The (s,S) Policy for the Production/Inventory
System with Compound Poisson Demands
Hyo-Seong Lee
M.M. Srinivasan
Technical Report 88-2
April 1988
Department of Industrial and Operations Engineering
The University of Michigan, Ann Arbor 48109-2117

Abstract
A production/inventory system is considered for items produced by a single production facility.
The demand for the item is assumed to arrive according to a compound Poisson process. The
processing time required to produce an item is assumed to follow an arbitrary distribution. An
(s,S) policy is considered in which production stops at the instant that the inventory level is raised
to S and production begins again at an inspection point when the inventory level is observed to
have dropped to or below s for the first time. The time interval between two successive inspection
points during a non-production period is a random variable which follows an arbitrary
distribution. Under a cost structure which includes a set-up cost, a linear holding cost and a linear
backorder cost, an expression for the expected cost per unit time for given control values is
obtained, and using a convex property of the cost function, a procedure to find the optimal (s,S)
policy is presented. The continuous review (s,S) policy with compound Poisson demands is also
discussed.

1. Introduction
There has been a considerable amount of work on the (s,S) inventory control policy.
Most of these studies treat inventory systems in which as many items as needed can be
replenished all at once. While this is a realistic assumption in many cases, in some real
manufacturing systems, the production facility can only produce one item at a time, and hence
inventory can only be replenished on an item-by-item basis. Motivated by this fact, in this
paper we analyze production/inventory systems where inventory is replenished on an itemby-item basis.
In the inventory system considered here a single production facility produces a single
type of item. The demand for the item is assumed to follow a compound Poisson process:
demand arrives according to a Poisson process with rate X and the size of each demand, X, is a
random variable following an arbitrary discrete distribution. The processing time for
producing (replenishing) an item is assumed to be an independent and identically distributed
random variable U which follows an arbitrary distribution. For stability of the system, we
assume that XE(X)E(U)<1. If a demand occurs, it is supplied directly from the inventory, if
there is a sufficient inventory of items available. If the inventory is not available, the demand
is backordered. When the "on hand inventory" is not sufficient to meet the demand, only the
unsatisfied portion of the demand is backordered. The relevant costs which are to be traded off
in this paper include a set-up cost, a holding cost and a backorder cost. We assume that
inventory holding costs are incurred linearly over time with respect to the inventory level;
backorder costs are incurred linearly over time with respect to the backorder level; a set-up cost
is incurred each time the production facility is turned on. The objective is to find a
production/inventory policy to minimize the expected cost per unit time in the long run.
Since (s,S) policies are known to be effective in a variety of inventory situations[2,10,11],
the operating policy considered in this paper is an (s,S) policy. The characteristics of the (s,S)
policy considered in this paper will be described below.
1

As soon as the inventory level is raised to S, the production facility is turned off and a
non-production period begins. During the non-production period, the inventory level is
inspected only at certain points in time (inspection points). The time interval between two
successive inspections during the non-production period is a random variable following an
arbitrary distribution. The non-production period lasts until an inspection point is
encountered at which the inventory level is observed to have first reached or dropped below s. At
this point, the production facility is turned on and a production period begins. During the
production period, inventory is replenished on an item-by-item basis. If the inventory level
reaches S, the production period ends and the next non-production period begins, which initiates
another cycle in the (s,S) policy system (refer to figure 1).
X represents inspection point
S
-S 4-i C l Fs1 —--- 1
non-production production period
period
Fig. 1. The (s,S) Policy with Compound Poisson Demands
and Irregular Inspection Intervals
The random inspection interval policy in this paper would be used, for example, in
situations where the production facility is used for some secondary work during a nonproduction period. Suppose each secondary work takes a random amount of time. Then, the
most reasonable inspection policy in this situation would be to inspect the inventory level at the
end of each such secondary work, to determine whether the facility can be used for the
2

secondary work once more, or whether it should be switched back to the primary mode of
operation.
Since the replenishment in our model is made on an item-by-item basis, it is easy to
note that our model has an analogy with queueing systems. In fact, the work on control of
queueing systems, for instance, the papers by Heyman[4], Yadin and Naor[12], Bell[2], Sobel[8]
and Lee and Srinivasan[5], have a close relationship with production/inventory systems.
Using this inventory-queueing analogy, some studies have been done on (s,S) policies for the
production/inventory systems in which the inventory is replenished item-by-item.
Gavish and Graves[3] considered a continuous review (s,S) production/inventory
system with unit Poisson demand arrivals and deterministic processing times. They
developed a fairly efficient procedure to find the optimal (s,S) values by using an underlying
M/D/1 queueing system. Tijms[9] studied a similar continuous review (s,S) system with
arbitrary processing times and under a more general cost structure. To find the optimal control
values, he used a denumerable state Semi-Markov decision process. However, as pointed out by
Gavish and Graves[3], the computational time using a Markov decision process approach for
solving such problems is much higher than that using an intelligent search technique. For the
batch demand case, Altiok[1] considered a problem with a compound Poisson demand process
and phase-type unit processing times. This model uses an underlying birth and death process
to obtain the steady state probability of each inventory level for a given policy and found
optimal control values by using a search technique. However, in this paper no special properties
for the cost function were proved, that would assist in the search for the optimum.
Lee and Srinivasan[6] developed an algorithm to find the optimal control values for the
continuous review (s,S) system with unit Poisson demands and arbitrary processing times. In
this paper, some properties on the convexity and unimodality of the components of the cost
function were proved. These properties result in the development of a very simple and effective
search algorithm. In particular, the solution found by this algorithm is always guaranteed to
be a global optimum.
3

The (s,S) policy in this paper extends the work of Lee and Srinivasan[6] to the situation
where both bulk arrivals and arbitrary inspection intervals are considered. The general
approach used in this paper is basically similar to that used in the earlier work. However, the
analysis for this problem is much more complicated since the inventory level at either an
inspection point or the beginning of a production period does not have a deterministic value. To
overcome this difficulty, we will use some results derived by Lee and Srinivasan[5].
In the following sections, an expression for the expected cost per unit time for given control
values will be derived and after that an efficient procedure to find the optimal control values
will be presented. In addition to the policy which considers the inspection of the inventory at
random points, the situation where the inventory level is continuously monitored during the
non-production period will be also discussed.
2. Analysis of the Model
2.1. Notation
For analytical convenience, we set r=S-s, and throughout the paper we will use (r,S)
instead of (s,S) to describe a policy. Thus, the (r,S) policy represents the policy with S-r as a
lower control value and S as an upper control value. Also, for notational convenience,
throughout this paper, for any discrete random variable (r.v.) D used in the analysis, we adopt
the following notation:
di = P(D=i},
00
D(z) = Xdi z, the probability generating function (p.g.f.) of A,
i=O
d(1) = E(D),
d(2) = E(D(D-1)).
Similarly, for any continuous r.v. C used in the analysis, C(.) and C*(.) will denote the
cumulative density function (c.d.f.) and Laplace Stieltjes Transform (LST) of C respectively.
4

