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P. Arikaran, Dr. V. Jayabalan,& R. Senthilkumar
International Journal of Engineering (IJE), Volume (4) : Issue (1)
44
Analysis of Unequal Areas Facility Layout Problems
P.Arikaran arivarsini_2006@yahoo.com
Research scholar,
Dept. of Manufacturing, Anna University,
Chennai 600 025 India
Professor & H.O.D.,Dept. of Mechanical Engineering,
M.N.M. Jain Engineering College,
Chennai 600097, phone: 9444154329
Dr. V. Jayabalan jbalan@annauniv.edu
Professor, Dept. of Manufacturing
and Controller of Examinations, Anna University,
Chennai 600 025 India
R. Senthilkumar
rskumar1967@yahoo.com
Assistant Professor, Dept.of Mechanical Engineering,
M.N.M. Jain Engineering College,
Chennai 600097, India
______________________________________________________
Abstract
The facility layout design has been regarded as the key to improve plant
productivity, which are relevant to both manufacturing problems; various
optimization approaches for small problems and heuristic approaches for the
larger problems have been proposed to elucidate the problem. Unequal area
facility layout problems comprise a class of extremely difficult and widely
applicable optimization problems, arising in many diverse areas. There are
many variations on the basic formulation, involving alternative objective
functions, side constraints, distance metrics, cost measures, and facility
shapes. Various techniques were applied after finding the solution through
traditional methods to get much improved optimum solutions. Different
heuristics were used to solve the unequal area facility layout problems. Multi-
objective approaches are the norm and developing facility layout software using
meta-heuristics such as simulated annealing (SA), genetic algorithm (GA), ant
colony algorithm (ACO), and concurrent engineering is prevailing nowadays.
Sometimes hybrid approaches were used by applying combination of above
techniques i.e. combining high level genetic algorithm with simulated annealing
or genetic algorithm followed by simulation techniques to get the better
solutions. Application of these facility lay out designs includes construction
sites, manufacturing industry and service industries.and service sectors.
Facility Layout Problems (FLPs) are known to be NP-hard
Keywords—Facility layout, unequal area , Hybrid methods , Genetic Algorithm, Automated layout
Manufacturing Industries, construction sites.
P. Arikaran, Dr. V. Jayabalan,& R. Senthilkumar
International Journal of Engineering (IJE), Volume (4) : Issue (1)
45
1.INTRODUCTION
The static facility layout problem (SFLP) is a well-researched problem of finding
positions of departments on the plant floor such that departments do not overlap while some
objective is optimized. The most commonly used objective is minimizing material handling cost
(i.e., minimizing the sum of the product of the flow of materials, distance, and transportation
cost per unit per distance unit for each pair of departments). When material flows between
departments change during the planning horizon, the problem becomes the dynamic facility
layout problem (DFLP).
A solution to the FLP is a block layout that specifies the relative location and the dimensions of
each department. Once a block layout has been achieved, a detailed layout can be designed
which specifies department locations, aisle structures and input/output point locations [3, 6, 7].
Two types of approaches for finding provably optimal solutions for the FLP have been proposed
in the literature. The first type are graph-theoretic approaches that assume that the desirability
of locating each pair of facilities adjacent to each other is known. Initially, the area and shape of
the departments are ignored, and each department is simply represented by a node in a graph.
Adjacency relationships between departments can now be represented by arcs connecting the
corresponding nodes in the graph. The objective is then to construct a graph that maximizes
the weight on the adjacencies between nodes. We refer the reader to [4] for more details. The
second type are mathematical programming formulations with objective functions based on an
appropriately weighted sum of centroid-to-centroid distances between departments. Exact
mixed integer programming formulations were proposed for the above type was shown in
[8,9].
2. UNEQUAL AREA LAYOUTS
The unequal-areas facility layout problem (FLP) is concerned with finding the optimal
arrangement of a given number of non-overlapping indivisible departments with unequal area
requirements within a facility. The block layout design problem with unequal areas, which was
originally formulated by Armour and Buffa in the early 1960s, is a fundamental optimization
problem encountered in many manufacturing and service organizations. Different methods
were discussed for solving unequal area problems in Literature. These methods are
summarized under the various topics to have a understanding of Unequal area problem.
