Ann Oper Res (2016) 242:321–354 DOI 10.1007/s10479-014-1641-3
A hybrid genetic algorithm for sequence-dependent disassembly line balancing problem Can B. Kalayci · Olcay Polat · Surendra M. Gupta
Published online: 19 June 2014 © Springer Science+Business Media New York 2014
Abstract For remanufacturing or recycling companies, a reverse supply chain is of prime importance since it facilitates in recovering parts and materials from end-of-life products. In reverse supply chains, selective separation of desired parts and materials from returned products is achieved by means of disassembly which is a process of systematic separation of an assembly into its components, subassemblies or other groupings. Due to its high productivity and suitability for automation, disassembly line is the most efficient layout for product recovery operations. A disassembly line must be balanced to optimize the use of resources (viz., labor, money and time). In this paper, we consider a sequence-dependent disassembly line balancing problem (SDDLBP) with multiple objectives that requires the assignment of disassembly tasks to a set of ordered disassembly workstations while satisfying the disassembly precedence constraints and optimizing the effectiveness of several measures considering sequence dependent time increments. A hybrid algorithm that combines a genetic algorithm with a variable neighborhood search method (VNSGA) is proposed to solve the SDDLBP. The performance of VNSGA was thoroughly investigated using numerous data instances that have been gathered and adapted from the disassembly and the assembly line balancing literature. Using the data instances, the performance of VNSGA was compared with the best known metaheuristic methods reported in the literature. The tests demonstrated the superiority of the proposed method among all the methods considered.
C. B. Kalayci · O. Polat Department of Industrial Engineering, Pamukkale University, Denizli 20070, Turkey e-mail:
[email protected] O. Polat e-mail:
[email protected] S. M. Gupta (B) Department of Mechanical and Industrial Engineering, Northeastern University, 360 Huntington Avenue, 334 SN, Boston, MA 02115, USA e-mail:
[email protected] URL: http://www.coe.neu.edu/~smgupta/
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Keywords Reverse supply chain · Disassembly · Assembly · Sequence-dependent disassembly line balancing · Metaheuristics · Hybrid genetic algorithm · Variable neighborhood search
1 Introduction In reverse supply chains, which are used for product recovery, environmental laws and directives are as important as cost minimization and profit maximization (Wang and Gupta 2011). Product recovery for end-of-life (EOL) products is achieved through remanufacturing, refurbishing or recycling. Gungor and Gupta (1999) and Ilgin and Gupta (2010) provide extensive reviews of product recovery. Out of these, remanufacturing is most desirable as it is the most profitable and environmentally friendly option. One of the goals of a remanufactured item is to make it as good as new. This way the remanufacturer can offer new products’ warranties for the remanufactured products to consumers (Ilgin and Gupta 2012). This is not the case with refurbished products where the goal is to keep the products functional by changing and/or repairing some of the components. Of course, with recycled products, the identity of the original product is lost and only the materials recovered from that product (instead of virgin materials) are used to produce new products. The most critical and time consuming step of remanufacturing is disassembly. Disassembly tasks were traditionally carried out using manual labor. However, firms are embracing automated disassembly systems at an increasing rate because of their desire to reduce electronic scrap, increase productivity levels and reduce labor costs. To that end, disassembly lines are most suitable for disassembly operations. A disassembly line must be balanced to optimize the use of resources (viz., labor, money and time). Disassembly operations have unique characteristics and cannot be considered as the reverse of assembly operations. In an assembly line, the quality and quantity of components used at the stations can be controlled by imposing strict conditions. However, no such conditions can be imposed on EOL products moving on a disassembly line. In a disassembly environment, the flow process is divergent; a single product is broken down into many subassemblies and parts. The flow process is convergent in an assembly environment. There is also a high degree of uncertainty in the structure, quality, reliability and the condition of the returned products in disassembly. Additionally, some parts of the product may be hazardous and may require special handling which affects the utilization of disassembly workstations. Since disassembly tends to be expensive, disassembly line balancing becomes significant in minimizing resources invested in disassembly and maximizing the level of automation. Disassembly line balancing problem (DLBP) is a multi-objective problem that was first described by Gungor and Gupta (2002) and was mathematically proven to be NP-complete by McGovern and Gupta (2007) making the goal to achieve the optimal balance computationally expensive. Exhaustive search works well enough in obtaining optimal solutions for small sized instances; however its exponential time complexity limits its application on large sized instances. Thus, an efficient search method needs to be employed to attain a (near) optimal condition with respect to objective functions. Although some researchers have formulated the DLBP using mathematical programming techniques (Altekin and Akkan 2012; Altekin et al. 2008; Koc et al. 2009), it quickly becomes unsolvable for a practical sized problem due to its combinatorial nature. For this reason, there is an increasing trend for using metaheuristic techniques such as genetic algorithms (GA) (McGovern and Gupta 2007), ant colony optimization (ACO) (Agrawal and Tiwari 20080; Ding et al. 2010; Kalayci and Gupta 2013a; McGovern and Gupta 2005a), simulated annealing (SA) (Kalayci and Gupta 2013d; Kalayci
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et al. 2012), tabu search (TS) (Kalayci and Gupta 2013e), artificial bee colony (ABC) (Kalayci and Gupta 2013b) and particle swarm optimization (PSO) (Kalayci and Gupta 2013c). See (McGovern and Gupta 2011)for more information on DLBP. Scholl et al. (2008) introduced the concept of sequence dependency in assembly lines and this concept was extended to the disassembly line balancing problem by Kalayci and Gupta (2013a). In this paper we consider a sequence-dependent DLBP (SDDLBP), which is a generalization of the DLBP, and propose to solve it using a hybrid genetic algorithm (HGA) approach that combines a variable neighborhood search method with genetic algorithm (VNSGA). We test its performance on numerous data instances that have been gathered and adapted from the disassembly and the assembly line balancing literature and compare the performance of VNSGA with the best known metaheuristic methods reported in the literature and demonstrate the superiority of the proposed method among all the methods considered. The rest of the paper is organized as follows: problem definition and formulation is given in Sect. 2. Section 3 describes the proposed VNSGA for the multi-objective SDDLBP. The computational experience to evaluate its performance on numerical examples and the comparisons are provided in Sect. 4. Finally some conclusions are given in Sect. 5.
