An inventory approach for a Local Collection Point in

Int. J. Sustainable Society, Vol. 1, No. 1, 2008

An inventory approach for a Local Collection Point in reverse supply chains Qinglong Gou, Lihong Ren and Liang Liang School of Management, University of Science and Technology of China, Hefei, Anhui 230036, PR China E-mail: [email protected] E-mail: [email protected] E-mail: [email protected]

Zhimin Huang* School of Business, Adelphi University, Garden City, New York, NY 11530, USA and School of Business, Renmin University of China, Beijing 100872, PR China Fax: +516 877 4607 E-mail: [email protected] *Corresponding author Abstract: Product recovery has received increasing attention in recent years. For a product recovery system, collection of used products is one of the key concerns. This article focuses on a special Local Collection Point (LCP) which handles a single type of used products. This special LCP can either to recover the used products and resell them in second-hand markets, or to deliver them to the Centralised Returns Center directly. For such a reverse system, we initiate and develop a joint inventory policy to improve its efficiency. The arrival process of used products, the handling/repairing time and the customer arrival process for repaired products are all assumed random. By formulating the system into a continuous-time Markov chain, the system’s stationary distribution and its profit formulation are obtained. An algorithm for optimal inventory policies is developed. Managerial implications of the system characters and policies are discussed through the article. Keywords: Green Supply Chain Management; GrSCM; inventory model; Local Collection Point; LCP; production recovery; reverse supply chains. Reference to this paper should be made as follows: Gou, Q., Ren, L., Liang, L. and Huang, Z. (2008) ‘An inventory approach for a Local Collection Point in reverse supply chains’, Int. J. Sustainable Society, Vol. 1, No. 1, pp.55–84.

Copyright © 2008 Inderscience Enterprises Ltd.

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Q. Gou et al. Biographical notes: Qinglong Gou is a Lecturer in the School of Management at University of Science and Technology of China (USTC). He received his BS in Economics, MS and PhD in Management Science and Engineering from USTC in 2000, 2003, 2007, respectively. His research interests are mainly in supply chain management and reverse logistics. He has published articles in Int. J. Information Technology and Decision Making and other journals. Lihong Ren is a Lecturer at the University of Science and Technology of China (USTC). She received her BS in Economics from USTC. Now, she is a graduate student in Management Science and Engineering at the USTC. Her primary interests lie in the green supply chain management. Liang Liang is a Professor of Management Science and Engineering at the University of Science and the Technology of China (USTC). He received his PhD in System Engineering at the Southeast University. His research interests are mainly in data envelopment analysis, supply chain management, vendor managed inventory, reverse logistics and multi-criteria decision making analysis. He has published articles in Operational Research, European Journal of Operational Research, Int. J. Production Economics, Omega, Journal of Operational Management, Int. J. Production Research, Int. J. Information Technology and Decision Making, Int. J. Global Energy Issue, Computers and Operations research, Expert Systems with Applications, and other journals. Zhimin Huang is a Professor of Operations Management at the Adelphi University and a Guest Professor at the Renmin University of China. He has received his PhD from The University of Texas at Austin. His research interests are mainly in supply chain management, data envelopment analysis, distribution channels, game theory, chance constrained programming theory and multi-criteria decision making analysis. He has published articles in Naval Research Logistics, Decision Sciences, Journal of Operational Research Society, European Journal of Operational Research, Journal of Economic Behavior and Organization, Optimization, Omega, Journal of Mathematical Analysis and Applications, Computers and Operations Research, Int. J. Production Economics and other journals.

1

Introduction

Due to the escalating deterioration of environment (e.g. diminishing raw material resources, overflowing waste sites and increasing levels of pollution), environmental protection has become a major issue in our modern society. With the awareness of the importance of sustainable economic development and environmental protection, product recovery has received increasing attention from both academic researchers and practitioners. According to the level of disassembly and the quality required as well as the resulting products, the recovery options can be divided into three categories: 1

direct re-use or re-sale

2

product recovery management including repairing, refurbishing, remanufacturing, cannibalisation and recycling

3

waste management including incineration and land filling (Thierry et al., 1995).

An inventory approach for a Local Collection Point in reverse supply chains

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As illustrated by Fleischmann et al. (2000), most product recovery systems are involved in the following series of activities, i.e. 1

collection of used products from the consumer

2

determining the condition of the used products by inspection and/or separation to find out whether they are recoverable

3

reprocessing or reconditioning the used products to capture their remaining value

4

disposal of the unrecoverable used products

5

redistribution of the recovered products.

Among the series of activities of product recovery, collection refers to bringing the used products from customers to a recovery point. Collection has become a key concern in companies involved in product recovery. First, collection is indeed the first activities of product recovery and triggers the other activities of the recovery system. Secondly, there exist considerable uncertainties in the collection activities, including the quantity, timing and quality of the returned products. Thirdly, the collection activities are critical in determining the economic viability of the entire recovery process. As stated by Goggin, Reay and Browne (2000), “in many instances, actual recovery process activities are economically viable but the entire business is not, due to collection costs”.

Therefore, the Local Collection Point (LCP) which is also called as a collection center or a collector by researchers, plays a crucial role in the product recovery. Generally, for companies involved in product recovery, strategic problems include the following two important issues: 1

the collection issue, i.e. how to obtain used products from consumers

2

the recovery issue, i.e. where to handle the returned products.

For the collection issue, different types of LCPs are utilised for different products. The LCPs may be local authorities, retailers, commissioned transport firms, distributors and charity organisations, which have a long- or short-term business relationship with the recovers (Goggin, Reay and Browne, 2000). No matter what the specific types are, the LCPs are always located in and familiar with local markets. There are two alternatives to resolve the recovery issue. One is to use in-house distribution centers and the other one is to utilise Centralised Returns Centers (CRCs). In the first situation, forward and reverse distribution services are combined, while in the second situation CRCs are often independent. As argued by Gooley (2003), it is better to handle reverse logistics with an independent CRC for the following reasons: 1

A stand-alone CRC is separate from the forward supply chain, and consequently this gives the company the benefit of a high degree of focus on reverse logistics.

2

Due to economies of scale, the CRC improves efficiency in sorting and repacking activities.

3

Because of the higher volumes, the CRC is able to purchase some specialised assets.

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For instance, in the General Motors’ new reverse supply chain system, dealers send all returned products to a single handling facility. The second question is which activities the LCPs should be responsible for. Most LCPs are responsible only for collection activities, i.e. bringing the used products to the Centralised Returns Center (CRC), but do not recover the used products due to the following reasons: 1

special technologies or assets may be needed for the recovery

2

emissions may occur in the recovery process

3

the LCP cannot gain the economy of scale.

