Linear Programming Models for the User and System Optimal Dynamic Network Design Problem: Formulations,Comparisons and Extensions

In this paper we formulate a network design model in which the traffic flows satisfy dynamic user equilibrium conditions for a single destination. The model presented here incorporates the Cell Transmission Model (CTM); a traffic flow model capable of capturing shockwaves and link spillovers. Comparisons are made between the properties of the Dynamic User equilibrium Network Design Problem (DUE NDP) and an existing Dynamic System Optimal (DSO) NDP formulation. Both network design models have different objective functions with similar constraint sets which are linear and convex. Numerical demonstrations are made on multiple networks to demonstrate the efficacy of the model and demonstrate important differences between the DUE and DSO NDP approaches. In addition, the flexibility of the approach is demonstrated by extending the formulation to account for demand uncertainty. This is formulated as a stochastic programming problem and initial test results are demonstrated on test networks. It is observed that not accounting for demand uncertainty explicitly, provides sub-optimal solution to the DUE NDP problem.     

A goal programming model for joint decision making of inventory lot-size,supplier selection and carrier selection

In this paper, we address a problem in which a storage space constrained buyer procures a single product in multiple periods from multiple suppliers. The production capacity constrained suppliers offer all-unit quantity discounts. The late deliveries and rejections are also incorporated in sourcing. In addition, we consider transportation cost explicitly in decision making which may vary because of freight quantity and distance of shipment between the buyer and a supplier. We propose a multi-objective integer linear programming model for joint decision making of inventory lot-sizing, supplier selection and carrier selection problem. In the multi-objective formulation, net rejected items, net costs and net late delivered items are considered as three objectives that have to be minimized simultaneously over the decision horizon. The intent of the model is to determine the timings, lot-size to be procured, and supplier and carrier to be chosen in each replenishment period. We solve the multi-objective optimization problem using three variants of goal programming (GP) approaches: preemptive GP, non-preemptive GP and weighted max–min fuzzy GP. The solution of these models is compared at different service-level requirements using value path approach.     

需求与物流网络不确定下的应急救援选址问题

针对应急物流中需求与物流网络的不确定性特征,对应急救援中应急需求和物流网络均不确定条件下的应急配送中心选址问题进行研究,以成本最小化为目标,建立基于集合覆盖的应急救援设施选址的随机规划模型,采用期望值法和随机模拟两种方法处理数学模型中的不确定性。通过算例与仿真研究,获得应急救援下的配送中心选址最优方案。结果表明,相对于传统的期望值方法处理随机参数,随机模拟方法具有较明显优势。     

Multi-objective reliability-redundancy allocation problem using particle swarm optimization

This paper considers the multi-objective reliability redundancy allocation problem of a series system where the reliability of the system and the corresponding designing cost are considered as two different objectives. Due to non-stochastic uncertain and conflicting factors it is difficult to reduce the cost of the system and improve the reliability of the system simultaneously. In such situations, the decision making is difficult, and the presence of multi-objectives gives rise to multi-objective optimization problem (MOOP), which leads to Pareto optimal solutions instead of a single optimal solution. However in order to make the model more flexible and adaptable to human decision process, the optimization model can be expressed as fuzzy nonlinear programming problems with fuzzy numbers. Thus in a fuzzy environment, a fuzzy multi-objective optimization problem (FMOOP) is formulated from the original crisp optimization problem. In order to solve the resultant problem, a crisp optimization problem is reformulated from FMOOP by taking into account the preference of decision maker regarding cost and reliability goals and then particle swarm optimization is applied to solve the resulting fuzzified MOOP under a number of constraints. The approach has been demonstrated through the case study of a pharmaceutical plant situated in the northern part of India.     

An α-Fuzzy Goal Approximate Algorithm for Solving Fuzzy Multiple Objective Linear Programming Problems

Multiple conflicting objectives in many decision making problems can be well described by multiple objective linear programming (MOLP) models. This paper deals with the vague and imprecise information in a multiple objective problem by fuzzy numbers to represent parameters of an MOLP model. This so-called fuzzy MOLP (or FMOLP) model will reflect some uncertainty in the problem solution process since most decision makers often have imprecise goals for their decision objectives. This study proposes an approximate algorithm based on a fuzzy goal optimization under the satisfactory degree α to handle both fuzzy and imprecise issues. The concept of a general fuzzy number is used in the proposed algorithm for an FMOLP problem with fuzzy parameters. As a result, this algorithm will allow decision makers to provide fuzzy goals in any form of membership functions.     