2 Problem definition and formulation The objective of the DLBP is to efficiently utilize the resources aiming to find the minimum number of disassembly workstations required, optimally assigning the disassembly tasks to the workstations, and improving the layout and material-handling features of the disassembly line (Gungor and Gupta 2002). The main difference between regular DLBP and SDDLBP lies in the task time interactions. Task time interactions between two tasks occur when they do not have any precedence relationship, thus providing a choice in the order in which they may be performed. In a regular DLBP, the sequence in which tasks without precedence relationship between them are performed is not important. However, when sequence-dependent setup times between tasks are present in a disassembly line, the sequence in which precedence independent tasks are performed may matter. Thus, in SDDLBP, whenever precedence free tasks interact, their task times may be influenced based on the order in which they are performed. This may happen because one component may hinder the other component and thus may require additional movements and/or prevent it from using the most efficient or convenient disassembly process. For example, in an EOL computer disassembly process, the removal of a hard disk may take 12 s if the floppy drive, which is normally placed on top of the hard disk, had already been removed. However, if the floppy drive is still in its slot, it may prolong the disassembly of the hard disk by, say, 4 s, increasing the disassembly time of the hard disk to 16 s. This is because of the additional movements required due to hindrance which may prevent the use of the most efficient disassembly process. Thus, there is a sequence dependency between the hard disk and the floppy drive. Simply stated, for precedence free sequence dependent tasks i and j, disassembly time of task i is affected by task j if task j is yet to be performed in the disassembly sequence (i and j are interacting tasks). Moreover, these sequence-dependent time increment calculations add complexity to the problem making it harder to solve because the algorithm has to calculate idle times after every iteration to allocate tasks to workstations which in turn requires a dynamic matrix structure. These additional nuances make matters worse by adding layers of computations and prolong the time required to reach near optimal solutions. The problem addressed in this paper SDDLBP augments sequence-dependent time considerations to the classical DLBP in the following way: whenever a precedence free task j
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Fig. 1 Precedence relationships (solid line arrows) and sequence dependent time increments (dashed line arrows) for the 8-part PC example
Table 1 Knowledge database for the PC example
Part
Task
Time
Hazardous
Demand
PC top cover
1
14
No
360
Floppy drive
2
10
No
500
Hard drive
3
12
No
620
Back plane
4
18
No
480
PCI cards
5
23
No
540
RAM modules
6
16
No
750
Power supply
7
20
No
295
Motherboard
8
36
No
720
is assigned to be done before task i is finished, sdi j time units has be added to compute the total processing time if task i and task j are sequence dependent. Illustrative example The precedence relationships (solid line arrows) and sequence dependent time increments (dashed line arrows) for an 8 part PC disassembly process are illustrated in Fig. 1 and their knowledge database is given in Table 1. Additional data required for sequence dependencies are as follows: sd23 = 2, sd32 = 4, sd56 = 1, sd65 = 3. This example is taken from Kalayci and Gupta (2013a). For a feasible sequence 1, 3, 2, 5, 6, 8, 7, 4, since part 3 is disassembled before part 2, sequence dependency sd23 = 2 takes place because when part 3 is disassembled, the obstructing part 2 is still not taken out, i.e., the part removal time for part 3 (t3 = 12) is increased which results in t3 = t3 + sd23 = 14. Similarly, since part 5 is disassembled before part 6, sequence dependency sd65 = 3 takes place because when part 5 is disassembled, the obstructing part 6 is still not taken out, i.e., the part removal time for part 5 (t5 = 23) is increased which results in t5 = t5 + sd65 = 26. Since our investigation is concerned with solving the SDDLBP which is an extension of the DLBP, some assumptions have to be made in order to exclude elements of minor relevance and to focus on those aspects that are of paramount interest. Based on Gungor and Gupta (2002), major assumptions of our mathematical model are as follows:
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• • • • • • • • • • • •
325
A single product type is to be disassembled on a disassembly line, The supply of the EOL product is infinite, The exact quantity of each part available in the product is known and constant, A disassembly task cannot be divided between two workstations, Each task has to be assigned to at least and at most one sequence and one workstation, Each part has an assumed associated resale value which includes its market value and recycled material value, The line is paced, Part removal times (task processing time) are deterministic, constant, and discrete, Each product undergoes complete disassembly even if the demand is zero, All products contain all parts with no additions, deletions, modifications or physical defects, The sum of part removal times of all the parts assigned to a workstation must not exceed the cycle time, The precedence relationships among the parts must not be violated.
The SDDLBP investigated in this paper is concerned with a paced disassembly line for a single model of product that undergoes complete disassembly. In this content, the particular model investigated in this paper seeks to fulfill four objectives according to a preemptive lexicographic perspective. These objectives were considered from the highest priority to the lowest priority as follows: Objective 1: minimize the number of workstations; Objective 2: minimize the total idle time by evenly distributing the idle times among all workstations; Objective 3: remove hazardous components as early as possible in the disassembly sequence; Objective 4: remove high demand components before low demand components in the disassembly sequence, respectively. According to definitions above, a mathematical formulation for the SDDLBP based on (McGovern and Gupta 2007; Scholl et al. 2008) is proposed with the following notation: Sets & indices: i, j ∈ N (i, j) ∈ S D i ∈ SDj k, v ∈ K r ∈ I Pi
Set of tasks Set of interacting tasks Set of task j interacting with a given task i Set of workstations, where upper bound number of workstations is |N | Set of task i that immediately precede task r in the precedence network
Parameters: c hi di ti sd ji
Cycle time (maximum time available at each workstation) Hazard value; 1 if part i is hazardous, else 0. Demand quantity of part i requested Part removal time (task processing time) of part i Sequence dependent time increment influence of task j on task i
Decision variables: yk m Tk ti wi jk xik zi
1, if station k is used; 0, otherwise. Number of used workstations Total processing time requirement in workstation k Updated part removal time of i considering sequence dependent time increment 1, if task j is assigned to station k and executed before task i;0, otherwise. 1, if task j is assigned to station k;0, otherwise. Sequence of task i
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The model formulation is given as follows: min f 1 = m min f 2 = (c − Tk )2
(1) (2)
k∈K
min f 3 =
zi · h i
(3)
z i · di
(4)
i∈N
min f 4 =
i∈N
Subject to:
xik = 1 ∀i ∈ N
(5)
Tk ≤ c · yk ∀k ∈ K ti · xik ∀k ∈ K Tk =
(6)
k∈K
i∈N
ti = ti +
sdi j · wi jk ∀k ∈ K
(7) (8)
i∈S D j
x jk +
wi jk ≤ wi jk + 1 (i, j) ∈ S D, k ∈ K
k∈K
x jk ≤ wi jk +
w jik
(i, j) ∈ S D, k ∈ K
(9) (10)
k∈K
wi jk ≤ x jk ∀ j ∈ N , i ∈ S D j , k ∈ K wi jk + w jik = 1 ∀ (i, j) ∈ S D k∈K
(12)
k∈K
xiv ≤
xr k ∀v ∈ K , i ∈ N , r ∈ I Pi
k∈K |k≤v i∈N ti
m=
(11)
c
≤ m ≤ |N |
yk
(13)
(14) (15)
k∈K
zi =
wi jk + 1 ∀i ∈ N
(16)
k∈K j∈N
xik , wi jk ∈ {0, 1} ∀i ∈ N , (i, j) ∈ S D, k ∈ K yk , m, Tk , ti , z i ≥ 0
(17)
