- Email: [email protected]
Multi-attribute decision making using constrained criteria
Cornput. & 00s
ResL’ol
4. PP 139.145.
Pergamon Pres.
1977
Punted m Great Britain
MULTI-ATTRIBUTE DECISION MAKING USING CONSTRAINED CRITERIA STAN SCHENKERMAN* College of Business Administration,
Umversity
of Bridgeport.
Bridgeport.
Connecticut
06602, U.S.A.
Scope and purpose-This paper illustrates a simple, practicable, and intuitive approach to decision making in the face of multiple attributes. Presented within the context of a mini-case involving a three-member decision-making group, the approach keeps the attributes distinct and requires only computations easily done with a hand-held calculator. Because the approach enhances, rather than impairs insight, it encourages the active participation of even non-mathematically inclined decision makers.
Abstract-A previous paper discussed a constrained-decision-criteria approach to multiple criteria decision making in a uni-attribute framework. This paper illustrates the extension of that approach to the multi-attribute case. In contrast to the usual methods employed with multiple attributes, the extension avoids amalgamation and preserves the distinct character of each attribute. Decision maker insight is thereby enhanced. Furthermore. the computationally simple solution techniques of the uni-attribute case are directly applicable. Thus the approach, which is practicable and intuitively appealing, is suited to the active participation of even mathematically unsophisticated decision makers.
INTRODUCTION
The mini-case The setting is decision making with multiple attributes. The scenario, highly simplified but surely recognizable, involves three characters: Finance, Marketing, and Manufacturing. Each is a scalar thinker; each views the universe in terms of a single attribute. They are grappling with a six-act, six-state decision problem. Finance, vitally concerned with the bid price of the firm’s common stock, judges every proposal by its impact on profit. Furthermore, he often reproaches himself for having foresight less acute than his 20-20 hindsight. His profit payoff and opportunity loss matrices for the subject problem are given in Tables 1 and 2. (Profit is represented by Attribute 1.) Marketing believes his is a sacred mission-to increase the firm’s small market share: It matters not what profit be, Nor Cost, increased alarmingly. All is but that jewel so rare, A growing, dominant Market Share. Table 3 shows his evaluation of the market share payoffs (Attribute 2). Manufacturing tries to maximize his year-end bonus by being tyrannically cost conscious. Demanding economic production lots and tight inventory management, he compulsively measures all results against the best that could have been achieved. His opportunity loss matrix is given in Table 4. (Attribute 3 represents manufacturing plus inventory costs.) Background In form at least, this scenario (to which we will return subsequently) provides difference between uni- and multi-attribute decision making. Compared to the case, multi-attribute decision making is complicated by incommensurability among profit dollars, for example, being different from cost dollars, both being different *Stanley Schenkerman
the essential uni-attribute attributesfrom market
is Associate Professor of Quantitative Analysis in the College of Business Administration, of Bridgeport. He holds: B.E.E. from Citv Colleae of the Citv Universitv of New York. M.E.E. from New York University, and both M.S. in Industrial Management and ch.D. in Opeiations Research from the’Polytechnic Institute of New York. He is a licensed Professional Engineer (New York). Prior to joining the University of Bridgeport, he was in engineering and management in the aerospace industry. He has published over a dozen articles in iechnical,biofessional, and academic, journals. His present research areas focus on robust decision making under uncertainty and multi-attribute decision making. University
C A 0 a., Vol 4. No 2-E
139
140
STAN SCHENKERMAN
Table I. Profit payoff matrix (attribute
1) in $,OOO
Table 2. Profit opportunity loss matrix (attribute 1) in $,OOO
Slate
1
2
Act
0.10
0.15
I
50 40 30 20 10 0
50 55 45 35 2s 15
2 3 4 s 6
Table
Act I 2 3 4 5 6
4 Probs 0.35 0.25
5 0.10
0.05
Expctd payoff
50 55 60 65 55 45
50 55 60 65 70 60
50 55 60 65 70 75
50.00 53.50 54.75 52.25 44.50 35.25
3
50 55 60 50 40 30
3. Market-share payoff (attribute 2) in %
1 10 10 10 IO IO IO
2 9 11 11 11 II 11
6
matrix
Act I 2 3 4 5 6
Table
State 4 3
5
6
Act
7 9 II 13 13 13
6 8 IO 12 14 14
5 6 9 11 I3 15
2 3 4 5 6
8 10 12 I2 12 12
I
2
0 10 20 30 40 so
5 0 IO 20 30 40
State 3 4 IO s 0 10 20 30
15 10 5 0 IO 20
5
6
20 IS 10 5 0 IO
25 20 15 IO 5 0
4. Manufacturing-cost regret matrix (attribute 3) in $,OOO State 3
I
2
0
0
50 120 170 240 300
0 60 120 170 260
0 0 70 130 180
IO
4
5
6
0 0 0 0 90
0 0 0 0 0
000 0 0 0 80 140
share. Thus, each state-action pair yields a vector-valued consequence. Even with a utility function for each attribute, the result is a utility vector instead of the scalar utility of the uni-attribute case. These vectors cause the comphcations. Comparing scalars is easy; comparing vector-valued functions is not. The usual approaches for coping with multi-attribute decision problems employ amalgamation schemes-schemes effecting a weighted combination of the individual attributes. By using such an amalgamation, a problem in multiple attributes (a problem in vectors) is reduced to a problem with a single surrogate attribute (a probtem in scalars), which then can be treated by widely available uni-attribute techniques [2-j]. Amalgamation is a valid technique if a suitable surrogate measure actually can be determined and if the surrogate truly reflects the decision maker’s preferences. Of course, the decision maker must still be capable of specifying a decision criterion that accords with his definition of “best”. When the amalgamation scheme results in a surrogate attribute whose consequences are expressible in scalar utility terms, the normative definition of best is optimization of expected utility. There are no fundamental conceptual (or behavioral) problems-just those operational problems attending