heuristics for an industrial car sequencing problem considering paint and assembly shop objectives

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Computers & Industrial Engineering 55 (2008) 295–310

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Heuristics for an industrial car sequencing problemconsidering paint and assembly shop objectives

Alexandre Joly *, Yannick Frein

Institut National Polytechnique de Grenoble, Industrial Engineering Department, G-SCOP Laboratory, France

Received 31 July 2007; received in revised form 5 November 2007; accepted 18 December 2007Available online 25 December 2007

Abstract

The aim of this paper is to study the problem of sequencing a set of vehicles within an industrial environment consid-ering the assembly shop objectives, but also the objectives of the paint shop. The first approach is to solve the problem witha mono-objective function. One heuristic (a progressive, construction-sequence algorithm) and three meta-heuristics (sim-ulated annealing, variable neighbourhood search and an evolutionary algorithm) are described and compared. As themono-objective approach has limited possibilities, a multi-objective heuristic is finally presented and tested. Because ofthe industrial context of this research, the computation time is a decisive factor to select the appropriate heuristic.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Car sequencing problem; Paint and assembly shops; Meta-heuristics; Mono- and multi-objective optimisation

1. Introduction

In order to be competitive, car makers have to provide many kinds of vehicles with an increased number ofoptions. However, due to production line costs, different vehicles are assembled using the same lines, which areknown as ‘‘mixed-model assembly lines”.

In an automotive assembly plant, production lines go through three shops: the body shop, the paint shopand the assembly shop. Workloads depend on vehicle characteristics. At the assembly shop, some operationsneed more time or specific resources and cannot be achieved in the cycle time required by the production rate.These longer operations cannot be successively carried out without a risk of slowing down production and oneof the objectives of the assembly shop is to level resource utilization over time in order to guarantee a smoothassembly. This also ensures that the components consumption will be as constant as possible. To allow the

0360-8352/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.cie.2007.12.014

* Corresponding author. Tel.: +33 3 70 04 72 64.E-mail addresses: alexandre_ [email protected] (A. Joly), [email protected] (Y. Frein).

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296 A. Joly, Y. Frein / Computers & Industrial Engineering 55 (2008) 295–310

suppliers to produce and deliver their components just in time, the sequence entering the assembly shop is sentto the suppliers several days before. To ensure good synchronisation, assembly shop objectives have to bereached when building the sequence.

Other objectives that must be reached when building the sequence concern the paint shop. Indeed colourchangeovers are expensive because painting tools have to be cleaned. To minimise these costs, vehicles withthe same colour have to be painted successively.

The aim of this paper is to define the vehicle scheduling that minimises manufacturing costs, considering theassembly and the paint shop objectives.

It is innovative to deal, as we do, with a real-life industrial problem. We do not only consider the assemblyshop objectives as numerous works, but also the paint shop objectives. The strategy used to consider the paintshop objectives, called densification, is also original and has never been used in the literature to the best of ourknowledge. To solve our industrial problem, several heuristics and recent meta-heuristics are proposed andcompared. As the mono-objective approach has limited possibilities, a multi-objective heuristic is finally pre-sented and tested.

The second section presents a literature review. In the third section we present the problem formulation. InSection 4 we first solve the problem with a mono-objective approach with four heuristics: a progressive con-struction-sequence algorithm, a variable neighbourhood heuristic, a simulated annealing, and finally an evo-lutionary algorithm. Then in Section 5 we present a multi-objective approach to solve the same problem.Finally, the last section makes comparisons and gives the conclusions. Many algorithms are analysed in thispaper and for sake of brevity or because of industrial secrecy, some details have been omitted, but nothing,according to us, that could lead to misunderstanding.

2. Literature review

Numerous studies have been devoted to mixed-model scheduling. Most of them focus on the assembly shopwhere stakes are very high for the automotive industry. For example, the sequencing method developed byToyota Motor Corporation, the Goal Chasing, as described by Monden (1983), tries to keep part consump-tion, as far as possible, constant in minimising the gap between real consumption and ideal consumption.Miltenburg (1989) pursues the same goal but with different consumption levels. The mixed-model sequencingapproach used at Huyndai Motor Company and presented by Duplaga, Hahn, and Hur (1996) is also a mod-ified version of Toyota’s Goal Chasing. Other works deal with the problem of sequencing in buffers (Muhl,Charpentier, & Chaxel, 2003). This problem is known as the dynamic JIT sequencing problem (Garcia-Sabat-er, 2001).

