A decomposition approach for the integrated vehicle-crew-roster problem with days-o_ pattern

Transcription

1 A decomposition approach for the integrated vehicle-crew-roster problem with days-o_ pattern Marta Mesquita, Margarida Moz, Ana Paias e Margarida Pato CIO Working Paper 2/2012

2 A decomposition approach for the integrated vehicle-crew-roster problem with days-off pattern Marta Mesquita 1, Margarida Moz 2, Ana Paias 3 and Margarida Pato 2 1 CIO and ISA-UTL,Tapada da Ajuda, Lisboa, Portugal marta@math.isa.utl.pt 2 CIO and ISEG-UTL,Rua do Quelhas 6, Lisboa, Portugal mmoz@iseg.utl.pt,mpato@iseg.utl.pt 3 CIO and DEIO, Faculdade de Ciências da Universidade de Lisboa, BLOCO C6, Piso 4, Lisboa, Portugal ampaias@fc.ul.pt February 3, 2012 Abstract The integrated vehicle-crew-roster problem with days-off pattern aims to simultaneously determine minimum cost vehicle and daily crew schedules that cover all timetabled trips and a minimum cost roster covering all daily crew duties according to a pre-defined days-off pattern. This problem is formulated as a new integer linear programming model and is solved by a heuristic approach based on Benders decomposition that iterates between the solution of an integrated vehicle-crew scheduling problem and the solution of a rostering problem. Computational experience with data from two bus companies in Portugal and data from benchmark vehicle scheduling instances shows the ability of the approach for producing a variety of solutions within reasonable computing times as well as the advantages of integrating the three problems. Keywords: transportation, vehicle and crew scheduling, driver rostering, Benders decomposition. 1 Introduction Operational planning is an important phase in the planning of activities within public transit companies. At this stage, scheduling for both the vehicles and the drivers is defined so that transport demand in a specific area is satisfied. 1

3 Due to the great complexity of these problems, most transit companies use computer-aided systems for tackling them on a sequential basis. Firstly, the vehicle scheduling problem is solved, secondly the crew scheduling and, lastly, the driver rostering. The vehicle scheduling problem takes as input a set of timetabled trips and produces the vehicle schedules that cover all timetabled trips (or trips for short). A vehicle schedule is the sequence of trips to be assigned to a vehicle on a day. The crew scheduling problem defines the daily crew duties that cover the vehicle schedules. A crew duty is the sequence of tasks to be assigned to a single driver on a day. Finally, for a planning horizon of several days, crew duties are assigned to the drivers of the company leading to a roster that must comply with labor and institutional norms. A roster is the sequence of crew duties assigned to each driver throughout the planning horizon. The sequential approach takes the output of the current problem as the input of the next one. Consequently, one cannot guarantee that the final result is the best solution to the overall problem since there is a high dependency among these three problems. In order to overcome this drawback, it is important to develop integration methodologies. The integrated vehicle-crew-roster problem aims to simultaneously assign drivers of a company to vehicles and assign vehicles to a set of pre-defined timetabled trips that cover passenger transport demand in a specific area, during a planning horizon. Despite its computational burden, some work has been reported on the integration of all or some of these problems expecting to outperform the corresponding sequential approaches. [11], [10], [18], [2], [20] have developed efficient algorithms for the integrated vehicle-crew scheduling problem. Crew-rostering integration has been devised by [4], [7], [8] and [12] albeit within other transport contexts (railway and air crews) and by [5] for airport staff. Advantages of integrating these three problems were pointed out in [16] through a statistical study over different solutions resulting from a sequential approach where rosters were produced, satisfying labor and internal company laws, without any pre-defined pattern for working or rest periods. In that work, some statistical tests revealed no correlation between the quality of integrated vehicle-crew solutions and the corresponding rostering solutions, hence highlighting the need to develop methodologies that address these three problems in an integrated form. Considering those conclusions and a preliminary study presented in [17], this paper proposes a new heuristic approach with embedded column generation and branch-and-bound techniques within a Benders decomposition to solve the integrated vehicle-crew-roster problem with days-off pattern (VCRPat). The VCRPat solution consists of daily vehicle and crew schedules and a roster that complies with a days-off pattern in use at some Portuguese public transit companies for a specific group of drivers that share the same rest periods pattern but not necessarily the same type of work. Some special features of the VCRPat mathematical formulation are exploited leading to a decomposition algorithm that iterates between the solution of daily integrated vehicle-crew scheduling problems and the solution of a rostering problem. The resulting heuristic algorithm seems to be a promising alternative tool to the sequential approach. 2

