The Income Statement of Sony Manufacturing Company and other necessary details have been summarized below: Particulars Product X Product Y Total (Rs) : : : : Sales units 10,000 10,000 20,000 Sales…

a. Weighted Average Contribution Margin per Unit

The contribution margin per unit is calculated as:

$$\text{Contribution Margin per Unit}=\frac{\text{Total Contribution Margin}}{\text{Total Units Sold}}$$

Given:

• Total contribution margin = Rs. 70,000
• Total units sold = 20,000

$$\text{Weighted Average Contribution Margin per Unit}=\frac{70,000}{20,000}=\text{Rs. 3.50 per unit}$$

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b. Overall Break-Even Point (BEP) in Units and Amount

The break-even point (BEP) is where total revenue equals total costs (fixed + variable). It can be calculated using:

$$\text{BEP (units)}=\frac{\text{Total Fixed Costs}}{\text{Weighted Average Contribution Margin per Unit}}$$

Given:

• Total fixed costs = Product-wise fixed cost (Rs. 30,000) + Joint fixed cost (Rs. 22,000) = Rs. 52,000
• Weighted average contribution margin per unit = Rs. 3.50

$$\text{BEP (units)}=\frac{52,000}{3.50}=\text{14,857 units (approx.)}$$

To find the BEP in rupees, multiply by the average selling price per unit:

$$\text{Average Selling Price per Unit}=\frac{\text{Total Sales Revenue}}{\text{Total Units Sold}}=\frac{400,000}{20,000}=\text{Rs. 20 per unit}$$

$$\text{BEP (amount)}=14,857\times 20=\text{Rs. 297,140 (approx.)}$$

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c. Linear Programming Model for Profit Maximization

Given constraints:

• Raw material: 5,000 kg available (next year)
• Labour hours: 4,000 hours available (next year)

Decision Variables:

Let:

• $x$ = Number of units of Product X to produce
• $y$ = Number of units of Product Y to produce

Objective Function (Maximize Profit):

From the income statement:

• Contribution margin per unit of X = Rs. 40,000 / 10,000 = Rs. 4 per unit
• Contribution margin per unit of Y = Rs. 30,000 / 10,000 = Rs. 3 per unit

$$\text{Maximize }Z=4x+3y$$

Constraints:

1. Raw Material Constraint:

   • Product X uses 10 kg per unit (since Rs. 100,000 / 10,000 units = Rs. 10 per unit at Rs. 5/kg → 2 kg? Correction: Re-examining the data:
     • For Product X: Total material cost = Rs. 100,000 for 10,000 units → Rs. 10 per unit
     • Given material cost is Rs. 5 per kg, so: $$\text{Material per unit of X}=\frac{10}{5}=2\text{ kg/unit}$$
     • For Product Y: Total material cost = Rs. 50,000 for 10,000 units → Rs. 5 per unit $$\text{Material per unit of Y}=\frac{5}{5}=1\text{ kg/unit}$$
   • Total material constraint: $$2x+y\le 5,000$$

2. Labour Constraint:

   • Product X uses 6 DLH per unit (since Rs. 60,000 / 10,000 units = Rs. 6 per unit at Rs. 6/DLH → 1 DLH per unit? Correction:
     • For Product X: Total labour cost = Rs. 60,000 for 10,000 units → Rs. 6 per unit
     • Given labour rate is Rs. 6 per DLH, so: $$\text{Labour per unit of X}=\frac{6}{6}=1\text{ DLH/unit}$$
     • For Product Y: Total labour cost = Rs. 120,000 for 10,000 units → Rs. 12 per unit $$\text{Labour per unit of Y}=\frac{12}{6}=2\text{ DLH/unit}$$
   • Total labour constraint: $$x+2y\le 4,000$$

3. Non-negativity Constraints: $$x\ge 0,y\ge 0$$

Final Linear Programming Model:

$$\text{Maximize }Z=4x+3y$$ Subject to: $$2x+y\le 5,000\text{(Material constraint)}$$ $$x+2y\le 4,000\text{(Labour constraint)}$$ $$x,y\ge 0$$

Graphical Solution (Corner Points Method):

1. Find intersection points of constraints:

   • Solve $2x+y=5,000$ and $x+2y=4,000$: $$\{\begin{array}{l}2x+y=5,000\\ x+2y=4,000\end{array}$$ Multiply the second equation by 2: $$2x+4y=8,000$$ Subtract the first equation: $$3y=3,000⟹y=1,000$$ Substitute back: $$2x+1,000=5,000⟹x=2,000$$ Intersection point: $(2,000,1,000)$

2. Evaluate objective function at corner points:

   • $(0,0)$: $Z=0$
   • $(0,2,000)$: $Z=4(0)+3(2,000)=6,000$
   • $(2,500,0)$: $Z=4(2,500)+3(0)=10,000$
   • $(2,000,1,000)$: $Z=4(2,000)+3(1,000)=11,000$

Optimal Solution:

• Produce 2,000 units of X and 1,000 units of Y for maximum profit of Rs. 11,000.

Verification of Constraints at Optimal Point:

• Material: $2(2,000)+1,000=5,000$ kg (fully utilized)
• Labour: $2,000+2(1,000)=4,000$ hours (fully utilized)

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