Computers & Industrial Engineering 99 (2016) 97–105
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Computers & Industrial Engineering
journal homepage: www.elsevier.com/locate/caie
Shortest path problem of uncertain random network
Yuhong Sheng a, Yuan Gao b,⇑
a
b
College of Mathematics and System Science, Xinjiang University, Urumqi 830046, China
State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, 100044 Beijing, China
a r t i c l e
i n f o
Article history:
Received 27 October 2015
Received in revised form 2 July 2016
Accepted 9 July 2016
Available online 14 July 2016
Keywords:
Shortest path problem
Chance theory
Uncertain random variable
Uncertain random network
a b s t r a c t
The shortest path problem is one of the most fundamental problems in network optimization. This paper
is concerned with shortest path problems in non-deterministic environment, in which some arcs have
stochastic lengths and meanwhile some have uncertain lengths. In order to deal with path problems in
such network, this paper introduces the chance theory and uses uncertain random variable to describe
non-deterministic lengths, based on which two types of shortest path in uncertain random networks
are defined. To obtain the shortest paths, an algorithm derived from the Dijkstra Algorithm is proposed.
Finally, numerical examples are given to illustrate the effectiveness of the algorithms.
Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction
As one of the fundamental problems in network optimization,
the shortest path problem concentrates on finding a path with
minimum distance, time or cost from the source to the destination.
Some other optimization problems, such as transportation, communications and supply chain management, can be considered as
special cases of the shortest path problem. Since the late 1950s,
the shortest path problem and problems stemming from it have
been widely studied, and some successful algorithms have been
proposed, such as Bellman (1958), Dijkstra (1959), Dreyfus
(1969), and Floyd (1962).
In traditional networks, the length(weight) of each arc is deterministic. However, because of failure, maintenance, or other reasons, arc lengths are non-deterministic in many situations. Some
researchers believed that these non-deterministic phenomena conform to randomness, so they introduced probability theory into
network optimization problems and used random variables to
describe the non-deterministic lengths. Random network was first
investigated by Frank and Hakimi (1965) in 1965 for modeling
communication networks with random capacities. From then on,
random networks have been well developed and widely applied.
Many researchers have done lots of work on random shortest path
problems, such as Frank (1969), Nie and Wu (2009), Chen, Lam,
Sumalee, and Li (2012), and Zockaie, Nie, and Mahmassani
(2014). In these literature, the wights of arcs were regarded as random variables, and corresponding models were studied. However,
⇑ Corresponding author.
E-mail addresses: [email protected] (Y. Sheng), [email protected]
(Y. Gao).
http://dx.doi.org/10.1016/j.cie.2016.07.011
0360-8352/Ó 2016 Elsevier Ltd. All rights reserved.
such probability-based methods can effectively work only when
probability distribution functions of non-deterministic phenomena
are exactly obtained. That is to say, if there are not enough observational data to support the non-deterministic phenomena, the
probability distribution functions estimated via statistical methods
will not be close to real situations, which indicates that it is not
suitable to employ random variables to model non-deterministic
phenomena in these cases.
In the cases that there is little or no observational data for nondeterministic phenomena, a feasible and economic way is to estimate the data by experts based on their subjective information
and experiences. As a matter of course, additivity and some other
specific properties of probability calculus fall away when there
are only expert empirical data, since human factors are so imprecise and there is almost no rigorous additive property can be discovered (Kóczy, 1992). To describe expert data in arc lengths,
uncertainty theory (Liu, 2007, 2010) was introduced into network
optimization problems and uncertain variables were employed to
model arc lengths. Uncertain network was first explored by Liu
(2009) for modeling project scheduling problem with uncertain
duration times. In 2011, Gao (2011) deduced the uncertainty
distribution function of the shortest path length, and studied the
a-shortest path and the most measure shortest path in uncertain
networks. In 2014, Han, Peng, and Wang (2014) studied maximum
flow problems in networks with uncertain capacities, and designed
a 99-algorithm to calculate the uncertainty distribution function of
maximum flow. In 2015, Gao, Yang, Li, and Kar (2015) and Gao and
Qin (in press) investigated uncertain graphs, and proposed efficient
algorithms to calculate the distribution function of the diameter
and the edge-connectivity of an uncertain graph. Besides, the
98
Y. Sheng, Y. Gao / Computers & Industrial Engineering 99 (2016) 97–105
uncertain minimum cost flow problem was investigated by Ding
(2014), and the Chinese postman problem was studied by Zhang
and Peng (2012).
In some real cases, uncertainty and randomness simultaneously
appear in a system. Specifically, for some non-deterministic phenomena, we have enough observational data to obtain their probability distribution functions; while for others, we can only
estimate them by expert data. In order to deal with this kind of systems, Liu (2013) proposed chance theory in 2013. Further, Liu
(2014) introduced the chance theory into networks, which involves
both random arc lengths and uncertain arc lengths. In the paper,
Liu proposed a concept of uncertain random network, in which
the lengths of some arcs are random variables and other lengths
are uncertain variables. Furthermore, Liu studied the chance distribution function of minimum length from source node to destination node in uncertain random networks. Besides, models for
maximum flow problems in uncertain random networks were constructed by Sheng and Gao (2014).
