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Cost benefits from standardization of the packaging glass bottles
Computers & Industrial Engineering 62 (2012) 693–702
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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
Article history: Available online 3 December 2011 Keywords: Glass bottle Inventory pooling Reverse logistics Standardization
a b s t r a c t This study deals with a recycling system with two competing brewers. It is assumed that they coordinate their manufacturing operations through standardization of their glass bottles for easy implementation of extended 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 become no longer necessary. Secondly, cost reduction is achieved through streamlining of collection and reuse processes. Finally, under the stochastic demand of glass bottles their inventory holding costs and lost sales cost are reduced via inventory pooling. Through the development of the mathematical models we determine an optimal operation policy of the two brewers that maximizes the sum of benefits obtained 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 function value and decision variables. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction 1.1. Background In classical logistics systems the material and related information flow are managed in forward direction, i.e., from raw materials to the final products delivered to the customers. In reverse logistics, backward flow is managed, i.e., the used and reusable parts and products return from the customers to the producers. In this way natural resources can be saved for future generations and firms can contribute to the sustainable development efforts (Dobos & Richter, 2004). There are several reasons why heightened attention has been paid to reverse logistics in the past decades: Firstly, producers and consumers became more environmentally conscious, and started to realize that it is time to abandon the ‘throw – away age’. Secondly, tighter legislation in some countries forced producers to take back products after use and either recover them or dispose of them properly. Thirdly, some producers realized that recovery operations can lead to additional profits (Teunter & Vlachos, 2002). Extended producer responsibility (EPR) is a strategy designed to promote the integration of environmental costs associated with products throughout their life cycles into the market price of the products. Under EPR, firms are obligated to meet a given take back quota for the end of used products, and certain amount of penalty will be charged if it is breached (OECD, 1999). Fifteen countries in ⇑ Corresponding author. Tel.: +82 42 350 3113; fax: +82 42 350 3110. E-mail address: [email protected] (H. Hwang). 0360-8352/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2011.11.026
Europe, including Germany, the United Kingdom, France and Hungary, four countries in Asia including Korea, Japan, Taiwan and Australia, and countries in Latin America including Mexico and Brazil, have introduced the EPR system. The number of items that are controlled by extended producer responsibility is increasing as the industries become more complex and the laws and regulations on environmental issues are tightened (Ko & Hwang, 2009). Glass bottle, as a representative of packaging materials, is one of the most important items that need to be controlled by EPR legislation. And also, in terms of carbon dioxide (CO2) emission reduction, higher reuse rate of used bottles and more cost-effective inventory policy becomes significant (Hekkert, Joosten, & Worrell, 1998). Recently, seven major brewing companies in Korea standardized the shapes and colors of their bottles to avoid the costly sorting and exchange procedures in retrieving empty bottles. They reported that in 2007 alone, about 24 billion glass bottles were standardized, which resulted in an annual cost saving of approximately US 40 million dollars. The standardization also enabled the brewers to enjoy the benefit of inventory pooling, the practice of using a common pool of stock for satisfying two or more sources of random demands. To be more specific, it refers to an arrangement in which different companies or stocking points share their inventories and has been proven to be an effective strategy in improving companies’ logistical performances while reducing the total system cost at same time. In this arrangement, lateral transshipments are used to satisfy the demand of a company that is out 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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companies. In this paper we investigate the sources of cost reductions through the development of mathematical model and then discuss desirable management policies of government and business circle for more effective environment protection efforts. 1.2. Problem description We deal with a reverse logistics system with two brewers, A and B, and consider the system in two situations, before and after standardization of glass bottles. Fig. 1 depicts the model with before standardization and shows the flow of glass bottles before standardization, where products are sold in two different types of glass bottles. For the development of models the following assumptions are made: The demand of glass bottles during a unit time interval is satisfied by either reused bottles or newly purchased bottles. The demand follows a normal distribution with known mean and variance. The market consists of two segments, one with large orders such as bars and restaurants, and the other with smaller orders such as individual customers. All the bottles sold to the first market segment can be retrieved in its entirety at the cost of a given unit handling commission chd while the return rate of used bottles from the second segment is dependent on the unit buy-back price. That is, the retrieval rate from individual customer can be expressed as a function of the amount brewers are willing to pay cb. Through inspection and sorting processes, some bottles corresponding to a proportion of d are found to be in wrong hands and thus those bottles are to be exchanged between the brewers incurring the exchange cost. If the obligatory take back quota b is breached, then brewers have to pay the unit penalty cost cpnt for the unsatisfied amount. The brewers adopt order-up-to S inventory policy for serviceable bottles. 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 bottles are jointly collected with an identical buy-back price which are to 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 exchange process do not exist. Furthermore, through inventory pool-
ing they can expect to have a further reduction in inventory cost and lost sales cost while incurring the associated transportation cost. The decision variables are the order-up-to level of each brewer, the joint unit buy-back price, and the allocation ratio of the collected bottles. In this paper, through the development of the mathematical models we determine analytically the cost benefits obtained from standardization. The benefit is measured by the difference in the sum of each brewer’s cost. 1.3. Previous studies A deterministic EOQ-type reverse logistics model was first addressed by Schrady (1967). In his model, it is assumed that manufacturing and remanufacturing rate is infinite without waste disposal, where a single manufacturing and more than one remanufacturing cycles occur. The generalized model of the Schrady’s for the case of finite recycling rate was presented by Nahmias and Rivera (1979). Also, a multi product extension of these models was examined by Mabini, Pintelon, and Gelders (1992). A model with different inventory holding cost rates for manufactured and remanufactured items was studied by Teunter (2001). In these models, the return rate of used products is considered as a given parameter. Dobos and Richter (2000 and 2003) and Richter and Dobos (1999) introduced the concept of waste disposal in the recycling models, where the return rate is considered as decision variable. One of the common unrealistic assumptions in the EOQ based models is that all units manufactured and remanufactured are of good quality. Dobos and Richter (2006) extended their models considering quality of the bought back products, where some part of returned products are discarded due to the poor quality condition. Jaber, Goyal, and Imran (2008) studied a model, where the rate of defective items decreases as the number of shipments increases by the effect of learning. Lee, Gen, and Rhee (2009) proposed the remanufacturing system with three-stage logistics network model for minimizing the total costs considering multistage and multi-product. They suggested hybrid genetic algorithm as a solution methodology. Liu, Kim, and Hwang (2009) modeled a production inventory system with rework, where a stationary demand can be satisfied with both new product and reworked
Fig. 1. The model with before standardization.
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Fig. 2. The model with after standardization.
defective product. Ko and Hwang (2009) developed a model, where the return rate of used products is treated as a function of the unit buy-back price for used products and the minimum allowed quality level. In this model, newly manufactured products and remanufactured products can have different retail prices. The stochastic version of reverse logistics systems is also proposed by various authors as well. Cho and Parlar (1991) surveyed the literature about the optimal maintenance and replacement models for multi-unit systems. Kelly and Silver (1989) studied a system, where recovered items can be immediately reused with the assumption of random returns and a fixed lead time. Tang and Grubbstrom (2005) studied a manufacturing/remanufacturing system with stochastic lead times and a constant demand. In his model, it is assumed that there are two supply sources for replenishing serviceable inventories. The two-location inventory system, where inventory pooling occurs through lateral transshipment is first addressed by Gross (1963). In his model, the optimal ordering and transshipment quantities for the system that minimizes the system costs are determined. Karmarkar and Patel (1977) and Karmarkar (1979) generalized the two-location problem for the case of multi location inventory distribution systems. Targas (1989) studied the effect of inventory pooling via lateral transshipment on the service levels realized at the stocking points, and, conversely, the effect of service level constraints on the optimal order-up-to level and transshipment policy. Recently, Wong, Oudheusden, and Cattrysse (2007) dealt with the cost allocation problem with spare parts inventory pooling.
clsi chi creui cinsi cex ctrAB ctrbs cpnt r(cbi)
ai b
c di b ASi Si Di R0i Ri Iþ i I I Qi f(yi)
li ri
unit lost sales cost unit inventory cost of serviceable bottles unit cost for reuse unit inspection cost of used bottles unit exchange cost unit transportation cost between two brewers A and B unit transportation cost between brewer and storage area unit penalty cost for unsatisfied take-back-quota return rate of used bottles ratio of large order in demand rbligatory take-back-quota allocation ratio of collected bottles fraction of collected bottles that requires exchange parameter of the return rate function amount sold order-up-to level demand quantity of the product number of collected bottles before exchange number of collected bottles after exchange on-hand inventory of bottles amount of shortage of bottles ordering quantity of new bottles density function of brewer i’s demand of unit time period mean value of brewer i’s demand of unit time period standard deviation of brewer i’s demand of unit time period
2. Notations and assumption 2.1. Notations i cpi cbi chdi
i = A (brewer A), B (brewer B) unit purchasing cost of new bottles unit buy-back price of used bottles unit handling commission
Fig. 3 shows the return rate function in an exponential form. It is assumed that the brewers can collect a larger quantity of used bottles if they pay higher unit buy-back price. The brewers are obligated by law to pay the handling commission per a used bottle to the middle distribution channels which are in charge of selling and recovery of the products. Thus the handling commission can be considered as the low bound of unit buy-back price. Also, purchasing new bottles is preferable to buying back used bottles if
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Fig. 3. The return rate of used bottles.
the sum of unit buy-back price and unit reuse cost is higher than the unit new-bottle purchasing cost. 2.2. Assumptions 1. The demand of each brewer during unit time period is an independent and identically distributed random variable and follows normal distribution with known mean and variance. 2. The return rate of used bottles is dependent on the unit buyback price. 3. The bottles sold in a given period are available for reuse for the 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.
