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This paper investigates the issue of channel coordination for a two echelon supply chain consisting of one manufacturer and one retailer. In this supply chain, the demand is sensitive to promotional efforts/sales teams’ initiatives. A production-inventory model is developed that considers the procurement cost per unit as a function of the production rate. To resolve the issues of channel coordination and promotion-based demand, a variety of centralized coordinating systems are explored. An analytical method is employed to achieve optimal production rate, production lot size, backlogging and the initiatives of sales teams so that the profits of both manufacturer and retailer are maximized. Some numerical examples are solved in order to better understand the proposed production-inventory model. The results of a sensitivity analysis are also provided. Finally, some conclusions and future researches are included.

Inventory theory and, in particular, the economic order quantity (EOQ) inventory model were first introduced by Harris (1913). Since then, many variants of the EOQ inventory model have appeared in the inventory literature. Some of these variants correspond to EOQ models for supply chains. A supply chain is comprised of vendors/suppliers, assemblers/manufacturers, distribution centers, retailers and final customers. It is important to mention that a supply chain composed of two stages (i.e., one manufacturer and one retailer) is considered a complex system. The success of any supply chain system depends on its level of collaboration and integration. In this direction, the study of supply chains from collaborative and integrative perspectives has recently attracted the attention of industrial engineers. Previous research efforts exist in the inventory literature that have shown that when the supply chains are vertically integrated their channels profits are maximized (Wu et al., 2009, Li et al., 2009, Ru and Wang, 2010, Sana, 2011a, Sana, 2011b, Sana, 2012, Sana, 2013 and Wu, 2013). Björk (2008) has found an analytical solution of an economic production quantity model while the demand is crisp and the cycle time is fuzzy in nature. In a subsequent paper, Björk (2012) has investigated a fuzzy multi-item economic production quantity model to obtain analytical solution (i.e., optimal size of some production batches) under uncertain cycle times which are handled with triangular fuzzy numbers. Also, there are several research works that study the integration of supply chain systems with multi-stages. These studies have shown that the costs of the channels in integrated supply chains are minimized (Cárdenas-Barrón, 2007, Cárdenas-Barrón et al., 2012a, Cárdenas-Barrón et al., 2012b, Ben-Daya et al., 2013, Cárdenas-Barrón and Porter, 2013 and Cárdenas-Barrón and Treviño-Garza, 2013). In an oligopolistic marketing environment, the managers of business organizations have a lot of pressure to sell products to downstream channel members. Generally speaking, vendors persuade their customers with sales teams’ initiatives or promotional strategies, i.e., free gifts, delay in payments, discounts, displays, packaging, special services and advertising, among others. In this direction, Goyal and Gunasekaran (1995) have proposed an integrated production-inventory-marketing model to determine the optimal economic production lot size and economic order quantity of raw materials in a multi-stage production system considering the effect of different marketing policies such as the price per unit product and advertising frequency on the demand of a perishable product. On the other hand, Nair and Tarasewich (2003) have obtained the optimal design of promotional efforts such as free gifts, discounts and special services. Later, Krishnan et al. (2004) have shown that pricing, displays, free goods and advertising are essential actions to achieve maximum revenues. Subsequently, Sun (2005) has found that there exists a relation between the customers and different types of promotions that results in a solid impact on stronger brands. Afterwards, Szmerekovsky and Zhang (2009) have developed the pricing options and two-tier advertising actions between one manufacturer and one retailer when customers’ demand is dependent on the retail price and advertising by both players. Xie and Wei (2009) and Xie and Neyret (2009) have investigated the optimal collaborating advertising strategies and equilibrium pricing in a two-layer supply chain. Additionally, Ramanathan and Muyldermans (2010) have studied the effect of promotional efforts on the sales of soft drinks. Recently, Sana (2010) has presented a multi-item EOQ model for perishable and ameliorating products when the time varying demand is dependent on the enterprise's initiatives such as advertising and salesmen's initiatives. Furthermore, Sana, 2011a and Sana, 2011b has developed EOQ models for similar products when the demand of the end customers depends on the stock level, selling price and sales teams’ initiatives. In this paper, we develop a production-inventory model for a two echelon supply chain consisting of one manufacturer and one retailer. In the proposed model, the procurement cost per unit is assumed to be a convex function of production rate and the demand of final customers is an increasing function of sales teams’ initiatives. The stockout situation is also considered at the retailer channel. The cost of initiatives by sales teams is an increasing nonlinear function of the sales teams’ initiatives. This cost is shared by the manufacturer and the retailer. The profits per unit quantity of the manufacturer and the retailer are formulated by trading off the cost and profit parameters. Three strategies are discussed for a centralized supply chain system. The rest of the paper is organized as follows. Section 2 introduces the assumptions and notations that are considered in the development of the production-inventory model. In Section 3,the production-inventory model is formulated and analyzed. Section 4 illustrates the production-inventory model with numerical examples and a sensitivity analysis is provided in Section 5. Finally, Section 6 provides some conclusions and future research directions.

In an oligopolistic marketing environment, promotional effort/sales teams’ initiatives increase the demand of the final customer resulting in higher demand for each member of the supply chain. In a collaborative system, both the manufacturer and the retailer share the cost for sales teams’ initiatives helping to make the decision to share their profits. We have discussed the production-inventory model for three possible strategies for a supply chain working in a collaborative manner. The first strategy is joint profit maximization. The second strategy is the manufacturer determines the optimal production rate and production lot size while the retailer determines the optimal shortage level. The third strategy is that the manufacturer determines the optimal production rate and the retailer sets the optimal order size (production lot size) and the shortage level. In all three cases, the profits per unit quantity are maximized by increasing one unit of the effort/initiatives of the sales teams. Then, the total profits are considered to achieve our goals, i.e., the optimal strategy at which the total profit of the collaborating system is at a maximum. It is observed that the second strategy is more profitable than the other two strategies. In this strategy, the costs of holding inventory and procurement are minimized by the manufacturer as he controls the production rate and production lot size, whereas the shortage levelis optimized by the retailer resulting in a minimum penalty for shortages. As a whole, the second strategy (Case 2) is more profitable than the other strategies although the second strategy follows the Stakelberg approach, i.e., the manufacturer is the decision maker. In such cases, the joint profit is shared by both parties according to their target profits and their investments in the costs of the channel. The major contribution of the proposed production-inventory model is the assembly of variable procurement cost, sales teams’ initiatives, sensitive demand, backlogging, variable production rate and production lot size in one production-inventory model for a two echelon supply chain. Although the inventory theory contains many models, this work is an addition to this line of research. As far as the authors’ knowledge goes, this type of model in a production-inventory system has not yet been published in supply chain modeling. The present article may be extended further in many ways. One immediate extension can be done for multi-item products in a multi-echelon supply chain system. Stock out situations at the manufacturer may be incorporated further in the proposed paper. The major limitations of the proposed production-inventory model are that the production rate and the demand rate are deterministic in nature. These limitations may be relaxed in future extensions of the present production-inventory model and these will be some interesting issues that one can deal with in the near future.