Q.A company has two grades of inspectors, I and II to undertake quality control inspection. At least 1,500 pieces must be inspected in an 8-hour day. Grade I inspector can check 20 pieces in an hour with an accuracy of 96%. Grade II inspector checks 14 pieces an hour with an accuracy of 92%. Wages of grade I Inspector are Rs. 5 per hour while those of Grade II Inspector are Rs. 4 per hour. Any error made by Inspector costs Rs. 3 to the company. If there are, in all, 10 grade I inspectors and 15 Grade II inspectors in the company, find the optimal assignment of inspectors that minimizes the daily inspection cost. Formulate the above problem as linear programming problem.

Concept: Linear Programming Formulation

We need to minimize the total daily cost (wages + error cost) subject to inspection capacity and availability constraints.

Step 1 – Decision Variables:

Let $x_1$ = number of Grade I inspectors assigned, $x_2$ = number of Grade II inspectors assigned.

Step 2 – Objective Function (minimize daily cost):

Each Grade I inspector works 8 hours at Rs. 5/hour (wage = Rs. 40) and checks $20 \times 8 = 160$ pieces with 4% error rate, causing error cost $= 160 \times 0.04 \times 3 = \text{Rs. } 19.2$. Total cost per Grade I inspector = $40 + 19.2 = 59.2$.

Each Grade II inspector works 8 hours at Rs. 4/hour (wage = Rs. 32) and checks $14 \times 8 = 112$ pieces with 8% error rate, causing error cost $= 112 \times 0.08 \times 3 = \text{Rs. } 26.88$. Total cost per Grade II inspector = $32 + 26.88 = 58.88$. …

Why Linear Programming Captures This Problem

The company faces a resource-allocation decision: how many inspectors of each grade should work to meet the inspection quota while keeping costs down. Costs come from two sources—wages paid and penalties for errors—and both scale linearly with the number of inspectors assigned. The constraints (minimum pieces inspected, maximum inspectors available) are also linear inequalities. This structure is exactly what linear programming handles: optimize a linear objective over a feasible region defined by linear constraints.

The key insight is to express the total daily cost per inspector type as the sum of wage cost and expected error cost, then multiply by the number of inspectors assigned.

Step-by-Step Formulation

1. Define the decision variables

Let $x_1$ = number of Grade I inspectors assigned for the day.

Let $x_2$ = number of Grade II inspectors assigned for the day.

Both $x_1$ and $x_2$ must be non-negative and, in practice, integers (though the LPP formulation itself treats them as continuous; integer constraints would make it an ILP).

2. Calculate the inspection throughput constraint

Each Grade I inspector works 8 hours and checks 20 pieces/hour, so one Grade I inspector inspects $8 \times 20 = 160$ pieces per day.

Each Grade II inspector works 8 hours and checks 14 pieces/hour, so one Grade II inspector inspects $8 \times 14 = 112$ pieces per day.

The company requires at least 1,500 pieces inspected daily:

3. Determine the cost structure

Wage cost:

Grade I: Rs. 5/hour $\times$ 8 hours = Rs. 40 per inspector per day.

Grade II: Rs. 4/hour $\times$ 8 hours = Rs. 32 per inspector per day.

Error cost:

Grade I has 96% accuracy, so 4% of pieces are incorrectly inspected. Each error costs Rs. 3.

One Grade I inspector inspects 160 pieces/day, so expected errors = $0.04 \times 160 = 6.4$ pieces.

Error cost per Grade I inspector per day = $6.4 \times 3 = \text{Rs. } 19.2$.

Grade II has 92% accuracy, so 8% error rate.

One Grade II inspector inspects 112 pieces/day, so expected errors = $0.08 \times 112 = 8.96$ pieces.

Error cost per Grade II inspector per day = $8.96 \times 3 = \text{Rs. } 26.88$.

Total daily cost per inspector:

Grade I: $40 + 19.2 = \text{Rs. } 59.2$

Grade II: $32 + 26.88 = \text{Rs. } 58.88$

4. Write the objective function

We want to minimize the total daily inspection cost:

5. State the availability constraints

The company has at most 10 Grade I inspectors:

The company has at most 15 Grade II inspectors:

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