Models for inverse minimum spanning tree ... - Semantic Scholar
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Journal of Intelligent & Fuzzy Systems 27 (2014) 2691–2702 DOI:10.3233/IFS-141384 IOS Press
Models for inverse minimum spanning tree problem with fuzzy edge weights Jingyu Zhanga , Jian Zhoub,∗ and Shuya Zhongb a Department b School
of Clinical Decision Support Solutions, Philips Research North America, NY, USA of Management, Shanghai University, Shanghai, China
Abstract. An inverse minimum spanning tree problem is to make the least modification on the edge weights such that a predetermined spanning tree is a minimum spanning tree with respect to the new edge weights. In this paper, a type of fuzzy inverse minimum spanning tree problem is introduced from a LAN reconstruction problem, where the weights of edges are assumed to be fuzzy variables. The concept of fuzzy α-minimum spanning tree is initialized, and subsequently a fuzzy α-minimum spanning tree model and a credibility maximization model are presented to formulate the problem according to different decision criteria. In order to solve the two fuzzy models, a fuzzy simulation for computing credibility is designed and then embedded into a genetic algorithm to produce some hybrid intelligent algorithms. Finally, some computational examples are given to illustrate the effectiveness of the proposed algorithms. Keywords: Minimum spanning tree, inverse optimization, fuzzy programming, genetic algorithm
1. Introduction The inverse minimum spanning tree (IMST) or inverse spanning tree problem is a type of inverse optimization problems. In an IMST problem, there is a connected graph G = (V, E) with n vertices, m edges, and an edge weight vector c. The objective of IMST problem is to modify the weight vector c so that a predetermined spanning tree T 0 is a minimum spanning tree of graph G with respect to the new weight vector x, and simultaneously the total modification of weights is a minimum. The IMST problem was first studied by Zhang et al. [26] with partition constraints. Since then, much work has been done on the IMST related problems because many practical problems can be handled in this framework. Guan and Zhang [9] considered a class ∗ Corresponding
author. Jian Zhou, School of Management, Shanghai University, Shanghai 200444, China. E-mail: zhou jian@ shu.edu.cn.
of inverse constrained bottleneck problems under the weighted l∞ norm. In these inverse problems, two weight vectors under bound restrictions were modified so that a given candidate solution became an optimal one and the deviation of the weights, measured by the weighted l∞ norm, was minimum. Wang et al. [23] also discussed a class of inverse dominant problems under the weighted l∞ norm in the area of combinatorial optimization. Similarly, Yang and Zhang [24] considered some inverse min-max or max-min network problems not only under the weighted l∞ norm but also the weighted l1 norm, while Guan et al. [7] initially studied an inverse problem of optimization problems with combined minmax-minsum objective functions. Besides, Wang [22] proposed two models for the partial inverse most unbalanced spanning tree problem under the weighted Hamming distance and the weighted l1 norm. Liu and Yao [18] also brought the Hamming distance into the weighted inverse maximum perfect matching problems. In the practical research area of inverse optimization, Farag´o et al. [4] applied the
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inverse shortest path problem to high-speed telecommunication networks and especially considered the case when no weighting exists that could make each given solution exactly optimal. Sever [19] adopted a learning algorithm based on a numerical solution of a system of linear equations which was derived from a wellknown regularization model, and applied to the task of reconstruction of an inhomogeneous object as pattern recognition for the inverse problems. For solving the classical IMST problem and some of its derivatives, considerable effort has been devoted to designing efficient algorithms with lower computational complexity as far as possible. Sokkalingam et al. [20] developed a specific implementation of the successive shortest path algorithm that run in O(n3 ) time for the IMST problem. Subsequently Ahuaj and Orlin [2] improved the previous O(n3 ) time algorithm to a more efficient O(n2 log n) time algorithm, and Hochbaum [11] continued the improvement in the complexity of algorithm to make the run time O(nm log2 n). He et al. [10] presented some strongly polynomial algorithms for the weighted IMST problems under the Hamming distance in accordance with the method for minimum-weight node cover problem on bipartite graph. Recently, Guan and Zhang [8] proposed a linear O(n) time algorithm for the inverse 1-median problem on trees under the weighted l∞ norm, as well as two polynomial time algorithms with time complexities O(n log n) and O(n) for the inverse problem under the weighted bottleneck-Hamming distance. Since in some practical circumstances, the edge weights cannot be explicitly determined, some endeavor was done to deal with the IMST related problems with indeterminate information by considering the edge weights as random variables (see, e.g., [21, 27]). However, the approximation by random variables does not work when there are not enough data for estimation or when the edge weights possess an unobservable nature. Hence some different kinds of variables have been also suggested for describing the indeterminate edge weights. For example, fuzzy variables were brought into the quadratic minimum spanning tree problem in [5] and the maximum spanning tree problem in [14]. Eshghi and Nematian [3] considered the quadratic minimum spanning tree problem with fuzzy random edge weights. Katagiri et al. [12] characterized the edge weights of a graph as random fuzzy variables in a minimum spanning tree problem. Recently, Zhang et al. [28, 29] discussed the inverse minimum spanning tree problem by assuming all the edge weights as uncertain variables. See Zhou et al. [30, 32, 34] for more details.
In this paper, an IMST problem with fuzzy edge weights is considered from the practical application angle. Specifically, in a LAN reconstruction problem [6], the parameters of bandwidth and net speed are both attached to edges. Obviously, the net speeds of LAN are not fixed variables with respect to the bandwidths due to the changing environments. In this situation, it is natural to represent the net speeds as fuzzy numbers by some expert knowledge. This is actually an IMST problem with fuzzy weights, and the classic IMST algorithms become powerless to such a problem. In order to solve this type of IMST problem, Zhou et al. [31] proposed a conception of fuzzy α-minimum spanning tree, which is essentially a transformation of the classical minimum spanning tree. Based on this notion, Zhou et al. [33] developed a fuzzy programming model with some chance constraints to formulate an IMST problem with fuzzy weights. In view of the deficiency of the definition in [31], in this paper, we define a new concept of fuzzy α-minimum spanning tree from a novel angle, and then provide a fuzzy α-minimum spanning tree model and a credibility maximization model for formulating this problem in practical situations according to different decision criteria. In this paper, the IMST problem with fuzzy weights is referred to as the fuzzy inverse minimum spanning tree (FIMST) problem. The rest of this paper is organized as follows. In Section 2, we briefly review the classic IMST problem, and the concepts of fuzzy variable and credibility of fuzzy event. Section 3 introduces the application backgrounds and mathematical description of the FIMST problem, and defines a fuzzy α-minimum spanning tree. Following that, two fuzzy programming models are presented for the FIMST problem in Section 4. In order to solve the proposed fuzzy models, a fuzzy simulation for computing credibility and a genetic algorithm are combined to produce some hybrid intelligent algorithms in Section 5. Section 6 illustrates the efficiency of the proposed algorithms by some numerical experiments.
2. Preliminaries In order to describe the FIMST problem more clearly, a classic concept of minimum spanning tree, a path optimality condition, and the mathematical formulation of the IMST problem are reviewed in this section. We also introduce the concept of fuzzy variable and credibility of fuzzy event for our purpose.
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2.1. Classic inverse minimum spanning tree problem Let G = (V, E) denote a connected graph consisting of the vertex set V = {v1 , v2 , . . . , vn } and the edge set E = {1, 2, . . . , m}. A spanning tree T = T (V, S) of G is a connected acyclic subgraph containing all vertices. For simplicity, we denote a spanning tree T by its edge set S throughout this paper. Definition 1. (Minimum Spanning Tree) Given a connected graph G = (V, E) with edge weights ci , i ∈ E = {1, 2, . . . , m}, a spanning tree T 0 is said to be a minimum spanning tree if ci ≤ (1) cj i∈T 0
j∈T
holds for any spanning tree T . Example 1. A graph G with 4 vertices and 6 edges is shown in Figure 1. When (c1 , c2 , c3 , c4 , c5 , c6 ) = (1, 3, 4, 2, 2, 3), the unique minimum spanning tree is T 0 = {AB, AD, AC} with i∈T 0 ci = c1 + c4 + c5 = 5. When (c1 , c2 , c3 , c4 , c5 , c6 ) = (3, 1, 2, 3, 4, 1), there are two minimum spanning trees: 0 T10 = {AB, BC, BD}, and T2 = {BC, AD, BD}, with c = c = 5. i∈T10 i i∈T20 i The classic IMST problem is to find some new edge weights such that a given spanning tree T 0 is a minimum spanning tree with respect to the new edge weights and the total change of the edge weights is minimized. Example 2. A graph G with 6 vertices and 10 edges is shown in Figure 2, where ci denotes the weight on edge i, and the solid line represents a predetermined spanning tree T 0 . The goal of the classic IMST problem is to find
Fig. 1. Graph G for Example 1.
