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Independent Demand InventorySystems

UNIT 15 INDEPENDENT DEMAND

INVENTORY SYSTEMSObjectives

After going through this unit, you will be able to learn

What is an independent demand system•

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What is a production-distribution system

The need for inventory

To develop a deterministic model

What are holding and ordering costs (or setup costs)

Meaning of economic order quantity, reorder point, lead-time, average inventory,stock cycle.

How the model behaves around the optimum point

To perform sensitivity analysis

What is finite production rate and how it affects the basic model

The concept of planned shortages

What is shortage cost and how it affects the basic model

Effect of quantity discount on order quantity

What is material cost

What is carrying charge

Effect of constraints on working capital and warehouse space

What is an Optimal Policy curve

Single Period ModelMultiple Period Models

What is uncertainty in demand and what is a stochastic model

What are overstocking and under-stocking costs

What are shortage costs and what is backlogging

What are expected profits, shortages and demand

What is demand during LT and what is variability

What is a service level

Where to stock and how to determine ROPWhat are safety stocks

Why selective control

What is ABC and VED analysis

Structure

15.1 Introduction

15.2 Basic Inventory Model

15.3 Model Sensitivity

15.4 Gradual Replacement Model

15.5 Basic Model with Backlogging

15.6 Bulk Discount Model

15.7 Independent Demand System for Multiple Products19

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15.8 Models with Uncertain Demand

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Materials Planning

15.9 Selective Control of Inventory

15.10 Summary

15.11 Self-Assessment Exercises

15.12 Further Readings

15.1 INTRODUCTIONInventory items can be broadly. classified into two types: (a) Independent definedinventory items and (b) Dependent demand inventory items. The former is based onthe items own usage history and statistical variations in demand, while the later is

based on the production schedules of end items of which the inventoried item is a part. Hence, the former is based on a replenishment philosophy while the later is based on a requirements philosophy. In this chapter we will be discussing theindependent demand inventory system.

Fig. 15.1: A Production-Distribution System

Figure 15.1 shows a production-distribution system (where the output is obtained by processing certain inputs). As we move along the arrows from input to output wecome across the various stock points:

a) Raw materials & supplies : Raw materials are needed to start the production process. Hence they are the inputs to the system.

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There may be instances when there is a temporary increase in productionrequirements. If raw material is not available, production stops. Hence a rawmaterial inventory is essential.

Raw materials are usually transported to the factory site. Quite often there aredelays in supply time which may disrupt operations if we don't have rawmaterial inventory on hand.

Sometimes suppliers of raw materials give bulk discounts which attracts thecompanies into stocking raw material even if it is not needed.

b) In-process inventories : During the production process raw materials aretransformed into semi-finished products which are next converted into finishedgoods. For example, in Figure 15.1, during transformation the raw material is

partly processed on A and then it moves into B.

If the processed material from A is not available, unit B operations get disrupted.

This may happen if A breaks down. Hence we have a stock point between A andB. We call this work-in-process (WIP) inventory.

If B breaks down, we may notice a WIP build up in between A and B. This isunwanted inventory. This situation may, however, affect other downstream

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operations if we don't have a stock point immediately after B. This would be ourfinished goods inventory. As we see here, inventories are needed for de-couplingthe sequential stages in the production system to maintain smooth flow.

c) Finished goods inventories : As we saw above, in-process inventories areconverted to. finished goods which go into the factory warehouse.

If the goods are not available in the warehouse the activities of thedistributors would suffer unless they have an inventory to depend on.

Similarly the retailer functioning suffers if the distributor fails to meet therequirements.

Consumers have a wide range of preferences. Both distributors and retailerscarry a wide range of items to meet their needs. Because of these wideranging needs their stocking policies also differ.

Even though the problems faced at each stage are different the basic questions askedall through while determining inventory policies are:

How much to order, and

When to order

We will try to address some of the complexities mentioned above with help ofseveral inventory models. First we will make several assumptions (see Figure 15.4)to define a idealized situation with the help of a basic inventory model. Then we willrelax the assumptions and move towards more practical situations.

15.2 BASIC INVENTORY MODEL

Let us take a situation from figure 15.1. We said raw materials are procured from

suppliers. The company (which receives the supply) has a certain annual requirement.To satisfy this need the company resources (which cost money) are used to sendorders to the supplier. The supplier processes the order (again consuming time,resource and money) and ships (either by road, rail or sea) the ordered quantity backto the company. To satisfy the annual requirement, if we order one item each timethen the cost of ordering goes up. On the other extreme, if we decide to order theentire annual requirement all at once, the cost of carrying inventory goes up becausewe are faced with the problem of stocking die whole lot. The ordering cost mayinclude clerical costs, follow-up costs ....etc. The carrying cost is due to the hugecapital (money) that is tied up. Each unit of item held in inventory for each unit oftime costs us money. Thus, if we want to reduce the annual ordering cost, we wouldlike to order in large bulks which would then increase the holding cost. Figure 1.5.2

and Table 15.1 show an example taking these two extreme situations.

