the 7th Online World Conference on Soft Computing in
Industrial Applications (WSC7)
PROCESS-ORIENTED PLANT LAYOUT DESIGN
USING A FUZZY SET DECOMPOSITION
ALGORITHM
E. Avineria, P. Vaishb
aTechnion
bOracle
- Israel Institute of Technology, Faculty of Civil Engineering,
Haifa 32000, Israel,
E-mail : avineri@internet-zahav.net
Software India ltd., 5th Floor, Cyber Gateway, Hi-Tec City,
Madha Pur, Hyderabad - 500081, India,
E-Mail: vaishprakhar@yahoo.com
1 INTRODUCTION
Efficient plant layout is critical to achieve and maintain manufacturing
competitiveness.
20% to 50% of total operating cost is spent on Material Handling (MH)
and an appropriate facilities design can reduce this cost by at least 10%
to 30%
Process-oriented layout:
physical arrangement of given number of machines (or facilities) within
a given configuration.
1 INTRODUCTION
Facility Layout Problem (FLP) considers the amount of load or number
of trips multiplied with distance-related costs, and looks for a solution
which would minimize the value of function F:
M
M
F = ∑∑ (traffic)ij * (distance) f(i)f(j)
i =1 j =1
where (traffic)ij is the traffic between machines i and j; (distance)f(i)f(j) is
the distance between machine locations and M is the number of
machines.
1 INTRODUCTION
Rectilinear Distance Model
Distance between i and j is:
dij = | xi - xj | + | yi - yj |
The number of different layout patterns, in which M machines are placed
(in N locations or cells (N≥M)
N!
( N − M )!
NP-completeness of the FLP: impractical to search for optimal solutions.
Fuzzy sets theory was found to be a powerful method of approximate
reasoning.
2. MATHEMATICAL FORMULATION
Connection matrix [Mij], mij≥0
If mij>0,mjk>0, and mik=0 then no mass moves from i to k. But indirectly
mass moves from machine i to machine k via machine j.
The distance between machines i and j, dij, is inversely proportional to mij
because more is the mass nearer the machines should be. Similarly djk is
inversely proportional to mjk.
dik ≤ dij+djk.
1/mik ≤ 1/mij + 1/mjk
mik ≥ {mij*mjk}/{mij+mjk}
2. MATHEMATICAL FORMULATION
This forms the principle of fuzzy set algorithm that mass transfer between
no machines is zero as assumed in conventional layout algorithms.
Σjmij all mass transported from machine i to other machines
Σimij all mass transported to machine j from all other machines
Pmax= Max{Σjmij , Σimij }
Thus:
mik/Pmax ≥ (mij/Pmax * mjk/Pmax}/{mij/Pmax + mjk/Pmax)
∀ i,j | mij+mjk > 0
(*)
2. MATHEMATICAL FORMULATION
A fuzzy relation matrix M’ can be defined.
Each of its elements represents the membership value of a pair of
machines to the fuzzy set “machines that should be placed close to
each other”, as calculated by (*) and its extensions. In other words,
The higher the membership value is, the nearer the machines should be
placed.
The total mass transfer between machines i and j is m’ij+m’ji. By adding
the element min{m’ij, m’ji} to the element max{m’ij, m’ji} we will get the
matrix M’’ij.
The larger the value of m’ij+m’ji, the closer the machines i and j should
be placed.
2. MATHEMATICAL FORMULATION
Applying the fuzzy decomposition algorithm to M’’ij, the decomposition
of fuzzy relation may be summarized as:
R= Σαα*Rα
0<α≤1; α1 >α2 ! Rα1 ⊂ Rα2
where Rα denotes a relation between all machine pairs with m’’ij >α, and
α*Rα means all the elements of relation Rα multiplied with α.
The basic structure of the algorithm is as follows:
2. MATHEMATICAL FORMULATION
Step 1: Construct the modified fuzzy relation matrix M’’ij.
