eoq and jit

A COMPARISON OF THE ECONOMIC ORDER QUANTITY AND QUICKRESPONSE INVENTORY REPLENISHMENT METHODS

by

Walter Zinn The Ohio State University

and

John M. CharnesThe University of Kansas

Logistics managers have at their disposal at least two well-known competing methods ofdetermining order quantity: Economic Order Quantity (EOQ) and Quick Response (QR). Eachmethod takes a distinct approach and makes substantially different recommendations for orderquantity. Although both methods are discussed extensively in the literature and included in logis-tics and operations management textbooks, a comprehensive comparison of the two methods isnot currently available in the logistics literature. Accordingly, the goal of this paper is to comparethe EOQ and QR methods in order to propose a set of rules by which managers can select the mostappropriate method to use under various circumstances.

The EOQ method, introduced by Ford W. Harris (1915), balances the cost of ordering an item with the inventory holding cost for that item. Given a known level of annual demand, a firmmust balance the cost of ordering smaller quantities more frequently to minimize holding cost,against the cost of making a smaller number of larger-quantity purchases to minimize orderingcost. The EOQ determines the optimal order quantity that minimizes the joint cost of ordering andholding inventory.

In contrast, the QR method considers only the inventory holding cost and ignores the order-ing cost. QR is a general term that describes the criteria used to determine the order quantity in several well-known rapid-replenishment inventory management methods such as Just-In-Time or Continuous Replenishment. The QR order quantity is just enough to support operations until thenext delivery. Therefore, the QR quantity is the product of the daily demand and the time betweendeliveries, in days. Thus, each inventory replenishment method ignores factors that are included inthe other method. While the EOQ method ignores the time between deliveries in the determinationof order quantity, the QR method ignores three variables included in the EOQ: the cost of an order,the product’s unit value, and the unit cost of holding inventory. Note that there are EOQ models

JOURNAL OF BUSINESS LOGISTICS, Vol. 26, No. 2, 2005 119

available that include safety stock in the computation of the EOQ. However, these models are out-side the scope of this research in that we consider base stock only.

This comparison is timely, as the drive to reduce inventory investment is leading firms towardsuse of QR and away from the EOQ. One reason for inventory reduction is that a growing numberof firms today are increasingly concerned with the risk associated with stocking the wrong inven-tory and not just the cost of holding it. This risk is currently not included explicitly in either the EOQor QR. As a result, some have questioned the value of the EOQ and argue that it has outlived its use-fulness (Woolsey 1988). On the other hand, the EOQ remains a part of business school curricula andcontinues to be used by some managers. Which method is better? Should firms always movetowards QR? Are there circumstances under which managers should favor the EOQ over QR? Ourgoal is to answer these questions.

The equations used to compute the EOQ and QR quantities are below.

where: QEOQ= Quantity delivered for the EOQ methodd = Average daily demand in unitsP = Cost of an orderH = Average annual cost of holding inventoryV = Unit product value

QQR = td

where:QQR = Quantity delivered for the QR methodt = Time between deliveries, in days

We make the comparison between the EOQ and QR methods with the following assumptions:

1. Both methods are applied in a continuous review inventory system with deterministicorder quantity and deterministic time between deliveries where product is made tostock.

2. The scope of the comparison is limited to base or cycle stock.

3. The cost of an order (P) is assumed to be the same under both inventory replenishmentmethods.

The third assumption is needed because the cost of an order is measured differently wheneach method is applied. The cost of an order is defined in the literature to be all out-of-pocket costsincurred when an order is issued, starting with product search and negotiation. They also include ordertransmittal, receiving product, placing it in storage, and processing the invoice for payment (Stockand Lambert 2001). Freight charges and freight audit costs are also included (Lambert and Bennion1986). In the EOQ method, product search and negotiation costs are higher because each order is

HV

PdQEOQ

720=

120 ZINN AND CHARNES

negotiated independently. In contrast, in the QR method product search and negotiation costs arelower because individual orders are issued under a long-term supply contract. This reduces thenumber of suppliers included in the search and limits the number of items to be negotiated with theselected supplier. On the other hand, the QR method has a higher cost of receiving and handling prod-uct, and of freight and audit costs because the QR implies a larger number of smaller orders than theEOQ method. Thus, the differences in the cost of an order between the two methods are assumedto cancel out.

A final issue to note before comparing the costs of the EOQ and the QR methods is the rela-tionship between QQR and QEOQ. The EOQ order quantity, QEOQ, is always greater than or equal tothe QR quantity, QQR, because in the QR method the firm receives only enough inventory to coverthe demand until the next delivery. If there is any less, the inventory will run out prior to the nextdelivery. Recall that we assume a fixed quantity and fixed time between deliveries replenishmentsystem.

The next section presents a review of related work comparing the EOQ with Just-In-Time(JIT) methods. This is followed by a section examining reasons for the shift by practitioners towardQR and away from the EOQ. A quantitative comparison between the two inventory replenishmentmethods follows. The final two sections contain a discussion of results and our conclusions and man-agerial implications.

