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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

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

This problem is a variation of the integrated production-distribution 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 production and distribution operations that are intimately linked with no intermediate inventory. Even though the research on integrated production-distribution scheduling models is fairly recent, much excellent work have been done in the last decade. Vehicle routing method is an efficient way to serve multiple customers by a shipment. However, only few literature studied the models with vehicle routing method (e.g., [

The rest of the paper is organized as follows. In Section

In this section, we first summarize the notations we used in our model and introduce some new notations in the following list:

We then present some straightforward optimality properties for the problem. Proofs are omitted.

There exists an optimal solution for the problem in which

there is no idle time between the jobs processed on each machine in the processing facility,

the departure time of each shipment is the completion time of the last job included in the shipment,

jobs that are processed on the same machine and delivered in the same shipment are processed consecutively on that machine.

In this section, we study two special cases of the problem with

In the problem with

Clearly, our problem with

The problem with

There exists an optimal solution for the problem with

If

If

If

If

Since the above results are straightforward, we omit the proofs. Obviously, it is easy to construct the following polynomial-time exact algorithm for the problem with

When

We first introduce some straightforward optimality properties for this case, which will be used in the sequel. Proofs are easy, and we leave them to the reader.

There exists an optimal solution for the case

In the following, we present a polynomial-time heuristic, denoted as H1, for the case and analyze its worst-case performance. In heuristic H1, similar to heuristic MMLS, we also let

In heuristic H1, most of the computing time is used for sorting jobs. Hence, the time complexity of the heuristic is

There are three cases associated with

In Step 9 of the heuristic, we select the best of the three solutions; hence

As mentioned before, except the case with

In heuristic H2, most of the computing time is used for sorting jobs in Step 1. Hence, the time complexity of the heuristic is

It is clear that

For the case with

If

In

There are two subcases associated with

When

When

Therefore, in both cases, we can conclude that

If

By Theorem

In this paper, we have studied a parallel machine scheduling model with batch delivery to two customers. The objective is to minimize the tradeoff between the maximum arrival time of the jobs and total distribution cost. Computational complexity of various cases of the problem has been clarified, and algorithms for these cases are provided. More specifically, we first provided a polynomial time heuristic with worst-case ratio bound of

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors thank the AE and the anonymous referees for their helpful comments and suggestions. This research was supported in part by the National Natural Science Foundation of China (71501051), the Humanities and Social Sciences Research Foundation of Ministry of Education of China (13YJC630239), and the Foundation for Distinguished Young Teachers in Higher Education of Guangdong Province (YQ201403).