An Algorithm for Lower Reliability Bounds of Multistate Two-Terminal ...

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An Algorithm for Lower Reliability Bounds of Multistate Two-Terminal Networks Sarintip Satitsatian and Kailash C. Kapur Abstract—Network reliability is extensively used to measure the degree of stability of the quality of infrastructure services. The performance of an infrastructure network and its components degrades over time in real situations. Multi-state reliability modeling that allows a finite number of different states for the performance of the network and its components is more appropriate for the reliability assessment, and provides a more realistic view of the network performance than the traditional binary reliability modeling. Due to the computational complexity of the enumerative methods in evaluating the multi-state reliability, the problem can be reduced to searching lower boundary points, and using them to evaluate reliability. Lower boundary points can be used to compute the exact reliability value and reliability bounds. We present an algorithm to search for lower boundary points. The proposed algorithm has considerable improvement in terms of computational efficiency by significantly reducing the number of iterations to obtain lower reliability bounds.

a component state vector; number of states of the system state of the system when a component state vector is , component state space;

equivalence class ; for , a set of all LBP of state ; iff for all and there exists at least one such that state of arc , a random variable component state vector, a random vector

Index Terms—Lower boundary points, multistate reliability, network reliability.

,

ACRONYM1 LBP lower boundary point MIP Minimal improvement path LRB lower reliability bound SV seed vector

NOMENCLATURE LBP A vector is an LBP of state iff , and implies that . Path A path in a network is a finite sequence ,

NOTATION a directed flow network with a set of nodes and a set of arcs the arc is incident to the initial node and the terminal node and nodes and are adjacent to each other, and ,

, are the specified source node and and sink node respectively; the maximum capacity allowed at arc is a vector of flow in which denotes a quantity of material flowing in arc ; number of states of arc , current state of arc that is the amount of flow allowed to be sent through arc ,

Manuscript received August 29, 2005; revised November 1, 2005. Associate Editor: L. Cui.. The authors are with the Industrial Engineering, University of Washington, Seattle, WA 98195 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TR.2006.874922 1The

singular and plural of an acronym are always spelled the same.

, where the initial node, and respectively;

, is is the terminal node of . for and .

ASSUMPTIONS 1) Each node is perfectly reliable. 2) The capacity of each arc is a non-negative integer-valued random variable according to a given distribution. 3) The capacities of different arcs are statistically independent. 4) All flows in the network obey the flow conservation law. is non-decreasing. 5) , and are not empty. 6) I. INTRODUCTION NFRASTRUCTURE networks, such as computer and communication networks, power transmission and distribution networks, transportation networks, oil/gas production networks, and logistic networks, can be considered as flow networks whose arcs have independent, discrete, limited, and multi-valued random capacities. The reliability of such a network is defined as the probability that the required demands can be supplied from a source to a sink through multi-state links. The binary reliability assessment is insufficient to evaluate networks with multi-state behavior. To avoid incorrect

