- Email: [email protected]
Integrated Optimization of Mixed-Model Assembly Line Balancing and Buffer Allocation Based on Operation Time Complexity
Available online at www.sciencedirect.com Available online at www.sciencedirect.com
ScienceDirect ScienceDirect Available online atonline www.sciencedirect.com Procedia CIRP 00 (2019) 000–000 Available at www.sciencedirect.com
ScienceDirect ScienceDirect
Procedia CIRP 00 (2019) 000–000
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
ProcediaProcedia CIRP 00CIRP (2017) 81000–000 (2019) 1040–1045
52nd CIRP Conference on Manufacturing Systems 52nd CIRP Conference on Manufacturing Systems
www.elsevier.com/locate/procedia
Integrated Optimization of Mixed-Model Assembly Line Balancing and Integrated Optimization ofBased Mixed-Model Assembly Line Balancing and 28th CIRP Design Conference, May 2018, Nantes, Buffer Allocation on Operation Time France Complexity Buffer Allocation Based on Operation Time Complexity a,* a A new methodology the Lei functional anda,physical Xuemeito Liuanalyze , Mingliang , Qingfei Zeng Aiping Lia architecture of a,* a a a Xuemei , Mingliang Lei Qingfei Zeng210804, , Aiping School of Mechanical Engineering,oriented Tongji,University, Shanghai China Li identification existing products for Liu an assembly product family a
a School of Mechanical Engineering, Tongji University, Shanghai 210804, China * Corresponding author. Tel.: +86-185-2135-4437. E-mail address: [email protected]
Paul Stief *, Jean-Yves Dantan, Alain Etienne, Ali Siadat
* Corresponding author. Tel.: +86-185-2135-4437. E-mail address: [email protected] École Nationale Supérieure d’Arts et Métiers, Arts et Métiers ParisTech, LCFC EA 4495, 4 Rue Augustin Fresnel, Metz 57078, France
Abstract
* Corresponding author. Tel.: +33 3 87 37 54 30; E-mail address: [email protected]
Abstract This paper presents a new approach to coupling line balancing and buffer allocation for stochastic mixed-model assembly line. Three types of complexity are defined, that are time diversity complexity for single station, time diversity complexity for a whole line, and time stochastic This paper presents a new approach to coupling line balancing and buffer allocation for stochastic mixed-model assembly line. Three types of fluctuations complexity. An integrated optimization model for line balancing and buffer allocation is established with the objectives of Abstract complexity are defined, that are time diversity complexity for single station, time diversity complexity for a whole line, and time stochastic maximizing productivity, minimizing complexity and total buffer capacity. Then an improved genetic algorithm is applied to solve the model. fluctuations complexity. An integrated optimization model for line balancing and buffer allocation is established with the objectives of Finally, effectiveness of the method is verified by an example of a mixed-model assemblyisline. Inmaximizing today’sthe business environment, the trend towardsand more product andThen customization unbroken. Due to this development, of productivity, minimizing complexity total buffervariety capacity. an improved genetic algorithm is applied to solvethe theneed model. agile and reconfigurable production systems emerged to cope with various products and product Finally, the effectiveness of the method is verified by an example of a mixed-model assembly line.families. To design and optimize production © 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license systems as well as to choose the optimal product matches, product analysis methods are needed. Indeed, most of the known methods aim to © 2019 The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/3.0/) analyze product or one product family physical Different families, may differ largely in terms of the number and © 2019 The Authors. Published by Elsevier Ltd. This islevel. an open accessproduct article under the however, CC BY-NC-ND license This is aan open access article under the on CCthe BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/) Peer-review under responsibility of the scientific committee of the 52nd CIRP Conference on Manufacturing Systems. nature of components. This fact impedes an efficient comparison and choice of appropriate product family combinations for the production (http://creativecommons.org/licenses/by-nc-nd/3.0/) Peer-review under responsibility of the scientific committee of the 52nd CIRP Conference on Manufacturing Systems. system. A new methodology is proposed to analyze existing products in view ofConference their functional and physicalSystems. architecture. The aim is to cluster Peer-review under responsibility of the scientific committee of the 52nd CIRP on Manufacturing Keywords: mixed-model assembly line; line balancing; complexity; buffer allocation; integration optimization these products in new assembly oriented product families for the optimization of existing assembly lines and the creation of future reconfigurable assembly Based on Datum Chain, the physical buffer structure of the integration products isoptimization analyzed. Functional subassemblies are identified, and Keywords:systems. mixed-model assembly line;Flow line balancing; complexity; allocation; a functional analysis is performed. Moreover, a hybrid functional and physical architecture graph (HyFPAG) is the output which depicts the similarity between product families by providing design support to both, production system planners and product designers. An illustrative example of a nail-clipper is used to explain the proposed methodology. An industrial case study on two product families of steering columns of 1. Introduction thyssenkrupp Presta France is then carried out to give a first industrial evaluation of the proposed approach. ©1. 2017 The Authors. Published by Elsevier B.V. Introduction ALBP which deals 2018. with assembly of many models in the Mixed-model assembly ofline (MMAL) is a type of CIRP Peer-review under responsibility the scientific committee of the 28th Design Conference
same [2] was of themany first models to study assembly line where severallinedistinct models a basic ALBP line. whichThomopolous deals with assembly in the the Mixed-model assembly (MMAL) is of a type of MMALBP, which treated MMALBP as a single assembly allocation same line. Thomopolous [2] was the first to study the assembly line where several distinct models of a basic line balancing problem by asconverting different are the two problems in balancing the designand of the mixed-model MMALBP, which treated(SALBP) MMALBP a single assembly product are core assembled. Line buffer allocation models into a single product through combined precedence assembly line. Usually, the assembly line is balanced first, line balancing problem (SALBP) by converting different are the two core problems in the design of the mixed-model diagram.into Haqa Asingle N [3] product proposedthrough a hybridcombined genetic algorithm to and then the buffer allocation. After theline balancing scheme is precedence assembly line. Usually, the assembly is balanced first, 1.determined, Introduction ofmodels the product range and characteristicsassembly manufactured and/or optimize the balance of multi-variety lines. When the processing time of each station may not be diagram. Haq A system. N [3] proposed a hybridthe genetic to and then the buffer allocation. After the balancing scheme is assembled inthe this In this context, main algorithm challenge in generatingthe initial ofpopulation, theassembly hierarchical position exactly the same, so buffers need to be set station betweenmay stations to optimize balance multi-variety lines. When determined, the processing time of each not be Due to the fast development in the domain of modelling and analysis is now not only to cope with single weight method used population, to improve the its quality, avoidposition invalid avoid downtime. The change station time hastoa generating the is initial hierarchical exactlyproduction the same,and so buffers need totrend be setof stations communication an ongoing ofbetween digitization and products, a limited product range or existing productInfamilies, search space and improve search efficiency. actual large impact on the buffer allocation and the line productivity. weight method used to improve its quality, avoidtoinvalid avoid production downtime. The change are of station time has a digitalization, manufacturing enterprises facing important but also to bethere ableisto analyze and to compare products define production, may be stochastic fluctuations in processing So it is difficult to get the best results by serial optimization search space and improve search efficiency. Inexisting actual large impactinontoday’s the buffer allocation and the line aproductivity. challenges market environments: continuing new product families. can be observed that classical time of taskthere duemay toItbe human fatigue, machine failure, and mode that balancing firstthe and then bufferbyallocation [1]. production, stochastic fluctuations in processing So it is difficult to get best results serial optimization tendency towards reduction of product development times and product families are regrouped in function of clients or features. logistics speed. So the stochastic assembly line balancing is The assembly line balancing problem (ALBP) is the time of task due oriented to human fatigue, machine failure, and mode thatproduct balancing first andInthen buffer there allocation shortened lifecycles. addition, isstations an [1]. increasing However, assembly product families are hardly totasks. find. proposed to deal with uncertain processing time of the assignment of assembly tasks to different with logistics speed. Sofamily the stochastic assembly linemainly balancing is The of assembly line balancing problem (ALBP) the demand customization, being at the same time in or a is global On the[4] product level, products differ in two Hamta dealt with a multi-objective single-model ALBP various constrains satisfied, while optimizing one more proposed to deal with uncertain processing time of the tasks. assignment of assembly tasks to different stations with competition with all over the world. This trend, main characteristics: (i) the number of components (ii) the with flexible processing times, the processing timeand of tasks is objectives. The competitors mixed-model lineone balancing Hamta [4] dealt with a mechanical, multi-objective single-model ALBP variousis constrains satisfied, whileassembly optimizing or micro more which inducing the development from macro to type of components (e.g. electrical, electronical). between the lower and upper bounds, and then proposed problem (MMALBP) can be seen as a particular case of the with flexiblemethodologies processing times, the processing time ofproducts tasks isa objectives. Thein mixed-model balancing markets, results diminished lotassembly sizes due line to augmenting Classical considering mainly single thealready lower and upperproduct bounds, families and thenanalyze proposedthea problemvarieties (MMALBP) can be seen as a particular case of [1]. the product (high-volume to low-volume production) orbetween solitary, existing 2212-8271 © 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license To cope with this augmenting variety as well as to be able to product structure on a physical level (components level) which (http://creativecommons.org/licenses/by-nc-nd/3.0/) 2212-8271 ©under 2019responsibility The optimization Authors. of Published by Elsevier Ltd. an open access article under the CC BY-NC-ND license an efficient definition and identify potentials in This existing causes regarding Peer-reviewpossible the scientific committee ofthe theis52nd CIRP Conference on difficulties Manufacturing Systems. (http://creativecommons.org/licenses/by-nc-nd/3.0/) production system, it is important to have a precise knowledge comparison of different product families. Addressing this product are assembled. Line balancing and buffer Keywords: Assembly; Design method; Family identification
Peer-review under responsibility of the scientific committee of the 52nd CIRP Conference on Manufacturing Systems.
