Dynamic stochastic lot sizing with forecast evolution in rolling‐horizon planning

Forel, Alexandre; Grunow, Martin Article — Published Version Dynamic stochastic lot sizing with forecast evolution in rolling‐horizon planning Production and Operations Management Provided in Cooperation with: John Wiley & Sons Suggested Citation: Forel, Alexandre; Grunow, Martin (2022) : Dynamic stochastic lot sizing with forecast evolution in rolling‐horizon planning, Production and Operations Management, ISSN 1937-5956, Wiley, Hoboken, NJ, Vol. 32, Iss. 2, pp. 449-468, https://doi.org/10.1111/poms.13881 This Version is available at: https://hdl.handle.net/10419/287847 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by-nc/4.0/ Received: 1 September 2021 Accepted: 24 August 2022 DOI: 10.1111/poms.13881 ORIGINAL ARTICLE Dynamic stochastic lot sizing with forecast evolution in rolling-horizon planning Alexandre Forel1,2Martin Grunow1 1TUM School of Management, Technical University of Munich, Arcisstraße 21, Munich, Germany 2Advanced Optimization in a Networked Economy (AdONE), Technical University of Munich, Arcisstraße 21, Munich, Germany Correspondence Alexandre Forel, TUM School of Management, Technical University of Munich, Arcisstraße 21, 80333 Munich, Germany. Email: [email protected] Handling Editor: Panos Kouvelis Funding information Deutsche Forschungsgemeinschaft, Grant/Award Number: 277991500/GRK2201 Abstract Academic approaches considering demand uncertainty in lot sizing are seldom used in practice. Industry typically implements deterministic models and accounts for uncertainties by using a rolling-horizon planning framework with frequent forecast updates. This paper bridges this gap by proposing a stochastic lot-sizing methodology adapted to rolling-horizon processes. Using the martingale model of forecast evolution (MMFE), we are able to anticipate the forecast updates from rolling-horizon planning in stochastic lot sizing. Our formulation is extended with production recourse to reflect the replanning flexibility of rolling-horizon planning. Extensive simulations on both synthetic and real-world data show the value of forecast evolution models. Forecast evolution models reduce actual costs by 14%on average compared to traditional deterministic planning. The advantage of the extended model with production recourse depends on several factors including capacity, correlation, and uncertainty. Sensitivity analyses show that recourse can reduce costs by an additional 3%on average and up to 10%in specific settings. Using real-world and synthetic data, we provide the first analysis of the value of additive and multiplicative MMFE-based planning models when the true forecast evolution process is unknown. We show that, contrary to the existing consensus, the additive model performs more robustly than the multiplicative model on a wide array of problem settings. KEYWORDS additive and multiplicative martingale model of forecast evolution, forecast evolution, lot sizing, recourse, rolling horizon 1INTRODUCTION Demand uncertainty has been studied extensively in stochastic lot sizing using probability distributions to model the uncertain demand. However, the use of these models in industry has been limited. A major shortcoming is that they cannot be integrated properly into the periodic planning processes that manufacturing companies use to update demand forecasts. Thus, the substantial information technology (IT) support, human resources, and time dedicated to forecasting are ignored. In fact, previous research on stochastic lot sizing has neglected the value of forecasts in generating demand Accepted by Panos Kouvelis, after two revisions. This is an open access article under the terms of the Creative Commons Attribution-NonCommercial License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited and is not used for commercial purposes. © 2022 The Authors. Production and Operations Management published by Wiley Periodicals LLC on behalf of Production and Operations Management Society. distributions altogether. Stochastic lot-sizing approaches suitable for industry adoption should exploit the data contained in historic forecasts. Demand distributions must be updated dynamically based not only on the latest demand realizations but also on forecasts to fit rolling-horizon processes. The use of forecast evolution models can bridge between stochastic lot-sizing models and rolling-horizon planning in industry. Different from demand distributions used in traditional stochastic lot sizing, they model demand uncertainty as encountered in rolling-horizon planning. The martingale model of forecast evolution (MMFE) developed by Graves et al. (1986) and Heath and Jackson (1994) models future forecast changes as a stochastic process. Two methods for modeling the forecast evolution process according to the Prod Oper Manag. 2023;32:449–468. wileyonlinelibrary.com/journal/poms 449 450 FOREL AND GRUNOW Production and Operations Management MMFE have been proposed. The additive model measures the difference between successive forecasts and assumes that these differences follow a multivariate normal distribution. The multiplicative model measures the ratio between successive forecasts and assumes that this ratio follows a log-normal distribution. While it has been argued that the multiplicative model is more relevant when demand fluctuates over time (Hausman, 1969; Heath & Jackson, 1994), extensive comparisons of the two MMFE models are still missing. In particular, the cost of modeling error, that is using the additive or multiplicative model when the true process is unknown, has not been evaluated so far. Hence, MMFE models can be estimated directly from the history of past demand and successive forecasts revisions routinely collected in industry. Demand forecasting is typically an organizational alignment step that is part of the sales and operations planning processes. Hence, the forecasts observed in each planning period can be a mix of expert judgmental forecasts and forecasts obtained from forecasting algorithms. Chen and Lee (2009) review how the MMFE generalizes several classical prediction models such as autoregressive moving average models. Here, the MMFE parameters can be determined exactly. Still, the strength of MMFE lies in its ability to integrate a combination of model-based quantitative forecasts and expert-based judgmental forecasts (Heath & Jackson, 1994). In this context, MMFE model parameters have to be estimated from historical data. Yet, despite the central role of data, the application of MMFE to real-world cases is rare and many questions remain open regarding the applicability and value of forecast evolution models. Lot-sizing approaches suitable for rolling-horizon planning must not only account for the forecast updating process but also for the ability of planners to adapt production plans. Ignoring this replanning opportunity leads to overly conservative decisions and ultimately higher costs. In the stochastic lot-sizing literature, the replanning opportunity has been captured by introducing recourse production decisions that react to demand observations. With MMFE-based models, recourse decisions respond not only to the realized demand but also to forecast updates for the entire horizon, providing richer information and representing industrial planning processes. Lot-sizing approaches that capture this planning flexibility while maintaining computational tractability are lacking. Moreover, even though rollinghorizon schemes shape planning processes in manufacturing companies, only limited attention has been given to the performance of stochastic lot-sizing models in rolling-horizon planning. Hence, the value of these methods compared to traditional deterministic planning is not always clear. This work is motivated by our collaboration with a large producer of chemicals used in agriculture that manages expensive multipurpose equipment in the face of an uncertain demand. Demand has a yearly seasonal