production inventory system with random supply interruptions statue and random lead times

Acta Mathematica Scientia 2011,31B(1):117–133

http://actams.wipm.ac.cn

PRODUCTION INVENTORY SYSTEM WITH

RANDOM SUPPLY INTERRUPTIONS STATUE

AND RANDOM LEAD TIMES∗

Hou Yumei (��)1,2 Liu Wenyuan (��)1

Zhang Qiang (��)2 Wu Fengqing (�)1

1. College of Economics and Management, Yanshan University, Qinhuangdao,066004, China;

2. College of Management and Economics, Beijing Institute of Technology, Beijing 100080, China

E-mail: [email protected]

Abstract This article analyzes a continuous-review inventory system with random sup-

ply interruptions and random lead time which may be interrupted by a random number

of supplier’s OFF periods. The inventory with constant demand rate is managed by a

(r; q1, q2, · · · , qm) policy and supplies from an unreliable sole supplier. By renewal theory

and matrix Geometric method, the long-run average cost function is obtained and some im-

portant properties of the function are proved. Furthermore, performance of the inventory

is derived.

Key words Markov processes; inventory theory; supply chain management; PH-

distribution; order-up-to policy

2000 MR Subject Classification 90B05

1 Introduction

How to efficiently manage a production and inventory system with random supply inter-

ruptions is a problem in supply chain management and has led to considerable researches in

recent years [1]. Supply interruptions which are called OFF represent the periods when a sup-

plier cannot produce and deal with inventory orders due to some unpredictable events. When

the supply is available, the period is called ON. This unreliable supplier switches intermittently

between ON and OFF. The aim of the article is to address the issue of how to determine the

optimal policy in the inventory system subject to supply interruptions.

We consider a continuous-review inventory system with random lead time, random supply

interruptions, and backorders. More specifically, the supplier’s status alternates between a

PH-distributed ON period and an exponential OFF period. The inventory system is controlled

∗Received October 25, 2007; revised July 4, 2008. This work was supported by the National Natural Science

Foundation of China (71071134 and 71001073). It is also funds by Hebei Science and Technology Research and

Development Program (10457202D-3) and 2010 Social Development of Research Subject of Hebei Province

(201005006)

118 ACTA MATHEMATICA SCIENTIA Vol.31 Ser.B

by an order-up-to policy: whenever the inventory level drops to a reorder point r, an order

is placed and the inventory level will be raised to r + qi (i = 1, 2, · · · , m) at an order arrival

instant (see [2] for commonly used inventory policies). If an order placed during the supplier’s

ON period, the order is dealt with and is processed immediately and is finished after a random

lead time. In contrast, if an order is placed during the supplier’s OFF period, the order is put

on hold for the rest of the OFF period and will be dealt with at the beginning of the subsequent

ON period. Hence, the supplier breakdowns not only affect the acceptance (start of processing)

of a new coming order but also interrupt the lead time of an in-process order. Moreover, we

assume that the maximum number of outstanding orders (either on hold or in process) at any

time is limited to one.

The important features of our model have both random lead time and supply interruptions.

The features make our model more realistic than any other supply chain systems because in

reality suppliers are not always available due to unexpected events such as equipment break-

downs, material shortages, price inflation, strikes, embargoes, and political crises. In addition,

the inventory lead time is usually non-zero and uncertain due to production and/or transporta-

tion capacity constraints. Considering all the reasons, it is worthy of studying an inventory

system with supply interruptions and random lead time.

There are two streams of researches related to our study. The first area is to model the

presence of a failure-prone production facility in single- or multi-stage production system where

the continuous flow of a product is subjected to random disruptions caused by the failure and

restoration [3–11]. In particular, [7] gave an extensive review of this line of researches. While

these studies addressed the machine failures, they did not pay much attention to the issue of

lead time broken off by the supply OFF. This is focused on in our research.

The second area is the research dealing with the impact of the uncertainty in supply process

of inventory management. [12] indicated the need for developing models with the supplier’s

uncertainty. Since then, the supplier’s uncertainty issue was addressed in many studies such

as [13–25]. Most of these researches, however, focused on the unreliable supplier with zero

lead time [19, 26, 27], which assumes that supply switching from ON to OFF and it has no

impact on the productive processing of a previous order. In other words, upon accepting an

order, the supplier will process and deliver the order no matter if the supplier’s working state

change or not during the lead time. While this assumption is over-simplification to most real

inventory systems. In generally, the processing of any outstanding order stops when the supplier

interruption occurs. So [20] presented an inventory model with non-zero lead time affected by

supply interruptions by using an approximation approach. Recently, [28] presented an exact

analysis on the model with random demand, lost sales, and supply interruptions affecting

both the order dealing with and the productive process of pending order. However, due to

the complexities, the work mainly uses a computational approach to search the best inventory

policy without proving the structural property of the cost function. Extending [28], we consider

the model with PH-distribued ON period and random lead time which is affected by supply

interruptions. Our work also generalizes [21] by including the non-zero lead time.

Our main contribution is two-fold: (i) providing an exact performance analysis of the

system by the matrix analytical method and the renewal reward theorem; (ii) investigating

theoretically the properties of the average cost function of the system.

