Chapter 12.3. Technical Issues in Inventory Management
1. Net Inventory Positions in <Q, r> Systems with Non-Stationary Poisson Demand
Processes

-
Journal of the Korean Operations Research Society, Vol. 6, No. 2, Oct., 1981
-
C. S. Sung
Introduction
In both continuous-review and periodic-review non-stationary inventory systems, the non-stationary
Poisson demand process and the associated inventory position processes were proved being mutually
independent of each other, which lead to the probability distribution of the corresponding net inventory
position process in the form of a finite product sum of those two process distributions. It is also discussed
how theses results can correspond to analytical stochastic inventory cost function formulations in terms of
the probability distributions of the processes.

Problem
Inventory systems are operated largely based on some operating policies concerning review systems
and ordering rules. The so-called transactions-reporting (continuous-review) systems and periodic-review
systems are commonly used for inventory system review. In both inventory systems the inventory
position
{IPt : t  0} totally depends upon the demand process {N t : t  0} Therefore, once it is
verified that
{IPt u } and {D(t u ,t ] } where D(t u ,t ]  N t  N t u for a lead time u>0, are mutually
independent of each other, the analysis of net inventory process
{NIS t : t  0} will become
straightforward, from which the cost process can be immediately derived whose average one may seek.
The net inventory process is defined as NIS t  IPt u  D( t u ,t ]
The primary objective of this study is to prove that
{IPt u : t  u  0} and {D(t u ,t ] } are mutually
independent of each other, even in the case of non-standard inventory models with non-stationary Poisson
demand processes.

Proof of Mutual Independence
From the point of view of the mathematical theory of probability a stochastic process is best defined as
a famility
X (t ); t T 
of random variables, where the parameter set T is called the index set of the
process.
When demands arrive at time points
t1 , t2 ,, (0  t1  t2 ) , the successive inter-arrival times
X i ; i  1
X 1  t1 , X 2  t 2  t1 ,, X n  t n  t n 1 ,. Let N t be cumulative
demand by time t , t  0 . Then N t ; t  0 is a discrete-valued continuous-parameter stochastic
are defined as
process with sample paths increasing in unit steps.
IPt at time t totally depends upon the demand process Dt ; t  T . If an
is started with IP0  r  i (i  1,2,, Q) at time t  0 , then
An inventory position
inventory
system
IPt   r  j (j  1,2,, Q) at time t   0 can be reached after the (i  j )  or
i  (m 1)Q  (Q  j); m  1,2,
total number of order placements by time
demand materialization by time
t   , where m denotes the
t   and
(i  j )   max 0, i  j.
Suppose now that we consider the sequence of events consisting of the times at which an order in the
amount of Q is placed and received in the constant lead time
elapsed between the
.
Defining
Yk to be the time
(k  1) th and k th orders, the sequence of random variables Yk ; k  1,2,
forms a modified renewal process in which the distribution functions are given by




PY1  y1   P S i  y 1  Fi ( y1 )  P N y1  i ,
 i  the initial stock over the reorder point r.
i

th
where  S i  the renewal epoch of the i demand and so equal to  X k
k 1
F ()  the n - fold convolutio n of the identical distributi on F of X ,
i
 i
and likewise,


PYk  y k   PS Q  y k   P N yk  Q  FQ ( y k ) , for k  2,3,,

since Yk  y k   ( S i  ( k 1) Q  S i  ( k  2 ) Q )  y k

 S Q  y k  for k  2,3,.
Thus, a new renewal process
Wm ; m  0,1,2,
is defined such that
W0  Y0  0
m
Wm   Yk  S i  ( m 1) Q , m  1,2,3,,
k 1
where “ m  0 ” means that no order is placed yet.
Let (t     ) and
m be, respectively, particular values of the time T and the serial number
M of the last order placed no later than t   . If we assume that IPt   r  j ( j  1,2,, Q)
at time t   , then we see that (Q  j ) demands are further needed in the time interval
(t     , t   ] , for   0 , since the inventory position at time t    is r  Q immediately
after the m
Let
th
order is placed at time
t    .
Z t  be the time from t   until the first demand subsequent to t   , that is,
Z t   S N t   1  (t   ),
where
S N t   t    S N t   1 .
Z t  will be the residual or excess waiting time at epoch t   . Then, the distribution
function of Z t  can be determined by use of the renewal equation for m(t )  EN t .
The variable
t   Z be the time point at which the first demand occurs after time t   . Then,
the random variable Z t  may have different distribution from those of X i s. However, the
Furthermore, let
distribution of D( t  ,t ] is determined by partitioning in accordance with the time
the first demand occurs after the time
t   Z at which
t   and the time interval (t    Z , t ] in which k 1
demands occur. For the explicit form of the distribution of
Z t  
and
D
( t  , t ]
, refer to Sung [2].
PIPt   x is a function of PN t   y, which
is determined by N t  . Thereby, we shall prove that given
From the preceding discussions we can see that
means the inventory position
IPt 
IP0  r  i (i  1,2,, Q) at time t  0 , for non-stationary Poisson demand process, the
distribution of
IPt  is independent of that of D( t  ,t ] , even though D(t  ,t ]  N t  N t  .
Theorem 2.1
For the continuous-review <Q, r> inventory system with backorders allowed, constant lead time
  0,
demands occurring in accord with a non-stationary Poisson process with finite mean, and with IP0 = r + i
(i = 1,2,…Q),
PIPt   r  j , Dt  ,T   k   PIPt   r  jPDt  ,T   k ,
for j = 1,2,…Q and k = 0, 1, 2,…
Theorem 2.2
For the periodic-review inventory systems of <nQ, r, T> and <R, r, T> with the same restrictions placed in
Theorem 2.1 and

