CHAP04 Sensitivity

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Transcript of CHAP04 Sensitivity
S26 : Chapter 4  Sensitivity Analysis & The Simplex Method
Chapter 4  Sensitivity Analysis & The Simplex Method : S1
Chapter 4
Sensitivity Analysis & The Simplex Method
Excel Solutions1.Howie's numbers are correct. Factory overhead is not a variable cost  it is a sunk cost that must be paid regardless of which type of hot tub is produced.
2.By contradiction, suppose that a variable assumes an optimal value that is somewhere between its upper and lower limits and that the reduced cost for this variable is not zero. This implies that if this variable is increased or decreased by one unit, the absolute value of the change in the objective function will be equivalent to the absolute value of the reduced cost. Increasing the value of this variable by one unit will either increase or decrease the value of the objective function while decreasing the value of the variable will have the opposite effect. Thus, by increasing or decreasing the value of this variable the objective function can be improved. Therefore, the solution could not have been optimal.
3.See file: Prb4_3.xlsmMicrosoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Adjustable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$C$4Value: X15041E+302.8
$D$4Value: X204.66666666724.6666666671E+30
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$E$8Used:100201E+3010
$E$9Used:151.333333333151515
a.The objective function coefficient for X1 can decrease by 2.8 (to 1.2) or increase by any amount without changing the optimal solution.
b. The optimal solution is unique. None of the allowable increase or decrease values for the objective coefficients are zero.
c.It has to increase by at least 4.666667
d.If X2 were forced to equal 1, the optimal objective function value would be approximately 20  4.67 = 15.33.
e.An increase of 10 in the RHS value of the second constraint is within the allowable range of increase for the shadow price of this constraint. Therefore, if the RHS for the second constraint increases from 15 to 25 the new objective function value would be approximately 20 + 1.33 10=33.33.
f.The new reduced cost for X2 would be 2  (4 0 + 1 1.333) = 0.67. Therefore, it would be profitable to increase the value of X2 and the current solution would no longer be optimal but we cant say what the new solution would be without resolving the problem.4.See file: Prb4_4.xlsmMicrosoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Adjustable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$C$4Value: X112021E+300
$D$4Value: X200401E+30
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$E$10Used:1202101E+30
$E$8Used:12081E+3020
$E$9Used:122121E+3010
a.Constraint 2 is binding. (Note that this constraint is reflected by the third row in the Constraints section of the report.) b.There is an alternate optimal solution. Variable X2 is at its lower bound (of zero) but also has a reduced cost of zero. This indicates that the value of X2 could be increased while having zero impact on the optimal objective function value.
c.There is no way to answer this question directly from the sensitivity report. We know that the optimal solution would change since the allowable decrease in the objective function value of X1 is zero, but we cannot tell what the new optimal solution would be. If we resolve the revised model the solution is X1=2, X2=5.
d.It can decrease by any amount without changing the solution.
e.Constraint 2 is the only binding constraint. Therefore, we would want to increase its RHS value before any other.
5.See file: Prb4_5.xlsm Microsoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Adjustable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$C$4Value: X10351E+303
$D$4Value: X20131E+301
$E$4Value: X310424
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$F$8Produced:2221E+301
$F$9Produced:20111E+30
a.The allowable decrease is 4, so the smallest value of the objective for X3 without changing the solution is 0.b.The new objective would be unbounded.
c.Increasing the RHS of the first constraint reduces the feasible region. Therefore, the objective function would be increased (made worse) at a rate of 2 per unit increase in the RHS value. Thus, the new objective function value would be 4 + 2 5 = 14.
d.4  21 = 2
e.Yes, 1/3 + 1/4 = 7/12 < 100%
6.a. MIN: 260X13 + 220X14 + 290X15 + 230X23 + 240X24 + 310X25
S.T.:
X13 + X14 + X15 ( 20
X23 + X24 + X25 ( 20
X13 + X23 ( 10
X14 + X24 ( 15
X15 + X25 ( 10
Xij ( 0
b. See file: Prb4_6.xlsmc. See below (Total cost = $8,600)Microsoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Adjustable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$C$10Eustis Miami0502601E+3050
$D$10Eustis Orlando100220200
$E$10Eustis Tallahassee1002900310
$C$11Clermont Miami10023050230
$D$11Clermont Orlando50240020
$E$11Clermont Tallahassee003101E+300
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$C$12Shipped Miami1023010510
$D$12Shipped Orlando152401555
$E$12Shipped Tallahassee103101055
$F$10Eustis Used20202055
$F$11Clermont Used150201E+305
d. No.
e.No, alternate optima exist.Capacity
MiamiOrlandoTallahasseeUsedAvailable
Eustis01552020
Clermont10051520
Shipped101510
Demand101510
Total Cost$8,600
f. The solution would not change. The current solution uses only 15 of the 20 tons of capacity available at Clermont.
g. Reducing the capacity in Eustis would increase costs by $20 per unit yielding an objective function value of $8,600+205 = $8,700
h. Every additional ton of concentrate shipped from Eustis to Miami would increase costs by $50.
7.See file: Prb4_7.xlsm
Total Profit $32,500
Microsoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Adjustable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$C$4Generators13002501E+30150
$D$4 Alternators02251502251E+30
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$E$8Wiring:26012526020260
$E$9Testing:13001401E+3010
a. There is 0 excess wiring capacity and 10 excess units of testing.
b.$32,500 + $12510 = $33,750
c.$225
d.Using the 100% rule: 50/150 + 75/225 = 2/3 < 100%. The solution would not change.
e.Let X = the increase in the price of alternators. We can be sure the optimal solution will not change as long as 25/150 + X/225 < 1 or X < 225(125/150) = 187.5. So the maximum profit on alternators would be $150+$187.5 = $337.5.
f.The new reduced cost on alternators would be 150  1251.5  02 = 37.5. Thus, it would still be unprofitable to produce alternators and the solution would not change.