the elicitation of the decision maker’s utility function, the assessment of the state distribution, and the specification of attribute weights. (The first two problems also attend uni~attribute decision making; the last, only the muftiattribute case.) There are, however, those managerial decision makers for whom amalgamation is an unsatisfactory approach. In effect, the weights needed to combine the attributes are statements of acceptable tradeoffs among incommensurables. Such decision makers may be reluctant or unable to specify these tradeoff regimes, either explicitly or implicitly. They may prefer to keep the attributes distinct instead of obscuring them in an amalgamation which is most likely nonintuitive. Certainly, when the decision maker is a group, rather than an individual, the difficulty is compounded even further. In order to arrive at a single surrogate, a group utility function is required; yet group utility functions do not now exist, at least in any practicable sense. Even in the uni-attribute case, there is experimental evidence that individual decision makers will not rely on a single decision criterion[l]. They may not have the mathematical sophistication to appreciate the implications of utility and the maximization of expected utility. They may feel that utility cannot adequately account for adverse outcomes; they may be skeptical about using expectation for nonrecurring events. Often they take on overly con-
Multi-attribute decision making using constrained criteria
141
servative approach, e.g., maximin payoff, foregoing the benefits of a more realistic decision criterion. If they accept a single criterion it is only after, or in conjunction with, the satisficing of certain security levels. Clearly, multiple attributes aggravate this situation even more. A previous paper[6] described an alternative to maximizing expected utility, an alternative that such decision makers might find more acceptable. That approach, which addresses the uni-attribute case, has the effect of allowing the decision maker to consider multiple decision criteria. Specifically, his anxieties and aspirations are embodied in a mathematical program whose deterministic and/or chance constraints, when satisfied, assure acceptable security and attainment levels. The decision maker is thus freed to pursue the greater gains afforded by a more propitious, less conservative criterion. When this approach is used, certain decision criteria (e.g., maximin payoff, minimax regret) lose their raisons d’etre-the security they would otherwise provide being assured by the constraints. The action space under consideration is reduced thereby, as the few most acceptable acts are brought into sharp focus. Overview
In this paper the constrained-criteria approach of that previous paper is applied to the multi-attribute case. With this approach (which, in contrast to the classical methods, avoids amalgamation schemes): The distinct character of each attribute is preserved throughout. Thus the need to specify tradeoff regimes is eliminated and decision maker insight is enhanced. Satisficing security and aspiration levels may be prescribed for each attribute and in each attribute’s terms. The simple uni-attribute solution techniques (which are amenable to manual computation) may be employed even with multiple attributes. The effects of proposed problem changes are easily seen. Decision maker involvement is encouraged; the approach is intuitively appealing. Of course, these benefits are even more important in group decision situations. After reviewing the uni-attribute constrained-criteria approach we show the minor modifications necessitated by multiple attributes. Then we illustrate the approach by continuing our three-character mini-case. We conclude with some remarks about informational demands and decision maker participation. SUMMARY OF UNI-ATTRIBUTE
CASE
In order to make this paper self-contained, we summarize the uni-attribute case discussed in [61. The quintessence of that approach is the formulation of the decision problem as a mathematical programming model in which: The decision maker embodies his anxieties and aspirations in a set of deterministic and/or chance constraints. The feasible region resulting therefrom represents an acceptable set of actions. The decision maker is thus free to choose a propitious decision criterion as the objective function to be optimized over the feasible set of acts. Alternately, he can impose tighter restrictions repeatedly until the feasible set cannot be reduced further. Formally, let k = 1,2, . . . , K denote K disjoint and exhaustive acts, K finite; Vk denotes the feasible set of acts. j= 1,2 1..., J denote J disjoint and exhaustive states of nature, J finite. akj = the payoff of Act k under State j, finite. the OppOrtUnity loss (regR?t) of Act k under State j, finite. pj = the State j probability, where 0 5 pj 5 1, V j; Zjpj = 1. b, (n = 1,2, . . .) = appropriate security levels prescribed by the decision maker. q,,, (m = 1,2, . . .) = appropriate chance-constraint probabilities prescribed by the decision maker. Then, the deterministic constraints, which must be satisfied for each act in the feasible set, &j
rkj
=
r,i
Minj(akj)2bl,
Vk
(1)
142
STAN SCHENKERMAN
Maxi (rkj)= bz,
tf k
(2)
respectively assure at least a minimum payoff level, at most a maximum regret. The analogous chance constraints are Prob(c,rbl)rqi,
Probhi
5
b2) 2 q2,
Vk
(3)
vk
each of which can be generalized into a partial or total probability distribution. Constraints can also be imposed on expectaton and on Hurwicz value:
where 0 5 c s 1 and c is the Hurwicz coefficient of optimism. By selecting one or more constraint types and prescribing acceptable values for the requisite b’s and q’s, the decision maker explicates his anxieties and aspirations. With his security concerns assuaged, the decision maker is free to optimize a propitius decision criterion. Suitable objective functions include (but are certainly not limited to) the Bayesian (maximization of expected payoff, minimization of expected opportunity loss) and probabilistic functions structured after (3) and (4), e.g., Maximizevr Prob (akj 2 b,) Maximizevr, Prob (F&j5 b6).
