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Periodic review lost-sales inventory models with compound Poisson demand and constant lead times of any length
European Journal of Operational Research 220 (2012) 106–114
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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
Periodic review lost-sales inventory models with compound Poisson demand and constant lead times of any length Marco Bijvank a,⇑, Søren Glud Johansen b a b
Rotterdam School of Management, Erasmus University, Burgemeester Oudlaan 50, 3062 PA, Rotterdam, The Netherlands Department of Mathematics, Aarhus University, Ny Munkegade 118, DK-8000 Aarhus C, Denmark
a r t i c l e
i n f o
Article history: Received 29 March 2010 Accepted 20 January 2012 Available online 28 January 2012 Keywords: Inventory Lost sales Periodic review Base stock Heuristics
a b s t r a c t In almost all literature on inventory models with lost sales and periodic reviews the lead time is assumed to be either an integer multiple of or less than the review period. In a lot of practical settings such restrictions are not satisfied. We develop new models allowing constant lead times of any length when demand is compound Poisson. Besides an optimal policy, we consider pure and restricted base-stock policies under new lead time and cost circumstances. Based on our numerical results we conclude that the latter policy, which imposes a restriction on the maximum order size, performs almost as well as the optimal policy. We also propose an approximation procedure to determine the base-stock levels for both policies with closed-form expressions. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction In order to make a supply chain more flexible and responsive to customer requests in a cost-efficient manner, the involved parties have to cooperate and integrate activities. Usually a vendor determines the lead time experienced by a retailer, whereas the retailer has to decide on the replenishment policy; i.e., the frequency to review the inventory levels and the amount of items to order from the vendor at these review instants to anticipate on the stochastic demand during the lead time. Once the retailer has decided on both aspects, this information can be shared with the supplier(s) to anticipate on future order moments and the order size distribution. Ultimately, this can result in improvements for the entire supply chain. Traditionally, multi-echelon systems are used to model inventory control in the supply chain (see, e.g., Graves, 1996; Gallego and Özer, 2003; Parker and Kapuscinski, 2004; Gürbüz et al., 2007). In such models it is (almost) always assumed that excess demand is backordered. However, due to the increased competition in the retail environment, customers are not willing to wait anymore and excess demand is lost in many practical settings. Since the optimal replenishment policy for lost-sales inventory systems cannot be characterized by an easy-to-compute policy, this is interesting to investigate (Zipkin, 2008b; Bijvank and Vis, 2011). The goal of this paper is to develop and compare mathematical models for lost-sales inventory systems with different replenishment policies. The inventory models should be able to vary the ⇑ Corresponding author. Tel.: +31 104082199. E-mail address: [email protected] (M. Bijvank). 0377-2217/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2012.01.041
review period length in relation to the lead time to study the impact it has on the performance measures such as the expected total costs and service level. In real life most suppliers dispatch their delivery trucks to retailers on a regular basis based on the order frequency of the retailer. The fixed cost incurred for each order is shared among all items ordered at the same supplier. As a result, the fixed order cost for each item is negligible. The total order costs over a time period only depend on the order frequency. Longer review periods result in lower order costs compared to the costs over a similar time period with shorter review periods. Besides order costs, the total costs for a retailer consist of holding costs for keeping units on stock and penalty costs when a customer demand is not satisfied. The influence of the review period length on these costs is not as trivial as for the order costs, since they depend on the replenishment policy. For base-stock policies, Sezen (2006) illustrates by simulation the impact of the review period length on the average on-hand inventory levels and the fill rate in settings with fractional lead times (i.e., lead times shorter than a review period). The numerical results show that the variability in the demand process is the most important factor to set the duration of a review period. No costs are considered, and no analytical procedure is proposed to determine the length of a review period or on how to set the base-stock level. In order to study the impact of the review period length, no assumptions should be imposed on the relationship between the lead time and the review interval. We are the first authors, to our knowledge, to develop such models when excess demand is lost. In the literature, it is assumed that either the lead time is an integral multiple of the review period (e.g., Zipkin, 2008a;
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Huh et al., 2009 and the references therein) or a fraction of the review period (Kapalka et al., 1999; Chiang, 2006, 2007). In practice, the lead time is a constant and mainly determined by the supplier. The lead time should, however, also include the time for a retailer to transship the delivered items to the shelves. Therefore, the lead time is not strictly related to the review period length in real inventory systems. To relax the previously mentioned assumptions on the lead time, the objective function should be adjusted accordingly. In models where the lead time is an integer multiple of the review period the holding costs are charged after the demand in a review period has occurred (Zipkin, 2000). In other words, demand is assumed to occur periodically at the beginning of such a period and the holding costs are charged over the average on-hand inventory level (which remains constant over the review period), or equivalently, demand occurs at any point in time and the holding cost driver is the ending inventory of a review period. As mentioned by Rosling (2002) it is more realistic that demand arrives in continuous time, and the holding cost driver should be the average on-hand inventory level. Chiang (2006, 2007) divides the review period in smaller sub-periods, where the holding cost driver is the ending inventory of each sub-period. When such sub-periods are small enough, this approach approximates a system with holding costs charged over the average inventory. In this paper, we model the demand and costs to incur in continuous time without any sub-periods. This enables a retailer to compare the impact on the average costs when the review period length varies in relation to the lead time. To derive exact expressions for the average inventory levels we restrict the demand to be compound Poisson. Otherwise, approximation expressions should be used (Bijvank, 2009). We only consider the undiscounted setting, contrary to Chiang (2006, 2007) and Zipkin (2008a,b) who also include a discount rate. A second contribution of this paper is the study of the optimal replenishment policy as well as base-stock policies for the more general inventory models as discussed above. Since there is no fixed order cost per item, base-stock policies are commonly used in such situations. For a recent overview and comparison of different replenishment policies, we refer to Zipkin (2008a). He shows that pure base-stock policies (PBSPs) perform badly with respect to a cost minimization function, especially when the penalty cost for a lost-sales occurrence is low. However, Huh et al. (2009) show that the best PBSP becomes asymptotically optimal as the lostsales penalty increases. Johansen and Thorstenson (2008) introduce restricted base-stock policies (RBSPs), which impose an upper limit on the order sizes. In particular, we derive closed-form expressions that approximate the performance of PBSPs. Such approximations can be used to determine near-optimal values of the base-stock level. Furthermore, we propose a formula to set the maximum order size in the RBSP. This paper is organized as follows. In Section 2, we develop the inventory models for different replenishment policies under the new cost and lead time circumstances when the review period length is given. Besides the PBSP and RBSP, we also model the optimal replenishment policy. Even though these models give exact results, the computational effort can be quite excessive, especially for relatively long lead times. In Section 3 we derive closed-form expressions to approximate the performance measures of interest for lost-sales inventory systems with the PBSP, such as the fraction of demand lost, the average inventory on hand and the average costs. We also indicate how these approximations can be used to set the base-stock level and maximum order quantity in the RBSP. In Section 4 we compare the performance of both base-stock policies with the optimal costs. Based on our numerical results we conclude that the costs associated with the best base-stock level for the RBSP deviate on average less than 0.5% from the minimal costs, while the approximation procedure results in average
cost deviations of about 1–2% from the optimal costs. Section 5 contains our concluding remarks. 2. Different replenishment policies In order to model the base-stock and optimal replenishment policies we first introduce the general framework and notations in Section 2.1. The optimal replenishment policy is modeled as a Markov chain in Section 2.2. Only minor modifications are necessary to model the PBSP and RBSP in Section 2.3 and Section 2.4, respectively. 2.1. Notation and assumptions The time between two reviews is called a review period. Its length is denoted by R. At each review instant an order is issued. The order arrives after a constant lead time L. Let T denote the start of a considered review period, and t the time of the order delivery within the same period. Define r ¼ L mod R. Note that t = T + r. The number of full review periods from time T until the delivery of the order issued at time T equals l = (L r)/R. Hence, L = lR + r. Fig. 1 illustrates the notation based on an example where R 6 L < 2R (i.e., l = 1). The special case when the lead time is an integer multiple of the review period is described by Zipkin (2008a) and Johansen and Thorstenson (2008) (i.e., r = 0, l = L/R and t = T), whereas Chiang (2006, 2007) assumes fractional lead times (i.e., L < R and l = 0). Demand is assumed to be independent and identically distributed over time. We consider a Poisson customer arrival process with rate k. In practice it is common to observe that the customer demand size follows a delayed geometric distribution with mean l = 1/(1 h), where l P 1 (Johnston et al., 2003). We denote the total demand of n customers as a random variable Xn, where
PðX 1 ¼ dÞ ¼ ð1 hÞhd1 ;
d P 1;
ð1Þ
and more general
8 1 h; > > > > > > < hPðX n ¼ d 1Þ; PðX n ¼ dÞ ¼ hPðX n ¼ d 1Þ þ ð1 hÞPðX n1 ¼ d 1Þ; > > > > ð1 hÞPðX n1 ¼ d 1Þ; > > : 0;
n ¼ 1; d ¼ 1 n ¼ 1; d > 1 n ¼ 2;...;d 1 n¼d n > d:
For any s P 0, the demand during a time period of length s is a random variable Ds with probability mass function gs(d), where
8 ks > if d ¼ 0 PðX n ¼ dÞeks ðkn!sÞ ; if d > 0: : n¼1
In the special case of a pure Poisson demand process, l = 1 and P(Xn = n) = 1. We also define
G0s ðsÞ ¼ PrðDs < sÞ ¼
s1 X
g s ðdÞ;
d¼0 s X
G1s ðsÞ ¼ E½ðs Ds Þþ ¼
G0s ðdÞ;
d¼1
Fig. 1. An example of the time framework for reviews and order deliveries where R 6 L < 2R.
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with (A)+ = max{A, 0}. We denote the set of non-negative integers by N0 , and the set of all integers between m and n by Nm;n ¼ fi 2 N0 jm 6 i 6 ng ¼ fm; m þ 1; . . . ; ng. Furthermore, N1þl 0;s is defined as an (1 + l)-dimensional vector space over N0;s . This notation is used to model the inventory system as a Markov chain. Its state at time T (just after ordering) is denoted (i, y), where i is the actual inventory on hand and y is a vector with components yk ; k 2 N0;l . Component yk is the amount ordered at time T (l k)R to be delivered at time T (l k)R + L = t + kR (see also Fig. 1). In particular, yl can be represented as a function a(i, y0, . . . , yl1), which is prescribed by the replenishment policy. We let F(y) denote the vector obtained from y by removing its first component. The state of the Markov chain at time t is denoted (j, z), where j is the updated on-hand inventory level and z = F(y). We denote the actual demand from time T to time t and from t to T + R by dr and dRr, respectively. As a result, j = (i dr)+ + y0. To complete the Markov chain description, we have to specify the one-step transition probabilities between the different states of the inventory system. The transition probabilities from state (i, y) at time T to state (j, F(y)) at time t are denoted by P(i,y),(j,F(y)). By pðj;zÞ;ði;z;yl Þ we denote the transition probabilities from state (j, z) at time t to state (i, z, yl) at time T + R where yl is the amount ordered at this time. It is easily seen that
P ði;yÞ;ðidþy0 ;FðyÞÞ
8 if 0 6 d < i > < g r ðdÞ; ¼ 1 G0r ðiÞ; if d ¼ i > : 0; otherwise
and
pðj;zÞ;ðjd;z;aðjd;zÞÞ
8 if 0 6 d < j > < g Rr ðdÞ; ¼ 1 G0Rr ðjÞ; if d ¼ j > : 0; otherwise:
ð2Þ
ð3Þ
i1 i X i i1 þ Hði 1Þ HðiÞ ¼ þ PðX 1 ¼ jÞHði jÞ ¼ h k j¼1 k k
i1 X
g s ðjÞHði jÞ
ð4Þ
ð5Þ
j¼0
and the expected demand lost during such a period is
E½ðDs iÞþ ¼ E½Ds i þ E½ði Ds Þþ ¼ kls i þ G1s ðiÞ:
j1 X
g Rr ðdÞV n1 ðj d; z; aðj d; zÞÞ
d¼0
þ 1 G0Rr ðjÞ V n1 ð0; z; að0; zÞÞ; V n ði; yÞ ¼ cr ðiÞ þ
i1 X
ð7Þ
g r ðdÞv n ði d þ y0 ; FðyÞÞ
d¼0
ð8Þ
The notation introduced in this section for the different time instants of the Markov model is summarized in Table 1. The valueiteration algorithm (Tijms, 2003) with accuracy number e repeats to increase n by one and compute the value functions in Eqs. (7) and (8) until Mn mn < e, where
mn ¼ minfV n ði; yÞ V n1 ði; yÞg; ði;yÞ
M n ¼ maxfV n ði; yÞ V n1 ði; yÞg: ði;yÞ
When this value-iteration algorithm is stopped after the n-th iteration, then (mn + Mn)/2 cannot deviate more than 100e% from the long-run average costs per review period. The numerical results reported in Section 4 are computed with e = 1E4. 2.2. Optimal policy
because it takes on average 1/k time units before a customer arrives and (i X1)+ units remain on stock after this demand arrival. Note that the first equality in Eq. (4) is based on the memoryless property of the Poisson customer arrival process, whereas the other equalities are simplifications based on the delayed geometric demand size distribution. For other distributions, the first equality in Eq. (4) can be used to compute H(i) iteratively for i = 1, 2, . . .. However, it is straightforward to replace the (commonly observed) distribution in Eq. (1) by others. The mean time-weighted inventory during a period of length s for a given initial inventory level of i units is
Hs ðiÞ ¼ HðiÞ
v n ðj; zÞ ¼ cRr ðjÞ þ
þ 1 G0r ðiÞ v n ðy0 ; FðyÞÞ:
We want to compute the total expected costs over a review period, and express the expected costs incurred over a period of length s by cs(i) when the on-hand inventory level equals i at the beginning of this period and no order delivery occurs during this period. The cost function cs(i) consists of the expected holding and penalty costs. First, we let H(i) be the mean time-weighted stockholding until the inventory is fully depleted given an initial inventory level of i units. By definition H(0) = 0. For compound Poisson demand where the customer demands follow a delayed geometric distribution,
ði þ 1Þi ði 1Þi h ; ¼ 2k 2k
By h and p we denote the unit holding cost per unit time and the unit penalty cost for each lost demand. Consequently, cs(i) = hHs(i) + pE[(Ds i)+]. Notice, when h = 1 and p = 0 this expresses the average inventory, and when h = 0 and p = 1 this expresses the average demand lost. Furthermore, we note that previous papers compute the average holding costs as hE[(i Ds)+] (see, e.g., Chiang, 2006, 2007; Zipkin, 2008a), which is the unit holding costs after the demand has occurred in a period of length s instead of the average holding costs when demand occurs in continuous time. We can compute the performance measures of interest for this inventory system by value iteration. Our focus is on the long-run average costs. We let Vn(i, y) denote the total expected costs incurred over the time interval from time T to time T + nR when the system is in state (i, y) at time T and the system incurs no costs at and after time T + nR. Moreover, we let vn(j, z) denote the total expected costs incurred over the time interval from time t to time T + nR when the system is in state (j, z) at time t and the system incurs no costs at and after time T + nR. Consequently, V0(i, y) = 0 and, recursively for n = 1, 2, . . .