LITERATURE REVIEW

Fazel (1997) computed the cost difference between the EOQ and Just-In-Time (JIT) methodsand estimated the demand and purchase price indifference points between the EOQ and JIT. He devel-oped a model to estimate the demand above which EOQ is less costly than JIT and the maximumunit purchase price below which JIT is preferable.

In a follow-up study, Fazel, Fischer, and Gilbert (1998) suggested that JIT should be used at lower levels of demand whereas EOQ should be used at higher levels of demand. They also identified factors that affect the cost difference between JIT and EOQ-based solutions. These twostudies considered the cost but not the risk of holding inventory.

Schneiderjans and Cao (2000 and 2001) expand on the Fazel, Fischer, and Gilbert (1998)model by adding as a variable the reduction in space requirements caused by JIT. They conclude thata JIT method is almost always preferable to an EOQ-based method, whether the demand is high orlow, because JIT always requires less warehouse space than the EOQ method.

Ramasesh (1990) developed a decision model to help companies decide whether to movefrom an EOQ method to a JIT method. The model compares the fixed cost of initiating a JIT method(such as the development of sources of supply, installation of quality assurance procedures, nego-tiating long-term contracts, etc.) with the savings obtained from operating a JIT method that has alower cost than the traditional EOQ method. Ramasesh concludes that the incentive to move awayfrom the EOQ is greater when the demand is large and the order cost is low.


Instead of choosing between the two methods, Johnson and Stice (1993) propose a “hybrid”that they call “Not Quite JIT.” Their idea is to alleviate some of the costs and risks associated withJIT by proposing a method that combines a lower level of inventory with JIT deliveries. The authorslist ordering and transaction costs as some that are higher in the JIT methods. They also list the poten-tial for stockouts and the inflexibility associated with being locked into long-term contracts as risksinherent to JIT.

FROM THE EOQ TO QR

In recent years, a growing number of firms adopted the QR method in an attempt to significantlyreduce or even eliminate inventory. As noted earlier, the QR order quantity is less than or equal tothe EOQ quantity. The goal of QR is to minimize only the cost of holding inventory, while the goalof the EOQ is to minimize the joint cost of ordering and holding inventory. QR is therefore subop-timal because it disregards the cost of ordering. Thus, adopting QR apparently counters the notionof integrated logistics in that it aims to minimize a single cost. Why, then, are firms increasingly adopt-ing QR? We see two main reasons: the risk of holding inventory and the decline in the cost of an order.Each is discussed.

The Risk Related to Holding Inventory

When holding inventory, firms incur the risk that inventory will lose value as a result ofchanges in product technology or in consumer preferences. For example, computer retailers hold-ing a large inventory incur the risk of having to sell product at a significant discount if the inven-tory is made obsolete by a new wave of technological innovation. Similarly, product proliferationin industries such as automotive, packaged goods, or apparel offer consumers a wide choice ofproducts. This increases the likelihood that firms will hold the wrong inventory because the num-ber of choices available complicates the task of forecasting consumer preference. Apparel mer-chandisers, for instance, often sell product at large discounts to reduce the inventory of low demanditems. Thus, by purchasing smaller quantities per order, retailers shift the risk of holding inventoryto suppliers.

Unless the risk of holding inventory is made explicit, the EOQ method will always appear tobe preferable to the QR method. As mentioned earlier, the reason is that the EOQ method opti-mizes the joint cost of ordering and holding inventory while the QR method minimizes only the hold-ing cost. Thus, the need to manage the risk of holding inventory may substantially explain the shiftfrom the EOQ to the QR observed in many companies.

Our operational definition of risk (R) is the percent decline in the value of a unit of product result-ing from changes in product technology or consumer preference. For example, R has a value of .3if a computer part is sold at a 30 percent discount after it is made obsolete by a newly introducedsubstitute part. While the determination of R for each product is beyond the scope of this research,one could estimate it on the basis of historical discount values or with the help of subject matter experts.


The Decline in the Cost of an Order

A second reason for the increased adoption of QR is the decline in the cost of an order (P).Advances in computer technology triggered a decline in the cost of processing information, whichin turn has led to a significant decline in the cost of an order. As a result, the value of the EOQ declined(Jones 1991). This helps to explain the decreased usage of the EOQ because it makes the cost of anorder considerably less important than the inventory holding cost, thereby reducing the importanceof considering the trade-off between the two costs.

