Multi-valued Decision Diagrams Research Articles

A Branch-and-Price Algorithm Enhanced by Decision Diagrams for the Kidney Exchange Problem

Problem definition: Kidney paired donation programs allow patients registered with an incompatible donor to receive a suitable kidney from another donor, as long as the latter’s co-registered patient, if any, also receives a kidney from a different donor. The kidney exchange problem (KEP) aims to find an optimal collection of kidney exchanges taking the form of cycles and chains. Methodology/results: We develop the first decomposition method that is able to consider long cycles and long chains for projected large realistic instances. Particularly, we propose a branch-and-price framework in which the pricing problems are solved (for the first time in packing problems in a digraph) through multivalued decision diagrams. We present a new upper bound on the optimal value of the KEP, obtained via our master problem. Computational experiments show superior performance of our method over the state of the art by optimally solving almost all instances in the PrefLib library for multiple cycle and chain lengths. Managerial implications: Our algorithm also allows the prioritization of the solution composition, for example, chains over cycles or vice versa, and we conclude, similar to previous findings, that chains benefit the overall matching efficiency and highly sensitized patients. Funding: This work was supported by NSERC Discovery Grant (RGPIN-2021-02609). Supplemental Material: The e-companion is available at https://doi.org/10.1287/msom.2022.0192 .

Reliability Analysis on electric propulsion System Combining GO-FLOW Methodology with Multi-Valued Decision Diagrams

The electric propulsion system of an electric aircraft is the core power of the whole aircraft, which provides assurance for the safe flight of the aircraft. There are multiple stages in the flight process of electric aircraft, as well as different failure behaviors of system components at different stages. The electric motor system is a k-out-of-n system. Therefore, it is not possible to accurately describe the electric motor system with the existing operators when using the GO-FLOW method for electric propulsion system reliability calculations. In order to solve this problem, this paper models the electric motor system using the multi-valued decision diagram method (MDD method), based on which a new operator 43 is proposed to simulate the electric motor system, and then builds the GO-FLOW model and calculates the overall reliability based on the distributed electric propulsion system schematic. The example results show that the method can calculate the reliability of the distributed electric propulsion system more realistically.

Computing the reliability of mixed uncertain random [formula omitted]-out-of-[formula omitted] systems with multiple possible states

This paper presents an uncertain random multi-state k-out-of-n system (urm-k-out-of-n system, for short), where both random components and uncertain components coexist. States of the system and components display multiplicity and are defined as uncertain random variables. Since reliability is a crucial index to reflect system performance, this paper proves some formulas to calculate the reliability of urm-k-out-of-n systems in accordance with chance theory, which have two special cases, random systems and uncertain systems. Later, a mixed algorithm for multi-valued decision diagram (MDD) and binary search is designed to assess the reliability index. Compared to the general algorithm, it operates in significantly less time when the system becomes complex.

Efficient performability analysis of dynamic multi-state k-out-of-n: G systems

A dynamic multi-state k-out-of-n: G system, denoted by DMS(k, n, G) is a system where the system and its components exhibit multiple performance levels, and the system has different requirements on the number of working components in different states. DMS(k, n, G) abounds in both industrial and military applications. In this paper, a novel and efficient analytical method based on multi-valued decision diagrams (MDDs) is proposed for performability assessment of DMS(k, n, G) with non-identical components. Unlike existing approaches where multiple combination operations between models of lower performance levels are needed for constructing the model of a higher performance level, the proposed MDD generation algorithm constructs the system performability MDD in a top-down manner by considering multiple system state requirements simultaneously. A smart home lighting control system is analyzed to demonstrate the application of the proposed method. A detailed case study of an oil supply system is provided to verify the correctness of the proposed method and illustrate component sensitivity analysis. Complexity analysis and comprehensive empirical studies are performed to demonstrate that the efficiency of the proposed model construction is greatly improved as compared to the existing method, enabling fast model generation and efficient analysis of large-scale DMS(k, n, G).