There are less works concerning paint shop objectives. For example, let us mention (Guerre-Chaley, Frein,& Bouffard Vercelli, 1995; Lahmar, Ergan, & Benjaafar, 2003), who focus on dynamic scheduling in order toproduce vehicles having the same colour successively.

And there are some recent works that do focus on multiple objectives. McMullen has greatly contributed tothe field of mixed-model scheduling (McMullen, 2001; McMullen & Frazier, 1998; McMullen, Tarasewich, &Frazier, 2000). In McMullen (2001), the author compares different heuristics: simulated annealing, Tabusearch and genetic algorithms in order to build a sequence. However, tests are carried out with academic datasets of up to 100 products. Hyun, Kim, and Kim (1998) also proposes a genetic algorithm to solve mixed-model scheduling problems considering the number of setups, the usage rate and the total utility work. Thelargest size problem considered has only 20 different products, representing 1012 to 1035 different solutions.In the tests presented here the sequence includes 640 vehicles which can have up to 11 options and one outof fourteen colours. Six hundred and forty is the average number of vehicles produced daily by an assemblyline in an automotive plant that must be sequenced each day. An automotive plant can have up to three dif-ferent assembly lines.

In this paper, we consider the objectives of the paint and assembly shops and propose a multi-objectiveapproach to solve scheduling problems with real industrial data. The aim is to provide an efficient solutionmethod according to the objectives of both shops in a short, yet reasonable time. Moreover, we use an originalapproach to minimise the changeovers cost at the paint shop.

The next section presents the problem formulation.

A. Joly, Y. Frein / Computers & Industrial Engineering 55 (2008) 295–310 297

3. Problem formulation

The sequence has to comply with the objectives of the assembly shop and of the paint shop. The first part of thissection describes the objective function used to evaluate the quality of the sequence for the assembly shop, whilethe second part presents a new objective function to evaluate the quality of the sequence for the paint shop.

3.1. The assembly shop objectives

To consider the assembly shop objectives, we use one function described in Miltenburg (1989) and used innumerous works (Hyun et al., 1998; Inman & Bulfin, 1992; McMullen, 2001). This objective function repre-sents the sum of the squared differences between the real consumption of an option and its ideal consumptionrate for each option and for each vehicle (see Fig. 1). Minimising the consumption ensures smooth options anda stable flow. Miltenburg (1989) presents three others objective functions which seek to minimise the variationand give similar schedules.N Total number of vehicles to sequenceI Total number of options availableni Number of options of type i to produce i = 1, 2, . . . , I

Wi Weight of the option i. i = 1, 2, . . . , I

p Position of the vehicle. p = 1, 2, . . . ,N

xi,p Number of vehicles with the option i between the positions 1 and p. xi,p = 0, 1, . . . ,N � 1

The minimisation of the usage rate is IS ¼PN

p¼1

PIi¼1

W i xi;p�pniNð ÞPI

i¼1W i

2

IS denotes the smoothing function for the assembly shop. IS is an objective function measuring the variabil-ity in the resource usage.

Let us write IS ¼PN

p¼1ISV with ISV the smoothing function for a particular vehicle.

3.2. The paint shop objectives

Each automotive plant has a buffer at the entrance of the paint shop that allows resequencing of vehicles inorder to paint successive vehicles with the same colour. This resequencing is more efficient if vehicles with thesame colour are close to one another when entering the buffer. However this dynamic resequencing requiresanother buffer at the entrance of the assembly shop in order to rebuild the initial sequence. Further detailsconcerning resequencing are given in Bernier and Frein (2004) and Ding and Sun (2004).

There are a number of strategies for preparing the sequence for the paint shop.

NNumber of vehicles

To sequence

Number of vehicles with option i to produce

nii

xip

p.n /N

0ni /N

p

Gap tominimize

x

i

Fig. 1. Gap to minimise for a best usage rate.


The first one would be to minimise the changeover costs when building the planned sequence by using, forexample, an objective function such as the one used by McMullen et al. (2000). This objective is severe: itrequires vehicles with same colour to be sequenced successively.

The second one is to move these vehicles as close as possible to one another when building the plannedsequence, even if vehicles with the same colour are not necessarily adjacent. This sequence refers to the orderin which the cars enter the buffer at the entrance of the paint shop.

