The leading approach to the simplex method, a widely used technique for balancing complex logistical constraints, can't get any better. In 1939, upon arriving late to his statistics course at UC Berkeley, George Dantzig--a first-year graduate student--copied two problems off the blackboard, thinking they were a homework assignment. He found the homework "harder to do than usual," he would later recount, and apologized to the professor for taking some extra days to complete it. A few weeks later, his professor told him that he had solved two famous open problems in statistics. Dantzig's work would provide the basis for his doctoral dissertation and, decades later, inspiration for the film .

Budget 2025: What's the best and worst that could happen for Labour? Any big red box moment is risky. Now the chancellor's big choices are out there, what's the best-case scenario for Reeves and Starmer, and what's the worst that could happen next? On the positive side of the ledger, Labour MPs have gone off to their constituencies in a better mood this week. That is in large part down to the chancellor's decision to scrap the limit on bigger families getting some extra benefits.

The classic Traveling Salesman Problem (TSP) and V ehicle Routing Problem (VRP) represent fundamental NP-hard combinatorial optimization challenges. In these routing problems, an agent commences from a designated node and fulfills specific task requisites. The primary objective is to minimize the total travel time or cost of access.

Such decaying rate is proved to be best achievable.

Ranking and selection (R&S) aims to identify the alternative with the best mean performance among $k$ simulated alternatives. The practical value of R&S depends on accurate simulation input modeling, which often suffers from the curse of input uncertainty due to limited data. Distributionally robust ranking and selection (DRR&S) addresses this challenge by modeling input uncertainty via an ambiguity set of $m > 1$ plausible input distributions, resulting in $km$ scenarios in total. Recent DRR&S studies suggest a key structural insight: additivity in budget allocation is essential for efficiency. However, existing justifications are heuristic, and fundamental properties such as consistency and the precise allocation pattern induced by additivity remain poorly understood. In this paper, we propose a simple additive allocation (AA) procedure that aims to exclusively sample the $k + m - 1$ previously hypothesized critical scenarios. Leveraging boundary-crossing arguments, we establish a lower bound on the probability of correct selection and characterize the procedure's budget allocation behavior. We then prove that AA is consistent and, surprisingly, achieves additivity in the strongest sense: as the total budget increases, only $k + m - 1$ scenarios are sampled infinitely often. Notably, the worst-case scenarios of non-best alternatives may not be among them, challenging prior beliefs about their criticality. These results offer new and counterintuitive insights into the additive structure of DRR&S. To improve practical performance while preserving this structure, we introduce a general additive allocation (GAA) framework that flexibly incorporates sampling rules from traditional R&S procedures in a modular fashion. Numerical experiments support our theoretical findings and demonstrate the competitive performance of the proposed GAA procedures.

From Snowpiercer to The Day After Tomorrow, countless movies and series have put forward their vision of how climate change might reshape the world. Worryingly, scientists predict that the reality might be far more shocking than anything imagined by a Hollywood studio. Now, artificial intelligence (AI) reveals what this might look like. With Google's ImageFX AI image generator, MailOnline has used the latest scientific research to predict how the world will be in 2100. As greenhouse gas levels continue to increase, scientists predict that entire cities will be plunged under water.