Fig. 4. Inventory related components in case 1 and 2.
The expected amount of collected bottles after exchange:
E½RA jDA 6 SA ¼ ð1 dA ÞE½R0A jDA 6 SA þ dB E½R’B where
E½R0A jDA 6 SA ¼ faA þ rðcbA Þð1 aA ÞgE½DA jDA 6 SA
3. Mathematical model For our objective function eleven different cost elements are considered. They are the inventory holding, lost sales cost, buyback cost of used bottles, inspection cost, handling commission paid to the market segment with large orders, reuse cost, exchange cost, transportation cost between two brewers, transportation cost between the brewers and storage area, purchasing cost of new bottles, and penalty cost related to obligatory take-back quota. Two models are developed in this section, the model with before standardization depicted in Fig. 1 and the model with after standardization in Fig. 2.
E½R0B ¼ faB þ rðcbB Þð1 aB ÞgE½ASB E½ASB ¼ PðDB 6 SB ÞE½DB jDB 6 SB þ PðDB > SB ÞSB The expected quantity of new bottles purchased:
E½Q A jDA 6 SA ¼ SA E½IþA E½RA jDA 6 SA The total expected cost
C 1A ðSA ; cbA Þ
¼
0
ð3Þ
(SA, cbA)) of case 1 becomes
þ cbA rðcbA Þð1 aA ÞE½DA jDA 6 SA
cex dA E½R0A jDA 6 SA þ ðctrAB þ cex ÞdB E½R0B þ creuA E½RA jDA 6 SA þ cpA E½Q A jDA 6 SA þ cpnt MaxfbE½DA jDA 6 SA E½RA jDA 6 SA ; 0g
The total expected cost is formulated considering the relative size 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-back price, inspection cost, exchange cost, remanufacturing cost, penalty cost (if the take-back quota is breached), and purchasing cost of new bottles. In this case, no shortage occurs. Let Iþ A be the amount of inventories incurred after satisfying demand. Then (SA Iþ A RA) number of new bottles is purchased to stock up to the target inventory level SA, where
Z
chA E½IþA
(C 1A
þ chdA aA E½DA jDA 6 SA þ cinsA E½R0A jDA 6 SA
3.1. Model with Before Standardization
E½IþA ¼
ð2Þ
ð4Þ
Case (2) DA > SA The expected cost of case 2 consists of lost sales cost, buy-back price, inspection cost, exchange cost, remanufacturing cost, penalty cost (if the take-back quota is breached), and new-bottle purchasing cost. In this case, SA number of bottles are sold and lost sales I A may occur. The order size of new bottles becomes SA RA, where
E½IA ¼
Z
1
SA
ðyA SA Þf ðyA ÞdyA
ð5Þ
The expected amount of collected bottles after exchange:
SA
ðSA yA Þf ðyA ÞdyA
ð1Þ
E½RA jDA > SA ¼ ð1 dA ÞE½R0A jDA 6 SA þ dB E½R0B
ð6Þ
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where
E½R0A jDA > SA ¼ faA þ rðcbA Þð1 aA ÞgSA E½R0B ¼ faB þ rðcbB Þð1 aB ÞgE½ASB E½ASB ¼ PðDB 6 SB ÞE½DB jDB 6 SB þ PðDB > SB ÞSB The expected quantity of new bottles purchased:
E½Q A jDA > SA ¼ SA E½RA jDA > SA The total expected cost of case 2,
C 2A ðSA ; cbA Þ
¼
ð7Þ C 2A
(SA, cbA) can be expressed as
clsA E½IA
þ cbA rðcbA Þð1 aA ÞSA þ chdA aA SA þ cinsA E½R0A jDA > SA cex dA E½R0A jDA > SA
Fig. 6. Inventory related components of case 2 in the model with after standardization.
þ ðctrAB þ cex ÞdB E½R0B þ creuA E½RA jDA > SA þ cpA E½Q A jDA > SA þ cpnt MaxfbSA E½RA jDA 6 SA ; 0g ð8Þ Combining Eqs. (4) and (8), the total expected cost of brewer A can be written as:
C A ðSA ; cbA Þ ¼ PðDA 6 SA Þ C 1A ðSA ; cbA Þ þ PðDA > SA Þ C 2A ðSA ; cbA Þ
ð9Þ
Similarly, the total expected cost of brewer B is
C B ðSB ; cbB Þ ¼ PðDB 6 SB Þ C 1B ðSB ; cbB Þ þ PðDB > SB Þ C 2B ðSB ; cbB Þ
E½RB jDA 6 SA and DB 6 SB ¼ aB E½DB jDB 6 SB þ ð1 cÞE½TRjDA 6 SA and DB 6 SB where
E½TRjDA 6 SA and DB 6 SB ¼ rðcb Þfð1 aA ÞE½DA jDA 6 SA þ ð1 aB ÞE½DB jDB 6 SB g
ð10Þ
We are to find the best order-up-to level S (Si), and the unit buyback price of each manufacturer (cbi) which minimize the total expected cost. (i = A, B)
The total expected cost of case 1 for the system, (C 1AþB (SA, SB, cb,
c)) can be formulated as C 1AþB ðSA ; SB ; cb ; cÞ ¼ chA E½IþA þ chB E½IþB þ ðcb þ ctrbs ÞE½TRjDA 6 SA and DB 6 SB þ chdA aA E½DA jDA 6 SA
3.2. Model with After Standardization
þ chdB aB E½DB jDB 6 SB þ creuA E½RA jDA 6 SA
Compared to the current practice model, the cost terms for inspection and exchange are no longer needed in the objective function while the transportation cost between brewers and storage 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 buyback price, remanufacturing cost, inventory cost, penalty cost (if the 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 in þ Fig. 5, ðIþ A ; I B Þ P 0. Thus the expected inventory of each brewer can be written as:
and DB 6 SB þ creuB E½RB jDA 6 SA
E½IþA ¼
E½IþB ¼
Z
SA
ðSA yA Þf ðyA ÞdyA
ð11Þ
ðSB yB Þf ðyB ÞdyB
ð12Þ
0
Z 0
SB
The expected amount of return:
E½RA jDA 6 SA
and DB 6 SB E½RB jDA 6 SA
and DB 6 SB Þ þ cpnt MaxfbðE½DA jDA 6 SA þ E½DB jDB 6 SB Þ E½RA jDA 6 SA E½RB jDA 6 SA
and DB 6 SB
and DB 6 SB ; 0g
ð15Þ
Case (2) DA > SA and DB > SB The total expected cost of two brewers consists of buy-back price, remanufacturing cost, lost sales cost, penalty cost (if the take-back quota is breached in terms of total sales amount), and new-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 the total expected costs of case 2, C 2AþB (SA, SB, cb, c), becomes
C 2AþB ðSA ; SB ; cb ; cÞ ¼ clsA E½IA þ clsB E½IB þ ðcb þ ctrbs ÞE½TRjDA > SA and DB > SB þ chdA aA SA þ chdB aB SB
E½RA jDA 6 SA and DB 6 SB ¼ aA E½DA jDA 6 SA þ cE½TRjDA 6 SA and DB 6 SB
and DB 6 SB þ cp ðSA þ SB E½IþA E½IþB
ð13Þ
þ creuA E½RA jDA > SA
and DB > SB
þ creuB E½RB jDA > SA
and DB > SB
þ cp ðSA þ SB E½RA jDA > SA E½RB jDA > SA
and DB > SB
and DB > SB Þ
þ cpnt MaxfbðSA þ SB Þ E½RA jDA > SA and DB > SB E½RB jDA > SA
and DB > SB ; 0g ð16Þ
Fig. 5. Inventory related components of case 1 in the model with after standardization.
