Approximation and visualization of Pareto frontier: Interactive Decision Maps technique

1. Approximation and visualization of Pareto frontier:Interactive Decision Maps technique Alexander V. Lotov Dorodnicyn Computing Centre of Russian Academy of Sciences, and Lomonosov Moscow State University, Russia Part 1

2. Plan of the talk 1. Few words concerning multi-objective optimization 2. Pareto frontier methods 3. Visualization: why it is needed? 4. Interactive Decision Maps (IDM) technique for visualization of Pareto frontier: software demo 5. Mathematics of the IDM technique in convex case 6. Real-life applications of the IDM 7. IDM in Web Participatory Decision Support 8. IDM in the non-convex case 9. Current studies

3. Detailed information on the IDM technique is given in the book Lotov A.V., Bushenkov V.A., and Kamenev G.K. Interactive Decision Maps. Approximation and Visualization of Pareto Frontier. Kluwer Academic Publishers, 2004.

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. Decision maker (DM) is needed to select a unique solution from the set of Pareto-optimal solutions

9. Decision maker Usually the DM is a convenient abstraction only since many different people (advisers, experts, analysts, various stakeholders) influence (or try to influence) the decision. However, it has sense, and so the concept of the DM is permanently used in the MOO field.

10. Classification of MCDA methods according to the role of the Decision Maker MCDA no-preference methods a posteriori preference (Pareto frontier) methods a priori preference methods interactive methods

11. Preference modeling: Aren’t the questions too complicated? • Tversky A. Intransitivity of preferences. Psychological Review, 1969, n 76. • Larichev O. Cognitive Validity in Design of Decision-Aiding Techniques. Journal of Multi-Criteria Decision Analysis, 1992, v.1, n 3.

12. Structure of a mental model (result of experimental psychological studies)

13. Important feature of the three-levels 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.

14. Coordinating the levels A large part of human mental activities is related to the coordination of the levels. To settle the conflict between the levels, time is required. Psychologists assure that sleeping is used by the brain to coordinate the mental levels. (Compare with the proverb: ``The morning is wiser than the evening''). In his famous letter on making a tough decision, Benjamin Franklin advised to spend several days to make a choice. It is known that group decision and brainstorming sessions are more effective if they last at least two days.

15. Coordinating the levels in MOO problems Thus, to settle the conflict between the levels of one’s mental model in finding a balance between different objectives in a multi-objective optimization problem, he/she needs to keep information on the problem in his/her brains for a sufficiently long time. Such opportunity is provided by a posteriori (Pareto frontier) methods. The absence of method-related time pressure is an important advantage of them. In contrast, other approaches require fast answers to the questions on preferences.

16. Pareto frontier methods Pareto frontier methods are devoted to approximation of the Pareto set and informing the DM concerning it. In contrast to preference-oriented methods, Pareto frontier methods do not ask multiple questions concerning the preferences, but only inform the DM on the problem.

17. The first Pareto frontier method in MCO: generating the Pareto frontier in linear bi-criterion problem (S.Gass and T.Saaty, 1955). Parametric LP methods may be used for solving where changes from 0 to 1.

18. Graph was provided to DM! The feasible criterion values are provided along with the tradeoff rates, which are the most important decision information.

19. Two main problems must be solved in the framework of the Pareto frontier methods in the case of m > 2 • How to approximate the Pareto frontier • How to inform the stakeholders about the Pareto frontier

20. Two basic ways for informing a stakeholder about the Pareto frontier • By providing a list of the criterion points that belong to the Pareto frontier • By visualization of the Pareto frontier

21. Selecting from a large list of criterion points (more than a dozen) with more than two criteria is too complicated for a human being, see • Larichev O. Cognitive Validity in Design of Decision-Aiding Techniques. Journal of Multi-Criteria Decision Analysis, 1992, v.1, n 3.

22. VISUALIZATION — why it is needed? 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”..

23. Visualizationcan influence all levels of human thinking and simplify by this the process of coordination.

24. Requirements that must be satisfied by a visualization technique To be effective, a visualization technique must satisfy some requirements, which include (i) simplicity, that is, visualization must be immediately understandable, (ii) persistence, that is, the graphs must linger in the mind of the beholder, and (iii) completeness, that is, all relevant information must be depicted by the graphs.

25. Example: Goal identification with visualization

26. Usual goal programming

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

28. What really happens Then, by using some distance function, the closest point of the set Z=f(X) is found. Z=f(X) z0 z* 0

29. Goal method based on visualization

30. Tradeoff information helps to specify the goal f(x*) f(x1) f(x2)

31. Goal identification at the Pareto frontier Once again, criterion tradeoff information is important for decision maker for identification of the preferable non-dominated feasible criterion point (goal). It can be done directly at the non-dominated frontier by using the computer mouse.

