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97

Multiple criteria decision making: eight

concepts of optimality1

Milan ZelenyGraduate School of Business Administration,

GBA 626E, Fordham University at Lincoln Center,

New York, NY 10023, USA

E-mail: [email protected]

Multiple Criteria Decision Making (MCDM) is firmly

rooted in an alternative concept of optimality where mul-

tiple (rather than single) criteria characterize the notion of

the best (or optimal), as is prevalent in the areas of eco-

nomics, engineering, management and business. These are

often constrained problems where search for an optimal so-

lution requires some form of evaluating criteria performance

tradeoffs. Because there are no tradeoffs along a single cri-

terion, optimality is an essentially multi-criteria concept. In

this paper we extend and develop the notion of optimum

as a balance among multiple criteria. We introduce eight

different, separate and mutually irreducible optimality con-cepts in a classification scheme where the traditional single-

objective optimality is only a special case. These eight op-

timality concepts provide a useful initiatory framework for

the future MCDM research and applications.

Keywords: Optimality, optimization, optimal design, linear

programming, multiple criteria decision making

1. Introduction

Any criterion (measure, yardstick) is characterizedby a score (or a range of scores) of performance that

is most preferred by the decision maker within a given

context. Such contextually most preferred scores are,

if feasible, also optimal.

The difficulty arises when the most preferred scores

are infeasible, i.e., when there are explicit or implicit

constraints on decision alternatives or criteria that pre-

1The author is obliged to Prof. Alan Singer of Christchurch,NZ, for his encouragement and insights into the formalism andphilosophy of the optimality concepts.

Milan Zeleny is currently Professor ofManagement Systems at Fordham Uni-versity in New York City. He alsoserved on the faculties of the Uni-versity of South Carolina, CopenhagenSchool of Economics, European Insti-tute for Advanced Studies in Manage-ment (Brussels), and for ten years atColumbia University School of Busi-ness. He holds Dipl.Ing. from thePrague School of Economics, and anM.S. and Ph.D. from the University ofRochester. He is listed in Whos Who(Marquis editions) in Science and En-

gineering, in America, in the East, and in the World. He hasserved on editorial boards of International Journal of Operationsand Quantitative Management; Journal of the International Strate-gic Management; Operations Research; Computers and Operations

Research; Future Generations Computer Systems; Fuzzy Sets andSystems; General Systems Yearbookand is the Editor-in-Chief of

Human Systems Management. Also OR/MIS Editor and contributorof five entries in the International Encyclopedia of Business and

Management. Among Zelenys books are Multiple Criteria Deci-sion Making (McGraw-Hill), Linear Multiobjective Programming

(Springer-Verlag), Autopoiesis, Dissipative Structures and Sponta-neous Social Orders (Westview Press), MCDM-Past Decades andFuture Trends (JAI Press), Autopoiesis: A Theory of Living Orga-nization (Elsevier North-Holland), and Uncertain Prospects Rank-ing and Portfolio Analysis (Verlag Anton Hain). Zeleny publishedover 300 papers and articles, ranging from operations research, cy-bernetics and general systems, to economics, history of science, to-tal quality management, and simulation of autopoiesis and artificiallife (AL).

vent the achievement of the most preferred perfor-

mance.When the most preferred (optimum) is infeasible, it

can only be approached through maximization or min-

imization with respect to constraints. When the con-

straints are fixed and there is only a single criterion,we have a case of simple computation.

If we relax at least some of the constraints or con-

sider multiple criteria then the situation turns multi-dimensional: tradeoffs emerge, criteria scores have to

be balanced and computational maximization or min-

imization becomes inadequate.

When there is only a single criterion, no matterhow comprehensive or complex, its maximization or

minimization with respect to constraints is sufficient.

When there are multiple criteria, more complex, mul-

Human Systems Management 17 (1998) 97107ISSN 0167-2533 / $8.00 1998, IOS Press. All rights reserved

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98 M. Zeleny / Multiple criteria decision making

Fig. 1. Single-objective optimality.

tiple forms of optimality and optimization have to be

explored. In this paper we introduce eight such opti-

mality concepts.