4 For airline operations, Benders decomposition methods have already been proposed by [6] and [14] for the integrated aircraft routing and crew scheduling problem and by [15] for the integrated aircraft routing, crew scheduling and flight re-timing problem. Also in the same application context, [19] presented a Benders decomposition approach to deal with the integrated fleet assignment, aircraft routing and crew scheduling problem. This paper is organized as follows: Section 2 introduces the VCRPat in detail, Section 3 presents a mathematical formulation, Section 4 gives a description of the decomposition approach, Section 5 shows computational results taken from sets of real and benchmark based instances and Section 6 presents some conclusions. 2 Problem definition During a planning horizon H of 7 weeks, partitioned into 49 days, drivers from a set M must be assigned to a fleet of vehicles from a set D of depots in order to perform a set of timetabled trips. The location and the number of vehicles available in each depot is known. Let N h be the set of trips to be performed on day h. For each trip, the starting and ending times and locations are known. Trips i and j are compatible if the same vehicle can perform both in sequence. Between compatible trips i and j and between trips and depots, a deadhead trip may occur where the vehicle runs without passengers. There are three types of deadhead trips: those between the ending location of a trip and the starting location of a compatible trip, those from a depot to the starting location of a trip (pull-out) and those from an ending location of a trip to a depot (pull-in). Without loss of generality and in order to simplify the notation, depots will be seen as special trips. Let I h be the set of pairs of compatible trips (i, j) on day h. The sequence of timetabled trips and deadhead trips performed by a vehicle on day h H defines a vehicle schedule. Each vehicle schedule starts and ends at the same depot and is subdivided at points defined by time and location, called relief points, where a change of driver may occur. Two consecutive relief points define a task, that is, the smallest amount of work to be assigned to the same vehicle and crew. Depots as well as the ending location of timetabled trips are potential relief points. Hence, task (i, j) corresponds to the deadhead from trip i to trip j followed by trip j. A crew duty is a daily combination of tasks that respects labor law, union contracts and internal rules of the company. These rules depend on the particular situation under study and usually constrain the maximum and minimum spread (time elapsed between the beginning and ending of a crew duty), the maximum working time without a break, break duration. Drivers may switch between two vehicles if the relief points are close enough in time and walking distance. Let L h be the set of crew duties of day h and L h ij Lh the set of crew duties covering task (i, j), on day h. Each crew duty has to be assigned to a driver who works according to a pre- 3

5 Table 1: Cyclic days-off pattern Monday Tuesday Wednesday Thursday Friday Saturday Sunday Table 2: Schedule set S week 1 week 2 week 3 week 4 week 5 week 6 week 7 S S S S S S S defined days-off pattern. The specific pattern considered in this study is used in some Portuguese public transit companies for specific groups of drivers. For a time horizon of 7 weeks (49 days), Table 1 displays such a pattern through a 0 1 matrix, where 0 stands for day-off and 1 for workday. Each row of the matrix corresponds to a weekday. Seven consecutive columns correspond to the 7 weeks of the time horizon, being the last day in column i (i = 1,..., 6) followed by the first day in column i + 1 and the last day in column 7 followed by the first day of column 1. Since a 49 days schedule may begin in row 1 of any column, this days-off pattern leads to a schedule set S with 7 schedules (Table 2). Each schedule S i, i = 1,..., 7 defines a sequence of working and rest periods, containing 1 week with 1 day-off, 5 weeks with 2 consecutive days-off and 1 week with 3 consecutive days-off. Every period of 3 consecutive days-off includes Saturday and Sunday and if it also includes a Friday then it will be followed by 5 consecutive workdays. The pattern in Table 1 is cyclic only in what concerns the days-off. Hence any number of drivers may be assigned to each schedule, getting each driver the same number of days off and the same type of rest periods, at the end of the 49 days. A roster is an assignment of drivers to schedules of S covering all the crew duties defined for the planning horizon where each driver is assigned to at least one schedule and each crew duty is covered by exactly one schedule. The cyclic days-off pattern satisfies some compulsory constraints: the minimum number of days-off per week (1 day), the minimum number of consecutive days-off (2 4

6 days), the minimum number of Sundays/weekends off in the planning period (2 days) and the maximum number of consecutive workdays (6 days). Besides these constraints, drivers must rest a minimum number of hours between two consecutive crew duties and, consequently, a crew duty starting in the morning cannot be assigned to a driver that had worked on a crew duty starting in the evening of the previous day. Moreover, for a roster to be well accepted in the company it should attain balanced workload, meaning that different spread duty types should be equitably distributed among drivers, thus leading to a fair distribution of overtime as well as total working time. The VCRPat aims at simultaneously determining a minimum cost set of vehicle schedules that daily covers all timetabled trips, a minimum cost set of crew duties that daily covers all vehicle schedules and a minimum cost balanced roster for the planning horizon. 3 Mathematical formulation Decision variables, related to the scheduling of vehicles, are defined as zij dh = 1 if a vehicle from depot d performs trips i and j in sequence on day h, or 0 represent, respectively, the pull-out from, related with fuel consumption and/or vehicle maintenance, is associated with the deadhead trip from i to j. Daily vehicle schedules are covered with crew duties. Decision variables wl h are defined for feasible crew duties, therefore the rules defining the feasibility of the crew duties are handled in the model through L h. Let wl h = 1 if crew duty l is selected on day h, or 0 otherwise. A cost e l is assigned to each crew duty l. Cost e l usually includes a fixed cost (for example, a driver s salary) and operational costs related to overtime, evening periods and others. In the rostering process, each crew duty in the solution is assigned to a driver who works according to one of the seven pre-defined days-off schedules. and z dh id otherwise. In particular, zdj dh depot d to trip j and the pull-in from i to depot d. A vehicle cost c d ij Let x m s = 1 if driver m is assigned to schedule s, or 0 otherwise, and let yl mh = 1 if driver m performs crew duty l on day h, or 0 otherwise. Management often wishes to know the minimum workforce required to operate the fleet of vehicles, so as to transfer drivers to other departments of the company or to replace those absent. Such policy results in minimizing crew duty costs e l associated with variables wl h and costs rm associated with x m s. To avoid infeasible sequences of crew duties in a roster, for each day h, the set of crew duties, L h, is partitioned into early duties, L h E, starting before 3:30 p.m. and late duties, L h A, starting after 3:30 p.m. Moreover, according to the spread, set L h is also partitioned, into short duties, L h T, which have a maximum spread of 5 hours (without a break), normal duties, L h N, with spread ]5, 9] hours, and long duties, L h O, with spread ]9, 10.75] hours (with overtime). Rosters should attain balanced workload among drivers which can be measured by the number of crew duties of each spread type assigned to the same driver. For this purpose, variables η T and η O that represent, respectively, the maximum number of short 5