2212-8271 © 2019 The Authors. Published by Elsevier Ltd. This is an©open article Published under theby CC BY-NC-ND 2212-8271 2017access The Authors. Elsevier B.V. license (http://creativecommons.org/licenses/by-nc-nd/3.0/) Peer-review under responsibility of scientific the scientific committee theCIRP 52ndDesign CIRPConference Conference2018. on Manufacturing Systems. Peer-review under responsibility of the committee of the of 28th 10.1016/j.procir.2019.03.248
2
Xuemei Liu et al. / Procedia CIRP 81 (2019) 1040–1045 Xuemei Liu et al. / Procedia CIRP 00 (2019) 000–000
new algorithm based on the combination of particle swarm optimization (PSO) algorithm with variable neighborhood search (VNS) to solve the problem. Hop [5] used fuzzy time to describe the phenomenon of task time fluctuation, establishes fuzzy binary linear programming balance model, and developed fuzzy heuristic method to solve this model. Cakir [6] proposed a modified simulated annealing algorithm to obtain the pareto-optimal solution for the problem of stochastic assembly line balancing with the goal of minimizing the smoothness index and design cost. As an effective method for measuring uncertainty, complexity theory is applied to measure the uncertainty of production systems. At present, the research on complexity of production system mainly focuses on the analysis and measurement of complexity features by means of information theory and entropy measurement. Representatives include Frizelle, Efthymiou [7] and so on. Xiaowei Zhu [8] established a station complexity model based on the change of different products for the mixed-model assembly line, which provided a new perspective for the sorting optimization of the mixed line products. Jaber [9] studied the changes in system complexity and information entropy for different product batches. Buffer allocation problem (BAP) is a key issue in manufacturing system design, and it is also a NP-hard combinatorial optimization problem. The buffer allocation optimization not only determines the buffer capacity, but also allocates it to the appropriate position. The reference [10] presented a comprehensive survey on BAP in manufacturing system, and the review aimed to provide an overview of recent advances in the BAP and present ideas for future research. Lei Li [11] decomposed the long production line into different parts and proposed a fast algorithm to solve the BAP in unreliable production lines. A large number of scholars have carried out a lot of research on the key issues such as line balancing and buffer allocation in the design of mixed-model assembly line, and proposed related theories and algorithms for analysis and solution. However, there are few studies on optimization of line balancing and buffer allocation simultaneously for stochastic mixed-model assembly lines, especially considering the time diversity of mixed-model products, and fluctuations in processing time. In view of the above deficiencies, an optimization method is proposed which can simultaneously deal with both balancing and buffer allocation problems. Taking the productivity, the complexity of the assembly line and the number of buffers as the optimization goal, considering the precedence relationship constraints of the tasks, the integrated optimization scheme is obtained through the improved genetic algorithm. And a case study is taken to show the effectiveness and efficiency of this method. The reminder of this paper is organized as follows. Section 2 presents the problem description; Section 3 defines three types of complexity and the optimization model is presented in section 4; Section 5 presents a case study; Finally the conclusion and future research direction are provided in section 6.
1041
2. Problem description Given the number of assembled product models, the set of various tasks for different product models, the precedence relationship between the tasks, the processing time of each task, and limit of each buffer capacity are known. The problem is to assign tasks to the given stations respecting to precedence constraint, to define the buffers capacity among stations, and to balance the workload of each station, maximize the productivity, minimize the total buffer capacity. The problem-related parameters are defined as Table1: Table 1. List of parameters. Parameter
definition
M
Number of product models
m
Index of product models, (m=1,…,M)
Dm
Demand for each model
D
Total demand, ( D = å Dm )
M
m=1
K
Number of stations
k
Index of stations, (k=1,…,K)
C
The cycle time
N
Number of tasks
i,j
Index of tasks, state, etc
O
Set of tasks, (O=1,…,N)
tim
Processing time of task i for model m
Tkm
Time of model m at station k
Bk
Buffer capacity of the station k
Sk
Set of tasks for station k
Y
Matrix of task precedence
3. Measurement of complexity Complexity is defined as a measure of uncertainty. At present, information entropy is one of the most important means to study the complexity of manufacturing systems. This paper proposes a method for measuring the complexity of mixed-model assembly lines based on information entropy theory. The following describes the calculation method of information entropy: The discrete random variable X has n possible values (x1,x2,…,xn), its probability is (p1, p2, ... ,pn), then the information entropy of the variable X can be defined as n
E ( X ) = -å pi log 2 pi
(1)
i =1
n
Where, pi ³ 0 , å pi = 1 . If X represents a system, xi and pi i =1
denote the n states that may occur in the system and their corresponding probabilities, E(X) characterizes the information entropy value of system X. For the mixed-model assembly line, the difference in the combination of tasks, the different processing time of the same task for different models, and the processing time fluctuation of the task will lead to the diversity and volatility of the time of the station. The state of the station time can be
Xuemei Liu et al. / Procedia CIRP 81 (2019) 1040–1045 Xuemei Liu et al. / Procedia CIRP 00 (2019) 000–000
1042
analysed by the state of the task time. So, this paper defines three types of complexity, including time diversity complexity for single station, time diversity complexity for a whole line, and time stochastic fluctuations complexity. 3.1 Time diversity complexity for single station Since multiple product models are produced on the assembly line at the same time, the tasks assigned to each station are not necessarily the same for different product models. Even if the assignment of the tasks is identical, there may be a difference in time for the same tasks of different product models. So different work times performed in the station depend on the type of assembly model, which in turn results in multiple states of the workstation. Without considering the time fluctuation of task, we can define the time diversity complexity for single station that is a measurement of the uncertainty of the station processing time. It depends on the various time states that may occur at each station and their probability of occurrence. Firstly, the introduction of a 0-1 variable Wik means that the task i is assigned to the workstation k, and
3
