We consider a single parallel-batching machine scheduling problem with delivery involving both batching scheduling and common due date assignment. The orders are first processed on the single parallel-batching machine and then delivered in batches to the customers. The batching machine can process several orders at the same time. The processing time of a production batch on the machine is equal to the longest processing time of the orders assigned into this batch. A common due date for all the orders in the same delivery batch and a delivery date for each order need to be determined in order to minimize total weighted flow time. We first prove that this problem is NP hard in the strong sense. Two optimal algorithms by using dynamic programming are derived for the two special cases with a given sequence of orders on the machine and a given batching in the production part, respectively.

A wide variety of practical problems are closely related to the batching production and delivery in batches considered in this paper. In a supply chain of many industries, production operation and delivery operation are two key operational functions. Many industries first produce products and transport their finished products to the customers directly without holding intermediate inventories. For example, there exist the operations with production of steel ingots and delivery of the finished ingots in the iron and steel industry. The steel ingots need to heat to a high temperature in a soaking pit in order to keep the steady performance of the steel ingots. The soaking pit heats several ingots simultaneously where the soaking pit can be viewed as a parallel-batching machine. Then the steady ingots finished in the soaking pit are directly transported to the corresponding customers in batches. This is the motivation for the considered problem. All ingots in the same shipment for delivery to the customer have a common due date. The common due date is a decision variable. Production operation and delivery operation are linked together without any intermediate step since no inventory is involved. The customer does not want to accept his order earlier than the due date. Thus, the orders are finished before the due date, and they have to be storage in inventory for the industries. It is important to coordinate these two operations and schedule them jointly in order to achieve optimal inventory level in the iron and steel industry. Hence, the coordination between production and delivery has become more practical and become one of the most important topics. In such supply chain application, it is well understood that coordinated production and delivery scheduling can significantly reduce the inventory level and the customer service level for the decision maker.

Motivated by the above-described applications in the supply chain, we address a coordinated scheduling problem of production on a single batching machine and delivery operation in batches. The orders are first processed on the single batching machine and then delivered to the customers directly without intermediate. In the production part, the production schedule specifies the processing sequencing and batching of the orders on the machine. Completed orders are delivered in batches to the customers by homogeneous vehicles. We assume that each delivery shipment can only carry up to a number of orders. The delivery schedule specifies how many batches to use, how to batch, and the delivery date of each batch from the machine. The production batches may be partitioned into delivery batches to determine the delivery date. A common due date for the same delivery batch has to be determined. A delivery batch has the same delivery date and the same due date. The problem is to find a coordinated schedule of production batching and delivery batching such that an objective function that takes into account the inventory level is optimized. The inventory level is measured by a function of the times when the finished orders are delivered to the customers. A common due date for the orders in the same delivery batch is the flow time of each order in this batch. Therefore, the objective is to minimize total weighted flow time of orders.

The described problem in this paper falls into two categories: batch delivery and batching scheduling. Many researchers have made contributions in the coordinated scheduling area with production and delivery. We briefly discuss some work related to the problems where the objective is to minimize the sum of the total flow time and delivery cost. Cheng and Kahlbacher [

Batching is an important feature of many practical industries in the supply chain. The parallel-batching scheduling problems considered in this paper are motivated by burn-in operations in the very large-scale integrated circuit manufacturing. The detail surveys for the batching scheduling problems can be seen in [

The problem formulated above combines three types of decisions: scheduling, batching, and due date assignment. The independent and simultaneously available orders have to be scheduled and partitioned into batches during processing and delivery. After the orders in a batch finished processing on the parallel-batching machine, they must be delivered to the customers in batches within limited delivery capacity. In a real supply chain, the managers may not stipulate the delivery for each specific order. In such a coordinated system, the linkage between order production scheduling and delivery dispatching of finished orders is extremely important in order to determine the common due date for each delivery batch. Our work differs from the above where we study not only the batching and sequencing schedule of orders on the parallel-batching machine, but also the batching for the finished orders in the delivery part where the delivery date and the common due date for each delivery batch need to be determined.

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

In this paper, a coordinated scheduling problem on a single parallel-batching machine with batch delivery is addressed in order to determine the common due date for each delivery. The problem can be described as follows.

All the orders

In the production part, all orders and the machine are available at time 0. The parallel-batching machine has a capacity limit

In the delivery part, each delivery shipment will transported by a dedicated vehicle. Due to limited vehicle capacity

The vehicle is stationed at the processing facility at time 0 and must go back to the facility once it finishes a delivery batch. A delivery vehicle can depart from the processing facility only when all the orders to be delivered have finished processing.

All orders are not interruption. When the processing of a batch is executed, it cannot be interrupted, and other orders cannot be introduced into the batch. Orders processed in a batch have the same starting time and completion time on the machine.

