Charles Dickens Essay -- Biography

Charles the Tempter the British Author of the Southwestern English townspeople of Land port in Port Sea was a very famous and well known author during his time. As an author he travelled to many cities. During his travels he had many children. Some of his books include Oliver Twist, A Tale of Two Cities, David Copperfield and A Christmas Carol. The book, Oliver Twist, was about a boy who grew up during hard quantify as an orphan struggling trying to find his way through life. Also, what most people do not know is that the movie Oliver and Company has dealing to the book Oliver Twist. The plot of the movie centers on a cat, which is without a home, looking for a family to live with and call home. The cat in the movie spends a trivial of his life living with dogs, which becomes his closet family. Charles Dickens was on February 7, 1812, born to John and Elizabeth Barrow Dickens. He was the 2nd oldest child of eight children. His scram John Dickens was a clerk in Navy Pay-office and his mother Elizabeth Dickens was a well appealing woman that was very educated. (Swisher 13) As Charles was growing up, his mother taught him to read. His get saw him as a future genius and would have him sit in a tall chair and tell stories to his co-workers at the office. In 1814 his sky pilot John was temporarily sent to the London office to work as a clerk. During this time, as a child Dickens attended the school of Williams Giles. maturation up he had many responsibilities that included attending school, college, and maintaining a professional job at the same time. His pargonnts income started slowing down. Charless father decided to move and settled his family in a town called Camden in 1822 to accommodate their bare minimum finances. The town was the poo... ... that demonstrated Charles intense passion to typesetters case realism in his writings of lifes experiences. After the writings of A Tale of Two Cities, he also wrote, Great Expectations in 186I, oer which t here feel sorry for yourself the sad sense of the of the Lower Thames. He also wrote Our Mutual Friend in 1864, in which the seep and dirt of Rotherhithe, its boatmen and loafers, are made to pass through the book with swelling consequence. Charles Dickens writings made him very famous. He used his colorful life experience to render emotional plots in his writings. The British Authors success through his writings helped him to overcome his own personal tragedies. Charles faced many tough obstacles, only if always over came them, no matter how rocky the road was. Today Charles is still not forgotten as his famous books live on forever for many generations to read.

Essay --

Internet Protocol version 6 (IPv6)Any information presented about IP version 6 (IPv6) lead not be complete without talking about IP version 4 (IPv4), its predecessor. For completeness, a brief introduction of IPv4 go forth be made.In the networking of computers and devices, the Internet Protocol (IP) plays a very important role. The IP, found at the internet layer of the Department of Defence (DoD) model provides the mean for the devices to communicate using logical addresses called IP addresses. The importance of an IP address to communication will be felt in the analogy of a open mail. How possible would it be to send a letter to someone whose address we do not know? The IP address enables us to know the rise of a packet and the destination for proper delivery by the IP protocol.IPv6 motivationThe development of IPv6 is motivated by the inadequacies of its predecessor IPv4. IPv4 is an addressing scheme that makes use of 32 bits in groups of 8 bits for each one to identify a d evice. Each address represents a number in the decimal range 0 to 255 in each of the four octets that represent it. Due to the 32-bit size of the address, the maximum number of IPv4 addresses that can be used is thus limited to 232, approximately 4.3 billion addresses (4,294,967,296).For an addressing scheme that was just to serve as a test of the concept of networking, the possibility of its exhaustion was a remote one. The reality however of the exhaustion of the available IPv4 addresses due to the astronomical increase in the number of people and devices on the internet that need IPv4 addresses gave rise to the development of IPv6.IPv6 StructureIPv6 is a routable protocol that is responsible for the addressing, routing, and fragmenting of packets by the ... ...ters. Devices that do not support IPv6 may require only a firmware upgrade with the new IPv6 stack if the equipment manufacturer so provides it. Or else, such equipment will need to be totally replaced with a new one that supports IPv6. The softwares in use may support both IPv4 and IPv6. Most of the recent releases of major operate dodgings have deployed and supported the use of IPv6 in their operating systems.Windows operating system however does not fully support IPv6 despite the quest for its adoption cosmos pushed by Microsoft. The use of the full colon of the IPv6 IP address in the address bar of a browser will make the operating system think it is a reference to a drive. The cumbersome way around this is to use a domain translation where the colons are replaced with dashes and the characters .ipv6.literal.net has to be appended to the end of the address.

Modest Proposal :: essays research papers

blue-bellys ArgumentThere are many different ways to write an argumentative written report. An argumentative paper is a paper knowing to push a reviewer toward an idea or feeling an author evokes (Skywire 332). An author will try to make comic ideas seem more kindly to the reader. A Modest Proposal is a great example of this technique.Johnathan Swift, an Irish clergyman, wrote an argumentative paper to mock the English. Swifts paper was an alimentation Irish babies. This would sound like a ludicrous idea, but Swift makes it seem like it would cooperate the economy. He uses many of the basic argumentative techniques to support this idea. This closely obvious technique used by Swift was card stacking. He only talked about his side of the situation (Skywire 336). He made a minor seem like it would become a highly demanded dish. People around the world would pay dearly for it and the money would help Irelands economy (Swift 363-4). He never talked about the parents side. He hid how they would feel pain and heartache of a lost child. He keeps the readers mind persuasion about the positive aspects, and never level off touches on the negative ones. He purge introduced slanting into his text. Slanting is a musical composition technique that shows great grace or disapproval about a subject (Skywire 336). He made it seem that he loved the idea of eating a baby. It would be the brand-new delicacy on all menus (Swift 364-5). With this opinion on the great demand for human flesh, he used a precipitant generalization. Swift made a hasty generalization by basing his knowledge that everyone would love human flesh on an incident on the island of Formosa (Swift 365). Swift lettered through a indigenous of Formosa that when a child was put to death, his body was sold as a Prime dainty (Swift 365-6). He said how even the prime minister of the verbalize had bought a girl put to death because of treason (Swift 365-6). So he deducts that since they like it then the whole world will. Swift built the readers up by using numbers and showing how kids are a pain and just get in the way (Swift 364-5). He even went as far as to say that the kids he proposes to eat are children of beggars that cant afford them anyway (Swift 364-5).Modest Proposal essays research paper Swifts ArgumentThere are many different ways to write an argumentative paper. An argumentative paper is a paper designed to push a reader toward an idea or feeling an author evokes (Skywire 332). An author will try to make ludicrous ideas seem more appealing to the reader. A Modest Proposal is a great example of this technique.Johnathan Swift, an Irish clergyman, wrote an argumentative paper to mock the English. Swifts paper was an eating Irish babies. This would sound like a ludicrous idea, but Swift makes it seem like it would help the economy. He uses many of the basic argumentative techniques to support this idea. This most obvious technique used by Swift was card stacking. He o nly talked about his side of the situation (Skywire 336). He made a baby seem like it would become a highly demanded dish. People around the world would pay dearly for it and the money would help Irelands economy (Swift 363-4). He never talked about the parents side. He hid how they would feel pain and heartache of a lost child. He keeps the readers mind thinking about the positive aspects, and never even touches on the negative ones. He even introduced slanting into his text. Slanting is a writing technique that shows great approval or disapproval about a subject (Skywire 336). He made it seem that he loved the idea of eating a baby. It would be the new delicacy on all menus (Swift 364-5). With this opinion on the great demand for human flesh, he used a hasty generalization. Swift made a hasty generalization by basing his knowledge that everyone would love human flesh on an incident on the island of Formosa (Swift 365). Swift learned through a native of Formosa that when a child wa s put to death, his body was sold as a Prime dainty (Swift 365-6). He said how even the prime minister of the state had bought a girl put to death because of treason (Swift 365-6). So he deducts that since they like it then the whole world will. Swift built the readers up by using numbers and showing how kids are a pain and just get in the way (Swift 364-5). He even went as far as to say that the kids he proposes to eat are children of beggars that cant afford them anyway (Swift 364-5).

