cost benefits from standardization of the packaging glass bottles

Computers & Industrial Engineering 62 (2012) 693–702

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Computers & Industrial Engineering

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Cost benefits from standardization of the packaging glass bottles

Young Dae Ko, Injoon Noh, Hark Hwang ⇑Department of Industrial and Systems Engineering, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Republic of Korea

a r t i c l e i n f o a b s t r a c t

Article history:Available online 3 December 2011

Keywords:Glass bottleInventory poolingReverse logisticsStandardization

0360-8352/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.cie.2011.11.026

⇑ Corresponding author. Tel.: +82 42 350 3113; faxE-mail address: [email protected] (H. Hwang

This study deals with a recycling system with two competing brewers. It is assumed that they coordinatetheir manufacturing operations through standardization of their glass bottles for easy implementation ofextended producer responsibility (EPR). Immediate benefits from the standardization are three folds.Firstly, the sorting and exchange processes of the bottles collected for reuse by each brewer becomeno longer necessary. Secondly, cost reduction is achieved through streamlining of collection and reuseprocesses. Finally, under the stochastic demand of glass bottles their inventory holding costs and lostsales cost are reduced via inventory pooling. Through the development of the mathematical modelswe determine an optimal operation policy of the two brewers that maximizes the sum of benefitsobtained from standardization. Numerical examples are solved to show the validity of the model.Sensitivity tests are also performed to examine the effects of system parameters on the objective functionvalue and decision variables.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

1.1. Background

In classical logistics systems the material and related informa-tion flow are managed in forward direction, i.e., from raw materialsto the final products delivered to the customers. In reverse logis-tics, backward flow is managed, i.e., the used and reusable partsand products return from the customers to the producers. In thisway natural resources can be saved for future generations andfirms can contribute to the sustainable development efforts (Dobos& Richter, 2004). There are several reasons why heightened atten-tion has been paid to reverse logistics in the past decades: Firstly,producers and consumers became more environmentallyconscious, and started to realize that it is time to abandon the‘throw – away age’. Secondly, tighter legislation in some countriesforced producers to take back products after use and either recoverthem or dispose of them properly. Thirdly, some producers realizedthat recovery operations can lead to additional profits (Teunter &Vlachos, 2002).

Extended producer responsibility (EPR) is a strategy designed topromote the integration of environmental costs associated withproducts throughout their life cycles into the market price of theproducts. Under EPR, firms are obligated to meet a given take backquota for the end of used products, and certain amount of penaltywill be charged if it is breached (OECD, 1999). Fifteen countries in

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: +82 42 350 3110.).

Europe, including Germany, the United Kingdom, France andHungary, four countries in Asia including Korea, Japan, Taiwanand Australia, and countries in Latin America including Mexicoand Brazil, have introduced the EPR system. The number of itemsthat are controlled by extended producer responsibility is increas-ing as the industries become more complex and the laws andregulations on environmental issues are tightened (Ko & Hwang,2009). Glass bottle, as a representative of packaging materials, isone of the most important items that need to be controlled byEPR legislation. And also, in terms of carbon dioxide (CO2) emissionreduction, higher reuse rate of used bottles and more cost-effectiveinventory policy becomes significant (Hekkert, Joosten, & Worrell,1998).

Recently, seven major brewing companies in Korea standard-ized the shapes and colors of their bottles to avoid the costlysorting and exchange procedures in retrieving empty bottles. Theyreported that in 2007 alone, about 24 billion glass bottles werestandardized, which resulted in an annual cost saving of approxi-mately US 40 million dollars. The standardization also enabledthe brewers to enjoy the benefit of inventory pooling, the practiceof using a common pool of stock for satisfying two or more sourcesof random demands. To be more specific, it refers to an arrange-ment in which different companies or stocking points share theirinventories and has been proven to be an effective strategy inimproving companies’ logistical performances while reducing thetotal system cost at same time. In this arrangement, lateral trans-shipments are used to satisfy the demand of a company that isout of stock from other company with surplus on-hand inventory.This study is motivated by the experience of the Korean brewing

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694 Y.D. Ko et al. / Computers & Industrial Engineering 62 (2012) 693–702

companies. In this paper we investigate the sources of cost reduc-tions through the development of mathematical model and thendiscuss desirable management policies of government and busi-ness circle for more effective environment protection efforts.

1.2. Problem description

We deal with a reverse logistics system with two brewers, Aand B, and consider the system in two situations, before and afterstandardization of glass bottles. Fig. 1 depicts the model withbefore standardization and shows the flow of glass bottles beforestandardization, where products are sold in two different typesof glass bottles. For the development of models the followingassumptions are made: The demand of glass bottles during a unittime interval is satisfied by either reused bottles or newly pur-chased bottles. The demand follows a normal distribution withknown mean and variance. The market consists of two segments,one with large orders such as bars and restaurants, and the otherwith smaller orders such as individual customers.

