Operations Research. Multicriteria programming exercise.

Transcription

1 Operations Research Multicriteria programming exercise.

2 The lexicographical method Example: Determine: - graphically set X P of all feasible solutions, - set X N of all non-dominated solutions and - graphically and numerically lexicographically optimal solution of the MLP problem: 2 z 1 = 5x 1 + x z 2 = x 2 5 max z 3 = x 1 + x 2 under conditions x 1 + x 2 4, (1) x 1 2x 2 3, (2) x 1 0, x 2 0. (3),(4) Consider the order of importance: 1. z 3, 2. z 2, 3. z 1.

3 The lexicographical method Figure: MLP problem, graphical view of sets X P, X N

4 The lexicographical method Set X P of all feasible solutions of the MLP problem is quadrangle OABC, X P = 4-angleOABC, where O = [0, 0], A = [3, 0], B = [11/3, 1/3], C = [0, 4]. The set X P is marked (by crosshatch) in Figure. There are also marked normal vectors n 1, n 2 of objective functions. In solving problem MLP by lexicographical method first determines the set X 1 of all optimal solutions with respect to z 3 (the most important objective function). Set X 1 a whole line segment BC: X 1 = BC. In addition, we are looking at this line segment a maximum of the objective function z 2 (the second the most important objective function). This maximum becomes a single point, thus a set X 2 = C is a one-element and therefore it is the only compromise solution. We write:

5 The lexicographical method X opt,lex = C = (0, 4); z = (z 1(C), z 2(C), z 2(C)) = (4, 4, 12). Because of the solutions both monocriterial problems (due to the objective function z 1, respectively z 2), it is obvious that a set X N of all non-dominated solutions of MLP problemis is a whole line segment BC: X N = BC. Graphical views of sets X P, X N, as well as the compromise solution, see Figure.

6 The lexicographical method Numerically we proceed as follows: We convert inequalities to equations by introducing slack variables x 1, x 2, annuls a objective equation, the other is evident from the table. First, of course, maximize the most important objective function.

7 Z.p. x 1 x 2 x 1 x 2 b i p = b i a ik pro a ik > 0 x x 2 (1) z max 3 z max 2 z max 1 x 1 0 (3) x z max 3 z max 2 z max x x ( 1 ) z MAX z max z MAX 1 x x z max 3 z MAX 2 z MAX

8 The lexicographical method Of course, even numerically, we came to the same compromise solution as graphically: X opt = C = (0, 4); z = (z 1(C), z 2(C), z 2(C)) = (4, 4, 12).

9 Example: Determine: - graphically set X P of all feasible solutions, - set X N of all non-dominated solutions and - compromise solution by the method of aggregation of objective functionsof the MLP problem:» z1 = 3x 1 + x 2 max z 2 = 10x 1 x 2 under conditions 3x 1 + 2x 2 24, (1) x 1 + x 2 9, (2) x 1 2, (3) x 1 0, x 2 0. (4), (5) Consider both of the objective function equally important.

10 Solution: The set X P of all feasible solutions of the MLP problem is a 5-angle ABCDE, X P = 5-angleABCDE, where A = [2, 0], B = [8, 0], C = [6, 3], D = [4, 5], E = [2, 5]. The set X N of all non-dominated solutions of the MLP problem is broken line B C D E A, ie unification of line segments X N = BC CD DE EA. The set X opt of all compromised solutions of the MLP problem is a line segment AE, because the aggregated objective function for weights v 1 = v 2 = 0, 5 is z = 7 x1 max. Thus, 2 X opt = AE.

11 Example: Solve the MLP problem by the method of goal programming: " # z1 = x 1 max z 2 = x 2 under conditions x 1 + 2x 2 6, (1) 2x 1 + x 2 6, (2) x 1 0, x 2 0, (3),(4) A target values take the ideal values of objective functions.

12 Solution: The ideal values of objective functions, and thus also target values are: h 1 = z 1 = z 1,max = 3, h 2 = z 2 = z 2,max = 3. Compromised solution of MLP problem is: x opt = (2, 2), and z opt = (2, 2).

13 Example: MLP problem " z1 = x 1 + 2x 2 # z 2 = x 1 under conditions 2x 1 x 2 12, (1) 2x 1 + x 2 20, (2) x 1 0, x 2 0, (3),(4) solve by method of goal programming for h 1 = 40; h 2 = 8.

14 Solution: Compromised solution of MLP problem is: x opt = (0, 20), and z opt = (40, 0).

15 Operations Research Multicriteria programming exercise.