Simple assembly line balancing problem under task deterioration

Article history: Received 18 December 2010 Received in revised form 18 February 2011 Accepted 21 February 2011 Available online 22 February 2011 This paper introduces the effect of task deterioration in simple assembly line balancing problem. In many realistic assembly lines, a deterioration task is considered when a task is started earlier than the assigned time since the station time is constant and the earliness of the task does not reduce the cycle time. This phenomenon is known as deteriorating tasks. Therefore, we seek an optimal assignment and schedule of tasks in workstations, in order to minimize the number of stations for a given cycle time, which is known as SALBP-1. For this purpose, a mathematical model is proposed. Since the pure SALBP-1 is proved to be NP-hard and considering task deterioration complicates problem further, we propose a genetic algorithm for solving such problem. Several well-known test problems are solved to study the performance of the proposed approach. © 2011 Growing Science Ltd. All rights reserved


Introduction
The initial study in assembly line balancing (ALB) problem is in assigning tasks to work concentrating on precedence relationship among the tasks to optimize some measurable performance (Erel & Sarin, 1998).Bryton (1954) conducted the first attempt in assembly line balancing and Salveson (1955) aimed to publish the first work in ALB problem.The objective functions could be categorized in two distinctive points of view, including minimizing cycle time in constant work stations, which is defined in (SALBP-2), and minimizing the workstation in fixed cycle times, which is defined in (SALBP-1) (Becker & Scholl, 2006).The workstation is a set of tasks that are to be assigned to some resources and the cycle time is defined as the time between completions of successive products.Kara et al. (2009), Özcan and Toklu (2009), Scholl and Becker (2006), Lapierre et al.(2006), Kim et al.(2009), Levitin et al.(2006), Andre´s et al.(2008) and Toksarı et al.(2008) addressed the ALB problem with different objective functions and the cycle time is considered to be fixed in most of these papers.Although in a real situation, the cycle time could be increased due to tardiness and lag caused by the precedence workstation, Gupta and Gupta (1990) and Browne and Yechialli (1990) made the first endeavor in this field by considering a dependent function on starting time for processing of elements in workstation.A fire fighting example is introduced by Kunnathur and Gupta (1990), in which the time for necessary measures to fight the fire increases if the actions are started by delay.Other similar examples for the aforementioned could be found in line production systems, financial and accounting management, etc. Owing to the wide application of the processing time in different conditions, a wide variety of models based on time for processing have been developed by Ji andCheng (2009, 2008), Wang et al. (2009aWang et al. ( , 2009bWang et al. ( , 2009c)), Low et al. (2008) and Yin et al. (2010).Especially, in line-balancing problem, Toksar (2010) considered machine-human fault and considered aspects altogether to provide an optimal solution in (SALBP-1).A comprehensive review of literature of the assembly-line balancing problem reveals that there is no hypothesis that considers task-deterioration.Considering the fact that this characteristic aids in applying more realistic definitions of SALBP compared with traditional models and formula in processing time of deteriorating tasks, the hypothesis is of great importance.The problem description is asserted in section 2, which is followed by mathematical modeling in section 3. A developed genetic model for SALBP is presented in section 4 and ultimately two well-known examples are presented and solved by mathematical and developed GA and the results are compared for two methods for small and large-scale problems.In small-scale problem, the mathematical modeling is proposed to provide the optimum solution and for large-scale problems, the comparison is executed while the effects of deterioration are not considered.

SALBP under task deterioration
The assembly line is defined as a group of workstations, which is a place in which some specific operations are performed on products, and certain task scheduled for each workstation.Passing time in workstation for products is known as cycle time, (C) and equivalent to the time interval between successively completed production units.The number of tasks is shown by N and should be assigned in workstations.The processing time function is presented as the following equation, Task-deterioration is considered in Eq. ( 1).P j is the task time of task j, a j is the fixed time in processing time, b j is the deterioration growth rate and T j denotes the delay in processing time and is defined as a gap time from the beginning up to starting of the tasks.The assumption of lag-time at the beginning time of tasks in this paper differs from the specific scheduling problem due to precedence relationship among the tasks.The tasks are considered available from the beginning in scheduling problem, although in assembly line the precedence relationships and the nature of tasks result in preventing to use this assumption, which is typically used in scheduling problem.The Eq. ( 1) is not used in this field and some adjustment seems to be necessary.Consequently, the developed form of Eq. ( 1) is presented in Eq. (2) which considers the available time and start time of each task, separately, ( ) The start-time of task is denoted by ST j and the available-time is presented by AV j .In assembly line, each task can be started only when all predecessors have finished and the available-time is known as the time when all the related predecessors are finished.The delay-time of the task is modeled as the difference between the start-time and available-time.

