We consider the scheduling problems arising when two agents, each with a set of nonpreemptive jobs, compete to perform their respective jobs on a common processing resource. Each agent wants to minimize a certain objective function, which depends on the completion times of its jobs only. The objective functions we consider in this paper are maximum of regular functions (associated with each job), number of late jobs, and total weighted completion times. We obtain different scenarios, depending on the objective function of each agent, and on the structure of the processing system (single machine or shop). For each scenario, we address the complexity of various problems, namely, finding the optimal solution for one agent with a constraint on the other agent's cost function, finding single nondominated schedules (i.e., such that a better schedule for one of the two agents necessarily results in a worse schedule for the other agent), and generating all nondominated schedules.

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This paper studies a topical and economically significant capacitated network design problem that arises in the telecommunications industry. In this problem, given point-to-point communication demand in a network must be met by installing (loading) capacitated facilities on the arcs: Loading a facility incurs an arc specific and facility dependent cost. This paper develops modeling and solution approaches for loading facilities to satisfy the given demand at minimum cost. We consider two approaches for solving the underlying mixed integer program: a Lagrangian relaxation strategy, and a cutting plane approach that uses three classes of valid inequalities that we identify for the problem. We show that a linear programming formulation that includes these inequalities always approximates the value of the mixed integer program at least as well as the Lagrangian relaxation bound. Our computational results on a set of prototypical telecommunication data show that including these inequalities considerably improves the gap between the integer programming formulation and its linear programming relaxation: from an average of 25% to an average of 8%. These results show that strong cutting planes can be an effective modeling and algorithmic tool for solving problems of the size that arise in the telecommunications industry.

The network loading problem (NLP) is a specialized capacitated network design problem in which prescribed point-to-point demand between various pairs of nodes of a network must be met by installing (loading) a capacitated facility. We can load any number of units of the facility on each of the arcs at a specified arc dependent cost. The problem is to determine the number of facilities to be loaded on the arcs that will satisfy the given demand at minimum cost. This paper studies two core subproblems of the NLP. The first problem, motivated by a Lagrangian relaxation approach for solving the problem, considers a multiple commodity, single arc capacitated network design problem. The second problem is a three node network; this specialized network arises in larger networks if we aggregate nodes. In both cases, we develop families of facets and completely characterize the convex hull of feasible solutions to the integer programming formulation of the problems. These results in turn strengthen the formulation of the NLP.

The paper studies the problem of predicting traffic equilibrium (TE) in a transportation network within the framework of decision making among discrete choices in a probabilistic and uncertain environment. Conventional approaches to predict TE typically assume that travel times are deterministic and perceived accurately by the travelers; and some new TE models have considered probabilistic travel times or inaccurate perceptions but not both. In our generalized model, which we refer to as GTESP, the travel time on each route is random and each traveler perceives, possibly inaccurately, a travel time probability distribution for each route which may vary from traveler to traveler. Each traveler uses a disutility function of travel time to evaluate each route and chooses that route which minimizes his expected disutility. GTESP is difficult to solve for general nonlinear disutility functions. However, special cases—in particular when arc travel times are statistically independent and the disutility functions to evaluate route travel times are linear, exponential, or quadratic—are solvable, at least approximately. GTESP is general in the sense that most existing TE models can be shown to be special or limiting cases of GTESP. Furthermore, this paper demonstrates, with illustrative examples, that GTESP appears to capture travelers' risk-taking behavior more realistically than existing TE models.