complexity of the time combination of the tasks assigned to the station. However, it does not balance the complexity of the working time between the stations. So, it is necessary to measure the diversity of assembly time of the entire line, which is time diversity complexity for a whole line. From the perspective of the whole line, the state of the assembly line cannot simply be seen as assembling different models. There may be differences between the working status of each station and the assembled model. So, it is necessary to distinguish it from the single-station complexity, and use another model to measure the time diversity complexity for the whole line. In the mixed-model assembly line, the time of different models at the same station or different stations are not necessarily the same, but it basically falls within a certain range, and there is a deviation in the probability of occurrence in different ranges, which can be used to measure the complexity. Divide the processing time of all models at all stations into r intervals (s1, s2,…,sr ), representing r kinds of states. And then count the probability of each interval (pl1, pl2,…,plr). The sum of the probabilities of all states is 1, that is r
ì1, i is assigned to k Wik = í î0, i is not assigned to k
(2)
å p
lj
j =1
(7)
= 1, plj ³ 0
But not all states will appear. Counting all states whose probability is not 0, the total number of states that appear is n. So, the time diversity complexity for a whole line HL is
Then N
(3)
Tkm = å Wik tim i =1
If the standard processing time of the two models at station k is the same, that is, Tkm1=Tkm2, (m1,m2=1,2,…,M) the two models are regarded as the same state. The station k has n time states during the assembly process, and the probability of each state is ( P1ks , P2 ks , …, Pnks ). And then Piks =（1-µ）M
DJ Tkm
å ( D T m=1
, i = (1, 2,..., n - 1)
m km
n
H L = - ln(n + 1)å Pli log 2 Pli
(4)
)
Where, µ is the failure rate of station k, Pnks=µ. DJ is the total number of different models with the same standard working time in the station k (E.g. Tkm1=Tkm2, then DJ=Dm1+Dm2). If different models have different station time, then DJ=Dm. So, the time diversity complexity for k-workstation Hk-S is n
H k -S = - ln(n + 1)å PikS log 2 PikS
(5)
i =1
K
H S = å H i -S
3.3 Time stochastic fluctuations complexity In order to effectively deal with the time fluctuation of the stochastic assembly line, the distribution of the actual time of the tasks is taken as the state of calculating the complexity of the station. For the model m, the standard processing time of the task i is known as tim, and some actual processing time samples are statistically obtained, compared with the standard time, and divided into the following n intervals (s1,s2,…,sn), respectively representing n kinds of states. The probability that the task i of the model m appears in the n intervals is (pmi1, pmi2, ..., pmin), the number of task in the k-station is nk, the state of the time of the task i is ji, and the probability that the task i appears in the state ji is pmij . The operation on the station is i
carried out in sequence, and the probability of a certain state of the station is( pm1 j1 pm 2 j2 pmnk jn ).The sum of the probabilities of all states is 1, that is
Then, the total time diversity complexity for single station HS is (6)
i =1
(8)
i =1
k
(9) Then, the time stochastic fluctuations complexity of model m in k-workstation is
3.2 Time diversity complexity for a whole line The single-station complexity takes into account the
(10)
4
Xuemei Liu et al. / Procedia CIRP 81 (2019) 1040–1045 Xuemei Liu et al. / Procedia CIRP 00 (2019) 000–000
The impact of different model on the assembly process is equal to the proportion of their demand. The demand proportion for model m is: lm =
Dm
(11)
M
å D
m
m=1
1043
So, the optimization model for line balancing and buffer allocation is ìmax PR ï ímin B ïmin H-s î
(17)
s.t.
(18) (19) (20) (21) (22)
So, the time stochastic fluctuations complexity for kworkstation is M
Hk
= å lm H
(12)
km
m =1
4. Optimization model for line balancing and buffer allocation For the time stochastic fluctuations complexity, if the complexity difference between stations is too large, some stations may deviate greatly from the standard processing time, resulting in changes of the bottleneck station, which further causes the station to be idle and overloaded frequently, and even cause blockage. So it is necessary to consider the time fluctuation complexity of the station, reduce the difference in complexity between the stations, and make the processing time uncertainty of each station tend to be balanced. The objective function of the difference in time stochastic fluctuations complexity between stations is: HB =
K
å ( H k =1
k
- H max )
2
(13)
Where the Hmax is H max = Max{H1 , H 2 ,..., H K }
(14)
Y = ( yij ) N ´N (i, j = 1, 2, ×××, N )
Constraint (18) assures that each task must be assigned to the station; the constraint (19) assures that each task is assigned only once; constraints (20) indicate that each station must be assigned to at least a task; constraint (21) indicates that the buffer capacity of the station must be lower than the upper limit of the buffer capacity, and the last station does not set the buffer; Constraint (22) represents the matrix of precedence between tasks. 5. Case study We now present a mixed-model assembly line case study to verify the proposed method. There are three kinds of models produced in this mixed-model assembly line, which are symbolised with A, B, and C. The demand of these three models is DA=900, DB=1200, DC=600. A total of 39 tasks, the precedence relationship between the tasks is known, and it is shown in Fig. 1. The processing time of each task for each model is known and it is shown in Table 2. 3
1
2
4
5
7
15
6
14
16
13
So, the time complexity model for stochastic mixed-model assembly line is H -s = aH S + bH L +cH B
(15)
Where, a,b, and c are weight coefficients, which satisfy a+b+c=1. They are obtained by the subjective weighting method combined with the actual planning requirements. This paper considers the interaction between station time variation and buffer allocation caused by the task allocation scheme. The simulation software is used to obtain the assembly line productivity PR, PR and the total buffer capacity B are taken as the objective functions to ensure the assembly line performance. The total buffer capacity B is defined as follows K -1
B = å Bk k =1
(16)
17
12
8
18
19
11
23
24
25
27
26
29
9
10
22
33
28
30
20
21
34
35
32
31
37
36
39
38
Fig.1. The combined precedence diagram.
The mixed-model assembly line has 6 serial stations, the failure rate of each station is 0.02, and the maximum buffer capacity configured for a single station is 5, and the time fluctuation data of each tasks are known. According to the analysis of the actual time fluctuation data, the fluctuation time is divided into four intervals, and then counting the probability that they appear in each interval. The time fluctuation probability of model A is shown in Table 3 (Model B, model C are not shown here due to the length of the paper).
