Planning and Scheduling in Supply Chains: an Overview of Issues in Practice

Planning and Scheduling in Supply Chains: An Overview of Issues in Practice

Stephan Kreipl • Michael Pinedo SAP Germany AG & Co.KG, Neurottstrasse 15a, 69190 Walldorf, Germany Stern School of Business, New York University, 40 West Fourth Street, New York, New York 10012 his paper gives an overview of the theory and practice of planning and scheduling in supply chains. TIt ﬁrst gives an overview of the various planning and scheduling models that have been studied in the literature, including lot sizing models and machine scheduling models. It subsequently categorizes the various industrial sectors in which planning and scheduling in the supply chains are important; these industries include continuous manufacturing as well as discrete manufacturing. We then describe how planning and scheduling models can be used in the design and the development of decision support systems for planning and scheduling in supply chains and discuss in detail the implementation of such a system at the Carlsberg A/S beerbrewer in Denmark. We conclude with a discussion on the current trends in the design and the implementation of planning and scheduling systems in practice. Key words: planning; scheduling; supply chain management; enterprise resource planning (ERP) systems; multi-echelon inventory control Submissions and Acceptance: Received October 2002; revisions received April 2003; accepted July 2003.

1. Introduction taking into account inventory holding costs and trans- This paper focuses on models and solution ap- portation costs. A planning model may make a dis- proaches for planning and scheduling in supply tinction between different product families, but usu- chains. It describes several classes of planning and ally does not make a distinction between different scheduling models that are currently being used in products within a family. It may determine the opti- systems that optimize supply chains. It discusses the mal run length (or, equivalently, batch size or lot size) architecture of decision support systems that have of a given product family when a decision has been been implemented in industry and the problems that made to produce such a family at a given facility. If there are multiple families produced at the same fa- have come up in the implementation and integration cility, then there may be setup costs and setup times. of systems in supply chains. In the implementations The optimal run length of a product family is a considered, the total cost in the supply chain has to be function of the trade-off between the setup cost minimized, i.e., the stages in the supply chain do not and/or setup time and the inventory carrying cost. compete in any form with one another, but collaborate The main objectives in medium term planning in- in order to minimize total costs. This paper focuses volve inventory carrying costs, transportation costs, primarily on how to integrate medium term planning tardiness costs, and the major setup costs. However, models (e.g., lot sizing models) and detailed schedul- in a medium term planning model, it is typically not ing models (e.g., job shop scheduling models) into a customary to take the sequence dependency of single framework. setup times and setup costs into account. The se- A medium term production planning model typi- quence dependency of setups is difﬁcult to incorpo- cally optimizes several consecutive stages in a supply rate in an integer programming formulation and can chain (i.e., a multi-echelon model), with each stage increase the complexity of the formulation signiﬁ- having one or more facilities. Such a model is de- cantly. signed to allocate the production of the different prod- A short term detailed scheduling model is typically ucts to the various facilities in each time period, while only concerned with a single facility, or, at most, with

