Problem $3.$ You work for the R&D group at XYZ Manufacturing Company. Recently you were

presented with an equipment purchase problem. The company is going to introduce five new

products $($Products A$,$ B$,$ C$,$ D$,$ and E$),$ and can produce these products on five different

machines $($Machines $1,2,3,4,$ and $5).$ In this problem, your task will be to decide which

machine$($s$)$ to purchase and what product to produce on which machine$($s$).$ The demand for

the products and the purchase costs for the machines are given below:

The products and machines are such that Product A can be produced on Machine $1$;

Product $B$ can be produced on either Machine $1$ or $2$; Product $C$ can be produced on

Machine $1,2$ or $3$; Product $D$ can be produced on Machine $1,2,3$ or $4$; and Product E can

be produced on any of the five machines. The following table shows the unit production

cost for each product$-$machine pair.

Which machines should the XYZ Manufacturing Company buy and how should the

products be allocated to the machines? Assume there are no capacity constraints and all

demands must be met.

Hint $1$: Notice from the cost matrix that if$,$ for example, XYZ buys Machines $1$ and $2,$ then all

products except for A will be produced on Machine $2,$ since it is clearly cheaper to do so$.$

Hint $2$: One can find the analogy if we treat products A$,$ B$,$ C$,$ D$,$ E as demand in periods $1,2,$

$3,4,5.$ This is also equivalent to treating $5$ machines as if successive $5$ periods. This is

possible because the jobs which can be done by the machines are inclusively increasing.