1. Duality Theory

2. The Essence
Every linear program has another linear program associated with it: Its ‘dual’
The dual complements the original linear program, the ‘primal’
The theory of duality provides many insights into what is happening ‘behind the scenes’

3. Primal and Dual Primal Decision variables: x max Z = cx s.to Ax  b n
Decision variables: y
min W = yb
s.to yA  c m
y  0 n

4. Primal and Dual - Example Standard Algebraic Form
Max Z = 3x1+ 5x2
s. to x1 ≤ 4
2x2 ≤ 12
3x1+ 2x2 ≤ 18
x1,x2 ≥ 0
Min W = 4y1+ 12y2+ 18y3
s. to y1+ 3y3 ≥ 3
2y2+ 2y3 ≥ 5
y1, y2, y3 ≥ 0

5. Primal and Dual - Example Matrix Form

6. Symmetry Property
For any primal problem and its dual problem, all relationships between them must be symmetric because…

7. Duality Theorem
The following are the only possible relationships between the primal and dual problems:
If one problem has feasible solutions and a bounded objective function (and so has an optimal solution), then so does the other problem, so both the weak and the strong duality properties are applicable
If one variable has feasible solutions but an unbounded objective function (no optimal solutions), then the other problem has no feasible solutions
If one variable has no feasible solutions, then the other problem either has no feasible solutions or an unbounded objective function

8. Relationships between Primal and Dual
Weak duality property If x is a feasible solution to the primal, and y is a feasible solution to the dual, then cx  yb
Strong duality property If x* is an optimal solution to the primal, and y* is an optimal solution to the dual, then cx* = y*b

9. Complementary Solutions
At each iteration, the simplex method identifies
x, a BFS for the primal, and
a complementary solution y, a BS for the dual (which can be found from the row coefficients under slack variables)
For any primal feasible (but suboptimal) x, its complementary solution y is dual infeasible, with cx=yb
For any primal optimal x*, its complementary solution y* is dual optimal, with cx*=y*b

10. Complementary Slackness
Associated variables between primal and dual
Primal
(original variable) xj
(slack variable) xsn+i
Dual
ysm+j (surplus variable)
yi (original variable)
Complementary slackness property: When one variable in primal is basic, its associated variable in dual is nonbasic
Primal
(m variables) basic
(n variables) nonbasic
Dual
nonbasic (m variables)
basic (n variables)

11. Primal and Dual Primal Z=cx Dual W=yb superoptimal suboptimal
(optimal) Z*
W* (optimal)
suboptimal
superoptimal

12. Insight B-1A B-1 B-1A B-1 B-1b B-1b cBB-1A-c cBB-1 cBB-1b xB yA-c y yb
B.V. Z
Original Variables
Slack Variables
r.h.s. x1 x2 … xn
xsn+1
xsn+m 1
cBB-1A-c
cBB-1
cBB-1b xB
B-1A
B-1
B-1b
B.V. Z
Original Variables
Slack Variables
r.h.s. x1 x2 … xn
xsn+1
xsn+m 1
yA-c y yb xB
B-1A
B-1
B-1b

13. A More Detailed Look at Sensitivity
We are interested in the optimal solution sensitivity to changes in model parameters
Coefficients aij, cj
Right hand sides bi
If we change the model parameters, how does it affect
Feasibility?
If violated, then primal infeasible, but may be dual feasible
Optimality?
If violated, then dual infeasible, but may be primal feasible

14. General Procedure for Sensitivity Analysis
Revise the model
Calculate changes to the original final tableau
Covert to the proper form using Gaussian elimination
Feasibility Test: Is the new right-hand-sides  0 ?
Optimality Test: Is the new row-0 coefficients  0 ?
Reoptimization
If feasible, but not optimal, continue solving with the primal simplex method
If not feasible, but satisfies optimality, continue solving with the dual simplex method
If feasible and satisfies optimality, then calculate new x* and Z*

15. Example of Sensitivity Analysis
Original model
Max Z = 3x1+ 5x2
s. to x1 ≤ 4
2x2 ≤ 12
3x1+ 2x2 ≤ 18
x1,x2 ≥ 0
Revised model
Max Z= 4x1+ 5x2
s. to x1 ≤ 4
2x2 ≤ 24
2x1+ 2x2 ≤ 18
x1,x2 ≥ 0
Original final simplex tableau
Basic variable Z x1 x2 s1 s2 s3
r.h.s. 1
3/2 36
1/3
-1/3 2
1/2 6
cBB-1A-c
cBB-1
cBB-1b
B-1A
B-1
B-1b
What happens when A, b, c changes?

