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An Integrated Partial Backlogging Inventory Model
« on: April 23, 2011, 09:10:56 am »
An Integrated Partial Backlogging Inventory Model having Weibull Demand and Variable Deterioration rate with the Effect of Trade Credit
Author : P.K. Tripathy,  S. Pradhan
International Journal of Scientific & Engineering Research, IJSER - Volume 2, Issue 4, April-2011
ISSN 2229-5518
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Abstract - Demand considered in most of the classical inventory models is constant, while in most of the practical cases the demand changes with time. In this article, an inventory model is developed with time dependent two parameter weibull demand rate whose deterioration rate increases with time. Each cycle has shortages, which have been partially backlogged to suit present day competition in the market. Also the effect of permissible delay is also incorporated in this study. The total cost consists of ordering cost, inventory holding cost, shortage / backordering cost, lost sale cost and deterioration cost are formulated as an optimal control problem using trade credit policy. Optimal solution for the model is derived and the trade credit on the optimal replenishment policy are studied with the help of numerical examples.

Index Terms- Inventory, Shortages, partial backlogging, weibull demand, trade credit , variable deterioration, replenishment.

In the classical inventory economic order quantity (EOQ) model, it was tacitly assumed that the customer must pay for the items as soon as the items are received. However, in practices or when the economy turns sour, the supplier frequently offers its customers a permissible delay in payments to attract new customer. In todayís business transaction, it is frequently observed that a customer is allowed some grace period before settling the accounts with the supplier or the producer. The customer does not have to pay any interest during this fixed period but if the payment gets delayed beyond the period interest will be charged by the supplier. This arrangement comes out to be very advantageous to the customer as he may delay the payment till the end of the permissible delay period. During the period, he may sell the goods, accumulate revenues on the sales and earn interest on that revenue. Thus it makes economic sense for the customer to delay the payment of the replenishment account upto the last day of the settlement period allowed by the supplier on the producer. Similarly for supplier, it helps to attract new customer as it can be considered some sort of loan. Furthermore, it helps in the bulk sale of goods and the existence of credit period serves to reduce the cost of holding stock to the user, because it reduces the amount of capital invested in stock for the duration of the credit period. So, the concept of permissible delay in payments beneficial both for supplier and customer.

In some real life situation there is a part of demand which can not be satisfied from the current inventory, leaving the system in stock out. In these systems two situations are mainly considered, all customers wait until the arrival of the next order (Complete back order case) or all customers leave the system (lost sales case). However, in practical, some customers are able to wait for the next order to satisfy their demands during the stock out period, while others do not wish to or can not wait and they have to fill their demands from other sources. This situation is modeled by consideration of partial back- ordering in the formulation of the mathematical model. Wee (1999) developes a determinatic inventory model with quantity discount, pricing and partial backlogging when the product in stock deteriorates with time according a weibull distribution. Teng (2002) presents on EOQ model under the conditions of permissible delay. Chen et al. (2003) establish an inventory model having weibull deterioration and time varying demand. Wu et al. (2003) considered an inventory model where deteriorating rate and demand rate are follows the Chenís model (2003) where shortages are permitted. Papachristos and Skouri (2003) present a production inventory model with production rate, product demand rate and deteriorating rate, all considered as functions of the time. Their model follows shortages and the partial rate is a hyperbolic function of the time up to the order point. They propose an algorithm for finding the solution of the problem. Abad (2003) considers the problem of determining the optimal price and lot size for reseller in which unsatisfied demand is partially backordered. There are several interesting papers related to partial backlogging and trade credits viz. park (1982), Jamal et al. (1997) , Lin et al. (2000), Dye et al. (2007) and their references.

With this motivation, in this paper an attempt is made to formulate an inventory model in time dependence of demand follows a two parameters weibull type with time   variable deterioration where unsatisfied demand is partially backlogged and delay payment is allowed. With the help of this model, supplier can easily extract its order quantity, cycle period for shortages as well as payment time to reduce the total inventory cost.

   To develop the proposed model, the following notations and assumptions are used in this article.

2.1   Notations
I(t)   : Inventory level at time t
Q (t)   : Order quantity at time t = 0
A, h,C1,
C2, C3   : Ordering cost, holding cost, deteriorating cost, shortage cost for backlogged items and the unit cost of lost sales respectively. All of the cost parameters are positive constants.
Ie   :   Interest which can be earned
Ic   :   Interest charges which invested in inventory, 
M   :   Permissible delay in setting the amounts, 0<M< T
T   :   Length of the replenishment cycle
T1   :   Time when inventory level comes down to zero, 0 < T1< T
TIC1   :   Total inventory cost when M < T1
TIC2   :   Total inventory cost when T1, < M < T

2.2   Assumptions:
1.   The inventory system involves only one item
2.   Replenishment occurs instantaneously on ordering i.e. lead time is zero and it takes place at an infinite rate.
3.   The demand rate of any time t is   two parameter weibull type, where   are called scale and shape parameter respectively.
4.   Q (t) = vt is the variable deterioration rate, where 0 < v <<1.
5.   Shortages are allowed and unsatisfied demand is backlogged at the rate of 

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