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1 Computers and Chemical Engineering 33 (2009) Contents lists available at ScienceDirect Computers and Chemical Engineering journal homepage: Integration of production planning and scheduling: Overview, challenges and opportunities Christos T. Maravelias, Charles Sung Department of Chemical and Biological Engineering, University of Wisconsin-Madison, 1415 Engineering Dr., Madison, WI, 53706, USA article info abstract Article history: Received 9 October 2008 Received in revised form 8 May 2009 Accepted 1 June 2009 Available online 11 June 2009 Keywords: Supply chain management Production planning Scheduling Mixed-integer programming We review the integration of medium-term production planning and short-term scheduling. We begin with an overview of supply chain management and the associated planning problems. Next, we formally define the production planning problem and explain why integration with scheduling leads to better solutions. We present the major modeling approaches for the integration of scheduling and planning decisions, and discuss the major solution strategies. We close with an account of the challenges and opportunities in this area Elsevier Ltd. All rights reserved. 1. Introduction The supply chain (SC) of a manufacturing company is a network of facilities and distribution options that performs the following functions: procurement of raw materials, transformation of raw material into finished products, and distribution of finished products to customers. The goal is to achieve high customer satisfaction level at low cost (Christopher, 1998; Chopra & Meindl, 2001; Shapiro, 2006). Chemical supply chains in particular contain large opportunities to reduce cost: they are complex interconnected systems that change constantly, and their activities represent a significant portion of total cost to serve customers (Ferrio & Wassick, 2008). Tayur (2003) noted that inventories in US supply chains can be reduced substantially, without affecting customer satisfaction levels, leading to significant savings. Inventory levels can be reduced if the efficiency of the SC as a whole is improved. Higher efficiency can be achieved through proper coordination of material, financial and information flows across the SC (Grossmann, 2005; Stadtler, 2005; Varma, Reklaitis, Blau, & Pekny, 2007). The planning problems that have to be solved to achieve this coordination cover a wide range of activities, from procurement and production to distribution and sales, and a wide range of time scales from long-term (strategic) to short-term (operational) decisions (Fig. 1). Corresponding author. Tel.: ; fax: address: maravelias@wisc.edu (C.T. Maravelias). Strategic (long-term) planning determines the structure of the supply chain (e.g. facility location). Medium-term (tactical) planning is concerned with decisions such as the assignment of production targets to facilities and the transportation from facilities to warehouses to distribution centers. Finally, short-term planning is carried out on a daily or weekly basis to determine the assignment of tasks to units and the sequencing of tasks in each unit. At the production level, short-term planning is referred to as scheduling. However, due to interconnections between different levels of the supply chain, there are numerous trade-offs between decisions made at the various nodes of the SC. To achieve globally optimal solutions therefore the interdependencies between the different planning functions should be taken into account, and planning decisions should be made simultaneously. In other words, planning problems should be integrated. In this paper, we specifically review approaches for the integration of medium-term production planning and short-term scheduling (Shah, 2005). In Section 2, we review production planning and present the standard lot-sizing formulation that is often used in production planning systems. In the next section, we discuss why integration with scheduling is necessary, review the major approaches to process scheduling, and discuss the implementation of production planning solutions. In Sections 4 and 5, we review the different modeling approaches and discuss the main solution strategies developed to solve the integrated models effectively. We close with a discussion of open challenges in this area and some promising research directions /$ see front matter 2009 Elsevier Ltd. All rights reserved. doi: /j.compchemeng

2 1920 C.T. Maravelias, C. Sung / Computers and Chemical Engineering 33 (2009) Nomenclature Indices i I k K n N t T product (item) resource small-bucket (scheduling) time period of length n big-bucket (planning) time period of length t Sets D i N t X t Y t set of direct successors of item i scheduling time periods in planning period t domain of planning variables in period t domain of scheduling variables in period t Parameters c i setup cost for item i C kt capacity of resource k during period t d it demand for product i at end of time period t h i holding cost of product i M it upper bound on production of item i during period t p i production cost per unit of item i r ii amount of item i required to make one unit of successor item i D i ik capacity requirement of resource k per unit of item i ˇik capacity requirement of resource k to setup for item i i lead time for item i, in units of time periods Variables Ch t total holding cost in time period t Cp t total production cost in time period t CT total overall cost (objective function) P it production amount (target) of item i in period t S it inventory level of item i at the end of time period t Y it =1 if item i is produced in time period t Functions f(p it ) generic function for constraining feasibility of production targets g(p it ) generic function for defining cost to meet production targets 2. Production planning 2.1. Problem statement The objective in production planning is to fulfill customer demand at minimum total (i.e. production + inventory) cost. Formally, we are given: Fig. 1. Supply chain planning matrix (modified from Meyr, Wagner, & Rohde, 2002). We are interested in the integration of medium-term production planning and shortterm scheduling (highlighted). See also Fleischmann, Meyr, and Wagner (2002). If the demand cannot be satisfied in every period, then two variants are considered. In the first one, unsatisfied demand is backlogged and a backlog cost is paid until the backlogged demand is satisfied. In the second one, unsatisfied demand is discarded at cost. Production planning is often represented as a network problem, with a node for each item and time period, and arcs for the