This paper will further study the shortest path problem within
the framework of chance theory. The contributions of this paper
are threefold. First, we propose an algorithm, which is derived from
Dijkstra Algorithm, to calculate the chance distribution function of
the minimum length from source node to destination node in an
uncertain random network. Note that in Liu (2014), only a theoretical formula was proposed to calculate the chance distribution
function of the minimum length, which is difficult to use in practice. The proposed algorithm in this paper numerically calculates
the chance distribution function.
Second, two types of shortest paths in uncertain random networks are first proposed, which are simply named as type-I shortest
path and type-II shortest path, respectively. In an uncertain random
network, the length of a path is an uncertain random variable,
instead of a constant. As a result, the traditional concept of shortest
path is not suitable in uncertain random networks any more. The
definitions of type-I and type-II shortest paths are both based on
minimizing deviations between the chance distribution function
of the length of a path and the chance distribution function of
the minimum length from source node to destination node. The
deviations between chance distribution functions are presented
in Section 4 in detail.
Third, optimization models are respectively constructed to
model the proposed types of shortest paths. The properties of the
models are investigated, based on which an algorithm is developed
to solve the model. The effectiveness of the algorithm is illustrated
by several numerical examples.
It should be pointed out that in this paper, we don’t distinguish
risk-averse decision-makers and risk-prone users decision-makers,
that is, we assume that all the decision-makes have the same perspective on uncertainty. Besides, all the uncertain lengths and random lengths are assumed to be independent throughout this
paper. The remainder of this paper is organized as follows. In Section 2, some basic concepts of uncertainty theory and chance theory
used throughout this paper are introduced. Section 3 proposes an
algorithm to calculate the chance distribution function of the minimum length from the source node to the destination node in uncertain random networks. In Section 4, two types of shortest paths in
uncertain random networks are proposed and investigated in details,
and then an algorithm is designed to search the corresponding paths.
In Section 5, a numerical example is given to illustrate the proposed
model and algorithm. Section 6 gives a brief summary to this paper.
deterministic phenomena. Specifically, if we have enough historical or experimental data for the non-deterministic phenomena,
then these phenomena can be described by random variables;
while if there are only expert empirical data to estimate the nondeterministic phenomena, it is better to use uncertain variables
to describe these phenomena.
In this section, we first review some concepts of uncertainty
theory, including uncertain measure, uncertain variable and operational law. Then we introduce some useful definitions and properties about chance theory, such as uncertain random variable and
chance distribution function.
2.1. Uncertainty theory
Let C be a nonempty set, and L be a r-algebra over C. Each element K 2 L is designated a number MfKg. Liu (2007) proposed
four axioms to ensure that the set function MfKg satisfy certain
mathematical properties.
Axiom 1 (Normality). MfCg ¼ 1.
Axiom 2 (Duality). MfKg þ MfKc g ¼ 1 for any event K.
Axiom 3 (Subadditivity). For every countable sequence of events
fKi g, we have
(
)
1
1
X
[
Ki 6
MfKi g:
M
i¼1
i¼1
Axiom 4 (Product Axiom). Let ðCi ; Li ; Mi Þ be uncertainty spaces for
i ¼ 1; 2; . . .. Then the product uncertain measure M is an uncertain
measure satisfying
(
)
1
1
^
Y
Ki ¼ Mi fKi g;
M
i¼1
i¼1
V
where is the maximum value operator, and Ki are arbitrarily chosen events from Li for i ¼ 1; 2; . . ., respectively.
Note that the first three axioms of uncertainty theory also hold
in probability theory, while the product axiom of uncertainty theory is totally different from that of probability theory, which leads
to the difference of these two theories.
To describe a quantity with uncertainty, uncertain variable is
defined analogous to the definition of random variable.
Definition 1 Liu (2007). An uncertain variable n is a measurable
function from an uncertainty space ðC; L; MÞ to the set of real
numbers, i.e., for any Borel set B of real numbers, the set
fn 2 Bg ¼ fc 2 CjnðcÞ 2 Bg
is an event.
Let x 2 R and B ¼ ð1; x. Then UðxÞ ¼ Mfn 2 Bg ¼ Mfn 6 xg is
called the uncertainty distribution function of n. The inverse function U1 is called the inverse uncertainty distribution function of n
if it exists and is unique for each a 2 ð0; 1Þ. Inverse uncertainty distribution function plays a crucial role in operations of independent
uncertain variables.
The following theorem presents the operational law of independent uncertain variables, which gives the uncertainty distribution
function of n ¼ f ðn1 ; n2 ; . . . ; nn Þ.
2. Preliminaries
Recall that uncertainty theory and probability theory are two
different mathematical tools to describe and model non-
Theorem 1 Liu (2010). Let n1 ; n2 ; . . . ; nn be independent uncertain
variables with uncertainty distribution functions U1 ; U2 ; . . . ; Un ,
respectively. If f is strictly increasing function, then
Y. Sheng, Y. Gao / Computers & Industrial Engineering 99 (2016) 97–105
n ¼ f ðn1 ; n2 ; . . . ; nn Þ
has a chance distribution function
is an uncertain variable with uncertainty distribution function
UðxÞ ¼
UðxÞ ¼
min Ui ðxi Þ:
sup
f ðx1 ;x2 ;...;xn Þ¼x 16i6n
Using inverse uncertain distribution function, the operation law
can be formulated as
Theorem 2 Liu (2010). Let n1 ; n2 ; . . . ; nn be independent uncertain
variables with uncertainty distribution functions U1 ; U2 ; . . . ; Un ,
respectively. If f ðn1 ; n2 ; . . . ; nn Þ is strictly increasing with respect to
n1 ; n2 ; . . . ; nm and strictly decreasing with respect to nmþ1 ; nmþ2 ; . . . ; nn ,
then n ¼ f ðn1 ; n2 ; . . . ; nn Þ is an uncertain variable with an inverse
uncertainty distribution function
1
1
1
U1 ðaÞ ¼ f U1
1 ðaÞ; . . . ; Um ðaÞ; Umþ1 ð1 aÞ; . . . ; Un ð1 aÞ :
Obviously, the inverse function of U1 is the uncertain distribution function of n, i.e., U.
