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Forecast facilitated lot-for-lot ordering in the presence of autocorrelated demand
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Computers & Industrial Engineering 54 (2008) 840–850 www.elsevier.com/locate/caie
Forecast facilitated lot-for-lot ordering in the presence of autocorrelated demand Layth C. Alwan a, John J. Liu b, Dong-Qing Yao
c,*
a
c
School of Business Administration, University of Wisconsin-Milwaukee, WI 53201, USA b Department of Logistics, Hong Kong Polytechnic University, Hong Kong, PR China Department of Management, College of Business and Economics, Towson University, 8000 York Road, Towson, MD 21252, USA Received 2 February 2005; accepted 4 October 2007 Available online 4 November 2007
Abstract With consistent effort in setup reduction as encouraged by JIT principle, lot-for-lot ordering is gaining popularity in MRP applications. A lot-for-lot order is an immediate copy of the MPS (master production schedule) – direct reflection of demand forecasts. Since all levels of MRP plans are based on MPS, the accuracy of MRP is highly dependent of the accuracy of demand forecasting. In this paper, we are concerned about the impact of forecasting to the performance of a lot-for-lot MRP system when there is notable variability and autocorrelation in the underlying demand process (e.g., an AR(1) process). Specifically under a stationary AR(1) demand, we examine the performance of the MRP based on the most common EWMA forecast model, and then compare it with a minimum mean square error (MSE) forecast model. The notable findings of this study include: (1) MRP performance differs noticeably under the two different forecasting models. (2) The MSE-optimal forecasting performs no worse than the EWMA forecasting in all aspects of MRP applications. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Lot-for-lot ordering; MRP; Forecasting; AR(1) process; Monte Carlo simulation
1. Introduction As evidenced by a recent empirical study (Jonsson & Mattsson, 2003), lot-for-lot ordering is the most frequently applied method in production planning and by users of materials requirements planning (MRP). This fact is not surprising since lot-for-lot has much appeal. As noted out by Hopp and Spearman (2000), lot-forlot ordering is not only simple but also it is consistent with a just-in-time philosophy of making only what is needed. Hopp and Spearman (2000) further point out that lot-for-lot ordering is a preferable policy in manufacturing environments where setup times (costs) are minimal. In its simplicity, a lot-for-lot ordering system calls for releasing an order at any given time period t to match the net requirements for period t + L where L representing the lead time. In practical application, companies *
Corresponding author. Tel.: +1 410 704 6185; fax: +1 410 704 3236. E-mail address: [email protected] (D.-Q. Yao).
0360-8352/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2007.10.014
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do not know future demand with certainty and, thus, must rely on forecasting for production planning. It is thus natural to consider the impact of the forecasting method on the performance of the production ordering system. A number of authors have provided certain insights on the effects of forecasting bias on MRP performance (Biggs & Campion, 1982; Enns, 2002; Lee & Adam, 1986). These studies, however, do not make explicit assumptions about the nature of the demand process nor do they consider specific forecasting methods. In supply-chain research, the effects of different forecasting methods applied to different demand patterns have been studied (Alwan, Liu, & Yao, 2003; Chen et al., 2000a; Graves, 1999). In particular, these studies demonstrated the bullwhip effect is dependent on both the correlative structure of the demand process and the relative accuracy of the forecasting method. The accuracy of all lower levels of MRP is dependent of the accuracy of master production scheduling (MPS), which is determined by the demand forecasting; Therefore to study the effects of forecasting on top level plays pivotal role in all other lower levels of MRP planning. This paper provides distinct contributions to MRP research. First, we consider, the performance of a MRP system in the presence of a stationary correlative demand process, namely, a first-order autoregressive process or AR(1) process. Second, we study the system’s MRP performance when a common, but simplified, forecasting scheme – the exponential weighted moving-average (EWMA) method – is employed in comparison to an optimal forecasting scheme. In the area of forecasting, an ‘‘optimal’’ forecast model is traditionally meant to imply that the forecast model has minimum mean square forecast errors. For clarity, we will refer to such a model as a mean square error optimal (or MSE-optimal) forecast model. The remainder of this paper is organized as follows: Section 2 describes the lot-for-lot MRP framework and introduces the AR(1) demand process. Section 3 formulates the order release quantities in relationship to the forecasting scheme employed. Section 4 describes a simulation study conducted to gain insights on the relative performance of the MRP systems given the specific forecasting method used. Section 5 offers our concluding remarks. 2. Lot-for-lot MRP under AR(1) demand The implementation of an MRP system calls for the determination each period of a planned order release (POR) for the top MRP level. There are a number of approaches for determining the value of POR for any given period. As noted, one common approach for planned order release determination is based on a lot-forlot ordering policy. With a lot-for-lot policy, the level of the planned order release for any given period is set to the net requirement L periods later, i.e., L is the manufacturing lead time. We will assume that the L is fixed and known. There are a number of potential stochastic processes that can be assumed for the demand process, ranging from a simple independent and identically distributed (IID) process to a nonstationary process. In reality, there is empirical evidence that few real-world processes exhibit pure IID behavior (Alwan, 2000). One correlative demand process that has been frequently studied in the supply-chain literature is the first-order autoregressive model, AR(1) (Chen, Drezner, Ryan, & Simchi-Levi, 2000a, 2000b; Kahn, 1987; Lee, Padmanabhan, & Whang, 1997, 2000). The AR(1) process is often studied because pure first-order correlative effects commonly occur in real-world processes (Alwan, 2000). Furthermore, even for processes that do not follow a pure AR(1) process, an AR(1) model is a flexible model that can reasonably capture lagged effects so commonly found in demand processes. In the following subsections, we describe the AR(1) process in more detail, present the forecasting models to be considered, and present the order quantities to be examined in this paper. 2.1. Demand process Suppose the system is faced with an AR(1) demand process given as follows: d t ¼ s þ /d t1 þ et ;
t ¼ 0; 1; 2; . . .