Note that, if Eq. (8) is removed and ti is replaced by ti in this mathematical formulation, SDDLBP reduces to the basic DLBP. The first objective given in Eq. (1) is to minimize the number of workstations for a given cycle time (Battaïa and Dolgui 2013; Baybars 1986; Gurevsky et al. 2012). It rewards the minimum number of workstations, but allows the unlimited variance in the idle times between workstations because no comparison is made between station times. Thus, it does not force to minimize the total idle time of workstations. The second objective given in Eq. (2) is to aggressively ensure that idle times at each workstation are similar, though at the expense of generating a non-linear objective function (McGovern and Gupta 2007). The method is
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computed based on the minimum number of workstations required as well as the sum of the square of the idle times for all the workstations. This penalizes solutions where, even though the number of workstations may be minimized, one or more have exorbitant amount of idle times when compared to the other workstations. It also provides for leveling the workload between different workstations on the disassembly line. Therefore, a resulting minimum performance value is the more desirable solution indicating both a minimum number of workstations and similar idle times across all workstations. As the third objective (see Eq. 3), a hazard measure was developed to quantify each solution sequence’s performance, with a lower calculated value being more desirable (McGovern and Gupta 2007). This measure is based on binary variables that indicate whether a part is considered to contain hazardous material (the binary variable is equal to 1 if the part is hazardous, else 0) and its position in the sequence. A given solution sequence hazard measure is defined as the sum of hazard binary flags multiplied by their position number in the solution sequence, thereby rewarding the removal of hazardous parts early in the part removal sequence. As the fourth objective (Eq. 4), a demand measure was developed to quantify each solution sequence’s performance, with a lower calculated value being more desirable (McGovern and Gupta 2007). This measure is based on positive integer values that indicate the quantity required of a given part after it is removed (or 0 if it is not desired) and its position in the sequence. A solution sequence demand measure is then defined as the sum of the demand value multiplied by the position of the part in the sequence, thereby rewarding the removal of high demand parts early in the part removal sequence. The constraint given in Eq. (5) ensure that all tasks are assigned to only one workstation which represents the complete assignment of each task. The constraint given in Eq. (6) ensures that the work content of a workstation does not exceed the cycle time. Equation (7) calculates total work content of a workstation by considering updated part removal time. Equation (8) updates part removal times of assigned tasks by considering sequence dependent time increments. The constraints (9)–(12) ensure that the ordering defined for the subset of interacting tasks is unique (Scholl et al. 2008). The interactions are symbolized by dashed lines, the direct precedence relations by solid arcs as given in Fig. 1. Since the precedence relationships play a major role in the disassembly of a product, we need to ensure that the precedence relations among the tasks are not violated. The constraints given in Eq. (13) are designed to do just that. This constraint set allows the assignment of task i to station k if and only if all of its predecessors are already assigned somewhere between stations 1 through k, thus making sure that the precedence constraints are preserved in the sequence. Equation (14) guarantees that the number of workstations is between theoretical minimum and maximum number of workstations. Thus, it is obvious that the minimum number of workstations cannot exceed the number of parts and should be greater than theoretical number of workstations that represent total updated part removal time divided by cycle time. Equation (15) calculates the number of workstations defined in the first objective function. Equation (16) finds sequence of tasks. Finally, Eq. (17) defines variable domains.
3 Proposed hybrid genetic algorithm approach (VNSGA) DLBP was proven to be NP-complete (McGovern and Gupta 2007). Since SDDLBP is a generalization of DLBP (by setting all sequence dependent time increments to zero, SDDLBP reduces to DLBP), SDDLBP is NP-complete too. Since SDDLBP falls into the NP-complete class of combinatorial optimization problems, when the problem size increases, the solution space is exponentially increased and an optimal solution in polynomial time cannot be found
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Fig. 2 Flow diagram of VNSGA
as it can be time consuming for optimum seeking methods to obtain an optimal solution within this vast search space. Therefore, it is necessary to use alternative methods in order to reach (near) optimal solutions faster. Metaheuristics such as GA and HGA (hybrid genetic algorithm) seem to be particularly suited for this task because they process a set of solutions in parallel, possibly exploiting similarities of solutions by recombination that provides an alternative to traditional optimization techniques to locate (near) optimum solutions in complex landscapes. In this paper we propose a HGA approach that combines a VNSGA. Here we use a task based representation (Kalayci and Gupta 2013b). The length of the chromosome is defined by the number of tasks and each gene of the chromosome represents a task. Tasks are assigned to workstations using next fit algorithm according to the task sequence in the chromosome, as long as the predetermined cycle time is not exceeded. Once the cycle time is exceeded, a new work station is opened for assignment, and the procedure is repeated until there are no more tasks to assign. Flow diagram of proposed VNSGA is depicted in Fig. 2. 3.1 Initial population and solution construction strategy Initial solutions are randomly generated for the HGA such that each solution is assured to be feasible to force the HGA search in the feasible solution space. Heuristics that are used to feed HGA by an initial feasible solution are given as follows: Greatest Ranked Positional Weight, Longest Processing Time, Shortest Processing Time, Greatest Number of Immediate Successors, Greatest Number of Successors, Random Priority, Smallest Task Number, Greatest Average Ranked Positional Weight, Smallest Upper Bound, Smallest (Upper Bound Divided by the Number of Successors), Greatest (Processing Time Divided by the Upper Bound),
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Ann Oper Res (2016) 242:321–354 Table 2 Solution construction strategy
329 Step
Description
1
Start
2
According to the precedence constraints construct the available task set
3
According to the cycle time construct the candidate task set
4
If the set of candidate task is null, go to step 6
5
Select the task randomly from the candidate task set and assign the task to the current workstation; go back to step 2
6
If the set of available task is null, go to step 8
7
Open a new workstation, go back to step 2
8
Stop the procedure
Smallest Lower Bound, Minimum Slack, Minimum (Number of Successors Divided by Task Slack), Greatest Number of Immediate Predecessors (Baykasoglu 2006). A repair function is applied to ensure that each created solution is feasible. The strategy of building a feasible balancing solution is the key issue to solve the DLBP. We use station-oriented procedure for a solution constructing strategy in which solutions are generated by filling workstations successively one after the other (Ding et al. 2010).The procedure is initiated by the opening of a first station. Then, tasks are successively assigned to this station until more tasks cannot be assigned and a new station is opened. In each iteration, a task is randomly chosen from the set of candidate tasks to assign to the current station. When no more tasks may be assigned to the open station, this is closed and the following station is opened. The procedure finalizes when there are no more tasks left to assign. In order to describe the process to build a feasible balancing solution, available task and candidate task are defined as follows: a task is an available task if and only if it has not already been assigned to a workstation and all of its predecessors have already been assigned to a workstation. A task is a candidate task if and only if it belongs to the set of available task and the idle time of current workstation is higher than or equal to the processing time of the task. The generation procedure of a feasible balancing solution is given in Table 2. 3.2 Elitism strategy and variable neighborhood search Sometimes good chromosome can be lost after crossover or mutation operators are applied. Often the GA will rediscover these lost improvements in a subsequent generation but there is no guarantee. To combat this we can use a popular feature known as elitism. Elitism involves copying a small part of the population, unchanged, into the next generation. This can sometimes have a dramatic impact on performance by ensuring that the genetic algorithm does not waste time rediscovering previously discarded partial solutions. Therefore, in this study an elitism strategy that saves best members of the population to the next generation is applied. A predefined elitism rate parameter decides how many fittest members of the population will be saved for the next generation. In addition, a variable neighborhood search (VNS) approach (Mladenovi´c and Hansen 1997) is integrated to hybrid genetic algorithm framework as an improvement to the algorithm. VNS uses the idea of systematically changing the neighborhoods in order to improve the
