Mathematical models for the grain warehouse location problem: a systematic literature reviewAbstract
Abstract The decision-making process for determining the location of grain storage facilities is essential for the entire food supply chain, reducing post-harvest losses, assisting in food security, and optimizing logistics, especially in countries with high agricultural production and storage deficits, such as Brazil. This paper analyzes mathematical models focused on the location problem of silos through a systematic literature review (SLR) following the PRISMA protocol. Bibliometric analyses were performed using Bibliometrix. The SLR highlighted that, although scarce, existing models satisfactorily optimize the location problem, particularly in developing countries. India was the primary research center, leading in citations and global partnerships. Disruption scenarios were considered in only one work, emphasizing the potential costs of unexpected logistical problems. Some models integrated social (job creation, lost workdays, congestion, accidents, noise, community well-being) and environmental (CO2 emissions, grain loss) indicators. All models addressed strategic and tactical planning scope, integrating sustainability and costs, while only two included operational planning. This research guides new models by outlining practices and decisions to optimize logistics and strengthen food supply chains through location models focused on grain warehouses, informing both future academic work and practical applications in logistics and public policy. Future research should integrate stochastic approaches, multiple warehouse and grain types, expanded social and environmental sustainability metrics, logistical failure assessments, and international logistics connectivity.
Keywords:
Grain warehouse; Logistics; Operations research; Localization; PRISMA
Resumo
Resumo O processo de tomada de decisão para determinar a localização de instalações de armazenagem de grãos é essencial para toda a cadeia de suprimentos de alimentos, reduzindo perdas pós-colheita, auxiliando na segurança alimentar e otimizando a logística, especialmente em países com alta produção agrícola e déficits de armazenagem, como o Brasil. Este artigo analisa modelos matemáticos focados no problema de localização de silos por meio de uma revisão sistemática da literatura (RSL) seguindo o protocolo PRISMA. Análises bibliométricas foram realizadas utilizando o pacote Bibliometrix. A RSL destacou que, embora escassos, os modelos existentes otimizam satisfatoriamente o problema de localização, particularmente em países em desenvolvimento. A Índia foi o principal centro de pesquisa, liderando em citações e parcerias globais. Cenários de interrupção foram considerados em apenas um trabalho, enfatizando os custos potenciais de problemas logísticos inesperados. Alguns modelos integraram indicadores sociais (criação de empregos, dias de trabalho perdidos, congestionamento, acidentes, ruído, bem-estar da comunidade) e ambientais (emissões de CO2, perdas de grãos). Todos os modelos abordaram o escopo de planejamento estratégico e tático, integrando sustentabilidade e custos, enquanto apenas dois incluíram o planejamento operacional. Esta pesquisa orienta novos modelos, delineando práticas e decisões para otimizar a logística e fortalecer as cadeias de suprimentos de alimentos por meio de modelos de localização focados em armazéns de grãos, assim servindo como base tanto para trabalhos acadêmicos futuros quanto para aplicações práticas em logística e políticas públicas. Pesquisas futuras devem integrar abordagens estocásticas, múltiplos tipos de armazéns e grãos, métricas ampliadas de sustentabilidade social e ambiental, avaliações de falhas logísticas e conectividade logística internacional.
Palavras-chave:
Armazém de grãos; Logística; Pesquisa operacional; Localização; PRISMA
1 Introduction
Agriculture is crucial to any nation’s well-being, growth, and economic power, especially in countries with high agricultural activity, such as Brazil. This importance also drives technical and scientific development focused on agriculture and agribusiness, significantly increasing food production in these countries and globally (Shukla et al., 2017). Among the various agricultural products, Brazil stands out globally as a reference in grain production, being a leader in soybeans and advancing in corn, with increasingly more technology that provides greater product quality and productivity (Basso et al., 2024; Contini & Alves, 2023). The storage sector is a vital part of the grain supply chain’s logistics activities and, consequently, in the economic aspects of grain production (Filippi et al., 2022; Singh et al., 2018).
Grain warehouses play strategic roles in logistics and production operations, encompassing inventory controls and business opportunities from price variations during harvest and season peculiarities (Singh et al., 2024). Therefore, understanding the supply and demand dynamics for grains also allows for analysis of the costs associated with transporting the products, which are a major grain price component and are highly related; these transportation costs – freight rates – tend to be lower during the off-season and higher during periods of high demand for transportation services (Beirão et al., 2021; Hyland et al., 2016; Kussano & Batalha, 2012; Le Cotty et al., 2017).
Furthermore, warehouses provide the potential for economic advantage in various logistics scenarios and are important for maintaining product quality, especially for perishable products, such as grain commodities (Haque et al., 2025). Storage ensures the proper maintenance of the grains’ physical and chemical aspects, such as moisture and prevention of mechanical damage, from the harvest in the fields to the sale to consumers (Brandão et al., 2019). Driven by operational requirements and economic advantages, the establishment of warehouses has become a competitive factor in the grain supply chain, marked by disputes over the rights to use the physical space of silos (Filippi et al., 2022).
