A lost-sales continuous-review inventory system with emergency ordering

Int. J. Production Economics 58 (1999) 93—112

A lost-sales continuous-review inventory system with emergency ordering Esmail Mohebbi, Morton J.M. Posner* Department of Mechanical and Industrial Engineering, University of Toronto, 5 King+s College Road, Toronto Ont., Canada M5S 3G8 Received 29 October 1996; accepted 27 April 1998

Abstract While existing lost-sales continuous-review inventory models with a provision for emergency ordering under an (s, S) or (s, Q) policy assume that regular and emergency orders are placed at levels s and 0, respectively, we develop an exact model with non-unit-sized demands where orders of sizes Q and Q are placed at reorder levels s and s . Using     a level-crossing methodology, we derive the stationary distribution of the inventory level (stock on hand) and present explicit expressions for the average cost rate functions with/without a service level constraint. Some numerical results, in comparison with the case with no provision for emergency ordering, are presented.  1999 Elsevier Science B.V. All rights reserved. Keywords: Inventory systems; Multiple ordering levels; Emergency orders; Level-crossing methodology; Lost sales

1. Introduction It is often seen in practice that suppliers offer different means for supplying their goods and services at different costs. As such, customers have the option of choosing between a regular delivery (shipped, for instance, by a truck) or an emergency or a priority rushed delivery (shipped by airplane) for a higher charge. Hence, as far as inventory management is concerned, it may be beneficial to invoke a policy involving both regular and emergency orders — particularly when shortages may be very costly. Some real life applications of such a policy can be found in the highly competitive retail industry where a surge in demand for a specific product can lead to a sudden depletion of stock and, as a result, trigger an emergency replenishment order to avoid stockout. Another area of application includes certain classes of cyclic manufacturing systems (e.g., chemicals or steel) where high set-up costs force manufacturers to cluster their products, based on product formulations and/or production processes, into a number of product families which are scheduled for production periodically. Clearly, in such systems, since each product can only be run within its own family, ensuring the availability of raw material by the due date (possibly through emergency ordering, if necessary)

* Corresponding author. Fax: #1 416 978 3453; e-mail: [email protected]. 0925-5273/99/$ — see front matter  1999 Elsevier Science B.V. All rights reserved PII: S 0 9 2 5 - 5 2 7 3 ( 9 8 ) 0 0 2 0 8 - 4

94

E. Mohebbi, M.J.M. Posner/Int. J. Production Economics 58 (1999) 93–112

to support manufacturing of all items scheduled within the production run is of extreme importance; any failure in this regard can readily result in excessive set-up and/or shortage costs in the near future. Inventory systems with emergency orders have been the subject for a number of studies in the literature. In the context of continuous-review systems (see [1,2] for a review of periodic-review models), an early study by Morse [3] addressed an (s, Q) inventory system with Poisson demand and lost sales in which orders of sizes Q and s (Q's'0) are placed at levels s and zero, respectively. Assuming that lead times for each order size are identical and exponentially distributed, he constructed a Markov model of the process (where it would be non-Markovian in nature under the stated set of assumptions) and presented the steady state probabilities for the inventory level. He then used these results to work out a net profit function. However, there appears to be some confusion due to the lack of an adequate description of the order of replenishments when two orders (one of each size) are outstanding. Bhat [4] clarified the issue by adding the assumption that the emergency order of size s is always assumed to be delivered before the regular order of size Q — while lead times remain independent — to the Morse model. Moinzadeh and Nahmias [1] considered a policy that involves placing regular and emergency orders for quantities Q and Q at reorder levels s and s (s (s ), respectively.       Under the assumption of full backlogging of the excess demand, constant lead times of different lengths, and at most one order of each type outstanding, they developed an approximate expression for the average cost per unit time and presented a procedure for determining the policy parameters. In a later work, Moinzadeh and Schmidt [2] presented a steady state analysis of an (S!1, S) inventory system with unit Poisson demand, fixed lead times for regular and emergency orders, and multiple orders outstanding. Their analysis included both backorders and lost sales cases. Kalpakam and Sapna [5] studied an (s, S) inventory system with compound Poisson demands and lost sales in which two types of replenishment orders of sizes *(S!s) and )s are placed, either at epochs where the inventory level (on hand) falls below s or reaches zero, to raise the inventory position (on hand#on order) to S. Assuming no more than one order of each type outstanding at any time, they allowed replenishment rates for each of these orders to be chosen from a given set, and developed a semi-Markov decision model to determine the optimal rates that minimize the long-run expected cost rate. Elsewhere [6], they developed a lost sales (s, Q) model with unit-sized renewal demands which, in addition to placing a regular order of size Q (Q's) with exponentially distributed lead time at level s, allows for the possibility of an instant replenishment of size s with probability p (0)p)1) at the time of a stockout. Their analysis was further extended in [7] to a more general case where the mean replenishment rates for orders of sizes Q and s, placed at levels s (s'0) and zero, respectively, are assumed to be Markovian and depend on the number and type of the outstanding orders. Again, allowing for a maximum of one order of each size outstanding at any time, they formulated the long-run average cost rate function for the system. More recently, Moinzadeh and Aggarwal [8] extended the results in [2] to multiechelon inventory systems. Considering the above, it seems that incorporation of multiple supplying modes into the area of continuous-review inventory systems with lost sales — excluding the special case of (S!1, S) policies — has been limited to (s, Q) or (s, S) policies with regular and emergency orders being placed at reorder levels s and zero. This paper advances the treatment of existing lost-sales models with non-unit-sized demands to the more general case of determining the optimal values of the control policy parameters under an (s , Q , s , Q )     policy, as introduced by Moinzadeh and Nahmias [1], such that the average cost rate of the inventory system is minimized. We use a system-point (SP) method of level crossings [9,10] to obtain the stationary distribution of the inventory level in a lost-sales continuous-review inventory system with compound Poisson demand and nonidentical exponentially distributed lead times. This distribution will then be used to formulate two cost minimization models with/without a service level constraint. The rest of this paper is organized as follows: In Section 2 we present our mathematical model, with explicit expressions for the long-run average cost per unit time derived in Section 3. Section 4 displays a variety of numerical results, and some concluding remarks are given in Section 5.