2.1 Tree structure Model
The facility layout design has been regarded as the key to improve plant productivity, which are
relevant to both manufacturing and service sectors. A tree structure model has been proposed
for representing the unequal-area facility layout by[10]. Each facility has a different rectangular
shape specified by its area and aspect ratio. In this layout problem, based on the assumption
that the shop floor has enough space for laying out the facilities, no constraint is considered for
a shop floor. Objectives are minimizing total part movement between facilities and total
rectangular layout area where all facilities and dead spaces are enclosed. Using the genetic
code corresponding to two kinds of information, facility sequence and branching positions in the
tree structure model, a genetic algorithm has been applied for finding non-dominated solutions
in the two-objective layout problem. [10] used three kinds of crossover (PMX, OX, CX) for the
former part of the chromosome and one-point crossover for the latter part. Two kinds of layout
problems have been tested by the proposed method. The results demonstrated that the
presented algorithm was able to find good solutions in enough short time.
2.2 Genetic Search
P. Arikaran, Dr. V. Jayabalan,& R. Senthilkumar
International Journal of Engineering (IJE), Volume (4) : Issue (1)
46
[23] used Genetic search for solving construction site-level unequal-area facility layout
problems. A construction site represents a conflux of concerns, constantly calling for a broad
and multi-criteria approach to solving problems related to site planning and design. As an
important part of site planning and design, the objective of site-level facility layout is to allocate
appropriate locations and areas for accommodating temporary site-level facilities such as
warehouses, job offices, workshops and batch plants. Depending on the size, location and
nature of the project, the required temporary facilities may vary. The layout of facilities can
influence on the production time and cost in projects. [23]described a construction site-level
facility layout problem as allocating a set of predetermined facilities into a set of predetermined
places, while satisfying layout constraints and requirements. A genetic algorithm system, which
is a computational model of Darwinian evolution theory, was employed to solve the facilities
layout problem. A case study was presented to demonstrate the efficiency of the genetic
algorithm system in solving the construction site-level facility layout problems
2.3 Hybrid Method
[12] presented the solution of the unequal area problem by hybridizing the meta-heuristic
methods i.e. Genetic Algorithm (GA) and Simulated Annealing (SA). (SA) is a related global
optimization technique that traverses the search space by testing random mutations on an
individual solution. A mutation that increases fitness is always accepted. A mutation that lowers
fitness is accepted probabilistically based on the difference in fitness and a decreasing
temperature parameter. In SA parlance, one speaks of seeking the lowest energy instead of the
maximum fitness. SA can also be used within a standard GA algorithm by starting with a
relatively high rate of mutation and decreasing it over time along a given schedule.
[12] could be used in future as a reference for those researchers interested in tackling this
challenging unequal facility layout problem. A mathematical model was developed for the
unequal size facility layout problem with fixed flow between departments. The orientations of
the departments with various sizes were considered to minimize the distance traveled by
people, material, and other supporting services in the safest and most effective manner. Some
of the constraints considered in the modeling were the restricted areas, reserved department
locations, and also the irregularity of the shapes of manufacturing layout. This paper has also
presented the use of hybrid algorithm (GA - SA) as a general methodology to solve the facility
layout problem under consideration.
A hybrid optimization approach was presented in [21] for the layout design of unequal-area
facilities. Simulated annealing was used to optimize a randomly generated initial placement on
an “extended plane” considering the unequal-area facilities enclosed in magnified envelop
blocks. An analytical method was then applied to obtain the optimum placement of each
envelop block in the direction of steepest descent. Stepwise reduction of the sizes of the
envelop blocks allowed controlled convergence in a multi-phase optimization process. The
presented test problems include two large size benchmark problems of 50 and 100 facilities of
unequal areas. The results indicated a significant improvement over previously published
techniques for unequal-area facilities and could yield solutions of the same quality as obtained
by PLANOPT, a general-purpose layout optimization program based on pseudo-exhaustive
search.
2.4 Convex Optimization Framework
[13] presented a convex-optimization-based framework for efficiently finding competitive
solutions for this problem. The framework is based on the combination of two mathematical
programming models. The first model is a convex relaxation of the layout problem that
establishes the relative position of the departments within the facility, and the second model
uses semi-definite optimization to determine the final layout. Aspect ratio constraints, frequently
used in facility layout methods to restrict the occurrence of overly long and narrow departments
P. Arikaran, Dr. V. Jayabalan,& R. Senthilkumar
International Journal of Engineering (IJE), Volume (4) : Issue (1)
47
in the computed layouts, are taken into account by both models. It suggested that using
ellipsoids instead of circles to approximate the initial positions of departments could provide
better results and Ellipsoids would likely provide more realistic estimations of department
positions, since departments in real-world applications are not square-shaped. Further work
also included adjusting the
ϕ
(the parameter that can control what the desired smallest length
or width should be in each department’s layout) and potentially using a different value
ϕ
i for
each department. Finally, different combinations of first stage and second stage models from
past papers tested to get the over all results.
