1. Introduction to Operations Research
Duality Theory

2. 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.

3. 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

4. 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

5. 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

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

7. 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

8. 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

9. 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.

10. 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