Base-stock policy for the lost-sales inventory system

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Aug 17, 2004 - 1 Introduction. We consider a single-item inventory system with periodic review. ... As explained by Zipkin (2000, §9.6.5) it is essentially impossible to solve exactly the cost model for the ... They also provide a tractable basis.
Base-stock policy for the lost-sales inventory system with periodic review Søren Glud Johansena, ∗, Anders Thorstensonb August 17, 2004

a

Department of Operations Research, University of Aarhus, Ny Munkegade, Bldg. 530, DK-8000 Aarhus C, Denmark b Department of Accounting, Finance and Logistics, Aarhus School of Business, Fuglesangs All´e 4, DK-8210 Aarhus V, Denmark Abstract Periodic review systems in which unfilled demands are treated as lost sales are of importance as building blocks for inventory control and coordination, particularly in the retail sector. When replenishments are provided by a single supplier, it is also natural to assume that orders do not cross in time, i.e., lead times are sequential. We suggest an approximate cost model in order to find good base-stock policies for such a setting, when lead times are gamma distributed, demand is compound Poisson, and a standard cost structure is used. The approximation is based upon a distribution for the inventory on order specified in analogy with the continuous-review case. Little’s formula is then used to find the approximate average inventory on hand and the average lost sales. The base-stock level prescribed by our approximate cost model is easy to find. Several alternative approximate models are suggested or obtained from the literature. In a numerical study we evaluate the approximate models by means of simulation. It is shown that the standard approximations might perform quite badly in some cases, and that our suggested approximation outperforms the alternative approximations for the set of parameter variations tested. Surprisingly, the base-stock levels obtained from our approximation also work well when orders are allowed to cross in time because lead times are independent. Keywords : Approximations; Base-stock; Compound Poisson; Lost sales; Periodic review; Sequential supply system. ∗

Corresponding author. Tel.: +45 8942 3547; fax: +45 8613 1769; e-mail: [email protected]

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1

Introduction

We consider a single-item inventory system with periodic review. Recent developments with access to point of sale information and e-business solutions integrated with Enterprise Resource Planning systems justify continuous rather than periodic review. However, implementation and maintenance of such information systems may require investments. Furthermore, while the access to information may be (almost) continuous, there is often only a limited number of replenishment opportunities, say, once or perhaps twice a day. For many systems a periodic-review model will therefore provide a better representation. Rao (2003) also emphasizes that periodic-review policies may be the preferred choice because they facilitate coordination of order epochs in multi-item or multi-echelon systems. This is an important point, since the single-item model is often a building block for more complex systems. In this paper, we assume that the system under study is controlled by a base-stock policy and that demand is met to the extent that available stock permits. Unsatisfied demand is lost permanently by the system, as is often the case in the retail sector. It can also describe a setting in which demand when the system is out of stock is satisfied by emergency orders outside the normal inventory control system. The inventory position (stock on hand and on order) is monitored at each review epoch, when according to the policy a replenishment order is placed to restore the base-stock s. Hence, several orders may be outstanding at a time if the lead time exceeds the review interval. This policy with review interval R is commonly referred to as an (R, S) policy, where S = s. Rao (2003) gives an extensive analysis of this policy, but his analysis is restricted to the backordering case only. Demand is assumed to be either a pure or a compound Poisson stochastic variable. In some of the models used for comparisons it will be approximated initially by a continuous distribution, and in one case the distribution-free or maximal approximation will be applied (Zipkin, 2000, §6.4.2). The lead time of a replenishment order is gamma distributed and all the units ordered at the same epoch have the same lead time. Our focus is on the case where orders do not cross in time, which is realistic when a single supplier is used. Single sourcing has often been the favored strategy under concepts such as Just-inTime, Supply Chain Management, etc. If replenishments come from different suppliers, then order crossover is common in practice according to Robinson et al. (2001). For the inventory system with backorders, they then recommend computing the base stock by the shortfall distribution, derived by assuming independent lead times, rather than by the lead-time demand distribution. Although in our model we assume that orders do not cross in time, we simulate the effect of independent lead times as well in our numerical study. Persson (1995, p.17) points out that managers may be reluctant to having several orders outstanding simultaneously, since there is a tendency in practice for time between 2

orders to be longer than the lead time. If this reluctance is in some part caused by lack of understanding about the effects, models that take several outstanding orders into account might be useful as decision support tools. As explained by Zipkin (2000, §9.6.5) it is essentially impossible to solve exactly the cost model for the considered, periodic-review inventory system with lost sales. Little is known about the structure of the true optimal policy, and it is probably very complex and difficult to implement. Morton (1971) and Nahmias (1979) suggest a myopic policy which aims to maintain a constant stockout probability. The myopic policy is determined as the solution to an optimization problem at each review epoch and is therefore computationally more demanding than a stationary policy. Other options are the periodic (r, s) and (r, q) policies, where r, s and q denote the reorder point, the order-up-to level and the order quantity, respectively. However, performance expressions for these policies are available only under very restrictive assumptions. For example, Johansen and Hill (2000) give (approximate) expressions for the (r, q) policy with continuous demand and a fixed cost of placing a replenishment order when r < q. This inequality implies that there never will be more than one order outstanding at any time. As explained by Johansen (2001), who studies the single-item system with Poisson demand, the global minimum average cost cannot in general be found in the class of base-stock policies. However, such policies are attractive in many practical settings because they are inexpensive and easy to implement. They also provide a tractable basis for comparisons. Downs et al. (2001) study the periodic model with multiple items, constant lead times, resource constraints and lost sales. They assume, consistent with the growth of information technologies to monitor retail inventories and process orders, that the cost of placing an order is negligible because the investment in the information system is treated as a sunk cost. Hence, there is no penalty for small orders. This argument also presumes that the transportation cost structure is such that small orders are not penalized. Therefore, the authors employ base-stock policies, and they prove that the average cost of each item is convex in its base-stock. Downs et al. also propose a finite-period, base-stock approach that involves two phases. In the first phase, a nonparametric statistical method is used for each item to calculate a closed-form estimate of its expected average cost as a function of the base stock. A linear program designed to minimize the sum of the average costs subject to linear resource constraints is solved in the second phase. When demand is pure or compound Poisson, as assumed in this paper, the cost functions estimated in the first phase could alternatively be computed from our Eq. (9). Another example is provided by the inventory system treated by Frank et al. (2001). In their system there is both deterministic demand, that must be met immediately in each period, and stochastic demand, that when not satisfied