77 Kreipl and Pinedo: Planning and Scheduling in Supply Chains 78 Production and Operations Management 13(1), pp. 77–92, © 2004 Production and Operations Management Society a single stage. Such a model usually takes more de- There is an extensive literature on supply chain tailed information into account than a planning management. Many papers and books focus on sup- model. It is typically assumed that there are a given ply chain coordination; a significant amount of this number of jobs and each one has its own parameters work has an emphasis on inventory control, pricing (including sequence-dependent setup times and se- issues, and the value of information; see Simchi-Levi, quence-dependent setup costs). The jobs have to be Kaminsky, and Simchi-Levi (2000), Chopra and scheduled in such a way that one or more objectives Meindl (2001), and Stadtler and Kilger (2000). There is are minimized, e.g., the number of jobs that are also an extensive literature on production planning shipped late, the total setup time, and so on. and scheduling theory. A significant amount of re- Clearly, planning models differ from scheduling search has been done on the solution methods appli- models in a number of ways. First, planning models cable to planning and scheduling models; see Shapiro often cover multiple stages and optimize over a medium (2001). Planning models and scheduling models have term horizon, whereas scheduling models are usually often been studied independently from one another in designed for a single stage (or facility) and optimize over order to obtain elegant theoretical results. Planning a short term horizon. Second, planning models use more models are often based on (multi-echelon) inventory aggregate information, whereas scheduling models use theory and lot sizing; see Zipkin (2000), Kimms (1997), more detailed information. Third, the objective to be Drexl and Kimms (1997), Muckstadt and Roundy minimized in a planning model is typically a total cost (1993), and Dobson (1987, 1992). Scheduling models objective and the unit in which this is measured is a typically focus on how to schedule a number of jobs in monetary unit; the objective to be minimized in a sched- a given machine environment in order to minimize uling model is typically a function of the completion some objective. For general treatises on scheduling, times of the jobs and the unit in which this is measured see Bhaskaran and Pinedo (1992), Brucker (1998), is often a time unit. Nevertheless, even though there are Pinedo (2002), and Pinedo and Chao (1999). For appli- fundamental differences between these two types of cations of scheduling to supply chain management, models, they often have to be incorporated into a single see Hall and Potts (2000) and Lourenco (2001). Some framework, share information, and interact extensively research has been done on more integrated models in with one another. the form of hierarchical planning systems; this re- Planning and scheduling models may also interact search has resulted in frameworks that incorporate with other types of models, such as long term strategic planning and scheduling; see Bowersox and Closs models, facility location models, demand manage- (1996), Barbarosoglu and Ozgur (1999), Dhaenens- ment models, and forecasting models; these models Flipo and Finke (2001), Shapiro (2001), and Miller are not discussed in this paper. The interactions with (2002). For examples of descriptions of successful in- these other types of models tend to be less intensive dustrial implementations, see Haq (1991), Arntzen, and less interactive. In what follows, we assume that the Brown, Harrison, and Trafton (1995), Hadavi (1998), physical settings in the supply chain have already been and Shepherd and Lapide (1998). established; the configuration of the chain is given, and This paper is organized as follows. The second section the number of facilities at each stage is known. describes and categorizes some of the typical industrial Supply chains in the various industries are often not settings. The third section discusses the overall frame- very similar and may actually give rise to different works in which planning models as well as scheduling sets of issues and problems. This paper considers ap- models have to be embedded. The fourth section de- plications of planning and scheduling models in sup- scribes a standard mixed integer programming formu- ply chains in various industry sectors. A distinction is lation of a planning model for a supply chain. The fifth made between two types of industries, namely the section covers a typical formulation of a scheduling continuous manufacturing industries (which include problem in a facility in a supply chain. The sixth section the process industries) and the discrete manufacturing describes an actual implementation of a planning and industries (which include, for example, automotive scheduling software system at the Danish beerbrewer and consumer electronics). Each one of these two main Carlsberg A/S. The last section presents the conclusions categories is subdivided into several subcategories. and discusses the impact of the Internet on decision This categorization is used because of the fact that the support systems in supply chains. planning and scheduling procedures in the two main categories tend to be different. We focus on the frameworks in which the planning and scheduling models 2. Supply Chain Settings and have to be embedded; we describe the type of infor- Configurations mation that has to be transferred back and forth be- This section gives a concise overview of the various tween the modules and the kinds of optimization that types of supply chains. It describes the differences in is done within the modules. the characteristics and the parameters of the various Kreipl and Pinedo: Planning and Scheduling in Supply Chains Production and Operations Management 13(1), pp. 77–92, © 2004 Production and Operations Management Society 79 categories. It first describes the various different in- single machine and parallel machine scheduling mod- dustry groups and their supply chain characteristics els. If it operates according to mts, then it may follow and then discusses how the different planning and a so-called s-S or Q-R inventory control policy. If it is scheduling models analyzed in the literature can be a mixture of mto and mts, then the scheduling policies used in the management of these chains. One can become a mixture of inventory control and detailed make a distinction between two types of manufactur- scheduling rules. ing industries, namely: Discrete Manufacturing. The discrete manufacturing (I) Continuous manufacturing industries (e.g., the industry sector is quite diverse and includes the auto- process industries), motive industry, the appliances industry, and the pc (II) Discrete manufacturing industries (e.g., cars, industry. From the perspective of planning and sched- semiconductors). uling, a distinction can be made between three differ- These two industry sectors are not all-encompassing; ent types of operations in this sector. The reason for the borderlines are somewhat blurry and may overlap. making such a distinction is based on the fact that However, planning and scheduling in continuous planning and scheduling in these three segments are manufacturing (the process industries) often have to quite different. deal with issues that are quite different from those in (II-a) Primary converting operations (e.g., cutting discrete manufacturing. and shaping of sheet metal), Continuous Manufacturing. Continuous manufactur- (II-b) Main production operations (e.g., production ing (process) industries often have various types of of engines, pcbs, wafers), and different operations. The most common types of op- (II-c) Assembly operations (e.g., cars, pcs). erations can be categorized as follows: Primary Converting Operations in Discrete Manufac- (I-a) Main processing operations, turing (II-a). Primary converting operations are some- (I-b) Finishing or converting operations. what similar to the finishing operations in the process Main Processing Operations in Continuous Manufac- industries. These operations may typically include turing (I-a). The main production facilities in the pro- stamping, cutting, or bending. The output of this op- cess industries are, for example, paper mills, steel eration is often a particular part that is cut and bent mills, aluminum mills, chemical plants, and refineries. into a given shape. There are usually few operations In paper, steel, and aluminum mills, the machines take done on such an item, and the routing in such a facility in the raw material (e.g., wood, iron ore, alumina) and is relatively simple. The final product of a primary produce rolls of paper, steel, or aluminum, which converting facility is usually not a finished good, but afterwards are handled and transported with special- basically a part or piece made of a single material ized material-handling equipment. Machines that do (boxes, containers, frames, stamped body parts of cars, the main processing operations typically have very and so on). Examples of the types of operations in this high startup and shutdown costs and usually work category are stamping plants that produce body parts around the clock. A machine in the process industries for cars, and plants that produce epoxy boards of also incurs a high changeover cost when it has to various sizes for the facilities that produce Printed switch over from one product to another. Various Circuit Boards. The planning and scheduling proce- methodologies can be used for analyzing and solving dures under II-a may be similar to those under I-b. the models for such operations, including cyclic However, they may be here more integrated with the scheduling procedures and Mixed Integer Program- operations downstream. ming approaches. Main Production Operations in Discrete Manufacturing Finishing Operations in Continuous Manufacturing (II-b). The main production operations are those op- (I-b). Many process industries have some form of fin- erations that require multiple different operations by ishing operations that do some converting of the out- different machine tools, and the product (as well as its put of the main production facilities. This converting parts) may have to follow a certain route through the usually involves cutting of the material, bending, fold- facility going through various work centers. Capital ing, and possibly painting or printing. These opera- investments have to be made in various types of ma- tions often (but not always) produce commodity-type chine tools (lathes, mills, chip fabrication equipment). items, for which the producer has many clients. For For example, in the semiconductor industry, wafers example, a finishing operation in the paper industry typically have to undergo hundreds of steps. These may produce cut size paper out of the rolls that come operations include oxidation, deposition, and metalli- from the paper mill. The paper finishing business is zation, lithography, etching, ion implantation, pho- often a mixture of Make-To-Stock (mts) and Make-To- toresist stripping, and inspection and measurements. Order (mto). If it operates according to mto, then the It is often the case that certain operations have to be scheduling is based on customer due dates and se- performed repeatedly and that certain orders have to quence-dependent setup times. This leads often to visit certain workcenters in the facility several times, Kreipl and Pinedo: Planning and Scheduling in Supply Chains 80 Production and Operations Management 13(1), pp. 77–92, © 2004 Production and Operations Management Society i.e., they have to recirculate through the facility. In Table 1 semiconductor and Printed Circuit Board manufactur- Product ing, the operations are often organized in a job shop Sector Processes Time horizon Clock-speed differentiation fashion. Each order has its own route through the (I-a) planning long-medium low very low system, its own quantity (and processing times), and (I-b) planning/scheduling medium/short medium/high medium/low its own committed shipping date. An order typically (II-a) planning/scheduling medium/short medium very low represents a batch of identical items that requires se- (II-b) planning/scheduling medium/short medium medium/low quence-dependent setup times at many operations. (II-c) scheduling short high high Assembly Operations in Discrete Manufacturing (II-c). The main purpose of an assembly facility is to put different parts together. An assembly facility typically There are some basic differences between the pa- does not alter the shape or form of any one of the rameters and operating characteristics of the facilities individual parts (with the possible exception of the in the two main categories described above. Several of painting of the parts). Assembly operations usually do these differences have an impact on the planning and not require major investments in machine tools, but do scheduling processes, including the differences in (i) require investments in material handling systems (and the planning horizon, (ii) the clock-speed, and (iii) the possibly robotic assembly equipment). An assembly level of product differentiation. operation may be organized in workcells, in assembly (i) The planning horizon in continuous manufactur- lines, or according to a mixture of workcells and as- ing facilities tends to be longer than the planning sembly lines. For example, pcs are assembled in work- horizon in the discrete manufacturing facilities. In cells, whereas cars and TVs are typically put together continuous as well as in discrete manufacturing the in assembly lines. Workcells typically do not require planning horizons tend to be shorter more down- any sequencing, but they may be subject to learning stream in the supply chain. curves. In assembly operations that are set up in a line, (ii) The so-called “clock-speed” tends to be higher the sequencing is based on grouping and spacing heu- in a discrete manufacturing facility than in a continu- ristics combined with committed shipping dates. The ous manufacturing facility. A high clock-speed im- schedules that are generated by the grouping and spac- plies that existing plans and schedules often have to be ing heuristics typically affect not only the throughput of changed or adjusted; that is, planning and scheduling the line, but also the quality of the items produced. is more reactive. In continuous as well as in discrete Supply chains in both continuous and discrete man- manufacturing, the clock-speed increases the more ufacturing may have, besides the stages described downstream in the supply chain. above, additional stages. In a supply chain in a process (iii) In discrete manufacturing, there may be a sig- industry, there may be a stage preceding Stage I-a in nificant amount of mass customization and product which the raw material is being gathered at its point of differentiation. In continuous manufacturing, mass- origination (which may be a forest or a mine) and customization does not play a very important role. taken to the main processing operations. There may The number of SKUs in discrete manufacturing tends also be a distribution stage following stage I-b. A to be significantly larger than the number of SKUs in company may have its own distribution centers in continuous manufacturing. The number of SKUs different geographical locations, where it keeps cer- tends to increase more downstream in the supply tain SKUs in stock for immediate delivery. The com- chain. pany may also ship directly from its manufacturing These operating characteristics are summarized in operations to customers. A supply chain in a discrete Table 1. Because of these differences, the planning and manufacturing industry also may have other types of scheduling issues in each one of the sectors can be stages. There may be a stage preceding stage II-a in very different. Table 2 presents a summary of the which raw material is being collected at a supplier model types that can be used in the different catego- (which may be an operation of the type I-b) and ries as well as the corresponding solution techniques. brought to a primary converting operation. There may Note that problems that have continuous variables also be a stage following stage II-c which would con- may lead to Mixed Integer Programming (mip) formu- sist of distribution operations (e.g., dealerships). lations, whereas problems that have only discrete vari- Supply chains in both continuous and discrete man- ables may lead to pure Integer Programming (ip) ufacturing may have several facilities at each one of formulations (or Disjunctive Programming formu- the stages, each one feeding into several facilities at lations). However, a discrete problem in which certain stages downstream. The configuration of an entire variables assume large values (i.e., the number of units chain may be quite complicated: For example, there to be produced) may be replaced by a continuous may be assembly operations that produce subassem- problem, resulting in a Mixed Integer Programming blies that have to be fed into a production operation. formulation rather than a pure Integer Programming Kreipl and Pinedo: Planning and Scheduling in Supply Chains Production and Operations Management 13(1), pp. 77–92, © 2004 Production and Operations Management Society 81