All the bottles sold to the first market segment can be retrievedin its entirety at the cost of a given unit handling commission chd

while the return rate of used bottles from the second segment isdependent on the unit buy-back price. That is, the retrieval ratefrom individual customer can be expressed as a function of theamount brewers are willing to pay cb. Through inspection and sort-ing processes, some bottles corresponding to a proportion of d arefound to be in wrong hands and thus those bottles are to beexchanged between the brewers incurring the exchange cost. Ifthe obligatory take back quota b is breached, then brewers haveto pay the unit penalty cost cpnt for the unsatisfied amount. Thebrewers adopt order-up-to S inventory policy for serviceable bot-tles. The target inventory level S and the unit buy-back price cb

of each brewer become decision variables.Fig. 2 depicts the model with after standardization and shows

the reverse logistics system after standardization. Used bottlesare jointly collected with an identical buy-back price which areto be allocated to brewer A and B with the ratio of c and 1 � c,respectively. Note that differing from Fig. 1, the inspection and ex-change process do not exist. Furthermore, through inventory pool-

Fig. 1. The model with be

ing they can expect to have a further reduction in inventory costand lost sales cost while incurring the associated transportationcost. The decision variables are the order-up-to level of eachbrewer, the joint unit buy-back price, and the allocation ratio ofthe collected bottles. In this paper, through the development ofthe mathematical models we determine analytically the cost ben-efits obtained from standardization. The benefit is measured by thedifference in the sum of each brewer’s cost.

1.3. Previous studies

A deterministic EOQ-type reverse logistics model was first ad-dressed by Schrady (1967). In his model, it is assumed that manu-facturing and remanufacturing rate is infinite without wastedisposal, where a single manufacturing and more than one reman-ufacturing cycles occur. The generalized model of the Schrady’s forthe case of finite recycling rate was presented by Nahmias andRivera (1979). Also, a multi product extension of these modelswas examined by Mabini, Pintelon, and Gelders (1992). A modelwith different inventory holding cost rates for manufactured andremanufactured items was studied by Teunter (2001). In thesemodels, the return rate of used products is considered as a givenparameter. Dobos and Richter (2000 and 2003) and Richter andDobos (1999) introduced the concept of waste disposal in therecycling models, where the return rate is considered as decisionvariable. One of the common unrealistic assumptions in the EOQbased models is that all units manufactured and remanufacturedare of good quality. Dobos and Richter (2006) extended their mod-els considering quality of the bought back products, where somepart of returned products are discarded due to the poor qualitycondition. Jaber, Goyal, and Imran (2008) studied a model, wherethe rate of defective items decreases as the number of shipmentsincreases by the effect of learning. Lee, Gen, and Rhee (2009)proposed the remanufacturing system with three-stage logisticsnetwork model for minimizing the total costs considering multi-stage and multi-product. They suggested hybrid genetic algorithmas a solution methodology. Liu, Kim, and Hwang (2009) modeled aproduction inventory system with rework, where a stationarydemand can be satisfied with both new product and reworked

fore standardization.

Fig. 2. The model with after standardization.

Y.D. Ko et al. / Computers & Industrial Engineering 62 (2012) 693–702 695

defective product. Ko and Hwang (2009) developed a model, wherethe return rate of used products is treated as a function of the unitbuy-back price for used products and the minimum allowedquality level. In this model, newly manufactured products andremanufactured products can have different retail prices.

The stochastic version of reverse logistics systems is also pro-posed by various authors as well. Cho and Parlar (1991) surveyedthe literature about the optimal maintenance and replacementmodels for multi-unit systems. Kelly and Silver (1989) studied asystem, where recovered items can be immediately reused withthe assumption of random returns and a fixed lead time. Tangand Grubbstrom (2005) studied a manufacturing/remanufacturingsystem with stochastic lead times and a constant demand. In hismodel, it is assumed that there are two supply sources for replen-ishing serviceable inventories.

The two-location inventory system, where inventory poolingoccurs through lateral transshipment is first addressed by Gross(1963). In his model, the optimal ordering and transshipmentquantities for the system that minimizes the system costs aredetermined. Karmarkar and Patel (1977) and Karmarkar (1979)generalized the two-location problem for the case of multi locationinventory distribution systems. Targas (1989) studied the effect ofinventory pooling via lateral transshipment on the service levelsrealized at the stocking points, and, conversely, the effect of servicelevel constraints on the optimal order-up-to level and transship-ment policy. Recently, Wong, Oudheusden, and Cattrysse (2007)dealt with the cost allocation problem with spare parts inventorypooling.

