A generalized interactive goal programming procedure

1993 Compurers OpsRes. Vol. 20.No.7,pp.747-753. Printed in Great Britain. All rights reserved

A GENERALIZED

GARY

0305-0548/93 $6.00+0.00 Copyright 0 1993Pergamon Press Ltd

INTERACTIVE GOAL PROGRAMMING PROCEDURE R. REEvEs’tf and SCOTT R. HEDIN’§

‘Department of Management Science, University of South Carolina, Columbia, SC 29208 and ‘College of Business Administration, Gonzaga University, Spokane, WA 99258, U.S.A. (Received August 1991; in revised form October 1992)

Scope and Purpose-Interactive

goal programming (IGP) can be an important aid in many decision making situations involving multiple and conflicting objectives. The assumptions underlying IGP, however, may be overly restrictive and not always consistent with decision maker (DM) preferences. An IGP procedure which is more general in its assumptions regarding DM preferences than previous approaches is developed and illustrated with examples. It provides DMs with more information and allows for more flexibility in considering tradeoffs and adjusting goal target levels without appreciably increasing either the computational burden of the procedure or the information processing and evaluation requirements of the DM. Abstract-This paper generalizes the interactive sequential goal programming (ISGP) procedure of Masud and Hwang [l]. A fundamental ISGP assumption concerning the preemptive relative importance of achieving adjusted goal target levels is shown to be unnecessarily restrictive and is relaxed. The modified procedure is illustrated with examples.

INTRODUCTION

Goal programming (GP) is one of the most popular and widely utilized approaches to multiple criteria decision making (MCDM) [2,3]. Although GP, especially in its preemptive priority form, as opposed to its weighted form, has engendered a substantial amount of controversy [4]. Aside from the preemptive priority issue, another major criticism of GP, as well as the more general class of MCDM prior preference techniques, is their required prior definition of preferences. A decision maker (DM) is not likely to be able to provide clear and accurate preference information in an information void prior to model solution. This criticism has motivated the development of a variety of interactive MCDM procedures [S] in which DM preferences are allowed to evolve iteratively as a DM learns more about the tradeoffs involved and the solution space of the problem under consideration. In interactive GP (IGP), DMs are asked to specify target levels or bounds for each objective. Given this information, solutions are generated and DMs are asked to consider them in revising these levels or bounds. New solutions are generated and the process continues until a solution which is satisfactory to the DM is identified. This paper focuses on the interactive sequential goal programming (ISGP) procedure of Masud and Hwang [l]. ISGP does not require any prior preference information or assume a preemptive priority structure. However, preemptive priorities are used at a higher level as a part of the overall interactive process. To generate additional solutions, DMs must be willing to loosen some goal levels, while tightening others. ISGP assumes that achieving goal levels which have been loosened is preemptively more important to the DM than achieving those which have been tightened. Clearly, this assumption about DM preferences may not hold for all DMs in all decision making situations. tG. R. Reeves is a Professor of Management Science at the University of South Carolina. He received his doctorate in Operations Research from Washington University. His interests include applied management science, multiple criteria decision making, mathematical programming and production-operations management. His research on these subjects has appeared in Management Science, Interfaces, Decision Sciences, Computers & Operations Research, the European Journal of Operational Research, IIE Transactions, and other journals. :Author for correspondence. $S. R. Hedin is an Assistant Professor of Operations Management at Gonzaga University and a Ph.D. candidate in Management Science at the University of South Carolina. His research interests include multicriteria decision making, manufacturing accounting systems and shop floor control for flexible manufacturing systems. 747

GARYR. REEVESand SCOTT R. HEDIN

748

The purpose of this paper is to generalize the ISGP [l] procedure. The major generalization relaxes the assumption about the preemptive relative importance of achieving adjusted goal levels. The generalized procedure provides DMs with more information and allows for more flexibility in considering tradeoffs and adjusting goal levels without appreciably increasing either the computational burden of the procedure or the information processing and evaluation requirements of the DM. INTERACTIVE

SEQUENTIAL

GOAL

PROGRAMMING

A MCDM problem involving K objectives can be stated as: maxCfi(x), _&(x), . . . Jdxll subject to

(1)