pattern, which is especially challenging due to uncertainties in both the volume and timing of the peak selling season. In a similar setting, Schlapp et al. (2022) study a stylized model without forecast evolution and production constraints. However, since capacity is limited, production often starts ahead of the peak season, which can lead to substantial on-hand inventory. Moreover, expensive cleaning operations have to be conducted each time the equipment is set up for a different product family. The company’s planning problem thus exhibits the key trade-off between demand satisfaction, inventory costs, and setup costs that is captured by a lot-sizing problem. Because early forecasts often have poor accuracy, planning is implemented in a rolling-horizon fashion to benefit from frequent forecast updates. We contribute to the state of the art in the following ways. 1. We elevate modeling demand uncertainty from distributions to MMFE in lot-sizing models to account for the central role of forecast evolution processes in real-world rolling-horizon planning. We show that both the additive and the multiplicative MMFE-based lot-sizing models can be solved efficiently using existing linearization techniques. The stochastic planning models can be solved to optimality without resorting to approximations for capacity allocation. Our modeling approach covers important real-world considerations including fixed setup costs, multiple products sharing limited capacity, and a complex correlation structure of the forecast updates. While we focus on the classical production planning problem of lot sizing, our solution approach is applicable to a wide range of problem settings. By showing that MMFE can be applied to rich production planning problems, we aim to foster the adoption of forecast evolution models in research and industry. 2. We demonstrate the value of forecast evolution for lotsizing models in rolling-horizon planning on synthetic and real-word data. We show that stochastic models based on forecast evolution consistently outperform deterministic models in rolling horizon. On average, they reduce overall costs by around 14%. In contrast, stochastic models that only account for demand uncertainty but ignore forecasts and their evolution fail to reduce costs compared to simple deterministic models. These results clearly show that the evolution of forecasts must be considered in effective decision-support systems for rolling-horizon planning. 3. We assess the strengths and weaknesses of the additive and multiplicative models. We analyze the performance when the true forecast evolution process is unknown but has to be estimated from data. We show that the additive MMFE is more robust in a wide array of problem settings, even when demand fluctuates over time. The multiplicative model, on the other hand, lacks robustness to unknown forecast evolution processes and can even lead to significant cost increases compared to the deterministic benchmark. The superior performance of the additive MMFE, also observed on real-world data, refutes the previous consensus on the suitability of the two forecast evolution models. 4. We develop an extended model that allows production recourse and measure the value of recourse in repeated rolling-horizon simulations. We show that production recourse leads to around 3%cost savings on average. We DYNAMIC STOCHASTIC LOT SIZING WITH FORECAST EVOLUTION 451 Production and Operations Management also identify key parameters that influence the value of recourse such as the correlation of forecast updates of different products and time periods. When the forecast evolution process is positively correlated over products and negatively correlated over time periods, the value of recourse can be up to 10%. In our extended model, a high value of recourse can be obtained with small scenario trees, allowing for computationally efficient implementations. In the following section, a brief review of related literature is presented. In Section 3, we introduce the additive and multiplicative MMFE and describe how they can be used to dynamically update the demand distributions over the horizon. In particular, we recall how to obtain the distributions of demand and cumulative demand from the forecast evolution process and analyze the effect of forecast update correlation on the variance of the cumulative demand for additive and multiplicative MMFE. Section 4provides the MMFEbased lot-sizing formulation. We then introduce a multistage formulation that allows production recourse with a scenariobased representation of demand uncertainty. In Section 5, we assess the value of forecast evolution models and the value of recourse in extensive rolling-horizon simulations using synthetic and real-world data. Our findings are summarized in Section 6, where we also provide suggestions for future research. 2 LITERATURE REVIEW In this section, we review the literature on stochastic lot sizing and forecast evolution. We locate our work at the intersection of the two research streams and highlight gaps in the existing literature. 2.1 Stochastic lot sizing and rolling-horizon planning The analysis of the value of adapting lot-sizing decisions in rolling-horizon planning can be traced back to Bookbinder and Tan (1988), who introduce different strategies to update decisions. Using the static strategy, decisions are determined all at once and fixed over the planning horizon. The dynamic strategy, on the other hand, allows decisions to be adapted as new information is observed in rolling horizon. The authors emphasize that dynamic planning approaches are especially relevant when demand distributions are dynamically updated in rolling horizon. Dynamic strategies can be implemented through scenario-based formulations in which production decisions are set as recourse variables. Escudero et al. (1993) present several lot-sizing formulations that allow increasing levels of recourse in a multistage scenario tree. Brandimarte (2006) investigates the value of scenario-based stochastic lot sizing in rolling-horizon planning by means of repeated simulations. They show that scenario models allow good performance through recourse decisions but require long computation times. Recently, Thevenin et al. (2020) use a combination of heuristics and advanced sampling techniques to implement dynamic strategies in a multiechelon lot-sizing context. Scenario-based models are notoriously hard to solve. To improve computational performance, Helber et al. (2013) develop piecewise-linear approximations (PLAs) of the expected inventory and backlog functions and show that they outperform scenario-based formulations without recourse. These formulations have proved flexible and have been used in several production planning settings. Rossi et al. (2015)use PLA to determine the parameters of near-optimal production policies. De Smet et al. (2020) include sequence-dependent changeovers in a lot-sizing and scheduling problem. Tempelmeier and Hilger (2015) and van Pelt and Fransoo (2018) introduce fill-rate service-level constraints. Sereshti et al. (2021) extend this work showing that PLAs can be used to formulate many types of service-level constraints in stochastic lot sizing. However, PLA methods may lead to overly conservative production plans as they do not allow for recourse decisions. To incorporate the replanning opportunity in lot-sizing problems, Tavaghof-Gigloo and Minner (2021) integrate a heuristic in an extended PLA formulation and investigate its benefits in rolling-horizon simulations. A significant limitation of the above-cited works is that they assume the demand distributions to be known. Yet, demand distributions are seldom available in practice. This problem was discussed by Klabjan et al. (2013) who propose nonparametric approaches to estimate demand distributions from past observations. Still, this work ignores forecasts and their updates stemming from the rolling-horizon processes. We contribute to this research stream in two ways. First, we show that forecast evolution models can provide demand distributions that are dynamically updated in rolling-horizon planning and readily integrated in lot-sizing models using existing methods. Second, we extend existing PLA formulations to allow production recourse over discrete scenarios. Thus, we combine the strengths of PLA and scenario methods to allow flexible decisions while ensuring fast computation. 