Math 157

| Course Design GuideCollege of Natural SciencesMTH/157 Version 3Math for Elementary Teachers II| Copyright  © 2011, 2009, 2007 by University of Phoenix. All rights reserved. Course Description This course is the second in a two-part series designed for K–8 preservice teachers to address the conceptual framework for mathematics taught in elementary school. The focus of Part Two will be on measurement, geometry, probability, and data analysis. The relationship of the course concepts to the National Council of Teachers of Mathematics Standards for K–8 instruction is also addressed. PoliciesFaculty and students/learners will be held responsible for understanding and adhering to all policies contained within the following two documents: University policies: You must be logged into the student website to view this document. Instructor policies: This document is posted in the Course Materials forum. University policies are subject to change. Be sure to read the policies at the beginning of each class. Policies may be slightly different depending on the modality in which you attend class. If you have recently changed modalities, read the policies governing your current class modality. Course MaterialsBillstein, R. , Libeskind, S. , & Lott, J. W. (2010). A problem solving approach to mathematics for elementary school teachers (10th ed. ). Boston, MA: Addison-Wesley. All electronic materials are available on the student website. Week One: Data Analysis| | Details| Due| Points| Objectives| 1. 1 Use appropriate statistical methods to analyze data. 1. 2 Develop predictions based on data. | | | Course Preparation| Read the course description and objectives. Read the instructor’s biography and post your own. | | | Reading| Read Ch. 9 of A Problem Solving Approach to Mathematics for Elementary School Teachers. | | Reading| Read Ch. 10 of A Problem Solving Approach to Mathematics for Elementary School Teachers. | | | Reading| Read the Associate Level Mat erial: Using MyMathLab ®. | | | Reading| Read this week’s Electronic Reserve Readings. | | | Participation| Participate in class discussion. | | 10| Discussion Questions| Respond to weekly discussion questions. | | 10| IndividualMyMathLab ® Orientation| Complete the Orientation Assignment located in MyMathLab ®. | | 45| Week Two: Probability| | Details| Due| Points| Objectives| 1 2. 3 Apply basic concepts of probability. | | | Reading| Review Ch. of A Problem Solving Approach to Mathematics for Elementary School Teachers. | | | Reading| Review Ch. 10 of A Problem Solving Approach to Mathematics for Elementary School Teachers. | | | Nongraded Activities and PreparationSpinner Activity| View the Spinner Activity Animation located on the student website. | | | IndividualText Problems 1| Complete Text Problems 1 located in MyMathLab ®. | | 70| IndividualProbability Games| Resources: http://www. betweenwaters. comAccess to the Probability Games on the Between Waters websit e by using the following directions:Go to http://www. betweenwaters. omScroll down and click on Probability Games. Locate the Coin Flip and Dice Roll games. Click Play under each activity to play the games. Play both the Coin Flip and Dice Roll games. After you have played the games, write a 350- to 700-word paper describing your experience. Include the following in your paper:What did you learn about how probabilities are determined? What method might be the most difficult concept for children to learn and why? Post your paper as an attachment. | | 100| ————————————————- ————————————————- Week Three: Introduction to Geometry| Details| Due| Points| Objectives| 2 3. 4 Apply characteristics and properties of two- and three-dimensional geometric shapes in problem so lving. 3. 5 Identify geometric figures and shapes based on mathematical arguments. 3. 6 Use visualization, spatial reasoning, and geometric modeling to solve problems. | | | Reading| Read Ch. 11 of A Problem Solving Approach to Mathematics for Elementary School Teachers. | | | Reading| Read this week’s Electronic Reserve Readings. | | | Participation| Participate in class discussion. | | 10| Discussion Questions| Respond to weekly discussion questions. | 10| ————————————————- ————————————————- Week Four: Introduction to Geometry, Continued| | Details| Due| Points| Objectives| 3 4. 7 Apply characteristics and properties of two- and three-dimensional geometric shapes in problem solving. 4. 8 Identify geometric figures and shapes based on mathematical argumen ts. 4. 9 Use visualization, spatial reasoning, and geometric modeling to solve problems. | | | Reading| Review Ch. 11 of A Problem Solving Approach to Mathematics for Elementary School Teachers. | | IndividualText Problems 2| Complete Text Problems 2 located in MyMathLab ®. | | 35| IndividualGeometry Manipulatives| Prepare an activity involving a geometric manipulative designed to teach a geometric concept to an elementary school student. You may create your own activity or modify an existing activity; if you are modifying an existing activity, however, ensure your sources are properly cited. Create a handout including the following information:A detailed description of your activity, which must include the application of the characteristics and properties of the hosen geometric shapeInstructions for conducting the activityMaterials neededNational Council of Teacher of Mathematics standards addressed| | 100| —————————à ¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€Ã¢â‚¬â€- ————————————————- Week Five: Applications of Geometry| | Details| Due| Points| Objectives| 4 5. 10 Specify locations using coordinate geometry. 5. 11 Describe spatial relationships using coordinate geometry. 5. 12 Use symmetry to analyze mathematical situations. | | | Reading| Read Ch. 12 of A Problem Solving Approach to Mathematics for Elementary School Teachers. | | | Reading| Read Ch. 4 of A Problem Solving Approach to Mathematics for Elementary School Teachers. | | | Reading| Read this week’s Electronic Reserve Readings. | | | Participation| Participate in class discussion. | | 10| Discussion Questions| Respond to weekly discussion questions. | | 10| Nongraded Activities and PreparationAnimations| View the following animations located on the student website:Grapher AnimationTransformations AnimationLady Bu g Transformation Animation| | | ————————————————- ————————————————- Week Six: Applications of Geometry, Continued| | Details| Due| Points|Objectives| 5 6. 13 Specify locations using coordinate geometry. 6. 14 Describe spatial relationships using coordinate geometry. 6. 15 Use symmetry to analyze mathematical situations. | | | Reading| Review Ch. 12 of A Problem Solving Approach to Mathematics for Elementary School Teachers. | | | Reading| Review Ch. 14 of A Problem Solving Approach to Mathematics for Elementary School Teachers. | | | IndividualText Problems 3| Complete Text Problems 3 located in MyMathLab ®. | | 85| IndividualTessellation Patterns| Resource: Associate Level Material: Appendix ACreate a tessellation pattern using theMicrosoft ® Paint program, the GeoGebra website, a Microsoft ® PowerPoint ® presentation, or other means available to you, or you may draw something by hand. Ask your instructor for assistance if needed. Use color and shading to create a visually-pleasing tessellation. Write a 350- to 700-word paper including the following:An explanation of why you chose the tessellated figureThe type of transformation used and whyThe actual tessellation or a picture of the created tessellation * Format your paper consistent with APA guidelines. | | 100| ————————————————- ————————————————-Week Seven: Applications of Measurement| | Details| Due| Points| Objectives| 6 7. 16 Identify the relevant attributes of objects when solving problems. 7. 17 Apply appropriate techniques, tools, and formulas to determ ine measurements. | | | Reading| Read Ch. 13 of A Problem Solving Approach to Mathematics for Elementary School Teachers. | | | Reading| Read this week’s Electronic Reserve Readings. | | | Participation| Participate in class discussion. | | 10| Discussion Questions| Respond to weekly discussion questions. | | 10| ————————————————- ———————————————— Week Eight: Applications of Measurement, Continued| | Details| Due| Points| Objectives| 7 8. 18 Identify the relevant attributes of objects when solving problems. 8. 19 Apply appropriate techniques, tools, and formulas to determine measurements. | | | Reading| Review Ch. 13 of A Problem Solving Approach to Mathematics for Elementary School Teachers. | | | IndividualText Problems 4| Complete Text Problems 4 locate d in MyMathLab ®. | | 40| IndividualReflective Paper| Prepare a 700- to 1,050-word paper synthesizing the major concepts addressed in this course.Include the following in your paper:Summarize the major mathematical concepts of the course. Explain how the concepts learned in this course are relevant to the characteristics of a professional mathematics teacher. Determine how the course concepts have influenced your ideas and philosophy of teaching. Recommend changes to the practice of mathematics instruction based on your learning experiences in the MTH/156 and MTH/157 courses. Format your paper consistent with APA guidelines. | | 100| ————————————————- ————————————————-Week Nine: Mathematical Connection| | Details| Due| Points| Objectives| 8 9. 20 Synthesize the mathematical concepts addressed in this course. | | | CapstoneParticipation| Participate in class discussion. | | 10| Capstone Discussion Questions| Respond to weekly discussion questions. | | 10| Final ProjectFinal Exam| Complete the Final Exam located in MyMathLab ®. | | 225| ————————————————- Optional Discussion Questions Week One Discussion Questions How do all the branches in a tree diagram illustrate the counting principle or generate all possible outcomes?Explain your answer. * When a student is taught how to find the mean of a set of data, why might they have a difficult time accepting the answer? Provide an example. Week Three Discussion Questions How might you involve children in learning geometric concepts? Which geometric concept do you think will be most difficult for children to learn and why? * Why is three-dimensional geometry important? What difficultie s might students have when working in three-dimensional geometry? Week Five Discussion Questions Why do some children have difficulty with rotational symmetry?What methods can you use to help them understand rotational symmetry? * Research the flag for the state or country in which you live. Determine the number of lines of symmetry in the flag, and describe the lines of symmetry you discover. What concept might you use this activity for in an elementary school setting? Week Seven Discussion Questions Accurate measurement of the volume of different shapes is an important mathematical concept. Review the following scenario and respond: * A student read about Volkswagen packing in the 1960s. She was interested in knowing the maximum number of students that fit into a Volkswagen car.How might you help her estimate an answer in a reasonable way? Explain. * What are one to three activities that helped you understand the concept of area? How did these activities help you understand the co ncept? Might the same activities help children understand the concept? Explain. Week Nine Discussion Questions What two mathematical concepts that you have learned in this course do you feel will be the most beneficial to you in the classroom? Why? * * Select one mathematical concept you have learned in this course and provide a brief example of how you could incorporate it into a lesson in the classroom.What steps would you take to ensure students understand the concept? Copyright University of Phoenix ® is a registered trademark of Apollo Group, Inc. in the United States and/or other countries. Microsoft ®, Windows ®, and Windows NT ® are registered trademarks of Microsoft Corporation in the United States and/or other countries. All other company and product names are trademarks or registered trademarks of their respective companies. Use of these marks is not intended to imply endorsement, sponsorship, or affiliation. Edited in accordance with University of Phoenix ® edi torial standards and practices.

