# Management of Attainable Tradeoffs between Conflicting Goals

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2 1034 JOURNAL OF COMPUTERS, VOL. 4, NO. 10, OCTOBER 2009 goals. Novel approaches to coping with endogenous uncertainties and catastrophic risks are discussed in Section IV. Finally, the requirements for transparency and public understanding are summarized in Section V. II. DECISIONS AND OUTCOMES Rational decision-making always involves analysis of relations between alternative decisions and the corresponding consequences. One distinguishes two types of problems: 2 There is a given set of (at least two) discrete alternatives. One has to decide which should be selected. In some situations a ranking of alternatives is additionally required. Further on we refer to this type of problem as Discrete Alternative (DA) choice problem. Typical examples include: selecting a car, a house, a project. A decision is composed of a set of value(s) selected from an infinite 3 set of feasible decisions. Such a set is typically given implicitly, i.e., by a specification of the relations between decisions, optionally also involving other factors that need to be considered when making the decision. A simple (in terms of number of decision variables) example: decide the amount of kerosene to be tanked in an aircraft. A complex problem: decide a portfolio of structural and financial instruments for integrated management of catastrophic flood risks. Although there are methods and tools specialized for each of these two types of problems, there are also many common methodological issues. Therefore it is worth to consider both of them in terms of the mathematical programming, which provides a powerful analytical framework for analysis of different approaches to decisionmaking. A mathematical model describes the modeled problem by means of variables that are abstract representations of those elements of the problem which need to be considered in order to evaluate the consequences of implementing a decision (usually represented by a vector composed of many variables). More precisely, such a model is typically developed using the following concepts: Decisions (inputs) x, which are controlled by the user; External decisions (inputs) z, which are not controlled by the user; Outcomes (outputs) y, used for measuring the consequences of the implementation of inputs; Auxiliary variables introduced for various reasons (e.g., to simplify model specification, or to allow for easier computational tasks); and Relations between decisions x and z, and outcomes y; such relations are typically presented in the form: where F( ) is a vector of functions. y = F(x, z ), (1) 2 We discuss here only a single decision-maker support; therefore we refrain from considering issues related to group decision-making. 3 Or at least large enough to practically exclude analysis of each individual alternative. 2 N O 7 I A H = J D A = J E? A O. N Figure 1. Structure of the use of a mathematical model for decisionmaking support. A rational decision-making support aims at finding values of decision variables x which will result in a solution of the problem that best fits preferences of the decision-maker. Such preferences can be represented by a preferential structure P (x, y), which typically induces partial ordering of solutions obtained for different combinations of values of decisions. Thus, the basic function of decision-making support is to help the decision-maker find values for his/her decision variables x which will result in a solution of the problem that best fits his/her preferences represented by P (x, y). A typical decision problem has an infinite number of solutions therefore the relations (1) need to be represented by a mathematical model. One should note that also the discrete alternative choice problem can be represented as an algebraic model. This is particularly needed if values of criteria for (possibly many) alternatives must be computed from parameterized complex relations, see e.g., [1], and/or for problems with a large number of alternatives. A structure of the use of a model for decision-making support is illustrated in Figure 1. Such a support is composed of two stages: Development and maintenance of a model that adequately represents relations (1); Organizing a process of the model analysis in which the user can specify and modify his/her preferences upon combining their own experience and intuition with learning about the problem from the analyses of various solutions. These two stages are briefly summarized below. A. Modeling process Modeling is a network of activities, often referred to as a modeling process. Such a process should be supported by a modeling technology that is a craft of a systematic treatment of modeling tasks using a combination of pertinent elements of applied science, experience, intuition, and modeling resources. The latter being composed of knowledge encoded in models, data, and modeling tools. In most publications which deal with modeling, small problems are used as an illustration of the modeling methods and tools presented. Often, these can also be applied to large problems. This is especially true for the DA type of problems for which the model development stage consists of selecting sets of alternatives and attributes, adapting an existing data management utility, and O