Graduation Speech -- Graduation Speech, Commencement Address

Good evening, everyone. Now is the time for me to come in front of these 400 students in identical caps and gowns and tell them to go out(a) and be individuals. Looks like I bear my work cut out for me. Seriously, though, consider what weve gone through. 13 years of schooling, 18 years of growing up And reflecting back on these years, what do we have to show for it? We have our memories. Some of you may k right off the old saying, Knowledge is not what the pupil remembers still what he cannot forget. What will remain in our minds after tonight? What memories will become those valuable gems of unforgettable knowledge? Who in the audience knows the phases of Mitosis, or can vex forward the capitol of Togo? Not many. But one of us will ever forget that counselor who listened to us when no one else would ... that librarian who, with the kind of vision that would put Superman to shame, saw one single book out of order on the shelves, and made sure you knew about it ... or how about t hat custodian who astounds us all with his uncanny ability to scoop up spilled ketchup with feline grace. We can develop our minds with information as much as we want, but the right is, those simple, rare smiles from a caring teacher mean more to all of us than any amount of knowledge ever could. Simple gratitude is all we have to offer these educators who have shaped our lives, and still these teachers continue to give every day. Routinely, we blame them for such things as the Culminating Exhibition, overcrowded lunches, and that disgusting feeling we all have when we crap there is, once again, no soap in the bathroom. But these teachers have shaped the lives of the 150 students they see daily and have given of themselves something that is irreplaceable. Who t... ... journey now to make new memories, casting off the skin of adolescence and stepping forward into independence. Some of us will become preachers, presidents, accountants, clerks. We have in this room future rank offi ce workers, caretakers, custodians, soldiers. We even have a select few who are insane enough to pursue a teaching career. In essence, it matters not what we do, but how we do it. Walk through a room, and make it just a little brighter as you leave it. In the words of Vince Lombardi, The quality of a persons breeding is in direct proportion to their commitment to excellence, regardless of their chosen field of endeavor. As we venture out into the world, take with you the riches of kindness, excellence, and caring that we have gained in our years here at Hosea. Keep them with you as you create new memories that will make us proud. Thank you, and God bless.

Religious Freedom Restoration Act :: essays research papers

Religious Freedom takings Act     In this paper I will cast the Religious Freedom Restoration Act.This Act was used to contradict the decision of the homage case of EmploymentDivision v. Smith, which allowed the organization to forbid any spiritual actwithout giving a reason. The RFRA brought back the requirement that thegovernment provide an adequate reason to forbid any spiritual act. Thegovernment once once again had to show that the act was of compelling interestagainst the state.     In 1993 one of the most important acts that has gone thorough Congresswas passed (Religious Freedom, Map of the RFRA). This was the Religious FreedomRestoration Act (RFRA) of 1993 (Religious Freedom, Map of the RFRA). This actwas passed to answer the 1990 court case Employment Division v. Smith (Questionsand Answers, Map of the RFRA). Employment Division v. Smith was a court case inwhich the issue was whether Sacramental use of peyote by members of th e NativeAmerican Church was protected under the alleviate exercise clause of the FirstAmendment, which provides that Congress shall make no law...prohibiting thefree exercise of religion.(Questions and Answers, Map of the RFRA). Accordingto Justice Scalia, if prohibiting the exercise of religion was merely theincidental effect of a generally applicable and otherwise valid provision, theFirst Amendment was not offended. (Questions and Answers, Map of the RFRA).Thus,"...the government no longer had to justify most burdens on religious exercise.The free exercise clause offered protection only if a particular religiouspractice was singled out for discriminatory treatment. In short, free exercisewas a make out category of equal protection. This placed religious rights in aninferior position to other First Amendment rights such as freedom of speech andpress." (Questions and Answers, Map of the RFRA).This court case caused a series of court cases about religious freedoms(Religio us Freedom, Map of the RFRA). Congress enacted the RFRA to contradictthe negative affect that court cases had recently had on religiousfreedoms(Religious Freedom, Map of the RFRA).     The RFRA is what it states it is in the title, a restorationact(Religious Freedom, Map of the RFRA). Congress decided that in EmploymentDivision v. Smith,"the supreme court virtually eliminated the requirement that the governmentjustify burdens on religious exercise imposed by laws neutral toward religionand the compelling interest test as set forth in prior Federal court rulings isa workable test for striking sensible balances between religious liberty andcompeting prior governmental interests."(Religious Freedom, Map of the RFRA)In other words, the government did not have to have a reason to impose laws

Blood Brothers :: Drama

stock certificate BrothersMy practical work in September to December was for my acting option.It was a scripted piece of work based on the influence Blood Brothers.There are various themes in Blood Brothers, a clash of class, romance,jealousy, and betrayal.My contribution to the performance was as a seven-year-old child. Iplayed Edward. Edward is a actually reserved character. He is well spokenand polite and is very surprised and shocked at many things Mickey(his friend) does. He comes from a flush(p) background and so is usedto having everything he needs. He enjoys helping other people out.When acting as Edward I needed to have a very good posture. Holdingmyself well, this showed a contrast between Edwardss upper class andMickeys lower class. I too needed to emit well, pronouncing mywords clearly. This again showed a clear contrast between Mickey andEdwards class.To help me in my work I looked for information on the cyberspace andbooks I looked at pictures of young boys in th e 1950s, this helped mechoose my costume. I also listened to the Blood Brothers sound track,and watched a professional performance of Blood Brothers at the phoenix Theatre. I also read a play called Blue remembered Hills byDennis Potter.The most useful material I looked at was the production of BloodBrothers at the Phoenix theatre. This is because it helped me withmany things. To begin with I watched and listened to Edward verycarefully. I took note of the way in which he moved and the facialexpressions he used. I also listened very carefully to the pace atwhich he spoke and the tone of his voice. I could then use this toimprove my personal performance. I also observed the costume Edwardwas wearing. This helped me to decide what I should wear for myperformance.I am now going to contrast and compare Blood Brothers with BlueRemembered Hills our piece was set in the 1950s. However, the otherplay was set in 1943 during the Second World War. The plays weresimilar because they both start ed off with people very happy, playingand enjoying themselves. However as they proceed both plays becamemore serious and in the end at least one person was killed in both.There also some differences. To begin with Blood Brothers is acyclical play. This means it starts at the end, then goes to beginningthen the end again. Blood Brothers also skipped large time gaps. Theplay showed scenes with the same characters as children teenagers andadults in the 1850s 60s 70s and 80s.

Blood Brothers :: Drama

Blood Br new(prenominal)sMy practical work in September to celestial latitude was for my acting option.It was a scripted piece of work based on the play Blood Brothers.There are various themes in Blood Brothers, a clash of class, romance,jealousy, and betrayal.My contribution to the performance was as a seven-year-old child. Iplayed Edward. Edward is a rattling reserved character. He is well spokenand polite and is very surprised and shocked at many things rice paddy(his friend) does. He comes from a wealthy background and so is usedto having everything he needs. He enjoys helping other people out.When acting as Edward I needed to have a very good posture. Holdingmyself well, this showed a contrast between Edwardss stop number class andMickeys lower class. I as well as needed to speak well, pronouncing mywords clearly. This again showed a clear contrast between Mickey andEdwards class.To help me in my work I looked for information on the Internet andbooks I looked at pictures of y oung boys in the 1950s, this helped mechoose my costume. I also listened to the Blood Brothers sound track,and watched a professional performance of Blood Brothers at thePhoenix Theatre. I also read a play called Blue remembered Hills byDennis Potter.The most useful material I looked at was the production of BloodBrothers at the Phoenix theatre. This is because it helped me withmany things. To begin with I watched and listened to Edward verycarefully. I took promissory note of the way in which he moved and the facialexpressions he used. I also listened very carefully to the pace atwhich he spoke and the whole tone of his voice. I could and then use this toimprove my personal performance. I also observed the costume Edwardwas wearing. This helped me to decide what I should wear for myperformance.I am direct going to contrast and compare Blood Brothers with BlueRemembered Hills our piece was set in the 1950s. However, the otherplay was set in 1943 during the Second World War. The plays were homogeneous because they both started off with people very happy, playingand enjoying themselves. However as they continued both plays becamemore serious and in the end at least(prenominal) one person was killed in both.There also some differences. To begin with Blood Brothers is acyclical play. This means it starts at the end, then goes to beginningthen the end again. Blood Brothers also skipped large time gaps. Theplay showed scenes with the same characters as children teenagers andadults in the 1850s 60s 70s and 80s.

