An Improved Sum-of-Disjoint-Products Technique for ... - IEEE Xplore

IEEE TRANSACTIONS ON RELIABILITY, VOL. 64, NO. 4, DECEMBER 2015

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An Improved Sum-of-Disjoint-Products Technique for Symbolic Multi-State Flow Network Reliability Wei-Chang Yeh, Senior Member, IEEE

Abstract—Many real-world systems, such as electric power and transportation, satisfy the flow conservation law, and can be considered to be multistate flow networks (MFN). Reliability is a popular index for evaluating MFN performance, and the sum-of-disjoint products (SDP) technique is a popular tool for evaluating MFN reliability. An improved SDP (iSDP) is proposed in this study by adding simplification procedures to reduce the number of multiplications and summations to increase the efficiency of the SDP; the validity and effectiveness of the iSDP is also demonstrated in this study. The iSDP's computational complexity is also analyzed, and one benchmark example is given to illustrate how MFN reliability is determined using the proposed iSDP method. In addition, computational experiments with two MFN are conducted to test the performance of the iSDP method, along with comparisons to the existing best-known SDP method. Index Terms—Multistate flow network, network reliability, simplification procedure, sum-of-disjoint products technique.

MP

Multi-state flow network, which satisfies the conservation law Binary-state flow network, which satisfies the conservation law Minimal Path

MC

Minimal Cut

BFN

-MP

-Minimal Path

-MC

-Minimal Cut

DP

Disjoint Products

SDP

Sum-of-DP Technique

RSDP

Recursive SDP

iSDP

Improved SDP

GCF

Greatest Common Factors

,

forms of complements of GCF among all products or elements in elements in vector

but not in

elements of without counting the arcs in . network reliability such that units of flow can be successfully transferred from nodes 1 to number of all -MP between nodes 1 and probability of event

occurs

largest elements in set smallest elements in set is a state vector, where

, where

, and

where

, and

,

where

NOTATIONS number of elements in set , e.g., number of nodes in number of arcs

current state (capacity value) of

th -MP, where

ACRONYMS AND ABBREVIATIONS MFN

connected network with the node set , the arc set , and the vector , which denotes the maximum capacity of arcs. th arc

is the

number of vertices Manuscript received December 20, 2013; revised December 10, 2014, April 07, 2015, and May 19, 2015; accepted May 24, 2015. Date of publication September 03, 2015; date of current version November 25, 2015. This work was supported in part by the National Science Council of Taiwan, R.O.C. under grant NSC101-2221-E-007-079-MY3. Associate Editor: M. Zuo. The authors is with the Integration and Collaboration Laboratory, Department of Industrial Engineering and Engineering Management, National Tsing Hua University, Hsinchu 300, Taiwan (e-mail: [email protected]). Digital Object Identifier 10.1109/TR.2015.2452573

,

iff

for all

iff

for some

iff in

and

, , and

for at least one and

, where

0018-9529 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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, , and is the maximum state or maximum capacity of component , and

for all

NOMENCLATURE Flow conservation law MP MC Dominated Parent vector Sibling vectors -MP

Reliability

NP-hard

#P-hard

The net flow into and out of a node, not including the source and target nodes, is equal to zero. An edge set such that if any edge is removed from this set, then the remaining set is no longer an MP. An edge set such that if any edge is removed from this set, then the remaining set is no longer an MC. If vectors , then is dominated by . is a parent vector of , where . vectors with the same parent vector A system-state vector is a -MP iff the total flow from nodes 1 to is , and no other distinct -MP, say , s.t. in . The probability that the required amount of flow (say ) can be transmitted successfully from nodes 1 to , i.e., for all -MP . A problem is NP-hard iff there is an NP-complete problem that is polynomial time Turing-reducible to such a problem [1], [2]. A problem is #P-hard iff there is a #P-complete problem that is polynomial time Turing-reducible to such problem [1], [2].

