Expected Value Model for Optimal Assignment Problem with
Uncertain Profits
Bo Zhang1 , Jin Peng2,∗
1
Faculty of Mathematics and Statistics, Huazhong Normal University
Hubei 430079, China
2
Institute of Uncertain Systems, Huanggang Normal University
Hubei 438000, China
Abstract
This paper employs uncertain programming to deal with optimal assignment problem with uncertain profits. Within the framework of uncertain programming, a concept of expected optimal
assignment for uncertain optimal assignment problem is proposed, and then an expected value model
is constructed. Taking advantage of properties of uncertainty theory, this model can be transformed
into a corresponding deterministic form which can be solved by Kuhn-Munkres algorithm.
Keywords: optimal assignment problem, uncertainty theory, uncertain programming, Kuhn-Munkres
algorithm
1
Introduction
Optimal assignment problem is usually met by decision maker. Assume that in a company, there are m
workers and n jobs (m ≥ n), and any worker can be assigned to any job. If the profits of the workers in
different jobs are different, how can the decision maker find an assignment such that the total profit of the
workers is maximized? This problem is called optimal assignment problem (OAP). Usually, the numbers
of workers and jobs are equal, i.e., m = n. However, the assignment problem can be made rather more
flexible than it first appears. Suppose there are four workers and three jobs. Then a fourth dummy job
can be invented, perhaps called “sitting still doing nothing”, with a profit of 0 for the worker assigned to
it. The assignment problem can then be solved in the usual way.
In early period, assignment problem was investigated in deterministic environment. Many efficient
approaches such as Kuhn-Munkres algorithm [15][30], auction algorithm [1] have been developed. However, in application, some uncertain factors will appear because of the lack of history data, insufficient
information or some other reasons. As a result, it is not suitable to employ classical algorithms in these
situations.
Some researchers employed probability theory to study optimal assignment problem. The research
of random assignment problems dated back to Donath [6], who investigated the limiting behavior of the
linear assignment problem by solving randomly generated instances. Later, Walkup [34] established an
upper bound for the expected cost of optimal assignment, which is one of the first results concerning
the behavior of random linear assignment problem. In 1982, Burkard and Fincke [3] presented the first
asymptotic results for the expected optimal value of random quadratic assignment problem. For more
research of random assignment problem, we may consult Pardalos and Ramakrishnan [31], Kellerer and
Wirsching [14], Linusson and Wästlund [17], etc.
∗ Corresponding
author. E-mail addresses: pengjin01@tsinghua.org.cn.
1
In 1965, Zadeh [39] proposed the concept of fuzzy sets. Since then, many researchers employed fuzzy
concepts to assignment problem. For instance, Chen [4] proposed a fuzzy assignment model that did
not consider the differences of individuals, and also proved some theorems. In 1994, Herrera et al. [11]
investigated the fuzzy matching problem using linguistic variables. Lin and Wen [16] proposed an efficient
algorithm based on the labeling method for the fuzzy assignment problem. Many other researchers, such
as Feng and Yang [7], Majumdar and Bhunia [29], Ye and Xu [38], Liu and Gao [18], Kagade and Bajaj
[13], have done a lot of work in this field.
However, if uncertain factor comes from the decision maker’s empirical estimation, it is not suitable
to employ random variable or fuzzy variable to describe the uncertain factor. Fortunately, in order to
deal with human data, such as empirical estimation, uncertainty theory was proposed by Liu [19] in
2007. Liu [27] said that “When the sample size is too small (even no-sample) to estimate the probability
distribution, some domain experts are invited to evaluate the belief degree of each event. In this situation,
