Showing papers on "Assignment problem published in 1980"
Book•
01 Nov 1980
TL;DR: In this article, the linear sum assignment problem and the cardinality matching problem are used to solve the Chinese Postman Problem. But the linear Sum Matching Problem is not one of the problems we consider in this paper.
Abstract: 1. The Linear Sum Assignment Problem.- 2. The Linear Bottleneck Assignment Problem.- 3. The Cardinality Matching Problem.- 4. The Sum Matching Problem.- 5. The Bottleneck Matching Problem.- 6. The Chinese Postman Problem.- 7. Quadratic Assignment Problems.- 8. QAP Heuristic 1: The method of increasing degree of freedom.- 9. QAP Heuristic 2: Cutting plane and exchange method.- 10. General Subroutines.
203 citations
TL;DR: This paper presents a breadth-first branch and bound algorithm which differs from the method of Smith, Srinivasan and Thompson in the selection of the subtour to be split, in the ordering of the arcs in the selected subtour, and in the computation of different partial lower bounds and in different data structures to facilitate the updating of the cost matrix.
Abstract: Many algorithms have been developed for the optimal solution of the asymmetric travelling salesman problem: the most efficient ones are based on the subtour elimination approach. This paper presents a breadth-first branch and bound algorithm which differs from the method of Smith, Srinivasan and Thompson in the selection of the subtour to be split, in the ordering of the arcs in the selected subtour, in the computation of different partial lower bounds and in different data structures to facilitate the updating of the cost matrix. Extensive computational results considering random problems with up to 240 vertices are presented for various ranges of the coefficients of the cost matrix.
190 citations
TL;DR: In this paper, the authors present a new formulation of the quadratic assignment problem by transforming the objective function into a linear objective function by introducing a number of new variables and constraints and the resulting problem is a 0-1 linear integer program with a highly specialized structure.
Abstract: In this paper we present a new formulation of the quadratic assignment problem. This is done by transforming the quadratic objective function into a linear objective function by introducing a number of new variables and constraints. The resulting problem is a 0-1 linear integer program with a highly specialized structure. This permits the use of the partitioning scheme of Benders where only the original variables need be considered. The algorithm described thus iterates between two problems. The master problem is a pure 0-1 integer program, and the subproblem is a transportation problem whose optimal solution is shown to be readily available from the master problem in closed form. Computational experience on problems available in the literature is provided.
139 citations
TL;DR: R H U first ( s ) nex t (s) last ( s) as the row ass igned to c o l u m n y ( j -1 ) as the label o f c o L R , = 0, row i is un labe led (i = 1 , . . . , n);
Abstract: R H U first ( s ) nex t (s) last ( s ) as the row ass igned to c o l u m n y ( j -1 . . . . . n); as the label o f c o l u m n j ; i fLCj = 0, c o l u m n j is un labe led ( ] = 1 . . . . . n); as the label o f r ow i; i f L R , = 0, row i is un labe led (i = 1 , . . . , n); as the a s s ignmen t cost; as the se t con ta in ing the c o l um ns co r respond ing to the unass igned zero e l emen t s o f r ow i o f the cos t ma t r ix (i -1, . . . , n); as the se t con ta in ing the cu r r en t no t comple te ly -exp lo red rows; as the se t con ta in ing the unass igned rows; as t he f irst e l e m e n t o f set s; as t he e l e m e n t fol lowing the las t cons idered e l emen t o f se t s; as the las t e l e m e n t of set s.
126 citations
TL;DR: A sufficient condition for optimality is presented which implies that a global optimum can be obtained by successively optimizing at most N + 1 objective functions for the linear program, where N is the number of time periods in the planning horizon.
Abstract: A dynamic model for the optimal control of traffic flow over a network is considered. The model, which treats congestion explicitly in the flow equations, gives rise to nonlinear, nonconvex mathematical programming problems. It has been shown for a piecewise linear version of this model that a global optimum is contained in the set of optimal solutions of a certain linear program. This paper presents a sufficient condition for optimality which implies that a global optimum can be obtained by successively optimizing at most N + 1 objective functions for the linear program, where N is the number of time periods in the planning horizon. Computational results are reported to indicate the efficiency of this approach.
102 citations
TL;DR: A simple algorithm for a chairman assignment is given which guarantees a small discrepancy and the situation that not only states form unions, but also unions form federations, etc., with one overall organization is investigated.
101 citations
TL;DR: This paper presents a new algorithm for solving the assignment problem based on a scheme of relaxing the given problem into a series of simple network flow transportation problems for each of which an optimal solution can be easily obtained.