However, even with the importance of storage already recognized and established by the stakeholders in the supply chain, Brazil’s storage capacity does not keep pace with its grain production. Data from the Brazilian National Supply Company (Brasil, 2024a, b) indicated a total static capacity – for bulk solid and conventional – of approximately 207 million tons in 2024, compared to an estimated grain production of 322 million tons for the 2024/25 harvest. Certain regions, such as Brazil’s southern region, have a satisfactory capacity to meet the supply chain’s needs better; still, the other producing regions do not have the same quality infrastructure, resulting in a storage deficit that compromises the entire chain (Brasil, 2024a, b).
It is also necessary to understand the static capacity of a warehouse: it is the total amount, in mass, that the physical structure can support at once (Peixoto et al., 2022; Silva et al., 2016). Another metric used in this area is dynamic storage capacity, which measures the storage capacity over specific periods according to product turnover, considering static capacity, storage period, and average retention time (Maia et al., 2013; Silva & Santos, 2019).
The warehouses also have different categories based on their purposes, with different physical structures, storage arrangements, and types of products. Vertical metal silos focus on the storage of grains or bulk products, employing conveyors to transport grains from various transport modes into the silo and using gravity for discharge, while also offering rapid construction (Dornelas et al., 2021). Likewise, metal silos can be arranged in groups known as silo batteries, consisting of sets of silos in designated locations or supply centers (Brandão et al., 2019). Finally, within the main categories of silos, there are bag silos: these are temporary versions also aimed at grain storage, consisting of low-cost, high-density polyethylene for short-term storage (Krzyzanowski et al., 2023).
Determining new warehouse locations is a central challenge in improving the country’s storage capacity. Locations are even more significant in the case of Brazil, where road transport predominates, characterized by poor physical conditions of the highways, high freight costs, and even a lack of security (A. L. R. de Oliveira et al., 2015; Peixoto et al., 2022). Therefore, the efficient location of grain warehouses can relieve the costs generated by the country’s inadequate road infrastructure, increasing Brazilian competitiveness and, consequently, the profitability and resilience of the supply chain.
Locating warehouses requires considering various factors to minimize costs related to the flow and transportation of a specific volume of goods from producers to customers (Jacyna-Gołda & Izdebski, 2017). This decision requires substantial information from those in charge, including demographic, economic, and logistical data covering movement routes, supply and demand, available labor, and various other factors (Gergin et al., 2023). The decision on the location of the warehouse can be mainly categorized at the strategic and tactical levels: evaluating public policies and stock balancing at the national level, to the flow of grains, inventory rules, and expected production volume (Mogale et al., 2020a).
Hence, it is vital to understand the intrinsic characteristics of agricultural warehouses and their relationships with the economic landscape of grain production, sales, and transportation. With knowledge of these factors, it is possible to seek solutions to mitigate the negative aspects of this supply chain, especially those relevant to Brazil: predominance of road transportation, high transportation costs, poor quality of roads, and storage deficits. Among the tools that help allocate warehouses is mathematical modeling, a tool derived from operations research that enables analyzing scenarios and finding optimal solutions for potential locations of new storage structures (Mogale et al., 2018; Oliveira et al., 2022).
First, this type of model maps out the possible nodes and routes along which goods can be moved, commonly represented by cities or points of interest and roads, with the central objective of minimizing transportation costs while meeting customer demand or maximizing profit (Amaral et al., 2012; Ehmke, 2012; Nakandala et al., 2016). After constructing this logistical network, the possibilities for opening warehouses are added, along with the effects this new center will have on the transportation network, from which the model will indicate optimal locations based on the created scenarios, which may include additional factors such as disruption situations or pursuit of sustainability (Hosseini-Motlagh et al., 2020; Maiyar & Thakkar, 2019).
To investigate the use of mathematical models in the strategic decision-making process of locating and opening agricultural warehouses, the following research questions were established for the systematic literature review (SLR):
(RQ1) How is scientific literature characterized and distributed in mathematical modeling for the location problem of grain warehouses?
(RQ2) Who are the leading researchers, and what are their contexts for optimizing mathematical models focused on the location problem of grain warehouses?
(RQ3) What are the fundamental elements of the mathematical models that assist in the location problem of grain warehouses?
(RQ4) What are the gaps and recommended future research for mathematical models applied to the decision-making process for the location problem of grain warehouses?
2 Methodology
This SLR gathers and synthesizes research on mathematical programming models in operational research for locating agricultural warehouses, primarily for grains and other similar agricultural products. To achieve this objective, the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) protocol was used, a renowned method in the area of SLRs that ensures transparency, rigor, and reproducibility of the structured analyses of the scientific literature (Page et al., 2021).
The scientific productions found were submitted to and interpreted through bibliometric analyses, a set of quantitative methods used to measure and examine academic publications based on their bibliographic information, indicators, and other variables. The analyses were conducted using the Bibliometrix package, an open-source tool built in the R language that includes the main bibliometric analyses, as well as adaptations to the interface (“biblioshiny”) to facilitate its use and interpretation by the scientific community (Aria & Cuccurullo, 2017).
Thus, the main guiding question of the research was based on the research questions and adjusted to guide the formulation of the search string: “What are the mathematical programming models for the location of grain warehouses?”. The inclusion and exclusion criteria were defined as follows: (1) only articles submitted to peer review, excluding conference bibliographic materials and books, among others; (2) only articles in English or Portuguese; (3) the terms of the search string should appear in the title, abstract, and keywords. The year of publication was not considered due to the search’s highly specific nature, which significantly reduces the number of records. As for the databases, two international platforms were used, Scopus and Web of Science (WoS), employing the same search string with technical adaptations equivalent to the syntax of each database. Table 1 presents the complete information about the search strings, their inclusion and exclusion criteria in each database, and the search dates and filters.