However, as Gooley (2003) argued, handling returned goods at the originating distribution center also may offer significant savings if the transportation costs is really high. Specifically, for some special industries such as TV set, computer and so on, the used products have a relatively high value, the recovered products can be sold easily in a second-hand market, and the transportation cost is relatively high. Therefore, the LCPs of such industries always have certain recovery capability. After recovering the used products, LCPs sell the recovered products in a second-hand market to earn a relatively higher profit. This article is focusing on this type of LCPs which handles single type of used products. Besides its original collection function of bringing the used product from customers to the CRC, the LCP considered here has a certain recovery capability and can sell the repaired products in a second-hand market. The LCP has two possible channels handling the used products after they are collected from customers: 1

The R-Channel(Repair and Resale): to recover these used products and resell them in a local second-hand market with a relatively higher price.

2

The D-channel(Delivery Directly): to deliver them to the CRC directly with a relatively lower price.

Different revenue and costs are occurred for different channels and thus it is necessary to manage those two channels properly. In fact, the special bi-channel reverse system has been utilised in China. As a developing country with a population of 1.3 billion, China faces the largest environmental pressure. First, along with the fast economical development, more and more waste has being produced. Secondly, though more and more Chinese enterprises have recognised the importance of Green Supply Chain Management (GrSCM) and tried to put it into practice, most of these enterprises are lack of experience and necessary tools as well as management skills, the adoption of GrSCM is rather low (Zhu and Sarkis, 2004). As a result, constructing the recycle networks is more likely the government behaviour. As a specific example for used electric and electronic products, now, the Guangdong government is setting up a recycle network in Zhujiang Delta which consists of a CRC and multiple LCPs. According to the environmental plan, the LCPs will spread over the whole Zhujiang Delta, with at least one collection point in each town.

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Further, there is a large population of rural migrant workers in the Zhujiang Delta. The rural migrant workers are generally from villages of Western China, they normally have temporary jobs, and their wages are relatively low. Therefore, they would prefer used products than brand new ones. Considering a large second-hand market of used electric and electronic products, the LCPs always have certain repairing or recovery capability. When they receive used electronic products, they would rather repair and then resell them at a second-hand market than deliver them to the CRCs. Although this type of special bi-channel LCPs is common in China, literatures have paid little attention on it because its scales is relatively low, most of LCPs have just two or three employees. For this special bi-channel reverse system, we introduce a joint inventory control and product recovery policy. The policy is controlled by two variables, i.e. the maximum inventory level S1 for un-repaired products in the D-Channel and S2 for repaired products in the R-Channel. Our objective is to find the optimal S1 and S2 to maximise the system-wide profits. To achieve this objective, we model the system as a continuous-time Markov chain and thus, obtain the system profit formulation. Also, an algorithm for searching optimal solutions is introduced and the solution trends are analysed. The article is organised as follows. Literatures are reviewed in Section 2. Models are developed in Section 3 and the system characters are analysed in Section 4. An algorithm for searching the optimal solution is introduced in Section 5. Solution trends are analysed in Section 6. Finally, the concluding remarks are in Section 7.

2

Literature review

The environmental issues have gained attention for a long time and the respective literature may date back to the 1970s. For instance, as early as 1977, Böttcher and Rembold (1977) studied the optimal location of facilities in a regional system of solidwaste and waste-water disposal, Heyman (1977) intended to optimise the disposal policies for a single item inventory system with product returns. Along with the in-depth study of environmental problems, the following tendencies have been noticed: 1

The discussion about the environment has evolved from end-of-pipe control towards waste prevention at the source by redesigning products and processes.

2

There is a shift from the effect-directed approaches towards the preventive approaches, where the latter suggest an integrated supply chain approach.

By analysing these tendencies carefully, Bloemhof-Ruwaard, van Beek and Hordijk (1995) indicated the necessity of introducing the operational research into the study of environmental issues. Furthermore, the authors advocated integrating the environmental issues into the supply chain modelling and foresaw the possible directions in this area, including waste management, product recovery management, source-directed product management and so on. After the study of Bloemhof-Ruwaard, van Beek and Hordijk (1995), the green supply chain was first introduced as a formal concept by Manufacturing Research Center of the Michigan University in 1996 (Handfield, 1996).

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With a main idea of reducing costs while helping the environment, the GrSCM which is also called the Green Sustainable Supply Chain by some researchers, is defined as “integrating environmental thinking into supply chain management, including product design, material sourcing and selection, manufacturing processes, delivery of the final product to the consumers as well as end-of-life management of the product after its useful life” (Zhu and Sarkis, 2004; Srivastava, 2007).

As illustrated by Srivastava (2007), the study of GrSCM was mainly concentrated on two areas: green design and green operations. The literature on green design emphasised on environmental conscious design and life-cycle assessment/analysis of the product as well as developing an understanding of how design decisions affect a product’s environmental compatibility (Navin-Chandra, 1991; Glantschnig, 1994; Madu, Kuei and Madu, 2002). Similarly, green operations involve all operational aspects related to reverse logistics and network design, green manufacturing and remanufacturing and waste management. Some key issues on green operations include: integrating remanufacturing with internal operations (Ferrer and Whybark, 2001), understanding the effects of competitions among remanufacturers (Majumder and Groenevelt, 2001), integrating product design, product take-back and supply chain incentives (Guide and van Wassenhove, 2001, 2002) and integrating remanufacturing and reverse logistics with supply chain design (Goggin and Browne, 2000; Fleischmann et al., 2001; Savaskan, Bhattacharya and Van Wassenhove, 2004; Chouinard, D’Amours and Ait-Kadi, 2005). We can see that most key issues of green operations mentioned above belong to the research of reverse logistics or product recovery, where the reverse logistics is one of the five key activities for establishing a reverse supply chain (Guide and van Wassenhove, 2002) and comprises network design with aspects of product acquisition (i.e. collection) and remanufacturing (Fleischmann et al., 1997). In fact, the main issues in this area include: 1

reverse network design

2

inventory control and production planning for a product recovery system.