8.See file: Prb4_8.xlsmMicrosoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Adjustable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$C$5Razors2400701040
$D$5Zoomers42004053.335
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$E$9Polymer Used90032900200300
$E$10Production Time Used240022400200800
$E$11Total Production Used66007001E+3040
$E$12Product Mix Used18003001E+30480
a. No, it could decrease by $40 without changing the solution.
b. It could. At a profit of $35 there could be alternate optimal solutions.
c. Each additional unit of polymer (up to 200 more) would increase profit by $32 per unit, each unit decrease of polymer (up to 300 less) would decrease profit by $32 per unit.
d. This is a nonbinding constraint. At the optimal solution, this constraint is 40 units below its RHS value (700) so changing the constraint RHS value (within its allowable limits) would not change the solution or allow for additional profit.
e. The allowable increase on the shadow price for labor is 200 units. So we cant say what profit would result from a 300 unit increase in labor without resolving the problem.9.See file: Prb4_9.xlsmTotal Profit = $26,740
Microsoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Adjustable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$B$19Acres to Plant Watermelons60025628.566.33333333
$C$19Acres to Plant Cantaloupes400284.599.528.5
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$F$11Water Used per hour)6,000 1 600015001000
$F$19Acres Planted per hour)1001991002020
a.The profit per acre for watermelons can drop by $66.33 (to $189.67 per acre). That would occur if the price of watermelons dropped to about $2.26 per unit.
b.The profit per acre of cantaloupes would have to increase by $99.50 ($384 per acre). That would occur if the price of cantaloupes dropped to about $1.33 per unit. c.60/66.33 + 50/99.5 = 1.407 > 100%. Therefore we cannot guarantee that the solution is still optimal.
d.The farmer should lease all 20 acres and be willing to pay up to $199 per acre (assuming he can use his own water on the additional 20 acres or otherwise water it as needed for free).
10. See file: Prb4_10.xlsmTotal Profit = $26,000
Microsoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Adjustable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$C$5Qty Doors200500300233.3333
$D$5Qty Windows400400350150
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$E$9Cutting Used40350404013.33333
$E$10Sanding Used40300406.66666720
$E$11Finishing Used500601E+3010
a.No, the objective coefficient on doors could increase by $300 (to $800) without changing the optimal solution.b.Yes, the objective coefficient on windows can only decrease by $150 (to $250) without changing the optimal solution.c.The shadow price (marginal value) is $0 because there is a surplus of this resource.d.$350*20=$7000.e.The shadow price on sanding is $300 / unit (within the allowable increase/decrease). The allowable decrease is 20 units. So if another company was willing to pay $400 for 15 hours of capacity, Sanderson would come out ahead by $1,500 (i.e., (400300)*15). It would be profitable to Sanderson to offer up to 20 hours of their sanding capacity for this purpose. If the other company wanted 25 hours of capacity, the LP model would need to be modified to see if this would be profitable. (And that analysis shows it would not be economicallybeneficial to do so as total profit would drop from $26,000 to $25,000.)11. See file: Prb4_11.xlsm Total cost = $453,300
Microsoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Adjustable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$B$6  Make Model 13,000 0.00 50 4 57
$C$6  Make Model 2550 0.00 83 14 8
$D$6  Make Model 3900 0.00 130 8 137
$B$7  Buy Model 10 4.00 61 1E+30 4
$C$7  Buy Model 21,450 0.00 97 8 14
$D$7  Buy Model 30 8.00 145 1E+30 8
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$B$13# Available Model 13,000 57.00 3000 380 2900
$C$13# Available Model 22,000 97.00 2000 1E+30 1450
$D$13# Available Model 3900 137.00 900 211.1111 900
$E$17  Wiring Used9,525 0.00 10000 1E+30 475
$E$18  Harnessing Used5,000 (7.00)5000 633.3333 1100
a.No.
b.The cost of making model 1 slip rings can increase by $4 without changing the solution.
c.No. The allowable decrease on the objective coefficient for buying model 2 slip rings is $14.
d.No. There is presently a surplus of 475 hours in the wiring department. Overtime would only add to this surplus.
e.Yes. Harnessing represents a binding constraint with a shadow price of $7. Each additional unit of this resource (up to 633.33) will reduce costs by $7. Since workers are paid an additional $6 per hour for overtime, the company could save $633.33.
f.See spider chart in the file. Costs are most sensitive to reductions in the harnessing requirements for model 1 slip rings.
12.See file: Prb4_12.xlsm
Total Revenue = $220,290
Microsoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Adjustable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$C$4Units Produced Country405.800.003501E+3050
$D$4Units Produced Contemporary173.910.00450751266.666667
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$C$4Units Produced Country405.800.000289.85507251E+30
$D$4Units Produced Contemporary173.9165.220238.0952381114.8148148
$E$8Router Used 956.520.0010001E+3043.47826087
$E$9Sander Used 2,000.00110.14200090.909090912000
$E$10Polisher Used 1,275.360.0015001E+30224.6376812
a. No. There is already a surplus of routing capacity.
b. Yes. They should be will to pay $110.14 per unit (above and beyond what they already pay) for up to ~90 additional units.
c. Nothing. There is already a surplus of polishing capacity.
d. The price would have to decrease by more than $1266.67. Currently, the company only makes $450 on this item. So this price reduction would actually result in the company selling this item at a loss. The constraint requiring at least 30% of the production to consist of contemporary tables creates this anomaly.
13.See file: Prb4_13.xlsmMicrosoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Decision Variable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$B$6Tahoe23.571428570.00450300.000000750.00000007
$C$6Pacific150.00115042.85714296714.2857144
$D$6Savannah40.714285710.00800100.000000116.66666682
$E$6Aspen07.1428571434007.1428571431E+30
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$F$10Glue Used6,0000.428571600015002590.909091
$F$11Pressing Used7,5007.142857750033001500
$F$12Pine chips Used30,0000.1428573000027333.333338250
$F$13Oak chips Used33,2140.000000625001E+3029285.71429
a.No.
b. Yes.
c. No, the price on Tahoe panels can go down by $50 without changing the optimal solution.d.Yes, the allowable increase on the profit coefficient for pallets of Aspen panels is $7.14, so the solution would change.e.This is a binding constraint with a shadow price of ~$7.14 and an allowable increase of 3,300. They should be willing to pay ~$7143 for 1,000 additional units of pressing capacity.f.Pine chips are a binding constraint worth ~$0.14 per pound. Reducing the amount available by 5,000 pounds would reduce profit by about $714. If they could sell the same 5,000 pounds for $1,250 they should do so as they would be ahead by $1250  $714.g.Profit is most sensitive to changes in pressing capacity.
h.As pressing capacity increases from 5000 to 6000, we make fewer Tahoes and more Savannahs. Beyond 6000, we start making Pacifics and fewer Savannahs.
