Optimal Replenishment Strategy under Combined Criteria

Abstract: This article focuses on determining an optimal replenishment policy for items with three-parameter Weibull distribution deterioration where it represents the time to deterioration of a product. It is also observed that the demand of a consumer product usually varies with its cost and hence, the demand rate should be taken as price dependent. Holding cost is a linear function of time. Here replenishment strategy is developed under combined criteria of demand, deterioration and holding cost. The applicability of the model lies in the management of inventories of deteriorating products and for the particular items for which demand falls due to increase of its cost. Numerical illustrations and sensitivity analysis are provided to illuminate the effect of change of model parameters.


I. Introduction
Inventory management for deteriorating items is very challenging task in all types of business and making proper inventory decision is a key factor of success. Deterioration makes product demand and value dull. So it is a trending research area for all the researchers of inventory area. Deterioration makes product demand and value dull.
The instantaneous rate of deterioration of inventory at time t is defined by where  is the location parameter, ≥  ;  is the scale parameter and  is the shape parameter; represents the three-parameter Weibull Distribution deterioration rate at any time 0  t . Three parameter Weibull distribution is applicable for items with any initial value of rate of deterioration and which start deterioration only after a definite period of time. Ghare and Schrader [11] considered the inventory model for exponentially decaying inventory. Covert and Philip [13] obtained an inventory model for deteriorating items by taking the two-parameter Weibull distribution deterioration with constant demand and holding cost. Philip [3] developed a generalised inventory model of a three-parameter Weibull distribution deterioration and constant demand. Jalan et al. [1] and Chakrabarty et al. [20] made an extension of model of Covert and Philip [13] and Philip [3] by including a three-parameter Weibull distribution deterioration, time-dependent demand rate and shortages in the inventory. Wu et al. [4,5], Deng [12] derived an inventory model for deteriorating items by following the Weibull distribution with time-varying and ramp type demand respectively. Banerjee and Agrawal [14], Ghosh and chaudhuri [15], Sanni and Chukwu [19] obtained a solution for an inventory model with Weibull deterioration and various types of deterministic demand (namely, trended and time quadratic). Ghosh et al. [16] developed a production model with Weibull demand. Goyal and Giri [17], Manna and Chaudhuri [18], Sahoo and Tripathy [6] did an attempt in his paper to obtain the optimal ordering quantity of deteriorating items for time-dependent deterioration rate and demand. Tripathy and Pradhan [9] introduced a model with Weibull distribution deterioration and power demand inventory. Tripathy  The current study focuses on a certain kind of demand pattern where demands of the products are dependent upon its selling price. This inventory model is applicable for deteriorating items and assumes that deterioration follows a three-parameter Weibull distribution. The unique advantage of the functional form of Weibull distribution accommodates with both increasing and decreasing deterioration. Holding cost is also time-varying. An analytical solution of the model is betalked and illustrated with the help of several numerical illustrations and sensitivity of the optimal solution is also examined.

II. Assumptions and Notations
The model is developed with the following assumptions and notations i. ii.
represents the three-parameter Weibull distribution deterioration rate at any time iii. c is the constant purchase cost per unit item. iv. A is the ordering cost per cycle. v.
, where  >0 represents the inventory holding cost per unit of item per unit time. vi.
T is the total cycle length.
vii. Shortages are not allowed.
viii.  is the profit rate.
is the inventory level at any time t .

III. Mathematical Model
be the inventory level at any time t is governed by the following differential equation.
The inventory gradually depletes over time due to two key factors of demand and deterioration. The first term of the right hand side of equation (1) indicates the depletion of inventory per unit time due to deterioration only, while the second term represents that due to demand only. The negative sign indicates that the inventory level decreases over time due to these two factors.
If there is no decay, the differential equation of the inventory level ) ( 2 t I at any time t can be written as The solution of equation (5) is ( 1 t I be the initial stock levels at time 0 and t respectively, the amount of stock depleted in time t due to both demand and deterioration is ) Similarly when there is no decay, the amount of stock depleted in time t due to demand only is As of now loss to the system is due to either decay or demand, the order quantity in each cycle is as follows.
The total cost per cycle is (9) Hence the average system cost is Now profit can be written as a function of the cycle length and price as given below (11) Our aim is to determine the values of T and p that minimize the ) , ( p T C  and maximize the ) , (14) Solution of equations (12) and (13)

IV. Numerical Illustrations & Sensitivity Analysis
This section presents and solves three numerical illustrations to explain how the solution procedure works in support of  , ( p T  are all slightly sensitive to changes in the parameter β and  The observation from above example is that the total cost function is strictly convex. Thus, the optimal value of T can be obtained with the help of total cost function of the model where the total cost per unit time of the inventory system is minimum.

V. CONCLUSION
The classical economic order quantity model assumes a pre-determined constant demand rate and no effects on deterioration of items. In reality, not only demand varies with time, but also variable costs are affected by demand of product. In the proposed model, it represents a deterministic inventory model for deteriorating inventory model with three parameter weibull distribution deterioration rate and demand rate is function of selling price. The holding cost is time varying and linear function of time.
Furthermore, three numerical illustrations are provided in support of theory. Sensitivity analysis for parameters is discussed to assess the effects of optimum solutions with the change of parameters. The derived model is suitable for items with any initial value of the rate of deterioration and also which start deteriorating only after certain period of time. In real situation, almost demands of frequently used products are dependent on selling price for which this model is quite applicable.
A future research may be considered to extend the model under stochastic demand, random or non-instantaneous deterioration and credit policy.

ACKNOWLEDGMENT
The authors express their thanks to anonymous reviewer for the fruitful suggestions to develop this work.