production, demand satisfaction, and inventory (see Fig. 2). The network representation can be extended to include backlog arcs General formulation If we assume that demand can always be satisfied, then a general formulation for production planning is given in (PP). Feasible production targets are modeled via functions f(p it ) in Eq. (RC), production cost Cp t in period t is calculated via function g(p it ) in (PC), holding cost Ch t is calculated in Eq. (HC), and the material balance for item i at the end of period t is expressed in Eq. (MB). min CT = (Cp t + Ch t ) t T s.t. f (P it ) 0 t (RC) Cp t = g(p it ) t (PC) Ch t = h i S it t (HC) i S it = S i,t 1 + P it d it i, t (MB) P it,s it 0 i, t (PP) Generic functions f(p it ) and g(p it ) depend on the characteristics of the process network and often involve a large number of constraints. The former defines the set of feasible production amounts P it, while the later expresses the production cost as a function of P it. To accurately provide feasibility and production cost information, detailed models with additional variables are used. Among the various production planning methods, their major modeling differences lie in the modeling of resource constraints (RC) and production cost constraints (PC). (i) A planning horizon divided into a set T of time periods. (ii) A set I of products (items) with holding cost h i, and customer demand d it for product i I due at the end of time period t T. (iii) Resource capacities. (iv) Production costs. The optimization decisions include: (i) Production amount (target) P it of item i I in period t T. (ii) Inventory level S it of item i at the end of period t. Fig. 2. Flows in production planning (shown here for item i).

3 C.T. Maravelias, C. Sung / Computers and Chemical Engineering 33 (2009) Lot-sizing formulation A formulation that is used to address production planning problems in the Operations Research (OR) literature is the multi-item capacitated lot-sizing formulation with setup times and costs. Resource constraints f(p it ) 0 are expressed via the two constraints in Eq. (RC1), where the binary variable Y it is equal to 1 if item i is produced in period t, ik is the capacity requirement of resource k K per unit of item i, ˇik is the setup for item i in resource k, C kt is the capacity of resource k during period t, and M it is an upper bound on production of item i during period t. ik P it + ˇik Y it C kt t, k (RC1) i i P it M it Y it i, t The production costs are calculated in (PC1), where c i and p i are the setup and unit production costs for item i: Cp t = (c i Y it + p i P it ) t (PC1) i By replacing Eqs. (RC) and (PC) with Eqs. (RC1) and (PC1) we obtain the basic production planning formulation (PP1), which can be extended to address multi-level production problems, using the ideas of lead time and bill of materials. In this case, Eq. (MB) is replaced by Eq. (MB1), S it = S i,t 1 + P i,t i dit + i, t (MB1) i D i r ii P i t where i is the lead time for item i, D i is the set of direct successors of i, and r ii is the amount of i required to make one unit of i D i (Pochet & Wolsey, 2006). Numerous reformulation results and decomposition methods (e.g. column generation, Lagrangian relaxation/decomposition) have been proposed to improve the solution of lot-sizing-based production planning problems (Miller, Nemhauser, & Savelsbergh, 2003). To obtain more accurate production targets, the above formulations have also been extended to include overtime, product substitutes, productivity and capacity utilization. These extensions form the basis of many production planning systems (Pochet & Wolsey, 2006). 3. Integration with scheduling 3.1. Why is integration necessary? The recent trend towards product customization and diversification in the chemical industry has led to multi-product facilities, which are often complex process networks (batch mixing/splitting) with multiple utilities, and sequence-dependent changeover times and costs. At the same time, it is imperative that facilities be able to respond to demand fluctuations. This implies that existing assets have to be utilized close to their capacity. The production planning of heavily loaded units subject to complex operational constraints is a challenging task because production targets have to be feasible while being close to system limits. To address this challenge, researchers and practitioners in the process systems engineering (PSE) community have proposed production planning methods that incorporate scheduling submodels (Bassett, Pekny, & Reklaitis, 1996; Grossmann, Van den Heever, & Harjunkoski, 2002). In terms of formulation (PP), this means that more detailed formulations are used for the resource constraints in Eq. (RC) and the production cost in Eq. (PC). These formulations can be grouped into three categories: (a) detailed scheduling models, (b) relaxations/aggregations of scheduling models, and (c) surrogate models derived through off-line analysis of the manufacturing facilities. Since the resulting integrated planning scheduling models may be hard to solve, several solution strategies have also been proposed. These strategies can be broadly classified into: (a) hierarchical methods, (b) iterative methods, and (c) full-space methods. The first two decompose the integrated problem into a master and a slave subproblem, while the last considers the integrated problem. However, MIP decomposition approaches (e.g. Lagrangian relaxation/decomposition) can be used to solve a full-space model. Since almost any modeling approach can in theory be combined with any solution strategy, we review these separately. We start with modeling approaches in Section 4 and continue with solution strategies in Section Scheduling approaches Before we discuss how scheduling models are used in integrated formulations, we briefly review the major approaches to scheduling in PSE. In general, scheduling is a decision-making process that concerns the allocation of limited resources to competing tasks over time with the goal of optimizing one or more objectives (Pinedo, 2002). While there are many different classes of scheduling problems, the major scheduling decisions most often are the assignment of production tasks to processing units and the sequencing and timing of tasks on each unit. Thus, the general scheduling problem can be posed as follows: Given are: (i) Production facility data; e.g., processing unit and storage vessel capacities, utility availability, unit connectivity. (ii) Detailed production recipes; e.g. stoichiometric coefficients, processing times, processing rates, utility requirements. (iii) Production costs; e.g. raw