No.1 Hou et al: PRODUCTION INVENTORY SYSTEM WITH RANDOM SUPPLY 119

The rest structure of this article is as follows. Section 2 presents the model formulation

and preliminaries. In Section 3, the long-run average cost function is developed based on the

renewal reward theorem. In Section 4, the properties of the average cost function are revealed

and the partial convexity of the cost function is discussed. Section 5 comes the conclusion.

2 Formulation and Preliminaries

We consider an inventory system with a non-perishable single item and a constant demand

rate. Without loss of generality, the demand rate is assumed to be one unit per unit time.

The demand is met immediately if the inventory is available and is backlogged otherwise. The

replenishment of the inventory comes from a single monopolistic supplier who may be ON

or OFF for a random length. The supplier’s OFF is exponential and the supplier’s ON has

PH-type distribution. For the sake of convenience, we introduce the PH-distribution firstly.

2.1 Introduction of PH-distribution

A continuous probability distribution F (·) is a phast type distribution (PH-distribution)

if it is the distribution of the time until absorption in a finite-state Markov Process with a

single absorbing state, that is, there exists a probability vector (α, αm+1) and an infinitesimal

generator of the form ⎡⎣G g0

0 0

⎤⎦ ,

such that F (t) = 1− α exp(Gt)e for t ≥ 0.

The m×m matrix G is nonsingular and has negative diagonal elements (Gii < 0 (1 ≤ i ≤

m)) and nonnegative off-diagonal elements (Gij ≥ 0 (i �= j)). The vector g0 is nonnegative and

satisfies

Ge + g0 = 0. (2.1)

The pair (α,G) is called a representation of PH-type distribution, where α = (α1, α2, · · · , αm)

satisfyingm∑

i=1

αi = 1.

2.2 The Model

The supplier’s ON periods are independent and identically distributed (i.i.d) random vari-

ables, denoted by X , with a PH-distribution. This PH-distribution has an irreducible rep-

resentation (α,G) of order m. The supplier’s OFF periods are also i.i.d. random variables,

denoted by Y , which has an exponential distribution with parameter μ.

It is well known that the PH-distribution with an irreducible representation (α,G) can be

considered as the distribution of the time-to-absorption of an (m + 1)-state Continuous-Time

Markov Chain (CTMC) with a state space Φ = {1, 2, · · · , m, 0}, where state 0 is an absorbing

state. The infinitesimal generator of this CTMC is given by⎡⎣G g0

0 0

⎤⎦ ,

where G e + g0 = 0, and e is a column vector with all components being one with appropriate

dimension. The distribution function of X is

FX(t) = 1− α exp(Gt)e, t ≥ 0, (2.2)


the mean of X is E(X) = −αG−1e (See [29, 30] for details about PH distribution).

To model the availability status of the supplier, a CTMC with m + 1 states, denoted by

{Y(t), t ≥ 0}, is constructed. Status i (1 ≤ i ≤ m) corresponds to ”stage” i of the ON period

(or phase i of PH-distribution) and state 0 corresponds to the OFF period, that is,

Y(t) =

⎧⎨⎩

i, if the supplier is in phase i of the ON period at time t;

0, if the supplier is in OFF at time t.

Obviously, the state space of the CTMC {Y(t), t ≥ 0} is Φ and its infinitesimal generator is

given by

G̃ =

⎡⎣ G g0

μα −μ

⎤⎦ .

Let pij(t) = Pr{Y(t) = j |Y(0) = i}, i, j ∈ Φ. The transition probability matrix of Y(t),

defined as P(t) = [ pij(t) ], i, j ∈ Φ, is given by

⎧⎨⎩P(t) = exp(G̃t),

P(0) = α.(2.3)

If an order is placed during the ON period, it is processed and filled after a random lead

time, which may be interrupted by a random number of OFF periods. In contrast, if an order

is placed during the OFF period, it is put on hold for the rest of the OFF period. The on-hold

order is then processed in a regular manner when the supplier becomes available. The lead

time, denoted by L, represents the preparation of an order (production and/or transportation

time) and is assumed to be exponentially distributed with parameter θ. The lead time may

be interrupted by a supplier’s OFF period and resumed from that interruption point when the

supplier becomes available again. Note that an order may be delayed due to both lead time

and the supplier interruptions. The order delay is defined as the time interval between order

placement and order arrival, and depends on the status of the supplier at the order placement

instant. Therefore, an order is called a type i order if it is placed during the phase i of the ON

period and is denoted by Lisum , where 1 ≤ i ≤ m. Clearly, Li

sum is the sum of the lead time L

and a random number of OFF periods or interruptions during L. When an order arrives, the

inventory level is raised to the target level qi + r units. The order quantity qi is determined

at its arrival epoch and depends on phase i when the order starts to process. With such a

(r; q1, q2, · · · , qm) policy, it is reasonable to assume that the maximum number of outstanding

orders is one.

Furthermore, we assume that X , Y , and L are mutually independent. For most prac-

tical systems, the supplier’s ON period is much longer than the supplier’s OFF period and

lead time. Thus, it is reasonable to use PH-distribution for ON period and exponential dis-

tributions for OFF period and lead time. A cost structure imposed on the system consists of

order cost $K/order, linear holding cost of $ch/unit/time, and backorder cost (or penalty cost)

cs/unit/time.

2.3 Order Delay

Above all, We focus on the property of type i order delay.


Consider a new CTMT {Y(t), t ≥ 0} with state space Ω = Φ ∪ { 0 } and infinitesimal

generator

Q̂ =

⎡⎣S s0

0 0

⎤⎦ .

