1. 3 points [You should study the self-study material on the standard forms of LP, avail-...

 

 

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1. 3 points [You should study the self-study material on the standard forms of LP, avail- able in the Canvas/Files.] Put the following linear programming problem in standard form, that is, standard inequality form. (Do not solve it.) minimize 1 - 3x2 - x3 subject to ₁ - 2x2 + x3 = 3 -x1 + x₂ ≥ 1 X1 ≥ 0 2 unconstrained X3 ≥ 1. 2 Reduction to standard form vbmda-1158 LPs in non-standard form can be reduced to standard form. For example: min 3x12x2 + x3 +1 Here are the reduction rules to use: Rule x=a min f = max(-f) max(f+const) max f f(x) ≥a -f(x) ≤-a a and --a x ≥a ⇒ I'=x-a≥0 x≤a ⇒x=a-x>0 No restriction on a → I s.t.1+2 ≥-3 2x1 + x₂ ≤2 1 + 2 + 3 = 4 Τ1 Σ -2 2 ≤ 3 The objective function then becomes: x,x+,x->0 The inequalities become: -3x1 + 2x2x3 = -3(₁-2) +2(3-2)-3+x3 =-3x₁2x2x3 + x3 +12 vilamponi burbusie Changes max-3x1 + 2x2-3-1 max -3x1 + 2x2 3 1-₂ ≤3 x1 + x2 + x3 ≤ 4,-21-22-23 ≤-4 Replace z with a -2, then 12-2 x 20 Replace 22 with 3-2, then x2 ≤3 Replace 23 with 2-3 and 3, ₂20 (the +12 can be dropped) T₁-T2 ≤3 (1-2)-(3-x₂) ≤3 ₁+2 ≤8 2x1 + x2 ≤22(x-2)+(3-x2) ≤ 2 ⇒2x1-x2 ≤3 1 + x2 + x3 ≤ 4(x-2)+(3-₂) + (x - 3) ≤ 4 x₁-x₂+x-x≤3 -21-22-23 ≤-4-(₁-2) +-(3-2)-(-x-x3) ≤-4 -x₁+x₂-x+x≤-3 20 The resulting standard form is: pianot villamponi od tamojt dopraful max - 3x12x2-x3+xz s.t. x₁ + x2 ≤8 2x1-x2 ≤3 x₁-x₂+x-x3 ≤3 - x₁ + x₂-x² + x3 ≤ -3 x1, x2, x3, 20 After finding an optimal solution to this LP problem, we get the optimal solution to the original problem by using the reverse transformations: x₁ = x₁-2 rd & driw dairenosib of 'I sau sw tadt stol x2 = 3-2 19dow) to anot add bas egneds f'assob & axtos mort 18olo yllau ai boen ai ted x3 = x3 x3 odni sotilaupsal moet vos stated as sure will voitonut svitido sda jadi cela stol and am ni nattinw ad neds nao mat viileupas bisbaste od ddiw land of op v usod aktileupe notov oilT colderiny 19to 101 stan smise (eoldalov asala basqqs ew 190) bas min 3x12x2 + x3 + 1 = -max -3x1 + 2x2 - 3 - 1 = 1-max(-3x1 + 2x2x3) 1. 3 points [You should study the self-study material on the standard forms of LP, avail- able in the Canvas/Files.] Put the following linear programming problem in standard form, that is, standard inequality form. (Do not solve it.) minimize 1 - 3x2 - x3 subject to ₁ - 2x2 + x3 = 3 -x1 + x₂ ≥ 1 X1 ≥ 0 2 unconstrained X3 ≥ 1. 2 Reduction to standard form vbmda-1158 LPs in non-standard form can be reduced to standard form. For example: min 3x12x2 + x3 +1 Here are the reduction rules to use: Rule x=a min f = max(-f) max(f+const) max f f(x) ≥a -f(x) ≤-a a and --a x ≥a ⇒ I'=x-a≥0 x≤a ⇒x=a-x>0 No restriction on a → I s.t.1+2 ≥-3 2x1 + x₂ ≤2 1 + 2 + 3 = 4 Τ1 Σ -2 2 ≤ 3 The objective function then becomes: x,x+,x->0 The inequalities become: -3x1 + 2x2x3 = -3(₁-2) +2(3-2)-3+x3 =-3x₁2x2x3 + x3 +12 vilamponi burbusie Changes max-3x1 + 2x2-3-1 max -3x1 + 2x2 3 1-₂ ≤3 x1 + x2 + x3 ≤ 4,-21-22-23 ≤-4 Replace z with a -2, then 12-2 x 20 Replace 22 with 3-2, then x2 ≤3 Replace 23 with 2-3 and 3, ₂20 (the +12 can be dropped) T₁-T2 ≤3 (1-2)-(3-x₂) ≤3 ₁+2 ≤8 2x1 + x2 ≤22(x-2)+(3-x2) ≤ 2 ⇒2x1-x2 ≤3 1 + x2 + x3 ≤ 4(x-2)+(3-₂) + (x - 3) ≤ 4 x₁-x₂+x-x≤3 -21-22-23 ≤-4-(₁-2) +-(3-2)-(-x-x3) ≤-4 -x₁+x₂-x+x≤-3 20 The resulting standard form is: pianot villamponi od tamojt dopraful max - 3x12x2-x3+xz s.t. x₁ + x2 ≤8 2x1-x2 ≤3 x₁-x₂+x-x3 ≤3 - x₁ + x₂-x² + x3 ≤ -3 x1, x2, x3, 20 After finding an optimal solution to this LP problem, we get the optimal solution to the original problem by using the reverse transformations: x₁ = x₁-2 rd & driw dairenosib of 'I sau sw tadt stol x2 = 3-2 19dow) to anot add bas egneds f'assob & axtos mort 18olo yllau ai boen ai ted x3 = x3 x3 odni sotilaupsal moet vos stated as sure will voitonut svitido sda jadi cela stol and am ni nattinw ad neds nao mat viileupas bisbaste od ddiw land of op v usod aktileupe notov oilT colderiny 19to 101 stan smise (eoldalov asala basqqs ew 190) bas min 3x12x2 + x3 + 1 = -max -3x1 + 2x2 - 3 - 1 = 1-max(-3x1 + 2x2x3)

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1 Answer The given linear programming problem is minimize x 1 3x 2 x 3 subject to x ... View the full answer