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Journal of Industrial & Management Optimization

2008 , Volume 4 , Issue 4

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Decomposition branch and bound algorithm for optimization problems over efficient sets
Nguyen Van Thoai
2008, 4(4): 647-660 doi: 10.3934/jimo.2008.4.647 +[Abstract](76) +[PDF](196.8KB)
Abstract:
The problem of optimizing some given function over the efficient set is one of the most interesting and important concepts in multicriteria decision making. As the efficient set is in general nonconvex, even for the case of linear multicriteria programming problems, optimizing over the efficient set belongs to a typical problem class of multiextremal optimization problems, which can have local optima different from global optima.
    In this article, we consider the case where the multicriteria programming problem is linear. Characterizing the set of efficient solutions by some constraint of 'reverse convex' type in the space of criteria, we formulate the problem of minimizing a function $f$ over the efficient set as a global optimization problem with a special structure. For the resulting problem, a decomposition branch and bound based algorithm is then proposed, in which the branching procedure is performed in the criteria space. Convergence properties of the algorithm are discussed, and preliminary computational results are reported.
Optimal traffic signal control for an $M\times N$ traffic network
Yinfei Li and Shuping Chen
2008, 4(4): 661-672 doi: 10.3934/jimo.2008.4.661 +[Abstract](87) +[PDF](155.5KB)
Abstract:
This paper discusses the optimal traffic control signal setting for an $M \times N$ rectangular road traffic network. By introducing the concepts of synchronization rate and non-synchronization degree, a mathematical model is constructed and an optimization problem is posed. Then, a new iterative algorithm is developed to solve this optimal traffic control signal setting problem. Convergence properties for this iterative algorithm are established. Finally, a numerical example is solved to illustrate the effectiveness of the method.
Dynamic power price problem: An inverse variational inequality approach
Junfeng Yang
2008, 4(4): 673-684 doi: 10.3934/jimo.2008.4.673 +[Abstract](141) +[PDF](189.4KB)
Abstract:
This paper considers an optimal control perspective on dynamic power price problem where the load on the power-grid is controlled via price. The optimal regulatory price is characterized by inverse variational inequality in which the function value and the control variable are in the opposite positions of the classical variational inequality. Discrete and continuum models with load constraints are developed and existence theorems are established under quite reasonable assumptions. Preliminary numerical results also show the feasibility of the proposed models.
On the sensitivity of desirability functions for multiresponse optimization
Caglar S. Aksezer
2008, 4(4): 685-696 doi: 10.3934/jimo.2008.4.685 +[Abstract](67) +[PDF](169.8KB)
Abstract:
Desirability functions have been one of the most important multiresponse optimization technique since the early eighties. Main reasons for this popularity might be counted as the convenience of the implementation of the method and it's availability in many experimental design software packages. Technique itself involves somehow subjective parameters such as the importance coefficients between response characteristics that are used to calculate overall desirability, weights used in determining the shape of each individual response and the size of the specification band of the response. However, the impact of these sensitive parameters on the solution set is mostly uninvestigated. This paper proposes a procedure to analyze the sensitivity of the important characteristic parameters of desirability functions and their impact on pareto-optimal solution set. The proposed procedure uses the experimental design tools on the solution space and estimates a prediction equation on the overall desirability to identify the sensitive parameters. For illustration, a classical desirability example is selected from the literature and results are given along with the discussion.
An approach to controlled mechanical systems based on the multiobjective optimization technique
Vadim Azhmyakov
2008, 4(4): 697-712 doi: 10.3934/jimo.2008.4.697 +[Abstract](45) +[PDF](231.6KB)
Abstract:
This paper discusses the multiobjective optimization techniques for a class of optimal control problems in mechanics. We deal with constrained nonlinear control systems described by the Euler-Lagrange or Hamilton equations and study the variational structure of the solution of the corresponding boundary-value problems. We also reduce the original ''mechanical'' problem to an auxiliary multiobjective optimization problem. This approach makes it possible to apply the effective theoretical and computational results from multiobjective programming to the original problem. We consider first order computational schemes for optimal control problems governed by mechanical systems and examine some illustrative examples.