2. Notations and assumption

2.1. Notations

i
i = A (brewer A), B (brewer B) cpi unit purchasing cost of new bottles cbi unit buy-back price of used bottles chdi unit handling commission
clsi
unit lost sales cost chi unit inventory cost of serviceable bottles creui unit cost for reuse cinsi unit inspection cost of used bottles cex unit exchange cost ctrAB unit transportation cost between two brewers A and B ctrbs unit transportation cost between brewer and storage
area
cpnt unit penalty cost for unsatisfied take-back-quota r(cbi) return rate of used bottles ai ratio of large order in demand b rbligatory take-back-quota c allocation ratio of collected bottles di fraction of collected bottles that requires exchange b parameter of the return rate function ASi amount sold Si order-up-to level Di demand quantity of the product R0i number of collected bottles before exchange Ri number of collected bottles after exchange Iþi on-hand inventory of bottles I�I amount of shortage of bottles Qi ordering quantity of new bottles f(yi) density function of brewer i’s demand of unit time
period
li mean value of brewer i’s demand of unit time period ri standard deviation of brewer i’s demand of unit time
period

Fig. 3 shows the return rate function in an exponential form. It isassumed that the brewers can collect a larger quantity of used bot-tles if they pay higher unit buy-back price. The brewers are obli-gated by law to pay the handling commission per a used bottle tothe middle distribution channels which are in charge of sellingand recovery of the products. Thus the handling commission canbe considered as the low bound of unit buy-back price. Also, pur-chasing new bottles is preferable to buying back used bottles if

Fig. 3. The return rate of used bottles.

Fig. 4. Inventory related components in case 1 and 2.


the sum of unit buy-back price and unit reuse cost is higher than theunit new-bottle purchasing cost.

2.2. Assumptions

1. The demand of each brewer during unit time period is anindependent and identically distributed random variableand follows normal distribution with known mean andvariance.

2. The return rate of used bottles is dependent on the unit buy-back price.

3. The bottles sold in a given period are available for reuse forthe immediately succeeding period.

4. The reused bottles are as good as new ones.5. At the end of each period new bottles are ordered up to the

target inventory level and replenished instantaneously.6. Lost sales occur if the demand exceeds the inventory level S.7. Penalty cost occurs for the unsatisfied amount of take back

quota.

3. Mathematical model

For our objective function eleven different cost elements areconsidered. They are the inventory holding, lost sales cost, buybackcost of used bottles, inspection cost, handling commission paid tothe market segment with large orders, reuse cost, exchange cost,transportation cost between two brewers, transportation costbetween the brewers and storage area, purchasing cost of new bot-tles, and penalty cost related to obligatory take-back quota. Twomodels are developed in this section, the model with before stan-dardization depicted in Fig. 1 and the model with after standardi-zation in Fig. 2.

3.1. Model with Before Standardization

The total expected cost is formulated considering the relativesize of Si and Di, i.e., DA 6 SA and DA > SA (shown in Fig. 4).

Case (1) DA 6 SA

The expected cost of case 1 consists of inventory cost, buy-backprice, inspection cost, exchange cost, remanufacturing cost, pen-alty cost (if the take-back quota is breached), and purchasing costof new bottles. In this case, no shortage occurs. Let IþA be theamount of inventories incurred after satisfying demand. Then(SA � IþA � RA) number of new bottles is purchased to stock up tothe target inventory level SA, where

E½IþA � ¼Z SA

0ðSA � yAÞf ðyAÞdyA ð1Þ

The expected amount of collected bottles after exchange:

E½RAjDA 6 SA� ¼ ð1� dAÞE½R0AjDA 6 SA� þ dBE½R’B� ð2Þ

where

E½R0AjDA 6 SA� ¼ faA þ rðcbAÞð1� aAÞgE½DAjDA 6 SA�E½R0B� ¼ faB þ rðcbBÞð1� aBÞgE½ASB�E½ASB� ¼ PðDB 6 SBÞE½DBjDB 6 SB� þ PðDB > SBÞSB

The expected quantity of new bottles purchased:

E½Q AjDA 6 SA� ¼ SA � E½IþA � � E½RAjDA 6 SA� ð3Þ

The total expected cost (C1A (SA,cbA)) of case 1 becomes

C1AðSA; cbAÞ ¼ chAE½IþA � þ cbArðcbAÞð1� aAÞE½DAjDA 6 SA�

þ chdAaAE½DAjDA 6 SA� þ cinsAE½R0AjDA 6 SA�� cexdAE½R0AjDA 6 SA� þ ðctrAB þ cexÞdBE½R0B�þ creuAE½RAjDA 6 SA� þ cpAE½Q AjDA 6 SA�þ cpntMaxfbE½DAjDA 6 SA� � E½RAjDA 6 SA�;0g ð4Þ