(8)
Solving the w&attribute model Solving the model is usually quite simple and proceeds in two phases: (1) finding a feasible set of acts, and (2) optimizing the objective function over that feasible set. The feasible set consists of those acts not violating any constraint. It is obtained by evaluating each act not previously eliminated against each constraint until either a violation eliminates that act or the act is found not to violate any constraint. An efficient order is: ~terministic constraints, Types (1) and (2)-by inspection. Chance constraints, Types (3) and (4)-by summing the pj over all appropriate states and comparing the sum with the right side. Types (5) and (@-by computing the left-side expression and comparing with the right side. Phase 2 requires selecting that act from the feasible set that optimizes the objective function. This selection is accomplished by exhaustive evaluation of the objective function for each act in the feasible set, exactly as in a conventional evaluation of the decision criterion. Alternately, the decision maker can tighten the constraints, or impose additional constraints, to further restrict the feasible set. This process can be repeated until the feasible set consists of only one act (or until any further restriction would eliminate all remaining acts). EXTENSION TO ~ULTIFLE ATTRIBUTES
The payoff (opportunity loss) matrix of a managerial decision problem is a two-dimensional array of commensurate elements, e.g., the units of each element being profit dollars. Such a matrix represents a single attribute. Consider a three-dimensional array comprised of a set of payoff (opportunity loss) matrices-each matrix corresponding to a single attribute. This array is the multi-attribute extension of the uni-attribute matrix. Using the uni-attribute approach, an independent set of deterministic and chance constraints can be formulated for each attribute. Then, the collection of these independent constraint sets may be treated as one enlarged constraint set, denoted the “grand set”. Neither the numbers nor the types of constraints need be the same for each constraint set. In fact, it is possible for some attributes to have empty constraint sets.
Multi-attribute decision making using constrained criteria
143
The objective function for the multi-attribute model may be handled as follows: For each attribute, myopically choose a decision criterion to be the objective function, i.e. choose each objective function without regard to the remaining attributes. Select the most propitious as the grand objective function. Prescribe satisficing Ievels for the others and incorporate them as additional cons~aints in the grand set. Alternately, as in the uni-attribute case, the feasible set can be reduced by tightening existing constraints or imposing additional constraints. Analytic extension Extending the variables and equations of the uni-attribute case to the multi-attribute case requires only one new variable (to index the attributes) and the modification of several others to include the index. Let i = 1,2,. . . , 1 denote the I attributes, I finite. Then, akj becomes a,, the payoff for Attribute i; rkj becomes rkji,the opportunity loss for Attribute i; b, becomes bni, a prescribed security level for Attribute i. Observe that security levels must have the same units as the corresponding attributes. Thus, the b,,, are indexed by attribute. In contrast, the q,,,, being probabilities, are dimensionless and do not require attribute indexing. Therefore, the multi-attribute versions of uni-attribute equations (l)-(S) are, respectively, Minj (au) 2 bli,
Vk
(1’)
Maxi (rkji)= bzi,
Vk
(2’)
Prob(&jc?b,,)Lg,,
Vk
(3’)
PrOb (rkji5 bzi) 2 q2,
Vk
(4’)
Vk
Cjakjipj2 bx, c Maxi (okii)+ (1
- C)
Mini (a&i)2 b4ir V k
(6’)
byi)
(7’)
Maximize, Prob (f&i 5 b6i).
(8’)
Maximize, Prob
(akji
2
Equations (1’-6’) are constraints and may be imposed for any number of attributes. ~uations (7’-8’) are candidate grand objective functions; at most one of these wouid be used and that only for a single attribute. Solution strategy As in the uni-attribute case, solving the model proceeds in two phases: (I) finding a feasible set of acts, and (2) optimizing the objective function over that feasible set. Phase 1, checking each act for feasibility, may be done one constraint type at a time, each attribute seriatim, using the methods previously described. An act is eliminated as soon as a violation is found. If not eliminated, the act is admitted to the feasible set. Clearly, if the feasible set is empty, i.e. if every act of the initial action space violates some constraint in the grand set, the prescribed satisficing levels are too stringent (in combination at least) and therefore unattainable. This is more likely to happen in the multi- than in the uni-at~ibute case. Phase 2 is accomphshed exactly as in the uni-attribute case: select the optimal act by exhaustively evaluating the (uni-attribute) objective function for each act in the feasible set. It should be observed that while Phase 1 has been slightly complicated by the multiple attributes, Phase 2 has not. In general, it may even be less tedious than in the uni-attribute case because of the reduced set of feasible acts over which evaluations are required. MEANWHILE
. . our heroes have been engaged in much discussion. They finafly agree that, for the firm’s 1
benefit, expected profit should be maximized. This agreement, however, has not been wrested easily; it is contingent upon each obtaining certain assurance for his own vested interest.
