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A simple algorithm for reliability evaluation in dynamic networks with stochastic transit times a
a
b
Hassan Salehi Fathabadi , Somayeh Khezri & Salman Khodayifar a
School of Mathematics, Statistics Computer Science, University of Tehran, Tehran, Iran
b
School of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Gava zang, Zanjan, Iran Published online: 10 Mar 2015.
To cite this article: Hassan Salehi Fathabadi, Somayeh Khezri & Salman Khodayifar (2015): A simple algorithm for reliability evaluation in dynamic networks with stochastic transit times, Journal of Industrial and Production Engineering, DOI: 10.1080/21681015.2015.1015460 To link to this article: http://dx.doi.org/10.1080/21681015.2015.1015460
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Journal of Industrial and Production Engineering, 2015 http://dx.doi.org/10.1080/21681015.2015.1015460
A simple algorithm for reliability evaluation in dynamic networks with stochastic transit times Hassan Salehi Fathabadia, Somayeh Khezria and Salman Khodayifarb* a
School of Mathematics, Statistics Computer Science, University of Tehran, Tehran, Iran; bSchool of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Gava zang, Zanjan, Iran (Received November 2014; revised January 2015; accepted January 2015) In a network flow, transit time of an arc is the time span that a unit of flow takes to travel through this arc. In most real-world systems, such as road traffic, communication networks, pipeline systems, transit time of an arc is not constant and may take a value, randomly, from among several possible values. In such systems, reliability is the main concern. Given a demand d, time threshold T, and budget B, we define the reliability as the probability that d units of flow can be sent from the source to the sink under time horizon T and budget B. In this paper, we propose a simple algorithm to evaluate the reliability of networks in which the transition times are stochastic variable.
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Keywords: network flow; stochastic variable; transit time; reliability
1. Introduction Flows variations over time (sometimes called dynamic flows) are important features in network flow problems arising in various real applications such as road or air traffic control, production/distribution systems, and communication networks. Examples and further applications can be found in the literature such as [1,3,4,15,16,19]. This class of network flows was first introduced by Ford and Fulkerson [3]. In their paper on the subject, they introduced flows with transit times and described a polynomial time algorithm to solve the maximum flow over time, also called the maximum dynamic flow problem. However, flows over time are significantly harder than their static counterparts. For example, both minimum cost flows over time and fractional multicommodity flows over time are NP-hard, even for very simple series–parallel networks. In the class of dynamic network flows, quickest path problem is to find a path (named quickest path) with minimum transmission time to send a given amount of flow from the source to the sink, where each arc has two attributes: the capacity and the transit time. Traditionally, in analysis of network flow, we assume that the network is deterministic, namely arc’s capacity, transmission’s cost on the arcs, supply or demand of nodes, and so on, are definite. However in many real-world systems, it is not true and at least one character of network is stochastic. Instance, the arc’s capacity may be variable due to failure, maintenance, etc. Such a network is called a stochastic-flow network in many papers [2,5–10,12–14,17]. In such networks, the system capacity namely the maximum value of flow from the source to the sink is not fixed. Hence, many authors [2,5–7,11,18,20] proposed methods to calculate two performance indices, the *Corresponding author. Email:
[email protected] © 2015 Chinese Institute of Industrial Engineers
probability that the upper bound of the system capacity is d and the probability that the lower bound of the system capacity is d, for a level d. Lin [12,13], extend the quickest path problem from the deterministic case to the stochastic case. He evaluates the probability that the stochastic-flow network can send d units of flow from the source to the sink within a given time horizon T. As you know, transit time of an arc is the time it takes for a unit of flow to travel through this arc. However, in many real networks such as pipeline systems, the transit time of arcs is stochastic variables. That is, these times are random variables that take their values according to a probability distribution. In this paper, we examine the network that the transit time of arcs is independent stochastic variables with certain probability distributions. We evaluate the probability that the network can send d units of flow from the source to the sink under time horizon T. 1.1. Contribution In summary, we define the reliability as the probability that d units of flow can be sent from the source to the sink within T units of time. Many authors studied the stochastic-flow network, instance Lin [7] studied the stochastic-flow network in which each arc or node has several capacities and may fail. Given the demand d, the system reliability is the probability that the maximum flow of the network is not less than d. He proposed an algorithm to generate all lower boundary points for d in terms of simple paths. Lin [13] extends the quickest path problem to a stochasticflow network in which the capacity of each arc is variable. He evaluates the system reliability that d units of data can be sent from the source to the sink under both time
2
H.S. Fathabadi et al. Suppose that pj ¼ ðaj1 ; aj2 ; . . .; ajnj Þ is a simple path (a path without any repetition of nodes). The following notations are used throughout this paper. U(pj) is the capacity of pj which is defined as: U ðpj Þ ¼ minfuji j1 i nj g
(1)
C(d, pj) is the total cost for carrying d units of flow through pj, which is defined as: Cðd; pj Þ ¼
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Figure 1. An example network.
