- Email: [email protected]
High-Mix Low-Volume Flow Shop Manufacturing System Scheduling
Proceedings of the 14th IFAC Symposium on Information Control Problems in Manufacturing Bucharest, Romania, May 23-25, 2012
High-Mix Low-Volume Flow Shop Manufacturing System Scheduling Juraj Svancara, Zdenka Kralova Institute of Control and Industrial Informatics Faculty of Electrical Engineering and Information Technology Slovak University of Technology in Bratislava Ilkovičova 3, 812 19 Bratislava, Slovak Republic e-mail: [email protected] Abstract: The paper deals with the production scheduling optimization in High-Mix Low-Volume (HMLV) flow shop manufacturing system. Three new heuristic production scheduling algorithms balancing the machines assignment to the production orders are proposed. The goal of the algorithms is to find a production schedule that minimizes the total production time of the set of jobs - makespan. An example case study, where the proposed scheduling algorithms are verified and compared with basic solutions using the simulation model of the HMLV manufacturing system created in Witness simulation software, is presented. Keywords: Production scheduling, HMLV manufacturing systems, Flow shop, Simulation 2. PRODUCTION SCHEDULING IN HMLV
1. INTRODUCTION Customization of final products and almost unpredictable behaviour of market demands significantly affect the process of production operative control, planning and scheduling. The main objective of production management is to increase the adaptability of production system resources. This process is complex as it involves various inputs such as resource utilization, resource limitation, product variety, product characteristics, and market demand (Neoh, 2004). Thus, production planning and scheduling play an important role in production resources adaptability optimization especially in High-Mix Low-Volume (HMLV) manufacturing environments which need to produce more product variety with lower volume in the dynamic production environment. For HMLV manufacturing environments, product type variety can be as high as 600 different products or greater, and incoming order quantities may range anywhere from 01000 for individual products within the mix over a particular time horizon (Mahoney, 1997). Products are typically segmented into product families - products arranged into groups according to the identical PCB - printed circuit board.
Production scheduling is the process of assigning manufacturing resources over the time to the set of manufacturing processes in the process plan. It determines the most appropriate time to execute each operation, taking into account the temporal relationships between manufacturing processes and the capacity limitations of the shared manufacturing resources. The assignments also affect the optimality of a schedule with respect to criteria such as cost, tardiness, or throughput (Shen, 2006). In summary, production scheduling is an optimization process where limited resources are allocated over the time horizon among both parallel and sequential activities and its goal is to optimize one or more objectives. This optimization process is becoming increasingly important for HMLV manufacturing enterprises to increase their productivity and profitability through greater shop floor agility to survive in a globally competitive market. The primary objectives of scheduling in the HMLV manufacturing environments are as follows (Mahoney, 1997):
This paper provides three new heuristic production scheduling algorithms that minimize the total production time in HMLV manufacturing systems considering variety of product families and randomly changing market demand. The rest of the paper is organised as follows. Production scheduling in HMLV systems is given in Section 2. Problem formulation is given in Section 3. Section 4 provides the algorithm proposals and Section 5 algorithms implementation. Experimental case study is presented in Section 6. Finally, conclusions are summarised in Section 7.
978-3-902661-98-2/12/$20.00 © 2012 IFAC
•
on time delivery
•
minimum lead times
•
minimum total production time - makespan
•
maximum production resources utilization
Production scheduling deals with two main sub-problems: 1. Assign the operations to the specific production resources. This objective is known as the production loading (forward loading, backward loading). The goal of the loading is to specify the production resources capacity and to split the capacity utilization evenly. Loading uses rough processing and setup times and gives only rough production schedule.
145
10.3182/20120523-3-RO-2023.00130
INCOM 2012, May 23-25, 2012 Bucharest, Romania
2. Specify the sequence in which the operations are processed. This objective is known as the production sequencing. In this step of production scheduling the exact processing and setup times are used and an accurate production schedule for short time horizon is proposed. A well-known production scheduling is the classical problem where a set of jobs and a set of machines are given. Each machine can handle at most one job at a time. Each job consists of a chain of operations that needs to be processed during an uninterrupted time period of given length on a given machine. The purpose is to find the best schedule, i.e., an allocation of the operations to time intervals on the machines that has the minimum total duration required for completion of all of the jobs. The total number of possible solutions for a classical scheduling problem with n jobs and m machines is ݊Ǩ .
characteristics in behavior of a system, as it is capable of combining the relevant elements of the system according to the actual logic of the operations, which can help reflect the real behavior of the system (Guan, 2010). Simulation analysis has been proved as a necessity by several studies, as due to the highly uncertain manufacturing environments, it is hard to build mathematical models, which consider the random character of manufacturing systems (Bley, 2000, Svancara 2010, Zulch, 2002).
