ISSN:

1547-5816

eISSN:

1553-166X

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## Journal of Industrial & Management Optimization

July 2014 , Volume 10 , Issue 3

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2014, 10(3): 665-689
doi: 10.3934/jimo.2014.10.665

*+*[Abstract](1887)*+*[PDF](1066.5KB)**Abstract:**

The current study proposes a sequential approach for humanitarian logistics in natural disasters based on the Bayesian group information updates (GIU). First, a dynamic time-dependent nonlinear model without GIU is proposed. Then, two losses are addressed to explain the influence of a disaster on supply, demand, and humanitarian logistics. The two losses include losses caused by the mismatch between supply and demand in affected areas and the time losses caused by logistics processes under emergency conditions. Therefore, a multi-period humanitarian logistics planning model with GIU is established based on the model without GIU using Bayesian theory. Then, the model with GIU is revised into a single-objective model, and then a matrix-coding-based genetic algorithm is developed to solve the revised model. Finally, the proposed methodology is applied to the humanitarian logistics problems of emergency response encountered during the Wenchuan Earthquake in China. Computational results show that the proposed methodology can generate specific logistics plans for allocating relief resources according to updated information. Therefore, emergency planners can gain insights for humanitarian logistics planning in natural disaster response by inputting their own sets of data.

2014, 10(3): 691-700
doi: 10.3934/jimo.2014.10.691

*+*[Abstract](1263)*+*[PDF](331.7KB)**Abstract:**

This paper considers a single-machine scheduling and due date assignment problem in which the processing time of a job depends on its position in a processing sequence and jobs can be rejected by incurring penalties. The objective is to minimize the sum of the scheduling criterion of the accepted jobs and the total penalty of the rejected jobs. We first consider the problem with the common due date assignment method where the scheduling criterion is a cost function that includes the costs of earliness, tardiness, and due date assignment. We provide a polynomial-time algorithm to solve the problem. We then provide a unified model for solving the single-machine scheduling problem with rejection and position-dependent processing times. Finally, we extend the results to the setting involving various due date assignment methods.

2014, 10(3): 701-715
doi: 10.3934/jimo.2014.10.701

*+*[Abstract](1252)*+*[PDF](840.6KB)**Abstract:**

Vehicle routing problem is a classic combinational optimization problem, which has been attracting research attentions in logistics and optimization area. Conventional static vehicle routing problem assumes the logistics information is accurate and timely, and does not take into account the uncertainties, which is therefore inadequate during practical applications. In this paper, a vehicle initial routing optimization model considering uncertainties is proposed, the vehicle capacity, customer time-window, and the maximum travelling distance as well as the road capacity are considered. In the cyber-physical logistics system background, a routing adjustment model is proposed to minimize the total distribution cost considering the road congestion, and the static and dynamic models are proposed for traffic information transmission network to quantitatively analyse the impact of the traffic information transmission delay on the vehicle routing optimization. The learnable genetic algorithm is adopted to solve the initial routing optimization model and the routing adjustment model. The simulation results have verified its effectiveness.

2014, 10(3): 717-741
doi: 10.3934/jimo.2014.10.717

*+*[Abstract](1183)*+*[PDF](481.6KB)**Abstract:**

In this paper, to design a piecewise linear contractual function, we consider to solve the single-level nonconvex programming with integral operator which is equivalent to the principal-agent bilevel programming model with continuous distribution. A modified constraint shifting homotopy method for solving the Karush-Kuhn-Tucker system of the discrete nonconvex programming is proposed and the global convergence from any initial point in shifted feasible set is proven under some mild conditions. A simple homotopy path tracing algorithm is given and is implemented in Matlab. For some typical risk averse utility functions and the typical distribution functions which simultaneously satisfy monotone likelihood ratio condition and convexity of the distribution function condition, some numerical tests to design the piecewise linear contract are done by our homotopy method as well as by using

*fmincon*in Matlab, LOQO and MINOS and, as a comparison, the piecewise constant contracts are also designed by solving the single-level nonconvex programming which is equivalent to the principal-agent bilevel programming model with corresponding discrete distributions. Numerical tests show that: to design a piecewise linear contract, which is much better than a piecewise constant contract, it needs only to solve a much lower dimensional optimization problem and hence needs much less computing time. Numerical experiences also show that the modified constraint shifting homotopy method is feasible and robust.

