New scheduling rules for a dynamic flexible flow line problem with sequencedependent setup times
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Abstract
In the literature, the application of multiobjective dynamic scheduling problem and simple priority rules are widely studied. Although these rules are not efficient enough due to simplicity and lack of general insight, composite dispatching rules have a very suitable performance because they result from experiments. In this paper, a dynamic flexible flow line problem with sequencedependent setup times is studied. The objective of the problem is minimization of mean flow time and mean tardiness. A 0–1 mixed integer model of the problem is formulated. Since the problem is NPhard, four new composite dispatching rules are proposed to solve it by applying genetic programming framework and choosing proper operators. Furthermore, a discreteevent simulation model is made to examine the performances of scheduling rules considering four new heuristic rules and the six adapted heuristic rules from the literature. It is clear from the experimental results that composite dispatching rules that are formed from genetic programming have a better performance in minimization of mean flow time and mean tardiness than others.
Keywords
Scheduling Dynamic flexible flow line Simulation Heuristics Genetic programmingIntroduction
Scheduling involves the allocation of resources over a period of time to perform a collection of tasks (Baker 1974). It is a decisionmaking process that plays an important role in most manufacturing and service industries (Pinedo 1995). The hybrid flow line (HFL) scheduling problem is defined in the literature, e.g., Kianfar et al. (2012) and GómezGasquet et al. (2012). In HFL, there are g stages and there is at least one stage with more than one machine where the jobs arrive continuously during time and pass the stages sequentially from stage one through g with the same order. If the jobs skip some stages, it is called flexible flow line (FFL) scheduling problem, e.g., Kurz and Askin (2004), Quadt and Kuhn (2005) and Kia et al. (2010).
Salvador (1973) proposed for the first time a definition of the HFL problem for minimizing makespan. He presented a branch and bound method to solve the problem. A doublestage hybrid flow shop problem, with one machine in stage two, is examined for minimizing makespan by Gupta (1988). He proved that the problem was NPhard and developed a heuristic rule for it. Therefore, FFL is NPhard. Sawik (1993) proposed a heuristic rule to minimize makespan for a limited buffer FFL problem and later proposed a new rule for the same problem with no inprocess buffer (Sawik 1995). Kia et al. (2010) proposed two new scheduling rules with sequencedependent setup times (SDST) considering nonzero job arrival times for a dynamic FFL problem. Although several papers have been written on the extent of hybrid flow shop and hybrid flow line, most of them are limited to a special case of a double stage (e.g., Gupta 1988; Guinet et al. 1996) or to a particular framework of machines in each stage (e.g., Kochhar and Morris 1987; Sawik 1993, Sawik 2002; Mirabi et al. 2014; MalekiDarounkolaei et al. 2012). The papers that were in FFL with nonzero job arrival times focused on the heuristic rules that were rarely seen, except Kia et al. (2010). Jolai et al. (2012) proposed a novel hybrid metaheuristic rule in FFL with SDST. A new dispatching rule for twostage FFL is proposed by Li et al. (2013). Finally, a comprehensive survey in scheduling problems is presented by Allahverdi (2015) and also Neufeld et al. (2016).
In this paper, scheduling rules are studied to solve a dynamic flexible flow line scheduling problem with SDST using simulation. We used genetic programming to create new composite dispatching rules and a discreteevent model that examined the performance of scheduling rules in terms of mean flow time and mean tardiness.
The rest of the paper is organized as follows. The definition of the problem is given in “Problem definition”. “Problem mathematical modeling” introduces a mathematical model for considering the problem. The next section “Scheduling rules” presents those rules adapted from the literature. The genetic programming framework that is illustrated for generating composite dispatching rules is represented in “Genetic programming”. The next section is the “Simulation model”, and in “Experiment design” the details of experiments are designed and presented for scheduling rules. “Experiment results” provide the results and analyses, and the final section is the “Conclusion”.
Problem definition

The number of stages is g and also there is at least one stage with more than one machine.

All jobs visit all stages through stage g, while skipping some stages is possible.

Machines that are placed in each stage are identical.

All machines are always available.

