ISSN:
1547-5816
eISSN:
1553-166X
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Journal of Industrial & Management Optimization
April 2011 , Volume 7 , Issue 2
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2011, 7(2): 283-289
doi: 10.3934/jimo.2011.7.283
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Abstract:
In this paper, by using some analytic techniques, several sufficient conditions are given to ensure the passivity of continuous-time recurrent neural networks with delays. The passivity conditions are presented in terms of some negative semi-definite matrices. They are easily verifiable and easier to check computing with some conditions in terms of complicated linear matrix inequality.
In this paper, by using some analytic techniques, several sufficient conditions are given to ensure the passivity of continuous-time recurrent neural networks with delays. The passivity conditions are presented in terms of some negative semi-definite matrices. They are easily verifiable and easier to check computing with some conditions in terms of complicated linear matrix inequality.
2011, 7(2): 291-315
doi: 10.3934/jimo.2011.7.291
+[Abstract](1069)
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Abstract:
The operation of the electrorheological clutch is simulated by a nonlinear parabolic equation which describes the motion of electrorheological fluid in the gap between the driving and driven rotors. In this case, the velocity of the driving rotor is prescribed on one part of the boundary. Nonlocal nonlinear boundary condition is given on a part of the boundary, which corresponds to the driven rotor A problem on optimal control of acceleration or braking of the driven rotor is formulated and studied. Functions of time of the angular velocity of the driving rotor and of the voltages are considered to be controls. In the case that the clutch acts as an accelerator, the energy consumed in the acceleration of the driven rotor is minimized under the restriction that at some instant, the angular velocity and the acceleration of the driven rotor are localized within given regions. In the case of braking, the energy production is maximized. The existence of a solution of optimal control problem is proved and necessary optimality conditions are established.
The operation of the electrorheological clutch is simulated by a nonlinear parabolic equation which describes the motion of electrorheological fluid in the gap between the driving and driven rotors. In this case, the velocity of the driving rotor is prescribed on one part of the boundary. Nonlocal nonlinear boundary condition is given on a part of the boundary, which corresponds to the driven rotor A problem on optimal control of acceleration or braking of the driven rotor is formulated and studied. Functions of time of the angular velocity of the driving rotor and of the voltages are considered to be controls. In the case that the clutch acts as an accelerator, the energy consumed in the acceleration of the driven rotor is minimized under the restriction that at some instant, the angular velocity and the acceleration of the driven rotor are localized within given regions. In the case of braking, the energy production is maximized. The existence of a solution of optimal control problem is proved and necessary optimality conditions are established.
2011, 7(2): 317-345
doi: 10.3934/jimo.2011.7.317
+[Abstract](1386)
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Abstract:
We consider a class of stochastic nonlinear complementarity problems. We propose a new reformulation of the stochastic complementarity problem, that is, a two-stage stochastic mathematical programming model reformulation. Based on this reformulation, we propose a smoothing-based sample average approximation method for stochastic complementarity problem and prove its convergence. As an application, a supply chain super-network equilibrium is modeled as a stochastic nonlinear complementarity problem and numerical results on the problem are reported.
We consider a class of stochastic nonlinear complementarity problems. We propose a new reformulation of the stochastic complementarity problem, that is, a two-stage stochastic mathematical programming model reformulation. Based on this reformulation, we propose a smoothing-based sample average approximation method for stochastic complementarity problem and prove its convergence. As an application, a supply chain super-network equilibrium is modeled as a stochastic nonlinear complementarity problem and numerical results on the problem are reported.
2011, 7(2): 347-364
doi: 10.3934/jimo.2011.7.347
+[Abstract](1458)
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Abstract:
To apply the traditional marginal-cost pricing to drive a user equilibrium of the oligopolistic game to the system optimum, it requires to classify the users into different classes and then charge discriminatory tolls across user classes. By realizing the difficulty of discriminating users when they differ in some unobservable ways, Yang and Zhang investigated existence of anonymous link tolls for transportation networks recently. In this paper, we consider the anonymous link tolls for the oligopolistic game with nonseparable, nonlinear and asymmetric cost functions with fixed demands. With similar techniques developed by Yang and Zhang, we first prove the existence of anonymous link tolls to decentralize the system optimum to a user equilibrium. Then, by deriving some bounds on the so-called price of anarchy, we analyze the efficiency of such a toll strategy when the tolls are considered as part of the system cost.
