With the adoption of the United Nations’ Sustainable Development Goals, the focus on improving supply chain sustainability and proper order distribution has become a critical problem. This study proposes a novel two-stage algorithm that involves supplier sustainability to assess the system reliability of a supply chain. System reliability, which gauges the probability of the supply chain successfully delivering a designated amount of goods to the market while considering supplier sustainability and production capacity, is an essential performance indicator used to evaluate supply chain capability and allocate orders. We establish a multistate sustainable supply chain network, where each node symbolizes a market, assembler, warehouse, or supplier, and each connecting edge signifies a carrier. The proposed two-stage algorithm first integrates a Z-number-based indifference threshold-based attribute ratio analysis (called Z-ITARA) and the reference ideal method (called Z-RIM) to assess supplier and order allocation sustainability. Afterward, sensitivity analysis is adopted to assign the flow pattern, and the changes in system reliability are observed. To demonstrate the effectiveness of the proposed algorithm, a real case of an audio corporation between China and Taiwan is studied.

As modern society increasingly relies on the stable operation of the electric power system (EPS), it is essential to evaluate the performance of the EPS and identify the critical infrastructure to ensure a reliable power supply. Since infrastructures of the EPS, including power plants, substations, and transmission lines, can operate at multiple power capacity levels due to failures or maintenance, the power capacity should be seen as multistate. In this way, the EPS is modeled as the multistate electric power network (MEPN) where power plants, substations, and transmission lines are denoted as sources, nodes, and branches, respectively. Supply reliability, defined as the probability that the MEPN can satisfy demand, is served as a performance index to measure the MEPN capability. To calculate supply reliability, an algorithm is developed based on the concept of lower power capacity patterns. Furthermore, a process called infrastructure importance measurement is introduced to systematically analyze the impact of individual infrastructure failures on supply reliability to rank the most critical infrastructure. In particular, those components in each infrastructure are not required to be identical, enhancing its applicability to the real-world system. A practical EPS in Taiwan is illustrated to provide feasibility for the system manager to evaluate the MEPN supply reliability and identify the critical infrastructures.

Air transport plays an important role in supporting the global tourism industry. An unstable or uncertain air transport network leads to passenger dissatisfaction; therefore, a performance assessment is crucial to the management processes of travel agencies. Flights in an air transport network may be delayed by several factors when the travel time and cost significantly affect passenger satisfaction. This study evaluates reliability as the probability of successfully fulfilling multiple demands under the budget and time constraints with the impacts of flight delays, demonstrating the ability to satisfy passengers. Employing the minimal path concept, this study first models a multiple-demand multistate air transport network considering flight delays (MSAD) then identifies feasible minimal paths under the budget and time constraints and their maximum time-delay vectors. Reliability then is derived from all flow vectors that enable transport as required and their corresponding time-delay vectors. This study aims at an efficient algorithm that provides managers of travel agencies with insight into their MSAD and a basis for managerial decisions.

Many real-world problems, such as power line transmission planning, computer network management, water pipe planning, and traffic flow planning, can be modeled using multistate network systems. One of the most interesting aspects of these problems is that one typically wishes to know the reliability of such a plan, namely the network reliability. To evaluate the reliability of a specific network system, the three-stage method (TSM) is typically applied. TSM consists of 1) finding all minimal paths, 2) finding all$d$-system vectors, and 3) calculating the union probability for reliability, where 1) is one of the most challenging problems discussed in the literature. This article proposes an improved merge search method that combines the advantages of both enumeration and searching technologies. A step-by-step exploration of the proposed method is presented and several benchmark comparisons are performed using well-known algorithms to demonstrate the efficiency of the proposed method. The results show that the computation time of the proposed algorithm can save as much as 10 times against the well-known methods in the literature.

