The Generalized Median Tour Problem: Modeling, Solving and an Application

We introduce, formulate, and solve the Generalized Median Tour Problem, which is motivated in the health supplies distribution for urban and rural areas. A region comprises districts that must be served by a specialized vehicle visiting its health facilities. We propose a distribution strategy to serve these health facilities efficiently. A single tour is determined that visits a set of health facilities (nodes) composed of disjoint clusters. The tour must visit at least one facility within each cluster, and the unvisited facilities are assigned to the closest facility on the tour. We minimize the sum of the total tour distance and the access distance traveled by the unvisited facilities. Efficient formulations are proposed and several solution strategies are developed to avoid subtours based on branch & cut. We solve a set of test instances and a real-world instance to show the efficiency of our solution approaches.


I. INTRODUCTION
Humanity is frequently exposed to different types of natural disasters or sanitary crises worldwide, such as earthquakes, tsunamis (e.g., Chile 1960 and 2010, Japan 2011 and Indonesia 2004), tornadoes (e.g., Katrina 2005 and Irma 2017), and pandemics (e.g., Covid-19, H1N1, SARS, MERS, etc.). Therefore, an efficient supply chain management is crucial to assure the distribution of essential goods (food and water for people and animals) and health supplies (medical samples, tests, vaccines, blood, and face masks). In the case of natural disasters, the main challenge, after it occurs, is to redesign, repair, or completely restore the distribution networks. On the other hand, in a sanitary crisis, the need for adapting and supplying the existing health facilities is required to cover and protect affected inhabitants and regions in a short time. Thus, the aim is to avoid large transportation distances and times, while yielding an effective, efficient, and fast distribution of the required health supplies.
The associate editor coordinating the review of this manuscript and approving it for publication was Donghyun Kim .
In the case of critical health supplies (e.g., blood or vaccines), distribution traceability and items conservation are relevant to ensure their quality and effectiveness. Hence, the number of stops or transfer stations should be reduced (crossdocking or intermediate facilities). Simultaneously, special and expensive transportation modes are usually required (e.g., airplanes, ambulances, and special trucks), which provide proper conditions inside the vehicles, including cooling, security, and stability [1]- [3]. This issue is particularly relevant and challenging for geographically isolated rural areas in undeveloped and developing countries, given the limited, or even rudimentary nature of their local transportation systems.
Accordingly, this research is motivated by the challenge of designing a distribution network for essential health supplies for a set of mixed urban/rural districts or regions under a sanitary crisis of a pandemic or epidemic. Under these scenarios, political authorities are forced to establish lock-downs and cordons sanitaires in specific regions, usually coinciding with districts or municipalities. Furthermore, the distribution network must allow a fast distribution of the health supplies to a set of health facilities VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ that can receive and deliver these supplies to the involved inhabitants. An inherent constraint that arises in these scenarios, associated with the political and administrative organization matters, states that each non-selected facility must be assigned only to facilities at the same district to which it belongs. This constraint ensures the duty for the distribution process to specific related sanitary and political authorities at the districts. Besides, it facilitates the distribution process, avoiding travels between different districts, and thus helping to observe lockdowns and cordons sanitaires.
Summarizing, at least one of the existing health facilities at each district has to be selected for a first-stage distribution, and every non-selected facility must be assigned to a single selected facility in the same district. The first-stage distribution consists of a visit sequence considering only the selected facilities at all involved districts from a known specific depot (e.g., an airport or a regional hospital). At the same time, the second stage-distribution process is defined as direct trips between the selected health facilities and the assigned (nonselected) facilities. Hence, the distribution system must be optimally designed to minimize the total time of the firststage distribution process plus the total time associated with the second-stage distribution process.
A relevant additional practical concern is observed, which requires that a truck that visits the selected facilities at all districts may enter and exit each district only once, ensuring proper and safe handling of the supplies inside the truck and proper traceability of the distribution process. Fig. 1 presents a representation of a southern area of Chile (generated in Google My Maps ), containing 12 districts (framed areas with red lines), where the circles represent the health facilities, and the grey circles represent the selected facilities for the first distribution stage. Thus, the health supplies arrive at the main airport (yellow circle) and subsequently are distributed by a refrigerated truck (black lines) to the selected points on the tour. Next, all remaining facilities are assigned to a selected facility in its district (brown lines). Note that it is possible to select more than one health facility at each district to be part of the truck's route. Finally, the objective is to minimize the total travel time of the vehicle and the travel time between the selected facilities and the other non-selected health facilities.
In order to address the previously discussed distribution network design problem, this paper aims at introducing, modeling, and solving the Generalized Median Tour Problem (GMTP). In this novel problem, the nodes (health facilities) are grouped into clusters (districts), and a single tour or route must visit all the clusters, such that the nodes within each cluster that are not in the tour must be assigned to the closest node within the tour. Also, the tour must visit each cluster exactly one time, and the tour visits at least one node per cluster, like the Generalized Traveling Salesman Problem [4], GTSP, and Insular Traveling Salesman Problem [5], InTSP.
Similar applications of this problem arise when: both construction and access costs (or distances) assumed by clients are significant; there is no capacity constraint for the vehicle, and the clients are grouped into disjoints clusters. The studied problem may be applied in food delivery in rural areas, water delivery systems in humanitarian logistics, parcel delivery, school bus routing, telecommunication network design, etc.
A distinctive feature of the GMTP in comparison to the InTSP (where the ship only can visit the docks of the islands) is that the route can access all the demand nodes in the GMTP. Additionally, the InTSP considers that each cluster may be visited (enter and exit) more than once. Notice that if the access distances between clusters in the GMTP are arbitrarily high, then the problem is reduced to the Median Tour Problem, MTP [6].
Note that GMTP is NP-Hard. If the access distance is negligible, the problem is also reduced to the Generalized Traveling Salesman Problem, which is NP-Hard. Furthermore, if the access distances are extremely high, the solution implies to visit all the nodes for each cluster, obtaining the Clustered Traveling Salesman Problem (CTSP).
The paper is organized as follows. Section II presents an updated literature review of related problems. Section III introduces a MIP formulation for the problem assuming at least one node visited per cluster. Section IV exposes the numerical results to solve the GMTP and a real application about health supplies in a region in southern Chile. Section V presents conclusions and future work.

