Minimum spanning tree problem of uncertain random network

Transcription

1 DOI 1.17/s Minimum spanning tree problem of uncertain random network Yuhong Sheng Zhongfeng Qin Gang Shi Received: 29 October 214 / Accepted: 29 November 214 Springer Science+Business Media New York 214 Abstract Minimum spanning tree problem is a typical and fundamental problem in combinatorial optimization. Most of the existing literature is devoted to the case with deterministic or random weights. However, due to lack of data, a proportion of edge weights have to be estimated according to experts evaluations, which may be considered as uncertain variables. This paper focuses on the case where some weights are random variables and the others are uncertain variables. The concept of an ideal chance distribution is introduced and its expression is given based on the uncertainty distributions and probability distributions. A model is formulated to find a minimum spanning tree whose chance distribution is the closest to the ideal one. Finally, a numerical example is given to illustrate the modelling idea of the study. Y. Sheng Department of Mathematical Sciences, Tsinghua University, Beijing 184, China sheng-yh12@mails.tsinghua.edu.cn Y. Sheng College of Mathematical and System Sciences, Xinjiang University, Ürümqi 8346, China Z. Qin School of Economics and Management, Beihang University, Beijing 1191, China qin@buaa.edu.cn G. Shi School of Information Science and Engineering, Xinjiang University, Ürümqi 8346, China G. Shi (B Department of Computer Sciences, Tsinghua University, Beijing 184, China shigang122@163.com Keywords Minimum spanning tree Chance theory Uncertain random network Ideal chance distribution Introduction The minimum spanning tree problem aims to find a spanning tree with least total weight (cost, distance, etc. in a connected graph. The pioneer work was initiated by Borüvka (1926 to minimize the transmit cost in an electricity network for Moravia. Actually similar cases also arise in many other applications of designing communication network, transportation network, logistics network, etc. This problem has been widely studied and many efficient algorithms have been presented when the edge weights are deterministic, such as Kruskal (1956 algorithm, Prim (1957 algorithm and Dijkstra (1959 algorithm. Nowadays, the problem has become a typical and fundamental problem in combinatorial optimization. In general, a model with inaccurate parameters may produce ineffective solutions. For example, consider designing a communication network and meanwhile assuming deterministic weights. Since in practice the network is inevitably affected by collisions, congestions and interferences, the obtained optimal solution may no longer be a minimum spanning tree. Therefore, it is necessary to introduce indeterminacy into the edge weights. The early works mainly focus on handling randomness, i.e., regarding edge weights as random variables. Seemingly the first study is the random network proposed by Frank and Hakimi (1965, in which the flow in an edge is a random variable rather than a fixed number. Similarly, Ishii et al. (1981 generalized the minimum spanning tree problem toward a stochastic version by considering the edge costs as random variables, and Ishii and Matsutomi (1995 extended the work to the situation in which

2 parameters of probability distributions are unknown and estimated by a confidence region from statistical data. Frieze (1985 analyzed the probabilistic properties of the length of a minimum spanning tree when the lengths of edges are independent identically distributed non-negative random variables. Instead of considering the randomness of edge weights, Bertsimas (199 introduced a so-called probabilistic minimum spanning tree in which some points are present with certain probability and found a closed-form expression for the expected length of a given spanning tree. Stochastic network optimization models work very well when there are enough data to estimate the probability distributions of edge weights. However, in reality, data are