A unified model for state feedback of discrete event systems II: Control synthesis problems
Qiying Hu, Chen Xu and Wuyi Yue
2008, 4(4): 713-726 doi: 10.3934/jimo.2008.4.713 +[Abstract](105) +[PDF](192.3KB)
Abstract:
This paper studies control synthesis problems in a new model framework for discrete event state feedback control systems. The new model framework consists of a basis model as well as concurrent models. We study relationships between the basis model and the concurrent models from the perspective of a predicate being controllable and synthesizable. We derive a linear order for the models, and moreover, show that they are equivalent under certain conditions. Based on this, three conditions are presented for synthesizing a predicate completely. Finally, the optimal control synthesis problem for both the basis model and the concurrent models is studied.
Some characterizations and applications on strongly $\alpha$-preinvex and strongly $\alpha$-invex functions
Cai-Ping Liu
2008, 4(4): 727-738 doi: 10.3934/jimo.2008.4.727 +[Abstract](65) +[PDF](149.7KB)
Abstract:
In this paper, a new class of generalized convex functions is introduced, which is called the strongly quasi $\alpha$-preinvex functions. Some properties of strongly $\alpha$-preinvex functions are studied. We establish the relationships among the strongly quasi $\alpha$-preinvex functions, strongly quasi $\alpha$-invex functions and strongly quasi $\alpha\eta$-monotonicity under some suitable conditions. As applications, a class of perturbed variational-like inequality problems is introduced, some relationships between the perturbed variational-like inequality and optimization problems are established under the assumptions of strongly $\alpha$-invex functions.
Guaranteed descent conjugate gradient methods with modified secant condition
Shishun Li and Zhengda Huang
2008, 4(4): 739-755 doi: 10.3934/jimo.2008.4.739 +[Abstract](63) +[PDF](192.2KB)
Abstract:
Conjugate gradient methods are typically used to solve large scale unconstrained optimization problems. Recently, Hager and Zhang (2006) proposed two guaranteed descent conjugate gradient methods. In this paper, following Hager and Zhang (2006), we will use the modified secant condition given by Zhang et al.(1999) to present two new descent conjugate gradient methods. An interesting feature of these new methods is that they take both the gradient and function value information. Under some suitable assumptions, global convergence properties for these methods are established. Numerical comparisons with the Hager-Zhang methods are given.
A selection problem for a constrained linear regression model
Shaoyong Lai and Qichang Xie
2008, 4(4): 757-766 doi: 10.3934/jimo.2008.4.757 +[Abstract](54) +[PDF](171.3KB)
Abstract:
Selecting a good estimate for a constricted linear regression model is investigated by using the generalized information criterion. Some asymptotic properties of the selection procedure with the model average technique are established. It is shown that the selection procedure is asymptotically efficient in the sense that a fitted estimate asymptotically obtains the minimum average squared error from a class of model average estimators.
Sequential characterization of solutions in convex composite programming and applications to vector optimization
Radu Ioan Boţ, Anca Grad and Gert Wanka
2008, 4(4): 767-782 doi: 10.3934/jimo.2008.4.767 +[Abstract](66) +[PDF](219.3KB)
Abstract:
When characterizing optimal solutions of both scalar and vector optimization problems usually constraint qualifications have to be satisfied. By considering sequential characterizations, given for the first time in vector optimization in this paper, this drawback is eliminated. In order to establish them we give first of all sequential characterizations for a convex composed optimization problem with geometric and cone constraints. Then, by means of scalarization, we extend them to the vector case. For exemplification we particularize the characterization in the case of linear and set scalarization.
A power penalty approach to american option pricing with jump diffusion processes
Kai Zhang, Xiaoqi Yang and Kok Lay Teo
2008, 4(4): 783-799 doi: 10.3934/jimo.2008.4.783 +[Abstract](58) +[PDF](241.4KB)
Abstract:
This paper is devoted to develop a power penalty method for pricing the American option model where the underlying asset is assumed to follow a jump diffusion process. With the help of the linear complementarity problem and variational inequalities, we propose a power penalty approach for a partial integro-differential complementarity problem, which is the mathematical model of pricing the American option with a jump diffusion process. The convergence analysis of the power penalty approach is established. Finally, based on the finite element discretization, a numerical scheme is developed to solve the penalized problem and the numerical tests are designed to illustrate the efficiency of this method.
