Coordinating a three-level supply chain with learning-based continuous improvement

Int. J. Production Economics 127 (2010) 27–38

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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe

Coordinating a three-level supply chain with learning-based continuous improvement$ Mohamad Y. Jaber a,n, Maurice Bonney b, Alfred L. Guiffrida c a

Department of Mechanical and Industrial Engineering, Ryerson University, Toronto, ON, Canada, M5B 2K3 Nottingham University Business School, University of Nottingham, Nottingham, NG8 1BB, UK c Department of Management and Information Systems, Kent State University, Kent, Ohio 44242, USA b

a r t i c l e in fo

abstract

Article history: Received 8 April 2010 Accepted 12 April 2010 Available online 20 April 2010

Learning curve theory has been widely used as a managerial tool to describe and model product and process improvement. This paper investigates a three-level supply chain (supplier–manufacturer– retailer) where the manufacturing operations undergo a learning-based continuous improvement process. Improvements in the manufacturer’s operation are characterized by enhanced capacity utilization, reductions in set-ups times, and improved product quality through the elimination of rework. As a result of these continuous improvements, the manufacturer can justify a production policy that is based on more frequent, smaller lot size production. For this production policy to be practical and not sub-optimal to the supply chain, the manufacturer must integrate its lot-sizing models with the replenishment policies of its upstream raw material suppliers and the demand requirements of its downstream customers (retailers). Mathematical models that achieve chain-wide lot-sizing integration are developed and solution procedures for the models are illustrated by numerical examples. The results demonstrate that learning-based improvements in set-up time and rework allow retailers to order in progressively smaller lot sizes as the manufacturer offers larger discounts and profits and that the entire supply chain benefits from implementing learning-based continuous quality improvements. The results also demonstrate that forgetting effects lead to increases in supply chain costs. & 2010 Elsevier B.V. All rights reserved.

Keywords: Learning Forgetting Set-up reduction Lot-sizing Product quality Supply chain coordination Quantity discounts

1. Introduction The modern market for products is dynamic, global and competitive. This environment imposes pressures on companies to deliver quality products at competitive prices when they are required. Product life cycles are shortening requiring companies to reduce the time from concept to market. Together, these pressures compel companies to be responsive to market changes, efficient and flexible. The late 1990s and the beginning of this millennium was a period of intense interest in supply chain management providing sustainable competitive advantage for companies (Dell and Fedman, 1999). Effective supply chain management involves the integration of functions such as production, purchasing, materials management, warehousing and inventory control, distribution,

$ Paper selected from presentations at the 12th International Symposium on Logistics ‘‘Sustainable Collaborative Global Supply Chains’’, Budapest, Hungary, July 8–10, 2007. The selection process has been managed by Professor Kulwant S. Pawar, Nottingham University Business School, and Professor Chandra Lalwani, Hull Business School, UK. n Corresponding author. Tel.: + 1 416 979 5000x7623; fax: +1 416 979 5265. E-mail addresses: [email protected], [email protected] (M.Y. Jaber), [email protected] (M. Bonney), [email protected] (A.L. Guiffrida).

0925-5273/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ijpe.2010.04.010

shipping, and transport logistics. This integration is needed within the operations of specific supply chain members and, more importantly, across all members of the supply chain. To maintain sustainable competitiveness, operations within the supply chain will benefit from continuous improvement programmes that include fostering organizational learning. Historically, learning curve theory has been applied to a diverse set of management decision areas such as inventory control, production planning and quality improvement. These decision areas exist both within the individual organizations of the supply chain and, as a result of the interdependencies among chain members, across the supply chain as a whole. By using established learning models to model these learning effects, management may utilize capacity, manage inventories and coordinate production and distribution better throughout the chain. The lot-sizing problem with learning and forgetting effects in production has received considerable attention from researchers and a detailed review of this literature is found in Jaber and Bonney (1999). Jaber and Bonney (2003) also investigated the effects that learning and forgetting in set-ups and in product quality have on the economic lot-sizing problem. In recent years, the lot-sizing problem with learning and forgetting has been investigated within the context of the economic manufacture quantity model (e.g., Balkhi, 2003; Chiu et al., 2003; Chiu and

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M.Y. Jaber et al. / Int. J. Production Economics 127 (2010) 27–38

Chen, 2005; Jaber and Guiffrida, 2007; Alamri and Balkhi, 2007; Jaber and Bonney, 2007; Jaber et al., 2009) and to a lesser extent in conjunction with the Joint Economic Lot-Sizing Problem (JELSP) by Nanda and Nam (1992, 1993). The JELSP forms the basis of a two-level supply chain with order coordination between the chain members. Nanda and Nam (1992) developed a joint manufacturer–retailer inventory (twolevel supply chain) model for the case of a single buyer. Production costs were assumed to reduce according to a power form learning curve (Wright, 1936) with forgetting effects caused by breaks in production. A quantity discount schedule was proposed based on the change of total variable costs of the buyer and manufacturer. To meet the demand of the buyer, the manufacturer considers either a lot-for-lot (LFL) production policy (e.g., Banerjee, 1986; Goyal and Gupta, 1989), or a production quantity that is a multiple of the buyer’s order quantity (Lee and Rosenblatt, 1985; Goyal and Gupta, 1989). Nanda and Nam (1992) assumed a LFL policy, and did not specify the form of the forgetting curve. They extended their work in a subsequent paper (Nanda and Nam, 1993) to include multiple retailers. This paper develops the work of Nanda and Nam (1992) and Jaber and Bonney (2003) to investigate a joint replenishment inventory model for a three-stage (supplier–manufacturer–retailer) decentralized supply chain with the manufacturer encountering learning and forgetting effects in set-ups, production, and product quality. The objective of the research is to fill gaps in the literature with respect to quality improvement in the supply chain. Our model integrates research from the fields of economic lot sizing in supply chains and learning theory to construct a quality based supply chain model. Quality improvement in supply chain management has not been addressed in detail within the supply chain literature (Sila et al., 2006). The majority of research on quality improvement in supply chains is based on two dimensions (i) empirical survey based research that attempts to identify the importance of quality improvement in the supply chain as well as identifying the key factors that support the need for quality in the supply chain (see for example; Lo and Yeung, 2006; Kuei et al., 2001; Kanji and Wong, 1999) or (ii) the development of two-stage supply chain decision models for implementing quality issues into supply chains (see for example Zhu et al., 2007). In this paper we present a decision model for analyzing quality improvement. Our model bridges the current gap in supply chain quality management by introducing a quantitative decision model for investigating quality improvement in a more realistic three-stage (supplier–manufacturer– retailer) environment. Munson and Rosenblatt (2001) were the first to model a threelevel supply chain. They assumed that all parameters were deterministic and that: (i) the retailer orders a single product according to its economic order quantity (EOQ), (ii) the manufacturer optimises its lot-sizing policy according to the lumpy order pattern, which is an integer multiple of the retailer’s order quantity, and (iii) an integer multiple of the manufacturer’s order quantity the supplier orders based on the lumpy ordering pattern of the manufacturer. Munson and Rosenblatt (2001) further assumed that the manufacturer is the most influential player in the supply chain who offers quantity discounts to the retailer to entice him/her to order in larger quantities than the retailer’s economic order quantity. Quantity discounts were computed in the model (e.g., $/unit) as the difference in holding and ordering costs between the retailer’s old ordering policy (no coordination) and its new ordering policy (with coordination) divided by the annual demand. Jaber et al. (2006) extended the work of Munson and Rosenblatt (2001) by assuming a price discount approach, a price-dependent demand and profit sharing scenarios. This paper, like Jaber et al. (2006), adopts a centralized decision-making

process for coordinating the supply chain model and assumes that the savings (increased profits) arising from coordination will be shared among the players in the supply chain. The coordination of decisions within a supply chain can be broadly classified into two types of decision-making structures: centralized or decentralized. A centralized decision-making process assumes a unique decision-maker managing the whole supply chain with an objective to minimise (maximise) the total supply chain cost (profit) whereas a decentralized decision-making process involves multiple decision makers who have conflicting objectives. A casual review of the literature of recent supply chain publications illustrates that both orientations (centralized and decentralized) routinely appear in the literature. Recent literature on SC management classified by type of decision-making process is tabulated below: Models with centralized decision making

