11 Chapter 3 Solutions

 

Case Problem 1: Product Mix

Note to Instructor: The difference between rele relevant vant and sunk costs is critical. critical. The cost of the shipment of nuts is a sunk cost. cost. Practice in aapplying pplying sensitivity sensitivity analysis to a business decisi decision on is obtained. You may want to suggest that sensitivity analyses other than the ones we have suggested be undertaken. 1.

Cost per pound of ingredients Almo Almond ndss Brazil Filb Fi lbeert rtss Pecans Waln Wa lnut utss

$7 $750 500/ 0/60 6000 00 = $1.2 $1.25 5 $7125/7500 = $.95 $6 $675 750/ 0/75 7500 00 = $.9 $.90 0 $7200/6000 = $1.20 $787 $7875/ 5/75 7500 00 = $1.0 $1.05 5

Cost of nuts in three mixes:

2.

Regular Regul ar m mix: ix:

.15($1.25) .15($1.25) + .25($.95 .25($.95)) + .25($90) .25($90) + .10($1.20 .10($1.20)) + .25($ .25($1.05) 1.05) = $1 $1.0325 .0325

Deluxe Del uxe mix

.20 .20($1 ($1.25 .25)) + .20 .20($. ($.95) 95) + .20($ .20($.90 .90)) + .20( .20($1. $1.20) 20) + .20($ .20($1.0 1.05) 5) = $1.07 $1.07

Holiday Holid ay mix: mix:

.25($ .25($1.25) 1.25) + .15($.95 .15($.95)) + .15($.90) .15($.90) + .25 .25($1.2 ($1.20) 0) + .20($1.05) .20($1.05) = $1. $1.10 10

Let

 R = pounds of Regular Mix produced  D = pounds of Deluxe Mix produced  H  =  = pounds of Holiday Mix produced

Note that the cost of the five shipments of nuts is a sunk (not a relevant) cost and should not affect the decision. decision. However, this information may may be useful to management management in future pricing and purchasing decisions. A linear programming model for the optimal product mix is given. The following linear programming model can be solved to maximize profit contribution for the nuts already purchased.

Max s.t.

1.65 R

+

2.00 D

+

2.25 H 

0.15 R 0.25 R 0.25 R 0.10 R 0.25 R    R

+ + + + +

0.20 D 0.20 D 0.20 D 0.20 D 0.20 D

+ + + + +

0.25 H  0.15 H  0.15 H  0.25 H  0.20 H 

 



 



 



 

 

 



   H 

MGTC74W07



 

   D  

 

 



6000 7500 7500 6000 7500 10000 3000 5000

Almonds Brazil Filberts Pecans Walnuts Regular Deluxe Holiday

R,  D,  H    0

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Chapter 3 – 11th ed.

 

The solution found using The Management Scientist  is  is shown below. Objective Function Value =

 

61375.000

Variable -------------R D H

Value --------------17500.000 10624.999 5000.000

Constraint

Slack/Surplus

-------------1 2 3 4 5 6 7 8

--------------0.000 250.000 250.000 875.000 0.000 7500.000 7624.999 0.000

Reduced Costs -----------------0.000 0.000 0.000

Dual Prices -----------------8.500 0.000 0.000 0.000 1.500 0.000 0.000 -0.175

OBJECTIVE COEFFICIENT RANGES   Variable  -----------  R D H

Lower Limit --------------1.500 1.892 No Lower Limit

Current Value --------------1.650 2.000 2.250

Upper Limit --------------2.000 2.200 2.425

Current Value --------------6000.000 7500.000 7500.000 6000.000 7500.000

Upper Limit --------------6583.333 No Upper Limit No Upper Limit No Upper Limit 7750.000

RIGHT HAND SIDE RANGES   Constraint  -----------  1   2   3   4   5    

MGTC74W07

6 7

Lower Limit --------------5390.000 7250.000 7250.000 5125.000 6750.000 No Lower Lower Limit Limit No

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10000.000 3000.000

17500.000 10624.999

Chapter 3 – 11th ed.

 

 

8

-0.000

5000.000

9692.307

3.

From tthe he dual pr prices ices it it can be se seen en that additi additional onal almond almondss are wor worth th $8.50 p per er poun pound d to TJ. Additional walnuts are worth $1.50 per pound. pound. From the slack vari variables, ables, we see that additional Brazil nut, Filberts, and Pecans are of no value since they are already in excess supply.

4.

Yes, pu purcha rchase se the almon almonds. ds. The du dual al price price sho shows ws that that eac each h pound iiss wort worth h $8.50; the du dual al price is applicable for increases up to 583.33 pounds.

Resolving the problem by changing the right-hand side of constraint 1 from 6000 to 7000 yields the following optimal optimal solution. The optimal solution solution has increased in value by $4958.34. Note that only 583.33 pounds of the additional additional almonds were used, but that the increase in profit contribution more than justifies the $1000 cost of the shipment. Objective Function Value =

 

66333.336

Variable -------------R D H

Value --------------11666.667 17916.668 5000.000

Reduced Costs -----------------0.000 0.000 0.000

Constraint -------------1 2 3 4 5 6 7 8

Slack/Surplus --------------416.667 250.000 250.000 0.000 0.000 1666.667 14916.667 0.000

Dual Prices -----------------0.000 0.000 0.000 5.667 4.333 0.000 0.000 -0.033

OBJECTIVE COEFFICIENT RANGES   Variable  -----------  R D H

Lower Limit --------------1.000 1.976 No Lower Limit

Current Value --------------1.650 2.000 2.250

Upper Limit --------------1.750 3.300 2.283

Current Value

Upper Limit

RIGHT HAND SIDE RANGES   Constraint

MGTC74W07

Lower Limit

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Chapter 3 – 11th ed.

 

 -----------  1   2   3   4   5   6   7  

8

5.

MGTC74W07

--------------6583.333 7250.000 7250.000 4210.000 7250.000 No Lower Limit No Lower Limit

--------------7000.000 7500.000 7500.000 6000.000 7500.000 10000.000 3000.000

0.002

5000.000

--------------No Upper Limit No Upper Limit No Upper Limit 6250.000 7750.000 11666.667 17916.668 15529.412

From th thee dual prices prices it is clear clear that there there is no advantag advantagee to not sati satisfyi sfying ng the orde orders rs for the Regular and Deluxe mixes. However, it would be advantageous to negot negotiate iate a decrease in the Holiday mix requirement.

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Chapter 3 – 11th ed.