Post on 28-Nov-2014
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