Driver Scheduling Research Articles

The Trip Scheduling Problem

The hours of service (HOS) regulations of the department of transportation severely restrict the set of feasible driver schedules. So much so that establishing whether a sequence of full truckload transportation requests, each with a dispatch window at the origin, can feasibly be executed by a driver is no longer a matter of simple forward simulation. We consider this problem and prove that the feasibility of a driver schedule can be checked in polynomial time by providing an O(n3) algorithm for establishing whether a sequence of full truckload transportation requests, each with a dispatch window at the origin, can be executed by a driver.

An Evolutionary Squeaky Wheel Optimisation Approach to Personnel Scheduling

The quest for robust heuristics that are able to solve more than one problem is ongoing. In this paper, we present, discuss and analyse a technique called Evolutionary Squeaky Wheel Optimisation and apply it to two different personnel scheduling problems. Evolutionary Squeaky Wheel Optimisation improves the original Squeaky Wheel Optimisation's effectiveness and execution speed by incorporating two extra steps (Selection and Mutation) for added evolution. In the Evolutionary Squeaky Wheel Optimisation, a cycle of Analysis-Selection-Mutation-Prioritization-Construction continues until stopping conditions are reached. The aim of the Analysis step is to identify below average solution components by calculating a fitness value for all components. The Selection step then chooses amongst these underperformers and discards some probabilistically based on fitness. The Mutation step further discards a few components at random. Solutions can become incomplete and thus repairs may be required. The repairs are carried out by using the Prioritization to first produce priorities that determine an order by which the following Construction step then schedules the remaining components. Therefore, improvement in the Evolutionary Squeaky Wheel Optimisation is achieved by selective solution disruption mixed with interative improvement and constructive repair. Strong experimental results are reported on two different domains of personnel scheduling: bus and rail driver scheduling and hospital nurse scheduling.

Locating drivers in a trucking terminal network

We consider the problem of determining the home locations, or domiciles, of truck drivers for a less-than-truckload carrier. Domiciling decisions are complex, in part due to regulations and union rules restricting driver schedules, but have a significant impact on the operating costs of less-than-truckload carriers. We present an iterative scheme, using driver dispatch technology in each iteration, to allocate drivers to terminals and to determine drivers’ bids so as to satisfy union requirements. Computational experiments demonstrate the value of the iterative scheme and quantify the impact of union rules on the number of drivers required (and the resulting operating costs).

Truck Driver Scheduling and U.S. Hours of Service Regulations

In the United States of America driving and working hours of truck drivers must comply with the hours of service regulations imposed by the U.S. Department of Transportation. Generating work plans complying with these regulations for a given sequence of locations to be visited is a non-trivial task if the service time at these locations is time constrained. For a planning horizon of one week this paper presents an O(\lambda^3) algorithm for problems in which each location must be visited within one of multiple time windows if the gap between these time windows is at least 10 hours. This situation occurs, for example, if, because of opening hours of docks, handling operations can only be performed weekdays between 8 hours and 22 hours. Furthermore, this paper presents a heuristic approach for scheduling driving and working hours of truck drivers for planning horizons of more than one week.

An Evolutionary Squeaky Wheel Optimization Approach to Personnel Scheduling

The quest for robust heuristics that are able to solve more than one problem is ongoing. In this paper, we present, discuss and analyze a technique called Evolutionary Squeaky Wheel Optimization and apply it to two different personnel scheduling problems. Evolutionary Squeaky Wheel Optimization improves the original squeaky wheel optimization's effectiveness and execution speed by incorporating two additional steps (selection and mutation) for added evolution. In the evolutionary squeaky wheel optimization, a cycle of analysis-selection-mutation-prioritization-construction continues until stopping conditions are reached. The aim of the analysis step is to identify below average solution components by calculating a fitness value for all components. The selection step then chooses amongst these underperformers and discards some probabilistically based on fitness. The mutation step further discards a few components at random. Solutions can become incomplete and thus repairs may be required. The repair is carried out by using the prioritization step to first produce priorities that determine an order by which the following construction step then schedules the remaining components. Therefore, improvements in the evolutionary squeaky wheel optimization is achieved by selective solution disruption mixed with iterative improvement and constructive repair. Strong experimental results are reported on two different domains of personnel scheduling: bus and rail driver scheduling and hospital nurse scheduling.

