Math Meth Oper Res (2003) 57 : 329–344
A transfer/clearing inventory model under sporadic review Oded Berman1, Dmitry Krass1, David Perry2 1 Rotman School of Management, University of Toronto, 105 St. George Street, Toronto, Ontario, Canada M5S 3E6 (e-mail:
[email protected]) 2 Visiting Professor from Department of Statistics, University of Haifa, Israel 31905 Manuscript received: August 2002/Final version received: October 2002
Abstract. We consider two locations in tandem of an inventory model, socalled the bu¤er and the store. The content levels in both locations are controlled simultaneously by transferring inventory from the bu¤er location to the store location. Under a certain cost function that takes into account the trade-o¤ between the transferal set-up costs, the proportional holding and unsatisfied demand costs, the control policy of ‘‘when’’ and ‘‘how much’’ to transfer is restricted to the case of clearings under conditions of sporadic review. The objective is to find the optimal production rate or the optimal rate of the clearing process in heavy tra‰c in both locations. The purpose of this study is two-fold: we are interested both in the performance analysis of the relevant cost functionals and also in optimization and sensitivity analysis. Key words: Clearing; continuous, periodic and sporadic reviews: transferal setup costs; holding costs; heavy tra‰c; Martingale
1 Introduction We consider the heavy tra‰c version of a stochastic demand inventory system operating in two locations. The first location is a production facility, called the bu¤er, in which one machine produces continuously and uniformly a certain product. The controller also owns a retail facility called the store, at a di¤erent location (e.g., in a shopping center). The facilities face independent stochastic demand processes. The inventory level at the store is replenished from time to time by transfers from the bu¤er. We assume that backlogging is not allowed at both locations (i.e., demands that cannot be satisfied directly from inventory are lost), and that at each location there is a certain cost associated with lost sales. We assume that under conditions of heavy tra‰c and in the absence of
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any control, the content level of the bu¤er fluctuates as a reflected Brownian motion (RBM ) with drift mb and di¤usion parameter sb2 . Note that the content level is generated by reflection on the total production minus the total demand (satisfied or unsatisfied), with 0 acting as the reflecting barrier. It is assumed that returns can occur at the store (and, possibly, at the bu¤er as well), and thus during arbitrary intervals of times the demand minus returns in the store, or demand minus production in the bu¤er, can be positive or negative. In between positive jumps of the content level in the store (which are also the time intervals in between clearings in the bu¤er), the store’s content level is assumed to fluctuate as an RBM with drift ms and diffusion parameter ss2 . Assuming that the proportional holding costs in the bu¤er are more expensive than that at the store (or alternatively, that the sale prices and/or lost demand costs at the store are higher), there exists a strong motivation to transfer inventory from the bu¤er to the store. We assume that each incidence of transfer of inventory from the bu¤er to the store is associated with certain fixed transferal costs. The transfer policy is strongly influenced by these costs. For example, if the transferal costs are negligible, the bu¤er may be superfluous – and the production may be transferred immediately to the store. However, a more natural assumption is that the transferal cost is significant. Whenever this cost is su‰ciently high, a natural and intuitive management policy is to clear the bu¤er entirely at every transfer. In this study we restrict attention to ‘‘complete clearance policies’’. Thus, the controls available to the decisions maker are the production rate at the bu¤er location and the rate of inventory transfers from the bu¤er to the store. As noted above, the content level in the bu¤er is controlled as a clearing regenerative process such that between successive clearings it is an RBM. The content level in the store is a jump di¤usion process where each positive jump is the amount cleared from the bu¤er and transferred to the store. For a typical realization see Figure 1. By composing the given Brownian component together with the controlled compound jump component, the content level in the store fluctuates as a reflected jump di¤usion process whose law is determined by the rule of the control clearing process (see Remark 1 below). It is interesting to note that while increasing the clearing rate the expected content level in the bu¤er decreases (as expected). But, the e¤ect of the clearing rate on the content level in the store is not clear (see Section 5). Nevertheless, there exists a certain trade-o¤ between the