Study on disruption management scheduling problem of flow shop under supply chain environment

This paper presents a disruption scheduling model for an environment of proportional two-machine no-wait flow shop. To achieve the objects of minimization of weighted sum of makespan and minimization of weighted sum of tardiness, we introduce a revised PSO algorithm which is designed with a neighborhood search structure. According to the experiment, the effectivity of the method proposed is proven.


Introduction
In reality, unexpected events are often inevitable to cause disruption to flow shop system.They lead to failure of system control.Therefore, we need to carry out a scheduling recovery plan to minimize the lost.
Scheduling recovery problem is practically useful in production and processing.Many of them are proven NPhard problems [1] and the problem in this research is one of them.There are abundant researches about the global static scheduling and periodic rolling scheduling [2].However, random disruption is more common in production and processing practical environment.In disruption management problem, we should pay more attention to the combination of the initial scheduling objective and the new scheduling plan instead of global optimization.Lee [3] proposes two methods for unfinished jobs: one is to arrange them to other machines with extra cost and another is to wait till the recovery.Liou [4] have studied the disruption management problem in a single machine.Bo [5] establishes the scheduling model based on SPT rule.
This paper studies the disruption management recovery problem in the environment of proportional two-machine no-wait flow shop.We propose a disruption management model pred-mgt, and solve it with the HDPSO algorithm.A case study has been done to testify our research.

Problem description
In this research, we assume that that in the supply chain, there is a supplier known as M1 and a manufacturer known as M2.A job is processed only once at either M1 or M2, and it must be processed continuously in the supply chain.M1 or M2 can process only one job at a time, and a job's process time at M1 is proportional to that at M2.The process time of any job at M2 is longer than that at M1.The job being disrupted has to be reprocessed from the start of the supply chain.The processing environment can be named as proportional two-machine no-wait flow shop environment.The following notations are used : ), S means feasible process schedule, means the set of feasible process schedule.

The initial scheduling scheme
The objective of the initial scheduling scheme is the minimization of weighted sum of makespan, which is shown as . It has been proven that with the WSPT rule, the objective can be accomplished.Therefore, the initial scheduling can be described as F nwt F S in which the best process schedule isS , and the objective value is

Disruption
In reality, unexpected events may cause disruption in supply chain.Some of them like power outages can impact all the members of the supply chain.Under this S for the initial scheduling scheme will be not the best even infeasible.Hence, we need to reschedule the remaining jobs ^' 1,2, , , , In figure 1, the first half shows the initial scheduling scheme S , while the second half shows the disruption management strategy with the interference of disruption.The disruption occurs during 1 2 t t , and the duration is notated as t t

G
. The start time after the disruption is set as 0 t .We can see that job 4 and the jobs after are divided into two parts, one is processed before the disruption while the other after.Obviously, the condition ^1 0 1 2 j j t min t p p j Jc !+ + | should be met, so that certain job or jobs can be scheduled in the window between job 3 and the disruption.The remaining jobs defined as a job set J c are renumbered from 1 to nc according to the WSPT rule.
In this situation, two objectives are taken consideration.One is the initial objective, the minimization of weighted sum of makespan , and the other is the recovery objective, the minimization of weighted sum of tardiness

Problem Model
The remaining jobs influenced by the disruption should be divided into two parts.Therefore, we define . . 1

A solution based on HDPSO algorithm
In this research, we propose the Hybrid Discrete Particle Swarm Optimization (HDPSO) algorithm to solve the pred mgt model.The basic thought of the HDPSO algorithm is described as follow: Initialization: The position of every particle in the population is represented by a ' n -dimensional vector 1 2 randomly initialized with an even-distributed random number in (0,1).
Particle evolution: The update formula of particle position consists of i P t (pbest), g P t (gbest) and N f (neighborhood search strategy).The particle evolution strategy is designed as follow: In the evolution strategy: as the part that they cross with each other, otherwise, the position of particle remains unchanged

Neighborhood search design
To make up the weakness of the local search of HDPSO algorithm, we propose the neighborhood search operator.
There are two neighborhood structures in this research, insert and swap.
Insert: Choose a job processed before 1 t and a job processed after 2 t randomly.Insert the latter one into the position of the former one, and the jobs between them all move afterwards for one position.
Swap: Choose a job processed before 1 t and a job processed after 2 t randomly.Swap them with each other.
After the implementation of the two neighbourhood structures, if in the new order, the completion time of jobs before 1 t is less than 1 t and the objective evaluation becomes better, the operation of insert and swap is allowed, otherwise it's prohibited.

Experiment design
In this research, processing speed of 1 M and 2 M is set The number of initial jobs ready to be processed is 50.Jobs are sorted with the WSPT rule.

Experiments analysis
In this study, we use six classic indicators to evaluate the performance of the algorithm.For AQ, HDPSO algorithm gives better consideration of approximation and dispersion than HDPSO1 algorithm.
To sum up, HDPSO algorithm has a better comprehensive performance than HDPSO1 algorithm.In other words, neighborhood search structure has a great improvement of the algorithm to solve this problem.

Conclusion
In this study, a recovery model for disruption in proportional two-machine no-wait flow shop environment has been developed, in order to minimize the weighted sum of makespan and the weighted sum of tardiness.It's solved using the HDPSO algorithm which is revised from PSO algorithm.Moreover, a neighborhood search structure with insert and swap is developed to improve the performance of the algorithm.
The environment and disruption mentioned in our study is less practical than that in real life like that disruptions do occur synchronously and so on.These conditions need our further research in the future.
the scheduling problem with disruption can be shown as following with three-parametric method:

1 jp
is set as a random production time array, so 2 j p is easy to get.The weight of jobs is w set as a random weight array.During the case, six group experiments are carried out.They are differentiated with the different disruption window.For each group, 10 times of experiments are carried out.To demonstrate the usefulness of neighborhood search, we compare the HDPSO algorithm above with the HDPSO1 algorithm (a special HDPSO algorithm without neighborhood search).The parameters of the algorithm: the number of particles in the population is 40.The probability of particle mutation

Table . 1
They are ONVG, CM, Algorithms performance indicator comparison