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330 Table 3 The VNS procedure
Ann Oper Res (2016) 242:321–354 Step
Description
1
Take initial solution π 0 from HGA algorithm as an input
2
Accept initial solution π 0 as the best solution π 1 to start shaking procedure
3
If VNS iteration limit is reached, go to Step 11
4
Apply a shaking operator randomly to the current best solution π 1 and obtain a π 2 solution in the neighborhood of the current best solution π 1 If solution π 2 is not better than current best solution π 1 , go to Step 7
5 6
Save solution π 2 as current best solution π 1 and go back to Step 3
7
Accept solution π 2 as solution π 3 for starting local search procedure
8
Apply a local search operator to the solution π 3 and obtain a solution π 4
9
If solution π 4 is better than current best solutionπ 1 , save π 4 as current best solution π 1 and go to Step 3 If VNS iteration limit is not reached, go to Step 8
10 11
Stop VNS procedure and send best found solution π 1 to the HGA algorithm
current solution and aims to further explore the solution space, which may not be explored by a simple local search technique (Hansen et al. 2010). Shaking and local search operators are used in the implementation of the VNS. The shaking operator decides the search direction of the VNS from a set of neighborhoods. After a set of preliminary experiments, we observed that the probability of escaping from local optima increases when the shaking operator is integrated with local search rather than using a single shaking operator. Therefore, each solution obtained by the shaking operator is further evaluated with the local search operator in order to explore new promising neighborhoods of the current solution. In this study, we implemented the Variable Neighborhood Descend (VND) algorithm as the local search operator. The VND algorithm aims to combine the set of neighborhoods (m-max) in a deterministic order expecting that using more than one neighborhood structure results in better solutions. After each shaking operation, the VND algorithm allows a maximum of n-max trials for possible improvements. At the end of the VND algorithm, if there is an improvement, then the shaking operations start from the first operation. Otherwise, shaking continues with the next operation. After reaching the maximum number of shaking operations (k-max), the search procedure continues with the first operation in the new iteration. The VNS procedure is given in Table 3. 3.3 Fitness evaluation and selection GA aim at finding the fittest chromosome over a set of generations. The fitness functions provide a measure of an individual’s performance in the search space. The proposed VNSGA algorithm tries to minimize the fitness functions according to the priority of first (4), second (5), third (6) and fourth (7) objective functions, respectively. Therefore, when two solutions with the same number of workstations (m) are found, the solution with less total idle time is considered as a better solution and when both the m and the total idle time values of both
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Fig. 3 Crossover operators
solutions are equal to each other, then hazard measure and demand measure values are respectively checked for comparison according to a pre-emptive goal programming (lexicographic) perspective. In our preliminary experiments, roulette wheel selection technique performed better than tournament selection technique. Therefore, the individuals for mating are selected by roulette wheel technique. The implementation of this technique is carried out by generating the cumulative probability distribution over the list of individuals using a probability proportional to the fitness of the individual. Although the first objective dominates other objectives according to the lexicographic perspective considered in this paper, using this objective in scaling fitness values would not be a smart choice since it only takes discrete values. When there are alternate optimal solutions having same objective value, this objective function does not give a strong distinction between the alternate solutions. To overcome this difficulty, in this study, instead we used the second objective as the fitness function which behaves in parallel to the first objective. Thus, first objective has still dominance over other objectives. In the selection scheme, which scales the fitness values of the members within the population so that the sum of the rescaled fitness values equals to 1. To select a parent, first, uniform random number within the interval (0, 1) is generated, and then the member whose cumulative rescaled fitness value is greater than the generated number is selected as the parent. 3.4 Crossover operator Three part fragment reordering crossover (Akpınar and Bayhan 2011) and precedence preservative crossover (Kongar and Gupta 2005) are used as crossover operators in this study. In fragment reordering crossover, two points, which cut each of the parent into three parts (head, middle and tail), are generated randomly. As demonstrated in Fig. 3a, Parent 1 is recombined with Parent 2 in order to form new children to ensure that the resulting offspring are always feasible. In the disassembly line balancing problem, recombination must guarantee feasibility because of the precedence constraints. The first offspring keeps the head and tail parts of the first parent. The middle part is filled in by adding all missing tasks in the order in which they are contained in the second parent. The other offspring is built based on the head and tail parts of the second parent and its middle part is filled in by adding the missing tasks in the order in which they are contained in the first parent. Both of the generated offspring become feasible as their middle part is also filled according to the precedence feasible order. In precedence preservative crossover operator (see Fig. 3b), in addition to the two strings representing the chromosomes of the parents, an additional mask vector is added. This mask
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vector of length n is randomly filled with ones or zeros that defines the order in which the operations are successively drawn from Parent 1 and Parent 2.The algorithm starts by initializing an empty offspring. The operations corresponding to zeros in the mask vector is directly added to the offspring in the same position. After an operation is selected from the first parent, it is deleted in the second parent. Next, the remaining operations are added to the offspring in the order of appearance in the second parent. 3.5 Neighborhood structures and mutation operators In this study, a number of neighborhood structures [swap, insert, one point left operator (OPLO), one point right operator (OPRO), two point right operator and precedence based uniform operator] are employed as mutation operators (Fig. 4). In order to avoid redundant moves, in the neighborhood structures, only feasible moves are allowed, i.e. those which do not violate the precedence constraints. Figure 4 illustrates the different neighborhood structures. Each neighborhood structure changes the initial configuration. Swap operation (SO) randomly selects two tasks in the line sequence, swaps their positions and returns a new feasible solution (Fig. 4a).
