Peer-review under responsibility of the scientific committee of the 49th CIRP Conference on Manufacturing Systems doi: 10.1016/j.procir.2016.11.072 Procedia CIRP 57 2016 416 – 421 Sci
Trang 12212-8271 © 2016 The Authors Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license
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Peer-review under responsibility of the scientific committee of the 49th CIRP Conference on Manufacturing Systems
doi: 10.1016/j.procir.2016.11.072
Procedia CIRP 57 ( 2016 ) 416 – 421
ScienceDirect
49th CIRP Conference on Manufacturing Systems (CIRP-CMS 2016) Hybrid multi-objective optimization method for solving simultaneously the line balancing, equipment and buffer sizing problems for hybrid assembly
systems Jonathan Oesterlea,*, Thomas Bauernhansla,b, Lionel Amodeoc
a Fraunhofer Institute for Manufacturing Engineering and Automation (IPA), Nobelstrasse 12, 70569 Stuttgart, Germany
b Institute of Industrial Manufacturing and Management (IFF), University of Stuttgart, Nobelstrasse 12, 70569 Stuttgart, Germany
c Charles Delaunay Institute (ICD-LOSI), University of Technology of Troyes, STMR, UMR CNRS 6279, 12 rue Marie Curie, 10010 Troyes, France
* Corresponding author Tel.: +49(0)711/970-1199; fax: +49(0)711/970-1009 E-mail address: jno@ipa.fhg.de
Abstract
In today’s dynamic and uncertain markets, companies are required to regularly renew their product and process platforms through new production technologies and factory infrastructure This results in shortening products’, processes’ and factories life cycle, engendering in return an increase of the complexity of assembly planning tasks, which are seen as increasingly uncertain and complex to control This article presents a hybrid multi-objective optimization algorithm aiming at solving simultaneously the line balancing, equipment selection and buffer sizing problem under consideration of capacity and cost-oriented objectives The proposed algorithm is compared to two classical evolutionary algorithms, the NSGA2 and SPEA2
© 2015 The Authors Published by Elsevier B.V
Peer-review under responsibility of Scientific committee of the 49th CIRP Conference on Manufacturing Systems (CIRP-CMS 2016)
Keywords: Assembly Line Design problem; balancing problem; equipment selection; buffer sizing; Mixed Model Line; Multiobjective
1 Introduction
Since product mass customization became a viable strategy
in the mid-1990s, there has been tremendous market pressure
on companies to deliver personalized products and services to
customers with mass production efficiency, costs and quality
[1] Assembly lines, which are the most commonly used
assembly systems and allow meeting these cost and efficiency
requirements permit the assembly of products by workers
with limited training and dedicated machines and/or robots
Due to the high investment and running costs involved, the
design and re-design of such lines is highly important and a
number of crucial decisions have to be made, including
product design, process selection, line layout configuration,
line balancing and buffer sizing Usually, due to their
complexity, these problems are considered successively at one
time [2,3] However, successively addressing the previous
steps will most likely not lead to a global optimum of the
whole system The last two crucial steps are the line balancing
and buffer sizing In the former case, the tasks are assigned to workstations such that the efficiency of the line is optimized The effects of the unreliability of machines and/or robots, e.g starvation and blockage, disrupting the material flow in the assembly line, are limited through buffer inclusion Additionally, they help to smooth and balance the material flow between stations However, the inclusion of buffers requires additional capital investment, floor space of the line, in-process inventory [4,5] and also increases the lead time Changeable, dynamic and uncertain markets forces companies to regularly renew their product and process platforms through new production technologies and factory infrastructure in order to fit explicitly the requirements of individual customers This results in shortening not only products life cycles but also factories’ and processes’ This consistently affects the complexity of assembly planning projects, which are seen as increasingly uncertain, complex, dynamic and difficult to control These characteristics features are mostly related to the choice of the right level of
© 2016 The Authors Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license
( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).
