logistics coordinators

Abstract: We study the problem of scheduling jobs on a serial-batch machine with the aim of minimising the total weighted late work. In a serial-batch setting, jobs within a batch are processed sequentially, and none are removed from the machine until the last job in the batch completes its processing. The processing time of a batch is the sum of the processing times of the jobs within it, and the completion time for each job in the batch is equal to the makespan of the jobs in the batch. When a new batch begins, a constant setup time is required for the machine. We show that minimising the total weighted late work in this environment is $NP$-hard even if all jobs have a common due date and unit weight. For the general problem, we present a pseudo-polynomial time dynamic programming algorithm. Additionally, we explore two special cases, i.e., one with a common due date and another with an agreeable condition among due dates, processing times and weights. For both special cases, we develop specialised pseudo-polynomial time dynamic programming algorithms. The proposed approaches are equipped with specialised acceleration techniques to enhance their computational performance. The extended experiments demonstrate that the dynamic programming algorithms outperform Gurobi in time efficiency.

Abstract: We study strategyproof mechanisms for building a pathway between two regions of a line segment separated by an obstacle. Each of the $n$ agents has a private location within its region and may use either its original route to a facility or the new pathway, whose traversal cost is a fraction $k\in[0,1)$ of its length. We seek strategyproof (SP) and group-strategyproof (GSP) mechanisms that approximately minimize maximum cost or social cost. After characterizing optimal pathways for both objectives, we establish a tight deterministic maximum-cost approximation ratio of $\frac{2}{1+k}$ and a deterministic social-cost upper bound of $\frac{n}{1+k(n-1)}$, together with complementary lower bounds. Both upper bounds are achieved by GSP mechanisms. We then study randomized mechanisms under strategyproofness in expectation. A power-proportional mechanism achieves a social-cost approximation ratio at most $5$, independent of $n$ and $k$, with a tight guarantee of $3$ for this mechanism when $k=0$. We prove randomized lower bounds of $\frac{3+2k}{2+3k}$ for maximum cost and $\max\big\{1,\frac{285}{263+385k}\big\}$ for social cost, the latter for $n\ge7$. Finally, we improve several bounds for the real-line pathway model of [Chan and Wang, AAMAS 2023]. Our deterministic maximum-cost lower bound of $2$ matches the upper bound obtainable from [Qin, Fang, and Liu, COCOA 2024]. We strengthen the deterministic social-cost lower bound from $\frac32$ to $2$ under SP and to $\max\{2,n-1\}$ under GSP. For randomized social cost, we sharpen the guarantee of Chan and Wang's proportional mechanism from $6$ to $3$ and raise their lower bound from $1.02$ to $\frac{285}{263}\approx1.08365$ for $n\ge7$.