Improved Peel-and-Bound: Methods for Generating Dual Bounds with Multivalued Decision Diagrams

Decision diagrams are an increasingly important tool in cutting-edge solvers for discrete optimization. However, the field of decision diagrams is relatively new, and is still incorporating the library of techniques that conventional solvers have had decades to build. We drew inspiration from the warm-start technique used in conventional solvers to address one of the major challenges faced by decision diagram based methods. Decision diagrams become more useful the wider they are allowed to be, but also become more costly to generate, especially with large numbers of variables. In the original version of this paper, we presented a method of peeling off a sub-graph of previously constructed diagrams and using it as the initial diagram for subsequent iterations that we call peel-and-bound. We tested the method on the sequence ordering problem, and our results indicate that our peel-and-bound scheme generates stronger bounds than a branch-and-bound scheme using the same propagators, and at significantly less computational cost. In this extended version of the paper, we also propose new methods for using relaxed decision diagrams to improve the solutions found using restricted decision diagrams, discuss the heuristic decisions involved with the parallelization of peel-and-bound, and discuss how peel-and-bound can be hyper-optimized for sequencing problems. Furthermore, we test the new methods on the sequence ordering problem and the traveling salesman problem with time-windows (TSPTW), and include an updated and generalized implementation of the algorithm capable of handling any discrete optimization problem. The new results show that peel-and-bound outperforms ddo (a decision diagram based branch-and-bound solver) on the TSPTW. We also close 15 open benchmark instances of the TSPTW.

System-Level Dependability Analysis of Bitcoin under Eclipse and 51% Attacks

Bitcoin is an electronic cryptocurrency developed based on Blockchain technology. With its decentralized feature, it has become incredibly popular since its invention. However, the Bitcoin network suffers from 51% attacks, where if malicious attackers’ control over half of the computing power, they are able to rewrite the network. The attackers are capable of doing so by initiating the Eclipse attack first, which aims to monopolize all communications from and to a controlled Bitcoin node. In this paper, we model and analyze the dependability of the Bitcoin network subject to the Eclipse and 51% attacks. We propose a hierarchical model that encompasses a continuous-time Markov chain method for the node-level dependability analysis and a multi-valued decision diagram method for the system-level dependability analysis. Detailed case studies on Bitcoin systems with homogeneous and heterogeneous nodes are conducted to demonstrate the proposed model and investigate the impacts of several critical parameters on Bitcoin network dependability.

Performance evaluation of non-repairable cyclic phased-mission systems using evidential reasoning rule and multi-valued decision diagrams

Phased-mission systems (PMS) often perform different missions that have multiple consecutive, non-overlapping phases. Many recent studies have focused on the reliability analysis for PMS. However, not much attention has been paid to the PMS performance evaluation and the phase cycle characteristic of PMS. In this paper, a performance evaluation method of non-repairable cyclic phased-mission systems (CPMS) is proposed. Considering the uncertainty caused by the change of phase duration and component performance degradation, the average duration of each phase and the fuzzy state probability vector of each component are obtained by the evidential reasoning rule (ER rule). Then, the multi-valued decision diagrams (MDD) manipulation rules suitable for cyclic phase conditions are proposed, and the component state composition path generation method based on cycle-end component states is developed. On the above basis, MDD models for CPMS (CPMS-MDD) in different states are established, and the system performance is evaluated by the continuous time Markov chains (CTMC). A phase duration adjustment strategy is further proposed based on the evaluation results. The effectiveness and correctness of the proposed method are verified by case studies on performance evaluation of satellite turntable system. Comparative study shows the concise and efficient of the proposed method.

