3.5 Cluster-First Route-Second Heuristic
3.5.3 Logic-Based Benders Decomposition Scheme
The idea of classical Benders Decomposition is to split the original (compact) model into two parts, namely a restricted master problem and a subproblem by separating the variables and using LP duality. Logic-Based Benders Decomposition [72] aims at extending this approach to a "logical split" of the problem into a master problemand a subproblem where the subproblem yields Benders cuts throughlogical deduction. This means, Benders infeasibility cuts are added to the master problem whenever the solution of a subproblem is infeasible and the master problem is then resolved with these cuts.
The Assignment Problem (AP)
deals with assigning stations to vehicles and serves as the master problem of this approach.
Much work on the AP in conjunction with Vehicle Routing Problems (VRP)is done with heuristic approaches (e.g., Genetic Algorithms) as it can be tough to find good routing costs approximation which can be formulated efficiently inside a MIP. In fact it turned
out to be the biggest challenge to effectively encode the routing costs approximation into the MIP model. We need a lower bound on the TSP, because if we would overestimate the routing costs, obviously the subproblem would always be feasible as the optimal TSP tour would be lower than the estimation. Thus, our algorithm would terminate regardless whether the solution is optimal or not. Schuijbroek et al. [140] use aMaximum Spanning Star which they have proved to be an upper bound on the routing costs for their problem. However, in our case we need a lower bound, and thus, we ended up by using a single commodity flow formulation for computing a minimum spanning tree (MST) for approximating the routing costs in our MIP model. It has several advantages:
It is a standard and well-known problem of graph theory, so it can be formulated very well inside the MIP and can be solved relatively efficiently. As preprocessing we calculate an upper boundω on the assignment per cluster. Thus, we define a variablet= 0. Then, we seek for the pickup station with the least travel cost from the depot and the delivery station with the least travel cost to the depot and add these costs tot. Afterwards, we look for any lowest cost edges to add them also totas long ast <ˆt. The number of nodes added during this procedure is an upper bound on the maximum assigned stations per cluster. This helps us to get some bound and our MIP model does not become quadratic (see Eq. 3.43 in the MIP).
Letxvl∀v∈V0, l∈Lbe 1 ifvis assigned to vehiclel, 0 otherwise,yluv∀(u, v)∈A0, l∈L be 1 if there exists an edge fromutovin the MST for vehiclel, 0 otherwise,fuvl ∀(u, v)∈ A0 the flow between nodesu and v for the MST in clusterl andhl be the routing costs approximation for vehiclel. Moreover, we define the constantτ as the scaling factor for the MST in the objective function. This scaling factor is used to neglect any influence of the MST in the objective function because we only want to maximize the total assigned stations. The MST in the objective function is only used for routing costs approximation.
Then our MIP model is formally defined as follows:
max X
3.5. Cluster-First Route-Second Heuristic X
(u,v)∈A0
yluv= X
v∈V
xvl ∀l∈L (3.44)
yluv≤xvl ∀l∈L,(u, v)∈A0 (3.45)
yluv≤xul ∀l∈L,(u, v)∈A0 (3.46)
hl≥ X
(u,v)∈A
yluv·tuv ∀l∈L (3.47)
hl≤ˆt ∀l∈L (3.48)
xvl, yluv∈ {0,1} ∀v∈V0,(u, v)∈A0, l∈L (3.49) hl, fuvl ∈R+ ∀(u, v)∈A0, l∈L (3.50) The objective function (3.37) maximizes the stations assigned to vehicles and minimizes the approximated costs by the spanning trees computed for each cluster. Note that not all stations may be assigned to a cluster as it is not possible to serve all stations within the given time limit. Inequalities (3.38)–(3.40) constitute the assignment of stations to the vehicles whereas inequalities (3.41)–(3.46) represent the spanning tree polytope.
