Comparison of wait time approximations in distribution networks using (R,Q)-order policies
CComparison of wait time approximations in distribution networks using(R,Q)-order policies
Christopher Grob a, ∗ , Andreas Bley b a Volkswagen AG, Wolfsburger Straße, 34219 Baunatal, Germany b Universit¨at Kassel, Heinr.-Plett-Straße 40, 34132 Kassel, Germany
Abstract
We compare different approximations for the wait time in distribution networks, in which all warehouses usean (R,Q)-order policy. Reporting on the results of extensive computational experiments, we evaluate thequality of several approximations presented in the literature. In these experiments, we used a simulationframework that was set-up to replicate the behavior of the material flow in a real distribution network ratherthan to comply with the assumptions made in the literature for the different approximation. First, we usedrandom demand data to analyze which approximation works best under which conditions. In a second step,we then checked if the results obtained for random data can be confirmed also for real-world demand datafrom our industrial partner. Eventually, we derive some guidelines which shall help practitioners to selectapproximations which are suited well for their application. Still, our results recommend further testing withthe actual application’s data to verify if a chosen wait time approximation is indeed appropriate in a specificapplication setting.
Keywords:
Inventory Management, Wait time approximations, Distribution System, (R,Q)-order policy,Supply Chain Management ∗ Corresponding author
Email addresses: [email protected] (Christopher Grob), [email protected] (AndreasBley) a r X i v : . [ s t a t . A P ] J a n . Introduction Over the last decades multiple approximations for wait time in distribution networks that employ (R,Q)-order policies have been introduced in the literature. While each publication reports some numerical resultsand comparison, no comprehensive study analyzing experimentally which approximation works best in whichsituation has been published until now. With this paper we try to provide such a comparison.Our motivation for this study arose during the work on an algorithm to determine optimal reorder pointsin distribution networks with (R,Q)-inventory policy (Grob & Bley (2018)). This generic algorithm workswith a wide range of inventory models and wait time approximations. In our computational experimentswe observed that the optimal reorder points depend heavily on the chosen wait time approximation and,furthermore, that the quality of each approximation heavily depends on the characteristics of the actualnetwork. Wait time is the stochastic delays in deliveries to local warehouses caused by stock-outs at thecentral warehouse. This naturally led to the question which wait time approximations works best in whichsituation and, if possible, to derive some simple rules and guidelines for practitioners to select a suitableapproximation in a real-world application. The need for such a comparison was also confirmed by thereviewers of that paper.The traditional approach used in many multi-echelon optimization publications is to approximate thestochastic effective lead time by its mean, as first done in the METRIC model by Sherbrooke (1968). Deuer-meyer & Schwarz (1981) later applied METRIC to divergent multi-echelon systems with batch ordering.Svoronos & Zipkin (1988) presented several methods to derive the exact distributions for the case of identicallocal warehouses and Poisson distributed demands. As these approximations are computationally rathercostly, Svornos and Zipkin also proposed approximations for the mean and the variance of the wait time innetworks with Poisson distributed demands. Axs¨ater (2003) derived wait time approximations using a nor-mal approximation. All of these approaches are based on Little’s law and therefore lead to identical averagewait times for all local warehouses. In many practical applications, however, the frequencies and sizes ofcustomer demands as well as the order quantities of local warehouses vary widely, which makes this approachunsuitable. Andersson & Marklund (2000) therefore proposed a heuristic approach, which adjusts the waittimes derived from Little’s law for each local warehouse based on average demand and order quantity, butwhich neglects the variance of the wait times.Three more recent publications introduce approximations that provide mean and variance of the waittime individually for each local warehouse and that do not assume Little’s law. Kiesm¨uller et al. (2004)proposed an approximation for an n -level network based on a mixed Erlang distribution of the aggregatedemand. Berling & Farvid (2014) derived several methods to estimate the wait time in a 2-level networkand their numerical analysis shows good results for an approximation based on a log-normal distribution.Grob & Bley (2018) proposed to approximate the aggregate lead time demand at the central warehouse bya negative Binomial distribution and then used a variant of the approximation of Kiesm¨uller et al. (2004) toestimate the wait time.The remained of this paper is organized as follows. In Section 2 we present our inventory model andintroduce the necessary formulas to set-up and evaluate our simulations. In Section 3 we review the differentwait time approximations from the literature that are considered in this study. The simulation framework isdescribed in Section 4, while the experimental results are discussed in-depth in Section 5.
2. Inventory model
In our model, we consider a two-level distribution network. A central warehouse is supplied with partsfrom an external source. The external supplier has unlimited stock, but the lead time is random with knownmean and variance. The central warehouse distributes the parts further on to n local warehouses, whichfulfill the stochastic stationary customer demands. Figure 1 shows a sample network. Let the inventoryposition be defined as stock on hand plus outstanding orders minus backorders. The central as well as thelocal warehouses order according to a continuous review ( R, Q )-inventory policy: Whenever the inventoryposition is R or below R , a given lot size Q is ordered from the predecessor in the network until the inventoryposition is above R again.The local warehouses i = 1 , , ..., n order from the central warehouse 0, whereas the central warehouseorders from the external supplier. If sufficient stock is on hand, the order arrives a random lead time withknown distribution, L or L i , respectively, later. In this paper, we assume complete deliveries. If stock isinsufficient to fulfill the complete order, a backorder is placed and the (complete) order is fulfilled as soon as2 xternal supplier Central warehouse Local warehouses Customer demand Figure 1: Example of a 2-level distribution network enough stock is available again. Orders and backorders are served first-come-first-serve. If a stock-out occursat the central warehouse, orders of local warehouses have to wait for an additional time until sufficient stockis available again. This additional time is modeled as a random variable, the wait time W i . The effectivestochastic lead time therefore is L effi = L i + W i . L i and W i are assumed to be independent random variables.We also assume unlimited stock at the supplier. Hence, the wait time of orders of the central warehouse is 0.The local warehouses fulfill external customer demand. Unfulfilled customer orders are backordered untilsufficient stock is on hand and are served on a first-come-first-serve basis, again.The following table summarize our notation: R i reorder point of warehouse i , Q i order quantity of warehouse i , p i price of the considered part at warehouse i ,¯ β i fill rate target of warehouse i , µ i expected customer demand at warehouse i during one time unit, σ i standard deviation of customer demand at warehouse i during one time unit, L i lead time of warehouse i , W i ( R ) wait time of local warehouse i , which depends on the central reorder point, L effi ( R ) effective lead time of local warehouse i , defined as L i + W i ( R ), D i ( t ) demand at the local warehouse i during a time period of length t , and I i inventory level of warehouse i .We assume that customer demands arrive according to a compound Poisson process. The probability foran order of size k at node i is denoted by pdf K i ( k ). If the order size K i follows a logarithmic distribution,the demand during a given time period (for example the lead time) is negatively binomial distributed.Rossetti & ¨Unl¨u (2011) have studied the adequacy of different distributions and distribution selection rules.They considered a single warehouse that uses ( R, Q )-order policy and performed simulations to determinehow suitable certain distributions and distribution selection rules are to model lead time demand. Theyconclude that the negative binomial distribution has “excellent potential”, especially if demand variability ishigh. Also, our industrial partner has obtained promising results in previous studies by using the negativebinomial distribution to model lead time demand.In order to derive a formula for the fill rates at the local warehouses, we first need to estimate the demandduring the effective lead time D i ( L effi ). The mean and the variance of this so-called lead time demand are E [ D i ( L effi )] = µ i E [ L effi ] andVar[ D i ( L effi )] = σ i E [ L effi ] + µ i V ar [ L effi ] . The probability that the inventory level, which is the stock on hand minus the backorders, at node i is equal3o j can be expressed as P r ( I levi = j ) = 1 Q i R i + Q i (cid:88) l = R i +1 P r ( D i ( L effi ) = l − j ) . This follows from the uniformity property of the inventory position (Axs¨ater (2006)).If only complete deliveries are permitted, an order can only be fulfilled if the inventory level j is at leastas big as the order size k . With this approach, also referred to as order fill rate , i.e., the fraction of ordersthat can be fulfilled directly from stock on hand, we obtain β i ( R i ) = R i + Q i (cid:88) k =1 R i + Q i (cid:88) j = k pdf K i ( k ) · P r ( I lev = j )= R i + Q i (cid:88) k =1 R i + Q i (cid:88) j = k pdf K i ( k ) 1 Q i R i + Q i (cid:88) l = R i +1 P r ( D i ( L effi ) = l − j ) . (2.1)In order to calculate the central depot’s fill rate, we have to determine the mean and the variance ofthe central lead time demand first. Let L be a random variable denoting a time period and let pdf L ( l )denote the probability density function of L . We assume that the support of this function is positive. Dueto the uniformity property of the inventory position, ( I P osi ( t ) − R i ) is uniformly distributed on (0 , Q i ]. Theprobability of depot i placing at most k orders, conditioned on L = l , is δ i ( k | L = l ) = Q i (cid:88) x =1 Q i P r ( D i ( l ) ≤ kQ i + x −
