Role of batch size in scheduling optimization of flexible manufacturing system using genetic algorithm
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Abstract
Flexible manufacturing system (FMS) readily addresses the dynamic needs of the customers in terms of variety and quality. At present, there is a need to produce a wide range of quality products in limited time span. Ontime delivery of customers’ orders is critical in maketoorder (MTO) manufacturing systems. The completion time of the orders depends on several factors including arrival rate, variability, and batch size, to name a few. Among those, batch size is a significant construct for effective scheduling of an FMS, as it directly affects completion time. On the other hand, constant batch size makes MTO less responsive to customers’ demands. In this paper, an FMS scheduling problem with n jobs and m machines is studied to minimize lateness in meeting due dates, with focus on the impact of batch size. The effect of batch size on completion time of the orders is investigated under following strategies: (1) constant batch size, (2) minimum part set, and (3) optimal batch size. A mathematical model is developed to optimize batch size considering completion time, lateness penalties and setup times. Scheduling of an FMS is not only a combinatorial optimization problem but also NPhard problem. Suitable solutions of such problems through exact methods are difficult. Hence, a metaheuristic Genetic algorithm is used to optimize scheduling of the FMS.
Keywords
Flexible manufacturing system (FMS) Scheduling optimization Batch size, due dates Completion time Genetic algorithm (GA)Introduction
An FMS is a highly automated production system consisting of a group of computer numerical control machine tools, linked by material handling system and controlled by a distributed control system. An FMS can simultaneously process medium variety and medium size volumes of products (Browne et al. 1984). An FMS is a sophisticated production system to respond dynamic variations of contemporary market including lead time reduction, flexibility to respond market variations and higher productivity (Atmani and Lashkari 1998). The major problems that an FMS has to face include designing, planning, scheduling and controlling. Among these, scheduling problem is a major challenge (Stecke 1985). An FMS’s scheduling differs from a conventional job shop scheduling due to routing flexibility of parts (Jain and Elmaraghy 1997).
Scheduling is the assignment of resources over time to perform tasks. In conventional scheduling system, only one resource is considered, which processes the parts. An FMS scheduling differs from conventional job shop scheduling. The complexity of FMS scheduling is due to the flexibility of an FMS in terms of machine, product, operation and routing (Browne et al. 1984). Scheduling of an FMS is effected by several factors including orders arrivals, due dates and batch size (Liu and MacCarthy 1996). The production management has to set policies whether to handle periodic or continuous orders of the customers. Meeting customers’ due dates play key role in make to order system and the FMSs are mostly suited for MTO system. Due dates are effected by various factors including arrival rate, variability and batch size. Among these factors, batch size is significant factor for effective scheduling of an FMS as it directly affects due dates. In addition, constant batch size makes MTO less responsive to customers’ demands (Beemsterboer et al. 2017).
Extensive literature has been studied to address scheduling of an FMS. The scheduling of an FMS is NPhard problems, and is solved by heuristics to find near optimal solution (Baker 1974). Close conformance to due dates is accomplished by minimizing mean and maximum tardiness. In MTO system, minimization of tardiness is significant. Nidhiry and Saravanan (2014) addressed FMS scheduling problem to minimize idle time of the machines and total penalty cost. Sixteen computer numericcontrol machine tools were used to process eighty different parts in an FMS. The results of the proposed heuristic were compared with cuckoo search and particle swarm optimization (PSO). The results depicted that the proposed approach outperformed cuckoo search and PSO.
Keung et al. (2003) proposed intelligent hierarchical control technique for an FMS. The paper aimed to optimize machine utilization as well as to balance capacity of tool magazine. The proposed model was assessed based on two benchmarks: earliness and tardiness penalty costs. Jerald et al. (2005) determined optimal Scheduling of three FMS with eight, fifteen and sixteen machines. The objective was to optimize idleness of the machine and penalty costs. Saravanan and Haq (2008) applied scattersearch approach to optimize FMS scheduling, considering multiple objectives, including minimization of machine idle time and total penalty costs in case of exceeding the due dates.