Fig. 15.2: Two Extreme Situations

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Table 15.1: Two extreme situations

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Materials Planning

(Note: Situation B is much better than A. But then is this the best?)

To find out if there is a still better solution, let us depend on a graphical modelshowing the various costs as Q varies from 0 to 12000. By substituting differentvalues for Q we plot the results to give the holding cost curve, the ordering cost curveand the total cost curve (Figure 15.3). The minimum point on the total cost curve isgiven by Q0.

Figure 15.3 : The Economic Order Quantity

For this value of Q0 (see Fig 15.3):

Annual holding cost = Annual ordering cost

This Q0 is the optimal value of Q and is known as the economic order quantity (EOQ). This is the quantity for which the cost of inventory is minimum. This EOQformula is known as Wilson’s Lot Size Formula Substituting the expression for Q0 inthe total cost expression and simplifying it we get:

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The same expression for Q. can be obtained with the help of differential calculuswhere we minimize the total cost expression w.r.t. the variable Q. Figure 15.4 showsthe basic inventory model along with its assumptions.

Fig. 15.4: Basic Inventory Model

The economic order quantity tells us how much, to order in each cycle. But we wouldalso like to know when to place the order. As we know, there is a certain time neededto process an order by both the customer and the supplier. The time between placingan order and its arrival is known as the lead time (Figure 15.4). This time (shown inthe x-axis) tells uswhen we should be placing the order during each stock cycle. In other words, as wecan see from the y-axis, there is just enough stock to last the lead time. This level ofstock is known as the reorder level or reorder point (ROP). Figure 15.5. elaboratesthe concept of ROP and it's relationship with lead time.Reorder point (ROP) = demand during lead time (ddlt)

= demand rate (d) * lead time (LT)ROP =d*LT

In our example:

Annual requirement (R) = 12000 (given)

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Materials Planning

[Note: If LT is greater than 2t but less than 3t, then there will be two ordersoutstanding. If it is greater than 3t but less than 4t, then three orders will be

outstanding and so on]

Thus if 3t > LT > 2t (say 5 weeks)

then ROP =d*LT-2*Qo

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The assumptions in Fig 15.4 make the model somewhat trivial and remove it tar fromreality. Real life situations are quite complex. For example we might be faced withthe cases of non-instantaneous replenishment, shortages, uncertain demand patterns,resource constraints and so on. These cases can be handled by modifying theassumptions in the basic model. Before discussing these other models let us firststudy the sensitivity of this model.

Activity A

A manufacturer carries stock of an item with an annual demand of 30,000 units.Although the inventory manager cannot estimate setup cost (s) or holding cost (h)

precisely. She feels that the ratio of the two is somewhere between 100 to 1 and 150to 1; that is 5/h=100 to s/h=150. Calculate EOQ on both conditions.

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Activity B

With annual demand of 30,000 units, s/h ratio of 100 to 1, and a lead time of tendays, what recorder point should a Macro company use? Macro is open for business

250 days per year, and sales are assumed to occur at a constant rate. What wouldhappen if the lead time sometimes went up to 15 days?

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15.3 MODEL SENSITIVITY

The optimal quantity Qo = 490 seems to be a rather odd lot. Suppose we are orderingthese unite in a truck which can carry only 250 units at a time. In that case we would

be interested to know how much extra it would cost us to order 250 units each time.

Or for that matter we have a truck which can carry 1000 units say. We would like toknow how this would reflect in our costs.

Activity C

How sensitive is the optimal Q to the s/h ratio? If s/h doubles or triples, what happensto Q*? s is the setup cost and h is the holding cost.

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Note:

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Total cost is lower than that for the basic model.

If then Q . That means we need a fully dedicated supply system to

meet the demand. There is only one initial setup (a dependable system with high C

d → → ∞

p )and the cost minimization process forces the lot size Q to be extremely highapproaching . There is no inventory build-up.∞

Activity E

Explain how the situation of the finite production (gradual replenishment) note

inventory differs from the simple lot size situation. What impact does the cost of theitem have on each situation? Explain.