1) for k = 0 , , , . . . . (2.2)By integrating or summing over all possible times l , we obtain δ i ( k ) = (cid:90) ∞ l =0 (cid:32) Q i (cid:88) x =1 Q i P r ( D i ( l ) ≤ kQ i + x − · pdf L ( l ) (cid:33) d l (2.3)for a continuous distribution of the lead time (cp. (Ross, 2014, p.115)) or the analogous sum for a discretedistribution. Equation (2.2) or (2.3), respectively, imply that the probability that exactly k subbatch ordersof size Q i are placed from depot i is s ordi ( k ) = δ i (0) , if k = 0 ,δ i ( k ) − δ i ( k − , if k > , , otherwise.Using these terms, the mean and the variance of the lead time demand at the central warehouse can beexpressed as E [ D ( L )] = N (cid:88) i =1 µ i E [ L ] and (2.4)Var[ D ( L )] = N (cid:88) i =1 (cid:34) ∞ (cid:88) k =0 ( µ i E [ L ] − kQ i ) s ordi ( k ) (cid:35) . (2.5)Note that, if the common subbatch size q = gcd ( Q , . . . , Q n ) of all local order quantities is greater than 1,the above calculations should be done in units of q , as this leads to a better the approximation of the fillrate.Similar to Berling & Marklund (2013), we fit standard distributions to the mean and the variance oflead time demand using the following distribution selection rule: If V ar [ D ( L )] > E [ D ( L )], we use thenegative binomial distribution (which only exists if this condition holds) and use (2.1) to calculate the fillrate. In all other cases, we propose to use the gamma distribution. This same rule is applied to choose thedistributions for the local warehouses. 4 . Wait time approximations In this section we present the four wait time approximations that we consider in our analysis. We haveselected one METRIC-type approximation by Axs¨ater (2003) as a representative for this class of approxima-tions and the three approximations proposed by Berling & Farvid (2014), by Kiesm¨uller et al. (2004) and byGrob & Bley (2018).For the sake of simplicity, all approximations are presented only for the 2-level case. Note that theapproximation by Berling and Farvid is only valid for 2-levels, while the approximation by Kiesm¨uller et al.is also applicable for the more general n -level case. Axs¨ater introduced his approximation for the 2-level case,but he also sketched an extension to n -level case at the end of his paper. Also the approximation by Grob &Bley has been introduced for 2 levels and can, in principle, be extended to the n -level case, but the qualityof the approximation is rather bad for more than 2 levels. For benchmark and comparison purposes a METRIC-type approximation based on Little’s law is used. Itwas originally introduced by Sherbrooke (1968). Deuermeyer & Schwarz (1981) were the first to adapt it forbatch ordering systems and it was widely used in research publications since then. Deuermeyer and Schwarzassumed identical demand and order quantities at local warehouses. If these parameters are not identicalamong the local warehouses, the formula only yields the average delay for all warehouses. This means thatone may get significant errors if the order quantities of local warehouses differ substantially. Nevertheless,this approach is still commonly used as an approximation for the expected wait time. Due to their simplicity,the METRIC-type approximations are probably the most widely used approach in practical applications.There exist some approaches which also model the variance of the average delay. Svoronos & Zipkin(1988) derive the variance based on an extension of Little’s law for higher moments, but it is only exact if theorder quantity of all local warehouses is 1. Axs¨ater (2003) derives the variance of the average delay based ona normal approximation of lead time demand.We have chosen to use Axs¨ater’s approximation as the representative for METRIC-type approximations.He derives mean and variance based on Little’s law, the normal approximation of lead time demand, constantcentral lead time, and a continuous uniform approximation of the (discrete) central inventory position. Here,the wait times are identical for all local warehouses, therefore we drop the subscript i of W i . Under theseassumptions we obtain E [ W ] = E [( I ) − ] (cid:80) ni =1 µ i = G ( k ( R )) (cid:80) ni =1 µ i n (cid:88) i =1 σ i E [ L ] , and (3.1) V ar [ W ] = (cid:18) E [ W ] G ( k ( R )) (cid:19) (1 − Φ( k ( R ))) − ( E [ W ]) k ( R ) G ( k ( R )) − ( E [ W ]) , (3.2)where G ( · ) is the normal loss function and k ( R ) = R + Q − (cid:80) ni =1 ( µ i E [ L ]) (cid:80) ni =1 σ i E [ L ] . Axs¨ater compares the approximation to the exact solution for a problem instance with very low demand,as the exact solution is computable in this case. He concludes that the deviations of the approximation fromthe exact solution would be acceptable in most practical situations.
Kiesm¨uller et al. (2004) derive approximations for the first two moments of the wait time. They considera compound renewal customer demand process and use the stationary interval method of Whitt (1982) tosuperpose the renewal processes with mixed Erlang distributions. Their model only applies for non-negativecentral reorder points. They introduce an additional random variable O i for the actual replenishment ordersize of warehouse i and approximate the first moment as E [ W i ] ≈ E [ L ] Q (cid:18) E (cid:20)(cid:16) D ( ˆ L ) + O i − R (cid:17) + (cid:21) − E (cid:20)(cid:16) D ( ˆ L ) + O i − ( R + Q ) (cid:17) + (cid:21)(cid:19) , (3.3)5here ˆ L is random variable representing the residual lifetime of L with distribution function F ˆ L ( y ) = 1 E [ L ] (cid:90) y (1 − F L ( z )) dz. (3.4)The second moment of the wait time can be approximated by E [ W i ] ≈ E [ L ] Q (cid:18) E (cid:20)(cid:16) D ( ˜ L ) + O i − R (cid:17) + (cid:21) − E (cid:20)(cid:16) D ( ˜ L ) + O i − ( R + Q ) (cid:17) + (cid:21)(cid:19) , (3.5)where ˜ L is a random variable with distribution function F ˜ L ( y ) = 2 E [ L ] (cid:90) y (cid:90) ∞ x ( z − x ) dF L ( z ) dx. (3.6)For further details of the calculation we refer the reader to the original paper by Kiesm¨uller et al. (2004).Kiesm¨uller et al. report that the performance of their approximation is excellent if order quantities arelarge. This applies especially for the order quantities of the local warehouses. The error of the approximationreportedly also decreases with increasing demand variability.Unfortunately, the approximation by Kiesmuller et al. yields negative value for the variances in somesituations, an observation that was also made by Berling & Farvid (2014). Furthermore, the two-momentfitting technique using the mixed Erlang distribution may lead to serious numerical difficulties if the squaredcoefficient of variation is very low. This happens, for example, in situations where the order quantity of thelocal warehouse is very large compared to lead time demand. Berling & Farvid (2014) estimate mean and variance of the wait time by replacing stochastic lead timedemand with a stochastic demand rate. They offer several methods to approximate this demand rate assumingthat the central lead time is constant.We have chosen to use method ”E5” from their paper, which uses an approximation based on the log-normal distribution. Berling and Farvid obtained good results with this method in their numerical analysis.Additionally, it is easy to implement and it does not require challenging computations.Berling and Farvid compared their methods to Kiesm¨uller et al. (2004), Axs¨ater (2003), Andersson &Marklund (2000), and Farvid & Rosling (2014). Their method showed superior results for the cases consideredin this study. Kiesm¨uller et al., however, state that their approximation works best with highly variabledemand and large order quantities, while Berling and Farvid use very low order quantities (compared to thedemand rate) and a demand process with low variance. Thus, it is not surprising that the results obtainedare in favor for Berling and Farvid’s approach.Berling and Farvid’s approximation does not work if the order quantity of a local warehouse is larger thanthe sum of the order quantity of the central warehouse and the central reorder point. In this case, a negativeexpected wait time is calculated. However, this scenario is not very common. We recommend setting thewait time equal to the central lead time in such a case.
Grob & Bley (2018) propose to approximate the aggregate lead time demand at the central warehouseby a negative Binomial distribution and then compute the first two moments of the wait time analogousto Kiesm¨uller et al. (2004). The negative Binomial distribution is a flexible distribution and suitable tomodel lead time demand. Also, this approach does not face numerical instabilities and negative variances,as encountered by Kiesm¨uller et al.In the industrial application considered in Grob & Bley (2018) and in this paper, the order quantity islarge compared to lead time demand. We therefore assume that the actual order size is always equal to theorder quantity. If order quantities are too small for this assumption to hold, it is easy to incorporate thechanges into the model. Adapting equations (3.3) and (3.5), one can express the first two moments of thewait time as E [ W i ] ≈ E [ L ] Q (cid:18) E (cid:20)(cid:16) D ( ˆ L ) + Q i − R (cid:17) + (cid:21) − E (cid:20)(cid:16) D ( ˆ L ) + Q − ( R + Q ) (cid:17) + (cid:21)(cid:19) (3.7)6nd E [ W i ] ≈ E [ L ] Q (cid:18) E (cid:20)(cid:16) D ( ˜ L ) + Q i − R (cid:17) + (cid:21) − E (cid:20)(cid:16) D ( ˜ L ) + Q i − ( R + Q ) (cid:17) + (cid:21)(cid:19) , (3.8)where ˆ L and ˜ L are defined as in (3.4) and (3.6).We determine mean and variance of D i ( ˆ L ) and D i ( ˜ L ) with the help of (2.4) and (2.5), replacing L withthe respective random variables. Instead of deriving the real distribution function, we then fit the mean andvariance to a negative Binomial distribution. If the variance is smaller than the mean, the negative Binomialdistribution is not defined. This can only happen, if local order quantities and the coefficient of variation oflocal demand are small. In this case, we propose using the gamma distribution instead.For a discrete distribution with positive support, such as the negative Binomial distribution, it is easy toshow that E [( X − z ) + ] = E [ X ] − z (cid:88) x =0 xf X ( x ) − z (1 − F X ( z )) , z ≥ . (3.9)A similar equation can be derived for the gamma distribution.Given the issues with the original Kiesm¨uller et al. wait time approximation, our motivation was todevelop a more stable approach for use in practical applications. The negative binomial approximation,although inspired by Kiesm¨uller et al.’s approximation, does not suffer from the same instabilities and itis easier to compute. However, the negative binomial approximation has issues if Q i (cid:29) Q . In this case,the approximate value for the variance may be negative as well, but this case is a very unlikely setting inreal-world applications.
