A) what is the maximum price the company would be willing to pay for additional

Question

A) what is the maximum price the company would be willing to pay for additional pears? B) for what quality range of pears is the shadow price valid C) what is the marginal value of peaches? D) for what quanity range of peaches is the shadow price? E) what is the range of the profit margin for fruit cocktail? F) what is the range profit margin for fruit delight?
The Sunshine Food Processing Company produces three canned fruit products mixed fruit (x1), fruit cocktail (x2), and fruit delight (x3). The main 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 linear programming formulation. MAX Z 10x1 + 6x2 + 8x3 (profit, S) ST 20x1 + 10x2+ 16x3 320 (pears, lb) 10x1 +20x2 +16x3 400 (peaches, lb) xi + 2x2 + 2x3 S 43 (mixing, hr) xi+ x2+ x3 60 (canning, hr) 2xi tx2+ x3 S40 (packaging, hr) xi2 0 The optimal simplex tableau is as follows: 0 Ci 10 6 8 10 X 10 8/15 1/15 -1/30 0 008 Zi 106 128/15 7/15 1/15 000 176 Ci-Zi O 0 -8/15 -7715 -1/15 0 0

Explanation / Answer

A) what is the maximum price the company would be willing to pay for additional pears?

maximum profit is $176

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B) for what quality range of pears is the shadow price valid

Maximum pears available is 320 lb so range is 0 to 320 lb

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C) what is the marginal value of peaches?

Maximum available peaches are 400 lb, so margin is 400 lb.

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D) for what quanity range of peaches is the shadow price?

Available peaches are 400 lb, so peaches above 400 lb will cause shadow price.

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E) what is the range of the profit margin for fruit cocktail?

maximum profit is earned at cocktail (x2)=16

and profit on one cocktail is $6 so profit on 16 cocktail is 16*6=96

hence profit margin is $96

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F) what is the range profit margin for fruit delight?

maximum profit is earned at fruit delight (x3)=0

hence profit margin is $0

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