Monte Carlo Methods for Reliability Evaluation of ... - Semantic Scholar

IEEE TRANSACTIONS ON RELIABILITY, VOL. 60, NO. 1, MARCH 2011

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Monte Carlo Methods for Reliability Evaluation of Linear Sensor Systems Qingyu Yang and Yong Chen

Abstract—A linear sensor system is defined as a sensor system in which the sensor measurements have a linear relationship to source variables that cannot be directly measured. Evaluation of the reliability of a general linear sensor system is a #P problem whose computational time increases exponentially with the increment of the number of sensors. To overcome the computational complexity, Monte Carlo methods are developed in this paper to approximate the sensor system’s reliability. The crude Monte Carlo method is not efficient when the sensor system is highly reliable. A Monte Carlo method that has been improved for network reliability, known as the Recursive Variance Reduction (RVR) method, is further adapted for the reliability problem of linear sensor systems. To apply the RVR method, new methods are proposed to obtain minimal cut sets of the linear sensor system, particularly under the conditions where the states of some sensors are fixed as failed or functional. A case study in a multistage automotive assembly process is conducted to demonstrate the efficiency of the proposed methods.

number of source variables

Index Terms—Linear sensor system, matroid theory, Monte Carlo method, sensor system reliability.

component state vector, where denotes the state of the th sensor if the th sensor works ( if the th sensor fails) and

matrix obtained from after removing the rows in that correspond to the measurements from the failed sensors minimum allowable number of working sensors number of Monte Carlo samples linear sensor system, where denotes the th sensor a minimal cut set of the sensor denotes the th system, where sensor in

ACRONYMS RVR

Recursive Variance Reduction

MRRS

Minimal Rank Reduction Set

RREF

Reduced Row Echelon Form

MCS-RC

Minimal Cut Set under Restricted Conditions

a set of sensors whose states are set as failed a set of sensors whose states are set as working system state ( system works and

MCS-NRC Minimal Cut Set that is Not under a Restricted Condition

if the sensor if it fails)

system reliability of a linear sensor system, NOTATION

,

estimator of using the crude Monte Carlo method and the RVR method, respectively

,

sample variance of respectively

sensor measurements source signals matrix, each row corresponds to a sensor measurement; each column corresponds to a source variable number of sensors in a linear sensor system Manuscript received June 14, 2009; revised April 27, 2010; accepted August 04, 2010. Date of publication January 17, 2011; date of current version March 02, 2011. This research was supported by the National Science Foundation Grants CMMI-0528735 and CMMI-0726939. Associate Editor: S. H. Xu. Q. Yang is with the Department of Industrial and System Engineering, Wayne State University Detroit, Detroit, MI 48202 USA (e-mail: [email protected]). Y. Chen is with the Department of Mechanical and Industrial Engineering, University of Iowa, Iowa City, IA 52242 USA. Digital Object Identifier 10.1109/TR.2010.2103970 0018-9529/$26.00 © 2011 IEEE

and

,

reliability of sensor , rank of a matrix cardinality of a set sample size of Monte Carlo method sample space of the crude Monte Carlo method, which consists different component state of vectors for a system with components.

306


set of all component state vectors that result in the system failure known component state vectors that ensure the functioning of the system known component state vectors that result in the system failure, unknown component state vectors, ,

probability of respectively probability of

and

,

, which equals

matrix obtained from after deleting columns corresponding to the working sensors and pivoting columns corresponding to the failed sensors ,

computational time of the crude Monte Carlo method and the RVR method, respectively performance comparison between the RVR method and the crude Monte Carlo method, i.e.,

I. INTRODUCTION Sensor systems are commonly used to investigate physical phenomena. Often, the variables representing physical phenomena, herein referred to as source variables, cannot be directly measured. The available sensor measurements are usually combined responses of the source variables. If there is a linear relationship between the sensor measurements and the source variables, the sensor system is called a linear sensor system. With further consideration of sensor noises, (1) where

is the vector of sensor measurements; is the vector of source variables, the values of which are of interest but cannot be directly measured by the sensor is the vector of sensor noises, which, in this system; paper, is assumed to be normally distributed with unknown conis a known constant matrix, indistant variance; and cating the linear relationship between the sensor measurements and the source variables. The matrix is assumed to be of full column rank, i.e., . Linear sensor systems have broad applications in areas including manufacturing fault diagnosis, array signal processing, calibration of wireless sensor networks, and sensor systems in electric power systems. Linear sensor systems are widely deployed in complex multistage manufacturing systems to measure dimensional deviations of selected points on parts,

