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In this paper, for effective multi-echelon supply chains under stochastic and fuzzy environments, an inventory management framework and deterministic/stochastic-neuro-fuzzy cost models within the context of this framework are structured. Then, a numerical application in a three-echelon tree-structure chain is presented to show the applicability and performance of proposed framework. It can be said that, by our framework, efficient forecast data is ensured, realistic cost titles are considered in proposed models, and also the minimum total supply chain cost values under demand, lead time and expediting cost pattern changes are presented and examined in detail.

Supply chain inventory management (SCIM) is an integrated approach to the planning and control of inventory, throughout the entire network of cooperating organizations from the source of supply to the end user. SCIM is focused on the end-customer demand and aims at improving customer service, increasing product variety, and lowering costs (Giannoccaro, Pontrandolfo, & Scozzi, 2003). Most manufacturing enterprises are organized into networks of manufacturing and distribution sites that procure raw material, process them into finished goods, and distribute the finish goods to customers. The terms “multi-echelon” or “multi-level” production/distribution networks are also synonymous with such networks (or supply chains (SCs)), when an item moves through more than one step before reaching the final customer (Ganeshan, 1999 and Rau et al., 2003). Fig. 1 shows a multi-echelon system consisting of a number of suppliers, plants, warehouses, distribution centers and customers (Andersson and Melchiors, 2001, Axsater, 1990 and Axsater, 2003). Full-size image (33 K) Fig. 1. A multi-echelon inventory system. Figure options The analysis of multi-echelon inventory systems that pervades the business world has a long history (Chiang & Monahan, 2005). Given the importance of these systems, many researchers have studied their operating characteristics under a variety of conditions and assumptions (Moinzadeh & Aggarwal, 1997). Since the development of the economic order quantity (EOQ) formula by Harris in 1913, researchers and practitioners have been actively concerned with the analysis and modeling of inventory systems under different operating parameters and modeling assumptions (Routroy & Kodali, 2005). Research on multi-echelon inventory models has gained importance over the last decade mainly because integrated control of supply chains consisting of several processing and distribution stages has become feasible, through modern information technology (Diks and de Kok, 1998, Kalchschmidt et al., 2003 and Rau et al., 2003). Clark and Scarf (1960) were the first to study the two-echelon inventory model (Bollapragada et al., 1998, Chiang and Monahan, 2005, Diks and de Kok, 1998, Dong and Lee, 2003, Rau et al., 2003, Tee and Rossetti, 2002 and van der Vorst et al., 2000). They proved the optimality of a base stock policy for the pure serial inventory system and developed an efficient decomposing method to compute the optimal base stock ordering policy. Bessler and Veinott (1965) extended the Clark and Scarf (1960) model to include general arborescent structures. The depot-warehouse problem was addressed by Eppen and Schrage (1981) who analysed a model with stockless central depot (van der Heijden, 1999). Several authors have also considered this problem in various forms (Bollapragada et al., 1998, Dong and Lee, 2003, Moinzadeh and Aggarwal, 1997, Parker and Kapuscinski, 2004, Tee and Rossetti, 2002, van der Heijden, 1999 and van der Vorst et al., 2000). Sherbrooke (1968) constructed the METRIC (Multi-Echelon Technique for Recoverable Item Control) model, which identifies the stock levels that minimize the expected number of backorders at the lower echelon subject to a budget constraint. Thereafter, a large set of models that generally seek to identify optimal lot sizes and safety stocks in a multi-echelon framework were produced by many researchers. In addition to analytical models, simulation models have also been developed to capture the complex interactions of the multi-echelon inventory problems. For detailed literature review of multi-echelon models please see Taskin Gumus and Guneri (2007). After a detailed literature review about the title, it can be seen that there are several deficiencies and rough assumptions related to research technique, echelon number, inventory policy, demand and lead time assumptions, and objective function (Taskin Gumus, 2007). In this paper, some of these deficiencies are eliminated and some of the assumptions are expanded about the titles listed above. Many researches have studied these problems as well as emphasized the need of integration among SC stages to make the chain effectively and efficiently satisfy customer requests (e.g. (Towill, 1996)). Beside the integration issue, the uncertainty has to be dealt with in order to define an effective SC inventory policy. In addition to the uncertainty on supply (e.g. lead times) and demand, information delays associated with the manufacturing and distribution processes characterize SCs (Giannoccaro et al., 2003). In the market, the participants of a supply chain not only face the uncertainties of product demands and