ð6Þ
To find an optimal policy and the corresponding costs for the inventory system described in Section 2.1, we model the system as a Markov chain with an infinite state space. More specifically, T 2þl the state space at time and at time t T equals S ¼ ði; yÞ 2 N0 t 1þl it is S ¼ ðj; zÞ 2 N0 . The value function Vn(i,y) at time T is specified by Eq. (8). At time t, the one-step transition probabilities depend on the entire state description. Therefore, we use the following relative value function at time t
Table 1 Notation to model the inventory system as a Markov chain at a review instant (time T) or order deliver (time t). Time T
Time t
State vector Inventory level Orders outstanding One-step transition probabilities
(i,y) i = (j DRr)+ y = (y0, y1, . . . , yl) P(i, y), (j, F(y))
(j,z) j = (i Dr)+ + y0 z = F(y) pðj;zÞ;ði;z;yl Þ
Value function
Vn(i, y)
vn(j, z)
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v n ðj; zÞ ¼ cRr ðjÞ þ
j X
and the state space at time t is
g Rr ðj iÞ minfV n1 ði; z; yl Þg
(
yl P0
i¼1
S ts
þ 1 G0Rr ðjÞ minfV n1 ð0; z; yl Þg:
¼
ðj; zÞ 2
N1þl 0;s j
yl P0
To illustrate the form of the optimal policy, we use an example in which the demand is pure Poisson with k = 5 and l = 1. Furthermore, L = 1.5R, h = 1, and p = 19. Let yl = a⁄(i, z) denote the optimal order quantity when the on-hand inventory level equals i units and the outstanding order quantities are z just before ordering at time T. Notice that in this example at most one order is outstanding at time T. Hence, z = y0 in this example. An optimal policy can be found with a value-iteration algorithm, and is provided in Table 2. The first column represents the on-hand inventory level i, whereas the first row represents the size of the outstanding order y0. This optimal policy results in the minimal costs C⁄ = 9.63 per review period and corresponds to a fill rate of 98.05%. The optimal policy of Table 2 is consistent with the properties derived in the literature. Chiang (2006) proves that the optimal order quantity is nonincreasing in the inventory position, and the related rate of decrease is less than one, which has also been observed for lost-sales inventory systems with continuous review (Johansen and Thorstenson, 1993, 1996). Zipkin (2008b) proves by induction that the optimal cost function is L-natural convex (i.e., convex, submodular and it contains a property related to diagonal-dominance). This implies that the optimal order quantities are monotone decreasing in the inventory position and the optimal order quantities are more sensitive to recent orders. 2.3. Pure base-stock policy (PBSP)
þ
l X
) zk 6 s :
ð10Þ
k¼1
The one-step transition probabilities from state ði; yÞ 2 S Ts at time T are given by Eq. (2), whereas the probabilities from state ðj; zÞ 2 S ts P at time t are expressed by Eq. (3) with aði; zÞ ¼ s i lk¼1 zk . Similarly, Eqs. (7) and (8) are used as backward recursion for all ðj; zÞ 2 S ts and all ði; yÞ 2 S Ts , respectively, to compute the expected total costs for a given value of base-stock level s. We use a bisection method without upper or lower bounds to find the best base stock, which we denote by s. It is our experience that the total costs are a quasi-convex function of the base-stock level. Notice that the convexity result proved by Downs et al. (2001) and Chiang (2006) does not include our inventory system in general. Consequently, previous results cannot be used directly. 2.4. Restricted base-stock policy (RBSP) As mentioned in Section 2.3, the PBSP and the optimal policy coincide in numerous states of the Markov chain. However, the rate of decrease of the optimal order quantity in the inventory position is less than one (see Section 2.2), whereas this rate equals one in the PBSP. Therefore, in the RBSP we include a maximum order quantity (the dark gray area in the example of Table 2). This policy with base stock s and upper limit q on the order sizes prescribes to issue a replenishment order at each review time T of size
( yl ¼ min s i
l1 X
) yk ; q :
k¼0
The PBSP with base-stock level s prescribes to issue a replenishment order at each review time T, such that the inventory position (inventory on hand plus inventory on order) equals base-stock leP vel s. Hence, yl ¼ s i l1 k¼0 yk . If we compare this policy to the optimal policy of the example discussed at the end of Section 2.2, a large number of states result in the same order quantities when s = 18 in this example (the light gray areas in Table 2). The state space of the Markov chain for the PBSP at time T is
( S Ts
¼
ði; yÞ 2
2þl N0;s i
þ
l X
) yk ¼ s
The state space of the Markov chain at time T is S Ts;q ¼ Ys;q ðiÞ, where
( y2
Ys;q ðiÞ ¼
k¼0
Table 2 The optimal order quantities a⁄(i, y0) when the on-hand inventory level equals i units and the outstanding order has size y0 at a review instant. The order quantities in the PBSP with s = 18 coincide with a large part of the optimal policy a⁄(i, y0) (light gray), whereas more quantities are included in the RBSP with s = 18 and q = 7 (dark gray).
k¼0
Zs;q ðjÞ ¼
z2
X l
Nl0;q
k¼1
i¼0 fig
) yk 6 s i ;
whereas the state space at time t equals S ts;q ¼ where
( ð9Þ
X l
N1þl 0;q
Ss
Ss
j¼0 fjg
Zs;q ðjÞ,
) zk 6 s j :
Eqs. (7) and (8) n o are used for the RBSP with aði; zÞ ¼ min P s i lk¼1 zk ; q to compute the total expected costs. Based on enumeration we can determine the best values of s and q. These optimal values are denoted by s⁄ and q⁄, respectively. 3. Approximation model As discussed in Section 2, the numerical computations of the performance measures can be performed by value iteration. This can, however, require quite a computational effort. As shown by Johansen and Thorstenson (2008), there are closed-form expressions to approximate the value of the performance measures for a lost-sales inventory system with a PBSP and geometric distributed demand per review period. We derive new closed-form expressions to approximate the performance measures for inventory systems controlled by a PBSP under the more general cost and lead time circumstances as described in Section 2. Based on these expressions, we can find the value of base-stock levels that minimize the approximate total costs. We extend this approach to the RBSP at the end of this section. For the PBSP, the inventory position at review time T is equal to base-stock level s. All the orders outstanding at that time arrive before or at time T + L. Orders issued after T arrive at time
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T + R + L or later. Therefore, we consider the time interval [T + L, T + R + L), which is called a shifted review period by Kapalka et al. (1999). In case of backordering, the net inventory at time s 2 [T + L, T + R + L) equals s minus the demand during the time interval from T to s. Because the net inventory cannot be negative in our lost-sales model, we correct the demand distribution with a factor ~cs . Therefore, we approximate the probability that the inventory on hand at time s 2 [T + L, T + R + L) equals i as
p~ s ðiÞ ¼
~cs g sT ðs iÞ; if 0 < i 6 s 1 ~cs GsT ðsÞ; if i ¼ 0;
eI þ ¼ s
~ TþL ðiÞ ¼ ~cs E½ðs DL Þþ ¼ ~cs G1L ðsÞ: ip
ð11Þ
i¼0
Similarly, the approximation for the expected inventory on hand ~ 1 just before the delivery at time T + R + L is eI s ¼ cs GLþR ðsÞ. Moreover, þ e e the average order size is approximated as I s I s . This amount represents the long-run average demand satisfied during a (shifted) review period. As a result, the approximation for the long-run fraction of demand lost is 1
1
e s ¼ 1 ~cs GL ðsÞ GLþR ðsÞ A klR
ð12Þ
and the approximate average inventory on hand is
eI s ¼ 1 R
s X
~cs ½g L ðs iÞ g LþR ðs iÞHðiÞ:
e Based on Little’s formula, we set it equal to eI þ s I s L=R and get
~cs ¼
s : ðl þ 1Þ G1L ðsÞ G1LþR ðsÞ þ G1ðlþ1ÞR ðsÞ
ð14Þ
The value of the base-stock level that minimizes the approximate expected total costs equals
where we derive the value of factor ~cs at the end of this section. Hence, the approximate expected inventory on hand at time T + L is s X
i Rrh r s ~cs G1LþRr ðsÞ þ s ~cs G1LþRr ðsÞ eI þs eI s R R
Rr 1 1 ~ GL ðsÞ G1LþR ðsÞ : ¼ s cs Gðlþ1ÞR ðsÞ þ R
ð13Þ
i¼1
The last aspect of the approximation procedure is to determine ~cs . We apply Little’s formula (Stidham, 1974) and equate the approximate average inventory on order with the approximate average amount ordered per unit time multiplied by L. Fig. 2 illustrates how the inventory position, inventory on hand and inventory on order change during the shifted review period. An order is placed at time T + L + R r = T + (l + 1)R. Consequently, the inventory on order during the interval [T + L, T + (l + 1)R) is constant, and it is also constant during the interval [T + (l + 1)R, T + L + R). Based on our approximations, the expected inventory on hand at review time T + L + R r is approximated as ~cs G1LþRr ðsÞ. Consequently, the average inventory on order during interval [T + L + R r, T + L + R) is approximated as s ~cs G1LþRr ðsÞ. The approximate average inventory on order during [T + L, T + L + R r) e is eI þ s I s units less (i.e., the approximate average order size). Therefore, the average inventory on order is approximated as
e s g; ~s ¼ argminfheI s þ pkl A sP0
where heI s approximates the expected holding costs per unit time e s approximates the expected costs (see Eq. (13) for eI s ) and pkl A e s ) for a given base due to lost sales in a unit time (see Eq. (12) for A stock s. Notice the resemblance between our approximation procedure for the PBSP in case of lost sales and the performance expressions in case of backordering (Zipkin, 2000). When ~cs is 1, the expressions derived in this section are exactly those as for the backorder model since no correction has to be made for lost sales. Furthermore, lims!1 ~cs ¼ 1 based upon this observation about the backorder model. In this section, we have discussed a procedure to set the value of base-stock level s based on closed-form expressions that approximate the long-run behavior of a periodic review inventory system with lost sales in case a PBSP is applied. The same base-stock level can be used for the RBSP. However, the value of q should also be specified. As indicated by Fig. 2, any order placed after time T is delivered at or after time T + L + R. Therefore, the inventory position at time T should cover the demand over L + R time units. The inventory position at time T is s, which also represents the maximum inventory position. Therefore, at most s/(L + R) units can be demanded per unit time in order to satisfy all the demand. This results in a maximum order quantity of sR/(L + R) units per review period. We propose to set q equal to this quantity rounded to the nearest integer and rounded up in case of a tie. The performance of these approximation procedures are analyzed in the next section. 4. Numerical results The goal of this section is threefold. First, we compare the performance of the different replenishment policies. Second, we compare different approaches to set suitable values of base-stock level s and maximum order size q. Third, we show how the models of Section 2 can be used to perform a sensitivity analysis on the review period length. 4.1. Comparison of replenishment policies
Fig. 2. The inventory position (solid line), inventory on hand (dashed line) and inventory on order (dotted line) for a specific time interval [T + L, T + L + R).