E. Mohebbi, M.J.M. Posner/Int. J. Production Economics 58 (1999) 93–112

95

2. Mathematical model Consider a single-item inventory system where demands in random batch sizes occur according to a Poisson process with rate j ('0). Specifically, for the sake of analytical simplicity in the current exposition of the model, we take demand sizes to be exponentially distributed with mean k\. 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. Lead times for both regular and emergency orders are assumed to be independent and exponentially distributed with means p\ and p\, respectively. Normally, we expect p p .     The control policy requires placing a regular order of size Q and an emergency order of size Q at epochs   of inventory level downcrossing reorder levels s and s (0)s (s ), respectively — provided that there is no     order of the same type outstanding. This clearly implies that a maximum number of one order of each type can be outstanding at any point in time, assuming that 0)s (Q and s (Q . While the form of an     optimal control policy for this class of models remains an open question, our choice for the control policy mainly follows Moinzadeh and Nahmias [1] in light of the fact that (s, Q) policies are simple and easy to implement, and provide more analytical facility in our modeling process. Nevertheless, as far as overshooting the reorder level due to a random-sized demand is concerned, it should be pointed out that as long as there is no large overshoot of the reorder level, the application of (s, Q)-type policies is reasonably justified. However, in the case of a large overshoot, as noted by Hadley and Whitin [11], the order quantity can be set to s#Q rather than Q in the implementation of the control policy thereby avoiding severe stockout disruptions. Moreover, in some practical situations where there are limitations in terms of coordination, packing and transportation, quantized ordering policies are more likely to be adopted. We note that as far as the relationship between the order quantities Q and Q are concerned, the following   four cases can be distinguished: Case 1: s #Q )s ,    Case 2: Q )s and s #Q 's ,      Case 3: Q 'Q 's ,    Case 4: Q *Q 's .    In this paper, we limit our description of the modeling process to Case 1. The other three cases can be treated in the same way with minor modifications (see [12] for details). Let +¼(t), t*0, denote the inventory level at time t. It is readily verified that for the above set of assumptions, ¼(t)3º,[0, s #Q), where Q"Q #Q . Moreover, ¼(t) can be characterized as a regen   erative process with its regenerative points defined at epochs, for instance, at which the inventory level reaches level Q from below. (Note that due to the compound Poisson demand process and independent exponentially distributed lead times, the time intervals between these epochs constitute a renewal process.) Thus, there exists a stationary distribution characterizing the limiting behavior of the inventory level process. We, therefore, write ¼"lim ¼(t) in distribution, and let f and F denote the corresponding stationary R density and distribution functions, respectively. Our primary goal in this section is to derive the stationary density f to enable cost optimization. To achieve this, we apply a level-crossing methodology. The methodology is based on establishing a series of balance equations by equating the rates at which the sample function tracings of ¼ enter and depart carefully chosen subsets of states from º. These equations are solved along with a proper form of a normalizing equation, resulting in f. According to the control policy, there may be zero or one regular order of size Q , or one emergency order  of size Q (although with a low probability of p /(p #p )), or two orders (one of each type) outstanding at    

96

E. Mohebbi, M.J.M. Posner/Int. J. Production Economics 58 (1999) 93–112

Fig. 1. A typical sample path of the inventory process.

any time. Hence, in the long run, the evolution of the ¼ process can be viewed as the aggregation of processes ¼ , ¼ , ¼ , and ¼ , each representing that portion of the SP tracing of ¼ corresponding to zero, one     regular, one emergency, and two outstanding order(s), respectively. This interpretation of the state space is addressed by the terms pages 0, (1, 1), (1, 2), and 2 in the terminology of the level-crossing methodology. A typical sample path of the inventory process for Case 1 is depicted in Fig. 1. Analogously, F and f can also be partitioned into F , F , F , F , and f , f , f , f , respectively.         We now establish the balance equations for each and every page, in the manner described before, starting with page 2, where ¼ 3[0, s ]. Invoking the PASTA property [13] we obtain:   Q

Q>/ j e\I?f (a) da#j e\I? dF (a)#j e\I? dF (a)#j e\I? dF (a)"(p #p ) f ,        ? ?Q ?Q ?Q



Q>/



Q>/





(1)

E. Mohebbi, M.J.M. Posner/Int. J. Production Economics 58 (1999) 93–112

97

Q>/ Q>/ eI?\Uf (a) da#j e\I?\U dF (a)#j e\I?\U dF (a)    ?U ?Q ?Q Q

j







Q>/ #j e\I?\U dF (a)"(p #p )F (w), w3(0, s ],      ?Q



Q

(2)

Q>/ [e\I?\U!e\I?\Q] f (a) da#j [e\I?\U!e\I?\Q] dF (a)   ?U ?Q

j





U

U

"j



e\I?\Qf (a) da#(p #p )   

?Q



f (a) da, w3(s , s ],   

(3)

?Q

where f "dF (0),Pr(¼ "0) denotes the probability mass at level 0 on page 2. The left-hand side (LHS)    of Eq. (1) represents the entry rate of the SP into level 0 due to demand occurrences while the system is in state a on page 2 with a3(0, s ] or on page (1, 1) with a3(s , s #Q ), or on page (1, 2) with a3(s , s #Q ),        or on page 0 with a3(s , s #Q), and whose sizes are large enough to clear the system out of its stock. On the   right-hand side (RHS) of Eq. (1), (p #p ) f  corresponds to the departure rate from level 0 as the result of    receiving either regular or emergency outstanding orders. Eq. (2) extends the same balance flow rate structure as in Eq. (1) to interval [0, w) on page 2. Similarly, the first and the second terms on the LHS of Eq. (3) describe the entrance rates into interval (s , w) on page 2 from state levels a above, a3[w, s ) on   page 2 and a3(s , s #Q ) on page (1, 2), respectively, due to demand arrivals whose sizes are just large    enough to take the SP into that interval without downcrossing level s . As before, the RHS of Eq. (3) refers to  the departure rate of the SP from the designated interval as the result of demand arrivals (i.e., the first term) or replenishments of type 1 or 2 (the second term). Note that since s (s , there are no transitions from pages   (1, 1) and 0 into interval (s , w) on page 2.  In turning our attention to page (1, 2), where ¼ 3(s , s #Q ), it should be kept in mind that the     assumption of p p makes the possibility of observing this portion of the sample function tracings very   unlikely in practice. However, from a theoretical point of view, we have Q>/ U  j [e\I?\U!e\I?\Q ] dF (a)"j e\I?\Qf (a) da#p F (w), w3(s , Q ),       ?U ?Q

(4)

p f "(j#p ) f /,    

(5)