From a logistics point of view, product recovery creates a reverse flow of goods which originates at the customer zones. After the used products have been collected by LCPs, they should be delivered to a point of recovery, i.e. the CRC. Thus, the reverse network design mainly focuses on where should the LCPs and the recovery centers be located. Examples of such facility location-allocation models include Fleischmann et al. (2001), Jayaraman, Patterson and Rolland (2003), Lu and Bostel (2005), Salema, Póvoa and Novais (2006), Min, Ko and Ko (2006) and Aras and Aksen (2008). By taking forward and reverse flow into account, Fleischmann et al. (2001), Lu and Bostel (2005) and Salema, Póvoa and Novais (2006) formulated discrete facility location-allocation models into mixed-integer liner programs. However, they did not take the collection facilities into account which were incorporated by Jayaraman, Patterson and Rolland (2003), Min, Ko and Ko (2006) and Aras and Aksen (2008). Inventory control on reverse systems has played a significant role in the product recovery management. There are two types of models in the literature: deterministic models and stochastic models. Examples for the former include Mabini, Pintelon and Gelders (1992) and Richter (1996a,b), and the latter include Pierskalla and Voelker (1976), Cohen, Nahmias and Pierskalla (1980), Cho and Parlar (1991), Inderfurth (1997),

An inventory approach for a Local Collection Point in reverse supply chains

61

Var der Laan, Dekker and Salomon (1996), Kiesmüller and Scherer (2003), Mahadevan, Pyke and Fleischmann (2003) and DeCroix and Zipkin (2005). As illustrated by Fleischmann et al. (1997), most of these studies were based on an appropriate control mechanism which integrated the return flow of used products into the producers’ materials planning. Kiesmüller and Scherer (2003) provided a method for the exact computation of the parameters which determined the optimal periodic policy for a product recovery system. Fleischmann and Kuik (2003) considered a single inventory point facing independent stochastic demand and item returns, and analysed the item returns impact on inventory control. Mahadevan, Pyke and Fleischmann (2003) focused on the production control and inventory management in the remanufacturing context, where a remanufacturing facility that received a stream of returned products according to a Poisson process was involved in the traditional forward supply chain. DeCroix and Zipkin (2005) considered an inventory system with an assembly structure where uncertain returns from customers were involved. A recent inventory control paper incorporating the collection facilities was Gou et al. (forthcoming), in which a joint policy consisted of the inventory control, delivery and handling decisions of used products was introduced for a reverse system with a single CRC and multiple LCPs. Utilising the knowledge of stochastic process, the authors formulated the CRC’s inventory process into a M/Mr/1/’ queue (a queueing system with Poisson arrivals and bulk services) and an algorithm for searching optimal solution was also presented. In our article, we introduce an inventory policy for a special LCP with bichannels of handling used products. By considering the used product arrival process, the inventory process, the used product repairing process and the selling process simultaneously, the proposed policy can be utilised to improve the efficiency and effectiveness of the LCPs.

3

Model development

We assume that the LCP receives a stream of a single type of used products from consumers which follows a Poisson process with an average arrival rate of Ȝ (see Figure 1). For those used products, the LCP can either sell them directly to the CRC at a relatively lower price w1, or repair them and then resale them with a relatively higher price w2. For convenience, we call the two ways of handling the used products as a D-Channel and a R-Channel respectively. Furthermore, the following are assumed: 1

A LCP can only repair a single product once a time and the repairing time for each product is exponentially distributed with an average length of 1/μ.

2

The customer’s arrival for repaired products follows a Poisson process with an average rate of v. Noting that all the repaired products may have been sold out before a customer arrival, backlog is not allowed.

3

Since the R-Channel is just a complementary of the D-Channel, O > > P and O > > v are assumed.

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Figure 1

Product flow of a Local Collection Point

The inventory control and product return handling policy is determined by two variables S1 and S2, where S1 is the maximum inventory level for un-repaired products in the D-Channel and S2 is for repaired products in the R-Channel. The policy is described as follows: 1

Once the inventory level for un-repaired products reaches its maximum level S1, the LCP delivers/sells all of them to the CRC immediately (at a price of w1).

2

When a repairing process is finished, the LCP will stop repairing used products either under the condition that there is no un-repaired products inventory in its warehouse, or that the inventory of repaired products reaches its maximum level S2.

The cost structure is: 1

h1. the D-Channel inventory holding cost per unit time per unit product

2

h2. the R-Channel inventory holding cost per unit time per unit product

3

c. the acquisition cost for each unit of used product

4

r. the repairing cost per unit product per unit time

5

AD. the delivery cost for each batch of used product from the LCP to the CRC.

The revenue structure is: 1

w1. the price for each unit of un-repaired product

2

w2. the price for each unit of repaired product.

The objective is to decide the optimal S1 and S2 to maximise the system average profit. Let I1 be the inventory level of un-repaired products in the D-Channel, I2 the inventory level of repaired products in the R-Channel, and J the quantity of used products in repairing process. Note that J takes values on 0 or 1, where the value of 1 refers that the LCP is in a repairing process and the value of 0 means otherwise. Then, (I1,J,I2) illustrates LCP’s basic system state. Following the assumptions and the inventory control and product handling policy discussed here, we can obtain the LCP’s state transition diagram shown in Figure 2. In Figure 2, all the possible system states are illustrated. Furthermore, the one-way arrow between two states implies that the system may transfer from one state to another in a short time, and the number [ is the transition rate, where [ = O, P or Q.

An inventory approach for a Local Collection Point in reverse supply chains Figure 2

63

The system state transition diagram

Given current system state (I1(t), J(t), I2(t)), the system state for any later time t + 't ('t > 0) is just related to the current state and has no relationship with that before time t. Thus, the system forms a continuous-time Markov chain. Since the state space is finite and ergodic, the system can reach a stable state. Let p(i1 , j ,i2 ) be the probability that the system is at the state of I1 = i1, J = j, I2 = i2 for any time t, from Figure 2 we have:

O p(0,0,0)

vp(0,0,1)

(1)

(O  v) p(0,0,i2 )

vp(0,0,i2 1)  P p(0,1,i2 1)

(O  v) p(0,0, S2 )

P p(0,1, S2 1)  O p( S11,0,S

(O  P ) p(0,1,0)

2

1, ! , S2  1,

O p(0,0,i2 )  O p( S1 1,1,i2 )  vp(0,1,i2 1)  P p(1,1,i2 1) , i2

1, ! , S2  1

O p(i1 1,1,0)  vp(i1 ,1,1) , i1 1, ! , S1  1

(O  P  v) p(i1 ,1,i2 )

O p(i1 1,1,i2 )  P p(i1 1,1,i2 1)  vp(i1 ,1,i2 1) i1

1, ! , S1  2, i2

1, ! , S2  2

(O  P  v) p( S1 1,1,i2 )

O p( S1  2,1,i2 )  vp( S1 1,1,i2 1), i2 1, ! , S2  2,

(O  P  v) p(i1 ,1, S2 1)

O p(i1 1,1, S2 1)  P p(i1 1,1, S2  2)  vp(i1 1,0, S2 )

(2) (3)

)

O p(0,0,0)  O p( S11,1,0 )  vp(0,1,1)

(O  P  v) p(0,1,i2 ) (O  P ) p(i1 ,1,0)

i2

i1 1,! , S1  2,

(4) (5) (6) (7) (8)

(9)

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Q. Gou et al. (O  P  v) p( S1 1,1, S2 1) (O  v) p(i1  0, S2 )

O p( S1  2,1, S2 1)

(10)

O p(i1 1,0, S2 )  P p(i1 ,1, S2 1) , i1 1, ! , S1  1.

