Spencer Enterprises is attempting to choose among a series of new investment alternatives. The potential investment alternatives, the net present value of the future stream of returns, the capital requirements, and the available capital funds over the next three years are summarized as follows:

Capital Requirements $($ $ $)$

Alternative Net Present Value $($ $ $)$ ?Year $1$ ?Year $2$ ?Year $3$

Limited warehouse expansion $4,0003,0001,0004,000$ ?

Extensive warehouse expansion $6,0002,5003,5003,500$ ?

Test market new product $10,5006,0004,0005,000$ ?

Advertising campaign $4,0002,0001,5001,800$ ?

Basic research $8,0005,0001,0004,000$ ?

Purchase new equipment $3,0001,000500900$ ?

Capital funds available $7,0008,75010,500$ ?

$($ a $)$ ?Develop and solve an integer programming model for maximizing the net present value.

Max x $1+$ ? x $2+$ ? x $3+$ ? x $4+$ ? x $5+$ ? x $6$ ?

s $.$ t $.$ ?

?x $1+$ ? x $2+$ ? x $3+$ ? x $4+$ ? x $5+$ ? x $6$ ? $ Year $1$

?x $1+$ ? x $2+$ ? x $3+$ ? x $4+$ ? x $5+$ ? x $6$ ? $ Year $2$

?x $1+$ ? x $2+$ ? x $3+$ ? x $4+$ ? x $5+$ ? x $6$ ? $ Year $3$

x $1,$ ?x $2,$ ?x $3,$ ?x $4,$ ?x $5,$ ?x $6=0,1$ ?

Select the correct answers below.

? Optimal Solution:

x $1$ ?

x $2$ ?

x $3$ ?

x $4$ ?

x $5$ ?

x $6$ ?

? Net Present Value: $

?

$($ b $)$ ?Assume that only one of the warehouse expansion projects can be implemented. Modify your model of part $($ a $).$ ?Which of the following constraint must be added to the model?

$($ i $)$ ?x $1=$ ?x $2$

$($ ii $)$ ?x $1+$ ?x $2>=1$

$($ iii $)$ ?x $1+$ ?x $2<=1$

$($ iv $)$ ?None of the above.

?

$($ c $)$ ?Suppose that, if test marketing of the new product is carried out, the advertising campaign also must be conducted. Modify your formulation of part $($ b $)$ ?to reflect this new situation. Which of the following constraint must be added to the model in part $($ b $)?$

?

$($ i $)$ ?x $3>=$ ?x $4$

$($ ii $)$ ?x $4>=$ ?x $3$

$($ iii $)$ ?x $3+$ ?x $4<=1$

$($ iv $)$ ?x $3+$ ?x $4>=1$