2. Eight concepts of optimality

That, which is given, fixed or determined a priori

cannot be subsequently altered and therefore is not a

subject of optimization.What is not given can still to be altered, selected

or chosen and it is therefore, by definition, subject tosubsequent optimization. Consequently, different op-

timality concepts can be derived from the distinction

between what is given and what is yet to be (opti-

mally) determined.

We have chosen to demonstrate each of the eight

concepts in five complementary ways: (a) simple,

real-life examples with a defined set of discrete alter-

natives; (b) formal (mathematical) statement of a gen-

eral optimization problem; (c) graphical representa-

tion of two-dimensional linear-programming problem;

and (d) simple numerical problem.

2.1. Single-objective optimality

This refers to conventional maximization (or opti-

mization). Although not a tradeoff balancing prob-

lem, it should be included at least for the sake of

completeness or as a special case of the bona fide

optimization.

In order to maximize a single criterion, it is suf-

ficient to perform technical measurement and search

tasks. Once X and f were formulated and fixed,the optimum (e.g., maximum) is found by compu-

tation and no decision or conflict resolution processes

are needed. The search for optimality is reduced

to scalarization, assigning each alternative a number

(scalar) and then searching out the highest-numbered

alternative.

(1a) From a list of five places (X), find the onethat is the cheapest (Min f) for vacations. From alist (X) of affordable (e.g., under $500) transportationmodes between L.A. N.Y.C., find the one that isfastest (Max f), or safest, or cheapest, and so on.

(1b) Formally, we are given a set X of objects-of-choice (alternatives) and a function f, mappingX onto a well-ordered set of real numbers R, i.e.,f: X R. We then solve Maxf(x) subject tox X. To solve this problem, select x X. Ifx satisfies f(x) f(x) for all x X, then fis maximized at x and x is the maximal solutionfor x X.

(1c) Linear-programming maximization problem is:

Max f=cx

s.t. Ax b, x 0,

where c Rn and A Rmn are coefficient vector

and matrix of dimensions 1 n and m n, respec-tively, b Rm is m 1-dimensional vector ofgivenresources and x Rn is n 1 vector of decisionvariables (solutions).

In two dimensions (solution xj , j = 1, 2), the ob-jective functionf is a straight line, f=c1x1+ c2x2,to be maximized over the convex polygonXof linearinequalitiesai1x1+ai2x2 bi, i= 1, 2, . . . , m, thatis, the intersection of half-planes representing prob-

lem constraints. The maximal solutionx is unique,except under special mathematical conditions.This sit-

uation is graphically outlined in Fig. 1.

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M. Zeleny / Multiple criteria decision making 99

Fig. 2. Multiobjective optimality.

(1d) Consider the following linear-programming

problem with two variables and five resource con-

straints:

Max f=400x1+300x2

subject to

4x1 20,

2x1+ 6x2 24,

12x1+ 4x2 60,

3x2 10.5,

4x1+ 4x2 26.

The maximal solution isx1 =4.25,x2 =2.25, and

f(x) = 2375. Observe that everything is givena priori and no market prices of resources are nec-

essary. However, if p1 = 30, p2 = 40, p3 = 9.5,p4 = 20 and p5 = 10 were the respective marketprices ($/unit) of the five constrained resources, to-

tal cost of the resource portfolio (20, 24, 60, 10.5, 26)would beB = 2600.

2.2. Multiobjective optimality

The real optimality, being distinct from above sim-

ple computations, involves balancing multiple crite-

ria and their tradeoffs. This corresponds to the vec-

tor optimization of Maxf1(x), Max f2(x), . . . andMax fk(x), i.e., simultaneously and subject to x X.

The concurrent maximization of individual objec-

tive functions should be non-scalarized, separate and

independent, i.e., not subject to aggregation, like

forming and maximizing a superfunction U{f1(x),f2(x), . . . , f k(x)}. Such aggregation would effec-

tively reduce the multiobjective optimality to a single-

objective maximization.

(2a) From a list of five places (X), select the onethat is the cheapest (Min f1) and safest (Max f2) forvacations. From a list (X) of available (under $500)transportation modes between L.A. N.Y.C., find theone that is the fastest (Max f1) and safest (Max f2)and cheapest (Min f3), and so on.

