A rich vehicle routing problem arising in the replenishment of automated teller machines

Article History: Received 29 December 2017 Accepted 26 July 2018 Available 31 July 2018 This paper introduces, models, and solves a rich vehicle routing problem (VRP) motivated by the case study of replenishment of automated teller machines (ATMs) in Turkey. In this practical problem, commodities can be taken from the depot, as well as from the branches to efficiently manage the inventory shortages at ATMs. This rich VRP variant concerns with the joint multiple depots, pickup and delivery, multi-trip, and homogeneous fixed vehicle fleet. We first mathematically formulate the problem as a mixed-integer linear programming model. We then apply a Geographic Information System (GIS)-based solution method, which uses a tabu search heuristic optimization method, to a real dataset of one of the major bank. Our numerical results show that we are able to obtain solutions within reasonable solution time for this new and challenging practical problem. The paper presents computational and managerial results by analyzing the trade-offs between various constraints.


Introduction
In logistics operations, fulfilling consumer demands for diverse and premium products is an important challenge [1].The classical vehicle routing problem (VRP) aims to determine an optimal routing plan for a fleet of homogeneous vehicles to serve a set of customers, such that each vehicle route starts and ends at the depot, each customer is visited once by one vehicle, and some side constraints are satisfied.Many variants and extensions of the VRP have intensively studied in the literature.For further details about the VRP and its variants, we refer the reader to Laporte [2] and Toth and Vigo [3].
Over the last years, several variants of multiconstrained VRPs have been studied, forming a class of problems known as Rich VRPs.Lahyani et al. [4] presented a comprehensive and relevant taxonomy for the literature devoted to Rich VRPs.The authors have investigated 41 articles devoted to rich VRPs in detail, and developed an elaborate definition of RVRPs.
Karaoglan et al. [5] studied aircraft routing and scheduling for cargo transportation in an airline company in Turkey.Karagul and Gungor [6] studied the mixed fleet VRP to optimize the distribution of the tourists who have traveled between the airport and the hotels in Turkey.The authors developed a Savings algorithm, a Sweep algorithm and a random permutation alignment.Furthermore, a genetic algorihm and random search algorithms algorithms are also developed.Van Anholt et al. [7] introduced, modeled, and solved a rich multiperiod inventory-routing problem with pickups and deliveries motivated by the replenishment of automated teller machines (ATMs) in *Corresponding Author the Netherlands.The authors first decomposed the problem into several more manageable subproblems by means of a clustering procedure, and they simplified the subproblems by fixing some variables.Valid inequalities are then generated to strength the resulting subproblems.An efficient branch-and-cut algorithm is then developed.Karagul et al. [8] used 2-Opt based evolution strategy for travelling salesman problem.
A variant of the VRP known as multiple depots VRP, in which more than one depot is considered, studied by many researchers.Crevier et al. [9] considered the multiple depots VRP with interdepot routes.Braekers et al. [10] proposed exact and meta-heuristic approach for a general heterogeneous dial-a-ride problem with multiple depots.Contardo and Martinelli [11] developed a new exact algorithm for the multiple depots VRP under capacity and route length constraints.For further details about the multiple depots VRP and its variants, we refer the reader to the review paper of Montoya et al. [12].Another interesting variant is multi-trip VRP, in which vehicles can perform several trips per day, because of their limited number and capacity [13][14][15].
An important family of routing problems is pickup-and-delivery problems (PDPs) in which goods have to be transported from different origins to different destinations.In one-to-one variant of PDPs, each customer demand consists of transporting a load from one pickup node to one destination node.Many exact and heuristic methods are developed for this problem variant which is usually referred to as the pickup-and-delivery VRP.Xu et al. [16] studied a rich PDP with many side constraints.Sigurd et al. [17] considered the transportation of live animals.For further details about the PDPs and its variants, we refer the reader to Battarra et al. [18], Berbeglia et al. [19], Koç and Laporte [20], and Parragh et al. [21,22].
In recent years, Geographical Information System (GIS)-based solution methods used to solve several optimization problems.Casas et al. [23] developed an automated network generation procedure for routing of unmanned aerial vehicles in a GIS environment.Bozkaya et al. [24] studied the competitive multi-facility location-routing problem and presented a hybrid heuristic algorithm.The method is applied on a case study arising at a supermarket store chain in the city of Istanbul.The authors used genetic algorithm for the location part, and tabu search of GIS-based solution method for the VRP part.Samanlioglu [25] developed a multi-objective location-routing problem and described a mathematical model.The author used a GIS software to obtain the data related to the Marmara region of Turkey.Yanik et al. [26] considered the capacitated VRP with multiple pickup, single delivery and time windows, and proposed a hybrid metaheuristic approach.The method integrates a genetic algorithm for vendor selection and allocation, and a GIS-based solution method which uses a modified savings algorithm for the routing part.Krichen et al. [27] studied the VRP with loading and distance constraints and used a GIS solution method to solve the problem.Our study is motivated by the problem faced by one of the major bank of Turkey operating in the city of Gaziantep.We consider a multi-depot, multi-trip, pick-up and delivery with homogenous vehicle fleet.We first define this new problem and presented a mathematical formulation.We then used a GIS-based solution approach employs a tabu search algorithm which can be used to store, analyze and visualize all data as well as model solutions in geographic format.We considered a real dataset of the bank and applied our GIS-based solution approach.We finally provide several managerial and policy insights by exploring the trade-offs between various constraints.The remainder of this paper is structured as follows.Section 2 presents the problem definition and mathematical formulation.Section 3 describes the solution approach.Section 4 presents a case study with input data.Section 5 presents the solutions we propose.Finally, Section 6 provides our conclusions.