Abstract: This paper presents a new algorithm for solving the assignment problem. The algorithm is based on a scheme of relaxing the given problem into a series of simple network flow transportation problems for each of which an optimal solution can be easily obtained. The algorithm is thus seen to be able to take advantage of the nice properties in both the primal and the dual approaches for the assignment problem. The computational bound for the algorithm is shown to be 0n3 and the average computation time is better than most of the specialized assignment algorithms.
99 citations
TL;DR: This part examines algorithmic approaches for calculating the flow patterns resulting from the different modes of assignment, based on remodeling the elastic-demand TAP as an equivalent assignment problem in an expanded network.
Abstract: Part I of this study reviewed the formulation of the traffic assignment problem TAP in a network and identified its underlying rationale. This part examines algorithmic approaches for calculating the flow patterns resulting from the different modes of assignment. An efficient methodology for solving the elastic-demand TAP is based on remodeling it as an equivalent assignment problem in an expanded network. The variable-demand TAP is then tranformed into a fixed-demand TAP, with a trip table consisting of the potential demands, and can be solved by available fixed-demand assignment algorithms. Three alternative transformations are described.
96 citations
TL;DR: In this article, the authors present a review of the traffic assignment problem with elastic demands, originally described by Beckmann, McGuire and Winsten, and examine algorithmic approaches for calculating the resulting flow patterns.
Abstract: This study reviews, in two parts, the formulation, interpretation, and solution methodology of the traffic assignment problem with elastic demands, originally described by Beckmann, McGuire and Winsten. This paper Part I defines the different possible modes of traffic assignment in a network and identifies the economic rationales of the associated extremum formulations. The accompanying paper Part II examines algorithmic approaches for calculating the resulting flow patterns. The assignment problem is characterized as an interaction between users and operators of the transportation system, analogous to the interaction between consumers and producers in the marketplace. The notions of “user-optimization” and “system-optimization” thus acquire congruent interpretations, expressed by a systemwide maximization objective.
85 citations
TL;DR: An algorithm to solve the m-source, n-destination assignment problem in expected time O(mn log n) under the assumption that the edge costs are independent random variables and the costs of the edges incident with any given source are identically distributed is given.
Abstract: We give an algorithm to solve the m-source, n-destination assignment problem in expected time O(mn log n) under the assumption that the edge costs are independent random variables and the costs of the edges incident with any given source are identically distributed. The algorithm achieves its efficiency through an unusual application of priority queues.
82 citations
TL;DR: An optimization algorithm for the traffic assignment problem is presented and it was seen that when stopping rules were based on insufficiently tight tolerances, substantial errors could appear in the equilibrium traffic assignment.
Abstract: An optimization algorithm for the traffic assignment problem is presented. It is based on determining, for the flow corresponding to each origin at a time, a set of disjoint circuits with the most negative total marginal cost, followed by a unidimensional optimization along these circuits. The proposed method was compared to other well known approaches on randomly generated networks. In most cases the proposed method proved to be highly efficient, particularly when appropriate tighter stopping rules were used. In this sense, it was seen that when stopping rules were based on insufficiently tight tolerances, substantial errors could appear in the equilibrium traffic assignment.
TL;DR: The most interesting features are that the model explicitly takes into account capacity constraints on the buses, and that the distribution of trips between different zone is influenced by the frequencies of the bus lines.
Abstract: Assume that a bus network is given, i.e. we are given a network of streets on which certain bus lines have been set up. Let the total number of buses be given. Assume furthermore that the total demand for bus transportation is given in the form of the marginal totals of an origin-destination matrix, i.e. the total demand for travel from certain origins as well as the total demand for travel to certain destinations is given. Problem: Determine the complete travel pattern and decide which bus frequencies to use on the various lines. The problem is formulated as a non-linear programming problem in the form of a compound minimization problem. The most interesting features are that the model explicitly takes into account capacity constraints on the buses, and that the distribution of trips between different zone is influenced by the frequencies of the bus lines. An iterative algorithm to solve this problem is developed. The algorithm converges to stationary points. This scheme employs as a component an algorithm for the combined distribution- assignment problem in traffic planning which is developed by using decomposition. The model has been tested on the bus network in the town of Linkoping (80,000 inhabitants). The model suggests certain actions which are in agreement with the actions actually taken by the bus operator.
TL;DR: A technique to significantly reduce the dimensionality of the optimization problem is presented, and the errors introduced by the conversion of an essentially discrete problem into a continuous one are estimated and bounded.