enabling automatic diagnosis of manufacturing process faults [1]–[3]. In these processes, the sensor measurements are given in terms of dimensional deviations from the nominal; the source variables indicate the process faults, e.g., the deviation of locator pins in an assembly process; and matrix is determined by process design information including product geometry, and tooling layout. Another application of linear sensor systems is in array signal processing [4]–[6]. In array signal processing, sensors, e.g. microphones or radar, are organized in patterns, or arrays, to detect the location of source variables, e.g. speakers or transmitters. Assuming the number of source variables is given, and the signals occupy a narrow range of frequencies, known as narrow-band, a model such as given by (1) can be applied, where indicates the vector of signals received by the array sensors, indicates the vector of source variables, and matrix is determined by the steering for the sensors toward source variables. A linear sensor model is also used in wireless sensor calibration for localization [7], [8], which can be described as follows: in an off-line setting, the true distance between sensor nodes can be measured by an independent, accurate method; then, a mathematical model mapping the measured distance to the true distance is established to estimate calibration parameters. This procedure can be modeled by (1), indicating the vector of true distances, indicating with consisting the vector of calibration parameters, and matrix of measured distance and 0/1 entries. A fourth application of linear sensor systems is in an electrical power system [9]–[11], where the sensor measurements include physical variables such as power flows and power injection, indicates the vector of sources signals’ features, e.g., voltage magnitudes and phase angles, and matrix is determined by the design information such as the position of the sensors. Due to sensor failures, a linear sensor system might lose its capability to identify the source variables. Therefore, it is important to model and evaluate the reliability of a linear sensor system during its design phase. In the literature, the reliability problem of linear sensor systems was investigated when considering different sensor failure modes. In [12], the reliability of a linear sensor system was studied when the sensor has a failure mode of sensor precision degradation, under which situation the system’s working conditions are the same as those of the so-called series-weighted- -out-ofsystems [13]. When applying the existing algorithms to evaluate the exact system reliability, however, an exponential computational time is needed. To overcome the computational difficulty, a weighted interval method is developed in [12] to efficiently evaluate the upper bound and lower bound of system reliability. In contrast, the reliability of a linear sensor system is modeled in [14] when considering catastrophic sensor failures (i.e., when absolutely no useful information was obtained from the failed sensor). Under this situation, evaluating the exact reliability of the linear sensor system reliability is a #P complete problem [14], which is at least as challenging as the corresponding NP complete problem. As a result, the computational complexity increases exponentially with the increment of the number of sensors. In practice, the number of sensors in a linear sensor system could be very large; for example, the coordinate sensor systems applied in complex manufacturing processes generally consist