proportional holding costs in the store and the transferal cost. In general, optimal control transfer policy should be determined according to the two criteria of ‘‘how much’’ and ‘‘when’’ to transfer. The ‘‘how much’’ part has already been fixed by assuming that each transfer is a clearing. The ‘‘when’’ criterion is determined according to the law between clearings. A natural control policy is a stationary control transferal policy (SCTP). That is, we assume that the compound clearing process is a certain compound renewal process. Specifically, let t be the generic interrenewal cycle between clearings. Three natural types of SCTP’s can be identified: (i), t ¼ tq for some content level q > 0, (ii), t ¼ t0 for some t0 > 0, and (iii), t ¼ tðlÞ @ expðlÞ, independent of the history of both content processes. The first rule t ¼ tq resembles a continuous review inventory policy. In this case the controller ‘‘sees’’ the content level of the bu¤er continuously over time. The production starts at 0 and the bu¤er is cleared whenever level q is reached at tq . The sec-
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331
Fig. 1. Content Levels at the Bu¤er and the Store Locations
ond rule t ¼ t0 corresponds to a periodic review system. Here, the bu¤er is cleared according to a pre-determined schedule t0 ; 2t0 ; 3t0 ; . . . : Under the third rule t ¼ tðlÞ, the bu¤er is cleared at random times according to the random visits of the controller. We call this type of SCTP a ‘‘sporadic review’’ policy. A complete randomness means that the controller clears the bu¤er in accordance with a Poisson process with rate l, ðtðlÞ @ expðlÞÞ, where l is a control parameter. We assume complete randomness for sporadic review policies in the remainder of the paper. The inventory system with only one location (i.e., the bu¤er) under the ‘‘continuous’’ and ‘‘periodic’’ review control policies t ¼ tq and t ¼ t0 , respectively, has already been considered in Berman and Perry (2001). The sporadic review control policy t ¼ tðlÞ, as well as the two-location inventory system under the heavy tra‰c conditions for all the above three SCTP’s have not been studied at all, to the best of our knowledge. While the analysis of Berman and Perry (2001) can be applied directly to analyze the content level of the bu¤er in our model, unfortunately it seems that the stochastic analysis of the store under continuous and periodic reviews is intractable for optimization since closed form expressions for the cost of this system are not available. It appears that the content level process generated in the store can be interpreted as the work of a GI =G=1-type queue with two types of customers (see e.g., Perry and Stadje, 1999). More precisely, it is that of the GI =D=1 queue, with D ¼ q, under the continuous review control policy, and that of the D=G=1-type queue with D ¼ t0 under the periodic review con-
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trol policy. Thus, closed-form expressions cannot be derived for these two cases. Surprisingly, under the sporadic review control policy t ¼ tðlÞ @ expðlÞ (or alternatively, if the bu¤er is cleared in accordance with a Poisson process with rate l) the work process generated in the store is a jump di¤usion process of the M=M=1-type queue (the formal proof will be given in Section 3). Namely, it is a reflected Le´vy process composed of a BM and a compound Poisson components with exponential (n) jumps where the rate n is formally given in Section 2. The heavy tra‰c approximations of such systems are generally modeled by a Brownian motion (BM) or by reflected BM (RBM) when negative inventory is not allowed. For examples and discussions of such models see Bather (1966), Puterman (1975), Browne and Zipkin (1991), Asmussen and Perry (1998), Perry (1997) and Perry and Stadje (1999, 2002). Because of the technical considerations discussed above, we generally confine our analysis to sporadic review policies in the remainder of the paper. We note that this class of policies may also be attractive under some circumstances for practical reasons quite unrelated to the technical ones. Note that the sporadic review policies require much less commitment from the controller (only the clearing rate needs to be specified), whereas the controller has to monitor the system continuously under continuous review policy, and commit to a rigid schedule in the periodic review case. Thus, sporadic review policies may be attractive in situations where the controller splits time between several subsystems. Another setting where sporadic review policies may be natural is when the opportunities for clearings arise infrequently (e.g., where partial car load shipments are used to transfer inventory, and the opportunities for shipments arise randomly, rather than according to some pre-fixed schedule). As mentioned earlier, in the bu¤er location the inventory content fluctuates due to changes in the volume of production, demand and return items. Inventory