NASA Case

The U.S. space agency, NASA, must deal constantly with such decision problems in choosing how to divide its limited budgets among many competing missions proposed. 1 Table 11.2 shows a fictitious list of alternatives.

We must decide which of the 14 indicated missions to include in program plans for the 2000-2024 era. Thus it should be clear that the needed decision variables are

\(x_{j} \begin{cases}1 & \text { if mission } j \text { is selected } \\ 0 & \text { otherwise }\end{cases}\)

TABLE 11.2 Proposed Missions in NASA Example

Budget Constraints

The budget constraints that give capital budgeting problems their name limit project expenditures in particular time periods.

Budget requirements in Table 11.2 span five time periods: 2000-2004, 2005- 2009,2010-2014,2015-2019, and 2020-2024. We form budget constraints for each of these periods by summing project decision variables times their needs.

\(\begin{array}{lll}

6 x_{1}+2 x_{2}+3 x_{3}+1 x_{7}+4 x_{9}+5 x_{12} & \leq 10 & (2000-2004) \\

3 x_{2}+5 x_{3}+5 x_{5}+8 x_{7}+5 x_{9}+8 x_{10} & & (2005-2009) \\

+7 x_{12}+1 x_{13}+4 x_{14} & \leq 12 & \\

8 x_{5}+1 x_{6}+4 x_{10}+2 x_{11}+4 x_{13}+5 x_{14} & \leq 14 & (2010-2014) \\

8 x_{6}+5 x_{8}+7 x_{11}+1 x_{13}+3 x_{14} & \leq 14 & (2015-2019) \\

10 x_{4}+4 x_{6}+1 x_{13}+3 x_{14} & \leq 14 & (2020-2024)

\end{array}\)

SAMPLE EXERCISE 11.4: FORMULATING BUDGET CONSTRAINTS

A department store is considering 4 possible expansions into presently unoccupied space in a shopping mall. The following table shows how much (in millions of dollars) each expansion would cost in the next two fiscal years, and the required floor space (in thousands of square feet).

Using decision variables

\(x_{j} \begin{cases}1 & \text { expansion } j \text { is selected } \\ 0 & \text { otherwise }\end{cases}\)

formulate implied constraints on investment funds and floor space assuming that 10 million dollars are available in each of the two years and that the expansion cannot exceed 17 thousand square feet.

Modeling: The three required budget constraints are

\(\begin{array}{ll}

1.5 x_{1}+5.0 x_{2}+7.3 x_{3}+1.9 x_{4} \leq 10 & \text { (year 1 budget) } \\

3.5 x_{1}+1.8 x_{2}+6.0 x_{3}+4.2 x_{4} \leq 10 & \text { (year 2 budget) } \\

2.2 x_{1}+9.1 x_{2}+5.3 x_{3}+8.6 x_{4} \leq 17 & \text { (floor space) }

\end{array}\)

Modeling Mutually Exclusive Choices

Capital budgeting problems often come with other constraints besides simple budget limits. For example, two or more proposed projects may be mutually exclusive. That is, at most one of them can be included in a solution.

Table 11.2 indicates three such conflicts. Possibilities \(j=4\) and 5 represent alternative timing for the same mission. Technologies for missions \(j=8\) and 11 are incompatible. Numbers \(j=9\) and 14 involve two different ways to accomplish the SETI program.

Such incompatibilities are easily modeled with 0-1 decision variables (11.6). For our NASA example, the result is

For our NASA example, the result is

\(\begin{aligned}

&x_{4}+x_{5} \leq 1 \\

&x_{8}+x_{11} \leq 1 \\

&x_{9}+x_{14} \leq 1

\end{aligned}\) \[\]

Modeling Dependencies between Projects

Another characteristic relationship between projects arises when one project depends on another. We cannot choose such a dependent project unless we also include the option on which it depends.

We can model dependencies among projects as easily as mutual exclusiveness.

Variable \(x_{j}\) cannot \(=1\) unless \(x_{i}\) does too.

Table 11.2 shows that mission \(j=11\) depends on mission 2 in our NASA example, and also that mission \(j=3\) must be chosen if any of projects \(4, \ldots, 7\) is. Implied constraints are

\(\begin{aligned}

&x_{11} \leq x_{2} \\

&x_{4} \leq x_{3} \\

&x_{5} \leq x_{3} \\

&x_{6} \leq x_{3} \\

&x_{7} \leq x_{3}

\end{aligned}\)

To complete formulation of our version of NASA's decision problem, we need an objective function. Like almost all public agencies, NASA has many. They may try to maximize the intellectual gains of selected missions, maximize the direct benefit to life on earth, and so on.

We will assume that a weighted sum of these different objective functions tan be used to estimate a value for each mission. Resulting values are included in Table 11.2. Combining with previous elements, we obtain the NASA example model:

Price: $15.92

Solution: The downloadable solution consists of 10 pages, 592 words and 5 charts.
Deliverable: Word Document