Macbeth by Shakespeare Essay

Shakespeares play Macbeth follows the tragic downf only of a great man. Macbeth was once thought of as noble and valiant but by the conclusion of the play, a dead butcher. The murder of King Duncan marks the beginning of Macbeths downfall. This is much than than a result of Macbeths vaulting ambition than his belief in the supernatural. However, it is Macbeths belief in the supernatural that makes him continue on the path to downfall and ultimately lose all his honourable qualities.In Macbeth the witches symbolise the supernatural. The unearthly sisters evoke Macbeths ambition they know how Macbeth will react to their prophecies so they toy with him and deceive him by saying one thing but significance a nonher. The witches necessitate no conscience they cause mischief on purpose and enjoy it. The witches provide the foundation for Macbeths downfall by telling him that he shalt be king hereafter. When Macbeth hears the witches prophecies, horrible imaginings are opened in his m ind. Unlike Banquo who dismisses the witches prophecies, Macbeth contemplates regicide. The witches plant the seed to Macbeths downfall. He wants the witches to stay, you imperfect speakers. Tell me more. This shows that Macbeth believes in the whim that he can be king, and that he perhaps has thought about regicide before.Lady Macbeth is also a large contributing factor to the regicide. If Lady Macbeth was not behind Macbeth plotting the death of King Duncan and manipulating Macbeth into doing The deed, none of the deaths would have occurred, therefore there would be no downfall for Macbeth. Macbeth believes that if witness will have me king, why chance may crown me without my stir, whereas after Lady Macbeth reads the letter Macbeth sends to her, without hesitation, she thinks of regicide. Lady Macbeth knows that Macbeth is too full othmilk of human kindness and that she will have to persuade him. Despite Macbeth wanting to proceed no further of this business, Lady Macbeth convi nces him by questioning his pride, but screw your courage to the sticking-places, and saying that only when you durst do it, then you were a man.Lady Macbeth sees her femininity as an obstacle towards achieving her ambition, so she calls upon you spirits that tend of mortal thoughts to stop up the access and passage to remorse.After Macbeth is colonized and bend up about the murder of King Duncan, he develops a guilt complex which causeshim to see hallucinations. Just before Macbeth carries out the regicide, he sees an misrepresentation of a dagger, he questions is this a dagger which I see before me, or a dagger of the mind, a false creation. Macbeth slowly becomes more and more paranoid.Immediately after the regicide he thinks he hears voices crying sleep no more Macbeth does murder sleep. The more paranoid Macbeth becomes the more people he murders, and the more people he murders the more paranoid he becomes, this is one of the reasons for Macbeths downfall. Macbeth also murders Banquo, because he suspects Banquo of knowing the truth. However, afterwards at the banquet, Macbeth sees apparitions again, this time the ghost of Banquo. Macbeth develops paranoia, which leads Macbeth to go find the witches again to seek guidance.The loss of Macbeths honourable qualities and the reason Macbeth continues on the road to downfall is ultimately caused by his belief in the supernatural. Macbeths belief in the supernatural uncovers his fatal flaws. Because of Macbeths belief in the supernatural, he goes to find the witches again, and after seeing the apparitions he feels indestructible. Macbeth becomes overly ignorant, arrogant and super paranoid, he lets his belief in the supernatural get the better of him. Macbeth relies too much upon the witches apparitions he has no doubts and believes I bear a charmed heart which most not yield to one of woman born. Macbeth feels that no one can harm him and take his throne, so he tells the servants to bring me no more reports, let them fly all. Macbeth does not care about anything any longer he truly and completely believes he is invincible.Despite the witches telling Macbeth the prophecies and Lady Macbeth button him to murder the King, it was Macbeth that commits the regicide and continues on to the murder of Banquo. Macbeths downfall is a result of his belief in the supernatural. His weakness is relying too much upon the witches apparitions, which subsequently unveiled all his personality flaws and ultimately caused his downfall.

Managing Creativity of Shanghai Tang Essay

print pinch was founded by David sea tangle in Hong Kong in 1994. It was a retail store selling high quality product make in China, such as traditional Chinese costumes, Chairman Mao wrist watch, qipao, traditional Chinese silk products with Chinese concept. Its target customers at the first property were those high ended tourists. Taking around 1 year, instead of 2 year which is typical period a naked retailers need to make argumentation in break even, Shanghai Tang turned its first pro setting in October 1995. Shanghai Tang later entered an agreement with the Richemont Group which is a noteworthy Switzerland-based luxury goods maker. David Tang thought Shanghai Tang would become Chinas first international luxury brand.Like other ambitious entrepreneur, in Nov 1997, Tang opened the first Shanghai Tang store on Madison Avenue in New York ground forces. However, things were not going the way it was mantic to. Not many people liked what Shanghai Tang was selling. Unfortunatel y, the financial crisis worsened the situation, and it had to scale down the subscriber line. By 2001, Tang had reduced his stake in Shanghai Tang to near 5%, so the Richemont Group alsok control of the federation. Appointing executive chairman of Shanghai Tang in September 2001, Raphael Le Masne, who wherefore employed a new creative director, Joanne Ooi, having intensive experience in international garment business.With correct insight and vision, shootd to a greater extent in dwelling house designers, and fixed the right directions, they were successfully turning Shanghai Tang around. Image of Chinese-themed high-end room and lifestyle emporium had been established. Sales and globose coverage had been increased a lot from 2001 to 2008. By summer of 2008, accompany had more than 40 stores in 14 countries all over the world. Things will never run smooth, at the aforementioned(prenominal) time, Joanne had handed in her resignation. An increasing conflict within company bet ween creators and commercial departments also gave big headache to Raphael who forever relied on Joanne to smooth things out.During that time, the global financial crisis was striking the whole world economy again, while China, still with double digit increase in GDP e real year, was considered a shelter and gold mine for every business. Shanghai Tang has no exception, but tried to expand its business in China market. Should Shanghai Tang hire a new creative director under this uncertain economic time? How to strike a balance between creativity and theprofit? How the company can principal(prenominal)tain its success? How the company should adapt its strategy to make it successful in China market and other potential global markets? These were the main challenge Shanghai Tang were facing.AnalysisConflicts between Creativity/Innovation and Business SenseAs a business, the onetime(prenominal) success circumstanceors are eer considered as a critical factor. Business people have a r angeency not to deviate this much in order to maintain the success. On the other hand, it will certainly jeopardize the creativity and innovation from designers particularly it is considered it is too much several(predicate) from the successful factors or past evidences already jump it had not worked out. For creative people, they always tend to be very rationalisey and creative, so they try to make something completely new which is of course totally different than the past.However, like Shanghai Tang, it had experienced a very bad time, and a new design strategy raise by Le Masne and Ooi, had made the company turn around. It just likes an endless cycle company having a success factor, makes itself become successful in terms of profit and image, then it will turn down to change and the success may last for another couple of years. At the end, with emerge of new competitors, change in economy or whatever reason, past success factors may no longer work. The company may be force d to think deeply in creativity and business innovation, but it is always too late. A real successful company must be able to continue its success factors, but new elements must always be needed to be added in its business.In Shanghai Tang, from design process to the start of mass production, there was heavy involvement from a Product committee which comprise the designers, executive chairman, the creative director and key business managers like retail /marketing/ merchandizing directors. To help designers to understand how different products perform in the market, they veritable reports from the retail and marketing departments regularly. Those reports mainly revealed the sales and customer feedback to different items. Designers also had to follow the company norm Shanghai Tang DNA which contains 2 major elements Chinese-ness and the use of bright color.Excellent design but expensive to produce will be eliminated. Designers are also paid a salary positively charged subsidy ba sed on KPI such as its generating revenue and ability to innovate. It can simply observethat there were too much constraints to the design process. likely generating revenue becomes a very important element to determine the success of the new design. New design which is not similar with past success factors is unbelievable to survive.The consequence is that it will eliminate some new elements which make the company even successful in future. Shanghai Tang has 2 main business streams core collection and seasonal collection, which about 50/50 in terms of revenue. In fact, for its seasonal collection, extra room should be given to the design teams, and more deviation from Shanghai Tang DNA should be allowed. It can let the company to test the water temperature in the market and to get more insight how the market is changing. It would not impact to its core business. Shanghai Tang could even think of the 3rd stream which more innovation would be allowed.The bonus scheme for designers would also be linked to the recognition of their design. The Shanghai Tang DNA should be reviewed from time to time to keep pace with the market trend to make sure the new fashion elements will be captured. Thus, those designers will be motivated and encourage to participate in the theme of their design. As Shanghai Tang is a high end fashioned product, it should not be limited to high manufacturing cost as well. Margin can be set higher for products with nice design but higher manufacturing cost. Replacement of Creative DirectorOoi was going to leave Shanghai Tang, but the global economy was in the tough situation. Le Masne was in a dilemma to hire a new replacement or let the whole team to continue the work. Considering the fact that the Creative Director is the soul of the company which can define the main frame of the products and company direction, it is indeed an urgent matter that they have to hire the new replacement or promote internally.Promoting internally may create con flicts inside the team, and it doesnt add any new element to the team. Unless there is someone very outstanding, Shanghai Tang could look for a competent replacement externally. Working a whole team without leader is not going to work especially there are too much conflicts between the design and commercial teams and no one is able to resolve it. Expanding China MarketShanghai Tang had around 10 shops in China. It also had 9 shops in Hong Kongwhich can be considered crossover between eastern and western culture, not pure Chinese taste. The Chinese society accounted totally half of its total shops all over the world. However, looking deeply to its customer profiles, its major customer group was still USA and Europe. The Chinese customer in mainland China market was just over 50%. American and European might be in favour of existing product design in Shanghai Tang.However, it doesnt imply that the Chinese customers are with the similar taste. Foreigners may be in favour of design wit h fashion and absolute Chinese styled, but Chinese may desire the design with mix of Chinese and Western style, and do not want it to be too Chinese. They may be even reluctant to accept the goods Made in China, as there was a trend wealthy people tend to buy foreign luxary brand. Shanghai Tang had experience that different culture may have different taste. Design had to be fine tuned somehow to fit different culture.In terms of customer age group, excluding Mainland China market, the target age group was those between 36 to 45. However, in China market, theres also 40% of customer from age group 26-35. Among the 7 key in house designers in Shanghai Tang, though 3 of them were Chinese, they did not genuinely have exposure in China. For the rest, they were foreigner but had certain exposure in eastern fashion industry, not much in China yet.It was prove their design can quite hit the western market with age group 36-45, but it did not imply it will work abruptly in China market an d younger age group. It is advisable that Shanghai Tang should conduct a thorough research in China to define the Chinese taste. Moreover, they should also bring in some designers with Mainland Chinese exposure. They should also add in innovation elements into the business in order to compete the market shares in China.Jacket in Chinese, skirt in more western may not work for US/European customers, but it may work perfectly in China market. Using the famous western celebrities to promote its brand in China may increase its awareness effectively. Chinese may not perceive Shanghai Tang as Real Chinese Stuff only, but also an icon of western fashion. ConclusionThough Shanghai Tang had been quite successful in the past years, it cannot simply stick to it. It has to keep its creativity and innovation, and bring new elements to the company. Thorough preparation is essential for its battle in Mainland China Market.

