Discrete Dynamics in Nature and Society

Volume 2018, Article ID 6091946, 15 pages

https://doi.org/10.1155/2018/6091946

## Inventory and Production Dynamics in a Discrete-Time Vendor-Managed Inventory Supply Chain System

^{1}School of Business Administration, Zhongnan University of Economics and Law, Wuhan, China^{2}School of Management, Huazhong University of Science and Technology, Wuhan, China^{3}School of Automation, Huazhong University of Science and Technology, Wuhan, China

Correspondence should be addressed to Hongwei Wang; nc.ude.tsuh@gnawwh

Received 18 June 2018; Accepted 1 September 2018; Published 19 September 2018

Academic Editor: Lu Zhen

Copyright © 2018 Yongchang Wei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper presents some analytical results on production and order dynamics in the context of a discrete-time VMI supply chain system composed of one retailer and one manufacturer. We firstly derive the lower bound and upper bound on the range of inventory fluctuations for the retailer under unknown demand. We prove that the production fluctuations can be interestingly smoothed and stabilized independent of the delivery frequency of the manufacturer used to satisfy the retailer’s demand, even if the retailer subsystem is unstable. The sufficient and necessary stability condition for the whole supply chain system is obtained. To further explore the production fluctuation problem, the bullwhip effect under unknown demand is explored based on a transfer function model with the purpose of disclosing the influences of parameters on production fluctuations. Finally, simulation experiments are used to validate the theoretical results with respect to inventory and production fluctuations.

#### 1. Introduction

Vendor-managed inventory (VMI) is a well-known collaborative program [1–3], in which a retailer propagates inventory information and point of sales data to its supplier, while the upstream supplier makes replenishment decision for both the retailer and itself. Since it was pioneered by Wal-Mart in the early 1980s [4], VMI has been successfully implemented by many leading firms from different industries, such as Glaxosmithkline [5], Nestle and Tesco [6], Boeing and Alcoa [7], and Shell Chemical [8].

The benefits of VMI for both suppliers and retailers have been extensively studied in the literature [9–11]. For vendors, the VMI program improves decision-making flexibility since they gain the right to determine the amount and timing of the order for retailers, which further leads to reduced bullwhip effect [12] and better utilization of manufacturing capacity, as well as better synchronization of replenishment planning [13]. For retailers, VMI saves ordering cost and inventory cost and ensures the customer service level through signing VMI contracts. For example, a supplier will be penalized once the inventory of retailer exceeds certain ranges [14]. To successfully implement the VMI program, how to make use of the flexibility of decision-making in inventory control to benefit both upstream and downstream members is a highly challenging problem.

Most of the existing literature has focused on game theory or optimization models [15–19], while this paper explores the impact of replenishment parameters on the inventory and production dynamics [20], which are closely related to the total costs. Significant costs can be incurred by production fluctuations due to frequently switching production quantities up and down, which further complicate the activities in labor force arrangement, capacity adjustment, and equipment maintenance and management. However, in a dynamic system with uncertain demand, deriving optimal solutions from an optimization model incorporating all these cost components is impractical. Furthermore, in practice, through information sharing even with order batching [21–24], VMI has become one of the frequently employed programs for eliminating bullwhip effect [23, 25, 26]. This implies that production can be smoothed and transportation cost can be saved via the implementation of VMI program. However, we notice that the majority of existing bullwhip literature has ignored the interaction between inventory and production fluctuations.

In a VMI system, the upstream supplier has the flexibility to determine the amount and timing of the replenishment order for the retailer. However, as mentioned, the supplier might be penalized once the retailer’s inventory exceeds the predetermined range. Therefore, in order to reduce penalty cost, the supplier should know how the replenishment parameters for the retailer affect the range of inventory fluctuations of the retailer. In addition, the supplier should also consider the production or ordering cost and inventory cost for the whole system when designing its own replenishment policies. In this paper, we aim to answer three questions for the implementation of VMI: How the replenishment rules designed by the supplier for itself and retailer affect the stability, inventory fluctuations, and production smoothing for the entire system under uncertain demand? What is the trade-off between inventory fluctuations and production fluctuations? Specifically, what actions can be taken to limit inventory and order fluctuations. This question will be addressed in terms of stability. As investigated in the literature, stability is a fundamental problem for any dynamical system including supply chain systems [21, 27, 28]. After being disrupted, the state of a stable system will return to its steady state gradually. Actually, a stable supply chain system tends to be resilient in the presence of demand disruptions [29]. Furthermore, stability analysis simplifies the parameters selection and performance optimization. For a stable supply chain system, stability limits the fluctuation of inventory and order in a certain range after the sudden change of demand. By contrast, unstable designs will cause undesirable fluctuations for order and inventory, which leads to high cost due to inventory accumulation or stockouts. How the replenishment parameters affect the magnitude of production smoothing for the manufacturer.