threshold T and budget B. He is proposed an algorithm to generate all lower boundary points for (d, T, B), and then evaluates the system reliability in terms of such points. In this paper, we examine the network that the transit time of arcs is independent stochastic variables with certain probability distributions such as pipeline systems. We present a mathematical model for the presented problem and define Maximal Admissible Transition Vectors (MATV) and present an algorithm to generate all MATVs and finally, evaluate the reliability. The organization of this paper is as follows: In Section 2, we present a mathematical model of the problem and define MATV. In Section 3, we present an algorithm to generate all MATVs and then evaluate the reliability. In Section 4, we examine a numerical example. In Section 5, we obtain the computational complexity of the proposed algorithm. 2. Preliminary definitions Assume that G = (N, A, L, U, C) is a flow network with a source node s and a sink node t. N denotes the set of nodes, A = {ai|1 ≤ i ≤ n} denotes the set of arcs, U = {ui| 1 ≤ i ≤ n} with ui denoting the capacity of ai, C = {ci| 1 ≤ i ≤ n}with ci denoting the transition cost of ai, and L = {li|1 ≤ i ≤ n} with li denoting the minimal transition time of ai. The transit time of arc ai is denoting by τi, that takes possible values 0 ti1 ti2 . . . tiri ¼ T , where tij is an integer for j = 1, 2, … , ri. The vector τ = (τ1, τ2, … , τn) is called the transition vector. Such a network flow is assumed to further satisfy the following assumptions: (1) Each node is perfectly reliable. (2) The transit time of each arc is an integer-valued random variable. (3) The transit times of different arcs are statistically independent. (4) All flows are sent through multiple simple paths simultaneously. At the first, we remember two mathematical definitions for inequality. Suppose that X = (x1, x2, … , xn) and Y = (y1, y2, … , yn). We say that Y ≥ X, if yi ≥ xi for all i ¼ 1; 2; . . .; n. If Y ≥ X and there exist at least one i such that yi > xi, we say that Y > X.
nj X
ðd:cji Þ
(2)
i¼1
T τ(pj) is the transit time of flow, as much as U(pj), through pj, when the transition vector is τ. T τ(pj) is computed as:
T s ðpj Þ ¼
nj X
sji
(3)
i¼1
kðd; s; pj Þ is the total transmission time of d units of flow through pj under the transition vector τ. Clearly, ( s if d U ðpj Þ T ðpj Þ l m kðd; s; pj Þ ¼ (4) d T s ðpj Þ U ðp if d [ U ðpj Þ jÞ where d xe is the smallest integer such greater than x. Suppose that p1 ; p2 ; . . .; pk are k disjoint simple paths and we wish to send d1 ; d2 ; . . .; dk units of flow through p1 ; p2 ; . . .; pk ; respectively. Let U ¼ fðd1 ; . . .; dk ÞjCðd1 ; p1 Þ þ . . . þ Cðdk ; pk Þ B; d1 þ þ dk ¼ dg: and let ρ(d1, … , dk, τ, B) be the minimum time needed to send d1 ; d2 ; . . .; dk units of flow through p1 ; p2 ; . . .; pk , respectively, under the transition vector τ and budget B. Hence, for ðd1 ; . . .; dk Þ 2 U: qðd1 ; d2 ; . . .; dk ; s; BÞ ¼ maxfkðd1 ; s; p1 Þ; . . .; kðdk ; s; pk Þg (5)
