Two-warehouse inventory model for deteriorating items with price-sensitive demand and partially backlogged shortages under inflationary conditions

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Introduction
Demand and price are perhaps some of the most fundamental concepts of inventory management and they are also the backbone of a market economy. The law of demand states that, if all other factors remain at a constant level, the higher the price, the lower is the quantity demanded. As a result, demand of very high priced products will be on decline. Hence the price of the product plays a very crucial role in inventory analysis. In recent years, a number of industries have used various innovative pricing strategies viz., creative pricing schemes on internet sales, two-part tariffs, bundling, peak-load pricing and dynamic pricing, to boost the market demand and to manage their inventory effectively. The Moreover in the prevailing economy, the effects of inflation and time value of money cannot be ignored; as it increases the cost of goods. When the general price level rises, each unit of currency buys fewer goods and services; consequently, inflation is also a decline in the real value of money -a loss of purchasing power in the medium of exchange which is also the monetary unit of account in the economy. Further, from a financial standpoint, an inventory represents a capital investment and must compete with other assets for a firm's limited capital funds. And, rising inflation directly affects the financial situation of an organization. Thus, while determining the optimal inventory policy the effect of inflation should be considered. In the past many authors have developed different inventory models under inflationary conditions with different assumptions. In 1975, Buzacott developed an economic order quantity model under the impact of inflation. Bierman and Thomas (1977) proposed the EOQ model considering the effect of both inflation and time value of money. (Yang, 2004) developed an inventory model for deteriorating items with constant demand rate under inflationary conditions in a two warehouse inventory system and fully backlogged shortages. Several other researchers have worked in this area like (Jaggi et al., 2006;Dey et al., 2008;. Recently,  presented the effect of FIFO and LIFO dispatching policies in a two warehouse environment for deteriorating items under inflationary conditions with fully backlogged shortages. The characteristic of all of the above articles is that the unsatisfied demand (due to shortages) is completely backlogged. However, in reality, demands for foods, medicines, etc. are usually lost during the shortage period. Generally it is observed for fashionable items and high-tech products with short product life cycle, the willingness for a customer to wait for backlogging during a shortage period is diminishing with the length of the waiting time. Hence, the longer the waiting time, the smaller the backlogging rate. (Abad, 1996) first developed a pricing and ordering policy for a variable rate of deterioration with partially backlogged shortages. Later to reflect this phenomenon, (Yang, 2006) modified (Yang, 2004) model for partially backlogged shortages. Dye et al. (2007) modified the (Abad, 1996) model taking into consideration the backorder cost and lost sale. Shah and Shukla (2009) also developed a deterministic inventory model for deteriorating items with partially backlogged shortages.
Further, (Yang, 2012) extended (Yang, 2006) model for the three-parameter Weibull deterioration distribution. Recently,  explored the effect of FIFO and LIFO dispatching policies in a two warehouse inventory system for deteriorating items with partially backlogged shortages. This paper aims to develop an inventory model for deteriorating items in a two warehouse system with price dependent demand under inflationary conditions. Moreover, the model considers partially backlogged shortages, where the backlogging rate decreases exponentially as the waiting time increases. Further, we have investigated the application of FIFO and LIFO dispatching policies in different scenarios in the model. The main purpose of the present model is to determine the optimal inventory and pricing strategies, so as to maximize the total average profit of the system. Finally, numerical examples and sensitivity analysis have been presented to illustrate the applicability of FIFO and LIFO dispatch policies in different scenarios. These findings eventually serve as a ready reckoner for the organization to take appropriate decision under the prevailing environment.

Assumptions and Notations
The following assumptions and notations have been used in this paper.

Assumptions:
1. The demand rate D(P), is assumed to be dependent on the selling price and of form,  

Model description and analysis
In the present study demand is assumed to be a decreasing function of selling price given by   where k and e are positive constants. Shortages are allowed to accumulate in the model but are partially backlogged. Moreover a two warehouse inventory model has been devised, where the OW has a fixed capacity of W units and the RW has unlimited capacity. The units in RW are stored only when the capacity of OW has been utilized completely. However, in such a scenario organization has an option to adopt either FIFO or LIFO dispatching policy. The following sections discuss the model formulation for both the policies.