PressingTahoePacificSavannahAspenProfit
$G$11$B$6$C$6$D$6$E$6$F$7
5,00042.85714286028.571428570$42,143
5,50038.57142857035.714285710$45,929
6,00034.28571429042.857142860$49,714
6,50030.71428571542.142857140$53,286
7,00027.142857141041.428571430$56,857
14.See file: Prb4_14.xlsmMicrosoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine) Decision Variable Cells
CellNameFinalValueReducedCostObjectiveCoefficientAllowableIncreaseAllowableDecrease
$C$11Feed 1153.25023.251E+30
$D$11Feed 29.50.0002.51E+301.500000033
$E$11Feed 38.50.00034.50000016.5000002
Constraints
CellNameFinalValueShadowPriceConstraintR.H. SideAllowableIncreaseAllowableDecrease
$F$6Min A 980.0064341E+30
$F$7Min B 802.50805.59
$F$8Min C 320.0016161E+30
$F$9Min D 2000.00128721E+30
$F$6Max A 980.001281E+3030
$F$7Max B 800.001601E+3080
$F$8Max C 322.25326.3333333333.666666667
$F$9Max D 2000.002561E+3056
a.No.b. Yes.
c.Feed 1 is at its upper bound, increasing it further (if possible) would reduce the objective function; hence its negative reduced cost. Feeds 2 and 3 are between the upper and lower bounds and thefore have reduced costs of zero. (If there reduced costs were not zero, that would indicate further imporvements could be made in the objective by changing their values.)
d.A cost increase of $3 for Feed 3 is within the range of its allowable increase; so the optimal solution would not change but the objective function would change (increase) by $3*8.5=$25.50.e.If possible, they should reduce the minimum requirement for nutrient B. It is binding at its lower bound and loosening that requirement would improve the objective by $2.5 per unit loosened (up to 9 units).
f.If possible, they should increase the maximum requirement for nutrient C. It is binding at its upper bound and loosening that requirement would improve the objective by $2.25 per unit loosened (up to 6.333 units).
g. All the costs coefficients vary in a positive linear fashion with total cost with changes in the cost of Feed 1 having the greatest impact.
15.See file: Prb4_15.xlsmMicrosoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Objective Cell (Max)
CellNameFinal Value
$F$6Total Profit6925
Decision Variable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$B$5Quantity Product 16.670.00908.33333E093.7500006
$C$5Quantity Product 241.250.001206.66666781.11111E08
$D$5Quantity Product 39.170.0015030.00000530.000003
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$E$9Machine 1 Hrs Used10504001E+30295
$E$10Machine 2 Hrs Used240152406.6666666718.3333333
$E$11Machine 3 Hrs Used32003203.3333333327.5
$E$12Total Labor Hrs Used665566550.41666675
a.No.
b.The shadow price is $0 indicating that additional units of this resource will not help increase profit as this is a nonbinding constratint.
c.The shadow price is $5, indicating that additional units of labor would help to increase profits provided they can be obtained at no more than $5 per hour above the current labor cost.d.The allowable decrease on the objective function coefficient for product 2 is $0. So if that coefficient is an estimate it has no wiggle room in the downward direction without affecting the optimal solution.
16.See file: Prb4_16.xlsmMicrosoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Objective Cell (Min)
CellNameFinal Value
$H$7Cost / lb Total5.85
Decision Variable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$C$6Amount (lbs) Raisins0.7502.50.20000001E+30
$D$6Amount (lbs) Grain0.101.51E+300.500000
$E$6Amount (lbs) Chocolate0.1021E+300.500000
$F$6Amount (lbs) Peanuts0.9503.51E+300.333333
$G$6Amount (lbs) Almonds0.1031E+302.500000
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$H$10 Vitamins (grams) Total47.50407.51E+30
$H$11 Minerals (grams) Total150.5150.10.5
$H$12 Protien (grams) Total12.90102.91E+30
$H$13 Calories Total738.50600138.51E+30
$H$6Amount (lbs) Total20.82520.0689650.013245
$C$6Amount (lbs) Raisins0.75000.651E+30
$D$6Amount (lbs) Grain0.10.500.0333330.1
$E$6Amount (lbs) Chocolate0.10.500.050.125
$F$6Amount (lbs) Peanuts0.95000.851E+30
$G$6Amount (lbs) Almonds0.12.500.0250.0833333
$C$6Amount (lbs) Raisins0.75001E+300.25
$D$6Amount (lbs) Grain0.1001E+300.9
$E$6Amount (lbs) Chocolate0.1001E+300.9
$F$6Amount (lbs) Peanuts0.95001E+300.05
$G$6Amount (lbs) Almonds0.1001E+300.9
a.No.b.Yes, it has a maximum allowable increase of $0.20 (or to $2.70).
c.No, it has a maximum allowable decrease of $0.33 (or to $3.17).d.The constraint on the mineral content of the mix is binding and, therefore, would lead to a reduction in the price of the mix if this constraint could be loosened.17.See file: Prb4_17.xlsmTotal Profit = $215,000
Microsoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Adjustable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$B$4Quantity Made HyperLink50043.666666675343.666666671E+30
$C$4Quantity Made FastLink1000048161E+30
$D$4Quantity Made SpeedLink1500033151
$E$4Quantity Made MicroLink225003215.272727273
$F$4Quantity Made EtherLink5009.666666667389.6666666671E+30
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$G$13PC Board (sq in) Used60,5000800001E+3019500
$G$14Resistors Used100,0000100000105006000
$G$15Memory chips Used30,0007.53000020002333.333333
$G$16Assembly Hours Used3,688050001E+301312.5
$G$17Fast/Hyper>2 Used0160500875
a. The constraints on the number of resistors and memory chips are binding. Also the constraint requiring twice as many Fastlink cards as Hyperlink cards is binding.
b. Hyperlinks. The company would earn $43.67 for each Hyperlink it could avoid producing.
c. Yes. Profits would increase by $7.51,000=$7,500.
d. The solution would change if the objective coefficients on the Speedlink decreased by $1 or if the objective coefficient on the Microlink increased by $1. Thus, the objective coefficients on these products would be of greatest concern.
e.See the file. Total profit appears to be most sensitive to changes in the unit profit for Microlinks.
