Introduction to Operations Research

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