Red Brand Canners is a medium sized company that cans and distributes a variety of fruit and vegetable products under private brands in the western states. The company makes three different tomato products including whole tomatoes, tomato juice and tomato paste. They also distribute Choice peach halves, peach nectarine and cooking apple products.
As part of their discussion over the amount of tomato products to pack in a particular season , it was observed that 3 million pounds of tomatoes are available for packing and distributing. This included two kinds of tomatoes ‘Grade A’ which constituted 20% of available crop and ‘Grade B’ being the remaining.
The problem can be divided into the following parts:
(a) How to allocate 3 million pounds of tomatoes in order to maximize profits? (b) Should the company purchase additional Grade ‘A’ tomatoes? if so, what is the maximum amount the company can purchase and also formulate the new optimum product mix
(A) Optimal Solution:
The objective is to maximize the profit by obtaining an optimal mix of Grade A and Grade B tomatoes. Hence, the problem can be formulated algebraically as below:
si : selling price
li : pounds of tomatoes used
vi : variable costs per case of ith product
xai : Amounts of Grade A in pounds used in ith product.
xbi : Amounts of Grade A in pounds used in ith product.
The problem can be solved using excel solver solution of the same has been given in Appendix A. Also, solving the problem using Lingo, we obtain:
Maximum Profit: 53340 $
xa1: 525000 lbs
xb1: 175000 lbs
xa2: 75000 lbs
xb2: 225000 lbs
xa3: 0.000000 lbs
xb3: 2000000 lbs
(B) Coopers Solution:
Cooper agreed that the company would do well on the tomato crop, and that the incremental profit on whole tomatoes was greater than the incremental profit on any other tomato product.
Based on the availability of 600,000 pounds of grade “A” tomatoes of 9 points per pound quality, grade “B” tomatoes of 5 points per pound quality can be mixed to generate whole tomatoes of 8 points quality.
Let X denote the pounds of grade “B” tomatoes that can be mixed: (600,000*9+X*5)/(600,000+X)=8.
0.0822*(xa1 + xb1)+ 0.066*(xa2 + xb2)+ 0.078*( xa3 + xb3)-180000; xa2 +xa3 = 0;
Maximum Profit: 14960 $
xa1: 600000 lbs
xb1: 200000 lbs
xa2: 0 lbs
xb2: 0 lbs
xa3: 0 lbs
xb3: 2000000 lbs
0.0067*(xa1 + xb1) - 0.0045*(xa2 + xb2) + 0.0048*(xa3 + xb3); xa2 +xa3 = 0;
xa2 +xa3 = 0;
Maximum Profit: 84000 $
xa1: 0 lbs
xb1: 0 lbs
xa2: 250000 lbs
xb2: 750000 lbs
xa3: 350000 lbs
xb3: 1650000 lbs
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 tomato juice as the next profitable item and so the remaining 400,000 lbs of grade B tomatoes and 600,000 lbs of grade “A” tomatoes should be used for making tomato juice.
A fundamental shortcoming in the analysis is that the fact that the grade A tomatoes implicitly cost the company more than the 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 and, hence, their purchase price is a “sunk” cost. It does not make sense to penalize the production of whole tomatoes because of the cost already incurred in purchasing Grade ‘A’ tomatoes
(D) Decision to purchase additional 80,000 pounds of Grade A tomatoes at 8.5 Cents per pound:
The issue is whether or not to buy 80,000 additional pounds of grade A tomatoes. This would increase the amount of available grade A tomatoes from...