the belief degree usually has much larger variance than the long-run frequency and we should deal with
it by uncertainty theory.” It is too adventurous if we deal with the belief degree by probability theory,
because it may lead to counterintuitive results.
This paper concerns about uncertain optimal assignment problem (UOAP), in which profit is uncertain. We regard profit as uncertain variable. In order to construct an uncertain model, we first introduce
the concept of expected optimal assignment for optimal assignment problem with uncertain profits. After that, we construct the expected value model for uncertain optimal assignment problem. Under the
framework of uncertainty theory, the model can be transformed into a corresponding deterministic form.
This provides an effective method to find a solution to the expected value model, that is, we can employ
any classical algorithm to find an optimal assignment on the deterministic problem. In this paper, we
employ Kuhn-Munkres algorithm.
The paper is organized as follows: Section 2 presents some basic concepts and properties selected from
uncertainty theory, and then describes the history and application of uncertain programming. In Section
3, uncertain optimal assignment problem is described, then the concept of expected optimal assignment
is proposed. Section 4 constructs the expected value model and derives the method to find expected
optimal assignment for uncertain optimal assignment problem. Section 5 illustrates the proposed method
by an example. The last section concludes this paper with a brief summary.
2
Preliminaries
In order to model human language like “about 50km”, “low speed”, and “big size”, uncertainty theory
was founded by Liu [19] in 2007. Recently, much research work has been done on the development of
uncertainty theory. For example, uncertain calculus was initialized by Liu [21] to deal with differentiation
and integration of functions of uncertain processes. Based on uncertain calculus, Liu [20] introduced
uncertain differential equation. After that, Chen and Liu [5] proved the existence and uniqueness theorem
for uncertain differential equations. Furthermore, uncertain inference was introduced by Liu [21] via
conditional uncertain measure. Later, Gao et al. [8] derived some expressions of Liu’s inference rule
for uncertain systems. In addition, uncertainty theory was also applied to uncertain statistics (Liu [25],
Wang et al. [35][36]), uncertain risk analysis and uncertain reliability analysis (Liu [24]), uncertain logic
(Liu [26]), uncertain finance (Liu [21], Peng and Yao [32]), and uncertain control (Liu [23], Zhu [40]). To
explore the recent developments of uncertainty theory, the readers may consult [25].
2.1
Uncertainty Theory
In this section, we introduce some concepts and results from uncertainty theory, which will be used
throughout in this paper.
2
Let Γ be a nonempty set, L a σ-algebra over Γ. Each element Λ ∈ L is called an event. For any
Λ ∈ L, a set function M{·} is said to be an uncertain measure if it satisfies the following three axioms
[19]:
(1)(Normality Axiom) M{Γ} = 1;
(2)(Duality Axiom) M{Λ} + M{Λc } = 1 for any Λ ∈ L;
(3)(Subadditivity Axiom) For every countable sequence of events {Λi }, we have
(∞ )
∞
[
X
M
Λi ≤
M{Λi }.
i=1
i=1
The triplet (Γ, L, M) is called an uncertainty space. In order to obtain an uncertain measure of
compound event, the fourth axiom called product axiom was presented by Liu [21].
(4)(Product Axiom) Let (Γk , Lk , Mk ) be uncertainty spaces for k = 1, 2, · · · , n. Then the product
uncertain measure M is an uncertain measure on the product σ-algebra L1 × L2 × · · · × Ln satisfying
( n
)
Y
M
Λk = min Mk {Λk }
k=1
1≤k≤n
where Λk ∈ Lk , for k = 1, 2, · · · , n.
Definition 2.1 [19] 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 2.2 [19] The uncertainty distribution Φ : < → [0, 1] of an uncertain variable ξ is defined by
Φ(x) = M{γ ∈ Γ|ξ(γ) ≤ x}
for any real number x. The inverse function Φ−1 is called the inverse uncertainty distribution of ξ.
Example 2.1: The zigzag uncertain variable 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


