A robust optimization approach for the road network daily maintenance routing problem with uncertain service time

A robust optimization approach for the road network daily maintenance routing problem with uncertain service time

Transportation Research Part E 85 (2016) 40–51 Contents lists available at ScienceDirect Transportation Research Part E journal homepage: www.elsevi...

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Transportation Research Part E 85 (2016) 40–51

Contents lists available at ScienceDirect

Transportation Research Part E journal homepage: www.elsevier.com/locate/tre

A robust optimization approach for the road network daily maintenance routing problem with uncertain service time Lu Chen a,⇑, Michel Gendreau b, Minh Hoàng Hà b, André Langevin b a b

School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China Département de Mathématiques et de Génie Industriel, École Polytechnique de Montréal and CIRRELT, H3C 3A7, Canada

a r t i c l e

i n f o

Article history: Received 1 April 2015 Received in revised form 19 October 2015 Accepted 16 November 2015

Keywords: Arc routing problem Robust optimization Uncertainty Chance-constrained programming Branch-and-cut Monte Carlo simulation

a b s t r a c t This paper studies the robust optimization approach for the routing problem encountered in daily maintenance operations of a road network. The uncertainty of service time is considered. The robust optimization approach yields routes that minimize total cost while being less sensitive to substantial deviations of service times. A robust optimization model is developed and solved by the branch-and-cut method. In computational experiments, the behavior of the robust solutions and their performance are analyzed using Monte Carlo simulation. The robust optimization model is also compared with a classic chanceconstrained programming model. The experimental analysis provides managerial insights for decision makers to determine an appropriate routing strategy. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction The problem studied in this paper concerns the routing problem faced by a road network monitoring service, where each day a part of the road network needs to be monitored by a fleet of vehicles. Monitoring operations include visually checking the operational status of each road segment, evaluating the function of the auxiliary facilities, reporting the defects of the road and so on. An estimated service time is associated with each request for monitoring service on a road segment. A travel time is associated with each segment of road. The problem consists of determining a set of minimum cost monitoring routes where each required segment of road is serviced on one of the routes. In the case of road monitoring, the estimated service time can be radically different from the actual service time, due to road conditions, accidents, technician’s skills, and so on. This uncertainty on one road segment can result in a large delay for road segments scheduled later for the same monitoring vehicle. Therefore, it becomes important to construct vehicle routing strategies that will be efficient in presence of uncertainty in service time. This work is motivated by a real transportation service application where each day a network of high-speed freeways needs to be monitored by vehicles from different maintenance stations in a large urban setting. In this application, about 50% of the monitoring requests have estimates for service time that significantly differ from actual monitoring times. The problem has been addressed by Chen et al. (2014) as a variant of the Capacitated Arc Routing Problem (CARP), in which strong assumptions were made regarding the distribution of service times. Chen et al. (2014) focus on optimizing expected total service cost and analyze the performance of a routing policy based on expected objective value. In this paper, we use the robust optimization methodology to formulate the problem, which allows us to consider the risks associated with extreme

⇑ Corresponding author. http://dx.doi.org/10.1016/j.tre.2015.11.006 1366-5545/Ó 2015 Elsevier Ltd. All rights reserved.

L. Chen et al. / Transportation Research Part E 85 (2016) 40–51

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outcomes. The objective is to obtain a solution that optimizes the worst-case value over all data uncertainty while not being overly conservative. We also present a comparison with a chance-constrained method to illustrate the differences with the robust optimization approach. The differences are important in determining which strategy is better to apply under a specific situation. This paper is organized as follows. Section 2 presents a brief literature review. The robust formulation of the problem is described in Section 3. A branch-and-cut algorithm is presented in Section 4 to solve the robust optimization model optimally. Section 5 reports experimental results. Finally, Section 6 concludes the paper and proposes future research directions.