II. LITERATURE REVIEW
The GMTP is motivated by the InTSP, and it is an extension of the GTSP and the MTP. The InTSP, recently introduced by [5], involves maritime and ground transportation costs with a bi-objective perspective. This problem consists of defining optimal sequences for collecting the waste generated inside a set of islands. Each island has a known set of docks or ports that potentially may be used to collect the generated waste. Besides, there is a ship departing from a fixed, known port. The ship must collect the waste from all the islands and returns to the starting port, having enough capacity to transport all waste on a single trip. Moreover, the waste generated on each island is transported inside the island to one or more docks where the ship will arrive, allowing an island to be visited more than once. Consequently, if each island (cluster) is visited once (in one or more docks), then this problem is similar to the GMTP. On the other hand, in the GTSP (introduced by [7]), the nodes form a set of clusters, and the objective is to build a minimum cost tour that starts and ends in a depot, visiting a single node of each cluster. If the distance or access cost between nodes inside the same cluster is 0, then the GMTP becomes the GTSP.
Some related vehicle routing problems that address the demand allocation to the visited nodes along the route is found in the extensive facility location problem literature [8]- [10]. Recently, [11], provide an updated review afterward commented and referenced by three specialized researchers [12]- [14], who provide remarkable future research directions. This family of the network design problems is characterized as connected structures aimed at serving a set of clients. Such structures may have the shape of a path, tree, tour, or sub-graph, and the demand nodes that are embraced by these topologies must be assigned to a node that belongs to the structure. There are vehicle routing applications where it is not possible to visit all clients directly, by economic, practical, or geographical considerations. In this case, the clients that are not visited must travel to the closest visited customer. For example, the Median Tour Problem (MTP) [6], determines a tour that must visit just p of the n possible nodes of the network. Two objectives are minimized: the total distance of the tour and the total travel distance for the nodes, reaching their closest stop on the tour. A closest problem to the MTP are the Ring Star Problem (RSP) and the Median Cycle Problem (MCP), where a single cycle through a set of nodes, minimizing the cost of the cycle, and the assignment costs of the nodes that are not in the cycle [9], [15]. An extensive review of transportation problems where the objective is to design a single cycle in a non-directed network is presented in [16].
As previously mentioned, another related problem is the GTSP, introduced by [7]. In this problem, the nodes form a set of clusters, and the objective is to build a minimum cost tour that starts and ends in a depot, visiting a single node of each cluster [17]. This problem has several real applications. [18] suggest that a wide variety of optimization problems can be modeled as a GTSP, e.g., the design of postbox collection routes, goods distribution by sea, etc. [19], propose an integer formulation for the GTSP. [20] introduce facets that define valid inequalities for the GTSP, and [4] develop a branch & cut algorithm to the symmetric case of the GTSP. Since the GTSP is a NP-Hard problem, there have been a lot of articles that develop heuristics, providing good feasible solutions to the problem in a short time [21]- [23].
A significantly related variant is the Clustered Traveling Salesman Problem (CTSP), where the nodes are separated in clusters, and all the nodes of each cluster must be visited consecutively before departing to another cluster or the depot. If the salesman visits a cluster, it cannot leave the cluster until all clients have been served [24]. Applications of this problem cover a wide range of areas. Several algorithms have been developed for solving the CTSP: approximation algorithms [25], [26], tabu search [27], Lagrangian relaxation [28]; genetic algorithms, [29], Grasp, [30]; hybrid heuristics [31]; memetic algorithms, [32]; and other heuristics [33]. A special case of the CTSP is the Ordered Clustered Traveling Salesman Problem (OCTSP), where a visit sequence to the clusters is defined a priori, [34], [35].
According to the previous review, the GMTP is a new combinatorial problem. Table 1 details the differences with closest problems presented in literature. The clients are separated into clusters that must be visited by a tour. The problem consists of determining a tour or a simple cycle that visits each cluster once, and the nodes within a cluster could be visited at least once. Some nodes may be out of the tour, which must be assigned to their closest node on the selected tour. Two types of distances are considered: the total distance of the tour, and the total access distance (sum of the access distances to reach the nearest node on the tour). The GMTP is an extension of the cycle location problems, using the mini-sum access criteria, such as the RSP, MCP, MTP, among others, where the nodes are grouped in disjoint clusters.