The first part (warehouse OR silo) searches for silos or specific storage facilities related to the agricultural sector, avoiding purely industrial ones or those that do not encompass the chosen theme, such as general goods depots. The second part (grain OR soy OR rice OR corn OR wheat OR barley OR sorghum) specifies which goods the warehouses must focus on. In addition to the general word grain, examples of certain grains highly relevant to the sector were also included, as using a term as general as “food” would lead to multiple unwanted studies.
The third part (location OR localization) indicates that scientific productions must address location issues, assisting in the decision-making process for opening and determining the location. Lastly, “model*” clarifies that any mathematical programming models are sought, but since these have various names, the generic term “model*” was chosen with the wildcard symbol “*,” indicating that any suffixes will be accepted. It was not necessary to specify “mathematical” because this type of location problem is already well developed in the operational research field so that the previous terms would return the desired studies; by keeping the search broader, other relevant types of models could also be found, and it would be up to the screening process to decide whether they would be included in the SLR.
2.1 Data analysis
Descriptive and advanced analyses were performed to separate, visualize, and interpret the data. The descriptive analyses used frequencies and simple classifications: overview of production, countries, the most cited articles, international collaboration, and word cloud. Among the more advanced analyses involving correlation matrices and other complex techniques, the following were used: three-fields diagram (Sankey diagram), co-occurrence, co-citation, and thematic mapping. To better analyze the selected research, terms extracted from the abstract in bigrams (sequences of two words) were used for co-occurrence analyses and thematic mapping. This methodology was based on previous research that already focuses on the analysis of the bigrams and trigrams present in the abstract, identifying common terms that clarify the context but are often not included in the keywords, such as methodological terms and approaches (Hassan & Duarte, 2024).
2.1.1 Word cloud and Three-fields plot
A word cloud, or weighted list, visualizes the most recurrent terms within a collection of analyzed texts or articles, showing the most prominent terms in an easy-to-interpret graph. The size of each word corresponds to its frequency, and larger terms reflect higher prevalence in the sources (Alkhammash, 2023). Concurrently, smaller terms have lower frequency but retain analytical value, potentially signaling emerging research trends or niche topics that need further exploration (Mulay et al., 2020).
The three-fields plot relates three distinct variables or categories – with logical relationships – chosen by the researcher, using a Sankey diagram as the visual structure; often, the left variables are intellectual roots or origins, and the right ones are research contents, such as keywords. These three elements are connected by lines that indicate the horizontal connections (flow) between them and are separated by columns representing the chosen categories or sections. The larger the rectangle of the columns, the greater the number of articles associated with that field (Rusydiana, 2021).
2.1.2 Co-occurrence and co-citation network
Co-occurrence networks use mathematical structures to represent and relate objects in a network format, utilizing vertices – nodes or points – connected by edges – lines or connections – of varying sizes that represent relationships. The first step in creating networks is to form a correlation matrix of the terms, thus allowing for the grouping and interpretation of the diagonal and non-diagonal relations of the items in the matrix; the size of the vertex is based on the diagonal relationship of the elements, and the size of the edges on the non-diagonal relationship (Aria & Cuccurullo, 2017). The co-occurrence network of words is considered a conceptual analysis of the evaluated terms, identifying the overall scenario and how groups and terms relate, which enables the inference of precursors and key concepts from the matrix used (Sedighi, 2016). This study conducted a co-occurrence analysis based on terms extracted from the abstracts of articles; the terms were separated into bigrams, indicating that pairs of directly adjacent words were considered unique entries for the data analysis (Aria et al., 2022).
Co-citation networks, on the other hand, are an analysis of the intellectual structure of the relationships of the references – authors, documents, or sources – of the evaluated articles, also based on a correlation matrix; co-citation occurs when two entries cite a third entry from the matrix (Aria & Cuccurullo, 2017). This map helps understand patterns in references – such as highlights in certain research topics – and identify seminal works or authors that were not included in the SLR itself but are strongly mentioned by the included articles, thereby considered important in the research topic (Calabretta et al., 2011). In the present research, a co-citation network was created with the authors, aiming to find authorship relationships between the articles and the seminal authors in this field of research.
2.1.3 Thematic map
Thematic mapping is a method that mixes a clustering algorithm and keyword network to identify domain themes, using extracted terms and classifying them into groups, positioning them according to centrality (X-axis) and density (Y-axis) (Alkhammash, 2023; Aria & Cuccurullo, 2017). The interpretation of the map is presented by Cobo et al. (2011), who separates the graph into four quadrants: (1) motor themes (upper right quadrant) have high density and centrality, are well-developed topics, and remain important to the research area; (2) niche themes (upper left quadrant) have high density and low centrality, are well-developed subjects but with limited importance, being relevant only in specific discussions; (3) emerging or declining themes (lower left quadrant) have low centrality and low density, indicating underdeveloped and marginal themes, which represent early-stage emerging themes or themes that are in the process of disappearing; (4) Basic themes (lower right quadrant) have high centrality and low density, representing important themes for the area that have not yet been developed.
