Graduate School of Information, Production and Systems, Waseda University 13. Layout Design and Cellular Manufacturing DesignSoft Computing Lab.WASEDA UNIVERSITY , IPS 213. Layout Design andCellular Manufacturing Design1. Single-row Machine Layout Problem (s-MLP) 1.1 s-MLP Formulation1.2 s-MLP Representation1.3 s-MLP Evaluation1.4 s-MLP Numerical Example2.Multi-row Machine Layout Problem (m-MLP)2.1 m-MLP Formulation 2.2 m-MLP Representation2.3 m-MLP Initialization2.4 m-MLP Evaluation2.5 m-MLP Genetic Operations2.6 m-MLP Numerical Example3. m-MLP in Fuzzy Environment 3.1 Fuzzy Clearance3.2 Fuzzy m-MLP3.3 Fuzzy m-MLP Formulation and Representation3.4 Fuzzy m-MLP Feasibility3.5 Fuzzy m-MLP Evaluation3.6 Fuzzy m-MLP Numerical ExampleSoft Computing Lab.WASEDA UNIVERSITY , IPS 313. Layout Design andCellular Manufacturing Design4. Fuzzy Facility Layout Problems4.1 FLP Formulation and Fuzzy Interflow4.2 FLP Representation and Initialization4.3 FLP Genetic Operations and Constructing a Layout4.4 FLP Evaluation4.5 Fuzzy FLP Numerical Example5. Cellular Manufacturing Design5.1 Introduction to CMD5.2 Major Issues on CMD5.3 Mathematical Formulation5.4 Genetic Representation and Operations5.5 Evaluation and Overall Procedure5.6 Numerical ExamplesSoft Computing Lab.WASEDA UNIVERSITY , IPS 41. Single-row Machine Layout Problem (s-MLP)1.1 s-MLP Formulation1.2 s-MLP Representation1.3 s-MLP Evaluation1.4 s-MLP Numerical Example1. Multi-row Machine Layout Problem (m-MLP)2. m-MLP in Fuzzy Environment 3. Fuzzy Facility Layout Problems4. Cellular Manufacturing Design13. Layout Design andCellular Manufacturing DesignSoft Computing Lab.WASEDA UNIVERSITY , IPS 5Optimum arrangement of Physical Facilities such as machines or departments, is a criteria area in manufacturing environment.Design Criterion is considered as the minimizing material handling cost. Heuristic Technique is the most promising approach for solving the practical size Facility Layout Design (FLD) problems.The layout of machines in a flexible machining system is typically determined by the type of material handling devices used. The most used material handling devices are as follows:Material handling robotAutomated Guided Vehicle (AGV)Gantry robot Layout Design ProblemKusiak, A. and S. Heragu: "The facility layout problem, European Journal of Operational Research, Vol.29, pp. 229-251, 1987.1. s-MLPSoft Computing Lab.WASEDA UNIVERSITY , IPS 6Machine/Facility Layout Design (M/FLD) Problem based on Genetic Algorithms:Cohon, J., S. Hegde, and N. Martin: "Distributed genetic algorithms for the floor-plan design problem, IEEE Transactions on Computer-Aided Design, Vol.10, pp. 483-491,1991.Tam, K.: "Genetic algorithms, function optimization, facility layout design, European Journal of Operational Research, Vol. 63, pp. 322-346,1992.Tate, D. and A. Smith: "Unequal-area facility layout by genetic search, IIE Transactions, Vol. 27, pp. 465-472, 1995.Tate, D. and A. Smith: "Genetic approach to quadratic assignment problem, Computers andOperations Research, Vol. 22, pp. 73-83, 1995. Cheng, R. and M. Gen: "Genetic search for facility layout design under interflows uncertainty, Japanese Journal of Fuzzy Theory and System, Vol. 8, No. 2, pp. 335-346, 1996.Cheng, R. and M. Gen: "Genetic algorithms for multi-row machine layout problem, in Gen M. and R. Cheng: GeneticAlgorithm and Engineering Design, John Wiley & Sons, New York, NY, 19971. s-MLPSoft Computing Lab.WASEDA UNIVERSITY , IPS 7Recently, the interest in application of Genetic algorithms to facility layout design has been growing rapidly:David W. and S. Alice: "Penalty guided genetic search for reliability design optimization, Computers and Industrial Engineering, Special Issue on Genetic Algorithms, Vol.30, No. 4, pp.895-904, 1996. David W. A. E. Smith, and T. David: "Adaptive Penalty Methods for Genetic Optimization of Constrained Combinatorial Problems, INFORMS Journal on Computing, Vol. 8, No. 2, pp.173-182, 1996.Schnecke V. and O. Vornberger: "Hybrid Genetic Algorithms for Constrained Placement Problems, IEEE Transactions on Evolutionary Computation, Vol. 1, No. 4, pp. 266277, 1997.Rajasekharan, M. B. A. Peters, and T. Yang: "A Genetic Algorithm for Facility Layout Design in Flexible Manufacturing Systems, International Journal of Production Research, Vol. 36, No.1, pp. 95-110, 19981. s-MLPSoft Computing Lab.WASEDA UNIVERSITY , IPS 81. s-MLPIn order to model the single-row machine layout problem, the following assumptions are made:machines are rectangular in shape orientation of machines is knownfor example, all machines are to be oriented lengthwiseFig. 13.1 Illustration of Parameters and Decision Variables Assumption: Kusiak, A.: Intelligent Manufacturing System, Prentice-Hall, Englewood Cliffs, NJ, 1990.milVlidijljxixjbjmjbiSoft Computing Lab.WASEDA UNIVERSITY , IPS 91.1 s-MLP FormulationNotation: n isthe number of machines. fijthe frequency of trips between machines i and j. cij the handling cost per unit distance traveled between machine iand j. li the length of machine i. dij the minimum clearance between machines i and j. bithe width of machine i. xithe distance between the center of machine i and the vertical reference line lV.n i xn i j n i i d l l x xx x f ciij j i j ij inini jij ij,..., 1 , 0,..., 1 , 1 ,..., , ) (21. t s.min11 1 + + + + Soft Computing Lab.WASEDA UNIVERSITY , IPS 101.2 s-MLP RepresentationThe essential problem in s-MLP can be viewed as the sequencing problem of machines, so it can be solved in two separate steps:sequencing machinesgenerating actual layoutRepresentation:A straightforward way to encode the machine layout into a chromosome for a single-row case is to use the permutation of machines.Generally, for an n-machine problem, a chromosome vk is given as follows:where mik represents a machine which is in the ith position of kth chromosome.Genetic Operators:Here we use PMX crossover and inverse mutation.] ... [2 1knk kkm m m v Soft Computing Lab.WASEDA UNIVERSITY , IPS 111.3 s-MLP EvaluationEvaluation Function:According to the sequence and the geometric requirements of machines, we can calculate the x-axis coordinates of all machines as follows:Then we can calculate the total cost for the kth chromosome as follows:Because the layout design is a minimization problem, we must convert the objective function value of each chromosome to the fitness value, such that a fitter chromosome has a larger fitness value.The conversion is done by the following evaluation function:] ... [1 1knk kx x x + 11 1nini jkjki ij ij kx x f c fkkfv eval1) ( Soft Computing Lab.WASEDA UNIVERSITY , IPS 12Algorithm for s-MLP procedure: GA for s-MLPbegint 0;initialize encoding P(t) with permutation of machines;evaluate with permutation decoding P(t);while (not termination condition) doPMX crossover P(t) to yield C(t);inverse mutation P(t) to yield C(t);evaluate C(t);select P (t+1) from P(t) and C(t) ; t t+1; endendSoft Computing Lab.WASEDA UNIVERSITY , IPS 131.4 s-MLP Numerical ExampleThe test problem, given by Kusiak, isa six-machine layout problem. the machine size, the frequency matrix, the cost matrix, and the clearance between machines for the problem are as follows:Machine SizesMachine iDimension ( li bi ) 1 5.0 3.0 2 2.0 2.0 3 2.5 2.0 4 6.0 3.5 5 3.0 1.5 6 4.0 4.0