Katherine and Bianca Essay

Kate and Bianca clearly do not get on with each other, when Bianca comes onto the scene she is harsh straight away, the third thing that she says is commenting on Bianca’s behavior. â€Å"A pretty peat! It is best put finger in the eye and she knew why†. Kate calls Bianca a spoilt child and then goes on to say that the best thing that Bianca could do if she could not think of an excuse would be to make herself cry so that Kate got the blame. Bianca mocks Kate by using words that have a hidden depth â€Å"sister, content you in my discontent† After the comment from Katherine â€Å"A pretty peat† Bianca tells Kate not to feel bad, and then goes on to talk sweetly to Baptista. This leads me on to talking about Bianca’s speech. She says what she thinks is right in front of her father, even if she does not really mean it. â€Å"What you command me to do I will do†, in this she is talking to Baptista, she is pleasant and obedient. She also says later on â€Å"so well I know my duty to my elders†. At the start of the play people would have thought that this remark was of her politeness but when it gets into the play some people think that it is suggesting something, as she is saying it to Kate, it may be a sarcastic hint that she is older than Bianca and not as fair as she is. Bianca always comes across as if she is saying sweet and kind things but underneath the original meaning it is almost as if there is a hidden depth to her speech and it goes back to the original meaning behind the play of deception and disguise. After Bianca’s’ wooing scene she says â€Å"farewell, sweet masters both, I must be gone† she comes across to be flirting with them, but she is not interested in either of them and is confusing them. Shakespeare gets the point across about the two sisters by using how others react to them and also what others say about them, either to them or about them. Kate does not have a very good reputation and so people are used to what she is like. They are horrified at some of the things that she says but not at all surprised because everyone knows Kate as the Shrew. When Petruchio first meets Kate he thinks that he can tame her; â€Å"For I am he am born to tame you, Kate†, he thinks that underneath the ill-tempered behaviour and the bad mannered speech that there is something else there that no one can see, he thinks that it needs bringing out. He wants to have a wife so he thinks that she will be a perfect challenge. Baptista gives up hope on Kate because he thinks that she will never calm down and be like Bianca, he asks her â€Å"Why, how now, daughter Katherine in your dumps? † This is because she has been shouting so much that she is left feeling low spirited and out of temper. Petruchio woos Kate and she tries to fight back, he says â€Å"Twas told me you were a rough cay and sullen, and now I find report a very liar† he tells her that she is beautiful and how she is perfect and all she does is fight back.