Generalized Confidence Constraints

In robust optimization, finding a solution that solely respects the constraints is not enough. Usually, the uncertainty and unknown parameters of the model are represented by random variables. In such conditions, a good solution is a solution robust to most-likely assignments of these random variables. Recently, the Confidence constraint has been introduced by Mercier-Aubin et al. in order to enforce this type of robustness in constraint programming. Unfortunately, it is restricted to a conjunction of binary inequalities In this paper, we generalize the Confidence constraint to any constraint and propose an implementation based on Multi-valued Decision Diagrams (MDDs). The Confidence constraint is defined over a vector of random variables. For a given constraint C, and given a threshold, the Confidence constraint ensures that the probability for C to be satisfied by a sample of the random variables is greater than the threshold. We propose to use MDDs to represent the constraints on the random variables. MDDs are an efficient tool for representing combinatorial constraints, thanks to their exponential compression power. Here, both random and decision variables are stored in the MDD, and propagation rules are proposed for removing values of decision variables that cannot lead to robust solutions. Furthermore, for several constraints, we show that decision variables can be omitted from the MDD because lighter filtering algorithms are sufficient. This leads to gain an exponential factor in the MDD size. The experimental results obtained on a chemical deliveries problem in factories – where the chemicals consumption are uncertain – shows the efficiency of the proposed approach.

Symbolic Verification of Current-State Opacity of Discrete Event Systems Using Petri Nets

Given a discrete event system, it is said to be opaque if an intruder who can partially observe system behavior cannot infer that the system is necessarily in a predefined secret. A discrete event system is current-state opaque if an intruder is never able to verify whether the current state of the system is within the secret states. This article addresses the verification of current-state opacity of a discrete event system modeled with Petri nets, where the secret is defined as a set of states. The existing methods cannot deal with large-sized systems due to the curse of dimensionality. Multi-valued decision diagrams (MDDs) are widely accepted as an efficient and compact data structure for representing the extremely large reachability sets of Petri nets. We show that MDDs can be used in Petri nets with partial observable transitions and propose an approach for the verification of current-state opacity for bounded Petri nets. It is shown by experimental studies that the symbolic approach is practically more efficient than traditional techniques such as basis reachability graphs (BRGs).

Efficient Analysis of Resource Availability for Cloud Computing Systems to Reduce SLA Violations

Resource availability of cloud computing systems is vital for today's information infrastructures. However, the constituent computing nodes are not fault free. The availability requirements on the delivered computing resources should be defined clearly by using a formal, contractual agreement, known as the Service Level Agreement (SLA), between providers and customers. To reduce the risk of various SLA violations, an efficient analysis of resource availability is important. This paper proposes a new analytical approach based on multi-valued decision diagrams (MDD) for the efficient resource availability analysis of cloud computing systems with heterogeneous, multi-state computing nodes. Particularly, a novel and efficient MDD construction method is presented to generate compact MDD models encoding different amounts of cumulative computing resources. Two detailed case studies are performed to illustrate basics and application of the proposed approach to reduce SLA violations and guarantee the availability requirements on the delivered computing resources. Benchmark studies are further conducted to show efficiency of the proposed MDD-based approach as compared with the continuous-time Markov chains-based method and the universal generation function-based method.

A new method for analysis of Multi-State systems based on Multi-valued decision diagram under epistemic uncertainty

The development of the system's model is an important step in reliability analysis. The system's model in reliability analysis similar to other knowledge areas is used as an explanatory and research tool. In reliability analysis, this model is mathematical and allows us to determine quantitative characteristics to evaluate the behavior of the original system. Typically such a mathematical model approximates the behavior of the initial system and has some uncertainty. However, this uncertainty can increase significantly if the initial data for building the model is uncertain and incompletely specified. The uncertainty of the mathematical model also causes incorrect estimates of the system behavior and its reliability. Therefore, it is important to develop methods that allow us to take into account the uncertain nature of the initial data, first of all, take into account epistemic uncertainty in initial data. A new method for the development of a mathematical model of a Multi-State System (MSS) in the form of a Multi-Valued Decision Diagram based on incompletely specified and uncertain data is proposed. In other words, the proposed method takes into account the epistemic uncertainty of the initial data. The specific of this method is the use of Data Mining based classification procedures, in particular, the Fuzzy Decision Trees.

