Showing papers on "Assignment problem published in 1989"

Optimal strategies: A new assignment model for transit networks

Abstract: 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

Faster scaling algorithms for network problems

Abstract: This paper presents algorithms for the assignment problem, the transportation problem, and the minimum-cost flow problem of operations research. The algorithms find a minimum-cost solution, yet run in time close to the best-known bounds for the corresponding problems without costs. For example, the assignment problem (equivalently, minimum-cost matching in a bipartite graph) can be solved in $O(\sqrt {nm} \log (nN))$ time, where $n,m$, and N denote the number of vertices, number of edges, and largest magnitude of a cost; costs are assumed to be integral. The algorithms work by scaling. As in the work of Goldberg and Tarjan, in each scaled problem an approximate optimum solution is found, rather than an exact optimum.

Dynamic network traffic assignment considered as a continuous time optimal control problem

Abstract: Two continuous time formulations of the dynamic traffic assignment problem are considered, one that corresponds to system optimization and the other to a version of user optimization on a single mode network using optimal control theory. Pontryagin's necessary conditions are analyzed and given economic interpretations that correspond to intuitive notions regarding dynamic system optimized and dynamic user optimized traffic flow patterns. Notably, we offer the first dynamic generalization of Beckmann's equivalent optimization problem for static user optimized traffic assignment in the form of an optimal control problem. The analysis further establishes that a constraint qualification and convexity requirements for the Hamiltonian, which together ensure that the necessary conditions are also sufficient, are satisfied under commonly encountered regularity conditions.

Applying Integer Linear Programming to the Fleet Assignment Problem

Abstract: We formulated and solved the fleet assignment problem as an integer linear programming model, permitting assignment of two or more fleets to a flight schedule simultaneously. The objective function can take a variety of forms including profit maximization, cost minimization, and the optimal utilization of a particular fleet type. Several departments at American Airlines use the model to assist in fleet planning and schedule development. It will become one of 10 key decision modules for the next generation scheduling system currently being developed by American Airlines Decision Technologies.

Channel assignment in cellular radio

Abstract: Some heuristic channel-assignment algorithms for cellular systems are described. These algorithms have yielded optimal, or near-optimal assignments, in many cases. The channel-assignment problem can be viewed as a generalized graph-coloring problem, and these algorithms have been developed, in part, by suitably adapting some of the ideas previously introduced in heuristic graph-coloring algorithms. The channel-assignment problem is formulated as a minimum-span problem, i.e. a problem wherein the requirement is to find the minimum bandwidth necessary to satisfy a given demand. Examples are presented, and algorithm performance results are discussed. >

The auction algorithm for the transportation problem

Abstract: The auction algorithm is a parallel relaxation method for solving the classical assignment problem. It resembles a competitive bidding process whereby unassigned persons bid simultaneously for objects, thereby raising their prices. Once all bids are in, objects are awarded to the highest bidder. This paper generalizes the auction algorithm to solve linear transportation problems. The idea is to convert the transportation problem into an assignment problem, and then to modify the auction algorithm to exploit the special structure of this problem. Computational results show that this modified version of the auction algorithm is very efficient for certain types of transportation problems.

A Neural Network-Based Optimization Algorithm for the Static Weapon-Target Assignment Problem

Abstract: A neural network-based algorithm was developed for the static weapon-target assignment problem in ballistic missile defense. An optimal assignment policy is one which allocates targets to weapon platforms such that the total expected leakage value of targets surviving the defense is minimized. This involves the minimization of a nonlinear objective function subject to inequality constraints specifying the maximum number of interceptors available to each platform and the maximum number of interceptors allowed to be fired at each target as imposed by the battle management/command control and communications system. The algorithm consists of solving a system of ordinary differential equations whose trajectories are the assignment variables of the problem. Simulations of the algorithm on PC and VAX computers were carried out using a simple numerical scheme. In all the battle instances tested, the algorithm has proven to be stable and to converge to solutions very close to global optima. The time to achieve con...

