Decomposition-based bi-objective optimization for sustainable robotic assembly line balancing problems

Journal of Manufacturing Systems 55 (2020) 30–43

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Decomposition-based bi-objective optimization for sustainable robotic assembly line balancing problems

T

Binghai Zhou*, Qiong Wu School of Mechanical Engineering, Tongji University, Caoan Road 4800, Shanghai, China

ARTICLE INFO

ABSTRACT

Keywords: Multi-objective optimization Energy consumption models Robot Assembly line balancing MOEA/D

Due to the increasing greenhouse gas emissions and the energy crisis, the manufacturing industry which is one of the most energy intensive sector is paying close attention to the improvement of environmental performance efficiency. Therefore, in this paper the automated assembly line is balanced in a sustainable way which aims to optimize a green manufacturing objective (the total energy consumption) and a productivity-related objective (similar working load) simultaneously. A comprehensive total energy consumption of each processing stage was analyzed and modeled. To make the model more practical, a sequence-based changeover time and robots with different efficiencies and energy consuming rates are considered and optimized. To properly solve the problem, the proposed novel optimal solution takes the well-known MOEA/D as a base and incorporates a well-designed coding scheme and a problem-specific local search mechanism. Computational experiments are conducted to evaluated each improving strategies of the algorithm and its superiority over two other high-performing multiobjective optimization methods. The model allows decision makers to select more sustainable assembly operations based on their decision impacts in both productivity and energy-saving.

1. Introduction Owing to the continuous increasing energy consumption, global energy crisis and climate change, the sustainable development of modern society has become one of the most serious challenges facing human race. Economy, society, and environment are considered as the three pillars of sustainability. However, the industries have traditionally focused on the first aspect, i.e., improving the quality and efficiency of production, but the environmental or social aspects were with little regard. Seeing into industrial energy consumption, the production processes and manufacturing activities play the major roles, which are responsible for approximately 90 % of the total [1]. These facts inevitably led the researchers and manufactures to pay serious attention to improve energy efficiency and control greenhouse gas emissions in the industrial production process, instead of giving high priority to production efficiency and cost. Differ from the traditional manufacturing, the accelerating utilization of industrial robots in automatic production lines brings lots of benefits, but meanwhile consumes a significant amount of electrical energy. It motivates us to pay close attention to the energy efficiency in robotic assembly lines. According to the International Federation of Robotics, by 2018, global sales of industrial robots have accelerated reaching an all-time high of more than 380,000 units, twice as much as ⁎

in 2013. These industrial robots are used to perform various operations which are physically demanding, highly repetitive or high-risk, such as assembly, stamping, die casting, forging and welding, due to their high precision and capability. As a result, the electrical energy consumed by robots becomes one of the primary forms of energy consumption in the manufacturing. Fysikopoulos et al. indicated that the energy cost during a car manufacturing process contributes about 9–12 % of the total manufacturing cost, and 20 % reduction in energy consumption can result in about 2–2.4 % saving in the final manufacturing cost [2]. Therefore, considering energy efficiency in robotic lines will contribute significantly to both the economy and the environment, also making the enterprises more competitive and sustainable in the market. Nowadays, it is widely believed that there can be tremendous opportunities in developing novel optimization techniques and strategies for sustainable manufacturing, by simultaneously including both economic criterions and energy efficiency criterions (e.g., controlling GHG emissions, decreasing total costs and greening the industry) [3,4]. These challenges spread among the optimization of manufacturing activities in various levels such as machining processes (e.g. [5]), scheduling (e.g. [6]), production planning (e.g. [1]), line balancing, supply chain (e.g. [7]) and so forth. If the robotic assembly line, as one of the most cost intensive and energy related processes during manufacturing, was balanced in a sustainable way, the benefits could be enormous. In

Corresponding author. E-mail addresses: [email protected] (B. Zhou), [email protected] (Q. Wu).

https://doi.org/10.1016/j.jmsy.2020.02.005 Received 26 June 2019; Received in revised form 17 February 2020; Accepted 18 February 2020 0278-6125/ © 2020 The Society of Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.

Journal of Manufacturing Systems 55 (2020) 30–43

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this paper, we proposed an optimization method of a bi-objective Sustainable Robotic Assembly Line Balancing Problem (SRALBP), which put emphasis on the potential benefits for involving the energy efficiency criterion into traditional robotic line balancing problem, and established a more comprehensive model with energy consumption falling into four categories. Specifically, we considered the sequencebased changeover of fixtures and tools, which accounts for a non-ignorable part of both workstation time and energy consumption when a robot performs various tasks within a workstation. Since the traditional ALBP is an assigning problem, which does not regard the sequencing of tasks assigned to the stations, the proposed mathematical model also take the task sequence into consideration. The assembly line balancing problem (ALBP) is one of the most studied problems in the industrial engineering literature. In ALBP, a set of required tasks are assigned to a serious of workstations in order to produce a product and some objective functions should be optimized subjected to a set of constraints. General Assembly Line Balancing Problems (GALBPs) regard further specification, such as parallel stations (e.g., [8,9]), product diversity (e.g., [10,11]), stationary resources (e.g. [12]), equipment selection (e.g. [13]), or space constraints (e.g., [14]), among others. Extensive reviews on ALBP are done by Becker and Scholl [15], and Battaïa and Dolgui [16]. In manual assembly lines, the actual processing times for activities vary considerably and can hardly obtain the optimal balance; while the performance of robotic assembly lines is rather predictable, which gives rise for improving the system performance through appropriate line balancing and robot assignment [17]. Assembly lines using robots are called robotic assembly lines. Since there are different types of robots available in the market, which can execute the same task with different capabilities and efficiencies, the allocation of robots to workstations has great influence on the performance of assembly lines. First formulated by Rubinovitz and Bukchin based on ALBPs, the Robotic Assembly Line Balancing Problems (RALBPs) concern not only assigning tasks to workstations but also allocating the best fitting robot for each workstation so as to improve the productivity [18]. RALBPs can be classified into two types. In RALBP-I type, it aims at minimizing the number of workstations and the cycle time is fixed, while RALBP-II type deals with minimizing the cycle time with a given number of workstations [19]. A larger number of solution methods have been developed to solve the RALBP, including mathematical programming approaches, heuristic and meta-heuristic algorithms. Since assembly line balancing problem is NP-hard, the computational time of an exact method to find an optimum solution will be much greater than any heuristic method to yield a good near optimum solution [20]. For studies focused on single-objective robotic assembly line balancing problems, both optimum and heuristic algorithms can be found. A branch-and-bound algorithm designed by Rubinovitz et al. [18], an integer programming model with a simulation-based adjustment technique used by Tsai and Yao [21], and a cutting plane algorithm proposed by Kim and Park [22] were used to balance the robotic assembly line. However, the algorithm still requires huge amount of computational resources even heuristic rules are incorporated, which is only suitable for small problems. Gao et al. proposed a genetic algorithm which was hybridized with a local search for type-II RALBP [19]. Zhou et al. proposed a genetic algorithm with the mechanism of simulated annealing for balancing robotic weld assembly lines [23]. Nilakantan and Ponnambalam proposed bio-inspired search algorithms, PSO algorithm and a hybrid cuckoo search-PSO to minimize the cycle time [24]. During the last decades, more complex ALBPs with several conflicting objective functions have aroused extensive academic interests. Pareto Front based approximation methods and aggregative methods, especially meta-heuristic algorithms, have been used to solve the multiobjective problems. Yoosefelahi et al. formulated a multi-objective mixed integer linear programming model to minimize the cycle time, robot setup costs and robot costs [25]. An evolution algorithm was used to solve the problem. Rabbani et al. proposed a model to minimize

