Multiple criteria decision support - A review

European Journal of Operational Research 63 (1992) 361-375 North-Holland

361

Invited Review

Multi.pie criteria decision support A

review

*

Pekka Korhonen Helsinki School of Economics and Business Administration, Runeberginkatu 14-16, 00100 Helsinki, Finland Herbert Moskowitz

Krannert Graduate School of Management, Purdue University, West Lafayette, IN 47907, USA Jyrki Wal l eni us

Helsinki School of Economics and Business Administration, Runeberginkatu 14-16, 00100 Helsinki, Finland Received April 1992

Abstract: We provide a problem oriented review of multiple criteria decision research. We focus on

classifying multiple criteria decision making problems, and discussing how decision makers can be assisted (supported) in structuring and solving such problems. We do not review existing multiple criteria procedures, but do provide references to relevant software systems implementing such procedures for the computer. Both discrete alternative and continuous mathematical programming multiple objective problems are discussed. We conclude the paper by identifying exciting directions and promising areas for continued and future research in multiple criteria decision support. Keywords: Multiple criteria programming; Decision Support Systems; preference analysis; software

1. Introduction

Since Charnes and Cooper (1961) developed goal programming and Keeney and Raiffa (1976) developed the theory and methods for multiattribute utility assessment, multiple criteria deci* This paper has evolved from a tutorial presented by Professors Pekka Korhonen and Jyrki Wallenius at the TIMS/ ORSA meeting in Las Vegas, May 7-9, 1990. This paper was written while Pekka Korhonen was a Visiting Professor at the University of Georgia and Jyrki Wallenius a Visiting Professor at the Texas A&M University. Correspondence to: J. Wallenius, Helsinki School of Economics and Business Administration, Runeberginkatu 14-16, 00100 Helsinki, Finland.

sion making (MCDM) has become one of the most active, international, and interdisciplinary fields of research in management science and operations research. The purpose of this article is to review recent developments in this field. The main focus is on providing classifications of various types of MCDM problems and reviewing some representative approaches and systems designed to help decision makers (DM) formulate and solve such problems. Most of the software systems have currently been implemented on personal computers. Deterministic problems that are represented either using a discrete, countable (usually finite) set of decision alternatives or an uncountable set of decision alternatives, de-

0377-2217/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved

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scribed using a continuous mathematical programming framework, are covered. MCDM research in the 1970's emphasized the theoretical foundations of multiple objective mathematical programming and the development of procedures and algorithms for solving such problems - especially multiple objective linear programming problems and discrete problems. Many ideas originated from the theory of mathematical programming. The algorithms were programmed for mainframe computers (often in FORTRAN) and were used mainly for illustrative purposes. The systems were often of a prototypical nature, lacked user-friendly interfaces (appealing visuals), and documentation was nonexistent to sparse. Examples of such early prototypical systems were Dyer (1973) and Wallenius and Zionts (1976). During the 1980's, emphasis shifted away from multiple objective optimization towards providing multiple criteria decision support to DMs and practitioners. This shift implies that more and more research is focusing on capturing the DM's actual decision/choice behavior, instead of solving well-structured problems under hypothetical and unrealistic assumptions concerning the DM's preference structure and behavior. This emphasis on decision support has brought to light many important issues, such as: -the importance of developing appealing communication facilities to the DM (e.g., interfaces based on the use of spreadsheets, colors, graphical representations, windows, and on-line help capabilities, providing a simple grammar of the communication language); the realization that problem solving should not be seen in isolation; the organizational context is important; the f~ct that the entire process of decision making from problem identification to solution implementation should be supported. For instance, it is not realistic to assume that a DM is able to formulate a problem precisely prior to the solution process - and then solve it. It is essential that h e / s h e can approach a problem on a more evolutionary basis, in which several steps of redefinition and solution follow each other. Hence, the behavioral and pragmatic realism of decision tools has and is still increasing. Multiple Criteria Decision Support Systems (MCDSSs) allow us to analyze multiple criteria,

and to incorporate the DM's preferences over these criteria into the analysis. Such systems make it possible to perform 'what to do - to achieve' analyses. In other words - in contrast to traditional 'what - if' analysis - we operate with the consequences (criteria, objectives) and determine, what we want to achieve. Then the system will inform us, how to obtain what we want (technically, the values of the decision variables). MCDSSs also seek to support the modeling and structuring of decision problems, often making use of advanced visualization capabilities. Our review largely reflects the backgrounds, interests, and biases of its authors. Literature that provides a deeper background and additional references in the field include lgnizio (1976), Hwang and Masud (1979), Zeleny (1982), Yu (1985), Steuer (1986), Moskowitz and Bunn (1987), Lotfi and Teich (1989), Vanderpooten and Vincke (1989), Aksoy (1990), White (1990), Shin and Ravindran (1991), H. Wallenius (1991), OIson and Courtney (1991), Gardiner and Steuer (1992), and Dyer et al. (1992). This paper consists of 6 sections. In Section 1 we have discussed the purpose and scope of this article. In Section 2 we present the framework of our study and in Section 3 we develop a classification of different multiple criteria decision problems. Section 4 describes different principles for structuring and solving different classes of multiple criteria decision problems. Section 5 identifies several promising areas for continued and future research. Section 6 concludes the paper.

