Page 45 -

MANAGEMENT SCIENCE MODELS AND TECHNIQUES 25

of the real situation, and (b) a thorough and complex model that is the most accurate
mathematical representation of the real situation possible. Why might the model
described in part (a) be preferred by the manager?
6 Suppose you are going on a weekend trip to a city that is d kilometres away. Develop
a model that determines your round-trip fuel costs. What assumptions or
approximations are necessary? Are these assumptions or approximations acceptable
to you?
7 A food store in Glasgow specializes in selling organic produce to local restaurants. In
the summer it buys fresh raspberries (a Scottish delicacy) from two authorized
organic farms in Tayside – the McGregor Farm and the Campbell Farm. Raspberries
are supplied ready for sale in cartons containing ½ kilo. The McGregor Farm charges
the food store £0.20 per carton and can supply no more than 4000 cartons a week
during the short growing season. The Campbell Farm charges £0.25 per carton and
can supply no more than 3000 cartons a week. The food store anticipates being able
to sell cartons at £0.75.
Let x to represent the number of cartons each week shipped from the McGregor
Farm and y to represent the number of cartons each week shipped from the
Campbell Farm.
a. Write a mathematical expression to show the total number of cartons received each
week by the Glasgow food store.
b. Write a mathematical expression to show the total cost of cartons received each
week.
c. Write a mathematical expression to show the total profit made by the food store each
week from selling cartons to local restaurants.
d. The food store anticipates that local restaurants will buy no more than 5000 cartons a
week. Write this mathematically as a constraint.
e. Write mathematically the supply constraint for each farm.
f. Assuming the food store wants to maximize profit from selling raspberries, write out the
full mathematical model.
g. What key assumptions have you had to make for f?
8 For most products, higher prices result in a decreased demand, whereas lower prices
result in an increased demand. Let:
d ¼ annual demand for a product in units
p ¼ price per unit
Assume that a firm accepts the following price-demand relationship as being
realistic:
d ¼ 800 10p
where p must be between E20 and E70.
a. How many units can the firm sell at the E20 per-unit price? At the E70 per-unit
price?
b. Show the mathematical model for the total revenue (TR), which is the annual demand
multiplied by the unit price.
c. Based on other considerations, the firm’s management will only consider price
alternatives of E30, E40 and E50. Use your model from part (b) to determine the price
alternative that will maximize the total revenue.
d. What are the expected annual demand and the total revenue corresponding to your
recommended price?

Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has
deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.