We prefer the second strategy for two reasons:First, the assembly shop objectives are harder to reach if vehicles of the same colour are adjacent than if

these vehicles are only close to one another in the sequence.Second, the body shop process disrupts the planned sequence (Bernier & Frein, 2004) and the second strat-

egy is more robust than the first faced with these disruptions. Indeed the paint buffer allows to reorder vehiclesin order to minimise changeover costs.

We call densification the act to move the vehicles closer in the planned sequence to ensure better perfor-mances according to the paint shop objectives. The function to evaluate the densification is based on thesum of the inverse of the gaps between a vehicle with a particular colour and all other vehicles with the samecolour.N Total number of vehicles to sequenceC Total number of colours availablemi Number of vehicles of colour i to produce, i = 1, 2, . . . ,C

Bi Weight of colour i, i = 1, 2, . . . ,Cpi,j Position of the jth vehicle of colour i p = 1, 2, . . . ,N; j = 1, 2, . . . ,mi; i = 1, 2, . . . ,C

Given the jth vehicle of colour i (j = 1, 2, . . . ,mi), the proximity of other vehicles of the same colour isdefined by:

ID;i;j ¼Xmi

j0 ¼ 1

j0 6¼ j

1

jpi;j0 � pi;jj

To convert this indicator to a zero-one scale, we divide it by the best value. The best value of this indicatoris obtained when all the vehicles are contiguous.

We can obtain the same result wherever the set of vehicles of colour j is positioned in the list, provided theyare contiguous.

We then obtain ImaxD;i;j ¼

Pmi

j0 ¼ 1j0 6¼ j

1jj0�jj

We define the densification function for the paint shop as the weighted sum of the ratiosID;i;j

ImaxD;i;j

.

ID ¼

PCi¼1Bimi

Pmij¼1

ID;i;j

ImaxD;i;jPC

i¼1Bimi

We call ID the densification function for the paint shop. Let us write IDV the densification function for aparticular vehicle.

The best value of the objective function ID is 1, when all vehicles with the same colour are sequencedsuccessively.

4. Mono-objective tested optimisation methods

As mentioned above, the size of the industrial problem to solve is considerable. With 640 vehicles, 11options and 14 colours, there are many variables that involve a huge diversity: 3.19 101084 different sequencespossible in the list presented here. To simplify the problem we first consider only one global objective (IG) as aweighted sum of IS and ID.


ID is a function to maximise (with a value less than one) and since IG is a function to minimise, we perform1 � ID.

TableObject

IS

ID

IG

IG ¼a � IS þ b � ð1� IDÞ

aþ b

a, b are weights assigned with smoothing and densification functions.IS and ID do not have the same variation range. IS can be far greater than ID so, in order to balance both

functions, we choose a = 1 and b = 104. Determination of these weights greatly influences the exploration andbecause it is not easy to parameter these weights, the next part of this paper will deals with a multi-objectiveapproach.

In order to have a reference for our objective functions, we randomly sequenced vehicles one million times.Table 1 below gives the results.

Within the context of this research and with regard to the size of the target industrial application, heuristicsseem to be the best methods to solve our problem. However it is very difficult to determine the best heuristicwithout testing. Therefore, we implemented several different heuristics in order to compare them. These heu-ristics were evaluated considering the mono-objective function IG and the computation time required to findthe best solution.

The tested heuristics are:

4.1 A progressive construction-sequence algorithm (PCSA) selected due to its capacity to rapidly give asolution,

4.2 A neighbourhood search heuristic (NSH) selected due to its convergence ability,4.3 A simulated annealing (SA) selected due to its ability to move away from a local optimum, and4.4 An evolutionary algorithm (EA) selected due to its capacity of investigation.

A literature review and detailed discussion of greedy algorithms, local search algorithm, simulated anneal-ing or evolutionary algorithm are beyond the scope of this paper. However, the interested reader is referred toMichalewitcz and Fogel (2002) for an introduction to modern heuristics.

Tests presented in this paper were all performed on a Pentium IV processor with 2.6 GHz and 1 GB RAM.

4.1. Progressive construction-sequence algorithm (PCSA)

The algorithm used to progressively sequence the vehicles is inspired by the algorithm developed by Delavaland Castelain (1996) and Baratou (1998). This algorithm is a truncated branch and bound. For each vehicle tosequence, it calculates the function cost of all the vehicles remaining to sequence and chooses the one with thelowest function cost. See Castelain et al. (1995) for a detailed description concerning this heuristic.