2.2 Forecast evolution models Since the early analyses of forecast revision processes conducted by Hausman (1969) and Hausman and Peterson (1972), the MMFE has been applied to a wide variety of problems including defining supply contracts (Donohue, 2000), capacity planning (Boyacı & Özer, 2010), and inventory management (Bicer & Seifert, 2017; Iida & Zipkin, 2006; Özer & Wei, 2004; Wang & Tomlin, 2009; Wang et al., 2012). The aforementioned research focuses on determining optimal policies analytically but does not consider complex production planning settings such as managing multiple products with limited capacity and fixed setup costs for production. Further, it does not consider the rollinghorizon implementation of production plans. In particular, 452 FOREL AND GRUNOW Production and Operations Management unconditional production decisions that do not depend on demand scenarios should be determined over the short-term horizon. This provides a reference plan that can be communicated to upstream and downstream partners in the supply chain in each review period. Pinçe et al. (2020) apply the findings of Wang et al. (2012) on multiplicative MMFE for multiordering newsvendor to a real-world data set of a large product portfolio. They discuss the challenges of applying MMFE-based planning models from real-world data but do not investigate the out-of-sample value of their method when the true forecast evolution model is unknown. A second research stream studies the rolling-horizon implementation of forecast evolution models. Norouzi and Uzsoy (2014) determine the key properties of the uncertain demand under additive and multiplicative MMFE and derive the optimal base-stock policy for a single-product, uncapacitated planning problem with a chance constraint. Albey et al. (2015) extend this work with a heuristic that solves the multiproduct problem based on a predetermined capacity allocation. They evaluate the rolling-horizon performance of the MMFE model in a real-world case study in the semiconductor industry. Ziarnetzky et al. (2018) adapted the method to a multiplicative MMFE and evaluate it in rolling-horizon planning with synthetic data. Albey et al. (2016) combine the model with a genetic algorithm to allocate capacity to products. They show the benefits of the improved allocation in a simulation study under additive MMFE. Ziarnetzky et al. (2020) perform extensive rolling-horizon simulations to evaluate and compare the performance of the additive and multiplicative MMFE. The forecast evolution is set to follow either the additive or multiplicative MMFE and the forecast and production plan updates are performed in a rolling-horizon fashion. However, the ability of MMFE models to generalize when the true forecast evolution process is unknown has not been studied so far. We extend the research stream on MMFE by further relaxing the limiting assumptions of the model. We consider a general lot-sizing setting with multiple products, limited capacity, inventory holding costs, and fixed costs for setup operations. The model does not rely on a predetermined allocation of capacity and can be solved to optimality. Further, we provide insights into the strengths and weaknesses of the additive and multiplicative MMFE, analyze their ability to generalize from historical data, and evaluate performance through rolling-horizon simulations on both synthetic and real-world data. 3FROM FORECAST EVOLUTION TO DEMAND DISTRIBUTIONS In this section, we introduce the additive and multiplicative MMFE as formalized by Heath and Jackson (1994). For each model, we recall how the probability distributions underlying the uncertain demand can be deduced from the stochastic forecast evolution process over the planning horizon. This fully describes the dynamic updating of the demand distribution as new forecasts are observed in rolling horizon. Then, we show how to obtain the distributions of the cumulative demand over the horizon by adapting the results from Norouzi and Uzsoy (2014). This step is essential to derive linearized lot-sizing formulations that are tractable, as will be shown in Section 4. Finally, we analyze the effect of forecast update correlation on the cumulative demand variance for the additive and multiplicative MMFE. 3.1 Problem setting Consider the rolling-horizon planning of Kproducts with a horizon of Tperiods. In each review period, updated forecasts are observed and used to calculate a production plan. Let Fs∈ ℝ(K×T)be the forecast vector obtained at the beginning of period sgiven by Fs=[Fs 1,1,…,Fs 1,T,…,Fs K,1,…,Fs K,T]⊺, where Fs k,tis the forecast of product kin the t-th period of the planning horizon as seen in review period s. An initial forecast vector denoted by F1is available. In each review period, a new forecast is also obtained for the last period in the planning horizon. The demand observed at the end of period s is denoted by Ds k. After the demand has been observed, the forecast is no longer updated. 3.2 Additive MMFE The additive MMFE describes the evolution of the forecast vector by the relation Fs post =Fs+𝜺 s+1,(1) where the forecast update vector 𝜀s+1is observed at the beginning of review period s+1. The postupdate forecast vector Fs post =[Ds 1,Fs+1 1,1,Fs+1 1,2,…,Fs+1 1,T−1,…, Ds K,Fs+1 K,1,…,Fs+1 K,T−1]⊺contains both the demand Ds k observed at the end of period sfor all products kand the updated forecasts Fs+1 k,tof all products over the horizon. In period s+1, the planning horizon is rolled forward by one period. The forecast vector Fs+1is then composed of the updated forecasts in Fs post and the initial forecasts Fs+1 k,Tof all products kin the T-th period of the planning horizon. The forecast update vector follows a multivariate normal distribution 𝜀s∼(0,Σ). The covariance matrix Σ∈ ℝ(K×T,K×T)can be expressed as Σ=⎡⎢⎢⎢⎢⎢⎣ (𝜎1 1)2…𝜌 1,T 1,K𝜎1 1𝜎T K …𝜌 t1,t2 k1,k2𝜎t1 k1𝜎t2 k2… 𝜌T,1 K,1𝜎T K𝜎1 1…(𝜎T K)2 ⎤⎥⎥⎥⎥⎥⎦ ,(2) DYNAMIC STOCHASTIC LOT SIZING WITH FORECAST EVOLUTION 453 Production and Operations Management FIGURE 1 Demand and forecast observed at three successive review periods [Color figure can be viewed at wileyonlinelibrary.com] where 𝜎t kis the standard deviation of the t-th period of the forecast updating process for product k, and 𝜌t1,t2 k1,k2is the correlation between the forecast update of product k1at time t1 and product k2at time t2. The covariance matrix describes the uncertainty of the forecast updating process over the horizon and the correlation between the forecast updates of different products and time periods. 3.2.1 Demand distribution The demand follows the same updating process as the forecast and is given by Ds k=Fs k,1+𝜀 s+1 k,1. In any review period s, the demand for the t-th period in the planning horizon is subject to tforecast updates. As such, the demand in period s+t−1 as seen from period sfollows the relation Ds+t−1 k=Fs k,t+ t ∑ 𝜏=1 𝜀s+𝜏 k,t−𝜏+1.(3) Since the forecast update vectors 𝜀are independent and normally distributed, the demand in period s+t−1 follows a normal distribution Ds+t−1 k∼(Fs k,t, 𝜎k,t2), where  𝜎k,t2= ∑t 𝜏=1(𝜎𝜏 k)2is the residual uncertainty of the t-th period in the planning horizon. The residual uncertainty depends only on how far the demand period is in the planning horizon and is a direct measure of the forecast accuracy over the horizon. The demand and forecast revision process are illustrated in Figure 1for three review periods. 