 0,
 


P IPTk  r  j, DTk , Tk    m  P IPTk  r  j P DTk , Tk    m ,
for m, k = 1,2,… and j = 0, 1, 2,…Q (R-r for <R, r, T> )

Deriving Limit Distribution of Net Inventory Position Processes
For  Q, r  systems,
NIS t  IPt   D(t  ,t ] for t    0
 OH t  BOt
NIS t  OH t , if NIS t  0
 BO t , otherwise.
From the result of Theorem 2.1,
Q
P{NIS t  r  s}   P{IPt   r  j, D(t  ,t ]  j  s} ,
j 1
for s  Q, Q  1, ,0,1,2, 
Q
  P{IPt   r  j, D(t  ,t ]  j  s} ,
j 1
where
P{D(t  ,t ]  j  s}  P{D(t  ,t ]  j  s} , if j  s
=0
Now, consider the distribution of
, otherwise
IPt  processes. Under the assuming conditions discussed in
Theorem 2.1, it follows that
P{N t  n}  P{S n  t   }  P{S n1  t   } , n  1,2, 
 FS n (t   )  FS n 1 (t   )

t 
0
e
 m ( s  )
{m( s   )}n 1 {n  m( s   )}
 ( s   )ds
(n  1)!
n
where
m(t )  E{N t }
 (t ) 
d
m(t )
dt
f Sn (t   )e
 m ( t  )
{m( s   )}n1
 (s   )
(n  1)!
For  nQ, r , T  and  R, r , T  systems, such derivations can be made regarding the following
relations;
NISTk   IPTk  D(Tk ,Tk  ]  OH Tk   BOTk 

Conclusions
It is clear that the discussions in this paper can directly lead to some analytical cost mode constructions
of related stochastic inventory systems <Q, r>, <nQ, r, T> and <R, r, T> with demands occurring in accord
with a non-stationary Poisson process, having finite mean, in terms of the probability distributions OHt
and BOt.
Therefore, for the implementation of this work the only thing to do is stochastically to characterize the
mean value functions of demand processes under real non-stationary circumstances.
Finally, recall that this study was restricted to the inventory systems having non-stationary Poisson
demand processes. However, the subject associated with more general demand processes is still open to
question.
2. A simple closed form of the long-run distribution of stationary inventory
position processes in <R, r, T> inventory systems

-
Journal of the Operations Research Society of Japan, Vol. 25, No. 2, June, 1982
-
C. S. Sung
Introduction
A simple-and-general close form of the long-run distribution of stationary inventory position processes in
<R, r, T> inventory systems is formulated in recurrence relations, the computational procedure of which
is also applicable to determine the long-run distribution of non-stationary inventory position processes in
those systems as long as their limits exist. Moreover, its computation on computer is economically
plausible so that a significant contribution to the analytical study on such inventory problems is greatly
anticipated.

Problem
IP  OH  OO  BO,
IP  R
IP: inventory position
OH: on hand inventory
OO: on hand
BO: backorders
T  Tk 1  Tk , if IPTk  r
T : review time
Tk : kth review time kT, T0
{IPTk } : inventory position at a review time Tk
The process
{IPTk }k 1 just depends on the immediately preceding state of inventory level but not on
any further inventory history
D(Tk ,Tk 1 ] (k  1,2,) : the associated inter-review-period demand processes
PT ,ij  Pr{IP(k 1)T  r  j IPkT  r  i}
 P r {IP2T  r  j IPT  r  i} , for k  (0,1,2,)
and all
(i, j )  S *  {1,2,, R  r}
PT  {PT ,ij } : the associated stationary transition matrix defined on the state space
S  {r  1, r  2,, R}

: the inventory position immediately after the kth review
IPKT
 : lead time

Transition Matrix
(ⅰ) j  i
j  R  r,
and
(ⅱ) i  R  r
j  R  r,
and
(ⅲ) i  j  R  r ,
and
D( kT ,( k 1)T ]  i  j
and
and
D( kT ,( k 1)T ]  l  i
D( kT ,( k 1)T ]  {0, or
l  ( R  r )}
Otherwise, the transition probabilities are zeros
Theorem 1
If a finite N  N stationary transition matrix P is primitive, then the powers pm for m  1 approach a
constant stochastic matrix G such that each row of G is the unique probability vector
…, 
N
satisfying
The matrix
 P =
 =  1,  2,
and PG = GP = G.
PT is found primitive in Table 1, where  i representing Pr {D( kT ,( k 1)T ]  i} with the
constant review-period
T  Tk 1  Tk for k  (0,1,2,) and i  (0,1,2,) .