Also let hðd; s; BÞ denote the minimum transition time to send d units of flow, through p1 ; p2 ; . . .; pk from the source node to the sink node under the transition vector τ and budget B. Clearly, hðd; s; BÞ ¼ minfqðd1 ; . . .; dk ; s; BÞjðd1 ; d2 ; . . .; dk Þ 2 Ug (6) Now Rd,T,B is the system reliability, which is defined as: Rd;T ;B ¼ Prfsjhðd; s; BÞ T g
(7)
The inequality hðd; s; BÞ T means that the transit vector τ is capable to send d units of flow from the source to the sink under time horizon T and budget B. Such vector is called Admissible Transition Vector (ATV). A MATV is an ATV that satisfies:
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(1) hðd; s; BÞ T : (2) If τʹ > τ then hðd; s0 ; BÞ [ T . In the following Theorem, we show that the calculation of Rd,T,B is needful computation of all MATVs. Theorem 2.1. If τ is a MATV, then hðd; s0 ; BÞ T for any τʹ ≤ τ. Proof. Since τ is a MATV, there exists a ðd1 ; . . .; dk Þ such that qðd1 ; d2 ; . . .; dk ; s; BÞ T . Suppose τʹ is an arbitrary transition vector such that τʹ ≤ τ. Hence s0ji sji for any 1 ≤ i ≤ nj and 1 ≤ j ≤ k. Therefore, kðdj ; s0 ; pj Þ kðdj ; s; pj Þ for any 1 ≤ j ≤ k, and maxfkðd1 ; s0 ; p1 Þ; . . .; kðdk ; s0 ; pk Þg maxfkðd1 ; s; p1 Þ; . . .; kðdk ; s; pk Þg. Then ρ(d1, … , dk, τʹ, B) ≤ ρ(d1, … , dk, τ, B) and hðd; s0 ; BÞ T . Theorem 2.1 implies that Rd;T ;B ¼ Prfsjs sj ; sj is a MATV}. Suppose that there are m MATVs for (d, T, B). Let As = {τ|τ ≤ τs}, hence the system reliability is defined as: ( ) s¼m [ As : Rd;T ;B ¼ Pr (8) s¼1
Now according to the above Theorem, we propose an algorithm to determine all MATVs.
Theorem 3.1. All MATVs τ are generated by the algorithm MATV. Proof. We first claim that any transition vector from algorithm is a MATV. Suppose that τ obtained from the proposed algorithm. It is clear that θ(d, τ, B) ≤ T, since there exists (d1, … , dk) ∊ Φ such that kðd1 ; s; p1 Þ T ; kðd2 ; s; p2 Þ T ; . . .; kðdk ; s; pk Þ T : Suppose τ is not a MATV, then there exists a MATV such that τʹ > τ. Because of the transit time of arcs that is not in pj for any j = 1, … , k, is equal T, hence without loss of generality, we assume there exists an arc a1i ∊ p1 such that s01i [ s1i : Because of, τ2 is generated from the proposed algo3 rithm and L1 :¼ 4l l L1
d1 U ðp1 Þ
In the following Theorem, we show that the presented algorithm generates all MATVs.
