Vehicle Routing with Stochastic Supply of Crowd Vehicles and Time Windows

The growth of e-commerce has increased demand for last-mile deliveries, increasing the level of congestion in the existing transportation infrastructure in urban areas. Crowdsourcing deliveries can provide the additional capacity needed to meet the growing demand in a cost-effective way. We introduce a setting where a crowd-shipping platform sells heterogeneous products of different sizes from a central depot. Items sold vary from groceries to electronics. Some items must be delivered within a time window, whereas others need a customer signature. Furthermore, customer presence is not guaranteed, and some deliveries may need to be returned to the depot. Delivery requests are fulfilled by a fleet of professional drivers and a pool of crowd drivers. We present a crowd-shipping platform that standardizes crowd drivers’ capacities and compensates them to return undelivered packages back to the depot. We formulate a two-stage stochastic model, and we propose a branch and price algorithm to solve the problem exactly and a column generation heuristic to solve larger problems quickly. We further develop an analytical method to calculate upper bounds on the supply of vehicles and an innovative cohesive pricing problem to generate columns for the pool of crowd drivers. Computational experiments are carried out on modified Solomon instances with a pool of 100 crowd vehicles. The branch and price algorithm is able to solve instances of up to 100 customers. We show that the value of the stochastic solution can be as high as 18% when compared with the solution obtained from a deterministic simplification of the model. Significant cost reductions of up to 28% are achieved by implementing crowd drivers with low compensations or higher capacities. Finally, we evaluate what happens when crowd drivers are given the autonomy to select routes based on rational and irrational behavior. There is no cost increase when crowd drivers are rational and select routes that have a higher compensation first. However, when crowd drivers are irrational and select routes randomly, the cost can increase up to 4.2% for some instances.

Column generation for minimizing total completion time in a parallel-batching environment

This paper deals with the $$1|{p-\text {batch}, s_j\le b}|\sum C_j$$ 1 | p - batch , s j ≤ b | ∑ C j scheduling problem, where jobs are scheduled in batches on a single machine in order to minimize the total completion time. A size is given for each job, such that the total size of each batch cannot exceed a fixed capacity b. A graph-based model is proposed for computing a very effective lower bound based on linear programming; the model, with an exponential number of variables, is solved by column generation and embedded into both a heuristic price and branch algorithm and an exact branch and price algorithm. The same model is able to handle parallel-machine problems like $$Pm|{p-\text {batch}, s_j\le b}|\sum C_j$$ P m | p - batch , s j ≤ b | ∑ C j very efficiently. Computational results show that the new lower bound strongly dominates the bounds currently available in the literature, and the proposed heuristic algorithm is able to achieve high-quality solutions on large problems in a reasonable computation time. For the single-machine case, the exact branch and price algorithm is able to solve all the tested instances with 30 jobs and a good amount of 40-job examples.

Decomposition-Based Approaches for a Class of Two-Stage Robust Binary Optimization Problems

In this paper, we study a class of two-stage robust binary optimization problems with objective uncertainty, where recourse decisions are restricted to be mixed-binary. For these problems, we present a deterministic equivalent formulation through the convexification of the recourse-feasible region. We then explore this formulation under the lens of a relaxation, showing that the specific relaxation we propose can be solved by using the branch-and-price algorithm. We present conditions under which this relaxation is exact and describe alternative exact solution methods when this is not the case. Despite the two-stage nature of the problem, we provide NP-completeness results based on our reformulations. Finally, we present various applications in which the methodology we propose can be applied. We compare our exact methodology to those approximate methods recently proposed in the literature under the name [Formula: see text]adaptability. Our computational results show that our methodology is able to produce better solutions in less computational time compared with the [Formula: see text]adaptability approach, as well as to solve bigger instances than those previously managed in the literature. Summary of Contribution: Our manuscript describes an exact solution approach for a class of robust binary optimization problems with mixed-binary recourse and objective uncertainty. Its development reposes first on a reformulation of the problem, then a carefully constructed relaxation of this reformulation. Our solution approach is designed to exploit the two-stage and binary structure of the problem for effective resolution. In its execution, it relies on the branch-and-price algorithm and its efficient implementation. With our computational experiments, we show that our proposed exact solution method outperforms the existing approximate methodologies and, therefore, pushes the computational envelope for the class of problems considered.