robot purchasing costs, robot setup costs, sequence dependent setup costs, and cycle time for type II robotic mixed-model assembly line balancing problem [26]. They used NSGA-II and multi-objective particle swarm optimization (MOPSO) to solve the problem. Zhou and Wu used an improved immune clonal selection algorithm to solve the biobjective RALBP considering time and space constraints [27]. However, the studies on RALBP mentioned above so far has focused on traditional objectives, such as minimizing cycle time, minimizing number of workstations and maximizing assembly line efficiency. To the best of authors’ knowledge, researches which take into consideration the sustainability in the robotic assembly line systems is relatively limited. There exist only a few studies concerning energy consumption or carbon footprint, which will be discussed in this paragraph. Nilakantan et al. investigated the energy consumption in straight robotic assembly lines and developed two models to minimize the cycle time and energy consumption [28]. They utilized particle swarm optimization (PSO) to solve the problem. Nilakantan et al. minimized the energy consumption of a U-shaped robotic assembly line [29]. Li et al. subsequently investigated the reduction of total energy consumption in two-sided robotic assembly lines and developed a multi-objective restarted simulated annealing algorithm to obtain Pareto solutions [30]. Their results indicated that the optimization of line balancing and the minimization of energy consumption in some situations were conflicting. Zhou and Kang presented a mathematical model with three objectives of minimizing the cycle time, the sum of energy consumption, and the total cost of robots of assembly lines [31]. They developed a multi-objective hybrid imperialist competitive algorithm with nondominated sorting strategy to solve the problem. One of the limitations of the existing researches on RALBP lies in the lack of practical features of real-life manufacturing systems. It can be observed that the existing researches on energy-optimized assembly line problems are highly concentrated on optimizing the energy consumption during operating and standby process. However, the transport of workpieces between workstations and changeover between tasks are also non-negligible processes in the assembly line. On one hand, transportation and changeover also generate considerable energy consumption. On the other hand, the changeover time has significant influence on both workstation time and standby energy consumption. In many assembly lines, such as car body welding lines, the operation time for the tasks is comparatively short. The number of changeovers makes a big difference to the total workstation time. Therefore, a comprehensive energy optimization considering operating, standby, transportation and changeover during the balancing of robotic assembly lines urgently needs to be studied. Another limitation lies in the requirement of incorporating problemspecific strategies to guarantee satisfactory performance [32]. Notice that our robotic assembly line balancing problem SRALBP is a multiobjective optimization problem bounded by the industrial related constraints. When solving these constrained multi-objective optimization problems (CMOPs), it is important to maintain a balance among convergence, diversity and feasibility of a population. And there are two major aspects to achieve this balance, which are the multi-objective optimization method and the constraint-handling technique [33]. In terms of optimization method, multi-objective evolutionary algorithms (MOEAs) are widely used to solve multi-objective optimization problems in manufacturing systems [34–37, among others], since MOEAs can produce a set of well distributed non-dominated solutions in a single run. According to the selection strategy used in the evolutionary process, MOEAs can be classified into different types. Most of existing MOEAs are dominance-based MOEA, which uses a selection strategy based on Pareto domination. For instance, the most popular NSGA-II adopts a non-dominated sorting and elitism-preserving strategy [38]. Another type of MOEAs is the decomposition-based MOEA, which has attracted much attention in recent years. It decomposes a multi-objective optimization problem into a number of singleobjective optimization problems, and use different weight vectors and 31

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B. Zhou and Q. Wu

changeover may happen between tasks. Since each type of robots have considerably different energy consumption rates and performing capabilities for the same assembly tasks, total energy consumption and production efficiency of a robotic assembly line vary depending on its task sequence, task to workstation assignment and type of robot allotted. Thus, three kinds of decisions should be made in the sustainable robotic assembly line balancing:

aggregation functions. One of the major advantages of MOEA/D over Pareto dominance based MOEAs is that single objective local search techniques can be readily used in MOEA/D [39]. Therefore, a MOEA/D is adopted in this paper to solve the proposed SRALBP considering energy consumption and changeovers. Meanwhile, a local search mechanism is designed and implemented in the MOEA/D. By exploring the properties of proposed problem, more focuses on the optimization of energy consumption is placed during the design of the search mechanisms to guarantee more satisfactory performance. As for the constraint-handling technique, the majority of existing methods for ALBPs are feasibility-driven methods, where the feasible solutions are always better than infeasible solutions. However, the method may lead to being trapped in local optima. In this paper, an infeasibility-driven strategy is adopted to trade off the feasibility and convergence. A small proportion of infeasible solutions is maintained in the population to take full advantage of the useful information contained in the infeasible solutions. The improvement in search ability and convergence of the algorithm is verified through experiments. The aim of this paper is to effectively apply the enhanced MOEA/D and infeasibility-driven strategy to a practical robotic assembly line balancing problem, optimizing the comprehensive energy optimization in a robotic assembly line when balancing the work load and assigning the best fitting robots to workstations, so that the energy efficiency and sustainability of the assembly line can be improved. In comparison with the existing studies, the contributions of this paper are as follows:

(1) Assigning assembly tasks to each workstation, ensuring the work load of each station are similar and cycle time meets the constraint. (2) Allocating the best fit robot to each workstation considering energy efficiency and performing capabilities. (3) Sequencing the assembly operations assigned to each workstation to make the minimum number of changeovers. 2.1. Assumptions and notations Before introducing the mathematical model, some basic assumptions considered in this paper are introduced as follows: (1) The assembly tasks cannot be subdivided. (2) The precedence diagram and changeover relationships between activities are known. (3) A task can be allocated only when the cycle time and the precedence constraints are satisfied. (4) Only one robot can be assigned to a station but there is no limitation on the quantity of robots of any type; any task or robot can be assigned to any station. (5) The time taken to perform a task depends on the assigned robot and the time taken for a robot to perform a task is determined. (6) The energy consumption to perform a task or stand-by depends on the type of robots. (7) Material handling, loading and unloading, and other set-up energy consumption are not neglected in this study. Tool-changing time is considered and the changeover time of one robot between every task are equal and constant. (8) The robotic assembly line for a homogenous product is balanced.

