a study of the effect of stochastic inventory reorder point on shortage and delay

8/12/2019 A Study of the Effect of Stochastic Inventory Reorder Point on Shortage and Delay

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A Study of the Effect of Stochastic Inventory Reorder Point on

Shortage and Delay

CHENG Yongsheng1, JIA Li2

1. School of Business Administration, Jiangxi University of Finance and Economics, Nanchang,

Jiangxi, China

2. School of Foreign Languages, Jiangxi University of Finance and Economics, Nanchang, Jiangxi,

China

[email protected]

Abstract: This paper aims to study the effect of stochastic inventory reorder point on shortage and

delay. In the case of normally distributed demand, partial delivery and with (R, Q) reordering policy,

this paper presents the distribution of the amount of stock out of in a reordering cycle, mean and

variance at the given level of reorder point. The analytical function of the backordering time mean is

also worked out. The main findings of the current study show that shortage and backordering time arethe decreasing function of reorder pointR, which are tested by some numerical examples.

Keywords: reorder point, shortage, delay

1 IntroductionAmong the current studies of inventory control, most of them pursue the minimum of inventory cost as

the objective function and the inner optimization of the inventory system. This evidently does not

satisfy the strategic needs of enterprises. Inventory control should be aimed to maintain the inventory

at a favorable level to guarantee the realization of the organizations' goal. Accordingly, the inventory

control aim should be a set of diversified objectives. Suppose the objective is the pursuit of the cost

minimum at the satisfactory service level, the shortage and the backordering time could be the key

index for service. For example, some clients claim the shortage limited at a designated amount and thebackordering time at a given point. In time of shortage, they would like to wait within the said time

limit. Otherwise, the clients would turn to the other suppliers. Therefore, the operators should make

their inventory control policy in light of service.

Assume the formula of reorder point, shortage and delay to be expressed

as ),(2 DRfES= , ),(3 DRfE = by acknowledging the shortage and backordering time, the

operators will choose reorder point reversely to control inventory material. Under the favorable

condition of shortage and delay, an appropriate reordering policy R is made to control the inventory

materials. This paper aims at studying the effect of reorder point on shortage and delay.

A host of research[1]-[3] [6]-[8]

has been devoted to the construct of various inventory models, to the

studies of single cycle or multi-cycle, of single commodity or multi-commodity, of commodity with the

features such as corrosive, obsolete or stock loss, of one-echelon, two-echelon or multi-echelon, open

loop or closed loop, and to the studies of inventory control in the various specific situations. Themonographic research on shortage and delay is comparatively fewer. Song

Xu

Liu[4]

formulate

the iterative algorithm of shortage based on orders. Song [5]finds out the exact expression of shortage

concerning orders and upper and lower bound. When and Huang [9]construct the iterative algorithm of

working out distributors substantial need rate for components in the situation of permitting

transshipment, and they attain the approximate formula of average shortage of orders. [10] By

Markovian analysis, they construct the iterative algorithm of working out system stable probability, put

forward the approximate formula of shortage based on orders. Wang and Lin [11study the back ordering

and inventory policy based on the lead time and the price discount.


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Most of the inventory control models hypothesize the lead time of delivery (or purchase) as an external

constant. It, actually, is an internal variable. Generally speaking, lead time can be divided into

shipment time and back ordering time. Shipment time can be assumed as a constant; back orderingtime depends on the stock level. When the stock level is higher, the shortage is lower and back

ordering time is less. Accordingly, in the (R, Q) policy, the back ordering time relies on the reorder

point R. In the case of normally distributed demand, partial shortage and reorder policy (R Q), this

paper will explore the distribution of stock out, mean and variance of stock out at the given level of

reorder point. It will then discuss the effect of back ordering time mean, and these are tested by some

numerical examples.

2 Model assumptions and notation presentationThe retailer is assumed to meet the customer demand with a normal distribution and to replenish the

stock from the outside supplier. The normally distributed demand assumptions are widely used for the

universal existence and the adaptability of the normal distribution, which can be the simplification of

any other distribution. Transportation times for the delivery are assumed to be constant. Based on a

first-come, first-serve principle, continuous review installation stock (R

Q) policies, that is,

continuous review inventory position are made. When the inventory position at a considered

installation declines to or below the reorder pointR, a number of batches of size Qare ordered. Since

lead time is bigger than 0, inventory position is possibly out of stock before the arrival of the ordered.

Thus this leads to shortage. All stock outs are delayed orders and backordered. Accordingly, shortage

number and backordering time are not definite. This is a plausible policy not uncommon in practice,

such as in furniture business. For a further discussion of the assumptions, let us introduce the following

notation:

Q=batch size at retailer

L=lead time for an order to arrive at retailer

DL=demand within the lead time for an order to arrive at retailer

R=reorder point for retailer =standard deviation of demand per unit of time at retailer

It=inventory level at the time point t=average demand per time unit at retailer

=stochastic delay at the warehouse for retailer orders

)(x = density of standardized normal distribution function

)(x =cumulative distribution function of the standardized normal distribution function

)(xfL =density function of demand within the lead time

S =shortage within reordering cycle

T =operating cycle (e.g., annual)

3 Effect of shortage and delay3.1 ShortageAssume that at time 0, the inventory level declines to RI =0 . An orderQis triggered, and afterLtheorder arrives at the retailer. Here Qis bigger than the demand within the lead time, i.e, the batch size

at retailer is sufficient enough to satisfy the demand within the lead time. It can be concluded that the

demand within the lead time for an order to arrive at retailer DL fits a normal distribution on

),( 2LL . If RDL> ,shortage takes place. Shortage within reordering cycle can be expressed

as RDS L = , and the corresponding probability density function can be expressed


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as )()( Rsfsf Ls += . If RDL , shortage is 0, and probability function can be expressed

as )()()()0( L

LR

dxxfRDPSP

R

LL

===>

(1)

Fig. 1 Shortage probability density/ normal distribution

Within a reordering cycle, average shortage number can be expressed as

)](1)[(2

)()(

)()(

)()()(

2

2

2

)(

L

LRRLe

dxxfRdxxfx

dxxfRx

dxxfRxRDEES

L

LR

RLL

R

LR

LR

L

+=

=

=

==

+

(2)

Shortage number is the decreasing function of reorder pointR .

