corso mae metodi quantitativi per il management quantitative methods for management roma, 18...

Corso MAECorso MAEMetodi Quantitativi per il ManagementMetodi Quantitativi per il Management

Quantitative methods for ManagementQuantitative methods for Management

Roma, 18 settembre - 24 ottobre 2003

Prof. Gianni Di PilloProf. Laura Palagi

Dipartimento di Informatica e SistemisticaUniversita` 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

standard deluxeunit profit 10 15

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

3 standard deluxe6 grinding 4 27 polishing 2 5

Mathematical model for factory A

max 10 x1 + 15 x2

4 x1 + 4 x2 <= 75

4 x1 + 2 x2 <= 80

2 x1 + 5 x2 <= 60

x1 , x2 >= 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

5

10

15

20

25

30

35

40

40

45

5 10 15 20 25 30 35 40 40 45x1

x2 The constraint 4 x1 + 2 x2 <= 80 does not play any role in defining the feasible region4 x1 +

2 x2 =

80

All non negative points constitutes the set of the feasible solution for factory A

Any point in F is such that 4 x1 + 2 x2 < 80

F

The grinding constraint 4 x1 + 2 x2 <= 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

Microsoft Excel 9.0 Rapporto valoriFoglio di lavoro: [factoryA.xls]Foglio1Data di creazione: 28/09/2003 15.35.26

Cella obiettivo (Max)Cella Nome Valori originali Valore finale$C$10 PROFIT standard 250 225

Celle variabiliCella Nome Valori originali Valore finale$C$9 production standard 10 11,25$D$9 production deluxe 10 7,5

VincoliCella Nome Valore della cella Formula Stato Tolleranza$C$11 raw constraint standard 75 $C$11<=$E$11 Vincolante 0$C$12 grinding constraint standard 60 $C$12<=$E$12 Non vincolante 20$C$13 polishing constraint standard 60 $C$13<=$E$13 Vincolante 0

Excel’s Answer report: details

Celle variabiliCella Nome Valori originali Valore finale$C$9 production standard 10 11,25$D$9 production deluxe 10 7,5

Adjustable cells

Cella obiettivo (Max)Cella Nome Valori originali Valore finale$C$10 PROFIT standard 250 225

Target cell (max) Initial guess Final valuename

Excel’s Answer report: details

VincoliCella Nome Valore della cella Formula Stato Tolleranza$C$11 raw constraint standard 75 $C$11<=$E$11 Vincolante 0$C$12 grinding constraint standard 60 $C$12<=$E$12 Non vincolante 20$C$13 polishing constraint standard 60 $C$13<=$E$13 Vincolante 0

italian

english

ConstraintsCell Name Cell Value Formula Status Slack

$C$11 raw constraint standard 75 $C$11<=$E$11 Binding 0$C$12 grinding constraint standard 60 $C$12<=$E$12 Nonbinding 20$C$13 polishing constraint standard 60 $C$13<=$E$13 Binding 0

The information on binding and nonbinding constraint is of most interest when it pertains scarse resources.

Use Excel’s Answer report

ConstraintsCell Name Cell Value Formula Status Slack

$C$11 raw constraint standard 75 $C$11<=$E$11 Binding 0$C$12 grinding constraint standard 60 $C$12<=$E$12 Nonbinding 20$C$13 polishing constraint standard 60 $C$13<=$E$13 Binding 0

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)

Microsoft Excel 9.0 Rapporto sensibilitàFoglio di lavoro: [factoryA.xls]Foglio1Data di creazione: 28/09/2003 23.11.45

Celle variabiliValore ridotto oggettivo consentito consentito

Cella Nome finale Costo Coefficiente Incremento Decremento$C$9 production standard 11,25 0 10 5 4$D$9 production deluxe 7,5 0 15 10 5

VincoliValore ombra Vincolo consentito consentito

Cella Nome finale Prezzo a destra Incremento Decremento$C$11 raw constraint standard 75 1,666666667 75 15 27$C$12 grinding constraint standard 60 0 80 1E+30 20$C$13 polishing constraint standard 60 1,666666667 60 33,75 22,5

Excel produces also an optional sensitivity report

Excel’s Sensitivity report (Factory A)

Changing cellFinal reduced objective allowable allowable

Cella Nome value cost coefficient increase decrease$C$9 production standard 11,25 0 10 5 4$D$9 production deluxe 7,5 0 15 10 5