3.2.2 Demand covariance Although the forecast update vectors are observed independently in each review period, the update of different products and time periods described in Equation (1) can be correlated. It follows that the demand distributions of a product in different periods of the planning horizon may be correlated. In review period s, the covariance between the demands of product kin periods t1and t2of the planning horizon is given by 𝛾t1,t2 k=Cov (Ds+t1−1 k,Ds+t2−1 k) = min(t1,t2) ∑ 𝜏=1 𝜌t1−𝜏+1,t2−𝜏+1 k,k𝜎t1−𝜏+1 k𝜎t2−𝜏+1 k.(4) The demand correlation depends only on how many forecast update vectors are observed in which the two periods are both in the planning horizon. The covariance between demand observations in different periods is necessary to determine the distribution underlying the cumulative demand. 3.2.3 Cumulative demand distribution The cumulative demand of product kin period tof the planning horizon at review period s,CDs k,t=∑t 𝜏=1Ds+𝜏−1 k, is uncertain since demand is uncertain over the planning horizon. As a sum of correlated, normally distributed random variables, the cumulative demand CDs k,tfollows a normal distribution with mean ∑t 𝜏=1Fs k,𝜏 and variance ∑t t1=1∑t t2=1𝛾t1,t2 k. The variance of the cumulative demand depends only on the covariance of the demand distributions of the same product. The variance of the cumulative demand increases linearly with the forecast update correlation between two time periods. The cumulative demand distribution describes the demand uncertainty over the planning horizon. Determining the cumulative demand distributions allows the stochastic lotsizing problem to be solved with the formulation introduced in Section 4. 3.3 Multiplicative MMFE In the multiplicative MMFE, the forecast evolution process follows the relation Fs k,t=Fs−1 k,t+1⋅exp(𝜀s k,t+1),(5) 454 FOREL AND GRUNOW Production and Operations Management where the forecast update vector 𝜀sfollows a multivariate normal distribution 𝜀s∼(𝜇,Σ) and each marginal distribution is given by 𝜀k,t∼(−𝜎2 k,t 2,𝜎2 k,t). The forecast updating process is unbiased, as for the additive model. However, there is a key difference between the two models: In the additive model, forecast uncertainty depends only on the variance of the forecast update distribution, whereas in the multiplicative MMFE, the uncertainty associated with a forecast update is relative to the forecast value. Further, since the multiplicative MMFE is based on a log-normal distribution, the forecast evolution distribution has a heavier tail than the normal distribution underlying the additive MMFE. The variance of the forecast updating process depends both on the forecast update covariance matrix Σand on the forecast vector Fs. Because of these properties, the multiplicative MMFE has been described as more relevant in practice since forecasts tend to be reviewed in a relative manner. The multiplicative model can also be suitable when demand has significant fluctuations over time. In fact, applying a log-transformation is a well-known technique to achieve homogeneous variance when data are heteroscedastic such as time-series forecasting with nonstationary data. Yet, there remain many open questions on how to apply the multiplicative model using available forecast and demand data. In the numerical study in Section 5, we detail the estimation process of the multiplicative model from data and assess its performance in rolling-horizon planning. 3.3.1 Demand distribution The demand of product kin each review period sfollows the same relation as the forecast update so that Ds k=Fs k,1⋅ exp(𝜀s+1 k,1). From this relation and Equation (5), the demand in period s+t−1 as seen from review period sis given by Ds+t−1 k=Fs k,t⋅exp (t ∑ 𝜏=1 𝜀s+𝜏 k,t−𝜏+1).(6) The demand in period s+t−1 follows a log-normal distribution, log(Ds+t−1 k)∼(log(Fs k,t)− 𝜎k,t2 2, 𝜎k,t2), where  𝜎k,t2= ∑t 𝜏=1(𝜎𝜏 k)2is the residual uncertainty of the t-ahead period. The residual uncertainty in the log domain is independent of the review period, as for the additive model. However, demand variance depends on both the forecast update variance and the value of the forecast. 3.3.2 Demand covariance The demands of product kin periods t1and t2of the planning horizon in review period sare correlated with covariance 𝛾t1,t2 k=Cov (log (Ds+t1−1 k),log (Ds+t2−1 k)) = min(t1,t2) ∑ 𝜏=1 𝜌t1−𝜏+1,t2−𝜏+1 k,k𝜎t1−𝜏+1 k𝜎t2−𝜏+1 k.(7) The demand covariance can be deduced similarly as for the additive case by analyzing the covariance of the forecast evolution process in the log domain. The covariance of the demand periods is used to estimate the parameters of the distribution underlying the cumulative demand. 3.3.3 Cumulative demand distribution Contrary to the additive case, there is no closed-form expression for the cumulative demand since it is the sum of correlated log-normal distributions. However, it has been observed that the sum of log-normal distributions can be well approximated by a log-normal distribution. To estimate the cumulative demand distributions with multiplicative MMFE, we follow the approach of Norouzi and Uzsoy (2014) and apply the Fenton–Wilkinson approximation (FWA). The method is attractive because of its computational simplicity and overall high approximation quality over a wide range of parameters. The approximation is based on matching the first two moments of the approximating log-normal distribution with the moments of the sum of the correlated log-normal distributions (Abu-Dayya & Beaulieu, 1994). Following the moment-matching approximation, the cumulative demand CDk,tapproximately follows a log-normal distribution, log(CDk,t)∼(mk,t,vk,t), with parameters mk,t=2log(y1)−1 2log(y2) and vk,t= log(y2)−2log(y1), where y1=∑t 𝜏=1Fk,𝜏 and y2= t ∑ 𝜏=1 (Fk,𝜏)2exp ( 𝜎k,𝜏2) +2 t−1 ∑ i=1 t ∑ j=i+1 Fk,iFk,jexp (min(i,j) ∑ 𝜏=1 𝜌i−𝜏+1,j−𝜏+1 k,k𝜎i−𝜏+1 k𝜎j−𝜏+1 k).(8) This approximate cumulative demand distribution is used in Section 4to solve the stochastic lot-sizing problem. 3.4 Influence of forecast update correlation on the cumulative demand variance The variance of the cumulative demand has been shown to depend linearly on the forecast update correlation for the additive model. In the multiplicative model, although the relation between the forecast update correlation and the cumulative demand variance appears exponential, it is approximately linear over the relevant domain. Proposition 1. Under multiplicative MMFE, the variance of the cumulative demand of product k in period t, Var(CDk,t),is DYNAMIC STOCHASTIC LOT SIZING WITH FORECAST EVOLUTION 455 Production and Operations Management FIGURE 2 Evolution of variance with correlation coefficient for the (a) additive and (b) multiplicative MMFE [Color figure can be viewed at wileyonlinelibrary.com] approximately linear in the forecast update correlation 𝜌t1,t2 k,k for t1,t2≤t with slope given by 𝜕Var(CDk,t) 𝜕𝜌t1,t2 k,k ≈2𝜎t1 k𝜎t2 kexp (𝛽i)t−t2+1 ∑ i=1 Fk,t1+i−1Fk,t2+i−1, (9) where 𝛽i=∑t1+i−1 𝜏=1;𝜏≠i𝜌t1+i−𝜏,t2+i−𝜏 k,k𝜎t1+i−𝜏 k𝜎t2+i−𝜏 k. Proposition 1implies that ignoring the correlation between demand periods can lead to under- (resp. over-) estimation of the cumulative demand variance if the correlation is positive (resp. negative). The proof is provided in Supporting Information EC.1. The effect of the correlation coefficient is proportional not only to the variance but also to the forecast values. Thus, ignoring correlation has a greater impact for large forecasts. Moreover, Proposition 1suggests that the multiplicative model is more sensitive to estimation errors of correlation parameters than the additive model when forecasts are large. We analyze the evolution of the variance of the cumulative demand distribution with the forecast update correlation and compare the additive and multiplicative MMFE. We consider a single product planned over a horizon of T=2 periods and investigate the effect of forecast update correlation on the cumulative demand CD2. The forecast updating process is defined with standard deviation 𝜎1=𝜎 2=20 for the additive model and 𝜎1=𝜎 2=0.2 for the multiplicative model. The initial forecast in periods 1 and 2 are set equal F1=F2and chosen within the set {50,100,150}. The effect of time correlation on the variance of the cumulative demand in period 2 is shown in Figure 2for the additive and multiplicative models. The figure highlights the linear relationship between the forecast update correlation and the variance of the cumulative demand for both the additive and multiplicative cases. It further illustrates the impact of the forecast value on the variance of the cumulative demand for the multiplicative model. 