The shape or topography of the set of potential plants yields models in the plane, network location models, and discrete location or mixed-integer programming models, respectively.
For each of the subclasses distances are calculated using some metric.
2.11.3.1 Continuous location models
They are characterized by a continuous solution space which states, that each point in the space represents a feasible location. Further, the measurement of distances is carried out by a suitable metric (mainly by lp-standards). Continuous location models (models in the plane) are characterized through two essential attributes: (a) the solution space is continuous, that is, it is feasible to locate facilities on every point in the plane. (b) Distance is measured with a suitable metric. Typically, the Manhattan or right-angle distance metric, the Euclidean or straight-line distance metric, or the lp-distance metric is employed. Continuous location models require to calculate coordinates (x, y) Є Rp × Rp for P facilities. The objective is to minimize the sum of distances between the facilities and m given demand points. The subject of the Weber problem is to determine the coordinates (x, y) Є Rp × Rp of a single facility such that the sum of the (weighted) distances wkdk(x, y) to given demand points k Є K located in ( ak ; bk) is minimized. The corresponding optimization is given below:
𝑆𝑊𝑃 = min 𝑥, 𝑦 𝑝𝑗 =1(𝑤𝑘𝑑𝑘 𝑥𝑗, 𝑦𝑗 (2.6)
60 Where 𝑑𝑘 𝑥𝑗, 𝑦𝑗 = [𝑥– 𝑎𝑘 ]2+ [𝑥– 𝑏𝑘 ]2
This Simple Weber Problem has a century-long tradition for the case of 𝐾 =3 demand points.
An extended version of the above SWP that requires to locate p, 1 < p < 𝐾 facilities and to allocate demand to the chosen facilities denoted as Multi-source Weber Problem (MWP), is NP-hard. It can be modelled as the non-linear mixed-integer program
𝑀𝑊𝑃 = 𝑚𝑖𝑛 𝑘ℇ𝐾 𝑝𝑗 =1(𝑤𝑘𝑑𝑘(𝑥𝑗, 𝑦𝑗))𝑧𝑘𝑗 (2.7a)
S.t 𝑝𝑗 =1𝑧𝑘𝑗 = 1 ∀𝑘 ℇ 𝐾, (2.7b)
𝑧𝑘𝑗ℇ B, ∀𝑘 ℇ 𝐾, 𝑗 = 1, … , 𝑝, (2.7c)
𝑥, 𝑦 ℇℝ𝑝 (2.7d)
Where B = {0,1} and 𝑧𝑘𝑗 equals 1 if demand point k is assigned to facility j. Exact solution procedures reformulate the model as a set partitioning problem, the LP-relaxation of which can be solved by column generation.
2.11.3.2 Discrete or network location models
In network location models distances are computed as shortest paths in a graph. Nodes represent demand points and potential facility sites correspond to a subset of the nodes and to points on arcs.
The network location model corresponding to the continuous multi-source Weber model is called P-median problem. In the P-median problem p facilities have to be located on a graph such that the sum of distances between the nodes of the graph and the facility located nearest is minimized.
i. P-Median Model: Another important way to measure the effectiveness of facility location is by evaluating the average (total) distance between the demand points
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and the facilities. When the average (total) distance decreases, the accessibility and effectiveness of the facilities increases. The p-median location model involves the location of a fixed number p of facilities. The objective is to locate the p facilities in such a manner that the total weighted distance of serving all demand is minimised. Weighted distance for a demand point represents the amount of demand multiplied by the distance to the closest facility. For example, if demand is measured in terms of the number of trips that need to be made by users of the facility, then weighted distance represents the total mileage involved in going to the facility. For a fixed level of demand, minimising total weighted distance is equivalent to minimising average distance. This model form can address many different types of application, from locating schools and health clinics to locating road maintenance garages and emergency response vehicles. Because this model captures the essence of locating a set of facilities to serve an area by maximising accessibility, it has become a popular model for application (Church, 2002). In the p-median problem p facilities have to be located on a graph such that the sum of distances between the nodes of the graph and the facility located nearest is minimized. Let K denote the set of nodes, J ⊆ K the set of potential facilities, wkdkj the weighted distance between nodes k and j, yj a binary decision variable being equal to 1 if node j is chosen as a facility (0, otherwise), and xkj a binary decision variable reflecting the assignment of demand node k ℇ K to the potential facility site j. Then
𝑃𝑀𝑃 = 𝑚𝑖𝑛 𝑝𝑗 =𝐽(𝑤𝑘𝑑𝑘)𝑧𝑘𝑗 (2.8a)
S.t
𝑧𝑘𝑗 = 1 ∀𝑘 ℇ 𝐾
𝑝𝑗 =1 , (2.8b)
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𝑧𝑘𝑗 −𝑦𝑗 ≤ ∀𝑘 ℇ 𝐾, 𝑗 ℇ 𝐽 (2.8c)
𝑦𝑗 = 𝑝
𝑝
𝑗 =𝐽 (2.8d)
𝑧𝑘𝑗, 𝑦𝑗ℇ B, ∀𝑘 ℇ 𝐾, ∀𝑗 ℇ 𝐽, (2.8e)
formally describes the p-median problem. Constraints (2.8b) guarantee that demand is satisfied, inequalities (2.8c) couple the location and the assignment decision, and constraint (2.8d) fixes the number of selected facilities to p.