ASSUMPTIONS An MFN satisfies the following assumptions. 1. Each node is perfectly reliable. 2. All flow in the network obeys the conservation law. 3. The capacity of each arc is a non-negative integer-valued random variable based on a given distribution, and is assumed to be statistically independent. 4. The graph is connected; otherwise, the reliability is 0 if there is no path between the source and sink nodes. 5. The graph is free of self-loops.

I. INTRODUCTION

B

ECAUSE networks in today's world are ubiquitous, designing reliable networks and assessing their performance in a dynamic way is critical to achieve and maintain the required quality of service in industrial, domestic, and defense applications. Many real-world networks satisfy the flow conservation law, and can be treated as multistate flow networks (MFNs); these networks include computer and communication systems [3], power transmission and distribution systems [2], [5], [29], transportation systems, and oil and gas production systems [5], [6], to name a few. In recent years, network reliability theory has been applied extensively to MFNs for the design of new systems, and the improvement or further development of existing systems. Hence, evaluating an MFN's reliability is important, and many studies over the past several decades have been devoted to solving the related problems. It has long been known that nearly all network performance measurements are computationally intractable (i.e., NP-hard) [1], [2], [26]–[28]. The methods used to evaluate MFN reliability typically utilize a variety of tools for system modeling and reliability index calculation. Among the most popular tools are network-based algorithms based on either -minimal path ( -MP) or -minimal cut ( -MC) [16]–[25], where is the required quantity at the sink node. MFN reliability such that at least units of flow can be successfully sent from a source node to a sink node is defined as

(1) where is a non-negative, integer-valued flow requirement for a given problem, and is usually a random variable, whose distribution can be determined via continuous observation and prediction. Each state vector is called a -MP in (1). Once all -MPs have been found, the computation of (1) is examined; this topic has already been the subject of many studies. The most convenient technique discussed in the literature is the SDP method, which was initially discussed by Abraham [7]. It has been demonstrated that the SDP method has a greater efficiency than the other available methods [8]–[12], which include the state-space methods [2], the disjoint method [13], and the inclusion-exclusion methods [14], [15] in binary-state networks, in which arcs are either working or failed. The various versions of the existing SDP methods all involve a Boolean expansion and minimization after each DP is formed [7]–[12]. The recursive SDP (RSDP), which recursively implements SDP, is currently the most popular SDP method for use in MFN reliability problems [12]. As expected, the run times

YEH: AN IMPROVED SUM-OF-DISJOINT-PRODUCTS TECHNIQUE FOR SYMBOLIC MULTI-STATE FLOW NETWORK RELIABILITY

for all of the existing SDP methods find that DP increases exponentially as the number of -MP or -MC increases. Thus, a simple, efficient, effective SDP method is needed to evaluate the general MFN reliability after finding all -MPs. The purpose of this study is to develop a more efficient, intuitive algorithm than the existing SDP methods. This paper is organized as follows. Section II presents a formal introduction to the SDP and RSDP methods. Section III contains preliminary findings about the relationship between the different types of DP, the special characteristics of DP, and a discussion of the theorems and properties. Section IV presents the proposed iSDP method that improves the SDP method using the known -MP, along with a discussion of the method's efficiency and effectiveness. Additionally, in Section IV, the proposed iSDP method is demonstrated with a benchmark example to show how MFN reliability is computed using known -MPs. In Section V, a more comprehensive empirical study that compares the proposed iSDP method with the existing RSDP method illustrates the advantages and improvements of the iSDP method. Concluding remarks are provided in Section VI.

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3) Property 3: (7) If

, because

, and

(8) then we have the following property. 4) Property 4:

(9) Because rewrite Property 3 as follows. 5) Property 5:

, we can

II. THE SDP AND RSDP METHODS The basic premise of the SDP method will be illustrated and derived in detail in this section. Given that all -MPs have been found, , (1) can be rewritten to evaluate the probability of the union of events where the state vector is greater than or equal to at least one of the -MPs:

(10) Developed from the fundamental concepts of set theory, the formulation of the SDP method is 6) Property 6: (11)

(2) Equation (2) can be further revised by methodically counting the union of possibly non-disjointed sets in terms of the intersections of all subsets:

Zuo et al. [11] described an alternative procedure, the so-called RSDP method, for implementing SDP recursively. The basic idea of RSDP is that the probability of a union with vectors can be calculated by calculating the probabilities of several unions with vectors or less as summarized below; this concept is based on the SDP principle. Theorem 1:

(3) Because each product in (3) is disjointed,

where (12)

(4) , then a simplifying procedure If based on set theory can be used in the SDP or RSDP methods to reduce, whenever possible, the number of DPs in (4). 1) Property 1: (5) Because the capacity of each arc is assumed to be statistically independent, we have the following two properties. 2) Property 2:

(6)

(13) From the investigation of the RSDP method in [11], the RSDP method is more efficient than the other available methods when there are more than 15 arcs in the system. Hence, the RSDP method is selected for comparison with the proposed iSDP method. III. PRELIMINARIES Before introducing the proposed iSDP method to investigate its improvements to the SDP method when calculating MFN reliability, a number of useful properties and results are described in this section. The following comes directly from the definition

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TABLE I ALL VECTORS IN

and characteristics of , which plays a major role in the proposed iSDP method for any . Property 7: 1. , , and ; 2. is the last vector in , where ; 3. and are sibling vectors with the same parent vector ; 4. the total number of vectors in is , and the total vectors are in ; and 5. , and

Fig. 1. Example MFN. TABLE II PROBABILITY DISTRIBUTION OF THE ARC CAPACITIES IN FIG. 1

. For example, Table I lists all vectors including sibling and parent vectors in for and ,2,3,4,5. Note that the vectors grouped by dots are sibling vectors (e.g., , , and ), and the vector above the sibling vectors is the related parent vector (e.g., is the parent vector of , , and ). A. SDP Based On Based on , Theorem 1 is written in direct form without using the recursive formula in the following theorem. Theorem 2:

(14)

STEP 4. Let

, and ,

then go to STEP 3. STEP 5. Calculate STEP 6. If , let Otherwise, halt.

. , and go to STEP 2.

Consider the well-known bridge network shown in Fig. 1 with , , and , where nodes 1, and 4 are the specified source node, and sink node. , , and are all known 3-MPs. The capacity of each arc in the bridge network is given in Table II. The above algorithm is demonstrated below. Solution procedure:

The overall procedure for implementing Theorem 2, including the construction of , is summarized in the following Algorithm 1.

STEP 1. Let

, and

Algorithm 1. Find the exact symbolic MFN reliability based on known -MP and Theorem 2. Input: A connected graph with the node set , the edge set , a source node 1, a sink node , and all -MPs . Output: A network reliability . STEP 1. Let , and calculate . STEP 2. Let , , and . STEP 3. If no sibling vectors in , go to STEP 5.

(15) where

(16)


TABLE III ACCUMULATED PROBABILITY OF THE ARC CAPACITIES IN FIG. 1

(17)

(18) (19) STEP 2. Let STEP 3. Because STEP 4. STEP 4. Let

,

, and . are sibling vectors, go to

and ,

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Thus, for ; however, is obtained recursively using ; and the proposed is obtained directly without any information about . B. Remove Dominated Vectors The following property directly follows from set theory to improve Theorems 1 and 2 by removing these dominated elements. 1) Property 8: If , then and all of its offspring can be removed from without affecting the final reliability. Proof: Consider first. From Property 7, we have

(20) (26) and go to STEP 3. STEP 3. Because no sibling vectors are present in go to STEP 5. STEP 5. Let

,

, we have

If

;

can be removed from

.

Thus, Property 8 is true for . Assuming that , , and is true, then , and all of its offspring can be removed from . If if otherwise

(21) STEP 6. Because

, and then we have According to the above assumption, we also have

, halt.

(27) .