THE TOTAL COST UNDER THE EOQ AND QR

As noted above, the purpose of this research is to compare the total cost of ordering and hold-ing inventory under the alternatives of selecting either the EOQ or QR as a method to determine orderquantity. In the case of the QR, the total ordering and holding costs are:

(1)

where:

CQR = Total cost if the firm buys the QR quantity

In the case of the EOQ, the total cost function is similar, except for a term added to capture themonetary expression of the risk of holding inventory. We call this term the Expected Loss of Value(ELV). The ELV is the product of the number of units of inventory at risk and the expected loss perunit. The inventory at risk is the difference between the EOQ order quantity and the QR orderquantity (QQR-QEOQ). Recall that QQR is the minimum amount that a firm may order so that QEOQ

always equals or exceeds QQR. Therefore, the greater the quantity ordered beyond QQR, the higherthe inventory at risk. The potential loss per unit of inventory at risk is the product of the value of eachunit (V) and risk (R), the expected discount needed to sell the unit. The risk, R, varies from 0 to 1,where 1 means 100 percent discount and 0 means no discount. Thus:

ELV = (QEOQ–QQR)VR (2)

where:ELV = Expected loss of valueR = Risk (expected discount needed to sell the unit)

The total cost of ordering, holding and assuming risk in the EOQ method is:

(3)

where:CEOQ = Total cost if firm buys the EOQ quantity

( )VRQQVHQ

Q

dPC QREOQ

EOQ

EOQEOQ −++=

2

360

VHQ

Q

dPC QR

QRQR 2

360 +=


A COST COMPARISON OF THE EOQ AND QR

In this section, we compare the total cost of using the EOQ vs. QR. Our goal is not just to com-pute the lowest cost, but principally to identify the impact of different variables on the decision. Thiswill ultimately enable managers to understand the differences between the two inventory replenishmentmethods and to select one method or the other in a given situation.

The object of the comparison is to compute the difference between the total costs of adoptingthe EOQ vs. QR. The difference is denoted as �C = CQR–CEOQ. When �C > 0, firms should adopt the EOQ method. In contrast, firms should adopt the QR method whenever �C < 0. The costdifference, �C, can also be expressed as a function of the six variables defined previously in Equations 1 and 3:

(4)

The above equation has five terms, each corresponding to an identified cost. The first twoterms represent the cost of ordering (OCQR) and the holding cost (HCQR) in the QR method. The next two terms represent the ordering (OCEOQ) and holding (HCEOQ) costs in the EOQ method. Thelast term represents the expected loss of value (ELV).

We begin our cost comparison by selecting realistic ranges of values for each of the variables.These values are based on a search of the literature for values of the same variables used in relatedresearch projects. The input values are listed in Table 1 and the justification for their selection is givenin the next subsection of the paper. This is followed by the data analysis, which is presented in twostages in the results section. In the first stage, we investigate the general relationship between eachof the 6 variables considered here and the cost differential �C. The general relationship is catego-rized by the changes in the value of �C with respect to each of the 6 variables. The general relationshipis also examined with respect to the five cost terms included in Equation 4, viz., OCQR, HCQR,OCEOQ, HCEOQ, and ELV. At the end of this stage, we establish whether the general relationshipbetween each of the six variables and �C is positive or negative.

However, because of interactions among the six variables, the general relationship may not hold for the entire range of values considered. Therefore, in the second stage, we examine howinteractions affect the general relationship between �C and the six variables. This is done by taking the partial derivative of �C with respect to each of the variables to see whether changes ina second variable alter the sign of the derivative. For example, assume that the derivative of �C withrespect to the order cost (P) is positive. The goal of the analysis in the second stage is to determinewhether the sign changes from positive to negative when the values for the other five input variablesare changed.

VRQQVHQ

Q

dPVH

Q

Q

dPC QREOQ

EOQ

EOQ

QR

QR

)(2

360

2

360 −−−−+=∆


Values for Input Variables

The literature review identified a reasonable medium value and/or range for each variableincluded in the cost comparison. The low and high values for each variable were determined by sys-tematically increasing and decreasing the medium value by 50%. The variables and the valuesselected are given in Table 1.

TABLE 1

VALUES FOR INPUT VARIABLES

Variable Low Medium High

R .175 .35 .525

P 60 120 180

d 125 250 375

V 37.5 75 112.5

t 1.5 3 4.5

H .125 .25 .375

The medium value of R corresponds to a 35% discount to sell a product that has lost value dueto changes in product technology or in consumer preference. A survey of websites selling discon-tinued computer parts reveals that the price differential with regularly available items is between 15and 68%, which is a reasonable match with the selected range for R. Ranges for the remainingvariables are available in published research. The cost of an order (P) varies from 40 to 175 dollars(Fazel 1997; Fazel, Fischer, and Gilbert 1998; Lambert and Bennion 1986); daily demand (d) from50 to 500 units (Evers 1997, 1999; Schwarz and Weng 1999; Zinn, Marmorstein, and Charnes1992); and product value (V) from 4 to 150 dollars (Johnson and Anderson 2000; Ramasesh 1990).The time between deliveries (t) varies from 6 hours to 21 days, a range with a higher medium valuethan the one adopted in this research (Closs et al. 1998; Evers 1997, 1999; Schwarz and Weng1999; Tyagi and Das 1997; Zinn, Marmorstein and Charnes 1992). However, given that the natureof this research is to compare the EOQ with QR (which assumes a relatively short time between deliv-eries), we opted for a shorter range for t. Finally, the inventory holding cost (H) averages about 25%per year (Fazel 1997; Johnson and Anderson 2000; Ramasesh 1990). To be consistent with ourpractice of determining the range of an input variable by adding plus or minus 50% to the mediumvalue, we selected the range displayed in Table 1.


