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Model I : Purchasing Model without Shortage

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 c1 = 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) = cs . D / Q + c1 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,

 

c1 = 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, cs = $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 c5              = $ 100

Number of items per order      = 500

Annual demand, D     = 500 x 4 = 2000.

c1 = Procurement price x inventory carrying cost per year

= 50 * 0.20 = $ 10

 

Total annual cost of the existing inventory policy

= D/Q.cs + 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.