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Enhancing Rolling Horizon Production Planning Through Stochastic Optimization Evaluated by Means of Simulation
\email{[email removed]}
\email{[email removed]}
\email{[email removed]}
\email{[email removed]}
\affil[1]{Institute of Production and Logistics Management, Johannes Kepler University Linz, Altenbergerstra\ss e 69, 4040 Linz, Austria}
\affil[2]{Department for Production and Operations Management, University of Applied Sciences Upper Austria, Wehrgrabengasse 1, 4400 Steyr, Austria}
\abstract{ Production planning must account for uncertainty in a production system, arising from fluctuating demand forecasts. Therefore, this article focuses on the integration of updated customer demand into the rolling horizon planning cycle. We use scenario-based stochastic programming to solve capacitated lot sizing problems under stochastic demand in a rolling horizon environment. This environment is replicated using a discrete event simulation-optimization framework, where the optimization problem is periodically solved, leveraging the latest demand information to continually adjust the production plan. We evaluate the stochastic optimization approach and compare its performance to solving a deterministic lot sizing model, using expected demand figures as input, as well as to standard Material Requirements Planning (MRP). In the simulation study, we analyze three different customer behaviors related to forecasting, along with four levels of shop load, within a multi-item and multi-stage production system. We test a range of significant parameter values for the three planning methods and compute the overall costs to benchmark them. The results show that the production plans obtained by MRP are outperformed by deterministic and stochastic optimization. Particularly, when facing tight resource restrictions and rising uncertainty in customer demand, the use of stochastic optimization becomes preferable compared to deterministic optimization. }
Fluctuating demand forecasts complicate supply chain management in various industrial sectors, necessitating medium-term production planning to mitigate customer-driven uncertainty. In these dynamic environments, effective medium-term production planning requires rolling horizon planning and regular updates of customer demand, inventory levels, and available capacity Stefansson2009. Customer demand fluctuates over time, and digitalization promotes periodic exchange of demand information between suppliers and customers in the form of demand forecasts Norouzi.2014. Business customers typically provide long-term demand forecasts, which are regularly adjusted to ensure precise anticipation of demand, a practice known as the forecast evolution model, necessitating dynamic adjustments to production plans Heath.1994. While medium-term planning can use deterministic or stochastic mathematical optimization models, industry often employs simpler approaches like material requirements planning (MRP) Orlicky.1975, Hopp.2011. Stochastic optimization is well-suited for dynamic settings as it incorporates uncertainties into the decision-making process, enhancing the robustness and adaptability of production plans.
A suitable concept to implement the described problem domain in one application framework is the combination of discrete-event simulation with an optimization solver. Simulation provides a controlled environment for assessing various production strategies and their potential outcomes before operational implementation, enhancing decision-making and resource utilization Negahban.2014. It can model diverse probabilistic behaviors relevant to production systems under varying operational conditions, such as changing demand and shop floor uncertainties Law.2014, thereby helping to increase the overall efficiency and performance of manufacturing systems Sahin.2013. Combining simulation and optimization systematically addresses the complexities of real-world production systems. The simulation component accurately represents the production system over time, incorporating stochastic influences. The optimization component solves an optimization problem based on the current system status. Although the optimization component doesn't fully capture all stochastic aspects, such as shop floor uncertainties, its interaction with the simulation helps mitigate these effects. Additionally, incorporating some stochastic parameters, like uncertain customer demands, into the optimization model further reduces stochastic impacts.
Other researchers have demonstrated the potential of simulation-optimization for medium-term planning. gansterer_simulation-based_2014 utilized simulation-optimization to determine good planning parameters for MRP under four distinct demand scenarios. almeder2009simulation combined discrete-event simulation and optimization to identify an operation plan for a supply chain network based on a case study. In grunow_dynamic_stochastic_2023, rolling-horizon planning and forecasting are integrated into a capacitated lot-sizing formulation, and numerical results are computed using simulation.
While closely related to our work, these studies differ in several aspects. Our work applies a rolling horizon forecast evolution approach for demand generation, continuously updating forecasts until the due date and incorporates demand variation into the stochastic optimization model. Consequently, the optimization always receives the latest demand information from the simulation. Additionally, our approach contrasts with other works by not optimizing MRP planning parameters like safety stock, lot policy, and planned lead time. Instead, we compare the best identified MRP configurations to the results of deterministic and stochastic optimization. Standard MRP, the de facto industry standard for medium-term production planning, is used as a benchmark, providing practical insights into the optimization performance. We have implemented MRP within our simulation framework to facilitate this comparison.
The article's central focus is to evaluate rolling horizon planning via solving stochastic capacitated lot sizing problems as a medium-term planning strategy in a dynamic context using simulation. To address the challenges of uncertain environments where customer demands change and shop floor processing is stochastic, we introduce a simulation-optimization framework. This framework combines discrete-event simulation with two-stage stochastic programming in a rolling horizon environment, ensuring optimization problems are solved to optimality throughout the simulation runtime.
To systematically evaluate the performance of the proposed simulation-optimization framework, we first use an elementary multi-item and multi-stage production system comprising two end items, each consisting of one component. Subsequently, we introduce more complex multi-item multi-stage production system structures to demonstrate the applicability of our approach. These evaluated production systems allow us to conduct an in-depth analysis of the effects of frequent demand updates on rolling horizon planning decisions. To increase complexity, we consider three different customer behaviors, fifteen different demand fluctuations, stochastic setup times, and up to four different shop floor loads to analyze planning behavior under congestion.
Our objective is to minimize overall costs in a multi-item, multi-stage production system, encompassing setup, production, inventory, and backlog costs. Through a comprehensive simulation study, we benchmark deterministic and stochastic optimization against standard MRP, highlighting the potential effectiveness and competitiveness of the optimization results. The overall costs are composed of end item inventory, component inventory, work-in-progress (WIP) for end items and components, and tardiness costs.
The theoretical contributions of this article are fivefold:
The practical contributions of this article are twofold: it enhances decision-making by integrating discrete-event simulation with stochastic optimization, providing a practical framework that helps managers make more informed decisions under uncertain demand conditions. Additionally, it improves resource utilization by optimizing production schedules, thereby reducing overall production costs in dynamic manufacturing environments.
Based on the context and findings discussed in the introduction, the following research questions are derived to guide the remainder of the article.
The remainder of this article is structured as follows: Section (ref) reviews literature related to the simulation-optimization approach in medium-term production planning. Section (ref) introduces the simulation-optimization framework, detailing medium-term planning, short-term planning, shop floor processing, the optimization model, and the forecast model. Section (ref) describes the simulation study, including the production systems and experimental setup. Section (ref) presents and discusses the numerical results for an elementary production system, while Section (ref) investigates a more realistic use case. Finally, Section (ref) concludes with a summary and future research directions.
In this section we review works that combine simulation and optimization in order to solve difficult decision problems in the production industry, especially lot sizing. Several works consider single-item production environments, e.g. campuzano-bolarin_rolling_2020 investigate variable lead times for a single-item single-echelon system. They evaluate lot sizing techniques, namely Silver-Meal and Wagner-Within, in a system dynamics simulation and propose a rolling horizon planning approach. Also vargas_master_2011 study rolling horizon planning for a one product uncapacitated master scheduling problem under stochastic demand. They show that explicitly considering demand stochasticity within the lot sizing model is superior to adding safety stocks in terms of costs.
The case of several items is considered in brandimarte_multi-item_2006, where a plant-location-based lot sizing formulation for multi-item single-echelon production systems under multi-stage demand uncertainty is proposed. The production plans obtained by solving the stochastic models are compared to the results of a deterministic model. The respective production decisions are applied in a rolling horizon manner. Within the simulation, customer demand is assumed to follow a normal distribution and respectively scenario trees are sampled according to this assumption. Multiple production stages and demand updates, as they are used within our work, are not considered.
gansterer_simulation-based_2014 present different approaches to efficiently determine important MRP parameters as they are used in hierarchical production planning, namely planned lead time, safety stock and lot sizes. After analyzing a regular grid over the possible parameter settings in order to find the most suitable setting, they apply a variable neighborhood search procedure to find the best setting in a shorter amount of time. While this work exclusively focuses on planning by means of MRP with well chosen parameters, we additionally solve mathematical lot sizing models and investigate the obtained production plans.
