Supply chain coordination model with controllable lead time and service level constraint

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Abstract

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.

Highlights

► A supply chain model with controllable lead time and service level is developed. ► Changes of lead time on optimal order policies and benefits are examined. ► Changes of service level on optimal order policies and benefits are examined. ► A price discount mechanism is proposed to coordinate the supply chain. ► Managerial implications are highlights.

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 (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, 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 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.

Section snippets

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.

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 shortages of each order cycle is E(X-r)+=r+(X-r)dF(x)=δL

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, δ = 7 units/week, hs = $30/unit/year , S = $1000/set-up, k = 1, β = 0.5, θ = 1%, that is, the service level constraint 1  θ = 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

A price discount coordination mechanism

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 Table 2, Table 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

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

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.

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