Abstract: This paper presents a computer system configuration design problem in which the objective is to select the CPU speed, the capacities of secondary storage devices, and the allocation of a set of files across the secondary storage devices so as to maximize the system throughput subject to a cost constraint. It is shown that any relative maximum of this complex nonlinear programming problem is also a global maximum. A technique to significantly reduce the dimensionality of the optimization problem is presented along.with an example to illustrate the model's usefulness. The well-known file assignment problem is shown to be a subproblem of this model, and an example is given which demonstrates this fact. Finally, the errors introduced by the conversion of an essentially discrete problem into a continuous one are estimated and bounded.
01 Jan 1980
TL;DR: In this article, a binary branch and bound algorithm for the exact solution of the Koopmans-Beckmann quadratic assignment problem is described which exploits both the transformation and the greedily obtained approximate solution described in a previous paper by the author.
Abstract: In this paper a binary branch and bound algorithm for the exact solution of the Koopmans-Beckmann quadratic assignment problem is described which exploits both the transformation and the greedily obtained approximate solution described in a previous paper by the author. This branch and bound algorithm has the property that at each bound an associated solution is obtained simultaneously, thereby rendering any premature termination of the algorithm less wasteful.
TL;DR: It is shown how several urban location models can be derived from the mathematical framework based on the network equilibrium problem by reinterpreting the zone-to-zone trip variable.
Abstract: Recent advances in network equilibrium modeling provide efficient algorithms for solving the urban trip assignment problem. These models can be extended to incorporate the trip distribution problem with two types of variable demand functions. By reinterpreting the zone-to-zone trip variable, these models can be viewed as urban location models. This paper synthesizes these results, and shows how several urban location models can be derived from the mathematical framework based on the network equilibrium problem.
TL;DR: In this article, a bounding technique based on the extraction from the Koopmans Beckmann quadratic assignment (QAP) formulation is proposed to solve a large linear assignment problem (which can then be solved optimally), leaving as a residual problem as small as possible.
01 Jan 1980
TL;DR: Using results about the algebraic manipulation of fuzzy numbers, computationally attractive algorithms for fuzzy data are provided.
Abstract: Often in real-case problems, all the numerical data are not precisely known and the nature of the uncertainty is possibilistic14 rather than probabilistic. Then, the data are said fuzzy. The adaptation of an ordinary algorithm (appropriate to precise data) to fuzzy data is not always straightforward. Theoretically, the direct application of the extension principle of fuzzy set theory solves this problem, but not generally in a computationally attractive manner. Practically, the case of forecasting algorithms, where the result may be fuzzy is different from this of decision algorithms where the result must be precise. For an illustrative purpose, we successively deal with the PERT, assignment, travelling salesman and transportation problems. Using results about the algebraic manipulation of fuzzy numbers, computationally attractive algorithms for fuzzy data are provided.
01 Jan 1980
TL;DR: Weakly admissible transformations are introduced for solving algebraic assignment and transportation problems, which cover so important classes as problems with sum objectives, bottleneck objectives, lexicographical objectives and others as mentioned in this paper.
Abstract: Weakly admissible transformations are introduced for solving algebraic assignment and transportation problems, which cover so important classes as problems with sum objectives, bottleneck objectives, lexicographical objectives and others. A transformation of the cost matrix is called weakly admissible, if there are two constants α and β in the underlying semigroup that for all feasible solutions the composition of α and the objective value with respect to the original cost coefficients is equal to the composition of β and the objective value with respect to the transformed cost coefficients. The elements α and β can be determined by shortest path algorithms. An optimal solution for the algebraic assignment problem can be found after at most n weakly admissible transformations, therefore the proposed method yields an O(n 3) algorithm for algebraic assignment problems.
TL;DR: For the distance matrix of symmetric traveling salesman problems a simple transformation into an equivalent asymmetric one is given to yield sharper lowerbounds and less subtours from the transformed distance matrix.
Abstract: For the distance matrix of symmetric traveling salesman problems a simple transformation into an equivalent asymmetric one is given. Assignment algorithms yield sharper lowerbounds and less subtours from the transformed distance matrix. This implies a better performance for traveling salesman algorithms based on the assignment relaxation.
TL;DR: In this paper, the Koopmans-Beckmann linear program is decomposable and the primal subprogram constitutes a simple linear assignment problem whose optimal solution set always contains at least one assignment of plants to locations.
TL;DR: This paper recasts the multiple data path assignment problem solved by Torng and Wilhelm by the dynamic programming method into a minimal covering problem following a switching theoretic approach and achieves minimal cost solutions by assigning weights to the bus-compatible sets present in the feasible solutions.
Abstract: This paper recasts the multiple data path assignment problem solved by Torng and Wilhelm by the dynamic programming method [1] into a minimal covering problem following a switching theoretic approach. The concept of bus compatibility for the data transfers is used to obtain the various ways of interconnecting the circuit modules with the minimum number of buses that allow concurrent data transfers. These have been called the feasible solutions of the problem. The minimal cost solutions are obtained by assigning weights to the bus-compatible sets present in the feasible solutions. Minimization of the cost of the solution by increasing the number of buses is also discussed.