ð1Þ
where dt is the observed demand in period t, s > 0, j/j < 1, and {et} is an independent and identically distributed normal (IIDN) process with mean 0 and variance r2e . The condition of j/j < 1 ensures that the process is stationary. It is useful to note that ld = E(dt) = s/(1 /) and r2d ¼ Varðd t Þ ¼ r2e =ð1 /2 Þ for any t. In this
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paper, we assume / is known and fixed. In practice, if / is unknown, we may first obtain unbiased estimates of ^ and then substitute the demand parameter with the corresponding parameter estimate. demand parameter, /, As noted earlier, the AR(1) model is widely used in supply-chain management research (e.g., Chen, Ryan, & Simchi-Levi, 2000b; Lee, So, & Tang, 2000). One of the primary reasons for the focus on the AR(1) model is that it possesses good dynamic characteristics. Namely, by varying the / parameter, we are able to study the effects of processes which are random, nonrandom but stationary, or even nonstationary processes. This flexibility allows us to gain practical insights for many real demand patterns. By varying the values of /, one can represent a wide variety of process behaviors. When / = 0, we have an IIDN process with mean s and variance r2e . For 1 < / < 0, the process is negatively correlated and will exhibit period-to-period oscillatory behavior. For 0 < / < 1, the demand process will be positively correlated which is reflected by as wandering or meandering sequence of observations. As j/j approaches 1, the process approaches nonstationary behavior, most notably, the random walk model – ARIMA(0, 1, 0) – is a special case of the AR(1) model when / = 1. As pointed out by Graves and Willems (2000), varying a stationary demand model is an important exercise for gaining fundamental insights into the relationship between variables such as inventory and orders relative to demand characterization. 2.2. Forecast models Smoothing methods, such as moving averages and exponential smoothing are widely employed for forecasting purposes in many production and operations management applications, largely because of their simplicity and ease of implementation. As such, most researchers of supply-chain management (SCM) problems requiring a forecast model have based their studies on either the moving-average method (e.g., Chen et al., 2000a) or the exponential weighted moving-average (EWMA) method (e.g., Chen et al., 2000b). Given the close connection between the moving-average method and the EWMA method, we will only focus on one method (namely, the EWMA method) in this paper. The EWMA model can be expressed as follows: F tþ1jt ¼ ad t þ ð1 aÞF tjt1
ð2Þ
where 0 < a < 1 is the smoothing constant and Ft+1jt is the forecast of period t + 1 made at the end of period t. It should be noted that the forecasts for periods t + i (i = 1, 2, . . .) made at time t are equal, that is, Ft+ijt = Ft+1jt for i = 1, 2, . . .. Hence, the forecasts for all lead times will follow a horizontal line parallel to the time axis. Even though the EWMA method, and to a lesser extent the moving-average method, has flexibility for adapting to a variety of correlated demand processes, it is MSE optimal for only one underlying time-series model, namely, a first-order integrated moving average, denoted by ARIMA(0, 1, 1) or IMA(1, 1) (e.g., Graves, 1999). An ARIMA(0, 1, 1) process is a nonstationary process that can be interpreted as a randomwalk trend plus a random deviation from the trend. Thus, under no circumstance is the EWMA method MSE optimal for a stationary AR(1) process. This fact opens up consideration of employing an MSE-optimal forecast scheme for the assumed AR(1) process. By recursively applying (1), it is easy to show that: d tþi ¼ s þ /s þ þ /i1 s þ /i d t þ /i1 etþ1 þ /i2 etþ2 þ þ etþi ¼
1 /i s þ /i d t þ /i1 etþ1 þ /i2 etþ2 þ þ etþi 1/
ð3Þ
For a general ARIMA process, it can be shown that the minimum mean square error forecast for period t + i is the conditional expectation of dt+i given current and previous observations dt, dt-1, dt-2,. . . (see Box, Jenkins, & Reinsel, 1994). In the case of an AR(1) process, this implies the MSE-optimal forecast function is given by E(dt+ijdt). Since E(et+ijdt) = 0 for i = 1, 2, . . ., it immediately follows that for an AR(1) process, the MSE-optimal forecast function is given by: F tþijt ¼
1 /i s þ /i d t ; 1/
for i ¼ 1; 2; . . .
ð4Þ
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In contrast to the two previous methods, this forecast function is not a horizontal projection into the future. Instead the forecasts revert back towards the overall mean level of s/1 /. The MSE-optimal forecast function reflects the fact that the AR(1) process is stationary and has the property of conditional mean reversion; that is, even though the process can be expected to wander away (below or above) from the overall mean it is also expected to eventually return back to the overall mean. The moving-average and EWMA methods fail to capture this mean reversion property of a stationary AR(1) process. To illustrate the dynamics of time series and forecasting forecasts, consider an AR(1) process with the following parameters: s = 50, / = 0.8, and re = 10. In Fig. 1, we show 50 consecutive observations randomly generated from such an AR(1) process. Using the statistical package Minitab, we found the optimal EWMA model and projected its forecasts into the future as shown in Fig. 1. In addition, we show in Fig. 1 the optimal MSE forecasts based on the AR(1) model. As can be seen from Fig. 1, the AR(1) projected forecast revert back to the long-run mean while the EWMA forecasts are horizontally ‘offset’. The implication in an operations management setting is that the EWMA model will over forecast demand potentially resulting in unnecessary costs such as additional inventory holding costs. On the flip side, if current demand of an AR(1) process was below the long-run mean, the EWMA forecasts will be horizontally offset on the low side resulting in other potential costs such as shortage and backlog costs. One argument often presented against the use of optimal methods is that their implementation is more difficult than the simple smoothing methods when parameters are unknown. It is pointed out that to implement optimal methods requires statistical skills in time-series modeling, including knowledge of model identification, model estimation, and tests for model adequacy, that are beyond the skill set of a typical operations manager. However, we believe that the industrial use of more sophisticated time-series models is steadily growing because of two reasons. First, the requirement of intense statistical training, often referred to as 6r training, is increasingly becoming commonplace (Hoerl, 1998). At corporations like GE, Motorola, and Allied Signal, organizational cultures are being developed in which there is a strong desire from employees throughout the organization to learn and implement advanced statistical techniques. Indeed, the authors of this paper can report that seminars in time-series analysis are part of the regular continuing education program at GE-Medical Systems and are required to be taken by all supply-chain managers. Second, modern computational tools are readily available to make possible automated implementation of time-series modeling including the general class of ARIMA models. These programs are designed to automate model identification, model fitting, and forecasting (see Shumway, 1986). 2.3. Planned order releases As noted earlier, the accuracy of all levels of MRP depends on the MPS. In addition, once the demand of top MRP level is determined, demands of lower levels of MRP are certain, and can be derived based on the bill Future 280
Optimal EWMA Forecast
Demand
270 Optimal MSE Forecast
260 250
Mean
240 230 220 1
7
14
21
28
35
42
49
56
63
70
Time Fig. 1. AR(1) process and forecasting (/ = 0.8).
