An Inventory Model Of Decaying Products

In this paper, an order degree stock list for deteriorating points with life clip has been discussed. Deficits are considered and to the full backlogged. The demand rate has been considered of the signifier that first it increases linearly, so it remains changeless and eventually it decreases to zero in the deficit period. Such type of demand rate is more realistic. This theoretical account is interesting as it deals with broad assortment of consumer demand behaviours. Cost minimisation technique has been used to work out the stock list job under consideration.

1. Introduction

Several points such as nutrient points, veggies and pharmaceuticals decay with clip when kept in stock list. Therefore it is of import to analyze the stock list of the points which deteriorate with clip. Several writers have presented the stock list theoretical accounts of disintegrating points. Ghare & A ; Schrader ( 1963 ) ab initio considered the impact of impairment for a changeless demand. Later on Shah & A ; Jaiswal ( 1977 ) , Aggarwal ( 1978 ) , Covert & A ; Philips ( 1973 ) and Goyal & A ; Giri ( 2001 ) developed the stock list theoretical accounts of deteriorating points.

Mandal & A ; Pal ( 1998 ) studied an order degree theoretical account for an stock list that deteriorates at a changeless rate where as demand is a ramp type over clip and obtained approximative solution for little impairment rate. Wu & A ; Ouyang ( 2000 ) besides studied the same type of stock list job. Such type of demand seems to be more realistic as they represent the demand behaviour of new trade names of consumer goods coming to the market. Wu & A ; Ouyang used the Newton-Raphson method to work out the numerical job of minimising the entire stock list cost. They besides provided a sum-up of the literature on the subject. Warburton & A ; Jazia ( 2007 ) presented a theoretical account of an order degree stock list system incorporating the points that deteriorates with clip with ramp type of demand. They obtained the solution in footings of Lambert W map. They besides presented some belongingss of the optimum solution in which preciseness issues emerge.

In this paper, an order degree stock list for deteriorating points with life clip has been discussed. Deficits are considered and to the full backlogged. The demand rate has been considered of the signifier that first it increases linearly, so it remains changeless and eventually it decreases to zero in the deficit period. Such type of demand rate is more realistic. This theoretical account is interesting as it deals with broad assortment of consumer demand behavior. Cost minimisation technique has been used to work out the stock list job under consideration.

2. Premises and Notations

1. The premises considered here are as follows:
2. The stock list is replenished sporadically with period T.
3. Deficits are allowed and to the full backlogged.
4. A changeless fraction of the on-hand stock list deteriorates per unit clip. Deterioration of points starts after clip which is the life clip.
5. The demand rate is assumed of the signifier

The notations used in this chapter are as follows:

1. I ( T ) = Stock degree at any clip T.
2. = Deterioration rate.
3. = The life clip after which the impairment of points starts.
4. T= The period of stock list rhythm.
5. C= The entire sum cost during the stock list rhythm T.
6. = The keeping cost per unit per unit clip.
7. = The deficit cost per unit per unit clip.
8. = The impairment cost per unit per unit clip.
9. Q= The maximal stock degree after the backlogging.

3. Mathematical Model and Analysis

We assume that the stock list starts with a positive value I ( 0 ) =Q after carry throughing the backorders. We develop the stock list theoretical account in the undermentioned two instances:

CaseI:

Case II:

Case I: When life clip lies in the interval.

In this instance, the stock list diminutions due to the demand up to the life clip. During the period the stock list lessenings due to both the consumer demand and impairment. In this period the demand rate is additive map of clip, i.e. it increases with clip. During the period the demand rate becomes changeless and the stock degree reduces to zero at. During the period the demand rate lessenings with clip and in this period there is deficit of points which is to the full backlogged. The stock list theoretical account under consideration has been presented in figure 1 given below:

The needed differential equations regulating the stock list are given by

The boundary conditions are given by

The solution of equation ( 1 ) is given by

Using the boundary status, we have

Therefore

The solution of equation ( 2 ) is given by

Using the continuity of I ( T ) at, we have

Therefore

The solution of equation ( 3 ) is given by

Using the boundary status, we have

ThereforeAA

The solution of equation ( 4 ) is given by

Using the boundary status, we have

Therefore

Since the stock list is uninterrupted, the equations ( 6 ) and ( 7 ) must fit at. Therefore we have

The entire stock list cost TC is the amount of keeping cost, deficit cost and impairment cost. Therefore we have

where, and represent the unit keeping cost, deficit cost and impairment cost and, and are the figure of points held in the stock list, the figure of points backordered and the figure of points deteriorated severally.