The Reactions of Hoover and Roosevelt to the Great Depression Assignment

The Reactions of Hoover and Roosevelt to the Great Depression - Assignment Example The Smoot-Hawley Tariff Act (1930) however has been seen by historians and economists today though as something that actually made the problems associated with the Great Depression worse, not better. Those economists that believe that the economy can only benefit from lowering prices point to tariffs as a way of increasing prices rather than dropping them. Roosevelt, on the other hand, campaigned on a balanced budget and a promise not to intervene with the economy. However, once he was elected he went ahead and expanded some of Hoover’s programs and created some of his own. The minimum wage act and the Davis-Beacon Act (1931) meant a reduction is price flexibility which slowed the economy even further. The New Deal which was a program implemented under Roosevelt’s reign was actually two different deals. The first one which ran from 1933 – 1935 was aimed at inputting money at the top of the economy so that the people at the bottom benefitted from the trickle-down effect. The Agricultural Adjustment Act (1933) for example paid farmers to reduce their production. No one was actually able to explain why that would help children in the poverty end of the scale who were going without food or the countless numbers of tenants and sharecroppers who were evicted and left without a job but it did make the larger (wealthier) farmers happy. As far as helping alleviate the impact of the Depression however, it was a non-starter – consumer demand fell because of course there were fewer people with money to spend. The National Industry Recovery Act (1933) was instrumental in setting up a centralized planning scheme that would encourage businesses to set prices that would drive weaker and smaller businesses out of the marketplace completely. Again this might have benefitted larger businesses but the smaller ones were still forced to close and unemployment figures still continued to rise, meaning there was less money being spent in the economy.  

Public Relations - interview with a facility manager of the nursing Essay

Public Relations - interview with a facility manager of the nursing home - Essay Example So I presented myself in Mr. Russell's chamber in the nursing home. He appeared to be an amicable man of perfection and he answered all my questions with patience. According to Mr. Russell, Mother Mary Nursing Home was very small as its former infrastructure is concerned. There were just 20 beds and 2 chambers for doctors. Today the whole scenario presents above 200 beds and 15 private chambers for the doctors. Like any other nursing home or hospital there are both outdoor and indoor facilities. People can come for personal check ups by the physicians who attend the outdoor section. On the other hand, there are also facilities for the patients who are directly admitted in the nursing home. As a Facility Manager, it is Mr. Russell's duty to look after whether the patients are getting all the facilities for which the authority has promised to them. He mentions some of the facilities that are unique in this nursing home and he further adds that these unique features have helped the nursing home to ensure its popularity. First of all there is an insurance facility every patient, which is very rare in the nursing homes. The patient can apply for the insurance with a certain amount of money that has to be deposited in the nursing home bank. This also helps the patients to get facilities in case they are admitted in future. Insurances are also helpful if someone from their families is admitted there. Mother Mary Nursing Home keeps all the tracks of the past records of the patients who are admitted here. These computerized records are very helpful to the doctors in case someone is admitted again in the future. Physicians can get all the details about the medical history of the patient and this helps in the process of diagnosis. There is free breakfast in every morning for the patients and tea and coffee for the visitors. The nursing home authority makes it a point so that no issue regarding the hygiene of the patient can be raised. The foods are healthy and applicable for all the patients no matter what disease they are suffering from. There are also special arrangements for the patients whose conditions are more critical than the others. All the patients are kept in separate rooms and there is no dormitory in the nursing home where privacy of the patients may get disturbed. Attendants are very regular in their job. All the rooms are cleaned up from time to time. Hygiene is in the culture of the nursing home and this is why patients are highly satisfied with the kind of services they are provided with in this organization. Politeness and cooperative nature of the attendants is focused so that the patients never feel lonely in this building. This homelike situation is the main thing which has brought success to Mother Mary Nursing Home. When asked about the financial expenditure of the nursing home Mr. Russell clearly said that they always try that services from this organization should be within the reach of the middle class people who arrive here for treatment. Of course some of the bills may seem to be expensive to them but when someone looks back to the kind of atmosphere and extra facilities which he/she had

Critical Thinking and Ethics Essay Example | Topics and Well Written Essays - 750 words

Critical Thinking and Ethics - Essay Example However, personal ethics can change with a period of time and changing experiences though the morals and values forming the foundation of ethics may continue to be the same. Social responsibility is concerned with the behavior considering the society as well as variables surrounding the individual e.g. drinking or smoking decisions, decisions on bribery, decisions on city cleanliness etc. ("Personal Ethics and Life") Since, man is a social animal and cannot be divided from the society, it would be correct to say that the personal ethics need an evaluation as well as a serious thought for linking them with social responsibility. One needs to be very clear as to how personal ethical standards are related with the social background. It is necessary to analyze personal ethics as the actions of an individual in a wider context do have an indirect impact on the society at large. These are established in life's earlier stages from home, school, and church and later in life they influence the work place, city and nation too. Therefore, it is essential not to mistake personal opinions with universal principles in the light of thoughtful and socially accepted personal ethical standards. Here in this brief write up we would take an example that a project manager (PM) faces in his day to day life. Such an issue cannot be solved conveniently on the basis of theory or formal academic approach. It needs prudence, situational approach and morality to arrive at a solution which is ethically correct. Now, what is morality differs from person to person and is a debatable topic. Let us view the situation and the possible action as well as the basis for the action. Suppose the PM is uncomfortable as international project requirements contradict the domestic customs and laws, while they are well accepted in the foreign country. This leads to contradiction with the permissible practices in domestic environment. Here, the PM needs to decide what correct is ethically and what is not. Would the payment made to foreign officials be considered a bribe or "facilitation o0f business processes". Here is a way to solve the ethical dilemma. One can check himself on the ethics yardstick by setting out some standards to decide a right approach. Are the rules that are followed by you, usually understood as a part of the task Do you have sufficient comfort defending your action personally in front of public Would you be comfortable if your friends, relatives, and family are aware of your action Would you be ok if somebody does this same act with you or your loved ones Would the impact of a similar action be positive on the society, if everyone starts following the same practice Is the act not resulting in harm to any part, if not doing positive If the answers to all the above questions are yes, then the act can be considered ethically correct. (The manager and the negotiator) Conclusion The writings of "Critical Thinking and Ethics; From Theory to Application" give an insight on the ethical aspects of psychology. These writings highlight the importance of self evaluation and analysis of personal behavior with respect to the social circumstances making

EDUCATION Essay Example | Topics and Well Written Essays - 250 words

EDUCATION - Essay Example Online education is the solution for students who want to enroll in a prestigious university, but could not, because of distance, time and the complexity involved in balancing other relevant responsibilities. Through online education, a student with various roles as a professional, mother, or part-time worker, could enjoy the benefits of convenience through accessing course modules at their most accessible time. Further, one could exercise skills in time management to prioritize diverse social responsibilities as a family member or an employee, in addition to being an active and collaboratively participating student. Interaction with colleagues and instructors are eminently made easy. Concurrently, online education assists students through multidimensional support systems such as video conferencing options, sending correspondences through emails, accessing online library, and counseling services through the university’s live support site, among others. The unlimited support sy stems provided by online education is truly a remarkable advantage that is readily available to those who opt for this innovative technological breakthrough.

