Goal Programming Approach for Solving Interval MOLP Problems

Transcription

1 In the name of Absolute Power & Absolute Knowledge Goal Programming Approach for Solving Interval MOLP Problems S Rivaz, MA Yaghoobi and M Hladí 8th Small Worshop on Interval Methods Charles University Prague, Czech Republic, June 9-11, 2015

2 Motivation In many real-world situations, the parameters of decisionmaing problems are not deterministic In this context, it is convenient to extend the traditional mathematical programming models incorporating their uncertainty One of the tools for tacling vagueness is interval programming in which every uncertain parameter is determined with a closed interval Moreover, many real-world problems inherently impose the need to investigate multiple and conflicting objective functions This paper treats multiobjective linear programming problems with interval objective functions coefficients and target intervals 2 / 24 In the name of Absolute Power & Absolut

3 MOLP Problems with Interval Objective Functions Coefficients and Target Intervals c j x j = t, = 1,, p, st Ax b, c j C j, = 1,, p, j = 1,, n, t T, = 1,, p, x 0, where C j is the closed interval [c l j, cu j ] representing a region the coefficients c j possibly tae T is the closed interval [t l, tu ] representing a region the target value t possibly tae The constraints of the problem are as in conventional MOLP problem 3 / 24 In the name of Absolute Power & Absolut

4 The Wor of Inuiguchi & Kume (1991) Possible Deviation D = [d l, du ] of [ n cl j x j, n cu j x j] from T [t l n cu j x j, t u n cl j x j] if t l n cu j x j 0, D = [0, ( n cu j x j t l ) (t u n cl j x j)] if t l n cu j x j < 0 < t u n cl j x j, [ n cl j x j t u, n cu j x j t l ] if t u n cl j x j 0, where denotes the maximum 4 / 24 In the name of Absolute Power & Absolut

5 0 λ 1, w 0 and p =1 w = 1 5 / 24 In the name of Absolute Power & Absolut

6 The Wor of Inuiguchi & Kume (1991) st min λ p =1 c u j x j + d l w (d l + du+ ) + (1 λ)vl dl+ = t l, = 1,, p, c l j x j + d u d u+ = t u, = 1,, p, d l, dl+ d l, du, du+ Ax b, + du+ v l, = 1,, p, x j 0, j = 1,, n,, vl 0, = 1,, p 6 / 24 In the name of Absolute Power & Absolut

7 The Wor of Inuiguchi & Kume (1991) st d l min λ c u j x j + d l p w v + (1 λ)v u =1 dl+ = t l, = 1,, p, c l j x j + d u d u+ = t u, = 1,, p, Ax b, d l+ v, = 1,, p, d u v, = 1,, p, v v u, = 1,, p, x j 0, j = 1,, n, v u 0,, du+, v 0, = 1,, p, dl+, du 7 / 24 In the name of Absolute Power & Absolut

8 The Wor of Inuiguchi & Kume (1991) Necessary Deviation E = [e l, eu ] of [ n cl j x j, n cu j x j] from T [t l n cl j x j, t u n cu j x j] if t u n cu j x j t l n cl j x j 0, [0, (t u n cu j x j) ( n cl j x j t l )] if t l n cl j x j < 0 < t u n cu j x j, E = [ n cu j x j t u, n cl j x j t l ] if t l n cl j x j t u n cu j x j 0, [ n cl j x j t l, n cu j x j t u ] n if cu j x j t u n cu j x j t l 0, [0, ( n cu j x j t u ) (t l n cl j x j)] n if cl j x j t l < 0 < n cu j x jt u, [t u n cu j x j, t l n cl j x j] n if cl j x j t l n cu j x j t u 0 8 / 24 In the name of Absolute Power & Absolut

9 The Wor of Inuiguchi & Kume (1991) st min λ ((e l p =1 w ((e l + eu+ ) (e l+ c l j x j + e l + eu )) + (1 λ)ul el+ = t l, = 1,, p, c u j x j + e u e u+ = t u, = 1,, p, Ax b, + eu+ ) (e l+ + eu )) ul, = 1,, p, x j 0, j = 1,, n, e l, el+, eu, eu+, ul 0, = 1,, p, where denotes the minimum 9 / 24 In the name of Absolute Power & Absolut

10 The Wor of Inuiguchi & Kume (1991) st e l min λ c l j x j + e l p w u + (1 λ)u u =1 el+ = t l, = 1,, p, c u j x j + e u e u+ = t u, = 1,, p, e l Ax b, + el+ u, = 1,, p, e u + e u+ u, = 1,, p, u u u, = 1,, p, x j 0, j = 1,, n, u u 0,, eu+, u 0, = 1,, p, el+, eu 10 / 24 In the name of Absolute Power & Absolut

11 Our Aim Minimizing the total distance of planned intervals, [ n cl j x j, n cu j x j], from target intervals, [t l, tu ], = 1,, p In order to do that, we use the following distance concept between two intervals 11 / 24 In the name of Absolute Power & Absolut

12 The Distance Between Two Intervals Definition Let [a l, a u ] and [b l, b u ] be two intervals Then, the distance between them is defined as (Moore, 2009): d([a l, a u ], [b l, b u ]) = max{ a l b l, a u b u } It is easy to chec that the distance d, defined in the above, introduces a metric on the space of closed intervals in the set of real numbers 12 / 24 In the name of Absolute Power & Absolut