2014, 10(3): 743-759
doi: 10.3934/jimo.2014.10.743

*+*[Abstract](1543)*+*[PDF](799.9KB)**Abstract:**

The classical augmented Lagrangian method (ALM) is an efficient method for solving convex optimization with linear constraints. However, the efficiency of ALM, to some extent, is hinged by the computational efforts on solving the resulting subproblems. For the convex optimization with some favorable structures, e.g., either the objective function is separable or the matrices in linear constraints are well-posed, a relaxation to the subproblems of ALM can substantially result in solutions with closed-form. Unfortunately, the relaxation skill can not be extended directly to the generic convex optimization without special structures, particularly for the case of objective function with coupled variables. In this paper, by further relaxing the resulting subproblems of ALM, we propose several novel augmented Lagrangian-based proximal point algorithms. Algorithmically, the next iterate is produced by integrating the predictor, which is obtained in either primal-dual or dual-primal order, with the current iterate. Numerical results demonstrate the promising performances of the proposed algorithms.

2014, 10(3): 761-776
doi: 10.3934/jimo.2014.10.761

*+*[Abstract](1153)*+*[PDF](450.3KB)**Abstract:**

This paper deals with the optimization of a hydrothermal problem that considers a non-smooth Lagrangian $L(t ,z,z^{\prime})$. We consider a general case where the functions $L_{z^{\prime}}(t ,\cdot,\cdot)$ and $L_{z}(t ,\cdot ,\cdot)$ are discontinuous in $\{(t,z,z^{\prime})/z^{\prime}=\phi(t,z)\}$, which is the borderline point between two power generation zones. This situation arises in problems of optimization of hydrothermal systems where the thermal plant input-output curve considers the shape of the cost curve in the neighborhood of the valve points. The problem shall be formulated in the framework of nonsmooth analysis, using the generalized (or Clarke's) gradient. We shall obtain a necessary minimum condition and we shall generalize the known result (smooth transition) that the derivative of the minimum presents a constancy interval. Finally, we shall present an example.

Solving structural engineering design optimization problems using an artificial bee colony algorithm

2014, 10(3): 777-794
doi: 10.3934/jimo.2014.10.777

*+*[Abstract](2202)*+*[PDF](325.2KB)**Abstract:**

The main goal of the present paper is to solve structural engineering design optimization problems with nonlinear resource constraints. Real world problems in engineering domain are generally large scale or nonlinear or constrained optimization problems. Since heuristic methods are powerful than the traditional numerical methods, as they don't requires the derivatives of the functions and provides near to the global solution. Hence, in this article, a penalty guided artificial bee colony (ABC) algorithm is presented to search the optimal solution of the problem in the feasible region of the entire search space. Numerical results of the structural design optimization problems are reported and compared. As shown, the solutions by the proposed approach are all superior to those best solutions by typical approaches in the literature. Also we can say, our results indicate that the proposed approach may yield better solutions to engineering problems than those obtained using current algorithms.

2014, 10(3): 795-815
doi: 10.3934/jimo.2014.10.795

*+*[Abstract](1249)*+*[PDF](459.7KB)**Abstract:**

This paper focuses on stochastic investment games between two investors with incorporating the influence of the macro economical environment that modeled by a Markov chain with $d$ states. There are two correlated assets are available to two investors, each investor can only invest into one of assets and his opponent choose to invest the other one. The dynamic of the two assets are driven by two drifted Brownian motion with coefficients specified by the functions of the Markov chain. Thus the system considered in this paper is controlled SDEs with random coefficients. Only one payoff function is available to both investors, one investor wants to maximize the expected payoff function, while his opponent wants to minimize the quantity at the same time. As results, the existence of the saddle point of the game, a couple of equations satisfied by the value functions and optimal policies for both investors are derived. Based on finite-difference method and weak convergence theory, a vector-valued Markov chain is constructed for approximating the underlying risky process weakly, which enables us to obtain the value function and optimal policies numerically. To some extend, we can view this paper as a further research of the problems proposed in Wan [23].

