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Journal of Industrial & Management Optimization

2012 , Volume 8 , Issue 3

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An approximation algorithm for the $k$-level facility location problem with submodular penalties
Gaidi Li , Zhen Wang and  Dachuan Xu
2012, 8(3): 521-529 doi: 10.3934/jimo.2012.8.521 +[Abstract](62) +[PDF](334.4KB)
In this paper, we consider the $k$-level facility location problem with submodular penalties ($k$-FLPSP). We propose a primal-dual $6$-approximation (combinatorial) algorithm for the $k$-FLPSP.
Optimal investment with a value-at-risk constraint
Jingzhen Liu , Lihua Bai and  Ka-Fai Cedric Yiu
2012, 8(3): 531-547 doi: 10.3934/jimo.2012.8.531 +[Abstract](40) +[PDF](432.8KB)
We consider constrained investment problem with the objective of minimizing the ruin probability. In this paper, we formulate the cash reserve and investment model for the insurance company and analyze the value-at-risk ($VaR$) in a short time horizon. For risk regulation, we impose it as a risk constraint dynamically. Then the problem becomes minimizing the probability of ruin together with the imposed risk constraint. By solving the corresponding Hamilton-Jacobi-Bellman equations, we derive analytic expressions for the optimal value function and the corresponding optimal strategies. Looking at the value-at-risk alone, we show that it is possible to reduce the overall risk by an increased exposure to the risky asset. This is aligned with the risk diversification effect for negative correlated or uncorrelated risky asset with the stochastic of the fundamental insurance business. Moreover, studying the optimal strategies, we find that a different investment strategy will be in place depending on the Sharpe ratio of the risky asset.
Portfolio optimization and risk measurement based on non-dominated sorting genetic algorithm
Ping-Chen Lin
2012, 8(3): 549-564 doi: 10.3934/jimo.2012.8.549 +[Abstract](31) +[PDF](627.8KB)
This paper introduces a multi-objective genetic algorithm (MOGA) in regard to the portfolio optimization issue in different risk measures, such as mean-variance, semi-variance, mean-variance-skewness, mean-absolute-deviation and lower-partial-moment to optimize risk of portfolio. This study introduces a PONSGA model by appling the non-dominated sorting genetic algorithm (NSGA-II) to maximize both the expected return and the skewness of portfolio as well as to simultaneously minimize different portfolio risks. The experimental results demonstrated that the PONSGA approach is significantly superior to the GA in all performances, examined such as the coefficient of variation, Sharpe index, Sortino index and portfolio performance index. The statistical significance tests also showed that the NSGA-II models outperformed the GA models in different risk measures.
A Max-Min clustering method for $k$-means algorithm of data clustering
Baolan Yuan , Wanjun Zhang and  Yubo Yuan
2012, 8(3): 565-575 doi: 10.3934/jimo.2012.8.565 +[Abstract](99) +[PDF](168.0KB)
As it is known that the performance of the $k$-means algorithm for data clustering largely depends on the choice of the Max-Min centers, and the algorithm generally uses random procedures to get them. In order to improve the efficiency of the $k$-means algorithm, a good selection method of clustering starting centers is proposed in this paper. The proposed algorithm determines a Max-Min scale for each cluster of patterns, and calculate Max-Min clustering centers according to the norm of the points. Experiments results show that the proposed algorithm provides good performance of clustering.
Single-period inventory model with discrete stochastic demand based on prospect theory
Wei Liu , Shiji Song and  Cheng Wu
2012, 8(3): 577-590 doi: 10.3934/jimo.2012.8.577 +[Abstract](28) +[PDF](400.2KB)
This paper studies a single-period inventory (newsvendor) problem with discrete stochastic demand. In general, most of the previous works are based on the expected profit/cost criterion or expected utility criterion. We consider the effect of irrational factor under uncertainty and therefore incorporate prospect theory into inventory model. Our objective is to maximize the overall value of the prospect, which can be calculated by using the value function and the weighting function. For any given initial inventory level, it can be shown that a state-dependent order-up-to policy is optimal. Further, the optimal policy has a simple structure, and the retailer can easily decide whether to place an order or not. Moreover, the impacts of parameters on the optimal policy are illustrated through numerical experiments.
A neighboring extremal solution for an optimal switched impulsive control problem
Canghua Jiang , Kok Lay Teo , Ryan Loxton and  Guang-Ren Duan
2012, 8(3): 591-609 doi: 10.3934/jimo.2012.8.591 +[Abstract](33) +[PDF](424.3KB)
This paper presents a neighboring extremal solution for a class of optimal switched impulsive control problems with perturbations in the initial state, terminal condition and system's parameters. The sequence of mode's switching is pre-specified, and the decision variables, i.e. the switching times and parameters of the system involved, have inequality constraints. It is assumed that the active status of these constraints is unchanged with the perturbations. We derive this solution by expanding the necessary conditions for optimality to first-order and then solving the resulting multiple-point boundary-value problem by the backward sweep technique. Numerical simulations are presented to illustrate this solution method.