management with returning items are well studied in the literature related to recovery management problems under conditions of stationary demand, where returns are independent of previous sales. The reader can refer to the articles by Van der Laan et al. (1996) and Thierry et al. (1995). Dynamic lot size models with returning items and disposal are discussed by Beltran and Krass (2002) and by Beltran et al. (1998). In section 2 we describe the dynamics of the model in its two locations and define the cost functionals which lay the groundwork for optimization. In sections 3 and 4 the stochastic behavior of the content levels, both in the bu¤er and in the store, is analyzed. In section 5 we introduce two optimization problems; one dealing with choosing the optimal production rate given a known clearing rate, and the other dealing with selecting the optimal production rate given a known clearing rate. Numerical sensitivity analysis of the problems is also included in Section 5. 2 The model We consider the heavy tra‰c version of a production inventory (PI ) system of a firm operating in two locations. In both locations negative inventory is not allowed. In the first location (the bu¤er), a certain product is continuously produced, stored and demanded. The content level at time t is generated by the total amount produced minus the total amount demanded and satisfied up
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to time t. Sporadically, the bu¤er is cleared in accordance with a Poisson process of rate l, but in between clearings the content level fluctuates according to an RBM with drift mb (positive or negative) and variance sb2 . The RBM represents the heavy tra‰c approximation of the continuous time reflected random walk generated by the deterministic production minus the random demand (see e.g., Berman and Perry, 2001). At time t ¼ tðlÞ (tðlÞ @ expðlÞ independently of the history of the content level up to tðlÞ), the bu¤er is cleared and the entire inventory is transferred to another location (the store). The transfer of inventory from the bu¤er is motivated either by the fact that the sale price in the store is higher and/or the holding costs plus the unsatisfied demand costs in the bu¤er are lower. The clearing process in the bu¤er is a regenerative process because neither production nor demand never stop. We designate the content level process of the bu¤er by W ¼ fW ðtÞ : t b 0g. The total amount of clearings transferred from the bu¤er is presented as the batch input of process arriving in the store (as will be seen in the sequel, the compound process of these batches is a compound Poisson process whose jumps are exponential). There is no production in the store. The content level is generated by the batch arrivals transferred from the bu¤er minus random demands plus returns. The resulting content level process V ¼ fV ðtÞ : t b 0g is generated by reflection on the superposition between a ðms < 0; ss2 Þ BM and the compound Poisson process Y ¼ fY ðtÞ : t b 0g where Y ðtÞ ¼ S1 þ þ SNðtÞ , N ¼ fNðtÞ : t b 0g is a Poisson process with rate l and S1 ; S2 ; . . . are i:i:d:, random variables that represent the amount of the clearings transferred. Remark 1. As was already mentioned and will be seen in Theorem 1 below, Si is exponentially distributed ðnÞ where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mb2 2l mb n¼ þ sb4 sb2 sb2 is the absolute value of the negative root of the quadratic equation jðaÞ l ¼ a 2 sb2 =2 mb a l: Thus, V is a stable process if and only if l=n þ ms < 0. This implies that any optimization on the control parameter l is subject to the restriction l<
2ms ðms þ mb Þ : sb2
ð1Þ
Note that the latter condition for stability is independent of ss2 . In fact, if ss2 ¼ 0, V is the work process of the M=M=1 queue with arrival rate l and service mean 1=n. Also, ms < 0 implies that for stability mb < jms j. We assume that the controller incurs a set-up cost R for each clearing transferred. Also, she continuously incurs holding costs at a rate proportional to the content levels of W and V. The constants of proportionality (holding costs) being hb and hs ; respectively. The Processes Lb ¼ fLb ðtÞ : t b 0g and Ls ¼ fLs ðtÞ : t b 0g are the local time processes at 0, respectively (see e.g., Berman and Perry, 2001). Therefore, pb dLb ðtÞ and ps dLs ðtÞ represent the infinitesimal unsatisfied demand costs in the bu¤er and in the store, respectively.
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Given the processes Y; N; W; V; Lb and Ls ; an interest rate b, and constants R; hb ; hs ; pb ; ps , the expected cost of the system is: ðy ðy bt H¼E R e dNðtÞ þ ebt ðhb W ðtÞ þ hs V ðtÞÞ dt 0
þ pb
0
ðy
ebt dLb ðtÞ þ ps
ðy
0
ebt dLs ðtÞ :
ð2Þ
0
It is easy to verify that ð y e
E
bt
¼ l=b:
dNðtÞ
0
The clearing process W is a regenerative process with cycle tðlÞ. Therefore, (see Berman and Perry, 2001) ! ð y ð tðlÞ lþb bt bt E E e W ðtÞ dt ¼ e W ðtÞ dt b 0 0 and ð y E
e
bt
dLb ðtÞ
0
lþb E ¼ b
!