Fig. 4 Neighborhood structures
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Insert operation (IO) randomly selects a position of a task and inserts it to another randomly selected position in the disassembly sequence (Fig. 4b). OPLO randomly selects a cut point in the disassembly sequence and keeps the positions of the tasks which are assigned after this cut point while reconstructing a new sub-sequence before the cut position according to the solution construction strategy (Fig. 4c). OPRO randomly selects a cut position in the disassembly sequence, reconstructs a new sequence starting from this position according to the solution construction strategy while keeping the positions of tasks as already assigned before the cut position (Fig. 4d). Two point cut operator (TPCU), also called scramble sublist operator (Tseng et al. 2008), randomly selects two cut positions in the disassembly sequence, reconstructs a new subsequence within these cut positions while not changing the positions of tasks located out of the two selected cut positions (Fig. 4e). Precedence based uniform operator (PBUO) randomly creates a binary vector which has the same length as a sequence that will decide which tasks will change their positions. It reconstructs a new sub-sequence for the tasks which corresponds to ones while keeping the positions of tasks which correspond to zeros in the binary vector (Fig. 4f).
4 Numerical results The proposed VNSGA was coded in MATLAB and C++ and tested on Intel Xeon E5-2650 2.00 GHz processor with 32 GB RAM. The performance of VNSGA was thoroughly investigated using numerous data instances that have been gathered and adapted from the disassembly and the assembly line balancing literature. Using the data instances, the performance of VNSGA was compared with the best known metaheuristic methods reported in the literature. The input data included the component definition and disassembly operational information. A full factorial design of experiments was performed to find the best value combination of parameters (elitism rate (er), crossover rate (cr) and mutation rate (mr)) to run VNSGA experiments. To that end, the algorithm was investigated using the levels of these parameters given in Table 4. After preliminary experiments, the parameter set with er = .05, cr = .8 and mr = .2 was determined to be the best combination. 4.1 Comparison of VNSGA with other algorithms using SDDLBP instances The performance of the proposed VNSGA was compared with other algorithms reported in the SDDLBP literature. Specifically, the performance of VNSGA was compared with that of the artificial bee colony (ABC) algorithm, ACO metaheuristics, HGA, PSO, river formation dynamics (RFD), SA and TS. All algorithms were executed 30 times (sample size) each in order to have sufficient statistical data for comparison since each algorithm had probabilistic and randomized features. Two SDDLBP scenarios were considered.
Table 4 The experiment of VNSGA parameters with levels using a full factorial design
Parameters
Levels Level 1 (low)
Level 2 (medium)
Level 3 (high)
er
.05
.1
.2
cr
–
.6
.8
mr
.1
.2
.4
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The first SDDLBP scenario was for a product consisting of n = 10 components while the second SDDLBP scenario was a cellular telephone instance with n = 25 components. The data for these instances were modified from McGovern and Gupta (2006) and Gupta et al. (2004) respectively. Kalayci and Gupta (2013a) added sequence-dependent part removal time influence on parts interacting with each other. See Kalayci and Gupta (2013a) for the knowledge database and precedence relationships of these products. For the first SDDLBP scenario, while the exhaustive search method was able to find optimal solution in 215t time on average, VNSGA was able to successfully find the optimal solution in under t time on average under the restriction of the system specifications. In fact, all the algorithms considered were able to find the optimal solution. Detailed comparison of the average, standard deviation, standard error and confidence interval values for each objective are given in Tables 5 and 6. Table 5 shows that ABC, ACO, GA, PSO, RFD, SA, TS and VNSGA methods were able to find the optimal solutions for 10-part product disassembly in 2.40t, 5.36t, 5.30t, 0.81t, 3.62t, 0.59t, 0.87t and 0.65t time, respectively. In Table 6, we can see the standard error values calculated within 95 % confidence interval which is demonstrated in Fig. 5. As can be seen in Fig. 5; the proposed VNSGA was one of the fastest algorithms to reach optimal solutions. As a result, for a small sized benchmark SDDLBP, all of the methods were able to reach optimal solutions very quickly as compared to the exhaustive search method. For the second SDDLBP scenario, due the vast search space (25!), the exhaustive search in this case is prohibitive and thus, the optimal solution is unknown. According to our rough calculations, exhaustive search method would require more than one year to find an optimal solution even if we were to use a super computer and a faster programming language to solve the problem! Therefore, the best solutions found by the algorithms could be accepted as near optimal solutions since optimality cannot be guaranteed. In Table 5, we can see that all of the methods were able to find 10 as the minimum number of workstations (first objective). In Table 6, since standard error value of each algorithm for the first objective is zero, it points out that all algorithms would eventually stick to the same objective value in all experiments. Figure 6 demonstrates this situation. Therefore, our next stop is objective 2 for comparing the performance of the algorithms. In Table 5, we can see that ABC, ACO, GA, PSO, RFD, SA and TS and VNSGA found 10.07, 17.77, 12.13, 13.97, 16.00, 11.70, 13.30 and 9.00 values respectively for the second objective of the 25-part instance, meaning that algorithms performed best in the order of VNSGA, ABC, SA, GA, TS, PSO, RFD and ACO on average for the second objective. However, according to Table 6, there is no statistically significant difference between GA, SA and TS approaches within 95 % confidence interval since their results overlap each other considering standard errors given in Table 6. The performance comparison of ABC, ACO, GA, PSO, RFD, SA, TS and VNSGA for the 25-part cellular telephone instance is demonstrated in Fig. 7. Here we can see that the proposed VNSGA algorithm provided significantly better results than all of other approaches for the second objective within 95 % confidence interval with its robust performance. In terms of speed, VNSGA also outperforms all of the algorithms according to the results provided in Tables 5 and 6 for the second scenario. The performance of the algorithm on the third objective is demonstrated in Fig. 8 within 95 % confidence interval according to the standard error values provided in Table 6. Please see Fig. 9 for the performance comparison of the fourth objective which strengthens the superiority of the VNSGA approach in terms of robustness. As a result, VNSGA outperforms other algorithms provided. The breakaway point of the performance comparison was objective 2 which tries to ensure that idle times at each workstation are similar in the disassembly line.