2. Literature review We review in the three following subsections respectively the works on the Arc Routing Problem (ARP) with stochastic parameters, on the VRP with stochastic times, and on robust optimization. 2.1. ARP with stochastic parameters Despite numerous publications dealing with efficient scheduling methods for the ARP, very few addressed the inherent stochastic and dynamic nature of problem parameters. Fleury et al. (2004) consider the CARP with stochastic demands. A memetic algorithm is adapted to handle the randomness of the demands. Fleury et al. (2005) evaluate the robustness of CARP solutions against demand fluctuations and examine how this robustness can be improved. Christiansen et al. (2009) address the CARP with stochastic demands that follow a Poisson distribution. The objective is to find a collection of routes with minimum expected cost. The problem is solved exactly by a Branch-and-Price algorithm, and the stochastic nature of the demand is incorporated into the pricing problem. Laporte et al. (2010) study the CARP with stochastic demands in the context of garbage collection. An adaptive large-scale neighborhood search heuristic is developed to construct a solution that takes into account the expected cost of recourse. Tagmouti et al. (2011) study a dynamic variant of the ARP, in which the service cost is time dependent. The problem is motivated from winter road gritting service. A variable neighborhood descent (VND) heuristics is developed and adapted to the dynamic situation, where the service time functions on the required arcs are updated according to weather reports. 2.2. VRP with stochastic times To the best of our knowledge, no work dealing with the ARP with stochastic travel times has appeared in the literature. There exist, however, some papers that address the Vehicle Routing Problem with Stochastic Travel Times (VRP-STT), the node routing counterpart of the arc routing problem with stochastic travel times. Gendreau et al. (forthcoming) survey the literature on VRP with stochastic elements, namely stochastic demands, stochastic customers, and stochastic service and travel times. Some papers treat the travel time as a random variable that follows a probability distribution. Kao (1978) proposes two solution approaches, based on dynamic programming and implicit enumeration, to solve the travelling salesman problem with stochastic travel times, a special case of the VRP-STT. The objective is to find a tour that has the greatest probability of completion by a specified deadline C. In a subsequent study by Sniedovich (1981), the dynamic programming approach is shown to be effective for obtaining close-to-optimal solutions when the problem is monotonic. Laporte et al. (1992) consider the VRP with stochastic service and travel times, in which vehicles incur a penalty proportional to the duration of their route in excess of a preset constant. A chance-constrained programming model and two different recourse strategies are developed. Kenyon and Morton (2003) study the same problem as Laporte et al. (1992). They provide conditions under which the stochastic problem reduces to a simpler deterministic problem and give bounds on the optimal solution. A branch-andcut algorithm is embedded within a Monte Carlo solution procedure for solving the VRP with stochastic travel and service times. Li et al. (2010) study the VRP with stochastic travel times and time windows. It is assumed in their work that the travel time is a continuous random variable with normal distribution. A chance-constrained programming model and a stochastic programming model with recourse are proposed to formulate the problem. Zhang et al. (2012) assume that the travel time on each arc follows a normal distribution, and formulate the problem as a chance-constrained programming model. A scatter search approach and a genetic algorithm are developed to solve this model. Jaillet et al. (2013) study the routing problems on networks with deadlines imposed on some nodes. Uncertain travel times are considered and characterized by exact distributions, or by a distributional uncertainty set incorporating ambiguity. Algorithms with decomposition techniques are developed to find optimal routing policies such that arrival times at nodes respect deadlines ‘‘as much as possible”. Queuing approaches have also been applied to model the stochastic travel time or travel speed of vehicles. Van Woensel et al. (2008) introduce the traffic congestion component in the standard VRP based on queuing theory. Results show that explicitly taking into account congestion during the optimization results in routes that are considerably shorter in terms of total travel time. In Lecluyse et al. (2009), the travel speed in each time period is also obtained by applying queuing theory to traffic flows.

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2.3. Robust optimization The robust optimization approach was introduced for convex optimization problems by Ben-Tal and Nemirovski (1998, 1999), and extended by Bertsimas and Sim (2004). The methodology has then been applied in different settings, such as portfolio optimization (El-Ghaoui et al., 2003; Goldfarb and Iyengar, 2003), production scheduling (Alem and Morabito, 2012; Varas et al., 2014), and supply chain management (Bertsimas and Thiele, 2004; Ben-Tal et al., 2005). In transportation and routing, List et al. (2003) apply robust optimization for fleet planning under uncertainty. The authors explore the tradeoff between the expected cost of a solution and a measure of the risk associated with extreme outcomes of uncertain variables. Sungur et al. (2008) apply a robust optimization approach to solve the VRP with demand uncertainty. The objective is to obtain routes that minimize transportation costs while satisfying all demands in a given bounded uncertainty set. Computational results showed that the robust solution can protect from unmet demand while incurring a small additional cost over deterministic optimal routes. Souyris et al. (2012) consider the problem of dispatching technicians to service distributed equipment. Estimates of the service times can be subject to a significant amount of uncertainty. A robust optimization model is proposed to formulate the problem as a vehicle routing problem with soft time windows. The model is solved by brand-and-price. Erera et al. (2010) consider duration constraints when solving the vehicle routing problem with stochastic demands. A set of vehicle routes is first selected before the uncertain customer demands are realized. Once the demand values are realized, additional travel time is needed due to recourse actions. A robust optimization approach is applied to minimize the sum of expected route durations over all possible realizations. Agra et al. (2013) address the robust vehicle routing problem with time windows. Two formulations are proposed based on different robust approaches. Each formulation uses different robust optimization tools to handle the uncertainty. The two formulations are compared on a test bed composed of maritime transportation instances. Gounaris et al. (2013) derive robust optimization counterparts of different deterministic Capacitated VRP (CVRP) with uncertain customer demands. Robust rounded capacity inequalities are proposed and used to expedite the solution of the robust CVRP. In this paper, the road network daily maintenance routing problem is formulated as an Arc Routing Problem with Stochastic Service Times (ARP-SST) and robust optimization approach is applied to solve this problem. To the best of our knowledge, there have been no attempts to tackle ARPs with stochastic times, except our previous work (Chen et al., 2014). Furthermore, robust optimization has never been applied to an ARP. 3. Robust optimization formulation The robust optimization methodology assumes that the uncertain parameters belong to a given bounded uncertainty set. An important question then is how to formulate a robust problem that is not more difficult to solve than its deterministic counterpart. In this section, we first identify the deterministic ART-SST formulation, as well as the definition of uncertainty sets of service time. Then, the derivation of the robust optimization formulation is presented. 3.1. Notations The mathematical formulation for the problem is defined on a directed graph G = (V, A), with a node set V including a depot (node 0) and an artificial node 00 , and an arc set A. Node 00 is only connected to the depot. Define R  A as a set of required arcs that must be serviced once. Each arc a 2 R has an uncertain service time. The service times may be affected by the road conditions, accidents, technician’s skills, and so on. The problem is to find a set of vehicle routes, such that: (1) each arc in R is serviced exactly once; (2) each route starts and ends at the depot; (3) when a required arc is not served during the day with the available fleet, it will be scheduled for the next day, with a high penalty. The following notations are used in the mathematical formulation: K O(v) I(v) S A(S) B da k ta sa

q h f

set of vehicles set of arcs leaving node v, v 2 V set of arcs entering node v, v 2 V subset of arcs set of arcs whose two end nodes are from subset S the maximum allowed working duration of each vehicle during each planning horizon travel distance of arc a, a 2 A unit travel cost travel time on arc a, a 2 A service time on arc a, a 2 R unit penalty cost for those arcs that are not served and will be scheduled for the next day unit cost for excess duration of work fixed cost for using a vehicle

