The α-maximum Flow Model with Uncertain Capacities

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1 International April 25, 2013 Journal7:12 of Uncertainty, WSPC/INSTRUCTION Fuzziness and Knowledge-Based FILE Uncertain*-maximum*Flow*Model Systems c World Scientific Publishing Company The α-maximum Flow Model with Uncertain Capacities Sibo Ding Uncertainty Operations Research Laboratory, School of Management Henan University of Technology, Zhengzhou , China dingsibo@haut.edu.cn Uncertain theory is a new tool to deal with the maximum flow problem with uncertain arc capacities. This paper investigates uncertain maximum flow problem and presents the uncertainty distribution of the maximum flow. Uncertain α-maximum flow model is formulated. It is proved that there exists an equivalence relationship between uncertain α-maximum flow model and the classic deterministic maximum flow model, which builds a bridge between uncertain maximum flow problem and deterministic maximum flow problem. Furthermore, some important properties of the model are analyzed, based on which a polynomial exact algorithm is proposed. Finally, a numerical example is presented to illustrate the model and the algorithm. Keywords: Maximum flow problem; Uncertainty theory; Uncertain Programming; Generic preflow-push algorithm. 1. Introduction The maximum flow problem is one of the core issues of network optimization and has been widely studied. This problem was first investigated by Fulkerson and Dantzig 1. Then, Ford and Fulkerson 2 solved it using augmenting path algorithm. Motivated by a desire to develop a method with improved worst-case complexity, Dinic 3 introduced the concept of layered networks. His algorithm proceeds by augmenting flows along directed paths from source to sink in the layered network. Edmonds and Karp 4 also independently proposed that the Ford and Fulkerson algorithm augments flow along shortest paths. Until this point all maximum flow algorithms were augmenting path algorithms. However, the augmenting path algorithm could be slow because it might perform a large number of augmentations. In order to reduce the number of augmentations, Karzanov 5 introduced the first preflow-push algorithm on layered networks. Goldberg and Tarjan 6 constructed distance labels instead of layered networks to improve the running time of preflow-push algorithm. They described a very flexible generic preflow-push algorithm that performs push and relabel operations at active nodes. Their algorithm can examine active nodes in any order. In practice, flow capacities may change over time in communication network or transportation network, and one may assume network arcs have different values (capacity, cost) that are random variables with known probability distributions. As an extension of deterministic maximum flow problem, the stochastic maximum flow problem has been investigated extensively. Frank and Hakimi 7 assumed that each branch Crresponding author 1

