solving a robotic assembly line balancing problem using efficient hybrid methods

J Heuristics (2014) 20:235–259DOI 10.1007/s10732-014-9239-0

Solving a robotic assembly line balancing problem usingefficient hybrid methods

Slim Daoud · Hicham Chehade · Farouk Yalaoui ·Lionel Amodeo

Received: 20 July 2012 / Revised: 31 May 2013 / Accepted: 17 February 2014 /Published online: 9 March 2014© Springer Science+Business Media New York 2014

Abstract In this paper we are studying a robotic assembly line balancing problem. Thegoal is to maximize the efficiency of the line and to balance the different tasks betweenthe robots by defining the suitable tasks and components to assign to each robot. We areinterested in a robotic line which consists of seizing the products on a moving conveyorand placing them on different location points. The performances evaluations of thesystem are done using a discret event simulation model. This latter has been developedwith C++ language. As in our industrial application we are bounded by the executiontime, we propose some resolution methods which define the suitable component andpoint positions in order to define the strategy of pick and place for each robot. Thesemethods are based on the ant colony optimization, particle swarm optimization andgenetic algorithms. To enhance the quality of the developed algorithms and to avoidlocal optima, we have coupled these algorithms with guided local search. After that, anexact method based on full enumeration is also developed to assess the quality of thedeveloped methods. Then, we try to select the best algorithm which is able to get thebest solutions with a small execution time. This is the main advantage of our methodscompared to exact methods. This fact represents a great interest taking in considerationthat the selected methods are used to manage the functioning of real industrial roboticassembly lines. Numerical results show that the selected algorithm performs optimallyfor the tested instances in a reasonable computation time and satisfies the industrialconstraint.

S. Daoud · H. Chehade · F. Yalaoui · L. AmodeoARIES Packaging, Technopole de l’Aube en Champagne, Rosires, Francee-mail: [email protected]

S. Daoud · H. Chehade (B) · F. Yalaoui · L. AmodeoICD, LOSI, Troyes University of Technology, UMR 6281, CNRS,Troyes, Francee-mail: [email protected]

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Keywords Robotic assembly line balancing · Metaheuristics · Guided local search

1 Introduction

Using robots, the flexibility and the automation of assembly lines can be enhanced. Therobotic assembly line balancing (RALB) problem is based on a balanced distributionof work between robots with an attempt for optimal assignments task. In this paper,we present an industrial application of robotic assembly line balancing.

The contribution of our work is to develop a system which is able to manage thefunctioning of a real robotic system in which we are supposed to balance the tasksbetween the robots and to maximize the efficiency of line. The efficiency is defined asthe number of seized components of products by each robot. As we have an industrialconstraint that arose out of the execution time, we are unable to apply an exact methodthat requires a large execution time for this system. Metaheuristics were thus therelevant solution. In one hand, we present some search techniques based on ACO,PSO and GA and in the second one we enhance the performances of the algorithms bycoupling them with a guided local search to our RALB in order to escape from localoptimal solutions.

The motivation of this work is first based on an industrial request to find the bestand balanced distribution of components and location points for each robot in orderto maximize the efficiency of the line. The second motivation is based of the lack ofworks deal with the robotic assembly line balancing for maximizing the efficiency.

The remainder of this paper is organized as follows. The second section presents theproblem description which is the type E of robotic assembly line balancing problemsand the problem formulation. The different search techniques and the guided localsearch are presented in the third section. Computational experiments and numericalresults are presented in the fourth section before ending the paper by a conclusion andperspectives for future works.

2 State of the art

As mentioned before, we are interested in a robotic assembly system which consistsof seizing components of products and assembling them on different location pointson a moving conveyor belt by pick and place robots.

In this literature review, we discuss about many topics such as: multi-robot assemblycell using pick and place robots, robotic assembly line balancing problem and theapplication of the developed resolution techniques in robotic systems.

First, many researches were interested with the multi-robot assembly cells usingpick and place robots. These kinds of robots have been widely applied in automatedequipments and press fabrication industries as they could satisfy some special perfor-mances requirements. There are different types of robots. Huang et al. (2007) havestudied the time-minimum trajectory planning problem of the diamond robot. For scararobot, Taylan Das and Canan Dlger (2005) have proposed a mathematical modelingand a dynamic simulation referring to the experiment data available. Kelaiaiaa et al.

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Solving a robotic assembly line balancing problem 237

(2012) have presented a methodology of dimensional design of parallel delta robotsbased on the genetic algorithm SPEA-II.

May et al. (1989) have proposed a flexible architecture for the control systemof multiple robots coordination. Bozma and Kalalioglu (2012) have studied the muti-robot coordination in pick and place operations. This problem is similar to our researchwork which arose out of a real time factory automation problem where there aremultiple robots working in parallel. The goal is to define a multi-robot coordinationstrategy. For that reason, they have proposed an approach on non cooperative gametheory where the robot decides its tasks based on the local observation of the conveyorbelt and the decision of the robot neighbor. Lee and Lee (2002) have developed a severalautomata modeling for the supervisory control logic for a multi-robot assembly cell.Ho and Ji (2009) have presented a mathematical model and a heuristic for a schedulingproblem of PCB components on a sequential pick and place machines.

Another important feature taken in consideration our paper is the robotic assem-bly line balancing. Rubinovitz and Bukchin (1991) were the first to introduce therobotic assembly line balancing (RALB). They have suggested a branch and boundmethod to solve SALB which aims to balance the workload of different robots. Kimand Park (1995) have proposed an integer programming formulation and a cuttingplane algorithm for the robotic assembly line balancing. Tsai and Yao (1993) havepresented an approach which provides the number and the type of robot for roboticassembly lines. This approach is based on an integrated capacity planning procedure.Khouja et al. (2000) have developed two-stage methodology to perform robotic assem-bly cells. Nicosia et al. (2002) have proposed a dynamic programming algorithm inorder to minimize the cost of the workstation. Their problem aims to assign all tasks tonon identical workstations subject to precedence constraints. Bukchin and Rabinow-itch (2006) have generalized this approach by developing an optimal solution basedon a back traking branch and bound algorithm. Levitin et al. (2006) have dealt with thetype II of robotic assembly line balancing (RALB-II) problem and proposed a geneticalgorithm GA which aims to assign tasks to workstations and to select the best robottype for each workstation. Yoosefelahi et al. (2012) have presented a new formulationof the robotic assembly line balancing problem type II. This latter aims to minimizethe cycle time, the robot setup costs and the robot costs. The authors have developedthree versions of multi-objective evolution strategies to solve this problem.

