Analysis of Constant Deteriorating Inventory Management with Quadratic Demand Rate

This paper investigates inventory-production systems where items fo llow constant deterioration. The objective is to develop an optimal policy that min imizes total average cost. The quadratic demand technique is applied to control the problem in order to determine the optimal production policy, holding cost and cost of deterioration. Sensitivity analysis is conducted to study the effect of the cost parameters on the objective function.


Introduction
The purpose of the present paper is to give a new dimension to the inventory literature on time varying demand patterns. Researchers have extensively discussed various types of inventory models with linear trend (positive or negative) in demand. The main Limitation in linear time-vary ing demand rate is that it implies a uniform change in the demand per unit t ime. This rarely happens in the case of any co mmodity in the market. In recent years, so me models have been developed with a demand rate that changes exponentially with time. Demands for spare parts of new aeroplanes, computer chips of advanced computer mach ines, etc. decrease very rapidly with t ime. So me modellers suggest that this type of rapid change in demand can be represented by an exponential function of time. The present authors feel that an exponential rate of change in demand is extraord inarily high and the demand fluctuation of any commodity in the real market cannot be so high .A realistic approach is to think o f accelerated growth (or decline) in the demand rate in the situations cited above and it can be best represented by a quadratic function of time. Thus, this paper has the scope of direct application in the very practical situations noted above.
Goods deterio rate and their value reduces with t ime. Elect ron ic p roducts may beco me obsolete as technology changes. Fashion tends to depreciate the value of clothing over time. Batteries die out as they age. The effect of time is even more critical for perishable goods such as foodstuff and cigarettes. The effect of deterioration and time/age is that the classical inventory model has to be readjusted K. Heng, J.
Labban, R. Linn (1) In general, deterioration is defined as decay, damage, spoilage, evaporation, obsolesce, pilferage, loss of utility or loss of marg inal value of a co mmodity that results in decrease of usefulness from the original one. The decrease or loss of utility due to decay is usually a function of the on-hand inventory. It is reasonable note that a product may be understood to have lifet ime, which ends when utility reaches zero.
The continuously decaying/deterioration of items is classified as age-dependent ongoing deterioration, and age-independent ongoing deterioration. Blood, fish, strawberry are so me of the examp les of the fo rmer wh ile alcohol, gasoline and radioactive chemical and grain products are examp les of the latter H. Wee (4).
Haip ing and Wang (7) developed an economic policy model for deteriorat ing items with time proportional demand. Donaldson (8) derived an analytical solution to the problems of obtaining the optimal nu mber of replenish ments and the optimal replen ishment times of an EOQ model with a linearly time dependent demand pattern, over a finite time horizon. Zangwill (9) developed a discrete-in-time dynamic programming algorithm to solve an inventory model by allo wing the inventory levels to be negative where the demand pattern is time dependent. Follo wing The approach of Donaldson (8), Murdeshwar (6,)Sahu and Sukla (10) has tried to derive an exact solution for a fin ite horizon inventory model to obtain the optimal nu mber of replenishments, optimal replen ishment times and the optimal times at which the inventory level falls to zero, assuming the demand rate to be linearly t ime dependent and shortages. Hamid (3) presented a heuristic model for determining the ordering schedule when inventory items are subject to deterioration and demand changes linearly over time and obtained an optimal replenishment cycle length. Goswami and Chaudhuri (1) presented an EOQ model for deteriorating items with shortage and linear t rend in demand. Bradshaw and Erro l (2), published a paper in which they derived unbounded control policies for a class of linear time invariant production-inventory systems.
This paper investigates inventory-production systems where items follow constant deterioration. The objective is to develop an optimal policy that minimizes the cost associated with inventory and production rate. The quadratic demand technique is applied to control the problem in order to determine the optimal production policy. Sensitivity analysis is conducted to study the effect of the cost parameters on the objective function.

Assumptions and Notations
The follo wing assumptions and notations have been used in developing the model.
Here a stands for the initial demand rate and b for the positive trend in demand.
(ii) The production rate Say of the on-hand inventory deteriorates per unit time.
(iii) The lead-time is zero and shortages are not allowed. (v) C is the total average cost for the production cycle and S is the stock level reached in the cycle.

(vi)
The set up cost is not considered in this model because it is taken to be fixed for the whole cycle time.
(vii) Planning horizon is finite.

Mathematical Formulation and Solution
Let q be the inventory level at any time t ) 0 ( The differential equations governing the system in the interval The stock level in itially is zero. Production begins just after t=0, continues up to Using the value of ( ) t R , the two equations (1) and (2) take the form The solution of equation (3)  Similarly, the solution of equation (4) also is (neglecting the powers of θ greater than 1) Fro m (6) and (7) we get the relation Now the average holding cost becomes Now substituting the value of S fro m (9) and simp lifying we get The average cost due to deterioration in the total cycle t ime is

Conclusions
In this article, a determin istic inventory model has been proposed for deteriorating item with quadratic demand rate, where shortages are not allo wed. The goal o f the paper is to incorporate the deterioration phenomenon together into an inventory model over a fin ite planning horizon. This paper investigates inventory-production systems where items follow constant deterioration. The objective is to develop an optimal policy that minimizes total average cost. The quadratic demand technique is applied to control the problem in order to determine the optimal production policy, holding cost and cost of deterioration.