(11)

Let k = i1 + j + i2(S2 + 1) + 1, then k takes value on the set of {1, 2, }, (S1 + 1) (S2 + 1)  1} and has a one-to-one relation with the system states (i1, j, i2). Thus G k is the system state index. Denote S k p(i1 , j ,i2 ) and S (S1 , S 2 , ! , S N ) , where N = (S1 + 1)(S2 + 1)  1, the relations presented by Equations (1)–(11) can be translated into Equation (12), i.e. G SQ 0 (12) where Q {q( k1 ,k2 ) } is the infinitesimal generator of the continuous-time Markov chain. Furthermore, we have G Se 1

(13) T

where e = (1, 1, }, 1) . From Equations (12)–(13), the stationary distribution can be easily calculated. We utilise Example 1 to illustrate a special case of the considered system. Example 1. A special case of the system Consider the case that O = 10, P = 4, Q = 2, S1 = 3, S2 = 3. The system states are shown Table 1. Table 1

The system states of Example 1 (0,0,0)

(0,1,0)

(1,1,0)

(2,1,0)

(0,0,1)

(0,1,1)

(1,1,1)

(2,1,1)

(0,0,2)

(0,1,2)

(1,1,2)

(2,1,2)

(0,0,3)

(1,0,3)

(2,0,3)

–

The infinitesimal generator is

Q

§ 10 ¨ ¨ ¨ ¨ ¨ ¨2 ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ¨ ©

10

· ¸ ¸ ¸ 14 10 4 ¸ 14 4 ¸ ¸ 12 10 ¸ 16 10 4 ¸ ¸ 16 10 2 4 ¸ ¸ 16 2 10 4 ¸ 12 10 2 ¸ ¸ 16 10 2 4 ¸ 16 10 2 4 ¸ 16 2 10 4 ¸ ¸ 12 10 2 ¸ 12 10 ¸¸ 2 2 10 12 ¹¸

14 10 10 2

4

and the corresponding stationary distribution is as Table 2.

An inventory approach for a Local Collection Point in reverse supply chains Table 2

65

The stationary probability of Example 1 0.0033

0.0275

0.0268

0.0246

0.0166

0.0529

0.0502

0.0383

0.0443

0.0951

0.0883

0.0552

0.1601

0.1628

0.1541

–

Once S k

p(i1 , j ,i2 ) is calculated, the average inventory level of un-repaired products is

given by

¦ (i p

H1

1 (i1 , j ,i2 ) )

(14)

i1 , j ,i2

and that of repaired product is

¦ (i p

H2

2 (i1 , j ,i2 ) ).

(15)

i1 , j ,i2

Furthermore, the busy ratio for the repairing facility is

N

¦ ( jp

(i1 , j ,i2 ) ).

(16)

i1 , j ,i2

To formulate the shortage probability of the R-Channel, we utilise the fact that the R-Channel’s shortage happens only under the condition that ‘the current R-channel’s inventory level is zero and a customer arrives before the end of the current repairing process’. For any time point t, denote A1 = {J(t) = 1, I2(t) = 0}, A2 = {J(t) = 0, I2(t) = 0}, and A = A1 + A2. Obviously, the event A represents the fact that the inventory level of repaired products is zero, A1 represents that the inventory level of repaired product is zero and LCP is repairing a used product, whereas A2 refers that the inventory level of repaired products is zero and LCP’s repairing facility is idle. According to the inventory management and repairing policy, A2 = {I1 = 0, J = 0, I2 = 0}, and thus P( A1 )

¦

p(i1, j ,i2 )

(17)

i2 0, j 1

P( A2 )

¦

p(i1, j ,i2 )

(18)

i1 0, j 0,i2 0

Furthermore, assuming the event A happens in time t. Denote X the time interval of the returned product arrivals, Y the time of a repairing process, and Z the time interval of customer arrivals. Also, let BR be the event of R-Channel’s shortage, then BR

{Y ! Z | A1}  { X  Y ! Z | A2 }

(19)

where X, Y, Z follow exponential distribution with average length of 1/O, 1/P, 1/Q, respectively.

66

Q. Gou et al. From the independence between variables X, Y, Z and A1, A2, we have P( BR )

P ( A1 ) P (Y ! Z )  P ( A2 )( X  Y ! Z )

(20)

Also, it is easy to obtain f

P(Y ! Z )

³ ve

 vt

v

˜ e  P t dt

(21)

P v

0

f

P( X  Y ! Z )

³ ve

 vt

P( X  Y ! t )dt

0

f

³ ve 0

 vt

§ t · ¨1  P e  P x ˜ (1  e O (t  x ) dx)dt ¸ ¨ ¸ © 0 ¹

³

(22)

P · v § ˜ 1 . P  v ¨© O  v ¸¹ Combine Equations (17), (18), (20)–(22), we have the R-Channel’s shortage probability as

I

v P vi

2

¦

p(i1 , j ,i2 ) 

0, j 1

v § P · ˜ ¨1  P  v © O  v ¸¹ i

1

¦

p(i1 , j ,i2 )

(23)

0, j 0,i2 0

Specifically, while O > > P and O > > Q, we have

I|

v

¦p

P vi

2

(i1 , j ,i2 )

(24)

0

Note that Q (1  I) is the actual sale rates of the R-Channel, the final used products rate that be conducted by the D-Channel is O  Q (1  I). Then, the average interval between two successive transportations from the LCP to the CRC is E (T )

S1

O  v(1  I )

(25)

and the average transportation cost per unit time is d

AD E (T )

AD (O  v(1  I )) S1

(26)

Let G1, G2 stand for the average revenue per unit time of the D-Channel and the R-Channel respectively, then G1

w1 (O  v(1  I ))

(27)

G2

w2 v(1  I )

(28)

Therefore, the average profit per unit time for the system is PI

G1  G2  (h1 H1  h2 H 2  rN  d  cO ),

(29)

An inventory approach for a Local Collection Point in reverse supply chains

67

i.e. PI

w1 (O  v(1  I ))  w2 v(1  I )  h1

¦ i p

1 (i1 , j ,i2 )

i1 , j ,i2

r

¦

i1 , j ,i2

jp(i1 , j ,i2 ) 

 h2 ¦  i2 p(i , j,i ) 1

2

i1 , j ,i2

AD (O  v(1  I ))  Oc S1

(30)

where the value of I is given by Equation (23). Equation (30) formulates the system average profit. In Equation (30), O, P, Q, w1,w2, h1, h2, AD, c and r are the system parameters, the stationary distribution p(i1 , j ,i2 ) and the R-Channel’s shortage probability I are the intermediate variables, which are implicit functions of the decision variables (S1, S2). Thus, our objective is to find the optimal (S1, S2) in order to maximise the profit.

