Problem A$-3($Algo$)$

A manufacturing firm has discontinued production of a certain unprofitable product line. Considerable excess production capacity was created as a result. Management is considering devoting this excess capacity to one or more of three products: X$1,$ X$2,$ and X$3.$

Machine hours required per unit are

PRODUCT

MACHINE TYPE X$1$ X$2$ X$3$

Milling machine $457$

Lathe $670$

Grinder $1004$

The available time in machine hours per week is

MACHINE HOURS PER WEEK

Milling machines $700$

Lathes $602$

Grinders $80$

The salespeople estimate they can sell all the units of X$1$ and X$2$ that can be made. But the sales potential of X$3$ is $90$ units per week maximum.

Unit profits for the three products are

UNIT PROFITS

X$1$ $ $39$

X$230$

X$329$

a$.$ Set up the equations that can be solved to maximize the profit per week. $($Leave no cells blank $-$ be certain to enter $"0"$ wherever required.$)$

s$.$t

b$.$ Solve these equations using the Excel Solver. $($Leave no cells blank $-$ be certain to enter $"0"$ wherever required. Round your answers to the nearest whole number. Use your ROUNDED decision variable values to compute total profit.$)$

c$.$ How much of each constraint or resource is unused? $($Leave no cells blank $-$ be certain to enter $"0"$ wherever required. Round your answers to the nearest whole number. Use your ROUNDED decision variable values to compute the unused resources.$)$

d$-1.$ Would the machines work at capacity?

d$-2.$ Will X$3$ be at maximum sales capacity?

multiple choice

No

Yes

e$.$ Suppose that an additional $250$ hours per week can be obtained from the Grinder by working overtime. The incremental cost would be $$1\mathrm{.}75$ per hour. What is the shadow price for one additional hour of overtime in the Grinder Department? If this same shadow price must be maintained and if working overtime must remain profitable, should all additional $250$ overtime hours be utilized? $($Round "shadow price" to $2$ decimal places.$)$