Studies in Systems, Decision and Control 254 Tofigh Allahviranloo Uncertain Information and Linear Systems Studies in Systems, Decision and Control Volume 254 Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other Of particular value to both the contributors and the readership are the short publication timeframe and the world-wide distribution and exposure which enable both a wide and rapid dissemination of research output ** Indexing: The books of this series are submitted to ISI, SCOPUS, DBLP, Ulrichs, MathSciNet, Current Mathematical Publications, Mathematical Reviews, Zentralblatt Math: MetaPress and Springerlink More information about this series at http://www.springer.com/series/13304 Tofigh Allahviranloo Uncertain Information and Linear Systems 123 Togh Allahviranloo Faculty of Engineering and Natural Sciences Bahỗeehir University Istanbul, Turkey ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-030-31323-4 ISBN 978-3-030-31324-1 (eBook) https://doi.org/10.1007/978-3-030-31324-1 © Springer Nature Switzerland AG 2020 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland To My Father And My Late Teacher, Prof G R Jahanshahloo Preface In this book, I tried to introduce and apply the uncertain information or data in several types to analyze the linear systems These versions of information are very applicable in our applied science The initial subjects of this book point out the important uncertainties to use in real-life problem modeling Having information about several types of ambiguities, vagueness, and uncertainties is important in modeling the problems that involve linguistic variables, parameters, and word computing Nowadays, most of our real-life problems are related to decision making at the right time, and therefore, we should use intelligent decision science Clearly, every intelligent system needs real data in our environment to have an appropriate and flexible mathematical model Most of the mentioned problems can be modeled by mathematical models, and a system of linear equations is their final status that must be solved The newest versions of uncertain information have been discussed in this book This book has been prepared for all undergraduate students in mathematics, computer science, and engineering involved with fuzzy and uncertainty Especially in industrial engineering and applied mathematics in the field of optimization, one of the most important subjects is the linear systems with uncertainty Istanbul, Turkey September, 2019 Tofigh Allahviranloo vii Contents Introduction 1.1 Introduction 1 9 9 12 14 16 21 24 31 35 49 56 61 61 61 68 70 92 109 119 141 Uncertainty 2.1 Introduction to Uncertainty 2.2 Uncertainty 2.2.1 Distribution Functions 2.2.2 Measurable Space 2.2.3 Uncertainty Space 2.2.4 Uncertainty Distribution Functions 2.2.5 Uncertain Set 2.2.6 Membership Function 2.2.7 Level Wise Membership Function or Interval Form 2.2.8 Arithmetic on Intervals Form of Membership Function 2.2.9 Distance Between Uncertain Sets 2.2.10 Ranking of Uncertain Sets Uncertain Linear Systems 3.1 Introduction 3.2 Uncertain Vector and Matrix 3.3 The Solution Set of an Uncertain Linear System 3.4 Solution Sets of Uncertain System of Linear