RESULTS

Recall that the data analysis is done in two stages. In the first stage, we determine the generalrelationship between each of the six variables considered here and the cost differential �C. This isdone by plotting �C versus each of the six variables to see whether the general relationship for eachof the six variables with respect to �C is positive or negative. A second goal is to break down thegeneral relationship into the five cost terms defined above: OCQR, HCQR, OCEOQ, HCEOQ, andELV. This is followed by the second stage, where we examine how interactions affect the generalrelationships between �C and the six variables.

The General Relationships Among �C, Input Variables and Cost Terms

The relationships among the cost differential (�C), the five cost terms and each of the input vari-ables are illustrated in Figures 1 through 6. Each figure has cost on the vertical axis and one of theinput variables on the horizontal axis. For consistency, the value specified for each of the otherfive input variables is the lowest in the range shown in Table 1. For example, Figure 1 shows thatthe general relationship between �C and the variable risk (R) is negative. The values used to prepareFigure 1 are the medium value for variable R and the low value for the other five input variables.Note that OCEOQ and HCEOQ appear as the same line in all six figures because, by definition, order-ing and holding costs are equal when the EOQ quantity is ordered.

Risk

The relationship between R and �C is negative. The higher the risk, the less likely it is that afirm will adopt the EOQ, because the level of inventory held under the QR method is lower than inthe EOQ method. Firms manage risk by keeping inventory as low as possible. In the equation for�C, R impacts only the Expected Loss of Value (ELV) term, which has a negative sign. Figure 1 illus-trates that ELV is the only cost term that varies with risk.

Similarly, for low risk products managers are usually better off ordering EOQ quantitiesbecause it minimizes the joint cost of ordering and holding inventory, while the QR minimizes theholding cost only. At the extreme, when risk is zero, the EOQ is always the preferred method.


FIGURE 1

RELATIONSHIPS BETWEEN RISK, COST DIFFERENTIAL, AND COST TERMS

Cost of an Order

The effect of P on �C is generally positive. There is an incentive to place fewer and larger ordersas the cost of an order increases. This favors the adoption of the EOQ because it is the only methodthat considers the cost of an order as a variable in determining order size. In the equation for �C,P impacts most cost terms but, as Figure 2 shows, its greatest impact is on the ordering cost underthe QR method (OCQR).

The impact of OCQR on the choice between QR and the EOQ should not be underestimated.There will be serious consequences if a firm switches from the EOQ to QR without first managingto bring down the cost of an order (P). Adopting the QR when P is high will result in a very highcost structure, because the QR requires more orders per year, which causes the annual cost of ordering to grow accordingly.

-10000

-5000

0

5000

10000

15000

20000

0.175 0.350 0.525

Risk (R )

Cost OCEOQ

HCEOQ

OCQR

HCQR

ELV

�C


FIGURE 2

RELATIONSHIPS BETWEEN THE COST OF AN ORDER, COST DIFFERENTIAL, AND COST TERMS

Daily Demand

The general relationship between d and �C is negative. As illustrated in Figure 3, when d is small,�C tends to be large, which favors the selection of the EOQ because the OCQR is relatively largeand also constant with respect to d. As d increases, �C decreases and there is a greater tendency toselect the QR. This is because HCQR, OCEOQ, and HCEOQ grow as d grows. Therefore, QR is morelikely to be the best alternative for items with high daily demand, while the EO is likely to be thebest alternative for items with low daily demand.

0

10000

20000

30000

40000

50000

60.0 120.0 180.0

Cost of an Order (P )

Cost OCEOQ

HCEOQ

OCQR

HCQR

ELV

�C


FIGURE 3

RELATIONSHIPS BETWEEN DAILY DEMAND, COST DIFFERENTIAL, AND COST TERMS

Product Value

The general relationship between V and �C is negative. As shown in Figure 4, OCQR remainsrelatively large and is independent of V. Therefore, when V is small, the EOQ tends to be the pre-ferred inventory replenishment method. As V increases in value, HCQR, OCEOQ, and HCEOQ increasein value as well. The increase in OCEOQ and HCEOQ offsets OCQR and the growth in HCQR, result-ing in a decline in �C. Thus, QR becomes the lowest cost inventory replenishment method as Vincreases.