Xuemei Liu et al. / Procedia CIRP 81 (2019) 1040–1045 Xuemei Liu et al. / Procedia CIRP 00 (2019) 000–000
1044
5
Table 2. The processing time of tasks. time/s
i
A
B
C
1
150
150
150
2
120
120
120
3
480
0
4
360
240
5
1080
6
540
7 8
time/s
i
time/s
i
A
B
C
A
B
C
14
720
780
780
15
150
150
150
27
300
300
300
28
120
0
0
480
16
360
360
360
29
60
60
60
0
17
360
360
360
30
180
180
180
960
1200
18
150
150
150
31
180
180
180
540
540
19
180
180
180
32
60
60
60
240
0
240
20
180
180
180
33
240
240
240
150
150
150
21
1080
1020
1140
34
120
120
120
9
150
150
150
22
480
480
480
35
120
120
120
10
420
420
420
23
540
420
420
36
60
60
60
11
0
180
0
24
300
300
300
37
120
120
120
12
300
300
300
25
360
360
360
38
60
60
60
13
0
540
540
26
300
300
300
39
60
60
60
Table 3. The time fluctuation probability of model A. i
pi1
pi2
pi3
pi4
i
pi1
pi2
pi3
pi4
0
0.23
0.73
0.04
14
0
0.23
0.73
2
0
0.2
3
0.07
0.1
0.78
0.02
15
0
0.08
0.6
0.03
16
0
0.1
4
0
5
0
0
0.98
0.02
17
0
0.08
0.86
0.06
18
0
6
0
7
0
0.2
0.78
0.02
19
0
0
0.94
0.06
20
0.25
8 9
0
0.23
0.73
0.04
21
0
0.2
0.78
0.02
22
10 11
0
0
0.94
0.06
0
0.2
0.78
0.02
12
0
0.2
0.78
13
0
0.1
0.88
1
i
pi1
pi2
pi3
pi4
0.04
27
0
0.23
0.73
0.04
0.86
0.06
28
0
0.2
0.78
0.02
0.88
0.02
29
0
0.1
0.84
0.06
0.2
0.78
0.02
30
0
0.2
0.78
0.02
0.08
0.86
0.06
31
0
0
0.94
0.06
0.1
0.88
0.02
32
0.07
0.4
0.5
0.03
0.4
0.32
0.03
33
0
0.2
0.78
0.02
0
0.23
0.73
0.04
34
0
0.1
0.84
0.06
0
0.1
0.84
0.06
35
0
0.1
0.84
0.06
23
0
0.1
0.9
0
36
0
0.23
0.73
0.04
24
0.25
0.3
0.42
0.03
37
0
0.12
0.86
0.02
0.02
25
0
0.1
0.84
0.06
38
0
0.2
0.78
0.02
0.02
26
0
0.08
0.86
0.06
39
0
0.1
0.87
0.03
The genetic algorithm is used to solve the model of this paper. The parameters of the genetic algorithm are set as: population size popSize=200, elite selection probability Ps=0.2, crossover rate Pc=0.3, mutation rate Pm=0.5, genetic iteration number Nnew=100. The optimal tasks assignment and buffer allocation scheme is shown in Table 4. Table 4. Integrated optimization scheme. k
tasks
Hk-S
Hk
Bk
1
[1,3,2,8,9,10,11,12,15]
1.068587
5.3741
1
2
[4,5,13,18]
0.962887
5.223
3
3
[6,20,21]
2.135096
5.2691
4
4
[7,14,22,26,34]
2.147251
4.8465
4
5
[12,17,19,23,24,37]
1.024515
4.911
3
1.025033
4.921
/
6
[25,27,28,29,30,31, 32,33,35,36,38,39]
In this scheme, the productivity of the line is 46.9643/h through simulation, the time diversity complexity for the whole line is 1.55558bit, the index of difference in time
stochastic fluctuations complexity between stations is 0.34928, and the total buffer capacity is 15. In order to verify the superiority of integrated optimization, integration optimization and serial optimization methods are now compared. The serial optimization is balancing first, and then the buffer allocation. The results are shown in Table 5. Table 5. Integrated optimization and serial optimization scheme comparison. B
PR （ /h
）
HS
HL
Integrated optimization
15
46.9643
8.363369
1.55558
Serial optimization
18
46.6391
8.449385453
1.735126457
The final solution has a large advantage in the total number of buffer capacity while maintaining a high level of productivity. Integrated optimization can obtain a complete scheme including balancing and buffer allocation at the same
6
Xuemei Liu et al. / Procedia CIRP 81 (2019) 1040–1045 Xuemei Liu et al. / Procedia CIRP 00 (2019) 000–000
time, and avoids the problem that serial optimization may not obtain global optimal solution due to premature phenomenon. Therefore, integrated optimization has a large advantage over serial optimization in terms of computational efficiency and optimization results. This paper considers the impact of stochastic fluctuations in processing time. For the problem of stochastic assembly line balance, the common solution is to assume that the processing time obeys the normal distribution, and the equilibrium rate is the highest and the average time of the station is balanced under the condition that the station meets a certain completion rate [12]. In order to verify the effectiveness of the method in this paper, compare it with the method in the literature. Using the method in the reference [12], with the highest balance ratio and the balance of work time, the optimal tasks assignment and buffer allocation scheme are as follows: the tasks in station 1-6 is {1,9,7,8,18},{11,20,15,26,2,3,13},{12,10,4,5,6},{21,14,16,17 ,22},{34,37,23,14,33,24,28,25},{27,29,30,31,32,35,36,38,39 }. The buffer capacity is {2,4,3,3,4}. The Table 6 shows the comparison between the method in this paper and the method in literature. Table 6. Method comparison. Station
Hk in this paper
in literature [12]
1
5.3741
5.8085
2
5.223
5.469
3
5.2691
4.5887
4
4.8465
5.3192
5
4.911
4.8465
6
4.921
5.6296
The index of difference
0.34928
0.683134
PR(/h)
46.9643
43.7182
Compared with the integrated optimization scheme, although the literature method has a relatively low time fluctuation complexity of the 3rd station, the time fluctuation complexity of the 1st and 6th stations is relatively high, which also leads to an increase in the index of difference in time stochastic fluctuations complexity between stations. The increase and imbalance of the time fluctuation complexity will bring the fluctuation of the standard working time of the station, which will adversely affect the production of the assembly line. From the perspective of productivity level, it can be seen that the comprehensive balance scheme considering the stochastic fluctuation factors of the assembly time is more excellent, and has a 7.42% improvement effect compared with the scheme in literature. 6. Conclusion This research deals with simultaneous line balancing and buffer allocation decisions for stochastic mixed-model assembly line. Three types of complexity are defined, which
1045
including time diversity complexity for single station, time diversity complexity for a whole line, and time stochastic fluctuations complexity. An integrated optimization model for line balancing and buffer allocation is established with the objectives of maximizing productivity, minimizing complexity and total buffer capacity. The accurate productivity of the line is obtained by the simulation software, and the improved genetic algorithm is used to solve the model. The case shows that the method can obtain the overall scheme of line balancing and buffer allocation in one optimization process. While maintaining a high balance rate and productivity, the station complexity is more balanced and the total buffer capacity is smaller. This paper only studies the serial assembly line, and does not consider parallel stations. he different structure of the line may also affect the complexity of the assembly line. In further research, assembly lines for different structure will be studied. Acknowledgement This work is supported by National Key R&D Program of China (2018YFB1700902). References [1] Tiacci L. Simultaneous balancing and buffer allocation decisions for the design of mixed-model assembly lines with parallel workstations and stochastic task times. International Journal of Production Economics. 2015, 162(4): 201-215. [2] Thomopoulos N T. Mixed Model Line Balancing with Smoothed Station Assignments. Management Science. 1970(9): 593-603. [3] Haq A N, Rengarajan K, Jayaprakash J. A hybrid genetic algorithm approach to mixed-model assembly line balancing. The International Journal of Advanced Manufacturing Technology. 2006(3-4): 337-341. [4] Hamta N, Ghomi S M T F, Jolai F, et al. A hybrid PSO algorithm for a multi-objective assembly line balancing problem with flexible operation times, sequence-dependent setup times and learning effect. International Journal of Production Economics. 2013, 141(1): 99-111. [5] Hop N V. A heuristic solution for fuzzy mixed-model line balancing problem. European Journal of Operational Research. 2006, 168(3): 798810. [6] Cakir B, Altiparmak F, Dengiz B. Multi-objective optimization of a stochastic assembly line balancing: A hybrid simulated annealing algorithm. Computers & Industrial Engineering. 2011, 60(3): 376-384. [7] Efthymiou K, Pagoropoulos A, Papakostas N, et al. Manufacturing systems complexity: An assessment of manufacturing performance indicators unpredictability. Cirp Journal of Manufacturing Science & Technology. 2014, 7(4): 324-334. [8] Zhu X, Hu S J, Koren Y, et al. Modeling of Manufacturing Complexity in Mixed-Model Assembly Lines. Journal of Manufacturing Science and Engineering. 2008, 130(5): 649-659. [9] Jaber M Y. Lot sizing with permissible delay in payments and entropy cost. Computers & Industrial Engineering. 2007, 52(1): 78-88. [10] Demir L, Tunali S, Eliiyi D T. The state of the art on buffer allocation problem: a comprehensive survey. Journal of Intelligent Manufacturing. 2014, 25(3): 371-392. [11] Li L, Qian Y L, Yang Y M, et al. A fast algorithm for buffer allocation problem. International Journal of Production Research. 2016, 54(11): 113. [12] Zhou L, Song H, Han Y. Stochastic production line load balancing based on genetic algorithm. Mechanical Manufacturing. 2003, 41(3): 23-25.