Two strategies were tested: A strategy for the assembly shop objectives (SAS) and a strategy for the paintshop objectives (SPS).

SAS: A strategy that favours the respect of the assembly shop objectives.The vehicle with the lowest ISV is chosen each time. If several vehicles have the same ISV, the algorithmsequences the one with the same colour as the last sequenced. This strategy provides a sequence with anexcellent smoothing objective function for the whole sequence (IS). We call this strategy SAS (strategyfor the assembly shop).

1ive function values with 106 lists randomly sequenced

Worst value Average value Best value

97 048 12 438 19580.1539 0.1708 0.191610.53 2.073 1.026


SPS: A strategy that favours the respect of the paint shop objectives.This strategy aims at successively sequencing the vehicles with the same colour. For each sub-sequence ofvehicles having the same colour, vehicles are sequenced minimising ISV. This algorithm provides the bestvalue for the densification objective function for the whole sequence (ID = 1). We call this strategy SPS(strategy for paint shop).

These algorithms give solutions in less than 1 s. Results are presented in Table 2. The reference is the aver-age value obtained after the same set of vehicles were randomly sequenced 106 times.

4.2. The variable neighbourhood search heuristic (VNS)

This method starts with an initial solution and consists of generating a nearby solution by applying an ele-mentary transformation. The new solution is then evaluated with the objective function and replaces the initialsolution if the function cost is lower. This procedure is repeated until no other neighbour could improve thesolution. Heuristic performance depends on the initial solution and on the neighbourhood.

4.2.1. The initial solution

Three initial solutions were tested. A solution obtained with SAS, a solution obtained with SPS and finallya random solution.

4.2.2. The neighbourhood

The neighbourhood requires an elementary transformation and two vehicles to perform the transforma-tion. First we describe the different possibilities for defining the neighbourhood and then we present the testsperformed to select the most efficient one.

4.2.2.1. Transformation operators. Four elementary transformations were tested. See Table 3 for a schematicdescription of these operators.

– Swap, which consists of swapping the position of two different vehicles.– Movement, which consists of moving a vehicle from an initial position to another one.– Reverse, which consists of reversing a part of the sequence.– One of the three elementary transformations above randomly chosen.

4.2.2.2. Strategies for applying the transformation. Three different methods were tested to determine the initialvehicle to swap, move, or reverse at the beginning of the section.

Table 2Objectives functions values obtained with the progressive sequence algorithm

IS ID IG

Average (106 lists) 12 438 0.1708 2.073SAS 121 0.309 0.702SPS 5471 1 0.547

Table 3Transformation operators

Before After

Swap between 3 and 7 1 2 3 4 5 6 7 8 9 1 2 7 4 5 6 3 8 9Movement of 3 towards 7 1 2 3 4 5 6 7 8 9 1 2 4 5 6 3 7 8 9Reverse between 3 and 7 1 2 3 4 5 6 7 8 9 1 2 7 6 5 4 3 8 9


– The vehicle with the worst ISV value (A1).– The vehicle with the worst IDV value (A2).– Random choice between the vehicle with the worst value of ISV and the one with the worst value of IDV

(A3).

Three different methods were also tested to determine the other vehicle to swap, the vehicle to move closeror the last vehicle of the section to reverse.

– A vehicle with the same colour as the initial vehicle (B1).– A vehicle differing in up to three options from the initial vehicle (B2).– A randomly selected vehicle (B3).

4.2.2.3. Comparison of the four operators. The four different operators were first applied to the followingvehicles:

– A vehicle randomly chosen between the vehicle with the worst value of ISV or the one with the worst valueof IDV (A3).

– Another randomly selected vehicle (B3).

Table 4 gives the IG values obtained with these four transformation operators.The random operator gives the best IG with three different initial solutions. This operator has then been

selected to be the transformation operator for our selected NSH. The solution space is so vast that the heu-ristic needs to explore many different solution areas and the random operator is the one which allows the mostgreatly differing regions to be visited. Other tests were performed with A1, A2, B1 and B2. In each test, therandom operator provides the best solution.

4.2.2.4. Comparison of the nine strategies. We tested the random transformation operator and different lists ofvehicles in 60 s. Table 5 shows the results. The rows represent the different strategies used to select the initialvehicle and the columns represent the three strategies used to select the other vehicle for the transformation.