XEX

ISGP [l] consists of an initialization phase followed by alternating evaluation and iteration phases (Fig. 1). Initialization involves generating the K ideal solutions by optimizing each objective individually over the feasible region. This provides upper and lower bounds, f* and f* respectively, on the value of each objective over the set of ideal solutions. DMs are then asked to set initial goal target levels, b, for each objective within this range. An initial primary solution (PS) is generated by minimizing the sum of the underachievement of the goal levels over the feasible region. Nondominance of the resulting solution is assured here and throughout the procedure by including the maximization of the sum of the overachievement of the goals as the lowest priority objective

Senerste idesl solutions Specify lnitisl goal Welt Generate Pf Generate

AS

Fig. 1. ISGP.

749

A generalized interactive goal programming procedure

in all optimization problems solved. Differenc’es in the magnitudes or ranges of the objectives can be taken into account by assigning normalizing weights to the underachievement and overachievement deviational variables in the goal constraints. An initial set of alternative solutions (AS) is generated by constraining each objective individually to be no worse than its target level while minimizing the sum of the underachievement of the remaining target levels over the feasible region. This information (objective value bounds, PS and AS, and current goal target levels) is organized into a tradeoff table and presented to the DM for evaluation. If the DM is satisfied with any current solution, the procedure terminates. If not, the DM is asked to modify his/her target levels relative to the current PS, and the process continues with the iteration phase. Consistency checks are built into the process such that if a DM wishes to improve one or more objective values, he/she must be willing to worsen at least one other objective value. The iteration phase consists of generating a new PS and a new set of AS based on the modified goal target levels, updating the tradeoff table, and returning to the evaluation phase. It is here that ISGP distinguishes preemptively between target levels which have been loosened and those which have been tightened by the DM during the previous evaluation phase. Minimizing the sum of the underachievement of the targets which have been loosened is considered preemptively more important than minimizing the sum of the underachievement of those which have been tightened in both the PS and the AS. The PS and AS problems solved during the iteration phase of ISGP are

PS=min{(Cjd,~),(Zid,~),(Ei,j-d+

-df)}

(2)

subject to &(x) + w,d, - w,d: = b,

h=l,...,K

(3)

xEX,deD

and AS,=min{ (Z:j,j+kd,y), (Ci,izLd;), (Xi,j-d+

-df

)}

(4)

subject to L(x) - w,d: = b, fh(x) + w,d, - w,,d,+ = b,

h=l,...

,K, h#k

(5)

xgX,dED

for k = 1,. . , K, where i and i index the sets of goal target levels which have been loosened and tightened, respectively, from the previous iteration, w,,= b,,-f*,,, and D = {did-*d+ =O, 0~ d- < 1, d+ 20).

GENERALIZATION

ISGP can be generalized to (1) simplify the initialization phase and (2) generate additional solutions during the iteration phase (Fig. 2). It is not necessary to involve the DM at all during initialization. More importantly, the preemptive priority assumption of the PS and AS problems solved during the iteration phase is unnecessarily restrictive and may not be representative of actual DM preferences. A less restrictive PS and additional AS can be identified. With regard to initialization, ISGP requires that the DM specify initial target levels for each objective prior to the generation of the initial PS and AS. But there is no need to involve the DM at all at this stage of the process. The ideal solutions can be used as the initial AS, AS, = {max fk(x)Ix~X}

k=l,...,K

(6)

and the initial PS can be generated as the compromise solution which minimizes the sum of the underachievements of the individual goal levels from the ideal, PS=(minE,f,*-f,(x)lx~X}. This simplifies the overall process and removes the DM entirely from the initialization phase.

(7)

GARYR. REEVESand Scorr R. HEDIN

Fig. 2. Generalized IGP.