Besides the best PBSP with base-stock level s and the best RBSP with base-stock level s⁄ and maximum order size q⁄, we consider the optimal base-stock level sBO in case excess demand is backordered and the base-stock level ~s when the approximation procedure of Section 3 is used. Furthermore, we also investigate three other RBSPs with base stocks s set equal to ~s,s and sBO. Their max~; q and qBO, respectively) are set imum order sizes q (denoted q equal to sR/(L + R) rounded to the nearest integer (rounded up in case of a tie). We also report the average costs C⁄ per unit time for the optimal replenishment policy, which is computed according to Section 2.2. We illustrate the performance of the base-stock policies and optimal policy for a test bed where R = 1 and L/R is ranging from
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M. Bijvank, S.G. Johansen / European Journal of Operational Research 220 (2012) 106–114 Table 3 Results for pure and compound Poisson demand when the average demand per period is 5 and R 6 L < 2R. k, l
L
p
C⁄
~s
s
sBO
s⁄,q⁄
CI~s
CIs
CIsBO
CI~s;q~
CIs;q
CIsBO ;qBO
CIs ;q
5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1
1 1 1 1 1 1 1 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.75 1.75 1.75 1.75 1.75 1.75 1.75
4 9 19 39 59 79 99 4 9 19 39 59 79 99 4 9 19 39 59 79 99 4 9 19 39 59 79 99
6.19 7.79 9.10 10.30 10.91 11.39 11.67 6.27 7.96 9.41 10.64 11.34 11.76 12.16 6.35 8.12 9.63 10.99 11.68 12.18 12.52 6.42 8.27 9.87 11.27 12.03 12.53 12.93
11 13 15 16 17 17 17 12 14 16 17 18 18 19 13 15 17 18 19 20 20 14 16 18 20 21 22 22
11 13 15 16 17 17 18 12 15 16 18 19 19 19 13 16 18 19 20 21 21 15 17 19 21 22 22 23
12 14 15 17 17 18 18 14 15 17 18 19 19 20 15 17 18 20 21 21 21 16 18 20 21 22 23 23
12,5 13,7 15,7 16,9 17,9 17,9 18,9 13,5 15,6 16,7 18,8 19,8 19,9 19,9 14,5 16,6 18,7 19,8 20,8 21,8 21,9 15,5 17,6 19,7 21,7 22,8 22,8 23,8
1.74 0.95 0.62 0.27 0.25 0.18 1.89 3.38 2.45 0.94 1.94 0.75 3.19 0.21 4.72 3.71 2.33 4.26 3.19 1.01 2.79 5.01 4.56 3.58 1.80 1.09 0.41 0.85
1.74 0.95 0.62 0.27 0.25 0.18 0.21 3.38 2.27 0.94 0.67 0.68 0.31 0.21 4.72 2.42 1.48 0.61 0.48 0.61 0.34 4.98 2.47 1.30 0.95 0.85 0.41 0.64
3.07 1.98 0.62 1.56 0.25 0.56 0.21 8.65 2.27 1.81 0.67 0.68 0.31 0.68 8.47 4.72 1.48 1.65 2.28 0.61 0.34 7.73 3.78 2.81 0.95 0.85 1.81 0.64
0.36 0.17 0.16 0.08 0.07 0.07 1.85 0.80 1.34 0.23 1.66 0.61 3.36 0.32 0.98 2.07 1.47 4.70 3.01 0.89 2.92 1.14 2.85 2.74 1.72 0.76 0.11 0.74
0.36 0.17 0.16 0.08 0.07 0.07 0.08 0.80 0.77 0.23 0.17 0.27 0.17 0.32 0.98 0.15 0.26 0.11 0.11 0.23 0.16 0.09 0.29 0.18 0.35 0.34 0.11 0.28
1.22 0.93 0.16 1.29 0.07 0.37 0.08 4.29 0.77 1.05 0.17 0.27 0.17 0.44 3.92 2.67 0.26 0.98 1.68 0.23 0.16 3.65 1.95 1.30 0.35 0.34 1.29 0.28
0.34 0.17 0.16 0.07 0.07 0.07 0.08 0.15 0.24 0.23 0.17 0.27 0.10 0.06 0.08 0.15 0.26 0.11 0.11 0.23 0.11 0.09 0.29 0.18 0.32 0.34 0.11 0.28
2.5,2 2.5,2 2.5,2
1 1 1
4 9 19
9.02 12.17 15.00
12 15 19
12 15 19
13 17 20
12,5 16,7 19,9
2.38 1.56 0.85
2.38 1.56 0.85
3.82 3.06 1.92
0.53 0.67 0.29
0.53 0.67 0.29
1.87 1.93 1.23
0.35 0.35 0.27
2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2
1 1 1 1 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.75 1.75 1.75 1.75 1.75 1.75 1.75
39 59 79 99 4 9 19 39 59 79 99 4 9 19 39 59 79 99 4 9 19 39 59 79 99
17.59 19.03 20.02 20.76 9.13 12.43 15.42 18.15 19.65 20.69 21.47 9.23 12.66 15.79 18.66 20.23 21.31 22.13 9.32 12.87 16.13 19.12 20.77 21.89 22.75
21 23 24 25 12 16 20 23 24 26 26 13 18 21 24 26 27 28 14 19 23 26 28 29 30
22 23 24 25 12 17 20 23 25 26 27 13 18 22 25 27 28 29 14 19 23 26 28 29 30
22 24 25 26 14 18 21 24 26 27 28 16 20 23 26 28 29 30 17 21 25 28 29 31 31
22,11 23,13 24,13 25,14 13,5 17,7 20,9 23,11 25,11 26,12 27,13 14,5 18,7 22,8 25,10 27,11 28,12 29,12 15,5 20,6 23,8 27,10 28,11 30,11 30,12
0.73 0.30 0.24 0.23 3.39 2.29 1.17 0.69 1.01 0.39 0.80 4.31 2.57 1.71 1.28 0.84 0.85 0.67 4.85 2.91 1.71 1.11 0.77 0.72 0.59
0.58 0.30 0.24 0.23 3.39 2.05 1.17 0.69 0.52 0.39 0.37 4.31 2.57 1.57 0.88 0.77 0.58 0.54 4.85 2.91 1.71 1.11 0.77 0.72 0.59
0.58 0.76 0.62 0.85 4.96 3.13 1.60 1.03 1.26 0.99 1.13 8.29 5.08 2.58 1.62 1.77 1.39 1.46 8.49 4.77 3.73 2.32 1.08 1.82 0.74
0.52 0.12 0.14 0.11 1.17 1.09 0.42 0.34 0.86 0.18 0.72 1.14 0.52 0.94 0.88 0.66 0.68 0.59 1.18 0.75 0.54 0.66 0.40 0.44 0.38
0.28 0.12 0.14 0.11 1.17 0.61 0.42 0.34 0.23 0.18 0.20 1.14 0.52 0.46 0.31 0.35 0.28 0.28 1.18 0.75 0.54 0.66 0.40 0.44 0.38
0.28 0.52 0.43 0.68 1.30 1.25 0.57 0.53 0.90 0.69 0.86 3.06 2.62 1.23 0.88 1.24 1.02 1.12 3.34 2.38 2.20 1.47 0.57 1.32 0.41
0.28 0.12 0.10 0.11 0.37 0.36 0.42 0.32 0.23 0.18 0.19 0.44 0.52 0.45 0.31 0.35 0.28 0.28 0.62 0.63 0.54 0.51 0.36 0.40 0.34
0 to 3.75. Since the cost function is linear in the parameters p and h, one of them is a redundant parameter. We set h at unity and the penalty cost p has values of 4, 9, 19, 39, 59, 79 and 99. We use two demand distributions, pure and compound Poisson with an average demand of 2, 5 and 10 units per review period. For the compound Poisson demand process, customer demands are delayed geometric and we consider 4 parameter settings for (k, l): (1, 2), (2.5, 2), (5, 2) and (2.5, 4). As a result, we consider 336 problem instances for pure Poisson demand and 448 problem instances for compound Poisson demand. Table 3 provides detailed results when R 6 L < 2R, as in Fig. 1. For the different base-stock levels, the table shows the cost increases (CIs) in percentages of the optimal costs C⁄. We observe
that the base-stock levels found with the approximation procedure are smaller than the optimal base-stock levels ð~s 6 sÞ. The backorder model, however, finds base-stock levels larger than the optimal values ðs 6 sBO Þ. This is the case for all test instances. The table also shows that the best RBSP results in policies with almost the same average costs as the optimal policy. We can conclude the same for . The results are not decisive whether the RBSP specified by s and q the approximation procedure or the backorder model performs better. The cost increases CI~s;q~ for pure Poisson demand are irregular, whereas they are at most about 1% for compound Poisson demand. The results for all the different problem settings are summarized in Tables 4 and 5 for pure and compound Poisson demand,
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Table 4 The average results for pure Poisson demand with k 2 D ¼ f2; 5; 10g, lead time L 2 L ¼ f0:25ij0 6 i 6 15g and penalty cost p 2 P ¼ f4; 9; 19; 39; 59; 79; 99g. k
L
p
C⁄
D
L
P
10.82
2.58
1.65
3.47
1.74
0.44
1.92
0.30
2 5 10
L L L
P P P
6.14 10.37 15.94
1.85 2.46 3.44
1.69 1.68 1.57
3.31 3.71 3.38
0.87 1.55 2.81
0.56 0.40 0.37
1.57 2.25 1.96
0.33 0.30 0.27
D D D D D D D D D D D D D D D D
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75
P P P P P P P P P P P P P P P P
8.32 8.85 9.29 9.69 10.05 10.37 10.66 10.92 11.17 11.39 11.60 11.80 11.99 12.17 12.33 12.49
0.00 1.72 1.99 1.41 0.93 2.73 3.12 2.37 1.87 3.38 3.67 3.22 2.75 3.91 4.23 4.01
0.00 0.24 0.55 0.66 0.66 1.21 1.48 1.61 1.53 2.11 2.36 2.48 2.32 2.81 3.18 3.20
0.00 0.58 0.69 1.17 1.10 1.93 2.70 2.65 3.43 4.60 4.42 5.43 5.38 6.71 7.03 7.72
0.00 2.11 2.01 1.35 0.52 2.28 2.36 1.37 1.01 2.35 2.40 1.90 1.33 2.37 2.41 2.09
0.00 0.29 0.07 0.18 0.19 0.31 0.35 0.31 0.41 0.42 0.54 0.76 0.69 0.77 0.88 0.90
0.00 0.23 0.17 0.49 0.56 0.92 1.29 1.31 1.90 2.42 2.32 2.99 3.27 3.99 4.21 4.67
0.00 0.02 0.06 0.11 0.11 0.20 0.17 0.26 0.28 0.36 0.46 0.50 0.55 0.56 0.60 0.60
D D D D D D D
L L L L L L L
4 9 19 39 59 79 99
6.72 8.57 10.19 11.62 12.40 12.92 13.31
5.16 3.68 2.58 1.98 1.79 1.53 1.37
4.47 2.73 1.59 0.94 0.73 0.57 0.52
9.19 5.65 3.33 2.12 1.55 1.41 1.04
2.18 2.17 1.82 1.62 1.64 1.45 1.30
0.86 0.73 0.43 0.28 0.32 0.24 0.24
4.41 3.22 1.91 1.27 0.99 0.97 0.68
0.40 0.47 0.34 0.25 0.24 0.20 0.20
CI~s
CIs
CIsBO
CI~s;q~
CIs;q
CIsBO ;qBO
CIs ;q
Table 5 The average results for compound Poisson demand with ðk; lÞ 2 D ¼ fð1; 2Þ; ð2:5; 2Þ; ð2:5; 4Þ; ð2; 5Þg, lead time L 2 L ¼ f0:25ij0 6 i 6 15g and penalty cost p 2 P ¼ f4; 9; 19; 39; 59; 79; 99g. k, l
L
p
C⁄
CI~s
CIs
CIsBO
CI~s;q~
CIs;q
CIsBO ;qBO
CIs ;q
D 1,2 2.5,2 5,2 2.5,4
L L L L L
P P P P P
22.82 11.03 17.39 25.25 37.63
1.85 1.53 1.87 2.17 1.85
1.74 1.53 1.81 1.80 1.82
3.33 2.45 3.60 3.91 3.36
0.78 0.67 0.71 1.06 0.69
0.59 0.65 0.59 0.51 0.61
1.77 1.08 1.99 2.28 1.74
0.42 0.49 0.41 0.35 0.42
D D D D D D D D D D D D D D D D
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75
P P P P P P P P P P P P P P P P
17.84 18.90 19.81 20.60 21.32 21.95 22.51 23.03 23.52 23.97 24.38 24.77 25.14 25.49 25.82 26.14
0.00 0.31 0.72 0.83 0.88 1.36 1.69 1.85 1.80 2.28 2.59 2.74 2.66 3.05 3.40 3.49
0.00 0.21 0.47 0.71 0.85 1.18 1.50 1.72 1.76 2.14 2.43 2.61 2.61 2.96 3.31 3.40
0.00 0.33 0.81 1.13 1.45 2.03 2.73 2.99 3.36 4.01 4.67 5.16 5.16 6.02 6.54 6.89
0.00 0.27 0.55 0.44 0.36 0.68 0.78 0.80 0.70 0.99 1.05 1.13 1.02 1.14 1.28 1.35
0.00 0.11 0.19 0.25 0.31 0.43 0.45 0.57 0.63 0.74 0.75 0.87 0.91 0.99 1.08 1.13
0.00 0.13 0.35 0.50 0.81 0.97 1.30 1.49 1.79 2.08 2.48 2.79 2.95 3.31 3.58 3.87
0.00 0.06 0.11 0.19 0.24 0.31 0.38 0.44 0.48 0.56 0.58 0.61 0.63 0.67 0.69 0.72
D D D D D D D
L L L L L L L
4 9 19 39 59 79 99
12.07 16.66 20.92 24.87 27.05 28.54 29.68
4.29 2.97 1.92 1.26 0.98 0.82 0.74
4.26 2.86 1.78 1.10 0.85 0.73 0.61
7.83 5.46 3.41 2.19 1.72 1.43 1.27
1.23 1.02 0.85 0.70 0.61 0.55 0.53
1.13 0.81 0.61 0.46 0.41 0.39 0.33
3.45 2.81 1.93 1.34 1.11 0.94 0.86
0.49 0.52 0.49 0.42 0.37 0.33 0.30
respectively. In the first row we take the average over all problem instances. The other rows report average values for specific values of either the average demand, the lead time or the penalty cost.
