Question
In: Operations Management
The Sun food co. produces three canned fruit products. The mail ingredients in each product are...
| The Sun food co. produces three canned fruit products. The mail ingredients in each product | |||||||||
| are pears and peaches. Each product is produced in lots and must go through three processes - | |||||||||
| mixing, canning, and packaging. The resource requirements for each product and each process | |||||||||
| are shown in the following LP formulation. | |||||||||
| Maximize Z = 10 x1 + 6 x2 + 8 x3 (profit, $) | |||||||||
| subject to | |||||||||
| 20 x1 + 10 x2 + 16 x3 <= 320 (pears, lbs) | |||||||||
| 10 x1 + 20 x2 + 16 x3 <= 400 (peaches, lbs) | |||||||||
| x1 + 2 x2 + 2 x3 <= 43 (mixing, hr) | |||||||||
| x1 + x2 + x3 <= 60 (canning, hr) | |||||||||
| 2 x1 + x2 + x3 <= 40 (packaging, hr) | |||||||||
| x1, x2, x3 >= 0 (non-negativity conditions) | |||||||||
| A) | What is the maximum price the company would be willing to pay for additional pears? | ||||||||
| How much could be purchased at that price? | |||||||||
| B) | What is the marginal value of peaches? Over what range is this price valid? | ||||||||
| C) | The company can purchase a new mixing machine that would increase the available | ||||||||
| mixing time from 40 to 60 hours. Would this affect the optimal solution? | |||||||||
| D) | The company can also purchase a new packaging machine that would increase the available | ||||||||
| packaging time from 40 to 50 hours. Would you recommend this? Explain. | |||||||||
| E) | If the manager were to attempt to secure additional units of only one of the resources, | ||||||||
| which should it be and why? | |||||||||
Solutions
Expert Solution
Solution using Solver is as follows
Formula:
E2 =SUMPRODUCT(B2:D2,$B$9:$D$9) copy to E2:E7
Sensitivity report is as under
A) Refer sensitivity report, Shadow price of pear constraint is 0.467. Therefore, Maximum price the company would be willing to pay for additional pears is 0.467
Allowable increase is 80 . So a maximum of 80 should be purchased at that price.
B) Marginal value of peaches is 0.067 (shadow price of peaches constraint)
It is valid over range of 400-240 = 160 to 400+30 = 430 (refer allowable increase and decrease in sensitivity report)
D) No, There is already unutilized capacity availableon mixing machine. Therefore, it is not beneficial to purchase new machine.
Also refer sensitivity report, shadow price of mixing is 0, which means it will not increase the profit.
D) No, There is already unutilized capacity available on packaging machine. Therefore, it is not beneficial to purchase new packaging machine.
Also refer sensitivity report, shadow price of packaging is 0, which means it will not increase the profit.
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