Long-run distribution 유도
Following is the system of equations the computation will start with ;
R
(1)

i  r 1
 i  1, and
PT* ,
from
1 R   0 R1   R1
 2 R  1 R 1   0 R2   R  2
 3 R   2 R1  1 R   0 R3   R3
 R2r  R   R3r  R1    1 r 3   0 r  2   r 2
 R1r  R   R2r  R1    1 r  2   0 r 1   r 1
i
(2)
(1   0 ) R i    j  R i  j ,
i  1,2,, R  r  1
for
j 1
(3)
letting
(1   0 )
 R i
 Ki
R
i
 R i  j
 R i
 j
,
R
R
j 1
and
(1   0 ) K i    j K i  j ,
j 1
i  1,2,, R  r  1
Ko  1
i
(4)
for
for
i  1,2,, R  r  1
Ki 
(5)
1
1  0
i

j 1
j
K i j ,
i  1,2,, R  r  1
for
Thence,
1
R

i  r 1
i
R 1 r
{
K
i 1
 1} i
i
Hence,
R 
(6)
1
1
,
R 1 r
K
i 1
i
 R i  K i  R ,

and
for i  1,2,, R  1  r
Long-run expected average inventory cost 계산
Net inventory position process
NIS kT u  IPkT  D( kT ,kT u ] for t    0
 OH kT u  BO kT u , and
(7)
NIS kT u  OH kT u ,
if
hence
NIS kT u  0
 BO kT u , otherwise,
where
  u  T 
R r
Pr {NIS kT u  r  s}   Pr {IPkT  r  j, D( kT ,kT u ]  j  s} ,
j 1
for s  R  r , R  r  1, ,0,1,2, 
R r
  Pr {IPkT  r  j}Pr {D( kT ,kT u ]  j  s} ,
(8)
for 0  u  T
j 1
where
Thus
Pr {D( kT ,kT u ]  j  s}  Pr {D( kT ,kT u ]  j  s},
Pr {OH kT u
js
 0 , otherwise
 x}  Pr {NIS kT u  x},
for x  0,1,2,
R r
(9)
if
  Pr {IPkT  r  j}Pr {D( kT ,kT u ]  r  j  x}
j 1

(10)
E{OH kT u }   x  Pr {OH kT u  x}
x 0
R r
r j
  Pr {IP  r  j} x  Pr {D( kT ,kT u ]  r  j  x}

kT
j 1
x 0
R r
r j
j 1
x 0
  Pr {IPkT  r  j} (r  j  n)  Pr {D( kT ,kT u ]  n}
Similarly, since
Pr {BOkT u  x}  Pr {NIS kT u   x},
for
R r
r j
j 1
x 0
x  1,2,
  Pr {IPkT  r  j} x  Pr {D( kT ,kT u ]  r  j  x}

(11)
E{BOkT  u }   x  Pr {BOkT  u  x}
x 1
R r
r j
j 1
x0
  Pr {IPkT  r  j} x  Pr {D( kT ,kT u ]  r  j  x}
R r
r j
j 1
x 0
  Pr {IPkT  r  j}[ E{D( kT ,kT u ] }  (r  j )   (r  j  n)  Pr {D( kT ,kT u ]  n}]
BO kT  BO ( kT ,( k 1)T  ]  BO ( kT ,kT  ]
BO kT : the number of backorder incurred between kT  
inventory position of r  j immediately after the kth review
and (k  1)T   , given the
BO ( kT ,( k 1)T  ] : the number of backorders at time (k  1)T   , before items ordered at ( k  1)T (if
order is made) are delivered
BO ( kT ,kT  ] : the number of backorders at time kT   , after items ordered at time kT (if order is
placed)
Therefore, from the result of Eq. (11),
R r


EBO kT    Pr IPkT  r  j E D( kT ,( k 1)T  ]  D( kT ,kT  ) 
j 1
(12)
r j

  (r  j  n)Pr D( kT ,( k 1)T  ]  n Pr D( kT ,kT  ]  n.
n 0

For ordering cost consideration, let POD (k ) be the probability that an order will be placed at a given

review time (k  1)T (k  0,1,2, ) . It follows that given IPkT  r  j ( j  1,2,  , R  r ) ,
R r

 

Pr IP( k 1)T  r   Pr IPkT  r  j Pr IP( k 1)T  r | IPkT  r  j .
j 1
Under the assumption that inter-review-period demands are independent,
POD (k )  Pr IP( k 1)T  r
R r



