Note to Instructor: The difference between relevant and sunk costs is critical. The cost of the shipment of nuts is a sunk cost. Practice in applying sensitivity analysis to a business decision 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
Almonds $7500/6000 = $1.25
Brazil
Filberts
Pecans
$7125/7500 = $.95
$6750/7500 = $.90
$7200/6000 = $1.20
Walnuts $7875/7500 = $1.05
Cost of nuts in three mixes:
Regular mix: .15($1.25) + .25($.95) + .25($90) + .10($1.20) + .25($1.05) = $1.0325
Deluxe mix .20($1.25) + .20($.95) + .20($.90) + .20($1.20) + .20($1.05) = $1.07
Holiday mix: .25($1.25) + .15($.95) + .15($.90) + .25($1.20) + .20($1.05) = $1.10
2. 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. However, this information may be useful to 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 +
0.15
R +
0.25
R
0.25
R
0.10
R
0.25
R
R
+
+
+
+
2.00
D
0.20
D
0.20
D
0.20
0.20
D
D
0.20
D
R , D , H  0
D
+
+
+
+
+
+
2.25
H
0.25
H
0.15
H
0.15
0.25
H
H
0.20
H
H
 6000
 7500
 7500
 6000
 3000
 5000
Almonds
Brazil
Filberts
Pecans
 7500 Walnuts
 10000 Regular
Deluxe
Holiday
MGTC74W07 Page 1 of 3 Chapter 3 – 11 th ed.

The solution found using The Management Scientist is shown below.
Objective Function Value = 61375.000
Variable Value Reduced Costs
-------------- --------------- ------------------
R 17500.000 0.000
D 10624.999 0.000
H 5000.000 0.000
Constraint Slack/Surplus Dual Prices
-------------- --------------- ------------------
1 0.000 8.500
2 250.000 0.000
3 250.000 0.000
4 875.000 0.000
5 0.000 1.500
6 7500.000 0.000
7 7624.999 0.000
8 0.000 -0.175
OBJECTIVE COEFFICIENT RANGES
Variable Lower Limit Current Value Upper Limit
------------ --------------- --------------- ---------------
R 1.500 1.650 2.000
D 1.892 2.000 2.200
H No Lower Limit 2.250 2.425
RIGHT HAND SIDE RANGES
Constraint Lower Limit Current Value Upper Limit
------------ --------------- --------------- ---------------
1 5390.000 6000.000 6583.333
2 7250.000 7500.000 No Upper Limit
3 7250.000 7500.000 No Upper Limit
4 5125.000 6000.000 No Upper Limit
5 6750.000 7500.000 7750.000
6 No Lower Limit 10000.000 17500.000
7 No Lower Limit 3000.000 10624.999
8 -0.000 5000.000 9692.307
3. From the dual prices it can be seen that additional almonds are worth $8.50 per pound to
TJ. Additional walnuts are worth $1.50 per pound. From the slack variables, we see that additional Brazil nut, Filberts, and Pecans are of no value since they are already in excess supply.
4. Yes, purchase the almonds. The dual price shows that each pound is worth $8.50; the dual price is applicable for increases up to 583.33 pounds.
MGTC74W07 Page 2 of 3 Chapter 3 – 11 th ed.

Resolving the problem by changing the right-hand side of constraint 1 from 6000 to 7000 yields the following optimal solution. The optimal solution has increased in value by
$4958.34. Note that only 583.33 pounds of the 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 Value Reduced Costs
-------------- --------------- ------------------
R 11666.667 0.000
D 17916.668 0.000
H 5000.000 0.000
Constraint Slack/Surplus Dual Prices
-------------- --------------- ------------------
1 416.667 0.000
2 250.000 0.000
3 250.000 0.000
4 0.000 5.667
5 0.000 4.333
6 1666.667 0.000
7 14916.667 0.000
8 0.000 -0.033
OBJECTIVE COEFFICIENT RANGES
Variable Lower Limit Current Value Upper Limit
------------ --------------- --------------- ---------------
R 1.000 1.650 1.750
D 1.976 2.000 3.300
H No Lower Limit 2.250 2.283
RIGHT HAND SIDE RANGES
Constraint Lower Limit Current Value Upper Limit
------------ --------------- --------------- ---------------
1 6583.333 7000.000 No Upper Limit
2 7250.000 7500.000 No Upper Limit
3 7250.000 7500.000 No Upper Limit
4 4210.000 6000.000 6250.000
5 7250.000 7500.000 7750.000
6 No Lower Limit 10000.000 11666.667
7 No Lower Limit 3000.000 17916.668
8 0.002 5000.000 15529.412
5. From the dual prices it is clear that there is no advantage to not satisfying the orders for the
Regular and Deluxe mixes. However, it would be advantageous to negotiate a decrease in the Holiday mix requirement.
MGTC74W07 Page 3 of 3 Chapter 3 – 11 th ed.