Table 2 Figure 1 Planning and Scheduling in Supply Chains

Sector Models Solution techniques

(I-a) Lot sizing models (multi-stage); Mixed Integer Programming cyclic scheduling models formulations (I-b) Single machine scheduling models; Batch scheduling; mixtures parallel machine scheduling of inventory control rules models and dispatching rules (II-a) Single machine scheduling models; Batch scheduling and Parallel machine scheduling dispatching rules models (II-b) Flow Shop and Job Shop Scheduling Integer Programming Models with specific routing formulations; shifting patterns bottleneck heuristics; dispatching rules (II-c) Assembly Line Models; Workcell Grouping and Spacing Models Heuristics; Make-to- Order/Just-In-Time formulation. Planning models typically result in Mixed Integer Programming formulations with a mix effective and timely manner. Of course, this overall of continuous and discrete variables. Scheduling mod- objective forces each one of the individual stages to els usually do not have any continuous variables; they formulate its own objectives. may have continuous variables when preemptions Since planning and scheduling in a global supply and job splitting are allowed. When there are few chain requires the coordination of operations in all discrete variables, it makes a lot of sense to solve the stages of the chain, the models and solution tech- Linear Programming relaxation of the mip. The solu- niques described in the previous section have to be tion may provide a useful lower bound and may give integrated within a single framework. Different mod- indications regarding the structure of the optimal so- els that represent successive stages have to exchange lutions of the mip. If the formulation of the problem is information and interact with one another in various a pure Integer Program (which is often the case with a ways. A continuous model for one stage may have to scheduling problem), then solving the linear relax- interact with a discrete model for the next stage. ation typically does not provide a significant amount Planning and scheduling procedures in a supply of benefit. chain are typically used in various phases: a first Examples of applications of planning and schedul- phase involves a multi-stage medium term planning ing in continuous manufacturing can be found in Haq process (using aggregate data), and a subsequent (1991), Murthy et al. (2001), and Rachlin et al. (2001). phase performs a short term detailed scheduling at Examples of planning and scheduling in discrete man- each one of those stages separately. Typically, when- ufacturing are described in Arntzen, Brown, Harrison, ever a planning procedure has been applied and the and Trafton (1995), De Bontridder (2001), and Van- results have become available, each facility can apply daele and Lambrecht (2001). its scheduling procedures. However, scheduling pro- In the following four sections, we discuss frame- cedures are usually applied more frequently than works for planning and scheduling in supply chains, planning procedures. Each facility in every one of we present some examples of planning and schedul- these stages has its own detailed scheduling issues to ing models that have formed a basis for several sys- deal with; see Figure 1. tems that have been implemented in practice, and we If successive stages in a supply chain belong to the describe an actual implementation. These four sec- same company, then it is usually the case that these tions have been inspired primarily by the design of stages are incorporated into a single planning model. systems developed and implemented by SAP Ger- The medium term planning process attempts to min- many AG; see Braun (2001), Braun and Groenewald imize the total cost over all the stages. The costs that (2000), and Strobel (2001). have to be minimized in this optimization process include production costs, storage costs, transportation 3. Frameworks for Planning and costs, tardiness costs, non-delivery costs, handling Scheduling in Supply Chains costs, costs for increases in resource capacities (e.g., The main objective in a supply chain or production scheduling third shifts), and costs for increases in distribution network is to produce and deliver fin- storage capacities. ished products to end consumers in the most cost- In this medium term optimization process, many Kreipl and Pinedo: Planning and Scheduling in Supply Chains 82 Production and Operations Management 13(1), pp. 77–92, © 2004 Production and Operations Management Society input data are only considered in an aggregate form. Figure 2 Data Aggregation and Constraint Propagation For example, time is often measured in weeks or months rather than days. Distinctions are usually only made between major product families, and no distinctions are made between different products within one family. A setup cost may be taken into account, but it may only be considered as a function of the product itself and not as a function of the sequence. The results of this optimization process are daily or weekly production quantities for all product families at each location or facility as well as the amounts scheduled for transport every week between the locations. The production of the orders requires a certain amount of the capacities of the resources at the various facilities, but no detailed scheduling takes place in the medium term optimization. The output consists of the allocations of resources to the various product families, the assignment of products to the various facilities in each time period, and the inventory levels of the finished goods at the various locations. As stated be- product within a family is taken into account. The fore, in this phase of the optimization process, a dis- minor setup times and setup costs in between differ- tinction may be made between different product fam- ent products from the same family are taken into ilies, but not between different products within the account as well as the sequence dependency. same family. The model is typically formulated as a The factory is now not a single entity; each product Mixed Integer Program. Variables that represent has to undergo a number of operations on different quantities that have to be produced are often contin- machines. Each product has a given route and given uous variables. The integer (discrete) variables are processing requirements on the various machines. The often 0-1 variables; they are, for example, needed in detailed scheduling problem can be analyzed as a job the formulation when a decision has to be made shop problem and various techniques can be used, whether or not a particular product family will be including: produced at a certain facility during a given time (i) dispatching rules, period. (ii) shifting bottleneck techniques, The output of the medium term planning process is (iii) local search procedures (e.g., genetic algo- an input to the detailed (short term) scheduling pro- rithms), or cess. The detailed scheduling problems typically at- (iv) integer programming techniques. tempt to optimize each stage and each facility sepa- The objective takes into account the individual rately. So, in the scheduling phase of the optimization due dates of the orders, sequence-dependent setup process, the process is partitioned according to: times, sequence-dependent setup costs, lead times, (i) the different stages and facilities, and as well as the costs of the resources. However, if two (ii) the different time periods. successive facilities (or stages) are tightly coupled So, in each detailed scheduling problem the scope is with one another (i.e., the two facilities operate considerably narrower (with regard to time as well as according to the jit principle), then the short term space), but the level of detail taken into consideration scheduling process may optimize the two facilities is considerably higher; see Figure 2. This level of detail jointly. It actually may consider them as a single is increased in the following dimensions: facility with the transportation in between the two (i) the time is measured in a smaller unit (e.g., days facilities as another operation. or hours); the process may be even time continuous, The interaction between a planning module and a (ii) the horizon is shorter, scheduling module may be intricate. A scheduling (iii) the product demand is more precisely defined, module may cover only a relatively short horizon and (e.g., one month), whereas the planning module (iv) the facility is not a single entity, but a collection may cover a longer horizon (e.g., six months). After of resources or machines. the schedule has been fixed for the first month (fix- The product demand now does not consist, as in the ing the schedule for this month required some input medium term planning process, of aggregate de- from the planning module), the planning module mands for entire product families. In the detailed does not consider this first month any more; it as- scheduling process, the demand for each individual sumes the schedule for the first month to be fixed. Kreipl and Pinedo: Planning and Scheduling in Supply Chains Production and Operations Management 13(1), pp. 77–92, © 2004 Production and Operations Management Society 83

