Multiple replenishment orders in a continuous-review inventory system with lost sales

Operations Research Letters 30 (2002) 117 – 129

Operations Research Letters www.elsevier.com/locate/dsw

Multiple replenishment orders in a continuous-review inventory system with lost sales Esmail Mohebbia; ∗ , Morton J.M. Posnerb a Department

of Industrial and Management Systems Engineering; University of Nebraska-Lincoln; 175; Nebraska Hall; Lincoln, NE 68588-0518; USA b Department of Mechanical and Industrial Engineering; University of Toronto; Toronto; Ont.; M5S 3G8 Canada Received 1 December 2000; accepted 20 December 2001

Abstract For a continuous-review inventory system with lost sales, non-unit-sized demand, multiple replenishment orders outstanding, and split deliveries, we derive the stationary distribution of the on-hand inventory using a level-crossing methodology. This distribution is then used to formulate an exact expression for the long-run average cost per unit time. Some numerical c 2002 Elsevier Science B.V. All rights reserved. results are also presented.  Keywords: Inventory; Lost sales; Multiple replenishments; Splitting orders

1. Introduction In this paper we attempt to link two areas of inventory control theory, namely, lost-sales continuous-review inventory models with multiple orders outstanding, and inventory models with split deliveries. Rapid advancements in computer and information technology along with the crucial role of the element of responsiveness in establishing wining strategies for today’s modern supply chains has boosted interest in continuous-review inventory control policies. At the same time, intense competition in most retailing industries has made the lost sales case more pertinent than ever. Lost sales models are also used to describe situations where demands during stockout periods are satis7ed through an alternative source of supply (external to the regular replenishment channel) for an additional cost. The analytic treatment of continuous-review inventory systems with lost sales and positive lead times, however, when more than one replenishment order may be outstanding at any time is commonly acknowledged as a di8cult task (see [7] for a detailed discussion). A glance at the current literature on inventory models, as we will show later in this section, clearly indicates that this problem area in general remains largely unexplored. On the other hand, owing to the growing popularity of supply chain management in practice during the past few years, the idea of order splitting as an e;ective way of reducing inventories while maintaining acceptable service levels has received much attention in the literature. The order-splitting concept can be observed in two di;erent scenarios. Under the 7rst scenario (also referred to as the multiple-sourcing problem in the inventory literature) a replenishment order triggered at a reorder point is split among multiple suppliers and the resulting split orders are placed simultaneously. The second ∗

Corresponding author. E-mail address: [email protected] (E. Mohebbi).

c 2002 Elsevier Science B.V. All rights reserved. 0167-6377/02/$ - see front matter
PII: S 0 1 6 7 - 6 3 7 7 ( 0 2 ) 0 0 1 0 8 - 6

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E. Mohebbi, M.J.M. Posner / Operations Research Letters 30 (2002) 117 – 129

