Z-Advanced numbers processes

Abstract

Decisions are made based on information. Information must be reliable in order to be useful and in many subjects this confidence depends on some conditions. Thus, the Z-Advanced number (ZA-number) consists of several parts each of which play their own roles. The ZA-number is classified in two main states. The first state that is, ZA<inf>One</inf>((y<inf>i</inf>,A<inf>y<inf>i</inf> </inf>),B,C) contains three members. The first part consists of i-th members in the form of (y<inf>i</inf>,A<inf>y<inf>i</inf> </inf>) where each of the A<inf>y<inf>i</inf> </inf> is the restriction for y<inf>i</inf> corresponding itself. The second part, that is, B is a restriction (constraint) for unknown real values x such that y<inf>i</inf> occurs absolutely sure. The third part, that is C is a scale of confidentiality related to the second component, subject to first component. What is important to keep in mind is that A and B in most cases, are based on perception. And therefore, they are inherently imprecise requirements. Inaccuracies of A and B can be achieved by the use of simplifying assumptions about A and B used, hypotheses that aim at simplifying the complexity of calculations with Z and increasing the informative results of calculations. Second mode ZA<inf>Two</inf>((y<inf>1,i</inf>,A<inf>y<inf>1,i</inf> </inf>),(y<inf>2,j</inf>,A<inf>y<inf>2,j</inf> </inf>),…,(y<inf>n,z</inf>,A<inf>y<inf>n,z</inf> </inf>),B,C) contains (n+2) members. The first part consists of i-th members in the form of A<inf>y<inf>i</inf> </inf>. That each of A<inf>y<inf>1i</inf> </inf> is the approximate value for y<inf>1i</inf> corresponding itself. The second part consists of j-th member in the form of A<inf>y2j</inf> that each of A<inf>y2j</inf> is the limit for y<inf>2j</inf> corresponding themselves that are dependent on A<inf>y<inf>1i</inf> </inf> prior to themselves. And in this way, the (n)h part consists of Z-numbers in the form of A<inf>y<inf>z</inf> </inf> that each one of A<inf>y<inf>1z</inf> </inf> is the approximate value for y<inf>1z</inf> corresponding themselves that are dependent on A<inf>y</inf> prior to itself. The (n+1)th part that is, B is the restriction on unknown variable x that has real value, can have, under the condition that the y<inf>i</inf> have certainly occurred. The (n+2)th part, that is C, is a scale of confidentiality related to (n+1)th component on the condition that (n) th component is dependent on the first component. The extended version of ZA-numbers is as follow ZA<inf>One</inf>((y<inf>i</inf>,A<inf>y<inf>i</inf> </inf>),B,C) ZA<inf>Two</inf>((y<inf>1,i</inf>,A<inf>y<inf>1,i</inf> </inf>),(y<inf>2,j</inf>,A<inf>y<inf>2,j</inf> </inf>),…,(y<inf>n,z</inf>,A<inf>y<inf>n,z</inf> </inf>),B,C) which will be discussed in details. The main aim of this article is to introduce the concept (idea) of ZA-numbers and determine the framework of calculations methods with these numbers. © 2019 Elsevier B.V., All rights reserved.