On maximizing the expected terminal utility by investment and reinsurance
Lin Xu, Rongming Wang and Dingjun Yao
2008, 4(4): 801-815 doi: 10.3934/jimo.2008.4.801 +[Abstract](77) +[PDF](205.7KB)
Abstract:
In this paper, optimal problems for the insurer who can invest on risky market and purchase reinsurance are considered. The surplus process of the insurer is a kind of perturbed classical risk model with stochastic premium income. The investment return generating process of the risky market is a drifted Brownian motion plus a compound Poisson process. The objective function in this paper is to maximize the expected utility of wealth of the insurer at terminal time, say $T$. By solving the Hamilton-Jacobi-Bellman equations related to our optimal control problems, the closed form expression for optimal strategy and the value function is derived, which indicates that the value function for an insurer to purchase both investment and reinsurance is always better than the one for the insurer to purchase only either investment or reinsurance.
Bounds on delay start LPT algorithm for scheduling on two identical machines in the $l_p$ norm
Ling Lin, Dong He and Zhiyi Tan
2008, 4(4): 817-826 doi: 10.3934/jimo.2008.4.817 +[Abstract](70) +[PDF](152.9KB)
Abstract:
In this paper, we consider the scheduling problem on two identical machines with objective to minimize the sum of the $l_{p}$ norm of every machine's load. We present the worst-case ratio of $DSL(l)$ for any integer $l$, here $DSL(l)$ first assigns the $l-1$ largest jobs optimally, then assigns the remaining jobs by $LS$ rule. It follows that $DSL(l)$ is an all-norm $\frac{l+2}{l+1}$ -approximation algorithm. Improved tight bound is given for $l=7$ in the $l_{2}$ norm.
Supply chain partnership for Three-Echelon deteriorating inventory model
Jonas C. P. Yu, H. M. Wee and K. J. Wang
2008, 4(4): 827-842 doi: 10.3934/jimo.2008.4.827 +[Abstract](73) +[PDF](312.5KB)
Abstract:
The different facilities in a supply chain usually develop their partnership through information sharing and strategic alliances to achieve the overall benefit of the system. In this study, we propose a supply chain network system with two producers, a single distributor and two retailers. Each retailer has a deterministic demand rate. A mathematical model of deteriorating item is developed to consider a vertical integration of the producer, the distributor and the retailer and a horizontal integration of the producers. We show how the integrated approach to decision making can achieve global optimum. Numerical examples and a sensitivity analysis are given to validate the proposed system.
Price and delivery-time competition of perishable products: Existence and uniqueness of Nash equilibrium
Xiaolin Xu and Xiaoqiang Cai
2008, 4(4): 843-859 doi: 10.3934/jimo.2008.4.843 +[Abstract](77) +[PDF](212.2KB)
Abstract:
We consider such a problem: Multiple sellers sell a family of substitutable perishable products in a common market with correlative random demands. Due to the perishable nature of a product, delivery time directly affects its freshness level, which in turn affects its demand. Each seller's demand is formulated as a function of all sellers' prices and delivery-times. To maximize his own expected profit, each seller needs to set his selling price and delivery time simultaneously, by taking into account his competitors' reactions. We present three models to address different practical situations. In Model I, we assume that each seller faces constant operating and purchasing costs, but is subject to a given service reliability constraint. In contrast, we assume delivery-time dependent operating and purchasing costs without service reliability constraints in Model II and Model III respectively. We establish the existence of a price and delivery-time equilibrium, under mild conditions in Model I and II, and restrictive conditions in Model III. A novel method is adopted to establish the uniqueness of the equilibrium in Model I, and an iterative procedure is designed to compute the equilibrium prices and delivery-times in Model III.

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