scenario (also known as the split- or staggered-delivery problem) refers to a situation where a replenishment order placed on a single supplier at a reorder point is delivered in a number of shipments which may or may not have the same size. Both scenarios have been extensively studied in the literature. Our review of the relevant literature starts with references that pertain to lost-sales continuous-review inventory models with positive lead times in which more than one replenishment order may be outstanding at any given point in time—that is, we focus on models which allow for placing new replenishment orders while awaiting the arrivals of orders that have already been triggered. Excluding the special case of base-stock or (S − 1; S) policy (which requires placing an order for the exact quantity of the satis7ed demand whenever a demand occurs), very little has appeared in the literature which incorporates the possibility of having more than one replenishment order outstanding into the general class of (s; Q) or (s; S) continuous-review models with lost sales and non-zero lead times. To our knowledge, the earliest work in this area is Morse [18] in which he derived the steady-state probabilities as well as some performance measures for a lost-sales (s; Q) inventory system with unit-sized Poisson demand and exponential lead times. Letting s ¿ Q in this model he allowed for more than one replenishment order to be outstanding at any time. More recently, Kalpakam and Arivarignan [12] presented a numerical procedure for computing the same type of measures for a similar model with unit demands described by a renewal process and lead times following an exponential distribution. Hill [9,10] also presented a steady-state analysis of an (s; Q) inventory system with unit Poisson demands and lost sales in which at most two orders are allowed to be outstanding. His analysis included both cases of constant and exponential lead times; but as noted by himself, involved a considerable amount of computation. By comparison, the literature on the issue of splitting orders is fairly rich. Most of the research in the area has been focussed on splitting orders under a dual- or multiple-sourcing scenario as described above (see [20,8,13,19,14,3,6,21,5] among others). Some of the main advantages of multiple sourcing commonly acknowledged in these studies include reduction in the safety stock, and savings in the holding and the shortage costs at the expense of a possible increase in ordering cost. Among those studies related to the split-delivery problem are Moinzadeh and Lee [17], Liao and Yang [15], Chiang and Chiang [4], and Janssen et al. [11]. Common advantages of the split-delivery systems, as noted in the literature, include lowering the safety stocks, savings in terms of the inventory holding, shortage and ordering costs while taking advantage of the quantity discounts, and facilitating the scheduling operations on the vendors’ side through sharing information about the future orders between customers and their suppliers. The main contribution of this paper is twofold: First, it extends the existing literature on lost-sales continuousreview inventory systems with unit-sized demand, positive lead time, and multiple replenishment orders to include the exact treatment of a case involving a non-unit-sized demand process with the possibility of many orders outstanding. Second, it incorporates the concept of delivery splitting into the analysis of lost-sales inventory systems with multiple reorder levels and variable lead times. More speci7cally, We employ a system-point (SP) method of level crossings [2] to derive the stationary distribution of the on-hand inventory for a continuous-review inventory system distinctly characterized by (i) non-unit-sized demand, (ii) multiple orders outstanding, (iii) split deliveries, (iv) variable lead times, and (v) lost sales. This probability distribution is then used to formulate a cost minimization problem. This paper is organized as follows. In Section 2 we provide a detailed description of the model. The level-crossing formulation of the problem follows in Section 3. In Section 4 we derive an exact expression for the long-run average cost per unit time, and present some numerical results. Section 5 contains our concluding remarks. 2. Model description Consider a continuous-review inventory system in which the demand process is described by a compound Poisson stream: Demands of random sizes arise according to a Poisson process with rate . Demand sizes are

E. Mohebbi, M.J.M. Posner / Operations Research Letters 30 (2002) 117 – 129

119

independent and identically distributed (i.i.d.) random variables, which for purposes of analytical simplicity in the current exposition are assumed to follow an exponential distribution with mean −1 . Our modeling approach, however, can also accommodate more general compounding distributions using numerical means of solution. All demands during stockouts—including the excess demand when the size of an order is larger than the stock-on-hand—are considered to be lost. The control policy is a special case of the well-known (s; nQ) policy. According to an (s; nQ) policy in a lost-sales environment whenever the inventory position (stock-on-hand + stock-on-order) declines to or below the reorder level s, an order consisting of a su8cient number of batches of size Q is placed so that the resulting inventory position immediately after ordering is between s and s + Q. Let n = s=Q denote the smallest integer greater than s=Q. Then, it should be clear that due to the lost-sales assumption, the largest possible size for a single replenishment order is nQ. Incorporating the concept of delivery splitting, we allow for each replenishment order of size iQ (i = 1; 2; : : : ; n) to be delivered in i batches of size Q. We also note that in analytical treatments of lost-sales inventory systems, it is often preferred to express the control policy in terms of the inventory level (stock-on-hand) since, unlike the backorder case, demands occurring while the system is out of stock have no reMection on the inventory position. Hence, an alternative description for the control policy in terms of the inventory level is as follows: The policy requires placing orders of size Q for every downcrossing of level s − iQ (i = 0; 1; 2; : : : ; n − 1) when tracing the inventory level process. This readily implies that when the inventory level is between max(0; s − iQ), and s − (i − 1)Q, there are i (i = 0; 1; : : : ; n) orders, each of size Q, outstanding. Clearly, when s ¡ Q (or equivalently, n = 1), the commonly used (s; Q) policy with a maximum of one order outstanding at any time is resulted. It should be pointed out that in addition to the analytical advantage of applying such a policy in our modeling process, we 7nd that in many practical situations quantized ordering is preferred where limitations exist in terms of packaging, transportation, and coordination. Furthermore, the (s; nQ)-type models are increasingly regarded as building blocks in modeling supply chains in which order quantities between various links must equal an integer multiple of a 7xed quantity. Finally, we assume that lead times are i.i.d. random variables following an exponential distribution with mean −1 . 3. Problem formulation Let {W (t); t ¿ 0} denote the inventory level at time t. Clearly, W (t) ∈ U ≡ [0; s + Q], and as mentioned earlier, when W (t) ∈ [max(0; s − iQ); s − (i − 1)Q], there are i (i = 0; 1; : : : ; n) orders outstanding at the time t, each of size Q. Furthermore, the memoryless property of exponentially distributed demand sizes implies that the W (t) process regenerates itself at epochs of downcrossing of level s − (n − 1)Q, and consequently, it is time stationary. We, therefore, write W = limt→∞ W (t) in distribution, and let f and F denote the associated stationary density and distribution functions, respectively. Our primary goal in this section is to derive the stationary distribution of the inventory level. To achieve this, we apply a level-crossing methodology, based on establishing a series of balance equations, by equating the rates at which sample function tracings of W enter and depart carefully chosen subsets of states from U . These equations, along with a proper form of the normalizing equation, are then solved to obtain the stationary density f. Recall that the control policy allows for up to i (i = 0; 1; : : : ; n) orders, each of size Q, to be outstanding at any time. Thus, in the long run, the evolution of the W process can be viewed as the aggregation of processes Wi , with i = 0; 1; : : : ; n being a subset indicator for that portion of the sample function tracing of W for which there are i orders outstanding. This interpretation of the state space in the terminology of the level-crossing methodology is expressed by using the term page i. A typical sample function tracing of the inventory process is depicted in Fig. 1. Accordingly, F and f can also be partitioned into Fi and fi ; (i = 0; 1; : : : ; n), respectively. Hence, Fi (w) = Pr(Wi 6 w); is the probability of stock-on-hand not exceeding w when there are i orders outstanding, and fi (w) = Fi (w).