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of material (BOM). Therefore, in this research, we focus on studying the effects of different forecasting techniques on the top MRP level under lot for lot ordering policy. A lot-for-lot policy calls for the level of the planned order release to be set to the net requirement of a future period. In general, the net requirement of any given period is equal to the gross requirements of that period minus ending inventory status of the previous period. If the net requirement is negative, then this implies inventory is sufficient to cover the gross requirement which, in turn, implies a zero planned order release. It will be convenient to define the following notation: xt Actual ending inventory status by end of period t. Iþ Actual ending on-hand inventory by end of period t = (xt)+. t I Actual ending back-order level by end of period t = (-xt)+. t PORt+1 Order released at the beginning of period t + 1. Given dt is the observed demand in period t + i and a lead time of L, which is assumed certain in this paper, it can be recognized that actual inventory status is dictated by the following balance equation: xtþi ¼ xtþi1 þ PORtþiL d tþi
ð5Þ
In real-life applications, standing at period t, future gross requirement and future inventory status for period t + i are typically unknown. The projected gross requirement would simply be the forecast of demand L periods in the future. Using forecasts for future demand, (5) converts into an equation for projected inventory status: ^xtþijt ¼ ^xtþi1jt þ PORtþiL F tþijt ;
for i ¼ 1; 2; . . . ; L
ð6Þ
where ^xtjt ¼ xt . As noted earlier, if the net requirement is negative, then the planned order release would be set to zero. In terms of a projected gross requirement, this implies that planned order release would be defined as: PORtþ1 ¼ ðF tþLþ1jt ^xtþLjt Þþ
ð7Þ
By recursively applying (6) in conjunction with (7), we find: !þ Lþ1 L X X PORtþ1 ¼ F tþijt PORtiþ1 xt i¼1
ð8Þ
i¼1
Consider now the implementation of an EWMA forecasting scheme. Given that the forecasts are simply associated horizontal forecast function set to the value of Ft+1jt, we find that (8) is given by: !þ L X PORtþ1 ¼ ½L þ 1F tþ1jt PORtiþ1 xt ð9Þ i¼1
In terms of an MSE-optimal forecasting scheme, we can use (4) to find: Lþ1 Lþ1 X X 1 /i ðL þ 1Þð1 /Þ /ð1 /Lþ1 Þ /ð1 /Lþ1 Þ i dt s þ / dt ¼ F tþijt ¼ s þ 2 ð1 /Þ 1/ ð1 /Þ i¼1 i¼1
ð10Þ
Replacing (10) into (8), we find: PORtþ1 ¼
K 1 þ K 2d t
L X
!þ PORtiþ1 xt
ð11Þ
i¼1
where K1 ¼ K2 ¼
ðL þ 1Þð1 /Þ /ð1 /Lþ1 Þ ð1 /Þ2 /ð1 /Lþ1 Þ ð1 /Þ
s
ð12Þ ð13Þ
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3. Study of MRP performance under different forecasting schemes We evaluated MRP performance under different forecasting scenarios by Monte Carlo simulation. To simulate an AR(1) process, it is commonly recommended that a warm-up period be allowed so that the AR(1) process stabilizes and is free of any start-up effects. In our case, we used the first 50 periods for warming up. Upon warming up, the experiment was designed with a 150 period rolling MRP horizon. This warming up period plus the 150 period rolling horizon constituted one replication in our study. All of our final results are based on simulations of 8000 replications that brought us a high-level accuracy in terms of low standard errors of estimate. In our experiment, we consider three forecasting schemes in the MRP implementation. First, we consider an EWMA forecasting scheme with a wide range of a values. We also consider the best possible EWMA model as measured by smallest mean square error. Such an optimal EWMA model would be associated with optimal smoothing constant which we denote by a*, which can be obtained by varying a. Generally, a* increases in /. Actually, for certain range of /, we can obtain the following relationship between a* and / if our objective is to minimize the long-term forecast error, i.e., to minimize lim ðVarðF tþ1 d tþ1 ÞÞ. t!1
Proposition 1. If 13 < / < 1, a ¼ 3/1 2/ Proof. See Appendix. Under two different forecasting scenarios – EWMA (with optimal smoothing constant a* and non-optimal a) and optimal MSE, we compared the top level of MRP performance – total cost, service level, and fill rate. In order to compare total costs under two different forecasting techniques, we introduced the following two parameters: h: holding cost per unit per period of time. b: backlog cost per unit per period of time. Thus total cost is computed with the following equation: TC ¼
200 X
½hI þ t þ bI t
ð14Þ
t¼51
In addition to the total costs, we are interested in average service level ðSLÞ and average fill rate ðFRÞ. Service level is defined as the probability that available inventory met actual demand, P200 SLt ð15Þ SL ¼ t¼51 150 where SLt ¼
1;
if xt P 0;
0;
otherwise
Average fill rate is defined as: P200 FRt FR ¼ t¼51 150
ð16Þ
where FRt ¼
minðd t ; I t þ PORtL Þ dt
In this paper, we also study the effect of different parameters, such as L, a, and /, on the system performance. Furthermore, Enns (2002) suggests that study of MRP performance when demand is uncertain should be reported in terms of a measure of variability relative to mean level. In particular, Enns (2002) suggests the coefficient of variation (CV) which in our case would be:
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sffiffiffiffiffiffiffiffiffiffiffiffi re pffiffiffiffiffiffiffiffi 1/2 rd re 1 / ¼ ¼ CV ¼ s l s 1þ/
ð17Þ
1/
4. Simulation results Through numerous simulations we have done, we have observed the following simulation results: Observation 1. Everything being equal, TC EWMA P TC EWMA P TC MSE , which suggests the MSE is always better than EWMA in terms of the total inventory and backlog costs. In addition, EWMA with optimal a is better than non-optimal a. For convenience, we will measure the relative improvement rate (IR) as follows: IR ¼
Total cost under EWMA Total cost under MSE TC EWMA TC MSE ¼ Total cost under EWMA TC MSE
ð18Þ
We tried different combination of h and b (h > b, h = b, and h < b). One interesting result is that regardless of h and b, the index of IR is unchanged given the same CV, / and lead time L. Furthermore, IR is also independent of CV. It suggests that IR only depends on the lead time L and /. As long as L and / is fixed, MSE has constant relative improvement rate over EWMA with optimal a. Fig. 2 shows IR with regard to lead time L with different /. From the Fig. 2, it is obvious that IR decreases in L. When L is very large (e.g., L = 40 for / = 0.7), IR is close to 0. The result indicates that EWMA performs well if the lead time is large enough, which may justify why EWMA is widely used in practice. Observation 2. We are also interested in the relationship between / and a*, and our simulation results revealed that: (a) For a fixed L, the larger the /, the larger the a*. This is to be expected and confirmed by the proposition because larger / means the demand process is more fluctuated, which requires larger smoothing constant a* in order to make more accurate forecasting. (b) For a fixed /, the larger the L, the smaller the a* (Fig. 3). One explanation is as follows: larger L makes the demand less sensitive to the MRP planning and lead to smaller a*. Fig. 3 demonstrated the relationship between / and a* for different L. Observation 3. We studied the performance of EWMA and MSE in terms of average service rate ðSLÞ and average fill rate ðFRÞ for different cases, the simulation results reveal that
50
IR (%)
40
φ = 0 .9
30 20
φ = 0 .7 φ = 0 .5
10
φ = 0 .3 0 1
2
3
4
5
6
7
8
L Fig. 2. IR vs. lead time under optimal MSE and EWMA with optimal a.
L.C. Alwan et al. / Computers & Industrial Engineering 54 (2008) 840–850
847
0.9 L=1
0.8 0.7
L=2
0.6
α
∗
L=6
0.5
L=4
0.4 0.3 0.2 0.1 0.0 0.3
0.4
0.5
0.6
0.7
0.8
0.9
φ Fig. 3. Relationship between / and a* for different lead time.