6 1038 JOURNAL OF COMPUTERS, VOL. 4, NO. 10, OCTOBER 2009 G ) α * α = + J E K K I 2 + G G ) >, EI? HA JA 2 Figure 3. Limitations of the weighted-sum approach. The value of α determines the ratio of: (1) improvement (measured in the percentage of the corresponding criterion value) of one criterion, and (2) its compensation by worsening the other criterion. In other words, a stronger preference for q 1 (than for q 2 ) implies a smaller value of α; therefore to compensate (in terms of keeping the ASF value unchanged) an improvement (worsening) of q 2 by say x%, the value of q 1 has to change by αx%. The interpretation of values of α appears to be natural, and therefore the linear aggregation is believed to support intuitive and robust analysis. To show that this is not true let us consider two values of α, namely those corresponding to the slopes represented by α 1 and α 2 showninfig.3a.foranyα<α 1 solution A will be selected, and for α 1 <α<α 2 and α>α 2 solutions B and C, respectively. For the other two cases (α = α 1 and α = α 2 ) there is no unique solution (thus any solution from either segment AB or BC can be selected). Thus the weights used for linear criteria aggregation support poorly examination of the Pareto set. In some situations the same solution is returned even if substantial changes of weights were made; e.g., the cheapest solution A will be selected not only when the cost criterion q 1 is infinitely more important than the emission q 2 but also if q 1 is only slightly more important than q 2. In other situations a tiny change in preferences (represented by α 1 in Fig. 3a) results in a substantially different solution. 9 The second serious flaw of the weighted sum approach is that many Pareto-efficient solutions are never shown to the user. This is illustrated for two types of problems in Fig. 3. For a continuous LP model (Fig. 3a) a typical LP solver will return a solution corresponding to a vertex, therefore infinite number of solutions located between vertices A, B, C will never be shown. For discrete problems only efficient solutions located on a convex cover defined by the alternatives will be found by this method (e.g., solutions B, C, E, F in Fig. 3b will never be shown to the user). Limiting the user choice to only a subset of Pareto solutions is not only an ethical problem of restricting the user sovereignty. Actually, many of the solutions excluded from the analysis can represent the 9 In many problems the distance between some vertices is large. *, + -. / G tradeoffs between criteria that are preferred by the user, see e.g., examples in [12]. To illustrate this point let us consider two maximized criteria and three alternatives: A = {0, 1}, B = {0.49, 0.49}, C = {1, 0}. Assume that a user selects α =1(i.e., assigns equal weights to both criteria). Solution B corresponds perfectly to these preferences, but the weighted-sum approach never finds it. The third shortcoming is due to a common belief that using weights is intuitive because there is always a positive correlation between increasing the weight for a criterion and the corresponding improvement of the criterion value. Simple examples (cf e.g., [12], [13]) show that also this is not true. In addition to the fundamental problems summarized above, the linear aggregation has a number of other drawbacks, including double-counting (of dependent criteria), and a tacit assumption of a fully compensatory character of criteria (with constant substitution rates for the whole ranges of criteria). 10 More details can be found e.g., in [4], [11], [12], and [13]. B. Reference point methods Most of the successful multicriteria analysis methods are based on the two-step approach: define a reference (aspiration, goal) point composed of the desired values of all criteria; further on we denote the reference point as RFP; find a Pareto solution that is (in a sense) closest to this point. The displaced ideal point method [14] uses the Utopia (marked by U in Fig. 2) as the RFP, and the family of methods known as Goal Programming (originating from [15]) assumes that the RFP is defined by the user. Two types of methodological problems had to be solved in order to effectively represent user preferences in this two-stage procedure: first, a sequential specification of RFPs; second, the selection of a measure for the distance between the RFP and the Pareto set. This led to the development of theory, software and application of the aspiration-based decision support-summarized in [16]; their relations to the Goal Programming are summarized in [17]. Another stream of the developments in this field is outlined in [13]. A comprehensive discussion of the theoretical background of the reference point methodology, tools for their implementation as well as a detailed presentation of several applications can be found in [7]. Here we only outline the basic features of this approach. The Utopia point (composed of best values of all criteria, and marked by the letter U in Fig. 2) is an obvious initial RFP. However, in most practical applications the Utopia point is far away from the Pareto set; the Utopia point is therefore a clearly unrealistic goal. Thus for an effective multicriteria analysis it is essential to represent the user preferences using two mechanisms in a concerted way: first, sequential specifications of RFPs supporting 10 If one of the criteria is expressed in monetary units, then this is equivalent to accepting the principles of monetarization (i.e., that all criteria can be converted to monetary units).