materials, utilities, cleaning, etc. (iv) Production targets or orders with due dates. Our goal is to determine: (i) The allocation of resources (equipment units and utilities) to processing tasks. (ii) The sequencing and timing of tasks on processing units. Typical objective functions include the minimization of makespan, lateness and earliness, as well as the minimization of total cost. Finally, batching decisions (i.e., the number and size of batches) are often treated as planning decisions (and thus provided to the scheduling problem) but can also be viewed as part of the scheduling problem. Scheduling formulations are most often expressed as mixedinteger programming (MIP) models, leading to an integrated planning scheduling MIP model. However, other modeling and solution paradigms can be used, such as constraint programming (CP) (Baptiste, Le Pape, & Nuijten, 2001; Hooker, 2006). In this case, however, the integrated model will be a hybrid MIP/CP model, which will require a special solution strategy. Scheduling formulations can be broadly classified into: (i) network-based formulations for general processes and (ii) batchbased formulations for sequential processes. The former is used to address problems in complex process networks with batch mixing/splitting and recycle streams, whereas the latter is used for single-stage, multi-stage and multi-purpose processes where batches are processed sequentially and where batch splitting/mixing are not allowed and there are no recycle streams. Network-based approaches can be further classified into discrete-time formulations (Kondili, Pantelides, & Sargent, 1993; Pantelides, 1994) where the time horizon is divided a priori,

4 1922 C.T. Maravelias, C. Sung / Computers and Chemical Engineering 33 (2009) Fig. 3. Broad classification of proposed scheduling approaches. Shown approaches can be further classified; e.g. precedence-based approaches can be divided into immediate and global precedence formulations. possibly into equal sub-periods; continuous-time formulations (Schilling & Pantelides, 1996; Zhang & Sargent, 1996) where the time horizon is partitioned as part of the optimization; and mixedtime formulations (Maravelias, 2005) where the time grid is fixed but the durations of the tasks are variable. Continuous-time formulations are further subdivided into approaches that employ common (Mockus & Reklaitis, 1999; Castro, Barbosa-Povoa, & Matos, 2001; Maravelias & Grossmann, 2003; Sundaramoorthy & Karimi, 2005) and unit-specific time grids (Giannelos & Georgiadis, 2002; Ierapetritou & Floudas, 1998). Network-based formulations consider batching and scheduling decisions simultaneously and can be readily extended to account for utility and storage constraints. In batch-based formulations batches are treated as discrete entities moving through the different stages of the process, thus preserving batch identity. Most batch-based formulations employ sequencing variables and constraints rather than a time grid. Traditional sequential approaches do not consider batching decisions, which means that the batching problem has to be solved prior to the scheduling one. They can be further classified into slot-based (Pinto & Grossmann, 1995), precedence-based (Cerda, Henning, & Grossmann, 1997; Gupta & Karimi, 2003; Mendez, Henning, & Cerda, 2001), and formulations that use resourcetask network (RTN) ideas (Castro & Grossmann, 2005; Castro, Grossmann, & Novais, 2006). Finally, Prasad and Maravelias (2008) and Sundaramoorthy and Maravelias (2008a,b) recently developed sequential approaches that consider batching and scheduling simultaneously. A broad classification of scheduling methods is given in Fig. 3. A thorough review of scheduling problems, modeling approaches and solution strategies can be found in Mendez, Cerda, Grossmann, Harjunkoski, and Fahl (2006). Note that in this approach we use two time grids: a big-bucket planning grid (weeks or months) and a small-bucket scheduling grid (hours), with the former being the scheduling horizon of the latter. If a discrete-time scheduling grid is used, then linking constraints can be readily expressed (see Fig. 4), but when a continuous-time scheduling approach is followed, this is not trivial because the end of each planning period cannot be assigned a priori to a scheduling period/event/slot. The same is true for aggregations and relaxations of scheduling models. Also, since both market and production environments are dynamic, production targets should be constantly updated to respond to disturbances such as raw material delivery delays, equipment failures, production delays, demand changes, etc. Thus, a production planning solution P it is typically used only for a few early periods, after which problem data is updated, and the problem is re-solved over a new planning horizon. In other words, by its very nature production planning is solved in a rolling horizon manner. It is important to note here though that this iterative implementation is not related to the modeling or solution strategy, i.e. a production planning solution is implemented in a rolling horizon fashion regardless of how it is obtained. Nevertheless, as we will discuss later, this iterative implementation has inspired rolling horizon-based solution approaches. In this context, the integration of planning and scheduling is necessary primarily in order to obtain a feasible and (near) optimal production planning solution; not to obtain a scheduling solution that will be readily implemented Implementation From an operational point of view, production targets P it obtained by solving the production planning problem are used as inputs to scheduling, thus leading to a hierarchical flow of information from production planning to scheduling. However, as explained above, to obtain feasible and (near) optimal production targets, process capacity and cost information have to be communicated from the scheduling to the production planning formulation via the integration of some form of a scheduling model. In principle, the linking of the two problems can be achieved via constraints that enforce that production targets for each planning period are equal to the orders due at the end of the corresponding scheduling horizon. Fig. 4. Integrated formulation using a discrete-time scheduling submodel. A planning model is expressed for big-bucket time grid (t T), while scheduling variables and constraints (e.g. assignment and resource) are expressed for small-bucket time grid (n N). Communication is achieved via constraints linking production targets with demands and production costs between the two levels at the end of each planning period.

