328

CHAPTER 8 Linear Programming Modeling Applications: With Computer Analyses in Excel and QM for Windows

See our Internet home page at www.prenhall.com/render homework problems 8-24 to 8-28.

for additional

On Monday, September 13, 1999, Mitchell Gordon, vice president of operations at Red Brand Canners, asked the controller, the sales manager, and the production manager to meet with him to discuss the ~mount of tomato products to pack that season. The tomato crop, which had been purchased at planting, was beginning to arrive at the cannery, arid packing operations would have to be started by the following Monday. 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. William Cooper, the controller, and Charles Myers, the sales manager; were the first to arrive in Gordon's office. Dan Tucker, the production manager, came in a few minutes later and said that he had picked up Produce Inspection's latest estimate of the quality of the incoming tomatoes. According to the report, about 20% of the crop was grade A quality and the remaining portion of the 3-million-pound crop was grade B. Gordon asked Myers about the demand fortomato products for the coming year. Myers replied that they could sell all of .the whole canned tomatoes they could produce. The expected demand for tomato juice and tomato paste, on the other hand, was limited. The sales manager then passed around the latest demand forecast, which is shown in ~6. He reminded the group that the selling prices had been set in light of the longterm marketing strategy of the company and that the potential ·saleshad been forecast at these prices. Bill Cooper, after looking at Myers's estimates of demand, said that it looked like the company "should do quite well [on

the tomato crop] this year." With the new accounting system that had been set up, he had been able to compute the contribution for each product, and according to his analysis, the incremental profit on whole tomatoes was greater than the incremental profit on any other tomato product. In May, after Red Brand had signed contracts agreeing to purchase the grower's production at an average delivered price of 6 cents R~r pound, Cooper had computed 'the tomato products' contributions (see Table 8.7). • Dan Tucker brought to Cooper's attention that although there was ample production capacity, it was impossible to produce all whole tomatoes because too small a portion of the tomato crop was grade A quality. Red Brand used a numerical scale to record the quality of both raw produce and prepared products. This scale ran from 0 to 10, the higher number rerresenting better quality. According to this scale, grade A tomatoes averaged nine points per pound and grade B tomatoes averaged five points per pound. Tucker noted that the minimum average input quality was eight points per pound for canned whole tomatoes and six points per pound for juice. Paste could be made entirely from grade B tomatoes. This meant that whole-tomato production was limited to 800,000 pounds, Gordon stated that this was not a real limitation. He had been recently solicited to purchase 80,000 pounds of grade A tomatoes at 8 cents per pound and at that time had turned down the offer. He felt, however, that the tomatoes were still available. Myers, who had been doing some calculations, said that although he agreed that the company "should do quite well this

A

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Yz

.

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·Demand Forecasts 24-2

Yz 24-2 Yz
24-2Yz
. !/

whole tomatoes choice peach halves peach nectar (~U\ cc"' J cooking apples

i.;i~
c.,

4-:00 5.40 4.60 4:50 4.90 3.80

800,000 10,000
5,000

24-272 tomato juice 24-2

50.000 15,000 80,000

Yz

24-2 Yz tomato .paste 2. l. 1- '< BD·/, )(

3)

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Case Study

329

Product Item Profitability

1.18 Variable overhead Variable selling Packaging material 0.24

lAO
0.32 0.30 0.56 1.80 4.38 1.02 0.70 0.32

1.27 0.23 0040 0:60 1.70 4.20 0.40 0.52 (0.12)

1.32 0.36 0.85 0.65 1.20 4.38 0.12 0.21 (0.09)

0.70 0.22 0.28 0.70 0.90 2.80

0.54 0.26 0.38 0.77 1.50 ; 3.45 0.35 0.23 0.12

OAO
0.70 l.08 3.60 0040 0.28 0.12

Fruit"
Total variable costs Contribution Less allocated overhead Net profit

UO
0.75 0.35

* Product usage is as follows:
Product Whole tomatoes Peach halves Peach nectar Tomato juice Cooking apples Tomato paste Pounds per Case 18 18 17 20 27 25