sometimes not available or difficult to obtain. In such a case, the estimation of edge weights may be based on evaluations of domain experts, which brings subjective uncertainty. In other words, edge weights are assumed to be general uncertain quantities rather than random variables. An alternative tool is the uncertainty theory founded by Liu (27 to deal with such indeterministic quantities. In this framework, several researchers extended network optimization problems to their uncertain counterparts by considering some parameters as uncertain variables. For example, Liu (29b first formulated a project scheduling network model with uncertain durations. Gao (211 studied the shortest path problem with uncertain arc lengths. On the assumption of uncertain edges, various uncertain minimum spanning problems are developed according to practical requirements, for instance, inverse minimum spanning tree problem (Zhang et al. 213, quadratic minimum spanning tree problem (Zhou et al. 214a and path optimality conditions of uncertain minimum spanning tree (Zhou et al In practice, uncertainty and randomness often simultaneously appear in a complex network. For example, some edge weights are considered as random variables when we obtain their data, and the others are considered as uncertain variables when data are not available and they are subjectively provided. In order to describe this complex situation, Liu (213a introduced the concept of chance measure and uncertain random variable. Uncertain random variable not only can degenerate into a random one, but only can degenerate into an uncertain one. Thus, it provides a more general description for indeterministic quantity. After that, Liu (213b derived some formulas of expected value. Guo and Wang (214, Sheng and Yao (214 gave some formulas of variance. In the aspect of application, several researchers further developed the approach such as Liu (213b introducing uncertain random programming. Zhou et al. (214b introduced uncertain random multi-objective programming, Qin (213 introduced uncertain random goal programming, and Ke et al. (215 introduced uncertain random multilevel programming, respectively. In addition, Gao and Yao (215 proposed uncertain random process and uncertain random renewal process. Yao and Gao (215 studied uncertain random alternating renewal process. Liu (214 introduced an uncertain random network jointly composed of random and uncertain edge weights. Then Sheng and Gao (214 derived the chance distribution of the maximum flow, and Sheng and Gao (215 discussed the shortest path problem of an uncertain random network. Following the framework of uncertain random network, this paper is devoted to studying the minimum spanning tree problem in the simultaneous presence of random and uncertain weights. The main objectives of the paper are to (1 define an ideal chance distribution associated with an uncertain random network and derive its expression, and (2 establish an optimization model to seek a minimum spanning tree for an uncertain random network. The associated algorithms are designed to calculate the ideal chance distribution and seek the minimum spanning tree, respectively. The remainder of the paper is organized as follows. Section Preliminaries reviews the preliminaries on uncertainty theory, chance theory and uncertain random network. Section Ideal chance distribution aims to defining and deriving an ideal chance distribution and section Minimum spanning tree model presents a model and algorithm to solve the proposed model. A numerical example is presented in section A numerical example to illustrate the effectiveness. Finally, section Conclusions gives a brief summary to the paper. Preliminaries In this section, we will review the basic preliminaries on uncertainty theory, chance theory and uncertain random network. Uncertainty theory In order to rationally handle belief degrees, uncertainty theory was introduced by Liu (27 and subsequently investigated by several scholars. The first fundamental concept is uncertain measure defined by Liu (27. Let L be a σ - algebra on a nonempty set Ɣ. A set function M : L [, 1] is called an uncertain measure if it satisfies the following properties: Normality, i.e., M{Ɣ} =1 for the universal set Ɣ; Duality, i.e., M{ } +M{ c } = 1 for any event ; Subadditivity, i.e., for every countable sequence of events 1, 2,, we have { } M i M { i }. i=1 i=1 The triple (Ɣ,L, M is called an uncertainty space. If (Ɣ k, L k, M k are all uncertainty spaces, then Liu (29a defined a product uncertain measure M satisfying