5 C.T. Maravelias, C. Sung / Computers and Chemical Engineering 33 (2009) Modeling approaches 4.1. Detailed scheduling models In this section we outline two natural approaches to the integration of scheduling formulations with model (PP). However, other approaches can also be followed. In the first approach, we replace resource and production cost constraints in Eqs. (RC) and (PC) for each planning period with a scheduling submodel and use linking constraints as described in the previous section. This leads to the following general formulation (PP2): min c t x t t s.t. A I t x t b I (I) (PP2) t A II t x t + B t y t b II t, t (II) x t X t,y t Y t where x t are the planning variables for planning period t, y t are the scheduling variables in the submodel used for planning period t, Eq. (I) includes Eqs. (MB) and (HC), and Eq. (II) corresponds to the scheduling submodel of choice and the linking constraints. In the second approach, we consider a monolithic scheduling model over the entire planning horizon. In this case, planning variables can be replaced by scheduling variables and Eqs. (MB) and (HC) can be eliminated. If point t of the big-bucket planning grid corresponds to point n t of the small-bucket scheduling grid, then planning variable P it can be replaced by the scheduling shipment variable at n t, planning inventory S it can be replaced by the corresponding scheduling inventory variable at point n t, etc. If N t = {n t 1 +1, n t 1 +2,..., n t } is the set of small-bucket periods within planning (big-bucket) period t, then scheduling variables can be grouped into subsets Y t : Y = t Y t. The integrated problem therefore can be written as follows: min c t y t t s.t. A t y t b I (I) (PP3) t B t y t y t Y t b II t, t (II) Note that in formulation (PP3) the blocks are linked via a set of linking constraints rather than a set of linking variables as in (PP2). These constraints typically involve activities occurring near the boundaries between planning time buckets. For example the material balance at point n t + 1 includes inventory variables at n t and n t + 1 which belong to subsets Y t and Y t+1, respectively Relaxed and aggregated scheduling formulations Though integrated models with detailed scheduling formulations can in principle provide optimal production targets, they most often result in large MIP models that cannot be solved to optimality. One way to overcome this limitation is through use of advanced solution strategies, a topic we review in the next section. Another option is to develop an approximation of the original model which provides some short-term information while being easier to solve. The approximate model is obtained by removing some of the constraints, or by aggregating some of the decisions of the original scheduling formulation. A common strategy is to keep job assignment constraints and variables but discard sequencing constraints and variables (Harjunkoski & Grossmann, 2002; Jain & Grossmann, 2001; Maravelias, 2006; Roe, Papageorgiou, & Shah, 2005; Romero, Badell, Bagajewicz, & Puigjaner, 2003). Aggregation reduces the number of variables and constraints in a formulation by combining them in various ways (Bassett, Pekny, et al., 1996). Wilkinson, Shah, and Pantelides (1995) studied a scheduling model in which the time horizon was originally comprised of many sub-periods. They proposed aggregating sub-periods into larger periods (i.e. planning periods) with the restriction that variables and constraints near period boundaries remained disaggregated. Vancza, Kis, and Kovacs (2004) proposed aggregation of standard data by time, resource capacity, and operations. Simplified models can also be generated from analyzing the scheduling model in regards to production campaigns. Birewar and Grossmann (1990) proposed using an aggregate traveling salesman problem (TSP) LP for sequencing batches within a multi-period planning model in order to estimate the time requirements in the production planning of parallel lines of flowshop plants. Sukoyo, Matsuoka, and Muraki (2004) also discussed enumerating mixedproduct campaigns and then using these as units of allocation in the production planning problem. Wellons and Reklaitis (1991) proposed using optimization to identify cyclic campaigns with dominant production rates. Their simplified model constrained total campaign time plus setup time to be less than one planning period. Henriques (2006) pointed out that, to be efficient, production rates should be constrained between rate of minimum cost and rate of maximum production. Erdirik-Dogan and Grossmann (2007) developed a planning model for single and two-stage parallel batch reactors in which the effect of sequence-dependent changeovers for scheduling is anticipated by modeling TSP constraints into the proposed model Off-line surrogate models An alternative method for generating an accurate but computationally tractable description of the resource constraints and production costs of a facility is to carry out off-line calculations. The goal of this approach is to generate constraints that define the feasible region of the scheduling model and the production cost as functions of production targets P it only. The generation of these constraints can be resource-intensive, but once they are generated off-line they can be incorporated in the integrated formulation without further computation. For example, Wan, Pekny, and Reklaitis (2006) proposed a simulation-based optimization approach to create a support vector for use as a surrogate model. Sung and Maravelias (2007) presented a projection-based method that uses off-line computations based on a detailed scheduling model to generate the convex hull of feasible production targets and a convex underestimation of total production cost. These approximating functions are expressed in terms of the planning variables P it (i.e. the scheduling variables are projected out), yet provide all the necessary information to solve the production planning problem effectively (Fig. 5). They also proposed maintaining a repository of known-to-be-feasible solutions for later use in a rolling horizon algorithm Hybrid modeling for rolling horizon approaches A compromise between modeling accuracy and computational burden is to use detailed scheduling models for a few early periods and a relaxation, aggregation or surrogate formulation for late periods. As explained in the previous section, production targets are continuously updated, which in some cases implies that approximate solutions in later periods do not significantly affect the quality of the solution. This hybrid approach lends itself to rolling horizon solution methods.

