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Journal of Industrial & Management Optimization

2009 , Volume 5 , Issue 4

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Levitin-Polyak well-posedness of generalized vector quasi-equilibrium problems
M. H. Li , S. J. Li and  W. Y. Zhang
2009, 5(4): 683-696 doi: 10.3934/jimo.2009.5.683 +[Abstract](40) +[PDF](191.1KB)
In this paper, Levitin-Polyak well-posedness for two classes of generalized vector quasi-equilibrium problems is introduced. Criteria and characterizations of the Levitin-Polyak well-posedness are investigated. By virtue of gap functions for the generalized vector quasi-equilibrium problems, some equivalent relations are obtained between the Levitin-Polyak well-posedness for optimization problems and the Levitin-Polyak well-posedness for generalized vector quasi-equilibrium problems. Finally, a set-valued version of Ekeland's variational principle is derived and applied to establish a sufficient condition for Levitin-Polyak well-posedness of a class of generalized vector quasi-equilibrium problems.
On second order symmetric duality in nondifferentiable multiobjective programming
Xinmin Yang
2009, 5(4): 697-703 doi: 10.3934/jimo.2009.5.697 +[Abstract](40) +[PDF](109.7KB)
In this paper, we point out an inconsistency between assumptions and results on the second order strong and converse duality in a recent paper of I. Ahmad ( Information Sciences 173 (2005) 23-34). We then provide appropriate modifications to rectify this deficiency.
Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications
Ling Yun Wang , Wei Hua Gui , Kok Lay Teo , Ryan Loxton and  Chun Hua Yang
2009, 5(4): 705-718 doi: 10.3934/jimo.2009.5.705 +[Abstract](47) +[PDF](191.2KB)
In this paper, we consider a class of optimal control problems involving time delayed dynamical systems and subject to continuous state inequality constraints. We show that this type of problem can be approximated by a sequence of time delayed optimal control problems subject to inequality constraints in canonical form and with multiple characteristic time points appearing in the cost and constraint functions. We derive formulae for the gradient of the cost and constraint functions of the approximate problems. On this basis, each approximate problem can be solved using a gradient-based optimization technique. The computational method obtained is then applied to an industrial problem arising in the study of purification process of zinc sulphate electrolyte. The results are highly satisfactory.
Random time ruin probability for the renewal risk model with heavy-tailed claims
Yuebao Wang , Qingwu Gao , Kaiyong Wang and  Xijun Liu
2009, 5(4): 719-736 doi: 10.3934/jimo.2009.5.719 +[Abstract](40) +[PDF](241.7KB)
In this paper, we investigate the asymptotic behavior of the random time ruin probability for the renewal risk model with heavy-tailed claim sizes. Under the assumption that the claim sizes are independent and long-tailed, we give the equivalent conditions on asymptotic behavior for the random time ruin probability, where the independent or dependent structure among the inter-arrival times is not needed. While, under the assumption that the claim sizes are of some negative dependence structure and consistently varying tails, we obtain the sufficient condition of asymptotic behavior for the random time ruin probability which will require some negative dependence structure among the inter-arrival times.
Optimal control of piecewise affine systems with piecewise affine state feedback
Changzhi Wu , Kok Lay Teo and  Volker Rehbock
2009, 5(4): 737-747 doi: 10.3934/jimo.2009.5.737 +[Abstract](54) +[PDF](181.9KB)
In this paper, we consider a class of optimal control problems involving piecewise affine (PWA) systems with piecewise affine state feedback. We first show that if the piecewise affine state feedback control is assumed to be continuous at the switching boundaries, then the number of switching amongst PWA systems is finite. On this basis, this optimal control problem is transformed into a discrete valued optimal control problem. For this discrete valued optimal control problem, we introduce the time scaling transform to convert it into an equivalent constrained optimal parameter selection problem, for which it can be solved by existing optimal control techniques for optimal parameter selection problems. A numerical example is solved so as to illustrate the proposed method.
The hot-rolling batch scheduling method based on the prize collecting vehicle routing problem
Tao Zhang , W. Art Chaovalitwongse , Yue-Jie Zhang and  P. M. Pardalos
2009, 5(4): 749-765 doi: 10.3934/jimo.2009.5.749 +[Abstract](42) +[PDF](289.2KB)
This paper studies a hot-rolling batch scheduling problem, which is a challenging problem commonly arising in the iron-steel industry. This problem deals with forming steel strips into rolling units and determining the rolling sequences while minimizing changes in characteristics (e.g., width, thickness and rigidity) of all neighbor steel strips and maximizing the machine utilization. Based on technical rules used in iron-steel production practice, we formulate this problem as a prize collecting vehicle routing problem, which is a hard combinatorial multi-objective optimization problem. We develop a new heuristic approach to solve this problem by enhancing the framework of particle swarm optimization (PSO). The key features of our approach are the utilization of number mapping function, which allows the PSO algorithm to deal with discrete problems, and the employment of tabu search at the end of every PSO iteration. We investigate and measure the performance of the proposed approach using three real life datasets obtained from a well-known iron-steel production company in China. The results suggest that our approach is very efficient and effective in providing high-quality and practical solutions, and our approach outperforms traditional PSO and tabu search algorithms based on these datasets.