,`

.|0 31 12 9 28 9031 0 21 41 24 6212 21 0 14 12 219 41 14 0 72 8028 24 12 72 0 4090 62 21 80 40 0] [ijf

,`

.|0 4 8 3 3 54 0 5 3 2 48 5 0 5 5 63 3 5 0 2 43 2 5 2 0 45 4 6 4 4 0] [ijc

,`

.|0 2 1 1 1 12 0 3 1 1 21 3 0 1 1 11 1 1 0 1 11 1 1 1 0 11 2 1 1 1 0] [ijdSoft Computing Lab.WASEDA UNIVERSITY , IPS 141.4 s-MLP Numerical ExampleThe evolutionary environment of our implementation is given as follows:popSize=20, maxGen=50, pM=0.4, pC=0.2The best chromosome is listed as follows: Generation the best solution occurred:17 Cost: 19531.00The sequence of machines:[613524]Fig. 13.2 single-row layout for the example problem m6m1 m4 m2m3m5vrlhrlSoft Computing Lab.WASEDA UNIVERSITY , IPS 151.4 s-MLP Numerical ExampleEvolutionary Process0.000060.000050.000040.000030.000020.00001010 20 30 40 50Fitnessaverage fitnessbest fitnessFig. 13.3 Evolutionary process for test problemSoft Computing Lab.WASEDA UNIVERSITY , IPS 1613. Layout Design andCellular Manufacturing Design1. Single-row Machine Layout Problem 2. Multi-row Machine Layout Problem (m-MLP)2.1 m-MLP Formulation 2.2 m-MLP Representation2.3 m-MLP Initialization2.4 m-MLP Evaluation2.5 m-MLP Genetic Operations2.6 m-MLP Numerical Example1. M-MLP in Fuzzy Environment 2. Fuzzy Facility Layout Problems3. Cellular Manufacturing DesignSoft Computing Lab.WASEDA UNIVERSITY , IPS 171. m-MLPThe essential of this problem comprises three different tasks:allocate machines to rows (to determine y coordinates).find the best positions of machines within each row (to determine x coordinates).The first one is a combinatorial optimization problem. Although second one is the single-row layout problem, it is easy to see that the best solution for each row may not be good for global solution of the problem due to the existence of traffic cost among rows.Thus we cannot simply handle this problem as several single-row layout problems. Heragu, S. and A. Kusiak: "Machine layout problem in flexible manufacturing systems, Operations Research, Vol. 36, pp. 258-268, 1988.Soft Computing Lab.WASEDA UNIVERSITY , IPS 182.1 m-MLP FormulationNotation:Let the decision variable zik ben

is the number of machines.m the number of rows.fijthe frequency of trips between machines i and j.cij the handling cost per unit distance traveled between machines i and j.lithe length of machine i.l0the separation between two adjacent rows.dijthe minimum clearance between machines i and j.bi the width of machine i.xi the distance between the center of machine i and the vertical reference line lV.yi the distance between the center of machine i and the horizontal reference line lH.'otherwiserow to allocated is machine if, 0, 1 k izikmjmimkdikd0jFig. 13.4Illustration of parameters, decision variables, and reference lineslkxiykl0lVlHSoft Computing Lab.WASEDA UNIVERSITY , IPS 192.1 m-MLP FormulationThe multiple-row machine layout problem with unequal area can be formulated as a mixed-integer programming problem:( )m k n i zn i y xm i n zn i zn i z k l yn j i d l l z z x xy y x x f ciki inkikmkikikmkiij j i jk ik j ij i j inini jij ij, , 1 , , , 1 , 1 , 0, , 1 , 0 ,, , 1 ,, , 1 , 1, , 1 , ) 1 (, , 1 , , ) (21t. s.min111011 1 < + + + + Soft Computing Lab.WASEDA UNIVERSITY , IPS 202.2 m-MLP RepresentationRepresentation:Cheng, R. and M. Gen: "Genetic algorithms for multi-row machine layout problem, in Gen M. and R. Cheng: GeneticAlgorithm and Engineering Design, John Wiley & Sons, New York, 1997.For the multiple-row machine layout problem, a representation scheme can be viewed as an extended permutation representation, which contains three lists of separator/ machine symbol/ neat clearance.For n machines and two-row case, the representation is sketched as follows:}] ..., , , { }, ..., , , { , [2 12 1ni i i i i inm m m s t. requiremen rowtwo the to according part two into list the separate to position cutting the denotesand machines between clearance neat the denotesposition th the in machine represent wheresm mj m mij ij ijij ij..1 Soft Computing Lab.WASEDA UNIVERSITY , IPS 212.2 m-MLP RepresentationCalculation of x-axis Coordinates:Suppose that machines mk and mi are arranged as shown in this Figure, the net clearance and x-axis position can be calculated as follows:( )k kkkk i ikik ikiki il d xl l d x xd l21210+ + + + + + mkmi(1/2)li(1/2)dki (1/2)dki ilki(1/2)lkxkxiFig. 13.5 Neat Clearance and Decision Variablesi k dlmkikii iand machines between clearance required themachines two between separationmachine with associated clearance neat wherelVlHSoft Computing Lab.WASEDA UNIVERSITY , IPS 222.2 m-MLP RepresentationCalculation of y-axis Coordinates: to determine based on the separations of rows.The separation between rows can be predetermine according to features of the material handling systemLet us consider the two-row case.If we suppose that the position of the first row is 0, then the y-axis coordinates can be calculated as follows:lVFig. 13.6 Illustration of y-axis coordinatesmi1mi2mi3mi4mi5mi6llH'row second the in is ifrow first the in is ifiiim lmy,, 00Soft Computing Lab.WASEDA UNIVERSITY , IPS 232.3 m-MLP Initializationprocedure: Initializationbegin i 0 while (i popSize) do generate separator randomly; generate machine list randomly; check the feasibility of a chromosome; if the chromosome is feasible then generate neat clearance list randomly; i i + 1; elsedelete selected list; endendInitialization of Multi-row Machine Layout:Soft Computing Lab.WASEDA UNIVERSITY , IPS 242.3 m-MLP Initializationprocedure: Machine Permutationbegin i 0; 0 {m1,