We use the following notation.
X = batch size of a demand, r.v.,
V = inspection interval during the non-production period, r.v.,
A = number of demand requests which arrive during V, r.v.,
B = number of items demanded during V, r.v.,
U = processing time to produce a unit item, r.v.,
G = number of demand requests which arrive during U, r.v.,
H = number of items demanded during U, r.v.,
X*i = i-fold convolution of X,
*(i)
*i = P(i-fold convolution of X is j),
Yr = total number of items demanded during a non-production period when r=S-s is used
as a control value, r.v.,
I(r,S) = inventory level when production period begins, when r and S are used as control
values, r.v.,
K = set-up cost,
ch = holding cost/item/unit time,
cb = back order cost/item/unit time,
CN(r,S) = expected cost during a non-production period when r and S are used as control
values,
Cp(r,S) = expected cost during a production period,
C(r,S) = CN(r,S) + Cp(r,S),
L(r) = expected length of a cycle when r is used as a control value,
TC(r,S) = expected cost per unit time.
5

2.2 General Approach
Our first objective is to obtain an expression for TC(r,S), the expected cost per unit time
for given control values r and S. To obtain this, we use a regenerative process theorem, since
the epoch marking the start of a cycle forms a regeneration point. It follows that what we have is
a renewal reward process, and thus from the renewal reward theorem (see, for example,
Ross[7]), the expected cost per unit time, when r and S are used as control values, is obtained by
TC(r,S) = (r) + K (2.1)
L(r)
Note that we use L(r) in equation (2.1) instead of L(r,S) to denote the expected cycle time when r
and S are used as control values. This is because the expected cycle time is a function of r alone
as will be explained later. Since the term C(r,S) is expressed as a sum of CN(r,S) and Cp(r,S), to
obtain TC(r,S) by using equation (2.1), we need to find the terms CN(r,S) and Cp(r,S) as well as
the term L(r). We now show how the terms CN(r,S), Cp(r,S) and L(r) are obtained.
2.3. Computing the terms ai, bi, gi, and hi:
To obtain CN(r,S) and Cp(r,S), we need to know ai, gi, bi and hi as well as the first two
moments of B and H. Hence, we will first describe how these terms are calculated.
By definition,
00
ai = i xt dV(t), (2.2.a)
0
00
gi= Xte-Xt dU(t). (2.2.b)
0
However, for the computation of ai and gi, we use the following equivalent expressions:
ai =!.* (i) (2.3.a)
gi(= )U*(i)(h) (2.3.b)
gi i!
6

where V*(i)(o) and U*(i)(0) are the ith derivatives of the LSTs of V and U with respect to 0. Once
the terms ai and gi have been calculated, the probability that j items are demanded during an
inspection interval (or a processing time) is obtained by
J *(i)
J *(i)
hj = E gix. j=0,1,2,...... (2.4.b)
i=0 J
The moments of B and H are obtained by using the following result, stated as lemma
2.1. Since lemma 2.1 can be easily proved, we omit the proof here.
Lemma 2.1. The probability generating function of B and H are given by
B(z) = V*(X-XX(z)). (2.5.a)
H(z) = U*(-XX(z)). (2.5.b)
From equation (2.5), we can obtain the first and second factorial moments of B and H as
b(1) = E(B) = XE(V)x(1), (2.6.a)
b(2) = E(B(B-1)) = (Xx(1))2E(V2) + Xx(2)E(V), (2.6.b)
h() = E(H) = XE(U)x(1), (2.6.c)
h(2) = E(H(H-1)) = (Xx(1))2E(U2) + Xx(2)E(U). (2.6.d)
2.4. Computing the term CN(rS):
In this section, we will compute the term CN(r,S), the expected total cost during a nonproduction period when r and S are used as control values. Let EVk be the expected cost incurred
during an inspection interval V initiated with k inventories. Then, the expected cost during a
non-production period when r and S are used as control values is obtained by the following
recursive equation:
7

r-1
CN(r,S) = EVS + bjCN(r-j,S-j). (2.7)
j=O
The explanation of equation (2.7) is as follows.
For any value of r and S, the (r,S) policy must have at least one inspection. Therefore, CN(r,S) is
expressed by the expected cost during the interval upto the first inspection point plus the expected
cost from this inspection point onwards until the end of non-production period. The first term
EVS in (2.7) represents the expected cost during the interval upto the first inspection point, and
the second term represents the expected cost from this point onward until the end of nonproduction period. The explanation for the second term is based on the following reasoning:
1) If the number of items demanded during the interval upto the first inspection point is more
than or equal to r, the non-production period ends here, so that there is no further cost
incurred during the non-production period.
2) Suppose j items are demanded during the interval upto the first inspection point. Then, the
inventory level at the point of the first inspection becomes S-j. If j is less than r, the inventory
level at the first inspection point is greater than the lower control value s and the nonproduction period does not end at this point. Since the inventory level at this point is S-j, an
expected cost of CN(r-j,S-j) is incurred from this point onward until the end of the nonproduction period. Consequently, since the probability that j items are demanded during the
interval upto the first inspection point is bj, the second term of (2.7) is obtained.
If we solve equation (2.7) for CN(r,S), we obtain
r-1
CN(r,S) = l-bEVs+ bjCN(r-j,S-j)} (2.8)
j=1
where an empty sum is defined to be 0.
In equation (2.8), we see that the term CN(r,S) can be computed if the value of EVS is known.
The term EVS can be obtained by using the following lemma, stated as lemma 2.2. The proof of
lemma 2.2 is given in appendix A.
8