Specifically, we attempt to explore the inventory and production fluctuations in the context of a VMI supply chain system composed of a manufacturer and a retailer. The manufacturer uses a reorder point policy to manage the retailer’s inventory and the APIOBPCS (automatic pipeline, inventory, and order based production control system) for production control [12, 30]. The contributions of this paper lie in threefold. Firstly, we analytically studied the impact of reorder point, delivery frequency, lead time, and demand characteristics on the range of the retailer’s inventory under uncertain demand by extending a more generalized replenishment policy. Secondly, we derived the necessary and sufficient stability conditions of the VMI system for arbitrary lead times by considering the interaction between the retailer subsystem and the manufacturer subsystem, which are significant due to the development of global supply chains and complicated manufacturing process [31]. We prove that production fluctuations are independent of the replenishment parameters in the reorder point policy. Besides, we demonstrate that although the production of the upstream manufacturer can be perfectly smoothed without paying extra attention to the quantity and frequency of replenishment for the retailer, the parameters in the reorder point policy have significant influences on the inventory fluctuations for both retailer and manufacturer. Finally, to further quantify the production smoothing magnitude, the impact of all the decision parameters on the robust bullwhip metric is further studied based on a transfer function model. The results of this research provide insightful guidelines on the implementation of VMI program.

The remainder of this paper is organized as follows. Section 2 introduces related literature. Section 3 describes the VMI model in terms of the retailer subsystem and manufacturer subsystem. Section 4 studies the stability and inventory oscillations for the retailer subsystem. The production smoothing effect and inventory fluctuation for the manufacturer are studied in Section 5. Section 6 focuses on the robustness of the bullwhip effect problem under uncertain demand. This paper is concluded in Section 7.

#### 2. Literature Review

Over the past decades, the advantages of VMI program have been extensively discussed. The shift of the authority of inventory control from downstream enables the supplier to choose the time and quantity for each replenishment order and also the route to transport the required goods to multiple retailers. It was well recognized that VMI brings more advantages over information sharing alone [4, 32–35]. In Fry et al. [14], a (z, Z) type contract was included in a VMI supply chain and the authors found that such a supply chain performs better than traditional supply chains. Lee et al. [3] studied the benefits and coordination problem for four business scenarios: traditional system, integrated system, VMI with stockout-cost sharing, and VMI with fixed transfer payments. Ben-Daya et al. [15] compared the benefits for three distinct supply chains, two of which were in the context of vendor-managed inventory consignment partnership. However, the supplier and retailer might belong to different companies or a vertically integrated company. The result showed that flexibility of the supplier has significant influences on the benefits of the vendor. Dong et al. [36] examined the benefits of the VMI program via empirical analysis, from which the results showed that inventory and stockout reductions might be realized at different times following the implementation of VMI. That is, inventory reduction initially may be the major benefit to distributor from VMI, while the benefits of stockout reduction may be realized after the first year. Lu et al. [9] studied the impact of two kinds of overconfidence on the inventory decision-making from a behavioral perspective. The study demonstrated behavioral factors might, to some extent, enhance the supplier’s profit through continuous effort.

Although the opportunities of VMI are evident in many industries, the benefit allocation and coordination problem has been argued continuously [3, 37, 38]. A number of authors argued that the majority of the benefits from VMI tend to flow to the supplier, rather than to the retailer [32]. Lee and Cho [39] examined the problem of designing a VMI contract with consignment stock and stockout-cost sharing in a inventory system between a supplier and a retailer. The result showed that the retailer may not always benefit from VMI. However, Mishra and Raghunathan [40] found that VMI intensifies the competition among manufacturers of competing brands, thus providing benefits to retailers. Pan and So [41] analyzed the interaction among the assembler and two component suppliers under a VMI contract. One supplier has uncertainty in the supply process, in which the actual number of components available for assembly is equal to a random fraction of the production quantity. Under deterministic and stochastic demand, the optimal component prices offered by the assembler are derived.

In addition to the previous literature, optimal replenishment policies of VMI supply chain systems with multiple retailers have been studied frequently over the recent years [17, 42–45]. Nevertheless, we see that the existing literature has ignored the dynamic fluctuations of order and inventory, which can be hardly incorporated into the previous optimization or game-theory based models. Although our VMI model is similar to an early work by Disney and Towill [12], we generalized their model by allowing for free adjustment of the amount and timing of replenishment frequency. Another important difference is that we pay attention to the trade-off between the production smoothing and inventory fluctuations by providing some analytical results, while their work is focused on simulation experiments.

