Answers to Red Brand Canners Case Part I 1. Based on the availability of 600,000 pounds of grade ³A´ tomatoes (grade 9), one can mix in some grade ³B´ tomatoes (grade 5) to generate a mix of grade 8. Let X denote the pounds of grade ³B´ tomatoes that can be mixed in. in. Then: (600,000*9+X*5)/(600,000+X)=8. Solving this, yields X=200,000 lbs for a total weight of 800,000 lbs. 2. Cooper¶s suggestion restricts the usage of tomatoes to merely 800,000 lbs (as shown in 1). The leftover leftover tomatoes could be used used profitably to at least make tomato paste. It might might also be more profitable to make use of o f the A tomatoes to produce some tomato juice as well, and end up with a mix products that together produce the most profi pro fit. t. 3. In Exhibit 3, Myers attempt to prorate the unit cost per tomato (18 cents) based on the quality of tomato (grade A or B). The first first equation (1) indicates indicates that the sum sum of 600,000 lb lb multiplied by the unit cost per lb for grade A tomatoes (Z) and 2,400,000 2,400, 000 lb multiplied by the unit cost per lb for grade B tomatoes (Y) equals the total cost paid (3,000,000 (3,000, 000 lbs multiplied by 18 cents per lb). lb). The second equation (2) defines defines the relative relationship between the unit prices for grade A and B tomatoes to matoes based on the relative ³quality´ points po ints for the two grades. Solving the two equations equations yields the values for the unit unit prices for the two grades. Based on this one can state stat e that the unit cost per lb for tomatoes of o f ³quality of 1´ is equal to the value of Z/9 or Y/5. This is then used to find the the adjusted fruit cost. For example, example, the cost per case of whole tomatoes would be: (Z/9) $/lb *8*18 lb/case or $4.47 per case. Similarly, for tomato juice the cost per case would be: (Z/9) $/lb * 6 * 20 lb/ case = $3.72 per case. Since Myers believes that tomato paste is the most profitable option, he would like to sell as much tomato paste that demand allows, which is 80,000 cases or 2,000,000 lbs (80,000 cases*25lb per case). Beyond that Myers ranks ranks tomato juice as the next profitable profitable item and so the remaining 400,000 lbs of grade B tomatoes and 600,000 lbs of grade ³A´ tomatoes should be used for making tomato to mato juice. A fundamental shortcoming in the analysis is that the fact that the grade A tomatoes implicitly cost the company more than the t he grade B tomatoes has nothing with the current task at hand which is to maximize the operating profit for the season given the resources (tomatoes) at hand. Indeed, tomatoes have already been purchased purchased and, hence, their purchase price is a ³sunk´ cost. It does not make make sense to penalize the production of whole whole tomatoes because of the cost already incurred in purchasing grade A tomatoes. 4. Let wA= weigh of tomato grade gr ade A allocated to produce whole who le tomatoes (lbs), wB=weight of tomato grade B allocated to produce whole tomatoes to matoes (lbs), jA=weight of tomato grade g rade A allocated to produce whole who le tomatoes (lbs), jB=weight of tomato grade B allocated to produce whole tomatoes (lbs), pA=weight of tomato grade A allocated allocated to produce whole tomatoes to matoes (lbs),
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pB=weight of tomato grade B allocated to produce whole tomatoes (lbs). The formulation that maximizes net income for the tomato manufacturing operation is given below. First, the objective function is to maximize the generated income from selling the various tomato products. The income from selling a case of a product type is the contribution shown in Exhibit 2 plus the fruit cost. For each product type the income per pound is obtained by dividing the income per case by the weight (lb) per case. Then the total income for the product is the unit income per lb multiplied by the weight (in lbs) of tomatoes used in producing the product type. The objective function is then: Maximize ((1.20+3.24)*100)/18)*(wA+ wB) + ((0.36+3.60)*100)/20)*( jA+ jB) + ((1.05+4.50)*100)/25)*( pA+ pB) The first type of constraint is that the mix of to mato grades used should ensure the required scores for the whole tomatoes and the tomato juice products. No constraint is necessary for the tomato paste product as using either tomato grade A or B would meet the score required for tomato paste. For the whole tomatoes product, the combined score is 9*wA + 5* wB should be 8*(w A+ wB) or wA- 3wB 0. Similarly, for the tomato juice product, 9*j A + 5* jB should be 6*(j A+ jB) or 3jA - jB 0. Another set of constraints is needed to limit production of each type of tomato product so as to not to exceed demand. Finally, the usage of tomatoes should be limited to the amounts available of each grade. The complete formulation is then: Maximize ((1.20+3.24)*100)/18)*(wA+ wB) + ((0.36+3.60)*100)/20)*( jA+ jB) + ((1.05+4.50)*100)/25)*( pA+ pB) wA- 3wB 0 (whole tomatoes quality constraint) 3jA - jB 0 (tomato juice quality constraint) wA+ wB 800,000*18 (whole tomatoes production less than demand) jA+ jB 50,000*20 (tomato juice production less than demand) pA+ pB 80,000*25 (tomato paste production less than demand) wA+ jA+ pA 600,000 (grade A tomatoes usage less than available amount) wB+ jB+ pB 2,400,000 (grade B tomatoes usage less than available amount) wA, wB, jA, jB, pA, pB 0 5. In order to capture the possibility of ordering more grade A tomatoes, let eA denote the extra weight of grade A tomatoes to be purchased. The formulation can be modified to capture the cost (in the objective function) of purchasing the extra grade A tomatoes and the corresponding increased availability of such tomatoes (grade A to matoes usage constraint). The new formulation is: Maximize ((1.20+3.24)*100)/18)*(wA+ wB) + ((0.36+3.60)*100)/20)*( jA+ jB) + ((1.05+4.50)*100)/25)*( pA+ pB) ± 0.255eA wA- 3wB 0 (whole tomatoes quality constraint) 3jA - jB 0 (tomato juice quality constraint) wA+ wB 800,000*18 (whole tomatoes production less than demand) jA+ jB 50,000*20 (tomato juice production less than demand)
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pA+ pB 80,000*25 (tomato paste production less than demand) wA+ jA+ pA - eA 600,000 (grade A tomatoes usage less than available amount) wB+ jB+ pB 2,400,000 (grade B tomatoes usage less than available amount) 0 e A 80,000 (maximum amount of extra grade A tomatoes available) wA, wB, jA, jB, pA, pB, eA 0
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