Here, state i (1 ≤ i ≤ m), represents status i of an ON period; state 0 represents an OFF period;

and state 0, which is the absorbing state of {Y(t), t ≥ 0}, corresponds to the end of the order

delay. We now specify the elements of Q̂. Note that {Y(t), t ≥ 0} leaves state i (1 ≤ i ≤ m)

either at the end of the lead time L or at epoch which the supplier status changes. Thus, given

{Y(t), t ≥ 0} being in state i, the sojourn time in state i has the exponential distribution with

rate −Gii +θ; then it may make a transition into absorbing state 0 with rate θ or into transient

state 0 with rate g0i . In addition, a transition from 0 to a transient state i , 1 ≤ i ≤ m, happens

with rate μαi. Consequently, we have

S =

⎛⎝G − θI g0

μα −μ

⎞⎠ , (2.4)

where I is an identity matrix with appropriate dimension and

s0 =

⎛⎝ θe

0

⎞⎠ .

A type i order delay is the time interval from the epoch of a type i order being placed to the

epoch of the order arrival. Correspondingly, this order delay is the first passage time from state

i (1 ≤ i ≤ m) to absorbing state 0 of {Y(t), t ≥ 0}. Therefore, a type i order delay Lisum has a

PH-distribution with a PH-representation (eti,S, e), where et

i is a row vector with one for its

ith-component, zero for others and appropriate dimension.((·)t is the transpose of vector (·))

To prove that a type i order delay has a PH-distribution, there needs some preliminaries.

The first one is the construction of Lisum. The second is a method to derive the inverse of a

block matrix.

First, let X(i) be the time interval between the epoch of a type i order placement and the

end of current ON period. Hence,

FX(i)(t) = 1− eti exp(Gt)e.

We can define a delayed PH-renewal process as

Ni(t) = sup

{n : X(i) +

n∑j=1

Xj ≤ t

},

where Xj (j = 1, 2, 3, · · ·) is an i.i.d sequence with FX(t) (t ≥ 0) given by (2.2). Let Ni be the

number of supply interruptions during the i order delay. Hence, Ni = Ni(L).

Lemma 2.1 For 1 ≤ i ≤ m, Ni is determined by the inequalities

X(i) +

Ni−1∑j=1

Xj ≤ L < X(i) +

Ni∑j=1

Xj (Ni ≥ 1), (2.5)


and type i order delay Lisum is determined by

Lisum =

⎧⎪⎪⎨⎪⎪⎩

L, Ni = 0,

L +

Ni∑j=1

Yj , Ni > 0,(2.6)

where {Yj , j ≥ 0} is an i.i.d sequence with FY (t) = 1− e−μt (t ≥ 0).

The second preliminary is Theorem 2.2.4 of [31] (page 23).

Lemma 2.2 If B =

⎛⎝B11 B12

B21 B22

⎞⎠ and B11 is non-singlar, then,

B−1 =

⎛⎝B−1

11 + B−111 B12B

−122.1B21B

−111 −B−1

11 B12B−122.1

−B−122.1B21B

−111 B−1

22.1

⎞⎠ ,

where B22.1 = B22 − B−121 B12.

With these results, we prove that a type i order delay Lisum has PH-distribution with a

PH-representation (eti,S, e).

Theorem 2.3 A type i order delay Lisum (1 ≤ i ≤ m) has the PH-distribution with cdf

FLisum

(x) = 1− eti exp(xS)e (x ≥ 0).

Proof According to Lemma 2.1, the Laplace transform of FLisum

(x) is

f̃FLi

sum(s) = E[exp (−sLi

sum)] =θ

s + θ

s + μ− sα(θI − G)−1g0 − setiI − G)−1g0

s + μ− sI − G)−1g0. (2.7)

The calculation of (2.7) may be refers to Appendix A. Using Lemma 2.2, we have

eti(I − S)−1S0 =

θ

s + θ


s + μ− sI − G)−1g0. (2.8)

Integrating (2.7) and (2.8), the Laplace transform of FLisum

(x) is f̃FLi

sum(s) = et

i(I − S)−1S0.

Hence, Lisum has PH-distribution.

Because Lisum has PH-distribution, we verify that the nth moment of a type i order delay is

E[ (Lisum)n ] = (−1)nn!(et

iS−ne) (n ≥ 1,1 ≤ i ≤ m) [29]. Furthermore, by the similar technique,

we can also show that the sum L0isum = Y + Li

sum (1 ≤ i ≤ m) or a type 0 order delay has a

PH-distribution (denoted by FL0isum

(x)) with an irreducible representation (eti,Ri, e), where

Ri =

⎡⎢⎢⎣−μ μ et

i 0

0 G − θI g0

0 μα −μ

⎤⎥⎥⎦ , (2.9)

and E[Lisum] = −et

1R−1i e.

To derive the Long-run average cost, conditional probability qij denoted that a type i order

is delivered in status j of the ON period, is needed, that is, qij = Pr{Y(Lisum) = j/Y(0) = i}.

Let Q= [qij ](1 ≤ i, j ≤ m). Then, we have the following result.

