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Graduate Program in Business Information Systems

Graduate Program in Business Information Systems. Linear Programming: Sensitivity Analysis and Duality Aslı Sencer. Shadow Prices and Opportunity Costs. LP solution answers the tactical question, i.e., how much to produce Suppose the focus is on resources rather than the products, i.e.,

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Graduate Program in Business Information Systems

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  1. Graduate Program in Business Information Systems Linear Programming: Sensitivity Analysis and Duality Aslı Sencer

  2. Shadow Prices and Opportunity Costs • LP solution answers the tactical question, i.e., how much to produce • Suppose the focus is on resources rather than the products, i.e., • Each resource has a shadow price that reflects the true impact of scarcity. • To find these, we need a transformation of the primal problem which is referred to as the dual problem. BIS 517- Aslı Sencer

  3. Ex:Redwood Furniture Product Mix problem (revisited) : number of tables produced in a period : number of chairs produced in a period Optimal Solution: Xt=4 tables, Xc=9 Chairs Profit*=$96 Optimal Solution: Xt=4, Xc=9 Profit*=$96 BIS 517- Aslı Sencer

  4. Increasing the Available Resources • What happens if the available wood is increased by 1 ft? • Need to resolve LP with the new constraint • which yields XT*=4.05, XC*=8.975, P*=$96.10 BIS 517- Aslı Sencer

  5. Graphical Representation Xc 15 Xt=4.05, Xc=8.975 P=$96.10 Constraint 1 11 NEW OPTIMAL SOLUTION (4,9) Constraint 2 Xt 10 22 BIS 517- Aslı Sencer

  6. Shadow Price and Opportunity Cost Optimal profit in the new problem is $96.10-$96=$0.1 greater! SHADOW PRICE • Shadow price is the marginal value of a resource. • Shadow price is the opportunity cost of not increasing the resource. BIS 517- Aslı Sencer

  7. Question? • Question 1: How much should the DM be willing to pay for a unit increase in wood resource? • Answer: Infact, the DM should not pay more than $0.1 for a unit increase in the current wood capacity of $300 BIS 517- Aslı Sencer

  8. Question? • Question 2: If the wood resource is to be increased by 100ft (i.e., it will be 400 ft now), what will be the new optimal profit? • Answer: Can not tell directly! Shadow prices are valid only for certain ranges of change in the available resources. BIS 517- Aslı Sencer

  9. Question? • Why do you think it is so? Xc 15 Constraint 1 11 (4,9) NEW OPTIMUM Constraint 2 Xt 10 22 BIS 517- Aslı Sencer

  10. The Dual Problem: Technical Approach DUAL PROBLEM PRIMAL PROBLEM • For any primal solution Xt, Xc (not necessarily optimal), • there is a corresponding dual solution Uw, UL. • If the primal solution is not optimal, then dual solution is infeasible! • If they are both feasible then both solutions are optimal and P*=C*! BIS 517- Aslı Sencer

  11. Dual Problem: Economical Interpretation • Primal problem: Production Manager’s perspective: Optimize resource allocation to maximize Total Profit. • Dual problem:Economist’s perspective: Optimize resource allocation to minimize aggregate value of increasing any resource by one more unit. BIS 517- Aslı Sencer

  12. Dual formulation If for any productMarginal opportunity cost > Marginal return Do not produce Marginal opportunity cost < Marginal return Produce more Marginal opportunity cost = Marginal return Current production level is optimal BIS 517- Aslı Sencer

  13. Sensitivity Analysis Using Excel Solver BIS 517- Aslı Sencer

  14. Questions? • If available wood is 310ft, what is the new optimal solution? • 310-300=10ft increase is required • From the sensitivity analysis allowable increase is 360, so shadow prices are valid! • Pnew*=96+10*0.1=$97. The optimal solution is found by solving • If the available wood is 700ft what is the new optimal solution? • 700>300+360=660, so a new solution will exist. We need to resolve it with new constraint! BIS 517- Aslı Sencer

  15. Questions? • If the unit profit of a table is decreased to $5, new optimum? • Current value is 6, thus $1 decrease is required. • In the sensitivity table, allowable decrease is 2. • So current solution is still optimal. • Xt=4, Xc=9 and P=5(4)+8(9)=$92 • Would you hire an extra labor for 10 hrs at a total cost of $5? • In the sensitivity table, allowable increase is 40, so dual prices are valid. • increase in optimal profit=0.6(10)=$6. Net saving=$6-$5=$1>0, so hire labor! BIS 517- Aslı Sencer

  16. Questions? • How is the optimal solution found in this case? • The optimal solution is found by solving BIS 517- Aslı Sencer

  17. Pricing new products using shadow prices • Making a desk would divert resources from tables and chairs, and fewer would be made. • Redwood evaluates new products: • Bench having profit of $7, needing 25 board feet of wood and 7 hours of labor. • Planter box having profit of $2, needing 10 board feet of wood and 2 hours of labor. • The opportunity costs for one of each are: • Bench: $.10(25) + .60(7) = $6.70 (< $7). Make it, because doing so increases P by $.30/unit. • Planter box: $.10(10) + .60(2) = $2.40 (> $2). Do not make. Resources are too valuable. BIS 517- Aslı Sencer

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