T
d1 U ðp1 Þ
m
m5 is the largest integer such that
T therefore
Suppose L0j ¼ 4l
T
l
Step 1:
For j = 1, …, k, find the largest dj such that
nj P
l lji
i¼1
Generate all non-negative integer solutions of
Step 3:
For each solution (d1, … , dk) do
Step 3.1:
For j = 1, …, k, do find the transmission cost nj P Cðdj ; pj Þ ¼ ðdj :cji Þ i¼1 k P Cðdj ; pj Þ [ B then go Step 3.5 If
k P
m
dj U ðpj Þ
T:
dj ¼ d and dj dj :
j¼1
j¼1
Step 3.3:
For j =2 1, … 3 , k, do Lj :¼ 4l
T
m5
dj U ðpj Þ
Generate all non-negative integer solutions of nj P lji0 ¼ Lj ; lji0 lji : i¼1
Step 3.4:
Step 3.5:
m
d1 U ðp1 Þ
[T
Therefore, or there exists an arc au not in pj for any j = 1, … , k such that s0u \T or there exists 1 ≤ j ≤ k such
x :¼ null; s :¼ 0:
Step 3.2:
0 i¼1 sli
m5 for any j ¼ 1; . . .; k:
Algorithm MATV Step 0:
Step 2:
Pn 1
hence ρ(d1, d2, … , dk, τʹ, B) > T. Since τʹ is a MATV, hence there exists ðd10 ; . . P .; dk0 Þ 2 U nj 0 0 0 0 0 such that qðd1 ; d2 ; . . .; dk ; s ; BÞ T ; therefore i¼1 sji l d0 m j U ðpj Þ T for any j ¼ 1; . . .; k: 2 3 d0 j U ðpj Þ
3. Main result
3
For each set of solutions s :¼ s þ 1; ss is obtained according to 0 l if ai 2 pj ; j ¼ 1; . . .; k sji ¼ ji T o:w For g = 1 to g = s–1 do If τs ≥ τg then delete τg from ω and x :¼ x [ fss g: If τs < τg return. Otherwise ω: = ω [ {τs}. Next (d1, … , dk).
4
H.S. Fathabadi et al.
j = 1, … , k. Suppose Lj ¼ 4l
Table 1. The arc’s data of the example network.
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Arc
Capacity
Transit time
Probability
Cost
a1
2
3
a3
5
a4
2
a5
2
0.05 0.1 0.05 0.05 0.75 0.2 0.1 0.7 0.15 0.85 0.3 0.1 0.2 0.4 0.25 0.05 0.05 0.05 0.6 0.1 0.2 0.1 0.6 0.1 0.1 0.05 0.75 0.05 0.05 0.05 0.85 0.3 0.1 0.1 0.5 0.05 0.1 0.1 0.55 0.2 0.85 0.15
3
a2
4 3 2 1 0 4 3 2 4 3 4 3 2 1 4 3 2 1 0 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1 4 3 2 1 0 4 3
a6
2
a7
3
a8
4
a9
2
a10
3
a11
1
2
4 2 3
4
k. Hence
nj P
3 T dj U ðpj Þ
m5 for all j = 1, …,
sji ¼ Lj ; for all j ¼ 1; . . .; k: Therefore, there
i¼1
exists an arc au not in pj, for all j = 1, … , k, such that τu < T (otherwise τ ∊ ω). And define τʹ = (τ1, τ2, …, τu + 1, … , τn) that τʹ > τ and θ(d, τʹ, B) = T. It contradicts that τ is a MATV. Or there exists 1 ≤ j ≤ k such that kðdj ; s; pj Þ\T and, there exists an arc au ∊ pj such that τu < T, define τʹ = (τ1, τ2, … , τu + 1, … , τn), then τʹ > τ and θ(d, τʹ, B) = T. It contradicts that τ is a MATV. Case 2: Suppose that θ(d, τ, B) < T, then there exists (d1, … , dk) ∊ Φ such that ρ(d1, d2, … , dk, τ, B) < T, and kðd1 ; s; p1 Þ\T : Hence there exists an arc au ∊ p1 such that τu < T, define τʹ = (τ1, …, τu + 1, …, τn), therefore τʹ > τ and θ(d, τʹ, B) ≤ T. That contradicts that τ is a MATV. 4. Numerical example
3
1
2
4
3
We examine the network in Figure 1 and use the above algorithm for search all MATVs. Let d = 9, T = 4, B = 80, p1 = {a1, a2, a4}, p2 = {a5, a6, a7}, and p3 = {a8, a9, a10}. The capacity and the probabilistic transit time of each arc are shown in the Table 1. The largest integers l m P d dj ; j ¼ 1; 2; 3;such that la U ðpj j Þ T are a2pj
d1 ¼ 2; d2 ¼ 4; d3 ¼ 4. From Table 2, three MATV are generated from the algorithm. Set A7 ¼ fsjs s7 g; A8 ¼ fsjs s8 g and A9 = {τ|τ ≤ τ9}. In the calculating process, PrðA7 Þ ¼ Prfsjs (0, 3, 4, 1, 0, 1, 1, 1, 1, 0, 4)g ¼ Prfs1 0g Prfs2 3g Prfs3 4g Prfs4 1g