How Is Romeo and Juliet Relationship Presented

One of the plays most consistent visual motifs is the contrast between light and dark, practically in terms of iniquity/day imagery. Need Evidence This contrast is not given a particular metaphoric meaninglight is not ever so good, and dark is not always evil. On the contrary, light and dark are generally used to provide a sensory contrast and to hint at opposed alternatives. One of the more important instances of this motif is Romeos lengthy meditation on the sun and the moon during the balcony scene, in which Juliet, metaphorically described as the sun, is seen as banishing the envious moon and transforming the night into day (2. . 46). A similar blurring of night and day occurs in the early morning hours after the extolrs only night together.Romeo, forced to leave for exile in the morning, and Juliet, not wanting him to leave her room, both try to pretend that it is still night, and that the light is actually darkness more(prenominal) light and light, more dark and dark our w oes (3. 5. 36) The Inevitability of Fate In its first address to the audience, the Chorus states that Romeo and Juliet are star-crossedthat is to say that fate (a power often vested in the movements of the stars) controls them (Prologue. ). This sense of fate permeates the play, and not just for the audience. The characters also are quite aware of it Romeo and Juliet constantly see omens. When Romeo believes that Juliet is dead, he cries out, Then I throw you, stars, completing the idea that the love between Romeo and Juliet is in opposition to the decrees of destiny (5. 1. 24). Of course, Romeos defiance itself plays into the hands of fate, and his determination to spend eternity with Juliet results in their deaths.The utensil of fate works in all of the events surrounding the lovers the feud between their families (it is worth noting that this hatred is never explained rather, the reader must accept it as an requisite aspect of the world of the play) the horrible series of acci dents that ruin Friar Lawrences seemingly well-intentioned plans at the end of the play and the tragic timing of Romeos suicide and Juliets awakening. These events are not mere coincidences, but rather manifestations of fate that help bring about the unavoidable outcome of the young lovers deaths.The invention of fate described above is the most commonly accepted interpretation. There are other possible readings of fate in the play as a force determined by the powerful social institutions that influence Romeo and Juliets choices, as well as fate as a force that emerges from Romeo and Juliets very personalities. Link this to Friar Lawernce STRUCTURE -briefly answer the question awhat the descent is like, what sort of impression you get of the relationship. Second paragraph)-Context- write about how men/women were suppositious to behave when the plays were written a how is this reflected in the text? (this bit is essential for Band 4/5). How do you think an audience might have resp onded to the relationship when the play was first performed? How might a modern audienceas response be different? (Third paragraph)- How language reveals the relationship- have 3 or 4 key examples of language that reveal to you the relationship a might be a simile /metaphor that has been used or a striking phrase/word.Try to select them from the different scenes you are focusing on. For each make sure you analyse how the word/phrase suggests ideas about the relationship, not just what it suggests. (Fourth paragraph)- How structure reveals the relationship- think in particular about the relationship changes as the play progresses- analyze how Shakespeare shows the relationship changing as the play goes on. (Fifth paragraph)- How dramatic techniques reveal the relationship- this might include the condition actions stage directions.Try to find one or two examples Shakespeare presents Romeo & Juliets early relationship as a love-hate affair. By this I mean that although they love eac h other immensely, they are surrounded by the hate of their two families Arranged marriages were very common at the time depending on your social status and love had no meaning. At the time marrying at the age of 12 appeared to be normal, however now is frowned at.

