An Inventory Model for Items with Two Parameter Weibull Distribution Deterioration and Backlogging

In this paper an inventory model is developed with time dependent power pattern demand and shortages due to deterioration and demand. The deterioration is assumed to follow a two parameter Weibull distribution. Three different cases with complete, partial, no backlogging are considered. The optimal analytical solution of the model is derived. Suitable numerical example has been discussed to understand the problem. Further sensitivity analysis of the decision variables has been done to examine the effect of changes in the values of the parameters on the optimal inventory policy.


Introduction
Deterioration of items in an inventory is a common phenomenon in business situations.This is due to the fact that the items in the inventory become obsolete, devalued, decay or damaged depending on the type of goods.As a consequence of the deterioration shortages may occur.Hence deterioration factor has to be given importance while determining the optimal policy for an inventory model.
Whitin [1] was the first to consider the deterioration of inventory items, he dealt with the deterioration of fashion goods at the end of prescribed storage period.Ghare and Schrader [2] later formulated a mathematical model with a constant deterioration rate.Covert and Philip [3] then extended Ghare and Schrader's model for variable rate of deterioration by assuming two parameter Weibull distribution functions.
Resently, researchers are analyzing the effect of deterioration and the variations in the demand rate with time in supply chain and logistics.Dave and Patel [4] derived a lot size model for constant deterioration of items with time proportional demand.Sachan [5] modified Dave and Patel's [4] model.Goel and Aggarwal [6] formulated an order-level inventory system with power-demand pattern for deteriorating items.Datta and Pal [7] presented an EOQ model with the demand rate dependent on instantaneous stock displayed until a predefined maximum level of inventory L is achieved.After this level is re-ached, the demand rate becomes constant (D(t) = a[I(t) b ] for I(t) > L and D(t) = aL b for 0  I(t)  L).Chang and Dye [8] developed an EOQ model with a similar power demand and considered partial backlogging of orders.He stated that if longer the waiting time smaller the backlogging rate would be.So the proportion of the customers who would like to accept backlogging at time t decreases with the waiting time for the next replenishment.In this situation the backlogging rate is defined as where t i is the time at which the i th replenishment is being made and δ is the backlogging parameter.Several researchers have extended their idea to different situations considering various deterioration rates and time value of money.Valuable models in this direction are the models of S. R. Singh, and T. J. Singh [9], Tarun Jeet Singh, Shiv Raj Singh and Rajul Dutt [10], C. K. Tripathy and L. M. Pradhan [11], etc.In continuation with these developments an inventory model for Weibull deteriorating items is developed in this paper with power pattern time dependent demand.Shortages are allowed with backlogging of orders.An analytical solution of the mode is derived and illustrated with the help of numerical examples.The sensitivity analysis of the optimal solution is carried out with respect to changes in various parametric values.These changes are depicted in the Tables 1-3 in Section 4.  K i : The total cost per time unit.

Assumptions
 The inventory consists of only one type of items.
(1 )/ n n   The expression for demand rate is  The variable deterioration rate (t) is assumed to follow the two parameter weibull distribution function (i.e.) (t) = αβt β1 , where α is the scale parameter, α > 0; β is the shape parameter β > 0; t is the time to deterioration, t > 0. The replenishment rate is infinite. The lead-time is zero or negligible. The planning horizon is infinite. During the stock out period, the backlogging rate is variable and is dependent on the length of the waiting time for the next replenishment.The proportion of the customers who would like to accept the backlogging at time "t" is with the waiting time (T  t) for the next replenishment i.e., for the negative inventory the backlogging rate is the backlogging parameter and t 1  t  T.

Mathematical Model
With above assumptions, the on-hand inventory level at any instant of time is exhibited in Figure 1.

