The Philbrick Company has two plants on opposite sides of

the United States. Each of these plants produces the same two prod$-$

ucts and then sells them to wholesalers within its half of the coun$-$

try. The orders from wholesalers have already been received for the

next $2$ months $($February and March$),$ where the number of units

requested are shown below. $($The company is not obligated to com$-$

pletely fill these orders but will do so if it can without decreasing its

profits.$)$

Plant $1$ Plant $2$

Product February March February March

$13,6006,3004,9004,200$

$24,5005,4005,1006,000$

Each plant has $20$ production days available in February and $23$

production days available in March to produce and ship these prod$-$

ucts. Inventories are depleted at the end of January, but each plant

has enough inventory capacity to hold $1,000$ units total of the two

products if an excess amount is produced in February for sale in

March. In either plant, the cost of holding inventory in this way is

$$3$ per unit of product $1$ and $$4$ per unit of product $2.$

Each plant has the same two production processes, each of

which can be used to produce either of the two products. The pro$-$

duction cost per unit produced of each product is shown below for

each process in each plant. Plant $1$ Plant $2$

Product Process $1$ Process $2$ Process $1$ Process $2$

$1$ $$62$ $$59$ $$61$ $$65$

$2$ $$78$ $$85$ $$89$ $$86$

The production rate for each product $($number of units produced per

day devoted to that product$)$ also is given for each process in each

plant below.

Plant $1$ Plant $2$

Product Process $1$ Process $2$ Process $1$ Process $2$

$1100140130110$

$2120150160130$

The net sales revenue $($selling price minus normal shipping

costs$)$ the company receives when a plant sells the products to its

own customers $($the wholesalers in its half of the country$)$ is $$83$ per

unit of product $1$ and $$112$ per unit of product $2.$ However, it also is

possible $($and occasionally desirable$)$ for a plant to make a shipment

to the other half of the country to help fill the sales of the other plant.

When this happens, an extra shipping cost of $$9$ per unit of product $1$

and $$7$ per unit of product $2$ is incurred.

Management now needs to determine how much of each prod$-$

uct should be produced by each production process in each plant

during each month, as well as how much each plant should sell of

each product in each month and how much each plant should ship

of each product in each month to the other plant$$s customers. The

objective is to determine which feasible plan would maximize the

total profit $($total net sales revenue minus the sum of the production

costs, inventory costs, and extra shipping costs$).$

$($a$)$ Formulate a complete linear programming model in algebraic

form that shows the individual constraints and decision varia$-$

bles for this problem.

C $($b$)$ Formulate this same model on an Excel spreadsheet instead.