Peer-review under responsibility of the scientifi c committee of the 49th CIRP Conference on Manufacturing Systems
Trang 2automation (e.g fully, semi-automatic or manual) and
equipment regarding technical data and engendered product
costs, which in turn affects the buffer location and size in the
assembly line
In the views of significant uncertainty, the ability to plan
the most flexible and economic assembly system by taking
product and process alternatives is highly important This
article presents a holistic planning method for a mixed-model
line under consideration of product, processes and resources
alternatives, aiming at optimizing capacity- and cost-oriented
objectives This planning method addresses simultaneously
the process selection, line balancing and buffer sizing
2 State-of-the-Art
The Assembly Line Balancing Problem (ALBP) consists in
finding a feasible line balancing, i.e., an assignment of tasks
to station such that precedence constraints and possible
further restrictions are fulfilled Due to different conditions in
industrial manufacturing, assembly line systems and
corresponding ALBPs have been extensively studied and
different classification schemes and state-of-the-art have been
proposed The most recent reviews of ALBPs are those
provided by Becker and Scholl [6,7] Baybars [8] proposed a
common classification scheme, which distinguish between
the: (i) Simple Assembly Line Balancing Problem, and the (ii)
Generalized Assembly Line Balancing Problem (GALBP) In
the former case, only one single product is processed, and the
problem is restricted by precedence relations and cycle time
constraints In the latter case, problems involving e.g parallel
workstations, parallel tasks, unequally equipped workstations,
problems involving sequence-dependent or stochastic
processing times and problems considering mixed and
multi-model lines can be found Battaia and Dolgui [2] provided a
taxonomy of ALBPs ALBPs can also be classified into (i)
capacity-oriented objectives and (ii) cost-oriented objectives
Hazir et al [9] extended the classification of Boysen et al by
incorporating cost and profit aspects
Two different approaches have been proposed to
incorporate processing alternatives into ALBP [10] The
former one is known as the equipment selection problem and
is based on the assumption that there is a fixed set of
equipment (exactly one of each) that has to be selected and
assigned to a station The latter consists in assigning processes
to tasks In addition to line balancing, for each task exactly
one processing alternative has to be chosen out of a set of
possible ones These alternatives are determined through task
requirements concerning either technological alternatives (e.g
gluing, clinching) or resource alternatives (e.g machines or
manpower) Approaches dealing with processing alternatives
can be found here [11–14] Capacho and Pastor [15] considers
alternative variants that an assembly process may admit Each
assembly variant is represented by a subgraph and determines
the tasks required to assemble a part of a particular product
Up to now, the problem has been defined and modelling in a
restricted version and an extended version [16,15] This
problem, also known as the Alternative Subgraphs Assembly
Line Balancing Problem (ASALBP) considers alternative
assembly precedence subgraphs that involve either the same
or different set of tasks Not only heuristic methods have been developed and tested comprehensively [17,18] but also an exact method [19]
The buffer allocation problem (BAP) aims at allocating a certain amount of buffer, among intermediate buffer locations