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Define the following decision variables: =1, if arc a is serviced by vehicle k; 0, otherwise. a 2 R, k 2 K. =1, if vehicle k is used; 0, otherwise. k 2 K. number of times arc a is traversed (without servicing) by vehicle k, a 2 A, k 2 K. =1, if arc a is assigned to the next day; 0, otherwise. k 2 K. overtime work for vehicle k, k 2 K.

xak zk yak

va

Dk

3.2. Deterministic ARP-SST formulation The deterministic ARP-SST problem can be formulated as follows. P1

X XX X X min D ¼ min ðD1 þ D2 þ D3 þ D4 Þ ¼ min f zk þ k  da  yak þ h Dk þ q v a da k2K

k2K a2A

k2K

! ð1Þ

a2R

Subject to:

X xak ¼ 1  v a ; k2K

X

a2Oðv Þ

X

ðxak þ yak Þ 

8a 2 R X

ð2Þ

ðxak þ yak Þ ¼ 0;

a2Iðv Þ

ðxak þ yak Þ P xbk ;

8v 2 V; k 2 K

8S  V; 0 R S; 2 6 jSj 6 jVj  1; b 2 AðSÞ \ R; k 2 K

ð3Þ ð4Þ

a2OðSÞ

1 X xak ; 8k 2 K jRj a2R X X sa xak þ t a yak  Dk 6 B; zk P

a2R

ð5Þ

8k 2 K

xak ; zk ; v a 2 f0; 1g;

8a 2 R; k 2 K

yak P 0; integer; 8a 2 A; k 2 K

Dk P 0;

ð6Þ

a2A

8k 2 K

ð7Þ ð8Þ ð9Þ

The objective function (1) minimizes the total cost D, which is the sum of the fixed cost D1, the deadheading travel cost D2, the cost for overtime work D3, and the penalty for not servicing the required arcs in time D4. Constraints (2) ensure that each arc a in the set of the required arcs is serviced exactly once, or an arc is not serviced when va takes the value of 1. Constraints (3) are the network flow conservation constraints. Constraints (4) are connectivity constraints. They ensure that if a subset S of arcs contains a required arc serviced by vehicle k, then there must be an arc traversed by vehicle k going from a node in S to a node outside of S (vehicle k crosses the border of S). Constraints (5) represent the fact that a vehicle is used only if it serves at least one required arc. Constraints (6) ensure that the total working duration (travel time plus service time) of each vehicle minus the overtime work Dk is less or equal to the maximum working duration B. From the equation, the driver’s working duration may exceed B by Dk. Constraints (7)–(9) define the properties of the variables. 3.3. Robust counterpart formulation The robust optimization framework is applied in order to construct the robust counterpart of the model P1 considering the uncertainty of service times. The objective is to find an efficient routing solution that is insensitive to the uncertainty in service time. 3.3.1. Uncertainty sets We assume that the service time on each arc belongs to an uncertainty set, without additional distribution assumptions. Since the uncertainty of the service times affects the feasibility of a solution, robust optimization seeks to obtain a solution that will be feasible for any realization taken by the service times. However, complete protection from adverse realizations comes at the expense of a severe deterioration in the objective. Therefore, the uncertain parameter sets over which the worst cases are computed should be chosen to achieve a trade-off between system performance and protection against uncertainty. Here, we consider that the service time s is uncertain and belongs to a bounded set U. The uncertainty set is constructed based on deviations around the nominal time value sa for each arc a (a 2 R). Thus, for each arc, the service time takes values in the interval [sa ; sa þ cmax ], where cmax is the maximum deviation of the service time on arc a. a a

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In addition, we assume that the uncertainty does not concentrate on any one route. Thus, we impose Ck as a budget of uncertainty to restrict the cumulative deviation on route k to be less than or equal to Ck . S k Thus, the uncertainty set U ¼ U where: k2K

( jRj

k

U ¼

s2R

: sa ¼ sa þ na cmax ; 0 6 na 6 1; a

X 8a 2 R; na cmax 6 Ck a

)

a2R

3.3.2. Robust optimization formulation The robust counterpart is obtained by replacing Constraints (6) in P1 by the following constraints.

X X sa xak þ t a yak  Dk 6 B; a2R

8k 2 K;

0

8s 2 U k

ð6 Þ

a2A 0

Note that there is such a constraint for each vehicle and each element in the uncertainty set U. Constraints (6 ) can be replaced by the following:

X max sa xak s2U k

!

