On the Meaningfulness of Optimal Solutions to Scheduling Problems: Can an Optimal Solution be Nonoptimal?

Abstract

We consider the problem of finding an optimal schedule for jobs on a single machine when there are penalties for both tardy and early arrivals. We point out that if attention is paid to how these penalties are measured, then a change of scale of measurement might lead to the anomalous situation where a schedule is optimal if these parameters are measured in one way, but not if they are measured in a different way that seems equally acceptable. In particular, we note that if the penalties measure utilities or disutilities, or loss of goodwill or customer satisfaction, then these kinds of anomalies can occur, for instance if we change both unit and zero point in scales measuring these penalties. We investigate situations where problems of these sorts arise for four specific penalty functions under a variety of different assumptions. The results of the paper have implications far beyond the specific scheduling problems we consider, and suggest that considerations of scale of measurement should enter into analysis of conclusions of optimality both in scheduling problems and throughout combinatorial optimization.M any practical problems involve the search for an optimal schedule. We consider the problem of scheduling n jobs on a single machine in which each job has a specified due date or completion time and a penalty is applied for a completion time different from the desired one. In many practical problems, a penalty is applied only for tardy completions; while more generally, a penalty is applied to both early and tardy completions, perhaps in a different way. The interest in scheduling problems where penalties are applied to early arrivals as well as late arrivals is closely tied to the concept of "just-in-time" production, the goal being to have "the right amount of materials of the right quality at the right time in the right place to produce the right quantity of items demanded by the next step of the production" (Cheng 1990). Because penalties can be applied to early arrivals, we allow the machine to lie idle and we schedule without preemption, i.e., we do not allow a job to be interrupted once it is started. The penalties we study involve weighting factors that weight the deviations from desired completion times.Often these weights are not uniquely determined. For instance, two weight assignments may be equally acceptable if they both give rise to the same ordering of the items or if one is related to the other by a change of scale. Typically weights are measured using some scale of measurement, and we examine the effect on the solution to a scheduling problem if we make admissible changes of scale. We show that in some cases such changes can transform an optimal solution into a nonoptimal one, and we systematically describe those situations when this anomaly occurs. More precisely, we show that the conclusion of optimality can be meaningless in a technical sense that we shall make precise. The main point of this paper is to show that considerations of scale change need to play a role in analysis of scheduli...

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]).…”

Meaningless statements

“…Typically weights are measured using some scale of measurement, and we can ask about the effect on the solution to a scheduling problem if we make admissible changes of scale. Mahadev, Pekeč, and Roberts [17] show that in some cases such changes can transform an optimal solution into a non-optimal one, and systematically describe those situations when this anomaly occurs.…”

Sensitivity analysis for some scheduling problems

“…Guinand, et al [4] presented a sensitivity analysis for the problem of scheduling trees with communication delays on two identical machines to minimize the makespan. The sensitivity of an optimal schedule to linear translation of the cost parameter vector was studied by James, et al [5] , Mahadev, et al [6] and Hall, et al [7] made sensitivity analysis for scheduling problems that are solvable in polynomial time such as 1|| w j C j and 1||L max , and also examined the computational complexity of sensitivity analysis for intractable scheduling problems.…”