Behnam Malakooti, Ph.D., P.E.

A Decision Support System for

** Ranking DIscrete ALternatives **

RADIAL is a software package developed for the IBM-PC

Designed by B. Malakooti and programmed by E. Tandler

Department of Systems Engineering and

Center for Automation and Intelligent Systems Research

Case Western Reserve University

Cleveland, Ohio 44106

We develop an automated decision support system for

the IBM-PC for selecting the most Preferred discrete

multiple criteria alternative or for ranking a set

of alternatives.

In the process of decision-making, the decision-maker must choose some course of action among the various alternatives. In nearly all decision-making problems, there are several conflicting criteria for judging the possible alternatives. The main concern of the decision-maker is to fulfill his goals while satisfying the constraints of the system.

In this paper we introduce RADIAL, a software package that assists the decision-maker in the selection of the most preferred discrete multiple-criteria alternative or in the ranking of a set of alternatives. RADIAL consists of six decisions support subroutines, providing the user with array tools for solving Multiple Criteria Decision-Making (MCDM) problems. Although the subroutines vary in their degree of sophistication, they are all very simple to use, making RADIAL ideal for those with little or no background in MCDM. At the same time, the user has the opportunity to change all pertinent parameter values and to control the sequence of operations when applicable, making RADIAL a powerful instrument in the hands of MCDM experts as well. Section 2 contains brief descriptions of how the subroutines work and a few examples. Section 3 describes other features of RADIAL. Section 4 discusses the results obtained by the package. Some final remarks are in Section 5. For detailed discussions of the subroutines, the reader may refer to [1]. References [2, 3, 4, 5, and 6] illustrate applications of the subroutines to production/manufacturing problems. For further references on multiple criteria decision making, see [7].

RADIAL consists of six subroutines. In the following descriptions of these subroutines, let us assume that we are given a set of n alternatives, which are denoted by X1, X2, ... Xn, and that each alternative has m criteria (objectives). Let xk,i , Xk,2, Xk ,m denote the values of the criteria of an alternative Xk.

2.1 Q-RALO (Quick Ranking b@ Lexicogravhic Ordering): In Q-RALO, the user simply lists the criteria in order of importance. The program then ranks the alternatives according to the order specified by the user. The alternative having the highest value for the most. Important criterion is ranked #1, the one having the second highest is ranked #2, and so forth. Ties are broken according to the values of the next-most important criterion. For example. let A, the set of alternatives, be

X1 = (8, 1, 6)

X2 = (8, 6, 3)

X3 = (5, 8, 3)

X4 = (6, 8, 2)

Suppose the user feels that Criterion #2 is the most important criterion, followed by #3, then #1. The highest ranking alternative is then X3 , followed by X4 , X2 , and X1

2.2 Q-RITA (Quick Rankina by Identification of Trial Alternatives): Q-RITA obtains the closest feasible alternatives to a trial alternative (also called a goal, ideal, aspiration, or target alternative). The program shows how this alternative compares with the feasible alternatives, indicating (for each criterion) how many alternatives have higher values than the trial alternative and how many have lower values. Q-RITA then lists the alternatives, if any, that (a) are dominated by (i.e. inefficient with respect to) the trial alternative, (b) dominate it, and (c) are identical to it. The alternatives are then ranked according to the differences between the values of their criteria and those of the corresponding criteria of the trial alternative. Large positive differences (big improvements) and small negative differences (minimal sacrifices) are sought. The desirability of each alternative is determined by the lowest among the differences associated with each criterion.

Let Xt = xt, 1 , xt, 2 , ...t xt,m denote the values of the trial alternative, where there are m criteria. If there are n alternatives, then for each k = 1, 2, ..., n. alternative Xk is associated with

Uk = min { xk, i - xt, i }, i = 1, 2, ..., m (1)

The alternatives are then ranked according to their respective U, the one with the highest U being ranked #1. For example, if the set of alternatives is set A above, and the trial alternative is Xt = (8, 5, 4), then the highest-ranking alternative is X2, followed by X4, X3, and Xi.