Z

= cost

per pound of grade A tomatoes in cents

Marginal Analysis of Tomato Products

}[ = cost per pound of grade B tomatoes in cents (600,000 lb x Z) + (2,400,000 Ib x Y) = (3,000,000 lb x 6)
(1) (2)

Z
9

y
5

Z = 9.32 cents per pound Y = 5.18 cents per pound

Selling price Variable cost (excluding tomato cost)

$4.00 2.52 $1.48

$4.50 3.18 $1.32 1.24 $0.08 1.95·

-- :

..

--

~r~5]\30

Tomato cost Marginal profit

1.49 ($0.01)

330

CHAPTER 8 Linear Programming

Modeling Applications: With Computer Analyses in Excel and QM for Windows Discussion Questions L Structure this problem verbally, including a written descrip-, \ion of the constraints and objective. What are the 2ecision variables? 2. Develop a mathematical formulation for Red Brand's objective fu~ction and constraints. 3. Solve the problem and discuss the results.
Source: "Red Brand Canners" revised with permission of S·tanford University. Graduate School of Business. Copyright © 1969 and 1977 by the Board of Trustees of the Leland Stanford Junior University.

year," it would not be by canning whole tomatoes. It seemed to him that the tomato cost should be allocated on the basis of quality and quantity rather than by quantity only, as Cooper had done. Therefore, he had recomputed the marginal profit on this basis (see Table 8.8), and from his results had concluded that ~ Brand should use 2 million pounds of the grade B tomatoes for ~e, and the remaining 400,000 pounds of grade B tomatoes and all of the grade A tomatoes for juice. If the demand expect'!tions were realized,a contribution of $48,000 would be made on this year's tomatO crop.

2.
The workload in many areas of bank operations has the characteristics of a nonuniform distribution with respect to time of day. For example, at Chase Manhattan Bank in New York, the number of domestic money transfer requests received from customers, if plotted against time of day, would appear to have the shape of an inverted U curve with the peak around 1 P.M. For efficient 'use of resources, the personnel available should, therefore, vary correspondingly. Figure 8.2 shows a typical workload curve and corresponding personnel requirements at different hours of the day. A variable capacity can be achieved effectively by employing part-time personnel. Because part-timers are not entitled to all the fringe benefits, they are often more economical than fulltime employees. Other considerations, however, may limit the extent to which part-time people can be hired in a given department. The problem is to find an optimum workforce schedule that would meet personnel requirements at any given time and also be economical. Some of the factors affecting personnel assignment are listed here: 1. By corporate policy, part-time personnel hours are limited to a maximum of 40% of the day's total requirement.

3. ;),

,5,

fl,

Full-time employees work for 8 hours (L hour for lunch included) per day. Thus, a full-timer's productive time is 35 hours per week. Part-timers work for at least 4 hours per day but less than 8 hours and are not allowed a lunch break. .Fifty percent of the full-timers go to lunch between 11 A.M. and noon, and the remaining 50% go between noon and 1 P.M. The shift starts at 9 A.M. and ends at 7 P.M. (i.e .• overtime is limited to 2 hours). Any work left over at 7 P.M. is considered holdover for the next day. A full-time employee is not allowed to work more than 5 hours overtime per week. He or she is paid at the normal rate for overtime hours-not at one-and-a-haLf times the normal rate applicable to hours in excess of 40 per week. Fringe benefits are not applied to overtime hours.

In addition, the following costs are pertinent:
'I

a,

The average cost per full-time personnel hour (fringe benefits included) is $10.1 L Workforce Requirements

9-10 10-11 11-12 12-1 1-2 2-3 3--4 4-5 5--6 6-7

A.M.

14 25 26 38 55 60 51 29 14 9

P.M.