6 1924 C.T. Maravelias, C. Sung / Computers and Chemical Engineering 33 (2009) scheduling submodels back to the master problem, then the methods are iterative. If the integrated formulation contains detailed scheduling submodels for each planning period, then its solution provides all the necessary information. However, these models are hard to solve and require advanced solution methods. We will refer to these methods as full-space methods Hierarchical methods Fig. 5. Projection-based method of Sung and Maravelias (2007). (a) Complex process network with six processing units (U1 6), 10 tasks (T1 10), and 13 chemicals (2 feeds F1 2, 8 intermediates S1 6 and INT1 2, and 3 products A C). (b) Projection onto (P At, P Bt, P Ct)-space of the feasible region of scheduling model describing process network in (a). 5. Solution strategies Solution strategies for the integrated planning scheduling problem can be classified into three categories as shown in Fig. 6. In decomposition methods the problem is decomposed into a master (high-level) subproblem used to determine production targets and a slave (low-level) subproblem with detailed scheduling. The production targets or other high-level decisions are used as inputs to the slave subproblem. If the flow of information is only from the master subproblem towards the slave subproblem(s) then the methods are hierarchical. If there is a feedback loop from the In hierarchical methods, the master problem provides a set of high-level decisions, such as production targets and selection of tasks (McKay, Safayeni, & Buzacott, 1995). This information is then fed as input to the lower-level scheduling subproblem with the goal of obtaining a complete scheduling solution (Amaro & Barbosa-Póvoa, 2008). If a feasible schedule with predicted production amounts does not exist, then a feasible schedule in the neighborhood of this is sought out so as to have a globally feasible solution (Kelly & Zyngier, 2008). To provide more realistic predictions, Subrahmanyam, Pekny, and Reklaitis (1996) proposed using an approximate scheduling model at the master level, an approach that is widely followed. Furthermore, given an infeasible set of production targets a number of heuristic rules (e.g. shifting, partial batch mixing/splitting) can be applied to obtain a feasible schedule or a posteriori improve the schedules obtained (Bassett, Pekny, et al., 1996; Grunow, Günther, & Lehmann, 2002). Yan, Xia, Zhu, Liu, and Guo (2003) discussed a hierarchical approach, where they first solve the production planning problem in the presence of aggregate capacity constraints to get production amounts and then use tabu-search to ensure feasibility at the lower level. Hierarchical decomposition can also be used within a rolling horizon framework, where detailed scheduling models are used for a few early periods and aggregate models are used for later periods. The production targets for the early periods are exact and thus directly implemented, while the targets for the later periods are updated as the horizon rolls (Dimitriadis, Shah, & Pantelides, 1997; Wu & Ierapetritou, 2007). Rolling horizon solution methods are grouped here as hierarchical because there is no information going back to the master problem during the solution process. Lin, Floudas, Modi, and Juhasz (2002) presented a three level integrated model for medium-term multi-stage production scheduling. The first level is solved to optimize the length of the rolling horizon window and product inclusion. The second level is solved to include additional products until the minimum operational levels of some units are met. The final level is a scheduling model solved to maximize throughput for chosen products. Then, the first level is solved again to choose the next rolling horizon window and this continues until the overall horizon is visited. Van den Heever and Grossmann (2003) presented a hierarchical decomposition scheme for an integrated production planning and reactive scheduling problem for hydrogen supply. The upper level is solved to determine production amounts and feed/energy prices in the presence of a simplified pipeline model and the lower level is solved in a rolling horizon fashion to determine unit operation in the presence of a detailed pipeline model. Finally, Honkomp, Mockus, and Reklaitis (1999) and Sand and Engell (2004) discussed hierarchical approaches that employ rolling horizon methods to address problems under uncertainty Iterative methods Fig. 6. Solution strategies for integrated production planning and scheduling. In the absence of detailed resource constraints and production costs, the production targets or task-unit assignments obtained by the first solution of the master problem are likely to be infeasible or suboptimal. Instead of trying to find feasible schedules that are in the vicinity of these decisions, iterative methods attempt to