The results of Tables 4 and 5 show that the costs deviate relatively more when the lead time increases. However, this result is less clear when s and q are based on the approximation procedure for the RBSP in case of pure Poisson demand. The cost increases in this particular setting vary a lot. Furthermore, our approximation procedure for the lost-sales model performs on average better than the backorder model (for the PBSP and RBSP). When the values of the inventory control parameters are based on the backorder model, the cost increases can even run up to 20% for the PBSP and to 11–15% for the RBSP. The results in both tables show that on average the relative cost increase for using a (restricted) base-stock policy decreases when the penalty cost increases. Furthermore, the cost increase for using in the RBSP is about the same as the cost increase for using s and q the best values of s and q. We also observe that the costs for the best RBSP (s⁄, q⁄) deviate on average less than 0.5% from the optimal costs, with an observed maximum deviation of 2.0%. Therefore, we conclude that a RBSP performs excellent. The RBSP outperforms the PBSP in all test problems. To conclude our comparison of the different policies, we compare the computational effort to find the best values for each of the policies. Table 6 reports the average number of iterations (Nopt) and the average CPU time (CPUopt) in seconds to find the optimal policy, as well as the average CPU time in seconds to find the best PBSP and RBSP (denoted by CPUs and CPUs ;q , respectively). All numerical results have been performed on a 2.0 Ghz AMD Opteron 246 processor running Linux and C++. The fourth and fifth line present the results when the average demand is 10, where (k, l) is (10, 1) for pure Poisson demand on line 4, and (5, 2) and (2.5, 4) for compound Poisson demand on line 4 and 5, respectively. We do not present the CPU time for the approximation procedures, since this is within milliseconds. It becomes clear that the computational effort to find the best RBSP can be very large (up to over 3 hours for a single problem instance). This illustrates that the approximation procedure is very efficient. Furthermore, we note that the computational effort to find the best PBSP can be improved significantly when the observation ~s 6 s 6 sBO is used. 4.2. Impact of review period length In this section, we perform a sensitivity analysis to understand how the costs of the optimal replenishment policy changes in response to changes to the review period length. We consider the same cost setting as in the example of Section 2.2 (i.e., p = 19 and h = 1). Set L = 1.5 and let R range between 0.5 and 2.5. Fig. 3 presents the results for the different demand distributions. It is clear that a longer review period results in higher costs due to higher average inventory levels to cover a larger time period with demand uncertainty without reordering. It is interesting to see that the rate of increase is less when there is more variance in the demand. This is consistent with the simulation results of Sezen (2006), who concludes that the variability in the demand process is the most important factor to set the duration of a review period. Our models of Section 2 give an analytical procedure to support these observations for more general inventory systems as studied by Sezen (2006). 5. Conclusions and future research In this paper, we considered a lost-sales inventory model with periodic reviews. Such inventory systems are commonly seen in a retail environment. A retailer has to decide about the review frequency next to the replenishment quantities. We developed a general model for inventory systems where lead times and review periods can be of any length. In particular, we studied systems with a cost objective function consisting of the expected holding and
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Table 6 The computational effort to evaluate the exact models of Section 2 for pure and compound Poisson demand with k l 2 D ¼ f2; 5; 10g, lead time L 2 L ¼ f0:25ij0 6 i 6 15g and penalty cost p 2 P ¼ f4; 9; 19; 39; 59; 79; 99g. kl
L
p
Poisson Nopt
Compound Poisson CPUopt Mean
a
CPUs Max
Mean
CPUs ;q Max
Mean
Nopt Max
CPUopt
CPUs
CPUs ;q
Mean
Max
Mean
Max
Mean
Max
D
L
P
14.45
7.17
151.99
5.20
110.98
30.86
893.70
17.19
29.14
496.89
23.39
689.37
281.93
12016.78
2 5 10 (1) 10 (2)
L L L L
P P P P
19.00 12.85 11.50
0.92 1.81 18.76
4.51 11.97 151.99
0.06 0.78 14.76
0.41 7.91 110.98
0.05 2.32 90.22
0.37 22.70 893.70
25.98 15.38 12.82 14.57
1.42 4.31 41.81 69.02
7.65 38.63 400.81 496.89
0.11 1.94 31.42 60.09
0.99 20.01 372.01 689.37
0.19 7.64 238.97 880.91
1.94 87.26 3027.17 12016.78
D D D D D D D D D D D D D D D D
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75
P P P P P P P P P P P P P P P P
8.52 9.05 9.43 9.86 10.95 11.71 12.43 13.38 14.67 15.19 16.19 17.29 19.43 20.33 20.76 22.00
0.00 0.00 0.00 0.00 0.01 0.02 0.03 0.03 0.19 0.43 0.60 0.79 9.05 24.03 33.94 45.53
0.01 0.01 0.01 0.01 0.02 0.06 0.08 0.05 0.39 1.15 1.81 2.46 21.73 81.70 117.84 151.99
0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.01 0.18 0.32 0.46 0.56 10.07 17.29 23.68 30.64
0.00 0.01 0.01 0.00 0.01 0.02 0.03 0.03 0.56 1.12 1.55 1.68 40.29 65.33 91.11 110.98
0.00 0.00 0.01 0.01 0.18 0.14 0.19 0.26 3.26 3.58 4.87 6.43 69.66 97.01 135.28 172.95
0.01 0.02 0.02 0.04 0.67 0.62 0.84 1.15 14.08 17.30 22.77 30.02 332.02 494.22 697.72 893.70
10.96 11.79 12.04 12.46 13.39 14.32 15.25 16.25 17.61 18.18 19.29 20.54 21.57 22.71 23.82 24.82
0.00 0.00 0.00 0.00 0.02 0.09 0.08 0.08 0.58 2.26 2.97 3.82 31.48 118.95 141.87 164.02
0.01 0.01 0.01 0.01 0.08 0.28 0.22 0.29 2.14 10.44 13.14 15.50 110.40 432.29 474.83 496.89
0.00 0.00 0.00 0.00 0.01 0.02 0.03 0.03 0.61 1.21 1.57 2.18 38.62 99.13 99.69 131.16
0.00 0.01 0.00 0.01 0.04 0.08 0.10 0.12 3.00 4.92 6.63 10.12 176.21 519.41 534.36 689.37
0.02 0.04 0.05 0.07 0.53 1.10 1.41 1.75 14.56 29.36 36.37 43.76 504.39 1033.95 1272.42 1571.08
0.09 0.23 0.31 0.40 3.29 6.99 8.57 10.84 99.64 200.38 244.77 298.35 4074.24 8097.17 10359.09 12016.78
D D D D D D D
L L L L L L L
4 9 19 39 59 79 99
16.17 15.31 14.60 14.13 13.85 13.56 13.52
4.72 5.48 6.62 7.70 8.35 8.85 8.44
77.19 92.89 117.11 129.25 141.48 151.79 151.99
3.12 3.81 5.06 5.51 6.29 6.15 6.47
63.03 61.44 93.90 88.15 95.67 96.50 110.98
6.04 12.18 21.23 32.88 41.41 48.65 53.65
86.75 193.36 349.94 530.33 689.98 803.43 893.70
17.61 17.84 17.63 17.11 16.86 16.67 16.59
7.38 15.92 26.92 36.71 39.19 38.38 39.48
91.27 239.91 427.76 496.89 477.42a 434.88a 431.64a
3.21 8.34 18.18 32.58 41.11 29.37 30.95
38.21 131.25 329.08 516.68 689.37 396.38 476.80
5.63 24.82 85.21 239.62 385.77 544.04 688.40
66.82 306.48 1162.05 3865.65 6904.26 9975.66 12016.78
Due to memory shortage, we had to restrict the state space. As a result, the actual CPU time is higher.
Fig. 3. The expected total costs per unit time for the optimal policy (on left) and the best PBSP (on right) where h = 1, p = 19, L = 1.5 and R 2 [0.5; 2.5].
penalty costs per unit time. This general structure makes it possible for any retailer to determine the review period length, contrary to the existing literature. In practice, pure base-stock policies are often implemented because of their simplicity. In contrast to inventory systems with backorders, such replenishment policies are known to be suboptimal when excess demand is lost. Pure base-stock policies (PBSPs) based on a backorder model perform bad with average cost deviations up to 4–5%. Therefore, the replenishment policy should be adjusted to incorporate lost sales, and we proposed a restricted base-stock policy (RBSP) which limits the order size to a maximum amount q = sR/(L + R). From numerical results we concluded that the costs for the best values of s and q
deviate on average less than 0.5% from the optimal costs. We also developed an approximation procedure to set near-optimal values of s and q with closed-form expressions. The results for the approximation procedure show irregular cost increases compared to the optimal costs for pure Poisson demand, whereas they are at most about 1% for compound Poisson demand. Our recommended policy is the RBSP specified by the base-stock level s of the best PBSP and equal to sR=ðL þ RÞ rounded (up in case of a tie) to the nearest q integer, because the average cost for this policy is similar to the best values of s and q for the RBSP. There are several interesting areas to extent this work. One is to see whether the RBSP is also near optimal when there is a product
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substitution between related products. Another direction for future research is to consider a service level restriction instead of a penalty cost for each unsatisfied demand. Such a penalty cost can be difficult to quantity in practice, since it includes loss of goodwill. The transportation and fixed order cost could also be included to show the impact of the length of the review period.
References Bijvank, M. 2009. Service inventory management: solution approaches for inventory systems without backorders. Ph.D. thesis, VU University, Amsterdam, The Netherlands. Bijvank, M., Vis, I.F.A., 2011. Lost-sales inventory theory: A review. European Journal of Operational Research 215, 1–13. Chiang, C., 2006. Optimal ordering policies for periodic-review systems with replenishment cycles. European Journal of Operational Research 170, 44–56. Chiang, C., 2007. Optimal ordering policies for periodic-review systems with a refined intra-cycle time scale. European Journal of Operational Research 177, 872–881. Downs, B., Metters, R., Semple, J., 2001. Managing inventory with multiple products, lags in delivery, resource constraints, and lost sales: a mathematical programming approach. Management Science 47, 464–479. Gallego, G., Özer, Ö., 2003. Optimal replenishment policies for multiechelon inventory problems under advanced demand information. Manufacturing and Service Operations Management 5, 157–175. Graves, S.C., 1996. A multiechelon inventory model with fixed replenishment intervals. Management Science 42, 1–19. Gürbüz, M.C., Moinzadeh, K., Zhou, Y.P., 2007. Coordinated replenishment strategies in inventory/distribution systems. Management Science 53, 293–307.
