9th International Conference on Industrial Engineering and Industrial Management

XXI International Conference on Industrial Engineering and Operations Management

International IIE Conference 2015

Aveiro, Portugal. July 6-8, 2015

On the on-hand stock estimation in a lost sales

context and periodic review policy

Abstract Traditionally, inventory literature assumes that unfilled demand can be

backordered for the next replenishment cycle. However, there are a lot of practical

situations where, if an item is out of stock, backordering assumption in not appli-

cable and unfilled demand is lost. The main problem of the lost sales case is the

mathematical difficulty of its treatment. This paper focuses on the estimation of on

hand stock levels just after the order arrives, i.e. at the beginning of the cycle. On

one hand, we present a review of the existing literature on the on hand stock esti-

mation in lost sales case. On the other hand, we propose a new close-form ap-

proach to compute the probability vector associated to the on hand stock levels at

the beginning of the cycle for periodic review systems and discrete demands. Nu-

merical results show that our approximation presents low deviations and over-

comes other estimation methods.

Keywords: on hand stock, lost sales, periodic review, discrete demand

1 Introduction

There are two possible customer's responses when an item is temporarily out of

stock: (i) unfilled demand is backordered and filled as soon as the replenishment

order arrives or (ii) unfilled demand is lost. Although there are many real life

situations where backordering assumption is not applicable, only a handful of in-

ventory papers study optimal policies for systems with lost sales. This is mainly

because backordering models are easier to formulate and simpler to analyze

[(Hadley and Whitin 1963), (Zipkin 2008), (Bijvank and Vis 2012)]. However, the

assumption of excess demand being lost is of practical importance in sectors such

as retailing (Gruen et al. 2002), service sector (Diels and Wiebach 2011) or on-

line commerce (Breugelmans et al. 2006).

Recently, lost sales models have received more attention in inventory research

and there is a proliferation of publications under this context [(Bijvank and Vis

2011), (Bijvank et al. 2014)]. Lost sales problem was formulated long time ago by

(Karlin and Scarf 1958) but it is still very hard to formulate and deal with. In in-

2

ventory systems the demand is satisfied with the available stock at the beginning

of the cycle, i.e. just after the order arrives. At this point, if we assume backorder-

ing, the probability associated to each stock state can be easily computed as the

difference between the inventory position at the replenishment moment (which is

equal to the base stock, S) and the demand during the lead time. However, apply-

ing the same reasoning in a lost sales context could lead to negative net stocks, so

that finding the on hand stock probability vector becomes a challenge. Therefore,

the key question in lost sales models is to know accurately the on hand steady

probability vector at the beginning of the cycle ( ( )RP OH ) because not only ser-

vice measures but also performance metrics must be computed based on them.

To the best of our knowledge, only (Cards et al. 2006) propose an exact ex-

pression to compute ( )RP OH for lost sales context, periodic review policy and

discrete demand. This expression applies only when there is just one outstanding

replenishment order at every time, however it is not a close formula what compli-

cates its implementation in practical environments. The main goal of this paper is

to propose a new close-form approximation to compute the probability of on hand

stock levels for lost sales context and periodic review (R, S). Furthermore, we pre-

sent a thorough review of current existing methods and analyze their performance.

The rest of the paper is organized as follows. Section 2 introduces the notation

and basic assumptions of this paper. Section 3 describes current methods to com-

pute the on hand stock probability vector and proposes a new approximation. Nu-

merical results are presented in Section 4. Finally, Section 5 highlights the most

relevant conclusions of this work and presents further researches.

2 Basic Notation and Assumptions

In general, the periodic review, base stock (R, S) system places replenishment or-

ders every R fixed time periods to raise the inventory position to the base stock S.

The replenishment order arrives after a constant lead time L. Figure 1 shows an

example of the evolution of the on hand stock and the inventory position for the

lost sales case. Notation used in it and in the rest of the paper is as follows:

S = base stock (units),

R = review period and replenishment cycle corresponding to the time be-

tween two consecutive deliveries (time units),

L = lead time for the replenishment order (time units),

OHt = on hand stock in time t from the first reception (units),

IPt = inventory position in time t from the first reception (units),

NSt = net stock in time t from the first reception (units),

Dt = accumulated demand during t consecutive periods (units),

ft() = probability mass function of Dt,

Ft() = cumulative distribution function of Dt,

3

X+ = maximum {X, 0} for any expression X,

E[X] = expected value of expression X.

