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(Interactive?) Pareto Frontier Visualization ( Chapter 7(?) of the Dagstuhl book)

(Interactive?) Pareto Frontier Visualization ( Chapter 7(?) of the Dagstuhl book). A.V. Lotov Dorodnicyn Computing Centre of Russian Academy of Sciences, and Lomonosov Moscow State University, Russia. Concerning my research after the Dagstuhl meeting in November 2004.

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(Interactive?) Pareto Frontier Visualization ( Chapter 7(?) of the Dagstuhl book)

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  1. (Interactive?) Pareto Frontier Visualization(Chapter 7(?) of the Dagstuhl book) A.V. Lotov Dorodnicyn Computing Centre of Russian Academy of Sciences, and Lomonosov Moscow State University, Russia

  2. Concerning my research after the Dagstuhl meeting in November 2004 • My department has won a tender of Russian Agency for Science and Innovations for developing hybrid methods for approximating and interactive Pareto frontier visualization for the case of 3 to 8 criteria for non-linear models given as black box and characterized by several hundreds inputs. Approximation quality measures and stopping rules must be provided. Software must be implemented at a combination of personal computer and multiprocessor systems. The job was completed in October 2006. • Since 2006, we take part in a project for Web support of participatory municipal budget planning in Spain supported by the government of Madrid. Support is based on Pareto frontier visualization in Web.

  3. Why visualization of the Pareto frontier as a whole is needed? (Is it an illustration or practical decision support tool)?

  4. Notation X = feasible set in decision space, Z=f(X) = feasible set in criterion space Pareto domination Non-dominated (efficient, Pareto) set

  5. Feasible set in criterion space Z=f(X)

  6. Pareto domination (minimization case)

  7. Non-dominated (Pareto) frontier Z=f(X)

  8. Visualizationis a transformation of symbolic data into geometric information. About one half of human brain’s neurons is associated with vision, and this fact provides a solid basis for successful application of visualization for transformation data into knowledge. ”A picture is worth a thousand words”.

  9. Visualization for illustration of usual goal programming

  10. Goal identification - 1 • DM has to identify the goal (without information on the set Z=f(X)). z* 0

  11. Goal identification - 2 Then, by using some distance function, the closest point of the set Z=f(X) is found. Z=f(X) z0 z* 0

  12. Visualization in decision support In MCDA problems, visualization can provide geometric information concerning both the feasible criterion values and the objective tradeoffs: • total objective tradeoff • local tradeoff rate for a smooth frontier

  13. Non-dominated (Pareto) frontier and the objective tradeoff rate f(x*) f(x1) f(x2)

  14. Goal identification at the Pareto frontier For a decision maker, criterion tradeoff information is important for identification of a preferable non-dominated feasible criterion point (goal) directly at the non-dominated frontier by using the computer mouse. Such a goal can be used as a reference point that is close to the Pareto frontier

  15. Pareto frontier and the feasible goal Z=f(X)

  16. A feasible goal (or its neighborhood) can be used as the starting information in various procedures, say, in rules formulation or in selecting a part of Pareto frontier for subsequent study

  17. Can the user identify a goal? Real-life decision making proves that people like to use the goal approach. It means that they are able to identify a preferable criterion point. Visualization of the Pareto frontier provides an opportunity to identify the feasible non-dominated goal. Due to the feasibility, the traditional problem of specifying the distance between the goal and the feasible criterion set vanishes.

  18. One quotation In a general bi-criterion case, it has a sense to display all efficient decisions by computing and depicting the associated criterion points; then, decision maker can be invited to identify the best point at the compromise curve. B.Roy Decisions avec criteres multiples. Metra International, v.11(1), 121-151 (1972)

  19. Thus, the question is: is it possible and is it profitable to visualize the Pareto frontier in the case of more than two-three criteria?

  20. Visualization and psychological aspects of decision making

  21. Psychological aspects of thinking

  22. Important feature of the three-level model of human mentality: The three levels have different pictures of the reality, and much efforts of the human mental activity is related to coordination of the levels. The conflict between mental levels may result in non-transitive answers concerning their preference.

