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1 Introduction

Location analysis has deep roots in regional science and represents a classic method in the discipline. Location analysis, in general, concerns the organization or arrangement of goods, resources, services or activities in space. Such analysis can be used to answer questions of why activities/phenomena occur at certain places and how to best locate goods/services to achieve certain purposes. Early location analysis work can be traced back to Johann Heinrich von Thünen, Walter Christaller, August Lösch, Alfred Weber and Harold Hotelling, among others. von Thünen (1826) proposed a location theory to explain the principles that account for different agricultural land uses by linking locational rent with agricultural production and transportation costs. Focusing on factory location, Weber (1909) was interested in finding the best site on the continuous plane that minimizes transportation costs, equivalent to profit maximization under production, labor supply and demand assumptions. Hotelling (1929) examined the location strategies of two firms and their price setting considering demand distribution, transportation costs and competition. Using a linear city/market, Hotelling showed that with fixed pricing and production costs both firms would ideally locate at the halfway point, with each capturing/serving half the total market. Going beyond a single area or region, Christaller (1933) conceived of human settlements as a system and developed central place theory to explain the spatial organization of villages, towns and cities. Building upon the interrelations of economic activities between places, this suggests that settlement patterns reflect a hexagon-shaped hierarchy, with centers and their associated hinterlands. Lösch (1941) expanded on central place theory to allow for sophisticated spatial arrangements that considered economies of scale and specialization. These pioneering studies have laid the fundamental foundation for the field of location analysis by connecting locational choices to various economic activities.

Since these pioneering studies, locational analysis has flourished in regional science and beyond. One stream of application and development has sought to verify, extend and refine associated location theory. For example, Alonso (1964) extended von Thünen’s agricultural land use theory to the urban setting and developed bid-rent models of land use distribution as a function of the distance from the central business district. Modern agriculture location theory has also evolved to account for more realistic conditions (see Lucas and Chhajed 2004). Similarly, central place theory has been extended to examine city size (Beckmann 1958), hierarchy of villages (von Böventer 1963), and shopping centers (Eaton and Lipsey 1982), and account for customer shopping behavior (Ghosh and McLafferty 1987) and agglomeration effects (Fischer 2011; Mulligan et al. 2012). Models have also been used to interpret, test and/or verify various aspects of central place theory as well as gain insights into underlying processes (Beaumont 1987; Curtin and Church 2007).

Another stream of activity has involved specification and solution of supporting mathematical models. Initial work was devoted to solving and extending the Weber problem (Wesolowsky 1993). Although the Weber problem appears rather simple, solving the problem exactly has been challenging given the continuous nature of the problem, where a firm (or firms) can be sited anywhere in geographic space. Early studies focused on the geometric characteristics of the problem and used a mechanical analogue device known as the Varignon frame. Later, iterative algorithms, including the well-known Weiszfeld algorithm (Weiszfeld 1937), were developed for model solution. Various extensions have also been made to the Weber problem by introducing alternative distance metrics, including multiple facilities, and allowing stochastic demand, among others (Drezner et al. 2002). The Weber problem has also served as the inspiration for a range of contemporary modeling efforts, some of which will be discussed in Sect. 12.2.

Building upon the foundation laid by the above work, location analysis and modeling have evolved in terms of theoretical development and empirical application. While originally focused on descriptive characteristics associated with why and how activities/communities are organized in space, the field has advanced to be more prescriptive in nature through the assistance of making locational decisions for various purposes. A significant number of location models have been developed to support real-world applications at the urban and regional scale for both the public and private sectors. The following section briefly reviews the field with a focus on a selected number of models and applications. This is followed by a discussion of the challenges in location analysis. Looking forward, we highlight future research directions associated with emerging applications, big spatial data and ways to address computational challenges. Finally, concluding remarks are given.

2 Analytical Approaches

As suggested above, much of the underlying economic and spatial theory associated with location analysis has historically been descriptive in nature, seeking to develop a better understanding of existing patterns and observed conditions. Examples include bid-rent curves, regression models based on proximity to a city center and distance-decay oriented interaction models. Evolving computing capabilities have enabled description to be carried out using mathematical models, and also support prescriptive decision making about where to best locate goods and services in combination with responsible resource allocation.

