A column generation algorithm for the estimation of origin–destination matrices in congested traffic networks

The general problem of estimating origin–destination (O–D) matrices in congested traffic networks is formulated as a mathematical programme with equilibrium constraints, referred to as the demand adjustment problem (DAP). This approach integrates the O–D matrix estimation and the network equilibrium assignment into one process. In this paper, a column generation algorithm for the DAP is presented. This algorithm iteratively solves a deterministic user equilibrium model for a given O–D matrix and a DAP restricted to the previously generated paths, whose solution generates a new O–D trip matrix estimation. The restricted DAP is formulated via a single level optimization problem. The convergence on local minimum of the proposed algorithm requires only the continuity of the link travel cost functions and the gauges used in the definition of the DAP.     

Environmentally friendly facility location with market competition

Many traditional facility location models assume spatial monopoly where market competition is ignored. Since facility locations affect the firm’s market exposure and subsequently its profit, accounting for the impact of the location decisions on customers while anticipating the reaction of competitor firms is essential. In this paper, we introduce a competitive facility location problem where market prices and production costs are determined through the economic equilibrium while explicitly considering competition from other firms. In order to accommodate for the growing efforts on limiting carbon emissions, the presented model includes constraints on the amount of carbon emissions that are due to transportation, while allowing carbon trading. The problem is formulated as a mixed integer non-linear model. Through numerical examples, we illustrate the effect of market competition on the location decisions and discuss the impact of emission limits and carbon trading on customers.     

Improving solution of discrete competitive facility location problems

We consider discrete competitive facility location problems in this paper. Such problems could be viewed as a search of nodes in a network, composed of candidate and customer demand nodes, which connections correspond to attractiveness between customers and facilities located at the candidate nodes. The number of customers is usually very large. For some models of customer behavior exact solution approaches could be used. However, for other models and/or when the size of problem is too high to solve exactly, heuristic algorithms may be used. The solution of discrete competitive facility location problems using genetic algorithms is considered in this paper. The new strategies for dynamic adjustment of some parameters of genetic algorithm, such as probabilities for the crossover and mutation operations are proposed and applied to improve the canonical genetic algorithm. The algorithm is also specially adopted to solve discrete competitive facility location problems by proposing a strategy for selection of the most promising values of the variables in the mutation procedure. The developed genetic algorithm is demonstrated by solving instances of competitive facility location problems for an entering firm.     

非线性互补约束规划问题的一个新的QP-free算法

本文研究了非线性互补约束均衡问题.利用互补函数以及光滑近似法,把非线性互补约束均衡问题转化为一个光滑非线性规划问题,得到了超线性收敛速度,数值实验结果表明本文提出的算法是可行的.     

A two-phase ant colony algorithm for multi-echelon defective supply chain network design

Supply chain system is an integrated production system of a product. In the past researches, this system was often assumed to be an equilibrium structure, but in real production process, some members in this system usually cannot effectively complete their production task because of the losses of production, which will reduce the performance of the whole supply chain production system. This supply chain with the losses of production is called the defective supply chain (DSC) system. This research will discuss the partner selection and the production–distribution planning in this DSC network system. Besides the cost of production and transportation, the reliability of the structure and the unbalance of this system caused by the losses of production are considered. Then a germane mathematical programming model is developed for solving this problem. Due to the complex problem and in order to get a satisfactory near-optimal solution with great speed, this research proposes seeking the solution with the solving model based on ant colony algorithm. The application results in real cases show that the solving model presented by this research can quickly and effectively plan the most suitable type of the DSC network and decision-making of the production–distribution. Finally, a comparative numerical experiment is performed by using the proposed approach and the common single-phase ant colony algorithm (SAC) to demonstrate the performance of the proposed approach. The analysis results show that the proposed approach can outperform the SAC in partner selection and production–distribution planning for DSC network design.     

Mathematical Programming with System of Equilibrium Constraints

In this paper, we study the mathematical program with system of equilibrium constraints. This problem contains bilevel program with system of equilibrium constraints, semi-infinite program with system of equilibrium constraints, mathematical program with Nash equilibrium constraints, mathematical program with system of mixed variational like inequalities constraints. We establish the existence theorems of mathematical program with system of equilibrium constraints under various assumptions.     

