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Supply chain coordination model with controllable lead time and service level constraint
Computers & Industrial Engineering 61 (2011) 858–864
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Supply chain coordination model with controllable lead time and service level constraint q Yina Li, Xuejun Xu, Fei Ye ⇑ School of Business Administration, South China University of Technology, Guangzhou 510640, China
a r t i c l e
i n f o
Article history: Received 13 April 2010 Received in revised form 1 March 2011 Accepted 27 May 2011 Available online 13 June 2011 Keywords: Controllable lead time Supply chain Service level constraint Price discount mechanism
a b s t r a c t We consider the coordination issue in a decentralized supply chain composed of a vendor and a buyer in this paper. The vendor offers a single product to the buyer who is faced with service level constraint. In addition, lead time can be reduced by added crashing cost. We analyze two supply chain inventory models. The first one is developed under decentralized mode based on Stackelberg model, the other one is developed under centralized mode of the integrated supply chain. The solution procedures are also provided to get the optimal solutions of these two models. Finally, a price discount mechanism is proposed to induce both the vendor and the buyer to accept the centralized model. The feasibility and efficiency of the proposed models are manifested by numerical examples and some managerial implications are highlighted. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The theoretical and practical studies on Japanese Just-in-Time (JIT) have resulted in considerable efforts in reduction on lead time and inventory-related costs simultaneously (Ben-Daya & Hariga, 2003). In JIT philosophy, lead time can be reduced by effective methods and added crashing cost (Axsater, 2011). It is quite different from the traditional Economic Order Quantity (EOQ) literature that viewing lead time as a prescribed constant or a stochastic variable. Time-Based Competition (TBC), which focuses on management of time, specially lead time, has becoming one of the most important modes to gain competitive advantage. Lead time reduction can lower safety stock, improve customer service level, realize supply chain quick response to customer requirements, all of which combines to help companies to gain and maintain competitive advantage in existing and new markets (Tersine & Hummingbird, 1995). Liao and Shyu (1991) were the first to consider a continuous review model in which order quantity is predetermined and lead time is a unique decision variable. Then more and more inventory model literature considering lead time reduction issue has been developed, which can be mainly divided into two broad categories. The first category focused on considering how to find the optimal inventory policy to minimize the total relevant cost of either single firm or supply chain, including the stock out cost, which is used an exact value to express. For example, Ben-Daya and Raouf q
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(1994) extended Liao and Shyu (1991) model by viewing both lead time and order quantity as decision variables. Ouyang, Yen, and Wu (1996) further treated the shortages to be a mixture of backorders and lost sales. Pan, Hsiao, and Lee (2002) considered lead time crashing cost to be a function of both order quantity and reduced lead time. Pan and Yang (2002) viewed lead time as a decision variable and obtained a lower joint expected cost and shorter lead time of integrated supply chain. Ouyang, Wu, and Ho (2004) extended Pan and Yang (2002) model by further considering the reorder point as one of the decision variables and shortages permitted. Yang and Pan (2004) minimized the total expected cost of integrated supply chain by simultaneously optimizing the order quantity, lead time, process quality and number of deliveries. Chang, Ouyang, Wu, and Ho (2006) developed an integrated supply chain model with controllable lead time and ordering cost reduction. Ouyang, Wu, and Ho (2007) formulated and solved an integrated inventory model involving imperfect production process with controllable lead time. Chandra and Grabis (2008) optimized the total inventory and procurement cost in a variable lead-time inventory system where lead-time is a dependent function of procurement cost. Lin (2009) considered an integrated model where lead time and ordering cost can be reduced by additional investment. Ye and Xu (2010) developed an asymmetric Nash bargaining model to get the best cost allocation ratio to coordination the benefits of both parties in a decentralized supply chain with controllable lead time. It should be noted that all of the first category models considering controllable lead time assumed the stock out cost can be expressed by an exact value. However, the second category researchers took the opinion that, in many practical situations,
Y. Li et al. / Computers & Industrial Engineering 61 (2011) 858–864
the stock out cost is very difficult to identify by an exact value for there will always be some intangible loss such as the damage of a company’s reputation and credibility, the potential delay to other parts of inventory system, and so on. And the stock out will negatively impact customers’ satisfactions since they are unable to get the products they want at the time they want them, resulting in higher customer dissatisfaction and lower customer loyalty. Hence, they used a service level constraint to replace the stock out cost in the objective function of the models in the first category, requiring a certain proportion of demand should be met in each cycle. Ouyang and Wu (1997) used a service level constraint to replace the stock out cost in the objective function of Ouyang et al. (1996) model to bind the stock out level each cycle. Chu, Yang, and Chen (2005) studied the same model as Ouyang and Wu (1997), while improved the algorithm to get the optimal solutions. In a recent paper, Jha and Shanker (2009) proposed a two-echelon integrated supply chain inventor model with controllable lead time and service level constraint. However, Jha and Shanker (2009) model focused on integrated channel in which assuming a central planner maximizes overall channel performance, with the power to impose a globally optimal inventory policy on each party. Although such a joint optimal order and production policy leads to a significant total cost reduction comparing to independently derived policies, there is an additional set of problems involved in implementing joint policies (Sucky, 2006). For example, a multiform supply chain without a central planner will have incentive conflicts because different parties in a supply chain generally have different and often conflicting objectives. Therefore, how to make effective coordination mechanisms that retaining healthy partnerships among independent chain parties and improving the chain value is critical to the supply chain success. In this paper, we make two major contributions to the present literature on supply chain optimization problem with controllable lead time. First, different from the first category studies that minimizing the total relevant expected cost of single firm or supply chain, which using an exact value to express the stock out cost, we use a service level constraint in place of the stock out cost to bound the stock out level per cycle and avoid the difficulty to estimate the exact value of stock out cost in real situations. Secondly, different from the service level approach model in the second category studies, we extend Ouyang and Wu (1997) model considering service level constraint and controllable lead time from the perspective of single buyer into supply chain perspective. And we also relax the assumption in Jha and Shanker (2009) model from the perspective of supply chain that long-term strategic partnership between the vendor and the buyer was well established and they could cooperate with each other to obtain an optimal integrated joint policy under centralized decision. In contrast to Jha and Shanker (2009), we assume the vendor and the buyer representing different benefit entities and take their individual rationalities into consideration, examine coordinated decision in a decentralized (two-echelon) supply chain that consists of one buyer and one vendor, with controllable lead time and service level constraint. In addition, in order to induce both parties to accept the centralized decision model and realize the Pareto dominance of the entire supply chain system, an effective price discount mechanism is developed to coordinate benefits between the vendor and the buyer and make both of them realize Pareto improvement through coordination. The remainder of the paper is organized as follows. Section 2 introduces assumptions and notations. In Section 3, two different inventory models, decentralized model and centralized model, with controllable lead time and service level constraint are proposed. The procedures are also suggested to get the optimal solutions. Numerical examples and sensitive analysis of different service level constraints are provided to illustrate the results of
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the proposed models in Section 4 and Section 5 propose a price discount mechanism to coordination the profits between the vendor and the buyer. Finally, conclusions and further research areas are provided in Section 6. 2. Notations and assumptions To establish the proposed models, the following notations and assumptions are used. Additional notations and assumptions will be listed later when needed. 2.1. Notations The decision variables are Q The buyer’s order quantity L The length of lead time m The number of lots in which the product is delivered from the vendor to the buyer in one production cycle, a positive integer a The ratio of price discount offered by the vendor. a 2 ½0; 1 The other related parameters are D Average demand per year P The vendor’s production capability per year (P > D) p The buyer’s retail price w The vendor’s wholesale price c The vendor’s unit production cost A The buyer’s ordering cost per order S The vendor’s setup cost per set-up hr The buyer’s unit holding cost per year hs The vendor’s unit holding cost per year h Proportion of demand that are not met from stock. h 2 ½0; 1 b The fraction of the demand during the shortage period that will be backordered. b 2 ½0; 1
2.2. Assumptions 1. There is a single vendor and a single buyer that representing different entities in a decentralized supply chain. 2. The buyer adopts a continuous review inventory policy and makes replenishments whenever the inventory level falls to the reorder point r. 3. The reorder point r = expected demand during lead time + safety stock. The demand X during lead time L is assumed to be normally distributed with pffiffiffi pffiffiffi mean DL and standard deviation d L. That is, r ¼ lL þ kd L, where k is the safety factor. 4. The buyer orders a lot of size Q and the vendor manufactures the product in lots of size mQ with a finite production rate P (P > D) and ship in quantity Q to the buyer over m lots. 5. The buyer faces with a service level constraint not lower than 1 h. 6. Shortages will be partly backordered with the fraction b. 7. The lead time has n mutually independent components. The ith component has a minimum duration ai and normal duration bi, and a crashing cost per unit time ci. Furthermore, for convenience, we arrange ci such that c1 6 c2 6 6 cn . Then, it is clear that the reduction of lead time should be first on component 1 (because it has the minimum unit crashing cost), and then component 2, and so on. P 8. If we let L0 ¼ nj¼1 bj and Li be the length of lead time with components 1; 2; . . . i crashed to their minimum duration, then Li is P P P P expressed as Li ¼ ij¼1 aj þ nj¼iþ1 bj ¼ nj¼1 bj ij¼1 ðbj aj Þ
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Y. Li et al. / Computers & Industrial Engineering 61 (2011) 858–864
Pi
¼ L0 j¼1 ðbj aj Þ, i ¼ 1; 2; . . . n. And the lead time crashing cost R(L) per cycle for a given L 2 ½Li ; Li1 is given by P RðLÞ ¼ ci ðLi1 LÞ þ i1 j¼1 cj ðbj aj Þ. 9. All information are common information to both the vendor and the buyer.