(a) Service level is always close to 50% for both optimal MSE and EWMA with optimal a (see Fig. 4). Therefore in terms of service level, there is no significant difference between two forecasting techniques. Since it is desirable to have higher service level, this result is interesting because it is counterintuitivemore advanced (optimal) forecasting technique does not mean higher service level. In this research, the service level is around 50% because we do not consider safety stock. If we need to achieve higher service level, then appropriate mechanism can be designed. For example we can set safety re re stock SS ¼ z pffiffiffiffiffiffiffiffi , where z is decided by the service level, and pffiffiffiffiffiffiffiffi is the standard deviation of the 2 2 1/
1/
demand process. (b) In terms of average fill rate ðFRÞ, MSE always outperforms EWMA* for all cases. In addition, the larger the CV, the lower the fill rate (see Fig. 5). This can be explained as follows: larger CV means more uncertain demand, which makes forecasting more difficult, leads to lower fill rate. (c) The larger the L, the lower the fill rate (see Fig. 6). Larger L makes forecasting less effective in MRP planning, thus lead to lower fill rate. Observation 4. Finally, we examined IR of EWMA with different a in terms of MSE. Fig. 7 shows the relationship between IR and EWMA with different a. Again, the larger the lead time L, the less effective the forecasting technique. Furthermore there exists an optimal a* that can minimize the total inventory costs, and a* can be obtained from the proposition or the lowest IR in the curves.
0.55
MSE
0.54
EWMA
Service Level
0.53 0.52 0.51 0.50 0.49 0.48 0.47 0.46 0.45 0.1
0.2
CV Fig. 4. Service level ðSLÞ vs. CV (L = 1, / = 0.9).
0.3
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MSE
0.8
Fill Rate
EWMA
0.7
0.6
0.1
0.2
0.3
CV Fig. 5. Fill rate vs. CV (L = 8, / = 0.9).
0.95 MSE EWMA
Fill Rate
0.85
0.75
0.65
0.55 1
2
3
4
5
6
7
8
L Fig. 6. Fill rate vs. L (CV = 0.3, / = 0.7).
70
IR (%)
60
L=1
50 L=4
40
L = 10
30 20
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
α
0.9 1.0
Fig. 7. IR vs. a (/ = 0.7).
5. Concluding remarks In this paper, we examined the performance of MRP with lot-for-lot ordering size under two different forecasting techniques – EWMA and MSE. Specifically, we studied the effects of demand forecasting on the top level of MRP plan because the accuracy of all levels of MRP is highly related to the top level, which is further determined by demand forecasting. We evaluated total cost, fill rate and service rate under different scenarios.
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In terms of total cost, we found MSE is always better than EWMA with optimal a. One interesting result is that relative improvement rate is constant given the same CV, /, and L. In addition, we examined the relationship between the optimal a and other parameters (L and /). The simulation results showed that the optimal a increases in / for a fixed L and decreases in L for a fixed /. For service level, MSE and EWMA are both close to 50% if safety stock is not considered; however MSE always has higher fill rate than EWMA does. Furthermore, we found the fill rate decreases in CV and the lead time L. Simulation results also revealed that the performance of EWMA is close to MSE with optimal a in terms of total inventory cost if the lead time is large enough, which may explain the widespread use of EWMA in practice. The research can be extended in several directions. First, after the top MRP level is planned, all lower levels of MRP can be exploded based on bill of material (BOM) information. Therefore, we can further investigate the effects of different forecasting on lower levels of MRP planning. Second, all the simulation results in this paper are obtained under the assumption of AR(1) demand. We may extend the demand to nonstationary demand pattern such as IMA(1) model, then compare the performance of MRP under different forecasting techniques. Third, we may extend the ordering policy from lot-for-lot to other ordering size, even considering safety stock for different cases. Appendix Proof of the Proposition: VarðF tþ1 d tþ1 Þ ¼ Var
t X
! að1 aÞ
i1
að1 aÞ
i1
d tiþ1 d tþ1
i¼1
¼ Var
t X i¼1
d tiþ1 þ Varðd tþ1 Þ 2Cov
t X
!! að1 aÞ
i1
d tiþ1 ; d tþ1
i¼1
It can be obtained that r2e 2/ð1 aÞ r2e lim ðVarðF tþ1 d tþ1 ÞÞ ¼ 1 þ þ 2 2 t!1 1 ð1 aÞ/ 1 /2 1 ð1 aÞ 1 / " # t X /i i1 2 lim að1 aÞ r2 2 e t!1 1 / i¼1 a2 r2e 2/ð1 aÞ r2e 2ar2e / ¼ 1 þ þ 2 2 2 2 1 ð1 aÞ/ 1 ð1 aÞ/ 1/ 1/ 1 ð1 aÞ 1 / r2e a a 2/ð1 aÞ 2a/ þ þ1 ¼ 1 /2 2 a 2 a 1 ð1 aÞ/ 1 ð1 aÞ/ r2e a 3a/ þ a2 / ¼ þ 1 1 /2 ð2 aÞð1 / þ a/Þ a2
To minimize lim ðVarðF tþ1 d tþ1 ÞÞ, it is equivalent to minimize f(a), where t!1
f ðaÞ ¼
a 3a/ þ a2 / þ1 ð2 aÞð1 / þ a/Þ
By some algebra, we can simplify f ðaÞ ¼
2 2/ ð2 2/Þ þ ð3/ 1Þa a2 /
Since (2 2/) + (3/ 1)a a2/ = (2 a)(1 / + a/) > 0, we only need to Max½a2 / þ ð3/ 1Þa þ a ð2 2/Þ. . It can be solved that if / > 0, a ¼ 3/1 2/
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Since 0 < a* < 1, it is necessary that 13 < / < 1. . Q.E.D Therefore we prove that when 13 < / < 1, a ¼ 3/1 2/ References Alwan, L. C. (2000). Statistical process analysis. Burr Ridge, IL: McGraw Hill/Irwin. Alwan, L. C., Liu, J. J., & Yao, D. Q. (2003). Stochastic characterization of upstream demand processes in a supply chain. IIE Transactions, 35, 207–219. Biggs, J. R., & Campion, W. M. (1982). The effect and cost of forecast error bias for multiple-stage production-inventory systems. Decision Sciences, 13(4), 570–584. Box, G. E., Jenkins, G. M., & Reinsel, G. C. (1994). Time series analysis: Forecasting and control (3rd ed.). Englewood Cliffs, NJ: Prentice Hall. Chen, F. Z., Drezner, Z., Ryan, J. K., & Simchi-Levi, D. (2000a). Quantifying the bullwhip effect in a simple supply chain: The impact of forecasting, lead times, and information. Management Science, 46, 436–443. Chen, F., Ryan, J. K., & Simchi-Levi, D. (2000b). The impact of exponential smoothing forecasts on the bullwhip effect. Naval Research Logistics, 47, 269–286. Enns, S. T. (2002). MRP performance effects due to forecast bias and demand uncertainty. European Journal of Operational Research, 138, 87–102. Graves, S. C. (1999). A single-item inventory model for a nonstationary demand process. Manufacturing and Service Operations Management, 1, 50–61. Graves, S. C., & Willems, S. P. (2000). Optimizing strategic safety stock placement in supply chains. Manufacturing and Service Operations Management, 2, 68–83. Hoerl, R. W. (1998). Six sigma and the future of the quality profession. Quality Progress, 35–42. Hopp, W. J., & Spearman, M. L. (2000). Factory physics (2nd ed.). Boston, MA: McGraw-Hill/Irwin. Jonsson, P., & Mattsson, S.-A. (2003). The implications of fit between planning environments and manufacturing planning and control methods. International Journal of Operations & Production Management, 23, 872–900. Kahn, J. (1987). Inventories and the volatility of production. The American Economic Review, 77, 667–679. Lee, T. S., & Adam, E. E. (1986). Forecasting error evaluation in material requirements planning (MRP) production-inventory systems. Management Science, 32(9), 1186–1205. Lee, H. L., Padmanabhan, P., & Whang, S. (1997). Information distortion in a supply chain: The bullwhip effect. Management Science, 43, 546–558. Lee, H. L., So, K. C., & Tang, C. S. (2000). The value of information sharing in a two-level supply chain. Management Science, 46, 626–643. Shumway, R. H. (1986). AUTOBOX (Version 1.02). The American Statistician, 40, 299–300.