of a production line to optimize some specific objective, e.g throughput of the assembly line, minimum total buffer size The solution approaches used to solve the BAP involve both a generative and evaluative method While the former aims at searching for an optimal solution, the latter aims at evaluating various performance measures by means of analytical methods and/or simulation A comprehensive survey about the BAP has been provided by Demir et al [4] While the BAP has been widely studied in the literature, only a few works are focused
on solving the multiobjective BAP [20] Example of works taking only respectively one criterion into consideration are [21,22] and several criteria [23,20,24]
So far, both single and multi-objective assembly line balancing problems and buffer sizing problems have almost always been investigated separately in literature, suffering from the lack of simultaneous considerations Tiacci [25] addresses the problem of simultaneously balancing a line and allocated buffers with stochastic task times and parallel workstations
Despite the huge amount of research done over the last years, there is still a vast bridge between the methods provided by the literature and the current industrial problems and market features, engendering a difficult practical use of these methods Indeed, despite the direct simplicity of the available models, some recent characteristic features of the present day situation engendered shortcomings on the current methods Indeed, the requirement of quality engenders the need to not only select and plan the most reliable system, with e.g low maintenance effort, low material waste and a low number of deficient products but also the most economic one While most of the studies consider equipment costs on a high level, other product costs elements (e.g breakdowns, quality) are not examined Furthermore, most of the studies previously listed, only optimize either one capacity-oriented or cost-oriented objective However, since most real-life decision and planning situations involve multiple conflicting criteria, a multi-criteria optimization model that considers both time and cost based criteria, conjugated with a robust cost model, would better reflect the current industrial needs Last but not least, in order to achieve a global optimum of the whole assembly system, the process selection, the line balancing and buffer sizing have to be addressed simultaneously, which has,
to the best knowledge of the authors, not been addressed yet
3 Problem description
There are ܲ models of a product, a set of tasks ܸ comprised of a set ܵܩ of components’ and processes’ alternatives and a set of equipment ܧ For each task ݆, there is
a set of available equipment ܧ with different properties, e.g task processing times, costs, scrap rate The problem is to select tasks and equipment and assign them to workstations in order to minimize idle time between models and workstations Additionally, the buffers have to be allocated between stations
Trang 3such that the throughput of the line is maximized and the total
unit product costs, resulting from the line balancing and
buffer sizing are minimized Using the node and precedence
graph representation of Scholl et al [19], following notation
will be used The set of nodes ܸ ൌ ሼͳǡ ǥ ǡ ܰሽ consists of sets:
(i) ܸ, set of real tasks, (ii) ܸ௦, set of entry tasks (iii) ܸ௧, set of
terminal tasks and (iv) ܸௗ, set of dummy tasks Each task ݆ א
ܸ has a set of available resources ܧ={ܴ ܹሽ, where, ܴ
and ܹ respectively represent the set of automatic and manual
resources Each equipment ݈ א ܧ has a some tasks specific
and unspecific properties, such as a task processing time for
task ݆ and model , ݐ, a scrap rate ݏݎ and ܥ் and ܮ்
respectively the initial purchasing price and useful life of