2.2. Chance theory
The chance space is the product of ðC; L; MÞ ðX; A; PrÞ, in
which ðC; L; MÞ is an uncertainty space and ðX; A; PrÞ is a probability space.
Definition 2 Liu (2013). Let ðC; L; MÞ ðX; A; PrÞ be a chance
space, and let H 2 L A be an uncertain random event. Then the
chance measure of H is defined as
ChfHg ¼
Z
1
Prfx 2 XjMfc 2 Cjðc; xÞ 2 Hg P rgdr:
0
Liu (2013) proved that a chance measure satisfies normality,
duality, and monotonicity properties, that is (i) ChfC Xg ¼ 1;
(ii) ChfHg þ ChfHc g ¼ 1 for any event H; (iii) ChfH1 g 6 ChfH2 g
for any real number set H1 H2 . Besides, Hou (2014) proved the
subadditivity of chance measure, that is,
Ch
(
1
[
i¼1
)
Hi
6
1
X
ChfHi g
i¼1
for a sequence of events H1 ; H2 ; . . ..
Definition 3 Liu (2013). An uncertain random variable is a
function n from a chance space ðC; L; MÞ ðX; A; PrÞ to the set of
real numbers such that fn 2 Bg is an event in L A for any Borel
set B of real numbers.
Definition 4 Liu (2013). Let n be an uncertain random variable.
Then its chance distribution function is defined by
UðxÞ ¼ Chfn 6 xg
for any x 2 R.
The chance distribution function of a random variable is just its
probability distribution function, and the chance distribution function of an uncertain variable is just its uncertainty distribution
function. The following theorem presents the operational law of
uncertain random variables.
Theorem 3 Liu (2013). Let g1 ; g2 ; . . . ; gm be independent uncertain
random
variables
with
probability
distribution
functions
G1 ; G2 ; . . . ; Gm , respectively, and let s1 ; s2 ; . . . ; sn be uncertain variables. Then the uncertain random variable
n ¼ f ðg1 ; g2 ; . . . ; gm ; s1 ; s2 ; . . . ; sn Þ
99
Z
Rm
Fðx; y1 ; . . . ; ym ÞdG1 ðy1 Þ . . . dGm ðym Þ
where Fðx; y1 ; . . . ; ym Þ is the uncertainty distribution function of uncertain variable f ðy1 ; y2 ; . . . ; ym ; s1 ; s2 ; . . . ; sn Þ for any real numbers
y1 ; y2 ; . . . ; ym .
3. The chance distribution of the minimum length
In this section, we introduce the definition of uncertain random
networks and the chance distribution function of the minimum
length from source node to destination node. An algorithm for calculating the chance distribution function is proposed. First, some
assumptions are listed as follows.
(1) There is only one source node and only one destination node.
(2) The length of each arc ði; jÞ is finite.
(3) The length of each arc ði; jÞ is either a positive uncertain variable or a positive random variables.
(4) All the uncertain variables and the random variables are
independent.
Assume that the node set of the uncertain random network is
N ¼ f1; 2; . . . ; ng, the uncertain arc set is U and the random arc
set is R, i.e.,
U ¼ fði; jÞjarc ði; jÞ has uncertain lengthg;
R ¼ fði; jÞjarc ði; jÞ has random lengthg:
In this paper, all deterministic arcs are regarded as special
uncertain arcs. Let nij denote the length of arc ði; jÞ; ði; jÞ 2 U [ R.
Then nij are uncertain variables if ði; jÞ 2 U, and nij are random variables if ði; jÞ 2 R. Since the length of each arc is assumed to be
finite, we have aij 6 nij 6 bij , where aij and bij are the lower bound
and upper bound of nij respectively. For simplicity, we define
n ¼ fnij g, for ði; jÞ 2 U [ R.
Definition 5 Liu (2014). Assume N is the set of nodes, U is the set
of uncertain arcs, R is the set of random arcs, and n is the set of
uncertain and random lengths. Then the quartette ðN ; U; R; nÞ is
called an uncertain random network.
The uncertain random network becomes a random network
(Frank & Hakimi, 1965) if all arc lengths are random variables;
and becomes an uncertain network (Liu, 2010) if all arc lengths
are uncertain variables.
In a deterministic network ðN ; A; WÞ, where A is the set of arcs
and W ¼ fwij jði; jÞ 2 Ag is the set of arc lengths, the shortest path
from the source node to the destination node can be easily found
by the Dijkstra Algorithm (Dijkstra, 1959). Obviously, the length
of the shortest path, i.e., f ðWÞ, is an increasing function of length
wij , for all ði; jÞ 2 A. Actually, it is easy to verify that
f ðw1 Þ 6 f ðw2 Þ;
where w1 ¼ fwij jði; jÞ 2 Ag; w2 ¼ fwij jði; jÞ 2 Ag, and wij 6 wij , for all
ði; jÞ 2 A.
