To formulate the Linear Programming Problem (LPP) for this problem, let's proceed step by step.

Step 1: Define the Decision Variables

Let:

• $x$: Number of skilled labourers employed per day
• $y$: Number of semi-skilled labourers employed per day

Step 2: Objective Function

The goal is to minimize the total daily cost for wages and rejection losses.

• Daily wage cost per labourer:
  Skilled labour: $240 \, \text{Rs}$
  Semi-skilled labour: $160 \, \text{Rs}$

• Daily rejection loss per labourer:
  Rejection rate for skilled labour = 5% (0.05)
  Rejection rate for semi-skilled labour = 10% (0.10)
  Cost of rejecting an assembly = $25 \, \text{Rs}$

1. Assemblies rejected by skilled labour per day = $0.05 \times 80 = 4$ assemblies

   • Cost of rejection per skilled labour = $4 \times 25 = 100 \, \text{Rs}$

2. Assemblies rejected by semi-skilled labour per day = $0.10 \times 60 = 6$ assemblies

   • Cost of rejection per semi-skilled labour = $6 \times 25 = 150 \, \text{Rs}$

Thus, the total daily cost (wages + rejection loss) is:
$\text{Minimize } Z = 240x + 100x + 160y + 150y$
Simplifying:
$Z = 340x + 310y$

Step 3: Constraints

1. Production Constraint
   Each skilled labour produces 80 assemblies, and each semi-skilled labour produces 60 assemblies.
   The required weekly production = $1,86,000 \, \text{assemblies}$.
   Daily production requirement = $\frac{1,86,000}{6} = 31,000 \, \text{assemblies}$.
   Thus:
   $80x + 60y \geq 31,000$

2. Semi-Skilled Labour Limit Constraint
   The company wants to limit the number of semi-skilled labourers to utmost 400 per day:
   $y \leq 400$

3. Non-Negativity Constraint
   Labourers cannot be negative:
   $x \geq 0, \, y \geq 0$

Final LPP Formulation

$\text{Minimize } Z = 340x + 310y$

Subject to:

1. $80x + 60y \geq 31,000$
2. $y \leq 400$
3. $x \geq 0, \, y \geq 0$

Let me know if you need me to solve the LPP or explain any steps further!

Related Questions

1. How is the rejection rate incorporated into the objective function?
2. How can this LPP be solved using the graphical method?
3. What changes in the formulation if there is no limit on semi-skilled labourers?
4. How do we interpret the optimal solution of this LPP?
5. What happens if the weekly production target increases to 2,00,000 assemblies?

Tip

In LPPs, clearly defining decision variables and interpreting constraints is critical to correct formulation.