Theorem 2.4 The conditional probability matrix is

Q = θ[θI − (G + g0α)]−1. (2.10)


Proof Let A = −(G + g0α− θI). According to (2.1), we have

−Gii =

m∑j=1,j �=i

Gij + g0i (i = 1, 2, · · · , m)

and

θ + [−Gii] ≥

m∑j=1,j �=i

Gij + g0i ≥

m∑j=1,j �=i

Gij + g0i αj .

Thus, the matrix A is strictly diagonally dominant whichm∑

j=1

αj = 1. According to Theorem

2.3 on Page 137 of [32], the matrix A is non-singular.

Given state i (i = 1, · · · , m), the next transition of {Y(t), t ≥ 0} will be either to state l

with probability Gil/(−Gii + θ), l �= i, or to state 0 with probability g0i /(−Gii + θ). Using the

exponential distribution property, qij satisfies the following equation:

qij =∑l�=i

Gil

−Gii + θqlj +

g0i

−Gii + θ

m∑l=1

αlqlj , i �= j.

For the i = j case, besides the two kinds of transitions in the i �= j case, the next transition

may be to the absorbing state 0, therefore

qii =θ

−Gii + θ+∑l�=i

Gil

−Gii + θqli +

g0i

−Gii + θ

m∑l=1

αlqli.

Or, equivalently,

θqij =

m∑l=1

Gilqlj + g0i

m∑l=1

αlqlj , i �= j,

θqii = θ +

m∑l=1

Gilqli + g0i

m∑l=1

αlqli.

In matrix form, we have

θQ = θI + GQ+ g0αQ.

Solving this matrix equation gives (2.10).

With the Lamma above, it is shown that the expected order delay is longer for a stochastic

longer OFF period by using the stochastic order argument [33].

3 Average Cost Function

The long-run average cost function can be developed by applying the renewal reward

theorem to a regenerative cycle of the inventory process. This cycle is defined as the time

interval between two consecutive instants at which the inventory level drops to r and the

supplier is OFF. Moreover, the cycle can be classified according to the type of the first order

in the cycle. Thus, the probability of having a type i cycle is αi (1 ≤ i ≤ m).

Let Ti(q1, q2, · · · , qm) and Ci(r; q1, q2, · · · , qm) be the length of type i cycle and the cost

incurred during type i cycle, respectively. Denoting their means by T̄i(q1, q2, · · · , qm) and


C̄i(r; q1, q2, · · · , qm), we obtain the expected length of an arbitrary cycle as T̄ (q1, q2, · · · , qm) =

αT̄ and the expected cost during an arbitrary cycle as C̄(r; q1, q2, · · · , qm) = αC̄, where

C̄ = (C̄1(r; q1, q2, · · · , qm), C̄2(r; q1, q2, · · · , qm), · · · , C̄m(r; q1, q2, · · · , qm))t,

T̄ = (T̄1(q1, q2, · · · , qm), T̄2(q1, q2, · · · , qm), · · · , T̄m(q1, q2, · · · , qm))t.

According to the renewal reward theorem [32], the long-run average cost per unit time can be

formed as follows:

AC(r; q1, q2, · · · , qm) =C̄(r; q1, q2, · · · , qm)

T̄ (q1, q2, · · · , qm)=

αC̄

αT̄. (3.1)

3.1 Expected Cycle Time

To compute T̄ (q1, q2, · · · , qm), we define the sub-cycle as the interval between two consec-

utive instants at which the inventory level drops to r. According to the single outstanding

order assumption, there is only one order in each sub-cycle. Moreover, a type i residual cycle,

denoted by TiR(q1, q2, · · · , qm), is defined as the time required to complete the cycle when the

inventory level drops to r and the supplier is in status i, 1 ≤ i ≤ m. Let CiR(r; q1, q2, · · · , qm)

be the costs incurred in a type i residual cycle. Denote the means of TiR(q1, q2, · · · , qm) and

CiR(r; q1, q2, · · · , qm) by T̄iR(q1, q2, · · · , qm) and C̄iR(r; q1, q2, · · · , qm), respectively. Let

T̄R = (T̄1R(q1, q2, · · · , qm), T̄2R(q1, q2, · · · , qm), · · · , T̄mR(q1, q2, · · · , qm))t

and

C̄R = (C̄1R(r; q1, q2, · · · , qm), C̄2R(r; q1, q2, · · · , qm), · · · , C̄mR(r; q1, q2, · · · , qm))t.

Now, if a type i ≥ 1 order is placed at time 0 (start of a sub-cycle), then, Y(Lisum + qi) = n

implies that the supplier status is n at the end of the sub-cycle (0 ≤ n ≤ m). In particular,

Y(Lisum + qi) = 0 indicates that a new regenerative cycle starts at the end of the sub-cycle.

Theorem 3.1 The mean of an arbitrary cycle, T̄ (q1, q2, · · · , qm), is given by

T̄ (q1, q2, · · · , qm) = μ−1 + αT̄R, (3.2)

where the column vector T̄R satisfies

T̄R =[I −H

]−1· (E[L1

sum] + q1, E[L2sum] + q2, · · · , E[Lm

sum] + qm)t, (3.3)

where

H =((G P̂(q1))1·, (G P̂(q2))2·, · · · , (G P̂(qm))m·

)t, P̂(qk) = [pij(qk)], (1 ≤ i, j, k ≤ m),

which is the submatrix of P(qk) = [pij(qk)] (0 ≤ i, j ≤ m), and (G P̂(qk))k· is the kth row of

the matrix (G P̂(qk)) (See Equation (2.3) for the definition of pij(t)).