ConstraintsFinal shadow constraint allowable allowable

Cella Nome value prices R.h. side increase decrease$C$11 raw constraint standard 75 1,666666667 75 15 27$C$12 grinding constraint standard 60 0 80 1E+30 20$C$13 polishing constraint standard 60 1,666666667 60 33,75 22,5

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

5

10

15

20

25

30

35

40

40

45

5 10 15 20 25 30 35 40 40 45x1

x2

F

Actual optimal solution

max 10 x1 + 15 x2Actual objective function

If the coefficient of x1 = standard stays between 10

+ 5 = 15- 4 = 6

Allowable increase

Allowable decrease

10 x1 + 15 x2

15 x1 + 15 x2

6 x1 + 15 x2

The same happens changing coefficient x2

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)


Cell Name value prices R.h. side increase decrease$C$11 raw constraint standard 75 1,666666667 75 15 27$C$12 grinding constraint standard 60 0 80 1E+30 20$C$13 polishing constraint standard 60 1,666666667 60 33,75 22,5

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

A B C D E2 factory A3 standard deluxe4 unit profit 10 155 raw material 4 46 grinding 4 27 polishing 2 589 production 11,66667 7,333333

10 PROFIT 226,666711 raw constraint 76 76 raw availability12 grinding constraint 61,33333 80 max grinding13 polishing constraint 60 60 max polishing

Add 1 unit to raw availabilityAdd 1.66 to the profit




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.

production 10,91667 7,833333PROFIT 226,6667raw constraint 75 75 raw availabilitygrinding constraint 59,33333 80 max grindingpolishing constraint 61 61 max polishing

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)

89 production 17,5 5

10 PROFIT 25011 raw constraint 90 100 raw availability12 grinding constraint 80 80 max grinding13 polishing constraint 60 60 max polishing

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

production 9,6875 8,125PROFIT 218,75raw constraint 71,25 75 raw availabilitygrinding constraint 55 55 max grindingpolishing constraint 60 60 max polishing

Decrease the grinding availability more than 20 (e.g. 80-25=55)

Solution change (worst !)

final Shadow r.h.s allowable allowableCella Nome value price increase decrease$C$11 raw constraint standard 71,25 0 75 1E+30 3,75$C$12 grinding constraint standard 55 1,25 55 5 31$C$13 polishing constraint standard 60 2,5 60 7,5 32,5

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 x3 + 15 x3

4 x3 + 4 x3 <= 45

5 x3 + 3 x4 <= 60

5 x3 + 6 x4 <= 75

x3 , x3 >= 0

Let analyze Factory B production problem

Left hand side (l.h.s.) Right hand side (r.h.s)

Excel’s Answer reportMicrosoft Excel 9.0 Rapporto valoriFoglio di lavoro: [factoryB.xls]Foglio1Data di creazione: 29/09/2003 0.01.25

Cella obiettivo (Max)Cella Nome Valori originali Valore finale$C$10 PROFIT standard 112,5 168,75

Celle variabiliCella Nome Valori originali Valore finale$C$9 production standard 0 0$D$9 production deluxe 7,5 11,25

VincoliCella Nome Valore della cella Formula Stato Tolleranza$C$11 raw constraint standard 45 $C$11<=$E$11 Vincolante 0$C$12 grinding constraint standard 33,75 $C$12<=$E$12 Non vincolante 26,25$C$13 polishing constraint standard 67,5 $C$13<=$E$13 Non vincolante 7,5

Geometric interpretation

5 x3 +

3 x4 =

60

5 10 15 20 30 40 50x3

5

10

15

20

30

40

50x4

4 x3 +

4 x4 =

45

5 x3 + 6 x

4 = 75

Two constraints 5 x3 + 6 x4 <= 75 and 5 x3 + 3 x4 <=

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 x3 + 6 x4 < 75 and 5 x3 + 3 x4 < 60

The grinding and polishing constraints are always nonbinding

Excel’s Sensitivity report (Factory B)

Microsoft Excel 9.0 Rapporto sensibilitàFoglio di lavoro: [factoryB.xls]Foglio1Data di creazione: 29/09/2003 0.01.38

Celle variabiliValore ridotto oggettivo consentito consentito

Cella Nome finale Costo Coefficiente Incremento Decremento$C$9 production standard 0 -4,999999999 10 4,999999999 1E+30$D$9 production deluxe 11,25 0 15 1E+30 4,999999999