Huh, W.T., Janakiraman, G., Muckstadt, J.A., Rusmevichientong, P., 2009. Asymptotic optimality of order-up-to policies in lost sales inventory systems. Management Science 55, 404–420. Johansen, S.G., Thorstenson, A., 1993. Optimal and approximate ðQ ; rÞ inventory policies with lost sales and gamma-distributed lead time. International Journal of Production Economics 30–31, 179–194. Johansen, S.G., Thorstenson, A., 1996. Optimal ðr; Q Þ inventory policies with Poisson demands and lost sales: discounted and undiscounted cases. International Journal of Production Economics 46–47, 359–371. Johansen, S.G., Thorstenson, A. (2008). Pure and restricted base-stock policies for the lost-sales inventory system with periodic review and constant lead times. Technical report, Department of Operations Research, Aarhus University. Johnston, F.R., Boylan, J.E., Shale, E.A., 2003. An examination of the size of orders from customers, their characterisation and the implications for inventory control of slow moving items. Journal of the Operational Research Society 54, 833–837. Kapalka, B.A., Katircioglu, K., Puterman, M.L., 1999. Retail inventory control with lost sales service constraints, and fractional lead times. Production and Operations Management 8, 393–408. Parker, R.P., Kapuscinski, R., 2004. Optimal policies for a capacitated two-echelon inventory system. Operations Research 52, 739–755. Rosling, K., 2002. Inventory cost rate functions with nonlinear shortage costs. Operations Research 5, 1007–1017. Sezen, B., 2006. Changes in performance under various lengths of review periods in a periodic review inventory control system with lost sales. International Journal of Physical Distribution and Logistics Management 36, 360–373. Stidham, S., 1974. A last word on L ¼ kW. Operations Research 22, 417–421. Tijms, H.C., 2003. A First Course in Stochastic Models. Wiley, Chichester. Zipkin, P.H., 2000. Foundations of Inventory Management. McGraw-Hill, New York. Zipkin, P.H., 2008a. Old and new methods for lost-sales inventory systems. Operations Research 56, 1256–1263. Zipkin, P.H., 2008b. On the structure of lost-sales inventory models. Operations Research 56, 937–944.
Contents lists available at SciVerse ScienceDirect
European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor
Production, Manufacturing and Logistics
Periodic review lost-sales inventory models with compound Poisson demand and constant lead times of any length Marco Bijvank a,⇑, Søren Glud Johansen b a b
Rotterdam School of Management, Erasmus University, Burgemeester Oudlaan 50, 3062 PA, Rotterdam, The Netherlands Department of Mathematics, Aarhus University, Ny Munkegade 118, DK-8000 Aarhus C, Denmark
a r t i c l e
i n f o
Article history: Received 29 March 2010 Accepted 20 January 2012 Available online 28 January 2012 Keywords: Inventory Lost sales Periodic review Base stock Heuristics
a b s t r a c t In almost all literature on inventory models with lost sales and periodic reviews the lead time is assumed to be either an integer multiple of or less than the review period. In a lot of practical settings such restrictions are not satisfied. We develop new models allowing constant lead times of any length when demand is compound Poisson. Besides an optimal policy, we consider pure and restricted base-stock policies under new lead time and cost circumstances. Based on our numerical results we conclude that the latter policy, which imposes a restriction on the maximum order size, performs almost as well as the optimal policy. We also propose an approximation procedure to determine the base-stock levels for both policies with closed-form expressions. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction In order to make a supply chain more flexible and responsive to customer requests in a cost-efficient manner, the involved parties have to cooperate and integrate activities. Usually a vendor determines the lead time experienced by a retailer, whereas the retailer has to decide on the replenishment policy; i.e., the frequency to review the inventory levels and the amount of items to order from the vendor at these review instants to anticipate on the stochastic demand during the lead time. Once the retailer has decided on both aspects, this information can be shared with the supplier(s) to anticipate on future order moments and the order size distribution. Ultimately, this can result in improvements for the entire supply chain. Traditionally, multi-echelon systems are used to model inventory control in the supply chain (see, e.g., Graves, 1996; Gallego and Özer, 2003; Parker and Kapuscinski, 2004; Gürbüz et al., 2007). In such models it is (almost) always assumed that excess demand is backordered. However, due to the increased competition in the retail environment, customers are not willing to wait anymore and excess demand is lost in many practical settings. Since the optimal replenishment policy for lost-sales inventory systems cannot be characterized by an easy-to-compute policy, this is interesting to investigate (Zipkin, 2008b; Bijvank and Vis, 2011). The goal of this paper is to develop and compare mathematical models for lost-sales inventory systems with different replenishment policies. The inventory models should be able to vary the ⇑ Corresponding author. Tel.: +31 104082199. E-mail address: [email protected] (M. Bijvank). 0377-2217/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2012.01.041
review period length in relation to the lead time to study the impact it has on the performance measures such as the expected total costs and service level. In real life most suppliers dispatch their delivery trucks to retailers on a regular basis based on the order frequency of the retailer. The fixed cost incurred for each order is shared among all items ordered at the same supplier. As a result, the fixed order cost for each item is negligible. The total order costs over a time period only depend on the order frequency. Longer review periods result in lower order costs compared to the costs over a similar time period with shorter review periods. Besides order costs, the total costs for a retailer consist of holding costs for keeping units on stock and penalty costs when a customer demand is not satisfied. The influence of the review period length on these costs is not as trivial as for the order costs, since they depend on the replenishment policy. For base-stock policies, Sezen (2006) illustrates by simulation the impact of the review period length on the average on-hand inventory levels and the fill rate in settings with fractional lead times (i.e., lead times shorter than a review period). The numerical results show that the variability in the demand process is the most important factor to set the duration of a review period. No costs are considered, and no analytical procedure is proposed to determine the length of a review period or on how to set the base-stock level. In order to study the impact of the review period length, no assumptions should be imposed on the relationship between the lead time and the review interval. We are the first authors, to our knowledge, to develop such models when excess demand is lost. In the literature, it is assumed that either the lead time is an integral multiple of the review period (e.g., Zipkin, 2008a;
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Huh et al., 2009 and the references therein) or a fraction of the review period (Kapalka et al., 1999; Chiang, 2006, 2007). In practice, the lead time is a constant and mainly determined by the supplier. The lead time should, however, also include the time for a retailer to transship the delivered items to the shelves. Therefore, the lead time is not strictly related to the review period length in real inventory systems. To relax the previously mentioned assumptions on the lead time, the objective function should be adjusted accordingly. In models where the lead time is an integer multiple of the review period the holding costs are charged after the demand in a review period has occurred (Zipkin, 2000). In other words, demand is assumed to occur periodically at the beginning of such a period and the holding costs are charged over the average on-hand inventory level (which remains constant over the review period), or equivalently, demand occurs at any point in time and the holding cost driver is the ending inventory of a review period. As mentioned by Rosling (2002) it is more realistic that demand arrives in continuous time, and the holding cost driver should be the average on-hand inventory level. Chiang (2006, 2007) divides the review period in smaller sub-periods, where the holding cost driver is the ending inventory of each sub-period. When such sub-periods are small enough, this approach approximates a system with holding costs charged over the average inventory. In this paper, we model the demand and costs to incur in continuous time without any sub-periods. This enables a retailer to compare the impact on the average costs when the review period length varies in relation to the lead time. To derive exact expressions for the average inventory levels we restrict the demand to be compound Poisson. Otherwise, approximation expressions should be used (Bijvank, 2009). We only consider the undiscounted setting, contrary to Chiang (2006, 2007) and Zipkin (2008a,b) who also include a discount rate. A second contribution of this paper is the study of the optimal replenishment policy as well as base-stock policies for the more general inventory models as discussed above. Since there is no fixed order cost per item, base-stock policies are commonly used in such situations. For a recent overview and comparison of different replenishment policies, we refer to Zipkin (2008a). He shows that pure base-stock policies (PBSPs) perform badly with respect to a cost minimization function, especially when the penalty cost for a lost-sales occurrence is low. However, Huh et al. (2009) show that the best PBSP becomes asymptotically optimal as the lostsales penalty increases. Johansen and Thorstenson (2008) introduce restricted base-stock policies (RBSPs), which impose an upper limit on the order sizes. In particular, we derive closed-form expressions that approximate the performance of PBSPs. Such approximations can be used to determine near-optimal values of the base-stock level. Furthermore, we propose a formula to set the maximum order size in the RBSP. This paper is organized as follows. In Section 2, we develop the inventory models for different replenishment policies under the new cost and lead time circumstances when the review period length is given. Besides the PBSP and RBSP, we also model the optimal replenishment policy. Even though these models give exact results, the computational effort can be quite excessive, especially for relatively long lead times. In Section 3 we derive closed-form expressions to approximate the performance measures of interest for lost-sales inventory systems with the PBSP, such as the fraction of demand lost, the average inventory on hand and the average costs. We also indicate how these approximations can be used to set the base-stock level and maximum order quantity in the RBSP. In Section 4 we compare the performance of both base-stock policies with the optimal costs. Based on our numerical results we conclude that the costs associated with the best base-stock level for the RBSP deviate on average less than 0.5% from the minimal costs, while the approximation procedure results in average
cost deviations of about 1–2% from the optimal costs. Section 5 contains our concluding remarks. 2. Different replenishment policies In order to model the base-stock and optimal replenishment policies we first introduce the general framework and notations in Section 2.1. The optimal replenishment policy is modeled as a Markov chain in Section 2.2. Only minor modifications are necessary to model the PBSP and RBSP in Section 2.3 and Section 2.4, respectively. 2.1. Notation and assumptions The time between two reviews is called a review period. Its length is denoted by R. At each review instant an order is issued. The order arrives after a constant lead time L. Let T denote the start of a considered review period, and t the time of the order delivery within the same period. Define r ¼ L mod R. Note that t = T + r. The number of full review periods from time T until the delivery of the order issued at time T equals l = (L r)/R. Hence, L = lR + r. Fig. 1 illustrates the notation based on an example where R 6 L < 2R (i.e., l = 1). The special case when the lead time is an integer multiple of the review period is described by Zipkin (2008a) and Johansen and Thorstenson (2008) (i.e., r = 0, l = L/R and t = T), whereas Chiang (2006, 2007) assumes fractional lead times (i.e., L < R and l = 0). Demand is assumed to be independent and identically distributed over time. We consider a Poisson customer arrival process with rate k. In practice it is common to observe that the customer demand size follows a delayed geometric distribution with mean l = 1/(1 h), where l P 1 (Johnston et al., 2003). We denote the total demand of n customers as a random variable Xn, where
PðX 1 ¼ dÞ ¼ ð1 hÞhd1 ;
d P 1;
ð1Þ
and more general
8 1 h; > > > > > > < hPðX n ¼ d 1Þ; PðX n ¼ dÞ ¼ hPðX n ¼ d 1Þ þ ð1 hÞPðX n1 ¼ d 1Þ; > > > > ð1 hÞPðX n1 ¼ d 1Þ; > > : 0;
n ¼ 1; d ¼ 1 n ¼ 1; d > 1 n ¼ 2;...;d 1 n¼d n > d:
For any s P 0, the demand during a time period of length s is a random variable Ds with probability mass function gs(d), where
8 ks > if d ¼ 0
In the special case of a pure Poisson demand process, l = 1 and P(Xn = n) = 1. We also define
G0s ðsÞ ¼ PrðDs < sÞ ¼
s1 X
g s ðdÞ;
d¼0 s X
G1s ðsÞ ¼ E½ðs Ds Þþ ¼
G0s ðdÞ;
d¼1
Fig. 1. An example of the time framework for reviews and order deliveries where R 6 L < 2R.
