AN EOQ INVENTORY MODEL USING RAMP TYPE DEMAND WITH DETERIORATION AND SHORTAGES INVENTORY MODEL USING RAMP TYPE DEMAND WITH DETERIORATION AND

: In this paper an EOQ Inventory model is developed in which inventory is depleted not only by demand also by deterioration, here demand rate is a ramp type function of time also the in this model shortages are allowed. The model is solved analytically by enumerating two possible shortages models to obtain the optimum solution


Introduction
In earlier days classical inventory model like Harris [1] assumes that the depletion of inventory is due to a constant demand rate. But subsequently, it was noticed that depletion of inventory may take place due to deterioration also, and then the problem of decision makers is how to control and maintain inventories of deteriorating items. Many researchers like Ghare and Schrader [2], Goyal et al. [3], Covert and Philip [4], Aggrawal and Jaggi [5], Cohen [6], Mishra [7] are very important in this connection. As the time progressed, several other researchers developed inventory models with deteriorating items with time dependent demand rates. In this connection, related works may refer to Ritchie [8], Hariga [9], S.P.Singh and V.K.Sehgal [10] Ghose and Chaudhari [11], Donaldson [12], Silver [13], Datta and pal [14], Deb and Chaudhari [15], Pal and mandal [16], Goel and Aggrawal [17]. Mishra [7] developed a two parameter Weibull distribution deterioration for an inventory model. This was followed by many researchers like S.P.Singh and G.C.Panda [18] Dev and Patel [19], Shah and Jaiswal [20], Giri et al. [21] etc. In the present paper, we drive the three EOQ inventory models for items that deteriorate at a Weibull rate, assuming the demand rate with a ramp type function of time. Shortages are allowed. The demand rate for such items increases with time up to certain time and then ultimately stabilizes and becomes constant. Finally numerical examples are proposed to demonstrate our developed model and the solution procedure.

Assumptions and Notations
The fundamental assumptions and notations used in this paper are given as follows: When β = 1, Z(t) becomes a constant which is the case of an exponential decay. When β< 1, the rate of deterioration is decreasing with time t and β > 1, it is increasing with t. 9) Shortages are allowed and completely backlogged. 10) S is the maximum inventory level of each ordering cycle. 11) The demand rate D(t) is assumed to be a ramp type function of time: Where H(t-μ) is the well known Heaviside's function defined as follows; H(t-μ) = 1, t ≥ μ = 0, t < μ

Mathematical Models and Solutions
The objective of the inventory problem here is to find the optimal order quantity so as to keep the total relevant cost minimum. Based on whether the inventory starts with shortages or not, there are two possible models under the assumptions described in section 2.

Model I : Inventory Starts Without Shortages
In the subsection, we will analyze the inventory starts without shortages. Replenishment is made at time t = 0 when the inventory level is at its maximum, S. Due to reasons of market demand and deterioration of the items, the inventory level gradually diminishes during the period (0, t1) and finally falls to zero at t = t1. Shortages are allowed during the period (t1, T) which are completely backlogged. The total number of backlogged items is replaced by the next replenishment.
The inventory level I(t) of the system at any time t over [0, T] can be described by the following equations: The boundary condition are I(t1) = 0 and I(0) = S By assumptions of (h) and (k) in section (2) and assuming µ < t1, the two governing equations (1) and (2) becomes: Now by using the boundary conditions (3), the solution of above three equations are respectively given as: When 0 < α << 1, we neglect the second and higher terms of α, equation (7) and (8) Since I(t1) = 0, we get from equation (8) with neglecting second and higher order terms of α as: Evaluating the above two integrals and using equation (12), we get as The total number of inventory holding during the period [0, t1] is Evaluating the above integrals and neglecting the second and higher order of α, we get as: The total shortage quantity during the interval [t1, T] is Then the average total cost per unit time is given by ( ) Now substituting the expressions for Dt, I1, I2 given by the equations (13), (14) and (15) respectively and then eliminating S by the equation (12), we get as: To minimize the average total cost per unit time, the optimal value of 1 t , say Since (0) < 0 and (T) > 0, then (0).(T) < 0. So there exits one solution t1 = t1 *  (0, T) of equation (3.1.17), which can be easily solved by Newton-Raphson method.
Also, Substituting t1 = t1 * in equation (12), we find the optimum value of S, given as: Again the total amount of backorder at the end of the cycle is ( ) . so the optimal value of Q is given by [Singh et . al., Vol.5 (Iss.2: SE) And the minimum value of the average total cost ( ) as from equation (16).

Model II : Inventory Starts with Shortages
In this subsection, we have considered the deterministic inventory model for deteriorating items, where the inventory is allowed to start with shortages. Here the two situations may arise, depending on procurement time 1 t . (i)  , neglecting the second and higher order terms of α, the solutions of the differential equations (22)-(24) are respectively given as: As discussed in subsection III.1,

( )
The optimal value of 1 t for the minimum average total cost per unit of time is the solution Provided that the value of 1 t satisfies the condition ( ) For different values of the various parameters, Eq.(34) can be solved numerically using the Newton-Raphson method.
Furthermore, the total backorder amount for the entire cycle is Therefore the optimal order quantity, Q * is ( ) In this situation, the two governing Eqs. (20) and (21), become The solution of the differential equations (36)-(38) with the boundary conditions I(0) = 0 and I(T) = 0 are ( ) To minimize 3 C T the optimal value of 1 t can be obtained by solving the equation Which also satisfies the condition ( ) The total backorder amount for the cycle is 2 Therefore the optimal order quantity Q * is

Numerical Examples
Example: To illustrate the theory developed above, the following numerical example has been considered.
Let the input parameters are as follows: Here Model I denote that the inventory model starts without shortages and Model II denote that the inventory model starts with shortages, and then applying the procedure described in previous section, the optimal solution for Model I and Model II are those given in  From above Table it is found that the minimum average total cost per unit of time and the optimal order quantity both are smaller in the case when the inventory starts with shortages.

Conclusions
We have analyzed two order-level inventory models for deteriorating items with a ramp type demand function of time and deterioration rate is assumed to follow a two-parameter Weibull distribution. Analytical solutions of the model are discussed and are illustrated with the help of two numerical examples.