2.3 Q-RAW (Quick Rankine by Assessment of Weizhts): Q-RAW assumes a linear utility function and ranks the alternatives using the objective weights, i.e. the relative importance of the criteria. Let the objective weights be denoted by W = w1, W2, Wm. The utility (desirability) of an alternative Xk is the sum of its weignted objective values, that is

The user specifies whether the objective weights are assessed by direct entry or by tradeoffs with marginal rate of substitution questions.

2.3.1 In the direct entry method, the user simply enters each objective weight. For example, given set A above, he may enter the weights W = (0.05, 0.60, 0.35), in which case the highest ranked alternative is X3, followed by X4, X2, and X1.

2.3.2 In the tradeoff method, the user is presented with two alternatives, one of which is missing a value for one criterion. The user is to provide the value that would, in his judgment, make the two alternatives equally good. After several such questions are answered, the weights are assessed by a linear programming approach. For example, the two alternatives could be as follows:

Alternative A = (6.5, 4.5, 4.0)

Alternative B = (6.2, 4.5, ???)

In going from Alternative A to Alternative B. Criterion #2 unchanged, but Criterion #1 has been decreased from 6.5 to 6.2. To offset this decrease, an increase in Criterion 93 is offered. The user specifies the amount of this increase by replacing "???" with the numerical value that would make him indifferent between Alternatives A and B. If this value is, for instance, 4.1, then Q-RAW derives the LP constraint

(6.5-6.2)wl - (4.1-4.0)W3 - I1,3 = 0 (3)

where I1,3 is an inconsistency compensation variable. The linear program assesses the objective weights by minimizing the sum of all the inconsistency compensation variables. For a problem with m criteria, there are m(m-l)/2 possible tradeoff questions, although generally no more than m+l need be answered for accurate results.

2.4 RASP (Ranking Alternatives with Strength of Preference): RASP is similar to Q-RAW, differing only in the method assessing the objective weights. In RASP, the user makes paired comparisons of alternatives and indicates the strength of his preference for one alternative over the other. He uses ratings of A, E, I, O, and U to indicate weather the strength of his preference is Absolutely important (A), Especially 3.mportant (E), Important (I), of Ordinary importance (O), or Unimportant (U). A linear programming technique is then used to calculate W = Wl, W2, .., Wm, Whereupon the alternatives are ranked according to the linear utility function. Suppose, for example, that set A above is a subset of the given set of alternatives. Let the objective weights be W = (0.05, "0.60, 0.35); the decision maker does not know these weights explicitly, but he has enough of an intuitive grasp of them to be able to respond to the paired comparison questions. The first question would be Xi vs. X2; the decision-maker would prefer X2 with strength of preference "A" (this can be verified by substituting values into Equation 2). Next, he would prefer Xa to X2 with a rating of "I", and X3 to X4 with "U" RASP then yields W = (0.04, 0.58, 0.38), a close approximation of the true weights. Using Equation 2, RASP then ranks the entire set of alternatives accordingly. This method was applied in [2 and 3]. See [3] for details of the method.

2.5 GAS (Gradient-based Alternative Selection): GAS uses several heuristic techniques to eliminate sub-optimal alternatives until only the most preferred alternatives remain. The user answers paired comparison questions, some with strength of preference. GAS uses this information to generate the gradient at a given point, thus finding the best direction in which to explore new alternatives, and perform a one-dimensional search in the direction of the gradient in order to find the most preferred alternative as quickly as possible. Meanwhile, it uses other techniques to eliminate sub-optimal alternatives. After the most preferred alternative is found, the user may continue answering questions to find the second-most preferred alternative. He may continue further to find other good alternatives. GAS is a highly sophisticated decision support subroutine, and an illustration of it would be to lengthy to include; for full detail of the method, the reader is referred to [1]. For applications of method, see [4 and 5]

2.6 COES (Computerized Oven-Ended Search: Like GAS, COES uses the concepts of gradienm assessment by strength of preference and one-dimensional searching to explore alternatives. This program does not rank alternatives; it merely asks the user paired comparison questions, retaining the preferred alternative until a better one is found. The user may continue the procedure as long as he desires. COES makes no assumptions regarding the structure or the existence of the utility function. See [4 and 5].