A New Approach to Stable Matching Problems

Abstract: It is shown that Stable Matching problems are the same as problems about stable configurations of X-networks. Consequences include easy proofs of old theorems, a new simple algorithm for finding a stable matching, an understanding of the difference between Stable Marriage and Stable Roommates, NP-completeness of Three-party Stable Marriage, CC-completeness of several Stable Matching problems, and a fast parallel reduction from the Stable Marriage problem to the Assignment problem.

A parallel algorithm for the quadratic assignment problem

Abstract: In this paper we present a parallel branch and bound algorithm for the solution of quadratic assignment problems. The algorithm has been coded in VS/FORTRAN and run on an IBM 3090-400E vector multiprocessor. The algorithm is exact and finds all the solutions. Computational results on a variety of test problems are presented.

The placement optimization program: a practical solution to the disk file assignment problem

Abstract: In this paper we describe a practical mathematical formulation and solution of the so-called “File Assignment Problem” (FAP) for computer disks. Our FAP solution has been implemented in a PL/I program known as the Placement Optimization Program (POP). The algorithm consists of three major components — two heuristic optimization models and a queueing network model. POP has been used in validation studies to assign files to disks in two IBM MVS complexes. The resulting savings in I/O response times were 22% and 25%, respectively. Throughout the paper we shall emphasize the real-world nature of our approach to the disk FAP, which we believe sets it apart from previous attempts.

Transit assignment to minimal routes: an efficient new algorithm

Abstract: The mathematical formulation of the transit assignment problem to minimal routes is presented. Then, an efficient solution algorithm is derived which dramatically reduces the execution time when compared with the traditional Chriqui and Robillard's algorithm and its heuristic simplifications

A Lagrangian Relaxation Approach To The Classroom Assignment Problem

Abstract: This paper describes one of the algorithms in a new timetabling and student registration system that was implemented at the University of Waterloo (Ontario, Canada) in May, 1987. We are given a set of class meetings that has already been assigned to specific days and time periods of the week, and a collection of available classrooms on campus. We must determine an acceptable assignment of classes to rooms based on a variety of factors that measure the desirability of a particular assignment.The problem is subdivided into two separate components. Given a function Cij that represents the cost of assigning class i to room j, we present a heuristic algorithm based on a Lagrangian Relaxation for solving the multiple period assignment problem [MPAP]. We then describe a cost function model that will be shown to reasonably reflect the general preference structure of the faculty and the administration. Computational results on 1400 courses at the University of Waterloo are discussed.

Reconfiguration of VLSI arrays by covering

Abstract: In VLSI arrays, redundant cells are added as spares. The proposed approach is applicable as an offline technique at either production time and/or run time. Reconfiguration is implemented by index mapping using a sequence of two operators. A reconfiguration algorithm which utilizes index mapping is proposed. This algorithm uses a new approach to the assignment problem. It is shown that reconfiguration of fault-free cells is equivalent to a covering of faulty cells by spare cells. It is also established that in a two-dimensional array the optimal spare assignment is given by a maximum matching. This translates to a maximum flow. It is shown that a variation of the matching preserves the optimality of the assignment, while reducing the time complexity of the reconfiguration algorithm Characterization theorems for index mapping and simulation results to substantiate the practicality of the approach are presented. >

A new approach to the pin assignment problem

Abstract: A study, motivated by the goal of integrating the placement and routing steps in the physical design of VLSI circuits, of the pin assignment problem for macrocells is discussed. It is assumed that the macrocells have already been placed. It is also assumed that the design of the macrocells is still soft in that although the pins in a cell have a fixed relative order, they can be shifted around the boundary of the cell. An algorithm to determine how the pins are to be shifted so that the weighted sum of the lengths of the connecting wires is minimal is developed. Good experimental results were obtained. >

