برنامه ریزی تولید چند دوره ای و طراحی دسته ای نیروگاه ها تحت عدم قطعیت

A two-stage stochastic multiperiod LGDP (linear generalized disjunctive programming) model was developed to address the integrated design and production planning of multiproduct batch plants. Both problems are encompassed considering uncertainty in product demands represented by a set of scenarios. The design variables are modeled as here-and-now decisions which are made before the demand realization, while the production planning variables are delayed in a wait-and-see mode to optimize in the face of uncertainty. Specifically, the proposed model determines the structure of the batch plant (duplication of units in series and in parallel) and the unit sizes, together with the production planning decisions in each time period within each scenario. The model also allows the incorporation of new equipment items at different periods. The objective is to maximize the expected net present value of the benefit. To assess the advantages of the proposed formulation, an extraction process that produces oleoresins is solved.

Batch processes have been widely studied throughout the last years due to their particular suitability for the production of large number of low-volume, high-value products in the same facility (Barbosa-Póvoa, 2007). Usually, at the stage of conceptual design of a batch plant, there are parameters, either external or internal to the process, which are subject to considerable uncertainty. These market and technical parameters include, for instance, product demands, raw materials availability, prices of chemicals, reaction constants, efficiencies, etc. This work is focused on multiproduct batch plants where several products are produced following the same sequence of processing stages. A special feature of these facilities is their ability to meet production requirements and maximize profits given uncertainties in the market demands for the products. In dealing with optimization under uncertainty, three research philosophies have been employed over the last years: stochastic programming, fuzzy programming and stochastic dynamic programming (for a short overview see Sahinidis, 2004). Most of the existing approaches that address the effect of uncertainty into batch process optimization have applied stochastic programming (Acevedo and Pistikopoulos, 1998, Aguilar-Lasserre et al., 2009, Cao and Yuan, 2002, Cui and Engell, 2010, Ierapetritou and Pistikopoulos, 1996, Liu and Sahinidis, 1996, Maravelias and Grossmann, 2001, Petkov and Maranas, 1997, Subrahmanyam et al., 1994 and Wu and Ierapetritou, 2007). Stochastic programming deals with optimization problems whose uncertain parameters are modeled either by continuous probability distributions or by a finite number of scenarios. The approach using scenario analysis has been considerably exploited in the literature and has proven to provide reliable and practical results for optimization under uncertainty (Alonso-Ayuso et al., 2005, Escudero et al., 1993, Gupta and Maranas, 2003, Karuppiah et al., 2010, Liu and Sahinidis, 1996, Shah and Pantelides, 1992 and Subrahmanyam et al., 1994). In this paper, the uncertainty in product demands is tackled by the scenario approach. In general, the two-stage stochastic programming strategy has been considered an effective and widely used method for addressing the optimization problems under uncertainty. In this approach, decision variables are explicitly classified according to whether they are implemented before or after a random event occurs. First-stage (here-and-now) decisions must be made before the uncertain parameters reveal themselves while second-stage (wait-and-see) decisions, also called recourse actions, are made after the outcome of the random events is observed. Thus, through recourse actions, stochastic models consider corrective measures that can be taken after the realization of some uncertain parameters. The two most common objective functions in the literature are the expected cost/profit of the problem. In the area of batch processing, there are significant contributions on design and planning under uncertainty. Approaches tackling the production planning problem include Liu and Sahinidis (1996) who presented a two-stage model for the process planning and process capacity expansion with random variables that assume values from both discrete and continuous probability. Petkov and Maranas (1997) extended the multiperiod planning and scheduling model for multiproduct plants introduced by Birewar and Grossmann (1990) including uncertain product demands. Wu and Ierapetritou (2007) proposed a multi-stage stochastic programming formulation for the simultaneous solution of production planning and scheduling problems using a rolling horizon strategy. With regard to the batch plant design under uncertainty, Shah and Pantelides (1992) presented a stochastic formulation for the design with uncertain product requirements considering different scenarios. Subrahmanyam et al. (1994) addressed the design and scheduling of batch process through a multiperiod model. The problem is split into two stages: in the first, the design is obtained without considering scheduling constraints, while, in the second stage, a detailed scheduling model is solved. Cao and Yuan (2002) addressed the problem of the optimal design of batch plants with uncertainty in product demands considering different operating modes of parallel units for different products. Alonso-Ayuso et al. (2005) proposed an approach for the product selection and plant sizing problems under uncertainty. Aguilar-Lasserre et al. (2009) developed a multi-objective optimization problem for the design of batch plants with uncertain market demands for products. Even though many contributions dealing with the design of batch plants under uncertainty have been published, the simultaneous optimization of the design and planning decisions with capacity expansion of the plant has not been sufficiently studied including all the elements considered in this article. Therefore, the goal of this work is to propose a scenario-based approach for the simultaneous design and production planning of multiproduct batch plants under uncertain demands over a multiperiod context. From the design perspective, both kinds of unit duplications, in series and in parallel are considered. A two-stage stochastic model is proposed, where capacity expansion is admitted. New in parallel units working out-of-phase can be added in different time periods. The selection of the number of units in series can be only made in the first time period. First-stage decisions consist of design variables (mainly Boolean variables) that allow determining the batch plant structure. Second-stage decisions consist of planning variables (continuous variables) to determine the production, purchases, and inventories of raw materials and products for each period throughout the time horizon under each scenario, given the plant structure decided at the first-stage. The design alternative of duplicating units in series in a unit operation has been recently introduced in general models for multiproduct batch plants by Moreno and Montagna (2007). As they remarked, this kind of duplication is only used in specific unit operations and the trade-offs introduced in the process depend on the operation. One important characteristic of this work is that generalized disjunctive programming (GDP) has been employed in order to formulate the multiperiod stochastic linear model. GDP has been introduced as an alternative model to the mixed-integer programming (MIP), where discrete decisions and constraints are represented through disjunctions and logical propositions (Lee and Grossmann, 2000 and Vecchietti et al., 2003). The remaining parts of this article are organized as follows. First, the problem of design and planning with capacity expansion under uncertainty is described in Section 2. The two-stage stochastic linear generalized disjunctive programming (LGDP) formulation is developed in Section 3. Section 4 is devoted to the LGDP model reformulation into a mixed-integer linear programming MILP model. Considering a batch plant producing oleoresins as a motivating example, numerical results are presented using the proposed model in Section 5. Finally, some concluding remarks are summarized in Section 6.