For an uncertain random network ðN ; U; R; nÞ, since the length
of each arc is non-deterministic, the concept of the shortest path
is different from the deterministic network. With the length taking
different values, the shortest path from the source node to the destination node varies, which indicates that we may not find a path
that is always the shortest. However, the chance distribution function of the minimum length can be obtained by the following
method.
100
Y. Sheng, Y. Gao / Computers & Industrial Engineering 99 (2016) 97–105
Theorem 4 Liu (2014). For uncertain random network ðN ; U; R; nÞ,
assume the uncertain lengths nij has uncertainty distribution function
!ij for any ði; jÞ 2 U, and the random lengths wij has probability
distribution function Gij for any ði; jÞ 2 R, respectively. Then the
chance distribution function of the minimum length from the source
node to the destination node is
UðxÞ ¼
Z
þ1
Z
þ1
...
0
Fðx; yij ; ði; jÞ 2 RÞ
0
Y
dGij ðyij Þ;
ð1Þ
ði;jÞ2R
where Fðx; yij ; ði; jÞ 2 RÞ is determined by its inverse uncertainty
distribution function
F 1 ða; yij ; ði; jÞ 2 RÞ ¼ f ðcij ; ði; jÞ 2 U [ RÞ;
(
cij ¼
1
ij ð
!
aÞ; if ði; jÞ 2 U
yij ;
if
ði; jÞ 2 R
Fig. 1. Network ðN ; U; R; nÞ for Example 1.
ð2Þ
ð3Þ
and f is the length of shortest path of the deterministic network
ðN ; U [ R; WÞ, where W ¼ fcij g. Obviously, f can be calculated by the
Dijkstra Algorithm.
The following theorem is obvious.
Theorem 5. Let P be a path from the source node to the destination
node of an uncertain random network ðN ; U; R; nÞ and let WðzÞ be the
chance distribution function of the length of path P. Then we always
have
WðxÞ 6 UðxÞ
where UðxÞ is the chance distribution function of the minimum length
from the source node to the destination node.
Note that formula (1) is only a theoretical formula to calculate
UðxÞ, which is difficult to use. In order to calculate the chance distribution function of the minimum length from source node to destination node in an uncertain random network, we propose the
following algorithm:
UðxÞ ¼
Z
þ1
0
and a 2 A, calculate F 1 ða; yij ; ði; jÞ 2 RÞ according to formulas (2) and (3). Then, obtain the discrete form of
Fðx; yij ; ði; jÞ 2 RÞ.
Step 3. Obtain the uncertainty distribution function of
Fðx; yij ; ði; jÞ 2 RÞ from its discrete form via linear
interpolation.
Step 4. Input Fðx; yij ; ði; jÞ 2 RÞ into formula (1), then obtain the
chance distribution function UðxÞ.
Obviously, Algorithm 1 is only an approximate method. However, the smaller the steps of the partitions are, the more precise
chance distribution function Algorithm 1 obtain. To illustrate Algorithm 1, we present the following example.
Example 1. Uncertain random network ðN ; U; R; nÞ has 4 nodes
and 4 arcs, which is shown in Fig. 1. Assume that the uncertain
lengths s24 ; s34 have regular uncertainty distribution functions
!24 ; !34 , and the random lengths n12 ; n13 have probability distribution functions G12 ; G13 , respectively. Then the minimum length
from node 1 to node 4 is an uncertain random variable, that is,
ðn12 þ s24 Þ ^ ðn13 þ s34 Þ. According to Theorem 4, the chance distribution function of the minimum length can be obtained by
þ1
0
Fðx; y12 ; y13 ÞdG12 ðy12 ÞdG13 ðy13 Þ;
where Fðx; y12 ; y13 Þ is the uncertainty distribution function of uncertain variable ðy12 þ s24 Þ ^ ðy13 þ s34 Þ. Given a 2 ð0; 1Þ, the inverse
uncertainty distribution function of Fðx; y12 ; y13 Þ can be calculated
by
F 1 ða; y12 ; y13 Þ ¼ ðy12 þ !24 ðaÞÞ ^ ðy13 þ !34 ðaÞÞ:
1
1
For simplicity, assume that the uncertain lengths follow the linear distribution function, i.e., s24 Lð2; 5Þ; s34 Lð3; 4Þ, and the
random lengths follow the uniform distribution function, i.e.,
n12 Uð1; 3Þ; n13 Uð2; 3Þ. For given a 2 ð0; 1Þ, the inverse uncertainty distribution function of s24 and s34 are respectively
!1
24 ðaÞ ¼ 2 þ 3a;
!1
34 ðaÞ ¼ 3 þ a:
The probability distribution function of n12 and n13 are
respectively
8
0;
>
>
>
>
>
>
<
G12 ðy12 Þ ¼
>
>
>
>
>
>
:
Algorithm 1.
Step 1. Discretize the length of each random arc. To be more precise, for random arc ði; jÞ 2 R, give a partition Pij on interval ½aij ; bij with step D1 ¼ 0:01. Then, random length nij
takes value in Pij .