Case (3) DA 6 SA and DB > SB Fig. 7 shows the inventory status of the two brewers, A with positive inventory and B with lost sales. Through inventory pooling, brewer A can help B by supplying the bottles in inventory with B. As results, both the inventory holding cost of A and the lost sales cost of B can be reduced while transportation costs between two brewers occur.
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Fig. 7. Inventory related components of case 3 in the model with after standardization.
Case (3–1) E½Iþ A > E½IB The expected lost sales of brewer B becomes zero, and the ex pected inventory of brewer A decreases to E½Iþ A E½I B . The ex pected amount of lateral transshipment is E½IB , and the total expected 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 the take-back quota is breached in terms of total sales amount), and new-bottle purchasing cost. The sum of the total expected costs of case 3–1, (C 31 AþB (SA, SB, cb, c)), becomes
þ C 31 AþB ðSA ; SB ; c b ; cÞ ¼ c hA fE½I A E½I B g þ ðcb þ c trbs ÞE½TRjDA 6 SA
and DB > SB þ chdA aA E½DA jDA 6 SA þ chdB aB fSB þ E½IB g þ ctrAB E½IB þ creuA E½RA jDA 6 SA
and DB > SB
þ creuB E½RB jDA 6 SA
and DB > SB
þ cp ðSA þ SB fE½IþA E½IB g E½RA jDA 6 SA and DB > SB E½RB jDA 6 SA and DB > SB Þ þ cpnt MaxfbðE½DA jDA 6 SA þ SB þ E½IB Þ E½RA jDA 6 SA
and DB > SB E½RB jDA 6 SA
and DB > SB ; 0g E½Iþ A
ð17Þ
E½I B
Case (3–2) þ The expected lost sales of brewer B decreases to E½I B E½IA , and the expected inventory of brewer A becomes zero. The expected amount of lateral transshipment is E½Iþ A , and the total expected sales amount is increased by the amount of E½Iþ A . Total expected cost of two brewers consists of buy-back price, remanufacturing cost, lost sales cost (B), penalty cost (if the take-back quota is breached in terms of total sales amount), and new-bottle purchasing cost. The sum of the total expected costs of case 3–2, (C 32 AþB (SA, SB, cb, c)), becomes þ C 32 AþB ðSA ; SB ; c b ; cÞ ¼ c lsB fE½I B E½I A g þ ðcb þ c trbs ÞE½TRjDA 6 SA
and DB > SB þ chdA aA E½DA jDA 6 SA and DB > SB
þ creuB E½RB jDA 6 SA
and DB > SB
þ cp ðSA þ SB E½RA jDA 6 SA
DB > SB E½RB jDA 6 SA
þ C 41 AþB ðSA ; SB ; cb ; cÞ ¼ c hB fE½I B E½IA g þ ðc b þ c trbs ÞE½TRjDA > SA and DB 6 SB þ chdA aA fSA þ E½IA g þ chdB aB E½DB jDB 6 SB þ ctrAB E½IA þ creuA E½RA jDA > SA and DB 6 SB þ creuB E½RB jDA > SA and DB 6 SB þ cp ðSA þ SB fE½IþB E½IA g E½RA jDA > SA and DB 6 SB E½RB jDA > SA and DB 6 SB Þ þ cpnt MaxfbðSA þ E½IA þ E½DB jDB 6 SB Þ E½RA jDA > SA and DB 6 SB E½RB jDA > SA and DB 6 SB ; 0g ð19Þ þ Case (4–2) E½I A > E½I B þ The expected lost sales of brewer A decreases to E½I A E½I B , and the expected inventory of brewer B becomes zero. The expected amount of lateral transshipment is E½Iþ B , and the total expected sales amount is also increased by the amount of E½Iþ B . The total expected cost of case 4–2 for the two brewers (C 42 AþB (SA, SB, cb, c)):
þ C 42 AþB ðSA ; SB ; cb ; cÞ ¼ c lsA fE½IA E½IB g þ ðc b þ c trbs ÞE½TRjDA > SA
and DB 6 SB þ chdA aA fSA þ E½IþB þ chdB aB E½DB jDB 6 SB þ ctrAB E½IþB þ creuA E½RA jDA > SA
and DB 6 SB
þ creuB E½RB jDA > SA
and DB 6 SB
þ cp ðSA þ SB E½RA jDA > SA E½RB jDA > SA
and DB 6 SB
and DB 6 SB Þ
þ cpnt MaxfbðSA þ SB Þ E½RA jDA > SA DB 6 SB E½RB jDA > SA
and
and DB 6 SB ; 0g ð20Þ
¼ PðDA 6 SA and DB 6 SB Þ C 1AþB ðSA ; SB ; cb ; cÞ þ PðDA > SA
and DB > SB
E½RB jDA 6 SA and DB > SB Þ þ cpnt MaxfbðSA þ SB Þ E½RA jDA 6 SA
þ Case (4–1) E½I A E½I B The expected lost sales of brewer A becomes zero, and the ex pected inventory of brewer A decreases to E½Iþ B E½I A . The expected amount of lateral transshipment is E½I , and the total expected sales A amount is increased by the amount of E½I A as well. The total expected cost of case 4–1 for the two brewers (C 41 AþB (SA, SB, cb, c)):
Combining the four cases, the total expected cost of the two brewers can be found and
þ chdB aB fSB þ E½IþA g þ ctrAB E½IþA þ creuA E½RA jDA 6 SA
Fig. 8. Inventory related components of case 4 in the model with after standardization.
and DB > SB Þ C 2AþB ðSA ; SB ; cb ; cÞ þ PðDA 6 SA ; DB > SB and SA DA P DB SB Þ
and
and DB > SB ; 0g ð18Þ
Case (4) DA > SA and DB 6 SB Fig. 8 shows the inventory related components of case 4. Note that by interchanging A with B in Figs. 8 and 7 can be obtained. Thus the cost functions in case 4 can be developed quite easily from those in case 3.
31 ðSA ; SB ; cb ; cÞ þ PðDA 6 SA ; DB > SB and SA C AþB
DA < DB SB Þ C 32 AþB ðSA ; SB ; c b ; cÞ þ PðDA > SA ; DB 6 SB and DA SA 6 SB DB Þ 41 ðSA ; SB ; cb ; cÞ þ PðDA > SA ; DB 6 SB and C AþB 42 DA SA > SB DB Þ C AþB ðSA ; SB ; cb ; cÞ
ð21Þ
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We want to find the order-up-to level S (SA, SB), and the joint unit buy-back price (cb), and the allocation ratio (c) in a way to minimize the total expected cost of Eq. (21). 4. Solution procedure The equilibrium solutions of Eqs. (9) and (10) are developed based on the repeated game model integrated with Tabu search algorithm (refer to Fig. 9). And Eq. (21) in the model with after standardization is solved by Tabu search. 4.1. Repeated game model In game theory, a repeated game is defined to be an extensive type of game which consists of some number of repetitions of base game known as well-studied 2-person games. Generally, when the repeated game performs a sufficiently long time, an equilibrium solution can be obtained (Benoit & Krishna, 1985). For the model with before standardization, we firstly assume that the values of the order-up-to level (SB), and the unit buy-back price (cbB) of brewer B are given as an initial solution. And then, by Tabu search the best solution (SA, cbA) is found that minimizes the total expected cost of brewer A. Similarly, with the values of SA and cbA of brewer A determined in previous step, brewer B finds the best order-up-to level (SB), and the unit buy-back price (cbB) using Tabu search. Repeating the above procedure we obtain an equilibrium solution. 4.2. Tabu search Tabu search is a higher level heuristic procedure for solving optimization problems, designed to guide other methods to escape the trap of local optimality. Tabu search has obtained optimal and near optimal solutions to a wide variety of classical and practical problems in applications ranging from character recognition to neural networks. It uses flexible structures memory, conditions for strategically constraining and freeing the search process, and memory functions of varying time spans for intensifying and diversifying the search (Glover, 1990). The Tabu search starts at an initial point and then moves to one of the neighborhood points that give the best result of the objective function at each iteration. And this move continues until a stopping criterion has been met. The method forbids points with certain attributes with the goals of preventing cycling and guiding the search towards unexplored regions of the solution space. This is done using an important feature of the Tabu search method called tabu list which consists of the latest moves made. In its simplest form, tabu search requires the following elements:
Initial point Neighborhood generation method Tabu list Aspiration criterion Stopping criterion
The Tabu search algorithm starts at an initial point, as ðS0A ; S0B ; c0b ; c0 Þ in the case of the model with after standardization, and it goes through a number of iterations until the stopping criterion is met. At each iteration, r numbers of directions are randomly generated, and line search is performed for each direction. The point which gives the best objective value among the r candidate neighborhood points is selected as the next point if the direction is non-tabu, or it satisfies the aspiration criterion. After moving to the next point, the direction of the latest move is newly stored in the tabu list. This procedure is repeated until the stop criterion
Fig. 9. The solution procedure.