The train driver recovery problem—A set partitioning based model and solution method

The need to recover a train driver schedule occurs during major disruptions in the daily railway operations. Based on data from the Danish passenger railway operator DSB S-tog A/S, a solution method to the train driver recovery problem (TDRP) is developed. The TDRP is formulated as a set partitioning problem. We define a disruption neighbourhood by identifying a small set of drivers and train tasks directly affected by the disruption. Based on the disruption neighbourhood, the TDRP model is formed and solved. If the TDRP solution provides a feasible recovery for the drivers within the disruption neighbourhood, we consider that the problem is solved. However, if a feasible solution is not found, the disruption neighbourhood is expanded by adding further drivers or increasing the recovery time period. Fractional solutions to the LP relaxation of the TDRP are resolved with a constraint branching strategy using the depth-first search of the Branch & Bound tree. The LP relaxation of the TDRP possesses strong integer properties. We present test scenarios generated from the historical real-life operations data of DSB S-tog A/S. The numerical results show that all but one tested instances produce integer solutions to the LP relaxation of the TDRP and solutions are found within a few seconds.

Truck Driver Scheduling and Regulation (EC) No 561/2006

Since April 2007 working hours of truck drivers in the European Union are regulated by regulation (EC) No 561/2006. According to the new regulations, road transport undertakings must organise the work of drivers in a way that drivers are able to comply with the regulations and can be made liable for infringements committed by the drivers. Although of particular importance in long-distance haulage, regulations on working hours of truck drivers have only received very little attention in the scheduling literature. This paper presents a method for scheduling work weeks of truck drivers which fully considers regulation (EC) No 561/2006 for journeys made on roads in the European Union. It is proven that, given a sequence of locations to be visited within specified time windows, the approach finds a schedule complying with the regulation if such a schedule exists.

Driver scheduling problem modelling

The Drivers Scheduling Problem (DSP) consists of selecting a set of duties for vehicle drivers, such as bus, train and boat drivers or plane pilots, for the transportation of passengers or goods. This is a complex problem as it involves several constraints related to labour and company rules and may also entail different evaluation criteria and objectives. The ability to develop an adequate model for this problem, which can represent the real problem as closely as possible is an important research area. In this paper we present new mathematical models for the DSP which embody the very complexity of the drivers scheduling problem, besides demonstrating that the solutions generated by these models can easily be implemented in real situations. On the strength of extensive passenger transportation experience in bus companies in Portugal, we propose and test new alternative models to formulate the DSP. These models are based on Set Partitioning/Covering models. Moreover, they also take into account the bus operator issues and the user’s standpoint and environment.

Multi-phase dynamic constraint aggregation for set partitioning type problems

Dynamic constraint aggregation is an iterative method that was recently introduced to speed up the linear relaxation solution process of set partitioning type problems. This speed up is mostly due to the use, at each iteration, of an aggregated problem defined by aggregating disjoint subsets of constraints from the set partitioning model. This aggregation is updated when needed to ensure the exactness of the overall approach. In this paper, we propose a new version of this method, called the multi-phase dynamic constraint aggregation method, which essentially adds to the original method a partial pricing strategy that involves multiple phases. This strategy helps keeping the size of the aggregated problem as small as possible, yielding a faster average computation time per iteration and fewer iterations. We also establish theoretical results that provide some insights explaining the success of the proposed method. Tests on the linear relaxation of simultaneous bus and driver scheduling problems involving up to 2,000 set partitioning constraints show that the partial pricing strategy speeds up the original method by an average factor of 4.5.

Critical time windowed train driver relief opportunities

Train driver scheduling is the problem of constructing an efficient schedule of driver shifts, each of which contains a sequence of work on one or more trains separated by breaks. Relief opportunities (ROs) occur when trains stop at a station. While relieving on arrival at a train station is the preferred practice in the UK rail industry, considering relieving at other times within the full window of relief opportunities (WRO) at a stop might allow for a schedule optimization algorithm to build better schedules. However, simply expanding each WRO into ROs at individual minutes within the WRO would exponentially increase the complexity of the combinatorial optimization problem. A rational approach would be to be selective in considering the WROs when applying Generate-and-Select (GaS); this could either take the form of a pre-processing stage to GaS, or that of augmenting (or even replacing) the generation phase of GaS.