18.See file: Prb4_18.xlsm
Total Cost = $2,9750,000
Microsoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Adjustable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$B$4Number to Make Electric30,0000557.0000000031E+30
$C$4Number to Make Gas10,0000851014.00000001
$B$5Number to Buy Electric07671E+307.000000003
$C$5Number to Buy Gas5,00009514.0000000110
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$B$10Total Available Electric30,00060300002000010000
$C$10Total Available Gas15,00095150001E+305000
$D$14Production Used10,00025100008004000
$D$15Assembly Used14,0000150001E+301000
$D$16Packaging Used4,000050001E+301000
a. The current solution remains optimal until the cost of producing electric trimmers increases by $7 to $62.
b. The solution would not change. The allowable increase is $10.c. The shadow price in the assembly area is zero. There is already a surplus of assembly capacity so the company should not pay anything to acquire more of it.d. The company should be willing to pay $25 per production hour above and beyond what it is already
paying. Production capacity is a binding constraint.
e.See the file. Total profit is most sensitive to the cost of making the electric trimmer.f.Every 100 unit increase in production capacity results in 250 more gas trimmer being made instead of bought.
19.See file: Prb4_19.xlsmMicrosoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Objective Cell (Max)
CellNameFinal Value
$C$13Total profit Baskets1526.5
Decision Variable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$C$6Baskets302.51.59502E072.23607E07
$D$6Juice8701.752.23607E071.59502E07
$C$7Baskets002.501E+30
$D$7Juice22501.751E+302.11058E07
$C$8Baskets11402.51.11803E075.31672E08
$D$8Juice18601.755.31672E081.11803E07
$C$9Baskets07.08329E162.501E+30
$D$9Juice10001.751E+307.97508E08
$C$10Baskets7502.51E+302.23607E07
$D$10Juice07.4476E161.757.4476E161E+30
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$C$17Actual Wghtd Grade Baskets7200.6015310
$D$17Actual Wghtd Grade Juice14950.60103.636363610
$E$6Total900.859069.090909096.666666667
$E$7Total2251.45225207.272727320
$E$8Total3002.0530020207.2727273
$E$9Total1002.651006.66666666769.09090909
$E$10Total753.257524812
a. $1,526.5b. Yes. Some students might already have this solution. You tend to get a different (alternate) optimal solution depending on whether you use the Standard LP engine or the Gurobi engine. Either way, to make sure the solver is using the most grade 5 oranges for fruit baskets, students should try to maximize that quantity while holding the optimal profit at $1526.5 as a constraint.Pounds used for
GradeBasketsJuice
1090
261.5163.5
30300
455.544.5
5750
c. Yes. The shadow price on grade 4 oranges is $2.65 with an allowable increase in the resource of 6.6667 (in 1000s).d.See file. Increases in the requirements for the fruit baskets have the least negative impact on profit.20.See file: Prb4_20.xlsmMicrosoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Objective Cell (Max)
CellNameFinal Value
$E$16Total Profit 5012.5
Decision Variable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$E$5Regular150.000.003.751E+300.0000001
$F$5Supreme0.000.007.7501E+30
$E$6Regular95.650.005.252.09091E082.0000001
$F$6Supreme254.350.009.251E+302.09091E08
$E$7Regular54.350.003.251.76923E072.09091E08
$F$7Supreme195.650.007.252.09091E081.76923E07
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$E$11Actual Octane Level Regular97.66670.0000907.6666666671E+30
$F$11Actual Octane Level Supreme97.00000.0000972.7777777784.888888889
$E$8Available Regular3003.253005020.90909091
$F$8Available Supreme4507.254502011.36363636
$G$5 Used1500.515054.3478260950
$G$6 Used350235054.3478260950
$G$7 Used25003001E+3050
a. Yes, as reflected by the allowable increases and decreases of zero on an objective coefficients.
b. Note that the sensitivity information cannot be used to answer this (and the following) questions. Add a constraint holding the objective at its optimal value and then maximize E11 and F11 separately. Regular octane rating = 97.67, supreme octane rating = 97.
c. Regular octane rating = 90.0, supreme octane rating = 102.11.
d. The solution in part c should be implemented. The company would receive negative publicity if the answer in part b were implemented and the public learned that the inexpensive regular gas had a higher octane rating than the more expensive supreme gas.
e. This cannot be answered from the given information. Based on the above sensitivity report, they should be willing to buy an additional ~54 barrels at this price, but beyond that the shadow price it not guaranteed to hold.
21.See file: Prb4_21.xlsmTotal Cost = $730
Microsoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Adjustable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$C$10Location 1 Location 3026541E+3026
$D$10Location 1 Location 41001751E+30
$E$10Location 1 Location 5102325
$F$10Location 1 Location 6503052
$C$11Location 2 Location 3902425
$D$11Location 2 Location 405181E+305
$E$11Location 2 Location 5901952
$F$11Location 2 Location 605311E+305
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$C$12Actual Location 390101E+301
$D$12Actual Location 410111011
$E$12Actual Location 51051041
$F$12Actual Location 650101E+305
$C$12Location 3 Received90541E+30
$D$12Location 4 Received100551E+30
$E$12Location 5 Received100551E+30
$F$12Location 6 Received52511
$G$10Location 1 Shipped16281611
$G$11Location 2 Shipped18241814
a.
Yes. None of the allowable increases or decreases on the objective function coefficients are zero.
b. Location 6.
c.
$26
d.
The allowable increase for the RHS of the last constraint is 1 so the shadow price of this constraint will not hold if the RHS is increased from 5 to 8. Thus, this question cannot be answered with the information available in this sensitivity report.
22.See file: Prb4_22.xlsm
Total Cost = $702
Microsoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Adjustable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$C$10Location 1 Location 3031541E+3031
$D$10Location 1 Location 4100175.00000010.0000001
$E$10Location 1 Location 505231E+305
$F$10Location 1 Location 650300.00000017.0000001
$C$11Location 2 Location 380247.00000015.0000001
$D$11Location 2 Location 400180.00000015.0000001
$E$11Location 2 Location 5100195.00000011E+30
$F$11Location 2 Location 600311E+300
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$C$12Actual Location 380101E+302
$D$12Actual Location 41061030
$E$12Actual Location 51051032
$F$12Actual Location 650101E+305
$C$12Actual Location 380531E+30
$D$12Actual Location 4100551E+30
$E$12Actual Location 5100551E+30
$F$12Actual Location 657530
$G$10Location 1 Shipped15231503
$G$11Location 2 Shipped18241823
a.
No. At least one of the allowable increases or decreases on the objective function coefficients is zero.
b. $31.c. $31.