ScienceDirect ScienceDirect Available online atonline www.sciencedirect.com Procedia CIRP 00 (2019) 000–000 Available at www.sciencedirect.com
ScienceDirect ScienceDirect
Procedia CIRP 00 (2019) 000–000
www.elsevier.com/locate/procedia www.elsevier.com/locate/procedia
ProcediaProcedia CIRP 00CIRP (2017) 81000–000 (2019) 1040–1045
52nd CIRP Conference on Manufacturing Systems 52nd CIRP Conference on Manufacturing Systems
www.elsevier.com/locate/procedia
Integrated Optimization of Mixed-Model Assembly Line Balancing and Integrated Optimization ofBased Mixed-Model Assembly Line Balancing and 28th CIRP Design Conference, May 2018, Nantes, Buffer Allocation on Operation Time France Complexity Buffer Allocation Based on Operation Time Complexity a,* a A new methodology the Lei functional anda,physical Xuemeito Liuanalyze , Mingliang , Qingfei Zeng Aiping Lia architecture of a,* a a a Xuemei , Mingliang Lei Qingfei Zeng210804, , Aiping School of Mechanical Engineering,oriented Tongji,University, Shanghai China Li identification existing products for Liu an assembly product family a
a School of Mechanical Engineering, Tongji University, Shanghai 210804, China * Corresponding author. Tel.: +86-185-2135-4437. E-mail address: [email protected]
Paul Stief *, Jean-Yves Dantan, Alain Etienne, Ali Siadat
* Corresponding author. Tel.: +86-185-2135-4437. E-mail address: [email protected] École Nationale Supérieure d’Arts et Métiers, Arts et Métiers ParisTech, LCFC EA 4495, 4 Rue Augustin Fresnel, Metz 57078, France
Abstract
* Corresponding author. Tel.: +33 3 87 37 54 30; E-mail address: [email protected]
Abstract This paper presents a new approach to coupling line balancing and buffer allocation for stochastic mixed-model assembly line. Three types of complexity are defined, that are time diversity complexity for single station, time diversity complexity for a whole line, and time stochastic This paper presents a new approach to coupling line balancing and buffer allocation for stochastic mixed-model assembly line. Three types of fluctuations complexity. An integrated optimization model for line balancing and buffer allocation is established with the objectives of Abstract complexity are defined, that are time diversity complexity for single station, time diversity complexity for a whole line, and time stochastic maximizing productivity, minimizing complexity and total buffer capacity. Then an improved genetic algorithm is applied to solve the model. fluctuations complexity. An integrated optimization model for line balancing and buffer allocation is established with the objectives of Finally, effectiveness of the method is verified by an example of a mixed-model assemblyisline. Inmaximizing today’sthe business environment, the trend towardsand more product andThen customization unbroken. Due to this development, of productivity, minimizing complexity total buffervariety capacity. an improved genetic algorithm is applied to solvethe theneed model. agile and reconfigurable production systems emerged to cope with various products and product Finally, the effectiveness of the method is verified by an example of a mixed-model assembly line.families. To design and optimize production © 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license systems as well as to choose the optimal product matches, product analysis methods are needed. Indeed, most of the known methods aim to © 2019 The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/3.0/) analyze product or one product family physical Different families, may differ largely in terms of the number and © 2019 The Authors. Published by Elsevier Ltd. This islevel. an open accessproduct article under the however, CC BY-NC-ND license This is aan open access article under the on CCthe BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/) Peer-review under responsibility of the scientific committee of the 52nd CIRP Conference on Manufacturing Systems. nature of components. This fact impedes an efficient comparison and choice of appropriate product family combinations for the production (http://creativecommons.org/licenses/by-nc-nd/3.0/) Peer-review under responsibility of the scientific committee of the 52nd CIRP Conference on Manufacturing Systems. system. A new methodology is proposed to analyze existing products in view ofConference their functional and physicalSystems. architecture. The aim is to cluster Peer-review under responsibility of the scientific committee of the 52nd CIRP on Manufacturing Keywords: mixed-model assembly line; line balancing; complexity; buffer allocation; integration optimization these products in new assembly oriented product families for the optimization of existing assembly lines and the creation of future reconfigurable assembly Based on Datum Chain, the physical buffer structure of the integration products isoptimization analyzed. Functional subassemblies are identified, and Keywords:systems. mixed-model assembly line;Flow line balancing; complexity; allocation; a functional analysis is performed. Moreover, a hybrid functional and physical architecture graph (HyFPAG) is the output which depicts the similarity between product families by providing design support to both, production system planners and product designers. An illustrative example of a nail-clipper is used to explain the proposed methodology. An industrial case study on two product families of steering columns of 1. Introduction thyssenkrupp Presta France is then carried out to give a first industrial evaluation of the proposed approach. ©1. 2017 The Authors. Published by Elsevier B.V. Introduction ALBP which deals 2018. with assembly of many models in the Mixed-model assembly ofline (MMAL) is a type of CIRP Peer-review under responsibility the scientific committee of the 28th Design Conference
same [2] was of themany first models to study assembly line where severallinedistinct models a basic ALBP line. whichThomopolous deals with assembly in the the Mixed-model assembly (MMAL) is of a type of MMALBP, which treated MMALBP as a single assembly allocation same line. Thomopolous [2] was the first to study the assembly line where several distinct models of a basic line balancing problem by asconverting different are the two problems in balancing the designand of the mixed-model MMALBP, which treated(SALBP) MMALBP a single assembly product are core assembled. Line buffer allocation models into a single product through combined precedence assembly line. Usually, the assembly line is balanced first, line balancing problem (SALBP) by converting different are the two core problems in the design of the mixed-model diagram.into Haqa Asingle N [3] product proposedthrough a hybridcombined genetic algorithm to and then the buffer allocation. After theline balancing scheme is precedence assembly line. Usually, the assembly is balanced first, 1.determined, Introduction ofmodels the product range and characteristicsassembly manufactured and/or optimize the balance of multi-variety lines. When the processing time of each station may not be diagram. Haq A system. N [3] proposed a hybridthe genetic to and then the buffer allocation. After the balancing scheme is assembled inthe this In this context, main algorithm challenge in generatingthe initial ofpopulation, theassembly hierarchical position exactly the same, so buffers need to be set station betweenmay stations to optimize balance multi-variety lines. When determined, the processing time of each not be Due to the fast development in the domain of modelling and analysis is now not only to cope with single weight method used population, to improve the its quality, avoidposition invalid avoid downtime. The change station time hastoa generating the is initial hierarchical exactlyproduction the same,and so buffers need totrend be setof stations communication an ongoing ofbetween digitization and products, a limited product range or existing productInfamilies, search space and improve search efficiency. actual large impact on the buffer allocation and the line productivity. weight method used to improve its quality, avoidtoinvalid avoid production downtime. The change are of station time has a digitalization, manufacturing enterprises facing important but also to bethere ableisto analyze and to compare products define production, may be stochastic fluctuations in processing So it is difficult to get the best results by serial optimization search space and improve search efficiency. Inexisting actual large impactinontoday’s the buffer allocation and the line aproductivity. challenges market environments: continuing new product families. can be observed that classical time of taskthere duemay toItbe human fatigue, machine failure, and mode that balancing firstthe and then bufferbyallocation [1]. production, stochastic fluctuations in processing So it is difficult to get best results serial optimization tendency towards reduction of product development times and product families are regrouped in function of clients or features. logistics speed. So the stochastic assembly line balancing is The assembly line balancing problem (ALBP) is the time of task due oriented to human fatigue, machine failure, and mode thatproduct balancing first andInthen buffer there allocation shortened lifecycles. addition, isstations an [1]. increasing However, assembly product families are hardly totasks. find. proposed to deal with uncertain processing time of the assignment of assembly tasks to different with logistics speed. Sofamily the stochastic assembly linemainly balancing is The of assembly line balancing problem (ALBP) the demand customization, being at the same time in or a is global On the[4] product level, products differ in two Hamta dealt with a multi-objective single-model ALBP various constrains satisfied, while optimizing one more proposed to deal with uncertain processing time of the tasks. assignment of assembly tasks to different stations with competition with all over the world. This trend, main characteristics: (i) the number of components (ii) the with flexible processing times, the processing timeand of tasks is objectives. The competitors mixed-model lineone balancing Hamta [4] dealt with a mechanical, multi-objective single-model ALBP variousis constrains satisfied, whileassembly optimizing or micro more which inducing the development from macro to type of components (e.g. electrical, electronical). between the lower and upper bounds, and then proposed problem (MMALBP) can be seen as a particular case of the with flexiblemethodologies processing times, the processing time ofproducts tasks isa objectives. Thein mixed-model balancing markets, results diminished lotassembly sizes due line to augmenting Classical considering mainly single thealready lower and upperproduct bounds, families and thenanalyze proposedthea problemvarieties (MMALBP) can be seen as a particular case of [1]. the product (high-volume to low-volume production) orbetween solitary, existing 2212-8271 © 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license To cope with this augmenting variety as well as to be able to product structure on a physical level (components level) which (http://creativecommons.org/licenses/by-nc-nd/3.0/) 2212-8271 ©under 2019responsibility The optimization Authors. of Published by Elsevier Ltd. an open access article under the CC BY-NC-ND license an efficient definition and identify potentials in This existing causes regarding Peer-reviewpossible the scientific committee ofthe theis52nd CIRP Conference on difficulties Manufacturing Systems. (http://creativecommons.org/licenses/by-nc-nd/3.0/) production system, it is important to have a precise knowledge comparison of different product families. Addressing this product are assembled. Line balancing and buffer Keywords: Assembly; Design method; Family identification
Peer-review under responsibility of the scientific committee of the 52nd CIRP Conference on Manufacturing Systems.
