An integrated multi-stage supply chain inventory model under an infinite planning horizon and continuous price decrease

Abstract

This paper studies an integrated inventory model in a supply chain that involves procurement, production and delivery activities. The model is studied in an environment where products experience continuous price decrease and planning is performed in an infinite time horizon. In this model, a manufacturing facility purchases a fixed-quantity of raw materials from an outside supplier, processes the materials, and delivers a fixed-quantity of finished products to a customer periodically. In order to take advantage of the decreasing price trend, customers demand frequent deliveries of small lots of finished products, and this inventory management strategy has been used by many successful companies in technology-related industries. Therefore, the ultimate intention of this research is to study and model the inventory system for high-tech companies whose products are experiencing continuous price decrease. This model is used to determine an optimal economic lot size model for raw material procurement, production setup and finished goods delivering under an infinite planning horizon. Two efficient algorithms are developed in this paper to solve this nonlinear model and the test results consistently indicate that ordering of raw materials and delivery of finished goods should be frequent in small lots for low ordering and shipment costs. Finally an operational schedule is provided to show the implementation procedure of the model.

Introduction

The continuous globalization of economy and increasing competition have forced companies to rely more and more on effective supply chains to improve their overall performance. Integration among two or more different business entities is an important way to gain competitive advantages as it lowers overall supply chain cost. According to Lambert and Cooper (2000), “supply chain management deals with total process excellence and represents a new way of managing the business and relationships with other members of the supply chain.” The challenging issue in the current supply chain management is to figure out how to accomplish the cross-functional coordination among the parties of various businesses. Supply chain integration can take place either through intra- or inter-company cooperation or combination of both. The intra-company cooperation considers integrating a manufacturer’s raw material procurement with its production, whereas the inter-company cooperation includes integration between the manufacturer’s production and the customer’s ordering.

Integrated inventory models can be developed either for an integrated vendor–buyer (IVB) system (inter-firm cooperation) or an integrated procurement-production (IPP) system (intra-firm cooperation) or combination of both IVB and IPP systems. It is explained by Lee (2005) that IVB systems coordinate the customer and the manufacturer in deciding the quantities of the ordering lot size and production batch size, but they do not include the raw material procurement, whereas IPP systems determine the raw material procurement lot size and the production batch size to minimize the total cost without considering the customer’s ordering quantity or the inventory holding cost. The model which combines IVB and IPP systems together prescribes economic raw material procurement lot size, production batch size and customer’s ordering lot size.

Golhar and Sarker, 1992, Jamal and Sarker, 1993, and Sarker and Parija (1994) report some important results on IPP inventory models. Integrated inventory models developed by Nanda and Nam, 1992, Lu, 1995, Goyal, 1995, Hill, 1997, Hill, 1999, Vishwanathan, 1998 and Goyal and Nebebe (2000) cover IVB systems without taking the raw material procurement into consideration. Lu (1995) developed an optimal policy for a single-vendor single-buyer problem in which the delivery quantity to the customer is identical in all replenishments. Goyal (1995) and Hill (1997) removed the restriction of identical shipments and allowed delivering all available vendor inventories to the customer. Their models showed that ‘deliver what is produced’ is better than ‘identical delivery quantity.’ However, Vishwanathan (1998) discussed that none of the strategies explained by Lu, 1995, Goyal, 1995, Hill, 1997 obtains the best results for all possible problem parameters. Hill, 1999, Goyal and Nebebe, 2000 kept working on IVB systems to obtain a better optimal solution while considering alternative policies.

More recently, Lee (2005) proposed an integrated inventory model for a single-manufacturer, single-buyer supply chain problem by combining IVB and IPP systems together. Therefore, the joint economic lot sizes of the manufacturer’s raw material ordering, manufacturing batch, and customer’s ordering are generated by the model. Lee (2005) indicates that there is no existing literature considering both IPP systems and a customer’s ordering quantity and inventory carrying cost together and emphasized the importance of the combined systems in the inventory control model to lower total cost of the supply chain. Banerjee, Kim, and Burton (2007) studied a single-product, three-stage inventory model involving a single-supplier, a single-manufacturer and multiple retailers. In their work they assumed that all retailers have the same replenishment cycle which basically makes the case of multiple retailers the same as a single-retailer case. They also assumed that the relationship among the three lot sizes, replenishing, and manufacturing and supplying is always a factor of an integer multipliers. Khouja (2003) had actually studied the three-stage inventory model earlier than Banerjee et al. (2007). However, the model proposed by Khouja (2003) is based on the assumption that the cycle times of the three stages are the same, or coordinated by integer multipliers or powers of two. The common unrealistic assumption of the above integrated inventory models is that the unit cost is constant over the planning horizon. However, especially for the successful companies in high-tech industries this is not a reasonable assumption. The price of the components and finished goods decreases continuously during their life cycle.

When some of the recent works in the area of inventory models under price change are examined, it can be realized that studies of Khouja and Park, 2003, Teng and Yang, 2004, Khouja et al., 2005, Teng et al., 2005, Khouja and Goyal, 2006, Mungan et al., 2010 and Glock (2011) are some of the important ones. All of the studies mentioned above consider only one side of the supply chain system. Either the economic order quantity (EOQ) model or the economic production quantity (EPQ) model is taken into consideration instead of looking at a larger picture by integration. An approximate closed-form expression for the optimal cycle time for a product whose unit cost is decreasing continuously by constant percentage over a finite planning horizon is studied by Khouja and Park (2003). Teng and Yang (2004) examined another EOQ model with partial backlogging during a finite planning horizon.

Recently, Teng et al. (2005) relaxed the traditional EPQ model and allowed time varying cost and demand. In other research done in 2005 by Khouja et al., an efficient algorithm is developed for solving the joint replenishment problem for products that may be experiencing unit cost increase or decrease in the EOQ model. More recently, Khouja and Goyal (2006) removed the restriction of having equal cycle times and allowed cycle times of varying length for the EOQ model in order to improve the model of Khouja and Park (2003).

Conclusively, the change in unit cost attracts the attention of a few researchers in recent years for the inventory model, but most of them only considered one side of the supply chain, which is either the customer or the vendor side. However, nowadays integration of operations in a supply chain is really essential in order for a company or companies to be successful in the competitive market. Unfortunately, most of the researchers who studied the price change did not consider this key issue of the supply chain management. Our work presented in this paper incorporates both ends of the supply chain, the vendors, and the customers in a coordinated supply chain system with products experiencing continuous unit cost decrease.