3 Results
3.1 PRISMA diagram
Figure 1 illustrates scientific publications’ identification, screening, and inclusion flow according to the PRISMA protocol. In the initial search from Scopus and Web of Science databases, 78 and 59 records were returned, totaling 137. Of these, 33 were removed for not being scientific articles, five for not being in English or Portuguese, and 36 for being duplicates between the databases. Sixty-three articles advanced to the initial screening, of which 48 were excluded after reading the title and abstract. The full versions of the remaining 15 articles were obtained, meaning no article reading was hindered by the level of access to the journals. Therefore, four articles were removed after a complete reading of the text: three were excluded because their models did not consider opening new warehouses, and one was excluded for using a model type outside the SLR’s scope.
Therefore, the SLR obtained 11 articles in the final sample. This quantity is relatively low compared to other SLRs, which is explained by the highly specialized scope of this research and the low number of works in the sector. The selected articles are presented in Appendix 1.
3.2 Overview
Articles were published between 2005 and 2023, with an average age since publication of 6.73 years across eight different journals, totaling 33 authors, with only one article having a single author. There was significant international collaboration, with 45.45% of the articles containing authors from different countries. The documents also had a high average citation count of 31.64, representing the importance in the academic community. Complementing the information on international co-authorship, Figure 2 presents the collaboration map between countries based on the corresponding author and co-authors. The country with the highest interaction is India, which collaborated with Brazil, Portugal, Switzerland, and the United Kingdom, with the last two having more collaborations. Brazil also collaborated with Portugal, and the United Kingdom with Switzerland, but at lower levels than those presented with India at the center.
According to Figures 2 and 3, it becomes evident that India is the center of work integrations and article relevance. India had 267 citations in total, a significantly larger amount than the second-place Iran, with only 36 citations, followed by the United Kingdom (18 citations), Ghana (11 citations), and Brazil (10 citations).
Furthermore, the number of global citations separated by article (name, year, and journal) can be observed in Figure 4. The main author, Mogale D. G., a researcher from India, stands out with four articles in the SLR, two of which had a high number of citations, with 87 and 76, respectively. Consequently, this author can be considered one of the most important for the scope of this SLR, with high relevance in collaboration in mathematical modeling for warehouse location. Similarly, the corresponding author, Maiyar L. M., from India, is notable for his 52 citations. As discussed in Section 3.3, he is the main information integrator within the author network.
Figure 5 presents the three-fields plot that relates three properties of the articles: country of origin, journal, and author’s keywords. Countries with more publications publish in varied journals, but the keywords are interconnected, meaning there is a pattern of identification for the issues in this sector, such as “agribusiness” and variations of the word optimization (“optimization” and “optimisation”) that change according to the type of English. The combination of the keywords “food grain supply chain” was not included in the initial searches. Still, it proved to be a potential identifier of the issue addressed in this work, requiring further research to understand what other aspects are part of this area beyond the location of grain warehouses. Lastly, due to the authors’ countries, the term “developing countries” is frequently mentioned, as several articles focused on the local issues of each author, who are consequently residents of developing countries.
3.3 Co-citation
Figure 6 presents the seminal authors cited by the articles of the SLR, which have been grouped into three clusters. In the blue cluster, two authors stand out – who also have contributions in this SLR but do not share co-authorship – Maiyar L. M. and Mogale D. G., with betweenness (centrality metric) scores of 268 and 51, respectively; the high values indicate that these authors are important for connecting and sharing information among the other nodes, thus being essential for efficient information flow. Both authors are from India and focus on optimization models in the agricultural supply chain, primarily in logistical operations, considering demand, supply, stock capacity, and transportation modes. Another author included in the blue cluster is Ahumada O., a highly recognized researcher from Mexico in mathematical planning models in the food supply chain, with proposals for operational, tactical, and strategic models. A significant example of his contributions is a reference literature review in the sector (Ahumada & Villalobos, 2009), with over 700 citations.
In the green cluster, the prominent authors include Mohammed A. and Validi S. from the United Kingdom, Soto-Silva from Chile, and Soysal M. from Turkey. This group is broader, with a similar scope – but wider – than the blue cluster, including research on horizontal collaboration and the distribution of perishable products, sustainability in the supply chain, and supply chain resilience. Finally, the last cluster, in red, is smaller and includes authors An K. from China, Ge H. from the United States, Asgari N. from the United Kingdom, and Gholamian M. R. from Iran. The red cluster primarily focuses on optimization, complexity, risk mitigation, and other supply chain-related factors, considering multiple factors and agents. Asgari N. stands out in researching allocation models and the opening of facilities in different areas – such as ports, urban services, and warehouses for wheat – considering costs, distance, sustainability, supply chain robustness, and service coverage, among others.
3.4 Word cloud, co-occurrence network, and thematic map
Figure 7 presents the word cloud of the identified bigrams, with weights based on the frequency of the terms: larger words represent higher frequency. The terms “supply chain” and “food grain” were the most cited, indicating that the type of modeling problem in this work is situated within the areas of food supply chains, specifically grains.