, if b ≤ x ≤ c



2(c − b)




1,
if x ≥ c.
Definition 2.3 [19] Let ξ be an uncertain variable. Then the expected value of ξ is defined by
Z +∞
Z 0
E[ξ] =
M{ξ ≥ r}dr −
M{ξ ≤ r}dr
0
−∞
provided that at least one of the two integrals is finite.
If ξ has an uncertainty distribution Φ, then the expected value may be calculated by
Z +∞
Z 0
E[ξ] =
(1 − Φ(x))dx −
Φ(x)dx.
0
−∞
3
Theorem 2.1 [25] Let ξ1 , ξ2 , · · ·, ξn be independent uncertain variables with uncertainty distributions
Φ1 , Φ2 , · · ·, Φn , respectively. If the function f (x1 , x2 , · · · , xn ) is strictly increasing with respect to
x1 , x2 , · · · , xm and strictly decreasing with respect to xm+1 , xm+2 , · · · , xn , then
ξ = f (ξ1 , ξ2 , · · · , ξn )
is an uncertain variable with inverse uncertainty distribution
−1
−1
−1
Ψ−1 (α) = f (Φ−1
1 (α), · · · , Φm (α), Φm+1 (1 − α), · · · , Φn (1 − α)).
Example 2.2: Let ξ1 and ξ2 be independent uncertain variables with uncertainty distributions Φ1 and
Φ2 , respectively. Since the function
f (x1 , x2 ) = x1 − x2
is strictly increasing with respect to x1 and strictly decreasing with respect to x2 , the inverse uncertainty
distribution of the difference ξ1 − ξ2 is
−1
Ψ−1 (α) = Φ−1
1 (α) − Φ2 (1 − α).
Example 2.3: Let ξ1 , ξ2 , ξ3 are independent uncertain variables with uncertainty distributions Φ1 , Φ2 , Φ3 ,
respectively. Since the function
f (x1 , x2 ) = (x1 + x2 )/x3
is strictly increasing with respect to x1 , x2 and strictly decreasing with respect to x3 , the inverse uncertainty distribution of (x1 + x2 )/x3 is
−1
−1
Ψ−1 (α) = (Φ−1
1 (α) + Φ2 (α))/Φ3 (1 − α).
Furthermore, assume the function f (x1 , x2 , · · · , xn ) is strictly increasing with respect to x1 , x2 , · · · , xm
and strictly decreasing with respect to xm+1 , xm+2 , · · · , xn . It has been proved by Liu and Ha [28] that
the uncertain variable ξ = f (ξ1 , ξ2 , · · · , ξn ) has an expected value
Z
E[ξ] =
0
1
−1
−1
−1
f (Φ−1
1 (α), · · · , Φm (α), Φm+1 (1 − α), · · · , Φn (1 − α))dα.
It can be calculated that the expected value of the linear uncertain variable ξ ∼ L(a, b) is E[ξ] =
(a + b)/2, the expected value of the zigzag uncertain variable ξ ∼ Z(a, b, c) is E[ξ] = (a + 2b + c)/4, and
the expected value of the normal uncertain variable ξ ∼ N (e, σ) is E[ξ] = e.
Theorem 2.2 [25] Let ξ and η be independent uncertain variables with finite expected values. Then for
any real numbers a and b, we have
E[aξ + bη] = aE[ξ] + bE[η].
2.2
Uncertain Programming
Founded by Liu [22] in 2009 and refined by Liu [25] in 2010, uncertain programming is a type of mathematical programming involving uncertain variables.
Assume that x is a decision vector, ξ is an uncertain vector, f (x, ξ) is an objective function, and gj (x, ξ)
are constraint functions, j = 1, 2, · · · , p. In order to obtain a decision with minimum expected objective
4
value subject to a set of chance constraints, Liu [22] proposed the following uncertain programming
model,


min E[f (x, ξ)]


 x
(2.1)
subject to :




M{gj (x, ξ) ≤ 0} ≥ αj , j = 1, 2, · · · , p.
A key problem in the research area of uncertain programming is how to solve the model like (2.1). Fortunately, under the framework of uncertainty theory, the uncertain programming model (2.1) is equivalent
to the crisp model

Z 1

−1
−1
−1


min
f (x, Φ−1
 x
1 (α), · · · , Φm (α), Φm+1 (1 − α), · · · , Φn (1 − α))dα

0
(2.2)
subject to :





−1
−1
−1
gj (x, Φ−1
j = 1, 2, · · · , p
1 (αj ), · · · , Φk (αj ), Φk+1 (1 − αj ), · · · , Φn (1 − αj )) ≤ 0,
where f (x, ξ1 , ξ2 , · · · , ξn ) is strictly increasing with respect to ξ1 , ξ2 , · · · , ξm and strictly decreasing with
respect to ξm+1 , ξm+2 , · · · , ξn , and gj (x, ξ1 , ξ2 , · · · , ξn ) is strictly increasing with respect to ξ1 , ξ2 , · · · , ξk
and strictly decreasing with respect to ξk+1 , ξk+2 , · · · , ξn .
After converting the uncertain programming model to a crisp model, we may solve it by simplex
method, branch-and-bound method, genetic algorithm, neural networks, tabu search, and so on.
In order to show the applications of the model as mentioned above, the models of uncertain programming on uncertain system reliability design, uncertain machine scheduling problem, uncertain vehicle
routing problem, uncertain facility location problem, uncertain project scheduling problem and so on,
were given by Liu [22], respectively. In addition, Bhattacharyya et al. [2] introduced the concept of
multiple objective uncertain optimization problems. In 2009, Yan [37] provided two models for portfolio
selection in which the securities are assumed to be uncertain variables. After that, Huang [12] introduced
a risk curve and developed a mean-risk model for uncertain portfolio selection. Furthermore, two models
of economic order quantity for inventory based on uncertain theory was provided by Rong [33]. In addition, Gao [9] investigated the shortest problem with uncertain arc lengths, and given the α-shortest path
model and the most shortest path model in an uncertain network, and Gao [10] proposed two uncertain
models for single facility location problem.
3
Problem Description
An optimal assignment problem with n workers and n jobs can be represented in a profit matrix P , where
pij indicate the profit made by worker i on job j. The problem is to find an assignment that maximizes
the total profit of the workers,

