Advances in Operations Research

Volume 2018, Article ID 8720643, 13 pages

https://doi.org/10.1155/2018/8720643

## Multiobjective Optimization for Multimode Transportation Problems

^{1}Lab-STICC UMR CNRS 6285, University of Brest, Brest, France^{2}Dalarna University, Falun, Sweden

Correspondence should be addressed to Pascal Rebreyend; es.ud@brp

Received 5 December 2017; Accepted 15 April 2018; Published 7 June 2018

Academic Editor: Alessandra Oppio

Copyright © 2018 Laurent Lemarchand et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We propose modelling for a facilities localization problem in the context of multimode transportation. The applicative goal is to locate service facilities such as schools or hospitals while optimizing the different transportation modes to these facilities. We formalize the School Problem and solve it first exactly using an adapted -constraint multiobjective method. Because of the size of the instances considered, we have also explored the use of heuristic methods based on evolutionary multiobjective frameworks, namely, NSGA2 and a modified version of PAES. Those methods are mixed with an original local search technique to provide better results. Numerical comparisons of solutions sets quality are made using the hypervolume metric. Based on the results for test-cases that can be solved exactly, efficient implementation for PAES and NSGA2 allows execution times comparison for large instances. Results show good performances for the heuristic approaches as compared to the exact algorithm for small test-cases. Approximate methods present a scalable behavior on largest problem instances. A master/slave parallelization scheme also helps to reduce execution times significantly for the modified PAES approach.

#### 1. Introduction

Localization of facilities such as public services (schools, hospitals, etc.) is an important problem for social planners and policymakers. Most of the time, this problem is formulated as the (single objective) -median problem which is a central problem in Operations Research (see, for example, [1, 2] for surveys). The -median was introduced by Hakimi [3] who describes its basic properties. Its basic variant can be defined as follows: given a set of demand nodes, distance values for each pair of nodes, and a fixed number of facilities, locating each facility at one of the nodes, while minimizing the sum of distances from each node to its closest facility.

Recent developed algorithms solve the single objective version exactly for instances of thousands of nodes (e.g., 25.000 nodes in [4]). However, for a policymaker, considering additional objectives would be useful when solving -median problems on real cases, leading to different variants; e.g.,(i)*dispersion problem*: the -dispersion problem consists of spanning the facilities by maximizing the minimal distance between two of them. This objective function is suitable to locate business franchises and also when locating obnoxious facilities [5].(ii)*-center problem*: the -center problem [6] aims at minimizing the maximal distance of demand nodes from their facility, or the average distance of a fraction of those that are the farthest from their closest facility, e.g., 5% farthest of them. This problem formulation can be applied to locate emergency services such as fire stations.(iii)*multimode transportation location (MTL) problem*: in many real cases, transportation can be done by different means (by foot, bike, car, buses, etc.) depending on criteria like a threshold on the distance to the nearest facility. As an example, pupils are going to school by foot (category A) or by public transportation (category B), with a threshold on the distance defining the 2 categories, e.g., 2 km. The objective is then to minimize both the mean distance for those of category A and the number of people in category B.

In the following we will focus on the MTL problem with multiple objectives. The practical context is to optimize location of schools. This School Problem is a typical multimode transportation problem since, depending the distance, pupils can go to school by foot or by bus. MTL problems can be seen as multiobjective optimization problems if means of transportation have impact on each other. Optimizing the cost for one of them can degrade the other objective value. When considering a multiple objective optimization problem (MOO), a single solution optimizing all of the objectives simultaneously rarely exists. Let be an objective function mapping solutions , the search space, to the objective space , with . MOO algorithms look for solutions such that is optimized (in the sequel, we consider minimization). is a point of the objective space , with each of being one of the objective function values to be minimized. Many approaches rely on the* dominance* concept to choose, among a set of solutions , those ones that represent the best trade-offs of objectives within the search space. We say that a solution * dominates* another one if and . It is denoted as . Namely, is as good as on all objectives and better than for at least one of them. Solutions that are not dominated by any member of are* efficient* solutions and constitute the* Pareto set *. The set of points corresponding to the efficient solutions is the* Pareto front *. In the sequel, we say for short that a point dominates if .

The goal of MOO algorithms is generally to determine or approximate points of and associated solutions.

