ارائه مدل برنامه ریزی خطی مختلط عدد صحیح برای طراحی همزمان و برنامه ریزی نیروگاه های جریان کارگاهی

Models representing batch plants, especially flowshop facilities where all the products require the same processing sequence, have received much attention in the last decades. In particular, plant design and production scheduling have been addressed as disconnected problems due to the tremendous combinatory complexity associated to their simultaneous optimization. This paper develops a model for both design and scheduling of flowshop batch plants considering mixed product campaign and parallel unit duplication. Thus, a realistic formulation is attained, where industrial and commercial aspects are jointly taken into account. The proposed approach is formulated as a Mixed Integer Linear Programming model that determines the number of units per stages, unit and batch sizes and batch sequencing in each unit in order to fulfill the demand requirements at minimum investment cost. A set of novel constraints is proposed where the number of batches of each product in the campaign is an optimization variable. The approach performance is evaluated through several numerical examples.

Batch processes are characterized by their flexibility and ability to produce low-volume products, sharing the same equipment. The main classification of batch processes is based on the production path involved for products manufacture: flowshop or multiproduct batch plants are employed when all the products require all the stages following the same sequence of operations, while, in jobshop or multipurpose batch plants, products can follow different processing sequences, not necessarily employing all the stages. In this paper, the study is focused on flowshop or multiproduct batch plants. The general design problem of this type of facilities consists of determining: (a) the plant configuration, i.e. the number of parallel units required for each stage and, sometimes, the assignment of intermediate storage between stages; (b) the unit and storage vessel sizes; and (c) the number and size of batches for all the products, in order to optimize an economic performance measure while satisfying constraints on the production requirements in the available time horizon. This problem has been generally formulated as a mixed-integer non linear programming (MINLP) model [1]. On the other hand, taking into account that all products, usually with similar recipes, are processed using the same stages, production must be scheduled in order to improve the plant performance and avoid large inventory levels. According to Papageorgiou and Pantelides [2], the campaign mode operation is particularly appropriate for plants working under stable demand patterns over long planning horizons. The plant can be operated with mixed product campaigns (MPC), where in each campaign various batches of different products are manufactured and the same batches arrangement is cyclically repeated over the time horizon. In this case, several decisions must be made at the scheduling level: the number of batches of each product involved in the production campaign and their sequencing in order to optimize a suitable performance measure. This problem represents an important challenge given the combinatorial nature of scheduling decisions. Most formulations for scheduling belong to the set of NP-complete problems [3] and, despite significant advances in optimization approaches, there is still a number of major challenges and questions that remain unsolved [4]. When the plant design is not a priori provided, the problem becomes worse, because both the number of batches of each product and the available equipment are unknown. In this last case, in order to simplify the model, most of the formulations assume single product campaigns (SPC), where all batches of a given product are manufactured before switching to another product. However, this proposal is not appropriate for the production or commercial points of view. The multistage nature of a batch plant allows four different storage options: (i) unlimited intermediate storage (UIS); (i) finite intermediate storage (FIS); (iii) no intermediate storage (NIS); (iv) zero wait (ZW). In both the NIS and ZW modes, there is no storage between stages, while for UIS and FIS modes intermediate storage is provided. Intermediate materials can wait in storage with unlimited capacity in UIS and limited capacity in FIS, or in the current processing unit in NIS. In the ZW mode, the batch must be immediately transferred to a unit of the downstream stage after being processed. In this work, a detailed MILP mathematical formulation for the simultaneous design and scheduling of flowshop plants is addressed. Unlike the previous proposed approaches, the number and size of batches of each product in the campaign, the number and size of process units for each stage and the production sequence in each unit are model variables. Considering that the number of units in each stage and the number of batches in the campaign are unknown, the scheduling formulation represents a great challenge from the modeling point of view. A slot-based continuous-time representation for modeling the assignment of batches to units and their sequencing is employed. Taking into account the combinatorial nature of the scheduling problem, appropriate constraints must be included in the model to assess the cycle time of the production campaign, fulfilling demands over the time horizon. In order to attain a general formulation, multistage facilities are assumed, including out of phase unit duplication at every stage. The unit sizes are restricted to discrete values and an upper bound for the number of batches of each product in the campaign is provided, in order to keep the linearity of the problem. Therefore, the proposed MILP model represents a novel approach where the simultaneous optimization of plant design and scheduling considering MPCs can be solved to global optimality with reasonable computational effort. The approach performance is assessed through the several examples where different values for the key problem parameters are analyzed. This paper is organized as follows. In the next section a literature review is presented. In Section 3 the problem description and its main assumptions are stated; in Section 4 the mathematical formulation is described in detail, while in Section 5 several examples are solved in order to illustrate the proposed approach. Finally, conclusions and future works are discussed in Section 6.

This article presents a new approach for simultaneous design and scheduling of flowshop plants. Until now, previous works had generally solved these problems resorting to decoupled models, without taking into account the trade-offs between them. Usually, when design problem is formulated, they assumed SPC and, thus, the formulation is strongly simplified. However, this kind of solutions is not suitable from the commercial point of view since large product stocks are required. The approach proposed in this work addresses the joint design and scheduling of flowshop plants with MPC production mode. For batch plant design, a maximum number of out of phase duplicated units is considered, and batch unit sizes are selected according to a set of available discrete sizes, as usually found in commercial and industrial practices. For scheduling, a maximum number of batches for each product in the campaign composition is established. In this way, the nonlinearities in the mathematical model are avoided. Therefore, the proposed formulation is addressed as a MILP one, which determines the global optimal solution for the simultaneous flowshop plant design and scheduling. Taking into account the combinatorial complexity of scheduling models, appropriate preordering constraints have been developed and incorporated in the proposed formulation. This aims at reducing alternative solutions, search space, and thus computational effort. In short, the proposed formulation simultaneously determines plant configuration, including the number of units at each stage and their sizes, and the number and size of the batches of every product. These batches are produced through MPCs that are cyclically repeated over the time horizon. The campaign configuration includes the allocation of batches to the selected units and their sequencing. This formulation has been applied to several problems in order to highlight the trade-offs among the different involved decisions and the mathematical characteristics of the model. Also, the performance has been assessed. Thus, a new modeling strategy has been presented to solve an optimization problem not previously approached with the assumptions here adopted. From the mathematical point of view, future works will be focused on computational performance including major problems.