A cloud–fog computing system is modeled as a network topology where each edge represents a transmission line, and each node represents a hub, an Internet of Things (IoT) device, a fog server, or a cloud server. Such a system with several capacities at each edge is called a stochastic-flow cloud–fog computing network (SCCN). There are three phases of transmitting data: the first is to transmit data from the IoT devices to the fog servers. The second and third phases are transmitting a part of the data from the fog servers to the cloud servers for the further operations and back to the IoT devices for the immediate operations, respectively. Based on the transmission mechanisms, two demands, including initial demands and processed demands, should be satisfied simultaneously, where an initial demand (a processed demand) is outing from an IoT device (a fog server). An algorithm is developed to evaluate reliability, the probability of successfully transmitting the data through the SCCN, by elucidating the flow relationship among the IoT devices, edge servers, and cloud servers. Furthermore, another approach to adjust the flow assignment is also proposed for the case that demand may not be an integer. A large real case is provided to validate the applicability and scalability of the proposed methodology.

To improve the success probability of a mission execution, scheduled checkpointing is often implemented to save completed portions of the mission task so that a system can resume the mission execution effectively after its restoration whenever the system failure occurs. This paper considers a repairable computing system subject to the scheduled checkpointing. The checkpointing intervals are deterministic, but can be even or uneven. The system repair time is fixed while the system time-to-failure can follow any arbitrary type of distributions. The maximum number of repairs is specified by a certain threshold value. A multi-valued decision diagram (MDD)-based analytical approach is proposed to evaluate the exact success probability of a mission execution for the considered repairable system. The proposed approach enables generating a compact mission MDD model where identical subMDD models can be merged to improve computational efficiency and reduce storage requirement. The MDD model, once being constructed, can be reused for system reliability evaluations using different input parameter values. A benchmark study is presented to show the efficiency of proposed MDD approach. A case study is performed to illustrate the application of the proposed MDD approach to facilitate decision making about proper system design and parameter selection.

A reliability evaluation method is proposed for production systems by applying the capacitated-flow production network (CFPN) model. In particular, this article considers in-line stockers to monitor and to avoid blockage and starvation in the CFPN. First, the minimal capacity that all cells (group of machines) should provide to meet demand is generated. Second, status of stocker is monitored via the proposed “stocker volume check table” to calculate the volume usage of a stocker, and the stocker nonutilization is then obtained. Finally, the system reliability is derived in terms of the minimal capacity vector and stocker nonutilization. An important contribution of this article is to avoid complicated dependency calculation when multiple stockers are considered. A practical case of printed circuit board production is further studied to illustrate the applicability of the proposed method.

As transport is a vital component in the modern tourism industry, determining reliable-performance tourism transport networks is critical for tour operators and travel agencies. The performance of a tourism transport network in this study is investigated by evaluating system reliability, which is the probability of successfully transporting requested passengers between multiple origins and multiple destinations within a time constraint when tardiness may occur. A stochastic multiple-origin-destination tourism transport network with tardiness (SMTNT) is formulated for system reliability evaluation. Identify all minimal travel paths (MTP) and corresponding maximum tardiness vectors that successfully connect each origin–destination pair under a time constraint. System reliability is derived from all flow vectors enabling delivery via MTPs a requested number of passengers at different tardiness levels. The proposed algorithm is successful in handling a combination of time constraints in the case of tardiness with multiple origin–destination pairs. Furthermore, it helps travel agencies obtain an insight into the performance of their SMTNT. An example describing the working of the proposed algorithm with an analysis of system reliability is provided. The results of the study can be used for improving the algorithm in the future.

Network reliability evaluation for a stochastic flow network (SFN) can be computed in terms of all lower boundary points (LBPs). Although several algorithms have been presented in the literature as solutions for the LBP problem, the efficiency is always occurring at the present for the large SFN. In this paper, a binding technique generating binding-flow vectors is therefore developed to narrow down the searching range of flows for demand. This technique is able to reduce the loading of the candidate comparison procedure greatly. An algorithm based on this binding technique is proposed to solve the LBP problem. We compare the performance of the proposed algorithm to well-known algorithms in terms of CPU time, through three benchmark networks and a practical case. The experimental results show the functional efficiency of our proposed binding LBP algorithm.

