Asymmetric weighted voting systems

The weighted voting system (WVS) studied consists of n units, each of which provide a binary decision (0 or 1) or abstain from voting. Each unit has its own individual weight. System output is 1 if the cumulative weight of all 1-opting units is at least a pre-specified fraction τ of the cumulative weight of all non-abstaining units. Otherwise, system output is 0. The system input is either 0 or 1. Every unit is characterized by the probability of making decisions 0 and 1 and by probability of abstaining for each input. The system fails if the system output is not equal to its input.In this paper, an asymmetric WVS is suggested in which each voting unit has two weights. The first weight is applied when the unit's decision is 0 and the second weight is applied when the unit's decision is 1. The asymmetry of unit weights allows the WVS designer to take advantage of the knowledge of statistical asymmetry of voting units (asymmetric probabilities of making correct decisions with respect to the input). The paper presents an algorithm for asymmetric WVS reliability evaluation. This algorithm is based on using a universal generating function technique.For a system consisting of voting units with given reliability characteristics, one can maximize the entire system reliability by choosing proper unit weights and threshold value. An algorithm is suggested which finds the optimal unit weights and the threshold. A genetic algorithm is used as the optimization tool. An example is presented in which the superiority of asymmetric WVS over regular symmetric one is demonstrated.     

Weighted voting systems: reliability versus rapidity

The weighted voting system (WVS) consists of n units that each provide a binary decision (0 or 1) or abstain from voting. Each unit has its own individual weight. System output is 1 if the cumulative weight of all 1-opting units is at least a pre-specified fraction τ of the cumulative weight of all non-abstaining units. Otherwise, system output is 0. The system input is either 0 or 1. Every unit is characterized by probability of making decisions 0 and 1 and by probability of abstaining for each input. The system fails if its output is not equal to its input.This paper shows that if the WVS consists of units that need different time to produce their outputs, the decision time of the entire system depends on the distribution of unit weights and on the value of τ. It shows also that a tradeoff exists between the system reliability and its rapidity.An algorithm that finds the system parameters maximizing its reliability under constraint imposed on the expected system decision time is suggested. Illustrative examples are presented.     

Reliability optimization for weighted voting system

The voting system studied consists of n units, each of which provides a binary decision (0 or 1), or abstains from voting. The system output is 1 if the cumulative weight of all 1-opting units is at least a pre-specified fraction τ of the cumulative weight of all non-abstaining units. Otherwise the system output is 0.The existing method for evaluating the reliability of a weighted voting system can not be applied to real systems without imposing some restrictions because of its combinatorial complexity. In this paper a method is suggested which allows the reliability of weighted voting system to be exactly evaluated without imposing constraints on unit weights or threshold value. The approach is based on using a universal generating function technique. Using the suggested method an optimization procedure is developed for system reliability maximization by choosing proper unit weights and threshold values. A genetic algorithm is used as the optimization tool. Examples are presented.     

Optimal unit grouping in weighted voting systems

The voting system studied consists of n units that each provide a binary decision (0 or 1) or abstain from voting. Each unit has its own individual weight. System output is 1 if the cumulative weight of all 1-opting units is at least a pre-specified fraction τ of the cumulative weight of all non-abstaining units. Otherwise system output is 0.For a system consisting of voting units with given reliability characteristics one can maximize the entire system reliability by choosing proper unit weights and threshold values τ. In this paper it is shown that additional system reliability improvement can be achieved by grouping units in voting subsystems and tallying the weighted votes of these subsystems (groups) to make a final decision. An algorithm is suggested which finds the optimal element grouping as well as unit weights within each group, weights of the groups and corresponding system thresholds. The approach is based on using a universal generating function technique for evaluating system reliability. A genetic algorithm is used as the optimization tool. Examples are presented.     

Analysis and optimization of weighted voting systems consisting of voting units with limited availability

The voting system studied consists of n voting units each either providing a binary decision (0 or 1) or abstaining from voting. The system output is 1 if the cumulative weight of all 1-opting units is at least a pre-specified fraction τ of the cumulative weight of all non-abstaining units. Otherwise, the system output is 0.In this paper, we study the effect of limited availability of the voting units on the entire voting system reliability. Two different types of systems are considered. In the system of type 1, the absence of unit output (unit unavailability) is interpreted by the system as abstention from voting. In the system of type 2, the unavailable state of the voting unit and its abstention from voting can be distinguished and the system parameters can be adjusted to optimize its performance for each combination of available units.There are two ways to improve reliability of weighted voting system consisting of units with the given output probability distribution: optimization of system parameters (weights of units and threshold factor value) and units availability enhancement (for example, by choosing proper maintenance policy). This paper shows a method of incorporating information about units' availability into a procedure for determining the optimal system parameters. It also presents a method for determining indices that measure importance of voting units availability for both types of systems. These indices indicate voting units for which efforts of availability enhancement are the most beneficial from the entire system reliability improvement point of view.The approach is based on using a universal generating function technique and optimization procedure presented in [5]. Examples are presented.     