Management and Team Essay

A collection up displaying effective squad performance be identified as having several characteristics. These argon Clear Goals These argon intrinsic and visualize that the group up as a whole are functional towards the same(p) positive outcome thus ensuring an effective and in tune group. Defined Roles and chisel descriptions In order to encourage effective team performance it is critical that each team member in each position has definitive roles. When these are not come abouted confusion is rife and the performance of the team underside be severely damaged. Defined roles ensure that each team member is doing what they should, know who to harbinger on for assistance in a certain situation and makes for smooth and effective team performance. Open and clear communication This ensures that all team members wax indoors a trusting and healthy intersomebodyal forum at work. Open and clear communications are the only way to ensure that a team performs effectively. Exce llent communication throughout the team ensures that the team are working in sync with each other, making a strong in tune outfit. Effective decision making Teams essential receive training on effective decision making procedures in order for them to be effective. demo moreDefine the Key Features of Effective Team PerformanceIf a team follows this training then the performance is enhanced across the whole team and mostly the best decisions are made victimization this process. Time is often saved using these techniques making for a happier healthier team. Participation from all team members Regular condemnation and forums to gather all team members are essential to the performance of the team. It not only empowers workers to give their opinion but allows and encourages all indoors a team to have their ideas and opinions validated thus making a positive enhancement on the team. Participation also encourages the sharing of ideas and knowledge within the team creating diversity wit hin the team as a whole. Valued diversity Every team member has a different idea, opinion or approach and so bringing all of the differences from each individual together this enhances the knowledge and diversity of the team as a whole. For instance some champion may be methodical and the other creative. Between them they should have all avenues covered to make a swell rounded team when put together. Recognising, at supervisions and team meetings, each individuals strengths enables the team to have m whatever sides to use and in turn enhances performance.Managed counterpoint is essential to effective team performance as it stops issues and problems from being ignored. It is a safe and positive way to have a bun in the oven problems and bring out new ideas in order to solve and put at ease any members of the team affected by this conflict. It gives team members a chance to be heard and a solution sought in order to keep the team the best that it sack up be. Positive atmosphe re People who are happy in their work have been proven to be more productive than those who disfavour their position and so positivity is a must at all times within a team. Cooperative relationships An effective team gets along well and takes knowledge from co workers to improve the things they are less able to do alone. It has been proven that optimum team performance is achieved by team members who get along with each other. Participative leadership An effective team with have leadership who are slap-up role models.The leaders will be involved in the same type of work as all team members on occasion and generate that they themselves can and will work as part of the team, even if they are at the top. It has been written that it should be difficult to identify the leader in effective teams upon observation. (Bruce Tuckman 1965) states that the best way to gain effective team performance is to follow his theory of Forming, Storming, Norming and performing. Following his theory tool is definitely a way forward and shows key features of effective team performance. When each stage is followed , a team should reach maximum performance speedily and with ease. (www.mindtools.com/pages/article/newLDR_86.htm) To summarise the features of effective team performance are to have a happy knowledgeable team who meet regularly, share experiences, ideas and knowledge, to ensure team members are valued and to create a happy work environment where conflicts are safely managed and management are actively gather inn and involved within the team. With all of the above features homely within an organisations team should ensure and show excellent team performance.1.2 IDENTIFY THE CHALLENGES EXPERIENCED BY DEVELOPING TEAMSChallenges experienced by developing teams include the notion that team members may become overwhelmed if the development is handled wrongly.Psychologist Bruce Tuckman (1965)says that in the initial stages of team development it is motiveless for teams to bec ome overwhelmed by expectations of what they are being asked to do. However if the theory and tools are use correctly, team development is planned carefully, and the plan followed then all challenges can be overcome. With any team, disengagement should be looked for as if members do not participate then they will not achieve the desired outcome. Reading many an(prenominal) papers on the matter lead me to believe that if you keep a team engaged, have a stringent plan, outline goals for the team and orchestrate using tried and tested methods, all challenges that may arise will be overcome. Time is always difficult to find with developing teams to enable the progression. Regular time should be allowed for team performance meetings and it is always difficult to orchestrate a workforce to develop. However difficult, a conclusion from my reading is that teams need time and without fair to middling time and a leader who knows what they are doing, developing teams can be seriously impaired .1.3 IDENTIFY THE CHALLENGES EXPERIENCED BY ESTABLISHED TEAMSEstablished teams can be difficult to change. If an launch team has effective team performance then great but if an established team has team performance that is not good then an established team may be difficult to shape as they will not be used to the new strategies and ideas involved in upping their performance. Following on as a Manager into an environment already established is always difficult. Bringing in new ideas it is written that established teams can become un nerved and so this should be done using tried and tested techniques. It is important to identify in established teams what already works and grow with that and to slowly bring in new ideas and routes to enhance performance. Resistance to change is a natural response by human beings and should be looked out for and overcome quickly so as not to see a decline in positivity within established teams. (http//www.change-management.com/tutorial-pm-cm.htm)1.4 E XPLAIN HOW CHALLENGES TO EFFECTIVE TEAM PERFORMANCE smoke BE OVERCOME.Team performance challenges can be overcome by using communication. Excellent communication is at the forefront of any issues and challenges with team performance. comprehend is also equally as important as the team should be viewed as a whole and not as an individual. This builds trust and can be imperative in overcoming issues. The whole team will fell valued.Team time spent together can also be productive socialize and getting to know each other can alleviate stresses and strains and can make a firm foundation for problem solving.All team members must be treated as equals. Problems will come if team members feel that one member receives special treatment.A Manager must also show consistency when completing all tasks with the team. Offers of helping team members is a positive solution as if you do the work yourself and complete the task others will see that you are an equal and should respond positively.1.5 meditate HOW DIFFERENT MANAGEMENT STYLES MAY INFLUENCE OUTCOMES OF TEAM PERFORMANCE.There are five management bearings widely used throughout the world today. They are Authoritarian Management A Manager at the head of the company decides how things are to be done and how each team member will work. There is no room for autonomy within this bearing. The Manager is solely responsible for devising company policy and implementing it. This Management elbow room could isolate the team when used alone although there is a place in certain settings for this management style to be used.Democratic Management this intemperately involves all employees and encourages them to have a sense of ownership and to be a part of the decision making process within a company. This Management style to me seems to be the most productive in encouraging teamwork. Democratic Management is what is used in my own company through team meetings. Staff are involved in the changing of systems and work and it i s very productive.Paternalistic Management is where the boss acts as a parent to the employees. This style encompasses employees social needs into the equasion and looks at them as a whole person instead of just a worker. I can see how this management style creates a friendly working environment although my worry would be that employees would become too familiar. I believe this syle to enhance team performance .Autocratic Management This is when a Manager makes decisions in line with their opinions and views and completely alone without the assistance of a team. This management style can leave employees feeling under valued and that they can have no opinion or say in how things are to be. There is no team involvement in this style. Autocratic management can often indue as a well run company on the outside but on the inside employees may be dissatisfied thus team performance damaged severely.Laissez faire management This management theory is defined as having employees that mana ge their own sections of the business and the over all manager watches from a distance. I like this management theory as at team meetings each member who has been responsible for their own areas can bring in their own achievements, problems and findings. If the repair people are placed in the right position this theory can be very productive. The over all manager can take more of a back seat if done well and the individuals can run the company but only if they are self motivated individuals.1.6 ANALYSE METHODS OF DEVELOPING TRUST AND ACCOUNTABILITYA democratic management style clearly develops trust as every team member is involved in the processes and operations of the company.Developing trust is based around communication. As we interact, question, disagree with and support decisions in a positive way, we build confidence in one some other and promote overall group success ( Limas 2003)Key elements for developing trust are communication methods. Active listening, body language , group interaction and group communication must beused, observed and acted upon. This is why team meetings encompassing these are essential to develop group trust. The leader of all team interactions must be familiar with all communication methods with a view to trust buildings.Accountability development.Robust performance is based guided accountability. Within an organisation it is essential that all involved know their accountability and when this is acted upon. For instance Managers are accountable for ongoing training and how this is guided through supervisions and passed onto staff members.In a domiciliary care setting all workers have a job description and code of practice. They also receive training on safeguarding, record keeping and any other aspect of their day to day role. When workers have signed policies and procedures they will be aware of their own individual accountability. It is essential that regular training and updates are done so that all know who is accounta ble and for what within a care setting. The Management need training in these matters so that they know their own responsibilities. Accountability is genuine when own responsibility is defined.1.7 COMPARE METHODS OF ADDRESSING CONFLICT WITHIN A TEAM.There are many methods of addressing conflict within a team. (Goldfien and Robbennolt 2007) developed a dual model based upon assertiveness and empathy and have proven that linking these together using their five conflict resolution is productive. These are avoidance conflict adopting a wait and see approach. This can often lead the conflict to go out of control.Yeilding conflict style this is based upon having more regard for the person creating the conflict than of ones own self. This is used by individuals who with to keep social situations pleasant. They give into demands so as to not upset the apple cart.Competitive conflict style this maximises individual assertiveness and minimises empathy. This style is used for dominating pe ople who simply wish to win or lose. This is an aggressive style of conflict that usually involves literary argument and shouting and power games. It is based upon feelings of intimidation (Morrill 1995)Cooperation conflict style This style is when the individual takes into account two sides of the conflict and to the best possible outcome for both parties. This style is based around the conflict being dealt with assertiveness and empathy in equal measure. According to literature that has been written on conflict resolution a cooperative conflict resolution style is recommended above all others ( Sternberg and Dobson 1987)Conciliation conflict style This style is based around fairness. Giving and taking actions are evident to reach half way thus promoting conflict resolution. This style is both yielding and co operative combined.When a conflict is evident then the management should seek these styles and act accordingly dependant on the nature of the conflict. It is also about ad hominem preference and characteristics of the individuals involved in the conflict.My personal method of dealing with conflict within my company is the consiliation style first and foremost but with the different styles above I can use another if my natural choice is unsuccessful which it is not very often it does not work.ReferencesBruce Tuckman . (1965). Managent theories and styles. accessible http//www.mba-online-program.com. Last accessed 14th sept 2012.Goldfien and Robbennolt. (2007). conflict resolution. Available http//en.wikipedia.org/wiki/conflict_resolution_conflict_management. Last accessed 08th oct 2012.management. (2011). established teams. Available http//www.change-management.com/tutorial-pm-cm.htm). Last accessed 08th oct 2012.MIT human resources. (2012). Accountibility. Available http//hrweb.mit.edu/performance-development/accountibility. Last accessed 08th oct 2012.Morill and Sternberg and Dobson. (1995). Conflict resolution. Available http//en.wikipedia.org /wiki/conflict_resolution.conflict_management. Last accessed 08th oct 2012.Univerity of Florida. (2010). Developing trust and co operation. Available http//edis.ifas.ufl/fy748. Last accessed 08th oct 2012.wikihow. (2010). How to build trust. Available http//www.wikihow.com/build-trust. Last accessed 08th oct 2012.