is satisfied. Generally in Tabu search, two kinds of stop criterion have been used: One in terms of a total elapsed number of iterations, and the other finding the last best solution again. The former stop criterion is adopted in this study (see Fig. 10). Step 1. Initialization Choose the number of random search direction to be used at each iteration (r) Choose the maximum number of iteration (MAX_ITER) Choose an initial point x0 ¼ ððS0A ; S0B ; c0b ; c0 ÞÞ Let 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 that i ATCðxj þ k d Þ ¼ min ATCðxj þ ki d Þ 16i6r 2.2 Check tabu list If (d 2 TLÞorðd 2 TL and ATCðxj þ k d Þ< Best value), go to step 2.3. Otherwise choose second best solution and repeat step 2.2 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 criterion If 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 an example 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 both models. The results are consistent with our expectation. Compared with the model with before standardization, standardization causes a substantial amount of cost reduction, i.e.,
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Fig. 10. The basic procedure of Tabu search.
Table 1 The result of the model with before standardization.
Brewer A Brewer B Sum
S
cb
Return rate
Total expected cost
990,000 389,900 1379,900
82.43 82.43 NA
0.8999 0.8999 NA
82,692,884 28,983,294 111,676,179
Table 2 The result of the model with after standardization.
Without inv. pooling With inv. pooling
SA + SB
cb
Return rate
Total expected cost
1378,500 1313,500
82.50 82.48
0.9001 0.9000
100,984,662 100,737,565
9.574% = (111,676,179–100,984,662)/111,676,179 = 0.09574. Note that Inventory pooling further reduces the total cost from 9.574% with standardization alone to 9.795% = (111,676,179– 100,737,565)/111,676,179 = 0.09795. Standardization have only slight 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 standardization, brewers no longer pay the inspection cost of the collected bottles and transportation cost due to the exchange process. Inventory pooling has an effect of slightly decreasing the order-up-to level, which implies that the two brewers can run their business with s smaller inventory. 5.2. Sensitivity tests The sensitivity analysis was performed on the system parameters which we think are closely related with the interests of the government as well as the brewers. For the government point of view we investigated the effect of obligatory take back quota and the penalty cost on the actual return rate. The standard deviation
Fig. 11. Actual return rate vs. obligatory take-back quota.
of demand, the proportion of the market segment of large orders, and unit transshipment cost were selected for the study reflecting the brewers’ interests. Since the sensitivity analysis show similar results in all three situations, here we present them only for the model with after standardization with lateral transshipment for the sake of brevity. Fig. 11 shows the change of the actual return rate for various values of the obligatory take-back quota. It can be observed that the actual return rate remains approximately at the level of 0.85 if the quota is not larger than 0.8. When the obligatory takeback-quota increases beyond 0.8 the actual return rate increases that calls for the brewers to pay higher unit buy-back price to avoid penalty cost. The relationship between the actual return rate and the penalty cost is shown in Fig. 12. We tested the penalty cost from 0 to 360 with an increment of 30. It is observed that the actual return rate converges to the level of the obligatory take-back quota (=0.9) as the penalty cost increases. Figs. 13 and 14 show the changes of the total expected cost and the sum of the order-up-to-level (SA + SB) as the standard deviation
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Fig. 12. Actual return rate vs. penalty cost.
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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. 13. Total expected cost vs. standard deviation of demands.
Fig. 17. Total expected cost vs. standard deviation of demands. Fig. 14. Order-up-to-level vs. standard deviation of demands.
of demands varies. Larger standard deviation of demand could incur more chances of having lost sales and higher inventories. As expected, the total expected cost increases as the standard deviation of demands increases. Also, the order-up-to-level increases in order to reduce the chances of lost sales. Figs. 15–17 show the responses of the actual return rate, buyback price and the total expected cost for the changing sales rate in the market segment with large orders such as from bars and restaurants. When the sales rate for large orders increase, the brewers can easily collect used bottle from those places without much effort. Especially, when the ratio is more than 0.8 which implies that enough used bottles are coming from the bars and restaurants, they are in no need of paying more than the unit handling commission to individual consumer for used bottle. Thus the total expected cost decreases. Fig. 18 shows the change of the sum of the order-up-to-level for the various values of the unit transshipment costs. It can be observed that lower unit transshipment cost triggers intensifying
Fig. 18. Order-up-to-level vs. unit transshipment cost.
inventory pooling effect. The sum of the order-up-to-level is reduced due to the low unit transshipment cost that makes the coordination between two brewers much easier.
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6. Conclusions Motivated by a report that standardization of glass bottles resulted in a substantial amount of cost reduction in Korean beverage industry, this study analyzes the cost benefits of a recycling system with two competing brewers. For the analysis, we develop the mathematical models in two situations, a case with before standardization (the model with before standardization) and another case with after standardization (the model with after standardization). To facilitate the analysis it is assumed that the demand of each brewer is a random variable following normal distribution with known mean and variance, and the return rate of used bottles is a function of the unit buy-back price. We find that the inspection and exchange process become no longer needed after standardization, which contributes to a reduction of the total expected cost. Additional cost reduction is possible through inventory pooling via lateral transshipment between two brewers. The second case is further examined by two sub-cases, with and without inventory pooling. An equilibrium solution of the first case is found based on the repeated game approach integrated with Tabu search. For the second case only Tabu search algorithm is utilized. To illustrate the model, a hypothetical problem is solved. The results confirm our expectation that standardization of products for common use can yield a substantial amount of cost benefits. As further studies our model could be improved by regarding the bottles collected during a given period as a weighted sum of the glass bottles produced during recent finite number of past periods. Another possibility is to extend the current model to the case of more than two brewers for the development of generic model. We hope that the study results would contribute in developing policies for preserving clean environment by the government officials as well as business managers. Acknowledgement This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2007313-D00909). References Benoit, J. P., & Krishna, V. (1985). Finitely repeated games. Econometrica, 53(4), 905–922. Cho, D. I., & Parlar, M. (1991). A survey of maintenance models for multi-unit systems. European Journal of Operational Research, 51(1), 1–23. Dobos, I., & Richter, K. (2000). The integer EOQ repair and waste disposal modelfurther analysis. Central European Journal of Operations Research, 8, 173–194.
Dobos, I., & Richter, K. (2003). A production/recycling model with stationary demand and return rates. Central European Journal of Operations Research, 11(1), 35–46. Dobos, I., & Richter, K. (2004). An extended production/recycling model with stationary demand and return rates. International Journal of Production Economics, 90(3), 311–323. Dobos, I., & Richter, K. (2006). A production/recycling model with quality consideration. International Journal of Production Economics, 104(2), 571–579. Glover, F. (1990). Tabu search: A tutorial. Interfaces, 20(4), 74–94. Gross, D. (1963). Centralized inventory control in multi location supply systems, Multistage Inventory Models and Techniques. Stanford: Stanford University Press. Hekkert, M., Joosten, L., & Worrell, E. (1998). CO2 emission reduction by improved use of packaging materials. ConAccount Workshop Ecologizing Societal Metabolism, 175, 182. Jaber, M. Y., Goyal, S. K., & Imran, M. (2008). Economic production quantity model with imperfect quality subject to learning effect. International Journal of Production Economics, 115(1), 143–150. Karmarkar, U. S. (1979). Convex/stochastic programming and multi location inventory problems. Naval Research Logistics Quarterly, 26, 1–19. Karmarkar, U. S., & Patel, N. R. (1977). The one-period N-location distribution problem. Naval Research Logistics Quarterly, 24, 559–575. Kelly, P., & Silver, E. A. (1989). Purchasing policy of new containers considering the random returns of previously issued containers. Naval Research Logistics Quarterly, 24, 385–405. Ko, Y. D., & Hwang, H. (2009). Efficient operation policy in a closed-loop tire manufacturing system with EPR. International Journal of Industrial Engineering and Manufacturing Systems, 8(3), 162–170. Lee, J. E., Gen, M., & Rhee, K. G. (2009). Network model and optimization of reverse logistics by hybrid genetic algorithm. Computers & Industrial Engineering, 56, 951–964. Liu, N., Kim, Y., & Hwang, H. (2009). An optimal operating policy for the production system with rework. Computers & Industrial Engineering, 56, 874–887. Mabini, M. C., Pintelon, L. M., & Gelders, L. F. (1992). EOQ type formulation for controlling repairable inventories. International Journal of Production Economics, 28(1), 21–33. Nahmias, S., & Rivera, H. (1979). A deterministic model for a repairable item inventory system with a finite repair rate. International Journal of Production Research, 17(3), 215–221. OECD (1999). Extended producer responsibility: A guidance manual for governments. France: Paris. Richter, K., & Dobos, I. (1999). Analysis of the EOQ repair and waste disposal model with integer setup numbers. International Journal of Production Economics, 59(1– 3), 463–467. Schrady, D. A. (1967). A deterministic inventory model for repairable items. Naval Research Logistics Quarterly, 14(3), 391–398. Tang, O., & Grubbstrom, R. W. (2005). Considering stochastic lead times in a manufacturing/ remanufacturing system with deterministic demands and returns. International Journal of Production Economics, 93–94, 285–300. Targas, G. (1989). Effects of pooling on the optimization and service levels of twolocation inventory systems. IIE Transactions, 21(3), 250–257. Teunter, R. H. (2001). Economic order quantities for recoverable item inventory system. Naval Research Logistics, 48(6), 484–495. Teunter, R. H., & Vlachos, D. (2002). On necessity of a disposal option for returned items that can be remanufactured. International Journal of Production Economics, 75(3), 257–266. Wong, H., Oudheusden, D. V., & Cattrysse, D. (2007). Cost allocation in spare parts inventory pooling. Transportation Research Part E, 43(4), 370–386.