Models with decentralized decision making

¨ Uster et al. (2008) Zou et al. (2008) Majumber and Srinnvasan (2006) 4) Schwart et al. (2006) 5) Van Hoesel et al. (2005)

1) Disney et al. (2008) 2) Bernstein et al. (2006) 3) Mishra et al. (2007)

1) 2) 3)

¨ u¨ et al., (2005) 4) Gull 5) Huang and Iravani (2005)

The remainder of the paper is organized as follows. The next section, Section 2, provides a brief description of the learning– forgetting process. Section 3 describes the notation and assumptions. Section 4 develops a mathematical programming model with its sub-cost functions and its solution procedure. Section 5 provides numerical examples and discusses the results. Section 6 summarises and concludes the paper.

2. The learning and forgetting process Most researchers and practitioners accept the Wright (1936) learning curve because of its simple mathematical form and its ability to fit empirical data quite well (e.g., Yelle, 1979; Jaber, 2006) and so it is understandable that most works that have investigated the economic order (manufacture) quantity model with learning and forgetting use the power form learning curve, usually written as yx ¼ y1 xb

ð1Þ

where yx is the time to produce the xth unit, y1 is the time to produce the first unit, x is the cumulative quantity produced, and b is the learning exponent; where 0obo1, b ¼  log(f)/log(2), and f is the learning rate expressed as a percentage. The learn–forget curve model (LFCM) developed by Jaber and Bonney (1996) will be adopted in this article. The LFCM has been shown to be a potential model to capture the learning–forgetting process. Interested readers may refer to Jaber and Bonney (1997), ¨ Jaber et al. (2003), and Jaber and Sikstrom (2004a; 2004b) for further background. The LFCM suggests that the forgetting curve is of a power form similar to (1) with the forgetting exponent in cycle i, fi, computed as fi ¼

bð1bÞlogðui þ ni Þ   log 1 þB=tðui þni Þ

ð2Þ

where 0rfi r1, ni is the number of units produced in cycle i up to the point of interruption, B is the break time to which total forgetting occurs, b is the learning curve exponent in (1), and ui is the number of units remembered at the beginning of cycle i,

M.Y. Jaber et al. / Int. J. Production Economics 127 (2010) 27–38

where ui + 1 rui +ni. That is, if the learning process is interrupted for a time of at least B, then performance reverts to a threshold value, usually equivalent to y1, t(ui +ni) denotes the time to produce ui + ni units (equivalent units of cumulative production accumulated by end of cycle i), and is computed from (1) as Z ui þ ni þ 1=2 uX i þ ni tðui þ ni Þ ¼ y1 xb ffi y1 xb dx x¼1

¼

ui þ 1=2

i y1 h ðui þ ni þ 1=2Þ1b ðui þ 1=2Þ1b 1b

c1

manufacturer’s fixed cost (e.g. material cost) for reworking one defective unit. manufacturer’s cost per unit of time spent for reworking defective items. demand rate measured from the retailing end of the chain, assumed to be constant. order quantity for player j, where j ¼s, m, r. cycle time for player j, where j ¼s, m, r and Tm ¼ lmTr and Ts ¼ lsTm ¼ lslmTr. Note that the manufacturer’s cycle consists of two segments, a production and inventory accumulation segment, t1, and an inventory depletion segment, t2, where Tm ¼t1 + t2 (Fig. 1). an integer multiplier to adjust the order quantity of player i to that of j where iaj and lj ¼1,2,3,y. For example, Qm ¼ lmQr and Qs ¼glsQm ¼glslmQr. Note that the manufacturer delivers n shipments of size Qr to the retailer during t1, the manufacturer’s production segment of Tm, and k shipments of size Qr during t2, the manufacturer’s inventory depletion segment of Tm, such that lm ¼n + k (Fig. 1).

c2 D Qj Tj

ð3Þ

The number of units remembered at the beginning of cycle i +1 is given from Jaber and Bonney (1996) as

lj

f =b

ui þ 1 ¼ ðui þ ni Þð1 þ fi =bÞ $i i ð4Þ Pi1 where u1 ¼0, ui r j ¼ 1 ni , and $i is the number of units that would have accumulated if production had not ceased for di units of time. $i is computed from (3) as    1=ð1bÞ 1b  $i ¼ tðui þ ni Þ þ di ð5Þ T1 Total forgetting occurs when ui + 1 ¼0. However, from (4), ui + 1-0 when di-B, which may be a large finite value. Given the findings of Anderlohr (1969), McKenna and Glendon (1985), and Globerson et al. (1998), the assumption of total forgetting might not be unrealistic. Nanda and Nam (1992) represented the number of units P remembered at the beginning of cycle i as ui ¼ Z i1 j ¼ 1 ni , where Z is the percentage of learning retention (1  forgetting) after production breaks but neither specified how Z behaves when short or long production breaks are encountered, nor did they provide an empirical validation of their assumption.

3. Notation and assumptions This paper considers a single product with no shortages occurring, that delivery from stocks has zero lead-time, and an infinite planning horizon. Like Munson and Rosenblatt (2001), the paper investigates a three-level supply chain in which all model parameters are assumed to be deterministic and the channel members use the same lot-sizing policies: (i) the retailer orders according to its economic order quantity (EOQ), (ii) the manufacturer optimises its lot-sizing policy according to a lumpy pattern of orders, which is an integer multiple of the retailer’s order quantity, and (iii) the supplier orders according to the resulting lumpy ordering pattern of the manufacturer, which is an integer multiple of the manufacturer’s order quantity. The following notation is used: j

hj 0 hm Aj K~ i

g

r c‘

a subscript identifying a specific player in a supply chain; j ¼s, m, r where s¼supplier, m ¼manufacturer, r¼ retailer) holding cost for player j, where j ¼s, m, r. manufacturer’s holding cost of raw material received from its supplier. order cost per cycle for player j ¼s, m, r. manufacturer’s set-up cost for production cycle i due to knowledge decay or forgetting, where K1 r K~ i r Kmin with K1 being the cost of the first set-up (i¼1) and Kmin being the minimum set-up cost attained. usage ratio (i.e., 1 for 1, 2 for 1, etc) of raw material in one unit of a manufacturer’s finished product. probability of the process going out-of-control. manufacturer’s labour cost per unit of time.