With regard to the generation of additional solutions, both the PS and AS, during the iteration phase, there is no reason to assume that achieving revised goal target levels which have been loosened is preemptively more important to a DM than achieving those which have been tightened. DMs might have different attitudes toward risk and making tradeoffs in the levels of achievement of various goals in different problem situations or in different stages of the interactive process. A DM’s attitude toward risk could be classified as risk averse, risk neutral or risk seeking. The ISGP ph~osophy of preemptively minimizing the underachievement of goal target levels which have been loosened before those which have been tightened, is closest to that of a risk averse DM. It assumes that since some targets have been loosened, the DMs top priority is to not give up more than these loosened amounts, no matter how much improvement could be achieved with respect to the goals which have been tightened, But a DM could just as easily have a more risk seeking attitude of first wishing to minimize the underachievement of the target levels which have been tightened, no matter how much the achievement of those which have been loosened is worsened. A DM could be risk neutral and simply wish to achieve the revised target levels without distinguishing preemptively between those which have been loosened or tightened. It is also possible that a DM’s attitude toward risk could change during the course of the interactive process. However, ISGP does not allow for any of these latter possibilities. A DM may not be aware of his/her risk preference, it may not be clearcut, its accurate elicitation could be difficult, and it could change during the interactive process. As a result, the iteration phase of ISGP can be generalized to generate additional AS consistent with other attitudes toward risk. This will expose DMs to a wider range of solutions and allow for more flexibility in the decision making process without appreciably increasing the computational burden or the DM’s information processing requirements. Specifically, let the PS be the risk neutral solution which seeks to minimize the sum of the

A

underachievement

751

generalizedinteractivegoal programmingprocedure

of all of the revised goal target levels without regard to direction of change, (Ci,j-d+ -dj’)}

PS=min((&d,),

(8)

subject to (3) and expand the AS to include all risk neutral (RN) (Ei,j-dz

ARNS,=min((~.,,,,,d,),

-di+)}

(9)

subject to (5), risk averse (RA), ARAS,=min{(Xj,j+,d,r),

(Ci,izkd;), (Ci,j-d’

-dj’)}

(10)

-dj+)}

(11)

subject to (5), and risk seeking (RS), ARSS,=min{ (Ci,i+ kd;), (Cj,j, kdj), (Xi,j-d,? subject to (5), solutions for k = 1, . . . , K, where h indexes the entire set of goals and j and i index the sets of goals whose levels have been loosened and tightened, respectively.

AN EXAMPLE

For purposes of illustration, the generalized procedure will be applied to the same problem as in the original ISGP paper [l]. The problem involves determining the amounts of several food groups to include a diet to meet nutritional requirements and quantity restrictions, while attempting to minimize three conflicting objectives: cost, cholesterol and carbohydrates. The complete problem statement is included as an appendix. The application of the original and generalized ISGP procedures to the example problem through one iteration is summarized in Tables 1 and 2, respectively. The DM is removed entirely from the initialization phase of the generalized procedure, resulting in different initial primary and alternative solutions. This could result in the DM setting different goal levels in the evaluation phase. However, for purposes of comparing the first iteration of the two procedures the same goal levels are used. More solutions are generated during the iteration phase of the generalized procedure, however, some of these are duplicates. These solutions with duplicates removed are shown in Table 3. If only one goal level is loosened or tightened from the previous PS, as is the case with the second goal, then the risk neutral, risk averse and risk seeking AS will be the same for that goal. Other

Table 2. Generalized

IGP example

Table 1. ISGP example

Initialization I’ f* b PS A% AS, AS, Evaluation h Iteration PS A% AS, AS,

fl

L

h

2.26 3.47 2.50 2.56 2.50 3.07 2.96

8.44 71.53 15.00 27.47 35.12 15.00 71.53

93.34 380.39 100.00 376.60 354.98 380.39 100.0

2.40

40.00

15.00

2.89 2.40 2.89 3.47

40.00 45.24 40.00 47.88

227.52 345.96 227.52 150.00

1

Initialization AS, AS, A% PS

2.26 3.47 3.08 2.97

67.83 8.44 71.53 28.55

28 I .67 380.39 93.34 257.53

Evaluation b

2.40

40.00

150.00

Iteration 1 PS ARNS. ARNS; ARNS, ARAS, ARAS, ARAS, ARSS, ARSS, ARSS,

2.89 2.40 2.89 3.12 2.40 2.89 3.47 2.40 2.89 2.67

40.00 45.24 40.00 50.84 45.24 40.00 47.88 69.65 40.00 70.33

221.52 345.96 227.52 150.00 345.96 227.52 15O.cNl 228.80 227.52 150.00

GARY

152 Table 3. Generalized

fi Iteration PS AS AS AS AS AS

1

2.89 2.40 2.40 2.61 3.12 3.47

IGP+ompressed table

h 40.00 45.24 69.65 70.33 50.84 47.88

R.