Several medium-term planning approaches aim at additionally including short-term decisions within the planning procedure. One example is the integrated lot sizing and scheduling problem, which is studied in curcio_adaptation_2018. Detailed scheduling decisions are taken into account in the optimization model. The authors investigate static robust and two-stage stochastic optimization approaches as approximation strategies to the multi-stage setting. Customer demand is assumed to follow a log-normal distribution. The integrated lot sizing and cutting stock problem under demand uncertainty is studied in curcio_integrated_2022. The authors apply a robust and two-stage stochastic optimization approach in a rolling horizon manner, in order to adapt the models to the multi-stage stochastic setting. However, including short term decisions within the medium term planning model leads to a strong increase in model size and makes it more difficult to obtain a solution quickly.
quezada_multi-stage_2020 consider multi-stage stochastic programming for an uncapacitated remanufacturing environment. Among other parameters, customer demand is assumed to be uncertain. Scenario trees are used to represent this uncertainty and demands are sampled from a uniform distribution. The authors develop a branch and cut algorithm and apply the proposed multi-stage stochastic model in a rolling horizon manner. Capacity restrictions are not considered, even though they constitute a major limitation in real-world environments. For this reason we investigate a capacitated setting and analyze the effect of different resource utilizations.
simonis_simulationoptimization_2023 investigate simulation-optimization for tablet production in the pharmaceutical industry. In this work, optimization is not applied in a rolling horizon setting. Instead, the authors solve the lot sizing problem for the full planning horizon, simulate the obtained production plan, use the simulation results to update the optimization model and start over the next optimization run. This is done until the optimization result does not change anymore. Customer demand is assumed to be uncertain and a variable neighborhood search procedure to solve the lot sizing problem in a generalized uncertainty framework is proposed.
The forecast evolution model is considered in the work of grunow_dynamic_stochastic_2023. They study the differences of using the additive and multiplicative version of forecast evolution in dynamic lot sizing under demand uncertainty. In their approach the authors investigate a multi-stage demand uncertainty setting, however modify the model by splitting the time horizon in two parts. A piecewise-linear approximation is used to determine production lots in the short term, whereas a scenario-tree representation is used to model the demand uncertainty for the later periods. Instead of focusing on different variants of the forecast evolution modeling approach, we analyze different customer update behaviours in combination with varying levels of resource utilization and investigate the suitability of three planning strategies.
almeder2009simulation investigate simulation and optimization in a supply chain network context. They consider several suppliers, production facilities and customers, while modelling the material flow via different transportation modes. In their work, optimization and simulation are applied iteratively, each considering a different level of details, using mixed integer linear programming (MILP) models to obtain decision rules for a discrete-event simulation environment. The authors point out, that with this approach it is possible to treat nonlinearities, complex structures, stochasticity in the simulation environment, while the optimization is able to provide optimal solutions for a simplified version of the problem. In our work, we also follow this idea, however, instead of exclusively using deterministic optimization models, we also make use of stochastic models. This allows us to explicitly consider future demand variations in the planning procedure.
For a broad overview of the full spectrum of simulation-optimization methods we refer to available survey papers, such as figueira_hybrid_2014. The authors explore the characteristics of the different approaches and provide a systematic classification and taxonomy. Also Juan.2015 review literature that combines optimization techniques with simulation in order to make better decisions in uncertain real world environments. For general literature on the lot sizing problem we refer to the comprehensive surveys of karimi_capacitated_2003, jans_modeling_2008 and quadt_capacitated_2008.
Even though mathematical optimization models for lot sizing problems have already been applied to stochastic demand situations in a rolling horizon setting, to the best of our knowledge, we are the first to investigate different customer update behaviours and system utilizations. This allows to derive important managerial insights regarding the suitability of different planning approaches. Moreover, we evaluate the proposed planning strategies not only by drawing random demand realizations, but by using discrete-event simulation. This allows to consider additional uncertainties on the shop floor, such as stochastic setup times, in the planning procedure.
This section details the elements of the created simulation model, starting with demand generation using forecast evolution. It then covers the implemented planning approaches, namely MRP, as well as the used deterministic and stochastic optimization models. Finally, it describes how these components interact during a simulation run.
To evaluate the influence of customer demand on a production system, it is effective to use a range of varying customer demand scenarios. In a typical supply chain setting, customers provide and regularly update long-term demand forecasts. An effective method to assess the impact of these forecast uncertainties is to generate customer demand using the additive Martingale Model of Forecast Evolution (MMFE) method Heath.1994,Norouzi.2014.
For the applied MMFE the variable $D_{itb}$ in Equation (ref) denotes the demand forecast for item $i$ for due date $t$, which is provided $b$ periods before delivery. The variable $F_{it}$ represents the long-term forecast of end item $i$ for due date $t$. This is a constant model parameter for end items and set in a way that the expected order amount $E[D_{it0}]$ is for example 200 units. The forecast update horizon, denoted by $T$, is the period during which forecast updates are performed. For periods beyond the forecast horizon $t > T$, customers provide the long-term forecast $F_{it}$. Within the applied simulation model for $T$ the constant value of \( {{12}}\) periods is set.
The random update term applied during $b \leq T$ is denoted as $\varepsilon_{itb}$ in Equation (ref). {It} is modelled as a truncated normally distributed random variable with mean $0$ and a standard deviation $\sigma_{\varepsilon}$, which is defined as the $\alpha$ fraction of the long-term forecast $F_{it}$. In this setting, $\alpha$ represents the level of unsystematic forecast error. The unsystematic forecast error is the unpredictable, random deviation in forecasting outcomes Zeiml.2019,Altendorfer.Felberbauer.2023. Note that for the forecast updates $\varepsilon_{itb}$, a truncated normal distribution is necessary to avoid negative demand forecast values, i.e., $\varepsilon_{itb} > (-D_{it,b+1})$ and $\varepsilon_{itb} < D_{it,b+1}$. Setting $\varepsilon_{it0}=0$ indicates the final order amount, with no more updates in the delivery period. The expected order amount $E[D_{it0}]$ is set to the corresponding demand behavior quantities for the end items in the numerical study.
{We test} different values for $\alpha$ in combination with three different customer behaviors. More detailed information on the investigated customer behaviours can be found in Section (ref).
Medium-term planning is part of hierarchical production planning and involves making decisions that have a significant impact on the overall production capacity and the allocation of resources over a duration of several weeks or months Luo.2022. The medium-term planning approach of MRP provides detailed material replenishment and capacity plans to cover a planning horizon extending over a few months DOLGUI2007269. Advantage of standard MRP is the algorithmic simplicity of iteratively repeating steps, which always provides an applicable production plan independent of the production system complexity. One disadvantage of standard MRP is the missing capacity limitation for a planning period due to the decoupling of the time-phasing procedure (which relies on fixed planned lead times) from the time-dependent workloads in individual departments Zijm.1996. From a manufacturing perspective, this leads to plans, which may be infeasible due to missing machine capacity. MRP is also prone to the bullwhip effect due to planning nervousness introduced by changes in the Master Production Schedule (MPS) Rahman.2020. The concept of MRP is known for its textbook description like Hopp.2011, Orlicky.1975 or Vollmann.1997 using tables and detailed verbal descriptions of the according steps. In contrast Bregni.2013 are representing MRP in mathematical terms. Regardless of its formalization, MRP is an approach aimed at finding planning solutions that effectively align both internal and external supply and demand, while explicitly taking into consideration the lead times involved. The primary objective is to keep the projected inventory levels low for end items and components within the planning horizon, based on the selected lot-sizing policy. MRP additionally ensures that the projected inventory on hand remains above or equal to the established safety stock level. There also exist further developments of MRP like the concept of demand driven MRP (DDMRP), which enhances the traditional MRP approach to better handle dynamic demand and supply chain variability. The core idea of MRP, however, remains the same Ptak.2018. The following procedure describes standard MRP as it is implemented in our discrete-event simulation model. At the beginning, all customer demands that fall within the predetermined planning horizon are considered as part of the standard MRP process. The process of MRP involves the following steps: (i) The gross requirements are applied to perform netting, (ii) The predetermined lot sizing policies "fixed order quantity" (FOQ) or "fixed order period" (FOP) are applied, (iii) Time phasing is done, typically through backward scheduling with fixed planned lead times, (iv) The gross requirements for necessary components and raw materials are calculated through the bill of materials (BOM) explosion step. These steps are executed for the specified items in the BOM, starting with the end items that have BOM level 0, and progressing until the final BOM level is reached. The work of Hopp.2011 provides detailed information on standard MRP, and the simulation framework was implemented accordingly.
