CMA Intermediate · Operations Management and Strategic Management · Optimum Allocation of Resources - LPP
Sundaram Metals maximises profit Z = 5x + 4y subject to: 6x + 4y ≤ 24 (machine hours) and x + 2y ≤ 6 (labour hours), x, y ≥ 0. The optimal solution is x = 3, y = 1.5 with Z = 21. What is the shadow price per machine hour?
The shadow price per machine hour is ₹0.75. Solving the dual equations 6u + v = 5 and 4u + 2v = 4 gives u = 0.75 and v = 0.5. The dual objective 24×0.75 + 6×0.5 equals 21, matching the primal optimum.
- A₹0.50
- B₹0.75Correct
- C₹1.00
- D₹1.50
Explanation
Both constraints bind. Dual: 6u + v = 5 and 4u + 2v = 4. From the first, v = 5 - 6u; substituting gives 4u + 10 - 12u = 4, so u = 0.75 and v = 0.5. Check: 24(0.75) + 6(0.5) = 18 + 3 = 21, equal to Z. The value ₹0.50 is the labour-hour shadow price, not the machine-hour one.
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