4

Analyses of the system characters

In this section, we change S1 and S2 over a wide range of values in order to obtain their effects on the main system characters. The system characters that are considered include: 1

The D-Channel’s shortage probability M when the R-Channel requests a used product from it.

2

The R-Channel’s shortage probability I when a customer arrives LCP.

3

The D-Channel’s average inventory level H1.

4

The R-Channel’s average inventory level H2.

5

The busy ratio N of the LCP’s repairing facility.

4.1 The D-channel’s shortage probability M According to the inventory control and repairing policy, the R-Channel will request a used product from the D-Channel when any one of the following two events happens: 1

B1. The R-Channel has just completed a repairing process and the inventory level of recovered products is lower than S2.

2

B2. A customer arrival makes the repaired product inventory level fall from S2 to S2  1.

When the R-Channel requires a used product from the D-Channel, whereas the D-Channel’s used product inventory level is zero, shortage will happen. Let BD(t) denote the event that a D-Channel shortage occurs at time t. To make this clear, we separate time point t into two points, t and t+. Then, the shortage process can be described as follows: a request occurs from the R-Channel at time t (i.e. B1(t) or B2(t) happens) and the

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D-Channel’s inventory level is zero at the same time, then a D-Channel’s shortage happens in time t+ (i.e. BD(t) happens). Therefore, the shortage probability is

M

¦

p(i1 , j ,i2 )  p(0,0, S2 )

(31)

i1 0, j 1,i2 d S2  2

Change the value of (S1, S2), we obtain the trend of the D-Channel’s shortage probability as in Figure 3. 1

The D-Channel’s shortage probability M falls down rapidly as S1 increases.

2

M has an increasing tendency as S2 increases. However, when S2 is large enough (S2 t 5), M tends to be stable.

4.2 The R-channel’s shortage probability I As shown in Figure 4, the R-Channel’s shortage probability I has the following tendencies: 1

As the increasing of S1 and S2, I has a decreasing trend.

2

The change of S2 has a relatively larger impact on I.

Figure 3

The D-channel’s shortage probability M

Figure 4

The R-channel’s shortage probability I

An inventory approach for a Local Collection Point in reverse supply chains

¦p

Let q

( i1 , j ,i2 )

69

, which stands for the probability that the R-Channel’s inventory level

i2 0

is zero. Combining it with Equation (24), we have

I|

v q. P v

(32)

Equation (32) illustrates that I has an approximately linear relation with q. Therefore, the trend of I can be illustrated from that of q. Assume that the R-Channel can always get the requested used products from D-Channel at any time, i.e. the D-Channel’s shortage probability is zero. Furthermore, let us treat the event of the completion of a repairing process as a ‘customer arrival’, and the customer’s purchase of a recovered product as the ‘completion of a service’, then the R-Channel forms a M/M/1/N queueing system in which the customer’s average arrival rate is P, the average service time is 1/Q, and the system capacity is N = S2. For the M/M/1/N queueing system, conclusions of queueing theory illustrate that the probability that there is no customer in the system is

q

­ 1 U U z1 ° S 1 °1  U 2 ® 1 ° U 1 °¯ S2  1

(33)

U

P / v.

(34)

where

However, due to the D-Channel’s shortage, the R-Channel can not always get its request from the D-Channel immediately. Assume that the D-channel’s shortage occurs, the customer’s arrival time is X + Y, where X is the time needed for a used product’s arrival to LCP, and Y is the handling time for it. Thus, the average ‘customer arrival’ rate of the D-Channel is P0, which satisfies Equation (35), i.e. 1

P0

(1  M )

1

P

§1 1· M ¨  ¸ ©O P¹

1

P



M , O

(35)

or,

P0

PO O  PM

(36)

Substituting the P in Equation (34) by P0, or letting

U

PO (O  PM )v

,

(37)

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70 Figure 5

The D-channel’s average inventory level H1

then Equation (33) presents the accurate probability that the R-Channel’s inventory is zero. Note that 1 U 1 U

S2 1

1 1  U  "  U S2

,

(38)

we know that q decreases as S2 increases and the tendency of q has nothing to do with U. This explains why the R-Channel’s shortage probability decreases as S2 increases. Specifically, if U < 1, letting S2 o + f, we have q = 1  U. This explains why I reduces to a stable value eventually as S2 increases under the condition of Q > P. Conversely, when U > 1, the denominator of the right side of Equation (38) increases rapidly as S2 increases, and this leads that fact that I decreases rapidly to 0 as S2 increases under the condition that Q < P. In addition, Section 4.1 illustrates that the D-Channel’s shortage probability M decreases as S1 increases. Furthermore, when S2 is fixed, Equations (37)–(38) imply that the decrease of M will lead to the increase of U and the decrease of q. Figure 4 also illustrates the following trends: when S1 is large enough, its impact on M decreases rapidly; and so does q. The above two facts not only explain why R-Channel’s shortage probability I declines as S1 increases, but also explain why this impact declines rapidly with the increasing of S1.

4.3 The D-channel’s average inventory level H1 The trend of the D-Channel’s average inventory level H1 is shown in Figure 5. As shown in Figure 5, 1

As S1 increases, the D-Channel’s average inventory level H1 increases linearly.

2

Along with the increase of S2, the D-Channel’s average inventory level H1 has a declining trend, but the influence is no more than 3%.

3

Along with the increase of v, the D-Channel’s average inventory level H1 has a declining trend.

4

The D-Channel’s average inventory level H1 is approximately to and less than (S1  1)/2.

