4.7 Sensitivity analysis in Linear Programming
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1 4.7 Sensitivity analysis in Linear Programming Evaluate the sensitivity of an optimal solution with respect to variations in the data (model parameters). Example: Production planning max n j n a j p j ij x x j j b i x j profit availability i-th resource i m j n E. Amaldi Foundations of Operations Research. Politecnico di Milano
2 4.7. Geometric interpretation Example: x max x + x ½ x + x x + x 4 x, x 4 optimal solution ½ x + x = 4/ x * = 4/ with z * = 8/ 4 x x + x = 4 E. Amaldi Foundations of Operations Research. Politecnico di Milano
3 Variation of the objective function coefficients x If c x + x = z with ½ c 4 optimal solution x * 4/ = with z 4/ * = 8/ x + x = ½ x + x = 4 x x + x = 4 max x + x ½ x + x x + x 4 x, x E. Amaldi Foundations of Operations Research. Politecnico di Milano
4 Variations of the right-hand-side terms 4 x x * / x * =, z * = / + 8/ = / 8/ optimal solution x * ½ x + x = + 4/ 4/ x * = with z * = 8/ 4 x x + x = 4 max x + x ½ x +x + x + x 4 x, x E. Amaldi Foundations of Operations Research. Politecnico di Milano 4
5 Definition: The shadow price of the i-th resourse = maximum price the company is willing to pay to buy an additional unit of i-th resource. Shadow price of the first resource = / 8/ = / = z * E. Amaldi Foundations of Operations Research. Politecnico di Milano
6 4.7. Sensitivity analysis: algebraic form Evaluate the sensitivity of an optimal solution when the model parameters (c j, a ij, b i ) vary. Given min c T x and optimal basic solution x * (P) Ax = b x x B* = B - b x N* = Within which limits the basis B remains optimal? Conditions: ) B - b feasibility ) c T N = c T N c T B B - N T optimality Topic of the 4th comupter laboratory session. E. Amaldi Foundations of Operations Research. Politecnico di Milano 6
7 Variation of right-hand-side terms b b+ δ k e k with e k = k k n The basis B with the basic solution remains optimal as long as x * = B- (b+ δ k e k ) c N do not vary B - (b+ δ k e k ) B - b -δ k B - e k m inequalities that define an interval of variation for δ k E. Amaldi Foundations of Operations Research. Politecnico di Milano 7
8 The basis B remains optimal, but the optimal basic feasible solution changes. The objective function values goes from c T B B - b to c T B B - (b+ δ k e k ) z * = c T B B - (δ k e k ) = δ k y * k y *T shadow price! z * b k optimal value of the dual variable E. Amaldi Foundations of Operations Research. Politecnico di Milano 8
9 Variation of the cost coefficients Given c c+ δ k e k abasisb remains optimal as long as c T N c T N c T B B - N In such a case, the optimal basic (feasible) solution does not change: x * B = B - b x * N = E. Amaldi Foundations of Operations Research. Politecnico di Milano 9
10 If x k is a nonbasic variable c T N = c T N c T B B - N T con c T B = c T B c T N = (c T N + δ k e T k) c T B B - N = (c T N c T B B - N ) + δ k e T k = c T N + δ k e T k T δ k -c k Reduced cost = max decrease of c k for which B remains optimal ( larger decrease c k < ) z * = since z * = c T B B - b= c T B x * B E. Amaldi Foundations of Operations Research. Politecnico di Milano
11 If x k is a basic variable c T N = c T N c T B B - N T with c T N = c T N c T N = c T N (c T B + δ k e T k) B - N = (c T N c T B B - N ) - δ k e T k B - N c T N δ k T k T c T N δ k T k where T k k-the row of B - N n-m inequalities that define a variation interval for δ k E. Amaldi Foundations of Operations Research. Politecnico di Milano
12 In such a case z * = δ k e T k B - b= δ k x * k x * B z * c k Similar analysis for the coefficients a ij E. Amaldi Foundations of Operations Research. Politecnico di Milano
13 Example: sensitivity analysis Initial tableau: initial basis min c T x Ax = b x x x x x 4 x x 6 -z x 4 x x 6 6 b I E. Amaldi Foundations of Operations Research. Politecnico di Milano
14 Optimal tableau: c T N c T N c T B B - N pivoting operations premultiply by B - x x x x 4 x x 6 -z 7/ 7/ 6/ / x / / / -/ x 8/ / -/ / x B - b B - N B - I = B - E. Amaldi Foundations of Operations Research. Politecnico di Milano 4
15 E. Amaldi Foundations of Operations Research. Politecnico di Milano Thus x B = 6 x x x B B B - b= 4 8 c B = x N = 4 x x x c N = y T = c T B B - = 6 Optimal dual solution:
16 E. Amaldi Foundations of Operations Research. Politecnico di Milano 6 Variations of b : b b+ δ k e k e k = k x * B = B- (b+ δ k e k ) remains optimal as long as B - (b+ δ k e k ) B - b - δ k B - e k k =,, k = : δ 4 8 -/ δ 4 k = : -4 δ k = : -4 δ
17 E. Amaldi Foundations of Operations Research. Politecnico di Milano 7 Variations of c : c c+ δ k c k For x k basic, B remains optimal if and only if c T N δ k e T k B - N c N = 6 7 B - N = reduced cost δ -7/ δ 4-6/ δ -/ For x k nonbasic, B remains otpimal if and only if δ k -c k
18 k = : δ k = : k = 6: δ / / δ 6 7/ E. Amaldi Foundations of Operations Research. Politecnico di Milano 8
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