ð tðlÞ e
bt
dLb ðtÞ :
0
Given lsb2 < 2ms ðms þ mb Þ the inventory process V is a regenerative process by Remark 1. There are many possibilities for defining a cycle. For example, let V ð0Þ ¼ 0 and take Tl ¼ infft > tðlÞ : V ðtÞ ¼ 0g as an arbitrary cycle for V. Note that Tl is the time between two visits of state 0 under the restriction that each cycle includes at least one jump. The fact that V is a regenerative process enables us to define the expected values of the arguments in (2) in terms of cycle functionals. Thus, ÐT ð y Eð 0 l ebt V ðtÞ dtÞ bt e V ðtÞ dt ¼ E 1 yðbÞ 0 and ð y E 0
ebt dLs ðtÞ
¼
Eð
Ð Tl 0
ebt dLs ðtÞÞ 1 yðbÞ
where yðbÞ ¼ EðebTl Þ. According to the discussion above, the cost function (2) can be expressed in terms of the cycle functinals as follows (see e.g., Berman and Perry, 2001) H ¼ Rl=b þ hb A1 ðl þ bÞ=b þ pb A2 ðl þ bÞ=b þ hs A3 =ð1 yðbÞÞ þ ps A4 =ð1 yðbÞÞ
ð3Þ
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where !
ð tðlÞ
def
A1 ¼ E
e
bt
e
bt
W ðtÞ dt ;
ð4Þ
0
!
ð tðlÞ
def
A2 ¼ E
dLb ðtÞ ;
ð5Þ
0 def
A3 ¼ E
ð Tl
ebt V ðtÞ dt
ð6Þ
0
and def
A4 ¼ E
ð Tl e
bt
dLs ðtÞ :
ð7Þ
0
To compute the functionals A1 ; A2 ; A3 and A4 we use the martingale M ¼ fMðtÞ : t b 0g as will be discussed next. 3 The bu¤er The main tool of our analysis is the martingale M ¼ fMðtÞ : t b 0g (a variant of which has been used in Berman and Perry (2001), for the continuous and periodic control policies t ¼ tq and t ¼ t0 , respectively). If X ðtÞ is a Le´vy process with no negative jumps and exponent jðaÞ ¼ log EeaX ð1Þ , U ¼ fUðtÞ : t b 0g is an adapted process with bounded variation on finite intervals and ZðtÞ ¼ X ðtÞ þ UðtÞ, then the process M where MðtÞ ¼ jðaÞ
ðt
eaZðsÞ ds þ eaZð0Þ eaZðtÞ a
0
ðt
eaZðsÞ dUðsÞ
ð8Þ
0
is a martingale. By applying the optional sampling theorem (with the stopping time t) to the martingale M we get the fundamental identity ð t aZðsÞ e ds ¼ eaZð0Þ þ EeaZðtÞ jðaÞE 0
þ aE
ðt
eaZðsÞ dUðsÞ:
ð9Þ
0
The result given in the next theorem is known (see e.g., Asmussen and Perry, 1998). However, the proof introduced here is di¤erent and shorter. Theorem 1. Let t ¼ tðlÞ @ expðlÞ, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mb2 2l mb x¼ þ þ sb4 sb2 sb2
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and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mb2 2l mb n¼ þ : sb4 sb2 sb2 Then, Lb ðtðlÞÞ @ expðxÞ, W ðtðlÞÞ @ expðnÞ and Lb ðtðlÞÞ and W ðtðlÞÞ are independent. Proof. The content level process W is a regenerative process such that in between cycles fW ðtÞ : t a tðlÞg is a ðmb ; sb2 Þ RBM stopped at tðlÞ. The exponent associated with the content level of the bu¤er is jb ðaÞ ¼ ð1=2Þsb2 a 2 mb a: By definition W ðtÞ ¼ X ðtÞ þ Lb ðtÞ where X ðtÞ is a (mb ; sb2 ) BM. Setting UðtÞ ¼ Lb ðtÞð1 þ b=aÞ for some b > 0 in (9), we get ZðtÞ ¼ W ðtÞ þ ðb=aÞLb ðtÞ. Obviously, U is an adapted process with bounded variation on finite intervals so that the process jb ðaÞ
