Control and Optimization in Applied Mathematics
Volume:5 Issue: 1, Spring- Summer 2020

In this paper, ‎we develop general necessary optimality conditions of the KKT types for non-smooth continuous-time optimization problems with inequality constraints‎. ‎The primary instrument in our study is the concept of a convexificator‎. ‎Based on this concept‎, ‎non-smooth versions of the Mangasarian-Fromovitz constraint qualification are presented‎. ‎Then‎, ‎we derive optimality conditions for this problem under weak assumptions‎. ‎Indeed‎, ‎the constraint functions and the objective function that exist in this problem are not necessarily differentiable or convex.

Linear programming problems have exact parameters‎. ‎In most real-world‎, ‎we are dealing with situations in which accurate data and complete information are not available‎. ‎Uncertainty approaches such as fuzzy and random can be used to deal with uncertainties in real-life‎. ‎Fuzzy and stochastic theories cannot be used if the number of experts and the level of experience is so low that it is impossible to extract membership functions or the number of samples is small‎. ‎To solve these problems‎, ‎the grey system theory is proposed‎. ‎In this paper‎, ‎a linear programming problem in a grey environment with resources in interval grey numbers is considered‎. ‎Most of the proposed methods for solving grey linear programming problems become common linear programming problems‎. ‎However‎, ‎we seek to solve the problem directly without turning it into a standard linear programming problem for the purpose of maintaining uncertainty in the original problem data in the final solution‎. ‎For this purpose‎, ‎we present a method based on the duality theory for solving the grey linear programming problems‎. ‎This method is more straightforward and less complicated than previous methods‎. ‎We emphasize that the concept presented is beneficial for real and practical conditions in management and planning problems‎. ‎Therefore‎, ‎we shall illustrate our method with some examples in different situations.

A fuzzy distance measure is introduced in this paper to evaluate the fuzzy distance between two fuzzy numbers‎. ‎For this purpose‎, ‎alpha-values of fuzzy numbers are used to develop an integral-based fuzzy distance measure‎. ‎The properties of the proposed fuzzy distance measure are verified‎. ‎The proposed fuzzy distance measure is also compared with other fuzzy distance measures.

The relief logistics and humanitarian supply chain in academic literature refer to the process of planning‎, ‎execution‎, ‎and effective controlling of the flow of costs and information and storage of necessary goods and materials from the point of origin to consumption with the primary purpose of reducing and relieving the affected people suffer. This paper discusses a multi-objective model for multi-period location-distribution-routing problems considering the evacuation of casualties and homeless people and fuzzy paths in relief logistics‎. ‎Firstly‎, ‎an uncertain multi-objective model of the problem was developed based on uncertain parameters of demand‎, ‎time‎, ‎and transport capacity‎, ‎and then‎, ‎using the fuzzy programming method‎, ‎uncertain parameters of the problem were controlled‎. ‎As the problem is NP-hard and GAMS software has not able to solve the model in larger sizes‎, meta-heuristic algorithms of NSGA-II and MOPSO were used to solve the problem.

In this paper‎, ‎we present a new approach to solving stochastic differential equations and the Vasicek equation by using Brownian wavelets and multiple Ito-integral‎. ‎Firstly‎, ‎the calculation of the multiple Ito-integral based on the structure of Brownian motion is presented and the error of Ito-integrate computation is minimized under this condition‎. ‎Then‎, ‎the Brownian wavelets 1D and 3D based on coefficients Brownian motion are introduced‎. ‎After that‎, ‎a system of linear and nonlinear equations of coefficients Brownian motion is obtained such that by solving this system the approximate solution of the Vasicek equation is obtained‎. In the last section, ‎some numerical examples are given.

Estimating the target parameter while the prior distribution function is known‎, ‎and several observations which are provided by the sensor node is the main goal in this paper‎. ‎In wireless sensor networks (WSN)‎, ‎nodes sense the environment and send data to a sink node called Fusion Center (FC)‎. ‎FC collects data and estimates the observed parameter with user-defined precision‎. ‎The proposed algorithm increases network lifetime and has an efficient estimation process‎. ‎For this purpose‎, ‎the proposed algorithm schedules node’s activity and determines the multihop path between nodes and FC‎. ‎Simulation and performance analysis demonstrates proposed algorithm fulfills its goals.