Synchronous Usage of Parameterized Achievement Scalarizing Functions in Interactive Compromise Programming

Transcription

1 Synchronous Usage of Parameterized Achievement Scalarizing Functions in Interactive Compromise Programming Yury Nikulin and Volha Karelkina University of Turku, Department of Mathematics and Statistics MCDM 2013, Malaga, 17 June 2013 Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

2 Outline 1 Introduction 2 Basic notations and definitions 3 Achievement scalarizing functions 4 Directing interactive solution process 5 Three-objective median location problem 6 Numerical experiment 7 References 8 Future Research Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

3 Introduction Interactive process The basic steps in interactive algorithms find an initial feasible solution, interact with the decision maker, and obtain a new solution (or a set of new solutions). If the new solution (or one of them) or one of the previous solutions is acceptable to the decision maker, stop. Otherwise, go to the previous step. Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

4 Introduction Two major requirements are set for a scalarizing function (Y. Sawaragi, H. Nakayama and T. Tanino, Theory of multiobjective optimization, Academic Press, Orlando, 1985) correctness: every solution found by means of scalarization should be (weakly) Pareto optimal, and completeness: it should be able to cover the entire set of Pareto optimal solutions. Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

5 Basic notations and definitions f(x) = (f 1 (x), f 2 (x),..., f m (x)) x X min f i(x), i N m = {1, 2,...,m} x X A set of minima of the i th objective function If M i (X) = arg min x X f i(x), i N m m M i (X), i=1 then there exists at least one solution which delivers a minimum for all objectives. Such a solution can be called an ideal solution. Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

6 Basic notations and definitions Pareto optimality P m (X) A decision vector x X is Pareto optimal if there exists no x X such that f i (x) f i (x ) for all i N m and f j (x) < f j (x ) for at least one index j. Slater optimality Sl m (X) A decision vector x X is weakly Pareto optimal if there exists no x X such that f i (x) < f i (x ) for all i N m. Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

7 Basic notations and definitions Ideal and nadir objective vectors f I = (f I 1,...,f I m ) and f N = (f N 1,..., f N m ) f I i f N i = min x P m (X) f i(x), i N m, = max x P m (X) f i(x), i N m. Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

8 Achievement scalarizing functions Achievement scalarizing functions A. P. Wierzbicki, The use of reference objectives in multiobjective optimization. Lecture notes in economics and mathematical systems, 177 (1980) s R : R m R The scalarized problem min s R(f(x)) (1) x X 1 Increasing, if for any y 1, y 2 R m, yi 1 yi 2 for all i N m, then s R (y 1 ) s R (y 2 ). 2 Strictly increasing, if for any y 1, y 2 R m, yi 1 < yi 2 for all i N m, then s R (y 1 ) < s R (y 2 ). 3 Strongly increasing, if for any y 1, y 2 R m, yi 1 yi 2 for all i N m and y 1 y 2, then s R (y 1 ) < s R (y 2 ). Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

9 Achievement scalarizing functions Optimality conditions for ASFs Theorem 1 Let s R be strongly (strictly) increasing. If x X is an optimal solution of problem (1), then x is (weakly) Pareto optimal. 2 If s R is increasing and the solution of (1) x X is unique, then x is Pareto optimal. Theorem If s R is strictly increasing and x X is weakly Pareto optimal, then it is a solution of (1) with f R = f(x ) and the optimal value of s R is zero. Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

10 A parameterized ASF Achievement scalarizing functions s q R (f(x),λ) = max I q N m: I q =q { i I q max[λ i (f i (x) f R i ), 0] where q N m and λ = {λ 1,...,λ m },λ i > 0, i N m. for q N m : s q R (f(x),λ) 0; q = 1 : sr 1(f(x),λ) = max i N m max[λ i (f i (x) fi R ), 0] = sr (f(x),λ); q = m : sr m(f(x),λ) = i N m max[λ i (f i (x) fi R ), 0] = sr 1(f(x),λ). }, min x X sq R (f(x),λ). (2) Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

11 Achievement scalarizing functions Yu. Nikulin, K. Miettinen and M. M. Mäkelä, A new achievement scalarizing function based on parameterization in multiobjective optimization, OR Spectrum, 34 (1) (2012) Theorem Given problem (2), let f R be a reference point such that there exists no feasible solution whose image strictly dominates f R. Also assume λ i > 0 for all i N m. Then among the optimal solutions of problem (2) is a weakly Pareto optimal solution. Theorem Given problem (2), let f R be a reference point. Also assume λ i > 0 for all i N m. Then among the optimal solutions of problem (2) there exists at least one Pareto optimal solution. Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

12 Directing interactive solution process The case of three objectives m = 3 { s q R (f(x),λ) = max I q {1,2,3}: I q =q i I q max[λ i (f i (x) f R i ), 0] where q = 1, 2, 3 and λ = (λ 1,λ 2,λ 3 ), λ i > 0, i N 3. }, f R = f I = (f I 1, f I 2, f I 3 ) Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

13 Directing interactive solution process q = 1 sr {max[λ 1 (f(x),λ) = max 1 (f 1 (x) f1 R ), 0], max[λ 2(f 2 (x) f2 R ), 0], } max[λ 3 (f 3 (x) f3 R ), 0] } = max {λ 1 (f 1 (x) f1 I ),λ 2(f 2 (x) f2 I ),λ 3(f 3 (x) f3 I ) ; Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

