Re Modelling EOQ

Re-modelling EOQ and JITpurchasing for performance

enhancement in the ready mixedconcrete industries of Chongqing,

China and SingaporeWu Min and Low Sui Pheng

Department of Building, National University of Singapore, Singapore

Abstract

Purpose – To develop just-in-time (JIT) purchasing threshold value (JPTV) models for ready mixedconcrete (RMC) suppliers to decide whether or not to switch from an economic order quantity (EOQ)approach to a JIT purchasing approach for the purchase of their raw materials, when a price discountis offered.

Design/methodology/approach – The existing economic order quantity (EOQ) with a pricediscount versus the JIT purchasing cost comparative models neglect some important cost componentsunder the inventory management systems, for example, the out-of-stock costs and the impact ofinventory policy on product quality and production flexibility. In addition, these models do notempirically study the capability of an inventory facility to hold the EOQ-JIT cost indifference point’samount of inventory. These models suggest that the JIT purchasing approach is always preferred tothe EOQ approach when the JIT purchasing approach can capitalize on physical plant space reduction.The JPTV models developed in this study overcome the two limitations of the existing EOQ and JITpurchasing cost comparative models.

Findings – By developing the JPTV models, this study suggests that the theoretical advantages ofJIT purchasing may have been overstated.

Originality/value – The field studies conducted in the RMC industries in Chongqing, China andSingapore supported the propositions in this study. The JPTV models, if adopted, would help toenhance performance in the RMC industries in other cities as well.

Keywords Economic order quantities, Just in time, Purchasing, Concretes, China, Singapore

Paper type Research paper

1. IntroductionThe economic order quantity (EOQ) approach can be regarded as the conventionalmethod for purchasing materials. EOQ is the quantity of material within an order thatminimizes the total costs that are required to order and hold inventory (Harris, 1915;Peterson and Silver, 1979; Fazel, 1997). This approach leads to placing of large-sizedand infrequent orders (Schonberger, 1982). This approach was originallyconceptualized by Harris (1915) in 1915. Fazel et al. (1998) suggested that Harris’(1915) model can be modified to incorporate different price discount schemes to betterreflect the industry practice. The EOQ approach has been a fundamental technique forinventory management decisions and continues to be the starting point in the

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Received July 2004Revised November 2004Accepted December 2004

International Journal of Productivityand Performance ManagementVol. 54 No. 4, 2005pp. 256-277q Emerald Group Publishing Limited1741-0401DOI 10.1108/17410400510593811

development of many subsequent inventory purchasing models (Ray and Chaudhuri,1997; Charabarty et al., 1998).

Materials can also be purchased in a just-in-time (JIT) fashion (Low and Chan, 1997;Monden, 1998), which advocates smaller-sized and more frequent orders. Thisapproach was originally explored by Kanzler as a method for reducing inventory levelsat the Fordson Tractor Plant in the 1920s (Peterson, 2002). The JIT purchasing methodis an important technique of the JIT philosophy, which is regarded as one of the mostimportant productivity enhancement management innovations of the twentiethcentury in the manufacturing industry (Schonberger, 1982). The JIT approach providesthe right materials, in the right quantities and quality, just in time for production(Vokurka and Davis, 1996).

Although it was originally introduced in the manufacturing industry, theimplementation of the JIT purchasing approach has been extended beyond that. Itssuccessful implementation in many industries prompted many companies that are stillusing the EOQ approach to ponder whether they should switch to the JIT purchasingapproach for managing their inventories.

1.1. Research problemReady mixed concrete (RMC) is an important building material that is widely usedtoday in the construction industry (Anson et al., 2002; Wang et al., 2001). RMC mixingis a prototypical example of JIT construction processes (Tommelein and Li, 1999).Inspired by the success achieved through the implementation of the JIT philosophy inthe manufacturing industry, Wu and Low (2003) and Low and Wu (2005a, b) surveyedthe implementation status of JIT purchasing in the RMC industries in Singapore andChongqing, China. The surveys found that both the EOQ approach and the JITpurchasing approach were adopted by companies to manage the procurement of rawmaterials in the RMC industry.

The JIT purchasing approach results in an advantage of maintaining relativelysmaller inventories and thereby can reduce the plant space. Hence, the existing EOQwith a price discount versus JIT purchasing cost comparative models concluded thatthe JIT purchasing approach was always preferred to the EOQ approach, provided thatthe JIT purchasing approach could experience and capitalize on this physical plantspace reduction (Schniederjans and Cao, 2000). However, this conclusion was notconsistent with the industry practice, leading to a need to re-examine the presentlyavailable EOQ versus JIT purchasing cost comparative models. In a closer examinationof these models, it could be found that these models had two limitations. First, thesemodels did not theoretically nor empirically ascertain the capability of an inventoryfacility to hold the EOQ-JIT cost indifference point’s amount of inventory. TheEOQ-JIT cost indifference point is the level of demand at which the costs of the EOQwith a price discount system equal that of the JIT purchasing system. Inventoryfacility is defined as a physical plant space where raw materials, goods or merchandiseare stored. In the RMC industry, an inventory facility can be a storehouse, a warehouse,an aggregates depot, a sand yard or a cement terminal. Second, these models alsoignored some important cost components of the inventory purchasing systems. Thesecosts included, but not limited to, the impact of the inventory purchasing policy onproduct quality, production flexibility and organizational competitiveness (Zhu et al.,1994; Cheng and Podolsky, 1996) and out-of-stock costs (Johnson and Stice, 1993; Low

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257

and Chong, 2001; Singh, 2003;). Because of these two limitations, these models cannotclearly explain the inventory purchasing approaches adopted in the RMC industry.

1.2. Research aim and objectivesBased on the above background, the aim of this study is to develop JIT purchasingthreshold value (JPTV) models for a RMC supplier. These JPTV models envisageovercoming the two limitations of the presently available EOQ with a price discountversus JIT purchasing cost comparative models. The JPTV models help the RMCsupplier to decide whether the company should switch to the JIT purchasing approachif the RMC supplier is still adopting the EOQ approach to manage the procurement ofhis raw materials. In the process, this would help to enhance the overall performance ofthe RMC industry worldwide.