Modeling the reliability of threshold weighted voting systems

In many applications, ranging from target detection to safety monitoring systems, we are interested in determining whether or not to accept a hypothesis based on the information available. In this paper we model the reliability of threshold weighted voting systems (WVS) with multi-failure-modes, where a general recursive reliability function of the WVS is presented. We also develop approximation formulas for calculating the reliability of WVS based on a large number of units. We also develop reliability functions of time-dependent threshold weighted voting systems, where each unit is a function of time. Finally, the optimal stopping time that minimizes the total cost of the systems subject to a reliability constraint is discussed.     

Analyzing the behavior and reliability of voting systems comprising tri-state units using enumerated simulation

Voting is a common technique used in combining results from peer experts, for multiple purposes, and in a variety of domains. In distributed decision making systems, voting mechanisms are used to obtain a decision by incorporating the opinion of multiple units. Voting systems have many applications in fault tolerant systems, mutual exclusion in distributed systems, and replicated databases. We are specifically interested in voting systems as used in decision-making applications.In this paper, we describe a synthetic experimental procedure to study the behavior of a variety of voting system configurations using a simulator to: analyze the state of each expert, apply a voting mechanism, and analyze the voting results. We introduce an enumerated-simulation approach and compare it to existing mathematical approaches. The paper studies the following behaviors of a voting system: (1) the reliability of the voting system, R; (2) the probability of reaching a consensus, P_c; (3) certainty index, T; and (4) the confidence index, C. The configuration parameters controlling the analysis are: (1) the number of participating experts, N, (2) the possible output states of an expert, and (3) the probability distribution of each expert states. We illustrate the application of this approach to a voting system that consists of N units, each has three states: correct (success), wrong (failed), and abstain (did not produce an output). The final output of the decision-making (voting) system is correct if a consensus is reached on a correct unit output, abstain if all units abstain from voting, and wrong otherwise. We will show that using the proposed approach, we can easily conduct studies to unleash several behaviors of a decision-making system with tri-state experts.     

Reliability-centered predictive maintenance scheduling for a continuously monitored system subject to degradation

This paper tries to integrate sequential imperfect maintenance policy into condition-based predictive maintenance (CBPM). A reliability-centered predictive maintenance policy is proposed for a continuously monitored system subject to degradation due to the imperfect maintenance. It is assumed that the system hazard rate is a known function of the system condition and then can be derived directly through CBPM. A hybrid hazard rate recursion rule based on the concept of age reduction factor and hazard rate increase factor is built up to predict the evolution of the system reliability in different maintenance cycles. Whenever the system reliability reaches the threshold R, an imperfect preventive maintenance (PM) is performed on the system. The optimal reliability threshold R is determined by minimizing the cumulative maintenance cost per unit time in the residual life of the system which is based on simulation. Finally, a discussion is presented to show how the optimal results depend on the different cost parameters.     

A reliability allocation method for agricultural machinery based on AHP-IFM

In the product design phase, the available product failure data are limited, and the weight allocation method is often used to assign reliability targets to each unit. The integrated factors method (IFM) can calculate the reliability allocation weights considering multiple influencing factors simultaneously, but it cannot reflect the difference in the importance of each factor and each unit. The analytic hierarchy process (AHP) can calculate the relative importance weights of each factor and each unit. Combining the AHP with the IFM can make the IFM more adaptable to the system and more accurate for reliability allocation. However, the current combination method can cause two problems: the invalidation of the influencing factor weights and the imbalance of the unit weights. To address these two shortcomings, the AHP-IFM proposed in this paper introduces a weight weakening factor and exponentially corrects the unit weights for units, which can better apply the relative importance weights of each influencing factor and each unit to the reliability allocation. The effectiveness of the AHP-IFM is verified by comparison with existing methods and data. Finally, an AHP-IFM applicable to agricultural machinery is proposed, and the reliability allocation of a no-till seeder is used as a case to verify the feasibility of the AHP-IFM.     