Equations in Interval Parametric Format 3.5 The System of Linear Equations with Uncertain RHS 3.6 Uncertain Complex System 3.7 An Approach to Find the Algebraic Solution for Systems with Uncertain RHS 3.8 An Estimation of the Solution of an Uncertain Systems with Uncertain RHS ix x Contents 3.8.1 Interval Gaussian Elimination Method Allocating Method for the Uncertain Systems with Uncertain RHS 3.10 Allocating Method for the Fully Uncertain Systems 3.10.1 Allocating Method for the Fully Uncertain Systems (Non-symmetric Solutions) 3.11 LR Solution for Systems with Uncertain RHS (Best Approximation Method) 3.12 LR Solution for Systems with Uncertain RHS (Distance Method) 143 154 163 173 178 184 211 211 211 213 229 230 232 234 245 246 247 250 3.9 Advanced Uncertainty and Linear Equations 4.1 Introduction 4.2 The Uncertain Arithmetic on Pseudo-octagonal Uncertain Sets 4.2.1 The Uncertain Arithmetic Operations on Pseudo-octagonal Uncertain Sets 4.2.2 Solving Uncertain Equation as A ỵ X ẳ B 4.2.3 Solving Uncertain Equation as A Á X ¼ B 4.2.4 Solving Uncertain Equation as A X ỵ B ẳ C 4.3 Combined Uncertain Sets 4.3.1 Ranking of Combined Uncertain Sets 4.3.2 Distance Between Combined Uncertain Sets 4.3.3 Ranking Method Based on Expected Value 4.3.4 Advanced Combined Uncertain Sets (ACUSs) Bibliography 255 Chapter Introduction 1.1 Introduction Let’s start with a sentence from ‘Albert Einstein’: As for the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they not refer to reality Since the mathematical laws point to reality, this point is not conjectured with certainty, and since it speaks decisive mathematical rules, it does not refer to reality and is far from reality In fact, uncertainty has a history of human civilization and humanity has long been thinking of controlling and exploiting this type of information One of the most ancient and obscure concepts has been the phrase “luck” Evidence of gambling is said to have been obtained in Egypt in 3500 BC and found similar to the current dice there The gambling and dice have acted an important role in developing the theory of probability In the 15th century, Gerolamo Cardano was one of the most knowledgeable individuals in the field of formal operations of algebra In his “Game of Chance”, he presented his first analysis of lucky laws In this century, Galileo Galilei has also solved such problems in numerical form In 1657, Christiaan Huygens wrote the first book on probability entitled “On the calculation of chance games.” This book was a real birthday of probability The theory of probability started mathematically by Blaise Pascal and Pierre de Fermat in the 17th century, which sought to solve mathematical problems in certain gambling issues From the seventeenth century, the theory of probability was constantly developed and applied in various disciplines Nowadays, the possibility in most engineering and management fields is an important tool, and even its use in medicine, ethics, law, and so on In this regard, Pascal says: It’s great that science was © Springer Nature Switzerland AG 2020 T Allahviranloo, Uncertain Information and Linear Systems, Studies in Systems, Decision and Control 