برنامه ریزی تولید یکپارچه و بهینه سازی برنامه ریزی از صنعت فرآیند چند محصولی چندسایتی

The current manufacturing environment for process industry has changed from a traditional single-site, single market to a more integrated global production mode where multiple sites are serving a global market. In this paper, the integrated planning and scheduling problem for the multisite, multiproduct batch plants is considered. The major challenge for addressing this problem is that the corresponding optimization problem becomes computationally intractable as the number of production sites, markets, and products increases in the supply chain network. To effectively deal with the increasing complexity, the block angular structure of the constraints matrix is exploited by relaxing the inventory constraints between adjoining time periods using the augmented Lagrangian decomposition method. To resolve the issues of non-separable cross-product terms in the augmented Lagrangian function, we apply diagonal approximation method. Several examples have been studied to demonstrate that the proposed approach yields significant computational savings compared to the full-scale integrated model.

Modern process industries operate as a large integrated complex that involve multiproduct, multipurpose, and multisite production facilities serving a global market. The process industries supply chain is composed of production facilities and distribution centers, where the final products are transported from the production facilities to distribution centers and then to retailers to satisfy the customers demand. In current global market, spatially distributed production facilities across various geographical locations can no longer be regarded as independent from each other and interactions between the manufacturing sites and the distribution centers should be taken into account when making decisions. In this context, the issues of enterprise planning and coordination across production plants and distribution facilities are important for robust response to global demand and to maintain business competitiveness, sustainability, and growth (Papageorgiou, 2009). As the pressure to reduce the costs and inventories increases, centralized approaches have become the main policies to address supply chain optimization. An excellent overview of the enterprise-wide optimization (EWO) and the challenges related to process industry supply chain is highlighted by Grossmann (2005). Varma, Reklaitis, Blau, and Pekny (2007) described the main concepts of EWO and presented the potential research opportunities in addressing the problem of EWO models and solution approaches. Supply chain optimization can be considered an equivalent term for describing the enterprise-wide optimization (Shapiro, 2001) although supply chain optimization places more emphasis on logistics and distribution, whereas enterprise-wide optimization is aimed at manufacturing facilities optimization. Key issues and challenges faced by process industry supply chain are highlighted by Shah, 2004 and Shah, 2005. Traditional supply chain management planning decisions can be divided into three levels: strategic (long-term), tactical (medium-term), and operational (short-term). The long-term planning determines the infrastructure (e.g. facility location, transportation network). The medium-term planning covers a time horizon between few months to a year and is concerned with decisions such as production, inventory, and distribution profiles. Finally, short-term planning decision deals with issues such as assignment of tasks to units and sequencing of tasks in each unit. The short-term planning level covers time horizon between days to a few weeks and at production level, is typically refer to as scheduling. Wassick (2009) proposed a planning and scheduling model based on resource task network for an integrated chemical complex. He considered the enterprise-wide optimization of the liquid waste treatment network with their model. Kreipl and Pinedo (2004) discussed issues present in modeling the planning and scheduling decisions for supply chain management. For a multisite facilities, the size and level of interdependences between these sites present unique challenges to the integrated tactical production planning and day-to-day scheduling problem and these challenges are highlighted by Kallrath (2002a). For further elucidation of various aspects of planning, the reader is directed to the work of Timpe and Kallrath (2000) and Kallrath (2002b). A simple network featuring the multisite facilities is given in Fig. 1, where multiple products may be produced in individual process plants at different locations spread across geographic region and then transported to distribution centers to satisfy customers demand. These multisite plants produce a number of products driven by market demand under operating conditions such as sequence dependent switchovers and resource constraints. Each plant within the enterprise may have different production capacity and costs, different product recipes, and different transportation costs to the markets according to the location of the plants. To maintain economic competitiveness in a global market, interdependences between the different plants, including intermediate products and shared resources need be taken into consideration when making planning decisions. Furthermore, the planning model should take into account not only individual production facilities constraints but also transportation constraints because in addition to minimizing the production cost, it's important to minimize the costs of products transportation from production facilities to the distribution center. Thus, simultaneous planning of all activities from production to distribution stage is important in a multisite process industry supply chain (Shah, 1998).Wilkinson, Cortier, Shah, and Pantelides (1996) proposed an aggregated planning model based on the resource task network framework developed by Pantelides (1994). Their proposed planning model considers integration of production, inventory, and distribution in multisite facilities. Lin and Chen (2007) developed a multistage, multisite planning model that deals with routings of manufactured products demand among different production plants. They simultaneously combine two different time scales (i.e. monthly and daily) in their formulation by considering varying time buckets. Verderame and Floudas (2009) developed an operational planning model which captures the interactions between production facilities and distribution centers in multisite production facilities network. Their proposed multisite planning with product aggregation model (Multisite-PPDM) incorporates a tight upper bound on the production capacity and transportation cost between production facilities and customers distribution centers in the supply chain network under consideration. A multisite production planning and distribution model is proposed by Jackson and Grossmann (2003) where they utilized nonlinear process models to represent production facilities. They have exploited two different decomposition schemes to solve