AES Corporation management (Organizational Behavior) Essay

AES Corporation management (Organizational Behavior) - Essay Example Finance management is crucial and the fact that we have been able to survive the current financial pressures proves our financial strength. We immediately took steps to reduce capital expenditure, disposed off assets, liquidated equity to meet the margin calls, retrenched, and withdrew from risky business areas. We are aware that AES may not be able to access the capital market and has to rely on the internally generated funds. Besides, as per the analysts report we may not be able to command a fair value for the assets that we put up for sale but the directors have already taken additional steps to provide a more substantial liquidity cushion. This will definitely leave us a better-capitalized and stronger company with less earnings volatility. Bidding power contracts is not an issue because people have the expertise to sustain competition like Shell and Bechtel. The focus now should not be on the investors, but on attaining liquidity. All the other fronts have been attacked simultaneously. Organizational changes have already been made with a view to enhance operating performance, further reduction of operating costs, revenue enhancements. Two special offices – the Cost Cutting Office and the Turnaround Office would assist in better coordination on management of expenses. It would also assist in taking prompt decisions to dispose off or retain businesses. Apart from these, what else can a new board of directors do? Failures of companies like Kellogg and Apple, who shared the vision of alternative type of enterprise.

Occupational Therapy Assistant Essay Example | Topics and Well Written Essays - 250 words

Occupational Therapy Assistant - Essay Example The decision I made has been coupled by a lot of adjustments I have to make in my life so as to be the perfect person I want to be. Being that my occupation involves interaction with people of different ages, I am learning how to handle the old and young patients who might be put under my care. The old need special treatment as well as the young. Since the work as an occupational therapist assistant may mean working in hospitals, outpatient clinics, schools, rehabilitation centers, nursing homes and even mental health facilities, I am currently adapting to different lives and different people. I know it is not a walk in the park working in a mental health facility, but yet again, I understand how much these patients need someone like me to look after them. For this reason, having understood the difficulty involved in working under these conditions, I have trained myself to be adaptable to any circumstances that might be associated with my

Educational orientation Essay Example for Free

Educational orientation Essay According to the journal for American association for counseling and development (2008) and Zweigenhaff and Domhoff (2003) the African Americans value very much education and are ready to make sacrifices to achieve the education qualifications. Though they have that desire and willingness to have a good education background they are faced with a number of challenges that affect their performance in school. According to Rovai et al (2007) African American students as compared to the other students have lower performance standards and this is raising concern because this area has not been given much consideration to determine what could be the actual reasons behind their low grades. Lincoln et al (1990) and Henderson and Sumler (1999) discusses some of the reasons why these students do not perform well as others as to include difficulties to integrate and accept the various people from different cultural backgrounds they encounter in school and where they live. The hip hop culture is also said to contribute to the low grades because more of this students have indulged themselves in weird lifestyles that cannot allow them to have enough time with their studies. As argued in Cross (2005) and Wayne (2005) most of the African American students come from average family background and when they go to school with the white American they are seen by the fellow white students as being inferior which psychologically will affect them in their education as they find that they are discriminated. This creates a gap between them that and limits the way they will associate with the others and their presence in the school is threatened. The other reasons for their low grade as discussed in smiley (2006) include the poverty that they live in that makes them struggle to meet their basic needs and making them lose enough concentration to the education and even at times dropping out of school to try life elsewhere. The other reasons according to Obiakor (2002) and Ashe (2002) leading to low academic performance among the African Americans include the way they select the schools that they go to which might have low standards, having different curriculums that may not cover all the necessary topics and choosing areas of study that they are not competent in and also some of the teaching staff may not be giving them support because they tend to display behavior that show no much concern to their education. How to address the problem. Ogbu (2003) and Wright et al (2001) suggests that the teachers and the schools have a responsibility to encourage and support all the students they have under their care to perform well. He also argues that this will be done by helping the students to change the various negative attitudes they have towards their teachers, other students and the education materials like books. Obiakor et al (2002) suggests that the learning environment needs to be made conducive for all the students so that nobody feels threatened by the other because it will help the students settle in their education. As argued by Allen et al (1998) and Wayne (2005) the parents and guardians are encouraged to give moral support to the students and they should encourage them to develop interest in the religion as it offers psychological support. References A. P. Rovai, Louis B. Gallien, Helen R. Stiff (2007): Closing the African American achievement gap in higher education. National association for college admission counseling. Retrievedonline http://www. nacacnet. org/PUBLICATIONSRESOURCES/BOOKREVIEWS Alex B. Henderson, Janice Sumler (1999). Freedom’s odyssey Clark Atlanta university press. Allen K, Stelzer, P Wielkiewicz, M (1998). The ecology of leadership: adapting to challenges of a changing world. The journal of leadership. Bertram D. Ashe (2002). From within the frame. Routledge publishers Charles E. Lincoln, Lawrence H. Mamiya (1990). The black church in the African American experience. Duke university press. Cross T. (2005). The persisting racial gap in college student graduation rates. The journal of higher education. Festus E. Obiakor, Bridgie Alexis ford (2002). Creating successful learning environments for African American learners with exceptionalities. Corwin press Faye Z. Belgrave, KevinW. Allison (2005). African American psychology. Sage publishers. .J. Hale (2001). Learning while black. JHU publishers. Journal of counseling and development by American association for counseling and development vol 79 2008. Ogbu J (2003). Black American students in an affluent suburb: a study of academic disengagement. Lawrence Erlbaum publishers New Jersey. Tavis Smiley (2006): The covenant with black America. Third world press. W. Wayne (2005). African Americans and the color line in Ohio. Ohio university press. R. Zweigenhaft, G. Domhoff (2003). Blacks in the white elite. Rowman and Littlefield. Richard Wright, A. Chapman, Malcolm (2001). Black voices. Signet classic publishers