1.3. Research hypothesisThe research hypothesis for this study is that the EOQ approach is preferred to the JITpurchasing approach in managing material’s procurement when the additional costsresulting from JIT purchasing approach is high and when the annual demand isextremely low or high. This is evident even when the JIT operation can experience andcapitalize on physical plant space reduction.

1.4. Research strategy and assumptions in this studyThe research strategy of this study is to expand on only one of the assumptions of theplatform created earlier by Fazel et al. (1998) and Schniederjans and Cao (2000) todevelop the JPTV models for the RMC industry. It should be noted that the paper isbased on a study by Harris (1915), as the models of Schniederjans and Cao (2000) weredeveloped based on the models of Fazel et al. (1998), which were, in turn, developedfrom the work of Harris (1915). It should also be noted that the assumptions made byFazel et al. (1998) and Schniederjans and Cao (2000) were also presented in Fazel (1997)and Schniederjans and Cao (2001). This is because the first two studies were largelysimilar to the later two studies, except for the consideration of the price discount factor.

This study is based on nine assumptions as shown in Table I. The first eightassumptions were made by the previous researchers. Assumption No. 9 was made bythe authors. It assumes that the inventory physical storage costs under the EOQapproach, for example, rental, utilities and personnel salaries are linearly related to theaverage inventory level. Assumption No. 9 was expanded from assumption No. 3. Thisis to expand the “total carrying costs” in the classical EOQ model to include theinventory physical storage costs. This assumption is possible when selecting aninventory purchasing approach either for a new entrant for for an existing company.The rationale for this assumption is presented later.

2. Revised EOQ with a price discount model for the RMC industryWhile comparing the cost difference between the EOQ with a price discount versus JITpurchasing approach, Fazel et al. (1998) developed their EOQ-JIT cost indifferencepoint models which suggested that JIT purchasing was economical only when theannual demand was low (Schniederjans and Cao, 2000). Arguing that the inventoryphysical storage cost reduction resulting from JIT purchasing was ignored in themodels of Fazel et al. (1998), Schniederjans and Cao (2000) developed their own

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EOQ-JIT cost indifference point models. Both Schniederjans and Cao’s (2000) and Fazelet al.’s (1998) EOQ-JIT cost indifference point models were based on the classical EOQmodel and a price discount scheme proposed by Fazel et al. (1998). To develop theJPTV model for the RMC industry, the total annual cost under the classical EOQ modeland the price discount scheme proposed by the previous researchers are discussedbelow.

2.1. The total annual costs under the EOQ modelThe total annual costs under the classical EOQ model. Based on assumptions No. 1 toNo. 6 in Table I, the total annual cost of using an EOQ system for inventory ordering(TCE ) is the sum of the inventory ordering cost, inventory carrying cost, and the cost ofthe actual purchased units, and is given by:

TCE ¼kD

Qþ

Qh

2þ PED ð1Þ

where Q is the fixed order quantity, h is the annual cost of carrying one unit ofinventory in stock, k is the cost of placing an order, D is the annual demand for the

Assumptions made by previous researchers1 Ordering cost under the EOQ system is fixed per order (Harris, 1915)2 Carrying cost under the EOQ system for the inventory item is constant on a per

unit basis (Harris, 1915)3 Total carrying costs (excluding the inventory physical storage costs) under the

EOQ system are linearly related to the average quantity held in inventory(Harris, 1915)

4 Annual demand for the item is known and constant (Harris, 1915)5 The unit price per item under the EOQ or JIT purchasing system remain

constant (Fazel et al., 1998)6 Orders under the EOQ system are arranged in such a way that the succeeding

delivery arrives at the time that the quantity from the previous delivery hasjust been depleted (Harris, 1915); or “safety stock costs required in an EOQmodel are balanced out by other costs incurred in the use of a JITmodel”(Schniederjans and Cao, 2001, p. 112). Hence, safety stocks under theinventory systems are not considered

7 The order under the EOQ system are raised at the optimal economic orderquantity (Fazel et al., 1998)

8 The materials suppliers of the JIT companies produce and store their productsin large batches and respond to the JIT challenge by delivering them in smallqauntities. The carrying and ordering costs (e.g. storage, inspection,transportation, preparation of purchasing orders for each delivery, etc.) of theJIT company were mainly transferred to the material suppliers and solelyreflected in the unit price to the JIT company. Therefore, the unit price underthe JIT purchasing system, which includes the portion of carrying and orderingcosts that are passed onto the the buyer, is higher than the unit price under theEOQ system (Fazel et al., 1998)

Assumptions made by the authors9 The inventory physical storage costs under the EOQ system, for example,

rental, utilities and personnel salaries are linearly related to the averageinventory level

Table I.Assumptions made in the

study


259

item, PE is the purchase price per unit, D=Q is the annual ordering frequency, Q=2 isthe annual average inventory level in the inventory facility. The first ratio is theinventory ordering cost item. The second ratio is the inventory carrying cost item. Thelast item is the annual purchasing cost component. It should be noted that based onassumption No. 6 in Table I, which was agreed by Fazel et al. (1998) and Schniederjansand Cao (2000), the costs of not having stock are not considered in Equation (1).Although the risk of no stock or late delivery and the cost associated with it can varyfrom place to place and between industries, this still needs to be recognized and will beconsidered later.

It should also be noted that although Fazel et al. (1998) and Schniederjans and Cao(2000) used the term “the total annual cost of an inventory item under an EOQ system”to refer to “TCE” in Equation (1), the so called “fixed cost”, for example, rental, utilities,and personnel salary, were excluded from the inventory carrying cost item. This wasan important assumption agreed by Fazel et al. (1998), and Schniederjans and Cao(2000). This is assumption No. 3 in Table I. This assumption was expanded to becomeassumption No. 9 in Table I, that is to say, to expand the “total carrying costs” in theclassical EOQ model to include the inventory physical storage cost.