Max
s:t:
DS ps ðmÞ ¼ ðw cÞD mQ hs Q2 m 1 DP 1 þ 2D P 8 > Max pr ðQ ; LÞ ¼ ðp wÞD DðAþRðLÞÞ > Q >
< > > > :
hr s:t:
pffiffi d LWðkÞ h
Q 2
pffiffiffi pffiffiffi þ kd L þ ð1 bÞd LWðkÞ
6Q
3. The model
ð4Þ
3.1. The buyer’s inventory model According to the above setting, the buyer’s annual expected profit equals to the annual revenue minus annual total cost, which is consisted of annual purchasing cost, annual ordering cost, annual holding cost and annual lead time crashing cost. The buyer’s annual revenue equals to pD, the annual purchasing cost, annual ordering cost and annual lead time crashing cost will be equal to wD, DA/Q, DR(L)/Q, respectively. And since the expected R þ1 shortages of each order cycle is EðX rÞþ ¼ r ðX rÞdFðxÞ ¼ pffiffiffi d LWðkÞ, where WðkÞ ¼ /ðkÞ k½1 UðkÞ, and /; U are the probability density function and cumulative distribution function of the standard normal distribution, respectively. With backorder ratio b, pffiffiffi the expected number of backorders per cycle is bd LWðkÞ and the pffiffiffi expected demand lost is ð1 bÞd LWðkÞ. Hence, the average pffiffiffi pffiffiffi on-hand inventory for the buyer is Q2 þ ð1 bÞd LWðkÞ þ kd L p ffiffi ffi p ffiffi ffi and annual holding cost equals to hr Q2 þ ð1 bÞd LWðkÞ þ kd L . Consistent with the previous literature, we define the service level should meet the requirement that the actual proportion of demands not met from stock should not exceed the desired value of h. Therefore, the buyer will decide the optimal order quantity and lead time to maximize his own annual expected profit, subject to a service level constraint:
Max
pr ðQ ; LÞ ¼ ðp wÞD DðAþRðLÞÞ Q hr
s:t:
pffiffi d LWðkÞ Q
Q 2
pffiffiffi pffiffiffi þ kd L þ ð1 bÞd LWðkÞ
In order to solve this kind of problem, similar to literature Ouyang and Wu (1997), Chu et al. (2005), Jha and Shanker (2009), we ignore the service level constraint in Eq. (4) first. From Eq.(4), we can easily get that pr(Q, L) is concave with respect to order quantity Q and convex with respect to lead time L (for the details of proof, please refer to the published paper dealing with controllable lead time, e.g., Ouyang et al., 2004). Hence, for each Li ði ¼ 0; 1; . . . ; n, we can get the optimal order quantity under leader–follower relationship without the service level constraint will be:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DðA þ RðLi ÞÞ Qi ¼ hr
ative Alogrithm 1 is suggested to get the buyer’s optimal order quantity and lead time with service level constraint under decentralized decision mode.
ð1Þ Algorithm 1
6h
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i ÞÞ Step1: For each Li ði ¼ 0; 1; . . . ; n compute Q i ¼ 2DðAþRðL using hr Eq. (5). pffiffiffi pffiffiffi
d Li wðkÞ d Li wðkÞ Step2: Compute , set xi ¼ max Q i ; , for h h i ¼ 0; 1; 2; . . . ; n. Step3: Set pr ðx ; L Þ ¼ Maxi¼0;1;2;...n pr ðxi ; Li Þ, and ðx ; L Þ is buyer’s optimal decision.
pr ðQ ; LÞ ¼ ðp wÞD DðAþRðLÞÞ Q hr
s:t:
pffiffi d LWðkÞ h
ð5Þ
And the buyer’s maximum annual expected profit will occur at the end points of the interval ½Li ; Li1 , i ¼ 1; 2; . . . n. (We denote the optimal lead time here to be L ). Now considering the service level constraint, for each pffiffiffi d Li WðkÞ Li ði ¼ 0; 1; . . . ; n, if 6 Q i , then Qi is the local maximum of h pr ðQ i ; Li Þ and the service level constraint does not take effect. If pffiffiffi pffiffiffi d Li WðkÞ d Li wðkÞ > Q , then is the local maximum of pr ðQ i ; Li Þ so that i h h the service level per cycle can be met. Hence, for fix L 2 ½Li ; Li1 , the pffiffiffi
d Li wðkÞ buyer’s optimal order quantity will be max Q i ; . The iterh
According to Chu et al. (2005), Eq. (1) can be transformed into:
Max
Q 2
pffiffiffi pffiffiffi þ kd L þ ð1 bÞd LWðkÞ
ð2Þ
6Q
3.2. The vendor’s inventory model Similar to most of the literature dealing with controllable lead time with the same setting for the vendor (e.g., Ouyang et al., 2004), the vendor’s annual expected profit is represented by:
DS Q D 2D hs m 1 1þ ps ðmÞ ¼ ðw cÞD mQ 2 P P
ð3Þ
3.3. Leader–follower relationship When the buyer and the vendor belong to different entities, they will decide the optimal policies to maximize their own expected profits separately. Among the many possible uncooperative gaming assumptions, we consider the popular leader–follower (i.e. Stackelberg) structure, where the buyer is the Stackelberg follower who sets the order quantity Q and lead time L to maximize its own annual expected profit first and the vendor is the leader who determines the corresponding number of lots m to maximize its own annual expected profit accordingly. Under such a setting, the Stackelberg model between the vendor and the buyer is given by:
The vendor will decide the optimal number of lots m reacts to the buyer’s optimal order quantity x and optimal lead time L . For any given L 2 ½Li ; Li1 and order quantity, ps ðmÞ is concave in m, since
@ 2 ps ðx ; mÞ 2DS ¼ 3 <0 @m2 m x
ð6Þ
Therefore, the vendor’s optimal value of m ¼ m is obtained when
ps ðx ; m Þ P ps ðx ; m þ 1Þ ps ðx ; m Þ P ps ðx ; m 1Þ
ð7Þ
From Eq. (7) we can easily get m is the first integer satisfying
2DS D 2DS 2 6 6 h ðx Þ 1 s m ðm þ 1Þ P m ðm 1Þ
ð8Þ
3.4. Long-run cooperation and the centralized solution Consider now the situation in which both the buyer and the vendor belong to the classic single decision-maker system and
Y. Li et al. / Computers & Industrial Engineering 61 (2011) 858–864
there is a central planner makes all decisions to maximize the total expected profit of the entire supply chain. Under such situation, the integrated inventory of supply chain under centralized mode is given by
psc ðQ ; L; mÞ ¼ ðp cÞD DðmAþSþmRðLÞÞ mQ
Max
pffiffiffi pffiffiffi þ kd L þ ð1 bÞd LWðkÞ hs Q2 m 1 DP 1 þ 2D P
hr pffiffi d LWðkÞ h
s:t:
Q 2
ð9Þ
6Q
By ignoring the service level constraint, we can also easily get that psc ðQ ; L; mÞ is concave with respect to order quantity Q and convex with respect to lead time L (please also refer to the published paper dealing with controllable lead time, e.g., Ouyang et al., 2004, for the details of proof). Hence, for each Li ði ¼ 0; 1; . . . ; n, we can get the optimal order quantity under centralized decision mode without service level constraint will be:
( Qi ¼
2DðA þ mS þ RðLÞÞ hr þ hs m 1 DP 1 þ 2D P
)12 ð10Þ
And the maximum annual expected profit of the entire supply chain will occur at the end points of the interval ½Li ; Li1 , i ¼ 1; 2; . . . n. (We denote the optimal lead time here to be L ). As the service level constraint is taken into consideration, similar to that of decentralized decision mode, we can also obtain that for fix L 2 ½Li ; Li1 , the optimal order quantity of the centralized pffiffiffi
d Li wðkÞ . decision mode will be max Q i ; h 2
ðQ ;L;mÞ ¼ m2DS And since @ psc@m 2 3 Q < 0, then psc ðQ ; L; mÞ is concave in m for fixed Q and L 2 ½Li ; Li1 , which indicate that there must be an optimal m to meet Eq. (11):
psc ðQ ; L; m Þ P psc ðQ ; L; m þ 1Þ psc ðQ ; L; m Þ P psc ðQ ; L; m 1Þ
ð11Þ
Hence, the iterative Alogrithm 2 is suggested to find the optimal values of Q, L and m under centralized decision mode. Algorithm 2 Step 1: Start with m = 1. Step 2: For each Li ði ¼ 0; 1; . . . ; n, compute Qi using Eq. (10). pffiffiffi pffiffiffi
d Li wðkÞ d Li wðkÞ Step 3: Compute , set x ¼ max Q ; , i i h h i ¼ 0; 1; 2; . . . ; n. Step 4: For each set of ðxi ; Li ; mÞ, compute psc ðxi ; Li ; mÞ using Eq. (9), i ¼ 0; 1; 2; . . . ; n. Step 5: Set psc x then m ; Lm ; m ¼ Maxi¼0;1;...;n psc ðxi ; Li ; mÞ, xm ; Lm ; m is the optimal solution for any given m. Step 6: Set m ¼ m þ 1, repeat Step 2 to Step 5 to get psc x m ; Lm ; m . Step 7: If psc x m ; Lm ; m P psc xm1 ; Lm1 ; m 1 , go to Step 6, otherwise go to Step 8. Step 8: Set ðx ; L ; m Þ ¼ x then m1 ; Lm1 ; m 1 , ðx ; L ; m Þ is the optimal solution set for the centralized decision model. 4. Numerical example In order to illustrate the proposed model, let us consider an inventory system with the following characteristics: D = 600 units/year, p = $100/unit, w = $90/unit, c = $80/unit, hr = $20/unit/year, P = 3000 units/year, A = $200/order, d = 7 units/