p11

 p
 21
P =
 ..
 .

pn1
p12
···
p1n
p22
..
.
···
p2n
..
.
pn2
· · · pnn
..
.




.



In classical deterministic optimal assignment problem, each profit pij is a positive crisp value. Many
effective algorithm can be employed to solve the deterministic optimal problem, such as Kuhn-Munkres
Algorithm.
Following example illustrates how to obtain the optimal assignment by Kuhn-Munkres algorithm.
Example 3.1: Assume there are four workers and four jobs, and the profit matrix is
5

3 2

 2 3

P =
 2 3

6 4
2
4
5
5
4


4 

.
3 

3
After employing of the Kuhn-Munkres algorithm to the profit matrix P , an optimal assignment A is
presented in Table 1,
Table 1: An Optimal Assignment A
worker
1
2
3
4
job
4
2
3
1
profit
4
3
5
6
whose total profit of the workers is 4 + 3 + 5 + 6 = 18.
However, in practical, because of the lack of history data, some uncertain factors will appear. In this
situation, the profit data can only be obtained from the decision maker’s empirical estimation. Then how
can we deal with these uncertain factors? Fortunately, uncertainty theory provides a new tool to deal
with uncertain information, especially expert data and subjective estimation.
In this paper, the uncertain factor for optimal assignment problem is the profits for workers. We
employ uncertain variable to describe the profit of each worker, that is, each profit pij is replaced by a
nonnegative uncertain variable ξij . Before starting model construction, some notations and assumptions
are listed in Table 2.
Table 2: List of Notations and Assumptions
ξij
P̃
xij
Ã
uncertain variable of the profit made by worker i on job j,
and all the uncertain variables ξij are positive and independent
profit matrix with uncertain profit ξij
zero and one decision variable on profit ξij
an assignment for P̃
Clearly, Ã is an assignment if and only if
 X
xij = 1 for i = 1, 2, · · · , n




 1≤j≤n
X
xij = 1 for j = 1, 2, · · · , n



1≤i≤n


xij = {0, 1}
for i, j = 1, 2, · · · , n.
Remark 1: xij = 0 means that worker i is not assigned to job j; xij = 1 means that worker i is assigned
to job j. The first constraint requires that every worker is assigned to exactly one job, and the second
constraint requires that every job is assigned exactly one worker.
Different from that in a deterministic profit matrix P , the optimal assignment profit is an uncertain
variable in an uncertain profit P̃ . Then, for uncertain profit matrix P̃ , how can we obtain the optimal
assignment?
Expected value is the average value of uncertain variable in the sense of uncertain measure, and
represents the size of uncertain variable. In order to obtain the decision with maximum expected return,
we give the concept of expected optimal assignment for uncertain optimal assignment problem.
6
Definition 3.1 Let P̃ be an uncertain profit matrix, Ã∗ an assignment. Then Ã∗ is called expected
optimal assignment (EOA) if
E[w(Ã∗ )] ≥ E[w(Ã)]
holds for assignment Ã, where w(Ã∗ ) stands for the total profit of expected optimal assignment Ã∗ and
w(Ã) stands for the total profit of assignment Ã.
As is mentioned before, the optimal assignment profit is an uncertain variable in uncertain profit
matrix P̃ . How can we get expected optimal assignment? In Section 4, we will give the answer.
4
Expected Value Model
Within the framework of uncertainty theory, the concept of expected optimal assignment in uncertain
phenomena is proposed in above section.
In this section, we will construct expected valued model by the concept of expected optimal assignment. Fortunately, taking advantage of some properties of uncertainty theory, this model can be
transformed into a corresponding deterministic form.
Theorem 4.1 Let P̃ be an uncertain profit matrix. Then, the expected optimal assignment of P̃ is just
the optimal assignment of P , where P a profit matrix, and pij = E[ξij ].
Proof According to Definition 3.1, the expected value model can be presented as following:
X

max E[
xij ξij ]