The extension of assumption No. 3 to assumption No. 9 is feasible when selecting aninventory purchasing approach either for a new entrant or for an existing companywhich is considering switching from the EOQ approach to the JIT purchasingapproach. For a new entrant, the amount of inventory facility, utilities and the numberof logistics staff can be easily adjusted based the average inventory level, because theinventory facilities have not been built yet. For an existing EOQ company, the excessinventory facility can be rented out if this company switches to a JIT purchasingapproach as observed by Schniederjans and Cao (2000). The logistics staff may bereallocated to other posts.

It should be noted that the inventory physical storage cost under the EOQ system isnot necessarily a linear function with the average inventory level (this is assumptionNo. 9). For instance, let us put ourselves in the position of a warehouse man storing cubicboxes. All the boxes are identical. The area of each surface of each box is 1m2. Let us lookat a simple scenario of three boxes and then extend it to a more complicated scenario of nboxes. For three boxes, the total floor area occupied may be 1m2, 2m2 or 3m2. This isshown in Figure 1. For n boxes, the total floor area occupied may be 1m2, 2m2, 3m2 . . .n2 1ð Þm2, or nm 2. This is shown in Figure 2. Hence, the total floor area taken up by n

boxes does not have to be nm2, which is the logical extension of assumption No. 9.Figures 1 and 2 show that this assumption is biased against the high inventory levelEOQ system if the annual cost of holding one unit of inventory under the high inventory

Figure 1.Floor area occupied bythree cubic boxes

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level EOQ system is assumed to be the same as that under the low inventory level EOQsystem, as the n boxes may be stored in a floor area that is smaller than nm2.

Indeed, a higher and more compact inventory facility is usually built to save floorarea and reduce the inventory physical storage costs in the RMC industry when theaverage inventory level increases. This is because the inventory physical storage costsper unt in a higher and more compact inventory facilty are usually lower than that inthe smaller sized inventory facility, provided that the other conditions, such as rentalrate, utility rate and labor rate remain unchanged.

Hence, the assumption made by the authors (assumption No. 9) tended to be quiteconservative for the high inventory level EOQ system when compared with a JITpurchasing system, provided that the annual cost of holding one unit of inventory instock in the high inventory level EOQ system is determined based on the average valueof inventory facilities or that of a low inventory level EOQ system. This study showsthat it is still possible for the EOQ system to be more cost effective than the JITpurchasing system even when the JIT operation can experience physical plant spacereduction; and an unfavourable assumption is made against the EOQ system.Definitely, Gaither (1996) also suggested that the inventory physical storage costsshould be included as a component of the “total carrying costs”.

The extension of assumption No. 3 in Table I to assumption No. 9 suggests thatthere are reasons to include all components of inventory carrying costs into “h”, whencomparing an EOQ system with a JIT purchasing system to select the inventorypurchasing policy. Nevertheless, it is essential to note that the amount of inventoryfacility and the number of logistics staff are usually fixed during the operation stageunder the EOQ system. Hence, the extension of “h” may not be possible during theoperation stage. Therefore, the models developed in this study are suitable only forselecting an inventory purchasing approach.

The total annual costs under the revised EOQ model. For the purpose of comparingthis study with that of Fazel et al.’s (1998) and Schniederjans and Cao’s (2000) models,the same terminology “the total annual cost using an EOQ system for inventoryordering” is still used in this study to refer to “TCE”. The “expanded annual cost ofcarrying one unit of inventory” under the revised EOQ model is symbolized as “H” inthe later discussion. When “h” is expanded to become “H”, the total costs under therevised EOQ model (TCEr) are given by:

Figure 2.Floor area occupied by n

cubic boxes


261

TCEr ¼kD

Qþ

HQ

2þ PED ð2Þ

Since “H” in Equation (2) is greater than “h” in Equation (1), TCEr is also greater thanTCE in Equation (1) in the classical EOQ model.

2.2. Price discount schemeExisting price discount models. The cost incurred by suppliers is usually a decreasingfunction of the size of the delivery lot; the delivery price of an inventory is thus usuallya decreasing function of the order quantity (Fazel et al., 1998). Goyal and Gupta (1989)concluded that there were three basic types of price discount models, namely:

(1) two-part tariff;

(2) two-block tariff; and

(3) all unit quantity discount.

In the two-part tariff price discount model, “the buyer is required to pay a fixed chargeand a uniform price p for all units purchased. Although the buyer pays the samemarginal unit price for all quantities, the average price paid is a monotonicallydecreasing function of quantity purchased” (Goyal and Gupta, 1989, p. 263). In thetwo-block tariff price discount model, “the price of a unit, p1, is maintained up to aquantity x, the per unit price p2 is charged for all units in excess of quantity (p1f p2)”(Goyal and Gupta, 1989, p. 263). In the all unit quantity discount model, when “a buyerbuys less than a quantity x, the price of all units is decreased” (Goyal and Gupta, 1989,p. 263). Britney et al. (1983a, b), Dolan (1987) and Wilcox et al. (1987) suggested severalvariations within each of the three basic types of price quantity discount models.

Fazel et al.’s (1998) price discount scheme. The price discount scheme proposed byFazel et al. (1998) had two assumptions:

(1) For quantities below a certain level (Qmax) the delivery price was a decreasing,continuous and linear function of the order quantity.

(2) Beyond Qmax, however, the price stayed at its minimum (PminE ), which was the

lowest price the supplier would charge, no matter how large the order quantitywas.

