Corso MAE Metodi Quantitativi per il Management Quantitative methods for Management Roma, 18 settembre - 24 ottobre 2003 Prof. Gianni Di Pillo Prof. Laura Palagi Dipartimento di Informatica e Sistemistica Universita` di Roma “La Sapienza”

Solver reports LP software packages provide more information than the optimal values of decision variables and of the objective function Reports Answer Sensitivity Limits This information can be required right after Solver has found an optimal solution

The production problem of factory A Factory A makes two products: standard and deluxe Each unit of product yields the following profit Each product requires 4 Kg of raw material Factory A is allocated 75 Kg of raw material Factory A has a grinding capacities of 80 hours per week and polishing capacity of 60 hours per week The grinding and polishing times in hours for a unit of each type of product of factory A are

Mathematical model for factory A max 10 x x 2 4 x x 2 <= 75 4 x x 2 <= 80 2 x x 2 <= 60 x 1, x 2 >= 0 Constraints are inequalities At the optimal solution they may be satisfied as = binding constraint < nonbinding constraint When the constraint is nonbinding the difference between the two sides is called slack Left hand side (l.h.s.) Right hand side (r.h.s)

Geometric interpretation x1x1 x2x2 The constraint 4 x x 2 <= 80 does not play any role in defining the feasible region 4 x x 2 = 80 All non negative points constitutes the set of the feasible solution for factory A Any point in F is such that 4 x x 2 < 80 F The grinding constraint 4 x x 2 <= 80 is always nonbinding This is not true in general In particular it is not binding in the optimal solution

Using the Solver We have also here the information on the value of the constraint

Excel’s Answer report

Excel’s Answer report: details Adjustable cells Target cell (max)Initial guessFinal value nam e

Excel’s Answer report: details italian english The information on binding and nonbinding constraint is of most interest when it pertains scarse resources.

Use Excel’s Answer report From the table above, we see that the raw material and the polishing constraints are binding, but there are 20 unit (hours) of slack in the grinding constraint. This information can be used together with the other Solver reports

Excel’s Sensitivity report (italian) Excel produces also an optional sensitivity report

Excel’s Sensitivity report (Factory A)

Excel’s Sensitivity report (top part) If a decision variable is positive (an activity is being performed at a positve level), then the columns of allowable increase and decrease indicate how much more or less profitable this activity would have to be before the current optimal solution would no longer be optimal In the top part there is a separate line for each changing cell (i.e. for each decision variable)

Geometric interpretation for factory A x1x1 x2x2 F Actual optimal solution max 10 x x 2 Actual objective function If the coefficient of x 1 = standard stays between = = 6 Allowable increase Allowable decrease 10 x x 2 15 x x 2 6 x x 2 The same happens changing coefficient x 2 It is a “rotation” around the optimal point

Reduced costs in Excel’s Sensitivity report If a decision variable is zero, the it is evidently not profitable to include this activity in the optimal mix The reduced cost (or dual value) of an activity indicates how much more profitable each unit of this activity would have to be before it would be optimal to include in the optimal mix In this case, variables are positive, and the corresponding reduced costs are zero !

Excel’s Sensitivity report (botton part) In the bottom part, there is a line for each constraint (not including simple lower or bound on the decision variables) If a constraint is binding, then the company has used all the available resource and it might consider buying more of it How much the company will be willing to pay for each extra unit of material ?

Shadow prices in Excel’s Sensitivity report The shadow prices indicates how much extra unit of resource is worth in terms of increasing the total profit The shadow price is the change in the objective value for unit change in the availability of the resource In Factory A problem, the shadow price for the raw material is 1.6. This means that each extra unit (kg) of raw material (on top of the 75 available) would add 1.6 Euro to the total profit.