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anew weight vector x = (x1 , x2 , . . . , x10 ) to minimize 10 i=1 |xi − ci | provided that the predetermined spanning tree T 0 is a minimum spanning tree with respect to x. So, how to judge whether the predetermined spanning tree is a minimum spanning tree with respect to a weight vector? Before providing the mathematical description of the IMST problem, some concepts are introduced as follows. First, we refer to the edges in the predetermined spanning tree T 0 as tree edges, and the edges not in T 0 as non-tree edges. For convenience, denote the set of non-tree edges as E\T 0 . A path in the graph G is called a tree path if the path only contains tree edges. It is clear that there must be a unique tree path between any two vertices of graph G. It follows directly that there is a unique tree path between the two vertices of any non-tree edge j, called tree path of edge j and denoted by Pj . Example 3. In Figure 2, the set of non-tree edges is E\T 0 = {6, 7, 8, 9, 10}, and the tree path of non-tree edge BD is AB-AE-DE, i.e., P9 = {1, 3, 5}. Moreover, Ahuja et al. [1] proved an equivalent condition of minimum spanning tree, referred to as the path optimality condition, which characterizes the minimum spanning tree by a set of constraints on the non-tree edges and their tree paths. Lemma 1. (Ahuja et al. [1], Path Optimality Condition) Given a connected graph G = (V, E) with edge weights xi , i ∈ E = {1, 2, . . . , m}, a spanning tree T 0 is a minimum spanning tree if and only if xi − xj ≤ 0,
j ∈ E\T 0 , i ∈ Pj
(2)
where E\T 0 is the set of non-tree edges, and Pj is the tree path of edge j. According to Lemma 1, the classic IMST problem can be formulated as
Fig. 2. Graph G for Example 2.
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⎧ m ⎪ ⎪ min |xi − ci | ⎪ ⎪ ⎨ x
3.1. Application backgrounds
i=1
⎪ subject to: ⎪ ⎪ ⎪ ⎩ xi − xj ≤ 0,
(3) j ∈ E\T 0 , i ∈ Pj
where ci and xi are the original and new weights of edge i, i ∈ E, respectively. Some efficient algorithms are available to solve this model (see, e.g., [2, 11, 20]). Note that in [20] other types of objective functions are considered as well for the IMST problems. 2.2. Fuzzy variables Since its introduction in 1965 by Zadeh [25], fuzzy set theory has been well developed and applied in a wide variety of real problems. In the following, we briefly review the concepts of possibility, necessity, and credibility of fuzzy event. Let be a nonempty set, P() the power set of , and Pos a possibility measure. Then the triplet (, P(), Pos) is called a possibility space. A fuzzy variable is defined as a function from a possibility space (, P(), Pos) to the set of real numbers. Suppose that ξ is a fuzzy variable with membership function µ. Then the possibility, necessity, and credibility of a fuzzy event {ξ ≥ r} can be represented by Pos{ξ ≥ r} = sup µ(u),
Many reconstruction problems in practice can be transformed to FIMST problems. In order to see that, let us consider a LAN reconstruction problem as follows. Much research work shows that the spanning tree structure is the best topology for telecommunication network designs [13], especially in computer network systems. LANs are commonly used as a communication infrastructure that meets the demands of users in a local environment. These computer networks typically consist of several LAN segments connected via bridges. Suppose that there is an old LAN, in which several service centers are interconnected via bridges. For some reasons (e.g., the tremendous network congestion), the bandwidths on bridges must be modified. The decision-maker hopes that a predetermined spanning tree becomes a minimum spanning tree with respect to the net speed on each bridge in order to ensure high speed between some main service centers. Also the total bandwidth modification should be minimized so as to diminish the cost of reconstruction. Apparently the net speeds are related to bandwidths. It is natural to describe the net speed on a bridge as a fuzzy number instead of a deterministic one with respect to bandwidths of bridges when there are no former statistical data in some cases. This is a typical inverse minimum spanning tree problem with fuzzy weights, i.e., an FIMST problem.
u≥r
Nec{ξ ≥ r} = 1 − sup µ(u),
3.2. Mathematical description
1 Cr{ξ ≥ r} = ( Pos{ξ ≥ r} + Nec{ξ ≥ r} ) . 2
In order to provide a mathematical description of the FIMST problem, the following notations are used:
u 0 for some j ∈ E\T 0 and i ∈ Pj }). 5.2. Hybrid intelligent algorithm procedure Genetic algorithms (GA) have demonstrated considerable success in providing good solutions to many complex optimization problems. Due to the fuzziness of weight vector ξ(x), a fuzzy simulation is designed for calculating the credibility U, and then embedded into a GA to produce some hybrid intelligent algorithms for solving models (7) and (11) efficiently. As an illustration, the following steps show how the hybrid intelligent algorithm for model (7) works. Step 1. Initialize pop size feasible chromosomes k ) for k = 1, 2, . . . , pop size Vk = (x1k , x2k , . . . , xm from the potential region [l1 , u1 ] × [l2 , u2 ] × . . . × [lm , um ] uniformly, whose feasibility is checked by Cr{ξi (Vk ) − ξj (Vk ) ≤ 0, j ∈ E\T 0 , i ∈ Pj } ≥ α, k = 1, 2, . . . , pop size, respectively, where the credibilities are calculated by the fuzzy simulation designed above. Step 2. Calculate the objective values O(Vk ) = m k i=1 |xi −ci | for all chromosomes Vk , k = 1, 2, . . . , pop size, respectively. Step 3. Compute the fitness of all chromosomes Vk , k = 1, 2, . . . , pop size assuming the chromosomes have been rearranged from good to bad according to their objective values O(Vk ). Step 4. Select the chromosomes for a new population by spinning the roulette wheel according to the fitness of all chromosomes. Step 5. Update the chromosomes by crossover operation, where the feasibility of new chromosomes is checked by the fuzzy simulation designed above. Step 6. Update the chromosomes by mutation operation, where the feasibility of new chromosomes is checked by the fuzzy simulation. Step 7. Repeat the second to fifth steps for N times. Step 8. Report the best chromosome V ∗ as the optimal solution. A similar hybrid intelligent algorithm has also been designed for solving model (11) effectively. More detailed procedure will follow a specific example and be illustrated clearly in Section 6.