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week, hs = $30/unit/year , S = $1000/set-up, k = 1, b = 0.5, h = 1%, that is, the service level constraint 1 h = 99%. The lead time has three components with the data shown in Table 1. Through applying the previously developed algorithms, we obtain the results of leader–follower relationship and long-run cooperation situation, which are shown in Tables 2 and 3, respectively. From Table 2, we can easily find that the buyer will select the optimal order quantity x ¼ 116:62 units, and optimal lead time L ¼ 4 weeks to get his own maximum expected profit pr ð116:62; 4Þ ¼ $3397:91. Then using Eq. (8), we can get the vendor’s optimal number of lots m ¼ 2 and the vendor’s maximum expected profit ps ð2Þ ¼ $1678:24. Thus, the total expected profit of the entire supply chain is psc ð116:62; 4; 2Þ ¼ $5076:15 . From Table 3, we get the optimal order quantity x ¼ 237:53 units, optimal lead time L ¼ 4weeks and optimal number of lots m ¼ 1, thus, the total expected profit of the entire supply chain is psc ð237:53; 4; 1Þ ¼ $5532:67, which is 8.99% higher than that of leader–follower relationship in this case. Based on this case, we further analyze the variation of the results under centralized and decentralized model by considering different service level constraints, which is shown in Table 4 and Figs. 1–5. Table 4 and Figs. 1–5 show the impacts of the change of service level constraint on the expected profits of different supply chain parties, optimal order quantity and optimal lead time. From Table 4 and Figs. 1–5, we can clearly see the following managerial implications in both cases. (1) From Fig. 1 we can clearly see that under centralized decision model, the expected profit of the entire supply chain will be decreased with the increase of the service level. That is, the higher the service level is, the lower the supply chain’s expected profit will be. Especially, if the service level has been set to a certain high level, to further upgrade the service level should be at great cost. In our case, when the service level is 99.7%, the supply chain’s expected profit equals to $5129.92. But if we need to upgrade the service level to be 99.8%, the supply chain’s expected profit will be decreased to $3688.68, which is decreased by 28%. Obviously, it is not advisable to upgrade the service level from 99.7% to 99.8% from the cost perspective. That is, there will be a balance to ‘‘trade-off’’ as a company sets the target of service level, 100% service level is not the best choice for all companies. (2) From Figs. 2 and 3, we can clearly observe that in both the decentralized and centralized decision models, the buyer’s expected profit will be decreased with the increase of the service level, while the vendor’s expected profit will be increased with the increase of the service level. Moreover, though we find the supply chain’s expected profit will be improved in the centralized model comparing to that of decentralized model in Fig. 1, the buyer’s expected profit under centralized model, however, will be lower than that of decentralized model, which indicates that the key to realize the maximum expected profit of the entire supply chain is to develop reasonable coordination mechanism to make both the buyer and the vendor’s expected profits under centralized model will not lower than that of decentralized model, that is, achieve Pareto improvement. (3) From Fig. 4 we can see that the optimal order quantity will be increased with the increase of the service level in both centralized and decentralized models, which indicates that high service level should be at the expense of high inventory level. (4) From Fig. 5 we can see that the optimal lead time will be shorter under high service level than that of lower service level in both centralized and decentralized models. For instance, as the service level is increased from 97% to
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Y. Li et al. / Computers & Industrial Engineering 61 (2011) 858–864
Table 1 Lead time data. Lead time component i
Normal duration bi (days)
Minimum duration ai (days)
Unit crashing cost ci ($/day)
1 2 3
20 20 16
6 6 9
0.4 1.2 5.0
of the entire supply chain. The price discount coordination mechanism is well established and applied to previous inventory literature to achieve supply chain coordination (Hu & Munson, 2010; Li & Liu, 2006; Monahan, 1984; Shin & Benton, 2007; Weng, 1995). Considering the vendor offer the following price discount scheme:
w0 ¼ 99.5%, the optimal lead time under decentralized model will be shortened from 6 weeks to 3 weeks. This also helps to illustrate that shorter lead time is an effective way to realize quick customer response. In addition, we can see that for each service level, there exists an optimal lead time to maximize the buyer’s expected profit (decentralized decision model) and the supply chain’s expected profit (centralized decision model). That is, assuming the lead time to be controllable will get more profit improvement comparing to the fixed lead time studies.
Though the centralized model can yield more profit than the decentralized one from the perspective of the entire supply chain (as shown in Fig. 1), the vendor and the buyer might not be willing to behave as in the centralized model unless both of them can gain more profits than that of Stackelberg model. Comparing the results of Tables 2 and 3, we can easily find that the buyers’ expected profit under centralized model is lower than that of Stackelberg model, as is shown in Fig. 2, which means if there is no reasonable incentive mechanism, the buyer may not be willing to shift to the centralized decision model due to his individual rationality. So the key to induce both the buyer and the vendor to accept the centralized model is to develop reasonable coordination mechanism to make both of them earn more benefits from centralized model than that of decentralized model. In this section, we develop a price discount mechanism to coordinate the supply chain benefit. The objective of this coordination mechanism is to help to not only meet both the vendor and the buyer’s individual rationalities, but also realize Pareto dominance
L – L
Q < x ;
w;
ð12Þ
ð1 aÞw; Q P x ; L ¼ L
to encourage the buyer to make decisions to maximize the expected profit of the entire supply chain. Then the buyer’s profit will be
DðA þ RðL ÞÞ x pffiffiffiffiffiffi pffiffiffiffiffiffi x hr þ kd L þ ð1 bÞd L WðkÞ 2
p0r ðx ; L Þ ¼ ðp ð1 aÞwÞD
ð13Þ
The condition that the buyer accept the price discount scheme is that the price compensation the buyer gains must be more than his increased cost, that is, p0r ðx ; L Þ P pr ðx ; L Þ, therefore
aP
5. A price discount coordination mechanism
pr ðx ; L Þ p0r ðx ; L Þ wD
¼ a1
ð14Þ
And the vendor’s expected profit will be
DS hs mx x D 2D 1þ m 1 P P 2
p0s ðx Þ ¼ ðð1 aÞw cÞD
ð15Þ
The vendor will accept the price discount when p0s ðx Þ P ps ðx Þ, that is
a6
p0s ðx Þ ps ðx Þ wD
¼ a2
ð16Þ
Hence, as a1 6 a 6 a2 , the vendor and the buyer are willing to accept the price discount mechanism since both of their expect profits will be improved comparing with that of decentralized model. The optimal order quantity will shift from x to x , the optimal lead time will shift from L to L , and the wholesale price shift from w to ð1 aÞw. Then the Pareto dominance of the entire supply chain will be achieved.
Table 2 Summary of the results under decentralized model (1 h = 99%).
a
pffiffiffi
Li (weeks)
Qi (units)
d
8 6 4a 3
109.54 111.07 115.52 124.27
164.93 142.83 116.62 101.00
Li wðkÞ h
(units)
xi (units)
The buyer’s expected profit ($)
m (lots)
The vendor’s expected profit ($)
The supply chain’s expected profit ($)
164.93 142.83 116.62a 124.27
3210.67 3350.81 3397.91a 3261.94
2a
1678.24
5076.15
The optimal solutions.