In this paper, next to standard MRP, we also obtain production plans by formulating and solving MILP models, which we integrate into the developed simulation-optimization framework. Our models are based on the multi-item multi-echelon capacitated lot sizing formulations of thevenin_material_2021. Table (ref) provides the necessary notation.
The task is to produce a set of items $\mathcal{I}$ over a fixed discrete time horizon $\mathcal{H}$, using capacitated production resources $\mathcal{K}$, in order to satisfy customer demand for these items, which underlies some uncertainty. The goal is to minimize the total costs, consisting of setup, production, inventory, backlog and lost sales costs. Items can be separated into end items $\mathcal{I}_e$, that face external customer demand, and components $\mathcal{I}_c$ that are needed for the production of end items, as well as other components. The operation structure is represented by the BOM $R_{ij}$, defining the number of units of component $i$ necessary to produce one unit of item $j$. Each item $i \in \mathcal{I}$ has a lead time $L_i$ that needs to pass, before it can be further processed. In order to produce an item $i$, all necessary resources $\mathcal{K}_i$ need to be set up. For each of these resources $k$ the setup procedure comes at a setup cost of $s_{ik}$ and consumes a setup time of $t_{ik}$ units of the $C_{kt}$ units that are available in period $t$. Furthermore each produced unit of item $i$ comes at production cost $v_i$ and consumes additional production time $p_{ik}$ of resource $k$. At the beginning of the planning horizon there are $\hat{I}_{i0}$ units of item $i$ available in the initial inventory. During the planning horizon, in each period $t$ arriving production orders $\hat{I}_{it}$ of item $i$ may become available. These orders have been started in earlier periods before the start of the planning horizon and can be used for further processing, as soon as they arrive. Production capacity that is occupied by these arriving orders is already excluded from the available capacity $C_{kt}$. In each period of the planning horizon $\mathcal{H}$ there is customer demand for end items $\mathcal{I}_e$, which underlies some uncertainty. Demand of a certain period can either be satisfied by items that are on inventory or items that are finished in the respective period. Storing items on inventory comes at holding costs $h_i$. In the case where not enough items are available to satisfy the demand, backlog costs $b_i$ per period need to be paid until the demand is fulfilled. Backlog not satisfied in the last period of the planning horizon is considered as lost sales. These cannot be fulfilled later and incur lost sales costs $e_i$, which are usually much higher than backlog costs. Allowing lost sales guarantees a feasible solution, even in the case where the production capacity is not sufficient to fulfill all customer orders during the planning horizon. The task is to determine setups and production quantities, that define a production plan respecting the requirements of the operation structure and production system, while minimizing the total resulting costs.
Despite the fact, that the underlying production environment faces several uncertainties, such as stochastic customer demands and setup times, it is reasonable to solve a deterministic optimization model at every stage of the rolling horizon setting. Due to frequent updates that the optimization part receives from the simulation environment, the relevant information adapts over time. Decisions made in the deterministic model are based on the latest available information update, without considering possible changes in the future. Table (ref) shows the variables used in the deterministic optimization model, which is formulated in (ref) - (ref).
such that
The objective (ref) is to minimize the total costs over the planning horizon, which consist of setup, production, inventory, backlog, and lost sales costs. While production and inventory costs incur for all items, backlog costs only incur for end items. Backlog in the last period of the planning horizon is denoted separately as lost sales. Constraints (ref) and (ref) define the inventory balance for end items and components respectively, taking into account lead times. Constraints (ref) make sure that items can only be produced if the suitable setup is made in the respective period. The capacity constraints (ref) restrict the combined consumption of resource capacity by setup time and production time for each period and resource. Finally, the variable domains are specified in (ref) - (ref).
Although due to frequent updates, uncertainty in the parameters can be partly considered in a rolling horizon style deterministic optimization, it still comes with several disadvantages. Production lots are connected via several production stages, which respectively have certain lead times. In order to effectively consider uncertainty in the customer demands, fluctuations need to be anticipated long before the actual due date. Reaction times in production systems with several echelons are usually longer than one period, meaning that a demand update shortly before the actual due date cannot be compensated anymore by updated optimization runs. The used mathematical model should make use of the available information on the parameter's stochasticity at every call. To realize this, a suitable approach is two-stage scenario-based stochastic optimization, see e.g. birge_louveaux_2011. It takes into account the stochasticity by sampling discrete scenarios from an assumed underlying probability distribution and incorporating a metric over these scenarios, such as the expected value, into the optimization model. In the following, we denote a scenario by $\omega$ and corresponding demand realization as $D_{it}^{\omega}$, which represents the realized demand of end item $i \in \mathcal{I}_e$ in period $t \in \mathcal{H}$ for scenario $\omega$. The set of all scenarios included in the optimization model is denoted as $\Omega$, with the number of scenarios being the cardinality $|\Omega|$ of this set. Within the optimization model each scenario $\omega$ is weighted with a probability of $p_{\omega}$. In the following, we describe a stochastic version of the model presented in Section (ref), using two-stage scenario-based stochastic programming. In two-stage models decision variables are split into first stage and second stage decisions. While first stage decisions are fixed before the uncertainty is realized, second stage decisions can be adjusted after the uncertain parameters are revealed. Decisions variables of the first stage can be implemented here-and-now, while variables of the second stage are scenario dependent and are therefore referred to as wait-and-see decisions.
{There are different possibilities to model the discussed lot sizing problem in a two-stage setting. In the two-stage stochastic programming model of thevenin_material_2021 the setup and quantity decisions form the first stage decisions, while inventory and backlog are put into the second stage. Quantities are therefore assumed to be fixed over the whole planning horizon, while the scenario dependent inventory and backlog variables are used to estimate the costs related to the discrete demand scenarios. schlenkrich_capacitated_2024 propose a model that assumes flexibility in production quantities and only fix setup decisions in the first stage, allowing to adapt production quantities as soon as demand is known. Since we iteratively solve the optimization model in a rolling horizon manner, a mix between these two approaches is desirable. Production orders at the beginning of the planning horizon are made here-and-now, while production quantities of later periods are naturally flexible, because the optimization model will be solved again with updated information at a later point in time. In order to adequately reflect this setting within the two-stage optimization model, we introduce an additional parameter $\tilde{T}$ representing the period up to which production quantities are assumed to be fixed within the model. Instead of fixing the production quantities $Q_{it}$, associated with item $i \in \mathcal{I}$, for all periods $t \in \mathcal{H}$ in the first decision stage, we only put a part of the quantity decisions $Q_{it}$ with $t \leq \tilde{T}$ in the first stage, while moving the decisions for the remaining periods $t \in \{\tilde{T}+1,\ldots, T\}$ to the second stage. This reflects the planning behaviour in the rolling horizon setting and allows the model to consider the possibility to adjust production orders, as soon as new demand information is available. It is necessary to put at least the production quantities of the initial period $t=1$ in the first stage, because these decisions are directly translated into production orders within the simulation and therefore need to be decided here and now. The parameter $\tilde{T}$ is set to a value between $1$ and $T$, where $\tilde{T} = 1$ results in a model with very high flexibility and $\tilde{T} = T$ in a model that assumes production quantities to be fixed for the whole planning horizon.}
Table (ref) shows the used decision variables for the stochastic model, which is formulated in (ref)-(ref).
such that
{ The objective (ref) of the stochastic optimization model is to minimize the total resulting costs over the planning horizon consisting of setup costs, as well as the expected {production}, inventory, backlog and lost sales costs over all scenarios $\omega \in \Omega$. The inventory balance constraints for end items and components need to hold for all scenarios $\omega \in \Omega$ and are transformed to (ref) and (ref). Here we note that production quantity variables, as well as inventory and backlog variables are equipped with a scenario index $\omega \in \Omega$. In order to guarantee that quantity variables for periods up to $\tilde{T}$ are actually implementable here and now, we introduce non-anticipativity constraints (ref) to force quantity variables $Q_{it}^{\omega}$ with $t \leq \tilde{T}$ to have the same value for all scenarios $\omega, \omega' \in \Omega$. These constraints make the corresponding decision variables equivalent to first stage variables. Setup and resource capacity constraints need to hold for each scenario $\omega \in \Omega$ and are transformed to (ref) and (ref). While variable domains for setups (ref) remain unchanged from the deterministic model, the domains for quantity, inventory and backlog variables are modified to (ref),(ref) and (ref). }egin{comment} { The proposed model (ref)-(ref) is modified as follows: Production quantity decision variables $Q_{it}$ are equipped with an scenario index $\omega \in \Omega$ analogously to the inventory and backlog variables. The inventory balance constraints (ref) and (ref) are modified to (ref) and (ref).