The strategy to apply depends on the initial list. We finally selected the strategy that transforms the initialvehicle which has the worst value of ISV or the worst IDV with another vehicle randomly chosen. This strategyprovided effective solutions with all initial lists.

Finally the neighbourhood search heuristic used is a member of the variable neighbourhood search (VNS)family. A VNS is a technique originally proposed by Mladenovic and Hansen (1997). VNS is one of the fewmeta-heuristics that have been proposed as a means to combine efficient local optimisation procedures (e.g.gradient descent) with heuristics having the ability to cope with local optima. VNS works by exploring

Table 4Comparison of the four operators with IG

Swap Move Reverse Random

Random list 0.702 0.737 0.808 0.695

SAS 0.644 0.667 0.639 0.602

SPS 0.323 0.296 0.308 0.275

Table 5Performance of the different strategies

Same colour (B1) Same options (B2) Random vehicle (B3)

Worst ISV (A1) Good with SAS Not good AverageWorst IDV (A2) Not good Good with SPS AverageWorst ISV or worst IDV (A3) Good Good Good


increasingly distant neighbourhoods (Lalonde, 2002). In our case, we increase the neighbourhood size whenchoosing the initial vehicle for the transformation. If no transformation with the vehicle having the worstISV gives a better solution, the vehicle having the second worst ISV or IDV is chosen and so on until a bettersolution is found.

4.3. A simulated annealing algorithm (SA)

This algorithm is a classical optimisation meta-heuristic used in the literature reviewed above. It is also aneighbourhood search heuristic which allows an escape from local optima by accepting solutions that are notnecessarily better than the last ones found. The probability of preserving a worse solution depends on the cur-rent temperature. See Michalewitcz and Fogel (2002) for further details about SA.

4.3.1. Neighbourhood definition

The vehicle with the worst ISD or the worst IDV is selected to perform the transformation with another ran-domly selected vehicle. The operator is also randomly selected between a swap, a movement or a reverse.

4.4. The evolutionary algorithm (EA)

Evolutionary algorithms are inspired from biological evolution. The EA searches the solution space bymaintaining a set of solutions called the population. Each solution represents an individual. The aim is to com-bine solutions in order to obtain new ones. A new individual is generated by combining two other individualslike a crossover between two parents. Mutation can also sometimes occur in each new generation.

The main steps are detailed below and an explanation concerning the choice of parameters is given. Asmight be expected, the EA performance depends on chosen parameters.

4.4.1. Generation of initial solutions

The initial solutions are generated with SAS, SPS or are randomly sequenced.

4.4.2. Parent selection

Parents are selected according to a roulette wheel: the better the function objective, the greater the prob-ability to be selected.

4.4.3. The crossover operator

To determine the crossover operator for our application case, we compare four well-known crossoveroperators:

– Order based crossover.This crossover consists of building an offspring in choosing a sub-sequence from one parent and in preserv-ing the relative order of vehicles from the other parent (Davis, 1995).

– Partially mapped crossover operator.This crossover builds an offspring by choosing some vehicles from one parent and preserving the order andposition of as many vehicles as possible from the other parent (Goldberg & Lingle, 1985).

– Cycle crossover.This crossover builds an offspring in such a way that each vehicle and its position comes from one of theparents (Oliver et al., 1987).

– Uniform order crossover.This crossover builds an offspring by choosing a sub-sequence from one parent. Vehicles in this sub-sequence are permuted with vehicles of the same position in the other parent sequence (Davis, 1991).

A schematic description of these crossover operators is given in Table 6.To select one of these operators, tests have been carried out with the same initial lists, the same parameters

and the same computation times. Our conclusion is that no particular crossover operator seems to be more

Table 6Different crossover operators

Parent 1 – P1 Parent 2 – P2 Offspring

Order based – Sub-sequence 4-5-6 from P1 1 2 3 4 5 6 7 8 9 4 2 1 3 6 5 8 7 9 2 1 3 4 5 6 8 7 9Partially mapped – Vehicles selected: 2-4-8 in P1 1 2 3 4 5 6 7 8 9 4 2 1 3 6 5 8 7 9 1 2 3 4 6 5 7 8 9Cycle – Sub-sequence 1-2-3-4 from P1 1 2 3 4 5 6 7 8 9 4 2 1 3 6 5 8 7 9 1 2 3 4 6 5 8 7 9

Uniform sub-sequence 4-5-6 from P1 1 2 3 4 5 6 7 8 9 3 2 1 4 5 6 8 7 9


efficient in our application cases than any of the other crossover operators. Nonetheless, the cycle crossoveroperator is less efficient due to its production of very similar offsprings and thus does not allow for explorationof all solution space while the partially mapped operator requires slightly less computation time than theothers.