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Ann Oper Res (2016) 242:321–354 Table 5 Average and standard deviation values for objectives for the 10 parts instance
335 Objective
Method
Scenario 1 Average
f1
f2
f3
t
Standard deviation
Average
Standard deviation
ABC
5.00
0.00
10.00
0.00
ACO
5.00
0.00
10.00
0.00
GA
5.00
0.00
10.00
0.00
PSO
5.00
0.00
10.00
0.00
RFD
5.00
0.00
10.00
0.00
SA
5.00
0.00
10.00
0.00
TS
5.00
0.00
10.00
0.00
VNSGA
5.00
0.00
10.00
0.00 2.16
ABC
67.00
0.00
10.07
ACO
67.00
0.00
17.77
1.41
GA
67.00
0.00
12.13
2.56
PSO
67.00
0.00
13.97
1.96
RFD
67.00
0.00
16.00
0.00
SA
67.00
0.00
11.70
1.82
TS
67.00
0.00
13.30
1.70
VNSGA
67.00
0.00
9.00
0.00
ABC
5.00
0.00
80.00
1.14
ACO
5.00
0.00
82.80
1.32 0.73
GA
5.00
0.00
79.77
PSO
5.00
0.00
80.63
2.46
RFD
5.00
0.00
80.60
0.62
SA
5.00
0.00
83.43
3.22
TS
5.00
0.00
83.10
2.87 0.00
VNSGA f4
Scenario 2
5.00
0.00
80.00
ABC
9605.00
0.00
925.57
5.07
ACO
9605.00
0.00
949.37
6.31
GA
9605.00
0.00
924.90
2.40
PSO
9605.00
0.00
932.50
11.53
RFD
9605.00
0.00
939.83
2.29
SA
9605.00
0.00
940.93
12.40
TS
9605.00
0.00
941.30
12.51
VNSGA
9605.00
0.00
925.00
0.00
2.40
1.73
124.60
146.79
ABC ACO
5.36
5.61
244.67
161.49
GA
5.30
4.62
156.37
138.27
PSO
0.81
0.16
40.74
24.71
RFD
3.62
3.45
222.25
119.24
SA
0.59
0.71
297.91
141.19
TS
0.87
1.05
273.02
142.14
VNSGA
0.65
0.35
35.55
20.02
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Table 6 Standard error and confidence interval values for objectives for the 10 parts instance Objective
Method
Scenario 1 Standard error
f1
f2
f3
f4
t
123
Scenario 2 95 % Confidence
Interval
Standard error
95 % Confidence
Interval
ABC
0.00
5.00
5.00
0.00
10.00
10.00
ACO
0.00
5.00
5.00
0.00
10.00
10.00
GA
0.00
5.00
5.00
0.00
10.00
10.00
PSO
0.00
5.00
5.00
0.00
10.00
10.00
RFD
0.00
5.00
5.00
0.00
10.00
10.00
SA
0.00
5.00
5.00
0.00
10.00
10.00
TS
0.00
5.00
5.00
0.00
10.00
10.00
VNSGA
0.00
5.00
5.00
0.00
10.00
10.00
ABC
0.00
67.00
67.00
−0.71
9.35
10.78
ACO
0.00
67.00
67.00
−0.46
17.30
18.23
GA
0.00
67.00
67.00
−0.84
11.29
12.97
PSO
0.00
67.00
67.00
−0.64
13.32
14.61
RFD
0.00
67.00
67.00
0.00
16.00
16.00
SA
0.00
67.00
67.00
−0.60
11.10
12.30
TS
0.00
67.00
67.00
−0.56
12.74
13.86
VNSGA
0.00
67.00
67.00
0.00
0.00
0.00
ABC
0.00
5.00
5.00
−0.38
79.62
80.38
ACO
0.00
5.00
5.00
−0.44
82.36
83.24
GA
0.00
5.00
5.00
−0.24
79.53
80.01
PSO
0.00
5.00
5.00
−0.81
79.83
81.44
RFD
0.00
5.00
5.00
−0.20
80.40
80.80
SA
0.00
5.00
5.00
−1.06
82.37
84.49
TS
0.00
5.00
5.00
−0.94
82.16
84.04
VNSGA
0.00
5.00
5.00
0.00
0.00
0.00
ABC
0.00
9605.00
9605.00
−1.67
923.90
927.23
ACO
0.00
9605.00
9605.00
−2.07
947.29
951.44
GA
0.00
9605.00
9605.00
−0.79
924.11
925.69
PSO
0.00
9605.00
9605.00
−3.79
928.71
936.29
RFD
0.00
9605.00
9605.00
−0.75
939.08
940.59
SA
0.00
9605.00
9605.00
−4.08
936.85
945.01
TS
0.00
9605.00
9605.00
−4.12
937.18
945.42
VNSGA
0.00
9605.00
9605.00
0.00
0.00
0.00
ABC
−0.90
1.50
3.30 −48.29
76.31
172.89
ACO
−2.92
2.44
8.27 −53.12
191.55
297.80
GA
−2.41
2.90
7.71 −45.49
110.88
201.86
PSO
−0.08
0.72
0.89
−8.13
32.61
48.87
RFD
−1.79
1.83
5.41 −39.23
183.02
261.47
SA
−0.37
0.22
0.96 −46.45
251.46
344.36
TS
−0.55
0.33
1.42 −46.76
226.26
319.78
VNSGA −0.12
0.52
0.77
−7.16
28.39
42.71
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Fig. 5 Time performance comparison of ABC, ACO, GA, PSO, RFD, SA, TS and VNSGA for the 10 parts instance
Fig. 6 f 1 Performance comparison of ABC, ACO, GA, PSO, RFD, SA, TS and VNSGA for the 25-part instance
4.2 Comparison of VNSGA with other methods on DLBP instance The performance of the proposed VNSGA was also compared with the DLBP instances reported in the literature (see Table 7). As seen in the table, for the 25-part DLBP data instance, the VNSGA approach outperforms other methods. Therefore, VNSGA approach can be labeled as a superior technique compared to the ones reported in the literature. 4.3 Comparison of VNSGA with other methods on modified ALBP instances In order to further extend numerical tests for VNSGA, benchmarking problem sets in Scholl and Klein (1999) were used. Information about these benchmarking problems is organized in
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Fig. 7 f 2 Performance comparison of ABC, ACO, GA, PSO, RFD, SA, TS and VNSGA for 25-part instance
Fig. 8 f 3 Performance comparison of ABC, ACO, GA, PSO, RFD, SA, TS and VNSGA for 25-part SDDLBP instance
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Fig. 9 f 4 Performance comparison of ABC, ACO, GA, PSO, RFD, SA, TS and VNSGA for 25-part SDDLBP instance Table 7 Comparison of the averaged results for 25-part DLBP instance Publication
Approach
Average performance f1
f2
f3
Best performance f4
f1
f2
f3
f4
Ding et al. (2010)
MOACO
9
11
–
874.3
9
11
–
838
McGovern and Gupta (2005b)
H-K
10
137
84
943
–
–
–
– –
McGovern and Gupta (2005b)
ACO
9
10.2
87.2
924.4
–
–
–
Tuncel et al. (2012)
RL
–
–
–
–
9
9
97
862
Kalayci et al. (2012)
SA
9
9
85.5
918.5
9
9
81
853 853
Kalayci et al. (2011)
ABC
9
9
83.9
887.4
9
9
81
Kalayci and Gupta (2011)
GA
9
9
84
874.3
9
9
82
868
Kalayci and Gupta (2012)
PSO
9
9
83.5
889.07
9
9
80
857
VNSGA
9
9
