Procedia Computer Science 20 ( 2013 ) 83 – 89
1877-0509 © 2013 The Authors. Published by Elsevier B.V.
Selection and peer-review under responsibility of Missouri University of Science and Technology doi: 10.1016/j.procs.2013.09.243
ScienceDirect
Complex Adaptive Systems, Publication 3 Cihan H. Dagli, Editor in Chief
Conference Organized by Missouri University of Science and Technology 2013- Baltimore, MD
Multi-objective Evolutionary Algorithm with Strong Convergence
of Multi-area for Assembly Line Balancing
Problem with Worker Capability
Wenqiang
Zhang
a, Weitao Xu
aand Mitsuo Gen
b*
aCollege of Information Science and Engineering, Henan University of Technology, China bFuzzy Logic Systems Institute, Japan and Dept. of IE&EM, National Tsing Hua University, Taiwan
Abstract
Multiobjective assembly line balancing with worker capability (moALB-wc) is a realistic and important issue from classical assembly line balancing (ALB) problem involving conflicting criteria such as the cycle time, the total worker cost, and/or the variation of workload. This paper proposes a multiobjective evolutionary algorithm (MOEA) with strong convergence of multi-area (MOEA-SCM) to deal with moALB-wc problem considering minimization of the cycle time and total worker cost, given a fixed number of station limit. It adopts special fitness function strategy considering dominating and dominated relationship among individuals and hybrid selection mechanism so as to the individuals could converging toward the multiple areas of Pareto front. Such ability to strong convergence of multi-area could preserve both the convergence and even distribution performance of proposed algorithm. Numerical comparisons with various problem instances show that MOEA-SCM could get the better convergence distribution performance than existing MOEAs.
Keywords: assembly line balancing, worker capability, evolutionary algorithm;strong convergence of multi-area; multiobjective optimization
1.Introduction
The assembly line balancing (ALB) problem determines the assignment of various tasks to an ordered sequence of stations, while optimizing one or more objectives without violating restrictions imposed on the line in a manufacturing system. The traditional ALB problem is to process one model with tasks by allocating the tasks into stations to generate an optimal solution by considering the task time and task cost is the same for each worker. However, in the modern industries, according to different work capability, the skill level of a worker in a given task differs greatly among workers, i.e., the task times depend on the worker since they have different skill and capability[1]. Furthermore, operating the line causes short-term operating costs such as wages, material, set-up,
inventory and incompletion costs, especially, the labor cost or worker cost is the main cost in most manufacturing companies. Therefore, the manufacturing company managers have hired more non-regular employees to reduce the permanent worker cost. However, the workers with high skill level need to be paid more salaries although the
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© 2013 The Authors. Published by Elsevier B.V.
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assignment of them into assembly line could increase the line efficiency. How to allocate the proper workers to proper stations to obtain the best efficiency of the line and reduce the total cost is also a problem in multiobjective assembly line balancing with worker capability (moALB-wc).
The problem of balancing of assembly lines has been extensively examined in the literature and a number of review studies have been published [2-6]. Both exact and heuristic procedures have been developed to solve this
problem. Pinto et al. considered ALB problem where several alternatives are available in the choice of processing at each station[7]. Graves and Holmes suggested an algorithm for assignment of activities and equipment to assembly
line stations, satisfying the annual production rate [8]. Hopp et al. set out a case in which workers can vary in speed
and are benchmarked by defining the speed factor of each worker relative to a “standard worker”[9]. Zhang et al.
proposed a random key-based approach to deal with ALB problem with worker allocation [10]. However, most of the
above research works on ALB with considering worker capability prefer single objective optimization rather than multiobjective optimization.
Multiobjective optimization problem (MOOP) is a practical, important but very intractable optimization problem in which two of more conflicting objectives should be considered together, and many Pareto-optimal solutions with incommensurable quality are generated for decision makers. Multiobjective evolutionary algorithms (MOEAs) have been recognized to be well-suited for solving MOOPs [11, 19]. As two classical MOEAs, non-dominated sorting
genetic algorithm-II (NSGA-II) [12] and strength Pareto evolutionary algorithm 2 (SPEA2) [13] have been proven to
be able to get better quality in solving MOOPs. NSGA-II can get better quality owing to its Pareto ranking and crowding distance mechanism. SPEA2 depends on raw fitness assignment mechanism and density mechanism. However, these two algorithms both need CPU time. Zhang and Fujimura proposed an improved vector evaluated genetic algorithm with archive (IVEGA-A) that combined VEGA and Pareto-based scale-independent fitness function (GPSI-FF)-based archive mechanism for solving the process planning and scheduling problem [14].