Contents lists available at SciVerse ScienceDirect
Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
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
Article history: Available online 3 December 2011 Keywords: Glass bottle Inventory pooling Reverse logistics Standardization
a b s t r a c t This study deals with a recycling system with two competing brewers. It is assumed that they coordinate their manufacturing operations through standardization of their glass bottles for easy implementation of extended 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 become no longer necessary. Secondly, cost reduction is achieved through streamlining of collection and reuse processes. Finally, under the stochastic demand of glass bottles their inventory holding costs and lost sales cost are reduced via inventory pooling. Through the development of the mathematical models we determine an optimal operation policy of the two brewers that maximizes the sum of benefits obtained 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 function value and decision variables. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction 1.1. Background In classical logistics systems the material and related information flow are managed in forward direction, i.e., from raw materials to the final products delivered to the customers. In reverse logistics, backward flow is managed, i.e., the used and reusable parts and products return from the customers to the producers. In this way natural resources can be saved for future generations and firms can contribute to the sustainable development efforts (Dobos & Richter, 2004). There are several reasons why heightened attention has been paid to reverse logistics in the past decades: Firstly, producers and consumers became more environmentally conscious, and started to realize that it is time to abandon the ‘throw – away age’. Secondly, tighter legislation in some countries forced producers to take back products after use and either recover them or dispose of them properly. Thirdly, some producers realized that recovery operations can lead to additional profits (Teunter & Vlachos, 2002). Extended producer responsibility (EPR) is a strategy designed to promote the integration of environmental costs associated with products throughout their life cycles into the market price of the products. Under EPR, firms are obligated to meet a given take back quota for the end of used products, and certain amount of penalty will be charged if it is breached (OECD, 1999). Fifteen countries in ⇑ Corresponding author. Tel.: +82 42 350 3113; fax: +82 42 350 3110. E-mail address: [email protected] (H. Hwang). 0360-8352/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2011.11.026
Europe, including Germany, the United Kingdom, France and Hungary, four countries in Asia including Korea, Japan, Taiwan and Australia, and countries in Latin America including Mexico and Brazil, have introduced the EPR system. The number of items that are controlled by extended producer responsibility is increasing as the industries become more complex and the laws and regulations on environmental issues are tightened (Ko & Hwang, 2009). Glass bottle, as a representative of packaging materials, is one of the most important items that need to be controlled by EPR legislation. And also, in terms of carbon dioxide (CO2) emission reduction, higher reuse rate of used bottles and more cost-effective inventory policy becomes significant (Hekkert, Joosten, & Worrell, 1998). Recently, seven major brewing companies in Korea standardized the shapes and colors of their bottles to avoid the costly sorting and exchange procedures in retrieving empty bottles. They reported that in 2007 alone, about 24 billion glass bottles were standardized, which resulted in an annual cost saving of approximately US 40 million dollars. The standardization also enabled the brewers to enjoy the benefit of inventory pooling, the practice of using a common pool of stock for satisfying two or more sources of random demands. To be more specific, it refers to an arrangement in which different companies or stocking points share their inventories and has been proven to be an effective strategy in improving companies’ logistical performances while reducing the total system cost at same time. In this arrangement, lateral transshipments are used to satisfy the demand of a company that is out 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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companies. In this paper we investigate the sources of cost reductions through the development of mathematical model and then discuss desirable management policies of government and business circle for more effective environment protection efforts. 1.2. Problem description We deal with a reverse logistics system with two brewers, A and B, and consider the system in two situations, before and after standardization of glass bottles. Fig. 1 depicts the model with before standardization and shows the flow of glass bottles before standardization, where products are sold in two different types of glass bottles. For the development of models the following assumptions are made: The demand of glass bottles during a unit time interval is satisfied by either reused bottles or newly purchased bottles. The demand follows a normal distribution with known mean and variance. The market consists of two segments, one with large orders such as bars and restaurants, and the other with smaller orders such as individual customers. All the bottles sold to the first market segment can be retrieved in its entirety at the cost of a given unit handling commission chd while the return rate of used bottles from the second segment is dependent on the unit buy-back price. That is, the retrieval rate from individual customer can be expressed as a function of the amount brewers are willing to pay cb. Through inspection and sorting processes, some bottles corresponding to a proportion of d are found to be in wrong hands and thus those bottles are to be exchanged between the brewers incurring the exchange cost. If the obligatory take back quota b is breached, then brewers have to pay the unit penalty cost cpnt for the unsatisfied amount. The brewers adopt order-up-to S inventory policy for serviceable bottles. 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 bottles are jointly collected with an identical buy-back price which are to 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 exchange process do not exist. Furthermore, through inventory pool-
ing they can expect to have a further reduction in inventory cost and lost sales cost while incurring the associated transportation cost. The decision variables are the order-up-to level of each brewer, the joint unit buy-back price, and the allocation ratio of the collected bottles. In this paper, through the development of the mathematical models we determine analytically the cost benefits obtained from standardization. The benefit is measured by the difference in the sum of each brewer’s cost. 1.3. Previous studies A deterministic EOQ-type reverse logistics model was first addressed by Schrady (1967). In his model, it is assumed that manufacturing and remanufacturing rate is infinite without waste disposal, where a single manufacturing and more than one remanufacturing cycles occur. The generalized model of the Schrady’s for the case of finite recycling rate was presented by Nahmias and Rivera (1979). Also, a multi product extension of these models was examined by Mabini, Pintelon, and Gelders (1992). A model with different inventory holding cost rates for manufactured and remanufactured items was studied by Teunter (2001). In these models, the return rate of used products is considered as a given parameter. Dobos and Richter (2000 and 2003) and Richter and Dobos (1999) introduced the concept of waste disposal in the recycling models, where the return rate is considered as decision variable. One of the common unrealistic assumptions in the EOQ based models is that all units manufactured and remanufactured are of good quality. Dobos and Richter (2006) extended their models considering quality of the bought back products, where some part of returned products are discarded due to the poor quality condition. Jaber, Goyal, and Imran (2008) studied a model, where the rate of defective items decreases as the number of shipments increases by the effect of learning. Lee, Gen, and Rhee (2009) proposed the remanufacturing system with three-stage logistics network model for minimizing the total costs considering multistage and multi-product. They suggested hybrid genetic algorithm as a solution methodology. Liu, Kim, and Hwang (2009) modeled a production inventory system with rework, where a stationary demand can be satisfied with both new product and reworked
Fig. 1. The model with before standardization.
