Stabilizedcycle strategy for a multiitem, capacitated, hierarchical production planning problem in rolling schedules
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Abstract
Little research has been done on hierarchical production planning systems (HPPS) in the context of rolling schedules with servicelevel constraints. Here, we adapt the stabilizedcycle strategy, which has initially been created for the master planning level (Meistering and Stadtler in Prod Oper Manag 26:2247–2265, 2017), to a twolevel, multiitem, capacitated (shortmediumterm) HPPS with demand uncertainty. For each planning level, we present extensions for mixedinteger programming models from literature (CLSPL, PLSP) and introduce anticipation functions, as well as linking constraints. In a computational study, we analyze the performance of the HPPS with different rolling schedule strategies: the periodbased, the orderbased, and the stabilizedcycle strategy. It turns out that the stabilizedcycle strategy dominates the periodbased strategy for all studied instances. For some instances, the stabilizedcycle strategy even dominates the orderbased strategy; while in remaining instances, the stabilizedcycle strategy provides nondominated solutions with a significant smaller downside deviation from servicelevel agreements and only a minor increase of costs.
Keywords
Stabilizedcycle Hierarchical planning Rolling schedules Demand uncertainty Fill rate1 Introduction
Many manufacturing companies use computerbased planning systems (e.g., advanced planning systems (APS), enterprise resources planning (ERP) or manufacturing resources planning (MRP II) systems) to support strategic, tactical, and operational decisions. Planning systems not only require consideration of interdependencies between several planning tasks, but also must incorporate data uncertainty to react flexibly to data changes. This helps to minimize costs and to ensure servicelevel agreements. Shortmediumterm production planning problems (e.g., lotsizing and scheduling in a maketostock environment) are especially susceptible to demand uncertainty (e.g., lastminute canceled or rushed orders). However, current commercial mixedinteger programming (MIP) solvers cannot provide (even near) optimum solutions for such a problem in a total model, while successively solved partial models ignore most interdependencies between decisions. In a compromise between interdependencies and variety of decisions in a model, hierarchical planning systems (HPS) are used to meet requirements of shortmediumterm production planning problems (Fleischmann et al. 2015).
The impact of mediumterm planning decisions on actual costs and service levels of the shortterm planning level is rarely studied for a hierarchical production planning system (HPPS) facing demand uncertainty and executed in rolling schedules. Here, we define an HPPS as the compilation of an hierarchical planning concept in a software system including mathematical (e.g., MIP) models, coordination mechanism and (optional) organizational specifications for implementation (e.g., rolling schedules). This article presents an HPPS solving multiitem, capacitated production planning and singlemachine scheduling problems applied in rolling schedules. The HPPS aims to reach minimal costs and minimal downside deviations of actual service levels from target service levels at the end of a finite reporting period (i.e., 1 year). A potential application area of the HPPS might be the consumer goods industry. However, models within the paper are idealized standard models which can be adapted in straightforward manner since these are linear MIP models.
In Meistering and Stadtler (2017), a new strategy named “stabilizedcycle” has been presented to better cope with demand uncertainty in singlelevel, mediumterm production planning problems. In this article, we extend their work by additionally considering a shortterm production planning problem resulting in a short–mediumterm HPPS. As in Meistering and Stadtler (2017), the servicelevel definition used is the productspecific fill rate [e.g., the proportion of a product’s demand directly fulfilled from stock (Minner and Transchel 2010)], which must be reached at the end of a finite reporting period. Such a servicelevel definition is in line with findings from a theoretical (Thomas 2005) and a practical (Wieland et al. 2012) point of view.
The paper is structured as follows. Section 2 includes a literature review of planning strategies for planning systems considering demand uncertainty and HPPS. In Section 3, the concept of multiitem, capacitated HPPS is presented. The MIP models used within the HPPS are introduced in Sect. 4. Section 5 contains a computational study and Sect. 6 summarizes results and provides an outlook for future research.
2 Literature review
According to Bookbinder and Tan (1988), three planning strategies exist for dealing with demand uncertainty in production planning systems: the static uncertainty strategy, the static–dynamic uncertainty strategy and the dynamic uncertainty strategy. In the static uncertainty strategy, all setup and lotsize decisions are made once at the beginning of the planning interval. All decisions in the planning interval are realized and cannot be revised later. In the static–dynamic uncertainty strategy, setup decisions are made at the beginning of the planning interval, while lotsize decisions are made at the beginning of each period, when the initial inventory is known. In the dynamic uncertainty strategy, only decisions concerning the current period are made considering the current period’s information. This strategy provides farfromoptimum solutions, especially if the ratio between setup and inventoryholding costs is high (Bookbinder and Tan 1988).
Rolling schedules are an alternative to better cope with data uncertainty and are widely used in APS, ERP or MRP IIdriven production planning (Stadtler and Fleischmann 2012; Fleischmann et al. 2015). In rolling schedules a plan is generated for a finite planning interval, but only decisions for periods in the frozen horizon are realized (Stadtler et al. 2012). Decisions for periods beyond the frozen horizon are preliminary and can be revised later. After the replanning interval has elapsed, information is updated and a new plan is generated. The determination of the frozen horizon length can either be period, order or servicelevel based (Meistering and Stadtler 2017). Thus, there are at least three rolling schedule strategies: the periodbased, the orderbased and the stabilizedcycle strategy. In the periodbased strategy the frozen horizon is set to a given number of periods (often one period), while the frozen horizon in the orderbased strategy is product specific and set to a given number of order cycles (Sridharan and Berry 1990). Note that, the period and the orderbased strategy can be regarded as special cases of the static uncertainty strategy where only the first periods/order cycles in the planning horizon are fixed. In the stabilizedcycle strategy, no fixed frozen horizon needs to be determined in advance (Meistering and Stadtler 2017). However, a products current order cycle is kept as long as the service level is under control. Meistering and Stadtler (2017) show that, for a singlelevel, multiitem, capacitated production planning system with demand uncertainty, all of them are superior to the static uncertainty strategy. Hence, we use rolling schedules as one element of the HPPS.
APS use an HPS architecture to support strategic, tactical, and operative decisions. According to Schneeweiss (2003), an HPS consists of at least two linked planning levels with one subproblem each: a top and a base level. Commonly, an anticipated base level is used at the top level to predict a possible baselevel reaction, which in turn influences toplevel decisions. While the top level instructs the base level (top–down signal), the base level considers the instructions and may provide reactions to the top level (bottom–up influence). After final decisions are made at the base level, these are implemented in the object system (e.g., the shop floor). Subsequently, expost feedback is provided to the planning system. According to Fleischmann and Meyr (2003), an HPPS is characterized by three major features (top–down): an increasing level of detail, a decreasing planning horizon and an increasing planning frequency.
The first HPPS has been presented in Hax and Meal (1975). Here, the authors divide all production planning tasks into separate levels. The coordination of isolated levels is defined by instructions. Rolling schedules are used for replanning, assuming certainty of demand. Further research on this type of HPPS has been done, for example by Bitran and Hax (1977), Bitran et al. (1981) and Bitran et al. (1982). According to Gelders and Van Wassenhove (1982), it is not sufficient to solve problems for each level individually and that effective coordination mechanisms are needed to obtain good results for HPPS. A large amount of research has been done for HPPS implemented in MRP systems (e.g., Andersson et al. 1981; Axsaeter and Joensson 1984; Meal et al. 1987). However, shortcomings of MRP systems (e.g., noncompliance with capacity constraints) are well known. Thus, we omit an extensive review of articles dealing with HPPS in MRP environments. Instead, we focus on HPPS as an APS or an add on to MRP II, ERP systems, which are more relevant to our production planning problem. More general literature reviews of theoretical and operational HPS application can be found in Steven (1994) and MacKay et al. (1995).
A detailed analysis of HPPS is documented in Stadtler (1988). Here, the architecture of HPPS is analyzed, regarding its practical application to lotsizing problems. Focusing on lotsizing problems, Stadtler (1988) develops the concept of effective lotsize demand. In a computational study including rolling schedules and partial demand uncertainty, Stadtler (1988) shows that effective lotsize demands increase the precision of estimating inventory levels. This results in more suitable directives for shortterm planning and, in turn, decreases costs.
