A Location-Arc Routing Problem (LARP) is a practical problem, while a few mathematical programming models have been considered for this problem. In this paper, a mixed non-linear programming model is presented for a multi-period LARP with the time windows under demand uncertainty. The time windows modeling in the arc routing problem is rarely. To the best our knowledge, it is the first time that the robust LARP model is verified and an optimal solution is presented for it. For this purpose, the CPLEX solver is used for solving cash supply problems of a bank as a case study. These problems are discrete and large-size. Therefore, the method LRP and LARP models can be used to solve these problems. The comparing results of the LRP and LARP models prove that the LARP has a better performance regarding timing and optimal solution. Furthermore, comparing the results of deterministic and robust LARP models for this case study shows the validity of the robust optimization approach.

Keywords: Location-arc routing problem; Time windows; Multiple periods; Robust optimization; Demand Uncertainty.Introduction

Routing problems had two major categories:

the node routing problem that customers are located on the node

the arc routing problems that customers are located on the arc or edge of a graph (Laporte, Nickel, & da Gama, 2015).

The Vehicle Routing Problem (VRP) and Traveling Salesman Problem (TSP) are node routing problem. The Rural Postman Problem (RPP) and Chinese Postman Problem (CPP) are arc routing problems. In a CPP, all arcs of a graph are served while in an RPP only a subset of arcs or edges is required to be served (Lopes, Plastria, Ferreira, & Santos, 2014). The Capacitated Arc Routing Problem (CARP) was introduced by (Golden & Wong, 1981) and is an expansion of the RPP; however, it is a complex problem because of considering multiple vehicles with capacity constraints. In CARP the customer demands are on arcs (edges) instead of nodes and the goal is to reduce the traveling costs of all links.

A Location-Arc Routing Problem (LARP) is a combination of two well-known problems:

CARP

Location-Allocation Problem (LAP) (Ghiani, Laganá, Laporte, & Musmanno, 2007).

The LAPR is similar to the CARP. The only major difference between LARP and CARP is that besides routing, the best locations for depots are determined. Therefore, a binary decision variable is added to the mathematical model that has made it even more complicated. As far as we know, only is presented six mathematical models for LARP until now. These models are not verified and are solved with heuristic or metaheuristic method. For example, in the model of (Lopes et al., 2014) each customer has required one vehicle, and this is not practical. Also, (Huber, 2016) formulate two objectives functions and don’t present constraints. Therefore, in this paper presents a new multi-period LARP under uncertain demand.

The constraints of time windows, time limitation for vehicles, and using the vehicle for some customer are added to the model. The time windows modeling in arc routing problem has a different structure than the node routing problem, since a required arc with a time window after servicing, can be used as a deadheading arc without a time window as a route. (Çetinkaya, Karaoglan, & Gökçen, 2013) claim it is not possible to modeling time windows in arc routing problem and should be transform it’s to node routing problem. (Macedo, Alves, de Carvalho, Clautiaux, & Hanafi, 2011) is not modeled time window and solving with metaheuristic methide. (Lystlund & Wøhlk, 2012) proposed a mathematical model for this purpose and we inspiration it. The model is linearized and solved with CPLEX solver in GAMS. For uncertainty is used Bertsimas method.

The rest of the paper is structured as follows. Section 2 is reviewed the CARP and LARP. In Section 3, the robust deterministic linear model is presented. The new model is validated by comparing the results of LARP and LRP for solving a bank case study. The validation of the robust model is also performed in Section 4. The final conclusions are presented in Section 5.

Literature review

There are a few articles in the LARP and proximity with the CARP, in which this section reviews some studies on the CARP and the LARP.

CARP review

The review of the CARP from 2000 to 2018 as summarized in Tab. 1. The most studies have focused on solving basic CARP under deterministic conditions. For an uncertain condition, only (Fleury, Lacomme, Prins, & Ramdane-Chérif, 2005) used a stochastic method for the CARP under uncertainty. Also, (Fleury, Lacomme, Prins, & Ramdane-Cherif, 2005) and (Mei, Tang, & Yao, 2010) applied a robust approach. In most studies, the LARP model has been solved using heuristic or meta-heuristic methods. Other studies proposing solutions based on exact methods have applied either some techniques for transforming the CARP to the CVRP or a heuristic to build an initial solution for decision variables. Furthermore, the CARP with time window is rarely and all of the research are presented a nonlinear mathematical model except the model (Lystlund & Wøhlk, 2012). All of them have not verifed proposed model and not presented exact solution.

