ISSN:
1547-5816
eISSN:
1553-166X
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Journal of Industrial & Management Optimization
January 2008 , Volume 4 , Issue 1
Special Issue on Optimization and Optimal Control with Applications
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2008, 4(1): i-ii
doi: 10.3934/jimo.2008.4.1i
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Abstract:
Optimization and optimal control problems arise in diverse areas such as traditional engineering, natural resource utilization, financial engineering, vehicle route selection and supply chain planning. This special issue contains eight full-length papers reflecting the recent advances in the numerical solution of some complex optimization and optimal control problems. Most of these results were orally presented in the special session entitled 'Optimization and Optimal Control with Applications' of the AIMS' 6th International Conference on Dynamical Systems, Differential Equations and Applications held at University of Poitiers, France from June 25th - 28th, 2006.
Optimization and optimal control problems arise in diverse areas such as traditional engineering, natural resource utilization, financial engineering, vehicle route selection and supply chain planning. This special issue contains eight full-length papers reflecting the recent advances in the numerical solution of some complex optimization and optimal control problems. Most of these results were orally presented in the special session entitled 'Optimization and Optimal Control with Applications' of the AIMS' 6th International Conference on Dynamical Systems, Differential Equations and Applications held at University of Poitiers, France from June 25th - 28th, 2006.
2008, 4(1): 1-15
doi: 10.3934/jimo.2008.4.1
+[Abstract](1408)
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Abstract:
Oscillatory inputs have been observed to increase the yield of chemical reactors beyond the level possible by steady inputs. To obtain the optimal inputs, iterative dynamic programming is well suited, because a very large number of time stages can be used without encountering computational problems. To observe the benefits of oscillatory inputs, the effects of the initial state and the final state can be eliminated by normalizing the average yields with respect to the yield from a shorter final time. Two examples show that optimal oscillatory control policy can improve the yield substantially. The third example shows that there are situations where oscillatory behaviour is optimal, but the benefits are negligible. The optimal control policies can be readily established with iterative dynamic programming with the use of a large number of time stages of flexible length.
Oscillatory inputs have been observed to increase the yield of chemical reactors beyond the level possible by steady inputs. To obtain the optimal inputs, iterative dynamic programming is well suited, because a very large number of time stages can be used without encountering computational problems. To observe the benefits of oscillatory inputs, the effects of the initial state and the final state can be eliminated by normalizing the average yields with respect to the yield from a shorter final time. Two examples show that optimal oscillatory control policy can improve the yield substantially. The third example shows that there are situations where oscillatory behaviour is optimal, but the benefits are negligible. The optimal control policies can be readily established with iterative dynamic programming with the use of a large number of time stages of flexible length.
2008, 4(1): 17-32
doi: 10.3934/jimo.2008.4.17
+[Abstract](1469)
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Abstract:
This paper is concerned with the second order nonlinear impulsive evolution differential equations perturbed by unbounded operator on Banach space. Discussing the perturbation of time-varying operator matrix and constructing the corresponding evolution system generated by operator matrix we introduce the reasonable mild solution of second order nonlinear impulsive evolution differential equations and prove the existence of mild solutions. Existence of optimal controls for a Lagrange problem of systems governed by the second order nonlinear impulsive evolution equations is also presented. An example is given for demonstration.
This paper is concerned with the second order nonlinear impulsive evolution differential equations perturbed by unbounded operator on Banach space. Discussing the perturbation of time-varying operator matrix and constructing the corresponding evolution system generated by operator matrix we introduce the reasonable mild solution of second order nonlinear impulsive evolution differential equations and prove the existence of mild solutions. Existence of optimal controls for a Lagrange problem of systems governed by the second order nonlinear impulsive evolution equations is also presented. An example is given for demonstration.
2008, 4(1): 33-52
doi: 10.3934/jimo.2008.4.33
+[Abstract](1407)
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Abstract:
In this paper we present a novel method for short term forecast of time series based on Knot-Optimizing Spline Networks (KOSNETS). The time series is first approximated by a nonlinear recurrent system. The resulting recurrent system is then approximated by feedforward $B$-spline networks, yielding a nonlinear optimization problem. In this optimization problem, both the knot points and the coefficients of the $B$-splines are decision variables so that the solution to the problem has both optimal coefficients and partition points. To demonstrate the usefulness and accuracy of the method, numerical simulations and tests using various model and real time series are performed. The numerical simulation results are compared with those from a well-known regression method, MARS. The comparison shows that our method outperforms MARS for nonlinear problems.
