionmaker can be minimized. We develop the proposed methodology with an introduction to the notation used in the following discussion.

Let X' and X d denote the sets of feasible solutions to problems (BCLP) and (BCIP) respectively, in the decision variable space. Hence, X d G X'. An element of X " is denoted as x. The objective function vector is defined for all x E X' , and is denoted by f ( x ) = (C 'x ,C2x) ' . The set Z' = { f ( x ) l x E X') denotes the set of feasible solutions to problem (BCLP) in the objective function space. An element of Z' is denoted as z , and its components are denoted as z I and z,, respectively. Specific objective function vectors are denoted using superscripts on z , such as Z' for instance.

The initial step consists of relaxing the integrality constraints and solving problem (BCLP) by the bicriteria linear programming algorithm. If the optimal solution to problem (BCLP) is integer, then it is also optimal to problem (BCIP). Otherwise, a branch-and-bound search for the integer optimum is conducted. This phase is similar to the branch-and-bound search used in solving single objective integer programming problems. To begin with, an initial incumbent integer solution is obtained using a heuristic procedure (see Loulou and Michaelides [161). The initial incumbent can be any feasible integer solution, since the optimal solution will subsequently be obtained by the algorithm. Although it is desirable to start with an incumbent that is efficient with respect to the set X d , the computational effort involved in finding such a solution can be considerable. The initial incumbent is denoted as z I , and the set Z' is partitional into two subsets SI and S, as

and

S, = (212 E zc, Z ] < 2 ; - E , z , z z ; )

where E is a small positive quantity. These two partitions are mutually exclusive and collectively exhaustive of the solutions that either dominate the incumbent or are effi-

cient with respect to it. The union of SI and S, is nonconvex, and hence the search is conducted separately in each set. The motivation for this partitioning strategy comes from our initial computational results which showed that the set of feasible solutions to a candidate problem generated in the branch-and-bound search after fixing some variables is almost entirely dominated by the incumbent. The partitions represent' simple bounds on the objective functions. Since the objectives are functions of the decision variables, these bounds are expressed as constraints on the decision variables and added to the problem tableau. The search is first conducted in partition SI (which contains the solutions that dominate the incumbent), followed by a search in S,.

The branch-and-bound search in each partition begins from the solution to problem (BCLP) already obtained. The last-in, first-out (LIFO) strategy is used in this search. The branch-and-bound search generates candidate problems for investigation by branching on a fractional variable from a given solution. A candidate problem can be fathomed at this stage if the solution obtained from branching is already optimal to its bicriteria linear relaxation and is less preferred than the incumbent. A fathoming test termed Fathoming Test 1 is used for this purpose, and is described in a subsequent section. If a candidate problem cannot be fathomed, then the usefulness of solving its bicriteria linear relaxation against continuing to branch from the current solution is assessed. This involves a strategic assessment of the current candidate problem, and this is also described in a subsequent section. If the bicriteria linear relaxation is solved, then its solution can be used to fathom the candidate problem under either of the following conditions.

1 ) The decisionmaker prefers the incumbent to the optimal solution to the bicriteria linear relaxation.

2) The optimal solution to the bicriteria relaxation is integer, and is preferred to the incumbent. In this case, the incumbent is updated by designating the new integer solution found as the incumbent.

On the other hand, if the optimal solution to the bicriteria linear relaxation is continuous and is preferred to the incumbent, then branching is continued from the candidate problem. A fathoming test termed Fathoming Test 2 is used to determine this, and is also described later. While conducting the search in SI, it is possible to obtain integer solutions that dominate the incumbent. In this case, the partitions are tightened as follows. Given that the sets SI and S, are obtained by partitioning Z' using the integer solution Z I , and that a new integer solution z2 that dominates z 1 has been found at a candidate problem, the partitions SI and S, are tightened using z 2 follows:

and

398

The tightened partitions are enforced in all subsequent search, and the problem (BCIP) is solved by completing the search in the two partitions separately.

