基于多目标遗传算法的项目调度及其仿真研究
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摘要
项目管理至少可以追溯到4500年以前埃及金字塔以及1800年以前南美玛雅墓的建成，它们依靠的是最简单最原始的工具。后来慢慢发展到始于美国杜邦公司的CPM技术、始于美国海军开发“北极星导弹计划”的PERT技术。时至今日，大量关于项目管理的研究如雨后春笋般涌现：从原理、算法到系统（包括相关的软件），从简单到复杂，以充分帮助管理人员对各种项目进行计划、安排、监督和控制。
项目调度问题，是项目管理的一个重要研究领域，自20世纪60年代被提出之后，就引起了各行各界专家和学者们的广泛关注。随着现代化技术广泛应用于生产，具有较高自动化水平的生产系统，在使生产过程更加合理、高效运行的同时，也使得项目调度问题变得更加困难，其复杂性往往超出人脑的决策能力。
大多数项目调度问题是一类组合优化问题，计算复杂性理论已经证明大多数组合优化问题是NP-hard问题。传统的运筹学求解方法往往无法在多项式时间范围内寻求到这些NP-hard问题的最优解，随之人们开始尝试在多项式时间范围内求解的近似算法即启发式方法。
目前对复杂的项目调度问题的求解算法主要集中在对元启发式方法即智能优化方法的研究。智能优化算法是模拟某一自然现象或过程而建立起来的具有高度并行、自组织、自学习与自适应特征的适于复杂的高度非线性问题求解的算法。这类算法包括模拟煺火、禁忌搜索、粒子群算法、蚁群算法和进化计算等等。
理论和算法是为应用服务的，建筑行业，作为项目管理应用的主要应用行业之一，也针对项目管理提出更多更高的要求。为了帮助建筑行业的施工单位更好的控制施工过程，增强项目的可预见性，避免不必要的损失，虚拟施工技术因此而产生。
鉴于以上的考虑，本文以多模式资源约束的项目调度及其应用研究为对象，以多目标遗传算法为手段展开研究。目前主要存在如下的问题：
（1）尽管智能优化算法能同时处理一组解，以及很好的逼近非凸或不连续的最优前端面，然而对设计的参数进行动态调整和自适应改变依然是值得探讨的方向。
（2）多目标遗传算法的求解质量和求解效率还有待于提高。基于Pareto的多目标遗传算法在每一次迭代时都要构造Pareto最优解集，有必要寻找构造Pareto最优解集的最少时间复杂度。
（3）项目时间成本质量历来是项目管理的三大控制点，决定着项目的成败。然而更多的研究却集中在时间费用的优化，而缺少对质量因素的考虑。或者即使考虑了质量，而往往限定其中的一个或两个因素作为常量来考虑。因此建立合理的项目时间成本质量的多目标优化模型并加以解决是当务之急。
（4）在实际的项目调度过程中更多的存在着不确定性因素，这些不确定因素使得项目调度在本质上成为一个动态的过程。对不确定性因素的研究更加贴近实际，如何针对考虑不确定性因素的问题建立随机或者模糊网络模型并采用合适的机制去解决也是我们亟待思考的问题。
因此，本文结合这些问题，具体做了如下方面的研究工作：
（1）在探讨组合优化问题、计算复杂性等相关原理的基础上，总结和比较了各种优化算法的优缺点，指出元启发式方法即智能优化算法是目前研究的趋势。在与实际应用相关的难解组合优化问题方面，智能优化方法使得在合理的时间范围内明显提高了找到高质量解的能力，尤其对于大型的或了解甚少的问题而言，智能优化算法的作用更加显著。
（2）结合组合优化理论，介绍了资源约束的项目调度的一般模型、规范分类以及基于不同求解方法的已有文献，指明多模式资源约束的项目调度是可行的研究方向。
（3）研究了多目标遗传算法及其基本概念：非支配集构造、种群保留机制以及遗传操作等等。对多目标遗传算法作了合理的改进，对分布系数采用多项式逼近获得交叉系数的概率分布，找出了交叉系数合适的改进方案，同时证明了种群大小的下限。
（4）分别以时间、成本和质量进行建模分析，指出了各个因素之间的相互关系及其研究思路，同时形成时间-成本-质量的多目标权衡数学模型。并用多目标遗传算法加以求解及其仿真实现，最后比较了不同方法的优劣，从收敛性和分布性方面证明了改进的方法的有效性。同时给出了基于不同偏好的建筑项目调度问题的决策者决策过程。
（5）对项目调度中涉及到的随机变量的分布、数字特征进行分析，指出了PERT网络分析法往往低估了项目完成时间，给项目的完成带来不必要的损失。研究了随机多目标决策问题的数学模型及其等价问题的处理，提出了随机条件下多目标多模式资源约束的项目调度建模、算法分析，并给出了仿真结果。
（6）基于虚拟建筑机理的分析，对虚拟原型（CVP）系统涉及的集成平台、开发环境、设计步骤等方面进行研究，说明项目调度及其优化理论在虚拟建筑中的应用，并以一个实例进行应用分析。
论文的主要创新点：
（1）修正了NSGA-Ⅱ算法中原有的固定交叉系数，使得该交叉系数能够动态调整和自适应改变，并给出算法中种群大小的下限值并加以证明。
（2）将多目标遗传算法应用到考虑时间成本质量权衡问题的多目标多模式资源约束的项目调度问题中，给出了优化机制分析，并从收敛性和分布性方面证明方法的有效性。
（3）基于活动时间的随机变化引起活动成本和质量的动态改变，结合机会约束规划设计了概率转换机制和随机数模拟，以提高多目标遗传算法求解多目标多模式资源约束的随机项目调度问题的能力。
总体来说，就算法而言，本文以多目标遗传算法的NSGA-Ⅱ算法为着眼点，提出了两种修改策略，并通过实验证明了算法的有效性，充实了多目标优化算法理论。就项目调度而言，对项目的绩效评估综合考虑了质量因素，丰富了常规的时间费用分析理论；对项目调度的研究考虑了随机因素，弥补了不确定性网络分析理论。