4.4.4. The mutation operator

The same operators as the transformation operators used in NSH were implemented and tested. The dif-ferent mutation operators provide similar results but the random operator seems to be slightly better. Fig. 2shows a comparison between these different operators. This test was performed in 60 s with the same EAparameters (initial population = 10; offspring = 5, mutation rate = 0.2). Several tests were performed with dif-ferent values of the parameters to define the good settings: the mutation rate, the initial population and thenumber of offsprings.

4.4.5. Evaluation of the population

Each individual is evaluated according to the objective function IG. When a new offspring is generated, wecheck that this offspring does not already exist in the population to avoid consanguinity and to avoid fallinginto local optima.

4.4.6. Selection of surviving population

Individuals staying in the next generation are chosen according to a roulette wheel selection. The better theindividuals, the greater the probability for them to survive in the next generation.

4.5. Comparison between the different mono-objective heuristics

The different heuristics were compared with three different initial lists and with the four optimisation meth-ods: PCSA, VNS, SA and EA. The objective function to minimise is IG.

One of the industrial specifications is to have the possibility of using heuristics for dynamic resequencing.When disruptions occur in the process, some vehicles may no longer be produced. In order to carry on pro-ducing vehicles, others are sequenced instead. For example, if the supplier of aluminium wheels is no longerable to supply his components, vehicles with standard wheels will be produced instead. It follows that a newset of vehicles will have to be sequenced. As there are several assembly lines (up to 3), and as the main vehicleflow includes several days of production (until 5), the computation per day and per line is limited to 120 s.Thirty minutes is therefore the longest time required. However, we performed tests during 300 s in order tocompare heuristic convergence.

4.5.1. Input parameters

Termination condition. Computation time is 300 sInitial list. Vehicles were initially randomly sequenced or with PCSA (SAS, SPS)Heuristic convergences are represented in Fig. 3 (the initial list sequenced randomly), Fig. 4 (the initial list

sequenced with SAS), and Fig. 5 (the initial list sequenced with SPS). Table 7 gives the results obtained withthe three different optimisation methods.

Comparison of the convergence with 4 different mutation operators

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 2500 5000 7500 10000 12500 15000 17500 20000

Number of evaluation

IG :

Obj

ectiv

e fu

nctio

n to

min

imiz

eREVERSE IG = 0.76

SWAP IG = 0.75

MOVE IG = 0.75

RANDOM IG = 0.73

Fig. 2. Comparison of four mutation operators.

0.5

0.7

0.9

1.1

1.3

1.5

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300Computation time in seconds

Obj

ectiv

e fu

nctio

n to

min

imiz

e (IG

)

SIMULATED ANNEALING IG=0.68

VARIOUS NEIGHBORHOOD SEARCH IG=0.63

EVOLUTIONARY ALGORITHM IG=0.52

Comparison of the convergence between 3 optimisation methods sequencing a random list of 640 vehicles

Fig. 3. Comparison between VNS, SA and EA with an initial list sequenced randomly.


Fig. 3 shows that the three optimisation methods give similar results before 80 s. Although the VNS isslightly better. After 80 s, the evolutionary algorithm provides far better solutions whereas simulated anneal-ing stagnates.

Fig. 4 shows that simulated annealing is the least efficient method as shown in Fig. 3. We assume VNS givesbetter solutions than SA because it allows a very oriented search and avoids falling into local optima. IndeedVNS is able to jump from local optima to another space region when changing neighbourhoods whereas SAloses time before escaping from a local solution. When computation time is lower than 30 s, VNS provides thebest solution whereas after 30 s EA takes the lead. Moreover, it seems that EA would have provided even bet-ter solutions if computation time were longer.

Fig. 5 confirms results shown in Fig. 4. EA is better after 15 s and SA is less efficient.Results given in Table 7 are those obtained after 300 s. The best value is always provided with EA. The

indicators of the initial solutions are also given. We can see the considerable influence of the initial solution.

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

Computation time in seconds

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8O

bjec

tive

func

tion

to m

inim

ize

(IG)




Comparison of the convergence between 3 optimisation methods sequencing a liste initially sequenced with SAS

Fig. 4. Comparison between VNS, SA and EA with an initial list sequenced with SAS.

