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Optimal Facility Location in Hierarchical Systems with Mathematical Optimization

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A facility location problem considers a set of spatially distributed customers and a set of facilities that serve customer demands to determine the locations of facilities based on a given set of candite locations. More specifically, a facility location problem answers the following questions: Which facilities should be opened? Where should the opened facilities be established?  Which customers should be served from which facility to maximize the service level or minimize the total costs?

One of the key features of many practical facility location problems is that different types of facilities with different roles (e.g., production, distribution, warehousing) work seamlessly together where a natural material flow, like a hierarchy, exists between them. The facilities of the same type are considered as an echelon or a layer, which defines a level in the hierarchy of facilities. In other words, a hierarchical system is referred to the system in which facilities interact with one another throughout a multi-level hierarchy to provide services or products for customers. In such a hierarchy, customers are assumed to be at the lowest level of the hierarchy.  Accordingly, the decision-makers of hierarchical systems should decide about the best locations for their interacting facilities in a multi-layer structure. The most important hierarchical systems are healthcare systems, postal delivery services, production-distribution systems, solid waste management networks, banking systems, and education systems.

There is a unique material flow in each hierarchical system. Take a three-level healthcare system including local clinics, hospitals, and regional hospitals as an example. In this hierarchical system, patients are first served by a local clinic for the diagnosis or treatment of common diseases. When a patient is directed to a hospital, he/she might receive an advanced diagnosis, advanced treatment, or simple surgery operations. Finally, some patients might need to go to regional hospitals for specialized surgery operations or intensive treatment. Thus, regional hospitals may accept patients from both hospitals or local clinics accepted.

Due to the complexity of material flows in hierarchical systems, like the mentioned three-level healthcare system, making optimal decisions about the location of facilities are so important because location decisions in such systems significantly affect the service level and total cost of the system. Therefore, decision makers’ location decisions should lead to cost minimization and service availability maximization. Since the facility location problem is a well-established use case of Mathematical Optimization, the decision-makers of hierarchical systems can use this technology to instantly make the best location decisions that meet these goals.

Mathematical Optimization, or Mathematical Programming, is a powerful prescriptive analytics methodology that enables decision-makers to instantly identify the best decision out of a large number of alternatives (e.g., millions of possible decisions) that leads to the best possible result according to prespecified criteria such as profitability, service level, resource utilization, etc. Mathematical Optimization includes mathematical modeling methods that capture the key features of a complex business problem including business rules, objectives, and decisions as data-driven mathematical models. To ensure these models accurately represent all aspects of the business problem, they are validated against historical and current data. Then, they are solved by powerful optimization solvers. Indeed, the optimization solvers are powerful computational engines that read optimization models and then deliver the best decision, also known as the optimal solution. Mathematical Optimization has many advantages including but not limited to considering interdependencies of complex systems, supporting what-if scenario analysis, avoiding personal bias, and significant flexibility in constantly changing business environments.

Decision-makers can take benefit from different types of mathematical optimization models developed for hierarchical systems. One type of these optimization models is the Hierarchical Facility Location Problem (HFLP) model. HFLP models determine the locations of the facilities of a hierarchical system to meet one or more than one objective. There are two main objectives for facility location optimization in hierarchical systems: cost minimization and service coverage maximization. One of the most well-known objectives for HFLP is the covering objective. For example, Hierarchical Maximal Covering Location Problem (HMCLP) maximizes the number of the demands covered by the facilities of a hierarchical system. HMCLP is used when the number of facilities is not enough to cover all demands.

Adopting policies that increase the reliability of systems is very important. The reason for this importance is that some facilities fail when a natural disaster (e.g., floods, earthquakes, bad climate conditions) or man-made disaster (e.g., congestion, war, labor disruptions,) occurs. Therefore, costs of systems increase or the number of the demands covered by their facilities decreases. For example, hurricane Katrina interrupted the production of 1.4 million barrels of oil per day in the Gulf of Mexico in 2005. Therefore, it is very important to consider reliability in facility location planning so that facilities operate efficiently not only in normal conditions but also when a disruption occurs.

The nature of facilities in many crucial services is hierarchical.  Therefore, increasing the reliability of hierarchical systems is very important. This importance is because many hierarchical systems play a vital role in providing important services for societies. For instance, healthcare systems, emergency medical services (EMS), telecommunications systems, and production-distribution systems are hierarchical systems.

Disruptions can decrease the number of covered demands due to the failures of facilities. Therefore, considering reliability in HMCLP can maximize the number of the demands covered by facilities when a disruption occurs. To that end, the risk of disruptions of facilities is included in the mathematical optimization model of the HMCLP problem. As a result, the optimization model determines the locations of facilities of a hierarchical system from a set of candidate locations to maximize the number of the demands covered by them when a disruption occurs.

In conclusion, making the best location decisions for the facilities of hierarchical systems are crucial because they significantly affect the cost, service level, and reliability of the hierarchical systems. However, the complexity of these systems such as the unique features of the material flow between their facilities makes it impossible to instantly determine the best locations of the facilities that lead to minimum costs and maximum service level. Therefore, decision-makers should take the advantage of mathematical optimization models including but not limited to HMCLP and HFLP to instantly obtain the optimal locations of their facilities that guarantee the profitability and reliability of their hierarchical systems. 

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