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is treated as lost sales. To facilitate the analysis, the assumption of zero lead times is introduced. With non-zero lead times, our approximate model might be a useful building block for this system. Hadley and Whitin (1963, §5-2) suggest, based on approximations different from ours, how to fix the base-stock s in a periodic-review system under a standard cost structure, when unsatisfied demand is either backordered or lost. Their suggestions are described in §2.4 and §2.5 and are investigated in our numerical study. Our approach to finding a good approximate model is based upon first specifying a distribution for the inventory on order that is analogous to the exact distribution for a related continuous-review model. Little’s formula is then applied to find the approximate average inventory on hand and the average lost sales. We also design two new alternative approximations, one for backordering and one for lost sales, based on the concept of extended lead times (Axs¨ater, 1993 and Rao, 2003). The numerical study shows that our approximate model based on Little’s formula outperforms the alternative approximations, especially the standard textbook approximations. An unexpected result is that the base-stock levels obtained from our approximation also work well when order crossing caused by independent lead times is allowed. The rest of this paper is organized as follows. In §2 we specify the models to be compared. After presenting initial notation, our approximate model is first specified with constant lead times and then extended to the case with stochastic lead times. Next, for the alternative approximations we consider both backorder models and lost-sales models. This section is concluded by showing how the demand probabilities can be computed. Our numerical study is presented in §3. First, we describe the design of the study. Then, the results from the base-case analysis are presented, followed by a presentation of the results from base-case parameter variations. Finally, our conclusions are stated in §4.

2 2.1

Modeling Notation

The lead time L of a replenishment order is gamma distributed with shape parameter α = (E[L])2 /V [L]. We assume that α ≥ 1 and that the lead times of different orders do not cross in time. The times from the demand epochs until the next review epoch, are realizations of a random variable U which is uniformly distributed over the interval from 0 to R (Axs¨ater, 1993 and Rao, 2003). We call L 0 = U + L the extended lead time. Note that E[L0 ] = R/2 + E[L]. Customer demands are Poisson with rate λ. The size of each customer demand is distributed as Y , where Y is either one always (pure Poisson demand) or delayed geometric (stuttering Poisson demand with Y − 1 being geometric). The delayed geometric distribution has been found suitable for the pattern of demand sizes for both fast- and 4

slow-moving items (Johnston et al., 2003). By D T we denote the generic variable for the demand during an interval of random length T . Observe that E[D T ] = λE[Y ]E[T ] and GT (s) = E[max{s − DT , 0}] =

s−1 X

x=0

Pr{DT ≤ x}.

Below we use this notation with T equal to L, L + R and L 0 . By h and π we denote the unit holding cost per unit time and the unit shortage cost, respectively. The cost π equals the price at which units are sold, plus the unit penalty cost incurred when demand is lost, less the variable ordering cost per unit. We assume that order set-up costs are either negligibly small or can be regarded as ’sunk’, and may thus be ignored.

2.2

Constant lead times

First we consider the case with constant lead times being an integer number of review intervals. After units (if any) are delivered and ordered at a review epoch, the inventory on order, IO(s), stays constant until the next review epoch. If unsatisfied demand is backordered, IO(s) is distributed as DL (Zipkin, 2000, §6.7.3). When unsatisfied demand is lost, as assumed in this paper, IO(s) is never larger than the base-stock s. We approximate it as the random variable IO1 (s) with probability mass function ( P1 (s) Pr{DL = x| DL < s}, x = 0, 1, . . . , s − 1 Pr{IO 1 (s) = x} = (1) 1 − P1 (s), x = s, where P1 (s) is our approximation of Pr{IO(s) < s}. Johansen (2003) has investigated base-stock policies for the lost-sales inventory system with continuous review, Poisson demand and Erlangian lead times which do not cross in time. For that system DL has the negative binomial distribution, and the distribution of the inventory on order is specified exactly by Eq. (1). Moreover, PASTA (Wolff, 1982) and Little’s formula (Stidham, 1974) can be used to determine P 1 (s) as the exact fill rate. We use Eq. (1) to obtain the following approximation of the average inventory on order IO 1 (s) = P1 (s)E[DL | DL < s] + (1 − P1 (s))s = s − P1 (s)

GL (s) . Pr{DL < s}

(2)

Eq. (1) also provides that the approximate average demand satisfied per unit time is S1 (s) = P1 (s)

GL (s) − GL+R (s) . R Pr{DL < s}

(3)

Note that the average amount ordered per unit time equals the average demand satisfied per unit time. Therefore, Little’s formula yields that the exact (but unknown) average inventory on order can be expressed as the product of S(s) and the expected lead times 5

of the orders, where S(s) is the exact (but unknown) average demand satisfied per unit time. Assuming that IO 1 (s) = S1 (s)E[L], we obtain from Eqs. (2) and (3) that P1 (s) =

Pr{DL < s}Rs GL (s)(E[L] + R) − GL+R (s)E[L]

(4)

S1 (s) =

[GL (s) − GL+R (s)]s . GL (s)(E[L] + R) − GL+R (s)E[L]

(5)

and

¯ Let I(s) denote the exact (but unknown) long-run average inventory on hand. Our approximation of the average inventory on hand at the beginning of a review period (after any receipt of stock) is I¯1+ (s) = s − IO 1 (s) = s − S1 (s)E[L], whereas our approximation of the average inventory on hand at the end of a review period (before any receipt of stock) is I¯1− (s) = I¯1+ (s) − S1 (s)R = s − S1 (s)(E[L] + R).