Fransoo (1993) discovers that traditional uncapacitated lotsizing algorithms result in cycle times which are in conflict with capacity concerns. Hence, the author develops a twolevel hierarchical model for the flow process industry under the assumption of uncertain, stationary demands. In the HPPS, cycle time and lotsize decisions are considered at the top level, since they determine the longterm output. At the base level, shortterm production order scheduling is used to control the service levels of individual products. Note that no costs minimization takes place at the base level. This helps to stabilize operational schedules. In a case study, Fransoo (1993) illustrates the application of the HPPS to a realworld company, but no general evidence nor any major profit increase caused by the HPPS is mentioned.
Relatively new research on HPPS dealing with uncertainties is documented in Gebhard (2009). A robust twolevel HPPS, based on the structure of an APS, implemented in rolling schedules is developed to better cope with uncertainties. At the top level, a robust, aggregated model is used to determine longterm production plans and capacities per month. At the base level, a robust model formulation of the capacitated lotsizing problem defines weekly lot sizes. Sequencing and demand during a week is not considered. In a computational study, Gebhard (2009) shows that the robust HPPS is advantageous to a deterministic HPPS with static safety stocks. However, due to high complexity of scenariobased robust optimization models, the study only considers three replans for each instance. Thus, it demonstrates that robust optimization can be applied to HPPS, but generalizations are limited.
Seipl (2009) studies an HPPS which, in contrast to Gebhard (2009), consists of deterministic MIP models in a threelevel HPPS. While the top level considers lotsizing decisions per month for a planning interval of a year, the medium level tackles lotsize decisions for weekly periods for a planning interval of 12 weeks. To cope with demand uncertainty, Seipl (2009) uses static safety stocks. Rolling schedules with periodbased frozen horizons are used at the medium level. Thus, weekly lotsize decisions can be adjusted to actual inventories. The base level dedicates machine scheduling for frozen periods of the medium level. Instructions from upper to lower levels are provided through a top–down signal. Seipl (2009) studied interdependencies between different input (e.g., production cycle lengths, penalty costs, safety stocks) and output (e.g., costs, mean lot sizes, fill rates) parameters. In a computational study, Seipl (2009) finds out that actual fill rates at the end of a finite reporting period vary largely with a large portion of downside deviations. This finding remains consistent, irrespective of input parameters, and leads to a low level of trust in HPPS.
Guimaraes (2013) uses a hierarchical approach similar to an APS to solve production planning problems in the beverage industry. The HPPS is structured in a short–mediumterm planning (lotsizing and scheduling) problem and a longterm planning (capacity coordination) problem. While each of the decision levels is studied separately, a holistic analysis of the HPPS is omitted. The short–mediumterm planning problem, including sequencedependent setup costs, is tackled through a newly formulated model of the capacitated lotsizing and scheduling problem, with sequencing decisions (CLSD) implemented in rolling schedules with periodbased frozen horizons. The model integrates short and mediumterm production planning in a single problem with a detailed plan at the beginning of the planning interval and a raw estimation of capacity utilization and costs in later periods. Demand is assumed to be seasonal and known with certainty. Hence, beside the endofhorizon effect in rolling schedules, demand uncertainty is excluded.
A comparison of an integrated production planning system (IPPS) to an HPPS is made by Vogel et al. (2017). The studied HPPS considers aggregate production planning and master production scheduling (MPS), applied in rolling schedules. Thus, machine scheduling is not considered by Vogel et al. (2017). While demand is assumed to be known with certainty at the beginning of the planning interval, it is assumed to be uncertain in later periods. In a computational study, Vogel et al. (2017) demonstrate that the IPPS is superior to the HPPS for the studied instances.
The literature review shows that much research has been done in the field of HPPS. However, most of the studied HPPS only use optimization models at the top level, while baselevel decisions result from disaggregating toplevel decisions. While Guimaraes (2013) ignores demand uncertainty, most of the studied HPPS assume certain demand at the beginning of a planning interval and uncertain demand in later periods (e.g., Stadtler 1988; Vogel et al. 2017). Only few HPPS consider demand uncertainty at the level at which it occurs: the shortterm production planning and scheduling level (e.g., Fransoo 1993; Seipl 2009). However, the HPPS of Fransoo (1993) can only be applied to planning problems with stationary demands and assumes lost sales. Thus, nonfulfilled demand from the past does not influence current production planning decisions. The HPPS of Seipl (2009) can be applied to production planning problems with nonstationary demand. Furthermore, Seipl (2009) assumes that nonfulfilled demand from the past is backordered and must be fulfilled with the next goods receipt, prior to current period demands. Thus, the production planning problem considered in this article is closely related to the one studied in Seipl (2009).
To the best of our knowledge, no HPPS exists that minimizes costs and downside deviations of fill rates for a multiitem, capacitated, production planning and scheduling problem with demand uncertainty.The novelty of our paper is that we show how standard MIP model formulations (e.g., the CLSPL and the PLSP) have to be adapted to become an HPPS that keeps setup and holding costs low while taking into account fill rate constraints evaluated over a given reporting period. While the toplevel planning tasks have already been addressed in Meistering and Stadtler (2017) in the context of the CLSP with rolling schedules, it is not clear if the stabilizedcycle strategy also works in a hierarchical setting. Especially, we now assess the fulfillment of the fill rate constraints at the bottom planning level based on daily demands. Furthermore, we present an extensive computational study of a hierarchical production planning system with service level constraints in rolling schedules.
3 A capacitated hierarchical production planning system for rolling schedules
In this section, we introduce a twolevel HPPS for the capacitated, multiitem production and scheduling problem. According to the proposal of Stadtler (1988), the section is structured as follows. First, the planning problem is analyzed and decomposed into subproblems (Sect. 3.1). Second, planning models and planning strategies for each subproblem are proposed (Sect. 3.2). Third, a mechanism for coordinating subproblems is introduced (Sect. 3.3).
3.1 Analysis and decomposition of the planning problem
To analyze the capacitated, multiitem production planning problem and to decompose it into subproblems, we make use of the supply chain planning matrix presented in Stadtler and Fleischmann (2012). The matrix classifies and decomposes supply chain planning tasks subject to their business functions (e.g., procurement, production, distribution or sales) and according to their planning horizons (e.g., long, medium or shortterm). Production planning and scheduling tasks are classified in the production section with a medium and a shortterm planning horizon. The aim of these tasks is to find lot sizes and production sequences for machines at the lowest level of data aggregation, so they can be used for shop floor control. Thus, the lotsizing and machine scheduling problem defines the subproblem of the base level in our HPPS. However, lotsizing and machine scheduling decisions depend on available machine capacities, which are defined at the master planning level. Additionally, seasonal demands usually cannot be foreseen at the base level due to the shortterm planning horizon. At the top level, decisions regarding available machine capacities and seasonal inventories are made with a mediumterm planning horizon. These decisions are provided to the base level.
3.2 Planning models and strategies
After subproblems are defined, planning models and strategies for the subproblems must be chosen. According to Suerie and Stadtler (2003), planning models are differentiated regarding the characteristics of periods. If the length of a period is chosen so that several setup activities can take place in a single period, the model is characterized as a bigbucket model. The period length of a smallbucket model is rather short, permitting at most one setup per period. Subsequently, periods of a bigbucket (smallbucket) model are named macroperiods (microperiods). Here, we distinguish between planning strategies for replanning at the beginning of a macroperiod and for replanning during a macroperiod. While the planning strategy for replanning at the beginning of a macroperiod concerns both levels, the planning strategy for replanning during a macroperiod only affects the base level. A rolling schedule strategy is used for replanning at the beginning of a macroperiod and the static–dynamic uncertainty strategy is used for replanning during a macroperiod.
For determining production sequences on a single capacitated machine and defining the production volume per product and microperiod for a shortterm planning interval, we propose the proportional lotsizing and scheduling problem (PLSP)—a smallbucket model—presented in Haase (1994). The PLSP determines the sequence and lot sizes for multiitems on a single, capacitated machine while considering setup and inventoryholding costs.
Using the CLSPL at the top level, some sequencing decisions of the PLSP at the base level are anticipated. This makes the CLSPL a predestined model in conjunction with the PLSP in an HPPS. However, using the standard CLSPL model formulation of Suerie and Stadtler (2003) is insufficient in the context of rolling schedules and demand uncertainty. Hence, we extend it in accordance with the proposals of Meistering and Stadtler (2017) for the CLSP model formulation and add further anticipation functions. Following the implementation of decisions of the PLSP in the object system, these are realized and current data are provided to the planning system by expost feedback prior to each replan in rolling schedules, see Fig. 1.
3.3 Coordination of models and levels
Data aggregation of the capacitated, multiitem hierarchical production planning system
Top level  Base level  