2.

Framework

2.1. Definhions Multiple Criteria Decision Making (MCDM): A single DM is to choose among a countable (usually finite) or uncountable set of alternatives using two or more (multiple) criteria. When the values of the criteria are assumed to be known with certainty, the MCDM-problem is called deterministic; otherwise it is nondeterministic (or stochastic).

Multiple Criteria Decision Support Systems (MCDSS): Decision Support Systems that help structure and solve MCDM problems.

Negotiations~group decision making: A natu-

P. Korhonen et al. / Multi-criteria decision support

ral extension of M C D M , when more than one decision agent (DM) is present. Multiattribute Utility Theory (MAUT): Focuses on structuring of multicriteria or multiattribute alternatives, usually in the presence of risk or uncertainty, and on techniques for assessing individuals' utilities/values and subjective probabilities. M A U T is sometimes subsumed under M C D M , but is often treated separately when risks or uncertainties have a significant role in the definition and assessment of alternatives (Dyer et al., 1992, Section 4).

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dependent on the assumptions made concerning the feasible set X, the objective functions, the decision variables, etc. If the n u m b e r of elements of X is countable, then the decision problem is discrete. The number of elements of set Q is then also countable, and the problem can be presented, for instance, using the following notation: "max" i~l

qi

where I = {1, 2 . . . . . n} is the index set consisting of the indices referring to the alternatives, qi ~ Q c R k, which is a point in the criterion space, and n

2.2. A mathematical problem formulation

Q = [,.J {qi}. i=1

Consider the following mathematical programming problem:

where x ~ R n is a vector of decision variables, fi, i = 1, 2 . . . . . k, are objective functions, and X is the set of feasible decision alternatives. The set of feasible solutions in the criterion space R k is called the feasible region and denoted by Q:

The number of alternatives n may be infinite or finite. If n is finite, but very large, we usually make the additional assumption that all alternatives are not known at the beginning of the decision process. Unless mentioned, we assume that n << ~. We also refer to a discrete problem by stating that the alternatives are defined explicitly. Sometimes it is convenient to enumerate all alternatives and then formulate a discrete problem as follows:

Q={f(x)

"max"

t

max

q=f(x)

s.t.

x~X

= ( f l ( x ) ..... fk(X))

(2.1)

lx~X}.

Problem (2.1) seldom has a unique solution, i.e. an optimal solution that simultaneously maximizes all criteria. Instead, there are many 'reasonable' solutions, called efficient solutions. Any choice from among the set of efficient solutions is an acceptable and possible compromise solution, until we have additional information about the D M ' s preference structure. Although the types of problems addressed and the specifics of the approaches proposed vary, the ultimate goal is to

help a DM find the most preferred solution (consistent with his/her preference structure) for his/ her decision problem. Definition. A point x ° ~ X is efficient if and only if there does not exist another x ~ X such that fi(x) >fi(x °) for all i ~ K = {1, 2 . . . . . k} and fi(x) 4~fi(x °) for at least one i ~ K . A point in the criterion space that corresponds to an efficient point in the decision variable space is called a nondominated solution. The nature of the solution process is heavily

~

qixi

i-1

s.t.

~ xi

:

1,

x i = 0, 1.

i-1

If the number of elements of set X is not countable, the alternatives are usually defined using a mathematical model formulation, and the problem is called continuous. Therefore, in this case we say that the alternatives are defined implicitly. Note that the problem in which the decision variables are assumed to have only integer values, belongs to the first class, because all solutions can be enumerated (given that the constraint set is bounded). If the constraint set is bounded, the number of alternatives is finite, but usually so large that we cannot solve the problem by first enumerating the alternatives and then choosing the best alternative. Conceptually the multiple objective mathematical programming problem may be regarded as a value (utility) function maximization program: max s.t.

F(q)

q~Q,

P. Korhonen et al. / Multi-criteria decision support

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where v is a value (utility) function, which is a marginally increasing function in the criterion space and defined at least in the feasible region Q. Function v specifies the DM's preference structure over the feasible (nondominated) region. The role that the DM's value function plays in the analysis is a demarcating feature between various MCDM and MCDSS models. Generally, if the value function is estimated explicitly, the system is considered to be in the MAUT category. If the value function is implicit (assumed to exist but is otherwise unknown) or no such function is assumed to exist, the system is usually classified under MCDM (Dyer et al., 1992).