Less frequently, but also with great importance, there are terms such as: “mathematical model, decision support, grain storage, storage capacity, hub disruption, chain network.” These words represent the areas in which mathematical models are applied, the sectors that seek optimization, or which technical aspects are being modeled. The models are viewed as decision support tools that, in this case, aim to enhance the logistics and storage network of the supply chain. Within this sector, they can assess the physical storage capacity of grain in the stocks or potential scenarios of chain disruption, proposing options for opening new locations to minimize damage in the evaluated hypothetical situations.
With even lower frequency are various terms with certain important relationships, such as aspects of environmental and social sustainability (“social responsibility, social sustainability, sustainable food, environmental objectives”); those can be added to the model to influence the weights of decisions. There are also those bigrams related to the previous supply chain terms, merely demonstrating variations of the same topics (“grain supply, grain silo, wheat supply, transportation cost, food supply, storage facilities”). Lastly, there are terms with aspects of the models (“objective function, based multi-objective, optimal solutions”) and with techniques used in the resolution and interpretation of the results (“swarm optimization, particle swarm, solution techniques, Pareto based, sensitivity analysis”).
Figure 8 contains the co-occurrence map of the bigrams in the abstract, showing the relationships between the terms and their proximities. Six clusters were created: three larger ones (blue, purple, and green) and three smaller ones (brown, orange, and red).
The blue cluster is a central cluster. It represents terms related to the supply chain, which includes the network of product supply, cost resilience, decision support, and the importance of considering social responsibility in the process. Technical modeling terms are also present, demonstrating the properties of the models that focus on this sector, such as multi-objective functions. Similarly, the purple group is also a central cluster, focusing more on the aspects of mathematical models related to grains, and which parameters or variables are being considered. For example, chain demand, storage capacity, total cost, and idle time are numerical attributes that are part of the model. Additionally, terms indicating social and environmental concerns, such as sustainable food, environmental goals, and social sustainability, are also present.
The green cluster addresses global logistics, also using technical terms related to models, but with the significant difference of focusing on opening centers (“hubs”) to improve overall efficiency and anticipate disruptions in the supply chain, and to choose strategic locations that minimize losses in unexpected cases. This grouping is strongly interconnected with the purple group, indicating that it may be an extension that has separated from the main group, possessing sufficient peculiarities to form a new set while maintaining strong ties with mathematical modeling terms.
The brown cluster has only two terms representing grain supply and, again, decision support, which integrates grain supply among the other groups, lacking significant specificity. The orange cluster stands out by including developing countries, so the storage capacities and costs are unique in adapting to these scenarios, which are also connected with the supply chain groups (blue) and grain supply (brown). Finally, the red group is isolated and emphasizes the importance of production in the state of Mato Grosso and its relationships with the international market, potentially indicating case studies in the sector within this geographical area and its relationships with exports and imports.
The analysis extends from the co-occurrence network presented in Figure 8 to the quadrant classification shown in Figure 9. Both figures map the conceptual structure of the research, but with different objectives and outputs. The co-occurrence network primarily identifies and displays the linkages between research terms, grouping them into clusters. The thematic map advances this initial structure by evaluating the clusters themselves, plotting them on a grid of centrality (relevance) and density (development) to classify each as a motor, niche, basic, or emerging theme (Cobo et al., 2011).
Due to these different methodologies and the minimum weights used for clustering, the resulting thematic groups may exhibit differences in terms and colors (Aria et al., 2022). Furthermore, the visual representation differs: the co-occurrence network graph emphasizes the interconnectivity between individual concepts, while the thematic map’s quadrant layout highlights the comparative role and maturity of each theme.
The topics of the two most significant clusters from the co-occurrence network (Figure 8), which focused on mathematical modeling related to grains (“mathematical model”) and decision support supply chains (“supply chain”), were classified as driving themes (Figure 9), indicating they are well-developed and relevant both to the scope of the work and to potential connections with other related themes. The topic of grain storage (“grain storage”) is positioned between niche and potential decline or emergence, requiring further temporal analysis to determine how the grouping behaves. One caveat is the difference between the storage capacity mentioned in the red group and grain storage: the former is a generic term, which, although it integrates the grain grouping, can also represent the storage of other types of products; the latter refers specifically to the use of warehouses for grains, which is the most desired by this study.
The international market cluster, which included cases from producers and other economic aspects, was categorized as a declining or emerging theme, likely due to the specificity of this sector and the types of work it encompasses, not being well developed, and not having high density due to the low amount of discussion. There is also an interesting aspect: the works most likely (higher probability according to the test) to be part of this cluster have Brazilian authorship, indicating that the articles from this country are concerned with the international market, specifically in improving the competitiveness of soybean exportation.
The last cluster, “mixed integer,” is the equivalent of the green “cluster” in Figure 8, representing mixed-integer models that assist in logistics more broadly and with concerns about the location of storage centers, classified among the basic and emerging or declining themes. Its location on the thematic map also requires a temporal analysis to understand the movement trend, but it can be stated that it is an area that should be observed for future research in the sector, as it has medium centrality with low density.
4 Discussion
Table 2 shows the articles in the SLR with the respective model classification according to the planning scope, variable classes, and model type. The categories and classifications were adapted to the warehouse localization scope, thus considering only the most relevant information for the analyzed sector.