Figure 3 Scheduling and Planning Horizons quence dependency of the setup times into account; setup times are estimated and embedded in the total production times. The total setup times in the detailed schedule may actually be higher than the setup times anticipated in the planning procedure.) If the results of the detailed scheduling process indicate that the input to the planning process has to be modified, then new input data for the planning process have to be generated and the planning process have to be redone. Second, there may be an exogenous reason necessi- tating a feedback from the detailed scheduling process to the medium term planning process. A major disruption may occur on the factory floor level, e.g., an important machine goes down for an extended period of time. A disruption may be of such a magnitude that its effects cannot be contained within the facility where it occurs. The entire planning process may be affected and therefore the scheduling processes at other facilities as well. So a framework with a feed- However, the planning module still tries to optimize back mechanism may allow the overall optimization the second up to the sixth month. Doing so, it con- process to iterate (see Figure 4). siders the output of the scheduling module as a The individual modules within the planning and boundary condition. However, it also may be the scheduling framework for a given chain may have case that the time periods covered by the detailed other interesting features. Two types of features that scheduling process and the medium term planning are often incorporated are decomposition features and process overlap; see Figure 3. so-called discretization features. Each feature can be A planning and scheduling framework for a supply activated and deactivated by the user of the system. chain typically must have a mechanism that allows Decomposition is often used when the optimization feedback from a scheduling module to the planning problem is simply too large to be dealt with effectively module; see Figure 4. This feedback mechanism en- by the routines available. A decomposition process ables the optimization process to go through sev- partitions the overall problem in a number of subprob- eral iterations. It may be used under various circum- lems and solves the (smaller) subproblems separately. stances: First, the results of the detailed short term At the end of the process, the partial solutions are put optimization process may indicate that the estimates together in a single overall solution. Decomposition used as input data for the medium term planning can be done according to: process were not accurate. (The average production (i) time; times in the planning processes do not take the se- (ii) available resources (facilities or machines); (iii) product families; and Figure 4 Information Flows Between Planning and Scheduling Systems (iv) geographical areas. Some of the decompositions may be designed in such a way that they are activated automatically by the system itself, and other decompositions may be designed in such a way that they have to be activated by the user of the system. Decomposition is used in medium term modules as well as in detailed scheduling modules. In medium term planning, the decomposition is often based on time and/or on product family (these may be internal decompositions activated by the system itself). The user may specify in a medium term planning process a geographical decomposition. In the detailed scheduling process, the decomposition is often machine-based (such a decomposition may be done internally by the system or imposed by the user). One type of discretization feature may be used when the continuous version of a problem (for exam- Kreipl and Pinedo: Planning and Scheduling in Supply Chains 84 Production and Operations Management 13(1), pp. 77–92, © 2004 Production and Operations Management Society ple, a linear programming relaxation of a more realis- Figure 5 A System with Three Stages tic integer programming formulation) does not yield sufficiently accurate results. To obtain more accurate results, certain constraints may have to be imposed on given variables. For example, production quantities are often not allowed to assume just any values, but only values that are multiples of given fixed amounts or lot sizes (e.g., the capacity of a tank in the brewing of beer). The quantities that have to be transported between two facilities also have to be multiples of a fixed amount (e.g., the size of a container). This type of discretization may transform the problem from a continuous problem (i.e., a linear program) to a discrete upstream stage (Stage 1) has two factories in parallel. optimization problem. They both feed Stage 2, which is a distribution center Another type of discretization can be done with (dc). Both Stages 1 and 2 can deliver to a customer, respect to time. It allows the user of the system to which is a part of Stage 3; see Figure 5. The factories determine the size of the time unit. If the user is only have no room for finished goods storage and the cus- interested in a rough plan, he may set the time unit to tomer does not want to receive any early deliveries. be equal to a week. That is, the results of the optimi- The problem has the following parameters and in- zation then only specify what is going to be produced put data. The two factories work around the clock; so their available weekly production capacity is 24 ϫ 7 that week, but will not specify what is going to be ϭ produced within each day of that week. If the user sets 168 hours. There are two major product families, F1 F the time unit equal to one day, the result will be and 2. As stated before, in the medium term planning process, all the products within a family are consid- significantly more precise. Besides specifying the sizes ered identical. The demand forecasts for the next four of the time units, a system may use time units of weeks are known (the unit of time being one week). In different sizes for different periods. The discretization this section, the subscripts and superscripts have the feature is often implemented in the medium term following meaning. planning modules. The first week of a three-month The subscript i (i ϭ 1,...,4),refers to time period i. planning period may be specified on a daily basis, the The subscript j (j ϭ 1, 2), refers to product family j. next three weeks may be determined on a weekly The subscript k (k ϭ 1, 2), refers to factory k. basis, and all activities beyond the first month are The subscript l (l ϭ 1, 2, 3) refers to stage l; planned based on a continuous model. Discretization l ϭ 1 refers to the two factories, with respect to time does not change the nature of the l ϭ 2 refers to the distribution center, and problem; if the problem is a linear program, then it l ϭ 3 refers to the customer. will remain a linear program. The superscript p refers to a production parameter. The APO system of SAP Germany enables the mod- The superscript s refers to a storage parameter. eler to activate and deactivate the discretization of The superscript ␶ refers to a transportation param- various types of constraints in order to improve the eter. performance of the optimization process. For example, The demand for product family j, j ϭ 1, 2, at the dc discretization may be used for daily and weekly time level (stage 2) by the end of week i, i ϭ 1,...,4,is buckets, but not for monthly time buckets in which denoted by D . The demand for product family j, Linear Programming is used without discretization. ij2 j ϭ 1, 2, at the customer level (stage 3) by the end of ϭ week i, i 1,...,4,is denoted by Dij3. Production 4. Medium Term Planning Models for times and costs are given: p ϭ Supply Chains cjk the cost to produce 1 unit of family j in factory k. p ϭ This section considers a standard medium term plan- tjk the time (in hours) to produce 1000 units of ning model for a supply chain. It does not present the family j in factory k. p model in its full generality; the notation needed for a The tjk is the reciprocal of the rate of production. more general model is simply too cumbersome. A Storage costs and transportation data include: s ϭ description is given of a model with many of the c2 the storage cost for one unit of any type in the relevant parameters having fixed values (in order to dc per week. ␶ ϭ simplify the notation). It also does not incorporate all ck2ؠ the transportation cost for a unit of any type of the features described in the previous section (e.g., from factory k to the dc. ␶ ϭ all the time units are of the same size). ckؠ3 the transportation cost for a unit of any type Consider three stages in series. The first and most from factory k to the customer. Kreipl and Pinedo: Planning and Scheduling in Supply Chains Production and Operations Management 13(1), pp. 77–92, © 2004 Production and Operations Management Society 85

