Inventory model with stochastic lead-time and price dependent demand incorporating advance payment

Inventory model for an item is developed in stochastic environment with price-dependent demand over a finite time horizon. Here, probabilistic lead-time is considered and shortages are allowed (if occurs). In any business, placement of an order is normally connected with the advance payment (AP). Again, depending upon the amount of AP, unit price is quoted, i.e., price discount is allowed. Till now, this realistic factor is overlooked by the researchers. In this model, unit price is inversely related with the AP amount. Against this financial benefit, the management has to incur an expenditure paying interest against AP. Taking these into account, mathematical expression is derived for the expected average profit of the system. A closed form solution to maximize the expected average profit is obtained when demand is constant. In other cases model is solved using generalized reduced gradient (GRG) technique and stochastic search genetic algorithm (GA). Moreover, results of the models without and with advance payment are presented and solved. The numerical examples are presented to illustrate the model and the results for two models obtained from two methods are compared in different cases. Also, some parametric studies and sensitivity analyses have been carried out to illustrate the behavior of the proposed model. It is observed that advance payment has positive effect on the system.     

Inventory models for defective items incorporating marketing decisions with variable production cost

This paper investigates the finite replenishment inventory models of a single product with imperfect production process. In this process, a certain fraction or a random number of produced items are defective. These non-conforming items are rejected or reworked or if they reached to the customer, refunded. Here, a generalised unit cost function is formulated incorporating the several factors like raw material, labour, replenishment rate and others factors of the manufacturing system. The rate of replenishment is considered to be a variable. The selling price of an unit is determined by a mark-up over the production cost. Optimum production of the product is suggested to have maximum profit using a gradient based mathematical programming technique for optimization. Finally, numerical examples are given to illustrate the results and the significant features of the production system. As a particular case, the result of the perfect system (without defective items) are obtained. Also, the effect of changes in the selling rate, defectiveness, production cost and other parameters on the optimal average profit are graphically presented. Some interesting decisions regarding production policy are established.     

Two storage inventory model with fuzzy deterioration over a random planning horizon

An inventory model for a deteriorating item with stock dependent demand is developed under two storage facilities over a random planning horizon, which is assumed to follow exponential distribution with known parameter. For crisp deterioration rate, the expected profit is derived and maximized via genetic algorithm (GA). On the other hand, when deterioration rate is imprecise then optimistic/pessimistic equivalent of fuzzy objective function is obtained using possibility/necessity measure of fuzzy event. Fuzzy simulation process is proposed to maximize the optimistic/pessimistic return and finally fuzzy simulation-based GA is developed to solve the model. The models are illustrated with some numerical data. Sensitivity analyses on expected profit function with respect to distribution parameter λ and confidence levels α₁ and α₂ are also presented.     

A transportation problem with fuzzy-stochastic cost

In this paper, we consider some transportation problems (TPs) with different types of fuzzy-stochastic unit transportation costs and budget constraints. These fuzzy stochastic costs are reduced to corresponding crisp ones in two different ways. For the first method, using the definition of α$\alpha$-cut of the fuzzy numbers, expectation is taken separately on both lower and upper α$\alpha$-cuts and then mean expectation is calculated with the help of signed distance. In the second procedure, we realize fuzzy random events (ξ?r)$(\xi ?r)$ and (ξ?r)$(\xi ?r)$ for the fuzzy random variable (ξ)$(\xi )$. Using credibility measure of these events, mean chances for the above fuzzy random events are calculated and then expectation is taken to get the crisp expressions. The reduced deterministic problems of the fuzzy stochastic TP are solved using a real coded genetic algorithm with Roulette wheel selection, arithmetic crossover and random mutation. Few numerical examples are demonstrated to find the optimal solutions of the proposed models.     