Z3. Three solution principles The multiple objective decision procedures always assume the intervention of a DM at some stage in the solution process. A popular way to involve the DM in the solution process is to use an interactive approach (Keeney and Raiffa, 1976; Steuer 1986). Based on the role that the value function plays in the analysis, we can distinguish the following three solution principles for solving various kinds of MCDM problems: 1. Assume the existence of a value function v, and assess it explicitly. 2. Assume the existence of a stable value function v, but do not attempt to assess it explicitly. Make assumptions of the general functional form of the value function. 3. Do not assume the existence of a stable value function v, either explicit, or implicit. The three solution principles lead to the following corresponding general approaches to solving MCDM problems.

value/utility function using elaborate interview techniques; - a s s e s s necessary probability functions if there exists uncertainty; check the internal consistency of the DM's responses; order the alternatives in terms of value/expected utility. In other words, once an explicit value/utility function has been assessed, this function is used to rank order (a finite number of) the decision alternatives or optimize over the feasible decision space (Churchman and Ackoff, 1954; Keeney and Raiffa, 1976; Farquhar, 1983; Fishburn, 1988). Interactive software implementing such systems on personal computers exists. -

-

-

-

m

a

k

e

a

n

k

Approach 2: Interactive articulation of preferences (a) Based on an implicit value function. This was a basic paradigm used in the interactive multiple criteria approach in the 1970's. DM's responses to specific questions were used to guide the solution process towards an 'optimal' or 'most preferred' solution (in theory), assuming that the DM behaves according to some specific (but unknown) underlying value function. The DM need not be aware of the existence of his/her underlying value function (Hwang and Masud, 1979; Steuer, 1986; Shin and Ravindran, 1991; White, 1990). The following steps typically characterize this approach: make assumptions of the underlying value function, in particular, its functional form; find an initial solution that is preferably feasible and efficient; - interact with the DM to find his/her reaction or response to the solution; update our knowledge about the underlying value function; generate an improved solution using the updated underlying value function; iterate until an 'optimal' solution has been identified or the DM is satisfied with the solution. Interactive software that implement such systems for a computer have often been developed by the authors of the above procedures for experimental purposes. (b) Based on no stable value function. These approaches are typically based on projecting a DM's aspiration levels regarding the objectives on the feasible region. This projection is -

-

-

-

Approach 1: Prior articulation of preferences This is the classical approach adopted in multiattribute utility or decision analysis. Using Keeney-Raiffa type of interaction (Keeney and Raiffa, 1976), the following steps can be identified: assumptions about the underlying value/utility function, in particular, its functional form; assess the parameters of the underlying

r

P. Korhonen et a L / Multi-criteriadecision support usually accomplished via minimizing so called achievement scalarizing functions (Wierzbicki, 1980; Steuer and Choo, 1983). No assumptions whatsoever are postulated about the DM's value function. Typically the following steps are included: present the DM with an efficient solution and provide h i m / h e r with as much information as possible about the nondominated region, in particular in the 'neighborhood' of the current solution; - ask the DM to provide preference information in the form of aspiration levels, weights, etc.; use the responses to generate a single nondominated solution or a set of nondominated solutions for the DM's evaluation; iterate until (1) the DM stops, or (2) some specified termination criteria for the search have been satisfied. In essence, this approach seeks to help the DM more or less freely search the set of efficient solutions. Interactive software that implement such systems for a computer have been developed (Lewandowski et al., 1989; Korhonen, 1987, 1988). When such interactive algorithms (2a or b) are applied to real-world problems, according to Shin and Ravindran (1991), the most critical factor is the functional restrictions placed on the criterion functions, constraints, and the unknown value function. Their paper presents information on the functional assumptions of each procedure. Another important factor is the preference assessment or interaction style. Shin and Ravindran list the following eight typical interaction styles: (a) binary pairwise comparisons; (b) pairwise comparisons; (c) vector comparisons; (d) precise local tradeoff ratios; (e) interval local tradeoff ratios; (f) comparative tradeoff ratios; (g) listing the indices of the criterion functions to be improved or sacrificed; also specifying amounts of improvement or sacrifice; (h) aspiration levels (reference points). Approach 3: Posterior articulation of preferences This approach typically generates nondominated solutions and presents them to the D M for evaluation. H e / s h e is then expected to choose the most preferred solution from this set. No assumptions of the underlying value function are

365

postulated. Typically, the following steps can be identified: generate an initial efficient solution; generate additional efficient solutions; - present such solutions in the criterion space to the DM and ask h i m / h e r to choose the best one. How the solutions (alternatives) are presented and visualized, depend on the individual procedure. Several such procedures have been implemented for the computer (e.g., A D B A S E system by Steuer, 1986).

3.

Classifications

of

M C D M

p r o b l e m s

The first classification of MCDM problems is discrete vs. continuous to represent the decision alternatives. An example of a discrete problem is the choice of a set of investment projects. An example of a continuous problem is a macroeconomic model, where equations are used to represent the behavior of an economy; the instruments are the key decision variables used to control economic policy (H. Wallenius, 1991). Both types of problems are common in practice. We use this classification to partition this section into two separate parts. 3.1. Discrete, explicitly defined alternatives From a practical perspective, discrete choice problems may be classified in at least the following ways: the number of alternatives is small vs. large; the number of criteria is small vs. large; values of the criteria are known vs. not known with certainty; - the alternatives are known vs. not known apriori; the criteria are explicitly vs. implicitly specified. These classifications will generate, in total, 32 ( = 25) different problem combinations. Not all have been extensively researched. The most common problems investigated in the literature are the following: (1) The number of alternatives is large, the number of criteria is small (less than 10), the criterion values are known with certainty, and the alternatives are known a priori. (An example:

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choice of consumer durables, say washing machines, from a large set of available models. The criteria that a buyer would consider would include cost, electricity consumption, water consumption, and the overall quality of the machine (Zeleny, 1982, pp. 210-211).) (2) As in (1) above, but the values of the criteria are not known with certainty. Usually, the outcomes of the criteria are modelled assuming some probability distribution. (An example: choice of location and development pattern for a commercial airport. The alternatives would consist of all possible location-development pattern combinations over time. The criteria would include cost, pollution levels, distance of the airport from downtown and major residential areas, etc. The outcomes would, however, be known only probabilistically (Keeney and Raiffa, 1976).) (3) Problems having a small number of a priori known alternatives evaluated using a (usually) large number of criteria often possessing a hierarchical structure. In addition to criteria having a hierarchical structure many of the criteria are 'soft'. (An example: choice of a mainframe computer from a preselected set. Typically, only a few alternatives are evaluated using several criteria. Many of the criteria are very difficult to express quantitatively, and they have a natural hierarchy (Korhonen, 1986).) (4) The number of alternatives is very large (infinite) or generation of alternatives is costly or in other respects difficult. Then it is realistic to assume that not all the decision alternatives are available at the beginning of the decision process. The DM is required to make two kinds of decisions: (i) to identify the most preferred alternative from the available set of alternatives, and (ii) to indicate whether to continue the search for a more preferred alternative or not. (An example: an organization recruiting an employee for a specific job. Initially the recruiting officer may receive a certain number of applications from interested people. The applicants' qualities would be evaluated using different criteria, such as education, work experience, impression from an on-site interview. Then the overall best applicant would be identified. The question that the recruiting officer must eventually answer is, whether to extend a n offer to one of the candidates, or whether to augment the initial sample (Korhonen et al., 1991).)

(5) The DM is not able or willing to explicitly specify h i s / h e r criteria. H e / s h e will make the decision by considering different characteristics (attributes) of the alternatives; comparisons between alternatives are usually conducted in a pairwise manner. (An example: choice of a home from a (small) set of potential homes. According to a recent real estate survey in the Bryan-College Station, Texas area, home buyers, on the average, look at 6 homes, before making a final choice. Some criteria (price, number of bedrooms) are easy to specify, but others are not (neighborhood, yard, plan, etc.) (Real Estate Center, Texas A & M University).)

3.2. Continuous, implicitly defined alternatives Again, from a practical perspective, continuous problems may be classified in at least the following ways: the underlying model is linear vs. nonlinear (i.e., the functional relationships of the model are linear vs. nonlinear with respect to the decisions/ actions) the variables are continuous vs. integer the number of criteria is small vs. large - the problem size (that is, the number of constraints a n d / o r variables) is large scale vs. small scale - the relationships between the variables are quantitative vs. qualitative the decision alternatives are known vs. not known a priori. These classifications will generate, in total, 64 different problem combinations. Not all have been extensively researched. The most common problems studied in the literature are the following: (1) The underlying model is linear, the variables are continuous, the number of criteria is small, the problem size is relatively small scale, the relationships between the variables are quantitative, and the decision space is known a priori. (An example: pricing alcoholic beverages in a state monopoly. Decision variables would represent price changes in different categories of alcoholic beverages, such as beer, wine, spirits, etc. The criteria would include the maximization of state revenues, minimization of pure alcohol consumption, and curbing of inflation due to increases in liquor prices (Korhonen and Soismaa, 1988).) -

-

-

-

P. Korhonen et al. / Multi-criteria decision support

(2) As in (1) above, with the exception that the underlying model is described using nonlinear constraint functions. (An example: inspection plan for a continuous production system. The decision variables would represent the rate at which random sampling of the production line is to take place, and the extent of 100% sampling (free of defectives) before switching back to random sampiing. The objectives would include the minimization of expected unit cost of inspection and replacement, and minimization of average outgoing percentage of defectives (Roy and Wallenius, 1991).) (3) As in (1) above, with the exception that some (or all) of the variables assume integer values. (An example: advertising media selection. The decision variables would represent the numbers of ads in each advertising medium. The criteria to be maximized would include the readership numbers partitioned into different categories (surrogate measures for the impact of an ad), and the cost of advertising to be minimized (Korhonen, Narula and Wallenius, 1989).) (4) As in (1) above, with the exception that the underlying model is relatively large scale. (An example: large scale forest sector models. The decision variables would represent different types of treatments subjected to different types of forest areas at different times. The criteria would include short run and long run profit, growth of forest, etc. (Soismaa, 1988).) (5) As in (1) above, with the exception that some or all of the relationships between variables are qualitative. (An example: identifying a marketing strategy for a software house. The decision variables would represent the different kinds of pure strategies available, such as 'foreign companies', 'domestic companies', foreign professionals', etc. The criteria would include maximization of short run and long run profits, minimization of workload, etc. (Korhonen and Wallenius, 1990).)