254, https://doi.org/10.1007/978-3-030-31324-1_1 4.3 Combined Uncertain Sets 243 Then, l3;l rị ẳ l1;l rị l2;l rị; l3;u rị ẳ l1;u rị l2;u rị; N3;l rị ẳ N1;l rị N2;l rị; N3;u rị ẳ N1;u rị N2;u rị Now we can define the difference as the following form as well, 9n3 ; n2 H n1 ẳ 1ịn3 Iff n2 ẳ n1 ỵ 1ịn3 Then l3;l rị ẳ l2;l rị l1;l rị; N3;l rị ẳ N2;l rị N1;l rị; l3;u rị ẳ l2;u rị l1;u rị; N3;u rị ẳ N2;u rị N1;u rị As like as the difference between two uncertain sets we have two cases as well and, Note Logically, If one of these cases is true then the Hukuhara difference exists and if n1 ÀH n2 exists in the first case and then n2 ÀH n1 also exist in second case Note Moreover, when two cases are the same then n3 ¼ 1ịn3 then n3 ẳ and n1 ẳ n2 So, in accordance with two cases and the definition of an interval, two end points of the difference can be defined as the following form, Case (i) È Â ÃÉ l3;l ¼ l1;l À l2;l ¼ uju l3;l ; l3;u ; È Â ÃÉ N3;l ¼ N1;l À N2;l ¼ uju N3;l ; N3;u ; È Â ÃÉ l3;u ¼ l1;l À l2;l ¼ max uju l3;l ; l3;u ; È Â ÃÉ N3;u ¼ N1;l À N2;l ¼ max uju N3;l ; N3;u ; And also, È Â ÃÉ l3;l ¼ l1;u À l2;u ¼ uju l3;l ; l3;u ; È Â ÃÉ N3;l ¼ N1;u À N2;u ¼ uju N3;l ; N3;u ; È Â ÃÉ l3;u ¼ l1;u À l2;u ¼ max uju l3;l ; l3;u ; È Â ÃÉ N3;u ¼ N1;u À N2;u ¼ max uju N3;l ; N3;u ; 244 Advanced Uncertainty and Linear Equations It is clear that, È É l3;l ¼ l1;l À l2;l ; l1;u À l2;u ; È É N3;l ¼ N1;l À N2;l ; N1;u À N2;u ; È É l3;u ¼ max l1;l À l2;l ; l1;u À l2;u ; È É N3;u ¼ max N1;l À N2;l ; N1;u À N2;u ; And, Â È É È Éà l3 ¼ l1;l À l2;l ; l1;u À l2;u ; max l1;l À l2;l ; l1;u À l2;u ; Â È É È Éà N3 ¼ N1;l À N2;l ; N1;u À N2;u ; max N1;l À N2;l ; N1;u À N2;u Now the generalized Hukuhara difference can be defined as follow Definition 4.20 (The Generalized Hukuhara for Combined Uncertain Sets) For two membership functions of two combined uncertain sets n1 ; n2 the generalized Hukuhara difference is defined as, 9n3 ; n1 gH n2 ẳ n3 , > > iị > > < > > > > : ðiiÞ & n1 ẳ n2 ỵ n3 , l1 ẳ l2 ỵ l3 N1 ẳ N2 ỵ N3 or& l2 ẳ l1 ỵ 1ịl3 n2 ẳ n1 ỵ 1ịn3 , N2 ẳ N1 ỵ 1ịN3 Such that in level wise form of n3 ẳ l3 ; N3 ị, ẩ ẫ ẩ Éà l3 ¼ l1;l À l2;l ; l1;u À l2;u ; max l1;l À l2;l ; l1;u À l2;u ; Â È É È Éà N3 ¼ N1;l À N2;l ; N1;u À N2;u ; max N1;l À N2;l ; N1;u À N2;u Similar to the uncertain sets, in some cases the gH-difference does not exist It is sufficient that both shifts into directions not exist Definition 4.21 (Hausdorff Distance of Combined Uncertain Sets) For two membership functions of uncertain sets n1 and n2 , DH ðn1 ; n2 Þ R ! is defined as follow, DH n1 ; n2 ị ẳ DH l1 ; N1 Þ; ðl2 ; N2 ÞÞ É È ¼ sup0 r fmax l1;l ðrÞ À l2;l ðrÞ; l1;u ðrÞ À l2;u ðrÞ ; É È max N1;l ðrÞ À N2;l ðrÞ; N1;u ðrÞ À N2;u ðrÞ g All of the following properties can be verified easily 4.3 Combined Uncertain Sets 245 The properties: Suppose that n1 ; n2 ; n3 ; n4 are combined uncertain sets and k R And n1 ÀgH n2 ; n1 ÀgH n3 exist Then, D H ð n1 ; n2 Þ ! D H n1 ; n2 ị ẳ , n1 ẳ n2 DH n1 ỵ n3 ; n2 ỵ n3 ị ẳ DH n1 ; n2 ị DH kn1 ; kn2 ị ẳ jkjDH n1; n2 ị DH n1 ÀgH n2 ; n1 ÀgH n3 ¼ DH ðn2 ; n3 Þ À Á DH ÀgH kn1 ; ÀgH kn2 ẳ jkjDH n1 ; n2 ị: Based on the previous properties on the uncertain sets these properties can be proved easily and in the similar