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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