2.5 Tabu search Method
Tabu search (TS) is similar to simulated annealing in that both traverse the solution space by
testing mutations of an individual solution. While simulated annealing generates only one
mutated solution, tabu search generates many mutated solutions and moves to the solution
with the lowest energy of those generated. In order to prevent cycling and encourage greater
movement through the solution space, a tabu list is maintained of partial or complete solutions.
It is forbidden to move to a solution that contains elements of the tabu list, which is updated as
the solution traverses the solution space.
[14] discussed a slicing tree based tabu search heuristic for the rectangular, continual plane
facility layout problem by incorporation of facilities with unequal areas and integrated the
possibility to specify various requirements regarding (rectangular) shape and dimensions of
each individual facility by using bounding curves which made possible to solve problems
containing facilities of fixed and facilities of flexible shapes at the same time. This paper
presented a procedure that calculated the layout corresponding to a given slicing tree on the
basis of bounding curves and integrated the tabu search to find the better results. [19]
proposed a heuristics for the dynamic facility layout problem with unequal-area departments.
The solution is improved using a tabu search heuristic. The heuristics were tested on some
instances from the DFLP and static facility layout problem (SFLP) literature. The results
obtained demonstrated the effectiveness of the heuristics.
2.6 Genetic Algorithm
[15] presented a genetic algorithm-based model for facility layout problems with unequal
departmental areas and different geometric shape constraints. Gene structures of the genetic
algorithm are used to represent layout of departments. The algorithm involved deriving an
initial assignment of departments to the given floor plan and then, possibly, improving the
solution quality through genetic algorithm mechanisms (i.e., exchange parts of layout). Since
genetic algorithm is parameter sensitive, the experiments indicated that crossover type,
mutation type, mutation probability, and population size which are the main parameters that
designers need to consider while designing facility layout with genetic algorithm. Guidelines for
such parameters were also given.
[11] presented a heuristic search methodology, based on genetic algorithms (GA), for unequal
area layout. this methodology was applied to several standard test problems from the literature,
and was showed that the GA method gave solutions which were much better than the best
previously reported solutions.[11] used penalty-directed search to find very good solutions to
problems with difficult-to-satisfy side constraints, and to perform multi-criterion optimization with
respect to cost measures that have been considered incommensurable in the past. The
methodology presented in [11] is not intrinsically restricted to layout problems, but could be
extended to other hard combinatorial problems. The GA/penalty method's ability to find
improved solutions to known problems, together with the ability to address problems with ill-
behaved cost functions, multiple objectives, and/or side constraints, constitutes a significant
contribution to the state of the art in facility layout. Furthermore, GA could be implemented to
P. Arikaran, Dr. V. Jayabalan,& R. Senthilkumar
International Journal of Engineering (IJE), Volume (4) : Issue (1)
48
take advantage of parallel hardware to an extent not possible for other heuristic optimization
methods.
[18] outlined a GA based algorithm for solving the single-floor facility layout problem with
departments of both equal and unequal sizes. The GA performance was evaluated using
several test problems available in the literature. The results indicated that GA may provide a
better alternative in a realistic environment where the objective is to find a number of
“reasonably good” layouts. The implementation also provided the flexibility of having fixed
departments and to interactively modify the layouts produced.
[24] gave a solution to the unequal area facilities layout problem by genetic algorithm. The
majority of the issued facilities layout problems (FLPs) minimize the material handling cost and
ignore other factors, such as area utilization, department shape and site shape size. These
factors, however, might influence greatly the objective function and should give consideration.
The research range of [24] was focused on the unequal areas department facilities layout
problem, and implement analysis of variance (ANOVA) of statistics to find out the best site size
of layout by genetic algorithm. The proposed module took the minimum total layout cost (TLC)
into account. TLC was an objective function combining material flow factor cost (MFFC), shape
ratio factor (SRF) and area utilization factor (AUF). In addition, a rule-based of expert system
was implemented to create space-filling curve for connecting each unequal area department to
be continuously placed without disjoint (partition). In this manner, there was no gap between
each unequal area department. The experimental results showed that the proposed approach
is more feasible in dealing with the facilities layout problems in the real world.