29

Fig. 1 illustrates the behaviour of the inventory for the three players (supplier–manufacturer–retailer) in the supply chain. The retailer orders according to its economic order quantity in lots of size Qr every Tr units of time indefinitely. The manufacturer treats Qr as an input parameter and finds the values of n and k that optimise its total cost, where n is the number of shipments of size Qr occurring during t1 and k is the number of shipments during t2. In Fig. 1, n¼2 and k¼3 and lm ¼n + k¼2+3¼5. After the manufacturer makes n shipments to the retailer, the retailer’s inventory attains a maximum level of kQr at which stage the manufacturer ceases production and depletes its inventory over t2, until its inventory level is zero, after which the manufacturer commences production and the cycle repeats itself indefinitely. Figs. A1 and A2 are enlargements of the production over t1 and depletion over t2. In each cycle, the manufacturer produces lm Q r units corresponding to lm shipments of size Qr each of which is delivered to the retailer in the manufacturing cycle Tm. Since each finished product uses g units of raw material, a shipment of size g lm Q r is delivered by the supplier at the beginning of each manufacturer cycle. In turn, the supplier replenishes its inventory by Qs ¼ g ls lm Q r every Ts units of time, where ls is chosen to minimise supplier’s total cost.

4. Three-level supply chain cost model This section develops a mathematical model of the three-level supply chain cost function. Sections 4.1–4.4, present the cost functions for the retailer, the manufacturer, the supplier, and the supply chain respectively.

4.1. Retailer’s cost function The retailer’s cost per unit of time is given as

cr ðQr Þ ¼

Ar D Qr þ hr Qr 2

ð6Þ

The retailer facing demand rate D orders its economic pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi order quantity Qr ¼ 2Ar D=hr every cycle (Tr ¼ Qr =D) from the manufacturer.

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4.2. Manufacturer’s cost function The manufacturer’s total cost per cycle is the sum of its set-up cost, raw material procurement cost, labour cost, holding cost and quality cost. 4.2.1. Set-up cost From Jaber and manufacturer cycle i ( K1 ðgi þ 1Þs , K~ i ¼ Kmin ,

Bonney (2003), the set-up cost for the is given as: if if

gi o ns gi Z ns

ð7Þ

and K~ i is the cost of the ith set-up due to knowledge decay or P aDtj forgetting, where gi ¼ i1 1 represents the cumulative j ¼ 0 di e experience in set-ups, where 1rirn, Dt ¼tr  te and te and tr are the times at which the information was encoded and retrieved, di is the amount of knowledge acquired in repetition i, similar to b in (1), s is the exponent of the learning curve, where s ¼ logðf=100Þ=log2 and f is the learning rate expressed as a percentage, and a is the forgetting exponent. For example, let d1 ¼1 unit of information, acquired in repetition 1 at time 0, te ¼0,

Inventory level

and a ¼ 0.05, where g1 ¼0. Assume now that the next set-up, i.e. set-up 2, occurs at time 10, tr ¼10, and di ¼1, where i¼1,2,3,y, then Dt ¼tr te ¼10  0 ¼10 and d1(Dt¼ 10)¼1  e  0.05  10 ¼0.6065 units of information are remembered (or 60.65% of set-up 1). If the cost for set-up 1 is 200, K1 ¼200 and s¼0.3219, then from (7), K2 ¼200  (0.6065 +1)  0.3219 ¼171.69. However, with no knowledge decay; i.e., a ¼0 where d1(Dt¼ 10)¼1, g1(dt ¼10)+1 ¼2 in (7) is 2 rather than 1.6065, and K2 would have been 160. Suppose now that the third set-up is performed at time 20, tr ¼ 20. Then, the knowledge remembered at the beginning of the third set-up is the sum of that remembered from the first set-up d1(Dt¼20) ¼1  e  0.05  20 ¼0.3679, where tr ¼20 and te ¼0, and from the second set-up d2(Dt ¼20–10) ¼1  e  0.05  (20–10) ¼ 0.6065, where tr ¼20 and te ¼10. Then, the knowledge remembered at the end of the third set-up is the sum of that remembered from the first and the second at time 20, and K3 ¼200  (0.3679+0.6065 + 1)  0.3219 ¼160.67. Note that when there is no decay in set-up knowledge, a ¼0, gn ¼ d(n–1). 4.2.2. Labour cost Like set-ups, it is assumed also that the production process is subjected to learning and forgetting effects. The production cost is

Supplier Ts

gλm (λs − 1)Qr

gλm (λs − 1)Qr

λs = 2 λm = n + k = 5

Time

Tm t1

t2

kQr

Manufacturer

n=2

k=3

Time Tr

Retailer Qr

Time supply chain Fig. 1. The inventory behaviour for the three players (supplier-manufacturer-retailer) in a supply chain.

M.Y. Jaber et al. / Int. J. Production Economics 127 (2010) 27–38

assumed to follow the Wright (1936) learning curve as described in (1), which has been found to be easier to implement and understand than the more complex models (e.g., Yelle, 1979; Jaber, 2006). The production cost is obtained by integrating (3) over the proper limits to give Z Qm þ ui þ 1=2 PCi,m ðlm ,Qr Þ ¼ c‘ y1 xb dx u þ 1=2

i i y1 h ðQm þui þ 1=2Þ1b ðui þ 1=2Þ1b 1b h i y1 ðlm Qr þ ui þ1=2Þ1b ðui þ1=2Þ1b ¼ c‘ 1b

¼ c‘

ð8Þ

At the start of a manufacturer production cycle, raw material is shipped to the manufacturer by its supplier every Tm units of time. Its inventory of raw material is replenished instantaneously by gqn + t ¼ g(n+ k)Q. The holding cost of raw material is given from (11) and (12) as HCi0 ðlm ,Qr Þ Z Tm Z t1 Z t1 0 ¼ h0m IðtÞdt ¼ hm IðtÞdt ¼ h0m ðgQm gqðtÞÞ dt 0 0 0 #

1=1b Z t1 " 1b 1b 0 ¼ hm g lm Qr  t þ ðui þ 1=2Þ þðui þ 1=2Þ dt yi 0  2b=1b y1 g 1b t1 þðui þ1=2Þ1b ¼ h0m g lm Qr t1 h0m 2b y1 þh0m

4.2.3. Holding cost of finished products The manufacturer’s holding cost is the sum of Eqs. (A.1) and (A.2) derived in the Appendix. Then, by substituting Tr ¼ Qr =D gives HCi,m ðlm ,Qr Þ