Q>/ j [e\I?\U!e\I?\/] f (a) da#p F (w!Q )     ?U



U



U



e\I?\/  dF (a)#p dF (a), w3(Q , s #Q ],       ?/ ?Q

"j



(6)

98

E. Mohebbi, M.J.M. Posner/Int. J. Production Economics 58 (1999) 93–112

U\/ Q>/ [e\I?\U!e\I?\Q\/] f (a) da#p j f (a) da    ?U ?Q





U

U



"j



e\I?\Q\/f (a) da#p  

?Q>/

f (a) da, w3(s #Q , s #Q ),     

(7)

?Q>/

where f /"dF (Q ),Pr(¼ "Q ) represents the probability mass associated with level Q on page       (1, 2). Eq. (4) displays the equilibrium flow rates for the interval (s , w) on this page. Namely, while the entry  rate into this interval is only due to demands (LHS), the departure rate (RHS) consists of the arrival rates of demands and outstanding emergency orders. There are no transitions from other pages into this interval on page (1, 2) for the set of assumptions considered here. As for Eq. (5), the term p f  on the LHS denotes the   entry rate into level ¼ "Q due to regular replenishments when ¼ "0. On the RHS, (j#p ) f / refers      to the departure rate from this level resulting from the arrivals of demands or outstanding emergency orders. Eqs. (6) and (7) characterize the same concept as in Eq. (4) for intervals [Q , w) and (s #Q , w) on page (1, 2),    respectively. However, unlike Eq. (4), there are transitions from intervals [0, w!Q ) and (s , w!Q ) on    page 2 into these intervals, respectively. The terms p F (w!Q ) and p U\/f (a) da on the LHS of Eqs. (6)     ?Q  and (7), respectively, refer to such transitions caused by regular replenishments on page 2. Similarly, for page (1, 1) with ¼ 3(s , s #Q ), we obtain:     Q>/ Q>/ [e\I?\U!e\I?\Q] dF (a)#j [e\I?\U!e\I?\Q] dF (a) j   ?Q ?U





U

"j



e\I?\Qf (a) da#p F (w), w3(s , Q ),     

(8)

?Q p f "(j#p ) f /,    

(9)

Q>/ Q>/ [e\I?\U!e\I?\/] f (a) da#j [e\I?\U!e\I?\/] dF (a)#p F (w!Q ) j      ?U ?Q





U

U  " e\I?\/ dF (a)#p dF (a), w3(Q , s #Q ],         ?/ ?/







(10)

Q>/ Q>/ 1 j [e\I?\U!e\I?\Q\/] f (a) da#j [e\I?\U!e\I?\Q\/] dF (a) Q>/Q   ?U ?Q

 

U\/ #p







U



U





f (a) da"j e\I?\Q \/ f (a) da#p f (a) da, w3(s #Q , s ] ,        ?Q ?Q>/ ?Q>/ 



(11)

99

E. Mohebbi, M.J.M. Posner/Int. J. Production Economics 58 (1999) 93–112

U\/ Q>/ [e\I?\U!e\I?\Q] f (a) da#p j f (a) da    ?U ?Q\/





U

U

"j



e\I?\Qf (a) da#p  

?Q



f (a) da, w3(s , s #Q ),    

(12)

?Q

with f /"dF ( Q ),Pr(¼ "Q ), and 1 being 1 if condition A is true and 0 otherwise.       Also, letting f /"dF (Q),Pr(¼ "Q), and q"1 if s #Q (s #Q and q"0 if s #Q 's #Q ,            the balance equations for page 0 with ¼ 3(s , s #Q) can be characterized as follows:    U Q>/ [e\I?\U!e\I?\Q] dF (a)"j e\I?\Qf (a) da, w3(s , min(s #Q , s #Q )], j        ?U ?Q





(13)

Q>/ [e\I?\U!e\I?\ Q>/ Q>/] dF (a)#qp F (w!Q ) j 1 Q>/$Q>/     ?U

 

U



e\I?\ Q>/ Q>/f (a) da, #(1!q)p F (w!Q )"j     ? Q>/ Q>/



w3(min(s #Q , s #Q ), max(s #Q , s #Q )] ,        

(14)

Q>/ j [e\I?\U!e\I?\ Q>/ Q>/] dF (a)#p [F (w!Q ) 1 Q>/$Q>/     ?U

 

!qF (s #Q !Q )]#p [F (w!Q )!(1!q)F (s #Q !Q )]            U



"j



e\I?\ Q>/ Q>/f (a) da, w3(max(s #Q , s #Q ), Q) ,     

? Q>/ Q>/

(15)

Q>/ j 1 [e\I?\U!e\I?\0] dF (a)#p F (w!Q )#p F (w!Q ) Q>/Q>/0        ?U

 

U

"j





e\I?\0f (a) da, w3(R, Q) , 

?0 p f /p #p f /p "j f /,       

(16)

(17)

100

E. Mohebbi, M.J.M. Posner/Int. J. Production Economics 58 (1999) 93–112

U\/ Q>/ U\/ [e\I?\U!e\I?\/] f (a) da#p j dF (a)#p dF (a)      ?U ?/ ?/







U

"j



e\I?\/ dF (a), w3(Q, s #Q],  

(18)

?/ Q>/ U\/ U\/ 1 j [e\I?\U!e\I?\Q\/] f (a) da#p f (a) da#p f (a) da Q>/Q      ?U ?Q>/ ?Q>/

 



U



"j





e\I?\Q\/f (a) da, w3(s #Q, s #Q ] ,    

?1>/

(19)

Q>/ U\/ U\/   j [e\I?\U!e\I?\Q \/ ] f (a) da#p f (a) da#p f (a) da          ?U ?Q ?Q >/ \/







U



"j

e\I?\Q\/f (a) da, w3(s #Q , s #Q).    