(2b) Formally, we are given a set X of alterna-tives and a set of k functions f1, f2, . . . , f k, map-ping Xonto a well-ordered set of real numbers R,i.e., fi: X R for all i. Define F: X Rk andF(x) = {f1(x), f2(x), . . . , f k(x)}.

Next, we seek a subset X ofX such that, for allx X and all x X X, the following is true:F(x) >B F(x), where >B marks an (balancing)ordering ofRk, andX is theset of optimal solutions.

(2c) Multiobjective linear-programming problem is:

Max F =C x

s.t. Ax b, x 0,

where C Rkn and A Rmn are coefficientmatrices of dimensions k n and m n, respec-tively, b Rm is m 1-dimensional vector ofgivenresources and x Rn is n 1 vector of decisionvariables (solutions).

In two dimensions (solution xj , j = 1, 2), objec-tive functions f1 and f2 (or more) are straight lines,fk =ck1x1+ ck2x2, k =1, 2. They are to be maxi-mized over the convex polygon Xof linear inequal-ities ai1x1 +ai2x2 bi, i = 1, 2, . . . , m. The so-lution set X is a set of nondominated solutions andthe general ordering >B becomes simply (largeror equal to). The situation is graphically displayed

in Fig. 2.

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(2d) Numerical example:

Max f1= 400x1+300x2, and

f2= 300x1+400x2

subject to

4x1 20,

2x1+ 6x2 24,

12x1+ 4x2 60,

3x2 10.5,

4x1+ 4x2 26.

The maximal solution with respect to f1 is x1 =

4.25, x2 =2.25,f1(4.25, 2.25) = 2375; maximal so-lution with respect to f2 is x

1 = 3.75, x

2 = 2.75,

f2(3.75, 2.75) = 2225. The set of optimal (nondom-inated) solutionsX includes the two extreme pointsabove and their connecting (feasible) line defined as

4x1 + 4x2 = 26. For example, 0.5(4.25, 2.25) +0.5(3.75, 2.75) = (4.0, 2.5) is another nondominatedpoint on this line. Total cost of the resource portfolio

remains at B = 2600.

2.3. Optimal system design: single criterion

Instead of optimizing a given system with respect

to selected criteria, humans often seek to construct an

optimal system of decision alternatives (optimal fea-

sible set), designed with respect to given criteria. A

single-criterion design is the simplest of all such con-

cepts, producing the best (optimal) set of alternatives

Xat which a single objective functionf(x) is maxi-mized, subject to the cost of design (affordability).

(3a) Design a list of affordable places (X) which as-sure the cheapest (Min f) vacations. Design a list (X)of affordable (under $500) transportation modes be-

tween L.A. N.Y.C., which assure the fastest (Max f)mode of transportation.

(3b) Formally, we design a feasible set of alterna-

tives where function freaches its maximum subjectto stipulated cost of the design. We seek (design) a

subset X of U (the universal set) so that for somefunction C:U R we have C(X) c, where c isstipulated (maximum) budget or cost of design, and

for all x X and all x U, the f(x) f(x) istrue. The levelc ofC(X) does not have to be fixeda priori.

(3c) Linear-programming optimal-design problem

is:

Fig. 3. Optimal system design: single criterion.

Max f=cx

s.t. Ax b, x 0,

pb B,

where c Rn and A Rmn are coefficient vectorand matrix of dimensions 1 n and m n, respec-tively, b Rm is m 1-dimensional unknown (tobe determined) vector of resources and x Rn isn1 vector of decision variables (solutions). Further,p Rm is 1 mvector of unit prices ofm resourcesandB is total available budget (or cost).

Solving the above problem means finding the opti-mal allocation of budget B so that the resulting (pur-chased) portfolio of resourcesb assures the feasibilityofx andf(x) = Max f(x).

In two dimensions (solution xj, j = 1, 2), we areto determine convex polygon X by finding afford-able bis for linear inequalities ai1x1 +ai2x2 bi,i = 1, 2, . . . , m, such that the objective function fis maximized at x X. Linear inequalities thusbecome linear equations and the resulting X (theirintersection) is reduced to a single point maximal

solutionx. This situation is graphically represented

in Fig. 3.(3d) Numerical example:

Max f=400x1+300x2

subject to

4x1 29.4,

2x1+ 6x2 14.7,

12x1+ 4x2 88.0,

3x2 0,

4x1+ 4x2 29.4.