In order to guarantee that quantity variables for periods up to $\tilde{T}$ are actually implementable here and now, we introduce non-anticipativity constraints (ref) to force variables to have the same value for all scenarios $\omega, \omega' \in \Omega$. } \end{comment}
{ To reduce the size of the stochastic model, in our implementation we follow thevenin_material_2021 by implicitly modelling the non-anticipativity constraints. For each item $i \in \mathcal{I}$ and $t \leq \tilde{T}$ we replace the set of quantity variables \{$Q_{it}^{\omega} | \omega \in \Omega\}$ by the single variable $Q_{it}$. By doing so, the non-anticipativity constraints are implicitly considered in the design of the decision variables. }
In this section, we introduce the developed simulation-optimization framework and elaborate on the interaction of its technical components. In general, discrete-event simulation stands as a well-established approach for modeling production systems, enabling the exploration and enhancement of performance across various system configurations Altendorfer.2016,Altendorfer.Felberbauer.2023,Enns.2001,Juan.2015. It can be used to mimic the behavior of a production system for a time-span of up to several years relying on a hierarchical planning approach like MRP Seiringer.Algorithm,Werth.2023, reorder points system (RPS) Seiringer.WSC23, constant work in progress (CONWIP) Xanthopoulos.2021, or in combination with solving an MILP model to obtain optimized production plans gansterer_simulation-based_2014. To effectively integrate production system simulation with optimization in a rolling horizon planning framework, several key requirements need to be addressed. On the one hand, up-to-date system and demand related information needs to be provided to the optimization component by the simulation environment. This information comprises the latest customer demand updates, available inventory and production orders currently being processed, as well as occupied resource capacities. On the other hand, the output of the optimization, e.g. the optimal values of the decision variables, have to be translated into production orders which can subsequently be processed in the simulation environment. In the following, the most important planning and interaction requirements are described in more detail.
Rolling horizon planning is dynamic, involving several critical steps. The process begins by starting the simulation with a limited runtime of $n$ periods. During the simulation, a rolling horizon planning window, which captures a timespan of $T$ periods from the current simulation time $\bar{t}$, is iteratively shifted by one discrete time period. In this context, $T$ denotes the length of the planning window. A shift of the rolling horizon window is performed by moving the observed interval from $(\bar{t},\bar{t}+T)$ to $(\bar{t}+1,\bar{t}+1+T)$. Each shift or increase of $\bar{t}$ to $\bar{t}+1$ triggers a new planning cycle, until the current simulation time $\bar{t}$ is equal to the runtime limit $n$.
After a new planning cycle has started, new customer orders are generated and existing orders are updated according to the forecast evolution model outlined in Section (ref). The demand forecast of an order for item $i$ and due date $t$ is denoted by $D_{itb}$, assuming that the current simulation time is $b$ periods before the due date, i.e. $t-\bar{t}=b$. After generating and updating customer demands, the simulation provides two types of parameters to the optimization component. First, general information on the production system setting is required. This comprises information on the components $\mathcal{I}_c$ and end items $\mathcal{I}_e$, as well as resources $\mathcal{K}$ and respective resource requirements of items ($\mathcal{I}_k$ and $\mathcal{K}_i$). Furthermore, BOM structure ${R}_{ij}$ and time related parameters, e.g. lead times ${L}_i$, setup times ${t}_{ik}$ and processing times ${p}_{ik}$, need to be provided. Finally, also cost-related parameters, e.g. setup ${s}_{ik}$, processing ${v}_{i}$, holding ${h}_{i}$, backlog ${b}_i$ and lost sales costs ${e}_i$, are part of the required general information and need to be passed to the optimization component. These parameters stay constant during the whole simulation runtime.
Next to the general information, the optimization component also requires dynamic parameters that change during simulation. These include the inventory at hand $\hat{I}_{i0}$ and information on arriving orders during the planning window $\hat{I}_{it}$, as well as available resource capacity ${C}_{kt}$ and finally, the updated customer demand forecasts ${D}_{itb}$. To provide the initial inventory $\hat{I}_{i0}$, the current on stock quantity of item $i$ at the current simulation time is reported. To identify the production orders $\hat{I}_{it}$ of item $i$ that will arrive in period $t$, all waiting and processing orders with planned end $t$ are selected and the respective order quantities are summed. To compute the available capacity ${C}_{kt}$ of a resource $k$ in period $t$, all planned and running production orders in the simulation are selected and their required capacity is subtracted from the initially available capacity of the upcoming period. For currently running production orders (at time $\bar{t}$), only the remaining capacity requirement between the current simulation time and the planned end date is taken into consideration. For waiting production orders, the full requirement is subtracted. In case the summed capacity requirement of the production orders in the simulation exceeds the capacity of a resource in the upcoming period, capacity of subsequent periods is consumed, meaning that $C_{kt}$ is reduced, respectively. Finally, the latest demand information for customer orders within the planning horizon is provided to the optimization component. Depending on the optimization approach, different parameters are necessary. For the deterministic optimization model it is sufficient to report the most recent demand forecasts $D_{itb}$, which are used as the deterministic demands $D_{it}$ in the model. The scenario-based stochastic optimization on the other hand, makes use of different scenarios $\omega \in \Omega$ and the respective scenario demands $D_{it}^{\omega}$ to generate production plans that perform well in the face of uncertain customer demand. In order to generate meaningful demand scenarios, we use historical demand forecasts $D_{it'b}$, where the due date $t'$ was already prior to the current simulation time $\bar{t}$, i.e. $t'<\bar{t}$, to estimate how much the actually realized demand at the due date will differ from its prediction. For this purpose, we compare the predicted quantity to the actual quantity when the order is due and track this deviation during the whole demand information horizon. We collect these deviations during the warm-up phase $z$ of the simulation, i.e. $0 \leq \bar{t} \leq z$, and calculate the mean deviation for each combination of end item $i$ and distance to due date $b$. This allows us to estimate the standard deviation $\sigma_{ib}$ of a truncated normal distribution, which is later used to draw the scenario demands $D_{it}^{\omega}$ for the stochastic optimization model. Assuming that the due date $t$ is reached in $b$ periods, the respective mean of the truncated normal distribution is modeled as the most current prediction $D_{itb}$, which represents the most up-to-date information available. To summarize, the scenario demands $D_{it}^{\omega}$ for end item $i$ at due date $t$ used in the stochastic optimization model are drawn from a truncated normal distribution with mean $D_{itb}$ and standard deviation $\sigma_{ib}$, assuming that period $t$ is $b$ periods from the current simulation time. For a more detailed explanation of this procedure we refer to Altendorfer.Felberbauer.2023.
After all necessary general and dynamic parameters are provided to the optimization component, the chosen model can be formulated and solved. The simulation environment is required to await the resolution of the optimization problem. After solving the model to optimality, the optimization component returns the optimal decision variables, i.e. the production quantities ${Q}_{it}$ to the simulation. Production quantities having a period index corresponding to the current simulation time $\bar{t}$ are used to generate production orders in the simulation, e.g. for $Q_{it} = 200$, a production order of 200 pieces for item $i$ is generated if $t=\bar{t}$. These production orders are released immediately at time $\bar{t}$, i.e. the planned start date is $\bar{t}$. The end date of each created production order of item $i$ is set to the sum of the planned start date and the respective planned lead time $L_i$. Finally, the orders are released and processed on the shop floor, where the released production orders of previous periods are processed as well. The current period is simulated until the beginning of the next discrete period, which triggers the next planning cycle. In the following section the technical interaction of the developed components is described in more detail.
The simulation model is provided in form of a discrete-event simulation model developed with Anylogic. In Figure (ref) a schematic representation of the simulation-optimization framework is shown.