Solving multiobjective -median instances is of course related to -median problem exact and heuristic resolution approaches but also to general approaches used for solving MOO problems, either exactly or approximately. The former are often based on Integer or Binary Programming (Multiobjective Integer Programming, MOIP), the latter on Evolutionary Multiobjective Algorithms (EMOA). Our main contribution is to develop and evaluate the two kinds of approaches, in order to be able to solve exactly medium size cases of the School Problem and approximately very large scale cases. We exploit some mathematical properties of our targeted problem in order to model it with MOIP and solve it exactly with an -constraint algorithm. Large test-cases are handled with 2 different general EMOA frameworks, namely, the* Pareto Archived Evolution Strategy* (PAES, [7]) and* Nondominated Sorting Genetic Algorithm **2* (NSGA2, [8]). We have modified the former and mixed it with a local search technique: we have adapted to the multiple objective case an efficient neighborhood evaluation procedure [9] developed for the -median problem. NSGA2 can also use our local search technique, as a postprocessing step. We show that, in many cases, for an equivalent computational effort, a well-known population-based approach such as NSGA2 is outperformed by the single individual method, PAES [7], thanks to our hybrid approach. Efficient parallelization helps for handling large test-cases. As shown in Section 2, similar approaches exist for MOO -median, but with different multiple objectives and local search algorithms. Furthermore, to our knowledge, no results have been presented for the parallelization of these approaches.

Section 2 introduces related work for multiple objective optimization problems embedding -median like formulation. Next, in Section 3, we formalize our bicriteria multimode transportation problem, with its specific objective functions. In Section 4, we present an exact approach for solving the problem, with an -constraint like technique. The problem can also be solved using popular MOO heuristic approaches like NSGA2 and PAES. We show how to adapt those frameworks to our problem solving, coupling them with an aggregation technique for performing local search. The MOO frameworks are presented in Section 5, and the local search, using limited neighborhood, is detailed in Section 5.2, along with its exploitation by the MOO methods. Evaluation and comparison of the proposed algorithms are realized with the hypervolume standard metric [10], over Beasley’s benchmark [11] in Section 6. Last, conclusion and perspectives are detailed in Section 7.

#### 2. Related Work

A few works extend the -median problem with multiple objectives. The -median problem is -hard [12], and MOO versions are also -hard since they embed the single objective version. Thus, if some exact algorithms exist, heuristic approaches are preferred for large test-cases.

The MOO -median problem with an additional facility cost objective is dealt with in [13]. Each facility is weighted by a building cost, and the goal is to minimize the sum of distances to locations (-median objective) and the sum of costs of the opened locations. The authors in [13] used two approaches. The first is an -constraint like formulation that is a mix of two-phases algorithms [14] and classical -constraint approach [15]. According to the authors, it leads to a close approximation of the full Pareto front. Even if its second objective is different (facility cost instead of number of pupils by public transportation), the technique used is similar to our approach concerning the -constraint problem formulation. However, since the number of nodes varies in our case for the distance count (for instance, pupils going to school by public transportation are not taken into account for the distance of demands from facilities), the method must be adapted, as it will be shown in Section 4. The second approach used in [13] helps to handle large test-cases and is based on MOGA framework, with a path relinking local search procedure for mixing solutions. Problems with uniform demand at each location (unitary problems) and with up to 400 demands and 20 facilities are processed. As MOGA, NSGA2, which we are testing, also uses a population-based approach.

In [16], the authors formulate a biobjective -median problem with the following objectives : minimal distance to the closest facility (-median traditional objective) and maximization of the minimal distance between facilities (-dispersion problem objective). They formulate and solve it as a MOIP, with an -constraint algorithm and also with a customized approach, based on threshold fixing for the minimal distance between facilities. Taking into account the threshold leads to additional constraints in a single objective subproblem. The algorithm iterates on subproblems, in order to provide the exact Pareto front of the biobjective initial problem. Once again, our second objective is different and implies an adaptation of the -constraint algorithm. Furthermore, the modified MOIP developed in [16] is specific to the dispersion problem, since it iterates on a threshold value of dispersion (minimal closeness of selected locations).