þ

a2R

X

ta yak  Dk 6 B;

00

8k 2 K

ð6 Þ

a2A 00

The maximization in (6 ) can be dualized to obtain:

X X sa xak þ min ak Ck þ bak a;b

a2R

!

a2R

þ

X t a yak  Dk 6 B;

8k 2 K

ð10Þ

a2A

s.t

ak cmax xak þ bak P cmax xak ; 8a 2 R; 8k 2 K a a ak P 0; 8k 2 K bak P 0;

8a 2 R;

ð11Þ ð12Þ

8k 2 K

ð13Þ 00

where ak and bak are the dual variables associated with the constraints of the maximization problem in (6 ). The minimization in (10) can be removed yielding the following set of constraints:

X X X sa xak þ ak Ck þ bak þ t a yak  Dk 6 B; a2R

a2R

0

8k 2 K

ð10 Þ

a2A

Using the binary condition on variables xak, Constraints (11) can be linearized as the following constraints:

ak þ

bak

cmax a

P xak ;

8a 2 R;

0

8k 2 K

ð11 Þ

Thus, the complete robust optimization formulation can be written as follows: P2

X XX X X min D ¼ minðD1 þ D2 þ D3 þ D4 Þ ¼ min f zk þ k  da  yak þ h Dk þ q v a da k2K

k2K a2A

k2K

!

a2R

Subject to

ð2—5Þ; ð7—9Þ; ð100 —110 Þ; ð12—13Þ: 4. Branch-and-cut algorithm In this section, we present a branch-and-cut algorithm to solve the robust model (P2). Model P2 is firstly solved with the connectivity constraints (4) relaxed. Then, if an illegal solution is detected, new constraints are added to exclude the illegal solution. This process continues until the solution obtained is feasible for the original model. In order to identify violated connectivity inequalities, both heuristic and exact procedures can be used. Though exact method can be applied by solving min-cut problems, it is quite time consuming as shown in Hà et al. (2014). Hence, a heuristic is developed to identify connectivity violations for a fractional solution obtained from the relaxed model. The detail of the heuristic, which is also called a separation procedure, is described as follows. Separation procedure: Given a fractional solution, the decision variables concerning the network flow are retrieved. That is all the variables {xak, yak}. A graph is induced by connecting those arcs with which xak + yak P s, where s is a predefined value, and 0 6 s < 1. The resulting graph is denoted as G0 . If G0 is not weakly connected, then the connectivity inequalities (defined in Constraints (4)) associated with each weakly connected component of G0 are checked for connectivity violation.

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In addition, the following symmetry breaking constraints are introduced to speed up the process of the branch-and-cut algorithm.

zk P zkþ1 ;

1 6 k 6 jKj  1

j X

j X

i¼1

i¼1

2ðjiÞ xik P

2ðjiÞ xi;kþ1 ;

ð14Þ

8j 2 R; 1 6 k 6 jKj  1

ð15Þ

Constraints (14) force the vehicles used to be the first ones in the list. Constraints (15) avoid the occurrence of equivalent solutions in which route indices are merely permuted. Interested readers are referred to Adulyasak et al. (2014) for detailed explanations on the symmetry breaking constraints. We now describe in detail the algorithm. We first relax the connectivity constraints (4) of the original model (P2), and solve a new linear program (P20 ). P20

X XX X X min D ¼ minðD1 þ D2 þ D3 þ D4 Þ ¼ min f zk þ k  da  yak þ h Dk þ q v a da k2K

k2K a2A

k2K

!

a2R

Subject to:

ð2—3Þ; ð5Þ; ð100 —110 Þ; ð12—13Þ; and 0 6 xak ; zk ; v a 6 1;

ð16Þ

yak ; Dk P 0;

ð17Þ

8a 2 R; k 2 K 8a 2 A; k 2 K

After (P20 ) is solved, the separation procedure is activated to search for violated constraints. The constraint set of the P20 is then updated by including the violated constraints and P20 is solved again. This process continues iteratively until the optimum found for P20 is feasible for constraints (4). Then, if there are fractional variables, we branch on one of the fractional variable to generate two new subproblems. If all the variables are integer, we explore another subproblem, i.e., for each subproblem, we repeat the previous procedure. The branch-and-cut algorithm is built around CPLEX 12.5 with the Callable Library. All the CPLEX parameters are set to their default values. The heuristic separation procedure for the connectivity constraints (4) is repeated with s = 0, 0.25, 0.5, and 0.75 respectively. The running time of the branch-and-cut algorithm is limited to one hour. 5. Experimental analysis The main motivation of robust optimization is to achieve a good trade-off between the level of robustness and the total cost. In this section, some experiments are conducted and analyzed using Monte Carlo simulation to evaluate how the solutions behave for different level of robustness. We also compare the robust solution with an alternative method to address service time uncertainty. All the tests were performed on a personal computer with a 2.53 GHz duo processor and 4.0 GB of RAM. 5.1. Problem settings for the robust model In order to conduct the experiments, a case was defined based on the real network of high-speed freeways in the city of Shanghai as shown in Fig. 1. The complete network has been partitioned into two sectors, each of which corresponds to the area serviced by a single depot. Sector 1 consists of 20 vertices and 52 arcs, and sector 2 consists of 17 vertices and 41 arcs. In each sector, around half of the arcs were selected and defined as required arcs (in red1 arrows) according to typical daily monitoring operations. According to the real monitoring operation of the company, the average travel speed of the vehicles without conducting monitoring service is about 30 km/h, and the average travel speed of the vehicles in service is about 15 km/h. Thus, for an arc of length da, the travel time (ta) is defined as da/30 (h) = 2da (min), and the service time (sa) is defined as da/15 (h) = 4da (min). Based on the observation of historical data, it is assumed that the service time on a required arc follows a normal distribution with a mean value (la) of 4da and a standard deviation (ra) of 0.5la. In the following experiments, the solutions are evaluated by simulating actual service times generated from normal distributions N(la, ra) for each required arc a. The maximum allowed working duration of each vehicle (B) is set to be 240 min (a shift). To be able to solve the robust optimization model, other problem settings are defined as follows: (1) Uncertainty sets of service times. For a given arc a, we define the nominal service time value sa equal to la, and the maximum deviation of the service time cmax as cmax = ura. Thus, u is the level of uncertainty considered in the robust a a model, and is defined the same for all the required arcs to avoid the combinatorial problem of instance generation,

1

For interpretation of color in Fig. 1, the reader is referred to the web version of this article.