1=1b n Qr X ð1bÞðj1ÞQr nðui þ 1=2ÞQr þ ðui þ 1=2Þ1b ¼ hm hm D i¼1 y1 D D

1=1b Qr ð1bÞnQr ðu þ 1=2ÞQr þðui þ 1=2Þ1b hm i 2D y1 D 2D

1=1b nðn1ÞQr2 ðt1 nQr =DÞ 1b hm þ hm t1 þ ðui þ 1=2Þ1b 2 y1 2D

ui ðt1 nQr =DÞ lm Qr þ hm nQr ðt1i nQr =DÞ hm 2 2 kðk þ1ÞQr2 hm kðt1 nQr =DÞQr ð9Þ þ hm 2D þ hm

where n and k are the number of lots of size Qr shipped to the retailer every Tr units of time in the manufacturer’s production and inventory depletion cycle, i.e.,Tm ¼(n + k)Tr ¼ lmTr , hm is the holding cost per unit of finished product per unit of time, and t1 is the production segment of Tm over which Qm ¼ lmQr units are produced. Note that t1 is then a function of lm and Qr, and its expression is given from (1) as Z lm Qr þ ui þ 1=2 i y1 h t1 ¼ y1 xb dx ¼ ðlm Qr þ ui þ 1=2Þ1b ðui þ1=2Þ1b 1b ui þ 1=2 ð10Þ 4.2.4. Holding cost of raw material The manufacturer procures its raw material from its supplier in lots of size gQm ¼glmQr where g is the usage ratio (i.e., 1 for 1, 2 for 1, etc) of raw material in one unit of a manufacturer’s finished product. The manufacturer replenishes its inventory of raw material instantaneously at the beginning of every cycle of length Tm. The manufacturer depletes its inventory of raw material during the time t1, the production segment of Tm, where the inventory level of raw material is given as ( gQm gqðtÞ, 0 rt r t1 IðtÞ ¼ ð11Þ 0, t1 r t rTm where q(t) is given from (3) as  1=1b 1b t þðui þ 1=2Þ1b ðui þ 1=2Þ, qðtÞ ¼ y1

where

0 r t rt1 ð12Þ

and gq(t) represents the total usage of raw material by finished products by time t. Note that q(0) ¼0 and q(t1) ¼Qm ¼ lmQr.

31

y1 gðui þ1=2Þ2b þh0m gðui þ1=2Þt1 2b

ð13Þ

4.2.5. Quality cost Adopting the approach of Porteus (1986), the expected number of defectives in a lot of size Qm, w, is estimated to be 2 w ¼ Qm ð1rÞð1ð1rÞQm Þ=r  rQm =2 (Porteus, 1986). As empirically validated in Jaber and Bonney (2003), the time to rework a defective item is assumed to follow a learning curve of the form described in (1) as rn ¼ r1 nb

ð14Þ

where rn is the time to rework the n th defective item, r1 is the time to rework the first defective item, and b is, like b, a learning slope. Note that this paper assumes the same type of defect is generated every time. This may not be true in practice as reworking different types of defective units may follow learning curves whose parameters (r1, b) are different. The total time to rework w units in a lot of size Qm ¼ lmQr is given by integrating (14) over the proper limits as Z w þ vi þ 1=2 i r1 h tðwÞ ¼ r1 nb dn ¼ ðwþ vi þ 1=2Þ1b ðvi þ 1=2Þ1b 1b vi þ 1=2 ð15Þ where vi is the experience in correcting defectives, is measured as the number of units remembered at the beginning of cycle i. The manufacturer’s quality cost (cost of reworking defective items) QCm ðwÞ ¼ c1 w þ c2 tðwÞ is given by QCi,m ðlm ,Qr Þ ¼ c1 r 2

ðlm Qr Þ2 r1 þ c2 2 1b

3 !1b ðlm Qr Þ2 1b 5 4 þ vi þ1=2  r ðvi þ 1=2Þ 2

ð16Þ

4.2.6. Manufacturer’s cost function The manufacturer’s total cost per unit time in cycle i is the sum of the set-up cost,K~ i , labour cost, PCi,m, holding cost of finished 0 products, HCi,m, holding cost of raw material, HCi,m, and quality cost, divided by the manufacturer’s cycle length, Tm ¼ Qm =D ¼ lm Qr =D. This gives

ci,m ðlm ,Qr Þ 0 ¼ ðK~ i þ PCi,m ðlm ,Qr Þ þ HCi,m ðlm ,Qr Þ þ HCi,m ðlm ,Qr Þ þ QCi,m ðlm ,Qr Þ=T m " # K~ i D y1 D ðlm Qr þ ui þ1=2Þ1b ðui þ1=2Þ1b ¼ þ c‘ 1b l m Qr lm Qr

1=1b n X hm ð1bÞðj1ÞQr nðui þ 1=2Þ þ ðui þ 1=2Þ1b þ hm lm i ¼ 1 y1 D lm

þ



1=1b hm ð1bÞnQr ðu þ 1=2Þ þðui þ1=2Þ1b hm i 2lm 2lm y1 D

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1=1b nðn1ÞQr ðt1 nQr =DÞD 1b þ hm t1 þðui þ 1=2Þ1b 2lm Qr y1 2lm

u ðt1 nQr =DÞD D D kðkþ 1ÞQr n hm i þ hm ðt nQr =DÞ þhm 2lm Qr 2 lm 1i 2lm hm

D y1 gD ðt1 nQr =DÞ þ h0m gDt1 h0m 2lm ð2bÞlm Qr  2b=1b 1b y1 gDðui þ 1=2Þ2b  t1 þðui þ 1=2Þ1b þ h0m y1 ð2bÞlm Qr Dt1 ðlm Qr ÞD r1 D 0 þc2 þ hm gðui þ 1=2Þ þc1 r 2 ð1bÞlm Qr lm Qr 2 3 !1b 2 ð l Q Þ m r 1 b þ vi þ 1=2 4 r ðvi þ 1=2Þ 5 2 hm

5. Numerical examples and discussion of results

ð17Þ

4.3. Supplier’s cost function The supplier’s cost function per unit of time is the sum of the supplier’s order cost and the holding cost per cycle divided by the supplier’s cycle time, Ts ¼ lsQm ¼ lslmQr. This gives AD

s þ hs gQr lm ðls 1Þ=2 ð18Þ ls lm Qr 2 The term hs g ls ðls 1ÞQr2 lm =2D represents the supplier’s hold-

cs ðls , lm ,Qr Þ ¼

ing cost per cycle, and could be computed from Fig. 1. To illustrate, assume that the retailer orders from the manufacturer in lots of 500 units every two days, corresponding to D ¼250 units/day. In response, the manufacturer produces for lm ¼3 retailer cycles, corresponding to a manufacturer lot size Qm ¼ lmQr ¼ 3  500¼1500 units, where the manufacturer’s cycle time is 6 days; i.e., Tm ¼ lmTr ¼ lmQr/D ¼3  (500/250)¼ 6. Assuming that each unit of finished product uses g ¼2 units of raw material, and that the supplier orders for ls ¼2 manufacturing cycles, then the supplier’s lot size quantity is 6000 units of raw material with a replenishment every Ts ¼ lsTm ¼ 2  6¼12 days. At the beginning of its cycle, the supplier ships 3000 units of raw material to the manufacturer whereupon the suppliers inventory level reduces to 3000 and remains at this level for Tm ¼6 days. At time 6, the supplier ships the remaining 3000 units to the manufacturer and its inventory level reduces to zero and remains at this level for another 6 units of time. Assuming hs ¼1, then the supplier’s holding cost is 1  3000  6 +0 ¼ 2 18,000. Substituting these values in the term hs g ls ðls 1ÞQr2 lm = 2 2 2D ¼[1  2  2  (2 1)  (500)  (3) ]/(2  250) ¼ 18,000/cycle. 2 Dividing hs g ls ðls 1ÞQr2 lm =2D byTs ¼ lsTm ¼ lslmTr ¼ lslmQr/D, to get hs g Qr lm ðls 1Þ=2in (18). Jaber et al. (2006) discuss these points in detail. One may question how realistic it is to be left with no inventory for 6 units of time. This is true since the model was formulated with a deterministic demand rate. The model realism would be advanced by introducing stochastic demand and inventory service levels. We identify stochastic demand as a direction of our future research with respect to the model. 4.4. Supply chain cost function

