This paper provides a new methodology to solve bilinear, non-convex mathematical programming problems by a suitable transformation of variables. Schur's decomposition and special ordered sets (SOS) type 2 constrai...
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This paper provides a new methodology to solve bilinear, non-convex mathematical programming problems by a suitable transformation of variables. Schur's decomposition and special ordered sets (SOS) type 2 constraints are used resulting in a mixed integer linear or quadratic program in the two applications shown. While Beale, Tomlin and others developed the use of SOS type 2 variables to handle non-convexities, our approach is novel in two aspects. First, the use of Schur's decomposition as an integral part of the approximation step is new and leads to a numerically viable method to separate the variables. Second, the combination of our approach for handling bilinear side constraints in a complementarity or equilibrium problem setting is also new and opens the way to many interesting and realistic modi. cations to such models. We contrast our approach with other methods for solving bilinear problems also known as indefinite quadratic programs. From a practical point of view our methodology is helpful since no specialized procedures need to be created so that existing solvers can be used. The approach is illustrated with two engineering examples and the mathematical analysis appears in the Appendices. Journal of the Operational Research Society (2006) 57, 995-1004.
This paper investigates the F-policy queue using fuzzy parameters, in which the arrival rate, service rate, and start-up rate are all fuzzy numbers. The F-policy deals with the control of arrivals in a queueing system...
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This paper investigates the F-policy queue using fuzzy parameters, in which the arrival rate, service rate, and start-up rate are all fuzzy numbers. The F-policy deals with the control of arrivals in a queueing system, in which the server requires a start-up time before allowing customers to enter. A crisp F-policy queueing system generalised to a fuzzy environment would be widely applicable;therefore, we apply the alpha-cuts approach and Zadeh's extension principle to transform fuzzy F-policy queues into a family of crisp F-policy queues. This study presents a mathematical programming approach applicable to the construction of membership functions for the expected number of customers in the system. Furthermore, we propose an efficient solution procedure to compute the membership function of the expected number of customers in the system under different levels of alpha. Finally, we give an example of the proposed system as applied to a case in the automotive industry to demonstrate its practicality.
This paper examines linearly constrained bilevel programming problems in which the upper-level objective function depends on both the lower-level primal and dual optimal solutions. We parametrize the lower-level solut...
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This paper examines linearly constrained bilevel programming problems in which the upper-level objective function depends on both the lower-level primal and dual optimal solutions. We parametrize the lower-level solutions and thereby the upper-level objective function by the upper-level variables and argue that it may be non-convex and even discontinuous. However, when the upper-level objective is affine in the lower-level primal optimal solution, the parametric function is piece-wise linear. We show how this property facilitates the application of parametric programming and demonstrate how the approach allows for decomposition of a separable lower-level problem. When the upper-level objective is bilinear in the lower-level primal and dual optimal solutions, we also provide an exact linearisation method that reduces the bilevel problem to a single-level mixed-integer linear programme (MILP). We assess the performance of the parametric programming approach on two case studies of strategic investment in electricity markets and benchmark against state-of-the-art MILP and non-linear solution methods for bilevel optimisation problems. Preliminary results indicate substantial computational advantages over several standard solvers, especially when the lower-level problem separates into a large number of subproblems. Furthermore, we show that the parametric programming approach succeeds in solving problems to global optimality for which standard methods can fail.
This paper studies the single machine scheduling problem for the objective of minimizing the expected number of tardy jobs. Jobs have normally distributed processing times and a common deterministic due date. We devel...
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This paper studies the single machine scheduling problem for the objective of minimizing the expected number of tardy jobs. Jobs have normally distributed processing times and a common deterministic due date. We develop new approaches for this problem that generate near optimal solutions. The original stochastic problem is transformed into a non-linear integer programming model and its relaxations. Computational study validates their effectiveness by comparison with optimal solutions. (c) 2005 Published by Elsevier Ltd.
A method of Sequential Log-Convex programming (SLCP) is constructed that exploits the log-convex structure present in many engineering design problems. The mathematical structure of Geometric programming (GP) is combi...
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A method of Sequential Log-Convex programming (SLCP) is constructed that exploits the log-convex structure present in many engineering design problems. The mathematical structure of Geometric programming (GP) is combined with the ability of Sequential Quadratic Program (SQP) to accommodate a wide range of objective and constraint functions, resulting in a practical algorithm that can be adopted with little to no modification of existing design practices. Three test problems are considered to demonstrate the SLCP algorithm, comparing it with SQP and the modified Logspace Sequential Quadratic programming (LSQP). In these cases, SLCP shows up to a 77% reduction in number of iterations compared to SQP, and an 11% reduction compared to LSQP. The airfoil analysis code XFOIL is integrated into one of the case studies to show how SLCP can be used to evolve the fidelity of design problems that have initially been modeled as GP compatible. Finally, a methodology for design based on GP and SLCP is briefly discussed.