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

33. 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)

34. 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?Interactive Decision Maps technique answers: Yes, it is possible and profitable.

35. Interactive Decision Maps (IDM) technique

36. Edgeworth-Pareto Hull (EPH) of the feasible criterion set is It holds

37. Ideal point and Edgeworth-Pareto Hull P(Z) f(X) z*

38. Two spaces in MOO problems • Decision space: • Feasible set • Criterion space: • Feasible set in criterion space

39. The IDM technique is a tool for visualization of multi-objective (m>3) Pareto frontier. It is based on approximation of the EPH and subsequent interactive visualization of the Pareto frontier by using various collections of bi-objective slices of the EPH.

40. Decision map is a collection of bi-objective slices of the Pareto frontier (or EPH) in the case of three criteria. Interactive Decision Mapstechnique provides interactive visualization and animation of the decision maps. Thus, it is a tool for Pareto frontier visualization in the case of more than of three criteria.

41. An example problem for IDM software demonstration

42. The problem The problem of economic development of the region is studied. If the agricultural (to be precise, grain-crops) production would increase, it may spoil the environmental situation in the region. This is related to the fact that the increment in the grain-crops output requires irrigation and application of chemical fertilizers. It may result in negative environmental consequences, namely, a part of the fertilizers may find its way into the river and the lake with the withdrawal of water. Moreover, shortage of water in the lake may occur during the dry season.

43. Two agricultural zones are located in the region. Irrigation and fertilizer application in the upper zone (located higher than the lake) may result in a drop of the level of the lake and in the increment in water pollution. Irrigation and fertilizer application in the second zone that is located lower than the lake may also influence the lake. This influence is, however, not direct: irrigation and fertilizer application in the lower zone may require additional water release from the lake into the river (the release is regulated by a dam) to fulfill the requirements of pollution control at the monitoring station located in point A .

44. The model The model consists of three sub-models: • model of agricultural production; • water balances and constraints; • pollution balances and constraints. The production in an agricultural zone is described by a technological model, which includes N agricultural production technologies. Let xij, i=1,2...,N, j=1,2, be the area of the j-th zone where the i-th technology is applied. The areas xijare non-negative and are restricted by the total areas of zones

45. The i-th agricultural production technology in the j-th zone is described by the parameters aijk, k=1,2,3,4,5, given per unit area, where aij1 is production, aij2 is water application during the dry period, aij3 is fertilizers application during the dry period, aij4 is volume of the withdrawal (return) flow during the dry period, aij5 is amount of fertilizers brought to the river with the return flow during the dry period. Thus, one can relate the values of production, pollution, etc. to the distribution of the area among technologies in the zone where zj1 is production, zj2 is water application during the dry period, zj3 is fertilizers application during the dry period, zj4 is volume of the withdrawal (return) flow during the dry period, zj5 is amount of fertilizers brought to the river with the return flow during the dry period in the j-th zone.

46. The water balances are fairly simple. They include changes in water flows and water volumes during the dry period. The deficit of the inflow into the lake due to the irrigation equals to z12 z14. The additional water release through the dam during the dry period is denoted by d. It is supposed that the release d and water applications are constant during the dry season. Let T be the length of the dry period. The level of the lake at the end of the dry period is approximately given by L(T) = L  (z12 z14 + d)/, where L is the level without irrigation and additional release, and  is a given parameter. Flow in the mouth of the river near monitoring point A denoted by vAequals to vA0+(d  z22+ z24)/T, where vA0is the normal flow at point A. The constraint is imposed on the value of the flow vA vA*, where the value vA* is given. Thus, the following constraint is included into the model vA0+(d  z22+ z24)/T  vA*.

47. The increment in pollution concentration in the lake denoted by wL is approximately equal to z15/ , where  is a given parameter. The pollution flow (per day) at the point A denoted by wA is given by z25 /T + qA0, where qA0 is the normal pollution flow. It means that we neglect the influence of fertilizers application in the upper zone on pollution concentration in the mouth. Then, the value of wA equals to (z25 /T + qA0 ) / vA . The constraint wA≤wA* where wA* is given is transformed into the linear constraint (z25 /T + qA0 ) ≤wA* vA or z25 /T + qA0≤wA*(vA0+(d  z22+ z24)/T).

48. Decision variables and criteria The decision variables are allocations of land between different technologies in the agricultural zones as well as the additional water release through the dam. For the criteria, any collection of the variables of the model can be used. In the following demonstration, five economic and environmental criteria are used.

49. Computer demonstration: a simple example of regional water planning

50. Demonstration of software(IDM + Feasible Goals Method)