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with (A)+ = max{A, 0}. We denote the set of non-negative integers by N0 , and the set of all integers between m and n by Nm;n ¼ fi 2 N0 jm 6 i 6 ng ¼ fm; m þ 1; . . . ; ng. Furthermore, N1þl 0;s is defined as an (1 + l)-dimensional vector space over N0;s . This notation is used to model the inventory system as a Markov chain. Its state at time T (just after ordering) is denoted (i, y), where i is the actual inventory on hand and y is a vector with components yk ; k 2 N0;l . Component yk is the amount ordered at time T (l k)R to be delivered at time T (l k)R + L = t + kR (see also Fig. 1). In particular, yl can be represented as a function a(i, y0, . . . , yl1), which is prescribed by the replenishment policy. We let F(y) denote the vector obtained from y by removing its first component. The state of the Markov chain at time t is denoted (j, z), where j is the updated on-hand inventory level and z = F(y). We denote the actual demand from time T to time t and from t to T + R by dr and dRr, respectively. As a result, j = (i dr)+ + y0. To complete the Markov chain description, we have to specify the one-step transition probabilities between the different states of the inventory system. The transition probabilities from state (i, y) at time T to state (j, F(y)) at time t are denoted by P(i,y),(j,F(y)). By pðj;zÞ;ði;z;yl Þ we denote the transition probabilities from state (j, z) at time t to state (i, z, yl) at time T + R where yl is the amount ordered at this time. It is easily seen that
P ði;yÞ;ðidþy0 ;FðyÞÞ
8 if 0 6 d < i > < g r ðdÞ; ¼ 1 G0r ðiÞ; if d ¼ i > : 0; otherwise
and
pðj;zÞ;ðjd;z;aðjd;zÞÞ
8 if 0 6 d < j > < g Rr ðdÞ; ¼ 1 G0Rr ðjÞ; if d ¼ j > : 0; otherwise:
ð2Þ
ð3Þ
i1 i X i i1 þ Hði 1Þ HðiÞ ¼ þ PðX 1 ¼ jÞHði jÞ ¼ h k j¼1 k k
i1 X
g s ðjÞHði jÞ
ð4Þ
ð5Þ
j¼0
and the expected demand lost during such a period is
E½ðDs iÞþ ¼ E½Ds i þ E½ði Ds Þþ ¼ kls i þ G1s ðiÞ:
j1 X
g Rr ðdÞV n1 ðj d; z; aðj d; zÞÞ
d¼0
þ 1 G0Rr ðjÞ V n1 ð0; z; að0; zÞÞ; V n ði; yÞ ¼ cr ðiÞ þ
i1 X
ð7Þ
g r ðdÞv n ði d þ y0 ; FðyÞÞ
d¼0
ð8Þ
The notation introduced in this section for the different time instants of the Markov model is summarized in Table 1. The valueiteration algorithm (Tijms, 2003) with accuracy number e repeats to increase n by one and compute the value functions in Eqs. (7) and (8) until Mn mn < e, where
mn ¼ minfV n ði; yÞ V n1 ði; yÞg; ði;yÞ
M n ¼ maxfV n ði; yÞ V n1 ði; yÞg: ði;yÞ
When this value-iteration algorithm is stopped after the n-th iteration, then (mn + Mn)/2 cannot deviate more than 100e% from the long-run average costs per review period. The numerical results reported in Section 4 are computed with e = 1E4. 2.2. Optimal policy
because it takes on average 1/k time units before a customer arrives and (i X1)+ units remain on stock after this demand arrival. Note that the first equality in Eq. (4) is based on the memoryless property of the Poisson customer arrival process, whereas the other equalities are simplifications based on the delayed geometric demand size distribution. For other distributions, the first equality in Eq. (4) can be used to compute H(i) iteratively for i = 1, 2, . . .. However, it is straightforward to replace the (commonly observed) distribution in Eq. (1) by others. The mean time-weighted inventory during a period of length s for a given initial inventory level of i units is
Hs ðiÞ ¼ HðiÞ
v n ðj; zÞ ¼ cRr ðjÞ þ
þ 1 G0r ðiÞ v n ðy0 ; FðyÞÞ:
We want to compute the total expected costs over a review period, and express the expected costs incurred over a period of length s by cs(i) when the on-hand inventory level equals i at the beginning of this period and no order delivery occurs during this period. The cost function cs(i) consists of the expected holding and penalty costs. First, we let H(i) be the mean time-weighted stockholding until the inventory is fully depleted given an initial inventory level of i units. By definition H(0) = 0. For compound Poisson demand where the customer demands follow a delayed geometric distribution,
ði þ 1Þi ði 1Þi h ; ¼ 2k 2k
By h and p we denote the unit holding cost per unit time and the unit penalty cost for each lost demand. Consequently, cs(i) = hHs(i) + pE[(Ds i)+]. Notice, when h = 1 and p = 0 this expresses the average inventory, and when h = 0 and p = 1 this expresses the average demand lost. Furthermore, we note that previous papers compute the average holding costs as hE[(i Ds)+] (see, e.g., Chiang, 2006, 2007; Zipkin, 2008a), which is the unit holding costs after the demand has occurred in a period of length s instead of the average holding costs when demand occurs in continuous time. We can compute the performance measures of interest for this inventory system by value iteration. Our focus is on the long-run average costs. We let Vn(i, y) denote the total expected costs incurred over the time interval from time T to time T + nR when the system is in state (i, y) at time T and the system incurs no costs at and after time T + nR. Moreover, we let vn(j, z) denote the total expected costs incurred over the time interval from time t to time T + nR when the system is in state (j, z) at time t and the system incurs no costs at and after time T + nR. Consequently, V0(i, y) = 0 and, recursively for n = 1, 2, . . .
ð6Þ
To find an optimal policy and the corresponding costs for the inventory system described in Section 2.1, we model the system as a Markov chain with an infinite state space. More specifically, T 2þl the state space at time and at time t T equals S ¼ ði; yÞ 2 N0 t 1þl it is S ¼ ðj; zÞ 2 N0 . The value function Vn(i,y) at time T is specified by Eq. (8). At time t, the one-step transition probabilities depend on the entire state description. Therefore, we use the following relative value function at time t
Table 1 Notation to model the inventory system as a Markov chain at a review instant (time T) or order deliver (time t). Time T
Time t
State vector Inventory level Orders outstanding One-step transition probabilities
(i,y) i = (j DRr)+ y = (y0, y1, . . . , yl) P(i, y), (j, F(y))
(j,z) j = (i Dr)+ + y0 z = F(y) pðj;zÞ;ði;z;yl Þ
Value function
Vn(i, y)
vn(j, z)
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v n ðj; zÞ ¼ cRr ðjÞ þ
j X
and the state space at time t is
g Rr ðj iÞ minfV n1 ði; z; yl Þg
(
yl P0
i¼1
S ts
þ 1 G0Rr ðjÞ minfV n1 ð0; z; yl Þg:
¼
ðj; zÞ 2
N1þl 0;s j
yl P0
To illustrate the form of the optimal policy, we use an example in which the demand is pure Poisson with k = 5 and l = 1. Furthermore, L = 1.5R, h = 1, and p = 19. Let yl = a⁄(i, z) denote the optimal order quantity when the on-hand inventory level equals i units and the outstanding order quantities are z just before ordering at time T. Notice that in this example at most one order is outstanding at time T. Hence, z = y0 in this example. An optimal policy can be found with a value-iteration algorithm, and is provided in Table 2. The first column represents the on-hand inventory level i, whereas the first row represents the size of the outstanding order y0. This optimal policy results in the minimal costs C⁄ = 9.63 per review period and corresponds to a fill rate of 98.05%. The optimal policy of Table 2 is consistent with the properties derived in the literature. Chiang (2006) proves that the optimal order quantity is nonincreasing in the inventory position, and the related rate of decrease is less than one, which has also been observed for lost-sales inventory systems with continuous review (Johansen and Thorstenson, 1993, 1996). Zipkin (2008b) proves by induction that the optimal cost function is L-natural convex (i.e., convex, submodular and it contains a property related to diagonal-dominance). This implies that the optimal order quantities are monotone decreasing in the inventory position and the optimal order quantities are more sensitive to recent orders. 2.3. Pure base-stock policy (PBSP)
þ
l X
) zk 6 s :
ð10Þ
k¼1
The one-step transition probabilities from state ði; yÞ 2 S Ts at time T are given by Eq. (2), whereas the probabilities from state ðj; zÞ 2 S ts P at time t are expressed by Eq. (3) with aði; zÞ ¼ s i lk¼1 zk . Similarly, Eqs. (7) and (8) are used as backward recursion for all ðj; zÞ 2 S ts and all ði; yÞ 2 S Ts , respectively, to compute the expected total costs for a given value of base-stock level s. We use a bisection method without upper or lower bounds to find the best base stock, which we denote by s. It is our experience that the total costs are a quasi-convex function of the base-stock level. Notice that the convexity result proved by Downs et al. (2001) and Chiang (2006) does not include our inventory system in general. Consequently, previous results cannot be used directly. 2.4. Restricted base-stock policy (RBSP) As mentioned in Section 2.3, the PBSP and the optimal policy coincide in numerous states of the Markov chain. However, the rate of decrease of the optimal order quantity in the inventory position is less than one (see Section 2.2), whereas this rate equals one in the PBSP. Therefore, in the RBSP we include a maximum order quantity (the dark gray area in the example of Table 2). This policy with base stock s and upper limit q on the order sizes prescribes to issue a replenishment order at each review time T of size
( yl ¼ min s i
l1 X
) yk ; q :
k¼0
The PBSP with base-stock level s prescribes to issue a replenishment order at each review time T, such that the inventory position (inventory on hand plus inventory on order) equals base-stock leP vel s. Hence, yl ¼ s i l1 k¼0 yk . If we compare this policy to the optimal policy of the example discussed at the end of Section 2.2, a large number of states result in the same order quantities when s = 18 in this example (the light gray areas in Table 2). The state space of the Markov chain for the PBSP at time T is
( S Ts
¼
ði; yÞ 2
2þl N0;s i
þ
l X
) yk ¼ s
The state space of the Markov chain at time T is S Ts;q ¼ Ys;q ðiÞ, where
( y2
Ys;q ðiÞ ¼
k¼0
Table 2 The optimal order quantities a⁄(i, y0) when the on-hand inventory level equals i units and the outstanding order has size y0 at a review instant. The order quantities in the PBSP with s = 18 coincide with a large part of the optimal policy a⁄(i, y0) (light gray), whereas more quantities are included in the RBSP with s = 18 and q = 7 (dark gray).
k¼0
Zs;q ðjÞ ¼
z2
X l
Nl0;q
k¼1
i¼0 fig
) yk 6 s i ;
whereas the state space at time t equals S ts;q ¼ where
( ð9Þ
X l
N1þl 0;q
Ss
Ss
j¼0 fjg
Zs;q ðjÞ,
) zk 6 s j :
Eqs. (7) and (8) n o are used for the RBSP with aði; zÞ ¼ min P s i lk¼1 zk ; q to compute the total expected costs. Based on enumeration we can determine the best values of s and q. These optimal values are denoted by s⁄ and q⁄, respectively. 3. Approximation model As discussed in Section 2, the numerical computations of the performance measures can be performed by value iteration. This can, however, require quite a computational effort. As shown by Johansen and Thorstenson (2008), there are closed-form expressions to approximate the value of the performance measures for a lost-sales inventory system with a PBSP and geometric distributed demand per review period. We derive new closed-form expressions to approximate the performance measures for inventory systems controlled by a PBSP under the more general cost and lead time circumstances as described in Section 2. Based on these expressions, we can find the value of base-stock levels that minimize the approximate total costs. We extend this approach to the RBSP at the end of this section. For the PBSP, the inventory position at review time T is equal to base-stock level s. All the orders outstanding at that time arrive before or at time T + L. Orders issued after T arrive at time
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T + R + L or later. Therefore, we consider the time interval [T + L, T + R + L), which is called a shifted review period by Kapalka et al. (1999). In case of backordering, the net inventory at time s 2 [T + L, T + R + L) equals s minus the demand during the time interval from T to s. Because the net inventory cannot be negative in our lost-sales model, we correct the demand distribution with a factor ~cs . Therefore, we approximate the probability that the inventory on hand at time s 2 [T + L, T + R + L) equals i as
p~ s ðiÞ ¼
~cs g sT ðs iÞ; if 0 < i 6 s 1 ~cs GsT ðsÞ; if i ¼ 0;
eI þ ¼ s
~ TþL ðiÞ ¼ ~cs E½ðs DL Þþ ¼ ~cs G1L ðsÞ: ip
ð11Þ
i¼0
Similarly, the approximation for the expected inventory on hand ~ 1 just before the delivery at time T + R + L is eI s ¼ cs GLþR ðsÞ. Moreover, þ e e the average order size is approximated as I s I s . This amount represents the long-run average demand satisfied during a (shifted) review period. As a result, the approximation for the long-run fraction of demand lost is 1
1
e s ¼ 1 ~cs GL ðsÞ GLþR ðsÞ A klR
ð12Þ
and the approximate average inventory on hand is
eI s ¼ 1 R
s X
~cs ½g L ðs iÞ g LþR ðs iÞHðiÞ:
e Based on Little’s formula, we set it equal to eI þ s I s L=R and get
~cs ¼
s : ðl þ 1Þ G1L ðsÞ G1LþR ðsÞ þ G1ðlþ1ÞR ðsÞ
ð14Þ
The value of the base-stock level that minimizes the approximate expected total costs equals
where we derive the value of factor ~cs at the end of this section. Hence, the approximate expected inventory on hand at time T + L is s X
i Rrh r s ~cs G1LþRr ðsÞ þ s ~cs G1LþRr ðsÞ eI þs eI s R R
Rr 1 1 ~ GL ðsÞ G1LþR ðsÞ : ¼ s cs Gðlþ1ÞR ðsÞ þ R
ð13Þ
i¼1
The last aspect of the approximation procedure is to determine ~cs . We apply Little’s formula (Stidham, 1974) and equate the approximate average inventory on order with the approximate average amount ordered per unit time multiplied by L. Fig. 2 illustrates how the inventory position, inventory on hand and inventory on order change during the shifted review period. An order is placed at time T + L + R r = T + (l + 1)R. Consequently, the inventory on order during the interval [T + L, T + (l + 1)R) is constant, and it is also constant during the interval [T + (l + 1)R, T + L + R). Based on our approximations, the expected inventory on hand at review time T + L + R r is approximated as ~cs G1LþRr ðsÞ. Consequently, the average inventory on order during interval [T + L + R r, T + L + R) is approximated as s ~cs G1LþRr ðsÞ. The approximate average inventory on order during [T + L, T + L + R r) e is eI þ s I s units less (i.e., the approximate average order size). Therefore, the average inventory on order is approximated as
e s g; ~s ¼ argminfheI s þ pkl A sP0
where heI s approximates the expected holding costs per unit time e s approximates the expected costs (see Eq. (13) for eI s ) and pkl A e s ) for a given base due to lost sales in a unit time (see Eq. (12) for A stock s. Notice the resemblance between our approximation procedure for the PBSP in case of lost sales and the performance expressions in case of backordering (Zipkin, 2000). When ~cs is 1, the expressions derived in this section are exactly those as for the backorder model since no correction has to be made for lost sales. Furthermore, lims!1 ~cs ¼ 1 based upon this observation about the backorder model. In this section, we have discussed a procedure to set the value of base-stock level s based on closed-form expressions that approximate the long-run behavior of a periodic review inventory system with lost sales in case a PBSP is applied. The same base-stock level can be used for the RBSP. However, the value of q should also be specified. As indicated by Fig. 2, any order placed after time T is delivered at or after time T + L + R. Therefore, the inventory position at time T should cover the demand over L + R time units. The inventory position at time T is s, which also represents the maximum inventory position. Therefore, at most s/(L + R) units can be demanded per unit time in order to satisfy all the demand. This results in a maximum order quantity of sR/(L + R) units per review period. We propose to set q equal to this quantity rounded to the nearest integer and rounded up in case of a tie. The performance of these approximation procedures are analyzed in the next section. 4. Numerical results The goal of this section is threefold. First, we compare the performance of the different replenishment policies. Second, we compare different approaches to set suitable values of base-stock level s and maximum order size q. Third, we show how the models of Section 2 can be used to perform a sensitivity analysis on the review period length. 4.1. Comparison of replenishment policies
Fig. 2. The inventory position (solid line), inventory on hand (dashed line) and inventory on order (dotted line) for a specific time interval [T + L, T + L + R).