tools Each automatic resource ݈ א ܴ has an average failure
rate ߤǡ ሺߤൌ ͳȀܯܶܶܨሻ , an average repair rate
ݎǡ ሺݎ ൌ ͳȀܯܴܶܶሻ , and an average planned and
unplanned downtime ܦ Additionally it has an average
energy consumption ݁, a useful life ܮ, and ܥா the initial
capital investment Each manual resource ݈ א ܹ has a
standard wage ݓ, and one-time personal costs ܿ Each task
݆ א ܸ has a wage rate ߱ and material costs ܿ Each buffer
position has an initial capital investment ܥ௨ and a useful life
ܮ௨
The total task time of task ݆ performed by equipment ݈is
calculated by using ߙ, which represents the probability of
occurrence of model
ݐൌ σ ߙݐ
The mathematical model requires two assignment
variables, namely ݔ (for ݆ א ܸ , ݇ ൌ ͳǡ ǥ ǡ ݉ഥ , and ݈ א ܧ)
and ܻ ( ൌ ͳǡ ǥ ǡ ܤǡand ݍ ൌ ͳǡ ǥ ǡ ݉ഥ െ ͳ)
ݔ ൌ ൝
ͳ݂݅ݐܽݏ݆݇݅ݏܽݏݏ݅݃݊݁݀ݐݏݐܽݐ݅݊݇ݓ݅ݐ݄
݁ݍݑ݅݉݁݊ݐ݈
Ͳݐ݄݁ݎݓ݅ݏ݁
ܻ ൌ ቄͳ݂ܾ݅ܽݑ݂݂݁ݎݏ݅ݖ݁݅ݏܽݏݏ݅݃݊݁݀ݐܾݑ݂݂݁ݎݍ
Ͳݐ݄݁ݎݓ݅ݏ݁
(2)
In order the reduce the number of variables of ݔ, the
earliest and latest station, ܧԢ and ܮᇱ to which a task ݆ can be
assigned, can be computed using the relation proposed by
Scholl et al [19] and extended by relation ሺͳሻ, where ܲכሺ݆ሻ
and ܨכሺ݆ሻ represent respectively the list of all predecessors
and successors of a modified graph, in which all subgraphs
are replaced by a fictive task representing a lower bound on
the total task time, ݐݏሺݒሻ ൌ
אௌீሼݐݏሺܸሺ݃ሻሻ This allows to identify the subset ܤൌ ሼ݆ א ܸȁ݇ א ܵܫሽ representing
potential tasks assignable to station ݇, where ܵܫൌ ሾܧᇱ
ǡ ܮᇱ
ሿ, based on the new task time ߬ൌ ݐΤܥܶ
ܧԢൌ ඃ߬ σఢכ ሺሻ߬ඇ,ܮᇱ
ൌ ݉ഥ ͳ െ ඃ߬ σఢிכ ሺሻ߬ඇ (3)
The decisions that have to be taken address four related
issues: the design problem, (i) where components and
processes have to be selected, (ii) and where equipment for a
given task has to be selected, (iii) the balancing problem,
where the tasks have to be assigned to workstations, (iv) the
buffer allocation problem, where the buffer size between each
stations will be determined
The assumptions of the problem are listed below:
x Automatic resources on each stations are unreliable and
each station is separated by an intermediate finite buffer
x Breakdowns only occur when resources are operating
x The last station is never blocked
x Work pieces move through buffers with zero transit time and follow a First-In-First-Out strategy
x The operating time and repair time between failures are exponentially distributed
x If a breakdown occurs, unit in process will be scrap
x The setup time is included in the task processing time
4 Mathematical Model
The mathematical formulation of this problem can be stated as follows The objective function to be optimized (see (4)) depends on three objectives: (i) the minimization of the idle time in the line, (ii) the minimization of the total unit costs ܥ, and (iii) the maximization of the throughput rate ܧ
ܯܼ݅݊ଵൌ σ ഥ ሾܥܶ െ σאೖσאாೕݔ
σאೖσאாೕݔݐ ܥܶ, ݇ ൌ ͳǡ ǥ ǡ ݉ ഥ (10)
σאೖݔܽ ܣ, ݇ ൌ ͳǡ ǥ ǡ ݉ ഥ, ݈ א ܧ (11)
σאௌூ݇ݔ σאௌூೕ݇ݔ,
݆ ൌ ܸ െ ܸ௦, ݆ א ܨหܵܫת ܵܫ് , ݈ א ܧ (12)
σאௌூݔെ σאௌூೕݔ ݉ഥሺͳ െ σאௌூೕݔሻ,
݅ ൌ ܸ௦, ݆ א ܨȁܵܫת ܵܫ് , ݈ א ܧ (13)
σאௌூݔ σאௌூೕݔ, ݅ ൌ ܲ, ݆ א ܸ െ ܸ௧, ݈ א ܧ (14)
σאிσאௌூೕݔെ σאௌூݔൌ Ͳ, ݅ א ܸ௦, ݈ א ܧ (15)
σאೕσאௌூݔെ σאௌூೕݔൌ Ͳ, ݅ א ܸ௧, ݈ א ܧ (16)
σ σ ഥ ିଵሺܻܾ
ݔא ሼͲǡͳሽ, ݆ א ܸ, ݇ א ܵܫǡ ݈ א ܧ
ܻא ሼͲǡͳሽǡ א ͳǡ ǥ ǡ ܤݍ=ͳǡ ǥ ǡ ݉ ഥ െ ͳ (19)
Constraints (8) and (9) indicate that the throughput rate ܧ and the product cost ܿ are function of the balancing and buffer allocation, namely the selection of tasks, resources and assignment to stations and the sizes and locations of buffers Constraint (10) aims at verifying that the assignment of tasks