In this paper we present a novel method for short term forecast of time series based on Knot-Optimizing Spline Networks (KOSNETS). The time series is first approximated by a nonlinear recurrent system. The resulting recurrent system is then approximated by feedforward $B$-spline networks, yielding a nonlinear optimization problem. In this optimization problem, both the knot points and the coefficients of the $B$-splines are decision variables so that the solution to the problem has both optimal coefficients and partition points. To demonstrate the usefulness and accuracy of the method, numerical simulations and tests using various model and real time series are performed. The numerical simulation results are compared with those from a well-known regression method, MARS. The comparison shows that our method outperforms MARS for nonlinear problems.
2008, 4(1): 53-66
doi: 10.3934/jimo.2008.4.53
+[Abstract](1390)
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The Joint Replenishment Problem (JRP) is a multi-item inventory problem. The objective is to develop inventory policies that minimize total cost (comprised of holding and setup costs) over the planning horizon. In this paper we consider the extension of this problem to the multi-buyer, multi-item version of the JRP. We propose and test a mixed simulated annealing-genetic algorithm (SAGA) for the extended problem. Tests are conducted on problems from a leading bank in Hong Kong. Results are also compared to a pure GA approach and several interesting observations are made on the value of such meta-heuristics.
The Joint Replenishment Problem (JRP) is a multi-item inventory problem. The objective is to develop inventory policies that minimize total cost (comprised of holding and setup costs) over the planning horizon. In this paper we consider the extension of this problem to the multi-buyer, multi-item version of the JRP. We propose and test a mixed simulated annealing-genetic algorithm (SAGA) for the extended problem. Tests are conducted on problems from a leading bank in Hong Kong. Results are also compared to a pure GA approach and several interesting observations are made on the value of such meta-heuristics.
2008, 4(1): 67-79
doi: 10.3934/jimo.2008.4.67
+[Abstract](1223)
+[PDF](172.3KB)
Abstract:
The stabilization problem of a nonuniform Timoshenko beam with nonlinear locally distributed feedback controls is considered. By means of nonlinear semigroup theory, energy-perturbation approach and piecewise multiplier method it is shown that the energy of the closed loop system decays exponentially or in the rate of negative power of time.
The stabilization problem of a nonuniform Timoshenko beam with nonlinear locally distributed feedback controls is considered. By means of nonlinear semigroup theory, energy-perturbation approach and piecewise multiplier method it is shown that the energy of the closed loop system decays exponentially or in the rate of negative power of time.
2008, 4(1): 81-94
doi: 10.3934/jimo.2008.4.81
+[Abstract](1744)
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Abstract:
The optimal portfolio problem under a VaR (value at risk) constraint is reviewed. Two different formulations, namely with and without consumption, are illustrated. This problem can be formulated as a constrained stochastic optimal control problem. The optimality conditions can be derived using the dynamic programming technique and the method of Lagrange multiplier can be applied to handle the VaR constraint. The method is extended for inventory management. Different from traditional inventory models of minimizing overall cost, the cashflow dynamic of a manufacturer is derived by considering a portfolio of inventory of raw materials together with income and consumption. The VaR of the portfolio of assets is derived and imposed as a constraint. Furthermore, shortage cost and holding cost can also be formulated as probabilistic constraints. Under this formulation, we find that holdings in high risk inventory are optimally reduced by the imposed value-at-risk constraint.
The optimal portfolio problem under a VaR (value at risk) constraint is reviewed. Two different formulations, namely with and without consumption, are illustrated. This problem can be formulated as a constrained stochastic optimal control problem. The optimality conditions can be derived using the dynamic programming technique and the method of Lagrange multiplier can be applied to handle the VaR constraint. The method is extended for inventory management. Different from traditional inventory models of minimizing overall cost, the cashflow dynamic of a manufacturer is derived by considering a portfolio of inventory of raw materials together with income and consumption. The VaR of the portfolio of assets is derived and imposed as a constraint. Furthermore, shortage cost and holding cost can also be formulated as probabilistic constraints. Under this formulation, we find that holdings in high risk inventory are optimally reduced by the imposed value-at-risk constraint.
2008, 4(1): 95-105
doi: 10.3934/jimo.2008.4.95
+[Abstract](1328)
+[PDF](158.7KB)
Abstract:
We consider a class of path design problems which arise when an object needs to traverse between two points through a specified region. The path must optimise a prescribed criterion such as risk, reliability or cost and satisfy a number of constraints such as total travel time. Problems of this type readily arise in the defence, transport and communication industries. We specifically look at the problem of determining an optimal (in terms of minimizing the overall probability of detection) transit path for a submarine moving through a field of sonar sensors, subject to a total time constraint. A computational strategy along with results are presented.
We consider a class of path design problems which arise when an object needs to traverse between two points through a specified region. The path must optimise a prescribed criterion such as risk, reliability or cost and satisfy a number of constraints such as total travel time. Problems of this type readily arise in the defence, transport and communication industries. We specifically look at the problem of determining an optimal (in terms of minimizing the overall probability of detection) transit path for a submarine moving through a field of sonar sensors, subject to a total time constraint. A computational strategy along with results are presented.
