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A major issue in supply chain inventory management is the coordination of inventory policies adopted by different members in a supply chain including suppliers, manufacturers, distributors, etc. This paper presents a fractal-based approach for inventory management in order to minimize inventory costs and smooth material flows between supply chain members while responsively meeting customer demand. Within this framework, each member in the supply chain is defined as a self-similar structure, referred to as a fractal. A fractal-based echelon does not indicate a functional level or composition of supply chain members but indicates a group of multi- or hetero-functional fractals. The basic fractal unit (BFU) consists of five functional modules including an observer, an analyzer, a resolver, an organizer, and a reporter. The application of the fractal concept into inventory management makes it easy to intuitively understand and manage supply chain inventories because similar functional modules can be iteratively applied to an inventory management system. More specifically, we apply the fractal concept to a vendor managed inventory (VMI) model, referred to as fractal-based VMI (fVMI), where a vendor assumes responsibility for maintaining inventory levels and determining order quantities for his buyers. In this paper, we develop mathematical models for the analyzer and resolver to effectively manage supply chain inventories. For validating the proposed approach, a comprehensive simulation model, representing two VMI initiatives including traditional VMI and fVMI, is constructed and used for comparative analyses of case studies.

A supply chain is a complex network consisting of several organizations with different objectives for the production and distribution of products according to customer demand. Especially, supply chain management (SCM) is concerned with finding the best strategy for the entire supply chain (SC) by coordinating different enterprises along the logistics network or establishing business partnerships (Simchi-Levi et al., 2003). Many researchers have studied ways to optimize SCs so that manufacturers, distributers, and suppliers can maximize their profits. In order to find the best strategy in this complex network, intensive communication, and coordination between trading partners is required so that material and information flow along the SC can be optimized (Sari, 2008). Among various SCM issues, inventory management is to a greater extent relevant when the entire supply chain, namely a network of procurement, transformation or production, and delivery firms, is considered. According to the literature, inventory usually represents from 20% to 60% of the total assets of manufacturing firms (Giannoccaro et al., 2003). Supply chain inventory management (SCIM) is focused on end-customer demand and aims at improving customer service while lowering relevant costs (Verwijmeren et al., 1996). Inventory management policies prove critical in determining the profit of each supply chain members. The main considerations regarding SCIM policies include (i) nature of optimization (e.g., local, global), (ii) control type (e.g., centralized, distributed), (iii) nature of review of inventory levels (e.g., periodic, continuous, hybrid), (iv) type of demand function (e.g., linear, distribution), and (v) responsibility for inventory control (e.g., self-managed, vendor-managed), as illustrated in Table 1. Table 1. Main considerations affecting SCIM policies. Considerations for SCIM Exemplary choices Nature of optimization Local; global Control type Centralized; distributed Nature of review of inventory levels Periodic; continuous; hybrid Type of demand function Probability distribution; linear Responsibility of inventory control Self-managed; vendor-managed Table options An inventory policy can possess local or global goals (Axsäter and Juntti, 1996). Regarding the pursuit of local goals, the SC inventory policy results from a collection of local policies in which every SC member tends to make decisions on its own inventory individually based on local performance criteria. Several effective incentive mechanisms including quantity discounts, profit sharing, buybacks, etc., can be applied to align members' goals or profits in the supply chain. On the contrary, regarding the pursuit of global goals, the SC inventory policy tends to make decisions to optimize the entire inventory based on global performance criteria. There are two different strategies for managing SC inventory including centralized and distributed (or decentralized) inventory control (Petrovic et al., 1999). Under centralized control, a central decision-maker determines the policy that minimizes the entire SC cost using a high degree of coordination and communication between the SC members. However, under distributed control, each SC member monitors the status of their own local inventory and places orders to their predecessors based on their own performance criteria. With the help of several incentive mechanisms, distributed inventory policies, which are adopted in most cases in SCs, can achieve performance levels almost as high as those realized through centralized policies. Inventory management policies can also differ in terms of the manner of reviewing inventory levels. Under a periodic-review control policy, the inventory status is reviewed at every stage at a constant time interval. At each review, a replenishment order can be issued in order that the inventory status should meet the target level. In this case, the optimal quantity for replenishment, Q⁎, has