2 2 Sibo ding in communication network has a random capacity and attempted to find the probability of a flow between vertices. Frank and Frisch 8 considered how to determine the maximum flow probability distribution in networks where each capacity is a continuous random variable. Furthermore, Doulliez 9 studied multiterminal network with discrete probabilistic branch capacities. In addition, some researchers have tried to give lower and upper bounds on the expected maximum flow. Onaga 10 derived an upper bound in general undirected or directed networks, while Carey and Hendrickson 11 presented a efficient method to find a lower bound in general directed networks. After that, Nagamochi and Ibaraki 12 provided necessary and sufficient conditions for Carey and Hendrickson s lower bound. In reality, however, there exists indeterminacy about the parameters (capacities, costs) of maximum flow problems. That indeterminacy cannot be described by random variable because no samples are available. For instance, when networks subject to extreme events such as earthquakes, it is impossible to get probability distribution of arc capacities. If we insist on using probability theory to deal with indeterminacy, counterintuitive results will occur 13. But experts can estimate, based on their experience, the belief degree that arc capacities are less than or equal to a given value. In order to deal with this kind of human uncertainty, Liu 14 founded uncertainty theory and refined it 15. Since then, uncertainty theory and its application have experienced explosive growth. Peng and Iwamura 16 derived sufficient and necessary condition for uncertainty distribution. Liu and Ha 17 developed a formula for calculating the expected values of monotone functions of uncertain variables. In order to deal with mathematical programming involving with uncertain parameters, Liu 18 first proposed uncertain programming theory to model uncertain optimization problems. After that, Liu and Yao 19 introduced an uncertain multilevel programming for modeling uncertain decentralized decision systems, and Liu and Chen 20 developed an uncertain multiobjective programming and an uncertain goal programming. Some attempts have been made to collect expert s experimental data and get uncertain distribution. Liu 21 suggested the principle of least squares to estimate the unknown parameters of uncertainty distribution. Moreover, when a number of experts are available, Wang, Gao and Guo 22 applied the Delphi method to determine the uncertainty distributions. Chen and Ralescu 23 used B-spline method to estimate the uncertainty distribution. Other particular care has been taken to investigate uncertain graph and uncertain network. The connectedness index of uncertain graph was proposed by Gao and Gao 24. Zhang and Peng 25 suggested a method to calculate Euler index of uncertain graph. Gao 26 computed cycle index of uncertain graph. As an important contribution, Liu 21 first introduced uncertainty theory into network optimization. He studied project scheduling problem with uncertain duration times. Furthermore, Liu 27 assumed uncertainty and randomness simultaneously appear in a complex network, and put forward the concept of uncertain random network. Gao 28 gave an equivalence relation between the uncertain α-shortest path and the deterministic shortest path. And, Han and Peng 29 studied the uncertain maximum flow problem and provided a numerical solution method. This paper constructs an uncertain α-maximum flow model, analyzes properties of the model, and designs a new polynomial-time exact algorithm. The remainder of this paper is organized as follows. In section 2, some basic concepts and properties of uncertainty theory used throughout this paper are introduced. In section 3, Uncertain α-maximum flow model is formulated and its properties are analyzed. In section 4, an optimal algorithm is developed to solve the model. A numerical example is presented to illustrate the algorithm in Section 5. Section 6 gives a conclusion to this paper.

3 Uncertain α-maximum Flow Model 3 2. Preliminaries Uncertainty theory is a branch of mathematics for modeling human uncertainty. In order to deal with human uncertainty, Liu 14,15 presented four axioms: (1) normality axiom, (2) duality axiom, (3) subadditivity axiom, and (4) product axiom. In this section, we introduce some fundamental concepts and properties of uncertainty theory, which will be used throughout this paper. Definition Let Γ be a nonempty set, L a σ-algebra over Γ, and M an uncertain measure. Then the triplet (Γ, L, M) is called an uncertainty space. Definition An uncertain variable is a measurable function ξ from an uncertainty space (Γ, L, M) to the set of real numbers, i.e., for any Borel set B of real numbers, the set {ξ B} = {γ Γ ξ(γ) B} is an event. Definition The uncertainty distribution Φ of an uncertain variable ξ is defined by Φ(x) = M{ξ < x} for any real number x. The zigzag uncertainty distribution ξ Z(a, b, c) has an uncertainty distribution 0, if x a (x a)/2(b a), if a x b Φ(x) = (x + c 2b)/2(c b), if b x c 1, if x c. Definition An uncertainty distribution Φ(x) is said to be regular if it is a continuous and strictly increasing function with respect to x at which 0 < M(x) < 1, and lim x Φ(x) = 0, lim x + Φ(x) = 1. Definition The uncertain variables ξ 1, ξ 2,, ξ n are said to be independent if { n } n M (ξ i B i ) = M{ξ i B i } i=1 for any Borel sets B 1, B 2,, B n of real numbers. i=1 Theorem Let ξ 1, ξ 2,, ξ n be independent uncertain variables with regular uncertainty distributions Φ 1, Φ 2,, Φ n, respectively. If f is a strictly increasing function, then ξ = f(ξ 1, ξ 2,, ξ n )