Since, our problem is bounded by the industrial constraint related to the executiontime, we should develop an approach which is able to solve the problem in a shortcomputational times. We opt to metaheuristics taking in consideration that those latterhave proved their efficiency in solving, in short computational times, different kindsof combinatorial optimization problems. These metaheuristics are based on ACO, GAand PS. Then, guided local search is developed to enhance the search ability of thedeveloped algorithms.

The first method is based on ant colony optimization. Ant colony optimization(ACO) has been defined by Dorigo (1992). This algorithm is based on the behaviorof ants while searching for food source. In fact, ants deposit a chemical substanceon their ways. This chemical substance is called pheromone and it aims to guidethe ants while looking for the optimal solutions. Nowadays, scientists are applyingant colony algorithms to solve assembly line balancing problems and also RALB

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problems. Ze-qiang et al. (2007) have presented an improved ant colony optimizationto solve type 1 of the simple assembly line balancing problem. Simaria and Vilarinho(2009) have studied Two-sided assembly lines in which workers perform assemblytasks in both sides of the line. They have proposed a mathematical model and anant colony optimization to solve the problem. The main goal of the problem is tominimize the number of workstations. Yagmahan (2011) has dealt with the mixed-model assembly line balancing problem. He has developed a multi-objective ant colonyoptimization in order to minimize the number of stations for a given cycle time. Forrobotic assembly lines, Sharma et al. (2008) have used the ACO for the generation ofoptimized robotic assembly sequences by minimizing the energy function. Liang etal. (2012) have developed an ant colony optimization for on-line scheduling and duedate determination.

The second method is based on the particle swarm optimization which was intro-duced by Kennedy and Eberhart (1995). This algorithm, gleams with the behaviorof swarm intelligence in nature. Jian-sha et al. (2009) have proposed a hybrid PSOalgorithm for solving type 2 of an assembly line balancing problem using PSO globalsearch capability. Dongyun et al. (2010) have proposed the method of linearly decreas-ing the inertia weight of the PSO to solve on assembly line problem. Qiu-gao (2010)has proposed to solve the mixed-model assembly line balancing PSO and simulatedannealing algorithms. His results show that the PSO-SA algorithm is very efficient andensures a higher assembly line balancing ratio. Chutima and Chimklai (2012) havepresented a hybrid PSO algorithm to solve a multi-objective two-sided mixed modelassembly line balancing problem. This kind of PSO is hybridized with a negativeknowledge which employs the knowledge of the relative positions of different parti-cles in generating new solutions. Dou et al. (2011) have presented a PSO algorithmenhanced by the reduced variable neighborhood search which is used to perform alocal search within the neighbors of the best particle. This algorithm aims to solve theassembly line balancing problem of type 1.

The last method developed for our problem is the genetic algorithm (GA) initiallydefined by Holland (1975). Akpinar and Bayhan (2011) have proposed a hybrid geneticalgorithm to solve a mixed model assembly line balancing problem of type 1. This latteraims to minimize the number of workstations, maximize the workload smoothnessbetween workstations and maximize the workload smoothness within workstations.Gao et al. (2009) have presented a type II of robotic assembly line balancing (rALBII) problem, in which each task has to be assigned to each workstation and eachworkstation needs one available robot. They have proposed an innovative genetic(GA) hybridized with local search based on different neighborhood structures. Chenet al. (2012) have developed a grouping genetic algorithm for assembly line balancingof sewing lines with different labor skill levels in garment industries. Chu and Beasley(1998) have proposed a genetic algorithm for the mutidimensional knapsack problem.

As mentioned before, to enhance the performances of the developed algorithms,such as ACO, PSO and GA, we have coupled these letters by a guided local search(GLS). Voudouris and Tsang (1996) were the first to develop the GLS for combinato-rial optimization problems to avoid local optima by changing the objective functions.This is done by increasing penalties in an augmented objective function. Tseng et al.(2007) have proposed a guided local search to improve the performance of a memetic

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algorithm in order to guide it out of local minima and to solve assembly sequence plan-ning. Chehade et al. (2009a) have developed an ant colony optimization with guidedlocal search for selecting machines and sizing buffers in assembly lines. Meanwhile,guided local search was applied in many research works such as facility layout prob-lems by Yalaoui et al. (2009) and quadratic assignment problems as described by Haniet al. (2007).

This paper proposes a metaheuristics to solve a robotic assembly lines balancingof type E. We have added GLS procedure to escape from the local optima and toenhance the performances of the solutions. At our knowledge, the only paper whichinvestigates on real time factory automation problem is Bozma and Kalalioglu (2012).This problem is similar to our problem. Meanwhile, we add the balancing loadingof the lines for our studies. We may note that a few work which deal with compar-ison of metaheuristics in pick and place robotic system or in assembly line balanc-ing. Suren (2012) has presented an application and comparison of ACO and PSOin manufacturing automation which uses a pick and place robot material handlingsystem. Faisae Rashid et al. (2011) have reviewed the most frequently soft comput-ing approaches, (ACO, PSO and GA), for assembly sequence planning and assemblyline.