way 4.3.1 Ranking of Combined Uncertain Sets In my opinion working with these types of uncertainties needs more than other ones to have an ordering Because in real life most of the phrases between humans are combined uncertain This is why we are going to discuss about one of the methods to rank of these information Actually, a generalization of the methods for uncertain sets can be applied to order the combined uncertain sets Here are main axioms of ranking of combined uncertain sets To this end, the inequality ow two combined sets should be defined Main Axioms for Ranking of Combined Uncertain Sets Any ranking method should have the following properties for any f; g and n, f If If If f f f f g and g f then f % g g and g n then f n g then f ỵ n g ỵ n: For these sets, the union of supports of two uncertain sets is defined as support of a membership function of combined uncertain set Definition 4.22 (Level Wise Ranking) The support of a combined uncertain set n ẳ l; N ị is shown by Snị and is dened as, Snị ẳ Slị [ SðNÞ where SðlÞ; SðNÞ are the supports of two uncertain sets l; N Now the following axioms are added to the previous axioms maxfsuppðfÞg\minfsuppðgÞg ) f g maxfsuppðfÞg minfsuppðgÞg ) f g: 246 Advanced Uncertainty and Linear Equations One of the simple ranking method for ordering of uncertain sets in level wise form has now been discussed 4.3.2 Distance Between Combined Uncertain Sets As we noticed before, the discussion about distance is reasonable and we are going to consider that Now for any combined uncertain set n ẳ l; N ị, each set is an uncertain set and each one can have the following own definitions and properties in the Borel set As a generalization of definition of distance for uncertain set as an Expected value of their Hukuhara difference We again have, Dðn1 ; n2 ị ẳ E jn1 H n2 jị Subject to the Hukuhara difference exists On the other hand we have, n1 À H n2 ¼ n3 iff n ¼ n2 ỵ n3 And based on the linearity property of the expected value, En1 ị ẳ En2 ị ỵ En3 Þ Then Eðn3 Þ ¼ Eðn1 Þ À Eðn2 Þ To evaluate the expected value of a combined uncertain set, we can show it by, E nị ẳ Elị þ EðNÞ In this case we suppose that the probability density function is approximated by the closest triangular or trapezoidal membership functions And we all know that there is a relation between two expected values EðlÞ; EðNÞ Because the probability density function N is defined using the membership function of l And Elị ẳ x0;l ỵ Zỵ x0 ENị ẳ x0;N ỵ Zỵ x0 lðxÞdx À Zx0 lðxÞdx À1 NðxÞdx À Zx0 NðxÞdx À1 4.3 Combined Uncertain Sets 247 So, x0;l ỵ x0;N ỵ E nị ẳ Zỵ 1 lxị ỵ Nxịdx x0 Zx0 lxị ỵ Nxịdx Now the distance can be dened, Dn1 ; n2 ị ẳ E jn1 H n2 j ẳ n3 ị Denition 4.23 (Expected Value in Level Wise Form of Membership Function) For the combined uncertain set and its level wise membership function The expected value of the uncertain set is defined as follow, E ð nị ẳ Z1 inf nẵr ỵ sup nẵrịdr ẳ Z1 inf ẵlẵr; Nẵr ỵ supẵlẵr; Nẵrịdr ẳ Z1 ll rị ỵ Nu rịịdr Example 4.24 For any combined uncertain set with Triangular uncertain components, n ẳ l; N ị, such that l ¼ ðl1 ; l2 ; l3 Þ and N ¼ ðN1 ; N2 ; N3 Þ The expected value is, E nị ẳ Elị ỵ ENị l1 ỵ N1 ỵ 2l2 ỵ N2 ị ỵ l3 