مدل های تصادفی و راه حل های عددی برای برنامه ریزی تولید با برنامه های کاربردی برای صنعت کاغذ

This work is concerned with models and numerical algorithms for production planning of systems under uncertainties. Using stochastic processes to describe the system dynamics, we model the random demand and capacity processes by two finite-state continuous-time Markov chains. We seek the optimal production rate by minimizing an expected cost of the system. Discretizing the Hamilton–Jacobi–Bellman (HJB) equations satisfied by the value functions and using an approximation procedure yield the optimal solution, which allows us to make production decisions sequentially throughout the process lifespan. Three case studies are presented. Using demand data collected from a large paper manufacturer, the optimal production policies of the paper machine are obtained for different machine capacity and demand processes.

The integration of planning and scheduling in optimization has received increasing attention in recent years (e.g. Birewar and Grossmann, 1990, Coxhead, 1994, Papageorgiou and Pantelides, 1996a, Papageorgiou and Pantelides, 1996b and Petkov and Maranas, 1997; Sand, Engell, Märkert, Schultz & Schulz, 2000; Rodrigues, Latre & Rodrigues, 2000; Lasschuit and Thijssen, 2003; Neiro and Pinto, 2003). This trend follows the realization in industry that planning and scheduling decisions need to be considered simultaneously to gain a competitive advantage (see e.g. Shobrys and White, 2000). Two of the major challenges towards this integration are dealing with the different time scales and with the problem size of the resulting optimization model. In this work, we address these challenges by considering the integration of production planning and reactive scheduling in the optimization of a hydrogen supply network. An additional challenge that arises is the modeling and solution of the pipeline system. Pipeline flow models are difficult to solve due to the presence of nonlinear functions such as absolute values, signs (+ or −), and flow transitions that result in discontinuities and other non-convexities. It is, however, important to include these details in the model to ensure feasible flows and to satisfy customer demands and minimum contract pressure levels. While numerous articles have been published on either planning or scheduling (for a short review see Shah, 1998, and Grossmann, Van den Heever & Harjunkoski, 2001), only a few have addressed the integrated problem. Shobrys and White (2000) emphasize the importance of integrated planning, scheduling and control in industry, and address some non-technical challenges towards this integration, such as human and organizational behavior. On a technical level, Grossmann et al. (2001) discuss the role of optimization methods in achieving integration. These authors emphasize the importance of time representation, present a general disjunctive formulation for integrating planning and scheduling, as well as some techniques for dealing with the large size of integrated models. Regarding the time representation, one approach is to define a very large scheduling problem that spans the whole planning horizon and defining longer lengths for future time periods to yield a formulation where the immediate future includes more detail than the distant future. Bassett, Dave, Doyle, Kudva, Pekny, Reklaitis, Subramanyam, Miller and Zentner (1996) present such a model and show that it cannot be solved in the full space due to its size and that some type of decomposition is required. To this end, they propose a decomposition scheme where an aggregate planning level is solved, and separate detailed scheduling problems are subsequently solved for each planning period. Their decomposition algorithm ties in with a second approach to time representation for integration, namely to define an aggregate planning problem from which information is passed to a detailed scheduling model. Birewar and Grossmann (1990) propose such models for the simultaneous planning and scheduling of multipurpose batch plants. In their approach, planned production levels can be met within the available cycle time using an aggregate scheduling model. A third method for dealing with the different time scales is to use a rolling horizon approach where only a subset of the planning periods include the detailed scheduling decisions with shorter time increments. The detailed planning/scheduling period moves as the model is solved in time, thus the term ‘rolling horizon’. When such a planning/scheduling model is solved in real-time, the first planning period is often a detailed scheduling model while the future planning periods include only planning decisions and this is also the approach we use in this work. Dimitriadis, Shah and Pantelides (1997) presented RTN-based rolling horizon algorithms for medium term scheduling of multipurpose plants. Sand et al. (2000) use a rolling horizon approach, in combination with a Lagrangean relaxation algorithm, for the solution of a two-level hierarchical