Simplex Method and Reduced Costs, Duality and Marginal Costs Frdric - PowerPoint PPT Presentation

Simplex Method and Reduced Costs, Duality and Marginal Costs Frédéric Giroire FG Simplex 1/17

** Simplex Method and Reduced Costs, Strong Duality Theorem ** FG Simplex 2/17

Simplex - Reminder Start with a problem written under the standard form. + + Maximize 5 x 1 4 x 2 3 x 3 Subject to : + + ≤ 2 x 1 3 x 2 x 3 5 4 x 1 + + 2 x 3 ≤ 11 x 2 + + ≤ 3 x 1 4 x 2 2 x 3 8 ≥ x 1 , x 2 , x 3 0 . FG Simplex 3/17

Simplex - Reminder Write the Dictionary: = − − − x 4 5 2 x 1 3 x 2 x 3 = − − − x 5 11 4 x 1 x 2 2 x 3 = − − − x 6 8 3 x 1 4 x 2 2 x 3 = + + z 5 x 1 4 x 2 3 x 3 . Basic variables: x 4 , x 5 , x 6 , variables on the left. Non-basic variable: x 1 , x 2 , x 3 , variables on the right. A dictionary is feasible if a feasible solution is obtained by setting all non-basic variables to 0. FG Simplex 4/17

Simplex - Reduced Costs Write the Dictionary: = − − − x 4 5 2 x 1 3 x 2 x 3 = − − − x 5 11 4 x 1 x 2 2 x 3 = − − − x 6 8 3 x 1 4 x 2 2 x 3 = + + z 5 x 1 4 x 2 3 x 3 . We call Reduced Costs the coefficients of z . The reduced cost of x 1 is 5, of x 2 is 4 and of x 3 is 3. Reminder: If all reduced cost are non-positive, the solution is optimal and the simplex algorithm stops. FG Simplex 5/17

Simplex - Reduced Costs Relationship between reduced costs, c = ( c 1 ,..., c n ) and optimal solution of the dual problem π = ( π 1 ,..., π m ) . If we consider a general LP: ∑ n Maximize j = 1 c j x j ∑ n ≤ ( i = 1 , 2 , ··· , m ) Subject to: j = 1 a ij x j b i ≥ ( j = 1 , 2 , ··· , n ) x j 0 Lemma: When the simplex algorithm finishes, we have: m ∑ c j = c j − π i A ij i = 1 FG Simplex 6/17

Simplex - Reduced Costs We consider a general LP: ∑ n Maximize j = 1 c j x j ∑ n ≤ ( i = 1 , 2 , ··· , m ) Subject to: j = 1 a ij x j b i (1) ≥ ( j = 1 , 2 , ··· , n ) x j 0 We introduce the following notations, A and B. c T x Maximize Ax = b Subject to: x ≥ 0 The method of the simplex finishes with an optimal solution x and an associated basis. Let B ( 1 ) ,..., B ( m ) be the indices of basic variables. We define B = [ A B ( 1 ) ... A B ( m ) ] the matrix associated to the basis. We have x B = B − 1 b FG Simplex 7/17

Simplex - Reduced Costs c T x Maximize Subject to: Ax = b x ≥ 0 By studying what happens during a step of the simplex method, we can get the following expression for the reduced cost of variable x j c j = c j − c T B B − 1 A j . FG Simplex 8/17

Simplex - Reduced Costs When the method of the simplex finishes, the reduced costs are non-positive. c T − c T B B − 1 A ≤ 0 T . Let π be such that π T = c T B B − 1 FG Simplex 9/17

Simplex - Reduced Costs We get c T − c T B B − 1 A ≤ 0 T . c T − π T A ≤ 0 T . π T A ≥ c T . A T π ≥ c . ⇒ π is a feasible solution of the dual problem: π T b Minimize A T π ≥ c Subject to: (2) π ≥ 0 FG Simplex 10/17

Simplex - Reduced Costs Moreover, the value of p equals the value of the optimal value of the primal: π T b = c T B x B = c T x B B − 1 b = c T ⇒ π is an optimal solution of the dual problem (by the weak duality theorem). Theorem [Strong Duality]: If the primal problem has an optimal solution, x ∗ = ( x ∗ 1 ,..., X ∗ n ) , then the dual also has an optimal solution, y ∗ = ( y ∗ 1 ,..., y ∗ n ) , and j = ∑ c j x ∗ b i y ∗ ∑ i . j i FG Simplex 11/17