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E. Mohebbi, M.J.M. Posner / Operations Research Letters 30 (2002) 117 – 129

Fig. 1. A typical sample function tracing of the inventory process.

We now establish the balance equations, in the manner described before, for each and every page i (i = 0; 1; : : : ; n). Since the special case of n = 1 has already been studied in detail [16], here we focus on n ¿ 1. Considering the applicability of the PASTA property [22] under our set of assumptions, for page n, we have Wn ∈ [0; s − (n − 1)Q] with corresponding equations
s−(n−1)Q n−1
s−(i−1)Q  − e fn () d +  e− dFi () = n fn0 ; (1)  =0




s−(n−1)Q

=w

i=0

e

−(−w)

fn () d + 

=s−iQ

n−1
 i=0

s−(i−1)Q

=s−iQ

e−(−w) dFi () = n Fn (w);

w ∈ (0; s − (n − 1)Q]; (2)

where fn0 = dFn (0) = Pr(Wn = 0) denotes the probability mass associated with level 0 on page n. The left-hand side of (1) represents the SP entry rate into level 0 on page n due to demands whose sizes exceed the stock-on-hand, , on page n with  ∈ (0; s − (n − 1)Q], or on page i (i = 0; 1; : : : ; n − 1) with  ∈ (s − iQ; s − (i − 1)Q)]. The term n fn0 on the right-hand side of (1) refers to the departure rate from level 0 on page n as the result of receiving a replenishment order of size Q from among the n currently outstanding. Eq. (2) describes the equilibrium Mow rates for the interval [0; w) on page n. The 7rst term on the left-hand side of (2) represents the entrance rate into that interval from above on page n— caused by demands requesting more than  − w units, when  ∈ [w; s − (n − 1)Q]. The second term on the left-hand side of this equation represents the same type of entrance rate from page i (i = 0; 1; : : : ; n − 1), where  ∈ (s − iQ; s − (i − 1)Q], into the designated interval. The departure rate from the interval [0; w) on page n—due to arrivals of replenishment orders—is denoted by n Fn (w) on the right-hand side of (2).

E. Mohebbi, M.J.M. Posner / Operations Research Letters 30 (2002) 117 – 129

121

Turning our attention to page n − 1, where Wn−1 ∈ (s − (n − 1)Q; s − (n − 2)Q], we not only must consider the transition rate from page i (i = 0; 1; : : : ; n − 2) due to demands, but we must also incorporate the transition rate from page n as the result of replenishments. Thus, these equations can be formed as


s−(n−2)Q

[e

=w


=

−(−w)

−e

−(−s+(n−1)Q)

] dFn−1 () + 

n−2
 i=0

w

=s−(n−1)Q

s−(i−1)Q

=s−iQ

e−(−s+(n−1)Q) fn−1 () d + (n − 1) Fn−1 (w);

[e−(−w) − e−(−s+(n−1)Q) ] dFi ()

w ∈ (s − (n − 1)Q; Q);

Q ; n fn0 = [ + (n − 1) ]fn−1




s−(n−2)Q


=

n−2


=Q

e

s−(i−1)Q

=s−iQ

i=0 w

(4)