Y.D. Ko et al. / Computers & Industrial Engineering 62 (2012) 693–702
695
Fig. 2. The model with after standardization.
defective product. Ko and Hwang (2009) developed a model, where the return rate of used products is treated as a function of the unit buy-back price for used products and the minimum allowed quality level. In this model, newly manufactured products and remanufactured products can have different retail prices. The stochastic version of reverse logistics systems is also proposed by various authors as well. Cho and Parlar (1991) surveyed the literature about the optimal maintenance and replacement models for multi-unit systems. Kelly and Silver (1989) studied a system, where recovered items can be immediately reused with the assumption of random returns and a fixed lead time. Tang and Grubbstrom (2005) studied a manufacturing/remanufacturing system with stochastic lead times and a constant demand. In his model, it is assumed that there are two supply sources for replenishing serviceable inventories. The two-location inventory system, where inventory pooling occurs through lateral transshipment is first addressed by Gross (1963). In his model, the optimal ordering and transshipment quantities for the system that minimizes the system costs are determined. Karmarkar and Patel (1977) and Karmarkar (1979) generalized the two-location problem for the case of multi location inventory distribution systems. Targas (1989) studied the effect of inventory pooling via lateral transshipment on the service levels realized at the stocking points, and, conversely, the effect of service level constraints on the optimal order-up-to level and transshipment policy. Recently, Wong, Oudheusden, and Cattrysse (2007) dealt with the cost allocation problem with spare parts inventory pooling.
clsi chi creui cinsi cex ctrAB ctrbs cpnt r(cbi)
ai b
c di b ASi Si Di R0i Ri Iþ i I I Qi f(yi)
li ri
unit lost sales cost unit inventory cost of serviceable bottles unit cost for reuse unit inspection cost of used bottles unit exchange cost unit transportation cost between two brewers A and B unit transportation cost between brewer and storage area unit penalty cost for unsatisfied take-back-quota return rate of used bottles ratio of large order in demand rbligatory take-back-quota allocation ratio of collected bottles fraction of collected bottles that requires exchange parameter of the return rate function amount sold order-up-to level demand quantity of the product number of collected bottles before exchange number of collected bottles after exchange on-hand inventory of bottles amount of shortage of bottles ordering quantity of new bottles density function of brewer i’s demand of unit time period mean value of brewer i’s demand of unit time period standard deviation of brewer i’s demand of unit time period
2. Notations and assumption 2.1. Notations i cpi cbi chdi
i = A (brewer A), B (brewer B) unit purchasing cost of new bottles unit buy-back price of used bottles unit handling commission
Fig. 3 shows the return rate function in an exponential form. It is assumed that the brewers can collect a larger quantity of used bottles if they pay higher unit buy-back price. The brewers are obligated by law to pay the handling commission per a used bottle to the middle distribution channels which are in charge of selling and recovery of the products. Thus the handling commission can be considered as the low bound of unit buy-back price. Also, purchasing new bottles is preferable to buying back used bottles if
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Fig. 3. The return rate of used bottles.
the sum of unit buy-back price and unit reuse cost is higher than the unit new-bottle purchasing cost. 2.2. Assumptions 1. The demand of each brewer during unit time period is an independent and identically distributed random variable and follows normal distribution with known mean and variance. 2. The return rate of used bottles is dependent on the unit buyback price. 3. The bottles sold in a given period are available for reuse for the 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.
Fig. 4. Inventory related components in case 1 and 2.
The expected amount of collected bottles after exchange:
E½RA jDA 6 SA ¼ ð1 dA ÞE½R0A jDA 6 SA þ dB E½R’B where
E½R0A jDA 6 SA ¼ faA þ rðcbA Þð1 aA ÞgE½DA jDA 6 SA
3. Mathematical model For our objective function eleven different cost elements are considered. They are the inventory holding, lost sales cost, buyback cost of used bottles, inspection cost, handling commission paid to the market segment with large orders, reuse cost, exchange cost, transportation cost between two brewers, transportation cost between the brewers and storage area, purchasing cost of new bottles, and penalty cost related to obligatory take-back quota. Two models are developed in this section, the model with before standardization depicted in Fig. 1 and the model with after standardization in Fig. 2.
E½R0B ¼ faB þ rðcbB Þð1 aB ÞgE½ASB E½ASB ¼ PðDB 6 SB ÞE½DB jDB 6 SB þ PðDB > SB ÞSB The expected quantity of new bottles purchased:
E½Q A jDA 6 SA ¼ SA E½IþA E½RA jDA 6 SA The total expected cost
C 1A ðSA ; cbA Þ
¼
0
ð3Þ
(SA, cbA)) of case 1 becomes
þ cbA rðcbA Þð1 aA ÞE½DA jDA 6 SA
cex dA E½R0A jDA 6 SA þ ðctrAB þ cex ÞdB E½R0B þ creuA E½RA jDA 6 SA þ cpA E½Q A jDA 6 SA þ cpnt MaxfbE½DA jDA 6 SA E½RA jDA 6 SA ; 0g
The total expected cost is formulated considering the relative size 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-back price, inspection cost, exchange cost, remanufacturing cost, penalty cost (if the take-back quota is breached), and purchasing cost of new bottles. In this case, no shortage occurs. Let Iþ A be the amount of inventories incurred after satisfying demand. Then (SA Iþ A RA) number of new bottles is purchased to stock up to the target inventory level SA, where
Z
chA E½IþA
(C 1A
þ chdA aA E½DA jDA 6 SA þ cinsA E½R0A jDA 6 SA
3.1. Model with Before Standardization
E½IþA ¼
ð2Þ
ð4Þ
Case (2) DA > SA The expected cost of case 2 consists of lost sales cost, buy-back price, inspection cost, exchange cost, remanufacturing cost, penalty cost (if the take-back quota is breached), and new-bottle purchasing cost. In this case, SA number of bottles are sold and lost sales I A may occur. The order size of new bottles becomes SA RA, where
E½IA ¼
Z
1
SA
ðyA SA Þf ðyA ÞdyA
ð5Þ
The expected amount of collected bottles after exchange:
SA
ðSA yA Þf ðyA ÞdyA
ð1Þ
E½RA jDA > SA ¼ ð1 dA ÞE½R0A jDA 6 SA þ dB E½R0B
ð6Þ
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where
E½R0A jDA > SA ¼ faA þ rðcbA Þð1 aA ÞgSA E½R0B ¼ faB þ rðcbB Þð1 aB ÞgE½ASB E½ASB ¼ PðDB 6 SB ÞE½DB jDB 6 SB þ PðDB > SB ÞSB The expected quantity of new bottles purchased:
E½Q A jDA > SA ¼ SA E½RA jDA > SA The total expected cost of case 2,
C 2A ðSA ; cbA Þ
¼
ð7Þ C 2A
(SA, cbA) can be expressed as
clsA E½IA
þ cbA rðcbA Þð1 aA ÞSA þ chdA aA SA þ cinsA E½R0A jDA > SA cex dA E½R0A jDA > SA
Fig. 6. Inventory related components of case 2 in the model with after standardization.