3 Problem description

The problem comes from a pick and place robotic assembly line which is composedof a moving conveyor and robots in order to get a final assembled product. The mainobjective of this work is to balance all tasks for each robot and to maximize theline efficiency E by maximizing the gripping products. This robotic assembly line isillustrated in Fig. 1 and it is composed of I serial pick and place robots. Theses latters,seize the J components (on the left side of the figure) from the moving conveyor beltand places them in specific locations (right side of the figure). This operation whichis the robot motion is called in the rest of the paper the assigned tasks. Furthermore,the assembly of the final product requires the execution of K assigned tasks. The finalassembled product is composed of n layers and for each layer we have K1 components∀ α = 1, . . . , n . We note Kα,β the position of the component in the final assemblyproduct. For the first layer, β is laying between 1 and K1. After that, β is betweenKα+1 and Kα ,(∀ α = 1, . . . , n − 1 ).

The order in which the assigned tasks have to be performed is ensured by precedenceconstraints. Figure 2 presents an example of a final assembled product and shows theprecedence constraints between all the components marked by the different layers.

The following assumptions are stated to clarify the context in which the problemarises. These assumptions were formulated by Rubinovitz and Bukchin Rubinovitzand Bukchin (1991) to which we have added some extra ones:

1. The precedence relationships are due to technological assembly constraints2. The duration of an assembling task is deterministic and cannot be subdivided3. The duration of an assembling task depends on the assigned robot and the assigned

location points4. The robot activity (assembling task) is limited by the throughput rate

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Fig. 1 Robotic systems description

Fig. 2 Precedence constraints of final assembled product

5. A single robot is assigned to each station6. One type of product is considered in this problem

4 Resolution method

This section presents the methods developed to solve the robotic assembly line bal-ancing. The idea is to satisfy the industrial constraint about the execution time require-ments in order to manage the functioning on real time the robotic systems. For thisreason, we are unable to apply an exact method that requires a large execution time.Metaheuristics are one of the relevant solutions. We present an application of the antcolony optimization (ACO), particle swarm optimization (PSO) and genetic algorithm(GA) for the RALB problem. The adopted algorithms, in similar problems like SALBP,prove their efficiency to solve this type of problems.

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We present in second times the guided local search (GLS) and its application todifferent metaheuristics applied to our problem. The mean advantages of the GLS arethe ability of enhancing the quality of the developed algorithms and avoiding the localoptima. As explained in Rubinovitz and Bukchin (1991), the GLS is based on theincreasing of penalties of the augmented objective function. These penalties allow toavoid the local minima and to get the optimal solutions. After that, an exact methodbased on full enumeration is also developed to assess the quality of the differentdeveloped metaheuristics.

4.1 Proposed methodology

4.1.1 Ant colony optimization (ACO)

The first method is an ant colony optimization algorithm. Its methodology is inspiredfrom the ant behavior to search the way of food source. This algorithm is thenbased on pheromones (chemical substances) by using matrix showing the quantityof pheromones between the different points visited by each ant allowing then to lookfor the optimal solutions.

Solution encoding: In order to apply the algorithm to our problem, we have adoptedthe encoding presented in Table 1 for the robotic assembly line with K M locationpoints, J components and I robots. This table shows the encoding adopted for thelocation points and components. Each location points might be assigned to a robotamong I robots and to a component among J components. This gleams that eachassigned components j ( j = 1, . . . , J ) is picked by robot i (i = 1, . . . , I ) and placedon a deposit point k (k = 1, . . . , K M). As mentioned in the introduction, this robotmotion is called the assigned task. The algorithm procedure is as follows: first, eachant is deposited randomly on a starting point corresponding to the location points toassign to a robot and the component. Then, each ant will move from a point to anotherand at the end the fitness function is computed which is related to the number of seizedcomponents by each robot.

Tours construction: Parallel ants have been used to assign the robots and componentsfor each location points. An ant fi chooses to move from a point r to another point

Table 1 Encoding representations for ACO

Location points 1 Components number: 1 2 … J

Robot number: 1 2 … I

Location points 2 Components number: 1 2 … J


. . . . . .

. . . . . .

Location points KM Components number: 1 2 … J


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s based on standard equations for tours construction. In Eq. 1, q is a random numbergenerated between 0 and 1, q0 is a parameter (0 ≤ q0 ≤ 1) which determines therelative importance of exploitation against exploration. S∗ is a random variable chosenbased on a probability given by Eq. 2. ηr,s is the quantity of pheromone between thepoints r and s and ηr,s is a static value used as a heuristic of innate desirability tochoose s starting from r . ηr,s is also called the ant visibility to choose a point startingfrom another point.

s ={

arg maxu∈Jk (r)[ταr,u[ηβ

r,u] if q ≤ q0

s∗ otherwise.(1)

Parameters α and β are used to determine the relative importance of pheromonesversus the visibility. Jk(r) is the set of points not yet visited by ant k.

s∗ =⎧⎨⎩

[ταr,u [ηβ

r,u ]∑u∈Jk (r) [τα

r,u [ηβr,u ] if s ∈ Jk(r)

0 otherwise.(2)

Pheromone updates: There are two types of pheromone updates. The first one isthe local pheromone update which is applied when all ants have finished their tours.It is computed based on the pheromone evaporation rate (0 ≤ ρ ≤ 1) and τ0 which isthe initial quantity of pheromone. The local pheromone is computed based on Eq. 3.

τr,s = (1 − ρ)τr,s + ρτ0 (3)

The second one is the global pheromone update while being based on the non-dominated solutions obtained at each generation. The global pheromone is computedbased on Eq. 4.

τr,s = (1 − ρ)τr,s + ρ�τr,s (4)

We have modified the value of �r,s which is a parameter which consists of supportingor reinforcing the best solutions found so far. The value of �r,s is computed accordingto Eq. 5.

�τr,s = τ 0r,s + X

(1 − Lgb

Ugw

)(5)

Where τ0 is the value of the first pheromone, X is constant, Lgb is the value of theglobally best tour from the beginning of the trial and Ugw is the value of the globallyworst tour. With these updates of the pheromone matrix, arcs which have the highestpheromone quantities attract more ants in the next step. Algorithm 1 shows the globalstructure of the ACO algorithm.

