Brooklyn Cabinets is a manufacturer of kitchen cabinets. The two cabinetry styles manufactured by Brooklyn are contemporary and farmhouse. Contemporary style cabinets sell for $90 and farmhouse style cabinets sell for $85. Each cabinet produced must go through carpentry, painting, and fiishing processes. The following table summarizes how much time in each process must be devoted to each style of cabinet. Hours per process
Style Carpentry Painting Finishing
Contemporary 2.0 1.5 1.3
Farmhouse 2.5 1.0 1.2
Carpentry costs $15 per hour, painting costs $12 per hour, and fiishing costs $18 per hour, and the weekly number of hours available in the processes is 3000 in carpentry, 1500 in painting, and 1500 in fiishing. Brooklyn also has a contract that requires the company to supply one of its customers with 500 contemporary cabinets and 650 farmhouse style cabinets each week.
Let
x = the number of contemporary style cabinets produced each week
y = the number of farmhouse style cabinets produced each week
a. Develop the objective function, assuming that Brooklyn Cabinets wants to maximize the total weekly profit.
b. Show the mathematical expression for each of the constraints on the three processes.
Hours available in carpentry: x1 + y1 ≤
Hours available in painting: x2 + y2 ≤
Hours available in fiishing: x3 + y3 ≤
c. Show the mathematical expression for each of Brooklyn Cabinets' contractual agreements.

a) Objective Function:

P = $18.6x + $13.9y

b)

1. 2x + 2.5y 3000 For Carpentry

2. 1.5x +1.0 y 1500 For Painting

3. 1.3x + 1.2y 1500 For Finishing

c)

x 500

y 650

Explanation:

Data Given:

Contemporary Style Cabinets Sells For = $90

Farmhouse Style Cabinets Sell For = $85

Hours per process:

For Contemporary:

Time for Carpentry = 2.0

Time for Painting = 1.5

Time for Finishing = 1.3

Similarly,

for Farmhouse Cabinet style:

Time for Carpentry = 2.5

Time for Painting =  1.0

Time for Finishing = 1.2

Costing for Processes:

Carpentry = $15/hr

Painting = $12/hr

Finishing = $18/hr

Availability of hours in the week:

Carpentry = 3000 hours

Painting = 1500 hours

Finishing = 1500 hours

Orders for Cabinets in the week:

Contemporary Cabinets = 500 units/week

Farmhouse Style = 650 units/week  

Suppose,

x = number of contemporary style cabinets

y = number of Farmhouse cabinets

Step 1:

We need to calculate the total cost of the cabinets first.

Cost of Contemporary Style:

(2 x $15) + (1.5 x $12) + (1.3 x $18) = $71.4/cabinet

Similarly,

Cost of Farmhouse Style:

(2.5 x $15) + (1 x $12) + (1.2 x $18) = $71.1/cabinet

We know that the selling price of Contemporary and Farmhouse cabinets is $90 and $85 respectively. So, we can calculate the profit of both the cabinets.

For Contemporary Style:

Profit = ($90-$71.4) =$18.6

For Farmhouse Style:

Profit = ($85-$71.1)= $13.9

a) Objective Function for the maximization of the profit:

We know that x represents contemporary style and y represents farmhouse style. So, the profit is the basically the objective function. So.,

Objective Function:

P = $18.6x + $13.9y

b) Mathematical Expression for the constraints, which are:

1. 2x + 2.5y 3000 For Carpentry

2. 1.5x +1.0 y 1500 For Painting

3. 1.3x + 1.2y 1500 For Finishing

c) Mathematical Expression for Contracts:

x 500

y 650