7 C.T. Maravelias, C. Sung / Computers and Chemical Engineering 33 (2009) close the information loop from the scheduling subproblems to the master problem. The goal of such feedback is to find the true optimal high-level decisions. This can be achieved via the addition of integer cuts that exclude previously found solutions. Therefore, different solutions can be found by the master problem and evaluated by the lower-level subproblem. In addition, the master problem can provide an increasing lower bound while the subproblem can provide an upper bound. Thus, iterative methods can lead to optimal solutions if solved until the gap is closed. Papageorgiou and Pantelides (1996) proposed an integrated campaign planning and scheduling model where each higher level time period is made up of cyclic campaigns. The higher level problem is solved to fix a set of binary variables, i.e. campaigns are chosen. The lower level problem is used to generate campaigns. At the end of each iteration, an integer cut is added to prevent the same set of binary variables from being found again. Erdirik-Dogan and Grossmann (2006) proposed an integrated planning and scheduling model for scheduling continuous tasks on a single machine. The higher level is solved to fix a set of binary variables, i.e. assign tasks to time periods. At the end of each iteration, several different types of integer cuts are added to prevent the same or similar sets of binary variables from being found again. Stefansson, Shah, and Jensson (2006) presented a three-level integrated model and an iterative decomposition approach for multi-stage scheduling of orders in the pharmaceutical industry. The three levels involve campaign planning, campaign planning and order scheduling, and scheduling with setups. Wu and Ierapetritou (2007) discussed a combination of hierarchical and iterative decomposition schemes for a problem under uncertainty, with an outer hierarchical decomposition and an inner iterative decomposition. Many researchers have also proposed iterative methods, where a master MIP model is solved to obtain task-unit assignments and a constraint programming subproblem to find feasible sequences. Hooker, Ottosson, Thorsteinsson, and Kim (1999) and Jain and Grossmann (2001) considered the single-stage problem, while Harjunkoski and Grossmann (2002) studied multi-stage processes. Chu and Xia (2005) proposed for a resource-constrained scheduling problem a two level IP/CP iterative method that adds Bender s cuts to the high-level problem. Maravelias and Grossmann (2004) and Roe et al. (2005) also proposed hybrid methods for network processes. In general, iterative methods have a lot of promise, however their application has thus far been very formulation-specific Full-space methods The methods of this class consider the integrated problem, where a detailed scheduling model is used for the modeling of resource constraints and production costs. The first approach is the solution of the full-space model using standard mathematical programming methods (Bassett, Pekny, et al., 1996; Blanco, Masini, Petracci, & Bandoni, 2005; Kallrath, 2002; Kelly, 2004; Papageorgiou & Pantelides, 1996). Karimi and McDonald (1997) developed two formulations for planning and scheduling. In the first, tasks are assigned to unit-specific slots. However, the planning problem interacts with scheduling-defined production amounts via period boundaries (e.g. demand fulfillment), so slots in this formulation are disaggregated by time period as well. In the second formulation, slots are explicitly indexed by unit and by planning period. Joly, Moro, and Pinto (2002) proposed an integrated model for a refinery. The planning problem defined refinery topology and operating points, while the scheduling problem managed crude oil unloading from pipelines, transferring to storage tanks, and charging into units. Neiro and Pinto (2005) considered a multi-period formulation for refinery production planning. Scheduling models by themselves, however, are hard to solve despite only covering a horizon of several days or weeks. If the same level of detail is maintained, therefore, the integration with production planning that extends the horizon to several months results in computationally intractable models. Hence, the second approach is to use heuristic methods such as simulated annealing (Reklaitis, 2000) and genetic algorithms (Berning et al., 2004; Yan & Zhang, 2007). The third approach is to decompose the integrated formulation by exploiting its structure. The decomposed subproblems can then be solved iteratively using standard mathematical programming methods such as Bender s decomposition (Benders, 1962; Geofrion, 1972) or Lagrangian relaxation/decomposition (Everett, 1963; Fisher, 1981; Guignard & Kim, 1987). Note that these iterative methods are different from the ones presented in the previous subsection in that they consider the original integrated planning scheduling formulation. Bender s decomposition is typically used when the incidence matrix of the problem consists of blocks of constraints with a set of linking variables, as in (PP2). The main idea is to solve a relaxed master problem that involves only the linking variables, fix the values of these variables to obtain decoupled subproblems, and use the solution of these subproblems to improve the formulation of the master problem. The structure of the incidence matrix and an outline of the method for model (PP2) are given in Fig. 7a. Lagrangian relaxation is typically used when a problem consists of a set of easy constraints and a set of hard constraints whose removal (and addition as a penalty term in the objective) leads to an easy problem. In the case of production planning, this situation arises when a single scheduling problem is used over multiple planning periods, leading to a formulation that has structure similar to model (PP3). The hard constraints are the constraints linking blocks corresponding to consecutive periods. The structure of model (PP3) and the outline of the algorithm are given in Fig. 7b. Lagrangian Fig. 7. Mathematical programming decomposition schemes for integrated MIP formulations.