3 { } M k = M k { k } i=1 k=1 where k are arbitrarily chosen events from L k for k = 1, 2,, respectively. The second fundamental concept is uncertain variable which is defined by Liu (27 as a measurable function from an uncertainty space (Ɣ, L, M to the set of real numbers, i.e., for any Borel set B of real numbers, the set {ξ B} ={γ Ɣ ξ(γ B} is an event. In order to describe an uncertain variable, Liu (27 uses the concept of uncertainty distribution which is defined by M{ξ x} for any x R. Actually, inverse uncertainty distribution also plays an important role in uncertainty theory, which is the inverse function 1 of the uncertainty distribution when is regular. The uncertainty distribution of a monotonous function of uncertain variables is given as follows. Lemma 1 (Liu 21 Let ξ 1,ξ 2,,ξ n be independent uncertain variables with uncertainty distributions 1, 2,, n, respectively. If f (ξ 1,ξ 2,,ξ n is strictly increasing with respect to ξ 1,ξ 2,,ξ m and strictly decreasing with respect to ξ m+1,ξ m+2,,ξ n, then ξ = f (ξ 1,ξ 2,,ξ n is an uncertain variable with an inverse uncertainty distribution ( 1 (α = f 1 1 (α,, 1 m (α, 1 m+1 (1 α,, 1 n (1 α. In particular, if f is a strictly increasing function with ξ 1,ξ 2,,ξ n, then ξ = f (ξ 1,ξ 2,,ξ n has an uncertainty distribution sup f (x 1,x 2,,x n =x min i(x i. 1 i n Liu and Ha (21 provied an expected value formula to the case of function of uncertain variables. Lemma 2 (Liu and Ha 21 Letξ 1,ξ 2,,ξ n be independent uncertain variables with uncertainty distributions 1, 2,, n, respectively. If f (ξ 1,ξ 2,,ξ n is strictly increasing with respect to ξ 1,,ξ m and strictly decreasing with respect to ξ m+1,,ξ n, then the uncertain variable ξ = f (ξ 1,ξ 2,,ξ n has an expected value E[ξ] = 1 ( f Chance theory 1 1 (α,, 1 m (α, 1 m+1 (1 α,, 1 n (1 α dα. Let (Ɣ, L, M be an uncertainty space and (, A, Pr be a probability space. The product (Ɣ, L, M (, A, Pr is called a chance space. Let L A be an uncertain random event. In order describe the possibility of happening, Liu (213a defined a chance measure by Ch{ } = 1 Pr{ω M{γ Ɣ (γ, ω } r}dr. Further, Liu (213a proved that a chance measure satisfies the properties of normality, duality, and monotonicity, that is (1 Ch{Ɣ } =1; (2 Ch{ }+Ch{ c }=1for and event ; (3Ch{ 1 } Ch{ 2 } for any two events 1 2. In addition, Hou (214 proved that the subadditivity of chance measure also holds, i.e., { } Ch i Ch{ i } i=1 i=1 for any sequence of events 1, 2,. Definition 1 (Liu 213a An uncertain random variable is a measurable function ξ from a chance space (Ɣ, L, M (, A, Pr to the set of real numbers, i.e., {ξ B} is an event for any Borel set B. Random variables and uncertain variables are special cases of uncertain random variables. The sum and the product of a random variable and an uncertain variable are both uncertain random variables. In order to describe uncertain random variables, Liu (213a presented a definition of chance distribution. Definition 2 (Liu 213aLet ξ be an uncertain random variable. Then its chance distribution is defined by Ch{ξ x} for any x R. Lemma 3 (Liu 213b Let η 1,η 2,,η m be independent uncertain variables with probability distributions 1, 2,, m, respectively, and let τ 1,τ 2,,τ n be uncertain variables. Then the uncertain random variable ξ = f (η 1,η 2,,η m,τ 1,τ 2,,τ n has a chance distribution F(x, y 1,, y m d 1 (y 1 d m (y m R m where F(x, y 1,, y m is the uncertainty distribution of uncertain variable f (y 1, y 2,, y m,τ 1,τ 2,,τ n for any real numbers y 1, y 2,, y m. Lemma 4 (Liu 213b Let η 1,η 2,,η m be independent uncertain random variables with probability distributions, 1, 2,, m, respectively, and let τ 1,τ 2,,τ n be uncertain variables, then the uncertain random variable ξ = f (η 1,η 2,,η m,τ 1,τ 2,,τ n has an expected value E[ξ] = E[ f (y 1,, y m,τ 1,,τ n ] R m d 1 (y 1 d m (y m

