Introduction to inventory control

Marco BijvankRotterdam School of Management, Erasmus University, Burgemeester Oudlaan 50, 3062

PA, Rotterdam, The Netherlandse-mail: mbijvank@rsm.nl

Inventory is the number of products or resources held available in stock by an organizationand can include raw materials, work-in-process, component parts, and finished products.The inventory of manufacturers, distributors, and wholesalers is clustered in warehouses.Retailers keep their inventory either in a warehouse or in a store accessible to customers.Many types of inventory exist:

safety stock is the amount of inventory kept on hand to protect against uncertaintiesin customers demand and supply of items. The reason to keep this type of inventoryis because demand and lead times are not always known in advance and have to bepredicted. This type of inventory is also called buffer stock.

seasonal stock is the inventory built up to anticipate on expected peaks in sales orsupply, such that the production rate can be stabilized. This is also called anticipationstock.

cycle stock consists of the inventory waiting to be produced or transported in batchesinstead of one unit at a time. Reasons for batch replenishments include economies ofscale and quantity discounts.

decoupling stock is used to decouple the output of two inter-dependent workstationsbecause of different processing rates, set-up times or machine breakdowns. This permitsthe separation of decision making.

congestion stock results from items that share the same production equipment. Conse-quently items have to wait for workstations to become available and inventory is builtup.

pipeline stock includes inventory in transit between different parties of the supply chain.This is also called work-in-progress.

Not every type of inventory is kept at all parties in the supply chain. For example,decoupling stock and congestion stock are mainly kept in a manufacturing environment,whereas safety stock and pipeline stock are more important in a retail environment. In orderto handle the different types of inventory, control systems have to be developed. Accordingto Hax and Candea [1984], an inventory control system is a coordinated set of rules andprocedures that allows for routine decisions on when and how much to order of each item inorder to meet customer demand. A replenishment policy specifies how to decide upon these

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periodicreview

HHHH

continuousreview

single-periodPP

PPP

multi-period

constantdemand

HHH

time-varyingdemand

stochastic demand```

``````

deterministic demand

single-item```

``````

multi-item

inventory control

Figure 1: A classification of inventory control systems.

two decision variables. A classification of inventory control systems is given in Figure 1 andwill be explained below.

Most inventory systems concern single-item systems and consider one type of product ata time. In multi-item inventory systems a number of products are considered simultaneouslybecause of limited capacity availability, economies of scale for joint replenishments or otherreasons. The classification for single-item systems is described in the remainder of thissection. A similar classification can be made for multi-item systems and is therefore notincluded in the figure. See for more details Zipkin [2000]. Furthermore, we talk about aninventory model when it represents the inventory system. A model can include assumptionsand, therefore, it is a simplification of reality.

There is a distinction between predictable and unpredictable demand. If the demandin future periods can be forecasted with considerable precision, it is reasonable to use aninventory policy that assumes that all forecasts will always be accurate. This is the case indeterministic inventory models. However, when demand cannot be predicted very well, itbecomes necessary to use a stochastic inventory model where the demand in any time periodis a random variable rather than a known constant. Models in which demand is known (orforecasted) and constant over a planning horizon are called classical lot size models. A well-known example is the economic order quantity (EOQ) model introduced by Harris [1913].When demand is not constant but still predictable, the models are called dynamic lot sizemodels. Examples of techniques to determine the order size for such models are the Wagner-Within algorithm [Wagner and Whitin, 1958] and the Silver-Meal heuristic [Silver and Meal,1973].

In real life demand is mostly not known in advance. Therefore, a probability distributionis used to describe the behavior of the demand in stochastic inventory models. A specialclass of inventory control systems is concerned with products which have a very limitedperiod before it can no longer be sold. Examples are perishables (like food and flowers) oritems with a limited useful life (like newspapers and fashion). For such inventory systemsno decision has to be made regarding the order moment when the replenishment should take

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place, but only the order size has to be determined for a single time period. Such models arecalled single-period inventory models or newsboy models. In multi-period inventory modelsit should also be determined when replenishment orders are triggered.