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300Computation time in seconds

Obj

ectiv

e fu

nctio

n to

min

imiz

e (IG

)




Comparison of the convergence between 3 optimisation methods sequencing a liste initially sequenced with SPS

Fig. 5. Comparison between VNS, SA and EA with an initial list sequenced with SPS.

Table 7Results of optimisation with VNS, SA and EA

Random list PCSA: SAS PCSA: SPS

IS ID IG IS ID IG IS ID IG

Random 17322 0.17 2.56 SPS 5471 1 0.55 SAS 121 0.31 0.7

VNS 371 0.41 0.63 VNS 1043 0.87 0.23 VNS 362 0.53 0.51

SA 330 0.35 0.69 SA 994 0.82 0.28 SA 309 0.5 0.53

EA 301 0.5 0.53 EA 764 0.87 0.21 EA 310 0.64 0.39


Optimisation methods that start with solutions obtained by a progressive sequence construction algorithm(SAS and SPS) provide good solutions. Starting with a randomly sequenced list, it requires nearly 300 s forthe EA to provide the same performance as that provided by SPS.


Some other lists were tested using the same optimisation heuristics as those described above. In each case,EA provided the best performance in term of IG when computation time was greater than 90 s.

The last point to make concerns the objective function. The search orientation is given by the weightsassigned to the densification and to the smoothing objective functions. Since it is not easy to parameter theseweights, we are working on a multi-objective approach.

5. Multi-objective approach

5.1. Multi-objective evaluation

A multi-criteria approach requires the evaluation of sequences with respect to ID and IS separately. Thuseach sequence has two criteria. The best solutions are the non-dominated ones. A solution is dominated ifthere is at least one solution which is better in regards to both criteria.

Fig. 6 illustrates dominated and non-dominated solutions. The graphic to the left represents solutions con-sidering their ID and IS criteria. The hatched area includes dominated solutions. The four solutions, which arenot dominated, form the Pareto optima in the graphic to the right. T’Kindt and Billaut (2002) gives detailedexplanations concerning multi-criteria scheduling.

In order to find the best solutions, we implement a multi-objective evolutionary algorithm.

5.2. Multi-objective evolutionary algorithm (MOEA)

As a Pareto requires to keep several solutions, this meta-heuristic is well-suited for multiple objectivesresearch.

We used the ‘‘e-multi-objective evolutionary algorithm” recently proposed by Deb, Mohan, and Mishra(2003), for its ability to find well-converged and well-distributed solutions with a much smaller computationeffort than a number of state-of-the-art multi-objective genetic algorithms.

Mutation and crossover operators are those used by the evolutionary algorithm described in Section 4.4 butthe evaluation and selection are different.

Evaluation is carried out separately with ID and IS.Selection of individuals is based on the concept of e-dominance: the objective space is divided into hyper-

boxes. A solution e-dominates another if its hyperbox dominates the other.In Fig. 7, we can see that solution A e-dominates C whereas it is not true in terms of Pareto domination. If

two solutions share the same box like A and B in Fig. 7, we will choose the one closest to the bottom rightangle which will have the best value in terms of densification and smoothing. Applying the e-dominance con-cept ensures that solutions will be well-distributed throughout the objective space.

5.3. Results

Tests were performed using the same parameters as those used with the mono-objective evolutionary algo-rithm. The MOEA starts with solutions initially sequenced with PCSA (SAS and SPS). Fig. 8 presents theresults obtained after 10, 60 and 300 s.

Smoothing indicator. IS

Densification Indicator ID

Densification indicator ID

Smoothingindicator IS

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Fig. 6. Dominated and non-dominated solutions.

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eFig. 7. Illustration of the e-dominance concept (for minimising smoothing and maximising densification).

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10 secondes60 seconds300 secondsSASSPSBest value obtained with mono-objective EABest value obtained with SABest value obtained with VNS

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Dominated solutions obtained with Multi Objective Evolutionary Algorithm within different computation times

Fig. 8. Dominated solutions obtained with MOEA after 300 s.


As we can observe in Fig. 8, on the left hand side of the graphic, the best solution according to the smooth-ing function is the one provided by SAS. The multi-objective does not reach a solution with a lower IS.

However MOEA provides a solution as good as the SPS solution according to the densification function(on the right of the graphic, where ID = 1), but with a better smoothing indicator value (IS = 5212 withMOEA and 5471 with SPS).