4 Fig. 1 A simple uncertain random network where E[y 1,, y m,τ 1,,τ n ] is the expected value of uncertain variable f (y 1,, y m,τ 1,,τ n for any real numbers y 1, y 2,, y m. In particular, if η is a random variable and is τ an uncertain variable, then it follows Lemma 1 that E[η + τ] =E[η]+ E[τ]. Uncertain random network The term network is a synonym for a weighted graph, where the weights may be understood as cost, distance or time and so on. Assume that in a network some weights are random variables and others are uncertain variables. In order to model this type of network, Liu (214 presented the following concept. Definition 3 (Liu 214 Assume N is the collection of nodes, U is the collection of uncertain edges, R is the collection of random edges, and W is the collection of uncertain and random edge weights. Then the quartette (N, U, R, W is said to be an uncertain random network. The uncertain random network degenerates to a random network (Frank and Hakimi 1965 when all weights are random variables, or degenerates to an uncertain network (Liu 21 when all weights are uncertain variables. We present a simple example to interpret the concept. Figure 1 shows an uncertain random network (N, U, R, W of order 4 in which N ={1, 2, 3, 4}, U ={(1, 2, (3, 4}, R ={(1, 4, (2, 3}, W ={w 12,w 14,w 23,w 34 }. Ideal chance distribution For a network, we call it a deterministic network if its edge weights are all deterministic rather than random, uncertain or uncertain random. Let (N, E, W be a deterministic network, in which E is the collection of edges and W is the collection of deterministic weights. Many classical algorithms such as Kruskal (1956 algorithm and Prim (1957 algorithm can be used to find a minimum spanning tree of the network. The weight of the minimum spanning tree is a function with respect to the weight w ij of edge (i, j for (i, j E. We denote the function by f ({w ij (i, j E} or f (W. Assume that W 1 ={wij 1 (i, j E} and W 2 ={wij 2 (i, j E} are two collections of deterministic weights. If wij 1 w2 ij for all (i, j E, it is easy to verify that f (W 1 f (W 2. That is, the function f is an increasing function with respect to weights w ij. Next we turn to discuss an undirected uncertain random network (N, U, R, W. First we list some assumptions as follows: (1 The undirected uncertain random network is connected; (2 The weight of each edge (i, j U R is finite; (3 The weight of each edge (i, j U R is either a positive uncertain variable or a positive random variables; (4 All the uncertain weights are independent in the sense of uncertainty measure; (5 All the random weights are independent in the sense of probability measure. Without loss of generality, we assume that uncertain weight τ ij for (i, j U are defined on an uncertainty space (Ɣ, L, M and random weight ξ ij for (i, j R are defined on probability space (, A, Pr. Then for any given γ Ɣ and ω, τ ij (γ, (i, j U and ξ ij (ω, (i, j R are all crisp numbers. That is to say, if we know the realizations of random and uncertain edge weights, then the network (N, U, R, W is essentially a deterministic one and its minimum spanning tree may be obtained. In spite of the minimum spanning tree may not be unique, all the minimum spanning trees have the same weight for a given network. We still denote the weight of the minimum spanning tree by f (τ ij (γ, (i, j U; ξ ij (ω, (i, j R, which implies that f can be regarded as a function of γ and ω. In such a way, each uncertain random network (N, U, R, W corresponds to a relationship f which is a function of uncertain weights and random weights. It follows from the operational law that f (τ ij,(i, j U; ξ ij,(i, j R is also an uncertain random variable. The chance distribution of f (τ ij,(i, j U; ξ ij,(i, j R is called an ideal chance distribution associated with uncertain random network (N, U, R, W. Note that the ideal chance distribution is unique for a given uncertain random network. Next theorem gives how to calculate an ideal chance distribution. Theorem 1 Let (N, U, R, W be an uncertain random network. Suppose that the uncertain weights τ ij have regular uncertainty distributions ϒ ij for (i, j U, and the random weights ξ ij have probability distributions ij for (i, j R, respectively. Then the ideal chance distribution associated with the uncertain random network (N, U, R, W is