Equalities (3.38) state that the depot 0 is included in every cluster. Inequalities (3.39) ensure that each station is assigned to at most one vehicle, and equalities (3.40) state that the number of assigned pickup stations must be equal to the number of assigned delivery stations for each cluster. The spanning tree is modeled by a single commodity flow formulation where the outgoing flow of the root node for each cluster must be the number of nodes assigned to that cluster which is ensured by equalities (3.41). For all other nodes except the root node, equalities (3.42) define that every node must consume 1 flow. Inequalities (3.43) restrict the maximum flow value of each cluster. Furthermore, these inequalities define the flow to be 0, if the edge is not chosen by the MST (i.e., yuvl = 0). Equalities (3.44) state that the total number of selected edges for the MST must be the number of stations assigned to that cluster. Equalities (3.45) and (3.46) state that the edge (u, v) can only be chosen by the MST, if both nodes uandv are contained in the cluster l. Equalities (3.47) are used to assign the approximated routing costs for each vehiclel∈L to the variablehl. Inequalities (3.48) ensure that the approximated routing costs lie within the time budget of each vehicle.
We can extract the assignment of stations from thex-variables. Note that the assignment may not necessarily be feasible as we only approximate the routing costs. The MST is a lower bound on the TSP, and therefore, the optimal routing costs may be higher than the time budget.
Traveling Salesman Problems
have to be solved in our subproblems. After solving the AP we know which stations have to be serviced by which vehicle. Thus, we have to solve a TSP for each vehicle separately on the respective subgraph. As the TSP is already a well-studied problem, we use the State-Of-The-Art TSP solverConcorde [30]. However, Concorde is designed
for solving only symmetric TSP instances on the complete graph. Thus, we transform our asymmetric instance into a symmetric one as shown in [77]. Infeasible edges (i.e., pickup to pickup and delivery to delivery stations) are modeled by very high routing costs so that these edges will not be used. Because of the maximum time limit of 480min in our experiments, we get rather small TSP instances with about 25 stations for each subproblem. Concorde is able to solve those instances within milliseconds.
By usingConcordeand translating the result back, we retrieve an alternating tour through all stations of the cluster starting and ending at the depot with minimal costs. We have to check if the costs of the optimal TSP tour are within the common time limit ˆt of the vehicle. If this is the case, we have found a feasible solution. If this is not the case, we know that we have an infeasible assignment, because it is not possible to traverse all stations within the given time limit. Thus, we add a cut for this assignment and resolve the AP.
If all subproblems are feasible which means that the tours of the vehicles lie within their time budget, we have found a feasible and optimal solution to our problem and the algorithm terminates.
Benders infeasibility cuts
are generated whenever we find a cluster which is not feasible due to the time limit of the vehicles. As we deal with a homogeneous vehicle fleet we can add this cut for each vehicle. Let C be the the set of all cuts, then the Benders cut is formally defined as:
P
v∈cxvl ≤ |c| −1 ∀c∈C, l∈L. This means all setsS ⊆V0 where∃c∈C : S ⊇care discarded.
For improving the cuts we use the following additional heuristic: When we examine an infeasible assignment, we try to make the infeasible set smaller to cut off more infeasible assignments. Thus, we look for two stations, one pickup and one delivery station which have the least travel costs in the assignment. We remove these stations and try to resolve the TSP using Concorde. If the assignment stays infeasible we have found a better cut, and we can prune more infeasible assignments for the next run of the AP.
3.5.4 Computational Results
In this section we compare the Logic-Based Benders Decomposition with the effective and powerfulVariable Neighborhood Search (VNS)from [125]. For comparison, the VNS gets also the pre-processed instances as input so that it produces alternating tours picking up as many bikes as possible and delivering as many bikes as possible at every station.
As benchmark set we chose |V| ∈ {10,20,30,60}, |L| ∈ {1,2} and ˆt ∈ {120,240,480}.
The instances have been taken from [125] but now we only consider the type of the station rather than the exact fill levels.
3.5. Cluster-First Route-Second Heuristic The algorithm was implemented in C++ using GCC 4.8.2. For running our tests we used
a single core of an Intel XEON E5540 with 2.53 GHz and 3 GB of memory. As MIP solver for the AP and Concorde we used CPLEX 12.6.