-4000

0

4000

8000

12000

16000

125.0 250.0 375.0

Daily Demand (d )

Cost OCEOQ

HCEOQ

OCQR

HCQR

ELV

�C


FIGURE 4

RELATIONSHIPS BETWEEN PRODUCT VALUE, COST DIFFERENTIAL AND COST TERMS

Time Between Deliveries

Within the range of values considered in this research, the general relationship between the timebetween deliveries and �C is negative and non-linear. The slope of �C becomes less negative as tincreases. This implies that as t increases firms are more likely to adopt QR. This implication mayappear counterintuitive, since QR is usually associated with shorter time between deliveries. Infact, results show that adopting QR with a short time between deliveries usually results in a very highordering cost because the short time between deliveries means more frequent deliveries. Theserelationships are illustrated in Figure 5.

If we were to extend the range of t beyond the values considered in the research, the slope of�C would become positive. This occurs because OCQR is the dominant cost factor. The cost ofordering in the QR method is high whenever t is short. As t increases, OCQR drives �C down. Atthe extreme, when t is sufficiently large, OCQR ceases to be the dominant cost factor and is mitigatedby a combination of a growing HCQR and decreasing ELV, which eventually drives �C back up.

-4000

0

4000

8000

12000

16000

37.5 75.0 112.5

Product Value (V )

Cost OCEOQ

HCEOQ

OCQR

HCQR

ELV

�C


FIGURE 5

RELATIONSHIPS BETWEEN TIME BETWEEN DELIVERIES, COST DIFFERENTIAL AND COST TERMS

Inventory Holding Cost

Figure 6 shows that the inventory holding cost (H) has a minimal impact on the cost differen-tial (�C). This is because there are 2 major effects of H on �C and they cancel each other out. Onthe one hand, as H increases, OCEOQ also increases because QEOQ is smaller. Recall that H is in thedenominator of the equation to compute the QEOQ quantity. In addition, HCEOQ increases becausethe holding cost increases with the inventory holding cost (H). On the other hand, as H increases,the ELV decreases because QEOQ carries a negative sign in the computation of the ELV. Thus, as shownin Figure 6, these effects cancel out, rendering H as an unimportant factor to consider when decid-ing between QR and the EOQ. This is a somewhat counterintuitive conclusion because the inven-tory holding cost is often used as a justification to switch from the EOQ to QR.

-4000

0

4000

8000

12000

16000

1.5 3.0 4.5Time Between Deliveries (t )

Cost OCEOQ

HCEOQ

OCQR

HCQR

ELV

�C


FIGURE 6

RELATIONSHIPS BETWEEN INVENTORY HOLDING COST, COST DIFFERENTIAL AND COST TERMS

The Effect of Interactions on the General Relationships between DC and Input Variables

In the second stage of the analysis we examine how interactions affect the general relationshipbetween �C and each of the six input variables. This is done by taking the partial derivative of �Cwith respect to each of the variables, in turn. The algebraic signs of the partial derivatives correspondto the general relationships described above for the ranges of values specified in Table 1. Note thatthe partial derivative of �C with respect to P is positive throughout its range, while the sign of thepartial derivative with respect to H varies from slightly positive to slightly negative. The partial deriv-atives with respect to each of the four remaining variables are negative. Recall also that in this sec-ond-stage analysis, the partial derivative of �C is evaluated at the medium value of the variable withrespect to which the partial derivative is taken, and at the low values of the other five input variables.We investigate interactions through a sensitivity analysis in which we increase the values of the vari-ables held constant in the partial derivative, and then observe whether there are changes in thesigns of the partial. If a change in sign occurs, we conclude that there is an interaction between inputvariables, and proceed to explain it.

Two of the input variables, H and R, are not included in the interaction analysis. The reason forexcluding H is that the general relationship is essentially flat and therefore there is no sign tochange. The reason for excluding R is that the derivative of �C with respect to R is independent ofR. The partial derivative of �C with respect to R is shown below, after Equation 4 is first reproducedfor convenience and then manipulated to include the expressions for QQR and QEOQ.

0

4000

8000

12000

16000

0.125 0.250 0.375

Inventory Holding Cost (C )

Cost OCEOQ

HCEOQ

OCQR

HCQR

ELV

�C


(4)

The partial derivative of �C with respect to R is clearly independent of R:

(5)

Cost of an Order

The partial derivative of �C with respect to P is presented in Equation 6 below. This is followedby Table 2, which shows values of the derivative �C with respect to P for various levels of the fiveother input variables. For example, the number 182 at the top right corner of the table is the valueof the partial derivative of �C with respect to P when the input value for P is high and the values forthe other input variables (t, R, d, V, and H) are low. Recall that input values are taken from Table 1.