Proof For a type i residual cycle starting at time 0, conditioning on the state of {Y(t),

t ≥ 0} after Lisum + qi time units (recall that the demand rate is one and the inventory level

reaches r at this instant) and using the renewal argument, we obtain

TiR(q1, q2, · · · , qm)

=

⎧⎪⎨⎪⎩

Lisum + qi, if Y(Li

sum + qi) = 0,

Lisum + qi + TnR(q1, q2, · · · , qm), if Y(Li

sum + q) = n �= 0,for 1 ≤ i ≤ m. (3.4)


Note that {Y(t), t ≥ 0} is a CTMC with the conditional probability matrixQ. For i (1 ≤ i ≤ m)

and n (0 ≤ n ≤ m) we have

Pr{Y(Li

sum + qi) = n}

=

m∑j=1

Pr{Y(Li

sum) = j |Y(0) = i}

Pr{Y(Li

sum + qi) = n |Y(Lisum) = j

}

=

m∑j=1

qijpjn(qi). (3.5)

Because the length of Lisum depends on the supplier’s state j at the type i order receipt instant,

we define the conditional expectation of type i order delay and denote it by Ej[Lisum] (j =

1, 2, · · · , m). By the law of the total probability and (3.4), we have

T̄iR(q1, q2, · · · , qm)

=m∑

j=1

(Ej [L

isum] + qi

)qijpj0(qi) +

m∑n=1

m∑j=1

(Ej [L

isum] + qi + T̄nR(q1, q2, · · · , qm)

)qijpjn(qi)

= E[Lisum] + qi +

m∑n=1

m∑j=1

qijpjn(qi) T̄nR(q1, q2, · · · , qm). (3.6)

Note thatm∑

n=0

m∑j=1

qijpjn(qi) = 1 andm∑

j=1

Ej[Lisum]qij = E[Li

sum]. Rewriting (3.6) in matrix form

gives

T̄R = (L1sum + q1, L

2sum + q2, · · · , L

isum + qm)t +HT̄R. (3.7)

Because [I −H] is non-singular, hence we have (3.3). Due to the memoryless property of the

exponential distribution, the length of type i cycle is composed of one OFF period and one

type i residual cycle with probability αi (1 ≤ i ≤ m). Thus,

T̄ (q1, q2, · · · , qm) = 1/μ +

m∑i=1

αiT̄iR(q).

Hence, we have (3.2).

3.2 Expected Cycle Cost

We define ci(r) as the cost incurred during Lisum, c0i(r) as the cost incurred during the

interval of one OFF period plus Lisum (1 ≤ i ≤ m). Let B(r, qi) be the inventory holding cost

during the interval when the inventory drops from qi + r to r units, with an expression as

B(r, qi) =chq2

i

2+ chqir. (3.8)

Lemma 3.2 The means of ci(r) and c0i(r) (1 ≤ i ≤ m) are given, respectively, by

c̄i(r) = K + ch

(et

i exp(Sr)S−2e− retiS−1e− et

iS−2e

)− cse

ti exp(Sr)S−1e (3.9)

and

c̄0i(r) = K + ch

(et1 exp(Rir)R

−2i e− ret

1R−1i e− et

1R−2i e

)− cse

t1 exp(Rir)R

−1i e. (3.10)


Proof See Appendix.

For the special case of r = 0, (3.9) and (3.10) yield, for 1 ≤ i ≤ m,

c̄i(0) = K + csE[Lisum], c̄0i(0) = K + cs(1/μ + E[Li

sum]), (3.11)

which indicates that re-ordering at zero inventory level results in only ordering cost and back-

order cost during Lisum or OFF period plus Li

sum. Together with Proposition 3.4 ([34]), we can

see that: the costs incurring in one order delay (c̄i(0) and c̄0i(0), i = 1, 2, · · · , m) decrease in

parameter μ monotonously.

Theorem 3.3 The expected cost incurred in an arbitrary cycle C̄(r, q) is given by

C̄(r; q1, q2, · · · , qm) = αC̄, (3.12)

where C̄ satisfies

C̄ =

⎛⎜⎜⎜⎜⎜⎜⎝

c̄01(r) − c̄1(r)

c̄02(r) − c̄2(r)...

c̄0m(r) − c̄m(r)

⎞⎟⎟⎟⎟⎟⎟⎠

+[I −H

]−1

⎛⎜⎜⎜⎜⎜⎜⎝

c̄1(r) + B(r, q1)

c̄2(r) + B(r, q2)...

c̄m(r) + B(r, qm)

⎞⎟⎟⎟⎟⎟⎟⎠

. (3.13)

Proof For a type i residual cycle starting at t = 0, conditioning on the state of {Y(t), t ≥

0} at the end of the first sub-cycle and using the renewal argument, we have

CiR(r; q1, q2, · · · , qm)

=

⎧⎪⎨⎪⎩

ci(r) + B(r, qi), if Y(Lisum + qi) = 0,

ci(r) + B(r, qi) + CnR(r; q1, q2, · · · , qm), if Y(Lisum + qi) = n �= 0.