8 1926 C.T. Maravelias, C. Sung / Computers and Chemical Engineering 33 (2009) Fig. 8. Production planning scheduling integration matrix. Proposed methods are classified by modeling approach (rows) and solution approach (columns). Decomposition methods are based on the planning scheduling hierarchy ( )or the mathematical programming structure ( ). decomposition is a variation of Lagrangian relaxation in which complicating variables that appear in multiple constraints are disaggregated into copies. Copies of the same variable are related by linking constraints, which are relaxed by the decomposition. Kelly and co-workers (Kelly, 2002; Kelly & Zyngier, 2008) discussed decomposition methods for scheduling problems that can be extended to address the integrated problem. Chen and Pinto (2008) applied Lagrangian relaxation and Lagrangian/surrogate relaxation (Narciso & Lorena, 1999) on a continuous flexible process network model The integration matrix The modeling approaches and solution strategies outlined above can be combined in various ways, leading to the methods shown in the integration matrix in Fig. 8. Approximations of scheduling formulations (i.e. aggregations or relaxations of scheduling models and surrogate models) can be used in the master problem of both hierarchical and iterative strategies. Detailed scheduling models result in standalone formulations that can be solved directly or using iterative mathematical programming techniques. We note again that in practice all methods are implemented in a rolling horizon fashion. 6. Challenges and opportunities Despite efforts in this area, the integration of production planning and scheduling remains a hard problem. The major challenges that are specific to this problem lie in the development of computationally effective scheduling formulations for complex process networks, the communication between the master and slave subproblems in iterative schemes, and the development of hybrid methods that exploit complementary strengths of different solution techniques. Challenges in the areas of uncertainty and data integration are also very important, but are not discussed here since they are not problem-specific Modeling of complex process networks Existing scheduling formulations, especially for batch processes, are computationally expensive. Since scheduling models are a component of the integrated problem, the development of better scheduling formulations and tighter aggregations/relaxations will enhance the solution of the integrated problem. Interestingly, the focus in PSE scheduling literature has been on the development of MIP models whose efficiency is most often assessed by the computational requirements for a few instances. The development of scheduling models has been guided by empirical (computational) rather than theoretical results. On the other hand, the method of Fig. 9. Tightening of the feasible region P = P I P II of the LP-relaxation of the original formulation via the development of a reformulation for set P II. Since P II and P II* contain the same integer points, the reformulation M* has the same set of solutions as the original formulation M, but it is tighter. choice in OR literature has been the derivation of theoretical results concerning the structure and tightness of MIP formulations. The main idea in this approach is to generate a tighter formulation via the tightening or reformulation of a subset of constraints. Given a general MIP formulation (M): max{c T x : A I x b I,A II x b II,x X} =max{c T x : x P I,x P II,x X} or equivalently, max{c T x : x P I P II,x X}, (M) our goal is to derive a tighter formulation for the second set of constraints that will lead to a tighter polyhedron P II* P II and thus a tighter formulation (M*) (see Fig. 9): max{c T x : x P I P II,x X} (M*) The development of theoretical results and/or reformulations for existing and new MIP scheduling models has the potential to lead to effective solution strategies for the integrated problem. Gaglioppa, Miller, and Benjaafar (2008) recently presented, using echelon inventory, a new family of valid inequalities for a multitask, multi-stage, planning and scheduling problem. Pochet and Warichet (2008) used strengthening techniques and the analysis of small polytopes to strengthen the initial formulation of a continuous-time MIP formulation for cyclic scheduling and discuss MIP-based heuristic methods. Maravelias and Papalamprou (2009) developed polyhedral results for a discrete-time formulation for continuous multi-stage processes. An alternative approach is the extension of existing lot-sizingbased or other approaches to address problems in the chemical industry (Sahinidis & Grossmann, 1991) or the development of new lot-sizing formulations that account for some of the complicating features of process networks (Suerie, 2006; Sung & Maravelias, 2008). The advantage of this approach is that existing results and decomposition strategies for lot-sizing formulations can be readily exploited (Pochet & Wolsey, 2006). For example, decomposition approaches that decompose the problem by item into a set of multiperiod singe-item problems, rather than by time period, have not be studied extensively in the PSE literature Iterative methods The computational efficiency of iterative methods depends on (a) how fast we can solve the two subproblems, and (b) the number of iterations necessary to obtain the optimal solution and to prove optimality. Problem-specific algorithms can be used to achieve the former (see next subsection). For the latter, the tightness of the

