An EOQ model with exponentially increasing demand for deteriorating items under two levels of storage

Introduction

The term inventory includes both the manufactured products that are available for sale and the raw materials utilized in the manufacturing process. Inventory is among the most valuable assets possessed by a business, as inventory turnover is one of the principal contributors to revenue generation and, consequently, shareholder returns. Inventory is classified as a current asset on the balance sheet of the company. Raw materials, finished goods, and work-inprogress constitute the three categories of inventory. Inventory models for deteriorating items subject to exogenous demands have been studied by various authors in different periods. Most of the authors have assumed a constant rate of deterioration and also constant demand. These include Shah and Jaiswal (1977), Aggarwal and Goel (1984), Nahimias (1991), Aggarwal (1978), Aggarwal and Veena Jain (2001) [1, 2, 3] . Decay in radio active elements, spoilage in food grain storage, Pilferages form on hand inventory etc., are continuous in time and they are approximately proportional to the on-hand inventory. Efforts in analysing mathematical models of inventory in which a constant or variable proportion of the on-hand inventory gets deteriorated per unit of time have been studied. Ghare and Schrader (1963) and Emmons (1968) considered the case of a constant fraction of the on-hand inventory gets deteriorated per unit of time. This work has been extended by Covert and Philip (1973), who developed EOQ model for units with a variable rate of deterioration of items in the inventory. In all these models, shortages are not allowed and the demand is assumed to be deterministic. Shah (1977) has generalised the work of Ghare and Schrader (1963) and Covert and Philip (1973) have developed an inventory model by considering general deteriorating function with shortages. Two probabilistic models for deteriorating items with general deteriorating function are developed by Shah and Jaiswal (1977). Teng and Chang (2005) discussed an EOQ model for deteriorating items where demand depends on stock and price [4, 5, 6, 10, 11, 12] . We now develop a single period inventory model with exponentially increasing demand for deteriorating items under L2 -system like the one discussed by Hartley (1976), Sarma & Sastry (1988) [7, 9] . Whenever initial on-hand inventory exceeds the storage capacity of the own warehouse (OW), the excess stock will be kept in a rented warehouse (RW). The goods kept in RW are prone to a lower rate of deterioration than the goods kept on OW. This is one of the reasons for a costly holding cost in RW. We now study the effect of stock deterioration in both warehouses on the policy variables when the demand occurs with exponentially increasing during the period and determine the optimal order level.

Notations and Assumptions of the Model

We adopt the following notations in carrying out the mathematical model of the inventory system.

Model description and analysis

Let Q (t) be the inventory level at any time t (0 ≤ t ≤ T) and demand rate r (t) assumed to be deterministic and is increasing exponentially with time 't'.

Further let f (t) = ae αt , 0 ≤ α ≤ 1, a > 0

The inventory is depleting due to the simultaneous occurrence of demand and the deterioration.

Situation in RW

Let QR (t) denote the on-hand inventory at time t in RW. Then the differential equation that describes the state of inventory is given by

The solution of (1) with boundary condition at t = 0, QR (t) = Z is given by

Stock loss due to deterioration in the interval (0, tw) is given by

Using ( 5) in ( 4), we get

Loss of stock due to deterioration till time tw is given by

Situation in OW

During the interval (0, t1) the stock kept in OW is subjected to both deterioration and consumption. The inventory level QO (t) at time 't' of the system in OW satisfies the equation

The solution of (11) with the boundary condition at t = t1, QO (t) = W is given by

The solution of (12) with boundary condition at t = t1, Q (t1) = 0 is given by

Stock loss due to deterioration in the interval (0, t1) is given by

Using ( 17) in ( 16), we get

Loss of stock due to deterioration within cycle T till time t1 is

Backlogged demand i.e., B (t1) is given by

The total cost per unit time C (T, tw, t1), is the sum of the ordering cost, inventory holding cost and shortage cost, i.e.,

For F = H in the above equation, the model reduces to

This is same as in Aggarwal and Veena Jain (2001). Our objective is to minimise the cost function C (T, tw, t1) which is a differentiable function. This assumption is meaningful since the units kept in RW will be consumed first as the holding cost of the RW is higher than the holding cost of OW. So, the length of time used by RW is less than the length of time used by OW. Obviously, the total length of the cycle (T) is more than the length of the time t1.

From this assumption the cost function which is a three variable now becomes a function of single variable, i.e. T and the cost function of the entire inventory system become

To minimise the cost function C (T), the necessary condition is given by 0

The optimal value of T says T * can be obtained from equation (29) and optimal order quantity q* can be obtained from (22). The sufficient condition for minimising C (T) is

Sensitivity Analysis of the Model:

As an illustration to the above model, we consider a hypothetical system with the following parameters: C3 = 200, a = 50, α = 0.1, β = 0.3, γ = 0.8, θ1 = 0.05, θ2 = 0.02, C = 15, F = 6, H = 4, and π = 25. The equation (29) generally does not yield to a closed form solution. However, one can find the approximate solution by using Taylor series expansion for exponential function and ignoring second and higher order terms αθ2, αθ1, αβ, αβγ, etc., a quadratic equation in T will be obtained which can be solved to yield the approximate value of T and near optimum order quantity can be obtained from (22).

The following table shows the variations in the cycle length (T), cost and optimal order quantity for different values of '  '.

Observations

From the table 1, it is observed that for fixed values of deteriorating rates in OW and RW as the demand increases, optimum ordering quantity increases significantly and the associated cost is also increasing and the use of rented warehouse (RW) in time units is decreasing. This is quite expected result since the large amount of demand will empty the RW fastly.

The following table shows the variations in the cycle length (T), cost and optimal order quantity for different values of 'β'.

Observations

It is observed from the above table that as β increases, the usage of rented warehouse increases in time units. However, the optimum ordering quantity and the associated cost marginally increases.

Observations

From the table 3, it can be seen that as γ increases there is a marginal increase in the usage of rented warehouse in time units and the marginal increase can be seen in the economic order quantity and there is considerable decrease in the inventory total cost.

Discussion

In this paper an economic order quantity for deteriorating items when demand increases exponentially with time under two levels of storage is presented. For mathematical simplification certain assumptions are made in equation ( 27), which is justifiable since the total cycle length is always greater than the length of time used by OW and RW.

References

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