Case (2) DA > SA

The expected cost of case 2 consists of lost sales cost, buy-backprice, inspection cost, exchange cost, remanufacturing cost, pen-alty cost (if the take-back quota is breached), and new-bottle pur-chasing cost. In this case, SA number of bottles are sold and lostsales I�A may occur. The order size of new bottles becomes SA � RA,where

E½I�A � ¼Z 1

SA

ðyA � SAÞf ðyAÞdyA ð5Þ

The expected amount of collected bottles after exchange:

E½RAjDA > SA� ¼ ð1� dAÞE½R0AjDA 6 SA� þ dBE½R0B� ð6Þ

Fig. 6. Inventory related components of case 2 in the model with afterstandardization.


where

E½R0AjDA > SA� ¼ faA þ rðcbAÞð1� aAÞgSA

E½R0B� ¼ faB þ rðcbBÞð1� aBÞgE½ASB�E½ASB� ¼ PðDB 6 SBÞE½DBjDB 6 SB� þ PðDB > SBÞSB

The expected quantity of new bottles purchased:

E½Q AjDA > SA� ¼ SA � E½RAjDA > SA� ð7Þ

The total expected cost of case 2, C2A (SA,cbA) can be expressed as

C2AðSA; cbAÞ ¼ clsAE½I�A � þ cbArðcbAÞð1� aAÞSA þ chdAaASA

þ cinsAE½R0AjDA > SA� � cexdAE½R0AjDA > SA�þ ðctrAB þ cexÞdBE½R0B� þ creuAE½RAjDA > SA�þ cpAE½Q AjDA > SA� þ cpntMaxfbSA � E½RAjDA 6 SA�;0g

ð8Þ

Combining Eqs. (4) and (8), the total expected cost of brewer Acan be written as:

CAðSA; cbAÞ ¼ PðDA 6 SAÞ � C1AðSA; cbAÞ þ PðDA > SAÞ � C2

AðSA; cbAÞ ð9Þ

Similarly, the total expected cost of brewer B is

CBðSB; cbBÞ ¼ PðDB 6 SBÞ � C1BðSB; cbBÞ þ PðDB > SBÞ � C2

BðSB; cbBÞ ð10Þ

We are to find the best order-up-to level S (Si), and the unit buy-back price of each manufacturer (cbi) which minimize the total ex-pected cost. (i = A,B)

3.2. Model with After Standardization

Compared to the current practice model, the cost terms forinspection and exchange are no longer needed in the objectivefunction while the transportation cost between brewers and stor-age area is newly introduced. There are four cases to be considered.

Case (1) DA 6 SA and DB 6 SB

The sum of total expected cost of each brewer consists of buy-back price, remanufacturing cost, inventory cost, penalty cost (ifthe take-back quota is breached in terms of total sales amount),and new-bottle purchasing cost.

Since the demand of each brewer does not exceed S as shown inFig. 5, ðIþA ; IþB ÞP 0. Thus the expected inventory of each brewer canbe written as:

E½IþA � ¼Z SA

0ðSA � yAÞf ðyAÞdyA ð11Þ

E½IþB � ¼Z SB

0ðSB � yBÞf ðyBÞdyB ð12Þ

The expected amount of return:

E½RAjDA 6 SA and DB 6 SB� ¼ aAE½DAjDA 6 SA� þ cE½TRjDA 6 SA

and DB 6 SB� ð13Þ


E½RBjDA 6 SA and DB 6 SB� ¼ aBE½DBjDB 6 SB� þ ð1� cÞE½TRjDA 6 SA

and DB 6 SB�

where

E½TRjDA 6 SA and DB 6 SB� ¼ rðcbÞfð1� aAÞE½DAjDA

6 SA� þ ð1� aBÞE½DBjDB 6 SB�g

The total expected cost of case 1 for the system, (C1AþB(SA, SB, cb,

c)) can be formulated as

C1AþBðSA; SB; cb; cÞ ¼ chAE½IþA � þ chBE½IþB � þ ðcb þ ctrbsÞE½TRjDA 6 SA

and DB 6 SB� þ chdAaAE½DAjDA 6 SA�þ chdBaBE½DBjDB 6 SB� þ creuAE½RAjDA 6 SA

and DB 6 SB� þ creuBE½RBjDA 6 SA

and DB 6 SB� þ cpðSA þ SB � E½IþA � � E½IþB �� E½RAjDA 6 SA and DB 6 SB� � E½RBjDA 6 SA

and DB 6 SB�Þ þ cpntMaxfbðE½DAjDA 6 SA�þ E½DBjDB 6 SB�Þ � E½RAjDA 6 SA and DB 6 SB�� E½RBjDA 6 SA and DB 6 SB�;0g ð15Þ