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Sector 1 Depot 1

Depot 2 Sector 2

Fig. 1. Network of high-speed freeways in the city of Shanghai.

and to facilitate the analysis of the results. Three levels of uncertainty are considered in our experiments. That is, u 2 {0.4, 1.2, 2}. The deterministic model is recovered by setting u = 0. (2) The budget of uncertainty. Different approaches have been proposed to determine the budgets of uncertainty in order to represent different levels of conservatism. For example, Adida and Perakis (2006) use time-dependent linear functions, pffiffiffiffiffi and Souyris et al. (2012) define the budgets as m  cmax , where m is the estimated number of customers serviced by a each technician. We consider in this experiment the following expression for the budgets of uncertainty: P n max Ck ¼ e  jRj , where n is the estimated number of customers serviced on each route. The advantage of this a2R ca approach is that it allows varying the level of conservatism using parameter e, e 2 [0, 1]. This allows us to consider the non-protection case (e = 0), the full protection case (e = 1), and those intermediate cases. (3) Cost parameters. The unit travel cost k is set to be 5, and the unit cost for excess duration of work h is set to be 10. The unit penalty cost for delayed service q is set to be 20. The size of the fleet, |K|, is 5. The fixed cost of employing a vehicle f is set to be 200. 5.2. Sensitivity analysis In this section, we conduct a sensitivity analysis by varying the following parameters: (1) the level of robustness; and (2) the fleet size |K|. 5.2.1. The level of robustness Note that the robustness of a solution is determined by a combination of u and e. First, a solution with different level of robustness is obtained for each sector by solving the robust model P2 with a given u and e. A complete solution is given by variables {xak, yak, zk, va, Dk}, which define not only the routes of each vehicle k but also the amount of overtime work and delay. Then, the solution {xak, yak, zk, va, Dk} is evaluated on 1000 simulated scenarios with service time for each required arc drawn from the normal distribution N(la, ra). The objective of these simulation experiments is to analyze how the solutions behave in terms of total cost and service level. Fig. 2 illustrates the average value from the 1000 simulations of the objective function (1) in total cost D, which is composed of the fixed cost D1, the deadheading travel cost D2, the cost for overtime work D3, and the penalty for not servicing the required arcs in time D4. Please note that the service cost over arcs in R can be viewed as a sunk cost in the routing decision, since each required arc must be serviced once. The first observation is that the total cost decreases as the level of conservatism (e) increases. This is expected for the robust formulation, as the robust counterpart tries to protect the system against the uncertainty. However, the total cost will increase as e further increases. And this increase is dramatic when the level of uncertainty is high (u = 2). This is due to a large portion of penalty cost (D4) incurred under the environment with a high level of uncertainty. This tendency is the same for both sectors. It is interesting to notice that taking more uncertainty into account does not necessary imply more expensive solution. For example, as u increases from 0.4 to 1.2, the total cost decreases for both sectors. Since the robust optimization objective function incorporates penalty terms for delayed service and overtime work, the performance of the robust solutions is also evaluated in terms of service level. Given a solution {xak, yak, zk, va, Dk}, define d as the ratio of delayed service to total service requirement.

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(b) Sector 2

(a) Sector 1 Fig. 2. Average objective function.

P ma Sa d ¼ Pa2R a2R Sa Thus, the service level is given by 1  d. The average service level of the solutions is compared in Fig. 3. It is observed from the results that: (1) For Sector 1, the service level of the robust solutions is higher than 90%; for Sector 2, the service level of the robust solutions is higher than 80%. (2) The solutions that take less uncertainty into account (with smaller u) have a lower service level. However, the service level starts to decrease as e exceeds 0.5. And the service level drops sharply when both u and e are of large values. The results in Figs. 2 and 3 are useful to define appropriate values for parameters u and e in the robust formulation (P2), in order to find a robust solution that is both insensitive to the uncertainty in service time and is moderate in total cost increase. It is also noted that when the level of robustness is not high, less vehicles are employed to provide monitoring services. That is, some vehicles stay idle at the depot. In the next section, the sensitivity analysis with respect to the number of vehicles is given to see if it is beneficial to have a routing with more slack. 5.2.2. The number of vehicles The routing problems in the two sectors are solved again with a constant fleet size. The fleet size |K| for sector 1 is set to be 5, 6, and 7, respectively. The fleet size |K| for sector 2 is set to be 4, 5, and 6, respectively. The instances are solved by manually setting zk = 1, for each k 2 K. The solutions have similar trends for different level of variability. Therefore, we present the computational results only for those instances with u equal to 1.2. Fig. 4 shows the sensitivity analysis of the objective function (total cost) with respect to the number of vehicles. Fig. 5 illustrates the sensitivity analysis of the service level with respect to the number of vehicles. It is observed from the results that: (1) When the level of conservatism is low (e < 0.6), having more vehicles is more costly. However, having more vehicles may cost less when the level of conservatism is high.