[e−(−w) − e−(−Q) ]fn−1 () d

=w

+

(3)

−(−Q)

[e−(−w) − e−(−Q) ] dFi () + n Fn (w − Q)


dFn−1 () + (n − 1)

w

=Q

dFn−1 ();

w ∈ (Q; s − (n − 2)Q]

(5)

Q with fn−1 = dFn−1 (Q) = Pr(Wn−1 = Q) corresponding to the probability mass at level Q on page n − 1. Eq. (3) is concerned with the interval (s − (n − 1)Q; w) on page n − 1. The 7rst (second) term on the left-hand side of (3) corresponds to the entrance rate into that interval due to demands—when the system is in state  ∈ (w; s − (n − 2)Q] ( ∈ (s − iQ; s − (i − 1)Q]) on page n − 1 (i = 0; 1; : : : ; n − 2)—whose sizes are just large enough to take the SP into the interval without downcrossing level s − (n − 1)Q. Note that there is no transition from page n into the designated interval on page n − 1. The right-hand side of (3) represents the departure rate from that interval as the result of demand occurrences (7rst term) or replenishments (second term). The equilibrium mass Mow balance for level Wn−1 = Q on page n − 1 is characterized by (4), in which n fn0 represents the transition rate from level Wn = 0 on page n into page n − 1 due to receipts of Q denotes the departure rate from this level resulting from arrivals replenishment orders, and [ + (n − 1) ]fn−1 of demands or replenishment orders. Eq. (5) carries the same concept as in (3) over to the interval [Q; w) on page n − 1 except that the term n Fn (w − Q) on the left-hand side of (5) refers to the transition rate from page n into that interval as the result of replenishments. Similarly, for each and every page i (i = 1; : : : ; n − 2) with Wi ∈ (s − iQ; s − (i − 1)Q] we obtain




s−(i−1)Q

=w

+

[e−(−w) − e−(−s+iQ) ] dFi ()

i−1
 j=0


=

w

=s−iQ

s−( j−1)Q

=s−jQ

[e−(−w) − e−(−s+iQ) ] dFj () + (i + 1) Fi+1 (w − Q)

e−(−s+iQ) fi () d + i Fi (w);

w ∈ (s − iQ; (n − i)Q);

(6)

122

E. Mohebbi, M.J.M. Posner / Operations Research Letters 30 (2002) 117 – 129 (n−i−1)Q (i + 1) fi+1 = ( + i )fi(n−i)Q ;




s−(i−1)Q

=w

+

[e−(−w) − e−(−(n−i)Q) ]fi () d

i−1
 j=0


=

(7)

s−( j−1)Q

=s−jQ

w

=(n−i)Q

[e

−(−w)

−e

−(−(n−i)Q)

e−(−(n−i)Q) dFi () + i





] dFj () + (i + 1)

w

=(n−i)Q

dFi ();

w−Q

=(n−i−1)Q

dFi+1 ()

w ∈ ((n − i)Q; s − (i − 1)Q];

(8)

where fi(n−i)Q = dFi ((n − i)Q) = Pr(Wi = (n − i)Q) denotes the mass probability associated with level (n − i)Q on page i. Observe that pursuing the same line of reasoning applied earlier, (6) – (8) describe the balancing condition for subsets of states characterized by (s − (i − 1)Q; w); Wi = (n − i)Q, and [(n − i)Q; w) on page i (i = 1; : : : ; n − 2), respectively. Finally, the balance equations for page 0 are
s+Q
w −(−w) −(−s)  [e −e ] dF0 () + F1 (w − Q) =  e−(−s) f0 () d; w ∈ (s; nQ); (9) =w

=s

f1(n−1)Q = f0nQ ;


s+Q

=w

=

[e




−(−w)

w

=nQ

(10)

−e

−(−nQ)


]f0 () d +

e−(−nQ) dF0 ();

w−Q

=(n−1)Q

w ∈ (nQ; s + Q]

dF1 () (11)

with f0nQ = dF0 (nQ) = Pr(W0 = nQ) representing the probability mass at level nQ on page 0. Clearly, (9) – (11) have the same structure as (6) – (8) except that there are no transitions from (into) page 0 into (from) other pages due to replenishment orders (demands) arrivals. The level-crossing formulation of the model is completed by the normalizing equation: n
s−(i−1)Q  dFi (w) = 1: (12) i=0

max(0; s−iQ)

