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DATA ENVELOPMENT ANALYSIS 193

5 Kunz and Sons manufactures two products used in the heavy equipment industry. Both
products require manufacturing operations in two departments. The following are the
production time (in hours) and profit contribution figures for the two products.

Labour-Hours

Product Profit per Unit, E Dept. A Dept. B
1 25 6 12
2 20 8 10

For the coming production period, Kunz has available a total of 900 hours of labour that
can be allocated to either of the two departments. Find the production plan and labour
allocation (hours assigned in each department) that will maximize the total contribution to
profit.

6 The North Somerset Police’s Department schedules police officers for eight-hour shifts.
The start times for the shifts are 8:00 A.M., noon, 4:00 P.M., 8:00 P.M., midnight and 4:00 A.M.
An officer beginning a shift at one of these times works for the next eight hours. During
normal weekday operations, the number of officers needed varies depending on the time
of day. The department staffing guidelines require the following minimum number of
officers on duty:

Time of Day Minimum Officers on Duty

8:00 A.M.–Noon 5
Noon–4:00 P.M. 6
4:00 P.M.–8:00 P.M. 10
8:00 P.M.–Midnight 7
Midnight–4:00 A.M. 4
4:00 A.M.–8:00 A.M. 6

Determine the number of police officers that should be scheduled to begin the eight-hour shifts
at each of the six times (8:00 A.M., noon, 4:00 P.M., 8:00 P.M., midnight and 4:00 A.M.) to minimize
the total number of officers required. (Hint:Let x 1 ¼ the number of officers beginning work at
8:00 A.M., x 2 ¼ the number of officers beginning work at noon, and so on.)
7 Reconsider the Welte Mutual Funds problem from Section 4.5. Define your decision
variables as the fraction of funds invested in each security. Also, modify the constraints
limiting investments in the oil and steel industries as follows: no more than 50 per cent of
the total funds invested in stock (oil and steel) may be invested in the oil industry, and no
more than 50 per cent of the funds invested in stock (oil and steel) may be invested in the
steel industry.
a. Solve the revised linear programming model. What fraction of the portfolio should be
invested in each type of security?
b. How much should be invested in each type of security?
c. What are the total earnings for the portfolio?
d. What is the marginal rate of return on the portfolio? That is, how much more could be
earned by investing one more euro in the portfolio?

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