A new criterion for the inexact logarithmic-quadratic proximal method and its derived hybrid methods

To solve nonlinear complementarity problems, the inexact logarithmic-quadratic proximal (LQP) method solves a system of nonlinear equations (LQP system) approximately at each iteration. Therefore, the efficiencies of inexact-type LQP methods depend greatly on the involved inexact criteria used to solve the LQP systems. This paper relaxes inexact criteria of existing inexact-type LQP methods and thus makes it easier to solve the LQP system approximately. Based on the approximate solutions of the LQP systems, a descent method, and a prediction–correction method are presented. Convergence of the new methods are proved under mild assumptions. Numerical experiments for solving traffic equilibrium problems demonstrate that the new methods are more efficient than some existing methods and thus verify that the new inexact criterion is attractive in practice.     

Development of a new approach for deterministic supply chain network design

This paper proposes a mixed integer linear programming model and solution algorithm for solving supply chain network design problems in deterministic, multi-commodity, single-period contexts. The strategic level of supply chain planning and tactical level planning of supply chain are aggregated to propose an integrated model. The model integrates location and capacity choices for suppliers, plants and warehouses selection, product range assignment and production flows. The open-or-close decisions for the facilities are binary decision variables and the production and transportation flow decisions are continuous decision variables. Consequently, this problem is a binary mixed integer linear programming problem. In this paper, a modified version of Benders’ decomposition is proposed to solve the model. The most difficulty associated with the Benders’ decomposition is the solution of master problem, as in many real-life problems the model will be NP-hard and very time consuming. In the proposed procedure, the master problem will be developed using the surrogate constraints. We show that the main constraints of the master problem can be replaced by the strongest surrogate constraint. The generated problem with the strongest surrogate constraint is a valid relaxation of the main problem. Furthermore, a near-optimal initial solution is generated for a reduction in the number of iterations.     

A novel algorithm for area traffic capacity control with elastic travel demands

A non-linear area traffic control system with limited capacity is considered in this paper. Optimal signal settings and link capacity expansions can be determined while trip distribution and network flow are in equilibrium. This problem can be formulated as a non-linear mathematical program with equilibrium constraints. For the objective function a non-linear constrained optimization program for signal settings and link capacity expansion is determined. For the constraint set the elastic user equilibrium traffic assignment obeying Wardrop’s first principle can be formulated as a variational inequality. Since the constrained optimization problem is non-convex, only local optima can be obtained. In this paper, a novel algorithm using a non-smooth trust region approach is proposed. Numerical tests are performed using a real data city network and various example test networks in which the effectiveness and robustness of the proposed method are confirmed as compared to other well-known solution methods.     

A stochastic model for risk management in global supply chain networks

With the increasing emphasis on supply chain vulnerabilities, effective mathematical tools for analyzing and understanding appropriate supply chain risk management are now attracting much attention. This paper presents a stochastic model of the multi-stage global supply chain network problem, incorporating a set of related risks, namely, supply, demand, exchange, and disruption. We provide a new solution methodology using the Moreau–Yosida regularization, and design an algorithm for treating the multi-stage global supply chain network problem with profit maximization and risk minimization objectives.     

An approximation algorithm for the location of dairy enterprises under time constraints

The facility location problem constitutes a major part of the strategic planning of a dairy enterprise. The decision making process for such a long term investment is rather complicated, with many of the issues involved belonging to the NP – Hard class of algorithms and therefore the challenge for operational researchers who attempt to mathematically model the problem is significant. A number of models have appeared in the last few years in an attempt to find a solution to the problem. This paper introduces a model pertaining to saturated and competitive markets, which include a high investment risk as well as a small margin for incorrect decisions of the newcomers. Time is a crucial element, especially for sensitive products with a limited life span cycle. An approximation algorithm is used and the solution is then compared with the optimal one. Emphasis was given to the running time of the software program, both in the designing and the implementation of the algorithm.     

一个带产品定价约束的竞争选址双层规划模型及其求解算法

以大型连锁卖场的选址为研究背景,提出了一个在竞争环境下使获利最大的竞争选址定价双层规划模型,其中上层模型做出选址决策,下层模型确定产品的纳什均衡价格.将设施效用引入到模型中,用指数效用函数来刻画顾客的购物行为偏好,首次证明了不合作状态下双方价格均衡解的存在性和唯一性,并给出了求解最优设施点设置方案和价格均衡解的算法思想及数值算例.     