An important performance measure to determine utilization of an FMS is in optimizing makespan and completion time. Chan et al. (2005) addressed an FMS problem in two different situations. In first case, only three machines were under consideration, whereas in second scenario thirtythree machines were accommodated. The objective was to optimize makespan. Reddy and Rao (2006) addressed scheduling of machines as well as Automated Guided Vehicles (AGVs) simultaneously in an FMS to optimize makespan and mean flow time. The proposed approach was practiced for an FMS having six machines with two guided vehicles. The results depicted that the presented algorithm has diverse solutions. Kim et al. (2007) solved scheduling problems in an FMS with the objective to optimize makespan in an FMS having ten machines. The results obtained were compared with other techniques and it was found that the proposed algorithm outperformed in terms of quality of solution and convergence speed. Multimode Resource Constrained Project Scheduling Problem was solved using hybrid genetic algorithm. Feasible schedules were based on the minimum project completion time (Lova et al. 2009).
Udhayakumar and Kumanan (2010) performed multiobjective scheduling of tasks of AGVs using ant colony optimization (ACO). The objective was to maximize AGVs utilization which has also been validated on several problems. Machines and AGVs scheduling in flexible manufacturing system was also addressed by Chaudhry et al. (2011) using genetic algorithm spreadsheets to minimize total completion time. Zhang et al. (2012) proposed a model for flexible job shop scheduling considering transportation constraints and bounded processing time to minimize makespan and storage. Different types of problems were tested including sequencing as well as bounded processing times. The proposed model outperformed said types of scheduling problems. Raj et al. (2014) presented combined machines and tools scheduling for multimachine in an FMS to achieve best optimal sequences with the aim to minimize makespan. A metaheuristic GA was developed to minimize makespan of the scheduling problem. The parameters of the GA were determined by Taguchi orthogonal array. Analysis of variance was performed to examine significant factors affecting makespan (Candan and Yazgan 2015).
In MTO system, batch size plays a significant role for effective scheduling of an FMS due to its direct effect on completion time. Furthermore, constant batch size makes MTO less responsive to customers’ demands. A few researchers have addressed batch size problem in an FMS scheduling. Machine utilization of a flexible manufacturing cell was improved using hierarchic approach. In first stage, batches of parts were determined. In second stage, batches were sequenced and scheduled. A fuzzybased approach was proposed to solve machine loading problem in an FMS. The sequence of jobs was determined by evaluating the contribution of each job in terms of batch size and operation processing time (Vidyarthi and Tiwari 2001). Cheng et al. (2012) developed a mixed integer linear programming model to address scheduling problem of parallel batch processing of jobs to minimize makespan and completion time using polynomial time algorithm. Geng and Yuan (2018) investigated the unbounded parallel batching machine to minimize makespan and completion time using pareto optimization. Matin et al. (2017) proposed PSO to optimize batch size in flow shop to minimize makespan and completion time. Different batch size compositions were used on machines and an optimal batch size of all jobs on each machine was obtained. Ying and Lin (2018) addressed hybrid flow shop scheduling problem to minimize makespan using selftuning iterated greedy algorithm.
Scheduling of an FMS is NPhard problem and it is more complex as compared to classical job shop scheduling problems. Different heuristicbased approaches including cuckoo search, PSO, GA and artificial neural network have been used in an FMS scheduling (Sankar et al. 2003; Jerald et al. 2005; Burnwal and Deb 2013; Sadeghian and Sadeghian 2016). A hybrid multiobjective GA was presented to optimize makespan, AGV travel time and penalty cost due to jobs lateness. The proposed algorithm proved feasibility and effectiveness for all the objectives (Umar et al. 2015). A nondominated sorting biogeographybased approach to solve multiloading–unloading problem in an FMS was addressed to minimize makespan and total earliness (Rifai et al. 2018).
The main research gap identified in the literature is the lack of considering the effect of batch size on due dates and makespan in MTO system in an FMS scheduling. In this paper, an FMS scheduling problem is addressed to minimize lateness in meeting due dates. A mathematical model is developed for optimizing the batch size. GA is used to optimize lateness of each order and penalty cost. Then, multiobjective Pareto optimization is performed for three batch size strategies to evaluate its effect on completion time. These strategies are then compared with each other to investigate completion time.
The rest of the paper is organized as follows. Section 2 discusses mathematical model description. Section 3 gives insight of technique used and Sect. 4 presents results and discussion. In Sect. 5, conclusions and future work recommendations are presented.
Description of model
Machine distribution in each FMC
Flexible manufacturing cell (FMC)  Machines 