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6. Computational examples As to the two hybrid intelligent algorithms above, since they do not belong to the traditional classical algorithms, we generally do not concern with their efficiency by considering the computational complexity but care about their effectivity. In order to illustrate the effectiveness of the hybrid intelligent algorithms, we propose some numerical examples performed on a personal computer. Consider a LAN reconstruction problem with 6 service centers and 10 bridges showed in Figure 4, where the vertex set is V = {A, B, C, D, E, F }, the edge set is E = {1, 2, . . . , 10}, and the solid lines represent the predetermined spanning tree, denoted by T 0 . There are three weights on each bridge, where ci and xi represent the original and new bandwidths on bridge i, respectively, and ξi denotes the fuzzy net speed. The values of ci and ξi are given in Table 1. Here the fuzzy net speed ξi is assumed be only with respect to the bandwidth xi on bridge i, i.e., ξi = ξi (xi ) for the sake of simpleness. Example 5. If we want to minimize the total modification of bandwidths with a given confidence level α = 0.8 so as to diminish the cost of reconstruction, we
have the following fuzzy 0.8-minimum spanning tree model, ⎧ 10 ⎪ ⎪ ⎪ min ⎪ |xi − ci | ⎪ ⎪ ⎪ x i=1 ⎪ ⎨ subject to: ⎪ ⎪
⎪ ⎪ Cr ξi (xi ) − ξj (xj ) ≤ 0, j ∈ E\T 0 , i ∈ Pj ≥ 0.8 ⎪ ⎪ ⎪ ⎪ ⎩ 80 ≤ xi ≤ 180, i ∈ E (13) where the non-tree edge set E\T 0 = {6, 7, 8, 9, 10}, and Pj is the tree path of edge j ∈ E\T 0 . In order to solve model (13), the hybrid intelligent algorithm designed in Section 5 has been run with 10000 cycles in fuzzy simulation and 10000 generations in GA. The detailed heuristic procedures are described as follows. Representation structure We use a vector V = (x1 , x2 , . . . , x10 ) as a chromosome to represent a solution x. Initialization process We define an integer pop size as the number of chromosomes, and initialize pop size feasible chromosomes V1 , V2 , . . . , Vpop size uniformly from [80, 180]10 . The feasibility of chromosome Vk = k ) is checked by (x1k , x2k , . . . , x10 Cr{ξi (Vk ) − ξj (Vk ) ≤ 0, j ∈ E\T 0 , i ∈ Pj } ≥ α, k = 1, 2, . . ., pop size, respectively, where the credibilities are calculated by the fuzzy simulation designed in Section 5.1. Computing objective values In order to compute the objective values of all the chromosomes, we calculate
Fig. 4. The FIMST problem for computational examples.
O(Vk ) = Table 1 Weights for the FIMST problem in Figure 4 Edge i 1 2 3 4 5 6 7 8 9 10
Original weight ci
Fuzzy weight ξi (xi )
120 40 30 150 150 60 60 40 170 130
(x1 − 4, x1 , x1 + 4) (x2 − 8, x2 , x2 + 8) (x3 − 12, x3 , x3 + 12) (x4 − 16, x4 , x4 + 16) (x5 − 20, x5 , x5 + 20) (x6 − 24, x6 , x6 + 24) (x7 − 28, x7 , x7 + 28) (x8 − 32, x8 , x8 + 32) (x9 − 36, x9 , x9 + 36) (x10 − 40, x10 , x10 + 40)
10
|xik − ci |
i=1 k ), k = 1, 2, . . . , pop size, for Vk = (x1k , x2k , . . . , x10 respectively.
Evaluation function Rearrange the chromosomes V1 , V2 , . . . , Vpop size from good to bad according to their objective values O(Vk ) with assumption that a chromosome with smaller objective value is better. Then compute the fitness Eval(Vk ) for all the chromosomes Vk , k = 1, 2, . . . , pop size, where Eval(Vk ) is defined by Eval(Vk ) = a(1 − a)k−1 , k = 1, 2, . . . , pop size.
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In this formula, a ∈ (0, 1) is a parameter in the genetic system.
For convenience, we still denote the new chromosomes by V1 , V2 , . . . , Vpop size .
Selection process The selection process is based on spinning the roulette wheel pop size times which can be stated as follows,
Mutation operation We give a parameter Pm as the probability of mutation in advance. Similar to the process of crossover operation, the mutation operation will be performed in the following way.
Step 1. Calculate the cumulative probability qk for each chromosome Vk , where q0 = 0, qk =
k
Eval(Vj ), k = 1, 2, . . . , pop size.
j=1
Step 2. Generate a random real number r in (0, qpop size ]. Step 3. Select the kth chromosome Vk (1 ≤ k ≤ pop size) such that qk−1 < r ≤ qk . Step 4. Repeat Steps 2 and 3 pop size times and obtain pop size copies of chromosomes. As a result, we obtain pop size chromosomes as a new population and still denote them by V1 , V2 , . . . , Vpop size . Crossover operation In order to renew the chromosomes V1 , V2 , . . . , Vpop size by crossover operation, a parameter Pc is given in advance, called the probability of crossover. Then the following procedures of crossover operation are run. Step 1. Generating a random real number r from the interval [0, 1], the chromosome Vk is selected as a parent if r < Pc . Repeat the above process for k = 1, 2, . . . , pop size and denote the selected parents by V1 , V2 , V3 , . . . Step 2. Divide V1 , V2 , V3 , . . . into the following pairs: (V1 , V2 ), (V3 , V4 ), (V5 , V6 ), . . .. Then do the crossover operating on each pair according to
Step 1. Repeat the following process for k = 1, 2, . . . , pop size: generating a random real number r from the interval [0, 1], the chromosome Vk is selected as a parent for mutation operation if r < Pm . Step 2. For each selected parent V = (x1 , x2 , . . . , x10 ), we randomly generate a number w from the interval [0, W], where W is an appropriate large positive number. Here we set W = 100. Then we can obtain a new chromosome V = V + w · d, where di is randomly generated from [−1, 1], i = 1, 2, . . . , 10, respectively. Step 3. Check the feasibility of V by fuzzy simulation in Section 5.1. If V is not feasible, we randomly generate a number w from [0, w] and redo Step 2 until a new feasible chromosome is obtained, which will replace the selected parent chromosome. We still denote pop size new chromosomes by V1 , V2 , . . . , Vpop size . Following selection, crossover, and mutation, the new chromosomes V1 , V2 , . . . , Vpop size are ready for their next evaluation. Adopting different environment parameters of GA and letting GA terminate after 10000 cyclic repetitions of the above steps, we obtain the computational results shown in Table 2, where “Modification” is the minimum modification of edge weights. It appears that all the optimal values in Table 2 differ little from each other. In order to account it, we present a parameter, called the percent error, i.e., (actual value − optimal value)/optimal value × 100%, where “optimal
X = λ · V1 + (1−λ) · V2 , Y = (1−λ) · V1 +λ · V2 where λ is a random number generated from the interval (0, 1). Step 3. Check their children’s feasibility by fuzzy simulation in Section 5.1. If the above two children are feasible, then we replace the two parents with them. If one of the two children is feasible, we keep it and redo the crossover operation to get another feasible one. Else, we redo the crossover operation until we get two feasible children chromosomes.