  Pr IPkT  r  j 1  Pr D( kT ,( k 1)T ]  j  1.
j 1
In addition, we need define some cost parameters for the formulation of out long-run expected average
annual inventory cost function.
$ A : a fixed ordering cost
$C : the unit cost of each item independent of the quantity ordered
$W : the cost of review
I : the inventory carrying charge constant to give rise to the instantaneous charge rate IC  x from
carrying the on hand inventory level of x .
B : the fixed cost per unit backordered
B̂ : the cost per unit year of the shortage
Thus, for the  R, r , T  type of inventory systems in which demands occurring when the system is
out of stock are backordered, the general long-run expected average annual inventory cost function
H R, r, T  can be derived from Eqs. (10), (11), (12) and (13) as follows;
H  R, r , T  
(14)
W 1
  A[ lim POD (k )]
k 
T T
1 T 
 IC   [ lim EOH kT  u ]du
T  k 
1
1 T 
 B  [ lim EBO kT ]  Bˆ   [ lim EBO kT  u ]du.
T k 
T  k 
Note:
It is know that under the above stated cost systems, the steady state  R, r , T  policy is
optimal when the backorders are allowed.
For a specific example of standard Poisson demand process with
t , the limit terms in Eq. (14) can be
determined as follows;
(15)
 i  Pr D( kT ,( k 1)T ]  i 
e  T (  T ) i
(i  0,1,2,; k  0,1,2,).
i!
 r  j  lim Pr IPkT  r  j
k 
( j  1,2,, R  1  r )
 K R  r  j   R , from Eq. (6)
(16)
e  T

1  e  T
where
R
Rr  j

n 1
 ( T ) n 

K R r  j n   R ,
 n! 
and K R  r  j  n can be finitely determined from both the recurrence relations of Eqs. (5) and
(6).
(17)
r j
Rr
 e u (u ) n 
lim EOH kT u     r  j  (r  j  n)
, from Eq. (10).
k 
n!
j 1
n 0


(18)
r j
Rr

 e   u (  u ) n 
lim EBO kT  u     r  j u  (r  j )   (r  j  n)
 ,
k 
n
!
j 1
n

0



from Eq. (11).
(19)
r j
Rr

 e   (T  ) ( (T   )) n e   ( ) n 
lim EBO kT     r  j T   (r  j  n)

 ,
k 
n
!
n
!
j 1
n

0



from Eq. (12).
(20)
Rr
 j 1 e T (T ) n 
lim POD (k )    r  j 1  
, from Eq. (13).
k 
n
!
j 1
n

0


Similar derivations can also be easily made for compound Poisson demand processes
with Poisson demand occurrences
distributed demand sizes
N t 
Dt 
associated
and mutually independent geometrically (or uniformly)
X i ; i  1,2,, Nt .
It is generally suggested to use a digital computer along with an appropriate search routine to find the
R * , r * and T * which minimize such long-run expected average annual cost
functions H R, r, T  . As another computational procedure of solving such  R, r , T  system for
optimal values of
R * , r * and T * , Dynamic Programming approach has been dealt with in Hadley and Whitin [1] under
$C and of the convex expected cost of carrying inventory and
backorders in the period from  to   T .
the assumptions of the constant unit cost

Conclusions
The computation for the long-run distribution
IP

KT

;T 0

k 1

of a stationary inventory position process
was completed recurrently. In view of the transition matrix PT constructed in Table 1,
the recurrent computation approach is realized as a general approach successfully applicable to determine
the long-run distribution of inventory positions associated with any kind of demand processes in <R, r, T>
inventory systems.
The closed form of the long-run distribution

determined in recurrent format appears much simpler
and more suitable to make practical applications than the complicated solution given in Hadley and
Whitin, and further seems to save much time in their computation on computer, so that its significant
contribution to the analytical study on such inventory problems is greatly expected. Moreover, this
approach can be directly applied to the analysis of non-stationary inventory position processes in those
inventory systems as long as their limit distribution exists.
3. Continuous review (s, S) inventory model with limited backlogging levels and
stochastic lead times
-
Journal of the Operations Research Society of Japan, Vol. 31, No. 2, June, 1988

C. S. Sung and S. M. Yang
Introduction
A (s, S) inventory policy is studied for a continuous review inventory model in which backloggings are
restricted at limited levels and stochastic lead times are allowed. The model assumes that demands occur
in a Poisson process with parameter  , where s is the reorder point and S is the order-up-to level. Lead
times, defined as the time from an order placement until its shipment and denoted by L, are exponentially
distributed with parameter  . If a demand occurs when the system is out of stock, the demand is
backlogged. However, if backlogging sales exceed a limited level b, the exceeding sales is lost. It is
further assumed that at most one order is outstanding The steady state probability distributions of the
inventory levels are derived so as to determine the long run expected average cost. Then, an optimal
solution is characterized and its computational procedure is presented.
The objective of this paper is to determine a limited (fixed) backlogging level along with s and S that
minimize the long-run overall expected average cost. The backlogging level provides an information of
whether each instantaneous demand will be backlogged or lost

