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Approximation and Visualization of Interactive Decision Maps Short course of lectures Alexander V. Lotov Dorodnicyn Computing Center of Russian Academy of Sciences and Lomonosov Moscow State University Lecture 2. Classes of methods for multi- objective (multi-criteria) optimization. A posteriori preference methods Plan of the lecture 1. Old simple-minded approaches 2. Main types of modern methods for multi-criteria optimization (MCO): 2a. No-preference methods 2b. A priori preference methods 2c. Interactive methods and possible functions of criteria 2d. A posteriori preference (generating the Pareto frontier) 3. Stability of the Pareto frontier and a posteriori preference methods for constructing a list of Pareto-optimal points Old simple MCO approaches A priori restrictions: DM selects the most important criterion and specifies restrictions for others: yi min * y f ( x), x X , yi ~i , i 1,2,..., m; i i* y A priori weights: DM specifies weights for all criteria and solves the problem: m wi yi max y f ( x), x X i 1 Classification of modern methods according to the role of the DM MCDA No-preference A posteriori methods methods A priori Interactive preference methods methods No-preference methods The opinions of the decision maker are not taken into account. The problem is solved by expert using some relatively simple method (say, single criterion optimization of some function of criteria with “objectively” specified parameters). The solution is presented to the decision maker who can accept or reject it. An example of no-preference methods For example, the following optimization problem can be solved: minimize h (y) = max { i (yi - yi*) : i=1,2,...,m} on Y, where i = 1/(yi** - yi*), i=1,2,...,m, and y** is the “worst” possible criterion point. For y**, one can take the criterion point, which coordinates are the worst feasible values of particular criteria, or the worst values of particular criteria over the Pareto frontier, or any other “bad” criterion point. Surely, the decision depends to a great extent on the “worst” point. By modifying the “worst” point, one can get any point of the Pareto frontier. A priori preference methods (constructing the decision rule) • Methods based on multi-attribute utility theory (MAUT) • Methods based on direct weighting of criteria • Methods based on complicated weighting procedures (Analytic Hierarchy Process) • Outranking methods (Electra, etc) • Methods based on heuristic concepts (say, goal identification) The simplest method (by Keeny and Raiffa) for approximation of indifference curves for value functions based on multi-attribute utility theory The case of an additive function v(y1, y2)=v1(y1)+v2(y2) y2 v2(y2)=2 y max v2(y2)=1 v=0 v1(y1)=1 v1(y1)=2 y1 Methods based on direct weighting of criteria DM specifies weights for all criteria and maximizes m U ( y ) wi yi min y f ( x), x X i 1 Important disadvantage of linear functions: decrement in the value of one criterion can be compensated by the value of another criteria Complicated weighting procedures (say, Analytic Hierarchy Process) Procedures consist of weighting and subsequent single-criterion optimization. The AHP method helps to develop weights. DM has to answer m(m-1)/2 questions concerning relative importance of criteria. The AHP methods helps to study quantitative criteria, too ELECTRE method French school (led by Prof. Bernard Roi) proposed interesting methods for constructing an outranking relation; these methods are affective in the case of a small number of alternatives Goal identification - 1 • DM has to identify the goal without information on the set Y=f(X). y2 0 y1 Goal identification - 2 Then, by using some distance function, the closest point of the set Y=f(X) is found. y2 y0 Y=f(X) 0 y1 Goal identification - 3 The goal programming is the most often used MCDA technique, but if the goal is distant from the feasible criterion set Y=f(X), the solution y0 depends mainly on the distance function, but not on the goal. Qualified experts feel the feasibility of criterion values and manage to identify the appropriate goals, which are close to the feasible criterion set. Interactive (iterative) methods Methods are based on interaction of the decision maker with the computer and consists in a finite number of iterations. At the first stage of an iteration, the decision maker specifies parameters of a function of the criteria. At the second stage, the computer solves the single- criterion optimization problem with the criterion function specified by the decision maker. General scheme of an interactive method After the k-th iteration, the vectory ( k ) f ( x ( k ) ) and some other auxiliary information must be provided to decision maker. • Stage 1. DM explores the information obtained at the k-th iteration and, may be, previous iterations. ( k 1) DM specifies parameters of a new optimization problem h( f ( x), (k 1) ) max while x X • Stage 2. Computer solves the problem; the new decision and criterion vector are computed . ( k 1) ( k 1) y f (x ) The simplest interactive method The method is based on application of the linear function h (y) = <c, y>. An iteration of the method consists of two steps: a) Computer finds the decision x0 from the set X that provides the maximum of the linear function h (f(x)) = <c, f(x)> over X with some given vector of parameters c <0; b) DM studies the optimal decision x0 , the criterion point y0 = f(x0). If DM is not satisfied with the decision, he/she changes the values of the parameters c and the method goes to the next iteration. What scalar functions of the criteria can be used? Let h(y) be a scalar function of criteria. Let y0 be the point of maximum of h(y) over Y. 1) Does it belong to the Pareto frontier? Answer. If y y results in h(y’’) > h(y’) (the scalar function is increasing in respect to Pareto domination), then y0 belongs to the Pareto frontier. 