Period  Macro  Micro 
Planning interval  \(T = \{t \vert t = 1,\ldots , {\bar{t}}\}\)  \(S = \{s \vert s = 1,\ldots , {\bar{s}}; \text{ with } t^{{\rm B}} \le {\bar{t}}\}\) 
Products  \(j = 1,\ldots ,{\bar{j}}\)  
Setup costs  Per setup activity  
Holding costs  Per unit and period t  Per unit and period s 
Capacity  Units per period t  Units per period s 
Demand  At the end of each period t  At the end of each period s 
Demand variation  Per period t  Per period s 
Safety stocks  Per cycle length in macroperiods \(\tau\)  Per cycle length in microperiods \(\tau\) 
4 Mathematical optimization models
In this section, mathematical optimization models used within the HPPS are introduced. Section 4.1 presents an extended CLSPL model formulation and Sect. 4.2 introduces an extended PLSP model formulation.
4.1 Top level: extended CLSPL model formulation
In the following section, we extend the CLSPL model formulation of Suerie and Stadtler (2003) as presented in Meistering and Stadtler (2017). First, we incorporate safety stocks depending on the production cycle length (subsequently called dynamic) to better cope with demand uncertainty. Therefore, safety stocks are determined internally by the model, corresponding to the actual production cycle length, which might vary in the planning interval. Note that safety stocks for different cycle lengths must be provided to the model as data. Thus, they must be determined in advance for every product j, period t and production cycle length \(\tau ^{\mathrm{T}}\). Second, the looking beyond the planning horizon approach of Stadtler (2000) is applied to the model formulation to reduce the truncated horizon effect. Third, softcapacity constraints are used to ensure feasibility, even if capacity is insufficient. As the complexity of the extended CLSPL is still high, we split the planning interval to reduce complexity. Only the first part (\(t=1,\ldots ,t^{\mathrm{L}}\)) considers linked lot sizes, while \(t=t^{\mathrm{L}}+1,\ldots ,{\bar{t}}\) neglects them. Thus, the toplevel model is a mixture of an extended CLSPL and an extended CLSP.
 \(J = \{j \vert j = 1,\ldots , {\bar{j}}\}\)