144
STAN~HENK~RMA~
Finance imperiously demands a minimum profit of $20,000 and a limit of 0.10 on the probability of a profit regret exceeding $25,000. Marketing is adamant about having at least an even chance of securing 10% of the market. Manufa~turing~ remaining suspicious but now dissauded from his insistence on zero cost regret, declares that his ulcer cannot tolerate a regret over $125,000. Thus, they have implicitly agreed on the following multi-attribute model:
where all feasible acts must satisfy Mini (akjl) zz20
(10)
Prob (rkjg> 25) zs0.10
(11)
Prob (a&j22 10%)2 0.50
(i2)
Max, (r&r
125.
(13)
From Tabte L, (IO) eliminates Acts 5 and 6. From Table 4, Act 4 violates (t3). From Table 3 and the state probabilities listed in Table 1, Act 1 violates (12). As seen from Table 2, (I 1) would have eliminated Acts 5 and 6 had they not been eliminated previously. The feasible set consists of Acts 2 and 3, with Act 3 maximizing expected profit. It should be noticed that, for these data, Act 3 is Bayes optimal on Attribute 1 even in the unconstrained case. Thus the grand set constraints do not affect optimality and the “discussion” could have been shortened considerably. If manufacturing had persisted in his demand for zero regret for Attribute 3, Acts 2 and 3 would have been eliminated. The feasible set would then be empty, indicating a too stringent grand set-an unattainable combination of satisficing levels.
The satisficing levels for each at~~bute can actually be tightened sign~~~antly without invalidating the selection of Act 3. The following is the most stringent grand set that leaves Act 3 feasible: Minj (akjl) 2 JO, V k Prob(r,l>
15)rO.t0,
Vk
Prob (Ukjz2 10%) 2 0.95, MaXi(Tkj3)C: 120,
Vk
V k.
Thus, as in the u&attribute case, the optimal decision with rn~~ti-attribute constraints tends to be inv~ia~t under wide changes in security level and probability prescriptions. We would anticipate that this insensitivity depends upon the number of attributes, the number of acts in the initial action space, and the number of states of nature. CONCLUDING
REMARKS
Although rarely a~knowledged~ the information demands necessary to develop actuat payoffs or opportunity losses in any application of decision theory are generafly not triviateven when considering only a single attribute. When dealing with multiple attributes, these demands become all the more significant, regardless of the multi-attribute technique used. In fact, those approaches which, unlike the one herein, amalgamate the attributes into a single surrogate, additionally require a rationalizable attribute weighting scheme. Thus, papers such as this, which present constructs and solution methods, leave to the practitioner and the decision maker the often laborious, always unglamorous tasks of satisfying what are sometimes voracious demands for information.
Multi-attribute
decision
making
using constrained
criteria
145
Decision maker involvement
The importance of having the decision maker play an active role in imposing his judgments and preferences on problem formulation and solution is pointed out in [5, p. 339, p. 3411. Because it is intuitive and enhances insight, the approach described in this paper and in [6] is particularly well suited to continual decision maker involvement. Indeed, he must actively participate in each phase: Specification of attributes. Development of alternative acts/strategies. Formulation of grand set constraints and prescription of satisficing levels. Formulation of objective functions, selection of the grand objective function, and prescription of satisficing levels for the remaining objective functions. Iterative: evaluation of solutions and modification of the model. The relatively simple solution techniques--even for the multi-attribute model-greatly facilitate and encourage multiple iterations. Each iteration is under the decision maker’s complete control. Perhaps the most important feature of this approach is its preservation of distinct attributes, Keeping the attributes distinct eliminates the need to specify tradeoff regimes among attributes. It enhances the decision maker’s insight; it contributes to his understanding. Above all, it makes the decision maker a more knowledgeable participant. REFERENCES D. W. Conrath. From statistical decision theory to practice: some problems with the transition, Mgmt Sci. 19. 873 (1973). R. T. Eckenrode. Weighting multiple criteria, Mgmt Sci. 12, 180 (I%.(). G. P. Huber, Multi-attribute utility models: a review of field and field-like studies, Mgmt Sci. 20, 1393 (1974). G. P. Huber. Methods for quantifying subjective probabilities and multi-attribute utilities, Dec. Sci. 5. 430 (1974). L. P. Ritzman. Decision analysis with multiple objectives. Proc. VI Annual Meeting of the American Institute for Decision Sciences, pp. 339-343 (1974). 6. S. Schenkerman. Constrained decision criteria, Dec. Sci. 6. 42 (1975). I. 1. 3. 4. 5.
