遠東學報二十一卷第二期
中華民國九十三年四月出版
網路最短路徑問題之決策分析
Decision Analysis of The Shortest Path Problem in A Network
龔榮源 國立海軍官校資訊管理系
林佳姿 遠東技術學院資訊管理系
莊宗南 國立海洋大學商船系
摘
要
在過去，網路的最短路徑問題對於許多應用的領域諸如通訊、要徑及運輸是
相當重要的，所以一直受到許多研究者的重視。對於決策而言，最短路徑長
度與最短路徑是最重要的資訊。傳統上，網路的弧長被視為明確值，因此其
最短路徑長度與最短路徑亦是明確的。然而在實際的應用上，弧長可代表成
本或時間，因而可將弧長視為模糊集合。在本文中，我們提出一些演算法來
處理模糊最短路徑問題，這些方法在實務上可提供決策者作參考。文中亦舉
例驗證所提出的演算方法。
關鍵詞 : 網路，模糊，最短路徑問題，決策
Jung-Yuan Kung, Department of information management, National Chinese Naval Academy
Chia-Tzu Lin, Department of information management, Far East College
Tzung-Nan Chuang, Department of Merchant Marine, National Taiwan Ocean University
ABSTRACT
In the past, the shortest path problem in a network has attracted attention from
many researchers for its importance to various applications such as communication,
routing, and transportation. The main outputs of the shortest path problem are the
shortest path and the shortest path length, and they are central information for
decision-making. Traditionally, the arc lengths in a network have often been
considered as exact values, and hence the outputs of the shortest path problem are
exact. However, in real-world applications, the arc lengths can represent cost or
time and it can be viewed as fuzzy. In this paper, some developed fuzzy algorithms
to treat the fuzzy shortest path problem are presented to be helpful to decision
makers in practices. Besides, some simulation results are given to show the
presented algorithms.
Keywords: Network, fuzzy, shortest path problem, decision-making.
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algorithm is not necessarily the actual shortest path
1. Introduction
In the past decades, many researches have
length. To avoid the shortcoming, Chuang and Kung
concentrated on the shortest path problem in a
[10] proposed an approach that combined fuzzy
network since it is important to many applications
minimum algorithm and a method based on the
such as scheduling, routing, communication, and
intersection area of fuzzy triangular area.
transportation [1]. The arc length in the path of
approach can get the shortest path length and the
network may represent the time, the cost, etc. in
shortest path simultaneously. In this paper, we
practice. Traditionally, the arc length is considered to
present these developed algorithms or approaches
be exact values. If there are two crisp path lengths in
and make detailed analysis. In addition, some
a network, say p1 and p2, then the shortest path length
simulation results are included to demonstrate the
in the network is the minimum of p1 and p2. That is,
presented algorithms. These can be helpful to
either p1 or p2 can represent the shortest path length.
decision-makers as they make decision.
Their
However, in real world, the arc length in a network
can be referred to as a fuzzy set. Let d1={0.2/2,
2. Some related fuzzy set
0.5/4} and d2={0.3/1, 0.4/3} denote two fuzzy path
lengths where A(x)/x represents that the element x
operations
has a membership function A(x). It is clear that both
The class of fuzzy sets is denoted by S. We
d1 and d2 cannot represent the minimum of d1 and d2.
denote X
Hence, one needs the fuzzy minimum algorithm to
determine the fuzzy shortest length.
and
= {x1 , x2 ,..., xk } to be the universal set
AF : X → [0,1] to be the membership
function of F ∈ S . Some related fuzzy set operations
Several algorithms were proposed to treat the
fuzzy shortest path problem [2-10]. Dubois and
are reviewed as follows [11]:
Prade [2] first handled the fuzzy shortest path
(1) The membership function of MIN(A,B) that
problem. They employed the fuzzy minimum
is the minimum of fuzzy sets A and B is given by
operator to find the shortest path length, while they
did not develop a method to decide the shortest path.