Computers & Industrial Engineering 54 (2008) 840–850 www.elsevier.com/locate/caie
Forecast facilitated lot-for-lot ordering in the presence of autocorrelated demand Layth C. Alwan a, John J. Liu b, Dong-Qing Yao
c,*
a
c
School of Business Administration, University of Wisconsin-Milwaukee, WI 53201, USA b Department of Logistics, Hong Kong Polytechnic University, Hong Kong, PR China Department of Management, College of Business and Economics, Towson University, 8000 York Road, Towson, MD 21252, USA Received 2 February 2005; accepted 4 October 2007 Available online 4 November 2007
Abstract With consistent effort in setup reduction as encouraged by JIT principle, lot-for-lot ordering is gaining popularity in MRP applications. A lot-for-lot order is an immediate copy of the MPS (master production schedule) – direct reflection of demand forecasts. Since all levels of MRP plans are based on MPS, the accuracy of MRP is highly dependent of the accuracy of demand forecasting. In this paper, we are concerned about the impact of forecasting to the performance of a lot-for-lot MRP system when there is notable variability and autocorrelation in the underlying demand process (e.g., an AR(1) process). Specifically under a stationary AR(1) demand, we examine the performance of the MRP based on the most common EWMA forecast model, and then compare it with a minimum mean square error (MSE) forecast model. The notable findings of this study include: (1) MRP performance differs noticeably under the two different forecasting models. (2) The MSE-optimal forecasting performs no worse than the EWMA forecasting in all aspects of MRP applications. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Lot-for-lot ordering; MRP; Forecasting; AR(1) process; Monte Carlo simulation
1. Introduction As evidenced by a recent empirical study (Jonsson & Mattsson, 2003), lot-for-lot ordering is the most frequently applied method in production planning and by users of materials requirements planning (MRP). This fact is not surprising since lot-for-lot has much appeal. As noted out by Hopp and Spearman (2000), lot-forlot ordering is not only simple but also it is consistent with a just-in-time philosophy of making only what is needed. Hopp and Spearman (2000) further point out that lot-for-lot ordering is a preferable policy in manufacturing environments where setup times (costs) are minimal. In its simplicity, a lot-for-lot ordering system calls for releasing an order at any given time period t to match the net requirements for period t + L where L representing the lead time. In practical application, companies *
Corresponding author. Tel.: +1 410 704 6185; fax: +1 410 704 3236. E-mail address: [email protected] (D.-Q. Yao).
0360-8352/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.cie.2007.10.014
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do not know future demand with certainty and, thus, must rely on forecasting for production planning. It is thus natural to consider the impact of the forecasting method on the performance of the production ordering system. A number of authors have provided certain insights on the effects of forecasting bias on MRP performance (Biggs & Campion, 1982; Enns, 2002; Lee & Adam, 1986). These studies, however, do not make explicit assumptions about the nature of the demand process nor do they consider specific forecasting methods. In supply-chain research, the effects of different forecasting methods applied to different demand patterns have been studied (Alwan, Liu, & Yao, 2003; Chen et al., 2000a; Graves, 1999). In particular, these studies demonstrated the bullwhip effect is dependent on both the correlative structure of the demand process and the relative accuracy of the forecasting method. The accuracy of all lower levels of MRP is dependent of the accuracy of master production scheduling (MPS), which is determined by the demand forecasting; Therefore to study the effects of forecasting on top level plays pivotal role in all other lower levels of MRP planning. This paper provides distinct contributions to MRP research. First, we consider, the performance of a MRP system in the presence of a stationary correlative demand process, namely, a first-order autoregressive process or AR(1) process. Second, we study the system’s MRP performance when a common, but simplified, forecasting scheme – the exponential weighted moving-average (EWMA) method – is employed in comparison to an optimal forecasting scheme. In the area of forecasting, an ‘‘optimal’’ forecast model is traditionally meant to imply that the forecast model has minimum mean square forecast errors. For clarity, we will refer to such a model as a mean square error optimal (or MSE-optimal) forecast model. The remainder of this paper is organized as follows: Section 2 describes the lot-for-lot MRP framework and introduces the AR(1) demand process. Section 3 formulates the order release quantities in relationship to the forecasting scheme employed. Section 4 describes a simulation study conducted to gain insights on the relative performance of the MRP systems given the specific forecasting method used. Section 5 offers our concluding remarks. 2. Lot-for-lot MRP under AR(1) demand The implementation of an MRP system calls for the determination each period of a planned order release (POR) for the top MRP level. There are a number of approaches for determining the value of POR for any given period. As noted, one common approach for planned order release determination is based on a lot-forlot ordering policy. With a lot-for-lot policy, the level of the planned order release for any given period is set to the net requirement L periods later, i.e., L is the manufacturing lead time. We will assume that the L is fixed and known. There are a number of potential stochastic processes that can be assumed for the demand process, ranging from a simple independent and identically distributed (IID) process to a nonstationary process. In reality, there is empirical evidence that few real-world processes exhibit pure IID behavior (Alwan, 2000). One correlative demand process that has been frequently studied in the supply-chain literature is the first-order autoregressive model, AR(1) (Chen, Drezner, Ryan, & Simchi-Levi, 2000a, 2000b; Kahn, 1987; Lee, Padmanabhan, & Whang, 1997, 2000). The AR(1) process is often studied because pure first-order correlative effects commonly occur in real-world processes (Alwan, 2000). Furthermore, even for processes that do not follow a pure AR(1) process, an AR(1) model is a flexible model that can reasonably capture lagged effects so commonly found in demand processes. In the following subsections, we describe the AR(1) process in more detail, present the forecasting models to be considered, and present the order quantities to be examined in this paper. 2.1. Demand process Suppose the system is faced with an AR(1) demand process given as follows: d t ¼ s þ /d t1 þ et ;
t ¼ 0; 1; 2; . . .