ResL’ol
4. PP 139.145.
Pergamon Pres.
1977
Punted m Great Britain
MULTI-ATTRIBUTE DECISION MAKING USING CONSTRAINED CRITERIA STAN SCHENKERMAN* College of Business Administration,
Umversity
of Bridgeport.
Bridgeport.
Connecticut
06602, U.S.A.
Scope and purpose-This paper illustrates a simple, practicable, and intuitive approach to decision making in the face of multiple attributes. Presented within the context of a mini-case involving a three-member decision-making group, the approach keeps the attributes distinct and requires only computations easily done with a hand-held calculator. Because the approach enhances, rather than impairs insight, it encourages the active participation of even non-mathematically inclined decision makers.
Abstract-A previous paper discussed a constrained-decision-criteria approach to multiple criteria decision making in a uni-attribute framework. This paper illustrates the extension of that approach to the multi-attribute case. In contrast to the usual methods employed with multiple attributes, the extension avoids amalgamation and preserves the distinct character of each attribute. Decision maker insight is thereby enhanced. Furthermore. the computationally simple solution techniques of the uni-attribute case are directly applicable. Thus the approach, which is practicable and intuitively appealing, is suited to the active participation of even mathematically unsophisticated decision makers.
INTRODUCTION
The mini-case The setting is decision making with multiple attributes. The scenario, highly simplified but surely recognizable, involves three characters: Finance, Marketing, and Manufacturing. Each is a scalar thinker; each views the universe in terms of a single attribute. They are grappling with a six-act, six-state decision problem. Finance, vitally concerned with the bid price of the firm’s common stock, judges every proposal by its impact on profit. Furthermore, he often reproaches himself for having foresight less acute than his 20-20 hindsight. His profit payoff and opportunity loss matrices for the subject problem are given in Tables 1 and 2. (Profit is represented by Attribute 1.) Marketing believes his is a sacred mission-to increase the firm’s small market share: It matters not what profit be, Nor Cost, increased alarmingly. All is but that jewel so rare, A growing, dominant Market Share. Table 3 shows his evaluation of the market share payoffs (Attribute 2). Manufacturing tries to maximize his year-end bonus by being tyrannically cost conscious. Demanding economic production lots and tight inventory management, he compulsively measures all results against the best that could have been achieved. His opportunity loss matrix is given in Table 4. (Attribute 3 represents manufacturing plus inventory costs.) Background In form at least, this scenario (to which we will return subsequently) provides difference between uni- and multi-attribute decision making. Compared to the case, multi-attribute decision making is complicated by incommensurability among profit dollars, for example, being different from cost dollars, both being different *Stanley Schenkerman
the essential uni-attribute attributesfrom market
is Associate Professor of Quantitative Analysis in the College of Business Administration, of Bridgeport. He holds: B.E.E. from Citv Colleae of the Citv Universitv of New York. M.E.E. from New York University, and both M.S. in Industrial Management and ch.D. in Opeiations Research from the’Polytechnic Institute of New York. He is a licensed Professional Engineer (New York). Prior to joining the University of Bridgeport, he was in engineering and management in the aerospace industry. He has published over a dozen articles in iechnical,biofessional, and academic, journals. His present research areas focus on robust decision making under uncertainty and multi-attribute decision making. University
C A 0 a., Vol 4. No 2-E
139
140
STAN SCHENKERMAN
Table I. Profit payoff matrix (attribute
1) in $,OOO
Table 2. Profit opportunity loss matrix (attribute 1) in $,OOO
Slate
1
2
Act
0.10
0.15
I
50 40 30 20 10 0
50 55 45 35 2s 15
2 3 4 s 6
Table
Act I 2 3 4 5 6
4 Probs 0.35 0.25
5 0.10
0.05
Expctd payoff
50 55 60 65 55 45
50 55 60 65 70 60
50 55 60 65 70 75
50.00 53.50 54.75 52.25 44.50 35.25
3
50 55 60 50 40 30
3. Market-share payoff (attribute 2) in %
1 10 10 10 IO IO IO
2 9 11 11 11 II 11
6
matrix
Act I 2 3 4 5 6
Table
State 4 3
5
6
Act
7 9 II 13 13 13
6 8 IO 12 14 14
5 6 9 11 I3 15
2 3 4 5 6
8 10 12 I2 12 12
I
2
0 10 20 30 40 so
5 0 IO 20 30 40
State 3 4 IO s 0 10 20 30
15 10 5 0 IO 20
5
6
20 IS 10 5 0 IO
25 20 15 IO 5 0
4. Manufacturing-cost regret matrix (attribute 3) in $,OOO State 3
I
2
0
0
50 120 170 240 300
0 60 120 170 260
0 0 70 130 180
IO
4
5
6
0 0 0 0 90
0 0 0 0 0
000 0 0 0 80 140
share. Thus, each state-action pair yields a vector-valued consequence. Even with a utility function for each attribute, the result is a utility vector instead of the scalar utility of the uni-attribute case. These vectors cause the comphcations. Comparing scalars is easy; comparing vector-valued functions is not. The usual approaches for coping with multi-attribute decision problems employ amalgamation schemes-schemes effecting a weighted combination of the individual attributes. By using such an amalgamation, a problem in multiple attributes (a problem in vectors) is reduced to a problem with a single surrogate attribute (a probtem in scalars), which then can be treated by widely available uni-attribute techniques [2-j]. Amalgamation is a valid technique if a suitable surrogate measure actually can be determined and if the surrogate truly reflects the decision maker’s preferences. Of course, the decision maker must still be capable of specifying a decision criterion that accords with his definition of “best”. When the amalgamation scheme results in a surrogate attribute whose consequences are expressible in scalar utility terms, the normative definition of best is optimization of expected utility. There are no fundamental conceptual (or behavioral) problems-just those operational problems attending the elicitation of the decision maker’s utility