L20 LP part 6 Homework Review Postoptimality Analysis Summary 1

H19 2

H19 cont’d 3

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

Transforming Process 1.Convert Max to Min, i.e. Min f(x) = Min -F(x) 2.Convert negative bj to positive, mult by(-1) 3.Add slack variables 4.Add surplus variables 5.Add artificial var’s for “=” and or “ ≥” constraints 6.Create artificial cost function, 5

Problem 8.58 (from lecture) 6

H19 cont’d 7

8

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

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

What-if analyses 11 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

Sensitivity Analyses 12 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?

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

Recall during Formulation 14 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?

Any approximations in our formulation? 15

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

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

Simplex LaGrange Multipliers 18 Constraint Type ≤ = ≥ slackeithersurplus c’ column“regular”artificial Find the multipliers in the final tableau (right side)

Prob “=“ “ ≥” reg. colsart’l col’s

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

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

Excel solution 22

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

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! 24

Summary Simplex method determines: Multiple solutions (think c’) Unbounded problems (think pivot a ij <0) Degenerate Solutions (think b j =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 theorem can answer simple resource limits questions 25