Managing projects dates back at least4500years. From building the pyramids inEgypt and the Maya temples in South America with the simplest tools, CPM first usedby America DuPont Company and PRET in Polaris Missile Project by America navy,till now, a large number of project management problems on models, algorithms andsystems including some software have been studied to help managers plan, schedule,supervise and control projects.
Project scheduling problems, as one of the most important research areas inproject management, dating back to the1960s, have received wide attention from allkinds of specialists and scholars. With the application of modern technology,automation production system makes production more convenient and efficient butmakes project scheduling more difficult almost beyond the wisdom of human.
Most project scheduling problems belong to a kind of combinational optimization.The computational complexity theory indicates that many combinatorial optimizationsare NP-hard. Traditional optimization methods in operations research are difficult tosolve these NP-hard with the polynomial computing time needed. Thus theapproximate algorithms, also called heuristic methods, are required.
At present, solving the most complex project scheduling problems mainly focuson the study of the meta-heuristic methods, that is, intelligence optimization algorithm,including simulated annealing, tabu search, particle swarm optimization, ant colonyoptimization and evolutionary computation etc.. Intelligence optimization algorithmmimics nature’s phenomenon or procedure to drive its search towards an optimalsolution, appropriate to much more complex nonlinear problems with thecharacteristics of parallel, self-organization, self-learning and self-adaption.
Anyway, the researched theories and algorithms serve for their application.Construction industry, as one of the main application area in project management, demands more aiming to itself. In order to help construction enterprises better controlconstructing and improve forecasting so as to avoid unnecessary loss, constructionvirtual prototype technology comes into being.
In view of the above mentioned factors, this dissertation tries its best to takemulti-mode resource-constrained project scheduling and its application as researchobjects, and multi-objective genetic algorithm as instrument. The main problems in theresearch are as follows.