Lemma 2.2
EVk is expressed, recursively, as
EVk = EVk-1 + dk(cb+ch) - E(V)cb, (2.9.a)
where
-k
dk = r { Z Pr(Xi+...+Xj-1< k-1, X1+...+Xj> k)j(1 - z ai)
j=1 i=0
k-1
+ Y Pr(Xi+...+Xj< k-l)(j+l)aj+l}, (2.9.b)
j=0
and
Pr(Xo < k-l) is defined to be 1.
For k< 0, EVk can be recursively expressed as
EVk = EVk+l + E(V)cb. (2.10)
For an initial value for equations (2.9) and (2.10), we use
EVo = x(l)XE(V2)cb. (2.11)
2
Note that since (2.9.a) and (2.10) are recursive equations, once EVk-1 for k>l or EVk+l for k<0
is obtained, EVk can be computed very easily, that is, only dk is needed to be calculated to obtain
EVk. To obtain P(X1+...+Xj-<1 k-1, X1+...+Xj> k) in lemma 2.2, we use the following
equation:
k-j+l *(,)
P(X1+...+Xj-i< k-l, X1+...+Xj> k) = xki P(X > i). (2.12)
i=1
2.5. Computing the term Cp(rS)
During the production period, the production completion epochs are the times at which the
inventory is replenished. To compute Cp(r,S), we restrict our attention only to these epochs. Let
fij denote the expected cost from the epoch at which the inventory level reaches i, to the epoch at
which the inventory level is raised to j (j>i) for the first time. If i is equal to j, fi4 is defined to
be 0. Then the expected total cost incured during the production period, Cp(r,S), is just fI(r,S),S.
9

To obtain fI(r,S),S we first need an expression for the probability that the inventory level
at the beginning of the production period is j when r and S are used as control values.
By conditioning on the number of items demanded during the first inspection interval, we can
obtain a recursive equation that yields P(I(r,S)=j) as
r-1
P(I(r,S) = j) = bk P(I(r-k,S-k)=j)) + b-j, for j<S-r. (2.13)
k=O
Let us define a new random variable B such that
P B =j) = P(B=j IB>1), j=1,2,....
Note that the first and second factorial moments of B are
i,() = b) (2.14.a)
1-be'
B. (2)=.) (2.14.b)
By solving equation (2.13) in terms of P(I(r,S)=j), we obtain
r-1
P(I(r,S)=j) = S bkP(I(r-k,S-k)=j) + bs-j. (2.15)
k=l
Using equation (2.15), after some algebra, we can now express Cp(r,S) recursively as stated in
the following lemma. The proof of lemma 2.3 is given in appendix B.
Lemma 2.3
r-1
Cp(r,S)= I bj Cp(r-j,S-j) + Cp (1,S). (2.16)
j=1
From lemma 2.3 and equation (2.8), we can also express C(r,S) in a similar manner as
Cp(r,S). Lemma 2.4 states this result. The proof of lemma 2.4 is omitted.
Lemma 2.4
r-1
C(r,S)=, bj C(r-j,S-j) + C(1,S). (2.17)
j=l
*
10

From lemma 2.4, we see that, for any value of r and S, C(r,S) can be computed recursively
starting from r=l. Hence, C(r,S) can be obtained if only the values of C(l,k) are known for any
EVk
k. Since CN(l,k) is just 1-b it can be easily calculated. Thus, if we can compute Cp(l,k), we
can compute C(r,S) for any value of r and S. Therefore, we will describe below how the term
Cp(l,S) can be obtained for any given S.
Computing the term Cp(l,S)
To compute the term Cp(l,S), we consider two possible cases:
(a) Case S>1:
In this case, the inventory level at the beginning of a production period can be either
positive, zero, or negative. That is, if B < S, the inventory level I(1,S) is positive and if B > S,
I(1,S) is zero or negative. In particular, we can find the probability that the inventory level at
the beginning of a production period will take on value k as
P(I(1,S) = k) = P( B = S - k) = bS-k, k= -o,...,S-1. (2.18)
Consequently, the probability that I(1,S) will be non-positive is
S-1
P(I(1,S) < 0} = 1- I bj. (2.19)
j=1
Thus, the term Cp(l,S) is expressed as
Cp(1,S) = fI(1,S),S
S-1 S-1
= P(I(1,S) = j)fj,s = ~ P(I(1,S) = j)(fj,s-l + fS-l,S)
J=_oo j=_oo
0 S-1
= ~ P(I(1,S) = j)fj,S-l +, P(I(1,S) = j)fj,S-l + fS-1,S
J=-oo j=1
0 S-1
=, P(I(1,S) =j)fj,o + P(I(1,S) < O}fo,S-1 + ~ P(I(1,S) = jfj,S-l + fS-1,S
J=-oo j=1
0 S-1 S-1
= ~ P(I(1,S) = j)fj,o + (1 - I bj } fo,S-1 + 2 bs-j fj,S-l + fS-1,S ~ (2.20)
J=-oo j=1 j=1
11

Suppose that we can compute the value of f. j for any i<j and i20. Then we see from
0
equation (2.20) that, except for the first term, namely, I P(I(1,S) = jfj,0 all other terms can be
j=-oo
calculated easily by simple operations, i.e., as sums of finite terms. The first term which
consists of infinite terms, can be calculated without adding these infinite terms by exploiting
the analogy between our production/inventory system and a queueing system. In particular,
we will use a result on the control of MX/G/1 queueing systems with server vacations that has
been obtained by Lee and Srinivasan [5]. This result is stated below as lemma 2.5. A proof of
lemma 2.5 can be found in [5].
Lemma 2.5 Consider an MX/G/1 queue with vacation model with exhaustive service
discipline. Let U be a service time for an individual customer and i(l),i(2) be the first and
second factorial moments of the number of customers when service begins. Then the mean
waiting time of an arbitrary unit during a busy period is represented by
Wb(i(l),i(2)) = E x(l)E(U2+x2)(E(2 (2.21)
2i()1) + 2(1-p)
where p = XE(X)E(U).
From lemma 2.5 and the well known fact that the mean number of customers served in a busy
1
period initiated with one customer in an MX/G/1 queueing system is 1-, we get the following
result stated as lemma 2.6.
Lemma 2.6: Suppose that the inventory level at the epoch when either a production period begins
or the processing of an item ends, is known to be non-positive, and that the first and second
factorial moments of the shortage size at this epoch is known to be i-(1) and i-(2) respectively.
Then the expected total backorder time from this epoch to the epoch when the inventory level is
raised to zero for the first time is expressed as
12