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Assuming that there is no R-Channel, for any time t, the D-Channel’s inventory level I1 takes values on set {0, 1, }, S1  1} and Pj( D ) P{I1 j , j 0,1, ! , S1  1} 1/ S1 . Thus, the D-channel’s average inventory level is H1 = (S1  1)/2. However, the R-Channel’s introduction changes the distribution of I1, as shown in Figure 6. Figure 6 illustrates the distribution of the D-Channel’s inventory level at any time t, by given that S1 = 15, S2 = 6, O = 10, P = 4 and v = 2 or 6, respectively. We can see from the figure that when the R-Channel is introduced, the probability that I1 takes the values on 1, 2, }, S1  4 almost does not change and is still about to 1/(S1  1). However, the probability that I1 equals to 0 increases significantly while the probability that I1 equals to S1  1 decreases significantly, whereas that of S1  2 and S1  3 also decreases in a certain extent. The change of the distribution eventually leads to the slight decrease of the D-Channel’s average inventory level. In fact, the introduction of the R-Channel extends the D-Channel’s average delivery cycle length from E(T0) = S1/O to E(T) = S1/(O  Q (1  I)). At the same time, suppose that at any time t, the D-Channel’s inventory level is i, the introduction of R-Channel makes the D-Channel’s inventory level return to i  1 with a certain probability in a short time 't('t > 0). Once the above event happens, the D-Channel’s inventory level will eventually back to i again at sometime of the same delivery cycle. Meanwhile, even if the system inventory level is i + 1, it may also drop down to i with a certain probability. The above two facts actually extends the average time length that ‘the D-Channel’s inventory level is i’ during a delivery cycle. Since the average delivery cycle and the average time that D-Channel’s inventory level is i increase synchronously, the probability that the D-Channel’s inventory level is i keeps stable. However, according to our policy, for the case of i = S1  1, once a used product arrives at the LCP, the D-Channel’s inventory level changes to 0 immediately and another delivery cycle begins. It means that there is no chance for the D-Channel’s inventory level falling from S to S  1. Due to the above missing possibility, the probability that ‘the D-Channel’s inventory level is S1  1’ declines. Therefore, the D-Channel’s average inventory level declines too. Figure 6

The probability distribution of D-channel’s inventory level

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In addition, during a delivery cycle, the expected time that the D-Channel’s inventory level stays in S1  1 is no less than 1/O (Here, 1/O is the average time for a used product arrival), then we have PS( D1) t {O  v(1  I )} / ( S O ) and 'pS1 1 d v(1  I ) / ( S O ) . Here ' pS1 1 1

is the probability reduction of the event ‘D-Channel’s inventory level is S1  1’, which caused by the introduction of the R-Channel. Thus, compared with that before the R-Channel’s introduction, the average inventory reduction of the D-Channel should satisfy 'H1 d 'pS1 1 ˜ ( S  1)

S  1 v(1  I ) ˜  v / O. S O

(39)

4.4 The R-channel’s average inventory level H2 Changing the value of S1 and S2, we have their effect on the R-Channel’s average inventory level H2 (see Figure 7): 1

with the increasing of S1, H2 has a slight increasing trend

2

as S2 increases, H2 increases linearly

3

the changing of v affects H2 significantly, i.e. the larger v is, the smaller H2 is.

The discussion in Section 4.2 implies that the R-Channel’s inventory process forms a M/M/1/N queueing system, in which the average customer arrival rate is P0 = PO/(O + PM), the average service time is 1/v, and the system capacity is N = S2. Note that the R-Channel’s average inventory level equals the average queueing length of the M/M/1/N queueing system, we obtain the R-Channel’s average inventory level from the conclusion of M/M/1/N queuing system as follows:

Ls

­ U ( N  1) U N 1  ° 1  U N 1 ®1  U ° N /2 ¯

U z1 U 1

where U

PO is given by Equation (37). (O  PM )v

Figure 7

The R-channel’s average inventory level H2

(40)

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We find in Equation (40) that Ls increases as N (or S2) increases, whatever U > 1, U = 1 or U < 1. Also, when N is fixed, Ls is an ascending formulation of U. Further, Section 4.1 illustrates that M decreases as S1 increases. Therefore, as S1 increases, U increases slightly, and LS (or H2) has slight ascending.

4.5 The R-channel’s busy ratio N The R-Channel’s busy ratio is shown in Figure 8. The left one is for the case that O = 10, P = 4, S2 = 6 and the right one is for O = 10, P = 4, S1 = 15. We can obtain the following trend: 1

the R-Channel’s busy ratio N increases slightly as S2 increases

2

when U = P/v < 1, the changing of S1 does not affect the system busy ratio and the Rchannel’s busy ratio is approximate to U

3

when U = P/v > 1, the R-Channel’s busy level N has a slight increasing trend with increase of S1 and approaches to 1.

In fact, for the M/M/1/N queueing system, the knowledge of queueing theory tells us that the probability that its queuing length equals N is

PN

­ 1 U UN ° N 1  1 U ® ° 1/ N ¯

U z1

(41)

U 1

As to the specific R-Channel discussed above, U given by Equation (37). Considering that PS2

PO / {(O  PM )v} in Equation (41) is

¦

p(i1 ,0, S2 ) , we have

¦

p(i1 ,0, S2 )

0 d i1 d S1 1

N

¦ ( jp

(i1 , j ,i2 ) )

i1 , j ,i2

1

¦

1

¦

p(0,0,i2 )  PS2 d 1  PS2

0 di2 d S 2 1

Figure 8

p(0,0,i2 ) 

0 d i2 d S2 1

The R-channel’s busy ratio N

0d i1 d S1 1

(42)

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where PS2 is given by Equation (41) if we set N = S2. In fact, Equation (42) presents an upper bound of system’s busy level. For example, when O = 10, P = 4, S2 = 6, the approximate upper bound of N is 0.496, 0.711, 0.833 and 0.934, with the cases of v = 2–5, respectively. From the analysis of Equations (41)–(42), we obtain that PN decreases as N increases, i.e. the larger S2 is, the larger N is.

5

An algorithm for searching the optimal solution

Assume that there is no R-Channel in the LCP and let PI0 be system profit before the introduction of the R-Channel. Since all used products are conducted by the D-Channel, we have PI 0

§ AD O h1 ( S1  1) ·  ¸  O c. 2 © S1 ¹

O w1  ¨

(43)

Obviously, the optimal decision is given by the standard Economic Order Quantity (EOQ) model, i.e. when * S10

2 AD O / h1 ,

(44)

the system obtains its maximum profit as PI 0*

O ( w1  c )  2 AD O h1  h1 / 2.

(45)

Once the R-Channel is introduced, the used products are divided and conducted by the D-Channel and the R-Channel. Let OD and OR be average of the final product rates conducted by the D-Channel and the R-Channel, respectively, we have

OD

O  v(1  I ), OR

v(1  I )

(46)

where I is the shortage probability of the D-channel given by Equation (23). Let PID and PIR be the profits of the D-Channel and the R-Channel, respectively, then AD OD S1

PI D

( w1  c )OD  h1 H1 

PI R

( w2  c)OR  h2 H 2  rN

where H1

¦ (i p

1 (i1 , j ,i2 ) ),

i1 , j ,i2

H2

(47) (48)

¦ (i p

2 (i1 , j ,i2 ) ).

i1 , j ,i2

From the assumptions that O > > P and O > > Q, we know that O > OD > > Q > OR, i.e. the D-Channel is still the priority of the optimal problem. The discussion of Section 4.3 illustrates that the introduction of the R-Channel can reduce the D-Channel’s inventory level. Furthermore, Equation (39) implies that S1  1 v S 1   H1  1 . 2 2 O

(49)

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Assume that the D-Channel’s objective is to maximise its profit, then, the D-Channel obtains its maximum profit PI D* when S1*

2 AD OD / h1 .

(50)

Furthermore, we have ( w1  c )OD 

h1 h hv  2 AD OD h1 d PI D* d ( w1  c)OD  1  2 AD OD h1  1 . 2 2 O

(51)

Note that 0 d I d 1 in Equation (46), we have

O  v d OD d O , 0 d OR d v.