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15.5 BASIC INVENTORY MODEL WITH

BACKLOGGING ALLOWED

We relax another assumption here by allowing shortages. We plan to Acceptshortages or backorders (orders to be satisfied at a later date forcing customers towait). Let us assume that we can make customers wait by giving them a price

discount. This becomes a cost to the company. Let C b be the cost to have one unit backordered for one year (Rs/unit/year). That is if we are asking a customerdemanding one unit to wait for a year then the cost (to the company) of his waitingfor one year is C b. Figure 15.8 show stock positions ranging from -B (demanddeliberately put on backorder list) to (Q-B), the maximum level of positive inventoryimmediately after a lot size of Q is delivered. When amount Q is delivered B units gotowards satisfying the backlogs. Intuitively, a low shortage cost would bring in atendency to accumulate backorders (because A reduces the total holding costs).

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Materials Planning

Activity F

A company orders is lot size of 2000 units. The holding cost per unit per year is $8,

and the back order penalty per unit per year is $15. What should be the optimalinventory held, and what should be the maximum backorder position?

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Activity G

With an annual demand of 2000 units, setup costs of $250, holding costs of $8 perunit per year and back order penalty costs of $24 per unit per year, what is theoptimal time between orders? Use 250 day working Year and specify the time indays.

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15.6 BULK DISCOUNT MODEL

In the basic model we assumed there are no quantity discounts and the material pricewas considered to be fixed (say p per unit).

Material cost in the basic model = (price/unit) * R

= p * R

and Holding cost which is a function of price,

Ch = f(price)

= constant * price

= Fh * p

where Fh is the carrying charge (expressed as a fraction of inventory value)

Hence, the optimal quantity, Q0 = 2* R*Cp/(Fh*p)

Quite often suppliers give us price breaks. For example the supplier in our examplesays that he would give us a discount on the unit price if we ordered in the followingquantity ranges as per the following price discount schedule:

If we want to take advantage of these price discounts then our order quantities willincrease. Note:

Because of the increased Q inventory costs of holding go up. But thenwhat is the net effect of all these changes? Is there an overall advantageor disadvantage?

To answer these questions, let us look at our changed cost expression. The total costexpression is as follows:

∴ Total Cost = ordering cost + holding cost + material cost

TC = C p* (R/Q) + pi * Fh * (Q/2) + pi*R

Because of the price breaks (pi) we have a step function here. We are interested infinding out the quantity Q that is going to minimize the total cost expression. Figure15.9 shows the total cost curve for the valid quantity ranges and the minimum cost inthis curve gives the optimal quantity. When drawn to scale the minimum cost will beRs 48520/= for a order quantity of 1000 units.

We have a general procedure for calculating the optimal quantity for the bulkdiscount model:

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Calculate the EOQ for each price break (in our example for p1, p2, p3)

2. Determine if these EOQs are feasible. A feasible EOQ must fall within thequantity range for the corresponding price break. If the EOQ falls outside thisrange, it is not feasible and hence is eliminated.

3. Calculate total cost for feasible EOQs and at price breaks.

4. Select the quantity yielding the lowest total cost.

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Materials Planning

Fig.15.9: Total cost curves for model with price breaks.

Table 15.2 and 15.3 below summarize our example:

Table 15.2: Calculation of valid EOQs

Pl = Rs 5.00 P2 = Rs 4.50 P3 = Rs 4.00

Valid range 0499 500-999 ≥ 1000

EOQ 489.90 516.40 547.72

Feasible ? es es no

Table 15.3: Total cost summary for the example problem

Valid lot sizes Q = 490 Q = 516 Q = 500 Q = 1000

first feasible second feasible first rice second rice EOQ) EOQ) break quantity) break quantity)

Unit price 5.00 4.50 4.50 4.00

Material Cost Rs 60,000.00Rs 54,000.00 Rs 54,000.00 Rs 48,000.00

(price*R)

cost 244.90 232.60 240.00 120.00R *C

Inventor 245.00 232.20 225.00 400.00

holding cost(Q/2)*price*Fh

Total cost Rs 60,489.90Rs 54,464.90 Rs 154,465.00 Rs 48,520.00

We find there is a net advantage in ordering 1000 units for a minimum totalcost of Rs 48520.

Also note: As observed earlier. for EOQs

ordering cost = holding cost

(within rounding off errors in our example)

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Activity H

A company annually orders one million pounds of a certain raw material for use in itsown business process with annual holding costs estimated at 35% of the purchasePrice of $50 per 100-Pound bag, the purchasing managers wants to decide an ordersize. Marginal paper work costs are $10 per order. For order of 500 bags or more.The purchase price falls to $45 per bag; for orders of 1000 bags or more, the price is$40 per bag. What is the optional order size?