Fig. 1 On hand stock and inventory position evolution in a periodic review and lost sales system

This paper considers the following assumptions: (i) time is discrete and is or-

ganized in a numerable and infinite succession of equi-spaced instants; (ii) the

lead time, L, is known and constant; (iii) the replenishment order is added to the

inventory at the end of the period in which it is received, hence these products are

available for the next period; (iv) demand during a period is fulfilled with the on

hand stock at the beginning of that period; (v) only one outstanding replenishment

order is launched within any period, which means that L

4

2.1 Exact calculation

(Cards, Miralles, & Ros 2006) derive and inductive expression to compute

( )RP OH , based on modelling the on hand inventory as an ergodic Markov chain

with a set of states {0, 1,..., S}. This probability vector is computed by means of

calculating the probability transition matrixes of the on hand stock levels between:

(i) the beginning of the cycle and the review moment (times 0 and R-L),

R L jiM m

= , and (ii) the review and the beginning of the next replenishment cy-

cle (times R-L and R), L kj

M s = .

Hence, R R L L

M M M= is the transition matrix between two consecutive re-

plenishment cycles. Considering the case where the Markov chain is regular,

limn

Rn

M M

= where all rows of M are the same vector v , which is the principal

left eigenvector and their components are positive, add up to one, and represents

the state probabilities of every feasible value of the on hand stock at the beginning

of any cycle, i.e. ( )Rv P OH= .

Note that the convergence process required to find M requires a huge compu-

tational effort and may be time consuming, mainly for large values of S, which

complicates its implementation in practical environments. However it is very use-

ful for reference purposes.

2.2 Non Stock Out Approximation

A close form approximation to compute ( )RP OH can be derived based on the as-

sumption that there is not stockout during the lead time. In this case, the stock bal-

ance at R can be easily computed as

[ ]R R L L R L R L L R L LOH OH D S OH OH D S OH S D+

= + + =

Then, we can define

( )( )

( )

0

1 1 0

L

R

L

f S i i SP OH i

F S i

< =

= (1)

so that the vector of on hand stock probabilities is

5

( ) ( )( ) ( ) ( )1 1 1 0R L L LP OH F S f S f (2)

Expression (2) is quite similar to the backlog case, obviously except for the

zero stock probability.

2.3 Bijvank and Johansen Approach

(Bijvank and Johansen 2012) study optimal replenishment policies and suggest a

close-form expression to approximate the average on hand stock level when de-

mand follows a pure and compound Poisson distribution. To that end, authors de-

velop an approximation to compute ( )RP OH as in backordering case but correct-

ing the demand distribution with a factor Cs in order to avoid negative net stocks.

( )( )

( )

0

1 0

S L

R

S L

C f S i i SP OH i

C F S i

< =

= (3)

In its derivation, authors assume that L can be any value, but following the as-

sumption of L

6

then, we need to know the value of OH0. We assume that: (i) we initialize the

system with an stock equal to the base stock, i.e. OH0=S and (ii) the on hand stock

at the review period is always positive, i.e. OHR-L>0. Under this assumptions

( )_R OOS R L R LOH S S D D = (7)

and therefore the probability vector in an out of stock situation is expressed as

( ) ( ) ( ) ( )( )0 1 1R R L R L R LOOSP OH f f i F S

(8)

We propose a new close-form approximation which combines both extreme

situations, taking into account that the probability of an stock out occurs during L

can be computed as FR(S). Then, the proposed approach to compute ( )RP OH is

( ) ( ) ( ) ( )( ) ( )1R R R R RNOOS OOSP OH F S P OH F S P OH + (9)

where ( )R NOOSP OH represents the probability vector when there is not an

stock out situation (expression (2)) and ( )R OOSP OH the probability vector when

always there is an stock out during L (expression (8)).

3 Numerical results

This section illustrates deviations which arise from using approximations instead

of the exact expression for ( )RP OH calculation. We assume that demand follows

a Poisson distribution with = 0.01, 0.1, 0.5, 1, 2, 5 and 10. The inventory policy

values considered are: L = 1, 2, 3, 4, 5; R = 5, 7 and S = 5, 7, 9. The total feasible

combinations result in 189 cases. We quantify the deviations computing the mod-

ule of the error vector with the following expression:

e ap

Deviatione

=

(10)

where e

is the exact ( )RP OH vector and ap

is the approximate vector obtained

with the Non Stock Out Approximation vector (NSO), Bijvank&Johansen ap-

proach (B&J) and the new approach we propose in this paper (Prop.Appr.). Table

7

1 presents the average and the standard deviation of each approximation for dif-

ferent values of DR+L.