(6)

TABLE 2

VALUES OF THE DERIVATIVE OF �C WITH RESPECT TO P FOR VARIOUS LEVELS OF INPUT VARIABLES

P

low med high

t short 139 169 182med 19 49 62long –21 9 22

R low 139 169 182med 81 127 148high 22 86 114

d low 139 169 182med 98 139 158high 66 117 139

V low 139 169 182med 98 139 158high 66 117 139

H low 139 169 182med 139 169 182high 133 165 179

PH

VdR

P

VHd

tP

C 180180360−−=

∂∆∂

[ ]H

VPdVtdQQV

R

CQREOQ

720−=−−=∂∆∂

VRtdRH

VPdVHPdVHtd

t

P

VRQQVHQ

Q

PdVH

Q

Q

PdC QREOQ

EOQ

EOQ

QR

QR

+−−+=

−−−−+=∆

720720

2

1360

)(2

360

2

360


As expected, the majority of the values in Table 2 carry a positive sign, since the general rela-tionship between �C and P is positive. The only exception is marked in bold italic font. The signof the relationship is negative when P is low and t is long. In this case, QR is the lower cost methodbecause OCQR is kept low by the combined high value of t and low value of P. Note, in Figure 2,that OCQR is the dominant cost in the relationship between �C and P.

Daily Demand

The partial derivative of �C with respect to d is in Equation 7 below. Table 3 shows values ofthe derivative �C with respect to d for various levels of the five other input variables.

The values are interpreted in the same way as Table 2, with the difference that we expect mostvalues to carry a negative sign.

(7)

TABLE 3

VALUES OF THE DERIVATIVE OF �C WITH RESPECT TO D FOR VARIOUS LEVELS OF INPUT VARIABLES

d

low med high

t short –35 –21 –15med –22 –7 –1long –8 6 12

R low –35 –21 –15med –53 –31 –21high –72 –41 –27

P low –35 –21 –15med –55 –35 –26high –70 –46 –35

V low –35 –21 –15med –42 –22 –13high –44 –19 –8

H low –35 –21 –15med –32 –17 –11high –31 –16 –9

VRtHd

PVR

d

VHPVHt

d

C+−−=

∂∆∂ 180180

2


The only two positive values in Table 3 (marked bold italic) are for medium and high d whent is long. They show an interaction between d and t whereby the firm is better off adopting theEOQ as demand increases and the time between deliveries is long. This is explained mostly bychanges in the Expected Loss of Value (ELV) in Equation 4. When t is long and d increases, the ELVwill decrease because QEOQ increases slightly and QQR increases substantially. The reduction in ELVmakes the EOQ more attractive, changing the sign of the slope of �C for medium and high valuesof d.

Product Value

The derivative of �C with respect to V is in Equation 8 below. Similar to the analysis abovefor the other input variables, Table 4 shows values of the partial derivative �C with respect to V forvarious levels of the five other input variables. The values are interpreted in the same way as Table3. As a recap, –48, the number on the top right corner of the table is the value of the partial deriva-tive of �C with respect to V when V has the high value indicated in Table 1 and the other input vari-ables (P, t, R, d, and H) have the low values indicated in Table 1.

(8)

Values in Table 4 are mostly negative, which is consistent with the results displayed in Figure 4. The two exceptions correspond to interactions. The slope of �C with respect to V turns pos-itive for the cases when t is long and V is medium or high. This is explained mostly by changes inthe Expected Loss of Value (ELV) in Equation 4. When t is long and V increases, the ELV willdecrease because QEOQ also increases slightly. The reduction in ELV makes the EOQ more attrac-tive, changing the slope of �C.

RtdVH

PdR

V

HPdHtd

V

C+−−=

∂∆∂ 180180

2


TABLE 4

VALUES OF THE DERIVATIVE OF �C WITH RESPECT TO V FOR VARIOUS LEVELS OF INPUT VARIABLES

V

low med high

P low –116 –69 –48med –183 –116 –87high –234 –153 –116

t short –116 –69 –48med –72 –25 –4long –27 20 41

R low –116 –69 –48med –178 –103 –70high –239 –137 –91

d low –116 –69 –48med –139 –72 –42high –145 –64 –27

H low –116 –69 –48med –105 –58 –37high –102 –53 –30

Time Between Deliveries

The partial derivative of �C with respect to t is in Equation 9 below. Note, as illustrated in Figure 5, that the input variable t has a non-linear effect on �C that is different from the previousinput variables examined in this research. As explained earlier, the slope of �C with respect to tcarries a negative sign when t is short and then bottoms out. For longer values of t that are beyondthe scope of this research, the slope of �C turns positive. Table 5 below is similar to the equivalenttables for other input variables included in the research. It shows values of the partial derivative of�C with respect to t for various levels of the five other input variables.

(9)

The time between deliveries (t) has interactions with every other input variable, always whent is long or medium. Note, in Figure 5 that �C attains a minimum at t = 4.5 days and that it crosseszero (i.e., the point of indifference between the EOQ and QR) at t = 2.3 days. Therefore, the higherthe value of variables V, H, d, and R, the shorter the time between deliveries at which it is advantageous to adopt the EOQ. Conversely, the larger the value of P, the longer the time betweendeliveries must be before it becomes preferable to adopt the EOQ.