Step 2: Set α; ∆α
Step 3: Let R1 be the set of all pairs of machines; Let R2 = ∅
Step 4: Set R3 = R1-R2 (the set of pairs that haven’t been considered yet)
Step 5: Set R2 be a subset of R3, including all the pairs of machines to be
considered for placing in the current iteration (m’’ij ≥ α)
Step 6: Place the machine pairs that are included in set R2;
Step 7: Set R1=R1∪R2 ; Set R3=R3-R2
Step 8 If R3 is an empty set (no more machines to be placed) then stop,
otherwise –Set α=α-∆α ;
Step 9: Go to Step 4.
3 EXAMPLE
A to F machines. Factory floor: 2 * 3.
The flow mass (in Kg./Hour) between machines:
From i
To j
A
B
C
D
E
F
Column_
Sum Σjmij
A
B
C
D
E
F
--0
20
40
0
0
60
10
--0
0
0
0
10
0
15
--0
32
0
43
0
0
20
--0
0
20
0
0
0
25
--15
40
0
0
0
5
0
--5
Thus: Pmax = Max{Σjmij , Σimij} = Max{60,70) = 70
Row_Sum
Σimij
10
15
40
70
28
15
3 EXAMPLE
the matrix Mij/Pmax :
From i
To j
A
B
C
D
E
F
A
B
C
D
E
F
--- 0.143
0
0
0
0
0
--- 0.214
0
0
0
0.286
0
--- 0.286
0
0
0.571
0
0
--- 0.357 0.071
0
0
0.457
0
--0
0
0
0
0
0.214 ---
3 EXAMPLE
the maximum value of the path lengths between all pairs of machines is
calculated. These values are summarized in the following matrix:
From i
To j
A
B
C
D
E
F
A
B
C
D
E
F
--0.122
0.286
0.571
0.176
0.029
0.143
--0.095
0.114
0.021
0.004
0.086
0.214
--0.201
0.457
0.146
0.014
0.122
0.286
--0.176
0.029
0.003
0.026
0.159
0.357
--0.214
0.001
0.008
0.057
0.071
0.011
---
3 EXAMPLE
By adding the element min(m’ij,m’ji) to the element max(m’ij,m’ji) we
will get the matrix M’’ij as following:
From i
A
B
C
D
E
F
Row_Sum
--0
0.371
0.585
0.179
0.030
1.166
0.265
--0
0
0
0
0.265
0
0.310
--0
0.616
0.203
1.128
0
0.237
0.486
--0
0
0.723
0
0.047
0
0.533
--0.226
0.805
0
0.011
0
0.101
0
--0.112
0.265
0.604
0.858
1.219
0.795
0.459
To j
A
B
C
D
E
F
Column_Sum
Σjm’’ij
Σim’’ij
3 EXAMPLE
When the total mass flow from/to a specific machine x, Σim’’ix + Σjm’’xj ,
is large, we should consider to place it in the most approachable position
Row_Sum
A
1.166
B
0.265
C
1.128
D
0.723
E
0.805
F
0.112
Column_Sum
0.265
0.604
0.858
1.219
0.795
0.459
1.431
0.869
1.986
1.942
1.600
0.571
Σim’’ij
Σjm’’ij
Σim’’ij + Σjm’’ij
3 EXAMPLE
E
C
?
?
?
?
alternative 1
?
C
?
?
E
?
alternative 2
First arrangement (α=0.6)
3 EXAMPLE
E
C
?
D
A
?
alternative 1
A
C
?
D
E
?
alternative 2
Second arrangement (α=0.5)
3 EXAMPLE
E
C
B
alternative 1
A
D
C
A
B
F
D
E
F
alternative 2
Final arrangement at the end of the process (α =0.3)
4 COMPARISON WITH THE OPTIMAL SOLUTION
The total cost (number of movements) of the final arrangement solution
recommended by the fuzzy set decomposition method is 247 for the first
alternative and 217 for the second one.
The second alternative is also the optimal solution of this problem.
5 CONCLUSION
Fuzzy set based algorithms can form a firm basis for the design of plant
layout.
They provide a basic reasoning for the machine layout in the plant which
otherwise are subject to a complex process.
QUESTIONS ?