The following is the variance of computing shortage S.

Given222 )(ESESS = (3)

We have

= 022 )( dssfsES S , in whichfs(s) is density of normal distribution function of shortage S

in the interval (0,

).

)0()(

2)()(

)()(2)(

)()2()()(

2222

2

22

2222


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In the formula above, fL(R) is density of standardized normal distribution function, and its value is

available. ESis the mean of shortage, P(S 0) is probability of shortage, which can be obtained from

(1) and (2).Therefore, the standard deviation of shortage Sis

222 )(ESESSs == (4)

Average reordering cycle is

Q, and the number of reordering within the time for operation is

Q

T.

In the whole operating cycle T, the shortage number isQ

TES . (5)

Shortage rate based on the shortage number can be computed by the following:

Q

ES

SP

=

=

=

demandreorderingin thedemandcyclereorderingin thenumbershortage

demandtotal

numbershortagetotal)(

(6)

Shortage in the reordering cycle has no correlation with the reordering batch size. However, shortage

rate computed by number has negative correlation with reordering batch size. With the increase of the

reordering batch size, the total shortage decreases and shortage rate also declines during the whole

operating cycle.

3.2 Delay

Assume that at time 0, the inventory level declines to RI =0 . An orderQis triggered, and afterLtheorder arrives at the retailer. Shortage even goes before the arrival of the commodity. Assume that at

time t the inventory position is expressed as LtIt


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Fig. 2 probability density/ normal distribution between 0-t

The discussion above shows that the distribution of the delay time is comparatively complex and the

precise computing of average delay time is also a tough task. Some software may be used for the

approximate computing. According to LITTER formula, the average delay time E can be computedin a practical and simplified way as:

L

ESE =~

(8)

4 Numerical resultsIn the following, we shall do the numerical tests based on the model discussed above. Assume =

4/per day, Q=20. = 4, L = 4, T= 360 days, the average reordering cycle is 5 days, the numericalresults can be obtained in Table 1:

Table 1: Numerical results

reorder

oint

R

shortage

rate

P

1

shortage

ES(2)

variance2

3

standard

deviation (4)

annual

shortage

(5)

shortage

rate

P

6

delay time

E~

8

11 73% 6.30 38.82 6.23 453.27 31% 1.57

12 69% 5.58 35.42 5.95 401.93 28% 1.40

13 65% 4.91 31.95 5.65 353.76 25% 1.23

14 60% 4.29 28.49 5.34 308.93 21% 1.07

15 55% 3.72 25.09 5.01 267.58 19% 0.93

16 50% 3.19 21.81 4.67 229.79 16% 0.80

17 45% 2.72 18.72 4.33 195.58 14% 0.68

18 40% 2.29 15.85 3.98 164.93 11% 0.57

19 35% 1.91 13.24 3.64 137.76 10% 0.4820 31% 1.58 10.91 3.30 113.93 8% 0.40

21 27% 1.30 8.87 2.98 93.27 6% 0.32

22 23% 1.05 7.11 2.67 75.55 5% 0.26

23 19% 0.84 5.62 2.37 60.55 4% 0.21

24 16% 0.67 4.38 2.09 47.99 3% 0.17

25 13% 0.52 3.36 1.83 37.61 3% 0.13

26 11% 0.40 2.55 1.60 29.14 2% 0.10

R demand

0

),( 21 ttN

])(,)[(2

2 ttttN ++

t

probability density/distribution


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Table 1 illustrates that with reorder point increasing, shortage rate, shortage and its variance decline

and backordering time also decreases. Noticeably shortage rate based on reordering cycle is different

from that on shortage number. For example, if 16=R , the former is 50%; that means the annualreordering number is 72, and stockouts happen for 36 times before the commodity arrival; but the latteris only 16%. If shortage rate is required to be less than 60% and delay time to be less than one day,

reorder point can be chosen as 15=R . This is a satisfactory inventory policy.

5 ConclusionThe inventory control aim should be a set of diversified objectives, but the shortage and the

backordering time could be the key index for service. With the diversified inventory objectives, the

operators, by acknowledging the shortage and backordering time, will choose reorder point reversely to

control inventory material. Under the favorable condition of shortage and delay, an appropriate

reordering policyRis made to control the inventory materials. This paper aims at studying the function

relationship between reorder point R, shortage and back ordering time. In the case of normallydistributed demand, partial delivery and reorder policy (R, Q), this paper has explored the distribution

of stock out and delay, and has worked out the formula of the mean and the variance of stock out and

delay. The results show that the amount of stock out and the back ordering time is the decreasing

function of reorder point, and these are tested by some numerical examples.

References

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a study of the effect of stochastic inventory reorder point on shortage and delay

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