4 4 Sibo ding is an uncertain variable with inverse uncertainty distribution Ψ 1 (α) = f(φ 1 1 (α), Φ 1 2 (α),, Φ 1 n (α)). In reality, we can easily obtain strictly increasing functions f. Therefore, through Theorem 1, we can transform an indeterminacy model into a deterministic one by to its equivalent deterministic form M{f(ξ 1, ξ 2,, ξ n ) x} α, x f(φ 1 1 (α), Φ 1 2 (α),, Φ 1 n (α)). 3. Mathematical formulation This paper concerns uncertain maximum flow problem. The goal is to maximize the total flow sent from the source node to the sink node not exceeding the capacities on any arc and keeping flow balance in every node. In classic deterministic maximum flow problem, a flow network is 5-tuple N = (V, A, c, s, t), with source node s, sink node t, a finite node set V = {1, 2,, n} and arc set A = {(i, j) i, j V }, together with a nonnegative real-valued capacity function c defined on its arc set A. Denote u = {u ij (i, j) A} as the set of arc capacities. Then in network N, from the source s to the sink t, the maximum flow is a function of u, which is denoted as f. Given u, f(u) can be found. Denote x = {x ij (i, j) A} as the set of flow on arc (i, j). A flow is feasible if it satisfies conservation condition x sj x js = f, (i, j) A j:(s,j) A j:(j,s) A x ij x ji = 0, (i, j) A j:(i,j) A j:(t,j) A x tj j:(j,i) A j:(j,t) A x jt = f, (i, j) A and capacity constraint 0 x ij u ij, where f is the flow in the network N. In classic deterministic maximum flow problem, capacities of arcs are scrip values. Unfortunately, cases such as this which capacities of arcs are scrip values are rare. Especially, networks are quite large, and it is impossible to describe them explicitly. But, if there are enough data available, random maximum flow models may be considered as network models. However, in most cases, we cannot get enough data or data is invalid because of change in conditions. For example, due to impact of unexpected accidents on traffic flow, we cannot use probabilistic rules to describe the complexity of the networks. In this situation, the capacity data can only be obtained from the decision-maker s subjective estimation. Thus, it is unsuitable to regard subjective estimation data as random variables. In this paper, we employ uncertain variables to describe the capacities of arcs. We consider the maximum flow problem subject to the following assumptions: (1) The network is directed. (2) All capacities are nonnegative rational numbers (All computers store capacities as rational numbers and we can always transform rational numbers to integer numbers by multiplying them by a suitably large number). (3) The network does not contain a directed path from source node s to sink node t composed only of infinite capacity arcs. (4) The network does not contain parallel arcs.

5 Uncertain α-maximum Flow Model 5 Define ξ = {ξ ij (i, j) A}. We can denote the network with uncertain capacities of arcs as N = (V, A, ξ, s, t). The maximum flow is f(ξ). As a function of ξ, f is also an uncertain variable. Sometimes, the decision-maker assumes that the flow should satisfy some chance constraints with at least some given confidence level α. Then we have the following definition. Definition 1. A flow x is the α-maximum flow from source s to sink t if max{f M{ξ x} α} max{f M{ξ x } α} for any flow x from source s to sink t, where α is a predetermined confidence level. Chance constrained programming offers a powerful tool for modeling uncertain decision systems. The essential idea of chance constrained programming of α maximum flow model is to optimize the flow value in network with predetermined confidence level subject to capacity chance constraints. In order to find α-maximum flow, we propose the following uncertain α-maximum flow model. max f subject to : x ij f, i = s, x ji = 0, i V {s, t}, j:(i,j) A j:(j,i) A f, i = t, M{ξ ij x ij } α, (i, j) A x ij 0, (i, j) A, (1) where α is a predetermined confidence level provided by the decision-maker. In classic deterministic model, the maximum flow is obtained by polynomial algorithms. It implies that if there is one way to transform uncertain α-maximum model into its crisp equivalent, then we can solve the model in deterministic environment by polynomial algorithms. Therefore, we need to convert the chance constraints M{ξ ij x ij } α, (i, j) A into its crisp equivalent. In order to deign algorithm for α-maximum flow model, we first introduce property of the chance constraint. Lemma 1. According to Theorem 1, M{ξ ij x ij } α, for (i, j) A can be transformed as x ij Φ 1 ij (α). Proof. Denote the uncertainty distribution of ξ ij as Φ ij, for (i, j) A. According to Theorem 1, for any 0 < α < 1, M{ξ ij Φ 1 ij (α)} = α. Since uncertainty distribution is an increasing function, from M{ξ ij x ij } α = M{ξ ij Φ 1 ij (α)},