7 and long duties assigned to a driver during H are associated with penalties λ T and λ O. The VCRPat can be formulated through the following integer linear programming model: min ( h H d D d D c d ijz dh ij (i,j) I h + l L h e l w h l ) + m M s S r m x m s + λ T η T + λ O η O (1) z dh ij = 1, j N h D, h H (2) i:(i,j) I h ij ji = 0, j N h, d D, h H (3) i:(i,j) I h z dh i:(j,i) I h z dh z dh di ν d, d D, h H (4) i N h wl h zij dh 0, (i, j) I h, h H (5) l L h d D ij m M y mh l w h l = 0, l L h, h H (6) x m s 1, m M (7) s S yl mh a h s x m s 0, m M, h H (8) l L h s S yl mh + 1, m M, h H {1}, f g {E, A}(9) l L h f h H l L h t l L (h 1) g y m(h 1) l y mh l η t 0, m M, t {T, O} (10) z dh ij {0, 1}, (i, j) I h, d D, h H (11) wl h {0, 1}, l L h, h H (12) yl mh {0, 1}, l L h, m M, h H (13) x m s {0, 1}, s S, m M (14) η T, η O 0 and integer. (15) The objective function, a linear combination of all objectives, measures the quality of the solution in terms of vehicle and driver costs, as well as roster balancing. Constraints (2) and (3) describe the vehicle scheduling problem. Constraints (2) state that each timetabled trip is performed, exactly once, by a vehicle that comes directly from a depot or from the ending location of another timetabled trip. Constraints (2) together with (3) ensure that each vehicle returns to its source depot. Constraints (4) are depot capacity constraints 6

8 where ν d is the number of vehicles available at depot d. Constraints (5) link vehicle and crew duty variables ensuring that each task in a vehicle schedule is covered by, at least, one crew duty. Equalities (6) impose that each crew duty, in a solution, must be assigned to one driver. Constraints (7) state that each driver is assigned to one of the seven days-off schedules or is available for other service in the company. Constraints (8), where parameter a h s = 1 if h is a workday on schedule s, impose coherence between the assignment of a crew duty and the schedule assigned to each driver. Inequalities (9) forbid the sequence late-early duties to ensure that drivers rest a given minimum number of hours between consecutive duties. Furthermore, (9) impose a day-off period between changes of duty types which also forbid the sequence early-late. Inequalities (10) calculate the maximum number of short/long duties assigned to a driver. Constraints (10) along with the two last terms of the objective function ensure the integrality of variables η T and η O. The days-off schedules defined by the matrix [a h s ] already satisfy some of the obligatory rostering constraints. Note that, without this matrix additional rostering constraints are needed to model the rosters as may be seen in [16] where the authors presented a mathematical formulation for the integrated vehiclecrew-roster problem without any underlying pattern. A decomposition approach is suggested by the three combinatorial structures included in the above mathematical formulation, (1)-(15): an integer multicommodity network flow problem related with the daily scheduling of vehicles; a set covering structure defining the set of crew duties that daily cover the vehicle schedules; a mixed covering-assignment problem with additional constraints that defines the roster covering all (daily) crew duties for the planning horizon H. 4 Decomposition approach In 1962, Benders proposed a decomposition algorithm for solving large single objective mixed integer linear programming problems [1]. The algorithm alternates between a primal subproblem and a master problem. The subproblem is a restriction of the original problem where some decision variables values are fixed. In each iteration, the solution of the dual subproblem is used to construct Benders cuts. These cuts are added to the master problem and the resulting solution provides the values for the primal variables that are considered fixed in the subproblem. This method guarantees the convergence to the optimum under specific hypotheses later generalized by [9]. Such technique has been applied to different combinatorial optimization contexts (see [3]). The VCRPat may naturally be decomposed into a master problem and a subproblem. In fact, different temporal scheduling problems may be identified in the mathematical formulation for VCRPat: H daily integrated vehicle-crew scheduling problems, with variables z and w for the vehicle schedules and for the crew duties, respectively; and a rostering problem for the whole planning horizon, with variables y, x and η. These temporal scheduling problems share 7