23. See file: Prb4_23.xlsm
Total Cost = $3,011,360
Microsoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Adjustable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$B$15Macon Tacoma00.55381E+300.55
$C$15Macon San Diego01.838.251E+301.8
$D$15Macon Dallas00.2537.251E+300.25
$E$15Macon Denver01.137.51E+301.1
$F$15Macon St. Louis0137.61E+301
$G$15Macon Tampa12000037.30.10.85
$H$15Macon Baltimore6000037.150.250.1
$B$16Louisville Tacoma600039.350.10.05
$C$16Louisville San Diego01.0539.41E+301.05
$D$16Louisville Dallas00.1391E+300.1
$E$16Louisville Denver00.839.11E+300.8
$F$16Louisville St. Louis14400038.50.050.15
$G$16Louisville Tampa00.239.41E+300.2
$H$16Louisville Baltimore00.339.351E+300.3
$B$17Detroit Tacoma400041.30.050.05
$C$17Detroit San Diego00.9541.251E+300.95
$D$17Detroit Dallas10800040.850.10.55
$E$17Detroit Denver12600040.250.051E+30
$F$17Detroit St. Louis00.0540.51E+300.05
$G$17Detroit Tampa00.141.251E+300.1
$H$17Detroit Baltimore12000410.10.25
$B$18Phoenix Tacoma5800038.150.050.15
$C$18Phoenix San Diego14200037.150.150.05
$D$18Phoenix Dallas00.1537.851E+300.15
$E$18Phoenix Denver00.9381E+300.9
$F$18Phoenix St. Louis00.9538.251E+300.95
$G$18Phoenix Tampa00.7538.751E+300.75
$H$18Phoenix Baltimore01.0538.91E+301.05
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$B$19Total Shipped Tacoma6800085001E+301700
$C$19Total Shipped San Diego142000145001E+30300
$D$19Total Shipped Dallas108000135001E+302700
$E$19Total Shipped Denver126000.0512600400300
$F$19Total Shipped St. Louis144000180001E+303600
$G$19Total Shipped Tampa120000150001E+303000
$H$19Total Shipped Baltimore7200090001E+301800
$B$19Total Shipped Tacoma6800168001700300
$C$19Total Shipped San Diego1420001160026001E+30
$D$19Total Shipped Dallas108000.5510800400300
$E$19Total Shipped Denver1260001008025201E+30
$F$19Total Shipped St. Louis144000.1514400600300
$G$19Total Shipped Tampa120000.8512000400300
$H$19Total Shipped Baltimore72000.77200400300
$I$15Macon Produced1800036.4518000300400
$I$16Louisville Produced1500038.3515000300600
$I$17Detroit Produced2500040.325000300400
$I$18Phoenix Produced2000037.15200003002600
a. Yes.
b. Macon. Each additional unit of capacity there increases costs by $36.45 (which is the cheapest way to increase capacity).
c. Yes. The allowable increase in the objective coefficient for this variable is $0.05. So an $0.08 increase in shipping cost (from $1.90 to $1.98) should (and does) result in a new optimal solution.d. Yes. Several of the 80% constraints are binding. Relaxing these constraints would allow the company to distribute the products it can make in a less costly way to meet more of the demand that currently is not being met because of the 80% constraints. As long as the product sells for the same price in all markets, lowering/relaxing the 80% constraints should result in more profit.
e. This would cause the total cost to increase by $1 per unit or $1700 in total. So the company would likely want to pass this cost on to the distributor. 24.See file: Prb4_24.xlsmTotal Cost = $44,067.73
Microsoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Adjustable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$C$10Newspaper Newsprint588.240.0021.50.5198412724.64285714
$D$10Newspaper Packaging11.760.00263.1428571430.51984127
$E$10Newspaper Print Stock0.0018.14151E+3018.14285714
$C$11Mixed Paper Newsprint0.000.5525.751E+300.550420168
$D$11Mixed Paper Packaging71.430.0028.250.5504201681.357142857
$E$11Mixed Paper Print Stock428.570.0025.51.35714285726.39285714
$C$12White Office Paper Newsprint0.001.8723.751E+301.871848739
$D$12White Office Paper Packaging300.000.0026.751.5510204081E+30
$E$12White Office Paper Print Stock0.001.5527.51E+301.551020408
$C$13Cardboard Newsprint0.001.3124.51E+301.306722689
$D$13Cardboard Packaging397.780.0025.51.214843750.78125
$E$13Cardboard Print Stock0.0017.00171E+3017
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$C$14Pulp Available Newsprint500.0028.995001.651785714295.8482143
$D$14Pulp Available Packaging600.0036.436001.554621849278.4453782
$E$14Pulp Available Print Stock300.0037.703001.360294118243.6397059
$F$10Newspaper Used600.003.14600348.05672271.943277311
$F$11Mixed Paper Used500.000.89500348.05672271.943277311
$F$12White Office Paper Used300.004.21300327.58279781.828966881
$F$13Cardboard Used397.780.004001E+302.220888355
a. No.
b. Yes.
c. $0. There is a surplus of cardboard already.
d. Yes. The recycler currently pays $15 per ton. The shadow price is $3.14. So the recycler should be willing to pay up to $18.14 to acquire up to 348 additional tons.
e. Newspaper $28.99, Packaging Paper $36.43, Print Stock Paper $37.70.
f.
$1.87.
g. The yield on newsprint per ton of cardboard would have to increase from 0.0 to 24.499999/28.9915 = 0.845072. (This is the reduced cost of using cardboard for newsprint divided by the marginal value (shadow price) of newsprint.) First of all, information about changes to constraint coefficients is generally not directly available from the sensitivity report. Usually you'll want to use ad hoc sensitivity analysis (like a Solver table) to answer these types of questions. However, in this case, from the sensitivity report we know that the cost of converting 1 ton of cardboard into 0.8 tons of newsprint is $24.5. Also, from the sensitivity report we know that the marginal value of a ton of newsprint is ~$28.9915. So, if we spend $24.5 to get 0.8 tons of newsprint worth $28.9915 per ton, we'll have exchanged $24.5 for something that is worth 0.8*$28.9915 = ~$23.193  so we would have lost ~$1.307 in that transaction. That's why we aren't using cardboard to make newsprint. Also note that ~1.307 is the reduced cost in the sensitivity report for the variable that represents the amount of cardboard being converted to newsprint.
So, in this case, $24.50  0.8*$28.9915 turned out to be negative (that is, a bad deal where we lose value). Question g boils down to asking "What would the 0.8 in this equation have to be in order to get a nonnegative value?" So let's change the 0.8 factor into a variable called Q. Then, mathematically, we want to find the value of Q where $24.50Q*$28.9915=0.