(a) Sector 1

(b) Sector 2 Fig. 3. Average service level.

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(b) Sector 2

(a) Sector 1

Fig. 4. Sensitivity of objective function to the number of vehicles (u = 1.2).

(a) Sector 1

(b) Sector 2

Fig. 5. Sensitivity of service level to the number of vehicles (u = 1.2).

(2) A jump in total cost is observed from Fig. 4 when e moves from 0.5 to 0.6 for the curve with 5 vehicles. On the contrary, having more vehicles implies less sensitivity to uncertainty. (3) Having more vehicles implies better quality solutions in terms of service level (Fig. 5). The above observation can be attributed to the following facts: (1) Employing more vehicles implies routing with more slack. And the robust solutions amount to a clever management of the remaining vehicle capacity in such a way that the vehicles can easily collaborate by sharing their slacks in case of uncertainty. (2) Employing more vehicles can improve the service level by reducing the delayed monitoring service. However, further increase of the number of vehicles does not necessarily further improve the service level. The managerial implication of this analysis is twofold: (1) The result is very useful as a guide to determine the proper fleet size, which can provide protection for the worst case situation at a modest loss in the objective value. (2) The result also contains information on the robustness of a solution with a given fleet size. 5.3. Robust model (RM) versus chance-constrained programming model The intractability of chance-constrained problems using exact probability distributions has spurred recent interests in robust optimization in which data uncertainties are described using uncertainty sets. Ben-Tal et al. (2009) pointed out that the stochastic optimization approach (typically chance-constrained approach) is not always less conservative than the worst-case oriented robust optimization approach. In this section, we compare the robust solution with the solution obtained from the chance-constrained programming model (CCPM) to see if it is the case in our application. The CCPM replaces constraints (6) in P1 with a set of probabilistic constraints to ensure that the total working duration of each route is less than the upper bound B with probability a (Chen et al., 2014):

L. Chen et al. / Transportation Research Part E 85 (2016) 40–51

P

( ) X X sa xak þ t a yak 6 B P a; a2R

8k 2 K

49

ð18Þ

a2A 00

Unlike the uncertain constraints (6 ) in the RM, the above chance constraints do not include overtime work Dk. It is because the CCPM does not consider route failure (whenever the total working duration of the route exceeds B). Once a route failure occurs, there are different recourse actions to correct the route. Please refer to Chen et al. (2014) for discussion of different recourse strategies. 5.3.1. Comparison procedure The difference of the two models is in how the uncertain service time is defined. In the CCPM, the service time (sa) is considered to follow a normal distribution. The mean value and the standard deviation of the distribution are assumed to be values from historical data. That is, sa  N(la, ra2). In the RM, the level of uncertainty (u) is set to be 2. That is, the service time Sa 2 ½ua ; la þ 2ra . It is noted that in approximately 96% of the cases the actual service time is less than or equal to la + 2ra. In order to get comparable computational results, travel times are considered deterministic and equal to the mean value in the CCPM. The comparison procedure is explained as follows. The CCPM and the RM are applied respectively to solve the routing problem of Shanghai high-speed freeways (SHHSF) network given in Fig. 1. Then, Monte Carlo simulation is used to assess both the CCPM solution and the RM solution. Both solutions are evaluated on 1000 scenarios by simulating service time from normal distribution on the routes prescribed by the solutions. For each realization the total cost of the RM solution is calculated by summing up the four cost components, which are the fixed cost D1, the deadheading travel cost D2, the cost for overtime work D3, and the penalty for not servicing the required arcs in time D4. The total cost of the CCPM solution consists of the fixed cost D1, and the deadheading cost D2. In case that the total working duration exceeds the upper limit B, a cost for overtime work is incurred and included into the total cost of the CCPM solution. The complete procedure is illustrated in Fig. 6. The CCPM is also solved by the branch-and-cut algorithm (please refer to Chen et al., 2014) for detail description of the algorithm). We retrieved the solutions for experimental analysis after running the algorithm for an hour. 5.3.2. Computational results Fig. 7 compares the robust solutions with the CCPM solutions in terms of total cost. For robust solutions, the total cost is illustrated as a function of the level of conservatism for different level of uncertainty. The confidence level (a) in the CCPM model is set to be 70%, 80%, and 90% respectively. Each data line on the figure is the average value of 1000 service times’ realizations. From Fig. 7, we observed that: (1) Compared with the CCPM solutions, robust solutions are less costly. The chance-constrained approach is aimed at ‘‘hard” constraints. Thus, more vehicles are needed to satisfy the chance constraints. In robust solutions, violation of duration constraints is allowed with penalty. Thus, better schedules with fewer vehicles are obtained. Reduction in the total cost is observed in robust solutions, even in some cases where penalty cost for delaying service is needed. (2) In order to achieve the same service level, the CCPM solutions cost around 70% more than the RM solutions. (3) For Sector 1, the service levels of all the robust solutions are higher than 90%. This means that the robust solutions remain feasible in the CCPM (a 6 90%). For Sector 2, the result is similar. This observation makes it possible to apply the proposed robust optimization formulation as a tractable approach for obtaining a feasible solution in the chanceconstrained model.