Set of products
 \(T = \{t \vert t = 1,\ldots , {\bar{t}}\}\)

Set of macroperiods
 \({\text{bonus}}_{j,t,t+\tau }\)

(Negative) bonus payments for the last production cycle of each product j
 \(b^{\mathrm{T}}_{t}\)

Production capacity in macroperiod t
 \(d^{\mathrm{T}}_{j,t}\)

Expected demand of product j in macroperiod t
 \({\text{hc}}^{\mathrm{T}}_{j}\)

Inventoryholding costs per macroperiod t and product unit j
 \(i^{\mathrm{T}}_{j,0}\)

Initial inventory of product j
 \(\kappa _{j}\)

Production coefficient of product j
 \(l^{\mathrm{T}}_j\)

Last setup period of product j prior to the planning interval
 \(m^{\mathrm{T}}_{j,t}\)

Sufficiently large number for lotsize decisions (e.g., \((b_{t} + z^{\mathrm{T}}_{\mathrm{max}}) \cdot \kappa _j \;\;\; \forall j \in J, t \in T\))
 \({\text{pc}}^{\mathrm{T}}_{t}\)

Costs for an additional capacity unit in macroperiod t
 \({\text{sc}}_{j}\)

Setup costs of product j
 \({\text{ss}}^{\mathrm{T}}_{j,t,t+\tau }\)

Safety stocks for a cycle beginning in macroperiod t and a length of \(\tau\) macroperiods
 \(\tau ^{\mathrm{Tmax}}_{j}\)

Externally given productspecific upper bound for the TBO in macroperiods
 \(\tau _{j,t}^{\mu }\)

Myopic optimal TBO of product j in macroperiod t (e.g., determined by Groff’s heuristic)
 \(t^{\mathrm{L}}\)

The last macroperiod within the planning interval with linked lot sizes (\(t^{\mathrm{L}}\le {\bar{t}}\))
 \(T^{\mathrm{max}}\)

The last macroperiod considered beyond the planning horizon
 \(w^{\mathrm{T}}_{j,1}\)

Initial setup status, 1 if the resource is setup for product j, 0 otherwise
 \(z^{\mathrm{T}}_{\mathrm{max}}\)

Maximal additional capacity per macroperiod
 \({\text{BL}}^{\mathrm{T}}_{j,t}\)

Authorized backlogs of product j at the end of macroperiod t
 \(C^{\mathrm{T}}_{t}\)

Additional capacity units in macroperiod t
 \(I^{\mathrm{T}}_{j,t}\)

Physical inventory of product j at the end of macroperiod t
 \(QQ^{\mathrm{T}}_{j,t}\)

1, if period t is a singleitem production period of product j, 0 otherwise
 \(V^{\mathrm{T}}_{j,t,t+\tau }\)

1, if a setup for product j is scheduled in macroperiod t while the next setup is scheduled in macroperiod \(t+\tau\), 0 otherwise
 \(W^{\mathrm{T}}_{j,t}\)

1, if the setup state of product j is carried over from macroperiod \(t1\) to macroperiod t, 0 otherwise
 \(X^{\mathrm{T}}_{j,t}\)

Lot size of product j in macroperiod t
 \(Y^{\mathrm{T}}_{j,t}\)

1, if a setup for product j takes place in macroperiod t, 0 otherwise
 \(Z^{\mathrm{T}}\)

Total costs
4.2 Base level: extended PLSP model formulation
Next, we extend the PLSP model formulation of Haase (1994) by dynamic safety stocks and soft capacity constraints.
 \(J = \{j \vert j = 1,\ldots , {\bar{j}}\}\)

Set of products
 \(S = \{s \vert s = 1,\ldots , {\bar{s}}\}\)

Set of microperiods
 \(b^{\mathrm{B}}_{s}\)

Production capacity in microperiod s
 \(d^{\mathrm{B}}_{j,s}\)

