It seldom happens that future demand is not known exactly i.e., uncertain. We assume that the probability distribution of future demand is known which can be determined by some forecasting analysis. Probability distributions can be discrete or continuous. We will discuss some probabilistic models where only demand follows some probability distribution.

**(a) Model I (Discrete Case)**

**Assumptions.** Demand is instantaneous, total demand is filled at the beginning of the period, no setup cost, lead time is zero.

Let t = constant interval between orders.

Q = stock (in discrete units) for time t

d = estimated (random) demand with probability p(d).

Since the total demand is filled at the beginning of the period, the inventory position just after the demand occurs may be either positive (surplus) or negative (shortage).

When the demand does not exceed the stock Q i.e., d ≤ Q, the holding cost per unit time is

= (Q - d).c_{1}, d ≤ Q

When the demand d exceeds the stock Q i.e., d > Q, the shortage cost per unit time becomes

= O .c_{2}, d ≤ Q

The total expected cost per unit time is

For minimum c(Q), the condition

∆c(Q - 1) < c(Q)

**(a) Model I (Discrete Case)**

Assumptions. Demand is instantaneous, total demand is filled at the beginning of the period, no setup cost, lead time is zero.

t = constant interval between orders.

Q = stock (in discrete units) for time t

d = estimated (random) demand with probability p(d).

Since the total demand is filled at the beginning of the period, the inventory position just after the demand occurs may be either positive (surplus) or negative (shortage).

When the demand does not exceed the stock Q i.e., d ≤ Q, the holding cost per unit time is

= (Q – d) .c_{1}, d ≤ Q

= O.c_{1}, d ≥ Q.

When the demand d exceeds the stock Q i.e., d > Q, the shortage cost per unit time becomes

= 0.c_{2} , d ≤0

The total expected cost per unit time is

For minimum c(Q), the condition

Using the condition for minimum, we must haveThus combining thae optimum value of stock level Q can be obtained from the relationship

**Example 6.** A newspaper boy buys papers for $ 2 and sells them for $ 3 each. He cannot return the unsold newspapers. Daily demand has the following distribution

No. of customers 22 23 24 25 26 27 . 28 29 30 31

Probability 0.02 0.02 0.06 0.15 0.15 0.25 0.15 0.1 0.05 0.05

If each days demand is independent of the previous days, how many papers should be ordered each day ?

Solution. Here c_{1} = $ 2

c_{2} = $ (3 - 2) = $ 1

c_{2} /c_{1}+c_{2 }= 1 /2+1 =1/3 =0.33

Now we have to calculate the cumulative distribution i.e.,

No. of customer 22 23 24 25 26 27 28 29 30 31

Cumulative 0.02 0.04 0. 1 0.25 0.4 0.65 0.8 0.9 0.95 1

Since the value 0.33 lies between 0.25 and 0.4, the optimal Q* is taken as 26. i.e., the newspaper boy should order 26 newspaper per day to minimize the cost.

**(b) Model II (Continuous Case)**

Assumptions. The assumptions are same as in the discrete case expect that j{x) will be used as probability density function of demand x.

Proceeding in the same manner as in Model I (discrete case), the cost equation can be

formulated as follows :

The optimal value of Q can be obtained from dc(Q)/ dQ = 0 which gives

Thus we can find out the optimum value of Q, satisfying the above equation.

**Example 7.** A baking company makes a profit of $ 5 per kg. on each kg. cakes sold On the day it is baked. It disposes all cakes not sold on the date it is baked at a loss of $ 1.20 per kg. If demand is known to be rectangular between 2000 and 3000 kg., determine the optimal daily amount to be baked if the demand is instantaneous.

**Solution**. c_{1} = $ 5 , c_{2} = $ 1.20.

f(x) = 1/1000 , 2000 ≤ x ≤ 3000.

Also c2/c1+c2 =1.2/ 6.2=0.1935

Q - 2000 = 193.5

Q = 2193.5.

:. The company should bake 2193.5 kg. daily.

**(c) Model III (Discrete Case with Uniform Demand)**

Compare to Model I, the demand is uniform here rather than instantaneous and the other

assumptions are the same.

The optimum stock (i.e., EOQ) Q is obtained by the same relationship i.e.,

** (d) Model IV (Continuous Case with Uniform Demand)**

Compare to Model III, the demand is uniform here rather than instantaneous and the other assumptions are the same.

Here the optimum stock Q is obtained by the following relationship :

** PROBLEMS**

** **

**(Model I and III)**

1. A purchase manager places order each time for a Jot of 500 units of a particular item. From the available data the following results are obtained :

Ordering cost per order= $ 600

Cost per unit = $ 50

Annual demand = $ 1 000

Inventory carrying cost= 40%

Find out the loss to the organization due to his ordering policy.

