Vehicle routing with soft time windows and stochastic travel times : a column generation and branch-and-price solution approach

We study a vehicle routing problem with soft time windows and stochastic travel times. In this problem, we consider stochastic travel times to obtain routes which are both efficient and reliable. In our problem setting, soft time windows allow early and late servicing at customers by incurring some penalty costs. The objective is to minimize the sum of transportation costs and service costs. Transportation costs result from three elements which are the total distance traveled, the number of vehicles used and the total expected overtime of the drivers. Service costs are incurred for early and late arrivals; these correspond to time-window violations at the customers. We apply a column generation procedure to solve this problem. The master problem can be modeled as a classical set partitioning problem. The pricing subproblem, for each vehicle, corresponds to an elementary shortest path problem with resource constraints. To generate an integer solution, we embed our column generation procedure within a branch-and-price method. Computational results obtained by experimenting with well-known problem instances are reported

The path programming problem and a partial path relaxation

We introduce the class of path programming problems, which can be used to model many known optimization problems. A path programming problem can be formulated as a binary programming problem, for which the pricing problem can be modeled as a shortest path problem with resource constraints when column generation is used to solve its linear programming relaxation. Many optimization problems found in the literature belong to this class. We provide a framework for obtaining a partial path relaxation of a path programming problem. Like traditional path relaxations, the partial path relaxation allows the computational complexity of the pricing problem to be reduced, at the expense of a weaker linear programming bound. We demonstrate the versatility of this framework by providing different examples of partial path relaxations for a crew scheduling problem and vehicle routing problem

Energy-Efficient Location-Routing Problem with Time Windows with Dynamic Demand

Sustainability and energy savings have attracted considerable attention in recent years. However, in the traditional location-routing problem (LRP), the objective function has yet to minimize the distance traveled regardless of the amount of energy consumed. Although, distance is one of the major factors determining the energy consumption of a distribution network, it is not the only factor. Therefore, this paper explains the development of a novel formulation of the LRP that considers energy minimization, which is called the energy-efficient location-routing problem (EELRP). The energy consumed by a vehicle to travel between two nodes in a system depends on many forces. Among those, rolling resistance (RR) and aerodynamic drag are considered in this paper to be the major contributing forces. The presented mixed-integer non-linear program (MINLP) finds the best location-allocation routing plan with the objective function of minimizing total costs, including energy, emissions, and depot establishment. The proposed model can also handle the vehicle-selection problem with respect to a vehicles’ capacity, source of energy, and aerodynamic characteristics. The formulation proposed can also solve the problems with hard and soft time window constraints. Also, the model is enhanced to handle the EELRP with dynamic customers’ demands. Some examples are presented to illustrate the formulations presented in this paper

Workforce Scheduling with Order-Picking Assignments in Distribution Facilities

Scheduling the availability of order pickers is crucial for effective operations in a distribution facility with manual order pickers. When order-picking activities can only be performed in specific time windows, it is essential to jointly solve the order picker shift scheduling problem and the order picker planning problem of assigning and sequencing individual orders to order pickers. This requires decisions regarding the number of order pickers to schedule, shift start and end times, break times, as well as the assignment and timing of order-picking activities. We call this the order picker scheduling problem and present two formulations. A branch-and-price algorithm and a metaheuristic are developed to solve the problem. Numerical experiments illustrate that the metaheuristic finds near-optimal solutions at 80% shorter computation times. A case study at the largest supermarket chain in The Netherlands shows the applicability of the solution approach in a real-life business application. In particular, different shift structures are analyzed, and it is concluded that the retailer can increase the minimum compensated duration for employed workers from six hours to seven or eight hours while reducing the average labor cost with up to 5% savings when a 15-minute flexibility is implemented in the scheduling of break times

LOGIC AND CONSTRAINT PROGRAMMING FOR COMPUTATIONAL SUSTAINABILITY

Computational Sustainability is an interdisciplinary field that aims to develop computational and mathematical models and methods for decision making concerning the management and allocation of resources in order to help solve environmental problems. This thesis deals with a broad spectrum of such problems (energy efficiency, water management, limiting greenhouse gas emissions and fuel consumption) giving a contribution towards their solution by means of Logic Programming (LP) and Constraint Programming (CP), declarative paradigms from Artificial Intelligence of proven solidity. The problems described in this thesis were proposed by experts of the respective domains and tested on the real data instances they provided. The results are encouraging and show the aptness of the chosen methodologies and approaches. The overall aim of this work is twofold: both to address real world problems in order to achieve practical results and to get, from the application of LP and CP technologies to complex scenarios, feedback and directions useful for their improvement

Layered graph approaches for combinatorial optimization problems

Extending the concept of time-space networks, layered graphs associate information about one or multiple resource state values with nodes and arcs. While integer programming formulations based on them allow to model complex problems comparably easy, their large size makes them hard to solve for non-trivial instances. We detail and classify layered graph modeling techniques that have been used in the (recent) scientific literature and review methods to successfully solve the resulting large-scale, extended formulations. Modeling guidelines and important observations concerning the solution of layered graph formulations by decomposition methods are given together with several future research directions

Continuity of care in community midwifery

Continuity of care is often critical in delivering high quality health care. However, it is difficult to achieve in community health care where shift patterns and a need to minimise travelling time can reduce the scope for allocating staff to patients. Community midwifery is one example of such a challenge in the National Health Service where postnatal care typically involves a series of home visits. Ideally mothers would receive all of their antenatal and postnatal care from the same midwife. Minimising the number of staff-handovers helps ensure a better relationship between mothers and midwives, and provides more opportunity for staff to identify emerging problems over a series of home visits. This study examines the allocation and routing of midwives in the community using a variant of a multiple travelling salesmen problem algorithm incorporating staff preferences to explore trade-offs between travel time and continuity of care. This algorithm was integrated in a simulation to assess the additional effect of staff availability due to shift patterns and part-time working. The results indicate that continuity of care can be achieved with relatively small increases in travel time. However, shift patterns are problematic: perfect continuity of care is impractical but if there is a degree of flexibility in the visit schedule, reasonable continuity is feasible