We set τ = 0.001 for our experiments. The time limit for the Logic-Based Benders Decomposition was 1 hour and for the VNS we used half an hour. To compare to the VNS, we count the visited stations at the end of rebalancing. For every benchmark type we used 30 different instances and calculated the average.
Table 3.9 shows that small instances can easily be solved to optimality by the Logic-Based Benders Decomposition. Furthermore, all instances up to 30 stations using 2 vehicles and 8 hours of time budget have been solved to optimality. In overall, we obtained optimal solutions for 66% of the selected instance sets. In the third column of the Benders Decomposition we show the approximation quality of the MST for the TSP.
By approximation quality we measure how far off the estimated routing costs of the MST are compared to the optimal costs of the TSP retrieved by Concorde. Generally, it looks that approximation quality becomes better with bigger instances although we sometimes cannot solve all of the instances to optimality. In overall, we have achieved an approximation quality of 18% with respect to the TSP. The number of iterations and cuts are moderate for the selected instances.
When comparing the exact Logic-Based Benders Decomposition with the VNS, we noticed that differences between the exact algorithm and the metaheuristic become bigger when instance sets are larger although the VNS is most of the time not far off from the exact solution.
3.5.5 Conclusions and Future Work
We have proposed a novel problem definition for balancing BSS which is at least in some BSS practically highly relevant (i.e., Citybike Wien), but substantially simplifies the problem. To solve the problem by an exact approach we have come up with a Logic-Based Benders Decomposition scheme. The master problem is modeled as an Assignment Problem and the subproblems as Traveling Salesman Problems accordingly.
It is possible to solve these subproblems with State-Of-The-Art methods, likeConcorde, within milliseconds. Furthermore, we have shown that the Minimum spanning tree is a reasonable approximation for routing costs in the master problem. However, for bigger instances with more than 60 stations exact solutions are not always found within the time limit.
For future work it would be very interesting to investigate Branch-and-Check [150] as the subproblem can be solved very efficiently using Concorde. Furthermore, we would like to extend this approach to a heuristic Cluster-First Route-Second algorithm. If the AP would be solved heuristically, we could treat much bigger instances, probably even bigger instances than we have solved before (up to 700 stations in [125]). Moreover, as Concorde is very efficient for solving the TSP it would also be interesting to compare to
Instance set Logic-Based Benders Decomposition VNS
|V| |L| tˆmax Feasible [%] obj qual [%] #iterations #cuts tgtot [s] obj diff 10 2 120 100.00 6.93 23.15 12.97 15.58 0.39 4.93 -2.00 10 2 240 100.00 8.20 24.56 1.20 0.76 0.06 8.07 -0.13 10 3 120 100.00 7.93 26.55 9.60 14.84 0.32 7.20 -0.73 20 2 120 86.67 7.92 22.97 71.88 149.04 48.98 5.37 -2.56 20 2 240 60.00 15.67 13.76 18.56 63.38 0.56 10.00 -5.67 20 3 120 56.67 11.76 20.82 65.59 163.84 197.22 8.13 -3.63 20 3 240 100.00 16.87 18.26 3.57 3.74 1.43 15.90 -0.97 30 2 120 40.00 8.00 22.98 14.58 14.27 829.56 5.10 -2.90 30 2 240 6.67 21.00 13.61 1.50 0.71 2851.00 9.53-11.47 30 2 480 100.00 26.47 11.24 2.23 3.09 1.13 19.23 -7.23 30 3 120 10.00 12.00 24.14 1.00 0.00 3597.89 7.70 -4.30 30 3 240 30.00 23.78 14.39 8.00 5.96 7.79 17.53 -6.24 60 3 480 73.33 51.91 9.03 20.36 21.81 456.10 36.10-15.81
66.41 218.44 18.88 154.80-63.64
Table 3.9: Results from our computational tests showing Logic-Based Benders Decompo-sition
a Route-First Cluster-Second algorithm as these approaches got very efficient during the last decade [115].