An automated planning engine for biopharmaceutical production

We introduce an optimization-based production planning tool for the biotechnology industry. The industry’s planning problem is unusually challenging because the entire production process is regulated by multiple external agencies – such as the US Food and Drug Administration – representing countries where the biopharmaceutical is to be sold. The model is structured to precisely capture the constraints imposed by current and projected regulatory approvals of processes and facilities, as well as capturing the outcomes of quality testing and processing options, facility capacities and initial status of work-in-process. The result is a supply chain “Planning Engine” that generates capacity-feasible batch processing schedules for each production facility within the biomanufacturing supply chain and an availability schedule for finished product against a known set of demands and regulations. Developing the formulation based on distinct time grids tailored for each facility, planning problems with more than 27,000 boolean variables, more than 130,000 linear variables and more than 80,000 constraints are automatically formulated and solved within a few hours. The Planning Engine’s development and implementation at Bayer Healthcare’s Berkeley, CA manufacturing site is described.     

二层供应链网络均衡模型的研究

利用均衡理论和二层规划理论来研究供应链网络均衡问题。针对供应链网络中上下层成员之间具有的Stackelberg博弈特征以及同层成员之间具有的非合作博弈特征,构建了二层供应链网络的均衡模型,该模型实际上一个均衡约束的二层规划问题。此外,为了使得供应链网络在整体上实现最优,本文还在模型中引入回收契约以协调供应链网络。最后,利用罚函数法对模型进行了求解,算例分析说明了模型的合理性和有效性。     

The prediction–correction approach to nonlinear complementarity problems

This paper presents a prediction–correction approach to solving the nonlinear complementarity problem (NCP). Each iteration of the new method consists of a prediction and a correction. The predictor is produced by an inexact Logarithmic-Quadratic Proximal method; and then it is corrected by the Proximal Point Algorithm. Convergence of the new method is proved under mild assumptions. Comparison to existing methods shows the superiority of the new method. Numerical experiments including the application to traffic equilibrium problems demonstrate that the new method is attractive in practice.     

Lower and upper bounds for a two-stage capacitated facility location problem with handling costs

We study in this paper multi-product facility location problem in a two-stage supply chain in which plants have production limitation, potential depots have limited storage capacity and customer demands must be satisfied by plants via depots. In the paper, handling cost for batch process in depots is considered in a realistic way by a set of capacitated handling modules. Each module can be regards as alliance of equipment and manpower. The problem is to locate depots, choose appropriate handling modules and to determine the product flows from the plants, opened depots to customers with the objective to minimize total location, handling and transportation costs. For the problem, we developed a hybrid method. The initial lower and upper bounds are provided by applying a Lagrangean based on local search heuristic. Then a weighted Dantzig–Wolfe decomposition and path-relinking combined method are proposed to improve obtained bounds. Numerical experiments on 350 randomly generated instances demonstrate our method can provide high quality solution with gaps below 2%.     

竞争环境下截流问题的选址-价格模型

研究了竞争环境下考虑产品定价的截流设施选址问题。连锁企业在市场上新建设施时,市场上已有属于竞争对手的设施存在,在连锁企业新建设施位置确定之后,两个企业关于产品定价进行双寡头完全信息非合作博弈。定义了效用函数,引入Huff模型,以企业利润最大为目标,建立双层规划模型,证明了模型纳什均衡价格的存在性,并构造启发式算法对模型进行求解。算例分析表明,该算法求解结果较为理想,可用于大中型网络的规划选址问题。     

A capacitated competitive facility location problem

We consider a mathematical model similar in a sense to competitive location problems. There are two competing parties that sequentially open their facilities aiming to “capture” customers and maximize profit. In our model, we assume that facilities’ capacities are bounded. The model is formulated as a bilevel integer mathematical program, and we study the problem of obtaining its optimal (cooperative) solution. It is shown that the problem can be reformulated as that of maximization of a pseudo-Boolean function with the number of arguments equal to the number of places available for facility opening. We propose an algorithm for calculating an upper bound for values that the function takes on subsets which are specified by partial (0, 1)-vectors.     

Systems of generalized quasivariational inclusions problems with applications to variational analysis and optimization problems

In this paper, we study an existence theorem of systems of generalized quasivariational inclusions problem. By this result, we establish the existence theorems of solutions of systems of generalized equations, systems of generalized vector quasiequilibrium problem, collective variational fixed point, systems of generalized quasiloose saddle point, systems of minimax theorem, mathematical program with systems of variational inclusions constraints, mathematical program with systems of equilibrium constraints and systems of bilevel problem and semi-infinite problem with systems of equilibrium problem constraints. This research was supported by the National Science Council of the Republic of China.