2360

2 t

PVHdVRd

t

C−+=

∂∆∂


TABLE 5

VALUES OF THE DERIVATIVE OF �C WITH RESPECT TO T FOR VARIOUS LEVELS OF INPUT VARIABLES

t

short med long

P low –8487 –1287 47med –18087 –3687 –1020high –27687 –6087 –2087

d low –8487 –1287 47med –7373 –173 1160high –6260 940 2273

R low –8487 –1287 47med –7666 –466 867high –6846 354 1687

V low –8487 –1287 47med –7373 –173 1160high –6260 940 2273

H low –8487 –1287 47med –8194 –994 340high –7901 –701 633

CONCLUSIONS AND MANAGERIAL IMPLICATIONS

The goal of this paper is to compare the EOQ and the QR methods to replenish inventory, andthen propose a set of rules to help managers select the more appropriate method to use under vari-ous circumstances. Therefore, we began by writing cost equations for both the EOQ and QR and thenfound the difference, denoted by �C. The cost difference was analyzed both as a function of the sixindividual input variables composing the cost equations and also as a function of five cost termsobtained in the equation for �C (Equation 4). The analysis focused on determining the generalrelationship between �C and each input variable. The general relationships were then explained withrespect to each of the five cost terms. This was followed by an analysis of the effect of interactionsin the relationship between �C and each of the input variables.

The two main conclusions in this research are (1) the importance of reducing the cost of an order(P) whenever managers are considering switching from the EOQ to QR, and (2) the fundamentalrole of risk as justification for adopting the QR method. These two conclusions are supported byobserving in Figures 1 to 6 that OCQR and ELV are almost always the largest cost factors. Input variable P is a key determinant of the cost term OCQR. Similarly the risk (R) is a key determinantof the cost term ELV.


These two conclusions are further justified conceptually. The key difference between the EOQand the QR methods is that the former minimizes the joint cost of ordering and holding inventoryand the latter minimizes only the cost of holding inventory. Thus, the ordering cost in the QRmethod is managed independently of the inventory holding cost, which may have important con-sequences because the ordering cost (OCQR) is almost always the highest cost in Figures 1 to 6. Sim-ilar reasoning applies to risk, which is a key factor in determining that firms hold the right inventoryas opposed to merely the right level of inventory, but has so far not been explicitly included by oth-ers in models to determine order quantity.

The EOQ method is clearly the lowest cost method if the time between deliveries is short andthe order cost is high. This result is counterintuitive because QR is usually associated with a shorttime between deliveries. However, a short time between deliveries also increases order frequencyand therefore the cost of ordering. Thus, if QR is implemented in an environment where the cost ofan order P is high, the increase in ordering cost may negate any gains from the reduction in inven-tory. The cost of an order must be managed down before implementing the QR inventory replen-ishment method, or else using the QR method will be counterproductive.

The greater the risk related to holding inventory, the greater the advantage of adopting theQR, mostly because the risk represents a significant addition to the cost of holding inventory in theEOQ method. At the extreme, the EOQ method is always less costly when the risk is zero.

We reach the following additional conclusions:

1. The greater the cost of an order, the greater the advantage of adopting the EOQ. Thisis because the ordering cost increases proportionately when QR is adopted, as QRrequires more orders per year.

2. The greater the daily demand for a product, the greater the advantage of adopting QR.This is explained by the cost of ordering under the EOQ. An increase in demandincreases the ordering cost in EOQ, but is neutral with respect to the ordering cost in QR.

3. The greater the unit value of a product, the greater the advantage of adopting QR. Thisis partially explained by the impact of product value on the expected loss of value(ELV). The cost of the EOQ method grows as the ELV grows.

4. The time between deliveries, especially when short, has a significant upward impact onthe ordering cost in QR. Therefore, the shorter the time between deliveries, the greaterthe advantage of adopting the EOQ, although this result is also strongly affected by thecost of an order, as explained earlier.

5. The cost of holding inventory has little relevance to the choice between the EOQ andthe QR. Its impact on the cost difference between QR and the EOQ is negligible withinthe variable ranges considered in this research.


6. The interactions between variables were investigated by showing how changes in thevalue of one variable affect the partial derivative with respect to another variable.Changes in the sign of partial derivatives were observed always in cases where thetime between deliveries is medium or long, especially the latter. Results are summarizedin Table 6 below.

TABLE 6

SUMMARY OF RESULTS

QR is Preferred when EOQ is Preferred when• Time between deliveries is moderate • Time between deliveries is short and the cost of an order

or long and cost of an order is low is high• Cost of an order is medium or high

• Cost of an order is low • Cost of an order is low and time between deliveries is long• Risk is medium or high • Risk is low

• Risk is high and time between deliveries is long• Product value is high • Product value is low

• Product value is medium or high and time between deliveries is long

• Daily demand is high • Daily demand is low• Daily demand is high and time between deliveries is

medium or long

Table 6 shows that the EOQ should still be considered by managers, especially for the least impor-tant products, i.e., that have low risk, low value and low demand. It is crucial, however, that man-agers keep in mind that these conclusions are generalizations based on the values selected for thisresearch. Managers can determine the best approach to determine order quantity by substituting theirfirm’s appropriate values into Equation 4. Finally, this research has shown that despite the shiftobserved from the EOQ to QR inventory replenishment methods (explained by the simultaneousdecline in the cost of an order and the increase in the risk of holding inventory), use of the EOQ methodremains a viable option for managers.