VincoliValore ombra Vincolo consentito consentito

Cella Nome finale Prezzo a destra Incremento Decremento$C$11 raw constraint standard 45 3,75 45 5 45$C$12 grinding constraint standard 33,75 0 60 1E+30 26,25$C$13 polishing constraint standard 67,5 0 75 1E+30 7,5

Reduced costs in Excel’s Sensitivity reportCelle variabili

final Reduced objective allowable allowableCella Nome value cost Coefficient increase descrease$C$9 production standard 0 -4,999999999 10 4,999999999 1E+30$D$9 production deluxe 11,25 0 15 1E+30 4,999999999

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

5 10 20 30 40 50x3

5

10

15

20

30

40

50x4 PTOT = 10 x3 + 15 x4 15 x3 + 15 x4

Actual optimal solution

Change the equation of the profit

All the points in the segment from to

are optimal solution with the same value of PTOT


final shadow r.h.s. allowable allowableCella Nome value price increase decrease$C$11 raw constraint standard 45 3,75 45 5 45$C$12 grinding constraint standard 33,75 0 60 1E+30 26,25$C$13 polishing constraint standard 67,5 0 75 1E+30 7,5

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 x1 + 15 x2 + 10 x3 + 15 x4

4 x1 + 2 x2 <= 80

5 x3 + 3 x4 <= 60

2 x1 + 5 x2 <= 60

5 x3 + 6 x4 <= 75

4 x1 + 4 x2 + 4 x3 + 4 x4 <= 120

x1 , x2 , x3 , x4 >= 0

More than two variables: we can solve it with the Solver

Optimal solution for the company

A B C D E F2 COMPANY3 standard deluxe4 unit profit 10 15 data for the company5 raw material 4 46 factory B factory A7 grinding 5 3 4 2 data for the factories8 polishing 5 6 2 59 standard deluxe standard deluxe

10 company production 0 12,5 9,166667 8,33333333311 COMPANY PROFIT 404,1666712 factory B factory A13 grinding constraint 37,5 max grinding 60 grinding constraint 53 max grinding 8014 polishing constraint 75 max polishing 75 polishing constraint 60 max polishing 601516 raw constraint 120 120 raw availability

Optimal production: deluxe = 20.8, standard = 9.17

Profit = 404.16

Excel’s Answer reportMicrosoft Excel 9.0 Rapporto valoriFoglio di lavoro: [company.xls]Foglio1Data di creazione: 29/09/2003 0.19.17

Cella obiettivo (Max)Cella Nome Valori originali Valore finale

$C$11 COMPANY PROFIT standard 404,1666667 404,1666667

Celle variabiliCella Nome Valori originali Valore finale

$C$10 company production standard 0 0$D$10 company production deluxe 12,5 12,5$E$10 company production standard 9,166666667 9,166666667$F$10 company production deluxe 8,333333333 8,333333333

VincoliCella Nome Valore della cella Formula Stato Tolleranza

$C$16 raw constraint factory B 120 $C$16<=$E$16 Vincolante 0$C$13 grinding constraint factory B 37,5 $C$13<=$E$13 Non vincolante 22,5$C$14 polishing constraint factory B 75 $C$14<=$E$14 Vincolante 0$G$13 grinding constraint data for the factories 53,33333333 $G$13<=$I$13 Non vincolante 26,66666667$G$14 polishing constraint data for the factories 60 $G$14<=$I$14 Vincolante 0

Vincolante = binding

Excel’s Sensitivity report

Microsoft Excel 9.0 Rapporto sensibilitàFoglio di lavoro: [company.xls]Foglio1Data di creazione: 29/09/2003 0.19.17


Cella Nome Value cost Coefficient increase decrease$C$10 company production standard 0 -3,611111114 9,999999997 3,611111114 1E+30$D$10 company production deluxe 12,5 0 15 1E+30 4,333333336$E$10 company production standard 9,166666667 0 10 5 4$F$10 company production deluxe 8,333333333 0 15 10 5

ConstraintsFinal shadow Vincolo allowable allowable

Cella Nome Value price a destra increase decrease$C$16 raw constraint factory B 120 1,666666667 120 20 22$C$13 grinding constraint factory B 37,5 0 60 1E+30 22,5$C$14 polishing constraint factory B 75 1,388888889 75 33 30$G$13 grinding constraint data for the factories 53,33333333 0 80 1E+30 26,66666667$G$14 polishing constraint data for the factories 60 1,666666667 60 27,5 25

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) http://www.dis.uniroma1.it/~palagi

corso mae metodi quantitativi per il management quantitative methods for management roma, 18...

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