Case (2) DA > SA and DB > SB

The total expected cost of two brewers consists of buy-backprice, remanufacturing cost, lost sales cost, penalty cost (if thetake-back quota is breached in terms of total sales amount), andnew-bottle purchasing cost.

Since the demand of each brewer exceeds the inventory level S,lost sales occur for both brewers as shown in Fig. 6. The sum of thetotal expected costs of case 2, C2

AþB(SA, SB, cb, c), becomes

C2AþBðSA; SB; cb; cÞ ¼ clsAE½I�A � þ clsBE½I�B � þ ðcb þ ctrbsÞE½TRjDA > SA

and DB > SB� þ chdAaASA þ chdBaBSB

þ creuAE½RAjDA > SA and DB > SB�þ creuBE½RBjDA > SA and DB > SB�þ cpðSA þ SB � E½RAjDA > SA and DB > SB�� E½RBjDA > SA and DB > SB�Þþ cpntMaxfbðSA þ SBÞ � E½RAjDA > SA

and DB > SB� � E½RBjDA > SA and DB > SB�;0gð16Þ

Case (3) DA 6 SA and DB > SB

Fig. 7 shows the inventory status of the two brewers, A with po-sitive inventory and B with lost sales. Through inventory pooling,brewer A can help B by supplying the bottles in inventory withB. As results, both the inventory holding cost of A and the lost salescost of B can be reduced while transportation costs between twobrewers occur.




Case (3–1) E½IþA � > E½I�B �The expected lost sales of brewer B becomes zero, and the ex-

pected inventory of brewer A decreases to E½IþA � � E½I�B �. The ex-pected amount of lateral transshipment is E½I�B �, and the totalexpected sales amount is increased by the amount of E½I�B � as well.Total expected cost of two brewers consists of buy-back price,remanufacturing cost, inventory cost (A), penalty cost (if thetake-back quota is breached in terms of total sales amount), andnew-bottle purchasing cost. The sum of the total expected costsof case 3–1, (C3�1

AþB(SA, SB, cb, c)), becomes

C3�1AþBðSA; SB; cb; cÞ ¼ chAfE½IþA � � E½I�B �g þ ðcb þ ctrbsÞE½TRjDA 6 SA

and DB > SB� þ chdAaAE½DAjDA 6 SA�þ chdBaBfSB þ E½I�B �g þ ctrABE½I�B �þ creuAE½RAjDA 6 SA and DB > SB�þ creuBE½RBjDA 6 SA and DB > SB�þ cpðSA þ SB � fE½IþA � � E½I�B �g � E½RAjDA 6 SA

and DB > SB� � E½RBjDA 6 SA and DB > SB�Þþ cpntMaxfbðE½DAjDA 6 SA� þ SB þ E½I�B �Þ� E½RAjDA 6 SA and DB > SB� � E½RBjDA 6 SA

and DB > SB�;0g ð17Þ

Case (3–2) E½IþA � � E½I�B �The expected lost sales of brewer B decreases to E½I�B � � E½IþA �,

and the expected inventory of brewer A becomes zero. The ex-pected amount of lateral transshipment is E½IþA �, and the total ex-pected sales amount is increased by the amount of E½IþA �. Totalexpected cost of two brewers consists of buy-back price, remanu-facturing cost, lost sales cost (B), penalty cost (if the take-back quo-ta is breached in terms of total sales amount), and new-bottlepurchasing cost. The sum of the total expected costs of case 3–2,(C3�2

AþB(SA, SB, cb, c)), becomes

C3�2AþBðSA; SB; cb; cÞ ¼ clsBfE½I�B � � E½IþA �g þ ðcb þ ctrbsÞE½TRjDA 6 SA

and DB > SB� þ chdAaAE½DAjDA 6 SA�þ chdBaBfSB þ E½IþA �g þ ctrABE½IþA �þ creuAE½RAjDA 6 SA and DB > SB�þ creuBE½RBjDA 6 SA and DB > SB�þ cpðSA þ SB � E½RAjDA 6 SA and DB > SB�� E½RBjDA 6 SA and DB > SB�Þþ cpntMaxfbðSA þ SBÞ � E½RAjDA 6 SA andDB > SB� � E½RBjDA 6 SA and DB > SB�;0g ð18Þ

Case (4) DA > SA and DB 6 SB

Fig. 8 shows the inventory related components of case 4. Notethat by interchanging A with B in Figs. 8 and 7 can be obtained.Thus the cost functions in case 4 can be developed quite easilyfrom those in case 3.