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Fig. 4. Optimal system design: multiple criteria.

The right-hand sides (resource portfolio) have beenoptimally designed. Solving the above optimally de-signed system will yield x1 = 7.3446, x

2 = 0, and

f(x) = 2937.84. If market prices of the five re-sources, p1 = 30, p2 = 40, p3 = 9.5, p4 = 20and p5 = 10, remain unchanged, the total cost ofthe resource portfolio(29.4, 14.7, 88.0, 29.4)remainsB= 2600.

2.4. Optimal system design: multiple criteria

This optimal-system design involves multiple crite-

ria. As before, multiple criteria should not be scalar-ized into a superfunction U. Rather, all such criteriacompete independently or there is no need for theirseparate treatment.

(4a) Design a list of affordable places (X), whichassures the cheapest (Min f1) and the safest (Max f2)for vacations. Design a list (X) of affordable (under$500) transportation modes between L.A. N.Y.C.,which assures the fastest (Max f1) and safest (Max f2)and cheapest (Min f3), and so on.

(4b) Formally, we seek to design a subset X ofUso that for some function C:U R we have C(X)B F(x) is true. The>B isan (balancing) ordering ofRk andX is the set ofoptimal designs.

(4c) Linear multiobjective-design problem is:

Max F =C x

s.t. Ax b, x 0,

pb B,

whereC Rkn andA Rmn are coefficient ma-trices of dimensions k n and m n, respectively,b Rm is m 1-dimensionalunknown (to be deter-mined) vector of resources andx Rn isn1 vectorof decision variables (solutions). Further, p Rm is1 m vector of unit prices ofm resources and B istotal available budget (or cost).

Solving the above problem means finding the opti-

mal allocation of budget B so that the resulting (pur-chased) portfolio of resources b assures the feasibil-ity of x and F(x) at cost B and is as close aspossible to the metaoptimum design F at cost B,B B.

In two dimensions (solutionxj ,j = 1, 2), we are todetermine convex polygon X by finding affordablebis for linear inequalities ai1x1 +ai2x2 bi, i =1, 2, . . . , m, such that objective functions f1 and f2are straight lines,

fk= ck1x1+ ck2x2, k= 1, 2,

maximized at x X. Linear inequalities become

linear equations and resulting X (their intersection)is reduced to a single point, maximal solution x atcost B. This situation is graphically represented inFig. 4.

Instead of the set of nondominated solutions, we

now have a set (or family) of optimal system designs,

characterized by the same costB and perhaps varyingrelative importance of objective functionsf1 andf2.

(4d) Numerical example:

Max f1= 400x1+300x2, and

f2= 300x1+400x2

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Fig. 5. Single-objective valuation.

subject to

4x1 16.12,

2x1+ 6x2 23.3,

12x1+ 4x2 58.52,

3x2 7.62,

4x1+ 4x2 26.28.

The above is optimally designed portfolio of re-

sources. The maximal solution with respect to both

f1 and f2 is x1 = 4.03, x2 = 2.54, f1(4.03, 2.54) =

2375 and f2(4.03, 2.54) = 2225. This can be com-pared (for reference only) with the f1 and f2 perfor-mances in the earlier case of given right-hand sides.Assuming the same prices of resources, total costof this resource portfolio is B = 2386.74 2600.One can calculate even better performing portfolios

by spending the entire budget of 2600 (i.e., addi-

tional 213.26).

2.5. Optimal valuation: single criterion

All previously discussed optimization forms as-sume that decision criteria are given a priori. How-ever, in humandecision-making, different criteria are

continually being tried and applied, some are dis-carded, new ones added, until an optimal (properlybalanced) mix of both quantitative and qualitative cri-teria is identified. There is nothing more suboptimalthan engaging perfectly good means X towards un-worthy, ineffective or arbitrarily determined criteria(goals or objectives). Optimizing wrong criteria can-not yield optimal solutions.

If the set of alternativesX is given and fixeda pri-ori, we face a problem of optimalvaluation: Accord-

ing to what measure should the alternatives be evalu-

ated and ordered? According to criterionf1 or f2 orf3? Should most satisfactory vacations be measuredby cost? Entertainment value? Privacy conditions?