m2, , mn}; P ; while (i n) do pick up a machine m from 0 randomly; P P m ; 0 0m ; i i + 1;endoutput permutation list P ;endMachine Permutation: Let 0 denote the set of available machines and P denote the list of machine permutation, then the machine permutation is randomly generated as follows:Soft Computing Lab.WASEDA UNIVERSITY , IPS 252.3 m-MLP InitializationFeasibility Checking:Because of the existence of available working area restriction, we need to check whether the randomly generated machines permutation is feasible. Suppose the machine sequence for a row, say row 1, is as follows:[m1, m2,..., mk]Let L denote the restriction of working area and S1 denote the necessary space required, which is determined as follows:Then we compare S1 with L, the restriction of working area, if it is less than or equal to L, the randomly generated permutation is feasible; otherwise it is infeasible.lklkili ikiid d d l S0 10111 ,11+ + + +Soft Computing Lab.WASEDA UNIVERSITY , IPS 262.3 m-MLP InitializationNeat Clearance:Because of the existence of allowable space constraints, the neat clearance (real number) is randomly generated within an allocable region.Let L denote available space and L denote the length restriction of the working area. Then the initial available space can be calculated as follows:) ( 20 10111 ,1hkhnihi iniid d d l L L + + + +Soft Computing Lab.WASEDA UNIVERSITY , IPS 272.3 m-MLP InitializationLet denote the list of neat clearance for the row. Theoverall procedure is shown below:procedure: Neat Clearance Listbegin i 0; ; calculate initial available space L; while (i n) do pick up a neat clearance i within (0, L) randomly; i ; L L - i ; i i + 2; end output neat clearance list ;endSoft Computing Lab.WASEDA UNIVERSITY , IPS 282.4 m-MLP EvaluationEvaluation Function:In general, two kinds of illegal solutions may occur in the machine problem:1. overlapping of machines2. violation of working areaBecause the x-axis coordinates are represented as neat clearance, overlapping illegality will never occur in this encoding scheme.The violation of the working area can be measured in the following manner:For a given chromosome vk,letLk1 andLk2 be the necessary working areas required by machines which are arranged in the first row and second row, respectively, andlet Lku = max{Lk1, Lk2}thus the penalty coefficientis calculated as follows:' otherwiseif,0 , 0L LL LukukkSoft Computing Lab.WASEDA UNIVERSITY , IPS 292.4 m-MLP EvaluationThus the fitness function for chromosome vk is given as follows: where p is the big positive penalty value.fk the totaltravel cost among machines for chromosome vk ,which is determined as follows: popSize kP fv evalk kk..., , 2 , 1 ,1) ( +( ) + + 11 1nini jkjkikjki ij ij ky y x x f c fSoft Computing Lab.WASEDA UNIVERSITY , IPS 302.5 m-MLP Genetic OperationsCrossover: The basic element of our crossover procedure consists of three parts,a random way to determine separatoran ordinary PMX (partially mapped crossover) to manipulate machine permutation listan arithmetical crossover to manipulate neat clearance listprocedure: Crossoverbegin i 0; while (i popSize * pC) doselect two chromosomes randomly; generate a new separator; generate a new machine permutation with ordinary PMX; generate a new neat clearance list with arithmetical crossover;if the offspring is feasible then // check the feasibility i i + 1; else delete selected list;endendSoft Computing Lab.WASEDA UNIVERSITY , IPS 312.5 m-MLP Genetic Operations - CrossoverNew separator: The procedure for generating new separator contains two steps:determine the upper and lower bounds for a closed intervalselect an integer within the interval randomlyFor example, we have two parents shown as follows:p1=[{s1}, {m11, , mn1}, {11,, n1}]p2=[{s2}, {m12, , mn2}, {12,, n2}]The upper and lower bounds can be directly calculated as follows:sU= max{s1, s2} sL= min{s1, s2} Then we can make a closed interval with sU and sL as [sU, sL].The new separator is a randomly generated integer within this interval.Soft Computing Lab.WASEDA UNIVERSITY , IPS 322.5 m-MLP Genetic Operations - CrossoverNew neat clearance listSuppose there are two neat clearance lists:The new neat clearance is determined as follows:where1 and 2 are the randomly generated real number within theopen interval (0,1).The difference comparing with conventional one is that we require following relation holds for these two parameters: 1+ 2 , then we can enlarge the search space greatly, which is independent of initial search space. If we take conventional approach, the generated neat clearances between machines will be gradually decreasing along with the evolutionary process. In this case, search space is highly depended on the initial solution space.{11, 21, , n1}{12, 22, , n2}]i =1i1+ 2i2, i = 1, 2, , n1, 2 (0, 1)Soft Computing Lab.WASEDA UNIVERSITY , IPS 332.5 m-MLP Genetic Operations - MutationMutation: The mutation operator is designed with neighborhood technique to try to find an improved offspring.Firstly we give the definition of neighborhood for a given chromosome.Suppose the neat clearance list for a given chromosome is: {1, 2, , i , , n}And the ith gene i is selected for mutation.Let r be a given integer and then we divide the selected neat clearance i /r into 2r equal parts as follows: r jrrijijiii2 ..., , 3 , 2 ,11+ Soft Computing Lab.WASEDA UNIVERSITY , IPS 342.5 m-MLP Genetic Operations - MutationAfter getting r neat clearances, the set of neat clearances is listed below:The set of chromosome formed with above set of neat clearance lists together with the separator list and machine permutation list of the given chromosome are regarded as the neighborhood of the given chromosome.A chromosome is said to be 2r-optimum, if it is better than any others in the neighborhood. } ..., , ..., , , {} ..., , ..., , , {} ..., , ..., , , {1 122 112 1nkin in i Soft Computing Lab.WASEDA UNIVERSITY , IPS 352.5 m-MLP Genetic Operations - MutationThe proposed mutation is given as follows:procedure: Mutationbegin give an integer r; t 0; while (t popSize * pM) do pick up a gene t randomly; generate 2r neighbors of t; generate all neighbors of t; select the best neighbor as the offspring; t t + 1; endendSoft Computing Lab.WASEDA UNIVERSITY , IPS 36Algorithm for m-MLP procedure: GA for m-MLPbegint 0;initialize encoding P(t) with separator, machines list and neat clearance list;evaluate with permutation decoding P(t); // total cost and penalty to illegality.while (not termination condition) doPMX crossover to machine list, arithmetical crossover to neat clearance list and separator determined randomly, all for P(t) to yield C(t);neighbor search technique to mutation P(t) to yield C(t);evaluate C(t); // total cost and penalty to illegality.select P (t+1) from P(t) and C(t) ; t t+1; endendSoft Computing Lab.WASEDA UNIVERSITY , IPS 372.6 m-MLP Numerical ExampleThe test problem consists ofsix machines.The machine size, the frequency matrix, the cost matrix, and clearance between machines for the problem are as follows:Machine SizesMachine iDimension ( li bi ) 1 5.0 3.0 2 2.0 2.0 3 2.5 2.0 4 6.0 3.5 5 3.0 1.5 6 4.0 4.0

,`

.|0 31 12 9 28 9031 0 21 41 24 6212 21 0 14 12 219 41 14 0 72 8028 24 12 72 0 4090 62 21 80 40 0] [ijf

