IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 16, Issue 3 Ser. III (May – June 2020), PP 51-56 www.iosrjournals.org DOI: 10.9790/5728-1603035156 www.iosrjournals.org 51 | Page Optimal Pricing and Ordering Policy for Non-Instantaneous Deteriorating Items with Price and Stock Dependent Demand and Partial Backlogging 1* Vipin Kumar, 2 Anupama Sharma, 3 C.B.Gupta 1, 2 Dept. of Mathematics BKBIET, Pilani (Raj.) 3 LNM Institute of Information Technology, Jaipur (Raj.) *Corresponding author Abstract: In this paper we deals non-instantaneous deteriorating items with multivariable demand which depend on selling price and available stock level. The rate of deterioration is constant which start after a certain time because items are non-instantaneous. Shortages are permitted and partially backlogged with fixed rate. The aim to develop this model is to find the optimal ordering quantity and optimal selling price. To illustrate the proposed model, a numerical example is carried out. Keywords: Inventory, Deterioration, Multivariable demand, Partial Backlogging --------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 18-05-2020 Date of Acceptance: 03-06-2020 --------------------------------------------------------------------------------------------------------------------------------------- I. Introduction In this existing modern literature, numerous researcher giving their work by formulating different inventory models for deteriorating items by using different parameters to influence the model. Every existing product can't last life with the same utility value, that's why deterioration is the most important parameter for any inventory model. As we all are aware of the fact the many of items doesn't vanish immediately, usually there is a certain time period of this process of deterioration. By keeping this fact in mind, in the modern environment, non-instantaneous deterioration is taken as a key parameter for the research. Deterioration concept was first given by Ghare and Schrader (1963) in their study. Many researchers like Covert and Philip (1973), Dave and Patel (1981), Goyal & Giri (2001), etc. contribute significantly in this direction. Later, the non- instantaneous deterioration rate becomes the choice of researchers. Wu etal (2006) defined their inventory model for non-instantaneous deteriorating items. Dye (2013) used preservation technology on a non- instantaneous deteriorating stock model. Vipin et al. (2013) developed a two-warehouse inventory model for deteriorating items with ramp type demand. Tayal et al. (2014) illustrated an inventory model for deteriorating items with seasonal products. Jaggi et al. (2015) discussed credit financing in economic ordering policies for non-instantaneous decaying goods with price dependent demand under permissible delay in payments. Sanjay Sharma (2015) A Generalized EOQ Model for Time Dependent Deteriorating Items under Inflation .Vipin Kumar et al. (2015) developed a deterministicinventory model for Weibull deteriorating items with selling price dependent demand. Anupama Sharma et al. (2018) discussed an optimal inventory policy for deteriorating items with stock level and selling price dependent demand under trade credit. As in many situations, there can a possibility of stock out for the supplier, in such condition the customer can wait for the consignment to refill or can move to some other supplier for the same. In case the customer can wait for some time for their items to get delivered is termed as backlogging as it always for a span of time so it is called partial backlogging. In many inventory systems, the waiting time for the next replenishment would determine whether the backlogging would be accepted or not. Therefore, the normally backlogging rate is kept as variable and dependent on the time of waiting for the next replenishment. Wee (1993) gave economic production lot size model for deteriorating items with partial back-ordering. Teng (2004), Chaudhary (2010) gave their economic quality model with the concept of partial backlogging. Abad(1996) defined Optimal pricing and lot-sizing under conditions of perishability and partial back-ordering. Different optimal pricing and ordering policies were considered by many researchers in order to maximize the profit. Sheen (2008) developed dynamic pricing, promotion, and replenishment policies for deteriorating items with trade credit facilities. Later, Wu et. al (2009) and Goyal (2010) developed their model for Optimal Pricing and Ordering Policy for Non-Instantaneous Deteriorating Items. In this paper we deals non-instantaneous deteriorating items with multivariable demand which depend on selling price and available stock level. The rate of deterioration is constant which start after a certain time because items are non-instantaneous. Shortages are permitted and partially backlogged with fixed rate. The aim