(20)

?Q>/ Finally, the normalizing equation Q

Q>/ Q>/ Q>/ dF (w)# dF (w)# dF (w)# dF (w)"1     U UQ UQ UQ









(21)

completes our level-crossing formulation of the model. The above system of equations is solved in detail in Appendix A yielding the stationary distribution of the inventory level, which is summarized here as follows:



aeEIU, w3(0, s ],  f (w)"   aeEIU, w3(s , s ],   

(22)



w3(s , Q ), a eAIU,      #c IU, f (w)" !aeEIU\/ eA w3(Q , s #Q ],       !aeEIU\/#c eAIU, w3(s #Q , s #Q ),      

(23)

a eAIU,  !aeEIU\/#c eAIU,   f (w)"  !aeEIU\/#c eAIU,   !aeEIU\/#c eAIU,  

(24)



w3(s , Q ),   w3(Q , s #Q ],    w3(s #Q , s ],    w3(s , s #Q ),   

E. Mohebbi, M.J.M. Posner/Int. J. Production Economics 58 (1999) 93–112

101

f (w)" 



a w3(s , min(s #Q , s #Q )]       !qa eAIU\/!(1!q)a eAIU\/#c,    !a eAIU\/!a eAIU\/#c,    aeEIU\/!c eAIU\/!c eAIU\/#c,     aeEIU\/!c eAIU\/!c eAIU\/#c,     aeEIU\/!c eAIU\/!c eAIU\/#c,    

w3(min(s #Q , s #Q ), max(s #Q , s #Q )],         w3(max(s #Q , s #Q ), Q),     w3(Q, s #Q],  w3(s #Q, s #Q ],    w3(s #Q , s #Q),    (25)

a Pr(¼"0)"  , gk

(26)

ad Pr(¼"Q )"  ,  gk

(27)

ad Pr(¼"Q )"  ,  gk

(28)




c c a 1 Pr(¼"Q)"    1# . gN gk

(29)

(See Appendix A for values of all parameters used in the solution.)

3. Cost functions The stationary distribution of the inventory level derived in Section 2 can be used to obtain various measures in order to assess the performance of the system. One criterion commonly used in inventory control is the cost of operating the inventory system under a specified policy. In this regard, we consider two minimization problems: Problem 1

Problem 2,

min TCR(s , Q , s , Q )     s.t. 0)s (Q ,   s (s (Q ;   

min CR(s , Q , s , Q )     s.t. 0)s (Q ,   s (s (Q ,    0)SR(b;

where TCR(s , Q , s , Q ),long-run average sum of ordering, holding, and shortage costs per unit time,     CR(s , Q , s , Q ),long-run average sum of ordering and holding costs,     SR,stockout risk (probability of stockout at a demand occurrence), b,maximum allowable stockout risk in the system (0)b(1).

102

E. Mohebbi, M.J.M. Posner/Int. J. Production Economics 58 (1999) 93–112

Problem 2, in particular, with a service level constraint defined in the form of a maximum allowable stockout risk is of more interest in circumstances where it is difficult to obtain an estimate of shortage costs. Once again, invoking the PASTA property [13], the cost rate functions in the above two problems can be formulated as TCR(s , Q , s , Q )"j[(K #c Q )P #(K #c Q )P ]#hE(IL)#jnE(LS),            

(30)

CR(s , Q , s , Q )"j[(K #c Q )P #(K #c Q )P ]#hE(IL),            

(31)

where c " unit ordering cost for a regular (i"1) or emergency (i"2) type order, G K " fixed ordering cost for a regular (i"1) or emergency (i"2) type order, G h " holding cost per item per unit time, n " shortage cost per unit of lost sales, P " probability of placing a regular order (at a demand occurrence)  Q>/

"

P 

Q>/ e\IU\Q dF (w)# e\IU\Q dF (w),   UQ UQ





(32)

" probability of placing an emergency order (at a demand occurrence) Q>/ e\IU\Q dF (w)# e\IU\Q dF (w),   UQ UQ Q>/

"





(33)

E(IL) " average inventory level Q>/ "



w dF(w),

(34)

U E(LS) " average quantity of the lost sales  Q>/ "

 

kxe\IU>V dx dF(w),

(35)

V U using the memoryless property of the exponential distribution. Also, Q>/ SR" e\IU dF(w)"kE(LS).



(36)

U We impose no conditions on c and K , i"1, 2, although, it is reasonable to expect that G G c (c . Using Eqs. (22)—(29), explicit expressions for Eqs. (32)—(35) can be expressed in the following  

103

E. Mohebbi, M.J.M. Posner/Int. J. Production Economics 58 (1999) 93–112

forms:



1 eIQ e\I/ P " (1!e\I/)[a(e\EN IQ\)#a(e\EN IQ!e\EN IQ)]# (1!e\I/)+a (e\AN IQ!e\AN I/)     k gN cN  e\I/ #e\AN I/[c (1!e\AN IQ)#c (e\AN IQ!e\AN IQ)],# [a (e\AN I/!e\AN IQ)    cN  #c e\AN I/(e\AN IQ!1)#c (e\AN IQ!e\AN IQ>/)#c (e\AN IQ>/)!e\AN IQ)]    #a (e\IQ!e\I Q>/ Q>/)#c"e\IQ>/!e\IQ>/"#c(e\I Q>/ Q>/!e\I/)   



#ce\I/(1!e\IQ)#c(e\IQ>/!e\IQ>/)#ce\IQ(e\I/!e\I/)    #eIQ(e\I/f /#e\I/f /),  



eIQ e\I/ e\I/ (1!e\I/)[a(e\EN IQ!1)#a(e\EN IQ!e\EN IQ)]# +a (e\AN I/!e\AN IQ) P "     k gN cN  1 #e\AN I/[c (e\AN IQ!1)#c (e\AN IQ!e\AN IQ)],# (1!eI/)[a (e\AN IQ!e\AN I/)    cN  #c e\AN I/(1!e\AN IQ)#c (e\AN IQ>/!e\AN IQ)#c e\AN IQ(1!e\AN I/)]    #a (e\IQ!e\I Q>/Q>/)#c"e\IQ>/!e\IQ>/"#c(e\I Q>/Q>/!e\I/)    #ce\I/(1!e\IQ)#ce\I/(e\IQ>/!e\IQ)#ce\IQ(e\I/!e\I/)   



#eIQ(e\I/f /#e\I/f /),  



1 Q [a (eAIQ!eAI/)#c eAI/(1!eAIQ)#c eAI/(eAIQ!eAIQ)] E(IL)"    k c  Q # [a (eAIQ!eAI/)#c eAI/(1!eAIQ)#c (eAIQ>/!eAIQ)    c  #c eAIQ(1!eAI/)]#+a +[min(s #Q , s #Q )]!s,,#c"(s #Q )!(s #Q )"              #c+Q![max(s #Q , s #Q )],#cs (s #2Q)#c[(s #Q )!(s #Q)]           



#c(Q!Q#2s Q ) #Q f /#Q f /#Q f /,         



1 1 E(¸S)" (1#e\I/!e\I/!e\I/)[a(1!e\EN IQ)#a(e\EN IQ!e\EN IQ)]   k gN 1 # (1!e\I/)+a (e\AN IQ!eAN I/)#e\AN I/[c (1!e\AN IQ)#c (e\AN IQ!e\AN IQ)],    cN 

