# American Institute of Mathematical Sciences

ISSN:
1547-5816

eISSN:
1553-166X

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## Journal of Industrial & Management Optimization

2014 , Volume 10 , Issue 4

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2014, 10(4): 989-1000 doi: 10.3934/jimo.2014.10.989 +[Abstract](233) +[PDF](337.6KB)
Abstract:
This paper investigates how the intermediary firms can optimally determine the purchasing cycle length of a deteriorating product under return on inventory investment (ROII) maximization criterion. There are three key features differentiating this paper from the extant literature and being considered simultaneously in this paper, which are: 1) an alternative performance measurement (i.e., ROII) is proposed to formulate the inventory system; 2) the decision maker in this paper is an intermediary firm instead of a retailer; and 3) the deteriorating nature of products is considered. By incorporating the deteriorating nature of products and the special structure of the intermediary firm environments into the traditional economic order quantity model, the inventory problem encountered by the intermediary firm is mathematically formulated as a non-linear programming problem. Several interesting properties of the proposed inventory problem are developed and an efficient iterative algorithm is provided to search for the optimal solution. Also, the convergence of the iterative algorithm developed in this paper is proved. Finally, a numerical example is presented to illustrate the features of the proposed problem and the convergent search algorithm.
2014, 10(4): 1001-1018 doi: 10.3934/jimo.2014.10.1001 +[Abstract](180) +[PDF](644.6KB)
Abstract:
In the present paper, we move forward in the study of minimax fractional programming problem and establish sufficient optimality conditions under the assumptions of generalized $(H_p,r)$-invexity. Weak, strong and strict converse duality theorems are also derived for two types of dual models related to minimax fractional programming problem involving aforesaid invex functions. In order to show the existence of introduced class of functions, examples are given.
2014, 10(4): 1019-1030 doi: 10.3934/jimo.2014.10.1019 +[Abstract](183) +[PDF](769.9KB)
Abstract:
This paper established some new convergence results for the power penalty method for the nonlinear complementarity problem(NCP), and then applied the method to solve the general traffic assignment problem with elastic demand. The power penalty method approximates the NCP by a nonlinear equation containing a power penalty term. We prove that this method can handle general monotone NCPs. This result is important for the general traffic assignment problem with elastic demand because the associated NCP is often not $\xi$ monotone. This study considered the traffic assignment problem with symmetric and asymmetric link costs, as well as additive and non-additive route costs. We propose to use a column generation scheme based on the proposed power penalty method to solve this problem. Numerical results are provided to demonstrate the efficiency of the method.
Zhen Wang and Wei Wu
2014, 10(4): 1031-1039 doi: 10.3934/jimo.2014.10.1031 +[Abstract](204) +[PDF](315.1KB)
Abstract:
Higher order tensors are generalizations of matrices. In this paper, we extend the bounds for the greatest eigenvalue of positive square matrices to positive tensors, and give further results on the bounds for the greatest eigenvector of positive tensors.
2014, 10(4): 1041-1058 doi: 10.3934/jimo.2014.10.1041 +[Abstract](279) +[PDF](524.1KB)
Abstract:
In this paper, we consider a perturbed compound Poisson risk model with a randomized dividend strategy. Assume that decisions on paying off dividends are made at some random observation times. Whenever the observed value of the surplus process exceeds a given barrier $b$, the excess value will be paid off as dividends. We assume that the Laplace transform of the individual claim size belongs to the rational family. When the time intervals between successive decision times follow exponential distribution, we present explicit expressions for the Gerber-Shiu function. We also extend the exponential assumption to Erlang and discuss the solution procedure.
