Sequencing situations and games with non-linear cost functions under optimal order consistency

Jop Schouten,Alejandro Saavedra-Nieves,M Gloria Fiestras-Janeiro

doi:10.1016/j.ejor.2021.02.013

Abstract

This paper considers sequencing situations with non-linear cost functions under optimal order consistency. Specifically, we study sequencing situations with discounting cost functions and logarithmic cost functions of the completion time. In both settings, we show that the neighbor switching gains are non-negative and non-decreasing for every misplaced pair of players. We derive new conditions on the time-dependent neighbor switching gains in a sequencing situation under optimal order consistency to guarantee convexity of the associated sequencing game. Furthermore, we define two types of gain splitting rules for the class of sequencing situations under optimal order consistency. Each one of them is based on a procedure that specifies a path from the initial order to an optimal order, dividing the neighbor switching gains in every step among the two involved players. We prove that these allocations are stable under the same conditions that are required for convexity. These requirements are fulfilled for discounting and logarithmic sequencing situations, as well as in other settings, such as in sequencing situations with exponential cost functions.

Highlights

In this paper, we deal with one-machine sequencing situations with, in addition, an initial order specified
We focus on two distinguished concepts within cooperative game theory: the core and convexity
If optimal order consistency is satisfied for a particular sequencing situation, the optimal order can be determined after checking the neighbor switching gains of n(n−1) 2 pairs of players and rearranging at most n(n−1) 2 pairs of misplaced players, where n is the number of players in the sequencing situation

Summary

Introduction

We deal with one-machine sequencing situations with, in addition, an initial order specified. SaavedraNieves et al (2020) showed that, by imposing a set of conditions on the neighbor switching gains, the associated sequencing game of a sequencing situation under optimal order consistency is convex. This guarantees convexity for a much wider class of sequencing games and, in particular, for discounting and logarithmic sequencing games Another fundamental issue in this context is the definition of allocation rules for sequencing situations with non-linear cost functions under optimal order consistency. We divide the neighbor switching gains in every step of a path from the initial order to an optimal order among the two players involved using a distribution of weights not necessarily equal This leads to two different type of allocations, depending on the procedure that is used: the Gain Splitting Head rules (GSH-rules) and the Gain Splitting Tail rules (GST-rules).

Preliminaries

Convexity

Cost savings allocation rules

Specifying a path

Extending the GS-rules

Concluding remarks