The stationary distribution of the inventory level is the solution to the above system of equations. A detailed solution procedure is presented in the appendix, leading to the following forms (see the appendix for de7nitions and values of all the parameters):  n−1      (−1)n−j an−j en−j (w−(n−j)Q) + ca ; w ∈ (nQ; s + Q];    j=0 f0 (w) = (13) n−2       (−1)n−1−j bn−1−j en−1−j (w−(n−1−j)Q) + cb ; w ∈ (s; nQ);   j=0

E. Mohebbi, M.J.M. Posner / Operations Research Letters 30 (2002) 117 – 129

 n−i    n−j   Cn−j−1 (−1)n−j−i an−j en−j (w−(n−j−i)Q) ;     j=0      w ∈ ((n − i)Q; s − (i − 1)Q];

fi (w) =

i = 1; : : : ; n;

n−1−i   n−1−j     Cn−1−j−i (−1)n−1−j−i bn−1−j en−1−j (w−(n−1−j−i)Q) ;     j=0     w ∈ (s − iQ; (n − i)Q); i = 1; : : : ; n − 1;

Pr(W = 0) = fn0 ;

123

(14)

(15)

Pr[W = (n − i)Q] = fi(n−i)Q ;

i = 0; 1; : : : ; n − 1:

(16)

4. Cost function The stationary distribution of the inventory level can be readily used to formulate a variety of performance measures. Here, we focus on the average cost and let TCR(s; Q) denote the long-run average sum of ordering, holding, and shortage costs per unit time. Then, due to PASTA and the memoryless property of the exponential distribution, we can write TCR(s; Q) = Pord K + hE(IL) + $E(LS);

(17)

where K is the 7xed ordering cost; h the holding cost per unit per unit time; $ the shortage cost per unit of the lost sales; Pord the ordering probability (at a demand occurrence epoch); E(IL) the average inventory level; and E(LS) the average quantity of the lost sales (at a demand occurrence epoch). It can be readily veri7ed that n−1
s−(i−1)Q  Pord = e−(w−s+iQ) dFi (w); (18) w=s−iQ

i=0




E(IL) =

s+Q

w=0


E(LS) =

w dF(w);

s+Q




w=0

∞

x=0

(19)

xe−(w+x) d x dF(w):

Substituting (12) – (16) into (18) – (20) gives  n−1  jQ; s−(n−1−j)Q s (−1)n−j e−(n−j)Q an−j &− Pord = e Pn−j j=0

+

n−2  j=0

( j+1)Q (−1)n−1−j e−(n−1−j)Q bn−1−j &−s−(n−1−j)Q; Pn−1−j

s; nQ +cb &−1

 +

nQ; s+Q ca &−1

+ e(s−nQ) f0nQ

(20)

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E. Mohebbi, M.J.M. Posner / Operations Research Letters 30 (2002) 117 – 129

+

n−1 



 e

(s−iQ)

i=1

+

j=0

n−1−j ( j+1)Q Cn−1−j−i (−1)n−1−j−i e−(n−1−j−i)Q bn−1−j &−s−(n−1−j)Q; Pn−1−j



(s−nQ)

+e

n−1 

n−j s−(n−1−j)Q Cn−j−i (−1)n−j−i e−(n−j−i)Q an−j &−jQ;Pn−j



n−1−i  j=0

E(IL) =

n−i 

fi(n−i)Q

;

s+Q (−1)n−j e−n−j (n−j)Q an−j 'nQ; n−j

j=0

+

n−2 

nQ (−1)n−1−j e−n−1−j (n−1−j)Q bn−1−j 's;n−1−j

j=0

1 + {ca [s2 + 2sQ − (n2 − 1)Q2 ] + cb (n2 Q2 − s2 )} 2 +

n  n−i 

n−j s−(i−1)Q Cn−j−i (−1)n−j−i e−n−j (n−j−i)Q an−j '(n−i)Q; n−j

i=1 j=0

+

n−1 n−1−i   i=1

+

j=0

n−1 

n−1−j (n−i)Q Cn−1−j−i (−1)n−1−j−i e−n−1−j (n−1−j−i)Q bn−1−j 's−iQ; n−1−j

(n − i)Qfi(n−i)Q ;

i=0

 n−1 1  nQ; s+Q E(LS) = (−1)n−j e−n−j (n−j)Q an−j &− Pn−j  j=0

+

n−2  j=0

s; nQ (−1)n−1−j e−n−1−j (n−1−j)Q bn−1−j &− Pn−1−j

nQ; s+Q s; nQ + ca &−1 + cb &−1 +

+

n  n−i  i=1 j=0

+

e−(n−i)Q fi(n−i)Q + fn0

i=1

n−j (n−i)Q; s−(i−1)Q Cn−j−i (−1)n−j−i e−n−j (n−j−i)Q an−j &− Pn−j

n−1 n−1−i   i=1

n−1 

j=0

n−1−j s−iQ; (n−i)Q Cn−1−j−i (−1)n−1−j−i e−n−1−j (n−1−j−i)Q bn−1−j &− Pn−1−j

  