ðt
eaW ðsÞbLb ðsÞ ds þ 1 eaW ðtÞbLb ðtÞ
0
ða þ bÞ
ðt
eaW ðsÞbLb ðsÞ dLb ðsÞ
ð10Þ
0
is a martingale. Also, since Lb is a local time process W ¼ 0 as Lb increases, so that ðt
eaW ðsÞbLb ðsÞ dLb ðsÞ ¼
0
ðt
ebLb ðsÞ dLb ðsÞ:
0
Now apply the optional sampling theorem with the stopping time tðlÞ to (10) and recall that W ð0Þ ¼ Lb ð0Þ ¼ 0. We obtain by the optional sampling theorem (see e.g., Asmussen, 1987) ! ð tðlÞ
eaW ðsÞbLb ðsÞ ds
jb ðaÞE
¼ 1 þ EðeaW ðtðlÞÞbLb ðtðlÞÞ Þ
0
þ ða þ bÞE
!
ð tðlÞ e
bLb ðsÞ
dLb ðsÞ :
0
Let (W e ; Lbe ) be a pair of random variables whose joint law is the stationary distribution of the two dimensional regenerative process (W ðtÞ; Lb ðtÞ). We use the limit theorem for regenerative processes and PASTA (recall that tðlÞ @ expðlÞ), to obtain
A transfer/clearing inventory model under sporadic review
Eðe
aW ðtðlÞÞbLb ðtðlÞÞ
Þ ¼ ð1=EtðlÞÞE
337
!
ð tðlÞ e
aW ðsÞbLb ðsÞ
ds
0
¼ lE
!
ð tðlÞ e
aW ðsÞbLb ðsÞ
ds
0
or, after rearranging the terms ðjb ðaÞ lÞEðeaW ðtðlÞÞbLb ðtðlÞÞ Þ ¼ l þ lða þ bÞE
ð tðlÞ
! ebLb ðsÞ dLb ðsÞ :
ð11Þ
0
The right hand side of (11) is equal to 0 if and only if a ¼ x or a ¼ n because jb ðaÞ ¼ a 2 sb2 =2 mb a and x and n are the positive and the negative roots of the equation a 2 sb2 =2 mb a l ¼ 0; respectively. By inserting the positive root x in the left hand side of (11) we get ! ð tðlÞ
ebLb ðsÞ dLb ðsÞ
E
¼ 1=ðb þ xÞ:
0
Now factorize a 2 sb2 =2 mb a l ¼ ðsb2 =2Þða þ nÞða xÞ and use xn ¼ 2l=sb2 to obtain after some elementary algebra EðeaW ðtÞbLb ðtÞ Þ ¼
n x : nþa xþb
As n=ðn þ aÞ and x=ðx þ bÞ are the LTs of W ðtðlÞÞ and Lb ðtðlÞÞ, respectively, the proof is complete. 9 The next corollary follows immediately from Theorem 1. Corollary 1. (a), The successive amounts cleared from the bu¤er and transferred to the store are independent and exponentially ðnÞ distributed random variables. (b), The amounts of the unsatisfied demand in the bu¤er between successive clearings are independent and exponentially ðxÞ distributed random variables. (c), The i:i:d:, random variables in (a) and the i:i:d:, random variables in (b) are independent. To compute A1 in (4) we use another version of the martingale M. We set ZðtÞ ¼ W ðtÞ þ ðb=aÞt; then, UðtÞ ¼ W ðtÞð1 þ b=aÞ and clearly ðjb ðaÞ bÞ
ðt 0
eaW ðsÞbs ds þ eaW ð0Þ eaW ðtÞbt a
ðt 0
ebs dLb ðsÞ
ð12Þ
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is a martingale. By applying the optional sampling theorem with the stopping time tðlÞ to (12) we get ! ð tðlÞ
eaW ðsÞbs ds
ðjb ðaÞ bÞE
¼ 1 þ EðeaW ðtðlÞÞbtðlÞ Þ
0
þ aE
!