2014, 10(4): 1059-1070 doi: 10.3934/jimo.2014.10.1059 +[Abstract](301) +[PDF](300.9KB)
Abstract:
The purpose of this paper is to propose an effective linear programming technique for solving matrix games in which the payoffs are expressed with intervals and the choice of strategies for players is constrained, i.e., interval-valued constraint matrix games. Because the payoffs of the interval-valued constraint matrix game are intervals, its value is an interval as well. In this methodology, the value of the interval-valued constraint matrix game is regarded as a function of values in the payoff intervals, which is proven to be monotonous and non-decreasing. By the duality theorem of linear programming, it is proven that both players always have the identical interval-type value and hereby the interval-valued constraint matrix game has an interval-type value. A pair of auxiliary linear programming models is derived to compute the upper bound and the lower bound of the value of the interval-valued constraint matrix game by using the upper bounds and the lower bounds of the payoff intervals, respectively. Validity and applicability of the linear programming technique proposed in this paper is demonstrated with a numerical example of the market share game problem.
2014, 10(4): 1071-1090 doi: 10.3934/jimo.2014.10.1071 +[Abstract](201) +[PDF](577.3KB)
Abstract:
This paper studies a single-machine scheduling problem with the objective of minimizing the total tardiness for a set of independent jobs. The processing time of a job is modeled as a step function of its starting time and a specific deteriorating date. The total tardiness as one important objective in practice has not been concerned in the studies of single-machine scheduling problems with step-deteriorating jobs. To overcome the intractability of the problem, we propose a heuristic named simple weighted search procedure (SWSP) and a general variable neighborhood search algorithm (GVNS) to obtain near optimal solutions. Extensive numerical experiments are carried out on randomly generated test instances in order to evaluate the performance of the proposed algorithms. By comparing to the CPLEX optimization solver, the heuristic SWSP and the standard variable neighborhood search, it is shown that the proposed GVNS algorithm can provide better solutions within a reasonable running time.
2014, 10(4): 1091-1108 doi: 10.3934/jimo.2014.10.1091 +[Abstract](215) +[PDF](391.6KB)
Abstract:
In this paper, we propose a barrier function method for the generalized Nash equilibrium problem (GNEP) which, in contrast to the standard Nash equilibrium problem (NEP), allows the constraints for each player may depend on the rivals' strategies. We solve a sequence of NEPs, which are defined by logarithmic barrier functions of the joint inequality constraints. We demonstrate, under suitable conditions, that any accumulation point of the solutions to the sequence of NEPs is a solution to the GNEP. Moreover, a semismooth Newton method is used to solve the NEPs and sufficient conditions for the local superlinear convergence rate of the semismooth Newton method are derived. Finally, numerical results are reported to illustrate that the barrier approach for solving the GNEP is practical.
2014, 10(4): 1109-1127 doi: 10.3934/jimo.2014.10.1109 +[Abstract](318) +[PDF](996.4KB)
Abstract:
Minimizing VaR, as estimated from a set of scenarios, is a difficult integer programming problem. Solving the problem to optimality may demand using only a small number of scenarios, which leads to poor out-of-sample performance. A simple alternative is to minimize CVaR for several different quantile levels and then to select the optimized portfolio with the best out-of-sample VaR. We show that this approach is both practical and effective, outperforming integer programming and an existing VaR minimization heuristic. The CVaR quantile level acts as a regularization parameter and, therefore, its ideal value depends on the number of scenarios and other problem characteristics.
2014, 10(4): 1129-1146 doi: 10.3934/jimo.2014.10.1129 +[Abstract](288) +[PDF](662.8KB)
Abstract:
Both mathematical characteristics and computational aspects of dynamic optimization in finance have potential for extensions. Various proposed extensions are presented in this paper for dynamic optimization modelling in finance, adapted from developments in other areas of economics and mathematics. They show the need and potential for further areas of study and extensions in financial modelling. The extensions discussed and made concern (a) incorporation of the elements of a dynamic optimization model, (b) an improved model including physical capital, (c) some computational experiments. These extensions make dynamic financial optimisation relatively more organized, coherent and coordinated. These extensions are relevant for applications of financial models to academic and practical exercises. This paper reports initial efforts in providing some useful extensions; further work is necessary to complete the research agenda.