A taxonomy of voting schemes for data fusion and dependable computation

Voting is an important operation in the fusion of data originating from diverse sources and in the realization of ultrareliable systems based on multiple computation channels. Voting involves the derivation of an output data object from a collection of n input data objects, as prescribed by the requirements and constraints of a voting algorithm. The objects voted on can be quite complex in terms of content and, explicitly specified or implicit, structure. Regardless of implementation details (e.g., whether realized in hardware, software, or hybrid schemes) and object space properties, voting algorithms can be classified according to how they view the input and output data objects and how they handle the votes (weights) at input and output. A 16-class (binary 4-cube) categorization results from dichotomizing each of the above voter features. This categorization leads to an abstraction that helps in the study of voting algorithms with regard to the dependability level for the outputs and the speed at which they are obtained; viz, the quality and efficiency of the algorithms. The taxonomy is broad enough to cover, and detailed enough to distinguish among, a wide variety of commonly used voting algorithms in data fusion and dependable computation. It also provides insight into the relationships of various voting schemes and facilitates comparison and fine-tuning of such algorithms.     

Holistic reliability analysis of weighted voting systems from a multi-state perspective

The computation of the reliability of weighted voting systems is an important problem in reliability theory due to its potential application in security, target identification, safety and monitoring areas. Voting systems are used in a wide variety of applications where an acceptance or rejection decision has to be made about a binary proposition presented to the system. For these systems, it is of interest to obtain the probability so that based on the vote of decision-making units, the system aggregates these votes into the right decision when presented with such a proposition. This paper presents a holistic work on weighted voting system reliability by presenting modeling, computation, estimation and optimization techniques. The modeling part takes advantage of the structure of weighted voting systems to present a model of its reliability as a multi-state system. Next, based on the multi-state view of the system, an exact computational approach based on multi-state minimal cut and path vectors is introduced. The paper then acknowledges the computational complexity of the problem and provides a Monte Carlo simulation approach that estimates system reliability accurately and in an efficient computational time. Finally, an optimization heuristic that generates quasi-optimal solutions is presented that is able to solve the problem of maximizing the reliability of a weighted voting system based on a specified number of decision-making units with known reliability characteristics.     

A multiple system governed by a quasi-birth-and-death process

The system we consider comprises n units, of which one has to operate for the system to work. The other units are in repair, in cold standby, or waiting for repair. Only the working unit can fail. The operational and repair times follow phase-type distributions. Upon failure, it is replaced by a standby unit and goes to the repair facility. There is only one repairman. When one unit operates the system is up and when all the units are in repair or waiting for repair, the system is down. This system is governed by a finite quasi-birth-and-death process. The stationary probability vector and useful performance measures in reliability, such as the availability and the rate of occurrence of failures are explicitly calculated. This model extends other previously considered in the literature. The case with an infinite number of units in cold standby is also studied. Computational implementation of the results is performed via a numerical example, and the different systems considered are compared from the reliability measures determined.     

Construction of a Survivability Evaluation System for Military Aircraft

The evaluation indexes system of aircraft survivability is constructed for the first time from three aspects： susceptibility, vulnerability and combat resilience; the bargaining weight method is proposed to determine the weights of the indexes and evaluate aircraft survivability. The bargaining weight method brings different opinions into accord under the constraint of minimum loss, it can overcome the partial subjectivity in determining weights and evaluation, and has objectivity. The example testifies rationality and feasibility of the evaluation system.     

AHP‐IFM Target: An Innovative Method to Define Reliability Target in an Aerospace Prototype Based on Analytic Hierarchy Process

Reliability target definition is a crucial aspect of any reliability analysis. In literature, there are two types of analysis. The first one, called ‘bottom‐up’, goes back to the system's target using data of units through a fault tree analysis. Reliability data of components could be only partially available, particularly in the case of innovative systems. In the second type of analysis, called ‘top‐down’, starting from similar systems, the target of each unit is defined, by applying allocation techniques. Also, in this case, reliability data of similar systems might not be available, and the choice of the most appropriate technique could be tricky. The purpose of the present research is to combine the advantages of both usual approaches. The newly developed approach is based on the integrated factors method, whose values are adjusted trough a multicriteria method, the analytic hierarchy process, depending on the importance of each factor and each unit. The innovation of the proposed model consists in its dynamism, as most of the literature methods use constant weights for the factors involved in reliability allocation. No method takes into account the assignment of a different level of significance (weight) to different units of the system, simultaneously with the considered factors. The developed approach has been applied on an aerospace prototype system. The results show the goodness of the new method and its ability to overcome the problems noted in literature. Copyright © 2017 John Wiley & Sons, Ltd.     

