IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 8, Issue 2 (Sep. - Oct. 2013), PP 10-18 www.iosrjournals.org www.iosrjournals.org 11 | Page Development of An inventory model with Volume flexibility, random deterioration and increasing exponentially demand rate Dr. Ravish Kumar Yadav 1 , Ms. Pratibha Devi 2 1, Associate Professor, Department of Mathematics, Hindu College ,Moradabad 2, Research Scholar, Department of Mathematics, Hindu College ,Moradabad Abstract: A volume flexible inventory model is developed. This model is developed for deteriorating item by assuming that the deterioration rate is depend on a parameter and time. The production rate is variable, production cost become a function of production rate, unit cost depending upon material cost, labour cost and tool cost. The demand rate is increasing exponentially shortages are allowed. Cost minimization technique used to find out optimal values for different inventory variables. Key words: volume flexibility, deterioration I. Introduction: This study presents a production inventory model with a time dependent random deterioration function and increasing exponentially demand over a fix time horizon with the finite. The shortages allow and excess demand is backlogged. Expressions for optimal parameter are obtained .We also obtained Production scheduling period, maximum inventory level and total average cost. An inventory system the effect of deterioration plays an important role. Deterioration is derived as decay or damage such that the item cannot be used for its original propose. Foods, pharmaceuticals, chemicals, blood, drugs are a few examples of such items in which sufficient deterioration can take place during the storage period of the units and the importance of this loss must be taken into account when analyzing the system. In this paper we present a realistic inventory model in which the production rate is variable and demand is an exponentially increasing function time and deterioration is random function says that deterioration of an item depends upon the fluctuation of humidity, temperature, etc. It would be more reasonable and realistic if we assume the deterioration function  to depend upon a parameter " "  in addition to time t . When describing optimum policies for deteriorating items Ghare and Schrader (1963) proposed a constant rate of deterioration and constant rate demand. In recent year, inventory problem for deterioration items have been widely studied after Ghare and Schrader (1963), Covert and Philip (1973) formulated the model for variable deterioration rate with two parameters Weibull disturbation Goswami and Chaudhuri (1991), Bose et al (1995) assumed either instantaneous or finite production with different assumption on the pattern of deterioration. Balkhi and Benkheroot (1996) considered a production a production lot size inventory model with arbitrary production and demand rate depends on the time function. Bhunia and Maiti’s (1977) model to formulate a production inventory model. Chang and Deve (1999) investigated an EOQ model allow shortage and backlogging. It is assumed that the backlogging rate is variable and dependent on the length of waiting time for the next replenishment. Recently, many researchers have modified inventory policies by considering the “ time proportional partial backlogging rate” such as Wang (2002), Perumal (2002), Teng et al (2003), Skouri and Papachristos (2003) and Kun-Shan et al (2005) etc. In the Classical Economic Production Lot Size(EPLS) model, the production rate of a machine is regarded to be pre-determinded and inflexible1.Alder and Nanda, Sule, Axsater and Elmaghraby, Muth and Spearmann extended the EPLS model to situations where learning effects would induce an increase in the production rate. Proteus7, Rosenblat and Lee and Cheng considered the EPLS model in an imperfect production process in which the demand would exceed the supply. Schweitzer and Seidmann adopted, for the first time, the concept of flexibility in the machine production rate and discussed optimization of processing rates for a FMS (flexible manufacturing system). Obviously, the machine production rate is a decision variable in the case of a FMS and then the unit production cost becomes a function of the production rate. Khouja and Mehrez and Khouja extended the EPLS model to an imperfect production process with a flexible production rate. Silver, Moon, Gallego and Simchi Levi discussed the effects of slowing down production in the context of a manufacturing equipment of a family of items, assuming a common cycle for all the items. Gallego extended this model by removing the stipulation of a common cycle for all the items. But the above studies did not consider the demand rate to be variable. It is a common belief that large piles of goods displayed in a supermarket lead the customers to buy more. Volume flexibility is a major component in a FMS. The