Parallel Machine Scheduling with Batch Delivery to Two Customers

Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 247356, 6 pages http://dx.doi.org/10.1155/2015/247356 Research Article Parallel Machine Scheduling with Batch Delivery to Two Customers Xueling Zhong1 and Dakui Jiang2 1 Department of Internet Finance and Information Engineering, Guangdong University of Finance, Guangzhou 510520, China College of Management and Economics, Tianjin University, Tianjin 300072, China 2 Correspondence should be addressed to Xueling Zhong; Received 1 May 2015; Revised 22 August 2015; Accepted 31 August 2015 Academic Editor: Sergio Teggi Copyright © 2015 X. Zhong and D. Jiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In some make-to-order supply chains, the manufacturer needs to process and deliver products for customers at different locations. To coordinate production and distribution operations at the detailed scheduling level, we study a parallel machine scheduling model with batch delivery to two customers by vehicle routing method. In this model, the supply chain consists of a processing facility with 𝑚 parallel machines and two customers. A set of jobs containing 𝑛1 jobs from customer 1 and 𝑛2 jobs from customer 2 are first processed in the processing facility and then delivered to the customers directly without intermediate inventory. The problem is to find a joint schedule of production and distribution such that the tradeoff between maximum arrival time of the jobs and total distribution cost is minimized. The distribution cost of a delivery shipment consists of a fixed charge and a variable cost proportional to the total distance of the route taken by the shipment. We provide polynomial time heuristics with worst-case performance analysis for the problem. If 𝑚 = 2 and (𝑛1 − 𝑏)(𝑛2 − 𝑏) < 0, we propose a heuristic with worst-case ratio bound of 3/2, where 𝑏 is the capacity of the delivery shipment. Otherwise, the worst-case ratio bound of the heuristic we propose is 2 − 2/(𝑚 + 1). 1. Introduction To meet the soaring demands of electronic devices in recent years, manufacturers in China start building new factories to increase production capacities. Two different strategies are mainly adopted by these manufacturers, one is to build a new factory at the undeveloped land near current factory and the other is to place the new factory to a different region with lower labor cost. Take Foxconn Technology Group, the world’s largest electronics contractor manufacturer, for example, it not only built a new factory at Guanlan Technology Park after running one at Yousong Industrial District in Shenzhen city of China but also has been building many other factories at different regions of China. Clearly, it can share resources easily by adopting the former strategy and reduces production cost by adopting the latter one. Meanwhile, as a nonstandard parts supplier of a manufacturer adopting the former strategy, it should not only offer parts to the current factory but also provide parts to the new-built factory. In such applications, very little inventory of finished parts exists at any point of time since nonstandard parts are custom made and the supplier will not start production early before it receives orders from the manufacturer. Hence, the production and distribution operations of the supplier are linked immediately, and the close linkage between production and distribution necessitates coordinating production and distribution operations at the level of detailed scheduling. In this paper, we consider a parallel machine scheduling problem with batch delivery to two customers faced by the nonstandard part supplier in the above-described supply chain, which can be described as follows. There is a manufacturer, who has a set of 𝑚 > 2 identical parallel machines facility, 𝑀 = {1, 2, . . . , 𝑚}. At time zero, the manufacturer receives a set of 𝑛 jobs, 𝑁 = {1, 2, . . . , 𝑛}, from two customers 1 or 2, which are located at different locations in an underlying transportation network. Among jobs in 𝑁, jobs in the subset 𝑁𝑘 ⊆ 𝑁 are ordered by customer 𝑘, 𝑘 = 1, 2. Let 𝑛𝑘 = |𝑁𝑘 | denote the number of jobs in 𝑁𝑘 . It is easy to see that 𝑁 = 𝑁1 ∪ 𝑁2 and 𝑛 = 𝑛1 + 𝑛2 . Each job in 𝑁 should be processed onto one of the 𝑚 machines in manufacturer and 2 then delivered to its customer in batch without intermediate inventory. Associated with each job 𝑗 ∈ 𝑁 has a processing time of 𝑝𝑗 units. Job preemption is not allowed; that is, processing a job on a machine cannot be interrupted until it is finished. All jobs and machines are available at time 0. All the finished jobs ordered by customer 𝑘 need to be delivered in batch to customer 𝑘 by the vehicle, 𝑘 = 1, 2. Suppose that there are enough homogeneous vehicles available so that each vehicle will be used once and each delivery shipment will be transported by a dedicated vehicle. All the vehicles are stationed at the processing facility at time 0 and must go back to the facility once they complete a shipment. Each vehicle can carry up to at most 𝑏 jobs in one shipment. The transportation cost incurred by each batch consists of a fixed charge 𝑓 and a variable cost dependent on the particular route taken by the vehicle. We use 𝑐0𝑖 , 𝑐𝑖𝑗 , and 𝑐𝑖0 , respectively, to denote the variable cost for traveling from the processing facility to customer 𝑖 ∈ {1, 2}, from customer 𝑖 ∈ {1, 2} to customer 𝑗 ∈ {1, 2}, and from customer 𝑖 ∈ {1, 2} to the processing facility. The corresponding delivery times are denoted as 𝑡0𝑖 , 𝑡𝑖𝑗 , and 𝑡𝑖0 , respectively. We assume that 𝑡𝑖𝑖 = 𝑐𝑖𝑖 = 0, 𝑡𝑖𝑗 = 𝑡𝑗𝑖 , 𝑐𝑖𝑗 = 𝑐𝑗𝑖 , 𝑡0𝑖 + 𝑡𝑖𝑗 ≥ 𝑡𝑗0 , and 𝑐0𝑖 + 𝑐𝑖𝑗 ≥ 𝑐𝑗0 , 𝑖, 𝑗 = 0, 1, 2. A delivery vehicle can depart from the processing facility only when all the jobs to be delivered have completed processing. We use 𝐷𝑗 to denote the time when job 𝑗 ∈ 𝑁 arrives to its customer. Define 𝐷max = max{𝐷𝑗 | 𝑗 ∈ 𝑁} as the maximum arrival time to the two customers of jobs in 𝑁. The problem is to find a joint production and distribution schedule such that the tradeoff between maximum delivery time of the jobs and total distribution cost, that is, 𝛼𝐷max + (1 − 𝛼)𝑇, is minimized, where 𝑇 denotes the total distribution cost, 𝐷max measures customer service level, and 0 ≤ 𝛼 ≤ 1 is a given constant and represents the decision maker’s relative preference on customer service level and total distribution cost. Such an objective function is also adopted by Chen and Vairaktarakis [1]. This problem is a variation of the integrated productiondistribution scheduling models with batch delivery to multiple customers by vehicle routing method, which is always encountered in make-to-order or time-sensitive product supply chains. In these supply chains, finished jobs are often delivered to customers immediately or shortly after the production which lead to producti (...truncated)