Semi-Regenerative Inventory Process
Definition
(1)
Z t (t  0) denotes the random inventory level at time point t taking values on F={S, S-1,S-2,…,-b},
and
(2)
X n (n=0,1,2…) denotes the random inventory level at time period Tn (T0  0) (i.e., X n  Z Tn ),
Tn (n=1,2,3…) denotes the first ordering time point after Tn 1 if X n = i for i>s), or the arrival
time point of an order placed at Tn 1 if X n 1  s
where
Lemma 1
The stochastic process
( X , T )  {( X n , Tn ) : n  0,1,2,} is a Markov renewal process with state
space E,
Where E={S, S-1, …,S-s-b,s}
for S-s-b>s,
={S, S-1, …,s}
for S-s-b=s,
={S, S-1, …,s, …,S-s-b}
for S-s-b<s,
Lemma 2
Z  {Z t : t  0} is a semi-regenerative process with respect to the Markov renewal process (X,T) .

Steady-state Probability of Inventory Levels
In
chain X
order
to
characterize
the
transition
structure
of
the
imbedded
Markov
 { X n : n  0,1,2,} , let pij be the transition probability of moving from state i to j. Then,
pij  Pr[ X n 1  j X n  i ]  lim Q(i, j , t ) .
t 
Hence, for any i, j  E ,
1,
 (1  r )r S  j ,

pij   S  j
r ,
 0,
for
is
and
j  s,
for
is
and
j  S  i  b,
for
is
and
j  S  i  b,
otherwise,
Lemma 3
If b is infinitely large, then the Markov chain X is irreducible, aperiodic and non-null recurrent .
Lemma 4
If b is finite, then the Markov chain X is irreducible and non-null recurrent .
Lemma 5
The Markov renewal process (X, T) is aperiodic.
Lemmas 3, 4 and 5 lead to the next theorem.
Theorem 1
The Markov renewal process (X, T) is irreducible, aperiodic and non-null recurrent .
Theorem 2
For any j  F ,

p j  lim Pr[ Z t  j X 0  i]  {  i  K t (i, j )dt} /{{   i m(i)}
*
t 
It follows form Theorems that
for S  s  b  s;
iE
0
iE
 {(1  r S  j 1 ) /  } / q ( s, S , b),

 (1 /  ) / q ( s, S , b),
pij  
{r s  j /(   )} / q ( s, S , b),

(r s b /  ) / q ( s, S , b),
for
S  s  b 1  j  S ,
for
s  1  j  S  s  b,
for
for
 b  1  j  s,
j  b ,
for S  s  b  s;

{(1  r S  j 1 ) /  } / q ( s, S ),

{r s  j  r S  j 1  ( s  j )(1  r )r S  j } /{(   )q ( s, S ) AA},

pij   s  j
2 S  s  j 1
 (2 s  b  S )(1  r )r S  j } /{(   )q ( s, S ) AA},
{r  r

{r s  j  r S  j 1  (2s  b  S )(1  r )r S b } /{ q ( s, S ) AA},
where
for
s 1  j  S ,
for
S  s  b  1  j  s,
for
 b  1  j  S  s  b,
for
j  b ,
q( s, S , b)  ( s, S ) /   r S b /  ,
q ( s, S )  ( s, S ) /   r S  s /  ,
AA  1  r S b 1 .

Cost Structure
The fixed cost of ordering and the holding cost of on-hand inventory per unit per unit time are,
c0 and c1 . The backlogging cost per unit backlogging sale per unit time is
denoted by c2 and a fixed cost of lost sale is denoted by c3
respectively, denoted by
The expected cost associated with the (s, S) inventory policy having limited backlogging levels depends
on the steady-state probabilities of inventory levels and the amount of lost sales. Therefore, it is necessary
to figure out the expected amount of lost sales. Let E j (LS ) be the expected amount of lost sales during
a regeneration interval with initial state i  s , where If stands for lost sales. Then it holds that

