In this paper we define a new kind of mathematical programming problems. This kind, in which the decision set is a rough set, is called a rough programmingproblem. A rough optimal solution and a rough saddle point wi...
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In this paper we define a new kind of mathematical programming problems. This kind, in which the decision set is a rough set, is called a rough programmingproblem. A rough optimal solution and a rough saddle point will be characterized. Some illustrative examples are presented. (c) 2004 Elsevier B.V. All rights reserved.
A problem of scheduling jobs on parallel, identical machines under an additional continuous resource to minimize the makespan is considered. Jobs are non-preemtable and independent and all are available at the start o...
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A problem of scheduling jobs on parallel, identical machines under an additional continuous resource to minimize the makespan is considered. Jobs are non-preemtable and independent and all are available at the start of the process. The total amount of the continuous resource available at a time is limited and the resource is a renewable one. Each job simultaneously requires for its processing a machine and an amount (unknown in advance) of the continuous resource. The processing rate of a job depends on the amount of the resource allotted to this job at a time. The problem is to find a sequence of jobs on machines and, simultaneously, a continuous resource allocation which minimize the makespan. A heuristic approach to allocating the continuous resource is proposed. A tabu search algorithm to solve the considered problem is presented and the results for the algorithms with exact and heuristic procedures for allocating the continuous resource are compared on the basis of some computational experiments. Copyright (C) 2002 John Wiley Sons, Ltd.
A scheduling problem in a plastics forming plant is studied. This problem basically belongs to the class of unrelated parallel machine problems, but includes several restrictions which originate from the necessity to ...
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A scheduling problem in a plastics forming plant is studied. This problem basically belongs to the class of unrelated parallel machine problems, but includes several restrictions which originate from the necessity to use auxiliary equipments. We have proposed a method to solve them without relying upon dispatching rules by transforming the scheduling problem to a mathematical programming problem, and by representing feasible schedules by binary strings. This formulation enables to use the meta-heuristics (such as simulated annealing, genetic algorithms, etc.). In this paper, we propose a way for improving a search by meta-heuristics by modifying the representation of a schedule, i.e., not using the binary representation but using an alphabetical one. We actually carried out computational experiments for the problems of practical size, which shows that our methods can give satisfactory solutions to the considered scheduling problem
For nonlinear programmingproblems with equality constraints, Hestenes and Powell have independently proposed a dual method of solution in which squares of the constraint functions are added as penalties to the Lagran...
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For nonlinear programmingproblems with equality constraints, Hestenes and Powell have independently proposed a dual method of solution in which squares of the constraint functions are added as penalties to the Lagrangian, and a certain simple rule is used for updating the Lagrange multipliers after each cycle. Powell has essentially shown that the rate of convergence is linear if one starts with a sufficiently high penalty factor and sufficiently near to a local solution satisfying the usual second-order sufficient conditions for optimality. This paper furnishes the corresponding method for inequality-constrained problems. Global convergence to an optimal solution is established in the convex case for an arbitrary penalty factor and without the requirement that an exact minimum be calculated at each cycle. Furthermore, the Lagrange multipliers are shown to converge, even though the optimal multipliers may not be unique.
In this paper, the problem of minimizing a function f(x) subject to a constraint φ{symbol}(x)=0 is considered, where f is a scalar, x an n-vector, and φ{symbol} a q-vector, with q s is such that φ{symbol}(xs)=0, an...
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In this paper, the problem of minimizing a function f(x) subject to a constraint φ{symbol}(x)=0 is considered, where f is a scalar, x an n-vector, and φ{symbol} a q-vector, with q s is such that φ{symbol}(xs)=0, and at most N*=1+n -q if the starting point
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