Opportunities and limitations of AHP in multiobjective programming

Math1

Comput. Modding,

I I, pp. 206-209,

Vol.

0895.7177,88

1988

TOPICS

IN

THEORY

OF

$3.00 + 0.00

Pergamon Press plc

Printed in Great Britain THE

ANALYTIC

HIERARCHY

PROCESS

OPPORTUNITIES AND LIMITATIONS OF AHP IN MULTIOBJECTIVE PROGRAMMING

David

L. Olson

Department of Business Texas AbM University,

Analysis & Research College Station, TX

77843

USA

Ab_SILrat:l:. The Analytic Hierarchy Process is a useful tool for multiobjrctive decision making in its own right. In addition, it has the potential for expediting multiple objective programming analyses. Multiple objective proqramming techniques face the problem of a large (if not inEinite) number of alternatives. Generally, the problem involves optimizing some unexpressed utility function over a feasible region. AHP provides a Iuseful means of obtaining an initial linear approximation of t.his unexpressed utility function, with the potential of expediting the MOP analysi:;. Othl:r benefits include enhancinq decision maker learning through use of the consistcsncy measure. Problems discussed are the impact of comparing many objectives, the sensitivity of the consistency index, and l.he use of the eigen vector as a literal estimator of utility.

@Y~OL’&.

Analytic

Hierarchy

Process;

multiple

Consideration a model:

Management science is concerned with applying the scientific approach to business problems through development of a mathematical model of a problem, gathering sound data, and use of the model to aid decision making. The intent is to support human judqement. Many business decision problems can be supported by linear programming An obvious objective would be to models. profit Mximize within the constraints of A well known general formulation feasibility. would be:

st

2. -

i=l,..m 2 0 for

Max

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-

Max

2,

-

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2 c ,=I lJ

objectives

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%

ck,

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would

yield

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the

same

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set

A

consider common way to objective functions simultaneously additive linear utility function:

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same

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In (1)

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(31

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Utility. The concept of needing to balance considered. values has long been economic Jeremy Bentham and other economists discussed utility. Utility as an operational theory has been questioned by Georghescu-Roeqen (1954) and Simon (1979) Further, Morgenstern (1972). reported the phenomenon of satisficing, where makers were observed to accept many decision less than optimal solutions to many decision bounded March (1965) noted the problems. makers must rationality with which decision The direct development of utility often cope. is the objective of a strain of active research. has not resulted in an however, That approach, identifying specific decision easy means of

Jg, a11 xJ s bl

for

of multiple

means of product

J=l

2 J_, cJ xJ

x,

programming.

alternative investment choices, or balancing return with risk, or balancing quality with short run profit measures.

The Analytic Hierarchy Process ( AHP - Saaty, 1977, 1980) has proven t.0 5(~ vrry useful in assisting decision maker selection from a finite set of alternatives. This idea has been extended to decisions under conditions of multiple objectives as well. The purpose of this paper is to discuss the benefits and problems involved in using AHP as a means of developing a linear approximation of utility under conditions of multiple objectives in linear programming models.

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objective

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Many business decisions Multiple Objectives. objectives, requiring conflicting involve decision makers to reconcile tradeoffs among 1978; Zeleny, 1983; objectives (Cohen, these often become Such tradeoffs 1986). Steuer, balance the need to because of necessary different rates of return to such scales as cash or net present value of after tax profit, flow,

206

Proc. 6th Int.

maker utility. Regardless, provide a useful construct to decision makers balance multiple, objectives.

Conf

on Mathematical

utility does explain how conflicting

While utility could theoretically take on any shape, DeBreu (19751 defined a utility function as having the properties that more of a good is better than less (an increasing function), that more of a good contributes to utility but at a decreasing rate (convex), and that there are no satisfaction levels (continuous). Multinle Objective Prooramminq. Many decision problems supported by linear programming involve multiple Multiple objectives. objective programming is a class of techniques which seek solutions yielding optimal decision maker utility. This is done in a variety of ways. A key

concept in multiple objective analysis is that of nondominated solutions. For any number of a solution is nondominated if objectives, improvement on objective one would any necessitate diminished attainment on any other objective. On the other hand, a dominated solution would be one in which you could obtain a feasible solution which improved at least one objective without diminishing any others. Different modeling approaches to multiple objective problems can be viewed as different views of utility. A single objective linear programming model often views profit as a linear measure of utility. Multiattribute utility theory is concerned with identifying and defining utility. Multiple objective approaches (Hwang, et al. 1980; Evans 1984; Rosenthal 1985) have taken a variety of approaches to utility. Generating nondominated solution sets and allowing decision makers to select from that set infers maximum decision maker utility through choice. Unfortunately, this may involve a rather large number of nondominated solutions for decision maker inspection. Interactive techniques seek to incorporate decision maker utility through directed search for improved solutions assisted decision maker selection from among a by smaller subset of nondominated solutions. Desired linear number 1) 2) 3)

4)

5)

Features of MOLP. A multiple programming analysis should of features.