Table 3 Summary of the results under centralized model without coordination (1 h = 99%). m
Li (weeks)
Qi (units)
d
1
8 6 4a 3
235.34 235.89 237.53 240.90
2
8 6 4 3
129.61 130.13 131.67 134.82
a
a
The optimal solutions.
pffiffiffi
xi(units)
The supply chain’s expected profit ($)
The buyer’s expected profit ($)
The vendor’s expected profit ($)
164.93 142.83 116.62 101.00
235.34 235.89 237.53a 240.90
5468.70 5509.70 5532.67a 5483.96
2724.23 2760.95 2771.29 2697.30
2744.47 2748.75 2761.38 2786.66
164.93 142.83 116.62 101.00
164.93 142.83 131.67 134.82
4917.78 5107.96 5124.72 5006.20
3210.67 3350.81 3378.19 3253.68
1707.11 1757.15 1746.53 1752.52
Li wðkÞ h
(units)
Y. Li et al. / Computers & Industrial Engineering 61 (2011) 858–864
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Table 4 Summary of the results under different service levels. The buyers’ expected profit ($)
The vendor’s expected profit ($)
The supply chain’s expected profit ($)
391.78 1922.13 2610.85 2962.91 3335.20 3397.91 3421.44 3421.44 3421.44 3421.44
3296.89 3207.79 2866.20 2423.61 1755.41 1678.24 1632.93 1632.93 1632.93 1632.93
3688.68 5129.92 5477.04 5386.52 5090.61 5076.15 5054.37 5054.37 5054.37 5054.37
Results under centralized model without coordination 99.8% 3 1 504.98 391.78 99.7% 3 1 336.65 1922.13 99.6% 3 1 252.49 2610.85 99.5% 4 1 237.53 2771.29 99.2% 4 1 237.53 2771.29 99.0% 4 1 237.53 2771.29 98.5% 4 1 237.53 2771.29 98.0% 4 1 237.53 2771.29 97.5% 4 1 237.53 2771.29 97.0% 4 1 237.53 2771.29
3296.89 3207.79 2866.20 2761.38 2761.38 2761.38 2761.38 2761.38 2761.38 2761.38
3688.68 5129.92 5477.04 5532.67 5532.67 5532.67 5532.67 5532.67 5532.67 5532.67
Service level
Optimal lead time (weeks)
Optimal number of lots
Optimal order quantity (units)
Results under Stackelberg model 99.8% 3 1 504.98 99.7% 3 1 336.65 99.6% 3 1 252.49 99.5% 3 1 201.99 99.2% 4 2 145.78 99.0% 4 2 116.62 98.5% 6 2 111.07 98.0% 6 2 111.07 97.5% 6 2 111.07 97.0% 6 2 111.07
Fig. 3. The variation of the vendor’s expected profit under different service level constraints.
Fig. 4. The variation of the optimal order quantity under different service level constraints.
Fig. 1. The variation of the supply chain’s expected profit under different service level constraints.
Fig. 5. The variation of the optimal lead time under different service level constraints.
Fig. 2. The variation of the buyer’s expected profit under different service level constraints. Fig. 6. The vendor and the buyer’s expected profits under different price discount ratios.
Let us come back to the above numerical example with service level 1 h = 99%. Applying the previously developed price discount mechanism, we can get the range of price discount ratio is [1.16%, 2.01%]. Then the buyer’s expected profit will be changed within [3397.91, 3854.43], and the vendor’s expected profit will
be changed within [1678.24, 2134.76], both are better than that of decentralized model. And the expected profit of the entire supply chain keeps to be $5532.67, which indicates that the Pareto dominance of the supply chain is achieved.
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Moreover, the buyer and the vendor’s expected profit will be changed with the change of price discount ratio, as is shown in Fig. 6. Obviously, the vendor’s improved expected profit will be decreased with the increase of a, and the buyer’s improved expected profit will be increased with the increase of a. The exact value of a will be decided by the bargaining power between the vendor and the buyer within the range of [1.16%, 2.01%]. 6. Conclusions Many companies have recognized the significance of lead time as a competitive weapon and have used lead time as a means of differentiating themselves in the marketplace. Lead time is an important element in any inventory management system. In many practical situations, lead time is controllable by added crashing cost. This paper investigated how lead time and service level constraint affect the inventory model. In this paper, the decentralized and centralized models of supply chain inventory optimization with controllable lead time and service level constraint are proposed. The solution procedures to get the optimal solutions are suggested. At last, a price discount mechanism is developed to induce both the vendor and the buyer to accept the centralized model. The results of numerical example show that shortening lead time reasonably can improve the benefit of supply chain system comparing to that of uncontrollable lead time. Moreover, comparing to the decentralized model, the centralized model can get more profit from the perspective of the entire supply chain, while the key to realize the centralized model is to design suitable coordination mechanism to induce both the buyer and the vendor to accept it. The price discount mechanism proposed in this paper is effective. Other effective coordination mechanisms and the coordination mechanisms under asymmetric information situation dealing with controllable lead time problem can be the points of further research. Acknowledgements The authors greatly appreciate the anonymous referees for the valuable and helpful suggestions to improve the paper. This paper is supported by Natural Science Foundation of China (70971042, 71001041,71090403/71090400), Social Science Innovation Team Project of Guang Dong Province Universities (08JDTDXM63002), the Fundamental Research Funds for the Central Universities, SCUT (2009ZM0240, 2011ZM0037,2011SG003) and the Institute for Supply Chain Integration and Service Innovation. References Axsater, S. (2011). Inventory control when the lead-time changes. Production and Operations Management, 20(1), 72–81.
Ben-Daya, M., & Hariga, M. (2003). Lead time reduction in a stochastic inventory system with learning consideration. International Journal of Production Research, 41(3), 571–579. Ben-Daya, M., & Raouf, A. (1994). Inventory models involving lead time as decision variable. Journal of the Operational Research Society, 45(5), 579–582. Chandra, C., & Grabis, J. (2008). Inventory management with variable lead-time dependent procurement cost. Omega, 36(5), 877–887. Chang, H. C., Ouyang, L. Y., Wu, K. S., & Ho, C. H. (2006). Integrated vendor–buyer cooperative inventory models with controllable lead time and ordering cost reduction. European Journal of Operational Research, 170(2), 481–495. Chu, P., Yang, K., & Chen, P. (2005). Improved inventory models with service and lead time. Computers & Operations Research, 32(2), 285–296. Hu, J., & Munson, C. L. (2010). Incorporating quantity discounts and their inventory impacts into the centralized purchasing decision. European Journal of Operational Research, 201(2), 581–592. Jha, J. K., & Shanker, K. (2009). Two-echelon supply chain inventory model with controllable lead time and service level constraint. Computers & Industrial Engineering, 57(2), 1096–1104. Li, J., & Liu, L. (2006). Supply chain coordination with quantity discount policy. International Journal of Production Economics, 101(1), 89–98. Liao, C. J., & Shyu, C. H. (1991). An analytical determination of lead time with normal demand. International Journal of Operations and Production Management, 11(9), 72–78. Lin, Y. J. (2009). An integrated vendor–buyer inventory model with backorder price discount and effective investment to reduce ordering cost. Computers & Industrial Engineering, 56(4), 1597–1606. Monahan, J. P. (1984). A quantity discount pricing model to increase vendor profits. Management Science, 30(6), 720–726. Ouyang, L. Y., & Wu, K. S. (1997). Mixture inventory model involving variable lead time with a service level constraint. Computers & Operations Research, 24(9), 875–882. Ouyang, L. Y., Wu, K. S., & Ho, C. H. (2004). Integrated vendor–buyer cooperative models with stochastic demand in controllable lead time. International Journal of Production Economics, 92(3), 255–266. Ouyang, L. Y., Wu, K. S., & Ho, C. H. (2007). An integrated vendor–buyer inventory model with quality improvement and lead time reduction. International Journal of Production Economics, 108(1/2), 349–358. Ouyang, L. Y., Yen, N. C., & Wu, K. S. (1996). Mixture inventory model with backorders and lost sales for variable lead time. Journal of the Operational Research Society, 47(6), 829–832. Pan, J. C., Hsiao, Y. C., & Lee, C. J. (2002). Inventory model with fixed and variable lead time crash costs considerations. Journal of the Operational Research Society, 53(9), 1048–1053. Pan, J. C., & Yang, J. S. (2002). A study of integrated inventory with controllable lead time. International Journal of Production Research, 40(5), 1263–1272. Shin, H., & Benton, W. C. (2007). A quantity discount approach to supply chain coordination. European Journal of Operational Research, 180(2), 601–616. Sucky, E. (2006). A bargaining model with asymmetric information for a single supplier–single buyer problem. European Journal of Operational Research, 171(2), 516–535. Tersine, R. J., & Hummingbird, E. A. (1995). Lead-time reduction: The search for competitive advantage. International Journal of Operations and Production Management, 15(2), 8–18. Weng, Z. K. (1995). Channel coordination and quantity discounts. Management Science, 41(9), 1509–1522. Yang, J. S., & Pan, J. C. (2004). Just-in-time purchasing: An integrated inventory model involving deterministic variable lead time and quality improvement investment. International Journal of Production Research, 42(5), 853–863. Ye, F., & Xu, X. (2010). Cost allocation model for optimizing supply chain inventory with controllable lead time. Computers & Industrial Engineering, 59(1), 93–99.