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decision-making regarding network performance, multi-state reliability modeling and evaluation that consider the multi-state behavior of a network and its components should be implemented. The analysis of multi-state two-terminal network reliability may be the appropriate approach adopted to evaluate such infrastructure networks [1], [2]. There have been several approaches proposed for the reliability computation of the multi-state two-terminal network. An initial approach used to solve this problem is the complete enumeration method [3]–[6]. The method was computationally expensive because it related all possible combinations of component states to a system state. Patra & Misra [7], [8] proposed an enumerative procedure that used binary cut sets to generate component state vectors that potentially satisfied required demands. The infeasible component state vectors, which could not fulfill the required demands, were sorted out after the verification process. Several authors have presented other approaches to evaluate reliability in terms of the multi-state minimal path sets [9], [10] which has been known as lower boundary points (LBP) [4], [5], [11], upper critical connection vectors [3], or -MP [9], [12]. The exact reliability value of a particular state can be computed using the inclusion-exclusion principle if all LBP of the state are known [5], [13]. Xue [14] presented an algorithm to search for a family of LBP-candidates by applying discrete function theory, modular decomposition, and system enlarging. Lin et al. [9] proposed a method that generated a smaller size of a family of LBP-candidates. The proposed methods found LBP-candidates by solving sets of inequalities imposed by the structure of the network, and the flow-conservation law. The implicit enumeration method was used to solve the sets of inequalities to obtain LBP-candidates. The candidates were compared to each other, and the greater candidates were eliminated. All LBP were obtained after the comparison process. Note that the sets of inequalities could not be set up properly without knowing all binary minimal paths in advance, while searching for binary minimal paths was unfortunately an NP-hard problem. Lin [15] modified the method to be used for a multi-state network with node failure. Ramirez-Marquez et al. [10] proposed an algorithm to find potential LBP using the information sharing approach. The potential LBP were enumerated using the concept that a select number of network arcs shared information among each other. The number of the potential LBP exponentially grew as the number of arcs and the arcs states increased. Some of the enumerated potential LBP were not feasible, so the verification process was required. Similarly, as in Lin et al. [9], and Lin [15], all LBP were obtained after implementing the comparison process. For a large network with many LBP, the inclusion-exclusion principle may not be efficient to compute the exact reliability, and it may even consume more computational efforts than the complete enumeration method. Using the inclusion-exclusion principle, a subset of LBP can be used to compute lower reliability bounds (LRB) as presented in Hudson & Kapur [11]. At a certain number of LBP, computing reliability bounds gives partial information about the reliability value with less computational effort comparing to the complete enumeration method. Because it is not required to have all LBP to compute reliability bounds, the subset of LBP can be used to obtain partial

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information on the reliability value. The main purpose of this paper is to present an algorithm to find a subset of LBP for the computation of LRB. The LRB can be obtained using LBP with less computational effort compared to the existing algorithms. The proposed algorithm is applied to an example given in Lin et al. [9], and Ramirez-Marquez et al. [10] to compare computational efficiency. Another larger network example is presented to show the significant reduction of the reliability bounds computation time. II. MODEL FORMULATIONS, AND PRELIMINARIES consists of a finite set of nodes , A network and a set of arcs . Nodes, and arcs are numbered as , and . To avoid any confusion, we use , and to refer to a node, and an arc, respectively. The sets of nodes, and arcs can be also written as , and . Each arc is , and we use the notation assigned to a pair . The arc is said to be incident to the initial node , and the terminal node in which the nodes , and are said to be adjacent to each other. For a two terminal network, we assign , and to be a source node , and a sink node , respectively. For a fixed order of arcs, can be regarded denotes a quantity of material as a vector of flow in which flowing in the arc . is an integer value which ranges over a . The flow conservation law states that capacity interval the total flow into must be equal to the total flow out of , and . the flow is said to be conserved at node for all for all , and satisA flow in is feasible if fies the flow conservation law. In this paper, we consider a two-terminal network with one source node , and one sink node . The network performance under consideration is the maximal amount of quantity (maximal flow) which the network can carry from to with respect to capacities at all arcs. This problem is equivalent to the famous well-known max flow problem, where the objective is to maximize the flow from to in which the flow is conserved at all nodes but and , and the flow is feasible with respect to capacities at all arcs. The maximal flow is the net amount flowing from to , which equals the amount of flow departing at and arriving to . We consider a network consisting of perfectly reliable nodes, and multi-state arcs. The state of an arc corresponds to the capacity of the network component that is assumed to take on non-negative integer random variable with a known distribution. The capacities of arcs are statistically independent to each other. The maximum capacity (maximum state) of arc is denoted by . A component state vector indicates the current states of all arcs in a network, where denotes the current state of arc , . The state of a network is determined by the maximal amount of flow that can be sent from to under the capacities of arcs in . The yields the state of a network under , structure function where is the highest possible maximal flow (maximum state) of the network. We define an equivaas a set of component state vectors that have lence class as the maximal flow. Let be the component state space, .

SATITSATIAN AND KAPUR: AN ALGORITHM FOR LOWER RELIABILITY BOUNDS OF MULTISTATE TWO-TERMINAL NETWORKS

We have

for

, and

.