Problems similar to multiobjective -median have to also be examined in the context of disasters management [17], or tsunami exposure. In [18], the authors solve approximatively a 3-objective optimization problem for school localization. Similarly to our heuristic method, it uses the NSGA2 framework mixed with a local search procedure, but the objective is not exactly the same as with -median problems, since the number of facilities is not fixed but driven by a cost minimization objective. The second objective is related to tsunami risk exposure, and the third one takes into account both the distance to school and the number of pupils not able to go to school, as it is considered too far away (meaning, to go by foot). As we will see in the next section, this objective is in fact a mix of ours.

The -median problem and its multiobjectives variants have also some military applications. Localization of repairs, supply, or security facilities can be modelized and addressed as -median problems (see [19] for a survey). Common objectives are cost of localization, coverage of areas, and rapidity of deployment and also security of the localization that must be resilient. As an example, in [20], a multiobjective model, solved using a genetic annealing heuristic, takes into account cost and response time for Canadian forces prepositioning.

#### 3. The Multimode Transportation Problem Modelling

We want to optimize the location of services, such as schools, in such a way that users go by foot to the closest service if it is possible (according to a threshold on the distance they can walk) and others take public transports. In the following, we will focus on a particular case of multimode transportation problem, the School Problem. The School Problem is stated as follows: given a set of demand nodes holding pupils and a set of candidate nodes for locating schools, a value of , and distances between each couple (demand node, candidate node) (, ), define a set of locations for setting up schools (school nodes) among the candidate nodes with the following objectives:(i)The mean distance of demand nodes to the closest school node is minimized for those demand nodes that have a distance to closest school node less than the threshold .(ii)The total number of demand nodes with a distance to the closest school greater than is minimized.

The modelling of the School Problem can be stated as follows:where(i) is the number of demand nodes;(ii) is the number of candidate nodes (number of possible locations for a school);(iii) is the number of candidate nodes to be selected as school nodes;(iv) is the threshold on walking distance;(v) is the number of pupils located at demand node ;(vi) is the distance between demand node and candidate node ;(vii) is a decision variable, indicating if the pupils at node go to school by walk or not (category A nodes).;(viii) is a decision variable, indicating if the candidate node is selected or not as a school node;(ix) is a decision variable, indicating if the demand node closest school node is or not. If , we say that (demand node) is* covered* by (school node) .

Equations (1) and (2) reflect the objectives stated above, taking into account the number of pupils located at each demand node. Equation (3) fixes the number of selected school nodes. Equations (4) and (5) ensure that the single selected candidate node for pupils at a demand node is effectively a school node. Equation (6) ensures that pupils at a demand node are accounted to go walking if and only if a candidate node is selected for school location in the neighborhood of the node.

This formulation induces that candidate nodes are restricted to demand nodes, but it could be extended to an arbitrary set of candidate nodes. Also notice that distance data can correspond to Euclidean distances, with known geographical locations for the different nodes, or to shortest path values, computed with an algorithm as Floyd-Warshall if the nodes are embedded within a routing graph, with moving cost values on the edges (e.g., time by walking, distance, and transportation cost). In the latter case, the value and meaning of the threshold are to be adapted. The use of the threshold reflects the public Swedish policy where authorities offer free transportation to some pupils based on the walking distance to school. If , we speak about the unitary version of the School Problem.

We show in the next section how to model this problem as a MOIP and how to solve it with an ad-hoc technique.

#### 4. Exact Problem Formulation and Solving

It is possible to solve exactly multiple objective problems using -constraint approaches. These techniques require a linear formulation of the aimed problem. Since the first objective function of our model is nonlinear (1) and nonconvex, those techniques are not directly applicable. In Section 4.1, we present the problem as a multiobjective mixed integer linear program (MOMILP). Its resolution would allow for computing the optimal value of each objective (by removing other objectives from the formulation and using a MILP solver as CPLEX, [21]). However, even computing only the mean distance (our first objective) with this model required high execution times, as detailed in Section 4.1.

Therefore, we consider a simplified MOIP in Section 4.2 where the computation of the mean distance is replaced by a computation of the cumulative distance. Using this model, we can compute the Pareto front and associated solution set using an adapted -constraint [22]-like approach presented in Section 4.3. We show that this method provides the exact Pareto set for the problem stated in Section 3.

##### 4.1. MOMILP Modelling

We studied the translation of the School Problem to a MOMILP problem mainly to compute directly the minimum mean distance (1) using a MILP solver. Since this objective function is continuous, its linearization requires using continuous variables. Indeed, starting from the modelling of the School Problem presented in Section 3, we can linearize (1) using a new set of (continuous) variables , which replaces the set of (Boolean) variables . The semantics of variables is the following: if* demand node ** is covered by school node ** and ** is in category * (i.e., ) and, in this case, .