YANG AND CHEN: MONTE CARLO METHODS FOR RELIABILITY EVALUATION OF LINEAR SENSOR SYSTEMS

of hundreds of sensors, and the wireless sensor system may consist of thousands of sensors. Therefore, to evaluate the exact reliability of a large-size linear sensor system is time-consuming. In [14], efficient algorithms have been developed for exact reliability evaluation for two special linear sensor systems, but an efficient algorithm for reliability evaluation of a general linear sensor system has not been addressed in current literature. To overcome the computational complexity for the reliability evaluation of a general large-scale linear sensor system when considering catastrophic sensor failure, Monte Carlo methods are applied in this paper to efficiently approximate the reliability of linear sensor systems. The crude Monte Carlo method can be straightforwardly applied to the linear sensor system. The crude Monte Carlo method, however, does not efficiently obtain accurate estimates of linear system reliability when the system is highly reliable. In the network reliability literature, improved Monte Carlo methods [15]–[22] have been developed to increase the efficiency of system reliability evaluation, among which the Recursive Variance Reduction (RVR) method offers the best performance [20], [21]. To apply the RVR method for linear sensor systems, a critical requirement is to obtain minimal cut sets of the linear sensor system, especially if some sensors are fixed as failed or functional. This particular aspect has not been studied in the literature. In this paper, both the crude Monte Carlo and the RVR method are applied for the reliability evaluation of the linear sensor system. The remaining portions of this paper are organized as follows. Section II introduces the definition of the linear sensor system. Section III discusses the crude Monte Carlo method applied for linear sensor systems, and Section IV represents the RVR method. In Section V, the developed Monte Carlo methods are compared by using a case study. Finally, Section VI gives the conclusions. II. RELIABILITY DEFINITION OF A LINEAR SENSOR SYSTEM For many applications of linear sensor systems, it is required that each of the source variables be uniquely estimated, e.g., identify the location of all the speakers or transmitters in array signal processing [4]–[6]. Under this situation, the matrix is assumed to have full column rank. Matrix links sensor measurements to source variables. Each row of corresponds to a sensor measurement, and each column of corresponds to a source variable. Therefore, when a sensor fails catastrophically, the corresponding row in should be removed. After removal of the rows corresponding to the failed sensors, the rank of the remaining matrix may be less than the rank of matrix . As a result, the sensor system loses its capacity to estimate all the source variables due to the sensor failures. In addition, due to the existence of measurement noise, the sample size of the statistical method used to identify should be large enough to ensure the desired statistical efficiency. Therefore, it is often necessary to specify a minimum allowable number of working sensors, denoted by . Generally, can have a value between and . In this paper, the reliability of a linear sensor system is defined in Definition 1 as follows. Definition 1 (Reliability of a Linear Sensor System): Given the working probability of each sensor in a linear sensor system,

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and assuming the failure of sensors are -independent, the reliability of a linear sensor system is the probability that rank , and the number of the rows in is greater than or equal to , where is the matrix obtained from after removing the rows in that correspond to the failed sensors. III. CRUDE MONTE CARLO METHODS FOR RELIABILITY EVALUATION OF LINEAR SENSOR SYSTEMS In [14], it was proven that the reliability evaluation of a general linear sensor system is a #P complete problems, which is at least as hard as the corresponding NP complete problem. It is well accepted that an efficient algorithm that can be used to find the exact solution for a large-scale NP-hard problem does not exist. As a result, efficient approximation methods are essential to overcome the difficulty posed by the computational complexity. In general, there are two methods to approximate the system reliability. One is to evaluate the upper and lower bounds of the system reliability. Obtaining tight reliability bounds for a large-scale sensor system, however, still requires tremendous computational time. Using Monte Carlo methods is another way to approximate the reliability of complex systems. The crude Monte Carlo method is applied first to evaluate the reliability of linear sensor systems. Given a linear sensor system , and inconsisting of sensors, denote dicates the th sensor. Let , which is called a component state vector. Then indicates the states of the sendenotes the state of the th sensor ( if sors, where if fails); let be the state of sensor works, and the linear sensor system ( if sensor system works, and if fails). Both , and are Bernoulli random variables. The reliability of , denoted as , equals the expectation of , . The sampling plan of the crude Monte Carlo i.e. method consists of -independent samples. In each sample, -independent pseudorandom numbers are generated to simuwith probability late the component state vector , where ; and equaling the reliability of , denoted as , with probability equaling . Based on the generated , is obtained after removing the rows in that correspond to the simulated failed sensors whose component states equal zero. For the th sample, the system state, which is denoted by , is determined by based on Definition 1. After all the samples are generated, the system reliability can be estimated by an unbiased estimator , i.e., (2) and the sample variance of

is calculated as (3)

The crude Monte Carlo method is not able to efficiently obtain an accurate estimate of system reliability when the system is highly reliable. The inefficiency of the crude Monte Carlo method is illustrated by Fig. 1(a). For a large-scale system with components, the sample space, denoted by , of the crude

308


Fig. 1. (a) Sample space for crude Monte Carlo method. (b) Reduced sample space.

