Solvability And Its Applications Of Max-Plus Systems And Interval Max-Plus Systems

Max-plus systems and interval max-plus systems are two important study subjectsin max-plus algebra. The solutions of max-plus systems and interval max-plus systemsnot only have the theory significance, but also have important application values in thecontrol and optimization of flexible manufacturing, communication network and digitalcircuit. P Butkoviˇc gave the necessary and su?cient conditions of the solvability andthe unique solvability of max-plus systems. K Cechl′arov′a and R A Cuninghame-Greenproved a necessary and su?cient condition of the strong solvability of interval max-plussystems.The thesis is further discussing the solvability of max-plus systems and the strongsolvability of interval max-plus systems. Firstly, we introduce the concept of solvableentry of max-plus systems. We prove the necessary and su?cient conditions of the solv-ability and the unique solvability of max-plus systems o?ered by P Butkoviˇc using a newmethod and give a new necessary and su?cient condition of the unique solvability ofmax-plus systems. And then we describe the concepts of solvable interval and intervalstrong solution of interval max-plus systems and give the necessary and su?cient con-ditions of the strong solvability and the unique strong solvability of interval max-plussystems. Two polynomial algorithms are presented to decide the solvability of max-plussystems and the strong solvability of interval max-plus systems. Finally, we describe theapplications of the solvability of max-plus systems and the strong solvability of intervalmax-plus systems in communication network system and manufacturing system.This thesis is divided into four chapters, the structure is as follows:In Chapter 1, we introduce some basic concepts related to max-plus systems andinterval max-plus systems, such as max-plus algebra and its intervals, interval matrix,max-plus systems, interval max-plus systems, the interval operations, the matrix opera-tions and the interval matrix operations.In Chapter 2, we study the solvability and the unique solvability of max-plus systems.We introduce the concept of solvable entry of max-plus systems and give the feature ofsolvable entry. We prove the necessary and su?cient conditions of the solvability andthe unique solvability of max-plus systems o?ered by P Butkoviˇc using a new methodand provide a new necessary and su?cient condition of the unique solvability of max-plus systems. We propose a polynomial algorithm to decide the solvability of max-plussystems and give a numerical example.In Chapter 3, we research the strong solvability and the unique strong solvability ofinterval max-plus systems. We first introduce the concept of solvable interval of intervalmax-plus systems and consider the relation between solvable interval and solvable entry.Then, we prove a necessary and su?cient condition of the strong solvability of intervalmax-plus systems. It is proved that our criterion is equivalent to the criterion proposedby K Cechl′arov′a and R A Cuninghame-Green. In addition, we describe the concept ofinterval strong solution of interval max-plus systems and give the necessary and su?cientconditions of the unique strong solvability. Finally, we present a polynomial algorithmto decide the strong solvability of interval max-plus systems and provide a numericalexample.In Chapter 4, we discuss the application of the solvability of max-plus systems incommunication network system by setting up the multi-input and multi-output timesignal transmission network system model. We also discuss the application of the strongsolvability of interval max-plus systems in manufacturing system by building the workshopmulti-objective manufacturing system model.