Optimization Algorithms of PERT/CPM Network Diagrams in Linear Diophantine Fuzzy Environment

The idea of linear Diophantine fuzzy set(LDFS)theory with its control parameters is a strong model for machine learning and optimization under uncertainty.The activity times in the critical path method(CPM)representation procedures approach are initially static,but in the Project Evaluation and Review Technique(PERT)approach,they are probabilistic.This study proposes a novel way of project review and assessment methodology for a project network in a linear Diophantine fuzzy(LDF)environment.The LDF expected task time,LDF variance,LDF critical path,and LDF total expected time for determining the project network are all computed using LDF numbers as the time of each activity in the project network.The primary premise of the LDF-PERT approach is to address ambiguities in project network activity timesmore simply than other approaches such as conventional PERT,Fuzzy PERT,and so on.The LDF-PERT is an efficient approach to analyzing symmetries in fuzzy control systems to seek an optimal decision.We also present a new approach for locating LDF-CPM in a project network with uncertain and erroneous activity timings.When the available resources and activity times are imprecise and unpredictable,this strategy can help decision-makers make better judgments in a project.A comparison analysis of the proposed technique with the existing techniques has also been discussed.The suggested techniques are demonstrated with two suitable numerical examples.

Spherical Linear Diophantine Fuzzy Sets with Modeling Uncertainties in MCDM

The existing concepts of picture fuzzy sets(PFS),spherical fuzzy sets(SFSs),T-spherical fuzzy sets(T-SFSs)and neutrosophic sets(NSs)have numerous applications in decision-making problems,but they have various strict limitations for their satisfaction,dissatisfaction,abstain or refusal grades.To relax these strict constraints,we introduce the concept of spherical linearDiophantine fuzzy sets(SLDFSs)with the inclusion of reference or control parameters.A SLDFSwith parameterizations process is very helpful formodeling uncertainties in themulti-criteria decisionmaking(MCDM)process.SLDFSs can classify a physical systemwith the help of reference parameters.We discuss various real-life applications of SLDFSs towards digital image processing,network systems,vote casting,electrical engineering,medication,and selection of optimal choice.We show some drawbacks of operations of picture fuzzy sets and their corresponding aggregation operators.Some new operations on picture fuzzy sets are also introduced.Some fundamental operations on SLDFSs and different types of score functions of spherical linear Diophantine fuzzy numbers(SLDFNs)are proposed.New aggregation operators named spherical linear Diophantine fuzzy weighted geometric aggregation(SLDFWGA)and spherical linear Diophantine fuzzy weighted average aggregation(SLDFWAA)operators are developed for a robust MCDM approach.An application of the proposed methodology with SLDF information is illustrated.The comparison analysis of the final ranking is also given to demonstrate the validity,feasibility,and efficiency of the proposed MCDM approach.

Type Synthesis of Walking Robot Legs

Walking robots use leg structures to overcome obstacles or move on complicated terrains. Most robots of current researches are equipped with legs of simple structure. The specific design method of walking robot legs is seldom studied. Based on the generalized-function(GF) set theory, a systematic type synthesis process of designing robot legs is introduced. The specific mobility of robot legs is analyzed to obtain two main leg types as the goal of design.Number synthesis problem is decomposed into two stages, actuation and constraint synthesis by name,corresponding to the combinatorics results of linear Diophantine equations. Additional restrictions are discussed to narrow the search range to propose practical limb expressions and kinematic-pair designs. Finally, all the fifty-one leg structures of four subtypes are carried out, some of which are chosen to make up robot prototypes, demonstrating the validity of the method. This paper proposed a novel type synthesis methodology, which could be used to systematically design various practical robot legs and the derived robots.

Two-parameter families of uniquely extendable Diophantine triples

Let A and K be positive integers and ε∈ {-2,-1,1,2}. The main contribution of the paper is a proof that each of the D(ε~2)-triples {K, A^2 K+2εA,(A +1)~2 K + 2ε(A+1)} has uniqui extension to a D(ε~2)-quadruple. This is used to slightly strengthen the conditions required for the existencc of a D(1)-quintuple whose smallest three elements form a regular triple.

Multi-dimensional versions of a formula of Popoviciu

In this paper, an explicit formulation for multivariate truncated power functions of degree one is given firstly. Based on multivariate truncated power functions of degree one, a formulation is presented which counts the number of non-negative integer solutions of s×(s + 1) linear Diophantine equations and it can be considered as a multi-dimensional versions of the formula counting the number of non-negative integer solutions of ax + by = n which is given by Popoviciu in 1953.