Contents lists available at ScienceDirect
Computers & Industrial Engineering journal homepage: www.elsevier.com/locate/caie
Supply chain coordination model with controllable lead time and service level constraint q Yina Li, Xuejun Xu, Fei Ye ⇑ School of Business Administration, South China University of Technology, Guangzhou 510640, China
a r t i c l e
i n f o
Article history: Received 13 April 2010 Received in revised form 1 March 2011 Accepted 27 May 2011 Available online 13 June 2011 Keywords: Controllable lead time Supply chain Service level constraint Price discount mechanism
a b s t r a c t We consider the coordination issue in a decentralized supply chain composed of a vendor and a buyer in this paper. The vendor offers a single product to the buyer who is faced with service level constraint. In addition, lead time can be reduced by added crashing cost. We analyze two supply chain inventory models. The first one is developed under decentralized mode based on Stackelberg model, the other one is developed under centralized mode of the integrated supply chain. The solution procedures are also provided to get the optimal solutions of these two models. Finally, a price discount mechanism is proposed to induce both the vendor and the buyer to accept the centralized model. The feasibility and efficiency of the proposed models are manifested by numerical examples and some managerial implications are highlighted. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction The theoretical and practical studies on Japanese Just-in-Time (JIT) have resulted in considerable efforts in reduction on lead time and inventory-related costs simultaneously (Ben-Daya & Hariga, 2003). In JIT philosophy, lead time can be reduced by effective methods and added crashing cost (Axsater, 2011). It is quite different from the traditional Economic Order Quantity (EOQ) literature that viewing lead time as a prescribed constant or a stochastic variable. Time-Based Competition (TBC), which focuses on management of time, specially lead time, has becoming one of the most important modes to gain competitive advantage. Lead time reduction can lower safety stock, improve customer service level, realize supply chain quick response to customer requirements, all of which combines to help companies to gain and maintain competitive advantage in existing and new markets (Tersine & Hummingbird, 1995). Liao and Shyu (1991) were the first to consider a continuous review model in which order quantity is predetermined and lead time is a unique decision variable. Then more and more inventory model literature considering lead time reduction issue has been developed, which can be mainly divided into two broad categories. The first category focused on considering how to find the optimal inventory policy to minimize the total relevant cost of either single firm or supply chain, including the stock out cost, which is used an exact value to express. For example, Ben-Daya and Raouf q
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⇑ Corresponding author. Tel.: +86 20 87110984. E-mail address: [email protected] (F. Ye).
0360-8352/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2011.05.019
(1994) extended Liao and Shyu (1991) model by viewing both lead time and order quantity as decision variables. Ouyang, Yen, and Wu (1996) further treated the shortages to be a mixture of backorders and lost sales. Pan, Hsiao, and Lee (2002) considered lead time crashing cost to be a function of both order quantity and reduced lead time. Pan and Yang (2002) viewed lead time as a decision variable and obtained a lower joint expected cost and shorter lead time of integrated supply chain. Ouyang, Wu, and Ho (2004) extended Pan and Yang (2002) model by further considering the reorder point as one of the decision variables and shortages permitted. Yang and Pan (2004) minimized the total expected cost of integrated supply chain by simultaneously optimizing the order quantity, lead time, process quality and number of deliveries. Chang, Ouyang, Wu, and Ho (2006) developed an integrated supply chain model with controllable lead time and ordering cost reduction. Ouyang, Wu, and Ho (2007) formulated and solved an integrated inventory model involving imperfect production process with controllable lead time. Chandra and Grabis (2008) optimized the total inventory and procurement cost in a variable lead-time inventory system where lead-time is a dependent function of procurement cost. Lin (2009) considered an integrated model where lead time and ordering cost can be reduced by additional investment. Ye and Xu (2010) developed an asymmetric Nash bargaining model to get the best cost allocation ratio to coordination the benefits of both parties in a decentralized supply chain with controllable lead time. It should be noted that all of the first category models considering controllable lead time assumed the stock out cost can be expressed by an exact value. However, the second category researchers took the opinion that, in many practical situations,
Y. Li et al. / Computers & Industrial Engineering 61 (2011) 858–864
the stock out cost is very difficult to identify by an exact value for there will always be some intangible loss such as the damage of a company’s reputation and credibility, the potential delay to other parts of inventory system, and so on. And the stock out will negatively impact customers’ satisfactions since they are unable to get the products they want at the time they want them, resulting in higher customer dissatisfaction and lower customer loyalty. Hence, they used a service level constraint to replace the stock out cost in the objective function of the models in the first category, requiring a certain proportion of demand should be met in each cycle. Ouyang and Wu (1997) used a service level constraint to replace the stock out cost in the objective function of Ouyang et al. (1996) model to bind the stock out level each cycle. Chu, Yang, and Chen (2005) studied the same model as Ouyang and Wu (1997), while improved the algorithm to get the optimal solutions. In a recent paper, Jha and Shanker (2009) proposed a two-echelon integrated supply chain inventor model with controllable lead time and service level constraint. However, Jha and Shanker (2009) model focused on integrated channel in which assuming a central planner maximizes overall channel performance, with the power to impose a globally optimal inventory policy on each party. Although such a joint optimal order and production policy leads to a significant total cost reduction comparing to independently derived policies, there is an additional set of problems involved in implementing joint policies (Sucky, 2006). For example, a multiform supply chain without a central planner will have incentive conflicts because different parties in a supply chain generally have different and often conflicting objectives. Therefore, how to make effective coordination mechanisms that retaining healthy partnerships among independent chain parties and improving the chain value is critical to the supply chain success. In this paper, we make two major contributions to the present literature on supply chain optimization problem with controllable lead time. First, different from the first category studies that minimizing the total relevant expected cost of single firm or supply chain, which using an exact value to express the stock out cost, we use a service level constraint in place of the stock out cost to bound the stock out level per cycle and avoid the difficulty to estimate the exact value of stock out cost in real situations. Secondly, different from the service level approach model in the second category studies, we extend Ouyang and Wu (1997) model considering service level constraint and controllable lead time from the perspective of single buyer into supply chain perspective. And we also relax the assumption in Jha and Shanker (2009) model from the perspective of supply chain that long-term strategic partnership between the vendor and the buyer was well established and they could cooperate with each other to obtain an optimal integrated joint policy under centralized decision. In contrast to Jha and Shanker (2009), we assume the vendor and the buyer representing different benefit entities and take their individual rationalities into consideration, examine coordinated decision in a decentralized (two-echelon) supply chain that consists of one buyer and one vendor, with controllable lead time and service level constraint. In addition, in order to induce both parties to accept the centralized decision model and realize the Pareto dominance of the entire supply chain system, an effective price discount mechanism is developed to coordinate benefits between the vendor and the buyer and make both of them realize Pareto improvement through coordination. The remainder of the paper is organized as follows. Section 2 introduces assumptions and notations. In Section 3, two different inventory models, decentralized model and centralized model, with controllable lead time and service level constraint are proposed. The procedures are also suggested to get the optimal solutions. Numerical examples and sensitive analysis of different service level constraints are provided to illustrate the results of
859
the proposed models in Section 4 and Section 5 propose a price discount mechanism to coordination the profits between the vendor and the buyer. Finally, conclusions and further research areas are provided in Section 6. 2. Notations and assumptions To establish the proposed models, the following notations and assumptions are used. Additional notations and assumptions will be listed later when needed. 2.1. Notations The decision variables are Q The buyer’s order quantity L The length of lead time m The number of lots in which the product is delivered from the vendor to the buyer in one production cycle, a positive integer a The ratio of price discount offered by the vendor. a 2 ½0; 1 The other related parameters are D Average demand per year P The vendor’s production capability per year (P > D) p The buyer’s retail price w The vendor’s wholesale price c The vendor’s unit production cost A The buyer’s ordering cost per order S The vendor’s setup cost per set-up hr The buyer’s unit holding cost per year hs The vendor’s unit holding cost per year h Proportion of demand that are not met from stock. h 2 ½0; 1 b The fraction of the demand during the shortage period that will be backordered. b 2 ½0; 1
2.2. Assumptions 1. There is a single vendor and a single buyer that representing different entities in a decentralized supply chain. 2. The buyer adopts a continuous review inventory policy and makes replenishments whenever the inventory level falls to the reorder point r. 3. The reorder point r = expected demand during lead time + safety stock. The demand X during lead time L is assumed to be normally distributed with pffiffiffi pffiffiffi mean DL and standard deviation d L. That is, r ¼ lL þ kd L, where k is the safety factor. 4. The buyer orders a lot of size Q and the vendor manufactures the product in lots of size mQ with a finite production rate P (P > D) and ship in quantity Q to the buyer over m lots. 5. The buyer faces with a service level constraint not lower than 1 h. 6. Shortages will be partly backordered with the fraction b. 7. The lead time has n mutually independent components. The ith component has a minimum duration ai and normal duration bi, and a crashing cost per unit time ci. Furthermore, for convenience, we arrange ci such that c1 6 c2 6 6 cn . Then, it is clear that the reduction of lead time should be first on component 1 (because it has the minimum unit crashing cost), and then component 2, and so on. P 8. If we let L0 ¼ nj¼1 bj and Li be the length of lead time with components 1; 2; . . . i crashed to their minimum duration, then Li is P P P P expressed as Li ¼ ij¼1 aj þ nj¼iþ1 bj ¼ nj¼1 bj ij¼1 ðbj aj Þ
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Pi
¼ L0 j¼1 ðbj aj Þ, i ¼ 1; 2; . . . n. And the lead time crashing cost R(L) per cycle for a given L 2 ½Li ; Li1 is given by P RðLÞ ¼ ci ðLi1 LÞ þ i1 j¼1 cj ðbj aj Þ. 9. All information are common information to both the vendor and the buyer.