,`

.|0 4 8 3 3 54 0 5 3 2 48 5 0 5 5 63 3 5 0 2 43 2 5 2 0 45 4 6 4 4 0] [ijc

,`

.|2222220 2 1 1 1 12 0 3 1 1 21 3 0 1 1 11 1 1 0 1 11 1 1 1 0 11 2 1 1 1 0222222] [ijdSoft Computing Lab.WASEDA UNIVERSITY , IPS 382.6 m-MLP Numerical ExampleSeparation between two rows is 8.Width restriction of the working area is 22. Evolutionary environment of our implementation is given as follows:PopSize=20, maxGen=500, pM=0.4, pC=0.4, r =10The best chromosome is listed as follows:Generation the best solution occurred: 48Fitness: 0.5217Machines in row 1: [ 2 3 1 6 ] Machines in row 2: [ 4 5 ] Machine positions in row 1: [ 4.497.7412.4917.99 ] Machine positions in row 2: [ 5.0012.5 ] lVlHm4m5m2m3m1m6Fig. 13.7 Multiple-row layout for the example problemSoft Computing Lab.WASEDA UNIVERSITY , IPS 392.6 m-MLP Numerical ExampleEvolutionary Process0. 60. 50. 40. 30. 20. 10100 200 300 400 500FitnessEvolutionary processaverage fitnessbest fitnessFig. 13.8Evolutionary process for the example problemSoft Computing Lab.WASEDA UNIVERSITY , IPS 4013. Layout Design andCellular Manufacturing Design1. Single-row Machine Layout Problem (s-MLP)2. Multi-row Machine Layout Problem (m-MLP)3. m-MLP in Fuzzy Environment 3.1 Fuzzy Clearance3.2 Fuzzy m-MLP3.3 Fuzzy m-MLP Formulation and Representation3.4 Fuzzy m-MLP Feasibility3.5 Fuzzy m-MLP Evaluation3.6 Fuzzy m-MLP Numerical Example1. Fuzzy Facility Layout Problems2. Cellular Manufacturing DesignSoft Computing Lab.WASEDA UNIVERSITY , IPS 413. m-MLP in Fuzzy Environment We formulate fuzzy multi-row machine layout problem where the clearance between two adjacent machines is given as a fuzzy set.The membership function of the fuzzy clearance corresponds to the grade of satisfaction of separate distance. The objective functionTo maximize the minimum grade ofsatisfaction over machines and meanwhile minimize the total travel cost among machines.Soft Computing Lab.WASEDA UNIVERSITY , IPS 423.1 Fuzzy ClearanceFuzzy Clearance:The clearance dij between machines i and j is shown in this Figure:mimjvrllidijljxixjFig. 13.9 Clearance between Machinesij(xi,, xj ) the membership function of fuzzy clearance between two adjacent machine i and machine j. This represents the grade of satisfaction of the separated distance.dijl the least clearance for machines i andj.dijs the satisfactory clearance for machines i andj.Soft Computing Lab.WASEDA UNIVERSITY , IPS 433.1 Fuzzy Clearance - Membership FunctionThe membership function is defined as:( )( ) ( )( )' + lij j isij j ilijlijsijlij j i j isij j ij i ijx xx xd dd l l x xx xx x , 0,2 / 1, 1,( ) ( )lij j ilijsij j isijd l l d l l + + 2 / 1 , 2 / 1 Fig. 13.10 Membership function1ijxlijsijSoft Computing Lab.WASEDA UNIVERSITY , IPS 443.2 Fuzzy m-MLPThe essential of this problem comprises three different tasks:find a better allocation of machines to rowsfind a better sequence of machines within each rowfind a better position (x and y coordinates) for each machinesBecause the separation between rows can be predetermined according to the feature of material handling system, we can calculate y-axis coordinators based on the separations of rows.Instead of computing the y-axis directly, we treat it as how to allocate machines among rows. So we do not need consider the fuzzy clearance between machines.Soft Computing Lab.WASEDA UNIVERSITY , IPS 453.3 Fuzzy m-MLP Formulation and RepresentationFor multi-row case, the relation of working area restriction, machines and reference lines is illustrated in the right-side figure.Notation:li the length of machine i.bi the width of machine i.ijh the grade of satisfaction of horizontal separation between machines i and j.ijv the grade of satisfaction of vertical separation between machines i and j. di0lh the least clearance between machine i and right vertical reference line.di0lv the least clearance between machine i and upper horizontal reference line.W the width of working area.L the length of working area.mjmimkdijvdikhdj0lhdi0lhdi0lvlkbkdk0lvFig. 13.11Clearance between MachinesSoft Computing Lab.WASEDA UNIVERSITY , IPS 463.3 Fuzzy m-MLP Formulation and RepresentationWith the notation of fuzzy clearance, the multiple row machine layout problem with unequal area can be formulated as follows:( )) 10 ( , , 1 , 1 0) 9 ( , , 1 , 0 ,) 8 ( , , 1 2 / 1) 7 ( , , 1 2 / 1) 6 ( , , 1 , , 0 ) 1 () 5 ( , , 1 , , ) 1 ( ) , () 4 ( , , 1 , , ) , ( t. s.) 3 ( max) 2 ( max) 1 ( min001 1n j i or zn i y xn i W d b yn i L d l xn j i z zn j i z M x xn j i Mz x xy y x x f ciji ilvi i ilhi i iij ijvT ij j ivijhT ij j ihijvThTj i j ininjij ij + + + + + + + Soft Computing Lab.WASEDA UNIVERSITY , IPS 473.3 Fuzzy m-MLP Formulation and RepresentationThe objective function (1) is to minimize the total travel cost among machines.The objective functions (2) and (3) are to maximize the minimum grade of satisfaction over machines.Constraint (6) ensures that only one of the two constraints (4) and (5) hold.Constraints (7)and (8) ensure that machines are arranged within the restricted working area.Constraint (9) is a non-negativity constraint.Soft Computing Lab.WASEDA UNIVERSITY , IPS 483.3 Fuzzy m-MLP Formulation and RepresentationRepresentation and Calculation of x-axisFor n machines and two-row case, the representation is sketched as follows:Thenet clearance and x-axis position can be calculated as follows:}] ..., , , { }, ..., , , { }, [{2 1 2 1in i i in i im m m k ( )k klk k k i ilki k ilki ki il d x l l d x x d l21,21,0+ + + + + + i k dlmkikii iand machines between clearance required themachines two between separationmachine with associated clearance neat wheret. requiremen rowtowthe to accordingpart two into list the separate to position cutting the denotesand machines between clearance neat the denotesposition th the in machine represent wherekm mj m mij ij ijij ij..1 Soft Computing Lab.WASEDA UNIVERSITY , IPS 493.4 Fuzzy m-MLP FeasibilityFeasibility Checking:Because of the existence of available working area restriction, we need to check whether the randomly generated machines permutation is feasible. Suppose the machine sequence for a row, say row 1, is as follows: [m1, m2,..., mk]Let L denote the restriction of working area and S1 denote the necessary space required, which is determined as follows:Then we compare S1 with L, the restriction of working area, if it is less than or equal to L, the randomly generated permutation is feasible; otherwise it is infeasible.lklkili ikiid d d l S0 10111 ,11+ + + +Soft Computing Lab.WASEDA UNIVERSITY , IPS 503.5 Fuzzy m-MLP EvaluationEvaluation Function:The fitness function for chromosome vt is given as follows: where0 is an initial estimation of the bestobjective function value.ft is the travel cost among machines for chromosome vt, which is determined as follows: ( ) + + 11 1nini jtjtitjti ij ij ty y x x f c f11) , () (2 1201 +++ w wnx xwfw v evaltjti ijttSoft Computing Lab.WASEDA UNIVERSITY , IPS 513.6 Fuzzym-MLPNumerical ExampleTest problem is a 6-machine 2-row layout problem given by kusiak. the fuzzy clearances are considered as follows:[ ]