FMC1  1 
FMC2  3.6 
FMC3  5 
FMC4  2.4 
Assumptions

There are five different parts to be processed ensuring the presence of multitools in tool magazine.

Every part has a predefined processing sequence incorporating batch size, due dates and penalty costs in case of not meeting due dates. The processing time of every part is known and deterministic.

Every machine can perform only one operation at a time.

The sequence of operations for all parts is predetermined.

The sequencedependent setup times are considered.
Notations
Model’s notations
Notations  

k  Position of batch in sequence k = 1, 2, 3, …, K 
i, j  Parts i, j \(\in\) {1, 2, 3, …, I} 
m  Machine number m = 1, 2, 3, …, M 
o  Order number o = 1, 2, …O 
q _{ i}  Batch size of part i 
n _{ i}  Number of batches of part i 
d _{ i}  Demand of part i 
N  Total number of batches 
PC_{o}  Penalty cost of an order o 
LT_{o}  Lateness of an order o 
CT_{i}  Completion time of part i 
ST_{i}  Starting assembly time of an order in a sequence of part i 
RL_{o}  Release of an order o 
CT_{o}  Completion time of an order o 
DR_{o}  Estimated duration of an order o 
DDo  Due date of an order o 
MPS  Minimum part set 
Mathematical model
Equation (4) shows that one part i cannot be assigned to more than one machine. In Eq. (5), \({\text{CT}}_{o}\) shows completion time of order o and \({\text{DD}}_{o}\) is the corresponding due dates, \({\text{LT}}_{o}\) guarantees that lateness is a positive value. Equation (6) shows that demand of each part i must be satisfied. Total number of batches can be computed by Eq. (7). Equation (8) shows that starting time of an order o in sequence of part i must be synchronized with order release date. Equation (9) ensures that order production in sequence i + 1 can only be started once order in production sequence i is completed. Equation (10) represents completion time of order of part i.
Genetic algorithm
Genetic algorithm introduced by Holland was inspired by the concept of natural and biological evolution (Sivasankaran and Shahabudeen 2014). GA has proven a powerful tool for finding the near optimal solution of combinatorial problems that cannot be handled by exact methods due to their complexity (Zandieh et al. 2010). To apply genetic algorithm to solve real life optimization problems, two basic issues must be addressed: (1) Encoding the solution in the form of chromosomes and (2) fitness function evaluation (Hsu et al. 2005). In GA, the chromosomes encode candidate to find optimum results. Each result is represented by binary numbers as a string of 0 and 1 but other types of encodings are also possible in the GA (Chan et al. 2006). Brief description of the GA implementation procedure is presented in the following section.
GA implementation
Chromosome representation
Initial population
Fitness function evaluation
Each chromosome is evaluated against fitness function. Fitness function minimizes maximum lateness and completion time. Fitness function of each chromosome is calculated to choose best parents among all other chromosomes.
Crossover and mutation
Results and discussion
Processing time of parts on machines
M1  M2  M3  M4  M5  M6  

A  0  2  5  7  2  0 
B  3  2  0  4  0  3 
C  1  3  0  2  5  1 
D  4  0  4  0  1  5 
E  2  5  4  6  1  3 
Sequencedependent setup times
A  B  C  D  E  