Max
s:t:
DS ps ðmÞ ¼ ðw cÞD mQ hs Q2 m 1 DP 1 þ 2D P 8 > Max pr ðQ ; LÞ ¼ ðp wÞD DðAþRðLÞÞ > Q >
< > > > :
hr s:t:
pffiffi d LWðkÞ h
Q 2
pffiffiffi pffiffiffi þ kd L þ ð1 bÞd LWðkÞ
6Q
3. The model
ð4Þ
3.1. The buyer’s inventory model According to the above setting, the buyer’s annual expected profit equals to the annual revenue minus annual total cost, which is consisted of annual purchasing cost, annual ordering cost, annual holding cost and annual lead time crashing cost. The buyer’s annual revenue equals to pD, the annual purchasing cost, annual ordering cost and annual lead time crashing cost will be equal to wD, DA/Q, DR(L)/Q, respectively. And since the expected R þ1 shortages of each order cycle is EðX rÞþ ¼ r ðX rÞdFðxÞ ¼ pffiffiffi d LWðkÞ, where WðkÞ ¼ /ðkÞ k½1 UðkÞ, and /; U are the probability density function and cumulative distribution function of the standard normal distribution, respectively. With backorder ratio b, pffiffiffi the expected number of backorders per cycle is bd LWðkÞ and the pffiffiffi expected demand lost is ð1 bÞd LWðkÞ. Hence, the average pffiffiffi pffiffiffi on-hand inventory for the buyer is Q2 þ ð1 bÞd LWðkÞ þ kd L p ffiffi ffi p ffiffi ffi and annual holding cost equals to hr Q2 þ ð1 bÞd LWðkÞ þ kd L . Consistent with the previous literature, we define the service level should meet the requirement that the actual proportion of demands not met from stock should not exceed the desired value of h. Therefore, the buyer will decide the optimal order quantity and lead time to maximize his own annual expected profit, subject to a service level constraint:
Max
pr ðQ ; LÞ ¼ ðp wÞD DðAþRðLÞÞ Q hr
s:t:
pffiffi d LWðkÞ Q
Q 2
pffiffiffi pffiffiffi þ kd L þ ð1 bÞd LWðkÞ
In order to solve this kind of problem, similar to literature Ouyang and Wu (1997), Chu et al. (2005), Jha and Shanker (2009), we ignore the service level constraint in Eq. (4) first. From Eq.(4), we can easily get that pr(Q, L) is concave with respect to order quantity Q and convex with respect to lead time L (for the details of proof, please refer to the published paper dealing with controllable lead time, e.g., Ouyang et al., 2004). Hence, for each Li ði ¼ 0; 1; . . . ; n, we can get the optimal order quantity under leader–follower relationship without the service level constraint will be:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2DðA þ RðLi ÞÞ Qi ¼ hr
ative Alogrithm 1 is suggested to get the buyer’s optimal order quantity and lead time with service level constraint under decentralized decision mode.
ð1Þ Algorithm 1
6h
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i ÞÞ Step1: For each Li ði ¼ 0; 1; . . . ; n compute Q i ¼ 2DðAþRðL using hr Eq. (5). pffiffiffi pffiffiffi
d Li wðkÞ d Li wðkÞ Step2: Compute , set xi ¼ max Q i ; , for h h i ¼ 0; 1; 2; . . . ; n. Step3: Set pr ðx ; L Þ ¼ Maxi¼0;1;2;...n pr ðxi ; Li Þ, and ðx ; L Þ is buyer’s optimal decision.
pr ðQ ; LÞ ¼ ðp wÞD DðAþRðLÞÞ Q hr
s:t:
pffiffi d LWðkÞ h
ð5Þ
And the buyer’s maximum annual expected profit will occur at the end points of the interval ½Li ; Li1 , i ¼ 1; 2; . . . n. (We denote the optimal lead time here to be L ). Now considering the service level constraint, for each pffiffiffi d Li WðkÞ Li ði ¼ 0; 1; . . . ; n, if 6 Q i , then Qi is the local maximum of h pr ðQ i ; Li Þ and the service level constraint does not take effect. If pffiffiffi pffiffiffi d Li WðkÞ d Li wðkÞ > Q , then is the local maximum of pr ðQ i ; Li Þ so that i h h the service level per cycle can be met. Hence, for fix L 2 ½Li ; Li1 , the pffiffiffi
d Li wðkÞ buyer’s optimal order quantity will be max Q i ; . The iterh
According to Chu et al. (2005), Eq. (1) can be transformed into:
Max
Q 2
pffiffiffi pffiffiffi þ kd L þ ð1 bÞd LWðkÞ
ð2Þ
6Q
3.2. The vendor’s inventory model Similar to most of the literature dealing with controllable lead time with the same setting for the vendor (e.g., Ouyang et al., 2004), the vendor’s annual expected profit is represented by:
DS Q D 2D hs m 1 1þ ps ðmÞ ¼ ðw cÞD mQ 2 P P
ð3Þ
3.3. Leader–follower relationship When the buyer and the vendor belong to different entities, they will decide the optimal policies to maximize their own expected profits separately. Among the many possible uncooperative gaming assumptions, we consider the popular leader–follower (i.e. Stackelberg) structure, where the buyer is the Stackelberg follower who sets the order quantity Q and lead time L to maximize its own annual expected profit first and the vendor is the leader who determines the corresponding number of lots m to maximize its own annual expected profit accordingly. Under such a setting, the Stackelberg model between the vendor and the buyer is given by:
The vendor will decide the optimal number of lots m reacts to the buyer’s optimal order quantity x and optimal lead time L . For any given L 2 ½Li ; Li1 and order quantity, ps ðmÞ is concave in m, since
@ 2 ps ðx ; mÞ 2DS ¼ 3 <0 @m2 m x
ð6Þ
Therefore, the vendor’s optimal value of m ¼ m is obtained when
ps ðx ; m Þ P ps ðx ; m þ 1Þ ps ðx ; m Þ P ps ðx ; m 1Þ
ð7Þ
From Eq. (7) we can easily get m is the first integer satisfying
2DS D 2DS 2 6 6 h ðx Þ 1 s m ðm þ 1Þ P m ðm 1Þ
ð8Þ
3.4. Long-run cooperation and the centralized solution Consider now the situation in which both the buyer and the vendor belong to the classic single decision-maker system and
Y. Li et al. / Computers & Industrial Engineering 61 (2011) 858–864
there is a central planner makes all decisions to maximize the total expected profit of the entire supply chain. Under such situation, the integrated inventory of supply chain under centralized mode is given by
psc ðQ ; L; mÞ ¼ ðp cÞD DðmAþSþmRðLÞÞ mQ
Max
pffiffiffi pffiffiffi þ kd L þ ð1 bÞd LWðkÞ hs Q2 m 1 DP 1 þ 2D P
hr pffiffi d LWðkÞ h
s:t:
Q 2
ð9Þ
6Q
By ignoring the service level constraint, we can also easily get that psc ðQ ; L; mÞ is concave with respect to order quantity Q and convex with respect to lead time L (please also refer to the published paper dealing with controllable lead time, e.g., Ouyang et al., 2004, for the details of proof). Hence, for each Li ði ¼ 0; 1; . . . ; n, we can get the optimal order quantity under centralized decision mode without service level constraint will be:
( Qi ¼
2DðA þ mS þ RðLÞÞ hr þ hs m 1 DP 1 þ 2D P
)12 ð10Þ
And the maximum annual expected profit of the entire supply chain will occur at the end points of the interval ½Li ; Li1 , i ¼ 1; 2; . . . n. (We denote the optimal lead time here to be L ). As the service level constraint is taken into consideration, similar to that of decentralized decision mode, we can also obtain that for fix L 2 ½Li ; Li1 , the optimal order quantity of the centralized pffiffiffi
d Li wðkÞ . decision mode will be max Q i ; h 2
ðQ ;L;mÞ ¼ m2DS And since @ psc@m 2 3 Q < 0, then psc ðQ ; L; mÞ is concave in m for fixed Q and L 2 ½Li ; Li1 , which indicate that there must be an optimal m to meet Eq. (11):
psc ðQ ; L; m Þ P psc ðQ ; L; m þ 1Þ psc ðQ ; L; m Þ P psc ðQ ; L; m 1Þ
ð11Þ
Hence, the iterative Alogrithm 2 is suggested to find the optimal values of Q, L and m under centralized decision mode. Algorithm 2 Step 1: Start with m = 1. Step 2: For each Li ði ¼ 0; 1; . . . ; n, compute Qi using Eq. (10). pffiffiffi pffiffiffi
d Li wðkÞ d Li wðkÞ Step 3: Compute , set x ¼ max Q ; , i i h h i ¼ 0; 1; 2; . . . ; n. Step 4: For each set of ðxi ; Li ; mÞ, compute psc ðxi ; Li ; mÞ using Eq. (9), i ¼ 0; 1; 2; . . . ; n. Step 5: Set psc x then m ; Lm ; m ¼ Maxi¼0;1;...;n psc ðxi ; Li ; mÞ, xm ; Lm ; m is the optimal solution for any given m. Step 6: Set m ¼ m þ 1, repeat Step 2 to Step 5 to get psc x m ; Lm ; m . Step 7: If psc x m ; Lm ; m P psc xm1 ; Lm1 ; m 1 , go to Step 6, otherwise go to Step 8. Step 8: Set ðx ; L ; m Þ ¼ x then m1 ; Lm1 ; m 1 , ðx ; L ; m Þ is the optimal solution set for the centralized decision model. 4. Numerical example In order to illustrate the proposed model, let us consider an inventory system with the following characteristics: D = 600 units/year, p = $100/unit, w = $90/unit, c = $80/unit, hr = $20/unit/year, P = 3000 units/year, A = $200/order, d = 7 units/