Canonical duality solution for alternating support vector machine
Yubo Yuan
2012, 8(3): 611-621 doi: 10.3934/jimo.2012.8.611 +[Abstract](35) +[PDF](210.0KB)
Support vector machine (SVM) is one of the most popular machine learning methods and is educed from a binary data classification problem. In this paper, the canonical duality theory is used to solve the normal model of SVM. Several examples are illustrated to show that the exact solution can be obtained after the canonical duality problem being solved. Moreover, the support vectors can be located by non-zero elements of the canonical dual solution.
A common set of weight approach using an ideal decision making unit in data envelopment analysis
Saber Saati , Adel Hatami-Marbini , Per J. Agrell and  Madjid Tavana
2012, 8(3): 623-637 doi: 10.3934/jimo.2012.8.623 +[Abstract](57) +[PDF](321.2KB)
Data envelopment analysis (DEA) is a common non-parametric frontier analysis method. The multiplier framework of DEA allows flexibility in the selection of endogenous input and output weights of decision making units (DMUs) as to cautiously measure their efficiency. The calculation of DEA scores requires the solution of one linear program per DMU and generates an individual set of endogenous weights (multipliers) for each performance dimension. Given the large number of DMUs in real applications, the computational and conceptual complexities are considerable with weights that are potentially zero-valued or incommensurable across units. In this paper, we propose a two-phase algorithm to address these two problems. In the first step, we define an ideal DMU (IDMU) which is a hypothetical DMU consuming the least inputs to secure the most outputs. In the second step, we use the IDMU in a LP model with a small number of constraints to determine a common set of weights (CSW). In the final step of the process, we calculate the efficiency of the DMUs with the obtained CSW. The proposed model is applied to a numerical example and to a case study using panel data from 286 Danish district heating plants to illustrate the applicability of the proposed method.
An extended lifetime measure for telecommunications networks: Improvements and implementations
Zari Dzalilov , Iradj Ouveysi and  Tolga Bektaş
2012, 8(3): 639-649 doi: 10.3934/jimo.2012.8.639 +[Abstract](47) +[PDF](203.3KB)
Predicting the lifetime of a network is a stochastic and very hard task. Sensitivity analysis of a network in order to identify the weakest points in the network, provides valuable knowledge to draw an optimum investment strategy for the expansion of the networks for the network carriers. To achieve this goal, a new measure, called topology lifetime, was recently proposed for measuring the performance of a telecommunication network. This measure not only allows to perform a sensitivity analysis of the networks, but also it provides the means to compare the different topologies with respect to the ability of the network in supporting growth in network traffic before new capacity/facility is installed. This paper addresses some improvements upon the previously defined measures and presents the implementation results of the various lifetime measure methodologies. Computational analysis on some commonly used topologies show how the new measure can be utilized in assessing network performance.
Worst-case performance of the successive approximation algorithm for four identical knapsacks
Zhenbo Wang
2012, 8(3): 651-656 doi: 10.3934/jimo.2012.8.651 +[Abstract](24) +[PDF](288.6KB)
This paper studies the worst-case performance of the successive approximation algorithm for four identical knapsacks. The algorithm packs the knapsacks successively by using an exact algorithm on the remaining items for each single knapsack. We show that it is an $\frac{8}{11}$-approximation algorithm, and the bound is tight.
Integrated inventory model with stochastic lead time and controllable variability for milk runs
Xin Zhou , Liangping Shi and  Bingzhi Huang
2012, 8(3): 657-672 doi: 10.3934/jimo.2012.8.657 +[Abstract](47) +[PDF](317.5KB)
The current study deals with the lead-time variability reduction problem in a multi-supplier and single-buyer system with a milk-run delivery network. Under the assumption of finite-range stochastic lead time, we consider that the lead-time variance can be shortened at an extra crashing cost. Aiming to minimize the joint system costs, an integrated inventory model is formulated with a capacity constraint, and a solution procedure is developed to simultaneously optimize lead-time variance, replenishment cycle time, reorder point, and the integer numbers of shipment per production cycle for multiple suppliers. We also show that the buyer does a tradeoff between the crashing cost and the sum of holding and shortage costs for the decision making of the optimal lead-time variability. Numerical examples and sensitivity analysis are presented to validate the proposed model.
Optimal reinsurance-investment strategies for insurers under mean-CaR criteria
Yan Zeng and  Zhongfei Li
2012, 8(3): 673-690 doi: 10.3934/jimo.2012.8.673 +[Abstract](47) +[PDF](277.1KB)
This paper considers an optimal reinsurance-investment problem for an insurer, who aims to minimize the risk measured by Capital-at-Risk (CaR) with the constraint that the expected terminal wealth is not less than a predefined level. The surplus of the insurer is described by a Brownian motion with drift. The insurer can control her/his risk by purchasing proportional reinsurance, acquiring new business, and investing her/his surplus in a financial market consisting of one risk-free asset and multiple risky assets. Three mean-CaR models are constructed. By transforming these models into bilevel optimization problems, we derive the explicit expressions of the optimal deterministic rebalance reinsurance-investment strategies and the mean-CaR efficient frontiers. Sensitivity analysis of the results and a numerical example are provided.