Operating a Fleet of Electric Taxis Essay Example for Free

Operating a Fleet of Electric Taxis Essay Abstract. The deployment of electric taxi ? eets is highly desirable from a sustainable point of view. Nevertheless, the weak autonomy of this kind of vehicles requires a careful operation. The way of managing such a ? eet and the question of locating charging terminals for the vehicles are addressed in this paper. Methods for dealing with these tasks are proposed and their e? ciency is proved through simulations. 1. Introduction 1. 1. Context. Centrale OO 1 is a pioneering project aiming to deploy in Paris a ? eet of 100 % electric taxis. The company in charge of the management of the ? eet is the Soci? t? du Taxi Electrique Parisien (STEP).ee The deployment of such ? eets ? nds is main motivation in sustainable issues: electric vehicles release almost no air pollutants at the place where they are operated and have less noise pollution than internal combustion engine vehicles. However, the main drawback of an electric vehicle is its weak autonomy – 80 km in the case of the Centrale OO project. In taxi ? eet management, two kinds of requests can be di? erentiated: booking requests and opportunistic requests. The ? rst ones can be immediate or in advance of travel and have to be processed by the taxi dispatching system which assigns the request to a taxi. The opportunistic requests correspond to the traditional taxi services picking up passengers at cab-ranks or from the side of the road. Of course, this kind of requests is not processed by the dispatching system. The constraints of the management, as expressed by the STEP, are †¢ A taxi must never break down †¢ An opportunistic demand inside Paris and its suburbs must always be satis? ed (legal environment of Paris) †¢ The number of booking demands accepted has to be maximized The charging problem of the taxis must therefore be carefully addressed. At a tactical level, a good assignment of the trips to the taxis is crucial. We propose an e? cient way to manage the electric ? eet in real-time while taking into account the charging tasks. At a strategic level, the charging problem includes the determination of the best location for the charging terminals. The signi? cant initial investment (the cost of an electrical charging terminal is about 20. 000 euros) and the restricted vehicle autonomy give a high relevancy to the charging terminal location task. Indeed, a wrong placement may in e? ect lead to a poor ? eet management with vehicles having di?culties to charge the batteries due to charging terminals saturation or even with vehicles constantly running out of charge to keep operating. Our purpose is to propose a practical way for computing the â€Å"best† locations for the charging terminals. 1. 2. Model. We describe now formally the model we deal with in this paper. We derive also some elementary relations, which gives some informations on the capacity of a given system (in terms of number of trips that can be realized by unit of time). 1. 2. 1. Input description. A complete directed graph G = (V, A) models the network. The vertices are points in the city at which trips start and ? nish. They can moreover be used to locate vehicle charging terminals. The arcs model the possible trips. The duration of a trip is a random variable Ta of expectation ? a . The Key words and phrases. charging terminal location; electric vehicles; ? eet management system; mixed integer programming; simulation; taxi dispatching. This project has been funded by R? gion Ile de France. e 1 See the website http://taxioo. com/index. html for an artistic view. 1 hal-00721875, version 2 31 Jul 2012 demand for each possible trip a ? A is assumed to follow a Poisson process of rate ? a . Actually, this demand is split between a booking demand and an opportunistic demand, see Section 5 for a more accurate description. There are n taxis. A taxi consumes ? Wh by unit of time when it is moving. It stores ? Wh by unit of time when it is charging. The number of charging terminals is denoted by r. Several terminals can be located at the same vertex. ? 1. 2. 2. Elementary relations. Let us denote by ? a the average number of demands for a trip a that are ? a ? ?a . accepted by unit of time. We have ? 1 ? ? ? ?a ? a be the Let ? = ? a be the average number of trips accepted by unit of time and let ? = ? ? a? A a? A average duration of an accepted trip. ? The energy consumption of the system by unit of time is ? . The maximal rate of supply in energy is ? r. Therefore, we have the following inequality (1) ? ? ? ? r hal-00721875, version 2 31 Jul 2012 A second inequality can be derived by considerations on the time needed to perform the di? erent tasks. Let us consider a taxi over a time window of su? ciently large duration T . Denote by x the time during ? which it stores energy at a charging terminal. Over the time window, it spends in average T n unit of time with a customer on board. Therefore, we have ? T +x? T n During this duration x, it stores a quantity of energy that must cover in average the consumption over the time window. Hence ? ?T ? ? x n Combining these two inequalities leads to (2) ? (? + ? ) ? n Equations (1) and (2) can be summarized in the following inequality. (3) ? ? ? min ? r n , (? + ? )? Knowing the number of taxis, their e? ciency (encoded by ? ), the number of charging terminals, and their e? ciency (encoded by ?), then an upper bound of the number of trips that can be accepted by unit of time can be calculated. 1. 3. Plan. Section 2 is devoted to the literature review for the two problems addressed in this paper, namely ? eet management and charging terminal location. The following sections – Section 3 and Section 4 – detail the approaches proposed for each of these problems. Next, we describe a simulator that has been implemented for the evaluation of the proposed approaches (Section 5). The results of the experiments are described in Section 6. The paper ends with concluding remarks (Section 7). 2. Literature review 2. 1. Taxi dispatching. Traditional taxi dispatching systems are characterized by two principles. First, simple rules such as for example â€Å"nearest vehicle ? rst† or â€Å"least utilized ? rst† are used for dynamic vehicle assignment and second, the geographical space is usually divided into zones. In the literature, most of works on the topic basically focus on customer waiting time minimization by proposing improved methods for rule-based systems. In this context, Shrivastava et al. [SCMK97] describe a fuzzy model for rule selection and Alshamsi et al. [AAR09] propose a new technique for dynamically divide the dispatch areas. The recent apparition of transportation technologies (GPS, EDI, GIS) has widely increased the opportunities for ? eet management optimization. It is also the case for taxi dispatching. For example, Seow et 2 hal-00721875, version 2 31 Jul 2012 al. [SDL10] propose a collaborative model for taxi dispatching where a set of n taxis of the same zone are de? ned as the agents of the model and a set of n customers as the service-requests. The objective is then to maximize the total service quality solving a collaborative linear assignment problem. However, taxi dispatching is not the only aspect that can be optimized. For example, Lee et al. [LSP08] and Jia et al. [Jia08] use real-time vehicle information to propose a model for taxi relocation recommendation based on demand forecasting and a probability model for the design of taxi stops, respectively. Another approach for ? eet management optimization consists in modeling the problem as a variant of the Pick-up and Delivery Vehicle Routing Problem with Time Windows (PDVRPTW). The idea is to plan a set of routes satisfying known in advance customer requests. In the taxi management context, Wang et al. [WLC09] propose a bi-criteria two-phase method with an initial feasible assignment ? rst and a tabu search improvement later in order to minimize the number of vehicles and the sum of travel times for advanced bookings. However, the idea to block some vehicles only for advanced bookings might in some cases yield to a ? eet underutilization. Horn and al. [Hor02] and Meng et al. [MMYH10] try to ? ll the gap between simple non-optimized rule-based taxi dispatching systems and static routing approaches. The second paper describes a genetic network programming in order to ?nd the optimal balance between the waiting time and the detour time. The work of Horn [Hor02] is of particular interest in relation to the present work, proposing a taxi dispatching system architecture similar to our ? eet management system. He proposes a system for vehicle travel time minimization composed by a set of insertion algorithms to decide whether a new customer is accepted or not and a set of optimization mechanisms in order to improve the solution. However, some important di? erences exist between our work and these last two ? eet management systems. The ?rst di? erence is that in our case, the constraints related to the restricted autonomy of the vehicles have also to be taken into account by scheduling charging tasks in the routes of the vehicles. The second di? erence is that, unlike us, both articles deal with the multi-customer problem authorizing customers to share the same vehicle at the same time. 2. 