Optimizing the Choice of Multiyear Projects

The funds available in a company’s capital budgets for the next few years limit the selection and number of choices. The following example uses binary programming with Excel’s Solver tool to identify the set of choices that best satisfy the financial criterion of maximizing the set’s net present value when there are budgetary constraints over a number of years.

Binary programming restricts the values of specified cells to 0 or 1. These correspond to answers of “no” or “yes” (or “no, don’t do it” versus “yes, do it”) for identifying which choices are best.

Example 12.9: The executives of Goliath Industries are reviewing their capital budgets for the next three years. Table 12-1 lists the options before them and the CFO’s estimates of their net present values (NPVs) and their annual costs to complete. Note that the initial costs can extend over several years. The table also shows how much money the CFO expects will be available for capital expenditures during the next three years. (Note that the NPVs of all options are positive. Therefore, all are worthwhile investments.)

If Goliath chooses to build a new plant, it will not modernize the existing one. On the other hand, if the company decides not to build a new plant, it will modernize the existing one.

a. What options should Goliath choose, and why? b. What will be the net present values of the chosen options, how much of the available funds will be committed each year, and how much of the available funds will be left uncommitted?

Table฀12-1

Proposals for Capital Expenditures

Annual Costs

Year 2 Year 3

Modernize฀existing฀plant

0 0 Build฀new฀plant

$300,000 Expand฀distribution฀network

$20,000 Redesign฀existing฀Product฀A

0 0 Redesign฀existing฀Product฀B

$45,000 R&D฀on฀new฀Product฀X

$200,000 $75,000 R&D฀on฀new฀Product฀Y

$250,000 $200,000 Available฀funds

Solution: Figure 12-16 is a spreadsheet solution. Excel’s Solver tool was used to select the options that gave the maximum NPV for the choices, consistent with their costs and the budgets available for the next three years and the requirement either to build a new plant or modernize the existing one, but not both.

(Continued)

388 ❧ ® Corporate Financial Analysis with Microsoft Excel

Figure฀12-16

Optimum Solution for Goliath Industries

1 Example 12-9: GOLIATH INDUSTRIES

2 Input Data

Present

Annual Costs

Year 2 Year 3 4 Modernize existing plant

5 Build new plant

300,000 6 Expand distribution network

20,000 7 Redesign existing product A

8 Redesign existing product B

45,000 9 Develop new product X

200,000 $ 75,000 10 Develop new product Y

250,000 $ 200,000 11 Available funds

Choices 13 (1 = Yes,

Present

Annual Costs

Year 2 Year 3 15 Modernize existing plant

- $ - 16 Build new plant

- $ - 17 Expand distribution network

- $ - 18 Redesign existing product A

- $ - 19 Redesign existing product B

45,000 $ - 20 Develop new product X

- $ - 21 Develop new product Y

22 Present value and annual costs

23 Uncommitted funds

5,000 $ 100,000 24 Build OR modernize plant constraint

25 Sum of uncommitted funds

Key cell entries: B24: =B15+B16 B25: =SUM(D23:F23) C15: =$B15*C4, copy to C15:F21 C22: =SUM(C15:C21), copy to D22:F22 D23: =D11–D22, copy to E23:F23

Solver settings: Target cell is C22, to be maximized. Cells to vary are B15:B21. Constraints: B15:B21 = binary

D22:F22 <= D11:F11 B24 = 1 (Either modernize or build new plant.)

Options: Assume linear model

The decision variables are the binary values in cells B15:B21. These are 1 for an option that is selected and 0 for an option that is not selected. Enter trial values of 1 in these seven cells. By multiplying the decision variables in B15:B21 by the present values and annual costs for the options in C4:F10, the present values and annual costs for the selected options are calculated in C15:F21. The calculations are made by entering =$B15*C4 in Cell C15 and copying the entry to C15:F21. Options that are chosen have values of 1 in Cells B15:B21. Those that are not chosen have zero values in Cells B15:B21.

Note the logic of the entry =B15+B16 in Cell B24. When the Solver tool is executed, the value of this cell must equal one. This requires that either B15 or B16 must equal one, but not both: Either the existing plant must be modernized or a new one must be built.

The sums of the present values and annual costs for the selected options are calculated by entering =SUM(C15:C21) in Cell C22 and copying the entry to D22:F22. The uncommitted funds each year are

Capital Budgeting: The Basics ❧ 389

calculated by entering =D11-D22 in Cell D23 and copying the entry to E23:F23. The sum of the uncommitted funds is calculated in Cell B25 by the entry =SUM(D23:F23).