ỵ N3 ẳ Example 4.25 For any combined uncertain set with Triangular uncertain components, n ẳ l; N ị, such that l ẳ l1 ; l2 ; l3 ; l4 ị and N ¼ ðN1 ; N2 ; N3 ; N4 Þ The expected value is, E ð nÞ ¼ 4.3.3 Elị ỵ ENị l1 ỵ N1 ỵ l2 ỵ N2 þ l3 þ N3 þ l4 þ N4 ¼ Ranking Method Based on Expected Value The expected value ranks also the combined uncertain sets And the properties of ranking of combined uncertain sets can be displayed as well Based on the metric 248 Advanced Uncertainty and Linear Equations properties of the expected value, it should be easy to show that the properties are true by using a ‘S’ shaped sigmoid function Now for any combined uncertain set n, with the expected value Enị ẳ E l; N Þ, the ranking method can be defined as follow, < ỵ ea jElịj ; Rank nị ẳ 0; : ; ỵ eajElịj Elị [ Elị ẳ Elị\0 Such that a is the center of gravity of N R xNxịdx 0\a ẳ RX X NðxÞdx It is clear that RankðfÞ ðÀ1; 1Þ and the larger value of Rank the higher preference of uncertain set Definition 4.26 (Ranking) For two uncertain sets n1 and n2 , n1 n2 , Rankðn1 Þ\Rankðn2 Þ n1 % n2 , Rankn1 ị ẳ Rank n2 ị In this definition, the expected value plays an important role (a) Suppose that Eðl1 Þ and E ðl2 Þ have the same signs, E ðl1 Þ and E ðl2 Þ are positive and E l1 ị\Elị Then El1 ị ẳ jEl1 ịj and Elị ẳ jEl2 ịj Then jEl1 ịj\jE ðl2 Þj ) ÀjEðl2 Þj\ À jE ðl1 Þj 1 \ ỵ eajEl1 ịj ỵ eajEl2 ịj ) Rank n1 ị\Rank n2 ị ỵ eajEl2 ịj \1 ỵ eajEl1 ịj ) E l1 ị and Eðl2 Þ are negative and Eðl1 Þ\Eðl2 Þ Then ÀE l1 ị ẳ jE l1 ịj and E l2 ị ¼ jE ðl2 Þj Then ÀE ðl2 Þ\ À E ðl1 Þ and jE ðl2 Þj\jE ðl1 Þj ) ÀjE l1 ịj\ jEl2 ịj 1 \ ỵ eajEl1 ịj ỵ eajEl2 ịj ) Rank n1 ị\Rank n2 ị ỵ ejEl1 ịj \1 ỵ ejEl2 ịj ) 4.3 Combined Uncertain Sets 249 Now we see that, in two cases E ðl1 Þ\E ðl2 Þ ) Rank ðn1 Þ\Rank ðn2 Þ , n1 n2 (b) Suppose that Eðl1 Þ\0 and E ðl2 Þ [ have the different signs so, E ðl1 Þ\E ðl2 Þ Then E l1 ị ẳ jE l1 ịj and E l2 Þ ¼ jE ðl2 Þj If ÀE ðl1 Þ\ À E ðl2 Þ then jEðl1 Þj\ À jEðl2 Þj and we have ÀjE ðl1 Þj\jE ðl1 Þj\ À jEðl2 Þj ỵ eajEl1 ịj \1 ỵ eajEl2 ịj 1 ) \ \ ỵ eajEl1 ịj ỵ eajEl2 ịj ỵ eajEl2 ịj ) Rank n1 ị\Rank ðn2 Þ And again, E ðl1 Þ\E ðl2 Þ ) Rank ðn1 Þ\Rank ðn2 Þ , n1 n2 The following properties are true clearly ã If n ẳ l; N ị ẳ 0; 0; 0; 0ị; 0; 0; 0; 0ịị then Elị ẳ so Ranknị ẳ ã If n ẳ l; N ị ẳ 1; 1; 1; 1ị; 1; 1; 1; 1ịị then Elị ẳ so Rankfị ẳ 0:7310585 ã If n ẳ l; N ị ẳ 1; 1; 1; 1ị; 1; 1; 1; 1ịị then Elị ẳ Rankfị ẳ 0:7310585 ã If n ẳ l; N ị ẳ a; a; a; aị; a; a; a; aịị then Elị ẳ a so the Rank depends on the sign of a and it is, < ỵ eajaj ; a [ aẳ0 Rankfị ẳ 0; : ; a\0 ỵ eajaj For any n1 ; n2 and n3 combined uncertain sets the main properties of ranking method can be proved as the same as properties of uncertain sets • • • • n1 4n1 , Rankðn1 Þ Rankðn1 Þ If f4g and g4f then f % g Because, If f4g and g4n then f4n If f4g then f þ n4g þ n: Example 4.27 Let l1 ¼ l2 ¼ N1 and n1 ¼ ðl1 ; N1 Þ ¼ 0:1; 0:3; 0:3; 0:5ị; 0:1; 0:3; 0:3; 0:5ịị; n2 ẳ l2 ; N2 ị ẳ 0:1; 0:3; 0:3; 0:5ị; 0:2; 0:3; 0:3; 0:4ÞÞ are two combined uncertain sets with triangular membership functions 250 Advanced Uncertainty and Linear Equations First we should determine the expected values, El1 ị ẳ E l2 ị ẳ 0:3; a1 ẳ a2 ẳ 0:3 Now we calculate