To apply the traditional marginal-cost pricing to drive a user equilibrium of the oligopolistic game to the system optimum, it requires to classify the users into different classes and then charge discriminatory tolls across user classes. By realizing the difficulty of discriminating users when they differ in some unobservable ways, Yang and Zhang investigated existence of anonymous link tolls for transportation networks recently. In this paper, we consider the anonymous link tolls for the oligopolistic game with nonseparable, nonlinear and asymmetric cost functions with fixed demands. With similar techniques developed by Yang and Zhang, we first prove the existence of anonymous link tolls to decentralize the system optimum to a user equilibrium. Then, by deriving some bounds on the so-called price of anarchy, we analyze the efficiency of such a toll strategy when the tolls are considered as part of the system cost.
2011, 7(2): 365-383
doi: 10.3934/jimo.2011.7.365
+[Abstract](1638)
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Abstract:
In practical aggregate production planning (APP) decisions, the decision maker (DM) must simultaneously handle multiple conflicting goals that govern the use of the constrained resources. This study aims to present a two-phase fuzzy goal programming method to solve multi-objective APP problems with multiple products and multi-time periods. The designed fuzzy multi-objective linear programming model attempts to simultaneously minimize total costs, total carrying and backordering volume, and total rates of changes in labor levels with reference to inventory carrying levels, machine capacity, work-force levels, warehouse space and available budget. An industrial case is used to demonstrate the feasibility of applying the proposed method to real-life APP decisions. The contribution of this study lies in presenting a two-phase fuzzy goal programming methodology to solve multi-objective APP decision problems and provides a systematic decision-making framework that facilitates a DM to interactively adjust the search direction until the preferred efficient compromise solution is obtained.
In practical aggregate production planning (APP) decisions, the decision maker (DM) must simultaneously handle multiple conflicting goals that govern the use of the constrained resources. This study aims to present a two-phase fuzzy goal programming method to solve multi-objective APP problems with multiple products and multi-time periods. The designed fuzzy multi-objective linear programming model attempts to simultaneously minimize total costs, total carrying and backordering volume, and total rates of changes in labor levels with reference to inventory carrying levels, machine capacity, work-force levels, warehouse space and available budget. An industrial case is used to demonstrate the feasibility of applying the proposed method to real-life APP decisions. The contribution of this study lies in presenting a two-phase fuzzy goal programming methodology to solve multi-objective APP decision problems and provides a systematic decision-making framework that facilitates a DM to interactively adjust the search direction until the preferred efficient compromise solution is obtained.
2011, 7(2): 385-400
doi: 10.3934/jimo.2011.7.385
+[Abstract](1391)
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Abstract:
Given a task of tracking a trajectory, a recurrent neural network may be considered as a black-box nonlinear regression model for tracking unknown dynamic systems. An error function is used to measure the difference between the system outputs and the desired trajectory that formulates a nonlinear least square problem with dynamical constraints. With the dynamical constraints, classical gradient type methods are difficult and time consuming due to the involving of the computation of the partial derivatives along the trajectory. We develop an alternative learning algorithm, namely the weighted state space search algorithm, which searches the neighborhood of the target trajectory in the state space instead of the parameter space. Since there is no computation of partial derivatives involved, our algorithm is simple and fast. We demonstrate our approach by modeling the short-term foreign exchange rates. The empirical results show that the weighted state space search method is very promising and effective in solving least square problems with dynamical constraints. Numerical costs between the gradient method and our the proposed method are provided.
Given a task of tracking a trajectory, a recurrent neural network may be considered as a black-box nonlinear regression model for tracking unknown dynamic systems. An error function is used to measure the difference between the system outputs and the desired trajectory that formulates a nonlinear least square problem with dynamical constraints. With the dynamical constraints, classical gradient type methods are difficult and time consuming due to the involving of the computation of the partial derivatives along the trajectory. We develop an alternative learning algorithm, namely the weighted state space search algorithm, which searches the neighborhood of the target trajectory in the state space instead of the parameter space. Since there is no computation of partial derivatives involved, our algorithm is simple and fast. We demonstrate our approach by modeling the short-term foreign exchange rates. The empirical results show that the weighted state space search method is very promising and effective in solving least square problems with dynamical constraints. Numerical costs between the gradient method and our the proposed method are provided.