A fast retransmit, which reduces the sender's waiting time before retransmitting a lost segment, is applied to guarantee data integrity with no data loss in transmission. Based on fast retransmit, many application protocols have been enhanced and evolved to ensure quality of service and reduce data transmission time. One of application protocols is the multipath transmission control protocol which is catholically applied in modern computer networks. Communication lines used in this kind of network have different states, namely, failure, partial failure, and maintenance. Therefore, a computer network with a fast retransmit can be classified as stochastic and is called a stochastic-flow computer network with a fast retransmit (SCNFR). This paper assesses the system reliability of SCNFR for the successful transmission of demand d through k (k > 1) separate minimal paths (kSMiP) under the latency. An algorithm is proposed to find all minimal capacity vectors (MCVs) that satisfy the demand and latency. Then, the system reliability is computed based on all MCVs. We adopt two practical cases of the pan-European research and education network and the Taiwan academic network to explore the effectiveness of the algorithm. The results show that the system reliability can be classified as decision reference while the manager decides the better kSMiP with higher system reliability.

Completion time and cost in a project are affected by uncertainties, such as weather, human resources availability, equipment efficiency, etc. The activity durations of the project, thus, should be regarded as stochastic. Such a project is usually modeled as a stochastic project network (SPN). The SPN can be represented in the form of an activity-on-arc diagram, in which each activity has several possible durations and different costs. Every activity duration possesses a corresponding cost and probability. Based on the concept of minimal paths, two algorithms are utilized to find the upper and lower duration vectors under both time and budget constraints. For an SPN, the project reliability is defined as the probability that the project can be completed under both time and budget thresholds. Theoretically, it is difficult to compute project reliability using the upper and lower duration vectors directly because of the domination property among the duration vectors. Therefore, this paper proposes the derivation of project reliability interval depending on the relationships among the upper and lower duration vectors. The exact project reliability is proved to be contained within the interval. Two examples, including a practical case of disaster recovery system construction, are presented to demonstrate the scalability and practicability of the proposed solution procedure.

For a hybrid flow shop (HFS), due to the maintenance, partial failure, and possibility of failure, the number of machines at a workstation presents multiple levels, meaning that capacity of each workstation is stochastic. This study considers the stochastic capacity of each workstation and proposes a performance index based on system reliability to measure the probability that the HFS can complete demand d within time constraint T. The HFS is modeled as a stochastic-flow network, in which each arc is regarded as a workstation with stochastic capacity and each node as a buffer. An algorithm is then developed to find the lower capacity vectors that satisfy (d, T) and system reliability is evaluated. In addition, a practical example of an IC card manufacturing system is utilized to illustrate the proposed algorithm and show that such an index can provide enough information for managers to make decisions.

Production network performance and reliability are essential to satisfy customer orders in a timely manner. This paper proposes a statistical method for a production system to satisfy customer demand with a desired level of confidence, referred to as yield confidence, while simultaneously considering system reliability, defined as the probability that the amount of input can be processed based on the capacities of the individual workstations. The approach models a production system as a stochastic-flow production network, characterized by a discrete time Markov chain (DTMC), where one or more rework actions are possible. This model quantifies the probability that raw input is transformed into a finished product, which is subsequently used to calculate the amount of raw input needed to satisfy demand with a user-specified level of yield confidence. A pair of case studies, taken from the tile and circuit board industries, illustrates the assessment techniques as well as methods to identify workstation level enhancements that can improve network performance and reliability most significantly. Our results indicate that improving the reliability of workstations can enhance yield confidence because a lower volume of raw input can produce the desired volume of output, thereby minimizing the load placed on the production network.

From the viewpoint of service level agreements, data transmission accuracy is one of the critical performances for assessing Internet by service providers and enterprise customers. The stochastic computer network (SCN), in which each edge has several capacities and the accuracy rate, has multiple terminals. This paper is aimed mainly to evaluate the system reliability for an SCN, where system reliability is the probability that the demand can be fulfilled under the total accuracy rate. A minimal capacity vector allows the system to transmit demand to each terminal under the total accuracy rate. This study proposes an efficient algorithm to find all minimal capacity vectors by minimal paths. The system reliability can then be computed in terms of all minimal capacity vectors by the recursive sum of disjoint products (RSDP) algorithm.

