## 3 Literature Review
A special case is the hub location problem: In this problem, entities travel between predefined origin and destination (OD) nodes. These entities can either travel directly between OD nodes or through a hub. These hubs may facilitate the flow of entities, and reduce transportation costs due to economies of scale [9]. Clearly, the hub location is not restricted to supply chains. Many applications exist in the telecommunication, airline, and urban transportation sectors [46; 47]. The hub location involves several decisions: (1) Selecting a subset of hubs among a set of candidate locations, (2) determining inflows and outflows at the selected hubs, and (3) determine the transportation modes. A variant, closely related to this research, is the competitive hub location problem [24; 46; 48]. In the competitive hub location problem, a company intends to establish new hubs where there are already other competitors including the company's own hubs. The objective is generally to maximize profit, which involves two terms: (1) The company's revenue, which is proportional to the attracted demand, and (2) fixed and variable costs [46]. The mathematical problem of perishable food hubs has been little studied: Among the few works, [PERSON] et al. [9] proposed a MILP formulation for regional food hubs in the United States, which is solved in two phases with a linear relaxation in the first phase to filter our non-promising nodes and a second phase that included integer and binary variables. Later, [PERSON] et al. [49] presented a multi-objective perishable food hub location and scheduling problem for sustainable supply chains. Finally, [PERSON] et al. [50] presented a multi-objective formulation for the case of distribution of perishable foods considering emissions and shelf life. The last two references use Multi Objective Evolutionary Algorithms (MOEAs). Most works on food hub and food supply chains consider decisions based on centralized control of all links: Suppliers, hubs, and retailers [44; 49; 50; 51]. This is not realistic in the studied case: First, farmers in central Colombia have choices to sell and distribute their products. Second, in the literature of supply chains, the demand must be fully or partially satisfied [49; 52]. In this paper, we assume that the hub management makes decisions as to where to deliver the products based on some parameter of attractiveness of the trading point (i.e., the city wholesaler) as in [26; 27; 46] Normally, this decision has to do with several factors such as familiarity with the trading point, or with the probability of finding customers. We make the reasonable assumption that the demand of each city is far larger than the quantities that the hubs can deliver. After all, these regional hubs in the central region of Colombia are facilities located in small-to-medium size towns with limited capacity (the largest with around 600 tons) and for instance, the total demand of the country capital Bogota alone is over 40,000 tons per day. In the problem studied in this paper, we have a combination of centralized and decentralized decisions. The purchasing center has control on the operation of the facilities and of the distribution to the cities (i.e., outbound logistics); however, since farmers are not contractually bound by the food hubs, one goal is to attract farmers and therefore the agricultural production in the most profitable and sustainable way. As such, farmers are customers of the food supply chain system and therefore a portion of their yield is expected to be captured by the food hubs. Our work has both obvious similarities and important differences with the competitive hub location problem: On the similarities side, farmers may choose to travel to the food hubs or sell their products directly to middlemen, wholesalers, or to final customers as in [1; 6]. The decision will be made based again on some measure of attractiveness [7]. The objective function (maximizing profit) presented in this paper is consistent with those of the hub facility location literature (e.g., [27; 46; 53]).