Furthermore, under the consideration of the multiobjective characteristic of moALB-wc with simultaneously optimizing the two conflicting objectives, the cycle time and worker cost, seeking an optimal solution rapidly and effectively from all of the permutations, combinations of all of the tasks, manufacturing resources, human resources is very difficult. As for the MOEA, the special mechanisms need to be designed to increase the quality (both of convergence and distribution) and to reduce the computational time so that the heuristic algorithm can be applied easily on the real world ALB problem. In this study, we consider the moALB-wc and propose a new multiobjective evolutionary algorithm with strong convergence of multi-area (MOEA-SCM). The MOEA-SCM could converge to the center and two edges areas of Pareto front strongly and could both preserve the convergence rate and guarantee the better distribution performance.
The paper is organized as follows: Section 2 formulate the mathematical model for moALB-wc problem; Section 3 presents the detailed MOEA-SCM approach; Section 4 gives a discussion and analysis of numerical experiments results; finally, the conclusion and future work are given in Section 5.
2.Mathematical Formulation
In this study, the moALB-wc problem subjects to the following assumptions: A1. The precedence constraints among assembly tasks are known and constant.
A2. A worker is assigned to one station, and only processes the tasks assigned to that station. A3. A task cannot be split amongst two or more stations.
A4. The processing time of worker for each task is known.
A5. Material handling, loading and unloading times, set-up and tool changing times are negligible, or are included in the processing times.
A6. Task processing time differs among workers because of workers’ differences in work experience. A7. A worker can process all the tasks, and his/her work experience differs among tasks.
A8. Stations are located along a conveyor belt according to increased station index. A9. Workers have different worker cost according to the different work experience.
The moALB-wc problem concerns with the assignment of the tasks to stations and the allocation of the available workers for each station in order to minimize the cycle time and minimize the total cost under the constraint of precedence relationships. The notation used in the mathematical model can be summarized as follows:
Indices:
j, k: indices of task (j, k=1,2,…,n); i: index of station(i=1,2,…,m); w: index of worker (w=1,2,…, m).
Parameters:
n: number of tasks; m: number of stations/workers; djw: worker cost of worker w process task j;
tjw: processing time of task j by worker w; Suc(j): set of direct successors of task j; Pre(j): set of direct predecessors of task j; Si: set of tasks assigned to station i;
t(Si): processing time at station i, ( ) 1 1 ,
n m
i j w jw ij iw
t S t x y i;
ui: utilization of the station Si,
1 ( ) max ( ) i i i m t S i t S
u ; u: average utilization of all
stations, 1 1 m m i i u u . Decision Variables:
1, if task is assigned to station 1, if worker is working in station
, 0, otherwise 0, otherwise ij iw j i w i x y Mathematical Model: T 1 1 1 1 1 1 1 1 1 1 min max (1) min (2) s.t. , Pre( ), (3) 1 , (4) 1, (5) 1, (6) , 0,1 , , , (7) i n m jw ij iw j w i m m m jw iw i j S w m m ij ik i i m ij i m iw w m iw i ij iw c t x y d d y ix ix k j j x j y i y w x y i j w
The first objective (1) of the model is to minimize the cycle time of the assembly line. The second objective (2) is to minimize the total worker cost. Inequity (3) states that all predecessor of task j must be assign to a station, which is in front of or the same as the station that task j is assigned in. Equation (4) ensures that task j must be assigned to only one station. Equation (5) ensures that only one worker can be allocated to station i. Equation (6) ensures that worker w can be allocated to only one station and equation (7) represents the nonnegative restrictions.
3.Multiobjective Evolutionary Algorithm with Strong Convergence of Multi-area
3.1.Fitness Function
In recent MOEA, the Pareto domination relationship is usually used in calculating fitness function. NSGA-II and SPEA2 use rank and raw fitness as well as crowding distance as fitness function to evaluate the individuals. In this study, a Pareto dominating and dominated relationship-based fitness function (PDDR-FF) -based fitness function is proposed to evaluate the individuals. The PDDR-FF of an individual si is calculated by the following function:
( )i ( ) 1/ ( ( ) 1),i i 1, 2,...,
eval s q s p s i popSize (8)
The smaller value is better. If the individual belongs to nondominated one, its fitness value will not exceed one. The fitness value of individual which is dominated by other will exceed one. It is obvious that the nondominated individuals locating around the central area of Pareto font with bigger domination area will have smaller values (near to 0) than the edge points (near to 1).
