The Multi-State Inclusion-Exclusion Principle: Conventional and Improved Versions| Chapter 9 | Research Highlights in Mathematics and Computer Science Vol. 5

 This branch considers an emergent variant of the chaste problem of calculating the chance of the union of n occurrences, or, more precisely, the anticipation of the disjunction (ORing) of n indicator variables for these occurrences, i.e., the contingency of this disjunction being prepared one. The variant existing herein everything with multi-valued variables, accompanying the required possibility representing the stability of a multi-state delivery network (MSDN), whose binary system gain is a two-valued function delineated in terms of multi-costly component successes. The chapter explains a simple order for dealing with the aforementioned question in terms of a standard instance MSDN, where achievement is defined as the disjunction of prime implicants or slightest paths of the appropriate network. This method engages the multi-state inclusion-exclusion (MS-IE) standard, which is guide a multi-state generalization of the ANDing movement's idempotency property. The method is illustrated with a itemized symbolic instance of a real-world record of what happened, and it yields a more precise report of the same numerical advantage obtained earlier. The example manifests the MS-IE method's well-known imperfections and extreme inefficiency, but it again reveals the way to alternative methods in which aforementioned a flaw is (incompletely) mitigated. The building of a multi-state probability-ready expression(MS-PRE)  is a famous example of these plans.  Applying the MS-IE principle to the union of minority (factored or composite) pathways that are convinced (at a low cost) to PRE form is another potential approach. An creative technique for fusing the MS-PRE and MS-IE standard together is used as a tertiary candidate design in this article. The use of MS-PRE is restricted to 'shellable' disjointing of ORed terms, and the happening partially orthogonalized disjunctive form is afterward subjected to MS-IE. This new approach avoids the issues brought on by either MS-PRE or MS-IE and form the most of two together of them. The identical actual-world issue that was met with the usual MS-IE is effectively exploited to depict the method.

Author(s) Details:

Ali Muhammad Ali Rushdi,
Department of Electrical and Computer Engineering, King Abdulaziz University, P. O. Box 80200, Jeddah, 21589, Saudi Arabia.

Motaz Hussain Amashah,
Department of Computer Engineering and Networks, College of Computer Science and Engineering, University of Jeddah, P. O. Box 80327, Jeddah, Saudi Arabia.

Please see the link here: https://stm.bookpi.org/RHMCS-V5/article/view/9546

Reliability Analysis of a Multi-State Delivery Network through the Symbolic Derivation of a Probability-Ready Expression | Chapter 12 | Recent Advances in Mathematical Research and Computer Science Vol. 4

 The dependability of a multi-state delivery network (MSDN) with different suppliers, transfer stations, and marketplaces (depicted as vertices) connected by branches of multi-state capabilities, delivering a specific commodity or service between their end vertices is the subject of this chapter. We use a symbolic logic interpretation of network success to meet market demand while staying within budget and manufacturing capacity constraints, even while the network is deteriorating. This system success is a two-valued function described in terms of multi-valued component successes, and it has been derived in minimum form as the disjunction of prime implicants or minimal pathways of the relevant network in the literature. This chapter's key contribution is a systematic method for transforming this minimum expression into a probability-ready expression (PRE). The PRE notion has been effectively extrapolated from the two-valued logical domain to the multi-valued logical domain. This concept is crucial because it enables a one-to-one transformation of a random logical expression to its statistical expectation form by simply replacing all logic variables with their statistical expectations, as well as substituting arithmetic multiplication and addition for their logical counterparts (ANDing and ORing). This chapter's running example for showing PRE derivation is admittedly long and tiresome. The statistical expectation of the statement is that it will be equal to 1, and this is referred to as network dependability. The proposed method is demonstrated using a thorough symbolic example of a real-life case study, and it yields a more precise version of the identical numerical value obtained previously by another method. This chapter is part of a larger project to create educational materials for a variety of prospective methodologies for measuring multi-state reliability.

Author(S) Details

Ali Muhammad Ali Rushdi
Department of Electrical and Computer Engineering, King Abdulaziz University, Faculty of Engineering, P. O. Box 80200, Jeddah 21589, Kingdom of Saudi Arabia.

Motaz Hussain Amashah
Department of Computer Engineering and Networks, College of Computer Science and Engineering, University of Jeddah, P.O.Box 80327, Jeddah, Saudi Arabia.