A hybrid credibility-based fuzzy multiple objective optimisation to differential pricing and inventory policies with arbitrage consideration

In most markets, price differentiation mechanisms enable manufacturers to offer different prices for their products or services in different customer segments; however, the perfect price discrimination is usually impossible for manufacturers. The importance of accounting for uncertainty in such environments spurs an interest to develop appropriate decision-making tools to deal with uncertain and ill-defined parameters in joint pricing and lot-sizing problems. This paper proposes a hybrid bi-objective credibility-based fuzzy optimisation model including both quantitative and qualitative objectives to cope with these issues. Taking marketing and lot-sizing decisions into account simultaneously, the model aims to maximise the total profit of manufacturer and to improve service aspects of retailing simultaneously to set different prices with arbitrage consideration. After applying appropriate strategies to defuzzify the original model, the resulting non-linear multi-objective crisp model is then solved by a fuzzy goal programming method. An efficient stochastic search procedure using particle swarm optimisation is also proposed to solve the non-linear crisp model.     

Multi-objective optimization problem under fuzzy rule constraints using particle swarm optimization

In this paper, a fuzzy multi-objective programming problem is considered where functional relationships between decision variables and objective functions are not completely known to us. Due to uncertainty in real decision situations sometimes it is difficult to find the exact functional relationship between objectives and decision variables. It is assumed that information source from where some knowledge may be obtained about the objective functions consists of a block of fuzzy if-then rules. In such situations, the decision making is difficult and the presence of multiple objectives gives rise to multi-objective optimization problem under fuzzy rule constraints. In order to tackle the problem, appropriate fuzzy reasoning schemes are used to determine crisp functional relationship between the objective functions and the decision variables. Thus a multi-objective optimization problem is formulated from the original fuzzy rule-based multi-objective optimization model. In order to solve the resultant problem, a deterministic single-objective non-linear optimization problem is reformulated with the help of fuzzy optimization technique. Finally, PSO (Particle Swarm Optimization) algorithm is employed to solve the resultant single-objective non-linear optimization model and the computation procedure is illustrated by means of numerical examples.     

A two-phase approach for multi-objective programming problems with fuzzy coefficients

In this study, a two-phase procedure is introduced to solve multi-objective fuzzy linear programming problems. The procedure provides a practical solution approach, which is an integration of fuzzy parametric programming (FPP) and fuzzy linear programming (FLP), for solving real life multiple objective programming problems with all fuzzy coefficients. The interactive concept of the procedure is performed to reach simultaneous optimal solutions for all objective functions for different grades of precision according to the preferences of the decision-maker (DM). The procedure can be also performed to obtain lexicographic optimal and/or additive solutions if it is needed. In the first phase of the procedure, a family of vector optimization models is constructed by using FPP. Then in the second phase, each model is solved by FLP. The solutions are optimal and each one is an alternative decision plan for the DM.     

Using system dynamics for simulation and optimization of one coal industry system under fuzzy environment

In this paper, we have developed a model that integrates system dynamics with fuzzy multiple objective programming (SD-FMOP). This model can be used to study the complex interactions in a industry system. In the process of confirming sensitive parameters and fuzzy variables of the SD model, we made use of fuzzy multi-objective programming to help yield the solution. We adopted the chance-constraint programming model to convert the fuzzy variables into precise values. We use genetic algorithm to solve FMOP model, and obtain the Pareto solution through the programming models. It is evident that FMOP is effective in optimizing the given system to obtain the decision objectives of the SD model. The results recorded from the SD model are in our option, reasonable and credible. These results may help governments to establish more effective policy related to the coal industry development.     

Formulating the multi-segment goal programming

Goal programming (GP) is an important analytical approach devised to solve many real-word problems. However, the condition of multi-segment aspiration levels (MSAL) may exist in many marketing or decision management problems. The problem cannot be solved by current GP techniques. In order to improve the effective of GP and solve the multi-segment goal programming (MSGP) problem, this paper provides a new idea for programming the MSAL problem from multi-aspiration contribution levels viewpoint. This significantly improved the utility of GP in real application; in addition, two illustrative examples are included to demonstrate the solution procedure of the proposed model.     