RADIAL begins by duplicating the input data file (the "original data set") and removing all redundant and inefficient alternatives from this duplicate set (an alternative X is inefficient, if and only if there is another alternative Y such that the value of each criterion of Y is at least as high as its counterpart in X, and Y is not identical to X). Thus, the new data set contains only efficient alternatives and is termed the "efficient data set". The subroutines will accept either the original or the efficient data set as input; the main menu indicates which set is currently in use and provides the user with the option to switch to the other. In addition, either data set can be displayed onscreen in either its true form or the normalized form that is used for the calculations (in normalized form, each criterion is measured on a scale of 0 to 100).

RADIAL is designed to provide the user with as much flexibility as possible. The "Quick Ranking" subroutines (Q-RALO, Q-RITA, and Q-RAW) require only simple input, but values for certain parameters are needed in RASP, GAS, and COES. At the beginning of these three subroutines, the user is given the option to see the default values of the parameters, receive a brief explanation of their significance, and reset the values if he desires. In GAS, the user is also allowed to control the order in which the different phases of the method are carried out. Thus, if the user declines to change any of the parameters of GAS, the default values are used and the program is automatically controlled; the user only makes paired comparisons of alternatives, unencumbered by extraneous details and information, until the most preferred alternative is determined. He may instead reset some or all of the parameter values and opt for manual control. Then, at each paired comparison, the current phase of the method is reported and other information may be requested; alter each phase is completed, the user may choose which phase to enter next.

It should be obvious that, if it is valid to assume that one criterion can be favored to the exclusion of all others, the output generated by Q-RALO is completely accurate. The results yielded by Q-RITA depend on the trial alternative entered by the user; the closer the trial alternative to the best alternative, the more reliable the results. Since COES is simply an open-ended searching procedure. It does not generate output that can be evaluated for accuracy. Section contains simple examples demonstrating output of Q-RAW and RASP. To test GAS, we ran extensive computational experiments, assuming both quadratic and Chebyshev utility functions. The experiments showed that GAS almost always arrives at the optimal alternative, and in the remaining cases found alternatives that were very close to optimal. Moreover, GAS achieved its results without asking an excessive number of questions. For more details of our experiments and complete tabulations of the results, see [1].

The six different decision support subroutines in RADIAL give the user flexibility for solving Multiple Criteria Decision Making problems, and the ability to have the data listed in normalized form (with or without inefficient and redundant alternatives) is a convenient asset. The package is sophisticated but simple to use, making it ideal for MCDM novices and experts alike.

Professor B. Malakooti

Systems Engineering Department

Case Western Reserve University

Cleveland, Ohio 44106

1. Malakooti, B., "Solving Discrete Multiple Criteria Problems., A

Heuristic Interactive Approach," revisions required by IEEEE

Transactions on Systems, Man, and Cybernetics, 1984.

2. Malakooti, B., "A Methodology for the Automation of Medium or

Small Manufacturing Companies," Computers in Industry: An Inter:

national Journal, Vol. 7, 1985, pp. 333-341.

3. Malakooti, B., G. D'Souza, "Multiple Objective Programming for

the Quadratic Assignment Problem," International Journal, of

Production Research, Vol. 25, No. 2, pp. 285-300, 1987.

4. Malakooti, B., "A Decision Support System for Multiple Objective

Linear Programming Problems," revisions required by Journal of

Optimization Theory and Apylication (JOTA), 1985.

5. Malakooti, B., "An Interactive Manufacturing Planning Approach

with an Application to Computer Aided Facility Layout Selection,"

Larae Scale Systems: Theory and Avolications (In Press).

6. Malakooti, B., W. H. Balhorn, "Selection of Sampling Plans with

Multi-Attribute Defects in Computer-Aided Quality Control,"

International Journal of Production Research (In Press).

7. Steuer, R., Multiiple Criteria Optimization: Theory, Computation and Application, John Wiley, 1986.