2.1 Prescriptive Capabilities

While location theory has provided a comprehensive description/explanation of various activities, prescriptive capabilities have come to characterize more contemporary location analysis (Murray 2010). In these studies, determining the best locations for certain services or activities has proven beneficial for achieving overall efficiency. Modern location analysis has therefore been operationalized through development of mathematical models. Over the past few decades, literature on location models and associated applications are prolific. Summaries of work in this area can be found in articles including Chhajed et al. (1993), Brandeau and Chiu (1989), Owen and Daskin (1998); ReVelle and Eiselt (2005), Smith et al. (2009), Murray (2010), as well as books including Love et al. (1988); Drezner (1995), Daskin (1995), Drezner and Hamacher (2002), Church and Murray (2009), Farahani and Hekmatfar (2009), Eiselt and Marianov (2011), Laporte et al. (2015), and Eiselt and Marianov (2015). These reviews have focused on various aspects of the field and major achievements to date. This chapter will be forward-looking with elaboration on important future research areas in the field.

As noted previously, a location model has generally been conceived to be a bid-rent curve, regression model that includes distance and/or an interaction model represented as an equation. The prescriptive approach extends descriptive capabilities to allow for resource allocation and spatial decision making. In this sense, a contemporary location model therefore consists of one or multiple objective function(s) as well as a set of constraints. Objective functions are used to articulate the goal(s) that a particular problem aims to achieve. An objective function may reflect overall investment/operation costs or perhaps service benefits. These would then be optimized accordingly, with decisions made to produce the best objective function outcome. Constraints reflect the problem specific conditions that limit activities in some manner, necessarily establishing a mathematical linkage between decision variables.

Prescriptive oriented location models have been classified into different categories based upon a range of criteria. Categories of particular note include: continuous space, discrete space, network, stochastic, deterministic, single objective, multiple objective, number of facilities, service capacity, etc. Depending on how space is treated in a location model, it may be considered either continuous, discrete or network. The classic Weber problem is an example of a continuous problem as the factory to be located can be anywhere on the continuous plane. Alternatively, a discrete problem is one where there are only a finite number of candidate sites, identified a priori, and a finite number of objects to be served. Finally, a network problem could be discrete but may also be continuous, depending on whether siting could occur along arcs or if demand is distributed along arcs. Elaboration on these points and others follows in the subsequent sections.

2.2 Classic Models

There are a number of noteworthy location models that will serve to illustrate prescriptive capabilities. The location-allocation problem and its variants have arguably been among the most influential and widely relied upon prescriptive models. The location-allocation problem was formally introduced in the seminal work by Cooper (1963), extending the Weber problem to allow for multiple facilities to be sited on the continuous plane. Hakimi (1964) considered a network version of the problem where demand and service provision occur on a network with the objective to minimize the overall travel costs along the network. Demand is assumed to be at nodes, and facilities can be sited anywhere on the network. Although no specific solution method is provided, Hakimi proved that nodes on a network contain at least one optimal solution in the case of a network. Given this, the search for the best configuration of facility can be narrowed to the finite set consisting of only network nodes. This gives rise to the p-median problem: finding p sites among n predetermined points to serve discrete demand such that total travel cost is a minimum. ReVelle and Swan (1970) formulated the p-median problem. Location-allocation problems, especially the p-median problem, have been widely applied and extended to incorporate various problem specific conditions, including facility capacity, hierarchical structure, stochastic demand and competition. A summary of model development and application can be found in Mirchandani (1990), Marianov and Serra (2011), ReVelle et al. (2008), and Daskin and Maass (2015), among others.

Another category of prescriptive location models concerns regional coverage. Critical then is the notion of “coverage”, which is often defined based on whether demand can be served within a maximum acceptable travel distance/time. This coverage standard corresponds to the “range” concept introduced in central place theory. In contrast to location-allocation models, covering problems are driven by different performance criteria. Toregas et al. (1971) introduced the location set covering problem (LSCP) seeking to find the minimum number of facilities (and where to locate them) needed to provide complete coverage to a region. Recognizing that in many situations resources are not sufficient to ensure a full coverage of a region, Church and ReVelle (1974) proposed the maximal covering location problem (MCLP) to locate a limited number of facilities in order to achieve the greatest coverage of a region. These two classic covering problems have been extended to incorporate various coverage standards, redundant coverage, cooperative service provision and service capacity. A review of the covering problems and associated applications can be found in Schilling et al. (1993), Murray et al. (2010) and Farahani et al. (2012).