A  0  1  3  5  8 
B  2  0  3  4  6 
C  2  5  0  2  3 
D  4  7  2  0  8 
E  3  8  2  5  0 
The details of the orders of the customers
Order 1  Order 2  Order 3  Order 4  Order 5  

Order release date  4  1  6  10  14 
Duration  3  5  4  3  5 
Due date  12  8  13  15  22 
Penalty cost ($/day)  100  150  200  300  100 
Quantity A  10  20  30  10  30 
Quantity B  20  10  20  30  20 
Quantity C  0  30  10  10  10 
Quantity D  30  10  10  20  0 
Quantity E  20  20  40  0  10 
Optimization of lateness
Parameters values used in the GA
Sr. no.  Parameters  Parameters values 

1  Population size  300 
2  Generations  500 
3  Crossover probability  0.8 
4  Mutation probability  0.2 
Optimal scheduling of orders using the GA
2  1  3  4  5  

Start date  1  6  9  13  16 
Completion time  6  9  13  16  21 
Due date  8  12  13  15  22 
Lateness  0  0  0  1  0 
Penalty cost  0  0  0  $300  0 
Total penalty costs  $300 
In the second stage, the effect of batch size is investigated on total completion time of jobs. Three strategies are opted and then compared to analyze its effect on completion time. The strategies considered are: (1) Constant batch size, (2) Minimum Part Set, and (3) Variable batch size.
Effect of constant batch size on total completion time
Scenario#1: Effect of batch size on total completion time and setup time
Batch size  Number of batches  Optimum sequence  Completion time (Min)  Setup time (Min) 

Batch size = 5  A = 4, B = 2, C = 6, D = 2, E = 4  AAAAACCCCCCCCCCCCCCC AAAAAAAAAAAAAAABBBBBBBBBB CCCCCDDDDDDDDDDCCCCCCCCCC EEEEEEEEEEEEEEEEEEEE  364  96 
BBBBBBBBBBCCCCCCCCCC AAAAAAAAAAAAAAAAAAAA CCCCCCCCCCDDDDDDDDDD CCCCCCCCCCEEEEEEEEEEEEEEEEEEEE  382  90  
CCCCCCCCCCCCCCCCCCCC AAAAAAAAAAAAAAAAAAAA BBBBBBBBBBCCCCCDDDDDDDDDD CCCCCEEEEEEEEEEEEEEEEEEEE  398  78  
BBBBBBBBBBAAAAAAAAAA AAAAAAAAAACCCCCCCCCC CCCCCCCCCCDDDDDDDDDD CCCCCCCCCCEEEEEEEEEEEEEEEEEEEE  407  72  
AAAAAAAAAAAAAAAAAAAA BBBBBBBBBBCCCCCCCCCC CCCCCCCCCCCCCCCDDDDDDDDDD CCCCCEEEEEEEEEEEEEEEEEEEE  423  66 
Scenario #2: Effect of constant batch size on total completion time and setup time
Batch size  Number of batches  Optimum sequence  Completion time (Min)  Setup time (Min) 

Batch size = 10  A = 2, B = 1, C = 3, D = 1, E = 2  AAAAAAAAAABBBBBBBBBB DDDDDDDDDDCCCCCCCCCC CCCCCCCCCCCCCCCCCCCC EEEEEEEEEEEEEEEEEEEEAAAAAAAAAA  362  78 
AAAAAAAAAAAAAAAAAAAA BBBBBBBBBBDDDDDDDDDD CCCCCCCCCCCCCCCCCCCC EEEEEEEEEEEEEEEEEEEECCCCCCCCCC  393  72  
AAAAAAAAAAAAAAAAAAAA BBBBBBBBBBDDDDDDDDDD CCCCCCCCCCCCCCCCCCCC CCCCCCCCCCEEEEEEEEEEEEEEEEEEEE  401  60  
EEEEEEEEEEEEEEEEEEEE CCCCCCCCCCCCCCCCCCCC CCCCCCCCCCAAAAAAAAAAAAAAAAAAAA BBBBBBBBBBDDDDDDDDDD  429  60 
Effect of minimum part set strategy on total completion time
Effect of MPS strategy on total completion time
Sr. No.  MPS sequence  Optimum sequence  Completion time (Min)  Setup time (Min) 