Monte Carlo method is very large because it consists of difbe the set of all compoferent component state vectors. Let is only a nent state vectors that result in the system failure. small portion of if the system is highly reliable. As a result, one will have to take a very large number of random samples to obtain an from to get a sufficient number of samples in accurate estimate of the system failure probability. IV. RVR METHOD FOR RELIABILITY EVALUATION OF LINEAR SENSOR SYSTEMS As discussed in the previous section, in the network reliability literature, the RVR method outperforms other improved Monte Carlo methods. Thus, the RVR method is applied for the linear sensor systems in this paper. The main idea of the RVR method is to obtain sets of component state vectors before Monte Carlo sampling is performed. One set is illustrated in Fig. 1(b) as set (with probability ), which contains known component state vectors that ensure the system is working. The other set (with probability ), which is illustrated in Fig. 1(b) as set contains known component state vectors that result in system failure. Because the component state vectors in both and are known, the sampling space is then reduced from to , with probability . where Then, the system reliability can be calculated as

(4) The efficiency of the RVR method is improved because the variance of estimated system reliability decreases based on (5):

(5)

A. Algorithms to Find Minimal Cut Sets of Linear Sensor Systems A critical concern when applying the RVR Monte Carlo method for the linear sensor systems is to obtain minimal cut sets of the linear sensor system, particularly under the condition when the states of some sensors are fixed. In general, only a few

minimal cut sets need to be generated; thus the RVR method can efficiently approximate the reliability of the linear sensor systems. In the case of a linear sensor system, given that the states of some sensors are fixed as either functional or failed, a minimal set of sensors in the remaining sensors, whose simultaneous failure ensures system failure, is defined as a Minimal Cut Set under Restricted Conditions (MCS-RC). In the first step of the RVR method, none of the sensors are in a fixed state, i.e., both and equal . Obtaining a minimal cut set under this situation, which is called a Minimal Cut Set that is Not under a Restricted Condition (MCS-NRC), can be treated as a special case of finding a MCS-RC. In Section IV-A-1, a method is first developed to find a MCS-NRC. Based on this development, an algorithm is further proposed to obtain a MCS-RC in Section IV-A-2. 1) Method to Obtain a MCS-NRC: Based on the concept of the minimal cut set, and Definition 1, a MCS-NRC is a minimal set of sensors whose simultaneous failure results in at least one of the following conditions: (a) after removing the rows that correspond to all the failed sensors from , the rank of the reis reduced, i.e., ; or (b) the maining matrix is less than . number of rows in We first consider a minimal set of sensors whose simultaneous failure results in condition (a), which is defined in Definition 2. Definition 2: A Minimal Rank Reduction Set (MRRS) of a linear sensor system is defined as a minimal set of sensors whose simultaneous failure results in condition (a), i.e., . Result 1 as follows can be used to obtain an MRRS of a linear sensor system. The detailed proof is provided in the Appendix. Result 1: The set of sensors, which correspond to the nonzero elements in a nonzero row of the Reduced Row Echelon Form , is an MRRS; where is the transpo(RREF) of matrix sition of matrix . After obtaining an MRRS based on Result 1, the following Result 2 can be used to find a MCS-NRC of a linear sensor system. Result 2: Let be an MRRS. If the cardinality of is less , then is a MCS-NRC of the linear than or equal to with a cardinality sensor system. Otherwise, each subset of is a MCS-NRC of the linear sensor system. of Proof: Because any MRRS satisfies condition (a), must be a cut set of the linear sensor system. If the cardinality of is less than or equal to , the simultaneous failure of any proper subset of cannot result in condition (b). Based on the definition of MRRS, any proper subset of cannot satisfy is a minimal set of sensors that can condition (a) because satisfy condition (a). Therefore, any proper subset of cannot is a MCS-NRC be a cut set of the linear sensor system. So of the linear sensor system. If the cardinality of is larger than , a subset of with cardinality , denoted as , is obviously a minimal set of sensors whose simultaneous failure results in condition (b). Furthermore, for each proper subset of , the simultaneous failure of all the sensors cannot result in either condition (a) or condition (b). Thus is a MCS-NRC of the linear sensor system.