Risk-averse two-stage stochastic programming with an application to disaster management

Traditional two-stage stochastic programming is risk-neutral; that is, it considers the expectation as the preference criterion while comparing the random variables (e.g., total cost) to identify the best decisions. However, in the presence of variability risk measures should be incorporated into decision making problems in order to model its effects. In this study, we consider a risk-averse two-stage stochastic programming model, where we specify the conditional-value-at-risk (CVaR) as the risk measure. We construct two decomposition algorithms based on the generic Benders-decomposition approach to solve such problems. Both single-cut and multicut versions of the proposed decomposition algorithms are presented. We adapt the concepts of the value of perfect information (VPI) and the value of the stochastic solution (VSS) for the proposed risk-averse two-stage stochastic programming framework and define two stochastic measures on the VPI and VSS. We apply the proposed model to disaster management, which is one of the research fields that can significantly benefit from risk-averse two-stage stochastic programming models. In particular, we consider the problem of determining the response facility locations and the inventory levels of the relief supplies at each facility in the presence of uncertainty in demand and the damage level of the disaster network. We present numerical results to discuss how incorporating a risk measure affects the optimal solutions and demonstrate the computational effectiveness of the proposed methods.     

考虑区间约束的物流网络双层规划模型及算法

考虑物流网络需求的不确定性,利用区间参数度量不确定性变量与参数,建立区间需求模式下的物流网络双层规划模型,设计了一种含区间参数与变量的递阶优化遗传算法,通过定义问题求解的风险系数与最大决策偏差,给出适合物流网络结构的区间运算准则,实现模型的确定性转化。以区间松弛变量与0-1决策变量定义初始种群,通过两阶遗传操作运算,求解不同情景下双层规划目标的区间最优解与节点决策方案。算例测试表明算法求解的可操作性更强,求解结果具有区间最优解与情景决策的优越性。     

The methodology of multiple-criteria decision making in the optimization of an urban transportation system: case study of Poznań City in Poland

The methodology of multiple-criteria decision making applied to the optimization of an urban transportation system is presented in the paper. Three mathematical models of different complexity are constructed to optimize the allocation of vehicles to certain routes in a mass transit system. All models take into account both passengers' and operator's objectives. The optimization problems are formulated in terms of multiple-objective fuzzy linear programming and multiple-objective non-linear programming. The sensitivity and precision analysis of the models is carried out. Two interactive multi-objective mathematical programming procedures are utilized to solve the problems. They generate samples of Pareto-optimal compromise solutions and provide the decision maker (DM)with an effective tool that supports him/her in the decision making process. Finally, the DM selects the solution that best fits his or her expectations.     

Standard goal programming with fuzzy hierarchies: a sequential approach

This paper proposes a pragmatic model for multi-objective decision-making processes involving clusters of objectives which have a decisional meaning for the decision maker (DM). We provide the DMs with a comfortable tool that allows them to express their preferences both by comparing criteria of the same cluster and via the comparison between the different clusters. In standard goal programming the importance of the goals is modeled by the introduction of preferential weights or/and the incorporation of pre-emptive priorities. However, in many cases the DM is not able to establish a precise preference structure. Even in the case of precise weights the solution does not match necessarily the relative weights or, in the case of precise pre-emptive priority, the result could be very restrictive. In order to overcome these drawbacks, in this paper the normalized unwanted deviations are interpreted in terms of achievement degrees of the goals and fuzzy relations are used to model the relative importance of the goals. Thus, we show how several methodologies from the fuzzy goal programming literature can be tailored for solving standard GP problems. We apply this new modeling to problems where there is a “natural” clustering between goals of the same class. We address this situation by solving two phases; in the first one each class is handled separately taking into account the hierarchy of their goals and, in the second phase, we integrate the results of the first phase and the imprecise hierarchy of the different classes. We formulate a new goal programming model called as sequential goal programming with fuzzy hierarchy model. Because many real situations involve decision making in this environment, our proposal can be a useful tool of broad application. A numerical example illustrates the methodology.     

An Interactive Satisfying Method Based on Alternative Tolerance for Multiple Objective Optimization With Fuzzy Parameters

An interactive satisfying method based on alternative tolerance is presented for the multiple objective optimization problem with fuzzy parameters. Using the $alpha $ -level sets of the fuzzy numbers, all the objectives are modeled as the fuzzy goals, and the tolerances of the objectives are iteratively changed according to a decision maker for a satisfying solution. Via a specific attainable point programming model, the membership functions can be modified, and then, a lexicographic two-phase programming procedure is constructed correspondingly to find the final solution. In a special case, the objective constraint is added instead of changing the membership functions; therefore, the dissatisfying objectives for the decision maker can be improved step by step. The presented method not only acquires the $alpha $ -Pareto optimal or weak $alpha $-Pareto optimal solution of the fuzzy multiple objective optimization, but also satisfies the progressive preference of the decision maker. A numerical example shows its power.      