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15.7 INDEPENDENT DEMAND SYSTEMS FOR

MULTIPLE PRODUCTS

Suppose you walk into a company and find that there are five independent demanditems which need to be stocked in the warehouse. The annual requirements for eachitem and the price per unit is also known. The carrying charge F li = 0.2 and the cost ofordering C p = Rs 10 per order. The unit space occupied by different item types is alsogiven. The company is currently adopting an inventory policy of ordering each item

in lots thrice a year. The table below summarizes the existing data:

Table 15.4: Existing inventory policy of a certain company

There are several questions that come to mind now:

a) Questions regarding the existing policy (refer Table 15.5):

What is the total volume requirement for each type of item in this inventory ?(see Table 15.5, column 4)

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When all lots arrive at the same time, the total space is given by:

order qty * space required per unit

What is the average spate requirement (in m3) for each type ? What is the

total average space requirement ? (see Table,15.5, column 5)

Note: Average inventory = (Maximum + Minimum) ÷2

= (qty ordered + 0) s- 2

Hence, Average space = Average inv * space per unit

(see col 5)

Total average space =19000 m3

What is the total "rupee volume" (or rupee value) for each type ? (see Table15.5, col 6)

The rupee value becomes an aggregate measure of the inventory size. Thus differentsize and types of inventory item can be aggregated under one measure. The rupeevalue of each type is given by:

order quantity * price per unit

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What is the average inventory size (in rupees) for each type? What is thetotal average inventory (TI) in this case? (see Table 15.5, col 7)

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Materials Planning •

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Average inventory (in rupees) =Average inv * price per unit

Total average inventory (TI) =Rs 83500

What are the total orders for the current policy? (see Table 15.5, col 2) Eachitem is ordered three times a year.

Total orders (TO) =15What is the total inventory cost for this existing policy? (in other words, we wouldlike to know the cost of adopting this policy)

Total cost = total holding cost + total ordering cost

= [Avg inventory (in Rs)*Fh] + [(# orders)*C)]

= TI * F, + TO*Cp

= 83500 * 0.2+ 15 * 10 = Rs 16850

Table 15.5: Total volume and value of inventory items

Item orders Qty Tot Space(in m3)

Avg Space(in m3)

Tot Value(in Rs)

Avg Size(in Rs)

A 3 4000 2000 1000 20000 10000

B 3 7000 7000 3500 56000 28000

C 3 6000 6000 3000 36000 18000

D 3 3000 9000 4500 27000 13500

E 3 7000 14000 7000 28000 14000

15 19000 83500

b) Is there a better solution ? (refer Table 15.6):

Table 15.6: An optimal solution

ItemEOQ

(in units)#orders Avg space

(in m3)Avg size

(in Rs)

A 489.8979 24.4949 122.4745 1224.7 45

B 512.3475 40.9878 256.1738 2049.39

C 547.7226 32.86335 273.8613 1643.168

D 316.2278 28.4605 474.3416 1423.025

E 724.5688 28.98275 724.5688 1449.138

155.7893 1851.42 7789.465

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We know that EOQ is the quantity for which the inventory cost is minimum.If we take the EOQs for each type then is there a chance of minimizing thetotal cost?

Calculate the optimal quantities for each type using the EOQ formula. Thus

EOQ for each item is given by:

Qi = i p h iR *C /(F *p )2* where i = A,B,C,D,E

The EOQ values are shown in Table 15.6, column 2. Columns 4°and 5 givethe corresponding average space (in m3) and average size (in Rupees).

Column.3 gives -the number of orders (annual requirement , quantityordered) and total orders.

Total orders (TO) = 155 (compare with 15-of the existing policy)

What is the average size of the inventory (in rupees)?•

Average size = (EOQ/2) * price per unit (see col 5)

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Total average size (TI) = Rs 7789

(compare with Rs 83 500 of existing policy)

What is the average space requirement (in m3)?

Average space = (EOQ/2)*space per unit (see col 4)

Total avg space requirement = 1851.5 m3

(compare with 19000 m3 of existing policy)

What is the cost of this optimal policy?

Total cost = total holding cost + total ordering cost

= [Avg inventory (in Rs)*Fh] + [(# orders)*C p]

= TI * Fh + TO*CP

= 7789 * 0.2 + 155 * 10

= 1558+1555 = Rs 3108

When compared with Rs 16850 of existing policy we find a lot ofimprovement. Please note that the total ordering cost is almost equal to thetotal holding cost (within rounding off errors).

c) Now what if there arc resource constraints?

In real life there could be several resource constraints. For example, constraintson working capital, on storage space or on number of orders per annum. Tounderstand such situations, let us extend our example by imposing a constraint on

the amount of investment in inventory. We are told that the maximum we caninvest on an average in inventory is equal to Rs 6000. The current policy has anaverage investment in inventory of Rs 83500 which is very high. Even theoptimal policy with an average investment of 7789 is higher than this amount. Sowhat to do now?

