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Journal of Industrial & Management Optimization

2010 , Volume 6 , Issue 1

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Competitive risk management for online Bahncard problem
Lili Ding, Xinmin Liu and Yinfeng Xu
2010, 6(1): 1-14 doi: 10.3934/jimo.2010.6.1 +[Abstract](250) +[PDF](213.9KB)
Abstract:
In the Bahncard problem a traveler decides when to buy a Bahncard, i.e., a railway discount card of the German Deutsche Bundesbahn company, in an online setting. This problem is introduced by Fleischer and some optimal deterministic algorithms are presented with a fixed Bahncard price. In practice, however, travelers are trying to manage their risks by using some forms of rewards and their forecasting skills. We extend Fleischer's model to a new one in a risk management framework. For such an extended problem, we provide some flexible results which can be used by a traveler to obtain an optimal risk algorithm based on his risk tolerance and forecast. We further study another extention of the Bahncard problem with a fluctuated Bahncard price. We propose some algorithms and analyze their competitive ratios with and without risk, respectively. It turns out that a traveler can significantly improve his risk management performance by putting reasonable forecasts in conventional competitive analysis.
A numerical approach to infinite-dimensional linear programming in $L_1$ spaces
Satoshi Ito, Soon-Yi Wu, Ting-Jang Shiu and Kok Lay Teo
2010, 6(1): 15-28 doi: 10.3934/jimo.2010.6.15 +[Abstract](288) +[PDF](187.8KB)
Abstract:
An infinite-dimensional linear programming formulated on $L_1$ spaces, problem (P), is studied in this paper. A related optimization problem, general capacity problem (GCAP), is also mentioned in this paper. But we find that the optimal solution does not exist in problem (P). Thus, we approach the optimal value for problem (P) via solving the problem (GCAP). A proposed algorithm is shown that we solve a sequence of semi-infinite subproblems to approach the optimal value of problem (P). The error bound for the difference between the optimal value for problem (P) and optimal value for semi-infinite subproblem is also given in this paper. Finally, numerical examples are implemented and compared with discretization method to show our computational efficiency.
A recursive topographical differential evolution algorithm for potential energy minimization
M. Montaz Ali
2010, 6(1): 29-46 doi: 10.3934/jimo.2010.6.29 +[Abstract](234) +[PDF](264.8KB)
Abstract:
The problem of the determination of the minimum energy configuration of an arrangement of $N$ point particles under the interaction of their interatomic forces is discussed. The interatomic force is described by a classical many body potential, namely the Tersoff potential for silicon. We propose a global optimization algorithm for minimization of energy of clusters of particles using Tersoff potential. The algorithm combines the topographical differential evolution (TDE) with the modified recursive procedure of the recursive differential evolution (RDE) algorithm. It also introduces an initialization procedure for the population set. Two important features of the new algorithm are that it makes use of the \lq graph minima' for local search, and that the initial population set is generated with low function values. The global minima of clusters consisting of up to 20 particles are investigated. The new algorithm is compared with a recent genetic algorithm.
Necessary optimality conditions for switching control problems
Shahlar F. Maharramov
2010, 6(1): 47-55 doi: 10.3934/jimo.2010.6.47 +[Abstract](278) +[PDF](145.5KB)
Abstract:
In this paper we consider smooth and nonsmooth switching control problems. By applying Taylor incremental formula for the cost functional, first we obtain the necessary optimality condition for the smooth case. Then by applying the Freschet superdifferential and uniform upper subdifferentiability, we extend necessary condition to the nonsmooth switching control problems.
Analysis on a queue system with heterogeneous servers and uncertain patterns
Jau-Chuan Ke, Hsin-I Huang and Chuen-Horng Lin
2010, 6(1): 57-71 doi: 10.3934/jimo.2010.6.57 +[Abstract](304) +[PDF](204.3KB)
Abstract:
This work constructs the membership functions of the system characteristics of a heterogeneous-server queueing model with fuzzy customer arrival and service rates. The $\alpha$-cut approach is used to transform a fuzzy queue into a family of conventional crisp queues in this context. By means of the membership functions of the system characteristics, a set of parametric nonlinear programs is developed to describe the family of crisp heterogeneous-server queues. A numerical example is solved successfully to illustrate the validity of the proposed approach. By extending this model to the fuzzy environment, the system characteristics are expressed and governed by the membership functions, and more information is provided for use by designers and practitioners.