;

E. Mohebbi, M.J.M. Posner / Operations Research Letters 30 (2002) 117 – 129

125

Table 1 The e;ects of variations in system parameters on decision variables

Part (I): Variations in lead time  = 100; −1 = 50; K = 100; h = 1; $ = 8

Part (II): Variations in demand −1 = 50; −1 = 0:25; K = 100; h = 1; $ = 8

−1

s∗

Q∗

TCR(s∗ ; Q∗ )



s∗

Q∗

TCR(s∗ ; Q∗ )

0.13 0.17 0.25 0.50 0.67 1

1763.4 2135.6 2746.4 4520.4 5932.9 8150.0

756.8 819.1 760.0 1130.1 1483.2 2038.2

2456.1 2681.5 2959.1 3036.3 3613.7 4346.4

60 80 100 120 140 160

1783.0 2247.2 2746.4 3245.1 3744.2 4361.5

674.9 655.1 760.0 861.9 961.5 1087.9

2152.3 2535.9 2959.1 3371.7 3777.2 3826.8

Part (III): Variations in 7xed ordering cost  = 100; −1 = 50; −1 = 0:25; h = 1; $ = 8

Part (IV): Variations in unit shortage cost  = 100; −1 = 50; −1 = 0:25; K = 100; h = 1

K

s∗

Q∗

TCR(s∗ ; Q∗ )

60 80 100 120 140 160

2727.9 2732.6 2746.4 2762.3 2862.6 2868.9

682.0 719.5 760.0 797.8 1039.3 1077.0

2713.0 2825.3 2959.1 3086.1 3330.2 3423.5

$ 2 6 8 12 14 16

s∗ 2062.9 2616.0 2746.4 2926.1 2993.3 3051.0

Q∗ 888.2 753.0 760.0 768.1 770.7 772.7

TCR(s∗ ; Q∗ ) 2398.8 2838.5 2959.1 3127.4 3190.8 3245.5

where &*A; B =

'*A; B




=A


=

B

B

=A

e* d =

e

*

1 *B (e − e*A ); *

     1 1 1 *muB *A e −e : B− A− d = * * *

We note that due to the complex form of (17), application of calculus methods to establish the convexity of the cost functions in an analytical sense turns out to be impractical. Hence, the cost minimization process can be practically implemented using a nonlinear optimization software package. In this regard, we conducted local search to obtain optimal values s∗ and Q∗ for a number of sample problems which were set up to depict the e;ects of variations in system parameters on reorder level and order quantity. The results are summarized in Table 1. Several observations can be made throughout these results. In part (I) of the table, while the optimal reorder level s∗ and the associated average total cost rate both increase as −1 increases, the optimal order quantity Q∗ does not display such a monotonic pattern. This can be due to complex interactions that appear to exist among various components of the cost function. The same kind of increasing trend can be seen in parts (II) – (IV) of the table as the values of , K, and $ increase. However, the reorder level s∗ in parts (I), (II), and (IV) appears to be more sensitive than the order quantity Q∗ to increase in −1 ; , and $, whereas in part (III), variations in K seem to be mostly absorbed by Q∗ . This is intuitively justi7ed as the system responds to increase in the 7xed ordering cost by placing orders less frequently through increasing the lot size quantity.

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E. Mohebbi, M.J.M. Posner / Operations Research Letters 30 (2002) 117 – 129