The production batches may be partitioned into delivery batches to determine their delivery dates. A common due date for the same delivery batch has to be determined. A delivery batch has the same delivery date and the same due date.

Next, we present the notation that will be used throughout the paper:

Decision variables are as follows:

Here, the flow times of the orders assigned into the same delivery batch are equal to the common due date of the delivery batch. For convenience, we denote our problem by (P). Next, we present three general properties that will be useful throughout our study.

There exists an optimal schedule that production batches of the orders are processed continuously from time 0 without idle time on the single parallel-batching machine.

If there is any idle time between two continuous batches in the production part, the latter batch can be moved earlier to be processed on the machine. The objective value is not increased.

There exists an optimal schedule where the delivery date of a delivery batch is either the completion time of the last order on the machine which is included in the delivery batch or the available time of the vehicle immediately.

Assume that there exists a delivery batch such that its delivery date does not fit any above condition. That is neither at the completion time of the last order on the machine which is included in the delivery batch nor the available time of the vehicle immediately. Then we can move the delivery batch such that this delivery date can fit either of those conditions. It is easy to see that this delivery batch can be transported at that earlier time to the customer without increasing the total weighted flow time.

The delivery batch must contain all orders which are processed on the machine but not delivered within the scope of the delivery capacity to allow.

The finished orders on the parallel-batching machine should be delivered to the customer as early as possible. The delivery batch should contain orders as many as possible which are processed on the machine but not delivered.

In this section, we show that problem (P) is strongly NP hard by a reduction from 3-partition problem, which is known to be NP hard in the strong sense (see Garey and Johnson [

The following theorem shows the computational complexity of problem (P).

Problem (P) is strongly NP hard.

We next perform a polynomial time reduction from 3-partition problem.

Given a 3-PP instance, we construct an instance for (P) as follows.

There are

Assume that

Processing times and weights of orders as follows:

threshold:

We prove there exists a schedule for this instance with total weighted flow time less than or equal to

Note that the second term is minimized by setting

In this section, we consider two special cases of problem (P): (1) the problem where the order sequence on the machine in the production part is predetermined and (2) the problem where the batching on the machine in the production part is predetermined. Case (1) occurs when direct sequence is the production strategy used for the customer, while case (2) occurs when direct batching is the production strategy used for the producer. In practice, a direct sequence strategy will be used when the customer’s demands are relatively high, while it is more likely that a direct batching will be used when the production cost is relatively high.

In this case of the problem, the sequence of orders on the batching machine is predetermined

Let

Let

Define

Dynamic programming is as follows.

Initial conditions are as follows:

Recurrence relations are as follows:

Optimal solution value is

Algorithm DP1 solves problem (P-case 1) in

Due to the predetermined order sequence, the batching on the machine, the batching, and the delivery date of orders for the delivery need to be decided. From Lemma

The time complexity of Algorithm DP1 can be established as follows. Due to

In this case of the problem, the production batching of orders on the machine is predetermined

For problem (P-case 2), there exists an optimal schedule in which the production batches are processed in the nondecreasing

Consider an optimal schedule

Now consider the schedule

Comparing terms, it is clear that total weighted flow time of

Based on Lemma

For problem (P-case 2), there exists an optimal schedule in which the orders in the same production batch are sequenced in the longest weight rule of orders in the delivery part.

Assume that there exists an optimal schedule

Therefore, regardless of whether

The following dynamic solves the case of (P-case 2).

Let

Let

Dynamic programming is as follows.

Initial conditions are as follows:

Recurrence relations are as follows:

Optimal solution value is

Algorithm DP2 can solve problem (P-case 2) in

By Lemmas

The time complexity of Algorithm DP2 can be established as follows. Sorting the production batches in the nondecreasing

In this paper, we propose a coordinated scheduling problem with production operation and delivery operation in the supply chain with batching scheduling and common due date assignment. The orders are processed on a single parallel-batching machine and then the finished orders are transported in batches to the customer. The parallel-batching machine has a limited capacity and the number of orders in each delivery batch cannot exceed the delivery capacity. All orders in the same delivery batch to the customer have a common due date. The common due date of each delivery batch needs to be determined. Inventory level in the supply chain is measured by a function of the common due date for each delivery batch when the completed orders are delivered to the customers. The objective is to minimize total weighted flow time. We first prove that this problem is NP hard in the strong sense. For the two special cases with a given sequence of orders on the machine and a given batching in the production part, we derive two optimal algorithms based on the dynamic programming, respectively. A natural extension of this work would consider approximation algorithms for the problem with the objective of the total weighted flow time. Another extension of this work would consider other objective functions.

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

This research is partly supported by the National Natural Science Foundation of China (Grant no. 71101097 and Grant no. 71071100), the National High Technology Research and Development Program of China (863, Grant no. 2014AA041401), and the Liaoning BaiQianWan Talents Program (Grant no. 2014921043).