␶ ϭ cؠ23 the transportation cost for a unit of any type 4 2 4 2 3 2 ␶ s from the dc to the customer. ϩ ͸ ͸ cؠ z ϩ ͸ ͸ c q ϩ ͸ ͸ w Љv ␶ 23 ij 2 ij2 j ij2 t ϭ the transportation time from any one of the iϭ1 jϭ1 iϭ1 jϭ1 iϭ1 jϭ1 two factories to the dc, from any one of the two 3 2 2 2 factories to the customer, and from the dc to ϩ ͸ ͸ wٞv ϩ ͸ ␲v ϩ ͸ ␲v the customer; all transportation times are as- j ij3 4j2 4j3 iϭ1 jϭ1 jϭ1 jϭ1 sumed to be identical and equal to one week. The following weights and penalty costs are given: subject to the following weekly production capacity Љ ϭ wj the tardiness cost per unit per week for an order constraints: of family j products that arrive late at the dc. 2 wٞ ϭ the tardiness cost per unit per week for an order j ͸ tp x Յ 168 i ϭ 1,...,4; of family j products that arrive late at the cus- j1 ij1 jϭ1 tomer. ␲ ϭ the penalty for never delivering one unit of 2 ͸ p Յ ϭ product. tj2xij2 168 i 1,...,4; The objective is to minimize the total of the produc- jϭ1 tion costs, storage costs, transportation costs, tardiness subject to the following transportation constraints: costs, and penalty costs for non-delivery over a hori- Յ ϭ zon of four weeks. In order to formulate this problem yij1l UBj1l i 1,...,4; as a Mixed Integer Program, the following decision Ն ϭ ϭ variables have to be deﬁned: yij1l LBj1l or yij1l 0 i 1,...,4; x ϭ number of units of family j produced at plant k ijk Յ ϭ during period i. yij2l UBj2l i 1,...,4; y ϭ number of units of family j transported from Ն ϭ ϭ ijk2 yij2l LBj2l or yij2l 0 i 1,...,4; plant k to the dc in week i. ϭ 3 yijk3 number of units of family j transported from ͸ ϭ ϭ ϭ ϭ plant k to customer in week i. yijkl xijk i 1,...,4; j 1, 2, k 1, 2; ϭ lϭ2 zij number of units of family j transported from the dc to the customer in week i. ϭ 2 q0j2 number of units of family j in storage at the dc ͸ ϩ Յ ϩ ϭ ϭ yijk3 zij Diϩ1, j,3 vij3 i 1,...,3; j 1, 2; at time 0. ϭ ϭ k 1 qij2 number of units of family j in storage at the dc Յ ͑ ͒ ϭ in week i. z1j max 0, q0j2 j 1, 2; v ϭ number of units of family j that are tardy (have ij2 Յ ϩ ϩ ϭ not yet arrived) at the dc in week i. zij qiϪ1, j,2 yiϪ1, j,1,2 yiϪ1, j,2,2 i 2, 3, 4; ϭ v4j2 number of units of family j that have not been j ϭ 1, 2; delivered to the dc by the end of the planning horizon (the end of week 4). subject to the following storage constraints: ϭ ϭ ͑ Ϫ Ϫ ͒ ϭ v0j3 the number of units of family j that are tardy q1j2 max 0, q0j2 D1j2 z1j j 1, 2; (have not yet arrived) at the customer at time 0. ϭ q ϭ max ͑0, q Ϫ ϩ y Ϫ ϩ y Ϫ vij3 number of units of family j that are tardy (have ij2 i 1, j,2 i 1, j,1,2 i 1, j,2,2 not yet arrived) at the customer in week i. Ϫ D Ϫ z Ϫ v Ϫ ͒ jϭ1, 2 iϭ2, 3, 4; ϭ ij2 ij i 1, j,2 v4j3 the number of units of family j that have not been delivered to the customer by the end of subject to the following constraints regarding number the planning horizon (the end of week 4). of jobs tardy and number of jobs not delivered: There are various constraints in the form of upper ϭ ͑ Ϫ ͒ ϭ v1j2 max 0, D1j2 q0j2 j 1, 2; bounds UBjkl and lower bounds LBjkl on the quantities ϭ ͑ ϩ ϩ Ϫ Ϫ Ϫ ͒ of family j to be shipped from plant k to stage l. The vij2 max 0, Dij2 viϪ1, j,2 zij qij2 yiϪ1, j,1,2 yiϪ1, j,2,2 integer program can now be formulated as follows: j ϭ 1, 2; i ϭ 2, 3, 4; minimize ϭ ͑ ͒ ϭ v1j3 max 0, D1j3 j 1, 2; 4 2 2 4 2 2 4 2 2 v ϭ max ͑0, D ϩ v Ϫ Ϫ z Ϫ Ϫ y Ϫ ͸ ͸ ͸ p ϩ ͸ ͸ ͸ ␶ ϩ ͸ ͸ ͸ ␶ ij3 ij3 i 1, j,3 i 1, j i 1, j,1,3 cjkxijk ck2ؠyijk2 ckؠ3yijk3 ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ ϭ Ϫ ͒ ϭ ϭ i 1 j 1 k 1 i 1 j 1 k 1 i 1 j 1 k 1 yiϪ1, j,2,3 j 1, 2; i 2, 3, 4. Kreipl and Pinedo: Planning and Scheduling in Supply Chains 86 Production and Operations Management 13(1), pp. 77–92, © 2004 Production and Operations Management Society

Table 3 Table 5

Week 1 Week 2 Week 3 Week 4 Week 1 Week 2 Week 3

Di12 20,000 30,000 15,000 40,000 yi112 000 Di22 0 50,000 30,000 50,000 yi113 000 Di13 10,000 5,000 15,000 40,000 yi122 50,000 15,000 40,000 Di23 0 10,000 0 5,000 yi123 15,000 15,000 40,000 yi212 47,333 20,000 50,000 yi213 0 0 2,333 yi222 2,667 10,000 0 It is clear that most variables in this Mixed Integer yi223 10,000 0 2,667 Programming formulation are continuous variables.