3
Prfs5 0g Prfs6 1g Prfs7 1g Prfs8 1g Prfs9 1g Prfs10 0g Prfs11 4g ¼ 0:005508
Pnj 0 that i¼1 sji \L0j (otherwise τʹ is generated from the proposed algorithm and it contradicts that τ is generated from the proposed algorithm). In the first case, we define s00 ¼ ðs01 ; s02 ; . . .; s0u þ 1; . . .; s0n Þ: It is clear that τʹʹ > τʹ and θ(d, τʹʹ, B) ≤ T. That contradicts that τʹ is a MATV. In the second case, suppose ah ∊ pj such that s0h \T and define s00 ¼ ðs01 ; . . .; s0h þ 1; . . .; s0n Þ: It is clear that τʹʹ > τʹ and θ(d, τʹʹ, B) ≤ T. That contradicts that τʹ is a MATV. Hence τ is a MATV. Conversely, let τ be a MATV. Suppose to the contrary τ ∉ ω. We consider two the following cases: Case 1: Suppose that θ(d, τ, B) = T. Hence there exists (d1, … , dk) ∊ Φ such that ρ(d1, …, dk, τ, B) = T. Without loss of generality suppose that kðd1 ; s; p1 Þ ¼ T therefore or kðdj ; s; pj Þ ¼ T for all
PrðA8 Þ ¼ Prfsjs (0, 2, 4, 2, 0, 1, 1, 1, 1, 0, 4)g ¼ 0.00722925 PrðA9 Þ ¼ Prfsjs (1, 2, 4, 1, 0, 1, 1, 1, 1, 0, 4)g ¼ 0.0051408 PrðA7 \ A8 Þ ¼ Prfsjs (0, 2, 4, 1, 0, 1, 1, 1, 1, 0, 4)g ¼ 0.0048195 PrðA8 \ A9 Þ ¼ Prfsjs (0, 2, 4, 1, 0, 1, 1, 1, 1, 0, 4)g ¼ 0.0048195 PrðA7 \ A9 Þ ¼ Prfsjs (0, 2, 4, 1, 0, 1, 1, 1, 1, 0, 4)g ¼ 0.0048195 PrðA7 \ A8 \ A9 Þ ¼ Prfsjs (0, 2, 4, 1, 0, 1, 1, 1, 1, 0, 4)g ¼ 0.0048195
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Table 2. Result of the example. (d1, d2, d3)
(L1, L2, L3)
(1,4,4)
(4,2,2)
(2,3,4)
(4,2,2)
(2,4,3)
(4,2,2)
Therefore,by applying inclusion–exclusion, we have
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R9;4;80 ¼ PrðA7 Þ þ PrðA8 Þ þ PrðA9 Þ PrðA7 \ A8 Þ PrðA8 \ A9 Þ PrðA7 \ A9 Þ þ PrðA7 \ A8 \ A9 Þ ¼ 0.00823905
5. Computational complexity In the following, we calculate the computational complexity of the algorithm MATV. Theorem 5.1. The running time of the proposed algorithm is O(d2n). Proof. The complexity of the proposed algorithm is analyzed as follows. In Step 1, it takes at most O(n) time to find the largest assigned demands dj ; j ¼ 1; . . .; k: In Step 2, there are at most (d + 1) solutions of d1 + d2 + ⋯ + dk = d. For each (d1, d2, …, dk), it spends O(n) time to test the budget constraint (Steps 3.1,3.2) in the worst case. It subsequently takes at most O(n) time to test the time horizon and transform to transition vector of τ. The set ω contains at most (d + 1) elements. Step 3.4 needs O(dn) time to compare with other τ for each τg and O(d2n) time for all τg in the worst case. Hence, Step 3 needs O(d2n) time to generate all MATVs. In sum, it takes at most O(d2n) time to execute the proposed algorithm.
6. Conclusion In this paper, we have studied a network with stochastic transit times. In such networks, the system reliability is the probability that d units of flow can be sent through multiple simple paths under time horizon T and budget B. At first, we have defined MATVs for (d, T, B), the MATVs satisfying the requirements. We have shown that the system reliability can be computed in terms of all MATVs and then we have proposed an algorithm that generate all such vectors. The problem that, the capacity of an arc is varying over time is an interesting task to be addressed in the future.