9 C.T. Maravelias, C. Sung / Computers and Chemical Engineering 33 (2009) master problem and the generation of strong cuts are crucial. The master problem is typically a relaxation of a detailed scheduling model. If too many constraints are relaxed, the problem is easy to solve but admits too many solutions that are found infeasible by the subproblem; if few constraints are relaxed, it is tighter but harder to solve. A promising direction here is the relaxation of large number of constraints followed by tightening through preprocessing. For example, shortest-path algorithms can be used to determine time windows for the bottleneck units that can then be used to generate knapsack or cover inequalities that exclude infeasible assignments. The structure and features of the process network (e.g. storage policies) can also be used to derive tightening constraints (Burkard & Hatzl, 2005; Maravelias & Grossmann, 2004). The ability to generate strong cuts is probably the most important component of an iterative method. When a scheduling subproblem is found to be infeasible, we can generate an integer cut that excludes the current permutation of values of the integer variables (a no-good integer cut). However, the infeasibility of a subproblem is typically due to a small set of decisions. Hence, if no-good cuts are added, the same subset of decisions will appear in many solutions, leading to many iterations of infeasible subproblems. The key in overcoming this shortcoming is the development of methods that identify the smallest possible set of decisions that lead to infeasibility and generate integer cuts that exclude all the master problem solutions with this subset of decisions. This is a challenging task because mathematical programming methods do not return this minimal subset. However, there are methods for generating strong cuts that have not yet been studied extensively. First, if a subproblem is infeasible, we can break it down to smaller parts that can be analyzed to identify the source of infeasibility. Second, logic inference can be used to identify subsets of the current master solution that lead to infeasible subproblems. Third, the subproblem can be solved using methods that allow us to identify what decisions caused the infeasibility, e.g. CP. For the single-machine subproblem, the cuts proposed by Bockmayr and Pisaruk (2003) and Sadykov and Wolsey (2006) are effective. Also, the superset and subset integer cuts of Erdirik- Dogan and Grossmann (2006) and the single-unit cuts of Maravelias (2006) for the multi-stage subproblem can be effective, but more work is needed for complex process networks Hybrid methods The scheduling subproblem of many integrated methods involves only sequencing decisions. MIP methods are not effective in solving sequencing problems due to the modeling of the latter via big-m constraints. On the other hand, there are algorithms that have been specifically developed to address such problems, e.g. CP propagation algorithms (Baptiste et al., 2001; Hooker, 2006). Hence hybrid methods with a MIP master problem and sequencing subproblems solved using other methods can be potentially effective (Timpe, 2002). Nevertheless, existing hybrid methods address the minimization of assignment cost, a relatively easy problem because the objective function does not depend on the solution of the sequencing subproblem, and the minimization of makespan, but not the minimization of total (production + inventory) cost. Furthermore, most existing hybrid methods are iterative, meaning that (a) the master problem has to be solved repeatedly from scratch, and (b) the subproblem is solved only after the optimal solution of the master problem is found. A branch-and-bound method where the master MIP problem is solved as a relaxation of the integrated problem and the subproblem is solved whenever a solution of this relaxed problem is found (Fig. 10), has a number of advantages (Bockmayr & Kasper, 1998). First, the MIP does not have to be solved multiple times. Second, the Fig. 10. Hybrid branch-and-cut algorithm. The master problem (MP) is used as a relaxation; subproblem (SP) is solved whenever an integer solution of (MP) is obtained. subproblem is solved whenever an integer solution of the master problem is found, which means that integer cuts are added fast and a feasible solution can be found earlier. While the development of such a hybrid branch-and-cut algorithm is a challenging task, it has the potential to solve problems of industrial importance Off-line computations While off-line computations have been used to enhance the solution of control problems (Bemporad, Morari, Dua, & Pistikopoulos, 2002; Pannocchia, Rawlings, & Wright, 2007), they are still at their infancy for integrated planning scheduling problems. Nevertheless, the potential of this type of methods is enormous due to the nature of the integrated problem. A set of production targets is feasible if there is a detailed schedule that meets these targets. However, the existence of a feasible schedule and not the schedule itself is important. In other words, the scheduling part of the solution of an integrated approach serves more as a proof of correctness of the planning solution, rather than as a schedule that should be readily implemented. This proof of correctness for a given process network can be established off-line. One has to simply analyze the process network to determine what production targets are feasible and at what cost. This analysis has to be carried out once. Once the feasibility and cost information is available, it can be easily incorporated in the production planning formulation. The simple case of convex feasible region and convex cost function has been addressed by Sung and Maravelias (2007), while a first extension to the non-convex case is discussed in Sung and Maravelias (2009). Another opportunity in the area of off-line computations is the integration of previously acquired knowledge. The solution of an integrated model for a given process network involves the solution of essentially the same scheduling model several times (once for each period) with different data. Therefore, the production targets that are found to be infeasible in one period are likely to be infeasible for every other planning period as well. Similarly, the same targets are likely to be infeasible when solving a production planning problem for the same process network a few months later. If this facility-specific information is saved every time a planning problem is solved, an accurate model can be developed. This model identification exercise is something that has yet to be explored. Nevertheless, the form in which infeasibility information is saved and added back to the master problem remains a challenge.