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168 CHAPTER 4 LINEAR PROGRAMMING APPLICATIONS

Min 20DC þ 25EC þ 18DNC þ 20ENC
s:t:
DC þ EC þ DNC þ ENC ¼ 1000 Total interviews
DC þ EC 400 Households with children
DNC þ ENC 400 Households without children
DC þ EC DNC þ ENC 0 Evening interviews
0:4DC þ 0:6EC 0 Evening interviews
in households with children
0:6DNC þ 0:4ENC 0 Evening interviews
in households without children
DC; EC; DNC; ENC 0
The optimal solution to this linear program is shown in Figure 4.7. The solution
reveals that the minimum cost of E20320 occurs with the following interview schedule.

Number of Interviews

Household Day Evening Totals
Children 240 160 400
No children 240 360 600
Totals 480 520 1 000

Hence, 480 interviews will be scheduled during the day and 520 during the evening.
Households with children will be covered by 400 interviews, and households without
children will be covered by 600 interviews.
Selected sensitivity analysis information from Figure 4.7 shows a dual price of
19.2 for constraint 1. In other words, the value of the optimal solution will get
worse (the total interviewing cost will increase) by E19.20 if the number of inter-
views is increased from 1000 to 1001. Thus, E19.20 is the incremental cost of
obtaining additional interviews. It also is the savings that could be realized by
reducing the number of interviews from 1000 to 999.
The surplus variable, with a value of 200, for constraint 3 shows that 200 more
households without children will be interviewed than required. Similarly, the surplus
variable, with a value of 40, for constraint 4 shows that the number of evening
interviews exceeds the number of daytime interviews by 40. The zero values for the
surplus variables in constraints 5 and 6 indicate that the more expensive evening
interviews are being held at a minimum. Indeed, the dual price of five for constraint
5 indicates that if one more household (with children) than the minimum require-
ment must be interviewed during the evening, the total interviewing cost will go up
by E5.00. Similarly, constraint 6 shows that requiring one more household (without
children) to be interviewed during the evening will increase costs by E2.00.

4.5 Financial Applications

In finance, linear programming can be applied in problem situations involving
capital budgeting, make-or-buy decisions, asset allocation, portfolio selection, finan-
cial planning and many more. In this section, we describe a portfolio selection
problem and a problem involving funding of an early retirement programme.

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