2008, 4(1): 107-123
doi: 10.3934/jimo.2008.4.107
+[Abstract](1169)
+[PDF](239.6KB)
Abstract:
This paper presents a new basic model based on automatons for the state feedback control of discrete event systems (DES), including (repeated) concurrent DES. So, this new model unifies the Ramadge-Wonham framework and the controlled Petri nets, with or without concurrency or repeated concurrency. The repeated concurrent model under Ramadge-Wonham framework is first presented here. We study relationships between the concurrent models and the basic model. Based on this, we show that the uniqueness of the maximal permissive state feedback (PSF) of a predicate $P$ is equivalent to the weak interaction of $P$, which is also equivalent to that the set of PSF is closed under a disjunction. These results are also true for the concurrent systems, but the weak interaction may be difficult to be verified. Hence, we try to simplify the weak interaction by introducing concepts of cover, transitivity and local concurrently well-posedness (CWP). We show that the local CWP can ensure that the set of PSF for the concurrent systems equals that for the basic system.
This paper presents a new basic model based on automatons for the state feedback control of discrete event systems (DES), including (repeated) concurrent DES. So, this new model unifies the Ramadge-Wonham framework and the controlled Petri nets, with or without concurrency or repeated concurrency. The repeated concurrent model under Ramadge-Wonham framework is first presented here. We study relationships between the concurrent models and the basic model. Based on this, we show that the uniqueness of the maximal permissive state feedback (PSF) of a predicate $P$ is equivalent to the weak interaction of $P$, which is also equivalent to that the set of PSF is closed under a disjunction. These results are also true for the concurrent systems, but the weak interaction may be difficult to be verified. Hence, we try to simplify the weak interaction by introducing concepts of cover, transitivity and local concurrently well-posedness (CWP). We show that the local CWP can ensure that the set of PSF for the concurrent systems equals that for the basic system.
2008, 4(1): 125-142
doi: 10.3934/jimo.2008.4.125
+[Abstract](1308)
+[PDF](209.0KB)
Abstract:
By using the canonical dual transformation developed recently, we derive a pair of canonical dual problems for 0-1 quadratic programming problems in both minimization and maximization form. Regardless convexity, when the canonical duals are solvable, no duality gap exists between the primal and corresponding dual problems. Both global and local optimality conditions are given. An algorithm is presented for finding global minimizers, even when the primal objective function is not convex. Examples are included to illustrate this new approach.
By using the canonical dual transformation developed recently, we derive a pair of canonical dual problems for 0-1 quadratic programming problems in both minimization and maximization form. Regardless convexity, when the canonical duals are solvable, no duality gap exists between the primal and corresponding dual problems. Both global and local optimality conditions are given. An algorithm is presented for finding global minimizers, even when the primal objective function is not convex. Examples are included to illustrate this new approach.
2008, 4(1): 143-153
doi: 10.3934/jimo.2008.4.143
+[Abstract](1647)
+[PDF](181.3KB)
Abstract:
In this paper, we investigate the reformulation of the steady state security region problem for electrical power systems. Firstly, a simple security region problem with one changeable parameter is reformulated into a system of semismooth equations, which is composed by the normal power flow equations and an additional piecewise smooth equation. Then the semismooth Newton method and the smoothing Newton method can be applied to solve the problem. Preliminary numerical results show that the method is promising. Finally, by using the smoothing technique, a more complicated security region problem, the Euclidean security region problem, is reformulated as an equality constrained optimization problem. These works provide a possibility to implement on-line calculation of the security region of electrical power systems.
In this paper, we investigate the reformulation of the steady state security region problem for electrical power systems. Firstly, a simple security region problem with one changeable parameter is reformulated into a system of semismooth equations, which is composed by the normal power flow equations and an additional piecewise smooth equation. Then the semismooth Newton method and the smoothing Newton method can be applied to solve the problem. Preliminary numerical results show that the method is promising. Finally, by using the smoothing technique, a more complicated security region problem, the Euclidean security region problem, is reformulated as an equality constrained optimization problem. These works provide a possibility to implement on-line calculation of the security region of electrical power systems.
2008, 4(1): 155-165
doi: 10.3934/jimo.2008.4.155
+[Abstract](1917)
+[PDF](156.7KB)
Abstract:
In this paper, we obtain some stability results for the dual problem of a weak vector variational inequality problem. We establish the upper semicontinuity property of the solution set for a perturbed dual problem of a weak vector variational inequality problem. By virtue of a parametric gap function and a key assumption, we also obtain the lower semicontinuity property of the solution set for the perturbed dual problem. Some examples are given for the illustration of the necessity of our research on duality.
