Batch Scheduling for Work Centers with Multiple Manufacturing Machines under the Parallel-Sequence-transfer Mode

This paper, based on the parallel-sequence-transfer mode of batch jobs, aimed to implement research on the batch scheduling for work centers with multiple manufacturing machines in full detail. In order to arrange a precise product planning with batch jobs, obtain the maximum completing time with batch jobs, schedule manufacturing machines in each work center accurately, and get the actual number of manufacturing machines used to processing batch jobs in each work center, a new processing procedure of the processing time model under the parallel-sequence-transfer mode was designed. The new designed processing procedure was tested on a numeric example. The results show that the research in this thesis could not only arrange a precise production planning under the parallel-sequence-transfer mode, but also schedule manufacturing machines for each work center with multiple manufacturing machines accurately.


Introduction
The entire production planning management for modern manufacturing enterprises could be grouped into three categories: short-term planning management, medium-term planning management and longterm planning management [1].The workshop management and control of modern manufacturing enterprises is the core part of short-term planning management.The subsystem of the workshop management and control contains three key components: production planning, production scheduling and production monitoring.The production planning and scheduling is the emphasis and difficulty of the workshop management and control.The basic unit for assigning tasks of the workshop and arranging a detailed operational planning is work center.Work center that is composed of one or several identical manufacturing machines is the logical entity of the processing unit to perform the specific processing task [2].Making a production planning and scheduling is by nature an extremely difficult mission under such the circumstance that there are a lot of manufacturing machines in the production workshop, especially one certain work center is constituted by a few of identical manufacturing machines [3].In view of the difficulty of solving this issue, the batch scheduling for work centers with multiple manufacturing machines is our main work in this paper.The batch scheduling for work centers with multiple manufacturing machines is not only associated with the process routing of processing batch jobs, but also associated with operational organization modes that are employed to transfer batch jobs between each pair of work centers.Operational organization modes can be divided into three modes: sequence-transfer mode, parallel-transfer mode, and parallelsequence-transfer mode.In real-world manufacturing system, however, the sequence-transfer mode that could generate the longest maximum completing time of batch jobs is typically used to arrange a batch scheduling for work centers with multiple manufacturing machines.Thus, this paper concentrates on the parallel-sequence-transfer mode of batch jobs to improve the utilization of work centers as well as reduce the maximum completing time with batch jobs.
Scholars at home and abroad set out to research these academic fields earlier.The planning and scheduling problems of modern manufacturing enterprises can be classified into three levels: strategic level, factory level, and work center level.Moreover, the term of work center is a very significant concept in ERP system [4,5].Hence, there are a large number of research achievements on the scheduling of work center [6,7].The research achievements in the aspects of scheduling are as follows: Huo and Zhao [8] addressed a preemptive bi-criteria scheduling problem for m parallel machines with the objective of minimizing the total completing time subjecting to the makespan constraint.Jiang et al. [9] studied the distributed job shop scheduling problem, integrated multiple initialization rules, and proposed a collaborative search method to obtain three local reinforcement heuristic algorithms for the problem.Zhang et al. [10] established a multi-objective mixed integer linear programming model for the FJSP, aiming to minimize maximum completing time and total energy consumption.Homayouni et al. [11] constructed a mixed integer linear programming model considering transportation time for problem with transporting workpieces between processes.Zhang et al. [12] presented three formulas of makespan for a manual processing system under three operational organization modes.In operational organization modes research, Kong et al. [13][14][15] have been obtained a series of research achievements, including processing time models and handling time models of batch jobs, production scheduling of batch jobs and optimization algorithms of handling equipments and more.The above research achievements only dealt with the work centers with unitary manufacturing machine, but did not take the work centers with multiple manufacturing machines into account.And this together with the parallel-sequence-transfer mode is the main innovation of this paper.
The remainder of this paper is organized as follows.The problem description, assumptions, and notations are introduced in section 2. In section 3, a novel processing procedure of processing time model is designed.In section 4, a numeric example and the detailed analysis are performed, including the specific example and details, the precise production planning of batch jobs and the accurate schedule of manufacturing machines in each work center, the utilization of work centers with multiple manufacturing machines, and the comparison about the maximum completing time for batch jobs and utilization of each work center between three different operational organization modes.This paper ends with some conclusions and further research direction.

Problem Description, Assumptions and Notations
The research concentrates on the work centers with multiple manufacturing machines and the parallelsequence-transfer mode of batch jobs.Both an accurate production planning of batch jobs and a precise schedule of manufacturing machines in all work centers are to be obtained using the processing procedure of the processing time model designed in this paper.

