In this note we comment on whether the cost rate function of Model 2 of Rosling (2002) is exactInventory control; compound renewal process

Larsen, Christian

English

Research Papers in Economics

 
WORKING PAPER L-2007-03 
Christian Larsen 
Note: Comments on the paper by Rosling (2002)
 
Logistics/SCM  
Research Group   1
Note: Comments on the paper by Rosling (2002) 
 
Christian Larsen 
Department of Business Studies 
Aarhus School of Business, University of Aarhus 
Fuglesangs Allé 4 
DK-8210 Aarhus V 
Denmark 
e-mail: chl@asb.dk
tel.: +45 89486317 
 
Abstract In this note we comment on whether the cost rate function of Model 2 of 
Rosling (2002) is exact. 
 
Keywords: Inventory control, compound renewal process. 
 
 
 
1. Introduction 
 
In Rosling (2002) basic results for a compound renewal process are presented. These are 
subsequently applied to derive cost functions for various inventory systems having the 
common characteristics that all replenishment orders have constant lead-time L and all 
unfilled demands are backlogged, denoted Models 1 to 5. In the authors opinion the paper 
might contain a logical flaw concerning Model 2, whose cost function is claimed to be 
exact (or not explicitly claimed to approximate). Our reason for this claim is that Model 2 
is to be considered as a special case of Model 3 (which is explicitly acknowledged by 
Rosling (2002) to be approximate), when the length of the review period in the latter 
model converges to 0. We will therefore start deriving the cost function of Model 3, 
GM3(x,P), which measures the total cost incurred between two review instances. Contrary 
to Rosling (2002) we will explicitly introduce a variable P to denote the length of the 
review period. Then we analyze  {}
3
0
(, )
lim
M
P
Gx P
P
↓  and show that it is equal to GM2(x), 
the cost rate function of Model 2. Then we discuss the argumentation in Rosling (2002) 
for GM3(x,P) being approximate, and explain why, when taking Roslings argumentation 
literally, GM2(x) should then also be considered approximate. We illustrate our point with 
a numerical example which shows that the difference between GM2(x) and the simulated 
value (simulating a base stock system with constant inventory position x) is very small 
though. Throughout the note we try to use the notation of Rosling (2002) as closely as 
possible. 
 
 
 
   2
2. The renewal process 
 
The compound renewal process is characterized by a random variable T which denotes 
the time between customer request, where E[T] = 1/λ. It has distribution function Ω(t). 
The random variable T  is the residual time until the next customer request. It has density 
function 
%
() t ω % . The customer order size is given by a random variable X, which has 
distribution function Ψ(x). The random variables T(n) and X(n) are the n’th fold 
convolutions of T and X respectively, with respective distribution functions Ωn(t) and 
Ψn(x). Per definition Ω0(t) ≡ 1 and Ψ0(x) ≡ 1, that is T(0) ≡ 0 and X(0) ≡ 0.   is the 
distribution function of  , where 
() n t Ω %
( 1) TT n +− %
0() 1 t Ω ≡ % . For n =0,1,… define pn(t) = Ωn(t) - 
Ωn+1(t), and  .   1 () () () nn n pt t t + =Ω −Ω %% %
 
Lemma 1 
[]
0
1
() 0
()
() () 1 , 2 , . .
n
nn
pt n d
pt
ptp t n dt
λ
λ −
−= ⎧
= ⎨ −= ⎩
%  
 
Proof 
From Rosling (2002; Equations 2 and 3) it follows that  1 () () nn
d
tp
dt
λ − Ω= % t . Using the 
definitions of pn(t) and  ( ) n p t %  establishes the proof. 
 
 
Define the random variable D(t) to be the total demand in a time period of length t 
immediately following a demand instance. Define the random variable D(t)+1 = D(t) + X. 
Define the random variable  () t D %  to be the total demand in a randomly chosen time period 
of length t. Denote by F(t)(x),  F(t)+1(x),   their respective distribution functions. 
These are given as 
()() t Fx %
 
()
0
() ( ) () tn
n
n F xp t
∞
=
=Ψ ∑ x
n
 
 
()1 1
0
() ( ) () tn
n
F xp t
∞
++
=
=Ψ ∑ x
n
 
 
and 
 
()
0
() ( ) () tn
n
F xp t
∞
=
=Ψ ∑ % % x  
 
respectively. 
   3
The result of following lemma is stated verbally by Rosling (2002). It is of course 
intuitively very obvious. However, we give a formal proof. 
 