Verifying with the Solver Add 1 unit to raw availability Add 1.66 to the profit

Shadow prices in Excel’s Sensitivity report In Factory A problem, the shadow price for the polishing hours is 1.6. This means that each extra unit (hour) of polishing resource (on top of the 60 available) would add 1.6 Euro to the total profit. Add 1.66 to the profit Add 1 unit to polishing availability

Economic interpretation of shadow prices In Factory A problem, the grinding availability is not saturated (the constraint is non binding) and the shadow price for the grinding constraint is zero. The shadow prices for non binding constraints are always equal to zero. This makes economic sense: if the company already has more unit of a resources then it is using, the it certain is not willing to pay for more units.

Shadow prices are always valid ? Shadow prices analysis is valid only within certain ranges of changes in the resources. These ranges are given in Excel Sensitivity report (allowable increase and decrease) The allowable increase and decrease indicate the range in which the shadow price is relevant. Within the range every extra/loss of unit produce extra/loss profit (given by the corresponding shadow price) Beyond this range, it is difficult to say what will happen

Shadow prices are always valid ? The only way to find out how much less or more profitable is to change the amount and rerun the Solver Actually if we increase raw material up to 100 (>75+15) we gain only 50 Euro, that means that each extra unit is worth less than 1.66 (actually is 1)

Ranges for non binding constraint In the case of non binding constraint (the shadow prices are zero), the allowable increase is infinity because the shadow price remains zero no matter how many more units of resources are available. The allowable decrease is finite and it is exactly the amount of the slack (surplus) for the resource.

Ranges for non binding constraint For Factory A, the slack for the grinding constraint is 20: if the company has less than 20 unit of grinding hours, the current solution will no longer be optimal and the shadow price will change (becoming positive) to indicate that that resource is now a scarse one. To understand what will happen you need to run again the Solver

Verifying with the Solver Decrease the grinding availability more than 20 (e.g =55) Solution change (worst !) And also the shadow price !

Careful use of Excel’s Sensitivity Report All the analysis are valid only if we change only one input at time When we make a statement about an input, we are assuming that all the other are held constant. To see what happens when more than one input at time changes, you need to rerun the Solver.

Mathematical model for factory B max 10 x x 3 4 x x 3 <= 45 5 x x 4 <= 60 5 x x 4 <= 75 x3, x 3 >= 0 Let analyze Factory B production problem Left hand side (l.h.s.)Right hand side (r.h.s)

Excel’s Answer report

Geometric interpretation 5 x x 4 = x3x x4x4 4 x x 4 = 45 5 x x 4 = 75 Two constraints 5 x x 4 <= 75 and 5 x x 4 <= 60 do not play any role in defining the feasible region All non negative points constitutes the set of the feasible solution for factory B Any point in F is such that 5 x x 4 < 75 and 5 x x 4 < 60 The grinding and polishing constraints are always nonbinding

Excel’s Sensitivity report (Factory B)

Reduced costs in Excel’s Sensitivity report Factory B optimal mix does not include standard product The reduced cost of standard product is –5: the unit profit of the standard product must be increase of 5 unit up to 15 before standard product will be worth producing

Geometric view x3x x4x4 P TOT = 10 x x 4 15 x x 4 Actual optimal solution Change the equation of the profit All the points in the segment fromto are optimal solution with the same value of P TOT

Shadow prices in Excel’s Sensitivity report Grinding and polishing are non binding: the corresponding shadow prices are zero Raw material is a scarce resource: the shadow price is positive and the allowable increase and decrease stay in a finite range

Mathematical model for the multi plant max 10 x x x x 4 4 x x 2 <= 80 5 x x 4 <= 60 2 x x 2 <= 60 5 x x 4 <= 75 4 x x 2 +4 x x 4 <= 120 x 1, x 2, x 3, x 4 >= 0 More than two variables: we can solve it with the Solver

Optimal solution for the company Optimal production: deluxe = 20.8, standard = 9.17 Profit =

Excel’s Answer report Vincolante = binding

Excel’s Sensitivity report

References H.P. Williams, Model building in mathematical programming, John Wiley, 1999 W. L Winston and S. C. Albright, Practical Management Science, Duxbury Press, 1997 L. Palagi, Electronic version of the lectures (2004)