0≤i,j≤n

X



xij = 1 for i = 1, 2, · · · , n
 s.t.
(4.1)
1≤j≤n
X



x
=
1
for
j
=
1,
2,
·
·
·
,
n
ij




1≤i≤n


xij = {0, 1}
for i, j = 1, 2, · · · , n.
Note that, ξij , i, j = 1, 2, · · · , n, are independent, by Theorem 2.2, the model (4.1) can be equivalently
transformed to the following deterministic model:
X

max
xij E[ξij ]




0≤i,j≤n

X



xij = 1 for i = 1, 2, · · · , n
 s.t.
(4.2)
1≤j≤n
X



xij = 1 for j = 1, 2, · · · , n




1≤i≤n


xij = {0, 1}
for i, j = 1, 2, · · · , n.
In fact, the model (4.2) describes the optimal assignment in profit matrix P , whose elements pij are
E[ξij ]. The theorem is proved.
Example 4.1 Suppose the elements of uncertain profit matrix P̃ are all linear uncertain variables ξij ∼
L(aij , bij ), i, j = 1, 2, · · · , n, respectively. Then the expected value model can be transformed into the
following form:

X 1
 max

(aij + bij )xij


2


0≤i,j≤n

X


 s.t.
xij = 1 for i = 1, 2, · · · , n
(4.3)
1≤j≤n
X



xij = 1 for j = 1, 2, · · · , n




1≤i≤n



xij = {0, 1}
for i, j = 1, 2, · · · , n.
7
Example 4.2 Suppose the elements of uncertain profit matrix P̃ are all zigzag uncertain variables
ξij ∼ Z(aij , bij , cij ), i, j = 1, 2, · · · , n, respectively. Then the expected value model can be transformed
into the following form:

X 1
 max

(aij + 2bij + cij )xij


4


0≤i,j≤n

X


 s.t.
xij = 1 for i = 1, 2, · · · , n
(4.4)
1≤j≤n
X



xij = 1 for j = 1, 2, · · · , n




1≤i≤n



xij = {0, 1}
for i, j = 1, 2, · · · , n.
Example 4.3 Suppose the elements of uncertain profit matrix P̃ are all normal uncertain variables
ξij ∼ N (eij , σij ), i, j = 1, 2, · · · , n, respectively. Then the expected value model can be transformed into
the following form:
X