,`

.|0 . 1 0 . 0 0 . 2 8 . 0 6 . 2 5 . 1 0 . 2 0 . 10 . 1 0 . 2 0 . 0 6 . 1 2 . 2 5 . 0 0 . 1 0 . 10 . 1 8 . 0 6 . 1 0 . 0 4 . 0 5 . 1 0 . 1 0 . 10 . 1 6 . 2 2 . 2 4 . 0 0 . 0 0 . 1 0 . 1 0 . 10 . 1 5 . 1 5 . 0 5 . 1 0 . 1 0 . 0 6 . 0 0 . 10 . 1 0 . 2 0 . 1 0 . 1 0 . 1 6 . 0 0 . 0 0 . 16543216 5 4 3 2 10 0sijdr l[ ]

,`

.|0 . 2 0 . 0 0 . 4 2 . 2 6 . 5 8 . 1 0 . 7 0 . 20 . 2 0 . 4 0 . 0 0 . 5 8 . 2 0 . 6 2 . 6 0 . 20 . 2 2 . 2 0 . 5 0 . 0 1 . 1 0 . 2 0 . 4 0 . 20 . 2 6 . 5 8 . 2 1 . 1 0 . 0 0 . 4 0 . 5 0 . 20 . 2 8 . 1 0 . 6 0 . 2 0 . 4 0 . 0 8 . 2 0 . 20 . 2 0 . 7 2 . 6 0 . 4 0 . 5 8 . 2 0 . 0 0 . 26543216 5 4 3 2 10 0lijdr lMachine SizesMachine iDimension ( li bi ) 1 5.0 3.0 2 2.0 2.0 3 2.5 2.0 4 6.0 3.5 5 3.0 1.5 6 4.0 4.0Soft Computing Lab.WASEDA UNIVERSITY , IPS 523.6 Fuzzym-MLPNumerical ExampleThe evolutionary environment of our implementation is givenas follows:The restriction of working area is 22. The separation between rows is 8. We have got the best chromosome in the 38th generation listed as below:Sequence :264315x position:( 3.11,7.93,14.14,3.14,8.06,13.64 )separator:3cost:19660.56:0.875 pCpMpopSize maxGenw1w2 0.40.4 30200 0.5 0.5Soft Computing Lab.WASEDA UNIVERSITY , IPS 533.6 Fuzzym-MLPNumerical Examplem2m5m3m1m6 m4vrlhrlFig. 13.12 Multiple-row Layout for the Test ProblemFig. 13.13 Evaluation Process for Test Problembest fitnessaverage fitnessFitness0.60.40.2050 100 150 200Evolutionary processSoft Computing Lab.WASEDA UNIVERSITY , IPS 5413. Layout Design andCellular Manufacturing Design1. Single-row Machine Layout Problem (s-MLP)2. Multi-row Machine Layout Problem (m-MLP)3. M-MLP in Fuzzy Environment 4. Fuzzy Facility Layout Problems4.1 FLP Formulation and Fuzzy Interflow4.2 FLP Representation and Initialization4.3 FLP Genetic Operations and Constructing a Layout4.4 FLP Evaluation4.5 Fuzzy FLP Numerical Example1. Cellular Manufacturing DesignSoft Computing Lab.WASEDA UNIVERSITY , IPS 554. Fuzzy Facility Layout ProblemsThe flows among facilities may change from period to period due to the dynamic nature of businesses, growth, and demand fluctuation, and product mix.Unfortunately, changes in product mix, machine breakdowns, seasonal fluctuations, and demand are uncertain in nature.Under these circumstances, designers tend to obtain a satisfactory layout rather than an optimal layout.Kusiak, A. and S. Heragu: "The facility layout problem, EuropeanJournal of Operational Research, Vol.29, pp. 229-251, 1987.Rosenblatt, M.: The dynamics of plant layout, Management Science, Vol.32, pp. 76-86, 1986.Soft Computing Lab.WASEDA UNIVERSITY , IPS 564. Fuzzy Facility Layout ProblemsThere are two approaches proposed for modeling such uncertainties:1. flexible or probabilistic approach we must specify probability distribution for material flow2. robustness approach we must provide several demand scenarios and the optimal solutions for each scenario.Giving an exact probability distribution or giving some precise demand scenarios is as difficult as the optimal design of the facility layout itself.Cheng, R. and M. Gen: "Genetic search for facility layout design under interflows uncertainty, Japanese Journal of Fuzzy Theory and System,Vol. 8, No. 2, pp. 335-346, 1996.Soft Computing Lab.WASEDA UNIVERSITY , IPS 574.1 FLP Formulation and Fuzzy InterflowFormulation:There is a set of m facilities, denoted by {Mi}, i=1, 2, , m.Each facility is restricted to be rectangular and characterized by a triple (Ai, li, ui).Ai : the area of the facility li : the upper bound on the aspect ratioui : the lower bound on the aspect ratioRelationship among the height, width, and aspect ratio for a facility is like follows:A facility layout for given m facilities consists of a bounding rectangle, R, partitioned by horizontal and vertical line segments into m nonoverlapping rectangular regions, denoted by {ri}, i=1, 2, , m. Each region ri, with width xi and height yi, must be large enough to accommodate its facility Mi.m ..., , , i , uwhlm ..., , , i , A h wiiiii i i2 12 1 iwihih iw ihiwiAyxxAyi( )2 1/i i il / A w ( )2 1/i i iu / A wFig. 13.14 relationship between the height, width, and aspect ration for a facilitySoft Computing Lab.WASEDA UNIVERSITY , IPS 584.1 FLP Formulation and Fuzzy InterflowUncertainty of material flows among facilities can be represented as convex fuzzy number, which is called as fuzzy interflow.Here, a trapezoidal fuzzy number (TrFN) is used to represent to fuzzy interflow: A TrFN can be defined by a quadruple (a, b, c, d) and is shown as follows:Since a fuzzy number represents many possible real numbers that have different membership values, it is not easy to compare the final ratings to determine which alternative are preferred.a b c dx(x) Fuzzy Interflow:Kaufmann, A. and M. Gupta: Fuzzy Mathematical Models in Engineering and Management Science, North-Holland, Amsterdam, 1988.Soft Computing Lab.WASEDA UNIVERSITY , IPS 594.1 FLP Formulation and Fuzzy InterflowLee and Lis approach suggests the use of generalized mean and standard deviation based on the probability measures of fuzzy events to rank fuzzy number.When fuzzy number M is a TrFN, the generalized mean value with a uniform density is calculated as follows:The standard deviation is defined as follows:) ( 3)~(2 2 2 2d c b acd ab d c b aM m+ + + + + 24 3 43 34 3 4)~( ) (214 3 121) (3112 3 41)~( M m d c b ac d c dc db ca ab ba bM]]]

+ + ]]]]