The Solver tool is executed with a goal of maximizing the total present value in C22. The trial values of the decision variables entered in Cells B15:B21 are allowed to vary to achieve this goal, subject to the constraint that they are binary values of 1 and 0 (i.e., B15:B21=binary). (For some reason, it seems to work best to scroll down to bin, for binary, before entering the cell identities. An alternative is to enter the constraints B15:B21>=0, B15:B21<=1, and B15:B21=integer.) Additional constraints are imposed by requiring that capital expenditures each year are not more than the funds available (D22:F22<=D11:F11) and that either the existing plant must

be modernized or a new plant must be built (B24=1). Figure 12-17 shows the settings for these items. Figure฀12-17

Solver Settings for Optimizing the Choices for Capital Budgeting

Click the Options button on the Solver Parameters dialog box and select “Assume Linear Model,” as shown in Figure 12-18. Linear models can be solved faster than nonlinear ones. The linear model can be used here

Figure฀12-18

Solver Options Dialog Box with Assume Linear Model Selected

390 ® ❧ Corporate Financial Analysis with Microsoft Excel

because the entries in Cells C15:F23 are all linear functions of the decision variables in Cells B15:B21. (In other situations where Solver is used and the “Assume Linear Model” option has been chosen, an error message is given when the entries are not linear models of the decision variables and a solution is attempted. In such cases, go back and click off the choice.)

The solution shows that the best choices are to modernize the existing plant, redesign existing product B, and develop new product Y. These give a total present value of $1,075,000. There are uncommitted funds in each of the three years, with a total of $145,000 for all three years.

Example 12.10: How would the results for Example 12.9 change if the available funds were increased to $500,000 for year 1 and $600,000 for year 2?

Solution: A solution is obtained by copying Figure 12-17 to a new worksheet, editing Cells D11 and E11 with the new values $500,000 and $600,000, and executing Solver again. In fact, depending on the starting values in Cells B15:B21, two solutions can be obtained, both providing the same maximum present value of $1,650,000 in Cell C22. The first of these is shown in Figure 12-19. This solution was obtained with the starting values for the decision variables the same as those in Figure 12-17.

Figure฀12-19

Solution A: Starting Values in Cells B15:B21 Are Those Shown in Figure 12-17

1 Example 12-10 GOLIATH INDUSTRIES (First solution with increase in available funds)

2 Input Data

Present

Annual Costs

Year 2 Year 3 4 Modernize existing plant

$300,000 6 Expand distribution network

5 Build new plant

$20,000 7 Redesign existing product A

$45,000 9 Develop new product X

8 Redesign existing product B

$200,000 $75,000 10 Develop new product Y

Choices 13 (1 = Yes,

Present

Annual Costs

Year 2 Year 3 15 Modernize existing plant

$0 $0 16 Build new plant

$0 $0 17 Expand distribution network

$0 $0 18 Redesign existing product A

$0 $0 19 Redesign existing product B

$45,000 $0 20 Develop new product X

$200,000 $75,000 21 Develop new product Y

$250,000 $200,000 22 Present value and annual costs

$495,000 $275,000 23 Uncommitted funds

$105,000 $25,000 24 Build OR modernize plant constraint

25 Sum of uncommitted funds

(Continued)

Capital Budgeting: The Basics ❧ 391

An alert CFO might also seek a second goal of minimizing the annual costs—or, in other words, maximizing the sum of the uncommitted funds in Cell B25—while, at the same time, still achieving the first goal of maximizing the present value of the investments. To achieve this second goal while still achieving the first, we need to solve the example again with new Solver settings. The new target is Cell B25, which contains the sum of the uncommitted funds. The objective is to maximize this. A new constraint is added that requires the total present value in Cell C22 to equal $1,650,000, which is the value that satisfies the first goal. Figure 12-20 shows the new Solver settings.

Figure฀12-20

Solver Settings for Maximizing Uncommitted Funds with Same Total Present Value

Figure 12-21 shows the results when the spreadsheet is executed a second time with the new Solver settings. Note that the present value of the investments in Cell C22 equals $1,650,000, as before, and the sum of the uncommitted funds has been increased to $270,000. This is an increase of $75,000 over the earlier value of $195,000, a saving of $75,000 in cost over the earlier solution.

The best decision is to build the new plant, redesign existing products A and B, and develop new product Y. This provides a present value of $1,650,000 for the decisions and leaves $270,000 of uncommitted funds.

The existence of alternate solutions that satisfy an objective is not unusual. Rather than an annoyance, it is an opportunity. Two or more optimum solutions mean that CFOs have a choice of options—and an opportunity to invoke a second objective in addition to the first. It is sort of like having your cake and eating it too!

Optimizing the Choice of Multiyear Projects