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.

Maximize Z = c1x1 + … + cnxn

Subject to.

a11x1 + … + a1nxn ≤ b1

am1x1 + … + amnxn ≤ bm

x1 ≤ 0, …, xn ≤ 0

MaximizeZ = cx

Subject toAx ≤ b x ≥ 0

Max Z = cxSubject to Ax ≤ b and x ≥ 0

Min W = ybSubject to yA ≥ c and y ≥ 0

Primal Problem 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

MaximizeZ = 3x1 + 5x2

Subject to.

x1 ≤ 4

2x2 ≤ 12

3x1 + 2x2 ≤ 18

MinimizeW = 4y1 + 12y2 + 18y3

Subject to.

y1 + 3y3 ≥ 3

2y2 + 2y3 ≥ 5

Primal/Dual Problems for Wyndor Glass Co.

Primal Problem Dual Problem

Primal/Dual Problems for Wyndor Glass CoMatrix form:

2

153x

xZ

18

12

4

23

20

01

2

1

x

x

Max

18

12

4

321 yyyW

5323

20

01

321

yyy

Min

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

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

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.

Variable Descriptionxj Level of activity j

cj Unit profit from activity j

Z Total profit from all activitiesbi Amount of resource i available

aij Amount of resource i consumed by each unit of activity jyi Shadow price for resource I

W Value of Z