3_建模_精简 (2).ppt

1、Chapter 2,Overview of the Operations Research Modeling Approach,The usual phases of an OR study is the following:,Define the problem of interest and gather relevant data. Formulate a mathematical model to represent the problem. Develop a computer-based procedure for deriving solutions to the problem

2、 from the model. Test the model and refine it as needed. Prepare for the ongoing application of the model as prescribed by management. Implement,2.1 Defining the Problem and Gathering Data,Most practical problems are described in a vague, imprecise way.,The appropriate objectives constraints interre

3、lationships of the organization possible alternative courses of action time limits .,Problem definition,How to ascertain appropriate objectives,Identify the members of management who actually will be making the decisions and probe into this individuals thinking regarding the pertinent objectives. Co

4、ncern with the entire organization rather than that of only certain of its components,For Example:,For profit-making organizations: the objective may be:,In actual practice, a number of U.S. Corporations tends to adopt the goal of satisfied profits, combine with other objectives.,profit maximization

5、.,Gathering Relevant Data about the Problem,Because: to gain an accurate understanding of the problem to provide the needed input for the mathematical model,To install a computer-based management information system to collect the necessary data on an ongoing basis and in the needed form. To enlist t

6、he assistance of individuals in the organization to track down all the vital data. To spend considerable time to improve the precision of the data.,Discussion 1:,An OR study done for the San Francisco Police Department, resulted in the development of a computerized system for optimally scheduling an

7、d deploying police patrol officers. The new system provided annual savings of $11 million, an annual $3 million increase in traffic citation revenues, and a 20 percent improvement in response times.,Objects,In assessing the appropriate objectives for this study, three fundamental objectives were ide

8、ntified: 1. Maintain a high level of citizen safety. 2. Maintain a high level of officer morale. 3. Minimize the cost of operations.,Solution:,To satisfy the first objective, the police department and city government jointly established a desired level of protection. The mathematical model then impo

9、sed the requirement that this level of protection be achieved. Similarly, the model imposed the requirement of balancing the workload equitably among officers in order to work toward the second objective. Finally, the third objective was incorporated by adopting the long-term goal of minimizing the

10、number of officers needed to meet the first two objectives.,Discussion 2:,The Citgo Petroleum Corporation, optimized both refinery operations and the supply, distribution, and marketing of its products, thereby achieving a profit improvement of approximately $70 million per year. Data collection pla

11、yed a key role in this study.,A state of the art management database system(技术发展水平管理数据库系统) was developed and installed on a mainframe computer. Operations personnel were helped to input the data and then the data from the PCs were uploaded to the mainframe computer. Initially, a paper log(记事本) of er

12、ror messages generated 1 inch thick! Eventually, the number of error and warning messages (indicating bad or questionable numbers) was reduced to less than 10 for each new run.,2.2 Formulating A Mathematical Model,Why?,Represent the essence of the problem in mathematical model Reformulate the proble

13、m in a form that is convenient for analysis .,What is model ?,Model of the atoms Models of genetic structure Model of airplane/portraits/global .,形象模型：如地球仪、沙盘、风洞 模拟模型：建港口，模拟船只到达。学生模拟企业管理系统运行。 数学模型：用符号或数学工具描述现实系统。V=F（xi,yj,uk) G(xi,yj,uk)0,Models are invaluable for abstracting the essence of the subj

14、ect of inquiry, showing interrelationships, and facilitating analysis. Mathematical models are also idealized representations, but they are expressed in terms of mathematical symbols and expressions. For example, E=mc2 are familiar examples. Similarly, the mathematical model of a business problem is

15、 the system of equations and related mathematical expressions that describe the essence of the problem.,2.2 Constructing a Mathematical model,How to construct a model?,Expressed in terms of mathematical symbols and expressions that describe the essence of the problem. Some important terms: Decision

16、variables （决策变量） Objective function （目标函数） Constrains （约束条件）,2.2 Constructing a Mathematical model,The Two Mines Company own two different mines that produce an ore which, after being crushed, is graded into three classes: high, medium and low-grade. The company has contracted to provide a smelting