2) Method to Obtain a MCS-RC: Based on the method of obtaining a MCS-NRC discussed in the previous subsection, an algorithm is developed in this subsection to find a MCS-RC. The following two operations of a matrix will be used in the developed method. Operation 1: Delete a column in a matrix : remove column from matrix . Operation 2: Pivot a column in a matrix ; apply elementary row operations on so that column becomes a unit vector. (Only one entry in column is 1. The others are 0); then delete the row and column containing the unique non-zero entry in . Consider a linear sensor system represented by matrix . To obtain the MCS-RC, two operations are performed: for each sensor whose state is known as failed, delete the corresponding ; for each sensor whose state is known column in matrix . The as working, pivot the corresponding column in matrix new matrix obtained after all these operations is denoted by . After obtaining , the following Result 3 can be used to obtain a MCS-RC of the linear sensor system. The detailed proof of Result 3 is given in the Appendix. Result 3: The MCS-RCs of a linear sensor system , with a minimum allowable number of working sensors equaling , are the same as the MCS-NRCs of a new linear sensor system (which is represented by matrix ), with the minimum allow; where denotes able number of working sensors equal to the number of sensors whose states are fixed as functional in linear sensor system . For example, consider a linear sensor system with five sensors ) that is used to identify three source signals (i.e., (i.e., ). The transpose of matrix in (1), denoted as here, is given as

and is set to 4. Suppose sensor 5 is failed, and sensor 4 is working. We first delete column 5 from , and the remaining as follows. matrix is denoted by . Then, pivot column 4 of is transformed by subtracting row 2 from row 3, which First, as gives matrix

Second, delete row 2, and column 4 from

to get

as

According to Result 3, the MCS-RCs of (where sensor 5 is set as failed, and sensor 4 is set as working), are the same as the MCS-NRCs of a new sensor system that is represented by , whose minimum allowable number of working sensors changes . from 4 to Based on Results 1, 2, and 3, the method of finding a MCS-RC of a linear sensor system is given in Algorithm 1. Algorithm 1: Find a MCS-RC of a Linear Sensor System: Given the set of sensors whose states are set as failed (denoted

309

Fig. 2. RVR method applied to linear sensor system. (a) Initial step of RVR method. (b) Recursive step of RVR method.

as ), and the set of sensors whose states are set as working (denoted as ), a MCS-RC of a linear sensor system, denoted as , can be obtained as follows. 1) Set , which indicates the set of sensors , , and whose states are not fixed. Let . , the system is certain to fail. Then , 2) If return. 3) For each of the working sensors in , delete the corre; for each of the failed sensors in sponding column of , pivot the corresponding column of . The new matrix obtained after all these operations is denoted by . 4) Calculate the RREF of , denoted as . of that has the minimum number of 5) Find row nonzero entries, and this minimum number is denoted be the set of sensors corresponding to the as ; let , then nonzero entries in the th row of . If ; else . Note that in step 4 multiple rows of might have non-zero entries, corresponding to multiple minimal cut sets of the linear sensor system. Thus, how to select a MCS-RC that reaches the maximum efficiency needs to be considered. From (5), see that the improvement in variance of the RVR method depends on and ; larger values of or result in more values of is obtained improvement. In the developed RVR method, based on the minimal cut sets of the linear sensor system. Because a minimal cut set with large failure probability result in , a minimal cut set with maximum failure a large value of probability is preferred. Consider a minimal cut set of the linear , where is sensor system, denoted by the th sensor in that has failure probability . The can be calculated as failure probability of minimal cut set . Specifically, if all the sensors have the same reliability, a minimal cut set with smallest cardinality has the maximum failure probability. Thus, a MCS-RC with smallest cardinality is chosen when multiple MCS-RCs are obtained in step 4 of Algorithm 1. B. RVR Method for Reliability Evaluation of Linear Sensor Systems The RVR method applied for linear sensor systems is illustrated in Fig. 2. Fig. 2(a) shows the initial step of the RVR , is method. First, a NCS-NRC, denoted as obtained. The corresponding set of component state vectors is , in which denoted as the first zero elements indicate the failure states of sensors elements, denoted by in minimal cut , and the remaining , indicate the uncertain states of the other sensors. Each