16. The Changes in Coefficientsx2 x2
2x2 = 24
x1 = 4
x1 = 4
Z = 4x1+ 5x2
2x2 = 12
Z = 3x1+ 5x2 = 36
2x1+ 2x2 = 18
3x1+ 2x2 = 18 x1 x1

17. Example of Sensitivity Analysis
Calculate changes to the original final tableau
from original final tableau
use new coefficients

18. Example of Sensitivity Analysis
Basic variable Z x1 x2 s1 s2 s3
r.h.s. 1 48
- ½ 7 12 -3
Feasible?
Satisfies optimality criterion?

19. Example of Sensitivity Analysis
Simple dual simplex iteration
Basic variable Z x1 x2 s1 s2 s3
r.h.s. 1 48
- ½ 7 12 -3
Basic variable Z x1 x2 s1 s2 s3
r.h.s. 1

20. Changes in bi only (Case 1)
B.V. Z
Original Variables
Slack Variables
r.h.s. x1 x2 … xn
xsn+1
xsn+m 1
cBB-1A-c
cBB-1
cBB-1b xB
B-1A
B-1
B-1b
What is affected when only b changes?
Optimality criterion will always be satisfied
Feasibility might not

21. Changes in bi only Example Ato
B-1b=
cBB-1b=
Basic variable Z x1 x2 s1 s2 s3
r.h.s. 1
3/2
1/3
-1/3
1/2

22. Changes in bi only Example A
Simple dual simplex iteration
Basic variable Z x1 x2 s1 s2 s3
r.h.s. 1
3/2 54
1/3
-1/3 6
1/2 12 -2
Basic variable Z x1 x2 s1 s2 s3
r.h.s. 1

23. Changes in bi only Representing as differencesto
Can represent as
Then we can find
= change in r.h.s. of optimal tableau
= change in optimal Z value

24. Changes in bi only Allowable ranges
Assume only b2 changes
The current basis stays feasible (and optimal) as long as

25. Changes in coefficients of nonbasic variables (Case 2a)
B.V. Z
Original Variables
Slack Variables
r.h.s. x1 x2 … xn
xsn+1
xsn+m 1
cBB-1A-c
cBB-1
cBB-1b xB
B-1A
B-1
B-1b
What is affected when only c and A columns for a nonbasic variable change?
Feasibility criterion will always be satisfied
Optimality might not

26. Changes in coefficients of nonbasic variables Example A continued
From Example A (slide 22 – p241, Table 6.21), we had
Basic variable Z x1 x2 s1 s2 s3
r.h.s. 1
9/2
5/2 45 4
3/2
1/2 9 -3 -1 6
Change to
(cBB-1A - c)1= y* A – c =
(B-1A)1=

27. Changes in coefficients of nonbasic variables Allowable ranges
Assume only c1 changes
The current basis stays optimal (and feasible) as long as

28. Introduction of a new variable (Case 2b)
Assume the variable was always there, with ci=0, Aij=0
Can now assume it is a nonbasic variable at the original optimal, and apply the same approach as Case 2a
Example A
Max Z = 3x1+ 5x2
s. to x1 ≤ 4
2x2 ≤ 24
3x1+ 2x2 ≤ 18
x1,x2 ≥ 0
Example A with new variable
Max Z = 3x1+ 5x2 + 2x6
s. to x1 + x6 ≤ 4
2x2 ≤ 24
3x1+ 2x2 + 2x6 ≤ 18
x1,x2 ≥ 0

29. Other possible analyses
Changes to the coefficients of a basic variable
Utilize same approach as initial example, not much of a ‘special case’
Introduction of a new constraint
Would optimality criterion be satisfied?
Would feasibility criterion be satisfied?
Parametric analysis