Although the models are mathematically robust, their practical applicability depends on data availability and the cost of parameterization. Multiple models demonstrate this dependence by relying on structured data from national bodies, such as the Food Corporation of India (FCI) (Maiyar & Thakkar, 2019; Mogale, et al., 2020a, b; Mogale et al., 2018) or the Brazilian Institute of Geography and Statistics (IBGE) and the Information System for Freights (SIFRECA) (Mascarenhas et al., 2014; Milanez et al., 2016). The practical use of these models is limited by their need for reliable and updated logistical data, which is often scarce in developing nations (Essien et al., 2018).
In the Brazilian context, while national public entities like CONAB and IBGE, along with research groups such as the Group of Research and Extension in Agroindustrial Logistics (ESALQ-LOG), which publishes the SIFRECA, provide extensive data for modeling at the country level, obtaining detailed regional data remains a significant challenge. This inconsistency creates operational barriers to parameterization, as the quality and availability of logistical data can vary drastically across localities. States with a prominent agribusiness sector, such as Mato Grosso, often have abundant data, which stands in sharp contrast to other regions where such information is scarce. Moreover, not all relevant data is publicly available, creating an information asymmetry where inputs may only be accessible through private research or direct acquisition.
There were no significant differences regarding the level of planning, as models that aim to assist in the location and opening of warehouses need to combine strategic and tactical decisions. Strategic decisions about the opening of warehouses are long-term and require various deliberations from policymakers and key agents in the supply chains, assessing both the social, economic, and environmental benefits and consequences, as well as the balancing of inventory between regions and states (Mogale et al., 2018, 2023). Similarly, the tactical aspects of inventory rules, grain flow, expected production volumes, and demand are essential in the decision-making process for opening; thus, both levels of planning are intrinsically linked (Mogale et al., 2020a).
In addition, two studies also included sufficient actions to be incorporated at the operational planning level in the mathematical models. The model proposed by Mogale et al. (2018) included multiple operational transportation decisions focused on fleet allocation of vehicles and machinery, evaluating the types of trucks, availability of machinery at each warehouse, detailed inventory control, and travel and dwell times. Hosseini-Motlagh et al. (2020) considered the internal processing of wheat in factories with different types of flour production and levels of wheat grain quality. The proposed model also contained more operational variables for detailed inventory control, which varied by the grains and flours’ origin, destination, and storage time. Neither of the models solely focused on this planning scope, but added considerable optimizations for operational activities alongside tactical and strategic decisions.
Regarding the decision variables, all researchers included the possibility of opening a new warehouse, with simpler models predominantly considering logistical aspects such as route flow, supply, demand, transportation costs, storage capacity, and operational and warehouse opening costs (Foulds, 2005; Mascarenhas et al., 2014; Milanez et al., 2016). Some peculiarities are not visible in Table 2. A good example is the model proposed by Essien et al. (2018), which did not show major visible changes in the variables and parameters category, but it included unique aspects related to the international market, considering four groups of interest (market negotiators, airport exporters, port exporters, and producers or farmers), with different priorities, commercial capacities, and positive and negative weights.
Similarly, Steiner et al. (2017) did not consider the aspects evaluated in the variables category from Table 2. Still, it proposed an initial organization of municipalities into storage regions, performing a multi-objective task and going beyond simply considering the states already defined by the government, creating regions based on logistical demands. The first phase of the optimization involved delineating the region’s area by specifying the corresponding municipalities included, and the second part was to determine the number of silos to be constructed, along with their respective locations.
The warehouse categories were primarily proposed by Mogale’s articles (Mogale et al., 2020a, b, 2018, 2023), which studied the logistical scenario in India, which already has a supply chain with several warehouses types: the procurement centers (or Mandis) purchase directly from producers; the base silos located in acquisition or production states; the field silos located in consumer or demand states; and the regional silos located at a regional level that receive products and distribute them to consumer markets. Among the articles, there are some variations in nomenclature, such as the use of silos in some models and warehouses in others, but the central idea of all is the same.
Procurement centers send products to the storage facilities at the state supply bases, which, in turn, distribute them to the field storage structures in the demand states. These field storage facilities send the products to the regional storage structures, which then supply the products to purchasing markets or fair-price shops. The possibility of different types of grain warehouses opens new scenarios in location optimization problems, as the combination of different size silos or those with different logistical purposes creates new possibilities – in addition to greater flexibility – related to the efficient balance of goods, construction costs, and operational costs.
Five models considered sustainability weights, both social and environmental. The social indicators included: job opportunities created by the new silos; lost workdays due to the closure of existing silos; social costs from car congestion, car accidents, and road noise pollution generated by the silos’ activities; economic well-being of farmers and the general population (Hosseini-Motlagh et al., 2020; Maiyar & Thakkar, 2019; Mogale et al., 2023). The authors explained each assessment form and the methodologies and references for quantifying these social factors.
Furthermore, the environmental indicators considered: cost based on carbon dioxide (CO2) emissions from transportation, grain storage, and operations to establish a new silo; loss of grain in transportation and storage operational activities (Maiyar & Thakkar, 2019; Mogale et al., 2020a, b, 2023). Undoubtedly, the models utilized carbon dioxide emissions the most, which may have been derived from established methodologies for measuring and pricing these polluting emissions.