(1) In order to optimize the robotic assembly line toward a sustainable trend, a Type-I SRALBP considering the energy consumption of each process was analyzed and modeled, in which the objective is to minimize the total energy consumption and number of workstations simultaneously. (2) In the proposed model, the energy consumption was more comprehensively modeled, divided into the four parts: operating energy, standby energy, transport energy and changeover energy. Sequence-dependent changeover is incorporated to achieve a better modelling of the real robotic assembly lines balancing problem. Number of changeovers is optimized to reduce the unproductive time and reduce the number of workstations. (3) In this paper, an enhanced decomposition-based multi-objective algorithm (MOEA/D) has been proposed to address the proposed constrained multi-objective optimization problem. A constrainthandling technique which reserves informative infeasible individuals during the evolution process and an adaptive PBI method with regard to the infeasible individual distribution are integrated to improve its convergence and distribution. (4) Taking into account the features of the proposed SRALBP, a problem-specialized local search enhancement is integrated to strengthen the search capability. The dedicated energy-related local search further optimizes the total energy consumption and accelerate the search process.

The notations used in this paper are summarized in Table 1. 2.2. The model of total energy consumption objective function Basing on the observation of robotic assembly lines, the states of the whole assembly process include loading, assembly tasks operating by robots, tool changing, unloading, transporting between workstations, Table 1 Notations and explanations.

The rest of this paper is organized as follows. The description of a sustainable robotic assembly line balancing and the model formalization is carried out in Section 2. In Section 3, the detailed mechanism of the proposed enhanced MOEA/D is discussed. Computational experiments are carried out to validate and evaluate the performance of the algorithms in Section 4. Finally, conclusions are drawn and the prospect of future research is provided in Section 5. 2. Problem description and formulation The robotic assembly line consists of a set of workstations and a robot assigned to each workstation. The workpieces visit workstations successively as they are moved along the line by some kind of transportation system with a fixed energy consumption rate. A set of tasks have to be executed in an order satisfying the specified precedence constrains and

Notations

Explanation

i s CT r per cer ser te tir cr ps z ij

task (i = 1,2, 3, …, I ) station (s = 1,2, 3, …, UBm ) cycle time robot types (r = 1,2, 3, …, R ) operation energy consumption of the robot r per time unit changeover energy consumption of the robot r for one time standby energy consumption of the robot r per time unit transporting, loading and unloading energy consumption of station s processing time of task i by robot type r time for once changeover operation of robot type r number of changeovers in station s z ij is 1 if task i and task j are operated by the same tool, and 0

Pre (i ) xis yrs uij

32

otherwise set of immediate predecessors of task i xis is 1 if task i is assigned to station s and 0 otherwise yrs is 1 if robot type r is assigned to station s and 0 otherwise uij is 1 if task j is operated immediately after task i and 0 otherwise

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energy consumption of executing unloading and loading, and the transportation system such as a conveyor belt [40]. The transportation energy consumption can be calculated as follows:

TE = (m

2.2.4. Model of standby energy consumption The standby refers to the phase that the robot is waiting to process the next assembly operation. Standby energy consumption is the energy consumption of workstation running without assembling and robot waiting during changeovers, loading, unloading and idling. The standby energy consumption in workstation s can be calculated as follows:

Fig. 1. The energy consumption in a workstation.

and idling. An example of the energy consumption during the whole process in a workstation can be illustrated by Fig. 1. Hence in this study, according to their characteristics, the total energy consumption is divided into four categories: processing, changeover, transportation and standby.

R

SEs =

I

x is yrs tir ) ser

On the basis of the assumptions and models above, the mathematical formulation of SRALBP-I with energy saving and changeover consideration is given as follows: (6)

Min F = ( f1 , f2 ) UBm s=1

(1)

f2 (x ) = E =

2.2.2. Model of changeover energy consumption A tool or fixture change for the assigned robot may be required between the executions of two adjacent tasks depending on the processing method. The effect of task sequence in a workstation to changeover times is illustrated by Fig. 2. In order to model the changeover energy consumption, the number of changeovers in a workstation is introduced. The tasks assigned to station s are sequenced according to the precedence relationships and the principal of making the least time of changeovers. The changeover will happen if task j is operated immediately after task i and the same tool is required by both tasks, i.e.,

UBm

max x is

(PES + CES + SES ) + TE

UBm

sxjs s=1

sx is

(9)

x is = 1,

i

yrs = 1,

r

(10) (11)

s=1 I

R

R

x is yrs tir + i =1 r=1

(2)

x is xjs ,

I

I

yrs cr ( r=1

x is xjs uij z ij ) i=1 j =1

i, j = 1, …, I ,

s

(12) (13)

s

i , j = 1, …, I ,

CT ,

r,

s

(14)

The Eq. (7) and (8) are the objectives of the model to minimize the number of workstations and total energy consumption. The constrain given in Eq. (9) ensures that no precedence relationship is violated, that is, when a task j precedes another task i, it must be assigned to a station precedes another station which task i is assigned to, or they are assigned to a same station. Eq. (10) guarantees that all tasks must be assigned and every task is assigned to exactly one workstation. Eq. (11) ensures that exactly one robot can be assigned to each workstation. Eq. (12) ensures that the total workload time of every workstation, which is composed of operating time and changeover time, cannot exceed the

R

yrs cer ps

Pre (i )

UBm

The changeover energy consumption in workstation s can be calculated as follows:

r=1

j

s=1

x is , yrs, uij = 0 or 1,

CEs =

i,

UBm

ps = j

0,

s=1

uij

i

(8)

Subject to UBm

Notice that the first task in sequence is operated immediately after the last task. The number of changeovers ps in station s is given as follows:

s

(7)

i = 1,2, … , I

s=1

i, j , s

xis xjs uij z ij ,

(5)

i=1 r=1

f1 (x ) = m =

x is yrs tir per i =1 r=1

R

2.3. The mathematical model

R

x is xjs uij z ij = 1,

I

yrs (CT r=1

2.2.1. Model of processing energy consumption The processing energy consumption is the energy consumption of operating an assembly task using a tool by a robot. The consumption rate depends on the robot type. The processing energy consumption in workstation s can be calculated as follows:

PEs =

(4)

1) te

(3)

2.2.3. Model of transportation energy consumption The transportation energy consumption is the energy consumed from moving the workpiece along two adjacent workstations, including

Fig. 2. The effect of task sequence in workstation to changeover times.