Table 2 Solutions comparison of Example 5 Pop size Pc Pm 1 2 3 4 5 6 7 8 9 10
20 20 30 20 30 30 30 20 30 20
0.3 0.1 0.3 0.2 0.1 0.2 0.1 0.1 0.3 0.3
0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.1 0.1
a 0.05 0.05 0.05 0.10 0.05 0.10 0.10 0.10 0.08 0.08
Generation Modification Error(%) 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000
367.82 367.65 367.60 367.58 367.54 367.45 367.43 367.41 367.22 366.92
0.3 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0
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value” is the least one of all the ten minimal modifications obtained above. The last column named by “Error” in Table 2 is just this parameter. It follows from Table 2 that the percent error does not exceed 0.3% when different parameters are adopted, which implies that the hybrid intelligent algorithm is robust to the parameter settings and effective in solving the fuzzy α-minimum spanning tree model (13). We choose the least minimal modification from Table 2 as the optimal objective value of model (13), i.e., 366.92, whose corresponding optimal solution is
Table 3 Solution comparison of Example 6 Pop size Pc 1 2 3 4 5 6 7 8 9 10
30 20 20 20 30 30 20 30 20 30
0.2 0.3 0.1 0.2 0.3 0.1 0.3 0.1 0.1 0.3
Pm
a
0.2 0.2 0.3 0.2 0.2 0.3 0.1 0.2 0.2 0.1
0.10 0.05 0.10 0.10 0.05 0.10 0.08 0.05 0.05 0.08
Generation Credibility Error(%) 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000
0.9161 0.9157 0.9157 0.9154 0.9151 0.9150 0.9139 0.9135 0.9132 0.9131
0.3 0.3 0.3 0.3 0.2 0.2 0.1 0.04 0.01 0
x∗ = (83.98, 80.00, 80.00, 98.86, 120.98, 102.16, 110.86, 100.91, 170.00, 136.81). Example 6. If we want to maximize the credibility that the total modification of deterministic weights does not exceed 390, we have the following credibility maximization model, ⎧ 10 ⎪ ⎪ ⎪ ⎪ max Cr |xi − ci | ≤ 390 ⎪ ⎪ x ⎪ ⎪ i=1 ⎨ subject to: ⎪ ⎪ ⎪ ⎪ ξi (xi ) − ξj (xj ) ≤ 0, j ∈ E\T 0 , i ∈ Pj ⎪ ⎪ ⎪ ⎪ ⎩ 80 ≤ xi ≤ 180, i ∈ E (14) where the non-tree edge set E\T 0 = {6, 7, 8, 9, 10}, and Pj is the tree path of edge j ∈ E\T 0 . By using the principle of uncertainty, we may transfer model (14) to the following model, ⎧ 0 ⎪ max Cr ξ (x ) − ξ (x ) ≤ 0, j ∈ E\T , i ∈ P i i j j j ⎪ x ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ subject to: (15) 10 ⎪ ⎪ |x − c | ≤ 390 ⎪ i i ⎪ ⎪ i=1 ⎪ ⎪ ⎩ 80 ≤ xi ≤ 180, i ∈ E. A similar procedure of hybrid intelligent algorithm has been run for solving model (15) with different parameters in genetic system, and the result is showed in Table 3, where “Credibility” is the maximal credibility calculated. In order to measure the differentia of each other, the percent error is proposed and showed in Table 3 as “Error”. It follows from Table 3 that the percent error does not exceed 0.3% when different parameters are selected, which implies that the hybrid intelligent algorithm is robust to the parameter settings and effective in solving the credibility maximization model (15).
We choose the maximal credibility in Table 3 as the optimal objective value, i.e., 0.9131, whose corresponding optimal solution is x∗ = (95.19, 80.23, 80.01, 126.37, 137.52, 120.14, 141.37, 105.06, 170.00, 162.26).
7. Conclusion Inverse optimization problem theory is a subject extensively studied in the context of tomographic studies, seismic wave propagation, and in a wide range of statistical inference with prior problems. As a special type of inverse optimization problem, the inverse minimum spanning tree problem has been extensively discussed in the literature. An inverse minimum spanning tree problem is to make the least modification on the edge weights such that a predetermined spanning tree is a minimum spanning tree with respect to the new edge weights. Many reconstruction problems can be transformed to inverse minimum spanning tree problems. However, the applications of the inverse spanning tree problem encountered in practice usually involve some uncertain issues so that the edge weights cannot be explicitly determined. Thus it is necessary to investigate the inverse minimum spanning tree problem in the fuzzy environment. In this paper, a special type of fuzzy inverse minimum spanning tree problem is considered, i.e., inverse minimum spanning tree problem with fuzzy weights with respect to the deterministic new edge weights. In order to formulate this problem, a new concept of fuzzy α-minimum spanning tree is provided for FIMST problem. Subsequently, it is shown that the notion of the fuzzy α-minimum spanning tree can be characterized by a constraint on non-tree edges and their corresponding tree paths. By this characterization, referred to
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as the fuzzy path optimality condition. Based on the concept and fuzzy path optimality condition, a fuzzy α-minimum spanning tree model is proposed for the FIMST problem. Furthermore, a credibility maximization model is also presented to formulate the FIMST problem according to a different decision criterion. In the literature many efficient algorithms have been developed for solving the classic inverse minimum spanning tree problem. However, these algorithms become powerless to solve the two fuzzy models proposed above. In order to find the optimal new edge weights satisfying constraints, a fuzzy simulation for computing credibility is designed and then embedded into a genetic algorithm to produce two hybrid intelligent algorithms. These hybrid intelligent algorithms have been illustrated by two examples, whose results show that the algorithms are efficient for solving the proposed fuzzy models. Acknowledgements This work was supported in part by grants from the Ministry of Education Funded Project for Humanities and Social Sciences Research (No. 12JDXF005), the Innovation Program of Shanghai Municipal Education Commission (No. 13ZS065), and the Shanghai Philosophy and Social Science Planning Project (No. 2012BGL006). References [1] R.K. Ahuja, T.L. Magnanti and J.B. Orlin, Network Flows: Theory, Algorithms, and Applications, New Jersey, USA: Prentice Hall, Inc., 1993. [2] R.K. Ahuja and J.B. Orlin, A faster algorithm for the inverse spanning tree problem, Journal of Algorithms 34(1) (2000), 177–193. [3] K. Eshghi and J. Nematian, Special classes of mathematical programming models with fuzzy random variables, Journal of Intelligent and Fuzzy Systems 19(2) (2008), 131–140. ´ Szentesi and B. Szviatovszki, Inverse optimiza[4] A. Farag´o, A. tion in high-speed networks, Discrete Applied Mathematics 129(1) (2003), 83–98. [5] J. Gao and M. Lu, Fuzzy quadratic minimum spanning tree problem, Applied Mathematics and Computation 164(3) (2005), 773–788. [6] M. Gen and R. Cheng, Genetic Algorithms and Engineering Optimization, New York, USA: John Wiley & Sons, 2000. [7] X. Guan, P.M. Pardalos and X. Zuo, Inverse Max + Sum spanning tree problem by modifying the sum-cost vector under weighted l∞ Norm, Journal of Global Optimization (2014). DOI: 10.1007/s10898-014-0140-z [8] X. Guan and B. Zhang, Inverse 1-median problem on trees under weighted Hamming distance, Journal of Global Optimization 54(1) (2012), 75–82.
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[9] X. Guan and J. Zhang, Inverse constrained bottleneck problems under weighted l∞ norm, Computers and Operations Research 34(11) (2007), 3243–3254. [10] Y. He, B.W. Zhang and E.Y. Yao, Weighted inverse minimum spanning tree problems under Hamming distance, Journal of Combinatorial Optimization 9(1) (2005), 91–100. [11] D.S. Hochbaum, Efficient algorithms for the inverse spanningtree problem, Operations Research 51(5) (2003), 785–797. [12] H. Katagiri, K. Kato and T. Hasuike, A random fuzzy minimum spanning tree problem through a possibility-based value at risk model, Expert Systems with Applications 39(12) (2012), 10639–10646. [13] A. Kershenbaum, Telecommunication Network Design Algorithms, New York, USA: McGraw-Hill, 1993. [14] J.P. Linda and M.S. Sunitha, Fuzzy detour g-interior nodes and fuzzy detour g-boundary nodes of a fuzzy graph, Journal of Intelligent and Fuzzy Systems (2013). DOI: 10.3233/IFS131011 [15] B. Liu, Dependent-chance programming with fuzzy decisions, IEEE Transactions on Fuzzy Systems 7(3) (1999), 354–360. [16] B. Liu, Uncertainty Theory: An Introduction to its Axiomatic Foundations, Berlin, Germany: Springer-Verlag, 2004. [17] B. Liu and K. Iwamura, Chance constrained programming with fuzzy parameters, Fuzzy Sets and Systems 94(2) (1998), 227–237. [18] L. Liu and E. Yao, Weighted inverse maximum perfect matching problems under the Hamming distance, Journal of Global Optimization 55(3) (2013), 549–557. [19] A. Sever, A neural network algorithm to pattern recognition in inverse problems, Applied Mathematics and Computation 221 (2013), 484–490. [20] P.T. Sokkalingam, R.K. Ahuja and J.B. Orlin, Solving inverse spanning tree problems through network flow techniques, Operations Research 47(2) (1999), 291–298. [21] J.A. Torkestani and M.R. Meybodi, A learning automatabased heuristic algorithm for solving the minimum spanning tree problem in stochastic graphs, The Journal of Supercomputing 59(2) (2012), 1035–1054. [22] Q. Wang, Partial inverse most unbalanced spanning tree problem, Przeglad Elektrotechniczny 88(1B) (2012), 111–114. [23] Q. Wang, X. Yang and J. Zhang, A class of inverse dominant problems under weighted l∞ norm and an improved complexity bound for radzik’s algorithm, Journal of Global Optimization 34(4) (2006), 551–567. [24] X. Yang and J. Zhang, Some inverse min-max network problems under weighted l1 and l∞ norms with bound constraints on changes, Journal of Combinatorial Optimization 13(2) (2007), 123–135. [25] L.A. Zadeh, Fuzzy sets, Information and Control 8(3) (1996), 338–353. [26] J. Zhang, Z. Liu and Z. Ma, On the inverse problem of minimum spanning tree with partition constraints, Mathematical Methods and Operations Research 44(2) (1996), 171–187. [27] J. Zhang and J. Zhou, Models and hybrid algorithms for inverse minimum spanning tree problem with stochastic edge weights, World Journal of Modelling and Simulation 2(5) (2006), 297–311. [28] X. Zhang, Q. Wang and J. Zhou, A chance-constrained programming model for inverse spanning tree problem with uncertain edge weights, International Journal of Advancements in Computing Technology 5(6) (2013), 76–83. [29] X. Zhang, Q. Wang and J. Zhou, Two uncertain programming models for inverse minimum spanning tree problem,
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Industrial Engineering and Management Systems 12(1) (2013), 9–15. [30] J. Zhou, L. Chen and K. Wang, Path optimality conditions for minimum spanning tree problem with uncertain edge weights, International Journal of Uncertainty, Fuzziness & Knowledge-Based Systems (2014), in press. [31] J. Zhou, L. Chen and K. Wang, Fuzzy α-minimum spanning tree problem: Definition and solutions, Technical Report (2014).
