duality-2

duality-2 - Duality Primal Problem Max Z = 3X1 5X2 st X1 4...

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Duality Primal Problem Dual Problem Max Z = 3X1 + 5X2 st X1 4 2X2 12 3X1 + 2X2 ≤ 18 Min W = 4Y1 + 12Y2 + 18Y3 Y1 + 3Y3 ≥ 3 2Y2 + 2Y3 ≥ 5

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In matrix notation, in general terms Primal Dual Maximize Z = CX Min W = Yb st AX ≤ b YA ≥ C Economic Interpretation of solution variables under duality: The Yi values are “shadow prices.” They tell you how much the original objective function would increase if you had one more unit of the corresponding limited resource.
If the primal has n variables, the dual has n constraints. If the primal has m constraints, the dual has m variables. Each variable in the primal "maps" to a constraint in the dual. Each constraint in the dual "maps" to a variable in the primal. The shadow price in the primal is the value of "relaxing" a constraint by one unit. It is the value of the corresponding variable in the dual. The reduced cost in the primal is how much the objective function decreases if you let a non-basic variable enter at a level of one unit. It is the surplus (slack) in the dual.

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Facts about Primals and Duals 1) The optimal values of the primal and dual objective values are equal. 2) If a choice variable (Yi) in the dual is nonzero, then the corresponding slack or surplus variable in that row of the primal is zero. If Yi is zero, then Si is nonzero. That is Yi*Si = 0 And at least one of the two is nonzero. 3)
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duality-2 - Duality Primal Problem Max Z = 3X1 5X2 st X1 4...

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