Different inventory levels are considered to determine when an order has to be placed.The on-hand inventory level is the amount of physical inventory immediately available onthe shelves in a store or warehouse to meet customer demand. The occasion when theinventory level drops to zero is called a stock out. The demand exceeding the availablestock is referred to as excess demand. There are two ways to deal with this demand whenthere is a stock out. First, if the customer is willing to wait, the excess demand is helduntil the next delivery replenishes the inventory. This is called backlogging or backordering.Alternatively, the customer may not be willing to wait. In this case excess demand is lost,which is called lost sales. The on-hand inventory minus the backorders is called the netinventory. A positive net inventory represents the inventory on hand whereas a negative netinventory refers to a backlog. The inventory on order is the work-in-progress or the itemsordered but not yet delivered due to the lead time. When there are backorders, a part of theinventory on order is already reserved to meet customer demands from the past. Therefore,the inventory position is defined as the sum of the inventory on hand plus the inventory onorder minus the outstanding backorders (or backlog).

Replenishment Policies

How often the inventory status should be checked for replenishments is specified by the reviewinterval. This is the period that elapses between two consecutive times at which the stocklevel is known. Two types of review systems are widely used in business and industry. Eitherinventory is continuously monitored (continuous reviews) or inventory is reviewed at regularperiodic intervals (periodic reviews). The former type of control is often called transactionreporting, since continuous surveillance is not required but only at each transaction thatchanges the inventory position (e.g., demand or order delivery). Whether or not to order ata review time is determined by the reorder level. This is the inventory position at which avendor is triggered to place a replenishment order in order to maintain an adequate supplyof items to accommodate current and new customers. The mathematical notation for thereorder level equals either R or s dependent on the type of replenishment policy. It comprisesthe safety stock and the quantity of stock required to meet the average demand during thelead time plus the time until the next review moment. The lead time is the period of timebetween order placement and the delivery of the order such that the order is available forsatisfying customer demands. An order size can either be fixed or variable. The type ofreplenishment policies with variable order quantities are called order-up-to policies in whichthe order size is such that the inventory position is increased to an order-up-to level. Thislevel is denoted by S. Figure 2 shows the difference between continuous and periodic reviewswhen the order size is a fixed number and each customer demand equals one unit (also calledunit-sized demand). In the continuous review case, the order is placed immediately when theinventory position reaches the reorder level. In the periodic review case, the order placementhas to wait for the next review time after the inventory position has reached the reorder level.Figure 3 illustrates the concept of order-up-to policies in the case where customer demands

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lead time- -

R

6

time

inve

nto

ryle

vel

(a) continuous review

lead time-

review1 -

R

6

time

inve

nto

ryle

vel

(b) periodic review

Figure 2: The on-hand inventory level (solid line) and inventory position (dashed line) forreplenishment policies with a fixed order size under (a) continuous and (b) periodic review.

lead time- -

s

S 6

time

inve

nto

ryle

vel

(a) continuous review

lead time-

review1 -

s

S 6

time

inve

nto

ryle

vel

(b) periodic review

Figure 3: The on-hand inventory level (solid line) and inventory position (dashed line) fororder-up-to replenishment policies under (a) continuous and (b) periodic review.

are not always unit sized. Notice that the delay in the actual order placement can result inlarger order sizes in case of periodic reviews as compared to continuous reviews. There canbe fixed order costs incurred with each order. When no fixed order cost is charged, thereis no incentive not to place an order at each review moment in case of variable order sizes.Consequently, the reorder level does not play a role in order-up-to policies (i.e., it equalsS 1).

Based on these characteristics on the replenishment process we present the mathematicalnotation for the different types of replenishment policies in Table 1. The letter r specifies thelength of the review interval. The review interval length for continuous review systems is zeroand is therefore omitted. Furthermore, Q stands for fixed order quantities, S denotes theorder-up-to level, whereas R and s represent the reorder level for fixed-order-size policies andorder-up-to policies, respectively . Figure 2a and Figure 2b illustrate the (R,Q) policy forcontinuous and period reviews, respectively, whereas Figure 3a and Figure 3b give an exampleof the (s, S) policy for continuous and period reviews, respectively. A special class withinthe order-up-to policies are base-stock policies, in which the satisfied demand in between tworeview times is immediately ordered at the next review time. For such policies, the reorder