We can handle this situation by modifying the EOQ formula as follows:

EOQ formula for each item is given by:

Here we are attaching a price ( ) for using the limited resource (money in this case).In other words we pay a price for each unit of the limited resource in use. Thus, forcostly items the denominator is inflated more than for less costly items (e.g.

. For =0 the formula is same as the EOQ formula. As k increases

the lot size decreases as compared to the economic order quantity. Now = 0.05means we are paying 5 poise for investing each rupee in the inventory. We try to find

the value of in an iterative manner by increasing the value of in steps with the

objective of reducing the lot sizes, thereby reducing the amount invested ininventory. We do this until the upper limit on inventory is just met. See Table 15.7

below where we calculate optimal inventory sizes for different values of .

λ

( Dλ *p λ *p>

λ

) λ E

λ

λ

λ

Table 15.7: Order quantities and corresponding average inventory she in rupees for

different values of X

EOQ(1=.05)

Averagesize(Rs)

EOQ(1=.10)

Averagesize(Rs)

EOQ(1=.137)

Averagesize(Rs)

2190.89 1095.45 2000.00 1000.00 1887.02 943.51

3666.06 1833.03 3346.64 1673.32 3157.58 1578.79

293939 1469.69 2683.28 1341.64 2531.70 1265.85

2545.58 1272.79 2323.79 1161.90 2192.51 1096.26

2592.30 1296.15 2366.43 1183.22 2232.75 1116.37

6967.11 6360.07 6000.78

Thus at λ = 0.137 we are just able to satisfy the constraint on our working; capital.

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Activity I

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Materials Planning

Compute the corresponding space requirements for each of the different values of .Suppose there is a limitation on the average space available (being equal to 1750 m

λ 3

say). What is the modified formula now'? Try to find out how the order quantitieschange if there is this constraint on space instead of the working capital.

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…………………………………………………………………………………………d) What if we do not know the values of the carrying charge (Fh) and

cost of order (Cp)?

In most companies it is difficult to know the above mentioned values. Under suchcircumstances we depend on a tool known as the optimal policy curve which helps usfix inventories at an aggregate level. Each point in this curve is an optimal policy. Ifwe decided to keep the number of orders constant then we can move towards a bettersolution by decreasing the average investment in inventory or vice-versa. The curveexchanges total average inventory size (TI) with total orders (TO) in such a mannerthat

(TI)*(TO) = a constant (K)

where, K = i i

i

*p )2

1(R ∑

Thus, in our example (TI)*(TO) = (1/2)*(1557.893)2=1213515. It is the equation to arectangular hyperbola which is shown in Figure 15.10. Let us take the first optimal

policy calculated in (b) above. Our (TI)*(TO) = 7789.465* 155.7893 = 12135 15,same as the constant (K) above. As part of your practice problem try to verify thisresult again by taking the optimal policy in the resource constrained problemdiscussed in (c) above.

Fig. 15.10: Optimal policy curve

The optimal policy curve indicates that the existing policy having a TO=15 and TIequal to Rs 83500 is very close to the optimal policy point on the curve having a TIequal to Rs 80901 for a TO 15. Likewise we can verify that for a TO=155 the TI isRs 7829. Thus it is a handy tool that facilitates quick decision making.

15.8 MODELS WITH UNCERTAIN DEMAND

So tar we have been discussing only the deterministic models where the demand patterns are predicted with certainty. We also do not see any variations in the leadtime. But under real life situations there are a lot of variations (both in demand aswell as lead time). Here we

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depend on stochastic models. Stochastic models depend on historical data. Bylooking at past demand data we can assign weights to different levels of demand data.Higher weights indicate a greater frequency of occurrence or a greater likelihood ofthat event occurring (note: each level of demand is an event). Because of suchuncertainty in the demand pattern (similar uncertainties are there with the lead time)we look for expected outcomes, such as, expected demand. expected revenue.expected profit, expected inventory, expected shortages. and so on. We will considerthe following cases where we are faced with uncertainties because of the variations:

Case 1: A vendor stocks perishable items which, if not consumed, have to be eitherthrown or salvaged at the end of the stock period. Thus if demand falls shortof the quantity stocked we incur a cost of overstocking. Some examples:

A newsboy at the news-stand stocking newspapers for the day. These papers become useless the following day. At best they can be sold aswaste paper.

The local bakery stocking fresh bread for the day.

Monthly magazines stocked for a month.