Lemma 2 
() [] [ t ED t EX λ = % ]  
 
Proof:  
() ()
0 0
[] ( ) 1 ( ) tn n
n
E Dp t x
∞ ∞
=
⎡⎤
=− Ψ ⎢⎥
⎣⎦ ∑ ∫
% % d x  
 
From Lemma 1 it follows that 
 
() ( () 1
00 00
()1 ( ) ()1 ( ) tn n n n
nn
d ) E Dp t x d x p t x
dt
λλ
∞∞ ∞∞
+
==
⎡⎤ ⎡ ⎡⎤ =− Ψ − − Ψ ⎢⎥ ⎢ ⎣⎦ ⎣⎦ ⎣ ∑∑ ∫∫
% d x
⎤
⎥
⎦
]
 
 
  = λE[D(t)+1] - λE[D(t)] = λE[X] 
 
Thus   is a linear function in t.  Combining this with  () [ t ED %
{} () 0 lim 0 t t ED
↓ ⎡⎤ = ⎣⎦
%  
establishes the proof. 
 
 
 
3. Derivation of the cost expression in Model 3 
 
Consider a time point t0 where a review just has occurred bringing the inventory position 
at level x. Following the usual convention as measure of the total cost during the next 
review period we use the total cost incurred in the time interval (t0 + L, t0 + P + L]. 
Consider a time point t0 + L + τ with 0 < τ ≤ P. The expected on-hand inventory at this 
time point is 
() (,) ( )[ ] ( 1 () ) L
x
OH x x L E X F y dy τ τλ τ
∞
+ =− + + − ∫
%  
 
(note OH(x,τ) = 0 when x ≤ 0) while the expected back-log at this time point is 
 
() () (,) 1 () L
x
BLx F y d y τ τ
∞
+ =− ∫
%  
 
During this period the expected back-log increases with 
 
() ( ) () ( 0 ) () () LL P
x
BL P BL F y F y dy
∞
+ ⎡⎤ −= − ⎣⎦ ∫
%%  
   4
With probability   the net-inventory is non-positive at time point t () 1 L F τ + − % ( ) x 0 + L + τ. 
All in all we conclude that 
 
3
00
(, ) (,) (,)
PP
M Gx Ph O H xdp B L xd τ ττ =+ ∫∫ τ
)
 
 
   () ( () ( ) ( )
0
() () 1 ()
P
LL P L
x
F yF y d y b F x d τ π τ
∞
++ +− + − ∫∫
%% % . 
 
 
Therefore 
 
{} ()
3
() ( ) 0
(, )
lim ( ,0) (0) ( ) 1 ( )
M
tL P
tL x
Gx P d
hOH x pBL F y dy b F x
Pd t
π
∞
↓
=
⎡⎤ =+ − + − ⎢⎥ ⎣⎦ ∫
%%  
 
From Lemma 1 we have 
 
() () () ()1
0
() () ( ) () () tn n t t
n
dd
F yy p t F y F
dt dt
λ
∞
+
=
=Ψ = − ∑ % % y  
 
Therefore we get 
 
{} ()
3
() () ()1 0
(, )
l i m ( [] )( )1 ( ) ( ) ( )
M
LL L P
xx
Gx P
hx L EX h p F y d y F y F y d y
P
λλ π
∞∞
+ ↓ ⎡ ⎤ =− + + − + − ⎣ ⎦ ∫∫
%
 
     () () 1( L bF x +− % )
 
This is exactly the cost rate function, GM2(x), of Model 2 in Rosling (2002; Equation 12). 
We find it more logical to deduce the cost rate function of Model 2 in this way, instead of 
the direct way in Rosling (2002), where a perhaps subtle argument is used to explain the 
Bπ(x) term. 
 