(1) Though intelligence optimization algorithm can deal with a population ofsolutions, and better approach non-convex or discontinuous Pareto-optimal front, thedynamic adjustment and self-adaptive modification of the designed parameters are stillunder discussion.
(2) Solution quality and solving efficiency should be improved further. Based onPareto theory the multi-objective genetic algorithm must construct Pareto optimal setin every run, thus it is necessary to find the least time complexity to design the Paretooptimal set.
(3) Time, cost and quality are the important factors in managing projects and canbring the success or failure of the projects. More literatures focus on time-cost tradeoff,and few quality of a project is considered. Though in some studies project schedulingwith time, cost and quality considerations has been mentioned, researchers did not givean all-around design among the three but bounding one or two variables as constant. Itis urgent to build a suitable time-cost-quality tradeoff model and resolve it.
(4) Quite a number of uncertain factors exist in the project scheduling problems,which makes them form a dynamic procedure. The research on uncertain is morepractical. It is worth of our thinking to build stochastic or fuzzy network modelsaccording to uncertain state and resolve it.
So, with the problems above considered, the main research work and contributionof this dissertation includes:
(1) Based on the research on combinatorial optimization and computational complexity, the dissertation compares the positive and negative outcomes from allkinds of different optimization methods, and concludes that the intelligenceoptimization methods are becoming the research tend at present. In resolvingcombinatorial optimization which is quite hard to deal with, the intelligenceoptimization methods improve the ability to find high quality of solutions in thesuitable range of time. Especially, the intelligence optimization methods take muchmore effects on large-scale problems and problems seldom unknown by people.
(2) Integrating the theory of combinatorial optimization, the dissertationintroduces a general model of resource-constrained project scheduling problems andtheir classification, gives literatures review based on different methods. It alsoindicates that researching the multi-mode resource-constrained project schedulingproblem is feasible.
(3) The dissertation discusses the multi-objective genetic algorithm, including theconstruction of non-dominated sorting, the mechanics of crowding distance assignmentand genetic operations etc.. At the same time, we give a suggestion of self-adaptiveprocedure of multi-objective genetic algorithm by modifying the distribution index andthe population size.
(4) Based on analysis of minimizing time, minimizing cost and maximizingquality individually, the interrelation among them and the research scheme are pointedout before the multi-objective models on time-cost-quality tradeoff are formed. Thenthe multi-objective genetic algorithm is applied to the kind of models. Finally theresults gained by simulation show the validity of the applied method. Meanwhile, adecision-making scheme is given based on preference of time, cost and qualityindividually for scheduling a construction project.
(5) The dissertation gives a description on models and algorithms of projectscheduling problems under stochastic state, including distribution function and numbercharacteristics of the random variables. It also shows that PERT often undervalues theproject duration which will necessarily bring the risk for managing project. Then, the stochastic multi-objective problems and its mathematics models, resolving methods arestudied. Finally, simulations and conclusions on multi-objective multi-moderesource-constrained scheduling problems are made from the study.
(6) Based on the analysis of construction mechanism, the dissertation gives aconsideration on how to integrate the relative scheduling theory into constructionvirtual prototype system from the aspects of integrated platform, developmentenvironment, design steps, and a practical case is made and analyzed in the last.
The main innovative researches are included as follows:
(1) This dissertation modifies the fixed crossover distribution index, whichmakes the index be capable of self-adaptive adjustment. At the same time also provesthe lower limit of the population size.
(2) This dissertation applies multi-objective genetic algorithm to time-cost-qualitytradeoff problems which belong to the type of multi-mode resource-constrained projectscheduling problems, analyzes the optimization mechanism, and proves the validity ofthe improved algorithm from the convergence and distribution.
(3) Based on the dynamic change of activity cost and quality with the stochasticactivity duration, the dissertation designs a mechanism of probability conversion andsimulation of random number with integrating the chance-constrained programming,so as to improve the ability of resolving the multi-objective multi-moderesource-constrained stochastic project scheduling problems.