Linear Programming and Its Applications

Abstract: 0 Introduction.- I: Linear Programming.- 1 Geometric Linear Programming.- 0. Introduction.- 1. Two Examples: Profit Maximization and Cost Minimization.- 2. Canonical Forms for Linear Programming Problems.- 3. Polyhedral Convex Sets.- 4. The Two Examples Revisited.- 5. A Geometric Method for Linear Programming.- 6. Concluding Remarks.- Exercises.- 2 The Simplex Algorithm.- 0. Introduction.- 1. Canonical Slack Forms for Linear Programming Problems Tucker Tableaus.- 2. An Example: Profit Maximization.- 3. The Pivot Transformation.- 4. An Example: Cost Minimization.- 5. The Simplex Algorithm for Maximum Basic Feasible Tableaus.- 6. The Simplex Algorithm for Maximum Tableaus.- 7. Negative Transposition The Simplex Algorithm for Minimum Tableaus.- 8. Cycling.- 9. Concluding Remarks.- Exercises.- 3 Noncanonical Linear Programming Problems.- 0. Introduction.- 1. Unconstrained Variables.- 2. Equations of Constraint.- 3. Concluding Remarks.- Exercises.- 4 Duality Theory.- 0. Introduction.- 1. Duality in Canonical Tableaus.- 2. The Dual Simplex Algorithm.- 3. Matrix Formulation of Canonical Tableaus.- 4. The Duality Equation.- 5. The Duality Theorem.- 6. Duality in Noncanonical Tableaus.- 7. Concluding Remarks.- Exercises.- II: Applications.- 5 Matrix Games.- 0. Introduction.- 1. An Example Two-Person Zero-Sum Matrix Games.- 2. Linear Programming Formulation of Matrix Games.- 3. The Von Neumann Minimax Theorem.- 4. The Example Revisited.- 5. Two More Examples.- 6. Concluding Remarks.- Exercises.- 6 Transportation and Assignment Problems.- 0. Introduction.- 1. An Example The Balanced Transportation Problem.- 2. The Vogel Advanced-Start Method (VAM).- 3. The Transportation Algorithm.- 4. Another Example.- 5. Unbalanced Transportation Problems.- 6. The Assignment Problem.- 7. Concluding Remarks.- Exercises.- 7 Network-Flow Problems.- 0. Introduction.- 1. Graph-Theoretic Preliminaries.- 2. The Maximal-Flow Network Problem.- 3. The Max-Flow Min-Cut Theorem The Maximal-Flow Algorithm.- 4. The Shortest-Path Network Problem.- 5. The Minimal-Cost-Flow Network Problem.- 6. Transportation and Assignment Problems Revisited.- 7. Concluding Remarks.- Exercises.- APPENDIX A Matrix Algebra.- APPENDIX B Probability.- Answers to Selected Exercises.

Optimization of frequency assignment

Abstract: A systematic method of determining carrier frequency assignment is proposed to minimize the cochannel interference in satellite communication systems. For the mathematical treatment of the problem, discrete positioning of carriers is introduced to avoid the nonlinear expression inherent in interference evaluation. The proposed method converts this nonlinear problem into the well-known assignment problem and enables the development of an algorithm based on the branch-and-bound method. Extended models of the proposed method are presented for practical application. To demonstrate the effectiveness of the method, computation results are shown for two typical problems. In both of the examples, cochannel interference is significantly reduced as the result of optimization of frequency assignment. >

A Class of Dynamic Nonlinear Resource Allocation Problems

Abstract: : We consider a class of dynamic resource allocation problems a specific example of which is the Weapon-Target Assignment problem. This problem is concerned with the optimal assignment of resources in a military engagement. These problems are, in general, NP-Complete, so our aim is to provide insight into the problem and its solution. We will provide analytical results for simple cases of the problem. We will also provide sub-optimal algorithms, together with bounds on the objective function, under certain assumptions. The battle scenario of the military engagement being modeled is as follows. The offense launches a number of weapons ( the targets) which are aimed at valuable assets of the defense. The defense has a number of defensive weapons each of which can engage at most one target. The outcome of such an engagement is stochastic. In the static scenario all weapons are fired simultaneously. In the dynamic scenario some weapons are assigned and fired and the outcomes of these engagements are observed before further assignments are made.

