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

BIS 517- Aslı Sencer2 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 Resource used Resource Available Resource Left Wood300 0 Labor110 0

BIS 517- Aslı Sencer4 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 X T *=4.05, X C *=8.975, P*=$96.10

BIS 517- Aslı Sencer5 Graphical Representation Constraint 1 Constraint (4,9) XtXt XcXc NEW OPTIMAL SOLUTION X t =4.05, X c =8.975 P=$96.10

BIS 517- Aslı Sencer6 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 Sencer7 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 Sencer8 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 Sencer9 Question?  Why do you think it is so? Constraint 1 Constraint (4,9) XtXt XcXc NEW OPTIMUM

BIS 517- Aslı Sencer10 The Dual Problem: Technical Approach PRIMAL PROBLEM DUAL PROBLEM For any primal solution X t, X c (not necessarily optimal), there is a corresponding dual solution U w, U L. 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 Sencer11 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 Sencer12 Dual formulation If for any product Marginal 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 Sencer13 Sensitivity Analysis Using Excel Solver Adjustable Cells CellName Final Value Reduced Cost Objective Coefficient Allowable Increase Allowable Decrease $C$9Xt40662 $D$9Xc90844 Constraints CellName Final Value Shadow Price Current R.H. Side Allowable Increase Allowable Decrease $G$5<= LHS3000, $G$6<= LHS1100,

BIS 517- Aslı Sencer14 Questions?  If available wood is 310ft, what is the new optimal solution? =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> =660, so a new solution will exist. We need to resolve it with new constraint!

BIS 517- Aslı Sencer15 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 Sencer16 Questions?  How is the optimal solution found in this case? The optimal solution is found by solving

BIS 517- Aslı Sencer17 Pricing new products using shadow prices desk  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.