$12\mathrm{.}1-1.$ Reconsider the California Manufacturing Co$.$ example

presented in Sec. $12\mathrm{.}1.$ The mayor of San Diego now has con

tacted the company$$s president to try to persuade him to build a

factory and perhaps a warehouse in that city. With the tax incen

tives being offered the company, the president$$s staff estimates

that the net present value of building a factory in San Diego

would be $$7$ million and the amount of capital required to do this

would be $$4$ million. The net present value of building a ware

house there would be $$5$ million and the capital required would

be $$3$ million. $($This option would be considered only if a factory

also is being built there.$)$

The company president now wants the previous OR study

revised to incorporate these new alternatives into the overall

problem. The objective still is to find the feasible combination of

investments that maximizes the total net present value, given that

the amount of capital available for these investments is $$10$ million.

$($a$)$ Formulate a BIP model for this problem.The CALIFORNIA MANUFACTURING COMPANY is considering expansion by

building a new factory in either Los Angeles or San Francisco, or perhaps even in both

cities. It also is considering building at most one new warehouse, but the choice of loca

tion is restricted to a city where a new factory is being built. The net present value $($total

profitability considering the time value of money$)$ of each of these alternatives is shown

in the fourth column of Table $12\mathrm{.}1.$ The rightmost column gives the capital required

$($already included in the net present value$)$ for the respective investments, where the total

capital available is $$10$ million. The objective is to find the feasible combination of

alternatives that maximizes the total net present value.$$ TABLE $12\mathrm{.}1$ Data for the California Manufacturing Co$.$ example

Decision

Number

$1$

$2$

$3$

$4$

Yes$-$or$-$No

Question

Build factory in Los Angeles?

Build factory in San Francisco?

Build warehouse in Los Angeles?

Decision

Variable

Net Present

Value

x$1$

x$2$

x$3$

Build warehouse in San Francisco?

x$4$

$$9$ million

$$5$ million

$$6$ million

$$4$ million

Capital

Required

$$6$ million

$$3$ million

$$5$ million

$$2$ million

Capital available: $$10$ million