Consider a three-level supply chain where the supply chain members have the following parameters. The retailer’s demand is D ¼150,000 units per year, with an order cost of Ar ¼ $100, and a holding cost per unit per year of hr ¼$20. The manufacturer undergoes a learning-based continuous improvement process which results in a reduction in set-up, production and rework costs. The manufacturer’s initial set-up cost (prior to any learningbased improvements) per production cycle is K1 ¼$200 and the labour (production) cost per year is c‘ ¼$100,000. The production rate is 200,000 units per year, corresponding to a time (in years) to produce the first unit of y1 ¼1/200,000. The unit costs of holding finished products and raw materials are hm ¼$16, and 0 hm ¼$12, respectively. A single unit of a finished product requires g¼2 units of raw material. After completing production of the first unit, the probability of the manufacturer’s production process going out-of-control is 0.0001% (r ¼1/10,000). The time to rework the first defective is r1 ¼1/300,000 with the fixed cost (e.g. material cost) for reworking one defective unit of c1 ¼$20, and the annual cost labour cost of reworking defective items is c2 ¼$100,000. The supplier’s order cost per order is As ¼$400 with a holding cost per unit of raw material of hs ¼ $8. This section solves several numerical examples using the above input data for the cases of coordination and no coordination of orders among the players in the three-level supply chain described above. To illustrate, this paper starts with a numerical example that assumes no improvements in set-up, production and reworks take place at the manufacturer’s site. This implies that the respective learning rates for set-up, production and reworks are fset-up ¼ fprod ¼ frework ¼100% corresponding to learning exponents of s¼0, b¼0 and b ¼0, respectively. The numerical examples presented in this section are solved using EXCEL SOLVER enhanced by VISUAL BASIC code. The solution procedure used in this paper is to the same as that described in Jaber et al. (2006).

5.1. No coordination and no improvements When there is no coordination among the players in the supply chain, the retailer orders accordingpto its economic order quantity pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi in lots of size Qr ¼ 2Ar D=hr ¼ 2  100  150,000=20 ffi1225 every Tr ¼Qr/D ffi 0.0082 years at an annual cost of cr(1225)¼ $24,495. In turn, the manufacturer treats Qr ¼1225 as an input and optimises equation (17) by searching over the manufacturer’s lot size multiple lm such that

ci,m ðlm ,Qr Þ o ci,m ðlm 1,Qr Þ and

The total supply chain cost (objective function) is the sum of (6), (17) and (18). This can be written as a mathematical programming problem as follows: Minimize ci,sc ðls , lm ,Qr Þ, ¼ cr ðQr Þ þ ci,m ðlm ,Qr Þ þ cs ðls , lm ,Qr Þ,

Constraint (19a) guarantees two conditions, first that Qr is positive, and second that no shortages occur since Tr Z y1 Qr1b =ð1bÞ, where Tr ¼ Qr =D, which reduces to Qr =D Z y1 Qr1b =ð1bÞ ) Qr Z ðy1 D=ð1bÞÞ1=b .

ð19Þ

Subject to:  1=b Order quantity : Qr Z y1 D=ð1bÞ

ð19aÞ

Multipliers : lm , ls Z 1, are integers

ð19bÞ

ci,m ðlm ,Qr Þ o ci,m ðlm þ 1,Qr Þ for Qr ¼1225. This leads to ci,m (1,1225)¼ $263,227 per year which is the optimal policy for the manufacturer. Note that lm ¼n +k¼0 + 1¼1, means that no shipment (n ¼0) of size Qr ¼1225 is made during the manufacturer production time t1, where t1 oTr. The manufacturer orders raw material from the supplier in lots of glmQr ¼2  1  1225¼2450 units, every Tm ¼ lmTr ¼ lmQr/D ¼1  1225/150,000¼0.0082. In turn, the supplier treats glmQr as an input and optimises Eq. (18) by searching

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33

Table 1 Optimal policies for a three-level supply chain (supplier–manufacturer–retailer) with learning but no forgetting in production, reworks, and set-ups, with no coordination. Ex fprod (%)

1 2 3 4 5 6 7 8 9 10 11 12 13

100 90 80 70 100 100 100 100 100 100 90 80 70

frework

fset-up

(%)

(%)

100 100 100 100 90 80 70 100 100 100 90 80 70

100 100 100 100 100 100 100 90 80 70 90 80 70

ls lm Qr

2 2 2 2 2 2 2 2 2 2 2 2 2

1 1 1 1 1 1 1 1 1 1 1 1 1

1225 1225 1225 1225 1225 1225 1225 1225 1225 1225 1225 1225 1225

Annual cost

Manufacturer’s cost components

Supply chain

Retailer

Supplier Manufacturer Set up cost

Production cost

Quality cost

Finished product holding cost

Raw material holding cost

$309,767 $304,146 $302,479 $302,036 $308,033 $307,247 $306,914 $299,942 $291,717 $284,940 $292,587 $281,909 $274,356

$24,495 $24,495 $24,495 $24,495 $24,495 $24,495 $24,495 $24,495 $24,495 $24,495 $24,495 $24,495 $24,495

$22,045 $22,045 $22,045 $22,045 $22,045 $22,045 $22,045 $22,045 $22,045 $22,045 $22,045 $22,045 $22,045

$7,500 $2,136 $548 $127 $7,500 $7,500 $7,500 $7,500 $7,500 $7,500 $2,136 $548 $127

$186,774 $186,774 $186,774 $186,774 $185,040 $184,254 $183,921 $186,774 $186,774 $186,774 $185,040 $184,254 $183,921

$18,861 $19,387 $19,542 $19,584 $18,861 $18,861 $18,861 $18,861 $18,861 $18,861 $19,387 $19,542 $19,584

$1,102 $319 $85 $21 $1,102 $1,102 $1,102 $1,102 $1,102 $1,102 $319 $85 $21

$263,227 $257,606 $255,939 $255,496 $261,493 $260,707 $260,374 $253,402 $245,177 $238,400 $246,047 $235,369 $227,816

$48,990 $48,990 $48,990 $48,990 $48,990 $48,990 $48,990 $39,165 $30,940 $24,163 $39,165 $30,940 $24,163

Table 2 Optimal policies for a three-level supply chain (supplier–manufacturer–retailer) with learning in production, reworks, and set-ups, with coordination and no forgetting among the levels. Ex fprod (%)

1 2 3 4 5 6 7 8 9 10 11 12 13

100 90 80 70 100 100 100 100 100 100 90 80 70

frework

fset-up

(%)

(%)

100 100 100 100 90 80 70 100 100 100 90 80 70

100 100 100 100 100 100 100 90 80 70 90 80 70

ls lm Qr

3 3 3 3 3 3 3 3 4 4 3 4 4

1 1 1 1 1 1 1 1 1 1 1 1 1

667 668 668 667 669 671 671 607 555 507 611 559 511

Annual cost

Manufacturer’s cost components

Supply chain

Retailer

Supplier Manufacturer Set-up cost

Production cost

Quality cost

Finished product holding cost

Raw Material holding cost

$264,868 $259,577 $257,856 $257,346 $264,074 $263,643 $263,418 $246,140 $229,578 $214,610 $240,186 $221,701 $206,109

