On interactive sequencing situations with exponential cost functions

Abstract

This paper addresses interactive one-machine sequencing situations in which the costs of processing a job are given by an exponential function of its completion time. The main difference with the standard linear case is that the gain of switching two neighbors in a queue is time-dependent and depends on their exact position. We illustrate that finding an optimal order is complicated in general and we identify specific subclasses, which are tractable from an optimization perspective. More specifically, we show that in these subclasses, all neighbor switches in any path from the initial order to an optimal order lead to a non-negative gain. Moreover, we derive conditions on the time-dependent neighbor switching gains in a general interactive sequencing situation to guarantee convexity of the corresponding cooperative game. These conditions are satisfied within our specific subclasses of exponential interactive sequencing situations.

Sequencing Situations and Games With Non-Linear Cost Functions

“…We show that, by reusing the two different sets of conditions on the neighbor switching gains, both allocation rules prescribe a core-element of the associated sequencing game. In particular, we show that for discounting and logarithmic sequencing situations, the EGST-rule leads to a core-element, while for the three subclasses as defined in Saavedra-Nieves et al (2020), the EGSH-rule results in a core-element.…”

“…Following the lines as described in Saavedra-Nieves et al (2020), a (general) sequencing situation is represented by a tuple (N, σ 0 , p, c), where p = (p i ) i∈N and c = (c i ) i∈N summarize the processing times and cost functions, respectively. The set of all sequencing situations with player set N is denoted by SEQ N and a sequencing situation (N, σ 0 , p, c) is also denoted by…”

“…Lemma 2.1 [Saavedra-Nieves et al (2020)] Let (σ 0 , p, c) ∈ SEQ N be a sequencing situation and letσ ∈ Π(N ) be an optimal order. If…”

“…In particular, we deal with three specific examples of non-linear cost functions, namely exponential, discounting and logarithmic cost functions. Exponential sequencing situations were recently introduced and analyzed by Saavedra-Nieves, Schouten, and Borm (2020), while sequencing situations with discounting cost functions, i.e. discounting sequencing situations, were introduced by Rothkopf (1966).…”

“…First, we focus on convexity of the associated sequencing games. Saavedra-Nieves et al (2020) showed that, by imposing a set of conditions on the neighbor switching gains, the associated sequencing game of a (general) sequencing situation is convex. These conditions require the neighbor switching gains to be non-negative and non-decreasing for misplacements and nonpositive for non-misplacements.…”