A third category of prescriptive location models is center problems. The concern in this case involves locating one or more facilities/services so that the maximum distance from a demand to its closest sited facility is as short as possible. Differing from other location modeling approaches that focus on cost or system efficiency, center problems seek equality by ensuring that the worst access provided to any individual/place is as good as possible. The p-center problem was introduced by Hakimi (1964) and often assumes that facilities can be located anywhere in a region (continuous space) and that demand is concentrated at discrete points. Various algorithms have been developed to solve center problems, including a Voronoi diagram heuristic (Suzuki and Okabe 1995). The problem becomes a vertex p-center problem if the candidate facility sites are also restricted to predefined sites (Daskin 1995). The p-center problem has also been extended to consider service capacity, continuous demand and backup service provision. Refer to Drezner (2011), Tansel (2011) and Calik et al. (2015) for further discussion of center problems.

A fourth category of prescriptive location models is competitive demand approaches. Following the seminal work of Hotelling (1929), recognition of the need to address competition for service has arisen, with approaches developed to explicitly account for competition among sited facilities. In these problems, the location of additional firms will not only affect new markets but those of the competitors. Early theoretical studies have focused on modifying some of the economic assumptions made in Hotelling (1929) and examining associated equilibrium patterns. Subsequent competitive location models have shifted to account for market share consideration. Various conditions have been explored, including the type of service to be provided (e.g., convenience stores, shopping malls, gas stations, hotels), space (e.g., network, discrete location or continuous region), Nash and Stackelberg equilibria, consumers’ choices and market share delineation approaches. Refer to the work of Friesz et al. (1988), Serra and ReVelle (1995), Plastria (2001) and Drezner (2014) for more details.

It is conceivable that listing of categories could continue, likely numbering in the hundreds to account for the significant location model nuances. Rather than continue further, we leave it at the above major categories, but note that issues of dispersion (Goldman and Dearing 1975; Church and Garfinkel 1978; Moon and Chaudry 1984; Kuby 1987; Murray and Church 1995; Verter and Erkut 1995), hubs (O’Kelly 1986; Alumur and Kara 2008), interdiction (Scaparra and Church 2015), etc. are no less important or significant. However, due to space limitations, further review and discussion is not possible.

3 Challenges

There are numerous challenges confronting the use and application of location models. One issue noted here has to do with decision making processes unique to particular application contexts. A second issue concerns computing capabilities associated with solving structure models.

3.1 Application Contexts

Location analysis and modeling have been applied to solve a wide range of urban and regional problems. These applications include public facility siting (such as libraries, schools, post offices, and police stations), emergency facility placement (fire stations, ambulance), districting (political districting, service districting, police districting), healthcare facility and service planning, network design and routing (telecommunication, transportation), business locations (such as bank branches, retail facilities), military operations, agricultural management (production, storing, processing and distribution of agricultural products) as well as environmental problems (such as nature reserve site selection, wildlife management). A number of classic and modern applications have also been summarized in Lucas and Chhajed (2004) and Eiselt and Marianov (2011).

In a location model, mathematical abstraction is very critical as improper specification of the objective function or constraining relationships/conditions can result in locational decision making that is far from the best. The diverse applications of location analysis present challenges to problem formulation and model construction. Depending on the particular problem of interest, an existing location model might not be applicable, and constructing a new location model is sometimes necessary. Such a new model will involve identifying and formulating one or multiple goals and specifying the associated constraining conditions. Even for problems where an existing location modeling framework applies, oftentimes the existing model may need to be modified to account for application specific goals or constraining conditions, such as different cost functions, specific capacity requirement and special relationships among facilities or between demand and facilities. In other cases, when problems cannot be mathematically articulated or formulated, heuristic approaches will be needed to solve the problems approximately. These heuristic based approaches will be discussed below. Due to problem variety and complexity, constructing location models requires some level of creativity to accurately abstract real-world problems as well as the ability to link components/relationships mathematically.

3.2 Problem Solution

Beyond the abstraction process is the need for identifying, comparing and understanding alternative solutions. Decision making often involves a host of constitutes, particular for public section contexts. Different groups or individuals may have differing concerns and objectives. Further, they may have their own ideas about good alternatives to consider. Generating solutions remains a challenge. Understanding strengths and weaknesses and being able to communicate them is essential.