3. PROBLEM FORMULATION We consider HMLV flow shop manufacturing system with K stages in series, at stage s, s=1...K, there are Ms identical machines in parallel and an intermediate buffer between each of the production stages. (Fig. 1)
The problem becomes even more complex in the following situations (Shen, 2006). 1. When other manufacturing resources, such as operators and tools, are also considered during the scheduling process. For a classical job shop scheduling problem with n jobs, m machines, and k operators, the total number of possible solutions could be ሺ݊Ǩ ሻ . 2. When both process planning and manufacturing scheduling are to be done at the same time. Traditional approaches that treat process planning and manufacturing scheduling separately can result in suboptimal solutions for the two phases. Integrating the two phases into one optimization problem, by considering the constraints of both domains simultaneously, can theoretically result in a global optimal solution, but it increases the solution space significantly. 3. When unforeseen dynamic situations are considered. In a flow shop manufacturing environment, rarely do things go as expected. The system may be asked to include additional tasks that are not anticipated, or to adapt to changes to several tasks, or to neglect certain tasks. The resources available for performing tasks are subject to changes. Certain resources can become unavailable, and additional resources can be introduced. The beginning time and the processing time of a task are also subject to variations. A task can take more or less time than anticipated, and tasks can arrive early or late. Other uncertainties include power system failures, machine failures, operator absence, and unavailability of tools and materials. An optimal schedule, generated after considerable effort, may rapidly become unacceptable because of unforeseen dynamic situations on the shop floor and a new schedule may have to be generated. This kind of rescheduling problem is also called dynamic scheduling or real-time scheduling. Traditionally, analytical models are used in solving optimization and analytical problems in manufacturing systems. However, for the complexity and randomness inherent nowadays in the HMLV systems, it is difficult to establish accurate analytical model for them. Simulation technology can be an effective alternative in studying the
Fig. 1 Manufacturing system scheme We consider 10 different product families. Production orders are divided into batches - jobs. The processing times, batch sizes and setup times of jobs at the various stages depend on the actual product family number. Setup of the machine is required when the family number of job processing on the machine is changed. Setup times increase according to the number of machines processing jobs of the same product family (this fact is caused by material distribution complexity increasing). For the purpose of production scheduling (loading) optimization considering the high-mix and low-volume production orders divided into product families three new scheduling algorithms have been proposed. The main idea of the algorithms is balancing assignment of the machines to the production orders (families) according to the processing times or/and batch sizes or/and order sizes. The result of this assignment process is the creation of production streams, each representing one product family, that reduce the machines setup requirements. The goal of the algorithms is to find a production schedule that minimizes the total production time of the set of jobs - makespan. This idea can be illustrated by the following example. Consider 5 different production orders from 5 different product families and 6 production stages. Set of the production orders and specific icons of machines assigned to the production orders are depicted in Table 1.
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Table 1. Production orders
INCOM 2012, May 23-25, 2012 Bucharest, Romania
Scheduling algorithm assigns machines on each of the stages to the production orders that left to be processed on these stages according to the selected criterion. Step 1, Time = 0
4. ALGORITHM PROPOSALS The scheduling problem may be described in general as follows: K - number of production stages, s = 1…K
All production orders left to be processed on each of the stages. The production schedule (assignment of the machines to the production orders) in Time = 0 is depicted in Figure 2.
F - number of product families, f = 1…F Ms - number of machines at stage s ܱ - production order of product family f ܱ௦ - number of products of production order f that left to be processed on stage s ܬ - job - production orders divided into production batches, i = 1…Of / Bf ܤ - batch size of production order of product family f ܶ௦ - processing time of production order of product family f on stage s ܣ௦ - number of machines at stage s assigned to the jobs of product family f Since the priorities of the families are equal an assumption M s > F , ∀s have been made to ensure that at least one job
Fig. 2 Production schedule in Time = 0
of each family can be produced at each stage at a time. Step n, Time = N Production schedule is recalculated each time when new production order enters to the production system or when any of the production orders is completely processed at any of the stages. Production order number 3 is completely processed at stages number 1, 2, 3 and 4 and left to be processed at stages 5 and 6. Production order number 2 is completely processed at stages number 1, 2 and 3 and left to be processed at stages 4, 5 and 6. All other production orders still have to be processed at stages 1,2,3,4,5 and 6. Recalculated production schedule in Time = N is depicted in Figure 3.
Considering processing times, batch sizes and order sizes of the production orders, three different algorithms have been proposed. 1. Processing times algorithm This algorithm balances the assignment of the machines according to the processing times of production orders.
