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A general multiperiod model to optimize simultaneously production planning and design decisions applied to multiproduct batch plants is proposed. This model includes deterministic seasonal variations of costs, prices, demands and supplies. The overall problem is formulated as a mixed-integer linear programming model by applying appropriate linearizations of non-linear terms. The performance criterion is to maximize the net present value of the profit, which comprises sales, investment, inventories, waste disposal and resources costs, and a penalty term accounting for late deliveries. A noteworthy feature of this approach is the selection of unit dimensions from the available discrete sizes, following the usual procurement policy in this area. The model simultaneously calculates the plant structure (parallel units in every stage, and allocation of intermediate storage tanks), and unit sizes, as well as the production planning decisions in each period (stocks of both product and raw materials, production plans, policies of sales and procurement, etc.).

Nowadays, one of the most important challenges faced by business is the adjustment of the firm resources in order to satisfy market requirements subjected to fluctuations over time, mainly costs, prices, existences, demands, etc. In many industries, products have distinctive demand patterns that vary due to market or seasonal changes coupled with raw material supplies that also undergo changes. Because of these variations over time, there has been an increased interest in the development of multiperiod optimization models in recent years. Flexible production is receiving increased attention in the chemical processing industry. This flexible production leads to faster responses to the market fluctuations and is most commonly achieved in batch plants. In this work, efforts are focused on multiproduct batch production environment, where several different products are produced sharing the same equipment operating in the same sequence. A batch process refers to a general non-continuous process that consists of multiple stages employing a combination of identical parallel batch units. In a multiproduct batch plant each product is produced at a time. Batch units are characterized by a processing time and no simultaneous feed and removal is performed. Also, intermediate storage tanks may be available between successive stages of operation in order to decouple the production process. Fig. 1 shows a plant configuration of this type of industry.Most of the previous approaches in batch plants used to pose models that worked with a single long time period and constant conditions without considering variations due to seasonal or market fluctuations. Also, these previous efforts usually decouple the design and planning problems and solve only one problem making several assumptions over the other. In the design problem, the production requirement of each product and the total production time available are specified. A procedure is generated in order to determine the plant configuration and equipment sizes to minimize the capital cost. Different formulations have been developed and solved through different methodologies [1], [2], [3], [4] and [5]. Moreover, several approaches with varying degrees of detail have been introduced in the past years to solve the planning problem [6], [7] and [8]. Unlike previously cited works, a smaller number of articles have posed models for multiperiod scenarios. In general, these works follow the same trend only focusing on one problem at a time. Multiproduct batch facilities in a multiperiod scenario have been studied by Birewar and Grossmann [9] that presented a multiperiod linear programming model for production planning of batch plants that considers benefits and product inventory cost, but design decisions are not included in that approach. Voudouris and Grossmann [10] developed a cyclic MILP problem where synthesis, sizing and scheduling issues were integrated, including intermediate storage sizing and allocation. Van den Heever and Grossmann [11] considered a multiperiod nonlinear optimization model posed through general disjunctive programming for the design, and capacity expansion of general chemical process systems. They proposed two algorithms for the solution of the model in order to reduce the solution times of MINLP problems. Taking into account modeling and resolution difficulties in previous works in process industry, problems are generally decomposed into simpler steps: design, operation, planning, scheduling, etc. These problems, however, are related and they should be solved together, at least some of them. The trade-offs among them depend on several elements: time horizon, product lifetime, characteristics of facilities, supply policies, etc. Most of these works used to pose models working with only one time period with constant conditions. These alternatives can be improved if the problem elements can vary over time in a multiperiod context. Two main contributions have been addressed in this work. On one hand, concurrent design and production planning decisions for multiproduct batch plants have been simultaneously posed, so as to assess the trade-offs between them. On the other hand, the multiperiod effect has been explicitly taken into account. In this way, the changes caused by market and seasonal fluctuations in decision variables in every period are considered, using deterministic values proposed by the decision maker. In contrast to previous works, the optimal design is determined by considering the units available in discrete sizes which correspond to the real procurement of equipment. In order to get an MILP model, a linearization method is applied over bilinear terms. In this way, the original nonlinear and non-convex model is transformed to obtain a linear formulation that can be solved to global optimality with reasonable computational effort. In short, this general model handles deterministic seasonal variations of product demands and prices, the raw material and investment costs, takes into account discrete sizes of batch units and storage tanks, and considers inventories of both final products and raw materials. This is a valuable MILP model since it corresponds to a more realistic case. Decision makers can simultaneously assess different elements from the strategic and tactic points of view of the operations management. The implementation of the proposed model is demonstrated through its application in several examples. The paper is structured as follows. The main characteristics of the problem are discussed in Section 2. Section 3 presents a description of the proposed mathematical formulation. Illustrative examples are included in Section 4 and their results are discussed. Finally, some concluding comments are presented in Section 5.

A general model has been presented in this paper to simultaneously address the problems of multiproduct batch plant production planning and design over a multiperiod scenario. The original non-linear formulation has been transformed so as to obtain a mixed integer linear programming formulation which can be solved to global optimality. Several features can be stressed. The model considers design and production planning decisions at the same time. Previous efforts used to solve these problems separately which hinders the interactions between both types of decisions. Also, the model considers a multiperiod scenario. Then, seasonal and market variations can be taken into account. From the design point of view, the model considers several interesting elements: different configuration options are included (duplication of batch units, intermediate storage tanks allocation) and a real procurement policy is adopted, with units available in discrete units. From the production planning point of view, all the usual decisions are contemplated: inventories, sales, purchases, etc. In this first approach, deterministic fluctuations have been considered. This is an interesting formulation that allows managers to have a feedback about the impact of his or her decisions, considering interactions between design, commercial, production, sales and inventory policies simultaneously.