[32] J. Zhou, X. He and K. Wang, Uncertain quadratic minimum spanning tree problem, Journal of Communications 9(5) (2014), 385–390. [33] J. Zhou, Q. Wang and X. Zhang, The inverse spanning tree of a fuzzy graph based on credibility measure, Journal of Communications 8(9) (2013), 566–571. [34] J. Zhou, F. Yang and K. Wang, An inverse shortest path problem on an uncertain graph, Journal of Networks 9(9) (2014), 2353–2359.
Journal of Intelligent & Fuzzy Systems 27 (2014) 2691–2702 DOI:10.3233/IFS-141384 IOS Press
Models for inverse minimum spanning tree problem with fuzzy edge weights Jingyu Zhanga , Jian Zhoub,∗ and Shuya Zhongb a Department b School
of Clinical Decision Support Solutions, Philips Research North America, NY, USA of Management, Shanghai University, Shanghai, China
Abstract. An inverse minimum spanning tree problem is to make the least modification on the edge weights such that a predetermined spanning tree is a minimum spanning tree with respect to the new edge weights. In this paper, a type of fuzzy inverse minimum spanning tree problem is introduced from a LAN reconstruction problem, where the weights of edges are assumed to be fuzzy variables. The concept of fuzzy α-minimum spanning tree is initialized, and subsequently a fuzzy α-minimum spanning tree model and a credibility maximization model are presented to formulate the problem according to different decision criteria. In order to solve the two fuzzy models, a fuzzy simulation for computing credibility is designed and then embedded into a genetic algorithm to produce some hybrid intelligent algorithms. Finally, some computational examples are given to illustrate the effectiveness of the proposed algorithms. Keywords: Minimum spanning tree, inverse optimization, fuzzy programming, genetic algorithm
1. Introduction The inverse minimum spanning tree (IMST) or inverse spanning tree problem is a type of inverse optimization problems. In an IMST problem, there is a connected graph G = (V, E) with n vertices, m edges, and an edge weight vector c. The objective of IMST problem is to modify the weight vector c so that a predetermined spanning tree T 0 is a minimum spanning tree of graph G with respect to the new weight vector x, and simultaneously the total modification of weights is a minimum. The IMST problem was first studied by Zhang et al. [26] with partition constraints. Since then, much work has been done on the IMST related problems because many practical problems can be handled in this framework. Guan and Zhang [9] considered a class ∗ Corresponding
author. Jian Zhou, School of Management, Shanghai University, Shanghai 200444, China. E-mail: zhou jian@ shu.edu.cn.
of inverse constrained bottleneck problems under the weighted l∞ norm. In these inverse problems, two weight vectors under bound restrictions were modified so that a given candidate solution became an optimal one and the deviation of the weights, measured by the weighted l∞ norm, was minimum. Wang et al. [23] also discussed a class of inverse dominant problems under the weighted l∞ norm in the area of combinatorial optimization. Similarly, Yang and Zhang [24] considered some inverse min-max or max-min network problems not only under the weighted l∞ norm but also the weighted l1 norm, while Guan et al. [7] initially studied an inverse problem of optimization problems with combined minmax-minsum objective functions. Besides, Wang [22] proposed two models for the partial inverse most unbalanced spanning tree problem under the weighted Hamming distance and the weighted l1 norm. Liu and Yao [18] also brought the Hamming distance into the weighted inverse maximum perfect matching problems. In the practical research area of inverse optimization, Farag´o et al. [4] applied the
1064-1246/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved
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inverse shortest path problem to high-speed telecommunication networks and especially considered the case when no weighting exists that could make each given solution exactly optimal. Sever [19] adopted a learning algorithm based on a numerical solution of a system of linear equations which was derived from a wellknown regularization model, and applied to the task of reconstruction of an inhomogeneous object as pattern recognition for the inverse problems. For solving the classical IMST problem and some of its derivatives, considerable effort has been devoted to designing efficient algorithms with lower computational complexity as far as possible. Sokkalingam et al. [20] developed a specific implementation of the successive shortest path algorithm that run in O(n3 ) time for the IMST problem. Subsequently Ahuaj and Orlin [2] improved the previous O(n3 ) time algorithm to a more efficient O(n2 log n) time algorithm, and Hochbaum [11] continued the improvement in the complexity of algorithm to make the run time O(nm log2 n). He et al. [10] presented some strongly polynomial algorithms for the weighted IMST problems under the Hamming distance in accordance with the method for minimum-weight node cover problem on bipartite graph. Recently, Guan and Zhang [8] proposed a linear O(n) time algorithm for the inverse 1-median problem on trees under the weighted l∞ norm, as well as two polynomial time algorithms with time complexities O(n log n) and O(n) for the inverse problem under the weighted bottleneck-Hamming distance. Since in some practical circumstances, the edge weights cannot be explicitly determined, some endeavor was done to deal with the IMST related problems with indeterminate information by considering the edge weights as random variables (see, e.g., [21, 27]). However, the approximation by random variables does not work when there are not enough data for estimation or when the edge weights possess an unobservable nature. Hence some different kinds of variables have been also suggested for describing the indeterminate edge weights. For example, fuzzy variables were brought into the quadratic minimum spanning tree problem in [5] and the maximum spanning tree problem in [14]. Eshghi and Nematian [3] considered the quadratic minimum spanning tree problem with fuzzy random edge weights. Katagiri et al. [12] characterized the edge weights of a graph as random fuzzy variables in a minimum spanning tree problem. Recently, Zhang et al. [28, 29] discussed the inverse minimum spanning tree problem by assuming all the edge weights as uncertain variables. See Zhou et al. [30, 32, 34] for more details.