The four main components are simulation input (I), simulation model (II), planning method (III) and simulation output (IV). This is also the order in which the components are interacting. First the simulation input is provided to the simulation model which iteratively calls the planning method and finally stores the simulation output in a relational database, after the simulation run is finished. The simulation input comprises three types of input data:
A more detailed description of the required demand forecast information is presented in Section (ref). After the input data are available, the simulation model (II) simulates the production system until the end of the defined simulation runtime $n$. This requires the repeated call of the simulation components. First, the demand generation and customer update (4) component periodically generates the customer orders with the relevant information of item, quantity and due date. In this component the forecast evolution described in Section (ref) is implemented. The production order generation (5) component is called to generate production orders based on the input of a chosen planning method (III). The production order generation component can be seen as the interface for the planning interaction between the simulation model (II) and the planning method (III). In our framework, three planning methods are available, among which one is selected to be applied per simulation scenario. \textit{MRP (6)} is implemented in Anylogic directly, whereas, the \textit{deterministic optimization model (7)} and the \textit{stochastic optimization model (8)} are implemented in python, which uses CPLEX as MIP-solver. The python code is called from Anylogic during the simulation run and the simulation waits until the result is returned by the python module. For each of these three planning methods, the parameters lead time and safety stock must be defined, see the numbers (9), (10) and (11) in Figure (ref). For MRP, also one of the lot sizing policies, either FOP or FOQ, must be set before running the simulation. For the stochastic optimization model, also the number of scenarios is provided. Independent of the planning method, the computed release plan is used by the production order generation component to create new orders in the simulation. These orders are subsequently passed to the \textit{scheduling and release (12)} component. The production orders are sorted according to earliest due date (EDD) and are released to the corresponding machine queue. Their processing is then simulated in the \textit{job shop processing component (13)}. The respective amount of items of a finished production order are put on stock immediately after being finished and are available for \textit{customer delivery (14)}. Finished components are available for further processing of end items on the shop floor. To assess the service level, we check the difference between the due date of a customer order and its actual delivery. Delayed orders are fulfilled as soon as the required amount of items is available in stock. Finally, the simulation model provides two types of output. On the one hand, there are cost and order related key performance indicators (KPI), such as inventory, tardiness or overall costs over the whole simulation run per time unit [TU]. On the other hand, production related performance measures like utilization per machine or actual release dates of production orders are reported.
The simulation study aims at comparing the performance of MRP to deterministic and stochastic optimization in a rolling horizon planning environment. For this purpose the above described KPIs are computed. To ensure meaningful evaluation, each replication of the simulation ran for $n=400$ periods (equivalent to weeks), with the first 40 periods serving as a warm-up phase, i.e. $z=40$. After the warm-up phase, all statistics were reset, assuming that the simulation systems reached a steady state. This setting allowed an investigation of the production planning approach over the course of one year, considering a daily planning frequency.
Two sources of uncertainty were considered in the simulation. Firstly, uncertainty was introduced into the forecasting process for customer demand. Secondly, stochastic setup times were simulated assuming a log-normal distribution. The log-normal distribution is often used in production systems due to its positive skewness and ability to model variables with a lower limit of zero and large positive values. Its multiplicative nature simplifies analysis when dealing with variables that result from the multiplication of independent factors Limpert_Log_normal_distribution. The processing time of items and components, however, was assumed to be deterministic, assuming a stable production process. To capture the stochastic effects, 10 replications were used for each iteration. A replication corresponds to a completed simulation run for the predetermined runtime of $n$ periods.
In the following, the evaluated production system including the planning parameters, demand and shop loads are introduced. In Section (ref), the tested forecast scenarios are listed. We consider planned shop loads at the levels of: 85%, 90%, 95%, and 98%. These shop load levels are determined based on the set processing time. They allow a meaningful investigation of the production system behavior and represent realistic scenarios. A lower shop load than 85% will only marginally reflect effects of the tested planning parameters. For production companies, the economic target is also a high system shop load to reduce costs for production downtimes. The 98% shop load level is particularly interesting as it allows us to observe how the production system responds to tardiness caused by stochastic effects.
The elementary multi-item and multi-stage production system is illustrated in Figure (ref). Albeit the BOM consists of only three levels, it allows a meaningful comparison of optimization and MRP in the context of rolling horizon planning with forecast related demand uncertainty and production planning related performance indicators like tardiness, WIP level and inventory. It is important to recognize that smaller systems provide a clearer understanding of how planning parameter settings, influence outcomes Vidal.2022. The end items $10$ and $11$, at BOM level $0$ are produced on machine M2. The components $20$ and $21$ are produced on machine M1. For one unit of end item $10$, we need $1$ unit of component $20$. Likewise, we need $1$ piece of component $21$ for final product $11$. The raw material $100$, which is a purchased product, is needed for the components $20$ and $21$. This raw material is assumed to be always available and no stock out can occur for it.
For the production system, a base demand scenario has been established, in which customers consistently order on average 200 units of item 10 and 400 units of item 11 each period (week). This baseline demand scenario is diversified by incorporating various customer behaviors, as detailed in Section (ref), which enable a comparative analysis of these behaviors and their impact on production system performance. To compute planned shop loads of 85%, 90%, 95% and 98% for the demand scenario, the following settings are used: end item and component processing times in minutes per shop load are 1.56 minutes for 85%, 1.68 minutes for 90%, 1.8 minutes for 95% and 1.872 minutes for 98%. For all items a setup time of 144 minutes is applied. The value of 144 is based on a period capacity of 1440 minutes and a setup proportion of 10%. To parameterize the applied log-normal distribution for setup time, the coefficent of variance (CoV) is set to 20% leading to 28.8 minutes. This makes the production system capable of producing 600 unites per period (week). In the deterministic case, applying MRP, without forecast updates and a fixed setup time, with lot sizing policy FOP set to 1 and a planned lead time of 1 period, the planned shop load e.g. 98% is reached with service level 100%. These settings reflect the aimed production system behavior of one setup in every planning period.
As introduced in Section (ref), we model the uncertain behaviour of customer demand via the forecast evolution model. By varying the frequency of demand updates, their magnitude and respective distance to the due date, different demand behaviours can be modeled. In our numerical experiment, we study three customer behaviours. Customer Type A represents a very reliable customer that only once changes the long-term forecast with the beginning of the demand information horizon when $T$ is 12. Afterwards no further updates are submitted by Customer Type A. This mimics a deterministic demand setting for the optimization model, since there are no demand updates within the planning horizon. Customer Type B, on the other hand, can only give a rough estimation of the required amount in advance at the beginning of the demand information horizon, when $T$ is 12. The actual demands at the due dates can however vary significantly from the first submitted demand estimation, which is modelled by a second demand update right before the respective due date. Finally, Customer Type C changes demand information frequently and updates the required amount in every period of the planning horizon. Since the same $\alpha$-values are applied for Customer Type B and Customer Type C, the demand update pattern of Customer Type C represents the highest uncertainty among the three investigated customer types.
Furthermore, we study different demand variation factors $\alpha$. Variation factors represent the percentage of the long-term forecast by which the demand changes at every demand update step. The higher these $\alpha$ values are, the more demand uncertainty has to be absorbed by the production system. In our numerical experiment, we vary $\alpha$ within the range \{0.025, 0.05, 0.075, 0.1, 0.125\}, meaning that demands are changed between 2.5% and 12.5% of the long-term forecast at every update step. Through the analysis of these customer behaviors, our objective is to explore the managerial significance of demand information. This includes evaluating the impact of receiving daily updates on demand changes (or demand updates every period), the consequences of having a stable forecast but an uncertain demand realization (or only receiving a single update at the due date), and understanding the effects of the demand variation factor on decision-making processes.
In addition to customer types and demand variation factors, planning parameters for MRP and parameters for both deterministic and stochastic optimization are evaluated. Parameter ranges for this evaluation were chosen based on preliminary simulations.
For MRP, the parameters include safety stock (SS), lot policies such as fixed order period (FOP) with a number of periods (NP), fixed order quantity (FOQ) with lot size (LS), and planned lead time (LT). The ranges for SS are multiples of 0, 0.1, 0.2, 0.3, 0.4, 0.5, and 0.6 of the long-term forecast for the end item. For example, for item 10 with a forecast of 200 units, the safety stock (SS) range would be 0, 20, 40, 60, 80, 100, and 120 units. For FOP the NP values considered are 1, 2, 3, 4 and 5 and LT is set to 1, 2, 3 and 4 periods. For FOQ the LS is set to multiples of 0.5, 1.0, 1.25, 1.5, 1.75, and 2 times the long-term forecast. For example, for item 10 with a forecast of 200 units, the LS values would be 100, 200, 250, 300, 350, and 400 units.