$29,167 $29,133 $29,140 $29,147 $29,106 $29,074 $29,059 $30,824 $32,731 $34,993 $30,696 $32,552 $34,781

$25,667 $25,657 $25,659 $25,661 $25,650 $25,642 $25,638 $26,211 $26,902 $27,125 $26,166 $26,903 $27,104

$7,500 $2,339 $664 $172 $7,500 $7,500 $7,500 $7,500 $7,500 $7,500 $2,376 $709 $200

$101,663 $101,886 $101,841 $101,790 $101,257 $101,023 $100,894 $92,633 $85,023 $78,236 $92,582 $84,768 $77,851

$10,267 $10,565 $10,650 $10,670 $10,307 $10,328 $10,337 $9,355 $8,586 $7,901 $9,671 $8,966 $8,273

$600 $191 $56 $15 $602 $604 $604 $547 $502 $462 $179 $52 $15

$210,034 $204,787 $203,057 $202,538 $209,318 $208,927 $208,721 $189,105 $169,945 $152,492 $183,324 $162,246 $144,224

for the values ls such that

cs ðls , lm ,Qr Þ o cs ðls 1, lm ,Qr Þ and

cs ðls , lm ,Qr Þ o cs ðls þ1, lm ,Qr Þ for given values of lm ¼ 1 and Qr ¼1225. The optimal policy for the supplier is cs(2,1225)¼$22,045 per year. The total supply chain cost csc(2,1,1225)¼ cr(1225)+ cm(1,1225) + cs(2,1225) ¼$24,495 +$263,227+$22,045¼$309,767 (see example 1 of Table 1). 5.2. With coordination and no learning The numerical example of Section 5.1 is now solved assuming that the players in the supply chain coordinate their orders according to a centralised decision making policy. Optimising (19), for ls, lm, and Qr gives ls ¼3, lm ¼1, and Qr ¼667 corresponding to a supply chain total cost, of csc(3,1,667) ¼$264,867. The costs of the retail, manufacturer, and supplier are cr(667) ¼$29,167, cm(1,667)¼ $210,034, and cs(3,667) ¼$25,667, respectively (see example 1 of Table 2). Coordination reduces the supply chain cost provided that the coordination itself does not cost too much. In example 1, there was a 14.5% reduction in the supply chain cost from $309,767

$90,004 $89,806 $89,846 $89,891 $89,652 $89,472 $89,386 $79,070 $68,334 $58,393 $78,516 $67,751 $57885

under no coordination to $264,868 with coordination. An average cost reduction of 17.6% was found over the set of thirteen numerical experiments conducted. When learning-based improvement was confined to production only (experiments 1–4), the average cost savings was 14.7%. Similarly, for improvement in rework only (experiments 1, 5–7) the average cost savings was 14.2% while for set-up only (experiments 1, 8–10) the average cost savings was 21.2%. Under a uniform improvement rate of 10% for production, rework and set-ups (experiments 1, 11–13), an average cost savings of 21.3% resulted. The cost reductions cited are parameter specific; however the experiments conducted demonstrate that cost advantage is clearly possible as a result of lot size coordination within a supply chain. However, when players coordinate their orders, the savings are shifted to one or more players in the chain. To guarantee coordination, the players who may become disadvantaged because of coordination should be compensated by the players who benefit from the coordination (e.g., Munson and Rosenblatt, 2001; Jaber et al., 2006). In the numerical examples above, the manufacturer whose cost reduces from $263,227 to $210,034; a reduction of 20.21%, is the player who benefits from the coordination. On the other hand, the retailer and the supplier are disadvantaged. The retailer’s cost increases from $24,495 to $29,167, and the supplier’s cost from $22,045 to $25,667. For the coordination to be successful, the manufacturer has to compensate the retailer and the supplier by offering them unit discount

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for abiding with its order policy. This is similar to the method adopted by Munson and Rosenblatt (2001); however, they compensated the retailer but not the supplier. In this paper, we assume that both the retailer and the supplier are compensated. So, it is expected that the manufacturer compensates the retailer by $0.031 per unit (($29,167–$24,495)/150,000 ¼0.031) and the supplier by $0.024 per unit (($25,667 $22,045)/150,000¼ 0.024). This brings the retailer and supplier costs back to their original values for the case of no coordination, which are $24,495 and $22,045, respectively. Subsequently, the manufacturer’s cost will increase from $210,034 to $218,328 ($210,034+ ($29,167 $24,495) +($25,667  $22,045)¼$210,033 +$4,672+ $3,622¼$218,328). Like Jaber et al. (2006), a savings (profit) sharing scheme could be implemented so the players would share the savings from coordination. A simple profit sharing scheme could be implemented such that the cost for each player reduces by the same percentage as the chain. That is, the costs of the retailer, manufacturer and supplier reduce by 14.5%, so the retailer’s cost reduces from $24,495 to $20,944, the manufacturer’s cost reduces from $263,227 to $225,073, and the supplier’s cost reduces from $22,045 to $18,851. So, the unit discount that the manufacturer offers the retailer and the supplier can also offer an additional share of the profit that may take the form of unit discount. So, the manufacturer may offer both the retailer and the supplier unit discounts (compensation for losses and a share of the savings) of $0.055 per unit (($29,167 $20,944)/150,000 ¼0.0548) and $0.045 per unit (($25,667 $18,851)/150,000¼0.0454). The results are summarised in example 1 of Table 3.

5.3. Accounting for learning but with no forgetting The numerical example in Section 5.1 is now solved using different setup (fset-up ¼100%, 90%, 80%, and 70%), production (fprod ¼ 100%, 90%, 80%, and 70%), and rework (frework ¼100%, 90%, 80%, and 70%) learning rates. In these numerical examples, no forgetting is assumed in set-up (a ¼0), production and reworks (B-p). The learning rates were assumed to be in the range of 90–70%. This is consistent to what has been observed in practice (e.g.; Dutton and Thomas, 1984; Dar-El., 2000). When the manufacturer undergoes continuous improvement (learning in set-up, in production, and in reworks) its cost reduces by a maximum of 13.4% (from $263,227 to $227,816) when the manufacturer’s improvements are maximum (fset-up ¼ fprod ¼ frework ¼ 70%) as shown in Table 1. The manufacturer’s cost

in Tables 1 and 2 is the average of its costs for 10 consecutive 10 P cycles, i.e., ci,m ðlm ,Qr =10Þ, after which no significant reduction i¼1

is noticeable. However, the manufacturer may not benefit fully from such improvements if there is no coordination between the players in the supply chain. For example, when there is coordination and when the manufacturer’s improvements are maximum (fset-up ¼ fprod ¼ frework ¼70%), the manufacturer’s cost reduces by a maximum of 45.2% from $263,227 in example 1 in Table 1, to $144, 224 in example 13 in Table 2. With coordination and no improvements (fset-up ¼ fprod ¼ frework ¼100%) the manufacturer’s cost reduces by 20.20% (from $263,227 in example 1 in Table 1, to $210,034 in example 1 in Table 2). This shows that coordination allows the manufacturer to maximise the benefits from implementing continuous improvements (percentage increased from 13.4% to about 25% (45.20%–20.20%)). Results in Table 2 also show that the manufacturer’s cost, and subsequently that of the chain, is most sensitive to learning in set-ups (examples 8, 9 and 10) and least sensitive to learning in reworks (examples 5, 6, and 7). This is true since larger lots result in more defective items, and reducing defects may only be achieved by reducing the lot size through reducing the set-up cost. Results in Table 3 show (examples 11 and 13) that as improvement becomes faster (e.g., from fset-up ¼ fprod ¼ frework ¼90% to fset-up ¼ fprod ¼ frework ¼70%), the retailer orders in even smaller lots (lot size reduces from 611 to 511), the cost of the chain reduces and the manufacturer is able to offer larger unit discounts as a compensation to the retailer (from 0.0413 to 0.0686) and the supplier (from 0.0275 to 0.0337). Subsequently, the combined compensation and profit sharing increases too for the retailer (from 0.0706 to 0.1092) and the supplier (from 0.0538 to 0.0703).