the Rank ðn1 Þ and Rankðn2 Þ ẳ 0:522 ỵ ej0:09j Rankn2 ị ẳ ẳ 0:522 ỵ ej0:09j Rankn1 ị ẳ So, El1 Þ ¼ E ðl2 Þ ) Rank ðl1 Þ ¼ Rankðl2 Þ then g % f Definition 4.28 (Level Wise Ranking) For the following membership functions, n1 ẵr ẳ ẵl1 ½r; N1 ½r; n2 ½r ¼ ½l2 ½r; N2 ½r We sayn1 n2 iff l1 ½r l2 ½r & N1 ½r N2 ½r l1;u ðrÞ l2;u ðrÞN1;l ðrÞ N2;l ðrÞ; N1;u ðrÞ N2;u ðrÞ: Obviously all six conditions are true 4.3.4 iff È l1;l ðrÞ l2;l ðrÞ; Advanced Combined Uncertain Sets (ACUSs) We start this section by giving an example To this, first suppose that our information is conditional and their occurrence are connected to the occurrence of other supplementary information For instance, probably I will be at the university very likely about 11 P.M., but sure, after finishing my tasks very likely The happiness is, being at the university about 11 surely, and it is conditional to very likely finishing of the tasks Another example, you suppose that we would like to say that, when the individual iron in our body is likely about 29 lg The conditions are, • It is likely the body blood hemoglobin is about 10 and • It is likely the MCV factor is normal and • It is likely the MCH factor the blood is normal Then it is likely the body iron is about 29 lg It is clearly concluded that, this information is very incomplete and at the same time very complicated In this section, this information is defined as Advanced Combined Uncertain Sets (ACUS) An ACUS consists of several parts such that, each one plays its own role Basically, it contains three components and is shown by 4.3 Combined Uncertain Sets 251 ACUSðX; ðy; lY ðyÞÞ; g; qÞ: The first part consists of the form of ðy; lY ðyÞÞ where lY ðyÞ is the restriction on y in the set of random variables Y, as a membership function The second part, that is, g is a constraint for a number of unknown real values y such that yi occurs with a probability density and Z g ¼ probabilityðY is lY yịị ẳ R lY yịpY yịdy is g: ẳ f ðy; lY ðyÞÞ where R is defined as Y is probably p The third part, that is q is a scale of confidentiality related to the second component and considering the first component The important thing is, in most cases lY ðyÞ := lðyÞ and g inherently are imprecise requirements Definition 4.29 (ACUS) The general form of the ACUS mode is defined as follows ACUS ¼ ! \ k X; ðyi ; lðyi ÞÞ; g ðyi is lyi ịị; q ; iẳ1 i ẳ 1; ; k Subject to ! \ k Probability X is g yi is lyi ịị is q iẳ1 where lðyi Þ is an uncertain restriction on yi , and g, q are other uncertain sets where all can be approximated by trapezoidal membership functions The symbol \ is in the meaning and using all of ðyi is lðyi ÞÞ and g is conditional to all of ðyi is lðyi ÞÞ If all the conditions are true and in occurrence of g all of ðyi is lðyi ÞÞ are considered then for the last member the probability is, k ¼ n; ! \ n Prob X is g yi is lyi ịị ẳ ) iẳ1 q ¼ 1; It means the probability that y is g, equal to the value of one if all its conditions are met ! \ n Prob X is g yi is lyi ịị 6ẳ ) iẳ1 q 6¼ 1; k\n 252 Advanced Uncertainty and Linear Equations This means that the probability that yi is g, is not equal to the value of one if all its conditions are not established In fact, a