COMPARISON OF INVENTORY MODELS UNDER NON-INSTANTANEOUS DETERIORATION RATE WITH PROBABILISTIC DEMAND AND SHORTAGES

Abstract

INTRODUCTION As we assume that demand is undefined, it is interesting to explore how demand uncertainty affect performance of the system. A several researchers have deliberated inventory models assuming the demand to be persistent, time reliant, depending on price, stock dependent and exponential etc. under with and without shortages under different environment like delay in payment, with effect of inflation and taking variable holding cost. When we consider uncertain demand then our focus is on probabilistic demand where demand is considered as a random variable follows probabilistic distributions. But very effective part of the inventory system is deterioration which plays very crucial role. Many of the researchers studied inventory model with fixed deterioration and different deterioration like part wise, depend on time. The study of inventory models for declining items was first christened by Whitin[18]. A model with relentless rate of deterioration was originally established by Ghare as well as Schrader [7]. Raffat[14] categorized deterioration by time value of inventory. In the last few decades many researchers have done a lot of research work in studying and analyzing the different kinds of deteriorating inventory systems under different situations. From past few years the analysis of deteriorating inventory problems has expected a abundant deal of attention. Nahmias[12] divided the deteriorating inventory problems into given groups like fixed lifespan and random lifespan. Harris [10 ] first developed the EOQ formula with the help of differential calculus and same formula was again derived by Wilson[17]. Lots of works done in effect of deterioration. Covert and Philip [4] prolonged Schrader and Ghare continual deterioration rate to a two-parameter Weibull distribution. After years, there are quite a few remarkable papers allied to deterioration with and without shortages like Jaiswal and Shah[15], Giri along with Goyal [8], Bhunia with Maiti [2], Chung with Tsai[3] and Patelat.el. [13]. Sheikh and Patel [16] develop inventory model with altered deterioration rates. Aggarwal [1] deliberated an order level inventory model through continual rate of worsening. So many researcher are focus on probabilistic inventory models. Some of them have done their work with demand follows probabilistic distribution. Gupt[9] develop production model with probabilistic demand. Comparative study under different environment of inventory model was studied by Durai and Chakrabarti [5]. Multi-i tem inventory model with probabilistic demand derived by Kar, Roy and Maiti[11]. Fergany[6] studied probabilistic multi-item inventory form along with scarcities under limitations. Herein paper we pay attention to an inventory model with probabilistic demand follows Uniform, Triangular and exponential distribution with their mean. Non-instantaneous deterioration rate to be taken i.e. during the system deterioration depends on time. The model is represented with numerical example and a parametric analysis with tolerance limits. The goal of given analysis is to create relative inventory model that the firm gets enhanced profit comparative to profitable surroundings. An approach is accomplished, in which the entire profit in that path is maximized. Hence the optimal era of evaluation as well total optimal aim of inventory stage is resulted. NOTATIONS D(t):Demand x is a random variable follows probability distribution(Mean demand as .) A: Ordering cost C : Holding cost per unit time h C :scarcity cost per unit s C :Purchasing cost per unit c C :Promotion price per unit p T:Total cycle of in time θt:Deterioration rate during 0 ≤ t ≤t1 I(t):Inventory level at time t Q :Inventory level initially 1 Q :Inventory level at T 2 П:Profit ETP:Expected total profit ASSUMPTIONS (i) The demand rate of the goods is taken as probabilistic. (ii) Non-instantaneous deterioration rate is taken i.e. θt dependent on time. (iii) Shortages are allowed and totally backlogged. (iv) Single item inventory is considered. (v) Restoration or replacement of declining items are excluded. (vi) Holding cost is stable. DEVELOPMENT AND ANALYSIS OF THE MODEL Below diagram shows the nature of inventory at time 0 to T. The inventory level gradually diminishes due to demand and deterioration during the phase [0,t ] and ultimately falls to 1 zero at t = t . During the phase [t , T] shortages occur which is 1 1 completely backlogged. The inventory sets are describe by the differential equations: www.worldwidejournals.com 21 Shital S. Patel Department of Statistics , Veer Narmad South Gujarat University, Surat, Gujarat, India. PARIPEX INDIAN JOURNAL F RESEARCH | O January 202 Volume 10 | Issue 01 | 1 | PRINT ISSN No. 2250 1991 | DOI : 10.36106/paripex