Besides the best PBSP with base-stock level s and the best RBSP with base-stock level s⁄ and maximum order size q⁄, we consider the optimal base-stock level sBO in case excess demand is backordered and the base-stock level ~s when the approximation procedure of Section 3 is used. Furthermore, we also investigate three other RBSPs with base stocks s set equal to ~s,s and sBO. Their max~; q and qBO, respectively) are set imum order sizes q (denoted q equal to sR/(L + R) rounded to the nearest integer (rounded up in case of a tie). We also report the average costs C⁄ per unit time for the optimal replenishment policy, which is computed according to Section 2.2. We illustrate the performance of the base-stock policies and optimal policy for a test bed where R = 1 and L/R is ranging from
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M. Bijvank, S.G. Johansen / European Journal of Operational Research 220 (2012) 106–114 Table 3 Results for pure and compound Poisson demand when the average demand per period is 5 and R 6 L < 2R. k, l
L
p
C⁄
~s
s
sBO
s⁄,q⁄
CI~s
CIs
CIsBO
CI~s;q~
CIs;q
CIsBO ;qBO
CIs ;q
5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1 5,1
1 1 1 1 1 1 1 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.75 1.75 1.75 1.75 1.75 1.75 1.75
4 9 19 39 59 79 99 4 9 19 39 59 79 99 4 9 19 39 59 79 99 4 9 19 39 59 79 99
6.19 7.79 9.10 10.30 10.91 11.39 11.67 6.27 7.96 9.41 10.64 11.34 11.76 12.16 6.35 8.12 9.63 10.99 11.68 12.18 12.52 6.42 8.27 9.87 11.27 12.03 12.53 12.93
11 13 15 16 17 17 17 12 14 16 17 18 18 19 13 15 17 18 19 20 20 14 16 18 20 21 22 22
11 13 15 16 17 17 18 12 15 16 18 19 19 19 13 16 18 19 20 21 21 15 17 19 21 22 22 23
12 14 15 17 17 18 18 14 15 17 18 19 19 20 15 17 18 20 21 21 21 16 18 20 21 22 23 23
12,5 13,7 15,7 16,9 17,9 17,9 18,9 13,5 15,6 16,7 18,8 19,8 19,9 19,9 14,5 16,6 18,7 19,8 20,8 21,8 21,9 15,5 17,6 19,7 21,7 22,8 22,8 23,8
1.74 0.95 0.62 0.27 0.25 0.18 1.89 3.38 2.45 0.94 1.94 0.75 3.19 0.21 4.72 3.71 2.33 4.26 3.19 1.01 2.79 5.01 4.56 3.58 1.80 1.09 0.41 0.85
1.74 0.95 0.62 0.27 0.25 0.18 0.21 3.38 2.27 0.94 0.67 0.68 0.31 0.21 4.72 2.42 1.48 0.61 0.48 0.61 0.34 4.98 2.47 1.30 0.95 0.85 0.41 0.64
3.07 1.98 0.62 1.56 0.25 0.56 0.21 8.65 2.27 1.81 0.67 0.68 0.31 0.68 8.47 4.72 1.48 1.65 2.28 0.61 0.34 7.73 3.78 2.81 0.95 0.85 1.81 0.64
0.36 0.17 0.16 0.08 0.07 0.07 1.85 0.80 1.34 0.23 1.66 0.61 3.36 0.32 0.98 2.07 1.47 4.70 3.01 0.89 2.92 1.14 2.85 2.74 1.72 0.76 0.11 0.74
0.36 0.17 0.16 0.08 0.07 0.07 0.08 0.80 0.77 0.23 0.17 0.27 0.17 0.32 0.98 0.15 0.26 0.11 0.11 0.23 0.16 0.09 0.29 0.18 0.35 0.34 0.11 0.28
1.22 0.93 0.16 1.29 0.07 0.37 0.08 4.29 0.77 1.05 0.17 0.27 0.17 0.44 3.92 2.67 0.26 0.98 1.68 0.23 0.16 3.65 1.95 1.30 0.35 0.34 1.29 0.28
0.34 0.17 0.16 0.07 0.07 0.07 0.08 0.15 0.24 0.23 0.17 0.27 0.10 0.06 0.08 0.15 0.26 0.11 0.11 0.23 0.11 0.09 0.29 0.18 0.32 0.34 0.11 0.28
2.5,2 2.5,2 2.5,2
1 1 1
4 9 19
9.02 12.17 15.00
12 15 19
12 15 19
13 17 20
12,5 16,7 19,9
2.38 1.56 0.85
2.38 1.56 0.85
3.82 3.06 1.92
0.53 0.67 0.29
0.53 0.67 0.29
1.87 1.93 1.23
0.35 0.35 0.27
2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2 2.5,2
1 1 1 1 1.25 1.25 1.25 1.25 1.25 1.25 1.25 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.75 1.75 1.75 1.75 1.75 1.75 1.75
39 59 79 99 4 9 19 39 59 79 99 4 9 19 39 59 79 99 4 9 19 39 59 79 99
17.59 19.03 20.02 20.76 9.13 12.43 15.42 18.15 19.65 20.69 21.47 9.23 12.66 15.79 18.66 20.23 21.31 22.13 9.32 12.87 16.13 19.12 20.77 21.89 22.75
21 23 24 25 12 16 20 23 24 26 26 13 18 21 24 26 27 28 14 19 23 26 28 29 30
22 23 24 25 12 17 20 23 25 26 27 13 18 22 25 27 28 29 14 19 23 26 28 29 30
22 24 25 26 14 18 21 24 26 27 28 16 20 23 26 28 29 30 17 21 25 28 29 31 31
22,11 23,13 24,13 25,14 13,5 17,7 20,9 23,11 25,11 26,12 27,13 14,5 18,7 22,8 25,10 27,11 28,12 29,12 15,5 20,6 23,8 27,10 28,11 30,11 30,12
0.73 0.30 0.24 0.23 3.39 2.29 1.17 0.69 1.01 0.39 0.80 4.31 2.57 1.71 1.28 0.84 0.85 0.67 4.85 2.91 1.71 1.11 0.77 0.72 0.59
0.58 0.30 0.24 0.23 3.39 2.05 1.17 0.69 0.52 0.39 0.37 4.31 2.57 1.57 0.88 0.77 0.58 0.54 4.85 2.91 1.71 1.11 0.77 0.72 0.59
0.58 0.76 0.62 0.85 4.96 3.13 1.60 1.03 1.26 0.99 1.13 8.29 5.08 2.58 1.62 1.77 1.39 1.46 8.49 4.77 3.73 2.32 1.08 1.82 0.74
0.52 0.12 0.14 0.11 1.17 1.09 0.42 0.34 0.86 0.18 0.72 1.14 0.52 0.94 0.88 0.66 0.68 0.59 1.18 0.75 0.54 0.66 0.40 0.44 0.38
0.28 0.12 0.14 0.11 1.17 0.61 0.42 0.34 0.23 0.18 0.20 1.14 0.52 0.46 0.31 0.35 0.28 0.28 1.18 0.75 0.54 0.66 0.40 0.44 0.38
0.28 0.52 0.43 0.68 1.30 1.25 0.57 0.53 0.90 0.69 0.86 3.06 2.62 1.23 0.88 1.24 1.02 1.12 3.34 2.38 2.20 1.47 0.57 1.32 0.41
0.28 0.12 0.10 0.11 0.37 0.36 0.42 0.32 0.23 0.18 0.19 0.44 0.52 0.45 0.31 0.35 0.28 0.28 0.62 0.63 0.54 0.51 0.36 0.40 0.34
0 to 3.75. Since the cost function is linear in the parameters p and h, one of them is a redundant parameter. We set h at unity and the penalty cost p has values of 4, 9, 19, 39, 59, 79 and 99. We use two demand distributions, pure and compound Poisson with an average demand of 2, 5 and 10 units per review period. For the compound Poisson demand process, customer demands are delayed geometric and we consider 4 parameter settings for (k, l): (1, 2), (2.5, 2), (5, 2) and (2.5, 4). As a result, we consider 336 problem instances for pure Poisson demand and 448 problem instances for compound Poisson demand. Table 3 provides detailed results when R 6 L < 2R, as in Fig. 1. For the different base-stock levels, the table shows the cost increases (CIs) in percentages of the optimal costs C⁄. We observe
that the base-stock levels found with the approximation procedure are smaller than the optimal base-stock levels ð~s 6 sÞ. The backorder model, however, finds base-stock levels larger than the optimal values ðs 6 sBO Þ. This is the case for all test instances. The table also shows that the best RBSP results in policies with almost the same average costs as the optimal policy. We can conclude the same for . The results are not decisive whether the RBSP specified by s and q the approximation procedure or the backorder model performs better. The cost increases CI~s;q~ for pure Poisson demand are irregular, whereas they are at most about 1% for compound Poisson demand. The results for all the different problem settings are summarized in Tables 4 and 5 for pure and compound Poisson demand,
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Table 4 The average results for pure Poisson demand with k 2 D ¼ f2; 5; 10g, lead time L 2 L ¼ f0:25ij0 6 i 6 15g and penalty cost p 2 P ¼ f4; 9; 19; 39; 59; 79; 99g. k
L
p
C⁄
D
L
P
10.82
2.58
1.65
3.47
1.74
0.44
1.92
0.30
2 5 10
L L L
P P P
6.14 10.37 15.94
1.85 2.46 3.44
1.69 1.68 1.57
3.31 3.71 3.38
0.87 1.55 2.81
0.56 0.40 0.37
1.57 2.25 1.96
0.33 0.30 0.27
D D D D D D D D D D D D D D D D
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75
P P P P P P P P P P P P P P P P
8.32 8.85 9.29 9.69 10.05 10.37 10.66 10.92 11.17 11.39 11.60 11.80 11.99 12.17 12.33 12.49
0.00 1.72 1.99 1.41 0.93 2.73 3.12 2.37 1.87 3.38 3.67 3.22 2.75 3.91 4.23 4.01
0.00 0.24 0.55 0.66 0.66 1.21 1.48 1.61 1.53 2.11 2.36 2.48 2.32 2.81 3.18 3.20
0.00 0.58 0.69 1.17 1.10 1.93 2.70 2.65 3.43 4.60 4.42 5.43 5.38 6.71 7.03 7.72
0.00 2.11 2.01 1.35 0.52 2.28 2.36 1.37 1.01 2.35 2.40 1.90 1.33 2.37 2.41 2.09
0.00 0.29 0.07 0.18 0.19 0.31 0.35 0.31 0.41 0.42 0.54 0.76 0.69 0.77 0.88 0.90
0.00 0.23 0.17 0.49 0.56 0.92 1.29 1.31 1.90 2.42 2.32 2.99 3.27 3.99 4.21 4.67
0.00 0.02 0.06 0.11 0.11 0.20 0.17 0.26 0.28 0.36 0.46 0.50 0.55 0.56 0.60 0.60
D D D D D D D
L L L L L L L
4 9 19 39 59 79 99
6.72 8.57 10.19 11.62 12.40 12.92 13.31
5.16 3.68 2.58 1.98 1.79 1.53 1.37
4.47 2.73 1.59 0.94 0.73 0.57 0.52
9.19 5.65 3.33 2.12 1.55 1.41 1.04
2.18 2.17 1.82 1.62 1.64 1.45 1.30
0.86 0.73 0.43 0.28 0.32 0.24 0.24
4.41 3.22 1.91 1.27 0.99 0.97 0.68
0.40 0.47 0.34 0.25 0.24 0.20 0.20
CI~s
CIs
CIsBO
CI~s;q~
CIs;q
CIsBO ;qBO
CIs ;q
Table 5 The average results for compound Poisson demand with ðk; lÞ 2 D ¼ fð1; 2Þ; ð2:5; 2Þ; ð2:5; 4Þ; ð2; 5Þg, lead time L 2 L ¼ f0:25ij0 6 i 6 15g and penalty cost p 2 P ¼ f4; 9; 19; 39; 59; 79; 99g. k, l
L
p
C⁄
CI~s
CIs
CIsBO
CI~s;q~
CIs;q
CIsBO ;qBO
CIs ;q
D 1,2 2.5,2 5,2 2.5,4
L L L L L
P P P P P
22.82 11.03 17.39 25.25 37.63
1.85 1.53 1.87 2.17 1.85
1.74 1.53 1.81 1.80 1.82
3.33 2.45 3.60 3.91 3.36
0.78 0.67 0.71 1.06 0.69
0.59 0.65 0.59 0.51 0.61
1.77 1.08 1.99 2.28 1.74
0.42 0.49 0.41 0.35 0.42
D D D D D D D D D D D D D D D D
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75
P P P P P P P P P P P P P P P P
17.84 18.90 19.81 20.60 21.32 21.95 22.51 23.03 23.52 23.97 24.38 24.77 25.14 25.49 25.82 26.14
0.00 0.31 0.72 0.83 0.88 1.36 1.69 1.85 1.80 2.28 2.59 2.74 2.66 3.05 3.40 3.49
0.00 0.21 0.47 0.71 0.85 1.18 1.50 1.72 1.76 2.14 2.43 2.61 2.61 2.96 3.31 3.40
0.00 0.33 0.81 1.13 1.45 2.03 2.73 2.99 3.36 4.01 4.67 5.16 5.16 6.02 6.54 6.89
0.00 0.27 0.55 0.44 0.36 0.68 0.78 0.80 0.70 0.99 1.05 1.13 1.02 1.14 1.28 1.35
0.00 0.11 0.19 0.25 0.31 0.43 0.45 0.57 0.63 0.74 0.75 0.87 0.91 0.99 1.08 1.13
0.00 0.13 0.35 0.50 0.81 0.97 1.30 1.49 1.79 2.08 2.48 2.79 2.95 3.31 3.58 3.87
0.00 0.06 0.11 0.19 0.24 0.31 0.38 0.44 0.48 0.56 0.58 0.61 0.63 0.67 0.69 0.72
D D D D D D D
L L L L L L L
4 9 19 39 59 79 99
12.07 16.66 20.92 24.87 27.05 28.54 29.68
4.29 2.97 1.92 1.26 0.98 0.82 0.74
4.26 2.86 1.78 1.10 0.85 0.73 0.61
7.83 5.46 3.41 2.19 1.72 1.43 1.27
1.23 1.02 0.85 0.70 0.61 0.55 0.53
1.13 0.81 0.61 0.46 0.41 0.39 0.33
3.45 2.81 1.93 1.34 1.11 0.94 0.86
0.49 0.52 0.49 0.42 0.37 0.33 0.30
respectively. In the first row we take the average over all problem instances. The other rows report average values for specific values of either the average demand, the lead time or the penalty cost.