max
eij xij




0≤i,j≤n

X



xij = 1 for i = 1, 2, · · · , n
 s.t.
(4.5)
1≤j≤n
X



x
=
1
for
j
=
1,
2,
·
·
·
,
n
ij




1≤i≤n


xij = {0, 1}
for i, j = 1, 2, · · · , n.
5
Numerical Experiment
In the past, the methods of genetic algorithm, tabu search algorithm or some other Hybrid intelligent
algorithms were usually employed to find the optimal solution to uncertain programming problems.
However, the rigor of these methods are not good. Theorem 4.1 provides an effective method to obtain
expected optimal assignment in uncertain profit matrix P̃ . That is, we only need to employ KuhnMunkres algorithm to find the optimal assignment of a corresponding deterministic profit matrix P .
Roughly speaking, the method to obtain expected optimal assignment in uncertain profit matrix P̃
can be summarized as follows:
Step 1: Calculate E[ξij ], for each element ξij in P̃ ;
Step 2: Construct a deterministic profit matrix P , whose elements pij = E[ξij ];
Step 3: Employ Kuhn-Munkres algorithm to get the optimal assignment in profit matrix P .
The optimal assignment we get in Step 3 is just the expected optimal assignment in uncertain profit
matrix P̃ .
In this section, we give an example to illustrate the method as mentioned above. For the convenience
of description, we summarize the problem as follows. Suppose that a taxi firm has five drivers available,
and five customers wishing to be picked up as soon as possible. Now, the task for the decision maker is to
make the assignment plan such that the total profit (or time) of the drivers is maximized (or minimized).
At the beginning of the task, the decision maker needs to obtain the basic data, such as level of drivers’
understanding on the traffic condition, driving skill of each driver, and so on. In fact, since the plan is
made in advance, we usually cannot obtain these data exactly. In this situation, we usually obtain the
uncertain data by means of expert advice. Assume that all uncertain variables ξij are listed in Table 3.
Table 3: List of ξij
8
ξij
value of ξij
ξij
value of ξij
ξ11
L(2, 4)
ξ34
Z(3, 5, 6)
ξ12
Z(3, 4.5, 5)
ξ35
Z(1, 3, 5)
ξ13
Z(2, 3, 4)
ξ41
L(5, 7)
ξ14
Z(2, 3, 5)
ξ42
Z(4, 4.5, 5)
ξ15
N (4, 1)
ξ43
N (5, 2)
ξ21
L(1.5, 4.5)
ξ44
N (6, 1)
ξ22
Z(2, 3, 5)
ξ45
Z(3, 4, 5)
ξ23
N (2, 1)
ξ51
Z(2, 3, 5)
ξ24
Z(4, 5.5, 6)
ξ52
L(1, 5)
ξ25
N (3, 2)
ξ53
Z(3, 3.5, 5)
ξ31
Z(3, 4, 6)
ξ54
L(2.5, 5.5)
ξ32
Z(2, 3, 5)
ξ55
Z(2, 3, 4)
ξ33
L(3, 5)
−
−
Step 1: Calculate E[ξij ] of ξij listed in Table 4.
Table 4: List of E[ξij ]
ξij
E[ξij ]
ξij
E[ξij ]
ξ11
3
ξ34
4.75
ξ12
4.25
ξ35
3
ξ13
3
ξ41
6
ξ14
3.25
ξ42
4.5
ξ15
4
ξ43
5
ξ21
3
ξ44
6
ξ22
3.25
ξ45
4
ξ23
2
ξ51
3.25
ξ24
5.25
ξ52
3
ξ25
3
ξ53
3.75
ξ31
4.25
ξ54
4
ξ32
3.25
ξ55
3
ξ33
4
−
−
Step 2: From the data in Table 4, we construct a deterministic profit matrix