Note that, in [11], (22)

and

; and all of its offspring can be removed. Thus, is also true. Property 8 is always true. For example, in Table I can be expanded:

(23)

(28) If

, then (28) can be simplified: (29)

If

, then (28) can be simplified: (30)

(24) and (25)

Note that the RSDP method also includes a similar simplification technique; however, the RSDP method does not provide a clear explanation of it, particularly when it is used in the recursive form. Based on the above discussion, it is evident that this process is time-consuming and burdensome even if Property 8 is used in the SDP method to calculate the network reliability

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TABLE IV SUBGROUPS, IDENTIFIED WITH DOTTED RECTANGLES, IN CALCULATING

using Theorem 2. Thus, we now shift our focus to reduce the number of multiplications in each DP in the remainder of this study. C. Accumulated Probability The next property is implemented in the proposed iSDP method to reduce the numbers of multiplications and summations in calculating DPs, of which some states are at a lower level. 1) Property 9: , for all . To fully exploit Property 9, it is used with the calculated ; for example, the accumulated probability of the arc capacities in Fig. 1 and Table II is listed in Table III. If each is calculated in advance, and Property 9 is also used, then (16) can be simplified:

, and in Table IV, which has been simplified using Property 8. Note that this case may occur when . 3. If there are four or more vectors in , four vectors including the parent vector, two sibling vectors obtained from this parent vector, and a vector generated from the intersection of these two sibling vectors are grouped to calculate the probability of this subgroup. Note that all subgroups are also disjointed. For example, , are both subgroups in Table IV. 1) Property 11: If is a disjointed subgroup discussed in case 2, then

(31)

(34)

The numbers of multiplications and summations are thus reduced from 6 to 4, and 3 to 0, respectively, by Property 9, and pre-calculating . Even in this simple example, a significant improvement is made. 2) Property 10: (32)

, we have Proof: Because . Additionally, , and . Thus, this property is true. In the same way, we have Property 12. 2) Property 12: If is a disjointed subgroup in case 3, then

This section proposes a novel concept called greatest common factors (GCFs). The GCF of two or more items is the largest factor that divides evenly into each of the factors. For is the GCF of example, and . Thus,

(35)

D. The Greatest Common Factors

(33) The GCF explores some special relationships between the vectors, and plays a key role in reducing the number of multiplications in calculating the values of the DPs. It is also one of the key parameters in this study, and is used in the proposed iSDP repeatedly to reduce the number of multiplications. Theorem 2 can be simplified further after using the GCF concept. To find the GCFs easily, only the following three cases are considered in calculating . 1. If there is only one vector in , ; for example, . 2. If there are only two vectors in (i.e., one parent vector , and one vector generated from this parent vector), then ; for example,

For example, if , and , then the 2nd and 4th through 6th values are the same in both and ; appears in both and , where the underscore indicates a different value between the two. Thus,

(36) ; thus, (36) is simplified as Note that follows after implementing Property 9. (37)

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TABLE V COMPARISON OF THE NUMBERS OF MULTIPLICATIONS AND SUMMATIONS IN CALCULATING

Table Vlists the numbers of multiplications and summations using different simplification techniques, the RSDP method, and the iSDP method when calculating . Note that 1. The number of summations is equal to 2 to calculate each , , , , and . 2. There is no need to calculate , , , and because they have reached the maximum state of their capacity. The next theorem forms the basis of the iSDP method by integrating Properties 8 through 12 into Theorem 2. Theorem 3: (38) IV. PROPOSED ALGORITHM, AND AN EXAMPLE This section outlines the proposed iSDP method for computing the exact symbolic MFN reliability if all -MPs are known in advance. Theorem 3 is used repeatedly in the proposed algorithm to reduce the calculation burden. The detail of the proposed algorithm is described in the following steps.

Property 8 is implemented in step 3, if necessary. Properties 9, 11, and 12 are implemented in steps 0, 1, and 6 to increase the efficiency of calculating . Additionally, is calculated in step 6 based on Theorem 3. The correctness of the proposed iSDP follows directly from Theorem 3 and Properties 8 through 12. The primary complexity of the proposed algorithm is based on constructing and calculating for . The total number of vectors in

is at most 2, and the time

complexity to calculate each vector in is . Thus, we have the following theorem. Theorem 4: The proposed iSDP method locates all corresponding non-dominated DPs, and calculates with the time complexity . Because the MFN network reliability problem is NP-hard, and its computational difficulty grows exponentially with the network size, the example discussed in Section III using Fig. 1 is again selected to demonstrate the proposed iSDP method for calculating rather than investigating a larger network system. The capacity of each arc and the 3-MP of the bridge network are the same as those used in the example discussed in Section III.