2. Location issue. The location problem was originally de? ned by Webber when he considered how to position a single warehouse minimizing the total distance between the warehouse and a set of customers [Web29]. In 1964, Hakimi [Hak64] de? nes the P-median problem, the problem consists in determining the best location for a set of limited facilities in order to minimize the sum of the weighted distances between the clients and the facilities serving these clients. The problem increases its relevance during the last decades. High costs related to property acquisition and facility construction make facility location projects a critical aspect of strategic planning for a wide range of private and public ? rms. Indeed, the fact that facility location projects are long-term investments leads the researchers to focus on dynamic and stochastic location problems (see [OD98] for a review of this extension of the problem). Another important variant of the problem is the Capacitated Facility Location Problem (CFLP) where facilities have a constraining upper limit on the amount of demand they can satisfy. An extension of the CFLP closely to our problem is the Capacitated Facility Location Problem with Multiple facilities in the same site (CFLPM). In charging terminal location, the positions of the terminals are not the only decision variables, the number of terminals at each position have to be ? xed too. However, in some real-world applications, selecting the best location for distance minimization is not the best suitable choice. For example, in electric vehicle charging terminal location, like in other critical applications such as ambulance and ? re terminal location, the interest is to guarantee that the di? erent geographic zones are covered by a facility (closer than a previously ?xed covering distance). This class of problems are known as Covering Location Problems (see [WC74], [SVB93] and more recently [VP10] for a complete review of covering problems). In that context, the covering issue can be sometimes modelled as a problem constraint. However, if the covering distance is ? xed to a small value the problem might become unfeasible. The Maximal Covering Location Problem (MCLP) [CR74] locates the facilities in order to maximize the number of covered customers (customers with a distance to the nearest facility smaller than an initial ?xed distance). An extension of the problem is the maximal covering with mandatory closeness problem which imposes a maximal distance (less stringent than the covering distance) between the geographical zones and the nearest facility [CR74]. These covering models implicitly assume that if a geographical zone is covered by a facility then the facility will be always available to serve the demand. However, in some applications, when facilities have a ? xed capacity, being covered is not su? cient to guarantee the demand satisfaction. We ? nd 3 in the literature some models attempting to overcome this issue by maximizing the number of geographical zones covered by multiple facilities [DS81, HR86, GLS97]. 3. Fleet management We describe in this section two ways for managing the ? eet, a classical and rule-based one (Subsection 3. 1), and an improved one trying to address explicitly the charging issue (Subsection 3. 2). Let us ? rst introduce some notations. Let CRi be a booking customer request. Each customer request CRi is de? ned by a start time Si and an origin-destination pair Oi ? Di . The Si is ? xed by the customer when the customer request arrives. The completion time of a trip is Ci = Si + ? Oi Di , where ? Oi Di is the travel time between the origin and destination of the customer request CRi . Finally, let R : CTj be a taxi charging task scheduled on the charging terminal CTj . 3. 1. A classical rule-based taxi dispatching system. A taxi dispatching system based on the principles of the most common real-world systems (see for example [SCMK97], [LWCT04] or [AAR09]) is described in this section. The architecture of the current taxi dispatching systems are very similar to the system illustrated in Figure 1. The two main components of the system are (1) a customer acceptation mechanism deciding for each new customer if it is accepted (the accepted customers are inserted into a queue of customers) or rejected and (2) a rule-based mechanism assigning accepted customer requests (trips) to the free taxis. For each accepted trip i, the assigning process has to start a few minutes (? ) before the ? xed start time (Si ) in order to maximize the chances to ? nd a taxi to attend the demand. Once a trip is assigned to a taxi, the vehicle is automatically blocked and the taximeter begins counting. hal-00721875, version 2 31 Jul 2012 Customer Request Rule? based Customer Acceptation Mechanism Time? ordered queue of customers CRn CR2 CR1 Figure 1. Rule-based taxi dispatching system A rule for customer acceptation using the time windows for the trips already accepted is proposed. The idea is to limit the trips that have to be performed at the same time in order to minimize the number of not served customers and to establish a margin of k vehicles to attend opportunistic customers. For each new customer request CRnew the Algorithm 1 determines if it is accepted or not. Algorithm 1: Rule-based checking for customer acceptation for a margin of k vehicles L = {CR1 , CR2 , . . . , CRn }, list of already accepted customers CRnew , new booking customer request nC 0, number of trips performed at the same time than CRnew foreach CRi of L do if CRi is executed at the same time than CRnew ((Si ? Snew Ci ) or (Snew ? Si Cnew ) then Step 1: Increase the number of customers performed at the same time than CRnew (nC nC + 1) if condition to accept the customer (nC n ? k) then Step 2: Insert CRnew to the list of accepted customers L 4. Once the customer request CRi is accepted, it remains in the queue of customers until Si ? ? (? is usually ? xed around 20 minutes). At that moment, the system automatically starts looking for a free taxi having su? cient charge to operate the trip. If di? erent taxis are available, the system assigns the trip to the taxi minimizing the customer waiting time (a parameterizable not announced customer waiting time can be authorized). In the case of no vacant taxis are available, the system waits for a vehicle to become available. If the waiting time for any request exceeds the authorized maximal customer waiting time ?, the customer request is then canceled. Note that the number of unsatis? ed customers can be reduced by using a more restrictive rule for the customer acceptation mechanism. The main advantage of such a system where no future work is planned is the high degree of independence for taxi drivers. On the other hand, the drawbacks are the underutilization of the ? eet and the lost of e? ciency during the peak hours when most of the companies have to close their booking requests systems in order to avoid unsatis? ed customers. Indeed, some real-world systems do not integrate a customer acceptation mechanism leading, in rush hours, to unsatis?ed customers who had been initially accepted and they are ? nally served with an unannounced and, sometimes, intolerable delay or, eventually, never served at all. Furthermore, the charging tasks of the vehicles cannot be controlled leading to a poor ? eet management with vehicles having di? culties to charge the batteries due to charging terminals saturation. 