104

E. Mohebbi, M.J.M. Posner/Int. J. Production Economics 58 (1999) 93–112

1 # (1!e\I/)[a (e\AN IQ!e\AN I/)#c e\AN I/(1!e\AN IQ)#c (e\AN IQ>/!e\AN IQ)    cN  #c e\AN IQ(1!e\AN I/)]#a (e\IQ!e\I Q>/Q>/)#c"e\IQ>/!e\IQ>/"    #c(e\I Q>/Q>/!e\I/)#ce\I/(1!e\IQ)#e\I/[c(e\IQ>/!e\IQ)    1 #ce\IQ(1!e\I/)] # [e\I/f /#e\I/f /#e\I/f /#f ].      k



4. Numerical results Due to the complex form of Eqs. (30) and (31), investigating the convexity of the cost functions in an analytical sense turns out to be highly impractical. Thus, we pursued a four-dimensional numerical search — using the MATLAB optimization tool box — to find the optimal values of the control policy parameters. However, our experiments indicate the existence of local minima which in turn means any conclusions we make about the optimality of the obtained results are somewhat in doubt. Next, we set s "0 and Q "s    (the case considered in Refs. [3,7]), and conducted a two-dimensional search, leading to the same minimal point over a large grid of distant initial points. The results of our numerical experiments with the model are exhibited in Tables 1—7. Our primary goal in conducting these experiments — in addition to exploring the effects of variations in different parameters on system performance — was to investigate the improvement (if any) which might result from incorporating an emergency ordering policy in the performance of the inventory system. In this regard, we compare the results obtained from the present model with those derived from a model with only a regular mode of resupply under the same basic assumptions. (See Ref. [14] for details about the latter model.) Tables 1—5 contain the results for the special case with s "0 and Q "s , and Tables 6 and 7 correspond    to the general model with four decision variables s , Q , s , Q . Hence, the entries in the first five tables     consist of the optimal values of the control policy parameters — i.e., (s, Q), when no emergency orders are allowed, and (s , Q ), when an emergency mode of resupply is included in the system — and their associated   Table 1 The effect of variations in j on system performance j

n/h

Without emergency orders s

Q

With emergency orders TCR (s, Q)

s 

Q 

SV% TCR (s , Q , 0, s )   

20

10 20 30

2480.9 4052.7 4741.8

3030.0 4052.8 4741.9

13155.0 15287.0 16390.0

1132.5 2361.9 2940.7

3431.6 3460.4 3540.7

13230.0 14014.0 14470.0

N/A 8.33 11.72

50

10 20 30

6222.8 9933.3 11610.8

7312.2 9933.4 11610.9

32540.0 37731.0 40418.0

2553.4 5411.3 6786.2

8378.1 8384.9 8505.0

32529.0 34280.0 35298.0

0.03 9.15 12.67

100

10 20 30

12461.0 19732.9 23057.7

14443.0 19733.0 23057.8

64845.0 75135.0 80463.0

4880.0 10444.0 13145.0

16625.0 16598.0 16777.0

64672.0 68028.0 69977.0

0.27 9.46 13.03

105

E. Mohebbi, M.J.M. Posner/Int. J. Production Economics 58 (1999) 93–112 Table 2 The effect of variations in p on system performance  p 

n/h

Without emergency orders

With emergency orders

SV%

s

Q

TCR (s, Q)

s 

Q 

TCR (s , Q , 0, s )   

10

10 20 30

2480.9 4052.7 4741.8

3030.0 4052.8 4741.9

13155.0 15287.0 16390.0

1132.5 2361.9 2940.7

3431.6 3460.4 3540.7

13230.0 14014.0 14470.0

N/A 8.33 11.72

20

10 20 30

2480.9 4052.7 4741.8

3030.0 4052.8 4741.9

13155.0 15287.0 16390.0

1019.1 2074.3 2619.8

3418.4 3436.9 3465.3

13130.0 13787.0 14177.0

0.19 9.81 13.50

30

10 20 30

2480.9 4052.7 4741.8

3030.0 4052.8 4741.9

13155.0 15287.0 16390.0

970.3 1948.3 2465.8

3413.2 3429.8 3452.3

13088.0 13693.0 14055.0

0.51 10.43 14.25

Table 3 The effect of variations in K on system performance  K 

n/h

Without emergency orders

With emergency orders

SV%

s

Q

TCR (s, Q)

s 

Q 

TCR (s , Q , 0, s )   

80

10 20 30

2480.9 4052.7 4741.8

3030.0 4052.8 4741.9

13155.0 15287.0 16390.0

1116.1 2356.5 2937.6

3431.7 3460.0 3539.9

13221.0 14010.0 14467.0

N/A 8.35 11.73

90

10 20 30

2480.9 4052.7 4741.8

3030.0 4052.8 4741.9

13155.0 15287.0 16390.0

1124.4 2359.2 2939.2

3431.6 3460.2 3540.3

13225.0 14012.0 14468.0

N/A 8.34 11.73

100

10 20 30

2480.9 4052.7 4741.8

3030.0 4052.8 4741.9

13155.0 15287.0 16390.0

1132.5 2361.9 2940.7

3431.6 3460.4 3540.7

13230.0 14014.0 14470.0

N/A 8.33 11.72

cost rate values. As for the entries in Tables 6 and 7, due to the aforementioned difficulties encountered with a four-dimensional numerical search, the values obtained for (s , Q , s , Q ) do not necessarily represent the     optimal control policy parameters. However, these numbers, even though possibly suboptimal, can provide a clear understanding of how the addition of another mode of resupply can improve the performance of the system. Furthermore, it should be pointed out that while the experiments in Tables 1—4 and 6 were all conducted in the context of Problem 1, Tables 5 and 7 exhibit the results for Problem 2 — involving a constraint on the stockout risk. Under a basic parameter setting of j"20, k\"100, p\"1, p\"0.1, K "90, K "100, c "4.5,      c "7.5, h"1, and n"10, the control policy parameters and the average cost rate are derived while  changing the values of a particular parameter of interest for investigative purpose. The percentage of savings (if any) obtained by including emergency orders in the system are denoted by SV%.