1  AABCCCDEE  EECCCAABD  51  54 
2  EECDCCAAB  46  54  
3  AABDCCCEE  42  60  
4  AABDCCECE  41  90  
5  AACBDCCEE  40  102 
Total completion time and setup time for complete order
Sr. No.  Optimum sequence  Completion time (Min)  Setup time (Min) 

1  EECCCAABD  459  486 
2  EECDCCAAB  414  486 
3  AABDCCCEE  378  540 
4  AABDCCECE  369  810 
5  AACBDCCEE  360  918 
The results show that as completion time decreases, setup times drastically increase. The reason is that, in MPS strategy, a great number of setups are required as compared to batch production. For the sequence 3, completion time is 378 while setup time is 540 min. Even for the tradeoff solution, setup times are much higher than constant batch size results.
Effect of variable batch size on total completion time
Results of variable batch sizes
Batch  Optimum batch size  Optimum sequence  Completion time (Min)  Setup time (Min) 

1  A = (1,20) B = (2,5) C = (5,6) D = (10,1) E = (2,10)  AAAAAAAAAAAAAAAAAAAA BBBBBBBBBBCCCCCCCCCCCCCCC DDDDDDDDDDCCCCCCCCCC EEEEEEEEEECCCCC  353  78 
2  A = (2,10) B = (10,1) C = (1,30) D = (5,2) E = (2,10)  AACCCCCCCCCCAAAAAAAAAAAAAA CCCAAAABBBBBBBBBBCCCCCCC DDDDDCCCCCDDDDDCCCCC EEEEEEEEEEEEEEEEEEEE  359  150 
3  A = (5,4) B = (1,10) C = (10,3) D = (2,5) E = (5,4)  AAAAABBBBBCCCCCCCCCC AAAAAAAAAAAAAAABBBBB DDDDDDDDCCCCCCCCCCEEEEE DDCCCCCCCCCCEEEEEEEEEEEEEEE  364  156 
4  A = (10,2) B = (5,2) C = (10,3) D = (5,2) E = (10,2)  AAAAAAAAAACCCCCCCCCC CCCCCCCCCCAAAAAAAAAA BBBBBBBBBBDDDDDDDDDD CCCCCCCCCCEEEEEEEEEE EEEEEEEEEE  364  90 
5  A = (5,4) B = (2,5) C = (5,6) D = (10,1) E = (5,4)  AAAAABBBBBBCCCCCCCCCC CCCCCAAAAAAAAAAAAAAA BBBBDDDDDDDDDDCCCCC CCCCCCCCCCEEEEEEEEEE EEEEEEEEEE  364  96 
Comparison of three strategies of batch size
Conclusion
This research has presented an effort to optimize scheduling with objective of minimizing lateness and penalty costs in an FMS. To fulfill on time delivery of orders, the suitable batch size strategy selection and execution is significant as batch size directly affects completion time. Different batch sizes yield different completion time and sequencedependent setup times. Multiobjective Pareto optimization using GA is performed to investigate and compare the results of three batch size strategies: (1) constant batch size of 5 and 10. (2) MPS strategy (3) variable batch size. In first scenario, the batch size of each part is kept 5 and then 10. The larger batch size shows better results. Then, MPS strategy is addressed and results show that setup times are too high for the minimum value of completion time. In the last scenario, variable batch size is used to optimize conflicting objectives. All possible batch sizes of each part are evaluated using the GA and five best results are discussed. The results show that variable batch size strategy exhibits superior results as compared to constant and MPS batch size strategies. This work significantly contributes to an FMS to opt suitable batch size strategy in various demand patterns.
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