International
Journal7:12
of Uncertainty,
Fuzziness and Knowledge-Based
Systems
April 25, 2013
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FILE
Uncertain*-maximum*Flow*Model
c World Scientific Publishing Company
⃝
The α-maximum Flow Model with Uncertain Capacities
Sibo Ding∗
Uncertainty Operations Research Laboratory, School of Management
Henan University of Technology, Zhengzhou 450001, China
[email protected]
Uncertain theory is a new tool to deal with the maximum flow problem with uncertain arc
capacities. This paper investigates uncertain maximum flow problem and presents the uncertainty
distribution of the maximum flow. Uncertain α-maximum flow model is formulated. It is proved
that there exists an equivalence relationship between uncertain α-maximum flow model and the
classic deterministic maximum flow model, which builds a bridge between uncertain maximum
flow problem and deterministic maximum flow problem. Furthermore, some important properties
of the model are analyzed, based on which a polynomial exact algorithm is proposed. Finally, a
numerical example is presented to illustrate the model and the algorithm.
Keywords: Maximum flow problem; Uncertainty theory; Uncertain Programming; Generic preflow-push algorithm.
1. Introduction
The maximum flow problem is one of the core issues of network optimization and has been widely
studied. This problem was first investigated by Fulkerson and Dantzig 1 . Then, Ford and Fulkerson 2
solved it using augmenting path algorithm. Motivated by a desire to develop a method with improved
worst-case complexity, Dinic 3 introduced the concept of layered networks. His algorithm proceeds by
augmenting flows along directed paths from source to sink in the layered network. Edmonds and Karp 4
also independently proposed that the Ford and Fulkerson algorithm augments flow along shortest paths.
Until this point all maximum flow algorithms were augmenting path algorithms. However, the augmenting
path algorithm could be slow because it might perform a large number of augmentations. In order to
reduce the number of augmentations, Karzanov 5 introduced the first preflow-push algorithm on layered
networks. Goldberg and Tarjan 6 constructed distance labels instead of layered networks to improve the
running time of preflow-push algorithm. They described a very flexible generic preflow-push algorithm
that performs push and relabel operations at active nodes. Their algorithm can examine active nodes in
any order.
In practice, flow capacities may change over time in communication network or transportation network,
and one may assume network arcs have different values (capacity, cost) that are random variables with
known probability distributions. As an extension of deterministic maximum flow problem, the stochastic
maximum flow problem has been investigated extensively. Frank and Hakimi 7 assumed that each branch
∗ Crresponding
author
1
2
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in communication network has a random capacity and attempted to find the probability of a flow between
vertices. Frank and Frisch 8 considered how to determine the maximum flow probability distribution in
networks where each capacity is a continuous random variable. Furthermore, Doulliez 9 studied multiterminal network with discrete probabilistic branch capacities. In addition, some researchers have tried
to give lower and upper bounds on the expected maximum flow. Onaga 10 derived an upper bound in
general undirected or directed networks, while Carey and Hendrickson 11 presented a efficient method to
find a lower bound in general directed networks. After that, Nagamochi and Ibaraki 12 provided necessary
and sufficient conditions for Carey and Hendrickson’s lower bound.
In reality, however, there exists indeterminacy about the parameters (capacities, costs) of maximum
flow problems. That indeterminacy cannot be described by random variable because no samples are
available. For instance, when networks subject to extreme events such as earthquakes, it is impossible
to get probability distribution of arc capacities. If we insist on using probability theory to deal with
indeterminacy, counterintuitive results will occur 13 . But experts can estimate, based on their experience,
the belief degree that arc capacities are less than or equal to a given value. In order to deal with this
kind of human uncertainty, Liu 14 founded uncertainty theory and refined it 15 . Since then, uncertainty
theory and its application have experienced explosive growth. Peng and Iwamura 16 derived sufficient
and necessary condition for uncertainty distribution. Liu and Ha 17 developed a formula for calculating
the expected values of monotone functions of uncertain variables. In order to deal with mathematical
programming involving with uncertain parameters, Liu 18 first proposed uncertain programming theory
to model uncertain optimization problems. After that, Liu and Yao 19 introduced an uncertain multilevel
programming for modeling uncertain decentralized decision systems, and Liu and Chen 20 developed an
uncertain multiobjective programming and an uncertain goal programming.
Some attempts have been made to collect expert’s experimental data and get uncertain distribution.
Liu 21 suggested the principle of least squares to estimate the unknown parameters of uncertainty distribution. Moreover, when a number of experts are available, Wang, Gao and Guo 22 applied the Delphi
method to determine the uncertainty distributions. Chen and Ralescu 23 used B-spline method to estimate
the uncertainty distribution.
Other particular care has been taken to investigate uncertain graph and uncertain network. The
connectedness index of uncertain graph was proposed by Gao and Gao 24 . Zhang and Peng 25 suggested
a method to calculate Euler index of uncertain graph. Gao 26 computed cycle index of uncertain graph.
As an important contribution, Liu 21 first introduced uncertainty theory into network optimization.
He studied project scheduling problem with uncertain duration times. Furthermore, Liu 27 assumed
uncertainty and randomness simultaneously appear in a complex network, and put forward the concept
of uncertain random network. Gao 28 gave an equivalence relation between the uncertain α-shortest path
and the deterministic shortest path. And, Han and Peng 29 studied the uncertain maximum flow problem
and provided a numerical solution method. This paper constructs an uncertain α-maximum flow model,
analyzes properties of the model, and designs a new polynomial-time exact algorithm.
The remainder of this paper is organized as follows. In section 2, some basic concepts and properties
of uncertainty theory used throughout this paper are introduced. In section 3, Uncertain α-maximum
flow model is formulated and its properties are analyzed. In section 4, an optimal algorithm is developed
to solve the model. A numerical example is presented to illustrate the algorithm in Section 5. Section 6
gives a conclusion to this paper.
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2. Preliminaries
Uncertainty theory is a branch of mathematics for modeling human uncertainty. In order to deal
with human uncertainty, Liu 14,15 presented four axioms: (1) normality axiom, (2) duality axiom, (3)
subadditivity axiom, and (4) product axiom.
In this section, we introduce some fundamental concepts and properties of uncertainty theory, which
will be used throughout this paper.
Definition 1.
14
Let Γ be a nonempty set, L a σ-algebra over Γ, and M an uncertain measure. Then
the triplet (Γ, L, M) is called an uncertainty space.
Definition 2.
14
An uncertain variable is a measurable function ξ from an uncertainty space (Γ, L, M)
to the set of real numbers, i.e., for any Borel set B of real numbers, the set
{ξ ∈ B} = {γ ∈ Γ|ξ(γ) ∈ B}
is an event.
Definition 3.
14
The uncertainty distribution Φ of an uncertain variable ξ is defined by
Φ(x) = M{ξ < x}
for any real number x.
The zigzag uncertainty distribution ξ ∼ Z(a, b, c) has an uncertainty distribution


0,
if x ≤ a




(x − a)/2(b − a),
if a ≤ x ≤ b
Φ(x) =


(x + c − 2b)/2(c − b), if b ≤ x ≤ c




1,
if x ≥ c.
Definition 4.
21
An uncertainty distribution Φ(x) is said to be regular if it is a continuous and strictly
increasing function with respect to x at which 0 < M(x) < 1, and
lim Φ(x) = 0, lim Φ(x) = 1.
x→−∞
Definition 5.
15
x→+∞
The uncertain variables ξ1 , ξ2 , · · · , ξn are said to be independent if
{n
}
n
∩
∧
M
(ξi ∈ Bi ) =
M{ξi ∈ Bi }
i=1
i=1
for any Borel sets B1 , B2 , · · · , Bn of real numbers.
Theorem 1.
21
Let ξ1 , ξ2 , · · · , ξn be independent uncertain variables with regular uncertainty distribu-
tions Φ1 , Φ2 , · · · , Φn , respectively. If f is a strictly increasing function, then
ξ = f (ξ1 , ξ2 , · · · , ξn )
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is an uncertain variable with inverse uncertainty distribution
−1
−1
Ψ−1 (α) = f (Φ−1
1 (α), Φ2 (α), · · · , Φn (α)).
In reality, we can easily obtain strictly increasing functions f . Therefore, through Theorem 1, we can
transform an indeterminacy model into a deterministic one by
M{f (ξ1 , ξ2 , · · · , ξn ) ≤ x} ≥ α,
to its equivalent deterministic form
−1
−1
x ≥ f (Φ−1
1 (α), Φ2 (α), · · · , Φn (α)).
3. Mathematical formulation
This paper concerns uncertain maximum flow problem. The goal is to maximize the total flow sent
from the source node to the sink node not exceeding the capacities on any arc and keeping flow balance in
every node. In classic deterministic maximum flow problem, a flow network is 5-tuple N = (V, A, c, s, t),
with source node s, sink node t, a finite node set V = {1, 2, · · · , n} and arc set A = {(i, j)|i, j ∈ V },
together with a nonnegative real-valued capacity function c defined on its arc set A.
Denote u = {uij |(i, j) ∈ A} as the set of arc capacities. Then in network N , from the source s to the
sink t, the maximum flow is a function of u, which is denoted as f . Given u, f (u) can be found. Denote
x = {xij |(i, j) ∈ A} as the set of flow on arc (i, j). A flow is feasible if it satisfies conservation condition
 ∑
∑

xsj −
xjs = f, (i, j) ∈ A



j:(s,j)∈A
j:(j,s)∈A


 ∑
∑
xij −
xji = 0, (i, j) ∈ A

j:(i,j)∈A
j:(j,i)∈A


∑
∑



xtj −
xjt = −f, (i, j) ∈ A

j:(t,j)∈A
j:(j,t)∈A
and capacity constraint 0 ≤ xij ≤ uij , where f is the flow in the network N .
In classic deterministic maximum flow problem, capacities of arcs are scrip values. Unfortunately,
cases such as this which capacities of arcs are scrip values are rare. Especially, networks are quite large,
and it is impossible to describe them explicitly. But, if there are enough data available, random maximum
flow models may be considered as network models. However, in most cases, we cannot get enough data
or data is invalid because of change in conditions. For example, due to impact of unexpected accidents
on traffic flow, we cannot use probabilistic rules to describe the complexity of the networks. In this
situation, the capacity data can only be obtained from the decision-maker’s subjective estimation. Thus,
it is unsuitable to regard subjective estimation data as random variables. In this paper, we employ
uncertain variables to describe the capacities of arcs. We consider the maximum flow problem subject to
the following assumptions:
(1) The network is directed.
(2) All capacities are nonnegative rational numbers (All computers store capacities as rational numbers and we can always transform rational numbers to integer numbers by multiplying them by
a suitably large number).
(3) The network does not contain a directed path from source node s to sink node t composed only
of infinite capacity arcs.
(4) The network does not contain parallel arcs.
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Define ξ = {ξij |(i, j) ∈ A}. We can denote the network with uncertain capacities of arcs as N =
(V, A, ξ, s, t). The maximum flow is f (ξ). As a function of ξ, f is also an uncertain variable.
Sometimes, the decision-maker assumes that the flow should satisfy some chance constraints with at
least some given confidence level α. Then we have the following definition.
Definition 1. A flow x is the α-maximum flow from source s to sink t if
max{f |M{ξ ≤ x} ≤ α} ≥ max{f ′ |M{ξ ≤ x′ } ≤ α}
for any flow x′ from source s to sink t, where α is a predetermined confidence level.
Chance constrained programming offers a powerful tool for modeling uncertain decision systems. The
essential idea of chance constrained programming of α maximum flow model is to optimize the flow value
in network with predetermined confidence level subject to capacity chance constraints. In order to find
α-maximum flow, we propose the following uncertain α-maximum flow model.



max f





subject to :









f,


 ∑

∑
xij −
xji =
0,


j:(i,j)∈A
j:(j,i)∈A







−f,






M{ξij ≤ xij } ≤ α, (i, j) ∈ A




x ≥ 0, (i, j) ∈ A,
i = s,
∀i ∈ V − {s, t},
(1)
i = t,
ij
where α is a predetermined confidence level provided by the decision-maker.
In classic deterministic model, the maximum flow is obtained by polynomial algorithms. It implies
that if there is one way to transform uncertain α-maximum model into its crisp equivalent, then we can
solve the model in deterministic environment by polynomial algorithms. Therefore, we need to convert
the chance constraints
M{ξij ≤ xij } ≤ α, (i, j) ∈ A
into its crisp equivalent.
In order to deign algorithm for α-maximum flow model, we first introduce property of the chance
constraint.
Lemma 1. According to Theorem 1, M{ξij ≤ xij } ≤ α , for (i, j) ∈ A can be transformed as xij ≤
Φ−1
ij (α).
Proof. Denote the uncertainty distribution of ξij as Φij , for (i, j) ∈ A. According to Theorem 1, for any
0 < α < 1,
M{ξij ≤ Φ−1
ij (α)} = α.
Since uncertainty distribution is an increasing function, from
M{ξij ≤ xij } ≤ α = M{ξij ≤ Φ−1
ij (α)},
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we obtain
xij ≤ Φ−1
ij (α).
This proves the lemma.
Lemma 1 indicates that the chance constraint can be converted into its crisp equivalent, and we can
draw the following theorem.
Theorem 2. In network N = (V, A, ξ, s, t), ξij has a regular uncertainty distribution Φij , (i, j) ∈ A.
Then, α-maximum flow of N = (V, A, ξ, s, t) is just the maximum flow of the corresponding deterministic
network Ñ = (V, A, c, s, t), where the capacity of arc (i, j) ∈ A is Φ−1
ij (α).
Proof. By Lemma 1, model (1) can be easily converted into the following deterministic model:



max f






subject to :









f,


 ∑

∑
xij −
xji =
0,


j:(i,j)∈A
j:(j,i)∈A





−f,







xij ≤ Φ−1

ij (α), (i, j) ∈ A



x ≥ 0, (i, j) ∈ A
ij
i = s,
∀i ∈ V − {s, t},
(2)
i = t,
Thus, the solution to model (2) is just the maximum flow of deterministic network Ñ = (V, A, c, s, t),
where the capacity of arc(i, j) ∈ A is Φ−1
ij (α). Thus, We can obtain the maximum flow of Ñ = (V, A, c, s, t)
by using generic preflow-push algorithm. The theorem is proved.
Theorem 1 shows how to obtain α-maximum flow. Next, we further investigate the property of the
model for obtaining the inverse distribution of uncertain maximum flow.
Theorem 3. In network N = (V, A, ξ, s, t), ξij has a regular uncertainty distribution Φij . Then, the
inverse uncertainty distribution of f is determined by
Ψ−1 (α) = f (Φ−1
ij (α)|(i, j) ∈ A).
Proof. For model (1), the maximum flow f is a continnuous and increasing function with respect to
each capacity of arc. Obviously, increasing the capacity of each arc, we will get a greater flow. That is,
f (x) > f (y),
where x = {xij |(i, j) ∈ A} , y = {yij |(i, j) ∈ A}, and xij > yij . Thus, f is a strictly increasing function.
By Theorem 1 and Theorem 2, we can easily obtain the inverse uncertainty distribution of f with
respect to α. The theorem is proved.
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4. Solution algorithm
Generally, Monte Carlo simulation or heuristic algorithms are used to obtain solution to uncertain
programming with indeterminacy factors. A disadvantage of these methods in comparison to exact mathematical methods is that they usually provide only statistical estimates or approximate solutions, not
exact result. Theorem 2 provides a better way to obtain the α-maximum flow in uncertain network. We
only need to employ the generic preflow-push algorithm to find the maximum flow of the corresponding
deterministic network. Hence, based on Lemma 1 and Theorem 2, we can design the following optimal
solution algorithm for obtaining α-maximum flow.
Algorithm:
Step 1. Give a predetermined confidence level α and calculate Φ−1
ij (α), (i, j) ∈ A.
Step 2. Construct the corresponding deterministic network Ñ = (V, A, c, s, t), and set the capacity of
each arc uij equal to Φ−1
ij (α).
Step 3. Empoly the generic preflow-push algorithm to find α-maximum flow in network N .
This algorithm runs in O(n2 m) time (n is the number of nodes and m is the number of arcs) which
is the same as that of generic preflow-push algorithm.
5. Numerical example
In this section, we give an example to illustrate the algorithm. A network N = (V, A, ξ, s, t) is shown
in Fig.1. The α-maximum flow is f with distribution Ψ. The capacity of each arc (i, j) is listed in Table
1. For convenience, if ξij is a constant, we set Φ−1
ij (α) = c, for any α ∈ (0, 1). Thus, we can calculate
−1
Φ−1
ij (0.9) for each ξij . Values of Φij (0.9) are also listed Table 1. Then, we can obtain the following results:
(1) the α-maximum flow when α = 0.9;
(2) the uncertainty distribution of f (ξ).
Fig. 1. Network for example.
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Table 1. List of arc capacities and
Φ−1
ij (0.9).
arc (i, j)
ξij
Φ−1
ij (0.9)
(s, 1)
Z(14, 16, 18)
17.6
(1, 2)
6
6
(1, 3)
Z(9, 10, 11)
10.8
(2, t)
Z(10, 11, 12)
11.8
(3, t)
Z(10, 12, 14)
13.6
Using the data in Table 1, we construct the deterministic network Ñ = (V, A, c, s, t). It is shown in
Fig.2
Fig. 2. Network with upper bounds.
Then, we solve α-maximum flow problem using the generic preflow-push algorithm. We obtain the
final residual network given in Fig.3 (e(·) are excesses of nodes, d(·) are distance labels and rij is residual
capacity of any arc (i, j) ∈ A ).
Fig. 3. Final residual network.
Now the network contains no active node. The α-maximum flow in the network is shown in Fig.4 and
its value is 16.8.
Choosing different α and repeating the above process, we obtain the uncertainty distribution of f ,
which is listed in Table 2 and plotted in Fig.5.
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Fig. 4. The optimal solution to example.
Table 2. List of α-maximum flows.
α
xs1
x12
x13
x2t
x3t
f = Ψ−1 (α)
1
17
6
11
6
11
17
0.9
16.8
6
10.8
6
10.8
16.8
0.8
16.6
6
10.6
6
10.6
16.6
0.7
16.4
6
10.4
6
10.4
16.4
0.6
16.2
6
10.2
6
10.2
16.2
0.5
16
6
10
6
10
16
0.4
15.6
5.9
9.7
5.9
9.7
15.6
0.3
15.2
5.9
9.3
5.9
9.3
15.2
0.2
14.8
5.9
8.9
5.9
8.9
14.8
0.1
14.4
5.8
8.6
5.8
8.6
14.4
0
14
5.8
8.2
5.8
8.2
14
15
15.5
16
1
0.9
0.8
0.7
α
0.6
0.5
0.4
0.3
0.2
0.1
0
14
14.5
16.5
17
f
Fig. 5. Uncertainty distribution of f .
6. Conculsion
Interpretations of indeterminacy phenomena may vary from time to time or person to person. Uncertainty theory provides a new tool to deal with indeterminacy. Under the framework of uncertainty theory,
we present an extension to the classic maximum flow problem whose network capacities are uncertain
variables in stead of crisp values. The problem was formulated by uncertain α-maximum flow model.
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The relationship between the uncertain α-maximum flow model and its crisp equivalent is proved, and
the uncertainty distribution of the uncertain maximum flow is derived. Some important properties of
the model are analyzed, which help to develop a polynomial time exact algorithm. At last, an uncertain
α-maximum flow example is given and solved by the algorithm.
7. Acknowledgments
This work was supported by the National Natural Science Foundation of China Grant No.61273044
and Education Research Project No.YPGC2011-W03.
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