There could be instances where demand remains unfulfilled. These becomelost opportunities which are considered as the costs of understocking. These

problems are solved with the help of SINGLE PERIOD MODELS. Note that

a period could be a day, week, month or even a certain lead-time. For each ofthese periods we must have the historical demand data. Table 15.8 belowshows a couple of sample historical data:

Case 2: Items stocked during a stock cycle, if not consumed, can be used in thesubsequent cycle. If demand is unfulfilled then we end up having stockoutswhich result in either lost sales or backlogs. When customer demands arefilled at a later date, they are known as backlogs (i.e. customers are willing towait if they are given a discount on the price). These are items having longershelf life. For these problems we depend on MULTIPLE PERIOD MODELSwhere we have to determine the quantity to be stocked (Q) and the reorder

point (ROP). See Figure 15.11.

Figure 15.11: Multiple period modelSingle Period Model

Popularly known as the newsboy model. A newspaper vendor has to decide howmuch to stock so that he can maximize his expected profits at the end of the day. Thevendor depends on the following historical data (Table 15.9):

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Each newspaper costs him Rs 1.50 and he sells it at Rs 2.00. Unsold papers areuseless and are to be thrown. Hence the salvage value for unsold papers is zero. Thevendor has to decide how much to stock so that he can maximize his profits. Becauseof the uncertainty in demand, the quantity stocked could be either less than, equal toor greater than the demand.

if d > Qhe will face a penalty for understocking.This is his cost of understocking (C0).Which is the cost of profits foregone.

In our example,Co = sale price - cost price

= S - C

= 2 -1.50 = 50p

If d < QHe faces a penalty for overstocking. Thisis his cost of overstocking (Co). Which isthe purchase cost of the unsold units.

In our example,Co = C = Rs 1.50

The problem is where should he stock? He has the benefit of past experience (see thehistorical data available above). Suppose he stocks a quantity Q = 100. Table 15.10shows the purchase cost, revenue and profits for the various demand levels (eachdemand level is an event). For example, if demand = 75 then number sold = 75,revenue = 75 * 2.00=Rs 150 and since purchase cost = 100 * 1.50 = Rs 150, profit =revenue - cost = 0.

Table 15.10: The Newsboy problem

(d) prob (d) CostC*Q

Number sold Revenue(#sold*S)

profit(rev - cost)

75 .05 150 75 150 0

100 .10 150 100 200 50

125 .20 150 100 200 50

150 .30 150 100 200 50

175 .20 150 100 200 50

200 .10 150 100 200 50

225 .05 150 100 200 50

Because of the uncertainty in demand he is interested in the expected values ofrevenue and profits. His aim is to maximize expected profits.

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The objective is to maximize this expression. An incremental analysis approach isused to determine the optimal quantity Q. According to this analysis, for maximum

profit at Q:

Without getting into the details of this analysis, we state here the final result. Wewant the smallest value of Q which ensures that the following inequality remainsvalid:

The left hand side of this inequality shows the cumulative probability distribution ofdemand being less than or equal to Q. Thus prob (d ≤ 100) = 0 15, prob (d ≤ 5 75) =0.05, prob (d 225) = 1.00 and so on. The right hand side of this inequalityintroduces the service level concept taking into consideration the costs ofoverstocking and understocking. For example, if C

≤

u is very high as compared to C0 then the right hand side fraction also becomes very high (close to 1). In that case wewould like to maintain a high service level for satisfying the demand. That means wecannot afford to lose customers by understocking. Similarly, if C0 is high comparedto Cu we cannot afford to dump large quantities in the inventory. The service level isreduced and we can afford to lose customers.

In our example, C0 = Rs 1.50 and Cu = 2 -1.50 = 0.50Thus,

u

0 u

C 0.500.25

C C 1.50 0.50= =

+ +

Here we see a rather low service level, because the cost of understocking is quitesmall. The smallest value of Q for which the inequality is satisfied is 125. Note that,

prob (d 125) = 0.35 > 0.25 (see Table 15.11)≤Hence Q0 = 125 is the optimal quantity for which we can maximize the expected

profits:

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39


Then depending on historical data for demand during lead-time we try to fix the ROPusing the newsboy model concept. We find the largest ROP for which the followinginequality holds:

We also have the lead time historical demand data:

Table 15.12: Demand data

ddlt p(ddlt) p(ddlt≥ x)

50 .10 .10

40 .20 .3030 .30 .60

20 .20 .80

10 .10 .90.