The discount functions are mathematically defined as:

PE ¼

P0E ::::::::::::::::::::::::::::::::::::::::::::::::::::Q ¼ 0

P0E 2 pEQ:::::or:::::::

dPE

dQ¼ 2pE :::::::Q # Qmax

PminE :::::::::::::::::::::::::::::::::::::::::::::::::QfQmax

ð3Þ

whereP0E is the purchase price per unit when the order quantity equals zero, pE is a

constant representing the quantity discount rate. This price discount scheme was onevariation of the all unit quantity discount model, as the buyer paid the same unit pricefor every unit purchased (Fazel et al., 1998). It seems that this price discount schememay not fit well into the reality in the RMC industry on at least two counts. First, theinitial condition for the first-order differential equation:

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262

dPE

dQ¼ 2pE

in the price discount model (P0E ), may not even be a feasible price, as there may be a

minimum order size so that an infinitesimally small order size is not possible in theRMC industry. The lowest order quantity (Qmin) which one can order must be one withan unit price of P1

E , if the inventory is not divisible, for example, one truck of bulkPortland cement. Second, the intensive site studies carried out by the authors and thein-depth discussions with the general manager of a RMC supplier in Chongqing, Chinaand the production manger of a RMC supplier in Singapore suggested that a greaterprice discount can usually be offered by the raw materials suppliers, if the orderquantity was to be increased. There is no such things as Pmin

E . Nevertheless, there was amaximum order quantity limit for an individual RMC supplier. The maximum orderquantity limit for each individual RMC suppliers was usually governed by either theinventory carrying capacity of the inventory facility in the RMC batching plant or theproduction capacity of the raw material supplier, whichever was the lesser. Thegeneral manager of the RMC supplier in Chongqing and the production manger of theRMC supplier in Singapore made their suggestions based on their working experiencein the RMC industry.

A price discount scheme for the RMC industry. Based on the above background, tobetter reflect the realities in the RMC industries, a new price discount scheme issuggested. This price discount scheme is revised from that of Fazel et al.’s (1998) modeland is given below. It assumes that the delivery price per unit starts from PQmin

E , whereQmin is the lowest order quantity that can be placed. The delivery price is then adecreasing, continuous, and linear function of the order quantity for the rest of the orderquantities that are below a certain level (Qmax). Qmax is determined by the inventorycarrying capacity of the RMC supplier and the production capacity of the raw materialsupplier, whichever is the lesser. This discount scheme is graphically shown in Figure 3.

The discount function can be mathematically presented as:

PE ¼ PQmin

E 2 pE Q2 Qminð Þ forQmin # Q # Qmax ð4Þ

where pE is a constant.

2.3. Revised EOQ with a price discount model for the RMC industryThe revised EOQ with a price discount model for the RMC industry is developed byincorporating the price discount scheme given in Equation (4) into the revised EOQmodel. The total annual costs of the revised EOQ with a price discount model for theRMC industry is thus given by:

TCEr ¼kD

Qþ

QH

2þ PQmin

E þ pEQmin 2 pEQ� �

D forQmin # Q # Qmax ð5Þ

The optimum order quantity which minimizes the total cost under the revised EOQwith a price discount system (Q*

r ) can be derived by taking the first order derivativewith respect to Q of Equation (5) and setting it to zero, and is given by:


263

Q*r ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2kD

H 2 2pED

sforQmin # Q # Qmax ð6Þ

Equation (6) results in the total annual optimal cost under the revised EOQ with a pricediscount system as:

TCEr ¼ kD

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiH 2 2pED

2kD

rþ

H

2


H 2 2pED

s

þ PQmin

E þ pEQmin 2 pE


H 2 2pED

s" #D for Qmin

# Q # Qmax ð7Þ

Equation (7) is valid only when the inventory is ordered at its optimal economic orderquantity.

3. JPTV models3.1. EOQ-JIT cost difference function under the revised EOQ modelBased on assumption No. 8 in Table I, Fazel et al. (1998) suggested that the total annualcost under the JIT purchasing system (TCJ ) was the product of the unit price under theJIT purchasing system (PJ ) and the annual demand (D), and was given by:

TCJ ¼ PJD ð8Þ

As suggested earlier, this proposition did not consider the additional costs and benefitsresulting from JIT purchasing. Hence, the total annual cost under the JIT purchasingsystem for the JPTV models (TCJr) is proposed to be the product of the unit price under

Figure 3.The EOQ price discountscheme proposed for theRMC industry

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the JIT purchasing system (PJ ) and the annual demand (D) plus the additional costsand benefits resulting from JIT purchasing, and is given by:

TCJr ¼ PJD þ j ð9Þ

where j is the sum of the additional costs and benefits resulting from the JITpurchasing approach. The additional costs are mainly the increased out-of-stock costswhen inventory is purchased in a JIT fashion. The additional costs of JIT purchasingcontribute positively to j. The additional benefits are mainly improved quality,flexibility of production, reduced waste and increased organizational competitivenessunder the JIT purchasing system. The additional benefits of JIT purchasing contributenegatively to j.

It is essential to note that the cost of the inventory physical plant space reduction inthe JIT purchasing system has been assumed to take its maximum value and isconsidered in the total annual optimal cost under the revised EOQ system. This isbecause the maximum value of the inventory physical plant space reduction under theJIT purchasing system is the inventory physical plant space under the EOQ system;and the inventory physical plant space under the EOQ system has been considered as acomponent of inventory carrying costs and included in “H” in Equation (2). Theinventory physical plant space reduction was highlighted by Schniederjans and Cao(2000).

The total annual optimal cost under the revised EOQ with a price discount model(TCEr) has been given in Equation (7). Hence, the difference between the total annualoptimal cost under the revised EOQ with a price discount system and the total annualcost under the JIT purchasing system in Equation (9) (Zr) is:

Zr ¼ kD


2kD

rþ

H

2


H 2 2pED

s

þ PQmin

E þ pEQmin 2 pE


H 2 2pED

s" #D2 PJD2 j for Qmin

# Q # Qmax

ð10Þ

Setting Zr to zero, the roots of Zr Dð Þ ¼ 0 are the revised EOQ-JIT cost indifferencepoints (Dindr1 and Dindr2), given by:

Dindr1 ¼

kH 2 j PJ 2 PQmin

E 2 pEQmin

� �2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik 2H 2 2 2kHj PJ 2 PQmin

E 2 pEQmin

� �2 4pEkj 2

r

PJ 2 PQmin

E 2 pEQmin

� �2þ4pEk

forQmin # Q # Qmax

ð11Þ

where Dindr1 is the lower EOQ-JIT cost indifference point under the revised EOQ with aprice discount system for the RMC industry.