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order momentcontinuous review periodic review

order

size fixed (R,Q) (R,Q)

variableno fixed cost: (S 1, S) no fixed cost: (S 1, S)fixed cost: (s, S) fixed cost: (s, S)

Table 1: The notation for the types of replenishment policies most often applied in literatureand practice.

level s is equal to the order-up-to level minus one in case of discrete demand. When demandis continuous, the reorder level is equal to the order-up to level. Models with a base-stockpolicy are denoted as (S 1, S) policies. In continuous review models, base-stock policiesare also called one-for-one policies since every customer demand immediately triggers a neworder. Notice that the (R,Q) and (s, S) policy are identical for continuous review models ifall demand transactions are unit sized, s = R and S = R +Q. In that case, replenishmentsare always made when the inventory position is exactly at the reorder level. Consequently,the order size always equals Q or S s. See Figure 2a for an illustration.

Objective Function

To compare the performance of the different replenishment policies, the costs associatedwith each controlling system has to be minimized while simultaneously meeting a desiredcustomer service level. There are three types of inventory costs : (1) order costs associatedwith placing an order, (2) holding costs for carrying inventory until it is sold or used, and(3) penalty costs for unfulfilled customer demand. The order cost can consist of fixed costfor each time an order is placed and variable cost for each unit ordered. The holding cost ismainly the opportunity cost of the money invested in inventory. But it should represent allinventory carrying costs, including the cost of warehouse space, material handling, insuranceand obsolescence. The penalty cost is the cost of not having sufficient inventory to meetall customer demands. These shortage costs can be interpreted as the loss of customersgoodwill and the subsequent reluctance to do business with the firm, the cost of delayed orno revenue, and any possible extra administrative costs.

When demand is stochastic, shortages cannot be avoided. A service level is used in asupply chain to measure the performance of such inventory systems. The most commonmeasures of service are (1) service level, (2) service level, and (3) service level. Thefirst type of service level is an event-oriented criterion. It measures the probability that allcustomer demands are satisfied within a replenishment cycle. This definition is also calledthe cycle service level, since it measures the fraction of cycles in which a stock out occurs.The service level, or fill rate, is a quantity-oriented measure that represents the fractionof the demand satisfied directly from stock on hand. An example to illustrate the differencebetween the cycle service level and the fill rate is provided in Table 2. The total directlysatisfied demand from stock on hand is 12 units, while the total demand is for 15 units.

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order cycle inventory level demand satisfied demand1 4 5 42 6 8 63 3 2 2

Table 2: An example to show the difference between cycle service level and fill rate.

Therefore, the fill rate equals 12/15 = 80%. Since the demand exceeds the inventory levelin two out of three order cycles, the cycle service level equals 33.3%. The service levelincorporates the waiting time of the demands backordered. This service performance measureis the fraction of time during which there is no stock out. This service level definition is alsocalled the ready rate.

(R,Q) Policy with Continuous Reviews

When the inventory levels are monitored continuously, a fixed order size replenishment policyis most often used when there are fixed order costs charged per order. Since the EOQ-formulais robust in the sense that the cost function is rather flat around the optimal order size, thesize of the order is set according to this formula. Consequently, only the reorder level hasto be determined. First, consider a cycle service level to set this parameter. That is, theprobability that there is no out of stock occurence during a replenishment cycle. For an(R,Q) replenishment policy with continuous reviews, an order will arrive L time periodsafter the inventory position reaches the reorder level. Therefore, the cycle service levelis defined as the probability that demand during the lead time does not exceed R units;Pr(DL < R) . When demand follows a normal distribution this translates to

(R AV G LSTD L

) , (1)

where AV G and STD represent the average and standard deviation of the demand pertime period, respectively, and () is the cumulative distribution function of a standardnormal distribution. This latter quantity can be found in the appropriate look-up table (seeAppendix A). Equivalently to Equation (refeq:R),

R = AV G L+ z STD L, (2)

where z corresponds to the safety factor of a standard normal distribution with probability. Table 3 illustrates several values of the safety factor z for different values of .