The convergence of the above algorithm can easily be seen as follows. The algorithm follows the standard branch-and-bound search strategy, which has been shown to converge in the literature. The bicriteria linear relaxations are solved using the strategy of the method of Zionts and Wallenius [25], which also been shown to converge. We describe the major procedural components of the above framework in the discussion below.

Preference Assessment

The preference structure representation mechanism presented in this section is a specialization of the general scheme for multicriteria integer programming developed in Ramesh [19] to the bicriteria case. We first introduce the notation and definitions in the following discussion.

Given that the underlying utility function is pseudoconcave, we can define a convex cone as follows. Consider two distinct solutions z ' and z 2 such that neither dominates the other. Let the decisionmaker prefer z 2 to z ' . Then, the set of solutions falling on or dominated by the ray ( z l z = z ' + p ( z ' - z 2 ) , p > 0) is less preferred than z 2 , and no more preferred than 2'. The previous ray is a convex polyhedral cone generated from 2 ' in the direction ( z ' - z'). We will denote this cone as c [ z 2 + 2'1. z 2 is called the upper generator and z ' is called lower generator of the cone c [ z 2 > z ' ] . The symbol + is used to indicate the preference of z 2 over z ' . The theory underlying convex cones is developed in Korhonen er al. [l 11. In this research, we develop a specific application of convex cones as a preference structure representation mechanism in solving bicriteria integer programming problems.

The decisionmaker's pairwise judgements are elicited in two situations in the solution process. First, he is asked to evaluate decision alternatives while solving the bicriteria linear relaxation of a candidate problem. Second, his preference between the incumbent and the optimal solution to the bicriteria linear relaxation of a candidate problem is assessed. Convex cones can be derived from each such assessment. However, assessments involving the incumbent yield convex cones that dominate solutions that are less preferred than the incumbent. Hence these solutions need not be considered further in the search for the integer optimum. Accordingly, these cones are termed global cones, and can be used in identifying solutions inferior to the incumbent at every candidate problem. The global cones are ordered into a tree structure using the preference relationships among their generators. The tree is progressively built during the course of the solution process as follows.

In solving problem (BCIP), a sequence of successively preferred integer solutions (incumbents) is generated. This sequence as denoted as z' , i = 1 , . . . , k , where z k is the current incumbent at any given stage of the solution process. The global cones are ordered using this se-

Level: 2. . . . ..i;:' Fig. 1. Global representation.

quence. Consider a candidate problem whose optimal solution to its bicriteria linear relaxation is f. The decisionmaker's preference between f and z k is evaluated in testing the candidate problem for fathoming. The decision in this case yields a global cone as follows. If z k is preferred to f, then the global cone c [ z k + 21 is derived, provided that z k does not dominate 2. This cone is said to be a lower leaf at level k of the tree of global cones. If Z is preferred to z k , then the global cone c[f + z k ] is derived, provided that Z does not dominate z k . This cone is said to be an upper leaf at level k of the tree. Further- more, if 2 is integer, then the incumbent is updated by designating f as z k t ' . If the decisionmaker is indifferent between Z and z k , then no global cone can be derived. In the above process, if either z k dominates f or Z dominates z k , then the set of solutions to be eliminated are those simply dominated by zk . In this case, no convex cone is necessary.

The levels of the tree correspond to the incumbents generated. The structure of the tree for a particular realization is shown in Fig. 1. Arrows along the arcs lead from the more preferred to the less preferred solution. If a solution is dominated by any of these cones, then it is not preferred to the incumbent, and hence need not be considered further. Furthermore, it can easily be seen from the transitivity of preferences (which holds for a pseudoconcave utility function) that the cones c [ z C + z'l, c[z( ' + 2'3, c [ z d z U ] , c [ z" + 2'1, c [ z 2 + 2'1 and c [ z 2 + zhl are also global. In solving problem (BCIP), the global representation scheme is first set up using the initial incumbent obtained after solving problem (BCLP). The tree is subsequently developed during the course of the search process, and is employed in the fathoming tests and in the solution of the bicriteria linear relaxations of candidate problems.