to station is respecting the cycle time ܥܶ Constraint (11) assures that the required material space is not exceeding the available space ܣ at each station Relations (12) and (13) ensure that precedence constraints are respectively respected for all nodes Constraints (14)-(16) guarantee that all are at most assigned to one station Constraint (17) ensures that the total space of the buffers and assembly stations must not exceed the total available space for the assembly line Constraint (18) imposes that a unique size must be assigned to each buffer Constraint (19) defines the binary decision variables The objective function (9) aims at assessing the unit cost per part, given by the ratio of the total annual costs and the target net volume of defect-free parts ܸ௧ to be produced
ܿ ൌ ܥ௧௧ሺݔǡ ܻሻ ܸΤ ௧ (20)
Trang 4The total annual costs are provided by:
ܥ௧௧൫ݔǡ ܻ൯ ൌ ܥெ൫ݔ ൯ ܥ൫ݔ ൯ ܥைு൫ݔ ൯
ܥா൫ݔǡ ܻ൯ ܥ൫ݔǡ ܻ൯ ܥா൫ݔǡ ܻ൯
ܥ்ሺݔሻ
(21)
Where, ܥெ൫୨୩୪൯ǡ ܥ൫୨୩୪൯ , ܥைு൫୨୩୪൯ǡ ܥா൫୨୩୪ǡ ୮୯൯ǡ
ܥ൫୨୩୪ǡ ୮୯൯ǡ ܥா൫୨୩୪ǡ ୮୯൯ and ܥ்ሺ୨୩୪ሻ represent
respectively the total material costs, the labour costs, the
overhead costs, the energy costs, the annual building costs,
the equipment costs and the tooling costs Each of these terms
depends on either the line balancing results, the buffer
allocation results or both
ܥெሺݔሻ=σ ഥ σאೖݔܿ
Where,
ܸ௦௦=ܸ௧ ς ഥ ሺͳ െ ݎሻ
ୀଵ
ݎൌ ͳ െ ςאೖςאாೕݔሺͳ െ ݏݎሻ (23)
The total labour costs, ܥ, is provided by the sum of labour
costs used at each station ǡ ܿ The wage of a worker is
determined according to the most difficult task to be
performed in station ݇ [26] If ܯ represents the total number
of required workers, then:
ܥሺ௫ೕೖሻσ σאௐሺ݉ܽݔאೖሺݔ
ഥ
Overhead costs ܥைு are related to indirect labour required
to maintain production, which is modelled by a ratio of the
number of indirect ݎௗ workers, paid at wage rate ߱ௗ, for
each direct worker:
Taking the average planned and unplanned downtime ܦ of a
station ݇, and its probability of failure :
߬ ൌ ܥܶ൫ͳ σ ഥ
The energy costs ܥா are provided by the average energy
costs of each automatic equipment and buffer used at station
݇, ܿா In order to distribute building, tooling and equipment
costs over time, the capital recovery factor ܥܴܨ is used,
where ݆ represents either building, equipment, tooling, or
buffer
ܥܴܨൌ ݎሺͳ ݎሻೕΤሾሺͳ ݎሻೕെ ͳሿ (27)
Where ݎ is the annual discount rate and ܮ is the useful life in
number of years The annual building cost is computed given
one-off costs, the initial building capital investment ܥǡ of
size ݈ and width ݓ and running costs, e.g cost of energy for
lighting, heating, air conditioning, which can be calculated by
multiplying the space occupied by the assembly line with a
factor ݁, representing the annual energy cost for each m² The
one-off costs of equipment and tooling, ܥாܥ், are
provided by multiplying ܥܴܨ by the respective initial
investment ܥா and ܥ் Additionally, the costs associated to
buffer size are added to ܥா
ܥ൫ݔǡ ܻ൯ ൌ ൫σ σ ഥ ିଵሺܻܾ
ݓ ൈ ሺܥǡൈ ܥܴܨ௨ௗΤሺ݈ൈ ݓሻ ݁ሻ (28)
Where, ݈ and ݓ represent respectively the length and width of
a given station/buffer Finally, these annual costs can be used
to compute a unit cost per part
5 Resolution Method
Many methods exist for solving multiobjective optimization problems (MOP) Two main categories can be identified: (i) classical methods which use direct or gradient-based methods following some mathematical principles and (ii) non-traditional and population-based methods following some natural or physical principles Classical methods mostly attempt to scalarize multiple objectives and perform repeated applications to find a set of Pareto-optimal solutions, whereas population-based methods attempt to find a multiple Pareto-optimal solutions in a single simulation run [27] Since the ALBP and BAP are NP-hard [4,28], approximation methods are most suitable to rely on solving these problems The use