ii. P-Center Model In contrast to the P-median models which concentrate on optimizing the overall (or average) performance of the system, the P-center model attempts to minimize the worst performance of the system and thus addresses situations in which service inequity is more important than average system performance. In location literature, the P-center model is also referred to as the minimax model since it minimizes the maximum distance between any demand point and its nearest facility. The P-center model considers a demand point is served by its nearest facility and therefore full coverage to all demand points is always achieved. The problem asks for the center of a circle that has the smallest radius to cover all desired destinations. In the last several decades, the P-center model and its extensions have been investigated and applied in the context of locating facilities such as EMS centers, hospitals, fire station, and other public facilities. However, unlike the full coverage in the set covering models, which may lead to excessive number of facilities, the full coverage in the P-center model requires only a limited number (P) of facilities. The aim of p-center problem is to locate p facilities such that the maximum distance is minimized. Unfortunately, for the p-center problem we cannot restrict the set of potential facility sites to the
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set of nodes because the maximum of concave distance functions is no concave function any more. Fortunately, it suffices to consider a finite set of points on the arcs. These points can be determined as intersection points q for which the weighted distance widiq between q and node i ℇ K equals the weighted distance wkdiq between q and another node k ℇ K. Let J denote the set of intersection points. Then the discrete optimization model
𝑃𝐶𝑃 = min 𝑟 (2.9a)
S.t
𝑟 − 𝑝𝑗 =𝐽(𝑤𝑘𝑑𝑘)𝑧𝑘𝑗 ≥ 0 ∀𝑘 ℇ 𝐾 (2.9b) 𝑧𝑘𝑗 = 1 ∀𝑘 ℇ 𝐾
𝑝𝑗 =1 , (2.9c)
𝑧𝑘𝑗 −𝑦𝑗 ≤ 0 ∀𝑘 ℇ 𝐾, 𝑗 ℇ 𝐽 , (2.9d)
𝑦𝑗 = 𝑝
𝑝
𝑗 =𝐽 (2.9e)
𝑧𝑘𝑗, 𝑦𝑗ℇ B, ∀𝑘 ℇ 𝐾, ∀𝑗 ℇ 𝐽, 2.9f)
Formally describes the p-center problem which can be transformed into a sequence of covering problems.
iii. The covering model: The objective of covering models is to provide ―coverage‖
to demand points. A demand point is considered as covered only if a facility is available to service the demand point within a distance limit. Then the covering model
𝑆𝐶𝑃 = min 𝑝𝑗 =𝐽𝑦𝑗 (2.10a)
s.t 𝑝𝑗ℇ𝐽𝑎𝑘𝑗𝑦𝑗 ≥ 1 ∀𝑘 ℇ 𝐾, (2.10b)
𝑦𝑗ℇ B, ∀𝑘 ℇ 𝐾 (2.10c)
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With 𝑎𝑘𝑗 = 1 for 𝑤𝑘𝑑𝑘𝑗 < r and 𝑎𝑘𝑗 = 0 for 𝑤𝑘𝑑𝑘𝑗 ≥ r computes a set of at most p-centers with a radius smaller than r or shows that no such set exists.
The literature on covering problems is divided into two major parts: the location set covering problem (LSCP) and the maximal covering location problem (MCLP). LSCP is an earlier version facility location problem and it aims at locating the least number of facilities that are required to cover all demand points. Since all the demand points need to be covered in LSCP, regardless of their population, remoteness, and demand quantity, the resources required for facilities could be excessive. Recognizing this problem, the MCLP model that does not require full coverage to all demand points was developed. Instead, the model seeks the maximal coverage with a given number of facilities. The MCLP, and different variants of it, have been extensively used to solve various emergency service location problems.
2.11.3.3 Mixed-integer programming models
Starting with a given set of potential facility sites many location problems can be modelled as mixed integer programming models. Apparently, network location models differ only gradually from mixed integer programming models because the former ones can be stated as discrete optimization models. Yet network location models explicitly take the structure of the set of potential facilities and the distance metric into account while mixed-integer programming models just use input parameters without asking where they come from. A rough classification of discrete facility location models can be given as follows: (a) single- vs. Multistage models, (b) uncapacitated vs. capacitated models, (c) multiple- vs. single-sourcing, (d) single- vs. multi-product models, (e) static vs. dynamic models, and, last but not least, (f) models without and with routing options included.