a(i-(l),i-(2)) = Wb(i-(1),i-(2)) + E(U))
i-(2)E(U) + 2i-(1)
2(1-p) (2.22
where
X Ex()E(U2)+x(2)(E(U))2) E
Y= 2(-)p + E(U).
0
Now, using this result, we can compute the term,, P[I(1,S) = j)fj,0, as described below.
j=-oo
Let I'(1,S) be the shortage size at the beginning of production period given that the inventory
level at the beginning of production period is non-positive and let i-(1)(1,S) and i-(2)(1,S) be the
first and second factorial moments of I-(1,S). Then, if the condition is given that shortage
occurs at the beginning of the production period, the expected cost that is incurred from the
beginning of the production period to the time when the inventory level reaches 0 for the first
time, is given by a(i-(l)(1,S),i-(2)(1,S))cb. Since the probability that shortage occurs at the
0
beginning of the production period is P(I(1,S) < 0), the term F P(I(1,S) = j)fj,0 is computed by
j-'-OO
j=-oo
0
P(I(1,S) = j}fj,o = PI(1,S) < 0) a(i-(1)(1,S),i-(2)(1,S))cb, (2.23)
j=-oO
where i-()(1,S) and i-(2)(1,S) are given by
i-(l)(,S) = E( B I B >S) - S, (2.24.a)
i(2)(1,S) = E( B?2 I B>S) - E( B I B >S) - 2S E( B I B>S)+ S(S+1). (2.24.b)
In equation (2.24), the terms E( B I B >S) and E( B2 I B >S) are obtained from the following
equations:
S-1
E( B) = P B>S) E( B I B >S) + j, (2.25.b)
j=1
S-1
E( B2 )=P B>S)E( B2 I B >S) + X j2 j. (2.25.b)
j=1
13

(b) Case S<0:
In this case, to obtain Cp(l,S), we use an analogy between our production/inventory
system and an MX/G/1 queueing system. Observe that the length of the elapsed time from the
epoch where the inventory level reaches k to the epoch where the inventory level is raised to k+1
for the first time is equivalent to the busy period in an MX/G/1 queueing system that is initiated
with one customer. Note that, if S<0, the inventory level at the beginning of a production period
is always zero or negative. Using this observation and the well known fact that the expected
E(U)
length of a busy period in an MX/G/1 queueing system with mean service time E(U), is -(p)
where p = XE(U)x(1), we can obtain a recursive expression for Cp(l,k) as
E(U)
Cp(l,k) = Cp(l,k+l) + -— b, for k<0. (2.26)
The initial value for equation (2.26), namely, Cp(1,0), can be obtained using lemma 2.6 as
follows:
Cp(1,o) = (i-(1)(1,0),i-(2)(1,0))cb
=a(() b(2) )cb. (2.27)
Computing fk,k+l
Note from equation (2.20) that we can now compute the term Cp(l,S) if we can calculate
j-1
fij for i<j and i>0. Since the term fij can be expressed as fij = X fk,k+l, we can compute fij
k=i
if only we can compute fk,k+l for any k. Thus, we will describe the procedure to obtain fk,k+l,
for the case of k>0 since we do not need to calculate fk,k+l for k<0.
To this end, we first define EUk to be the expected cost incured during a unit processing
time initiated with k inventories. Then, fk,k+l is obtained by the following equation:
00
fkk+1 = EUk + ~ hj fk+l-j,k+1. (2.28)
j=0
14

The first term in (2.28) is the expected cost incurred during the processing time and the second
term in (2.28) is the expected cost incurred from the end of that processing time to the time at
which the inventory level is raised to k+1 for the first time. The term EUk, which is needed in
(2.28) can be obtained in the same way as EVk, that is, we can compute EUk by simply
replacing V and ai with U and gi respectively in lemma 2.2.
To obtain fk,k+l efficiently, we rewrite (2.28) as
k
fkk+l = EUk + z hj (fk+l-jk + fk,k+l)
j=1
00
+ hj (fk+l-j,0 + fO,k + fk,k+l) k>0. (2.29)
j=k+l
If we solve equation (2.29) for fkk+l, we obtain
1 k oo
fkk+l = -{EUk + ~ hj fk+l-j,k + X hj fk+l-j,o + f0,k P(H>k+l)). (2.30)
0 j=1 j=k+l
00
In equation (2.30), only the term k hj fk+l-j,0 is expressed as the sum of infinite elements.
j=k+l
This term, however, can be obtained without adding infinite terms in a manner similar to the
way we obtained the first term in equation (2.20), as will be described below.
Let I-(k) be the shortage size at the end of a processing time initiated with k inventories, given
that inventory level at the end of the processing time is non-positive, and let i-()(k) and
i-(2)(k) be the first and second factorial moments of I-(k). Then, similar to equation (2.23), the
00
term X hj fk+l-j,o is expressed as follows:
j=k+l
00, hj fk+1-j,o = P(H > k+1) a(i-(1)(k),i-(2)(k))cb, (2.31)
j=k+l
where i-()(k) and i-(2)(k) are obtained from
i-)(k) = E(H I H>k+1) - (k+1), (2.32.a)
i-(2)(k)= E(H2 I Hk+l) - E(H I Hk+1) - 2(k+1) E(H I Hk+1) +(k+l)(k+2). (2.32.b)
15

The terms E(H I H>k+1) and E(H2 I H'k+l) which are needed in equation (2.32) can be obtained
from
k
E(H) = E(H I H> k+l)P(H> k+1) +, j h. (2.33.a)
j=0
k
E(H2) = E(H2 I H> k+l)P(H> k+1} + T j2hj. (2.33.b)
j=0
As we see in equation (2.30), to obtain fk,k+l, the terms fO,k and fi,k for O<i<k should be
available beforehand. This implies that to obtain the term fk,k+l, we must calculate fi,i+l in the
order of i=0,1,2,...k, that is, f,i+1 is calculated recursively starting from i=0. This implies that
Cp(l,S) is calculated recursively starting from S=0. Since CN(1,S), which is nothing but EVS, is
also calculated recursively starting from S=0, we observe that C(1,S) is in turn calculated
recursively starting from S=0.
2.6. Computing L(r) and TC(r,S)
The expected cycle time in the (r,S) system can be obtained by again exploiting the
analogy between production/inventory systems and queueing systems. Let Yr be the total
number of items demanded during a non-production period when r=S-s is used as a control
value. Then, as described in section 2.5, the length of production period in our (r,S) system is
just the convolution of Yr busy periods in an MX/G/1 queueing system in which each busy period
is initiated with one customer. Therefore, the expected length of a production period, Lp(r), is
obtained as
E(Yr)E(U)
Lp(r) = ( (-p) (2.34)
where E(Yr) can be computed from
r-1
E(Yr) = bj E(Yr-j ) + b), r=1,2.. (2.35)
j=1
16

