Spatial Compromise Programming

1. Spatial Compromise Programming RESM 575 Spring 2011 Lecture 4

2. Last time • Weighting techniques • Point allocation • Ranking methods • Pairwise comparison • Class exercise • Lab: Criteria weighting for suitability models and spatial sensitivity

6. Review of question 3 • Why nonparametric? • Nonparametric tests: • Chi-square • U-test • Wilcoxon • Sign test • Kruskal-Wallis • Friedman • Mann-Whitney

8. Building suitability models, steps • Define the problem or goal • Decide on evaluation criteria • Normalize and create utility scales • Define weights for criteria • Calculate a ranking model result • Evaluate result

10. What is multiple objective decision making? • A form of decision analysis that seeks to analyze complex decision problems by dividing the problem into smaller understandable parts • Then, we integrate the parts in a logical manner to produce a meaningful solution

11. Generally speaking…. • Individuals, groups, and organizations, in their decision making efforts, • pursue multiple objectives • set multiple goals • evaluate their options according to multiple criteria and as a consequence experience conflict. • Decision making under these such conditions is characterized by incessant attempts at conflict resolution and the simultaneous attainment of goals.

12. Goal Programming (GP) • A form of linear programming that allows for consideration of multiple goals • GP can be used to determine the optimal solution to a multi-objective decision problem

13. Illustration (single objective decision making) Skilled labor Technical know how Build an auto of maximum horsepower Raw materials Energy “a purely technical problem”

14. Illustration (multiple objective decision making) Build the “best” auto Safety Price Depreciation rate Weight Reliability Size No single end or no single criterion, a multiple criteria decision problem or an economic problem according to Freidman’s definition Human value judgments, trade off evaluations, and assessments of criteria now become integral to the problem

15. Compromise programming (CP) • Similar to goal programming in that it uses the concept of minimum distance • A distance based technique that depends on the point of reference or “ideal” point • Attempts to minimize the “distance” from the ideal solution for a satisficing solution • The closest one to the ideal across all criteria is the compromise solution or compromise set

17. CP notes • The concept of non dominance is used in distance-based techniques to select the best solution or choice of alternative. • A solution is said to be non dominated if there exists no other feasible solution that will cause an improvement in a value of the objective or criterion functions without making a value of any other objective function worse (Tecle and Yitayew, 1990).

18. CP notes • The non dominance solution concept, originating with Pareto in 1906, has been one of the cornerstones of traditional economic theory. It is usually stated as the Pareto principle: “A state of the world A is preferable to a state of the world B if at least one person is better off in A and nobody is worse off. A state is said to be Pareto optimal or Pareto efficient when there is no other state in which one individual can obtain higher satisfaction without at the same time lowering the satisfaction of at least one other individual” (Just et al., 1982).

19. CP model

20. CP model

21. CP model notes • In (3), the parameter p can have values from zero to infinity and represents the concern of the decision maker over the maximum deviation (Tecle and Yitayew, 1990) (Duckstein and Opricovic, 1980).

22. CP model notes • The larger the value of p, the greater the concern becomes. For p = one, all weighted deviations are assumed to compensate each other perfectly. For p = two, each weighted deviation is accounted for in direct proportion to its size. As p approaches the limit of infinity, the alternative with the largest deviation completely dominates the distance measure (Zeleny, 1982).

23. Using the CP model • Assemble data for all evaluation criteria, this becomes the evaluation matrix

24. Evaluation Matrix More of Criteria 1 is preferred Lower costs for Criteria 2 is preferred Low Percentages for Criteria 3 are preferred

25. Using the CP model • Normalize the matrix based on rules, this becomes the payoff matrix More of Criteria 1 is preferred Lower costs for Criteria 2 is preferred Low Percentages for Criteria 3 are preferred 1 - (65/90)

26. Using the CP model • Find the best and worst for each alternative across the criteria from the payoff matrix which has been already normalized

27. Using the CP model • Integrate the criteria weights, f* and f** and values for the alternative into the CP model for a parameter value of p (1, 2, oo) Using criteria weights C1= .4, C2=.5, C3=.1 For Alternative A and p = 1 (.4) [(1.00-1.00)/(1.00-.500)] + .(.5) [(.722 – .277)/(.722-0)] + (.1)[(.58-0)/(.58 – 0)] Which is 0 + .3081 + .1 or .4081

28. For Alternative B and p = 2 (((.4) [(1.00-.800)/(1.00-.500)])2 + ((.5) [(.722 – .722)/(.722-0)])2 + ((.1)[(.58-.58)/(.58 – 0))2])1/2 Which is .1600 + 0 + 0 or .1600 For Alternative C and p = 1 (.4) [(1.00-.500)/(1.00-.500)] + .(.5) [(.722 – 0)/(.722-0)] + (.1)[(.58-.200)/(.58 – 0)] Which is .4 + .5 + .655 or 1.555

29. Results Therefore since B is the lowest value (closest to the ideal values across all the criteria, it would be the preferred alternative for the weights and when p = 1

30. Solving the CP model • The preferred alternative has the minimum Lp distance value for each p and weight set that may be used. • Thus, the alternative with the lowest value for the Lp metric will be the best compromise solution because it is the nearest solution with respect to the ideal point.

31. CP advantages • Simple conceptual structure • Simplicity makes it particularly useful for spatial decision problems in which decision makers tend to rely on their intuition and insight

32. CP limitation • Except at the two extremes where p = 0 and p = oo there is no clear interpretation of the various values of the parameter p. • Therefore, use different weights or values for p to test overall robustness of results

33. ArcGIS CP Extension