However, the transportation variables yijkl are subject to disjunctive constraints. In an alternative formulation of the problem, some integer (0–1) variables are tation decisions shown in Tables 4 and 5. The total cost needed to ensure that the continuous (transportation) of this solution is $3,004,950.20. variables yij1l are either 0 or larger than the lower If an additional constraint is added to this prob- bound LBj1l. Note that the Linear Programming relax- lem requiring the production lot sizes to be multi- ation of the formulation above (i.e., the formulation ples of 10,000 (such a constraint is fairly easy to without the disjunctive constraints) provides a valid formulate), then we obtain the solution in Tables 6 lower bound on the total cost. and 7. The total cost is in this case $3,029,672.00, The following numerical example illustrates an ap- which is indeed higher than the total cost without plication of the model described above. the production constraint that items have to be pro- Example 4.1. Consider the following instance of the duced in lots of 10,000. However, the increase is less problem described above. The production times and than 1%. The increased costs are mainly due to p ϭ p ϭ costs concerning factory 1 are: t11 1 hour and t21 excess production (the total production quantities p ϭ p ϭ 2 hours; c11 1 and c21 0.50. The production times now exceed the total demand quantities) and, con- p ϭ and costs concerning factory 2 are: t12 2 hours and sequently, additional transportation and storage p ϭ p ϭ p ϭ t22 3 hours; c12 0.50 and c22 0.25. costs. The storage cost for a unit of any type of product at It is clear that this formulation of this medium s the dc (c2) is 0.10 per unit per week. The transporta- term planning problem can be extended very easily ␶ ϭ ␶ ϭ tion costs are c12ؠ 0.10 per unit; c22ؠ 0.30 per unit; to more time periods, more factories at the ﬁrst ␶ ϭ ϭ ␶ ϭ ckؠ3 0.05 for k 1, 2, cؠ23 0.50 per unit. The stage, and more product families. An extension to forecast demand at the dc and from the customer for more stages may be a little bit more involved if there the two different product families are presented in is an increase in the complexity of the routing pat- Table 3. terns. From Factory 1 to the dc, there has to be each week at least a shipment of 10,000 units of product family 1 ϭ or otherwise nothing, i.e., LB112 10,000. From Fac- tory 2 to the dc, there has to be each week at most a Table 6 shipment of 10,000 units of product family 2, i.e., Week 1 Week 2 Week 3 Week 4 ␶ UB ϭ 10,000. The transportation time t is 1 week. 222 x 0000 Љ Љ i11 The tardiness cost w1 (w2) is $10.00 ($5.00) per unit x 60,000 20,000 60,000 0 ٞ ٞ i21 per week. The tardiness cost w 1 (w 2) is $20.00 ($15.00) x 70,000 30,000 80,000 0 ␲ i12 per unit per week. The penalty cost for not deliver- xi22 0 10,000 0 0 ing at all is $1000.00 per unit. The boundary condition

v0j3 is 0. Running these data through a Mixed Integer Pro- Table 7 gramming solver yields the production and transpor- Week 1 Week 2 Week 3

y 000 Table 4 i112 yi113 000 Week 1 Week 2 Week 3 Week 4 yi122 55,000 15,000 40,000 yi123 15,000 15,000 40,000 xi11 0000yi212 50,000 20,000 55,000 xi21 47,333 20,000 52,333 0 yi213 10,000 0 5,000 xi12 65,000 30,000 80,000 0 yi222 0 10,000 0 xi22 12,667 10,000 2,667 0 yi223 000 Kreipl and Pinedo: Planning and Scheduling in Supply Chains Production and Operations Management 13(1), pp. 77–92, © 2004 Production and Operations Management Society 87

5. Short Term Scheduling Models for setups; the indicator variable Iijk is1ifjobj is followed Supply Chains by job k on machine i, the indicator variable is 0 The short term scheduling problem for a facility in a otherwise. supply chain can be described as follows: The output This scheduling problem may be tackled via a num- of the medium term planning problem specifies that ber of different techniques, including a combination of dispatching rules, such as the Shortest Setup Time over the short term nj items of family j have to be produced. The scheduling problem can either be mod- (SST) first rule, the Earliest Due Date first (edd) rule, eled as a job shop (or flexible flow shop) that takes all and the Weighted Shortest Processing Time first the production steps in the facility into account, or as (wspt) rule. Other techniques may include genetic a somewhat simpler single (or parallel) machine algorithms or integer programming approaches. In scheduling problem that focuses only on the bottle- this phase, however, integer programming ap- neck operation. If the operations in a facility are well proaches are not often used because they are compu- balanced and the location of the bottleneck depends tationally quite intensive. on the types of orders that are in the system, then the Example 5.1. Consider the two factories described entire facility may have to be modeled as a job shop. If in the medium term planning process in the previous the bottleneck in the facility is a permanent bottleneck section. In the detailed scheduling process, the two (that never moves), then a focus on the bottleneck is factories may be scheduled independently from one justified. If the bottleneck stage is modeled as a par- another and the scheduling is done one week at a time. allel machine scheduling model, then the parallel ma- Consider Factory 1 with the two product families. The chines may not be identical. They may also be subject production process in this factory consists of various to different maintenance and repair schedules. steps, but one of these steps is the clear bottleneck. There is, of course, a close relationship between the This bottleneck consists of a number of resources in p time tjk in the medium term planning process and the parallel. Consider the operations of Factory 2 in the processing time of an order in the short term detailed example in the previous section and only on the first scheduling problem. The tp in the medium term plan- week of operations. The solution of the integer pro- jk ϭ ϭ ning process has to be estimated and may be a value gram yields xi12 65,000 and xi22 12,667. Of the anywhere in between the average processing time of 65,000 of product family 1, a total of 50,000 has to be an order at the bottleneck operation and the total shipped to the dc and the remainder has to go to the (estimated) throughput time of an order through the customer. Of the 12,667 of product family 2, a total of p dc facility. The tjk is a function of the processing times pij 2667 has to be shipped to the and the remaining as well as of the sequence-dependent setup times sijk. 10,000 has to go to the customer. Any given order cannot be released before all the Assume now that the following more detailed in- required raw material has arrived (these dates are formation is available (which was not taken into ac- typically stored in a Material Requirements Planning count in the medium term planning process). The time (mrp) system). That is, order j has an earliest possible unit in the scheduling process is 1 hour in contrast to starting time that is typically referred to as a release the 1 week in the medium term planning process (in date rj, a committed shipping date dj, and a priority actual implementations the time unit in the scheduling factor or weight wj. Dependent upon the manufactur- process can be made arbitrarily small). The scheduling ing environment, preemptions may or may not be horizon is 1 week. allowed. Every time a machine switches over from one Recall that 2 hours of the bottleneck resource are type of item to another type of item, a setup cost may required to produce 1000 units of Family 1 in Factory be incurred and a setup time may be required. If a 2, whereas 3 hours of the bottleneck resource are schedule calls for a large number of preemptions, a required for 1000 units of Family 2. This implies that, large number of setups may be incurred. based on these estimated production times, the The objective to be minimized may include the min- planned production takes the full capacity of the bot- imization of the total setup times on the machines at tleneck resource (in hours): the bottleneck as well as the total weighted tardiness, ¥ 65 ϫ 2 ϩ 12.667 ϫ 3 ϭ 168. which is denoted by wjTj. So the objective may be formulated as However, the 2 and 3 hours requirement of the ␣ ͸ ϩ ␣ ͸ bottleneck resource are only estimates. They are esti- 1 wj Tj 2 Iijk sijk , mates that are being used in the medium term plan- ␣ ␣ where the 1 and the 2 denote the weights of the two ning process in order not to have to make a distinction parts of the objective function. The first part of the between sequence-dependent setup times and run objective function is the total weighted tardiness, and times. The actual run times (or processing times), ex- the second part of the objective represents a total of all cluding any setup times are as follows: To produce in Kreipl and Pinedo: Planning and Scheduling in Supply Chains 88 Production and Operations Management 13(1), pp. 77–92, © 2004 Production and Operations Management Society