2014, 10(3): 817-826
doi: 10.3934/jimo.2014.10.817

*+*[Abstract](1251)*+*[PDF](293.9KB)**Abstract:**

The sensor network localization with uncertainties in anchor positions has been studied in this paper. We formulate this problem as a DC (difference of two convex functions) programming. Then, a DC programming based algorithm has been proposed to solve such a problem. Simulation results obtained by our proposed method are better performance than those obtained by existing ones.

2014, 10(3): 827-837
doi: 10.3934/jimo.2014.10.827

*+*[Abstract](1138)*+*[PDF](383.7KB)**Abstract:**

In this paper, we are concerned with the existence of solutions of a class of quasilinear elliptic hemivariational inequalities on unbounded domains. This kind of problems is more delicate due to the lack of compact embedding of the Sobolev spaces. By using the Clarke generalized directional derivatives for locally Lipschitz functions and some nonlinear function analysis techniques, such as the Ky Fan theorem for multivalued mappings, the theorem of finite intersection property, etc, we obtain the existence of solutions of the quasilinear elliptic hemivariational inequalities. Unlike those methods used in the references mentioned in this paper, we treat the case of unbounded domain by using the approximation of bounded domains.

2014, 10(3): 839-857
doi: 10.3934/jimo.2014.10.839

*+*[Abstract](1132)*+*[PDF](400.3KB)**Abstract:**

This paper analyzes a renewal input multiple working vacations queue with change over times under $(a,c,b)$ policy. The service and vacation times are exponentially distributed. The server begins service if there are at least $c$ units in the queue and the service takes place in batches with a minimum of size $a$ and a maximum of size $b~ (a\leq c \leq b)$. The steady state queue length distributions at arbitrary and pre-arrival epochs are obtained along with some special cases of the model. Performance measures and optimal cost policy is presented with numerical experiences for some particular values of the parameters. The genetic algorithm is employed to search the optimal values of some important parameters of the system.

2014, 10(3): 859-869
doi: 10.3934/jimo.2014.10.859

*+*[Abstract](1364)*+*[PDF](345.1KB)**Abstract:**

An alternating linearization method with inexact data, for the bilevel problem of minimizing a nonsmooth convex function over the optimal solution set of another nonsmooth convex problem, is presented in this paper. The problem is first approximately transformed into an unconstrained optimization with the help of a penalty function and we prove that the penalty function admits exact penalization under some mild conditions. The objective function of this unconstrained problem is the sum of two nonsmooth convex functions and in the algorithm each iteration involves solving two easily solved subproblems. It is shown that the iterative sequence converges to a solution of the original problem by parametric minimization theorem. Numerical experiments validate the theoretical convergence analysis and illustrate the implementation of the alternating linearization algorithm for this bilevel program.

2014, 10(3): 871-882
doi: 10.3934/jimo.2014.10.871

*+*[Abstract](1305)*+*[PDF](335.6KB)**Abstract:**

In this paper, we present an optimality condition which could determine whether a given KKT solution is globally optimal. This condition is equivalent to determining if the Hessian of the corresponding Largrangian is copositive over a set. To find the corresponding Lagrangian multiplier, two linear conic programming problems are constructed and then relaxed for computational purpose. Under the new condition, we proposed a local search based scheme to find a global optimal solution and showed its effectiveness by three examples.

2014, 10(3): 883-903
doi: 10.3934/jimo.2014.10.883

*+*[Abstract](1188)*+*[PDF](506.7KB)**Abstract:**

In this article, a nonlinear conjugate gradient method is studied and analyzed for finding the local solutions of two matrix optimization problems resulting from the decentralized static output feedback problem for continuous and discrete-time systems. The global convergence of the proposed method is established. Several numerical examples of decentralized static output feedback are presented that demonstrate the applicability of the considered approach.