Example 1. A television manufacturing company produces its own speakers, which areused in the production of its television sets. The television sets are assembled on a produc-tion line where one speaker is needed per set. The speakers are produced in batches becausethey do not warrant setting up a continuous production line, and relatively large quantitiescan be produced in a short time. Therefore, the speakers are placed into inventory untilthey are needed for assembly into television sets on the production line. Another reason tokeep inventory has to do with the fact that sales of TV sets have been quite variable. Con-

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0.75 0.8 0.85 0.9 0.92 0.94 0.95 0.96 0.98z 0.68 0.85 1.04 1.29 1.41 1.56 1.65 1.76 2.06

Table 3: The value of the safety factor z for different values for the cycle service level whendemand follows a normal distribution (see also Appendix A).

sequently, the demand for the speakers is quite variable as well.There is a lead time of 6 weeks between ordering a production run to produce speakers

and having speakers ready for assembly into television sets. The annual demand for speakersis a random variable that has a normal distribution with a mean of 96,000 units and astandard deviation of 6,930 units. To minimize the risk of disrupting the production lineproducing the TV sets, management has decided that the safety stock for speakers shouldbe large enough to avoid a stockout during this lead time for 95 percent of the time. Thecompany is interested in determining when to produce a batch of speakers and how manyspeakers to produce in each batch.

Several costs must be considered: Each time a batch is produced, a setup cost of $12,000is incurred. This cost includes the cost of tooling up, administrative costs, record keeping,and so forth. The production of speakers in large batches leads to a large inventory. Theestimated holding cost of keeping a speaker in stock is $0.30 per month. This cost includesthe cost of capital tied up in inventory. Since the money invested in inventory cannot beused in other productive ways, this cost of capital consists of the lost return (referred toas the opportunity cost) because alternative uses of the money must be forgone. Othercomponents of the holding cost include the cost of leasing the storage space, the cost ofinsurance against loss of inventory by fire, theft, or vandalism, taxes based on the value ofthe inventory, and the cost of personnel who oversee and protect the inventory.

The best replenishment policy is an (R,Q) policy with continuous reviews since product-line orders can be placed at all times. As mentioned before, the EOQ-formula is ratherrobust. Therefore, the best order quantity is given by Q =

2 AV GK/h. The setup

cost to produce the speakers is K = $12, 000, the unit holding cost is h = $0.30 per speakerper month. Since the holding cost is per month and the demand per year, we need to makesure that they are both expressed in the same time units. Consider years as time units. Theannual holding cost is $3.60 per speaker. As a result, Q =

2 96, 000 12, 000/3.60 =

25, 298.22 speakers. After comparing the total costs for Q = 25, 298 (that is $91, 073.5966per year) and for Q = 25, 299 (that is $91, 073.5967 per year), the production size shouldbe 25,298 speakers per batch.

The next step is to find the value of the reorder level R. The criterion used in thisexample is a cycle service level of 95%. The formula to use is the following: R = AV G L + z STD. The value of z corresponds to the value found in the lookup-table forthe cumulative density function of a standard normal distribution. This means z = 1.65.When we assume that there are 4 weeks in one month, then the lead time equals 1.5months. Consequently, the demand distribution is converted to monthly demand. To findthe standard deviation of the demand per month, it is not allowed to divide the standarddeviation over 12 months (similar to the average). Any algebraic calculations are onlyallowed based on the variance. The variance of the yearly demand is 6, 930 6, 930 =48, 024, 900. The variance per month is 4, 002, 075, and consequently, the standard devia-

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tion of the monthly demand is 2, 000.52 speakers. Furthermore, the average demand is8, 000 units per month. As a result,

R = AV G L+ z STD L

= 8, 000 1.5 + 1.65 2, 000.52

1.5

= 16, 042.71.

Since the reorder level represents a number of speakers, it should be rounded up to thenearest integer to guarantee that the service level is satisfied. In summary, the reoder levelshould be 16,043 speakers.

In Equation (2) it is assumed that lead times are constant. Whenever the lead time hasa mean and standard deviation denoted by AV GL and STDL, respectively, then

R = AV G AV GL+ z AV GL STD2 + AV G2 STDL2. (3)

Note that this expression is the same as Equation (2) when the lead time is constant (thatis, when AV GL = L and STDL = 0).