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4.1. Discrete framework

4. How to structure and solve M C D M problems?

Problem 1: Large set of alternatives and few criteria Generate a decision matrix, where the rows denote the decision alternatives, and the columns the criteria to be maximized (or minimized). At minimum, each column defines a preference rank order for the alternatives. Two different solution principles, which are not mutually exclusive, have been presented in the literature for dealing with this problem: (1) Alternatives are eliminated/filtered, until a single or just a few alternatives are left. (2) The DM is assisted in considering nondominated alternatives in a systematic manner, until h e / s h e stops or some prespecified termination conditions are fulfilled. The elimination may conveniently be carried out by sequentially introducing bounds for the criteria to exclude alternatives that fall outside these bounds. The Sequential Choice Procedure (SCP) by Larichev, Mechitov and Moshkovich (1990) is an example of such a system. However, some other support systems include this feature as optional. Since in the discrete case the dominated alternatives are easy to eliminate from the set of decision alternatives, the essential question becomes, how the DM can be helped in searching the set of nondominated alternatives. In problem 1 there are too many alternatives to allow simultaneous presentation of alternatives to the DM. One popular approach elicits the DM's aspiration levels for the criteria, and by some means projects the point corresponding to these aspiration levels a n d / o r the direction emanating from the current alternative and passing through the point defined by the aspiration levels, on the set of nondominated alternatives. The DM is then asked to choose his/her most preferred alternative from this set, and revise his/her aspiration levels, etc. Examples of such software systems are AIM (Lotfi, Stewart and Zionts, 1992) and VIMDA (Korhonen, 1988).

How would one go about structuring and solving the above problems discussed in Section 3? What solution principles have been used? What solution principles could be used? We use the classifications from above.

Problem 2: Values of criteria not known explicitly Generate a decision matrix a n d / o r decision tree indicating the chronology of the problem. Check whether assumptions of preferential independence and utility independence hold. Proceed to assess the DM's multiattribute value/utility

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function over the attributes (criteria) using midvalue splitting technique or lottery type questions, if there exists uncertainty (Keeney and Raiffa, 1976). Check the internal consistency of the responses. Assess all necessary probability functions (if there exists uncertainty), and rank order the alternatives using the expected value/utility criterion. This approach represents a classical multiattribute decision analysis. An example of an interactive software package that supports this kind of analysis is PCPDA (Kirkwood and Van der Felz, 1986).

Problem 3: Number of criteria large Generate a decision matrix. Now, a major difficulty stems from the presence of a large number of criteria (say, 20). How can we help the DM operate with so many criteria? Two popular approaches are: (1) Assume that the criteria have a hierarchical structure and consider them in clusters. (2) Provide a holistic representation of the problem/alternatives by visualizing the information in an appropriate way. A widely used method to deal with decision/ criterion hierarchies is the Analytic Hierarchy Process (AHP) (Saaty, 1980). The software package implementing the AHP is marketed under the name EXPERT CHOICE (Forman, Decision Support Software, Inc.). Various techniques are available for presenting multivariate data (such as Chernoff faces; Chernoff, 1972). A recently developed system (VICO) uses harmonious houses instead of faces, and presents to the DM a sequence of paired comparisons (houses), each visualizing a decision alternative. The DM is requested to indicate his/her preference between the two alternatives. The more preferred house is kept and is compared against another house. The search terminates when the DM has found his/her most preferred alternative (Korhonen, 1991). Problem 4: All alternatives are not known a priori Obtain a sample of alternatives. Generate a decision matrix based on this sample. Identify the best alternative in the sample using pairwise comparisons. Based on the degree of knowledge of a DM's underlying value/utility function, calculate the probability of finding more preferred alternatives, given that the sample is augmented with

additional alternatives. Based on this probability information and the cost of additional sampling, decide whether to augment the sample or not. A prototypical system implementing this idea has been developed in Korhonen et al. (1991).

Problem 5: Criteria not specified explicitly If the criteria are not known explicitly, a natural approach is to operate with (pairwise) comparisons. In this way, the graph describing the preference structure between the alternatives can be defined. The Outranking Method is based on this principle (Roy, 1973). Many software packages that implement this idea have been developed. The best known of them is the ELECTRE family pioneered by Roy and his associates (Roy, 1985). This approach is particularly popular in the French speaking MCDM community. See also ORESTE by Roubens (1982) and Pastijn and Leysen (1989) and PROMETHEE by Brans, Mareschal and Vincke (1984). 4.2. Mathematical programming framework Problem 1: Multiple Objective Linear Programming (MOLe) This is the most commonly studied problem in MCDM. Dozens of methods have been developed for solving MOLe problems. Many of them have also been implemented in interactive software packages for microcomputers. In early methods, a common feature was to operate with criterion weights, limiting consideration to efficient extreme points. Today, many systems are based on the use of aspiration level projections, where the projection is performed using Chebyshev-type achievement scalarizing functions. These functions can be controlled either by varying weights (keeping aspiration levels fixed) or by varying the aspiration levels (keeping weights fixed). The same idea was originally proposed in somewhat different form by Steuer and Choo (1983) and Wierzbicki (1980). Korhonen and Laakso (1986) proposed the idea of parameterizing the achievement scalarizing function, making it possible to project a direction on the efficient frontier, instead of one single point. Using these basic elements, the following different system variations have been developed: the system generates a finite set of solutions for a DM's evaluation; at each iteration the size -

P. Korhonen et al. / Multi-criteria decision support

of the search area is reduced (Steuer and Choo, 1983); - a DM freely specifies aspiration levels for the criteria, and the system projects them onto the set of efficient solutions (DIDAS family; Lewandowski et al., 1989); - a DM moves freely on the efficient frontier; this idea is implemented in the VIG system (Korhonen, 1987) under the name PARETO RACE (Korhonen and Wallenius, 1988).

Problem 2: Multiple objective nonlinear programming Develop a mathematical programming model of the underlying problem. From a DM's perspective, the most commonly used technique to solve the resulting mathematical programming model is the same as in (1) above: controlling the search of the efficient frontier by means of DM's aspiration levels. However, multiple objective nonlinear programming problems are much more complicated to solve than MOLP problems. A description of the entire mathematical structure of the underlying model is required, and to project aspiration levels onto the set of efficient solutions results in a nonlinear optimization problem. DIDAS-N by Kreglewski, Paczynski, Granat and Wierzbicki (1989) is a software package designed to solve such problems. See also Roy and Wallenius (1991).

Problem 3: Multiple objective integer linear programming Let your model incorporate also integer variables. Two different strategies can be used to solve multiple objective integer linear programming problems: (1) Operate purely with integer solutions. (2) Operate with continuous solutions, but generate a 'closest' integer solution when desired. Methods for generating efficient integer solutions have been studied extensively since Bitran (1977, 1979), but no one has yet developed an interactive algorithm with practical relevance (Zionts, 1979; Kiziltan and Yucaoglu, 1983; Gabbani and Magazine, 1986; Ramesh, Karwan and Zionts, 1986, 1989; Karaivanova, 1991).

369

lems by moving from one efficient solution to another, and allowing the DM to change his/her mind frequently. Each iteration takes some time, and therefore the solution process has to be planned in such a way that the total number of iterations is minimized. At each iteration, the method should gather as much information about the DM's preferences as possible and use this information as effectively as possible. For example, we might ask the DM to carefully consider the environment of the current solution to make sure that a new search direction is really good. Using this information we may then generate an efficient path that starts from the current solution and passes through the entire efficient frontier, until a boundary of the feasible region is reached. The most time consuming phase is the generation of new efficient solutions. In the future, this phase should probably be handled on a mainframe computer. The user would still interface with the system using a microcomputer (see, for example, Korhonen, Wallenius and Zionts, 1992). In large scale mathematical programming, technically speaking, pivoting is the critical operation in classical methods based on the use of the simplex method. Moving around on the efficient frontier requires a very large number of pivots. An interesting alternative approach would be to approximate the efficient frontier from inside using the ideas of interior point methods developed by Karmarkar (1984).

Problem 5: Qualitative multiple objectiue linear programming Problems where relationships between outcome variables and decision variables are qualitative is quite common in practice. However, to solve such problems using mathematical programming tools requires that all relationships are presented in quantitative form. The transformation of qualitative relationships into quantitative relationships can be performed using the AHP (Saaty, 1980). Then the problem can be solved using any appropriate MOLP package (Korhonen and Wallenius, 1990).

4.3. State-of-the-art of applications

Problem 4: Large scale multiple objective linear programming

Multiattribute decision analysis

Large scale MOLP problems cannot be solved in the same way interactively as small scale prob-

Corner and Kirkwood (1991) in a recent paper survey decision analysis applications in the opera-

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tions research literature. Their survey covers years 1970-1989. In total, 86 published case studies are reviewed. They all focus on analyzing decisions which use expected utility as the criterion for identifying the preferred decision alternative. To be included in this survey, a paper had to explicitly analyze alternatives for a decision problem using judgmental probabilities a n d / o r subjectively assessed utility functions. The applications were classified into five areas: energy, manufacturing and services, medical, public policy, and general. The authors emphasize that the applications cover a wide range of decision problems in both the public and private sectors. Many of the applications address strategic or policy decisions - providing counter examples to the often-stated criticism that operations research does not address strategic issues of central concern to top management.

Interactive mathematical programming White (1990) in a recent paper surveys published applications of mathematical programming multiple objective models, including goal programming, which use no a priori explicit value functions. The survey covers years 1955-1986. Judging from the length of the paper, it would appear that mathematical programming multiple objective methods, and goal programming in particular, are very popular approaches in practice. This is, however, not true. The bibliography classifies only 58 applications as implemented applications; namely applications where an actual problem of an actual organization is studied, using real data, in which actual DMs participate in the preference elicitation, and the results of which are implemented. Out of the 58 applications, 36 are goal programming applications. In actuality, only a handful of interactive multiple objective procedures have been applied and implemented in practice (see also Wallenius, 1991). During the last five years, there have been two important developments influencing the application potential of MCDM and MCDSS models; namely the exponential spread of personal computers and spreadsheet models (and spreadsheet based optimization). The impact of these significant developments is beginning to be noticed. Examples are the numerous applications of the AHP (Saaty, 1980) and the VIG systems (Korhonen, Siljam~iki and Wallenius, 1990).

5. A r e a s

for continued

and future research

While the past has been exciting for MCDM research, the future appears to be equally promising. Specific areas for new and continued research in multiple criteria decision support are presented below.