This paper presents a novel technique to measure the performance of a stochastic-flow manufacturing network (SMN) which violates the so-called flow conservation law due to the failure rates of stations. We address the mission reliability, the probability of demand satisfaction, as a performance indicator for the SMN while considering both the stochastic capacities and the multiple production lines. First, we construct a manufacturing system as an SMN through a graphical transformation, and decompose the transformed SMN into several paths for further analysis. Subsequently, two algorithms for different scenarios are designed to generate all minimal capacity vectors that stations should provide to satisfy the given demand. The first scenario is for the SMN with identical production lines in parallel. The second scenario is for distinct production lines with common stations in the SMN. We derive the mission reliability in terms of minimal capacity vectors by applying the recursive sum of disjoint products (RSDP) algorithm. A decision making issue is also discussed to decide a reliable production strategy.

An active research field is the evaluation of the reliability of a complex network. The most popular methods for such evaluation often use Minimal Paths (MP) or Minimal Cuts (MC) of the network. Although there are many algorithms developed to search for MP or MC, most of them are inefficient for searching a large network due to the combinatorial explosion problem. Another disadvantage is that the existing algorithms are applicable to specific counts of source and sink nodes (e.g., one-to-one, one-to-many, and so on). This article proposes a novel approach to search for all MP in a general flow network. The term “general” means that the approach can be used to search for all MP with multi-sources, multi-sinks in the network. The edges can be directed, undirected, or hybrid (mixed with directed and undirected arcs). Some benchmarks from the well-known algorithms in the literature are examined and compared. Moreover, the comprehensive tests are also performed with the grid networks, as well as the well-known networks in the literature to show the efficiency of the approach. A sample code is provided in the article for quick validation.

This paper proposes a performance indicator to evaluate the capability of a maintainable stochastic-flow network (MSFN). This MSFN is required to preserve a minimal service level so that it can senddunits of data or commodity from the source to the sink through multiple paths within time T. The proposed system reliability performance indicator quantifies the probability that a MSFN delivers a capacity level above the minimal service level with a budget no greater than B. Two procedures are integrated in the proposed algorithm: a procedure to estimate system reliability, and an adjusting procedure that utilizes the branch-and-bound approach for exact system reliability. The estimated system reliability with lower and upper bounds, and the exact system reliability, are then computed by applying the recursive sum of disjoint products (RSDP) algorithm.

This paper develops two techniques to analyse the performance of a stochastic-flow network (SFN) model, considering correlated failures. The first approach utilizes a correlated binomial distribution to characterize the failure behavior of the physical lines and routers internal to the individual edges and nodes in the network. The second employs a simulation technique, which can characterize correlated failures between every pair of physical lines and routers in the different edges and nodes comprising the network. Both approaches quantify the probability that a given amount of data can be sent from a source to a sink through this network. This probability that the network satisfies a specified level of demand is referred to as the SFN reliability. The techniques are demonstrated in the context of two case studies, including the Taiwan Academic Network, the backbone of the national computer network connecting all educational institutions in Taiwan. Experimental results demonstrate that correlation can produce a significantly negative impact on reliability, especially when there is a high level of network demand. The proposed approaches, thus, capture the influence of correlation on SFN reliability, offering methods to quantify the utility of reducing correlation.

This paper constructs a time-based multistate network composed of multistate edges to study network reliability. Each edge involves three attributes: variable capacity, lead time, and cost. The transmission time from the source to the sink is thus not fixed. Two problems are discussed in this paper. First, we evaluate the probability that the given amount of data can be sent through minimal paths simultaneously under both time threshold, and budget constraint. Such a probability we term network reliability. It can be treated as a performance index to measure the transmission ability of a complex multistate system. Then, the calculation procedures are proposed to make solution. To enhance network reliability, the network administrator decides the spare routing in advance to indicate the first and the second priority minimal paths. The second path will be responsible for the transmission duty if the first fails. The second problem is addressed to evaluate network reliability according to the spare routing.