5 (i, j R F(x; y ij, d ij (y ij, (1 where F(x; y ij,(i, j R is the uncertainty distribution of uncertain variable f (τ ij,(i, j U; y ij,(i, j R. Proof Let f (τ ij,(i, j U; ξ ij,(i, j R be the weight of ideal minimum spanning tree. From the definition of chance distribution of uncertain random variable and Lemma 3, we can obtain the ideal chance distribution as follows Ch{ f (τ ij,(i, j U; ξ ij,(i, j R x} = F(x; y ij,(i, j R d ij (y ij = F(x; y ij,(i, j R d ij (y ij where F(x; y ij,(i, j R is the uncertainty distribution of f (τ ij,(i, j U; y ij,(i, j R for any real numbers y ij,(i, j R. The theorem is completed. Remark 1 The uncertainty distribution F(x; y ij,(i, j R is determined by its inverse uncertainty distribution F 1 (α; y ij,(i, j R. By Lemma 1, for given α (, 1, we have F 1 (α; y ij,(i, j R = f (ϒij 1 (α, (i, j U; y ij,(i, j R which is just the weight of the minimum spanning tree of a determinacy network and may be calculated by using Kruskal s algorithm or Prim s algorithm. Remark 2 If the uncertain random network becomes a random network, then the ideal chance distribution is essentially the following probability distribution d ij (y ij. f (y ij, x Remark 3 If the uncertain random network becomes an uncertain network, then the ideal chance distribution is essentially an uncertainty distribution with the following inverse function 1 (α = f (ϒij 1 (α, (i, j U, α (, 1. Next we present two simple examples to illustrate the calculation of the ideal chance distribution. Example 1 Consider an uncertain random series network with four edges shown in Fig. 2. Assume that uncertain weights τ 1,τ 2 have regular uncertainty distributions ϒ 1,ϒ 2, Fig. 2 The uncertain random series network with four edges in Example 1 and random weights ξ 1,ξ 2 have probability distributions 1, 2, respectively. Then from Theorem 1, the ideal chance distribution associated with the network is F(x; y 1, y 2 d 1 (y 1 d 2 (y 2 where F(x; y 1, y 2 is determined by its inverse uncertainty distribution F 1 (α; y 1, y 2 = ϒ1 1 (α + ϒ 1 2 (α + y 1 + y 1. Example 2 Revisit the uncertain random network with four edges shown in Fig. 1 in which w 12 = τ 1,w 34 = τ 2 and w 14 = ξ 1,w 23 = ξ 2. We still assume that uncertain weights τ 1,τ 2 have regular uncertainty distributions ϒ 1,ϒ 2, and random weights ξ 1, ξ 2 have probability distributions 1, 2, respectively. Then from Theorem 1, the ideal chance distribution associated with the network is F(x; y 1, y 2 d 1 (y 1 d 2 (y 2 where F(x; y 1, y 2 is determined by its inverse uncertainty distribution F 1 (α; y 1, y 2 = (ϒ1 1 (α + ϒ 1 2 (α + y 1 (ϒ1 1 (α +ϒ 2 1 (α+y 2 (ϒ1 1 (α+y 1+y 2 (ϒ2 1 (α+y 1+y 2. Generally speaking, it is difficult to obtain an analytical expression of an ideal chance distribution. In practice we may use numerical techniques to calculate it. Next we propose the following algorithm to numerically calculate an ideal chance distribution. Algorithm 1 Step 1. For each random edge (i, j R, give a partition ij of interval [a ij, b ij ] with step =.1 and let random weight ξ ij only take values y ij in ij. Step 2. For given y ij ij for (i, j R and each α {.1,.2,,.99}, calculate F 1 (α; y ij,(i, j R by using Kruskal s algorithm or Prim s algorithm. Step 3. Derive the uncertainty distribution F(x; y ij,(i, j R via linear interpolation from its discrete form. Step 4. Input F(x; y ij,(i, j R into Equation (1 to gain the value of ideal chance distribution (x for each x. For a given uncertain random network, we can also obtain the expected value of uncertain random variable associated with the ideal chance distribution.