Problem definition and mathematical formulation
The problem is defined on a complete directed graph G = (N , A).The location of each ATM, branch, and the district office is represented by a node.N = {0} ∪ N b ∪ N c is a set of nodes in which "{0}" is the district office node, N b is a set of branch nodes, and N c is a set of ATM nodes.
Each ATM i ∈ N c has a demand q i and a service time p i .A fixed number of limited homogeneous vehicle fleet m is available.The index set of routes is denoted by R = {1, . . ., r, . ..}.
The capacity of a vehicle is denoted by Q.The maximum allowed working duration is T max for each vehicle.We use the real network distances when we computing the d ij values on each arc (i, j) ∈ A. Therefore, it is possible that i.e., asymmetric, which are illustrated in Figure 1.To formulate the problem, we define the following decision variables.Let x ijr be equal to 1 if a vehicle travels directly from node i to node j on route r ∈ R. Let f ijr be the amount of commodity flowing on arc (i, j) ∈ A by a vehicle on route r ∈ R.
In our problem, one considers a homogeneous fixed fleet of vehicles, as well as a set of ATMs with known demands.The demand of each ATM is expressed as money tray and each vehicle are designed to satisfy these specific ATM demands.
The bank has variable number of orders from ATMs which can be fulfilled by both district office and several branches.The journey of a vehicle which carries demanded money starts from district office and it begins to visit ATMs to load ordered money.If the money of a vehicle is finished before meeting the demand of ATMs, vehicle has two options.While the first option is to go back to district office, the second option is to go a branch to get money.Vehicles can perform several tours per day because of their limited number and capacity.The objective is to minimise the total en-route time of vehicles.Due to the minimization of total en-route time time in this problem, the vehicle visits district office or branch which is closer to it.This rich VRP variant is concerned with the joint multiple depots, pickup and delivery problems, multi-trip, and homogeneous fixed vehicle fleet.The mathematical formulation of the problem is given as follows: Minimize j∈N r∈R i∈N r∈R j∈N r∈R (i,j)∈A r∈R x ijr ∈ {0, 1} (i, j) ∈ A, r ∈ R (9) The objective function (1) minimizes the total enroute time.Constraints (2) bounds the number of vehicles.Constraints (3) and (4) ensure that each customer is visited exactly once.Constraints (5) impose that a vehicle cannot start route r + 1 before finish route r.Constraints ( 6) and ( 7) define the flows.Constraints (8) ensure that the total travel time cannot exceed the maximum allowed working duration.Finally, constraints ( 9) and ( 10) enforce the integrality and nonnegativity restrictions on the variables.

Solution approach
The mathematical formulation of the problem is a member of a rich VRP family [4], which is hard to solve optimally as it requires the joint solution several difficult subproblems.To overcome this barrier, we now present a GIS-based solution approach.
In practice, there are several commercial programs are available to solve the VRP and its extensive variations.GIS is a kind of system that provides spatial analyses and supports the decision-making activities by using various geographic data.It can also support logistic and marketing managers to evaluate placement options.Thus, GIS is used for replenishment of automated teller machines in Gaziantep [28].We used the ArcGIS 10.2 commercial package to solve our optimization problem and also for building our GIS-based decision support framework.The ArcGIS is frequently used in many broad areas where spatially-enabled data need to be stored, retrieved, analyzed, visualized and even served online [29].The ArcGIS is first used as a platform to store all problem data in geographic format.It visualizes all data as well as the solutions we obtain through our heuristic approach.
The software platform commercial package Ar-cGIS uses a tabu search heuristic algorithm to solve our defined problem.The solution method follows the classical tabu search principles such as non-improving solutions are accepted along the way.However, cycling of solutions are avoided using tabu lists and tabu tenure parameters [30].
In the last decades, tabu search heuristics are commonly used in VRP and its variants.It obtains quite competetive solutions and it is still an highly effective heuristic method [31][32][33].Initialization phase creates an origin-destination matrix of shortest travel costs between all locations that must be visited by a route.A feasible initial solution is then generated by inserting each location one at a time into the most suitable route.The improvement phase aims to obtain high quality solution by applying the following procedures.
• Changing the sequence nodes on a single route.
• Moving a single node from its current route to a better route.• Swapping two nodes between their respective routes.
Figure 2 shows the framework of the system proposed in a form of a diagram.