T fs A fs = round F M s ∀f , f ∈ F , O fs > 0 T ∑ fs f =1
(1)
Since the processing times are not integer values, proposed algorithm consider round value of the equation result. This fact can caused unequality between sum of the machines assigned to the production orders and total number of machines at stage s. For the purpose of equalization between sum of the machines assigned to the production orders and total number of machines, following sub-algorithms are used. F
If
∑A
T fs
fs
< M s then " Afs = A fs + 1" , f ,
= max
(2)
fs
> M s then " Afs = A fs − 1" , f , A fs = max
(3)
f =1
Afs
F
Fig. 3 Production schedule in Time = N
If
∑A f =1
Number of machines assigned to the specific production order is increased for the production stream, where Tfs / Afs is maximal or decreased, where Afs is minimal. The process of
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INCOM 2012, May 23-25, 2012 Bucharest, Romania
equalization assignment ends when sum of the machines assigned to the production orders is equal to the total number of machines. 2. Batch size algorithm This algorithm balances the assignment of the machines according to the processing times and batch sizes of production orders.
BT f fs Afs = round F M s ∀f , f ∈ F , O fs > 0 ∑ B f T fs f =1 F
If
∑A
B f T fs
fs
< M s then " Afs = A fs + 1" , f ,
fs
> M s then " Afs = A fs − 1" , f , A fs = max
Afs
f =1
Fig. 4 Graphical representation of the simulation model in Witness
(4)
Block diagram of the algorithms implementation is depicted in Figure 5.
= max (5)
F
If
∑A
(6)
f =1
3. Production order size algorithm This algorithm balances the assignment of the machines according to the processing times, batch sizes and number of products from a specific product family that left to be processed on the stages.
BT O f fs fs A fs = round F M s ∀f , f ∈ F , O fs > 0 BT O ∑ f fs fs f =1 F
If
∑A
fs
< M s then " Afs = A fs + 1" , f ,
B f Tfs O fs
f =1
Afs
(7)
Fig. 5 Scheduling algorithm implementation
= max (8) 6. CASE STUDY
F
If
∑A
fs
> M s then " Afs = A fs − 1" , f , A fs = max
In the example case study we have considered 10 production stages and 10 product families. Processing times of jobs from product families A-J at production stages 1-10 are shown in table 2, number of machines at each stage in table 3 and production orders are summarized in table 4. All input data were obtained from real manufacturing system.
(9)
f =1
5. IMPLEMENTATION For the purpose of the scheduling algorithms verification, the proposed algorithms have been implemented to the simulation model of HMLV flow shop manufacturing system created in Witness 3.0 simulation software. Simulation model considers the random character of processing and setup times by ± 10 % deviations, which make the simulation results even more credible. Production schedules are calculated using the functions, automatically modified according to the current algorithm chosen by user. Graphical representation of the simulation model in Witness simulation software is depicted in Figure 4 (example with 6 production stages).
148
Table 2. Processing times O\S
1
2
3
4
5
6
7
8
9
10
A
10.5
10.2
1.2
9.6
4.4
4.9
4.4
3.6
5.4
4
B
13.2
16
1.1
12.7
5.2
3.9
5.1
4.2
3.6
4.2
C
18.5
15.4
1.2
9.6
2.7
4.8
3.7
3.8
2.8
5.4
D
15
14.7
1.1
10.8
4.9
2.8
4.7
3.8
3.5
3.9
E
12.3
9.6
1.2
9
4.7
3.4
4.9
3.9
3.6
5.4
F
15.6
9.7
1.2
9.7
3.3
2.4
3.3
4.6
3.5
4.4
G
9.4
12
1.2
15
4
4.6
2.7
2.8
3.1
5.2
INCOM 2012, May 23-25, 2012 Bucharest, Romania
H
8.4
10.8
1.6
13.2
6.7
4.2
5.7
3.9
4.1
4.8
I
10.2
17.4
1.1
11.1
8.5
5.6
8.5
4.1
3.7
4.9
J
9.2
7.4
1.6
7.8
6.2
4.9
3.6
3.9
4.2
5.3
Sum of the machines assigned to the production orders is 28. Number of machines at stage 1 is 29. This unequality will be solved using equation (2) considering Tf1 / Af1 values shown in Table 6.
Table 3. Number of machines
Table 6. Tfs / Afs
S
1
2
3
4
5
6
7
8
9
10
M
29
30
11
27
19
18
18
17
18
18
T11/A11
T21/A21
T31/A31
T41/A41
T51/A51
T61/A61
2,6
3,3
3,1
3
3,1
3,1
Table 4. Production orders Number of machines assigned to the product family B is increased by 1.
