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L20 LP part 6

L20 LP part 6. Homework Review Postoptimality Analysis Summary. H19. H19 cont’d. Two-phase Simplex Method. Use artificial variables!. Phase I - finds a feasible basic solution Phase II- finds an optimal feasible basic solution , if it exists. Transforming Process.

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L20 LP part 6

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  1. L20 LP part 6 • Homework • Review • Postoptimality Analysis • Summary

  2. H19

  3. H19 cont’d

  4. Two-phase Simplex Method Use artificial variables! • Phase I - finds a feasible basic solution • Phase II- finds an optimal feasible basic solution, if it exists.

  5. 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,

  6. Problem 8.58 (from lecture)

  7. H19 cont’d

  8. H19 cont’d

  9. Simplex Method Identifies: • global solutions, if they exist • multiple solutions • unbounded problems • degenerate problems • infeasible problems

  10. Postoptimality Analysis • What happens to our results if: • The inputs change? • The conditions of the business objective change ? • The system/factory parameters change

  11. What-if analyses • Problem formulation: …. Make assumptions about • Resources/capacities, • Price coefficients • Cost coefficients • Also what would happen if conditions should change such as : • Environmental aspects • Conditions of use • Market conditions • Production capabilities • Determine Strategies to handle: • In case things change • To account for competitive reactions • Consider worst case possibilities

  12. Sensitivity Analyses how sensitive are the: a. optimal value (i.e. f(x) and b. optimal solution (i.e. x) … to the parameters (i.e. assumptions) in our model?

  13. Model parameters Consider your abc’s, i.e. A, b and c

  14. Recall during Formulation Constraints usually arise from : Laws of nature (e.g. F=ma) Laws of economics (e.g. profit=revenues-costs) Laws of man (e.g. max work week = 40 hrs) How accurate are our assumptions?

  15. Any approximations in our formulation?

  16. Abc’s of sensitivity analyses Let’s look at “b” first, i.e. “resource limits”

  17. Recall Relaxing constraints(i.e. adding more resources) Constraint Variation Sensitivity Theorem From LaGrange theory The instantaneous rate of change in the objective function with respect to relaxing a constraint IS the LaGrange multiplier!

  18. Simplex LaGrange Multipliers Find the multipliers in the final tableau (right side)

  19. Prob 8.58 reg. cols art’l col’s “=“ “≥”

  20. Excel “shadow prices” If Minimizing w/Excel…Reverse sign of LaGrange Multipliers! If Maximizing w/Excel… signs are the same as our tableaus

  21. Let’s minimize f even further Increase/decrease ei to reduce f(x)

  22. Excel solution

  23. Abc’s • Changes in cost coefficients, c • Changes in coefficient matrix A Often times it’s simpler to re-run LP Solver

  24. Ranging-limits on changes • RHS parameters, b • Cost coefficients, c • A coefficient matrix Yes, formulas are available… often times it’s much easier to just re-run your LP Solver!

  25. Summary • Simplex method determines: Multiple solutions (think c’) Unbounded problems (think pivot aij<0) Degenerate Solutions (think bj=0) Infeasible problems (think w≠0) • Sensitivity Analyses add important value to your LP solutions, can provide “strategies” • Sensitivity is as simple as Abc’s • Constraint variation sensitivity theoremcan answer simple resource limits questions

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