  23. To settle the conflict between levels, time is required. In his famous letter, Benjamin Franklin advised to spend several days to make a choice. Psychologists assure that sleeping is used by the brain to coordinate the mental levels. (Russian proverb: The morning is wiser than the evening).

  24. Pareto frontier methods and visualization Standard approach of the Pareto frontier methods: approximating the set P(Y) by a subset of its points and informing DM concerning such a list of points. However, selecting from large lists of multi-objective points is too complicated for a human being (O. Larichev. Cognitive Validity in Design of Decision-Aiding Techniques. Journal of Multi-Criteria Decision Analysis, 1992, 1(3).)

  25. Visualization of the Pareto frontier can help in transformation data into knowledge (in formation of mental picture of the MCDA problems ). Since visualization can influence all levels of thinking, it can support the search for a decision, that is not logically perfect, acceptable for all levels of human mentality.

  26. Pareto frontier visualization

  27. Preliminary remark:Stability (robustness) of the Pareto frontier and correctness of its approximation problem is not guaranteed

  28. Example: Slater S(Z) and Pareto P(Z) frontiers for the non-disturbed feasible set in criterion space A Z B C P(Z) S(Z)

  29. Stability (robustness) of the Pareto frontier P(Z) for the disturbed feasible set in criterion space A Z B P(Z) C

  30. If some natural requirements hold, the condition S(Z) = P(Z)where Z is the non-disturbed feasible set in criterion space, is the necessary andsufficient condition of stability of P(Z)to the disturbances of parameters. (Sawaragi Y., Nakayama H., Tanino T., 1985).

  31. Edgeworth-Pareto Hull Let Then

  32. Stability of the Edgeworth-Pareto Hull Edgeworth-Pareto Hull (EPH) Zp for the non- disturbed feasible set in criterion space Z A Zp Z B C

  33. Stability of the Edgeworth-Pareto Hull Disturbed EPH (Zp ) A Zp Z B C

  34. Normally, the Edgeworth-Pareto Hull is stable to the disturbances of parameters of the problem.In linear case, disturbances can even be estimated.

  35. Classification of the MCOproblems related to visualization procedures(in accordance to the number of decision alternatives and criteria)

  36. 0) bi-criterion case;1) finite number of decision alternatives: a) small number of non-dominated decision alternatives (not greater than a dozen); b) medium number of alternatives (not greater than about 1000) c) large number of alternatives (greater than about 1000).

  37. 2) infinite number of decision alternatives: a) approximation is given by a number of criterion points (both classical and EMO); b) tools for a convex case with polyhedral approximation: the case of more then three criteria.

  38. Visualization inspired by the bi-criterion case

  39. The first Pareto frontier method in MCO: generating the Pareto frontier in linear bi-criterion problem (S.Gass and T.Saaty, 1955). They used parametric LP method for the linear system where changes from 0 to 1.

  40. Picture was provided! The feasible criterion values are provided along with the objective tradeoffsincluding local tradeoff rates as well as the tradeoffs between any criterion points.

  41. The main problem in the two-criterion case is related to the methods for Pareto frontier approximation, which we do not discuss here.

  42. Visualization in the case m>3

  43. Decision maps The technique tries to use the advantages of the bi-criterion visualization (including visualization of local objective tradeoff rate and direct identification of the goal) in the case of three or four criteria. A series of values of the third (and, may be, fourth) criterion is specified (or several constraints on their values are imposed). Then, a series of bi-criterion graphs for the first and the second criteria is constructed and displayed in the same graph. Such an approach was known even in the 1970s.

  44. Recent example (from a paper of A.Mattson and A.Messac)

  45. Such graphs are known as the decision maps. They showlocal objective tradeoff rates for two criteria and total tradeoffs for criterion points in different bi-criterion graphs. Thus, decision maps provide graphic information on tradeoffs between all three (or even four) criteria. However, such decision maps cannot be displayed interactively since they require substantial time to be computed especially if the models of the decision situation are complicated.

  46. Visualization in the case of a finite number of alternatives

  47. Small number of non-dominated alternatives (not greater than a dozen)

  48. Bar chart (histogram): 4 alternatives, 6 criteria

  49. Value paths of 8 alternatives for 22 criteria

  50. Value paths of 48 alternatives

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