0 .10 1.00

Now Q0 = 490 (using the EOQ formula)C0 = Ch = Re 1 per unit per yearCu = Cs * (R/Q0 =10*(12000/490) = Rs 245 per unit per year

Note the relatively high cost of understocking. Intuitively, the ROP should be high.We cannot afford to lose customers and hence would rather dump the items in

inventory. To determine ROP, calculateu

0 u

C 10.0041

C +C 1 245= =

+

Which means the chances of a stockout should be very low. So we find ROP bychoosing the largest value for which the inequality is satisfied.

prob (d ≥ 50) = 0.10 >0.0041 (see Table 15.12 above)So ROP=50. Hence (Q, ROP) is equal to (490,50). According to this policy we placean order for 490 units when the stock level during any stock cycle comes down to 50units.

Situation B: The cost of shortages (Cs) is unknown

In real life situations it is difficult to quantify the cost of shortages. There are some

intangible negative aspects of shortages, backlogs and backorders. In these situationswe find Q as usual using the EOQ formula. To find ROP we use a. service levelconcept. Service level specification is a policy decision and gives a certain

probability of meeting the demand from inventory. For example we may have thefollowing policies specified by management:

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Let us take example and undertand both the above service level concepts. The data issame as in the example discussed in situation A. Except that C s is unknown. For thisdata our Q0 = 490 units. Table 15.13 shows the calculations.

40

Materials Planning

Table 15.13: Several level concept (Examples)

Situation C. Finding ROP with known demand data distributions

If the demand data follows some well defined distribution (e.g. a normal or a Poissondistribution) then we can find ROP and the safety stocks as follows. All we will be

doing here is to add safety stock to the expected demand during lead time to set theROP level. The service level defined by the managerial policies help us indetermining the safety stock. Now service level is the probability that the demandduring lead time will be available from inventory. Figure 15.12 shows therelationship between expected demand during lead-time, reorder point and safetystock. If service level is x then the risk of a stockout is given as (I-x) as shown in thefigure.For deterministic situations we said, ROP = demand rate * lead-time. Since bothdemand and lead-time are uncertain in nature. Demand varies because of theuncertainties in customer preferences. Lead-time could be affected by strikes,transportation difficulties and other supplier related uncertainties. The company facesstockouts if there is either a sudden surge in demand or there is an unnecessary delay

in the delivery of the order quantity. Therefore, when stocks are low, the companywould prefer to have a safety stock as a hedge against stcokouts. Hence, far

stochastic models,Reorder point = Expected demand during lead-time + Safety stock

ROP = E(ddlt) + SS

Fig. 15.12: Relationship between ROP and safety stock

Reorder point when demand is varying and lead-time is constant: We assume

that daily demand is uncertain, independent and normally distributed. Wesum the daily variances and then take the square root to get the standarddeviation.

•

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41


• Reorder point with variable lead--time and constant demand: Since lead-timeis varying we take the average lead-time and multiply with daily demand rateto go the average demand during lead-time. If the standard deviation of the

lead-time is σ then,LT

LTROP=d*LT z*d*σ+

Reorder point with variable , lead-time and demand: Here the total variance

in lead-time demand is the sum of variance due to demand fluctuations

( 2

d *LTσ ) plus the variance in the lead-time ( )2 2

LT *dσ . Thus,

•

2 2

d LTROP=d *LT z ( ) *LT ( ) (d)σ σ + +2

15.9 SELECTIVE CONTROL OF INVENTORY ITEMS

In real life situations we have a large number of items in the inventory. It may not be possible to review (which includes counting, placing orders, receiving stock) such alarge number of items. Review activities take time and cost money. Hence items areusually classified into important and less import, °it groups. The important ones getmore attention than the others. The goal is to keep stocks at low level while givinggood service. One of the selective control methods is known as ABC analysis . It isgoverned by the Pareto principle which segregates the vital few from the trivialmany. According to that principle it can be shown that around 10% of the inventoryitems would be accounting for around 70% to 80% of the inventory's total annualrupee value. Next 20% of the items account for around 20% 6f the annual rupeevalue and the remaining 70% around 10% of the value. For A class items closecontrol is required. For C class items routine checks at long intervals is adequate.Similarly we have a VED classification which stands for vital, essential anddesirable. Non availability of vital items would bring production to a halt. Hence theyneed to be stocked adequately. Stockout of essential items may result in expensive

procurement and stockout of desirable items may cause minor inconvenience. Moredetails of each inventory items are discussed in unit 14 and Block 6 of MS 5.

15.10 SUMMARYWe have discussed inventory control systems for independent demand items. Wehave tried to understand a production-distribution system and why there is a need forthe inventory function. Various inventory related costs have been mentioned. Thesecosts are traded-off while deciding how much to order and when to order. Modelsunder both deterministic and probabilistic situations have been presented.

15.11 SELF-ASSESSMENT EXERCISES

1. Consider the following types of items carried in a retail store: light bulbs, phonograph records, refrigerated drugs. Discuss the probable cost structure foreach of these items including, items cost, carrying cost, ordering cost and

stockout cost.2.