(11)


265

Dindr2 ¼

kH 2 j PJ 2 PQmin

E 2 pEQmin

� �þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik 2H 2 2 2kHj PJ 2 PQmin

E 2 pEQmin

� �2 4pEkj 2

r

PJ 2 PQmin

E 2 pEQmin

� �2þ4pEk

forQmin # Q # Qmax

ð12Þ

where Dindr2 is the upper EOQ-JIT cost indifference point under the revised EOQ with aprice discount system for the RMC industry. The models in Equation (11) and Equation(12) are the JPTV models developed for the first time in this study.

4. DiscussionEquation (11) and Equation (12) indicate that the sum of the additional costs andbenefits resulting from JIT purchasing (j) is an important factor that affects the lowerand upper EOQ-JIT cost indifference points. It should be noted that j varies fromindustry to industry and it may not be a constant even when the annual demandremains unchanged for the same manufacturers. However, there can be three scenariosfor j.

(1) j is equal to zero;

(2) j is greater than zero; and

(3) j is less than zero.

Hence, the discussion on the lower and upper EOQ-JIT cost indifference points ispresented below for the three scenarios.

4.1. j is equal to zeroWhen j equals zero, Equation (11) suggests that Dindr1 equals zero and Equation (12)suggests Dindr2 is positive. Furthermore, when j equals zero and when the pricediscount rate (pE ) is insignificant, it can be proved that the EOQ system is more costeffective than the JIT purchasing system, provided that the annual demand is greaterthan the upper EOQ-JIT cost indifference point (Dindr2). pE can be small, for instance,the price discount rate given in the hypothetical examples by Fazel et al. (1998) andSchniederjans and Cao (2000) was 4x1024 only.

When j equals zero, Dindr2 can be simplified as:

2kH

PJ 2 PQmin

E 2 pEQmin

� �2

þ4pEk

If Qminis also equal to zero, Dindr2 can be further simplified as:

2kH

PJ 2 POE

� �2

þ4pEk

(12)

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266

where:

P0E ¼ PQmin

E þ pEQmin

Let Dindr represent

2kH

PJ 2 POE

� �2þ4pEk

The cases in which both j and Qmin are equal to zero was the scenario that had beenstudied by Fazel et al. (1998) and Schniederjans and Cao (2000). Hence, it is essential tocompare the present study with Fazel et al.’s (1998) and Schniederjans and Cao’s (2000)models.

A comparison of the present study with that of Fazel et al.’s (1998) model

2kH

PJ 2 POE

� �2

þ4pEk

is similar to

2kh

PJ 2 POE

� �2

þ4pEk

¼ DindF

the EOQ-JIT cost indifference point proposed by Fazel et al. (1998). DindF can be derivedby substituting PE in Equation (1) with the second equation in Equation (3) and settingTCE in Equation (1) to equal TCJ in Equation (8). Since H is significantly greater thanh, the revised EOQ-JIT cost indifference point (Dindr2) is thus significantly greater thanthe EOQ-JIT cost indifference point proposed by Fazel et al. (1998) (DindF ), assumingthat the values of the other parameters, namely, k, PJ , P

0E and pE remain unchanged.

This is because some inventory physical storage costs were not accounted for in DindF

(Schniederjans and Cao, 2000). This finding suggests that the JIT purchasing systemcan still be cost effective even at a high level of annual demand, thus invalidating Fazelet al.’s (1998) conclusion that JIT purchasing was cost effective only at lower levels ofannual demand.

A comparison of the present study with that of Schniederjans and Cao’s (2000)model. The concept of the carrying capacity of an inventory facility can assist tocompare the present study with Schniederjans and Cao’s (2000) model. The carryingcapacity of an inventory facility is defined as the amount of inventory that can be heldby the inventory facility at a specific time assuming each unit of inventory item takesup a fixed amount of inventory facility (am2). Mathematically, it is the ratio betweenthe floor area of an inventory facility (NE ) and the area occupied by a unit of inventory(a), and can be presented as:

Qh ¼NE

að13Þ

where Qh is the carrying capacity of an inventory facility. In practice, to allow forflexibility, the size of the inventory facility is usually designed to be greater than the


267

size needed to hold the exact amount of optimal economic order quantity of inventory.Based on assumption No. 9 in Table I, it is reasonable to assume that the size of theinventory facility is b times the size which holds the optimal economic order quantityamount of inventory, which can be presented as:

Qh ¼ bQ*r ð14Þ

where b is the stock flexibility parameter and is greater than or equal to 1. It isimportant to note that the lower out-of-stock cost is the major issue in keeping firms onEOQ! Hence, b is often greater than 1. Substituting Qh in Equation (14) into Equation(13), would result in:

NE ¼ abQ*r ð15Þ

Equation (15) is the formula of the floor area of an inventory facility determined by theinventory optimal economic order quantity. Substituting Dindr into Equation (6), Q*

rindthe optimal economic order quantity at the revised EOQ-JIT cost indifference point(Dindr ) can be derived as:

Q*rind ¼

2k

PJ 2 P0E

ð16Þ

Substituting Q*rind for Q*

r in Equation (15), would result in the minimum area of theinventory facility under the revised EOQ with a price discount system that can housethe EOQ-JIT cost indifference point’s amount of inventory (NEdind) as:

NEdind ¼2abk

PJ 2 P0E

ð17Þ

Hence, the total cost under the revised EOQ with a price discount system will beequal to the total cost under the JIT purchasing system, provided that the floor areareaches

2abk

PJ 2 P0E

;

and the annual demand reaches

2kH

PJ 2 P0E

� �2

þ4pEk

;

and the order quantity is below Qmax. Meanwhile, the physical inventory plant spaceunder the revised EOQ with a price discount system can accommodate the EOQ-JITcost indifference point’s amount of inventory. For example, if F represents the annualcost to own and maintain a square meter of physical inventory plant space, 2abF isthen a component of H .