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.|+ +

+

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.|+ Lee, E. S. and R. Li, Comparison of fuzzy numbers based on the probability measure of fuzzy events, Operations Research, Vol. 15, pp. 887-896, 1988Soft Computing Lab.WASEDA UNIVERSITY , IPS 604.2 FLP Representation and InitializationRepresentation:A facility layout can be represented as a slicing structure constructed by recursively partitioning a R.+: operator of a horizontal cut, *: operator of a vertical cutSlicing structures comprising m given facilities (called operand) can be represented by slicing tree or polish expressions over the alphabet set = {1, 2, , m, +, *}.An example of a slicing structure is shown as follows:++*13244312*++431 2Fig. 13.15 A slicing structure and its representationsSoft Computing Lab.WASEDA UNIVERSITY , IPS 614.2 FLP Representation and Initialization** +1 2 3 +*4 561 234 56LayoutSlicing TreePossible prefixes:11 21 2 *1 2 * 31 2 * 3 41 2 * 3 4 51 2 * 3 4 5 61 2 * 3 4 5 6 +1 2 * 3 4 5 6 + +1 2 * 3 4 5 6 + + *******Chromosome:1 2 * 3 4 5 6 + + * *where + : horizontal cut * : vertical cutSoft Computing Lab.WASEDA UNIVERSITY , IPS 624.2 FLP Representation and InitializationInitialization:Guided random approach is proposed to initialize genetic system.It constructs a chromosome from left to right by picking one element at a time from the set until a complete chromosome is formed.Because the random permutation of operands and operators may yield an illegal chromosome, we must check its legality at each random picking.Concept of Prefix for a chromosome:For a given chromosome with a total size of (2m-1), a prefix is a partial expressing containing the first I elements of the chromosome with same order as they are in the chromosome:Soft Computing Lab.WASEDA UNIVERSITY , IPS 634.2 FLP Representation and InitializationProposition 1:For a given chromosome containing m operands and (m-1) operators, if the equation N(i) M(i) + 1, then the chromosome is legal polish expression.Corollary:For a given prefix , if it does not meet the condition of the above equation, it is impossible to develop a legal polish expressionfrom the prefix with a left-to-right generation procedure.where denote the set of all possible prefixes for a given chromosome.For a prefix , N( ) denote the total number of operands the prefix contains.M( ) the total number of operators the prefix containsSoft Computing Lab.WASEDA UNIVERSITY , IPS 644.2 FLP Representation and Initializationprocedure: Guided Random Initializationbegin1{*, +}; while (i popSize) do 0{1, 2,,m}; i ; select a random element 1 from 0; 0 0 1; i i 1;j1; while (j 2m-1) do if N(i)=M(i)+1 then select j from 0; else select j from 0 1; 0 0 j; i i j;jj+1; end ii+1;endendoperand = operator+1only operators can be bothYesNoSoft Computing Lab.WASEDA UNIVERSITY , IPS 654.3 FLP Genetic Operations and Constructing a LayoutGenetic Operations:Crossover:1 2 * 3 4 5 6 + + * *1 2 + 3 4 5 6 * * + *3 6 1 + 2 5 * * * + 4parent1:parent2:offspring:: Obtaining operands: Obtaining operatorsCohoon J., S. Hegde, and N. Martin: "Distributed genetic algorithms for the floor-plan design problem, IEEE Transactions on Computer-Aided Design,Vol. 10, pp. 483-491, 1991.Soft Computing Lab.WASEDA UNIVERSITY , IPS 664.3 FLP Genetic Operations and Constructing a Layout**+1 2 3 +*4 561 234 56+3*6 1+ *25 4*123546++ *1 2 3 **4 5612345 6(a) parent 1: (12*345*6++*) (a) parent 2: (361+254***+)(a) offspring 1: (12+345*6**+)Layouts after crossoverSoft Computing Lab.WASEDA UNIVERSITY , IPS 674.3 FLP Genetic Operations and Constructing a LayoutMutation:1 2 * 3 4 5 6 + + * *parent 1:offspring 1:1 2 * 3 4 5 6 + * + *1 2 * 3 4 5 6 + + * *parent 1:offspring 2:1 2 * 5 4 3 6 + + * *1 2 * 3 4 5 6 + + * *parent 1:offspring 1:1 2 + 3 4 5 6 + + * *(a) Swapping Mutation(b) Inverting Mutation(c) Altering MutationSoft Computing Lab.WASEDA UNIVERSITY , IPS 684.3 FLP Genetic Operations and Constructing a Layout** +1 2 3 +*4 561 234 56parent: (1 2 * 3 4 5 * 6 + + *)(a) After swapping : (1 2 * 3 4 5 * 6 + * +)1 234 56+**2 3 +*4 561(b) After inverting : (1 2 * 5 4 3 * 6 + + *) (c) After altering : (1 2 + 3 4 5 * 6 + + *)** +2 5 +*4 3611 254 361234 56*++2 3 +*4 561Layouts after MutationSoft Computing Lab.WASEDA UNIVERSITY , IPS 694.3 FLP Genetic Operations and Constructing a LayoutConstructing a Layout:Proposition 2:If a cut-pointi is the first position, scanning from right to left ranging from 2m-1 to 1, which lets the equation ni( ) = mi ( ) then at cut-point i the slicing tree can be separated into1. a left subtree containing the elements from 1 to i -1 of the given slicing tree2. a right subtree containing elements from i to 2m-2.Each subtree is a legal slicing tree.whereis a slicing tree with total size 2m-1.ian arbitrary cut-point in the tree (1 i 2m-2). ni( ) total number of operands contained in the right part from the cut-point to the right most part of the tree.mi( ) total number of operators contained in the same right part of the tree.Soft Computing Lab.WASEDA UNIVERSITY , IPS 704.3 FLP Genetic Operations and Constructing a Layoutprocedure: Construct (tree, length)begin i length; while (i > 0) doseparate tree to two subtrees; calculate xir, xil, yiu, yib;if the right subtree is a simple tree then calculate xir, xil, yiu, yib;i i - 2; if neighbor left tree is a simple tree then calculate xir, xil, yiu, yib; i i - 2;else construct (lefttree, length); i i length; end else construct (righttree, length); i i length; if neighbor left tree is a simple tree thencalculate xir, xil, yiu, yib; i i - 2;else construct (lefttree, length);i i length; endendendcalculate xi and yi using xir, xil, yiu, yib;endThe set of possible dimensions (xi , yi ) of region ri accommodating Mi can be determined along with the recursively separating processusing area requirement informationThe set of possible dimensions (xi , yi ) of region ri accommodating Mi can be determined along with the recursively separating processusing area requirement informationwhere xi = xir - xil,i = 1, 2, , m yi = yiu - yib,i = 1, 2, , mxir and xil denote the left andright boundary of region. yiu and yib denote the upper andbottom boundary of region.Soft Computing Lab.WASEDA UNIVERSITY , IPS 714.3 FLP Genetic Operations and Constructing a Layout1 234 56cut-point(1 2 * 3 4 5 * 6 + * +)n3 = m3 = 4+**2 3 +*4 561(1 2 * ) n1 = m1 = 4(3 4 5 * 6 + * )n1 = m1 = 4* *2 3 +*4 561

(4 5 * 6 +)n3 = m3 = 41 23+*4 56 (4 5 *)n1 = m1 = 41 236*4 5Constructing a layoutSoft Computing Lab.WASEDA UNIVERSITY , IPS 724.4 FLP EvaluationEvaluation:We evaluate each chromosome by the following function in which a penalty approach is adopted to handle violation for it. where m(Ck(vk)) denotes the generalized mean value for the total cost,

kthe total number of facilities which violate the aspect ratio constraints within the kth chromosome.P large penalty value.cij (vk) a trapezoidal fuzzy number to denote the fuzzy interflow between the facilities i and j. dij (vk)a real number to denote the Manhattan distance between centers of each pair of facilities i and j.P C mevalk k kk +)) (~(1) (vv mimjk ij k ij k kd c C1 1) ( ) (~) (~v v vSoft Computing Lab.WASEDA UNIVERSITY , IPS 734.5 Fuzzy FLP Numerical ExampleThe test problem contains 15 facilities and fuzzy interflow is represented as a trapezoidal fuzzy number.wherea is the optimistic estimation on material flow (the best case) d the pessimistic estimation (the worst case) b one average estimation (the near best case) c the other average estimation (the near worst case)a b c dSoft Computing Lab.WASEDA UNIVERSITY , IPS 744.5 Fuzzy FLP Numerical ExampleEvolutionary Environment of Our Experiment Population size (popSize)40 Crossover probability (pC) 0.4 Mutation Probability ( pM) 0.4 Maximum generation (maxGen) 200 Penalty value 5000ParametersSoft Computing Lab.WASEDA UNIVERSITY , IPS 754.5 Fuzzy FLP Numerical Example 1 1000.7 1 2 801 1 3 500.7 1.3 4 600.5 0.8 5 1200.9 1 6 400.6 1 7 200.7 1.4 8 401 1 9 1500.8 1.110 1200.5 1.511 500.7 1.112 100.8 1.213 200.95 1.514 300.75 1.2515 500.9 1.1Aspect RatioFacilityIdentificationAreaLower Bound Upper BoundTable. 13.1 Geometric Constraints of FacilitiesSoft Computing Lab.WASEDA UNIVERSITY , IPS 764.5 Fuzzy FLP Numerical ExampleTable. 13.2Fuzzy Interflow Among FacilitiesSoft Computing Lab.WASEDA UNIVERSITY , IPS 774.5 Fuzzy FLP Numerical ExampleTable. 13.3Fuzzy Interflow Among FacilitiesSoft Computing Lab.WASEDA UNIVERSITY , IPS 784.5 Fuzzy FLP Numerical ExampleTable. 13.4 Fuzzy Interflow Among FacilitiesSoft Computing Lab.WASEDA UNIVERSITY , IPS 794.5 Fuzzy FLP Numerical Example13141115121012345 6897+**+++ +**+*++*131411151210123456897Fig. 13. 16 Layout for the Best Chromosome Fig. 13. 17 Tree Representation Best chromosome: (13 1 + 2 6 8 3 + + 15 11 14 9 * + + * * 4 7 * 5 10 + 12 * * + *)The fuzzy cost measure of the best layout is C = (2946.91, 5841.40, 9561.81, 12613.53).~Soft Computing Lab.WASEDA UNIVERSITY , IPS 804.5 Fuzzy FLP Numerical Example We can use possibility theory and fuzzy integrals to interpret the fuzzy performance.Possibility that the layout will have total cost of 4394 units is 0.5The fuzzy integral of 4394 is 0.13which means that the fuzzy expectation of the layoutyielding total cost less than or equal to 4394 is 0.13The fuzzy integral of 9562 is 0.910.52947 4394 5841 9562 126141Possibility, when C = 4394~ Sax xx xx Id ) (d ) () (~Fuzzy Integral :Soft Computing Lab.WASEDA UNIVERSITY , IPS 814.5 Fuzzy FLP Numerical ExampleEvolutionary ProcessSoft Computing Lab.WASEDA UNIVERSITY , IPS 824.5 Fuzzy FLP Numerical ExampleAccording to the four cases, we can make four equivalent nonfuzzy problems using the same fuzzy data and solved by the proposed algorithms Best : (5 14 3 * 9 * * 15 10 + 12 13 * 1 * * 6 7 8 11 + * * 4 * 2 * + +) Near best:(6 7 + 8 * 9 1 2 + * * 5 15 11 13 * 10 3 * 14 * 4 12 * * + * * +)Near worst:(9 15 * 14 * 4 13 1 * * * 10 12 6 * + 11 3 8 7 + + 5 * 2 * * * +)Worst : (7 3 * 14 * 4 * 11 6 * 2 * + 12 * 9 13 8 * * + 5 * 10 15 1 + + *)Table. 13.5 Comparative ResultsSolutionsabcdFuzzy249758419562 12614 Best case279560689953 13215Near best case289550839625 12922Near worst case297158699581 12685Worst case3012625710024 13134Soft Computing Lab.WASEDA UNIVERSITY , IPS 834.5 Fuzzy FLP Numerical ExampleFig. 13.18 Layout and tree representation for the best case.Fig. 13.19 Layout and tree representation for the near-best case.Soft Computing Lab.WASEDA UNIVERSITY , IPS 844.5 Fuzzy FLP Numerical ExampleFig. 13.20Layout and tree representation for the near-worst case.Fig. 13.21 Layout and tree representation for the worst case.Soft Computing Lab.WASEDA UNIVERSITY , IPS 854.5 Fuzzy FLP Numerical ExampleFig. 13.22 Relative error with respect to the fuzzy solutionSoft Computing Lab.WASEDA UNIVERSITY , IPS 86ConclusionThe layout problem of machines is critical to designing an efficient flexible machining system. Usually, machine layout problem is treated without the consideration of imprecise data.We have discussed the conception of fuzzy clearance into multi-row machine layout problem and formulated fuzzy multi-row layout problembased on this concept.Genetic algorithms are applied to solve the fuzzy multi-row layout problem.Preliminary computational experiments demonstrated that genetic algorithms and fuzzy approach can be a promising way for multiple machine layout problems.Soft Computing Lab.WASEDA UNIVERSITY , IPS 87ConclusionFrom the comparative results we know thatFor an equivalent nonfuzzy case, we can obtained a layout will yield large costs than the solution obtained by the fuzzy approach.That is, the nonfuzzy approach can obtain a layout suitable for its considered case, while the fuzzy approach can get a reasonable solution suitable for all cases ranging from the best case to the worst case.Soft Computing Lab.WASEDA UNIVERSITY , IPS 8813. Layout Design andCellular Manufacturing Design1. Single-row Machine Layout Problem (s-MLP)2. Multi-row Machine Layout Problem (m-MLP)3. M-MLP in Fuzzy Environment 4. Fuzzy Facility Layout Problems5. Cellular Manufacturing Design5.1 Introduction to CMD5.2 Major Issues on CMD5.3 Mathematical Formulation5.4 Genetic Representation and Operations5.5 Evaluation and Overall Procedure5.6 Numerical ExamplesSoft Computing Lab.WASEDA UNIVERSITY , IPS 895.1 Introduction to CMD - Common configurations Common Configurations:Four common configurations of CMD when designingmost of manufacturing systems for a type of facility organization:1. Product Layout (Product-focused Line)2. Process Layout (Process-focused Job Shop)3. Group Technology (Cellular Manufacturing)4. Fixed Position It is used for large products such as ships, buildings, and airplanes because the size of the product makes it impractical to move the product between processing operations.All parts and processes, such as welding equipment, arebrought to the product.Soft Computing Lab.WASEDA UNIVERSITY , IPS 905.1 Introduction to CMD - Common configurations 1 10 100 0 10001010011000Product Layout(Product-Focused Line)Process Layout(Process-focused Job Shop)Number of part typesParts per hourFig. 