Two-echelon requirements planning with pricing decisions
Mehmet Önal and  H. Edwin Romeijn
2009, 5(4): 767-781 doi: 10.3934/jimo.2009.5.767 +[Abstract](52) +[PDF](231.3KB)
We consider a two-level uncapacitated lot-sizing problem where production, inventory carrying, transportation, and pricing decisions are integrated to maximize total profits. We show how this problem, under many different revenue functions and production, inventory holding, and transportation cost structures can be solved in polynomial time. As a byproduct, we develop polynomial-time algorithms for generalizations of single-level lot-sizing problems with pricing as well.
Optimality conditions for vector equilibrium problems with constraints
Qiu-Sheng Qiu
2009, 5(4): 783-790 doi: 10.3934/jimo.2009.5.783 +[Abstract](41) +[PDF](129.8KB)
In this paper, the necessary and sufficient conditions for weakly efficient solution of the vector equilibrium problems with constraints are obtained. The Kuhn-Tucker condition for weakly efficient solution of the vector equilibrium problems is derived. As interesting applications of the results in the paper, we obtain the optimality conditions for vector optimization problems with constraints.
Robust multiobjective dynamic programming: Minimax envelopes for efficient decisionmaking under scenario uncertainty
Matthew H. Henry and  Yacov Y. Haimes
2009, 5(4): 791-824 doi: 10.3934/jimo.2009.5.791 +[Abstract](29) +[PDF](377.2KB)
Many sequential decision problems are characterized by multiple objectives and can be formulated as multiobjective dynamic programs. A subset of these problems concerns systems that are only partially observable such that the system response to implemented policies is known to belong to a set of possible system responses but is not uniquely known prior to policy selection. A new methodology is developed to identify optimal strategies in finite-horizon multiobjective decision problems for systems of this type. These strategies will either be minimax efficient with respect to a partial ordering in the multiobjective space or, where minimax efficient strategies do not exist, minimax optimal with respect to a total ordering in a scalar space induced by decisionmaker preferences over the set of objectives. In formulating the dynamic program, system uncertainty is described by a finite set of scenario-specified system models, and the likelihood of any particular scenario is assumed to be unknown. By accounting for different scenarios, the multiobjective dynamic program and the resulting strategies are robust with respect to uncertainty in the underlying policy-response relationships. The solution concept is developed with assurances that the principle of optimality holds. An illustrative example demonstrates the methodology.
The modified cutting angle method for global minimization of increasing positively homogeneous functions over the unit simplex
Burak Ordin
2009, 5(4): 825-834 doi: 10.3934/jimo.2009.5.825 +[Abstract](27) +[PDF](146.2KB)
The paper deals with a method for global minimization of increasing positively homogeneous functions over the unit simplex, which is a version of the cutting angle method. A new approach for solving the auxiliary problem in the cutting angle method is proposed. In the method, the auxiliary problem is reformulated as a certain combinatorial problem. The modified version of the cutting angle method is also applied for Lipschitz functions that could be expressed as increasing positively homogeneous functions. We report results of numerical experiments which demonstrate that the proposed algorithm is very efficient in the search for a global minimum.
Modelling and optimal control for nonlinear multistage dynamical system of microbial fed-batch culture
Chongyang Liu , Zhaohua Gong , Enmin Feng and  Hongchao Yin
2009, 5(4): 835-850 doi: 10.3934/jimo.2009.5.835 +[Abstract](65) +[PDF](216.5KB)
In this paper, we propose a new controlled multistage system to formulate the fed-batch culture process of glycerol bio-dissimilation to 1,3-propanediol (1,3-PD) by regarding the feeding rate of glycerol as a control function. Compared with the previous systems, this system doesn't take the feeding process as an impulsive form, but a time-continuous process, which is much closer to the actual culture process. Some properties of the above dynamical system are then proved. To maximize the concentration of 1,3-PD at the terminal time, we develop an optimal control model subject to our proposed controlled multistage system and continuous state inequality constraints. The existence of optimal control is proved by bounded variation theory. Through the discretization of the control space, the control function is approximated by piecewise constant functions. In this way, the optimal control model is approximated by a sequence of parameter optimization problems. The convergence analysis of this approximation is also investigated. Furthermore, a global optimization algorithm is constructed on the basis of the above descretization concept and an improved Particle Swarm Optimization (PSO) algorithm. Numerical results show that, by employing the optimal control policy, the concentration of 1,3-PD at the terminal time can be increased considerably.