. If , and go to Step 1.

, go to Step 2.

B. Minimal Improvement Paths Algorithm (MIP Algorithm)

III. DEVELOPMENT OF THE ALGORITHM The algorithm presented in this paper is developed using the idea from the grand improvement procedure (Dinic-Karzanov approach) as presented in Rockafellar [16] to find the minimal improvement paths for the required flow. In this section, we first describe the multi-routing algorithm, and the minimal improvement paths algorithm (MIP algorithm) that will be used in the proposed algorithm. Next, we show that the MIP algorithm can be used to find an LBP of a multi-state two-terminal network. A. Multi-Routing Algorithm Using this algorithm, nodes in a network are classified to be in a sequence of sets where denotes the number of arcs that are needed so that a node can be reached from , and denotes the maximum number of arcs so that can be reached from . In other words,

(1) , and Note that , we denote For each all the possible ways by which .

Step 4 Update Else let

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for a two-terminal network. as a set of arcs that gives can be reached from nodes

(2) We can reach the sink node from the source node using at most arcs, and the path must pass consecutive nodes in using arcs in for . The path is considered to be one of the -paths. A multi-routing represents a whole family of all -paths. for , The multi-routing algorithm that yields for all is given as follows: and Initialize: Set , , and for all . , TERMINATE [the multirouting Step 1 If is obtained]. Else, let , and go to Step 2. Step 2 Let be a set of the first element in . If , go to Step 4. Else, go to Step 3. . Step 3 Let If there is an arc such that , and ; and there is an arc such that , and for . where is the — Let terminal node of , and is not in . , and go to Step 4. — Update Else, go to Step 4.

The MIP algorithm is used to trace back a way from the sink node to the source node by way of the multirouting that is generated along with a sequence of sets obtained from the multi-routing algorithm. We consider the amount departing at node as supplies that can send to others. We define the demands at as the amount that needs from other nodes to be conserved. Let indicate demands at node in which for any conserved node, based on the conservation law. We consider a two-terminal network in which the source node can provide , and the sink node has unlimited unlimited supplies, demands, . The algorithm begins with trying to satisfy the demands at as much as possible through any possible arcs with capacity constraints from any nodes adjacent to (nodes in ). The nodes that supply to then need some supplies ) to be conserved. This from their adjacent nodes (nodes in can be viewed as the demands at are transmitted through its adjacent nodes until the demands reach . After the demands reach , there might be some nodes that are not conserved after the transmitting demands process. Then, the excess demands at are rejected to any of their adany unconserved nodes in where the rejecting demands process bejacent nodes in gins with the nodes closest to . After that, the demands are rethrough down to by any possible transmitted from arcs with excess capacities. If the demands that are retransmitted , the demands are create any excess demands at the nodes in again rejected. This procedure is repeated until the termination condition is satisfied. The algorithm terminates upon reaching (necessarily from ), when no additional demands can to , and all the nodes in the sets be transmitted from are conserved, except at & . After the termination, we obtain that indicates the quantity that is required at arc to maximize the flow of the network under the given limited capacities at all arcs, as in for . If we reduce for any by any amount, the flow of the network that is maximized by will be reduced. as “the minimal improvement We denote paths vector (MIP vector)” where we can use at once to make a major improvement in the flow modification procedure to get a maximal flow under limited capacities at all arcs. The formal description of the MIP algorithm as described previously is given as follows. For the algorithm implementaas a set of recording pairs at for tion, we define in which is the demands transmitted to from . This is to record the amount of demands transmitted to any node in the order of arrival. Initialize: Set for all . (where is obtained from the multi-routing algorithm). Let , , and for all , and . Set for , and Step 1 [Transmitting the demands]

.