Product family
Time of arrival
Order size
Product family
Time of arrival
Order size
A
0
250
B
500
280
B
0
300
H
500
190
Final assignment of the machines at stage 1 to the production orders (families) in TIME = 0:
C
0
280
G
500
390
Table 7. Final machine assignments (Stage 1)
D
0
190
J
500
400
E
0
350
F
500
260
F
0
450
D
600
240
A
200
200
E
600
180
J
200
260
G
600
230
A11
A21
A31
A41
A51
A61
4
5
6
5
4
5
I
200
150
H
600
150
Final assignment of the machines at all stages to the production orders (families) in TIME = 0:
C
500
120
I
600
170
Table 8. Final machine assignments
A
500
310
B
600
230
In the following example we have used Processing times algorithm and input data as shown in Tables 1,2 and 3. Assignment of the machines at stage 1 to the production orders (families) in TIME = 0: Using equation (1) for product family A:
T A11 = round F 11 M 1 T ∑ f1 f =1 10, 5 .29 85,1
O\S
1
2
3
4
5
6
7
8
9
10
A
4
4
2
4
3
4
3
2
4
3
B
5
6
2
6
4
3
4
3
3
3
C
6
6
2
4
2
4
3
3
2
3
D
5
6
1
5
4
2
3
3
3
3
E
4
4
2
4
3
3
3
3
3
3
F
5
4
2
4
3
2
2
3
3
3
G
0
0
0
0
0
0
0
0
0
0
H
0
0
0
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0
0
0
0
J
0
0
0
0
0
0
0
0
0
0
Simulation experiments have been run using five different production scheduling approaches (1-5):
A11 = round
1 - Processing times algorithm
A11 = 4
2 - Batch size algorithm 3 - Production order size algorithm
Analogically for all of the product families where ܱ௦ Ͳ. Table 5. Machine assignments (Stage 1)
4 - Basic schedule 1 5 - Basic schedule 2
A11
A21
A31
A41
A51
A61
4
4
6
5
4
5
Basic schedules assign free machines to the first available job in the buffer. Job sequencing in buffers is random, when Basic schedules are used.
149
INCOM 2012, May 23-25, 2012 Bucharest, Romania
Four different sets of fifty simulation exper periments have been run. Average values of the total processing ng times (makespan) for 5 different scheduling approachess in four different experiment sets are summarized in Figuree 6. 6
ling algorithms were compared where the proposed scheduling with basic solutions, has been presented. pr
ACKNOWLE LEDGMENTS This work was supported by the th Scientific Grant Agency of the Ministry of Education of the th Slovak Republic Grant No. 1/1241/12.
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REFERE RENCES Bley, H.; Franke, C.; Ostermann, A., Methods and Tools for thee Building of Huge Simulation Models, Proceedings off the t 2nd CIRP International Seminar on Intelligent Computation Co in Manufacturing Engineering (ICME 2000),, 2000. 2
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Fig. 6 Total processing time (mak akespan) From the results in Figure 6 it is obviouss that th average values of the production makespan obtainedd using production schedules calculated by the proposed algor gorithms were better than values obtained using basic schedules les in all cases. All the results obtained using proposed algorithms alg are very similar but production schedule calculated ted by Production order size algorithm ensured the best value lue of the production makespan in all of the experiments. Proce cessing times of the production orders and batch sizes were ere identical in all experiments. For that reason all the savi aving of production makespan was caused only by redu eduction of setup requirements. Comparison of the setup req requirements in four experiments sets is shown in Figure 7.
Mahoney, R.M.: High-Mix Low-Volume L manufacturing. Hewlett-Packard Company, y, 1997. Neoh, S.C., Morad, N., Lim, Li Ch. P., Abdul-Aziz, Z.: Optimization of Product Mix ix Planning in High-Mix-LowVolume Industries Usingg Genetic G Algorithms, WSEAS Transactions on Systems, Issue Is 7, Volume 3, pp 25412545, 2004. Shen, W., Wang, L., Hao,, Q.: Q Agent-Based Distributed Manufacturing Process Plan lanning and Scheduling: A State of the Art Survey, IEEE Tr Transactions on Systems, Man and Cybernetics, Vol. 36, No.4, No 2006.
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Guan, Z., Wang, Ch., Huan ang, j., Wan, L., Shao, X., Optimization of manufactu cturing systems using genetic search and multi-resoluti ution simulation, 8th IEEE International Conference on Control and Automation, pp 1473-1480, 2010.
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Svancara, J., Kralova, Z.: Case se study on simulation analysis of a multiple product manuf nufacturing system. Proceedings of the 5th IFAC Internationa nal Conference on Management and Control of Production on and Logistics, MCPL 2010, Coimbra, Portugal.
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Zulch, G., Johnson, U., Fischer, er, J.: Hierarchical simulation of complex production systems ms by coupling of models, Int. J. Production Economics 77, 77 2002.