Reducing the transmission time is an imperative issue for many real-life systems such as computers, telecommunication networks, and transportation systems. For a deterministic flow network, the well-known quickest path problem is to find a path with minimum time for sending a specified amount of data through a single minimal path. A stochastic-flow network (SFN) composed by multistate arcs is more suitable to describe some real systems. The minimum transmission time through a SFN is thus not fixed. Allowing the data to be sent through two disjoint minimal paths simultaneously will shorten the transmission time. Hence, this paper is concerned with evaluating the reliability defined as when a SFN can transmit a given amount of data through a specified pair of minimal paths simultaneously under both time and budget constraints. In terms of subsets-union methods, a solution procedure is first proposed to calculate the reliability. Furthermore, a spare routing for boosting the reliability is established in advance to indicate the first pair, and the spare pairs of minimal paths. The corresponding reliability can be subsequently computed. An easy criterion is finally proposed to derive an ideal spare routing with higher reliability.

To minimize transmission time, the quickest path problem arises to find a path which sends a given amount of data from the unique source to the unique sink. Two deterministic attributes are involved in this problem: the capacity, and the lead time. However, in many real-life networks such as computer systems, or telecommunication systems, the arc capacity should be multistate due to failure, maintenance, and other such conditions. Such a network is named a multistate flow network, and the minimum transmission time is thus not fixed. We modify the quickest path problem to a multistate case. The new problem is to evaluate the probability thatdunits of data can be transmitted from the unique source to the unique sink under both the time, and budget constraints. In particular, the data are transmitted throughkseparate minimal paths simultaneously. Such a probability is named system reliability. An efficient algorithm is proposed to generate all minimal system states fulfilling the demand, time, and budget constraints. The system reliability is subsequently computed in terms of such system states. The optimalkminimal paths with the highest system reliability can further be obtained.

Network reliability evaluation for flow networks is an important issue in quality management. Many real-life systems can be modeled as stochastic-flow networks, in which each branch is multistate due to complete failure, partial failure, maintenance, etc. That is, each branch has several capacities with a probability distribution, and may fail. Hence, network reliability is the probability that a specified flow can be transmitted through the network successfully. Although there are many researches related to the evaluation of network reliability for a stochastic-flow network, how to assign a set of multistate components to the network so that the network reliability is maximal is never discussed. Therefore, this paper devotes to evaluating the optimal network reliability under components-assignments subject to a transmission budget, in which the transmission cost depends on each component's capacity. The network reliability under a components-assignment can be computed in terms of minimal paths, and state-space decomposition. Subsequently, we propose an optimization method based on a genetic algorithm. The experimental results show that the proposed method can be executed efficiently in a reasonable time.

This work devotes to the application of network methods for the reliability of a complex supply chain system, which is a set of several factories with supply-demand relationship. Two characters are considered in the proposed network: 1) nodes, and arcs all have multiple possible capacities, and may fail; and 2) the capacity weight varies with arcs, nodes, and types of commodity. The purpose of this paper is to study the systems reliability, in this case the possibility that a given quantity (d1,d2) of two types of commodities can be transmitted from the source factory to the destination factory simultaneously. Such a possibility can be treated as a performance index to measure the quality level of a supply chain network. The flow model is constructed by flow assignments, and capacity vectors. The (d1,d2) -MP, which represents the capacity of each arc/node, is a minimal capacity vector meeting the demand constraint. A simple algorithm in terms of minimal paths is first proposed to generate all (d1,d2)-MP. The system reliability can then be calculated efficiently in terms of (d1,d2)-MP. Time complexity of the proposed algorithm is also analyzed.

The quickest path (QP) problem is to find a path which sends a given amount of data from the source to the sink such that the transmission time is minimized. Two attributes are involved, namely, the capacity and the lead time. The capacity of each arc is assumed to be deterministic. However, in many real-life flow networks such as computer systems, telecommunication systems, etc., the capacity of each arc should be stochastic due to failure, maintenance, etc. Such a network is named a stochastic-flow network. Hence, the minimum transmission time is not a fixed number. We modify the QP problem to a stochastic case. The new problem is to evaluate the probability thatdunits of data can be sent from the source to the sink under both timeTand budgetBconstraints. Such a probability is named the system reliability. In particular, the data can be transmitted through two disjoint minimal paths (MPs) simultaneously. A simple algorithm is proposed to generate all (d,T,B)-QPs, and the system reliability can subsequently be computed. The optimal pair of MPs with highest system reliability could further be obtained.