33

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given cycle time. Eq. (13) ensures task i and task i must be assigned to the same station if task j is operated immediately after task i .

minSEsr . r

3.2. Constraint-handling techniques

3. Enhanced decomposition-based multi-objective algorithm

Although the coding scheme ensures the precedence constraints, it does not take the cycle time into consideration. The solution could be infeasible if the workstation time exceed the given cycle time. Since infeasible individuals with satisfactory objective values and small cycle time violations could possibly generate offspring which have better objective values, some informative infeasible individuals are maintained in the proposed A-MOEA/D-ID. The degree of cycle time constraint violation of an individual x is calculated as follows:

Considering the robotic assembly line balancing problem is wellknown as a NP-hard problem, a novel heuristic algorithm, which is built upon the framework of decomposition-based multi-objective algorithm (MOEA/D), is developed in this paper to resolve the proposed constrained multi-objective optimization problem (CMOP). Since the objectives conflict with each other, there does not exist a single solution that can optimize both objectives simultaneously, and a set of representative Pareto solutions are provided by the proposed algorithm. The MOEA/Ds apply a linear or nonlinear aggregation method to decompose a multi-objective optimization problem (MOP) into a number of single optimization subproblems (SOPs) and each SOP relates to one solution [41]. A set of uniformly distributed weight vectors is initially generated and the subproblem neighborhoods are used to obtain good population diversity and improve the balance between exploration and exploitation of the algorithm [42]. There are three commonly used decomposition approaches, including the weighted sum, Tchebycheff and boundary intersection approaches. In this paper, the penalty based boundary intersection (PBI) method is employed and improved to obtain better distributed solutions for our practical engineering optimization problems. In terms of the constraint-handling technique, trades off between the feasibility and convergence of a population are made simultaneously in this paper. Based on the analysis of specific features of the proposed SRALBP, a small proportion of infeasible solutions is maintained in the population to improve the convergence, and each subpopulation uses an adaptive constraint-handling method. The details of the proposed adaptive infeasibility-driven MOEA/D (A-MOEA/D-ID) are explained in the following sections.

V (x ) = max{0, maxSTs (x ) s

(15)

where STs (x ) is the workstation time of x in workstation s . Definition 1. A solution x is called feasible solution if V (x ) = 0 . Otherwise x is called infeasible solution. Then the proposed RALBP with energy consideration is transformed into an unconstrained multi-objective optimization problem with three objectives:

Min F = ( f1 , f2 , f3 )

(16)

f1 (x ) = m

(17)

f2 (x )=E

(18)

f3 (x )=V (x )

(19)

3.3. Weight vectors initialization and adjustment Since there is no prior knowledge about the Pareto front, a set of three-dimensional uniform distributed weight vectors w i = (w1i, w2i , w3i) for each subproblem i is generated to improve the uniformity of the approximated Pareto front obtained by the algorithm. The simplex lattice design method is adopted to set initial weight vectors:

3.1. Solution representation To avoid producing a narrow Pareto Front, a separator-based representation scheme expressing both tasks sequence and task-to-station assignments is adopted for the proposed A-MOEA/D-ID. A potential solution is expressed by randomly inserting several separators into the order of task genes satisfying the precedence relationship. Tasks are encoded by 1,2, 3, …, I , and since separators genes do not represent any specific task, they are encoded by number 0 to avoid confusion. Fig. 3 shows an example of the coding scheme. The details of decoding procedure are given as follows: Step 1. Allocate the tasks to the workstations according to the separators. Step 2. In every workstation, reorder the tasks to make the least time of changeovers without violating the precedence constraints. Step 3. For every type of robot r assigned to workstation s , calculate the station time STs r and station energy consumption SEs r . Step 4. Compare STs r with C , if STs r C , r R 0 . Assign the robot which performs the allocated tasks with the minSEsr to every station. If

R0 =

CT }

(20)

w1i + w2i + w3i = 1 wij

{0, H1 , H2 , …, HH }, j = 1,2, 3

(21)

the weight vector set is W = , where N = is the total number of weight vectors. However, according to Definition 1, the weight vectors dimension for constraint violation must be equal to zero to make the solution satisfy the cycle time constraint. To guide the individuals towards feasible region, w3i for each individual is dynamically adjusted and eventually becomes zero during the evolution in this paper. As the generation increases, individuals with smaller objective values and constraint violations are preferred to be selected. The values of weight vectors are updated as follows:

(w1,

w1i + w2i +

r R0

, assign the robot which performs the allocated tasks with the

wij wij

Fig. 3. The separator-based representation scheme.

34

w3i

=1

{0, H1 , H2 , …, HH }, i = 1,2 {0, H , 2H , …, HH }, i = 3

w 2, …, w N )

CH2 + 2

(22) (23) (24)

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d2 =

r(

gen 1 maxgen

)

(25)

where r is a uniformly distributed random number in (0, 1), gen is the current generation, maxgen is the maximum generation, is the updating factor. As illustrated by Fig. 4, the weight vectors in the early stage guide the search to a wide range, while by decreasing the value of w3i , a set of infeasible individuals with better objective values and smaller constraint violations can survive in the evolution. Thus, the proposed algorithm can obtain a population of both distributivity and diversity.

Generally, three aggregation functions, the Penalty Boundary Intersection (PBI), weight sum function and Tchebycheff function are used to decompose the MOP into a number of single-objective optimization subproblems in MOEA/D. In this study, the decomposition method used in A-MOEA/D-ID is the adaptive PBI method. 3.4.1. Basic PBI method In the PBI method, the optimization problem can be described as follows:

i

where

z *)T w

(F (x )

=

ni nmin n _infi nmax nmin ni

( i) =

(27)

w

(28)

(29)

where ni and n _infi are the number of solutions and infeasible solutions respectively, nmin and nmax are the minimum and maximum value of n , respectively. The relationship between the penalty factor of subproblem i and the density of infeasible i is as follows:

(26)

subject to x

d1 =

w ) w

3.4.2. Adaptive PBI method The penalty factor in the PBI method plays a significant role in the performance of MOEA/D. For practical engineering optimization problems with complex POFs, MOEA/D with a fixed PBI penalty factor can difficultly obtain well distributed solutions. To solve the problem, an adaptive penalty factor scheme, based on infeasible solutions, is designed in this paper. Since some infeasible solutions are preserved during the evolution, different subproblems with different infeasible solutions quantities should have different values. When it has a great deal of infeasible solutions, a smaller penalty factor can help to enhance the convergence speed; for a small number of infeasible solutions, a larger penalty factor will help to improve population diversity. We first define the density of infeasible solutions in the neighbor of subproblem i :

3.4. Adaptive PBI method

ming pbi (xw, z *) = d1 + d2

(z * + d1

> 0 is a predefined penalty parameter, w = (w1, w2, w3)T is the weight vector, z * = (z1*, z2*, …, z m* )T is the ideal point in the objective space, i.e., for each i = 1,2, 3, z i* = minfi (x ) . Since z i* is generally unknown before searching, z i* is replaced by fi (x ) with the smallest value during the searching process. The PBI decomposition method is illustrated by Fig. 5. It is can be observed that since d1 and d2 indicate the utopia distance and the perpendicular distance respectively between x and z * in direction w , they are able to represent the convergence and diversity respectively. Therefore plays a critical role in balancing convergence and diversity [43]. A small value of can accelerate convergence process, but it may result in a local optimum; while a large value of can improve the variety of population, but the convergence will be slowed down.