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week, hs = $30/unit/year , S = $1000/set-up, k = 1, b = 0.5, h = 1%, that is, the service level constraint 1 h = 99%. The lead time has three components with the data shown in Table 1. Through applying the previously developed algorithms, we obtain the results of leader–follower relationship and long-run cooperation situation, which are shown in Tables 2 and 3, respectively. From Table 2, we can easily find that the buyer will select the optimal order quantity x ¼ 116:62 units, and optimal lead time L ¼ 4 weeks to get his own maximum expected profit pr ð116:62; 4Þ ¼ $3397:91. Then using Eq. (8), we can get the vendor’s optimal number of lots m ¼ 2 and the vendor’s maximum expected profit ps ð2Þ ¼ $1678:24. Thus, the total expected profit of the entire supply chain is psc ð116:62; 4; 2Þ ¼ $5076:15 . From Table 3, we get the optimal order quantity x ¼ 237:53 units, optimal lead time L ¼ 4weeks and optimal number of lots m ¼ 1, thus, the total expected profit of the entire supply chain is psc ð237:53; 4; 1Þ ¼ $5532:67, which is 8.99% higher than that of leader–follower relationship in this case. Based on this case, we further analyze the variation of the results under centralized and decentralized model by considering different service level constraints, which is shown in Table 4 and Figs. 1–5. Table 4 and Figs. 1–5 show the impacts of the change of service level constraint on the expected profits of different supply chain parties, optimal order quantity and optimal lead time. From Table 4 and Figs. 1–5, we can clearly see the following managerial implications in both cases. (1) From Fig. 1 we can clearly see that under centralized decision model, the expected profit of the entire supply chain will be decreased with the increase of the service level. That is, the higher the service level is, the lower the supply chain’s expected profit will be. Especially, if the service level has been set to a certain high level, to further upgrade the service level should be at great cost. In our case, when the service level is 99.7%, the supply chain’s expected profit equals to $5129.92. But if we need to upgrade the service level to be 99.8%, the supply chain’s expected profit will be decreased to $3688.68, which is decreased by 28%. Obviously, it is not advisable to upgrade the service level from 99.7% to 99.8% from the cost perspective. That is, there will be a balance to ‘‘trade-off’’ as a company sets the target of service level, 100% service level is not the best choice for all companies. (2) From Figs. 2 and 3, we can clearly observe that in both the decentralized and centralized decision models, the buyer’s expected profit will be decreased with the increase of the service level, while the vendor’s expected profit will be increased with the increase of the service level. Moreover, though we find the supply chain’s expected profit will be improved in the centralized model comparing to that of decentralized model in Fig. 1, the buyer’s expected profit under centralized model, however, will be lower than that of decentralized model, which indicates that the key to realize the maximum expected profit of the entire supply chain is to develop reasonable coordination mechanism to make both the buyer and the vendor’s expected profits under centralized model will not lower than that of decentralized model, that is, achieve Pareto improvement. (3) From Fig. 4 we can see that the optimal order quantity will be increased with the increase of the service level in both centralized and decentralized models, which indicates that high service level should be at the expense of high inventory level. (4) From Fig. 5 we can see that the optimal lead time will be shorter under high service level than that of lower service level in both centralized and decentralized models. For instance, as the service level is increased from 97% to
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Table 1 Lead time data. Lead time component i
Normal duration bi (days)
Minimum duration ai (days)
Unit crashing cost ci ($/day)
1 2 3
20 20 16
6 6 9
0.4 1.2 5.0
of the entire supply chain. The price discount coordination mechanism is well established and applied to previous inventory literature to achieve supply chain coordination (Hu & Munson, 2010; Li & Liu, 2006; Monahan, 1984; Shin & Benton, 2007; Weng, 1995). Considering the vendor offer the following price discount scheme:
w0 ¼ 99.5%, the optimal lead time under decentralized model will be shortened from 6 weeks to 3 weeks. This also helps to illustrate that shorter lead time is an effective way to realize quick customer response. In addition, we can see that for each service level, there exists an optimal lead time to maximize the buyer’s expected profit (decentralized decision model) and the supply chain’s expected profit (centralized decision model). That is, assuming the lead time to be controllable will get more profit improvement comparing to the fixed lead time studies.
Though the centralized model can yield more profit than the decentralized one from the perspective of the entire supply chain (as shown in Fig. 1), the vendor and the buyer might not be willing to behave as in the centralized model unless both of them can gain more profits than that of Stackelberg model. Comparing the results of Tables 2 and 3, we can easily find that the buyers’ expected profit under centralized model is lower than that of Stackelberg model, as is shown in Fig. 2, which means if there is no reasonable incentive mechanism, the buyer may not be willing to shift to the centralized decision model due to his individual rationality. So the key to induce both the buyer and the vendor to accept the centralized model is to develop reasonable coordination mechanism to make both of them earn more benefits from centralized model than that of decentralized model. In this section, we develop a price discount mechanism to coordinate the supply chain benefit. The objective of this coordination mechanism is to help to not only meet both the vendor and the buyer’s individual rationalities, but also realize Pareto dominance
L – L
Q < x ;
w;
ð12Þ
ð1 aÞw; Q P x ; L ¼ L
to encourage the buyer to make decisions to maximize the expected profit of the entire supply chain. Then the buyer’s profit will be
DðA þ RðL ÞÞ x pffiffiffiffiffiffi pffiffiffiffiffiffi x hr þ kd L þ ð1 bÞd L WðkÞ 2
p0r ðx ; L Þ ¼ ðp ð1 aÞwÞD
ð13Þ
The condition that the buyer accept the price discount scheme is that the price compensation the buyer gains must be more than his increased cost, that is, p0r ðx ; L Þ P pr ðx ; L Þ, therefore
aP
5. A price discount coordination mechanism
pr ðx ; L Þ p0r ðx ; L Þ wD
¼ a1
ð14Þ
And the vendor’s expected profit will be
DS hs mx x D 2D 1þ m 1 P P 2
p0s ðx Þ ¼ ðð1 aÞw cÞD
ð15Þ
The vendor will accept the price discount when p0s ðx Þ P ps ðx Þ, that is
a6
p0s ðx Þ ps ðx Þ wD
¼ a2
ð16Þ
Hence, as a1 6 a 6 a2 , the vendor and the buyer are willing to accept the price discount mechanism since both of their expect profits will be improved comparing with that of decentralized model. The optimal order quantity will shift from x to x , the optimal lead time will shift from L to L , and the wholesale price shift from w to ð1 aÞw. Then the Pareto dominance of the entire supply chain will be achieved.
Table 2 Summary of the results under decentralized model (1 h = 99%).
a
pffiffiffi
Li (weeks)
Qi (units)
d
8 6 4a 3
109.54 111.07 115.52 124.27
164.93 142.83 116.62 101.00
Li wðkÞ h
(units)
xi (units)
The buyer’s expected profit ($)
m (lots)
The vendor’s expected profit ($)
The supply chain’s expected profit ($)
164.93 142.83 116.62a 124.27
3210.67 3350.81 3397.91a 3261.94
2a
1678.24
5076.15
The optimal solutions.