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Algorithm 1: Structure of the ACO algorithm for robotic assembly line balancing

Initialize the pheromone ratePlace randomly the N ants on the first location points to assign the components and robotsD P=1For D P=2 to the all location points (KM)

Choose the next points to visit (robot and components to select for the location points)based on equations 1 and 2 of tour construction

Check the precedence constraint of location points[Dp]If location points[DP]=Robot i then location points[DP+1]=Robot i or robot j

∀k ∈ pred(h) and ∀(i, j) ∈ 1, . . . , IEND ForAnt’s return to the starting pointLocal update of pheromones (equation 3)For k=1 to the number of ants

Evaluate the solutions (the number of seized products )END ForRetain the best solutionGlobal update of pheromones (equation 4)If the stopping condition is satisfied

Stop the algorithm and existElse

Return to the step of building a new tour for each antEnd If

4.1.2 Particle swarm optimization

Below, we describe the particle swarm optimization algorithm applied to solve the con-sidered robotic assembly line. This method is based on a population composed by dif-ferent particles. These latters change their position in a dimensional DomaineLimitresearch space. Each particle p has its vector position, its speed called vector velocity,and its best vector position ever visited (Pbest). For the swarm of particles, there isonly one best quality particle which is referred to vector global (Gbest). Thus at eachiteration, particles move by Pbest and Gbest to a new vector position.

Solution encoding: In this section we present the encoding of the particle swarmoptimization for this problem. Each particle in the population represents a solutionand a fitness function is computed and related to the number of seized components byeach robot. Each particle is composed of three parts, the first for location points, thesecond for robots number and the last concerns the components number. The size ofa particle is the sum of all location points K M . The position of a particle contains aninteger number that represents the indices of robot and components for each locationpoints. For the speed of the particle p, it represents the movement that occurs eachparticle for changing the robot number and components.

Particle Swarm Optimization operation: Individuals (particles) of a swarm showinclination to change their movements by using the information below:

– Best position of the particle p (local best) (Pbestp).– Best position of the particle group global best (gbestp).

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– Position of pth particle in i t th iteration xitp (i t = 1 · · · i termax and p = 1 · · · P).

– Speed of the particle p in iteration i t V itp .

Each individual speed changes according to Eq. 6:

V k+1i = wV k

i + C p(Rnd)(Pbesti − xki ) + Cg(Rnd)(Gbesti − xk

i )(modulo)

(DomaineLimit) (6)

Position of the particles changes by speeds as shown in Eq. 7

xk+1i = xk

i + V k+1i (7)

Pbesti − xki denotes the distance between the current vector position and personal

best position. The latter is calculated in using the best cost which corresponds to thisposition. The global best position Gbesti corresponds to the best cost found by a singleparticle in the population.

Algorithm 2 shows the structure of the PSO algorithm as described by Onut et al.(2008).

Algorithm 2: Structure of the PSO algorithm for robotic assembly line balancing

Generate initial solution (iter=1)randomly for all particles (p ∈ P) ForD P=1 to KM(total number of location points)

Check the precedence constraints of the location points [DP]If location points[DP]=Robot i then locations points[DP+1]=Robot i or Robot j

∀k ∈ pred(h) and ∀(i, j) ∈ 1...IEnd ForAssign the best for all particles with initial solutionFind the best among all particles and assign it to the GbestGenerate initial velocities (V i

pt) randomly for all particlesAdd velocities to the corresponding particles (xi

pt + 1)

While Number of iterations is not reachedDoFor p=1to P (total number of particles)

Update velocities (V ipt)

Modify the current positions (xipt)

If the solution is improved thenUpdate the current Pbest

End IfEnd ForIf the solution is improved then

Update the current GbestEnd IfEnd WhileReturn the best solution Gbest

4.1.3 Genetic algorithms

The solutions encoding for the GA is the same used for the PSO and which is presentedin Table 2. Therefore, a matrix with K M genes (chromosome) is proposed to code

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Table 2 Encoding representations for PSO

Particle P Points positions: 1 2 … KM Fitness

Robot number: 1,…,I 1,…,I … 1,…,I

Components number: 1,…,J 1,…,J … 1,…,J

Table 3 Example of thechromosome k 1 2 3 4 5

i 1 1 2 2 3

j 1 3 4 2 5

the problem. Each gene represents the location points of the final assembled productwhich contains the robots number and components number. In each gene containstwo integer values which represent respectively the robot number which is between 1and I and the component assigned to location points which is between 1 and J . Eachindividual is assigned a fitness related to the number of seized products. An exampleof the chromosome is presented in Table 3.

If we take in consideration the first gene in Table 1 which is in bold, it means thatthe first robot picks the first component and deposits it in the first deposit point in thefinal assembled product.

Genetic operations: In a genetic algorithm, it is very important to choose the favor-able operators such as crossover, mutation and selection.

– Fitness evaluation:The value provided by the fitness function is the number ofseized products which we are interested to maximize.

– Reproduction: The reproduction uses a crossing process which consists of obtain-ing two children by crossing one or more genes from parents. To solve our problem,a cross point is adopted (crossover 1X). We have opted for this type of crossoveroperator based on a previous work of Yalaoui et al. (2009) which show the advan-tages and the efficiency of this operator. In each location points, an exchange ofrobot and component number is randomly generated. As described in Fig. 3, thefirst part of parent 1 and the second part of the parent 2, which are colored in gray,are copied to generate a new solution (offspring 1). A similar procedure is appliedto generate the offspring 2. The crossover probability is Pc.

– Mutation: The objective of the mutation is to avoid the algorithm a local optimum.Two points are randomly selected from the chromosome. In our case, we choicerandomly a gene and random change of component is generated. The mutationprobability is Pm .