2014, 10(4): 1147-1168 doi: 10.3934/jimo.2014.10.1147 +[Abstract](298) +[PDF](434.9KB)
Abstract:
Successful supply chain management has to find an effective sourcing strategy to cope with uncertainty in both supply and demand. Most of existing literature deals with the problems related to the uncertain demand, however, in this paper, we discuss a model in which a firm is in the face of demand uncertainty and supplier reliability uncertainty at the same time prior to a single selling season. The firm has two instants to order from two suppliers (one supplier is completely reliable and the other is unreliable), while the unreliable supplier's reliability is uncertain at instant 1 and is completely observed at instant 2. To determine the cost-minimizing ordering strategies at both instants, the firm has to evaluate the trade-off between a more accurate forecast and a potentially higher unit cost at instant 2. We present the optimal supplementary order quantities with the realized reliability and give the optimal ordering policies and sourcing strategies at instant 1 under certain conditions. We find conditions under which the order quantity from the unreliable supplier at instant 1 is decreasing in the reorder costs at instant 2.
2014, 10(4): 1169-1189 doi: 10.3934/jimo.2014.10.1169 +[Abstract](332) +[PDF](459.3KB)
Abstract:
This paper addresses how different coordination mechanisms affect the information sharing behavior in a supply chain. We study information sharing in a make-to-stock supply chain under wholesale contract and revenue sharing contract. Under wholesale contract, we show that information sharing is always beneficial to the supplier and identify the conditions ensuring that information sharing is beneficial to the retailer. Under revenue sharing contract, information sharing is beneficial to the supplier, the retailer and the supply chain. This research indicates that whether sharing the demand information is beneficial depends on the coordination mechanism and parameters.
2014, 10(4): 1191-1208 doi: 10.3934/jimo.2014.10.1191 +[Abstract](209) +[PDF](761.5KB)
Abstract:
We study the multi-server machine interference problem with repair pressure coefficient and a modified Bernoulli vacation. The repair rate depends on the number of failed machines waiting in the system. In congestion, the server may increase the repair rate with pressure coefficient $\theta$ to reduce the queue length. At each repair completion of a server, the server may go for a vacation of random length with probability $p$ or may continue to repair the next failed machine, if any, with probability $1-p$. The entire system is modeled as a finite-state Markov chain and its steady state distribution is obtained by a recursive matrix approach. The major performance measures are evaluated based on this distribution. Under a cost structure, we propose to use the Quasi-Newton method and probabilistic global search Lausanne method to search for the global optimal system parameters. Numerical examples are presented to demonstrate the application of our approach.
2014, 10(4): 1209-1224 doi: 10.3934/jimo.2014.10.1209 +[Abstract](134) +[PDF](386.6KB)
Abstract:
In this work, we consider a variance-optimal hedging strategy for discretely sampled geometric Asian options, under exponential Lévy dynamics. Since it is difficult to hedge these instruments perfectly, here we choose to maximize a quadratic utility function and give the expressions of hedging strategies explicitly, based on the derived Föllmer-Schweizer decomposition of the contingent claim of geometric Asian options monitored at discrete times. The expression of its corresponding error is also given.
2014, 10(4): 1225-1234 doi: 10.3934/jimo.2014.10.1225 +[Abstract](209) +[PDF](352.2KB)
Abstract:
This paper deals with the lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem. Under new assumptions, which do not contain any information about solution mappings, we establish the lower semicontinuity of the solution mapping to a parametric generalized vector equilibrium problem by using a scalarization method. These results improve the corresponding ones in recent literature. Some examples are given to illustrate our results.