4. Set-up of simulation
In this section, we describe the simulation framework that we used in our experimental study. Figure 2shows the process diagram of this framework, which models a divergent inventory system where all warehousesuse a (R,Q)-order policy. Our goal was to simulate the behavior of real-world warehouses as realistic asnecessary, while keeping the simulation as simple as possible. Hence, we have chosen to simulate all processesbased on discrete time steps and a specified processing order of receiving shipments, fulfilling demands, andplacing new (back-) orders. In our application the discrete time step is one day, but it could also be any othertime unit. The final model was derived after multiple rounds of discussion and refinement involving severalpractitioners from the automotive industry. We are aware that this set-up may violate some assumptionsof the different wait time approximations. It can therefore also be seen as a test of robustness towards theperils of the real-world.
Figure 2: Schematic process diagram of the simulation
In the following, we describe the simulation and especially the process steps S1 to S4 of Figure 2 in moredetail.For each warehouse i in the network, we are given the input data shown in Table 1.7nput data NotationReorder points R i Order quantities Q i Expected transportation time E [ T i ]Variance of transportation time V ar [ T i ] Table 1: Warehouse input data needed for simulation
At the start of the simulation, the initial inventory on hand at warehouse i is set to R i + 1, no shipmentsare on their way, and all other variables are set to 0. Then, as soon as we have incoming demands at thelocal warehouses, new orders are triggered and propagated throughout the network as shown in Figure 2.Processing a time step, we first check in step S1 for each warehouse if there are any incoming shipmentsand add them to the inventory on hand.Then, in step S2 , we check for each warehouse if there are any outstanding backorders. In this step,we differentiate between local warehouses, which fulfill customer demand, and non-local warehouses, whichonly handle internal demand. For local warehouses, we use our inventory on hand to serve backorders firstcome - first serve as far as possible. For non-local warehouses, we use inventory on hand to fulfill orders ofsucceeding warehouses. In this case, if we fulfill an order of a successor, we also schedule the correspondingshipment to the successor. The transportation time t for this shipment is drawn at random according to agiven distribution. In our simulations, we used a gamma distribution, rounding the drawn random numbersto the nearest integers. The shipment is scheduled to arrive t days in the future at the successor. The minimaltransportation time possible is 1 day.Afterwards, in step S3 , we iterate over all local warehouses and try to fulfill the customer demand ofthat day first come - first serve. Here, our simulation allows for two modes of operation: The first one isbased on historical demand data, while the second one is based on artificial demand data drawn from randomdistributions. In the first case, we are given historical demand data for all local warehouses for a certainperiod of time and, for each day in this period, we try to fulfill all (historical) demand of that day in the orderit occurred. In the second case, we assume that demands arrive according to a compound Poisson process. Inthis case, the mean of the Poisson arrival process and the mean and the variance of the order size of each arrivalare given. We first draw the number of customers arriving from the Poisson distribution (Appendix C.4) andthen draw the order size for each customer. For the order sizes, the logarithmic distribution (Appendix C.2)is used. This leads to a negative binomial distribution of the lead time demand, so we can use the fill rateformulas (2.1) in this case. When using real-world data, estimates for the mean and the variance of demandare available from the inventory planning system.Finally, in the step S4 , we check if the inventory position (i.e., inventory on hand plus inventory on orderminus backorders) at each warehouse is less or equal to the reorder point. If this is the case, we place asmany backorders of size Q i at the preceding warehouse as necessary to raise the inventory position above R i again. If the preceding warehouse has enough inventory on hand, we fulfill the demand by scheduling ashipment as described in step S2 .Table 2 summarizes all parameters that can be configured in our simulation framework. It also shows thedefault settings that are used if not mentioned otherwise.We use a warm-up period, which should be a multiple of the longest transportation time in the network,to allow for a few order cycles to happen. We discard all measures obtained during this warm up period,except for the ones related to inventory, such as inventory on hand, inventory on order, and backorders.Parameter Input Default valueTransportation time distribution gammaInitial inventory pieces R + 1Warm-up period time -Runtime time -Demand type random or historical -Order size distribution logarithmic Table 2: Parameters of simulation
For each warehouse, we obtain the performance measures shown in Table 3 as the result of the sim-8lation. In addition, the order fill rate of each warehouse can be easily computed from these values asorders fulfilled / total orders.Performance measure DescriptionAverage inventory on hand Average inventory on hand over all time periodsAverage inventory on order Average of outstanding orders over all time periodsAverage backorders Average of backorders over all time periodsTotal orders Sum of incoming ordersOrders fulfilled Sum of incoming orders, that were fulfilled on the same dayWait time Mean and variance of times a warehouse had to wait forreplenishment orders from the time the order was made until itwas shipped Table 3: Performance measures for each warehouse of the simulation
5. Experimental results
In this section, we report on simulations that we have run to evaluate the accuracy of the different waittime approximations in our setting.In our experiments, we considered a 2-level network and prescribed several central fill rate targets. Wethen calculated the reorder points for all warehouses such that the given local fill rate targets are met. Thelocal reorder points depend on the the wait time, so different wait time approximations will lead to differentlocal reorder points. Figure 3 illustrates the process of calculating reorder points given a prescribed centralfill rate, a wait time approximation and local fill rate targets. We want to emphasize that for the sameprescribed central fill rate the resulting central reorder point is the same for all wait time approximations.The choice of approximation method only affects the local reorder points. Hence, the results are directlycomparable.
Figure 3: Process of calculating reorder points for the simulation given a prescribed central fill rate target, a wait time approx-imation and local fill rate targets
The presentation of our experimental results is split in two sections. First, we try to establish a generalunderstanding in which situation which approximation has the highest accuracy. For this purpose, we haveset up experiments using an artificial network and random data drawn from certain distributions. In a secondstep, we try to verify our findings using the network and historical demand data of a real-world industrialsupply chain. The second step is much more challenging than the first one, as the real-world demand datais “dirty”. Wagner (2002) coined this term to describe that real-world demand data usually contains manyirregularities, which often cannot be adequately represented by drawing random numbers based on a fixedprobability distribution. The real-world demand data therefore can be seen as a more challenging environmentand, consequently, a worse accuracy of the distribution-based wait time approximations should be expected.Nevertheless, any method that claims to have value in practical applications should be verified within apractical setting. To the best of our knowledge, we are the first to report on numerical tests of wait timeapproximations in a real-world setting with real-world demand data.To evaluate the accuracy of the wait time approximations regarding the mean and standard deviation,we use two measures which we will define in the following for the NB approximation and which will beanalogously used for all other approximations. Let µ NB ( t ) and σ NB ( t ) be the computed estimators of the9 B approximation for the mean and standard deviation of the wait time for test case t . Analogously, let µ SIMU ( t ) and σ SIMU ( t ) be the simulated mean and standard deviation of the wait time for test case t . InSection 5.1 we consider random demand data. Here, µ SIMU and σ SIMU are the averages of the simulatedvalues of all instances (in the respective group of instances) that were simulated. We calculate the error andthe absolute error of the mean of the wait time for the NB approximation for a set T of test cases asError = 1 | T | (cid:88) t ∈ T ( µ NB ( t ) − µ SIMU ( t )), and (5.1)Absolute Error = 1 | T | (cid:88) t ∈ T | µ NB ( t ) − µ SIMU ( t ) | . (5.2)In the same fashion, we calculate error measures for the standard deviation and all other approximationmethods. While eq. (5.1) conveys information about the average direction of the errors, eq. (5.2) conveysinformation about the average size of the errors. Our objective in the experiments with random demand data is two-fold: First, we want to evaluate theaccuracy of the given wait time approximations in different situations. In practice, however, managers donot care much about the wait time as a performance measure, they care much more if the resulting fill ratetargets are fulfilled. As second objective, we thus also want to find out if using the wait time approximationsto determine reorder points leads to small or big errors in the resulting fill rates. For this, we first computethe local reorder points for each test scenario and target fill rate using a wait time approximation, and then,in a second step, compare this with the actual fill rate observed for these reorder points in a simulation.
