Solution Pluralism and Metaheuristics

Transcription

1 1 / 27 Solution Pluralism and Metaheuristics Steven O. Kimbrough Ann Kuo LAU Hoong Chuin Frederic Murphy David Harlan Wood University of Pennsylvania University of Pennsylvania Singapore Management University Temple University University of Delaware July 28, 2011, Udine, IT, Metaheuristics International Conference

2 2 / 27 Outline 1 Introduction 2 Post-Solution Analysis of COMs 3 Extended Example GAP4_2 FoIs GAP4_2 IoIs 4 Further Areas of Opportunity 5 Discussion 6 End Matter

3 3 / 27 Recognizing this is a position paper... Distinction: the problem versus a model or representation of the problem. In OR, we don t solve the problem, we solve the model of the problem." So how do we solve the problem? Solution pluralism is a stance or default position, recommending finding many solutions for a model and using the resulting solution base (SoIs solutions of interest) for decision making.

4 4 / 27 How can models have more than one solution? Quadratic equation? We ll see examples, but 1 Solutions under different settings of parameters. 2 Solutions under different realizations of random variables. 3 In optimization models, solutions that are non-optimal for the model. 4 Combinations of the above.

5 5 / 27 Established examples Models with stochastic outputs; simulation models of various sorts. Sensitivity analysis of models; especially variance-based analyses, Monte Carlo methods. Robust optimization (in the uncertainty sense) Multi-objective optimization.

6 6 / 27 Post-solution analysis of constrained optimization models. Obtain a solution base from the neighborhood of the best solution(s), both feasible and infeasible. Example: Estimate the cost/benefit of altering constraint RHS values by making changes and evaluating solutions in the solution base. How do we generate a high-quality solution base? Positive results with FI-2Pop GA. In general, methods drawing on evolutionary computation are promising. Also, other kinds of population-based metaheuristics.

7 GAP4_2 FoIs GAP4_2 A representative GAP test problem. OR-Library, J.E. Beasley. uk/~mastjjb/jeb/info.html, [Beasley, 2009]. c machines, 30 jobs. 644 at optimum. Solved with FI-2Pop GA. FoIs: top 1000 feasible solutions, ranked by objective value. IoIs: top 1000 infeasible solutions, ranked by distance to feasibility. 7 / 27

8 GAP4_2 FoIs Can we find an optimal solution? Yes. Two! Optimal Solution A Optimal Solution B / 27

9 GAP4_2 FoIs Now let s compare them on their constraint slacks Comparison of Slacks Solution A= B= Decision makers may well prefer A over B or vice versa. 9 / 27

10 GAP4_2 FoIs Shadow-price-like questions Obj. Val A= B= Can the additional slack resources be usefully deployed? On which resources do we not have much opportunity for redeployment? (Ans: 2 & 5.) 10 / 27

11 GAP4_2 FoIs Shadow-price-like questions In both of my optimal solutions, constraint 1 has a slack of 2. Are there any good solutions available with a slack of at least 10? Yes. #23 has a slack on constraint 1 of 10 and an objective value of 640; #53, 11, 638; #97, 13, 636. What about constraint 2, which already has a slack of 5? #18, 641, 8; #68, 637, 15; #74, 637, 18. 3? At 0 or 1. #10, 642, 4; #39, 639, 8; #48, 638, 10. 4? At 1 or 2. #5, 643, 11. 5? At 0 or 1. #35, 639, 9. Plus, we can do combinations / 27

12 GAP4_2 FoIs Slacks for top 100 feasible solutions / 27

13 GAP4_2 FoIs Reduced-cost-like questions Job 25 is assigned to machine 2 in solution A and machine 3 in solution B. What s the best solution if we assign job 25 to machine 1? Answer: It has an objective value of 643. The solution is Its slacks: Job 25 to machine Obj. Val / 27

14 GAP4_2 IoIs Shadow-price-like questions Obj Is there opportunity to acquire more resources? To trade in slacks and surpluses? 14 / 27

15 GAP4_2 IoIs Shadow-price-like questions We re at 644, but we need to get to 651. Show me what I need to do. In the top 1000 IoIs there are 75 discovered solutions with objective values at or above / 27

16 GAP4_2 IoIs Shadow-price-like questions. 651? Obj A= B= Is there opportunity to acquire more resources? To trade in slacks and surpluses? 16 / 27

17 GAP4_2 IoIs Shadow-price-like questions. >651? Obj Is there opportunity to acquire more resources? To trade in slacks and surpluses? 17 / 27

18 GAP4_2 IoIs Reduced-cost-like questions Job 20 is assigned to machine 3 in solutions A and B, and to 3 (and sometimes 4) in all the FoIs. There are 57 solutions in the IoIs in which we assign job 20 to machine 2. What are the most promising of these solutions? 18 / 27

19 GAP4_2 IoIs Reduced-cost-like questions. 20 to 2? Obj Is there opportunity to acquire more resources? To trade in slacks and surpluses? Note: We do not find an opportunity with 20 assigned to 2 and z > / 27

20 20 / 27 GAP4_2 IoIs

21 Robust solutions under risk See Finding Robust-under-Risk Solutions for Flowshop Scheduling at MIC Constrained optimization models with imperfect knowledge of parameters. What if the parameters can be known under risk (in contrast to uncertainty)? One approach: 1 Obtain a solution base from the neighborhood of the best solution(s) to the static problem. 2 Evaluate the solutions in the solution base with Monte Carlo methods to find the most robust (to a given standard or measure). How do we get this solution base? Evolutionary computation. 21 / 27

22 22 / 27 Two-sided matching (stable matching) The problem(s)... Solution concept: stability Gale-Shapley and the deferred acceptance algorithm Wide use in practice. Issues: unfairness, non-simple matching conditions, etc. What about social welfare? Equity? Response: use evolutionary computation to find solution bases of stable and nearly stable solutions. Evaluate these with respect to other criteria. (See paper cited in position paper.)

23 23 / 27 Redistricting (Work in progress.) Problem: gerrymandering. Courts require that districts be nearly equal in population and connected. Leaves plenty of room for gerrymandering. Broadly agreed: compactness of districts (e.g., low circumference) is violated with gerrymandering. One approach to response: 1 Find a solution base by minimizing non-compactness, subject to connectivity and population size constraints. 2 Let the political process haggle over a suitable solution base (e.g., solutions within X% of the most compact). 3 If the political process fails, let the courts choose a solution at random from the solution base. (Or use some other scheme.)

24 24 / 27 Pattern 1 Develop a (more or less) conventional model for the problem to hand. 2 Define, pragmatically, the SoIs solutions of interest. 3 Obtain multiple solutions, a solution base for the SoIs, by using evolutionary computation. 4 Deliberate to decision (build another model?) from information derived from the solution base.

25 25 / 27 Future research Obviously, lots to explore here. What works well, what doesn t? How else can these ideas be applied? Exploring ways to use the solution base effectively. Vistualization? Suggest: A big switch, from focusing on solving a model, to obtaining the SoIs for the model. An important reason to use metaheuristics.

26 26 / 27 Beasley, J. E. (Accessed July 27, 2009). OR-Library. World Wide Web. info.html.

27 27 / 27 $Id: Udine-MIC-2011-pluralism-beamer.tex :23: