Multirow machine layout problem in fuzzy environment using genetic algorithms

Computm

Pergamon

ind. Engng Vol. 29, No. l-4, pp. 519-523.1995 Copyright 0 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 03604X352/95 $9.50 + 0.00

0360-8352(95)00127-l

Multirow Machine Layout Problem in Fuzzy Environment Using Genetic Algorithms Mitsuo Gen Kenichi Ida Chunhui Cheng Department of Industrial and Systems Engineering Ashiiga Institute of Technology, Ashikaga 326, Japan continuous opttiation and combinatorial optimiration. We propose an extended permutation representation as the coding scheme for multi-row machine layout problem, which contains three lists of repsr&or/machine rymbol/neat clearance. A new arithmetical crossover is defined to manipulate the neat clearance list. Mutation operator is designed with neighborhood technique to try to find an improved offspring. The performance of proposed algorithm is demonstrated by computer simulations.

This paper formulates fursy multirow Abstract: machine layout problem where the clearance between any two adjacent machines is given as a fuzzy set. The membership function of the fuzzy clearance corresponds to the grade of satisfaction of separate dii tance. The objective function considered here is to maximke the minimum grade of satisfaction over machines and meanwhile to minimize the total travel cost among machines. Genetic algorithms are investigated as a heuristic technique for solving the problem. The performance of proposed algorithm is demonstrated by computer simulations. Machine layout, Key words: and fu55y clearance.

1

2

genetic algorithms

2.1

Fuzzy Multi-row Layout Problem Fuzzy

Machine

clearance

The necessary clearance between machines is basically determined with respect to the technique aspects for a flexible machining system, such as to maximize safety level, to minimize noise or mutual interruptive level. During the design phase, it is hard to provide a precise value for the clearance between machines. In such case, we can represent the clearance as a fuzzy set . The clearance between machines denoted by &j is shown in Figure 1.

Introduction

The layout problem of machines is critical to designing an efficient flexible machining system. Many approaches have been reported to address this problem. Multirow machine layout is one of complex problems concerning efficient physical arrangement of machines under given multiple rows [l]. Usually, machine layout problem is treated without the consideration of imprecise data. But iu many practical cases, such situation is often encountered that the information gathered are of imprecise in nature, one must mahe his/her decision under fusry environment. Fu55y set theory provides a suitable technique for handling such a problem. The concept of fuzzy clearance is firstly introduced by Ida, Gen and Cheng in [2]. In this paper, we formulate furry multirow machine layout problem with this concept. The membership function of the fursy clearance corresponds to the grade of satisfaction of separate distance. The objective function considered here is to m aximke the minimum grade of satisfaction over machines and meanwhile to minimise the total travel cost among machines. Genetic algorithms have recently been applied to optimisation problems in diverse fields [g]. In thii paper, genetic algorithms are investigated as a heuristic technique for solving fursy multirow machine layout problem. We diiusse that the essential of the machine layout problem is the combining nature of

vd

Figure 1: Clearance between machines Let us denote the membership function of fussy clearance between any two adjacent machines i and j with bj (5i, 5j) which represents the grade of satisfaction of the separated distance. The membership function is defined as: Pij(%*zj)= (

I 519

1; (IZi-Zj

0;

1 Zi

-

I-f(Zi+lj)-~j)/(d~j-d~j);6~j 51 Zi ) Zi

-

Zj

Zj

2j

12

6fj

I< 6& I< 6~j

(‘I

520

17th International Conferenceon Computersand Industrial Engineering 61~ =

](l,

+ zi) + a,,I

1

(3)

where d~j is the least clearance for machines i and j and d[j is the satisfactory clearance for machines i and j . The membership function is illustrated in Figure 2.

The objective function (4) is to minimize the total travel cost among machines. The objective functions (5) and (6) are to maximize the minimum grade of satisfaction over machines. Constraint (9) ensures that only one of the two constraints (7) and (8) hold. Constraints(10) and (11) ensure that machines axe arranged within the restricted working area. Constralnt (12) is a nonnegativity constraint.

3

Multi-row

Ma-

The essential of this problem comprises three different tasks:

Figure 2: Membership function

2.2

GA for Fuzzy chine Layout

• find a better allocation of machines to rows • find a better sequence of machines within each

Mathematical model

row

For multi-row case, the working restriction is given by a couple, the width of area denoted with W and the length of area denoted with L. The relation of working area, machines and reference lines is illustrated in Figure 3.

..V

hd

Figure 3: Clearance between machine and reference line With the notation of fuzzy clearance, the multiple row machine layout problem with unequal area can be formulated as follows:

• find a better position (z and y coordinates) for each machines Because 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 rows. For the ease of explanation, we just consider two-row example but the proposed approach can be applied to general multirow case. 3.1

Representation

We use 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:

[{k}, { ~ , , , , ~ , , - . . , ,,~.}, { a , , , ~ , ~ , . . - , A,.}] max

i=I j=l h PT

max

#~

s.t.

j,,~(f,,=~)+M~,,>~,~.,

(5)

(6)

i,~=1,-..,-

(7)

~','i(~,, ,J) + M(1 - ~,i) > ~'~",i, j = 1,..., n(s) ~,j(1-~,i) =0, i , i = 1 , - . . , (9)

z, + ~-z,+d~0~< L, i = 1,...,n 1

I,

yi+~bi+dio<_W,

i: 1,--.,n

(10) (11)

where m% represents the machine in the j t h position, Aij denotes the neat clearance between machines mij_, and m~j. The separator k, generally 1 < k < n, denotes the cutting position to separate the fist into two parts according to the two row requirement. Suppose machines mk and m~ are arranged as shown in Figure 4. The net clearance and z-axis position can be calculated as follows:

(12)

Ai

_-

w h e r e l~ is the length of machine i, bl the width of machine i, #~ the grade of satisfaction of horisontal separation between machines i and j , p~j the grade of satisfaction of vertical separation between machines i and j , d ~ the least cleaxemce between machine i and right vertical reference line, and d ~ the least cleexanee between machine i and upper horizontal reference line.

zi

=

zh

=

zl,yi >_O, i = l , . . . , n

where A~ : Alh~ :

Alkl - d~i

d~o + Ah + ~lh

neat clearance associated with machine m~, sepaxation between two machines.

vrl] [

I~

17th International Conference on Computers and Industrial Engineering

521

Let L denote the restriction of working area and $1 denote the necessary space required, which is determined as follows: k

k-1

$1 = E 1, + E 4,,+1 + ~o + dl;,o i=1

Figure 4: Illustration of neat clearance and decision vaxiables 3.2

Initialization

The overall procedure of initialisation is shown below: procedure: inlti-li~ation begin i ~-- 0; w h i l e (i <_ pop_size) d o begin generate separator randomly; generate machine list randomly; check the feasibility of the offspring; i f feasible generate neat clearance list randomly; i4---i+1; else kill it; endif end end 3.2.1

machine permutation

Let H0 denote the set of available machines and P denote the list of machine permutation, then the machine permutation is randomly generated as follows: procedure: machine permutation

begin i ~-- 0; IIo ~ {ml, m 2 , . . . , m~};

P~; w h i l e (i _< n) d o begin pick up a machine rn' from H0 randomly; P ~-- P U r e ' ; Ho ~- Ho\m'; i ~-- i + 1 ; end return permutation list P; end

(13)

i=1

Then we compare $1 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. 3.2.3

neat clearance

Because of the existence of allowable space constraints, the neat cleaxanee (real number) is randomly generated within an allocable region. Let L t denote available space and L denote the length restriction of the working area. Then the initial available space can be calculated as follows: n

n-1

L' : 2L - ( E l~ + E i=1

d ~ + 1 + d~o + d~0) (14)

i=1

Let A denote the list of neat clearance for the row, The overall procedure is shown below: p r o c e d u r e : n e a t c l e a r a n c e llst begin i ~-- 0;

A ,.-- q~;

calculate initial available space L'; w h i l e (i S n) d o begin pick up s neat clearance 6i within (0, L') randomly; A .-- Au61; L' ~- L' - 6i; i~i+1; end return neat clearance list A; end 3.3

Evaluation

Function

Suppose we have a chromosome denoted with vz as follows:

A2,..., A,,}]

= [{,},

W e can get the travel cost a m o n g machines for tth chromosome as follows: n

n t

t

t

t

i=1 j = l

3.2.2

feasibility 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: [~rtl~ ~ r t 2 ~ . . . , 17tk]

Since we use a minimization objective function, we have to map the objective function to s fltnen function in order to ensure that fitter chromosome has larger value of the fitness function.

e.:(~,) = ~ , ~ + ~,_~ ~'J('l, =11,,+1 wt + w= = 1

(16) (17)

17th International Conference on Computers and Industrial Engineering

522

where A0 is a constant which can be dewed as an initial estimation of the best objective function value. After evaluating each chromosome, we make s roulette wheel and use it as the basic selection mechanism to reproduce next generation. In the roulette wheel selection, a fitter chromosome has a larger chance to be reproduced into next generation. 3.4

n e w n e a t c l e a r a n c e llst

3.4.2

We define a new arithmetical crossover to manipulate the neat clearance list. Suppose we have two parents' neat clearance lists as follows:

{a 1,

The new neat clearance associated with each machine can be calculated as follows:

Crossover

The basic element of our crossover procedure consists of three parts,

L~ = ~l/qtl 7!-ot2A~,

i = 1, 2 , . - . , n

a l , a~ E (0, 1) • a random way to determine separator • an ordinary P M X (partially mapped crossover) to manipulate machine permutation list • an arithmetical crossover to manipulate neat clearance list The scheme of the crossover operator is depicted below. p r o c e d u r e : crossover begin i~0; while (i < pop_size x pc) do sdect two chromosomes randomly; generate a new separator; generate a new machine permutation with ordinary PMX; generate a new neat clearance list with arithmetical crossover; check the feasibility of the offspring; i f feasible i*-- i + 1;, else kill it; endif end end 3.4.1

new separator

The procedure for generating new separator is rather simple, which contain two major steps: • determine the upper and lower bounds for a closed interval • select an integer within the interval randomly For example, we have two parents shown as follows:

= =

m ),1

where a l and a2 are the randomly generated real number within the open interval (0, 1). We should point out that the difference of proposed arithmetical crossover with conventional one is that the conventional arithmetical crossover is a linear combination of two parents' neat clearances, that is, the following relation holds for two parameters a l and a2. a l + a2 : 1 For our case, we require following relation holds for these two parameters: a l + a2 _< 2

3.5

Mutation

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: {At, A2,'- ",Ai,.. -, A , }

and the ith gene Ai is selected for mutation. Let k be a given integer and then we divide the selected neat clearance Ai into k equal parts as follows: A~,

=

--Ai k

(21)

A~,

=

A~ A~,- I ÷ --k--, j = 2, 3 , - . . , k

(22)

"

sU = max{s 1, s 2} sL = min{s 1, s 2} Then we can make a closed interval with su and at. as [s~, str]. The new separator is a randomly generated integer within this interval.

(20)

The reason is that 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. W e force these parameters follow the relation (20), then we can enlarge the search space greatly, which is independent of initial search space.

mr), {A1, A2,'" -,ZX }]

The upper and lower bounds can be directly calculated as follows:

(18) (19)

We can get k neat clearances. The set of neat clearsnce are listed below:

{al,

.--,

{A1, A2,'",AI~,''',an)

523

17th International Conference on Computers and Industrial Engineering The set of chromosome formed with above set of nest 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 k-optimum, if it is better t h a n any others in the neighborhood. The proposed m u t a t i o n comprises mainly three steps: firstly it selects a gene (a neat clearance) randomly; secondly it generates k neighbors of the selected gene; lastly it evaluates all neighbor layouts to select the best neighbor as the offspring. The scheme of the proposed mutation is depicted below. procedure: mutation begin give an integer k; i ~-- O; w h i l e (i <_pop_size X Pro) d o pick up a gene A~ randomly; generate k neighbors of Ai; evaluate all neighbor layouts; select the best neighbor as the offspring; i ~'-- i + 1; end end

4

[d~] =

5

I

2

3

4

5

6

r0

1.0 0.0 1.0 0.6 1.0 1.0 1.0 1.0 1.0 1.0 1.0 2.0

0.6

1.0

1.0

1.0

2.0

1.0 \

0.0 1.0 1.5 1.0 0.0 0.4 1.5 0.4 0.0 0.5 2.2 1.6 1.5 2.6 0.8

0.5 2.2 1.6 0.0 2.0

1.5 2.6 0.8 2.0 0.0

1.0 1.0 1.0 1.0

1 2 3 4 5 6

2.0

1 0.0 2.8 5.0 4.0 6.2 7.0

2 3 4 2.8 5.0 4.0 0.0 4.0 2.0 4.0 0.0 1.1 2.0 1.1 0.0 6.0 2.8 5.0 1.8 5.6 2.2

5 6 6.2 7.0 6.0 1.8 2.8 5.6 5.0 2.2 0.0 4.0 4.0 0.0

2.0 2.0 2.0 2.0 2.0

150

200

Figure 6: Evolution process for test problem

1 2 3 4 5 6

[0

I00 g~oluao~ry pr~ess

Test problem is a 6-machine 2-row layout problem given by Kusiak [1]. The fuzzy clearances are considered as follows:

[d'j] =

~.lltnrst

50

Example

lO

Figure 5: Multiple-row layout for the test problem

1.0

r0 2.0 \ 2.0 2.0 2.0 2.0 2.0

J

The evolutionary environment of our implementation is given as follows: n = 6, pop_size = 30, mazgen = 200, Pc : 0.4, P m : 0.4, k : 10, wt : 0.5 and w2 : 0.5. 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: 2 6 4 3 1 5 z position: ( 3.11, 7.93, 14.14, 3.14, 8.06, 13.64) separator: 3 The total travel cost is 19660.56 and average grade of satisfaction is 0.875. The layout based on the sequence is depicted in Figure 5. Figure 6 shows the evolution process for the test problem.

Conclusions

The layout problem of machines is critical to designing an ei~cient flexible machining system. Usually, machine layout problem is treated without the consideration of imprecise data. But in m a n y practical cases, one must make his/her decision under fuzzy environment. In this paper, we have introduced the conception of fuzzy clearance into multiple row machine layout problem and formulated fuzzy multirow machine layout problem based on this concept. Genetic algorithms are investigated as a heuristic technique for solving the problem. We proposed an extended permutation representation as the coding scheme for multi-row machine layout problem, which contains three lists of separator/machine sFrnbol/neat clearance. A new arithmetical crossover is defined to manipulate the neat clearance list. Mutation operator is designed with neighborhood technique to try to find an improved offspring. Preliminary computational experiments have been executed. The results demonstrated that genetic algorithms and fuzzy approach can be a promising way for multiple machine layout problems.

References [1] Kusiak, A., Intelligent Manufacturing Systems, Prentice Hall, New Jersey (1990). [2] Ida, K., M. Gen and C. Cheng, Fuzzy machine layout problem using genetic algorithms, Prec.

Fall Meeting of Japan Industrial Management Association (1994)199-200. Z, Genetic Algorithm + Data Structure = Evolution Programs, Second Edi-

[3] Michalewicz,

tion, Springer-Verlag (1994).