τ s1 s2 s3 s1 s2 s3 s1 s2 s3
¼ ð0; 3; 4; 1; 0; 1; 1; 1; 1; 0; 4Þ ¼ (0, 2, 4, 2, 0, 1, 1, 1, 1, 0, 4) ¼ ð1; 2; 4; 1; 0; 1; 1; 1; 1; 0; 4Þ ¼ ð0; 3; 4; 1; 0; 1; 1; 1; 1; 0; 4Þ ¼ ð0; 2; 4; 2; 0; 1; 1; 1; 1; 0; 4Þ ¼ (1, 2, 4, 1, 0, 1, 1, 1, 1, 0, 4) ¼ ð0; 3; 4; 1; 0; 1; 1; 1; 1; 0; 4Þ ¼ ð0; 2; 4; 2; 0; 1; 1; 1; 1; 0; 4Þ ¼ ð1; 2; 4; 1; 0; 1; 1; 1; 1; 0; 4Þ
τ∊Φ NO NO NO NO NO NO Yes Yes Yes
Notes on contributors Hassan Salehi Fathabadi is an Operations Research Professor in the School of Mathematics, Statistics and Computer Science, University of Tehran. He received his MSc in Computer Science from the Queen Mary College, University of London and his PhD in Operations Research from the University of Southampton, England. His main research and publications are in the areas of network flows, supply chains, and production planning. Somayeh Khezri received her BSc and MSc degrees in Mathematics from the University of kharazmi and the University of Tehran, in 2010 and 2012, respectively. Her special fields of interests include network flow optimization. Salman Khodayifar is an Operations Research Assistant Professor in School of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS). He received his BSc degree in Mathematics from the University of Mohaghegh Ardabili and his PhD and MSc degrees in Operations Research from the University of Tehran. His special fields of interests include network flow optimization, supply chains, distribution–production networks, and distribution generation (DG). References [1] Alexopoulos, C., “A note on state-space decomposition methods for analyzing stochastic flow networks,” IEEE Transactions on Reliability, 44, 354–357 (1995). [2] Aven, T., Reliability evaluation of multistate systems with multistate components, IEEE Transactions on Reliability, R-34, 473–479 (1985). [3] Ford, L. R. and D. R. Fulkerson, “Constructing maximal dynamic flows from static flows,” Operations Research, 6, 419–433 (1958). [4] Fonoberova, M. and D. Lozovanu, “Optimal dynamic flows in networks and applications,” The International Symposium The Issues of Calculation Optimization. Communications. Crimea, Ukraine, 292–293 (2007). [5] Hudson, J. C. and K. C. Kapur, “Reliability bounds for multistate systems with multistate components,” Operations Research, 33, 153–160 (1985).
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[6] Lin, J. S., C. C. Jane and J. Yuan, “On reliability evaluation of a capacitated-flow network in terms of minimal pathsets,” Networks, 25, 131–138 (1995). [7] Lin, Y. K., “A simple algorithm for reliability evaluation of a stochastic-flow network with node failure,” Computers & Operations Research, 28, 1277–1285 (2001). [8] Lin, Y. K., “Reliability of a stochastic-flow network with unreliable branches & nodes under budget constraints,” IEEE Transactions on Reliability, 53, 381–387 (2004). [9] Lin, Y. K., “Reliability evaluation for an information network with node failure under cost constraint,” IEEE Transactions on Systems, Man, and Cybernetics – Part A: Systems and Humans, 37, 180–188 (2007). [10] Lin, Y. K., “On a multicommodity stochastic-flow network with unreliable nodes subject to budget constraint,” European Journal of Operational Research, 176, 347–360 (2007). [11] Lin, Y. K., “On reliability evaluation of a stochasticflow network in terms of minimal cuts,” Journal of the Chinese Institute of Industrial Engineers, 18, 49–54 (2001). [12] Lin, Y. K., “Extend the quickest path problem to the system reliability evaluation for a stochastic-flow network,” Computers & Operations Research, 30, 567–575 (2003).
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