planning and scheduling problem where uncertainty has been included explicitly on the planning level. Other research on the integration of planning and scheduling includes that of Papageorgiou and Pantelides, 1996a and Papageorgiou and Pantelides, 1996b, who propose a bilevel decomposition approach for campaign planning and scheduling of multipurpose batch/semicontinuous plants with scheduling decisions aggregated on the planning level and some planning decisions fixed on the scheduling level. More work on the integration of planning and scheduling for batch plants was carried out by Rodrigues et al. (2000) who propose a multilevel decomposition approach where capacities are determined on the planning level and the scheduling level is based on the STN formulation. Das, Rickard, Shah and Macchietto (2000) discuss this integration by using the same basic data and variables in all models, while information is passed from the higher hierarchical to the lower levels as the models are solved in sequence. Bose and Pekny (2000) proposed a model predictive framework for the planning and scheduling of a consumer goods supply chain. They present an architecture where modules for demand forecasting, optimization, and simulation are integrated to deal with uncertainty in promotional demands. Petkov and Maranas (1997) address uncertainty in demand for batch plants by extending the model of Birewar and Grossmann. Reactive scheduling can be defined as the problem of updating a schedule dynamically as the constraints or assumptions on which it is based change. The problem at hand is therefore one of reactive scheduling, seeing that the scheduling decisions are adjusted as soon as updated pipeline measurements become available, and not only with each planning period. The reactive scheduling problem we consider in this work is different from the batch processing problems, in the sense that the plants and hydrogen pipeline do not operate in batches. Instead, there is a continuous supply of hydrogen and scheduling decisions involve basic on/off decisions to satisfy the demand as they become known. Coxhead (1994) gives a qualitative description of a similar concept for refinery planning and scheduling. In addition, the reactive scheduling considered here incorporates a complex model for the hydrogen pipeline network, to accurately determine a schedule that satisfies both pressure and flow constraints. The modeling of gas pipeline networks has been considered by a number of researchers in the past. Most of the work has been carried out for natural gas pipelines. Wong and Larson (1968) proposed a dynamic programming approach for the optimization of a natural gas pipeline system, but consider only a single compressor and single pipeline section and the input flow and internal pressures are fixed. Sood, Punk and Delmastro (1971) consider a similar problem and minimize the energy usage for operation of the compressors as it relates to flow and pressure. A single-period convex MINLP model for gas transmission pipeline synthesis was presented by Duran and Grossmann (1986) and solved for a superstructure of two wells leading to two demand points. This model includes discharge pressures, pipe dimensions, compressor location, and compressor power as variables. Martinez-Benet and Puigjaner (1988) consider the design of large-scale pipeline networks at steady state. While very large pipeline systems are solved in detail with their approach, the purpose of their work is design at steady state and they do not consider the operation of existing systems. Marques and Morari (1988) consider the on-line optimization of a natural gas transmission pipeline network, and contribute to work up to that time by using the compressor discharge pressure as manipulated variable and including the compressor constraints in their model. They link a dynamic pipeline simulator with an optimizing predictive control scheme over a moving horizon. Bullard and Biegler (1992) address discontinuous functions that arise in pipeline network models and present both continuous and mixed-integer versions of a smooth approximation for the max function associated with one-directional flow enforced by checkvalves. Their results show that the mixed-integer formulation requires significantly larger CPU times than the continuous formulation. Türkay and Grossmann (1998) present an example that follows the work by Bullard and Biegler (1992) and propose a disjunctive formulation to model the check valves. This formulation also requires discrete variables and has the potential of getting computationally taxing for large models, although it provides much tighter relaxations. These two sets of authors do not consider the compressor operation. Sun, Uraikul, Chan and Tontiwachwuthikul (1999) consider the optimization of natural gas pipeline operations, including the on/off status of compressors to satisfy demand while minimizing cost. They present an integrated expert system and operations research approach where the expert system is used to determine