This work formally defines the problem of fleet sizing with delays (FSD), where the option of delaying individual tasks within fleet sizing is considered. We prove that the FSD problem is NP-hard and solve a formulation of the FSD problem as a mixed integer linear problem (MILP). We then analyze the proposed method in detail in an abstract case and validate it in a case study of taxi rides in Manhattan. We show that fleet sizes can be decreased significantly and that the trade-off space of the number of required vehicles to execution time and added delay can be enlarged.

This paper studies the multi-robot task assignment problem in which a fleet of dispersed robots needs to efficiently transport a set of dynamically appearing packages from their initial locations to corresponding destinations within prescribed time-windows. Each robot can carry multiple packages simultaneously within its capacity. Given a sufficiently large robot fleet, the objective is to minimize the robots’ total travel time to transport the packages within their respective time-window constraints. The problem is shown to be NP-hard, and we design two group-based distributed auction algorithms to solve this task assignment problem. Guided by the auction algorithms, robots first distributively calculate feasible package groups that they can serve, and then communicate to find an assignment of package groups. We quantify the potential of the algorithms with respect to the number of employed robots and the capacity of the robots by considering the robots’ total travel time to transport all packages. Simulation results show that the designed algorithms are competitive compared with an exact centralized Integer Linear Program representation solved with the commercial solver Gurobi, and superior to popular greedy algorithms and a heuristic distributed task allocation method. Note to Practitioners—This work presents two group-based distributed auction algorithms for a sufficiently large fleet of robots to efficiently transport a set of dynamically appearing dispersed packages from their initial locations to corresponding destinations within prescribed time-windows. Each robot can carry multiple packages simultaneously within its capacity, and the objective is to minimize the robots’ total travel time to transport all the packages within the prescribed time-windows. The paper’s practical contributions are threefold: First, the multi-robot task assignment problem is formulated through a robot-group assignment strategy, which enables complex logistic scheduling for tasks grouped according to their distributions and time-windows. Second, we theoretically show that the multi-robot task assignment problem is an NP-hard problem, which implies the necessity for designing approximate task assignment algorithms. Third, the proposed group-based distributed auction algorithms are efficient and can be adapted for real scenarios.

Obtaining strong linear relaxations for capacitated covering problems constitutes a significant technical challenge. For one of the most basic cases, the relaxation based on knapsack-cover inequalities has an integrality gap of 2. We generalize the setting considering items that can be taken fractionally to cover a given demand, with a cost given by an arbitrary nondecreasing function (not necessarily convex) of the chosen fraction. We generalize the knapsack-cover inequalities and use them to obtain a polynomial $(2+\varepsilon)$-approximation algorithm. Our primal-dual procedure has a natural interpretation as a water-filling algorithm, which overcomes the difficulties implied by having different growth rates in the cost functions: when the cost of an item increases slowly at some superior segment, it carefully increases the priority of all preceding segments. We generalize our algorithm to the Unsplittable Flow-Cover problem on a line, also for fractional items with non-linear costs. We obtain a $4$-approximation in pseudopolynomial time ($4+\varepsilon$ in polynomial time), matching the approximation ratio of the classical setting. We also present a rounding algorithm with an approximation guarantee of 2. This result is coupled with a polynomial time separation algorithm that allows solving our linear relaxation up to a loss of a $(1+\varepsilon)$ factor.

Obtaining strong linear relaxations of capacitated covering problems constitute a significant technical challenge even for simple settings. For one of the most basic cases, the Knapsack-Cover (Min-Knapsack) problem, the relaxation based on knapsack-cover inequalities has an integrality gap of 2. These inequalities are exploited in more general problems, many of which admit primal-dual approximation algorithms. Inspired by problems from power and transport systems, we introduce a general setting in which items can be taken fractionally to cover a given demand. The cost incurred by an item is given by an arbitrary non-decreasing function of the chosen fraction. We generalize the knapsack-cover inequalities to this setting an use them to obtain a (2+ε)-approximate primal-dual algorithm. Our procedure has a natural interpretation as a bucket-filling algorithm which effectively overcomes the difficulties implied by having different slopes in the cost functions. More precisely, when some superior segment of an item presents a low slope, it helps to increase the priority of inferior segments. We also present a rounding algorithm with an approximation guarantee of 2. We generalize our algorithm to the Unsplittable Flow-Cover problem on a line, also for the setting of fractional items with non-linear costs. For this problem we obtain a (4+ε)-approximation algorithm in polynomial time, almost matching the 4-approximation algorithm known for the classical setting.