# Production Optimization in a Grain Facility through Mixed-Integer Linear Programming +
## 1 Introduction  
The fourth industrial revolution, including the digital transformation of production and business models, is characterized by the concept of Industry 4.0 [1]. Industry 4.0 integrates innovative approaches and new digital technologies, to take advantage of robotics, automation, data analytics, and computational intelligence to assist in better decision making and create added value. The main goal is to anticipate the demands of the market and customers, providing levels of efficiency and responsiveness that were not achievable in the traditional industrial model. In turn, the digital revolution also offers many opportunities for applying sustainable production and sustainable development approaches [2]. National and regional companies are required to adapt their production systems to consider the technological shift, for example in the agri-food sector [3]. Agricultural and food companies are encouraged to introduce smart technologies in manufacturing and production processes, promoting the shift from product-oriented business models to service-oriented business models [4; 5]. Artificial intelligence and optimization models are of paramount importance in the era of Industry 4.0. Although most media attention is devoted to conceive and develop new business models, there is still a large space for optimizing production operations, within the currently applied models. Specifically, in agriculture and related industries, information technologies are increasingly applied in problems such as production forecast, scheduling, planning, and distribution [6; 7; 8]. Following this line of research, this article introduces a Mixed-Integer Linear Programming (MILP) optimization model based on discrete time windows for a capacitated multi-product multi-line production scheduling problem. It allows modeling several constraints that are typical in the context of grain production, including limited storage space and shelf life of products. The model is extended by proposing a bounds-based scheme intended to accelerate convergence to the optimal solution. The model was tested with real data, taken from the top Uruguayan grain producer and exporter company and its soybean harvest in 2020. The goal was to plan the production of 17 products along 16 weeks, using two production lines. Storage, line cleaning times and limited shelf life constraints are part of this real instance. Significant cost reductions are achieved when compared to the ad hoc planning so far applied by the company. The tests also show that significant time reductions are achieved by applying the embedded bounds-based scheme proposed in this article. This article contributes toward a relevant research topic by proposing a MILP model for a relevant grain production problem. The problem is analyzed both theoretically and empirically, and the presented case study is one of the first reported research of applying ad hoc systematic optimization models for a real grain production company in Uruguay. The article extends our previous conference publication 'A case study of smart industry in Uruguay: grain production facility optimization' [9] presented at IV Ibero-American Congress on Smart Cities, Cancun, Mexico, December 2021. New content and contributions in this article include: (i) an expanded review of the related literature about optimization and the application of MILP models for grain production problems; (ii) the NP-completeness of the problem is demonstrated; (iii) a novel bounds-based scheme for speeding up convergence is introduced; and (iv) the real test case is run with 16 input sets where the original variables are altered to illustrate the performance of the algorithm under different scenarios. The article is organized as follows. Section 2 reviews relevant related works. Section 3 describes the addressed grain facility optimization problem and the proposed MILP formulation. Section 4 presents a study of the computational complexity of the optimization problem and computes useful lower bounds.