Preemption is not allowed.

There is no buffer constraint.

A machine can process at most one job at a time and a job can be processed by at most one machine at a time.

No job can be processed in one stage, except when the processing had been completed on the previous stage.

All jobs are not in the system from the beginning, but they can continuously enter over time.

The sequencedependent setup time is assumed for every job on each machine.

The setup time of every job on each machine is sequence dependent.

When a job comes into the system, its characteristics, i.e., processing times in each stage, setup times and due dates, are identified.

There is no priority between jobs.

There is no machine breakdown.
Problem mathematical modeling
In this section, the problem mathematical model is presented. The problem model is 0–1 mixed integer linear programming.
Notation
 n:

Number of jobs.
 \(i, j\):

Index of the number of job, \(i,\, j = 1, 2, \ldots , n\).
 g:

Number of stages in the shop.
 l:

Index of the stage, \(l = 1, 2, \ldots , g\).
 m ^{ l }:

Number of parallel machines at stage l.
 k:

Index of the machine, \(k = 1, 2, \ldots , m^{l}\).
 r _{ i }:

Arrival time of job i.
 d _{ i }:

Due date of job i.
 p _{ i } ^{ l } :

Processing time of job i at stage l.
 s _{ ij } ^{ l } :

Setup time from job i to job j at stage l.
 C _{ i } ^{ l } :

Completion time of job i at stage l.
 T _{ i }:

Tardiness of job i.
 F _{ i }:

Flow time of job i.
 X _{ ijkl }:

1 if job j is processed immediately after job i on machine k at stage l, otherwise 0.
Mathematical model
Eq. (1) shows the linear convex combination of the dual criteria problem. The objective function is minimization of mean flow time and mean tardiness. The constraints (2) and (3) calculate the mean flow time and mean tardiness value, respectively. The constraints (4) and (5) guarantee assigning only one job to each sequence position at each stage. Furthermore, the constraints (6) and (7) guarantee assigning one job to the first and last sequence position on each machine at each stage, respectively. It is clear that job 0 and job (n + 1) are not real jobs and just stated for formularization. The constraint (8) ensures that each job at each stage is processed once. The constraint (9) forces consistent sequence at each stage. The constraint (10) ensures that job processing cannot be started before release time of the job at the first stage. The constraint (11) forces that just at the first stage, the completion time for each job cannot be less than the sum of the release, processing time and setup time of that job. The constraint (12) states that at each stage on the particular machine, starting the processing of the next job before completing the previous job is not possible. The constraint (13) states that for each job, its processing at the next stage cannot be started before completing it at the previous stage. The constraint (14) states that completion time for each job at each stage is nonnegative. The constraint (15) determines the tardiness value for each job and constraint (16) states that the tardiness is nonnegative. The constraint (17) determines the flow time value for each job and, finally, the constraint (18) shows that the problem variables are binary.
Scheduling rules

Simple priority rules (SPR): These rules usually consist of just one parameter and are suitable for singleobjective problems such as process time and due date.