As a whole, in case of algorithms, the dissertation starts from the mathematicalmodel of multi-objective genetic algorithms, puts forward two kinds of modificationstrategy. As far as project scheduling is concerned, the dissertation considered thefactor of quality and stochastic environment. The experiments testify theireffectiveness and practicality. The contribution of the dissertation enhances thetheoretical basement of multi-objective optimization theory, enriches the time-costmodels and uncertain network models.
项目调度问题，是项目管理的一个重要研究领域，自20世纪60年代被提出之后，就引起了各行各界专家和学者们的广泛关注。随着现代化技术广泛应用于生产，具有较高自动化水平的生产系统，在使生产过程更加合理、高效运行的同时，也使得项目调度问题变得更加困难，其复杂性往往超出人脑的决策能力。
大多数项目调度问题是一类组合优化问题，计算复杂性理论已经证明大多数组合优化问题是NP-hard问题。传统的运筹学求解方法往往无法在多项式时间范围内寻求到这些NP-hard问题的最优解，随之人们开始尝试在多项式时间范围内求解的近似算法即启发式方法。
目前对复杂的项目调度问题的求解算法主要集中在对元启发式方法即智能优化方法的研究。智能优化算法是模拟某一自然现象或过程而建立起来的具有高度并行、自组织、自学习与自适应特征的适于复杂的高度非线性问题求解的算法。这类算法包括模拟煺火、禁忌搜索、粒子群算法、蚁群算法和进化计算等等。
理论和算法是为应用服务的，建筑行业，作为项目管理应用的主要应用行业之一，也针对项目管理提出更多更高的要求。为了帮助建筑行业的施工单位更好的控制施工过程，增强项目的可预见性，避免不必要的损失，虚拟施工技术因此而产生。
鉴于以上的考虑，本文以多模式资源约束的项目调度及其应用研究为对象，以多目标遗传算法为手段展开研究。目前主要存在如下的问题：
（1）尽管智能优化算法能同时处理一组解，以及很好的逼近非凸或不连续的最优前端面，然而对设计的参数进行动态调整和自适应改变依然是值得探讨的方向。
（2）多目标遗传算法的求解质量和求解效率还有待于提高。基于Pareto的多目标遗传算法在每一次迭代时都要构造Pareto最优解集，有必要寻找构造Pareto最优解集的最少时间复杂度。
（3）项目时间成本质量历来是项目管理的三大控制点，决定着项目的成败。然而更多的研究却集中在时间费用的优化，而缺少对质量因素的考虑。或者即使考虑了质量，而往往限定其中的一个或两个因素作为常量来考虑。因此建立合理的项目时间成本质量的多目标优化模型并加以解决是当务之急。
（4）在实际的项目调度过程中更多的存在着不确定性因素，这些不确定因素使得项目调度在本质上成为一个动态的过程。对不确定性因素的研究更加贴近实际，如何针对考虑不确定性因素的问题建立随机或者模糊网络模型并采用合适的机制去解决也是我们亟待思考的问题。
因此，本文结合这些问题，具体做了如下方面的研究工作：
（1）在探讨组合优化问题、计算复杂性等相关原理的基础上，总结和比较了各种优化算法的优缺点，指出元启发式方法即智能优化算法是目前研究的趋势。在与实际应用相关的难解组合优化问题方面，智能优化方法使得在合理的时间范围内明显提高了找到高质量解的能力，尤其对于大型的或了解甚少的问题而言，智能优化算法的作用更加显著。
（2）结合组合优化理论，介绍了资源约束的项目调度的一般模型、规范分类以及基于不同求解方法的已有文献，指明多模式资源约束的项目调度是可行的研究方向。
（3）研究了多目标遗传算法及其基本概念：非支配集构造、种群保留机制以及遗传操作等等。对多目标遗传算法作了合理的改进，对分布系数采用多项式逼近获得交叉系数的概率分布，找出了交叉系数合适的改进方案，同时证明了种群大小的下限。
（4）分别以时间、成本和质量进行建模分析，指出了各个因素之间的相互关系及其研究思路，同时形成时间-成本-质量的多目标权衡数学模型。并用多目标遗传算法加以求解及其仿真实现，最后比较了不同方法的优劣，从收敛性和分布性方面证明了改进的方法的有效性。同时给出了基于不同偏好的建筑项目调度问题的决策者决策过程。
（5）对项目调度中涉及到的随机变量的分布、数字特征进行分析，指出了PERT网络分析法往往低估了项目完成时间，给项目的完成带来不必要的损失。研究了随机多目标决策问题的数学模型及其等价问题的处理，提出了随机条件下多目标多模式资源约束的项目调度建模、算法分析，并给出了仿真结果。
（6）基于虚拟建筑机理的分析，对虚拟原型（CVP）系统涉及的集成平台、开发环境、设计步骤等方面进行研究，说明项目调度及其优化理论在虚拟建筑中的应用，并以一个实例进行应用分析。
论文的主要创新点：
（1）修正了NSGA-Ⅱ算法中原有的固定交叉系数，使得该交叉系数能够动态调整和自适应改变，并给出算法中种群大小的下限值并加以证明。
（2）将多目标遗传算法应用到考虑时间成本质量权衡问题的多目标多模式资源约束的项目调度问题中，给出了优化机制分析，并从收敛性和分布性方面证明方法的有效性。
（3）基于活动时间的随机变化引起活动成本和质量的动态改变，结合机会约束规划设计了概率转换机制和随机数模拟，以提高多目标遗传算法求解多目标多模式资源约束的随机项目调度问题的能力。
总体来说，就算法而言，本文以多目标遗传算法的NSGA-Ⅱ算法为着眼点，提出了两种修改策略，并通过实验证明了算法的有效性，充实了多目标优化算法理论。就项目调度而言，对项目的绩效评估综合考虑了质量因素，丰富了常规的时间费用分析理论；对项目调度的研究考虑了随机因素，弥补了不确定性网络分析理论。