Optimal service capacity in a multiple-server queueing system: A game theory approach
Wai-Ki Ching, Sin-Man Choi and Min Huang
2010, 6(1): 73-102 doi: 10.3934/jimo.2010.6.73 +[Abstract](318) +[PDF](266.7KB)
Abstract:
The economic behavior of service providers in a competitive environment is a very important and interesting research topic. A two-server service network has been proposed by Kalai et al. [14] for this purpose. Their model actually aims at studying both the role and impact of service capacity in capturing larger market share. The market share is important in maximizing individual's long-run expected profit. A Markovian queueing system of two servers was employed in their model for the captured problem. The obvious advantage of such a model is that it is mathematically tractable. They then further formulated the problem as a two-person strategic game and analyzed the equilibrium solutions. The main aim of this paper is to extend the results of their two-server queueing model to the case of a general multiple-server queueing model. Here we will focus on the case when the queueing system is stable. It is found that when the marginal cost of service capacity is low relatively to the revenue per customer, a unique Nash equilibrium exists, in which all servers choose the same service capacity and the expected waiting times are finite.
Congestion control with pricing in the absence of demand and cost functions: An improved trial and error method
Gang Qian, Deren Han and Hongjin He
2010, 6(1): 103-121 doi: 10.3934/jimo.2010.6.103 +[Abstract](208) +[PDF](225.8KB)
Abstract:
Without the information of the origin-destination demand function and users' valuation for travel time saving, the precise estimation of the road tolls for various pricing schemes must go in a trial-and-error manner, as suggested by [2] and [15], and recently realized by [6, 7, 11, 22, 24]. For a trial of the tolls pattern, the responses of the users can be observed and used to update the toll pattern for the next trial. Since getting the responses of the users is expensive, it is desirable to use the acquired information exhaustively; That is, we need to make the method converge to an approximate solution of the problem within as little number of changes as possible.
   In this paper, we propose to update the link tolls pattern in an improved manner, where the profit direction is the combination of two known directions. This combined manner makes the method more efficient than the method using solely one of them. We prove the global convergence of the method under suitable conditions as those in [6, 7, 24]. Some preliminary computational results are also reported.
Joint pricing and production planning for fixed priced multiple products with backorders
Lou Caccetta and Elham Mardaneh
2010, 6(1): 123-147 doi: 10.3934/jimo.2010.6.123 +[Abstract](271) +[PDF](362.8KB)
Abstract:
In this paper, we present a multiperiod model for production planning in a make-to-stock manufacturing system with constant pricing. We consider a multiproduct capacitated setting and introduce a demand-based model where the demand is a function of the price. There is an assumption that the production setup costs are negligible. A key part of the model is that backorders are allowed. As a result of this, the problem becomes a non linear programming problem with the nonlinearities in both the objective function and some constraints. We develop an algorithm that computes the global optimal production and pricing policy on a finite time horizon. We illustrate the application of the algorithm through a detailed numerical example.
A derivative-free method for solving large-scale nonlinear systems of equations
Gaohang Yu
2010, 6(1): 149-160 doi: 10.3934/jimo.2010.6.149 +[Abstract](275) +[PDF](188.6KB)
Abstract:
In this paper, a fully derivative-free method for solving large-scale nonlinear systems of equations is presented. It uses in a systematic way the well-known Polak-Ribière-Polyak (PRP) conjugate gradient direction as a search direction and employs a backtracking process to obtain a suitable stepsize. Assume that the nonlinear mapping is Lipschitz continuous, some global convergence results are established. A modification of this method which may allow the objective function's sufficiently nonmonotone behavior is also presented in this paper. Numerical comparisons using a set of large-scale test problems in the CUTE library show that the proposed methods are encouraging.
The viscosity approximation to the Hamilton-Jacobi-Bellman equation in optimal feedback control: Upper bounds for extended domains
Steven Richardson and Song Wang
2010, 6(1): 161-175 doi: 10.3934/jimo.2010.6.161 +[Abstract](340) +[PDF](256.3KB)
Abstract:
A number of numerical methods for solving optimal feedback control problems are based on the viscosity approximation to the Hamilton-Jacobi-Bellman (HJB) equation, with artificial boundary conditions defined on an extended domain. An upper bound for this extended domain is established, ensuring that the approximate solution converges to the viscosity solution of the HJB equation on some pre-defined domain of interest.