Using this semantics, objective (1) becomes linear, as shown in . Notice that, by removing the variables , we “forget”, for each demand node in category (i.e., with ), which school node covers it. It is possible to keep this information, but it is useless for the resolution of the problem (as long as there is at least one school).

To satisfy the semantics of the variables and specifically specify , we also add a continuous variable for each demand node and a continuous variable . The resulting model is as follows:

In this model, , and , ensure that : from and (and since are binaries), we have . By , we get , which gives the lower bound of .

From this result, states that when demand node is in category (i.e., ), (the minimization of ensures that only one is equal to and the other are zero). Equation implements Constraint (4), whereas the implementation of Constraint (6) is done by and (note that the former is needed for the first objective while the latter is needed for the second one).

We have tried to exploit this formulation for computing the minimal mean distance, with the CPLEX MILP solver [21], using the single objective function . In practice this approach did not work, except on our smallest example (100 nodes and 5 schools). On the other test-cases, CPLEX returned possibly nonoptimal solutions after reaching its memory limit. We think that this failure stems from two issues. The first issue is the variability of (and, as a result, the values of all nonzero ) which makes node selection a hard problem. The second issue is that when solving relaxed problems, with and partly continuous, enables putting even when is close to 1, since is very small. As a result, the algorithm needs to force almost all (and ) to integral values before getting a meaningful minimal bound. Since both issues are related to the computation of a mean distance, we considered a MOIP using the cumulative distance instead.

##### 4.2. MOIP Modelling

The MOIP formulation of the School Problem with the cumulative distance can be seen as a simplification of the MOMILP formulation of the previous section. In fact, we just need to state that . As a result, the variables are useless (, , and disappear). From this result, we propose the MOIP formulation for the School Problem, as follows:

Compared with the previous model, we replace by (12), (13), and (14), which ensure that demand node cannot be in if there is no school node at distance .

Notice that (13) and (15) both use , but with different meanings: the former ensures that a pupil cannot walk to a distant node (to minimize the second objective), while the latter prevents an artificial minimization of the cumulative distance by covering a demand by a distant node where a closer one is available.

As we have seen before, our School Problem is formalized only* approximately* by , since (9) uses the* cumulative* distance instead of the* mean* distance for pupils in category A. While the minimization of the cumulative distance can be compared to the -median problem (indeed, we can get the -median problem by adding for all and removing (13)), its exact resolution using MOIP solvers can be more difficult, as the relaxed problem (with continuous variables) has more solutions. To explain this issue, let us consider only two demand nodes (which are also candidate nodes) 1 and 2 such that and one location (). With the relaxed -median problem (), the solver gives directly the optimal cumulative distance , whatever the values of and . Using the relaxation of , the optimal solution sets all variable to except and which are set to 0 and the cumulative distance is 0.

Using this observation and the fact that the cumulative distance is still not the mean distance lead us to consider an adapted -constraint method where the first objective is (10).

##### 4.3. Exact Algorithm

-constraint [22] methods proceed as follows: a series of single objective (*i.e.,* Integer Program, IP) problems are solved, transforming all but one of the objectives of the MOIP considered into IP constraints. The constraint set is updated at each iteration to enforce the exploration of the whole objective space. In their paper [23], Ozlen and Azizoğlu introduce a recursive algorithm to generate a Pareto set for a MOIP problem. They use the set of already solved subproblems and their solutions to avoid solving a large number of IPs. In [14], a two phases algorithm is applied for the biobjective assignment problem. In the 2 objectives case, it first determines the extreme points in the objective space by discarding one objective at a time and solving the resulting single objective problem. Then, in a second phase, it partitions the objective space according to the range of values found at the first step and explores it by slices. It also combines the approach with heuristics specific to the assignment problem for enhancing the execution times.

An adapted -constraint algorithm which computes all nondominated points sorted by values (i.e., second objective) is presented in Algorithm 1 for solving . This approach uses the fact that given a nondominated point , other nondominated point such that must satisfy both and . Since the minimal cumulative distance is hard to compute (and not really useful as it may not be the minimal mean distance), we do not compute the extreme point associated with it.