(3.14)

Let Ej[ci(r)] be the conditional expectation of ci(r) given the type i order arrive at state j

(j = 1, 2, · · · , m). From (3.5), (3.14), and the total probability law, we have

C̄iR(r; q1, q2, · · · , qm) = c̄i(r) + B(r, qi) +

m∑n=1

m∑j=1

qijpjn(qi)C̄nR(r; r; q1, q2, · · · , qm). (3.15)

Note thatm∑

j=1

Ej [ci(r)]qij = c̄i(r). Rewriting (3.15) in matrix form, it gives

C̄R = (c̄1(r) + B(r, q1), c̄2(r) + B(r, q2), · · · , c̄m(r) + B(r, qm))t +HC̄R. (3.16)

Suppose that the first order of a cycle is placed in state i (i = 1, 2, · · · , m ) at t = 0.

Conditioning on the state at the end of the first sub-cycle and by the renewal argument, we

obtain

Ci(r; q1, q2, · · · , qm)

=

⎧⎨⎩

c0i(r) + B(r, qi), if Y(Lisum + qi) = 0,

c0i(r) + B(r, qi) + CnR(r; q1, q2, · · · , qm), if Y(Lisum + qi) = n �= 0.

(3.17)


Similarly, we can obtain

C̄ = (c̄01(r) + B(r, q1), c̄02(r)+, B(r, q2), · · · , c̄0m(r) + B(r, qm))t +HC̄R. (3.18)

Combining (3.17) and (3.18) gives (3.13). Furthermore,

C̄(r; q1, q2, · · · , qm) =m∑

i=1

αi C̄i(r; q1, q2, · · · , qm).

By setting appropriate values of cost parameters K, ch, and cs in C̄(r; q1, q2, · · · , qm), we

can compute the average number of orders, average inventory level, and average number of

stockouts per cycle or per unit time.

To simplify and explain the expression of the average cost function, we re-group different

types of costs. This needs some new notations: �ih(r) is the sum of ordering cost and holding

cost incurred in Lisum; �ip(r) is the backorder penalty cost incurred in Li

sum; �0ih(r) is the

sum of ordering cost and holding cost incurred during OFF period plus Lisum; and �0ip(r) is

the penalty cost incurred during OFF period plus Lisum. We can write

�ih(r) = K + ch

(et

i exp(Sr)S−2e− retiS−1e− et

iS−2e

),

�ip(r) = cseti exp(Sr)S−1e,

�0ih(r) = K + ch

(et1 exp(Rir)R

−2i e− ret

1R−1i e− et

1R−2i e

),

�0ip(r) = cset1 exp(Rir)R

−1i e.

According to Lemma 3.2, we have

c̄i(r) = �ih(r)−�ip(r), c̄0i(r) = �0ih(r)−�0ip(r).

Then, (3.13) can be rewritten as

C̄ =

⎛⎜⎜⎜⎜⎜⎜⎝

�01h(r) −�1h(r)

�02h(r) −�2h(r)...

�0mh(r) −�mh(r)

⎞⎟⎟⎟⎟⎟⎟⎠

+[I −H

]−1

·

⎛⎜⎜⎜⎜⎜⎜⎝

�1h(r) + B(r, q1)

�2h(r) + B(r, q2)...

�mh(r) + B(r, qm)

⎞⎟⎟⎟⎟⎟⎟⎠

+

⎛⎜⎜⎜⎜⎜⎜⎝

�1p(r)−�01p(r)

�2p(r)−�02p(r)...

�mp(r)−�0mp(r)

⎞⎟⎟⎟⎟⎟⎟⎠

+[I −H

]−1

·

⎛⎜⎜⎜⎜⎜⎜⎝

�1p(r)

�2p(r)...

�mp(r)

⎞⎟⎟⎟⎟⎟⎟⎠

. (3.19)

The first two terms correspond the ordering and holding costs, simply written as Δh, the last

two terms are the backorder cost, simply written as Δp. Thus, αΔh is the sum of the expected

ordering cost and holding cost and αΔp is the expected backorder cost in an arbitrary cycle.

Then, the long-run average cost function can be expressed as

AC(r; q1, q2, · · · , qm) =α

T̄ (q1, q2, · · · , qm)Δh +

α

T̄ (; q1, q2, · · · , qm)Δp. (3.20)


4 Properties

In this section, we reveal some properties of the average cost function. We start with the

convexity of c̄i(r) and c̄0i(r).

Lemma 4.1 (i) c̄i(r) and c̄0i(r) are convex (1 ≤ i ≤ m).

(ii) If r ≥ cs

ch, then,

∂c̄i(r)

∂r≥ 0,

∂c̄0i(r)

∂r≥ 0. (4.1)

Proof It follows from the constant demand rate that

ci(r) =

⎧⎪⎪⎨⎪⎪⎩

K + chrLisum −

1

2ch(Li

sum)2, Lisum < r,

K +1

2chr2 + cs(L

isum − r), Li

sum ≥ r,

1 ≤ i ≤ m. (4.2)

From (4.2), we have

c̄i(r) = K +

∫ r

0

(chry −

1

2chy2

)dFLi

sum(y)−

∫ r

0

(1

2chr2 + cs(y − r)

)dFLi

sum(y)

+

∫ ∞

0

(1

2chr2 + cs(y − r)

)dFLi

sum(y).

Taking the derivative of c̄i(r) with respect to r, we find

∂c̄i(r)

∂r= ch

∫ r

0

ydFLisum

(y) + (chr − cs)[1 − FLisum

(r)].