As mentioned earlier, predictive approaches for location analysis and modeling have mainly focused on identifying the best locational decision(s) for serving certain purpose(s). Solving these problems necessitates a search for the best set of locations, either in a continuous region or limited to predefined discrete sites. While a continuous problem usually means it is difficult to solve as there exists an infinite number of candidate sites to select from, searches confined to a finite number of sites may be nontrivial as well. In general, two strategies have been used to solve location models: exact methods and heuristic methods.

Exact methods are those producing a provably optimal solution. That is, solutions identified by these methods can be shown to be superior to all others, found in a process or not found. Enumerating all the possible solutions is sometimes relied upon, enabling identification and evaluation of associated objective function values. The method guarantees the best solution to be identified because all are explicitly considered. However, when the problem size grows in terms of the number of different configurations, solutions to consider, the computational requirements can be prohibitive, making enumeration impractical. Enumeration in the case of continuous space problems is generally infeasible given that an infinite number of siting configurations would need to be considered. For this reason, other exact methods have been developed, including linear programming, integer programming, branch-and-bound, dynamic programming, Lagrangian relaxation based methods as well as specialized algorithms that exploit geometric characteristics of certain problems (Elzinga and Hearn 1972; Matisziw and Murray 2009).

Irrespective of whether we have a continuous or discrete location problem, many are known to be NP-hard (Kariv and Hakimi 1979; Megiddo and Supowit 1984). This means that solving these problems exactly can be difficult or impossible, especially for large sized ones. For these problems as well as problems that are difficult to mathematically formulate, heuristic approaches are widely used for problem solution. Heuristic methods are often rule of thumb, ad-hoc strategies. Compared with exact methods, heuristic approaches can often solve a problem faster but problem solution quality is not known or guaranteed. Various heuristics have been used to solve location models, including the “alternate” method (Cooper 1963; Maranzana 1964), greedy based search (Church and ReVelle 1974), and vertex substitution or interchange (Teitz and Bart 1968). While many early heuristics focus on iterative improvement based on a local search neighborhood, high level modern metaheuristics represent a family of methods that often allow other solution spaces to be considered simultaneously, resulting in solutions less likely to be trapped in local optima (Brimberg et al. 2000). Modern metaheuristics have been widely applied to solve various location problems, including tabu search (Murray and Church 1995; Rolland et al. 1996), simulated annealing (Murray and Church 1996; Chiyoshi and Galvao 2000), and genetic algorithms (Bozkaya et al. 2002).

4 Looking Forward

The field of location analysis has evolved tremendously with continued visibility within and outside of regional science. Looking forward, we believe location analysis will continue to be essential for helping address future regional challenges. Future applications may require closer interaction/collaboration of researchers in location analysis with experts in other fields in order to enhance problem understanding and develop efficient problem solution strategies. Although GIS (geographic information system) continues to be recognized as important in location analysis, a wider adoption and integration of GIS into location analysis is expected. The advent of big data has the potential to revolutionize location analysis theoretically and practically. Additional insights gained from big data may help refine existing modeling frameworks and motivate novel solution approaches. With increased complexity and detail in location models due to big data, high performance computing will be an integral component of future analytical frameworks.

4.1 New Application Contexts

In years to come, location analysis will be used to help solve emerging challenges and issues at regional and national scales. Closer collaboration with scholars in other disciplines is expected for solving these challenges. One example concerns sustainable development. For example, moving towards a more sustainable environment, US EPA (2015) requires significant annual CO2 reductions: “22%–23% below 2005 levels in 2020; 28–29% below 2005 levels in 2025, and 32% below 2005 levels in 2030”. The CO2 reduction goal necessitates an increased use of renewable energy resources to substitute the conventional coal resources for future electricity generation. Solar has been identified as one of the important emerging renewable resources for future energy supply. Location analytical studies of future solar energy power plants and the distribution network presents an important application that will contribute to CO2 reduction goals. However, such an analysis requires collaboration with climate scientists, environmental experts, economists, and geographers to take into account future weather uncertainty, environmental impacts, economic development and population growth. Other likely applications relate to the challenges brought about by climate change. Extreme weather events such as droughts and floods are expected to occur more frequently in some local regions, leading to countless economic losses. Howitt et al. (2015) estimate that the recent drought in California has caused an economic loss of 2.7 billion dollars in 2015. Incorporating location analysis into efficient water allocation and flood mitigation strategies presents a sound way to help mitigate losses due to climate change. Of course, there are many other areas as well.

