Q.A company has two groups of inspectors, namely, group A and B, who are assigned to do a quality inspection work. It is required that at least 1800 pieces are inspected per 8-hour day. It is known that inspectors of group A can check pieces at the rate of 25 per hour with an accuracy of 98%, while inspectors of group B can check at the rate of 15 pieces per hour with an accuracy of 95%. The inspectors of group A and B are paid Rs. 40 and Rs. 30 per hour respectively to do the work. Each time an error is caused by any inspector, it costs a loss of Rs. 20 to the company. The company has 8 inspectors in group A and 10 in group B. The company wants to determine the optimal assignment of inspectors to minimise total inspection cost. Formulate an LPP.

Concept: Linear Programming Formulation – the goal is to minimise total cost (wages + error loss) subject to inspection rate, accuracy, and availability constraints.

Step 1: Define decision variables

Let $x$ = number of inspectors from group A assigned per day, $y$ = number from group B.

Each works 8 hours.

Step 2: Express the objective function

Wage cost: $40 \times 8x + 30 \times 8y = 320x + 240y$

Error loss per inspector:

• Group A: 25 pieces/hr × 8 hr × (2% error) × Rs. 20 = $25 \times 8 \times 0.02 \times 20 = 80$ per inspector
• Group B: 15 × 8 × 0.05 × 20 = 120 per inspector Total cost: $Z = (320+80)x + (240+120)y = 400x + 360y$

Step 3: Constraints

• Inspection requirement: $25 \times 8 \times x + 15 \times 8 \times y \geq 1800 \Rightarrow 200x + 120y \geq 1800$
• Availability: $x \leq 8$, $y \leq 10$ …

Why Linear Programming Works Here

The company faces a resource allocation problem with clear trade-offs. Group A inspectors are faster (25 pieces/hour) and more accurate (98%), but cost more per hour (Rs. 40). Group B inspectors are slower (15 pieces/hour), less accurate (95%), but cheaper per hour (Rs. 30). Each error costs Rs. 20.

The goal is to decide how many inspectors from each group to assign per 8-hour day so that:

• At least 1800 pieces are inspected daily.
• We don't exceed the available inspectors (8 in A, 10 in B).
• The total cost (wages + penalty for errors) is as low as possible.

Since all relationships are linear (costs add up, inspection rates multiply linearly with hours, error costs are proportional to pieces inspected), this is a textbook linear programming problem.

Step-by-Step Formulation

1. Define the decision variables

Let:

• $x$ = number of inspectors from group A assigned per day
• $y$ = number of inspectors from group B assigned per day

Both $x$ and $y$ must be non-negative integers in reality, but for an LPP we treat them as continuous variables (the integer restriction can be handled later if needed).

2. Express the inspection capacity constraint

Each group A inspector works 8 hours at 25 pieces/hour, so one inspector inspects $8 \times 25 = 200$ pieces per day.

Each group B inspector works 8 hours at 15 pieces/hour, so one inspects $8 \times 15 = 120$ pieces per day.

Total pieces inspected per day = $200x + 120y$.

We need at least 1800 pieces:

Divide through by 40 to simplify:

3. Express the availability constraints

Group A has 8 inspectors: $x \leq 8$

Group B has 10 inspectors: $y \leq 10$

Also, $x \geq 0$, $y \geq 0$.

4. Build the cost function (objective)

Cost has two components: wages and error penalty.

Wages per day:

Group A: Rs. 40/hour × 8 hours = Rs. 320 per inspector per day

Group B: Rs. 30/hour × 8 hours = Rs. 240 per inspector per day

Total wages = $320x + 240y$

Error penalty per day:

Group A accuracy = 98%, so error rate = 2%. Each inspector inspects 200 pieces/day, so errors per inspector = $200 \times 0.02 = 4$ pieces. Each error costs Rs. 20, so penalty per A inspector = $4 \times 20 = \text{Rs. }80$ per day.

Group B accuracy = 95%, so error rate = 5%. Errors per inspector = $120 \times 0.05 = 6$ pieces. Penalty per B inspector = $6 \times 20 = \text{Rs. }120$ per day.

Total error penalty = $80x + 120y$

Total cost (to minimise):

5. Write the complete LPP

Minimise:

Subject to:

$x \geq 0, \ y \geq 0$ …