Step 2. Denote A as a partition on interval ð0; 1Þ. For all yij 2 Pij
Z
if
y12 < 1
y12 1
;
2
if
1 6 y12 < 3
1;
if
y12 P 3;
and
8
0;
if
>
>
>
>
>
>
<
G12 ðy12 Þ ¼ y13 2; if
>
>
>
>
>
>
:
1;
if
y12 < 2
2 6 y12 < 3
y12 P 3:
Then, the chance distribution function of the minimum length
can be reformulated as
UðxÞ ¼
1
2
Z
1
3
Z
2
3
Fðx; y12 ; y13 Þdy12 dy13 ;
ð4Þ
where Fðx; y12 ; y13 Þ is determined by the inverse uncertainty distribution function
F 1 ða; y12 ; y13 Þ ¼ ðy12 þ ð2 þ 3aÞÞ ^ ðy13 þ ð3 þ aÞÞ;
ð5Þ
for each given a 2 ð0; 1Þ.
Recall that formula (4) is only a theoretical formula. Next, we
use Algorithm 1 to obtain the chance distribution function.
Give a partition P12 on interval ½1; 3 with step 0:2, and let random variable n12 take values in fy12 jy12 ¼ 1 þ 0:2 i; for i ¼
1; 2; . . . 10g; give a partition P13 on interval ½2; 3 with step 0:1
and let random variable n13 take values in fy13 jy13 ¼ 2þ
0:1 j; for j ¼ 1; 2; . . . ; 10g. For any y12 2 P12 ; y13 2 P13 , given
a 2 f0:1; 0:2; . . . ; 0:9g, we calculate F 1 ða; y12 ; y13 Þ by formula (5).
Table 1 lists the value of F 1 ða; y12 ; y13 Þ.
101
Y. Sheng, Y. Gao / Computers & Industrial Engineering 99 (2016) 97–105
Table 1
The inverse uncertainty distribution function F 1 ða; y12 ; y13 Þ.
UðxÞ ¼
Z
þ1
0
ROW
i
j
y12 = 1 + 0.2 ⁄ i
y13 = 2 + 0.1 ⁄ j
a
F1 (a; y12, y13)
1
2
...
10
11
12
...
20
...
91
92
...
100
1
1
...
1
2
2
...
2
...
10
10
...
10
1
2
...
10
1
2
...
10
...
1
2
...
10
1.2
1.2
...
1.2
1.4
1.4
...
1.4
...
3.0
3.0
...
3.0
2.1
2.2
...
3.0
2.1
2.2
...
3.0
...
2.1
2.2
...
3.0
0.1
0.1
...
0.1
0.1
0.1
...
0.1
...
0.1
0.1
...
0.1
3.5
3.5
...
3.5
3.7
3.7
...
3.7
...
5.2
5.3
...
5.3
101
...
200
1
...
10
...
1
...
10
...
1.2
...
3.0
...
2.1
...
3.0
...
0.2
...
0.2
...
3.8
...
5.6
...
801
...
900
1
...
10
1
...
10
1.2
...
3.0
2.1
...
3.0
0.9
...
0.9
5.9
...
6.9
Fixing y12 2 P12 and y13 2 P13 , we can obtain the uncertain dis-
tribution function Fðx; y12 ; y13 Þ form F 1 ða; y12 ; y13 Þ via linear interpolation. To be more precise, fixing i and j, let a take values from
f0:1; 0:2; . . . ; 0:9g, then we obtain F 1 ða; i; jÞ ¼ xk ; k ¼ 1; 2; . . . ; 9.
We use linear interpolation to obtain a continuous function of
the uncertain distribution function Fðx; 1 þ 0:2 i; 2 þ 0:1 jÞ as
follows:
Fðx; 1 þ 0:2 i; 2 þ 0:1 jÞ
8
0;
if
>
>
>
>
>
>
<
¼ ak þ ðakþ1 ai Þ ðxðxxxk Þ Þ ; if
kþ1
k
>
>
>
>
>
>
:
1;
if
Z
x 6 x1
xk 6 x 6 xkþ1 ; 1 6 k 6 9
x P x9
where ak ¼ 0:1 k.
At last, we obtain the chance distribution function of the minimum length, i.e.,
¼
¼
1
2
þ1
Fðx; y12 ; y13 ÞdG12 ðy12 ÞdG13 ðy13 Þ
0
Z
Z
3
1
2
3
Fðx; y12 ; y13 Þdy12 dy13
10 X
10
1X
Fðx; 1 þ 0:2 i; 2 þ 0:1 jÞ 0:2 0:1:
2 i¼1 j¼1
The chance distribution function of the minimum length from
node 1 to node 4 is shown in Fig. 2.
4. Models of the shortest path in uncertain random networks
In this section, we propose two concepts of shortest path in
uncertain random networks and then investigate their properties.
A path from the source node to the destination node is represented by p ¼ fxij ; ði; jÞ 2 U [ Rg, where xij ¼ 0 indicates that arc
ði; jÞ is not in the path and xij ¼ 1 indicates that arc ði; jÞ is in the
path. Then the length of the path fxij ; ði; jÞ 2 U [ Rg is
lðpÞ ¼
X
xij nij :
ði;jÞ2U[R
It is clear that lðpÞ is an uncertain random variable. Since uncertain random variables cannot be directly compared, we propose
the following two concepts of shortest path in an uncertain random network:
Definition 6. Let ðN ; U; R; nÞ be an uncertain random network, and
let UðzÞ be the chance distribution function of the minimum length
from the source node to the destination node. Assume P is the set
of paths from the source node to the destination node, and for each
p 2 P, the chance distribution function of its length is Wp ðzÞ. The
type-I deviation between the length of path p and the minimum
length is defined as
Dev 1 ðpÞ ¼
Z
þ1
0
UðzÞ Wp ðzÞ dz:
For path p 2 P, if
Fig. 2. The chance distribution function of the minimum length.
