We study a logistics scheduling problem where a manufacturer receives raw materials from a supplier, manufactures products in a factory, and delivers the finished products to a customer. The supplier, factory and customer are located at three different sites. The objective is to minimize the sum of work-in-process inventory cost and transport cost, which includes both supply and delivery costs. For the special case of the problem where all the jobs have identical processing times, we show that the inventory cost function can be unified into a common expression for various batching schemes. Based on this characteristic and other optimal properties, we develop an O(n) algorithm to solve this case. For the general problem, we examine several special cases, identify their optimal properties, and develop polynomial-time algorithms to solve them optimally.
The production logistics activity of enterprises is typically composed of three stages, namely supply, production and distribution. In recent years, much of the literature has studied logistics scheduling that integrates production scheduling and job delivery to customers. For example, the reader is referred to Lee and Chen (2001), Chang and Lee (2004), Chen and Vairaktarakis (2005), Hall and Potts (2005), Pundoor and Chen, 2005 and Pundoor and Chen, 2009, Chen and Pundoor (2006), and Wang and Cheng, 2007 and Wang and Cheng, 2009a. This line of research focuses on optimizing the total distribution cost and customer service level. On the other hand, for scheduling research that takes supply and production into consideration, Selvarajah and Steiner (2005) presented a polynomial-time algorithm to minimize the sum of total inventory holding cost and product batch delivery cost. Qi (2005) considered a logistics scheduling model that deals with material supply and job scheduling at the same time, where the objective is to minimize the sum of work-in-process (WIP) inventory cost and raw material supply cost.
There are research results on logistics scheduling that deals with all the three stages of supply, production and delivery. Hall and Potts (2003) considered multiple-production-stage scheduling with batch delivery in an arborescent supply chain. They analyzed the complexity of the problems and developed some dynamic programming algorithms. Wang and Cheng (2009b) considered a machine scheduling problem with supply and delivery of materials and products, where the warehouse, the factory and the customer are located at three different sites. The objective is to minimize the makespan. They did not take the WIP inventory cost into consideration.
In this paper we formulate a logistics scheduling model that considers production scheduling, raw material supply and product delivery at the same time. We assume that the supplier, manufacturer and customer are located at three different sites. Transportation service is provided by a third party, so transport vehicles are available at any time. The manufacturer needs to pay the third party for its service to transport materials to the factory, and deliver products to the customer. A job may be transported from the supplier's warehouse to the manufacturer's factory at any time just before it starts processing, and a product is available for delivery to the customer as soon as it finishes processing in the factory. On the other hand, since the cost of processing all the jobs in a planning period is normally fixed and independent of the production schedule used, we only consider the cost of holding intermediate inventory, which is in terms of the WIP inventory level of the factory. The problem under study is to find an optimal joint schedule for material supply, production scheduling, and job delivery so that the sum of WIP inventory cost and transport cost is minimized. This logistics scheduling problem models the practical situation where a single dominant firm controls both upstream and downstream stages in a supply chain. So the firm could, in the short run, simply optimize its own operational decisions regardless of the impact of such decisions on the other stages of the chain (see, e.g., Erengüc et al., 1999).
The rest of the paper is organized as follows. In Section 2 we formally describe our model and present the notation. In Section 3 we develop an optimal algorithm for the special case of the problem where all the jobs have identical processing times. In Section 4 we examine several special cases of the general problem, identify their optimal properties, and develop polynomial-time algorithms to solve these cases optimally. In the last section we conclude the paper and suggest topics for future research.
In this paper we studied a logistics scheduling problem with material supply and product delivery considerations. The objective is to minimize the sum of the WIP inventory cost and transport cost. When all the jobs have identical processing times, we showed that the expression of the WIP inventory cost function can be unified, and we proposed an O(n) optimal algorithm to solve this case. We also examined several special cases of the general problem, identified their optimal properties, and developed polynomial-time optimal algorithms to solve them.
As for future logistics scheduling research, researchers need to build a scheduling model that integrates the three stages, i.e., supply, production and distribution, of the typical logistics activity. Such a model has very different characteristics from those of the two-stage models found in the existing literature that consider only production and distribution, or supply and production. Consideration of various machine processing environments and objectives for a three-stage logistics scheduling model is worthy of future study, too.