to be calculated. Under a continuous-review control policy, a replenishment order is issued when the inventory position at the considered stage falls below a predetermined level, i.e., the reorder point. In that case, a fixed quantity, Q, is ordered. A hybrid control policy may also be applied for inventory management. The most common is the (s, S) policy. Under this policy, if the inventory position falls below the reorder point s, an order is issued to raise the stock up to the target level S. Inventory management policies are characterized based on the demand functions used. Most of the literature assumes that demand follows a certain probability distribution such as normal, Poisson, gamma, and so on. Some researchers use a linear demand function to simplify their models (Dong and Xu, 2002 and Nachiappan and Jawahar, 2007). Even though demand is the most important factor for inventory control, it is very difficult to forecast the exact demand in advance. The lack of visibility of real demand can and does cause a number of problems in a SC if it is not properly designed; even then fluctuations cannot be completely eliminated. One of the easiest ways to prepare for such problems is to apply probability distributions or linear functions for demand expectation. However, progressively shorter product life cycles as well as growing innovation rates make demand extremely variable and the collection of statistics, that are required by stochastic models, less and less reliable (Blackburn, 1991). Finally, inventory policies can be characterized based on the responsibility for inventory control. In traditional inventory control policies, each member is responsible for his own inventory control and production or distribution ordering activities. One fundamental characteristic and problem that all members in a traditional SC including retailers, distributors, and manufacturers must solve is just “how much to order the production system to make (or the suppliers to supply) to enable a SC echelon to satisfy its customers' demands.” Each member strives to develop local strategies for optimizing her own organizational goals without considering the impact of her strategies on the performance of other members. Upstream members do not know actual demand information from the market place because no information is shared between members (see Fig. 1(a)). SC members use only replenishment orders placed by their immediate downstream member to create demand forecasts and inventory plans. In other words, each echelon in the SC has information only about what their immediate customers want and not on what the end customer wants. Each member of SC, therefore, replenishes her own inventory by considering her local inventory position. Full-size image (29 K) Fig. 1. SC structure under SCIM and VMI. (a) Material and information flow under traditional SCIM policy and (b) material and information flow under VMI policy. Figure options In contrast to the traditional inventory control, many companies have been compelled to improve their SC operations by sharing demand and inventory information with their upstream and downstream members including customers (Disney and Towill, 2003a). Vendor managed inventory (VMI), also known as continuous replenishment, supplier-managed inventory, or consignment inventory, is a SC strategy where the vendor or supplier is given the responsibility of managing the customer's stock. VMI is one of the most widely discussed partnering initiatives for encouraging collaboration and information sharing between trading partners (Angulo et al., 2004). Under the VMI program, the retailer provides the vendor with access to its real-time inventory level through for instance POS (point of sale) data. The retailer may set a certain service level or spatial capacity for stocks, which are then taken into consideration by the vendor. Based on the information on retailers through POS systems for example, the vendor decides on the appropriate inventory level of each of the products and appropriate inventory policies to maintain those levels (see Fig. 1(b)). As a consequence, the retailer's role shifts from managing inventory to simply renting retail space (Simchi-Levi et al., 2003 and Mishra and Raghunathan, 2004). As illustrated in Fig. 1, the main difference between a traditional inventory control policy and a VMI program is who makes the decision to replenish retailer's inventory, i.e., whether or not the retailer places replenishment orders. When we consider a three-stage SC, which consists of three echelons including a manufacturer, a distributor, and a retailer, the distributor looks at its own inventory status and that of the retailer as well. On the other hand, all other echelons of the SC, i.e., between the distributor and manufacturer, operate in the same way as in traditional inventory management. VMI has become more popular in the grocery sector in the last 20 years due to the success of retailers such as Wal-Mart (Andel, 1996 and Waller et al., 1999). It was subsequently implemented by various leading companies in diverse industries including Electrolux (De Toni and Zamolo, 2005), Nestle and Tesco (Watson, 2005), Boeing (Micheau, 2005), etc. The popularity of VMI has led to the claim that VMI is the wave of the future and the concept will revolutionize the distribution channel (Burke, 1996; Cottrill, 1997). VMI is beneficial to both buyers and suppliers though the supplier may take a longer period of adjustment and reconfiguration before the benefits of VMI can be realized. VMI offers a competitive advantage for retailers with respect to higher product availability as