Answer to Question #283651 in Operations Research for Eden

Question

 A firm produces products A, B, and C, each of which passes through assembly and inspection departments. The number of person hours required by a unit of each product in each department is given in the following table.

   Person hours per unit of product

Product A

Product B

Product C

Assembly 

2

4

2

Inspection 

3

2

1

During a given week, the assembly and inspection departments have available at most 1500 and 1200 person-hours, respectively. if the unit profits for products A, B, and C are Birr 50, Birr 40, and Birr 60, respectively, determine the number of units of each product that should be produced in order to maximize the total profit and satisfy the constraints of the problem

Answer 1

In the given question, our decision variables are A, B and C. Now, let us assume that these products A, B and C are X1, X2 and X3 respectively.

Our objective is to maximize profits. Therefore, our objective function is maximize Z=50X1+40X2+60X3

Our constraints are on the available person hours and can be expressed mathematically as:

2X1+4X2+2X3≤1500

3X1+2X2+X3≤1200

We will now solve the above equations using simplex method.

We first introduce slack variables (S) to the inequalities for them to become equations as shown below:

2X1+4X2+2X3+S1=1500

3X1+2X2+X3+S2=1200

We then match the objective function Z to zero

Z-50X1-40X2-60X3=0

Next, we create an initial simplex table with the objective function at the bottom row as shown below:

From the table above, we need to identify the most negative value in the bottom row so as to determine the pivot column. In our case, -60 is the most negative value, hence values 2, 1 and -60 form the pivot column.

Next, we compute the quotients by dividing the entries in the C column(excluding the entry in the bottom row) with elements in the pivot column. In our case we divide 1500 by 2 to get 750 and 1200 by 1 to get 1200. Since the first row gives the smallest quotient, it becomes the pivot row and 2 becomes the pivot element.

We then conduct pivoting by first making all values in the pivot column zero and the pivot element one. We divide them with the pivot element as illustrated by the table below:

Now that there are no negative values in the bottom row, moving to another point would lower the value of the objective function, and so we stop there.

From the final table above, it is evident that zero units of both products A and B and 750 units of product C should be produced to generate a maximum profit of 45000 Birr.