Managing projects dates back at least4500years. From building the pyramids inEgypt and the Maya temples in South America with the simplest tools, CPM first usedby America DuPont Company and PRET in Polaris Missile Project by America navy,till now, a large number of project management problems on models, algorithms andsystems including some software have been studied to help managers plan, schedule,supervise and control projects.
Project scheduling problems, as one of the most important research areas inproject management, dating back to the1960s, have received wide attention from allkinds of specialists and scholars. With the application of modern technology,automation production system makes production more convenient and efficient butmakes project scheduling more difficult almost beyond the wisdom of human.
Most project scheduling problems belong to a kind of combinational optimization.The computational complexity theory indicates that many combinatorial optimizationsare NP-hard. Traditional optimization methods in operations research are difficult tosolve these NP-hard with the polynomial computing time needed. Thus theapproximate algorithms, also called heuristic methods, are required.
At present, solving the most complex project scheduling problems mainly focuson the study of the meta-heuristic methods, that is, intelligence optimization algorithm,including simulated annealing, tabu search, particle swarm optimization, ant colonyoptimization and evolutionary computation etc.. Intelligence optimization algorithmmimics nature’s phenomenon or procedure to drive its search towards an optimalsolution, appropriate to much more complex nonlinear problems with thecharacteristics of parallel, self-organization, self-learning and self-adaption.
Anyway, the researched theories and algorithms serve for their application.Construction industry, as one of the main application area in project management, demands more aiming to itself. In order to help construction enterprises better controlconstructing and improve forecasting so as to avoid unnecessary loss, constructionvirtual prototype technology comes into being.
In view of the above mentioned factors, this dissertation tries its best to takemulti-mode resource-constrained project scheduling and its application as researchobjects, and multi-objective genetic algorithm as instrument. The main problems in theresearch are as follows.
(1) Though intelligence optimization algorithm can deal with a population ofsolutions, and better approach non-convex or discontinuous Pareto-optimal front, thedynamic adjustment and self-adaptive modification of the designed parameters are stillunder discussion.
(2) Solution quality and solving efficiency should be improved further. Based onPareto theory the multi-objective genetic algorithm must construct Pareto optimal setin every run, thus it is necessary to find the least time complexity to design the Paretooptimal set.