function, the assessment of the state distribution, and the specification of attribute weights. (The first two problems also attend uni~attribute decision making; the last, only the muftiattribute case.) There are, however, those managerial decision makers for whom amalgamation is an unsatisfactory approach. In effect, the weights needed to combine the attributes are statements of acceptable tradeoffs among incommensurables. Such decision makers may be reluctant or unable to specify these tradeoff regimes, either explicitly or implicitly. They may prefer to keep the attributes distinct instead of obscuring them in an amalgamation which is most likely nonintuitive. Certainly, when the decision maker is a group, rather than an individual, the difficulty is compounded even further. In order to arrive at a single surrogate, a group utility function is required; yet group utility functions do not now exist, at least in any practicable sense. Even in the uni-attribute case, there is experimental evidence that individual decision makers will not rely on a single decision criterion[l]. They may not have the mathematical sophistication to appreciate the implications of utility and the maximization of expected utility. They may feel that utility cannot adequately account for adverse outcomes; they may be skeptical about using expectation for nonrecurring events. Often they take on overly con-
Multi-attribute decision making using constrained criteria
141
servative approach, e.g., maximin payoff, foregoing the benefits of a more realistic decision criterion. If they accept a single criterion it is only after, or in conjunction with, the satisficing of certain security levels. Clearly, multiple attributes aggravate this situation even more. A previous paper[6] described an alternative to maximizing expected utility, an alternative that such decision makers might find more acceptable. That approach, which addresses the uni-attribute case, has the effect of allowing the decision maker to consider multiple decision criteria. Specifically, his anxieties and aspirations are embodied in a mathematical program whose deterministic and/or chance constraints, when satisfied, assure acceptable security and attainment levels. The decision maker is thus freed to pursue the greater gains afforded by a more propitious, less conservative criterion. When this approach is used, certain decision criteria (e.g., maximin payoff, minimax regret) lose their raisons d’etre-the security they would otherwise provide being assured by the constraints. The action space under consideration is reduced thereby, as the few most acceptable acts are brought into sharp focus. Overview
In this paper the constrained-criteria approach of that previous paper is applied to the multi-attribute case. With this approach (which, in contrast to the classical methods, avoids amalgamation schemes): The distinct character of each attribute is preserved throughout. Thus the need to specify tradeoff regimes is eliminated and decision maker insight is enhanced. Satisficing security and aspiration levels may be prescribed for each attribute and in each attribute’s terms. The simple uni-attribute solution techniques (which are amenable to manual computation) may be employed even with multiple attributes. The effects of proposed problem changes are easily seen. Decision maker involvement is encouraged; the approach is intuitively appealing. Of course, these benefits are even more important in group decision situations. After reviewing the uni-attribute constrained-criteria approach we show the minor modifications necessitated by multiple attributes. Then we illustrate the approach by continuing our three-character mini-case. We conclude with some remarks about informational demands and decision maker participation. SUMMARY OF UNI-ATTRIBUTE
CASE
In order to make this paper self-contained, we summarize the uni-attribute case discussed in [61. The quintessence of that approach is the formulation of the decision problem as a mathematical programming model in which: The decision maker embodies his anxieties and aspirations in a set of deterministic and/or chance constraints. The feasible region resulting therefrom represents an acceptable set of actions. The decision maker is thus free to choose a propitious decision criterion as the objective function to be optimized over the feasible set of acts. Alternately, he can impose tighter restrictions repeatedly until the feasible set cannot be reduced further. Formally, let k = 1,2, . . . , K denote K disjoint and exhaustive acts, K finite; Vk denotes the feasible set of acts. j= 1,2 1..., J denote J disjoint and exhaustive states of nature, J finite. akj = the payoff of Act k under State j, finite. the OppOrtUnity loss (regR?t) of Act k under State j, finite. pj = the State j probability, where 0 5 pj 5 1, V j; Zjpj = 1. b, (n = 1,2, . . .) = appropriate security levels prescribed by the decision maker. q,,, (m = 1,2, . . .) = appropriate chance-constraint probabilities prescribed by the decision maker. Then, the deterministic constraints, which must be satisfied for each act in the feasible set, &j
rkj
=
r,i
Minj(akj)2bl,
Vk
(1)
142
STAN SCHENKERMAN
Maxi (rkj)= bz,
tf k
(2)
respectively assure at least a minimum payoff level, at most a maximum regret. The analogous chance constraints are Prob(c,rbl)rqi,
Probhi
5
b2) 2 q2,
Vk
(3)
vk
each of which can be generalized into a partial or total probability distribution. Constraints can also be imposed on expectaton and on Hurwicz value:
where 0 5 c s 1 and c is the Hurwicz coefficient of optimism. By selecting one or more constraint types and prescribing acceptable values for the requisite b’s and q’s, the decision maker explicates his anxieties and aspirations. With his security concerns assuaged, the decision maker is free to optimize a propitius decision criterion. Suitable objective functions include (but are certainly not limited to) the Bayesian (maximization of expected payoff, minimization of expected opportunity loss) and probabilistic functions structured after (3) and (4), e.g., Maximizevr Prob (akj 2 b,) Maximizevr, Prob (F&j5 b6).