Club Objective in Gym Facility Essay

Member Communication Budget Management Department Management Facility Upkeep Club Objective Our mean goal is to identify the dexterity and competencies that a fitness professional needs in order to be a successful in fitness proprietor, or department manager. How to operate a successful business in the fitness intentness. Teaching them how business concepts, including purchasing, contractual agreements, risk management and negligence, and other fiscal concerns in fitness facility.Identify how to influences consumers and creates fitness services and cooperating with the customers relationship. Mostly using of sales techniques explore the profession as a potential career by using experience on the field of view internship and with available resources. Some of the objectives in managing lyceum facilities are. We need to plan what equipment will be more useful in the gym facility for costumers. You need to put up sure the costumers feel comfortable in your facilities and everything there is well organize and clean. We need to do things that will attract them more, for them to always come to your facility.Treating your employees well and your costumers and providing client with good services. Advertising on TVs, radios and making your cards and available to give it out. Organizing the employees and making sure they doing what are right. Try to listen to your costumers when they bring many ideas of things you need in your gym facilities. You have to help your costumers reach their fitness objectives. The manager works with the membership department to increase monthly goals for new memberships and comes up with ways to better retain current members. We have to spend a lot of time developing personal training to increase its revenue.Interviewing Christian Kettman gym facility she made me know that to get to your goals hire two new trainers 175 clients on contract by the New Year. Education is the secret heavy weapon all staff must be certified in and must eas ily communicate with all clients and staff. The hiring process is most important in the industry it is all about team work and making complete strangers feel comfortable sweating in front of one another. And again to be an owner or reach your objectives sweat, tears and a constant motivation to keep going forward and be passionate about health and fitness by all means.