(3) Time, cost and quality are the important factors in managing projects and canbring the success or failure of the projects. More literatures focus on time-cost tradeoff,and few quality of a project is considered. Though in some studies project schedulingwith time, cost and quality considerations has been mentioned, researchers did not givean all-around design among the three but bounding one or two variables as constant. Itis urgent to build a suitable time-cost-quality tradeoff model and resolve it.
(4) Quite a number of uncertain factors exist in the project scheduling problems,which makes them form a dynamic procedure. The research on uncertain is morepractical. It is worth of our thinking to build stochastic or fuzzy network modelsaccording to uncertain state and resolve it.
So, with the problems above considered, the main research work and contributionof this dissertation includes:
(1) Based on the research on combinatorial optimization and computational complexity, the dissertation compares the positive and negative outcomes from allkinds of different optimization methods, and concludes that the intelligenceoptimization methods are becoming the research tend at present. In resolvingcombinatorial optimization which is quite hard to deal with, the intelligenceoptimization methods improve the ability to find high quality of solutions in thesuitable range of time. Especially, the intelligence optimization methods take muchmore effects on large-scale problems and problems seldom unknown by people.
(2) Integrating the theory of combinatorial optimization, the dissertationintroduces a general model of resource-constrained project scheduling problems andtheir classification, gives literatures review based on different methods. It alsoindicates that researching the multi-mode resource-constrained project schedulingproblem is feasible.
(3) The dissertation discusses the multi-objective genetic algorithm, including theconstruction of non-dominated sorting, the mechanics of crowding distance assignmentand genetic operations etc.. At the same time, we give a suggestion of self-adaptiveprocedure of multi-objective genetic algorithm by modifying the distribution index andthe population size.
(4) Based on analysis of minimizing time, minimizing cost and maximizingquality individually, the interrelation among them and the research scheme are pointedout before the multi-objective models on time-cost-quality tradeoff are formed. Thenthe multi-objective genetic algorithm is applied to the kind of models. Finally theresults gained by simulation show the validity of the applied method. Meanwhile, adecision-making scheme is given based on preference of time, cost and qualityindividually for scheduling a construction project.
(5) The dissertation gives a description on models and algorithms of projectscheduling problems under stochastic state, including distribution function and numbercharacteristics of the random variables. It also shows that PERT often undervalues theproject duration which will necessarily bring the risk for managing project. Then, the stochastic multi-objective problems and its mathematics models, resolving methods arestudied. Finally, simulations and conclusions on multi-objective multi-moderesource-constrained scheduling problems are made from the study.
(6) Based on the analysis of construction mechanism, the dissertation gives aconsideration on how to integrate the relative scheduling theory into constructionvirtual prototype system from the aspects of integrated platform, developmentenvironment, design steps, and a practical case is made and analyzed in the last.
The main innovative researches are included as follows:
(1) This dissertation modifies the fixed crossover distribution index, whichmakes the index be capable of self-adaptive adjustment. At the same time also provesthe lower limit of the population size.
(2) This dissertation applies multi-objective genetic algorithm to time-cost-qualitytradeoff problems which belong to the type of multi-mode resource-constrained projectscheduling problems, analyzes the optimization mechanism, and proves the validity ofthe improved algorithm from the convergence and distribution.
(3) Based on the dynamic change of activity cost and quality with the stochasticactivity duration, the dissertation designs a mechanism of probability conversion andsimulation of random number with integrating the chance-constrained programming,so as to improve the ability of resolving the multi-objective multi-moderesource-constrained stochastic project scheduling problems.
As a whole, in case of algorithms, the dissertation starts from the mathematicalmodel of multi-objective genetic algorithms, puts forward two kinds of modificationstrategy. As far as project scheduling is concerned, the dissertation considered thefactor of quality and stochastic environment. The experiments testify theireffectiveness and practicality. The contribution of the dissertation enhances thetheoretical basement of multi-objective optimization theory, enriches the time-costmodels and uncertain network models.
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