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1.1) Let

. If there exists such that , then let (from the multi-routing algorithm), and go to Step 1.2. Else go to Step 1.3. such that 1.2) If there exists for , and , calculate , and update , and . Let , and return to Step 1.1. Else go to Step 1.3. . If , then repeat Step 1.1. Else, 1.3) Let . go to Step 2 Step 2 [Inspecting for the un-conserved node] . 2.1) Let . 2.2) Let such that for 2.3) If there exists , and there exists such that all , then go to Step 3. Else, go to Step 2.4. . If , then repeat Step 2.2. 2.4) Let TERMINATE [the maximal flow Else is obtained]. Step 3 [Rejecting the demands] , . 3.1) Take the last element in , calculate ; 3.2) For , , and and update . , then replace back 3.3) If , and label as blocked. Go to Step to , delete from , and 4. Else go to Step 4. Step 4 [Retransmitting the demands] for , and 4.1) If there exists such that where , then go to Step 4.2. Else let , and go to Step 2.2. for 4.2) If there exists such that , , and is not “blocked”, . Update then calculate , , and . . Repeat Step 4.2. Let Else, go to Step 2. C. Preliminaries be a component state vector of a Let multi-state network with multi-state arcs. We denote as a network associated with in which the quantity is the quan. tity of material allowed to flow in an arc for all When the MIP algorithm is implemented, is limited to a capacity interval , that is . Lemma 1: Let be a component state vector of the multistate be a network associated with . For some network, and where the max flow of equals , if is the MIP vector of the network obtained by the MIP algorithm, is also the MIP vector of the network . , if the max flows of and are Proof: For some the same, the MIP algorithm will ignore the excess capacity at at the rejecting demand process. The MIP vector obtained will be the same from two different vectors. obtained from the Lemma 2: Let be the MIP vector of , the max flow of is less MIP algorithm. For any vector than the max flow of .

Proof: The MIP algorithm takes all the excess capacity from , and gives out the MIP vector, . The max flow obtained from , and are equal. Because is minimal, reducing any amount from any for all causes less max flow. obtained from the Theorem 1: If is the MIP vector of , then is one of the LBP of state MIP algorithm, and . Proof: For any vector , the max flow of is less by Lemma 2. Therefore, . For any vector than , for any vector by Lemma 1. D. Proposed Algorithm The rationale behind the algorithm is to reduce the search space by using the MIP algorithm to search for LBP. Then, we eliminate all vectors that are greater than LBP from the search space. For any state , we first identify a vector in state , and use the MIP algorithm to search for an LBP of state as described in Theorem 1. Let be a set of all LBP of state . . By The LBP obtained from can be written as Theorem 1, we have . Any vectors that are greater than can be eliminated from the search space. Each LBP of any state cannot be compared as less than or greater than each other. Let us consider the situation that we can find any two vectors, , and , that cannot be compared but are both in . Then, we can use the the same state MIP algorithm to search for two different LBP using , and . We name such vectors, , and , as “seed vectors.” If we can find the seed vectors that are in state , we can apply the MIP algorithm to the vectors to obtain different LBP of state as described previously. Seed vectors can be obtained by using the prior knowledge on binary minimal cuts. From the maximal flow-minimal cut theorem, the value of the maximal flow of a network is equal to the capacity of the minimal cut-set in the network [17]. At the beginning of the proposed algorithm, we identify all combinations of the states of the arcs in a given minimal cut. These combinations should give us the required amount of maximal flow (the state of the network). We obtain the seed vectors by using these combinations as capacities of the arcs in the minimal cut, and any arc that is not in the minimal cut is set at its maximum capacity . The vectors that are greater than the identified seed vectors are eliminated from the search space at this stage. We may use more than one minimal cut to eliminate more vectors from the search space. For any state , we denote a seed vector that is obtained as described previously. Given the seed vectors, the MIP algorithm is used to find LBP. Upon implementing the MIP algorithm, an MIP vector can be obtained. By Theorem 1, we show that the MIP vector obtained is one of the LBP of state if . Because is the MIP vector, the capacities of any minimal cuts are equal. Therefore, the capacity of (the state of ) can be obtained by finding the capacity of any minimal cuts. By Lemma 1, any vectors that are greater than the LBP are also eliminated from the search space at this stage. Note that we use only two binary minimal cuts in this algorithm. More binary minimal cuts can be used to get more seed vectors. The proposed algorithm is presented as follows.