Composite dispatching rules (CDR): These rules consist of the application of a combination of several SPRs, and when the machine becomes free then this CDR evaluates the queue and then chooses a job with the most priority level for processing on the machine.
If the CDR is made well, then it is proper for solving real multiobjective problems. In the literature, e.g., Barman (1997), it is clear that CDRs have a better performance than SPRs. Furthermore, Jayamohan and Rajendran (2000) stated that there were no rules with a good performance considering flow time and due date. So, we intend to indicate the efficiency of the proposed new CDRs and also compare them with the six scheduling rules in Kia et al. (2010) that are presented by considering two objectives of mean flow time and mean tardiness for the DFFL environment. The six scheduling rules in Kia et al. (2010) that are adapted for this study are as follows:
Hybrid shortest processing time and cyclic heuristics (HSPTCH): In this rule at each stage whenever a machine is free at the first step, jobs waiting in the queue for processing on the machine are arranged according to SPT. In the second step, a job that has minimum completion time on the machine relative to the other jobs waiting in the queue, according to the sequencedependent setup time of that job, is allocated to the machine.
Hybrid least work remaining and cyclic heuristics (HLWKRCH): In this rule at each stage whenever a machine is free at the first step, jobs waiting in the queue for processing on the machine are arranged according to the least total remaining process time. In the second step, jobs possessing minimum completion time on the machine, in relation to the other jobs waiting in the queue, is allocated to the machine according to the sequencedependent setup time of that job.
Hybrid earliest modified due date and cyclic heuristics (HEMDDCH): In this rule at each stage whenever a machine becomes free at the first step, jobs waiting in the queue for processing in the machine are arranged according to EMDD. In the second step, a job having the least completion time on the machine, in relation to the other jobs waiting in the queue, is allocated to the machine, according to the sequencedependent setup time of that job.
Hybrid Wilkerson & Irvin and cyclic heuristic (HW&ICH): In this rule at each stage whenever a machine is free at the first step, jobs waiting in the queue for processing in the machine are arranged according to W&I. In the second step, a job having the least completion time on the machine in relation to the other jobs waiting in the queue is allocated to the machine according to the sequencedependent setup time of that job.
In this paper, we focus on a computational method to make an effective CDR to solve a DFFL problem by a suitable algebraic combination of SPRs, but due to the width of the operator and parameter space, CDR’s efficiency evaluation is very difficult in comparison with applying SPRs manually. So, we used genetic programming to evaluate it.
Genetic programming
Genetic programming (GP) is one of the evaluation computing methods based on the survival and reproduction principle (Koza 1992). Each individual, i.e., computer program, is a syntax tree in the random initial population produced in GP including a set of function and terminals; thus, it is essential to select the function and terminal set accurately to create proper CDRs for solving DFFL problems. The function and terminal set and GP parameter setting are stated in the two following subsections.
Function and terminal set
There are various function and terminal sets that can affect the results’ quality and efficiency. Each of these terminals includes a dispatching rule that only a few of them are used due to the reduction of the search space.
Terminal set
Terminal  Terminal meaning 

ReleaseDate  Release date of a job (RD) 
DueDate  Due date of a job (DD) 
ProcessingTime  Processing time of a job for each operation (PT) 
CurrentTime  Current time (CT) 
RemainingTime  Remaining processing time of each job (RT) 
avgTotalProcTime  Average total processing time of each job (aTPT) 
Function set
Function  Function meaning 

+  Addition 
−  Subtraction 
×  Multiplication 
/  Division 
ADF(x1,x2)  Automatically defined function 
avgTotalProcTime  Average total processing time 
GP parameter setting
Choice of parameter values for GP
Parameter  Parameter values 

Population size  100 
Number of generations  200 
Creation type  Ramped half and half 
Maximum depth for creation  5 
Maximum depth for crossover  15 
Crossover probability  90% 
Swap mutation probability  3% 
Shrink mutation probability  3% 
Number of best rules copy to new generation  4 
It generates the first half randomly with a maximum depth of 5 and the second half with a variable depth between 1 to maximum depth. The population (rules) size is 100 and we generate it 200 times. We maintain variation via crossover, mutation, and the creation type ramped half and half. At each time, we arrange the generated population based on the performance measurement and then copy the four best rules in the following population to be preserved and not to be deleted in the next generation. The information of parameter values is summarized in Table 3.
Simulation model
Conditions like random arrival times, machine breakdowns and due date changes state a dynamic scheduling environment. The simulation is one of the ways to analyze the dynamic environment. In this paper, the arrival time of jobs is random, so a DFFL environment is present. A developed discreteevent simulation model is presented to evaluate the four best CDRs and the six heuristic rules. We use C++ programming language on the PC with 2.2 MHz CPU and 512 MB RAM.
Simulation parameters
Parameter  Value 