2212-8271 © 2019 The Authors. Published by Elsevier Ltd. This is an©open article Published under theby CC BY-NC-ND 2212-8271 2017access The Authors. Elsevier B.V. license (http://creativecommons.org/licenses/by-nc-nd/3.0/) Peer-review under responsibility of scientific the scientific committee theCIRP 52ndDesign CIRPConference Conference2018. on Manufacturing Systems. Peer-review under responsibility of the committee of the of 28th 10.1016/j.procir.2019.03.248
2
Xuemei Liu et al. / Procedia CIRP 81 (2019) 1040–1045 Xuemei Liu et al. / Procedia CIRP 00 (2019) 000–000
new algorithm based on the combination of particle swarm optimization (PSO) algorithm with variable neighborhood search (VNS) to solve the problem. Hop [5] used fuzzy time to describe the phenomenon of task time fluctuation, establishes fuzzy binary linear programming balance model, and developed fuzzy heuristic method to solve this model. Cakir [6] proposed a modified simulated annealing algorithm to obtain the pareto-optimal solution for the problem of stochastic assembly line balancing with the goal of minimizing the smoothness index and design cost. As an effective method for measuring uncertainty, complexity theory is applied to measure the uncertainty of production systems. At present, the research on complexity of production system mainly focuses on the analysis and measurement of complexity features by means of information theory and entropy measurement. Representatives include Frizelle, Efthymiou [7] and so on. Xiaowei Zhu [8] established a station complexity model based on the change of different products for the mixed-model assembly line, which provided a new perspective for the sorting optimization of the mixed line products. Jaber [9] studied the changes in system complexity and information entropy for different product batches. Buffer allocation problem (BAP) is a key issue in manufacturing system design, and it is also a NP-hard combinatorial optimization problem. The buffer allocation optimization not only determines the buffer capacity, but also allocates it to the appropriate position. The reference [10] presented a comprehensive survey on BAP in manufacturing system, and the review aimed to provide an overview of recent advances in the BAP and present ideas for future research. Lei Li [11] decomposed the long production line into different parts and proposed a fast algorithm to solve the BAP in unreliable production lines. A large number of scholars have carried out a lot of research on the key issues such as line balancing and buffer allocation in the design of mixed-model assembly line, and proposed related theories and algorithms for analysis and solution. However, there are few studies on optimization of line balancing and buffer allocation simultaneously for stochastic mixed-model assembly lines, especially considering the time diversity of mixed-model products, and fluctuations in processing time. In view of the above deficiencies, an optimization method is proposed which can simultaneously deal with both balancing and buffer allocation problems. Taking the productivity, the complexity of the assembly line and the number of buffers as the optimization goal, considering the precedence relationship constraints of the tasks, the integrated optimization scheme is obtained through the improved genetic algorithm. And a case study is taken to show the effectiveness and efficiency of this method. The reminder of this paper is organized as follows. Section 2 presents the problem description; Section 3 defines three types of complexity and the optimization model is presented in section 4; Section 5 presents a case study; Finally the conclusion and future research direction are provided in section 6.
1041
2. Problem description Given the number of assembled product models, the set of various tasks for different product models, the precedence relationship between the tasks, the processing time of each task, and limit of each buffer capacity are known. The problem is to assign tasks to the given stations respecting to precedence constraint, to define the buffers capacity among stations, and to balance the workload of each station, maximize the productivity, minimize the total buffer capacity. The problem-related parameters are defined as Table1: Table 1. List of parameters. Parameter
definition
M
Number of product models
m
Index of product models, (m=1,…,M)
Dm
Demand for each model
D
Total demand, ( D = å Dm )
M
m=1
K
Number of stations
k
Index of stations, (k=1,…,K)
C
The cycle time
N
Number of tasks
i,j
Index of tasks, state, etc
O
Set of tasks, (O=1,…,N)
tim
Processing time of task i for model m
Tkm
Time of model m at station k
Bk
Buffer capacity of the station k
Sk
Set of tasks for station k
Y
Matrix of task precedence
3. Measurement of complexity Complexity is defined as a measure of uncertainty. At present, information entropy is one of the most important means to study the complexity of manufacturing systems. This paper proposes a method for measuring the complexity of mixed-model assembly lines based on information entropy theory. The following describes the calculation method of information entropy: The discrete random variable X has n possible values (x1,x2,…,xn), its probability is (p1, p2, ... ,pn), then the information entropy of the variable X can be defined as n
E ( X ) = -å pi log 2 pi
(1)
i =1
n
Where, pi ³ 0 , å pi = 1 . If X represents a system, xi and pi i =1
denote the n states that may occur in the system and their corresponding probabilities, E(X) characterizes the information entropy value of system X. For the mixed-model assembly line, the difference in the combination of tasks, the different processing time of the same task for different models, and the processing time fluctuation of the task will lead to the diversity and volatility of the time of the station. The state of the station time can be
Xuemei Liu et al. / Procedia CIRP 81 (2019) 1040–1045 Xuemei Liu et al. / Procedia CIRP 00 (2019) 000–000
1042
analysed by the state of the task time. So, this paper defines three types of complexity, including time diversity complexity for single station, time diversity complexity for a whole line, and time stochastic fluctuations complexity. 3.1 Time diversity complexity for single station Since multiple product models are produced on the assembly line at the same time, the tasks assigned to each station are not necessarily the same for different product models. Even if the assignment of the tasks is identical, there may be a difference in time for the same tasks of different product models. So different work times performed in the station depend on the type of assembly model, which in turn results in multiple states of the workstation. Without considering the time fluctuation of task, we can define the time diversity complexity for single station that is a measurement of the uncertainty of the station processing time. It depends on the various time states that may occur at each station and their probability of occurrence. Firstly, the introduction of a 0-1 variable Wik means that the task i is assigned to the workstation k, and
3