Encoding Multi-Valued Decision Diagram Constraints as Binary Constraint Trees

Ordered Multi-valued Decision Diagram (MDD) is a compact representation used to model various constraints, such as regular constraints and table constraints. It can be particularly useful for representing ad-hoc problem specific constraints. Many algorithms have been proposed to enforce Generalized Arc Consistency (GAC) on MDD constraints. In this paper, we introduce a new compact representation called Binary Constraint Tree (BCT). We propose tree binary encodings to transform any MDD constraint into a BCT constraint. We also present a specialized algorithm enforcing GAC on the BCT constraint resulting from a MDD constraint. Experimental results on a large set of benchmarks show that the BCT GAC algorithm can significantly outperform state-of-the-art MDD as well as table GAC algorithms.

Reliability evaluation of phased mission system under multiple correlated failure modes

AbstractMany large complex equipment, such as spacecraft and satellite, are typical Phased Mission Systems (PMSs). Due to sophisticated structure and complex missions, the components and systems of PMS have multiple failure modes which are correlated. On the one hand, taking multiple correlated failure modes in to consideration can describe the multistate characteristics of the system more accurately and decouple the correlation between different phases of the system more clearly. On the other hand, its complexity makes the reliability evaluation of PMS more difficult. To solve this problem, this paper discusses the modeling method of multiple failure modes and their correlation, proposes an improved combination method of Multiple Failure Mode Tree (MFMT) and Multistate and Multivalued Decision Diagram (MMDD) model, and realizes the reliability evaluation of PMS in cross‐level (failure mode‐component‐system) and cross‐scale (single phase‐multiple phases). Finally, the proposed method is applied to evaluate the reliability of the satellite attitude orbit control system (AOCS) in order to verify its feasibility.

Combinatorial Reliability Evaluation of Multi-State System with Epistemic Uncertainty

Multi-state systems (MSSs) are common in real-word applications, in which a system and/or its components exhibit multiple stochastically-dependent states or performance levels. Such characteristic poses challenges to the reliability evaluation of MSSs. Multi-valued decision diagrams (MDDs) have been developed to address the reliability analysis of MSSs under the assumption that the failure-time parameters of system components are deterministic. However, due to epistemic uncertainty, it is often difficult or impossible to obtain the determinate values of the component parameters. Therefore, this paper aims to address the MDD-based reliability evaluation of MSSs with epistemic uncertainty by incorporating the interval theory and fuzzy set theory. The proposed methods are verified through a detailed case study of a high-speed train bogie system. The results show that the proposed methods can obtain practical reliability evaluation results reflecting the condition of epistemic uncertainty.

Reliability assessment of non-repairable multi-state phased mission systems with backup missions

Recently, the reliability of the multi-state phased mission systems (MS-PMSs) have been widely studied, which is used in the aerospace industry. Except for the phase dependency, the considering of the components/system multi-state behaviors in the PMSs also makes the reliability modeling more complicated. Meanwhile, the redundancy design can ensure high system reliability for the aerospace industry. The mission back-up design is always designed except for the redundant components, which can be reactivated when the working components fail. With the mission backup mechanism, some critical missions could be re-executed. To deal with these complicated situations, a new model based on multi-state multi-valued decision diagram (MMDD) is developed for the reliability modeling of MS-PMSs considering phase backups and multi-results. By using the PMS-MMDD method, the system reliability can be assessed efficiently. The correctness of the proposed method is verified by comparing with the MC simulation method. At last, the reliability of the propulsion system in a spacecraft is assessed to illustrate the proposed method.

Compiling capacitated single-item lot-sizing problem in a CostMDD

This paper deals with capacitated single-item lot sizing problem (CLSP) in an interactive support system context. The interaction is done thanks to queries to the system: The decision-maker makes partial choices and asks about the consequences in terms of costs but also about possible inventory levels. Hence, three fundamental queries are investigated: Find production plans with a cost less than K; Find possible plans with conditioning on a pair of variables (production and inventory); Find the production plans that simultaneously satisfy the previous queries. A knowledge compilation approach is used and composed of two phases: offline and online. Offline, a top-down algorithm computes cost multivalued decision diagrams. While online, we show how cost multivalued decision diagrams can improve the reactivity of answers to queries.The proposed approach is compared to classical constraint programming with CP Optimizer. Tests indicate that our algorithm is more efficient than CP re-solving in terms of computation time.