whether the volume of gas in the pipeline is sufficient to satisfy demands, and a fuzzy 0–1 linear program is used to determine which compressors to switch on or off. They consider a horizon of only 8 h and do not model the pipeline explicitly, but instead use one equation to approximate the volume of gas in the pipeline at an average pressure. None of the articles on pipeline optimization mentioned in this section consider long enough time horizons to handle the complex pricing of energy and feed implied by deregulation. Neither do they incorporate the possibility of using the pipeline as a storage device that requires good knowledge of the pipeline inventory and thus a detailed pipeline model. In most of these articles, the direction of flow is pre-specified, while flow in a supply network can be in either a positive or a negative direction. We address these issues by proposing two integrated multiperiod MINLP models for planning and reactive scheduling of the supply system. In addition, we propose a strategy for the integrated solution of these models and a Lagrangean decomposition-based heuristic to deal with the problem size at the planning level. Uncertainty in the demand forecast is partly dealt with in that the reactive scheduling allows changes in operation when demands are different from their predicted values and the planning model and monthly forecast are updated every 12 h as more information becomes known. To the best of our knowledge, the work presented in this paper is unique in combining daily or monthly planning decisions that are essential for the cost minimization of gas pipeline network operations, with hourly reactive scheduling of compressor operation, in an optimization framework. Isothermal conditions are assumed in this work and start-up and shut-down costs are not considered although these costs can in principle be included without much difficulty. In Section 2, we present the problem statement. This is followed by the proposed integrated solution methodology and the two optimization models for the planning and scheduling levels, respectively. Example 1 presents a small instance to demonstrate the proposed approach, and a larger instance that demonstrates the need for a specialized solution algorithm on the planning level. This is followed by a section on the proposed Lagrangean decomposition heuristic for the upper level planning model. Example 1 is revisited to demonstrate the performance of the proposed heuristic, while Example 2 demonstrates the performance of the proposed approach, including the planning heuristic, on the largest instance of the supply network. Our findings are summarized in the Section 9.

A methodology for the integrated production planning and reactive scheduling in the optimization of a hydrogen supply network was proposed. This method relies on a two-level decomposition where the upper level involves a multiperiod MINLP planning model and the lower level involves a multi/single period MINLP reactive scheduling model. The planning model determines feed and energy prices, as well as production targets and includes detailed economic calculations for each plant, with a simplified pipeline model. The scheduling model determines the compressor operation as actual demands become known, with fixed prices and production targets from the planning model and a detailed pipeline model. A Lagrangean decomposition heuristic is presented to address the computational effort involved in solving the planning model, while the reactive scheduling model is solved as sequential single period models. Information is passed between the planning and scheduling models as they are solved in a rolling horizon fashion each 12 or 1 h, respectively. Results show that the proposed Lagrangean decomposition heuristic shows more than an order of magnitude reduction in solution time compared to a full space approach using DICOPT++, while obtaining solutions of similar or better quality. In addition, the production levels from the scheduling level correspond exactly to those of the planning level, while the power consumption from the scheduling level is close to that of the planning level. Some discrepancies exist that we believe are due to (1) nonlinearities on the scheduling level; (2) relaxation of the load step variables on the planning level; and/or (3) solving the scheduling sequentially instead of simultaneously. These issues are the subject of future work. The proposed methodology has shown to successfully integrate the production planning and reactive scheduling levels, apart from the before mentioned discrepancies. In addition, the planning level addresses pricing issues that have never been considered before in the literature to yield significantly lower costs. Also, the scheduling level involves the most inclusive reactive scheduling model for gas pipeline distribution networks to date, in that the model incorporates plant models, compressor models, detailed pipeline dynamics, inventory calculations and the detailed large-scale pipeline network.