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Production Mix

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Submitted By nrichter
Words 452
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1.) Cost per pound of nuts:
Almonds $7500/6000 = $1.25
Brazil $7125/7500 = $.95
Filberts $6750/7500 = $.90

Pecans $7200/6000 = $1.20
Walnuts $7875/7500 = $1.05

Cost of nuts per mix:
Regular mix: .15($1.25) + .25($.95) + .25($.90) + .10($1.20) + .25($1.05) = $1.0325
Deluxe mix: .20($1.25) + .20($.95) + .25($.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 = number of regular mix produced
D = number of deluxe mix produced
H = number of holiday mix produced

The following linear programming model can be solved to maximize profit contribution for the nuts already purchased.
Max = 1.65R + 2D + 2.25H
.15R + .20D + .25H < = 6000 Almonds
.25R + .20D + .15H < = 7500 Brazil
.25R + .20D + .15H < = 7500 Filbert
.10R + .20D + .25H < = 6000 Pecans
.25R + .20D + .20H < = 7500 Walnuts R > = 10,000 Regular D > = 3000 Deluxe H > = 5000 Holiday

The optimal solution-using Lingo is $61,375. This would be your maximum profit using the objective function above. The profit from the Regular mix would be $17,500; the profit from the Deluxe mix would be $10,625; and the profit from the Holiday mix would be $5,000.

3.) Looking at the dual prices one can see that additional almonds are worth $8.50 per pound for TJ and additional walnuts are worth $1.50 per pound. Looking at the slack variables one can see that additional Brazil nuts, Filberts nuts, and Pecans are of no value since they are already in excess supply. 4.) I would purchase the additional 1000 pounds of almonds for $1000. By changing the right-hand constraint for the almonds from 6000 to 7000 you will get the optimal solution of $66,333.33. This is an overall increase in profit for the company. Profit will increase by $4958.33. Note that not all of the 1000 pounds of almonds will be used. There will be an excess of 416.667 pounds of almonds, but the increase in profit justifies the purchase of the 1000 pounds of almonds.
5.) Looking at the dual prices one can see that there is no advantage to not satisfying the orders for the Regular and Deluxe mixes. It would be smart for TJ to negotiate a decrease in the Holiday mix.
Conclusion
It would be of interest to the company to purchase an additional 1000 pounds of almonds for $1000. If the company does this they will not only increase their profit by $4958.33, but they will also have nuts to donate to the local charities.

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