Example 1 (continued). If the lead time would be stochastic with the same averageas before (that is 6 weeks), but with a standard deviation of 2 weeks, then the order sizewould remain the same (Q = 25, 298 speakers), but the reorder level becomes

R = 8, 000 1.5 + 1.65

1.5 2, 000.522 + 8, 0002 (2/4)2 = 19, 740. (4)

This means that the reorder level increases by 3,697 speakers, since the safety stock needsto hedge against more demand uncertainty during the lead time.

Besides the cycle service level, a commonly used definition for service is the fill rate,which corresponds to the fraction of demand satisfied immediately by stock on hand:

fill rate =expected demand satisfied immediately in a replenishment cycle

expected demand in a replenishment cycle

= 1 expected shortage in a replenishment cycleexpected demand in a replenishment cycle

. (5)

When excess demand is assumed to be backordered and the inventory level is continuouslymonitiored, a new order is placed whenever Q units have been demanded. This correspondsto the expected demand in a replenishment cycle. On the other hand, when excess demand isassumed to be lost this amount equals Q units plus the expected shortage in a replenishmentcycle (abbreviated as ESC). This means

backorder: fill rate = 1 ESCQ

lost sales: fill rate = 1 ESCQ+ ESC

For simplicity, only the backorder setting is discussed in the remainder of this document.

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Next, ESC needs to be calculated and is dependent on the reorder level. Since ESCrepresents the expected demand in excess of the reorder level in a replenishment cycle:ESC = E[max{DLR, 0}]. When the demand follows a normal distribution, this simplifiesto

ESC = STD L L(z), (6)

where L(z) is known as the loss function and

z =R AV G LSTD L . (7)

This latter expression was already derived in Equation (2). The loss function is given byL(z) = (z) z [1 (z)] and its values can be found in a table (see Appendix B).For instance, when the safey factor z equals 1.65, this corresponds to L(z) = 0.0206. Thismeans that for a cycle service level of 95%, the safety factor equals z = 1.65 and the expecteddemand to be exceeding the reorder level equals 0.0206 when demand would follow a standardnormal distribution during the lead time. Other values of L(z) are provided in Table 4.

z 0.68 0.85 1.04 1.29 1.41 1.56 1.65 1.76 2.06L(z) 0.1478 0.1100 0.0772 0.0465 0.0359 0.0255 0.0206 0.0158 0.0072

Table 4: The value of the loss function L(z) for different values for the safety factor z whendemand follows a normal distribution (see also Appendix B).

Note that the lead time is assumed to be constant. When the lead time is stochastic withmean AV GL and variance STDL2, then

ESC =AV GL STD2 + AV G2 STDL2 L(z) (8)

and

z =R AV G AV GL

AV GL STD2 + AV G2 STDL2 . (9)

This is exactly the same expression as Equation (7) when the lead time is constant (that is,when AV GL = L and STDL = 0).

When the values of the reorder level R and order quantity Q are determined, the amountof safety stock (SS) is given by

SS = R AV G AV GL, (10)and the average on-hand inventory level equals

inventory level =Q

2+ SS. (11)

Example 1 (continued). Lets consider the impact of stochastic lead times. Recall thatwith a constant lead time the reorder level shoud be 11, 291 units, whereas it should beequal to 12, 654 units when the lead time is stochastic. To compute the fill rate, the z-valueneeds to be determined first. However, for both scenarios this is already determined,namely

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z = 1.65. As a result, the loss function can be found in the standard normal loss table:L(z) = 0.0206. Lets first consider the fill rate when the reorder level equals 16, 043 units.

ESC = 2, 000.52

1.5 0.0206 = 50.47fill rate = 1 50.47

25, 298= 0.9980.

However, when the lead time is stochastic,

ESC =

1.5 2, 000.522 + 8, 0002 (2/4)2 0.0206 = 96.63fill rate = 1 96.63

25, 298= 0.9962.

This example illustrates two things: (1) when the fill rate is rather high, the lead timevariance is not of great influence, and (2) the fill rate is higher than the cycle service level.This latter observation is always true and is of importance for many practical settings.In most text books the cycle service level is used to define service level, whereas in manyreal-world applications the fill rate is used as service performance measure. When the textbook formulas are used, the reorder level is structurally set too high. Consequently, thesafety stock is higher as well as the average on-hand inventory level. This will result inhigher inventory costs, whereas the fill rate constraint would still be satisfied with a lowerreorder level.