17、plant with 12 tons of high-grade, 8 tons of medium-grade and 24 tons of low-grade ore per week. The two mines have different operating characteristics as detailed below.,One example,2.2 Constructing a Mathematical model,Mine Cost per day ($1000) Production (tons/day) High Medium Low X 180 6 3 4 Y 16

18、0 1 1 6 How many days per week should each mine be operated to fulfil the smelting plant contract?,2.2 Constructing a Mathematical model,Objective function,Minimize Z = 180 x + 160y subject to: 6x + y = 12 3x + y = 8 4x + 6y = 24 x = 0,Constrains,Decision variables,2.2 For Example:,An OR study done

19、for Monsanto Corp(孟山都公司). was concerned with optimizing production operations in Monsantos chemical plants to minimize the cost of meeting the target for the amount of a certain chemical product to be produced in a given month. The decision to be made are the dial setting for each of the catalytic r

20、eactors used to produce this products, where the setting determines both the amount produced and the cost of operating the reactor. How to create the mathematical model?,2.2 For Example: Formulating the model,For profit-making organizations: the objective may be:,Cost Minimization,Minimize,Subject t

21、o,For i=1,2,r,C ij = cost of reactor i at setting j P ij = production of reactor i at setting j T= production target R= number of reactors S = number of settings,Where,If reactor i is operated at setting j,Otherwise,ANOTHER EXAMPLES,Continental Airlines developed a detailed mathematical model for re

22、assigning crews to flights as soon as emergencies arise. In the first year of use, this model was applied four times to recover from major schedule disruptions. This led to savings of approximately $40 million. The Netherlands government agency responsible for water control and public works, commiss

23、ioned a major OR study to guide the development of a new national water management policy. The new policy saved hundreds of millions of dollars in investment expenditures and reduced agricultural damage by about $15 million per year.,Production Planning Problems（生产计划问题）,试列出下述产品规划问题的线性规划模型：某工厂生产A、B、C

24、三种产品，每吨利润分别为2000元、3000元、1000元；生产单位产品所需的工时及原材料如表所示。若供应的原材料每天不超过3吨，所能利用的劳动力日总工时是固定的，问如何制定日生产计划，使三种产品总利润最大？,解：第一步确定决策变量,为产品A的日产量,为产品B的日产量,为产品C的日产量,第二步明确约束条件,劳动力的约束条件为：,原材料的约束条件为：,第三步明确目标,总利润为：,产品规划问题的线性规划模型为：,How to deriving the solution ?,You might think it is very difficult, but it is very simple wit

25、h the help of software packages. Many algorithms help us to do it We will discuss it later in details What is optimizing? What is satisficing? ,2.3 Deriving solution from the model,What is satisficing ? Maybe you cannot find the word in dictionary? - satisficing = satisfactory + optimizing = good en

26、ough (Simon ) Who is Simon? Nobel Laureate and Eminent management scientist in economics. Herbert A Simon. What is the distinction of optimizing and satisficing? -Reflect the difference between theory and the realities . According to the Samuel Eilon : - Optimizing is the science of the ultimate; sa

27、tisficing is the art of the feasible”(优化是最终的科学,满意是可行的艺术),2.3 Deriving solution from the model,Herbert A Simon,1936年从芝加哥大学毕业，取得政治学学土学位。之后，从事了几年编辑和行政工作。1943年获得芝加哥大学政治学博士学位。 研究领域涉及认知心理学、计算机科学、公共行政、经济学、管理学和科学哲学等多个方向。 西蒙学识广博，是现今很多重要学术领域的创始人之一，如人工智能、信息处理、决策制定、问题解决、注意力经济、组织行为学、复杂系统等。他创造了术语“有限理性”（Bounded ra