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Fig. 7 Setup requirements nts 7. CONCLUSION The paper deals with the production schedu eduling optimization in High-Mix Low-Volume flow shop manu nufacturing systems. Three new heuristic production scheduling ling algorithms have been proposed and implemented in the simulation sim model of the HMLV manufacturing system crea reated in Witness simulation software. All three algorith rithms balance the machines assignment to the productionn orders (families) according to the selected criterion. An example exa case study,
150
High-Mix Low-Volume Flow Shop Manufacturing System Scheduling Juraj Svancara, Zdenka Kralova Institute of Control and Industrial Informatics Faculty of Electrical Engineering and Information Technology Slovak University of Technology in Bratislava Ilkovičova 3, 812 19 Bratislava, Slovak Republic e-mail: [email protected] Abstract: The paper deals with the production scheduling optimization in High-Mix Low-Volume (HMLV) flow shop manufacturing system. Three new heuristic production scheduling algorithms balancing the machines assignment to the production orders are proposed. The goal of the algorithms is to find a production schedule that minimizes the total production time of the set of jobs - makespan. An example case study, where the proposed scheduling algorithms are verified and compared with basic solutions using the simulation model of the HMLV manufacturing system created in Witness simulation software, is presented. Keywords: Production scheduling, HMLV manufacturing systems, Flow shop, Simulation 2. PRODUCTION SCHEDULING IN HMLV
1. INTRODUCTION Customization of final products and almost unpredictable behaviour of market demands significantly affect the process of production operative control, planning and scheduling. The main objective of production management is to increase the adaptability of production system resources. This process is complex as it involves various inputs such as resource utilization, resource limitation, product variety, product characteristics, and market demand (Neoh, 2004). Thus, production planning and scheduling play an important role in production resources adaptability optimization especially in High-Mix Low-Volume (HMLV) manufacturing environments which need to produce more product variety with lower volume in the dynamic production environment. For HMLV manufacturing environments, product type variety can be as high as 600 different products or greater, and incoming order quantities may range anywhere from 01000 for individual products within the mix over a particular time horizon (Mahoney, 1997). Products are typically segmented into product families - products arranged into groups according to the identical PCB - printed circuit board.
Production scheduling is the process of assigning manufacturing resources over the time to the set of manufacturing processes in the process plan. It determines the most appropriate time to execute each operation, taking into account the temporal relationships between manufacturing processes and the capacity limitations of the shared manufacturing resources. The assignments also affect the optimality of a schedule with respect to criteria such as cost, tardiness, or throughput (Shen, 2006). In summary, production scheduling is an optimization process where limited resources are allocated over the time horizon among both parallel and sequential activities and its goal is to optimize one or more objectives. This optimization process is becoming increasingly important for HMLV manufacturing enterprises to increase their productivity and profitability through greater shop floor agility to survive in a globally competitive market. The primary objectives of scheduling in the HMLV manufacturing environments are as follows (Mahoney, 1997):
This paper provides three new heuristic production scheduling algorithms that minimize the total production time in HMLV manufacturing systems considering variety of product families and randomly changing market demand. The rest of the paper is organised as follows. Production scheduling in HMLV systems is given in Section 2. Problem formulation is given in Section 3. Section 4 provides the algorithm proposals and Section 5 algorithms implementation. Experimental case study is presented in Section 6. Finally, conclusions are summarised in Section 7.
978-3-902661-98-2/12/$20.00 © 2012 IFAC
•
on time delivery
•
minimum lead times
•
minimum total production time - makespan
•
maximum production resources utilization
Production scheduling deals with two main sub-problems: 1. Assign the operations to the specific production resources. This objective is known as the production loading (forward loading, backward loading). The goal of the loading is to specify the production resources capacity and to split the capacity utilization evenly. Loading uses rough processing and setup times and gives only rough production schedule.
145
10.3182/20120523-3-RO-2023.00130
INCOM 2012, May 23-25, 2012 Bucharest, Romania
2. Specify the sequence in which the operations are processed. This objective is known as the production sequencing. In this step of production scheduling the exact processing and setup times are used and an accurate production schedule for short time horizon is proposed. A well-known production scheduling is the classical problem where a set of jobs and a set of machines are given. Each machine can handle at most one job at a time. Each job consists of a chain of operations that needs to be processed during an uninterrupted time period of given length on a given machine. The purpose is to find the best schedule, i.e., an allocation of the operations to time intervals on the machines that has the minimum total duration required for completion of all of the jobs. The total number of possible solutions for a classical scheduling problem with n jobs and m machines is ݊Ǩ .
characteristics in behavior of a system, as it is capable of combining the relevant elements of the system according to the actual logic of the operations, which can help reflect the real behavior of the system (Guan, 2010). Simulation analysis has been proved as a necessity by several studies, as due to the highly uncertain manufacturing environments, it is hard to build mathematical models, which consider the random character of manufacturing systems (Bley, 2000, Svancara 2010, Zulch, 2002).