Table 8 is 167.4 ϩ 16 ϭ 183.4. The shipment to the customer Job 1 2 3 4 leaves on time, but the shipment to the dc leaves late. Which one of these two schedules is preferable de- p 87.5 26.25 6.67 25 j pends on the weights ␣ and ␣ in the objective func- r 003636 1 2 j tion. dj 168 120 168 120 The results coming out of the detailed scheduling problem may be, for various reasons, not acceptable. When trying to minimize the makespan (in order to Factory 2 1000 units of Family 1, 1.75 hours of the ensure the production of the required quantities in the bottleneck resource is required, whereas 1000 units of one week), it may turn out that there does not exist a Family 2 requires 2.5 hours of the bottleneck resource. schedule that would complete the requested produc- To start producing units of Family 1, a setup of 16 tion within one week. The reason may be the follow- hours is required. To start producing units of Family 2, ing: the production times tp that were entered in the a setup of 6 hours is required. If each one of the jk medium term planning problem were estimated based products was to be produced in a single run in that on plant data, that include the average processing week, then the entire production could be done within times on the bottleneck machines, the expected 168 hours, since throughput times, the expected setup times, and so on. ϩ ϫ ϩ ϩ ϫ ϭ p 16 65 1.75 6 12.66 2.5 167.4. However, the value tjk did not represent an accurate cycle time, since the average production time may So, if there are not too many setup times, the original depend on the run length of the batches at the bottle- assumptions for the medium term planning model are neck. It may be that the schedule generated in the appropriate. detailed scheduling process has batch sizes that are However, the shipment to the customer is supposed very short and therefore an average production time to go on a truck at time 120 (after 5 days), whereas the that is larger than the estimates used in the medium shipment to the dc takes place at the end of the week term planning process. If there is a major discrepancy at time 168. All the raw material required to produce (i.e., the frequency of the setups is considerably higher Family 1 products are available at time 0, whereas the than usual), then a new estimate may have to be material necessary to produce family 2 products are developed for the tp in the medium term planning only available after 2 days, i.e., at time 48. jk process and the integer programming problem has to This problem can be modeled as a single machine be solved again. scheduling problem with the jobs having different release dates and sequence-dependent setup times and as objective 6. Carlsberg Denmark: An Example of ␣ ϩ ␣ ͸ a System Implementation 1 Cmax 2 wj Tj . There are many software vendors that sell custom- There are 4 different jobs (Table 8), with the following made solutions for supply chain planning and sched- processing times, release dates, and sequence-depen- uling. One of the largest companies in this field is sap, dent setup times. Each job is characterized by its fam- based in Walldorf (Germany). sap has a division that ily type and its destination. Jobs 1 and 2 correspond to develops its Advanced Planner and Optimizer (apo) jobs from the same family, so there is a 0 setup time if system. This supply chain planning and scheduling one job is followed by the other. If either job 1 or job 2 system has functionalities on various levels, including follows job 3 or 4, then a setup of 16 hours is required. (i) the tactical level and (ii) the operational level. If job 3 or 4 follows job 1 or 2, a setup of 6 hours is On the tactical level, the medium term planning required. scenarios for the global chain are monitored (from Two rules can be applied. One rule would follow distribution centers to plants and suppliers). The op- the Shortest Setup Time first rule (with ties being timizer automatically processes bills of materials broken according to the Earliest Due Date rule). This while taking capacities into account, and optimizes rule would generate the schedule 1, 2, 4, 3, since after transportation costs, production costs, storage costs, job 1 has been completed, job 2 has to be started since and revenues for demand. The sheer complexity of it has a 0 setup time. All jobs are then completed by this global view is handled by a rough-cut model that time 168. However, job 4 is completed tardy. It had to aggregates the time in buckets (e.g., day or week) and be shipped by time 120 and it is shipped by time 161. products and resources in families. The second rule follows the Earliest Due Date rule On the operational level, apo has a detailed sched- (with ties being broken according to the Shortest Setup uling model. At this level, the short term, day-to-day Time first rule). It results in the schedule 2, 4, 3, 1. operations are monitored, focusing especially on ex- Since there is an additional setup time, the makespan ceptions in supply chain operations. The optimizer Kreipl and Pinedo: Planning and Scheduling in Supply Chains Production and Operations Management 13(1), pp. 77–92, © 2004 Production and Operations Management Society 89