Similarly, the expected length of a non-production period, LN(r), can be computed from
E(Yr)
LN(r) = E( (2.36)
For more details on how equations (2.35) and (2.36) can be obtained, refer to Lee and
Srinivasan[5].
From (2.34) and (2.36), we can now obtain the expected length of a cycle as
E(Yr)
L(r) = Lp(r) + LN(r) = (lp) (2.37)
Note that, similar to the terms C(r,S), L(r) can be also computed recursively as
r-1 b(1)
L(r)= Z bjL(r-j) +.(2.38)
(l'p)Xx(1)
Since the term C(r,S) is obtained as C(r,S) = CN(r,S) + Cp(r,S), by substituting (2.36) and (2.37)
into (2.1), we finally obtain TC(r,S), the expected cost per unit time.
3. Finding the Optimal Control Values
In order to find the optimal control values (r*,S*), a two-dimensional search over the
integer parameter space must be made. Usually, the search for the optimal point in two
dimensional space is not easy. However, if we exploit some properties of the cost functions, we
can find a resonably efficient search procedure. Let us denote the optimal S value for a given r
by S*(r) and the optimal r value for a given S by r*(S). Then, we can prove the following
properties which enable us to devise an efficient search procedure. The proofs of these properties
are given in appendix C.
Propertv 1 For a given value of r, C(r,S) is convex with respect to S.
Property 2 If C(r,k) < C(r,k-1), then, C(r+l,k) < C(r+l,k-1).
Propertv 3 If S*(1)=n, then, S*(r) <n+r-1 for r>2.
Property 4 If S*(1)=n and S*(r)=k, then, k< S*(r+l) <n+r.
17

Property 5 S*(r) >0 for any r>l.
Now we are in a position to describe the search procedure. To find the optimal control
values (r*,S*), our algorithm makes use of properties 1-5 as well as the recursive nature of the
cost function C(r,S). Recall that C(r,S) is calculated recursively starting from r=1, S=0 and that
it can be calculated very quickly once we know C(iJ) for i<r-1 and j<S. Therefore our
algorithm starts with r=1 and finds C(1,S*(1)) first. To compute C(1,S*(1)), we simply compute
C(l,k) from k=0 up to the point k = S*(1) + 1, at which C(l,k) is first increased. Then, due to the
convexity of C(l,k), the minimum value found is at C(1,S*(1)). After we find C(1,S*(1)), we
compute TC(1,S*(1)) using equation (2.1).
Note that the search starts from k=0 since by property 5, C(l,k) for k<0 cannot be a
minimum. Once C(1,S*(1)) is found, we can find C(r,S*(r)) (and therefore, TC(r,S*(r))
sequentially in the order of r=2,3,4... using property 4. Note that from property 4, to find
C(r+l,S*(r+l)), we do not have to consider C(r+l,k) for k<S*(r): we just compute C(r+l,k)
starting from k=S*(r) until we encounter a point at which C(r+l,k) is first increased. Although
the upper bound for S*(r+l) in property 4 is S*(l)+r, our computational experiments show that
S*(r+l) is always either S*(r) or S*(r)+l. Thus, very few evaluations of C(r,k) are, in fact,
needed in order to obtain C(r,S*(r)) for each r.
The function TC(r,S*(r)) is very complex and we could not prove any
convexity/unimodality properties for the cost function TC(r,S*(r)). However, extensive
numerical tests suggest that TC(r,S*(r)) is unimodal with respect to r. (For the continuous
review (s,S) policy with unit Poisson demands, it has been proved [6] that TC(r,S*(r)) is
unimodal with respect to r.) Based on this observation, we can devise an extremely simple
search procedure: start with r=1 and find C(r,S*(r)) and TC(r,S*(r)); increase r by 1 and
compute C(r,S*(r)) and TC(r,S*(r)) for new r until TC(r,S*(r)) is first increased. Our
algorithm is fairly efficient since very few evaluations of C(r,k) yield C(r,S*(r)), and the term
C(r,k) can be computed very quickly using the values of C(ij) for i<k, which were previously
18

obtained. In fact, to obtain C(r,k), the new terms we should compute are only EVk and fk-i,k,
which are also easily calculated using EVj and fj-,j for j<k, that were obtained earlier.
Algorithm to find the optimal control values (r*,S*)
O. Set r=1, k=0.
Compute C(l,k)
1. Setk=k+l
Compute C(l,k)
If C(l,k) > C(l,k-1), then
S*(1)=k-1. Compute TC(1,S*(1)
go to step 2
else
go to step 1
endif
2. Set r=r+l, k=S*(r-1).
Compute C(r,k)
3. Setk=k+l
If C(r,k) > C(r,k-1), then
S*(r)=k-1. Compute TC(r,S*(r))
If TC(r,S*(r)) > TC(r-1,S*(r-1)) then
optimal control values have been found; (r-l,S*(r-1)).
stop
else
go to step 2
endif
else
go to step 3
endif
4. Numerical Examples
In order to verify the efficiency of our algorithm and to check the unimodality of
TC(r,S*(r)), we made extensive numerical tests. Among these, we present two examples below.
In example one, the inspection interval is assumed to follow a uniform distribution and the
processing time is assumed to follow an Erlang distribution. In example 2, the inspection
interval follows an exponential distribution and the processing time is assumed to follow a
special distribution, which is encountered frequently in manufacturing situations. The
19

processing time in example 2 has a deterministic value to if the production facility does not
fail during the processing time. However, if the production facility fails during the processing
time, which is assumed to occur with probability p, the processing time is the sum of t0 plus a
random repair time R which follows an exponential distribution with rate jI.
(example 1)
= 0.1, K=1000, ch= 1, cb= 20,
xi = 0.5, x2 = 0.3, x3 =0.2,
V follows a uniform distribution on [2,3] and
U follows a three stage Erlang distribution with mean 0.5.
(example 2)
X=0.1, K=1000, ch=l, cb=10
xi = 0.4, x2 = 0.4, x3 = 0.2,
V follows an exponential distribution with mean 3,
U = to with probability l-p,
= to + R with probability p,
where p=0.03, t0=1.2 and g=0.1.
The results of the policy comparisons for these two test examples are presented in tables
1 through 2. In each table, the values of r, s*(r), S*(r) and TC(r,S*(r)) are shown for each value
of r. As shown in the tables, in all these two examples, TC(r,S *(r)) shows a unimodal property
as was expected. Finally, although only three distributions for the inspection intervals and the
processing times are demonstrated in the examples, many other distributions can be
implemented easily if their Laplace-Stieltjes transform functions are well differentiable.