A dual ascent procedure for the set partitioning problem

The set partitioning problem is a fundamental model for many important real-life transportation problems, including airline crew and bus driver scheduling and vehicle routing. In this paper we propose a new dual ascent heuristic and an exact method for the set partitioning problem. The dual ascent heuristic finds an effective dual solution of the linear relaxation of the set partitioning problem and it is faster than traditional simplex based methods. Moreover, we show that the lower bound achieved dominates the one achieved by the classic Lagrangean relaxation of the set partitioning constraints. We describe a simple exact method that uses the dual solution to define a sequence of reduced set partitioning problems that are solved by a general purpose integer programming solver. Our computational results indicate that the new bounding procedure is fast and produces very good dual solutions. Moreover, the exact method proposed is easy to implement and it is competitive with the best branch and cut algorithms published in the literature so far.

Sleep Length as a Function of Morning Shift‐Start Time in Irregular Shift Schedules for Train Drivers: Self‐Rated Health and Individual Differences

Forty‐six male train drivers (mean age=46.5, SD=5.1) were recruited to participate in a diary study for 14 consecutive days with questions about their sleep and working hours. A polynomial mixed‐effect regression model showed a curvilinear relation (p <. 001) between shift‐start time and sleep duration for shifts starting at 03∶00–12∶00 h and with a near linear increase for ones starting between 04∶30 and 09∶00 h of approximately 0.7 h for every 1 h the shift was delayed. The longest sleeps were estimated at ∼8 h before shifts that started at ∼10∶00 h. The shortest sleeps were found for shifts that started before 04∶30 h and were estimated at ∼5 h. Individual differences were estimated with a random-effect standard deviation of 0.51 h, independent of shift‐start time (p =.005). One‐half of the between‐subject variance was explained by subjective health. A one‐step decrease in health was associated with a 26 min increase in sleep length. The results have practical implications for constructing shift schedules. Early morning shifts reduced sleep length substantially and should be mixed with later start hours to avoid the accumulation of sleep dept. Delaying the shift‐start past 10∶00 h had little effect on sleep opportunity; however, delaying shift‐start to between 04∶30 and 9∶00 h had a strong impact on sleep length, with 70% of the extra time used for sleep, suggesting large positive effects for this range of shift‐start times.

On assigning drivers for a home-delivery system on a performance basis

Consider a firm providing home-delivery of small parcels within a short period of time. The paper proposes a method to estimate the productivity of each driver on any fictitious route using linear regressions. The method separates the trip time in terms of the time spent: with customers, driving, and all other activities. Our tests show that productivity, measured in terms of number of customers visited per hour, vary across drivers. Even more, the same driver’s productivity vary for peak and off-peak demand periods considerably. This methodology should enhance the reliability of the driver scheduling programs, since it reduces the error and bias of the productivity offered by the assigned set of drivers. It shall also improve the reliability of the delivery time promised to the customer.

Effective search space control for large and/or complex driver scheduling problems

For real life bus and train driver scheduling instances, the number of columns in terms of the set covering/partitioning ILP model could run into billions making the problem very difficult. Column generation approaches have the drawback that the sub-problems for generating the columns would be computationally expensive in such situations. This paper proposes a hybrid solution method, called PowerSolver, of using an iterative heuristic to derive a series of small refined sub-problem instances fed into an existing efficient set covering ILP based solver. In each iteration, the minimum set of relief opportunities that guarantees a solution no worse than the current best is retained. Controlled by a user-defined strategy, a small number of the banned relief opportunities would be reactivated and some soft constraints may be relaxed before the new sub-problem instance is solved. PowerSolver is proving successful by many transport operators who are now routinely using it. Test results from some large scale real-life exercises will be reported.

Short-term work scheduling with job assignment flexibility for a multi-fleet transport system

In recent years, planning and management of operations of transportation companies has become increasingly complex as tighter financial constraints affect the ability to respond to changing demands for travel. This paper investigates how a firm can reduce its total labor cost, enhance the flexibility of its operations and improve worker productivity and utilization by determining the right mix of jobs in work schedules. Using a method we have developed to generate low cost work schedules for bus drivers of an inter-city transport system in India, we study the changes in worker productivity and utilization when a mix of primary and secondary jobs is built in work schedules. We also investigate other factors that can strongly influence worker productivity and utilization such as the amount of overtime in work schedules, the level of job assignment flexibility, and staff size. Test results and discussions of managerial implications are presented.