The GA algorithm is represented in algorithm 3.

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Fig. 3 The crossover operator

Algorithm 3: Structure of the GA algorithm for robotic assembly line balancing

Generate randomly the initial population of N chromosomesFor D P=1 to KM (total number of location points)

Check the precedence constraint of location points[DP]If location points[DP]=Robot i then location points[DP]=Robot i or robot j ∀k ∈

pred(h) and ∀(i, j) ∈ 1, ..., IEND ForFor each iteration Do

Evaluate the fitness of each individualIf stopping criteria satisfied (i ti = Ng?)

If i ti = Ng then StopElse Select two individuals as parents

Perform the crossover for two individuals based upon the crossover probabilityPerform the mutation for each gene of the one individual (offspring) based upon the

mutation probabilityInsert the parent offspring and offspring mutation in next populationSorting the next population such as selected the individuals which have the best fitnessUpdate the population

END ForEND If

4.1.4 The guided local search algorithm

In order to enhance the performance of the proposed algorithms, we propose to couplethem with the guided local search. This algorithm is able to avoid the local optima by the

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changing the objective function by increasing the penalties associated to the features.We have choosen this method since the lack of studies concerning the application ofthis specific algorithm (GLS) on robotic assembly line balancing and also taking inaccount the promising results in the works of Hani et al. (2007) and Chehade et al.(2009a). The GLS penalizes the features ft of each solution s which has the maximumvalue of the utility util(s, f t). The adaptation of the guided local search for the roboticassembly line balancing is realized while being based on the following analogy of thefeature ft of solution s which depends on the assignment of the components j andlocation points k of the final assembled product m for robot i . The relative cost isthe workload of each robot i . The variation of the objective function is due to themodification of penalties which are incremented by the guided local search for thefeatures that maximize the utility expression as defined in Eq. 8.

util(s, ft ) = It (s)ct

1 + pt(8)

Where:

– It (s): indicates if the feature ft is present in the current solution s (It (s) = 1) or(It (s) = 0)

– ct (s): indicates the cost function which gives the cost of having ft in s– pt : the penalty which is initially set to 0 and incremented in order to penalize the

occurrence of ft in local minima.

Therefore, the goal is to penalize the features which have highest costs. For thisreason, GLS uses an augmented cost function in order to avoid the local optimum. Theaugmented objective function h(s) is defined by the Eq. 9. The guided local search(GLS) can be presented in the following: starting from initial solution given by anydeveloped method, a local search method is applied to get the local minimum withrespect to the augmented cost function. If the objective function (not augmented) ofthis minimum better than the best objective function ever found, it is saved as the betterever found solution. Therefore, the configuration having the maximum utility wouldhave its corresponding penality increased. The GLS algorithm is given in algorithm 4.

h(s) = g(s) + λ′q∑

t=1

It (s)pt (9)

Where g(s) is the cost function, is the parameter for the diversification of the searchof solutions and q is the number of features defined over solutions. The cost of havingft in s, as described in Eq. 10:

c(i) =J∑

j=1

K∑k=1

M∑m=1

ti jkm xi jkm (10)

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The value of λ′ adapted to our problem is defined in Eq. 11 where J is the total numberof component and K M is the all deposit point for all final assembled products.

λ′ =∑J

j=1∑K

k=1∑M

m=1 xi jkm

2max(J, K M)(11)

Algorithm 4: Structure of the GLS

Step 1 Compute λ′Step 2 Set the best solution s′ to the initial solution sStep 3 Perform the local search with respect to the augmented cost functionStep 4 Set s∗ to the solution having the lower augmented cost functionStep 5 If the cost (s∗) < cost (s′), then replace s′ by s∗Step 6 Find the features of the s∗ having the maximum utility and increase thecorresponding penalty (pt =pt + 1Step 7 return to step 3 until a given stopping criterion (CPU time) is satisfiedStep 8 s′ is the best solution found for the original problem

Algorithm 5: Structure of the ACO with guided local search for robotic assembly linebalancing

Initialize the pheromone ratePlace randomly the N ants on the first location points to assign the components and robotsD P=1For D P=2 to the all location points (KM)

Choose the next points to visit (robot and components to select for the location points)based on equations 1 and 2 of tour construction

Check the precedence constraint of location points[Dp]If assembled task[k]=Robot i then location points[Dp+1]=Robot i or robot j ∀k ∈

pred(h) and ∀(i, j) ∈ 1, ..., IEND ForAnt’s return to the starting pointLocal update of pheromones (equation 3)For k=1 to the number of ants

Evaluate the solutionsEND ForPerform the guided local search GLSRetain the best solutionGlobal update of pheromones (equation 4)If the stopping condition is satisfied

Stop the algorithm and existElse

Return to the step of building a new tour for each antEnd If

Ant colony optimization with the guided local search algorithm: The algorithmgathering ant colony with the guided local search is summarized by the algorithm 5.

Particle Swarm optimization with the guided local search algorithm The algorithmgathering particle swarm optimization with the guided local search is summarized bythe algorithm 6.

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Genetic algorithm with the guided local search algorithm: The algorithm gatheringgenetic algorithm with the guided local search is summarized by the algorithm 7.

Algorithm 6: Structure of the PSO algorithm for robotic assembly line balancing

Generate initial solution (iter=1)randomly for all particles (p ∈ P) For D P=1 toKM(total number of location points)

Check the precedence constraints of the location points [DP]If location points[DP]=Robot i then locations points[DP+1]=Robot i or

Robot j ∀k ∈ pred(h) and ∀(i, j) ∈ 1...IEnd ForAssign the best for all particles with initial solutionFind the best among all particles and assign it to the GbestGenerate initial velocities (V i

pt) randomly for all particlesAdd velocities to the corresponding particles (xi

pt + 1)

While Number of iterations is not reachedDo For p=1to P (total number of particles)Update velocities (V i

pt)Modify the current positions (xi

pt)If the solution is improved then

Update the current PbestEnd IfEnd ForPerform the guided local search GLSIf the solution is improved then

Update the current GbestEnd IfEnd WhileReturn the best solution Gbest

Algorithm 7: Structure of the GA algorithm for robotic assembly line balancing

Generate randomly the initial population of N chromosomesFor D P=1 to KM (total number of location points)

Check the precedence constraint of location points[DP]If location points[DP]=Robot i then location points[DP]=Robot i or robot j ∀k ∈

pred(h) and ∀(i, j) ∈ 1, ..., IEND ForFor each iteration Do

Evaluate the fitness of each individual (the number of seized products )If stopping criteria satisfied (i ti = Ng?)

