Tokyo Spas is developing a production plan in an attempt to

Tokyo Spas is developing a production plan in an attempt to maximize profits from the manufacture of three models of hot tubs -- Standard, Classic, and Deluxe -- this season.
    These products are expected to yield estimated profits of $400, $345, and $600 each, respectively. Each hot tub produced requires the following primary resources: labor, piping, and pumps. Specifically, the Standard model requires approximately 15.5 labor hours, 14.5 feet of piping, and 1 pump per hot tub.
    Each Classic model hot tub requires around 10.5 labor hours, 20.0 feet of piping, and 1 pump, while the Deluxe model requires approximately 12.0 labor hours, 22.0 feet of piping, and 2 pumps per hot tub. Over the next 3 months, it appears that there will be 2,650 labor hours available, and Toyko has recently purchased 3,450 feet of piping 231 pumps that must meet their needs during these three months (i.e., this production season).

Please help formulate.

Solution

Here, let us assume

Qs = Quantity of standard models to be produced for profit maximization

Qc = Quantity of classic models to be produced for profit maximization

Qd = Quantity of deluxe models to be produced for profit maximization

Per model requirement of resources is tabulated as below

Standard

Classic

Deluxe

Labor ( Hours)

15.5

10.5

12

Piping   (Feets)

14.5

20

22

Pumps (Each)

1

1

2

Available resources

Labor ( Hours)

2650

Piping   (Feets)

3450

Pumps (Each)

231

The case can be formulated as

Maximize = 400 * Qs   +   345 * Qc + 600 * Qd

( Here 400 ,345 & 600 are the estimated profits in $ for standard, classic & deluxe models respectively)

Subject to

A .   (1/15.5) * Qs + (1/10.5) *Qc + (1/12) * Qd   2650 ( This is how we can formulate available labor hours with us and requirement of labor hours for each model)

B. ( 1/14.5 ) * Qs + (1/20) * Qc + (1/22) * Qd   3450( This is how we can formulate available piping with us and requirement of piping for each model)

C . 1 * Qs + 1* Qc + (1/2) *Qd 231 ( This is how we can formulate available pumps with us and requirement of pump for each model)

Final formulation will be

Maximize = 400 * Qs   +   345 * Qc + 600 * Qd

Subject to :

(1/15.5) * Qs + (1/10.5) *Qc + (1/12) * Qd 2650

( 1/14.5 ) * Qs + (1/20) * Qc + (1/22) * Qd 3450

1 * Qs + 1* Qc + (1/2) *Qd 231