2) But what about the opposite: can any point of the Pareto frontier can be found by optimization of the function h(y) over Y ? Well-known examples of scalar functions of criteria 1) Linear function h (y) = <c, y>, where c <0 (note that we consider the minimization problem!), is an increasing function in respect to Pareto domination; 2) Tchebycheff distance from the ideal point y* h (y) = max { i (yi - yi*) : i=1,2,...,m}, where i : i=1,2,...,m, are some non-negative coefficients, is a decreasing function in respect to Slater domination. Properties of the linear function 1) The maximum over Y of the linear function h (y) = <c, y>, where c <0, belongs to the Pareto frontier. However, the opposite is true only in the case of the convex EPH: any point of the Slater frontier can be the maximum over Y of the linear function h (y) = <c, y> with some non-positive c only if the EPH is convex. Example y2 P(Y) f(X) c y1 Properties of the Tchebycheff distance 2) The point y0 of the minimum over Y of the Tchebycheff distance h (y) = max { i (yi - yi*) : i=1,2,...,m}, where i : i=1,2,...,m, are some non-negative coefficients, belongs to the Slater frontier. Moreover, any point of the Slater frontier can be the minimum over Y of the Tchebycheff distance with some non-negative coefficients. Example y2 P(Y) f(X) y* y1 A posteriori preference methods A posteriori preference methods are based on approximating the Pareto frontier and informing the decision maker concerning it. A posteriori methods inform the DM about the Pareto optimal set without asking for his/her preferences. The DM has to specify a preferred Pareto point, i.e. non-dominated combination of criterion values, only after completing the exploration of the Pareto frontier. Thus, the single-shot specification of the preferred Pareto optimal objective point may be separated in time from the exploration phase. Two main problems that must be solved by the a posteriori preference methods: • How to approximate the Pareto frontier • How to inform the DM about the Pareto frontier In the case of two criteria, information of the DM is usually based on graphical display of the Pareto frontier. In the case of more, than two criteria, a list of objective points is usually provided to the DM. Question: is the problem of approximating the Pareto frontier stated correctly? Stability of the Pareto frontier-1 Example: Slater (weak Pareto) S(Y) and Pareto P(Y) frontiers for the non-disturbed feasible set in criterion space Y y2 A Y B C P(Y) S(Y) y1 Stability of the Pareto frontier-2 P(Y) for the disturbed feasible set in criterion space A Y B P(Y) C Stability of the Pareto frontier - 3 If some natural requirements hold, the condition S(Y) = P(Y) where Y is the non-disturbed feasible set in criterion space, is the necessary and sufficient condition of stability of P(Y) to the disturbances of parameters. (Sawaragi Y., Nakayama H., Tanino T., 1985). Stability of the Edgeworth-Pareto Hull - 1 Edgeworth-Pareto Hull (EPH) Yp for the non- disturbed feasible set in criterion space Y A Yp Y B C Stability of the Edgeworth-Pareto Hull - 2 Edgeworth-Pareto Hull (EPH) Yp for the disturbed feasible set in criterion space Y A Yp Y B C Stability of the Edgeworth-Pareto Hull - 3 If some natural requirements hold, the Edgeworth-Pareto Hull is stable to the disturbances of parameters of the problem. The first a posteriori preference method The first a posteriori preference method is the method for approximation of the set P(Y) in linear bi-criterion problems. It was introduced by S.Gass and T.Saaty in 1955 and is based parametric linear programming. Parametric LP problem for two criteria y1 (1 ) y2 min y Cx, Ax b where changes from 0 to 1. The problem is solved by using a method for solving the parametric LP problems In addition to the list of y2 objective points, picture was provided! y1 Different methods Restrictions-based method y2 y1 Restrictions-based method: formal description yi* min y f ( x), x X , yi li , i i*, p 0 ,1,...,Pi p The result: a large list of points of the Pareto frontier Weighted Tchebycheff metric as the distance from the ideal point y2 y* y1 Formal description The problems h( y) maxi ( yi yi *) : i 1,2,...,m min y f ( x), x X are solved for a large number of parameters i 0, i 1 i 1 m Result: a large list of Pareto points. Parametric LP methods for linear problems with m>2 Direct development of idea by Gass and Saaty: parametric LP methods for m>2 construct all Pareto vertices for a linear multi-objective problem using the movement from a vertex to another (see R.L.Steuer. Multiple-criteria optimization. NY: John Wiley, 1986). A very large list of vertices is provided to the DM (sometimes along with the efficient faces of the set Y). Evolutionary (including genetic) multiple criteria optimization y2 y1 Result: a large list of quasi-Pareto points. Approximation of the bi-criterion Pareto frontier by linear segments NISE (Cohon, 1978) y2 Picture is provided to the DM! y1 The preferred point of an approximation is identified by the DM. Lessons learned from bi-objective problems According to Bernard Roy, “In a general bi-criterion case, it has a sense to display all efficient decisions by computing and depicting the associated criterion points; then, DM can be invited to specify the best point at the compromise curve”. It is extremely important that, in bi-objective MOO problems, the graphs provide, along with Pareto optimal objective points, information about the objective tradeoffs. Tradeoff information helps to identify the most preferred point at the tradeoff curve. The question is: “How to apply this experience in the case of m>2 ?”