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Introduction to Operations Research

Duality Theory. Introduction to Operations Research. Every linear programming problem has associated with it another linear programming problem called the dual Major application of duality theory is in the interpretation and implementation of sensitivity analysis.

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Introduction to Operations Research

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  1. Duality Theory Introduction to Operations Research

  2. Every linear programming problem has associated with it another linear programming problem called the dual • Major application of duality theory is in the interpretation and implementation of sensitivity analysis. • Original linear programming problem will be referred to as the primal problem and a dual problem will be introduced. • Will assume that primal problem is in standard form. Duality theory

  3. Primal problem Maximize Z = c1x1 + … + cnxn Subject to. a11x1 + … + a1nxn ≤ b1 am1x1 + … + amnxn ≤ bm x1 ≤ 0, …, xn ≤ 0 Maximize Z = cx Subject to Ax ≤ b x ≥ 0

  4. Setting Up the Dual Problem • Coefficients in the objective function of the primal problem are the right-hand sides of the functional constraints in the dual problem • Right-hand sides of the functional constraints in the primal problem are the coefficients in the objective function of the dual problem • Coefficients of a variable in the functional constraints of the primal problem are the coefficients in a functional constraint of a dual problem • Parameters for functional constraints in either problem are the coefficients of a variable in the other problem • Coefficients in the objective function of either problem are the right-hand sides for the other problem Primal Problem Dual Problem Max Z = cx Subject to Ax ≤ b and x ≥ 0 Min W = yb Subject to yA ≥ c and y ≥ 0

  5. Dual Problem Primal/Dual Problems for Wyndor Glass Co. Maximize Z = 3x1 + 5x2 Subject to. x1 ≤ 4 2x2 ≤ 12 3x1 + 2x2 ≤ 18 Minimize W = 4y1 + 12y2 + 18y3 Subject to. y1 + 3y3 ≥ 3 2y2 + 2y3 ≥ 5 Primal Problem Dual Problem

  6. Dual Problem Primal/Dual Problems for Wyndor Glass Co Matrix form: Max Min

  7. Weak Duality Property • If x is a feasible solution for the primal problem and y is a feasible solution for the dual problem, then cx ≤ yb. • Wyndor Glass Co. Example: x1 = 3, x2 = 3, then Z = 24, y1 = 1, y2 = 1, y3 = 2, then W = 52 • Strong Duality Property • If x* is an optimal solution for the primal problem and y* is an optimal solution for the dual problem, then cx* = y*b. • Wyndor Glass Co. Example: x1 = 3, x2 = 3, then Z = 24, y1 = 1, y2 = 1, y3 = 2, then W = 52 • Symmetry Property • The dual of the dual problem is the primal problem Primal-Dual Relationships

  8. Complementary Solutions Property At each iteration, the simplex method simultaneously identifies a CPF solution x for the primal problem and a complementary solution y for the dual problem where cx = yb. If x is not optimal for the primal problem, then y is not feasible for the dual problem Wyndor Glass Co. Example: x1 = 0, x2 = 6, then Z = 30, y1 = 0, y2 = 5/2, y3 = 0, then W = 30. This is feasible for primal problem but violates constraint in dual problem Complementary Optimal Solutions Property At the final iteration, the simplex method simultaneously identifies an optimal solution for the x* primal problem and a complementary optimal solution y* for the dual problem cx* = y*b. The y* contains the shadow prices for the primal problem Primal-Dual Relationships

  9. If one problem has feasible solutions and a bounded objective function (optimal solution), then so does the other problem and both the weak and strong duality properties are applicable • If one problem has feasible solutions and an unbounded objective function (no optimal solution), then the other problem has no feasible solutions. • If one problem has no feasible solutions, then the other problem has either no feasible solutions or an unbounded objective function. Duality Theorem

  10. VariableDescription xj Level of activity j cj Unit profit from activity j Z Total profit from all activities bi Amount of resource i available aij Amount of resource i consumed by each unit of activity j yi Shadow price for resource I W Value of Z Economic Interpretation

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