Mathematical Problems in Engineering

Volume 2017 (2017), Article ID 4708135, 13 pages

https://doi.org/10.1155/2017/4708135

## A GRASP-Tabu Heuristic Approach to Territory Design for Pickup and Delivery Operations for Large-Scale Instances

^{1}Faculty of Engineering and Applied Sciences, Universidad de Los Andes Chile, Monseñor Álvaro Portillo 12455, Las Condes, Santiago, Chile^{2}Tecnologico de Monterrey, Escuela de Ingeniería y Ciencias, Eugenio Garza Sada 2501, 64849 Monterrey, NL, Mexico^{3}School of Computing, Informatics and Decision Systems Engineering, Arizona State University, Tempe, AZ 85287-8809, USA^{4}Universidad Autónoma de Nuevo León, Facultad de Ciencias Físico-Matemáticas, Av. Universidad s/n, 66450 San Nicolás de los Garza, NL, Mexico

Correspondence should be addressed to Neale R. Smith; xm.mseti@htimsn

Received 28 April 2017; Revised 28 September 2017; Accepted 2 October 2017; Published 29 October 2017

Academic Editor: Federica Caselli

Copyright © 2017 Rosa G. González-Ramírez 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 address a logistics districting problem faced by a parcel company whose operations consist of picking up and delivering packages over a service region. The districting process aims to find a partition of the service region into delivery and collection zones that may be served by a single vehicle that departs from a central depot. Criteria to be optimized are to balance workload content among the districts and to create districts of compact shape. A solution approach based on a hybrid procedure that combines elements of GRASP and Tabu Search (TS) is proposed to solve large-scale instances. Numerical experimentation is performed considering different instance sizes and types. Results show that the proposed solution approach is able to solve large-scale instances in reasonable computational times with good quality of the solutions obtained. To determine the quality of the solutions, results are compared with CPLEX solutions and with the current real solution to highlight the benefits of the proposed approach. Conclusions and recommendations for further research are provided.

#### 1. Introduction

In this study, we consider a logistic districting problem faced by a parcel company that operates in the metropolitan area of Monterrey, Mexico. This company is concerned with the delivery and pickup of packages within Monterrey. The latter is divided into districts for logistic purposes. Each district is served by a single vehicle that departs from a central depot, in which the packages are received. At the end of the day, the vehicle comes back to the depot. Currently, drivers take the route decisions in a dynamic manner during the day due to the variability of the demand. That is, it is assumed that the driver may receive new dispatch orders along the day.

The company’s current practice is to redesign the districting configuration every year and a half in order to account for demand variations. Although the day to day locations of the customers and the daily volume of demand are stochastic, the logistics planning managers consider that a reasonable approach is to use the data of a high workload day which is representative of the current and growing demand. The number of pickups and deliveries may vary, but, at the time, the company was considering data sets of about 1,100 customers, with about equal numbers of pickups and deliveries. In general, the days chosen by the company to serve as representative days have above-average demand and a geographical distribution of demand that is considered to be “typical,” that is, not unusually high or low in any area with respect to an average behavior.

Once a representative day is selected, the logistics manager identifies the locations of the customers on a map. Then, the districts are readjusted considering their estimated capacity. But this readjustment needs to consider the compact shape of the district, and all the required arrangements to balance workload content among all the districts must be made. It takes about three weeks for the managers to complete that process.

The managers of the company are mainly interested in balancing the workload and in the shape of the districts. Accordingly, we define two criteria to optimize during the districting process: workload balance and compactness. The former is important because it tends to minimize the number of vehicles needed and, consequently, it reduces capital investment, opportunity costs, and possibly the vehicle maintenance costs. The latter criterion is important because it tends to reduce travel distances within a district.

We propose a mathematical model for the districting problem abovementioned. Following the assumptions considered by the logistic planning manager, we assume that a representative day can be properly chosen. This approach is reasonable because it has been working well for the company over years. So, we adopt that assumption and provide a tool to support and make a more efficient districting design. Also, we propose a heuristic solution methodology that can find good districting configurations in large-scale instances. Note that the use of a heuristic algorithm is justified by the difficulty of the problem, which has been shown to be NP-Complete by [1].