The results of Tables 4 and 5 show that the costs deviate relatively more when the lead time increases. However, this result is less clear when s and q are based on the approximation procedure for the RBSP in case of pure Poisson demand. The cost increases in this particular setting vary a lot. Furthermore, our approximation procedure for the lost-sales model performs on average better than the backorder model (for the PBSP and RBSP). When the values of the inventory control parameters are based on the backorder model, the cost increases can even run up to 20% for the PBSP and to 11–15% for the RBSP. The results in both tables show that on average the relative cost increase for using a (restricted) base-stock policy decreases when the penalty cost increases. Furthermore, the cost increase for using in the RBSP is about the same as the cost increase for using s and q the best values of s and q. We also observe that the costs for the best RBSP (s⁄, q⁄) deviate on average less than 0.5% from the optimal costs, with an observed maximum deviation of 2.0%. Therefore, we conclude that a RBSP performs excellent. The RBSP outperforms the PBSP in all test problems. To conclude our comparison of the different policies, we compare the computational effort to find the best values for each of the policies. Table 6 reports the average number of iterations (Nopt) and the average CPU time (CPUopt) in seconds to find the optimal policy, as well as the average CPU time in seconds to find the best PBSP and RBSP (denoted by CPUs and CPUs ;q , respectively). All numerical results have been performed on a 2.0 Ghz AMD Opteron 246 processor running Linux and C++. The fourth and fifth line present the results when the average demand is 10, where (k, l) is (10, 1) for pure Poisson demand on line 4, and (5, 2) and (2.5, 4) for compound Poisson demand on line 4 and 5, respectively. We do not present the CPU time for the approximation procedures, since this is within milliseconds. It becomes clear that the computational effort to find the best RBSP can be very large (up to over 3 hours for a single problem instance). This illustrates that the approximation procedure is very efficient. Furthermore, we note that the computational effort to find the best PBSP can be improved significantly when the observation ~s 6 s 6 sBO is used. 4.2. Impact of review period length In this section, we perform a sensitivity analysis to understand how the costs of the optimal replenishment policy changes in response to changes to the review period length. We consider the same cost setting as in the example of Section 2.2 (i.e., p = 19 and h = 1). Set L = 1.5 and let R range between 0.5 and 2.5. Fig. 3 presents the results for the different demand distributions. It is clear that a longer review period results in higher costs due to higher average inventory levels to cover a larger time period with demand uncertainty without reordering. It is interesting to see that the rate of increase is less when there is more variance in the demand. This is consistent with the simulation results of Sezen (2006), who concludes that the variability in the demand process is the most important factor to set the duration of a review period. Our models of Section 2 give an analytical procedure to support these observations for more general inventory systems as studied by Sezen (2006). 5. Conclusions and future research In this paper, we considered a lost-sales inventory model with periodic reviews. Such inventory systems are commonly seen in a retail environment. A retailer has to decide about the review frequency next to the replenishment quantities. We developed a general model for inventory systems where lead times and review periods can be of any length. In particular, we studied systems with a cost objective function consisting of the expected holding and
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Table 6 The computational effort to evaluate the exact models of Section 2 for pure and compound Poisson demand with k l 2 D ¼ f2; 5; 10g, lead time L 2 L ¼ f0:25ij0 6 i 6 15g and penalty cost p 2 P ¼ f4; 9; 19; 39; 59; 79; 99g. kl
L
p
Poisson Nopt
Compound Poisson CPUopt Mean
a
CPUs Max
Mean
CPUs ;q Max
Mean
Nopt Max
CPUopt
CPUs
CPUs ;q
Mean
Max
Mean
Max
Mean
Max
D
L
P
14.45
7.17
151.99
5.20
110.98
30.86
893.70
17.19
29.14
496.89
23.39
689.37
281.93
12016.78
2 5 10 (1) 10 (2)
L L L L
P P P P
19.00 12.85 11.50
0.92 1.81 18.76
4.51 11.97 151.99
0.06 0.78 14.76
0.41 7.91 110.98
0.05 2.32 90.22
0.37 22.70 893.70
25.98 15.38 12.82 14.57
1.42 4.31 41.81 69.02
7.65 38.63 400.81 496.89
0.11 1.94 31.42 60.09
0.99 20.01 372.01 689.37
0.19 7.64 238.97 880.91
1.94 87.26 3027.17 12016.78
D D D D D D D D D D D D D D D D
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75
P P P P P P P P P P P P P P P P
8.52 9.05 9.43 9.86 10.95 11.71 12.43 13.38 14.67 15.19 16.19 17.29 19.43 20.33 20.76 22.00
0.00 0.00 0.00 0.00 0.01 0.02 0.03 0.03 0.19 0.43 0.60 0.79 9.05 24.03 33.94 45.53
0.01 0.01 0.01 0.01 0.02 0.06 0.08 0.05 0.39 1.15 1.81 2.46 21.73 81.70 117.84 151.99
0.00 0.00 0.00 0.00 0.00 0.01 0.01 0.01 0.18 0.32 0.46 0.56 10.07 17.29 23.68 30.64
0.00 0.01 0.01 0.00 0.01 0.02 0.03 0.03 0.56 1.12 1.55 1.68 40.29 65.33 91.11 110.98
0.00 0.00 0.01 0.01 0.18 0.14 0.19 0.26 3.26 3.58 4.87 6.43 69.66 97.01 135.28 172.95
0.01 0.02 0.02 0.04 0.67 0.62 0.84 1.15 14.08 17.30 22.77 30.02 332.02 494.22 697.72 893.70
10.96 11.79 12.04 12.46 13.39 14.32 15.25 16.25 17.61 18.18 19.29 20.54 21.57 22.71 23.82 24.82
0.00 0.00 0.00 0.00 0.02 0.09 0.08 0.08 0.58 2.26 2.97 3.82 31.48 118.95 141.87 164.02
0.01 0.01 0.01 0.01 0.08 0.28 0.22 0.29 2.14 10.44 13.14 15.50 110.40 432.29 474.83 496.89
0.00 0.00 0.00 0.00 0.01 0.02 0.03 0.03 0.61 1.21 1.57 2.18 38.62 99.13 99.69 131.16
0.00 0.01 0.00 0.01 0.04 0.08 0.10 0.12 3.00 4.92 6.63 10.12 176.21 519.41 534.36 689.37
0.02 0.04 0.05 0.07 0.53 1.10 1.41 1.75 14.56 29.36 36.37 43.76 504.39 1033.95 1272.42 1571.08
0.09 0.23 0.31 0.40 3.29 6.99 8.57 10.84 99.64 200.38 244.77 298.35 4074.24 8097.17 10359.09 12016.78
D D D D D D D
L L L L L L L
4 9 19 39 59 79 99
16.17 15.31 14.60 14.13 13.85 13.56 13.52
4.72 5.48 6.62 7.70 8.35 8.85 8.44
77.19 92.89 117.11 129.25 141.48 151.79 151.99
3.12 3.81 5.06 5.51 6.29 6.15 6.47
63.03 61.44 93.90 88.15 95.67 96.50 110.98
6.04 12.18 21.23 32.88 41.41 48.65 53.65
86.75 193.36 349.94 530.33 689.98 803.43 893.70
17.61 17.84 17.63 17.11 16.86 16.67 16.59
7.38 15.92 26.92 36.71 39.19 38.38 39.48
91.27 239.91 427.76 496.89 477.42a 434.88a 431.64a
3.21 8.34 18.18 32.58 41.11 29.37 30.95
38.21 131.25 329.08 516.68 689.37 396.38 476.80
5.63 24.82 85.21 239.62 385.77 544.04 688.40
66.82 306.48 1162.05 3865.65 6904.26 9975.66 12016.78
Due to memory shortage, we had to restrict the state space. As a result, the actual CPU time is higher.
Fig. 3. The expected total costs per unit time for the optimal policy (on left) and the best PBSP (on right) where h = 1, p = 19, L = 1.5 and R 2 [0.5; 2.5].
penalty costs per unit time. This general structure makes it possible for any retailer to determine the review period length, contrary to the existing literature. In practice, pure base-stock policies are often implemented because of their simplicity. In contrast to inventory systems with backorders, such replenishment policies are known to be suboptimal when excess demand is lost. Pure base-stock policies (PBSPs) based on a backorder model perform bad with average cost deviations up to 4–5%. Therefore, the replenishment policy should be adjusted to incorporate lost sales, and we proposed a restricted base-stock policy (RBSP) which limits the order size to a maximum amount q = sR/(L + R). From numerical results we concluded that the costs for the best values of s and q
deviate on average less than 0.5% from the optimal costs. We also developed an approximation procedure to set near-optimal values of s and q with closed-form expressions. The results for the approximation procedure show irregular cost increases compared to the optimal costs for pure Poisson demand, whereas they are at most about 1% for compound Poisson demand. Our recommended policy is the RBSP specified by the base-stock level s of the best PBSP and equal to sR=ðL þ RÞ rounded (up in case of a tie) to the nearest q integer, because the average cost for this policy is similar to the best values of s and q for the RBSP. There are several interesting areas to extent this work. One is to see whether the RBSP is also near optimal when there is a product
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substitution between related products. Another direction for future research is to consider a service level restriction instead of a penalty cost for each unsatisfied demand. Such a penalty cost can be difficult to quantity in practice, since it includes loss of goodwill. The transportation and fixed order cost could also be included to show the impact of the length of the review period.
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