ð1Þ
where dt is the observed demand in period t, s > 0, j/j < 1, and {et} is an independent and identically distributed normal (IIDN) process with mean 0 and variance r2e . The condition of j/j < 1 ensures that the process is stationary. It is useful to note that ld = E(dt) = s/(1 /) and r2d ¼ Varðd t Þ ¼ r2e =ð1 /2 Þ for any t. In this
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paper, we assume / is known and fixed. In practice, if / is unknown, we may first obtain unbiased estimates of ^ and then substitute the demand parameter with the corresponding parameter estimate. demand parameter, /, As noted earlier, the AR(1) model is widely used in supply-chain management research (e.g., Chen, Ryan, & Simchi-Levi, 2000b; Lee, So, & Tang, 2000). One of the primary reasons for the focus on the AR(1) model is that it possesses good dynamic characteristics. Namely, by varying the / parameter, we are able to study the effects of processes which are random, nonrandom but stationary, or even nonstationary processes. This flexibility allows us to gain practical insights for many real demand patterns. By varying the values of /, one can represent a wide variety of process behaviors. When / = 0, we have an IIDN process with mean s and variance r2e . For 1 < / < 0, the process is negatively correlated and will exhibit period-to-period oscillatory behavior. For 0 < / < 1, the demand process will be positively correlated which is reflected by as wandering or meandering sequence of observations. As j/j approaches 1, the process approaches nonstationary behavior, most notably, the random walk model – ARIMA(0, 1, 0) – is a special case of the AR(1) model when / = 1. As pointed out by Graves and Willems (2000), varying a stationary demand model is an important exercise for gaining fundamental insights into the relationship between variables such as inventory and orders relative to demand characterization. 2.2. Forecast models Smoothing methods, such as moving averages and exponential smoothing are widely employed for forecasting purposes in many production and operations management applications, largely because of their simplicity and ease of implementation. As such, most researchers of supply-chain management (SCM) problems requiring a forecast model have based their studies on either the moving-average method (e.g., Chen et al., 2000a) or the exponential weighted moving-average (EWMA) method (e.g., Chen et al., 2000b). Given the close connection between the moving-average method and the EWMA method, we will only focus on one method (namely, the EWMA method) in this paper. The EWMA model can be expressed as follows: F tþ1jt ¼ ad t þ ð1 aÞF tjt1
ð2Þ
where 0 < a < 1 is the smoothing constant and Ft+1jt is the forecast of period t + 1 made at the end of period t. It should be noted that the forecasts for periods t + i (i = 1, 2, . . .) made at time t are equal, that is, Ft+ijt = Ft+1jt for i = 1, 2, . . .. Hence, the forecasts for all lead times will follow a horizontal line parallel to the time axis. Even though the EWMA method, and to a lesser extent the moving-average method, has flexibility for adapting to a variety of correlated demand processes, it is MSE optimal for only one underlying time-series model, namely, a first-order integrated moving average, denoted by ARIMA(0, 1, 1) or IMA(1, 1) (e.g., Graves, 1999). An ARIMA(0, 1, 1) process is a nonstationary process that can be interpreted as a randomwalk trend plus a random deviation from the trend. Thus, under no circumstance is the EWMA method MSE optimal for a stationary AR(1) process. This fact opens up consideration of employing an MSE-optimal forecast scheme for the assumed AR(1) process. By recursively applying (1), it is easy to show that: d tþi ¼ s þ /s þ þ /i1 s þ /i d t þ /i1 etþ1 þ /i2 etþ2 þ þ etþi ¼
1 /i s þ /i d t þ /i1 etþ1 þ /i2 etþ2 þ þ etþi 1/
ð3Þ
For a general ARIMA process, it can be shown that the minimum mean square error forecast for period t + i is the conditional expectation of dt+i given current and previous observations dt, dt-1, dt-2,. . . (see Box, Jenkins, & Reinsel, 1994). In the case of an AR(1) process, this implies the MSE-optimal forecast function is given by E(dt+ijdt). Since E(et+ijdt) = 0 for i = 1, 2, . . ., it immediately follows that for an AR(1) process, the MSE-optimal forecast function is given by: F tþijt ¼
1 /i s þ /i d t ; 1/
for i ¼ 1; 2; . . .
ð4Þ
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In contrast to the two previous methods, this forecast function is not a horizontal projection into the future. Instead the forecasts revert back towards the overall mean level of s/1 /. The MSE-optimal forecast function reflects the fact that the AR(1) process is stationary and has the property of conditional mean reversion; that is, even though the process can be expected to wander away (below or above) from the overall mean it is also expected to eventually return back to the overall mean. The moving-average and EWMA methods fail to capture this mean reversion property of a stationary AR(1) process. To illustrate the dynamics of time series and forecasting forecasts, consider an AR(1) process with the following parameters: s = 50, / = 0.8, and re = 10. In Fig. 1, we show 50 consecutive observations randomly generated from such an AR(1) process. Using the statistical package Minitab, we found the optimal EWMA model and projected its forecasts into the future as shown in Fig. 1. In addition, we show in Fig. 1 the optimal MSE forecasts based on the AR(1) model. As can be seen from Fig. 1, the AR(1) projected forecast revert back to the long-run mean while the EWMA forecasts are horizontally ‘offset’. The implication in an operations management setting is that the EWMA model will over forecast demand potentially resulting in unnecessary costs such as additional inventory holding costs. On the flip side, if current demand of an AR(1) process was below the long-run mean, the EWMA forecasts will be horizontally offset on the low side resulting in other potential costs such as shortage and backlog costs. One argument often presented against the use of optimal methods is that their implementation is more difficult than the simple smoothing methods when parameters are unknown. It is pointed out that to implement optimal methods requires statistical skills in time-series modeling, including knowledge of model identification, model estimation, and tests for model adequacy, that are beyond the skill set of a typical operations manager. However, we believe that the industrial use of more sophisticated time-series models is steadily growing because of two reasons. First, the requirement of intense statistical training, often referred to as 6r training, is increasingly becoming commonplace (Hoerl, 1998). At corporations like GE, Motorola, and Allied Signal, organizational cultures are being developed in which there is a strong desire from employees throughout the organization to learn and implement advanced statistical techniques. Indeed, the authors of this paper can report that seminars in time-series analysis are part of the regular continuing education program at GE-Medical Systems and are required to be taken by all supply-chain managers. Second, modern computational tools are readily available to make possible automated implementation of time-series modeling including the general class of ARIMA models. These programs are designed to automate model identification, model fitting, and forecasting (see Shumway, 1986). 2.3. Planned order releases As noted earlier, the accuracy of all levels of MRP depends on the MPS. In addition, once the demand of top MRP level is determined, demands of lower levels of MRP are certain, and can be derived based on the bill Future 280
Optimal EWMA Forecast
Demand
270 Optimal MSE Forecast
260 250
Mean
240 230 220 1
7
14
21
28
35
42
49
56
63
70
Time Fig. 1. AR(1) process and forecasting (/ = 0.8).