3.5 Summary In this section, the multivariate forecast evolution process has been introduced for additive and multiplicative MMFE. The parameters of the resulting demand and cumulative demand distributions have been obtained. The cumulative demand distributions can be determined exactly for the additive model and approximately for the multiplicative model. Finally, we have analyzed the dependency of the cumulative demand variance on the forecast update correlation coefficient. In the next section, we derive efficient formulations for the stochastic lot-sizing problem based on the cumulative demand distributions estimated from the MMFE. 4 INTEGRATING FORECAST EVOLUTION IN STOCHASTIC LOT SIZING We integrate the additive and multiplicative MMFE in lot-sizing problems through the cumulative demand distributions derived in the previous section. We introduce the PLA formulation that can be solved efficiently and extend the model with scenario-based production recourse. The extended model combines the strengths of PLA and scenario methods, providing fast computations and flexible decisions. 4.1 Problem setting In each review period, the planner determines the production quantity Qk,tfor all Kproducts over the planning horizon of T periods. The products share the same equipment with limited capacity cap in each period. The planner aims to satisfy the uncertain demand while minimizing costs. The operational costs include inventory costs hckincurred at the end of each period and setup costs sckincurred each time a new product is set up. Unsatisfied demand is backordered and penalized with per-unit cost bck. The initial inventory is denoted by in0 k and can be positive or negative depending on whether there 456 FOREL AND GRUNOW Production and Operations Management is on-hand inventory or backlog. As demand is uncertain, the inventory Ik,tand backlog Bk,tat the end of each period are random variables. Since they depend on the production quantity, determining their expected value would require nonlinear constraints and lead to intractable formulations. 4.2 Linearization of inventory and backlog functions To obtain tractable formulations, the PLA method has been developed. It evaluates the first-order loss function at a selected number of breakpoints and determines the slope of the expected inventory and backlog between these breakpoints (Helber et al., 2013). Rossi et al. (2014) provide analytical bounds on the approximation error of PLA when the uncertain variable follows a normal distribution, which applies under additive MMFE. They show that the approximation error is small with only a few linearization points. The first-order loss function of a real variable xand random variable 𝜔with p.d.f. 𝜙and c.d.f. Φis defined as (x,𝜔)=𝔼[max(𝜔−x,0)]=∫+∞ x max(t−x,0) ⋅𝜙(t)dt =∫+∞ x (1−Φ(t))dt.(10) Let u=(uk,t,l)bethesetofL+1 breakpoints determined independently for each product and time period. The first breakpoint is set to uk,t,0=in0 k, which can be either positive or negative, and the last breakpoint is set to the highest inventory position attainable at the end of period twith full capacity utilization as uk,t,L=in0 k+cap ⋅t. The remaining breakpoints are set uniformly between these two bounds. For each segment, the slope of the expected inventory and backlog can be determined as Δl Bk,t=(uk,t,l+1,CDk,t)−(uk,t,l,CDk,t) uk,t,l+1−uk,t,l ,(11) Δl Ik,t=(uk,t,l+1,CDk,t)+uk,t,l+1−(uk,t,l,CDk,t)−uk,t,l uk,t,l+1−uk,t,l , (12) where CDk,tis the cumulative demand distribution of product kin period t. When demand forecasts evolve according to the MMFE, the cumulative demand distributions are updated over the planning horizon in each review period. Hence, the linearization procedure should also be conducted in each period. Section 3showed that the cumulative demand follows a normal and log-normal distribution for additive and multiplication MMFE, respectively. Calculating the slopes of the Lsegments of the expected inventory and backlog requires evaluating the first-order loss function K⋅T⋅(L+1) times in each review period. This evaluation is computationally cheap for a normal distribution since the first-order loss function of a normal variable can be expressed as a function of the first-order loss function of a standard normal (Rossi et al., 2014), and can thus be calculated offline. The calculation is more expensive for a log-normal variable since it requires evaluating many integrals as in Equation (10). Note also that the domain of the c.d.f. of a log-normal variable needs to be extended for negative values since the initial inventory in each period may be negative. The PLA method can be linked back to separable programming, a general framework to solve nonlinear problems using PLAs. Early approaches to solve the resulting model included an adaptation of the simplex algorithm (Bazaraa et al., 2006, section 11.3). We leave for future research to investigate how the special structure of PLA-based MILP could be exploited as a subclass of separable programming to further improve solution times. The PLA of two demand distributions following a normal and log-normal with equal mean and variance is shown in Figure 3. For the chosen parameter values, the functions for both distributions are very similar. The figure shows that the expected inventory and backlog functions can be already well approximated with only L=6 segments for both distributions. In practice, the feasible domain of the production variable Q∈[0,cap] can be large, especially when many products share the same production resource. The inventory and backlog functions are nonlinear only on a restricted part of this domain, as is also illustrated in Figure 3.In principle, techniques could be used to reduce the number of required breakpoints, and thus the calculation times (see, e.g., De Smet et al., 2020). Such approaches would require a search procedure for optimizing the placement of breakpoints. However, in our rolling-horizon problem, the nonlinear part may change for each planning period and the placement of breakpoints needs to be adapted accordingly. The benefit of reduced calculation times for the MILP is therefore reduced by the search time for the breakpoints. Hence, we use a large number of fixed, equidistant breakpoints in our numerical study to ensure that the nonlinear domain is always well covered. 4.3 Stochastic lot sizing without recourse The PLA formulation of the stochastic lot-sizing problem approximates the expected inventory and backlog with variables EIk,tand EBk,t, respectively. The key difference in our approach is that the cumulative demand distributions and their linearization are dynamically updated in each review period, following the results presented in Section 3. Additionally, we also adapt the formulation of van Pelt and Fransoo (2018) to account for penalty cost for backlogs. The formulation requires the introduction of auxiliary variables wk,t,lto measure the cumulative production from period 1 to tassociated with segment land binary auxiliary variables 𝜆k,t,lto ensure that the Lsegments are used consecutively. The model is formulated as the following mixed-integer linear program: DYNAMIC STOCHASTIC LOT SIZING WITH FORECAST EVOLUTION 463 Production and Operations Management FIGURE 7 Mean demand for (a) stationary, (b) random, and (c) seasonal patterns over simulation of eight periods as initial forecasts over the season length. The three demand patterns have the same average demand over the simulation length but different dynamics, which may impact the additive and multiplicative MMFE models differently. The forecast evolution models are set unbiased and uncorrelated with equal forecast update variance for all products and time periods. Low, medium, and high forecast uncertainty settings are defined with variance 𝜎2∈{100,400,700} for the additive model and 𝜎2∈{0.01,0.04,0.07} for the multiplicative model. Thus, the two forecast evolution models have the same variance under the average demand. 