Assignment problem

About: Assignment problem is a research topic. Over the lifetime, 7588 publications have been published within this topic receiving 172820 citations. The topic is also known as: marriage problem.

A Benders Decomposition Approach for the Locomotive and Car Assignment Problem

Abstract: One of the many problems faced by rail transportation companies is to optimize the utilization of the available stock of locomotives and cars. In this paper, we describe a decomposition method for the simultaneous assignment of locomotives and cars in the context of passenger transportation. Given a list of train legs and a fleet composed of several types of equipment, the problem is to determine a set of minimum cost equipment cycles such that every leg is covered using appropriate equipment. Linking constraints, which appear when both locomotives and cars are treated simultaneously, lead to a large integer programming formulation. We propose an exact algorithm, based on the Benders decomposition approach, that exploits the separability of the problem. Computational experiments carried on a number of real-life instances indicate that the method finds optimal solutions within short computing times. It also outperforms other approaches based on Lagrangian relaxation or Dantzig--Wolfe decomposition, as well as a simplex-based branch-and-bound method.

Nonconvexity of the dynamic traffic assignment problem

Abstract: ment of models of dynamic traffic flows on road networks. This difficulty is due to the fact that road traffic tends to behave in a first-in-first-out (FIFO) manner: that is, traffic which embarks on a road or other facility in period I exits from that facility (“on average”) before traffic which enters in any later time periods. The FIFO rquiremcnt does not cause a problem in static traffic assignment, but we show that it yields a nonconvex constraint set in dynamic assignment. cspccially if there are multiple destinations or commodities. We consider various formulations. each of which yields a nonconvex optimization problem which is a1 present computationally tractable only for relatively small-scale examples. The above FIFO problem arises even if there is no congestion. and even if travel demands are fixed. Further the problem arises whether we are modeling a system optimum or a user equilibrium, and whether we use an optimization formulation or a complemcntarity or variational inequality formulation. We make some suggestions for dealing with. or avoiding, the problem and for further research. A desirable but elusive goal of research in dynamic traffic assignment is to develop well-behaved multiperiod network models analogous to well-known static or single period models. In particular, many authors have remarked on the need to extend the present limited scope dynamic assignment models so as to be able to handle multiple destinations and multiple traffic types. However, the development of such dynamic models runs into an underlying obstacle or problem which does not appear to have been explicitly identified in the literature. The obstacle consists of a basic nonconvexity in the dynamic traffic flow problem. The purpose of this paper is to identify and draw attention to this nonconvexity problem. We suggest some ways in which one might attempt to deal with the problem, but we do not fully solve the problem or eliminate the difficulty. Instead, we identify the problem in the hope that this will motivate further research efforts to cope with, overcome or avoid it. Throughout we consider the dynamic traffic assignment formulated as a multiperiod network model. However, for brevity and because it is not essential for our purpose here, we do not set out any explicit form for the constraints or objective function of the network model. What we have to say will apply to a variety of existing and possible

Traffic equilibrium problem with route-specific costs: Formulation and algorithms

Abstract: Using a new gap function recently proposed by Facchinei and Soares [Facchinei, F., Soares, J., 1995. Testing a new class of algorithms for nonlinear complementarity problems. In: Giannessi, F., Maugeri, A. (Eds.), Variational Inequalities and Network Equilibrium Problems. Plenum Press, New York], we convert the nonlinear complementarity problem (NCP) formulation for the traffic equilibrium problem to an equivalent unconstrained optimization. This equivalent formulation uses both route flows and the minimum origin–destination travel costs as the decision variables. Two unique features of this formulation are that: (i) it can model the traffic assignment problem with a general route cost structure; (ii) it is smooth, unconstrained, and that every stationary point of the minimization corresponds to a global minimum. These properties permit a number of efficient algorithms for its solution. Two solution approaches are developed to solve the proposed formulation. Numerical results using a route-specific cost structure are provided and compared with the classic traffic equilibrium problem, which assumes an additive route cost function.

On the power assignment problem in radio networks

Abstract: Given a finite set S of points (i.e. the stations of a radio network) on a d-dimensional Euclidean space and a positive integer 1 ≤ h ≤ |S| - 1, the MIN d D h-RANGE ASSIGNMENT problem consists of assigning transmission ranges to the stations so as to minimize the total power consumption, provided that the transmission ranges of the stations ensure the communication between any pair of stations in at most h hops.Two main issues related to this problem are considered in this paper: the trade-off between the power consumption and the number of hops; the computational complexity of the MIN dD h-RANGE ASSIGNMENT problem.As for the first question, we provide a lower bound on the minimum power consumption of stations on the plane for constant h. The lower bound is a function of |S|, h and the minimum distance over all the pairs of stations in S. Then, we derive a constructive upper bound as a function of |S|, h and the maximum distance over all pairs of stations in S (i.e. the diameter of S). It turns out that when the minimum distance between any two stations is "not too small" (i.e. well spread instances) the upper bound matches the lower bound. Previous results for this problem were known only for very special 1-dimensional configurations (i.e., when points are arranged on a line at unitary distance) [Kirousis, Kranakis, Krizanc and Pelc, 1997].As for the second question, we observe that the tightness of our upper bound implies that MIN 2D h-RANGE ASSIGNMENT restricted to well spread instances admits a polynomial time approximation algorithm. Then, we also show that the same approximation result can be obtained for random instances. On the other hand, we prove that for h=|S|-1 (i.e. the unbounded case) MIN 2D h-RANGE ASSIGNMENT is NP-hard and MIN 3D h-RANGE ASSIGNMENT is APX-complete.