Search Strategy

The branch-and-bound search is conducted using the following strategy. Let z* denote the solution to a candidate problem obtained from branching. If z* is an integer solution, then the bicriteria linear relaxation of the candidate problem is solved. Otherwise, z* is tested for dominance by any of the global cones. If dominated, then the bicriteria linear relaxation is solved. Otherwise, branching is continued from z*. The rationale for this strategy is as follows. When z* is integer, it indicates that either z* could be a candidate for the next incumbent (if the

number of variables fixed so far is small) or that the candidate problem could be a candidate for fathoming (if the number of variables fixed so far is large). In either case, solving the bicriteria linear relaxation seems to be a strategic choice. In a similar manner, if z* is dominated by a global cone, it indicates that the solutions obtained by branching from z* also could be dominated by the global cones. This suggests that the candidate problem could be a candidate for fathoming. On the other hand, if z* is continuous and not dominated by any of the global cones, then it indicates that the decisionmaker may have to be asked many questions if the bicriteria linear relaxation is solved. This suggests the continuation of branching from the current candidate problem. Although other search strategies are possible (see Ramesh et al. [1986]), our computational experience shows that the proposed strategy is superior to those in Ramesh et al. [1986] in terms of both computation time and the number of questions.

Bicriteria Linear Relaxation

In solving the bicriteria linear relaxation of a candidate problem, certain cut constraints on the objective functions can be derived. These cuts are bounds on the objective functions, and the following results show how these cuts can be derived.

Proposition I : Let z“ and 2’’ be two efficient solutions to the bicriteria linear relaxation of a candidate problem. Let the decisionmaker prefer Z” to 2“. If at least one of the two solutions is integer, then the optimal integer solution to the candidate problem will satisfy the following constraints:

z 2 2 z ; otherwise

Proofi Consider the case when zf > z r . Therefore, any point y which is dominated by C [ Z ” + 2‘1 must satisfy y , < zf . Now, we will show that every efficient solution z to the bicriteria linear relaxation for which z I G z r is dominated by c [ z h + 2‘1.

Let there be an efficient solution z such that z I G z r and that z is not dominated by C [ Z ” + 2‘1. Let p > 0 be such that z , = zf + p ( z f - zf). Therefore, it follows that

2, > 2; + p( zz” - 22”). (1)

However, there exists a A E [0,1] such that

A.( z f , 22”)‘ + ( 1 - A ) .[( zf + p( z r - zf)),

( 2 ; + k ( 2; - 2 3 1 ‘ = ( z ; , z;)‘. (2) Hence it follows from (1) and (2) that

Since z‘ is efficient, a solution z satisfying the above conditions can not exist. Therefore, since at least one of

the two solutions z“ and 2’’ is integer, the optimal integer solution should satisfy the constraint z I 2 z r . Using a similar argument, the previous result can be shown for the case when z: < 27.

Proposition 1 applies to any underlying quasiconcave (and hence, pseudoconcave) utility function. If the utility function is linear, then Z” is also preferred to every solution on the facet containing Z” and Z“ in the objective function space. In this case, if Z” is integer, the cut can be tightened to z I 2 z f if zf > z r , and to z 2 2 z;, otherwise. However, in the case of nonlinear utility functions, the decisionmaker’s preference for an efficient tradeoff from Z” along the edge leading to z“ should be evaluated for this purpose. This is shown in the following result.

Proposition 2: Consider Proposition 1. Let z” be integer. If the decisionmaker likes the tradeoff from Z” along the edge leading to z“, then the following cut can be derived.

z I > zf otherwise

On the other hand, if he does not like the tradeoff, the following cut can be derived.

z 2 2 21 otherwise.

Proofi Consider the case when he likes the tradeoff. Therefore, there exists a point y on the edge joining Z“

and Z” such that he would prefer y to 2”. Consider the subcase when zf > zr . Hence we have zf > y , > z f . Ap- plying Proposition 1 to points y and z ” , the claim follows. Using similar reasoning, the above results can be shown when zf < z f .