(6)

¯ as To obtain a reasonably precise estimate, we approximate I(s) I¯1 (s) = (I¯1+ + I¯1− (s))/2 = s − S1 (s)E[L0 ].

(7)

This is consistent with the approach used by Rao (2003) and Hadley and Whitin (1963, §5-2). However, we observe that with discrete time it is more common to focus on the average inventory on hand at the end of the review periods (Zipkin, 2000, Ch. 9). If the latter definition of the average inventory is preferred, we only need to replace the approximation in Eq. (7) by the approximation in Eq. (6). The exact (but unknown) average demand lost per unit time is E[D R ]/R−S(s). Hence, the long-run average cost per unit time is ¯ + π(E[DR ]/R − S(s)). C(s) = hI(s)

(8)

Our approximation of this average cost is C1 (s) = πE[DR ]/R + hs − (hE[L0 ] + π)S1 (s).

(9)

Johansen (2003) has shown, by a sample-path analysis, that C(s) is convex in s for the system with continuous review, Poisson demand and Erlangian lead times which do not cross in time. His proof can easily be extended to show, for the system studied in this paper, that C(s) is convex and S(s) is concave in s. We cannot use the latter result directly to conclude that the expression in Eq. (5) is concave in s, because this expression is derived from the approximation in Eq. (1) of the distribution of the inventory on order. 6

But we conjecture that S1 (s) is concave in s. This is supported by our numerical study and the curve shown in Figure 1 (see §3.2). Therefore, we suggest to fix the base-stock as s1 = min {s | ∆S1 (s) < a1 } ,

(10)

where a1 = h/(hE[L0 ] + π) and ∆S1 (s) = S1 (s + 1) − S1 (s).

2.3

Stochastic lead times

If the shape parameter α of the gamma distributed lead times is finite, then L is stochastic, implying in general that the inventory on order is not constant during the review intervals. Hence, our motivation for the approximations in Eqs. (5) and (7) of the average demand satisfied per unit time and the average inventory on hand, respectively, does not apply directly. However, it is our experience from numerous numerical examples that the two approximations are good also in case of stochastic lead times. Therefore, for any shape parameter α ≥ 1, our approximation of the long-run average cost is specified by Eq. (9), where S1 (s) is specified by Eq. (5), and we suggest to fix the base-stock as s 1 in Eq. (10). Like Zipkin (2000, §7.4), we let the stochastic lead times be determined by an exogenous and sequential supply system. The evolution of the supply system is independent of our demands and orders because their contribution to the supplier’s overall workload is assumed to be negligible. Our simulation of the lead times in the sequential supply system is described in §3.1.

2.4

The backorder model

It is sometimes suggested that the lost-sales case can be readily represented by a backordering model. Therefore, suppose now that unsatisfied demand is backordered rather than lost. Then the time, from a customer demand until the delivery epoch of the replenishment order triggered by this demand, is distributed as the extended lead time L 0 . Like the ordinary lead times, the extended lead times do not cross in time. This observation provides that the (exact) performance measures derived for the inventory system with continuous review and lead times that do not cross (Zipkin, 2000, §7.4) also hold for the system with periodic review. The performance measures for the latter system, which we identify by subscript 2, are obtained by letting the lead times in the former system be distributed as L0 rather than as L. The average backorders are ¯2 (s) = E[max{DL0 − s, 0}] B and the average inventory on hand is ¯ 2 (s) = GL0 (s). I¯2 (s) = s − E[DL0 ] + B 7

PASTA (Wolff, 1982) provides that the long-run average demand backordered per unit time is λE[max{Y − max{s − DL0 , 0}, 0}] = λE[Y ] − FL0 (s), where FL0 (s) = λ

s−1 X y=0

Pr{Y > y} Pr{DL0 < s − y}

corresponds to what Feeney and Sherbrooke (1966) call the fills in their study of continuous review with independent lead times for orders triggered by different customers. The longrun average cost per unit time is C2 (s) = hI¯2 (s) + π[λE[Y ] − FL0 (s)], where π is the cost of each unit backordered. Hence C2 (s + 1) − C2 (s) = h Pr{DL0 ≤ s} − π∆FL0 (s), where ∆FL0 (s) = λ

s X y=0

Pr{Y > y} Pr{DL0 = s − y},

implying that a local minimum of the cost function is attained by fixing the base-stock as   ∆FL0 (s) h s2 = min s < . (11) Pr{DL0 ≤ s} π