In this paper, we obtain some stability results for the dual problem of a weak vector variational inequality problem. We establish the upper semicontinuity property of the solution set for a perturbed dual problem of a weak vector variational inequality problem. By virtue of a parametric gap function and a key assumption, we also obtain the lower semicontinuity property of the solution set for the perturbed dual problem. Some examples are given for the illustration of the necessity of our research on duality.
2008, 4(1): 167-181
doi: 10.3934/jimo.2008.4.167
+[Abstract](1384)
+[PDF](201.0KB)
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In this paper, we study the $\varepsilon$-generalized strong vector equilibrium problem ($\varepsilon$-GSVEP) and $\varepsilon$-extended vector equilibrium problem ($\varepsilon$-ESVEP) which can be regarded as approximate problems to the generalized strong vector equilibrium problems (GSVEP). Existence results for $\varepsilon$-GSVEP and $\varepsilon$-ESVEP are established. We also investigate various continuities of the solution mappings of $\varepsilon$-GSVEP and $\varepsilon$-ESVEP, respectively.
In this paper, we study the $\varepsilon$-generalized strong vector equilibrium problem ($\varepsilon$-GSVEP) and $\varepsilon$-extended vector equilibrium problem ($\varepsilon$-ESVEP) which can be regarded as approximate problems to the generalized strong vector equilibrium problems (GSVEP). Existence results for $\varepsilon$-GSVEP and $\varepsilon$-ESVEP are established. We also investigate various continuities of the solution mappings of $\varepsilon$-GSVEP and $\varepsilon$-ESVEP, respectively.
2008, 4(1): 183-197
doi: 10.3934/jimo.2008.4.183
+[Abstract](1255)
+[PDF](222.1KB)
Abstract:
Road pricing is considered one of the effective means to reduce traffic congestion and environmental damage, and it has been introduced in major highways of most countries. The road pricing problem can be formulated as a mathematical program with equilibrium constraints (MPEC) and the resulting MPEC can be solved efficiently by the implicit programming approach if the user's route costs are additive. However, route costs are generally nonadditive in the real world. In this paper we consider road pricing on the traffic equilibrium problem with nonadditive route costs based on users' disutility functions. We then show that this formulation can be reformulated as a mathematical program with strictly monotone mixed complementarity problem (MCP). Since a strictly monotone MCP has a unique solution for each upper level variable, we can apply the implicit programming approach to solve the resulting reformulation. We establish the differentiability of the resulting implicit function. Numerical experiments using various disutility functions and sample networks are done, and the results show that the implicit programming approach is robust to find a solution of the road pricing problem.
Road pricing is considered one of the effective means to reduce traffic congestion and environmental damage, and it has been introduced in major highways of most countries. The road pricing problem can be formulated as a mathematical program with equilibrium constraints (MPEC) and the resulting MPEC can be solved efficiently by the implicit programming approach if the user's route costs are additive. However, route costs are generally nonadditive in the real world. In this paper we consider road pricing on the traffic equilibrium problem with nonadditive route costs based on users' disutility functions. We then show that this formulation can be reformulated as a mathematical program with strictly monotone mixed complementarity problem (MCP). Since a strictly monotone MCP has a unique solution for each upper level variable, we can apply the implicit programming approach to solve the resulting reformulation. We establish the differentiability of the resulting implicit function. Numerical experiments using various disutility functions and sample networks are done, and the results show that the implicit programming approach is robust to find a solution of the road pricing problem.
2008, 4(1): 199-211
doi: 10.3934/jimo.2008.4.199
+[Abstract](1407)
+[PDF](233.4KB)
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The open source paradigm is often defined as a ''collaborative effort,'' implying that firms and consumers come together in a non-competitive climate. We show here that open source development can arise from a competitive climate. Under competition, we find that open source is the surplus maximizing outcome and can be in equilibrium if cost asymmetries are small. However, when cost asymmetries are large, contradictions between equilibrium and welfare maximization result. Considerations typical to public good problems arise, with issues of asymmetric contributions and free-riding. These issues should guide the firm's as well as the society's decisions to implement open source in particular environments. We analyze this problem in the framework of a dynamic duopolistic competition, with firms controlling their investments in software.
The open source paradigm is often defined as a ''collaborative effort,'' implying that firms and consumers come together in a non-competitive climate. We show here that open source development can arise from a competitive climate. Under competition, we find that open source is the surplus maximizing outcome and can be in equilibrium if cost asymmetries are small. However, when cost asymmetries are large, contradictions between equilibrium and welfare maximization result. Considerations typical to public good problems arise, with issues of asymmetric contributions and free-riding. These issues should guide the firm's as well as the society's decisions to implement open source in particular environments. We analyze this problem in the framework of a dynamic duopolistic competition, with firms controlling their investments in software.
2018 Impact Factor: 1.025
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