For some new regional applications, existing modeling frameworks can be used but might need substantial revisions to account for problem complexity. For example, interdiction approaches detailed in Scaparra and Church (2015) provide ways to identify critical components or locations in a region in order to prioritize fortification efforts when preparing for a future disaster. However, such location models have mainly focused on a certain type of service or facility. As for disaster management, many aspects need to be addressed simultaneously (such as lives, properties, transportation infrastructures, communication networks, etc.) and the consequent location analysis can be much more complicated. Scaparra and Church (2015) also noted that even though existing models are already complex, they have not been able to adequately address the interconnection of various components in a system. This also calls for interdisciplinary collaboration for a better understanding and modeling of interdependence and complexity of relevant elements in a region. Driven by the new applications, revisions of existing models or sometimes new modeling frameworks might be needed to address problem specific requirements and complexity. Overall, location analysis as an evolving field will continue to make contributions to regional science and help solve new regional challenges.

4.2 GIS

Location analysis often involves various types of data, ranging from demographics (e.g., population distribution), the built environment (e.g., transportation networks, land uses) to the natural environment (e.g., terrain information). Many of these data tend to be spatially explicit, but do give rise to various sorts of implicit information. For example, population is associated with specific cities in a region and roads connect certain places in an area. Given that GIS is a special information system designed to store, manage, process, analyze and display spatial and non-spatial data, there is a natural linkage between GIS and location analysis. In recent years, GIS has been increasingly used to support location analysis and has been widely recognized as important due to its powerful capabilities in data acquisition (as many data are readily available in the GIS form), management and processing. For example, GIS has been directly employed to conduct suitability analysis for various location decisions, including hospitals, roads and utility lines. Murray (2010) also highlighted the critical role of GIS in theoretical development of location analysis that goes beyond simple data support or manipulation. Reviews by Church (1999), Murray (2010) and Bruno and Giannikos (2015) all note the various contributions GIS has made to location analysis.

A wider adoption of GIS by location analysts and modelers will continue to help the field of location analysis advance. The integration of GIS into location analysis can further refine current models and broaden the applications. New location models or variants of existing models better reflecting a problem of interest might emerge due to finer details or alternative representation schemes available in GIS (Murray 2010). Also, GIS can be used to gain insights into the uncertainty associated with spatial data, scale, and modeling practices (Tong and Church 2012). Meanwhile, constructing location models requires certain level of mathematical skills, which is often beyond the knowledge of a general planner or analyst. The incorporation of location models in GIS software helps location analysis and modeling to reach a wider audience. In fact, some GIS commercial software has started to incorporate some of the classic location models. For example, the Esri ArcGIS software provides a location-allocation module that includes the p-median problem, location set covering problem and maximal covering location problem with optional considerations of service capacity and competition.

4.3 Big Data

Compared with decades ago when availability of locational data was an issue, big data has revolutionized the amount and detail of information available about human activities and the environment. Such data are collected through a range of technologies, such as cell phones, wearable devices, GPS, social media, cameras and various sensors, and provide an enormous amount of information about people’s movement and activities. For example, in 2014 New York City shared with the public the information about 173 million taxi trips. The data provided information about where and when individuals were picked up and dropped off. The unprecedented spatial-temporal coverage, as well as the richness and granularity of big data, allow researchers to gain new knowledge about human activities. It is estimated that big data will have a transformative impact on almost all fields (Shaw 2014). We also expect that the advent of big data will bring about new opportunities to further advance the field of location analysis.

We anticipate that the integration of big data into location analysis will enhance the resolution and accuracy of data input. Conventional data input in many location models relies upon field work or a number of data collection agencies, such as the Census Bureau. Often these data come in an aggregate form, e.g., total population at the census tract level, so how individuals are distributed within the aggregation unit is unknown. Depending on the specific aggregation scheme and scale used in the aggregation, solutions given by a location model may vary substantially (Francis et al. 2009). When continuous regional demand is assumed in location models, uniform or some theoretical distributions are often used, which may differ from where people are in reality. Such a discrepancy will also lead to solutions that may be far from the best. With increased data resolution and accuracy, big data has the potential to help location problems generate better results.