Expected demand of product j in microperiod s
 \(\epsilon\)

Small negative value
 \({\text{hc}}^{\mathrm{B}}_{s}\)

Inventoryholding costs per day and product unit j
 \(i^{\mathrm{B}}_{j,0}\)

Initial inventory of product j
 \(i^{\mathrm{B}}_{j,{\bar{s}}}\)

Final inventory of product j
 \(\kappa _{j}\)

Production coefficient of product j
 \(l^{\mathrm{B}}_j\)

Last setup of product j prior to the planning interval in microperiods
 \(m^{\mathrm{B}}_{j,s}\)

Sufficiently large number for production volume in microperiod s for product j (e.g., \((b^{\mathrm{B}}_{t}+z^{\mathrm{B}}_{\mathrm{max}})) \cdot \kappa _j \;\;\; \forall j \in J, t \in T\))
 \(m^{\mathrm{BL}}_{j,s}\)

Sufficiently large number for the backlog of product j in microperiod s
 \(m^{\mathrm{I}}_{j,s}\)

Sufficiently large number for the inventory of product j at the end of microperiod s
 \({\text{pc}}^{\mathrm{B}}_{s}\)

Costs for an additional capacity unit in microperiod s
 \({\text{sc}}_{j}\)

Setup costs of product j
 \({\text{sf}}^{\mathrm{T}}\)

Number of microperiods within a macroperiod
 \({\text{sf}}^{\mathrm{D}}\)

Number of microperiods within a day
 \({\text{ss}}^{\mathrm{B}}_{j,s,s+\tau }\)

Safety stocks for a cycle beginning in microperiod s and a length of \(\tau\) microperiods
 \(\tau ^{\mathrm{Bmax}}_{j}\)

Externally given productspecific upper bound for the TBO in microperiods
 \(w^{\mathrm{B}}_{j,0}\)

Initial setup state, 1 if resource is setup for product j, 0 otherwise
 \(z^{\mathrm{B}}_{\mathrm{max}}\)

Maximal additional capacity per microperiod
 \({\text{BL}}^{\mathrm{B}}_{j,s}\)

Authorized backlogs of product j at the end of microperiod s
 \(C^{\mathrm{B}}_{s}\)

Additional capacity units in microperiod s
 \(I^{\mathrm{B}}_{j,s}\)

Physical inventory of product j at the end of microperiod s
 \(N^{\mathrm{B}}_{j,t}\)

Indicates whether the inventory or the backlog is positive for product j at the end of microperiod s (= 1) or not (= 0)
 \(V^{\mathrm{B}}_{j,s,s+\tau }\)

1, if a setup for product j is scheduled in microperiod s while the next setup is scheduled in microperiod \(t+\tau\), 0 otherwise
 \(W^{\mathrm{B}}_{j,s}\)

1, if the setup state of product j is carried over from microperiod s to microperiod \(s+1\), 0 otherwise
 \(X^{\mathrm{B}}_{j,s}\)

Production volume of product j in microperiod s
 \(Y^{\mathrm{B}}_{j,s}\)

1, if a setup activity for product j takes place in microperiod s, 0 otherwise
 \(Z^{\mathrm{B}}\)

