Problem $1$: $(100$ points$)$ Hardgrave can manufacture four different products $($A$,$ B$,$ C$,$ D$)$ in any

combination. Each product requires time on each of four machines in minutes per pound of

products, as shown in the Table $1$:

Table $1$

PRODUCT

MACHINE A B C D

$1$

$2$

$3$

$4$

$5344$

$10652$

$6431$

$3832$

MAXIMUM DEMAND $400100150500$

Each machine is available $40$ hours per week. Products A$,$ B$,$ C$,$ and D may be sold at prices of $$9,$

$$7,$ $$6,$ and $$5$ per pound respectively. Material cost for each pound of product A is $$4$ and for each

pound of products B$,$ C$,$ and D is $$1.$ Given the maximum demand in the above table, to find the

optimal product mix to maximize profit for Hardgrave

a$)$ Formulate this model in algebraic form. $(30$ points$)$

b$)$ Formulate and solve a spreadsheet model for this problem. $(70$ points$)$