#### 3. Model Description and Analysis

##### 3.1. Description for the VMI System

Consider a periodic-review, single item VMI supply chain system composed of a retailer and a manufacturer. In a VMI program, the retailer should provide the manufacturer with demand and inventory information. On the basis of these information, the manufacturer makes replenishment decision to control the inventory for both the manufacturer and the retailer.

The timing of the events in each period is assumed as follows. At the beginning of each period , the manufacturer puts its newly finished work-in-progress (WIP) inventory into a warehouse. Assume that there is a fixed production lead time, , for the manufacturer to finish an order. At the same time, the retailer also receives its replenishment order from the manufacturer to recover its inventory. It takes the manufacturer periods to dispatch each replenishment order for the retailer. Note that one extra period is included in and to cope with the events order because the replenishment decisions are made at the beginning of each period. After putting final products into a warehouse, the manufacturer forecasts the demand for the coming period according to realized demand data. Then the manufacturer makes the production decision for itself and the replenishment decision for the retailer by reviewing the inventory information regarding the retailer’s inventory level, in-transient inventory, and reorder point, as well as the manufacturer’s own inventory level and WIP inventory. This is the essential difference in comparison with traditional supply chain models. To keep the model linear, we also assume that all the orders of the retailer which can not be satisfied immediately are backlogged. Finally, during the remaining time of each period, the retailer fulfils customer demand and any unfilled orders are also backlogged.

To be more general, we assume that the demand of the retailer is unknown but bounded by , in which and represent the lower bound and upper bound, respectively. The demand uncertainty is caused by a lot of factors, such as promotions, technological innovations, seasonal changes, and economical or political events. To cope with these uncertain factors, the manufacturer uses an adaptive reorder point policy to manage the retailer’s inventory. The reorder point is updated bywhere , , is a coefficient to smooth the reorder point while the parameter is used to adjust service level. We note that the reorder point satisfies as long as . Once the inventory position of the retailer, , is below the reorder point, an replenishment order will be triggered automatically or manually. The amount of the retailer’s replenishment order is determined by the following reorder point policy:in which the parameter is used to control the replenishment frequency for the retailer. Note that the reorder policy in Disney and Towill [12] is a special case when , which imposes a constraint on the amount of each order and limits the decision-making flexibility for the manufacturer to adjust the replenishment frequency to achieve economical scale. In our model, we add this parameter allowing for freely adjusting the frequency of replenishment for the retailer, which essentially affects the performance of the whole system. Actually, the retailer might determine the value for according to its inventory capacity and demand characteristics. We argue that the parameter plays a different role in contrast to the parameter in (1). The parameter is used to determine the replenishment frequency, while is used to set the reorder point and balance the trade-off between inventory cost and service level. In practice, if the distance between the manufacturer and the retailer is large, the manufacturer might choose a large to save transportation cost. Finally, we advocate that the parameter setting for and should consider all the cost components.

In a VMI system, a vendor makes replenishment decisions based on information sharing with respect to downstream member’s demand and inventory. In our periodic inventory model, the balanced equations for the retailer’s inventory level, , and in-transient inventory, , are, respectively, represented asandThe inventory position of the retailer is further obtained as . Similarly, the difference equations for the manufacturer’s inventory level and WIP are, respectively, represented asand

The exponential smoothing algorithm is accurate for short-term forecasting and easy to implement [30]; thus the manufacturer uses it to forecast the demand for the coming period. It should be noted that the change of reorder point should be incorporated into the forecasting process. The forecasting algorithm is obtained aswhere is the smoothing coefficient with . A small helps smoothing order fluctuations, while a large is more appropriate to respond to volatile demand.

The replenishment rule used by the manufacturer for production decision is called APIOBPCS [30]. In traditional supply chains, the replenishment decisions for upstream members are made by themselves, whereas the replenishment decision in a VMI model should exploit systematic information. For this purpose, we define system inventory as The replenishment rule for the manufacturer is represented as where is the parameter to establish the target value of inventory level , is the production lead time, is the desired value of WIP level, and and are two replenishment parameters for recovering the inventory level and the WIP level. As the demand is uncertain, the target level for inventory and WIP should be updated adaptively. Essentially, the replenishment rule (9) is a proportional controller with two feedback loops. The two parameters significantly increases the flexibility to meet practical requirements. However, the feedback mechanism, the inventory interaction between the retailer and the manufacturer, coupled with the production lead time, makes the dynamics of VMI system highly complicated.

