Answer to Question #175706 in Operations Research for Nomi

Question

The Pinewood Furniture Company produces chairs and tables from two resources—labor and wood. The company has 80 hours of labor and 36 board feet of wood available each day. Demand for chairs is limited to 6 per day. Each chair requires 8 hours of labor and 2 board feet of wood to produce, while a table requires 10 hours of labor and 6 board feet of wood. The profit derived from each chair is $400 and from each table, $100. The company wants to determine the number of chairs and tables to produce each day to maximize profit. Formulate a linear programming model for this problem and solve using the simplex method.

Answer 1

Profit per chair = $400

Profit per table = $100

Labor hours required to make a chair = 8 hours

Labor hours required to make a table = 10 hours

Wood used per chair = 2 board-ft.

Wood used per table = 6 board-ft.

Availability labor hours = 80 hours

Availability of wood = 36 board-ft.

Daily demand of chairs is limited to = 6 chairs per day

Formulate the linear programming model for the problem as shown below:

Let x and y represent chair and table respectively.

The problem is converted to canonical form by adding slack, surplus and artificial variables as appropriate

1. As the constraint-1 is of type '≤' we should add slack variable S1

2. As the constraint-2 is of type '≤' we should add slack variable S2

3. As the constraint-3 is of type '≤' we should add slack variable S3

After introducing slack variables

$Max\ Z_{profit}=400x+100y+0S_1+0S_2+0S_3$

subject to

$x+S_1=6$

$2x+6y+S_2=36$

$8x+10y+S_3=80\ \ and$

$x,y,S_1,S_2,S_3≥0$

Negative minimum $Z_j-C_j$  is -400 and its column index is 1. So, the entering variable is x.

Minimum ratio is 6 and its row index is 1. So, the leaving basis variable is S1.

∴ The pivot element is 1.

Entering =x, Departing =S1, Key Element =1

Negative minimum $Z_j-C_j$  is -100 and its column index is 2. So, the entering variable is y.

Minimum ratio is 3.2 and its row index is 3. So, the leaving basis variable is S3.

 ∴ The pivot element is 10.

Entering =y, Departing =S3, Key Element =10

Since all $Z_j-C_j≥0$

Hence, optimal solution is arrived with value of variables as :

x=6, y=3.2

$Max \ \ Z_{profit}=2720$

It can be observed that the highest value is at point C i.e. 2,720 which means the furniture company should make 6 chairs and 3.2 tables to maximize its profit.