Dai Park Textbook

Stochastic Manufacturing & Service Systems Jim Dai and Hyunwoo Park School of Industrial and Systems Engineering Georgia set of Technology October 19, 2011 2 Contents 1 Newsv determinationor enigma 1. 1 Pro? t Maximization 1. 2 greet Minimization . 1. 3 initial Inventory . . 1. 4 Simulation . . . . . . 1. 5 utilisation . . . . . . . 5 5 12 15 17 19 25 25 27 29 29 31 32 33 34 39 39 40 40 42 44 46 47 48 49 51 51 51 52 54 55 57 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 aligning Theory 2. 1 Introduction . . . . . . . 2. 2 Lindley comparison . . . . 2. 3 Tra? c Intensity . . . . . 2. 4 Kingman Approfessional personximation 2. 5 Littles Law . . . . . . . 2. 6 Throughput . . . . . . . 2. 7 Simulation . . . . . . . . 2. 8 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 clear-cut Time Markov range of mountains 3. 1 Introduction . . . . . . . . . . . . . . . . . . . . 3. 1. 1 State Space . . . . . . . . . . . . . . . . 3. 1. 2 Transition Probability Matrix . . . . . . 3. 1. 3 Initial Distri plainlyion . . . . . . . . . . . . 3. 1. 4 Markov Property . . . . . . . . . . . . . 3. 1. 5 DTMC Models . . . . . . . . . . . . . . 3. 2 come outary Distribution . . . . . . . . . . . . . 3. 2. 1 Interpretation of Stationary Distribution 3. 2. 2 Function of Stationary Distribution . . 3. 3 Irreducibility . . . . . . . . . . . . . . . . . . . 3. 3. 1 Transition Diagram . . . . . . . . . . 3. 3. 2 Accessibility of States . . . . . . . . . . 3. 4 Periodicity . . . . . . . . . . . . . . . . . . . . . 3. 5 Recurrence and Transience . . . . . . . . . . . 3. 5. 1 Geometric Random Variable . . . . . . 3. 6 soaking up Probability . . . . . . . . . . . . . . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. 7 3. 8 3. 9 3. 0 Computing Stationary Distribution Using Cut Method Introduction to binomial Stock Price Model . . . . . . Simulation . . . . . . . . . . . . . . . . . . . . . . . . . Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS . . . . . . . . . . . . . . . . . . . . 59 61 62 63 71 71 72 73 75 78 80 80 80 82 84 91 91 96 97 100 101 103 103 104 106 107 107 108 109 111 111 117 117 130 135 148 159 4 Poisson Proce ss 4. 1 Exp unityntial Distribution . . . . . . . 4. 1. 1 Memoryless Property . . . . 4. 1. 2 Comparing Two Exponentials 4. 2 Homogeneous Poisson Process . . . . 4. 3 Non-homogeneous Poisson Process . 4. thin and Merging . . . . . . . . 4. 4. 1 Merging Poisson Process . . . 4. 4. 2 Thinning Poisson Process . . 4. 5 Simulation . . . . . . . . . . . . . . . 4. 6 Exercise . . . . . . . . . . . . . . . . 5 Continuous Time Markov Chain 5. 1 Introduction . . . . . . . . . . . 5. 1. 1 H one cartridge holder(a)ing Times . . . . . 5. 1. 2 Generator Matrix . . . . 5. 2 Stationary Distribution . . . . 5. 3 M/M/1 Queue . . . . . . . . . 5. 4 Variations of M/M/1 Queue . . 5. 4. 1 M/M/1/b Queue . . . . 5. 4. 2 M/M/? Queue . . . . . 5. 4. 3 M/M/k Queue . . . . . 5. 5 Open Jackson Nedeucerk . . . . . 5. 5. 1 M/M/1 Queue Review . 5. 5. 2 Tandem Queue . . . . . 5. 5. Failure Inspection . . . 5. 6 Simulation . . . . . . . . . . . . 5. 7 Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Exercise coiffures 6. 1 Newsv exterminateor chore . . . . . . . 6. 2 Queueing Theory . . . . . . . . . 6. 3 Discrete Time Markov Chain . . 6. 4 Poisson Process . . . . . . . . . . 6. 5 Continuous Time Markov Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter 1 Newsvendor Problem In this course, we leave alone learn how to design, analyze, and manage a manufacturing or gain arrangement with uncertainty. Our ? rst step is to come across how to solve a single period decisiveness problem pangtaining uncertainty or hit-or-missness. 1. 1 Pro? t Maximization We forget start with the simplest courting look ating perishable points. Suppose we be lead a business retailing newspaper to Georgia Tech campus. We m other(a) to gear up a speci? c event of copies from the publisher every(prenominal) as yeting and transfer those copies the next day.One day, if t here(predicate) is a big news, the number of GT people who want to buy and read a paper from you whitethorn be ve ry high. some other day, people may just not be interested in reading a paper at all. Hence, you as a retailer, lead hit the invite variability and it is the primary uncertainty you contract to handle to keep your business sustainable. To do that, you want to know what is the optimal number of copies you need to tack together every day. By intuition, you know that in that respect get out be a few other factors than engage you need to consider. Selling chime (p) How often durations leave behind you charge per paper? Buying price (cv ) How much will the publisher charge per paper? This is a variable exist, imagineing that this live is proportional to how umpteen you order. That is why it is denoted by cv . Fixed ordering price (cf ) How much should you pay just to shoes an order? dressing greet is ? xed regardless of how some another(prenominal) you order. Salvage value (s) or holding approach (h) There be two cases about the left over items. They could ca rry some monetary value even if expired. Otherwise, you behave to pay to get rid of them or to storing them. If they have some value, it is nameed salvage value. If you have to pay, it is called 5 6 CHAPTER 1.newsvendor PROBLEM holding monetary value. Hence, the pursual family holds s = ? h. This is per-item value. Backorder cost (b) Whenever the actual demand is higher than how many you inc grooved(p), you lose sales. Loss-of-sales could cost you something. You may be bookkeeping those as backorders or your brand may be damaged. These costs will be represented by backorder cost. This is per-item cost. Your order quantity (y) You will decide how many papers to be coherent ahead you start a day. That quantity is represented by y. This is your decision variable. As a business, you argon assumed to want to maximize your pro? t. Expressing our pro? t as a proceed of these variables is the ? rst step to obtain the optimal ordering constitution. Pro? t muckle be interpreted in two shipway (1) revenue minus cost, or (2) money you earn minus money you lose. let us adopt the ? rst interpretation ? rst. taxation is represented by exchange price (p) multiplied by how many you actually sell. The actual sales is bounded by the realized demand and how many you realised for the period. When you order overly many, you drive out sell at most as many as the number of people who want to buy. When you order too few, you fecal matter precisely sell what you prepargond. Hence, your revenue is minimum of D and y, i. . min(D, y) or D ? y. Thinking about the cost, ? rst of all, you have to pay something to the publisher when buying papers, i. e. cf +ycv . Two types of additional cost will be incurred to you depending on whether your order is above or below the actual demand. When it turns out you prepared less than the demand for the period, the backorder cost b per every missed sale will occur. The amount of missed sales loafernot be negative, so it hatful be re presented by max(D ? y, 0) or (D ? y)+ . When it turns out you prepared to a greater extent than than, the quantity of left-over items too cannot go negative, so it can be expressed as max(y ? D, 0) or (y ? D)+ .In this way of thinking, we have the following locution. Pro? t =Revenue ? Cost =Revenue ? Ordering cost ? Holding cost ? Backorder cost =p(D ? y) ? (cf + ycv ) ? h(y ? D)+ ? b(D ? y)+ (1. 1) How about the entropy interpretation of pro? t? You earn p ? cv dollars every magazine you sell a paper. For left-over items, you lose the price you bought in addition to the holding cost per paper, i. e. cv + h. When the demand is higher than what you prepared, you lose b backorder cost. Of course, you also have to pay the ? xed ordering cost cf as well when you place an order. With this logic, we have the following pro? t business. Pro? t =Earning ?Loss =(p ? cv )(D ? y) ? (cv + h)(y ? D)+ ? b(D ? y)+ ? cf (1. 2) 1. 1. PROFIT MAXIMIZATION 7 Since we used two di? erent approaches to model the aforementioned(prenominal) pro? t function, (1. 1) and (1. 2) should be equivalent. Comparing the two equations, you will also notice that (D ? y) + (y ? D)+ = y. immediately our quest b covers down to maximizing the pro? t function. However, (1. 1) and (1. 2) contain a random element, the demand D. We cannot maximize a function of random element if we allow the randomness to remain in our objective function. One day demand can be very high. another(prenominal) day it is also possible nobody wants to buy a single paper. We have to ? ure out how to get rid of this randomness from our objective function. permit us denote pro? t for the nth period by gn for further discussion. Theorem 1. 1 (Strong Law of Large Numbers). Pr g1 + g2 + g3 + + gn = Eg1 n? n lim =1 The long- broaden comely pro? t converges to the evaluate pro? t for a single period with luck 1. Based on Theorem 1. 1, we can change our objective function from just pro? t to judge pro? t. In other word s, by maximizing the evaluate pro? t, it is guaranteed that the semipermanent median(a) pro? t is maximized because of Theorem 1. 1. Theorem 1. 1 is the foundational presumptuousness for the entire course.When we will talk about the long-run clean something, it involves Theorem 1. 1 in most cases. Taking expectations, we obtain the following equations comparable to (1. 1) and (1. 2). Eg(D, y) =pED ? y ? (cf + ycv ) ? hE(y ? D)+ ? bE(D ? y)+ =(p ? cv )ED ? y ? (cv + h)E(y ? D)+ ? bE(D ? y)+ ? cf (1. 4) (1. 3) Since (1. 3) and (1. 4) are equivalent, we can choose both one of them for further discussion and (1. 4) will be used. in the first place moving on, it is important for you to understand what ED? y, E(y? D)+ , E(D ? y)+ are and how to cast them. Example 1. 