objective have

207

Mmodeiling

This information is used as a clue this set. the relative weights for the combined to A pattern for weights objective function (3). focusing around the last selection is used to generate the next set of alternatives presented This set is also to the decision maker. The filtered to provide maximum dispersion. process continues for a predetermined number of or until decision maker satisfaction is steps, obtained. This process is quite workable in practice, and Steuer has presented a number of applications. two features which are There are, however, First, the unattractive with this method. requires that a set of solutions be technique filtered to obtain maximum dispersion (find the diverse solutions of m candidates). n most This is done automatically by Steuer’s code. that code is not widely available. However, the unattractive feature is that Another technique is entirely linear programming based, that only extreme point solutions are meaning of being produced by the technique. capable While this is not a problem in single objective not programming analyses, it does 1 inear utility the expected behavior of reflect functions. AHP has Analytic Hierarchy Process in MOLP. been applied as a means of identifying a linear approximation of utility for selection among a What is proposed is that set of alternatives. AHP be used as a means of identifying a working approximation for a utility function The eigen mathematical programming analysis. vector obtained from AHP would be the source This has been applied for the weights in (31. in O-l multiple objective programming analyses by Mitchell and Bingham (19861 and Bard(19861, and in a continuous multiple objective analysis by Olson, et al. (19861. The general procedure outlined in Olson, et al. was to begin by development of a payoff table, presenting decision makers with a view of the optimal solutions available for each objective For k objectives, this would yield k in turn. The payoff solutions (barring duplications). table would measure the attainment of each objective for each solution. PAYOFF TABLE

a

It should reflect utility, resulting in improved decision making. Undue burden upon decision makers should be avoided. Decision maker learning should be enhanced, especially through development of tradeoffs among objectives. Algorithmic support should be available, ideally without the need for custom computer code. Further, computer run time should be reasonable. The technique should work for models with more than two objectives.

Steuer ‘5 Method One interactive MOLP technique which’performs relatively well on this set of desireable features is Steuer’s method (Steuer 19771. The general operation of that techniqueis to generate a predetermined number of diverse nondominated solutions for decision maker consideration. The decision maker is asked for the preferred solution from

measures

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.Zk

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.z,

Zk

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z:

.

optimal

where objective ob]ective, obJectlve

Pi{ is

the

attainment

on

the

for the solution optimizing Z; is the optimal solution i.

2

z;

iN the for

J&

The decision maker can learn of the tradeoffs viewing this table. Adjustments to by eliminate the different scales of the k objective functions can be intuitively adjusted

Proc. 6th Znt. Conf.on Mathematical

208

for by the decision maker, or formal means of adjusting scale Olson, et al. can be used.

more a more in as given

AHP can then be applied to obtain the weights for (31. Without these weights, an additive lead combined objective function is liable to Further work on this to misleading solutions. AHP point can be obtained from the author. provides not only the approximated utility weights, but in addition provides the valuable feature of a measure of consistency (see Saaty 1980).

both of whom or in Bard, Bingham h Mitchell, dealt with large numbers of objectives. An added complication is the need to reflect when the importance to objectives, relative used to measure each objective in (31 scales was This point can be quite diverse. payoff addressed above in the section on the table. factor is the sensitivity of the Another Saaty consistency coefficient given by AHP. (19771 developed the measure: blllU

With the weights obtained, an initial solution to the MOLP can be obtained with linear programming. This yields a solution with attainments:

In addition, 2k new solutions can be obtained objective in by forcing improvement on each turn, developing a search pattern of two steps. For each of k objectives, a constraint is added :

z, z z;

l

*

+ can be varied

to some range (for example 511 and 25X) from the current attainment to the optimal attainment for objective i in order to generate 2k new solutions. The intent of this step is maker new alternatives objective by controlled solutions (including the presented to the decision selection.

to give the decision one which improve The 2ktl amounts. initial solution) are maker for preference

for a can be continued This procedure using the predetermined number of iterations, current last selected alternative as the solution for each step as in Steuer’s method, maker or until convergence (the decision selects the last current solution). If further investigation was desired, smaller grids around the current solution can be incorporated.

Modelling

as a measure of consistency, where $,,= is the maximum elgen value for the paired comparison matrix and n is the number of objectives compared. this measure should be less To be consistent, Manipulation of objective weights than .l. indicates that if all objectives are near egual the consistency measure will in importance, relatively consistent rankings of all require one if On the other hand, objectives. objective is much more important than the very little consistency among the other others, objective ratings is needed to pass the test This is at first sight a problem. limit. that it also reflects the concept However, lower rated objectives are not as crucial. Consider the following paired comparison where the relative ranking of objective objective 3 is allowed to vary: Objective Objective 1 2 3

Potential Problems W& &HJ. There potential difficulties with the AHP portion this analysis. Some potential difficulties with the rest of the analysis are addressed Olson, et al. The first difficulty is in the potential of compar lsons needed on the part decision maker. That number of comparisons is:

$ While paired

this number comparisons

is

are of in

number of the paired

(n-1)

potentially large, making has not proven difficult in

1

2

3

1

2 1

V

The relative importance objective 3 would have to the consistency test. Next,

matrix 2 to

3 1

of objective be 1:l to 1:4

2 to to pass

consider: Objective

The benefits expected from this approach are that special computer codes are not required (generally available linear programming codes can be used), decision maker learning is enhanced (both through use of the payoff table and AHP), and this technique has the capability of producing nondominated solutions that are not original constraint set corner points. Utility would be reflected through decision maker selection without the need to develop the full tradeoff function.

