a continuous review inventory model with order expediting

A continuous review inventory model with order

expediting

Aram SıtkıSezgi Çelik

Güncel Dörtköşe

Continuous review system Lead Time ‘L’ Fixed Duration ‘T’ Part of the duration can be either Ta or Te

Where Ta>Te; When inventory Reaches the level ‘r’ order

size ‘Q’ is released.

Abstract

Reorder point

If the net inventory at time T after the order release is equal to or smaller than ‘re’ , the order is expedited at a cost comprising a fixed cost per unit and the lead time is L=T+Te, otherwise L=T+Ta,

Abstract

Order Quantity Q

Reorder Point r

Order Expediting Point re

Decision Variables

To find inventory policy variables Q, r and re

That minimized Average Cost rate.

The Aim

When lead times are not negligible, random

demand during lead time might lead to shortage. If shortage implies high costs, stockouts can be reduced by issuing emergency orders.

Lead time usually comprises components such as order preparation, order delivery, manufacturing and transportation.

Introduction

In some cases, options exist for reducing the

duration of some of these components. For example; There are cases in which transportation can be carried out in either slower or faster mode such as by air and by truck. In most cases achieving the shorter lead time implies a cost premium.

Introduction

Introduction

However, rather than considering a second ( emergency) order we analyze the option of expediting the deliveryof the outstanding order at an additional cost. We consider a continuous review inventory system in which the lead time L includes a part of fixed duration. T and a part whose duration can be either Ta or Te where Ta>Te. When the net inventory reaches the reorder point ‘r’ an order of size ‘Q’ is released.

Introduction

If the net inventory at time T after the order

release is equal to or smaller than re, the order is expedited at a cost comprising a fixed cost Ke and a cost per unit ce and the lead time is L= T+Te Otherwise the lead time is L=T+Ta.

Therefore, the decision varibles consideret are the order quantity Q, the order reorder point r and the order expediting point re.

Introduction

Aim is to find the inventory policy variables Q,

r and re that minimize average cost rate.

Introduction

Q Order Quantity Ka Fixed ordering Cost per order Ke Fixed expediting cost per order Ca Acqusition cost per unit Ce Expediting cost per unit H inventory holding cost per unit time unit Π Penalty cost per unit short

Notation

B Expected number of units backordered in

a cycle µ Demand rate ( units demanded per unit of

time) H Expected on hand inventory per cycle in

units times time units. Ha Expected on hand inventory in cycles

without order expediting He Expected on hand inventory in cycles with

order expediting P Probability of expediting an order

Notation

I(t) Net inventory at time t after the release

of the normal order of the cycle X(t) Demand accumulated up to time t f(x,t) Probability density function of the

demand x during a time inveterval of size t ( f(x,t)=0 whenever x<0)

Notation

If I(t)≤re the inventory level at the end of the

constant part of the lead time is smaller or equal than the order expediting point re, then at an additional cost the the order is expedited, to be delivered at T+Te. Otherwise the order is delivered at T+Ta. We will state the problem in function R=r-re and r instead of in function of re and r. We will assume r≥0 and r≥re thus R≥0.

The Model

The probability of expediting an order is

therefore given by

p=P(I(T)’≤re) =P(X(T)≥r-re) .

The Model

The expected number of units backordered

per cycle B. To obtain B we will condition first on the demand x during the first component of the lead time, of length T:

B(x)=

The Model

We need to calculate the average demand

rate during the fixed duration part of the lead time (T) in the cases of expediting (λe) and no expediting (λa).

λa= E[X(T)|X(T) R] =

The model

λa= E[X(T)|X(T) R] = The following equivalence must hold. p λe+(1-p) λa= µ

The model

The expected on hand inventory per cycle is

h=pHe+(1-p)Ha

= - Q (Ta-p(Te-Ta)-+T)+ W

The model

Inventory Level

r

The Model

µ

λa

µ

T Ta

One cycle

Time

re

Expected inventory levels in cycles without order expediting

W= The expected length of the cycle isTt=T+(1-p) +p=

The Model

Q*= EQUATION (1)h-π EQUATION (2)

Equations

-++ EQUATION (3) EQUATION(4)

Equations

1. Set a value for and using Eq. (4) find thecorresponding value for Rmax; restricted to

positive integers.

2. For R = 0, 1….., Rmax repeat steps 3 to 7.

3. Set an initial value for Q (using for exampleEq. (1) with V = Ka and W = 0).

The Model

4. Repeat steps 5 and 6 until no changes

occur inQ and r:

5. Find r from Eq. (2).

6. Find Q from Eq. (1).

The Model

7. Set QR = Q and rR = r:

8. Find R* such that EC(QR* ; rR* ;R*)=

Min R EC(QR, rR,R); then QR* ; rR* ;R* is the solution of the algorithm.

The Model

Validation of the modelTable 1

Comparison of analytical and simulated result

Table 2Comparison of analytical and simulated results for a different back order

cost rate

Table 3Optimal inventory policies for different expediting costs

Table 4Optimal inventory policies for different values Te while holding T and Ta

constant

Table 5Optimal inventory policies for different values of Ta while holding T and Te

constant

Table 6Optimal inventory policies for different values of T while holding T+Ta and

Ta/Te constant

Table 7Optimal inventory policies for different values of Ta while holding T+Ta

and Te constant,for two values of Ke

In this paper we have developed a model to

find the optimal inventory policy when there is an expediting option.We have presented an algorithm to obtain the policy variables that attain a global minimal average cost rate when the inventory policy decision variables are integers.We have also discussed the case when the decision variables are real valued.

Conclusions

We have verified extensively the model by

simulation .In the numerical examples we have studied the behavior of our model with respect to the variation of some parameters.That analysis highlights how the proposed model be used ,not only to establish the inventory management policy for a given set of parameters but also to understand how the system would be affected by changes in these parameters, once the inventory policy is a adjusted accordingly.

Conclusion