Table 1.
Result of Example 1
*
r=S-s s*(r) S*(r) TC(r,S (r))
13 -1 12 18.2235
14 -1 13 17.8957
15 -1 14 17.6731
16 -1 15 17.5367
17 -1 16 17.4721
18* -1 17 17.4677
19 -1 18 17.5144
20 -1 19 17.6048
21 -1 20 17.7329
(* means the optimal policy)
Table 2.
Result of Example 2
r=S-s s*(r) S*(r) TC(r,S (r))
12 0 12 17.5078
13 -1 12 17.1587
14 -1 13 16.8800
15 -1 14 16.6971
16 -1 15 16.5934
17* -1 16 16.5558
18 -1 17 16.5742
19 -1 18 16.6403
20 1 19 16.7473
(* means the optimal policy)
2

5. Continuous Review Case
In this section, we consider the case in which the inventory level is continuously
monitored during the non-production period, so that the production period begins as soon as the
inventory level drops to or below the lower control value s. In this case, the general approach to
obtaining C(r,S) is basically the same as that outlined in the case of irregular inspection
intervals and the search procedure to find the optimal control values is identical to that
described in section 3 since all five properties in section 3 continue to hold. Therefore, in this
section, we will only discuss those expressions which are different from the corresponding
expressions in section 2.
The basic difference between the continuous review model and the irregular inspection
interval model is that, in the continuous review model, xj replaces bj in every equation
involving bj. Thus, to obtain CN(r,S) for example, the following equation is used instead of
equation (2.8):
r-1
CN(r,S) = EVS + A xjCN(r-j,S-j)), (5.1)
j=l
where EVS = S Ch; if S >0,
1
=-SCb' if S <O.
Note that the term EVS is the expected cost incurred from the instant the non-production period
begins to the instant the first demand arrives, and the second term represents the expected cost
incurred from the first demand arrival to the end of the non-production period.
To obtain Cp(r,S) and C(r,S), we use the following equations which correspond to
equations (2.16) and (2.17):
r-1
Cp(r,S) = x Cp(r-j,S-j) + Cp (1,S), (5.2)
j=l
r-1
C(r,S)= I xj C(r-j,S-j) + C(1,S). (5.3)
j=l
22

To obtain Cp(l,k) which is needed to obtain Cp(r,S), all the equations described in section 2 are
used with only slight modification. Besides replacing bj with xj here, the following expression
is used for P{I(1,S)=j):
P(I(1,S)=j)=P(X=S-j) =xSj. (5.4)
The expected cycle time can be obtained in the same way: instead of using equation (2.35), E(Yr)
r-1
must be calculated from E(Yr) = X xj E(Yrj ) + x(1); bj and b(1) should be substituted by xj
j=1
and x(1) in equation (2.38). All the other equations in section 2.6 except these two equations
continue to hold.
6. Conclusion
We have introduced (s,S) control policies for production/inventory systems with
compound Poisson arrival process and with irregular inspection intervals. We obtained an
expression for the expected cost per unit time for given control values and developed an
extremely simple, yet efficient search procedure to find the optimal control values. In devising
search procedures, convexity property as well as the recursive nature of the cost functions were
effectively exploited. We also showed that the same procedure can be directly applied to the
situation in which inventory level is continuously monitored during a non-production period.
23

Appendix A.
Proof of lemma 22
Let Vi be the length of an inspection interval given that the number of demand requests which
arrive during that inspection interval is i and let vi be E[Vi]. If we apply Bayes' formula, vi is
expressed as
00
i = E[ V IA=i ] = i (e-t t dV(t). (Al)
0
Using equation (A.1), we can obtain the following relationship after some algebraic
manipulations:
ajvj a_+1 (A2)
j+1 X
We can now obtain the term EVk by using equations (A.1) and (A.2). Note that the term EVk
consists of two components: the expected holding cost and the expected backorder cost during an
inspection interval. Let us denote by Dk the expected total holding time during an inspection
interval which is initiated with k inventories and denote by Ek the expected total backorder
time during the inspection interval initiated with k inventories. Then EVk is expressed as
EVk= ch Dk+ cb Ek. (A3)
We will first show how the expected holding time Dk is obtained. To this end, we will call the
inventory which is used for the ith demand during the inspection interval as the ith inventory.
Let di denote the expected holding time of the ith inventory during an inspection interval. Then
di is given by
i-i
di= L ajPr(Xl+...+Xj< i-1)vj
j=O
i
+ E [Pr(Xl+...+Xj-l< i-i, Xl+...+Xj> i)j.=,2,. (A.4)
j=l k=j
If we use equation (A.2), we can rewrite di as
- i-l
di =A[ Pr(Xl+...+Xj< i-l)(j+l)aj+
j=0

i oo
+ I {Pr(X1+...+Xj.l< i-l, X1+...+Xj> i)j E ak+l)} i=1,2, —. (A5)
j=l k=j
Thus the expected total holding time during the inspection interval initiated with k inventories
is just
k
k _
Dk= I di. (A6)
i=l
Let the expected time from the instant the ith demand arrives onwards until the instant the
inspection interval ends be ei. Then, Ek is expressed as
00
Ek = S ei. (A7)
i=k+l
Since the sum of di and ei is the expected length of an inspection interval, ei can be computed
from the following relationship:
ei = E(V) - di, i=1,2,, (A8)
with d = eo= 0.
Also from the renewal theorem, the expected total time from the instant each demand arrives
onwards to the instant the inspection interval ends is given by
x()E(V2) (A9)
ei 2'A.
i=0
From equations (A.7), (A.8) and (A.9), Ek is expressed as
00 _ k - x()E(V2)
Ek= ei- X ei= (2 - kE(V)- Dk}. (A10)
i=O i=O
Thus, from equation (A.3), EVk is given by
E-k= Xx(1)E(V2) kE(V)
EVk = Dk(ch + cb) + cb2 kE(V). (A.11)
Finally, from equation (A.11), EVk is expressed as
EVk = EVk-1 + dk(ch + cb) - E(V)cb. (A12)
For k<O, since dk = 0, from equation (A.12), EVk is expressed as
EVk = EVk+l + E(V)cb.
EVo, used as an initial value, can be obtained directly from equation (A.9)..

