Answer Happy

do the question above , example 17.1 given below which use in
this question
2. Consider the following multiobjective problem: x₁ + 2x₂, 2x₁ - x₂ maximize: subject to: X1, X₂ ≥ 0 2x₁ + x₂ ≤ 6 X₁ X₂ ≥ 1 a. Find the feasible decision set and payoff set (as Example 17.1 on p. 129)! Illustrate both! b. Solve the problem graphically with the weighting method with equal weights! c. Use & -constraint method assuming that the second objective has to be equal to or greater than 1!
Example 17.1 Decision space. maximize s. to x₁ + x₂, 1-2 #1, #2 20 3x1 + x₂ ≤ 4 21 +37₂ 4 129
X2 (0,4) (0, 4 3 (0,0) ,0) X1 Let f₁=2₁+2₂ f₂= 21-2₂ (+)x₁=¹ (-) 22 = 12² Construints: 3/1 + f2 2 +$2 2 + Payoffset: + X₁ X ₂ = 0 (1,1) - X ₂ = 1 (4,0) Game theory XI X1 + X₂ = 1 + = 0 X 2₂ f₁ + f₂20 ⇒ f22-f 2 h=1₂0 ⇒h<h 2 ≤43f1 +3f2 +f1-f2 ≤ 8 ⇒ 4f₁+2f2 ≤8 ⇒ 2f1 + f2 ≤4 2 ≤4f₁ + f2+3f₁-3f2 ≤8 ⇒ 4f1-2f₂ ≤8 ⇒ 2f1-f2 ≤ 4 2
Game theory fi $2 (0,4) (0,- 4 V Definition 17.4 A point f H is weakly nondominated, if there is no ƒ € H such that fi> fi for all i. Definition 17.5 A point f H is (strongly) nondominated, if there is no f H such that ≥f for all i, with strict inequality for at least one i. Example 17.2 In previous example all points of the linear segment connecting points (2,0) and (,) are weakly and strongly nondominated. V Note. If f is strongly nondominated, then it is also weakly nondominated. 17.1 Existence of nondominated solutions Example 17.3 No nondominated solution erists: 21,22 maximize s. to #1, #2 20 131