well as reductions in holding costs and some operational costs plus cash flow benefits (Benefield, 1987, Achabal et al., 2000 and Yao et al., 2007). On the other hand, vendors may gain opportunities to improve production and marketing efficiencies (Cottrill, 1997, Waller et al., 1999 and van der Vlist et al., 2007), and synchronize replenishment planning (Waller et al., 1999 and Çetinkaya and Lee, 2000) while reducing the bullwhip effect (Lee et al., 1997a, Lee et al., 1997b, Disney and Towill, 2001, Disney and Towill, 2003a and Disney and Towill, 2003b). However, a number of challenges may exist in practice that can potentially reduce the benefits gained from VMI or lead to failures in VMI programs. The readers can refer to the literature for some unsuccessful cases of VMI, such as Spartan Stores (Simchi-Levi et al., 2003) and Kmart (Fiddis, 1997). The objective of this paper is to propose a fractal-based conceptual framework for inventory management in order to minimize inventory costs and smooth material flows between SC members while responsively meeting customer demand. In this paper, the fractal concept has been adopted because of its intrinsic benefits in dealing with a dynamically changing SC environment including customer demand. By applying the fractal concept to SCIM, each member in the SC is defined as a self-similar structure, which is referred to as a fractal. We will discuss the concept, structure, functions, etc. of a fractal in the following sections in detail. In this paper, we also consider a VMI model, which consists of a three-stage SC including a manufacturer, a distributor, and a retailer. As shown in Fig. 2, however, we extend the concept of vendor-managed control to the relationship between distributors and the manufacturer. In other words, we maintain consistency in the relationships regarding “one vendor-multiple buyers,” and the vendor manages the buyer's inventories. For example, the manufacturer (vendor) manages the distributor's (buyer's) inventories, and the distributor (vendor) manages the retailer's (buyer's) inventories. We also consider transhipments between members in the same echelon to prevent backorders and ensure smooth material flow as in Fig. 2. Since we have applied the fractal concept to VMI, our model is referred to as fractal-based VMI (i.e., fVMI). According to the main considerations affecting SCIM policies as shown in Table 1, the proposed inventory policy in this paper can be classified as summarized in Table 2. Full-size image (22 K) Fig. 2. SC structure under fVMI. Figure options Table 2. Adopted SCIM policies. Considerations for SCIM Policy Nature of optimization Local Control type Distributed Nature of review of inventory levels Hybrid Type of demand function Probability distribution Responsibility of inventory control Vendor-managed Table options The remainder of this paper is organized as follows. Section 2 describes the concept, structure, and intrinsic characteristics of a fractal. A fractal-based framework for VMI is proposed in Section 3, followed by mathematical models in Section 4. A comparative study using simulation is discussed in Section 5 before concluding remarks are presented in Section 6.

In this paper, we have proposed an fVMI framework for inventory management in order to effectively manage inventory levels and minimize relevant costs. The proposed fVMI approach has advantageous features for managing VMI by adopting the fractal concept as described in Section 3.1 in detail. In the proposed framework, each member of the SC is defined as a fractal by itself, and any combination of members can also be defined as a fractal. The SC structure for fVMI is based on the structure for traditional VMI. However, two main differences exist between the two structures. Firstly, fVMI extends the concept of vender-managed control to the entire SC structure, while traditional VMI considers only the relationship between retailers and their immediate upstream members. Secondly, transhipment is permitted between members in the same echelon in fVMI. In fVMI, an (s, S) policy is used as a basis for inventory control. Under the (s, S) policy, a fractal optimizes the replenishment quantities between fractal-echelons, and the transhipment quantities within a fractal-echelon. To examine the efficiency of the fVMI framework, a comparative study based on simulations was conducted. In the simulation, two SCIM initiatives, including traditional SCIM, traditional VMI, and fVMI, were applied to an illustrative SC under three demand variations. The simulation results indicated that the cost savings derived from fVMI are significantly higher than those under traditional VMI for all levels of demand variability. The results also revealed that transhipment between retailers has positive effects on cost savings. In this study, we considered only the cost factors in SCIM. For further research, more factors such as the customer service levels of retailers and the bullwhip effects under fVMI can be considered and investigated to extend the proposed framework. Additional simulations also need to be conducted to compare the proposed framework with various optimization techniques that have not been considered in this paper. The adoption of the other characteristics of a fractal (i.e., self-organization including DRP, goal-orientation, etc.) into SCIM can also be considered as future research topics. For example, self-reconfiguration of supply networks can be dealt with to facilitate DRP under the fVMI framework, and autonomous goal generation of each fractal can be considered as another research issue with respect to the fVMI framework.