3. PROBLEM FORMULATION We consider HMLV flow shop manufacturing system with K stages in series, at stage s, s=1...K, there are Ms identical machines in parallel and an intermediate buffer between each of the production stages. (Fig. 1)
The problem becomes even more complex in the following situations (Shen, 2006). 1. When other manufacturing resources, such as operators and tools, are also considered during the scheduling process. For a classical job shop scheduling problem with n jobs, m machines, and k operators, the total number of possible solutions could be ሺ݊Ǩ ሻ . 2. When both process planning and manufacturing scheduling are to be done at the same time. Traditional approaches that treat process planning and manufacturing scheduling separately can result in suboptimal solutions for the two phases. Integrating the two phases into one optimization problem, by considering the constraints of both domains simultaneously, can theoretically result in a global optimal solution, but it increases the solution space significantly. 3. When unforeseen dynamic situations are considered. In a flow shop manufacturing environment, rarely do things go as expected. The system may be asked to include additional tasks that are not anticipated, or to adapt to changes to several tasks, or to neglect certain tasks. The resources available for performing tasks are subject to changes. Certain resources can become unavailable, and additional resources can be introduced. The beginning time and the processing time of a task are also subject to variations. A task can take more or less time than anticipated, and tasks can arrive early or late. Other uncertainties include power system failures, machine failures, operator absence, and unavailability of tools and materials. An optimal schedule, generated after considerable effort, may rapidly become unacceptable because of unforeseen dynamic situations on the shop floor and a new schedule may have to be generated. This kind of rescheduling problem is also called dynamic scheduling or real-time scheduling. Traditionally, analytical models are used in solving optimization and analytical problems in manufacturing systems. However, for the complexity and randomness inherent nowadays in the HMLV systems, it is difficult to establish accurate analytical model for them. Simulation technology can be an effective alternative in studying the
Fig. 1 Manufacturing system scheme We consider 10 different product families. Production orders are divided into batches - jobs. The processing times, batch sizes and setup times of jobs at the various stages depend on the actual product family number. Setup of the machine is required when the family number of job processing on the machine is changed. Setup times increase according to the number of machines processing jobs of the same product family (this fact is caused by material distribution complexity increasing). For the purpose of production scheduling (loading) optimization considering the high-mix and low-volume production orders divided into product families three new scheduling algorithms have been proposed. The main idea of the algorithms is balancing assignment of the machines to the production orders (families) according to the processing times or/and batch sizes or/and order sizes. The result of this assignment process is the creation of production streams, each representing one product family, that reduce the machines setup requirements. The goal of the algorithms is to find a production schedule that minimizes the total production time of the set of jobs - makespan. This idea can be illustrated by the following example. Consider 5 different production orders from 5 different product families and 6 production stages. Set of the production orders and specific icons of machines assigned to the production orders are depicted in Table 1.
146
Table 1. Production orders
INCOM 2012, May 23-25, 2012 Bucharest, Romania
Scheduling algorithm assigns machines on each of the stages to the production orders that left to be processed on these stages according to the selected criterion. Step 1, Time = 0
4. ALGORITHM PROPOSALS The scheduling problem may be described in general as follows: K - number of production stages, s = 1…K
All production orders left to be processed on each of the stages. The production schedule (assignment of the machines to the production orders) in Time = 0 is depicted in Figure 2.
F - number of product families, f = 1…F Ms - number of machines at stage s ܱ - production order of product family f ܱ௦ - number of products of production order f that left to be processed on stage s ܬ - job - production orders divided into production batches, i = 1…Of / Bf ܤ - batch size of production order of product family f ܶ௦ - processing time of production order of product family f on stage s ܣ௦ - number of machines at stage s assigned to the jobs of product family f Since the priorities of the families are equal an assumption M s > F , ∀s have been made to ensure that at least one job
Fig. 2 Production schedule in Time = 0
of each family can be produced at each stage at a time. Step n, Time = N Production schedule is recalculated each time when new production order enters to the production system or when any of the production orders is completely processed at any of the stages. Production order number 3 is completely processed at stages number 1, 2, 3 and 4 and left to be processed at stages 5 and 6. Production order number 2 is completely processed at stages number 1, 2 and 3 and left to be processed at stages 4, 5 and 6. All other production orders still have to be processed at stages 1,2,3,4,5 and 6. Recalculated production schedule in Time = N is depicted in Figure 3.
Considering processing times, batch sizes and order sizes of the production orders, three different algorithms have been proposed. 1. Processing times algorithm This algorithm balances the assignment of the machines according to the processing times of production orders.