While the models were validated internally, their external validation against real-world performance benchmarks varies. Most articles build their models using empirical data, emphasizing case studies in developing nations such as Brazil, India, and Iran. This is commonly achieved by using data from official government sources and field surveys, or by generating and solving multiple problem instances based on reliable reports. Although most studies present an optimized solution, few compare it directly to an established baseline. An example is the work by Essien et al. (2018), which compares its optimized network directly with the existing government network in Ghana, demonstrating a potential 16–34% reduction in transportation costs.
Additionally, while several models include important social and environmental indicators, they overlook several noteworthy operational and social realities. For example, none of the models explicitly addresses land-use conflicts, a relevant social issue in agricultural frontier expansion (Cáceres et al., 2020). Another important operational factor is agricultural seasonality. Multi-period models can effectively capture the temporal dynamics of harvest seasons (Hosseini-Motlagh et al., 2020; Mogale et al., 2023). Conversely, other models rely on annual production data, a simplification that ignores the logistical pressures during peak harvests and underestimates the required storage capacity for seasonal fluctuations.
Concerning the mathematical formulations, there was not much difference between the types of models, as only mixed-integer linear programming (MILP) or mixed-integer nonlinear programming (MINLP) models were used. However, within these model classifications, there is considerable diversity in their objective functions, constraints, and other components. Some models focus on a single objective of minimizing total costs, which typically include transportation, operational, and silo construction expenses. For instance, Foulds (2005) presents three evolving models that start by minimizing only transport costs and then progressively add variable and fixed construction costs. Other models adopt a multi-objective model to manage trade-offs, such as the model by Mogale et al. (2018) that minimizes both total network cost and total lead time.
Certain models integrate environmental and social indicators directly into their objective functions. One straightforward method involves formulating a single cost function that quantifies and sums logistics costs with environmental and social costs (Maiyar & Thakkar, 2019). In contrast, other models consider these dimensions as distinct, competing goals. To manage these trade-offs, multi-objective models balance economic costs with differing types of objectives; one includes environmental impacts and social indicators (Mogale et al., 2023), while another focuses on social responsibility and supply chain resiliency (Hosseini-Motlagh et al., 2020).
The complexity of the models extends to their formulation and specific rules. While all models include fundamental constraints for supply, demand, warehouse capacity, and flow balance, many introduce specialized rules to reflect real-world conditions. A notable example is the model proposed by Maiyar & Thakkar (2019), the only article that focused on potential disruptions of distribution centers and warehouses, assessing whether the costs of the supply network would increase if a storage node, either origin or destination, suddenly failed. It was found that, on average, if only one warehouse fails, the cost to meet demand would increase by 14%; however, if multiple warehouses fail, the total cost would increase as much as 40%.
Another clear case is the inclusion of constraints for the limited availability of heterogeneous capacitated vehicles – different types of trucks and rakes – at various echelons of the supply chain (Mogale et al., 2020a, b, 2018, 2023). Furthermore, only the work of Hosseini-Motlagh et al. (2020) did not use a completely deterministic model, incorporating stochasticity into the model’s components to assess uncertainties in the supply chain.
Finally, the SLR findings directly affect public policy, mainly in addressing Brazil’s persistent grain storage deficit. Government initiatives, such as the National Warehouse Program (Plano Nacional de Armazenagem or PNA) and its associated credit line, the Program for Construction and Expansion of Warehouses (Programa para Construção e Ampliação de Armazéns or PCA), aim to expand the country’s static capacity (Brasil, 2018). However, the success of these large investment programs depends on optimal facility placement, which is the problem addressed by the mathematical models in this SLR. Models that include sustainability factors can ensure that funded warehouses meet national environmental and social goals. Likewise, models that assess potential disruptions can help policymakers build a resilient supply chain, not just a larger one. These models provide a quantitative method for decision-making, helping create a network optimized for cost, efficiency, and resilience.
4.1 Research gaps and future scope
Going beyond the previously mentioned and evaluated perceptions of the models, the main future research proposals made by the authors of the articles in the SLR have also been summarized, including a brief overview and the percentage of articles that recommended a certain category (Figure 10). This summary allows for the assessment of topics related to location models that were not within the initial scope of the research, as they are improvements that the developers themselves deemed important.
Stochasticity is an additional aspect that can be incorporated into most models to account for uncertainties, such as global demand and transportation time forecasts for cargo, which vary according to internal and external factors; it brings models closer to reality because the real world is inherently uncertain. However, adding these probabilistic considerations also increases the complexity of the model, requiring more effort to develop an efficient tool. Regarding international logistics connectivity, adaptations have been proposed that would allow the model to be applied in neighboring countries, which could also imply mutual benefits through new access options to the sea.
Adding more sustainability considerations has also been suggested, both in works that already considered some aspects and wished to evaluate others, and in those that did not implement them in their models and proposed them as an evolution. In environmental variables, the water footprint of goods transported worldwide stands out, and post-harvest losses of food, a factor related to logistics, generate losses for the entire food system since all productive efforts have already been made. In social variables, food security is noted through the efficiency of the supply chain, as well as variables that can measure social satisfaction, awareness, and costs.