Consider the case when he does not like the tradeoff. Now consider the subcase when zp > zp. Clearly, z f is preferred to all the solutions along the edge between z” and 2”. Consider a point z = zh + 6(z‘ - zh), where 6 E

[0,1]. Since Z” is preferred to z , letting 6 -+ 0 and apply- ing Proposition 1 to points z and z” , the claim follows. Using similar reasoning, the case can be established when

The cuts derived according to Propositions 1 and 2 at a candidate problem can be enforced only at the candidate problems that are obtained by branching from it. We present the bicriteria linear programming algorithm in- corporating the above results and the proposed preference structure representation in the discussion below.

Let z* denote the solution to a candidate problem obtained from branching. Let A = {(A, 1 - A)lA E [0,1]} denote the set of feasible weights on the objectives. Initially, all the adjacent edges from z* that are efficient with respect to the set of feasible solutions to the bicriteria

zf < 2 ; .

linear relaxation of the candidate problem are identified. There can be at most two such edges in the objective function space for bicriteria problems. Let E denote the set of extreme point solutions these edges lead to. If the set E is empty, then the procedu-e is terminated with z* as the most preferred solution. 'thenvise, a solution is selected from E. Let this solutio1 be denoted as z". The solution z* is evaluated against z" as follows. To begin with, the utions z* and z" are tested for dominance by the global cmes. Let S * and S" denote the sets of global cone generators that are preferred to z* and z" , respectively. If S" U S* is not empty, then the following test is conducted. If z* is preferred to z", then we can derive a cone ( [ z* > 7 " ] , and if this cone dominates any solution ins" U S*, tl 'n it follows that z* can not be preferred to z" , since the dominated solution is already known to be preferred to either z* or z". If this test does not yield a result, then a similar test is conducted with the cone c [ z " + z * ] . If c [ z " + z * ] dominates any solution in S " U S*, then z" can not be preferred to z*. The decisionmaker is asked to choose between z* and z" only if either S" U S" is empty, or the above tests do not yield any result.

After obtaining the aforementioned decision, z* and z" are tested to determine whether they are integer solutions. If any of them is integer, then a cut is derived according to Proposition 1. If the preferred solution is integer, then the decisionmaker's preference for a tradeoff from the preferred solution along the edge leading to the less preferred solution in the above determination is explored. This is carried out as follows. Assume that z* is preferred to z" in the aforementioned determination. Consider a solution z ' = z* + E ( Z " - z * ) where E is a small positive quantity. The point z' is along the ray from z* in the direction ( z " - z*>, and is at a small positive distance from z*. The decisionmaker's preference between z* and z' is then assessed as above, using the tests of cone dominance. The decisionmaker is asked to ex- press his preference for the tradeoff, only if these tests do not yield any result. Based on this decision, a cut is derived according to Proposition 2. In our computational experiments, the value of E used in the above determination is chosen to be 0.001. In this case, the tradeoff preference is assessed in terms of a painvise comparison. Although this assessment is affected by the value of E , our computational results show that values up to 0.001 do not affect the correctness of this assessment.

In the aforementioned process, if z" is preferred to z* , then the set A is restricted by deriving the constraint ( A , I - A ) . ( z * - z " ) < - w where w is a small positive quantity. If 7* is preferred to z" , then the constraint ( A , 1 - A). (2" - z * ) < - w is derived. If z" is not preferred to z* , then it is dropped from E , and the above sequence of steps is repeated with another solution from E until either the set E is empty or a solution that is preferred to z y is found. If a solution preferred to z* is found, then it is denoted as z" and the procedure is continued as follows.

