Abstract
The recently-defined class of integer programming games (IPG) models situations where multiple self-interested decision makers interact, with their strategy sets represented by a finite set of linear constraints together with integer requirements. Many real-world problems can suitably be cast in this way, hence anticipating IPG outcomes is of crucial value for policy makers. Nash equilibria have been widely accepted as the solution concept of a game. Thus, their computation provides a reasonable prediction of games outcome. In this paper, we start by showing the computational complexity of deciding the existence of a Nash equilibrium for an IPG. Then, using sufficient conditions for their existence, we develop a general algorithmic approach that is guaranteed to return a Nash equilibrium when the game is finite and to approximate an equilibrium when payoff functions are Lipschitz continuous. We also showcase how our methodology can be changed to determine other types of equilibria. The performance of our methods is analyzed through computational experiments on knapsack, kidney exchange and a competitive lot-sizing games. To the best of our knowledge, this is the first time that equilibria computation methods for general IPGs have been designed and computationally tested.

Abstract
In this work, we study a flexible compensation scheme for last-mile delivery where a company outsources part of the activity of delivering products to its customers to occasional drivers (ODs), under a scheme named crowdshipping. All deliveries are completed at the minimum total cost incurred with their vehicles and drivers plus the compensation paid to the ODs. The company decides on the best compensation scheme to offer to the ODs at the planning stage. We model our problem based on a stochastic and dynamic environment where delivery orders and ODs volunteering to make deliveries present themselves randomly within fixed time windows. The uncertainty is endogenous in the sense that the compensation paid to ODs influences their availability. We develop a deep reinforcement learning (DRL) algorithm that can deal with large instances while focusing on the quality of the solution: we combine the combinatorial structure of the action space with the neural network of the approximated value function, involving techniques from machine learning and integer optimization. The results show the effectiveness of the DRL approach by examining out-of-sample performance and that it is suitable to process large samples of uncertain data, which induces better solutions.

Abstract
In this work, we study last-mile delivery with the option of crowd shipping. A company uses occasional drivers to complement its fleet in the activity of delivering products to its customers. We model it as a variant of the stochastic capacitated vehicle routing problem. Our approach is data-driven, where not only customer orders but also the availability of occasional drivers are uncertain. It is assumed that marginal distributions of the uncertainty vector are known, but the joint distribution is difficult to estimate. We optimize considering a worst-case joint distribution and model with a strategic planning perspective, where we calculate an optimal a priori solution before the uncertainty is revealed. A limit on the infea-sibility of the routes due to the capacity is imposed using probabilistic constraints. We propose an extended formulation for the problem using column-dependent rows and implement a branch-price-and-cut algorithm to solve it. We also develop a heuristic approximation to cope with larger instances of the problem. Through computational experiments, we analyze the solution and performance of the implemented algorithms.

Abstract
Kidney exchange programs (KEP) allow an incompatible patient-donor pair, whose donor cannot provide a kidney to the respective patient, to have a transplant exchange with another in a similar situation if there is compatibility. Exchanges can be performed via cycles or chains initiated by non-directed donors (NDD), i.e., donors that do not have an associated patient. The objective for optimization in KEP is generally to maximize the number of possible transplants. Following the course of recent approaches that consider a dynamic matching (exchanges are decided every time a pair or a NDD joins the pool), in this paper we explore two matching policies to find feasible exchanges: periodic, where the algorithm runs within some period (e.g each 3 month); and greedy, in which a matching run is done as soon as the pool is updated with a new pair or NDD. For each policy, we propose a matching algorithm that addresses the waiting times of pairs in a pool. We conduct computational experiments with the proposed algorithms and compare the results with those obtained when periodic and greedy matching aim at maximizing the number of transplants.

Abstract
A recent relevant innovation in last-mile delivery is to consider the possibility of goods being delivered by couriers appointed through crowdsourcing. In this paper we focus on the setting of in-store customers delivering goods, ordered by online customers, on their way home. We assume that not all the proposed delivery tasks will necessarily be accepted, and use logistic regression to model the crowd agents' willingness to undertake a delivery. This model is then used to build a novel compensation scheme that determines reward values, based on the current plan for the professional fleet's routes and on the couriers' probabilities of acceptance, by employing a direct search algorithm that seeks to minimise the expected cost.

Abstract
We study a setting in which a company not only has a fleet of capacitated vehicles and drivers available to make deliveries but may also use the services of occasional drivers (ODs) willing to make deliveries using their own vehicles in return for a small fee. Under such a business model, a.k.a crowdshipping, the company seeks to make all the deliveries at the minimum total cost, i.e., the cost associated with their vehicles plus the compensation paid to the ODs.We consider a stochastic and dynamic last-mile delivery environment in which customer delivery orders, as well as ODs available for deliveries, arrive randomly throughout the day, within fixed time windows.We present a novel deep reinforcement learning (DRL) approach to the problem that can deal with large problem instances. We formulate the action selection problem as a mixed-integer optimization program.The DRL approach is compared against other optimization under uncertainty approaches, namely, sample -average approximation (SAA) and distributionally robust optimization (DRO). The results show the effective-ness of the DRL approach by examining out-of-sample performance.