6 6 Sibo ding we obtain This proves the lemma. x ij Φ 1 ij (α). Lemma 1 indicates that the chance constraint can be converted into its crisp equivalent, and we can draw the following theorem. Theorem 2. In network N = (V, A, ξ, s, t), ξ ij has a regular uncertainty distribution Φ ij, (i, j) A. Then, α-maximum flow of N = (V, A, ξ, s, t) is just the maximum flow of the corresponding deterministic network Ñ = (V, A, c, s, t), where the capacity of arc (i, j) A is Φ 1 ij (α). Proof. By Lemma 1, model (1) can be easily converted into the following deterministic model: max f subject to : x ij f, i = s, x ji = 0, i V {s, t}, j:(i,j) A j:(j,i) A f, i = t, (2) x ij Φ 1 ij (α), (i, j) A x ij 0, (i, j) A Thus, the solution to model (2) is just the maximum flow of deterministic network Ñ = (V, A, c, s, t), where the capacity of arc(i, j) A is Φ 1 ij (α). Thus, We can obtain the maximum flow of Ñ = (V, A, c, s, t) by using generic preflow-push algorithm. The theorem is proved. Theorem 1 shows how to obtain α-maximum flow. Next, we further investigate the property of the model for obtaining the inverse distribution of uncertain maximum flow. Theorem 3. In network N = (V, A, ξ, s, t), ξ ij has a regular uncertainty distribution Φ ij. Then, the inverse uncertainty distribution of f is determined by Ψ 1 (α) = f(φ 1 ij (α) (i, j) A). Proof. For model (1), the maximum flow f is a continnuous and increasing function with respect to each capacity of arc. Obviously, increasing the capacity of each arc, we will get a greater flow. That is, f(x) > f(y), where x = {x ij (i, j) A}, y = {y ij (i, j) A}, and x ij > y ij. Thus, f is a strictly increasing function. By Theorem 1 and Theorem 2, we can easily obtain the inverse uncertainty distribution of f with respect to α. The theorem is proved.

7 Uncertain α-maximum Flow Model 7 4. Solution algorithm Generally, Monte Carlo simulation or heuristic algorithms are used to obtain solution to uncertain programming with indeterminacy factors. A disadvantage of these methods in comparison to exact mathematical methods is that they usually provide only statistical estimates or approximate solutions, not exact result. Theorem 2 provides a better way to obtain the α-maximum flow in uncertain network. We only need to employ the generic preflow-push algorithm to find the maximum flow of the corresponding deterministic network. Hence, based on Lemma 1 and Theorem 2, we can design the following optimal solution algorithm for obtaining α-maximum flow. Algorithm: Step 1. Give a predetermined confidence level α and calculate Φ 1 ij (α), (i, j) A. Step 2. Construct the corresponding deterministic network Ñ = (V, A, c, s, t), and set the capacity of each arc u ij equal to Φ 1 ij (α). Step 3. Empoly the generic preflow-push algorithm to find α-maximum flow in network N. This algorithm runs in O(n 2 m) time (n is the number of nodes and m is the number of arcs) which is the same as that of generic preflow-push algorithm. 5. Numerical example In this section, we give an example to illustrate the algorithm. A network N = (V, A, ξ, s, t) is shown in Fig.1. The α-maximum flow is f with distribution Ψ. The capacity of each arc (i, j) is listed in Table 1. For convenience, if ξ ij is a constant, we set Φ 1 ij (α) = c, for any α (0, 1). Thus, we can calculate Φ 1 ij (0.9) for each ξ ij. Values of Φ 1 ij (0.9) are also listed Table 1. Then, we can obtain the following results: (1) the α-maximum flow when α = 0.9; (2) the uncertainty distribution of f(ξ). Fig. 1. Network for example.