The remainder of this paper is organized as follows. Section 2 presents a literature review regarding districting problems. Section 3 presents the proposed mathematical model. Section 4 describes the heuristic algorithm and Section 5 presents the numerical results showing the suitability of the proposed algorithm to solve large-scale instances. Conclusions and recommendations for future research are given in Section 6.

#### 2. Literature Review

Despite the fact that this literature review is devoted to districting problems that involve routing decisions, it is worthy to mention one of the pioneer papers in districting area. Keeney [2] addresses the problem of partitioning a service area into districts. Each district corresponds to an existing facility that provides service to that particular district. Later, researchers keep paying attention to this topic, solving districting problems in diverse contexts, such as politics, sales territory alignment, schools, health care systems, emergency services, and logistics (see [3–6]).

As it can be inferred from above, some of the applications of districting problems involve an interrelation with routing decisions. This fact has motivated the researchers to merge both decision processes in the same problem. One of the initial findings appears in [7], in which the method for vehicle routing proposed in [8] is compared against a methodology based on a districting approach. Wong and Beasley [9] considered the Vehicle Routing Using Fixed Delivery Areas (VRFDA), which is closely related to districting problems. In that problem, a service area is divided into fixed subareas and the daily route in each of them may change from day to day. The authors propose a methodology for minimizing the traveled distance. A special case of the VRFDA is the Fixed Routes Problem (FRP) studied in [10], in which the service region is divided into subareas but each route does not vary from day to day.

Daganzo [11] proposes an approximation method for the design of multiple-vehicle delivery zones seeking tours of minimum total length. The objective in that paper is to explore the impact that the zone shape has on the expected length of each route. Later, in [12], a methodology in which the region is partitioned into zones of nearly rectangular shape elongated toward the source is presented. The number of points considered in the instances is large compared to the capacity of the vehicles; this fact complicates the problem’s resolution. Newell and Daganzo [13] analyze the districting of a region in which the underlying network of roads forms a dense ring-radial network. The authors proposed an approximation method for the design of multiple-vehicle delivery tours that minimizes the total traveled distance. The design of delivery zones for distributing perishable freight without transshipment is studied in [14]. Lei et al. [15] propose the multiple traveling salesperson and districting problem, which considers multiple periods and depots. The authors propose an adaptive large neighborhood search metaheuristic.

Regarding problems for designing delivery tours within districts, it can be mentioned [16], in which the design of multiple-vehicle delivery tours that satisfy time constraints for letter and parcel pickup is studied. The authors propose a methodology that involves partitioning the region into approximately rectangular delivery zones that are arranged into concentric rings around the depot using a continuous approximation of the model. Rosenfield et al. [17] study the problem of planning service districts with a time constraint. One of their main contributions is the derivation of analytical expressions to determine the optimal number of service districts for the US postal system. A methodology to design multidelivery tours associated with the servicing of an urban region of irregular shape is presented in [15]. The authors’ methodology is based on a sweep approach and assumes a rectangular grid structure. Novaes et al. [18] present a methodology for solving the same problem as [19], but they used a continuous approach to represent the region. Previous works are extended by [20] introducing some improvements to the ring-radial model. The authors present a special case of a Voronoi diagram and model the problem with a continuous approximation.

Muyldermans et al. [21, 22] address the problem of districting for salt spreading operations on roads. They assume that each district is served by a single facility aiming to minimize the deadheading distances and the number of vehicles required to service the region. The authors present a novel model based on a graph with demand at the nodes. Another districting problem modelled on a graph but considering the demand in the arcs is presented in [23], in which the concept of Eulerian districts is introduced.

Haugland et al. [24] consider the problem of designing districts for vehicle routing problems with stochastic demands. They presented a Tabu Search (TS) and multistart heuristics to solve the problem. Also, a multiobjective dynamic stochastic districting and routing problem is considered in [15]. In this problem, the customers of a territory stochastically evolve over the planning periods. Therefore, several objectives must be optimized via a coevolutionary algorithm. Another nondeterministic districting problem is studied in Sheu [25], in which an integrated fuzzy-optimization customer grouping based logistics distribution methodology is presented. The proposed method groups customers’ orders primarily based on the multiple attributes of customer demands. Then, the customer group-based delivery service priority is determined, and finally the routes for deliveries are established.