3. 2. The improved electric vehicle management system. An improved ? eet management system aiming to overcome the weakness of the rule-based taxi dispatching system is proposed in this section. The main objectives of the system are to maximize the number of accepted customers and to minimize the customer waiting time. One of the major issues is how to deal with opportunistic demand. Indeed, this kind of demand is unpredictable and must always be satis? ed, so free taxis must be at any moment able to satisfy the longest trip without running out of charge. This constraint makes the problem considerably more complex forcing the system to provide a mechanism ensuring the feasibility of the already accepted trips each time an opportunistic demand is accepted. The approach proposed consists in maintaining continuously a feasible planning for the taxis and the charging terminals (see Figure 2). Each time a customer asks for a trip, a simple insertion algorithm is run, at the end of which either the trip has been successfully inserted or not. The objective is to assign the customer to the taxi minimizing the customer waiting time (a parameterizable announced customer waiting time can be authorized). If none of the tried delays on the pick-up time leads to a feasible planning, a rescheduling algorithm allowing to reallocate the already accepted customers to the taxis is run. In all these processes, a key routine is often called, namely the charging task manager, which schedules the charging tasks of a taxi, given a planning for the other taxis and the charging terminals. Feasible planning:Temporal and autonomy? related constraints are satisfied Taxi 1 hal-00721875, version 2 31 Jul 2012 Taxi 2 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111 000 111 000 CR2 111 000 111 000 CR1 R : CT1 CR4 R : CT2 CR5 VEHICLES Customer Customer Request Acceptation Mechanism Feasible Planning Taxi 000000 n 111111 111111 000000 111111 000000 CR3 R : CT1 111111 000000 111111 000000 1111111111111111111111 0000000000000000000000 Taxi n Taxi 1 CT1 CHARGING TERMINALS CT2 Taxi 2 Figure 2. Customer acceptation mechanism of the electric vehicle management system In the case of an opportunistic demand, which is necessarily accepted, we follow exactly the same scheme except that there is no degree of freedom in the insertion process: the trip is inserted at the front of the planning of the taxi stopped by the customer, and the rescheduling algorithm is also run if it is necessary. 5 3. 2. 1. Insertion algorithm. This algorithm is the ?rst step in order to decide if a new trip CRnew is accepted or not. The objective is to assign the trip to the taxi minimizing the delay on the pick-up time (see Algorithm 2). The algorithm increasely tests the di? erent authorized pick-up times. Once the start time is ? xed, we sequentially try for each vehicle to insert the new request. First the scheduled charging tasks are removed. Then the new request is accepted only if it can be inserted with no constraint violation (the pick-up times of the rest of customers are respected and the current autonomy of the vehicle, without any charging task, is su? cient). In the case that the vehicle autonomy-related constraint is violated, a greedy algorithm trying to schedule a charging task between each pair of trips is proposed. After the charging tasks are inserted, if the taxi is able to perform the trips without running out of charge, then the customer request is also accepted. Algorithm 2: New request insertion algorithm for a maximal authorized delay of ? minutes V = {V1 , V2 , . . . , Vr }, list of taxis CRnew , new booking customer request accepted f alse, variable indicating if the new request is accepted st Snew , start time of the trip while st ? Snew + ?and accepted = f alse do foreach vi of V do Step 1: Delete the charging tasks of the vehicle vi if CRnew starting at st can be inserted in the route of the vehicle vi then if the vehicle autonomy-related constraint is satis? ed then Step 2: CRnew starting at st is inserted in the route of the vehicle vi (accepted true) else Step 3: Insert charging tasks for vi between each pair of trips if the vehicle autonomy-related constraint is satis? ed then Step 2: CRnew starting at st is inserted in the route of the vehicle vi (accepted true) if accepted = f alse then Step 4: Increase the pick-up time for the CRnew (st st + 1) hal-00721875, version 2 31 Jul 2012 3. 2. 2. Rescheduling algorithm. The rescheduling algorithm is proposed when the new customer is still not accepted after the insertion algorithm. As for the insertion algorithm, the goal is to ? nd a new feasible planning for the vehicles integrating the new request CRnew . The main di? erence is that the trips can be reassigned to di? erent vehicles. The problem without taking into account the autonomy-related constraints can be solved in polynomial time [NSZ02]. The idea is to convert the schedule of trips (without the charging tasks) into a graph and to verify using a max ? ow computation that all trips can be performed by the taxis. To construct the network two vertices are considered for each customer request CRi , the ? rst one vi represents the pick-up time and the second one vi the completion time of the customer request. Four dummy vertices are required: 0, 0 , a source s and a sink t. The arcs of the graph are (s, 0), (0 , t), all the (s, vi ), all the (vi , t), all the (vi , vi ), and all the (vi , vj ) such that the customer request CRj can be performed by the same taxi than the customer request CRi and after CRi , that means if Sj ?Ci + ? Di Oj . Except the arcs (s, 0) and (0 , t), they all have a capacity equal to 1. The arcs (s, 0) and (0 , t) have a capacity equal to n. A maximum ? ow computation in this directed graph determines the schedule feasibility and also proposes a new planning for the vehicles respecting the customers pick-up times. The max ? ow computation is integrated in the rescheduling algorithm in order to check the feasibility of the schedule for a given pick-up time st ? [Snew , Snew + ? ] and, if it is the case, to ? nd a reference planning (planning without charging tasks). A local search explores the neighborhood of the reference planning de? ned by the swap and the reallocation operators [Sav92]. Finally, for each explored planning respecting temporal constraints, the greedy algorithm for charging task scheduling is sequentially applied to the taxis that do not satisfy autonomy-related constraints (that is, taxis whose current charge is not enough to perform all 6 the trips assigned to them without adding charging tasks). If a feasible solution is found, the new customer is then accepted. 3. 2. 