14 Directing interactive solution process q = 2 sr {max[λ 2 (f(x),λ) = max 1 (f 1 (x) f1 R ), 0]+max[λ 2(f 2 (x) f2 R ), 0], max[λ 1 (f 1 (x) f1 R ), 0]+max[λ 3(f 3 (x) f3 R ), 0], } max[λ 2 (f 2 (x) f2 R ), 0]+max[λ 3(f 3 (x) f3 R ), 0] { = max λ 1 (f 1 (x) f1 I )+λ 2(f 2 (x) f2 I ), λ 1 (f 1 (x) f I 1 )+λ 3(f 3 (x) f I 3 ), λ 2 (f 2 (x) f I 2 )+λ 3(f 3 (x) f I 3 ) }; Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

15 Directing interactive solution process q = 3 sr {max[λ 3 (f(x),λ) = max 1 (f 1 (x) f1 R ), 0]+max[λ 2(f 2 (x) f2 R ), 0]+ } max[λ 3 (f 3 (x) f3 R ), 0] = λ 1 (f 1 (x) f1 I )+λ 2(f 2 (x) f2 I )+λ 3(f 3 (x) f3 I ); Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

16 Directing interactive solution process Corner coordinates q = 1 (α/λ 1 + f1 I,α/λ 2 + f2 I,α/λ 3 + f3 I) Top vertex coordinates q = 2 (α/2λ 1 + f1 I,α/2λ 2 + f2 I,α/2λ 3 + f3 I) Normal vector q = 3 (λ 1,λ 2,λ 3 ) Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

17 Directing interactive solution process Classification of the objective functions f i values are desired to be improved (i.e. decreased), f i values may be impaired (i.e. increased). A step of interactive process { λ c 1 i + 0.5λ c 1 λ c i = λ c 1 i i /c if f i is desired to be decreased 0.5λ c 1 i /c if f i is desired to be increased i = 1, 2, 3, c N Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

18 Problem parameters Three-objective median location problem A set of sectors that should be evacuated S, S = n, n N Each region i S, i = 1,...,n has a i habitants The number of candidate shelters E, E = l, l N The number of shelters to be located p E Path length from a sector i to a shelter j d ij R, j = 1,...,l Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

19 Problem parameters Three-objective median location problem Risk associated with a path from a sector i to a shelter j r ij (0, 1) Risk associated with a shelter j r j (0, 1) Capacity (number of individuals) allowed in a jth candidate shelter K j N Minimum number of individuals required for opening a jth shelter k j N Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

20 Three-objective median location problem Three objective p-median problem The distance required for the population to reach its shelter n l min a i d ij x ij i=1 j=1 The risk faced by the population as it travels to its shelter n l min a i r ij x ij i=1 j=1 Total risks associated with staying in the shelter n l min a i r j x ij i=1 j=1 Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

21 Constraints Three-objective median location problem l x ij = 1, i = 1,...,n j=1 (one evacuation path is chosen for each sector, with n the number of sectors) n a i x ij K j y j, j = 1,...,l i=1 (the maximum capacity for shelter j is not exceeded, with l the total number of candidate shelters) n a i x ij k j y j, j = 1,...,l i=1 (the minimum number of individuals required to open shelter j before it is opened) Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

22 Constraints Three-objective median location problem l y j = p j=1 (ensures p of the l candidate shelters are opened) x ij {0, 1}, i = 1,...,n, j = 1,...,l y j {0, 1}, j = 1,...,l Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

23 Numerical experiment Data set Distance matrix (d ij ) = here length d ij = 0 means that there is no path from a sector i to a shelter j; Number of individuals in each sector a = (5, 18, 21, 19, 29) Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

24 Numerical experiment Data set Risk associated with path from a sector i to a shelter j (r ij ) = Risk associated with a shelter j Capacity of a shelter j r = (0.2936, , , , ) K = (26, 25, 65, 40, 47) Minimum number of individuals required for opening shelter j k = (9, 8, 6, 9, 9) Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

25 Numerical experiment f I = (1353, , ) λ 1 = 1/f1 I, λ 2 = 1/f2 I, λ 3 = 1/f3 I λ = ( , , ) The most preferred solution λ = (0.01, , 20) MPS = (1487, , ) Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

26 Numerical experiment Results for q=1 iteration lambda mean value test , , , , , , , , , , , , Results for q=2 iteration lambda mean value test , , , , , , , , Results for q=3 iteration lambda mean value test , , , , , , , , , , , , , , , , , , , , , , , , , , , , Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

27 References References K. Miettenen, Nonlinear multiobjective optimization, Kluwer Academic Publeshers, Boston, Yu. Nikulin, K. Miettinen and M. M. Mäkelä, A new achievement scalarizing function based on parameterization in multiobjective optimization, OR Spectrum, 34 (1) (2012) A. P. Wierzbicki, The use of reference objectives in multiobjective optimization. In: G. Fandel and T. Gal (eds) Multiple criteria decision making theory and applications. MCDM theory and applications proceedings. Lecture notes in economics and mathematical systems, 177 (1980), Springer, Berlin, Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

28 Future Research Future Research M. Lugue, K. Miettinen, A. Ruiz, F. Ruiz, A two slope achievement scalarizing function for interactive multiobjective optimization, Computers and Operations Research 39 (2012), Self-tuning, no test for reference point achievability Tackling the case of strictly dominated reference point. Testing efficiency of bundle method based nondifferentiable solver. Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29

29 Future Research Thank You for Your Interest! Nikulin & Karelkina (UTU) PASFs in ICP Malaga, 17 June / 29