Behavioral aspects Psychologists and behavioral decision theorists have extensively studied human decision making. New theories and paradigms have been developed to explain decision and choice behavior (Kahneman and Tversky, 1979; Hogarth, 1987; Von Winterfeldt and Edwards, 1986). This behavioral (descriptive) research has had little impact on normative multiple criteria decision research. Relevant questions that need to be investigated include the following: - Does the information search pattern focus on alternatives (inter-alternative search) or attributes (criteria) (inter-attribute search)? Does this pattern change during the decision process? What decision rules do DMs use - initially, during the search process, in the final stage? - How do humans set aspiration levels for the criteria? In light of new information, how do they revise these aspiration levels? - What do DMs really mean when they state that one criterion is more important than another? For example: "Minimization of unemployment is more important than minimization of inflation". - What is the role of the starting point for different procedures? Do different starting points lead to different outcomes? If so, why? - What is the role of framing in MCDM (Tversky and Kahneman, 1981)? All these questions have significant implications for the design and development of MCDSSs. Utilization of advanced computer technology The utilization of advanced computer technology for decision research presents many challenges and opportunities, such as: - The possibilities of computer graphics in visualizing problems, alternatives, criteria. H. Wallenius (1991) cites four examples of recently published systems that use novel graphical interfaces. They are PARETO RACE (Korhonen and Wallenius, 1988), Korhonen's harmonious houses

P. Korhonenet al. / Multi-criteria decisionsupport (Korhonen, 1991), the system of holistic graphics by Kasanen, 0stermark and Zeleny (1991), and Lotov's Generalized Reachable Sets method (Lotov, 1984). Computer animation is yet another area of great promise. - New forms of interfaces, such as speech recognition. Dr. Ralph Gomory, a former Vice President of IBM, recently predicted that in less than a decade, speech recognition will be a practical reality. Networking of mainframe computers with personal computers to make it possible to solve truly large scale multiple objective mathematical programming problems. -

Linking of Multiple Criteria Decision Support Systems, Expert Systems, and Artificial Neural Networks Some interesting research, conducted by Ignizio (1991) at University of Houston, is currently underway to link MCDSS, Expert Systems, and Artificial Neural Networks. Ignizio functionally replicates neural networks via linear goal programming, with potential advantages over neural nets. Numerous applications in the real world can be envisioned. Modelling/formula tion In the early days of mathematical programming, considerable attention was paid to properly structuring and formulating real-world problems. This interest, however, faded, when researchers became fascinated with the development of new algorithms for solving various kinds of mathematical programming problems. Now, after a quarter of a century, problem formulation and modelling have begun to receive the attention they deserve. Several interesting research questions deal with problem formulation and modelling, for example: What is the usefulness of pluralistic modelling, where multiple modelling approaches to a problem are developed? Should we look at multiple problem representations? How should the results be amalgamated (Moskowitz and Bunn, 1987)? - W h a t role could Geoffrion's structured modelling play in helping problem formulation and modelling (Geoffrion, 1987)? - W h a t role could knowledge-based expert systems play in helping problem formulation and modelling? -

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- What is the desired level of detail or aggregation in a model? - How can we help DMs define the relevant objectives and alternatives? Robustness Precision in problem formulation, the DM's preferences/utilities, and subjective probabilities, have commonly been assumed in theory. This assumption has increasingly been challenged during the last five years. Areas for imprecision incorporation (robustness) include the following: (1) Problem formulation: see Korhonen et al. (1991) for an approach relaxing the assumption of a fixed set of decision alternatives. See also Sakawa (1983) for an example of the use of the fuzzy set logic to imprecise formulation. (2) Preferences and values/utilities: some DSSs allow DM's preferences to be imprecise, even intransitive. For example, ELECTRE (Roy, 1973), AHP (Saaty, 1980), VIG (Korhonen, 1987), MCRID (Moskowitz, 1990; Moskowitz, Preckel and Yang, 1991b), and PAIRS (Salo and H~im~il~iinen, 1991). (3) Subjective probabilities and/or utilities: Under the name ambiguity, vagueness in probability assessment has long been of concern to decision researchers. An example of a recent DSS that allows imprecision in probability and utility assessment is described in Moskowitz, Wong and Chu (1989) and Moskowitz, Preckel and Yang (1991a) under the name RID (Robust Interactive Decision Analysis). Theory of decomposition Imprecision in the assessment of utilities (and/or probabilities) discussed above under 'Robustness' raises the issue regarding error propagation. This has implications for determining how and to what degree one should structure and decompose a multiattribute value/utility function in an MCDM problem. Some initial theory and analysis have been developed on the decomposition of additive multiattribute utility functions (Ravinder, Kleinmuntz and Dyer, 1988; and Ravinder and Kleinmuntz, 1991). This work is being generalized and expanded by Moskowitz and his colleagues. More specifically, the parameters of the multiattribute utility function (relative importance weights and individual utility functions for each attribute) are treated as random

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variables. Then, under various informational conditions (regarding the attribute weights and attribute utilities), mathematical relationships are derived for the upper and lower bounds of the variance of the aggregate utility, for example, as a function of the number of attributes, the number of levels of a value tree hierarchy, etc. From this, determinations can be made regarding the number of attributes and value tree levels to be used to assess a multiattribute utility function, based on the goal of minimizing aggregate utility variance. Moreover, is it necessary to measure attribute weights precisely (e.g., will a rank order be adequate) or even at all; is it necessary to assess individual attribute utility functions precisely? This line of investigation should help to develop a theory of decomposition along with guidelines for application to MCDM problems.