8 8 Sibo ding Table 1. List of arc capacities and Φ 1 ij (0.9). arc (i, j) ξ ij Φ 1 ij (0.9) (s, 1) Z(14, 16, 18) 17.6 (1, 2) 6 6 (1, 3) Z(9, 10, 11) 10.8 (2, t) Z(10, 11, 12) 11.8 (3, t) Z(10, 12, 14) 13.6 Fig.2 Using the data in Table 1, we construct the deterministic network Ñ = (V, A, c, s, t). It is shown in Fig. 2. Network with upper bounds. Then, we solve α-maximum flow problem using the generic preflow-push algorithm. We obtain the final residual network given in Fig.3 (e( ) are excesses of nodes, d( ) are distance labels and r ij is residual capacity of any arc (i, j) A ). Fig. 3. Final residual network. Now the network contains no active node. The α-maximum flow in the network is shown in Fig.4 and its value is Choosing different α and repeating the above process, we obtain the uncertainty distribution of f, which is listed in Table 2 and plotted in Fig.5.

9 Uncertain α-maximum Flow Model 9 Fig. 4. The optimal solution to example. Table 2. List of α-maximum flows. α x s1 x 12 x 13 x 2t x 3t f = Ψ 1 (α) α f Fig. 5. Uncertainty distribution of f. 6. Conculsion Interpretations of indeterminacy phenomena may vary from time to time or person to person. Uncertainty theory provides a new tool to deal with indeterminacy. Under the framework of uncertainty theory, we present an extension to the classic maximum flow problem whose network capacities are uncertain variables in stead of crisp values. The problem was formulated by uncertain α-maximum flow model.

10 10 Sibo ding The relationship between the uncertain α-maximum flow model and its crisp equivalent is proved, and the uncertainty distribution of the uncertain maximum flow is derived. Some important properties of the model are analyzed, which help to develop a polynomial time exact algorithm. At last, an uncertain α-maximum flow example is given and solved by the algorithm. 7. Acknowledgments This work was supported by the National Natural Science Foundation of China Grant No and Education Research Project No.YPGC2011-W03. References 1. D. R. Fulkerson and G. B. Dantzig, Computations of maximum flow in networks, Naval Res. Log. Quart. 2 (4) (1955) G. B. Dantzig and D. R. Fulkerson, On the max-flow min-cut theorem of networks, in Linear Inequalities and Related Systems, eds. H. W. Kuhn and A. W. Tucker (Princeton University Press, Princeton, 1956) pp E. A. Dinic, Algorithm for solution of a problem of maximum flow in networks with power estimation,soviet Math. Dokl. 11 (1970) J. Edmonds and R. M. Karp, Theoretical improvements in algorithmic efficiency for network flow problems, J. ACM 19 (2) (1972) A. V. Karzanov, Determining the maximal flow in a network by the method of preflows, Soviet Math. Dokl. 15 (2) (1974) A. V. Goldberg and R. E. Tarjan, A new approach to the maximum flow problem, Proc. Of the 18th Annual ACM Symposium on the Theory of Computing, Berkeley, CA, May, 1986, pp H. Frank and S. L. Hakimi, Probabilistic flows through a communication network, IEEE Trans. on Circuit Theory 12 (3) (1965) H. Frank and I. T. Frisch, Communication, Transmission, and Transportation Networks (Addison-Wesley, Reading, 1971). 9. P. Doulliez, Probability distribution function for the capacity of a multiterminal network, Rev. Franc. Inform. Rech. Oper. 5 (V1) (1971) K. Onaga, Bounds on the average terminal capacity of probabilistic nets, IEEE Trans. Inf. Theory 14 (5) (1968) M. Carey and C. Henrickson, Bounds on expected performance of networks with links subject to failure, Networks 14 (3) (1986)

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