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of material (BOM). Therefore, in this research, we focus on studying the effects of different forecasting techniques on the top MRP level under lot for lot ordering policy. A lot-for-lot policy calls for the level of the planned order release to be set to the net requirement of a future period. In general, the net requirement of any given period is equal to the gross requirements of that period minus ending inventory status of the previous period. If the net requirement is negative, then this implies inventory is sufficient to cover the gross requirement which, in turn, implies a zero planned order release. It will be convenient to define the following notation: xt Actual ending inventory status by end of period t. Iþ Actual ending on-hand inventory by end of period t = (xt)+. t I Actual ending back-order level by end of period t = (-xt)+. t PORt+1 Order released at the beginning of period t + 1. Given dt is the observed demand in period t + i and a lead time of L, which is assumed certain in this paper, it can be recognized that actual inventory status is dictated by the following balance equation: xtþi ¼ xtþi1 þ PORtþiL d tþi
ð5Þ
In real-life applications, standing at period t, future gross requirement and future inventory status for period t + i are typically unknown. The projected gross requirement would simply be the forecast of demand L periods in the future. Using forecasts for future demand, (5) converts into an equation for projected inventory status: ^xtþijt ¼ ^xtþi1jt þ PORtþiL F tþijt ;
for i ¼ 1; 2; . . . ; L
ð6Þ
where ^xtjt ¼ xt . As noted earlier, if the net requirement is negative, then the planned order release would be set to zero. In terms of a projected gross requirement, this implies that planned order release would be defined as: PORtþ1 ¼ ðF tþLþ1jt ^xtþLjt Þþ
ð7Þ
By recursively applying (6) in conjunction with (7), we find: !þ Lþ1 L X X PORtþ1 ¼ F tþijt PORtiþ1 xt i¼1
ð8Þ
i¼1
Consider now the implementation of an EWMA forecasting scheme. Given that the forecasts are simply associated horizontal forecast function set to the value of Ft+1jt, we find that (8) is given by: !þ L X PORtþ1 ¼ ½L þ 1F tþ1jt PORtiþ1 xt ð9Þ i¼1
In terms of an MSE-optimal forecasting scheme, we can use (4) to find: Lþ1 Lþ1 X X 1 /i ðL þ 1Þð1 /Þ /ð1 /Lþ1 Þ /ð1 /Lþ1 Þ i dt s þ / dt ¼ F tþijt ¼ s þ 2 ð1 /Þ 1/ ð1 /Þ i¼1 i¼1
ð10Þ
Replacing (10) into (8), we find: PORtþ1 ¼
K 1 þ K 2d t
L X
!þ PORtiþ1 xt
ð11Þ
i¼1
where K1 ¼ K2 ¼
ðL þ 1Þð1 /Þ /ð1 /Lþ1 Þ ð1 /Þ2 /ð1 /Lþ1 Þ ð1 /Þ
s
ð12Þ ð13Þ
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3. Study of MRP performance under different forecasting schemes We evaluated MRP performance under different forecasting scenarios by Monte Carlo simulation. To simulate an AR(1) process, it is commonly recommended that a warm-up period be allowed so that the AR(1) process stabilizes and is free of any start-up effects. In our case, we used the first 50 periods for warming up. Upon warming up, the experiment was designed with a 150 period rolling MRP horizon. This warming up period plus the 150 period rolling horizon constituted one replication in our study. All of our final results are based on simulations of 8000 replications that brought us a high-level accuracy in terms of low standard errors of estimate. In our experiment, we consider three forecasting schemes in the MRP implementation. First, we consider an EWMA forecasting scheme with a wide range of a values. We also consider the best possible EWMA model as measured by smallest mean square error. Such an optimal EWMA model would be associated with optimal smoothing constant which we denote by a*, which can be obtained by varying a. Generally, a* increases in /. Actually, for certain range of /, we can obtain the following relationship between a* and / if our objective is to minimize the long-term forecast error, i.e., to minimize lim ðVarðF tþ1 d tþ1 ÞÞ. t!1
Proposition 1. If 13 < / < 1, a ¼ 3/1 2/ Proof. See Appendix. Under two different forecasting scenarios – EWMA (with optimal smoothing constant a* and non-optimal a) and optimal MSE, we compared the top level of MRP performance – total cost, service level, and fill rate. In order to compare total costs under two different forecasting techniques, we introduced the following two parameters: h: holding cost per unit per period of time. b: backlog cost per unit per period of time. Thus total cost is computed with the following equation: TC ¼
200 X
½hI þ t þ bI t
ð14Þ
t¼51
In addition to the total costs, we are interested in average service level ðSLÞ and average fill rate ðFRÞ. Service level is defined as the probability that available inventory met actual demand, P200 SLt ð15Þ SL ¼ t¼51 150 where SLt ¼
1;
if xt P 0;
0;
otherwise
Average fill rate is defined as: P200 FRt FR ¼ t¼51 150
ð16Þ
where FRt ¼
minðd t ; I t þ PORtL Þ dt
In this paper, we also study the effect of different parameters, such as L, a, and /, on the system performance. Furthermore, Enns (2002) suggests that study of MRP performance when demand is uncertain should be reported in terms of a measure of variability relative to mean level. In particular, Enns (2002) suggests the coefficient of variation (CV) which in our case would be:
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sffiffiffiffiffiffiffiffiffiffiffiffi re pffiffiffiffiffiffiffiffi 1/2 rd re 1 / ¼ ¼ CV ¼ s l s 1þ/
ð17Þ
1/
4. Simulation results Through numerous simulations we have done, we have observed the following simulation results: Observation 1. Everything being equal, TC EWMA P TC EWMA P TC MSE , which suggests the MSE is always better than EWMA in terms of the total inventory and backlog costs. In addition, EWMA with optimal a is better than non-optimal a. For convenience, we will measure the relative improvement rate (IR) as follows: IR ¼
Total cost under EWMA Total cost under MSE TC EWMA TC MSE ¼ Total cost under EWMA TC MSE
ð18Þ
We tried different combination of h and b (h > b, h = b, and h < b). One interesting result is that regardless of h and b, the index of IR is unchanged given the same CV, / and lead time L. Furthermore, IR is also independent of CV. It suggests that IR only depends on the lead time L and /. As long as L and / is fixed, MSE has constant relative improvement rate over EWMA with optimal a. Fig. 2 shows IR with regard to lead time L with different /. From the Fig. 2, it is obvious that IR decreases in L. When L is very large (e.g., L = 40 for / = 0.7), IR is close to 0. The result indicates that EWMA performs well if the lead time is large enough, which may justify why EWMA is widely used in practice. Observation 2. We are also interested in the relationship between / and a*, and our simulation results revealed that: (a) For a fixed L, the larger the /, the larger the a*. This is to be expected and confirmed by the proposition because larger / means the demand process is more fluctuated, which requires larger smoothing constant a* in order to make more accurate forecasting. (b) For a fixed /, the larger the L, the smaller the a* (Fig. 3). One explanation is as follows: larger L makes the demand less sensitive to the MRP planning and lead to smaller a*. Fig. 3 demonstrated the relationship between / and a* for different L. Observation 3. We studied the performance of EWMA and MSE in terms of average service rate ðSLÞ and average fill rate ðFRÞ for different cases, the simulation results reveal that
50
IR (%)
40
φ = 0 .9
30 20
φ = 0 .7 φ = 0 .5
10
φ = 0 .3 0 1
2
3
4
5
6
7
8
L Fig. 2. IR vs. lead time under optimal MSE and EWMA with optimal a.