2014, 10(3): 905-927
doi: 10.3934/jimo.2014.10.905

*+*[Abstract](1228)*+*[PDF](582.4KB)**Abstract:**

Index tracking is a popular way for passive fund management, which aims to reproduce the performance of a market index by investing in a subset of the constituents of the index. The formulation of index tracking with some realistic constraints always leads to an optimization problem that is very hard to solve. In this paper, we propose an approximate formulation to the original optimization problem and analyze the approximation error bound. It is shown that the approximation can be reasonably close to the original problem. We consider both cases where the mean absolute error and mean square error are used as the tracking error measurements. The mean absolute error measurement results in a mixed-integer linear programming problem and can be solved by some standard solvers, such as CPLEX. The mean square error measurement leads to a mixed-integer quadratic programming problem. An efficient hybrid heuristic method is given to solve this problem. We do some numerical experiments by the use of five data sets from OR-Library. The results are promising.

2014, 10(3): 929-943
doi: 10.3934/jimo.2014.10.929

*+*[Abstract](1685)*+*[PDF](622.3KB)**Abstract:**

We propose an optimal consensus design method for solving a finite-time optimal control problem involving a second-order multi-agent system. With this method, the optimal consensus problem can be modeled as an optimal parameter selection problem with continuous state inequality constraints and free terminal time. By virtue of the constraint transcription method and a time scaling transform method, a gradient-based optimization algorithm is developed to solve this optimal parameter selection problem. Furthermore, a new consensus protocol is designed, by which the consensus value of the system velocity can be chosen to be an arbitrary value. For illustration, simulation studies are carried out to demonstrate the proposed method.

2014, 10(3): 945-963
doi: 10.3934/jimo.2014.10.945

*+*[Abstract](1665)*+*[PDF](439.9KB)**Abstract:**

This paper studies the first-order cone constrained homogeneous quadratic programming problem. For efficient computation, the problem is reformulated as a linear conic programming problem. A union of second-order cones are designed to cover the first-order cone such that a sequence of linear conic programming problems can be constructed to approximate the conic reformulation. Since the cone of nonnegative quadratic forms over a union of second-order cones has a linear matrix inequalities representation, each linear conic programming problem in the sequence is polynomial-time solvable by applying semidefinite programming techniques. The convergence of the sequence is guaranteed when the union of second-order cones gets close enough to the first-order cone. In order to further improve the efficiency, an adaptive scheme is adopted. Numerical experiments are provided to illustrate the efficiency of the proposed approach.

2014, 10(3): 965-976
doi: 10.3934/jimo.2014.10.965

*+*[Abstract](1495)*+*[PDF](371.7KB)**Abstract:**

This paper focuses on a type of inverse linear second order cone programming (LSOCP) problems which require us to adjust the parameters in both the objective function and the constraint set of a given LSOCP problem as little as possible so that a known feasible solution becomes optimal one. This inverse problem can be formulated as a linear second order cone complementarity constrained optimization problem and is difficult to solve due to the presence of second order cone complementarity constraint. To solve this difficult problem, we first partially penalize the inverse problem and then propose the majorization approach to the penalized problem by solving a sequence of convex optimization problems with quadratic objective function and simple second order cone constraints. Numerical results demonstrate the efficiency of our approach to inverse LSOCP problems.

2014, 10(3): 977-987
doi: 10.3934/jimo.2014.10.977

*+*[Abstract](1535)*+*[PDF](370.3KB)**Abstract:**

Sample average approximation method is one of the well-behaved methods in the stochastic optimization. This paper presents a sample average approximation method based on a D-gap function for stochastic variational inequality problems. An unconstrained optimization reformulation is proposed for the expected-value formulation of stochastic variational inequality problems based on the D-gap function. An implementable sample average approximation method for the reformulation is established and it is proven that the optimal values and the optimal solutions of the approximation problems converge to their true counterpart with probability one as the sample size increases under some moderate assumptions. Finally, the preliminary numerical results for some test examples are reported, which show that the proposed method is promising.

2018 Impact Factor: 1.025

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