3.2.Main Framework of MOEA-SCM
The main framework of MOEA-SCM is shown in Fig. 1 where A(t) represents the archive at generation t and P(t)
A
A((tt))
1 1 1
copy the best
A(t)| individuals ntoA(ttt+1) from P(ttt+1)++ (At) by PDDR-FF value. rossov mutatio pops pops pops pops pops p piiizeizeizeize
p p P P((tt)) b b sub-pop-111 b b sub-pop-222 VEGA
generation t mating pool
cr m 1 popsize popsiiize P(t+1) generation enerationt t+1+ A A A((tt)) verrrrr on A(t+1) c |A in P b v 1 1 1 1 A A((tt))
archivet archivet archivet archivet+1
min fff1
minfff2
VEGA for Left-top area
VEGA for Right-bottom area PDDR-FF for Central area
Fig. 1 the framework of MOEA-SCM
In selection phase of MOEA-SCM, the PDDR-FF based sampling strategy has the advantage with the tendency converging toward the canter area of the Pareto front, but drawback to the edge region. It causes bad distribution performance. The VEGA prefers the edge rather than certer regions of Pareto front that it causes VEGA cannot achieve better distribution performance. So we combine these two mechanisms to improve the overall performance and reduce the computation time.
The solution procedure of one generation includes 3 phases.
Phase 1: Generating Mating Pool
Step 1: Selecting individuals into sub population 1 and 2 by VEGA (good for the edges area of Pareto front). In this step, individuals are selected with replacement according to objective 1 into sub population 1 while ignoring objective 2 until the size of the sub population 1 (half of population size) is reached. In the same manner, individuals are selected for objective 2 into sub population 2 without considering objective 1 until sub population 2 is full.
Step 2: Combining the sub populations and archiveA(t) to form the mating pool (good for the central area of Pareto front). After generating the sub populations in mating pool, the all individuals in archiveA(t)are also as parts of mating pool. As shown in Fig. 1, the sub-pop-1 stores the good individuals for one objective, and sub-population 2 holds the good individuals for the other objective. The archive saves the individuals with good PDDR-FF values. Therefore, in the mating pool, one-third of the individuals serve one objective, one-third the other objective, and the left one-third both the two objectives. The archive mechanism tries to cover the selection bias of VEGA. These three parts of the mating pool make the solutions converge to the Pareto front evenly.
Phase 2: Reproducting the new population
Arithmetical crossover (for task priority vector), weight mapped crossover (WMX) (for worker allocation vector), swap mutation are used to reproduce new individuals.
Phase 3: Updating the Archive
The individuals of A(t)and P(t+1)are combined to form a temporary archiveA’(t). Thereafter, the PDDR-FF values of all individuals in A’(t)are calculated and sorted. The smallest |A(t)|individuals in A’(t)are copied to form
A(t+1). This archive updating mechanism likes a elitist sampling strategy to keep the better individuals with better PDDR-FF values.
The strong convergence capability of VEGA and PDDR-FF ensures that the MOEA-SCM has the ability to converge to the true Pareto front both in central and edge areas.
3.3. Genetic Representation
As an illustrative example, an assembly line having 10 tasks, 4 stations and 4 worker is used. The precedence graph is shown in Fig. 2. The processing time of worker and cost data are shown in Table 1.
1 2 3 4 5 6 8 7 9 10
Fig. 2 A precedence graph for the example
Table 1. Data set of the example.
Task Predecessor Task Time Unit / Cost Unit
j Pre(j) w1 w2 w3 w4 1 Ø 17/83 22/78 19/81 13/87 2 Ø 21/79 22/78 16/84 20/80 3 Ø 12/88 25/75 27/73 15/85 4 {1} 29/71 21/79 19/81 16/84 5 {2,3} 31/69 25/75 26/74 22/78 6 {4} 28/72 18/82 20/80 21/79 7 Ø 42/58 28/72 23/77 34/66 8 {6} 27/73 33/67 40/60 25/75 9 Ø 19/81 13/87 17/83 34/66 10 {7,8,9} 26/74 27/73 35/65 26/74
The detailed genetic representation consists of three phases:
Phase 1: Creating a task sequence
Step 1.1: Generating a real number for each task as task priority in task priority vector (v1). The generated task
priority vector is shown in Fig. 3. 1
1 22 33 44 55 66 77 88 99 1010
)3.31 7.14 6.59 2.24 8.67 5.38 4.13 9.87 10.72 1.91 Fig. 3 The task priority vector
1
1 22 33 44 55 66 77 88 99 1010
9 2 3 5 7 1 4 6 8 10
Fig. 4 The task sequence vector
Step 1.2: Creating a task sequence (v2) by decoding method. At the beginning of creating a task sequence
procedure, we try to find a task for the first station. Task 1, 2, 3, 7, 9 are eligible for the position. From the task priority vector, the task 9 has the highest priority and is put into the task sequence. Then we delete task 9 off the precedence graph. By the same manner, the chromosome can easily be encoded. The generated task sequence vector is shown in Fig. 4.
Phase 2: Assigning worker to each station
Step 2.1: Encoding worker allocation vector (v3) by randomly assign the worker to each station (as shown in
Fig. 5).