of population-based optimization techniques, such as evolutionary algorithms (EA) is an appropriate approach for addressing MOP However, due to their global search nature, EAs are not so efficient as regards quickly and reliably leading the population toward the optimal front A most powerful mechanism can be obtained by combining the EA with a local search, which allows a balance between global and local search [29] Combining global and local search methods is known as memetic approach Despite a better accuracy for the final solution, memetic algorithms are also able to offer a better speed of convergence [30] The developed memetic algorithm is composed of a NSGA2 and a Simulated Annealing (SA), which will be explained in the next subchapters
5.1 Structure of the NSGA2
The solution encoding uses two different chromosomes In the first chromosome, three numbers are assigned that respectively represent the selected task, equipment, and workstation assignment To the second chromosome, two numbers are assigned, the buffer position and its size This attribution is done by taking the previously listed constraints into consideration
Initially, an initial population ܲ is created following classical ALBP heuristics [31] The production rate of a station ݇ in isolation, meaning its production rate when the station is not subject to starvation or blockage is given by and can be used to initiate its buffer allocation.:
ൌ ቀݎΤሺσאೖσאாೕݔݐሻቁ ሺߤൗ ݎሻ (29)
Where, ୩ its average repair rate, and Ɋ୩ its average failure Indeed, in order to avoid effects of starvation and blockage, stations with a poor production rate and thus a bigger chance
of breakdown should have a bigger buffer Therefore, the relative criticality ܥܴ of a buffer ݍ can be determined by: ܥܴൌ ݉݅݊ሼǡ ାଵሽିଵ σ ഥ ିଵ݉݅݊ሼǡ ାଵሽିଵ
ୀଵ
The initial buffer allocation is provided by multiplying ܥܴ
by the maximum buffer size ܤ at each location ݍ , for
ݍ ൌ ͳǡ ǥ ǡ ݉ഥ െ ͳ
Trang 5Fig 1 Solution encoding
The first population is evaluated through a discrete event
simulation, which calculates the throughput and total product
costs of each solution The population is then sorted based on
the Pareto non-domination concept In a second step, a child
population ܳ is created from ܲ by performing tournament
selection, crossover and mutation operations The two
populations ܲ and ܳ are combined to form a population ܴ
of size 2ܰ, where ܰ ൌ ȁܲȁ ൌ ȁܳȁ A non-dominated sorting
is used to classify the entire population ܴ, which is in turn
subdivided in several non-dominated front The new parent
population ܲଵ is filled with individuals of the best
non-dominated fronts Since ܴ is of size 2ܰ, while ܲଵ of size ܰ,
the last allowed front will be truncated by using niching
strategy to choose individuals from the last front which reside
in the least crowded region of this front These steps are
repeated until reaching a stopping criterion, e.g predefined
number of iterations
5.2 Simulated Annealing
The memetic NSGA2 procedure, based on an adaptation of
the PHC-NSGA2 [32], is explained in Fig 2 For any
generation ݐ ͳ, the local search procedure, which works on
both chromosomes by either swapping and/or changing tasks,
equipment assignment and buffer size, is applied for each
front ܨ of ܲ௧ The solutions selected for the local search,
representing the set ܮܵ are the least crowded solutions in the
objective space For each solution ݏ א ܮܵ, the SA procedure
will return a new solution After the local search, the new set
of solutions ܮ̴ܵ݊݁ݓ is added to ܨ A crowded distance in the
decision space is assigned to each solutions After applying
the local search for each front ܨ of ܲ௧, the new population ܳ௧
is generated
The general SA algorithm involves two main steps: (i)
selection of a proper annealing scheme consisting of