102
Z
þ1
Y. Sheng, Y. Gao / Computers & Industrial Engineering 99 (2016) 97–105
0
Z
UðzÞ Wp ðzÞ dz ¼ min Dev 1 ðpÞ
p2P
Z
þ1
¼ min
p2P
0
þ1
0
UðzÞ Wp ðzÞ dz;
then path p is called type-I shortest path in the uncertain random
network.
Definition 7. Let ðN ; U; R; nÞ be an uncertain random network, and
let UðzÞ be the chance distribution function of the minimum length
from the source node to the destination node. Assume P is the set
of paths from the source node to the destination node, and for each
p 2 P, the chance distribution function of its length is Wp ðzÞ. The
type-II deviation between the length of path p and the minimum
length is defined as
Z
þ1
0
fUðzÞ W1 ðzÞgdz ¼ 1:27;
fUðzÞ W2 ðzÞgdz ¼ 6:22:
Obviously, the first integral is smaller than the second one.
According to Definition 7, we find that the type-II shortest path
is Path1 with 0:11, i.e.,
max fUðzÞ W1 ðzÞg ¼ 0:11;
06z<þ1
max fUðzÞ W2 ðzÞg ¼ 0:38:
06z<þ1
Dev 2 ðpÞ ¼ max UðzÞ Wp ðzÞ :
Obviously, the first ‘‘max” value is smaller than the second one.
Typically, p ¼ fxij ; ði; jÞ 2 U [ Rg is a path from source node 1 to
destination node n if and only if
For path p 2 P, if
8
>
>
>
<P
06z<þ1
max UðzÞ Wp ðzÞ ¼ min Dev 2 ðpÞ ¼ min max UðzÞ Wp ðzÞ ;
p2P
06z<þ1
p2P 06z<þ1
then the path p is called type-II shortest path in the uncertain random network.
Remark 1. It should be pointed out that type-I deviation measures
the accumulative deviation between chance distribution function
U and chance distribution function Wp , while type-II deviation
measures the maximum deviation between U and Wp . Generally,
if path p1 has a larger type-I deviation than path p2 , then p1 has
a larger type-II deviation, and vice versa.
We still take the uncertain random network ðN ; U; R; nÞ presented in Example 1 as an example. There are two paths from node
1 to node 4, namely, Path1 (1 ! 2 ! 4) and Path2 (1 ! 3 ! 4). By
Algorithm 1, we can obtain the chance distribution function of
minimum length from node 1 to node 4, i.e., U. Similarly, we can
obtain the chance distribution function of Path1 and Path2, i.e.,
W1 and W2 , respectively. Fig. 3 shows the chance distribution function of minimum length, the length of Path1 and the length of
Path2. In Fig. 3, the symbol ‘‘M.L.” represents ‘‘the minimum
length”.
According to Definition 6, we find that the type-I shortest path
is Path1 with 1:27, i.e.,
8
i¼1
>
< 1;
x
x
¼
0;
26i6n1
ji
j:ði;jÞ2U[R ij
j:ðj;iÞ2calU[R
>
:
>
1; i ¼ n
>
>
:
xij ¼ f0; 1g; 8ði; jÞ 2 U [ R:
P
ð6Þ
According to Definition 6, the type-I shortest path
pI ¼ fxij ; ði; jÞ 2 U [ Rg from node 1 to node n is the optimal solution to the following model:
8
R þ1 minfxij ;ði;jÞ2U[Rg 0
UðzÞ Wp ðzÞ dz
>
>
>
>
>
s:t:
>
>
8
>
<
i¼1
>
< 1;
P
P
>
x
x
¼
0;
26i6n1
ji
>
j:ði;jÞ2U[R ij
j:ðj;iÞ2U[R
>
>
>
:
>
>
1; i ¼ n
>
>
:
xij ¼ f0; 1g; 8ði; jÞ 2 U [ R
ð7Þ
where UðzÞ is the chance distribution function of the minimum
nP
o
length from node 1 to node n and Wp ðzÞ ¼ Ch
ði;jÞ2U[R xij nij 6 z
is chance distribution function of the length of path
p ¼ fxij ; ði; jÞ 2 U [ Rg from node 1 to node n.
Considering the definition of chance distribution function, the
following theorem reformulates model (7).
Fig. 3. The chance distribution functions.
103
Y. Sheng, Y. Gao / Computers & Industrial Engineering 99 (2016) 97–105
Theorem 6. Let gij be independent random variables with probability
distribution functions Gij ; ði; jÞ 2 R, respectively, and let sij be independent uncertain variables with uncertainty distribution functions
!ij ; ði; jÞ 2 U, respectively. The model (7) can be reformulated as the
following model:
8
R þ1 R þ1
minfxij ;ði;jÞ2U[Rg 0
UðzÞ 1 !ðz yÞdGðyÞ dz
>
>
>
>
>
s:t:
>
>
8
>
<
i¼1
>
< 1;
P
P
>
xji ¼ 0;
26i6n1
>
j:ði;jÞ2U[R xij j:ðj;iÞ2U[R
>
>
>
:
>
>
1;
i¼n
>
>
:
xij ¼ f0; 1g; 8ði; jÞ 2 U [ R
ð8Þ
(
Z
UðzÞ ¼
þ1
Z
...