2.7 Swarm Optimisation
Layout of temporary facilities on a construction site is essential to enhancing productivity and
safety, and is a complex issue due to the unique nature of construction. [16] proposed a particle
swarm optimization (PSO)-based methodology to solve the construction site unequal-area
facility layout problem. A priority-based particle representation of the candidate solutions to the
layout problem was proposed. The particle-represented solution in terms of priorities should be
transformed to the specific layout plan with consideration of non-overlap and geometric
constraints. In addition, a modified solution space boundary handling approach was proposed
for controlling particle updating with regard to the priority value range. Computational
experiments were carried out to justify the efficiency of the proposed method and to investigate
its underlying performances. This paper claimed to provide an alternative and effective means
for solving the construction site unequal-area layout problem by utilizing the PSO algorithm.
2.8 Space Partitioning
[17] proposed a space partitioning method for facility layout problems with shape constraints A
heuristic algorithm was developed for the problems with the objective of minimizing the sum of
rectilinear distances weighted by flow amounts between the facilities. The suggested algorithm
was a simulated annealing algorithm in which a solution is encoded as a matrix that has
information about relative locations of the facilities on the floor. A block layout was constructed
by partitioning the floor into a set of rectangular blocks according to the information while
satisfying the areas of the facilities. [17] suggested three methods for the partitioning.
2.9 Ant System
Ant Colony Optimization (ACO) is a young metaheuristic algorithm which has shown promising
results in solving many optimization problems. To date, a formal ACO-based metaheuristic has
not been applied for solving Unequal Area Facility Layout Problems (UA-FLPs). [20] proposed
an Ant System (AS) (one of the ACO variants) to solve them. As a discrete optimization
algorithm, the proposed algorithm used slicing tree representation to easily represent the
P. Arikaran, Dr. V. Jayabalan,& R. Senthilkumar
International Journal of Engineering (IJE), Volume (4) : Issue (1)
49
problems without too restricting the solution space. It used several types of local search to
improve its search performance. It is then tested using several case problems with different
size and setting. Overall, the proposed algorithm showed encouraging results in solving UA-
FLPs.
2.10 MILP and MINLP optimization methods
[22]presented a new modelling framework for effectively finding global optimal solutions for
the block layout design problem with unequal areas. The most fundamental aspect of the
framework consists of an exact representation of the underlying area restrictions. Our
computational results consistently yield optimal solutions on several well-known test problems
from the published literature. Furthermore, different mixed-integer linear and mixed-integer
nonlinear optimization methods are compared. Our study indicates that the new modeling
framework together with simple constraints to avoid symmetric layout solutions can be
successfully used to find optimal layout solutions; therefore, seriously challenging other
optimization methods on this important class of hard, fundamental problems. The new modeling
framework may easily be applied in the context of the process plant layout and piping design
problems.
2.11 automated layout
[25] generated Automated layout of facilities of unequal areas. Common to the analytical
techniques for automated layout of rectangular facilities of unequal areas is the problem of too
rapid movement of the representative blocks to form a cluster. This phenomenon made the
converged designs too dependent on the initial layout and the order of movement of the blocks.
A concept of “controlled convergence” was introduced to solve this problem. Convergence is
controlled by carrying out the optimization with the “envelop blocks” of sizes much larger than
the actual facilities. The sizes of the envelop blocks were gradually reduced to the actual sizes
of the facilities through the optimization cycles. Test results were given to demonstrate the
effectiveness of the presented technique.
2.12 Mixed Integer Programming method
[26] proposed an -accurate model for optimal unequal-area block layout design by developing
a mixed-integer linear programming model for the block layout design problem with unequal
areas that satisfies the area requirements with a given accuracy. The basic aspect of the model
consists of an -accurate representation of the underlying non-convex and hyperbolic area
restrictions using cutting planes. The use of such a representation of the area restrictions gave
way to solve several challenging test problems to optimality with a guarantee that the final area
of each department was within an % error of the required area. Numerical results indicated
that the proposed model seriously challenge other optimization approaches on this important
class of hard, fundamental problems
2.13 Graph Theory method
[27] developed Graph theoretic heuristics for unequal-sized facility layout problem. It
considered the unequal-sized facility layout problem with the objective of minimizing total
transportation distance. The total transportation distance was defined as the sum of products of
flow amounts and rectilinear distances between facilities, where flow amount represents the
number of trips per time period between facilities. In the layout problem, it was assumed that
shapes of facilities are not fixed and that there was no empty space between facilities in the
layout. It proposed new graph theoretic heuristics for the problem. In the heuristics, an initial
layout was obtained by constructing a planar adjacency graph and then the solution was
improved by changing the adjacency graph (not the physical layout). Therefore, these
heuristics did not need an initial layout in advance, and sizes and locations of facilities did not
P. Arikaran, Dr. V. Jayabalan,& R. Senthilkumar
International Journal of Engineering (IJE), Volume (4) : Issue (1)
50
have to be considered in the improvement procedure. Computational results showed that the
proposed algorithms gave better solutions than those from CRAFT, which is one of the most
popular algorithms for unequal-sized facility layout problems.