A Study of Reliability of Multi‐State Systems with Two Performance Sharing Groups

The performance sharing can be widely seen in different kinds of engineering systems, such as meshed power distribution systems and interconnected data transmission systems. This paper presents a study of systems consisting of multi‐state units connected as two performance sharing groups, and the suggested methodology can be adapted for the case of three or more performance sharing groups. To be more general, the system unit is allowed to be in one single performance sharing group or both. Each unit has a random demand to satisfy, and the units can transmit capacity with each other given that the total performance transmitted in each performance sharing group does not surpass its maximum transmission capacity. An algorithm based on the universal generating function technique is proposed to evaluate the system reliability and the expected system performance deficiency. Copyright © 2016 John Wiley & Sons, Ltd.     

Some Considerations in Document Copying via Electrophotography

In designing equipment and materials suitable for electrophotographic reproduction of documents, it is necessary to consider very carefully each of the elements involved, including the design of mechanical components, the optical system, charging units, developing systems, fixing units, and the engineering of a recording medium which gives adequate performance. Each of these various units must be investigated not only from the standpoint of mechanically satisfactory functioning, but also with consideration for its interrelationship with the rest of the system. Some of the most important factors to be considered include contrast of the optical system and of the electrophotographic recording medium, selection of a suitable light source, proper design of a breakdown-free charging unit, reliability of the developing unit, and a high efficiency fixing unit. Compactness of equipment is also a major consideration, particularly for a desk-top office copier or similar small devices, if the electrophotographic copy is to be used as a lithographic master, additional problems are involved in adding this functional ability to that of good electrophotographic performance.     

Reliability of a Multi‐State Computer Network Through k Minimal Paths Within Tolerable Error Rate and Time Threshold

A computer system is usually modeled as a network topology where each branch denotes a transmission medium and each vertex represents a station of servers. Each branch has multiple capacities/states due to failure, partial failure, and maintenance. Such a network is named a multi‐state computer network (MSCN). From the viewpoint of quality management, transmission error rate and transmission time are both critical performance indicators to assess Internet quality for system managers and customers. Within both tolerable error rate and time threshold, the addressed problem is concentrated on an MSCN for computing the probability that d units of data can be sent through multiple minimal paths simultaneously. Such a probability is named system reliability. A solution procedure including an efficient algorithm based on MPs is proposed to derive the lower boundary vectors (LBVs) meeting the requirements. Then system reliability, which is represented as the probability of union of subsets, can be subsequently evaluated by the LBVs. Copyright © 2015 John Wiley & Sons, Ltd.     

Minimal cut sets of s–t networks with k-out-of-n nodes

In this paper, we extend traditional directed s–t network by letting nodes have k-out-of-n property: To generate output flows, a node must receive at least k flows from its n input links, where k is an integer assigned for the node and its value can be any number from 1 to n. To evaluate the system reliability, minimal cut sets for the extended network are defined for nodes. Under this definition, an extended network and its sink node have the same minimal cut sets. A new algorithm is designed to generate minimal cut sets for all nodes, starting with the source node and ending with the sink node. With different initializations, the algorithm can be applied for extended s–t networks with or without node failures.     

Modelling a reliability system governed by discrete phase-type distributions

We present an n-system with one online unit and the others in cold standby. There is a repairman. When the online fails it goes to repair, and instantaneously a standby unit becomes the online one. The operational and repair times follow discrete phase-type distributions. Given that any discrete distribution defined on the positive integers is a discrete phase-type distribution, the system can be considered a general one. A model with unlimited number of units is considered for approximating a system with a great number of units. We show that the process that governs the system is a quasi-birth-and-death process. For this system, performance reliability measures; the up and down periods, and the involved costs are calculated in a matrix and algorithmic form. We show that the discrete case is not a trivial case of the continuous one. The results given in this paper have been implemented computationally with Matlab.     

Reliability approximation using finite Weibull mixture distributions

The shape of measured or design life distributions of systems can vary considerably, and therefore frequently cannot be approximated by simple distribution functions. The scope of the paper is to prove that the reliability of an arbitrary system can be approximated well by a finite Weibull mixture with positive component weights only, without knowing the structure of the system, on condition that the unknown parameters of the mixture can be estimated. To support the main idea, five examples are presented. In order to estimate the unknown component parameters and the component weights of the Weibull mixture, some of the already existing methods are applied and the EM algorithm for the m-fold Weibull mixture is derived. The fitted distributions obtained by different methods are compared to the empirical ones by calculating the AIC and δ_C values. It can be concluded that the suggested Weibull mixture with an arbitrary but finite number of components is suitable for lifetime data approximation. For parameter estimation the combination of the alternative and EM algorithm is suggested.