Appendix B.
Proof of Lemma 2.4
Lemma 2.4
r-1
Cp(r,S)= I bj Cp(r-j,S-j) + Cp(l,S)
j=1
Proof Cp(r,S)= fI(r,S),S
S-r
= I P(I(r,S)=j) fj,s
j=-oo
S-r r-1
= ~ [ ~ bk P(I(r-k,S-k)=j} + bs-j] fj,S
j=-oo k=l
r-1 S-r S-r
= ~ bk I [P(I(r-k,S-k)=j)(fjs-k + fS-k,S)] + bs-jfj,S
k=l j=-0o j=-oo
r-1 S-r S-r
= X bk [ Cp(r-k,S-k) +, P(I(r-k,S-k)=j) fS-k,S] + bs-jfj,S
k=l j=-00 j=S-r
Since I P{I(r-k,S-k)=j) = 1, the equation becomes
j= —o
r-1 S-r
= X bk [ Cp(r-k,S-k) + fS-k,S] + ~ bs-jfj,s
k=l j=-oo
r-1 oo
= X bkCp(r-k,S-k) + I bjfs-j,s
k=l j=l
r-1
= X bkCp(r-k,S-k) + Cp(l,S).
k=l.

Appendix C.
Proofs of properties 1-5
Proof of Property 1 Since C(r,S) = CN(r,S) + Cp(r,S), note that for a given r,C(r,S) is convex in S
if CN(r,S) and Cp(r,S) are both convex in S. To prove the convexity of CN(r,S) and Cp(r,S), we
need the following properties stated as propositions a, b, and c.
Proosition a: EVk is convex with respect to k.
Proof To prove the convexity of EVk, we need to show that:
AEV(k) = (EVk+l - EVk) - (EVk - EVk-1) > 0, for all k. (C.1)
From equation (2.9), we obtain
AEV(k) = ( dk+l - dk)(ch +cb), for all k. (C.2)
Since dk is an expected holding time of the kth inventory, dk> dk-l for k>O and dk= dk-1=0 for
k<0 (or it can be proved from equation (2.9.a)). Also, since both ch and cb are positive, AEV(k) 2
0 for all k. Thus, Ek is convex with respect to k. A
Proposition b: EUk is convex with respect to k.
Proof The proof is obtained using an approach identical to that used in proposition a.
Proposition c: fk,k+l is convex with respect to k.
Proof Let Af(k) = (fk+l,k+2 - fk,k+l) - (fkk+l - fk-l,k). To prove the convexity of fk,k+l we
need to prove Af(k) > 0 for all k. We will prove this by induction on k.
For k<-2, Af(k) = 0. Suppose Af(k)>0 holds for k<n-1, for some n>0. We now prove that Af(k)>0
for k=n using the induction hypothesis. From equation (2.28), we have
00
fn,n+l = - EUn + S hj fn-j+l,n). (C.3)
0 j=1
By substituting (C.3) into Af(n), we obtain
1 co
Af(n) = I((EUn+i-EUn) - (EUn-EUn-1)) + I hj{ (fn-j+2,n+l - fn-j+l,n) - (fn-j+l,n - fn-j,n-1)}
= H-~~~~0 ~j=1
27

1 00
= ({(EUn+l-EUn) - (EUn-EUn-1)) + hi{ (fn,n+l - fn-l,n) - (fn-j+l,n-j+2 -fn-jn-j+l)}.
0 j=1
(C.4)
In equation (C.4), since EUn is convex, (EUn+i-EUn) - (EUn-EUn-1) > 0. Also, from the
induction hypothesis, (fn,n+l - fn-l,n) - (fn-j+l,n-j+2 - fn-j,n-j+l) 2 0 for all j21. Since hj > 0
for j20, it follows that Af(n) > 0.
So, by the principle of mathematical induction, Af(k)>0 for all k, hence, fk,k+l is convex with
respect to k.
Proposition d: For a given r, CN(r,S) is convex with respect to S.
Proof From equation (2.8), the term CN(r,S) is expressed as
r-1
CN(r,S) = (bEVS + ejCN(r-j,S-j)). (C.5)
j=1
To prove the convexity of CN(r,S) for given r, we need to show that
ACN(r,S) = ( CN(r,S+l) - CN(r,S) - I CN(r,S) - CN(r,S-1) ) > 0 for any given r. (C.6)
This will be proved by induction on r.
i) For r=l, from equation (2.8), we have
ACN(1,S) = CN(1,S+) -CN(,S) CN (,S) N(1,S) - CN(1,S-) )
1
- { (EVs+i - EVS) - (EVs - EVS-1)) (C.7)
Since EVk is convex with respect to k, ACN(1,S) > 0.
ii) Suppose ACN(r,S)>0 holds for rSn-1. We now prove ACN(r,S) > 0 for r=n.
ACN(n,S) = (CN(n,S+l) - CN(n,S)) - CN(n,S) - CN(n,S-1)
1
= - (EVS+I - EVS) - (EVs - EVS-1))
n-1
+ bj [ {CN(n-j,S+l-j) - CN(n-j,S-j) - CN(n-j,S-j) - CN(n-j,S-l-j) ]
j=1
(C.8)
Since EVk is convex, (EVS+i - EVS) - (EVS - EVS-1) > 0 and from the induction hypothesis,