3. SUMMARY
The various techniques such as simulated annealing (SA), genetic algorithm (GA), ant
colony algorithm (ACO), and concurrent engineering is prevailing nowadays. Sometimes
hybrid approaches were used by applying combination of above techniques i.e. combining high
level genetic algorithm with simulated annealing or genetic algorithm followed by simulation
techniques to get the better solutions. Application of these facility lay out designs includes
construction sites, manufacturing industry and service industries. Out of these techniques,
most commonly used techniques were discussed briefly to highlight the growth in unequal area
facility layout. There is a vast scope to improve the facility layout design to fulfill the dynamic
needs of Manufacturing and construction industries.
REFERENCES
[1] Armour, G.C. and E.S. Buffa, “A heuristic algorithm and simulation approach to relative
allocation of facilities.” Management Science, 1963, 9, 294-309
[2] K. Anstreicher, N. Brixius, J.-P. Goux, and J. Linderoth. Solving large quadratic assignment
problems on computational grids. Math. Program., 91(3, Ser. B):563–588, 2002.
[3] Tam, K.Y., “Genetic algorithms, function optimization, and facility layout design.” European
Journal of Operational Research, 1992 63, 322-346.
[4] Chutima, P., and Yiangkamolsing, C., Genetic Algorithms with Improvement Heuristic for
Plant Layout Problems. Research and Development Journal of the Engineering Institute of
Thailand,Vol.l0,No.2. 62-12. 1999.
[5] FrenchS, .,Sequencing and Scheduling: An Introduction to the Mathematics of the Job-
Shop. Ellis Horwood Ltd,1982.
[6l Sule, D.R., Manufacturing Facilities: Location, Planning, and Design. PWS Publishing
Company. 1994.
[7] Gen, M., and Chen, R., Genetic Algorithm and Engineering Design .John Wiley & Sons I.n c.
1997.
[8] R.D. Meller, V. Narayanan, and P.H. Vance. Optimal facility layout design. Oper. Res. Lett.,
23:117–127, 1999.
[9] B. Montreuil. A modeling framework for integrating layout design and flow network design.
Proceedings of the material handling research colloquium, pages 43–58, 1990.
[10] Terushige Honiden, Tree Structure Modeling and Genetic Algorithm-based Approach to
Unequal-area Facility Layout Problem by IEMS Vol. 3, No. 2, pp. 123-128, December 2004
[11] ] David M. Tate, Senior Member IIE and Alice E. Smith, Senior Member IIE1, Unequal
Area Facility Layout Using Genetic Search by Department of Industrial Engineering, University
of Pittsburgh--Accepted to IIE Transactions, April 1994
[12] Nurul Nadia Nordin, Zaitul Marlizawati Zainuddin, Sutinah Salim and Raja Rajeswari d/o
Ponnusamy, Mathematical Modeling and Hybrid Heuristic for Unequal Size Facility Layout
Problem Journal of Fundamental Sciences 5 (2009) 79-87
[13]Tam, K.Y., “Genetic algorithms, function optimization, and facility layout design.” European
Journal of Operational Research, 1992 63, 322-346.
[14]Tam, K.Y., “A simulated annealing algorithm for allocating space to manufacturing cells.”
International Journal of Production Research,1992 30, 63-87.
[15]Tate, D.M. and A.E. Smith, “Unequal-area facility layout by genetic search.” IIE
Transactions, 1995 27, 465-472.