76
825
9
9
76
825
Our approach
Bold values represent the best-so-far solutions
Table 8 by using the summary from Sprecher (1999) where ‘n’ is number of tasks, ‘tmin’ is minimal task time, ‘tmax’ is maximal task time, ‘tsum’ is the sum of task times, ‘OS’ is order strength (OS) [= number of all precedence relations/(n*(n − 1))], ‘TV’ is time variability ratio [= tmax/tmin], ‘div’ is degree of divergence of precedence graph and ‘conv’ is degree of convergence of precedence graph. These 21 different precedence graphs along with selected cycle time (hardest data according to the results provided in the literature in terms of speed to converge optimal solutions) are used to construct the disassembly line specific configurations. In addition to the selected cycle times and considered precedence relationships, hazard and demand data were added to modify the original simple assembly line balancing data. The probability of a part being hazardous was selected to be .25 while demand of each part was selected to be randomly
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Table 8 Characteristics of the benchmarking problems Author/set
tmin
tmax
tsum
OS
83
233
3,691
75,707
59.09
Arcus2
111
10
5,689
150,399
Barthol2
148
1
83
4,234
Bowman
8
3
17
Arcus1
n
TV
div
conv
15.84
0.73
0.73
40.38
568.90
0.63
0.63
25.80
83.00
0.74
0.72
75
75.00
5.67
0.88
0.80
Buxey
29
1
25
324
50.74
25.00
0.74
0.78
Gunther
35
1
40
483
59.50
40.00
0.78
0.76
Hahn
53
40
1,775
14,026
83.82
44.38
0.63
0.63
Heskiaoff
28
1
108
1,024
22.49
108.00
0.68
0.69
Jackson
11
1
7
46
58.18
7.00
0.77
0.77
Jaeschke
9
1
6
37
83.33
6.00
0.73
0.73
Kilbridge
45
3
55
552
44.55
18.33
0.67
0.69
Lutz1
32
100
1,400
14,140
83.47
14.00
0.76
0.82
Lutz2
89
1
10
485
77.55
10.00
0.75
0.75
Lutz3
89
1
74
1,644
77.55
74.00
0.75
0.75
Mansoor
11
2
45
185
60.00
22.50
0.79
0.91
Mertens
7
1
6
29
52.38
6.00
1.00
0.78
Mitchell
21
1
13
105
70.95
13.00
0.74
0.70
Mukherjee
94
8
171
4,208
44.80
21.38
0.51
0.51
Roszieg
25
1
13
125
71.67
13.00
0.74
0.69
Tonge
70
1
156
3,510
59.42
156.00
0.78
0.73
Wee-Mag
75
2
27
1,499
22.67
13.50
0.85
0.62
distributed between [1,100]. The knowledge database for each instance are given with respect to the columns of part number (#), hazard information (H ) and demand information (D) (see Appendix). These additional knowledge data that construct a bridge between assembly line balancing and disassembly line balancing literature are added to the Appendix of this paper. Thus, interested researchers will be able to make use of these data and compare their results with this study and other future studies. Computational results are grouped in two categories according to the number of tasks of each problem. In the first category (Table 9), problems with less than 80 tasks are listed while problems with more than 80 tasks are listed in the second category (Table 10). The computational results are given in Tables 9 and 10, respectively. 4.4 Comparison of VNSGA with Sabuncuoglu et al. (2000) HGA and Kilincci (2011) FSb on 70-task Tonge (1961) problem Sabuncuoglu et al. (2000) proposed a HGA for solving simple assembly line balancing type-I problem in which elitism strategy was implemented by using some concepts of SA approach. GAs are applicable to any kind of line balancing problem regardless of the OS. However, Sabuncuoglu et al. (2000) observed that GAs perform worse in problems with high OS, as in Tonge (1961) problem with 59.42 OS. If the number of precedence relationships increases, the possibility of generating offspring that are better than their parents decreases. Our VNSGA approach overcomes this issue with its systematic search process that is applied
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Table 9 Computational results (time limit: 1,000 s) for data sets in the first category with respect to all objective values Author/set
L B( f 1 )
c
m∗
f 1 (m)
f2
f3
f4
Bowman
20
5
5
5
149
14
2,068
Buxey
30
12
12
12
118
57
17,282
41
14
14
14
2,806
6
6
6
Gunther Hahn Heskiaoff Jackson
1,519
173
35,101
1.8734e+06
355
76,776
216
5
5
5
628
43
17,642
10
5
5
5
6
17
3,713
Jaeschke
7
7
7
7
26
19
2,142
Kilbridge
62
9
9
9
6
161
47,019
117
24,383
2,357
7
7
7
8.127e+05
Mansoor
Lutz1
94
2
2
2
5
18
2,722
Mertens
7
5
5
5
10
5
1,571
Mitchell
15
8
8
8
31
38
11,603
Roszieg
16
8
8
8
5
68
20,597
168
22
22
22
Tonge
Wee-Mag
2,152
563
1.1482e+05
170
22
22
22
3,002
575
1.1601e+05
173
22
22
22
5,196
550
1.1853e+05
179
21
21
21
3,459
563
1.1514e+05
182
20
20
20
968
593
1.157e+05
46
34
34
35
983
420
1.4573e+05
47
32
33
33
148
425
1.432e+05
49
31
32
32
189
364
1.479e+05
50
31
32
32
347
377
1.4702e+05
52
31
31
31
455
373
1.4699e+05
to the fittest chromosomes (elite members) and successful neighborhood structures. Kilincci (2011) proposed firing sequence backward (FSb) algorithm based on Petri net approach to solve simple assembly line balancing type-I problem. Comparison results with Sabuncuoglu et al. (2000) HGA and Kilincci (2011) FSb on the 70-task Tonge’s problem in terms of number of workstations are given in Table 11. Our VNSGA outperformed both methods on this problem.