(S 1, S) Policy with Periodic ReviewsA base-stock policy is often used when no fixed costs are charged for each order. In orderto keep the inventory level as low as possible, an order is placed at each review whereenough items are orderd to meet demand until the next order delivery. For an (S 1, S)replenishment policy with period reviews, the order that is placed at a review moment issuch that the inventory position equals S units after ordering. This order arrives after thelead time of L time periods. Since an order is placed at each review moment, the next orderwill arrive after r time periods. This means that the order-up-to level S should cover thedemand during L + r time periods. The question to be answered is how to set the value ofS.

First, lets use the cycle service level to determine the vaule of S. In an (S 1, S) policywith periodic reviews, this service level definition corresponds to the probability that thedemand during L + r time periods does not exceed S units; Pr(DL+r < S) . Whendemand follows a normal distribution this translates to

(S AV G (L+ r)STD L+ r

) , (12)

andS = AV G (L+ r) + z STD L+ r, (13)

where z corresponds to the safety factor of a standard normal distribution with probability. When these equations are compared to Equation (1) and (2), it becomes clear that they

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are similar except for the time period during which the inventory level needs to hedge againstdemand uncertainty.

Equation (13) assumes that lead times are constant. Similar to the previous section, astochastic lead time could be included where the average and standard deviation of the leadtime are denoted by AV GL and STDL, respectively. Consequently,

S = AV G (AV GL+ r) + z

(AV GL+ r) STD2 + AV G2 STDL2. (14)This is exactly the same expression as Equation (13) when the lead time is constant (thatis, when AV GL = L and STDL = 0).

Example 2. The central warehouse for a pharmacy orders antibiotics every two weeks(that is, every 14 days). The daily demand has an average of 2,000 boxes with a standarddeviation equal to 800 boxes. An order will arrive after exactly 5 days. Since the warehouseshould always have enough antibiotics on inventory, they want to provide their customerswith a service level of 99%. The manager of the warehouse would like to know how manyboxes to order with a 99% cycle service level.

Since no order costs are mentioned, it is best to order at each review moment to keepinventory holding costs as low possible. Consequently, the correct inventory replenishmentpolicy to consider for the warehouse is a base-stock policy. A 99% cycle service levelcorresponds to a z value equal to 2.33. The lead time is constant. As a result, Equation(13) should be used to set the base-stock level:

S = 2, 000 (5 + 14) + 2.33 8005 + 14 = 46, 124.99. (15)

To satisfy the service constraint it is best to round this number up to 46,125 boxes. Thismeans that the warehouse manager should order such an amount that the inventory position(that is, on-hand inventory minus backorders plus inventory on order) after ordering is equalto the base-stock level of 46,125 units.

Besides the cycle service level, the fill rate could also be used to set the base-stock levelS. See Equation (5) for a definition. Since an order is placed at each review moment, thereplenishment cycle equals r time units. Consequently, the expected demand in a replenish-ment cycle equals AV G r, whereas the expected shortage in a cycle (or ESC) is given byE[max{DL+r S, 0}]. Similar to the previous section,

ESC = STD L+ r L(z) (16)and

z =S AV G (L+ r)STD L+ r , (17)

for the particular inventory policy under study in this section. Whenever the lead time isstochastic, with a mean and standard deviation of AVGL and STDL, respectively, then

ESC =

(AV GL+ r) STD2 + AV G2 STDL2 L(z) (18)and

z =S AV G (AV GL+ r)

(AV GL+ r) STD2 + AV G2 STDL2 . (19)

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This is exactly the same expression as in Equation (17) when the lead time is constant (thatis, when AV GL = L and STDL = 0).

When the value of order-up-to level S is determined, the amount of safety stock (SS) isgiven by

SS = S AV G (AV GL+ r), (20)and the average on-hand inventory level equals

inventory level =r AV G

2+ SS. (21)

Example 2 (continued). As it turned out, the supplier of the antibiotics was not reallyreliable. Consequently, the warehouse manager wants to include the standard deviation ofthe lead time, which is 2 days. The average lead time turned out to be 6 days. Furthermore,the warehouse manager has read that cycle service level is not the only performance measurefor service towards customers, and decided to use a 99% fill rate instead.