0
Z
þ1
0
dGij ðyij Þ;
ði;jÞ2R
Y
X
GðyÞ ¼
Fðz; yij ; ði; jÞ 2 RÞ
Y
Z
Pr x 2 XjMfc 2 Cj
0
Z
Pr x 2 XjMf
¼
0
Z
min !ij ðxij r ij Þ:
x r ¼r
ði;jÞ2U ij ij
ði;jÞ2U
Proof. According to Theorem 4, the chance distribution function of
the minimum length from node 1 to node n can be written as
UðzÞ ¼
Z
þ1
0
Z
þ1
...
0
X
X
þ1
1
þ1
Z
¼
)
xij sij ðcÞ 6 zg P r dr
X
)
xij sij 6 zg P r dr
ði;jÞ2U
þ1
1
Mfs 6 z ygdGðyÞ
!fz ygdGðyÞ:
(
Z
Rjði;jÞ2Rj
X
! z
)
xij yij
ði;jÞ2R
Z
¼
Z
Mfy þ s 6 zgdGðyÞ ¼
Further, we can obtain
WðzÞ ¼
X
ði;jÞ2U
xij gij ðxÞ þ
0
¼
xij sij 6 z
Prfx 2 XjMfgðxÞ þ s 6 zg P rgdr
¼
Z
xij gij ðxÞ þ
ði;jÞ2R
1
X
ði;jÞ2U
ði;jÞ2R
(
1
xij gij þ
ði;jÞ2R
(
1
¼
X
xij nij 6 z ¼ Ch
)
jði;jÞ2Rj P
supP
x r ¼z
ði;jÞ2U ij ij
Y
dGij ðxij yij Þ
ði;jÞ2R
min !ij ðxij r ij Þ
x y
ði;jÞ2R ij ij
ði;jÞ2U
Y
dGij ðxij yij Þ:
ði;jÞ2R
where jði; jÞ 2 Rj denotes numbers of random edges.
The proof is completed. h
According to Definition 7, the type-II shortest path
pII ¼ fxij ; ði; jÞ 2 U [ Rg from node 1 to node n is the optimal solution to the following model:
xij yij 6y ði;jÞ2R
ði;jÞ2R
!ðrÞ ¼ P sup
(
ði;jÞ2U[R
R
dGij ðxij yij Þ;
X
WðzÞ ¼ Ch
1
where the distribution functions UðzÞ; GðyÞ and !ðrÞ can be obtained
by
)
Fðz; yij ; ði; jÞ 2 RÞ
Y
dGij ðyij Þ;
ði;jÞ2R
8
minfxij ;ði;jÞ2U[Rg max06z<þ1 fUðzÞ WðzÞg
>
>
>
>
>
s:t:
>
8
>
>
<
i¼1
>
< 1;
P
P
x
x
¼
0;
26i6n1
>
ij
ji
j:ði;jÞ2U[R
j:ðj;iÞ2U[R
>
>
>
>
:
>
>
1;
i¼n
>
>
:
xij ¼ f0; 1g; 8ði; jÞ 2 U [ R
ð9Þ
where Fðx; yij ; ði; jÞ 2 RÞ is determined by its inverse uncertainty distribution function
F 1 ða; yij ; ði; jÞ 2 RÞ ¼ f ðcij ; ði; jÞ 2 U [ RÞ;
cij ¼
8 1
>
< !ij ðaÞ; if
>
:
yij ;
if
ði; jÞ 2 U
ði; jÞ 2 R
and
probability
distribution
functions
Gij ; ði; jÞ 2 R
xij ¼ f0; 1g; 8ði; jÞ 2 R, the probability distribution function of the
P
random variable g ¼ ði;jÞ2R xij gij is
Z
X
Y
dGij ðxij yij Þ:
xij yij 6y ði;jÞ2R
ði;jÞ2R
And since sij are independent uncertain variables with uncertainty
distribution functions !ij and xij ¼ f0; 1g; 8ði; jÞ 2 U, we have the
uncertainty distribution function of the uncertain variable
P
s ¼ ði;jÞ2U xij sij by Theorem 2. That is
!ðrÞ ¼ Mfs 6 rg ¼ P sup
x r ¼r
ði;jÞ2U ij ij
tion of the length of path fxij ; ði; jÞ 2 U [ Rg from node 1 to node n.
Similarly, the following theorem reformulates model (9).
Theorem 7. The model (9) can be reformulated as the following
model:
and f is calculated by the Dijkstra algorithm for each given a.
Since gij ; ði; jÞ 2 R are independent random variables with
GðyÞ ¼ Prfg 6 yg ¼
where UðzÞ is the chance distribution function of the minimum
nP
o
length and WðzÞ ¼ Ch
ði;jÞ2U[R xij nij 6 z is chance distribution func-
min !ij ðxij r ij Þ:
ði;jÞ2U
According to Definition 1 and Theorem 3, we have
(
!
)
8
X
R þ1
>
>
>
minfxij ;ði;jÞ2U[Rg max06z<þ1 UðzÞ 1 ! z xij yij dGðyÞ
>
>
>
>
ði;jÞ2R
>
>
>
>
< s:t:
8
i¼1
> 1;
<
>
P
P
>
>
>
xji ¼ 0;
26i6n1
>
j:ði;jÞ2U[R xij j:ðj;iÞ2U[R
>
>
>
:
>
>
1;
i¼n
>
>
:
xij ¼ f0; 1g; 8ði; jÞ 2 U [ R
ð10Þ
where the distribution functions UðzÞ; GðyÞ and !ðrÞ can be obtained by
UðzÞ ¼
Z
Z
GðyÞ ¼
þ1
Z
þ1
...