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Step 1— [Identification of seed vectors]: Let be a set of all binary minimal cuts of a network. For , , identify all combinations of state such that ; and represent each combination using for , and is from 1 to the maximum number of possible combinations where for all otherwise.

(3)

Fig. 1. Network for example 1. TABLE I ARC STATE PROBABILITIES

Identify all combinations of state such that ; and represent each combination using for , and is from 1 to the maximum number of possible combinations where for all otherwise.

(4)

Identifyall combinations of and ; and for each combination, define for , and is from 1 to the maximum number of possible combinations where for all for all otherwise.

for where Let is number of all LBP of state . The reliability of a multistate network can be defined as the probability that a network capacity satisfies a specific demand level , that is

(5)

We define matrices , , and consisting of columns , , and on their -th, that has vectors -th, and -th rows respectively. Step 2—[Implementation of the MIP algorithm to obtain LBP of ]: Initialize 2.1) If , TERMINATE [LBP of are contained in ]. Else, go to Step 2.2. 2.2) If for any , then eliminate , and return to Step 2.1. Else, implement the MIP algorithm to obtain . and , go to 2.3) Check if Step 2.4. Else go to Step 2.5. , and repeat 2.4) is an LBP of state . Set Step 2.1. , and return to Step 2.1. 2.5) Eliminate IV. RELIABILITY BOUNDS COMPUTATION USING LOWER BOUNDARY POINTS Using the inclusion-exclusion principle, Hudson & Kapur [11] generate a sequence of lower reliability bounds (LRB) that are monotonically increasing with the value between 0 and 1. The first member of a sequence of LRB is computed using one LBP. The rest of the LBP are added up one by one to calculate the following members of the sequence. The final member of the sequence is the exact reliability value. We apply the Hudson & Kapur’s approach to compute LRB from the subset of LBP that is obtained from the proposed algorithm.

(6) We can compute LRB using the subset of LBP that is obtained from the proposed algorithm. We define as the lower reliability bounds of state , , where is the number of LBP obtained from the proposed algorithm. can be computed using the inclusion-exclusion principle as (7) where

. V. NUMERICAL EXAMPLES

A. Example 1 We demonstrate the proposed algorithm using a network as in Fig. 1. The network is first analysed in Lin [15]. Ramirez-Marquez et al. [10] use the same network to demonstrate their algorithm. Each node in the network is assumed to be perfectly reliable, and unlimited flow can be sent through all nodes. We and to obtain use two binary minimal cuts seed vectors. The capacities of arcs take integer values as , , , , , and . The maximal flow of the network takes any integer values from 0 to 5: . The probabilities of being in a particular state of all arcs (the capacities of arcs) are shown in Table I. The probabilities that the network is in a particular state (the maximal

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TABLE II NETWORK STATE PROBABILITIES

TABLE III LOWER RELIABILITY BOUNDS (LRB)

flow of the network) are computed by enumerating 2,304 component state vectors, and are shown in Table II. The following section illustrates how the proposed algorithm is executed. Step 1—[Identification of seed vectors]: List all the minimal cuts that we consider: Fig. 2. Network for example 2.

For

,

For

,

Step 1—[Implementation of the MIP algorithm to obtain LBP of state ]: After implementing the MIP algorithm, contains the LBP of state as follows. For

For

TABLE IV ARC PROBABILITIES

can be computed Using Equation (7), LRB of state using LBP obtained from the proposed algorithm as shown in Table III. Comparing to the algorithms presented in [9] and [15] that require all binary minimal paths, the subset of LBP can be obtained using the proposed algorithm without knowing all binary minimal cuts. Also, the proposed algorithm does not require the comparison process as in The proposed algorithm does not require all binary minimal cuts as in [9] and [15]. The proposed algorithm is superior for the LRB computation than the algorithm in Ramirez-Marquez et al. [10] because the number of seed vectors generated are significantly less than the potential LBP generated in Ramirez-Marquez et al. [10]. Therefore, the search space in the proposed algorithm is less. However, the proposed algorithm depends on the prior knowledge of binary minimal cut sets while the algorithm in Ramirez-Marquez et al. [10] can be executed without this prior knowledge. Note that, using two binary minimal cuts, the LRB obtained equal the exact reliability values. B. Example 2 Consider a network with 10 multi-state arcs, and perfectly reliable nodes (Fig. 2). The probabilities that arcs are in different states are shown in Table IV. There are 103, 680 component state vectors of the multistate network. These vectors are categorized into six different states, where the state number indicates the maximal flow that can be obtained from the arc capacities as in the component state vectors. The probability that a network