In order to compare the quality of the different wait time approximations for random demand data, weconsider a 2-level network, where warehouse 0 is the central warehouse. The demand characteristics of thedefault case are shown in Table 4. In this table, µ i and σ i denote the mean and the variance of the demandWare-house µ i σ i Q i ¯ β i E [ T i ] σ ( T i ) p i Table 4: Sample network characteristics for one day. From these values, one can easily derive the necessary parameters θ of the logarithmic order sizeand λ of the Poisson arrivals as θ i = 1 − µ i σ i and (5.3) λ i = − µ i (1 − θ i ) log ( θ i ) θ i . (5.4)We have chosen to represent daily demand by its mean and variance, as this representation is probably moreintuitive for practitioners. One can easily grasp the size and the variability of demand in the network.From this default case we derive our set of test cases by varying the parameters. Table 5 summarizes thetest cases we consider. If the variation typ is multiplicative, indicated by ‘m’ in Table 5, we multiply the10arameter by the variation value in the respective line of Table 4 . If the variation typ is absolute ‘a’, wereplace it. A special case is the number n of warehouses, where we vary the size of the network by consideringthe central warehouse 0 and n identical copies of warehouse 1. Note, that we only vary one parameter at atime and keep all other parameters fixed to the values shown in Table 4. Therefore, we have a total of 40test cases, including the base scenario as shown in Table 4.Parameter µ i σ i Q i , i > Q β i T p n
10 2,3,4,5,6,7,8,10,15,20 -
Table 5: Parameter variations and variation type, multiplicative (m) or absolute (a), for the creation of the test cases
Our objective is to test all wait time approximations over a wide range of central stocking quantities, fromstoring little at the central warehouse to storing a lot. We therefore prescribe different values for the centralfill rate target. Based on the prescribed central fill rate, we then calculate the central reorder point that isnecessary to fulfill the specified fill rate target as illustrated in Figure 3 to obtain sets of reorder points forthe simulation.The different wait time approximations suggest different ways to compute the central fill rate. To obtaincomparable results, we use the procedure described in Section 2 to approximate the central lead time demandand then calculate the fill rate using (2.1).In the simulations, we considered a time horizon of 2000 days and used a warm-up period of 500 days.Complementary numerical experiments revealed that 100 independent runs of the simulation for each instanceare sufficient to get reliable numerical results, so all reported simulation results are averages over 100 runsper instance. For further details on these tests we refer to Appendix A.In our initial experiment, we considered 5 scenarios where we prescribed a central fill rate of 20%, 40%,70%, 90% and 95%, respectively. We then calculated the central reorder point based on the prescribed centralfill rate and equation (2.1) using a binary search and the procedure described in Section 2. Finally, we ransimulations to check if the intended central fill rate was actually obtained. As shown in Table 6, especiallyin the high fill rate scenarios, the fill rates observed in the simulation are much lower than those anticipated.The 5 initial scenarios only cover the low and medium range of central stocking quantities. As our objectivePrescribed central fr Average simulated fr Name of scenario20% 10.86% low40% 44.77% medium low70% 56.06% /90% 63.59% /95% 67.80% medium highnew 95.16% high
Table 6: Prescribed central fill rate, average simulated fill rate and name of scenario is to cover a wide range of central stocking quantities in our tests, we manually adjusted the central reorderpoints for the base scenario and each test case in Table 5 (by trial and error with the help of the simulation)to create a new scenario that led to a high central fill rate of 95 .
16% in the simulation.In the remaining experiments, we then only consider the scenarios 20%, 40%, 95% and “new” shown inTable 6, which have been renamed for clarity to align with the central fill rates actually observed in thesimulations. Note that the “high” scenario is somewhat special: In this scenario, the central fill rate is sohigh that the chance an order has to wait at the central warehouse is really low. Hence, the effect of the waittime on the network is very small in this case. Nevertheless, we found it worthwhile to also consider this casefor completeness. 11or brevity, we refer to numbers obtained from the wait time approximations as computed numbers andto the numbers observed in the simulation as simulated numbers in the following.
In this section we discuss the results of the experiments with random data described in the previous section.We compare simulated and computed wait time results. Based on this analysis we give recommendationswhen to use which approximation. Then, we analyze if the local fill rate targets were satisfied and have alook at the inventory level for the different scenarios and approximations.
Wait time results.
Table 7 shows the average simulated and computed mean and standard deviation foreach scenario and approximation. The
KKSL approximation seems to perform best overall, while all otherapproximations seem to have great errors according to the simulated results. However, we will observe laterin this section that each approximation performs best for at least some scenarios and test cases. Hence,the aggregated results can only give a rough overview. Even more, some the aggregated results are stronglyinfluenced by few individual cases where an approximation did perform very poor. For example, the standarddeviation is badly overestimated by the NB approximation for a small number of parameter settings, althoughthis approximation performs very good for many other cases. In the aggregate view, this is not properlyrepresented. low medium low medium high highMean SD Mean SD Mean SD Mean SDSimulation 24.76 15.63 9.11 11.35 4.12 7.21 0.42 1.90 KKSL NB BF AXS
Table 7: Average wait time and standard deviation for different scenarios and wait time approximations in comparison to thesimulated results
In the following, we will analyze which approximation works best in which circumstances. When we referto certain parameters being high or low, this is always with respect to our base setting shown in Table 4.
Evaluation of the different approximations.
The
KKSL approximation has good accuracy for a wide rangeof input parameters. It is rather accurate if the central fill rate is medium to high and if there are many localwarehouses. It performs best if differences between local warehouses are small and if local order quantitiesare in a medium range, i.e., Q i /µ i ≈
20 . Consider for example Figure 4, which shows the errors for themean and the standard deviation of the wait time for the“medium high” scenario and on test instance, i.e.,the base parametrization shown in Table 4. While the approximation performs good overall, the differencesbetween local warehouses are large. The approximation is very good if the order quantity is neither too highnor too low, for example for warehouses 5 and 6, which have an order quantity of 150. The accuracy is worstfor warehouses with low demand and low order quantity.The NB approximation is very good for many input settings, especially at approximating the mean ofthe wait time. However, it badly overestimates the standard deviation if Q i /µ i or if Q /Q i are large. On theother hand, we observed very good results with the NB approximation when the central fill rate is mediumto high and when the local warehouses are heterogeneous.In scenarios where the local order quantity and the central lead time are very low BF is a suitableapproximation. In all other cases, it is not. We believe that the main reason for this is that it assumes avery smooth and nearly steady demand at the central warehouse. Berling & Farvid (2014) state that thisassumption is valid only for small Q i . In the numerical analysis presented in their paper, increasing Q i causeslarger errors. In our data, the ratio Q i /µ i is much larger than what Berling and Farvid considered. Moreover,our central lead times are much longer, so the system is not reset as frequently and the error of the linearapproach supposedly adds up.By design, the AXS approximation computes only the average of the mean and the standard deviationof the wait time over all local warehouses. Consequently, it is only suitable if differences between localwarehouses are not too large. Furthermore, it performs very good if the central fill rate is low or if thevariance of the demand is high. 12 E rr o r o f a pp r o x i m a t i o n Local warehouse MeanStandard deviation
Figure 4: Errors of
KKSL approximation for mean and standard deviation of wait time for the “medium high” scenario and thetest set of base parametrization of Table 4 for the different warehouses
Next, we will have a closer look at the effect of the network size on the quality of the approximations.For this experiment, we consider a network with n identical warehouses with the characteristics of warehouse1 in Table 4. Note that this construction favors approximations that benefit from homogeneous networks,namely KKSL and
AXS . We therefore focus on the trend and analyze how the accuracy of each individualapproximation changes if we increase the number of local warehouses.Figure 5 shows the average error of the computed from the simulated mean and standard deviation of thewait time for the “medium low” scenario. For the mean of the wait time, we see no clear trend for n < n ≥
10 the accuracy of the approximation increases for all methods except NB , which levels off for n > NB approximation: The approximationis bad for small networks, which is consistent with our previous analysis (as Q i µ i = 25), but the qualitydramatically improves with more local warehouses. This indicates that the mechanism that overestimatesstandard deviations for warehouses with those input parameters diminishes with increasing network size. Theother methods show a slightly improving quality with increasing network size. This behavior is very similarfor all other central fill rate scenarios. -25-20-15-10-5 0 5 4 6 8 10 12 14 16 18 20 A v e r a g e d e v i a t i o n Number of local warehousesKKSL NB BF AXS (a) Mean -20-10 0 10 20 30 40 50 60 70 80 90 100 4 6 8 10 12 14 16 18 20 A v e r a g e d e v i a t i o n Number of local warehousesKKSL NB BF AXS (b) Standard deviationFigure 5: Errors of mean and standard deviation of the wait time for the different approximations and network sizes in the“medium low” scenario
Finally, we look at the last test case, i.e., how do different wait time approximations perform if we changethe local fill rate target ( ¯ β i in Table 5). The local fill rate target has no effect on the accuracy of theapproximations in our numerical experiments. This is hardly surprising as the local fill rate target onlyinfluences the local reorder point. Therefore only the timing of orders changes but neither the quantity northe frequency of the ordering process. Local fill rates and inventory in network.
