For a single item inventory system, the demand is continuous, which is 10000 per year. The replacement is instantaneous and backorders (S units) per cycle are allowed as shown in the figure.

As soon as the quantity (Q units) ordered from the supplier is received, the backordered quantity is issued to the customers. The ordering cost is Rs. 300 per order. The carrying cost is Rs. 4 per unit per year. The cost of backordering is Rs. 25 per unit per year. Based on the total cost minimization criteria, the maximum inventory reached in the system is ________ (round off to nearest integer).

The correct answer is 1137.

1. Identify the given parameters from the problem and the diagram:
The provided diagram illustrates a classic saw-tooth inventory model with instantaneous replenishment and planned shortages (backorders allowed). The total quantity ordered is represented by Q, the backordered shortage quantity per cycle is S, and the vertical axis represents the inventory level over time. The maximum inventory level achieved in the system is given by the peak value, which is equal to $Q-S$ (referred to here as $Q_{\text{max}}$).

The numerical values given in the problem are:
• Annual demand rate, $D=10000$ units/year
• Ordering cost per order, $C_o=\text{Rs.}300$
• Inventory carrying (holding) cost, $C_h=\text{Rs.}4$ per unit per year
• Backordering (shortage) cost, $C_b=\text{Rs.}25$ per unit per year

2. Mathematical Formula:
For a total cost minimization criteria, the maximum inventory level ($Q_{\text{max}}$) in an inventory system with backorders is calculated as:

$$Q_{\text{max}}=\sqrt{\frac{2\times D\times C_o}{C_h}\times \frac{C_b}{C_b+C_h}}$$

3. Step-by-Step Calculation:
Substitute the given values into the formula:

$$Q_{\text{max}}=\sqrt{\frac{2\times 10000\times 300}{4}\times \frac{25}{25+4}}$$

Simplify the term representing the Economic Order Quantity (EOQ) without shortages:
$\frac{2\times 10000\times 300}{4}=\frac{6000000}{4}=1500000$

Now, multiply by the shortage factor:
$1500000\times \frac{25}{29}=\frac{37500000}{29}\approx 1293103.448$

Take the square root of the result to find $Q_{\text{max}}$:
$Q_{\text{max}}=\sqrt{1293103.448}\approx 1137.147\text{units}$

Rounding to the nearest integer, the maximum inventory reached in the system is 1137 units.