 
Rosling (2002) writes that Model 3 is only approximate. He writes: The approximation 
only concerns the first arrival in a period, so the effect should be small when the arrival 
intensity is not too low. We interpret this in the following way: as long as the average 
inter-arrival time between order requests 1/λ is small compared to the length of the 
review period P, then the approximation is almost exact. However when viewing Model 
2 as the limiting case (when P converges to 0) this requirement is bound to be violated, so 
therefore when taking the caution of Rosling (2002) literally then the cost rate function of 
Model 2 must be an approximation. We are however inclined to believe that Rosling 
(2002) is very cautious because for all practical purposes a review instance must be   5
considered as a randomly chosen time point with respect to the renewal process. So any 
differences between GM2(x) and the actual value should be very small. We illustrate this 
in the following section. 
 
 
 
 
4.  A numerical example 
 
Here we illustrate that GM2(x) is maybe only an approximation on a numerical example. 
from Zipkin (2000; Chapter 6). We have the cost parameters h = 0.25, p = 1.75, π = b = 
0. The lead-time L = 0.5. All customers demand 1 unit. We adapt the numerical example 
to a case of a non-Poisson arrival process by letting the time between order requests to be 
Erlang distribute with mean 1/3.6 and having k = 4 phases. Let γ = 14.4 be the rate of the 
underlying Poisson process. For this example the random variables D(L) and  () L D %  have 
probability distributions, see Cox (1962): 
 
 
1
()
0
()
()
()
ix k k
L
L
i
k
PD x e
ix k
γ γ
+ −
−
=
==
+ ∑ !
 
 
and 
 
 
1
0
() 1
0
()
0
!
()
()
1, 2,..
() !
i k
L
i
L ix k k
L
i
ki L
ex
ki
PD x
ki L
ex
ki x k
γ
γ
γ
γ
−
−
=
+ −
−
=
⎧ −
= ⎪
⎪ == ⎨
− ⎪ = ⎪ + ⎩
∑
∑
%  
 
 
We have developed a computer code, programmed in Visual Basic for Excel, to calculate 
the function GM2(x). We also have developed a simulation model, programmed in Arena 
version 9.0, of a base stock system with base stock level x, meaning that the inventory 
position is constantly at level x. The long run average cost of this inventory system 
should be equal to GM2(x). For the values of x = 0 to 5 our results are summarized in 
Table 1. 
 
<Table 1 about here> 
 
For all the values of x the differences between computed and simulated values are very 
small. What appears as a statistical significant difference can though be seen in the case x 
= 2. 
  
   6
 
5. Concluding remarks 
 
Based on the fact that Model 2 is a special case of Model 3, we have expressed a concern 
about the cost rate function of Model 2 being exact or not. More numerical analyses must 
be conducted in order to make an affirmative conclusion.  
 
 
 
References 
 
Cox  D.R.: Renewal theory. Methuen, London, 1962. 
 
Rosling K.: Inventory cost rate functions with nonlinear shortage costs. Operations 
Research, Vol 50, 2002, pp. 1007-1017. 
 
Zipkin P.H.: Foundations of inventory management. McGraw-Hill, Boston, USA, 2000. 
 
 
     7
Tables 
 
  
x GM2(x)  Sim Min Max 
0  3.1500 3.1511 3.1493 3.1523 
1  1.4521 1.4525 1.4509 1.4538 
2  0.4125 0.4140 0.4134 0.4145 
3  0.3305 0.3302 0.3299 0.3306 
4  0.5510 0.5507 0.5506 0.5510 
5  0.8000 0.7998 0.7997 0.8001 
Table 1: Results of an analysis of the numerical example. In column 2 the computed 
values of GM2(x) are stated. The simulation was replicated 5 times and each simulation 
lasted 250000 time units. Column 3 states the mean value and column 4 (5) is given the 
minimum (maximum) value across the replications. 
 Working Papers from Logistics/SCM Research Group
 
 
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ISBN 87-7882-181-9 
Department of Business Studies 
Aarhus School of Business 
University of Aarhus 
Fuglesangs Allé 4 
DK-8210 Aarhus V - Denmark 
 
Tel. +45 89 48 66 88 
Fax +45 86 15 01 88 
 
www.asb.dk  
 

Foundations of inventory management.

Renewal theory.

Note: Comments on the paper by Rosling (2002)

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Note: Comments on the paper by Rosling (2002)

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