6 Theorem 2 Let (N, U, R, W be an uncertain random network. Assume that uncertain weights τ ij have regular uncertainty distributions ϒ ij for (i, j U, and random weights ξ ij have probability distributions ij for (i, j R, respectively. Then the uncertain random variable ξ = f ( τ ij,(i, j U; ξ ij,(i, j R has an expected value E[ξ] = 1 f (ϒ 1 ij (i, j U; y ij,(i, j Rdα (α, d ij (y ij. Proof Since the uncertain random variable ξ = f (τ ij,(i, j U; ξ ij,(i, j R is a strictly increasing function with respect to τ ij,(i, j U, by Lemma 2, wehave E[τ ij,(i, j U; y ij,(i, j R] 1 ( = f (α, (i, j U; y ij,(i, j R dα. ϒ 1 ij It follows from Lemma 4 that we obtain E[ξ] = E[ f (τ ij,(i, j U; ξ ij (ω, (i, j R] d ij (y ij 1 = f ( ϒij 1 (α, (i, j U, y ij,(i, j R dα d ij (y ij. The theorem is completed. Example 3 In Example 1, by Theorem 2, we get the expected value associated with the ideal chance distribution as follows, E[ξ] = 1 (ϒ 1 1 (α + ϒ 1 2 (α + y 1 + y 1 dαd 1 (y 1 d 2 (y 2 = E[ξ 1 ]+E[ξ 2 ]+E[τ 1 ]+E[τ 2 ]. Minimum spanning tree model In a real-life network optimization problem, the found minimum spanning tree relies on the given optimization model. In this section, we will formulate an model for an uncertain random network and design an algorithm to seek a minimum spanning tree. Given a connected and undirected uncertain random network (N, U, R, W, where N = {1, 2,, n}, W = {τ ij,(i, j U; ξ ij,(i, j R}. Letx ij = represent edge (i, j / T and x ij = 1 represent edge (i, j T. Then, {x ij,(i, j U R} is a spanning tree if and only if x ij = n 1 R x ij S 1, 2 S n (2 R(S x ij ={, 1}, (i, j U R where U R(S denotes the collection of edges of induced subgraph from S N. The weight of a spanning tree {x ij,(i, j U R} is x ij τ ij + x ij ξ ij which is obviously an uncertain random variable. Its chance distribution is denoted by (z x ij,(i, j U R, i.e., (z x ij,(i, j U R = Ch x ij τ ij + x ij ξ ij z. Remark 4 Let (N, U, R, W be an uncertain random network and let (z be an ideal chance distribution associated with it. Assume that {x ij,(i, j U R} is a given spanning tree. Then we have (z x ij,(i, j U R (z. As stated in Ideal chance distribution section, each scenario (γ, ω corresponds to a minimum spanning tree and different scenarios perhaps correspond to different minimum spanning trees. If there exists an ideal minimum spanning tree which is a minimum one under each scenario, then it is the desired result. However, such an ideal minimum spanning tree may not really exist and moreover we also do not know which scenario will be realized in the future. Maybe a better alternative to find a spanning tree whose chance distribution is as close to the ideal chance distribution as possible. How to describe the closeness? We use the following definition. Definition 4 Let (N, U, R, W be an uncertain random network and (z an ideal chance distribution associated with it. Assume that {x ij,(i, j U R} is a spanning tree with chance distribution (z x ij,(i, j U R. If (z x ij,(i, j U R is the closest to the (z, i.e., { (z (z xij,(i, j U R } dz is minimum, then the spanning tree {x ij,(i, j U R} is called the minimum spanning tree in the uncertain random network.