In this work, a two-stage stochastic LGDP model has been formulated to address the design and production planning of multiproduct batch plants in presence of demand uncertainty. Several contributions can be emphasized in this article. The proposed model considered a dynamic context where variations in prices, product demands, costs, and raw materials availability due to seasonal or market fluctuations are taken into account. Therefore, the optimal structure initially adopted cannot be maintained during the plant lifetime. So, in this work, the plant configuration can be modified and new units can be added in order to fulfill new product requirements considering all the fluctuations. Design and production planning decisions are simultaneously considered. Many previous approaches prioritize decomposition of the problem in two levels: while in the first one the design is attained, in the second one the production is planned using the plant configuration previously obtained. Taking into account the proposed approach, critical trade-offs between design and production decisions are appropriately assessed as was shown in the example. The proposed approach using a LGDP optimization model is capable of handling different levels of decisions. Structural decisions (the duplication of units in parallel working out-of-phase), design decisions (unit sizes) and planning decisions (production, inventory, purchases, etc.) are appropriately represented using linear disjunctions. The disjunctive formulation of this problem allows for an easy and compact representation and visualization of the discrete choices posed. In order to obtain a linear model the size units are considered available in discrete sizes which correspond to the real procurement of equipment. In order to solve the LGDP model, the “big-M” reformulation was adopted to transform the LGDP model into a MILP one which can be solved to global optimality. Finally, the proposed model considers uncertainty in product demands represented by a set of scenarios. The formulation of the problem through scenarios allows for the simultaneous treatment of several variable elements: demands not only change along the time horizon but also they fluctuate taking into account the uncertain context. The design variables, such as the selection of equipment of standard size and the addition of new units in parallel in each time period, are independent of the scenarios, i.e., they are first-stage variables. On the other hand, production planning variables, which include working levels of the plants for each time period, are scenario-dependent variables, i.e., they are second-stage variables. The performance of the proposed formulation has been assessed through a motivating example dealing with a batch plant that produces vegetable extracts, particularly oleoresins. Several cases have been solved in reasonable computation times, showing the advantages of the presented model.