Simplex - Reduced Costs Moreover, the value of p equals the value of the optimal value of the primal: π T b = c T B x B = c T x B B − 1 b = c T ⇒ π is an optimal solution of the dual problem (by the weak duality theorem). Theorem [Strong Duality]: If the primal problem has an optimal solution, x ∗ = ( x ∗ 1 ,..., X ∗ n ) , then the dual also has an optimal solution, y ∗ = ( y ∗ 1 ,..., y ∗ n ) , and j = ∑ c j x ∗ b i y ∗ ∑ i . j i FG Simplex 11/17

Simplex - Reduced Costs Moreover, the value of p equals the value of the optimal value of the primal: π T b = c T B x B = c T x B B − 1 b = c T ⇒ π is an optimal solution of the dual problem (by the weak duality theorem). Theorem [Strong Duality]: If the primal problem has an optimal solution, x ∗ = ( x ∗ 1 ,..., X ∗ n ) , then the dual also has an optimal solution, y ∗ = ( y ∗ 1 ,..., y ∗ n ) , and j = ∑ c j x ∗ b i y ∗ ∑ i . j i FG Simplex 11/17

Simplex - Reduced Costs The Reduced Cost is - the amount by which an objective function coefficient would have to improve before it would be possible for a corresponding variable to assume a positive value in the optimal solution. FG Simplex 12/17

** Dual Variables and Marginal Costs ** FG Simplex 13/17

Signification of Dual Variables ∑ n ∑ m Max j = 1 c j x j Min i = 1 b i y i ∑ n ∑ m ≤ ( i = 1 , ··· , m ) ≥ ( j = 1 , ··· , n ) S. t.: j = 1 a ij x j b i S. t.: i = 1 a ij y i c j ≥ ( i = 1 , ··· , m ) ≥ 0 ( j = 1 , ··· , n ) y i 0 x j Signification can be given to variables of the dual problem: “The optimal values of the dual variables can be interpreted as the marginal costs of a small perturbation of the right member b .” FG Simplex 14/17

Signification of Dual Variables ∑ n ∑ m Maximize j = 1 c j x j Minimize i = 1 b i y i ∑ n ∑ m ≤ ( i = 1 , 2 , ··· , m ) ≥ ( j = 1 , 2 , ··· , n ) Subject to: j = 1 a ij x j b i Subject to: i = 1 a ij y i c j ≥ ( i = 1 , 2 , ··· , m ) ≥ ( j = 1 , 2 , ··· , n ) y i 0 x j 0 Dimension analysis for a factory problem: • x j : production of a product j (chair, ...) • b i : available quantity of resource i (wood, metal, ...) • a ij : unit of resource i per unit of product j • c j : net benefit of the production of a unit of product j FG Simplex 15/17

Signification of Dual Variables ∑ n ∑ m Maximize j = 1 c j x j Minimize i = 1 b i y i ∑ n ∑ m ≤ ( i = 1 , 2 , ··· , m ) ≥ ( j = 1 , 2 , ··· , n ) Subject to: j = 1 a ij x j b i Subject to: i = 1 a ij y i c j ≥ ( i = 1 , 2 , ··· , m ) ≥ ( j = 1 , 2 , ··· , n ) y i 0 x j 0 Dimension analysis for a factory problem: • x j : production of a product j (chair, ...) • b i : available quantity of resource i (wood, metal, ...) • a ij : unit of resource i per unit of product j • c j : net benefit of the production of a unit of product j unit of resource i/unit of product j euros/unit of product j euros/unit of resource i a y + ... + a y > c j n 1 j 1 j n → y i euro by unit of resource i . Unit value of resource i . FG Simplex 15/17

Signification of Dual Variables Theorem: If the LP admits at least one optimal solution, then there exists ε > 0, with the property: If | t i | ≤ ε ∀ i = 1 , 2 , ··· , m , then the LP ∑ n Max j = 1 c j x j ∑ n ≤ b i + t i ( i = 1 , 2 , ··· , m ) Subject to: j = 1 a ij x j (3) ≥ ( j = 1 , 2 , ··· , n ) . x j 0 has an optimal solution and the optimal value of the objective is m z ∗ + y ∗ ∑ i t i i = 1 with z ∗ the optimal solution of the initial LP and ( y ∗ 1 , y ∗ 2 , ··· , y ∗ m ) the optimal solution of its dual. FG Simplex 16/17

To be remembered • Definition of the Reduced Costs = − − − x 4 5 2 x 1 3 x 2 x 3 = 11 − 4 x 1 − − 2 x 3 x 5 x 2 = − − − x 6 8 3 x 1 4 x 2 2 x 3 = + + z 5 x 1 4 x 2 3 x 3 . • If all reduced cost are non-positive, the solution is optimal and the simplex algorithm stops. • Relationship between reduced costs, c = ( c 1 ,..., c n ) and optimal solution of the dual problem π = ( π 1 ,..., π m ) . When the simplex algorithm finishes, we have: c = c T − π T A FG Simplex 17/17