TL;DR: A new algorithm for solving quadratic assignment problems is presented, which employs a sequential search technique and improves the elements of this matrix, one by one, by solving a succession of linear assignment problems.
Abstract: A new algorithm for solving quadratic assignment problems is presented. The algorithm, which employs a sequential search technique, constructs a matrix of lower bounds on the costs of locating facilities at different sites. It then improves the elements of this matrix, one by one, by solving a succession of linear assignment problems. After all the elements of the matrix are improved, a feasible assignment is obtained, which results in an improved value for the objective function of the quadratic assignment problem. The procedure is repeated until the desired accuracy in the objective function value is obtained.
TL;DR: In this article, the authors studied a combined distribution and assignment problem with entropy objective and cost constraints and gave an algorithm based on Benders decomposition for its solution, which is a "dual" method.
Abstract: The combined distribution and assignment problem has been studied by many authors. The problems examined have been of different types. Some authors focus their attention on the user optimized combined distribution and assignment problem whereas others study a system optimized version. Apart from this difference there is also a difference in the way the distribution part and the assignment part are incorporated into the models. In this paper we study a combined distribution and assignment problem with entropy objective and cost constraints and give an algorithm based on Benders decomposition for its solution. The resulting solution method will be a “dual” method.
01 Jan 1980
TL;DR: The linear bottleneck assignment problem (LBAP) is strongly related to LSAP as mentioned in this paper, and it is associated with every permutation φ ∊ l n the costs of the costs.
Abstract: The Linear Bottleneck Assignment Problem (LBAP) is strongly related to LSAP. Here we associate with every permutation φ ∊ l n the costs
$$c\left( \varphi \right): = \mathop {\max }\limits_{i \in N} \;{c_{i,\varphi \left( i \right)}}$$
.
01 Jan 1980
TL;DR: In this article, a flight-to-gate assignment problem at airports in such a way as to minimize, or at least reduce, walking distances for passengers inside terminals is solved.
Abstract: This research solves the flight-to-gate assignment problem at airports in such a way as to minimize, or at least reduce, walking distances for passengers inside terminals. Two solution methods are suggested. The first is a heuristic algorithm which assigns the "most crowded" aircraft (i.e., most on-board passengers) to the best gate, while the second consists of formulating the problem as a linear program. A flight schedule of one day at Terminal No. 2 of Toronto International Airport is used to test and compare the two methods. The algorithm offers an assignment solution with a 27% reduction in the expected walking distance when compared to the original assignment at the airport. The linear program's assignment gives a 32% reduction. The heuristic algorithm is, therefore, only 5% suboptimal for the sample problem. In addition, its associated computational expenses, less than $10 per run, are by far cheaper than those of the linear program with expenses as high as $400 per run. Such excellent, or even acceptable, performance by the algorithm cannot be guaranteed for all problems. A strategy which helps decide when to use which approach is therefore suggested.
TL;DR: This paper recasts the multiple data path assignment problem solved by Torng and Wilhelm by the dynamic programming method into a minimal covering problem following a switching theoretic approach and achieves minimal cost solutions by assigning weights to the bus-compatible sets present in the feasible solutions.
Abstract: This paper recasts the multiple data path assignment problem solved by Torng and Wilhelm by the dynamic programming method [1] into a minimal covering problem following a switching theoretic approach. The concept of bus compatibility for the data transfers is used to obtain the various ways of interconnecting the circuit modules with the minimum number of buses that allow concurrent data transfers. These have been called the feasible solutions of the problem. The minimal cost solutions are obtained by assigning weights to the bus-compatible sets present in the feasible solutions. Minimization of the cost of the solution by increasing the number of buses is also discussed.
TL;DR: The algorithm exploits the zero nonzero structure of matrix A and uses optimum data structures and data manipulation methods to be useful in finding all optimum assignments in an n x n optimum assignment problem and generation of all digraphs that can be associated with an nx nsparse matrix.
Abstract: In this paper an algorithm is presented for listing all output sets for a large sparse square matrix A arising in large scale systems applications using network theory and the degree switching operations. The algorithm exploits the zero nonzero structure of matrix A and uses optimum data structures and data manipulation methods. The method is shown to be useful in finding all optimum assignments in an n x n optimum assignment problem and generation of all digraphs that can be associated with an n x nsparse matrix. The problem of testing whether there exists a set of vertex disjoint cycles of specified lengths in a network is shown to be NP-complete.