Using the new set of weights from the above restricted set A , a linear composite objective is constructed. A new efficient solution is generated by maximizing this objective. The new solution will be different from z* and those found to be less preferred than z* , since the weights that will generate composite objectives maximizing at these solutions have been eliminated by the constraints on the weights. Furthermore, Ramesh [19] has shown that inconsistency among the constraints on weights cannot arise while solving a bicriteria linear programming problem. Therefore, all these constraints are used in the entire procedure. However, the new solution could be the same as the current best solution zP. In this case, Z P is denoted as z*, and the above sequence of steps involving the exploration of adjacent efficient edges of z* are repeated. On the other hand, if the new solution is distinct from zp, then it is denoted as z 4 , and the decisionmaker's preference between Z P and z" is assessed using global cones as before. The decisionmaker is asked to choose between the two solutions only if these tests do not yield any result. Based on the preference, a constraint on weights is derived as before, and the procedure is repeated by designating the preferred solution as z*.

Fathoming Tests

The two fathoming tests used in the branch-and-bound search process are as follows.

Fathoming Test 1: This test is used to determine whether a candidate problem can be fathomed using the solution obtained from branching and the tree of global cones. The decisionmaker is not asked any questions in this test. Let z* denote the solution to the candidate problem obtained from branching. Initially, z* is tested for dominance by the global cones. If any global cone dominates it, then the incumbent is preferred to z* and the candidate problem is a candidate for fathoming. In this case, the adjacent efficient extreme point solutions of z* are determined. Let Z" and z h denote these solutions. Consider two solutions z' and 2'' such that z '= z* + E ( Z " - z * ) and z"= z* + E ( Z ~ - z*) , where E is a small positive quantity. The points z' and z" are along the rays from z* in the directions ( z " - z * ) and ( z h - z*) , respectively. If z* is preferred to both z' and z", then it is also optimal to the bicriteria linear relaxation of the candidate problem. In order to determine this, we first test z' and z" for dominance by the global cones. Let S * , S' and S" denote the sets of global cone generators that are preferred to z*,z ' , z" , respectively (note that S' and S" can be empty). z* is evaluated against z' as follows. If the cone c[z ' + z*] dominates any solution in S* U S', then clearly z' can not be preferred to I* since the dominated solution is already known to be preferred to either z* or z ' . If z' can not be preferred to z* , then z* is evaluated against z" by testing whether the cone c[z"+ z * ] dominates any solution in the set S* U S".

If z* is preferred to both z' and z", then the preference of the incumbent over z* indicates that the candi-

date problem can be fathomed. This test is similar to that used in solving the bicriteria linear relaxation of a candidate problem. If either z" is not dominated by any global cone, or if the above tests do not yield any result, then the candidate problem is not fathomed.

Fathoming Test 2: Let Z denote the most preferred extreme point solution to the bicriteria linear relaxation of a candidate problem. Initially, Z is tested for dominance by the global cones. This test generates the following two cases.

Case 1 (f is dominated by a global cone): In this case, the incumbent is preferred to Z, and the candidate problem is a candidate for fathoming. Therefore, Fathoming Test 1 is performed using Z. If the candidate problem can be fathomed using Test 1, then the decisionmaker's preferences for tradeoffs along the adjacent efficient edges of f (there will be at most two such edges) from Z are evaluated. If none of these tradeoffs are preferred, then the candidate problem is fathomed. Otherwise branching is continued from the candidate problem.

Case 2 (2 is not dominated by any global cone): In this case, the upper generators of the upper leaves at level k of the tree (where z k is the current incumbent) are tested for dominance by c [ z k + 21 (note that these solutions are already known to be preferred to z') . If any of these solutions is dominated, then clearly Z is at least as much preferred as z k . The decisionmaker is asked to choose between f and z k only if this test does not yield a result. If Z is preferred to z k and 2 is continuous, then branching is continued from the candidate problem. If 5 is preferred to z k and f is integer, then the incumbent is updated, and the decisionmaker's preferences for tradeoffs along the adjacent efficient edges of Z from f are evaluated. If none of the tradeoffs are preferred, then the candidate problem is fathomed. Otherwise branching is continued. If z k is preferred to 2, then Case 1 of the fathoming test is performed.