Multiple objective capital budgeting within the university

The problem of selecting capital projects in universities is compounded by the existence of conflicting multiple objectives on the part of different groups within the university community. Administrators, faculty members, students and politicians all have different goals and projects which they feel are important to the university and themselves. Because of the multiple objective dimension of the capital budgeting process goal programming becomes an appropriate solution approach. However, because of the indivisibility of some capital projects, mixed integer goal programming, a variation of the traditional GP model, must be employed. The solution approach is demonstrated via an illustrative case example. Model goals in the example (which are prioritized by the top administrative officers of the university) exist for capital budget and operating expenses, building and laboratory construction, accreditation, political interest and area performance.     

城市道路建设时序决策的鲁棒优化

为提高城市道路建设时序决策的鲁棒性,提出了城市道路建设时序决策优化的双 层规划模型。模型假定出行需求在一定范围内扰动,上层规划是在有限资金的约束下寻求各建设阶段的系统总出行时间与系统总出行时间对出行需求的灵敏度之间的综合最小值,下层规划为各建设阶段的随机用户均衡配流。文中推导出了系统总出行时间对出行需求灵敏度的计算式,并给出了模型的求解算法。最后以一个测试路网为例,对基于系统总出行时间、基于灵敏度、基于系统总出行时间与灵敏度综合出行时间的决策优化模型进行了计算分析,结果显示3种决策优化模型均可寻求到各自目标最优的城市道路建设时序,但在需求不确定的情景下基于灵敏度、基于系统总出行时间与灵敏度综合出行时间的决策优化结果更具鲁棒性。     

An SDP approach for multiperiod mixed 0–1 linear programming models with stochastic dominance constraints for risk management

In this paper we consider multiperiod mixed 0–1 linear programming models under uncertainty. We propose a risk averse strategy using stochastic dominance constraints (SDC) induced by mixed-integer linear recourse as the risk measure. The SDC strategy extends the existing literature to the multistage case and includes both the first-order and second-order constraints. We propose a stochastic dynamic programming (SDP) solution approach, where one has to overcome the negative impact of the cross-scenario constraints on the decomposability of the model. In our computational experience we compare our SDP approach against a commercial optimization package, in terms of solution accuracy and elapsed time. We use supply chain planning instances, where procurement, production, inventory, and distribution decisions need to be made under demand uncertainty. We confirm the hardness of the testbed, where the benchmark cannot find a feasible solution for half of the test instances while we always find one, and show the appealing tradeoff of SDP, in terms of solution accuracy and elapsed time, when solving medium-to-large instances.     

一类新的动态车辆调度问题的建模与算法

针对目前对于动态车辆调度问题的研究仅集中于考虑时间依赖或依概率变化的情形,在对原有动态车辆调度问题模型进行总结的基础上,综合考虑了时间依赖且网络依概率变化,以及结合带有时间窗和随机需求的情况,提出了新的问题模型,并提出求解该问题模型的多目标随机机会约束规划模型,设计了用遗传算法解决该模型的方案与步骤。实验结果表明,所提出的模型可有效地拟合交通状况,设计的算法可以有效地求解该模型。     

Multi-objective vehicle routing problem with time windows using goal programming and genetic algorithm

This paper presents a new model and solution for multi-objective vehicle routing problem with time windows (VRPTW) using goal programming and genetic algorithm that in which decision maker specifies optimistic aspiration levels to the objectives and deviations from those aspirations are minimized. VRPTW involves the routing of a set of vehicles with limited capacity from a central depot to a set of geographically dispersed customers with known demands and predefined time windows. This paper uses a direct interpretation of the VRPTW as a multi-objective problem where both the total required fleet size and total traveling distance are minimized while capacity and time windows constraints are secured. The present work aims at using a goal programming approach for the formulation of the problem and an adapted efficient genetic algorithm to solve it. In the genetic algorithm various heuristics incorporate local exploitation in the evolutionary search and the concept of Pareto optimality for the multi-objective optimization. Moreover part of initial population is initialized randomly and part is initialized using Push Forward Insertion Heuristic and λ-interchange mechanism. The algorithm is applied to solve the benchmark Solomon's 56 VRPTW 100-customer instances. Results show that the suggested approach is quiet effective, as it provides solutions that are competitive with the best known in the literature.