9 JOURNAL OF COMPUTERS, VOL. 4, NO. 10, OCTOBER of particular decisions are impossible, the preference structure among decisions can provide a stable basis for a relative ranking of alternatives. Thus the essence of identifying robust decisions is to perform comparative analyses of feasible decisions and to design robust policies with respect to the uncertainties and risks involved. Such decisions don t attempt to be optimal for any given scenario; they reflect trade-offs between being (1) possibly best for most likely situations, and (2) good enough for extreme events. A more detailed discussion of novel approaches to coping with uncertainties and risks can be found e.g., in [25], [26], [27]. V. TRANSPARENCY AND PUBLIC UNDERSTANDING By now it is commonly agreed that the provision of information is critical to public acceptance, and that in reality some commonly discussed problems are actually incorrectly understood. Selected issues of modeling for knowledge exchange are discussed in [28]. The relevance of this publication for policy making is illustrated e.g., by Sterman [29], who points out that although the Kyoto Protocol is one of the most widely discussed topics, most people believe that stabilizing emissions at near current rates will stabilize the climate. Recent debates on pension system reforms in several European countries also clearly show a wide misunderstanding of the consequences of population structure dynamics on economies in general and on pension systems in particular. These, and many other problems, can also be explained to the public by adapting relevant models for use in presentations that the public can understand. Unfortunately, various models developed for policy-making problems use different assumptions, and often different sets of data; therefore a comparative analysis of their results can at best be done and understood by a small community of modelers. The need for public access to knowledge pertinent to policymaking will certainly grow, see e.g., [30] for the discussion of access to environmental information; thus the role of models in public life will also grow accordingly. VI. CONCLUSIONS Development of models for complex problems does, and will, require various elements of science, craftsmanship, and art (see, e.g. [4] for a collection of arguments that supports this statement). Moreover, development and comprehensive analysis of a complex model requires collaboration of interdisciplinary teams, and thus a substantial amount of time and other resources. Therefore new modeling tools are needed for an effective support of collaborative modeling (both model development and exploitation) by interdisciplinary teams working at distant locations. SMT addresses these needs by supporting the development of models with complex structures and huge amounts of data, and diversified analyses of such models; moreover, it provides automatic documentation of the whole modeling process. Thus, SMT promotes modeling quality and transparency, which are critically important for model-based support of decision-making, especially in actual policy-making. Web-based modeling environments (like the SMT) not only support interdisciplinary modeling, but also provide opportunities for involving a larger audience in model analysis. In particular, a Web-based multicriteria analysis of discrete alternatives (called MCAA) has been recently developed; 16 it is currently being used for several applications, including analysis of future energy technologies (the latter to be done remotely by a large number of stakeholders). The composition of this paper was also motivated by the requirements of scientific support for decisionmaking. Each decision problem typically has a main focus. However, each non-trivial problem requires an integrated analysis of all relevant aspects, including the decision-making process, the data quality, uncertainties, and risks. Focusing on only one (even main) aspect and/or selecting too early a specific modeling paradigm is dangerous because it may neglect factors that can damage the quality of an incomplete analysis. Science-based support for policy-making is a process, and quality of the support is determined by its weakest element. This paper focuses mainly on multicriteria analysis of attainable goals. However, it also discusses recent developments in the modeling technology, integrated risk management, and transparency and public understanding. All these topics are relevant to supporting rational decision-making, therefore recent developments might be interesting for researchers and practitioners. ACKNOWLEDGMENTS The author gratefully acknowledges diversified contributions of Y. Ermoliev, T. Ermolieva, J. Granat, and A.P. Wierzbicki. Many discussions and joint activities on various modeling issues have contributed over many years to the development of the modeling methodology, novel methods for multicriteria analysis, and for integrated management of inherent uncertainties and catastrophic risks, and their applications to many diversified policymaking processes. Recent research on multicriteria analysis of discrete alternatives has been partly financially supported by the ECfunded Integrated Project NEEDS, and by the Austrian Federal Ministry of Science and Research. REFERENCES [1] M. Makowski, L. Somlyódy, and D. Watkins, Multiple criteria analysis for water quality management in the Nitra basin, Water Resources Bulletin, vol. 32, no. 5, pp , [2] M. Makowski, A structured modeling technology, European J. Oper. Res., vol. 166, no. 3, pp , draft version available from at/ marek/pubs/prepub.html. [3] A. Geoffrion, An introduction to structured modeling, Management Science, vol. 33, no. 5, pp , Readers interested in the MCAA are invited to visit in marek.

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