A Deterministic Inventory Model of Deteriorating Items with Two Rates of Production , Shortages , and Variable Production Cycle

A Continuous production control inventory model is developed for a deteriorating item having shortages and variable production cycle. It is assumed that the production rate is changed to another at a time when the inventory level reaches a prefixed level Q1 and continued until the inventory level reaches the level S > Q1 . The demand rate is assumed to be constant, and the production cycle T is taken as variable. The production is started again at a time when the shortage level reaches a prefixed quantity Q2. For this model, the total cost per unit time as a function of Q1, Q2, S, and T is derived. The optimal decision rules for Q1, Q2, S, and T are computed. The sensitivity of the optimal solution towards changes in the values of different system parameters is also studied. Results are illustrated by numerical examples.


Introduction
EOQ inventory models have long been attracting considerable amount of research attention.For the last fifteen years, researchers in this area have extended investigation into various models with considerations of item shortage, item deterioration, demand patterns, item order cycles, and their combinations.The control and maintenance of production inventories of deteriorating items with shortages have received much attention of several researchers in the recent years because most physical goods deteriorate over time.In reality, some of the items are either damaged or decayed or vaporized or affected by some other factors and are not in a perfect condition to satisfy the demand.Food items, drugs, pharmaceuticals, and radioactive substances are examples of items in which sufficient deterioration can take place during the normal storage period of the units, and consequently this loss must be taken into

Notations and Modeling Assumptions
The mathematical model in this paper is developed on the basis of the following notations and assumptions: i a is the constant demand rate; ii p 1 >a and p 2 > p 1 are the constant production rates started at time t 0 and at time t t 1 >o , respectively; iii C 1 is the holding cost per unit per unit time; iv C 2 is the shortage cost per unit per unit time; v C 3 is the cost of a deteriorated unit.C 1 , C 2 , and C 3 are known constants ; vi C 4 and C 5 are the constant unit production costs when the production rates are p 1 and p 2 respectively C 4 > C 5 ; vii Q t is the inventory level at time t ≥ 0 ; viii A is the setup cost; ix replenishment is instantaneous and lead time is zero; x T is the variable duration of production cycle; xi shortages are allowed and backlogged; xii C is the average cost of the system; xiii the distribution of the time to deterioration of an item follows the exponential distribution g t , where

2.1
θ is called the deterioration rate; a constant fraction θ 0 < θ 1 of the on-hand inventory deteriorates per unit time.It is assumed that no repair or replacement of the deteriorated items takes place during a given cycle.In this paper, we have considered a single commodity deterministic continuous production inventory model with a constant demand rate a.The production of the item is started initially at t 0 at a rate p 1 >a .Once the inventory level reaches Q 1 , the rate of production is switched over to p 2 > p 1 , and the production is stopped when the level of inventory reaches S > Q 1 and the inventory is depleted at a constant rate a.It is decided to backlog demands up to Q 2 which occur during stock-out time.Thus, the inventory level reaches Q 2 backorder level is Q 2 , the production is started at a faster rate p 2 so as to clear the backlog, and when the inventory level reaches 0 i.e., the backlog is cleared , the next production cycle starts at the lower rare p 1 .
We denote by 0, t 1 , the duration of production at the rate p 1 , by t 1 , t 2 , the duration of production at the rate p 2 , by t 2 , t 3 , the duration when there is no production but only consumption by demand at a rate a, by t 3 , t 4 , the duration of shortage, and by t 4 , T , the duration of time to backlog at the rate p 2 .The cycle then repeats itself after time T .The duration of a production cycle T is taken as variable.
This model is represented by Figure 1.

Model Formulation and Solution
Let Q t be the instantaneous state of the inventory level at any time t 0 ≤ t ≤ T ,then the differential equations describing the instantaneous states of Q t in the interval 0, T are given by the following:

3.1
The boundary conditions are neglecting higher powers of θ , 3.10b 3.11 From 3.5 , we have by 3.11 .

3.13
Again, from 3.6 , we have From 3.7 and Q T 0, we have 3.17 18 2a p 2 − a by using 3.6 and 3.7 ,

3.19
I T Total inventory carried over the period 0, T by 3.9 and 3.10a

3.25
Average cost of the system

3.28
The necessary conditions for C Q 1 , S, T to be minimum are

ISRN Applied Mathematics
Solving these and using 3.28 , we get the optimal values Q 1 * , Q 2 * , S * , and and T , respectively, which minimize C Q 1 , S, T provided they satisfy the following sufficient condition: If the solutions obtained from equations 3.30 , 3.31 , and 3.32 do not satisfy the sufficient condition 3.33 , we may conclude that no feasible solution will be optimal for the set of parameter values taken to solve equations 3.30 , 3.31 , and 3.32 .Such a situation will imply that the parameter values are inconsistent and there is some error in their estimation.

Sensitivity Analysis
Sensitivity analysis depicts the extent to which the optimal solution of the model is affected by changes or errors in its input parameter values.In this section, we study the sensitivity of

Concluding Remarks
In the present paper, we have dealt with a continuous production inventory model for deteriorating items with shortages in which two different rates of production are available, and it is possible that production started at one rate and after some time it may be switched over to another rate.It is assumed that the demand and production rates are constant and the distribution of the time to deterioration of an item follows the exponential distribution.Such a situation is desirable in the sense that by starting at a low rate of production, a large quantum stock of manufactured item, at the initial stage is avoided, leading to reduction in the holding cost.The variation in production rate provides a way resulting consumer satisfaction and earning potential profit.For this model, we have derived the average system cost and the optimal decision rules for Q 1 , Q 2 , S, and T when the deterioration rate θ is very small.Results are illustrated by numerical examples.However, success depends on the correctness of the estimation of the input parameters.In reality, however, management is most likely to be uncertain of the true values of these parameters.Moreover, their values may be changed over time due to their complex structures.Therefore, it is more reasonable to assume that these parameters are known only within some given ranges.

Table 1 :
Sensitivity Analysis of Example 4.1.

Table 2 :
Sensitivity Analysis of Example 4.2.

Table 3 :
Sensitivity Analysis of Example 4.3. is insensitive to changes in the values of parameters C 3 and θ, moderately sensitive to changes in the values of parameters C 1 , C 2 , C 5 , A, a, p 1 , and p 2 , and highly sensitive to changes in the value of parameter C 4 .iv Table 3 reveals that T * is insensitive to changes in the values of parameters C 2 , C 3 , and θ, p 2 , moderately sensitive to changes in the values of parameters C 1 , C 4 , C 5 , A, and a, and highly sensitive to changes in the value of parameter p 1 .v It can be seen that the optimum total cost C * is insensitive to changes in the values of parameters C 1 , C 2 , C 3 , θ, A, and p 2 and moderately sensitive to changes in the values of parameters C 4 , C 5 , a, and p 1 . *