We have been able to conclude only that the global minimum of C 2 (s) is always attained at s = s2 , when demand is pure Poisson. Then Y ≡ 1, implying that ∆F L0 (s) = λ Pr{DL0 = s}, and Pr{DL0 = s}/ Pr{DL0 ≤ s} is decreasing in s because DL0 is strongly unimodal (Keilson and Gerber, 1971 and Rosling, 2002). The latter property follows because DL0 is the convolution of the strongly unimodal variables D U and DL . Our derivation of the optimality condition in Eq. (11) generalizes a standard textbook result for the base-stock model (Silver et al, 1998, §8.3.1). Assuming that demand is continuous, Hadley and Whitin (1963, §5-2) approximate the long-run average demand backordered per review period as E[max{D L+R −s, 0}], and they approximate the average inventory on hand as s − E[D L0 ]. Therefore, their approximate long-run average cost per unit time is C2a (s) = h(s − E[DL0 ]) + πE[max{DL+R − s, 0}]/R. Consequently, they suggest to let the base-stock be the solution to Pr{DL+R > s} = hR/π. When demand is integer, this suggestion corresponds to fixing the base-stock as s2a = min{s| Pr{DL+R ≤ s} > (π − hR)/π}. 8

(12)

2.5

Alternative approximations

We now return to the case of lost sales, and consider three alternative approximations. First, Hadley and Whitin (1963, §5-2) approximate the average demand lost per review period as DL3 (s) = E[max{DL+R − s, 0}], and they approximate the average inventory on hand as I¯3 (s) = s − E[DL0 ] + DL3 (s). Therefore, their approximate long-run average cost per unit time is C3 (s) = h(s − E[DL0 ]) +

π + hR DL3 (s), R

and, when demand is continuous, they suggest to let the base-stock be the solution to Pr{DL+R > s} = hR/(π + hR). When demand is integer, this suggestion corresponds to fixing the base-stock as s3 = min{s| Pr{DL+R ≤ s} > π/(π + hR)}.

(13)

The second alternative approximation is based on the so-called ‘minmax distribution free’ approach. Applying this approach to the Hadley and Whitin approximation, Moon and Gallego (1994, Eq. (13)) suggest for the lost-sales model (for which β = 0 in their notation) fixing the (continuous) base-stock (which they call the ‘target inventory’) as p [π − hR] V [DL+R ] √ s4 = E[DL+R ] + , 2 hRπ where (Zipkin, 2000, page 286) V [DL+R ] = λE[Y 2 ](E[L] + R) + (λE[Y ])2 V [L]. When Y is delayed geometric, E[Y 2 ] = E[Y ](2E[Y ] − 1). A third alternative approximation can be derived from the fact that the periodic review model with lost sales can also be approximated by a continuous review model, where the lead times are specified by the extended lead time L 0 . For that model we define IO 0 (s) as the difference between the base-stock s and the inventory on hand. We approximate IO 0 (s) as the random variable IO0 (s) with probability mass function ( P0 (s) Pr{DL0 = x| DL0 < s}, x = 0, 1, . . . , s − 1 Pr{IO 0 (s) = x} = (14) 1 − P0 (s), x = s, 9

where P0 (s) is our approximation of Pr{IO 0 (s) < s}. Note that Eq. (14) is a simple modification of Eq. (1). Like in Eq. (2) we obtain IO 0 (s) = P0 (s)E[DL0 | DL0 < s] + (1 − P0 (s))s = s − P0 (s)

GL0 (s) . Pr{DL0 < s}

The approximate average demand satisfied per unit time is S0 (s) = P0 (s)

FL0 (s) . Pr{DL0 < s}

Using that IO 0 (s) = S0 (s)E[L0 ], which is an approximation because lost sales cause that E[L0 ] is not the exact mean of the extended lead times experienced, we obtain that P0 (s) =

Pr{DL0 < s}s FL0 (s)E[L0 ] + GL0 (s)

S0 (s) =

FL0 (s)s . FL0 (s)E[L0 ] + GL0 (s)

and

Because the approximate inventory on hand is s − S 0 (s)E[L0 ], the approximate average cost per unit time is C0 (s) = πλE[Y ] + hs − [hE[L0 ] + π]S0 (s), which is similar to Eq. (9). By the same argument as for Eq. (10), we conjecture that the cost function is minimized by the base-stock s0 = min {s | ∆S0 (s) < a1 } ,

(15)

where ∆S0 (s) = S0 (s + 1) − S0 (s).

2.6

Computation of probabilities

Because the number of customer demands is Poisson with rate λ, the number N L of demands during the gamma distributed lead time L has the negative binomial distribution with parameters α = (E[L]) 2 /V [L] and p = λ/(λ + α/E[L]) (Zipkin, 2000, §7.5.1.1). Hence, if V [L] > 0, Pr{NL = 0} = (1 − p)α and we have the useful recursion Pr{NL = n} =

α−1+n p Pr{NL = n − 1}, n = 1, 2, . . . n

If V [L] = 0, NL has the Poisson distribution with mean λE[L]. The number N R of demands during the review interval R has the Poisson distribution with mean λR, whereas 10

Table 1: Parameter values of the base case. Unit holding cost per review period Unit shortage cost Poisson demand rate Demand size Review interval Mean lead time Lead time variance

h=1 π = 20 λ = 10 Y =1 R=1 E[L] = 4 V [L] = 1

the number NU of demands during the uniformly distributed interval U has the probability mass function ∞ X (λR)m −λR Pr{NU = n} = e , n = 0, 1, . . . m+1 m=n

If Y ≡ 1, the lead time demand DL is distributed as NL , DL+R is the convolution of NL and NR , and DL0 is the convolution of NL and NU . It is therefore easy recursively to compute the expressions which specify the base-stocks suggested in Eqs. (10), (11), (12), (13) and (15). The same is true if E[Y ] > 1. Then Pr{D T = x| NT = n} is the same, say pn,x , for any positive T and in particular for T equal to L, L + R or L 0 . Because Y is assumed to be delayed geometric, p 0,0 = 1 and p1,1 = 1 − θ, where θ = 1 − 1/E[Y ]. Moreover, for x = 2, 3, . . .,   n=1  θp1,x−1 , pn,x = θpn,x−1 + (1 − θ)pn−1,x−1 , n = 2, . . . , x − 1   (1 − θ)px−1,x−1 , n = x.

