Inventory Management Systems with Hazardous Items of Two-Parameter Exponential Distribution

Problem statement: An exponentially decaying inventory was first devel oped by. The certain commodities were observed to shrink with ti me by a proportion which can be approximated by a negative exponential function of time. Here, the novelty we took into consideration was that a constant hazardous rate followed by the two-paramet er xponential distribution. Approach: We studied the inventory system with two-parameter exp onential distributed hazardous items in which production and demand rate were constant. Results: The mathematical model was developed with an exponential distribution hazardous item to obtain t he total cost per unit time of an inventory system. It was illustrated with the help of numerical example. The inventory controlled systems in terms of first order differential equations were solved numericall y. The effect of changes in the model parameters on decision variables and average total cost of an inv entory system were studied through numerical example. Conclusion: This study minimized the total cost for constant h azardous rate of exponential distribution.


INTRODUCTION
Inventory may be considered as an accumulation of a product that would be used to satisfy future demands for that product. It requires a policy to control that inventory. Since the pioneering research [4] inventory models are being treated by mathematical techniques. Until the nineteen fifties only inventory models with deterministic demands were considered. After the nineteen fifties, due to developments within the theory of stochastic processes, the research on inventory models with stochastic demands initiated. In all inventory models a general assumption is that products generated have indefinitely long lives. In general, almost all items deteriorate over time. There are many other products in the real world that are subject to a significant rate of deterioration. The impact of product deterioration should not be neglected in the decision process of production lot size. Deterioration can be classified as age-dependent on-going deterioration and age-independent on-going deterioration. Since long time, researchers are engaged in analyzing inventory models for deteriorating items such as volatile liquids, medicines, electronic components, fashion goods, fruits and vegetables. At the end of the storage period deterioration is studied [9] for the fashion goods industry. Earlier some researchers considered continuously decaying inventory for a constant demand [10] . An orderlevel inventory model was presented for deteriorating items with a constant rate of deterioration [12] . An orderlevel inventory model was also developed [1] by correcting and modifying the error [12] in calculating the average inventory holding cost. A variable deterioration rate of two parameter Weibull distribution was used to formulate a model with the assumptions of a constant demand rate and no shortages [3] . A model was developed for determining the production rate with deteriorating items to minimize the total cost function over a finite planning period [2] . An extended the model [6] , given [7] to deal with a case in which the finished product is also subject to a constant rate of decay. A multi-lot-size production-inventory system was considered for deteriorating items with constant production and demand rates [8] . This type of inventory control problems have been studied also by [5,11,[13][14][15] .
In this study we study the inventory system with two-parameter exponential distributed hazardous item in which production and demand rate are constant. An exponentially decaying inventory was first developed by [10] . The certain commodities were observed to shrink with time by a proportion which can be approximated by a negative exponential function of time. Here, the novelty we take into consideration is that a constant hazardous rate followed by the two-parameter exponential distribution. The probability density function of a two-parameter exponential distribution is given by: Where: µ = The location parameter θ>0 = The scale parameter The probability distribution function is: The hazardous rate of the on-hand inventory is given by: We are concerned with an inventory system for a single product subject to two-parameter exponential hazardous item to determine a production cycle time in this study. The objective of this study is to develop a mathematical model for obtaining an optimal purchase quantity for constant hazardous item associated with exponential distribution during the cycle time. To minimize the total cost per unit time of an inventoryproduction system is also our interest. The effect of changes in the model parameters on decision variables and total cost of an inventory-production system is studied through numerical example.

Basic model assumptions and notations:
The following are the assumptions applied in the proposed model: • A finite planning horizon is assumed • The production unit of the product is available and it can meet the demand • The inventory system deals with single item • The demand rate for the product is known and constant • Shortage is not allowed

Development of the model and solutions:
The objective of the inventory problem here is to determine the optimal order quantity so as to keep the total relevant cost as low as possible. Let x(t) be the on-hand inventory at any instant of time t(0≤t≤T). The state level of inventory amplify due to production rate and depletes due to demand rate and hazardous rate 1 θ of exponential distribution. The dynamics of the state equation of inventory level x(t) of the product at time t over period [0, T] is governed by the differential equation: with boundary conditions x(0) = 0 and x(T) = 0. The solution of differential Eq. 1 using boundary equation x(T) = 0 is: And assuming θ<1, an approximate value is obtained by neglecting those terms of degree greater than or equal to 2 in θ using Tailors series expansion of the exponential functions: The inventory holding cost in the system during production cost per unit time is: Using x(0) x = % and by (2) we obtain: The number of units that hazarded; H(T) during one cycle time is given by: So, cost of hazardous rate per unit time is: The production cost per unit time is given by: The set up cost per unit time is given by: Therefore by (3-6) the average total cost per unit time is given by: Our objective is to minimize the total cost per unit time K(T). The necessary condition for total cost K(T) to be minimum is: i.e., the second derivative is found to be positive. The inventory level x(t) in terms of the first-order differential Eq. 1 is solved numerically using the mathematical package MATLAB version 6.5. For this, Fig. 1 shows the convergence of the optimal inventory with unit time towards inventory goal level x(t) (say) about 3.25. Now we are going to display the Inventory holding cost, cost of hazardous rate and average total cost per unit time which are computed for the set of hazardous rates. Figure 2-5 show the inventory holding cost, cost of hazardous rate and total cost with the reference of hazardous rates given in Table 1.  Figure 2 shows that when hazardous rate increases, inventory holding cost per unit time increases at a certain level (say $247.02 per unit time) and then it gradually decreases. Figure 3 shows that when hazardous rate increases, cost of hazardous rate per unit time decreases. Figure 4 shows that when hazardous rate increases, average total cost per unit time increases at a certain level (say $78.13 per unit time) then it starts declining. Figure 5 shows that when hazardous rate increases and inventory holding cost increases, average total cost per unit time increases at a certain level (say $78.13 per unit time) then it abruptly decreases.

CONCLUSION
The optimal inventory control problem with exponential distributed hazardous item is studied and the mathematical model is developed for optimal purchase quantity. The average total cost per unit time is derived and is found be a relatively simple expression. However, this study minimizes the total cost for constant hazardous rate of exponential distribution.