9 constraint set (6) involving variables w and y. We propose a new heuristic approach based on Benders decomposition, that alternates between the solution of a master problem involving the z, w variables, a vehicle-crew scheduling problem for each day of H, and the solution of the corresponding subproblem involving the y, x, η variables, a rostering problem for H. The following sections describe the subproblem, the master problem and the decomposition algorithm for VCRPat, respectively. 4.1 The subproblem Fixing the values of variables z and w in VCRPat at z and w, given by the optimal solution of the master problem, one obtains the subproblem, SUB zw : h H( d D c d ijz dh ij (i,j) I h + l L h e l w h l ) + min m M m M s S r m x m s + λ T η T + λ O η O (16) y mh l = w h l, l L h, h H (17) (7) to (10), (13) to (15). (18) Note that, SUB zw is a rostering problem where (z, w) induces the sets of vehicle schedules and of crew duties. Let us denote by LSUB zw the linear programming relaxation of SUB zw where (13), (14) and (15) are replaced by yl mh 0, x m s 0 and η T, η O 0, respectively. The dual of LSUB zw is denoted by DLSUB zw being α, β, γ, δ and ζ the dual variables corresponding, respectively to (17), (7), (8), (9) and (10) in LSUB zw. If the number of drivers is enough to cover all crew duties given by w, while respecting all the other constraints, then LSUB zw has a feasible solution and thus an optimal solution (the feasible region is bounded along the optimization direction). In this case, a Benders cut is obtained from the optimal dual solution which corresponds to an extreme point of the feasible region of DLSUB zw. If the available drivers are not enough to cover all the crew duties in w then LSUB zw is infeasible and DLSUB zw is unbounded (there is always at least a feasible dual solution, the nil vector). In this case, the corresponding Benders cut is obtained from an extreme ray of the feasible region of DLSUB zw. The new cut is added, as a constraint, to the master problem. 4.2 The master problem Let P D and RD be, respectively, the set of extreme points and extreme rays, to be used in the current iteration of the decomposition algorithm. Suppose that P D has k1 extreme points and RD has k2 extreme rays. Then the master problem is MAST ER: min ϕ 0 (19) 8

10 ϕ 0 ( h H d D 0 h H c d ijz dh ij (i,j) I h + l L h (e l + α h lp q )w h l ) + τ q, q = 1,..., k1(20) l L h α h lr q w h l + ϱ q, q = 1,..., k2 (21) (2), (3), (4), (5), (11), (12) (22) where τ q = m M ϱ q = m M β m p q + β m r q + h H {1} m M f,g {E,A},f g h H {1} m M f,g {E,A},f g δ mhfg p q, δ mhfg r q, (α pq, β pq, γ pq, δ pq, ζ pq ) P D and (α rq, β rq, γ rq, δ rq, ζ rq ) RD. Constraints (20) and (21) defined from the Benders cuts involve variables related with all the days of the planning horizon and the resulting master problem is as difficult to solve as the original VCRPat. Moreover, the convergence results of the Benders algorithm to an optimal VCRPat solution do not apply here due to the combinatorial nature of the subproblem, SUB zw. In fact, Benders decomposition theory guarantees that an optimal solution for VCRPat y,x,η 0 is achieved, assuming it exists, where VCRPat y,x,η 0 is the problem obtained from relaxing the integrality constraints for the y, x, η variables in VCRPat. However, such solution might not be feasible for VCRPat due to the possibility of obtaining non-integer values for the y, x, η variables. Consequently, a non-exact approach is proposed where, in each iteration, the Benders cuts are relaxed into the MAST ER objective function, associated with non-negative multipliers u q and v q. A similar relaxation has been used, for instance, by [13] for tractors and trailers scheduling. In the current iteration, the relaxed master problem may be written as RMAST ER: k1 min ϕ 0 + q=1 u q ( ϕ 0 + h H d D c d ijz dh ij (i,j) I h + (e l + α h lp q )wl h + τ q ) + h H l L h k2 + v q ( α h lr q wl h + ϱ q ) (23) q=1 h H l L h (2), (3), (4), (5), (11), (12). (24) Any choice for the multiplier values (u q, v q ) that satisfy k1 q=1 u q = 1 converts (23) into: min h H ( d D c d ijz dh ij (i,j) I h + k1 k2 e h l wl h ) + u q τ q + v q ϱ q (25) l L h q=1 q=1 9

11 where k1 k2 e h l = e l + u q α h lp q + v q α h lr q. (26) q=1 Note that, a possible choice for (u q, v q ), that satisfies k1 q=1 u q = 1, is given by the dual variables associated with the Benders cuts constraints (20) in the linear relaxation of MAST ER. RMAST ER, rewritten as (25), (24) and (26), can be partitioned into H independent problems, one for each day of the planning horizon H. In each iteration of the decomposition procedure, crew duty costs in RM AST ER are updated with the multipliers and the DLSUB zw variables associated with constraint set (17), thus becoming dependent on the day h to which they relate. Usually, public transit companies satisfy two/three different transport demand types: one for Monday to Friday and one/two for weekends (Saturday/Sunday). Since days with the same demand type share the same set of trips, due to operational reasons they must have the same set of crew duties to be executed. Should the crew duty costs e h l in (25) not depend on the day h, as it occurs in practice, solving RMAST ER would reduce to solve two/three independent integrated vehicle-crew scheduling problems, one for each day-type. Therefore, to keep crew duty costs independent of the day h in which they occur, days are clustered according to the respective day-type demand. For each day-type, crew duty costs are penalized by considering a linear combination of all dual α variables values corresponding to the days h of this type. Then an integrated vehicle-crew scheduling problem is solved for each day type. Finally, the solution of RMAST ER is built by replicating for all days of H the solutions obtained for the corresponding day-types. In summary, Benders methodology is used to point out how and which crew duties should be penalized in order to obtain a set of vehicle-crew schedules that will lead to a better roster solution. q=1 4.3 The decomposition algorithm Algorithm 1 describes the non-exact Benders decomposition algorithm based on the proposals of the previous sections. The main body of the algorithm cycles up to a maximum of itermax iterations. In iteration k it solves RMAST ER k with the current sets of extreme points and extreme rays, P D k and RD k, (step 3), obtaining solution (z k, w k ), and then it solves LSUB (zk,w k ) (from step 4 to step 8). In step 3 an integrated vehicle-crew scheduling problem is solved for each day-type and the solution for RMAST ER k is built by replicating for all days of H the solutions obtained for the corresponding day-types. The integrated vehicle-crew scheduling problems are solved by the algorithm proposed in [18] which combines a heuristic column generation procedure with a branchand-bound scheme. The algorithm works in three phases. In phase 1, the set of tasks is defined and an initial set of crew duties is obtained. In phase 2, the linear programming relaxation of the model is solved using a column generation 10