2011, 7(2): 401-424
doi: 10.3934/jimo.2011.7.401
+[Abstract](1159)
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Abstract:
The household product industry often suffers from problems of high operational costs and/or lost sales from selling seasonal products since managers in different divisions/stages of the supply chain make their decisions independently. We are therefore motivated to propose an approach that integrates the operations in the three stages of a supply chain for the management of seasonal products. The goal of this study is to minimize the total costs incurred for the seasonal products in the whole selling period. In the first stage, we employ a newsvendor model to determine the optimal one-time-only order quantities of the seasonal products. Then, we utilize a clustering analysis in the second stage to categorize seasonal products into groups, and establish an optimal replenishment policy for the products in the same group. Finally, the third stage starts when the warehouse runs out of its inventory. We formulate a mathematical model to determine the optimal transshipment quantities among all of the retailers to minimize the expected total costs incurred for the rest of the selling period. Taking a household product company in Taiwan as our case study, we demonstrate the implementation of the proposed (integrated) approach. Based on the real-world data of 17 seasonal products, we show that our integrated approach significantly outperforms the company's current approach. Therefore, our integrated approach may serve as an effective decision support tool for the distribution and lateral transshipment of seasonal products in the household product industry.
The household product industry often suffers from problems of high operational costs and/or lost sales from selling seasonal products since managers in different divisions/stages of the supply chain make their decisions independently. We are therefore motivated to propose an approach that integrates the operations in the three stages of a supply chain for the management of seasonal products. The goal of this study is to minimize the total costs incurred for the seasonal products in the whole selling period. In the first stage, we employ a newsvendor model to determine the optimal one-time-only order quantities of the seasonal products. Then, we utilize a clustering analysis in the second stage to categorize seasonal products into groups, and establish an optimal replenishment policy for the products in the same group. Finally, the third stage starts when the warehouse runs out of its inventory. We formulate a mathematical model to determine the optimal transshipment quantities among all of the retailers to minimize the expected total costs incurred for the rest of the selling period. Taking a household product company in Taiwan as our case study, we demonstrate the implementation of the proposed (integrated) approach. Based on the real-world data of 17 seasonal products, we show that our integrated approach significantly outperforms the company's current approach. Therefore, our integrated approach may serve as an effective decision support tool for the distribution and lateral transshipment of seasonal products in the household product industry.
2011, 7(2): 425-434
doi: 10.3934/jimo.2011.7.425
+[Abstract](1486)
+[PDF](418.7KB)
Abstract:
We study an integrated market selection and inventory control problem that was initially proposed by Geunes et al. [Naval Research Logistics, 51(1):117-136, 2004]. This problem generalizes the classical EOQ problem by incorporating the market choice decisions. In this note, we further consider the problem with stochastic demand in which we assume the demand mean and variance are known for each market. We show that the problem can be formulated as an unconstrained nonlinear binary IP model. Its special structure leads to efficient solution algorithms and we summarize some interesting observations via numerical experiments.
We study an integrated market selection and inventory control problem that was initially proposed by Geunes et al. [Naval Research Logistics, 51(1):117-136, 2004]. This problem generalizes the classical EOQ problem by incorporating the market choice decisions. In this note, we further consider the problem with stochastic demand in which we assume the demand mean and variance are known for each market. We show that the problem can be formulated as an unconstrained nonlinear binary IP model. Its special structure leads to efficient solution algorithms and we summarize some interesting observations via numerical experiments.
2011, 7(2): 435-447
doi: 10.3934/jimo.2011.7.435
+[Abstract](1311)
+[PDF](349.1KB)
Abstract:
This paper establishes a convergence theory for an interior penalty method for a linear complementarity problem governing American option valuation. By introducing an interior penalty term, we first transform the complementarity problem into a nonlinear degenerated Black-Scholes PDE. We then prove that the PDE is uniquely solvable and its solution converges to that of the original complementarity problem. Furthermore, we demonstrate the advantages of the interior penalty method over exterior penalty methods by comparing it with an existing exterior penalty method.
This paper establishes a convergence theory for an interior penalty method for a linear complementarity problem governing American option valuation. By introducing an interior penalty term, we first transform the complementarity problem into a nonlinear degenerated Black-Scholes PDE. We then prove that the PDE is uniquely solvable and its solution converges to that of the original complementarity problem. Furthermore, we demonstrate the advantages of the interior penalty method over exterior penalty methods by comparing it with an existing exterior penalty method.
2011, 7(2): 449-465
doi: 10.3934/jimo.2011.7.449
+[Abstract](1067)
+[PDF](385.9KB)
Abstract:
In today's competitive market of the civil aviation industry, overbooking has been a common strategy for airlines to deal with uncertainty. However, while raising the overbooking level could recover partial losses caused by cancelation and no-show, this policy would bring more uncertainty into the system. As a solution, a new method of "transference" has recently been implemented by some major airlines in China. This method allows some of the overflowed passengers resulting from overbooking to board on a later flight with certain compensation. When it is properly implemented, airline companies could enjoy reduced uncertainty and improved revenue. In this paper, we build a model to depict this method, design a procedure to determine the optimal transferring quantity among flights of different departure times, analyze the overbooking level of each flight, and show improved revenue under the method of "transference". We also present a numerical example to highlight that our results may coincide with reality.