If i ti = Ng then StopElse

Select two individuals as parentsPerform the crossover for two individuals based upon the crossover probability PcPerform the mutation for each gene of the one individual (offspring) based upon the

mutation probability PmPerform the guided local search GLS

Insert the parent offspring and offspring mutation in next populationSorting the next population such as selected the individuals which have the best fitnessUpdate the population

END ForEND If

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250 S. Daoud et al.

4.2 Exact method

To verify the quality of the developed method with optimal solutions, we have devel-oped a full enumeration method which enumerates all the possible solutions and allowthen to select the optimal one of each tested instance. The exact method is called FEMin the rest of the paper.

5 Computational experiments

This section presents the experimental results for our problem by applying the differentdeveloped and tested methods (ACO, PSO, GA, ACO-GLS, PSO-GLS, GA-GLS,FEM) to different robotic assembly lines configurations. The industrial case consideredin this work is an automated packaging line dedicated for dairy food products. Basedon previous works for the applied algorithms that gave relatively good results, theparameters settings values were deduced by applying several tests. For example, foreach algorithm, the number of individuals, particles and ants were tested between 10and 100, and a compromise between the quality of the results and the convergence timeswas considered. The adopted parameters for ant colony optimization are based on aprevious work of Chehade et al. (2008), Chehade et al. (2009b). These parameters are:the number of ants N = 20, evaporation rate ρ = 0.6, q0 = 0.75, the parameters whichused to determine the relative importance of pheromones versus the visibility α = 0.7and β = 0.3 and finally the constant X = 7. For the PSO, the adopted parameters arebased on a previous work of Daoud et al. (2010). These parameters are: the number ofparticles = 20, the inertia factor of local best tour C p = 0.75, the inertia factor of globalbest tour Cg = 0.6 and the inertia w = 0.8. For the GA, the GA parameters values werefixed by applying pilot runs. The adapted parameters values are as follows: populationsize is 100, the crossover probability is 0.85 and the mutation probability is 0.1. In whatconcerns the guided local search GLS, its stopping criterion is the number of iterationwhich is fixed to 50. We note that the stopping criterion for all developed algorithmsis the number of generations. It is tested between 50 and 500 and fixed at 100.

We realize computational experiments by studying robotic assembly systems struc-tures which are different by i of robots, k the number of location points and n thenumber of layers of final assembled product. The first structure (S1) represents asystem with 2 robots, 4 or 8 location points and 2 or 4 layers whereas (S2), (S3),(S4) and (S5) refer respectively to systems with 4, 6, 8 and 10 robots with always thesame number of location points and layers. For each structure, we have tested fourinstances by changing at each time the number of location points and the number oflayers. Moreover, we have also tested the developed methods by changing the numberof picked product for the structures and instances mentioned above.

The parameters of the pick and place robots for all the tested structures are asthe follows: average cycle time of robots is 0.8 mm/s, the coverage area length is1,050 mm and the distance between two robots is 200 mm. The precedence diagram ofour problem which presents the precedence relations among the final product is shownabove in Fig. 2. The processing times of assembled task are generated according toa uniform distribution and depending on the layer. In our study, the processing time

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Table 4 Number of products for the robotic assembly line balancing with 50 products

Robot Location points Number of layers Number of products

FEM ACO-GLS PSO-GLS GA-GLS

(S1) 2 4 2 34 34 34 34

4 17 17 17 17

8 2 34 34 34 34

4 34 34 34 34

(S2) 4 4 2 50 50 50 50

4 32 32 32 32

8 2 50 50 50 50

4 50 50 50 50

(S3) 6 4 2 48 48 48 48

4 48 48 44 44

8 2 46 46 45 45

4 48 47 48 47

(S4) 8 4 2 50 50 50 50

4 50 50 47 48

8 2 50 49 48 48

4 50 48 48 47

for the first layer is between [1, 1.2] because the robot takes more time to deposit theproduct in the final assembled product and [0.8, 1] for the rest. The proportion betweenthe average workloads of different robots δ is imposed by our industrial partner andequal to 10 %. The required time to assemble J components is Cmax . For example, torequire 50 components, Cmax is equal to 21.6(s) for (S1) and (S2) and 11.2 (s) for(S3) and (S4).

The industrial application of our methods required an execution time < 1 s forall the structures. These methods will be installed on real case system to manage thefunctioning of a real industrial robotic system through a decision aid system developedwith C++.

The results are shown from Tables 4, 5, 6, 7, 8, 9, 10 and 11. These latters presentthe comparison results and the execution times for the exact method FEM and thedeveloped method with the different instances. The optimal number of non seizedproducts is presented in bold. The execution time for each method which respects theindustrial constraint is highlighted in all tables which related to the execution times ofthe considered methods.