The above analysis suggests that Schniederjans and Cao (2000) seem to haveoverlooked that it was possible for an inventory facility to hold the EOQ-JIT costindifference point’s amount of inventory when the floor area of an inventory facility

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reach NEind . Hence, another expression of this finding is that an EOQ based system canbe more cost effective than a JIT purchasing system when the size of the inventoryfacility is above NEind , and the magnitude of the annual demand is above the EOQ-JITcost indifference point Dindr , and the optimal economic order quantity is less than Qmax.As suggested earlier, NEind is the minimum area of the inventory facility that canaccommodate the EOQ-JIT cost indifference point’s amount of inventory under therevised EOQ with a price discount system.

4.2. j is greater than zeroWhen the additional costs of JIT purchasing are greater than the benefits of JITpurchasing, Equations (11) and (12) have three implications. The first is that the JITpurchasing approach may not be cost effective when the annual demand is low. This issuggested by Equation (11). The second is that the upper EOQ-JIT cost indifferencepoint (Dindr2) shifts to a lower value that is less than:

2kH

PJ 2 P0E

� �2

þ4pEk

:

This is suggested by Equation (12). The third is that the EOQ approach can always bepreferred to the JIT purchasing approach if the additional costs of JIT purchasing aresubstantially high. The analysis of the third implication is presented below.

Comparing Equation (11) with Equation (12), it can be found that the condition forthe lower cost indifference point (Dindr1) and the upper cost indifference point (Dindr2)are real is that the additional costs and benefits resulting from JIT purchasing (j) isbelow a specific value jð Þlimit , or:

j # jð Þlimit

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik 2H 2 PJ 2 PQmin

E 2 pEQmin

� �2

þ4pEk 3H 2

r2 kH PJ 2 PQmin

E 2 pEQmin

� �4pEk

ð18Þ

When j is greater than jð Þlimit , Dindr1 and Dindr2 are not feasible cost indifference points.In such a scenario, an EOQ system is always preferred to a JIT purchasing system.

4.3. j is less than zeroSetting Zr in Equation (10) to zero, Equation (10) can be rewritten as:

PJ 2 PQmin

E 2 pEQmin

� �D þ j ¼ kD


2kD

r

þH 2 2pED

2


H 2 2pED

sfor Qmin

# Q # Qmax

ð19Þ


269

When the additional benefits resulting from JIT purchasing, such as increased productquality and production flexibility, are considered, it is possible for j to be less thanzero. When j is less than zero and the price discount rate pE is insignificant, Dindr1 isnot a feasible cost indifference point. This is because, in such a scenario, it can beproved that the left hand side of Equation (19) is less than zero and the right hand sideof Equation (19) is greater than zero if the annual demand is equal to Dindr1 (Equation(11)). However, Dindr2 is still a feasible EOQ-JIT cost indifference point. Furthermore, itcan be proved that when j is less than zero, Dindr2 is greater than:

2kH

PJ 2 P0E

� �2

þ4pEk

:

This further suggests that JIT purchasing can probably be adopted in a much widerrange than that stipulated by the EOQ-JIT cost indifference point suggested by Fazelet al. (1998), provided that the benefits reaped from JIT purchasing, such asimprovements in quality and flexibility in production are greater than the increasedout-of-stock costs.

4.4. Three implications derived from the JPTV modelsThe JPTV models suggest that the EOQ and JIT purchasing approach are preferreddepending on the market conditions available in the locality. The JPTV models canhave three implications. First, when the additional costs of JIT purchasing cannot beignored and the annual demand is low, JIT purchasing may be an expensivealternative. Second, when the additional costs of JIT purchasing are substantially high,the EOQ purchasing approach may always be more cost effective than the JITpurchasing approach. Third, when the sum of the additional costs and benefitsresulting from JIT purchasing are insignificant, the JIT purchasing approach can bewidely adopted. However, it is still possible for the EOQ approach to be more costeffective than the JIT purchasing approach, provided that the annual demand issubstantially high. The components of the additional costs and benefits of JITpurchasing have been explained earlier. The development of the JPTV modelstheoretically suggests that the research hypothesis can be supported.

5. Case studyTemponi (1995) and Tommelein and Li (1999) observed that small companies usuallyhave difficulties in implementing JIT purchasing. There were two possible reasons forthis. The first is that the material supplier may not have an interest to act in a JITfashion when the annual demand was low. This is because the low annual demand ofthe buyer might mean insignificant profit reaped by the material supplier. The secondis that the material supplier might have difficulties in delivering the material to thesmall buyer in a JIT fashion, as his facilities have to be arranged to serve the largerbuyer, leading to high out-of-stock costs faced by the small buyer. Through surveys,Low and Wu (2005a, b) confirmed that because of the high out-of-stock risk resultingfrom JIT purchasing, none of the 20 registered RMC suppliers of the ChongqingConcrete Association procured their sand and aggregates in a JIT fashion. While thefirst and second implications of the JPTV models were supported by the studies ofTemponi (1995), Tommelein and Li (1999) and Low and Wu (2005a, b); the third

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implications of the JPTV models can be verified by a case study with information fromthe RMC industry in Singapore.

5.1. BackgroundThis case study pertained to cement purchasing by the cement division of a RMCsupplier in Singapore. The group of the company, in which this RMC supplier was asubsidiary, was incorporated in April 1973 and gained a listing in the Singapore StockExchange in 1979. The business scope of the group included the sale of RMC, dry mixmortar products and cement.

The cement division of this RMC supplier built two huge silos on Pulau Damar LautIsland and adopted an EOQ system to procure its cement from Japan. The designedcarrying capacity of each of the silos was approximately 25,000 tons. The average areaoccupied by one silo was about 2,800m2. The sum of the carrying capacities of the twosilos was approximately 40,000 tons (where safety and flexibility factor had beenconsidered. Stock flexibility parameter was b ¼ 1.25). Cement imported by the cementdivision of this RMC supplier was mainly transported using 40,000-ton cement carriers.This company placed an order approximately once a month for about 40,000 tons ofcement. The annual demand (D) was 520,000 tons in 2003.