5.2.2 MMFE models and benchmarks In practice, the true forecast evolution is unknown. To estimate the value of MMFE-based lot sizing when using a mismatched forecast evolution model, we run two distinct sets of simulations in which the forecast evolution process follows the assumptions of the additive and multiplicative MMFE. The mismatched model is estimated from a simulation of the true forecast evolution process over 1 million periods. The sampled forecast updates are measured according to the mismatched MMFE model and used to estimate its parameters. As in the previous real-world case study, the estimation procedure of the multiplicative model is not straightforward. When the forecast evolution process follows an additive MMFE, demand is normally distributed and demand observations may be zero (or negative, which is corrected to zero in all simulations of additive MMFE). It is also possible that a forecast with value of zero is updated to a positive forecast. These two cases, while frequently occurring in practical settings, are not compatible with the multiplicative MMFE. Thus, when estimating the parameters of the multiplicative MMFE, we remove all sampled forecasts that contain at least one zero value. For the setting with low uncertainty, this amounts to removing 0%,1%, and 41%of samples for the stationary, random, and seasonal patterns, respectively; 13%,26%, and 64%for the medium uncertainty setting; and 37%,50%, and 74%for the high-uncertainty setting. Clearly, more sample updates are removed from the data set as the demand pattern is more dynamic and as uncertainty increases. The high number of unusable samples is an important shortcoming of multiplicative MMFE since collecting data is an expensive process. Two benchmarks are introduced: (1) a deterministic model that uses the forecast as a point estimate and ignores uncertainty but observes the updated forecasts in each planning period and (2) a demand-driven stochastic model that ignores forecasts and their evolution and instead estimates demand distributions from historical data. For the stationary and random patterns, the demand-driven model estimates a stationary demand distribution. For the seasonal pattern, the demand-driven model estimates independent distributions for all periods in the season. The model estimates the parameters of normal and log-normal distributions from simulations of the forecast evolution process over 1 million periods using additive and multiplicative MMFE, respectively. These two models benchmark the planning practices of industry and the traditional stochastic lot-sizing literature, respectively. The performance of stochastic models is traditionally evaluated by measuring the value of the stochastic solution (VSS), which is based on a static evaluation of the deterministic and stochastic models, and the expected value of perfect information (EVPI), which measures the value of obtaining perfect forecasts (Birge & Louveaux, 2011). By comparing the performance of the stochastic models to the deterministic benchmark implemented in a rollinghorizon fashion, we extend the VSS to a more realistic setting in which the deterministic benchmark also benefits from updated forecasts. Further, the value of improving the forecasting process is measured by comparing the costs of MMFE-based models under the different uncertainty settings, providing a richer performance evaluation than the EVPI. 5.2.3 Results Each rolling-horizon simulation of the 36 instances is repeated 1000 times. Model performance is measured as the 464 FOREL AND GRUNOW Production and Operations Management TABLE 2 Simulation results when forecast evolution follows an additive MMFE process Demand Uncertainty cap Det. DD-stochastic Additive PLA (correct model) Multiplicative PLA (mismatched model) Extended Add. PLA (correct model) Stationary Low 300 4707.9 4711.9 (100.6%*) 4153.6 (88.6%*) 4118.4 (87.9%*) 4135.6 (88.2%*) 500 4676.2 4695.8 (100.9%*) 4152.0 (89.2%*) 4129.8 (88.7%*) 4131.0 (88.8%*) Medium 300 5312.4 5680.1 (108.2%*) 4652.2 (88.6%*) 5147.5 (98.0%*) 4593.8 (87.4%*) 500 5069.4 5722.9 (114.0%*) 4589.3 (91.3%*) 4974.9 (99.1%*) 4541.4 (90.4%*) High 300 5927.2 6326.7 (109.5%*) 4923.7 (85.2%*) 6689.6 (116.2%*) 4878.4 (84.4%*) 500 5421.0 6265.6 (117.2%*) 4811.5 (89.8%*) 5712.9 (106.8%*) 4770.0 (89.1%*) Random Low 300 4581.2 5365.6 (117.8%*) 4020.2 (88.2%*) 4122.6 (90.5%*) 4007.5 (87.9%*) 500 4515.3 5367.6 (119.5%*) 3986.4 (88.7%*) 4063.5 (90.5%*) 3969.7 (88.3%*) Medium 300 5336.4 6093.5 (116.0%*) 4568.2 (86.9%*) 5586.9 (106.4%*) 4501.4 (85.6%*) 500 5078.3 6009.9 (119.6%*) 4473.4 (89.0%*) 5134.8 (102.2%*) 4418.0 (87.8%*) High 300 6077.4 6645.8 (112.7%*) 4914.6 (83.3%*) 7209.2 (123.1%*) 4863.4 (82.4%*) 500 5525.4 6523.9 (120.1%*) 4749.0 (87.3%*) 5968.2 (109.8%*) 4712.2 (86.6%*) Seasonal Low 300 4598.6 4507.7 (99.1%*) 3968.4 (87.1%*) 5785.4 (127.3%*) 3932.9 (86.3%*) 500 4187.3 4229.0 (101.7%*) 3720.9 (89.4%*) 4589.2 (110.3%*) 3683.8 (88.5%*) Medium 300 5713.3 5771.8 (104.4%*) 4575.6 (82.4%*) 8225.2 (148.8%*) 4505.9 (81.0%*) 500 4957.1 5309.8 (108.5%*) 4222.4 (86.2%*) 5931.2 (121.1%*) 4159.6 (84.8%*) High 300 6609.7 6703.3 (107.7%*) 5066.8 (81.0%*) 9573.1 (156.0%*) 4970.3 (78.5%*) 500 5425.5 5913.5 (111.1%*) 4501.8 (84.4%*) 7911.5 (148.5%*) 4463.4 (83.7%*) Average 5206.6 5658.0 (108.7%*) 4447.2 (85.4%*) 5826.3 (111.9%*) 4402.1 (84.5%*) sum of realized inventory, backlog, and setup costs. The results under additive and multiplicative MMFE are presented in Table 2and Table 3, respectively, as the average of the costs over the 1000 repetitions. The statistical significance of all relative cost differences from the deterministic model is assessed using Student’s t-test. Statistical significance is indicated with the symbol (∗) for all relative values for which the associated p-value is strictly smaller than 5%. Our simulation results quantify the value of forecast evolution models compared to both traditional deterministic approaches typical in industry and stochastic models that focus solely on historical demand data and ignore forecast. The costs of the deterministic benchmark are especially high when capacity is tight, uncertainty is high, and demand fluctuates over time. It is also in these settings that the MMFE models with known forecast evolution provide large cost reductions. The stochastic, demand-driven model increases costs compared to the naive deterministic model for almost all instances. This can be explained by two reasons. First, the model is overly conservative since it accounts for the whole demand uncertainty for all periods in the planning horizon. Second, it is inaccurate because it only aims for the average observed demand and ignores the forecasts. This is especially true for the random demand pattern since, even though demand is stationary, the initial forecast values provide a lot of information on the final demand observations. On the other hand, the additive and multiplicative MMFE models reduce costs by 14%on average compared to the deterministic model when the forecast evolution process is known. The value of improving the forecasting process to reduce forecast uncertainty can be measured by comparing the costs of the correct MMFE model in different uncertainty settings. For instance, under additive MMFE in the seasonal setting with low capacity, the planner would be willing to pay up to 580 c.u. to reduce the forecast uncertainty from the high uncertainty to medium uncertainty, and up to 500 c.u. to reduce it further to the low uncertainty setting. Interestingly, the value of information appears higher in the simulation settings