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134 CHAPTER 3 LINEAR PROGRAMMING: SENSITIVITY ANALYSIS AND INTERPRETATION OF SOLUTION

20 The programme manager for Channel 10 would like to determine the best way to allocate the time
forthe11:00–11:30eveningnewsbroadcast.Specifically,shewouldliketo determinethenumber
of minutes of broadcast time to devote to local news, national news, weather and sports. Over the
30-minute broadcast, ten minutes are set aside for advertising. The station’s broadcast policy
states that at least 15 per cent of the time available should be devoted to local news coverage; the
time devoted to local news or national news must be at least 50 per cent of the total broadcast
time; the time devoted to the weather segment must be less than or equal to the time devoted to
the sports segment; the time devoted to the sports segment should be no longer than the total
time spent on the local and national news; and at least 20 per cent of the hould be devoted to the
weather segment. The production costs per minute are E300 for local news, E200 for national
news, E100 for weather and E100 for sports.
a. Formulate and solve a linear programme that can determine how the 20 available minutes
should be used to minimize the total cost of producing the programme.
b. Interpret the dual price for the constraint corresponding to the available time. What
advice would you give the station manager given this dual price?
c. Interpret the dual price for the constraint corresponding to the requirement that at least
15 per cent of the available time should be devoted to local coverage. What advice
would you give the station manager given this dual price?
d. Interpret the dual price for the constraint corresponding to the requirement that the time
devoted to the local and the national news must be at least 50 per cent of the total broadcast
time. What advice would you give the station manager given this dual price?
e. Interpret the dual price for the constraint corresponding to the requirement that the time
devoted to the weather segment must be less than or equal to the time devoted to the
sports segment. What advice would you give the station manager given this dual price?

CASE PROBLEM 1 Product Mix

J’s, Inc., makes three nut mixes for sale to gro- TJ’s accountant analyzed the cost of packaging
T cery chains located in India. The three mixes, materials, sales price per kilo and so forth, and
referred to as the Regular Mix, the Deluxe Mix and determined that the profit contribution per kilo is
the Holiday Mix, are made by mixing different per- E1.65 for the Regular Mix, E2.00 for the Deluxe Mix
centages of five types of nuts. and E2.25 for the Holiday Mix. These figures do not
In preparation for the next production period, TJ’s include the cost of specific types of nuts in the differ-
has just purchased the following shipments of nuts ent mixes because that cost can vary greatly in the
at the prices shown: commodity markets.
Customer orders already received are summar-
Type of Shipment Cost per ized here:
Nut Amount (kilos) Shipment ($)
Type of Mix Orders (Kilos)
Almond 6 000 7 500
Brazil 7 500 7 125 Regular 10 000
Filbert 7 500 6 750 Deluxe 3 000
Pecan 6 000 7 200 Holiday 5 000
Walnut 7 500 7 875
Because demand is running high, it is expected that
The Regular Mix consists of 15 per cent almonds, TJ’s will receive many more orders than can be
25 per cent Brazil nuts, 25 per cent filberts, 10 per satisfied.
cent pecans and 25 per cent walnuts. The Deluxe TJ’s is committed to using the available nuts to
Mix consists of 20 per cent of each type of nut, and maximize profit; nuts not used will be given to a local
the Holiday Mix consists of 25 per cent almonds, 15 charity. Even if it is not profitable to do so, TJ’s
per cent Brazil nuts, 15 per cent filberts, 25 per cent president indicated that the orders already received
pecans and 20 per cent walnuts. must be satisfied.

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