Mathematical formulation
The objective is to assign and schedule tasks to workstations so as to minimize the number of work stations for a given cycle time (SALBP-1).So there are N tasks (i ∈ {1…N}) that should be assigned and scheduled and it is evident that the minimum value of the number of stations is equal to and the maximum value of it equals the number of all tasks (each task is positioned in one separate station).Note that this model includes one assembly cycle, which means that every output product is essentially an output of the line.Additional parameters and variables are as follows, (1 ) ) 3) is the objective function of the model, which aims to minimize the number of workstations; Eq. ( 4) describes the workstation time limitation that must be less than cycle time.Constraint (5) denotes the total operating time for each station.Eq. ( 6) requires that each position in sequence must be filled with a task and the one-to-one correspondence between the tasks and positions is shown in Eq. ( 7).Eq. ( 8) calculates the number of workstations, and Eq. ( 9) ensures that the action of task assignments starts from the first workstation; as a result of such constraint, one can conclude that there is s no empty station between two active stations.Eq. ( 10) ensures that if a position is assigned to a station, the station must be necessarily selected and counted as the number of stations.Eq. ( 11) ensures that each position can be assigned only to one station.Eq. ( 12) denotes the task-time of position j.Eq. ( 13) calculates the start-time of the task in position j, and Eq. ( 14) counts for the starttime of station k in each cycle.Constraint (15) requires that the start-time of the task in position j and station k must be equal or more than the start-time of station.Eq. ( 16) denotes the equality of starttime of the task in position j and the task in position j in station k.Eqs.(17)(18)(19) are respectively in charge of the start time, completion time and available time for task i, and finally, constraint (20) guarantees the precedence relationships among the tasks.SALBP is classified in NP-hard problems and the considered problem in this paper with additional issue of task deterioration is more complicated.As known, the mathematical models are not able to solve this complex category of problems in the reasonable computational time for large-scale problems.Therefore, a GA is developed, which is known as a popular and the most practical metaheuristic algorithm for this field of problems.

Genetic algorithm for SALBP-1
Most of the problems dealing with the design of production systems are categorized as combinational and NP-Hard problems.The assembly-line balancing problems, among the combinational optimization problems, are also categorized as NP-Hard (Karp, 1972).In recent years, most researchers have used genetic algorithm to solve such problems (Kim et al. 2009, Chen et al. 2002).In genetic algorithm, the initial population of solutions, which appear as a set of feasible solutions (chromosomes), are selected randomly.The population is evaluated and the corresponding fitness function is calculated.The amount of fitness function provides a relative preference for the chromosomes; chromosomes with higher fitness, present a better solution.Taking steps toward optimization is achieved by generation, which is stated in the following form: 1. Selection: Some chromosomes are selected from the current population, note that chromosomes with higher fitness have more chance to be selected.2. Crossover: The selected chromosomes mate and generate offsprings 3. Mutation: Some chromosomes of the new population mutate This process is repeated until the solution is achieved (Goldberg 1989, Michalewicz 1996).In the developed GA, the number of iterations depends on the scale of the problems, which is considered 100×n (n is the number of all tasks) as this number of iterations has resulted in good outputs after testing several values for the number of iterations.The structure of proposed GA is described in following sections.

Representation and initial population
Each feasible solution is represented by a permutation of tasks, which satisfies the precedence relationships among the tasks.This type of representation clearly determines the sequence of tasks.The number of stations for each individual is calculated through the following statements.Beginning from the first positioned task, tasks are assigned to the current station such that their sequence is maintained.Whenever the sum of processing times in a station exceeds the cycle time, the task is assigned to the next station.Initial population is generated for a certain number such that the precedence relations are observed.We have chosen 50 through experimental evidences.The structure of a chromosome is depicted in Fig. 3.