L.C. Alwan et al. / Computers & Industrial Engineering 54 (2008) 840–850
847
0.9 L=1
0.8 0.7
L=2
0.6
α
∗
L=6
0.5
L=4
0.4 0.3 0.2 0.1 0.0 0.3
0.4
0.5
0.6
0.7
0.8
0.9
φ Fig. 3. Relationship between / and a* for different lead time.
(a) Service level is always close to 50% for both optimal MSE and EWMA with optimal a (see Fig. 4). Therefore in terms of service level, there is no significant difference between two forecasting techniques. Since it is desirable to have higher service level, this result is interesting because it is counterintuitivemore advanced (optimal) forecasting technique does not mean higher service level. In this research, the service level is around 50% because we do not consider safety stock. If we need to achieve higher service level, then appropriate mechanism can be designed. For example we can set safety re re stock SS ¼ z pffiffiffiffiffiffiffiffi , where z is decided by the service level, and pffiffiffiffiffiffiffiffi is the standard deviation of the 2 2 1/
1/
demand process. (b) In terms of average fill rate ðFRÞ, MSE always outperforms EWMA* for all cases. In addition, the larger the CV, the lower the fill rate (see Fig. 5). This can be explained as follows: larger CV means more uncertain demand, which makes forecasting more difficult, leads to lower fill rate. (c) The larger the L, the lower the fill rate (see Fig. 6). Larger L makes forecasting less effective in MRP planning, thus lead to lower fill rate. Observation 4. Finally, we examined IR of EWMA with different a in terms of MSE. Fig. 7 shows the relationship between IR and EWMA with different a. Again, the larger the lead time L, the less effective the forecasting technique. Furthermore there exists an optimal a* that can minimize the total inventory costs, and a* can be obtained from the proposition or the lowest IR in the curves.
0.55
MSE
0.54
EWMA
Service Level
0.53 0.52 0.51 0.50 0.49 0.48 0.47 0.46 0.45 0.1
0.2
CV Fig. 4. Service level ðSLÞ vs. CV (L = 1, / = 0.9).
0.3
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MSE
0.8
Fill Rate
EWMA
0.7
0.6
0.1
0.2
0.3
CV Fig. 5. Fill rate vs. CV (L = 8, / = 0.9).
0.95 MSE EWMA
Fill Rate
0.85
0.75
0.65
0.55 1
2
3
4
5
6
7
8
L Fig. 6. Fill rate vs. L (CV = 0.3, / = 0.7).
70
IR (%)
60
L=1
50 L=4
40
L = 10
30 20
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
α
0.9 1.0
Fig. 7. IR vs. a (/ = 0.7).
5. Concluding remarks In this paper, we examined the performance of MRP with lot-for-lot ordering size under two different forecasting techniques – EWMA and MSE. Specifically, we studied the effects of demand forecasting on the top level of MRP plan because the accuracy of all levels of MRP is highly related to the top level, which is further determined by demand forecasting. We evaluated total cost, fill rate and service rate under different scenarios.
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In terms of total cost, we found MSE is always better than EWMA with optimal a. One interesting result is that relative improvement rate is constant given the same CV, /, and L. In addition, we examined the relationship between the optimal a and other parameters (L and /). The simulation results showed that the optimal a increases in / for a fixed L and decreases in L for a fixed /. For service level, MSE and EWMA are both close to 50% if safety stock is not considered; however MSE always has higher fill rate than EWMA does. Furthermore, we found the fill rate decreases in CV and the lead time L. Simulation results also revealed that the performance of EWMA is close to MSE with optimal a in terms of total inventory cost if the lead time is large enough, which may explain the widespread use of EWMA in practice. The research can be extended in several directions. First, after the top MRP level is planned, all lower levels of MRP can be exploded based on bill of material (BOM) information. Therefore, we can further investigate the effects of different forecasting on lower levels of MRP planning. Second, all the simulation results in this paper are obtained under the assumption of AR(1) demand. We may extend the demand to nonstationary demand pattern such as IMA(1) model, then compare the performance of MRP under different forecasting techniques. Third, we may extend the ordering policy from lot-for-lot to other ordering size, even considering safety stock for different cases. Appendix Proof of the Proposition: VarðF tþ1 d tþ1 Þ ¼ Var
t X
! að1 aÞ
i1
að1 aÞ
i1
d tiþ1 d tþ1
i¼1
¼ Var
t X i¼1
d tiþ1 þ Varðd tþ1 Þ 2Cov
t X
!! að1 aÞ
i1
d tiþ1 ; d tþ1
i¼1
It can be obtained that r2e 2/ð1 aÞ r2e lim ðVarðF tþ1 d tþ1 ÞÞ ¼ 1 þ þ 2 2 t!1 1 ð1 aÞ/ 1 /2 1 ð1 aÞ 1 / " # t X /i i1 2 lim að1 aÞ r2 2 e t!1 1 / i¼1 a2 r2e 2/ð1 aÞ r2e 2ar2e / ¼ 1 þ þ 2 2 2 2 1 ð1 aÞ/ 1 ð1 aÞ/ 1/ 1/ 1 ð1 aÞ 1 / r2e a a 2/ð1 aÞ 2a/ þ þ1 ¼ 1 /2 2 a 2 a 1 ð1 aÞ/ 1 ð1 aÞ/ r2e a 3a/ þ a2 / ¼ þ 1 1 /2 ð2 aÞð1 / þ a/Þ a2
To minimize lim ðVarðF tþ1 d tþ1 ÞÞ, it is equivalent to minimize f(a), where t!1
f ðaÞ ¼
a 3a/ þ a2 / þ1 ð2 aÞð1 / þ a/Þ
By some algebra, we can simplify f ðaÞ ¼
2 2/ ð2 2/Þ þ ð3/ 1Þa a2 /
Since (2 2/) + (3/ 1)a a2/ = (2 a)(1 / + a/) > 0, we only need to Max½a2 / þ ð3/ 1Þa þ a ð2 2/Þ. . It can be solved that if / > 0, a ¼ 3/1 2/
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