Question

Seram Seramik Sdn. Bhd. received orders on décor lamp and vase made from clay with Perak Malay traditional design. Main resources used to produce both products are skill labour and clay specially purchased from Ipoh. Table 3.1 shows the requirements for both resources and estimated profit to produce each product.

The company has 10 skilled workers and clay always kept in stock 240kg per day for making both products. It works 8 hours per day.

i. Develop a complete Linear Programming model that represents all the above constraints, i.e. variables, objective function and constraint equations.

ii. Using graphical solution method, determine the mix quantities of lamp and vase to be produced by the company so that it maximizes the profit.

Answer 1

i.

Formulating as LP model

Decision variables:

Let the number of lamps be x.
Let the number of lamps be y.

 

Objective function:

Objective is to maximize total profit = 12x + 15y

 

Constraints:

1. Labour limit:
As there are 10 workers and each works 8 hours a day, total no. of hours = 8*10 = 80
2x + 8y <= 80

 

2. Clay limit:
4x + 6y <= 240

 

3. Non-negativity:
x, y >= 0

 

Formulated LP model:

Maximize z = 12x + 15y

 

subject to:
2x + 8y <= 80
4x + 6y <= 240
x, y >= 0

 

ii.

Plotting lines on graph:

From constraint 1,
2x + 8y = 80
or x/40 + y/10 = 1
Points: (40, 0) nd (0, 80)

From constraint 2,
4x + 6y = 240
or x/60 + y/40 = 1
Points: (60, 0) and (0, 40)

 

Graph:

 

The shaded region is the feasible region.

 

Considering corner points:

The feasible region has 3 corner points - P1 (0,0), P2(40,0) and P3(0,10).

z = 12x + 15y

z at P1 (z1) = 12*0 + 15*0 = 0+0 = 0
z at P2 (z2) = 12*40 + 15*0 = 480+0 = 480
z at P3 (z3) = 12*0 + 15*10 = 0+150 = 150

As z2 is greater than others, P2 (40, 0) is the optimal point.

Optimal mix: 40 lamps and 0 vases.