View Book:- https://stm.bookpi.org/RAMRCS-V4/article/view/4878

Complementation of Multi-State System Success to Obtain System Failure and Utilization of Both Boolean Functions in Checking Reliability Expressions | Chapter 11 | Novel Perspectives of Engineering Research Vol. 3

 This chapter addresses two important contributions that extend binary reliability techniques to multi-state reliability techniques: (a) the problem of complementation or inversion of the function of system success to that of system failure (or, equivalently, of deriving the logical minimal cutsets in terms of the logical minimal paths), and (b) the associated problem of hand-checking of a symbolic reliability expression, which relies on complete knowledge of the logical minimal paths. The research focuses on the resiliency of a multi-state distribution network with various suppliers and dwindling production capacity. It demonstrates two complementation techniques, one using multi-state De Morgan's rules and the other using the multi-state Boole-Shannon expansion. The chapter also presents a way for testing the stability of a multi-state system in terms of its logical minimal paths and logical minimal cutsets, which is a common use case for this complementation.

Author(S) Details

Ali Muhammad Ali Rushdi
Department of Electrical and Computer Engineering, King Abdulaziz University, Faculty of Engineering, P.O.Box 80200, Jeddah 21589, Saudi Arabia.

Motaz Hussain Amashah
Department of Computer Engineering and Networks, College of Computer Science and Engineering, University of Jeddah, P.O.Box 80327, Jeddah, Saudi Arabia.

View Book:- https://stm.bookpi.org/NPER-V3/article/view/4762

Multistate Reliability Evaluation of Communication Networks via Multi-Valued Karnaugh Maps and Exhaustive Search | Chapter 10 | Advanced Aspects of Engineering Research Vol. 1

 The reliability of two-terminal multistate flow networks is evaluated using two simple (yet useful) methods in this chapter. The Variable-Entered Karnaugh Map (VEKM) and the Multi-Valued Karnaugh Map (MVKM) are the two Karnaugh map variants used in these two methods (MVKM). These two versions are critical in providing not only the visual perspective needed to write better future applications, but also adequate verification tools. We compare these two types of map methods to the exhaustive search method, which ensures conceptual consistency at the cost of computational performance. Our goal is to assess the probability mass function (pmf) in a variety of situations, including flow from a source node to a sink node in a capacitated network with a multistate capacity model for the ties. Each network connection has a different capacity, which is thought to be mutually exclusive. The system's ability to successfully transmit at least a certain necessary system flow from the source (transmitter) to the sink (receiver) station is entirely dependent on its reliability. In order to achieve all successful states, the max-flow min-cut theorem is crucial. Two demonstrative examples with sufficient descriptions are given to illustrate the applicability of the proposed methods. In terms of pseudo-Boolean functions, the special case of a binary flow network is also considered.

Author (s) Details

Ali Muhammad Ali Rushdi
Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, P.O.Box 80200, Jeddah, 21589, Saudi Arabia.

Omar Mutab Alsalami
Department of Electrical Engineering, Faculty of Engineering, Taif University, Taif, Kingdom of Saudi Arabia.

View Book :- https://stm.bookpi.org/AAER-V1/article/view/336

Karnaugh-Map Analysis of a Commodity-Supply Multi-State Reliability System | Chapter 3 | Theory and Practice of Mathematics and Computer Science Vol. 8

 When at least km components are in state m or above for all m such that 1 m j, a multi-state k-out-of-n: G device has a multi-valued success greater than or equal to a certain value j (lying between 1 (the lowest non-zero output level) and M (the highest output level) This paper examines a commodity-supply scheme that serves as a gold standard for a non-repairable multi-state k-out-of-n: G system with non-identical independent components. Every instance of the multi-state system output is expressed as an explicit function of the system's multi-valued inputs. Our analysis yields a Multi-Valued Karnaugh Map (MVKM), which is a normal, special, and comprehensive representation of the multi-state system. We use “binary” entities to connect each instance of the output to the multi-valued inputs in order to construct this MVKM. These binary entities are represented by an eight-variable Conventional Karnaugh Map (CKM) that has been modified to a map with four four-valued variables each. Despite the fact that the maps used are relatively broad, they are still very useful due to their normal structure. There was no attempt to draw loops on the maps or find minimal formulas. The maps were merely useful tools. for combinatorial representation and jointly executing ANDing, ORing, and complementation operations The MVKM obtained can be used for symbolic analysis, yielding results that are numerically consistent with those previously obtained. The map is a valuable resource for computing a variety of Importance Measures for the system's components, as well as a useful tool for visualising several system resources.

Author(s) Details

Ali Muhammad Ali Rushdi
Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, P.O.Box 80200, Jeddah, 21589, Saudi Arabia.

Abdulghani Bakur Alsayegh
Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, P.O.Box 80200, Jeddah, 21589, Saudi Arabia.

View Book :- https://stm.bookpi.org/TPMCS-V8/issue/view/50