Concerning multiple objectives to be considered in a districting model, Tavares-Pereira et al. [26] consider a districting problem with multiple criteria and propose an evolutionary algorithm with local search to approximate the Pareto front. Also, a problem related to a bottled beverage distribution company is studied in [27]. Hence, a biobjective programming model is introduced considering dispersion and balancing with respect to the number of customers in each district as performance criteria. Two continuous location-districting models applied to transportation and logistics problems are developed in [28]. A Voronoi diagram is combined with an optimization algorithm for solving both proposed models.

In [29, 30], the same districting design problem for the operations of a parcel company is studied. In both references, the same mathematical model is proposed but in [30] a two-stage approach is considered. The second stage is related to a mathematical model that considers the solution of the first stage and, then, aims to minimize the dispersion of the workload in the districts. Small and medium size instances are solved via a hybrid algorithm. That algorithm is unable to efficiently solve large size instances. In [31], there is a review of the state of the art regarding modelling techniques and solution methods for problems in supply chain planning that handle districting or customer clustering.

The problem herein addressed is closely related to [29, 30]. However, there are significant differences among them. Particularly, in this paper we are proposing an adjusted heuristic that is capable of solving large-scale instances with low computational effort. In previous papers, only limited size instances are solved, but herein the proposed heuristic efficiently solves larger instances. The main differences between the algorithms is that the procedure for constructing feasible solutions, the neighborhoods considered, and the movements explored during the local search are modified. The latter changes allow the heuristic to solve large size instances. Moreover, instances with different structure are tested, and the heuristic algorithm shows to be robust. Specifically, a set of symmetric instances is considered in which the optimal solution is known due to their particular structure. These instances allow properly measuring the efficiency of the proposed heuristic.

Furthermore, other important characteristics of this research are that we aim to optimize the balance of workload content and compactness in a single objective function instead of a single objective (as usually). Also, parcel applications are scarce in the literature. Hence, this research is worthy due to parcel companies handling many customers and the proposed heuristic can efficiently solve these situations. The modeling structure proposed in this work also differs from others found in the literature, mainly in the fact that the districting problem is modeled as a graph in which demand occurs at the nodes.

In our situation under study, districting is a strategic decision that is not defined day to day and demand points are not fixed. We also distinguish between pickup and delivery operations. The only reviewed paper to do so is [16], but the objective is to minimize the expected length of the tours and the workload balance is addressed as a constraint. So, it differs from the model presented in this paper, in which we aim to balance workload content and also achieve districts of compact shape.

#### 3. Mathematical Formulation

Consider a connected, undirected graph , where is the set of vertices and is the set of edges. In general, the graph is not complete. All the edges , , , have a positive length and represent a real path between adjacent vertices and . Distances between vertices are edge lengths for adjacent pairs and shortest path distances for nonadjacent pairs. A district is defined as a subset of the vertices and is the district set. Each vertex represents a customer that may require either a pickup or a delivery. The depot is defined as the vertex .

We define and as the number of pickups/deliveries requested by each demand point , ; and and as the stopping time per pickups/delivery in each demand point , . As mentioned previously, the distance between a pair of demand points is denoted as . represents the average vehicle speed. and are continuous variables that represent the maximum workload content and compactness metrics, respectively. is a continuous auxiliary variable that takes the value of the travel time from the depot to the farthest point in district , .

The districting procedure seeks to optimize two criteria simultaneously: balancing workload among the districts and compactness of their shapes. Compactness is not defined uniquely for all the districting problems in the literature and it is generally defined according to the application context. For this problem, we define it as the distance between the two furthest apart vertices in a district, that is, , considering all vertices and belonging to a district. We employ a minimax objective in which the maximum compactness metric among the districts is minimized. Although the compactness metric that we employ could have the same value for districts of different shape (e.g., a long thin district), when combined with the objective to balance workload content, in practice it produces districts of approximately circular or square shape.

