1. INTRODUCTION:

Deteriorating items inventory models have been studied by many authors in past. It is well known that certain products such as medicine, volatile liquids, food stuff decrease under deterioration during their normal storage period. Therefore while determining the optimal inventory policy of such type of products the loss due to deterioration must be considered. Ghare and Schrader [9] first developed an EOQ model with constant rate of deterioration. Covert and Philip [8] extended this model by considering variable rate of deterioration. Shah [25]) further extended the model by considering shortages. The related work are found in [Nahmias [19], Raffat [22], Goyal and Giri [11], Wu et al. ([30], Ouyang et al. [20]].

Most of the existing literature in classical inventory model deal with single storage facility with the assumption that the available warehouse of the organization has unlimited capacity. But in actual practice many times the supplier provide price discounts for bulk purchases and the retailer may purchase more goods than can be stored in single warehouse (own warehouse). Therefore a rented warehouse RW is used to store the excess units over the fixed capacity W of the own warehouse. The rented warehouse is charged higher unit holding cost then the own warehouse, but offers a better preserving facility with a lower rate of deterioration.

Hartley [12] first developed a two warehouse inventory model. An inventory model with infinite rate of replenishment with two warehouse was considered by Sarma [24]. Pakkala and Achary [21] extended the two warehouse inventory model for deteriorating items with finite rate of replenishment and shortages. Other research work related to two warehouse can be found in, for instance [Benkherouf [2], Bhunia and Maiti [3], Kar et al. [14], Chung and Huang [7], Rong et al. [23]].

An economic order quantity model under condition of permissible delay in payments was first considered by Goyal [10]. The model was extended by Aggarwal and Jaggi [1]) for deteriorating items. Aggarwal and Jaggi’s [1] model was further extended by Jamal et al. [13] to consider shortages. An inventory model with varying rate of deterioration and linear trend in demand under trade credit was considered by Chang et al. [5]. Teng et al. [29] developed an optimal pricing and lot sizing model by considering price sensitive demand under permissible delay in payments. A literature review on inventory model under trade credit is given by Chang et al. [6]. Min et al. [16] developed an inventory model for exponentially deteriorating items under conditions of permissible delay in payments.

The effect of inflation and time value of money play important role in practical situations. Buzacott [4] and Mishra [17] simultaneously developed inventory model with constant demand and single inflation rate for all associated costs. Mishra [18] considered different inflation rate for different costs associated with inventory model with constant rate of demand. Yang [31] developed a two warehouse EOQ model for deteriorating items with partial backlogging and inflation. Singh et al. [27] considered a two warehouse inventory model for deteriorating items under inflation, time value of money and shortages. Singh et al. [28] developed an inventory model for non-instantaneous deteriorating items with stock dependent demand, inflation and partial back ordering with two warehouses. Singh et al. [26] considered a two-warehouse inventory model for deteriorating items under the condition of permissible delay in payments. Liang and Zhou [15] developed a two-warehouse inventory model for deteriorating items with constant rate of demand under conditionally permissible delay in payments.

In this paper we have developed a two-warehouse inventory model for deteriorating items with linear trend in demand with time varying holding cost under permissible delay in payments and inflation. Numerical examples are provided to illustrate the model and sensitivity analysis of the optimal solutions for major parameters is also carried out.

2. ASSUMPTIONS AND NOTATIONS:

Assumptions:

The following assumptions are considered for the development of two warehouse model.

1. Replenishment rate is infinite.

2. Lead time is zero

3. Shortages are not allowed.

4. OW has a fixed capacity W units and the RW has unlimited capacity.

5. The goods of OW are consumed only after consuming the goods kept in OW.

6. The demand rate D(t) is a linear function of time.

7. The unit inventory costs per unit in the RW are higher than those in the OW.

8. The retailer can accumulate revenue and earn interest after his/her customer pays for the amount of purchasing cost to the retailer until the end of the trade credir period offered by the supplier.

Notations:

The following notations are used for the development of the model:

D(t) : demand rate is a linear function of time t (a+bt, a>0, b>0)

Q : the replenishment quantity per replenishment

W : capacity of owned warehouse

α : the deterioration rate in OW, 0< α<1

β : the deterioration rate in RW, 0<β<1

A : replenishment cost per order for two warehouse system

c : purchasing cost per unit

p : selling price per unit

r : the discount rate

i : inflation rate

R=r-i : the net discount rate of inflation and it is a constant

HC(OW): holding cost per unit time is a linear function of time t (x₁+y₁t, x₁>0, 0<y₁<1) in OW

HC(RW): holding cost per unit time is a linear function of time t (x₂+y₂t, x₂>0, 0<y₂<1) in RW

T : length of inventory cycle

TCi: the total relevant cost per unit time (i=1,2,3)

I₀(t) : inventory level in OW at time t

I_r(t) : inventory level in RW at time t

t_r : the time at which the inventory level reaches zero in RW in two warehouse system

M : retailer’s trade credit period offered by the supplier in years which is as the fraction of the year

Ie : interest earned per Rs. per year

Ip : interest charged per Rs. in stock per year by the supplier

3. TWO WAREHOUSE MODEL:

At time t=0, a lot size of certain units enter the system. W units are kept in OW and the rest is stored in RW. The items of OW are consumed only after consuming the goods kept in RW. In the interval [0,tr], the inventory in RW gradually decreases due to demand and deterioration and it reaches to zero at t=tr. In OW, however, the inventory W decreases during the interval [0,tr] due to deterioration only, but during [tr, T], the inventory is depleted due to both demand and deterioration. By the time to T, both warehouses are empty as shown in figure describes the behaviour of inventory system.

6. CONCLUSION:

In this paper, we have developed a two warehouse inventory model for deteriorating items with time varying holding cost with inflation under permissible delay in payment. Shortages are not allowed. It is assume that rented warehouse holding cost is greater than own warehouse holding cost but provides a better storage facility and there by deterioration rate is low in rented warehouse. Numerical example and sensitivity analysis is also carried out.

7. REFERENCE:

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Received on 28.01.2013 Accepted on 05.02.2013

Research J. Science and Tech 5(1): Jan.-Mar.2013 page 113-119