13 The Distance Between the th Planned Interval and Its Target Interval ( = 1,, p) D = d([ c l jx j, c u jx j ], [t l, t u ]) = max( c l jx j t l, c u jx j t u ), = 1,, p, Using the deviational variables d l, dl+, du and d u+ as: c l jx j + d l d l+ = t l ; D can be written as follows: c u jx j + d u d u+ = t u ; D = max( d l d l+, d u d u+ ), = 1,, p 13 / 24 In the name of Absolute Power & Absolut

14 Our Model for Solving an Interval MOLP Problem with Target Intervals st min λ c l j x j + d l p w v + (1 λ)u (1) =1 dl+ = t l, = 1,, p, c u j x j + d u d u+ = t u, = 1,, p, d l Ax b, dl+ v, = 1,, p, d u d u+ v, = 1,, p, 14 / 24 In the name of Absolute Power & Absolut

15 Continued d l, dl+, du v u, = 1,, p, d l dl+ = 0, = 1,, p, d u du+ = 0, = 1,, p, x j 0, j = 1,, n, v 0, = 1,, p,, du+ 0, = 1,, p, u 0 The proposed model can be transformed to a linear programming problem It is obvious that () v, = 1,, p, can be substituted with two linear constraints () v and () v The nonlinear constraints d l dl+ = 0 and d u du+ = 0, = 1,, p, can be eliminated due to the dependency of columns of d l and d u, du+, when the, dl+ Simplex method is used for solving the model 15 / 24 In the name of Absolute Power & Absolut

16 New Model st min λ c l j x j + d l p w v + (1 λ)u (2) =1 dl+ = t l, = 1,, p, c u j x j + d u d u+ = t u, = 1,, p, d l Ax b, + dl+ v, = 1,, p, d u + d u+ v, = 1,, p, v u, = 1,, p, x j 0, j = 1,, n, v 0, = 1,, p, d l, dl+, du, du+ 0, = 1,, p, u 0 16 / 24 In the name of Absolute Power & Absolut

17 The Property of Model (2) Lemma Let model (2) be feasible then it has an optimal solution in which the constraints d l dl+ = 0 and du du+ = 0, = 1,, p, are satisfied 17 / 24 In the name of Absolute Power & Absolut

18 A Relationship Between Models (1) and (2) Theorem The optimal solution of model (1) can be obtained by solving model (2) 18 / 24 In the name of Absolute Power & Absolut

19 A Numerical Example In order to illustrate the proposed model, consider the following interval MOLP problem with interval targets which was proposed by Inuiguchi and Kume (1991): Optimize: st c 1 x 1 + c 4 y 1 = t 1, c 2 x 2 + c 5 y 2 = t 2, c 3 x 3 + c 6 y 3 = t 3, x 1 + x 2 + x 3 9, y 1 + y 2 + y 3 9, c 1 [2, 3], c 4 [5, 7], t 1 [28, 32], c 2 [2, 3], c 5 [1, 3], t 2 [25, 30], c 3 [4, 8], c 6 [2, 3], t 3 [31, 37], x j 0, y j 0, j = 1, 2, 3 19 / 24 In the name of Absolute Power & Absolut

20 Continued Applying the proposed model when λ = 1 2 and w 1 = w 2 = w 3 = 1 3, we have min 1 6 (v 1 + v 2 + v 3 ) u st 2x 1 + 5y 1 + d l 1 d l+ 1 = 28, 2x 2 + y 2 + d l 2 d l+ 2 = 25, 4x 3 + 2y 3 + d l 3 d l+ 3 = 31, 3x 2 + 7y 2 + d u 1 d u+ 1 = 32, 3x 2 + 3y 2 + d u 2 d u+ 2 = 30, 8x 2 + 3y 2 + d u 3 d u+ 3 = 37, x 1 + x 2 + x 3 9, y 1 + y 2 + y 3 9, d l 1 d l+ 1 v 1, 20 / 24 d u din u+ the name ofv Absolute, Power & Absolut

21 Continued d l, dl+ d l 2 d l+ 2 v 2, d u 2 d u+ 2 v 2, d l 3 d l+ 3 v 3, d u 3 d u+ 3 v 3, v u, = 1, 2, 3,, du x j 0, y j 0, j = 1, 2, 3, v 0, = 1, 2, 3,, du+ 0, = 1, 2, 3, u 0 The problem solution is x 1 = 0, x 2 = 45, x 3 = 45, y 1 = 5, y 2 = and y 3 = where D 1 = 3, D 2 = , D 3 = and D(x)= / 24 In the name of Absolute Power & Absolut

22 References Bitran, GR, Linear multiobjective problems with interval coefficients, Management Science, 26, (1980) Chinnec, JW, Ramadan, K, Linear programming with interval coefficients, Journal of Operational Research Society, 51, (2000) Fiedler, M, Nedoma, J, Rohn, J, Zimmermann, K, Linear optimization problems with inexact data, Springer-Verlag, New Yor (2006) Inuiguchi, M, Kume, Y, Goal programming problems with interval coefficients and target intervals, European Journal of Operational Research, 52, (1991) 22 / 24 In the name of Absolute Power & Absolut

23 References Jones, DF, Tamiz, M, Practical goal programming, Springer (2010) Moore, RE, Kearfott, RB, Cloud, MJ, Introduction to interval analysis, Society for Industrial and Applied Mathematics, Philadelphia (2009) Oliveira, C and Antunes, C H, Multiple objective linear programming models with interval coefficients - an illustrated overview, European Journal of Operational Research, 181, (2007) Oliveira, C and Antunes, C H, An interactive method of tacling uncertainty in interval multiple objective linear programming, Journal of Mathematical Sciences, 161, (2009) 23 / 24 In the name of Absolute Power & Absolut

24 Than You!