Total costs
4.3 Anticipation functions for the CLSPL
The main coordination problem between the planning levels of the HPPS addressed here is converting the products’ lot sizes into a feasible production sequence with minimal costs. While the PLSP takes the production sequence into account, the CLSPL—except for the first and the last scheduled products—neglects the production sequence within a macroperiod. The first and the last scheduled products are determined by the setup state carryover variables. All other products that are planned to be produced in a macroperiod are scheduled without any specific sequence.
As usual, the CLSPL assumes that demand is due at the end of a macroperiod. Hence, the production sequence during a macroperiod has no impact on demand fulfillment. However, the PLSP assumes that demand will be due at the end of a microperiod. Thus, the production sequence is important for demand fulfillment. Therefore, the aim of coordination is that CLSPL instructions provide the PLSP a sufficient degree of freedom for production sequencing. With a high degree of freedom, expected demands during a macro period can be fulfilled without multiple setup activities of one product in a macroperiod and/or without the need of additional capacity.
As there is often no further information on the demand pattern within a microperiod, we assume that demand in a macroperiod is evenly spread over its microperiods. This is an approximate anticipation function representing a compromise between the exactness of modeling the base level and the difficulty of modeling and solving the resultant model (along the lines of Schneeweiss 2003).
A high degree of freedom for production sequencing in the PLSP would be achieved by setting the minimal inventory for each product at the end of each macroperiod t such that it is sufficient to cover expected demands during the macroperiod \(t+1\). This leads to an identical number of setup activities in the CLSPL and the PLSP. However, inventoryholding costs and fill rates are unnecessarily high. Thus, a compromise between inventories and degree of freedom for production sequencing is needed to coordinate the CLSPL and the PLSP. We propose to add functions to the extended CLSPL model formulation for anticipating the (expected) minimal inventory for each product at the end of each macroperiod (\(t \le t^{\mathrm{L}}\)).
4.4 Linking constraints for the PLSP
4.5 Model adjustments for rolling schedules
In the periodbased strategy, the frozen horizon is set to a specific number of periods (Sridharan and Berry 1990). To gain high planning flexibility, the frozen horizon is usually set to the shortest possible number of periods for all products. According to Xie et al. (2003), it is beneficial to set the number of periods of the replanning interval equal to the number of frozen periods to reduce planning nervousness. With a replanning interval equal to the frozen horizon, \({\text{fh}}^{\mathrm{T}}_{j}\) has to be set to 0 for all products and replans.
In the orderbased strategy, the frozen horizon of product j is set to a number of production cycles (Sridharan and Berry 1990). Thus, for a multiitem lotsizing problem, the length of frozen horizons can differ between products. To enable replanning for each product in each replan, a replanning interval of one period is required. Hence, \({\text{fh}}^{\mathrm{T}}_{j}\) is defined as the last period of the \(X^{\mathrm{th}}\) production cycle if started in the first period, and is reduced by the number of periods of the replanning interval before every replan.
5 Computational study
We now analyze and compare the performance of the HPPS in rolling schedules with the stabilizedcycle, the orderbased and the periodbased strategy, in a computational study.
5.1 Test instances
Parameters of the data set attributed to Meistering and Stadtler (2017)
Seasonality factor, \({\text{sd}}^{\mathrm{T}}_{j,t}\)  Constant or seasonal 
Coefficient of variation of weekly demand, cv  0.2, 0.3 
Natural lengths of the production cycle \({\text{TBO}}_j\)  Mixed (2 / 3 / 5) 
Machine utilization (based on \(\mu _{j}\))  Low (70%), high (85%) 
Target fill rate, \(\beta ^{\mathrm{tar}}_j\)  95%, 98% 
The mean demand per week t over a reporting period is set to \(\mu ^{\mathrm{T}}_{j} = 1000\) for all products and instances. Here, instances with and without seasonal demand are studied. Given the seasonal factors (\({\text{sd}}^{\mathrm{T}}_{j,t}\)), the mean demand of product j in week t is \(d^{\mathrm{T}}_{j,t} = \mu ^{\mathrm{T}}_{j} \cdot {\text{sd}}^{\mathrm{T}}_{j,t}\), which is also the demand forecast for week t. At the base level, we assume that demand occurs at the end of each day. By assuming five days per week, we set the demand forecast of the last shift of a day to a fifth of \(d^{\mathrm{T}}_{j,w}\). All other shifts of a day presume a mean demand of zero. No bias and no updates of forecasts are considered. A forecast error only appears in the first period of a rolling schedule after lotsizing and sequencing decisions have been made (Meistering and Stadtler 2017). Given the coefficient of variation (cv), the daily standard deviation of the demand is \(\sigma ^{\mathrm{B}}_{j,s} = \sqrt{(d^{\mathrm{T}}_{j,w} \cdot {\text{cv}})^{2}/{\text{sf}}^{\mathrm{D}}}\). The realized demand \(d^{\mathrm{obs}}_{j,s}\) for product j at the end of the last shift of a day is calculated by a truncated normal distribution (\(\max \{0, {\text{NV}}(d^{\mathrm{B}}_{j,s}, \sigma ^{\mathrm{B}}_{j,s})\}\)).
Each instance is repeated 30 times (\({\bar{r}}=30\)). This is equivalent to a simulation of 30 years. Since every single year is taken as a sample (with identical initial inventories and setup states), we have 30 independent samples per instance. This should be large enough for statistical analysis (Sheskin 2011).
5.2 Rolling schedule strategies
Strategy settings used at the top level
Strategy  Top level  

Planning interval  Frozen horizon  Replanning  
Periodbased  12 periods (\(t^{\mathrm{L}} = 8\))  First week  First week 
Prderbased  12 periods (\(t^{\mathrm{L}} = 8\))  First cycle  First week 
Stabilizedcycle  12 periods (\(t^{\mathrm{L}} = 8\))  –  First week 
Strategy settings used at the base level
Strategy  Base level  

Planning interval  Frozen horizon for setups  Frozen horizon for lot sizes  Replanning  
Periodbased  60 shifts  15 shifts  Three shifts  Three shifts 
Orderbased  60 shifts  15 shifts  Three shifts  Three shifts 
Stabilizedcycle  60 shifts  15 shifts  Three shifts  Three shifts 
Objectives and constraints for top, base level and each planning strategy
The top level provides instructions to the base level for the first four weeks. Instructions generated by the CLSPL contain the available additional capacity per week, the target inventory for each product j at the end of the fourth week and allowed setup activities per week. For replans during a week, the static–dynamic uncertainty strategy is used in all strategies. Penalty costs for using a ’feasibility’ backlog (\({\text{BL}}^{\mathrm{FEA}}_{j,s}\)) in replans during a week are set to high costs (e.g., \({\text{pen}}^{\mathrm{FEA}}_j = {\text{pc}}^{\mathrm{B}}_s+1\)). This ensures that \({\text{BL}}^{\mathrm{FEA}}_{j,s}\) are only chosen by the model if there are no other means.
First, we study the rolling schedule strategy with periodbased frozen horizons. As can be seen from Table 3, only the first week’s decisions of the toplevel planning interval are fixed, while decisions for all other weeks are subject to revision. The replanning interval is set to the first week. In a replan at the base level at the beginning of a week, production volume and setup decisions are made for the upcoming 60 shifts. While all setup decisions within the first week are realized, the production volume might be revised by the daily replan. Note that setup activities are only allowed at the base level if they were planned by the top level.
Second, the orderbased rolling schedule strategy is considered. Here, productspecific, orderbased frozen horizons are used for freezing decisions at the top level. Therefore, a product’s frozen horizon is equal to the first production cycle length, if it started in the first period of the planning interval. Using the same replanning interval as the periodbased strategy, only decisions beyond a product’s frozen horizon may be revised in replans at the top level. The replanning strategy at the base level is similar to the one described in the periodbased strategy.
5.3 Computational results
The computational study has been executed on an Intel(R) Core(TM) i7 processor with a clock speed of 3.4 GHz, 16.0 GB RAM and 4 threads, using Windows 7 Professional and the same simulation environment as Meistering and Stadtler (2017). Models are implemented in XpressIVE (Version 1.24.12) and solved by the Xpress Optimizer (Version 29.01.10). The maximum computational time per replan and planning level is set to 100 s. While we observed that some replans at the beginning of a week do not yield optimal solutions within the computational time limit, all replans during a week reach optimal solutions. For the sake of completeness, mean optimality gaps per planning level for replans at the beginning of a week are presented in Tables 6, 7, 8 (i.e., CLSPL Gap and PLSP Gap). The optimization model of Tempelmeier (2011) is used in combination with the binary search heuristic of Manna and Waldinger (1987) to determine safety stocks.
Computational results of the periodbased strategy
Instance  \(c_{{\bar{r}}}\) ($)  \({\text{DD}}_{{\bar{r}}}\) (%)  CLSPL gap (%)  PLSP gap (%)  \(\overline{\text{ac}}\) (%)  