Ei ( LS )   k Pr[ ALS (Tn , Tn 1 )  k X n  i  s]
k 1

  k Pr[ ALS (0, T1 )  k X 0  i  s]
k 1

  k Pr[ D( L)  i  k  b]
k 1


  k  e x (ax) i  k b /(i  k  b)! e  x dx
k 1
0
 ( /  )r i b ,
where ALS (Tn , Tn 1 ) is the amount of lost sales during (Tn , Tn 1 ) .
Let EC ( s, S , b) be the overall expected average cost under the (s, S) inventory policy and given b.
S
0
EC( s, S , b)    j {c0  c3 E j ( LS )} /{  k m(k )}  c1  jp j  c2  jp j
j s
kE
*
j 0
*
j b
Then, EC ( s, S , b) functions specified over the distinct ranges of b follows.
For b  S  2s,
EC ( s, S , b)  [c0  c1{( S  s)( S  s  1) / 2  ( S  s) /   r s b ( S  s  b) /  }
 (c1  c 2 )(r s /  2  r s b /  2 )  c3r s b /  ] / q( s, S , b)
For S  2s  b  S  s,
EC ( s, S , b)  [c0  c1{( S  s)( S  s  1) / 2  ( S  s) /  2  r S  s ( s  b) /  2 }
 {c1 (s   ) /   c1r S s /    (c1  c2 )( r S  r 2 S s 1 ) / 
  (c1  c 2  c3 )r S b /   (2s  b  S )(c1  c 2 )r S 1
  (c 2  c3 )r 2 S  s b 1 /   c1 ( /   S  b  s)r S b 1
 ( c3  c 2 )( 2s  b  S )) r S b 1 } /( AA )] / q( s, S )
For S  s  b,
EC ( s, S , b)  [c0  c1{( S  s)( S  s  1) / 2  ( S  s) /  2  r S  s ( s  b) /  2 }
 {c1 (s   ) /   c1r S s /    (c1  c2 )r S / 
  (c3  c 2 /  )r S b  ( s  1   /  )(c1  c 2 )r S 1
  (c 2  c3 )r 2 S  s b 1 /    (2s  b  S )c3 r S b 1
 (2S  3s  b   /  )c2 r S b1} /( AA )] / q(s, S )
Now, a solution search procedure will be exploited for the optimal values of s, S and b with which the
overall expected average cost EC ( s, S , b) is minimized. In fact, the function EC ( s, S , b) is too
complicated to solve for the optimal values of s, S and b. However, EC ( s, S , b) can be characterized to
follow unimodal trends over each possible range of b.
Lemma 6
For
S  s  b , EC ( s, S ,) is either unimodal or monotonic.
Corollary 1
If
2c2 /   c3  0 , the optimal value of b is finite.
Lemma 7
For S  2s  b  S  s, EC ( s, S ,) is either unimodal or monotonic.
Lemma 8
For b  S  2s, EC ( s, S ,) is either unimodal or monotonic.
Lemmas 6, 7 and 8 provide the ranges of b and the cost relations under which there exists a finite b with
which the long-run overall expected average cost is minimized for fixed s and S. The existence of such a
finite b implies that the given (s, S) inventory model may be better than other (s, S) inventory models with
either complete backlogging or lost sales allowed.
The optimal value of f is upper bounded by
Let
S  2 (c0  c3 /  ) / c1
s * (b) and S * () denote the local optimal values of s and S, respectively, for a given problem
with a fixed b. Then, it can not be analytically verified but experienced in our various numerical problems
test that
EC ( s * (b), S * (b), b) is unimodal in b. If the unimodality holds and if 2c2 /   c3  0 ,
then it follows form Corollary that the optimal value b
*
is infinite.
Based on the above solution characteristics, a solution search procedure for problems satisfying
unimodality in b is described below:
Step1. Compute
W  2c2 /   c3 . If W  0 , go to Step 2.Otherwise, go to Step 3.
Step2. Let b   , and search
*
s * () and S * () . Then, stop the search procedure.
S.
*
Step4. Search s (b) , S () and compute EC ( s (b), S (b), b)
Step3. Select an arbitrary initial value for b, less than
*
*
*
Step5. Continue Steps 3 and 4 over b in a standard local descent search procedure by adjusting b in the
direction of descent until any further change in b value increases the total cost.

A Numerical Example
The examination starts with a base problem, where
  1.0 ,   .05 , c0  100.0 , c1  1.0 ,
c2  2.0 and c3  50.0 .The problem test is repeated with varying the given base problem parameters
one at a time.

Conclusions
This paper has examined a (s, S) inventory model with limited (fixed) backlogging levels and
stochastic lead times, where demand follows a Poisson process. Using the Markov renewal theory, the
steady-state probabilities of inventory levels are derived, upon which a long-run expected average cost
function is described. The cost function is then characterized as satisfying unimodality over certain ranges
of b when s and S are fixed. On the basis of characteristic, a solution search procedure is suggested and
illustrated with a numerical example.
4. Analysis of a multi-part spares inventory system subject to ambiguous fault
isolation

-
Journal of the Operations Research Society, Vol. 52, 2001
-
C. S. Sung and S. H. Kim
Introduction
This paper considers a multi-part spares Inventory model for a maintenance system composed of a sparesstocking centre and a repair centre. In the maintenance system, multi-part spares are jointly needed to
repair faulty end items determined by ambiguous fault isolation done by the built-in-test-equipment
(BITE). If any operating end item breaks down, then all associated parts should be replaced either
Iteratively (one at a time) or altogether. In the case of iterative replacement, failed parts are detected in the
field after fixing the broken end item. However, In the case of group (altogether) replacement, all
removed parts are sent to the repair centre where the failed part is detected and fixed. The repaired or nonfailed parts are then restocked at the spares-stocking centre. For the system, this paper is to derive the
exact expressions for the distribution function and the expected numbers of the backlogged end items
under the cannibalization policy. These expressions are then used in the optimization of the associated
spares' inventory level. Illustrative numerical examples are also presented.
Motivation: Previous works are all based on the assumptions that fault isolation is perfectly made in the
field. In other words, all the removed parts from the field are assumed failed. But, a serious increase in
maintenance burden and cost of spares due to the imperfect fault isolation of BITEs has been reported by
many in the literature.