Problem Description
There are n identical jobs that are to be processed in a set of work centers The i th work center includes i e manufacturing machines that are also identical.All jobs are to be processed under the parallel-sequence-transfer mode in the same sequence, from work center 1 WC to m WC .The objectives interested in this paper are to arrange a precise production planning of batch jobs under the parallel-sequence-transfer mode on the foundation of minimizing maximum completing time and schedule manufacturing machines for each work center reasonably on the premise of maximizing the utilization of work centers.
Because the parallel-sequence-transfer mode of batch jobs in previous literature was not defined in the context of this study, hence, an exact definition of the parallel-sequence-transfer mode of batch jobs in the work centers with multiple manufacturing machines is to be provided in full detail.
The parallel-sequence-transfer mode of batch jobs is a type of batch jobs' transfer pattern between each pair of work centers with multiple manufacturing machines in which some finished jobs in one certain work center will be transferred to the next adjacent work center to ensure continuous operation of each manufacturing machine in all work centers.The purposes of this operational organization mode is to minimize the maximum completing time of batch jobs and maximize utilization of each work center on the foundation of continuous operation.

Assumptions
Some requisite assumptions are treated in more detail as follows: • There are a few of identical manufacturing machines in one certain work center and all jobs are also the same type.• Batch jobs get available at time zero.
• All parameters are known without uncertainty, including the total number of batch jobs, the total number of work centers, the processing times and more.• The processing of manufacturing one job is non-preemptive.In other words, the processing of manufacturing one job is not interrupted until the completion of the job.• Two operations that are to be processed on the same manufacturing machine cannot overlap in time.
• Each job must be processed by only one manufacturing machine in one certain work center.
• The process routing of processing batch jobs could be attained from the process routing sheet.
• All manufacturing machines in each work center could run continuously without failure.
• The schedule of batch jobs in each work center is not left-justified when optimizing.

Notations
The following notations for this problem will be used throughout this paper.m : total number of work centers for processing identical batch jobs.i : index of work centers, {1, 2, , } im  .
n : total number of batch jobs that are to be processed.
. i e : maximum number of manufacturing machines belonging to the i th work center.
A i e : actual number of manufacturing machines used to perform the processing task in the i th work center, A ii ee  .k : index of manufacturing machines used to perform the processing task in the i th work center, {1, 2, , }  C : maximum completing time of batch jobs under the parallel-sequence-transfer mode in the i th work center.z   : round up to the smallest integer that is not less than z .i N : row vector of the total number of batch jobs processed on each manufacturing machine in the i th work center.
( , ) ik n : the total number of batch jobs processed on k th manufacturing machine in the i th work center.

P B
T : matrix of starting times for processing each job in all work centers.

P E
T : matrix of ending times for processing each job in all work centers.Step 1-2: Judge of , then go to Step 1-3.
Step 1-7: Judge of Step 2: The processing procedure of the intermediate processing time model in the i th ( 2 im  )

work center
Step 2-1: Judge of i : if im  , then go to Step 2-2; if im = , then set max ( , ) and the processing procedure of processing time model for batch jobs under the parallel-sequence-transfer in work centers with multiple manufacturing machines is terminated.
Step 2-11: Judge of k : If Step 3: The processing procedure of the processing time model in the i th ( 2 im  ) work center Step 3-1: , and then go to Step 3-2.

Specific Example and Details
There are ten identical jobs that are to be processed in a set of work centers , respectively.Time units of the processing times in all work centers are minutes.

The Precise Production Planning of Batch Jobs and the Accurate Schedule of Manufacturing Machines in Each Work Center
Using the processing procedure of the processing time model designed in section 3, we can arrange a precise production planning for batch jobs and obtain maximum completing time for batch jobs under the parallel-sequence-transfer mode.
The information of the actual number of manufacturing machines used to perform the processing task in the i th work center and the maximum completing time of batch jobs under the parallel- sequence-transfer mode are presented in table 1 in more detail.The information of the total number of batch jobs processed on each manufacturing machine used to perform the procession task in all work centers are derived from the processing procedure of the processing time model.
The accurate schedule of manufacturing machines used to perform the processing task in all work centers can also be obtained based on the processing procedure of the processing time model.
The working horizon of the first manufacturing machine in first work center is [0, 20].The working horizons of the second and third manufacturing machines in first work center are both [0,15].
The working horizons of the first and second manufacturing machines in second work center are both [8,23].
The working horizons of the first and the second manufacturing machines in third work center are both [11,32].The working horizons of the third and fourth manufacturing machines in third work center are both [18,32].
The working horizons of the first and the second manufacturing machines in fourth work center are both [28,34].The working horizons of the third and fourth manufacturing machines in fourth work center are both [30,34].