5. Conclusions In this paper we studied a lost-sales continuous-review (s; nQ) inventory system with compound Poisson demand and multiple reorder levels. Moreover, we adopted a scenario for split deliveries by allowing each replenishment order to be delivered in batches of size Q. Assuming that lead times are independent and exponentially distributed, we devised a level-crossing methodology to obtain the stationary distribution of the inventory level and formulated an exact expression for the long-run average cost rate function for the case where demand sizes follow an exponential distribution. We note that invoking results from Botta et al. [1], our approach has the capability of being extended to more general classes of lead time distributions, while preserving the analytical attractiveness of the exponential distribution. The modeling procedure can also accommodate more general compounding distributions, albeit by noting that the only regenerative point of the inventory level process will then occur at level 0. The resulting model can be solved practically using numerical means of solution. Finally, it should be pointed out that although in this paper we only discussed the case where the excess demand is lost, the model can be easily modi7ed to include the case where a demand that cannot be fully satis7ed from stock on hand is considered to be totally lost. However, in that case, due to the fact that there will be no probability masses at levels 0 and (n − i)Q; i = 0; 1; : : : ; n − 1, a numerical solution to the balance equations should be considered. Appendix—solution procedure The system of equations in Section 3 can be solved in a standard way using di;erential operators to obtain the general functional forms of the density f in all the intervals. Once the general solutions are obtained, a su8cient set of linearly independent relations must be developed in order to determine the values of all constant terms and the point probability masses involved in the solution. Hence, let D ≡ d=dw be a di;erential operator. Then, applying D D −  to (2), (3), (5), (6), (8), (9), and (11) yields ( + n )Dfn (w) − n fn (w) = 0;

w ∈ (0; s − (n − 1)Q];

( + (n − 1) )Dfn−1 (w) − (n − 1) fn−1 (w) = 0;

w ∈ (s − (n − 1)Q; Q);

(A.1) (A.2)

( + (n − 1) )Dfn−1 (w) − n fn (w − Q) − (n − 1) fn−1 (w) + n fn (w − Q) = 0;

w ∈ (Q; s − (n − 2)Q];

(A.3)

( + i )Dfi (w) − (i + 1) Dfi+1 (w − Q) − i fi (w) + (i + 1) fi+1 (w − Q) = 0; w ∈ (s − iQ; (n − i)Q);

i = 1; : : : ; n − 2;

(A.4)

( + i )Dfi (w) − (i + 1) Dfi+1 (w − Q) − i fi (w) + (i + 1) fi+1 (w − Q) = 0; w ∈ ((n − i)Q; s − (i − 1)Q];

i = 1; : : : ; n − 2;

(A.5)

Df0 (w) − Df1 (w − Q) + f1 (w − Q) = 0;

w ∈ (s; nQ);

(A.6)

Df0 (w) − Df1 (w − Q) + f1 (w − Q) = 0;

w ∈ (nQ; s + Q]:

(A.7)

E. Mohebbi, M.J.M. Posner / Operations Research Letters 30 (2002) 117 – 129

127

Let i = i =( + i ) and set Pi = 1 − i ; i = 1; : : : ; n. The general solutions to (A.1) and (A.2) can then be expressed as fn (w) = an en w ;

w ∈ (0; s − (n − 1)Q];

fn−1 (w) = bn−1 en−1 w ;

(A.8)

w ∈ (s − (n − 1)Q; Q);

(A.9)

respectively, with an and bn−1 being arbitrary constants. Let ˆi = (i + 1) =( + i ); i = 1; : : : ; n − 1. Then, substituting (A.8) into (A.3) leads to Dfn−1 (w) − n−1 fn−1 (w) + ˆn−1 Pn an en (w−Q) = 0;

w ∈ (Q; s − (n − 2)Q];

which has a general solution of the form fn−1 (w) = −nan en (w−Q) + an−1 en−1 w ;

w ∈ (Q; s − (n − 2)Q):

(A.10)

Thus, starting from (A.1) – (A.3)—with corresponding solutions (A.8) – (A.10)—equations (A.4) and (A.5) can be expressed recursively for i = 1; : : : ; n − 2; in the forms  Dfi (w) − i fi (w) + ˆi   Pi+1 fi+1 (w − Q)

+1(i¡n−2)

n−3−i  j=0

 

n−2−i 





ˆn−1−r  Pn+1−j fn+1−j (w − (n − 1 − j − i)Q)  = 0;

w ∈ (s − iQ; (n − i)Q);

r=j+1



Dfi (w) − i fi (w) + ˆi   Pi+1 fi+1 (w − Q)

+

n−2−i 

 

j=0

n−1−i 





ˆn−r  Pn−j fn−j (w − (n − j − i)Q)  = 0;

w ∈ ((n − i)Q; s − (i − 1)Q];

r=j+1

which upon using induction arguments, lead to the following general solutions: fi (w) =

n−1−i  j=0

fi (w) =

n−i 

n−1−j Cn−1−j−i (−1)n−1−j−i bn−1−j en−1−j (w−(n−1−j−i)Q) ;

n−j Cn−j−i (−1)n−j−i an−j en−j (w−(n−j−i)Q) ;

w ∈ (s − iQ; (n − i)Q);

w ∈ ((n − i)Q; s − (i − 1)Q];

