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Journal of Industrial & Management Optimization

2011 , Volume 7 , Issue 4

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Nonsmooth optimization over the (weakly or properly) Pareto set of a linear-quadratic multi-objective control problem: Explicit optimality conditions
Henri Bonnel and  Ngoc Sang Pham
2011, 7(4): 789-809 doi: 10.3934/jimo.2011.7.789 +[Abstract](58) +[PDF](422.4KB)
Abstract:
We present explicit optimality conditions for a nonsmooth functional defined over the (properly or weakly) Pareto set associated with a multi-objective linear-quadratic control problem. This problem is very difficult even in a finite dimensional setting , i.e. when, instead of a control problem, we deal with a mathematical programming problem. Amongst various applications, our problem may be considered as a response for a decision maker when he has to choose a solution over the solution set of the grand coalition $p$-player cooperative differential game.
Transient and steady state analysis of an M/M/1 queue with balking, catastrophes, server failures and repairs
Ahmed M. K. Tarabia
2011, 7(4): 811-823 doi: 10.3934/jimo.2011.7.811 +[Abstract](37) +[PDF](332.5KB)
Abstract:
Recently Krishna Kumar and Pavai [10] have obtained the transient distribution for the queue length of the system an M/M/1 queueing system with catastrophes, server failures using a direct technique. In this paper, we consider Krishna Kumar and Pavai [10] model with balking feature. Based on the generating function technique and a direct approach, transient and steady state analysis of the queue length is carried out Krishna Kumar and Pavai [10] model can be deduced from the new model. Moreover, some other special cases are shown as special cases of our solution.
Single-machine scheduling with stepwise tardiness costs and release times
Güvenç Şahin and  Ravindra K. Ahuja
2011, 7(4): 825-848 doi: 10.3934/jimo.2011.7.825 +[Abstract](40) +[PDF](413.0KB)
Abstract:
We study a scheduling problem that belongs to the yard operations component of the railroad planning problems, namely the hump sequencing problem. The scheduling problem is characterized as a single-machine problem with stepwise tardiness cost objectives. This is a new scheduling criterion which is also relevant in the context of traditional machine scheduling problems. We produce complexity results that characterize some cases of the problem as pseudo-polynomially solvable. For the difficult-to-solve cases of the problem, we develop mathematical programming formulations, and propose heuristic algorithms. We test the formulations and heuristic algorithms on randomly generated single-machine scheduling problems and real-life data sets for the hump sequencing problem. Our experiments show promising results for both sets of problems.
Uniform estimates for ruin probabilities in the renewal risk model with upper-tail independent claims and premiums
Yinghua Dong and  Yuebao Wang
2011, 7(4): 849-874 doi: 10.3934/jimo.2011.7.849 +[Abstract](51) +[PDF](436.4KB)
Abstract:
In this paper, we discuss a nonstandard renewal risk model, where the price process of the investment portfolio is modelled as a geometric Lévy process, the claim sizes and premium sizes form sequences of identically distributed and upper-tail independent random variables, respectively, the claim size and its corresponding inter-claim time satisfy a certain dependence structure described via a conditional tail probability of the claim size given the inter-claim time before the claim occurs, and there is a similar dependence structure between the premium size and the inter-arrival time before the premium is paid. When the claim-size distribution belongs to the extended-regular-varying class, we obtain a uniform tail asymptotics for stochastically discounted aggregate claims. Furthermore, assuming that the tail of the premium-size distribution is lighter than that of the claim-size distribution, the uniform estimates for the finite- and infinite-time ruin probabilities are presented respectively.
Optimal fleet composition via dynamic programming and golden section search
Ryan Loxton and  Qun Lin
2011, 7(4): 875-890 doi: 10.3934/jimo.2011.7.875 +[Abstract](31) +[PDF](389.7KB)
Abstract:
In this paper, we consider an optimization problem arising in vehicle fleet management. The problem is to construct a heterogeneous vehicle fleet in such a way that cost is minimized subject to a constraint on the overall fleet size. The cost function incorporates fixed and variable costs associated with the fleet, as well as hiring costs that are incurred when vehicle requirements exceed fleet capacity. We first consider the simple case when there is only one type of vehicle. We show that in this case the cost function is convex, and thus the problem can be solved efficiently using the well-known golden section method. We then devise an algorithm, based on dynamic programming and the golden section method, for solving the general problem in which there are multiple vehicle types. We conclude the paper with some simulation results.