3
4.25
3
3.25 4



 3
3.25
2
5.25 3 




.
P =
4.25
3.25
4
4.75
3




 6
4.5
5
6
4 


3.25
3
3.75
4
3
Step 3: Employ Kuhn-Munkres algorithm to the profit matrix P , the optimal assignment A is
presented in Table 5,
Table 5: The Optimal Assignment A
worker
1
2
3
4
5
job
2
4
3
1
5
profit
4.25
5.25
4
6
3
9
whose total profit of workers is 4.25 + 5.25 + 4 + 6 + 3 = 22.5.
According to Theorem 4.1, the expected optimal assignment in uncertain profit matrix P̃ is presented
in Table 6,
Table 6: The Expected Optimal Assignment Ã
worker
1
2
3
4
5
job
2
4
3
1
5
profit
4.25
5.25
4
6
3
whose total profit of workers is 4.25 + 5.25 + 4 + 6 + 3 = 22.5.
This means that in order to obtain the decision with maximum expected return, the decision maker
should assign his workers according to the optimal. The corresponding maximum profit is 22.5.
6
Conclusion
In practical application, some uncertain factors often appear in optimal assignment problem. This paper
concerns about uncertain optimal assignment problem, in which profit is uncertain. The uncertain factor
is described by uncertainty theory. In order to construct uncertain model, the concept of expected optimal
assignment is proposed. After that, the expected value model for uncertain optimal assignment problem
is proposed. With the help of uncertainty theory, the expected value model can be transformed into a
corresponding deterministic form, and then we can use Kuhn-Munkres algorithm to find its solution. At
last, an example is given to illustrate the method.
Acknowledgements
This work is supported by the National Natural Science Foundation (No.60874067), the Hubei Provincial
Natural Science Foundation (No.2010CDB02801), and the Scientific and Technological Innovation Team
Project (No.T201110) of Hubei Provincial Department of Education, China.
References
[1] Bertsekas D P, Auction Algorithms for Network Flow Problems: A Tutorial Introduction, Computational Optimization and Applications, Vol.1, No.1, 7-66, 1992.
[2] Bhattacharyya R, Chatterjee A, Kar S, Uncertainty Theory Based Novel Multi-Objective Optimization Technique Using Embedding Theorem with Application to R & D Project Portfolio Selection,
Applied Mathematics, Vol.1, 189-199, 2010.
[3] Burkard R E, Fincke U, The Asymptotic Probabilistic Behavior of Quadratic Sum Assignment
Problems, Zeitschrift für Operations Research, Vol.27, 73-81, 1982.
[4] Chen M S, On a Fuzzy Assignment Problem, Tamkang Journal of Mathematics, Vol.22, 407-411,
1985.
[5] Chen X, Liu B, Existence and Uniqueness Theorem for Uncertain Differential Equations, Fuzzy
Optimization and Decision Making, Vol.9, No.1, 69-81, 2010.
[6] Donath W E, Algorithm and Average-Value Bounds for Assignment Problems, IBM Journal of
Research and Development, Vol.13, No.4, 380-386, 1969.
10
[7] Feng Y, Yang L, A Two-Objective Fuzzy K-Cardinality Assignment Problem, Journal of Computational and Applied Mathematics, Vol.197, No.1, 233-244, 2006.
[8] Gao X, Gao Y, Ralescu D A, On Liu’s Inference Rule for Uncertain Systems, International Journal
of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol.18, No.1, 1-11, 2010.
[9] Gao Y, Shortest Path Problem with Uncertain Arc Lengths, Computers and Mathematics with
Applications, Vol.62, No.6, 2591-2600, 2011.
[10] Gao Y, Uncertain Models for Single Facility Location Problems on Networks, Applied Mathematical
Modelling, to be published.
[11] Herrera F, Lozano M, Verdegay J L, Applying Genetic Algorithms in Fuzzy Optimization Problems,
Fuzzy Systems & A.I.Reports and Letters, Vol.3, No.1, 39-52, 1994.
[12] Huang X X, Mean-risk Model for Uncertain Portfolio Selection, Fuzzy Optimization and Decision
Making, Vol.10, No.1, 71-89, 2011.
[13] Kagade K L, Bajaj V H, Fuzzy Method for Solving Multi-Objective Assignment Problem with
Interval Cost, Journal of Statistics and Mathematics, Vol.1, No.1, 01-09, 2010.
[14] Kellerer H, Wirsching G, Bottleneck Quadratic Assignment Problems and the Bandwidth Problem,
Asia-Pacific Journal of Operational Research, Vol.15, No.2, 169-177, 1998.
[15] Kuhn H W, The Hungarian Method for the Assignment Problem, Naval Research Logistics Quarterly, Vol.2, 83-97, 1955.
[16] Lin C, Wen U, A Labeling Algorithm for the Fuzzy Assignment Problem, Fuzzy Sets and Systems,
Vol.142, No.3, 373-391, 2004.
[17] Linusson S, Wästlund J, A Proof of Parisi’s Conjecture on the Random Assignment Problem,
Probability Theory and Related Fields, Vol.128, No.3, 419-440, 2004.
[18] Liu L, Gao X, Fuzzy Weighted Equilibrium Multi-Job Assignment Problem and Genetic Algorithm,
Applied Mathematical Modelling, Vol.33, No.10, 3926-3935, 2009.
[19] Liu B, Uncertainty Theory, 2nd Edition, Springer-Verlag, Berlin, 2007.
