Saddle Point Optimality Criteria of Interval Valued Non-Linear Programming Problem

The present paper aims to develop the Kuhn-Tucker and Fritz John criteria for saddle point optimality of interval-valued nonlinear programming problem.To achieve the study objective,we have proposed the definition of minimizer and maximizer of an interval-valued non-linear programming problem.Also,we have introduced the interval-valued Fritz-John and Kuhn Tucker saddle point problems.After that,we have established both the necessary and sufficient optimality conditions of an interval-valued non-linear minimization problem.Next,we have shown that both the saddle point conditions(Fritz-John and Kuhn-Tucker)are sufficient without any convexity requirements.Then with the convexity requirements,we have established that these saddle point optimality criteria are the necessary conditions for optimality of an interval-valued non-linear programming with real-valued constraints.Here,all the results are derived with the help of interval order relations.Finally,we illustrate all the results with the help of a numerical example.

Multi-objective vendor managed inventory system with interval type-2 fuzzy demand and order quantities

Purpose-The proposed IT2FMOVMI model intends to concurrently minimize total cost and warehouse space for the single vendor-retailer,multi-item and a consolidated vendor store.Regarding demand and order quantities with the deterministic and type-1 fuzzy numbers,we have also formulated the classic/crisp MOVMI model and type-1 fuzzy MOVMI(T1FMOVMI)model.The suggested solution technique can solve both crisp MOVMIand T1FMOVMIproblems.By finding the optimal ordered quantities and backorder levels,the Paretofronts are constructed to form the solution sets for the three models.Design/methodology/approach-A multi-objective vendor managed inventory(MOVMI)is the most recognized marketing and delivery technique for the service provider and the retail in the supply chain in Industry 4.0.Due to the evolving market conditions,the characteristics of the individual product,the delivery period and the manufacturing costs,the demand rate and order quantity of the MOVMI device are highly unpredictable.In such a scenario,a MOVMI system with a deterministic demand rate and order quantity cannot be designed to estimate the highly unforeseen cost of the problem.This paper introduces a novel interval type-2 fuzzy multi-objective vendor managed inventory(IT2FMOVMI)system,which uses interval type-2 fuzzy numbers(IT2FNs)to represent demand rate and order quantities.As the model is an NP-hard,the well-known meta-heuristic algorithm named NSGA-II(Non-dominated sorted genetic algorithm-II)with EKM(Enhanced Karnink-Mendel)algorithm based solution method has been established.Findings-The experimental simulations for the five test problems that demonstrated distinct conditions are considered from the real-datasets of SAPCO company.Experimental study concludes that T1FMOVMI and crisp MOVMI schemes are outclassed by IT2FMOVMI model,offering more accurate Pareto-Fronts and efficiency measurement values.Originality/value-Using fuzzy sets theory,a significant amount of work has been already done in past decades from various points of views to model the MOVMI.However,this is the very first attempt to introduce type-2 fuzzy modelling for the problem to address the realistic implementation of the imprecise parameters.

The Existence of Solutions of Elliptic Equations with Neumann Boundary Condition for Superlinear Problems

In this paper, we study and discuss the existence of multiple solutions of a class of non–linear elliptic equations with Neumann boundary condition, and obtain at least seven non–trivial solutions in which two are positive, two are negative and three are sign–changing. The study of problem (1.1): is based on the variational methods and critical point theory. We form our conclusion by using the sub–sup solution method, Mountain Pass Theorem in order intervals, Leray–Schauder degree theory and the invariance of decreasing flow.