ACUS-process can also be interpreted as follows q ¼ ProbðX is gjðy1 is lðy1 Þ ^ ðy2 is lðy2 Þ ^ Á Á Á ^ ðyn is lðyn ÞÞ So Z R lgj \ n iẳ1 yi is lyi ịị wị:pXjy1 ^^yn wịdw is q where pyjy1 ^ÁÁÁ^yn is the underlying (hidden) conditional probability density of Y The scalar product of lgj \ n iẳ1 yi is lyi ịị wị:pXjy1 ^^yn wị ẳ Pgj \ n iẳ1 yi is lyi ịị Is the probability measure of g \ ni¼1 ðyi is lðyi ÞÞ In fact we have lgj \ n i¼1 ðyi is lyi ịị wị:pXjy1 ^^yn wị ẳ Pgj \ n yi is lyi ịị Z iẳ1 ẳ lgj \ n yi is lyi ịị wị:pXjy1 ^^yn wịdw iẳ1 R Considering yi ị ẳ Ayi , more concretely we have, for two ACUSx1 ¼ À Á Á À À Á Á x1 ; y1 ; Ay1 ; g1 ; p1 ; ACUSx2 ¼ x; y2 ; Ay2 ; g2 ; p2 where 8À Á ÀÀ Á À Á À ÁÁ > y1 ; Ay1 ¼ y11 ; Ay11 & y12 ; Ay12 & & y1n ; Ay1n ; > > n < À Á P g1 ¼ /1 y1 ; Ay1 ẳ a0 ỵ Ay1i ; a0 ; and Ay1i are fuzzy numbers; > i¼1 R > >p ¼ l : R g1 jAy1 ðwÞ:pxjy ðwÞdðwÞ: And 8À Á ÀÀ Á À Á À ÁÁ > y2 ; Ay2 ¼ y21 ; Ay11 & y22 ; Ay22 & & y2m ; Ay2m ; > > n < À Á P g2 ¼ / y ; A y ¼ b0 ỵ bi Ay2i ; b0 ; bi and Ay2i are fuzzy numbers > i¼1 R > >p ¼ l : R g2 jAy2 ðwÞ:pxjy ðwÞdðwÞ: 4.3 Combined Uncertain Sets 253 we obtain ACUSx1 à ACUSx2 , where Symbol (*) in is an arbitrary operator as follows: R1 and R2 for ACUSx1 and ACUSx2 , first we define the rules R1 : R1 : If y1 is Ay1 then x1 is g1 & Probx1 is g1 ị ẳ p1 ; R2 : If y2 is Ay2 then x2 is g2 & Probx2 is g2 ị ẳ p2 So, Rwị ¼ ðR1 à R2 ÞðwÞ : if y is Ay then Z ẳ l^gjAy wị:pxjy wịdwị x is ^ g&^ p R ÀÀ Á Á where Ay ¼ / Ay1 ; Ay2 ; v ; i ¼ 1; ; n Now the concept for addition and multiplication is: ^gi wị ẳ sup ẩ w ẫ Ay ðvÞ; RðwÞ jv V ; ðwÞ U where / is an arbitrary function, depending on the type of issue, for example: ÀÀ Á Á X /P Ay1 ; Ay2 ; v ¼ ki Ayi ; ki f0; 1g i ÀÀ Á Á Y /Q Ay1 ; Ay2 ; v ¼ ki Aiyi ; ki f0; 1g i À À Á Á À À Á Á So for two ACUSx1 ¼ x; y1 ; Ay1 ; g1 ; p1 ; ACUSx2 ¼ x; y2 ; Ay2 ; g2 ; p2 we have: À Á ACUSx1 à ACUSx2 ¼ Ay1 à Ay2 ; R1 à R2 ; R01 à R02 Symbol (*) used in R1 à R2 and R01 à R02 are non-synonymous with the symbol (*) in Ay1 à Ay2 Let symbol (*) is the summation so Ay1 ỵ Ay2 is a fuzzy number: qị ẳ sup l vị ^ l q v ị ; ^ ẳ ỵ Ay2 Ay1 Ay2 v lR1 ỵ R2 wị ẳ sup lg1 jAy ðwÞ ^ lg2 jAy ðz À wÞ ; ^ ¼ min; lAy w Denoting these probability density functions as pRx1 jy1 and pRx2 jy2 Z pR1 ỵ R2 ẳ pRx1 jy1 wịpRx2 jy2 z wịdwị R z ¼ kjv 254 Advanced Uncertainty and Linear Equations So, ACUSx1 ỵ ACUSx2 ẳ Ay ; g; P ; where Ay ẳ Ay1 ỵ Ay2 ; > > n < À Á P ci Ay ; g ¼ gR1 ogR2 ¼ / Ay ¼ c0 þ > i¼1 R > : P ¼ P oP ¼ l ðwÞ:p ðwÞdðwÞ: R1 R2 xjy R gjAy & c0 ; ci and Ay are fuzzy numbers; c0 6¼ a0 ỵ b0 ; ci 6ẳ ỵ bi ; In conclusion, in this chapter we discussed about some new operations based on transmission of support on general uncertain sets and it means the new results are more general than usual computations on the special case of these types of uncertain sets The sets that we considered are the general form of triangular and trapezoidal uncertain sets The main advantage of these operations is, doing on the general uncertain sets makes less uncertainty in the results 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with certainty, and since it speaks decisive mathematical rules, it does not refer to reality and