These values are also applied to the components associated with the end items in the BOM. Selecting the same planning parameters for both end items and components is a suitable approach and allows a focused performance analysis of MRP in comparison to deterministic and stochastic optimization methods. In total, this results in 50,400 different MRP parameterizations, including the three customer types and demand parameters.
For deterministic and stochastic optimization, LT values of 1, 2, and 3 are evaluated, with SS values matching those in MRP. In stochastic optimization, the number of scenarios tested are {10, 20, 30 and 50. This leads to 1,260 different parameterizations for deterministic optimization and 5,040} for stochastic optimization. Each parameterization is simulated with 10 replications, resulting in a total of {567,000 individual simulation runs (MRP: 504,000; deterministic optimization: 12,600; stochastic optimization: 50,400).}
In this section, we present the results from the simulation study conducted to compare standard MRP against deterministic and stochastic optimization approaches. The discussed results represent the overall costs in cost units (CU) per period computed by the sum of finished goods inventory (FGI), Work in Progress (WIP) and tardiness per unit. For end items a FGI costing factor of {{2}} is applied, for the associated WIP a value of {{1}}. The inventory holding costs for components are {{1}} and for components WIP they are {{0.5}}. Tardiness costs for the end items is set to 38. The relation of end item WIP and end item tardiness costs represents a target service level of 95%. A service level of 0.95 means there is a 95% chance that the inventory on hand will satisfy customer needs, thereby eliminating the risk of running out of stock. The inventory costs of end items and components are twice the WIP value, as it is more costly to store end items or components. In the subsequent sections, we first present the baseline MRP results under medium demand variation ($\alpha=7.5\%$), which serve as reference values for benchmarking against deterministic and stochastic optimization. This is followed by an exploration of a suitable number of scenarios for the stochastic optimization model, as well as an investigating of the $\tilde{T}$ parameter. Subsequent sections delve into the effects of shop load congestion and the impact of forecast uncertainty. We then provide a comprehensive overview of costs and parameter variations across different resource utilizations, customer demand patterns, and levels of demand variation. Finally, a use case is presented, where we analyze a larger and more complex production system.
The well established medium-term planning approach of MRP is widely applied in industry and therefore serves as a benchmark. While the MRP data presented in Figure (ref) does not introduce novel MRP-specific discoveries, they do offer valuable perspectives on planning outcomes and validate the accuracy of the MRP simulation. Within Figure (ref), the minimum MRP overall costs associated with the simulated shop loads are illustrated. Each curve symbolizes distinct customer behaviors (reliable Customer Type A, volatile Customer Type B, nervous Customer Type C) subjected to a medium demand variation of 7.5%. Based on these behaviors, Customer Type A, characterized by the least uncertainty, invariably results in the lowest overall costs across all shop loads. The increasing demand uncertainty for Customer Types B and C corresponds to a noticeable increase in overall costs for every shop load. Meanwhile, the regular demand adjustments attributed to Customer Type C inject the highest unpredictability into the production system, culminating in the most pronounced overall costs across all shop loads.
The optimal planning parameters for Customer Type A, which result in the minimum total costs, have been identified as LT (Lead Time) \(= 1\), SS (Safety Stock) \(= 0\), and FOP (Fixed Order Period) \(= 1\). This strategy is effective due to Customer Type A's deterministic demand behavior, which enables the production system to counteract stochastic effects without needing safety stock, and by utilizing the minimum values for LT and FOP. However, when the shop load reaches \(98\%\), there is a slight increase in overall costs, as the higher production lead time causes increased tardiness within the production system.
For Customer Type B, characterized by volatile demand, employing FOQ consistently results in the minimum overall costs, effectively mitigating the impact of a higher shop load. Specifically, at shop loads of \(85\%\) and \(90\%\), the most cost-effective approach involves a modest safety stock combined with LT \(= 1\) and FOQ \(= 1.25\%\). At higher shop loads of \(95\%\) and \(98\%\), the best planning performance is achieved by increasing the lead time to 2 periods instead of using safety stock.
A similar trend is observed for the nervous Customer Type C. For a shop load of \(85\%\), combining a lead time of 1 with safety stock attains the lowest overall costs. For higher shop loads, however, it is more effective to replace safety stock with an increased lead time.
In conclusion, these identified parameter combinations are meaningful from a production logistics perspective. For reliable Customer Type A, with its largely deterministic demand, minor variations from the long-term forecast can still be efficiently managed within a lead time of 1, without safety stock, and by adopting a lot-for-lot approach as indicated by FOP 1. Similarly, for Customer Types B and C, who exhibit more volatile and fluctuating demand patterns, meaningful planning parameters are employed. The higher shop load necessitates compensation through either a safety stock or a longer lead time. An increased lead time helps avoid production delays caused by larger lot sizes, while a small safety stock can effectively buffer against demand fluctuations.
{
} A relevant parameter for the stochastic optimization model, introduced in Section (ref), is the number of demand scenarios included in the model. Including more scenarios leads to a better approximation of the stochastic demand distribution and should therefore provide a better performance of the approach. On the other hand, including more scenarios also negatively impacts the time required to solve the stochastic optimization model. Within this section we aim at finding a reasonable number of scenarios to include in our stochastic model and perform experiments with the number of scenarios in {\{10,20,30,50\}.} {A second important parameter for the stochastic optimization model is the number of periods with fixed, i.e. first stage, production quantities $\tilde{T}$, which is examined in more detailed in Section (ref). In the experiments conducted to determine a suitable number of scenarios we include the extreme cases $\tilde{T}=1$ and $\tilde{T}=12$, which represent the worst and best case in terms of model size respectively. More flexibility, i.e. $\tilde{T}=1$, leads to more second stage decision variables and therefore a larger model, while less flexibility, i.e. $\tilde{T}=12$, results in a smaller model.}
We evaluate the case of nervous customer update behaviour C with a medium demand variation of 7.5% in a production environment with 95% utilization. Figure (ref) shows the obtained total costs of the stochastic optimization model{s with $\tilde{T}=1$ and $\tilde{T}=12$} for an increasing number of included scenarios. Despite the fact that the performance of the deterministic model, as well as MRP are independent of the number of included scenarios, we report their performance in the investigated setting to put the stochastic performance into perspective. Additionally, on a second y-axis, the average runtime of the stochastic optimization model{s} is reported for one replication using the respective number of included scenarios. {Since the stochastic optimization model with $\tilde{T}=1$ is larger than the model with $\tilde{T}=12$ the increase in computational time is an expected result.} We see a strong improvement in total costs when including 20 instead of 10 scenarios, for which the stochastic approach{es} perform worse than the deterministic model. Increasing the number of scenarios to 30 further reduces the costs, however the improvement is significantly smaller. {For the model with $\tilde{T}=12$, the runtime has increased from 6 seconds for 10 scenarios to 17 seconds for 30 scenarios. In the case of the more flexible model with $\tilde{T}=1$, the runtime has increased from 86 seconds for 10 scenarios to 254 seconds for 30 scenarios. While the inclusion of further scenarios only results in slight cost reductions it significantly increases the average runtime for solving the stochastic models, especially in the case of $\tilde{T}=1$.} For this reason we have decided to include 30 scenarios in all following stochastic models, which serves as a reasonable trade-off between solution quality and runtime.
{
After determining a suitable number of scenarios for the stochastic optimization model we examine the number of periods $\tilde{T}$ up to which production quantities are assumed to be fixed, i.e. first stage decisions. Fixing the quantities for all periods ($\tilde{T}=12$) might be too conservative, however, the resulting stochastic models can be solved significantly faster than the more flexible models. Only fixing the quantities of the initial period ($\tilde{T}=1$) leads to larger optimization models, which are harder to solve. Moreover, these more flexible models might be too optimistic, because they assume that demand realizations for the whole remaining planning horizon are available after the initial period and all remaining quantities can be adjusted to this information. In order to investigate the impact of the level of flexibility in the model we repeat the experiments conducted in Section (ref) for utilizations of 85%, 90%, 95% and 98% with a fixed number of scenarios of 30 and test $\tilde{T} \in \{1,2,\ldots,12\}$. Figure (ref) shows the respective results. The stochastic model with the highest level of flexibility, i.e. $\tilde{T}=1$, results in the lowest total costs across all utilizations. Especially for higher utilizations also the model that assumes fixed production quantities for the whole planning horizon, i.e. $\tilde{T}=12$, leads to comparably low costs. Fixing a part of the quantity decisions, i.e. $\tilde{T} \in \{2,\ldots, 11\}$ was less successful for the investigated settings.