12 scheme where the vehicle variables z are explicitly considered and the crew variables w are implicitly considered. In fact, each feasible crew duty can be seen as a path in a network where the feasibility is established by using resources that are consumed along the network. The resource consumption is restricted by imposing time windows on the vertex of the network. The pricing problem is formulated as a resource-constrained shortest-path problem and is approximately solved using dynamic programming with a reduced state space that leads to a smaller but varied crew duty set, while maintaining the quality of the linear relaxation bound. This diversity of generated columns is an important factor since, whenever the solution of the linear programming relaxation is not integer, a third phase is needed where branch-and-bound techniques are used over the set of feasible crew duties generated at the root node. In step 4, exact standard algorithms are used to solve the linear programming relaxation of the rostering subproblem, LSUB (zk,w k ). If the LSUB (zk,w k ) solution is not integer and the corresponding value is less than the best upper bound (step 6), then branch-and-bound techniques are applied to obtain a roster (step 7). Solutions for the rostering subproblem involve a large number of binary variables and a great amount of resources are needed to obtain these solutions. To overcome such drawback, in step 7, different strategies have been incorporated into the branching process, yielding in most cases a feasible solution in short computing time. Depending on which subset of variables branching dichotomy is based, three branching strategies have been combined with variable fixing as follows: strategy 1: branching is performed over variables x and Σy; strategy 2: branching is performed over all integer variables while fixing to 1 the values of x and y such that x > a 0 and y > a 1 in LSUB (zk,w k ) optimal solution; strategy 3: branching is performed over variables y, while fixing to 1 the values of y such that y > a 2 in LSUB (zk,w k ) optimal solution. l L h t Notation Σy represents the vector of aggregated variables h H yl mh, m M, t {T, O}. In the first branching scheme, only variables x and the sums Σy are required to be integer. Consequently, in the final solution variables y might not be integer. To avoid obtaining non-integer solutions, a second branching scheme is considered where all x and y variables are required to be integer. In order to guarantee that integer solutions will be obtained in reasonable CPU time, the value of some variables is fixed using information given by the LSUB (zk,w k ) solution. As for the third strategy, branching over a subset of variables is combined with variable fixing. The thresholds a 0, a 1 and a 2 will be defined in the next section. Finally, the master problem solution (z k, w k ) - set of vehicle-crew schedules covering the planning horizon - along with the corresponding integer solution of the rostering subproblem (y k, x k, η k ) - roster for the horizon - lead to a feasible solution for the VCRPat and, consequently, to an upper bound on the 11

13 optimal value of VCRPat. In step 10 of Algorithm 1, the new feasible solution is included in the pool of VCRPat solutions.decision makers will further analyze the solutions of the pool, from a practical perspective, through different criteria. Algorithm 1 Decomposition algorithm pseudo-code Require: H, M, D, N h (h H), S, [a s h ](s S, h H), itermax 1: UB = + ; P D 1 = ; RD 1 = ; k = 1 {initialization} 2: repeat 3: solve RMAST ER k {that is, solve an integrated vehicle-crew scheduling problem for each day-type and build the solution for all days h H} let (z k, w k ) be this solution 4: solve LSUB (zk,w k ) and update P D k and RD k 5: if LSUB (zk,w k ) has a feasible solution, let (y k, x k, η k ) be its optimal solution and υ(y k, x k, η k ) the corresponding objective value then 6: if (y k, x k, η k ) is not integer and υ(y k, x k, η k ) UB then 7: apply branch-and-bound techniques 8: end if 9: if an integer solution (y k, x k, η k ) is available then 10: update UB and the pool of VCRPat solutions with (z k, w k, y k, x k, η k ) 11: end if 12: end if 13: k = k : until k itermax 5 Computational experiments A computational experiment was performed using instances based on real world data from two Portuguese public transit companies and also benchmark test instances. We have considered that all relaxed Bender cuts are equally penalized in the sense that in each iteration of the decomposition algorithm all the multipliers (u q, v q ) have the same value. Parameter itermax of Algorithm 1 was set to 10 as a result of previous experiments. The algorithms were coded in C, using VStudio 6.0/C++ and the programs ran on a PC Pentium IV 3.2 GHz. Linear programming relaxations and branch-and-bound schemes were solved with CPLEX 11.0 [21]. 5.1 Instances based on real data The input of each VCRPat instance includes the starting and ending times and locations for each timetabled trip and deadhead times between locations and depots. Two sets of real data instances were tested. The first set was derived from an urban bus service operating in the city of Lisboa and includes instances L122, L168, L224, L226, L238 with, respectively, 122, 168, 224, 226 and

