L19 LP part 5

1. L19 LP part 5 • Review • Two-Phase Simplex Algotithm • Summary • Test 3 results

2. Single Phase Simplex Method • Finds global solutions, if they exist • Identifies multiple solutions • Identifies unbounded problems • Identifies degenerate problems And, as we shall see in two-phase Simplex Method, it also… • identifies infeasible problems

3. Multiple Solutions Non-basic ci’=0 Non-unique global solutions, ∆f = 0

4. Unbounded problem Non-basic pivot column coefficients aij < 0

5. Degenerate basic solution One or more basic variable(s) = 0 Simplex method will move to a solution, slowly In rare cases it can “cycle” forever.

6. What’s next? Single-Phase Simplex Method handles “≤” inequality constraints But, LP problems have “=” and “≥ ” constraints! Such a tableau would not be feasible!

7. Simplex Method requires an initial basic feasible solution i.e. need canonic form to start Simplex “=” and or “≥” constraints or bj<0 (neg resource limits) cause infeasibility problems!

8. Can we transform to canonic form?

9. Basic Feasible Solution? Initial tableau requires 3 identity columns for 3 basic variables (m=3)! Missing third basic variable! Bad! Need “+1”, Bad!

10. Need two phases • Phase I - finds a feasible basic solution • Phase II- finds an optimal solution, if it exists. Two-phase Simplex Method using artificial variables!

11. What is so darn infeasible? Recall, in LaGrange technique, how we insured that an equation is not “violated”, i.e. feasible… We set hj=0 and gj+sj=0

12. Use Artificial Variables x5, x6 toObtaining feasibility! Use the simplex method to minimize an artificial cost function w (i.e. w=0).

13. Tableau w/Artificial Variables x5,x6 Canonic form, i.e. feasible basic solution!

14. Transforming Process • Convert Max to Min, i.e. Min f(x) = Min -F(x) • Convert negative bj to positive, mult by(-1) • Add slack variables • Add surplus variables • Add artificial var’s for “=” and or “≥” constraints • Create artificial cost function,

15. Table 8.17 after w=0, ignore art. var’s reduced cost Feasible when w=0

16. Two-Phase Simplex method Phase I • Transform infeasible LP to feasible using artificial variables. • Use Simplex Meth. To minimize artificial cost function (i.e. art. cost row). If w ≠ 0, problem is infeasible! Phase II • Use Simplex meth to min. reduced cost function (i.e.row) Ignore art. Var’s when choosing pivot columns!

17. Transformation example Ex8.65take out a sheet of paper…

18. Phase I canonical form

19. Summary • Need for initial basic feasible solution (i.e. canonic form) • Phase I – solve for min “artificial cost” use artificalvar’s for “=” and or “≥” constraints • Phase II –solve for min “cost” • Simplex method determines: Multiple solutions (think c’) Unbounded problems (think pivot aij<0) Degenerate Solutions (think bj=0) Infeasible problems (think w≠0)