13.23 Product Demand Volume versus Variety of Products or Parts A. Standridge: Modeling and Analysis of manufacturing Systems, John Wiley & Sons, New York, 1993Group Technology(Cellular Manufacturing)Soft Computing Lab.WASEDA UNIVERSITY , IPS 915.1 Introduction to CMD - Common configurations Product Layout (Product-focused Line):Machines are oriented such that the product flows from the first machine to second, from the second to the third and so on down the line.Raw material enters the front of the line. Upon completing processing at the last machine, the raw material has been converted into a finished product.Advantages of the product layouts are very low throughput time, low work-in-process inventories etc.T M T D G: machine: flow of materials or parts: departmentT: turning,M:milling,D: drilling,G: grindingmaterialpartsproductSoft Computing Lab.WASEDA UNIVERSITY , IPS 925.1 Introduction to CMD - Common configurations Process Layout (Process-focused Job Shop):Departments are composed of machines with similar capabilities that perform similar function.Highly skilled operators are typically required because successive batches assigned to a work center may require very different tooling and setup.materialpartsMM MM DD DD GG GGT T TBBBBM:milling,D: drilling,G: grinding, T: turning, B: boringmaterialpartsProductProductSoft Computing Lab.WASEDA UNIVERSITY , IPS 935.1 Introduction to CMD - Common configurations Group Technology (Cellular Manufacturing):It entails dividing the manufacturing facility into small groups or cells of machines, each cell being dedicated to a specified set of part types with similarity.Use of machines in a designated physical area for production ofa specific set of parts facilitates scheduling and control and reduces setup time, material handling, and throughput time.MTDMGBDDMM:milling,B: boring,T: turning,G: grinding,D: drillingmaterialpartsProductProductSoft Computing Lab.WASEDA UNIVERSITY , IPS 945.1 Introduction to CMD - Common configurations Throughput TimeWork in ProcessSkill LevelProduct FlexibilityDemand FlexibilityMachine Utilizationworker UtilizationUnit Production CostCharacteristicProductProcess Group Fixedlowlowchoicelowmediumhighhighlowlowlowmedium-highmedium-highmediummedium-highhighlowhighhighhighhighhighmedium-lowhighhighmediummediummixedhighmediummediummediumhigh A. Standridge: Modeling and Analysis of manufacturing Systems, John Wiley & Sons, Inc.,New York, 1993Table. 13.6 General Characteristics of Layout TypesSoft Computing Lab.WASEDA UNIVERSITY , IPS 955.1 Introduction to CMD - Group Technology Group Technology (GT) :A method identifying and exploiting the similarity among the attributes of a set of objects.A theory of management based on the principle that similar thing should be done similarly. things include product design, process planning, fabrication, assembly, and production control. It is this kind of efforts that small `focused factories` are being created as independent operating units within large facilities today.The central objective is to increase production efficiency by grouping various parts and products with similarity.It allows for the combined benefits of mass production while dealing with multi-product, small-lot-sized production.Soft Computing Lab.WASEDA UNIVERSITY , IPS 965.1 Introduction to CMD - Group Technology Cellular Manufacturing :Application of GT to organize cells that contain a set of machines to process a part family.Reasons for Establishing CMD:To reduce throughput timeTo reduce work-in-process inventoryTo improve part/product qualityTo reduce response time to customer ordersTo reduce move timeTo increase manufacturing flexibilityTo reduce unit production costSoft Computing Lab.WASEDA UNIVERSITY , IPS 975.1 Introduction to CMD - Group Technology Important Areas of GT Applications:Classification and codingProcess planningPart family and machine cell designGroup technology layout Group schedulingExamples of Automated CMD:Flexible Manufacturing System (FMS)Computer-Integrated Manufacturing (CIM)Just-In-Time (JIT) or Kanban SystemManufacturing Cell Design:Procedure of the machines and parts to form in GTMCD Problem is an NP-hard problemEvolutionary Search Methods, Simulated Annealing and GAsSoft Computing Lab.WASEDA UNIVERSITY , IPS 985.2 Major Issues on CMDThe design of cellular manufacturing systems consist of three major issues:1 Cell Formation... ... ......cell a cell b cell nmachines3 Cell Layoutcell ccell bcell a2 Machine Layout...cell iSoft Computing Lab.WASEDA UNIVERSITY , IPS 995.2 Major Issues on CMD - issue I1. Cell Formation: Goal of cell formation is to group machines into cells to minimize the intercell traffic, knowing that the number of machines in each cell is limited.Part Process Plan 1 M1 2 M2 3 M1 M2 M4 4 M3 5 M2 M4Process Plans for Parts12 3 4 5 11 1 2 11 131 4 11PartsMachinesPart/Machine MatrixSoft Computing Lab.WASEDA UNIVERSITY , IPS 1005.2 Major Issues on CMD - issue IOrganize cells that contain a set ofmachines to process a family of similarparts, while minimizingnumber of exceptional elements. Two machine cells: C1={1,3}, C2={2,4} Corresponding parts families: F1={1,4}, F2={2,3,5}PartsMachinesPartsMachines 14 352 111 3121114111234 5 1 11 2 1113 1 4 11Soft Computing Lab.WASEDA UNIVERSITY , IPS 1015.2 Major Issues on CMD - issue IPart Process Plan 1 M2 M5 2 M1 M7 3 M3 M4 M6 4 M3 M4 M6 5 M1 M7 6 M3 M4 M6 7 M2 M5Process Plans for Parts:123 4567 1 11 2 113 111 4 111 5 11 6 1117 11Part/Machine MatrixMachinesPartsSoft Computing Lab.WASEDA UNIVERSITY , IPS 1025.2 Major Issues on CMD - issue IPartsMachinesPartsMachines Three machine cells: C1={2,5}, C2={3,4,6}, C3={1,7} Corresponding parts families: F1={1,7}, F2={3,4,6}, F3={2,5} 1 23 4567 1 11 2 113 111 4 111 5 11 6 1117 11 1 73 46 25 2 115 113 111 4 111 6 1111 11 7 11Soft Computing Lab.WASEDA UNIVERSITY , IPS 1035.2 Major Issues on CMD - issue IIMachine Layout:Single Row Layout U-shape LayoutMulti Rows Layout Loop LayoutSoft Computing Lab.WASEDA UNIVERSITY , IPS 1045.2 Major Issues on CMD - issue IICell Layout:112320810517 21 1867116912419 1322141523cell acell ccell bSoft Computing Lab.WASEDA UNIVERSITY , IPS 1055.2 Major Issues on CMD - issue IIConsiderSimultaneously CellFormationMachineLayoutCellLayoutPrevious:Proposed:CellFormationMachine LayoutCell LayoutLocal optimalSoft Computing Lab.WASEDA UNIVERSITY , IPS 1065.3 Mathematical FormulationAssumption:Each machine has enough capacity to produce all parts.Shape of machine and cell is rectangle.Input Data:Number of machines.Area of each machine.Production volume & sequence of each part.Processing time on each machine for each part.Maximum number of cells.Maximum number of machines in a cell.Soft Computing Lab.WASEDA UNIVERSITY , IPS 1075.3 Mathematical Formulation - Objective IFormulation:One of the objective functions is total moves determined as the weighted sum of both intercell and intracell moves.cell b cell a + nkmpmqkpqMkpq kninjijCijk km d l n d z1 1 121 11 1min intercellintercellintracellintracell12 1 + Soft Computing Lab.WASEDA UNIVERSITY , IPS 1085.3 Mathematical Formulation - Objective Iwhere1 and 2 are weights attributed to the intercell and intracell moves respectively, 1 and 2 [0, 1], 1 +2 =1.n the number of cells.mk the number of machines in the cell k.dijCthe rectilinear distance between the centroids of the cell i and the cell j.nijthe total number of transportation between the cell i and the cell j. lkif a type of machine layout in the cell k is aloop layout, it takes e, otherwise it takes 1. e the move cost ratio of uni-direction/bi-direction per unit distance, e