4.4.1 Comparison of VNSGA with Chiang (1998) TS, Lapierre et al. (2006) TS and Kilincci (2011) FSb on 83-task Arcus1 problem and 111-task Arcus2 problem Chiang (1998) applied TS approach to simple assembly line balancing problem while Lapierre et al. (2006) proposed an improved version of TS for the same problem. Comparison of VNSGA with Chiang (1998) TS, Lapierre et al. (2006) TS and Kilincci (2011) FSb on 83task Arcus1 and 111-task Arcus2 problems in terms of number of workstations is given in Table 12. Our VNSGA performed as good as Lapierre et al. (2006) TS in finding optimal solutions while it performed better than Chiang (1998) TS in two instances and Kilincci (2011) FSb in one instance.
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Table 10 Computational results (time limit: 1,000 s) for data sets in the second category with respect to all objective values Author/set Arcus1
Arcus2
Barthol2
c
L B( f 1 )
m∗
f 1 (m)
f2
f3
f4
3,985
20
20
20
9.3416e+05
968
1.7082e+05
5,048
16
16
16
1.7643e+06
958
1.7153e+05
5,853
14
14
14
2.7918e+06
953
1.7096e+05
6,842
12
12
12
4.2611e+06
939
1.7015e+05
7,571
11
11
11
5.3723e+06
961
1.7289e+05
8,412
10
10
10
7.0937e+06
950
1.7097e+05
8,898
9
9
9
2.1423e+06
945
1.6912e+05
10,816
8
8
8
1.494e+07
948
1.6828e+05
5,755
27
27
27
2.5818e+06
1,849
3.2566e+05
7,520
20
21
21
2.9976e+06
1,804
3.2404e+05
8,847
18
18
18
4.3803e+06
1,791
3.2277e+05
10,027
16
16
16
6.3251e+06
1,809
3.2758e+05
1,0743
15
15
15
7.7572e+06
1,727
3.2192e+05
11,378
14
14
14
5.7581e+06
1,717
3.1781e+05
11,570
13
13
14
9.856e+06
1,745
3.2386e+05
17,067
9
9
9
1.1421e+06
1,685
3.2166e+05 5.6562e+05
85
50
–
52
906
2,991
89
48
–
50
1,174
2,788
5.6176e+05
91
47
–
49
1,179
2,926
5.6235e+05
95
45
–
47
1,279
2,741
5.5644e+05
Lutz2
15
34
34
34
63
921
1.9563e+05
Lutz3
150
12
12
12
2,050
960
1.8226e+05
Mukherjee
201
22
22
23
12,057
1,292
2.3422e+05
301
15
15
15
10,137
1,209
2.2954e+05
Table 11 Comparison of VNSGA with Sabuncuoglu et al. (2000) HGA on the 70-task problem of Tonge (1961) in terms of number of stations Author/set
Cycle time (c)
Optimal solution
Sabuncuoglu et al.’s HGA
Kilincci’s FSb
Our VNSGA
Tonge
170
22
23
22
22
173
22
23
22
22
179
21
22
22
21
182
20
22
21
20
Bold values represent the best-so-far solutions
4.4.2 Comparison of VNSGA with Sprecher (1999) AGSA on different problem sets Sprecher (1999) proposed an adapted general sequencing algorithm (AGSA) for solving simple assembly line balancing problem type I. Comparison of VNSGA with Sprecher (1999) AGSA on different problems in terms of number of workstations is given in Table 13. VNSGA performed better than Sprecher (1999) AGSA in one instance while it had similar performance on other data sets.
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Table 12 Comparison of VNSGA with Chiang (1998)) TS, Lapierre et al. (2006) TS and Kilincci (2011) FSb on 83-task Arcus1 and 111-task Arcus2 problems in terms of number of workstations Author/set Arcus1
Arcus2
c (cycle time)
Optimal solution
Chiang’s TS
Lapierre TS
Kilinci’s FSb
Our VNSGA
5, 048
16
17
16
16
16
5,853
14
14
14
14
14
6,842
12
12
12
12
12
7,571
11
11
11
11
11
8,412
10
10
10
10
10 9
8,898
9
9
9
9
10,816
8
8
8
8
8
5755
27
27
27
28
27
8,847
18
19
18
18
18
10,027
16
16
16
16
16
10,743
15
15
15
15
15
11,378
14
14
14
14
14
17,067
9
9
9
9
9
Bold values represent the best-so-far solutions
Table 13 Comparison of VNSGA with Sprecher (1999) AGSA on different problems in terms of number of workstations
Author/set c (cycle time) Optimal solution
Arcus2 Barthol2
Lutz2 Mukherjee
Bold values represent the best-so-far solutions
Sprecher (1999)’s AGSA
Our VNSGA
7, 520
–
21
21
11, 570
13
14
14
85
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52
52
89
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50
50
91
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49
49
15
34
34
34
201
22
23
23
46
34
35
35
47
–
33
33
49
32
32
32
50
32
32
32
52
31
32
31
5 Conclusions SDDLBP is a multi-objective NP-complete optimization problem. The main objective of this paper was to solve the SDDLBP which aimed to minimize the number of disassembly workstations, minimize the total idle time of all workstations by ensuring similar idle time at each workstation considering sequence dependent time increments, maximize the removal of hazardous components as early as possible in the disassembly sequence and maximize the removal of high demand components before low demand components. A fast,
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near-optimal, VNS integrated GA was developed and presented in this paper to solve multiobjective SDDLBP. To the best of our knowledge, the first application of VNS integrated GA in solving line balancing problems is addressed in this paper. The algorithms were tested on numerous scenarios with different cycle times from disassembly line balancing and assembly line balancing literature. Thus, a bridge between assembly line balancing and disassembly line balancing literature was constructed. The numerical results showed that the proposed VNSGA method generates as good as or superior solutions among other solution approaches for both assembly and DLBP reported in the literature. As a future research direction, since the proposed approach provides promising solutions, all of the data sets available in simple assembly line balancing literature can be further investigated with the proposed solution approach which may result in a comprehensive study for assembly lines or disassembly lines. Furthermore, the extensions of disassembly line balancing problems such as fuzzy and mixed model disassembly line balancing problems can be solved.
6 Appendix Arcus1 #
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