IV. COMPUTATIONAL RESULTS

The solution framework has been programmed in For- tran 77 and implemented on IBM 3081/GX at the State University of New York at Buffalo. The framework has been extensively tested with randomly generated problems. The decisionmaker's responses are simulated on the computer using different pseudoconcave utility functions. The overall performance has been measured in terms of the number of questions asked and the computation time. In particular, we have focussed on the number of questions because the computation times tend to be reasonable. We summarize the important results of the study in the following discussion.

The study has been conducted using a completely ran- domized block design with six levels in the number of variables (10, 20, 30, 40, 60, and 80) and four levels in the number of constraints (10, 20, 30 and 40). This is the class of problems for which results with single objective integer programming problems have been reported in the litera-

ture. Ten randomly generated problems have been solved in each category. The results show that the average number of questions is not significantly affected by the number of constraints, while it increases with the number of variables. The average number of questions on problems with 10, 20, 30,40, 60 and 80 variables varies in the ranges (8.9,12.1), (12.9,15.2), (13.2,18.5), (12.1,20.2), (16.2,22.4) and (21.6, 28.9), respectively. On the other hand, the computation times show more variability, and the average computation time varies between 1.97 CPU seconds to 129.44 CPU seconds of IBM 3081/GX in the class of problems tested.

The aforementioned results show that the number of questions and the computation times are reasonable. Fur- thermore, the computation times substantiate the viability of the method for implementation on high speed main- frame computers. However, the rapid advancement in computer technology suggests that the proposed methodology is viable for microcomputers as well. At the current state of this technology, we have 32-bit microprocessors with a nearly virtual storage capacity (which is mainly due to the tremendous decrease in memory costs), and cycle times in the nanoseconds range which equal the speeds of mainframes about two years ago. Therefore, we envisage high viability of this method in developing practical decision support systems for bicriteria integer programming problems on microcomputers.

The number of candidate problems fathomed by Fath- oming Test 1 ranges between 3.9% and 9.2% of the total number of candidate problems in the problems tested. The percentage of questions saved using the tree of global cones while solving the bicriterial linear relaxations ranges between 6.8% and 9.9% of the total number of potential questions. The total number of potential questions repre- sents the total number of decision situations at which the decisionmaker's responses to pairwise comparision questions or tradeoffs are required. The percentage of questions saved using the global cones in Fathoming Test 2 has been the most significant. This ranges between 22.94% and 33.13% of the total number of potential questions. This shows that a significant percentage of the solutions to the bicriteria linear relaxations of the candidate problems is dominated by global cones. Furthermore, we also experimented with the effects of enforcing the cut cos- traints on the objective functions and the partitioning strategy on the total number of questions and the CPU time. We solved the previous sets of problems under the following conditions: with and without the cut constraints, and with and without partitioning. In summary, the percentage of questions saved using the cut constraints ranges between 19.6% and 25.1%, and those saved using the partitioning strategy ranges between 15.4% and 21.9% of the total number of questions required without using them, respectively. There has been no significant change in the CPU time with regard to both the enforcement of the cut constraints and the partitions. More detailed computational results on the performance characteristics are given in Ramesh [19].

V. CONCLUSION [SI G. W. Evans, “An overview o f techniques for solving multiobjec- tivr mathematical programs,” Munugwnent Sci., vol. 30, no. 1 1 ,

~~

The bicriteria problem is an important problem in 19x4.

multiobjective optimization that has been studied extensively in the literature. Several practical applications of

[h] D. Gabbani and M. J . Magazine, “An interactive heuristic approach for multiobjective integer programming problems,” J . Oper. Res. Soc.. vol. 37, 1986.

this -problem have been proposkd, and the theoretical A. M. Geoffrion, “Solving bicriterion mathematical programs.” <)per. Res.. vol. 15, 1967.

nature of research on this problem are twofold. First, the .siunmaking - Tlieoy wid Applications, G . Fandel and T. Gal, Eds. bicriteria problem captures the nature of conflict among Berlin: Springer-Verlag, 19x0.