A full-Newton step interior-point algorithm for symmetric cone convex quadratic optimization
Yanqin Bai and  Lipu Zhang
2011, 7(4): 891-906 doi: 10.3934/jimo.2011.7.891 +[Abstract](54) +[PDF](381.2KB)
Abstract:
In this paper, we present a full-Newton step primal-dual interior-point algorithm for solving symmetric cone convex quadratic optimization problem, where the objective function is a convex quadratic function and the feasible set is the intersection of an affine subspace and a symmetric cone lies in Euclidean Jordan algebra. The search directions of the algorithm are obtained from the modification of NT-search direction in terms of the quadratic representation in Euclidean Jordan algebra. We prove that the algorithm has a quadratical convergence result. Furthermore, we present the complexity analysis for the algorithm and obtain the complexity bound as $\left\lceil 2\sqrt{r}\log\frac{\mu^0 r}{\epsilon}\right\rceil$, where $r$ is the rank of Euclidean Jordan algebras where the symmetric cone lies in.
An integrated approach for selection of service vendors in service supply chain
Fei Cheng , Shanlin Yang , Ram Akella and  Xiaoting Tang
2011, 7(4): 907-925 doi: 10.3934/jimo.2011.7.907 +[Abstract](48) +[PDF](419.5KB)
Abstract:
We explore the basic conceptual framework to imagine service supply chain that is a new research concern in supply chain management, and develop an integrated approach to optimize the selection of service vendors. Three insights arise on how service vendors can be obtained: (1) exploring multiple criteria decision-making problems with incomplete weight information; (2) identifying and eliminating criteria by information entropy method; (3) analyzing and calculating the final selection of qualified service vendors through a combining way between the fuzzy analytic hierarchy process and the multi-objective linear programming approach. Finally, an experiment implemented highlights effectiveness of the integrated approach.
Superconvergence property of finite element methods for parabolic optimal control problems
Chunjuan Hou , Yanping Chen and  Zuliang Lu
2011, 7(4): 927-945 doi: 10.3934/jimo.2011.7.927 +[Abstract](42) +[PDF](973.2KB)
Abstract:
In this paper, a finite element method for a parabolic optimal control problem is introduced and analyzed. For the discretization of a quadratic convex optimal control problem, the state and co-state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. As a result, it is proved in this paper that the difference between a suitable interpolation of the control and its finite element approximation has superconvergence property in order $O(h^2)$. Finally, two numerical examples are presented to confirm our theoretical results.
A new dynamic geometric approach for empirical analysis of financial ratios and bankruptcy
Alireza Bahiraie , A.K.M. Azhar and  Noor Akma Ibrahim
2011, 7(4): 947-965 doi: 10.3934/jimo.2011.7.947 +[Abstract](51) +[PDF](442.8KB)
Abstract:
This paper presents a complementary technique for the empirical analysis of financial ratios and bankruptcy risk using financial ratios. Within this new framework, we propose the use of a new measure of risk, the Dynamic Risk Space (DRS) measure. We provide evidence of the extent to which changes in values for this index are associated with changes in each axis's values and how this may alter our economic interpretation of changes in patterns and directions. In addition, this model tends to be generally useful for predicting financial distress and bankruptcy. This method would be a general methodological guideline associated with financial data, solving some methodological problems concerning financial ratios such as non-proportionality, non-asymmetry and non-scaled. To test the procedure, Multiple Discriminant Analysis (MDA), Logistic Analysis (LA) and Genetic Programming (GP) are employed to compare results by common and modified ratios for bankruptcy prediction. Classification methods outperformed using the DRS approach.
A variational problem and optimal control
Hang-Chin Lai , Jin-Chirng Lee and  Shuh-Jye Chern
2011, 7(4): 967-975 doi: 10.3934/jimo.2011.7.967 +[Abstract](35) +[PDF](316.1KB)
Abstract:
A variational problem involving two variables, the state and the control variables, is reduced to another variational problem in which the objective has no control variable, but the constrained identity has one. We then establish that the two problems are equivalent with the same optimal (state) solution under some conditions.
