Introduction to Operations Research Duality Theory

Duality theory 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.

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

Setting Up the Dual 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 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

Dual Problem Primal/Dual Problems for Wyndor Glass Co. Primal Problem 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

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

Primal-Dual Relationships 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. Symmetry Property The dual of the dual problem is the primal problem

Primal-Dual Relationships 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

Duality Theorem 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.

Economic Interpretation Variable Description 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