14 timetabled trips for Monday through Friday and vehicles from four depots. The second set was built from an urban bus service operating in the city of Porto and involves instances P108a, P108b, P156, P250 and P280 with, respectively, 108, 108, 156, 250 and 280 timetabled trips for Monday through Friday and vehicles from one depot. For all data sets, two types of demand for timetabled trips are considered, one for Monday through Friday and the other for weekend days. The number of timetabled trips for the weekends is around 40% of the number of timetabled trips for Monday through Friday. In order to satisfy the rules imposed by Portuguese labor law, union contracts and specific rules for both bus companies, some parameters have to be defined. Concerning the scheduling of the vehicles and crews: a penalty of 5000 is added to the cost of each pull-in and each pull-out trip in order to minimize the number of vehicle schedules; for each crew duty the minimum spread is set to 2 hours; the maximum spread is 5 hours for crew duties without a break (short duties), 9 hours for normal duties and 10 hours and 45 minutes for long duties; break times range from 1 hour to 2 hours and 20 minutes; the maximum duration allowed for a crew duty before a break occurs is 5 hours; 10(5) minutes is set for sign-in(sign-out) when the driver leaves(enters) the depot driving a vehicle; when the driver enters(leaves) the depot without a vehicle, the sign-in(sign-out) is 15 minutes; each crew duty cost is set to Depots as well as the ending location of timetabled trips are the potential relief points where a change of driver is allowed. In this context a driver can switch between two different vehicles at a relief point if and only if the location where he leaves the first vehicle is the same where he picks up the second one. Concerning the rostering, the parameters used to balance the different components of the objective function in the subproblem were set to r m = 5 (m M), λ O = 1 and λ T = 0.5. Such choice of parameters is explained by the main management goal of minimizing the number of drivers (penalized by r m ). Besides, minimizing the number of long duties (associated with parameter λ O ) is more relevant than minimizing short duties (associated with λ T ) due to overtime costs incurred by long duties. To compute the VCRPat objective function value, vehicle-crew costs are scaled to balance with roster costs. 13

15 5.2 Computational results In each iteration of the decomposition algorithm, vehicle-crew schedules for the planning horizon, a linear relaxation rostering solution and, eventually, a roster are obtained from the master and the subproblem, respectively. We have started in Section by studying the performance of the subproblem branch-andbound. Sections and present the decomposition algorithm results Branching strategies A comparison of the quality of the rosters and the time needed to obtain them was carried out for the real data instances under the three branching scenarios proposed in Section 4.3. The corresponding computational results are presented in Table 3. For each test problem and each branching strategy, 10 iterations of the decomposition algorithm were executed. Previous computational experience suggested to fix the thresholds a 0 and a 1, in strategy 2, at 0.75 and , respectively, and to fix a 2 at 0.85 in strategy 3. Column instance of Table 3 identifies the test instances. Column strategies displays the branching strategy, columns MILP and IntSol refer, respectively, to the number of times the branch-and-bound was performed and the number of times an integer solution, a roster, was obtained. The following four columns are related to the quality of the best integer solution obtained by the branch-andbound: the rostering component of the objective value ( BestSol ); the number of drivers assigned to work schedules ( driv ); the maximum number of short duties ( η T ) and the maximum number of long duties ( η O ) assigned to a single driver during the planning horizon of 49 days. The last column reads the total CPU time for the branch-and-bound, in seconds, excluding the root node CPU. Except for intance P280, strategy 3 attained the best performance concerning the number of times branch-and-bound was performed ( MILP ). Actually, for most cases, strategy 3 attained a good upper bound in early iterations. This feature makes false the condition in step 6 of Algorithm 1, thus skipping step 7, the branch-and-bound. As to the number of rosters obtained ( IntSol ), strategy 1 ( x, Σy int ) yielded the worse results for most instances. In fact, this strategy was the unique that did not produce integer solutions for some runs of MILP (see instances L224, P156, P250 and P280). As expected, branching over variables x m s (strategy 1) that define the assignment of each driver to a schedule is weaker than branching over variables yl mh that define the assignment of a driver to a specific crew duty in a specific day (strategy 3). Regarding the solution quality, strategy 3 yielded the best solution ( Best- Sol ) in nine instances out of 10. The results presented in the CPU column for the three strategies confirm that the branching strategy strongly affects the CPU time. For most cases, the smallest CPU times were obtained for strategy 3 whilst the largest ones were obtained for strategy 1. Strategy 2 produced the best results for instances L122, P108a and P108b which have the smaller number of timetabled trips. However, balancing the CPU 14