Best solutions obtained with mono-objective functions (VNS, SA and EA with a same computation time of300 s) are also represented in Fig. 8. On one hand, MOEA is less efficient than the mono-objective methods.On the other hand, it provides several solutions and provides the user with several possibilities according to hisstrategy. For example, if the resequencing buffers are small, the resequencing will not be very efficient. Theuser should therefore choose a solution in the left-side of the graphic. Whereas, if assembly shop objectiveshave priority, users should select amongst solutions located in the right-side of the graphic.

As we can observe in Fig. 8, there is no Pareto solution between ID = 0.5 and ID = 0.76. It seems that ID

can be improved without decreasing the performance of the smoothing objective. User should then choose asolution with an ID value of 0.77 rather than a solution with an ID value of 0.5.

Although MOEA requires lots of computation time, it could help the user set the weights of the mono-objective function. From an optimal Pareto, it is possible to determine the objective function weights to ori-entate the search towards a desired solution. If we consider that the set of optimal Pareto solutions forms acurve, it is then possible to plot a tangent from a particular point (the desired solution). The tangent coefficientgives the ratio between the smoothing and the densification weights.

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Dominated solutions obtained with Multi Objective Evolutionary Algorithm within different computation times

Fig. 9. Dominated solutions obtained with MOEA after 12 h.


Fig. 9 shows the Pareto solutions obtained with MOEA after 1, 2 and 12 h. The three solutions obtainedwith the single objective methods still refer to a computation time of 300 s. We can see that Pareto solutionsare better and that the last solutions are probably not the optimal solutions. However 12 h can be enough tohelp the user determine the weights.

6. Conclusions

Computation time is a main factor, decisive in obtaining the best solution. Within the context of thisresearch and because of the short available computation time, the use of advanced meta-heuristics, like var-iable neighbourhood search or evolutionary algorithm, is the most efficient solution to our problem.

A summary of computation time is given in Table 8. This table provides the appropriate heuristic accordingto the available computation time.

As mentioned before, the computation time is limited to 120 s. Furthermore, the tests suggest that themono-objective evolutionary algorithm starting with a PCSA or SPS sequence is the most appropriate heuris-tic to our problem. However, mono-objective approach implies the weight settings which need to be deter-mined with great care.

We suggest using the MOEA some times (once a week) to obtain a set of dominating solutions. This allowsthe user to observe the influence of one objective function over the other. It also helps to determine the appro-priate weights for the mono-objective EA that will be used daily to sequence the vehicles.

Following this study the different algorithms were again tested with several others parameters settings anddata sets regarding the different capacity of the production line (from 140 to 1200 vehicles to sequence eachday). The VNS was finally selected and industrialized to sequence the vehicles at PSA Peugeot Citroen. Appro-priate weights were defined after the tests. In the new sequencing system, parameters may be reset in order tomodify the chosen strategy (increase the densification or the smoothing). Fig. 10 shows the comparisonbetween the old industrial solution and solutions provided by the VNS with different initial lists (SAS orSPS) and different computation times (30 and 60 s).

Table 8Appropriate heuristic according to available time.

Available time (s) Appropriate heuristic

1 Progressive construction-sequence algorithm (PCSA)1–90 Initial solution sequenced with PCSA and optimisation with VNS>90 Initial solution sequenced with PCSA and optimisation with EA

Comparison between solutions provided by VNS and old industrial solution

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Industrial old solution obtained with 60 s MOEA with 12 hours

Solution obtained with VNS and SAS and 30 s Solution obtained with VNS and SPS and 30 s

Solution obtained with VNS and SAS and 60 s Solution obtained with VNS and SPS and 60 s

Fig. 10. Comparison between the old industrial solution and the solutions provided by VNS with different initial lists and differentcomputation times.


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Alexandre Joly is Doctor in Logistics and Industrial Engineering. He worked for PSA Peugeot Citroen in theCentral Production Department where he was in charge of applying best logistics principles in the different plantsof the group. He is now consultant in lean manufacturing.

Yannick Frein Professor at the Institut National Polytechnique de Grenoble, in the Industrial Engineering
Department. He is director of the G-SCOP laboratory. His research interests are in modelling and performanceevaluation of production systems, especially supply chains, and in dynamic scheduling, with especially applica-tions in the car industry.

heuristics for an industrial car sequencing problem considering paint and assembly shop objectives

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