þ ðctrAB þ cex ÞdB E½R0B þ creuA E½RA jDA > SA þ cpA E½Q A jDA > SA þ cpnt MaxfbSA E½RA jDA 6 SA ; 0g ð8Þ Combining Eqs. (4) and (8), the total expected cost of brewer A can be written as:
C A ðSA ; cbA Þ ¼ PðDA 6 SA Þ C 1A ðSA ; cbA Þ þ PðDA > SA Þ C 2A ðSA ; cbA Þ
ð9Þ
Similarly, the total expected cost of brewer B is
C B ðSB ; cbB Þ ¼ PðDB 6 SB Þ C 1B ðSB ; cbB Þ þ PðDB > SB Þ C 2B ðSB ; cbB Þ
E½RB jDA 6 SA and DB 6 SB ¼ aB E½DB jDB 6 SB þ ð1 cÞE½TRjDA 6 SA and DB 6 SB where
E½TRjDA 6 SA and DB 6 SB ¼ rðcb Þfð1 aA ÞE½DA jDA 6 SA þ ð1 aB ÞE½DB jDB 6 SB g
ð10Þ
We are to find the best order-up-to level S (Si), and the unit buyback price of each manufacturer (cbi) which minimize the total expected cost. (i = A, B)
The total expected cost of case 1 for the system, (C 1AþB (SA, SB, cb,
c)) can be formulated as C 1AþB ðSA ; SB ; cb ; cÞ ¼ chA E½IþA þ chB E½IþB þ ðcb þ ctrbs ÞE½TRjDA 6 SA and DB 6 SB þ chdA aA E½DA jDA 6 SA
3.2. Model with After Standardization
þ chdB aB E½DB jDB 6 SB þ creuA E½RA jDA 6 SA
Compared to the current practice model, the cost terms for inspection and exchange are no longer needed in the objective function while the transportation cost between brewers and storage 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 buyback price, remanufacturing cost, inventory cost, penalty cost (if the 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 in þ Fig. 5, ðIþ A ; I B Þ P 0. Thus the expected inventory of each brewer can be written as:
and DB 6 SB þ creuB E½RB jDA 6 SA
E½IþA ¼
E½IþB ¼
Z
SA
ðSA yA Þf ðyA ÞdyA
ð11Þ
ðSB yB Þf ðyB ÞdyB
ð12Þ
0
Z 0
SB
The expected amount of return:
E½RA jDA 6 SA
and DB 6 SB E½RB jDA 6 SA
and DB 6 SB Þ þ cpnt MaxfbðE½DA jDA 6 SA þ E½DB jDB 6 SB Þ E½RA jDA 6 SA E½RB jDA 6 SA
and DB 6 SB
and DB 6 SB ; 0g
ð15Þ
Case (2) DA > SA and DB > SB The total expected cost of two brewers consists of buy-back price, remanufacturing cost, lost sales cost, penalty cost (if the take-back quota is breached in terms of total sales amount), and new-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 the total expected costs of case 2, C 2AþB (SA, SB, cb, c), becomes
C 2AþB ðSA ; SB ; cb ; cÞ ¼ clsA E½IA þ clsB E½IB þ ðcb þ ctrbs ÞE½TRjDA > SA and DB > SB þ chdA aA SA þ chdB aB SB
E½RA jDA 6 SA and DB 6 SB ¼ aA E½DA jDA 6 SA þ cE½TRjDA 6 SA and DB 6 SB
and DB 6 SB þ cp ðSA þ SB E½IþA E½IþB
ð13Þ
þ creuA E½RA jDA > SA
and DB > SB
þ creuB E½RB jDA > SA
and DB > SB
þ cp ðSA þ SB E½RA jDA > SA E½RB jDA > SA
and DB > SB
and DB > SB Þ
þ cpnt MaxfbðSA þ SB Þ E½RA jDA > SA and DB > SB E½RB jDA > SA
and DB > SB ; 0g ð16Þ
Fig. 5. Inventory related components of case 1 in the model with after standardization.
Case (3) DA 6 SA and DB > SB Fig. 7 shows the inventory status of the two brewers, A with positive inventory and B with lost sales. Through inventory pooling, brewer A can help B by supplying the bottles in inventory with B. As results, both the inventory holding cost of A and the lost sales cost of B can be reduced while transportation costs between two brewers occur.
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Fig. 7. Inventory related components of case 3 in the model with after standardization.
Case (3–1) E½Iþ A > E½IB The expected lost sales of brewer B becomes zero, and the ex pected inventory of brewer A decreases to E½Iþ A E½I B . The ex pected amount of lateral transshipment is E½IB , and the total expected 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 the take-back quota is breached in terms of total sales amount), and new-bottle purchasing cost. The sum of the total expected costs of case 3–1, (C 31 AþB (SA, SB, cb, c)), becomes
þ C 31 AþB ðSA ; SB ; c b ; cÞ ¼ c hA fE½I A E½I B g þ ðcb þ c trbs ÞE½TRjDA 6 SA
and DB > SB þ chdA aA E½DA jDA 6 SA þ chdB aB fSB þ E½IB g þ ctrAB E½IB þ creuA E½RA jDA 6 SA
and DB > SB
þ creuB E½RB jDA 6 SA
and DB > SB
þ cp ðSA þ SB fE½IþA E½IB g E½RA jDA 6 SA and DB > SB E½RB jDA 6 SA and DB > SB Þ þ cpnt MaxfbðE½DA jDA 6 SA þ SB þ E½IB Þ E½RA jDA 6 SA
and DB > SB E½RB jDA 6 SA
and DB > SB ; 0g E½Iþ A
ð17Þ
E½I B
Case (3–2) þ The expected lost sales of brewer B decreases to E½I B E½IA , and the expected inventory of brewer A becomes zero. The expected amount of lateral transshipment is E½Iþ A , and the total expected sales amount is increased by the amount of E½Iþ A . Total expected cost of two brewers consists of buy-back price, remanufacturing cost, lost sales cost (B), penalty cost (if the take-back quota is breached in terms of total sales amount), and new-bottle purchasing cost. The sum of the total expected costs of case 3–2, (C 32 AþB (SA, SB, cb, c)), becomes þ C 32 AþB ðSA ; SB ; c b ; cÞ ¼ c lsB fE½I B E½I A g þ ðcb þ c trbs ÞE½TRjDA 6 SA
and DB > SB þ chdA aA E½DA jDA 6 SA and DB > SB
þ creuB E½RB jDA 6 SA
and DB > SB
þ cp ðSA þ SB E½RA jDA 6 SA
DB > SB E½RB jDA 6 SA
þ C 41 AþB ðSA ; SB ; cb ; cÞ ¼ c hB fE½I B E½IA g þ ðc b þ c trbs ÞE½TRjDA > SA and DB 6 SB þ chdA aA fSA þ E½IA g þ chdB aB E½DB jDB 6 SB þ ctrAB E½IA þ creuA E½RA jDA > SA and DB 6 SB þ creuB E½RB jDA > SA and DB 6 SB þ cp ðSA þ SB fE½IþB E½IA g E½RA jDA > SA and DB 6 SB E½RB jDA > SA and DB 6 SB Þ þ cpnt MaxfbðSA þ E½IA þ E½DB jDB 6 SB Þ E½RA jDA > SA and DB 6 SB E½RB jDA > SA and DB 6 SB ; 0g ð19Þ þ Case (4–2) E½I A > E½I B þ The expected lost sales of brewer A decreases to E½I A E½I B , and the expected inventory of brewer B becomes zero. The expected amount of lateral transshipment is E½Iþ B , and the total expected sales amount is also increased by the amount of E½Iþ B . The total expected cost of case 4–2 for the two brewers (C 42 AþB (SA, SB, cb, c)):
þ C 42 AþB ðSA ; SB ; cb ; cÞ ¼ c lsA fE½IA E½IB g þ ðc b þ c trbs ÞE½TRjDA > SA
and DB 6 SB þ chdA aA fSA þ E½IþB þ chdB aB E½DB jDB 6 SB þ ctrAB E½IþB þ creuA E½RA jDA > SA
and DB 6 SB
þ creuB E½RB jDA > SA
and DB 6 SB
þ cp ðSA þ SB E½RA jDA > SA E½RB jDA > SA
and DB 6 SB
and DB 6 SB Þ
þ cpnt MaxfbðSA þ SB Þ E½RA jDA > SA DB 6 SB E½RB jDA > SA
and
and DB 6 SB ; 0g ð20Þ
¼ PðDA 6 SA and DB 6 SB Þ C 1AþB ðSA ; SB ; cb ; cÞ þ PðDA > SA
and DB > SB
E½RB jDA 6 SA and DB > SB Þ þ cpnt MaxfbðSA þ SB Þ E½RA jDA 6 SA
þ Case (4–1) E½I A E½I B The expected lost sales of brewer A becomes zero, and the ex pected inventory of brewer A decreases to E½Iþ B E½I A . The expected amount of lateral transshipment is E½I , and the total expected sales A amount is increased by the amount of E½I A as well. The total expected cost of case 4–1 for the two brewers (C 41 AþB (SA, SB, cb, c)):
Combining the four cases, the total expected cost of the two brewers can be found and
þ chdB aB fSB þ E½IþA g þ ctrAB E½IþA þ creuA E½RA jDA 6 SA
Fig. 8. Inventory related components of case 4 in the model with after standardization.
and DB > SB Þ C 2AþB ðSA ; SB ; cb ; cÞ þ PðDA 6 SA ; DB > SB and SA DA P DB SB Þ
and
and DB > SB ; 0g ð18Þ
Case (4) DA > SA and DB 6 SB Fig. 8 shows the inventory related components of case 4. Note that by interchanging A with B in Figs. 8 and 7 can be obtained. Thus the cost functions in case 4 can be developed quite easily from those in case 3.