Doing a little algebra on that gives us Q = 24.5/28.9915 = 0.845072.
So if we could convert a ton of cardboard into about 0.845 tons of newsprint (instead of 0.8 tons of newsprint), it would become economically feasible to do so.
25. See file: Prb4_25.xlsm
Total Cost = $664,200
Microsoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Adjustable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$C$6Forward30006900.0000001
$D$6Forward Commodity25000501E+300.0000001
$E$6Forward185.71428570600.00000018.07692E08
$F$6Forward008001E+30
$C$7Center45000691E+306.66667E08
$D$7Center Commodity005001E+30
$E$7Center006001E+30
$F$7Center008001E+30
$C$8Rear006901E+30
$D$8Rear Commodity005001E+30
$E$8Rear1014.28571406000.0000001
$F$8Rear17000801E+300.0000001
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$C$9Loaded480069480022.5300
$D$9Loaded Commodity250050250022.52130
$E$9Loaded120060120022.5354.5454545
$F$9Loaded170080170027.27272727354.5454545
$G$6Forward weight2985.714286030001E+3014.28571429
$G$7Center weight4500060001E+301500
$G$8Rear weight2714.285714040001E+301285.714286
$I$6Forward volume85642.8571401450001E+3059357.14286
$I$7Center volume180000018000012000900
$I$8Rear volume154357.142901550001E+30642.8571429
$D$19Forward max < 00.000.0003022.5
$D$20Center max < 03134.290.0001E+303134.285714
$C$19Forward min > 0542.860.000542.85714291E+30
$C$20Center min > 0420.000.0004201E+30
a.No.
b.No.
c.$0. (Note that the allowable decrease for three different adjustable cells need to be considered.)
d.Total profit is most sensitive to changes in the marginal profit/price of commodity one.
26.See file: Prb4_26.xlsm
Total cost = $16,625
Microsoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Adjustable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$C$615030017550
$C$701253001E+30125
$C$810030015050
$C$902253001E+30225
$C$1001753001E+30175
$D$60505251E+3050
$D$70505251E+3050
$D$801505251E+30150
$D$903255251E+30325
$E$6Length of Lease (in months)0077550150
$E$7Length of Lease (in months)02257751E+30225
$E$8Length of Lease (in months)02757751E+30275
$F$65085012575
$F$701758501E+30175
$G$650975175125
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$H$6Avail.25300251E+3015
$H$7x Avail.1017510100
$H$8x Avail.20300201E+3010
$H$9x Avail.10751005
$H$10x Avail.5125555
a.Yes.
b.Because of the degeneracy, we cannot tell from the sensitivity report.
c.$425
d.The solutions in months 1 and 2 do not change within this region.
27. See file: Prb4_27.xlsm
a.
$H$21
$H$161265.471273.91282.31290.81299.21307.61309.91309.91309.91309.91309.9
b. 360,000 cf, Total profit ~$1,309,903c.~$42,140
d. At least $105,98028.a.MAX 4 X1 + 2 X2
ST2 X1 + 4 X2 + S1 = 20
3 X1 + 5 X2 + S2 = 15
X1, X2 , S1, S2 SYMBOL 179 \f "Symbol" 0
b.{X1, X2}, {X1, S1}, {X1, S2}, {X2, S1}, {X2, S2}, {S1, S 2}
c.
X1X2S1S2ObjectiveFeasible?
120150050no
25010020yes
310001540no
403806yes
50501010no
60020150yes
e.See table in c.
f.X1 = 5, X2 = 0
g.The second constraint is binding; the first is redundant.
29.a.MAX 2 X1 + 4 X2
ST1 X1 + 2 X2 + S1 = 8
1 X1 + 2 X2 + S2 = 12
1 X1 + 2 X2  S3 = 2
X1, X2 , S1, S2, S3 SYMBOL 179 \f "Symbol" 0
b.{X1, X2, S1}, {X1, X2, S2}, {X1, X2, S3}, {X1, S1, S2}, {X1, S1, S3},
{X1, S2, S3}, {X2, S1, S2}, {X2, S1, S3}, {X2, S2, S3}, {S1, S2, S3}
c.
X1X2S1S2S3ObjectiveFeasible?
1810200024no
21.333.3306.67010.67no
32500524yes
420101004yes
51202001024yes
6800201016no
7024808yes
80640424no
90404216yes
100081220no
e.See table in c.
f.There are alternate optimal solutions. One is given by X1 = 2, X2 = 5, S3 = 5.
g.The first and second constraints are binding.
30.a.MIN5 X1 + 3 X2 + 4 X3
STX1 + X2 + 2 X3  S1 = 2
5 X1 + 3 X2 + 2 X3  S2 = 1
X1, X2 , X3, S1, S2 SYMBOL 179 \f "Symbol" 0
b.{X1, X2}, {X1, X3}, {X1, S1}, {X1, S2}, {X2, X3},
{X2, S1}, {X2, S2}, {X3, S1}, {X3, S2}, {S1, S2}
c.
X1X2X3S1S2ObjectiveFeasible?
12.54.50001no
20.2501.125003.25no
30.2001.801no
42000910yes
500.51.25003.5no
600.3301.6701no
7020056yes
8000.5102no
9001014yes
10000210no
d.See table in c.
e.The optimal solution is X3 = 1, S2 = 1.
f.The first constraint is binding.
31.a.2X1 + 4X2 SYMBOL 163 \f "Symbol" 16
b. i.S=4
ii.S=0
iii.S=2
iv.S=4
32.a.31 + 4X2 SYMBOL 179 \f "Symbol" 12
b. i.S=3
ii.S=2
iii.S=13
iv.S=0
Case 41: A Nut Case
See file: Case4_1.xlsm.
1.No. The fixed costs should not be included in the calculation of unit profit margins.2.X1 = pounds of Whole product to produce
X2 = pounds of Cluster product to produce
X3 = pounds of Crunch product to produce
X4 = pounds of Roasted product to produce
MAX 1.93 X1 + 1.04 X2 + 1.15 X3 + 1.33 X4
ST
1 X1 + 1 X2 + 1 X3 + 1 X4 < 3600
2 X1 + 1.5 X2 + 1 X3 + 1.75 X4 < 3600
1 X1 + 0.7 X2 + 0.2 X3 + 0.00 X4 < 3600
2.5 X1 + 1.6 X2 + 1.25 X3 + 1.0 X4 < 3600
0.6 X1 + 0.4 X2 + 0.2 X3 + 1 X4 < 11000.4 X1 + 0.6 X2 + 0.8 X3 + 0 X4 < 800
1,000 < X1 < 99,999
400 < X2 < 500
0 < X3 < 150
0 < X4 < 200
3. See file: Case4_1.xlsm4.X1 = 1029, X2 = 400, X3 = 150, X4=200, Total profit = $2,839.765.
6.No.7.Yes.8.The company could increase profit by making fewer pounds of the Cluster product (it has a negative reduced cost).9.Each additional unit of the Roasted product adds about $0.55 to profit.