Also, evidence provided by big data will help us revise some assumptions made in existing location models to better reflect the reality. For example, in many location problems demand is often assigned to the closest facility when capacity allows. Insights gained from big data about individuals’ preferences can be incorporated into current location models to draw allocations more accurately. Also, in many location models demand is assumed to be fixed (often originate at home). Building on big data, location modelers can also take into account individual level movement dynamics into the locational decisions of the intended service. An incorporation of such travel-activity patterns in location analysis will significantly enhance modeling accuracy. Although in the past few decades a number of empirical studies have been conducted to examine individuals’ patron patterns as well as how trips are chained, most were based on conventional data collection such as travel diaries with a very limited number of individuals for a minimal number of days. Big data allows one to do such an examination with a much larger sample size for a longer time period.

While modern location analysis has mainly focused on prescription of the best locations for certain activities/services, the variety of big data offers researchers the opportunity to revisit the location theories. For example, geotagged big data mining can be used to help reveal meaningful distributions of activities or patterns in space. These findings may provide empirical evidence to verify or modify location theories. This also points to a new direction for future research.

Overall, big data will bring new opportunities to advance the field of location analysis. The large scale, fine resolution information provided by big data will help refine exiting models and inspire new approaches/models to better reflect real-world problems. Meanwhile, the large volume of big data will inevitably increase problem complexity tremendously, and solving the associated problems optimally may become extremely difficult. This will necessitate development of new efficient solution approaches and incorporation of high performance computing for problem solution. Additional discussion on these issues will be provided in the following section.

4.4 Efficient Solution Approaches and High Performance Computing

As mentioned earlier, solving problems exactly presents an important challenge in location analysis. The increased level of model complexity and big data will add more difficulty to problem solution. Solving these problems may involve tremendous amount of computation, especially for large sized problems. One the one hand, novel solution approaches will be needed to solve these problems efficiently. For example, considering that solving small sized problems are often much easier than large sized ones, novel strategies can be developed to decompose certain large problems into smaller sub-problems without sacrificing problem optimality. On the other hand, more efficient and effective heuristics will be continuously sought to solve large sized location problems approximately. Recent development of hybrid metaheuristics combining strengths of metaheuristics and classical solution techniques, such as branch and bound, has shown promise for location problem solution (Blum et al. 2008).

In addition to developing efficient approaches to solve problems, future efforts will be needed to focus on taking the advantage of high performance computing (HPC), which consists of a cluster of computers or processors known as nodes. In HPC, individual nodes can work together to solve complex problems more efficiently than can an individual computer. When solving a problem, workload of solving the entire problem needs to be divided and distributed to a number of nodes simultaneously in a parallel fashion. Using HPC to solve a problem requires an understanding of the computing hardware as the associated parallel architecture may differ, leading to different computational performances. More importantly, in HPC a scalable and efficient procedure is essential for performing the parallel computing. This often involves a customized process of division and synchronization of sub-tasks as well as information interchange (communication) among multiple processors. Studies have started to incorporate HPC to solve large sized, challenging location problems. For example, Redondo (2008) proposed evolutionary algorithm based heuristic approaches in a HPC setting to solve a competitive facility location problem on the continuous plane. However, to achieve the best HPC performance in terms of both solution quality and efficiency, the design of the parallel computing process can be challenging as it often varies with the problem to be solved and the specific solution approaches used. This points to an important area where more research is needed in the future.

5 Conclusions

Location analysis represents one of the core fields of regional science. Building upon the classic location theories, location analysis has evolved considerably in the few past decades. While early studies focused on an understanding of the distribution pattern and associated mechanism of human settlements and activities, contemporary location analysis has evolved to assist the locational decision making in various regional problems. Looking forward, the field of location analysis will continue to be relevant and influential in regional science. Location analysis will be used to help solve emerging issues concerning sustainability and environmental challenges. The big data age presents great opportunities for researchers to revisit location theories as well as further advance location modeling frameworks and the applications. We also anticipate a continued integration of GIS into location analysis for data support, model refinement and efficient problem solution. With increased problem complexity, future research will consist of development of computationally efficient solution approaches and an incorporation of high performance computing.