Which of the criteria captures best our values and pur-

poses? What specific criterion engages the available

means (X) in the most effective way?(5a) Select a single criterion (f), perhaps from a

list(f1, f2, f3, . . .), like entertainment, education, pri-vacy, cost, etc., which would best evaluate and order

a given list of affordable places (X) to secure mostsatisfactory or fulfilling (through Max for Min f) va-cations. Select a proper criterion (f) to evaluate a list(X) of affordable (under $500) transportation modesbetween L.A. N.Y.C., which assures the most sat-isfactory or the most effective (via Max f or Min f)mode of transportation.

(5b) Formally, we define Fto be the set of all func-tions f: X R. Let INT be a subset of (F U)of integrated problems in terms of relationships be-

tween criteriafand alternativesX. Select a functionf Fsuch that there isx Xfor allx Uand allf Fsuch that f(x) f(x), and(f, X) INT.

Because we have to select the most suitable cri-terion faccording to higher criteria, a metacriterionvalue complexV[f, X] must be invoked.

(5c) Linear-programming valuation problem: Find

fsuch that

Max f=cx

s.t. Ax b, x 0,

provides the best or most effective valuation of Xwith respect to a given complex of values V[f, X]andassures that optimal pattern(f, b , x) is preferred toany(fk , b , x

k), for all k.

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Fig. 6. Multiobjective valuation.

In two dimensions, we are asked to evaluate con-

vex polygon Xby selecting an objective function fso that f maximization at x X is preferred to fkmaximization for all k. This situation is graphicallyrepresented in Fig. 5.

(5d) Numerical example. In order to evaluateX,should we maximize f1 or f2? How do we select acriterion if only one is allowed (possible) or feasible?

Max f1= 400x1+300x2 or

Max f2= 300x1+400x2

subject to

4x1 20,

2x1+ 6x2 24,

12x1+ 4x2 60,

3x2 10.5,

4x1+ 4x2 26.

The maximal solution with respect to f1 is x1 =

4.25, x2 = 2.25, f1(4.25, 2.25) = 2375. Maximal

solution with respect to f2 is x1 = 3.75, x2 = 2.75,f2(3.75, 2.75) = 2225. Is 2375 off1better than 2225of f2? Only one of these valuation schemes can beselected.

2.6. Optimal valuation: multiple criteria

If the set of alternativesX is given and fixeda pri-ori, but a set of multiple criteria is still to be selected

for the evaluation and ordering ofX, we have a prob-lem of multiple-criteria valuation: Which setof cri-

teria best captures our value complex V[f, X]? Is it

(f1 and f2)? Or (f3 and f4)? Or perhaps (f1 and f2andf3)? Or some other combination?

(6a) Select a combination of criteria, perhaps from

a list (f1, f2, f3, f4, . . .), like entertainment, educa-tion, privacy, cost, etc., which would best evalu-

ate a given list of affordable places (X) in order tosecure most satisfactory or fulfilling (e.g., through

Max f1 and Min f2) vacations. Choose a set of cri-teria (f1, f2, . . . , f k) to evaluate a list (X) of af-fordable (under $500) transportation modes between

L.A. N.Y.C., assuring the most satisfactory or effec-tive (via Max f1 and Min f2) mode of transportation.(6b) Formally, we define Fk to be the set of all

functionsF: X Rk, whereF(x) = {f1(x), f2(x),. . . , f k(x)}. Let MINT be a subset of(Fk U), theset of multiple criteria problems, integrated in terms

of the relationships between criteria F and alterna-tivesX.

Select a functionF Fk such that there is x Xfor allx Uand allF Fk such thatF(x)>BF(x) and(F, X) MINT.

Because we have to identify the most suitable cri-

teria F, a metacriterion value complexV[F, X] must

be used.(6c) Linear multiobjective valuation problem is:

Find Fsuch that

Max F =C x

s.t. Ax b, x 0,

provides the best valuation of X with respect to agiven V[F, X]. Solving this problem means estab-lishing an optimal pattern(F, b , x).

In two dimensions, we are to evaluate convex

polygon Xby finding objective functions f1 and f2(straight lines, fk =ck1x1+ ck2x2, k =1, 2) so that

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their maximization atx Xis preferred to any other

pattern. This situation is graphically represented inFig. 6.