Ultimately, a planer does not care about wait time but is interestedin whether the fill rate targets at the local warehouses are met or violated. Additionally, as a secondary13bjective, she or he might prefer fill rates to be not much higher than the targets, as higher fill rates requireadditional stock.Table 8 gives an overview on the average deviation of the local warehouses’ fill rates from the fill ratetargets. It shows the average of all test cases except the text ones regarding the network size n and the fillrate targets ¯ β ( i ) (cp. table 5).Scenario KKSL NB BF AXS low 7.31% 9.04% -46.64% -7.04%medium low 3.36% 5.10% -13.83% -8.39%medium high 2.75% 1.70% -2.15% -6.24%high 5.13% 2.51% 5.19% 0.22%
Table 8: Average deviations from fill rate targets for local warehouses, average of all test cases except the test cases for thenetwork size n and the fill rate targets ¯ β ( i ) AXS and BF on average violate the fill rate constraints considerably, except for the “high” scenario,where the quality of the wait time approximation is hardly important. This confirms our previous findingthat these two wait time approximations are only suitable in some rather restrictive cases.For KKSL and NB , the fill rate constraints are – on average – fulfilled in most cases. However, for the“low” and the “medium low” scenario the observed average fill rates are much higher than the fill rate targets,indicating a substantial overstock.Figure 6 shows the average value of the reorder points and inventory levels at local warehouses. As bothbar charts show a very similar pattern, we discuss them together and more generically refer to stock insteadof reorder points and inventory levels. A v e r a g e v a l u e l o c a l r e o r d e r p o i n t s Scenario KKSLNBBFAXS (a) Reorder points A v e r a g e v a l u e l o c a l i n v e n t o r y l e v e l Scenario KKSLNBBFAXS (b) Inventory levelsFigure 6: Average value of reorder points and inventory levels for local warehouses for the different scenarios
Naturally, the local stock value is decreasing if more stock is kept at a central level.The stock situation for
AXS and BF reflects the violation of fill rate targets: Whenever the violation ishigh, stock is low. Both the fill rate and the stock levels indicate that these two approximations underestimatethe wait time.The much higher stock for the NB for the “low” scenario is striking. Contrary, for the “high” scenario,results based on NB have the lowest stock. The higher the fill rate, the more the stock situation for NB issimilar to the other approximations. The KKSL behaves more moderate than the NB approximation. Wealready found that the NB approximation dramatically overestimates the standard deviation of the wait timefor some input parameter constellations. For these instances, way too much stock is kept. If we remove thoseinstances from the analysis, the dramatic spike in the “low” scenario disappears. The NB approximationstill overstocks for low central fill rates, but the accuracy for higher central fill rates is much better. Onecould assume that the underestimation of the average fill rate shown in Table 8 is also highly influenced bythis and that NB may violate the fill rate constraints if these instances are excluded from analysis. However,this is not the case: Although the average fill rate is lower, the fill rate targets are still met, see Table 9.If we vary the local fill rate targets, the behavior of all approximations is not surprising: The higher thefill rate target, the more local stock is needed. The more accurate the approximation, the more precise thefill rate target is met and the less over- or understock, respectively, is observed.14 A v e r a g e v a l u e l o c a l r e o r d e r p o i n t s Scenario KKSLNBBFAXS
Figure 7: Average value of reorder points for local warehouses for the different scenarios excluding test cases not suitable for NB low medium low medium high highDeviation 8.46% 3.04% 1.15% 2.42% Table 9: Average deviation from fill rate target for local warehouses for NB approximation excluding test cases not suitable for NB Summary of the analysis based on random data.
The
KKSL approximation seems to be a good defaultchoice, unless local order quantities differ too much or the central fill rate is very low. A suitable alternativein many instances is the NB approximation, especially if the central fill rate is medium to high and if Q i µ i < NB often outperforms the KKSL approximation.The
AXS approximation should only be used if the characteristics of local warehouses are similar. If,additionally, the central fill rate is low or the variance of demand is very high, it is an excellent choice.Only for very low central lead time and local order quantities, BF should be chosen as approximation. The aim of the experiments presented in this section was to verify the findings for random data also ina real-world setting with real-world data. For this purpose, we considered a real-world distribution networkwith one central and 8 local warehouses and 445 parts from the total stock assortment in this network. For allparts, we were given fill rate targets, prices, order quantities and historical daily demand for all warehousesover a period of 1 year. The 445 parts were selected to make a small but representative sample of the totalstocking assortment of over 80,000 parts. For the majority of the parts, the variance of the demand is veryhigh compared to its mean.In the simulation runs, the first 100 days were used as a warm-up period and results from those dayswere discarded. The lead times were generated from gamma distributions with given means and variances.For all scenarios considered here, the same random seed was chosen. In order to mimic a realistic real-worldscenario, the reorder points were (re-)computed every 3 months based on (updated) demand distributions,order quantities, fill rate targets and prices. The demand distributions used as input in these computationswere also updated every 3 months based on the future demand forecast data from the productive inventoryplanning system.In the previous section, where we used artificial demand data, we followed an exploratory approach,manually searching for patterns in the data obtained in the computations and simulations. This was possible,because the size of the data set to be analyzed was sufficiently small. For the real-world data, this is notpossible anymore. Therefore, we have chosen to take a different approach here: We state the key findingsfrom the previous sections as hypotheses and try to confirm or falsify them here.The mean and the variance of the transportation time from the central warehouse to each local warehouseis the same for all parts, but it differs between local warehouses. For each local warehouse, the mean is between4 and 18 days and the variance is small, about 30% of the mean. The lead time from the supplier to thecentral warehouse depends on the supplier and is therefore not the same for all parts. The mean is between45 and 100 days. The variance is much higher, for some parts it is up to several factors larger than the mean.These characteristic values limit our analysis, so we cannot replicate all investigations from the previoussection.Also, the evaluation of the results contains an additional challenge. We constructed the simulationto mimic the behavior of a real-world distribution system, where demands change and reorder points are15ecalculated and adopted every 3 months. Every time the reorder points are updated, also the wait timeapproximation will change implicitly. The stock situation at the time of change, however, is determinedby the results of the past. In the simulation, it will take some time until the updated reorder points willbecome visible in the actual stock and, thus, in the actual wait time. This behavior distorts the comparisonof the approximation and the simulation, but this is something that would also happen if the approximationswere implemented in a real-world system. To cope with this difficulty, we will compare the mean and thestandard deviation of wait time computed by the respective wait time approximation, averaged over all re-computations, to the mean and the standard deviation observed in the simulation over the entire time horizonexcluding the warm-up period.We start by looking at the central fill rate in order to verify the findings from Table 6, namely that thesimulated fill rate is much lower than the prescribed fill rate for high fill rate scenarios. The results shownin Table 10 clearly confirm this observation. For the scenarios with a prescribed fill rate of 90% and 95%,the simulated fill rate is indeed much lower than those values. For low prescribed fill rates, the simulatedfill rate is higher than the prescribed value. Again we created an additional scenario by adjusting centralreorder points by trial and error to obtain a high central fill rate scenario named “new”, as similarly donefor Table 6. Prescribed Fill Rate Average Simulated Fill Rate20% 35.22%40% 41.87%70% 56.55%90% 70.30%95% 76.93%new 93.73%
Table 10: Prescribed central fill rate and average simulated value
In total we have 18,396 test cases. A test case contains all results of our experiments for each combinationof scenario, part number and warehouse. For example, the test case for scenario 20%, part “a” and warehouse“1” contains the results of the simulation as well as the a-priori calculated moments of the wait time for thefour different approximations.We use this big data set to test the following hypotheses, derived in the previous section for randomdemand data and specifically designed test cases. Note that our goal is not to confirm these hypotheses byany statistical means, but only to detect patterns in the data that either confirm or reject those findings.H1. The
KKSL approximation is suitable if the central fill rate is medium to high.H2. The
KKSL approximation performs best if differences between local warehouses are small and if localorder quantities are in a medium range.H3. The NB approximation overestimates the standard deviation significantly if Q i /µ i >
25, but it has a good accuracy if Q i /µ i ≤ NB approximation is good if the central fill rate is medium to high and when local warehouses areheterogeneous.H5. The error of the NB approximation of the standard deviation reduces if the network becomes larger.H6. The BF approximation performs good if local order quantities and central lead time are small.H7. The AXS approximation performs good if the network is homogeneous, the central fill rate is low orthe variance of demand is high.
Hypothesis H1:
KKSL is suitable if the central fill rate is medium to high.
While hypothesis H1 explicitlystates that
KKSL is suitable for medium to high central fill rates, this also implies that it is not as suitablefor low central fill rates. We will therefore analysis both situations, the accuracy of
KKSL for mediumto high central fill rates, corresponding to the 4 scenarios 70%, 90%,, 95% and “new” in Table 10, and itsaccuracy for low central fill rates, corresponding to scenarios 20%, 40%.16able 11 shows the ranking of the
KKSL approximation compared to the other approximations. Thecolumns “mean” and “standard deviation” show the respective percentages of the test cases, for which
KKSL produced the best, at least the second best, or the worst of the 4 approximations for the value of the mean orthe standard deviation of the wait time that was observed in the simulation. The column “combined” showsthe percentages of the test cases, where
KKSL produced the best, at least second best, or worst values forboth mean and standard deviation.For medium to high central fill rates,
KKSL is at least the second best approximation for the mean inabout 50% of the cases, but in 21.49% of the cases it is also the worst method. The accuracy is better forthe standard deviation. In more than 22% percent of the cases,
KKSL is best in approximating the meanand the standard deviation than any other method, as shown in the combined ranking. Only in less than 6%of all cases it is the worst overall method. Looking at the results for low central fill rates, we see that
KKSL performs even better.