Table 1 Deviation between exact and approximate methods to compute ( )RP OH

Average Standard deviation

NSO B&J Prop.Appr. NSO B&J Prop.Appr.

[0-0.1] 0.000 0.000 0.000 0.000 0.000 0.000

]0.1-1] 0.000 0.000 0.000 0.000 0.000 0.000

]1-5] 0.059 0.068 0.101 0.084 0.102 0.177

]5-10] 0.421 0.653 0.406 0.273 0.545 0.235

]10-20] 1.098 2.874 0.474 0.352 1.995 0.377

]20-50] 1.485 9.160 0.159 0.371 7.720 0.268

>50 1.420 21.469 0.006 0.094 11.801 0.021

Total 0.678 5.401 0.166 0.667 9.609 0.276

As can be observed, the B&J estimation presents the highest average and stan-

dard deviations whereas the new approach presented in this paper shows the low-

est. If we analyze the results for categories of DR+L we observe that, when demand

is low, all the approximations present a good performance. However, for high val-

ues of DR+L both the average and the standard deviations of NSO and B&J are in-

creasingly high.

4 Conclusions

We propose a new perspective to deal with lost sales systems. Most of the papers

on inventory control focus on proposing cost models or service metrics in lost

sales context, for which they need to know the on hand stock at the beginning of

the cycle. However, this work focuses directly on the calculation of the on hand

stock and proposes a close formula which can be applied to compute any other in-

ventory metric. We consider periodic review, (R, S) system, and discrete demands.

We observe that when DR+L is low, all approximations present low average and

standard deviations corresponding to cases which most of the demand has been

fulfilled. However, as numerical results point out, the proposed approximation

presents the best performance even for high values of DR+L.

Further researches should focus on analyzing the impact of these approxima-

tions when they are used to compute service measures. Furthermore, there is only

one exact expression which assumes that L

8

Acknowledgements

We gratefully acknowledge the help of Dr. Marco Bijvank, University of Calgary,

for providing useful comments to improve this work. All errors remain ours. This

research is part of a project supported by the Generalitat Valenciana, Ref.

GV/2014/006.

5 References

Bijvank, M., Huh, W.T., Janakiraman, G., & Kang, W. 2014. Robustness of Order-up-to Policies

in Lost-Sales Inventory Systems. Operations Research, 62, (5) 1040-1047

Bijvank, M. & Johansen, S.G. 2012. Periodic review lost-sales inventory models with compound

Poisson demand and constant lead times of any length. European Journal of Operational Re-

search, 220, (1) 106-114 available from: ISI:000302448100011

Bijvank, M. & Vis, I.F.A. 2011. Lost-sales inventory theory: A review. European Journal of Op-

erational Research, 215, (1) 1-13

Bijvank, M. & Vis, I.F.A. 2012. Lost-sales inventory systems with a service level criterion.

European Journal of Operational Research, 220, (3) 610-618

Breugelmans, E., Campo, K., & Gijsbrechts, E. 2006. Opportunities for active stock-out man-

agement in online stores: The impact of the stock-out policy on online stock-out reactions.

Journal of Retailing, 82, (3) 215-228

Cards, M., Miralles, C., & Ros, L. 2006. An exact calculation of the cycle service level in a

generalized periodic review system. Journal of the Operational Research Society, 57, (10)

1252-1255 available from: ISI:000241588300012

Diels, J.L. & Wiebach, N. 2011. Customer reactions in Out-of-Stock situations: Do promotion-

induced phantom positions alleviate the similarity substitution hypothsis? Berlin, SFB 649

Discussion paper 2011-021.

Gruen, T.W., Corsten, D., & Bharadwaj, S. 2002. Retail Out-of-Stocks: A Worldwide Examina-

tion of Extent Causes, Rates and Consumer Responses Washington, D.C., Grocery Manufac-

turers of America.

Hadley, G. & Whitin, T. 1963. Analysis of Inventory Systems Englewood Cliffs, NJ, Prentice-

Hall.

Karlin, S. & Scarf, H. 1958, "Inventory models of the Arrow-Harris-Marschak type with time

lag," In Studies in the mathematical theory of inventory and production, Standford, Ca.:

Standford University Press.

Zipkin, P. 2008. Old and New Methods for Lost-Sales Inventory Systems. Operations Research,

56, (5) 1256-1263 available from: ISI:000261236800016