10.Packaging is the only binding constraint (aside from the upper and lower variable bounds). Each additional unit of packaging material is worth $0.772 (above and beyond its normal variable cost).11.$0. It is a nonbinding resource.12.No, it could decrease by $0.305 without changing the optimal solution.13.
The Whole product is the only one that would be effected. The solution appears to be rather insensitive to changes in required roasting times.
14.
Note that scenarios 1 & 2 for Chocolate are infeasible.Case 42: Parket Sisters
(contributed by Jack Yurkiewicz, Pace University)
Let X1 = ballpoints produced, X2 = pencils produced, X3 = fountain pens produced
The LP formulation for this problem is:
MAX 3.0X1 + 3.0X2 + 5.0X3
ST1.2X1 + 1.7X2 + 1.2X3 SYMBOL 163 \f "Symbol" 1000
0.8X1 + 0.0X2 + 2.3X3 SYMBOL 163 \f "Symbol" 1200
2.0X1 + 3.0X2 + 4.5X3 SYMBOL 163 \f "Symbol" 2000
X1, X2, X3 SYMBOL 179 \f "Symbol" 0
See file: Case4_2.xlsm.
Total Profit = $2,766.67
The sensitivity report is shown below:
Microsoft Excel Sensitivity Report (Obtained Using Standard LP/Quadratic Engine)
Adjustable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$B$2Number to Produce: Ballpoints700.000.000032.0000.778
$C$2Number to Produce: Pencils0.001.383331.3831.00E+30
$D$2Number to Produce: Fountain pens133.330.000051.7502.000
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$E$6 Plastic Total Used:1000.001.16671000200.000466.667
$E$7 Chrome Total Used:866.670.000012001.00E+30333.333
$E$8 Stainless Steel Total Used:2000.000.80002000555.556333.333
1. The optimal weekly product mix is: 700 ballpoints, 0 pencils, and 133.33 fountain pens. We could round the fountain pens off to 133, but we shall not, in order to avoid rounding problems. The profit is $2,766.67.
2.No. None of the allowable increases or decreases on the constraints are zero.
3.Yes, the answer is unique. In the absence of degeneracy, we see that there are no variables which have a value of zero and simultaneously a zero reduced cost.
4. What is the marginal values for one more unit of each of the resources are: $1.17 for plastic, 0 for chrome, and $0.80 for stainless steel. These are the shadow prices for our three resources.
5. We can consider buying the 500 additional ounces of stainless steel because 500 is less than the maximum allowable increase of 555.56 ounces. Thus, the shadow price for stainless steel (and the other resources as well) remains the same. Now as to whether or not the deal is worthwhile, because the 80 cent shadow price for an additional ounce of stainless steel is more than the 60 cents the distributor is charging Parket (above which they ordinarily charge when the model was formulated), it pays to buy the 500 ounces they are willing to sell. Parket will be ahead 20 cents for each ounce they buy.
6. Because we are buying the 500 ounces, each ounce bought will increase our profit by 20 cents. Thus the 500 ounces will increase our profit by 500$0.20 = $100. Our new profit is then $2766.67 + $100 = $2866.67. Once we change the RHS of a constraint, we generally cannot tell what the new product mix will be. To get that, we must resolve the problem.
7.We cannot tell without resolving the problem. Five hundred additional ounces of plastic is above the allowable increase of 200 ounces. That is, we are guaranteed that the shadow prices given will remain the same so long as we have no more than 1200 (1000 + 200) ounces of plastic. Above that value, the shadow prices are different that those given. Thus, we cannot make any analysis at this point.
8. We can buy as many as 200 additional ounces of plastic before we run out of information from the computer output. Parket ordinarily pays $5.00 per ounce, and now the distributor is charging $6.00 per ounce, or $1.00 more than Parket paid. Still, the shadow price for plastic is $1.17 (or $1.1667 to be exact) which is more than the $1.00 the distributor is overcharging. Parket will still be ahead about 17 cents (or 16.6667 cents) for each ounce they buy. Buying the 200 ounces will increase their profit by (200)($0.17) = $34 (or $33.33 to be exact). The new profit will thus be $2766.67 + $33.33 = $2800.00. Again, we cannot tell what the new product mix will be without resolving the problem.
9.We can consider selling the 300 ounces of plastic because the allowable decrease of plastic is 466.67. That is, Parket can have as little as 533.33 ounces of plastic (1000  466.67) and still be assured that the shadow price plastic (and the others) will remain the same. Selling an ounce of plastic reduces Parkets profit by $1.17 (or $1.1667 to be exact), yet the company is willing to pay $6.50 per ounce. This represents a yield of $1.50 above what Parket ordinarily pays (they pay $5.00 per ounce, if you recall). Thus Parket will be ahead by ($1.50  $1.1667)300 = $100 if they sell the plastic. The new profit will be $2766.67 + $100 = $2866.67. Once again, we cannot tell what the new product mix will be unless we solve the problem.
10.Parket had ordered 1200 ounces of chrome, yet the optimal solution uses only 866.67 ounces. There are 333.33 ounces left over. That is why the shadow price for chrome is zero. If only 1000 ounces are delivered, Parket will still have 133.33 ounces of chrome left over. Because the shadow price is zero, and the 200 ounce shortage from the distributor (1200  1000) is less than the allowable decrease of 333.33 ounces, the shadow price is still zero. Thus the new profit will not change and remain at $2766.67. We do not have to solve the problem again here to get the new product mix. The product mix stays the same as: 700 ballpoints, zero pencils, and 133.33 fountain pens. The only difference is that the slack for chrome is now 133.33, instead of the original 333.33.
11. Price out the new design. It is Profit  Cost = $3  [(1.1)($1.17) + (2)($0) + (2)($0.80)] = $0.12, approximately. Because this value is positive, Parket should give the goahead for this new design.