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-

Objective 1 2 3

1

2

3

1

2 1

V

8 1

I ranking of objective In this case, any relative 2 to objective 3 greater than 1:l would pass The point might be that the consistency test. This minor details may not be that important. leads to the inference that consideration of many objectives is not necessary. Both tiitchell & Bingham and Bard dealt with of large number involving a analyses grouped the objectives, Both objectives. Bard, for combining similar areas of interest. reduced the paired comparison matrix instance, the This has from 10 objectives to 5. favorable feature of reducing the number of This would paired comparisons from 45 to 10. considered. at first glance reduce the factors of the original 10 objectives However, many If they of a similar nature to others. were (were contained the same functional tendencies statistical theory would highly correlated), indicate that one representative would suffice. A

last

point

is

that

AHP provides

a

linear

Proc. 6th Int. Conf. on Mathematical Modelling

of utility. It should be viewed as a gradient estimator of a nonlinear underlying utility function, and as such, can be expected

estimator

to change as the perspective of the decision maker changes. This point is addressed by Forman elsewhere in this proceedings. Conclusions. the tradeoffs in multiobjective workable means addition, a consistency is

A consistent means of explaining among objectives is a major need analysis. AHP provides a of obtaining this tradeoff. In valuable indicator of relative provided.

Steuer’s method is a well developed, workable interactive multiple objective technique. the search pattern provided by that However, technique is incapable of obtaining noncorner point efficient solutions due to the method used. Use of bounds on objectives through constraints would alleviate that problem, giving a multiple objective analysis the ability to reflect nonlinear utility. AHP can support this approach by providing a set of objective function weights which expedite the search in the direction of improved utility. Used directly as a linear estimator of utility, AHP would have the same feature as Steuer’s method, in that only original model corner points would be considered. A primary difficulty in applying AHP to multiobjective analyses is the potentially large number of paired comparisons asked of decision makers. However, paired comparisons have been demonstrated to be relatively easy in real applications. Further, f ecus upon important objectives would reduce the number of paired comparisons needed, and focus decision maker attention upon more important objectives. Probably more important is the need to eliminate the impact oE scale. It is difficult to give a relative importance of profit measured in millions oE dollars, with reduction of liability measured in lawsuits expected. This scalar complication can be reduced by focusing upon the concept of profit versus the concept of risk, and eliminating differences in scale by other means, such as the use of the range in objective values obtained from the payoff table as given in Olson, et al. AHP provides a valuable means of supporting multiple objective programming. It has proven very useful in comparing discrete alternatives. It should prove just as useful in supporting multiple objective programming analyses. REFERENCES J. F. (1986). A multiobjective methodology for selecting subsystem Hanaqement Science, automation options. 12, 1628-1641. Cohon, J. L. (1978). Multiobjective Prooramminq and Planninq. Academic Press, New York. The Theory of Value. The DeBreu, G. (1975). Colonial Press, Clinton, MA. An overview of techniques Evans, G. W. (1984). for solving multiobjective mathematical Manaqement Science, 30, 1268programs. 1282. Georghescu-Roegen, N. (1954). Choice, expectations and measurability. Quarterly Journal of Economics, 68, 503-541. Bard,

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C. L., A. S. M. Masud, S. R. Paidy and Mathematical programming K. Yoon (1960). with multiple objectives: A tutorial. Comnuters and Onerations Rm, 1, 5-31. March, J. G. (1978). Bounded rationality, ambiguity, and the engineering of choice. The Bell Journal of Economics, 2, 587-608. Mitchell, K. H. and G. Blngham (1986). Maximizing the benefits of Canadian Forces equipment overhaul programs using multlobjective optimization. INFOR.& 251-264. Horgenstern, 0. (1972). Thirteen critical points in contemporary economic theory: An interpretation. Journal of Economic Literature, & 1163-1169. Olson, D. L., M. Venkataramanan and J. Mote A technique using analytical (19861. hierarchv nrocess in multiobjectlve planning-mbdels. Socio-Economic Plannlnq Sciences, 20, 361-368. Rosenthal, R. E. (1985). Principles of multiobjective optimization. Decision Sciences 16 133-152. PI -I Saaty, T. L. (1977). A scaling method for priorities in hierarchical structures. Journal of Mathematical Psycholoqv, II, 234-281. Saaty, T. L. (1980). The Analytic Hierarchy Process McGraw-Hill, New York. Simon,‘A.)(1979). Rational decision making in business organizations. The American Economic Review, & 493-513. Steuer. A. E. (1977). An interactive multlole objective-linear programming procedure: TINS Studies in the Manaoement Sciences, b 225-239. Steuer, R. E. (1986). MUltiDle Criteria Ontlmization: Theory, Computation and AnDllcation. John Wiley C Sons, New York. Zeleny, H. (1983). Multinle Criteria Decision McGraw-Hill, New York. Makinq. Hwanq,