Effects of change of scale on optimality in a scheduling model with priorities and earliness/tardiness penalties

Abstract

consider the effect of changes of scale of measurement on the conclusion that a particular solution to a scheduling problem is optimal. The analysis in this paper was motivated by the problem of finding the optimal transportation schedule when there are penalties for both late and early arrivals, and when different items that need to be transported receive different priorities. We note that in this problem, if attention is paid to how certain parameters are measured, then a change of scale of measurement might lead to the anomalous situation where a schedule is optimal if the parameter is measured in one way, but not if the parameter is measured in a different way that seems equally acceptable. This conclusion about the sensitivity of the conclusion that a given solution to a combinatorial optimisation problem is optimal is different from the usual type of conclusion in sensitivity analysis, since it holds even though there is no change in the objective function, the constraints, or other input parameters, but only in scales of measurement. We emphasize the need to consider such changes of scale in analysis of scheduling and other combinatorial optimization problems. We also discuss the mathematical problems that arise in two special csses, where all desired arrival times are the same and the simplest case where they are not, namely the case where there are two distinct arrival times but one of them occurs exactly once. While specialized, these two examples illustrate the types of mathematical problems that arise from considerations of the interplay between scaletypes and optimisation.

Meaningless statements

“…If we use the same penalty function except to change h e and h t to be linear, then the conclusion of optimality of a schedule is meaningful under ratio scales. However, this turns out not to be the case under interval scales, as is also noted in [16].…”

“…Then assertion A implies assertion B, so, as noted in [16], it is meaningful to conclude that schedule D is optimal. Equation 5.1 arises in measurement theory in various contexts, and arose originally in psychophysics.…”

“…Thus, C is no longer optimal after a change of unit. As noted in [16], the priority functions h e and h t used in this example might be somewhat unrealistic. However, they illustrate the point.…”

“…Suppose it is also w-increasing in the sense that both h e and h t are strictly increasing in w and |C − d|-increasing in the sense that f(C, d) = A(|C − d|) for A an increasing function. Mahadev, Pekeč, and Roberts [16] show that in this case, a schedule is optimal if and only if it is attainable by a greedy algorithm. Moreover, any strictly increasing transformation of scale values from w i to φ(w i ) gives the same greedy solution.…”

Computer science and decision theory

“…Building models for consensus in this new context is another challenge that should be pursued. A related topic building on order relations concerns the issue of solving classic optimization problems when the costs are expressed in ordinal scales (not allowing an additive problem formulation), a problem receiving increasing attention in the planning and scheduling community of computer science (see [105,272,273,344,343,364]). Still another topic, returned to in Section 4.4, involves the connection between preferences and artificial intelligence, through the attempt to understand the interplay between reasoning and rationality (see for example [116]).…”