Although a few studies in the literature review have focused on the LARP, due to the practical applications and complexity of this problem, three review papers can be found on this issue. (Ghiani et al., 2007) surveyed three main applications of the LARP, namely, postal delivery, garbage collection and road maintenance, and also algorithms used to solve this problem. (M. Liu, Singh, & Ray, 2014) presented the LARP review and identified the research gap on this problem. Moreover, (Laporte et al., 2015) surveyed all studies on the LARP and emphasized on developing two presented models (Doulabi & Seifi, 2013; Lopes et al., 2014) and solved this problem with the exact method.

The first research on the LARP was carried out by (Levy & Bodin, 1989) for solving the routing problem at the post office in the USA. They used a Location-Allocation-Routing (L-A-P) method to solve the model. Based on this method, firstly, the depot location is determined and then required arcs be allocated depots. Finally, required arcs can be solved a Vehicle Routing Problem (VRP). In another method, called Allocate-Routing-Location, required arcs are first allocated to the depots, and then the depot location is determined based on the selected routes. According to the reports, this method has been more efficient than the previous methods (Ghiani et al., 2007). (Ghiani & Laporte, 2001) presented a linear multi-depot model for the LARP. Then, they transformed this model into the RPP and used a branch-and-cut algorithm for solving their model.

Only six studies have proposed a mathematical model for the LARP, and all have used a meta-heuristic algorithm for solving the deterministic models (Tab. 2). (Doulabi & Seifi, 2013) presented two Mixed-Integer Linear Programming (MILP) models considering the flow variables for single and multi-depot problems. They proposed a Simulated Annealing (SA) algorithm, which used an allocation-routing-location method at each iteration. It first builds a routing solution and then improves the depot locations. (Lopes et al., 2014) proposed a mathematical model and solved it by several heuristics. They tested different constructive heuristics combining Variable Neighborhood Search (VNS), Greedy Randomized Adaptive Search Procedure (GRASP) and Tabu Search (TS).

According to their results, the combination of TS and GRASP is the best case. The disadvantage of this model is that Constraint (7) in this paper implies that each customer is required an individual vehicle and this is not practical. The mathematical model in (Essink & Wagelmans, 2015) is the same as (Lopes et al., 2014); however, they are used hybrid TS-GRASP. (Riquelme-Rodríguez, Gamache, & Langevin, 2016) compared two methods for locating depots in the network. They proposed a non-linear model for a periodic LARP and used a heuristic method for solving this model. (Huber, 2016) presented a model with two objectives functions without formulating constraints. He using heuristic for solving benchmark instances. (Amini, Tavakkoli-Moghaddam et al. 2017) addressed an uncertain LARP and employing two scenario-based approaches. Performance of scenario-based models is evaluated with results of the numerical example.

Keywords: Location-arc routing problem; Time windows; Multiple periods; Robust optimization; Demand Uncertainty.Introduction

Routing problems had two major categories:

the node routing problem that customers are located on the node

the arc routing problems that customers are located on the arc or edge of a graph (Laporte, Nickel, & da Gama, 2015).

The Vehicle Routing Problem (VRP) and Traveling Salesman Problem (TSP) are node routing problem. The Rural Postman Problem (RPP) and Chinese Postman Problem (CPP) are arc routing problems. In a CPP, all arcs of a graph are served while in an RPP only a subset of arcs or edges is required to be served (Lopes, Plastria, Ferreira, & Santos, 2014). The Capacitated Arc Routing Problem (CARP) was introduced by (Golden & Wong, 1981) and is an expansion of the RPP; however, it is a complex problem because of considering multiple vehicles with capacity constraints. In CARP the customer demands are on arcs (edges) instead of nodes and the goal is to reduce the traveling costs of all links.

A Location-Arc Routing Problem (LARP) is a combination of two well-known problems:

CARP

Location-Allocation Problem (LAP) (Ghiani, Laganá, Laporte, & Musmanno, 2007).

The LAPR is similar to the CARP. The only major difference between LARP and CARP is that besides routing, the best locations for depots are determined. Therefore, a binary decision variable is added to the mathematical model that has made it even more complicated. As far as we know, only is presented six mathematical models for LARP until now. These models are not verified and are solved with heuristic or metaheuristic method. For example, in the model of (Lopes et al., 2014) each customer has required one vehicle, and this is not practical. Also, (Huber, 2016) formulate two objectives functions and don’t present constraints. Therefore, in this paper presents a new multi-period LARP under uncertain demand.