SHHSF network

CCPM sa ~ N(µa, σa2)

RM sa ∈[ µa, µa + 2 σa]

CCPM solution

RM solution

Monte Carlo Simulation Average value of the total cost of the CCPM solution

1,000 scenarios generated from N(µ a, σa2) Average value of the total cost of the RM solution

Performance comparison Fig. 6. Comparison procedure between the CCPM and the RM.

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α = 90% α = 90% α = 80%

α = 80%

α = 70%

α = 70%

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Fig. 7. Comparison of robust and chance-constrained solutions.

(4) The CCPM model is significantly more difficult to solve because special treatment of the chance constraints is needed. The intractability would be even higher with the consideration of the dependence of the stochastic parameters. 5.3.3. Comparison of the two approaches Based on the previous observation, we conclude that the robust optimization approach is superior to the chanceconstrained programming approach in the following aspects: (1) Although there have been studies addressing the relationship between robust optimization models and chance constraint programming models (Ben-Tal et al., 2009; Chen et al., 2010), there is no direct mathematical equivalence that could be established between the two formulations in this application. The reason is that in this study the uncertain service times belong to convex sets, where the uncertainty on each route is bounded. This allows us to capture the correlation between uncertain parameters, which is not achievable in our previous CCPM model. (2) The other advantage of assuming an upper bound on the total sum of service time deviations on each route is that it enables us to properly define the worst case. Thus, a robust solution can be obtained while not being overly conservative. And the desire to obtain a more robust solution can be easily achieved by enlarging the uncertainty set. While in the CCPM, although one can obtain more conservative solution by increasing the variation coefficient (standard deviation divided by mean value) of each distribution, the solution is usually very expensive. The reason is that the dependence of the probability mass of all realizations in CCPM is completely ignored. While the reality is that ‘‘bad things” should not all happen on one day. 6. Conclusion In this study, we propose the use of robust optimization to solve the routing problem of monitoring vehicles for road daily maintenance service with uncertain service time. A robust counterpart formulation has been developed to optimize the worst case value over all uncertain data that belong to a bounded set. To formulate the problem we follow the robust optimization literature for integer programming problems. The contributions here are to explore problem formulation that allows capturing the characteristics of the real problem and to preserve the linearity of the model structure. Thus, it allows us to adapt a branch-and-cut algorithm to efficiently solve instances of size relevant to real world problems. Our work has shown that robust optimization is an attractive alternative for solving routing problems under uncertainty because of the following practical advantages: (1) it does not require building detailed probability density functions of services times, which is not very practical for monitoring companies; and (2) it provides valuable information about the robustness and the risk aversion of the solutions that are helpful for the managers to make a decision. In the computational analysis, we have evaluated the practical qualities (total cost and service level) of the optimal solutions using Monte Carlo simulation. This analysis is very helpful for decision makers to understand how different level of robustness affects the behavior of the solutions, as well as the trade-off between loss of total cost and potential service level increase. In addition, the sensitivity analysis revealed the correlation between the fleet size and the robustness of the solution. The results demonstrate that employing more monitoring vehicles implies a more robust solution while not necessarily being more costly (sometimes even cheaper). In comparing the robust optimization approach against the chance-constraint programming method, we observed that the robust optimization approach was preferable in practice because of the following aspects: (1) even though delay is allowed in the robust model, the percentage of delayed service in the robust solutions is very limited; (2) the robust optimization approach is more effective to yield robust solutions with moderate cost increase, and the conservatism can be easily altered by adjusting the uncertainty set; and (3) the robust model is computationally more tractable than the chance constrained model.