(52)

Substitute Equation (52) into Equation (50), we have 2 AD (O  v) / h1 d S1* d 2 AD O / h1 .

(53)

For the LCP, the R-Channel’s introduction is valuable if there exists a feasible solution (S1, S2) leading to PI R  PI D t PI 0* . Substitute Equation (48) and the right side of Equation (51) into Equation (53), we obtain a necessary condition that a feasible solution exists as follows: h2 H 2  rN d ( w2  w1 )v 

h1v

O

 2 AD O h1  2 AD (O  v)h1

(54)

where H2 = LS is given by Equation (40) when N = S2. Note that LS is an ascending formulation of U, we know that H 2 t S2 / 2 when U t 1. Substitute H 2 t S2 / 2 into Equation (54) and take N > 0 into account, we obtain the upper bound of S2 under the condition of U d 1, i.e. S2 d

2( w2  w1 )v 2h1v  . h2 h2O

(55)

When U < 1, LS has an upper bound of U/(1  U), and we cannot obtain an upper bound of S2 from Equation (54). However, note that the increasing of S2 has almost no influence to the average inventory level of the repaired products when S2 is large enough, we can set an upper bound which is large enough in the algorithm for searching optimal solution. According to the above discussions, we propose an algorithm for searching the optimal solution as follows: Algorithm 1. Searching the optimal solution Step 1. Set the initial value of system parameters O, P, Q, w1, w2, c, h1, h2, AD, and an artificial upper bound M of S2, such as M = 50. Step 2. Use Equation (44) calculating the system optimal profit PI 0* before the introduction of the R-Channel.

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Step 3. Using Equation (53), calculating the upper and lower bound of S1, i.e. S1

2 AD O / h1 , S1

2 AD (O  v) / h1 .

Step 4. Calculate S2’s upper bound according to Equation (55), i.e. S2

­ 2( w2  w1 ) 2h1v ½ max ® M ,  ¾. h2 h2 O ¿ ¯

Step 5. For all possible solutions among the set {( S1 , S2 ) : S1 d S1 d S1 ,1 d S2 d S2 ; S1 , S2 are integers } use Algorithm 2 computing system profit PI , compare them and get the optimal solution.

Algorithm 2. Calculating the system profit Step 1. Set the initial value of system parameters O, P, Q, w1, w2, c, h1, h2, AD, r and that of the decision variable S1 and S2. Step 2. For the given O, P, Q, S1 and S2, use Algorithm 3 calculating the system characters, including the D-Channel’s average inventory level H1, the R-Channel’s average inventory level H2, system busy probability N and the R-Channel’s shortage probability I. Step 3. Utilise Equation (29) calculating the system profit, i.e. PI

G1  G2  h1 H1  h2 H 2  rN  d  cO ᧨

where G1, G2 and d are given by Equations (27),(28) and (26), respectively. Algorithm 3. Calculating the system characters Step 1. Set the initial value for parameters O, P, Q, S1 and S2. Step 2. Let N = (S1 + 1)(S2 + 1)  1 and set the one to one relationship between (i1, j, i2) and {1, 2, }, N}, i.e. to each possible system state (i1, j, i2), N = i1 + j + i2(S2 + 1) + 1. Step 3. For the given O, P, Q, S1 and S2, set the infinitesimal generator Q according to Equations (1)–(11). Step 4. Calculate the stationary probability of all the system states, i.e. to solve a linear K equation which consists of Equations (12)–(13) where the variable is S . Then utilise the K one-to-one relationship founded in Step 2 translating the stationary probability S to that of each state (i1, j, i2). Step 5. Utilise Equations (14)–(16) and (23) calculating the system characters, i.e. the D-Channel’s average inventory level H1, the R-Channel’s average inventory level H2, the system busy probability N and the R-Channel’s shortage probability I.

6

Results and analyses

In this section, we set the case that O = 20, P = 4, Q = 4, AD = 50, h1 = 8, h2 = 16, c = 50, r = 20, w1 = 80, w2 = 200 as a benchmark, change the system parameters’ value one by

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one, find the optimal solution for each case by utilising the algorithms in Section 5, and then obtain the solution tendency with respect to the change of system parameters.

6.1 The effects of prices: w1 and w2 Change the D-Channel’s price w1 in a large variation, we obtain the optimal solution trend shown in Figure 9. In Figure 9, S1* , S2* and PI* are the optimal solutions and the maximum profit of the considered system, and S0* and PI 0* are that before the introduction of the R-Channel. Figure 9 illustrates the following trends: 1

as the increasing of w1 , S2* has a decreasing tendency but S1* keeps stable

2

compared with that before the introduction of the R-channel, the D-channel’s optimal delivery batch S1* is a little smaller than that of the former S0*

3

as the increasing of w1, the additional profit brought by the introduction of the R-Channel reduces.

In fact, the smaller the price difference (w2  w1) between the two channels is, the smaller the additional profit per unit product that LCP get from R-Channel is. Thus, the value of keeping a higher repaired product inventory in R-Channel to avoid shortage decreases, i.e. S2* decreases. The above reason can also be utilised to analyse w2’s effects shown in Figure 10. Figure 9

The effect of the D-channel’s price w1

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Figure 10 The effect of the R-channel’s price w2

Figure 11 The effect of the repairing efficiency P

6.2 The effect of the repairing efficiency P Figure 11 shows the effect of the repairing efficiencyP on the optimal solution. As the repairing efficiency P increases, 1

the optimal delivery batch S1* of D-Channel declines slightly

2

the optimal maximum inventory level of repaired product S2* decreases rapidly

3

the system profit increases.

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In fact, Equation (50) presents a suggested value of the D-Channel’s delivery batch. Recalling Equations (32) and (46), we know that increase of P will lead to the increase of OR and the slight decline of OD. The latter fact makes the slight decline of S1 according to Equation (50). Furthermore, the increase of OR means that the R-Channel sells more used products and thus the system gains more profit. Meanwhile, Equation (37) implies that U increases rapidly as P increases. For a given S2, the increase of U leads to the increase of the R-Channel’s average inventory level. Therefore, a smaller S2 will be suggested to reduce the R-Channel’s inventory cost.