The annual cost of carrying one ton of cement (H ) was S$ 344/ton. The carrying costcould be split into cement check-in cost (hcheckin), cement storage costs (hstorage1, hstorage2and hstorage3) and cement check-out cost (hcheckout). The cement check-in cost was thedepreciation and operating cost of the facilities to unload cement from a cement carrier toa silo and hcheckin ¼ S$199 /year per ton. The cement storage costs were hstorage1 ¼ S$8/year per ton, hstorage2 ¼ S$ 14/year per ton, and hstorage3 ¼ S$ 23/year per ton. The cementstorage costs hstorage1 include insurance, cement spoilage cost and opportunity cost of theworking capital tied up with the cement purchased. The average interest rate ofgovernment bonds in Singapore in the year 2003 was used to compute the opportunitycost of the working capital tied up with the cement purchased. The cement storage costshstorage2 include the depreciation cost of the silo facilities, utilities, personnel salary andproperty tax. The cement storage costs hstorage3 include the land rental (2abF).

The cement check-out cost was the depreciation cost and operating cost of thefacilities, mainly cement trucks, to deliver cement from the silo to a RMC batchingplant and hcheckout ¼ S$ 100/year per ton. The cost of placing an order from Japan forthe 40,000 cement carrier was k ¼S$ 432,000 for transportation alone. Each ton ofcement took up a ¼ 0.112m2 of the inventory facility space. The annual cost to rent asquare meter of inventory facility was F ¼ S$ 84. If cement was purchased under a JITsystem in Singapore, the cost was PJ ¼ S$ 69/ton. The additional costs and benefitsresulting from JIT purchasing (j) was estimated to be zero. There were two reasons forthis. The first is that the transportation infrastructure system in Singapore was welldeveloped. Hence, the additional out-of-stock cost resulting from JIT purchasing ofcement from the local market was insignificant. The second is that although JITpurchasing might increase production flexibility and reduce waste, the stockpile oflarge amounts of cement in the EOQ system might help the cement division of the RMCsupplier to control the local market. The order was often raised one or two monthsbefore the departure of a cement carrier from Japan. The Japanese cementmanufacturers offered a few alternative pricing strategies. One pricing strategy wassuitable for the EOQ with a price discount system. Under this pricing strategy, the


271

lowest order quantity was Qmin ¼ 20,000 tons. The delivery price started at PQmin

E ¼ S$43.5/ton. For every additional ton ordered, the price would decrease bypE ¼ S$7:5 x1025 for the entire order lot. The discount was valid for orderquantities up to Qmax ¼ 50,000 tons, which was the maximum carrying capacity of thecement division of this RMC supplier.

It is essential to note that the information for this case study was collected throughinterviews with the overseas investment manager, the financial manager, theproduction manager and the customer service supervisor of the cement division of thissupplier in November 2003. It is also important to note that although the cementdivision of this RMC supplier ordered its cement in an EOQ fashion, the accountingsystem adopted by them did not exactly follow the EOQ approach. However, it wassuggested that the cost information could be structured as above to fit the EOQ model.In addition, the value of each parameter was the estimated average value. At this point,it should be noted that the examples given by Fazel et al. (1998, p. 107), andSchniederjans and Cao (2000, p. 291) were hypothetical.

5.2. EOQ-JIT cost indifference points derived from the JPTV modelsTo compare the present study with the study of previous researchers and to make theproblem simple, demand variability and safety stock were not considered in this casestudy. In addition, it was found that each ton of cement took up approximately 0.115m2

of the 100-ton silo and approximately 0.109m2 of the 25,000-ton silo. The annual cost ofholding one ton of cement in a 100-ton silo was slightly higher than S$344 and theannual cost of holding one ton of cement in a 25,000-ton silo was slightly lower thanS$344/year per ton. This shows that the annual cost of holding one ton of cement insilos can roughly be assumed to be a constant. Assumption No. 9 can roughly be met.Each order of cement was delivered by the 40,000 ton cement carrier. Hence, theordering cost under the EOQ system can be assumed to be fixed per order. The routineorder quantity of cement in the cement division of this RMC supplier was 40,000tons/order. This figure was between the minimum order quantity (20,000 tons/order)and the maximum order quantity (50,000 tons/order) under the revised EOQ with aprice discount model. Equation (6) was, therefore, suitable for computing the optimaleconomic order quantity. According to Equation (6), the optimal economic orderquantity was Q*

rd ¼ 41,097 tons/order. The optimal economic order quantity was closeto the routine order quantity. Hence, Equations (11) and (12) can be employed to deriveEOQ-JIT cost indifference points.

The lower EOQ-JIT cost indifference point (Dindr1) computed from Equation (11)was zero, as j was equal to zero. The upper EOQ-JIT cost indifference point (Dindr2)computed from Equation (12) was 421,224 tons. It should be noted that the EOQ-JITcost indifference point could be shifted to a value that was lower than 421,224tons/year, provided that j was greater than zero (for example, the additional costsresulting from JIT purchasing cannot be ignored); or it might be shifted to a value thatwas higher than 421,224 tons/year, provided that j was less than zero (for example, theimproved product quality and production flexibility resulting from JIT purchasing arecounted). Since the annual cement demand of the cement division of the RMC supplierIS (520,000 tons) was greater than the EOQ-JIT cost indifference point (421,224 tons),the cement needs to be ordered in the EOQ fashion at a lower price, rather than in theJIT fashion but at a higher price.

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A survey covering all the 15 registered members of the Ready-Mixed ConcreteAssociation of Singapore showed that five of the large companies purchased theircement by using the EOQ approach (Wu and Low, 2003). Besides the supply of cementto their own RMC batching plants, these companies also sold cement to the localconstruction market. The other ten RMC suppliers with a relatively smaller annualdemand purchased their cement in a JIT fashion.