Improvements related to the agricultural properties of the focus products, grains, have also been proposed. Post-harvest losses have already been mentioned in sustainability. However, grains still have unique relationships concerning perishability and monetary value, as the values of certain variables – such as moisture content – directly affect the product classification and, consequently, its selling price. Additionally, variability in planting and harvesting seasons and other planning aspects essential to grain production can be considered. Finally, regarding the types of grains, the development of multi-grain models is suggested, which expands the planning of all characteristics across multiple grains; this addition may also increase the model’s complexity and change the structure of the indices of the variables and parameters.
5 Conclusion
The SLR revealed few works on mathematical models focused on the decision-making process for the location problem of grain warehouses. However, the identified research adequately addresses the desired scenario, with different assumptions and techniques applicable to new formulations of operations research models. It is noted that the studies demonstrate methodological diversity and a clear focus on the challenges faced in the contexts of developing countries, among which India and its authors should be highlighted. The findings show a high presence of international collaborations, showcasing the search for solutions that align regional particularities with the potential for scaling and adaptation to other scenarios or countries.
All the models encompassed strategic and tactical planning levels, a result consistent with the type of decision to be made: to identify the best warehouse opening locations, the entire chain must be considered, with broad views and medium to long-term projections, as well as benefits and impacts on sustainability – social, environmental, and economic – robustness in disruption scenarios, costs, and improvements in routes throughout the region. Depending on the size of the warehouses, regional, state, and even national dynamics are affected; thus, their analysis requires higher levels of planning involving government participation and key players in the chain. Only two models also considered the operational level, controlling vehicle allocation, labor, and inventory in more detail. As for the model type, all used integer variables, of which five were MILP and six were MINLP.
Finally, this paper guides the formulation – or adaptation – of mathematical models that integrate high-impact decisions, sustainability, and the maximization of benefits or minimization of costs, enhancing logistical practices and strengthening the food supply chain and agribusiness in an increasingly complex scenario related to the location problem of grain warehouses. An application of this contribution is in public policy, where these models are decision-support tools that use a rigorous and efficient quantitative method to direct investments in national warehouse programs – such as the PNA/PCA – and enhance supply chain resilience.
Considering the information presented, the following considerations are proposed for future research:
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Incorporate stochastic elements to address uncertainties.
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Include multiple warehouse categories.
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Evaluate more sustainability variables, primarily social and environmental.
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Account for multiple grain types with distinct logistical, perishability, and quality traits.
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Assess the impacts of logistical failures, such as warehouse disruptions.
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Consider and integrate logistical structures from other countries as logistical possibilities.
Appendix 1 Eleven articles selected by the SLR with their corresponding identification, title, and journal.
| Articles | Title | Journal |
|---|---|---|
| Essien et al. (2018) | Decision support system for designing sustainable multi-stakeholder networks of grain storage facilities in developing countries | Computers and Electronics in Agriculture |
| Foulds (2005) | Dynamic network flow models of sustainable grain silo location | International Journal of Operational Research |
| Hosseini-Motlagh et al. (2020) | A novel hybrid approach for synchronized development of sustainability and resiliency in the wheat network | Computers and Electronics in Agriculture |
| Maiyar & Thakkar (2019) | Modelling and analysis of intermodal food grain transportation under hub disruption towards sustainability | International Journal of Production Economics |
| Mascarenhas et al. (2014) | Analysis of warehousing network for Mato Grosso´s soybeans: applying a localization model | Espacios |
| Milanez et al. (2016) | Optimization of Brazilian soybean exports: an application of a network design model | Espacios |
| Mogale et al. (2018) | Grain silo location-allocation problem with dwell time for optimization of food grain supply chain network | Transportation Research Part E: Logistics and Transportation Review |
| Mogale et al. (2023) | Designing a food supply chain for enhanced social sustainability in developing countries | International Journal of Production Research |
| Mogale et al. (2020a) | Modelling of sustainable food grain supply chain distribution system: a bi-objective approach | International Journal of Production Research |
| Mogale et al. (2020b) | Green food supply chain design considering risk and post-harvest losses: a case study | Annals of Operations Research |
| Steiner et al. (2017) | A multi-objective genetic algorithm based approach for location of grain silos in Paraná State of Brazil | Computers & Industrial Engineering |
Statement on Data Availability
The secondary data utilized in this research consist of bibliometric data extracted from scientific articles, which are in the public domain and accessible through their respective databases and scientific publishing platforms. If requested, the specific bibliometric dataset subjected to the tests described in this study, as well as the complete report of said tests, can be fully shared for verification and replication of results.
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Financial support:
This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.
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How to cite:
Rosa, A. G., Santos, A. P., Delgrossi, M. E., & Reis, S. A. (2025). Mathematical models for the grain warehouse location problem: a systematic literature review. Gestão & Produção, 32, e4925. https://doi.org/10.1590/1806-9649-2025v32e4925
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Editor-in-Chief
Pedro Munari
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Source: created by the authors (2025) with Bibliometrix.
Source: created by the authors (2025) with Bibliometrix.
Source: created by the authors (2025) with Bibliometrix.
Source: created by the authors (2025) with Bibliometrix.
Source: created by the authors (2025) with Bibliometrix.
Source: created by the authors (2025) with Bibliometrix.
Source: created by the authors (2025) with Bibliometrix.
Source: created by the authors (2025).