6.3 The effect of the customer arrival rate v Figure 12 shows the effect of the customer arrival rate on the optimal solution as follows: 1

As v increases, the system profit increases almost linearly at the beginning. However, the increase trend slows down when v is large enough.

2

As v increases, S1* declines slightly and S2* increases rapidly.

In fact, when the average customer arrival rate increases, more products can be sold from the R-Channel and thus the system can gain more profit. However, when the customer arrival rate is large enough, the R-Channel’s handling efficiency becomes a bottleneck of the system, and most requirements from customers can not meet. This is the reason why the increase trend of the system profit slows down when v is large enough. Figure 12 The effect of the customer arrival rate v

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Figure 13 The effect of R-channel’s inventory holding cost

6.4 The effect of R-channel’s inventory holding cost Figure 13 illustrates the effect of the R-Channel’s inventory holding cost. While the unit holding cost per unit time h2 increases 1

the D-Channel’s optimal delivery dispatch S1* keeps stable

2

the R-Channel’s maximum inventory level S2* decreases

3

the system profit decreases.

In fact, the increase of S2 can not only decrease the R-Channel’s customer loss rate, but also increase its average inventory level. Therefore, when the inventory holding cost for keeping a relatively higher inventory increases, a relatively lower S2 will be suggested. Furthermore, Equation (50) implies that the optimal D-Channel’s delivery batch S1 has no relation to h2.

7

Concluding remarks

Within the context of sustainable economic development, product recovery has received increasing attention. For companies involved in product recovery, collections of used products have become one of their key concerns (Guide, Teunter and Van Wassenhove, 2003) because the entire business of product recovery is very dependent on the efficiency and effectiveness of collections (Goggin, Reay and Browne, 2000). With a purpose of improving the collection efficiency, we focus on the inventory management of LCPs which handle a single type of used products in this article.

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Specifically, we introduce a joint inventory policy for a specific LCP. The LCP is assumed to have two channels handling the used product, i.e. 1

The R-Channel: to recover them and resell them in the local second-hand market with a relatively higher price.

2

The D-channel: to deliver them to the CRC directly with a relatively lower price.

The joint inventory policy proposed to the specific LCP consists of two decision variables, the maximum inventory level S1 for un-repaired products in the D-Channel, and S2 for repaired products in the R-Channel. By formulating the system into a continuous Markov chain, the stationary probability of each system state is calculated. Utilising those results, the system cost and profit are obtained. The purpose is to find the optimal (S1, S2) to maximise the system average profit. In addition, the system characters under the specific joint policy have been discussed. Related results are: 1

The D-Channel’s shortage probability falls down rapidly as S1 increases, while keeps more stable with the change of S2.

2

The R-Channel’s shortage probability is more dependent on R-Channel’s maximum inventory level S2: the larger S2 is, the minor the customer may lose from the shortage.

3

The D-Channel’s average inventory level H1 tends to, but less than (S1  1)/2. This fact shows H1 is highly dependent on the maximum inventory level of the D-Channel, which is also the delivery batch in this model.

4

The R-Channel’s average inventory level H2 is affected mainly by two factors, i.e. the R-Channel’s maximum inventory level S2, and the difference between the customer arrival rate and the repairing rate.

An algorithm for finding the optimal solution is developed, and the optimal solution trends are also analysed. The trend of the optimal solution is as follows. First, the introduction of the R-Channel can improve the profits of the LCPs. The larger the price difference between the two channels is, the more profit the LCP can get. Secondly, the optimal delivery batch from the LCP to the CRC is a little smaller than that before the introduction of the R-Channel. Furthermore, the optimal delivery batch keeps stable as the change of other parameters such as w1, w2, P, Q and h2. Thirdly, the optimal value of the maximum inventory level of the R-Channel S2 shows the following trends: 1

The larger the price difference between the two channels is, or the larger the customer arrival rates (for repaired products) P is, the larger the optimal S2 is.

2

The larger the repairing efficiency v is, or the larger the holding cost (per unit time per unit product) h2 is, the smaller the optimal S2 is.

The modelling framework presented in this article is exploratory revealing some limitations and several possibilities for future research. First, for modelling simplification, we assume that all the used productions and recovered products have the same obtain cost and resale price which may be far from the facts. In fact, the used products obtained from the consumer have a large variation in quality, brand, specification and so on. In China, there exists many specifications of products in TV set

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industry, and LCPs generally pay different prices for different used products with a fully consideration of the product’s quality, brand, specification and used-time. In addition, these variations finally affect the recovered product’s price and the customer’s acceptance degree in the second-hand market. Secondly, we do not consider the inspection process. As illustrated by Fleischmann et al. (2000), for most product recovery process, there exists an inspection process before recovering used product to determine the condition of the used products and find out whether they are recoverable. As a result of the inspection, some used products may be not recoverable or be not valuable to recover. Thus, further researches may add the inspection process into the considered model. Thirdly, for model simplification, we just consider the situation that a single type of product is involved in the considered LCP, whereas the LCP may handle multiple types of products in practices. Therefore, a more general inventory model for multiple types of used products may be formulated in future. However, the effects due to the former two limitations can be neglected when the following conditions are satisfied: 1

For the first limitation, when the differences between the resale prices and the obtain costs for all products do not vary largely.

2

For the latter limitation, when the inspection time is far shorter than that for repairing and the maximum inventory level for un-repaired product is relative high.

Therefore, in spite of the above limitations, our model is still useful in considering the inventory process, and the optimal inventory policy presented for the system is valuable. Another issue that could be interesting for future research is to consider the price strategy for the products. Specially, there exists a substitution between new products and recovered/remanufactured products (see Mitra and Webster, 2008), thus, the pricing decision is really important and should be considered in the further research. Furthermore, note that this article focuses on a specific bi-channel LCP, and the types of LCP (of used products) vary largely (for example, they may be local authorities, retailers, commission transport companies, distributors and so on), the inventory problem for other types of LCP should be studied in future. Finally, noting that the model proposed in this article is more theoretical which seems to be far from the practice, empirical verifications from the actually business may be studied later. In summary, with the consideration of the used product arrival process, the inventory process, the used product repairing process and the selling process simultaneously, this article introduces an inventory policy for a special bi-channel LCP. For this special policy, the inventory model is initiated, the system characters are analysed and the solution tendencies are also illustrated.

Acknowledgements The authors gratefully acknowledge two anonymous referees for their constructive comments and suggestions that were instrumental in improving this article. This research was supported in part by the National Natural Science Foundation of China (Grant #s: 70525001 and 70731003). Z. Huang would also like to acknowledge the Research Sabbatical Leave Grant of the Adelphi University for support of his research.

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