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3.1

Numerical study

Design of the study

In our numerical study we shall examine a base case and some variations thereof. Table 1 specifies the selected parameter values for the base case. The unit holding cost per review period is chosen as the monetary unit. The unit shortage cost is about twice as high as the corresponding cost figure often used as a benchmark value in backorder models. This is to reflect the fact that the shortage cost in the lost sales case also should include the unit contribution margin lost. The demand rate chosen is supposed to indicate a moderate order flow frequency. In the base case a pure Poisson demand process is assumed, since demand is for one unit at a time in this case. The review interval is taken as the time unit and the expected lead time is 4 time units, whereby several orders will typically be outstanding at a time. Note however that there is no requirement for lead time to be a multiple of 11

the review period, since lead time is modeled as a continuous, stochastic variable. In the base case the lead-time variance chosen is supposed to represent a moderate lead-time variation. We investigate settings where the shape parameter α of the gamma distributed lead times is an integer. Then the sequential lead times are Erlangian and can be seen as the sum of α independent phases, where the phases must be sequentially completed, and where the duration of each phase is exponentially distributed with mean E[L]/α. Hence the lead time of an order, placed when j ∈ {0, 1, . . . α} phases remain for the last order placed, is the sum of these phases and α − j additional phases. This observation facilitates simulation of the sequential lead times. It implies that, at any epoch, we can specify a virtual delivery epoch which is the sum of the current epoch, the remaining time of an active phase i ∈ {1, . . . α} and the total time of the phases indexed by {1, . . . α} \ {i}. When phase i is completed, the virtual delivery epoch is increased by a new exponential time generated for phase i, and the active phase is changed to the one indexed by i + 1 if i < α and 1 if i = α. The lead time of an order placed at review epoch t is then the difference between its virtual delivery epoch and t. We determine the base-stock levels from the approximate, analytical models presented in §2 and compute their associated performance measures from our approximate model in §2.2 and §2.3 . For comparison we then use simulation to estimate the true performance measures for the base-stock levels found. In the simulations we have attempted to record the average inventory on hand and the satisfied demand (or lost sales) as closely to their time-continuous values as possible. All simulation results reported are found by a model developed using the Arena simulation software (Version 5.00.02, Kelton et al., 2002). The results are based on 10 replications, where each replication runs the inventory system for 10,000 review periods following a 100 period long warm-up interval, initialized with no orders outstanding. The demand realizations are the same for all reported base-stock values.

3.2

Base case

Figure 1 illustrates for the base case approximate and simulated values of the average satisfied demand S(s) for a range of base-stock values s. The simulated sets of estimates are obtained from a simulation where the lead times are generated as Erlang distributed random variables that are either sequential (see §3.1), as assumed in our analytical models, or independent. According to Table 1 the Erlang distribution has mean E[L] = 4 and shape parameter α = E[L]2 /V [L] = 16 in the base case. As shown in Figure 1, the approximate values S0 (s) and S1 (s) are quite close and their plots also fairly closely follow that of the simulated estimates. Note, that the function S 1 (s) exhibits concavity, as conjectured in §2.2. For the range of base-stock values shown in Figure 1, the approximate values 12

S(s)

10.0





















 

















 



















































9.8 











































9.6































9.4 

 

















9.2

= Simulated value of S(s) with independent lead times























= S0 (s): Approximation with extended lead times



9.0 





= S1 (s): Our approximation















8.8

= Simulated value of S(s) with sequential lead times



s 55

60

65

70

75

80

Figure 1: Approximate and simulated values of S(s) for the base case.

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S1 (s) are closest to the simulated estimates with sequential lead times. For the high base-stock values they are very close to these estimates. For the lower base-stock levels they slightly overestimate the values from the simulation, though. This indicates the approximate nature of the values obtained from Eq. (5). Nevertheless, as shown by the results we report below, the approximation seems to work well enough for determining the best base-stock values. As expected, allowing order crossing caused by independent lead times gives higher estimated values of S(s) than those obtained when order crossing is not possible because lead times are sequential. This is the same effect as the one noted by Zipkin (1986) and Robinson et al. (2001). Figure 2 illustrates for the base case our approximate average cost C 1 (s), the exact and approximate average costs, C2 (s) and C2a (s), for the backorder model, and the average cost C3 (s) obtained by the lost-sales approximation suggested by Hadley and Whitin (H&W). In addition, Figure 2 includes the average cost approximation C 0 (s) derived by specifying the lead times as the extended lead time L 0 in the approximate lost-sales model with continuous review. (The cost C 2 (s) is not illustrated for s < 57 and the costs C 2a (s) and C3 (s) are not illustrated for s < 60 because they are larger than 50.) The base-stock values determined by Eqs. (15), (10), (11), (12) and (13) in the base case are s 0 = 63, s1 = 63, s2 = 70, s2a = 72 and s3 = 72. Using the ‘minmax distribution free’ approach in §2.5 gives s4 = 76 (rounded). Finally, Figure 2 also shows simulated estimates of C(s), for the base case under independent and sequential lead times, respectively. Two distinct groups of cost curves can be identified in Figure 2. On the one hand, there are the simulated estimates of C(s) together with the two approximations C 0 (s) and C1 (s). On the other hand there are the three approximations C 2 (s), C2a (s), and C3 (s). Except for the high base-stock levels giving very high fill rates, the latter group apparently gives quite bad estimates of the true average cost C(s). In addition, all the approximations in this group prescribe base-stock levels that are much too high compared to the best levels. Notably, the H&W lost-sales approximation C 3 (s) shows the worst performance in Figure 2, even worse than the corresponding backorder approximation C 2a (s). The approximation C2 (s) using backordering with sequential lead times is slightly better. Note also, that the base-stock value s4 = 76 prescribed by the ’minmax distribution free’ approach gives (as expected) the highest cost of all the base-stock values suggested. In the first group of estimates, the simulation with sequential lead times clearly gives a higher cost than the simulation with independent lead times, as expected. The two approximations C0 (s) and C1 (s) yield cost estimates that, except for the lowest base-stock values, lie between the two simulated estimates. The two approximate cost estimates are quite close, even though their difference seems to increase slightly for lower base-stock levels. However, the latter approximation seems to dominate the former, since its costs are closer to the simulated costs with sequential lead times, which is the focal setting here.