It is seen that if r ≥ cs

ch, then, ∂c̄i(r)

∂r≥ 0. So, the first inequality of (4.1) holds. Taking the

second order derivative, we obtain

∂2c̄i(r)

∂r2= ch[1− FLi

sum(r)] + csfLi

sum(r),

which is always positive, indicating that c̄i(r) is convex.

Similarly, we can prove the same properties for c̄0i(r).

By Lemma 4.1 we establish an important property for the average cost function.

Theorem 4.2 For a given vector (q1, q2, · · · , qm), AC(r; q1, q2, · · · , qm) is convex in r.

In addition, there is a unique optimal re-order level r∗ (q1, q2, · · · , qm), which satisfies 0 ≤ r∗

(q1, q2, · · · , qm) ≤ cs

ch.

Proof From (3.16) and (3.18), we have

C̄ =

⎛⎜⎜⎜⎜⎜⎜⎝

c̄01(r)

c̄02(r)...

c̄0m(r)

⎞⎟⎟⎟⎟⎟⎟⎠

+H(I −H)−1

⎛⎜⎜⎜⎜⎜⎜⎝

c̄1(r)

c̄2(r)...

c̄m(r)

⎞⎟⎟⎟⎟⎟⎟⎠

+(I −H)−1(B(r, q1), B(r, q2), · · · , B(r, qm)

)t

. (4.3)

Let (γ1, γ2, · · · , γm) = α(I −H)−1 and (β1, β2, · · · , βm) = αH(I −H))−1. Then, the expected

cost of a cycle can be written as

C̄(r; q1, q2, · · · , qm) =

m∑i=1

αic̄0i(r) +

m∑i=1

βic̄i(r) +

m∑i=1

γiB(r, qi), (4.4)


where βi ≥ 0 and γi ≥ 0. On the basis of Lemma 4.1 and (3.8), c̄0i(r), c̄i(r), and B(r, qi) are

all convex in r for fixed qi (i = 1, 2, · · · , m). Thus, given qi (i = 1, 2, · · · , m), C̄(r; q1, q2, · · · , qm)

is a linear combination of convex functions. That means C̄(r; q1, q2, · · · , qm) is convex in r.

Because T̄ (q1, q2, · · · , qm) is independent of r,

AC(r; q1, q2, · · · , qm) = C̄(r; q1, q2, · · · , qm)/T̄ (q1, q2, · · · , qm)

is convex in r. Hence, for a fixed vector (q1, q2, · · · , qm), an unique optimal re-order level

r∗(q1, q2, · · · , qm) exists [35].

Taking the partial derivative of AC(r; q1, q2, · · · , qm) with respect to r, we have

∂AC(r; q1, q2, · · · , qm)

∂r

=1

T̄ (q1, q2, · · · , qm)

{ m∑i=1

αi

∂c̄0i(r)

∂r+

m∑i=1

βi

∂c̄i(r)

∂r+

m∑i=1

γi(chqi + csr)

}. (4.5)

By Lemma 4.1, for fixed q1, q2, · · · , qm, if r ≥ cs

ch, then ∂AC(r,q)

∂r≥ 0, implying that AC(r; q1, q2,

· · · , qm) increases in r. Therefore, the optimal re-order level r∗(q1, q2, · · · , qm) satisfies

0 ≤ r∗(q1, q2, · · · , qm) ≤cs

ch

.

The upper bound cs/ch for r∗(q1, q2, · · · , qm) is intuitive. Note that the re-order point is

determined by the trade-off between the holding cost and backorder penalty cost. If we order

when r > cs/ch, that is, rch > cs , which implies that the holding cost rch for meeting one unit

demand is greater than the penalty cost cs for one unit shortage. So, it is better off to delay

the order until the inventory level is lower. Thus, the optimal re-order point must be less than

cs/ch.

Proposition 4.3 For given (r; q1, q2, · · · , qm) policy, AC(r; q1, q2, · · · , qm) is a linear func-

tion of K, ch and cs, respectively.

Proof Let

ui � eti exp(Sr)S−2e− ret

iS−1e− et

iS−2e

and

vi � et1 exp(Rir)R

−2i e− ret

1R−1i e− et

1R−2i e.

From (3.9), (3.10), (4.4), and (3.1), we obtain

AC(r; q1, q2, · · · , qm)

=1

T̄ (q1, q2, · · · , qm)

{K

(1 +

m∑i=1

βi

)+ ch

m∑i=1

(αivi + βiui)

−cs

[ m∑i=1

(αie

t1 exp(Rir)R

−1i e + βie

ti exp(Sr)S−1e

)]}+

m∑i=1

γiB(r, qi)

T̄ (q1, q2, · · · , qm). (4.6)

Therefore, for a given inventory policy, AC(r; q1, q2, · · · , qm) is a linear function of K, ch, and

cs, respectively.

According to Theorem 3.1, Theorem 3.3, and AC(r, q), we can derive the mean of the

ordering number per unit time. To get the ordering number per unit time, we find that the


average cost of one arbitrary cycle C̄(r, q) dependents on the order cost K, the holding cost ch,

and the penalty cost cs. So, we substitute C̄(K, ch, cs) for C̄(r, q). We also find that whenever

the inventory level drops to r, one order cost, one holding cost and one penalty cost are added to

the average cost of one arbitrary cycle C̄(K, ch, cs). So, C̄(1, 0, 0) records the ordering number

in one cycle. Similarly, C̄(0, 1, 0) records the sum of the inventory level and C̄(0, 0, 1) records

the sum of the backorder in one cycle. Let N̄ denote the mean of the ordering number per unit

time, IN denote the mean of the inventory level per unit time and B̄ denote the mean of the

backorder per unit time. Then from Theorem 3.3, we can deduce the following proposition.