For each district, the workload content is defined as the time required to perform all required pickups and deliveries and also includes an estimation of the time required to travel from the depot to a vertex in the district. The latter is approximated by the time required to travel from the depot to the farthest point in the district. Despite the fact that the closest vertex in the district or the centroid could also be employed to approximate the line haul distance from the depot, considering that the vehicle will visit all vertices in a district during the route, the farthest vertex is a reasonable metric. Additionally, the use of the farthest vertex simplifies the mathematical model. In order to balance the workload content among districts, the maximum workload among all districts is minimized. The following expression shows how the workload of a district is calculated:We propose a linear single objective mixed integer model in which a weighted sum of the compactness metric and the maximum workload content assigned to a district is minimized. Each metric is normalized with respect to an estimation of the optimal values of compactness and workload content ( and , resp.).

Since a feasible number of vehicles considered in the model are known a priori, the capacity of the vehicles is not explicitly considered in the model. To ensure that a feasible solution with respect to vehicle capacity is found, we balance district workloads and we keep the number of pickups and deliveries within certain limits. These limits will not allow pickups and deliveries to be intermixed, and this will release vehicle capacity by deliveries to allow subsequent pickups. The limits on the number of pickups and deliveries for each district are denoted by and , respectively.

The following decision variables are defined:The mathematical model is as follows:Equation (3) is the objective function that minimizes a weighted average of the maximum workload and compactness. Both terms in (3) are normalized and the relative weighting is given by , . Normalization parameters are estimated based on their optimal solution values as follows: is determined by estimating an average workload per district when it is approximately balanced by dividing the total workload of the region. is estimated as the traveling time between the two furthest points in a district, assuming that the service region is equally divided into districts of equal shape (assuming a circular shape for normalization parameters estimation). Constraints (4) guarantee that each demand point is assigned to exactly one district. Constraints (5) and (6) ensure that each district has a maximum number of pickups and deliveries, respectively. Constraints (7) guarantee that takes the value of the time from the depot to the farthest vertex of each district . Constraints (8) require that take the value of the maximum travel time between the vertices assigned to a district. Constraints (9) guarantee that takes the value of the maximum workload from among the districts. Constraints (10) are the binary and nonnegativity requirements.

#### 4. Solution Methodology

The redistricting problem has been shown to be NP-Complete by Altman (1997). Moreover, since constraints (5) and (6) impose a limit in the amount of pickups and deliveries assigned to a district, it turns out that even finding a feasible solution is a difficult task. Motivated by the difficulty of the problem, we propose a multistart heuristic that hybridizes GRASP and Tabu Search. The procedure seeks a feasible solution (partition of the customers into districts) of the model proposed in Section 3 aiming to obtain good partition quality in terms of the values of the objective function in which the weighted sum of compactness and work balance metrics are optimized. We will further describe the steps of the algorithm.

The general procedure consists of two phases: construction of a feasible initial partition and improvement by a local search. These phases are named as FIP-Construction and Local-Search, respectively. In the first phase, districts are conformed by partitioning all the customers into the districts in set . Hence, a solution of the proposed hybrid algorithm corresponds to a districting decision. The Local-Search procedure incorporates a Tabu Search (TS) short term memory, in which the recently visited partitions are labeled as Tabu active during a predefined number of iterations. The aspiration criterion allows a Tabu active move only if the resulting partition is better than the current best one. Among all the partitions created and improved, the overall best one is reported as the final output of the algorithm.

We propose two local search procedures that may be applied independently or in two different combinations. This results in four strategies that attempt to improve the current partition (set of districts) constructed during each iteration of the algorithm. A key concept is the* adjacency* of vertices and districts. A vertex is considered to be adjacent to a district if there exists at least one edge connecting the vertex with another one that already belongs to the district. Then, we restrict the allocation of the vertices only to its adjacent districts. Knowledge of the adjacency helps to avoid unnecessary evaluations that may result in long computational times in the local search and also to enhance compactness of the constructed partition.

The encoding of a partition is as follows: there is a list associated with each district, in which the index of the customers associated with it are included. Moreover, the quality of each partition will be measured as the weighted sum of both objectives considered in (3). Algorithm 1 shows the pseudocode of the procedure in which represents the best value of the objective function given by (3) and is the corresponding best partition found. The term* seed* will be used hereafter to denote a vertex chosen to initiate the construction of a district.* NSeeds* is the number of seed selection procedures used in the FIP-Construction method. We employ five different procedures to select a set of seeds. The procedures are used in sequential order, , during the iterations of the hybrid algorithm. In each iteration a feasible partition is attempted to be constructed and, if found, improved.