1
1 22 33 44
3 4 1 2
Fig. 5 worker allocation vector
1
1 22 33 44
1 4 7 9
Fig. 6 The breakpoint vector
Phase 3: Assigning tasks to each station
Step 3.1: Calculating the lower bound and the upper bound cycle time.
Step 3.2: Finding an optimal cycle time by bisection searching.
Step 3.3: Dividing the task sequence to form the breakpoint vector (v4) (as shown in Fig. 6). 3.4.Genetic Operators
Considering the above four vectors, we used arithmetical crossover (for task priority vector), weight mapped crossover (WMX) (for worker allocation vector), Swap mutation, and binary tournament selection operator (see Zhang et al. 2008 for a detailed discussion).
4. Experiments and Discussion
We employed Gunther’s problem data with 35 tasks and 6 stations, which is widely used in the ALB problem literature (Scholl, 1993)[15]. The number of workers is the same as the number of stations, 6 and the worker cost data
are generated randomly according to work level (see Zhang et al. 2008 for details).
All the simulation are performed on Pentium Dual-Core processor (2.70 GHz clock) and 2GB memory. The adopted parameters are listed as follows: population size, 100; maximum generation, 500; archive size, 50; crossover probability, 0.80 and 0.3; mutation probability, 0.40 and 0.1. MOEA-SCM, NSGA-II, and SPEA2 are run 30 times to compared the results with each other. It should be noted that the parameters of all 3 methods are the same, except for the size of archive. The archive sizes of MOEA-SCM is set to be half the population size, 50, while of NSGA-II and SPEA2 are set to be the same as the population size, 100.
Let SSjbe a solution set for each method. PF* is a known reference Pareto solutions. In this study, PF*in this study comes from combining all of the obtained Pareto set with 30 runs by 3 methods. The following three performance measures are considered.
Coverage C(S1,S2) is the percent of the individuals in S2 which are weakly dominated by S1[16]. The larger C(S1,S2) means that S1outperformsS2in convergence.
Generational distance GD(SSj) finds an average minimum distance of the solutions of SSj from PF*[17]. The
smaller GDof SSjmeans better SSjin approachingPF*.
Spacing SP(SSj) is the standard deviation of the closest distances of individuals bySSj [18] . Smaller SPmeans better distribution performance.
TheC, GDare used to verify convergence performance whileSPis used to check the distribution performance.
0.4 0.5 0.6
C(M OEA-SCM ,SPEA2) C(SPEA2,M OEA-SCM )
C
ove
ra
ge
(b) Coverage for M OEA-SCM and SPEA2
0.3 0.4 0.5 0.6 0.7
C(M OEA-SCM ,NSGA-II) C(NSGA-II,M OEA-SCM )
C
ove
ra
ge
(a) Coverage for M OEA-SCM and NSGA-II
3 4 5 6
M OEA-SCM NSGA-II SPEA2
A ve ra ge D is ta nc e (c) Average Distance 5 10 15 20
M OEA-SCM NSGA-II SPEA2
Sp ac in g (d) Spacing 25 30 35
M OEA-SCM NSGA-II SPEA2
C
P
U
tim
e
(e) CPU time
Fig. i 7 C, GD, SP and CPU times by 3 methods
The Fig. 7 show the numerical comparison of the box-and-whisker plots for C, GDandSPand the CPU times by 3 methods. The Fig. 7a and Fig. 7b demonstrate that the MOEA-SCM is better than other 2 methods on Cmeasure. The GDmeasure also indicates that MOEA-SCM can get slightly smaller median value than other 2 methods (Fig.
7c). The distribution performance SP indicates that MOEA-SCM is slightly better than NSGA-II and SPEA2
methods (Fig. 7d). From the comparisons of CPU time as shown in Fig. 7e, MOEA-SCM is much faster than NSGA-II and SPEA2.
In general, the convergence and distribution performance of MOEA-SCM is better than famous NSGA-II and SPEA2, and the efficiency is better.
5. Conclusions
In this study, a MOEA-SCM approach was proposed to solve moALB-wc. This approach mainly used the selection mechanism of VEGA and the PDDR-FF based archive strategy. These two mechanisms could converge to the multi-area of Pareto front strongly and obtained even distribution performance without special crowding distance mechanism. Meanwhile, the MOEA-SCM could reduce the CPU time rather than traditional MOEA. Numerical comparisons demonstrated that MOEA-SCM could achieve better performance both in efficacy and efficiency.
Acknowledgments
This research work is partly supported by the Education Dept. of Henan Province: Basic Research Program of Sci. and Tech. Key Project (No. 13A520203), the Plan of Nature Science Fundamental Research in Henan University of Tech. (No. 2012JCYJ04), the JSPS Grant-in-Aid for Scientific Research 862 (C) (No. 24510219) and the National Science Council, Taiwan (NSC101- 2811-E-007-004, NSC102-2811-E-007-005).
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