The experiments reported in Figure (ref) are performed without using safety stocks within the stochastic models. However, in order to guarantee a fair comparison with MRP, as well as deterministic optimization, we also consider the possibility for safety stocks within the stochastic optimization model. This option is included in a consecutive set of experiments in order to detect the most suitable $\tilde{T}$ values to be used in our numerical study. The results are reported in Figure (ref) and it shows that the presence of safety stocks reduces the effects of assuming flexibility in production quantities. However, we still identify $\tilde{T}=1$ to be the best choice for resource utilizations of 85% and 90%. For a utilization of 95%, a $\tilde{T}$ value of 4 periods is the best choice, while for the highest utilization of 98% the choice for 12 periods is most suitable. With rising utilization it therefore seems to be more efficient to increase the number of periods, in which production quantities are assumed to be fixed in the model. In the remaining part of the computational study we will use the best identified values for $\tilde{T}$, as well as the default value of 12, which is equivalent to fixing quantities of the whole planning horizon. In the case of 98% resource utilization, where $\tilde{T}=12$ is the best value, we additionally report results for the second best value, which is $\tilde{T}=1$.}
In this section, we study the effects of an increased shop load, which possibly leads to congestion on the shop floor. We report the cost reduction of deterministic and stochastic optimization models over standard MRP for Customer Types A, B and C in the case of medium demand variation (7.5%) and resource utilizations of 85%, 90%, 95% and 98%. For each of the three planning approaches we use the best performing parameter combination. The results are displayed in Figure (ref) for the deterministic, while Figure (ref) shows the results for the stochastic approach.
First of all, we note that the demand pattern of Customer Type A does not include updates, meaning that the stochastic optimization model is equivalent to the deterministic one, since all sampled scenarios represent the same demand. For this reason the performance of deterministic and stochastic optimization is equal for Customer Type A, i.e. the same cost reduction is reported in Figure (ref) and (ref). For low resource utilization, the optimization approaches are not able to generate an advantage over standard MRP in the case of Customer Type A. This is reasonable, since customer demand does not necessitate any short-term updates, and the low utilization allows the MRP approach to perfectly fulfill all demands on time. Only when considering high resource utilization, e.g. in the 98% case, we observe limitations of MRP. Standard MRP does not consider resource capacity constraints, which can be problematic in cases of tight capacities. This means that it is unable to recognize short term bottlenecks, which can lead to delayed customer orders, even if the final order quantity is known in advance. Using an optimization model instead allows to consider resource restrictions and to react to upcoming bottleneck situations by partly producing the required amounts in earlier periods. For customers of type A, in the case of medium demand variations and 98% utilization, this planning advantage can reduce the total costs by more than 10% compared to a standard MRP procedure.
Customer Type B changes demands last minute, meaning that the forecasted quantity does not match the final demand realization. In contrast to Customer Type A, the production plans obtained by the stochastic model differ from the plans resulting from the deterministic one. First of all, we observe a significant performance improvement of both optimization approaches compared to MRP along all utilizations. For lower utilizations both models, i.e. deterministic and stochastic, lead to similar cost reductions between 30% and 50%. The use of safety stocks allows the deterministic model to hedge against high demand realizations. While the best performing safety stock parameter for the deterministic model {lies between 30% and 40%} of the average demand, it is only 10% for the stochastic model. The limitations of the deterministic model with safety stocks become visible in the high utilization (98%) setting. While in this case stochastic optimization is able to outperform MRP by almost 50%, the deterministic model results in a lower reduction of 40%. {Additionally, for the stochastic optimization model we observe that for lower resource utilizations of 85% and 90% the most suitable choice for $\tilde{T}$ is 1, meaning that quantities should be assumed as flexible. For higher utilizations on the other hand, e.g. 95% and 98%, choosing $\tilde{T}$ as 12 is most suitable. This suggests that a more conservative modelling approach can be beneficial in cases of tight resource utilization.}
Finally, Customer Type C changes demand forecasts every period, which leads to severe instabilities in the production planning. Also for this update pattern both optimization approaches clearly outperform MRP along all utilizations. Again the deterministic models leverage high safety stocks to hedge against demand variations. While a safety stock of 20% of the mean demand leads to the best performance of the stochastic model, the deterministic model uses up to 60% in the high utilization setting. In case of tight resource capacities, the performance improvement of the deterministic approach over MRP reduces to 32% for 95% utilization and 22% for 98% utilization. Applying stochastic optimization on the other hand leads to a cost reduction of more than 40%, even in high utilization environments. This demonstrates the advantage of explicitly considering uncertainty in future demands within the optimization approach. {Again we observe that with rising utilization it is advantageous to increase the value of $\tilde{T}$, assuming a majority of the production quantities to be fixed within the model.}
In addition to the medium demand variation case (7.5%), also results for low demand variation (2.5%) and high demand variation (12.5%) are reported in Figures (ref) - (ref) in the Appendix. While the discussed differentiation between the optimization approaches is weaker in the low variation case, it is even stronger for high demand variations.
In order to assess the impact of increasing uncertainty in demand forecasts, we perform a series of experiments with demand variation values $\alpha$ in the range \{2.5%, 5%, 7.5%, 10%, 12.5%\}. Figure (ref) shows the cost reduction of stochastic optimization over deterministic optimization for a utilization of 95%.
First of all, we observe that for Customer Type B deterministic optimization outperforms the stochastic model in case of a low demand variation with 2.5%. Planning with the deterministic model and a safety stock of 10% of the mean period demand is enough to balance demand uncertainties and leads to a 6% cost reduction over the stochastic approach. With increasing demand variation however, we see the advantage of using stochastic optimization. The deterministic model leverages increasing safety stocks to hedge against the growing variability in the demands, using 30% for a demand variation of 7.5% and up to 60% for high variation of 12.5%. This increased safety stock weakens the performance of the deterministic approach and allows the stochastic model, which keeps the safety stock at a moderate level of 10%-20%, to result in reduced overall costs. While the performance improvement is 5% for medium demand variation of 7.5%, it is 14% in case of high variation with 12.5%.
Deterministic optimization could not outperform the stochastic approach for Customer Type C in the 95% utilization setting. Even for low demand variation of 2.5%, solving the stochastic model reduced the resulting overall costs by 5% compared to the deterministic version. This improvement further increased with higher demand variation leading to a 23% reduction in costs for high demand variation with 12.5%. Again we see that the deterministic model increases safety stocks to cope with high variability in the customer demands, storing up to 60% of the average demands as a buffer. For the stochastic approach, on the other hand, a lower safety stock of 30% is sufficient, due to its capability to anticipate possible variations in the upcoming periods.
Additionally, in Figure (ref) and Figure (ref) we report the results for the low utilization of 85% and high utilization of 98% case in the Appendix. The low utilization case demonstrates that deterministic optimization with safety stocks is sufficient for Customer Type B if enough resource capacity is available. It outperforms the stochastic approach for all demand variations. For Customer Type C solving the stochastic model is advantageous and cost improvements of up to 10% over the deterministic version are possible. In high utilization environments of 98%, stochastic optimization outperforms the deterministic model for both customer types B and C and over all demand variations. While improvements of up to 20% are possible for Customer Type B, a cost reduction of up to 30% can be reached for Customer Type C.
In this section, we summarize the results presented so far. Table (ref) provides an overview of the respective performance of the three approaches across different utilizations, customer demand update patterns and demand variations. We display the cost values and best performing parameter sets, out of the evaluated settings. Moreover for each planning situation, meaning a combination of customer type, demand variation and utilization, we report the relative cost difference ($\%\Delta$) of the best planning approach in percentage compared to standard MRP. The best performing model for each planning situation is highlighted in bold.