28、tionality）和“满意度”（satisficing），也是第一个分析复杂性架构（architecture of complexity）的人。 西蒙因其贡献和影响在他晚年获得了很多顶级荣誉，如：1975年的图灵奖、1978年的诺贝尔经济奖、1986年的美国国家科学奖章和1993年美国心理协会的终身成就奖。 作为科学家，他涉足的领域之多，成果之丰，影响之深远，令人叹为观止。,西蒙（19162001 ）主要著作 行政管理行为（1945）； 人类模型（1957）； 组织（1958）； 管理决策的新科学（1960）； 发明的模型（1977）； 思想模型（1979年）。,What is post o

29、ptimality analysis（后最优性分析 ）? - It is a very important tool of OR research. - Because the optimal solution for the original model may far from ideal for the really problem, additional analysis is needed. What kind of methods should be used? - What-if analysis widely used for this kind problem.,2.3 De

30、riving solution from the model,What is sensitivity analysis?（灵敏度分析） Post optimality involves conducting sensitivity analysis to determine which parameters of the model are most critical in determining the solution.,2.3 Deriving solution from the model,How to identify the sensitivity parameters? - in

31、 order to avoid distorting the output of the model we should assign the value of sensitivity parameters, or at least its range of likely values. We need to estimate the values.,2.3 Deriving solution from the model,2.4 Testing the model,Why?,Developing a large mathematical model, which is analogous（类

32、似） in some ways to developing a large computer program, it inevitably contains many bugs, so.,Try to find the bugs before we really use it for the work,How to test the model?,Model validation Use the retrospective test Documentation ,2.4 Testing the model,2.5 Prepare to apply the model,What shall we

33、 do in this phase? What happens after the testing phase has been completed and an acceptable model has been developed? -If the model is to be used repeatedly, the next step is to install a well documented system for applying the model as prescribed by management. This system will include the model,

34、solution procedure (including post optimality analysis), and operating procedures for implementation. - Then, even as personnel changes, the system can be called on at regular intervals to provide a specific numerical solution.,2.5 Prepare to apply the model,What is the “documented System”? -This sy

35、stem usually is computer-based. In fact, a considerable number of compute programs often need to be used and integrated. Databases and management information systems may provide up-to-date input for the model each time it is used, in which case interface programs are needed. In other cases, an inter

36、active computer-based system called a decision support system is installed to help managers use data and models to support (rather than replace) their decision making as needed. Another program may generate managerial reports (in the language of management) that interpret the output of the model and

37、 its implications for application,2.6 IMPLEMENTATION,What shall we do in the last phase: IMPLEMENTATION? This phase is a critical one because it is here, and only here, that the benefits of the study are reaped. Therefore, it is important for the OR team to participate in launching this phase, both

38、to make sure that model solutions are accurately translated to an operating procedure and to rectify any flaws in the solutions that are then uncovered.,2.6 IMPLEMENTATION,How to make it successful? The success of the implementation phase depends a great deal upon the support of both top management

39、and operating management. The OR team is much more likely to gain this support if it has kept management well informed and encouraged managements active guidance throughout the course of the study, Good communications help to ensure that the study accomplishes what management wanted and so deserves

40、implementation. They also give management a greater sense of ownership of the study, which encourages their support for implementation.,2.6 IMPLEMENTATION,What are the steps of implementation? The implementation phase involves several steps. First, the OR team gives operating management a careful ex

41、planation of the new system to be adopted and how it relates to operating realities.,Next, these two parties share the responsibility for developing the procedures required to put this system into operation. Operating management then sees that a detailed indoctrination is given to the personnel invo

42、lved, and the new course of action is initiated. If successful, the new system may be used for years to come. With this in mind, the OR team monitors the initial experience with the course of action taken and seeks to identify any modifications that should be made in the future.,2.6 IMPLEMENTATION,T

43、he last step is : obtain feedback. Throughout the entire period during which the new system is being used, it is important to continue to obtain feedback on how well the system is working and whether the assumptions of the model continue to be satisfied. When significant deviations from the original assumptions occur, the model should be revisited to determine if any modifications should be made in the sys