The system capacity of a deterministic flow network is the maximum flow from the source to the destination. In a single-commodity stochastic-flow network (branches all have several possible capacities, and may fail), the system reliability, the probability that the maximum flow is larger than or equal to a given demand, is an important performance index to measure the quality level of a network. In a two-commodity stochastic-flow network, different types of commodities are transmitted through the same network simultaneously, and compete for the capacities. We concentrate on the reliability problem for such a network subject to the budget constraint. This paper defines firstly the system capacity as a pattern. We propose a performance index, the probability that the system capacity is less than or equal to a given pattern subject to the budget constraint, to evaluate the system performance. A simple algorithm based on minimal cuts is proposed to generate all maximal vectors meeting the demand and budget constraints. The performance index can then be computed in terms of all such maximal vectors

Network analysis is an important approach to model real-world systems. System reliability, and system unreliability are two related performance indices useful to measure the quality level of a supply-demand system. For a binary-state network without flow, the system unreliability is the probability that the system can not connect the source and the sink. Extending to a limited-flow network in the single-commodity case, the arc capacity is stochastic, and the system capacity (i.e. the maximum flow) is not a fixed number. The system unreliability for (d+1), the probability that the upper bound of the system capacity equals d, can be computed in terms of upper boundary points. An upper boundary point is the maximal system state such that the system fulfills the demand. This paper concentrates on a multicommodity limited-flow network (MLFN) in which multicommodity are transmitted through unreliable nodes and arcs. Nevertheless, the system capacity is not suitable to be treated as the maximal sum of the commodity because each commodity consumes the capacity differently. We define the system capacity as a demand vector if the system fulfills at most such a demand vector. The main problem of this paper is to measure the quality level of a MLFN. We propose a new performance index, the probability that the upper bound of the system capacity equals the demand vector subject to the budget constraint, to evaluate the quality level of a MLFN. A branch-and-bound algorithm based on minimal cuts is presented to generate all upper boundary points in order to compute the performance index. The computational complexity of the proposed algorithm is analyzed

The quality of service is an important index to measure the performance of an information system. This paper constructs a stochastic-flow network to model the information system. In this network, each node and arc having a designated capacity will have different lower levels due to various partial and complete failures. The studied problem is to evaluate the possibility that a given amount of multicommodity can be sent through an information network under the cost constraint. Such a possibility, which is named the mission reliability, is an appropriate performance index to measure the quality level. The terminology "flow" represents the quantity of data transmitted via such a network, and "demand" represents the required data from clients. Based on the properties of minimal paths, a simple algorithm is first proposed to generate all lower boundary points for the demand; then, the mission reliability can be calculated in terms of such points. The lower boundary point for the demand is a minimal vector, which represents the capacity of each component (arc or node), such that the demand can be fulfilled. Extending the stochastic-flow network to the node failure case, another algorithm is proposed to calculate the mission reliability

System reliability evaluation for flow networks is an important issue in quality management. This paper concentrates on a stochastic-flow network in which nodes as well as branches have several possible capacities, and can fail. The possibility is evaluated that a given amount of messages can be transmitted through the stochastic-flow network under the budget constraint. Such a possibility, system reliability, is a performance index for a stochastic-flow network. A minimal path, an order sequence of nodes & branches from the source to the sink without cycles, is used to assign the flow to each component (branch or node). A lower boundary point for (d, C) is a minimal capacity vector, which enables the system to transmit d messages under the budget C. Based on minimal paths, an efficient algorithm is proposed to generate all lower boundary points for (d, C). The system reliability can then be calculated in terms of all lower boundary points for (d, C) by applying the inclusion-exclusion rule. Simulation shows that the implicit enumeration (step 1) of the proposed algorithm can be executed efficiently.