Answer to Question #297067 in Operations Research for Denver

Question

<e> 3. A firm can produce a good either by (i) a labor intensive technique, using 8 units of labor and 1 unit of capital or (ii) a capital intensive technique using 1 unit of labor and 2 units of capital. The firm can arrange up to 200 units of labor and 100 units of capital. Note that the firm produce goods X and Y in process land 1 and 2 respectively. Its objective is to maximize profit by selling the good.

i) Construct the objective function and the constant inequalities.

ii) By drawing the graphs of the linear constraints, find the optimal solutions.

 iii) Find the solution using the simplex methods.

Answer 1

Solution:

Let the no. of units produced for good X and Y be $x,y$ respectively.

(i):

Cost is missing for objective function.

So, we assume objective function is $Z=100x+150y$

Subject to constraints,

$8x+y\le 200 \\ x+2y\le 100 \\ x,y\ge 0$

(ii):

Corner points are:

O(0,0), A(25,0), B(20,40),C(0,50).

Now, putting them in Z.

Z at O(0,0), Z=0

Z at A(25,0), Z=2500

Z at B(20,40), Z=8000 (maximum)

Z at C(0,50), Z=7500

Maximum profit is 8000 at x=20, y=40.

(iii):

Simplex method:

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

After introducing slack variables

Max Z=100x1+150x2+0S1+0S2

subject to

8x1+x2+S1=200

x1+2x2+S2=100 and x1,x2,S1,S2≥0

Since all Zj-Cj≥0

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

x1=20,x2=40

Max Z=8000