Theorem 4.4 The mean of the ordering number per unit time is N̄ = C̄(1, 0, 0)/T̄ (q),

the mean of the inventory level per unit time is IN = C̄(0, 1, 0)/T̄(q), and the mean of the

backorder per unit time is B̄ = C̄(0, 0, 1)/T̄ (q).

5 Conclusion

We have studied a continuous review (r; q1, q2, · · · , qm) inventory system with supply in-

terruptions and random leadtime. To model such a system, we assume that the ON period is

PH-distributed and the leadtime is exponential. The long-run average cost AC(r; q1, q2, · · · , qm)

is obtained using the concept of multi-type order delays, the matrix analytic method, and the

renewal reward theorem. We have established some properties of the model and given a deeply

insights of it. The results offer practicing managers useful information in designing the best or-

der policy or understanding the cost effects of implementing a feasible policy under a stochastic

supply process.

Acknowledgments The authors are indebted to Dr. Q.M. He and H.Q. Zhang for

many helpful discussions and their contributions to this work. We thank for the help from

Huang Junfei, also.

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Appendix

Appendix A

Proof of formula (2.7)

According to (2.1), the probability density of the renewal number Ni is

Pr(Ni = n

)

=

⎧⎪⎪⎨⎪⎪⎩

Pr(X(i) ≥ L

), n = 0,

P r

(X(i) +

n−1∑j=1

Yj ≤ L < X(i) +

n∑j=1

Yj

), n > 0

=

⎧⎨⎩

1− eti(I − G)−1g0, n = 0,

eti(I − G)−1g0

[α(I − G)−1g0

](n−1)− et

i(I − G)−1g0[α(I − G)−1g0

](n), n > 0.

(A.1)

By the law of the total probability, the Laplace transform of the PH-distribution FLisum

is

f̃Lisum

(s) = E[e(−sLi

sum)]

= Pr(Ni = 0

)[e(−sL)

]+ Pr

(Ni = n

)[e(−sL+

n∑j=1

Yj)]

=θ

s + θ


s + μ− sI − G)−1g0. (A.2)

Hence, we have (2.7)

Appendix B

Proof of Lemma 3.2

Due to the constant demand rate, we have

ci(r) =

⎧⎪⎪⎨⎪⎪⎩

K + chrL̄i −1

2chL̄2

i , L̄i < r,

K +1

2chr2 + cs(L̄i − r), L̄i ≥ r,

1 ≤ i ≤ m. (B.1)

Taking expectation on (B.1), we have

c̄i(r) = K +

∫ r

0

(chry −

1

2chy2

)d[1− et

i exp (yS )e]

+

∫ ∞

r

(1

2chr2 + cs(y − r)

)d[1 − et

i exp (yS )e]

= K +

∫ r

0

(chry −

1

2chy2

)d[1− et

i exp (yS )e]

−

∫ r

0

(1

2chr2 + cs(y − r)

)d[1− et

i exp (yS )e]

+

∫ ∞

0

(1

2chr2 + cs(y − r)

)d[1 − et

i exp (yS)e]

= K +(chry −

1

2chy2

)[1− et

i exp (yS )e]|r0

−

∫ r

0

(chr − chy)[1− eti exp (yS )e]dy −

(1

2chr2 − csr

)[1− et

i exp (yS )e]|r0


−

∫ r

0

csy d[1− eti exp (yS )e]

(1

2chr2 − csr

)[1− et

i exp (yS )e]|∞0

+

∫ r

0

csy d[1− eti exp (yS )e]

= K +1

2chr2 [1− et

i exp (rS )e]− chr[y − eti exp (yS )S−1e]r0

+

∫ r

0

chy [1− eti exp (yS)e]dy −

(1

2chr2 − csr

)[1− et

i exp (rS )e]

−csy [1− eti exp (yS )e]r0 + cs

∫ r

0

[1− eti exp (yS )e]dy +

(1

2chr2 − csr

)+ csD̄i

= K +

{1

2chr2 [1 − et

i exp (rS )e]− chr2 + chreti exp (rS )S−1e− chret

iS−1e

}

+1/2r2 −

∫ r

0

chy eti exp (yS )edy −

(1

2chr2 − csr

)[1− et

i exp (rS )e]

−csr [1− eti exp (rS )e] + csr − cs

∫ r

0

eti exp (yS )edy +

(1

2chr2 − csr

)+ csD̄i

= K + cheti exp (Sr)S−2e− chret

iS−1e− cse

ti exp(Sr)S−1e− che

tiS−2e.

which gives (3.9).

Similarly, c0i(r) can be written as

c0i(r) =

⎧⎪⎪⎨⎪⎪⎩

K + chr(Y + Lisum)−

1

2ch(Y + Li

sum)2, (Y + Lisum) < r,

K +1

2chr2 + cs((Y + Li

sum)− r), (Y + Lisum) ≥ r,

1 ≤ i ≤ m. (B.2)

Taking the expectation on (B.2) gives (3.10).

production inventory system with random supply interruptions statue and random lead times

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