Case (4–1) E½I�A � � E½IþB �The expected lost sales of brewer A becomes zero, and the ex-

pected inventory of brewer A decreases to E½IþB � � E½I�A �. The expectedamount of lateral transshipment is E½I�A �, and the total expected salesamount is increased by the amount of E½I�A � as well. The total ex-pected cost of case 4–1 for the two brewers (C4�1

AþB(SA, SB, cb, c)):

C4�1AþBðSA; SB; cb; cÞ ¼ chBfE½IþB � � E½I�A �g þ ðcb þ ctrbsÞE½TRjDA > SA

and DB 6 SB� þ chdAaAfSA þ E½I�A �gþ chdBaBE½DBjDB 6 SB� þ ctrABE½I�A �þ creuAE½RAjDA > SA and DB 6 SB�þ creuBE½RBjDA > SA and DB 6 SB�þ cpðSA þ SB � fE½IþB � � E½I�A �g � E½RAjDA > SA

and DB 6 SB� � E½RBjDA > SA and DB 6 SB�Þþ cpntMaxfbðSA þ E½I�A � þ E½DBjDB 6 SB�Þ� E½RAjDA > SA and DB 6 SB� � E½RBjDA > SA

and DB 6 SB�;0g ð19Þ

Case (4–2) E½I�A � > E½IþB �The expected lost sales of brewer A decreases to E½I�A � � E½IþB �, and

the expected inventory of brewer B becomes zero. The expectedamount of lateral transshipment is E½IþB �, and the total expected salesamount is also increased by the amount of E½IþB �. The total expectedcost of case 4–2 for the two brewers (C4�2

AþB(SA, SB, cb, c)):

C4�2AþBðSA; SB; cb; cÞ ¼ clsAfE½I�A � � E½IþB �g þ ðcb þ ctrbsÞE½TRjDA > SA

and DB 6 SB� þ chdAaAfSA þ E½IþB �þ chdBaBE½DBjDB 6 SB� þ ctrABE½IþB �þ creuAE½RAjDA > SA and DB 6 SB�þ creuBE½RBjDA > SA and DB 6 SB�þ cpðSA þ SB � E½RAjDA > SA and DB 6 SB�� E½RBjDA > SA and DB 6 SB�Þþ cpntMaxfbðSA þ SBÞ � E½RAjDA > SA and

DB 6 SB� � E½RBjDA > SA and DB 6 SB�;0g ð20Þ

Combining the four cases, the total expected cost of the twobrewers can be found and

¼ PðDA 6 SA and DB 6 SBÞ � C1AþBðSA; SB; cb; cÞ þ PðDA > SA

and DB > SBÞ � C2AþBðSA; SB; cb; cÞ

þ PðDA 6 SA;DB > SB and SA � DA P DB � SBÞ

� C3�1AþBðSA; SB; cb; cÞ þ PðDA 6 SA;DB > SB and SA

� DA < DB � SBÞ � C3�2AþBðSA; SB; cb; cÞ

þ PðDA > SA;DB 6 SB and DA � SA 6 SB � DBÞ

� C4�1AþBðSA; SB; cb; cÞ þ PðDA > SA;DB 6 SB and

DA � SA > SB � DBÞ � C4�2AþBðSA; SB; cb; cÞ ð21Þ


We want to find the order-up-to level S (SA, SB), and the jointunit buy-back price (cb), and the allocation ratio (c) in a way tominimize the total expected cost of Eq. (21).

Fig. 9. The solution procedure.

4. Solution procedure

The equilibrium solutions of Eqs. (9) and (10) are developedbased on the repeated game model integrated with Tabu searchalgorithm (refer to Fig. 9). And Eq. (21) in the model with afterstandardization is solved by Tabu search.

4.1. Repeated game model

In game theory, a repeated game is defined to be an extensivetype of game which consists of some number of repetitions of basegame known as well-studied 2-person games. Generally, when therepeated game performs a sufficiently long time, an equilibriumsolution can be obtained (Benoit & Krishna, 1985). For the modelwith before standardization, we firstly assume that the values ofthe order-up-to level (SB), and the unit buy-back price (cbB) ofbrewer B are given as an initial solution. And then, by Tabu searchthe best solution (SA, cbA) is found that minimizes the totalexpected cost of brewer A. Similarly, with the values of SA and cbA

of brewer A determined in previous step, brewer B finds the bestorder-up-to level (SB), and the unit buy-back price (cbB) using Tabusearch. Repeating the above procedure we obtain an equilibriumsolution.