5.3. A comparison of the EOQ-JIT cost indifference pointsIf the annual cost of holding one unit of cement (H ) in Equation (12) was replaced bythe cement storage costs (hstorage1); and the additional costs and benefits resulting fromJIT purchasing was set to zero, and let P0

E ¼ PQmin

E 2 pEQmin, then Equation (12) wasconverted to be the formula for computing the EOQ-JIT cost indifference point asproposed by Fazel et al. (1998, p. 106). According to Fazel et al.’s (1998, p. 106) formula,the EOQ-JIT cost indifference point for cement purchasing was 9,796 ton/year.

Each ton of cement occupied at least 0.1m2 of the silo. Hence, JIT purchasing ofcement can take advantage of inventory physical plant space reduction. Based on theirown models, Schniederjans and Cao (2000) argued that when JIT operations couldcapitalize on inventory physical plant space reduction, “a JIT ordering system ispreferable to an EOQ system at any level of annual demand and with almost any coststructure” (Schniederjans and Cao, 2000, p. 294). Hence, the EOQ-JIT cost indifferencepoint would be þ1. The EOQ-JIT cost indifference points worked out from the modelsproposed by Fazel et al. (1998), Schniederjans and Cao (2000) and the analysis from thispaper are shown in Table II.

The batching capacity of the widely used batching plant was 90m3/hr in Singapore.Based on the estimation of the production manager of the RMC supplier in the casestudy, the average demand for cement of the 90m3/hr batching plant wasapproximately 40,500 ton/year in 2003. Each RMC supplier had at least one RMCbatching plant. Hence, the annual demand for cement of each of the RMC suppliersurveyed was at least 40,500 ton/year. This figure was significantly greater than theEOQ-JIT cost indifference point worked out from the models proposed by Fazel et al.(1998). Hence, the EOQ-JIT cost indifference point derived from the model of Fazel et al.(1998) suggests that all the RMC suppliers should operate in an EOQ fashion. However,as observed by Wu and Low (2003), ten RMC suppliers were purchasing their cementin a JIT fashion. These RMC suppliers used a number of 100-ton silos to store theirbuffer stock, as shown in Plate 1. The 100-ton silos were filled on a daily basis. Hence,the EOQ-JIT cost indifference point derived from the model of Fazel et al. (1998) wasnot supported by the cement purchasing process in the RMC industry in Singapore. Inthe meantime, the EOQ-JIT cost indifference point model proposed by Schniederjansand Cao (2000) suggested that all the RMC suppliers should operate in a JIT fashion, ascement purchasing can take advantage of physical plant space reduction. However, as

Fazel et al.’s

(1998)

Fazel et al.’s

(1998) model

Schniederjans and Cao

(2000) ’s model Authors’ model

EOQ-JIT cost indifference

point 9,796 (ton/year) þ1(ton/year) 421,224 (ton/year)

Table II.A comparison of the

EOQ-JIT cost indifferencepoints for cement

purchasing in the RMCindustry in Singapore


273

suggested earlier, the cement divisions of five big RMC suppliers were purchasing theircement in an EOQ fashion. To reap economies of scale, the cement division of theseRMC suppliers built a number of huge multi-cells silos on the Pulau Damar Laut island,as shown in Plate 2. Hence, the EOQ-JIT cost indifference point derived from the modelof Schniederjans and Cao (2000) was not supported by the cement purchasing processin the RMC industry in Singapore either. The EOQ-JIT cost indifference point proposedin this paper was supported by the case study.

This case study suggested that when the sum of the additional costs and benefits ofJIT purchasing were insignificant, the JIT purchasing approach could be widelyadopted. However, it was possible for the EOQ approach to be more cost effective thanthe JIT purchasing approach, provided that the annual demand was substantially high.This is the third implication of the JPTV models.

6. Conclusion and limitation of the studyThe research hypothesis formulated in this study is that the JIT purchasing approachis not always preferred to the EOQ approach in managing materials procurement, evenwhen the JIT operation can experience and capitalize on physical plant space reduction.This is the opposite of the prevailing understanding, established by previous studies.

The hypothesis has been developed from both theoretical and empirical grounds.The development of the JPTV models established the theoretical framework, while thecase studies in the RMC industries in Singapore provided the empirical ground to testthe hypothesis. The verification of the hypothesis suggests that the theoretical

Plate 1.100-ton cement silos

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274

advantages of JIT purchasing may have been overstated in the currently availableliterature, especially when the firms are purchasing to meet high and consistent levelsof demand and the JIT operation can take advantage of inventory physical plant spacereduction. The JPTV models developed in this study can be helpful to RMC suppliers toselect the purchasing approaches for their raw materials.

The technologies of concrete admixtures are often beyond the control of a RMCsupplier (Grace Construction Product, 2003). Hence, the focus of the JPTV models is onthe purchase of cement, sand and aggregates in the RMC industry. Although theobservations were made from Chongqing, China and Singapore, the general approachin this paper could be adopted in other countries. The models might be modified, asthere can be differences in other RMC markets, due to climate, social, economic andpolitical conditions. The JPTV models developed for the RMC industry can also haveimplications on the procurement of materials in other industries in which theconditions of the JPTV models can roughly be met, for example, the purchase ofliquefied petrochemical gas (LPG) by underground LPG terminal projects, and theorder of spare parts in helicopter companies and off-shore oil rigs. The JPTV modelscan also explicitly explain the difficulty in implementing JIT purchasing in smallcompanies whose annual demands are low (Temponi, 1995; Tommelein and Li, 1999) orin the developing countries where transportation infrastructures are under developedand the additional out-of stock costs cannot be neglected (Tanvir, 1986; Low and Wu,2005a, b). It is hoped that the models presented in this study would help to furtherenhance the performance of the RMC industry, as well as other industries, wherematerials procurement is concerned.

Plate 2.25,000-ton cement silos


275

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Documents

Re Modelling EOQ