106

E. Mohebbi, M.J.M. Posner/Int. J. Production Economics 58 (1999) 93–112

Table 4 The effect of variations in c on system performance  c 

n/h

Without emergency orders

With emergency orders

SV%

s

Q

TCR (s, Q)

s 

Q 

TCR (s , Q , 0, s )   

4.5

10 20 30

2480.9 4052.7 4741.8

3030.0 4052.8 4741.9

13155.0 15287.0 16390.0

1433.1 2167.4 2604.8

1433.2 2167.5 2604.9

11230.0 12380.0 13069.0

14.63 19.02 20.26

5.5

10 20 30

2480.9 4052.7 4741.8

3030.0 4052.8 4741.9

13155.0 15287.0 16390.0

1403.2 2282.9 2746.3

2069.1 2541.5 2876.3

12021.0 12990.0 13581.0

8.62 15.03 17.14

7.5

10 20 30

2480.9 4052.7 4741.8

3030.0 4052.8 4741.9

13155.0 15287.0 16390.0

1132.5 2361.9 2940.7

3431.6 3460.4 3540.7

13230.0 14014.0 14470.0

N/A 8.33 11.72

Table 5 The effect of a service level constraint on system performance b

0.05 0.01 0.001

Without emergency orders

With emergency orders

SV%

s

Q

CR (s, Q)

s 

Q 

CR (s , Q , 0, s )   

4556.6 7093.5 11022.4

4556.7 7093.6 11022.5

13408.0 17560.0 23537.0

1457.2 3999.2 7580.0

3431.3 4056.4 7580.1

12252.0 14162.0 18613.0

8.62 19.35 20.92

The effects of variations in j, p\, K and c for n/h"10, 20, and 30 in the special case of (s , Q , 0, s ) are       exhibited in Tables 1—4, respectively. As expected, the average cost rate TCR(s , Q , s , Q ) increases with     higher values of these parameters. However, the effect of variations in p\ and c , for a given ratio of n/h,   seems to be more significant than that of j and K . Also, we see that variations in the n/h ratio itself has  a major effect on the system cost. Furthermore, these results show that the savings obtained by adding an emergency mode of resupply to the system vary from N/A (i.e., uneconomical) for n/h"10 to 10—20% for n/h"30. These observations clearly propose that in circumstances where the shortage cost parameter is fairly large (as compared to other cost parameters), emergency orders can result in a reasonable amount of cost savings. Table 5 presents the effect of imposing a service level constraint b on the average cost for two systems, with and without a provision for emergency orders. Evidently, all other things being equal, the system with emergency orders provides the same target service level at a lower cost. Naturally, we expect to see a similar performance (if not better) by the (s , Q , s , Q ) model when relaxing     the constraints of s "0 and Q "s . The results for the basic parameter setting with n/h"10 are presented    in Tables 6 and 7. A glance at the savings obtained by applying such a four parameter control policy — although the computation may correspond to a local minimum — in comparison with applying the optimal (s, Q) policy justifies our expectation. In other words, there is a good potential for cost savings in applying an (s , Q , s , Q ) policy rather than a simplified (s , Q , 0, s ) policy. These observations, along with the       

107

E. Mohebbi, M.J.M. Posner/Int. J. Production Economics 58 (1999) 93–112 Table 6 The effect of variations in different parameters on system performance Without emergency orders

With emergency orders

s

Q

TCR (s, Q)

s 

Q

j 20 50 100

2480.9 6222.8 12461.0

3030.0 7312.2 14443.0

13155.0 32540.0 64845.0

2321.6 5583.6 10992.0

p  10 20 30

2480.9 2480.9 2480.9

3030.0 3030.0 3030.0

13155.0 13155.0 13155.0

K  80 90 100

2480.9 2480.9 2480.9

3030.0 3030.0 3030.0

c  4.5 5.5 7.5

2480.9 2480.9 2480.9

n 10 20 30

2480.9 4052.7 4741.8

SV% s 

Q 

TCR (s , Q , s , Q )    

2664.3 6305.3 12345.0

183.6 528.3 1118.0

485.6 952.5 1660.0

12839.0 31455.0 62429.0

2.40 3.33 3.74

2321.6 2254.4 2222.2

2664.3 2618.3 2604.0

183.6 159.6 145.4

485.6 452.4 436.6

12839.0 12737.0 12695.0

2.40 3.18 3.45

13155.0 13155.0 13155.0

2310.6 2316.3 2321.6

2655.8 2660.0 2664.3

196.7 189.9 183.6

464.2 475.2 485.6

12823.0 12831.0 12839.0

2.52 2.46 2.40

3030.0 3030.0 3030.0

13155.0 13155.0 13155.0

1481.4 1595.2 2343.1

1548.5 1670.9 2744.1

667.5 462.7 182.2

813.7 719.9 510.6

10760.0 11697.0 12841.0

18.21 11.08 2.39

3030.0 4052.8 4741.9

13155.0 15287.0 16390.0

2321.6 2728.9 2827.5

2664.3 2729.9 2828.5

183.6 597.3 730.3

485.6 655.4 731.3

12839.0 13310.0 13486.0

2.40 12.93 17.72



Table 7 The effect of variations in different parameters on system performance b

0.05 0.01 0.001

Without emergency orders

With emergency orders

s

Q

CR (s, Q)

s 

Q

4556.6 7093.5 11022.4

4556.7 7093.6 11022.5

13408.0 17560.0 23537.0

2354.6 2758.3 3207.9

2667.5 2759.3 3208.9



SV% s 

Q 

CR (s , Q , s , Q )    

214.2 644.8 1148.4

506.4 644.9 1149.4

11841.1 12914.0 13795.0

11.61 26.46 41.39

complexity of the cost function, suggest that a heuristic approach toward the four-parameter control policy should be pursued. To summarize, our results indicate that in the presence of a relatively large stockout cost or a higher standard for the service level, incorporating an emergency mode of resupply can be a very economically attractive alternative.

108

E. Mohebbi, M.J.M. Posner/Int. J. Production Economics 58 (1999) 93–112

5. Conclusions While existing lost-sales models with a provision for emergency ordering assume that regular and emergency orders are placed at reorder levels s and 0, respectively, in this paper we developed an exact model for the more general (s , Q , s , Q ) policy with compound Poisson demands, and exponential lead times.     Using a level-crossing methodology, we derived the stationary distribution of the inventory level which was then used to formulate two cost minimization problems with and without a service level constraint. Due to the complex form of the cost functions, a four-dimensional numerical search appeared impractical — suggesting that perhaps a heuristic approach should be preferred. However, the special case of s "0 and Q "s    can be well investigated through a two-dimensional search. The results of our experiments with this model, as compared to a simple (s, Q) model with no emergency orders, show that the inclusion of an emergency mode of resupply can be very economical in circumstances involving a fairly large shortage cost or a high target service level. In addition to the above, it should be pointed out that invoking results from Botta et al. [15], 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 demand compounding distributions. The resulting model can then be solved practically using numerical means of solution. Finally, it is worth mentioning that although in this paper we only discussed the case where the excess demand is lost, the model can be easily modified to include the case where a demand that cannot be fully satisfied 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, Q , Q and Q, a numerical   solution to the balance equations would seem to be the most practical.