(6d) Numerical example. How do we select a set

of criteria,f1,f2, or(f1,f2), that would best expressa given value complex?

Max f1= 400x1+300x2 or/and

Max f2= 300x1+400x2

subject to

4x1 20,

2x1+ 6x2 24,

12x1+ 4x2 60,

3x2 10.5,

4x1+ 4x2 26.

The maximal solution with respect to f1 is x1 =4.25, x2 = 2.25, f1(4.25, 2.25) = 2375. Maximalsolution with respect to f2 is x1 = 3.75, x

2 = 2.75,

f2(3.75, 2.75) = 2225. Should we use f1 or f2, orshould we use both f1 and f2 to achieve the bestvaluation of X? Only one of possible (single andmultiple criteria) valuation schemes is to be selected.

2.7. Optimal pattern matching: single criterion

Human decision making is obviously characterized

by a parallel search for both the criteria and alterna-tives in order to achieve their optimal interaction and

thus arrive at an optimal problem statement and its

optimal solution. Neither the criteria (f or F), northe alternatives (X) are givena prioriin a genuinedecision-making situation.

There is a problem formulation representing anoptimal pattern of interaction between alternatives

and criteria. It is this optimal, ideal or cognitive-

equilibrium problem formulation or pattern that is to

be approximated or matched as closely as possible

by decision makers. Single-objective matching of the

cognitive equilibrium is the simplest special case.(7a) Select a criterion (f), like entertainment, ed-

ucation, privacy, cost, etc., anddesign an affordable

list of places (X), which assures the most satisfac-tory or fulfilling2 (through Max for Min f) vacations.

2The notion of fulfilling or satisfying implies the search forbalance, harmony or fit between what one wants (or ought) to doand what one actually can do. We choose not only our means butalso our objectives and goals to achieve such balance.

Select a proper criterion (f) and design a list (X)

of affordable (under $500) transportation modes be-tween L.A. N.Y.C., which assures the most satisfac-tory (via Max f or Min f) mode of transportation.

(7b) Formally, we define F to be the set of allfunctions f: X R. Let INT be a subset of (F U) of integrated problems in terms of relationshipsbetween criteriaf and alternatives X.

Select (design) a subsetX ofUand functionf F such that for some function C :U R we haveC(X) < c, where c is stipulated (maximum) cost ofdesign there is x Xfor all x Uand all f Fsuch that f(x) f(x), and (f, X) INT.

Becausew have to select Xand identify the most

suitable criterion f, a metacriterion value complexV[f, X] must be used.

(7c) Linear single-criterion optimal matching prob-lem is: Find f andb such that

Max f=cx

s.t. Ax b, x 0,

matches as closely as possible the optimal but infea-sible (unaffordable) patternf and b, at pb B.

Solving the above problem means establishing theoptimal pattern (f, b, x) and its budgetary levelB, then finding the optimal allocation of actual bud-

get B so that the resulting (purchased) portfolio ofresourcesb, chosen objective function fand impliedsolution x assures that pattern (f,b,x) matches theoptimal pattern(f, b, x) as closely as possible.

In two dimensions, we are to determine convexpolygon X by finding affordable bis for linear in-equalities ai1x1 + ai2x2 bi, i = 1, 2, . . . , m, andan objective function f so that f maximization atx X costs B and it best matches f maximizedat x X, defined by bi s purchased for B

. Thesituation is graphically represented in Fig. 7.

(7d) Numerical example. Should we maximizef1orf2? How do we select a single criterion if only one

is allowed (possible) or feasible?Max f1= 400x1+300x2 or

Max f2= 300x1+400x2

subject to

4x1 29.4 or 0,

2x1+ 6x2 14.7 41.27,

12x1+ 4x2 88.0 27.52,

3x2 0 20.63,

4x1+ 4x2 29.4 27.52.

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Fig. 7. Optimal pattern matching: single criterion.

The above formulation describes two optimally de-

signed portfolios of resources with respect to f1 andf2, respectively. So, among the possible patternsare: (1) x1 = 7.3446, x

2 = 0; f1(x

) = 2937.84;B = 2600 and (2) x1 = 0, x

2 = 6.8783; f2(x

) =2751.32;B = 2600.