Fig. 4. The effective of iteration on weight vectors.

=1

(F (x )

1 1+ e

i

Fig. 6. An example for crossover operation.

Fig. 5. The PBI method.

35

(30)

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two cut points of both offspring. 3.5.2. Mutation operation A mutation operation that reorder a part of the tasks genes and reassign them to workstations is implemented. At the reorder stage, two points are chosen randomly and the separators inside the two mutation points are removed, and the task genes are reordered into a new sequence with respect to the precedence constraints. At the reassign stage, a new assignment of those tasks is carried out by randomly generating new separators between the tasks. The probability of the separator insertion depends on the current staWT tion filling rate, fr = CT , where WT is the current workstation time, CT is the given cycle time. An illustrative example of this mutation operation is in Fig. 7.

Fig. 7. An example for mutation operation.

Fig. 8. An example for the energy-based local search.

where

and

3.6. The energy-based local search

are the controlling factors.

In order to additionally inject diversity of search progress to obtain better distributed Pareto front approximations, as well as incorporate the problem information to the algorithm, a problem-dependent local search operation is utilized to further improve the energy efficiency. A portion of solutions are selected in each generation after the reproduction operation to go through the local search with equal probabilities. In the procedure, the task genes pairs from adjacent workstations are exchanged, with a cost of the interchange evaluated, so as to search the neighborhood of the individual. In terms of the cost, the energy consumption is concerned. The search progress can be illustrated by Fig. 8 and described as follows. For each workstation s, s = 1, …, S 1, start from the first station: Step 1. Find all the task pairs (x , y ) satisfying x s , y s + 1, and the operating sequence of x , y can be exchanged without violating precedence constraints. Step 2. Calculate the cost, which is the energy consumption increase caused by the interchange in workstation s and s + 1.

3.5. Reproduction operation Crossover and mutation operations modified based on the representation scheme are used to produce the offspring population. Parents are selected from some neighboring weight vectors. Offspring replaces its parent only if it has a better aggregation function value. 3.5.1. Crossover operation To ensure the obtained offspring satisfies the precedence restrictions, partially mapped crossover (PMX), which has been widely used in genetic algorithms for the ALBP is implemented and modified in this paper. Two parents are mated to generate two different off-springs, as illustrated in Fig. 6. The crossover operation procedure is described as follows: Step 1. Two random cut points are selected. Remove the separators inside the two cut points. Step 2. For the first offspring, the genes outside the random points are copied directly from the first parent; the task genes inside the two cut points are copied but in the order they appear in the second parent. Step 3. For the second offspring, the same mechanism is followed up but with the opposite parents. Step 4. Insert the separators inside the two cut points randomly into

(31)

cost x, y = E

Step 3. Exchange the task pair with the lowest negative cost mincost x , y . If cost x, y 0 for all the task pairs, the interchange will not be

x *,

y*

x,y

executed. Based on the descriptions for each component of A-MOEA/D-ID

Table 2 Parameter values. Algorithms

A-MOEA/D-ID

MO-PSO

Basic MOEA/D

Parameter

Value

Population size Size of neighborhood. Crossover probability Mutation probability Maximum generation Parameter of weight vectors update Parameter of local search Particle swarm size Velocity factor Inertia factor Crossover probability Mutation probability Maximum generation Population size Crossover probability Mutation probability Size of neighborhood. Maximum generation

36

Low

Middle

High

210 15 0.8 0.2 30 2 0.6 210 0.1 0.8 0.8 0.2 30 210 0.8 0.2 15 30

136 15 0.8 0.2 40 2 0.6 136 0.1 0.8 0.8 0.2 40 136 0.8 0.2 15 40

91 14 0.8 0.2 50 2 0.6 91 0.1 0.8 0.8 0.2 50 91 0.8 0.2 14 50

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given in previous subsections, the pseudo codes of proposed algorithm are summarized as follows.

4. Computational experiments

4.1. Problem instances and parameters

In order to evaluate the performance of the proposed A-MOEA/D-ID in this paper, we implemented our algorithms in MATLAB to solve a set of test problem instances with different features. First, we detail how the test instances are generated and specify the parameter values. Then, we introduce some multi-objective performance indicators used for the computational tests. The analyses on the effects of each proposed algorithm component are carried out, and the proposed algorithm is compared with the existing benchmark algorithm MO-PSO and the basic MOEA/D. Finally, managerial applications are discussed to get managerial insights.

Since no benchmark data set is available for the sustainable RALBP, the test problems used in this research are generated based on precedence graphs available in http://www.assembly-linebalancing.de/, which are benchmark data sets for SALBP-1. The cycle times (CT) in the data sets for each instance are determined by the method of Hoffmann [44]. Additional factors such as energy consumption rate, required tool for each task, and tool-changing time are generated in the following specifications. Two levels, four and six, of robot types were evaluated with a tool changing time randomly generated from uniform distributions U [15,25], and an energy consumption rate generated from the 37

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Table 3 Experimental results of different problems. Problem

I

R

BUXEY (CT =36)

29

4

LUTZ1 (CT = 2020 )

32

4

GUNTHER (CT =36)

KILBRID (CT = 79 )

BUXEY (CT = 36 )

LUTZ1 (CT =2020)

GUNTHER (CT =36)

KILBRID (CT =79)

HAHN (CT =2806)

TONGE (CT =234)

ARC83 (CT =4206)

MUKHERJE (CT =222)

HAHN (CT =2806)

TONGE (CT =234)

ARC83 (CT =4206)

MUKHERJE (CT =222)

35

45

29

32

35

45

53

70

83

94

53

70

83

94

4

4

6

6

6

6

4

4

4

4

6

6

6

6

¯ IGD ¯ HV N T ¯ IGD ¯ HV

N T ¯ IGD ¯ HV N T ¯ IGD ¯ HV N T ¯ IGD ¯ HV N T ¯ IGD ¯ HV N T ¯ IGD ¯ HV N T ¯ IGD ¯ HV N T ¯ IGD ¯ HV N T ¯ IGD ¯ HV N T ¯ IGD ¯ HV N T ¯ IGD ¯ HV N T ¯ IGD ¯ HV N T ¯ IGD ¯ HV N T ¯ IGD ¯ HV N T ¯ IGD ¯ HV N T

A0

A1

A2

MO-PSO

MOEA/D

0.101 0.059 11 24.740 10.173 0.041 6 38.007 1.335 0.025 9 19.229 4.881 0.073 10 34.702 4.350

1.296 0.055 8 20.109 12.367 0.022 5 34.719 2.494 0.017 6 15.315 9.803 0.054 7 30.578 6.285

1.435 0.034 9 22.668 21.356 0.037 4 32.328 2.081 0.020 8 18.336 7.269 0.054 8 28.776 6.973