with low capacity. This result contrasts with previous studies that found that advance information was not useful when utilization is high (Albey et al., 2015; Ziarnetzky et al., 2018,2020). This difference can be mainly explained by the fact that previous literature uses several approximations such as capacity allocation when determining the safety stocks of the different products. This severely restricts planning flexibility when utilization is high. Flexibility is even more important in our experiments since we consider products with different cost parameters whereas Ziarnetzky et al. (2018) and Albey et al. (2015) consider symmetric products. A detailed analysis of the effect of the simulation parameters and their interactions is given in Supporting Information EC.3. The cost of model misspecification is high for the multiplicative model. Indeed, Table 2shows that using a multiplicative forecast evolution model when the true process is additive can significantly increase costs even compared to traditional deterministic planning. On average, the costs of the multiplicative model are 12%larger and more than 50% when demand is seasonal and uncertainty is high as is the case DYNAMIC STOCHASTIC LOT SIZING WITH FORECAST EVOLUTION 465 Production and Operations Management TABLE 3 Simulation results when forecast evolution follows a multiplicative MMFE process Demand Uncertainty cap Det. DD-stochastic Additive PLA (mismatched model) Multiplicative PLA (correct model) Extended Mult. PLA (correct model) Stationary Low 300 4695.2 4801.1 (102.8%*) 4179.1 (89.5%*) 4151.7 (88.9%*) 4124.0 (88.2%*) 500 4616.2 4776.1 (104.0%*) 4190.7 (91.2%*) 4149.5 (90.3%*) 4117.3 (89.6%*) Medium 300 5452.8 6187.3 (115.4%*) 4852.8 (90.8%*) 4785.2 (89.8%*) 4691.1 (87.7%*) 500 5059.3 6082.8 (121.3%*) 4713.1 (94.2%*) 4644.4 (92.9%*) 4571.7 (91.3%*) High 300 6440.5 7368.2 (118.4%*) 5414.7 (87.6%*) 5469.3 (90.1%*) 5223.9 (84.6%*) 500 5494.7 7168.7 (132.1%*) 5035.5 (93.3%*) 5049.4 (94.2%*) 4909.8 (91.3%*) Random Low 300 4634.3 5566.5 (120.9%*) 4088.5 (88.8%*) 4018.2 (87.3%*) 3999.5 (86.8%*) 500 4524.1 5528.9 (122.9%*) 4019.0 (89.3%*) 3956.3 (88.0%*) 3925.1 (87.2%*) Medium 300 5576.9 6763.9 (124.2%*) 4851.8 (89.4%*) 4771.3 (88.4%*) 4625.9 (85.2%*) 500 5055.6 6677.1 (133.6%*) 4637.0 (92.9%*) 4534.6 (91.1%*) 4438.6 (89.0%*) High 300 6675.1 8012.7 (126.6%*) 5632.6 (89.5%*) 5596.4 (91.3%*) 5350.8 (85.1%*) 500 5575.4 7657.9 (139.4%*) 5048.3 (92.7%*) 5019.1 (92.9%*) 4856.8 (89.5%*) Seasonal Low 300 4806.4 4761.8 (100.8% ) 4171.4 (87.9%*) 4082.0 (86.4%*) 4021.5 (84.9%*) 500 4276.3 4510.2 (106.4%*) 3856.3 (90.8%*) 3749.0 (88.3%*) 3701.3 (87.2%*) Medium 300 6435.3 6623.6 (109.7%*) 5265.7 (85.9%*) 5286.7 (88.7%*) 5054.5 (82.1%*) 500 5162.3 6125.6 (120.9%*) 4635.4 (91.6%*) 4428.2 (88.1%*) 4331.7 (86.0%*) High 300 7762.7 8105.1 (116.3%*) 6169.5 (87.1%*) 6404.2 (95.0%*) 5977.3 (82.4%*) 500 5835.7 7329.8 (130.3%*) 5102.6 (90.5%*) 5024.9 (90.7%*) 4780.2 (85.5%*) Average 5448.8 6336.0 (116.3%*) 4770.2 (87.5%*) 4728.9 (86.8%*) 4594.5 (84.3%*) in our industry application. In contrast, the cost of model misspecification is low for the additive model. Table 3shows that when the true process is multiplicative, the additive model yields costs almost as low as the multiplicative model, suggesting that additive MMFE-based models are robust to errors in modeling the forecast evolution process. These results explain the superiority of the performance additive model for our industry case, in which the true forecast evolution process is unknown. 5.2.4 Value of recourse The value of recourse is defined as the difference between costs of the stochastic model without recourse and the extended stochastic model combining PLA and scenariobased recourse, as presented in Table 2and Table 3.Thevalue of recourse varies over the simulation settings similarly for both MMFE models. It is higher for more complex planning settings: When demand is dynamic, uncertainty is high, and capacity is limited. Overall, recourse is more beneficial under multiplicative MMFE. On average, the value of recourse is around 1.5%and 2.5%across all simulation settings and can reach 2.3%and 6.2%for the additive and multiplicative models, respectively. The detailed statistical analysis of the value of recourse is given in Supporting Information EC.4. It shows that these results are statistically significant at a p-value smaller than 5%. The distribution of the value of recourse is skewed so that in the majority of cases, observed costs are smaller than the average value. To further investigate the value of recourse in stochastic models and to identify settings in which it is most beneficial, we perform several sensitivity analyses. Impact of product and time correlation In Section 3, we have shown that positive (resp. negative) forecast time correlation was equivalent to a higher (resp. lower) cumulative demand variance for both MMFE models. For the extended model with recourse, correlation has an even larger impact since the recourse model can react to correlated forecast updates. We analyze the impact of the correlation structure on the value of recourse on the simulation setting with seasonal demand and cap =300. The influence of both product and time correlation is investigated. Product correlation is set constant over the horizon as 𝜌t,t 1,2=𝜌 k with 𝜌k∈{−0.6,0,0.6}. Time correlation is set between the first and second periods of the horizon for both products as 𝜌1,2 k,k=𝜌 twith 𝜌t∈{−0.6,0,0.6}. If both product and time correlation parameters are nonzero, then the first and second periods of the two products are also correlated. In this case, we set 𝜌1,2 1,2=𝜌 k⋅𝜌 t. The costs of the extended model with recourse relative to the costs of the model without recourse are presented in Table 4. The statistical significance of the relative cost is assessed with Student’s t-test and is shown with the symbol (∗)ifthep-value is below 0.05. The correlation structure 466 FOREL AND GRUNOW Production and Operations Management TABLE 4 Value of recourse for different correlation structures Additive MMFE Multiplicative MMFE 𝝆k=−0.6𝝆k=0𝝆k=0.6𝝆k=−0.6𝝆k=0𝝆k=0.6 𝜌t=−0.6 97.4 (*) 96.9 (*) 96.8 (*) 92.1 (*) 91.2 (*) 89.2 (*) 𝜌t=0 98.3 (*) 98.5 (*) 99.0 (*) 93.9 (*) 94.6 (*) 93.3 (*) 𝜌t=0.6 98.7 (*) 100.1 100.9 (*) 97.4 (*) 97.1 (*) 95.8 (*) has a strong impact on the value of recourse. Negative time correlation yields high value of recourse for both MMFE models, whereas positive time correlation leads to lower values than in the uncorrelated case. Recourse decisions can take advantage of negative time correlation by anticipating that forecast updates will compensate over time. Specifically, a forecast increase for the first period in the horizon might be compensated by a forecast decrease in the second period. Anticipating this effect leads to less conservative decisions. Hence, the largest improvements are observed when time correlation is negative and product correlation is positive. Here, a compensation over time occurs for both products and costs can be reduced by more than 10%compared to the stochastic model without recourse. This analysis shows the importance of including correlation in stochastic planning especially when using recourse models. Further, it confirms the trend that the multiplicative model benefits most from recourse. We also perform extensive sensitivity analyses of the available capacity and scenario structures. The sensitivity analysis of capacity shows that the value of recourse increases monotonously with the available capacity under additive MMFE. The value of recourse is larger under multiplicative MMFE and peaks when capacity is neither too limited nor too large. The sensitivity analysis of the scenario structure shows that scenario trees with an intermediate size, such as the one used throughout this section, are sufficient to benefit from recourse without substantial increase in computation times. Details on both sensitivity analyses are provided in Supporting Information EC.5. The value of flexibility is also studied in a broader context in Supporting Information EC.6 by comparing the value of recourse to the value of a free-return policy, which allows to liquidate excess inventory at no cost. The numerical results confirm that flexibility is more valuable under multiplicative MMFE than additive MMFE. Recourse and returns decisions are two flexibility levers that provide large cost reductions under multiplicative MMFE. Recourse decisions are more beneficial when capacity is tight whereas return decisions prove especially valuable when capacity is large. 