Constant  95/70/0.2  201,154.25  0.31  4.23  0.79  15.54 
95/85/0.2  204,176.70  0.10  5.60  1.10  19.24  
95/70/0.3  209,367.67  0.84  3.00  1.80  18.17  
95/85/0.3  216,325.25  0.56  4.33  4.28  20.44  
98/70/0.2  209,074.89  1.52  2.96  1.65  16.71  
98/85/0.2  215,259.95  0.84  4.25  1.78  17.36  
98/70/0.3  230,242.79  0.56  2.86  5.08  19.06  
98/85/0.3  238,826.72  0.86  4.40  8.17  21.85  
seasonal  95/70/0.2  202,963.49  0.13  3.91  0.75  18.23 
95/85/0.2  213,103.15  0.27  6.29  3.61  20.55  
95/70/0.3  211,191.49  0.55  2.98  1.48  19.39  
\(95/85/0.3^{*}\)  228,797.81  0.77  5.96  8.90  22.77  
98/70/0.2  210,908.06  1.07  3.02  1.49  18.45  
98/85/0.2  219,596.44  0.82  5.15  4.17  21.99  
98/70/0.3  229,411.40  0.89  2.94  5.21  21.68  
\(98/85/0.3^{*}\)  245,521.19  1.11  5.72  13.19  23.66  
Mean  217,870.08  0.70  4.23  3.97  19.69 
Computational results of the orderbased strategy
Instance  \(c_{{\bar{r}}}\) ($)  \({\text{DD}}_{{\bar{r}}}\) (%)  CLSPL gap (%)  PLSP gap (%)  \(\overline{\text{ac}}\)  

Constant  95/70/0.2  176,481.73  2.03  1.49  0.19  – 
95/85/0.2  176,928.68  2.38  2.70  0.21  –  
95/70/0.3  185,473.66  3.03  1.09  0.32  –  
95/85/0.3  185,843.00  2.81  2.14  0.55  –  
98/70/0.2  186,391.94  3.28  1.05  0.30  –  
98/85/0.2  188,963.68  2.69  2.14  0.45  –  
98/70/0.3  201,378.23  3.03  1.01  1.06  –  
98/85/0.3  205,845.48  2.73  2.43  2.55  –  
Seasonal  95/70/0.2  177,468.49  1.70  1.57  0.10  – 
95/85/0.2  181,717.08  2.29  3.95  2.30  –  
95/70/0.3  186,092.73  2.63  1.45  0.49  –  
\(95/85/0.3^{*}\)  195,414.52  3.27  4.15  5.75  –  
98/70/0.2  186,488.39  3.14  1.34  0.38  –  
98/85/0.2  194,255.53  3.09  3.57  3.73  –  
98/70/0.3  198,225.13  3.55  1.45  1.38  –  
\(98/85/0.3^{*}\)  212,108.09  3.49  4.28  9.52  –  
Mean  189,942.27  2.82  2.24  1.83  – 
Computational results of the stabilizedcycle strategy
Instance  \(c_{{\bar{r}}}\) [$]  \({\text{DD}}_{{\bar{r}}}\) (%)  CLSPL gap (%)  PLSP gap (%)  \(\overline{\text{ac}}\) (%)  

Constant  95/70/0.2  187,279.89  0.01  1.24  0.51  4.17 
95/85/0.2  193,948.93  0.02  2.55  1.09  4.50  
95/70/0.3  197,463.68  0.03  0.96  0.92  8.08  
95/85/0.3  202,662.12  0.05  2.16  2.26  6.40  
98/70/0.2  199,145.55  0.10  1.03  0.79  6.22  
98/85/0.2  203,940.04  0.14  2.12  1.80  5.85  
98/70/0.3  212,650.43  0.31  0.92  2.11  7.48  
98/85/0.3  217,841.00  0.21  2.49  4.65  6.99  
Seasonal  95/70/0.2  187,996.91  0.00  1.49  0.5  4.27 
95/85/0.2  196,113.42  0.00  3.84  4.24  3.63  
95/70/0.3  197,717.48  0.09  1.37  1.37  7.46  
\(95/85/0.3^{*}\)  211,708.52  0.09  4.00  7.56  5.93  
98/70/0.2  200,313.35  0.14  1.30  1.03  5.88  
98/85/0.2  209,525.61  0.13  3.40  4.43  5.55  
98/70/0.3  213,638.86  0.28  1.34  2.83  8.77  
\(98/85/0.3^{*}\)  231,382.41  0.32  4.04  11.50  8.21  
mean  203,958.01  0.12  2.14  2.97  6.21 
As shown in Tables 6, 7, 8, the stabilizedcycle strategy is the best strategy subject to \({\text{DD}}_{{\bar{r}}}\) for all instances. The orderbased strategy is the best strategy regarding \(c_{{\bar{r}}}\) for all instances. Both, the periodbased (\(\triangle \overline{c_{{\bar{r}}}} = 14.70\%\)) and the stabilizedcycle strategy (\(\triangle \overline{c_{{\bar{r}}}} = 7.38\%\)) result in higher costs compared to the orderbased strategy. However, the orderbased strategy does not ensure that the target fill rate is at least reached on average over all 30 repetitions and products. This results in rather high \({\text{DD}}_{{\bar{r}}}\). The stabilizedcycle strategy leads to lower \(c_{{\bar{r}}}\) and less \({\text{DD}}_{{\bar{r}}}\) than the periodbased strategy for all instances. Thus, the stabilizedcycle strategy is superior to the periodbased strategy. However, the evaluation of strategies is bicriteria meaning that it is impossible to trade off an increase of \(c_{{\bar{r}}}\) and a decrease of \({\text{DD}}_{{\bar{r}}}\). Hence, no clear conclusion regarding the stabilizedcycle and the orderbased strategy can be drawn without preference information of the decisionmaker.
Orderbased (\(\beta ^{\mathrm{tar}}_j = 0.98\)) vs. stabilizedcycle (\(\beta ^{tar}_j = 0.95\))
Instances  Orderbased \(\beta ^{\mathrm{tar}}_{j} = 0.98\)  Stabilizedcycle \(\beta ^{\mathrm{tar}}_{j} = 0.95\)  