CN(n-j,S+l-j) - CN(n-j,S-j) ) - { CN(n-j,S-j) - CN(n-j,S-l-j) ) >0 for j=l,....,n-1. So ACN(n,S)
>0. So by the principle of mathematical induction, ACN(r,S) >0 for all r, hence, CN(r,S) is
convex with respect to S for given r. U
Proposition e: For a given r, Cp(r,S) is convex with respect to S.
Proof From equation (2.24), the term Cp(r,S) is expressed as
S-r
Cp(r,S) = fI(r,S),S = P(I(r,S) = j)fj,S (C.9)
j=-OO
To prove the convexity of Cp(r,S) with respect to S, we need to show that
ACp(r,S) = { Cp(r,S+1) - Cp(r,S) I - { Cp(r,S) - Cp(r,S-1)) 2 0 for any r. (C.10)
From equation (C.9), this reduces to
S+1-r S-r
ACp(r,S) = ( I P(I(r,S+l) = j)fj,S+l - X P(I(r,S) = j)fj,S )
j=-oo J=-oo
S-r S-l-r
- ( P(I(r,S) = j)fj,S - ~ P(I(r,S-1) = j)fj,S-l )
j=-oo j=-oo
Using the fact that P(I(r,S+l) = j) = P(I(r,S) = j-1l, Cp(r,S) is expressed as
S+1-r S-r
ACp(r,S) = (, P[I(r,S+l) =j)(fj,s+l -fj-l,S) - ( ~ P(I(r,S) = j)(fj,S - fj-1,S-1))
j=-oo j=-oo
S+l-r
= E P(I(r,S+l) = j}( (fj,S+l - fj-1,S) - (fj-l,S - fj-2,S-1)). (C.11)
j=-oo
In equation (C.11), since fk,k+l is convex, (fj,S+l - fj-l,S) - (fj-1,S - fj-2,S-1) >0.
Thus ACp(r,S)>0 holds for any given r, hence Cp(r,S) is convex with respect to S. From
propositions d and e, the convexity of C(r,S) is proved. U
Propertv 2 If C(r,k) < C(r,k-1), then C(r+l,k) < C(r+l,k-1).
Proof If we apply lemma 2.4 to the given condition C(r,k)-C(r,k-1)<0, we obtain the following
inequality:
r-1, bj {C(r-j,k-j) - C(r-j,k-j-1)} + C(1,k) - C(1,k-1) < 0 (C.12)
j=1
If we continue to apply lemma 2.4 to inequality (C.12), it reduces to the following form:
r-1
S cj {C(,k-j- ) - C(,k-j-1) +C(,k) - C(k-l) < 0, (C.13)
j=l

where aj >0forj=l,...,r-1.
Note that since C(1j) is convex with respect to j, C(l,k)-C(l,k-1) is a non-decreasing function of
j. Thus, in order for the LHS of (C.13) to be negative, the following inequality must be satisfied:
C(l,k-j) - C(l,k-j-1) < 0 for j>r-1. (C.14)
Now under this condition, we must prove that
C(r+l,k) - C(r+l,k-1) < 0. (C.15)
If we apply lemma 2.4, inequality (C.15) becomes
r, bj {C(r+l-j,k-j) - C(r+-j,k-j-1)} + C(l,k) - C(l,k-1) < 0 (C.16)
j=1
We now compare each term of inequality (C.12) with that of (C.16). Since C(l,k-r)-C(1,k-r-1)<0,
inequality (C.16) is true if the following inequality holds for j=r-l,r-2,...,1:
C(r+1-j,k-j) - C(r+l-j,k-j-1) < C(r-j,k-j) - C(r-j,k-j-1), j=r-1,r-2,...,. (C.17)
We will prove this by induction on j.
(i) j=r-1
In this case inequality (C.17) is
C(2,k-r+l) - C(2,k-r) < C(l,k-r+l) - C(l,k-r). (C.18)
Since LHS of (C.18) is bi{(C(l,k-r) - C(l,k-r-1)} + C(l,k-r+l) - C(l,k-r) and noting that C(l,k-r) -
C(l,k-r-1)<0, inequality (C.18) proves to be true.
(ii) Assume inequality (C.17) holds for j=r-l,r-2,...,n+l. Under this induction hypothesis, we
will prove that (C.17) also holds for j=n, that is, we will show that
C(r+l-n,k-n) - C(r+l-n,k-n-1) < C(r-n,k-n) - C(r-n,k-n-1). (C.19)
Inequality (C.19) can be rewritten as
r-n
X bj {C(r+l-n-j,k-n-j) - C(r+-n-j,k-n-j-1)} + C(l,k-n) - C(l,k-n-1)
j=1
r-n-1
< I bj {C(r-n-j,k-n-j) - C(r-n-j,k-n-j-1)} + C(l,k-n) - C(l,k-n-1). (C.20)
j=1

From the induction hypothesis together with the fact that C(l,k-r) - C(l,k-r-1)<O, inequality
(C.20) is true. Therefore, from the principle of mathematical induction, inequality (C.17) is
true for j=r-l,r-2,...,1, and hence, inequality (C.16) is true. U
Propertv 3 If S*(1)=n, then, S*(r) <n+r-1 for r>2.
Proof Proof is done by induction on r.
(i) r=2
Note that proving inequality S*(2)<n+1 is equivalent to showing that
C(2,n+2) > C(2,n+1), (C.21)
which can be rewritten as
b1 C(l,n+l) + C(l,n+2) > bl C(l,n) + C(l,n+l). (C.22)
This inequality is true since C(l,k) is convex in k and S*(1)=n.
(ii) Assume that if S*(1)=n, then S*(j)<n+j-1 holds for j=l,...,r-1. We now prove that S*(r)<n+r1 under this induction hypothesis. Note that we have only to prove that
C(r,n+r) > C(r,n+r-1). (C.23)
Inequality (C.23) can be rewritten as
r-1
X bk {C(r-k,n+r-k) - C(r-k,n+r-l-k)} + C(l,n+r) - C(l,n+r-1) > 0. (C.24)
k=l
From the induction hypothesis, we have C(r-k,n+r-k) > C(r-k,n+r-l-k) for k=l,...,r-1, and
from the given condition that S*(1)=n, we have C(l,n+r)>C(1,n+r-1). Thus, inequality (C.24) is
true. Therefore, from the principle of mathematical induction, property 3 has been proved. -
Property 4 If S*(1)=n and S*(r)=k, then, k< S*(r+l) <n+r holds.
Proof Direct result of properties 2 and 3.
Propertv 5 S*(r) >0 for any r>l.
Proof Consider a policy (r,S) where S<0. Then, policy (r,0) always dominates this policy
since only non-positive inventory levels exist for these two policies and the expected shortage
level of policy (r,0) is always less than that of policy (r,S) by -S. Thus, S*(r) cannot be negative
for any r. U
a

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