Medium to high central fill rate Low central fill rateMeasurement Mean Standard deviation Combined Mean Standard deviation Combinedbest 28.80% 42.06% 22.19% 25.38% 48.25% 14.66%2nd or better 51.37% 71.33% 40.93% 56.80% 92.14% 49.77%worst 21.49% 9.17% 5.72% 10.36% 0.61% 0.18%
Table 11: Relative accuracy of the
KKSL approximation for scenarios with medium to high fill rate relative to the otherapproximations
Table 12 shows the absolute error of the approximated values from the values observed in the simulationfor the different methods, averaged over all test cases. Also these numbers support our observation that
KKSL is even better for low central fill rates than for medium to high central fill rates. For medium to highcentral fill rates, the accuracy of
KKSL for the mean is comparable to other methods, but its accuracy forthe standard deviation is better.Medium to high central fill rate Low central fill rateMean Standard deviation Mean Standard deviationSimulation 5.22 4.27 16.72 10.35KKSL 4.43 4.94 8.85 7.11NB 4.30 11.93 9.08 43.92AXS 4.37 6.36 9.86 15.11BF 4.86 5.72 11.29 6.93
Table 12: Average simulated values and absolute error of the approximations for test cases in different scenarios
In order to see if certain approximation tend to generally over- or underestimate the mean or the standarddeviation of the wait time, we also analyze the average positive or negative error of the approximations inTable 13 instead of the the averaged absolute errors. This additionally conveys information about the directionof the inaccuracies. For medium to high central fill rates, the mean is underestimated by all methods and
KKSL seems to perform especially bad. For the standard deviation, there is a general overestimationtendency in all approximations and
KKSL is very accurate.Medium to high central fill rate Low central fill rateMean Standard deviation Mean Standard deviationKKSL -2.31 1.83 -1.56 4.61NB -0.94 11.23 3.03 43.83AXS -2.11 4.33 3.25 14.92BF -2.01 2.73 -5.21 2.99
Table 13: Error of the approximations in different scenarios
In total,
KKSL seems to be a suitable choice for medium to high central fill rates. There are highinaccuracies, but these are not worse than with the other methods. For low central fill rates
KKSL was very17ccurate compared to the other methods for the real-world data. We find that
KKSL is suitable across allcentral fill rates and seems to be a good general choice relative to the other approximations.
Hypothesis H2:
KKSL performs best if differences between local warehouses are small and if local orderquantities are in a medium range.
In order to evaluate hypothesis H2, we have to more formally define whata small difference between local warehouses means. By talking about how different local warehouses are, werefer to the difference in order quantities as well as demand, i.e., the mean and variance of demand per timeunit. We formally define how much the value x i of an individual local warehouse differs from the mean of all x i as δx i := | x i − /n (cid:80) nj =1 x j | /n (cid:80) nj =1 x j , (5.5)where x i may be the order size Q i , the mean demand µ i or the variance of demand σ i .First, we analyze the case where the difference between warehouses is small only with respect to a measureand unrestricted with respect to the other measure, i.e., the three cases where δQ i ≤
20% for all i = 1 , ..., n , δµ i ≤
20% for all i = 1 , ..., n or δσ i ≤ i = 1 , ..., n . The reason for the latter 100% is that wehave to use a larger difference to have enough test cases in this class.Afterwards, we consider the case where the difference among the warehouses is small with respect to allthree measures. This combined evaluation of all three metrics, we have to use larger differences for δQ i and δµ i as well. There we consider all test cases for which δQ i ≤ ∩ δµ i ≤ ∩ δσ i ≤ i = 1 , ..., n .The second part of hypothesis H2 is that a local order quantity is in a medium range relative to the sizeof demand. Insights from Section 5.1.2 show that the ratio of Q i /µ i should be in the area of 20.We start by evaluating test cases where differences in order quantity and demand of local warehouses aresmall as defined above compared to test cases where those differences are large. For the large differences,we only evaluate test cases where δQ i >
20% for all i = 1 , ..., n for the order quantity and analogously forthe other two metrics. Therefore, in those test cases the respective metric of each local warehouse differssubstantially from its mean.Table 14 summarizes the results of this comparison. Again, it shows the proportion of cases for which KKSL was best, second or better, or worst of all approximations for the mean, the standard deviation or bothmeasures combined for different subsets of our test cases. With these results, the first part of Hypothesis H2cannot be confirmed. For our real-world data sets, the
KKSL approximation does not perform significantlybetter or worse if differences between local warehouses are larger or smaller. There is even some indicationthat the relative accuracy of
KKSL approximation does improve if differences are larger. The combinedmeasure of mean and standard deviation is better for large differences of order quantity and mean demandwhile it is slightly worse for the variance of demand. Especially for the mean demand
KKSL is the bestapproximation in 19 .
83% of the cases for large differences compared to 12 .
65% for small cases.
Diff. of . . . for Mean Standard deviation Combinedall i = 1 , ..., n Best 2nd or better Worst Best 2nd or better Worst Best 2nd or better Worst δQ i ≤
20% 33.95% 54.76% 25.19% 38.42% 71.19% 3.77% 17.47% 39.78% 1.88% >
20% 26.68% 52.93% 16.62% 45.01% 79.37% 6.71% 20.03% 44.51% 4.18% δµ i ≤
20% 24.38% 45.37% 34.88% 42.28% 70.06% 7.41% 12.65% 32.10% 5.86% >
20% 27.73% 53.34% 17.42% 44.16% 78.44% 6.29% 19.83% 44.12% 3.83% δσ i ≤ > δQ i ≤
40% and δµ i ≤
40% and 26.25% 45.80% 32.41% 37.66% 67.45% 7.22% 13.39% 29.92% 4.72% δσ i ≤ Table 14: Relative accuracy of
KKSL approximation compared to other approximations for test cases with small and largedifferences between local warehouses, with difference defined as in eq. (5.5)
KKSL approximation performs better or worse for order quantities in amedium range. t o
10 10 t o
20 20 t o
30 30 t o
40 40 t o
50 50 t o
60 60 t o
70 70 t o
80 80 t o
90 90 t o
100 100 t o
200 200 t o
300 300 t o
400 400 t o
500 500 t o
600 600 t o
700 700 t o
800 800 t o
900 900 t o m o r e t h a n A v e r a g e a b s o l u t d e v i a t i o n Ratio of order quantity and mean daily demandKKSL NB AXS BF
Figure 8: Absolute error of mean of the different approximations for different classes
Based on the findings using the real-world data, we cannot confirm hypothesis H2. However, note thatwe could not intensively test the combination of only small differences between local warehouses and of orderquantities in a medium range, because the number of test cases satisfying both conditions was too small.
Hypothesis H3: The NB approximation overestimates the standard deviation significantly if Q i /µ i > ,but it has a good accuracy if Q i /µ i ≤ . To test hypothesis H3 we first compare all test cases for which Q i /µ i >
25 to all test cases for which Q i /µ i ≤
25. We want to emphasize that this overapproximationonly occurs for the standard deviation and not for the mean. We look at the errors of the approximation,i.e., the difference between approximated standard deviation and simulated standard deviation, for theseresults. Table 15 shows the absolute average errir, separately for cases where Q i /µ i >
25 or Q i /µ i ≤ NB approximation. Average sd Average absolute errorCase NB KKSL AXS BF >
25 15450 6.35 22.97 5.73 9.37 6.18 ≤
25 306 3.80 3.55 2.59 4.73 3.31
Table 15: Absolute error of approximated standard deviation (sd) of wait time for test cases with different values of Q i /µ i For the majority of our test cases we have Q i /µ i >
25. (Note that the number of test cases does not addup to the 18,396 sets in total, because we have to exclude the sets for the central warehouse, where wait timedoes not apply. The results from the simulation with real-world data allows us to confirm hypothesis H3:The error is much larger for the standard deviation of the wait time if the NB approximation is used in thecase Q i /µ i >
25. For Q i /µ i ≤ NB performs similar to the other approximations. Note that the otherapproximations also get worse if Q i /µ i is large, but they still perform much better than NB then.A second interesting question is if the border is indeed rightly drawn at 25. Figure 9 shows the accuracyof the standard deviation predicted by the four approaches for different classes of Q i /µ i . (Detailed resultsincluding the number of observations in each class are shown in Table B.26 in the Appendix.) The errorfor the NB approximation gets large once Q i /µ i >
30. In all classes with Q i /µ i <
30, the quality of NB is comparable to the other approximations. So, 25 seems to be a good threshold value to decide whether NB should be used or not. Interestingly, the accuracy of AXS also becomes worse for increasing values of Q i /µ i , although at a much higher threshold. For KKSL and BF , we only see a light deterioration in theapproximation quality as Q i /µ i increases. 19 t o
10 10 t o
20 20 t o
30 30 t o
40 40 t o
50 50 t o
60 60 t o
70 70 t o
80 80 t o
90 90 t o
100 100 t o
200 200 t o
300 300 t o
400 400 t o
500 500 t o
600 600 t o
700 700 t o
800 800 t o
900 900 t o m o r e t h a n A v e r a g e a b s o l u t d e v i a t i o n Ratio of order quantity and mean daily demandKKSL NB AXS BF
Figure 9: Absolute error of standard deviation of the different approximations for different classes
We now focus on test cases where Q i /µ i ≤
25 and try to establish if the NB approximation can berecommended in these circumstances. Table 16 offers several insights: For the NB approximation, theapproximation of the standard deviation is indeed much more accurate for Q i /µ i ≤
25 than it is in general.The accuracy of
KKSL and
AXS does also significantly benefit from excluding results with high local orderquantity compared to mean demand. These three approximations all perform better than in the general caseshown in Table 13. The NB approximation performs on average worse than KKSL . It is also better than
AXS for the standard deviation, but worse than
AXS for the mean.
All senarios Medium to high central fill rate Low central fill rateMean Standard deviation Mean Standard deviation Mean Standard deviationSimulation 5.01 3.80 3.23 2.87 8.57 5.66KKSL -1.72 -0.16 -1.47 -0.24 -2.23 0.00NB -2.34 1.09 -2.12 -0.40 -2.79 4.07AXS 0.81 4.24 0.09 3.16 2.24 6.39BF -4.26 0.73 -2.58 1.42 -7.62 -0.67
Table 16: Average simulated values and error of the approximations for different scenarios, for test cases with Q i /µ i ≤ A similar analysis was done also with the average absolute error instead of the average positive andnegative error. In order to compare also the absolute sizes of the approximation errors. The results of thisanalysis, which are shown in Table B.28 in the Appendix, confirm the main observations from Table 16. Theremarkably good accuracy of
AXS in approximating the mean for medium to high central fill rate is due tocancel out effects.Additionally, we repeat the analysis done in Table 11 for the NB approximation and the limited subset ofresults with Q i /µ i ≤
25. Table 17 shows the results. Overall, the NB approximation seems to be suitable if Q i /µ i ≤
25. As we have already pointed out,
KKSL also benefits from this condition and therefore may bean overall better choice. Table 18 replicates the results for
KKSL . The share of cases where
KKSL is best orsecond best is much lower than for NB . We report the results for AXS in Table B.27 in the Appendix.