3. Charging task manager. As we have already seen, the insertion and the rescheduling algorithm constantly runs a greedy algorithm aiming to insert a charging task between each pair of successive trips of the same route. The algorithms proposed to determine if a new charging task can be integrated in a speci? c charging terminal planning are described in this subsection. The main feature of our problem is that the processing time of the new charging task is not ? xed, instead it is a decision variable de?ned between the interval limited by the minimal charging time for a vehicle pmin (customizable parameter) and the maximal charging time corresponding to the time necessary for a full charge. The problem to be solved by the charging terminal manager can be then formally stated as follows. A charging task Ri is de? ned by its time window [ri , di ], where ri is the earliest start time (earliest arrival time to the terminal) and di the latest end time (latest departure time from the terminal). Let pi be the decision variable corresponding to the processing time of the task Ri , then ri ? Si and Si + pi ? di , where Si is the e? ective start time of Ri . Given a feasible schedule of n charging tasks S n = {S1 , S2 , . . . , Sn , } for the charging terminals located at the same geographical position. We are given a new charging task Rn+1 with a time window [rn+1 , dn+1 ] and a processing time pn+1 inside the interval pmin ? pn+1 ? pmax . The problem consists in ? nding a new n+1 feasible schedule S n+1 = {S1 , S2 , . . . , Sn , Sn+1 } maximizing the processing time of the task Rn+1 (whence without changing the processing time of the other tasks). The mechanism tests ? rst a task insertion aiming to ? nd quickly a feasible solution. The complexity of the algorithm for task insertion maximizing the processing time of the new task is O(n) where the start times and completion times of the scheduled jobs are non-decreasing ordered. If no solution is found after the task insertion algorithm, a dichotomous algorithm allowing to reschedule the tasks is proposed in order to ? nd a solution maximizing the processing time of the new task. For each iteration of the algorithm, a satis? ability test based on constraint propagation involving energetic reasoning is ?rst triggered. The goal of the feasibility test is to detect an inconsistency indicating that it is not possible to ? nd a feasible schedule integrating the new task. Finally, if the energetic reasoning is not conclusive a local search algorithm is proposed in order to ? nd a solution. Satis? ability test: Energetic reasoning. A satis? ability test based on constraint propagation involving energetic reasoning is proposed [LE96]. A ? ctitious energy (which has nothing to do with the electricity) is produced by the charging terminals and it is consumed by the charging tasks. We determine the ? ctitious energy consumed by the tasks (Econsumed ) over a time interval ? = [t1 , t2 ] and we compare this ? ctitious energy with the available ? ctitious energy (Eproduced = m ? (t2 ? t1 )). The minimal ? ctitious energy consumed by the tasks in an interval ? = [t1 , t2 ] is: n+1 hal-00721875, version 2 31 Jul 2012 (4) Econsumed = i=1 max{0, min{pi , t2 ? t1 , ri + pi ? t1 , t2 ? di + pi }} If Econsumed Eproduced , it is then impossible to ? nd a feasible schedule S n+1 integrating the new task. The relevant intervals ? for a complete satis? ability analysis can be enumerate in O(n2 ). The test is restricted to the intervals [t1 , t2 ] speci? ed by {ri } ? {di } ? {ri + pi } ? {di ? pi } where the new task Rn+1 may consume (t1 ? dn+1 and t2 ? rn+1 ). Dichotomous algorithm. A dichotomous algorithm maximizing the processing time of the new task is described in this section (see Algorithm 3). A dichotomy is run on the processing time p as follows. For processing times p ? [pmin , pmax ], the satis? ability test based one the energetic reasoning indicates whether n+1 the necessary conditions are satis? ed or not. If it is the case, a local search mechanism tries to ? nd a feasible schedule. The parallel machine scheduling problem with time windows can be solved by a list scheduling algorithm. It means there exists a total ordering of the jobs (i. e. , a list) that, when a given machine assignment rule is applied, reaches the optimal solution. For our problem, this rule consists in allocating each task to the machine that allows it to start at the earliest (Earliest Start Time or EST rule). The local search mechanism proposed to solve the problem is based on this result. First, the tasks are ordered in a non-decreasing order of their due dates (Earliest Due Date or EDD rule), then the local search consists in 7 exploring di?erent permutations of the list de? ned by the insertion neighborhood (O(n2 )). For each list of task, the machines are assigned according to the EST rule in order to reach a feasible solution. If no feasible schedule is eventually found, the request is rejected. Algorithm 3: Dichotomous algorithm for processing time maximization min pmin max pmax n+1 n+1 Sbest ? while min ? max do Step 1: Fix the processing time p of the new task Rn+1 (p min+max ) 2 if Satisf iabilityT est() then Step 2: Sort the tasks according to the EDD rule Step 3: Local search using the insertion operator if a feasible schedule S n+1 = {S1 , S2 , .. . , Sn , Sn+1 } is found then Step 4: Update the lower limit (min p + 1) n+1 Step 5: Update the best solution (Sbest S n+1 ) else Step 6: No solution exists, update the upper limit (max p ? 1) else Step 7: No solution exists, update the upper limit (max p ? 1) n+1 if Sbest = ? then Step 8: No solution is found (return ? ) else n+1 Step 9: A feasible solution is found (return Sbest ) hal-00721875, version 2 31 Jul 2012 4. Electric vehicles charging terminal location The EV charging terminal location problem consists in determining the best locations of the charging terminals. The linear programming model has to take into account two important aspects. First, the charging terminals have to be conveniently spread over the geographical area in order to avoid remote geographical zones which di? cult taxi operability and ? eet management. The second aspect is that the model has to determine the number of charging points facilitating the charging process of the taxis by minimizing the risks of terminals saturation. For these purposes, we propose two models, one called the P -median model, the other the Demand-based model. V is the set of geographical points of the problem and J ? V is the set of potential locations where the charging terminals can be located. The number of terminals is limited to r. 4. 1. P -median model. Following Hakimi [Hak64], we de? ne xj to be the decision variables indicating if a facility is located to the point j and yij to be the variables indicating that the geographical point i is assigned to the facility located in j. The linear program minimizing the sum of the distances between clients and facilities can be written as follows. 8 (5) min i? V j? J distij yij s. t. (6) j? J yij yij xj j? J = 1 for all i ? V ? xj for all i ? V, j ? J ? r ? {0, 1} for all j ? J ? {0, 1} for all i ? V, j ? J (7) (8) (9) (10) xj yij hal-00721875, version 2 31 Jul 2012 4. 2. Demand-based model. Another approach consists in de? ning a model with two distances ? f ar and ? close as proposed by Church and ReVelle [CR74]. The idea is then to spread the terminals by ? xing a maximal distance (? f ar ) between the di? erent geographical zones and the nearest charging terminal and, at the same time, trying to maximize the demand that will be covered by a nearby charging terminal (? close ). We can then de? ne Jif ar (resp. Jiclose ) as the subset of points in J at distance less than ? f ar (resp. ?close ) from i ? V . Conversely, Vjclose is the set of points at distance less than ? close from the point j ? J. Let xj be the decision variable indicating the number of terminals located at point j ? J and yij to be the fraction of the demand di for i ? V covered by a charging terminal located in j at distance less than ? close from i. The linear programming model proposed to solve the problem called Demand-based model is the following. (11) max j? J i? Vjclose di yij s. t. (12) f j? Ji ar xj yij close j? Ji ? 1 for all i ? V ? 1 for all i ? V ? xj for all j ? J ? r ? Z+ for all j ? J ? R+ for all i ? V, j ? Jiclose (13) (14) i? Vjclose di yij xj j? J (15) (16) (17) xj yij The objective function (Eq. (11)) consists in maximizing the pointwise demand covered by a charging terminal considering the distance ? close . Eq. (12) imposes that a geographical zone i ? V must be covered at least for one charging terminal considering the distance ? f ar . Here the mandatory closeness is only required for the geographical zones closer than ? f ar from a potential charging terminal location in order to ? nd a solution even if this constraint is violated for some geographical zones. We stress that an adequately ? f ar make possible to spread the charging terminals over the geographical area. Eq. (13) speci? es that for each geographical zone i ? V the sum of the fractions of demand covered by a charging terminal considering the distance ? close has to be less or equal to the unit.