ð tðlÞ e
bs
dLb ðsÞ :
ð13Þ
0
By using again the limit theorem for regenerative processes and PASTA we get from (13) ! ð tðlÞ ðjb ðaÞ b lÞ aW ðtðlÞÞbtðlÞ bs Eðe Þ ¼ 1 þ aE e dLb ðsÞ : ð14Þ l 0 In a manner similar to that of Theorem 1 we define sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mb2 2ðl þ bÞ m b x~ ¼ 2 þ þ sb sb4 sb2 and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mb2 2ðl þ bÞ mb 2 n~ ¼ þ sb4 sb sb2 (x~ and ~ n are the positive and the negative roots of ðjb ðaÞ b lÞÞ to get A2 ¼ 1=x~:
ð15Þ
It follows from (14) and (15) that EðeaW ðtðlÞÞbtðlÞ Þ ¼
l n~ l þ b n~ þ a
ð16Þ
but Eðe
aW ðtðlÞÞbtðlÞ
Þ ¼ lE
!
ð tðlÞ e
aW ðsÞbs
ds
0
so that !
ð tðlÞ E
e
aW ðsÞbs
0
ds
¼
1 n~ : l þ b n~ þ a
ð17Þ
Taking derivative in (17) with respect to a and set a ¼ 0 we get A1 ¼
1 : ðl þ bÞ~ n
ð18Þ
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4 The store In this section we use the martingale ðjs ðaÞ bÞ
ðt
eaV ðuÞbu du þ 1 eaV ðtÞbt a
ðt
0
ebt dLs ðuÞ
0
in a manner similar to that of (12) for the analysis of the functionals A3 ¼ ÐT ÐT Eð 0 l ebt V ðtÞ dtÞ and A4 ¼ Eð 0 l ebt dLs ðtÞÞ. We assume that V ð0Þ ¼ 0 and lsb2 < 2ms ðms þ mb Þ which implies that V is a stable process. By theorem 1 the jumps of V are exponentially distributed (n) so that the exponent associated with V is n 2 2 js ðaÞ ¼ ð1=2Þss a ms a l 1 : ð19Þ nþa As mentioned, V is a regenerative process for which there are many possibilities to define a cycle. Arbitrarily, we define Tl ¼ inf ft > tðlÞ : V ðtÞ ¼ 0g as a generic cycle and generate the fundamental identity generated by the optional sampling theorem ðjs ðaÞ bÞE
ð Tl e
aV ðuÞbu
du
¼ 1 þ EðeaV ðTl ÞbTl Þ
0
þ aE
ð Tl e
bu
dLs ðuÞ :
ð20Þ
0
To compute yðbÞ ¼ EðebTl Þ we note first that V ðTl Þ ¼ 0 and that the equation js ðaÞ ¼ b has exactly three real roots (one positive and two negative), and designate the positive root by zðbÞ. By substituting a ¼ zðbÞ on the left hand side of (20) we get ð Tl bt e dLs ðtÞ : 1 ¼ yðbÞ þ zðbÞE 0
But, dLs ðtÞ ¼ 0 for all tðlÞ < t < Tl so that ! ð Tl ð tðlÞ bt bt e dLs ðtÞ ¼ E e dLs ðtÞ : E 0
0
By (15) the functional A4 ¼ E
!
ð tðlÞ e
bt
dLs ðtÞ
¼ 1=x~s ðbÞ
ð21Þ
0
where x~s ðbÞ is the positive root of js ðaÞ b l ¼ 0. We then have yðbÞ ¼ 1 zðbÞ=x~s ðbÞ:
ð22Þ
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O. Berman et al.