(8)
Solving the w&attribute model Solving the model is usually quite simple and proceeds in two phases: (1) finding a feasible set of acts, and (2) optimizing the objective function over that feasible set. The feasible set consists of those acts not violating any constraint. It is obtained by evaluating each act not previously eliminated against each constraint until either a violation eliminates that act or the act is found not to violate any constraint. An efficient order is: ~terministic constraints, Types (1) and (2)-by inspection. Chance constraints, Types (3) and (4)-by summing the pj over all appropriate states and comparing the sum with the right side. Types (5) and (@-by computing the left-side expression and comparing with the right side. Phase 2 requires selecting that act from the feasible set that optimizes the objective function. This selection is accomplished by exhaustive evaluation of the objective function for each act in the feasible set, exactly as in a conventional evaluation of the decision criterion. Alternately, the decision maker can tighten the constraints, or impose additional constraints, to further restrict the feasible set. This process can be repeated until the feasible set consists of only one act (or until any further restriction would eliminate all remaining acts). EXTENSION TO ~ULTIFLE ATTRIBUTES
The payoff (opportunity loss) matrix of a managerial decision problem is a two-dimensional array of commensurate elements, e.g., the units of each element being profit dollars. Such a matrix represents a single attribute. Consider a three-dimensional array comprised of a set of payoff (opportunity loss) matrices-each matrix corresponding to a single attribute. This array is the multi-attribute extension of the uni-attribute matrix. Using the uni-attribute approach, an independent set of deterministic and chance constraints can be formulated for each attribute. Then, the collection of these independent constraint sets may be treated as one enlarged constraint set, denoted the “grand set”. Neither the numbers nor the types of constraints need be the same for each constraint set. In fact, it is possible for some attributes to have empty constraint sets.
Multi-attribute decision making using constrained criteria
143
The objective function for the multi-attribute model may be handled as follows: For each attribute, myopically choose a decision criterion to be the objective function, i.e. choose each objective function without regard to the remaining attributes. Select the most propitious as the grand objective function. Prescribe satisficing Ievels for the others and incorporate them as additional cons~aints in the grand set. Alternately, as in the uni-attribute case, the feasible set can be reduced by tightening existing constraints or imposing additional constraints. Analytic extension Extending the variables and equations of the uni-attribute case to the multi-attribute case requires only one new variable (to index the attributes) and the modification of several others to include the index. Let i = 1,2,. . . , 1 denote the I attributes, I finite. Then, akj becomes a,, the payoff for Attribute i; rkj becomes rkji,the opportunity loss for Attribute i; b, becomes bni, a prescribed security level for Attribute i. Observe that security levels must have the same units as the corresponding attributes. Thus, the b,,, are indexed by attribute. In contrast, the q,,,, being probabilities, are dimensionless and do not require attribute indexing. Therefore, the multi-attribute versions of uni-attribute equations (l)-(S) are, respectively, Minj (au) 2 bli,
Vk
(1’)
Maxi (rkji)= bzi,
Vk
(2’)
Prob(&jc?b,,)Lg,,
Vk
(3’)
PrOb (rkji5 bzi) 2 q2,
Vk
(4’)
Vk
Cjakjipj2 bx, c Maxi (okii)+ (1
- C)
Mini (a&i)2 b4ir V k
(6’)
byi)
(7’)
Maximize, Prob (f&i 5 b6i).
(8’)
Maximize, Prob
(akji
2
Equations (1’-6’) are constraints and may be imposed for any number of attributes. ~uations (7’-8’) are candidate grand objective functions; at most one of these wouid be used and that only for a single attribute. Solution strategy As in the uni-attribute case, solving the model proceeds in two phases: (I) finding a feasible set of acts, and (2) optimizing the objective function over that feasible set. Phase 1, checking each act for feasibility, may be done one constraint type at a time, each attribute seriatim, using the methods previously described. An act is eliminated as soon as a violation is found. If not eliminated, the act is admitted to the feasible set. Clearly, if the feasible set is empty, i.e. if every act of the initial action space violates some constraint in the grand set, the prescribed satisficing levels are too stringent (in combination at least) and therefore unattainable. This is more likely to happen in the multi- than in the uni-at~ibute case. Phase 2 is accomphshed exactly as in the uni-attribute case: select the optimal act by exhaustively evaluating the (uni-attribute) objective function for each act in the feasible set. It should be observed that while Phase 1 has been slightly complicated by the multiple attributes, Phase 2 has not. In general, it may even be less tedious than in the uni-attribute case because of the reduced set of feasible acts over which evaluations are required. MEANWHILE
. . our heroes have been engaged in much discussion. They finafly agree that, for the firm’s 1
benefit, expected profit should be maximized. This agreement, however, has not been wrested easily; it is contingent upon each obtaining certain assurance for his own vested interest.