III. THE INTEGER PROGRAMMING MODEL FOR THE GENERALIZED MEDIAN TOUR PROBLEM
This Section aims at formulating de GMTP and also at proposing different valid cuts and strategies for solving the proposed GMTP formulation. Subsections III-A and III-B VOLUME 8, 2020 present the notation and the proposed IP formulation, respectively. Subsections III-C and III-D propose alternative valid inequalities for the GMTP, while Subsection III-E, establishes a combined strategy employing the two types of valid inequalities. Finally, Subsection III-F discusses some computational issues, and Subsection III-G describes the separation algorithm employed.

A. PROBLEM DESCRIPTION AND NOTATION
The GMTP consists in defining a single tour to serve a set of nodes, where the nodes belong to several disjoint clusters. Additionally, there is a single vehicle with enough capacity for serving all demands on a single tour. Thus, the tour must visit at least one node per cluster. Each non-visited node must be connected to one of the visited nodes at the same cluster.
Let G = (N , A) be an asymmetric complete graph, where N is the set of nodes, and A is the set of arcs. Let K be the set of disjoint clusters, where N p , with p ∈ K , is a subset of N . For the sake of simplicity, the cluster N 1 = {1} contains only the depot node. Let c ij be the travel distance over the arc (i, j) ∈ A. Also, every arc inside each cluster p, (i, j) ∈ A p with i, j ∈ N p , has an access distance d ij .
Let A be the set of access arcs, i.e., A = p∈K A p . Notice that, if the vehicle travels from node i to node j, it travels in a distance c ij . On the other hand, if a node i is not visited by the vehicle, it must be assigned to the visited node inside the cluster, traveling an access distance d ij .
The decision variables are defined as follows: Subject to: i∈N \{j} j∈N \{i} i∈N \{j} x ij = y jj ∀p ∈ K , j ∈ N p (8) y ij ≤ y jj ∀p ∈ K , i, j ∈ N p : i = j (9) i∈S j∈S\{i} The objective function (1) jointly minimizes the distance of the tour plus the access distance to the tour. Constraints (2) and (3) force the route to a single visit per cluster. Constraints (4) state the flow balance to each node. Constraints (5) assure a maximum of one visit per node. Constraint (6) assures that each cluster is visited at least once. Constraints (7) force the assignment of each node in a cluster to its closest node visited by the tour. Constraint set (8) indicates the assignment of node j to itself if the tour visits that node. Constraints (9) force the assignment of a node i to node j only if the node j is on the route. Constraints (10) preclude disconnected tours using flow variables. Constraint (11) states the binary nature of the variables. Note that the variables y ij can be relaxed as continuous.
The number of constraints in set 10 is O(2 |N | ), which makes this formulation intractable as it is. Thus, the next subsections present different strategies to deal with constraints (10) to solve the GMTP efficiently. First, a set of valid and tighter inequalities called Packing cuts are presented. Second, the Gavish and Graves [36] constraints are displayed. Third, a combination of the Gavish and Graves constraints and Packing cuts is used. Finally, we present a branch & cut scheme and separation algorithms for the required strategies.

D. GAVISH-GRAVES CONSTRAINTS
We adapted the Gavish-Graves constraints (GGC) for all nodes [36]. This set of constraints guarantee a connected tour. Therefore, we define f ij as a continuous variable indicating flow between nodes i and j, and the Gavish-Graves constraints are as follows: Constraints (13) -(16) replace constraints (10). Constraint (13) states the maximum flow that leaves the depot. Constraints (14) force the flow continuity for each node j. Constraints (15) assure if the arc (i, j) is not on the tour (i.e. x ij = 0), the flow variable f ij = 0. Constraints (16) assure the domain of variables f ij .

E. GAVISH-GRAVES CONSTRAINTS COMBINED WITH PACKING CUTS
We propose a combination of the Gavish-Graves constraints and packing cuts (GGC + Packing) for solving the GMTP. Packing cuts are used to break subtours S p strictly generated within clusters N p . Besides, the GGCs avoid subtours between clusters. Thus, we replace constraints (12) by constraints (17) as follows: Note that (17) is a particular case of (12).  Alternatively, for this case, we now redefine variables f pq as the flow when cluster p precedes cluster q in the tour, and the flow balance constraints are as follows: Thus, constraints (17) -(21) replace constraints (10). Constraint (17) deletes the subtours generated within each cluster. Constraint (18) states the flow that leaves the depot. Constraints (19) force the flow balance for each cluster q. Constraints (20) establish that if there is a flow between two clusters p and q, there is travel between both clusters. Constraints (21) assure the domain of variables f pq .

F. SOLUTION PROCEDURES
All mathematical models presented are solved by the commercial solver CPLEX 12.8. Nonetheless, The number of constraints in sets (12) and (17) are O(2 |N | ), which makes the formulations using packing constraints intractable as they are. As a consequence, we propose two approaches to deal with them.
The separation procedure follows the steps of the algorithm described in [38]. We apply this procedure to check the integer candidate solutions along the tree of the branch & bound algorithm, and to delete subtours within the branch & cut algorithm. This procedure is performed by employing CPLEX 12.8, which allows us to monitor the candidate solutions during the optimization process using its generic callbacks.