In this paper, an IMST problem with fuzzy edge weights is considered from the practical application angle. Specifically, in a LAN reconstruction problem [6], the parameters of bandwidth and net speed are both attached to edges. Obviously, the net speeds of LAN are not fixed variables with respect to the bandwidths due to the changing environments. In this situation, it is natural to represent the net speeds as fuzzy numbers by some expert knowledge. This is actually an IMST problem with fuzzy weights, and the classic IMST algorithms become powerless to such a problem. In order to solve this type of IMST problem, Zhou et al. [31] proposed a conception of fuzzy α-minimum spanning tree, which is essentially a transformation of the classical minimum spanning tree. Based on this notion, Zhou et al. [33] developed a fuzzy programming model with some chance constraints to formulate an IMST problem with fuzzy weights. In view of the deficiency of the definition in [31], in this paper, we define a new concept of fuzzy α-minimum spanning tree from a novel angle, and then provide a fuzzy α-minimum spanning tree model and a credibility maximization model for formulating this problem in practical situations according to different decision criteria. In this paper, the IMST problem with fuzzy weights is referred to as the fuzzy inverse minimum spanning tree (FIMST) problem. The rest of this paper is organized as follows. In Section 2, we briefly review the classic IMST problem, and the concepts of fuzzy variable and credibility of fuzzy event. Section 3 introduces the application backgrounds and mathematical description of the FIMST problem, and defines a fuzzy α-minimum spanning tree. Following that, two fuzzy programming models are presented for the FIMST problem in Section 4. In order to solve the proposed fuzzy models, a fuzzy simulation for computing credibility and a genetic algorithm are combined to produce some hybrid intelligent algorithms in Section 5. Section 6 illustrates the efficiency of the proposed algorithms by some numerical experiments.
2. Preliminaries In order to describe the FIMST problem more clearly, a classic concept of minimum spanning tree, a path optimality condition, and the mathematical formulation of the IMST problem are reviewed in this section. We also introduce the concept of fuzzy variable and credibility of fuzzy event for our purpose.
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2.1. Classic inverse minimum spanning tree problem Let G = (V, E) denote a connected graph consisting of the vertex set V = {v1 , v2 , . . . , vn } and the edge set E = {1, 2, . . . , m}. A spanning tree T = T (V, S) of G is a connected acyclic subgraph containing all vertices. For simplicity, we denote a spanning tree T by its edge set S throughout this paper. Definition 1. (Minimum Spanning Tree) Given a connected graph G = (V, E) with edge weights ci , i ∈ E = {1, 2, . . . , m}, a spanning tree T 0 is said to be a minimum spanning tree if ci ≤ (1) cj i∈T 0
j∈T
holds for any spanning tree T . Example 1. A graph G with 4 vertices and 6 edges is shown in Figure 1. When (c1 , c2 , c3 , c4 , c5 , c6 ) = (1, 3, 4, 2, 2, 3), the unique minimum spanning tree is T 0 = {AB, AD, AC} with i∈T 0 ci = c1 + c4 + c5 = 5. When (c1 , c2 , c3 , c4 , c5 , c6 ) = (3, 1, 2, 3, 4, 1), there are two minimum spanning trees: 0 T10 = {AB, BC, BD}, and T2 = {BC, AD, BD}, with c = c = 5. i∈T10 i i∈T20 i The classic IMST problem is to find some new edge weights such that a given spanning tree T 0 is a minimum spanning tree with respect to the new edge weights and the total change of the edge weights is minimized. Example 2. A graph G with 6 vertices and 10 edges is shown in Figure 2, where ci denotes the weight on edge i, and the solid line represents a predetermined spanning tree T 0 . The goal of the classic IMST problem is to find
Fig. 1. Graph G for Example 1.
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anew weight vector x = (x1 , x2 , . . . , x10 ) to minimize 10 i=1 |xi − ci | provided that the predetermined spanning tree T 0 is a minimum spanning tree with respect to x. So, how to judge whether the predetermined spanning tree is a minimum spanning tree with respect to a weight vector? Before providing the mathematical description of the IMST problem, some concepts are introduced as follows. First, we refer to the edges in the predetermined spanning tree T 0 as tree edges, and the edges not in T 0 as non-tree edges. For convenience, denote the set of non-tree edges as E\T 0 . A path in the graph G is called a tree path if the path only contains tree edges. It is clear that there must be a unique tree path between any two vertices of graph G. It follows directly that there is a unique tree path between the two vertices of any non-tree edge j, called tree path of edge j and denoted by Pj . Example 3. In Figure 2, the set of non-tree edges is E\T 0 = {6, 7, 8, 9, 10}, and the tree path of non-tree edge BD is AB-AE-DE, i.e., P9 = {1, 3, 5}. Moreover, Ahuja et al. [1] proved an equivalent condition of minimum spanning tree, referred to as the path optimality condition, which characterizes the minimum spanning tree by a set of constraints on the non-tree edges and their tree paths. Lemma 1. (Ahuja et al. [1], Path Optimality Condition) Given a connected graph G = (V, E) with edge weights xi , i ∈ E = {1, 2, . . . , m}, a spanning tree T 0 is a minimum spanning tree if and only if xi − xj ≤ 0,
j ∈ E\T 0 , i ∈ Pj
(2)
where E\T 0 is the set of non-tree edges, and Pj is the tree path of edge j. According to Lemma 1, the classic IMST problem can be formulated as
Fig. 2. Graph G for Example 2.
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⎧ m ⎪ ⎪ min |xi − ci | ⎪ ⎪ ⎨ x
3.1. Application backgrounds
i=1
⎪ subject to: ⎪ ⎪ ⎪ ⎩ xi − xj ≤ 0,
(3) j ∈ E\T 0 , i ∈ Pj
where ci and xi are the original and new weights of edge i, i ∈ E, respectively. Some efficient algorithms are available to solve this model (see, e.g., [2, 11, 20]). Note that in [20] other types of objective functions are considered as well for the IMST problems. 2.2. Fuzzy variables Since its introduction in 1965 by Zadeh [25], fuzzy set theory has been well developed and applied in a wide variety of real problems. In the following, we briefly review the concepts of possibility, necessity, and credibility of fuzzy event. Let be a nonempty set, P() the power set of , and Pos a possibility measure. Then the triplet (, P(), Pos) is called a possibility space. A fuzzy variable is defined as a function from a possibility space (, P(), Pos) to the set of real numbers. Suppose that ξ is a fuzzy variable with membership function µ. Then the possibility, necessity, and credibility of a fuzzy event {ξ ≥ r} can be represented by Pos{ξ ≥ r} = sup µ(u),
Many reconstruction problems in practice can be transformed to FIMST problems. In order to see that, let us consider a LAN reconstruction problem as follows. Much research work shows that the spanning tree structure is the best topology for telecommunication network designs [13], especially in computer network systems. LANs are commonly used as a communication infrastructure that meets the demands of users in a local environment. These computer networks typically consist of several LAN segments connected via bridges. Suppose that there is an old LAN, in which several service centers are interconnected via bridges. For some reasons (e.g., the tremendous network congestion), the bandwidths on bridges must be modified. The decision-maker hopes that a predetermined spanning tree becomes a minimum spanning tree with respect to the net speed on each bridge in order to ensure high speed between some main service centers. Also the total bandwidth modification should be minimized so as to diminish the cost of reconstruction. Apparently the net speeds are related to bandwidths. It is natural to describe the net speed on a bridge as a fuzzy number instead of a deterministic one with respect to bandwidths of bridges when there are no former statistical data in some cases. This is a typical inverse minimum spanning tree problem with fuzzy weights, i.e., an FIMST problem.
