To formulate the given problem as a linear programming problem, we first need to define the variables, the objective function, and the constraints based on the information provided.
Variables:
Let:
- \( x_1 \) = the number of units of foam produced (Product A)
- \( x_2 \) = the number of units of soap produced (Product B)
Objective Function:
We want to maximize the profit from producing foam and soap. The profit per unit for foam is birr 60, and for soap it’s birr 45. Thus, the objective function can be expressed as: \[ \text{Maximize } Z = 60x_1 + 45x_2 \]
Constraints:
We have constraints based on the availability of capital and labor:
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Capital Constraints: Each unit of foam requires 4 kg of capital, and each unit of soap requires 3 kg of capital. The total available capital is 100 kg. Thus, the constraint can be expressed as: \[ 4x_1 + 3x_2 \leq 100 \]
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Labor Constraints: Each unit of foam requires 4 hours of labor, and each unit of soap requires 3 hours of labor. The total available labor hours is 120. Thus, the constraint can be expressed as: \[ 4x_1 + 3x_2 \leq 120 \]
Non-negativity Constraints:
Both production quantities cannot be negative: \[ x_1 \geq 0 \] \[ x_2 \geq 0 \]
Summary of the Linear Programming Problem:
Putting it all together, we have the following linear programming formulation:
Objective Function: \[ \text{Maximize } Z = 60x_1 + 45x_2 \]
Subject to: \[ 4x_1 + 3x_2 \leq 100 \quad \text{(Capital Constraint)} \] \[ 4x_1 + 3x_2 \leq 120 \quad \text{(Labor Constraint)} \] \[ x_1 \geq 0 \quad \text{(Non-negativity for foam)} \] \[ x_2 \geq 0 \quad \text{(Non-negativity for soap)} \]
This linear programming model can be solved using various methods such as the Simplex method, graphical method, or by using software tools for optimization.
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