A case study
We now present a case study arising in one of the major bank operating in Turkey.The considered bank group is an integrated financial services group operating in every segment of the banking sector including corporate, commercial, small and medium-sized enterprises, payment systems, retail, private and investment banking together with its subsidiaries in pension and life insurance, leasing, factoring, brokerage, and asset management.As of September 2017, the bank group provides a wide range of financial services to its tens of million customers through an extensive distribution network of 942 domestic branches with 4,769 ATMs.
To manage the money flow between branches and ATMs, the bank group aims to speed up decisionmaking and implementation processes by establishing a well-designed logistic network.To do so, one district office, 12 branches and 53 ATMs which are located in the city of Gaziantep are considered to be designed.The locations of the ATMs, and district office and branches are illustrated in Figures 3 and 4, respectively.
Gaziantep with its 1,975,302 population in 2016 is the 8th most crowded city of Turkey and it is an important commercial and industrial center for Turkey.The considered stores are located in two districts which cover 85% of total population of Gaziantep.In total, there are 5 benchmark instances, i.e., GB-1, GB-2, GB-3, GB-4, and GB-5, which include all ATMs with different demands range from 5 to 45 money trays.Solving a network analysis problem in ArcGIS software, several parameters shown below have to be utilized in our study.Figure 5 shows an example of the user interface of ArcGIS of parameter entry.Table 1 presents the detailed information about benchmark instances.The first column shows the ATM number, while others present the daily demand.
• Vehicle number : Bank group has four vehicles.

Computational experiments and analyses
This section presents the results of the computational experiments.All experiments were conducted on a server with an Intel Core i7 CPU 3.07   We then present the results of the considered problem in the second part of the table .In each table, the first column shows the vehicle and its tour number.For example, "1/1" indicates that the first vehicle's first tour.The second and third columns show the start and end nodes of the vehicle tour, respectively.The other columns show the total number of orders, total  travel time (seconds), the total distance (km), total service time (seconds), and total en-route time (seconds), respectively.
Solution times for each instance are less than two seconds.Table 2 shows that all vehicles are used once.Tables 3, 4, and 5 show that vehicle 1 used two times, but vehicles 2, 3, and 4 only used once..01seconds for GB-1, GB-2, GB-3, GB-4, and GB-5, respectively.
When we compared our results with current one used by the bank for its daily operation, our results provided better solutions.On average, in terms of total distance, total travel times, and total en-route times, our method obtained 9.52%, 10.51%, and 10.65% better solutions.These results show that total distances are reduced when we relax each vehicle route starts and ends at the depot constraint.Similarly, total travel times and total en-route times are also reduced.Our results indicate that four vehicle are enough for satisfying ATM demands, and in general more than one vehicle tour is not necessary.

Conclusions
This paper has been motivated by the problem faced by one of the major banks of Turkey operating in the city of Gaziantep.We have defined a new rich vehicle routing problem which is concerned with the joint multiple depots, pickup and delivery, multi-trip, and homogeneous fixed vehicle fleet.We have presented a mathematical formulation for the problem.To obtain fast and good quality solutions, we have then used a GISbased solution approach employs a tabu search algorithm which can be used to store, analyze and visualize all data as well as model solutions in geographic format.We have considered a real dataset of the bank and have applied our GIS-based solution approach.We have finally provided several managerial and policy insights on results by exploring the problem.
Our results indicated that four vehicles are enough to satisfy the demand of ATMs for the bank and one vehicle tour is also enough for each vehicle in general.We have also shown that total distances are reduced if we do not consider each vehicle route starts and ends at the depot constraint.In a similar manner, total travel times and total en-route times are also reduced.Furthermore, the running times of the algorithm are so small that it can be used in practical bank operations.
For future studies, stochasticity and dynamism can be taken account in the problem definition, instead of using deterministic parameters.This would require new mathematical models and solution algorithms, such as stochastic optimization.Furthermore, new effective exact methods can be developed, such as Lagrangean relaxation to obtain lower bounds, or decomposition techniques to solve large size benchmark instances to optimality.

Figure 1 .
Figure 1.An example of asymmetric case from ATM 13 to 14, and from ATM 14 to 13.

Figure 2 .
Figure 2. The framework of the system proposed in a form of a diagram.

Figure 4 .
Figure 4. Locations of district office and branches.

Figure 5 .
Figure 5.An example of the user interface of ArcGIS of parameter entry.

Table 1 .
The detailed information about benchmark instances.

Table 2 .
The detailed results of instance GB-1.

Table 3 .
The detailed results of instance GB-2.

Table 4 .
The detailed results of instance GB-3.

Table 5 .
The detailed results of instance GB-4.

Table 6 .
The detailed results of instance GB-5.