##### 3.2. A Transfer Function Model via Transformation of Difference Equations

In the literature, the production smoothing behavior is usually characterized by bullwhip effect metrics [46]. To facilitate the study of production smoothing in terms of bullwhip effect, we develop a transfer function model by treating the demand of the retailer as input and the production of the manufacturer as output. Unlike existing methods in the literature, in which transfer functions are derived by combining the components of block graphs [30], we directly obtain the transfer function by formulating a state space model and manipulating the difference equations using z-transform [47]. This change avoids the complicated computation process of the well-known Mason’s gain formula [48].

From the replenishment rule (9), we obtain By the definition of system inventory, we can derive Substituting (6) and (11) into (10), we further have

From (12), it is very interesting to observe that for arbitrary demand processes, the parameter is irrelevant to the production dynamics. In traditional supply chains, order batching is frequently used to achieve transportation scale and reduce transportation cost. However, it is well recognized that order batching with low delivery frequency is one of the main causes of production fluctuations. In contrast to traditional supply chains, (12) demonstrates that, under the VMI program, the manufacturer can dispatch goods to the retailer with a low frequency to reduce transportation cost while not disrupting the execution of production schedules. In this sense, we conclude that transportation cost and production cost can be reduced simultaneously in a VMI system.

A state space model can be developed directly from (1), (7), and (12) by defining as the state vector, as input variable, and as the output variable. Define , a state space model is obtained as where and Using the z-transform in Jury [47], the state space model (13) is converted as Finally, we can obtain the following transfer function: As mentioned in Lalwani [49], state space model (13) lays a solid foundation for studying the controllability and observability problem, and it might open a door for studying nonlinear dynamics and incorporating time-varying lead time into the APIOBPCS model, which makes the model more realistic and the results appealing. In contrast to the existing literature, the transfer function (17) in this study is formulated in a matrix form, which facilitates the implementation in computers. More importantly, the derivation process is simpler than obtaining the transfer function by combining block graphs with Mason’s gain formula [48], which enables us study the production smoothing behavior in terms of worst-case bullwhip for unknown demand.

#### 4. Inventory Fluctuations of the Retailer

Before we discuss the dynamics of the VMI system, we shall firstly introduce the definition of stability. By stability, we mean that the output of a system is bounded if the input is bounded. The customer demand is treated as the input. Because the fluctuations of inventory and order are of great importance, we define that the retailer subsystem is stabilized if , or , is bounded. Note that is easily bounded because customer demand is bounded by nature. The manufacturer subsystem is stable if and are bounded.

Although the stability of the VMI system has been discussed in Disney and Towill [30], the obtained results are limited to small or specified lead times. Furthermore, the previous literature has ignored the interaction between the retailer and manufacturer. In particular, they failed to consider the dynamics of the retailer, which might be due to the fact that the model in Disney and Towill [30] only considers the special case , which simplifies the dynamics of the retailer subsystem.

In this section, we begin by studying the stability and inventory oscillation of the retailer. In the VMI program, the retailer may assume the risk of incurring high inventory cost or poor customer service level. To avoid such a problem, the retailer can increase its benefits by signing effective contractual agreements, which are characterized by the penalty cost once the retailer’s inventory beyond an upper bound or a lower bound. To reduce the penalty cost and maintaining satisfactory service level for the retailer, the manufacturer must clarify how the replenishment rule affects the range of inventory fluctuations of the retailer. To this end, as a first step, we shall now discuss the inventory fluctuation range of the retailer under uncertain demand with Theorem 1.

Theorem 1. *Assume that and . Define as the inventory level of the retailer at the end of each period, then the retailer’s inventory position satisfies and satisfies*

*Proof. *Let be the initial inventory of the retailer and be the reorder point in the first period. We set and . Because the inventory position is less than the reorder point, the manufacturer will not dispatch goods to the retailer until the retailer’s inventory position drops below the reorder point. The dynamics for the retailer’s inventory level and reorder point are depicted in Figure 1. Without loss of generality, we assume that for , for , and when . In the first stage , we have and . In the second stage , because the manufacturer should send the replenishment order to the retailer, which is given by . For , we have When and Because , the upper bound and lower bound for the inventory position are obtained by Inequality (23) implies that there exists a lower bound for once it drops below the reorder point. In the third stage , we still have and Further, we can obtain It means that once the inventory position is higher than the reorder point, it starts to decrease. The inventory level at the end of each period for the retailer is represented as Further, we directly obtain It completes the proof of Theorem 1.