In today's competitive market of the civil aviation industry, overbooking has been a common strategy for airlines to deal with uncertainty. However, while raising the overbooking level could recover partial losses caused by cancelation and no-show, this policy would bring more uncertainty into the system. As a solution, a new method of "transference" has recently been implemented by some major airlines in China. This method allows some of the overflowed passengers resulting from overbooking to board on a later flight with certain compensation. When it is properly implemented, airline companies could enjoy reduced uncertainty and improved revenue. In this paper, we build a model to depict this method, design a procedure to determine the optimal transferring quantity among flights of different departure times, analyze the overbooking level of each flight, and show improved revenue under the method of "transference". We also present a numerical example to highlight that our results may coincide with reality.
2011, 7(2): 467-482
doi: 10.3934/jimo.2011.7.467
+[Abstract](1341)
+[PDF](408.9KB)
Abstract:
In this paper, we propose a new class of smoothing functions which uniformly approximates the median function of three scalars. The proposed functions are the generalization of the smoothing function proposed by Li and Fukushima. Some favorable properties of the functions are investigated. By using the proposed functions, we reformulate the box constrained variational inequality problem (VIP) as a system of parameterized smooth equations, and then propose a smoothing Newton algorithm with a nonmonotone line search to solve the VIP. The proposed algorithm is proved to be globally and locally superlinearly convergent under suitable assumptions. Some numerical results for test problems from MCPLIB are also reported, which demonstrate that the proposed smoothing functions are valuable and the proposed algorithm is effective.
In this paper, we propose a new class of smoothing functions which uniformly approximates the median function of three scalars. The proposed functions are the generalization of the smoothing function proposed by Li and Fukushima. Some favorable properties of the functions are investigated. By using the proposed functions, we reformulate the box constrained variational inequality problem (VIP) as a system of parameterized smooth equations, and then propose a smoothing Newton algorithm with a nonmonotone line search to solve the VIP. The proposed algorithm is proved to be globally and locally superlinearly convergent under suitable assumptions. Some numerical results for test problems from MCPLIB are also reported, which demonstrate that the proposed smoothing functions are valuable and the proposed algorithm is effective.
2011, 7(2): 483-496
doi: 10.3934/jimo.2011.7.483
+[Abstract](1502)
+[PDF](372.7KB)
Abstract:
In this paper, we introduce a new kind of properly approximate efficient solution of vector optimization problems. Some properties for this new class of approximate solutions are established. Also necessary and sufficient conditions via nonlinear scalarizations are obtained for properly approximate solutions. And under the assumption of cone subconvexlike functions, we derive linear scalarizations for properly approximate efficient solutions.
In this paper, we introduce a new kind of properly approximate efficient solution of vector optimization problems. Some properties for this new class of approximate solutions are established. Also necessary and sufficient conditions via nonlinear scalarizations are obtained for properly approximate solutions. And under the assumption of cone subconvexlike functions, we derive linear scalarizations for properly approximate efficient solutions.
2011, 7(2): 497-521
doi: 10.3934/jimo.2011.7.497
+[Abstract](1293)
+[PDF](553.2KB)
Abstract:
In this paper, we present a new approach for finding a stable solution of a system of nonlinear equations arising from dynamical systems. We introduce the concept of stability functions and use this idea to construct stability solution models of several typical small signal stability problems in dynamical systems. Each model consists of a system of constrained semismooth equations. The advantage of the new models is twofold. Firstly, the stability requirement of dynamical systems is controlled by nonlinear inequalities. Secondly, the semismoothness property of the stability functions makes the models solvable by efficient numerical methods. We introduce smoothing functions for the stability functions and present a smoothing Newton method for solving the problems. Global and local quadratic convergence of the algorithm is established. Numerical examples from dynamical systems are also given to illustrate the efficiency of the new approach.
In this paper, we present a new approach for finding a stable solution of a system of nonlinear equations arising from dynamical systems. We introduce the concept of stability functions and use this idea to construct stability solution models of several typical small signal stability problems in dynamical systems. Each model consists of a system of constrained semismooth equations. The advantage of the new models is twofold. Firstly, the stability requirement of dynamical systems is controlled by nonlinear inequalities. Secondly, the semismoothness property of the stability functions makes the models solvable by efficient numerical methods. We introduce smoothing functions for the stability functions and present a smoothing Newton method for solving the problems. Global and local quadratic convergence of the algorithm is established. Numerical examples from dynamical systems are also given to illustrate the efficiency of the new approach.
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