Book Project on Multicriteria Optimization

1. Book Project on Multicriteria Optimization • 8) MO Procedures with Progressive Elicitation of Preferences • P. Wierzbicki, K. Miettinen, F. Ruiz • Trade-off based Methods

2. Trade-offs based methods The concept of trade-off • Objective: The change in one objective in relation to the change in another one, when moving from one feasible (efficient) solution to another one. • Subjective: How much of one objective does the DM consider desirable to sacrifice in order to improve another objective to a certain quantity.

3. Trade-offs based methods Definitions of trade-off

4. f* N Trade-offs based methods Graphical interpretation of the trade-off rates: the slope of the efficient frontier at the current iteration x*, f* f(S)

5. (f) = * f(S) f* N’ N Trade-offs based methods Graphical interpretation of the MRS: the slope of the indifference curve at the current iteration x*, f*

6. Trade-offs based methods How are the trade-offs used in interactive methods? • Almost all thetrade-off based interactive methods assume the existence of a value function of the DM, . • Type 1: The method calculates objective (efficient) trade-offs at the current iteration and the DM is asked to analyze them somehow. The next iteration is built on the base of this information. The stopping criterion is achieved if all trade-offs at the current solution are indifferent (or less preferred) for the DM. • Type 2: The DM is asked to give the MRS at the current iteration. This information is used to search increasing directions for . The stopping criterion is achieved when the MRS coincide with the objective trade-offs at the current solution.

7. f(S) f* N = N’ Trade-offs based methods Graphical interpretation of the stopping criterion: the efficient frontier and the indifference curve are tangent.

8. Trade-offs based methods How to determine objective trade-offs over the efficient frontier? • Linear case: The trade-offs are given by the shadow prices of the simplex tableau. Zionts and Wallenius (1976) propose a method to determine the efficient adjacent vertices.

9. If ij* are the optimal Kuhn-Tucker multipliers with respect to , then – ij* is a total or partial tradeoff rate (depending on whether there exist 0 multipliers or not) between fi and fj. Trade-offs based methods How to determine objective trade-offs over the efficient frontier? • Nonlinear case (2nd order conditions hold): Given a reference function fi, and a vector of bounds , we solve

10. where i = 1/(fih – zi*), z* is the ideal vector, and fh is the current iteration. Let i* be the optimal multiplier corresponding to . Then the normal vector is: Trade-offs based methods How to determine objective trade-offs over the efficient frontier? • Nonlinear case (no 2nd order conditions): We solve

11. and the value function Type 1 methods The Zionts-Wallenius method (Zionts and Wallenius, 1976) The method is based on linear approximations. We consider Given a guess h, the corresponding value function is maximized over the feasible set.

12. Type 1 methods The Zionts-Wallenius method (Zionts and Wallenius, 1976) • STEPS • Give an initial guess 1. Make h = 1. • Maximize the value function over the feasible set. Let xh be the optimal solution. • Determine all the adjacent efficient vertices not visited yet. • For each one, determine the corresponding trade-off. • Ask the DM about the desirability of all the trade-offs (“yes”, “no” or “indifferent”). • If all the answers are “no” or “indifferent”, stop. Else, find a new vector of weights h+1, such that  is increasing in the direction of the preferred trade-offs, decreasing in the direction of the non preferred ones, and constant in the direction of the indifferent ones. Make h = h+1. Go to step 2.

13. Type 1 methods The Interactive Surrogate Worth Trade-off (ISWT) method (Chancong and Haimes, 1978) • ASSUMPTIONS • The implicit value function is assumed to be continuously differentiable and monotonically decreasing. • The objective and the constraint functions are twice continuously differentiable. • The feasible region is compact. • Regularity and second order conditions hold at the efficient solutions.

14. Type 1 methods The ISWT method (Chancong and Haimes, 1978) • STEPS • Choose a reference function fi, and give a initial guess for 1. Make h = 1. • Solve Pi(h), and denote by xh its optimal solution. Determine the trade-offs making use of the optimal Kuhn-Tucker multipliers. • For each j i, the DM is asked to rank the corresponding trade-off in a scale from – 10 to 10 (surrogate worth). • If a stopping criterion is satisfied, stop. • Update h+1. Make h = h+1. Go to step 2.

15. Type 2 methods The Geoffrion, Dyer and Feinberg (GDF) method (Geoffrion, Dyer and Feinberg, 1972) • ASSUMPTIONS • The implicit value function is assumed to be continuously differentiable, monotonically decreasing and concave. • The objective functions are continuously differentiable and convex. • The feasible region is compact and convex.

16. Type 2 methods The GDF method (Geoffrion, Dyer and Feinberg, 1972) • STEPS • Choose an initial feasible point x1. Make h = 1. • Ask the DM to give the MRS at xh. • Solve a search direction problem for the value function. Denote by yh the corresponding optimal solution. • If yh = xh, stop. Else, make dh = yh – xh. • Calculate the optimal step-length th. • Make xh+1 = xh + thdh. Make h = h+1. Go to step 2.

17. Type 2 methods The Sequential Proxy Optimization Technique (SPOT) (Sakawa, 1982) • ASSUMPTIONS • The implicit value function is assumed to be continuously differentiable, monotonically decreasing and concave. • The objective functions are twice continuously differentiable and convex. • The feasible region is compact and convex. • Regularity conditions hold at the efficient solutions.

18. Type 2 methods The SPOT method (Sakawa, 1982) • STEPS • Choose a reference function fi, and give a initial guess for 1. Make h = 1. • Solve Pi(h), and denote by xh its optimal solution. Determine the trade-offs making use of the optimal Kuhn-Tucker multipliers, ijh. • The DM gives the MRS at xh, mijh. • Check the consistency of the MRS. If they are not consistent, go to step 3. • If a stopping criterion is achieved, stop. Else, let sjh = –(mijh – ijh), j i. • Select a proxy function and determine consistent parameters. • Determine the optimal step-length *. Let h+1 = h + *sh. Make h = h+1. Go to step 2.

19. Type 2 methods The PROJECT method (Luque, Yang and Wong, 2006) • ASSUMPTIONS • The implicit value function is assumed to be continuously differentiable, monotonically decreasing. • The objective functions are continuously differentiable. • The feasible region is compact. • Regularity conditions hold at the efficient solutions.

20. Type 2 methods The PROJECT method (Luque, Yang and Wong, 2006) • STEPS • Choose an initial efficient solution x1. Make h = 1. • Determine the normal vector, Nh, to the efficient frontier at xh. • The DM gives the MRS at xh, mh. • Calculate the projection of mh onto a tangent plane to the efficient frontier at xh, dfh. • If a stopping criterion is achieved, stop. Else, determine the optimal step-length, and set qih = fih + *dfih. • Project qh onto the efficient frontier, using a reference point scheme. Let x*, f* be the projection. • If the DM prefers f* to fh, then set xh+1 = x*, h = h+1 and go to step 2. Else, actualize the step-length * and go to step 6.

21. Thank you!