[20] Liu B, Fuzzy Process, Hybrid Process and Uncertain Process, Journal of Uncertain Systems, Vol.2,
No.1, 3-16, 2008.
[21] Liu B, Some Research Problems in Uncertainty Theory, Journal of Uncertain Systems, Vol.3, No.1,
3-10, 2009.
[22] Liu B, Theory and Practice of Uncertain Programming, 2nd Edition, Springer-Verlag, Berlin, 2009.
[23] Liu B, Uncertain Set Theory and Uncertain Inference Rule with Application to Uncertain Control,
Journal of Uncertain Systems, Vol.4, No.2, 83-98, 2010.
[24] Liu B, Uncertain Risk Analysis and Uncertain Reliability Analysis, Journal of Uncertain Systems,
Vol.4, No.3, 163-170, 2010.
[25] Liu B, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, SpringerVerlag, Berlin, 2010.
[26] Liu B, Uncertain Logic for Modeling Human Language, Journal of Uncertain Systems, Vol.5, No.1,
3-20, 2011.
11
[27] Liu B, Why is There a Need for Uncertainty Theory? Journal of Uncertain Systems, Vol.6, No.1,
3-10, 2012.
[28] Liu Y H, Ha M H, Expected Value of Function of Uncertain Variables, Journal of Uncertain Systems,
Vol.4, No.3, 181-186, 2010.
[29] Majumdar J, Bhunia A K, Elitist Genetic Algorithm for Assignment Problem with Imprecise Goal,
European Journal of Operational Research, Vol.177, No.1, 684-692, 2007.
[30] Munkres J, Algorithms for the Assignment and Transportation Problems, Journal of the Society of
Industrial and Applied Mathematics, Vol.5, No.1, 32-38, 1957.
[31] Pardalos P M, Ramakrishnan K G, On the Expected Optimal Value of Random Assignment Problems: Experimental Results and Open Questions, Computational Optimization and Applications,
Vol.2, No.3, 261-271, 1993.
[32] Peng J, Yao K, A New Option Pricing Model for Stocks in Uncertainty Markets, International
Journal of Operations Research, Vol.8, No.2, 18-26, 2011.
[33] Rong L X, Two New Uncertainty Programming Models of Inventory with Uncertain Costs, Journal
of Information & Computational Science, Vol.8, No.2, 280-288, 2011.
[34] Walkup D W, On the Expected Value of a Random Assignment Problem, SIAM Journal on Computation, Vol.8, 440-442, 1979.
[35] Wang X S, Gao Z C, Guo H Y, Delphi Method for Estimating Uncertainty Distributions, Information: An International Interdisciplinary Journal, to be published.
[36] Wang X S, Gao Z C, Guo H Y, Uncertain Hypothesis Testing for Expert’s Empirical Data, Mathematical and Computer Modelling, to be published.
[37] Yan L M, Optimal Portfolio Selection Models with Uncertain Returns, Modern Applied Science,
Vol.3, No.8, 2009.
[38] Ye X, Xu J, A Fuzzy Vehicle Routing Assignment Model with Connection Network Based on
Priority-Based Genetic Algorithm, World Journal of Modelling and Simulation, Vol.4, No.4, 257268, 2008.
[39] Zadeh L A, Fuzzy Sets, Information and Control, Vol.8, 338-353, 1965.
[40] Zhu Y, Uncertain Optimal Control with Application to a Portfolio Selection Model, Cybernetics
and Systems, Vol.41, No.7, 535-547, 2010.
Appendix
The Kuhn-Munkres algorithm is easier to describe if we formulate the problem using a bipartite graph.
Given a weighted complete bipartite graph G = (X, Y ) where edge xy has weight w(xy), find a matching
H from X to Y with maximum weight. In an application, X could be a set of workers, Y could be a set
of jobs, and w(xy) could be the profit made by assigning worker x to job y.
A feasible vertex labeling in G is a real-valued function l on X ∪ Y such that, for all x ∈ X and y ∈ Y ,
l(x) + l(y) ≥ w(xy).
If l is a feasible vertex labeling, we denote by Gl the subgraph of G which contains those edges where
l(x) + l(y) = w(xy).
12
The Kuhn-Munkres Algorithm
Start with an arbitrary feasible vertex labeling l, then determine Gl and choose an arbitrary matching
H in Gl .
Step 1: If H is complete for G, then H is optimal. Stop. Otherwise, there is some unmatched x ∈ X.
Set S = {x} and T = ∅.
Step 2: If NGl (S) 6= T , go to Step 3. Otherwise, NGl (S) = T . Compute
αl =
min {l(x) + l(y) − w(xy)}
x∈S,y∈T c
and construct a new labeling ˆl by


l(v) − αl



ˆl(v) =
l(v) + αl




l(v)
for v ∈ S
for v ∈ T
otherwise
Note that αl > 0 and NGl̂ (S) 6= T . Replace l by ˆl and Gl by Gl̂ .
Step 3: Choose a vertex y in NGl (S), not in T . If y is matched in H, say with z ∈ X, replace S by
S ∪ {z} and T by T ∪ {y}, and go to Step 2. Otherwise, there will be an H-augmenting path P from x
to y. Replace H by Ĥ = H∆E(P ) and go to Step 1.
Remark 2: In Step 2, the number of computations required to compute Gl̂ is clearly of order n2 . Since
the algorithm can cycle through Step 2 and Step 3 at most |X| times before finding an H-augmenting
path, and since the initial matching can be augmented at most |X| times before an optimal matching is
found, we see that the Kuhn-Munkres algorithm is an effective algorithm.
13