First-order optimality conditions for convex semi-infinite min-max programming with noncompact sets
Jinchuan Zhou , Changyu Wang , Naihua Xiu and  Soonyi Wu
2009, 5(4): 851-866 doi: 10.3934/jimo.2009.5.851 +[Abstract](57) +[PDF](214.6KB)
In this paper, based on some new characterizations of subderivative and subdifferential of sup-types functions, we develop the first-order necessary and sufficient optimality conditions for convex semi-infinite min-max programming problems in which the index sets are not necessarily compact.
A replenishment policy with defective products, backlog and delay of payments
Jun Li , Hairong Feng and  Mingchao Wang
2009, 5(4): 867-880 doi: 10.3934/jimo.2009.5.867 +[Abstract](39) +[PDF](212.3KB)
This paper develops a model to determine an optimal replenishment policy with defective items and shortage backlogging under conditions of permissible delay of payments. It is assumed that 100% of each lot are screened to separate good and defective items which are classified as imperfect quality and scrap items. Difference between unit selling price and unit purchase cost is also included in our mathematical model and analysis. Under this assumption, we model the retailer's inventory system as an expected profit maximization problem to determine the retailer's optimal inventory cycle time and optimal order quantity. Then, a theorem is established to describe the optimal replenishment policy for the retailer. Finally, numerical examples are given to illustrate the theorem and obtain some managerial phenomena.
New sufficient global optimality conditions for linearly constrained bivalent quadratic optimization problems
Yong Xia
2009, 5(4): 881-892 doi: 10.3934/jimo.2009.5.881 +[Abstract](34) +[PDF](189.0KB)
In this article, we obtain new sufficient global optimality conditions for bivalent quadratic optimization problems with linearly (equivalent and inequivalent) constraints, by exploring the local optimality condition. The global optimality condition can be further simplified when applied to special cases such as the $p$-dispersion-sum problem and the quadratic assignment problem.
The credibility premiums under generalized weighted loss functions
Limin Wen , Xianyi Wu and  Xiaobing Zhao
2009, 5(4): 893-910 doi: 10.3934/jimo.2009.5.893 +[Abstract](60) +[PDF](224.7KB)
In the classical credibility theory, almost all the credibility premium models are built on the basis of pure premium. However, the insurance practice demands that the premium must have a positive safety loading. In this paper, we consider the premium principle induced by a generalized loss function that can provide the premium principle with\ a positive safety loading. Under this generalized loss function, we derive its Bayes premium and two types of credibility premiums. Both credibility premiums are approximately convex combinations of the collective premium and some functions of the historical claims; while in a first case the function is linear in the historical claims and the corresponding credibility premium is not consistency, in the other one the function is taken as an empirical version of the individual premium and the corresponding credibility premium converges to the individual premium.
A filled function method for solving nonlinear complementarity problem
Liuyang Yuan , Zhongping Wan , Jingjing Zhang and  Bin Sun
2009, 5(4): 911-928 doi: 10.3934/jimo.2009.5.911 +[Abstract](46) +[PDF](207.0KB)
In this paper a filled function method is suggested for solving the nonlinear complementarity problem. Firstly, the original problem is converted into a corresponding unconstrained optimization problem by using the Fischer-Burmeister function. Subsequently, a new filled function with one parameter is proposed for solving unconstrained optimization problems. Some properties of the filled function are studied and discussed without Lipschitz continuity condition. Finally, an algorithm based on the proposed filled function for solving the nonlinear complementarity problem is presented. The implementation of the algorithm on several test problems is reported with numerical results.
A network simplex algorithm for solving the minimum distribution cost problem
I-Lin Wang and  Shiou-Jie Lin
2009, 5(4): 929-950 doi: 10.3934/jimo.2009.5.929 +[Abstract](48) +[PDF](490.1KB)
To model the distillation or decomposition of products in some manufacturing processes, a minimum distribution cost problem (MDCP) for a specialized manufacturing network flow model has been investigated. In an MDCP, a specialized node called a D-node is used to model a distillation process that connects with a single incoming arc and several outgoing arcs. The flow entering a D-node has to be distributed according to a pre-specified ratio associated with each of its outgoing arcs. This proportional relationship between arc flows associated with each D-node complicates the problem and makes the MDCP more difficult to solve than a conventional minimum cost network flow problem. A network simplex algorithm for an uncapacitated MDCP has been outlined in the literature. However, its detailed graphical procedures including the operations to obtain an initial basic feasible solution, to calculate or update the dual variables, and to pivot flows have never been reported. In this paper, we resolve these issues and propose a modified network simplex algorithm including detailed graphical operations in each elementary procedure. Our method not only deals with a capacitated MDCP, but also offers more theoretical insights into the mathematical properties of an MDCP.

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