AXS does not perform better than NB . In particular, it has very little middle ground, especially for the mean: Itis either the best approximation or the worst with a roughly even split.Overall, Hypothesis H3 is supported by our simulations with real-world data. Hypothesis H4: The NB approximation is good if the central fill rate is medium to high and when localwarehouses are heterogeneous. To evaluate hypothesis H4 we have to repeat the first part of the analysis ofH2 and for the subset of scenarios with medium to high fill rates, i.e., scenarios 70%, 90%, 95% and “new”in Table 10. Based on the findings for hypothesis H3, we only consider cases with Q i /µ i ≤
25. Because ofthese restrictions, however, we can rely only on a relatively small number of test cases in the following.One could argue that by only considering cases with Q i /µ i ≤
25 we exclude all test cases where the NB approximation has a bad accuracy. We would then naturally get good results. We have chosen to do so20ll scenarios Medium to high central fr Low central frMean sd Comb. Mean sd Comb. Mean sd Comb.Best 36.60% 38.89% 27.45% 41.18% 43.14% 29.90% 27.45% 30.39% 22.55%2nd or better 61.11% 60.78% 48.69% 61.76% 70.10% 55.88% 59.80% 42.16% 34.31%Worst 9.80% 17.32% 5.88% 11.27% 12.75% 7.35% 6.86% 26.47% 2.94% Table 17: Relative accuracy of the NB approximation for the mean, the standard deviation (sd) and a combined (comb.)compared to other approximations.Evaluated for different scenarios regarding the central fill rate (fr), only for test cases with Q i /µ i ≤ All scenarios Medium to high central fr Low central frMean sd Comb. Mean sd Comb. Mean sdn Comb.Best 14.38% 25.16% 8.17% 14.22% 25.98% 8.33% 14.71% 23.53% 7.84%2nd or better 56.54% 68.95% 38.89% 58.82% 69.12% 41.18% 51.96% 68.63% 34.31%Worst 11.11% 14.71% 9.80% 13.24% 15.69% 11.27% 6.86% 12.75% 6.86%
Table 18: Relative accuracy of the
KKSL approximation for the mean, the standard deviation (sd) and a combined (comb.)compared to other approximations. Evaluated for different scenarios regarding the central fill rate (fr), only for test cases with Q i /µ i ≤ to exclude the distorting effect of these test cases where the accuracy for the standard deviation of the NB approximation is really bad as established in the analysis of hypothesis H3. As we are looking at trends, i.e.,does the accuracy improve if warehouses are more heterogeneous and if the fill rate is higher, we are stillable to get general insights. However, we also repeated the analysis done here for all test cases. All findingsreported in the following are supported by this data as well.The results are summarized in Table 19, which has the same structure as Table 14. There is indicationthat the NB approximation should be used if differences between local order quantities δQ i are large as it ismore often more accuracte than the other approximations for mean and standard deviation of the wait time.For the differences in the mean of demand δµ i the relative accuracy is worse for larger differences. Forthe variance of demand results are mixed and there is no clear indication if NB approximation profits fromlarger differences. Diff. of . . . for Mean Standard deviation Combinedall i = 1 , ..., n Best 2nd or better Worst Best 2nd or better Worst Best 2nd or better Worst δQ i ≤
20% 37.50% 64.29% 19.64% 41.96% 69.64% 17.86% 25.00% 58.04% 13.39% >
20% 45.65% 58.70% 1.09% 44.57% 70.65% 6.52% 35.87% 53.26% 0.00% δµ i ≤
20% 50.00% 78.13% 7.81% 42.19% 81.25% 9.38% 31.25% 70.31% 6.25% >
20% 37.14% 54.29% 12.86% 43.57% 65.00% 14.29% 29.29% 49.29% 7.86% δσ i ≤ > Table 19: Relative accuracy of NB approximation compared to other approximations for test cases with small and largedifferences between local warehouses, for medium to high central fill rate scenarios and Q i /µ i ≤
25 and with difference definedas in eq. (5.5)
All in all, we have some supporting evidence that the NB approximation is good if the central fill rate ismedium to high, when local warehouses have a large difference in order quantity while the difference betweenthe mean demand should be rather small. This seems to hold also for the low central fill rate scenarios, buthere also the share of cases where NB is the worst approximation increases, as shown in Table 20. Hypothesis H5: The error of the NB approximation of the standard deviation reduces if the network becomeslarger. This hypothesis is based on the findings in Figure 5. There, we see a decreasing error of the NB approximation of the standard deviation. We have only limited means to test this in the real-world data,where we have 8 local warehouses. However, we do not have demand for all parts at all warehouses and,therefore, have results for which only a lower number of warehouses than 8 was considered.21 iff. of . . . for Mean Standard deviation Combinedall i = 1 , ..., n Best 2nd or better Worst Best 2nd or better Worst Best 2nd or better Worst δQ i ≤
20% 21.43% 53.57% 0.00% 28.57% 50.00% 21.43% 16.07% 35.71% 0.00% >
20% 34.78% 67.39% 15.22% 32.61% 32.61% 32.61% 30.43% 32.61% 6.52%
Table 20: Relative accuracy of NB approximation compared to other approximations for test cases with small and largedifferences between the order quantity of local warehouses, for low central fill rate scenarios and Q i /µ i ≤ Table 21 shows the error of the different approximations for all parts that ship to at most 3 local warehousesand for all parts that ship to all 8 local warehouses. Table B.29 in the Appendix shows the absolute error withsimilar findings. There is hardly a difference in the accuracy of the NB approximation between parts withat most 3 or exactly 8 local warehouses. We also checked if the share of cases where the NB approximationis the best or second best approximation changes (not presented here), but without finding any significantresults.Therefore, we cannot confirm hypothesis H5. The only approximation that seems to suffer from fewerwarehouses in the real-world data is AXS . We suspect that this is caused by the fact that this approximationconsiders only the mean across all warehouses and that variability is higher for fewer warehouses. ≤ Table 21: Mean of simulation and error of approximations for the standard deviation for results with different number of localwarehouses
Hypothesis H6: The BF approximation performs good if local order quantities and central lead time are small. Unfortunately, we cannot evaluate the two conditions of hypothesis H6 at the same time, as all test caseswith a relatively low local order quantity also have a relatively low central lead time. As the minimal meancentral lead time in our data set is about 45 days, it is questionable if we see effects at all. Nonetheless, wecan evaluate the two conditions separately and compare mean central lead times of about 45 days to about100 days.Table 22 shows the relative accuracy of the BF approximation for results with the two different leadtimes. The relative accuracy of the BF approximation is indeed better for the shorter central lead time.Mean L : ≈
45 days ≈
100 daysMeasurement Mean sd Combined Mean sd CombinedBest 23.78% 29.49% 12.99% 11.11% 35.42% 9.38%2nd or better 40.51% 62.41% 30.63% 30.56% 57.99% 22.57%Worst 41.75% 9.63% 7.65% 47.57% 9.38% 7.99%
Table 22: Relative accuracy of the BF approximation for the mean and standard deviation (sd), comparison for high and lowmean of central lead time compared to other approximations To compare results for low and high local order quantities, we proceed as for hypothesis H3. In Table 15we have already seen an improvement in the approximation of the standard deviation of BF for smallerlocal order quantities. Repeating the same analysis for the mean, find find that the quality of the BF approximation does not change. In fact, it stays remarkably constant. However, our limit of Q i /µ i ≤ BF profits from small order quantities and from smallercentral lead times. 22 ypothesis H7: The AXS approximation performs good if the network is homogeneous, the central fill rate islow or the variance of demand is high.
Hypothesis H8 consists of three conditions that we analyze separately.The statement concerning homogeneous networks is closely related to hypothesis H2. We therefore repeatthe analysis that we did for Table 14 in Table 23 for the
AXS approximation. We see that the accuracy of
AXS relative to the other approximations does not improve for the real-world data. In fact, and in contrastto to the hypothesis, the relative accuracy of
AXS decreases for more homogeneous networks. This indicatesthat the other approximations, especially
KKSL , benefit even more from similar warehouses.