Algorithm for Ranked Assignments with Applications to Multiobject Tracking

Abstract: A sequential algorithm is presented for obtaining a cost-ranked set of solutions to the assignment problem. Concurrently, a conservative bound can be calculated that indicates how many of the ranked set are better than other potential assignments that may have been missed. Applications to two important strategic defense initiative measurement-assignment problems, namely the cycle-to-cycle and the sensor-to-sensor problems, are demonstrated. The procedure is useful for initiating new object tracks as well as for assigning incoming measurements to established tracking filters. Knowledge of the ranked set of assignments, as opposed to a single optimum, is important because of measurement uncertainties. The prudent course may be to initiate several tentative tracks in certain close-call situations.

Hypercube implementation of the simplex algorithm

Abstract: Large, sparse, linear systems of equations arise frequently when constructing mathematical models of natural phenomena. Most often, these linear systems are fully constrained and can be solved via direct or iterative techniques. However, one important problem class requires solutions to underconstrained linear systems that maximize some objective function. These linear optimization problems are natural formulations of many business plans and often contain hundreds of equations with thousands of variables. Historically, linear optimization problems have been solved via the simplex method. Despite the excellent performance of the simplex method, the size of the optimization problems and the frequency of their solution make linear optimization a computationally taxing endeavor. This paper examines the performance of parallel variants of the simplex algorithm on the Intel iPSC, a message-based parallel system. Linear optimization test data are drawn from commercial sources and represent realistic problems. Analysis shows that the speedup obtained is sensitive to both the structure of the underlying data and the data partitioning.

Cellular network assignment processor using minimum/maximum convergence technique

Abstract: A cellular network assignment processor (10) for solving optimization problems utilizing a neural network architecture having a matrix of simple processing cells (12) that are highly interconnected in a regular structure. The cells (12) accept as input, costs in an assignment problem. The position of each cell (12) corresponds to the position of the cost in the associated constraint space of the assignment problem. Each cell (12) is capable of receiving, storing and transmitting cost values and is also capable of determining if it is the maximum or the minimum of cells (12) to which it's connected. Operating on one row of cells (12) at a time the processor (10) determines if a conflict exists between selected connected cells (12) until a cell (12) with no conflict is found in each row. The end result is a chosen cell (12), in each row, the chosen cells (12) together representing a valid solution to the assignment problem.

Layer assignment for VLSI interconnect delay minimization

Abstract: A formulation of the layer assignment problem for VLSI circuits is presented in which the objective is to minimize the interconnect delay by taking into account the resistance and capacitance of interconnect wires and contacts. For MOS circuits with two layers of interconnections the problem is shown to be equivalent to that of minimizing a weighted resistance of the corresponding RC network. This formulation readily handles wires with preassigned layers, such as power supply lines or module terminals. With user-defined weights assigned to selected nets, this method can be used to minimize critical path delays. The problem is shown to be NP-complete. A polynomial-time approximation algorithm, based on graph partitioning technique, is presented along with some experimental results. The layer assignment algorithm presented in this paper has been implemented in LISP and tested on several design examples with complexity ranging from tens to a few hundred nets. Computational complexity of this algorithm is on the order of O(n/sup 2/) for building the required data structure, and O(n/sup 1.5/) for actual layer assignment, where n is the number of wire segments in the routing. >

Interior-point methods in parallel computation

Abstract: Interior-point methods for linear programming, developed in the context of sequential computation, are used to obtain a parallel algorithm for the bipartite matching problem. The algorithm runs in O*( square root m) time. The results extend to the weighted bipartite matching problem and to the zero-one minimum-cost flow problem, yielding O*( square root m log C) algorithms. This improves previous bounds on these problems and illustrates the importance of interior-point methods in parallel algorithm design. >

Planning and scheduling in a flexible manufacturing system using a dynamic routing method for automated guided vehicles

Abstract: An approach that can dynamically solve the planning and scheduling problem in a flexible manufacturing system (FMS) is presented. This problem is formulated as the determination of an optimal routing assignment of p automated guided vehicles among m workstations in order to accomplish N tasks in an FMS. A useful task representation called workgraph is introduced to facilitate the latter computation; then the A* search algorithm, the minimax criterion, and source heuristic rules are used to solve this routing assignment problem dynamically. The approach obtains a near-optimal solution in moderate computation time, and, in addition, solves some dynamic situations so as to make the FMS more flexible. >