2014, 10(4): 1235-1259 doi: 10.3934/jimo.2014.10.1235 +[Abstract](262) +[PDF](498.2KB)
Abstract:
In the framework of dual risk model, Yao et al. [18](Optimal dividend and capital injection problem in the dual model with proportional and fixed transaction costs. European Journal of Operational Research, 211, 568-576) show how to determine optimal dividend and capital injection strategy when the dividend rate is unrestricted and the bankruptcy is forbidden. In this paper, we further include constrain on dividend rate and allow for bankruptcy when it is in deficit. We seek the optimal strategy for maximizing the expected discounted dividends minus the discounted capital injections before bankruptcy. Explicit solutions for strategy and value function are obtained when income jumps follow a hyper-exponential distribution, the corresponding limit results are presented, some known results are extended.
2014, 10(4): 1261-1277 doi: 10.3934/jimo.2014.10.1261 +[Abstract](205) +[PDF](430.8KB)
Abstract:
This paper considers the problem of simultaneously determining the price and inventory control strategies for deteriorating items. It is assumed that the rate of deterioration can be reduced by means of effective preservation technology investment and the demand rate is a function of selling price. The goal of this study is to maximize the total profit per unit time by simultaneously determining the optimal selling price, length of replenishment cycle and preservation technology investment. First, for a given preservation technology investment, we prove that the optimal selling price and the optimal length of replenishment cycle exist and are unique. Next, it is shown that the total profit per unit time is a concave function of the preservation technology investment. Then, an effective algorithm is designed to find the optimal joint policy. Finally, numerical examples to illustrate the solution procedure and some managerial implications are provided.
2014, 10(4): 1279-1296 doi: 10.3934/jimo.2014.10.1279 +[Abstract](289) +[PDF](447.3KB)
Abstract:
A new global optimization method combining genetic algorithm and Hooke-Jeeves method to solve a class of constrained optimization problems is studied in this paper. We first introduce the quadratic penalty function method and the exact penalty function method to transform the original constrained optimization problem with general equality and inequality constraints into a sequence of optimization problems only with box constraints. Then, the combination of genetic algorithm and Hooke-Jeeves method is applied to solve the transformed optimization problems. Since Hooke-Jeeves method is good at local search, our proposed method dramatically improves the accuracy and convergence rate of genetic algorithm. In view of the derivative-free of Hooke-Jeeves method, our method only requires information of objective function value which not only can overcome the computational difficulties caused by the ill-condition of the square penalty function, but also can handle the non-differentiability by the exact penalty function. Some well-known test problems are investigated. The numerical results show that our proposed method is efficient and robust.
2014, 10(4): 1297-1318 doi: 10.3934/jimo.2014.10.1297 +[Abstract](211) +[PDF](924.8KB)
Abstract:
To implement the optimal dispatch of distributed energy resources (DER) in the virtual power plant (VPP), a distributed optimal dispatch method based on ELM (Extreme Learning Machine) transformation is proposed. The joint distribution of maximum available outputs of multiple wind turbines in the VPP is firstly modeled with the Gumbel-Copula function. A VPP optimal dispatch model is then formulated to achieve maximum utilization of renewable energy generation, which can take into account the constraints of electric power network and DERs. Based on the Gumbel-Copula joint distribution, the nonlinear functional relationship between the wind power cost and wind turbine output is approximated using ELM. The approximated functional relationship is then transformed into a set of equality constraints, which can be easily integrated with the optimal dispatch model. To solve the optimal dispatch problem, a distributed primal-dual sub-gradient algorithm is proposed to determine the operational strategies of DERs via local decision making and limited communication between neighbors. Finally, case studies based on the 15-node and the 118-node virtual power plant prove that the proposed method is effective and can achieve identical performance as the centralized dispatch approach.
2014, 10(4): 1319-1321 doi: 10.3934/jimo.2014.10.1319 +[Abstract](147) +[PDF](209.9KB)
Abstract:
In this note, we obtain an important property from Condition $C$. Using the property, we can provide short proofs for some properties of (generalized) preinvex functions.

2016  Impact Factor: 0.994