Maintenance system Background
This paper considers a single-echelon maintenance system, composed of a spares-stocking centre (base)
and a repair centre (depot) serving for end items equipped with BITES. If an end item breaks down due to
its failed part, then any ambiguous fault isolation of the BITE will cause multi-parts to be removed and
replaced jointly with spares in the field. In fixing the end item, if there are not enough spares on hand to
replace any of the removed parts, then the whole end item will remain backlogged. For replacing such
multi-parts Jointly, two types of replacement procedures, including iterative and group replacement
procedures, will be considered together (but in a selective way, depending on the Jointly demanded part
types). When the iterative replacement (IR) procedure is applied, all the suspected parts are replaced with
spares In the field sequentially one at a time until every fault in the end item is fixed. Note that, during the
process of one-to-one part replacement, each removed part is immediately checked in the field if it is
faulty or not. Among all the associated removed (replaced) parts, each part found as good is then
restocked at the base, but every failed part is sent to the depot for repair. On the other hand, when the
group replacement (GR) procedure is applied, all the suspected faulty parts are replaced altogether with
spares in the field. All the associated removed parts are then directly sent to the depot where, by detailed
inspection using ATEs (automatic test equipments) equipped with sophisticated diagnostic software, every
failed part is screened out and put into the repair process. Such repaired or good parts are also restocked
at the base for reusage. See Figure 1.

Problem
For the proposed single-echelon maintenance system, this paper considers a continuous, infinite horizon,
order-for-order spares replenishment inventory model where ambiguous fault isolation, joint demand for
multi-parts, and cannibalization are all allowed. It is assumed that any breakdown end item contains only
one failed part.
For the model, the exact expressions for the distribution function and the expected number of backlogged
end items are derived and shown to be used in optimizing spares' inventories.

Model description (assumptions)

Job j : activity required for both fixing a failed end item and screening out its failed part. Each
job is represented by a combination (set) of various parts. Thus a job arrival will represent the
occurrence of a joint demand for multi-parts. Hence, the expected number of backlogged end
items can be obtained by computing the expected number of backlogged jobs (EBJ) for the end
items.

Replenishment time is assumed to be instantaneous.

One-for-one repair policy.

Jobs can be partitioned into two kinds of job sets including a set of job types to apply the IR
procedure and another set of job types to apply the GR procedure. After job arrived, required
process can be separated into two processes, one ‘fault-finding process’, and the other, ‘repair
process’.

Fault-finding process: required fault-finding time is assumed to be random variable whose
expected time is known for each job type. And there are unlimited numbers of servers.

Repair process: repair time for each part is assumed to be an i.i.d. random variable with finite
mean.

Cannibalization policy: Allow cannibalization which allows the unlimited and immediate
exchange of the operating and non-operating parts contained in all currently backlogged jobs.

Analysis of the system
Notations

J  [ J 1 ,..., J q ] a job matrix.

 j : occurrence rate of job j

L j : set of part types demanded by job j

S i : stock level of part i

T : Time for transporting the removed parts to the depot.

ri : the failure of part i of the end item is assumed to occur in a Poisson process with the
intensity rate ri .

hij : probability that job type j occurs due to the failure of part i. So, the arrival process of job
type j due to the failure of part i is represented by a Poisson process with the intensity rate being
denoted by rij .

Fi (t ) : total number of failures of part i in time interval (0, t ] . And Fij (t ) is total number of
occurrences of job type j occurs due to the failure of part i in (0, t ] . Then,
Those
hij  lim
t 
Fij (t )
Fi (t )
.
Fi (t ) and Fij (t ) can be obtained from the associated test results or field maintenance
reports.

Q : number of backlogged jobs in steady state

Ri : repair time of part i.

D j : fault-finding time for the parts associated with job j.

Oi f : number of part i being under the fault-finding process in steady state.

Aij (t ) : number of arrivals of job j due to the failure of part i in (0, t ] .

Z j (t ) : number of arrivals of job j in (0, t ] .