In order to find the correct base-stock level based on this new information, the expectedshortage needs to be computed first. Since the fill rate constraint is 99%, the followingexpression must hold:

fill rate = 1 ESCAV G r 0.99 = 1

ESC

2, 000 14 . (22)

This means that the expected shortage in a cycle of 14 days should be at most 280 boxes.ESC is expressed as

ESC =

(AV GL+ r) STD2 + AV G2 STDL2 L(z). (23)

Consequently,280 =

(6 + 14) 8002 + 2, 0002 22 L(z), (24)

such that L(z) = 0.0522. Since L(z) represents the loss function, we need to find the small-est value of z in the standard normal loss function that is smaller than 0.0522. Otherwise,the fill rate constraint is violated. This corresponds to a z value of 1.24. Finally, Equation(19) is used to find the value of the correct base-stock level:

1.24 =S 2, 000 (6 + 14)

(6 + 14) 8002 + 2, 0002 22 (25)

As a result,

S = 1.24

(6 + 14) 8002 + 2, 0002 22 + 2, 000 (6 + 14) = 46, 654.54. (26)

The new base-stock level becomes 46,655 boxes.

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(s, S) Policy with Periodic Reviews

When there are fixed cost involved with placing an order, it is no longer efficient anymoreto order at each review. As a result, a base-stock policy (as in the previous section) is notsufficient. A reorder level should be introduced to trigger the placement of an order (similarto the (R,Q) policy for continuous reviews). However, since orders can only be placed at areview moment, the inventory position can be lower than the reorder level at the momentan order is placed. The difference between the reorder level and the inventory position atorder placement is called the undershoot. Consequently, the order size equals the differencebetween order-up-to level S and reorder level s plus the undershoot. The average undershootis given by

AV GU =STD2 + r AV G2

2 AV G (27)and the variance of the undershoot by

STDU2 =3 r STD2 + r2 AV G2

3 AV GU2. (28)

Since the EOQ-formule is robust, it is common to set S s + AV GU equal to the EOQ.This means that once the reorder level is specified, the order-up-to level can be found easilyas well. Therefore, the focus in the remainder of this document is on how to set the reorderlevel.

Lets first consider the cycle service level to set the reorder level. Recall that the cycleservice level equals the probability of no stockout occurrence during a replenishment cycle.This means that the reorder level should cover the demand during the undershoot periodplus the lead time, since it takes another L time units before an order is delivered. Thistime period is also called the risk period. Consequently, Pr(Drisk < R) . The averagedemand during the risk period equals

AV GR = AV GU + AV G L, (29)and the variance of the demand during this time period equals

STDR2 = STDU2 + STD2 L. (30)Consequently, the same formulas as for the (R,Q) policy with continuous reviews could beused, but with a different time period during which the reorder level should hedge againstdemand uncertainty. Here, this time period is the risk period consisting of the lead timeplus the period of undershoot rather than just the lead time. As a result, AV G L shouldbe replaced by AV GR and STD L by STDR. This means that,

s = AV GR + z STDR. (31)When the lead time is not constant, AV GR and STDR need to be updated to

AV GR = AV GU + AV G AV GL, (32)STDR2 = STDU2 + AV GL STD2 + AV G2 STDL2. (33)

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Besides the cycle service level, the fill rate could also be used to set the reorder level s.Recall that this service level definition corresponds to the fraction of demand satisfied imme-diately by stock on hand. Similar to the (R,Q) policy with continuous reviews, the expecteddemand in a replenishment cycle is equal to the EOQ-formula (as explained above), whereasthe expected shortage in a cycle (or ESC) is given by E[max{Drisk s, 0}]. Consequently,

ESC = STDR L(z) (34)

and

z =s AV GRSTDR

. (35)

When the values of the reorder level s and order-up-to level S are determined, the amountof safety stock (SS) is given by

SS = s AV GR, (36)and the average on-hand inventory level equals

inventory level =S s+ AV GU

2+ SS. (37)