raw material supplies but also face the uncertainties of commodity prices and costs (Liu & Sahinidis, 1997). The first concern in incorporating uncertainties into supply chain modeling and optimization is the determination of suitable representation of the uncertain parameters (Gupta & Maranas, 2003). Three distinct methods are frequently mentioned for representing uncertainty (Gupta and Maranas, 2003 and Hameri and Paatela, 2005): First, the distribution-based approach, where the normal distribution with specified mean and standard deviation is widely invoked for modeling uncertain demands and/or parameters; second, the fuzzy-based approach, therein the forecast parameters are considered as fuzzy numbers with accompanied membership functions; and third, the scenario-based approach, in which several discrete scenarios with associated probability levels are used to describe expected occurrence of particular outcomes (Chen & Lee, 2004). A number of researches have been devoted to studying supply chain management under uncertain environments (Taskin Gumus & Guneri, 2007). For example, Gupta and Maranas, 2000 and Gupta et al., 2000 incorporate the uncertain demand via a normal probability function and propose a two-stage solution framework. A generalization to handle multi-period and multi-customer problems is recently proposed (Gupta & Maranas, 2003). Tsiakis, Shah, and Pantelides (2001) use scenario planning approach to describe demand uncertainties. Due to the potential of dealing with linguistic expressions and uncertain issues, fuzzy sets are used to handle uncertain demands and external raw material problems and in a later work, Petrovic, Roy, and Petrovic (1999) further consider uncertain supply deliveries. Giannoccaro et al. (2003) also apply fuzzy sets theory to model the uncertainties associated with both market demand and inventory costs. Despite their obvious negotiable and uncertain characteristics in real businesses, the product price is seldomly taken into account as a source of uncertainty in previous works (Chen & Lee, 2004). Instead, it is usually treated as known parameters. In this paper, demand, lead time and expediting cost uncertainties are emphasized and tried to be eliminated in a realistic way for successful inventory management in supply chains, under stochastic and fuzzy environments. Hence, an inventory management framework and deterministic/stochastic-neuro-fuzzy cost models within the context of this framework are structured. Then, a numerical application in a three-echelon tree-structure chain is presented to show the applicability and performance of the proposed framework.

After a detailed literature review about the title, it can be seen that there are several deficiencies and rough assumptions related to research technique, echelon number, inventory policy, demand and lead time assumptions, and objective function. In this paper, some of these deficiencies are eliminated and some of the assumptions are expanded about the titles listed above. The supply chain model developed in this paper remedies several deficiencies of similar model found in the literature. The basic deficiency is that demand and/or lead times are assumed to be deterministic, constant or to fit a probabilistic distribution. In our model, demand, lead time and expediting cost are determined by neuro-fuzzy calculations, a method that gives realistic results (Section 3.2). It is known that, the realism of demand and lead time is extremely important for cost calculations, purchasing decisions, production and inventory management applications, etc. in a supply chain. Also here, expediting cost pattern is forecasted by neuro-fuzzy calculations, too, that is usually assumed to be deterministic or not considered before. Another deficiency in literature is that limited echelons of a multi-echelon inventory system are usually considered. They rarely generalize their models to N echelon and usually consider two-echelon SCs. Similarly, they usually consider serial systems, instead of a tree conformation. In this paper, the models are generated for a three-echelon and tree-structured supply chain. Another assumption of published models is that late orders are delayed until the next order cycle arrives. Our model allows orders that arrive out of phase to be expedited. In this sense it more accurately reflects the real world, where expediting costs are tolerated to provide high service levels. Another innovation of our paper is that we employed a neural network to simulate our supply chain model. Conventional simulation software (ARENA, SLAM II, etc.) could not be used because our model incorporated neuro-fuzzy forecasting. As a result, it can be said that, our methodology and models ensure efficient forecast data, use realistic cost titles for three-echelon tree-structured supply chains, and help to eliminate several deficiencies encountered in literature via these specifications. For future research, the identity assumption for retailers and distributors can be eliminated. Also, the model can be expanded to consider multi product type and information channels (as Internet-enabled direct channel besides retail channel). Else, in addition to demand, lead times and expediting costs, other variables can be determined by neuro-fuzzy approximations to further increase the model’s fidelity to the real world.