Y j : number of job j’s being under the fault-finding process in steady state.
Since the whole arrival process of job j 'is a superposition of all the individual arrival processes of job j’s
incurred due to the failure of each part I in L j , the whole process can be represented by a Poisson
process with the intensity rate
Lemma 1. (a)

r .
iL j ij
Ai1 (t ) , Ai 2 (t ) , ..., Aiq (t ) are mutually independent.
(b) If
i  l , then regardless of job types, Aij (t ) and Alk (t ) are mutually independent.
Lemma 1(a) means that, although the arrival processes of different jobs are generated from the same part's
failure process, they are mutually independent. Lemma 1(b) means that, although any two arrival
processes represent the same job type, if those processes are generated from two different part's failure
processes, then they are mutually independent.
Lemma 2. The arrival processes of jobs Z1 (t ), Z 2 (t ),..., Z q (t ) are mutually independent.
Theorem 1. Y1 , Y2 ,..., Yq are mutually independent.
Theorem 2. P (Y  y ) 
q
 p{ y
j 1
Theorem 3. O
f
 JY .
j
, E ( D j ) j } .
Now it can be seen that the arrival process of failed part i's at the repair process is a Poisson process with
the rate
ri and the number of part i's being under the repair process can be interpreted as the number
of customers in M/G./  queuing system. Thus, the number of part i's being under the repair process has
a Poisson distribution.
Lemma 3. The steady state probability mass function of the number of part i's being under the repair
process is
p{x, (  iT  E ( Ri )ri )} where  i   jI rij / ri .
r
Theorem 4. The distribution function of Q in steady state is
n
ai
q
x 0
j 1
P(Q  m)  [[ p{x,  iT  E ( Ri )) ri }]   p{ y j , E ( D j ) j }]
y i 1
where   { y | Jy  S  m},
[a1 , a2 ,..., an ]T  S  m  Jy
Corollary 1. The expectation of Q is obtained as

E(Q)  m0{1  P(Q  m)} .

Optimization of spares inventories
Optimization problem is
Minimize E (Q )
n
Subject to
C S
i 1
i
i
K
Where K= total budget,
C i = unit price of part i for i  1,2,..., n, K and C i are non-
negative real values.
This programming is a nonlinear knapsack type of problem where the objective function is not separable
but dependent jointly on the stocking levels of multiple parts. This implies that the problem is NP-hard.
Therefore, a heuristic approach may be practical to consider. Theorem 5 shows that the solution space of
the problem can be reduced by considering a valid inequality, by which some solution search effort can be
saved.
Lemma
4.
If
S i0  S i* for i  N ' ( N '  N  {1,2,..., n}, N '   ) and
i  N  N ' , then [ E (Q)] s  s 0  [ E (Q)] s  s*
S i0  S i* for
where
S 0  (S10 , S 20 ,..., S n0 ), S *  (S1* , S 2* ,..., S n* ), and [ E (Q)] s  s 0 and [ E (Q)]s  s* are E(Q)
values at S  S
0
and S  S , respectively.
*
n
Theorem 5. A valid inequality for the solution space is specified as
C S
i 1
i
i
 K  C min , where
Cmin  min{ C1 , C2 ,..., Cn } and the solution vector S  (S1 , S 2 ,..., S n ) . Thus the associated
constraint can be modified as K  C min 
n
C S
i 1
i
i
K.
A greedy heuristic, called the marginal analysis (MA) approach, will first be considered. In the MA
approach, starting at S = (0,0‥‥ 0), the reduction of the objective function value E(Q) by increasing each
part type by one unit will be evaluated by selecting one part with the greatest marginal benefit ratio which
is defined as the reduced quantity of E(Q) divided by the unit price of the part. This part-selection process
continues iteratively until the budget is exhausted. In order to improve the MA approach, a modified
marginal analysis (MMA) approach can be exploited as evaluating the marginal benefit ratio by trying all
part types jointly demanded by each job type (called the joint increment of multiple parts) to be increased
together in single units as well as the single part treated in the MA approach. By experimental test given
in this paper, MMA approach may be regarded as an attractive approach.

Concluding remarks
For the inventory system, the exact expressions for the distribution function and the expected number of
the backlogged jobs are derived under the cannibalization policy. The increase of the expected number of
the backlogged jobs due to the ambiguous fault isolation of the BITE is shown to be significant in typical
sample case problems. In order to optimize the spares' inventory level, a greedy heuristic, called the
modified marginal analysis approach, is suggested and shown to perform well in an experimental test.
The results of this paper may be used in the following issues:

determining the optimal initial spare parts for an inventory system incorporating BITEs subject
to ambiguous fault isolation,

evaluating the spares' cost increase due to the ambiguous fault isolation of BITEs,

designing an acceptable level of accuracy for the BITEs which will minimize the total cost
including the procurement cost of those BITEs and the stocking cost of the initial spares to meet
a target service level of maintenance.
For further studies, the following issues may be interested in:

deriving any exact expressions of the expected number of backlogged jobs under any other
spares-using policies such as FCFS , instead of the cannibalization policy,

considering any multi-echelon inventory system where the depot is an upper-echelon facility
supplying spares to several bases (lower-echelon facilities).