Mathematical Model Considering Depreciation under Two Parameter Weibull Distribution Deterioration: Base for Computing Software

The present paper on inventory modeling is only a branch of Operation Research which may be treated as a branch of mathematics education. Inventory models with constant demand have been extensively studied in the inventory literature. In the present paper the previously studied research work is reconsidered taking price-dependent demand pattern in a power law form and a two-parameter Weibull distribution to represent the distribution of time to depreciation. The present paper includes the analytical development and solution of the proposed model. A numerical example has been discussed to illustrate the solution procedure and the sensitivity analysis of the solution is then carried out. Finally the convexity condition of e cost function and actual computational procedure of few important results have been shown in I, II, III Appendices. Sensitivity analysis depicts the extent to which the optimal solution of the model is affected by changes or errors in its input parameter values. The model forms a base for development of a computing software for real industry application.

I. Introduction

The usual inventory practice of constant demand with zero shortfalls to control stock with function as polynomial was first developed long back in 1915 and since then not much effort was made to consider irregular demand variations in study. After a gap of around 6 decades, it was Donaldson [1], who gave an exact policy for replenishment for a stock system with rate of demand that increases linearly at a fixed timeline and zero stockouts. But this method, being analytical in nature was too computationally and conceptually complex to be carried over in future research. This forced the researcher to study and explore a simplified model to solve such complex problem stated simply otherwise, Ritchie [6] silver [7], and Mitra et.al [5] all gave different methods, to provide an optimized result for replenishment policy computed without a complex algorithm. Also to note, the models studied did not consider the existing stock. And it is well known that all type of inventory does undergo, depreciation it varying rate of change and with different times. Patel and Dane [4] studied an EOQ based model to determine linear rate of demand and fixed rate of depreciation. They took the time periods for planning and replenishment to be finite and equal. Kashani - Bahari [2] developed a system with same assumptions but time periods for replenishment to be unequal. It is to develop a model for systemizing a replacement policy for stock with quadratic rate of demand, constant rate of depreciation and finite planning time. In this paper, a model is formulated by taking into consideration a quadratic function for demand and fixed rate of depreciation. The model is formulated and solved using numerical example and tested by sensitivity analysis. It is proposed to further develop a model to optimize the time for replacement of the further replacement cycles. The model is explained in terms of application to finite horizon a numerical example for adjustment of internals of replacement.