5.4. Accounting for learning and forgetting The numerical examples in Table 2 are replicated under the effect of forgetting with the assumption that total forgetting occurs if interruption in set-up, production, and reworks extends to periods of 1 year or longer (e.g., Globerson et al., 1998). The results are summarised in Table 4. Forgetting was shown to increase the manufacturer and the supply chain costs with no impact on the values of the decision variables ls, lm and Qr. This corroborates the findings summarised in Jaber and Bonney (1999) that forgetting is costly.

Table 3 Summary of results for the three-level supply chain for examples presented in Tables 1 and 2 with unit discounts. Ex

Compensation

Compensation and profit sharing

Annual cost

1 2 3 4 5 6 7 8 9 10 11 12 13

Unit discount

Annual cost

Unit discount

Supply chain

Retailer

Supplier

Manufacturer

Retailer

Supplier

Retailer

Supplier

Manufacturer

Retailer

Supplier

$264,868 $259,577 $257,856 $257,346 $264,074 $263,643 $263,418 $246,140 $229,578 $214,610 $240,186 $221,701 $206,109

$24,495 $24,495 $24,495 $24,495 $24,495 $24,495 $24,495 $24,495 $24,495 $24,495 $24,495 $24,495 $24,495

$22,045 $22,045 $22,045 $22,045 $22,045 $22,045 $22,045 $22,045 $22,045 $22,045 $22,045 $22,045 $22,045

$218,328 $213,037 $211,316 $210,806 $217,534 $217,103 $216,878 $199,600 $183,038 $168,070 $193,646 $175,161 $159,569

0.0311 0.0309 0.0310 0.0310 0.0307 0.0305 0.0304 0.0422 0.0549 0.0700 0.0413 0.0537 0.0686

0.0241 0.0241 0.0241 0.0241 0.0240 0.0240 0.0239 0.0278 0.0324 0.0339 0.0275 0.0324 0.0337

$20,944 $20,905 $20,881 $20,871 $20,999 $21,018 $21,023 $20,101 $19,277 $18,449 $20,108 $19,263 $18,402

$18,851 $18,815 $18,793 $18,784 $18,899 $18,917 $18,921 $18,091 $17,349 $16,604 $18,097 $17,337 $16,562

$225,073 $219,857 $218,182 $217,691 $224,176 $223,708 $223,474 $207,948 $192,952 $179,557 $201,981 $185,101 $171,145

0.0548 0.0549 0.0551 0.0552 0.0540 0.0537 0.0536 0.0715 0.0897 0.1103 0.0706 0.0886 0.1092

0.0454 0.0456 0.0458 0.0458 0.0450 0.0448 0.0448 0.0541 0.0637 0.0701 0.0538 0.0638 0.0703

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35

Table 4 Optimal policies for a three-level supply chain (supplier–manufacturer–retailer) with learning in production, reworks, and set-ups, with coordination and forgetting among the levels. Ex fprod (%)

2 3 4 5 6 7 8 9 10 11 12 13

90 80 70 100 100 100 100 100 100 90 80 70

frework

fset-up

(%)

(%)

100 100 100 90 80 70 100 100 100 90 80 70

100 100 100 100 100 100 90 80 70 90 80 70

ls lm Qr

3 3 3 3 3 3 3 4 4 3 4 4

1 1 1 1 1 1 1 1 1 1 1 1

668 668 668 669 670 671 607 555 507 610 559 511

Annual cost

Manufacturer’s cost

Supply chain

Retailer

Supplier Manufacturer Set-up cost

Production cost

Quality cost

Finished product holding cost

Raw material holding cost

$260,404 $258,448 $257,637 $264,301 $263,904 $263,631 $247,020 $230,900 $216,047 $242,085 $223,832 $208,018

$29,132 $29,133 $29,141 $29,120 $29,089 $29,070 $30,817 $32,709 $34,949 $30,706 $32,535 $34,734

$25,657 $25,657 $25,660 $25,654 $25,646 $25,641 $26,207 $26,892 $27,105 $26,167 $26,891 $27,082

$3,133 $1,230 $446 $7,500 $7,500 $7,500 $7,500 $7,500 $7,500 $3,171 $1,292 $500

$101,892 $101,886 $101,832 $101,393 $101,191 $101,037 $92,622 $84,971 $78,138 $92,674 $84,890 $77,919

$10,523 $10,624 $10,660 $10,297 $10,318 $10,330 $9354 $8581 $7891 $9621 $8929 $8249

$266 $112 $44 $602 $603 $604 $547 $501 $461 $246 $99 $38

$205,615 $203,658 $202,836 $209,527 $209,169 $208,920 $189,996 $171,299 $153,993 $185,212 $164,406 $146,202

6. Summary and conclusions This paper has investigated a three-level supply chain (supplier–manufacturer–retailer) where the manufacture undergoes a continuous improvement process. The continuous improvement process is characterized by reducing set-up times, increasing production capacity and eliminating rework as a result of learning-based improvements. The cases of coordination and no coordination were investigated using a set of numerical examples that addressed learning and forgetting in set-up times, production capacity and rework. Under the base case model (no learning or forgetting), coordination among channel members reduced total supply chain costs by 14.5% in comparison to no coordination among supply chain members. As learning-based improvement was introduced, coordination continued to dominate no coordination with respect to the level of reduction in the total supply chain costs. Experiments involving learning-based improvement in production only averaged a 14.7% reduction in total supply chain costs for the coordinated strategy. Average reductions of 14.2% and 21.2% were achieved for the cases of learning-based improvement in rework and set-ups respectively. When learning-based improvement was initiated simultaneously in production, rework and set-ups the average percent cost savings in the supply chain costs under coordination reached a maximum of average reduction of 21.3%. Traditionally, with coordination the manufacturer entices the retailer to order in larger lots than its economic order quantity. In this paper, the opposite was true. The manufacturer entices the retailer to order in smaller quantities than the retailer’s economic order quantity. As improvement becomes faster, the retailer is recommended to order in progressively smaller quantities as the manufacturer offers larger discounts and profits. The results also show that coordination allows the manufacturer to maximise the benefits from implementing continuous improvements. The numerical examples were also investigated for forgetting effects. It was shown that forgetting increases the supply chain cost under each possible improvement implementation. For example, when learning-based improvements were implemented simultaneously for production, set-ups and rework, forgetting increased the total supply chain costs by an average of nearly 1% but has no effect on the values of the decision variables forming the optimal policy. There are several aspects of this research that could be expanded. First, multiple types of defects could be introduced in which the rework time for a given class of defect is a function of the type of defect, Second, the limiting assumptions of

$89,801 $89,806 $89,854 $89,735 $89,557 $89,449 $79,973 $69,746 $60,003 $79,500 $69,196 $59,496

deterministic demand and zero lead time could be generalized to allow a stochastic demand during lead time interface for the lot-sizing decisions between channel members. Third, the total supply chain costs savings demonstrated by the set of numerical examples could be used as a benchmark for examining further economies in the cost structure of the supply chain that might result from the implementation of programs to better manage the interfaces between chain members. These issues might include vendor managed inventory, radio frequency identification, and cross docking. Lastly, the scope of the supply chain could be examined in an attempt to generalize the findings of the threestage chain to that of an n (n43) stage supply chain.