SATITSATIAN AND KAPUR: AN ALGORITHM FOR LOWER RELIABILITY BOUNDS OF MULTISTATE TWO-TERMINAL NETWORKS

TABLE V SYSTEM STATE PROBABILITIES

TABLE VI LOWER RELIABILITY BOUNDS (LRB), AND SEED VECTORS (SV) WITH DIFFERENT NUMBER OF BINARY MINIMAL CUTS USED

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The results show that the 2-cuts LRB can be obtained using extremely short amounts of time, however the LRB obtained at some particular states are not acceptable to estimate the network reliability. The 3-cuts LRB that gives informative reliability bounds can be obtained with significantly less computational time compared to the complete enumeration method. Using four binary minimal cuts, the LRB obtained equal the exact reliability value, however it consumes more computational time than the complete enumeration method. Note that the 4-cuts LRB computational time of state 3 is much longer than the complete enumeration due to the number of LBP. As shown in the results, the LRB value is improved when using more binary minimal cuts. VI. CONCLUSION

TABLE VII COMPUTATIONAL TIME (SECONDS) AND NUMBER OF LBPS

is in one of the six different states, and the number of vectors in each state, are shown in Table V. The exact reliability computation using the complete enumeration method is tedious due to many number of component state vectors. The computation of reliability bounds is considered as an alternative to provide information about how likely the network is in a particular state. Using different number of minimal cuts, we can compute LRB using LBP obtained from for all the proposed algorithm. Table VI shows , and the number of seed vectors (SV) obtained from the different number of binary minimal cuts used in the proposed , and algorithm. The first two minimal cuts used are . The third, and fourth minimal cuts are , respectively. and We implemented the proposed algorithm in a MATLAB program, and solved the example with a Pentium 4 2.80 GHz with 512 MB of RAM. The program execution time to obtain the exact reliabilities is 2,440 seconds. This time is used to generate all possible component state vectors, and to enumerate all of the probabilities so that the reliability values can be obtained. Table VII shows the LRB computational time for a different number of minimal cuts. This time includes the time to search for LBP with given seed vectors, and to calculate LRB using the inclusion-exclusion principle. The number of LBP used to calculate LRB are also shown in Table VII.

We propose an algorithm to search for a subset of lower boundary points (LBP), and use them to evaluate the reliability of a network in terms of lower reliability bounds (LRB). The advantage of our algorithm is to significantly reduce computation time in order to get partial information on the reliability value. Unlike any existing algorithms, the proposed algorithm gives the subset of LBP without any comparison process. The subset of LBP is used to compute LRB by the inclusion-exclusion principle. We show that the proposed algorithm is more efficient than the existing approaches for the reliability bounds computation. There are also some limitations of the algorithm. One is the assumption that the prior knowledge on some of the network binary minimal cuts is available. Another limitation is that the LRB computational time might exceed the complete enumeration method when computing the LRB using the inclusion-exclusion principle with high number of LBP. The results from the numerical examples show that the exact reliability values can be obtained without using all binary minimal cuts. Therefore, it is interesting as further research to improve the algorithm for the exact reliability computation. This is applicable for a network in which the number of LBP is not too large to implement the inclusion-exclusion principle. The LRB are improved when using more binary minimal cuts. Hence, another area of the further research is how binary minimal cuts should be selected so the LRB can be improved thoroughly. REFERENCES [1] R. Billinton and W. Zhang, “State extension for adequacy evaluation of composite power systems—applications,” IEEE Transactions on Power System, vol. 15, pp. 427–432, 2000. [2] A. Lisnianski and G. Levitin, Multi-State System Reliability: Assessment, Optimization, and Applications, A. P. B. M. Xie and T. Bendell, Eds. : World Scientific Publishing Co. Pte. Ltd. [3] T. Aven, “Reliability evaluation of multistate systems with multistate components,” IEEE Transactions on Reliability, vol. R-34, pp. 473–479, 1985. [4] R. A. Boedigheimer and K. C. Kapur, “Customer-driven reliability models for multistate coherent systems,” IEEE Transactions on Reliability, vol. 43, pp. 46–50, 1994. [5] J. C. Hudson and K. C. Kapur, “Reliability analysis for multistate systems with multistate components,” IIE Transactions, vol. 15, no. 2, pp. 127–135, 1983. [6] B. Natvig, “Two suggestions of how to define a multistate coherent system,” Advances in Applied Probability, vol. 14, pp. 434–445, 1982. [7] S. Patra and B. Misra, “Reliability evaluation of flow networks considering multistate modeling of network elements,” Microelectronics and Reliability, vol. 33, no. 14, pp. 2161–2164, 1996.