competing objectives in practical decision situations. The P. Hansen and M. Delattre, “Bicriterion cluster analysis as an exploration tool,” in Multiple Criteriu Problcwi Solving, S . Zionts,

resolution of this conflict requires the involvement of a Ed. Berlin: Springer-Verlag. 1977. decisionmaker in the solutions process. The decision- [IO] G. Klein. H. Moskowitz. and A. Ravindran, “Comparative evalua-

[7]

investigations are numerous. The reason for the extensive p, ..Bicriterion path problems,.. i n Mftltiple Deci-

[9]

tion o f prior versus progressive articulation of preference in bicri- maker has a set of efficient decision alternatives to choose from, unlike the Case Of Single objective optimization.

terion optimization,.3 [ I I ] P. Korhonen, J. Wallenius, and s. Zionts, “Solving the discrete

33, 1986,

This requires certain assumptions about the behavior of

methodology and the decisionmaker can work together to arrive at the most preferred solution. This has given rise to various approaches in solving such problems. Second, the bicriteria problem has several special properties which

multiple criteria problem using convex cones,’’ Management Sci., vol. 30, no. 11. 19x4.

dures for solving multicriteria mathematical programming problems (MMPP).” in Toward Interactive and Intelligent Decision Sup- port Systems, Y. Sawaragi, K. Inoue and H. Nakayama, Eds. Berlin: Springer-Verlag, 1986.

[131 K. D. Lawrence, G. R. Reeves, and s. M. Lawrence, “A multiple

the decisionmaker, in Order that a ’ystematic solution [I21 0, 1. Larichev and A. D. Nikiforov, “Analytical survey of proce-

can be effectively used in the solution process. Given the specialization in the structure of bicriteria problems, and the nature of conflict in decisionmaking, several re- searchers have considered decisionmaking from the perspective of bicriteria optimization. This characterization of bicriteria problems is the principal motivation behind this research.

In this paper, we have developed an efficient interactive solution framework for bicriteria integer programming. We have also provided the decisionmaking framework within which the proposed algorithm is developed. The proposed methodology employs an interactive assessment of the decisionmaker’s preference structure. This assessment is used within a branch-and-bound search process for the most preferred solution. Several new

objective shift allocation model,” IIE Trans., vol. 16, no. 4, 1984. S. M. Lee,, “Interactive integer goal programming: Methods and applications,” in Mulriple Criteria Problem Solving, S. Zionts, Ed. Berlin: Springer-Verlag, 1977. S . M. Lee and E. R. Clayton, “ A goal programming model for academic resource allocation,” Manugement Sci., vol. 18, no. 8, 1972. R. Loulou and E. Michaelides, “New-greedy-like heuristics for the multidimensional knapsack problem.” Oper. Res., vol. 27, no. 6, 1979. 0. Marcotte and R. M. Soland, “An interactive branch-and-bound algorithm for multiple criteria optimization,” Management Sci., vol. 32, no. I . 1986. H. Pasternak and U. Passy, “Bicriterion mathematical programs with Boolean variables,” in Multiple Criteriu Decision Making, J. L. Cochrane and M. Zeleny, Eds. Columbia, SC: University of South Carolina Press, 1982. R. Ramesh, Multicriteriu Integer Programming, unpublished doc- toral dissertation, Dept. Ind. Eng., Univ. Buffalo, 1985. R. Ramesh, M. H. Karwan, and S. Zionts, “ A Class of practical

41

[IS]

[I61

[I71

[I81

[I91

[20] concepts on bicriteria integer programming are intro-

solution process. The framework has been extensively tested with problems of various sizes. The practicality of this approach has been demonstrated by solving problems with up to 80 variables and 40 constraints. The results

interactive branch and bound algorithms for multicriteria integer programming,” Eur. J . Oper. Res., vol. 26, 1986.

by mixed integer programming,” in Multiple Criteria Decision Mak- ing, H. Thiriez and S . Zionts, Eds. Berlin: Springer-Verlag, 1976.

[22] H. Simon, Administrutive Behavior. New York: The Free Press, 1957.