1.217 0.042 6 23.784 13.512 0.029 4 34.027 1.794 0.019 6 14.870 7.457 0.066 6 31.085 7.236

3.059 0.028 5 19.274 47.517 0.006 3 30.255 3.824 0.015 4 13.180 11.630 0.047 4 26.802 8.236

0.063 8 40.569 43.214 0.523 14 22.626 34.274 0.094 15 37.387 10.040 0.023 11 34.684 16.894 0.014 7 25.989 29.315

0.043 6 36.167 49.264 0.369 11 16.364 40.264 0.063 12 33.635 17.844 0.012 7 28.480 21.208 0.009 5 24.407 135.052

0.053 7 34.163 53.625 0.403 12 20.367 40.747 0.060 12 34.627 90.000 0.005 8 27.273 22.889 0.008 6 24.353 177.993

0.051 6 35.935 50.628 0.374 12 14.943 41.629 0.058 13 34.946 32.423 0.012 7 30.592 17.179 0.011 5 21.624 146.540

0.221 5 33.601 65.163 0.204 9 15.274 53.624 0.014 9 28.639 100.577 0.001 4 24.061 32.683 0.007 3 20.245 514.704

0.435 15 25.345 35.329

0.610 8 96.934 32.530

0.019 8 51.619 43.352 0.731 14 38.482 84.793 0.150 9 29.646 35.079 0.720 11 102.452 83.245 0.120 13 72.351

0.274 13 21.496 48.376

0.240 5 61.376 53.157

0.012 6 44.852 54.253 0.528 12 30.492 105.842 0.103 8 27.367 56.835 0.341 8 82.435 124.293 0.060 10 63.194

0.395 14 24.563 56.362

0.210 6 55.530 53.722

0.015 6 41.600 52.646 0.529 11 29.491 101.384 0.110 7 26.639 72.624 0.324 9 73.526 113.834 0.090 12 61.295

0.342 13 23.597 45.286

0.250 5 60.777 57.480

0.011 4 43.122 59.359 0.639 11 31.582 99.638 0.009 6 26.853 66.908 0.410 8 79.270 132.452 0.080 10 60.235

0.153 11 18.958 63.563

0.160 3 46.223 63.405

0.010 4 40.115 73.626 0.313 10 26.326 120.269 0.004 5 23.468 103.520 0.142 6 55.275 183.432 0.030 8 48.852

(continued on next page)

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Table 3 (continued) Problem

I

R

ARC111 (CT =7916)

111

4

ARC111-2 (CT =5755)

111

4

BARTHOL2 (CT =133)

BARTHOL2-2 (CT =513)

ARC111 (CT =7916)

ARC111-2 (CT =5755)

BARTHOL2 (CT =133)

BARTHOL2-2 (CT =513)

Average

148

148

111

111

148

148

4

4

6

6

6

6

¯ IGD ¯ HV N T ¯ IGD ¯ HV

N T ¯ IGD ¯ HV N T ¯ IGD ¯ HV N T ¯ IGD ¯ HV N T ¯ IGD ¯ HV N T ¯ IGD ¯ HV N T ¯ IGD ¯ HV N T

¯ IGD ¯ HV N T

A0

A1

A2

MO-PSO

MOEA/D

58.927

78.692

137.328

93.346

202.040

0.017 7 34.514 8.022 0.190 10 209.476 30.935 0.028 9 103.522 43.214 0.429 15 84.245 284.250 0.052 10 42.734 11.341

0.013 5 30.345 10.164 0.130 6 122.808 45.148 0.024 7 87.467 62.625 0.294 12 79.482 524.520 0.035 7 37.238 16.253

0.014 6 30.683 16.318 0.080 7 115.916 48.886 0.018 8 75.113 69.463 0.231 13 77.524 682.460 0.033 7 37.462 18.163

0.012 5 31.668 12.403 0.110 5 112.946 46.544 0.021 6 74.987 59.246 0.318 11 76.325 495.250 0.042 8 36.289 15.362

0.010 3 30.302 43.012 0.020 4 77.859 67.708 0.013 5 72.582 82.452 0.132 8 60.352 729.050 0.194 5 33.623 34.264

0.022 10 79.964 584.847

0.240 16 228.345 75.727 0.054 14 133.787 65.257 0.197 11 67.140

0.019 6 66.864 1262.814

0.017 11 63.548 1186.692

0.120 13 163.626 98.868

0.100 14 158.346 100.864

0.028 10 114.757 65.257 0.197 11 67.140

0.026 12 110.738 118.238 0.119 8 53.065

0.017 7 68.492 1294.757

0.170 13 142.569 84.858 0.037 11 108.468 130.630 0.119 9 51.036

0.015 5 49.237 1416.727

0.050 8 92.432 153.868 0.011 9 98.473 120.104 0.130 8 50.791

algorithms and their operators are shown in Table 2. 4.2. Performance indicators To measure the performance of the multi-objective optimization algorithms, two widely used metrics are employed: inverted generational distance [45] and hypervolume [46]. The definitions are as follows. (1) Inverted Generational Distance (IGD): IGD is a metric which represents average distance from the ideal PF to the approximate PF achieved. It evaluates the performance on convergence and diversity simultaneously. A smaller IGD represents better performance with respect to both diversity and convergence. Let P * be a set of solutions in the ideal PF, and P denote the approximate PF achieved by the algorithm, then the IGD is calculated as follows:

Fig. 9. Obtained Pareto fronts for BUXEY(29,4) by A0, A1, A2, MO-PSO and basic MOEA/D.

IGD(P *, P ) =

uniform distribution U[0.5,1.5]. Type of tools are randomly generated for each task. The variability in activity performance times by different robot types is set 0.5 on the base of expected time in original instances. The instances scales are divided into three groups according to the number of tasks: small (I 50 ), medium (I = 51 to 100 ) and large (I = 101 to 150 ) datasets. We executed each algorithm 10 times with different random seeds, setting a maximum number of iterations as stopping criterion on a PC with an Intel Core i5 2.8 GHz CPU, 8 GB RAM. The parameters of the

x P*

d (x , P )

|P *|

(32)

where d (x , P ) is the Euclidean distance from P to its nearest member of the true Pareto front P * . (2) Hypervolume (HV): HV is Pareto compliant and measures both convergence and diversity of the obtained solutions along the PF. A larger HV means a 39

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Fig. 10. Boxplots of IGD and HV values for A0, A1, A2, MO-PSO and basic MOEA/D.

better dominance relation. It calculates the volume enclosed by the obtained set P and a reference point in the objective space. Let z r = (z1r , z2r , …, z mr)T be the reference point,

HV(P ) = volume( x P

[f1 (x ), z1r ]×…×[fm (x ), zmr])

where volume is the Lebesgue measure. 40

(33)

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(2)

Fig. 11. Pareto fronts obtained by A0 and A2 in BARTHOL2(148,4).