the large-scale nonlinear model using Lagrangian decomposition. In temporal decomposition, the inventory constraints between adjoining time periods are dualized in order to optimize the entire network for each planning time period. In spatial decomposition technique, interconnection constraints between the sites and markets are dualized in order to optimize each facility individually. They conclude that temporal decomposition technique performs far better than spatial decomposition technique. The traditional strategy to address planning and scheduling level decisions is to follow a hierarchical approach in which planning decisions are made first and then scheduling decisions are made using planning demand targets. However, this approach does not consider any interactions between the two decision making levels and thus the planning decisions may result in suboptimal or even infeasible scheduling problems. Due to significant interactions between planning and scheduling decisions levels in order to determine the global optimal solution it is necessary to consider the simultaneous optimization of the planning and scheduling decisions. However, this simultaneous optimization problem leads to a large problem size and the model becomes intractable when typical planning horizon is considered. For an overview of issues, challenges and optimization opportunities present in production planning and scheduling problem, the reader is referred to the work of Maravelias and Sung (2009). In recent years, the area of integrated planning and scheduling for single site has received much attention. Different decomposition strategies are developed to effectively deal with a large scale integrated model. One of the existing approaches follows a hierarchical decomposition method, where the upper level planning problem provides a set of decisions such as production and inventory targets to the lower level problem to determine the detailed schedule. If the solution of lower level problem is infeasible, an iterative framework is used to obtain a feasible solution (Bassett, Pekny, & Reklaitis, 1996). To further improve this approach, tight upper bounds on production capacity are implemented in upper level problem in presence of an approximate scheduling model or aggregated capacity constraints (Shapiro, 2001 and Shah, 2005). Another related idea is the one that follows a hierarchical decomposition within a rolling horizon framework. In this model detailed scheduling models are used for a few early periods and aggregated models are used for later periods (Dimitriadis et al., 1997, Li and Ierapetritou, 2010a, Verderame and Floudas, 2008 and Wu and Ierapetritou, 2007). A different decomposition strategy is based on the special structure of the large-scale mathematical programming model. The integrated planning and scheduling model has a block angular structure which arises when a single scheduling problem is used over multiple planning periods. The constraints matrix of the integrated problem has complicating variables that appear in multiple constraints. By making copies of the complicating variables, the complicating variables are transformed into complicating constraints (linking constraints) and these complicating constraints can be relaxed using the Lagrangian relaxation method. One major drawback of the Lagrangian relaxation (LR) is that there is duality gap between the solution of the Lagrangian relaxation method and original problem and to resolve this issue, augmented Lagrangian relaxation (ALR) method should be used (Li et al., 2008, Tosserams et al., 2006 and Tosserams et al., 2008). Li and Ierapetritou (2010b) applied augmented Lagrangian optimization method to integrated planning and scheduling problem for single site plants. One disadvantage of ALR method is the non-separability of the relaxed problem which arises due to the quadratic penalty terms present in the objective function. To resolve the issue of the non-separability, Li and Ierapetritou (2010b) studied two different approaches. The first approach is based on linearization the cross-product terms using diagonal quadratic approximation (DQA) (Li et al., 2008). However, in this approach, an approximation of the relaxation problem is solved and it may not lead to a global optimal solution of the original problem. In the second approach, Li and Ierapetritou (2010b) proposed a two-level optimization method which solves an exact relaxation problem. However, the proposed two-level optimization strategy requires a non-smooth quadratic problem to be optimized at every iteration. They conclude that DQA-ALO method is more effective than the two-level optimization method for the integrated planning and scheduling problems. Even though most companies operate in a multisite production manner, very limited attention has been paid on integrating planning and scheduling decisions for multisite facilities. The integrated planning and scheduling model for multisite facilities is important to ensure the consistency between planning and scheduling level decisions and to optimize production and transportation costs. Since the production planning and scheduling level deals with different time scales, the major challenge for the integration using mathematical programming methods lies in addressing large scale optimization models. The full-scale integrated planning and scheduling optimization model spans the entire planning horizon of interest and includes decisions regarding all the production sites and distribution centers. When typical planning horizon is considered, the integrated full-scale problem becomes intractable and a mathematical decomposition solution approach is necessary. In this work, we apply augmented Lagrangian relaxation method to solve the multisite production and distribution optimization problem. The paper is organized as follow. The problem statement is given in Section 2, whereas Section 3 presents the problem formulation. The general augmented Lagrangian method and its application to multisite facility is given in Section 4. The results of examples studied are shown in Section 5 and the paper concludes with Section 6.

This work addresses the problem of integrated planning and scheduling for multisite, multiproduct and multipurpose batch plants using the augmented Lagrangian method. The integrated multisite model is proposed by extending single site formulation of Li and Ierapetritou (2010b). The shipping costs from the production sites to distribution markets are taken into account explicitly in the integrated problem. Given the fixed demand forecast the model optimizes the production, inventory, transportation, and backorder costs. Temporal decomposition scheme was developed to address the large scale model resulting from multiperiod planning and scheduling problem. The augmented Lagrangian relaxation with diagonal approximation method allowed solution of the scheduling optimization problems into parallel. Three example problems solved to illustrate the advantages of applying the augmented Lagrangian decomposition scheme. With the proposed decomposition method, faster solution times were realized.