[23] J. Teghem and P. L. Kunsch, “Interactive methods for multiobiec-

duced in this paper, which Offer great efficiency in the [21] J. F. Shapiro, “Multiple criteria public investment decisionmaking

show that the method is viable for implementation in mainframes as well as microcomputers~ Possible of application are numerous. These include portfolio SekC-

tive integer programming,” in Large Scale Modeling and Interac‘tive Decision Analysis, G . Fandel, M. Grauer, A. Kurzhanski and A. P Wierzbicki, Ed. Ber1in:Springer-Verlag, 1986.

[24] L. N. van Wassenhove and L. F. Gelders. “Solving a bicriterion tion, project selection, warehouse location, production

more.

scheduling problem,” Eur. J . Oper. Res., vol. 4, 1980. S . Zionts and J. Wallenius, “An interactive multiple objective

utility functions,” Management Sci., vol. 29, no. 5. 1983.

[25] planning, strategic planning and Policy and many linear programming method for a class of underlying nonlinear

REFERENCES

[ I ] T. Arani, M. H. Karwan, and V. Lotfi. “ A Lagrangean relaxation approach to solve the second phase of the exam scheduling problem,” Eur. J . Oper. Res., vol. 34, 1988. H. P. Benson, “Vector maximization with two objective functions,” J . Optimization Theory and Appl., vol. 28, 1979. L. G. Chalmet, L. Lemonidis, and D. J. Elzinga, “An algorithm for the bicriterion integer programming problem,” Eur. J . Oper. Res., vol. 25, 1986.

141 S. de Samblanckx, P. Depraetere, and H. Muller, “Critical consid- erations concerning the multicriteria analysis by the method of Zionts and Wallenius,” Eur. J . Oper. Res., vol. IO, no. I , 1982.

R. Ramesh received the B.S. in chemical engineering in 1975 for the Indian Institute of Tech- nology, Madras, India. and the Ph. D. in industrial engineering in 1985 for the State University of New York at Buffalo, Buffalo, NY.

H e is an Assistant Professor in the Depart- ment of Management Science and Systems, School of Management, State University of New York, at Buffalo. His areas of research include mathematical programming, multicriteria decision analysis. network optimization, and opti-

121

[31

KAMESII et U / . : INTEKACTIVE MTTIIOI) F O K HIC'KITEKIA 1NTEC;F.K PKOC;KAMMIN<; 403

mization in data structures and databases. His articles on these topics have appeared in Management Science. ACM Transactiorls on Database Systems. Naval Research Logistics, and the Eirropean Joimmol qf Opera- tions Research.

more than 40 publications in journals, books and proceedings. He currently serves as Associate Editor of Naval Research Logistics, Com- puters & Operations Research, and I . I . E . Transactions.

Stanley Zionts received the Ph.D. in industrial administration in 1966 from Carnegie-Mellon University, Pittsburgh, PA.

He is Alumni Professor of Decision Support Systems and Chairman, Department of Manage- ment Science and Systems, School of Manage-

Mark H. Kaman received the B.E.S. and M.S.E. ment, State University of New York at Buffalo, Buffalo, New York. H e has worked with the European Institute for Advanced Studies in Management in Brussels, Belgium, the Ford Foundation in India, and U.S. Steel Corpora-

degrees in mathematical science and operations research in May 1974 and September 1974, respectively, from the Johns Hopkins University, Baltimore, MD, and the Ph.D. in operations research in 1976 from the Georgia Institute of tion. His areas of research include multiple criteria decisionmaking, Technology, Atlanta, GA. decision support systems, and mathematical programming. He is the

H e is Professor and Chair, Department of author of several books and proceedings and over eighty publications in Industrial Engineering, State University of New journals, books and proceedings. He currently serves as associate editor York at Buffalo. His areas of research include of Management Scrence and Naval Research Logistics journals and has mathematical programming, multiple criteria served as editor of special issues of each of those journals. H e is actively

decisionmaking, and vehicle routing and scheduling. He IS author of involved in the School of Management's China M.B.A. program.

an interactive method for bicriteria integer programming

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