\(c_{{\bar{r}}}\) [$]  \({\text{DD}}_{{\bar{r}}}\) (%)  \(c_{{\bar{r}}}\) ($)  \({\text{DD}}_{{\bar{r}}}\) (%)  
Constant  \(\cdot\)/70/0.3  201,378.23  1.06  197,463.68  0.03 
\(\cdot\)/85/0.3  205,845.48  0.92  202,662.12  0.05  
Seasonal  \(\cdot\)/70/0.3  198,225.13  1.44  197,717.48  0.09 
\(\cdot\)/85/0.3  212,108.09  1.42  211,708.52  0.09 
If the stabilizedcycle strategy is used, almost all products reach the target fill rate at the end of each reporting period. In only seven out of the 30 years, one product out of six misses the target fill rate by only a narrow margin. This results in the lowest mean downside deviations (\({\text{DD}}_{{\bar{r}}} = 0.03\%\)) among the three strategies. While the stabilizedcycle strategy results in lower mean costs than the periodbased strategy (\(\triangle c_{{\bar{r}}} = \,5.69\%\)), it leads to higher mean costs than the orderbased strategy (\(\triangle c_{{\bar{r}}} = 6.46\%\)) for the instance \(95\%/70\%/0.3\) with constant demand.
6 Conclusion
Based on the characterization of an HPPS proposed in Fleischmann and Meyr (2003), we developed an HPPS to tackle a capacitated, multiitem, shortmediumterm production planning and scheduling problem. In contrast to most studied HPPS, we address the lotsizing and machine scheduling decisions, assuming demand uncertainty, at the level at which it occurs: the shop floor. The aim of the HPPS is to provide production plans with a negligible small number of violations from a given target fill rate in a reporting period, while keeping setup and holding costs relatively low.
The HPPS is based on two levels: the master planning problem (top level) and the production planning and scheduling problem (base level). At each level, a deterministic MIP model formulation is used to solve the respective planning problem. While an extended CLSPL is used to determine the availability of additional capacity and endofperiod inventories at the top level, an extended PLSP is used to simultaneously define lot sizes and production sequences at the base level. Suitable anticipation functions for the CLSPL and linking constraints for the PLSP are provided. Using those functions and constraints, an appropriate coordination between the models is reached. Both extended model formulations are in line with the extensions presented in Meistering and Stadtler (2017). While rolling schedules are used as the planning strategy for replans at the beginning of each macroperiod, the static–dynamic uncertainty strategy is used for replans during the macroperiod at the base level.
The performance of three rolling schedule strategies is studied when being applied to the HPPS. The performance of the HPPS has been evaluated based on 16 instances, considering a reporting interval of 48 periods and six products, with and without seasonal demands. Most results from the mediumterm problem observed in Meistering and Stadtler (2017) have been confirmed for the HPPS.
As in the mediumterm problem, the orderbased strategy results in the lowest mean costs and the largest downside deviation. In contrast to the results of the mediumterm problem, in which the target fill rate could at least be reached on average over all products and all reporting periods for each instance, the results of the HPPS show that the target fill rate is not even reached on average for any of the studied instances.
The periodbased strategy leads to the highest mean costs in the HPPS. This is in line with the results observed for the mediumterm problem. Again, the fill rates of the HPPS differ from the results of the mediumterm problem. While in the mediumterm problem each product reached at least the target fill rate in every reporting period resulting in no downside deviation at all, meeting target fillrates for the HPPS cannot be ensured for the periodbased strategy.
The stabilizedcycle strategy leads to similar results for the HPPS as observed in the mediumterm problem. This means that almost no downside deviations of actual fill rates from the target fill rates exist at the end of a reporting period, while there is only a moderate increase of costs in comparison to the orderbased strategy. As the periodbased strategy results in higher costs and higher downside deviations than the stabilizedcycle strategy for all studied instances, the stabilizedcycle strategy is superior to the periodbased strategy. Since the evaluation of the performance is bicriterial, no direct comparison of the stabilizedcycle strategy and the orderbased strategy can be made without preference information of the decisionmaker. However, for some instances we are able to show that the stabilizedcycle strategy dominates the orderbased strategy. For the other instances, the stabilizedcycle strategy provides at least solutions with a smaller downside deviation and only a minor increase of costs compared to the orderbased strategy.
To be able to react more flexibly to uncertain demands during a macroperiod, future research should study rolling schedules at the base level instead of the static–dynamic uncertainty strategy. Since the presented mathematical optimization models are known to be NPhard (Florian et al. 1980), only small instance can be solved to optimality in reasonable computational times by commercial optimizers. Thus, either improved exact algorithms or metaheuristics should be sought to find good solutions in terms of quality and computational time, especially if realworld problems must be solved.
Notes
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