14

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= C3 (s): H&W’s lost-sales approximation = C2a (s): H&W’s backordering approximation = C2 (s): Backordering with sequential lead times = Simulated value of C(s) with sequential lead times = C1 (s): Our approximation = C0 (s): Approximation with the extended lead time = Simulated value of C(s) with independent lead times

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Figure 2: Approximate and simulated values of C(s) for the base case.

15

(This observation is also confirmed but not reported in the numerical cases studied in §3.3.) In terms of the base-stock levels prescribed, all the estimates in this group indicate similar levels, although the two approximations show a tendency to underestimate the best base-stock level. Based on these results, we conclude that the most interesting further comparisons are between the base stocks obtained from using C 1 (s), C2 (s), and C3 (s), respectively, evaluated by the performance measures estimated from the simulation.

3.3

Variations of the base case parameters

Under variations of the base case parameters, Table 2 reports the approximate values S1 (s) and C1 (s) for the base-stock values s1 , s2 , and s3 determined by Eqs. (10), (11), and (13), respectively. Table 2 also reports the corresponding simulated estimates of S(s) and C(s) under sequential and independent lead times. All simulation results have 95% confidence intervals which are quite small. The half-widths are all less than 1.5% of the estimated means and most are considerably smaller. The first case shown in Table 2 essentially repeats results for the base case from Figures 1 and 2. We then vary parameters one at a time by halving and doubling, respectively, the unit shortage cost π , the Poisson demand rate λ and the expected lead time E[L] in Cases 2-7. In Case 8 the lead-time is constant, whereas in Case 9 the lead-time variance V [L] is increased by a factor 4 compared to the base case. The last two cases, 10 and 11, consider two compound Poisson demand processes with the expected size of each customer demand E[Y ] equal to 2 or 4, but with the same average demand as in the base case. Parameters not mentioned in the table retain their base-case values. We first note that the base-stock level s 1 prescribed by our approximation is lower, in some cases considerably lower, than the levels s 2 and s3 found from the sequential backorder model and the H&W lost-sales approximation, respectively. The latter one in particular can suggest significantly higher levels. As regards the average satisfied demand S(s), the same conclusions apply as in Figure 1. Our approximate model gives values that are quite close to but in most cases slightly overestimates the simulated, sequential estimates. The simulated, independent estimates are always higher than the sequential ones, except in the case where lead times are constant, when they are equal, of course. Compared to the values obtained from our approximate model, the simulated, independent estimates are always higher, except in the short lead-time case (Case 6) and the constant lead-time case (Case 8) when they are equal or slightly lower. All the values reported correspond to fill rates from 91.5% (Case 2) and upwards. The last three columns in Table 2 show our approximate cost C 1 (s) and the simulated costs for the base-stock levels prescribed by the three approximations in the different cases. Our approximate cost gives a fairly accurate and consistent estimate of the simulated cost 16

Table 2: Approximate (appr.) and simulated results with sequential and independent lead times under variations in the base-case parameters. s appr. 1 base case 2 π = 10 3 π = 40 4 λ=5 5 λ = 20 6 E[L] = 2 7 E[L] = 8 8 V [L] = 0 9 V [L] = 4 10 λ=5 E[Y ] = 2 11 λ = 2.5 E[Y ] = 4

s1 = 63 s2 = 70 s3 = 72 s1 = 56 s2 = 64 s3 = 67 s1 = 69 s2 = 75 s3 = 77 s1 = 33 s2 = 36 s3 = 38 s1 = 124 s2 = 137 s3 = 140 s1 = 40 s2 = 43 s3 = 43 s1 = 105 s2 = 121 s3 = 130 s1 = 57 s2 = 62 s3 = 62 s1 = 70 s2 = 79 s3 = 90 s1 = 67 s2 = 75 s3 = 79 s1 = 73 s2 = 82 s3 = 89

9.62 9.84 9.87 9.24 9.66 9.76 9.81 9.92 9.94 4.80 4.89 4.93 19.33 19.71 19.77 9.80 9.90 9.90 9.31 9.72 9.85 9.80 9.94 9.94 9.22 9.53 9.75 9.50 9.75 9.83 9.29 9.59 9.75