MIN(A,B)(z) = sup min(A(x),B(y))
Later, Klein [7] proposed an improved algorithm that
(1)
where ‘sup’ operates under z=min(x,y).
can get the shortest path length as well as the shortest
(2) The membership function of A+B that is the
path. However, the assumption, made in their
addition of fuzzy sets A and B is given by
algorithm, that each arc lengths are 1 through a fixed
(A+B)(z)= sup min(A(x),B(y))
integer, was not reasonable and impractical. On the
(2)
where ‘sup’ operates under z=x+y.
other hand, as to continuous fuzzy shortest path
problem, based on multiple labeling method [3],
Okada and Soper [9] proposed a fuzzy algorithm to
3. Algorithms for the shortest path
deal with shortest path problem. Their method can
problem
offer non-dominated paths to decision makers, and
Suppose that there are N nodes in an acyclic
consequently provide the shortest path that they think
directed network. To find the fuzzy shortest path
for decision makers, whereas the path length that
length, Dubois and Prade [2] proposed the following
corresponds to the yielded shortest path by their
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recursion
中華民國九十三年四月出版
Example 1:
d(i)=MIN{Tij+d(j)}
(3)
Consider a classical network with fuzzy arc
where i is smaller than j and Tij is the fuzzy arc
lengths that is shown in Figure 1, we use Dubois and
length from node i to node j and d(i) is the fuzzy
Prade’s algorithm to find the shortest path length
shortest path length from node i to the destination
from the source node 1 to the destination node 6.
node N.
Let us show the above approach via the
following example.
D
2
A
4
F
C
1
6
B
3
G
5
E
Figure 1. A Classical Network.
In Figure 1, the arc lengths are A = {0.3/1,
0.5/13}}
0.4/3}, B = {0.5/2, 0.6/4}, C = {0.4/1, 0.7/2}, D =
={0.3/7,0.4/8,0.4/9,0.4/10}.
{0.5/2, 0.8/3}, E = {0.3/3, 0.5/4}, F = {0.4/4, 0.2/5},
and
As a result, the shortest path length is equal to
G = {0.6/2, 0.7/5}, respectively. The related
d(1). The value d(1), however, cannot correspond
results are given as follows: T12=A={0.3/1, 0.4/3},
to any path from the source node 1 to the destination
T13= B = {0.5/2, 0.6/4}, T23= C = {0.4/1, 0.7/2},
node 6. That is to say, all the path lengths,
T24=D = {0.5/2, 0.8/3}, T35= E = {0.3/3, 0.5/4},
associated with possible paths 1-2-4-6, 1-2-3-5-6, and
T46=F = {0.4/4, 0.2/5}, T56=G = {0.6/2, 0.7/5}; d(5)=
1-3-5-6, are not equal to d(1). Therefore, we are
T56=G = {0.6/2, 0.7/5},
unable to decide which path is the shortest one.
d(4)= T46= F = {0.4/4, 0.2/5},
Meanwhile, because each arc has different possible
d(3)=E+G={0.3/5,0.5/6,0.3/8,0.5/9},
length and not every arc length is 1 through a fixed
d(2)=MIN{D+d(4),C+d(3)}
integer, Klein’s algorithm [7] can not be applied to
this example.
□
=MIN{{0.4/6,0.4/7,0.2/8},
Suppose that the each arc length in the path
{0.3/6,0.4/7,0.5/8,0.3/9,0.4/10,0.5/11}}
is triangular fuzzy set, Chuang and Kung [10]
={0.4/6,0.4/7,0.2/8},
proposed the fuzzy shortest length heuristic
d(1)=MIN{A+d(2),B+d(3)}
procedure as follows:
=MIN{{0.3/7,0.3/8,0.4/9,0.4/10},
Fuzzy shortest length heuristic procedure
{0.3/7,0.5/8,0.3/9,0.5/10,0.5/11,0.3/12,
Input：di = (ai, bi, ci), i = 1, 2, ….., n, where di
denotes the triangular fuzzy length.
Output：dmin = (a, b, c) where dmin denotes the
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fuzzy shortest length.
Step 3: Let i=2.
Step 4: Calculate (a, b, c):
Step 1：Form the set U by sorting di in ascending
b=(96-24)/(20-10)=7.2;
order of bi; U = {U1, U2, …, Un}
a = min(4,6)=4;
where Ui = (a’i, b’i, c’i), i = 1, 2, ….., n.
c = min(15,12)=12.
Step 2：Set dmin = (a, b, c) = U1 = (a’1, b’1, c’1).
Step 5: Set dmin =(4,7.2,12)
Step 3：Let i = 2.
