Question

2. JOY leather, a manufacturer of leather Products, makes three types of belts A, B and C which are processed on three machines M1, M2 and M3. Belt A requires 2 hours on machine (M1) and 3 hours on machine (M2) and 2 hours on machine (M3). Belt B requires 3 hours on machine (M1), 2 hours on machine (M2) and 2 hour on machine (M3) and Belt C requires 5 hours on machine (M2) and 4 hours on machine (M3). There are 8 hours of time per day available on machine M1, 10 hours of time per day available on machine M2 and 15 hours of time per day available on machine M3. The profit gained from belt A is birr 3.00 per unit, from Belt B is birr 5.00 per unit, from belt C is birr 4.00 per unit. What should be the daily production of each type of belt so that the profit is maximum? a) Formulate the problem as LPM b) Solve the LPM using simplex algorithm. c) Interpret the shadow prices

Answer 1

Here is how you can formulate this problem as a linear programming model (LPM):

Let X1 be the number of belt A produced per day
Let X2 be the number of belt B produced per day
Let X3 be the number of belt C produced per day

The objective function is to maximize the profit, which is 3X1 + 5X2 + 4X3.

The constraints are as follows:

The time spent on machine M1 must be less than or equal to 8 hours per day:
2X1 + 3X2 <= 8
The time spent on machine M2 must be less than or equal to 10 hours per day:
3X1 + 2X2 + 5X3 <= 10
The time spent on machine M3 must be less than or equal to 15 hours per day:
2X1 + 2X2 + 4X3 <= 15
The number of belts produced must be positive:
X1, X2, X3 >= 0
To solve this LPM using the simplex algorithm, you can follow these steps:

Write the objective function and the constraints in standard form. In standard form, the objective function is to minimize the cost, and all constraints are in the form of "less than or equal to" a certain value.
Identify the decision variables and the objective function coefficients. In this case, the decision variables are X1, X2, and X3, and the objective function coefficients are 3, 5, and 4, respectively.
Set up the initial simplex tableau. The initial tableau consists of the decision variables, the objective function coefficients, and the constraints.
Choose a pivot element. The pivot element is the element in the objective function row that has the most negative coefficient. In this case, the pivot element is X1.
Find the pivot column. The pivot column is the column that corresponds to the pivot element. In this case, the pivot column is column X1.
Find the pivot row. The pivot row is the row that corresponds to the smallest positive ratio between the right-hand side (RHS) value and the corresponding pivot column element. In this case, the pivot row is the one corresponding to the constraint 2X1 + 3X2 <= 8.
Replace the pivot element with the reciprocal of itself. In this case, the pivot element is 2, so it is replaced with 1/2.
Use the pivot element to eliminate all other elements in the pivot column. In this case, you would use the pivot element (1/2) to eliminate the elements in the pivot column in the other rows.
Repeat steps 4 to 8 until you reach an optimal solution.
Interpret the shadow prices. The shadow price of a constraint is the change in the objective function value for a unit change in the right-hand side (RHS) value of the constraint, with all other variables held constant. Shadow prices can help you understand the value of the constraints in the LPM and how they contribute to the optimal solution.
I hope this helps! Let me know if you have any questions.