Utilization of Work Centers with Multiple Manufacturing Machines
For work centers with multiple manufacturing machines, the formula for calculating the utilization of the i th work center can be expressed by: 100% The utilization of each work center and the corresponding information are listed in table 2.

Comparison about Maximum completing time with Batch Jobs and the Utilization of Each Work Center among Three Different Operational Organization Modes
The most typical operational organization mode used in real-world manufacturing system is the sequence-transfer mode because of the simplification for arranging a production planning and controlling execution.However, maximum completing time with batch jobs under this mode is particular long, which leads to the poor responsiveness of customer requirements.In order to minimize the maximum completing time of batch jobs, the parallel-transfer mode could be employed, which leads to the poor utilization of each work center and a larger number of equipment shutdowns.The parallel-sequence-transfer mode that combines with the advantages of the above two operational organization modes can yield a better effect.
Table 3 shows the results with respect to the maximum completing time of batch jobs obtained for each operational organization mode.Table 3.The results with respect to the maximum completing time of batch jobs obtained for each operational organization mode.

Operational organization mode
The maximum completing time/(min) The percentage of the maximum completing time compared to the one under the sequence-transfer mode sequence-transfer mode 62 100% parallel-transfer mode 32 51.61% parallel-sequence-transfer mode 34 54.84% Figure 1 shows the results with respect to the utilization of each work center obtained for each operational organization mode.From the information of table 3 and figure 1, we could find the facts that the parallel-sequencetransfer mode could greatly reduce maximum completing time of batch jobs compared to the sequence-transfer mode and improve the utilization of work centers with multiple manufacturing machines compared to the parallel-transfer mode.

Conclusion
The work centers with multiple manufacturing machines have received lots of attention in the research society and have been also extensively used in the real-world manufacturing system.Some successful experiences show that the parallel-sequence-transfer mode can reduce the maximum completing time for batch jobs and improve the utilization of work centers with multiple manufacturing machines.Based on the above two ideas, a new processing procedure of processing time model under the parallel-sequence-transfer mode is designed to arrange a precise production planning with batch jobs, obtain the maximum completing time of batch jobs, schedule manufacturing machines in each work center accurately, and get the actual number of manufacturing machines used to processing batch jobs There is still a problem that is worthy of further discussion.It is the fact that only the identical batch jobs are considered, different types of batch jobs should also be taken into account.This problem should be further researched and explored.
j th processing task of the k th manufacturing machine.for processing the j th job in the i th work center.(, , )k P i j k B t: starting time of the j th processing task on the k th manufacturing machine in the i th work center.P i t : processing time per unit on any manufacturing machine in the i th work center.

3 . 1 :
of the i th work center for processing batch jobs.i P WC ET : ending time of the i th work center for processing batch jobs.Procedure of the Processing Time Model under the Parallel-sequence-transfer Mode A new processing procedure of the processing time model for identical batch jobs under the parallelsequence-transfer in work centers with multiple manufacturing machines can be grouped into three sub-steps: the processing procedures of the processing time model in the first work center, the intermediate processing time model in the i th ( 2 im  )work center and the processing time model in the i th ( 2 im  )work center.Step The processing procedure of the processing time model in the first work center Step 1otherwise.And then go to Step 1-2.
) k PP i j k E i j E tt = , and then judge of j : go to Step 1-4 if jn  , and let 1 k = and then go to Step 1-7 otherwise.
1 k = , and compare the values of n with i e : let A ii ee = if i ne  , and set A i en = otherwise.And then go to Step 2-3.

Step 2 - 5 :
and then judge of j : go to Step 2-5 if jn  , and let 1 k = and then go to Step 2-11 otherwise.Judge of k : If

.
The row vectors of the processing time per unit in all work centers and the maximum number of manufacturing machines belonging to the i th work center are

Figure 1 .
Figure 1.The results with respect to the utilization of each work center obtained for each operational organization mode.
mode parallel-transfer mode parallel-sequence-transfer mode in each work center.Finally, a numeric example is used to test the feasibility of the processing procedure of the processing time model designed in this paper.
and then go to Step 3-4.: if jn  , let jj = ; if jn  , let A i ke  , then go to Step 3-5; if A i ke  , and then go toStep 2.n

Table 1 .
The information of A i e and max C .

Table 2 .
The utilization of each work center and the corresponding information.