Number of jobs (n)  1450 
Number of stages in the shop (g)  8 
Number of parallel machines at stage l (m^{l})  Uniform [3–5] 
Processing time of job i at stage l (p _{ i } ^{ l } )  Uniform [20–60] 
Setup time of job i before job j at stage l (s _{ ij } ^{ l } )  Uniform between [4–12] and [8–24] 
Skipping probability of each job at each stage  0.00, 0.05 and 0.40 
Mean of interarrival time (a)  9–10 
 (I)
Initialization.
 (II)
Job data generation.
 (III)
Timing.
 (IV)
Events.
 (V)
Scheduling.
 (VI)
Status.
 (VII)
Report.
 Step 1:
Run module (I) and set type of scheduling rule.
 Step 2:
Run module (II).
 Step 3:
Run module (III) and set t = 0.
 Step 4:
Run module (IV) to determine the next event and advanced simulation time to the next event time.
 Step 5:
Run module (V) [according to the selected scheduling rule in step (I)] to schedule/reschedule the system.
 Step 6:
Run module (VI) to analyze the status of the model.
If the termination condition is not met, go to step 3.
 Step 7:
Run module (VII) to report the computational results.
Experiment design
Experimental settings for the scenarios
Scenario  Experimental setting  Purpose of investigation  

Shop utilization percentage (U)  Setup time ratio (%) (s)  
DFFLI  95  20  Base case—analyze the performance of scheduling rules 
DFFLII  95  20, 40  Analyze the effect of changing setup time ratio 
DFFLIII  95, 85  20  Analyze the effect of changing the mean interarrival time 
Experiment results
GPgenerated dispatching rules
Rule  Rule expression 

GP1  RD + 5PT + 2aTPT 
GP2  7aTPT + 11PT + 12RD 
GP3  3RD + 2DD + 3aTPT + PT2RD 
GP4  2DD + 8RD + 2aTPT−5PT 
As mentioned above, all experiments are conducted in ten iterations and the results of the scenarios are stated in the following three subsections considering the performance measurements.
Scenario results for the overall objective function
Overall objective function values for different scenarios
EMDD  W&I  HSPTCH  HLWKRCH  HEMDDCH  HW&ICH  GP1  GP2  GP3  GP4  

DFFLI  700.9  701.0  717.4  698.2  691.3  693.2  647.5  646.7  647.2  647.1 
DFFLII  854.3  852.8  860.1  861.3  833.9  834.1  832.9  832.7  833.3  833.2 
DFFLIII  638.0  637.4  657.0  633.3  628.3  629.4  627.0  626.2  626.9  626.5 
Scenarios’ results for the mean flow time
Mean flow time values for different scenarios
EMDD  W&I  HSPTCH  HLWKRCH  HEMDDCH  HW&ICH  GP1  GP2  GP3  GP4  

DFFLI  3502.7  3502.9  2966.8  2947.8  3454.7  3464.2  3235.5  3231.3  3234.2  3233.3 
DFFLII  4269.3  4261.8  3500.7  3567.1  4167.2  4168.1  4162.2  4161.5  4164.3  4164.0 
DFFLIII  3187.9  3185.1  2732.4  2683.6  3139.5  3145.2  3132.9  3129.1  3132.7  3130.3 
Scenario’s results for mean tardiness
Mean tardiness values for different scenarios
EMDD  W&I  HSPTCH  HLWKRCH  HEMDDCH  HW&ICH  GP1  GP2  GP3  GP4  

DFFLI  0.518  0.525  155.0  135.8  0.503  0.505  0.502  0.501  0.503  0.503 
DFFLII  0.585  0.611  200.0  184.9  0.574  0.590  0.570  0.561  0.568  0.569 
DFFLIII  0.495  0.505  138.1  120.7  0.499  0.506  0.500  0.495  0.498  0.496 
Conclusion
In this paper, we used the GP framework to obtain proper and effective CDRs for solving a DFFL problem. Finally, we created four evolved CDRs and proposed them for solving the DFFL problem. Experimental results indicated that the four proposed CDRs with 95% CI obtained a better solution in comparison with selected scheduling rules from the literature.
For future research, it is possible to consider the variation of the terminal set and ADF to develop the GP framework. It is also possible to use GP to find proper and effective CDRs for different objectives. Furthermore, it is useful to apply GP for other scheduling environments such as job shop or open shop and dynamic assumptions like machine breakdowns.
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