[16] Hong Zhang, and Jia Yuan Wang--Particle Swarm Optimization for Construction Site
Unequal-Area Layout Journal of Construction Engineering and Management, Vol. 134, No. 9,
September 2008, pp. 739-748
[17] KIM, JAE-GON; KIM, YEONG-DAE-A space partitioning method for facility layout problems
with shape constraints.by -IIE Transactions ,October 1, 1998
P. Arikaran, Dr. V. Jayabalan,& R. Senthilkumar
International Journal of Engineering (IJE), Volume (4) : Issue (1)
51
[18] ] Kochhar, J.S., Heragu, S.S. and Foster, B., "HOPE: A Genetic Algorithm for Unequal
Area Facility Layout Problem," Computers and Operations Research, 25, 583-594, 1998
[19] Alan R. McKendall Jr., Artak Hakobyan,Heuristics for the dynamic facility layout problem
with unequal-area departments European Journal of Operational Research, Volume 201, Issue
1, 16 February 2010, Pages 171-182
[20] Komarudin, Kuan Yew Wong, Applying Ant System for solving Unequal Area Facility
Layout Problems European Journal of Operational Research, Volume 202, Issue 3, 1 May
2010, Pages 730-746
[21] M. Mir, M. H. Imam, A hybrid optimization approach for layout design of unequal-area
facilities Computers & Industrial Engineering, Volume 39, Issues 1-2, February 2001, Pages
49-63
[22] Ignacio Castillo, Joakim Westerlund, Stefan Emet, Tapio Westerlund Optimization of block
layout design problems with unequal areas: A comparison of MILP and MINLP optimization
methods, Computers & Chemical Engineering, Volume 30, Issue 1, 15 November 2005, Pages
54-69,
[23] Heng Li, Peter E. D. Love, Genetic search for solving construction site-level unequal-area
facility layout problems Automation in Construction, Volume 9, Issue 2, March 2000, Pages
217-226
[24] Ming-Jaan Wang, Michael H. Hu, Meei-Yuh Ku, A solution to the unequal area facilities
layout problem by genetic algorithm Computers in Industry, Volume 56, Issue 2, February
2005, Pages 207-220
[25] M.Hasan Imam, Mustahsan Mir ,Automated layout of facilities of unequal areas Computers
& Industrial Engineering, Volume 24, Issue 3, July 1993, Pages 355-366
[26] Ignacio Castillo, Tapio Westerlund, An -accurate model for optimal unequal-area block
layout design Computers & Operations Research, Volume 32, Issue 3, March 2005, Pages
429-447
[27] J. -Y. Kim, Y. -D. Kim Graph theoretic heuristics for unequal-sized facility layout problems ,
Omega, Volume 23, Issue 4, August 1995, Pages 391-401
[28] S. Bock and K. Hoberg. Detailed layout planning for irregularly-shaped machines with
transportation
path design. Eur. J. Oper. Res., 177:693–718, 2007.
[29] L.R. Foulds. Graph Theory Applications. Springer-Verlag, New York, 1991.
[30] J.G. Kim and Y.D Kim. A space partitioning method for facility layout problems with shape
constraints. IIE Trans., 30(10):947–957, 1998.
[31] J.G. Kim and Y.D Kim. Branch and bound algorithm for locating input and output points of
departments on the block layout. J. Oper. Res. Soc., 50(5):517–525, 1999.
[32] K.Y. Lee, M.I. Rohb, and H.S. Jeong. An improved genetic algorithm for multi-floor facility
layout problems having inner structure walls and passages. Comput. Oper. Res., 32:879–899,
2005.
[33] Ibolya Jankovits; Chaomin Luo; M.F. Anjos; Anthony Vannelli, A Convex Optimisation
Framework for the Unequal-Areas Facility Layout Problem April 15, 2007, Manuscript Draft for
European Journal of Operational Research
[34] Scholz, Daniel Petrick, Anita Domschke, Wolfgang , A Slicing Tree and Tabu Search
based heuristic for the unequal area facility layout problem European Journal of Operational
Research. , 197 (August 2009,1) p-166-178
[35] Parames Chutima,J. ,Genetic Algorithm for Facility Layout Design with Unequal
Departmental Areas and Different Geometric Shape Constraints. Thammasa Itn Sc.Tech.,V
o1.6,N o.2, May-August 2001
[36] Ashwani Dhingra; Pankaj Chandra Hybrid Genetic algorithm for Multicriteria Scheduling
with sequence dependent set up time International Journal of Engineering Vol.3(5) Nov 2009 p-
520-530
[37] Science and Technology encyclopedia from internet