0
Fðz; yij ; ði; jÞ 2 RÞ
0
X
x r ¼r
ði;jÞ2U ij ij
dGij ðyij Þ;
ði;jÞ2R
Y
dGij ðxij yij Þ;
xij yij 6y ði;jÞ2R
ði;jÞ2R
!ðrÞ ¼ P sup
Y
min !ij ðxij rij Þ:
ði;jÞ2U
104
Y. Sheng, Y. Gao / Computers & Industrial Engineering 99 (2016) 97–105
In order to find the type-I and type-II shortest path in uncertain
random network ðN ; U; R; nÞ, we propose the following algorithm.
Algorithm 2.
Step 1. By Algorithm 1, calculate and save the value of chance distribution function of the minimum length from node 1 to
node n.
Step 2. Using the breadth first search algorithm (BFS), obtain all
paths in the uncertain random network.
Step 3. By Theorems 6 and 7, calculate type-I and type-II deviations between the chance distribution function of minimum length and that of each path obtained in Step 2
respectively.
Step 4. Compare the deviations, and choose the minimum value,
whose corresponding path is the shortest path.
5. Numerical example
In this section, we give an example to illustrate the algorithms
presented above. The uncertain random network ðN ; U; R; WÞ is
shown in Fig. 4, with 10 nodes and 15 arcs.
Denote the chance distribution function of the minimum length
from node 1 to node 10 as UðzÞ. The arc information is listed in
Table 2, in which nij are random variables, gij are uncertain variables. In Table 2, Uða; bÞ represents uniformly probability distribution function, Lða; bÞ represents linear uncertainty distribution
function and Zða; b; cÞ represents zigzag uncertainty distribution
Fig. 4. An uncertain random network ðN ; U; R; nÞ.
Table 2
Probability distributions and uncertainty distributions of arc lengths.
arc ði; jÞ
nij
arc ði; jÞ
gij
(1,2)
(2,5)
(1,3)
(3,5)
(6,9)
(6,10)
(7,9)
–
Uð14; 16Þ
Uð15; 18Þ
Uð16; 17Þ
Uð11; 13Þ
Uð12; 14Þ
Uð24; 25Þ
Uð8; 13Þ
–
(1,4)
(3,6)
(3,7)
(4,7)
(5,8)
(6,8)
(8,10)
(9,10)
Lð11; 13Þ
Zð14; 16; 17Þ
Lð12; 13Þ
Lð13; 14Þ
15
Lð11; 14Þ
Zð12; 15; 16Þ
Lð20; 22Þ
Table 3
Numerical result.
Paths
Path line
Type-I deviation
Type-II deviation
Path1
Path2
Path3
Path4
Path5
Path6
Path7
1 ! 2 ! 5 ! 8 ! 10
1 ! 3 ! 5 ! 8 ! 10
1 ! 3 ! 6 ! 8 ! 10
1 ! 3 ! 6 ! 10
1 ! 3 ! 6 ! 9 ! 10
1 ! 3 ! 7 ! 9 ! 10
1 ! 4 ! 7 ! 9 ! 10
4.48
1.83
3.54
0.44
10.18
4.39
1.03
0.92
0.57
0.69
0.15
0.99
0.90
0.33
function. If the length of arc ði; jÞ is a constant, we simply regard
it as a special uncertain variable. For example, the length of arc
ð5; 8Þ is 15, which is regarded as an uncertain variable.
Using the breadth first search algorithm, we can find all paths
from node 1 to node 10, which are listed in the second column
of Table 3. The chance distribution functions of these paths can
be obtained, which are shown in Fig. 5.
According to Theorems 6 and 7, we calculate the type-I and
type-II deviations of these paths, which are presented in the third
column and the fourth column of Table 3 respectively. According to
Definition 6, the type-I shortest path is Path4, i.e., 1 ! 3 ! 6 ! 10,
the type-I deviation of which is 0:44. According to Definition 7, the
type-II shortest path is Path4, i.e., 1 ! 3 ! 6 ! 10, the type-II deviation of which is 0:15.
If we sort the paths in column 1 of Table 3 in the ascending
order of type-I deviation and type-II deviation respectively, we
can obtain the same sequence of paths, which is consistent with
Remark 1.
Fig. 5. The chance distribution function of all paths.
Y. Sheng, Y. Gao / Computers & Industrial Engineering 99 (2016) 97–105
105
6. Conclusions
References
In an uncertain random network, some arc lengths are uncertain variables while some are random variables. Under the framework of chance theory, this paper designed an algorithm to
calculate the chance distribution function of minimum length from
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of the shortest path is no longer suitable in uncertain random networks, this paper proposed two types of shortest path in uncertain
random networks, namely type-I and type-II shortest paths, the
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optimal paths, an algorithm was designed, the effectiveness of
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The main drawback of this paper is that the first step of the
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Acknowledgements
The first author was supported by the National Natural
Science Foundation of China (Nos. 61462086 and 61563050),
and by Xinjiang University No. BS150206. The second author
was supported by the National Natural Science Foundation of
China (Nos. 71401008, 71401007 and 71571018), State Key
Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong
University (No. RCS2015ZZ003), and China Scholarship
Council.