Sewage Discharging in a Line: Global Optimization and Grand Cooperation

TL;DR

This paper models and optimizes cooperative sewage discharge along a river, demonstrating polynomial-time solutions, core stability, and introducing directional-convexity as a new concept for non-convex cooperative games.

Contribution

It formalizes the sewage discharge problem, develops optimal algorithms, proves core non-emptiness for the game, and introduces directional-convexity for non-convex cooperative games.

Findings

Optimal algorithms for sewage discharge problem in polynomial time

Core of the cooperative game is non-empty due to line structure

Directional-convexity as a sufficient condition for core non-emptiness

Abstract

Players cooperating in a line is a special while essential phenomenon in real life collaborating activities such as assembly line production, pipeline supply chain management and other streamlining operational settings. In this paper, we study the scenario of cooperative sewage discharge with multiple participants positioning in a line along a river such that the optimization decision and cooperation strategy are mutually affected by both upstream and downstream players. We make three main contributions accordingly: Firstly, we formalize the sewage discharge problem (SDP) for different groups of players, and use greedy strategy and dynamic programming to design the optimal algorithms to solve the SDP in polynomial time. Secondly, we show that the cooperative game defined on sewage discharge problem, referred to as SDG, has a non-empty core due to its special line-positioning structure. Therefore, a grand stable cooperation is guaranteed. Furthermore, inspired by the fact that the SDG is core non-empty while non-convex, we successfully identify a relaxed concept of convexity -- directional-convexity, which can also serve as a sufficient condition for a cooperative game having a non-empty core.