decreasing temperature with increasing iterations, and (ii) a
method generating a neighbor near the current search position
The transition probability scheme is different in
multiobjective optimization and choosing a proper transition
probability is difficult [33] The following geometric cooling,
which is widely employed for the annealing scheme, was
used:
The transition probability from state ݏ to ݏԢ is given by:
Begin
R t ՚ Pt Q t
F ՚ Fast_non-dominated_sort (Rt , M)
P t+1 ՚
i ՚ 1
While | P t+1|+|Fi|N do
Crowding_distance_assignment (F i , M)
P t+1 ՚ Pt+1 F i
i ՚ i + 1
EndWhile
Sort (F i , < n )
P t+1 ՚ Pt+1 F i [1: (N-|P t+1|)]
For each front F i אP t+1 do
LS ՚ Local_search_selection (Fi ) LS_new ՚
For ݏ א LS do
s=ݏ
T=ܶ
While max_iterations not reached
Generate a neighbor s’=N(s)
If s’ dominates s
move to s’
Else If s dominates s’
move to s’ with transition probability ܲ௧ሺ݂ሺݏሻǡ ݂ሺݏ ᇱሻǡ ܶሻ
Else If s and s’ do not dominate each other
move to s’
EndIf
T=annealing(T)
EndWhile
LS_new ՚ LS_new LS (s)
EndFor
F i ՚ F i LS_new Crowding_distance_assignment (F i , M) Sort(F i )
F i ՚ replace ( F i )
EndFor
Q t+1 ՚ make_new_pop (Pt+1 )
t ՚ t + 1
End
Fig 2 Basic iterations of the memetic NSGA2
Where, ሺǡ ᇱሻ is the cost criterion for the transition from state ݏ to ݏԢ and ܶ is the annealing temperature In our resolution method, following average cost criterion was used, where ܯ represents the number of objectives:
ܿሺݏǡ ݏᇱሻ ൌσమ ሺ௦ሻିሺ௦ᇱሻ
Where ୫ሺሻ represents the value of the objective function for
a given solution ݔ
6 Computational experiments
In order to assess the performances of the developed memetic NSGA2, it was compared with a NSGA2 and a SPEA2 by adapting benchmark problems, obtained from the webpage (www.assembly-line-balancing.de)for assembly line balancing research A Design of Experiments (DOE) was done in order to find the best combinations of initial parameters for the Simulated Annealing The found values are
ߙ ൌ ͲǤͻͷ , ܶൌ ͵ͲͲ , and max_iterations=100 The comparison of the algorithms was done using the Hypervolume (HV) indicator, proposed by Zitzler and Thiele [34], which measure the quality of a set of non-dominated solution in terms of convergence and diversity on a single scale The normalized HV obtained for each tested problem, which has to be maximized, is shown in Table 1 It seems to
be clear that the memetic NSGA2 outperformed the NSGA2 and SPEA2 on all experiments regarding the HV indicator
1 st
Chromosome
Tasks
Equipment
1 4 3 7
1 1 1 1
8 10
3 3
12 6
3 3 Assignation
Equipment
Tasks
number
1 2 1 4 1 1 3 6 4 3 3 3
1 2 4 5 3 7 8 9 11 10 12 6
2 nd
Chromosome
Buffer
1 2 3
…
Trang 6Table 1 Comparison of the memetic NSGA2 with the NSGA2 and SPEA2
Memetic NSGA2 NSGA2 SPEA2
HV calculated for 5 repeated experiments
7 Conclusion
So far, both single and multi-objective assembly line
balancing problems and buffer sizing problems have been
tackled separately in literature The proposed article presents a
hybrid multi-objective optimization method for solving
simultaneously the line balancing and buffer sizing problem
for hybrid assembly systems in order to optimized capacity
and costs-oriented objectives This algorithm outperformed
classical evolutionary algorithms, the SPEA2 and NSGA2, on
the tested benchmarks The proposed method combines both a
global and local search in order to better and faster converge
to the optimal Pareto front Future line of research could be to
investigate the influence of similarity in the process of
selection in order to reduce the amount of reconstruction
procedures in the crossover step or to investigate the effect of
other cost criterion in the probability transition of the
simulated annealing
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