A smoothing homotopy method based on Robinson's normal equation for mixed complementarity problems
Zhengyong Zhou and  Bo Yu
2011, 7(4): 977-989 doi: 10.3934/jimo.2011.7.977 +[Abstract](35) +[PDF](377.0KB)
Abstract:
In this paper, a probability-one homotopy method for solving mixed complementarity problems is proposed. The homotopy equation is constructed by using the Robinson's normal equation of mixed complementarity problem and a $C^2$-smooth approximation of projection function. Under the condition that the mixed complementarity problem has no solution at infinity, which is a weaker condition than several well-known ones, existence and convergence of a smooth homotopy path from almost any starting point in $\mathbb{R}^n$ are proven. The homotopy method is implemented in Matlab and numerical results on the MCPLIB test collection are given.
Multiple solutions for a class of semilinear elliptic variational inclusion problems
Rong Xiao and  Yuying Zhou
2011, 7(4): 991-1002 doi: 10.3934/jimo.2011.7.991 +[Abstract](37) +[PDF](355.3KB)
Abstract:
In this paper, by using a local linking theorem, we obtain the existence of multiple solutions for a class of semilinear elliptic variational inclusion problems at non-resonance.
Duality in linear programming: From trichotomy to quadrichotomy
Yanqun Liu
2011, 7(4): 1003-1011 doi: 10.3934/jimo.2011.7.1003 +[Abstract](43) +[PDF](337.8KB)
Abstract:
In this paper, we present a new approach to the duality of linear programming. We extend the boundedness to the so called inclusiveness, and show that inclusiveness and feasibility are a pair of coexisting and mutually dual properties in linear programming: one of them is possessed by a primal problem if and only if the other is possessed by the dual problem. This duality relation is consistent with the symmetry between the primal and dual problems and leads to a duality result that is considered a completion of the classical strong duality theorem. From this result, complete solvability information of the primal (or dual) problem can be derived solely from dual (or primal) information. This is demonstrated by applying the new duality results to a recent linear programming method.
Global convergence of an inexact operator splitting method for monotone variational inequalities
Zhili Ge , Gang Qian and  Deren Han
2011, 7(4): 1013-1026 doi: 10.3934/jimo.2011.7.1013 +[Abstract](63) +[PDF](393.1KB)
Abstract:
Recently, Han (Han D, Inexact operator splitting methods with self-adaptive strategy for variational inequality problems, Journal of Optimization Theory and Applications 132, 227-243 (2007)) proposed an inexact operator splitting method for solving variational inequality problems. It has advantage over the classical operator splitting method of Douglas-Peaceman-Rachford-Varga operator splitting methods (DPRV methods) and some of their variants, since it adopts a very flexible approximate rule in solving the subproblem in each iteration. However, its convergence is established under somewhat stringent condition that the underlying mapping $F$ is strongly monotone. In this paper, we mainly establish the global convergence of the method under weaker condition that the underlying mapping $F$ is monotone, which extends the fields of applications of the method relatively. Some numerical results are also presented to illustrate the method.
Nonconvex quadratic reformulations and solvable conditions for mixed integer quadratic programming problems
Ye Tian and  Cheng Lu
2011, 7(4): 1027-1039 doi: 10.3934/jimo.2011.7.1027 +[Abstract](41) +[PDF](372.5KB)
Abstract:
In this paper, we study a mixed integer constrained quadratic programming problem. This problem is NP-Hard. By reformulating the problem to a box constrained quadratic programming and solving the reformulated problem, we can obtain a global optimal solution of a sub-class of the original problem. The reformulated problem may not be convex and may not be solvable in polynomial time. Then we propose a solvability condition for the reformulated problem, and discuss methods to construct a solvable reformulation for the original problem. The reformulation methods identify a solvable subclass of the mixed integer constrained quadratic programming problem.
A trust-region filter-SQP method for mathematical programs with linear complementarity constraints
Chunlin Hao and  Xinwei Liu
2011, 7(4): 1041-1055 doi: 10.3934/jimo.2011.7.1041 +[Abstract](34) +[PDF](375.0KB)
Abstract:
A trust-region filter-SQP method for mathematical programs with linear complementarity constraints (MPLCCs) is presented. The method is similar to that proposed by Liu, Perakis and Sun [Computational Optimization and Applications, 34, 5-33, 2006] but it solves the trust-region quadratic programming subproblems at each iteration and uses the filter technique to promote the global convergence. As a result, the method here is more robust since it admits the use of Lagrangian Hessian information and its performance is not affected by any penalty parameter. The preliminary numerical results on test problems generated by the QPECgen generator show that the presented method is effective.

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