u≥r
Nec{ξ ≥ r} = 1 − sup µ(u),
3.2. Mathematical description
1 Cr{ξ ≥ r} = ( Pos{ξ ≥ r} + Nec{ξ ≥ r} ) . 2
In order to provide a mathematical description of the FIMST problem, the following notations are used:
u 0 for some j ∈ E\T 0 and i ∈ Pj }). 5.2. Hybrid intelligent algorithm procedure Genetic algorithms (GA) have demonstrated considerable success in providing good solutions to many complex optimization problems. Due to the fuzziness of weight vector ξ(x), a fuzzy simulation is designed for calculating the credibility U, and then embedded into a GA to produce some hybrid intelligent algorithms for solving models (7) and (11) efficiently. As an illustration, the following steps show how the hybrid intelligent algorithm for model (7) works. Step 1. Initialize pop size feasible chromosomes k ) for k = 1, 2, . . . , pop size Vk = (x1k , x2k , . . . , xm from the potential region [l1 , u1 ] × [l2 , u2 ] × . . . × [lm , um ] uniformly, whose feasibility is checked by Cr{ξi (Vk ) − ξj (Vk ) ≤ 0, j ∈ E\T 0 , i ∈ Pj } ≥ α, k = 1, 2, . . . , pop size, respectively, where the credibilities are calculated by the fuzzy simulation designed above. Step 2. Calculate the objective values O(Vk ) = m k i=1 |xi −ci | for all chromosomes Vk , k = 1, 2, . . . , pop size, respectively. Step 3. Compute the fitness of all chromosomes Vk , k = 1, 2, . . . , pop size assuming the chromosomes have been rearranged from good to bad according to their objective values O(Vk ). Step 4. Select the chromosomes for a new population by spinning the roulette wheel according to the fitness of all chromosomes. Step 5. Update the chromosomes by crossover operation, where the feasibility of new chromosomes is checked by the fuzzy simulation designed above. Step 6. Update the chromosomes by mutation operation, where the feasibility of new chromosomes is checked by the fuzzy simulation. Step 7. Repeat the second to fifth steps for N times. Step 8. Report the best chromosome V ∗ as the optimal solution. A similar hybrid intelligent algorithm has also been designed for solving model (11) effectively. More detailed procedure will follow a specific example and be illustrated clearly in Section 6.
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6. Computational examples As to the two hybrid intelligent algorithms above, since they do not belong to the traditional classical algorithms, we generally do not concern with their efficiency by considering the computational complexity but care about their effectivity. In order to illustrate the effectiveness of the hybrid intelligent algorithms, we propose some numerical examples performed on a personal computer. Consider a LAN reconstruction problem with 6 service centers and 10 bridges showed in Figure 4, where the vertex set is V = {A, B, C, D, E, F }, the edge set is E = {1, 2, . . . , 10}, and the solid lines represent the predetermined spanning tree, denoted by T 0 . There are three weights on each bridge, where ci and xi represent the original and new bandwidths on bridge i, respectively, and ξi denotes the fuzzy net speed. The values of ci and ξi are given in Table 1. Here the fuzzy net speed ξi is assumed be only with respect to the bandwidth xi on bridge i, i.e., ξi = ξi (xi ) for the sake of simpleness. Example 5. If we want to minimize the total modification of bandwidths with a given confidence level α = 0.8 so as to diminish the cost of reconstruction, we
have the following fuzzy 0.8-minimum spanning tree model, ⎧ 10 ⎪ ⎪ ⎪ min ⎪ |xi − ci | ⎪ ⎪ ⎪ x i=1 ⎪ ⎨ subject to: ⎪ ⎪
⎪ ⎪ Cr ξi (xi ) − ξj (xj ) ≤ 0, j ∈ E\T 0 , i ∈ Pj ≥ 0.8 ⎪ ⎪ ⎪ ⎪ ⎩ 80 ≤ xi ≤ 180, i ∈ E (13) where the non-tree edge set E\T 0 = {6, 7, 8, 9, 10}, and Pj is the tree path of edge j ∈ E\T 0 . In order to solve model (13), the hybrid intelligent algorithm designed in Section 5 has been run with 10000 cycles in fuzzy simulation and 10000 generations in GA. The detailed heuristic procedures are described as follows. Representation structure We use a vector V = (x1 , x2 , . . . , x10 ) as a chromosome to represent a solution x. Initialization process We define an integer pop size as the number of chromosomes, and initialize pop size feasible chromosomes V1 , V2 , . . . , Vpop size uniformly from [80, 180]10 . The feasibility of chromosome Vk = k ) is checked by (x1k , x2k , . . . , x10 Cr{ξi (Vk ) − ξj (Vk ) ≤ 0, j ∈ E\T 0 , i ∈ Pj } ≥ α, k = 1, 2, . . ., pop size, respectively, where the credibilities are calculated by the fuzzy simulation designed in Section 5.1. Computing objective values In order to compute the objective values of all the chromosomes, we calculate
Fig. 4. The FIMST problem for computational examples.
O(Vk ) = Table 1 Weights for the FIMST problem in Figure 4 Edge i 1 2 3 4 5 6 7 8 9 10
Original weight ci
Fuzzy weight ξi (xi )
120 40 30 150 150 60 60 40 170 130
(x1 − 4, x1 , x1 + 4) (x2 − 8, x2 , x2 + 8) (x3 − 12, x3 , x3 + 12) (x4 − 16, x4 , x4 + 16) (x5 − 20, x5 , x5 + 20) (x6 − 24, x6 , x6 + 24) (x7 − 28, x7 , x7 + 28) (x8 − 32, x8 , x8 + 32) (x9 − 36, x9 , x9 + 36) (x10 − 40, x10 , x10 + 40)
10
|xik − ci |
i=1 k ), k = 1, 2, . . . , pop size, for Vk = (x1k , x2k , . . . , x10 respectively.
Evaluation function Rearrange the chromosomes V1 , V2 , . . . , Vpop size from good to bad according to their objective values O(Vk ) with assumption that a chromosome with smaller objective value is better. Then compute the fitness Eval(Vk ) for all the chromosomes Vk , k = 1, 2, . . . , pop size, where Eval(Vk ) is defined by Eval(Vk ) = a(1 − a)k−1 , k = 1, 2, . . . , pop size.
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In this formula, a ∈ (0, 1) is a parameter in the genetic system.
For convenience, we still denote the new chromosomes by V1 , V2 , . . . , Vpop size .
Selection process The selection process is based on spinning the roulette wheel pop size times which can be stated as follows,
Mutation operation We give a parameter Pm as the probability of mutation in advance. Similar to the process of crossover operation, the mutation operation will be performed in the following way.
Step 1. Calculate the cumulative probability qk for each chromosome Vk , where q0 = 0, qk =
k
Eval(Vj ), k = 1, 2, . . . , pop size.
j=1
Step 2. Generate a random real number r in (0, qpop size ]. Step 3. Select the kth chromosome Vk (1 ≤ k ≤ pop size) such that qk−1 < r ≤ qk . Step 4. Repeat Steps 2 and 3 pop size times and obtain pop size copies of chromosomes. As a result, we obtain pop size chromosomes as a new population and still denote them by V1 , V2 , . . . , Vpop size . Crossover operation In order to renew the chromosomes V1 , V2 , . . . , Vpop size by crossover operation, a parameter Pc is given in advance, called the probability of crossover. Then the following procedures of crossover operation are run. Step 1. Generating a random real number r from the interval [0, 1], the chromosome Vk is selected as a parent if r < Pc . Repeat the above process for k = 1, 2, . . . , pop size and denote the selected parents by V1 , V2 , V3 , . . . Step 2. Divide V1 , V2 , V3 , . . . into the following pairs: (V1 , V2 ), (V3 , V4 ), (V5 , V6 ), . . .. Then do the crossover operating on each pair according to
Step 1. Repeat the following process for k = 1, 2, . . . , pop size: generating a random real number r from the interval [0, 1], the chromosome Vk is selected as a parent for mutation operation if r < Pm . Step 2. For each selected parent V = (x1 , x2 , . . . , x10 ), we randomly generate a number w from the interval [0, W], where W is an appropriate large positive number. Here we set W = 100. Then we can obtain a new chromosome V = V + w · d, where di is randomly generated from [−1, 1], i = 1, 2, . . . , 10, respectively. Step 3. Check the feasibility of V by fuzzy simulation in Section 5.1. If V is not feasible, we randomly generate a number w from [0, w] and redo Step 2 until a new feasible chromosome is obtained, which will replace the selected parent chromosome. We still denote pop size new chromosomes by V1 , V2 , . . . , Vpop size . Following selection, crossover, and mutation, the new chromosomes V1 , V2 , . . . , Vpop size are ready for their next evaluation. Adopting different environment parameters of GA and letting GA terminate after 10000 cyclic repetitions of the above steps, we obtain the computational results shown in Table 2, where “Modification” is the minimum modification of edge weights. It appears that all the optimal values in Table 2 differ little from each other. In order to account it, we present a parameter, called the percent error, i.e., (actual value − optimal value)/optimal value × 100%, where “optimal
X = λ · V1 + (1−λ) · V2 , Y = (1−λ) · V1 +λ · V2 where λ is a random number generated from the interval (0, 1). Step 3. Check their children’s feasibility by fuzzy simulation in Section 5.1. If the above two children are feasible, then we replace the two parents with them. If one of the two children is feasible, we keep it and redo the crossover operation to get another feasible one. Else, we redo the crossover operation until we get two feasible children chromosomes.
