A Company Wants to Build a New Factory in Either

Essay

A company wants to build a new factory in either Atlanta or Columbia. It is also considering building a warehouse in whichever city is selected for the new factory. The following table shows the net present value (NPV) and cost of each facility. The company wants to maximize the net present value of its facilities, but it only has $16 million to invest.
$$\begin{array} { c l c c } \text { Variable } & \text { Decision } & \begin{array} { c } \text { NPV } \\\text {( \$ millon )}\end{array} & \begin{array} { c } \text { Cost } \\\text {( \$ millon )} \end{array} \\\hline \mathbf { X } _ { 1 } & \text { Factory in Columbia } & 3 & 10 \\\mathbf { X } _ { \mathbf { 2 } } & \text { Factory in Atlanta } & 4 & 8 \\\mathbf { X } _ { 3 } & \text { Warehouse in Columbia } & 2 & 6 \\\mathbf { X } _ { 4 } & \text { Warehouse in Atlanta } & 1 & 5\end{array}$$ Based on this ILP formulation of the problem what is the optimal solution to the problem?
$\begin{array} { l l } \text { MAX: } & 3 \mathbf { X } _ { 1 } + 4 \mathbf { X } _ { \mathbf { 2 } } + 2 \mathbf { X } _ { \mathbf { 3 } } + \mathbf { X } _ { 4 } \\ \text { Subject to: } & 10 \mathbf { X } _ { 1 } + \mathbf { 8 } \mathbf { X } _ { \mathbf { 2 } } + 6 \mathbf { X } _ { \mathbf { 3 } } + 5 \mathbf { X } _ { \mathbf { 4 } } \leq 15 \\ & \mathbf { X } _ { 1 } + \mathbf { X } _ { \mathbf { 2 } } = 1 \\ & \mathbf { X } _ { \mathbf { 3 } } + \mathbf { X } _ { \mathbf { 4 } } \leq 1 \\ & \mathbf { X } _ { \mathbf { 3 } } - \mathbf { X } _ { 1 } \leq 0 \\ & \mathbf { X } _ { \mathbf { 4 } } - \mathbf { X } _ { \mathbf { 2 } } \leq 0 \\ & \mathbf { X } _ { \mathrm { i } } = 0,1 \end{array}$