Acknowledgements The authors wish to thank the anonymous referee for the insightful comments and suggestions which have improved the presentation of this paper. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada through grant OGP0004734.

Appendix A: Solution procedure In order to solve the system of equations in Section 2, we follow a standard procedure: First, using differential operators the integral equations are transformed into a set of differential equations which can be readily solved to obtain the general functional forms of the density f in all intervals. Second, a sufficient set of linearly independent relations is developed in order to determine the values of all constant terms and point probability masses involved in the solution. Hence, we start by letting 1D2,1d/dw2, and applying 1D21D!k2 to Eqs. (2)—(4), Eqs. (6)—(8), Eqs. (10)—(12), Eqs. (14)—(16) and Eqs. (17)—(20) to obtain (j#p #p )Df (w)!(p #p )kf (w)"0,      

w3(0, s ], 

(A.1)

(j#p #p )Df (w)!(p #p )kf (w)"0,      

w3(s , s ],  

(A.2)

(j#p )Df (w)!p kf (w)"0,    

w3(s , Q ),  

(j#p )Df (w)!p kf (w)!p Df (w!Q )#p kf (w!Q )"0, w3(Q , s #Q ],             

(A.3) (A.4)

E. Mohebbi, M.J.M. Posner/Int. J. Production Economics 58 (1999) 93–112

109

(j#p )Df (w)!p kf (w)!p Df (w!Q )#p kf (w!Q )"0, w3(s #Q , s #Q ], (A.5)               (j#p )Df (w)!p kf (w)"0, w3(s , Q ), (A.6)       (j#p )Df (w)!p kf (w)!p Df (w!Q )#p kf (w!Q )"0, w3(Q , s #Q ], (A.7)              1 +(j#p )Df (w)!p kf (w)!p Df (w!Q )#p kf (w!Q )"0, w3(s #Q , s ],, (A.8) Q>/Q              (j#p )Df (w)!p kf (w)!p Df (w!Q )#p kf (w!Q )"0, w(s , s #Q ), (A.9)              1 +jDf (w)!q[p Df (w!Q )!p kf (w!Q )]!(1!q)[p Df (w!Q ) Q>/$Q>/           !p kf (w!Q )]"0, w3(min(s #Q , s #Q ), max(s #Q , s #Q )],, (A.10)            +jDf (w)!p Df (w!Q )!p Df (w!Q )#p kf (w!Q )#p kf (w!Q )"0, 1              Q>/$Q>/ w3(max(s #Q , s #Q ), Q),, (A.11)     1 +jDf (w)!p Df (w!Q )!p Df (w!Q )#p kf (w!Q )#p kf (w!Q )"0, Q>/Q>/0              w3(R, Q),, (A.12) jDf (w)!p Df (w!Q )!p Df (w!Q )#p kf (w!Q )#p kf (w!Q )"0,              w3(Q, s #Q], (A.13)  +jDf (w)!p Df (w!Q )!p Df (w!Q )#p kf (w!Q )#p kf (w!Q )"0, 1              Q>/Q w3(s #Q, s #Q ],, (A.14)    jDf (w)!p Df (w!Q )!p Df (w!Q )#p kf (w!Q )#p kf (w!Q )"0,              w3(s #Q , s #Q), (A.15)    respectively. Let g"(p #p )/(j#p #p ), gN "1!g, c "p /(j#p ), cN "1!c , i"1, 2,     G G G G G d "p /(j#p ), and d "p /(j#p ). Eqs. (A.1)—(A.3) and Eq. (A.6) can be easily solved resulting the       following general solution forms: f (w)"aeEIU, w3(0, s ],    f (w)"aeEIU, w3(s , s ],     f (w)"a eAIU, w3(s , Q ),     f (w)"a eAIU, w3(s , Q ).     Substituting Eqs. (A.16) and (A.17) into Eqs. (A.4), (A.5), (A.7)—(A.9) leads to Df (w)!c kf (w)#d gN kaeEIU\/"0, w3(Q , s #Q ],         Df (w)!c kf (w)#d gN kaeEIU\/"0, w3(s #Q , s #Q ),          Df (w)!c kf (w)#d gN kaeEIU\/"0, w3(Q , s #Q ],         1 +Df (w)!c kf (w)#d gN kaeEIU\/"0, w3(s #Q , s ],, Q>/Q         Df (w)!c kf (w)#d gN kaeEIU\/"0, w3(s , s #Q ],        

(A.16) (A.17) (A.18) (A.19)

110

E. Mohebbi, M.J.M. Posner/Int. J. Production Economics 58 (1999) 93–112

with general solutions of the forms f (w)"!aeEIU\/#c eAIU, w3(Q , s #Q ],      

(A.20)

f (w)"!aeEIU\/#c eAIU, w3(s #Q , s #Q ),       

(A.21)

f (w)"!aeEIU\/#c eAIU, w3(Q , s #Q ],      

(A.22)

f (w)"!aeEIU\/#c eAIU, w3(s #Q , s ],      

(A.23)

f (w)"!aeEIU\/#c eAIU, w3(s , s #Q ],      

(A.24)

respectively. Similarly, using Eqs. (A.18)—(A.24) in Eqs. (A.10)—(A.15) results in 1 +Df (w)#qc ka eAIU\/#(1!q)c ka eAIU\/"0,   Q>/$Q>/    w3(min(s #Q , s #Q ), max(s #Q , s #Q )],,         1 +Df (w)#c ka eAIU\/#c ka eAIU\/"0,   Q>/$Q>/   