Figure 6 SAP-APO Optimizer Architecture ders for brewing and filling, different lot size constraints have to be taken into account. Production orders for the brewing have always a fixed lot size because the brewing tank has to be filled. If the demand quantity is higher than the fixed lot size, then additional production orders have to be created for the brewing process (each with the fixed lot size as the production quantity). Orders below the minimal lot size are increased to the minimal lot size and orders above the minimal lot size are either rounded up or down to the closest integer value. The filling resources have to be filled up to 100%. There is further a split in the business processes according to the sales volumes of the various products. There are three categories: A, B, and C. Category A are the fast movers and include the well-known brands Carlsberg Pils and Tuborg Green. Category C are the (more expensive) slow movers. schedules orders according to manufacturing con- Once the beer is bottled, it has to be transported straints that handle complex manufacturing environ- either to a central depot or to a local depot. Depending ments with alternative routings and resources, sec- on the different products and the quantities to be ondary resources, and multi-stage production. transported, either a direct delivery from the factory to Figure 6 shows the architecture of the optimizer in a local depot or a transport via the centralized depot is apo. For long and medium term planning, apo uses its better. Again, lot size constraints have to be taken into lp solvers (cplex). For detailed short term planning consideration when creating transport orders. The and scheduling, apo has various approaches, includ- transport durations depend, of course, on the origin ing Constraint Programming, Genetic Algorithms, and the destination. and Repair Algorithms. One of the main objectives of Carlsberg is to provide This section describes an implementation of the a given level of service to its customers. A typical way sap-apo system at the beerbrewer Carlsberg A/S in to achieve a given service level is to keep safety stocks Denmark. The modeling that forms the basis for this at the depots. The higher the safety stocks, the higher case is somewhat similar to the models described in the service level, but also the higher the inventory Sections 5 and 6 of this paper. Carlsberg Denmark costs. One function of a supply chain management A/S, the largest beerbrewer in Scandinavia, started in system is the computation of the lowest levels of 2001 a supply chain project with the objective to de- safety stocks that achieve the desired service levels. crease inventory costs, to optimize sourcing decisions, Carlsberg uses advanced safety stock methods to com- to increase customer service level, and in general to pute safety stock values for all its products at its change the business to a more demand-driven pro- central as well as at its local depots. These safety stock cess. Carlsberg selected apo. The scope of the project levels depend on the given service level, the demand gives an example of how a real-life implementation is forecast, the uncertainty in the forecast, the replenish- considering the planning and scheduling issues over ment lead time, and the typical lot sizes. several stages. The system is operational since the end The medium term planning module plans ahead for of 2002. 12 weeks, with the first 4 weeks in days and the The supply chain considered in the project consists remaining 8 weeks in weekly periods. Assuming a of three stages. The first stage is the production pro- given demand pattern (sales orders and forecasts), cess of the beer at two breweries with two and four apo creates a Mixed Integer Program, along the lines filling lines, respectively. Each filling line has a differ- described in Section 5, and tries to find a solution with ent capacity. The second stage consists of the central- minimum cost. The total costs are the sum of the ized warehouses, and the third stage consists of the production costs, the storage costs, the transportation local warehouses; see Figure 5. In the first stage, there costs, the late delivery (tardiness) costs, the non-deliv- are three production steps, namely brewing (and fer- ery costs, and the violation of the safety stock levels mentation), filtering, and filling of the beer. All three computed in the first step. Some of the costs men- steps have a limited capacity, but the bottleneck is tioned above can be specified in an exact way, such as usually the filling step. The resources for the filling the production and transportation costs. Other costs, operations at the two plants have different costs and such as storage, violation of safety stock, and late and processing times. When creating the production or- non-delivery costs, merely represent the priorities of Kreipl and Pinedo: Planning and Scheduling in Supply Chains 90 Production and Operations Management 13(1), pp. 77–92, © 2004 Production and Operations Management Society

Carlsberg. If, for example, Carlsberg considers the and the results that come out of the detailed schedul- safety stock in the local warehouses more critical than ing procedure. An attempt is made to fill up the trucks the safety stock in the central warehouse, then the cost with several products in order to ship mainly full assigned to the violation of safety stock for a product trucks. at the central warehouse is less than the cost of vio- A new medium term plan is generated every day. lating the safety stock of the same products at the local The daily run takes into account the most up-to-date warehouses. If neither safety stock can be maintained, capacity situation of all the available resources, the then the system will create a transport from the dc to results of the previous day detailed schedule, and the the local warehouse (provided the difference between most current demand forecast. Afterwards, another the costs of safety stock violations at the dc and at the detailed schedule and transportation plan are gener- local warehouse is higher than the transportation cost ated. from the dc to the local warehouse). Clearly, all cost The generation of the medium term plan was split types are strongly related with one another, and mod- into three Mixed Integer Programs, which were solved ifying one type of cost can have many unforeseen in consecutive runs. Each mip had between 100,000 consequences in the solution generated. Carlsberg de- and 500,000 variables and between 50,000 and 150,000 veloped its own cost model for storage costs; this constraints. Total running time was about 10–12 model, for example, takes into account the location hours. Each mip used product decomposition meth- occupied by a pallet, the maximum number of levels ods, which created 5 to 10 subproblems each. The pallets can be stacked, the number of products per generation of the subproblems has to take into account pallet, and the warehouse itself. Based on these pa- different priorities of the finished beer products and rameters for each product at each location, storage the fact that the same brewed and filtered beer type costs can be computed. may end up in different end products. The quality of The following constraints have to be taken into con- the solution is measured in business terms as well as sideration, namely the production times in the three in technical terms. Only by considering all dimensions production steps, the capacity of the bottling resources one can speak of a “good” or a “bad” solution. The on a daily or weekly level, the transportation times technical data and measures tend to be easy to collect between locations, the lot size constraints, the existing and understand; the more important business mea- stock, and the resource consumptions. sures are harder to understand and verify. The two The medium term plan is the result of various costs most important technical measures are (i) the differ- trade-offs and material consumption. The system gen- ence between the costs of the mip solution and the lp erates for the next 12 weeks the planned production relaxation solution, and (ii) the difference between the quantities for the three production steps in detail (in- overall delivery percentages of the mip solution and cluding the quantity of each product to be bottled on the lp relaxation solution. The difference between the each filling resource as well as the quantities to be costs was on average between 0.2% and 10%, but transported between the locations). sometimes shot up to 400%. A huge cost difference The short term scheduling starts its computations between the mip and the lp relaxation could occur in using the results obtained from the medium term a case where the lp could fulfill all demands, while the plan. The planned production orders for the first week mip could not fulfill all demands because of lot size that come out of the medium term planning system constraints. As unfilled demand brought about very are transformed into short term production orders on which a detailed scheduling procedure has to be applied. These production orders are then scheduled on Figure 7 Carlsberg-Denmark Planning System User Interface the filling resources by applying a genetic algorithm with as objective the minimization of the sum of the sequence-dependent setup times and the sum of the tardinesses. The due dates are specified by the medium term planning problem and are equal to the starting times of the transportation orders. It is possible that the results of the medium term plan are changed by the short term scheduling procedure (i.e., a different filling resource may be selected in the same plant). After the detailed scheduling has been completed, the transportation planning and scheduling is done. In this step, the trucks between the locations are filled based on the results of the planned transportation orders that come out of the medium term plan Kreipl and Pinedo: Planning and Scheduling in Supply Chains Production and Operations Management 13(1), pp. 77–92, © 2004 Production and Operations Management Society 91

Figure 8 Carlsberg-Denmark Scheduling System User Interface scheduling module, the unit may be an hour or a day; in the planning module, it may be a week or a month. Comparing the modeling that is done in practice for medium term planning processes with models that have been studied in the research literature, it be- comes clear that there are differences in emphasis. When multi-stage models are considered in the planning and scheduling research literature, there is more of an emphasis on setup costs (typically sequence- independent) and less of an emphasis on transportation costs; in the modeling that is done in practice, there is a very strong emphasis on transportation costs and less of an emphasis on setup costs. Incorporating setup costs as well as transportation costs in a multi- stage planning model may cause the number of variables to become prohibitively large.

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