Table 3 Summary of the results under centralized model without coordination (1 h = 99%). m
Li (weeks)
Qi (units)
d
1
8 6 4a 3
235.34 235.89 237.53 240.90
2
8 6 4 3
129.61 130.13 131.67 134.82
a
a
The optimal solutions.
pffiffiffi
xi(units)
The supply chain’s expected profit ($)
The buyer’s expected profit ($)
The vendor’s expected profit ($)
164.93 142.83 116.62 101.00
235.34 235.89 237.53a 240.90
5468.70 5509.70 5532.67a 5483.96
2724.23 2760.95 2771.29 2697.30
2744.47 2748.75 2761.38 2786.66
164.93 142.83 116.62 101.00
164.93 142.83 131.67 134.82
4917.78 5107.96 5124.72 5006.20
3210.67 3350.81 3378.19 3253.68
1707.11 1757.15 1746.53 1752.52
Li wðkÞ h
(units)
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Table 4 Summary of the results under different service levels. The buyers’ expected profit ($)
The vendor’s expected profit ($)
The supply chain’s expected profit ($)
391.78 1922.13 2610.85 2962.91 3335.20 3397.91 3421.44 3421.44 3421.44 3421.44
3296.89 3207.79 2866.20 2423.61 1755.41 1678.24 1632.93 1632.93 1632.93 1632.93
3688.68 5129.92 5477.04 5386.52 5090.61 5076.15 5054.37 5054.37 5054.37 5054.37
Results under centralized model without coordination 99.8% 3 1 504.98 391.78 99.7% 3 1 336.65 1922.13 99.6% 3 1 252.49 2610.85 99.5% 4 1 237.53 2771.29 99.2% 4 1 237.53 2771.29 99.0% 4 1 237.53 2771.29 98.5% 4 1 237.53 2771.29 98.0% 4 1 237.53 2771.29 97.5% 4 1 237.53 2771.29 97.0% 4 1 237.53 2771.29
3296.89 3207.79 2866.20 2761.38 2761.38 2761.38 2761.38 2761.38 2761.38 2761.38
3688.68 5129.92 5477.04 5532.67 5532.67 5532.67 5532.67 5532.67 5532.67 5532.67
Service level
Optimal lead time (weeks)
Optimal number of lots
Optimal order quantity (units)
Results under Stackelberg model 99.8% 3 1 504.98 99.7% 3 1 336.65 99.6% 3 1 252.49 99.5% 3 1 201.99 99.2% 4 2 145.78 99.0% 4 2 116.62 98.5% 6 2 111.07 98.0% 6 2 111.07 97.5% 6 2 111.07 97.0% 6 2 111.07
Fig. 3. The variation of the vendor’s expected profit under different service level constraints.
Fig. 4. The variation of the optimal order quantity under different service level constraints.
Fig. 1. The variation of the supply chain’s expected profit under different service level constraints.
Fig. 5. The variation of the optimal lead time under different service level constraints.
Fig. 2. The variation of the buyer’s expected profit under different service level constraints. Fig. 6. The vendor and the buyer’s expected profits under different price discount ratios.
Let us come back to the above numerical example with service level 1 h = 99%. Applying the previously developed price discount mechanism, we can get the range of price discount ratio is [1.16%, 2.01%]. Then the buyer’s expected profit will be changed within [3397.91, 3854.43], and the vendor’s expected profit will
be changed within [1678.24, 2134.76], both are better than that of decentralized model. And the expected profit of the entire supply chain keeps to be $5532.67, which indicates that the Pareto dominance of the supply chain is achieved.
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Moreover, the buyer and the vendor’s expected profit will be changed with the change of price discount ratio, as is shown in Fig. 6. Obviously, the vendor’s improved expected profit will be decreased with the increase of a, and the buyer’s improved expected profit will be increased with the increase of a. The exact value of a will be decided by the bargaining power between the vendor and the buyer within the range of [1.16%, 2.01%]. 6. Conclusions Many companies have recognized the significance of lead time as a competitive weapon and have used lead time as a means of differentiating themselves in the marketplace. Lead time is an important element in any inventory management system. In many practical situations, lead time is controllable by added crashing cost. This paper investigated how lead time and service level constraint affect the inventory model. In this paper, the decentralized and centralized models of supply chain inventory optimization with controllable lead time and service level constraint are proposed. The solution procedures to get the optimal solutions are suggested. At last, a price discount mechanism is developed to induce both the vendor and the buyer to accept the centralized model. The results of numerical example show that shortening lead time reasonably can improve the benefit of supply chain system comparing to that of uncontrollable lead time. Moreover, comparing to the decentralized model, the centralized model can get more profit from the perspective of the entire supply chain, while the key to realize the centralized model is to design suitable coordination mechanism to induce both the buyer and the vendor to accept it. The price discount mechanism proposed in this paper is effective. Other effective coordination mechanisms and the coordination mechanisms under asymmetric information situation dealing with controllable lead time problem can be the points of further research. Acknowledgements The authors greatly appreciate the anonymous referees for the valuable and helpful suggestions to improve the paper. This paper is supported by Natural Science Foundation of China (70971042, 71001041,71090403/71090400), Social Science Innovation Team Project of Guang Dong Province Universities (08JDTDXM63002), the Fundamental Research Funds for the Central Universities, SCUT (2009ZM0240, 2011ZM0037,2011SG003) and the Institute for Supply Chain Integration and Service Innovation. References Axsater, S. (2011). Inventory control when the lead-time changes. Production and Operations Management, 20(1), 72–81.
Ben-Daya, M., & Hariga, M. (2003). Lead time reduction in a stochastic inventory system with learning consideration. International Journal of Production Research, 41(3), 571–579. Ben-Daya, M., & Raouf, A. (1994). Inventory models involving lead time as decision variable. Journal of the Operational Research Society, 45(5), 579–582. Chandra, C., & Grabis, J. (2008). Inventory management with variable lead-time dependent procurement cost. Omega, 36(5), 877–887. Chang, H. C., Ouyang, L. Y., Wu, K. S., & Ho, C. H. (2006). Integrated vendor–buyer cooperative inventory models with controllable lead time and ordering cost reduction. European Journal of Operational Research, 170(2), 481–495. Chu, P., Yang, K., & Chen, P. (2005). Improved inventory models with service and lead time. Computers & Operations Research, 32(2), 285–296. Hu, J., & Munson, C. L. (2010). Incorporating quantity discounts and their inventory impacts into the centralized purchasing decision. European Journal of Operational Research, 201(2), 581–592. Jha, J. K., & Shanker, K. (2009). Two-echelon supply chain inventory model with controllable lead time and service level constraint. Computers & Industrial Engineering, 57(2), 1096–1104. Li, J., & Liu, L. (2006). Supply chain coordination with quantity discount policy. International Journal of Production Economics, 101(1), 89–98. Liao, C. J., & Shyu, C. H. (1991). An analytical determination of lead time with normal demand. International Journal of Operations and Production Management, 11(9), 72–78. Lin, Y. J. (2009). An integrated vendor–buyer inventory model with backorder price discount and effective investment to reduce ordering cost. Computers & Industrial Engineering, 56(4), 1597–1606. Monahan, J. P. (1984). A quantity discount pricing model to increase vendor profits. Management Science, 30(6), 720–726. Ouyang, L. Y., & Wu, K. S. (1997). Mixture inventory model involving variable lead time with a service level constraint. Computers & Operations Research, 24(9), 875–882. Ouyang, L. Y., Wu, K. S., & Ho, C. H. (2004). Integrated vendor–buyer cooperative models with stochastic demand in controllable lead time. International Journal of Production Economics, 92(3), 255–266. Ouyang, L. Y., Wu, K. S., & Ho, C. H. (2007). An integrated vendor–buyer inventory model with quality improvement and lead time reduction. International Journal of Production Economics, 108(1/2), 349–358. Ouyang, L. Y., Yen, N. C., & Wu, K. S. (1996). Mixture inventory model with backorders and lost sales for variable lead time. Journal of the Operational Research Society, 47(6), 829–832. Pan, J. C., Hsiao, Y. C., & Lee, C. J. (2002). Inventory model with fixed and variable lead time crash costs considerations. Journal of the Operational Research Society, 53(9), 1048–1053. Pan, J. C., & Yang, J. S. (2002). A study of integrated inventory with controllable lead time. International Journal of Production Research, 40(5), 1263–1272. Shin, H., & Benton, W. C. (2007). A quantity discount approach to supply chain coordination. European Journal of Operational Research, 180(2), 601–616. Sucky, E. (2006). A bargaining model with asymmetric information for a single supplier–single buyer problem. European Journal of Operational Research, 171(2), 516–535. Tersine, R. J., & Hummingbird, E. A. (1995). Lead-time reduction: The search for competitive advantage. International Journal of Operations and Production Management, 15(2), 8–18. Weng, Z. K. (1995). Channel coordination and quantity discounts. Management Science, 41(9), 1509–1522. Yang, J. S., & Pan, J. C. (2004). Just-in-time purchasing: An integrated inventory model involving deterministic variable lead time and quality improvement investment. International Journal of Production Research, 42(5), 853–863. Ye, F., & Xu, X. (2010). Cost allocation model for optimizing supply chain inventory with controllable lead time. Computers & Industrial Engineering, 59(1), 93–99.