(3)

(4)

(5)

Fig. 12. Solutions for different scenarios in HAHN(53,4).

Since we are working with real problems, the true Pareto fronts are not known. A pseudo-optimal Pareto set, which is an approximation of the true Pareto set, obtained by merging all the Pareto set approximations generated for each problem instance by any algorithm in any run will be considered. Besides, the reference points are specific for each problem instance by maximum values for the two objectives.

most diversely distributed set of solutions in average among the five algorithms. Moreover, the results also imply that the solutions found by A-MOEA/D-ID are closer to the pseudo-optimal Pareto fronts, which verifies that the solution sorting mechanism of A-MOEA/D-ID is more effective than that of MO-PSO and basic MOEA/D. Regarding the HV metric, the solutions achieved by the A-MOEA/DID are significantly better than those achieved by the other four algorithms, since the higher HV implies the better overall performance of the algorithm. Especially as the problem scales up, the advantage also increases. By comparing the performance of AMOEA/D-ID and basic MOEA/D, it is indicated that without the constraint-handling technique, the individuals can only be distributed among some narrow and disconnected feasible regions. As a result, most individuals can easily be trapped in one or a few of these feasible regions. In terms of the N indicator values, A-MOEA/D-ID is capable of finding more non-dominated solutions than MO-PSO and basic MOEA/D on average. The quantity of obtained non-dominated solutions also increases as the robot types grows, since the search space becomes larger. When it comes to the computational time, although the A-MOEA/DID spends more time on its sub-operators, it is able to solve the realistic problems within an acceptable computational time in view of its search ability and convergence performance. These results can also help us to find out if the removed components are necessary. By comparing the A-MOEA/D-ID with A1, it is observed that the simplified version without self-adaption of the penalty factor has its drawbacks on premature convergence due to the lack of diversity. In addition, compared to A2, the problem-dependent local search does help to push forward the Pareto front and improving the diversity according to the average IGD and HV metric.

To get a better intuition about the result of every independent run, the IGD and HV performance indicator values of every algorithms generated in the 10 runs have been represented by boxplots (see Fig. 10). In the figure, each box represents the distribution of IGD or HV values for a certain case by one algorithm (four instances of each problem scale are shown due to the lack of space, and similar results are obtained in every instance). It can be observed that the A-MOEA/D-ID outperforms the other four algorithms in most runs on both convergence and diversity. Thus, a conclusion can be made that the proposed A-MOEA/D-ID can work well on our SRALBP, and A-MOEA/D-ID do outperform the original MOEA/D in terms of the metrics of IGD, HV, N and computational time. In addition, the enhancing mechanisms show good performance and are much helpful to MOEA/D. The application prospect for the real-world engineering problems has been proved.

4.3. Experimentation and analysis of results First, a comparison of the proposed A-MOEA/D-ID with two limited variants is done to evaluate the performance of its components. Two most significant algorithm components, adaptive penalty factor and local search have been selected to be removed from A-MOEA/D-ID (A0), thus forming two variants (A1 and A2) of the A-MOEA/D-ID: A1: It differs from A0 in the absence of the self-adaption of the penalty factor , i.e., using a fixed value in the algorithm. plays a role in balancing convergence and diversity. A2: The difference between A0 and A2 lies in the problem specified local search operator, which is able to induce more diversity into the search mechanism. Secondly, the MO-PSO algorithm and the basic MOEA/D are compared with the complete A-MOEA/D-ID. According to the literature review, PSO algorithm has been widely used to solve assembly line balancing problems [24,47–50]. Since studies have shown the superiority of MOEA/D over NSGA-II in test instances with complicated PS shapes [39,51], NSGA-II is not chosen to be compared with. The same problem instances and performance indicators are used. Each case is performed 10 times by five algorithms respectively, and ¯ , HV ¯ , number of sothe average of every performance indicators IGD lutions, N , and computation time T is shown in Table 3. As an example, the obtained Pareto fronts for BUXEY(29,4) is shown in Fig. 9. The following observations can be made from the presented results.

4.4. Effect of the local search operator on energy saving The local search is a critical operator in the proposed algorithm and it influences the solution quality to a considerable extent. Focusing on the energy consumption, a typical example is shown in Fig. 11, which illustrates the Pareto front obtained by the complete algorithm A0 and local search removed version A2 in problem BARTHOL2(148,4). It can be evidently observed that the local search procedure has contributed significantly to improving the energy efficiency and the convergence of obtained solutions. Since the global search mechanism including selfadaption of the penalty factor remains the same for both algorithms, the diversity and quantity of solutions were almost similar for both algorithms. 4.5. Managerial applications

(1) According to the IGD metric, in most instances of the SRALBP, the AMOEA/D-ID clearly outperforms the MOEA/D and MO-PSO in term of the IGDs. It is indicated that the A-MOEA/D-ID can achieve the

In practical problems, the production efficiency of the assembly line and the energy efficiency are two crucial but conflict objectives, and it is important for the decision makers to determine the applicable 41

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balancing solution among the great number of feasible solutions. Since different decision makers may have different opinions and ambiguity may exist in the choice of the appropriate relative weights for each objective, all the values for both objectives should be presented. A preference vector w= (wm, wE )T can be used to denote the preference information of decision maker on number of workstations and energy consumption respectively, where wm + wE = 1. Instead of randomly generating the values of wm within [0,[1]], for better application, in this paper the values of wm and wE are equally divided into four intervals, i.e. [0, 0.25), [0.25, 0.5), [0.5, 0.75), [0.75, 1], corresponding to four application scenarios [52]. As an example, a medium-scale problem HAHN(53,4) is solved by A-MOEA/D-ID and the result is illustrated by Fig. 12. When the corresponding interval of preference vector is identified, the number of feasible solutions for the decision makers can be cut down, helping the manager to make the decision easier. Obviously, when greater importance is laid on the objective of minimizing total energy consumption, the right-hand intervals of the Pareto front should be selected. Meanwhile, the objective of workstation number becomes greater relatively. Put in a different way, when the tolerance range of workstation number is given by the decision maker, a solution with the lowest total energy consumption can be provided. Along with the referable applications for real situation production managements in this section, the proposed problem is believed to provides the managers a method to create a more environmentally friendly robotic assembly line on the basis of guaranteeing the production efficiency, thus making the enterprises more competitive and the industry more sustainable.

