توصيفگر ها :
جبر ماكس-مين , جبر جمعي ماكس , ماتريسي بازه‌اي , معادلات ماتريسي , معادلات ماتريسي بازه‌اي , حل‌پذيري سرارسري , حل‌پذيري قوي سراسري , حل‌پذيري ضعيف سراسري
چكيده فارسي :
در اين پايان‌نامه، ابتدا جبر ماكس و جبر جمعي ماكس معرفي مي‌شوند. سپس به بررسي حل‌پذيري معادله ماتريسي بازه‌اي A⊗X⊗C = B پرداخته و ويژگي‌هاي حل‌پذيري اين معادله را در جبر ماكس-مين مورد بررسي قرار مي‌‌دهيم. در اين پايان‌نامه، قضاياي مرتبط با هر نوع حل‌پذيري را ارائه و اثبات مي‌كنيم.
سپس، معادلات ماتريسي بازه‌اي در جبر جمعي ماكس بررسي مي‌شوند. بررسي مي‌كنيم كه تحت چه شرايطي دستگاه معادلات A⊗X⊗C = B در جبر جمعي ماكس حل‌پذير است و حل‌پذيري مرتبط با اين دستگاه معادلات را تحليل مي‌كنيم. همچنين، قضاياي مرتبط با اين نوع حل‌پذيري اثبات خواهند شد.
چكيده انگليسي :
In this thesis, we first introduce the fundamental concepts of max-algebra and max-plus algebra and then analyze the properties and characteristics of these two types of algebra. Subsequently, by examining matrices and vectors within these algebras, we demonstrate that the two algebras are isomorphic to each other. Additionally, we investigate the spectral properties in max algebra and proceed to explain and prove important theorems such as the Perron-Frobenius theorem and the fixed-point theorem.
In the following, we investigate the conditions under which interval matrix equations in max-min algebra are solvable. In max-min algebra, classical addition and multiplication operations are replaced with maximum and minimum operations. Max-min algebra is a triple (I,⊕,⊗), where I=[O,I] is a linearly ordered set with the least element O and the greatest element I and ⊕, ⊗ are binary operations defined as $a⊕b :=max{a,b}, a⊗b :=min{a,b}. Furthermore, the focus of this research is on interval matrices, where each element is represented as an interval instead of a fixed value, indicating a range of possible values. We introduce three types of solvability for these interval matrix equations in max-min algebra: strong universal solvability, universal solvability, and weak universal solvability. Subsequently, we examine the matrix equations
A⊗X⊗C = B , where A, B and C are interval matrices, and X is the unknown matrix that needs to be determined. The concept of strong universal solvability is defined as the existence of a matrix X that satisfies the equation for all possible values within the given intervals A, B and C. Additionally, universal solvability requires that for each interval matrix B, there exists a matrix $X$ that satisfies the equation for all interval matrices A and C. We then explain that weak universal solvability has a more flexible condition, where the equation must be satisfied for every specific combination of values within the intervals. The main advantage of max-min algebra is that it simplifies complex calculations of discrete systems and enables the analysis of more intricate problems.
The capability of solving interval matrix equations in max-plus algebra has been investigated. Max-plus algebra is an algebraic structure in which the classical addition and multiplication operations are replaced by the operations ⊕ and, ⊗ defined as a⊕b:=max{a,b}, a⊗b:= a+b. The focus of this dissertation is on equations of the form A⊗X⊗C = B, where A, B and C are given interval matrices, and X is the unknown matrix to be determined. We also examine the conditions under which the system of equations A⊗X⊗C = B, in max-plus algebra is solvable and analyze the solvability associated with this system of equations. Additionally, theorems related to this type of solvability will be proven. Practical applications of these findings are demonstrated through examples such as modeling transportation systems where various travel times and connections with uncertain times are represented as intervals. The results show how to determine suitable connection times to ensure passengers reach their destinations within specified times. Consequently, this study provides a comprehensive framework for solving interval matrix equations in max-plus algebra, which enhances the robustness and reliability of systems modeled by these equations.