G. SEPARATION ALGORITHM
This procedure is applied independently over two mathematical models. The first one, is stated by constraints (1) - (9), and (11), which we will refer as model A. The second model, is represented by constraints (1) -(9), (11), and (18) - (21), which we will refer as model B. This procedure deletes subtours by iteratively adding constraints (12) to A, and (17) to B.
When solving model A or B, once a candidate solution is found by a generic callback, we build a support graph G s (N , A s ) considering all variables x ij and y ij , whose associated values in the candidate solution are different from zero. In order to identify subtours, a super-node S is built to implement the shrinking technique [39]. LetN = N . The searching process begins in the depot (node 1), which is added in S. We include in S all nodes of the graph such that x ij = 1 : i ∈ S, j ∈N \ S or y ij = 1: j ∈ S, i ∈N \ S : (i, j) ∈ A . Once there are no more nodes to add to S and |S| < |N |, a subtour is identified. Then, to break the subtour S, the corresponding subtour elimination constraints are added into models A or B. The identified nodes in S are removed fromN , i.e.N \ S. Next, the set S is cleaned and the search starts again selecting a node i ∈N , which is added to S. This process is repeated untilN = ∅. The order of this algorithm is O (|N | 4 ).

IV. RESULTS
In this section, we detail the results of a computational results for the GMTP. The first subsection presents results for 37 test instances up to 299 nodes and 60 clusters. All instances were taken from [4], which are symmetric and Euclidean (i.e., c ij = c ji ; d ij = d ji ). For each test instance, the first node (node 1 or 0) is considered as a cluster, such that In the second subsection, we present results for a realworld instance. The models were coded and solved using Visual C++ and CPLEX 12.8. All experiments were run on a PC with 32GB of RAM and Processor Intel Core i7-7700. Table 2 shows the results using three subtour elimination strategies for the GMTP. The CPU time limit was set in 1 hour. UB is the best feasible integer solution found in the time limit. GAP is the integrality gap for each instance, and the CPU is the time reported for each instance. If the optimal solution is found, the GAP = 0, and the CPU time is reported. The instance label ANameB indicates that the instance has A clusters and B nodes. The column ''Nodes'' denotes the number of nodes explored within the branch & bound algorithm. Table 2, the optimal solution (GAP = 0.00%) is reported for most of the instances. Table 3 presents  summarized results of Table 2. The strategy of using packing cuts shows better behavior in all instances, which is reflected in the integrality GAP and CPU time when it is compared with the use of GGC constraints between clusters.

As observed in
Besides, the branch & cut strategy outperforms the other strategies in terms of the Integrality gap and computing time. However, the GGC + Packing Cuts strategy allows us to find a feasible solution in most cases, especially in the larger instances. It is important to note that not all instances reach the optimal solution or even a feasible solution within one hour, which can be explained by the combinatorial nature of this variant of the problem (GMTP).

B. REAL-WORLD INSTANCE
We solved a real-world instance in a southern region in Chile. The region comprises 23,890 km 2 and 1,557,414 inhabitants. The region contains three provinces: Arauco, Biobío, Concepción. Each province has a set of districts. The regional health services use the same provinces for budget and administrative purposes. Subsequently, each provincial health service is independent, and it is responsible for providing supplies to their districts. Consequently, each province is addressed and solved separately.