4.2.Selection mechanism
The popular Roulette wheel selection mechanism is used in this work (Sivanandam & Deepa, 2008).
After selection, the chromosomes are mated to produce a new population.

Genetic operators
The developed GA applies the popular precedence preservative crossover (PPX) (Sivanandam & Deepa 2008).After exerting crossover, some of individuals may undergo a mutation process, which selects a random task and exchanges it with preceding or following task randomly in a way that the precedence relationships are satisfied.This operator is executed with the probability of 0.1.

Computational results
First, the small example of Mertens, available at www.assembly-line-balancing.de, is investigated to examine the developed GA and the results are compared with the optimized solution.The precedence  1.The model results and the applied assignments from the genetic algorithm, which is derived from Mertens model, are shown in Table 2.The remaining results from the model and the genetic algorithm solution of Mertens and Jackson are presented in Table 3.In this table, the number of workstation is determined considering different cycle-times proposed for the model.Also, the genetic algorithm is run 20 times for each instance; and the maximum, the minimum and the standard deviation (SD) of the results are calculated and finally, the optimized solution of preceding examples are presented in the last column, which denotes the efficiency of the presented model and genetic algorithm.
Next, from Tables 1 to Table 4, Mertens example is solved for different cycle times (10, 15 and 18) considering task deterioration and using the presented model.Deterioration rate is considered the same for all tasks.As can be seen, the results obtained from the genetic algorithm are the same as the results of the presented model.Next, in order to evaluate the efficiency of genetic algorithm, some examples are solved for large-scale problems using the model without considering task deterioration, and the solutions are compared with the optimal solutions.The final solutions of the proposed model of this paper for large-scale models are the same as the one reported by Buxey and Killbridge.For Lutz model, also, the solution of genetic algorithm is close to the optimal one.Table 10 includes the results showing the efficiency of genetic algorithm.Finally, the large-sized examples are solved including the task deterioration rates (0.1 and 0.2) and using the developed genetic algorithm.The results are presented in Table 11.As shown in most of the examples for large-scale problems, the effect of task deterioration in increasing the number of workstation is significantly more than the case of small-scale problems.For example, by considering Tables 4 and Table 5, when the deterioration rate changes from 0.1 to 0.2 there is no increase in the number of stations for the small-scale problem of Mertens, while for the same change in deterioration rate for Lutz's example with cycle-time of 20, the number of workstations increases from 31 to 36 (Table 11).

Table 2
The results of SALBP-1 with the developed model and proposed GA without the effect of task deterioration for Mertens (Ct=10)

Table11
The results of SALBP-1 with the proposed GA with the effect of task deterioration for large-scale problems with task deterioration effect of 0.1 and 0.2

Conclusion
In this paper, the effect of task deterioration with a new concept in simple assembly-line balancing problem type1 (SALBP-1) is studied for the first time.A mathematical model and a genetic algorithm are proposed.Several well-known examples have been solved without the effect of deterioration for verifying the effectiveness of proposed model and GA, and then the same problems have been considered with the effect of task deterioration.The results indicated that by increasing the impact of task deteriorations, the number of workstations or the value of cycle time increase.Therefore, the increasing in the number of workstations and cycle time stimulate managers to keep both factors in minimum range while tasks are affected by deterioration.Finally, we have concluded that the existence of deterioration manages to increase the number of workstations depending on the task deterioration rates, but these increments can be kept in minimum ranges by optimum scheduling and sequencing of deteriorated tasks in workstations.

Fig. 1 .
Fig. 1.Precedence relations of a small sample Fig. 2. Sequence of tasks in stations for a small sample

jT
Designates the positions ({1,.., N}) of jobs in a sequence task in position j and assigned in station k, k The starting time of station k in each production cycle that final product exists from line, Z Number of occupied work stations (Z ∈ {Z min ,.., Z max }).The optimization model can be formulated as follows.Decision variables:

Table 3
The other results of SALBP-1 with proposed GA and developed model for the small-scale problems without task deterioration effect

Table 5
The results of SALBP-1 with the developed model and proposed GA with the effect of task deterioration for Mertens (Ct=10, deterioration=0.2)

Table 7
The results of SALBP-1 with the developed model and proposed GA with the effect of task deterioration for Mertens (Ct=15, deterioration=0.2)

Table 10
The results of SALBP-1 with proposed GA for large-sized problems without task deterioration effect