Example 3. The new head of the automotive section of Nichols Department Store has theresponsibility to ensure that reorder quantities for the various items have been correctlyestablished. He decides to test one item and chooses Michelin tires, XW size 185 x 14BSW. The placement of an order will cost $20, whereas the cost for each tire is $35. Theholding cost is 20% of the tire cost per year. Orders can only be placed once a week andthe delivery lead time is on average 5 days with a standard deviation of 1 day. The annualdemand follows a normal distribution with an average of 1,000 tires, whereas the standarddeviation of the daily demand is only 2 tires. Because customers generally wait in case ofa stockout and do not go elsewhere, the head of the section decided on a medium level ofservice, so he wants to ensure an 85% probability of not stocking out on this specific brandof tires. Assume the demand occurs 7 days per week, and one month equals 4.3 weeks.

Because there are fixed order cost charged with each order, these costs need to bebalanced with the holding cost. This means that the appropriate replenishment policyshould include a reorder level. Since orders can only be placed at fixed time intervals, itis a periodic replenishment policy. Therefore, the (s, S) policy with periodic reviews is thecorrect replenishment policy for this example. As mentioned before, the EOQ-formula isrobust in the sense that this amount gives a reasonable answer to the number of units toorder when an order is placed. In this example, lets use weeks as time unit. The averagedemand per week is 1, 000/(12 4.3) = 19.38 with a standard deviation of 22 7 = 5.29tires. The weekly holding cost is 0.2 35/(12 4.3) = 0.136. Consequently,

Q =

2 19.38 20

0.136= 75.593. (38)

Since this is the average order size, this value does not have to be rounded to an integervalue. The order size consists of the difference between the reorder level s and the order-

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up-to level S plus the average undershoot (that is, the amount of units below the reorderlevel when an order is placed). Since the demand follows a normal distribution, the averageundershoot equals

AV GU =5.292 + 1 19.382

2 19.38 = 10.41 tires, (39)whereas the standard deviation of the undershoot is given by

STDU =

3 1 5.292 + 12 19.382

3 10.412 = 6.69 tires. (40)

To compute the cycle service level, the demand during the risk period has to be considered.This equals the undershoot plus the demand during the lead time. The average demandduring the risk period is

AV GR = 10.41 + 19.38 (5/7) = 24.255, (41)

and the standard deviation of the demand during the risk period equals

STDR =

6.692 + (5/7) 5.292 + 19.382 (1/7)2 = 8.51. (42)

Since the restriction on the cycle service level is 85%, this corresponds to a z-value of 1.04.Finally, the reorder level can be computed by

s = 24.255 + 1.04 8.51 = 33.11. (43)

That is, the reorder level equals 34 tires. This vaue also specifies what the order-up-tolevel should be, since the result of the EOQ-formula (Q = 75.593) shoud be equal toS s+ AV GU . As a result,

S = Q + s AV GU = 75.593 + 34 10.41 = 99.18. (44)

This means that the order-up-to level should be set to 99 tires (the value is rounded to thenearest integer).For these specific values of the inventory replenishment policy (that is, s = 34 and S = 99),the fill rate is given by

fill rate = 1 STDR L(z)S s+ AV GU , (45)

where z = (34 24.255)/8.51 = 1.145. Note that this z value is not the same as for thecycle service level since the reorder level s is rounded in between. This z value correspondsto L(z) = 0.0627 and the fill rate equals

fill rate = 1 8.51 0.062799 34 + 10.41 = 0.9929. (46)

Even though the cycle service level is only 85%, the fill rate is rather high with 99.3%.

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References

F. Harris. How many parts to make at once. Factory, The Magazine of Management, 10:135136, 1913.

A.C. Hax and D. Candea. Production and Inventory Management. Prentice- Hall, EnglewoodCliffs, 1984.

E.A. Silver and H.C. Meal. A heuristic for selecting lot size quantities for the case of adeterministic time-varying demand rate and discrete opportunities for replenishments.Production and Inventory Management Journal, 2nd Quarter:6474, 1973.

H. Wagner and T.M. Whitin. Dynamic version of the economic lot size model. ManagementScience, 5:8996, 1958.

P.H. Zipkin. Foundations of inventory management. McGraw-Hill, New York, 2000.

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