This paper proposes an arlene scaling derivative-free trust region method with interior backtracking technique for bounded-constrained nonlinearprogramming. This method is designed to get a stationary point for such ...
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This paper proposes an arlene scaling derivative-free trust region method with interior backtracking technique for bounded-constrained nonlinearprogramming. This method is designed to get a stationary point for such a problem with polynomial interpolation models instead of the objective function in trust region subproblem. Combined with both trust region strategy and line search technique, at each iteration, the affine scaling derivative-free trust region subproblem generates a backtracking direction in order to obtain a new accepted interior feasible step. Global convergence and fast local convergence properties are established under some reasonable conditions. Some numerical results are also given to show the effectiveness of the proposed algorithm.
A method for solving probabilistic linearprogramming problems with exponential random variables is presented in this paper. Assuming that either some or all of the parameters are exponential random variables a transf...
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A method for solving probabilistic linearprogramming problems with exponential random variables is presented in this paper. Assuming that either some or all of the parameters are exponential random variables a transformation is presented to convert the probabilistic linearprogramming problem to a deterministic mathematical programming problem. A non-linear programming algorithm can then be used to solve the resulting deterministic problem. (C) 1998 Elsevier Science B.V. All rights reserved.
In this paper, a new method is presented for the optimisation of force distribution for combined traction/braking and cornering. In order to provide a general, simple and flexible problem formulation, the optimisation...
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In this paper, a new method is presented for the optimisation of force distribution for combined traction/braking and cornering. In order to provide a general, simple and flexible problem formulation, the optimisation is addressed as a quadratically constrained linearprogramming (QCLP) problem. Apart from fast numerical solutions, different driveline configurations can be included in the QCLP problem in a very straightforward fashion. The optimisation of the distribution of the individual wheel forces using the quasi-steady-state assumption is known to be useful for the study of the influence of particular driveline configurations on the combined lateral and longitudinal grip envelope of a particular vehicle-driveline configuration. The addition of the QCLP problem formulation makes another powerful tool available to the vehicle dynamics analyst to perform such studies.
The aim of this paper is to transform a multi-choice linearprogramming problem to a standard mathematical programming problem where the right hand side goals of some constraints are 'multi-choice' in nature. ...
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The aim of this paper is to transform a multi-choice linearprogramming problem to a standard mathematical programming problem where the right hand side goals of some constraints are 'multi-choice' in nature. For each of the constraint there may exist multiple number of goals, out of which exactly one is to be chosen. The selection of goals should be in such a manner that the combination of choices for each constraint should provide an optimal solution to the objective function. There may be more than one combination which will provide an optimal solution. However the problem cannot be solved by standard linearprogramming techniques. In order to solve the present multi-choice linearprogramming problem, this paper proposes a new transformation technique. Binary variables are introduced in the transformation technique to formulate a non-linear mixed integer programming model. Using standard non-linear programming software optimal solution of the proposed model can be obtained. Finally, a numerical example is presented to illustrate the transformation technique and the solution procedure. (C) 2009 Elsevier Inc. All rights reserved.
Cost-Safety tradeoff analysis is one of the most challenging tasks of structural maintenance. Undoubtedly, developing an economic and efficient schedule for structural maintenance and rehabilitation is highly acknowle...
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Cost-Safety tradeoff analysis is one of the most challenging tasks of structural maintenance. Undoubtedly, developing an economic and efficient schedule for structural maintenance and rehabilitation is highly acknowledged. While meta-heuristic optimization algorithms have been used widely to determine the best maintenance strategies to provide more economical structures, we present a mathematical programming model to overcome the limitations of previous studies. In this paper a Mixed Integer non-linear programming (MINLP) has been presented to find the optimal time of applying maintenance intervention in a deteriorating structure. While, considering the time value of money, postponing the maintenance actions will be more economic, this postponement may cause a decrease in the safety of structures. Due to this contradictory relation between the objectives, it is vital to find a reasonable trade-off between cost-safety. Our proposed approach considers different values of the discount rate of money. We apply our mathematical programming model to solve two optimization examples, which are found in the structural maintenance literature. It is shown that our proposed model is able to determine the optimal time of applying maintenance intervention to the structures with less total life cycle cost, and higher level of safety.
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