Reliability evaluation of demand-based warm standby systems with capacity storage

Warm standby is an energy-saving redundancy technique that consumes less energy than a conventional hot standby method. It can be naturally integrated with an energy storage technique to enhance system reliability. However, the integration approach of both techniques and the advantage it affords to system reliability have not been reported in literature. This study resolves this limitation by formulating a novel reliability model for demand-based warm standby systems with capacity storage. In this model, the chronological characteristics of warm standby components are explicitly explored before and after their activation. Moreover, different utilization sequences of warm standby components and storage components are embedded to analyze the effects of these sequences on system reliability. A multi-valued decision diagram is developed to evaluate system reliability considering successful activation probabilities of warm standby components. This diagram is applicable to arbitrary component lifetime distributions and warm standby systems. Numerical examples are presented to verify the application of the proposed methodology.

Revisiting the Complexity Analysis of Conflict-Based Search: New Computational Techniques and Improved Bounds

The problem of Multi-Agent Path Finding (MAPF) calls for finding a set of conflict-free paths for a fleet of agents operating in a given environment. Arguably, the state-of-the-art approach to computing optimal solutions is Conflict-Based Search (CBS). In this work we revisit the complexity analysis of CBS to provide tighter bounds on the algorithm's run-time in the worst-case. Our analysis paves the way to better pinpoint the parameters that govern (in the worst case) the algorithm's computational complexity. Our analysis is based on two complementary approaches: In the first approach we bound the run-time using the size of a Multi-valued Decision Diagram (MDD)--a layered graph which compactly contains all possible single-agent paths between two given vertices for a specific path length. In the second approach we express the running time by a novel recurrence relation which bounds the algorithm's complexity. We use generating functions-based analysis in order to tightly bound the recurrence. Using these technique we provide several new upper-bounds on CBS's complexity. The results allow us to improve the existing bound on the running time of CBS for many cases. For example, on a set of common benchmarks we improve the upper-bound by a factor of at least 2 to the power of 10,000,000.

Reliability assessment of stochastic networks with ER connectivity and ER dependency

In stochastic networks, nodes usually function dependently and interact with other nodes through connectivity links or dependency links. In this paper, the model for stochastic networks considering sub-networks with connectivity and dependency links of Erdös-Rényi (ER) topology is proposed, which is defined as networks with arbitrary pair of nodes randomly connected/depended by a constant probability. The reliability evaluation framework for the proposed networks is developed, where both of the extended multi-valued decision diagram (MDD) method and Monte Carlo simulation (MCS) are involved. The MDD method is proposed to assess the reliability of deterministic stochastic networks with ER connectivity and dependency, where arbitrary time to failure distributions of nodes are allowed. Based on the reliability evaluation for a stochastic network with a deterministic structure, the MCS is employed to achieve the reliability analysis of corresponding stochastic networks. Numerical examples are presented to demonstrate the proposed stochastic network model and reliability evaluation framework, where the probability distributions for the reliability of stochastic networks are provided.

Multiple-valued Decision Diagrams for Multi-state System Performability Analysis

Background: In a multi-state system, either the entire system or its components will have more than two states. Different kind of multi-state systems have been investigated, such as multistate series-parallel systems, multi-state k-out-of-n systems and multi-state linear consecutive-k-outof- n systems. Methods: Performability analysis is concerned with the calculation of probability distribution of different system performance levels. In this paper, Multi-valued Decision Diagram (MDD) based approaches are discussed to conduct the performability analysis of multi-state systems. Results: Examples are provided to illustrate the use of MDD approaches for three types of multistate systems, e.g. multi-state series-parallel systems, multi-state k-out-of-n systems and multi-state linear consecutive-k-out-of-n systems. Conclusion: The empirical results show that the MDD approaches have lower computational complexity than the traditional methods in model generation and evaluation.