Upper Hölder estimates of solutions to parametric primal and dual vector quasi-equilibria
Chunrong Chen and  Shengji Li
2012, 8(3): 691-703 doi: 10.3934/jimo.2012.8.691 +[Abstract](31) +[PDF](372.2KB)
In this paper, on one hand, we discuss upper Hölder type estimates of solutions to parametric vector quasi-equilibria with general settings, which generalize and extend the results of Chen et al. (Optim. Lett. 5: 85-98, 2011). On the other hand, combining the technique used for primal problems with suitable modifications, we also study upper Hölder type estimates of solutions to Minty-type parametric dual vector quasi-equilibria. The consequences obtained for dual problems are new in the literature.
On an exact penalty function method for semi-infinite programming problems
Cheng Ma , Xun Li , Ka-Fai Cedric Yiu , Yongjian Yang and  Liansheng Zhang
2012, 8(3): 705-726 doi: 10.3934/jimo.2012.8.705 +[Abstract](37) +[PDF](873.7KB)
In this paper, we study a new exact and smooth penalty function for semi-infinite programming problems with continuous inequality constraints. Through this exact penalty function, we can transform a semi-infinite programming problem into an unconstrained optimization problem. We find that, under some reasonable conditions when the penalty parameter is sufficiently large, the local minimizer of this penalty function is the local minimizer of the primal problem. Moreover, under some mild assumptions, the local exactness property is explored. The numerical results demonstrate that it is an effective and promising approach for solving constrained semi-infinite programming problems.
Note on "Cost analysis of the M/M/R machine repair problem with second optional repair: Newton-Quasi method"
Cheng-Dar Liou
2012, 8(3): 727-732 doi: 10.3934/jimo.2012.8.727 +[Abstract](23) +[PDF](289.8KB)
The machine repair problem has attracted considerable attention in the field of queuing systems, due to the wide range of difficulties it entails. Wang, Liao and Yen [K.H. Wang, C.W. Liao, T.C. Yen, Cost analysis of the M/M/R machine repair problem with second optional repair: Newton-Quasi method, Journal of Industrial and Management Optimization. 6 (2010), 197-207] [1] derived a cost model to determine the optimal number of the repairmen, the optimal values of the first essential repair rate, and the second optional repair rate while maintaining the system availability at a specified level. In their approach, a direct search method is first used to determine the optimal number of repairmen followed by the Newton-Quasi method to search for the two repair rates. However, this two stage search method restricts the search space and cannot guarantee global minimum solutions. In overcoming these limitations, this study employs a particle swarm optimization algorithm to ensure a thorough search of the solution space in the pursuit of global minimum solutions. Numerical results support the superior search characteristics of the proposed solution.
A sequential convex program method to DC program with joint chance constraints
Xiantao Xiao , Jian Gu , Liwei Zhang and  Shaowu Zhang
2012, 8(3): 733-747 doi: 10.3934/jimo.2012.8.733 +[Abstract](47) +[PDF](448.6KB)
In this paper, we consider a DC (difference of convex) programming problem with joint chance constraints (JCCDCP). We propose a DC function to approximate the constrained function and a corresponding DC program ($\textrm{P}_{\varepsilon}$) to approximate the JCCDCP. Under some mild assumptions, we show that the solution of Problem ($\textrm{P}_{\varepsilon}$) converges to the solution of JCCDCP when $\varepsilon\downarrow 0$. A sequential convex program method is constructed to solve the Problem ($\textrm{P}_{\varepsilon}$). At each iteration a convex program is solved based on the Monte Carlo method, and the generated optimal sequence is proved to converge to the stationary point of Problem ($\textrm{P}_{\varepsilon}$).
Lagrange multiplier rules for approximate solutions in vector optimization
Caiping Liu and  Heungwing Lee
2012, 8(3): 749-764 doi: 10.3934/jimo.2012.8.749 +[Abstract](47) +[PDF](364.0KB)
In Asplund space, Lagrange multiplier rules for approximate solutions of nonsmooth vector optimization problems are studied. The relationships between the vector and the scalar optimization problems are established. And the optimality conditions of approximate solutions for vector optimization are obtained. Moreover, the vector variational inequalities are considered by applying the partial results given in this paper.
Identification for systems governed by nonlinear interval differential equations
Tayel Dabbous
2012, 8(3): 765-780 doi: 10.3934/jimo.2012.8.765 +[Abstract](28) +[PDF](307.8KB)
In this paper we consider the identification problem for a class of systems governed by nonlinear time varying interval differential equations having unknown (interval) parameters. Using the fact that system output posses lower and upper bounds, we have developed two sets of ordinary differential equations that represent the behaviour of lower and upper bounds. Based on these differential equations, the interval identification problem is converted into an equivalent identification problem in which the unknown parameters are real valued functions. Using variational arguments, we have developed the necessary conditions of optimality for the equivalent problem on the basis of which the unknown lower and upper parameters (and hence the interval parameters) can be determined. Finally, we present some numerical simulations to illustrate the effectivness of the proposed technique.

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