144
STAN~HENK~RMA~
Finance imperiously demands a minimum profit of $20,000 and a limit of 0.10 on the probability of a profit regret exceeding $25,000. Marketing is adamant about having at least an even chance of securing 10% of the market. Manufa~turing~ remaining suspicious but now dissauded from his insistence on zero cost regret, declares that his ulcer cannot tolerate a regret over $125,000. Thus, they have implicitly agreed on the following multi-attribute model:
where all feasible acts must satisfy Mini (akjl) zz20
(10)
Prob (rkjg> 25) zs0.10
(11)
Prob (a&j22 10%)2 0.50
(i2)
Max, (r&r
125.
(13)
From Tabte L, (IO) eliminates Acts 5 and 6. From Table 4, Act 4 violates (t3). From Table 3 and the state probabilities listed in Table 1, Act 1 violates (12). As seen from Table 2, (I 1) would have eliminated Acts 5 and 6 had they not been eliminated previously. The feasible set consists of Acts 2 and 3, with Act 3 maximizing expected profit. It should be noticed that, for these data, Act 3 is Bayes optimal on Attribute 1 even in the unconstrained case. Thus the grand set constraints do not affect optimality and the “discussion” could have been shortened considerably. If manufacturing had persisted in his demand for zero regret for Attribute 3, Acts 2 and 3 would have been eliminated. The feasible set would then be empty, indicating a too stringent grand set-an unattainable combination of satisficing levels.
The satisficing levels for each at~~bute can actually be tightened sign~~~antly without invalidating the selection of Act 3. The following is the most stringent grand set that leaves Act 3 feasible: Minj (akjl) 2 JO, V k Prob(r,l>
15)rO.t0,
Vk
Prob (Ukjz2 10%) 2 0.95, MaXi(Tkj3)C: 120,
Vk
V k.
Thus, as in the u&attribute case, the optimal decision with rn~~ti-attribute constraints tends to be inv~ia~t under wide changes in security level and probability prescriptions. We would anticipate that this insensitivity depends upon the number of attributes, the number of acts in the initial action space, and the number of states of nature. CONCLUDING
REMARKS
Although rarely a~knowledged~ the information demands necessary to develop actuat payoffs or opportunity losses in any application of decision theory are generafly not triviateven when considering only a single attribute. When dealing with multiple attributes, these demands become all the more significant, regardless of the multi-attribute technique used. In fact, those approaches which, unlike the one herein, amalgamate the attributes into a single surrogate, additionally require a rationalizable attribute weighting scheme. Thus, papers such as this, which present constructs and solution methods, leave to the practitioner and the decision maker the often laborious, always unglamorous tasks of satisfying what are sometimes voracious demands for information.
Multi-attribute
decision
making
using constrained
criteria
145
Decision maker involvement
The importance of having the decision maker play an active role in imposing his judgments and preferences on problem formulation and solution is pointed out in [5, p. 339, p. 3411. Because it is intuitive and enhances insight, the approach described in this paper and in [6] is particularly well suited to continual decision maker involvement. Indeed, he must actively participate in each phase: Specification of attributes. Development of alternative acts/strategies. Formulation of grand set constraints and prescription of satisficing levels. Formulation of objective functions, selection of the grand objective function, and prescription of satisficing levels for the remaining objective functions. Iterative: evaluation of solutions and modification of the model. The relatively simple solution techniques--even for the multi-attribute model-greatly facilitate and encourage multiple iterations. Each iteration is under the decision maker’s complete control. Perhaps the most important feature of this approach is its preservation of distinct attributes, Keeping the attributes distinct eliminates the need to specify tradeoff regimes among attributes. It enhances the decision maker’s insight; it contributes to his understanding. Above all, it makes the decision maker a more knowledgeable participant. REFERENCES D. W. Conrath. From statistical decision theory to practice: some problems with the transition, Mgmt Sci. 19. 873 (1973). R. T. Eckenrode. Weighting multiple criteria, Mgmt Sci. 12, 180 (I%.(). G. P. Huber, Multi-attribute utility models: a review of field and field-like studies, Mgmt Sci. 20, 1393 (1974). G. P. Huber. Methods for quantifying subjective probabilities and multi-attribute utilities, Dec. Sci. 5. 430 (1974). L. P. Ritzman. Decision analysis with multiple objectives. Proc. VI Annual Meeting of the American Institute for Decision Sciences, pp. 339-343 (1974). 6. S. Schenkerman. Constrained decision criteria, Dec. Sci. 6. 42 (1975). I. 1. 3. 4. 5.