3.

4.

5.

Why stockout cost difficult to determine? Suggest an approach which might beused to estimate it.

A student was overheard saying. "The EOQ model assumptions are so restrictivethat the model would be hard to use in practice. Is it necessary to have adifferent model for each variation in assumptions why and why not?

Suppose you were managing a chain of retail department stores. The inventoryin each store is computerised, but these are a large number of different items. Asa top manager flow would you measure the overall inventory management

performance of each store. How would you use this information in yourrelationship with the individual store managers?

The owner of a hotel has 32 rooms is trying to determine whether to build anaddition incur stockouts, referring customers to competitor when demandexceeds supply. The cost of maintaining a hotel room averages $ 15/day. Atypical room rents for $45/ right. During the last six months, demand hasaveraged as follows:

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6. The speedy Grocery Store carries a particular brand of coffee which has thefollowing characteristics:

42

Materials Planning

Sales = 10 cases per seek

Ordering cost = $ 10 per order

Carrying Change = 30 per cent per year

Item cost = $ 60 per case

a) How many cases should be ordered at a time?

b) How offer will coffee be ordered?

c) What is the annual cost of ordering and carrying coffee?

d) What factors might cause the company to order a large or smaller amountthan the EOQ?

7. An appliance store carries a certain brand of TV which has the followingcharacteristics:

Average annual sales = 100 units

Ordering cost = 25 per order

Carrying cost = 25 per cent per year

Item cost = $ 400 per unit

Lead time = 4 days

Standard deviation of the daily demand = 0.l unit working days per year = 250

a) Determine the EOQ

b) Calculate the reorder point for a 95% service levels assessing normaldemand.

c) How tar apart would orders be placed on average?

8. For the following total cost TC, find the optimal order quantity Q*. A is aconstant. Is this a minimum on a maximum cost point? why?

TC = (27 + A) Q + +274

9. Inventory control is a national process in which decisions are offer made

irrationally. Explain.

10. Given a probability distribution of demand and a distribution of lead time, whatalternatives exists for finding the probability distribution of demand duringlead time? Select one attentive and explain how it works. Why is thedistribution of demand during lead time important?

11. An electrical motor housing has an annual usage rate of 75000 units/year , anordering; cost of $20, annual carrying' change of 15.4 per cent of the unit price.For lot sizes of fewer than 10,000 the unit price is 0.50, for 10,000 or more theunit price is $45. Delivery lead time is known with certainty to be two weeks.Determine the optional operating doctrine.

12. Demand for the local daily newspaper at a news stand is normally distributedwith a daily news of 210 copies and standard deviation of 70. A newspapersells for $25 cents and costs $20 to purchase. Day old newspapers are veryseldom requested, and

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13.

14.

therefore they are destroyed. What should he the news stand's daily order be tomaximise profits?

A bank purchases promotional ball point pens for $3 each. The company thatsupplies the pens suggests that if the imprinted pens were ordered is twice thequantity, a 35 per cent discount could be arranged. At present the bank orders100 pens every two months. Ordering costs are $l2, and bank's cost of money isl8 percent. What ordering policies should be followed?

You have a product whose average weekly sales are 600 units. By looking atthe records of past demand, you find that the demand patterns has followed thedistribution below:

WeeklyDemand Above

(in units)

Per cent of the timeDemand Above

400 100

450 90

500 79

550 64

600 50

650 22

700 8

750 3

800 0

The cost of carrying an item on inventory for one year is $1.30. Ordering cost is fixedat $72. Lead time is constant at one week. The stockout policy has been set to allow ' two stock outs/year on average. Determine die order quantity and the safety stock

that minimised the variable cost.

15.

The monthly demand for an item was recorded as follows:

Calculate the mean absolute deviation and the standard deviation of forecast error ifthe forecast is 300 per month. What safety stock would be required if orders arereceived monthly and if two stockout occurancies is twelve months are acceptable.

15.12 FURTHER READINGS

Adam, E.E. and Ebert, R.J. (1995) Production and Operation Management, fifthedition, Prentice-Hall of India Private Ltd, New Delhi.

Buffa, E. S., and Sarin, R. K. (1994) Modern Production/Operations Management,

John Wiley & Sons, Inc.

Chary, S. N,(1988) Production and Operations management ? Tata McGraw-Hill, New Delhi

Chase, R.B. and Aquilano, N. J. (1995) Production and Operations Management:

Manufacturing and Services, Richard D Irwin Inc.

Hax, A.C., and Candea, D. (1984) Production and Inventory , Management, Prentice-

H ll I I t C t l 2 d diti P ti H ll f I di N D lhi

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