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element can be either 0 or 1, i.e., , thus the carequals . Because the linear sensor system dinality of (set to at the beginning fails if all the sensors in fail, of the algorithm) is updated to . Then, if a sample falls into , the state of the linear sensor system is set to failure; other, will wise, the remaining component state vectors, i.e., be further divided into sets of component state vectors, denoted as , , in elements are 0, the th element equals 1, which the first and the remaining elements are . It can be easily verified that , and , for . For any element in , the states of the first sensors are known. Given these sensor ; states, if the sensor system is ensured to work, if the sensor system fails to work, ; otherwise, the system state cannot be determined. where the system state is unknown, If a sample falls in will be further divided into smaller subsets, which is illustrated in Fig. 2(b). First, a NCS-RC of the linear sensor system, denoted by , and the corresponding set of component state vec, are obtained. If the sample falls into , tors, denoted by the state of the linear sensor system is set to failure; otherwise, , will the remaining component state vectors in , i.e., subsets, i.e., , where be further divided into . This procedure continues, and the set of component state vectors is recursively divided into smaller subsets until the states of the linear sensor system can be determined for all the samples. The detailed steps of the RVR method applied for linear sensor systems are given in the following Algorithm 2, and a simple example to illustrate Algorithm 2 is provided in the Appendix. Algorithm 2. RVR Method Applied for Linear Sensor Systems: Consider as given the constant matrix in (1), the sample size , and a linear sensor system , where denotes the th sensor with reliability , . The system reliability, i.e., can be estimated by an unbiased estimator

works with certainty, i.e., and , then ; ; return. or 1.3. If fails with certainty, i.e., , then ; ; return. 2) Find a minimal cut set of the linear sensor system, denoted , where is the th sensor in , by . and 3) Divide the current sample space, i.e., the set of component state vectors that correspond to the set of sensors sets, denoted by whose states are not fixed, into , , and for , , . The first set, i.e., , corresponds to component state vectors where all of the sensors in the minimal cut set are set as failed, which results in the system , generate failure. For the remaining sets, , the number of samples, denoted as , , that fall in by using a random variable with multinomial dis. The probability tribution, i.e., , , is calculated as of , i.e., 1.2. If

(8) where and

is the reliability of

, i.e., the th sensor in , i.e.,

,

(9) 4) Recursively calculate and to : For Set , and Recursively call the procedure , and . 5) Compute and :

in

,

. . to calculate

(10)

(6) (11)

where is the number of samples that the system works. The sample variance of , denoted as , can be calculated as V. CASE STUDY (7) and

can be obtained by recursively calling Procedure ,where both and are initially set to at the . first step of calling procedure : Calculate and : Procedure 1) Set , which indicates the set of sensors be the subset of columns whose states are not fixed. Let corresponding to the sensors in set . All the of compose a matrix ; Let be the subset elements in of columns of corresponding only to the sensors in set . All the elements in compose a matrix . Both and are submatrices of . 1.1. If then ; ; return.

A sensor system used for a three-station sheet metal panel assembly process, as shown in Fig. 3, is used as a case study to examine and compare the efficiency of the proposed Monte Carlo methods. In this example, four parts (represented by rectangles in Fig. 3) are assembled and inspected through three assembly stations. In station I, two parts are assembled together. The subassembly of these two parts is moved to station II, and assembled with the other two parts. The four-part subassembly from station II is then positioned in station III for quality inspections. Each part or subassembly is positioned using two locators: a four-way locator, and a two-way locator. The four-way locator is positioned using a four-way hole (a round hole) on a part or subassembly to constrain the translational motion of the part or subassembly in both the and the directions. The two-way


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Fig. 3. Three-station panel assembly processes with coordinate sensors.

locator is positioned by a two-way hole or slot on the part or subassembly to constrain the part or subassembly motion only in the -direction so that rotational motions of the part or subassembly about the four-way locator are prohibited. The active part-holes of each stage, i.e., the part-holes used by locators to position parts or subassemblies at that stage, are also marked in Fig. 3. in Fig. 3, are A total of 12 locators, denoted as through used in this 3-station assembly process, including six four-way locators, and six two-way locators. The source variables in this process are potential process faults due to errors caused by the locators. Each of the four-way locators can have two potential process faults due to the dimensional locating errors in the -direction and in the -direction. Each 2-way locator has one potential fault: the dimensional locating error in the -direction. Therefore, 18 potential process faults (i.e., source variables) exist in this process as marked by numbers one through eighteen in Fig. 3. In this assembly process, the linear sensor system consists of 20 coordinate sensors, marked by through in Fig. 3. There are two types of sensors: the x-coordinate sensor (denoted by an arrow in the x-direction in Fig. 3), and the z-coordinate sensor (denoted by an arrow in the z-direction in Fig. 3); these sensors are used to measure the dimensional deviation of a part or subassembly in the x-direction, and in the z-direction, respectively. The crude Monte Carlo method, and the RVR method developed in Section III are used to evaluate the reliability of the linear sensor system that is used in this three-station assembly process. The approximate reliability obtained from these two Monte Carlo methods is very useful for comparing the reliability performance of different sensor system designs. When applying each method, nine scenarios are tested; is chosen to be 1000, 10,000, and 100,000; and the common reliability for all the sensors, denoted by , is set to 0.90, 0.95, and 0.99. In the literature about network reliability, multiplication of the estimated variance of system reliability and the computational time is commonly used as a criterion to compare the performance of var-