16 time and the solution quality, strategy 3 is the best choice in seven instances out of 10. Thus, in subsequent computational results, we adopted strategy 3 in step 7 of Algorithm Decomposition algorithm results - real data In this section we discuss computational results from the decomposition algorithm carried out on the set of 10 real based instances. These data always generated feasible LSUB zw, hence the Benders cuts were all generated from extreme points. Tables 4 and 5 show the solutions obtained from the relaxed master and from the subproblem. In these tables, for each test instance, the first row shows the first iteration solution, which corresponds to the sequential approach, and the second row describes the best result within the 10 iterations. Row improv highlights the improvement of the integrated approach over the sequential approach. Table 4 reports on the integrated vehicle-crew solutions for the two demand type days: Monday through Friday and weekend days in brackets. The second column shows the number of daily trips for Monday through Friday and for each weekend day. Column colgen reports on the number of generated crew duties in the first iteration and, in row improv, on the additional generated crew duties from iteration 2 to iteration 10. Columns vehicles and crews display, respectively, the number of daily vehicles and the number of daily crew duties from Monday through Friday and for the weekend, partitioned into short/normal/long. In rows improv, one can see that the number of additional crew duties generated is less than 17% of the number of crew duties generated in the first iteration. This shows that the decomposition algorithm requires a small additional effort, over the sequential algorithm, to generate new sets of crew duties that potentially lead to better integrated vehicle-crew-roster solutions. Although the solutions vary from iteration to iteration, the number of vehicles per day as well as the number of crew duties per day are the same in the first and in the best iteration, except for P108b where the number of weekend crew duties increases by one unit. Column crews shows, in general, a reduction in the number of short/long duties offset by an increase in the number of normal duties. By replacing short/long duties with normal duties, the decomposition algorithm fairly distributes work among the drivers. Moreover, for most instances, the number of long duties decreases, reducing overtime, which means cost savings for the company. Table 5 reports on the rosters obtained by the branch-and-bound (step 7 of Algorithm 1). Column M gives, for each instance, the number of drivers available for work. This number is set to 20% above the total number of crew duties of the planning horizon divided by 35 (working days per work schedule). Column totcrews shows the total number of crew duties partitioned into short/normal/long. These values are calculated from column crews in Table 4 by replicating 35 times the number of crew duties obtained for Monday through 15

17 Table 3: Rostering branching strategies instance strategies MILP IntSol BestSol driv η T η O CPU (s) L122 x,σy int all int + fix x,y y int + fix y L168 x,σy int all int + fix x,y y int + fix y L224 x,σy int all int + fix x,y y int + fix y L226 x,σy int all int + fix x,y y int + fix y L238 x,σy int all int + fix x,y y int + fix y P108a x,σy int all int + fix x,y y int + fix y P108b x,σy int all int + fix x,y y int + fix y P156 x,σy int all int + fix x,y y int + fix y P250 x,σy int all int + fix x,y y int + fix y P280 x,σy int all int + fix x,y y int + fix y

18 Friday plus 14 times the number of crew duties obtained for a weekend day. For column totcrews, row improv shows the savings in overtime cost for the planning horizon measured by the reduction in the number of long duties. Column driv presents the number of drivers assigned to work schedules and column η T /η O shows the maximum number of short/long duties assigned to a driver during the planning horizon. By comparing columns M and driv in Table 5 one can see that the number of drivers assigned to driving work is on average 5.38% below the number of drivers available. For these test instances, row improv in column totcrews shows significant savings in overtime costs corresponding to an average decrease of 7.2% on the number of long duties. Instance L238 is the exception with an increase of 28 long duties. This may be explained by one less driver assigned to work schedules which in turn may be due to the reduction on the number of late duties from 77 to 28 (see column early/late ). Note that, concerning instances L226 and P280 the algorithm reduced not only the number of drivers assigned to work ( 1 and 2, respectively), but also the number of long duties ( 119 and 140, respectively). The global reduction in the η T /η O values in rosters arises as a natural consequence of the decrease on the number of short/long duties in the RMAST ER solution. A lower bound for the η T (η O ) value may be calculated by dividing the total number of short (long) duties by the number of drivers assigned to driving work. Comparing the η T (η O ) values with the corresponding lower bounds, one may conclude that the algorithm attained the lower bound for η T in nine instances (for η O in seven instances). In most cases, the decomposition algorithm gives a better VCRPat solution than the sequential approach which corresponds to the first iteration of the algorithm. Actually, except for L224, P108a and P250, the decomposition approach improves on the first solution. In six out of the 10 instances, the best solution was obtained at the fifth or at a later iteration. As expected, Benders cuts adjust crew duty costs in RMAST ER problem, thus inducing better balanced sets of (daily) crew duties that lead to better rostering solutions in what concerns the number of drivers and/or the maximum number of short and long duties assigned to a driver. Table 6 reports on the CPU times in seconds spent to solve the relaxed master problem ( RM AST ER ), the linear relaxation of the subproblem ( LSU B ), the subproblem branch-and-bound ( SUB B ) and at last the total CPU time ( total ). For each instance, the first row concerns to the first iteration and the second row shows the total time for the 10 iterations. On average, the decomposition algorithm CPU time for the 10 iterations is 6.97 times the CPU spent by the sequential approach. In general, the subproblem CPU time contributes with the major part of the total CPU time. Instance P250 was the hardest one to solve, not only due to the large number of trips (250) but also because it has the largest percentage of late duties (12%) when compared with the other instances. This percentage is 7% for P280 and L122 and even lower for all the other instances. Note that, we have included in 17