Diff. of . . . for Mean Standard deviation Combinedall i = 1 , ..., n Best 2nd or better Worst Best 2nd or better Worst Best 2nd or better Worst δQ i ≤
20% 21.19% 53.63% 18.08% 13.32% 28.06% 31.50% 6.69% 17.42% 7.06% >
20% 21.28% 48.72% 24.33% 18.71% 43.28% 8.92% 7.05% 26.02% 4.44% δµ i ≤
20% 18.21% 44.44% 21.30% 11.42% 29.01% 19.44% 3.09% 12.35% 5.25% >
20% 21.33% 49.49% 23.53% 18.12% 41.49% 11.81% 7.08% 25.12% 4.78% δσ i ≤ > δQ i ≤
40% and δµ i ≤
40% and 20.47% 49.74% 20.87% 10.76% 28.48% 23.36% 3.02% 12.07% 3.94% δσ i ≤ Table 23: Relative accuracy of
AXS approximation compared to other approximations for test cases with small and largedifferences between local warehouses, with difference defined as in eq. (5.5)
A comparison of the different approximations for a low central fill rate is presented in Tables 16 and B.28 inthe Appendix. These results show that
AXS does not perform significantly better than other approximationsif central fill rates are low.Table 24 shows the average error of the approximation from the simulated values for different ratios ofvariance to mean of demand. (Table B.30 in the appendix shows the absolute errors). It seems that
AXS performs best if the ratio σ /µ is between 1 and 5 and its accuracy decreases for higher variance. σ /µ < σ /µ < σ /µ ≥ σ /µ ≥ Table 24: Average simulated values and errors of the approximations for the mean and standard deviation (sd), for test caseswith different values of σ /µ We did not find evidence to confirm hypothesis H8.
Summary of the analysis based on real-world data.
For our real-world data, the results concerning the accu-racy of the wait time approximations compared to the simulation are sobering. We often see large errors ofall considered approximations. We have (at least some) supporting evidence for hypothesis H1, H3, H4, andH6, but we can neither clearly confirm nor clearly reject the other hypotheses H2, H5, and H8.
6. Summary and Conclusion
In this paper, we presented the results of our extensive numerical experiments to analyze and comparethe quality of different wait time approximations proposed in the literature. In our study, we considered bothrandom as well as real-world demand data.Our experiments show that there is no generally “best” approximation, none of the wait time approxima-tions has a better accuracy than the others in all situations. In fact, it depends heavily on the characteristics23f demand and the network which approximation performs best. In our experiments, we also observed ratherlarge errors of the wait time approximations in some specific situations. This clearly shows room for improvedmodels, at least for these situations, and the need for further research in this area.Nevertheless, we were able to derive a few simple guidelines that describe which approximation is likely toperform accurate in which situation: The BF approximation should only be used if the local order quantitiesand central lead times are low. The AXS approximation is suitable for homogeneous networks, i.e., networksin which local warehouses are very similar with regards to demand structures and order quantities. The
KKSL and NB are generally good choices in all other circumstances. For the NB approximation, however,the ratio Q i /µ i is critical for the quality of the approximation of the standard deviation of wait time. NB should be used only if this ratio is smaller than 25.In order to decide which approximation to use in a real-world application, we recommend to pick oneor two likely candidates with the help of these guidelines and then perform additional simulations for thespecific demands and the network arising in this application. References
Andersson, J., & Marklund, J. (2000). Decentralized inventory control in a two-level distribution system.
European Journal of Operational Research , , 483–506.Axs¨ater, S. (2003). Approximate optimization of a two-level distribution inventory system. InternationalJournal of Production Economics , , 545–553.Axs¨ater, S. (2006). Inventory Control . International Series in Operations Research & Management Science(2nd ed.). Springer.Berling, P., & Farvid, M. (2014). Lead-time investigation and estimation in divergent supply chains.
Inter-national Journal of Production Economics , , 177–189.Berling, P., & Marklund, J. (2013). A model for heuristic coordination of real life distribution inventorysystems with lumpy demand. European Journal of Operational Research , , 515–526.Deuermeyer, B. L., & Schwarz, L. B. (1981). A model for the analysis of system service level in warehouse-retailer distribution systems: The identical retailer case. In L. B. Schwarz (Ed.), Multi-level produc-tion/inventory control systems: theory and practice (pp. 163–193). North Holland volume 16.Farvid, M., & Rosling, K. (2014). Customer waiting times in continuous review (nq,r) inventory systemswith compound poisson demand: Working paper. Working paper.Grob, C., & Bley, A. (2018). Comparision of wait time approximation in distribution networks using (r,q)-order policies: Working paper. Working paper.Kiesm¨uller, G. P., de Kok, T. G., Smits, S. R., & van Laarhoven, P. J. (2004). Evaluation of divergentn-echelon (s,nq)-policies under compound renewal demand.
OR-Spektrum , , 547–577.Ross, S. M. (2014). Introduction to Probability Models . Academic Press.Rossetti, M. D., & ¨Unl¨u, Y. (2011). Evaluating the robustness of lead time demand models.
InternationalJournal of Production Economics , , 159–176.Sherbrooke, C. C. (1968). Metric: A multi-echelon technique for recoverable item control. OperationsResearch , , 122–141.Svoronos, A., & Zipkin, P. (1988). Estimating the performance of multi-level inventory systems. OperationsResearch , , 57–72.Wagner, H. M. (2002). And then there were none. Operations Research , , 217–226.Whitt, W. (1982). Approximating a point process by a renewal process, j: two basic methods. Operationsresearch : the journal of the Operations Research Society of America , , 125–147.24 ppendix A. Number of instances needed We determined how many instances of each variation need to be run in the simulation to obtain dependableresults. For this, we ran a number of tests for all variations of one scenario. The results, which are summarizedin Table A.25 for the mean simulated fill rate as an example, are quite stable. Even if we look at meansimulated fill rates for an individual instance, we get only small deviations after 100 instances. We thereforedecided that 100 instances are sufficient. We repeated the analysis with the mean and the standard deviationof the wait time, and also regarding these performance measures 100 instances seemed to be sufficient to getreliable results.
Table A.25: Mean simulated fill rates over all variations and instances for local warehouses
Appendix B. Supplementary tables
Class No. observations Mean simulated NB KKSL AXS BFStandard deviation0 to 10 30 1.40 1.30 2.82 1.27 2.2310 to 20 156 3.63 3.16 2.89 4.77 3.3820 to 30 258 4.48 4.38 2.45 5.21 3.4930 to 40 348 4.80 7.00 3.32 5.53 4.2240 to 50 390 6.14 10.83 3.50 5.37 4.4150 to 60 804 7.27 15.06 4.43 5.91 4.9660 to 70 4416 7.26 22.26 4.49 6.65 5.3770 to 80 2772 7.59 23.94 4.87 7.74 5.6880 to 90 1902 6.83 24.85 5.17 8.60 5.9490 to 100 306 7.40 23.24 4.65 7.98 5.63100 to 200 2580 5.98 27.55 6.12 10.62 6.55200 to 300 528 3.66 26.77 8.98 15.40 9.37300 to 400 312 2.25 27.03 11.34 19.05 10.96400 to 500 210 2.72 26.49 11.87 19.55 11.05500 to 600 132 0.80 24.98 12.98 22.61 11.86600 to 700 126 1.53 28.15 13.34 21.71 12.35700 to 800 54 0.58 21.39 14.81 29.23 11.53800 to 900 90 0.65 25.57 12.95 23.90 9.95900 to 1000 30 2.79 19.61 14.87 30.54 8.92 > Table B.26: Absolute error of the standard deviation for test cases with different ratios of order quantity and mean daily demand,Analysis of H3
Table B.27: Relative accuracy of the
AXS approximation regarding the mean, standard deviation (sd) and combined (comb.)for different central fill rates (fr) and test cases with Q i /µ i ≤ All senarios Medium to high central fill rate Low central fill rateMean sd Mean sd Mean sdSimulation 5.01 3.80 3.23 2.87 8.57 5.66KKSL 3.18 2.59 2.49 2.32 4.56 3.14NB 3.04 3.55 2.41 2.09 4.31 6.46AXS 2.51 4.73 1.81 3.85 3.90 6.49BF 4.70 3.31 3.13 3.44 7.84 3.03
Table B.28: Average simulated values and absolute errors of mean and standard deviation (sd) of the approximations for differentscenarios, for test cases with Q i /µ i ≤ ≤ Table B.29: Mean of simulation and absolute error of the approximations for the standard deviation for test cases with differentnumber of local warehouses σ /µ < σ /µ < σ /µ ≥ σ /µ ≥ Table B.30: Average simulated values and absolute error of mean and standard deviation (sd) of the approximations, for testcases with different values of σ /µ ppendix C. Distributions In this section we define the required distributions in alphabetical order, as there are sometimes ambigu-ities in the definitions. We refer to the random variable as X , the probability mass function as f X ( x ) andthe distribution function as F X ( x ). Appendix C.1. Compound Poisson
Let N be a Poisson distributed random variable and X , X , X , . . . i.i.d random variables that are alsoindependent of N . Then Y = N (cid:88) i =1 X i (C.1)is compound Poisson distributed. Appendix C.2. Logarithmic distribution
Let θ ∈ (0 ,
1) be the shape parameter of the logarithmic distribution. The probability mass function is f X ( x ) = − θ x x ln(1 − θ ) for x = 1 , , , . . . (C.2)and the probability distribution function is F X ( x ) = − − θ ) x (cid:88) i =1 θ i i for x = 1 , , , . . . (C.3) Appendix C.3. Negative binomial distribution
The density function of the negative binomial distribution with parameters n > < p < f X ( x ) = Γ( n + x )Γ( n ) x ! p n (1 − p ) x for x = 0 , , , . . . (C.4)where Γ is the gamma function. The probability distribution function is F X ( x ) = 1 − I p ( k + 1 , r ) for x = 0 , , , . . . (C.5) Appendix C.4. Poisson distribution
Let λ > f X ( x ) = e − λ λ x x ! for x = 0 , , , . . . (C.6)and the probability distribution function is F X ( x ) = e − λ x (cid:88) j =0 λ j j ! for x = 0 , , , .., ..