ious Monte Carlo methods [17]–[20], [22]. In this case study, the RVR method developed for the linear sensor system is compared to the crude Monte Carlo method by using calculated as (12) where , and are the computational time of the crude Monte Carlo method, and the RVR method, respectively. , and are the sample variance of the crude Monte Carlo method, and the RVR method, respectively. indicates that the performance of the RVR method is better than that of the crude Monte indicates otherwise. The smaller the Carlo method; and value of is, the better the performance of the RVR method is when compared to the crude Monte Carlo method. Two different situations are considered for this three-stage panel assembly process. In the first situation, the twenty-dimen) from the twenty sensors are sional measurements (i.e., used to identify all 18 potential process faults at three stages ). Matrix in (1) is denoted by for this first (i.e., is given as the matrix case study situation; the transpose of is 18. shown at the bottom of the next page. The rank of Suppose is set to be 19. The performance of the Monte Carlo methods developed in this paper is presented in Table I. Observe that all the values in Table I are much less than 1. This result indicates that the performance of the RVR method is much better than that of the crude Monte Carlo method. When the ), the adlinear sensor system is highly reliable (i.e., vantage of the RVR method to the crude Monte Carlo method is the greatest. This result confirms the discussion in Section IV-B, which suggested that the crude Monte Carlo method is not efficient, especially when the system is highly reliable. In the second situation, only process faults at Stage I are considered for the fault diagnosis. Faults at the other two stages are assumed not to occur. Given the two 4-way locators and two 2-way locators used in Stage I, six potential process faults can ) by the 20 sensors (i.e., ). be identified (i.e., , consists of the first Matrix for this situation, denoted as

312

TABLE I PERFORMANCE OF THE RVR METHOD COMPARED TO CRUDE MONTE CARLO METHOD UNDER THE FIRST SITUATION


when the number of sensors is close to the minimum allowable number of working sensor. From Tables I and II, it can be seen that the estimation accuracy of the RVR method increases with the increment of the sample size . On the other hand, the computation time also increases with the increment of the sample size . Thus, in real applications, there is a tradeoff between estimation accuracy and computation time. VI. CONCLUSION

TABLE II PERFORMANCE OF THE RVR METHOD COMPARED TO CRUDE MONTE CARLO METHOD UNDER THE SECOND SITUATION

six columns of . Suppose is set to 7. Table II presents the performance of the RVR method compared to that of the crude Monte Carlo method. Similar to situation 1, the RVR method greatly outperforms the crude Monte Carlo method. Comparing Tables I and II, see that the performance of the RVR method for the second situation is better than that for the first situation. The reason is discussed as follows. It can be versensors is a cut set of the linear ified that any set of sensor system. Thus, the cut sets for the first situation generally have smaller cardinalities than those for the second situation. As a result, the linear sensor system has a lower reliability for the first situation than that for the second situation. As discussed in the paper, the efficiency of the RVR method increases with the increment of the system reliability. Thus, the performance of the RVR method for the second situation is better than that for the first situation. In general, when all the sensors have the same reliability, the proposed algorithm will be most efficient

Linear sensor systems have broad applications in areas such as manufacturing fault diagnosis, array signal processing, calibration of wireless sensor networks, and sensor systems in electrical power systems. To overcome the exponentially increased computational complexity for evaluating the exact reliability of a linear sensor system, Monte Carlo methods are applied in this paper to efficiently approximate the sensor system reliability. The crude Monte Carlo method is not efficient to obtain an accurate estimate of system reliability when the system is highly reliable. As a result, the RVR method in network reliability literature is adapted for the reliability problem of linear sensor systems. To apply the RVR method for linear sensor systems, new methods are proposed to find minimal cut sets of the linear sensor systems, particularly under the conditions where the states of some sensors are fixed as failed or functional. The developed Monte Carlo methods are applied to a case study, where a linear sensor system is used for fault diagnosis in a multistage automotive assembly process. The experiment results show that the RVR method is more efficient than the crude Monte Carlo method for reliability evaluation of linear sensor systems. The higher the sensor system reliability, the better the performance of the RVR method compared to the crude Monte Carlo method. The approximate reliability obtained from the developed Monte Carlo methods is very useful for comparing the reliability performance among various sensor system designs. In future research, we will further use simulation optimization methods to obtain the optimal design of a linear sensor system.