7 Based on the idea, we formulate the following optimization model to determine a minimum spanning tree for an uncertain random network, ( min (z (z xij,(i, j U R {x ij, R} dz s.t. x ij = n 1 R (3 x ij S 1, S N, 2 S n R(S x ij ={, 1}, (i, j U R From the definition of chance distribution, we have the following result. Theorem 3 Let (N, U, R, W be an uncertain random network with ideal chance distribution (z. Assume that uncertain weights τ ij have regular uncertainty distributions ϒ ij for (i, j U, and random weights ξ ij have probability distributions ij for (i, j R, respectively. The model (3 is equivalent to the following model: It follows from Lemma 3 that (z x ij,(i, j U R = Ch x ij τ ij + x ij ξ ij z = M x ij τ ij + y z R (y = M x ij τ ij z y (y. R Further, it follows from Lemma 1 that M x ij τ ij z y = sup min ij(x ij r ij x ij r ij = z y = ϒ z x ij y ij. (6 min {x ij, R} s.t. R (z R(S x ij ={, 1}, x ij = n 1 ϒ z x ij y ij d (y dz x ij S 1, S N, 2 S n (i, j U R (4 where (y = ϒ(r = x ij y ij y d ij (x ij y ij, ( sup min ϒ ij(x ij r ij. x ij r ij = r Proof Assume that {x ij,(i, j U R} is a spanning tree. Denote the probability distribution of x ij ξ ij by (y. From the assumption (5, we have (y = x ij y ij y d ij (x ij y ij. (5 Substituting Eqs. (5, 6 into the objective of Model (4, the theorem is completed. In order to seek the minimum spanning tree in an uncertain random network (N, U, R, W, we develop the following solution procedure. Algorithm 2 Step 1. Calculate the ideal chance distribution for the uncertain random network by using Algorithm 1. Step 2. Produce all the spanning trees in the uncertain random network by using the breadth-first search algorithm. Step 3. Calculate the chance distribution of the weight of each spanning tree according to Lemma 3. Step 4. Calculate the objective function of Model (4 for given spanning tree, and choose the minimum value of objective function, which corresponds to the desired minimum spanning tree. Algorithm 2 only gives the logical sequence of solving the proposed model. For large uncertain random networks,

8 Fig. 3 Uncertain random network Table 1 Uncertain distributions and probability distributions of edge weights edge(i, j τ ij edge(i, j ξ ij (1,2 L(14, 16 (1,4 U(11, 13 (2,4 L(15, 18 (2,3 U(14, 16 (3,4 Z(11, 13, 16 the efficiency may be low. Some efficient algorithms may be designed to improve the efficiency when real applications. A numerical example In this section, we will give an example to illustrate the proposed model in last section. We consider an uncertain random network (N, U, R, W with 4 nodes and 5 edges shown in Fig. 3. Assume the uncertain weights τ 12, τ 24, τ 34 have regular uncertainty distributions ϒ 12, ϒ 24, ϒ 34 and the random weights ξ 14, ξ 23 have probability distributions 14, 23, respectively. These parameters are listed Table 1 in which U represents uniformly distributed probability distribution, and L and Z represent linear uncertainty distribution and zigzag uncertainty distribution, respectively. From Theorem 1, we can obtain the ideal chance distribution associated with the network (N, U, R, W as follows F(x; y 14, y 23 d 14 (y 14 d 23 (y 23 where F(x; y 41, y 23 is determined by its inverse uncertainty distribution ( F 1 (α; y 14, y 23 = f ϒ12 1 (α, ϒ 1 24 (α, ϒ 1 34 (α, y 14, y 23 and f may be calculated by the Kruskal s algorithm or Prim s algorithm for each given α. Based on the graphical topology structure, all spanning trees are obtained and listed in Fig. 4. It is evident that the network in Fig. 3 has eight spanning trees. We may first calculate the ideal chance distribution (x associated with the network by Algorithm 1, which is shown in Fig. 5. Then according to the idea of Algorithm 2, we calculate the variation of the chance distribution of total weight of each spanning tree to the ideal one. All the results are listed in the second and the fourth columns in Table 2. Model (4 implies that Tree 6 is the desired minimum spanning tree since the variation between its chance distribution and the ideal one is minimum. In other words, the chance distribution of the weight of Tree 6 is the closest to the ideal one. In order to intuitively see this, Fig. 6 plots the shape of ideal chance distribution and the shapes of chance distributions Tree 1 Tree 2 Tree 3 Tree 4 Tree 5 Tree 6 Tree 7 Tree 8 Fig. 4 All spanning trees of uncertain random network in Fig. 3