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The proposed approach can be extended to solve more general problems with minor adjustment (uncertain travel time, heterogeneous vehicles, etc.). An interesting extension of this research would be to consider the possibility of adapting current routing to the realization of uncertain data later on, thus to achieve a better service quality by obtaining more efficient solutions with consideration of real time situation of the road network. Acknowledgements This work was supported by National Natural Science Foundation of China (Project number: 71271130) and the Fonds de recherche du Québec – Nature et Technologies. The authors would like to thank the anonymous reviewers for their valuable comments. References Adida, E., Perakis, G., 2006. A robust optimization approach to dynamic pricing and inventory control with no backorders. Math. Program. 107 (1–2), 97– 102. Adulyasak, Y., Cordeau, J.-F., Jans, R., 2014. Formulations and Branch-and-Cut algorithms for multi-vehicle production and inventory routing problems. INFORMS J. Comput. 26 (1), 103–120. Agra, A., Christiansen, M., Figueiredo, R., Hvattum, L.M., Poss, M., Requejo, C., 2013. The robust vehicle routing problem with time windows. Comput. Oper. Res. 40, 856–866. Alem, D.J., Morabito, R., 2012. Production planning in furniture settings via robust optimization. Comput. Oper. Res. 39 (2), 139–150. Ben-Tal, A., El Ghaoui, L., Nemirovski, A., 2009. Robust Optimization. Princeton University Press, Princeton, New Jersey. Ben-Tal, A., Nemirovski, A., 1998. Robust convex optimization. Math. Oper. Res. 23 (4), 769–805. Ben-Tal, A., Nemirovski, A., 1999. Robust solutions to uncertain programs. Oper. Res. Lett. 25 (1), 1–13. Ben-Tal, A., Golany, B., Nemirovski, A., Vial, J.P., 2005. Supplier-retailer flexible commitments contracts: a robust optimization approach. Manuf. Serv. Oper. Manage. 7 (3), 248–273. Bertsimas, D., Sim, M., 2004. The price of robustness. Oper. Res. 52 (1), 35–53. Bertsimas, D., Thiele, A., 2004. A robust optimization approach to supply chain management. In: Proceedings of IPCO X. Springer Lecture Notes on Computer Science, vol. 3064. Springer, Berlin, pp. 86–100. Chen, L., Hà, M.H., Langevin, A., Gendreau, M., 2014. Optimizing road network daily maintenance operations with stochastic service and travel times. Transp. Res. Part E 64, 88–102. Chen, W., Sim, M., Sun, J., Teo, C.P., 2010. From CVaR to uncertainty set: implications in joint chance-constrained optimization. Oper. Res. 58 (2), 470–485. Christiansen, C.H., Lysgaard, J., Wøhlk, S., 2009. A Branch-and-Price algorithm for the capacitated arc routing problem with stochastic demands. Oper. Res. Lett. 37, 392–398. El-Ghaoui, L., Oks, M., Oustry, F., 2003. Worst-case value-at-risk and robust portfolio optimization: a conic programming approach. Oper. Res. 51 (4), 543– 556. Erera, A.L., Morales, J.C., Savelsbergh, M., 2010. The vehicle routing problem with stochastic demand and duration constraints. Transp. Sci. 44 (4), 474–492. Fleury, G., Lacomme, P., Prins, C., 2004. Evolutionary algorithms for stochastic arc routing problems. In: Raidl, G.R. et al. (Eds.), EvoWorkshops. SpringerVerlag, Berlin, Heidelberg, pp. 501–512. Fleury, G., Lacomme, P., Prins, C., Ramdane-Chérif, W., 2005. Improving robustness of solutions to arc routing problems. J. Oper. Res. Soc. 56 (5), 526–538. Gendreau, M., Jabali, O., Rei, W., 2014. Stochastic VRP. In: Toth, P., Vigo, D. (Eds.), Vehicle Routing: Problems, Methods and Applications. MOS-SIAM Series on Optimization, second ed., vol. 18. SIAM, Philadelphia. Goldfarb, D., Iyengar, G., 2003. Robust portfolio selection problems. Math. Oper. Res. 28 (1), 1–38. Gounaris, C.E., Wiesemann, W., Floudas, C.A., 2013. The robust capacitated vehicle routing problem under demand uncertainty. Oper. Res. 61 (3), 677–693. Hà, M.-H., Bostel, N., Langevin, A., Rousseau, L.-M., 2014. Solving the close enough arc routing problem. Networks 63 (1), 107–118. Jaillet, P., Qi, J., Sim, M., 2013. Routing Optimization with Deadlines Under Uncertainty. Working Paper. . Kao, E.P.C., 1978. A preference order dynamic program for a stochastic traveling salesman problem. Oper. Res. 26 (6), 1033–1045. Kenyon, A.S., Morton, D.P., 2003. Stochastic vehicle routing with random travel times. Transp. Sci. 37 (1), 69–82. Laporte, G., Louveaux, F., Mercure, H., 1992. The vehicle routing problem with stochastic travel times. Transp. Sci. 26 (3), 161–170. Laporte, G., Musmanno, R., Vocaturo, F., 2010. An adaptive large neighbourhood search heuristic for the capacitated arc-routing problem with stochastic demands. Transp. Sci. 44 (1), 125–135. Lecluyse, C., Van Wosensel, T., Peremans, H., 2009. Vehicle routing with stochastic time-dependent travel times. 4OR: Quart. J. Oper. Res. 7 (4), 363–377. Li, X., Tian, P., Leung, S.C.H., 2010. Vehicle routing problems with time windows and stochastic travel and service times: models and algorithm. Int. J. Prod. Econ. 125, 137–145. List, G.F., Wood, B., Nozick, L.K., Turnquist, M.A., Jones, D.A., Kjeldgaard, E.A., Lawton, C.R., 2003. Robust optimization for fleet planning under uncertainty. Transp. Res. Part E 39, 209–227. Sniedovich, M., 1981. Analysis of a preference order traveling salesman problem. Oper. Res. 29 (6), 1234–1237. Souyris, S., Cortés, C.E., Ordóñez, F., Weintraub, A., 2012. A robust optimization approach to dispatching technicians under stochastic service times. Optimiz. Lett. 9, 1–20. Sungur, I., Ordóñez, F., Dessouky, M.M., 2008. A robust optimization approach for the capacitated vehicle routing problem with demand uncertainty. IIE Trans. 40 (5), 509–523. Tagmouti, M., Gendreau, M., Potvin, J.V., 2011. A dynamic capacitated arc routing problem with time-dependent service costs. Transp. Res. Part C 19, 20–28. Van Woensel, T., Kerbache, L., Peremans, H., Vandaele, N., 2008. Vehicle routing with dynamic travel times: a queueing approach. Eur. J. Oper. Res. 186, 990– 1007. Varas, M., Maturana, S., Pascual, R., Vargas, I., Vera, J., 2014. Scheduling production for a sawmill: a robust optimization approach. Int. J. Prod. Econ. 150, 37– 51. Zhang, T., Chaovalitwongse, W.A., Zhang, Y., 2012. Scatter search for the stochastic travel time vehicle routing problem with simultaneous pick-ups and deliveries. Comput. Oper. Res. 39, 2277–2290.