FIFO model formulation
The behaviour of the model over the time interval   0,T has been represented graphically in (Figure 1). Initially a lot size of Q F units enters the system. After meeting the backorders, S F units enter the inventory system, out of which W units are kept in OW and the remaining Z = (S F -W) units are kept in the RW. In this case as FIFO policy is being implemented, therefore the goods of the RW are consumed only after consuming the goods in OW. Starting from the initial stage till w t , the time the inventory in OW is depleted first due to the combined effect of demand and deterioration and the inventory level in RW also reduces from Z to 0 Z due to effect of deterioration. At time w t OW gets exhausted. Further, during the interval   1 , t t w depletion due to demand and deterioration will occur simultaneously in the RW and it reaches to zero at time 1 t . Moreover, during the interval   T t , 1 some part of the demand is backlogged and the rest is lost. The quantity to be ordered will be 1 ( ) During the time interval (0, w t ) the inventory level in the OW decreases due to the combined effect of both the demand and deterioration. The differential equation representing the inventory level in the OW during this interval is given by Noting that at Now, during the interval (0, w t ), the inventory level Z kept in RW also depletes to a level Z 0 due to the effect of deterioration. Hence, the differential equation below represent the inventory level in this interval is given by using the boundary condition   Z Again, during the time interval ( w t , 1 t ), the inventory level in RW decreases due to the combined effect of demand and deterioration both. The differential equation describing the inventory level this interval is given by using the boundary condition   0  . Hence, the shortage level at time t is represented by the following differential equation: After using the boundary condition   0 , the solution is given by Since, demand is considered as a function of selling price and shortages are partially backlogged. Hence, by using continuous compounding of inflation and discount rate, the present worth of the various costs during the cycle (0, T) is evaluated as follows: (a) Present worth of the ordering cost is Now, the present worth of the total average profit during the cycle (0, T), TP (S F , p) is thus given by the following expression: After substituting the values of these from Eqs. (12)(13)(14)(15)(16)(17)(18)Eq. (19) reduces to the present worth of the total average profit for the system Substituting the values of t 1 from Eq. (9), we get

Solution Procedure
Our objective is to maximize the present worth of total average profit. The necessary conditions for maximizing the present worth of total average profit are given by and using the initial condition   Noting that at Now, during the interval (0, w t ), the inventory W kept in OW also reduces from W to W 0 due to the effect of deterioration. Hence, the differential equation below represent the inventory level in this interval is given by After using the boundary condition   Again, during the time interval ( w t , 1 t ), the inventory level in OW decreases due to the combined effect of demand and deterioration both. The differential equation describing the inventory level this interval is given by Now at time 1 t inventory is exhausted in both the warehouses, so after time 1 t shortages start to accumulate. It is assumed that during the time (t 1 , T), only some fraction i.e.
Hence, the shortage level at time t is represented by the following differential equation: After using the boundary condition   0 Since, demand is considered as a function of selling price and shortages are partially backlogged. Hence, by using continuous compounding of inflation and discount rate, the present worth of the various costs during the cycle (0, T) is evaluated as follows:

(a) Present worth of the ordering cost is
(e) Present worth of the opportunity cost due to lost sales is Now, the present worth of the total average profit during the cycle (0, T), TP (S L , p) is thus given by the following expression: After substituting the values of these from Eqs. (35-41), Eq. (42) reduces to the present worth of the total average profit for the system Substituting the values of t w and t 1 from Eq. (26) and Eq. (32) respectively, we get

Solution Procedure
Our objective is to maximize the present worth of total average profit. The necessary conditions for maximizing the present worth of total average profit are given by          If the holding cost and the deterioration rate both are greater in OW than that of RW, then organization should adopt the FIFO policy; as it will be helpful for the decision maker to meet the demand from the OW first, in order to manage the high holding costs of OW.  If the holding cost in RW is higher than that of OW but the deterioration rate in RW is less than that of OW, then the results show that the cost associated with LIFO dispatching policy is less than the FIFO dispatching policy; LIFO policy is preferred.  Further, if the holding cost in both of the warehouses is equal but the deterioration rate in OW is larger than that of RW, then FIFO policy is recommended. It helps to sustain maximum freshness of the commodities for the consumer and reduce deterioration cost. So this shows that holding cost plays a dominating role in deterioration rate.  II. We study the effect of H and F on the both policies by taking different combinations of H and F, when deterioration rate in RW is greater (i.e.  = 0.06 and  = 0.1). Rest of the parameters are kept same.  LIFO policy is used if both the holding cost and deterioration rate in RW is high. It saves the organization from acquiring high holding costs for a longer period. So RW is vacated first i.e., items in RW are sold out first.  FIFO policy is adopted by the organization if holding cost in RW is comparative less than that of OW, even though the deterioration rate in OW is less than that of RW. This clearly suggests that holding cost plays a significant role in optimal decision making than deterioration rate.  However, if the holding cost in both the warehouses is same but deterioration rate in RW is high, then LIFO policy is recommended. As the items stored in RW are more prone to deterioration, therefore the RW is to be given priority over OW, so as to administer the loss due to deterioration. Table 3 summarises the finding for different rates of deterioration along with holding costs in both the warehouses in such a fashion which serve as a ready reckoner for the decision maker to arrive at appropriate policy decision. IV. Now we study the impact of R (inflation) on both policy selections, when the deterioration rates ( and) are in different combinations and rest of the parameters are to be kept same.  Table 4 suggests that:

III. Further,
 When inflation rate is increasing, then the present worth of total average profit decreases. It is apparent from the table that order quantity is more when the inflation is low, and it gradually declines with growing inflation. Since with mounting inflation, the prices are ought to rise, which results in stumpy demand. Thus in order to sustain expanding inflation rates the organization orders less, which also results in low profits.  From the table it is clearly visible that again deterioration rate plays a vital role in policy selection, rather than the inflation rate. When the holding cost are same in both the warehouse then the following observations are made with respect to the deterioration rate:  When the deterioration rate in OW is equal to that of RW, present worth of total average profit in both the policies is equal. Hence, the organization can adopt either LIFO or FIFO dispatching policy.  When the deterioration rate in OW is less than that of RW, present worth of total average profit in FIFO system is smaller than LIFO system. As the units in RW deteriorate rapidly, thus it is advisable to consume the goods of RW prior to that of OW.  On the other hand, if the deterioration rate in OW is more than that of RW, then present worth of total average profit in FIFO system is higher than that of LIFO system. Since in this case the items in RW are deteriorating at a slower rate, so operating OW prior to the RW is beneficial. Therefore FIFO policy is suggested which helps one to preserve the freshness of the commodities for the consumer.

V.
Here the impact of backlogging parameter δ is considered on the policy selection. Sensitivity analysis is performed by changing (increasing or decreasing) the backlogging parameter δ by 20% and 40%. All other parameters are remains same.  (Table 5) indicates that a decrease in backlogging parameter δ, i.e., an increase in backlogging rate, increases the order quantity which eventually results in higher profits. Since an increasing backlogging rate implies more of backlogged demand, hence from the order size, a major portion is utilized for satisfying the backlogged demand, which reduces the initial inventory for the organization and thus the inventory holding costs. Further as the deterioration rate is higher in OW, the FIFO dispatch policy is suggested.
VI. Now again we study the effect of k and e on both of the policies by taking different combinations of k and e and keeping all other parameters same as in case of base numerical.  Table 6 shows that:  For a fixed value of e (demand parameter), when the demand parameter k increases, then there is a sheer increase in the order quantity and hence the profit also increases. Obviously, as the demand parameter k is directly proportional to the demand, the rise in k escalates the demand, which forces the organization to order a large quantity.  Whereas, for a fixed value of k, an increase in demand parameter e would result in a lesser order quantity. Since e has an inverse effect on the demand, thus the order size decreases which eventually decreases the profit.
 The sensitivity analysis section helps the firm to identify and distinguish the parameters which influence the policy selection, and the parameters which influence the policy decision. It is evident from the tables 1, 2 and 3 that holding costs and deterioration rates in both the warehouses playa a major role in selecting the appropriate dispatching policy i.e., FIFO or LIFO. Whereas, the other parameters viz., inflation rate, backlogging rate and the demand parameters, do not play a role in policy selection. However these parameters suggest the firm to take appropriate policy decision i.e., the order quantity and the price for the product which may yield maximum profit in a particular case.

Conclusion
This paper has investigated the effect of FIFO and LIFO dispatching policies for deteriorating items in a two warehouse inventory system with price-sensitive demand under inflationary conditions. In addition, shortages are partially backlogged. The backlogging rate is considered to be an exponential decreasing function of the waiting time, since the willingness for a customer to wait for backlogging during a shortage period diminishes with the length of the waiting time. The developed models for both FIFO and LIFO dispatching policy jointly optimise the selling price and the initial inventory by maximizing the average profit.
The findings have been validated with the help of a numerical example. Moreover sensitivity analysis reveals the different parameters which influence the dispatching policy selection and policy decision. The policy selection i.e., FIFO or LIFO is only affected by the holding costs and the deterioration rates. However, the inflation rate, backlogging rate and the demand parameters, helps the decision maker to adopt appropriate inventory and pricing policy.
In future the model can be extended by incorporating some more practical situations, stock dependent demand, linear time dependent demand, trade credit policies and many more.