Table 2 Solutions comparison of Example 5 Pop size Pc Pm 1 2 3 4 5 6 7 8 9 10
20 20 30 20 30 30 30 20 30 20
0.3 0.1 0.3 0.2 0.1 0.2 0.1 0.1 0.3 0.3
0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.3 0.1 0.1
a 0.05 0.05 0.05 0.10 0.05 0.10 0.10 0.10 0.08 0.08
Generation Modification Error(%) 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000
367.82 367.65 367.60 367.58 367.54 367.45 367.43 367.41 367.22 366.92
0.3 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0
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value” is the least one of all the ten minimal modifications obtained above. The last column named by “Error” in Table 2 is just this parameter. It follows from Table 2 that the percent error does not exceed 0.3% when different parameters are adopted, which implies that the hybrid intelligent algorithm is robust to the parameter settings and effective in solving the fuzzy α-minimum spanning tree model (13). We choose the least minimal modification from Table 2 as the optimal objective value of model (13), i.e., 366.92, whose corresponding optimal solution is
Table 3 Solution comparison of Example 6 Pop size Pc 1 2 3 4 5 6 7 8 9 10
30 20 20 20 30 30 20 30 20 30
0.2 0.3 0.1 0.2 0.3 0.1 0.3 0.1 0.1 0.3
Pm
a
0.2 0.2 0.3 0.2 0.2 0.3 0.1 0.2 0.2 0.1
0.10 0.05 0.10 0.10 0.05 0.10 0.08 0.05 0.05 0.08
Generation Credibility Error(%) 10000 10000 10000 10000 10000 10000 10000 10000 10000 10000
0.9161 0.9157 0.9157 0.9154 0.9151 0.9150 0.9139 0.9135 0.9132 0.9131
0.3 0.3 0.3 0.3 0.2 0.2 0.1 0.04 0.01 0
x∗ = (83.98, 80.00, 80.00, 98.86, 120.98, 102.16, 110.86, 100.91, 170.00, 136.81). Example 6. If we want to maximize the credibility that the total modification of deterministic weights does not exceed 390, we have the following credibility maximization model, ⎧ 10 ⎪ ⎪ ⎪ ⎪ max Cr |xi − ci | ≤ 390 ⎪ ⎪ x ⎪ ⎪ i=1 ⎨ subject to: ⎪ ⎪ ⎪ ⎪ ξi (xi ) − ξj (xj ) ≤ 0, j ∈ E\T 0 , i ∈ Pj ⎪ ⎪ ⎪ ⎪ ⎩ 80 ≤ xi ≤ 180, i ∈ E (14) where the non-tree edge set E\T 0 = {6, 7, 8, 9, 10}, and Pj is the tree path of edge j ∈ E\T 0 . By using the principle of uncertainty, we may transfer model (14) to the following model, ⎧ 0 ⎪ max Cr ξ (x ) − ξ (x ) ≤ 0, j ∈ E\T , i ∈ P i i j j j ⎪ x ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ subject to: (15) 10 ⎪ ⎪ |x − c | ≤ 390 ⎪ i i ⎪ ⎪ i=1 ⎪ ⎪ ⎩ 80 ≤ xi ≤ 180, i ∈ E. A similar procedure of hybrid intelligent algorithm has been run for solving model (15) with different parameters in genetic system, and the result is showed in Table 3, where “Credibility” is the maximal credibility calculated. In order to measure the differentia of each other, the percent error is proposed and showed in Table 3 as “Error”. It follows from Table 3 that the percent error does not exceed 0.3% when different parameters are selected, which implies that the hybrid intelligent algorithm is robust to the parameter settings and effective in solving the credibility maximization model (15).
We choose the maximal credibility in Table 3 as the optimal objective value, i.e., 0.9131, whose corresponding optimal solution is x∗ = (95.19, 80.23, 80.01, 126.37, 137.52, 120.14, 141.37, 105.06, 170.00, 162.26).
7. Conclusion Inverse optimization problem theory is a subject extensively studied in the context of tomographic studies, seismic wave propagation, and in a wide range of statistical inference with prior problems. As a special type of inverse optimization problem, the inverse minimum spanning tree problem has been extensively discussed in the literature. An inverse minimum spanning tree problem is to make the least modification on the edge weights such that a predetermined spanning tree is a minimum spanning tree with respect to the new edge weights. Many reconstruction problems can be transformed to inverse minimum spanning tree problems. However, the applications of the inverse spanning tree problem encountered in practice usually involve some uncertain issues so that the edge weights cannot be explicitly determined. Thus it is necessary to investigate the inverse minimum spanning tree problem in the fuzzy environment. In this paper, a special type of fuzzy inverse minimum spanning tree problem is considered, i.e., inverse minimum spanning tree problem with fuzzy weights with respect to the deterministic new edge weights. In order to formulate this problem, a new concept of fuzzy α-minimum spanning tree is provided for FIMST problem. Subsequently, it is shown that the notion of the fuzzy α-minimum spanning tree can be characterized by a constraint on non-tree edges and their corresponding tree paths. By this characterization, referred to
J. Zhang et al. / Models for inverse minimum spanning tree problem with fuzzy edge weights
as the fuzzy path optimality condition. Based on the concept and fuzzy path optimality condition, a fuzzy α-minimum spanning tree model is proposed for the FIMST problem. Furthermore, a credibility maximization model is also presented to formulate the FIMST problem according to a different decision criterion. In the literature many efficient algorithms have been developed for solving the classic inverse minimum spanning tree problem. However, these algorithms become powerless to solve the two fuzzy models proposed above. In order to find the optimal new edge weights satisfying constraints, a fuzzy simulation for computing credibility is designed and then embedded into a genetic algorithm to produce two hybrid intelligent algorithms. These hybrid intelligent algorithms have been illustrated by two examples, whose results show that the algorithms are efficient for solving the proposed fuzzy models. Acknowledgements This work was supported in part by grants from the Ministry of Education Funded Project for Humanities and Social Sciences Research (No. 12JDXF005), the Innovation Program of Shanghai Municipal Education Commission (No. 13ZS065), and the Shanghai Philosophy and Social Science Planning Project (No. 2012BGL006). References [1] R.K. Ahuja, T.L. Magnanti and J.B. Orlin, Network Flows: Theory, Algorithms, and Applications, New Jersey, USA: Prentice Hall, Inc., 1993. [2] R.K. Ahuja and J.B. Orlin, A faster algorithm for the inverse spanning tree problem, Journal of Algorithms 34(1) (2000), 177–193. [3] K. Eshghi and J. Nematian, Special classes of mathematical programming models with fuzzy random variables, Journal of Intelligent and Fuzzy Systems 19(2) (2008), 131–140. ´ Szentesi and B. Szviatovszki, Inverse optimiza[4] A. Farag´o, A. tion in high-speed networks, Discrete Applied Mathematics 129(1) (2003), 83–98. [5] J. Gao and M. Lu, Fuzzy quadratic minimum spanning tree problem, Applied Mathematics and Computation 164(3) (2005), 773–788. [6] M. Gen and R. Cheng, Genetic Algorithms and Engineering Optimization, New York, USA: John Wiley & Sons, 2000. [7] X. Guan, P.M. Pardalos and X. Zuo, Inverse Max + Sum spanning tree problem by modifying the sum-cost vector under weighted l∞ Norm, Journal of Global Optimization (2014). DOI: 10.1007/s10898-014-0140-z [8] X. Guan and B. Zhang, Inverse 1-median problem on trees under weighted Hamming distance, Journal of Global Optimization 54(1) (2012), 75–82.
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