We may notice that for all tables the ACO-GLS, PSO-GLS and GA-GLS are ableto satisfy the industrial constraint. This means that each method has an execution timeless than 1 second for all structures. The objective of our study is to install the bestalgorithm between all the tested algorithms for different robotic assembly lines onreal case system. Therefore, our selection criterion is based on two parameters. Thefirst is to select the best algorithm which is able to provide an efficient solution withinrespect the industrial constraint. The second is to select the algorithm which proposesan efficient solution by maximizing the number of products that are not seized by

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Table 5 Execution time for the robotic assembly line balancing with 50 products

Robot Location points Number of layers Execution time (s)


(S1) 2 4 2 0.127 0.165 0.221 0.103

4 0.112

8 2 0.124

4 0.124

(S2) 4 4 2 0.254 0.334 0.361 0.256

4 0.267

8 2 0.278

4 0.289

(S3) 6 4 2 0.324 0.36 0.476 50.9

4 52.3

8 2 52.68

4 53.12

(S4) 8 4 2 0.4654 0.491 0.589 977.73

4 979.34

8 2 981.51

4 981.84

Mean 0.292 0.337 0.411 258.79

the robots. For this, we have used GAP of all developed method with FEM. G AP1

corresponds to the GAP in the number of seized products between the FEM and thedeveloped methods. G AP2 presents the GAP in execution times. These GAPS aredefined as follow:

G AP1AC O−GL S = R(F E M) − R(AC O − GL S)

R(F E M)∗ 100 % (12)

G AP1P SO−GL S = R(F E M) − R(P SO − GL S)

R(F E M)∗ 100 % (13)

G AP1G A−GL S = R(F E M) − R(G A − GL S)

R(F E M)∗ 100 % (14)

G AP2AC O−GL S = T (F E M) − T (AC O − GL S)

T (F E M)∗ 100 % (15)

G AP2P SO−GL S = T (F E M) − T (P SO − GL S)

T (F E M)∗ 100 % (16)

G AP2G A−GL S = T (F E M) − T (G A − GL S)

T (F E M)∗ 100 % (17)

Tables 4 and 5 show the comparison results and the execution times for the exactmethods and the different hybrid metaheuristics with the different tested instances forthe robotic assembly line balancing with 50 products. Let us take for example the first

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(S1) 2 4 2 68 68 68 68

4 68 68 68 68

8 2 68 68 68 68

4 68 68 68 68

(S2) 4 4 2 100 100 100 100

4 100 100 100 100

8 2 100 100 100 100

4 100 100 100 100

(S3) 6 4 2 98 98 98 98

4 98 98 98 98

8 2 100 100 99 98

4 98 97 97 97

(S4) 8 4 2 100 100 100 100

4 100 100 100 100

8 2 100 98 96 97

4 100 98 98 98

line of Table 4. It corresponds to structure (S1) with 2 robots, 4 or 8 location points and2 or 4 layers. We may notice that for this structure the numbers of seized componentsthat we have get with FEM is equal to 34. It is the same value for the optimal number ofseized components obtained by ACO-GLS, PSO-GLS and GA-GLS. In what concernsthe computational time, the average of the execution times gets with FEM is 258.79(s)against 0.292(s) for ACO-GLS, 0.337(s) for PSO-GLS, 0.411 (s) for GA-GLS. Fur-thermore, if we compare the execution times of the different developed methods withthose of exact method for all structures, we may notice that the developed methodpresents a big advantage. This advantage is more obvious when the size of the struc-tures is more important. We may notice that the ACO-GLS is faster from PSO-GLSand GA-GLS. It can reduce the execution time by 98 % from the exact method.

We may notice that for the execution time of FEM is not acceptable in the industrialcase since it is greater than 1 second. If we take in consideration all the instances ofTable 4, the ACO-GLS algorithm has succeeded to reach the optimal solutions thathave been obtained with the exact method for 13 instances over 16 tested ones. ForPSO-GLS, the algorithm has succeeded to reach 12 instances. For the GA-GLS hassucceeded to reach 11 instances over 16 tested ones. For the rest of the instances oflarger size (S3) and (S4), the difference is very small.

Tables 6 and 7 show the result of the comparison between ACO-GLS, PSO-GLS,GA-GLS algorithm and the FEM method to the robotic assembly lines balancing with100 products. We observe that for all instances, ACO-GLS has succeeded to reachthe optimal solutions for 13 instances. , PSO-GLS and GA-GLS have succeeded toachieve 12 instances. For the rest of the instances of a larger size, the difference isvery small.

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(S1) 2 4 2 0.169 0.22 0.295 0.12

4 0.17

8 2 0.21

4 0.27

(S2) 4 4 2 0.339 0.445 0.481 4.58

4 4.46

8 2 5.04

4 5.82

(S3) 6 4 2 0.432 0.58 0.635 59.91

4 59.96

8 2 61.47

4 64.09

(S4) 8 4 2 0.62 0.705 0.785 1053.57

4 1054.6

8 2 1055.78

4 1056.8

Mean 0.39 0.487 0.549 280.42

In what concerns the computational time presented in Table 7, the average of theexecution times gets with FEM is 280.42 (s) against 0.39 (s) for ACO-GLS, 0.487 (s)for PSO-GLS, 0.549 (s) for GA-GLS. Furthermore, if we compare the execution timesof the tested algorithms and those of the FEM method, we note that all algorithmsare very efficient compared to the exact method because they respect the industrialconstraint. We may notice that also for the ACO-GLS is faster than PSO-GLS andGA-GLS. This advantage is more obvious when the size of the structures is moreimportant.

In what concerns the computational time presented in Table 9, the average of theexecution times gets with FEM is 466.41(s) against 0.58(s) for ACO-GLS, 0.417(s)for PSO-GLS, 0.549(s) for GA-GLS. Furthermore, if we compare the execution timesof the tested algorithms and those of the FEM method, we note that all developedmethods respect the industrial constraint compared to the exact method.

Tables 8 and 9 present the comparison between the ACO-GLS, PSO-GLS, GA-GLS and FEM in order to better classify these methods and to show the effect ofthe guided local search for robotic assembly line balancing with 200 products. Wemay notice that for the 14 first instances, the ACO-GLS has succeeded to reach theoptimal solutions that have been obtained with the exact method. For the PSO-GLSand GA-GLS, they can reach 13 optimal solutions. We may notice that for the rest ofthe instances of a larger size (structures S3 and S4), the difference is very small asshowed by the Table 10. The biggest deviation between the solution obtained differentmethods and the optimal solution obtained by FEM corresponds to PSO-GLS and isequal to 3 %.