Step 6：Set i=3
Step 4：Calculate
Step 7：Repeat Step 4 to Step 6 until i =4.

b; if b ≤ a i'

'
b =  (b × bi ) − ( a × a i' )
'
 (b + b ' ) − ( a + a ' ) ; if b > a i
i
i

path length is equal to (4,7,12).
a = min (a, a’i);
examining the intersection area IA(i) of dmin and di
c = min (c, b’i).;
for i = 1, 2, 3. It is easy to yield IA(1)=3.56,
Step 5：Set dmin = (a, b, c).
IA(2)=1.63, IA(3)=1.04, and one can choose path
Step 6：Set i = i+1.
1-2-4-6 as the shortest path.
Complete the above procedure, the fuzzy shortest
Next, we want to decide the shortest path by
Step 7：Repeat Step 4 to Step 6 until i = n +1.
In addition, to decide the shortest path, they
A
2
used the idea that the much the intersection area of
B
Let us show the following example to
demonstrate Chuang and Kung’s algorithm.
4
F
D
1
two triangles is, the more alike they are.
C
3
E
6
5
G
Example 2:
Figure 2. A Classical Network.
A classical network with fuzzy triangular arc
lengths is shown in Figure 2.
In Figure 2, A=(1,3,4), B=(2,4,5),C=(1,2,3),
4. Conclusions
D=(2,4,6), E=(2,5,6), F=(2,3,8), and G=(3,5,7).
In the past, many researches have focused on the
From Figure 2, one can get the possible paths
shortest path problem in a network. Some algorithms
and the corresponding path lengths as follows:
path 1-2-4-6 with d1=A+C+F=(4,8,15),
were developed to deal with the fuzzy shortest path
path 1-2-5-6 with d2=A+D+G=(6,12,17),
problem. In this paper, we present these algorithms
path 1-3-5-6 with d3=B+E+G=(7,14,18).
and make detailed analysis. Besides, some simulation
results are given to show the presented algorithms.
Let us find the shortest path length by fuzzy
These can be helpful to decision-makers as they
shortest length heuristic procedure as follows:
Step 1:
make decision. In the future, we will also try to
U1=d1=(4,8,15), U2= d2=(6,12,17),
develop algorithms for shortest path problem based
on the arclength in a network with discrete and/or
U3= d3=(7,14,18).
continuous type of fuzzy sets.
Step 2:
dmin =(a, b, c)= U1=(a’1, b’1, c’1)= (4,8,15).
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5.References
[1]Jenson, P. and Barnes, J.(1980),” Network Flow
Programming,” John Wiley and Sons, New York.
[2]Dubois, D. and Prade, H.(1980),” Fuzzy Sets and
Systems: Theory and Applications,” Academic
Press, New York.
[3]Martins, E.Q.V.(1984),”On a multicriteria shortest
path problem,” Eur. J. Oper. Res., 16, 236-245.
[4]Ramik, J. and Rimanek, J.(1985),”Inequality
relation between fuzzy numbers and its use in
fuzzy optimization,” Fuzzy Sets and Systems,
16,123-138.
[5]Yager, R. (1986),” Paths of least resistance on
possibilistic production systems,” Fuzzy Sets and
Systems, 19, 121-132.
[6]Broumbaugh-Smith, J. and Shier, D.(1989),” An
empirical investigation of some bicriterion shortest
path algorithm,” Eur. J. Oper. Res., 43, 216-224.
[7]Klein, C.M.(1991),”Fuzzy shortest paths,” Fuzzy
Sets and Systems, 39,27-41.
[8]Lin, K. and Chen, M.(1994),” The fuzzy shortest
path problem and its most vital arcs,” Fuzzy Sets
and Systems, 58, 343-353.
[9]Okada, S. and Soper, T.(2000),” A shortest path
problem on a network with fuzzy lengths,” Fuzzy
Sets and Systems, 109, 129-140.
[10]Chuang, T.-N. and Kung, J.-Y.(2002),” A new
approach for the fuzzy shortest path problem,”
International conference on fuzzy systems and
knowledge discovery, 2, 381-384.
[11]Kaufmann, A.(1985),” Introduction to Fuzzy
Arithmetic,” Van Nostrand Reinhold, New Yor k.
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