Suppose that the value complex implies that the

chosen criterion should minimize the loss of the op-

portunity (i.e., not chosen) criterion, other things be-

ing equal. Choosingf1 would make f2 drop only to

80.08% of the opportunity performance while choos-ing f2 would make f1 drop to 70.24%. So,f1 has apreferable opportunity impact and the first pattern and

its resource portfolio should be selected.

A value complex indicating that resource quantities

used should be as small as possible (not only their

prices) would lead to choosing f2 and thus the secondpattern.

2.8. Optimal pattern matching: multiple criteria

Pattern matching with multiple criteria is more in-

volved and so far the most complex optimality con-

cept. In all matching concepts there is a need

to evaluate the closeness (resemblance or match) of

a proposed problem formulation (single- or multi-

criterion) to the optimal problem formulation (or pat-

tern).

(8a) Select necessary criteria (f1, f2, . . . , f k), likeentertainment, education, privacy, cost, etc., and

design an affordable list of places (X), whichassure the most satisfying or fulfilling (through

Max f1 and Min f2) vacations. Select proper cri-teria (f1, f2, . . . , f k) and design a list (X) of af-fordable (under $500) transportation modes between

L.A. N.Y.C., which assure the most satisfactory (viaMax f1 and Min f2) mode of transportation.

(8b) Formally, we define Fk to be the set of allfunctionsF: X Rk, whereF(x) = {f1(x), f2(x),. . . , f k(x)}. Let MINT be a subset of(Fk U), theset of multiple criteria problems, integrated in terms

of the relationships between criteria F and alterna-tivesX.

Select (design) a subset X ofU and functionF Fk such that for some function C:U Rwe have

C(X) < c, where c is stipulated (maximum) cost ofdesign there isx Xfor allx Uand allF Fk

such that F(x)>B F(x), and (F, X) MINT.

Because we have to identify the most suitable cri-

teria F, a metacriterion value complexV[F, X] mustbe used.

(8c) Linear multiobjective pattern matching prob-

lem is: Find F and b such that

Max F =C x

s.t. Ax b, x 0,

matches as closely as possible optimal but infeasible

(unaffordable) pattern F and b, at pb B.Solving the above problem means establishing the

optimal pattern(F, b, x)at its budgetary level B,then finding the optimal allocation of actual budgetBso that the resulting (purchased) portfolio of resources

b and chosen set of objective functions F imply solu-tionx, assuring that pattern(F,b,x) matches optimalpattern(F, b, x) as closely as possible.

In two dimensions, we are to determine convex

polygon X by finding affordable bis (X) for linearinequalitiesai1x1+ai2x2 bi, i= 1, 2, . . . , m, anddetermine objective functions f1and f2(straight lines,

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Fig. 8. Optimal pattern matching: multiple criteria.

fk = ck1x1+ ck2x2, k =1, 2) so their maximizationat x X, costing B, best matches f1 and f

2 maxi-

mized at x X, defined by bi s costing B. This

situation is graphically represented in Fig. 8.(8d) Numerical example. How do we select a set

of criteria,f1, f2 or (f1, f2), that would best expressour value complex?

Max f1= 400x1+300x2 or/and

Max f2= 300x1+400x2

subject to

4x1 29.4 or 0 or 19.98,

2x1+ 6x2 14.7 41.27 28.78,

12x1+ 4x2 88.0 27.52 72.48,

3x2 0 20.63 9.39,

4x1+ 4x2 29.4 27.52 32.50.

The above represents three optimally designed port-folios of resources with respect tof1, f2 and (f1, f2),

respectively. So, among the possible patterns are:(1) x1 = 7.3446, x2 = 0; f1(x

) = 2937.84;B = 2600, (2) x1 = 0, x

2 = 6.8783; f2(x

) =2751.32;B = 2600 and (3) x1 =4.996,x

2 =3.131;

f1(x) = 2937.84,f2(x) = 2751.32;B = 2951.96.If the value complex requires that B = 2600 is

not exceeded, we may match optimal pattern (3) atthat level by scaling it down by the optimum-path ra-tio r = 2600/2951.96= 0.88. The new pattern (3)is: x1 = 4.396, x

2 = 2.755; f1(x

) = 2585.30,f2(x

) = 2421.16; B = 2600. If producing bothproducts is valuable, then the choice could be themaximization of bothf1 andf2, mutatis mutandis.