19 the mathematical model (1)-(15) a set of constraints that forbids the sequence early-late duties which usually are modeled as soft constraints within mass transit companies. However, for the specific companies we are considering, they are hard constraints thus making more difficult the assignment of late duties Decomposition algorithm results - random data Although the main objective of this study is to address the specific situation that arises at two mass transit companies it is also important to assess the effectiveness and robustness of the proposed decomposition approach under different conditions. Therefore we present some results for the decomposition algorithm running with random data instances for the multiple-depot vehicle and crew scheduling problem that are available at huisman/instances.htm. Since the number of trips of the real data is around 200, we have selected the set of 10 instances trips200axx.txt and four depots. Each file trips200axx.txt gives the daily timetabled trips for Monday through Friday, being the weekend trips randomly selected, with probability 0.4, from the respective daily trips set. The same parameters described in Section 5.1, for the scheduling of vehicles and crews and the rostering, as well as 10 iterations of the decomposition algorithm, were considered. As in the case of real data, feasible LSUB zw were always obtained, hence the Benders cuts were all generated from extreme points. Tables 7 to 9 follow the same format of Tables 4 to 6. As for the real based instances, Algorithm 1 showed improvement on the results of the sequential approach. In general, the number of vehicles per day remained the same for all iterations. Even when the number of daily crew duties increased, a decrease in overtime was achieved by reducing the number of long duties, thus leading to more balanced rosters. In particular, looking at the RMAST ER solution for 200A10 instance we see that the best solution includes one more vehicle and two less daily crews than the first solution, although both solutions have the same costs in terms of the vehicle-crew objective component. But two less daily crew duties give the 70 less crew duties per period, as seen by comparing the figures in column early/late in table 8. This adjustment leads to less three drivers assigned to work schedules which represent significant cost savings for the integrated vehicle-crew-rostering problem even taking into account the increase of overtime. As to the CPU time, random data instances showed a greater computational effort than that of the real ones. In the random instances the major part of the time was spent in the relaxed master, while in the real data instances the major part was spent for obtaining the rosters. This may be explained by the fact that instances trips200axx.txt were specially devised for testing the integrated vehicle and crew scheduling problem. Moreover, these random instances have far less late duties than real based data thus turning the rostering integer solution easier to obtain. 18

20 Table 4: Decomposition algorithm - RMASTER daily solutions - real data instance trips iter colgen vehicles crews L (50) first 3676 (268) 9 (6) 1/8/8 (4/3/2) best 6 9 (6) 0/9/8 (3/5/1) improv +306 (+82) 0 1/+1/0 ( 1/+2/ 1) L (67) first 3944 (362) 17 (10) 4/14/20 (7/6/6) best 7 17 (10) 5/17/16 (5/9/5) improv +527 (+67) 0 +1/+3/ 4 ( 2/+3/ 1) L (89) first 6603 (1224) 18 (10) 0/9/30 (1/4/11) best 1 18 (10) 0/9/30 (1/4/11) improv +543 (+109) 0 0/0/0 (0/0/0) L (90) first 5901 (500) 16 (7) 0/8/26 (0/6/9) best 8 16 (7) 0/11/23 (0/7/8) improv +346 (+142) 0 0/+3/ 3 (0/+1/ 1) L (95) first 6015 (398) 22 (11) 5/13/36 (11/8/8) best 5 22 (11) 4/14/36 (8/9/10) improv +552 (+112) 0 1/+1/0 ( 3/+1/+2) P108a 108 (44) first 8929 (442) 6 (6) 0/4/7 (4/1/3) best 1 6 (6) 0/4/7 (4/1/3) improv +77 (+38) 0 0/0/0 (0/0/0) P108b 108(44) first 9358 (442) 6 (6) 0/2/9 (4/1/3) best 7 6 (6) 0/3/8 (3/3/3) improv +65 (+38) 0 0/+1/ 1 ( 1/+2/0) P (63) first 6103 (368) 15 (11) 3/9/16 (6/6/4) best 2 15 (11) 3/12/13 (6/6/4) improv +251 (+86) 0 0/+3/ 3 (0/0/0) P (102) first 6083 (724) 22 (15) 3/28/12 (10/8/8) best 1 22 (15) 3/28/12 (10/8/8) improv +319 (+150) 0 0/0/0 (0/0/0) P (114) first (1625) 20 (15) 0/9/30 (7/4/12) best (15) 0/13/26 (7/4/12) improv (+48) 0 0/+4/ 4 (0/0/0) 19

21 Table 5: Decomposition algorithm - RMAST ER and SUB - real data RMASTER solution roster instance M iter totcrews early/late driv η T /η O L first 91/322/ / /14 best 6 42/385/ / /12 improv / 2 L first 238/574/ / /16 best 7 245/721/ / /12 improv / 4 L first 14/371/ / /24 best 1 14/371/ / /24 improv 0 0 0/0 L first 0/364/ /0 49 0/25 best 8 0/483/ /0 48 0/21 improv / 4 L first 329/567/ / /18 best 5 252/616/ / /19 improv / + 1 P108a 17 first 56/154/ / /18 best 1 56/154/ / /18 improv 0 0 0/0 P108b 17 first 56/84/ / /23 best 7 42/147/ / /23 improv /0 P first 189/399/ / /16 best 2 189/504/ / /13 improv / 3 P first 245/1092/ / /9 best 1 245/1092/ / /9 improv 0 0 0/0 P first 98/371/ / /22 best 10 98/511/ / /20 improv / 2 20

22 Table 6: Decomposition algorithm - CPU time - real data instance iter RM AST ER LSU B SU B B total (s) (s) (s) (s) L122 first iters L168 first iters L224 first iters L226 first iters L238 first iters P108a first iters P108b first iters P156 first iters P250 first iters P280 first iters