2. An aircraft company uses rivets at an approximately constant rate of 5000 kg per year. The rivets cost $ 20 per kg. and the company personnel estimate that it costs $ 200 to place an order, and the carrying cost of inventory is 1 0% per year. How frequently should orders for rivets be placed and what quantities should be ordered for ?

3. The Inventory company, after an analysis of its accounting and production records, has determined that it uses $ 36000 per year of a component part purchased at $ 18 per part. The purchasing 2 cost is $ 40 per order, and its annual inventory carrying charges are 163% of the average inventory. Determine

(a) the most EOQ at one time.

(b) the most economic number of times to order per year.

(c) the average days' supply for ordering the most EOQ.

(year = 365 days)

4. A company uses 50,000 widgets per annum which costs $ 1 0 per piece to purchase. The ordering ad handling costs are $ 150 per order and carrying costs are 15% per annum. Find the EOQ. Suppose the company decides to make the widgets in its own factory and installed a machine which has capacity of 2,50,000 widgets per annum. What is the EOQ?

** **

**(Model III)**

5. A contractor has to supply 1 0000 bearing per day to an automobile manufacturer. He finds that when he starts production run, he can produce 25000 bearings per day. The cost of holding a

bearing in stock for a year is $ 2 and the setup cost of a production run is $ 1800. How

frequently should production run be made? (Assume 1 year = 300 working days)

6. In a paints manufacturing unit, the changeover from one type of paint to another is estimated to cost $ 1 00 per batch. The annual sales of a particular grade of paint are 20,000 litres and the

inventory carrying cost is $ 1 per litre. Given that the rate of production is 3 times the sales rate,

detem1ine the economic batch size and number of batches per year and total optimum yearly cost.

7. A product is sold at the rate of 30 pieces per day and is manufactured at a rate of 200 pieces per day. The set up costs of the machines are $ 300 and the holding cost is found to be $ 0.05

per piece day. Find optimum batch size, period of production and the optimum number of production run. (Assume I year = 365 days)

**(Model II and I)**

8. An item is to be supplied at a constant rate of 100 unit per day. The ordering cost for each supply is $ 20, cost of holding the item in inventory is $ 1 per unit per day while delay in the supply of the item induces a penalty of $ 3 per unit per day. Find the optimal policy (Q, /) and optimum shortage.

9. The demand for a product is 150 units per month and the items are withdrawn uniformly. The setup cost each time a production run is $. 12. The holding cost is $ 0.25 per item per month.

(a) Determine how often to make production run, if shortages are not allowed.

(b) Determine how often to make production run, if shortages cost $ 1 per item per month.

10.The demand for a certain item is uniform at a rate of 30 units per month The fixed cost is $ 10 each time a production run is made. The production cost is $ 2 per item and the holding cost is $ 0.3 per item per month. If the shortage cost is $ 1.5 per item per month, determine how often to make a production run and of what size should be?

**(Model IV)**

11. The demand for an item in a company is 24000 units per year, and the company can produce 2500 units per month. The one setup cost is $ 300 and the holding cost per unit per month is $ 0.3 and the shortage cost of one unit is $ 20 per month. Determine the optimum manufacturing quantity, the number of shortages and manufacturing time.

**(Probabilistic Models)**

12. Rework the Example 7, taking demand is uniform.

13. If the demand for a certain product has a rectangular distribution between 4000 and 500, find the optimal expected total cost if storage cost is $ 1 per unit and shortage cost is $ 7 per unit and the purchasing cost is $ 1 0 per unit, and the demand is instantaneous.

14. A certain children product is stocked by a company. The demand distribution is given below

Demand 0 10 20 30 40 50

Probability 0.1 0.20 0.35 0.2 0.1 0.05

Inventory carrying cost is $ 5, the storage cost is $ 20, find the economic order quantity.

**ANSWERS**

1. Loss is about $ 1 30 1 . 2. Q* = 1000 unit, n· = 5 times in a year

.

3. (a) Q* = 231, (b) n· = 9, (c) 40.6 days. 4. Q* = 1 0,000,000 and o· =3535.53.

5. Q* = 1 ,04,446, t" = I 0.44 days. 6. Q* = 2449.49, n• = 12.25, total cost= Rs. 2000.

7. Q* = 650.79. 8. Q*= 73.03, t• = 0.73 day, Q; = 18. 26.

9. (a) 120, (b) 1 34. 1 6. 10. Q* = 49, t* = 1 .63.

11. Q* = 4505.55, Q: = 13. 32, Manufacturing time = 1.8 month

.

13. Rs.49 187. t4. o· = 30 14. Q * =30