3. Conclusions

In Table 1 we summarize all eight basic optimal-

ity concepts according to a dual classification: sin-

gle versus multiple criteria compared with the level

or extent of the given, ranging from all-but to

none-except. The traditional concept of optimality,

characterized by too many givens and a single crite-

rion, naturally appears to be the farthest removed fromany practical conditions or circumstances for problem

solving.

When the decision criteria are fixed a priori, like in

the first four optimality concepts, their performance

achievements provide the measures of goodness. As

soon as the criteria themselves are to be selected (like

in the last four concepts), then a value complex V,or a metacriterion, is presumed and implied. In order

to avoid logically infinite regress (criteria for select-

ing criteria for selecting criteria . . .), value complexV must be anchored and integrated in fundamentalvalues that are broadly (at least temporarily) accepted

and not subject to choice (and thus optimization).

Should we design for speed or for safety? Should

we strive for both? Or should we aim at performance,

mileage and cost? How do we select the criteria them-

selves? Usually we take them as being given and thus

avoid the problem. Yet, they are not given because

somebody did have to select them first. What criteria

were used for selecting criteria?

Value complex determines that when we are in no

hurry and have children we might opt for safety, while

having no children, combined with a sense of ur-

gency or fear, could make us go for speed. Similarly,

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Table 1Eight concepts of optimality

Number of criteria

Given Single Multiple

Criteria & Traditional MCDMalternatives optimality

Criteria Optimal design Optimal designonly (De Novo programming) (De Novo programming)

Alternatives Optimal valuation Optimal valuationonly (Limited equilibrium) (Limited equilibrium)

Value complex Cognitive equilibrium Cognitive equilibriumonly (Matching) (Matching)

the choice between profits, quality and costs may be

guided by relying on standards of behavior and val-ues like not cheating the customer, serving the public

or satisfying the shareholder. Candor, trust, proper

recognition and reward could be other useful compo-

nents of the requisite value complex.

Value complex is based on principles but also

rooted in context or circumstances. Value complex

consists of mostly qualitative and difficult to measure

principles, ethics and rules, expressible only in im-

precise and fuzzy language rather than in crisp and

quantifiable functions.

There is a difference between constraints, goals and

objectives that is merely technical. Imposing a limit,bound or value on an objective will turn it into a goal

if approachable by degrees, or into a constraint if it is

to be strictly satisfied or adhered to. However, con-

straints are always stated individually and only rarely

(for purely technical reasons) summed up into mean-

ingless aggregates or superconstraints.

Yet, the same economic variables, the same con-

straints are freely combined, scalarized, added up,

weighted and aggregated as soon as their constrain-

ing values are relaxed or made approachable, as soon

as they become goals or objectives. This represents

a profound dilemma: how can criteria become addi-

tive and constraints not, even when they may haveidentical economic interpretations and contents?

To answer this dilemma, it has been proposed here

to broaden the traditional concept of optimality into

the eight different and mutually irreducible forms.

References

[1] P.G.W. Keen, The evolving concept of optimality, in: Multi-ple Criteria Decision Making, M.K. Starr and M. Zeleny, eds,TIMS Studies in the Management Science, Vol. 6, North-Holland Publishing Co., New York, 1977, pp. 3157.

[2] E.A. Galperin, Nonscalarized multiobjective global opti-mization, Journal of Optimization Theory and Applications75(1) (1992), 6986.

[3] M. Zeleny, Multiple Criteria Decision Making, McGraw-Hill, New York, 1982.

[4] M. Zeleny, Optimal system design with multiple criteria: DeNovo programming approach, Engineering Costs and Pro-duction Economics 10 (1986), 8994.

[5] M. Zeleny, Trade-offs-free management via De Novo pro-gramming, Int. Journal of Operations and Quantitative Man-agement1 (1995), 313.

[6] M. Zeleny, Cognitive equilibrium: a knowledge-based theoryof fuzziness and fuzzy sets, General Systems19(1991), 359381.

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