4.2. Tabu search

Tabu search is a higher level heuristic procedure for solvingoptimization problems, designed to guide other methods to escapethe trap of local optimality. Tabu search has obtained optimal andnear optimal solutions to a wide variety of classical and practicalproblems in applications ranging from character recognition toneural networks. It uses flexible structures memory, conditionsfor strategically constraining and freeing the search process, andmemory functions of varying time spans for intensifying and diver-sifying the search (Glover, 1990). The Tabu search starts at aninitial point and then moves to one of the neighborhood points thatgive the best result of the objective function at each iteration. Andthis move continues until a stopping criterion has been met. Themethod forbids points with certain attributes with the goals of pre-venting cycling and guiding the search towards unexplored regionsof the solution space. This is done using an important feature of theTabu search method called tabu list which consists of the latestmoves made. In its simplest form, tabu search requires the follow-ing elements:

� Initial point� Neighborhood generation method� Tabu list� Aspiration criterion� Stopping criterion

The Tabu search algorithm starts at an initial point, asðS0

A; S0B; c

0b; c

0Þ in the case of the model with after standardization,and it goes through a number of iterations until the stopping crite-rion is met. At each iteration, r numbers of directions are randomlygenerated, and line search is performed for each direction. Thepoint which gives the best objective value among the r candidateneighborhood points is selected as the next point if the directionis non-tabu, or it satisfies the aspiration criterion. After movingto the next point, the direction of the latest move is newly storedin the tabu list. This procedure is repeated until the stop criterion

is satisfied. Generally in Tabu search, two kinds of stop criterionhave been used: One in terms of a total elapsed number of itera-tions, and the other finding the last best solution again. The formerstop criterion is adopted in this study (see Fig. 10).

Step 1. InitializationChoose the number of random search direction to be usedat each iteration (r)Choose the maximum number of iteration (MAX_ITER)Choose an initial point x0 ¼ ððS0

A; S0B; c

0b; c

0ÞÞLet TL = Ø, Best value = ATC(x0), j = 0.Step 2. Perform Line Search

2.1 Generate r random direction, d1, d2, . . . , dr

Let k⁄ and d⁄ be such thatATCðxj þ k�d�Þ ¼ min

16i6rATCðxj þ kidiÞ

2.2 Check tabu list
If (d� 2 TLÞorðd� 2 TL and ATCðxj þ k�d�Þ< Best value), goto step 2.3.Otherwise choose second best solution and repeat step2.2
2.3 Update current point
Let xjþ1 ¼ xj þ k�d� and update tabu list.
If ATCðxj þ k�d�Þ < Best value then Best value =ATCðxj þ k�d�Þ

j = j + 1 and go to step 3.

Step 3. Check stopping criterionIf j = MAX_ITER, stop.Else go to Step 2.

5. Numerical example

5.1. Computational results

To illustrate the models and solution procedure we solved anexample problem with the following parameter values.

DA � N(800,000, 80,0002), DB � N(300,000, 30,0002)aA = aB = 0.48, b = 0.9, b = 0.02, chA = chB = 3, clsA = clsB = 300,

chdA = chdB = 50, cinsA = cinsB = 15, ctrAB = 20, cex = 50, ctrbs = 10, creuA = -creuB = 10, cpA = cpB = cp = 190, cpnt = 180, dA = dB = 0.03.

Tables 1 and 2 show the computational results for the bothmodels.

The results are consistent with our expectation. Comparedwith the model with before standardization, standardizationcauses a substantial amount of cost reduction, i.e.,

Fig. 10. The basic procedure of Tabu search.

Table 1The result of the model with before standardization.

S cb Return rate Total expected cost

Brewer A 990,000 82.43 0.8999 82,692,884Brewer B 389,900 82.43 0.8999 28,983,294Sum 1379,900 NA NA 111,676,179

Table 2The result of the model with after standardization.

SA + SB cb Return rate Total expected cost

Without inv. pooling 1378,500 82.50 0.9001 100,984,662With inv. pooling 1313,500 82.48 0.9000 100,737,565 Fig. 11. Actual return rate vs. obligatory take-back quota.