Furthermore, each province has a set of health facilities, such as hospitals, primary health centers, and rural health centers. All distances between each pair of nodes were determined using Google Maps . Notice that, the network is asymmetric, i.e. c ij = c ji and d ij = d ij . Fig. 4 shows a map (by Google My Maps ), where the three provinces are framed in red, and Table 4 shows the area (km 2 ), number of districts, and number of health facilities per province. Each district corresponds to a cluster, and each node is a health facility. Thus, a distribution route for health supplies is presented for each province. The depot is located near to the main airport in the region (node ''A'' in Fig. 4).
The objective function (1) minimizes the total distance of the tour plus the total distance traveled by nodes outside the tour. In some instances, the objectives might be in different units or magnitudes. In our problem, we add this feature to normalize the magnitude for both terms of the objective an make them comparable [40]. θ is the set of feasible solutions by (2)- (11), and Y ∈ θ is a feasible solution. α ∈ [0, 1] is a weight of 22. Notice that for a fixed α, the problem is still mono-objective, as shown in (22), where TP(Y ) is the total route distance, AS(Y ) is the total travel access distance for each solution Y , (TP min , AS min ) is the ideal point, and (TP max , AS max ) is the anti-ideal point. TP min is the minimum value for the shortest tour, and it is associated with the worst access distance AS max . Besides, AS min is the minimum value for the access distance, and it is associated with the worst tour distance TP max .
The results of the application of the model in the three provinces are presented in Table 5, where a set of optimal solutions is found by varying α from 0.1 to 1.0, using increments of 0.1 units. We used the Packing Cuts strategy to solve each instance. The notation is as follows: α is the weight for each objective; TP represents the total distance, in kilometers, of the tour; AS represents the total distance (in kilometers) traveled by the facilities not belonging to the tour; #St indicates the number of stops in the route (including the depot); #Cuts indicates the number of cuts found by the branch & cut algorithm; Time is the reported CPU time (in seconds) for each run, and GAP is the integrality Gap. Table 6 presents the average for cuts (#Cuts), CPU time (Time), and VOLUME 8, 2020 GAP (%). Notice that the instances of Concepción province take more time to solve due to the nodes are closer to each other. Table 7 indicates that the tour distance is longer in BioBío province because of the starting node is the airport and the facilities are more scattered. It should be noticed that the Biobío and Arauco areas are composed mainly of rural areas. It reflects the average distance from each health facility to the tour. As opposed, the tour distance is the shortest in Concepción, because this is a smaller and dense are. This fact implies that health facilities are closer to each other.    5 presents a set of optimal solutions for each province. The longer the tour, the shorter the total access distances. On the other hand, the shorter the tour, the longer the total access distances. Naturally, extreme solutions might not be practical. The tour distance would be the longest in the right extreme because all nodes are visited. In the left extreme, the total tour distance would be the shortest, but the number of nodes assigned is too high. Fig. 6 presents three solutions taken from Table 5. The first one (Fig. 6a represents the shortest route (like a GTSP solution). The second picture (Fig. 6b indicates the largest route, where all health facilities are visited. Finally, Fig. 6c presents an intermediate solution, with α = 0.5.