Declaration of Competing Interest None. Acknowledgements The authors would like to thank the anonymous referees for their constructive suggestions and comments. This work was supported by the National Natural Science Foundation of China [grant numbers 7147, 1135]. References [1] Salahi N, Jafari MA. Energy-Performance as a driver for optimal production planning. Appl Energy 2016;174:88–100. https://doi.org/10.1016/j.apenergy.2016.04. 085. [2] Fysikopoulos A, Anagnostakis D, Salonitis K, Chryssolouris G. An empirical study of the energy consumption in automotive assembly. Procedia Cirp 2012;3:477–82. https://doi.org/10.1016/j.procir.2012.07.082. [3] Koho M, Tapaninaho M, Heilala J, Torvinen S. Towards a concept for realizing sustainability in the manufacturing industry. J Ind Prod Eng 2015;32:33–43. https://doi.org/10.1080/21681015.2014.1000402. [4] Kishawy HA, Hegab H, Saad E. Design for sustainable manufacturing: approach, implementation, and assessment. Sustainability 2018;10:1–15. https://doi.org/10. 3390/su10103604. [5] Wang H, Zhong RY, Liu G, Mu W, Tian X, Leng D. An optimization model for energyefficient machining for sustainable production. J Clean Prod 2019;232:1121–33. https://doi.org/10.1016/j.jclepro.2019.05.271. [6] Che A, Wu X, Peng J, Yan P. Energy-efficient bi-objective single-machine scheduling with power-down mechanism. Comput Oper Res 2017;85:172–83. https://doi.org/ 10.1016/j.cor.2017.04.004. [7] Saavedra MMR, de O, Fontes CH, M. Freires FG. Renew Sustain Energy Rev 2018;82:247–59. https://doi.org/10.1016/J.RSER.2017.09.033. [8] Öztürk C, Tunalı S, Hnich B, Örnek A. Cyclic scheduling of flexible mixed model assembly lines with parallel stations. Int J Ind Manuf Syst Eng 2015;36:147–58. https://doi.org/10.1016/j.jmsy.2015.05.004. [9] Tiacci L. Mixed-model U-shaped assembly lines: balancing and comparing with straight lines with buffers and parallel workstations. Int J Ind Manuf Syst Eng 2017;45:286–305. https://doi.org/10.1016/j.jmsy.2017.07.005. [10] Akpinar Ş, Baykasoğlu A. Modeling and solving mixed-model assembly line balancing problem with setups. Part I: a mixed integer linear programming model. Int J Ind Manuf Syst Eng 2014;33:177–87. https://doi.org/10.1016/j.jmsy.2013.11.004. [11] Lopes TC, Michels AS, Sikora CGS, Magatão L. Balancing and cyclical scheduling of asynchronous mixed-model assembly lines with parallel stations. Int J Ind Manuf Syst Eng 2019;50:193–200. https://doi.org/10.1016/j.jmsy.2019.01.001. [12] Pearce BW, Antani K, Mears L, Funk K, Mayorga ME, Kurz ME. An effective integer program for a general assembly line balancing problem with parallel workers and additional assignment restrictions. Int J Ind Manuf Syst Eng 2019;50:180–92. https://doi.org/10.1016/j.jmsy.2018.12.011. [13] Ogan D, Azizoglu M. A branch and bound method for the line balancing problem in U-shaped assembly lines with equipment requirements. Int J Ind Manuf Syst Eng 2015;36:46–54. https://doi.org/10.1016/j.jmsy.2015.02.007. [14] Chica M, Bautista J, Cordón Ó, Damas S. A multiobjective model and evolutionary algorithms for robust time and space assembly line balancing under uncertain demand. Omega (United Kingdom) 2016;58:55–68. https://doi.org/10.1016/j.omega. 2015.04.003. [15] Becker C, Scholl A. A survey on problems and methods in generalized assembly line balancing. Eur J Oper Res 2006;168:694–715. https://doi.org/10.1016/j.ejor.2004. 07.023. [16] Battaïa O, Dolgui A. A taxonomy of line balancing problems and their solutionapproaches. Int J Prod Econ 2013;142:259–77. https://doi.org/10.1016/j.ijpe.2012. 10.020. [17] Levitin G, Rubinovitz J, Shnits B. A genetic algorithm for robotic assembly line balancing. Eur J Oper Res 2006;168:811–25. https://doi.org/10.1016/j.ejor.2004. 07.030. [18] Rubinovitz J, Bukchin J, Lenz E. RALB – a heuristic algorithm for design and balancing of robotic assembly lines. CIRP Ann Manuf Technol 1993;42:497–500. https://doi.org/10.1016/S0007-8506(07)62494-9. [19] Gao J, Sun L, Wang L, Gen M. An efficient approach for type II robotic assembly line balancing problems. Comput Ind Eng 2009;56:1065–80. https://doi.org/10.1016/j. cie.2008.09.027. [20] Gutjahr AL, Nemhauser GL. An algorithm for the line balancing problem. Manage Sci 1964;11:308–15. https://doi.org/10.1287/mnsc.11.2.308. [21] Tsai D-M, Yao M-J. A line-balance-based capacity planning procedure for seriestype robotic assembly line. Int J Prod Res 1993;31:1901–20. https://doi.org/10. 1080/00207549308956831. [22] Kim H, Park S. A strong cutting plane algorithm for the robotic assembly line balancing problem. Int J Prod Res 1995;33:2311–23. https://doi.org/10.1080/ 00207549508904817. [23] Zhou B, Wu Q. A novel optimal method of robotic weld assembly line balancing problems with changeover times: a case study. Assem Autom 2018;38:376–86. https://doi.org/10.1108/AA-02-2018-026.

5. Conclusions Motivated by the global energy crisis and upward trend of automatic production, this paper models a bi-objective sustainable robotic assembly line balancing problem which deals with both green manufacturing objective (the total energy consumption) and productivity-related objective (number of workstations with a given cycle time). The comprehensive and meticulous consideration of energy consumption in different states of robotic assembly line (processing, changeover, transportation and standby) will significantly contribute to the greening and sustainability of manufacturing process. An adaptive infeasibility-driven MOEA/D with a problem specified local search operator has been proposed to solve the problem. An adaptive penalty factor scheme has been incorporated to balancing the convergence and diversity of solutions. Aiming at further maximizing the energy efficiency, the problem-specific local search module was designed to further improve the performance of the algorithm. Computational experiments are designed and carried out to evaluate the effectiveness of the proposed algorithm. The experimental results show that the improving strategies are of great significance to the proposed algorithm and it outperforms MO-PSO and basic MOEA/D on both convergence and diversity. Managerial insights are provided to make tradeoffs between productivity goal and energy consumption objective, jointly making the whole robotic assembly line more sustainable. Future research may consider other aspects of sustainable manufacturing such as the greenhouse gas emission and the carbon footprint in the assembly procedure. Multi-objective optimization problems concerning productivity, energy and cost can be studied to give more reference to decision makers. Since the attention of this study was paid to single model, future work may include much complex real-world problems such as mixed-model assembly lines and two-sided assembly lines balancing problem. Besides, there are still much can be done to improve the algorithm. Since most of the existing studies proposed approximate methods and metaheuristics as solution approaches, some effective exact approaches can be proposed to solve small sized instances of the multi-objective ALBPs. Since the proposed enhancing mechanism is problem-specific, in our future works, the versatility of the algorithm can be exploited and application of A-MOEA/D-ID to other engineering optimization problems can be studied. 42

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