9.574% = (111,676,179–100,984,662)/111,676,179 = 0.09574. Notethat Inventory pooling further reduces the total cost from9.574% with standardization alone to 9.795% = (111,676,179–100,737,565)/111,676,179 = 0.09795. Standardization have onlyslight effect on the sum of the order-up-to–levels (SA + SB), i.e.,1,379,900 vs. 1,378,500. Under the model with after standardiza-tion, brewers no longer pay the inspection cost of the collected bot-tles and transportation cost due to the exchange process. Inventorypooling has an effect of slightly decreasing the order-up-to level,which implies that the two brewers can run their business with ssmaller inventory.

5.2. Sensitivity tests

The sensitivity analysis was performed on the system parame-ters which we think are closely related with the interests of thegovernment as well as the brewers. For the government point ofview we investigated the effect of obligatory take back quota andthe penalty cost on the actual return rate. The standard deviation

of demand, the proportion of the market segment of large orders,and unit transshipment cost were selected for the study reflectingthe brewers’ interests. Since the sensitivity analysis show similarresults in all three situations, here we present them only for themodel with after standardization with lateral transshipment forthe sake of brevity.

Fig. 11 shows the change of the actual return rate for variousvalues of the obligatory take-back quota. It can be observed thatthe actual return rate remains approximately at the level of 0.85if the quota is not larger than 0.8. When the obligatory take-back-quota increases beyond 0.8 the actual return rate increasesthat calls for the brewers to pay higher unit buy-back price to avoidpenalty cost.

The relationship between the actual return rate and the penaltycost is shown in Fig. 12. We tested the penalty cost from 0 to 360with an increment of 30. It is observed that the actual return rateconverges to the level of the obligatory take-back quota (=0.9) asthe penalty cost increases.

Figs. 13 and 14 show the changes of the total expected cost andthe sum of the order-up-to-level (SA + SB) as the standard deviation

Fig. 12. Actual return rate vs. penalty cost.

Fig. 13. Total expected cost vs. standard deviation of demands.

Fig. 14. Order-up-to-level vs. standard deviation of demands.

Fig. 15. Actual return rate vs. sales rate for bars and restaurants.

Fig. 16. Buy-back price vs. sales rate for bars and restaurants.

Fig. 17. Total expected cost vs. standard deviation of demands.

Fig. 18. Order-up-to-level vs. unit transshipment cost.


of demands varies. Larger standard deviation of demand couldincur more chances of having lost sales and higher inventories.As expected, the total expected cost increases as the standard devi-ation of demands increases. Also, the order-up-to-level increases inorder to reduce the chances of lost sales.

Figs. 15–17 show the responses of the actual return rate, buy-back price and the total expected cost for the changing sales ratein the market segment with large orders such as from bars andrestaurants. When the sales rate for large orders increase, thebrewers can easily collect used bottle from those places withoutmuch effort. Especially, when the ratio is more than 0.8 whichimplies that enough used bottles are coming from the bars and res-taurants, they are in no need of paying more than the unit handlingcommission to individual consumer for used bottle. Thus the totalexpected cost decreases.

Fig. 18 shows the change of the sum of the order-up-to-level forthe various values of the unit transshipment costs. It can be ob-served that lower unit transshipment cost triggers intensifying

inventory pooling effect. The sum of the order-up-to-level is re-duced due to the low unit transshipment cost that makes the coor-dination between two brewers much easier.


6. Conclusions

Motivated by a report that standardization of glass bottlesresulted in a substantial amount of cost reduction in Korean bever-age industry, this study analyzes the cost benefits of a recyclingsystem with two competing brewers. For the analysis, we developthe mathematical models in two situations, a case with beforestandardization (the model with before standardization) andanother case with after standardization (the model with after stan-dardization). To facilitate the analysis it is assumed that thedemand of each brewer is a random variable following normaldistribution with known mean and variance, and the return rateof used bottles is a function of the unit buy-back price. We find thatthe inspection and exchange process become no longer neededafter standardization, which contributes to a reduction of the totalexpected cost. Additional cost reduction is possible through inven-tory pooling via lateral transshipment between two brewers. Thesecond case is further examined by two sub-cases, with and with-out inventory pooling. An equilibrium solution of the first case isfound based on the repeated game approach integrated with Tabusearch. For the second case only Tabu search algorithm is utilized.To illustrate the model, a hypothetical problem is solved. The re-sults confirm our expectation that standardization of products forcommon use can yield a substantial amount of cost benefits. Asfurther studies our model could be improved by regarding the bot-tles collected during a given period as a weighted sum of the glassbottles produced during recent finite number of past periods.Another possibility is to extend the current model to the case ofmore than two brewers for the development of generic model.We hope that the study results would contribute in developingpolicies for preserving clean environment by the governmentofficials as well as business managers.

Acknowledgement

This work was supported by the Korea Research FoundationGrant funded by the Korean Government (MOEHRD) (KRF-2007-313-D00909).

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