Effect of Inflation on Two Storage Inventory Model with Time Dependent Deteriorating Items and Stock Dependent Demand

For deteriorating product, there always be a pressure on the company to maximize the profit. In the present model, an effort is made to develop a production inventory control model having two separate warehouses. In most of the study, the demand is assumed as time dependent which is not appropriate; but here authors have considered stock based demand. An approach is taken into consideration that deterioration rate different for different warehouses. KeywordsWarehouses, Inflation, Stock based demand, Variable rate of deterioration.


Introduction
Inventory is mostly based on the demand pattern, it may be deterministic or probabilistic. If the demand is more than inventory or items in stock, it leads to shortages or stock out. Shortage can be categorized into two different ways, first case is known as back order or backlogging in which demand is not fulfilled by the retailers. For example, if a customer goes to buy some books and if books are not available at that time then he can postpone purchasing for some time and he depend upon the shopkeeper to provide the books as soon as possible. Hence, shortages are allowed with partial backlogging case. The second one is called lost sales case where the demand is completely lost. For example, electronic equipment and gadgets etc. have very short life span due to innovative changes in the models and technology. In practise customer may be prepared to postpone purchase of some items such as fashionable item, electronic item etc. But for certain items such as medicines or food items customer will not wait and purchase them from alternative sources.

Introduction
Deterioration is the process of decay or damages to the items, such item may become unusable for original purposes. In the present study, authors have considered the deterioration as time dependent while in many studies, deterioration assumed constant which is not feasible. If authors look at the past then the study to justify the deteriorating inventory items was first time originally studied by Ghare (1963) where many phases of inventory control models was discussed and derived an updated form of Economic Order Quantity (EOQ) model and in this attempt author assume demand as exponentially decay. Covert and Philip (1973) developed an EOQ based model for deteriorating items under the effect of Weibull distribution with partial backlogging again, this study is more explore by Adler and Nanda (1974) for optimal lot size determinationmultiple product case and presented an inventory control model under the effects of the learning.
Under a new approach of linear trend in demand, Bierman and Thomas (1977) presented an inventory replenishment policy for inventory decisions under inflationary condition, Choi and Hwang (1986) developed and study continuously distributed time-lags and optimization of production planning problem. For finite uniform production rate and considering demand as linearly increasing with finite uniform production rate, Hong et al. (1990) discussed production policies to maximize the profit.
By considering, the effect of finite replenishment rate and shortages, Goswami and Chaudhuri (1991) presented an EOQ model for decaying items with an effect of linear increasing trend in demand. By considering two levels of storage, an attempt for inventory control product under decaying items and considering linear pattern in demand, was discussed by Goswami and Chaudhuri (1992). Bhunia and Maiti (1994) developed and presented a model based on two warehouses for a linear increasing pattern in demand. Time varying is one of the most important approach in the field of inventory also Giri and Chaudhuri (1997) developed heuristic inventory control model for spoilage items under time-varying demand and the effect, due to shortages are also considered.
To present inventory control techniques for two warehouses based deteriorating items, Bhunia and Maiti (1998) developed a model by considering time based demand. Lee and Ying (2000) analysed a model to control inventory and to optimize the profit for deteriorating products with two-warehouse by taking time based demands. Time varying approach also considered by Goyal and Giri (2003) to explain the production based inventory model where time dependent demand was assumed. Chellappan and Nataraja (2011) presented an model and gives a fuzzy rule by considering perishable pattern and presented a production-inventory system, For multi-warehouse based systems to determine a fixed and space leasing contracts, Hariga (2011) presented an inventory model, Singh and Saxena (2012) also explore many things in the field of inventory and presented a returned policy based model to justify backorders for a reverse logistics approach. Dem and Singh (2013) considering effect of quality and put an effort to establish a production model for highly deteriorating products. Forghani et al. (2013) presented a single inventory cycle based demand model by considering price adjustment. The concept of limited storage was a new technology in the area of inventory control. By considering limited storage, Sharma and Singh (2013a) presented an EPQ model of decaying product based inventory with linear exponentially increasing demand rate. Sharma and Singh (2013b) developed an inventory model by taking multivariate consumption rate for decaying items by assuming partial backlogging where the deterioration was considered as time dependent, Singh et al. (2016) explained the preservation technology and presented a model for deteriorating products under the effect of stock based demand with trade credit period. Sekar et al. (2017) investigated a model by applying a new concept of preservation technology and discussed an inventory control technique for deteriorating items under limited capacity storehouse, partial backlogging and the effect of inflation.
To present non-instantaneous deteriorating products, an inventory model with more effected assumptions under price sensitive demand developed by Rastogi et al. (2018) where partial backlogging of occurring shortages was also taken into account. By considering variable holding cost, Alfares and Ghaithan (2019) explained EOQ and EPQ production based inventory models for deteriorating items.
Present study is focused to develop an inventory model for two warehouses where deterioration assumed time dependent and the demand is assumed as stock dependent. For getting good results, different facilities, conditions and storage environment, here the holding cost of the inventory for both the warehouses are considered different.

Assumptions and Notations
During this study author assumes the following notations and assumptions: In the present model to gain maximum profit some modifying assumptions are made: (i) The demand rate D(t) which is taken as stock dependent is given by is taking as a parameter to present stock dependent consumption rate and a0  .
(ii) It is also assumes that the owned warehouse (OW) has a limited capacity of W units, whereas the rented warehouse (RW) has unlimited capacity. (iii) During the study it is also assumes that the goods that are kept in OW are consumed first, and then start the goods kept in RW. (iv) It is also assumes that, there has no deterioration in RW. No deterioration is taken or assumed at the supplier's end. (v) In the present study we have taken, inventory cost in RW is always higher in compare to OW. (vi) Lead time of the supplier is assumed as a random variable. (vii) Effect of inflation and time dependent value of money are taken into account. (viii) As to maximize the profit, unfulfilled demand is assumed backlogged where the fraction of shortages back-ordered is  , where 01   .

Mathematical Model Formulation
From the graphical representation of the model as mentioned in the Figure 1, authors can see that the inventory system. Here, out of Im total units, W units are move to OW and (Im -W) units are transfer to RW. As per the model, between the time intervals [0, ], authors can see that the inventory level is decreasing only in the RW as a combined effect of demand as well as deterioration. For OW, inventory level is decreasing only for deterioration as there is no demand during the interval [0, ] also between the time intervals [ , 1 ], the inventory level is start dropping (decaying) to zero due to overall effect of demand also during the same time interval deterioration also takes place. After completing one cycle, similar method is going on for the next cycle.
For RW the mathematical expression of the model can be expressed in the term of the following differential equations: The ordering quantity based on replenishment cycle, calculated as Now, to present the average profit of the retailer after end of one cycle, following elements should be counted:

(a) Cost of Holding the Inventory of the Retailer for RW
From the graphical representation of the model, it is obvious that to count holding cost of carrying inventory items. Authors should calculate by considering the effect of inflation as retailer is found to be in the possession of inventory between the intervals

(c) The Cost of Purchasing Items of the Retailer
Order is attempt by the retailer at t = 0, which mean that he can made the payment based his choice only. So, there will not be any effect on lead time taken by the supplier to make payment to retailer.

(d) Cost of Setup Inventory of the Retailer
It is obvious from the model that setup cost is always made by the retailer at initial time for each period, hence: SPC = A

(e) Shortage Cost of the Retailer
It can also be seen from the mathematical formulation of the model during the interval  