Now the sum accumulated figure of points held in the stock list over the interval is given by

The entire figure of backordered points is given by

The figure of points that deteriorate during the period is equal to the initial stock list sum Q minus the entire accumulative demand. Thus we have

Therefore the entire stock list cost is given by

The entire stock list cost therefore obtained may be considered as a map of the free parametric quantity, the backorder clip ; taking the other parametric quantities as fixed. The minimal entire cost can be obtained by distinguishing the cost map w. r. t. . Therefore we have

It can be shown that the 2nd derivative given by ( 17 ) is positive. Therefore equation ( 16 ) is the status of lower limit. The value of obtained from equation ( 16 ) gives the optimum value of ( denoted by ) . Consequently, the optimum values of Q ( denoted by ) and that of TC ( denoted by ) can be obtained.

Case II: When life timelies in the interval.

In this instance, the stock list diminutions due to the demand up to the life clip. During the period, the stock list lessenings due to the consumer demand. In this period the demand rate is additive map of clip, i.e. it increases with clip. During the period the demand rate becomes changeless and the stock degree reduces to zero at. During the period, the stock list lessenings due to the consumer demand merely while during the period, the stock list lessenings due to both the consumer demand every bit good as impairment. During the period the demand rate lessenings with clip and in this period there is deficit of points which is to the full backlogged. The needed differential equations regulating the stock list are given by

The boundary conditions are given by

The stock list theoretical account under consideration has been presented in figure 2 given below:

The solution of equation ( 18 ) is given by

Using the boundary status, we have

Therefore

The solution of equation ( 19 ) is given by

Using the boundary status at, we have

Therefore

The solution of equation ( 20 ) is given by

Using the boundary status, we have

Therefore

The solution of equation ( 21 ) is given by

Using the boundary status, we have

Therefore

Since the stock list is uninterrupted, the equations ( 24 ) and ( 25 ) must fit at. Therefore we have

Now the sum accumulated figure of points held in the stock list over the interval is given Bs

The entire figure of backordered points is given by

The figure of points that deteriorate during the period is equal to the initial stock list sum Q minus the entire accumulative demand. Thus we have

Therefore the entire stock list cost is given by

The equation ( 31 ) can be approximated as

The entire stock list cost therefore obtained may be considered as a map of the free parametric quantity, the backorder clip ; taking the other parametric quantities as fixed. The minimal entire cost can be obtained by distinguishing the cost map w. r. t. . Therefore we have

Again distinguishing the cost map w.r.t. , we get

The 2nd derivative given by ( 34 ) is positive. Therefore the value of given by ( 33 ) minimise the entire cost given by ( 32 ) . The value of obtained from equation ( 33 ) gives the optimum value of ( denoted by ) . Therefore we have

Consequently, the optimum values of Q ( denoted by ) and that of TC ( denoted by ) can be obtained.

4. Numeric Example

Keeping cost = Rs. 50 per unit per twelvemonth

Deficit cost = Rs. 200 per unit per twelvemonth

Decay cost = Rs 80 per unit per twelvemonth

Demand = 500 units

Life time= 0.25 twelvemonth

Fraction of stock list decay =0.001

Time twelvemonth

Replenishment clip T = 1 twelvemonth.

The optimum value of is given by twelvemonth. The optimum value of stock degree Q is given by. The optimum value of entire stock list cost is given by =Rs. 2092.08.

Decisions

In this paper, an order degree stock list theoretical account for deteriorating points with life clip has been discussed. Deficits are considered and to the full backlogged. Demand rate has been considered of dual incline type, i.e. first it increases linearly, so remains changeless and eventually decreases linearly to zero. The looks for entire costs have been obtained and the cost minimisation technique has been to obtain the optimum values of the parametric quantities.