T fs A fs = round F M s ∀f , f ∈ F , O fs > 0 T ∑ fs f =1
(1)
Since the processing times are not integer values, proposed algorithm consider round value of the equation result. This fact can caused unequality between sum of the machines assigned to the production orders and total number of machines at stage s. For the purpose of equalization between sum of the machines assigned to the production orders and total number of machines, following sub-algorithms are used. F
If
∑A
T fs
fs
< M s then " Afs = A fs + 1" , f ,
= max
(2)
fs
> M s then " Afs = A fs − 1" , f , A fs = max
(3)
f =1
Afs
F
Fig. 3 Production schedule in Time = N
If
∑A f =1
Number of machines assigned to the specific production order is increased for the production stream, where Tfs / Afs is maximal or decreased, where Afs is minimal. The process of
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equalization assignment ends when sum of the machines assigned to the production orders is equal to the total number of machines. 2. Batch size algorithm This algorithm balances the assignment of the machines according to the processing times and batch sizes of production orders.
BT f fs Afs = round F M s ∀f , f ∈ F , O fs > 0 ∑ B f T fs f =1 F
If
∑A
B f T fs
fs
< M s then " Afs = A fs + 1" , f ,
fs
> M s then " Afs = A fs − 1" , f , A fs = max
Afs
f =1
Fig. 4 Graphical representation of the simulation model in Witness
(4)
Block diagram of the algorithms implementation is depicted in Figure 5.
= max (5)
F
If
∑A
(6)
f =1
3. Production order size algorithm This algorithm balances the assignment of the machines according to the processing times, batch sizes and number of products from a specific product family that left to be processed on the stages.
BT O f fs fs A fs = round F M s ∀f , f ∈ F , O fs > 0 BT O ∑ f fs fs f =1 F
If
∑A
fs
< M s then " Afs = A fs + 1" , f ,
B f Tfs O fs
f =1
Afs
(7)
Fig. 5 Scheduling algorithm implementation
= max (8) 6. CASE STUDY
F
If
∑A
fs
> M s then " Afs = A fs − 1" , f , A fs = max
In the example case study we have considered 10 production stages and 10 product families. Processing times of jobs from product families A-J at production stages 1-10 are shown in table 2, number of machines at each stage in table 3 and production orders are summarized in table 4. All input data were obtained from real manufacturing system.
(9)
f =1
5. IMPLEMENTATION For the purpose of the scheduling algorithms verification, the proposed algorithms have been implemented to the simulation model of HMLV flow shop manufacturing system created in Witness 3.0 simulation software. Simulation model considers the random character of processing and setup times by ± 10 % deviations, which make the simulation results even more credible. Production schedules are calculated using the functions, automatically modified according to the current algorithm chosen by user. Graphical representation of the simulation model in Witness simulation software is depicted in Figure 4 (example with 6 production stages).
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Table 2. Processing times O\S
1
2
3
4
5
6
7
8
9
10
A
10.5
10.2
1.2
9.6
4.4
4.9
4.4
3.6
5.4
4
B
13.2
16
1.1
12.7
5.2
3.9
5.1
4.2
3.6
4.2
C
18.5
15.4
1.2
9.6
2.7
4.8
3.7
3.8
2.8
5.4
D
15
14.7
1.1
10.8
4.9
2.8
4.7
3.8
3.5
3.9
E
12.3
9.6
1.2
9
4.7
3.4
4.9
3.9
3.6
5.4
F
15.6
9.7
1.2
9.7
3.3
2.4
3.3
4.6
3.5
4.4
G
9.4
12
1.2
15
4
4.6
2.7
2.8
3.1
5.2
INCOM 2012, May 23-25, 2012 Bucharest, Romania
H
8.4
10.8
1.6
13.2
6.7
4.2
5.7
3.9
4.1
4.8
I
10.2
17.4
1.1
11.1
8.5
5.6
8.5
4.1
3.7
4.9
J
9.2
7.4
1.6
7.8
6.2
4.9
3.6
3.9
4.2
5.3
Sum of the machines assigned to the production orders is 28. Number of machines at stage 1 is 29. This unequality will be solved using equation (2) considering Tf1 / Af1 values shown in Table 6.
Table 3. Number of machines
Table 6. Tfs / Afs
S
1
2
3
4
5
6
7
8
9
10
M
29
30
11
27
19
18
18
17
18
18
T11/A11
T21/A21
T31/A31
T41/A41
T51/A51
T61/A61
2,6
3,3
3,1
3
3,1
3,1
Table 4. Production orders Number of machines assigned to the product family B is increased by 1.