Two storage inventory model in a mixed environment

Multi-item inventory models with two storage facility and bulk release pattern are developed with linearly time dependent demand in a finite time horizon under crisp, stochastic and fuzzy-stochastic environments. Here different inventory parameters—holding costs, ordering costs, purchase costs, etc.—are assumed as probabilistic or fuzzy in nature. In particular cases stochastic and crisp models are derived. Models are formulated as profit maximization principle and three different approaches are proposed for solution. In the first approach, fuzzy extension principle is used to find membership function of the objective function and then it’s Graded Mean Integration Value (GMIV) for different optimistic levels are taken as equivalent stochastic objectives. Then the stochastic model is transformed to a constraint multi-objective programming problem using Stochastic Non-linear Programming (SNLP) technique. The multi-objective problems are transferred to single objective problems using Interactive Fuzzy Satisfising (IFS) technique. Finally, a Region Reducing Genetic Algorithm (RRGA) based on entropy has been developed and implemented to solve the single objective problems. In the second approach, the above GMIV (which is stochastic in nature) is optimized with some degree of probability and using SNLP technique model is transferred to an equivalent single objective crisp problem and solved using RRGA. In the third approach, objective function is optimized with some degree of possibility/necessity and following this approach model is transformed to an equivalent constrained stochastic programming problem. Then it is transformed to an equivalent single objective crisp problem using SNLP technique and solved via RRGA. The models are illustrated with some numerical examples and some sensitivity analyses have been presented.     

Two-storage inventory model with lot-size dependent fuzzy lead-time under possibility constraints via genetic algorithm

Multi-item inventory models with stock dependent demand and two storage facilities are developed in a fuzzy environment where processing time of each unit is fuzzy and the processing time of a lot is correlated with its size. These are order-quantity reorder-point models with back-ordering if required. Here possibility and crisp constraints on investment and capacity of the small storehouse respectively are considered. The models are formulated as fuzzy chance constrained programming problem and is solved via generalized reduced gradient (GRG) technique when crisp equivalent of the constraints are available. A genetic algorithm (GA) is developed based on fuzzy simulation and entropy where region of search space gradually decreases to a small neighborhood of the optima and it is used to solve the models whenever the equivalent crisp form of the constraint is not available. The models are illustrated with some numerical examples and some sensitivity analyses have been done. For some particular cases results observed via GRG and GA are compared.     

Ru decorated Pt nanoparticles by a modified polyol process for enhanced catalytic activity for methanol oxidation

Ru decorated Pt nanoparticles on carbon nanotubes (MWCNTs) were prepared by a modified polyol synthesis method for enhanced catalytic activity for methanol oxidation. The characterizations for the electrocatalysts were carried out by transmission electron microscopy (TEM), X-ray diffraction (XRD), cyclic voltammetry (CV) and chronoamperometry (CA). The modified polyol synthesis method promoted position-controlled nucleation and growth of Ru atoms near Pt, and resulted in improved durability against catalyst poisoning compared to PtRu/MWCNTs prepared by common polyol method. This concept also allowed a high loading and dispersion of the catalyst on the carbon supports with few agglomerations of catalyst nanoparticles, resulting in high catalytic activity.     

XOR gate response in a mesoscopic ring with embedded quantum dots

We address XOR gate response in a mesoscopic ring threaded by a magnetic flux . The ring, composed of identical quantum dots, is symmetrically attached to two semi-infinite one-dimensional metallic electrodes and two gate voltages, viz, V_a and V_b, are applied, respectively, in each arm of the ring which are treated as the two inputs of the XOR gate. The calculations are based on the tight-binding model and the Green’s function method, which numerically compute the conductance–energy and current–voltage characteristics as functions of the ring-electrodes coupling strengths, magnetic flux and gate voltages. Quite interestingly it is observed that, for =₀/2 (₀=ch/e, the elementary flux-quantum) a high output current (1) (in the logical sense) appears if one, and only one, of the inputs to the gate is high (1), while if both inputs are low (0) or both are high (1), a low output current (0) appears. It clearly demonstrates the XOR behavior and this aspect may be utilized in designing the electronic logic gate.     

Interlayer force constants in high symmetry layer crystals

Symmetry adapted linear chain model has been used for the evaluation of compressional and shear interlayer force constants in MnCl₂ and GaSe from the observed zone centre and zone edge frequencies. The method is applicable to two major groups of high symmetry layer crystals having XAX and XAAX type layers.     

An empirical fit to estimated neutron emission cross sections from proton induced reactions

Neutron emission cross section for various elements from⁹Be to²⁰⁹Bi have been calculated using the hybrid model code ALICE-91 for proton induced reactions in the energy range 25 MeV to 105 Me V. An empirical expression relating neutron emission cross section to target mass number and incident proton energy has been obtained. The simple expression reduces the computation time significantly. The trend in the variation of neutron emission cross sections with respect to the target mass number and incident proton energy has been discussed within the framework of the model used.