(A.11)

(A.12)

j=0

respectively, for i = 1; : : : ; n − 2, where CrN = N !=r!(N − r)!, and the indicator function 1(A) has the value of 1 whenever the condition A is satis7ed, and is 0 otherwise. Setting i = 1 in (A.11) and (A.12) gives f1 (w) =

n−2 

(n − j − 1)(−1)n−j bn−1−j en−1−j (w−(n−2−j)Q) ;

w ∈ (s − Q; (n − 1)Q);

(A.13)

j=0

f1 (w) =

n−1  j=0

(n − j)(−1)n−j−1 an−j en−j (w−(n−1−j)Q) ;

w ∈ ((n − 1)Q; s]:

(A.14)

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E. Mohebbi, M.J.M. Posner / Operations Research Letters 30 (2002) 117 – 129

Substituting (A.13) and (A.14) into (A.6) and (A.7), respectively, results in n−2

1   Df0 (w) + (−1)n−j (n − j − 1)Pn−1−j bn−1−j en−1−j (w−(n−1−j)Q) = 0; P1

w ∈ (s; nQ);

j=0

n−1

1   (−1)n−j−1 (n − j)Pn−j an−j en−j (w−(n−j)Q) = 0; Df0 (w) + P1

w ∈ (nQ; s + Q];

j=0

whose general solutions are: f0 (w) =

n−2 

(−1)n−1−j bn−1−j en−1−j (w−(n−1−j)Q) + cb ;

w ∈ (s; nQ);

(A.15)

j=0

f0 (w) =

n−1 

(−1)n−j an−j en−j (w−(n−j)Q) + ca ;

w ∈ (nQ; s + Q]:

(A.16)

j=0

Clearly, our solution includes 2n + 1 unknowns (i.e., a1 ; : : : ; an ; b1 ; : : : ; bn−1 ; ca ; cb ), which along with the n + 1 point mass probabilities (i.e., fi(n−i)Q ; i = 0; : : : ; n), must yet be determined. Thus, a system of 3n + 2 linearly independent relations is required to complete the solution procedure. In this respect, substituting the general functional forms (A.8), (A.9), (A.10), (A.11), (A.12), (A.15), and (A.16) into the original balance equations (2), (3), (5), (6), (8), (9), (11), and comparing the coe8cients of common exponential terms and constant terms in each equation results in the following 2n + 1 relations: an fn0 − = 0; (A.17) n  n−i  an−j

Pn−j

j=0

+

n−j Cn−j−i (−1)n−j−i e−(n−j−i)Q e−Pn−j jQ + e−(n−i)Q fi(n−i)Q

n−1−i  j=0

n−i  an−j

Pn−j

j=0

+

bn−1−j n−1−j C (−1)n−j−i e−(n−1−j−i)Q e−Pn−1−j ( j+1)Q = 0; Pn−1−j n−1−j−i

i = 1; : : : ; n − 1;

(A.18)

n−j Cn−j−i (−1)n−1−j−i e−(n−j−i)Q e−Pn−j (s−(n−1−j)Q) + 1(i=1) cb e−s

n−1−i  j=0

bn−1−j n−1−j C (−1)n−1−j−i e−(n−1−j−i)Q e−Pn−1−j (s−(n−1−j)Q) = 0; Pn−1−j n−1−j−i

i = 1; : : : ; n − 1; (A.19)

n−1 

(−1)n−j

j=0

+

n−2 

an−j −(n−j)Q −Pn−j jQ e e + e−nQ (−cb + ca + f0nQ ) Pn−j

(−1)n−j

j=0 n−1  j=0

(−1)n−1−j

bn−1−j −(n−1−j)Q −Pn−1−j ( j+1)Q e e = 0; Pn−1−j

an−j −(n−j)Q −Pn−j (s−(n−1−j)Q) e e − ca e−(s+Q) = 0: Pn−j

(A.20) (A.21)

E. Mohebbi, M.J.M. Posner / Operations Research Letters 30 (2002) 117 – 129

129

An additional set of n relations are provided by (4), (7), and (10). The 7nal relation is due to the normalizing equation (12), which, using the general functional forms, can be expressed as n  cb (nQ − s) + ca (s − (n − 1)Q) + fi(n−i)Q = 1: (A.22) i=0

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