V. CONCLUSION AND FUTURE WORK
In this article, we propose, model, and solve a novel problem called Generalized Median Tour Problem (GMTP). In this problem, the nodes naturally conform clusters, and a single tour must visit each one of them, such that the nodes within each cluster that are not included in the tour must be assigned to the closest node on the tour. The solution minimizes the weighted sum of the tour distances (associated with the length of the tour), and the access distances (associated with the unvisited nodes to the selected nodes in the tour within each cluster).
Three solution strategies are proposed. The first one is based on employing a set of packing constraints within a branch & cut algorithm. The second one is based on using the Gavish and Graves constraints. The third method is based on the combination of these two first methods. The results show the suitability of the proposed and implemented procedures, solving benchmark instances up to 299 nodes.
We present and solve a real-world application for the health supplies distribution in the Biobío Region, Chile, from the main regional airport to the existing health facilities. Particularly, the analyzed cases are motivated by the relevant need for a fast distribution system and strategy for critical health supplies, such as vaccines for COVID-19 (or under other pandemic scenarios), and demanded blood under natural disasters (e.g., earthquakes). Also, we present a sensitivity analysis yielding different configurations for the distribution system, evidencing the suitability and convenience of the proposed methodology to support public authorities and decision-makers in the task of planning and designing such strategic distribution systems.
The tour could visit one or more nodes within each cluster according to the weights of the objective function. If the tour arcs are more significant than the access arcs, the tour will visit the smallest possible number of nodes per cluster. Oppositely, if the tour arcs are less relevant in comparison with the access arcs, the tour will visit more than one node within each cluster.
Future works may include the development of heuristic procedures to solve the different variants of the GMTP in a shorter CPU times. Other research directions may comprise the column generation approach and a branch & price strategy. Several GMTP extensions and variants can be explored, including multi-period and multi-vehicle formulations, capacitated vehicles, time-windows, and advanced transportation processes inside the clusters, such as consolidation strategies, sequencing with secondary vehicles, tree structures, among others. He currently works as a Professor and a Researcher with the Department of Industrial Engineering, Universidad Católica del Norte, Antofagasta, Chile. He has published his works in prestigious international journals on the fields of operational research, transportation research, industrial engineering, manufacturing, and operations management. His research interests include mixed integer programming models and algorithms for addressing real-world optimization problems in the industry, focusing on transport, logistics, and supply chain network design problems.