Operation Research

Executive Summary:

Red Brand Canners used to be a medium-size company canning and distributing a variety of fruit and vegetable products in the western part of USA in the 1960s. In one year, they signed an agreement at planting time, to purchase the entire tomato crop in a large field at an average delivered price of 6 cents per lb. At harvest time, Produce Inspection estimates that the total crop will be 3×106 lb, of which 20% is expected to be Grade A and the remaining portion expected to be Grade B. The company makes three different tomato products, and they set the selling prices of these products in light of the long-term marketing strategy of the company.

Introduction:

Moreover, Red Brand Canners uses a numerical scale to record the quality of both the raw produce and prepared products. The scale runs from 0 to 10 points, higher numbers representing better quality. On this scale, Grade "A" tomatoes were valued at 9 points/lb and Grade "B" tomatoes 5 points/lb.

Product (cases) SP TU VC DF MAIQR(per pound)

Whole Tomatoes 4 18 2.52 Unlimited 8

Tomato Juice 4.5 20 3.18 50,000 6

Tomato Paste 3.8 25 1.95 80,000 5

SP: Selling Price (Tk. /case), TU: Tomatoes (lb) used/case,

VC: Variable Costs (excluding tomato purchase costs) (Tk. /case), DF:

Demand Forecast (cases) at these prices and MAIQR: Minimum

Average Input Quality Requirement

Part I Questions & Answers:

Question No. 1.

Why Daiyan does states that the whole tomato production is limited to 8, 00,000 pounds? (I.e. where does the number 8, 00,000 come from?)

Answer To The Question No 1.

Based on the availability of 600,000 pounds of grade "A" tomatoes (grade 9), one can mix in some grade "B" tomatoes (grade 5) to generate a mix of grade 8. Let X denotes the pounds of grade "B" tomatoes that can be mixed 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 Pounds. Based these information's Dayan states that the whole tomato production is limited to 8, 00,000 pounds.

Question No. 2.

What is wrong with Wazed's suggestion to use the entire crop for whole tomatoes?

Answer To The Question No 2.

Wazed's suggestion restricts the usage of tomatoes to merely 800,000 Pounds (as shown in exhibit 1). The leftover tomatoes could be used profitably to at least make tomato paste. It might also be more profitable to make use of the "A" tomatoes to produce some tomato juice as well, and end up with mix products that together produce the most profit.

Question No. 3.

How does Mahbub compute his tomato costs in Exhibit 3? Who does he reach his conclusion to use 20, 00,000 pounds of "B" tomatoes for paste, the remaining 4, 00,000 pounds of "B" tomatoes, and the entire "A" in juice? What is wrong with Mahbub's reasoning?

Answer To The Question No 3.

In Exhibit 3, Mahbub attempt to prorate the unit cost per tomato (18 Pounds Per Case) based on the quality of tomato (Grade "A" or "B").

The first equation (1) indicates that the sum of 6,00,000 Pound multiplied by the unit cost per lb for grade "A" tomatoes (Z) and 2,400,000 Pound multiplied by the unit cost per lb for grade "B" tomatoes (Y) equals the total cost paid (3,000,000 Pounds multiplied by 18 cents per Pound).

The second equation (2) defines the relative relationship between the unit prices for grade "A" and "B" tomatoes based on the relative "quality" points for the two grades. Solving the two equations yields the values for the unit prices for the two grades.

Based on this one can state that the unit cost per pound for tomatoes of "quality of 1" is equal to the value of Z÷9 or Y÷5. This is then used to find the adjusted fruit cost. For example, the cost per case of whole tomatoes would be: (Z÷9) Tk. /pound×8×18 pound/case or Tk.4.47 per case. Similarly, for tomato juice the cost per case would be: (Z÷9) Tk. /pound×6×20 lb/ case = Tk. 3.72 per case.

Since Mahbub 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 pounds (80,000 cases×25pound per case). Beyond that Myers ranks tomato juice as the next profitable item and so the remaining 4, 00,000 pounds of grade "B" tomatoes and 6, 00,000 pounds 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.

Question No. 4.

Without including the possibility of the additional purchases suggested by Matin, formulate as an LP the problem of determining the optimal canning policy for this season's crop. Define your decision variables in terms of pounds of tomatoes. Express the objective function in Taka per pound.

Answer To The Question No 4.

Formulate as an LP the problem of determining the optimal canning policy for this Season's Crop.

The LP Solution:

Red Brand

Canners Case

AC BC AJ BJ AP BP

Objective function

Contribution margins Tk.0.247 Tk.0.247 Tk.0.198 Tk.0.198 Tk.0.222 Tk.0.222 Profit

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