Daily production capacity of a bearing manufacturing company is 30000 bearings. The daily demand of the bearing is 15000. The holding cost per year of keeping a bearing in the inventory is ₹ 20. The setup cost for the production of a batch is ₹ 1800. Assuming 300 working days in a year, the economic batch quantity in number of bearings is ______________ (in integer)

The correct answer is 40250.

Step-by-step Explanation:

To find the economic batch quantity (EBQ), we use the production inventory model (also known as the Economic Manufacturing Quantity model). Let us first list all the given values from the problem statement:

Daily production capacity (rate), p = 30,000 bearings/day
Daily demand rate, d = 15,000 bearings/day
Holding cost per bearing per year, C_H = ₹ 20
Setup cost per production batch, C_o = ₹ 1,800
Number of working days in a year = 300 days

Step 1: Calculate annual demand (D) and annual production capacity (P)

The annual demand is calculated by multiplying the daily demand by the number of working days in a year:
D = 15,000 × 300 = 4,500,000 bearings/year

Similarly, the annual production capacity is calculated by multiplying the daily production capacity by the number of working days in a year:
P = 30,000 × 300 = 9,000,000 bearings/year

Step 2: Apply the Economic Batch Quantity (EBQ) formula

The formula for Economic Batch Quantity (Q) is given by:

$$Q=\sqrt{\frac{2\times D\times C_o}{C_H}\times \left(\frac{P}{P-D}\right)}$$

Note that the term $\frac{P}{P-D}$ can also be simplified using the daily rates as $\frac{p}{p-d}$ because the number of working days cancels out.

Step 3: Substitute the values into the formula

$$Q=\sqrt{\frac{2\times \mathrm{4,500,000}\times \mathrm{1,800}}{20}\times \left(\frac{\mathrm{9,000,000}}{\mathrm{9,000,000}-\mathrm{4,500,000}}\right)}$$

Let us simplify the terms step-by-step:

First, evaluate the production-to-demand ratio component:
$\frac{\mathrm{9,000,000}}{\mathrm{9,000,000}-\mathrm{4,500,000}}=\frac{\mathrm{9,000,000}}{\mathrm{4,500,000}}=2$

Next, calculate the setup and holding cost component:
$\frac{2\times \mathrm{4,500,000}\times \mathrm{1,800}}{20}=2\times \mathrm{225,000}\times \mathrm{1,800}=\mathrm{810,000,000}$

Now, multiply the two parts under the square root:

$$Q=\sqrt{\mathrm{810,000,000}\times 2}$$

$$Q=\sqrt{\mathrm{1,620,000,000}}$$

$$Q\approx 40249.22$$

Rounding to the nearest integer as requested, we obtain:

Q ≈ 40250 bearings