S(s) simulated sequential 9.58 ± 0.03 9.82 ± 0.03 9.86 ± 0.03 9.15 ± 0.03 9.62 ± 0.03 9.74 ± 0.03 9.79 ± 0.03 9.91 ± 0.03 9.94 ± 0.03 4.78 ± 0.02 4.89 ± 0.02 4.93 ± 0.02 19.23 ± 0.05 19.67 ± 0.05 19.73 ± 0.05 9.76 ± 0.02 9.88 ± 0.02 9.88 ± 0.02 9.26 ± 0.03 9.71 ± 0.02 9.85 ± 0.02 9.75 ± 0.02 9.93 ± 0.02 9.93 ± 0.02 9.20 ± 0.02 9.52 ± 0.02 9.75 ± 0.02 9.48 ± 0.02 9.75 ± 0.02 9.84 ± 0.02 9.29 ± 0.04 9.60 ± 0.05 9.76 ± 0.05

simulated independent 9.73 ± 0.02 9.93 ± 0.02 9.96 ± 0.02 9.29 ± 0.02 9.77 ± 0.02 9.87 ± 0.02 9.91 ± 0.02 9.98 ± 0.03 9.99 ± 0.03 4.84 ± 0.01 4.93 ± 0.02 4.97 ± 0.02 19.57 ± 0.04 19.91 ± 0.04 19.94 ± 0.04 9.78 ± 0.02 9.90 ± 0.02 9.90 ± 0.02 9.64 ± 0.01 9.96 ± 0.03 10.00 ± 0.03 9.75 ± 0.02 9.93 ± 0.02 9.93 ± 0.02 9.83 ± 0.02 9.97 ± 0.03 10.01 ± 0.03 9.58 ± 0.02 9.83 ± 0.03 9.91 ± 0.03 9.36 ± 0.04 9.67 ± 0.05 9.81 ± 0.05

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appr. 27.21 29.01 30.09 22.07 23.88 25.44 32.26 33.73 34.81 15.43 16.16 17.20 50.35 54.01 55.67 19.59 20.36 20.36 39.72 43.88 49.32 16.83 18.36 18.36 44.07 45.62 51.08 34.33 36.05 38.10 45.33 46.94 50.21

C(s) simulated sequential 28.56 ± 0.21 29.66 ± 0.14 30.60 ± 0.13 23.37 ± 0.12 24.58 ± 0.10 25.95 ± 0.08 33.62 ± 0.32 34.36 ± 0.24 35.25 ± 0.22 15.94 ± 0.14 16.43 ± 0.10 17.36 ± 0.07 52.99 ± 0.20 55.25 ± 0.16 56.69 ± 0.20 20.62 ± 0.15 20.89 ± 0.11 20.89 ± 0.11 41.17 ± 0.25 44.34 ± 0.23 49.50 ± 0.32 18.24 ± 0.15 18.87 ± 0.10 18.87 ± 0.10 45.11 ± 0.46 46.28 ± 0.37 51.43 ± 0.31 35.29 ± 0.33 36.51 ± 0.28 38.35 ± 0.25 45.84 ± 0.59 47.19 ± 0.49 50.27 ± 0.42

simulated independent 25.00 ± 0.17 27.02 ± 0.13 28.33 ± 0.14 21.50 ± 0.13 22.53 ± 0.11 24.11 ± 0.11 28.48 ± 0.20 31.22 ± 0.17 32.75 ± 0.16 14.57 ± 0.14 15.32 ± 0.06 16.48 ± 0.04 44.74 ± 0.31 49.42 ± 0.22 51.59 ± 0.23 20.16 ± 0.17 20.41 ± 0.11 20.41 ± 0.11 30.68 ± 0.23 37.38 ± 0.27 45.27 ± 0.29 18.24 ± 0.15 18.87 ± 0.10 18.87 ± 0.10 29.58 ± 0.15 35.09 ± 0.19 45.16 ± 0.19 32.91 ± 0.30 34.60 ± 0.19 36.79 ± 0.15 44.22 ± 0.66 45.73 ± 0.49 49.10 ± 0.37

with sequential lead times. In the eleven cases our approximation underestimates the simulated costs by 1.1%-7.7%. The relative cost increase obtained by using the basestock levels prescribed by the backorder model with sequential lead times instead of using our approximation is in the range 1.3%-7.7%. Using the H&W lost-sales approximation the relative cost increase is in the range 1.3%-20.2%. The smallest cost increases occur in the case with short expected lead time (Case 6) and constant lead times (Case 8) and the largest increases occur for long lead times (Case 7) and low unit shortage cost (Case 2). We conclude that our approximation clearly outperforms the standard textbook approximations. In particular, using the standard lost-sales approximation can lead to significant cost increases. It is noteworthy, and somewhat surprising, that our approximation developed for the case of no order crossing, i.e., sequential lead times, also appears to work quite well for the case of order crossing caused by independent lead times. In general, the relative cost increases of using the alternative approximations to obtain the base-stock level instead of using our approximation are larger in most cases when lead times are independent than when they are sequential. Using the backorder model the relative cost increases are in the range 1.2%-21.8% and using the H&W lost- sales approximation the relative cost increases are in the range 1.2%-52.7%. Hence, using our approximation substantial cost savings can be obtained. The largest relative cost savings occur when the expected lead time is long (Case 7) or lead-time variability is high (Case 9). However, with independent lead times our cost approximation is less accurate on average and quite variable between cases. The relative cost deviations are in the range from -7.7% to +49%. In fact, the two largest cost savings using our approximation are obtained when the cost is misrepresented by 29.5% and 49%, respectively (Cases 7 and 9). Figures 3 and 4 take a closer look at Cases 5 and 8, respectively, using a format similar to that of Figure 2. Cases 5 and 8 were chosen because they represent cases when our approximation gives a relatively large misrepresentation of the simulated costs with sequential lead times. This precaution was taken so as not to overstate the results. Generally, the plots in these figures confirm the conclusions drawn from Figure 2. In this sense the figures help validate our suggested approximation. Figure 3 also emphasizes the effect of using our approximation in the case of independent lead times and order crossing. Even if the cost level is not approximated very well, the base-stock level found by our approximation is quite accurate. The more pronounced curvature of the simulated costs with independent lead times then accounts for the large relative cost savings obtained. Note that Figure 4 only includes one plot of simulated values, since Case 8 is the case with constant lead times. This case shows that even with constant lead times our approximation outperforms the standard textbook approximations.