Algorithm 2: The proposed iSDP. Solution procedure: Input: A connected graph with a node set , an edge set , a source node 1, a sink node , and all -MPs . Output: A network reliability . STEP 0. Calculate for , and . STEP 1. Let , and calculate based on Property 11. STEP 2. Let , , and . STEP 3. Let

STEP 4. If no DPs are sibling vectors within go to STEP 6. STEP 5. Let , and

then go to STEP 3. STEP 6. Calculate and let STEP 7. If , let Otherwise, halt.

STEP 0. Calculate shown in Table III;

for

, as ; and (see (16) and (31)).

STEP 1. Let , and (see (36) and (37)). STEP 2. Let , STEP 3. Because , let STEP 4. Because no elements within vectors, go to STEP 6. STEP 6. Let

, and . . are sibling

, then

based on Properties 11 and 12, . , and go to STEP 2.

(39) STEP 7. Because

, halt.

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TABLE VI RESULTS FOR THE DATA_10 DATASET WITH DIFFERENT VALUES OF

TABLE VII RESULTS FOR THE DATA_15 DATASET WITH DIFFERENT VALUES OF

V. COMPUTATIONAL EXPERIMENT Experimental results are provided to compare the performance of the RSDP method and the proposed iSDP method applied with known -MPs. Two -MP sets are generated randomly: data_10, and data_15. In these, , and in data_10; and , and in data_15. Three different SDPs are investigated: the RSDP method proposed in [11] without considering the simplification; the proposed iSDP method; and the iSDP* method, which is the proposed iSDP method without Property 9. These methods are programmed in MATLAB, and executed on an Intel CoreTM2 2.1 GHz PC with 2 GB of RAM. Note that iSDP* is similar to RSDP but with Property 8. To provide fair comparisons without being affected by the computer hardware and software, the total number of summations and multiplications required to find are also recorded, where the number of summations is also the number of DPs generated in the experiments. Note that there is no clear explanation as to whether the RSDP method implements the simplification only with a special DP or for every DP for all and ; the number of obtained DPs is thus under-estimated for the RSDP method. Tables VI and VII report the total number of multiplications and summations, and the CPU times using the three different SDP techniques for both data sets. From Tables VI and VII, the following observations are made. 1. The total number of multiplications and summations increases with the values of and , which meet the characteristics of an NP-hard problem. 2. The iSDP* method outperforms the RSDP method using 50% fewer multiplications and 50% less CPU time (in-

cluding CGF and the simplification) in the worst case. Thus, the proposed GCF concept and the proposed simplification (Property 8) method are effective in improving the SDP method. 3. The iSDP method outperforms the iSDP* method; thus, Property 9 is also effective at improving the SDP method, but the improvement is only 10%. As a result, Property 9 is not as significant as the GCF concept with regard to reducing the CPU time, and the number of multiplications. 4. The improved rate increases with , but decreases with . Thus, the efficacy of the proposed Properties 8 through 12 and the iSDP method are all proven based on the computational experiments and discussions in this section. VI. CONCLUSIONS Reliability is usually one of the most important indices in real systems. The SDP method has been shown to be efficient in evaluating MFN reliability in terms of known -MPs; however, determining the total number of enumerated DPs obtained from the known SDP method is tedious, and the total number of multiplications is also large. In this study, a new SDP-based algorithm called the iSDP method, which consists of the SDP method and the simplification procedure discussed in Properties 8 through 12 above, has been developed to overcome significant obstacles in the existing SDP method. The advantages of the iSDP method are based on the proposed algorithm, and are shown in numerical experiments. ACKNOWLEDGMENT The author wishes to thank the editor and reviewers for their constructive comments and recommendations, which have significantly improved the presentation of this paper.


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