REEVES

and SCOT-T R.

HEDIN

I

iteration

Table 4. Generalized

h Evaluation b

221.52 345.96 228.80 150.00 150.00 150.00

Iteration PS AS AS

IGP-iteration

2 table

/I

h

h

2.75

45.00

175.00

2.15 3.10 2.75

45.00 45.00 60.00

243.01 175.00 175.00

2

duplications are possible, for example, when the PS is able to achieve one or more goal targets completely, as is the case with the second goal. Given the potential for duplications, the increased information processing requirements on the DM due to the additional solutions should not be excessive, and these additional solutions should provide the DM with additional information and flexibility. Six candidate solutions are available to the DM after the first iteration of the generalized procedure (Table 3), as opposed to only three solutions previously (Table 1). Of these three additional solutions, one was generated using the risk neutral objective (9), while the other two were generated using the risk seeking objective (11) of the generalized procedure. They provide the DM with one additional tradeoff between cholesterol and carbohydrates for a given cost ($2.40) and two additional tradeoffs between cost and cholesterol for a given level of carbohydrates (150). This additional information should be helpful to DMs in adjusting goal target levels in arriving at a satisfactory solution. Once again, it is expected that given these additional solutions, DMs may establish different goal levels and explore different decision paths during the interactive solution process. However, for purposes of comparison, the same goal levels were used for the second iteration of both procedures. This time only one additional solution was generated (Table 4). The final solution in Table 4 is a new and risk seeking solution in which achieving the tightened targets for cost and carbohydrates is considered preemptively more important than achieving the loosened cholesterol target. This additional solution provides the DM with tradeoff information on cholesterol vs carbohydrates for a given level of cost and on cost vs cholesterol for a given level of carbohydrates which would be unavailable under the original procedure. CONCLUSIONS Interactive GP can be a powerful decision aid in situations involving multiple, conflicting objectives. Although the assumption of ISGP that a DM possesses a particular preemptive priority structure in making tradeoffs between objectives in all problem situations may be overly restrictive. A generalized interactive GP procedure which relaxes this assumption has been presented and illustrated with examples. It provides DMs with more information and allows for more flexibility in considering tradeoffs and adjusting goal target levels without appreciably increasing either the computational burden of the procedure or the information processing and evaluation requirements of the DM. APPENDIX Example Problem

minf,(x)=0.225x, +2.2x, +0.8x, +0.1x4+0.05x, minf2(x)= 10x, +20x,+ 120x, minf,(x)=24x1+27x,+15x,+l.lx,+52x,

+0.26x,

(Cost) (Cholesterol) (Carbohydrate-s)

subject to 720x,+107x,+7080x,+134xs+1000x,~5000 0.2x,+10.1x,+ 13.2x,+0.75x,+0.15x,+ 1.2~~2 12.5 344x,+1460x,+1040x,+75x,+17.4x,+240x,>2500 18x1+151x,+78x,+2.5x,+0.2x,+4x,~63

(Vitamin A) (Iron) (Calories) (Protein)

A generalized interactive goal programming procedure x, G6.0

(Milk, pints) (Beef, pounds) (Eggs, dozen) (Bread, ounces) (Lettuce and Salad, ounces) (Orange juice, pints)

x,Cl.O 13 go.25 x,< 10.0 xj < 10.0 x,<4.0 xj>O,j=i ,..., 6. REFERENCES 1. 2. 3. 4. 5.

A. S. Masud and C. L. Hwang, Interactive sequential goal programming. J. Res. Sot. 32,391-400 (1981). S. H. Zanakis and S. K. Gupta, A categorized bibliographic survey of goal programming. OMEGA 13.2 1t-222 ( 1985). C. Romero, A survey of generalized goal programming (197~1982). Europ. J. Opl Res. 25, 183-191 (1986). E. L. Hannan, An assessment of some criticisms of goal pro~amming. Compuiers Ops Res. 12, 525-541 (1985). Y. Aksoy, 1nteractivemultipleobj~tived~isionmaking:a bibljography(l965-19gg). A+rr Res. News 13,1-8 (1990).

753