Inventory Level before Shortage Period
During the period [0, t 1 ], the inventory depletes due to the demand and deterioration.Hence, the differential equation governing the inventory level I 1 (t) at any time t during the cycle [0, t 1 ] is given by with the boundary condition The solution of Equation (3.1) is given by The maximum positive inventory level is

Model I: (No Backlogging)
The state of inventory during the shortage period [t 1 ,T] is represented by the differential equation, with the boundary condition I 2 (t 1 ) = 0 at t = t 1 .
The solution of Equation (3.4) is given by The maximum backordered units are Hence, the order size during [0,T] is The total cost per replenishment cycle consists of the following cost components.
3.1.1.5.Purchase Cost per Cycle (I PC ) Hence, the total cost per time unit from (3.8), (3.9), (3.10), (3.11) is To minimize total average cost per unit time (K 1 ), the optimal value of t 1 can be obtained by solving the equation The value of t 1 obtained from (3.13) is used to obtain the optimal values of Q 1 and K 1 .Since the Equation (3.13) is nonlinear, it is solved using MATLAB.
The condition with the boundary condition I 2 (t 1 ) = 0 at t = t 1 .
The solution of Equation (3.4) is given by The maximum backordered units are 2 1 (1 ) Hence, the order size during [0,T] is

Cost Components
The total cost per replenishment cycle consists of the following cost components.

Backordered Cost per Cycle (
Hence, the total cost per time unit from (3.8), (3.9), (3.19), (3.20) To minimize total average cost per unit time (K 2 ), the optimal value of t 1 can be obtained by solving the eqution  uring the interval [t 1 ,T] stock out situation arises.the orders during this period are completely backlogged.The state of inventory during [t 1 ,T] can be repr with the boundary condition I 2 (t 1 ) = 0 at t = t 1 .

The solution of Equation (3.4) is given by
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The maximum backordered units are Hence, the order size d (3.28)

Cost Components
The total cost per replenishment cycle consists of the following cost components. 3.1.
) 1 2 Hence, the total cost per time unit from (3.8), (3.9), (3.29), (3.30), (3.31) is To minimize total average cost per unit 3 optimal value of t 1 can be obtained by solving the equation time (K ), the The value of t 1 obtained from (3.33) is used to obtain the optimal values of Q 3 and K 3 .Since the Equat is nonlinear, it is solved using MATLAB.
The condition  , is also satisfied for the value t 1 from (3.33).
(3.34) wing section.Sensitivity analysis is carried out with respect to backlogging and deterioration rate.

alysis
In this section the optimal value ( quantity ( i Q To illustrate and validate the proposed model, appropriate numerical data is considered and the optimal values are found in the follo parameter

Numerical Example and Sensitivity An
1 t  ), the optimal order  ) and the minimum to

Conclusion
In this paper, we have developed a deterministic inventtory model with power pattern demand and Weibull deterioration rate.This type of power pattern demand requires a differ t po where d is a positive constant, n may be any positive number, T is the planning horizon.

. Notations and Assumptions 2.1. Notations
I 2 (t): The level of negative inventory at time t, t 1  t  T.
the optimum time t 1 at which positive inventory is zero is 0.969997 time units and stock out period t 2 is of length is 0.030003 time units.This advices the retailer to y 6] V. P. Goel and S. P. Aggarwal, "Order Level Inventory System with Power Demand Pattern for Deteriorating bu 50 units which will cost a minimum of $ 658.28 (by rounding off i Q  ).The following observations have been made on the ba-Items," Proceedings of the All India Seminar on Operational Research and Decision Making, University of Delhi, New Delhi, 1981, pp.19-34.sis of the above table with the increase in scale parameter and shape parameter:  Increase in α results in decrease in inventory period, [7] T. K. Datta and A. K. Pal, "Order Level Inventory System with Power Demand Pattern for Items with Variable Rate of Deterioration," Indian Journal of Pure and Applied Mathematics, Vol. 19, No. 11, 1988, pp. 1043-1053.increase in order quantity and increase in total cost per time unit. Increase in β results in increase in inventory period, [8] H. J. Chang and C. Y. Dye, "An EOQ Model for Deteriorating Items with Time Varying Demand and Partial Backlogging," Journal of the Operational Research Society, Vol.50, No. 11, 1999, pp.1176-1182.decrease in order quantity and decrease in total cost per time unit.Hence in general 50% change in the parameters α, β results [9] S. R. Singh and T. J. Singh, "An EOQ Inventory Model with Weibull Distribution Deterioration, Ramp Type Demand and Partial Backlogging," Indian Journal of Mathematics and Mathematical Sciences, Vol. 3, No. 2, 2007, pp.127-137. ,[