complexity of the time combination of the tasks assigned to the station. However, it does not balance the complexity of the working time between the stations. So, it is necessary to measure the diversity of assembly time of the entire line, which is time diversity complexity for a whole line. From the perspective of the whole line, the state of the assembly line cannot simply be seen as assembling different models. There may be differences between the working status of each station and the assembled model. So, it is necessary to distinguish it from the single-station complexity, and use another model to measure the time diversity complexity for the whole line. In the mixed-model assembly line, the time of different models at the same station or different stations are not necessarily the same, but it basically falls within a certain range, and there is a deviation in the probability of occurrence in different ranges, which can be used to measure the complexity. Divide the processing time of all models at all stations into r intervals (s1, s2,…,sr ), representing r kinds of states. And then count the probability of each interval (pl1, pl2,…,plr). The sum of the probabilities of all states is 1, that is r
ì1, i is assigned to k Wik = í î0, i is not assigned to k
(2)
å p
lj
j =1
(7)
= 1, plj ³ 0
But not all states will appear. Counting all states whose probability is not 0, the total number of states that appear is n. So, the time diversity complexity for a whole line HL is
Then N
(3)
Tkm = å Wik tim i =1
If the standard processing time of the two models at station k is the same, that is, Tkm1=Tkm2, (m1,m2=1,2,…,M) the two models are regarded as the same state. The station k has n time states during the assembly process, and the probability of each state is ( P1ks , P2 ks , …, Pnks ). And then Piks =（1-µ）M
DJ Tkm
å ( D T m=1
, i = (1, 2,..., n - 1)
m km
n
H L = - ln(n + 1)å Pli log 2 Pli
(4)
)
Where, µ is the failure rate of station k, Pnks=µ. DJ is the total number of different models with the same standard working time in the station k (E.g. Tkm1=Tkm2, then DJ=Dm1+Dm2). If different models have different station time, then DJ=Dm. So, the time diversity complexity for k-workstation Hk-S is n
H k -S = - ln(n + 1)å PikS log 2 PikS
(5)
i =1
K
H S = å H i -S
3.3 Time stochastic fluctuations complexity In order to effectively deal with the time fluctuation of the stochastic assembly line, the distribution of the actual time of the tasks is taken as the state of calculating the complexity of the station. For the model m, the standard processing time of the task i is known as tim, and some actual processing time samples are statistically obtained, compared with the standard time, and divided into the following n intervals (s1,s2,…,sn), respectively representing n kinds of states. The probability that the task i of the model m appears in the n intervals is (pmi1, pmi2, ..., pmin), the number of task in the k-station is nk, the state of the time of the task i is ji, and the probability that the task i appears in the state ji is pmij . The operation on the station is i
carried out in sequence, and the probability of a certain state of the station is( pm1 j1 pm 2 j2 pmnk jn ).The sum of the probabilities of all states is 1, that is
Then, the total time diversity complexity for single station HS is (6)
i =1
(8)
i =1
k
(9) Then, the time stochastic fluctuations complexity of model m in k-workstation is
3.2 Time diversity complexity for a whole line The single-station complexity takes into account the
(10)
4
Xuemei Liu et al. / Procedia CIRP 81 (2019) 1040–1045 Xuemei Liu et al. / Procedia CIRP 00 (2019) 000–000
The impact of different model on the assembly process is equal to the proportion of their demand. The demand proportion for model m is: lm =
Dm
(11)
M
å D
m
m=1
1043
So, the optimization model for line balancing and buffer allocation is ìmax PR ï ímin B ïmin H-s î
(17)
s.t.
(18) (19) (20) (21) (22)
So, the time stochastic fluctuations complexity for kworkstation is M
Hk
= å lm H
(12)
km
m =1
4. Optimization model for line balancing and buffer allocation For the time stochastic fluctuations complexity, if the complexity difference between stations is too large, some stations may deviate greatly from the standard processing time, resulting in changes of the bottleneck station, which further causes the station to be idle and overloaded frequently, and even cause blockage. So it is necessary to consider the time fluctuation complexity of the station, reduce the difference in complexity between the stations, and make the processing time uncertainty of each station tend to be balanced. The objective function of the difference in time stochastic fluctuations complexity between stations is: HB =
K
å ( H k =1
k
- H max )
2
(13)
Where the Hmax is H max = Max{H1 , H 2 ,..., H K }
(14)
Y = ( yij ) N ´N (i, j = 1, 2, ×××, N )
Constraint (18) assures that each task must be assigned to the station; the constraint (19) assures that each task is assigned only once; constraints (20) indicate that each station must be assigned to at least a task; constraint (21) indicates that the buffer capacity of the station must be lower than the upper limit of the buffer capacity, and the last station does not set the buffer; Constraint (22) represents the matrix of precedence between tasks. 5. Case study We now present a mixed-model assembly line case study to verify the proposed method. There are three kinds of models produced in this mixed-model assembly line, which are symbolised with A, B, and C. The demand of these three models is DA=900, DB=1200, DC=600. A total of 39 tasks, the precedence relationship between the tasks is known, and it is shown in Fig. 1. The processing time of each task for each model is known and it is shown in Table 2. 3
1
2
4
5
7
15
6
14
16
13
So, the time complexity model for stochastic mixed-model assembly line is H -s = aH S + bH L +cH B
(15)
Where, a,b, and c are weight coefficients, which satisfy a+b+c=1. They are obtained by the subjective weighting method combined with the actual planning requirements. This paper considers the interaction between station time variation and buffer allocation caused by the task allocation scheme. The simulation software is used to obtain the assembly line productivity PR, PR and the total buffer capacity B are taken as the objective functions to ensure the assembly line performance. The total buffer capacity B is defined as follows K -1
B = å Bk k =1
(16)
17
12
8
18
19
11
23
24
25
27
26
29
9
10
22
33
28
30
20
21
34
35
32
31
37
36
39
38
Fig.1. The combined precedence diagram.
The mixed-model assembly line has 6 serial stations, the failure rate of each station is 0.02, and the maximum buffer capacity configured for a single station is 5, and the time fluctuation data of each tasks are known. According to the analysis of the actual time fluctuation data, the fluctuation time is divided into four intervals, and then counting the probability that they appear in each interval. The time fluctuation probability of model A is shown in Table 3 (Model B, model C are not shown here due to the length of the paper).
