Duality Theory

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

4. Primal Dual Primal and Dual - ExampleStandard 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 Dual Primal and Dual - ExampleMatrix 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, withcx*=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

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 thedual 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 cBB-1b cBB-1A-c cBB-1 B-1A B-1 B-1b What happens when A, b, c changes?

16. The Changes in Coefficients x2 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 • Feasible? • Satisfies optimality criterion?

19. Example of Sensitivity Analysis Simple dual simplex iteration

20. Changes in bi only (Case 1) • What is affected when only b changes? • Optimality criterion will always be satisfied • Feasibility might not

21. Changes in bi onlyExample A B-1b= cBB-1b= Change to

22. Changes in bi onlyExample A Simple dual simplex iteration

23. Changes in bi onlyRepresenting as differences Change to Can represent as = change in r.h.s. of optimal tableau Then we can find = change in optimal Z value

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

25. Changes in coefficients of nonbasic variables (Case 2a) • 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 variablesExample A continued • From Example A (slide 22 – p241, Table 6.21), we had (cBB-1A - c)1= y* A – c = (B-1A)1= Change to

27. Changes in coefficients of nonbasic variablesAllowable 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