9 proposed a concept of ideal chance distribution and gave its expression and calculation method. A model was formulated to seek a minimum spanning tree which is closest to the ideal one. Finally, the paper gave a numerical example to illustrate the modelling idea and the solution procedure. Acknowledgments This work was supported by National Natural Science Foundation of China Grant Nos , and Fig. 5 The shape of ideal chance distribution Table 2 Variation between chance distribution of the weight of each spanning tree and the ideal one Spanning trees Variation Spanning trees Variation Tree Tree5 5.6 Tree Tree6 2.8 Tree Tree7 3.7 Tree Tree Fig. 6 The shapes of chance distributions of all spanning trees of all the spanning trees. It is easy to see that the chance distributions of the weights of Tree 6 and Tree 7 are closer to the ideal one than other trees, which can also be seen from Table 2. Conclusions Indeterministic edge weights often appear in network optimization problem. Previously stochastic programming and uncertain programming are respectively used to model the minimum spanning tree problem for the case with sufficient data and the case without enough data. Different from these approaches, this paper considered a complex situation in which some weights are random variables and the others are uncertain variables. Chance theory and uncertain random programming were applied to find a minimum spanning tree for such an uncertain random network. This paper References Bertsimas, D. J. (199. The probabilistic minimum spanning tree problem. Networks, 2(3, Borüvka, O. (1926. O jistém problému minimálním. Práce Mor. Přírodovéd. Spol. v Brně, 3, Dijkstra, E. W. (1959. A note on two problems in connexion with graphs. Numerische Mathematik, 1(1, Frieze, A. M. (1985. On the value of a random minimum spanning tree problem. Discret Applied Mathematics, 1(1, Frank, H., & Hakimi, S. L. (1965. Probabilistic flows through a communication network. IEEE Transactions on Circuit Theory, 12, Gao, Y. (211. Shortest path problem with uncertain arc lengths. Computers and Mathematics with Applications, 62(6, Gao, J. W., & Yao, K. (215. Some concepts and theorems of uncertain random process. International Journal of Intelligent Systems, 3(1, Guo, H. Y., & Wang, X. S. (214. Variance of uncertain random variables. Journal of Uncertainty Analysis and Applications, 2, 6. Hou, Y. C. (214. Subadditivity of chance measure. Journal of Uncertainty Analysis and Applications, 2, 14. Ke, H., Su, T. Y., & Ni, Y. D. (215. Uncertain random multilevel programming with application to product control problem. Soft Computing (to be published. Ishii, H., Shiode, S., Nishida, T., & Namasuya, Y. (1981. Stochastic spanning tree problem. Discrete Applied Mathematics, 3(4, Ishii, H., & Matsutomi, T. (1995. Confidence regional method of stochastic spanning tree problem. Mathematical and Computer Modelling, 22(1, Kruskal, J. B. (1956. On the shortest spanning tree subtree of a graph and the traveling salesman problem. Proceedings of the American Mathematical Society, 7(1, Liu, B. (27. Uncertainty theory (2nd ed.. Berlin: Springer. Liu, B. (29a. Some research problems in uncertainty theory. Journal of Uncertain Systems, 3(1, 3 1. Liu, B. (29b. Theory and practice of uncertain programming (2nd ed.. Berlin: Springer. Liu, B. (21. Uncertainty theory: A branch of mathematics for modeling human uncertainty. Berlin: Springer. Liu, B. (214. Uncertain random graph and uncertain random network. Journal of Uncertain Systems, 8(1, Liu, Y. H., & Ha, M. H. (21. Expected value of function of uncertain variables. Journal of Uncertain Systems, 4(3, Liu, Y. H. (213a. Uncertain random variables: A mixture of uncertainty and randomness. Soft Computing, 17(4, Liu, Y. H. (213b. Uncertain random programming with applications. Fuzzy Optimization and Decision Making, 12(2, Prim, R. C. (1957. Shortest conection networks and some generalizations. Bell System Technical Journal, 36,

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