*Assumptions.* Demhand is known and unifom1, purchasing at equal interval, zero lead time, no shortages and instantaneous replenishment.

The corresponding model is shown in Fig. 6.1.

Let D = annual demand and c_{1} = holding cost/unit/year and time horizon = 1 year.

Total inventory over the time period t = Area of the first triangle

= ½ Q.t

Average inventory at any time = (1/2 Q.t ) / t = 1/2Q

Total cost = Ordering cost + Holding cost + Purchase cost (constant)

Minimize TC(Q) = c_{s} . D / Q + c_{1 }1/2 .Q +Constant

Time between orders

T* = Q*/ D

n* = optimum number of orders placed per year

= D / Q*

**Note.** if the holding cost is given as a percentage of average value of inventory held, then total annual holding cost,

c_{1} = c * 1, where c = unit cost

1 = % of the value of the average inventory.

The cost functions are shown in the Fig. 6.2

**Example 1.** A medical wholesaler supplies 30 bottles cough syrup each week to various shops. Cough syrups are purchased from the manufacturer in lots of 120 each for $ 1200 per lot. Ordering cost is $ 210 per order. All orders are filled the next day. The incremental cost is $ 0.60 per year to store a bottle in inventory. The wholesaler finances inventory investments by paying its holding company 2% monthly for borrowed funds. Suppose multiple and fractional lots also can be ordered.

How many bottles should be ordered and how frequently he should order ?

**Solution. **Consider 1 year = 52 weeks as working time.

Annual demand, D = 30 x 52 = 1560 .

Unit cost of purchases =1200/120 = $10

Ordering cost, c_{s }= $210

Inventory carrying cost = 0.6 + 10 *24/100 = $3 per unit/year

T* = = Q* / D = 467.33 / 1560 = 0.3 year

**Example 2.** A company purchases in lots of 500 items which is a 3 month supply. The cost per item is $ 50 and the ordering cost is $ 100. The inventory carrying cost is estimated at 20% of unit value. What is the total cost of the existing inventory policy? How much money could be saved by employing the economic order quantity?

**Solution.** Given c_{5 } = $ 100

Number of items per order = 500

Annual demand, D = 500 x 4 = 2000.

c_{1} = Procurement price x inventory carrying cost per year

= 50 * 0.20 = $ 10

Total annual cost of the existing inventory policy

= D/Q.c_{s} + Q/2.c = 2000/500*100+500/2*10 = $ 2900

Now

Then the corresponding annual cost

= 2000/200 x 100 + 200/2 *10 = $ 2000.

Hence by employing the economic order quantity, the company may save $ (2900 - 2000) = $ 900.