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(S1) 2 4 2 164 164 164 164

4 176 176 176 176

8 2 176 176 176 176

4 176 176 176 176

(S2) 4 4 2 200 200 200 200

4 200 200 200 200

8 2 200 200 200 200

4 200 200 200 200

(S3) 6 4 2 197 197 197 197

4 195 195 195 195

8 2 200 200 198 196

4 194 194 194 194

(S4) 8 4 2 200 200 200 200

4 200 200 200 200

8 2 200 198 194 196

4 200 195 195 195




(S1) 2 4 2 0.174 0.228 0.3 0.156

4 0.168

8 2 0.203

4 0.23

(S2) 4 4 2 0.362 0.48 0.509 3.174

4 3.338

8 2 3.503

4 3.699

(S3) 6 4 2 0.457 0.513 0.606 64.13

4 67.99

8 2 67.43

4 69.58

(S4) 8 4 2 0.675 0.72 0.847 1789.52

4 1791.53

8 2 1795.61

4 1802.34

Mean 0.417 0.485 0.565 466.41

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256 S. Daoud et al.




(S1) 2 4 2 182 182 182 182

4 182 182 182 182

8 2 191 191 191 191

4 193 193 193 193

(S2) 4 4 2 250 250 250 250

4 250 250 250 250

8 2 250 250 250 250

4 248 248 248 248

(S3) 6 4 2 250 250 250 250

4 250 250 250 250

8 2 249 249 249 249

4 247 247 246 246

(S4) 8 4 2 250 250 250 250

4 250 249 247 248

8 2 245 245 245 245

4 244 242 240 241

In what concerns the Table 10, it shows that for 14 first instances, the ACO-GLShas succeeded to reach the optimal solutions that have been obtained with FEM. Forthe PSO-GLS and GA-GLS, they can reach 13 optimal solutions over 16 tested ones.Furthermore, Table 11 presents the execution time. It shows that the ACO-GLS methodis faster than PSO-GLS and GA-GLS. But, all methods have satisfied the industrialconstraint about the execution time. This advantage is more obvious when the size ofthe structures is more important. In general, we may reduce the execution times by98 % if we apply the ACO-GLS, PSO-GLS and GA-GLS algorithms instead of theFEM method. We notice that the ACO-GLS is better than PSO-GLS and GA-GLS interms of rapidity and quality of obtained solutions.

Table 12 presents different GAP for the computational results and the executiontime for al tested instances of each robotic assembling lines balancing. Then, G AP1

corresponds to the GAP in the number of seized products between the FEM andthe developed methods. G AP2 presents the GAP in execution times. The average ofdifferent G AP1 for ACO-GLS, PSO-GLS and GA-GLS are respectively 0.306, 0.729and 0.752. We may notice that the greatest value of the different GAP corresponds ofthe PSO-GLS. In what concern the G AP2, we note that all the developed methodscan reduce the execution time until 99 % from the execution time of FEM.

If we analyze on globally the evolution of the average values of the GAP for eachtable, we note that for the 10 first instances all methods are able to get the optimalsolutions. For the rest, the different methods have a small deviation. We note also theadvantage of the ACO-GLS which has better performances than the PSO-GLS andGA-GLS. Note that taking in consideration the execution times, all developed methods

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(S1) 2 4 2 0.184 0.24 0.313 0.126

4 0.135

8 2 0.158

4 0.178

(S2) 4 4 2 0.42 0.529 0.56 3.379

4 3.631

8 2 3.8

4 3.9

(S3) 6 4 2 0.62 0.739 0.838 72.27

4 73.74

8 2 77.96

4 79.14

(S4) 8 4 2 0.794 0.877 0.907 2329.79

4 2333.32

8 2 2334.1

4 2335.36

Mean 0.504 0.596 0.654 603.18

Table 12 Differents GAP from full enumeration method

G AP1M1 G AP1

M2 G AP1M3 G AP2

M1 G AP2M2 G AP2

M3

50 products 0.578 1.736 1.881 99.87 99.87 99.841

100 products 0.341 0.545 0.545 99.86 99.86 99.86

200 products 0.227 0.422 0.422 99.91 99.89 99.91

250 products 0.08 0.214 0.16 99.91 99.9 99.91

Mean 0.306 0.729 0.752 99.89 99.886 99.882∗M1 : Ant colony optimization wi th the guided local search algori thm (AC O − GL S)∗M2 : Particle swarm optimization wi th the guided local search algori thm (P SO − GL S)∗M3 : Genetic algori thm wi th the guided local search algori thm (G A − GL S)

able to respect the industrial constraint. This fact represents a great interest becausethe applied method is used to manage the functioning of real industrial robotic system.

6 Conclusion

A robotic assembling line balancing problem has been studied in this paper. The objec-tive of our problem is to maximize the efficiency of the line which is calculated by thenumber of products seized by the robots. The computational results show the advan-tages of the different proposed methods. The ant colony optimization coupled withguided local search has been the most efficient method by getting optimal solutions

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for a large number of tested instances and with only a small deviation for the rest. Theother important advantage is its small execution time which satisfies the industrialconstraint. Accordingly, we can test and apply our algorithm on the real industrialcase. Several ways might be used to extend this work. One of the proposed directionsis to apply the fuzzy logic controllers since the parameters settings of all developedmethods are difficult. This latter, allows to enhance the parameters settings in order toimprove the search ability. As results, we can compare the effect of the guided localresearch and the fuzzy local search on the evolution of the fitness function. Anotherway of extension is a mathematical formulation for robotic assembly lines balancing.

Acknowledgments This research was supported by ARIES Packaging (France). The authors are alsovery grateful to the national association of technical research (ANRT) in France.

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