w3(max(s #Q , s #Q ), Q),,    

1 +Df (w)#c ka eAIU\/#c ka eAIU\/"0, w3(R, Q),, Q>/Q>/0      Df (w)!gkaeEIU\/#c kc eAIU\/#c kc eAIU\/"0, w3(Q, s #Q],        1 +Df (w)!gkaeEIU\/#c kc eAIU\/#c kc eAIU\/"0, w3(s #Q, s #Q ],, Q>/Q          Df (w)!gkaeEIU\/#c kc eAIU\/#c kc eAIU\/"0, w3(s #Q , s #Q),          whose corresponding general solutions are f (w)"!qa eAIU\/!(1!q)a eAIU\/#c, w3(min(s #Q , s #Q ), max(s #Q , s #Q )],             (A.25) f (w)"!a eAIU\/!a eAIU\/#c,    

w3(max(s #Q , s #Q ), Q),    

(A.26)

f (w)"!a eAIU\/!a eAIU\O#c, w3(R, Q),    

(A.27)

f (w)"aeEIU\/!c eAIU\/!c eAIU\/#c, w3(Q, s #Q],      

(A.28)

f (w)"aeEIU\/!c eAIU\/!c eAIU\/#c, w3(s #Q, s #Q ],        

(A.29)

f (w)"aeEIU\/!c eAIU\/!c eAIU\/#c, w3(s #Q , s #Q).        

(A.30)

Furthermore, Eq. (13) can be rewritten as Q>/

Q>/ e\I?\U dF (a)" e\I?\Q dF (a) w3(s , min(s #Q , s #Q )]. (A.31)        ?U ?Q Note that the RHS of the above equation is a constant with respect to w. Thus, multiplying both sides of Eq. (A.31) by e\IU and applying 1D2 yields





f (w)"a , w3(s , min(s #Q , s #Q )].       

(A.32)

E. Mohebbi, M.J.M. Posner/Int. J. Production Economics 58 (1999) 93–112

111

As mentioned earlier, we need to develop a sufficient number of linearly independent relations through which the values of the constant terms a, a, a , c , c , a , c , c (only if s #Q (s ), c , a , c (only               if s #Q Os #Q ), c, c, c (only if s #Q (s ), and c along with the point mass probabilities            f , f /, f /, and f / can be determined. In this regard, Eqs. (5), (9) and (17) provide three useful relations. More     relations can be obtained by substituting the general solution (A.16)—(A.30) and (A.32) back into original balance equations and comparing the coefficients of eIU in Eqs. (2)—(4), Eqs. (6)—(8), Eqs. (10)—(12), Eq. (13), +Eq. (15), Eq. (16),, Eqs. (18)—(20) as well as the constant terms in in Eq. (2) which (after some simplification) results in c  eAIQ>/!c"0,  cN 



1 Q>/Q

(A.33)



eAIQ (c !c )!c#c"0 ,     cN 

(A.34)

eEIQ (a!a)!c#1 c#1 (a #c)"0,    Q>/Q  Q>/Q   gN

(A.35)

a #c!c!k(f /#f /!f /)"0,      gN

(A.36)




q







a a  eAIQ#a !c #(1!q)  eAIQ#a !c     cN cN  

 

a a  eAIQ#  eAIQ#a !c "0, #1 Q>/Q>/0 cN   cN  

(A.37)

a c  eEIQ!  eAIQ>/"0, gN cN 

(A.38)

eEIQ eAIQ>/ (a!a)# (c !c )"0,     gN cN 

(A.39)

a eAI/ # (a !c )!kf /"0,    gN cN 

(A.40)

a c  eEIQ!  eAIQ>/"0, gN cN 

(A.41)





eAIQ 1 a # (c !c )"0 , Q>/Q    cN 

(A.42)

eAIQ>/ eEIQ (a!a)# (c !1 c !1 c )!1 a "0,    Q>/Q  Q>/Q  Q>/Q  gN cN 

(A.43)

a eAI/ # (a !c )!f /"0,    gN cN 

(A.44)

112

E. Mohebbi, M.J.M. Posner/Int. J. Production Economics 58 (1999) 93–112

a a  eEIQ!  eAIQ"0, gN cN  a eEIQ (a!a)#  eAIQ"0,   gN cN  a f !  "0.  gk

(A.45) (A.46) (A.47)

The last relation is due to the normalizing Eq. (21), which, using Eqs. (A.16)—(A.30) and (A.32) can be expressed as a [min(s #Q , s #Q )!s ]#c"(s #Q )!(s #Q )"#c[Q!max (s #Q , s #Q )]                 #cs #c(s !s !Q )#cQ #f #f /#f /#f /"1.           

(A.48)

References [1] K. Moinzadeh, S. Nahmias, A continuous-review model for an inventory system with two supply modes, Management Science 34 (1988) 761—773. [2] K. Moinzadeh, C.P. Schmidt, An (S!1, S) inventory system with emergency orders, Operations Research 39 (1991) 308—321. [3] P.M. Morse, Queues, Inventories and Maintenance, Wiley, New York, 1958. [4] U.N. Bhat, Elements of Applied Stochastic Processes, Wiley, New York, 1984. [5] S. Kalpakam, K.P. Sapna, A control policy of an inventory system with compound Poisson demand, Stochastic Analysis and Applications 11 (1993) 459—482. [6] S. Kalpakam, K.P. Sapna, A modified (s, S) inventory system with lost sales and emergency orders, Opsearch 30 (1993) 321—335. [7] S. Kalpakam, K.P. Sapna, A two reorder level inventory system with renewal demands, European Journal of Operational Research 84 (1995) 402—415. [8] K. Moinzadeh, P.K. Aggarwal, An information based multiechelon inventory system with emergency orders, Operations Research 45 (1997) 694—701. [9] P.H. Brill, M.J.M. Posner, Level crossings in point processes applied to queues: Single-server case, Operations Research 25 (1977) 626—647. [10] P.H. Brill, M.J.M. Posner, The system point method in exponential queues: a level crossing approach, Mathematics of Operations Research 6 (1981) 31—49. [11] G. Hadley, T.M. Whitin, Analysis of Inventory Systems, Prentice-Hall, Englewood Cliffs, NJ, 1963. [12] E. Mohebbi, M.J.M. Posner, Lost-sales continuous-review inventory systems with emergency ordering, Working paper C96-10, Department of Mechanical and Industrial Engineering, University of Toronto, 1996. [13] R.W. Wolff, Poisson arrivals see time averages, Operations Research 30 (1982) 223—231. [14] E. Mohebbi, M.J.M. Posner, A continuous-review inventory system with lost sales and variable lead time, Naval Research Logistics 45 (1998) 259—278. [15] R.F. Botta, C.M. Harris, W. Marchal, Characterization of generalized hyperexponential distribution functions, Stochastic Models 3 (1987) 115—148.