Group decision support Group Decision Support Systems have become a popular research area during the latter part of the 1980's. We provide several examples of exciting research in group decision and negotiation. (1) Development of novel systems to facilitate the communications between the group members and the negotiating parties (for example, Gear and Reed, 1988). (2) Determining how negotiators' utilities determine the structure of negotiation problems (for example, Mumpower, 1991). (3) Representing negotiation problems by means of a rule-based formalism (Kersten et al., 1991). (4) The development of negotiation support and group DSSs (for example, Jarke, Jelassi and Shakun, 1987; Nunamaker et al., 1989; Shakun, 1991). The purpose of such systems is either to help the individual parties to prepare for the actual negotiations or to help structure the negotiations so that a mutually beneficial solution can be found. For additional details, the reader is referred to a recent Focused Issue on Group Decision and Negotiation by Management Science (editors: Chatterjee, Kersten and Shakun, 1991). The area seems wide open for future research.

Fuzzy multiple criteria methodologies There exist many opportunities to apply fuzzy multiple criteria methodologies to management

and engineering problems; for example, in engineering design and in total quality management (e.g., Dhingra and Moskowitz, 1991; Hauser and Clausing, 1988). To illustrate, Quality Function Deployment (QFD) is a concept and approach for translating customer requirements into design or engineering characteristics of a product, and subsequently into the parts characteristics, process plans, and production requirements associated with its manufacture. The relationships between the customer attributes and the engineering characteristics, and among the engineering characteristics are typically vague and imprecise because of general inherent fuzziness in the system. Fuzzy theories and methodologies can be used in the QFD context to translate imprecise and vague information in the problem specification into fuzzy relationships; viz., fuzzy objective and constraint functions. The fuzzy constraints in QFD define a fuzzy feasible domain in the design space. A fuzzy modelling approach to QFD has been proposed, which allows a designer to consider tradeoffs among various customer attributes, while simultaneously taking into account the inherent fuzziness in any associated relationships (Kim et al., 1991). The modelling approach permits examination of the effects of system fuzziness and flexibility inherent or permitted in the system equations representing the design process. A mathematical formulation of the QFD process involves the use of fuzzy regression, multiattribute value/utility theory, and fuzzy multiple criteria optimization. This approach, which can be applied to a variety of design and management problems has raised a number of research issues: (1) What are the effects of possibility (uncertainty in the design parameters) and flexibility of system equations on design performance, as measured by overall customer value? (2) What are the characteristics and performance of fuzzy regression vis-a-vis statistical regression (ordinary least squares) as a function of the number of data points available, the quality of the data set, aptness of the statistical model specifications and assumptions (e.g., error term distribution, autocorrelation, etc.)? (3) What are the relationships among the Hvalue (similar to a confidence coefficient in statistics), type of membership function, and spread of estimated parameters in fuzzy regression?

P. Korhonen et al. / Multi-criteria decision support

(4) H o w r o b u s t a r e t h e d e s i g n p a r a m e t e r s to c h a n g e s in t h e m u l t i a t t r i b u t e v a l u e f u n c t i o n ? (In t h e c o n t e x t of Q F D , this h a s i m p l i c a t i o n s for e a s e o f m a n u f a c t u r a b i l i t y a n d p r o d u c t cycle t i m e reduction.) U n f o r t u n a t e l y , t h e use o f fuzzy m e t h o d o l o g i e s for M C D M in r e c e n t y e a r s has b e e n relatively l i m i t e d a n d u n p o p u l a r , with t h e e x c e p t i o n of J a p a n e s e scholars. H o w e v e r , it has a c l e a r a p p l i c a t i o n a n d p r o v i d e s an o p p o r t u n i t y for b o t h p r o m i s i n g t h e o r e t i c a l a n d m e t h o d o l o g i c a l res e a r c h a n d real w o r l d u s e a g e .

6. Conclusion W h a t is n e e d e d for t h e field o f m u l t i p l e criteria d e c i s i o n s u p p o r t to p r o s p e r ? W e c o n c l u d e with a q u o t e f r o m D y e r et al. (1992): " W e still n e e d simple, u n d e r s t a n d a b l e , a n d u s a b l e app r o a c h e s for solving M C D M a n d M A U T p r o b lems. Such a p p r o a c h e s will u n d o u b t e d l y b e built a r o u n d D e c i s i o n S u p p o r t Systems. T h e D e c i s i o n S u p p o r t Systems t h a t survive a n d a r e w i d e l y u s e d will have to b e u s e r - f r i e n d l y a n d have o t h e r g o o d qualities. S o f t w a r e t h a t has t h e a t t r i b u t e s o f easyto-use s p r e a d s h e e t s is m o s t d e s i r a b l e " .

Acknowledgment T h e a u t h o r s wish to t h a n k P r o f e s s o r S t a n l e y Z i o n t s , S U N Y Buffalo, for his c o m m e n t s .

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