Acknowledgment MY Jaber thanks the Social Sciences and Humanities Research Council (SSHRC) of Canada, Standard Grant Program, for funding his research.

Appendix A A.1. Part I: Holding cost during the manufacturer production segment The inventory level in a manufacturer production segment is described in Fig. A1. Approximate values for the average inventory level (AIL) for each portion in the production segment of the manufacturer’s cycle. AIL1 ¼ ð0 þ q1 Þ=2 where q1 is the manufacturer’s cumulative production by time Tr when the retailer makes the first withdrawal of Qr units and q1 ZQr. See Fig. A1 for illustration, where n¼2, and n represents the number of shipments of size Qr that occur during t1 the production segment of Tm. Similarly, AIL2 ¼ ðq1 Qr þ q2 Qr Þ=2 ¼ ððq1 þ q2 Þ=2ÞQr , where q2 is the manufacturer’s cumulative production by time 2Tr when the retailer makes a second withdrawal of Qr units and q2 4q1 ZQr. For n > 2 (not the case in Fig. A1), we have AIL3 ¼ ðq2 2Qr þ q3 2Qr Þ=2 ¼ ððq2 þ q3 Þ=2Þ2Qr , where q3 is the manufacturer’s cumulative production by time 3Tr when the retailer makes a withdrawal of Qr units and q3 4q2 4q1 ZQr, etc. Similarly, AILn ¼ ðqn1 ðn1ÞQr þ qn ðn1ÞQr Þ=2 ¼ ððqn1 þ qn Þ= 2Þðn1ÞQr , where qn is the manufacturer’s cumulative production by time nTr when the retailer makes the nth withdrawal of Qr units and qn 4? 4q5 4q4 4q3 4q2 4q1 ZQr. The manufacturer produces for a period of t1i units of time Qm ¼ lmQr units in cycle i, where it is assumed that the manufacturer produces in equal lots.

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M.Y. Jaber et al. / Int. J. Production Economics 127 (2010) 27–38

Inventory level

t1

2T r

q 2 − Qr

Qr Tr

q 2 − 2Qr

q1 Qr

AIL2

q1 − Qr

AIL1

Manufacturer's production time Fig. A1. Manufacturer’s inventory behaviour of its production segment with two (n¼ 2) retailer inventory withdrawals of size Qr for every retailer cycle of length Tr.

Qm − nQr = Qm − 2Qr

Qr

Inventory Level

t1 − n Qr D = t1 − 2 Qr D

Qr

Tr

Qr Tr

Manufacturer's Inventory Depletion Time Fig. A2. Manufacturer’s inventory behaviour of its inventory depletion segment with three (k¼ 3) retailer inventory withdrawals of size Qr for every retailer cycle of length Tr.

Therefore, the maximum inventory level attained in a manufacturer’s cycle is Qm nQr, where AILm ¼ ðqn nQr þ Qm nQr Þ =2 ¼ ððqn þ Qm Þ=2ÞnQ . The holding cost during the manufacturer’s production time is

HC1,m ¼ hm ðq1 þq2 þ q3 þ q4 þ q5 þ    þqn1 ÞTr þ hm Tr qn =2 hm ðQr þ2Qr þ 3Qr þ    þ ðn1ÞQr ÞTr

qn þ Qm nQr ðt1i nQr =DÞ þhm 2

HC1,m ¼ hm fAIL1 þ AIL2 þ AIL3 þ    þ AILn gTr þ hm AILm ðt1i nQr =DÞ

q þ q 

q þq  0 þq1 1 2 2 3 Tr þ hm Qr Tr þhm 2Qr Tr HC1,m ¼ hm 2 2 2

q þq 

q þ q  5 3 4 4 3Qr Tr þ hm 4Qr Tr þ    þhm 2 2

q  qn þ Qm n1 þ qn ðn1ÞQr Tr þhm nQr þhm 2 2 ðt1i nQr =DÞ, then

where q1 ¼



1=1b 1b Tr þ ðui þ 1=2Þ1b ui 1=2, y1

q2 ¼



1=1b 1b 2Tr þ ðui þ 1=2Þ1b ui 1=2 y1

M.Y. Jaber et al. / Int. J. Production Economics 127 (2010) 27–38



1=1b 1b 3Tr þ ðui þ1=2Þ1b ui 1=2, y1

1=1b 1b q4 ¼ 4Tr þ ðui þ1=2Þ1b ui 1=2, y1

1=1b 1b 5Tr þ ðui þ1=2Þ1b ui 1=2, q5 ¼ y1

1=1b 1b qn ¼ nT þ ðui þ 1=2Þ1b ui 1=2, y1

1=1b ui 1=2, with hm and ui Qm ¼ 1b=y1 t1i þ ðui þ 1=2Þ1b being the number of equivalent units of experience remembered P 1 at the beginning of cycle i, represented as 0 rui r ij ¼ ¼ 1 ðj1ÞQm . Total forgetting occurs when ui ¼0. q3 ¼

HCi,1,m ¼ hm T

n X 1b j¼1

y1

ðj1ÞT þðui þ 1=2Þ1b

1=1b



1=1b h 1b T nT þ ðui þ 1=2Þ 2 y1 ðui þ1=2ÞTr nðn1Þ ðt nQr =DÞ Qr Tr þ hm 1i hm hm 2 2 2

1=1b 1b ðui þ1=2Þ 1b ðt1i nQr =DÞ t þ ðui þ 1=2Þ hm  y1 1i 2

lm Qr nQr ðt1i nQr =DÞ þ hm ðA:1Þ 2 hm nðui þ 1=2ÞT þ

where Tr ¼ Qr =D:

A.2. Part II: Holding cost during the manufacturer inventory depletion segment The manufacturer ceases production by time t1i, when a maximum inventory level of Qm  nQr ¼(lm n)Qr is attained. As shown in Fig. A2, this maximum inventory level is depleted through regular deliveries of shipments of size Qr to the retailer every Tr units of time, where kQr ¼Qm  nQr. The holding cost during the manufacturer’s inventory depletion time is computed in a similar manner as in Jaber et al. (2006) as HCi2,m ¼ hm Qr ðk1ÞTr þ hm Qr ðk2ÞTr þ hm Qr ðk3ÞTr þ    þhm Qr Tr þ hm kQr ðt1i n nQr =DÞ, HCi2,m ¼ hm Qr kðk þ1ÞTr =2hm kQr ðt1i nQr =DÞ

ðA:2Þ

where Tr ¼ Qr =D:

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