By substituting (22) and (21) in (20) we get ð Tl 1 þ yðbÞ þ a=x~s ðbÞ aV ðuÞbu : E e du ¼ ðjs ðaÞ bÞ 0 Now q 1 þ yðbÞ þ a=x~s ðbÞ A3 ¼ qa ðjs ðaÞ bÞ
!
a¼0
¼ ð1=b 2 Þðx~s ðbÞ½1 yðbÞðms þ l=nÞ b=x~s ðbÞÞ:
ð23Þ
We now have all the functionals needed for optimization and sensitivity analysis. 5 Optimization and sensitivity analysis We present two optimization problems. In the first problem ðP1 Þ, we assume that the clearing rate l is fixed and there is a constant production rate p at the bu¤er which is the decision variable. In the second problem ðP2 Þ, we assume that the production rate is fixed and thus, given a known mb , the decision variable is the clearing rate l. We start with Problem (P1 ). Denote mb ¼ p mb0 where mb0 is the demand drift (sb2 is the di¤usion parameter of the demand). From the stability condition (1) p a mb0 þ jms j
lsb2 : 2jms j
ð24Þ
Thus, problem (P1 ) can be formulated as follows: min HðpÞ ¼ ðRl þ ðp mb1 ÞA1 ðpÞðl þ bÞ þ pb A2 ð pÞðl þ bÞÞ=b þ ðhs A3 ð pÞ þ ps A4 ðpÞÞ=ð1 yðbÞÞ; s:t:
0 a p a mb1 þ jms j
lsb2 2jms j
ðP1 Þ
where Ai ðpÞ 1 Ai i ¼ 1; 2; 3; 4, are given in (18), (15), (23) and (21), respectively, as functions of p. In problem (P2 ), mb is known, and the problem is min HðlÞ ¼ ðRl þ mb A1 ðlÞðl þ bÞ þ pb A2 ðlÞðl þ bÞÞ=b þ ðhs A3 ðlÞ þ ps A4 ðlÞÞ=ð1 yðbÞÞ; s:t: 0 a l a
2ms ðms þ mb Þ sb2
where Ai ðlÞ 1 Ai i ¼ 1; 2; 3; 4, as functions of l.
ðP2 Þ
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Table 1. Problems 1–19, in all problems R ¼ 30, mb0 ¼ 5, ms ¼ 3, b ¼ 1, l ¼ 1 Problem
hb
hs
sb2
ss2
pb
ps
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
1 5 10 100 5 5 5 5 5 5 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 5 10 100 5 5 5 5 5 5
.5 .5 .5 .5 .5 .5 .5 .1 1 2 .5 .5 .5 .5 .5 .5 .5 .5 .5
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 .1 .5 2
10 10 10 10 1 5 100 10 10 10 10 10 10 10 10 10 10 10 10
10 10 10 10 10 10 10 10 10 10 10 10 10 1 5 100 10 10 10
Table 2. Problems 20–38, in all problems R ¼ 30, mb ¼ 2, ms ¼ 3, b ¼ 1 Problem
hb
hs
sb2
ss2
pb
ps
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
1 5 10 100 5 5 5 5 5 5 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 5 10 100 5 5 5 5 5 5
.5 .5 .5 .5 .5 .5 .5 .1 1 2 .5 .5 .5 .5 .5 .5 .5 .5 .5
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 .1 .5 2
10 10 10 10 1 5 100 10 10 10 10 10 10 10 10 10 10 10 10
10 10 10 10 10 10 10 10 10 10 10 10 10 1 5 100 10 10 10
The objective functions HðlÞ and Hð pÞ are awkward and complicated. Both of them include the functionals zðbÞ and xs ðbÞ, the positive roots, respectively, of a third degree polynomial. We use numerical methods to compute them and apply the uniform search method (see e.g., Bazaraa, Sherali and Shetty (1993)). It turns out (based on extensive computational analysis) that
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Table 3. Results for Problems 1–19 lþb b
Problem
p
A1
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19
7.917 6.9222 6.245 3.864 6.473 6.694 7.917 6.328 7.155 7.491 6.770 6.349 5.780 6.230 6.438 7.917 6.686 6.729 6.849
1.539 1.077 .787 .169 .878 .975 1.559 .845 1.274 1.565 1.009 .826 .598 .776 .812 1.539 .976 .990 1.044
A2
lþb b
.162 .232 .319 1.474 .284 .256 .162 .059 .392 .638 .247 .302 .417 .322 .283 .162 .256 .252 .239
A3 1yð bÞ
A4 1yð bÞ
Cost
2.555 1.865 1.548 1.898 1.634 1.739 2.555 1.651 2.078 2.431 1.779 1.585 1.457 1.543 1.638 2.555 1.227 1.479 2.346
2.037 2.175 2.331 2.989 2.272 2.221 2.037 2.262 2.118 2.051 2.205 2.303 2.477 2.335 2.269 2.037 2.055 2.124 2.351
56.056 61.329 65.882 93.492 59.033 60.111 76.889 59.098 63.555 67.329 64.438 72.735 205.28 44.051 53.257 249.31 60.231 62.157 68.687