31 ðSA ; SB ; cb ; cÞ þ PðDA 6 SA ; DB > SB and SA C AþB
DA < DB SB Þ C 32 AþB ðSA ; SB ; c b ; cÞ þ PðDA > SA ; DB 6 SB and DA SA 6 SB DB Þ 41 ðSA ; SB ; cb ; cÞ þ PðDA > SA ; DB 6 SB and C AþB 42 DA SA > SB DB Þ C AþB ðSA ; SB ; cb ; cÞ
ð21Þ
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We want to find the order-up-to level S (SA, SB), and the joint unit buy-back price (cb), and the allocation ratio (c) in a way to minimize the total expected cost of Eq. (21). 4. Solution procedure The equilibrium solutions of Eqs. (9) and (10) are developed based on the repeated game model integrated with Tabu search algorithm (refer to Fig. 9). And Eq. (21) in the model with after standardization is solved by Tabu search. 4.1. Repeated game model In game theory, a repeated game is defined to be an extensive type of game which consists of some number of repetitions of base game known as well-studied 2-person games. Generally, when the repeated game performs a sufficiently long time, an equilibrium solution can be obtained (Benoit & Krishna, 1985). For the model with before standardization, we firstly assume that the values of the order-up-to level (SB), and the unit buy-back price (cbB) of brewer B are given as an initial solution. And then, by Tabu search the best solution (SA, cbA) is found that minimizes the total expected cost of brewer A. Similarly, with the values of SA and cbA of brewer A determined in previous step, brewer B finds the best order-up-to level (SB), and the unit buy-back price (cbB) using Tabu search. Repeating the above procedure we obtain an equilibrium solution. 4.2. Tabu search Tabu search is a higher level heuristic procedure for solving optimization problems, designed to guide other methods to escape the trap of local optimality. Tabu search has obtained optimal and near optimal solutions to a wide variety of classical and practical problems in applications ranging from character recognition to neural networks. It uses flexible structures memory, conditions for strategically constraining and freeing the search process, and memory functions of varying time spans for intensifying and diversifying the search (Glover, 1990). The Tabu search starts at an initial point and then moves to one of the neighborhood points that give the best result of the objective function at each iteration. And this move continues until a stopping criterion has been met. The method forbids points with certain attributes with the goals of preventing cycling and guiding the search towards unexplored regions of the solution space. This is done using an important feature of the Tabu search method called tabu list which consists of the latest moves made. In its simplest form, tabu search requires the following elements:
Initial point Neighborhood generation method Tabu list Aspiration criterion Stopping criterion
The Tabu search algorithm starts at an initial point, as ðS0A ; S0B ; c0b ; c0 Þ in the case of the model with after standardization, and it goes through a number of iterations until the stopping criterion is met. At each iteration, r numbers of directions are randomly generated, and line search is performed for each direction. The point which gives the best objective value among the r candidate neighborhood points is selected as the next point if the direction is non-tabu, or it satisfies the aspiration criterion. After moving to the next point, the direction of the latest move is newly stored in the tabu list. This procedure is repeated until the stop criterion
Fig. 9. The solution procedure.
is satisfied. Generally in Tabu search, two kinds of stop criterion have been used: One in terms of a total elapsed number of iterations, and the other finding the last best solution again. The former stop criterion is adopted in this study (see Fig. 10). Step 1. Initialization Choose the number of random search direction to be used at each iteration (r) Choose the maximum number of iteration (MAX_ITER) Choose an initial point x0 ¼ ððS0A ; S0B ; c0b ; c0 ÞÞ Let 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 that i ATCðxj þ k d Þ ¼ min ATCðxj þ ki d Þ 16i6r 2.2 Check tabu list If (d 2 TLÞorðd 2 TL and ATCðxj þ k d Þ< Best value), go to step 2.3. Otherwise choose second best solution and repeat step 2.2 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 criterion If 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 an example 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 both models. The results are consistent with our expectation. Compared with the model with before standardization, standardization causes a substantial amount of cost reduction, i.e.,
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Fig. 10. The basic procedure of Tabu search.
Table 1 The result of the model with before standardization.
Brewer A Brewer B Sum
S
cb
Return rate
Total expected cost
990,000 389,900 1379,900
82.43 82.43 NA
0.8999 0.8999 NA
82,692,884 28,983,294 111,676,179
Table 2 The result of the model with after standardization.
Without inv. pooling With inv. pooling
SA + SB
cb
Return rate
Total expected cost
1378,500 1313,500
82.50 82.48
0.9001 0.9000
100,984,662 100,737,565
9.574% = (111,676,179–100,984,662)/111,676,179 = 0.09574. Note that Inventory pooling further reduces the total cost from 9.574% with standardization alone to 9.795% = (111,676,179– 100,737,565)/111,676,179 = 0.09795. Standardization have only slight 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 standardization, brewers no longer pay the inspection cost of the collected bottles and transportation cost due to the exchange process. Inventory pooling has an effect of slightly decreasing the order-up-to level, which implies that the two brewers can run their business with s smaller inventory. 5.2. Sensitivity tests The sensitivity analysis was performed on the system parameters which we think are closely related with the interests of the government as well as the brewers. For the government point of view we investigated the effect of obligatory take back quota and the penalty cost on the actual return rate. The standard deviation
Fig. 11. Actual return rate vs. obligatory take-back quota.
of demand, the proportion of the market segment of large orders, and unit transshipment cost were selected for the study reflecting the brewers’ interests. Since the sensitivity analysis show similar results in all three situations, here we present them only for the model with after standardization with lateral transshipment for the sake of brevity. Fig. 11 shows the change of the actual return rate for various values of the obligatory take-back quota. It can be observed that the actual return rate remains approximately at the level of 0.85 if the quota is not larger than 0.8. When the obligatory takeback-quota increases beyond 0.8 the actual return rate increases that calls for the brewers to pay higher unit buy-back price to avoid penalty cost. The relationship between the actual return rate and the penalty cost is shown in Fig. 12. We tested the penalty cost from 0 to 360 with an increment of 30. It is observed that the actual return rate converges to the level of the obligatory take-back quota (=0.9) as the penalty cost increases. Figs. 13 and 14 show the changes of the total expected cost and the sum of the order-up-to-level (SA + SB) as the standard deviation
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Fig. 12. Actual return rate vs. penalty cost.
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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. 13. Total expected cost vs. standard deviation of demands.
Fig. 17. Total expected cost vs. standard deviation of demands. Fig. 14. Order-up-to-level vs. standard deviation of demands.
of demands varies. Larger standard deviation of demand could incur more chances of having lost sales and higher inventories. As expected, the total expected cost increases as the standard deviation of demands increases. Also, the order-up-to-level increases in order to reduce the chances of lost sales. Figs. 15–17 show the responses of the actual return rate, buyback price and the total expected cost for the changing sales rate in the market segment with large orders such as from bars and restaurants. When the sales rate for large orders increase, the brewers can easily collect used bottle from those places without much effort. Especially, when the ratio is more than 0.8 which implies that enough used bottles are coming from the bars and restaurants, they are in no need of paying more than the unit handling commission to individual consumer for used bottle. Thus the total expected cost decreases. Fig. 18 shows the change of the sum of the order-up-to-level for the various values of the unit transshipment costs. It can be observed that lower unit transshipment cost triggers intensifying
Fig. 18. Order-up-to-level vs. unit transshipment cost.
inventory pooling effect. The sum of the order-up-to-level is reduced due to the low unit transshipment cost that makes the coordination between two brewers much easier.
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6. Conclusions Motivated by a report that standardization of glass bottles resulted in a substantial amount of cost reduction in Korean beverage industry, this study analyzes the cost benefits of a recycling system with two competing brewers. For the analysis, we develop the mathematical models in two situations, a case with before standardization (the model with before standardization) and another case with after standardization (the model with after standardization). To facilitate the analysis it is assumed that the demand of each brewer is a random variable following normal distribution with known mean and variance, and the return rate of used bottles is a function of the unit buy-back price. We find that the inspection and exchange process become no longer needed after standardization, which contributes to a reduction of the total expected cost. Additional cost reduction is possible through inventory pooling via lateral transshipment between two brewers. The second case is further examined by two sub-cases, with and without inventory pooling. An equilibrium solution of the first case is found based on the repeated game approach integrated with Tabu search. For the second case only Tabu search algorithm is utilized. To illustrate the model, a hypothetical problem is solved. The results confirm our expectation that standardization of products for common use can yield a substantial amount of cost benefits. As further studies our model could be improved by regarding the bottles collected during a given period as a weighted sum of the glass bottles produced during recent finite number of past periods. Another possibility is to extend the current model to the case of more than two brewers for the development of generic model. We hope that the study results would contribute in developing policies for preserving clean environment by the government officials as well as business managers. Acknowledgement This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2007313-D00909). References Benoit, J. P., & Krishna, V. (1985). Finitely repeated games. Econometrica, 53(4), 905–922. Cho, D. I., & Parlar, M. (1991). A survey of maintenance models for multi-unit systems. European Journal of Operational Research, 51(1), 1–23. Dobos, I., & Richter, K. (2000). The integer EOQ repair and waste disposal modelfurther analysis. Central European Journal of Operations Research, 8, 173–194.
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