Introduction
A product-mix Linear Programming Problem (LPP) seeks the optimal allocation of scarce manufacturing capacities among competing products to achieve a specific economic objective. In this scenario, three machines—Milling, Lathe, and Grinder—represent limited resource capacities, and the objective is to maximize total contribution margin across products A, B, and C.
1. Mathematical Formulation of the LPP
Let $x_1, x_2, x_3$ represent the number of units produced of Products A, B, and C respectively.
- Objective Function: Maximize Profit $Z = 20x_1 + 6x_2 + 8x_3$
- Subject to Machine Capacity Constraints:
- Milling Machine: $8x_1 + 2x_2 + 3x_3 \le 250$
- Lathe: $4x_1 + 3x_2 + 0x_3 \le 150$
- Grinder: $2x_1 + 0x_2 + 1x_3 \le 50$
- Non-negativity Restrictions: $x_1, x_2, x_3 \ge 0$
2. Standard Form and Simplex Iterations
To solve via the Simplex method, introduce slack variables $S_1, S_2, S_3 \ge 0$ representing idle capacities of the Milling machine, Lathe, and Grinder respectively:
- $8x_1 + 2x_2 + 3x_3 + S_1 = 250$
- $4x_1 + 3x_2 + 0x_3 + S_2 = 150$
- $2x_1 + 0x_2 + 1x_3 + S_3 = 50$
The objective function in standard form becomes: Maximize $Z = 20x_1 + 6x_2 + 8x_3 + 0S_1 + 0S_2 + 0S_3$.
- Optimality Criterion: The simplex method proceeds iteratively by choosing the non-basic variable with the largest positive net evaluation $(C_j - Z_j)$ as the entering variable, and determining the leaving variable via the minimum non-negative ratio test. By definition, basic variables must yield $C_j - Z_j = 0$ in every valid tableau.
- Convergence to Optimality: Through sequential pivot operations, after entering $x_1$ and $x_3$, testing reveals that non-basic slack variables retain positive evaluation until the 4th iteration (pivoting on key element 4.5), which successfully drives all index row elements to satisfy $C_j - Z_j \le 0$.
3. Optimal Solution and Managerial Interpretation
- Optimal Product Mix: $x_1 = 0$ (Product A), $x_2 = 50$ (Product B), $x_3 = 50$ (Product C).
- Maximum Total Profit: $Z = 20(0) + 6(50) + 8(50) = ₹300 + ₹400 = ₹700$.
- Resource Utilization: At this optimum, the final values of slack variables are $S_1 = 0$, $S_2 = 0$, and $S_3 = 0$. All three machines are fully utilized with zero idle capacity.
- Managerial Insight: Product A is omitted from production despite its higher unit margin (₹20) because its excessive resource consumption across all three machines displaces more profitable combinations of Products B and C per bottleneck machine-hour.
Conclusion
The Linear Programming model demonstrates how prioritizing products solely based on gross unit margin can lead to sub-optimal operational decisions. By analyzing scarce resource trade-offs via Simplex optimization, the firm achieves a maximum profit of ₹700 with 100% resource utilization across all manufacturing centers.