Xuemei Liu et al. / Procedia CIRP 81 (2019) 1040–1045 Xuemei Liu et al. / Procedia CIRP 00 (2019) 000–000
1044
5
Table 2. The processing time of tasks. time/s
i
A
B
C
1
150
150
150
2
120
120
120
3
480
0
4
360
240
5
1080
6
540
7 8
time/s
i
time/s
i
A
B
C
A
B
C
14
720
780
780
15
150
150
150
27
300
300
300
28
120
0
0
480
16
360
360
360
29
60
60
60
0
17
360
360
360
30
180
180
180
960
1200
18
150
150
150
31
180
180
180
540
540
19
180
180
180
32
60
60
60
240
0
240
20
180
180
180
33
240
240
240
150
150
150
21
1080
1020
1140
34
120
120
120
9
150
150
150
22
480
480
480
35
120
120
120
10
420
420
420
23
540
420
420
36
60
60
60
11
0
180
0
24
300
300
300
37
120
120
120
12
300
300
300
25
360
360
360
38
60
60
60
13
0
540
540
26
300
300
300
39
60
60
60
Table 3. The time fluctuation probability of model A. i
pi1
pi2
pi3
pi4
i
pi1
pi2
pi3
pi4
0
0.23
0.73
0.04
14
0
0.23
0.73
2
0
0.2
3
0.07
0.1
0.78
0.02
15
0
0.08
0.6
0.03
16
0
0.1
4
0
5
0
0
0.98
0.02
17
0
0.08
0.86
0.06
18
0
6
0
7
0
0.2
0.78
0.02
19
0
0
0.94
0.06
20
0.25
8 9
0
0.23
0.73
0.04
21
0
0.2
0.78
0.02
22
10 11
0
0
0.94
0.06
0
0.2
0.78
0.02
12
0
0.2
0.78
13
0
0.1
0.88
1
i
pi1
pi2
pi3
pi4
0.04
27
0
0.23
0.73
0.04
0.86
0.06
28
0
0.2
0.78
0.02
0.88
0.02
29
0
0.1
0.84
0.06
0.2
0.78
0.02
30
0
0.2
0.78
0.02
0.08
0.86
0.06
31
0
0
0.94
0.06
0.1
0.88
0.02
32
0.07
0.4
0.5
0.03
0.4
0.32
0.03
33
0
0.2
0.78
0.02
0
0.23
0.73
0.04
34
0
0.1
0.84
0.06
0
0.1
0.84
0.06
35
0
0.1
0.84
0.06
23
0
0.1
0.9
0
36
0
0.23
0.73
0.04
24
0.25
0.3
0.42
0.03
37
0
0.12
0.86
0.02
0.02
25
0
0.1
0.84
0.06
38
0
0.2
0.78
0.02
0.02
26
0
0.08
0.86
0.06
39
0
0.1
0.87
0.03
The genetic algorithm is used to solve the model of this paper. The parameters of the genetic algorithm are set as: population size popSize=200, elite selection probability Ps=0.2, crossover rate Pc=0.3, mutation rate Pm=0.5, genetic iteration number Nnew=100. The optimal tasks assignment and buffer allocation scheme is shown in Table 4. Table 4. Integrated optimization scheme. k
tasks
Hk-S
Hk
Bk
1
[1,3,2,8,9,10,11,12,15]
1.068587
5.3741
1
2
[4,5,13,18]
0.962887
5.223
3
3
[6,20,21]
2.135096
5.2691
4
4
[7,14,22,26,34]
2.147251
4.8465
4
5
[12,17,19,23,24,37]
1.024515
4.911
3
1.025033
4.921
/
6
[25,27,28,29,30,31, 32,33,35,36,38,39]
In this scheme, the productivity of the line is 46.9643/h through simulation, the time diversity complexity for the whole line is 1.55558bit, the index of difference in time
stochastic fluctuations complexity between stations is 0.34928, and the total buffer capacity is 15. In order to verify the superiority of integrated optimization, integration optimization and serial optimization methods are now compared. The serial optimization is balancing first, and then the buffer allocation. The results are shown in Table 5. Table 5. Integrated optimization and serial optimization scheme comparison. B
PR （ /h
）
HS
HL
Integrated optimization
15
46.9643
8.363369
1.55558
Serial optimization
18
46.6391
8.449385453
1.735126457
The final solution has a large advantage in the total number of buffer capacity while maintaining a high level of productivity. Integrated optimization can obtain a complete scheme including balancing and buffer allocation at the same
6
Xuemei Liu et al. / Procedia CIRP 81 (2019) 1040–1045 Xuemei Liu et al. / Procedia CIRP 00 (2019) 000–000
time, and avoids the problem that serial optimization may not obtain global optimal solution due to premature phenomenon. Therefore, integrated optimization has a large advantage over serial optimization in terms of computational efficiency and optimization results. This paper considers the impact of stochastic fluctuations in processing time. For the problem of stochastic assembly line balance, the common solution is to assume that the processing time obeys the normal distribution, and the equilibrium rate is the highest and the average time of the station is balanced under the condition that the station meets a certain completion rate [12]. In order to verify the effectiveness of the method in this paper, compare it with the method in the literature. Using the method in the reference [12], with the highest balance ratio and the balance of work time, the optimal tasks assignment and buffer allocation scheme are as follows: the tasks in station 1-6 is {1,9,7,8,18},{11,20,15,26,2,3,13},{12,10,4,5,6},{21,14,16,17 ,22},{34,37,23,14,33,24,28,25},{27,29,30,31,32,35,36,38,39 }. The buffer capacity is {2,4,3,3,4}. The Table 6 shows the comparison between the method in this paper and the method in literature. Table 6. Method comparison. Station
Hk in this paper
in literature [12]
1
5.3741
5.8085
2
5.223
5.469
3
5.2691
4.5887
4
4.8465
5.3192
5
4.911
4.8465
6
4.921
5.6296
The index of difference
0.34928
0.683134
PR(/h)
46.9643
43.7182
Compared with the integrated optimization scheme, although the literature method has a relatively low time fluctuation complexity of the 3rd station, the time fluctuation complexity of the 1st and 6th stations is relatively high, which also leads to an increase in the index of difference in time stochastic fluctuations complexity between stations. The increase and imbalance of the time fluctuation complexity will bring the fluctuation of the standard working time of the station, which will adversely affect the production of the assembly line. From the perspective of productivity level, it can be seen that the comprehensive balance scheme considering the stochastic fluctuation factors of the assembly time is more excellent, and has a 7.42% improvement effect compared with the scheme in literature. 6. Conclusion This research deals with simultaneous line balancing and buffer allocation decisions for stochastic mixed-model assembly line. Three types of complexity are defined, which
1045
including time diversity complexity for single station, time diversity complexity for a whole line, and time stochastic fluctuations complexity. An integrated optimization model for line balancing and buffer allocation is established with the objectives of maximizing productivity, minimizing complexity and total buffer capacity. The accurate productivity of the line is obtained by the simulation software, and the improved genetic algorithm is used to solve the model. The case shows that the method can obtain the overall scheme of line balancing and buffer allocation in one optimization process. While maintaining a high balance rate and productivity, the station complexity is more balanced and the total buffer capacity is smaller. This paper only studies the serial assembly line, and does not consider parallel stations. he different structure of the line may also affect the complexity of the assembly line. In further research, assembly lines for different structure will be studied. Acknowledgement This work is supported by National Key R&D Program of China (2018YFB1700902). References [1] Tiacci L. Simultaneous balancing and buffer allocation decisions for the design of mixed-model assembly lines with parallel workstations and stochastic task times. International Journal of Production Economics. 2015, 162(4): 201-215. [2] Thomopoulos N T. Mixed Model Line Balancing with Smoothed Station Assignments. Management Science. 1970(9): 593-603. [3] Haq A N, Rengarajan K, Jayaprakash J. A hybrid genetic algorithm approach to mixed-model assembly line balancing. The International Journal of Advanced Manufacturing Technology. 2006(3-4): 337-341. [4] Hamta N, Ghomi S M T F, Jolai F, et al. A hybrid PSO algorithm for a multi-objective assembly line balancing problem with flexible operation times, sequence-dependent setup times and learning effect. International Journal of Production Economics. 2013, 141(1): 99-111. [5] Hop N V. A heuristic solution for fuzzy mixed-model line balancing problem. European Journal of Operational Research. 2006, 168(3): 798810. [6] Cakir B, Altiparmak F, Dengiz B. Multi-objective optimization of a stochastic assembly line balancing: A hybrid simulated annealing algorithm. Computers & Industrial Engineering. 2011, 60(3): 376-384. [7] Efthymiou K, Pagoropoulos A, Papakostas N, et al. Manufacturing systems complexity: An assessment of manufacturing performance indicators unpredictability. Cirp Journal of Manufacturing Science & Technology. 2014, 7(4): 324-334. [8] Zhu X, Hu S J, Koren Y, et al. Modeling of Manufacturing Complexity in Mixed-Model Assembly Lines. Journal of Manufacturing Science and Engineering. 2008, 130(5): 649-659. [9] Jaber M Y. Lot sizing with permissible delay in payments and entropy cost. Computers & Industrial Engineering. 2007, 52(1): 78-88. [10] Demir L, Tunali S, Eliiyi D T. The state of the art on buffer allocation problem: a comprehensive survey. Journal of Intelligent Manufacturing. 2014, 25(3): 371-392. [11] Li L, Qian Y L, Yang Y M, et al. A fast algorithm for buffer allocation problem. International Journal of Production Research. 2016, 54(11): 113. [12] Zhou L, Song H, Han Y. Stochastic production line load balancing based on genetic algorithm. Mechanical Manufacturing. 2003, 41(3): 23-25.