12. If the profit on ballpoints were to decrease to $2.50, down from the original assumed $3.00, this $0.50 decrease is still less than the allowable decrease of $0.78 for ballpoints. The optimal solution stays the same, but the total profit now drops by ($3.00  $2.50)(700 ballpoints) = $350. The new profit is $2416.67.
13. Price out the new felt tip pen: Profit Cost = Profit  [(1.8)($1.167) + (0.5)($0) + (1.3)($0.80)] = Profit 3.14. If we are to produce the felt tip then this amount should be nonnegative. Thus, Profit 3.14SYMBOL 179 \f "Symbol" 0, or
ProfitSYMBOL 179 \f "Symbol" $3.14. At exactly $3.14, we have alternative optima and would be indifferent between producing or not producing the felt tip pens.
14.It should be at least $4.38 ($3.00 plus the reduced cost of $1.38).
15.Currently it does not pay to make pencils; the profit of $3.00 is not high enough. Yet if Parket insists on making 20 pencils per week, their profit would be down by (20)($1.383) = $27.66. That is, they would have a total profit of $2766.67  $27.66 = $2739.
16.At $6.75, or $1.75 more than the assumed $5.00, we are just at the limit of the allowable increase for the profit of fountain pens. We have alternative optima at this point. Still, the optimal solution stays the same. The new profit is $2766.67 + ($1.75)(133.33 fountain pens) = $3000.
Case 43: Kamm Industries
See file: Case4_3.xlsm1.Total cost: $1,902,864Make
CarpetDobbiePanteraBuy
10na14,000
252,000na0
344,000na0
420,000na0
5077,5000
600109,500
70120,0000
8054,0755,925
97,50000
10069,5000
1122,041046,459
12083,0000
13010,0000
140381,0000
15064,0000
Microsoft Excel Sensitivity Report
Adjustable Cells
FinalReducedObjectiveAllowableAllowable
CellNameValueCostCoefficientIncreaseDecrease
$C$6Dobbie002.661E+300.0228248
$C$7Dobbie52,00002.550.05509591E+30
$C$8Dobbie44,00002.640.08058981E+30
$C$9Dobbie20,00002.560.02924811E+30
$C$10Dobbie001.611E+300.0080232
$C$11Dobbie001.621E+300.0173936
$C$12Dobbie001.641E+300.0248079
$C$13Dobbie001.481E+300.0040811
$C$14Dobbie7,50001.50.00710381E+30
$C$15Dobbie001.441E+300.0162417
$C$16Dobbie22,04101.640.00225230.00991
$C$17Dobbie001.571E+300.0043207
$C$18Dobbie001.491E+300.0044589
$C$19Dobbie001.311E+300.0033658
$C$20Dobbie001.511E+300.0019973
$D$10na77,50001.60.00802321E+30
$D$11na001.611E+300.0133342
$D$12na120,00001.610.00108431E+30
$D$13na54,07501.470.00543690.0008738
$D$14na001.51E+300.0071038
$D$15na69,50001.420.01624171E+30
$D$16na001.641E+300.0075932
$D$17na83,00001.560.00432071E+30
$D$18na10,00001.480.00445891E+30
$D$19na381,00001.30.00336581E+30
$D$20na64,00001.50.00199731E+30
$E$6na Buy14,00002.770.02282481E+30
$E$7na Buy002.731E+300.0550959
$E$8na Buy002.851E+300.0805898
$E$9na Buy002.731E+300.0292481
$E$10Buy001.761E+300.0415917
$E$11Buy109,50001.760.01333421E+30
$E$12Buy001.761E+300.0010843
$E$13Buy5,92501.590.00087380.0054369
$E$14Buy001.711E+300.0953073
$E$15Buy001.631E+300.0913951
$E$16Buy46,45901.80.00759320.0022523
$E$17Buy001.781E+300.0702167
$E$18Buy001.631E+300.0089908
$E$19Buy001.441E+300.0199327
$E$20Buy001.691E+300.0401119
Constraints
FinalShadowConstraintAllowableAllowable
CellNameValuePriceR.H. SideIncreaseDecrease
$F$6na Total14,0002.770140001E+3014000
$F$7na Total52,0002.6755200028234.69552000
$F$8na Total44,0002.7694400027251.54444000
$F$9na Total20,0002.7012000025055.64320000
$F$10Total77,5001.7187750054802.4166004.1565
$F$11Total109,5001.7601095001E+30109500
$F$12Total120,0001.75912000043575.3924774.123
$F$13Total60,0001.590600001E+305924.514
$F$14Total7,5001.615750030748.5027500
$F$15Total69,5001.5396950054711.5495994.2012
$F$16Total68,5001.800685001E+3046458.57
$F$17Total83,0001.7108300043322.9854746.4693
$F$18Total10,0001.6211000046018.6835041.8101
$F$19Total381,0001.42038100054045.1975921.1956
$F$20Total64,0001.6506400043292.6974743.1509
$I$21Used Hr/Yd323700.59903237012408.8065887.1342
$K$21Used na1726400.64271726401106.145310096.245
2.No. 3.Yes.
4. The shadow price for capacity on the Dobbie looms is 0.599 per hour. Losing one loom for the quarter results in 13(7242)=2158 lost hours of capacity and, as a result, increase costs by about $1,293.5.By the same reasoning, this would reduce costs by about $1,293.6.The shadow price for capacity on the Pantera looms is 0.6427 per hour. Losing one loom for the quarter results in 13(7242)=2158 lost hours of capacity and, as a result, increase costs by about $1,387.
7.By the same reasoning, this would reduce costs by about $1,387.
8.The shawdow prices for carpets 1, 6, 8 and 11 are the same as the marginal prices of buying these products. The allowable increase of 1E+30 indicate we can buy as much as we want at these prices.9.The company is paying $2.55 per yard to fill the order for carpet 2 on the Doobie loom. However, the demand for carpet 2 is $2.675 per yard. Thus, if the company did not have to fill the order for carpet 2 it could save not only the $2.55 per yard cost of carpet 2 but then be able to use the extra capacity on Dobbie loom to fill the remaining orders in a more efficient manner saving an additional $0.125 on every yard of carpet 2 it doesn't have to make.10.The shadow price of carpet 14 is the smallest at $1.42 per yard. So selling more of this carpet would produce the highest marginal profit for the company.
11.Yes. It's current cost is $2.77 with an allowable increase of $0.02282. So a price increase to $2.80 would cause the optimal solution to change. 12.No. It's current cost is $1.69 with an allowable decrease of $0.0401. So a price decrease to $1.65 would not cause the optimal solution to change.