A deterministic INVENTORY model with two levelsof storage (own warehouse and rented warehouse) withnon-instantaneous deteriorating items is studied. The supplieroffers the retailer a trade credit period to settle theamount. Different scenarios based on the deterioration andthe trade credit period have been considered. In this article, we have framed two models considering single warehouse(Model-I) and two warehouses (Model-II) for non-instantaneousdeteriorating items. The objective of this work is tominimize the total INVENTORY cost and to find the optimallength of replenishment and the optimal order quantity. Mathematical theorems have been developed to determinethe existence and the uniqueness of the optimal solution. Computational algorithms for the two different models aredesigned to find the optimal order quantity and the optimalcycle time. Comparison between the optimal solutions forthe two models is also given. Numerical illustrations andmanagerial insights obtained demonstrate the applicationand the performance of the proposed theory.

This paper studies a location– routing– INVENTORYproblem in a multi-period closed-loop supply chain withmultiple suppliers, producers, distribution centers, customers, collection centers, recovery, and recycling centers. In thissupply chain, centers are multiple levels, a price increasefactor is considered for operational costs at centers, INVENTORYand shortage (including lost sales and backlog) are allowed atproduction centers, arrival time of vehicles of each plant to itsdedicated distribution centers and also departure from themare considered, in such a way that the sum of system costs andthe sum of maximum time at each level should be minimized. The aforementioned problem is formulated in the form of abi-objective nonlinear integer programming model. Due tothe NP-hard nature of the problem, two meta-heuristics, namely, non-dominated sorting genetic algorithm (NSGA-II)and multi-objective particle swarm optimization (MOPSO), are used in large sizes. In addition, a Taguchi method is usedto set the parameters of these algorithms to enhance theirperformance. To evaluate the efficiency of the proposedalgorithms, the results for small-sized problems are comparedwith the results of the e-constraint method. Finally, fourmeasuring metrics, namely, the number of Pareto solutions, mean ideal distance, spacing metric, and quality metric, areused to compare NSGA-II and MOPSO.

The concept of super line graph was introduced in the year 1995 by Bagga, Beineke and Varma. Given a graph with at least r edges, the super line graph of index r, Lr(G), has as its vertices the sets of r-edges of G, with two adjacent if there is an edge in one set adjacent to an edge in the other set. The line completion number lc(G) of a graph G is the least positive integer r for which Lr(G) is a complete graph. In this paper, we  nd the line completion number of grid graph Pn  Pm for various cases of n and m.

In this paper some properties of a generalization of Fibonacci sequence are investigated. Then we solve the Diophantine equations x2±kxy-y2±x=0, where k is a positive integer, and describe the structure of solutions.

Many studies have been conducted in the last decades on cellular polypropylene (Cell-PP) films. Most of them focus on theoptimisation of the material for sensor applications. Processed under Gas Diffusion Expansion (GDE), Cell-PP films showhigh piezoelectric activity and low stiffness/density, properties that make them ideal for sensors. GDE increases the heightand decreases the length over height ratio (aspect ratio) of individual voids within the material. This change in voidmorphology, and eventually stiffness, results in a nonlinear piezoelectric response of these materials. In this study, a Cell-PP sample was tested under static, quasi-static and low-frequency compressive stress. The main aim is to evaluate itsmechanical and piezoelectric properties in the nonlinear region of its response over strain. The load– deflection curves aswell as the piezoelectric responses were obtained for stresses up to 270 kPa (engineering strain close to 0. 26). It is shownthat both the magnitude of the initial load and the strain rate have a critical effect on the creep/stress relaxation of the filmand eventually on its piezoelectric response. Finally, it is shown that under dynamic conditions, and for the sameengineering strain region, it is more relevant to present the piezoelectric response, in terms of strain rather than stress.

Facility location problem (FLP) is a mathematicalway to optimally locate facilities within a set of candidatesto satisfy the requirements of a given set of clients. This study addressed the uncapacitated FLP as it assuresthat the capacity of every selected facility is finite. Thus, even if the demand is not known, which often is the case, inreality, organizations may still be able to take strategicdecisions such as locating the facilities. There are differentapproaches relevant to the uncapacitated FLP. Here, thecuckoo search via Le´ vy flight (CS-LF) was used to solvethe problem. Though hybrid methods produce betterresults, this study employed CS-LF to determine first itspotential in finding solutions for the problem, particularlywhen applied to a real-world problem. The method wasapplied to the data set obtained from a department store inDavao City, Philippines. Results showed that applying CSLFyielded better facility locations compared to particleswarm optimization and other existing algorithms. Although these results showed that CS-LF is a promisingmethod to solve this particular problem, further studies onother FLP are recommended to establish a strong foundationof the capability of CS-LF in solving FLP.

In the present study, the effect of size distribution of magnetite nanoparticles in a PVDF matrix on the magnetic propertiesof PVDF? Fe3O4 nanocomposites was experimentally and theoretically investigated. The size distribution of nanoparticlesin polymer matrix and morphology of the nanocomposites were studied by the means of scanning electron microscopyand atomic force microscopy. It was found that when the Fe3O4 nanoparticles are introduced into the polymermatrix, their coagulation takes place. The increase in the size of the particles depends on their concentration in the polymermatrix, the type of polymer (polar, non-polar, its viscosity, etc. ), reaction temperatures, etc. In addition, when Fe3O4nanoparticles are introduced into the polymer network, the oxidation of the surface layer of particles occurs and themagnetic size decreases. Consequently, the reduced magnetic properties may also be observed. The hysteresis loops havebeen recorded in small magnetic field range. It was found that the magnetic hysteresis parameters depend on the size andconcentration of Fe3O4 nanoparticles. Theoretical calculations were compared with experimental results obtained fromM(H) measurements. The reasons of differences between theoretical and experimental results have been explained.