NOTES

Closs, David J., Anthony S. Roath, Thomas J. Goldsby, James A. Eckert, and Steven M. Swartz (1998),“An Empirical Comparison of Anticipatory and Response-Based Supply Chain Strategies,” Inter-national Journal of Logistics Management, Vol. 9, No. 2, pp. 21-34.

Evers, Philip T. (1999), “Filling Customer Orders from Multiple Locations: A Comparison of Pool-ing Methods,” Journal of Business Logistics, Vol. 20, No. 1, pp. 121-140.

Evers, Philip T. (1997), “Hidden Benefits of Emergency Transshipments,” Journal of BusinessLogistics, Vol. 18, No. 2, pp. 55-76.

Fazel, Farzaneh (1997), “A Comparative Analysis of Inventory Costs of JIT and EOQ Purchasing,”International Journal of Physical Distribution and Logistics Management, Vol. 27, No. 8, pp. 496-504.

Fazel, Farzaneh, Klaus Fischer, and Erika Gilbert (1998), “JIT Purchasing vs. EOQ with a Price Discount: An Analytical Comparison of Inventory Costs,” International Journal of ProductionEconomics, Vol. 54, No. 1, pp. 101-109.

Harris, Ford W. (1915), Operations and Cost, Chicago, IL: A.W. Shaw.

Johnson, Gene and James Stice (1993), “Not Quite Just-In-Time Inventories,” The National Pub-lic Accountant, Vol. 37, No. 3, pp. 26-29.

Johnson, M. Eric and Emily Anderson (2000), “Postponement Strategies for Channel Derivatives,”International Journal of Logistics Management, Vol. 11, No. 1, pp. 19-36.

Jones, Daniel J. (1991), “JIT and the EOQ Model: Odd Couple no More!” Management Account-ing, Vol. 72, No. 8, pp. 54-57.

Lambert, Douglas and Mark Bennion (1986), “Establishing A Minimum Order Policy,” Journal ofBusiness Logistics, Vol. 7, No. 2, pp. 91-109.

Ramasesh, Ranga (1990), “Recasting the Traditional Inventory Model to Implement Just-In-TimePurchasing,” Production and Inventory Management Journal, Vol. 31, No. 1, pp. 71-75.

Schneiderjans, Marc and Qing Cao (2001), “An Alternative Analysis of Inventory Costs of JIT andEOQ Purchasing,” International Journal of Physical Distribution and Logistics Management, Vol.31, No. 2, pp. 109-123.


Schneiderjans, Marc and Qing Cao (2000), “A Note on JIT Purchasing vs. EOQ with a Price Dis-count: An Expansion of Inventory Costs,” International Journal of Production Economics, Vol. 65,No. 3, pp. 289-294.

Schwarz, Leroy B. and Z. Kevin Weng (1999), “The Design of a JIT Supply Chain: The Effect ofLeadtime Uncertainty on Safety Stock,” Journal of Business Logistics, Vol. 20, No. 1, pp. 141-164.

Stock, James R. and Douglas M. Lambert (2001), Strategic Logistics Management, 4th Ed. New York:McGraw-Hill Irwin.

Tyagi, Rajesh and Chandrasekhar Das (1997), “A Methodology for Cost versus Service Trade-offsin Wholesale Location-Distribution Using Mathematical Programming and Analytic HierarchyProcess,” Journal of Business Logistics, Vol. 18. No. 2, pp. 77-100.

Woolsey, Gene (1988), “A Requiem for the EOQ: An Editorial,” Production and Inventory Man-agement Journal, Vol. 26, No. 3, pp. 68-72.

Zinn, Walter, Howard Marmorstein, and John Charnes (1992), “The Effect of AutocorrelatedDemand on Customer Service,” Journal of Business Logistics, Vol.13, No. 1, pp. 173-192.

ABOUT THE AUTHORS

Walter Zinn, Ph.D. Michigan State University, is Associate Professor of Logistics and Mar-keting at The Ohio State University. He is a former Systems Section Editor of the Journal of Busi-ness Logistics. Dr. Zinn’s research has been published in academic journals such as the Journal ofBusiness Logistics, European Journal of Operational Research, Journal of the Operational ResearchSociety, and The International Journal of Logistics Management. Professor Zinn is a member of theCouncil of Supply Chain Management Professionals.

John M. Charnes, Ph.D. University of Minnesota, is Professor in the Finance, Economics, andDecision Sciences Area of the School of Business at The University of Kansas, where he doesresearch and teaches courses in risk analysis, systems simulation, and applied statistics. His web-site is www.ku.edu/home/jcharnes.


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