5.3 Summary and recommendations The numerical study shows that integrating forecast evolution models in stochastic lot sizing can significantly improve planning quality compared to traditional deterministic approaches and stochastic methods based solely on demand data. The additive MMFE model is robust and performs well across all simulation instances: (1) when the forecast evolution process is known, (2) when it is unknown and estimated from mismatched updates, and (3) when it is learned from realworld historic data. Interestingly, the cost savings provided by stochastic models based on additive MMFE relative to the deterministic benchmark are similar on both the real-world and the synthetic data. On the other hand, the multiplicative model suffers from several limitations, which have been identified through our extensive numerical studies. First, the multiplicative model is particularly sensitive to model misspecification: When the forecast evolution model is unknown, the multiplicative model leads to a cost increase and often performs worse than traditional deterministic rolling-horizon planning. The performance deteriorates most when uncertainty is high, capacity is limited, and the demand is dynamic, for example, when there is a strong demand seasonality. This suggests that the relative forecast error measure, on which the multiplicative MMFE model is based, is more sensitive to a distributional error than the absolute measure underlying the additive model. On top of this limitation, the multiplicative model is more strongly impacted by estimation errors due to the variance being relative to the absolute value of the forecast as shown in Proposition 1. Thus, demand peaks and outliers can strongly impact the multiplicative model’s performance, which is clearly shown in the real-world case study. Further, due to its inability to include demand and forecasts values of zero, the multiplicative model overestimates forecast uncertainty when using historical data that include such periods. Thus, in contrast to the consensus in the MMFE literature stating that the multiplicative model better characterizes forecast revision processes, we advise to prioritize the implementation of the additive MMFE because of its robustness in a wide array of problem settings. In any case, we emphasize that the choice of the relevant MMFE model should not be based on an a priori goodness-of-fit analysis but instead on evaluating model performance through out-of-sample rolling-horizon simulations using historical data. The extended model with production recourse can provide consistent cost reductions with both real-world and synthetic data. Across all simulation settings, the value of recourse is higher for the multiplicative model. Still, recourse can consistently provide lower costs for the additive model. We have identified that the value of recourse is especially high when DYNAMIC STOCHASTIC LOT SIZING WITH FORECAST EVOLUTION 467 Production and Operations Management demand is dynamic, uncertainty is high and when forecast updates exhibit negative time correlation. The extended model with recourse requires managerial decisions as it impacts planning in several ways including slightly longer computation times and a reduced reference plan due to the presence of recourse decisions. In our analysis, we have provided initial guidelines to tune the model and find a good compromise between its advantages and limitations. 6CONCLUSION This paper proposed a methodology for a dynamic, stochastic, capacitated lot-sizing approach apt for use in rolling-horizon planning. For this purpose, we integrated forecast evolution models in tractable lot-sizing formulations. We have shown that cumulative demand distributions describing the forecast evolution can be integrated efficiently in stochastic lot-sizing models using existing linearization techniques. Rolling-horizon planning also allows for a revision of production plans. We therefore extended our approach with a scenario-tree representation of uncertainty to allow for production recourse. We quantified the value of forecast evolution models in a large-scale numerical study using both real-world and synthetic data. Forecast evolution models have been shown to provide significant cost reductions compared to both traditional deterministic methods used in industry and stochastic methods that use only demand history, which are common in the stochastic lot-sizing literature. On average, production recourse consistently reduces costs for both additive and multiplicative MMFE. Key parameters that impact the value of recourse such as product and time correlations have been identified through sensitivity analyses. This work proposes the first numerical comparison of additive and multiplicative MMFE in rolling-horizon planning when the true forecast evolution process is unknown. Previous literature stressed that the multiplicative MMFE better fits industry data than the additive MMFE, which is also observed in our analysis. However, we find that multiplicative MMFE-based planning performs poorly when the true forecast evolution process is unknown. Thus, we highlight that the MMFE model that best fits the historical data does not necessarily provide the best planning performance. The additive model, in contrast, performs robustly across all simulation instances. Future research could investigate how to make multiplicative forecast evolution models more robust to an unknown forecast updating process. The two main steps of defining a forecast evolution model may be challenged: (1) measuring forecast updates from data and (2) fitting a probability distribution to the updates. For instance, more robust multiplicative MMFE models could use novel approaches to measure the relative forecast updates or investigate alternative estimation techniques to find model parameters from data. The normality assumption, central to both additive and multiplicative forecast evolution models, could also be challenged by investigating other distributions or applying distributionfree methods. A last research direction pertains to the link between the revision of forecasts and the revision of planning decisions. It is known that frequent planning changes can cause nervousness in supply chains. Integrating forecast evolution in planning models may allow new methods to anticipate planning instability in rolling-horizon planning and derive replanning strategies yielding more stable plans. ACKNOWLEDGMENTS The authors thank the department editor as well as the anonymous senior editor and reviewers for their comments that substantially improved the manuscript. The work is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—277991500/GRK2201. ORCID Alexandre Forel https://orcid.org/0000-0002-9868-4804 REFERENCES Abu-Dayya, A. A., & Beaulieu, N. C. (1994). Outage probabilities in the presence of correlated lognormal interferers. IEEE Transactions on Vehicular Technology,43(1), 164–173. Albey, E., Norouzi, A., Kempf, K. G., & Uzsoy, R. (2015). Demand modeling with forecast evolution: An application to production planning. IEEE Transactions on Semiconductor Manufacturing,28(3), 374–384. Albey, E., Uzsoy, R., & Kempf, K. G. (2016). A chance constraint based multi-item production planning model using simulation optimization. In T. M. K. Roeder, P. I. Frazier, R. Szechtman, E. Zhou, T. Huschka, & S. E. 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