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[8] ——, “Evaluation of probability mass function of flow in a communcation network considering a multistate model of network links,” Microelectronics and Reliability, vol. 36, no. 3, pp. 415–421, 1996. [9] J. S. Lin, C. C. Jane, and J. Yuan, “On reliability evaluation of a capacitated-flow network in terms of minimal pathsets,” Networks, vol. 25, pp. 131–138, 1995. [10] J. E. Ramirez-Marquez, D. W. Coit, and M. Tortorella, A generalized multistate based path vector approach for multistate two-terminal reliability Under Review, IIE Transactions. [11] J. C. Hudson and K. C. Kapur, “Reliability bounds for multistate systems with multistate components,” Operation Research, vol. 33, no. 1, pp. 153–160, 1985. [12] W. C. Yeh, “Revised layered-network algorithm to search for all d-minpaths of a limited-flow acyclic network,” IEEE Transactions on Reliability, vol. 47, pp. 436–442, 1998. [13] E. El-Neweihi, F. Proschan, and J. Sethuraman, “Multistate coherent systems,” Journal of Applied Probability, vol. 15, pp. 675–688, 1978. [14] J. Xue, “On multistate system analysis,” IEEE Transactions on Reliability, vol. R-34, pp. 329–337, 1985. [15] Y. K. Lin, “A simple algorithm for reliability evaluation of a stochastic-flow network with node failure,” Computers and Operations Research, vol. 28, pp. 1277–1285, 2001. [16] R. T. Rockafellar, Network Flows and Monotropic Optimization. : Athena Scientific, 1998. [17] M. S. Bazaraa, J. J. Jarvis, and H. D. Sherali, Linear Programming and Network Flows, 2nd ed. : John Wiley and Sons, Inc, 1990.

Sarintip Satitsatian received the B.Eng. degree in mechanical engineering from the King Mongkut Institute of Technology at Ladkrabang (KMITL), Thailand, and the M.S. degree in industrial engineering from the University of Washington. She is currently working toward the Ph.D. degree in industrial engineering at the University of Washington. Her dissertation research focuses on multi-state network reliability modeling for infrastructure networks.

Kailash C. Kapur received the Ph.D. degree in industrial engineering from the University of California, Berkeley, CA, in 1969. He is a Professor of industrial engineering at the University of Washington, where he was also the Director from 1993 to 1999. He has coauthored the book Reliability in Engineering Design (John Wiley & Sons). He has written chapters on reliability and quality engineering for several handbooks. He has published over 60 papers in technical, research, and professional journals. His research focuses in the areas of quality engineering, design reliability, industrial experimental design, system optimization and control, and productivity improvement. He is a member of INFORMS, IIE, and ASQ. He is on the editorial board of journals such as Quality Engineering and TRANSACTIONS OF THE INSTITUTE OF INDUSTRIAL ENGINEERS.