For each planning approach we observe an increase in average overall costs with higher utilization. This is reasonable, since tight resource capacities restrict the opportunities of the planning methods and lead to costly delays. We summarize that the use of optimization models does not yield an advantage over standard MRP for reliable customers (type A) that do not change their demands in the short term, as long as resource utilization is moderately low (85% and 90%). We again highlight that the stochastic model is equivalent to the deterministic version for Customer Type A and we therefore only report the results of the deterministic model. In case of higher utilizations (95% and 98%) applying MRP is problematic even for the reliable Customer Type A if demand variation is high, leading to an irregular demand pattern that causes short term bottlenecks in the production system. The use of optimization approaches that consider the available resource capacities can counteract these bottlenecks and reduce the resulting costs by up to 13% in the 95% utilization case and up to 37% in the 98% utilization case.
Customer Type B updates demands once in the short term. This leads to delays if the amount of produced items is not sufficient to satisfy the realized demand and causes higher inventories in case of low demand realizations. We summarize that for Customer Type B the use of deterministic models using safety stocks is most efficient in case of low resource utilization (85% and 90%) across all demand variations. The availability of buffer capacity allows the method to counteract increasing demand variation by producing higher safety stocks. Significant cost reductions of up to 50% over standard MRP are possible. When considering higher utilization environments (95% and 98%) we observe a shift towards the stochastic optimization model. Tight capacity limitations are problematic for the creation of large safety stocks and cause a decline in the deterministic performance. Especially for large demand variations, planning by means of the stochastic optimization model outperforms the deterministic approach.
Finally, Customer Type C updates demands frequently representing the highest level of demand uncertainty. For this customer type we see a superior performance of the stochastic planning approach across all utilizations and demand variations. Despite relying on large safety stocks, the deterministic model is not able to provide production plans that are suited for the uncertain environment. Customer Type C highlights the ability of the stochastic optimization model to explicitly incorporate information on uncertain future demand realizations during the planning procedure. Leveraging stochastic optimization in highly uncertain demand settings, as they are modelled by Customer Type C, can lead to cost reductions of over 50% compared to standard MRP.
{Moreover, the results show that for lower resource utilizations it is advantageous to assume production quantities as flexible within the stochastic optimzation model, while for higher utilizations a more conservative approach of assuming them to be fixed is beneficial. This can be reasoned by the fact that assuming quantities as fully flexible can be overly optimistic. In the rolling horizon setting, quantities are actually decided one period after the other, meaning that in practice the assumption of full flexibility is only partially true. In combination with high resource utilization this is problematic because the model underestimates the tight resource restrictions.}
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In order to evaluate the proposed simulation-optimization framework in larger settings and verify its effectiveness in a more realistic environment, we propose an additional more complex multi-item multi-stage production system. Figure (ref) shows the BOM for the new production system consisting of 8 end items and 4 components produced on 3 different resources.
First, we again investigate the number of periods $\tilde{T}$ up to which production orders in the stochastic optimization model are assumed to be fixed. Since the larger production system includes a more complex BOM structure with multiple items sharing each of the three resources the most suitable value for $\tilde{T}$ might differ from the small production system. In order to evaluate the impact of this parameter on the performance of the stochastic model on this use case, we perform experiments with $\tilde{T} \in \{1,\ldots,12\}$ for Customer Type C with medium demand variation of 7.5%. The included number of scenarios in the stochastic model is again 30. For the more complex production system we test resource utilizations of 80%, 85% and 90%, since higher utilizations are hard to handle in this production setting. We report the respective results in Table (ref), providing cost values for MRP, the deterministic and stochastic optimization models, each with the respective best found parameters. For the stochastic model, we report the results for the default value $\tilde{T}$=12, as well as for the best found $\tilde{T}$. Table (ref) first shows results for the case where no safety stocks are applied in order to put the results including safety stocks into perspective. This is done because the task to identify the best safety stock is not trivial and it might be challenging in reality to decide on a suitable safety stock.
We observe that the stochastic optimization model clearly outperforms both, MRP and the deterministic model, in the case of no safety stocks. In fact, applying the deterministic model without safety stock leads to significantly larger costs, also compared to MRP. This is because MRP can compensate the lack of safety stock by adjusting the lot size policy. Applying the stochastic model in this case, leads to cost improvements between 35% and 54% compared to MRP.
The deterministic model benefits significantly from considering safety stocks. When choosing the most suitable safety stock, the deterministic approach outperforms MRP. When choosing a sufficient safety stock level, the performance of the deterministic model is similar to the stochastic optimization model and can lead to cost reductions of up to 65% compared to standard MRP. It is even able to outperform the stochastic model by around 6% in certain cases. The stochastic model, however, relies on significantly smaller safety stock levels, by dynamically creating safety buffers to counter demand variations. This can be beneficial in cases where large safety stocks should be avoided or efficiently determining the most suitable safety stock is difficult. When it comes to the most suitable choice of $\tilde{T}$ we observe the same outcome as for the small production system. For low resource utilization ($80\%$) it is beneficial to keep production quantities flexible ($\tilde{T}=1$), while for increasing utilization it is more suitable to consider fixed production quantities in the majority of the periods ($\tilde{T}=7$ for $85\%$ and $\tilde{T}=10$ for $90\%$). The practical use case evaluation has shown that using stochastic optimization models, as well as deterministic models with suitable safety stock, in a rolling horizon manner reduces overall costs by 31%-65%compared to standard MRP when facing production planning with fluctuating customer demands and stochastic setup times. A specifically interesting finding here is, that whenever safety stocks should be kept low, or might even be restricted based on storage limitations, the stochastic optimization provides a better performance than the deterministic one.}
In this work, we combine a {two-stage} stochastic optimization model for multi-item multi-echelon capacitated lot sizing with discrete-event simulation in order to generate a rolling horizon production planning framework. Making use of stochastic optimization allows to explicitly consider customer demand fluctuations in the lot sizing process. {Within the stochastic model we introduce a parameter that allows to consider the possibility to flexibly adjust production quantities in later periods and hence more accurately represents the underlying decision process.} We compare the stochastic planning approach to a deterministic optimization model using expected demands, as well as to a standard MRP approach. We evaluate the different planning methods by means of the developed discrete-event simulation environment and report the resulting overall costs.
From a managerial perspective, we conclude that the standard MRP approach is sufficient in case of production environments with low resource utilization and customer demands that underlie almost no uncertainty, meaning that the realized demands correspond to the forecasted values. In case of tight resource restrictions, the MRP approach is unable to anticipate capacity bottlenecks, caused by irregular demand patterns, even though demands do not change in the short term. For this scenario the use of optimization approaches helps to smooth the utilization of resources and counteracts the occurrence of bottlenecks.
In situations where customers change their forecasted demands last minute, standard MRP is not competitive with the use of optimization models. While solving a deterministic model using safety stocks works well for scenarios with low utilization, its effectiveness diminishes in situations with tight resource capacities. In contrast, using stochastic optimization takes into account demand uncertainty, creating a bridge to more effective production plans that remain robust even under high levels of utilization. This is especially true if customers frequently update their demand forecasts, implying a high level of uncertainty for future demands. {Regarding the choice of the most suitable approach to model flexibility in production quantities, we summarize that for lower resource utilizations, e.g. 85%, it is beneficial to model quantities fully flexible, while for increasing utilization, e.g. 95% and 98%, it is advantageous to use a more conservative modelling approach, assuming a large part of the quantities to be fixed.} The evaluation of a more complex production system sheds light on the effect of safety stocks in combination with the optimization approaches. Here, the deterministic optimization shows a comparable performance to the stochastic optimization (or even better) if high safety stocks are allowed. However, the stochastic optimization significantly outperforms the deterministic one if no or only low safety stocks are allowed.
For future work, we plan to replace the two-stage stochastic optimization approach with a more advanced multi-stage model that explicitly takes the possibility for adjustments in later periods into account. Moreover, we plan to evaluate more complex production systems by investigating the impact of the number of different items and production levels.
This research was funded in whole, or in part, by the Austrian Science Fund (FWF) [10.55776/P32954]. For the purpose of open access, the author has applied a CC BY public copyright licence to any Author Accepted Manuscript version arising from this submission.
The data that support the findings of this study are available in Zenodo at 10.5281/zenodo.10686729 upon reasonable request to Wolfgang Seiringer.