Product family
Time of arrival
Order size
Product family
Time of arrival
Order size
A
0
250
B
500
280
B
0
300
H
500
190
Final assignment of the machines at stage 1 to the production orders (families) in TIME = 0:
C
0
280
G
500
390
Table 7. Final machine assignments (Stage 1)
D
0
190
J
500
400
E
0
350
F
500
260
F
0
450
D
600
240
A
200
200
E
600
180
J
200
260
G
600
230
A11
A21
A31
A41
A51
A61
4
5
6
5
4
5
I
200
150
H
600
150
Final assignment of the machines at all stages to the production orders (families) in TIME = 0:
C
500
120
I
600
170
Table 8. Final machine assignments
A
500
310
B
600
230
In the following example we have used Processing times algorithm and input data as shown in Tables 1,2 and 3. Assignment of the machines at stage 1 to the production orders (families) in TIME = 0: Using equation (1) for product family A:
T A11 = round F 11 M 1 T ∑ f1 f =1 10, 5 .29 85,1
O\S
1
2
3
4
5
6
7
8
9
10
A
4
4
2
4
3
4
3
2
4
3
B
5
6
2
6
4
3
4
3
3
3
C
6
6
2
4
2
4
3
3
2
3
D
5
6
1
5
4
2
3
3
3
3
E
4
4
2
4
3
3
3
3
3
3
F
5
4
2
4
3
2
2
3
3
3
G
0
0
0
0
0
0
0
0
0
0
H
0
0
0
0
0
0
0
0
0
0
I
0
0
0
0
0
0
0
0
0
0
J
0
0
0
0
0
0
0
0
0
0
Simulation experiments have been run using five different production scheduling approaches (1-5):
A11 = round
1 - Processing times algorithm
A11 = 4
2 - Batch size algorithm 3 - Production order size algorithm
Analogically for all of the product families where ܱ௦ Ͳ. Table 5. Machine assignments (Stage 1)
4 - Basic schedule 1 5 - Basic schedule 2
A11
A21
A31
A41
A51
A61
4
4
6
5
4
5
Basic schedules assign free machines to the first available job in the buffer. Job sequencing in buffers is random, when Basic schedules are used.
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INCOM 2012, May 23-25, 2012 Bucharest, Romania
Four different sets of fifty simulation exper periments have been run. Average values of the total processing ng times (makespan) for 5 different scheduling approachess in four different experiment sets are summarized in Figuree 6. 6
ling algorithms were compared where the proposed scheduling with basic solutions, has been presented. pr
ACKNOWLE LEDGMENTS This work was supported by the th Scientific Grant Agency of the Ministry of Education of the th Slovak Republic Grant No. 1/1241/12.
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REFERE RENCES Bley, H.; Franke, C.; Ostermann, A., Methods and Tools for thee Building of Huge Simulation Models, Proceedings off the t 2nd CIRP International Seminar on Intelligent Computation Co in Manufacturing Engineering (ICME 2000),, 2000. 2
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Fig. 6 Total processing time (mak akespan) From the results in Figure 6 it is obviouss that th average values of the production makespan obtainedd using production schedules calculated by the proposed algor gorithms were better than values obtained using basic schedules les in all cases. All the results obtained using proposed algorithms alg are very similar but production schedule calculated ted by Production order size algorithm ensured the best value lue of the production makespan in all of the experiments. Proce cessing times of the production orders and batch sizes were ere identical in all experiments. For that reason all the savi aving of production makespan was caused only by redu eduction of setup requirements. Comparison of the setup req requirements in four experiments sets is shown in Figure 7.
Mahoney, R.M.: High-Mix Low-Volume L manufacturing. Hewlett-Packard Company, y, 1997. Neoh, S.C., Morad, N., Lim, Li Ch. P., Abdul-Aziz, Z.: Optimization of Product Mix ix Planning in High-Mix-LowVolume Industries Usingg Genetic G Algorithms, WSEAS Transactions on Systems, Issue Is 7, Volume 3, pp 25412545, 2004. Shen, W., Wang, L., Hao,, Q.: Q Agent-Based Distributed Manufacturing Process Plan lanning and Scheduling: A State of the Art Survey, IEEE Tr Transactions on Systems, Man and Cybernetics, Vol. 36, No.4, No 2006.
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Guan, Z., Wang, Ch., Huan ang, j., Wan, L., Shao, X., Optimization of manufactu cturing systems using genetic search and multi-resoluti ution simulation, 8th IEEE International Conference on Control and Automation, pp 1473-1480, 2010.
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Svancara, J., Kralova, Z.: Case se study on simulation analysis of a multiple product manuf nufacturing system. Proceedings of the 5th IFAC Internationa nal Conference on Management and Control of Production on and Logistics, MCPL 2010, Coimbra, Portugal.
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Zulch, G., Johnson, U., Fischer, er, J.: Hierarchical simulation of complex production systems ms by coupling of models, Int. J. Production Economics 77, 77 2002.
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Fig. 7 Setup requirements nts 7. CONCLUSION The paper deals with the production schedu eduling optimization in High-Mix Low-Volume flow shop manu nufacturing systems. Three new heuristic production scheduling ling algorithms have been proposed and implemented in the simulation sim model of the HMLV manufacturing system crea reated in Witness simulation software. All three algorith rithms balance the machines assignment to the productionn orders (families) according to the selected criterion. An example exa case study,
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