both HðlÞ and HðpÞ are unimodal and convex. Therefore, we also solve the problems using dichotomous search (the same optimal solutions with the two search methods were always obtained). To show the sensitivity analysis of ðP1 Þ and ðP2 Þ we solved 19 problems for each. The various parameters selected for problems ðP1 Þ and ðP2 Þ are summarized in Tables 1 and 2, respectively. Note that in problems 1–4 and 20–23, hb changes; in problems 2, 5–7, 21, 24–26 pb changes; in problems 2, 8–10, 21, 27–29 sb2 changes; in problems 1, 11–13, 20, 30–32 hs changes; in problems 11, 14–16, 30, 33–35 ps changes; and in problems 11, 17–19, 30, 36–38 ss2 changes. In Table 3 [4] we present in column 1 the run number, in column 2 the optimal p ½l ; in columns 3 and 5 the expected average inventory in the bu¤er and the store, in columns 4 and 6 the expected lost demand in the bu¤er and the store. Finally, in column 7 the optimal cost is shown. The following conclusions can be drawn from the numerical example: 1. When hb or hs increases (runs 1–4 or 1, 11–13) p decreases and (runs 20–23 and 20, 30–32) l increases (Hðp Þ and Hðl Þ increases). 2. When pb or sb2 increases (runs 5, 6, 2, 7 or 8, 2, 9, 10), p increases and (runs 24, 25, 21, 26 or 27, 21, 28, 29) l decreases (Hð p Þ and Hðl Þ increase). 3. When (runs 14, 15, 11, 16) ps increases, p increases. However, in contrast to the authors’ intuition (runs 31, 34, 30, 35), l increases as well. Perhaps, this result can be explained by the fact that when l increases the expected lost demand in the store decreases since even though the expected average inventory decreases clearing occurs more often. 4. When (runs 17, 18, 11, 19) ss2 increase, p increases as well. However, in contrast to the authors’ intuition (runs 36, 37, 30, 38) l increases as well. This can be explained by the fact that as ss2 increases for a fixed l both the
A transfer/clearing inventory model under sporadic review
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Table 4. Results for problems 20–38 lþb b
Problem
l
A1
20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
.133 .189 .268 1.622 .200 .195 .107 .198 .181 .169 .385 .534 .878 .301 .338 1.231 .371 .377 .402
1.891 1.797 1.692 .871 1.782 1.789 1.923 1.693 1.913 2.114 1.559 1.418 1.177 1.652 1.609 1.007 1.574 1.567 1.542
A2
lþb b
.132 .139 .147 .286 .140 .1396 .129 .029 .261 .470 .160 .176 .212 .151 .155 .248 .158 .159 .162
A3 1yð bÞ
A4 1yð bÞ
Cost
4.592 3.999 3.381 3.368 3.903 3.946 4.930 3.906 4.092 4.235 2.768 2.302 1.917 3.178 2.979 2.005 2.190 2.453 3.363
2.854 2.764 2.659 1.959 2.748 2.755 2.902 2.752 2.776 2.792 2.535 2.413 2.214 2.621 2.581 2.075 2.400 2.462 2.670
40.355 47.741 56.440 160.70 46.458 47.018 59.789 46.144 49.479 52.557 53.924 66.365 243.58 30.729 41.133 257.97 49.254 51.376 58.738
expected lost demand and the expected inventory in the store increase. Therefore, increasing l, that has an opposite e¤ect on the expected lost demand, neutralizes somewhat the increase in the expected lost demand and expected inventory level. 5. When hb and/or hs increase, the expected content levels in both the bu¤er and the store decrease when optimizing either according to p or l . 6. When pb ½ps increases, the expected lost demand in the bu¤er [store] decreases when optimizing either according to p or l . 7. When sb2 ½ss2 increases, the expected content level and the expected lost demand both in the bu¤er and store increase when optimizing either according to p or l .
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