The constraints of time windows, time limitation for vehicles, and using the vehicle for some customer are added to the model. The time windows modeling in arc routing problem has a different structure than the node routing problem, since a required arc with a time window after servicing, can be used as a deadheading arc without a time window as a route. (Çetinkaya, Karaoglan, & Gökçen, 2013) claim it is not possible to modeling time windows in arc routing problem and should be transform it’s to node routing problem. (Macedo, Alves, de Carvalho, Clautiaux, & Hanafi, 2011) is not modeled time window and solving with metaheuristic methide. (Lystlund & Wøhlk, 2012) proposed a mathematical model for this purpose and we inspiration it. The model is linearized and solved with CPLEX solver in GAMS. For uncertainty is used Bertsimas method.

The rest of the paper is structured as follows. Section 2 is reviewed the CARP and LARP. In Section 3, the robust deterministic linear model is presented. The new model is validated by comparing the results of LARP and LRP for solving a bank case study. The validation of the robust model is also performed in Section 4. The final conclusions are presented in Section 5.

Literature review

There are a few articles in the LARP and proximity with the CARP, in which this section reviews some studies on the CARP and the LARP.

CARP review

The review of the CARP from 2000 to 2018 as summarized in Tab. 1. The most studies have focused on solving basic CARP under deterministic conditions. For an uncertain condition, only (Fleury, Lacomme, Prins, & Ramdane-Chérif, 2005) used a stochastic method for the CARP under uncertainty. Also, (Fleury, Lacomme, Prins, & Ramdane-Cherif, 2005) and (Mei, Tang, & Yao, 2010) applied a robust approach. In most studies, the LARP model has been solved using heuristic or meta-heuristic methods. Other studies proposing solutions based on exact methods have applied either some techniques for transforming the CARP to the CVRP or a heuristic to build an initial solution for decision variables. Furthermore, the CARP with time window is rarely and all of the research are presented a nonlinear mathematical model except the model (Lystlund & Wøhlk, 2012). All of them have not verifed proposed model and not presented exact solution.

Although a few studies in the literature review have focused on the LARP, due to the practical applications and complexity of this problem, three review papers can be found on this issue. (Ghiani et al., 2007) surveyed three main applications of the LARP, namely, postal delivery, garbage collection and road maintenance, and also algorithms used to solve this problem. (M. Liu, Singh, & Ray, 2014) presented the LARP review and identified the research gap on this problem. Moreover, (Laporte et al., 2015) surveyed all studies on the LARP and emphasized on developing two presented models (Doulabi & Seifi, 2013; Lopes et al., 2014) and solved this problem with the exact method.

The first research on the LARP was carried out by (Levy & Bodin, 1989) for solving the routing problem at the post office in the USA. They used a Location-Allocation-Routing (L-A-P) method to solve the model. Based on this method, firstly, the depot location is determined and then required arcs be allocated depots. Finally, required arcs can be solved a Vehicle Routing Problem (VRP). In another method, called Allocate-Routing-Location, required arcs are first allocated to the depots, and then the depot location is determined based on the selected routes. According to the reports, this method has been more efficient than the previous methods (Ghiani et al., 2007). (Ghiani & Laporte, 2001) presented a linear multi-depot model for the LARP. Then, they transformed this model into the RPP and used a branch-and-cut algorithm for solving their model.

Only six studies have proposed a mathematical model for the LARP, and all have used a meta-heuristic algorithm for solving the deterministic models (Tab. 2). (Doulabi & Seifi, 2013) presented two Mixed-Integer Linear Programming (MILP) models considering the flow variables for single and multi-depot problems. They proposed a Simulated Annealing (SA) algorithm, which used an allocation-routing-location method at each iteration. It first builds a routing solution and then improves the depot locations. (Lopes et al., 2014) proposed a mathematical model and solved it by several heuristics. They tested different constructive heuristics combining Variable Neighborhood Search (VNS), Greedy Randomized Adaptive Search Procedure (GRASP) and Tabu Search (TS).

According to their results, the combination of TS and GRASP is the best case. The disadvantage of this model is that Constraint (7) in this paper implies that each customer is required an individual vehicle and this is not practical. The mathematical model in (Essink & Wagelmans, 2015) is the same as (Lopes et al., 2014); however, they are used hybrid TS-GRASP. (Riquelme-Rodríguez, Gamache, & Langevin, 2016) compared two methods for locating depots in the network. They proposed a non-linear model for a periodic LARP and used a heuristic method for solving this model. (Huber, 2016) presented a model with two objectives functions without formulating constraints. He using heuristic for solving benchmark instances. (Amini, Tavakkoli-Moghaddam et al. 2017) addressed an uncertain LARP and employing two scenario-based approaches. Performance of scenario-based models is evaluated with results of the numerical example.