APPENDIX PROOFS OF RESULT 1 AND RESULT 3, AND AN EXAMPLE TO ILLUSTRATE ALGORITHM 2.

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TABLE III RESULTS OF EACH STEP WHEN APPLYING ALGORITHM 2 FOR THE EXAMPLE

The proofs of Result 1 and Result 3 are based on a mathematical tool called matroid theory. For more details on matroid theory, as well as terminology such as vectorial matroid, dual matroid, circuit, cocircuit, and elongation, please refer to the references [23]–[25]. A. Proof of Result 1 By arranging the order of columns appropriately while keeping track of the column indices, each Reduced Row can be generally written in Echelon Form (RREF) of the form of

, where

is an

identity matrix

with

. The submatrix of the nonzero rows of is . Consider the vectorial matroid defined on , and its dual matroid denoted as . is the vectorial maBased on matroid theory [25], troid of , i.e., . For , i.e., each row of , let each nonzero row of RREF denote the set of columns corresponding indicates the to the nonzero elements of that row, where th column vector of . It is not difficult to see that the are linearly depencolumns in dent, and any subset of them is linearly independent. That is, is a circuit of the vectorial matroid , which is equivalent to the matroid . is a cocircuit (i.e., the circuit Therefore, of the dual matroid) of V( ). Based on matroid theory, is the minimal set of columns whose deletion from matrix results in the rank of reduced by 1. Thus, the corresponding set of sensor to is a MRRS.

where is the elongation of to height . Thus, each of the MCS-RCs of the linear sensor system is a set , which of sensors corresponding to a cocircuit of is the MCS-NRC of a linear sensor system represented by with the minimum number of working sensors equaling . C. An Example to Illustrate Algorithm 2 Consider a linear sensor system with three sensors (i.e., ) used to identify two source signals (i.e., ); is set to ; the 2; all the three sensor have the same reliability, i.e., sample size is chosen as 100; and the transpose of matrix in (1) is given as

When applying Algorithm 2, the results of each step are shown in Table III as follows. Based on (10) and (11), and are calculated as

B. Proof of Result 3 In our previous work [14], the MCS-NRC of a linear system is characterized as follows. For a linear sensor system with reliability defined as in Definition 1, each of its minimal cut sets , and is a set of sensors corresponding to a cocircuit of denotes the elongation of to vice versa; where height . Based on matroid theory, each of the MCS-RCs of the linear sensor system is a set of sensors corresponding to a cocircuit of ; where indicates the columns of corresponding to the set of working sensors with cardinality ; indicates the columns of corresponding to the set of is defined as the operation of deletion failed sensors ; is defined as the operation of contracto matriod , and tion to matriod . For the detailed description of deletion and contraction in a matroid, please refer to [25]. Based on matroid theory,

From (6), the estimated system reliability

and the sample variance of on (7) as

, i.e.,

is

, can be calculated based

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Qingyu Yang is an Assistant Professor in the Department of Industrial and System Engineering at Wayne State University. He received his Ph.D. in Industrial Engineering and a Master’s degree in Statistics from the University of Iowa in 2008 and 2007, respectively. He also has a Master’s degree in Intelligent System (2003) and the B.E. degree in Automatic Control (2000) from the University of Science and Technology of China (USTC). His research interests include statistical data analysis, complex system modeling, and system informatics.

Yong Chen is currently an Associate Professor in the Department of Mechanical and Industrial Engineering at the University of Iowa. He received the B. E. degree in computer science from Tsinghua University, China in 1998, the Master degree in Statistics and Ph.D. degree in Industrial & Operations Engineering, both from the University of Michigan in 2003. His research interests include quality and reliability engineering, pattern recognition, applied statistics, and simulation optimization. He is a member of the INFORMS, the ASA, and the IMS.