Global optimization algorithm for solving bilevel programming problems with quadratic lower levels
Paul B. Hermanns and Nguyen Van Thoai
2010, 6(1): 177-196 doi: 10.3934/jimo.2010.6.177 +[Abstract](276) +[PDF](220.1KB)
Abstract:
In this article, we propose a method for finding the global optimum of a class of nonlinear bilevel programming problems. The main idea of this method is to construct iteratively a sequence of points either ending at an optimal solution of the equivalent problem with a complementarity constraint, or converging to an optimal solution. The construction of such a sequence is performed by using a branch-and-bound scheme, together with some relaxation techniques, which are successfully applied in global optimization. Some illustrative examples and results on computational experiments are reported.
Cost analysis of the M/M/R machine repair problem with second optional repair: Newton-Quasi method
Kuo-Hsiung Wang, Chuen-Wen Liao and Tseng-Chang Yen
2010, 6(1): 197-207 doi: 10.3934/jimo.2010.6.197 +[Abstract](251) +[PDF](144.7KB)
Abstract:
This paper investigates the M/M/R machine repair problem with second optional repair. Failure times of the operating machines are assumed to be exponentially distributed with parameter $\lambda $. Repair times of the first essential repair and the second optional repair are assumed to follow exponential distributions. A failed machine may leave the system either after the first essential repair with probability $(1-\theta)$, or select to repair for second optional repair with probability $\theta$ $(0 \le \theta \le 1)$ at the completion of the first essential repair. We obtain the steady-state solutions through matrix-analytic method. A cost model is derived 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. We use the direct search method to deal with the number of repairmen problem and the Newton-Quasi method for the repair rate problem to minimize the system operating cost while all the constraints are satisfied.
A shadow-price based heuristic for capacity planning of TFT-LCD manufacturing
Tzu-Li Chen, James T. Lin and Shu-Cherng Fang
2010, 6(1): 209-239 doi: 10.3934/jimo.2010.6.209 +[Abstract](372) +[PDF](584.9KB)
Abstract:
This paper studies the capacity planning and expansion for the thin film transistor - liquid crystal display (TFT-LCD) manufacturing. Capacity planning is critical to TFT-LCD industry due to its complex product hierarchy and increasing product types; the coexistence of multiple generations of manufacturing technologies in a multi-site production environment; and the rapidly growing market demands. One key purpose of capacity planning is to simultaneously determine the profitable "product mix" and "production quantities" of each product group across various generation sites in a particular period and the optimal "capacity expansion quantity" of specific product groups at a certain site to improve the limited flexibility configurations through the acquisition of new auxiliary tools. This paper proposes a mixed integer linear programming model for capacity planning that incorporates practical characteristics and constraints in TFT-LCD manufacturing. A shadow-price based heuristic is developed to find a near-optimal solution. Preliminary computational study shows that the proposed heuristic provides good quality solutions in a reasonable amount of time. The proposed heuristic outperforms the traditional branch and bound method as the data size becomes large.
Penalty-based SAA method of stochastic nonlinear complementarity problems
Ming-Zheng Wang and M. Montaz Ali
2010, 6(1): 241-257 doi: 10.3934/jimo.2010.6.241 +[Abstract](235) +[PDF](210.9KB)
Abstract:
We consider a class of stochastic nonlinear complementarity problems. We first formulate the stochastic complementarity problem as a stochastic programming model. Based on this reformulation, we propose a penalty-based sample average approximation (in short, SAA) method for stochastic complementarity problem and prove its convergence. Finally, we report some numerical test results to show the efficiency of our method.
A potential reduction method for the generalized linear complementarity problem over a polyhedral cone
Fengming Ma, Yiju Wang and Hongge Zhao
2010, 6(1): 259-267 doi: 10.3934/jimo.2010.6.259 +[Abstract](233) +[PDF](146.2KB)
Abstract:
In this paper, we consider the generalized linear complementarity problem over a polyhedral cone arising in economics and engineering. For this problem, we first discuss its solution existence and then propose a potential reduction algorithm to solve it. The sparseness of the involved coefficient matrix is fully exploited in the computation of the algorithm and hence it has a relatively lower computational cost. The global convergence of the method is obtained under milder conditions. The given preliminary numerical experiments show the efficiency of the method.

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