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C(s)

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C3 (s): H&W’s lost-sales approximation = C2 (s): Backordering with sequential lead times = Simulated value of C(s) with sequential lead times = C1 (s): Our approximation = C0 (s): Approximation with the extended lead time = Simulated value of C(s) with independent lead times



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Figure 3: Approximate and simulated values of C(s) for Case 5 (λ = 20).

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Figure 4: Approximate and simulated values of C(s) for Case 8 (V [L] = 0).

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4

Conclusion

In this paper we have designed an approximate cost model for the single-item, periodic review, base-stock inventory control system in which unfilled demands are lost and replenishment lead times do not cross. Lead times are gamma distributed and demand is compound Poisson. The approximation is based on a distribution for the inventory on order motivated by an analogy with the corresponding inventory control system under continuous review. Using Little’s formula we have derived the approximate average inventory on hand and the average lost sales. For a standard cost structure the sum of the inventory holding and shortage costs can then be used to obtain a good base-stock level. This base-stock level is as easy to find as those typically prescribed by alternative, approximate cost models. We have considered four alternative approximations: The standard lost-sales and backordering approximations, the exact backordering model with sequential lead times and a lost-sales approximation using the concept of extended lead times. In a numerical study we have examined a base case and ten parameter variations. We used simulation to evaluate the different approximations. The lost-sales approximation using extended lead times appears to work almost as well as our suggested approximation. However, applying the base-stock levels prescribed by the standard approximations does not seem to work well in general. In some instances it gives relative cost increases compared to using our approximation that are as high as 10-20%. It is also noteworthy that both backordering approximations seem to work better than the standard lost-sales approximation. Moreover, in the case of independent lead times, although our approximation represents the total costs less well in this case, the base-stock levels it prescribes give higher relative cost savings than those obtained in the case of sequential lead times. Overall, the results from the numerical study show that determining the base stock as suggested by our approximation is far more reliable than using the alternative approximations tested.

References Axs¨ater, S., 1993. Optimization of order-up-to-S policies in two-echelon inventory systems, Naval Research Logistics 40, 245-253. 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 47:3, 464-479. Feeney, G.J., Sherbrooke, C.C., 1966. The (s − 1, s) inventory policy under compound Poisson demand, Management Science 12:5, 391-411.

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Frank, K.C., Zhang, R.Q., Duenyas, I., 2003. Optimal policies for inventory systems with priority demand classes, Operations Research, 51:6, 993-1002. Hadley, G., Whitin, T.M., 1963. Analysis of Inventory Systems. wood Cliffs.

Prentice-Hall, Engle-

Johansen, S.G., 2001. Pure and modified base-stock policies for the lost sales inventory system with negligible set-up costs and constant lead times, International Journal of Production Economics 71, 391-399. Johansen, S.G., 2003. Base-stock policies for the lost sales inventory system with Poisson demand and Erlangian lead times, accepted for publication in International Journal of Production Economics. Johansen, S.G., Hill, R.M., 2000. The (r, Q) control of a periodic-review inventory system with continuous demand and lost sales, International Journal of Production Economics 68, 279-286. 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 54:8, 833-837. Keilson, J., Gerber, H., 1971. Some results for discrete unimodality, Journal of the American Statistical Society 66, 386-389. Kelton, W.D., Sadowski, R.P., Sadowski, D.A., 2002. Simulation with Arena, Second Ed. McGraw-Hill, Boston. Moon, I., Gallego, G., 1994. Distribution free procedures for some inventory models, Journal of the Operational Research Society 45:6, 651-658. Morton, T.E., 1971. The near-myopic nature of the lagged-proportional-cost inventory problem with lost sales, Operations Research 19, 1708-1716. Nahmias, S., 1979. Simple approximations for a variety of dynamic leadtime lost-sales inventory models, Operations Research 27:5, 904-924. Persson, G., 1995. Logistics process redesign: Some useful insights, International Journal of Logistics Management 6:1, 13-26. Rao, U.S., 2003. Properties of the periodic review (R, T ) inventory control policy for stationary, stochastic demand, Manufacturing & Service Operations Management 5:1, 37-53.

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Robinson, L.W., Bradley, J.R., Thomas, L.J., 2001. Consequences of order crossover under order-up-to inventory policies, Manufacturing & Service Operations Management 3:3, 175-188. Rosling, K., 2002. Inventory cost rate functions with nonlinear shortage costs, Operations Research 50:6, 1007-1017. Silver, E.A., Pyke, D.F., Peterson, R., 1998. Inventory Management and Production Planning and Scheduling, Third Ed. Wiley, New York. Stidham, S., 1974. A last word on L = λW, Operations Research 22:2, 417-421. Wolff, R.W., 1982. Poisson arrivals see time averages, Operations Research 30:2, 223-231. Zipkin, P.H., 1986. Stochastic leadtimes in continuous-time inventory models, Naval Research Logistics Quarterly 33:4, 763-774. Zipkin, P.H., 2000. Foundations of Inventory Management. McGraw-Hill, Boston.

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