The theory of deterministic sequencing and scheduling has expanded rapidly during the past years. In this paper we survey the state of the art with respect to optimization and approximation algorithms and interpret these in terms of computational complexity theory. Special cases considered are single machine scheduling, identical, uniform and unrelated parallel machine scheduling, and open shop, flow shop and job shop scheduling. We indicate some problems for future research and include a selective bibliography.

/pdf/optimization-and-approximation-in-deterministic-sequencing-327pfewdb6.pdf

Optimization and Approximation in Deterministic Sequencing and Scheduling: a Survey

Urban Transportation Networks: Equilibrium Analysis With Mathematical Programming Methods

Introduction to linear and nonlinear programming

We survey and extend the results on the complexity of machine scheduling problems. After a brief review of the central concept of NP-completeness we give a classification of scheduling problems on single, different and identical machines and study the influence of various parameters on their complexity. The problems for which a polynomial-bounded algorithm is available are listed and NP-completeness is established for a large number of other machine scheduling problems. We finally discuss some questions that remain unanswered.

/pdf/complexity-of-machine-scheduling-problems-3kmjz4hlnd.pdf

Complexity of machine scheduling problems

We consider the problem of routing traffic to optimize the performance of a congested network. We are given a network, a rate of traffic between each pair of nodes, and a latency function for each edge specifying the time needed to traverse the edge given its congestion; the objective is to route traffic such that the sum of all travel times---the total latency---is minimized.In many settings, it may be expensive or impossible to regulate network traffic so as to implement an optimal assignment of routes. In the absence of regulation by some central authority, we assume that each network user routes its traffic on the minimum-latency path available to it, given the network congestion caused by the other users. In general such a "selfishly motivated" assignment of traffic to paths will not minimize the total latency; hence, this lack of regulation carries the cost of decreased network performance.In this article, we quantify the degradation in network performance due to unregulated traffic. We prove that if the latency of each edge is a linear function of its congestion, then the total latency of the routes chosen by selfish network users is at most 4/3 times the minimum possible total latency (subject to the condition that all traffic must be routed). We also consider the more general setting in which edge latency functions are assumed only to be continuous and nondecreasing in the edge congestion. Here, the total latency of the routes chosen by unregulated selfish network users may be arbitrarily larger than the minimum possible total latency; however, we prove that it is no more than the total latency incurred by optimally routing twice as much traffic.

/pdf/how-bad-is-selfish-routing-g644vu88rg.pdf

How bad is selfish routing

We describe a model for the transit assignment problem with a fixed set of transit lines The traveler chooses the strategy that allows him or her to reach his or her destination at minimum expected cost First we consider the case in which no congestion effects occur For the special case in which the waiting time at a stop depends only on the combined frequency, the problem is formulated as a linear programming problem of a size that increases linearly with the network size A label-setting algorithm is developed that solves the latter problem in polynomial time Nonlinear cost extensions of the model are considered as well

Optimal strategies: A new assignment model for transit networks

A class of production planning problems is considered in which known demands have to be satisfied over a finite horizon at minimum total costs. For each period, production and storage cost functions are specified. The production costs may include set-up costs and the production levels may he subject to capacity limits. The computational complexity of the problems in this class is investigated. Several algorithms proposed for their solution are described and analyzed. It is also shown that some special cases are NP-hard and hence unlikely to be solvable in polynomial time.

Deterministic Production Planning: Algorithms and Complexity

Erratum to authors' paper (Florian M., M. Klein. 1971. Deterministic production planning with concave costs and capacity constraints. Management Sci. 18 (1, September) 12–20.

Deterministic production planning with concave costs and capacity constraints.

An algorithm is developed for sequencing jobs on a single processor in order to minimize maximum lateness, subject to ready times and due dates. The method that we develop could be classified as branch-and-bound. However, it has the unusual feature that a complete solution is associated with each node of the enumeration tree.

On Scheduling with Ready Times and Due Dates to Minimize Maximum Lateness

This paper concerns a frequency-based route choice model for congested transit networks, which takes into account the consequences of congestion on the predicted flows as well as on the expected waiting and travel times. The paper builds on the results presented in Correa [Correa, J., 1999. Asignacion de flujos de pasajeros en redes de transporte publico congestionadas. Engineering thesis, U. de Chile, Santiago] and Cominetti and Correa [Cominetti, R., Correa, J., 2001. Common-lines and passenger assignment in congested transit networks. Transportation Science 35(3), 250–267], extending these to obtain a new characterization of the equilibria which allows us to formulate an equivalent optimization problem in terms of a computable gap function that vanishes at equilibrium. This new model formulation can deal with flow dependent travel times and is a generalization of the previously known strategy (hyperpath) based transit network equilibrium models. The approach leads to an algorithm which has been applied successfully on large scale networks. Computational results for transit networks originating from practice demonstrate the applicability of the proposed approach.

Michael Florian

Papers

Optimal strategies: A new assignment model for transit networks

Deterministic Production Planning: Algorithms and Complexity

Deterministic production planning with concave costs and capacity constraints.

On Scheduling with Ready Times and Due Dates to Minimize Maximum Lateness

A frequency-based assignment model for congested transit networks with strict capacity constraints: characterization and computation of equilibria