Simultaneous vehicle and driver scheduling: A case study in a limousine rental company

In this paper, we address a driver–vehicle scheduling problem in a limousine rental company. Given a set of trips to be covered, the goal consists in finding a driver–vehicle schedule that serves the maximum workload and optimizes several economic objectives while satisfying a set of imperative constraints. In this context, we propose a simultaneous scheduling of drivers and vehicles. The problem is modeled using the notion of partial consistent assignment. The solution approach is composed of two phases: the first one is based on constraint programming techniques and leads to the construction of an initial solution, improved in a second phase by a Simulated Annealing algorithm. Significant gains on the resulting solutions are systematically obtained in terms of quality, operational costs and elaboration time, compared to the current practice in the company.

A new hybrid heuristic for driver scheduling

This paper describes a new hybrid method based on the application of the Population Training Algorithm (PTA) and linear programming (LP) for generation of schedules for drivers in a public transport system. These methods are applied in an iterative way, where PTA is responsible for the generation of good columns (low cost and good covering of the tasks), and LP for solving a set partitioning problem formed by these columns. The PTA employs heuristics in fitness definition, guiding the population to settle down in search areas where the individuals cannot be improved by such heuristics. The driver schedules are represented by columns in a large-scale set partitioning problem, which are formed when solving the linear programming relaxation. The computational results are compared against a Simulated Annealing metaheuristic using randomly formed instances based on real problems.

Personnel scheduling in a complex logistic system: a railway application case

In complex logistic systems, such as transportation systems, dealing with personnel scheduling is a non-trivial task. Duties have to be created and assigned to workers in a way to optimize a certain objective function. In this paper, in particular, we consider the case of scheduling train drivers on a railway subnetwork. Train driver scheduling involves the construction of feasible duties from a set of trips to be carried out by a number of train drivers. Each duty consists of a sequence of trips to be carried out by a single train driver on a single day. The duties should be such that: each trip is covered by at least one duty, each duty satisfies feasibility constraints, additional constraints involving the complete schedule are satisfied, one or several objectives are met. In this paper we focus on minimizing the number of duties and on maximizing the robustness of the obtained schedule for outside disruptions. We present an implicit column generation solution approach. We describe a heuristic procedure to find an initial feasible solution together with a heuristic branch-and-price algorithm based on a dynamic programming algorithm for the pricing-out of columns. We tested our approach on the timetable of the Intercity train series 500, 700, 1600 and 1700 of NS Reizigers, the largest Dutch operator of passengers trains.

A dynamic driver management scheme for less-than-truckload carriers

This paper describes a scheme for the dynamic management of linehaul drivers developed for a large US less-than-truckload (LTL) carrier. Virtually all scheduling problems faced by transportation service providers are complicated by time-constrained vehicle operators that can be renewed only after resting. LTL driver scheduling is further complicated by the fact that trucking moves, unlike passenger airline flights or train dispatches, are not pre-scheduled. The technology developed in this paper combines greedy search with enumeration of time-feasible driver duties, and is capable of generating in a matter of minutes cost-effective driver schedules covering 15,000–20,000 loads and satisfying a variety of real-life driver constraints. Computational results justify the algorithmic design choices made in the development of the scheme, and a comparison with real-world dispatch data indicates that the technology produces driver schedules of very high quality.

A heuristic method for analyzing driver scheduling problem

A heuristic approach, ZEST for ESTimator, is developed to analyze bus driver scheduling problems and produce an estimate of the number of drivers required for a bus schedule. Based on the observation that the maximum number of drivers is needed in the morning and afternoon peaks, ZEST divides the driver scheduling problem into morning and afternoon subproblems, solves each subproblem separately, and, finally, combines the solutions. The key techniques in ZEST derive from manual scheduling operations that examine the critical decision points in a bus schedule that are vital for a good driver schedule and use these decision points to develop chains of meal breaks that dovetail one driver's meal break with another driver's. ZEST can be used as a standalone estimator of driver duties or as a component of other driver scheduling approaches