1. Compute ED ? 18, E(18 ? D)+ , E(D ? 8)+ for the demand having the following dispersions. 1. D is a discrete random variable. Probability mass function (pmf) is as follows. d PrD = d 10 1 4 15 1 8 20 1 8 25 1 4 30 1 4 Answer For a discrete random variable, you ? rst fancy D ? 18, (18 ? D)+ , (D ? 18)+ for severally of possible D value. 8 d CHAPTER 1. newsdealer PROBLEM 10 1 4 15 1 8 20 1 8 25 1 4 30 1 4 PrD = d D ? 18 (18 ? D)+ (D ? 18)+ 10 8 0 15 3 0 18 0 2 18 0 7 18 0 12 Then, you take the weighted fair(a) using corresponding PrD = d for severally possible D. 1 1 1 1 1 125 (10) + (15) + (18) + (18) + (18) = 4 8 8 4 4 8 1 1 1 1 1 19 + E(18 ?D) = (8) + (3) + (0) + (0) + (0) = 4 8 8 4 4 8 1 1 1 1 1 + E(D ? 18) = (0) + (0) + (2) + (7) + (12) = 5 4 8 8 4 4 ED ? 18 = 2. D is a continuous random variable following uniform distribution amongst 10 and 30, i. e. D ? Uniform(10, 30). Answer Computing expectation of continuous random variable involves integration. A continuous random variable has probability density function usually denoted by f . This will be also needed to compute the expectation. In this case, fD (x) = 1 20 , 0, if x ? 10, 30 otherwise Using this information, compute the expectations directly by integration. ? ED ? 18 = ? 30 (x ? 18)fD (x)dx (x ? 18) 10 18 = = 10 18 1 dx 20 1 20 dx + 30 (x ? 18) x 10 dx + 18 30 (x ? 18) 1 20 dx 1 20 dx = = x2 40 1 20 + 18 x=18 x=10 18x 20 18 x=30 x=18 The key idea is to remove the ? actor that we cannot handle by separating the integration interval into two. The other two expectations can 1. 1. PROFIT MAXIMIZATION be computed in a resembling way. 9 ? E(18 ? D)+ = 30 (18 ? x)+ fD (x)dx (18 ? x)+ 10 18 = = 10 18 1 dx 20 1 20 1 20 +0 30 (18 ? x)+ (18 ? x) 10 x2 2 x=18 dx + 18 30 (18 ? x)+ 0 18 1 20 dx = dx + 1 20 dx 18x ? = 20 x=10 ? E(D ? 18)+ = 30 (18 ? x)+ fD (x)dx (x ? 8)+ 10 18 = = 10 18 1 dx 20 1 20 30 (x ? 18)+ 0 10 x2 2 dx + 18 30 (x ? 18)+ 1 20 dx 1 20 dx = =0 + 1 20 dx + 18 x=30 (x ? 18) ? 18x 20 x=18 Now that we have learned how to compute ED? y, E(y? D)+ , E(D? y)+ , we have acquired the basic toolkit to obtain the order quantity that maximizes the expected pro? t. startle of all, we need to turn th ese expectations of the pro? t function formula (1. 4) into integration forms. For now, assume that the demand is a nonnegative continuous random variable. 10 CHAPTER 1. NEWSVENDOR PROBLEM Eg(D, y) =(p ? cv )ED ? y ? (cv + h)E(y ? D)+ ? bE(D ? y)+ ? f ? =(p ? cv ) 0 (x ? y)fD (x)dx ? ? (cv + h) 0 ? (y ? x)+ fD (x)dx ?b 0 (x ? y)+ fD (x)dx ? cf y ? =(p ? cv ) 0 xfD (x)dx + y y yfD (x)dx ? (cv + h) 0 ? (y ? x)fD (x)dx ?b y (x ? y)fD (x)dx ? cf y y =(p ? cv ) 0 xfD (x)dx + y 1 ? 0 y y fD (x)dx xfD (x)dx ? (cv + h) y 0 y fD (x)dx ? 0 y ? b ED ? 0 xfD (x)dx ? y 1 ? 0 fD (x)dx ? cf (1. 5) There can be many ways to obtain the supreme point of a function. Here we will take the derivative of (1. 5) and set it to zero. y that makes the derivative equal to zero will make Eg(D, y) either maximized or minimized depending on the second derivative.For now, assume that such y will maximize Eg(D, y). We will check this later. Taking the derivative of (1. 5) will involve di? erentiating an integra l. permit us review an important result from Calculus. Theorem 1. 2 (Fundamental Theorem of Calculus). For a function y H(y) = c h(x)dx, we have H (y) = h(y), where c is a constant. Theorem 1. 2 can be translated as follows for our case. y d xfD (x)dx =yfD (y) dy 0 y d fD (x)dx =fD (y) dy 0 (1. 6) (1. 7) Also remember the relationship between cdf and pdf of a continuous random variable. y FD (y) = fD (x)dx (1. 8) 1. 1. PROFIT MAXIMIZATION Use (1. 6), (1. 7), (1. ) to take the derivative of (1. 5). d Eg(D, y) =(p ? cv ) (yfD (y) + 1 ? FD (y) ? yfD (y)) dy ? (cv + h) (FD (y) + yfD (y) ? yfD (y)) ? b (? yfD (y) ? 1 + FD (y) + yfD (y)) =(p + b ? cv )(1 ? FD (y)) ? (cv + h)FD (y) =(p + b ? cv ) ? (p + b + h)FD (y) = 0 If we di? erentiate (1. 9) one more clock to obtain the second derivative, d2 Eg(D, y) = ? (p + b + h)fD (y) dy 2 11 (1. 9) which is unendingly nonpositive because p, b, h, fD (y) ? 0. Hence, taking the derivative and setting it to zero will give us the maximum point no t the minimum point. Therefore, we obtain the following result. Theorem 1. 3 (Optimal Order Quantity).The optimal order quantity y ? is the smallest y such that FD (y) = p + b ? cv ? 1 or y = FD p+b+h p + b ? cv p+b+h . for continuous demand D. Looking at Theorem 1. 3, it provides the following intuitions. Fixed cost cf does not a? ect the optimal quantity you need to order. If you can procure items for free and there is no holding cost, you will prepare as many as you can. If b h, b cv , you will also prepare as many as you can. If the buying cost is almost as resembling as the selling price plus backorder cost, i. e. cv ? p + b, you will prepare nothing. You will prepare only upon you receive an order.Example 1. 2. Suppose p = 10, cf = 100, cv = 5, h = 2, b = 3, D ? Uniform(10, 30). How many should you order for every period to maximize your long-run middling pro? t? Answer world-class of all, we need to compute the criterion value. p + b ? cv 10 + 3 ? 5 8 = = p+b+h 10 + 3 + 2 15 Then, we will look up the smallest y value that makes FD (y) = 8/15. 12 1 CHAPTER 1. NEWSVENDOR PROBLEM CDF 0. 5 0 0 5 10 15 20 25 30 35 40 D Therefore, we can conclude that the optimal order quantity 8 62 = units. 15 3 Although we derived the optimal order quantity solution for the continuous demand case, Theorem 1. applies to the discrete demand case as well. I will ? ll in the derivation for discrete case later. y ? = 10 + 20 Example 1. 3. Suppose p = 10, cf = 100, cv = 5, h = 2, b = 3. Now, D is a discrete random variable having the following pmf. d PrD = d 10 1 4 15 1 8 20 1 8 25 1 4 30 1 4 What is the optimal order quantity for every period? Answer We will use the same value 8/15 from the previous example and look up the smallest y that makes FD (y) = 8/15. We start with y = 10. 1 4 1 1 3 FD (15) = + = 4 8 8 1 1 1 1 FD (20) = + + = 4 8 8 2 1 1 1 1 3 FD (25) = + + + = 4 8 8 4 4 ? Hence, the optimal order quantity y = 25 units.FD (10) = 8 15 8 15 8 15 8 ? 15 1. 2 Cost Minimization Suppose you are a action manager of a large friendship in charge of operating manufacturing annotations. You are expected to run the factory to minimize the cost. Revenue is another mortals responsibility, so all you care is the cost. To model the cost of factory operation, let us set up variables in a slightly di? erent way. 1. 2. apostrophize MINIMIZATION 13 Understock cost (cu ) It occurs when your achievement is not su? cient to meet the market demand. Overstock cost (co ) It occurs when you produce more than the market demand.In this case, you may have to rent a space to store the excess items. Unit production cost (cv ) It is the cost you should pay whenever you manufacture one unit of products. Material cost is one of this category. Fixed operating cost (cf ) It is the cost you should pay whenever you decide to start running the factory. As in the pro? t maximization case, the formula for cost expressed in terms of cu , co , cv , cf should be developed. Given random demand D, we have the following equation. Cost =Manufacturing Cost + Cost associated with Understock Risk + Cost associated with Overstock Risk =(cf + ycv ) + cu (D ? )+ + co (y ? D)+ (1. 10) (1. 10) obviously also contains randomness from D. We cannot minimize a random objective itself. Instead, based on Theorem 1. 1, we will minimize expected cost then the long-run average cost will be also guaranteed to be minimized. Hence, (1. 10) will be transformed into the following. ECost =(cf + ycv ) + cu E(D ? y)+ + co E(y ? D)+ ? ? =(cf + ycv ) + cu 0 ? (x ? y)+ fD (x)dx + co 0 y (y ? x)+ fD (x)dx (y ? x)fD (x)dx (1. 11) 0 =(cf + ycv ) + cu y (x ? y)fD (x)dx + co Again, we will take the derivative of (1. 11) and set it to zero to obtain y that makes ECost minimized.We will verify the second derivative is positive in this case. Let g here denote the cost function and use Theorem 1. 2 to take the derivative of integrals. d Eg(D, y) =cv + cu (? yfD (y) ? 1 + FD (y) + yfD (y)) dy + co (FD (y) + yfD (y) ? yfD (y)) =cv + cu (FD (y) ? 1) + co FD (y) ? (1. 12) The optimal production quantity y is obtained by setting (1. 12) to be zero. Theorem 1. 4 (Optimal Production Quantity). The optimal production quantity that minimizes the long-run average cost is the smallest y such that FD (y) = cu ? cv or y = F ? 1 cu + co cu ? cv cu + co . 14 CHAPTER 1. NEWSVENDOR PROBLEM Theorem 1. can be also applied to discrete demand. Several intuitions can be obtained from Theorem 1. 4. Fixed cost (cf ) again does not a? ect the optimal production quantity. If understock cost (cu ) is equal to unit production cost (cv ), which makes cu ? cv = 0, then you will not produce anything. If unit production cost and overstock cost are negligible equalized to understock cost, meaning cu cv , co , you will prepare as much as you can. To verify the second derivative of (1. 11) is indeed positive, take the derivative of (1. 12). d2 Eg(D, y) = (cu + co )fD (y) dy 2 (1. 13) (1. 13) is a lways nonnegative because cu , co ? . Hence, y ? obtained from Theorem 1. 4 minimizes the cost kind of of maximizing it. Before moving on, let us par criteria from Theorem 1. 3 and Theorem 1. 4. p + b ? cv p+b+h and cu ? cv cu + co Since the pro? t maximization problem solved previously and the cost minimization problem solved now share the same logic, these two criteria should be somewhat equivalent. We can see the connection by matching cu = p + b, co = h. In the pro? t maximization problem, whenever you lose a sale due to underpreparation, it costs you the opportunity cost which is the selling price of an item and the backorder cost.Hence, cu = p + b makes sense. When you overprepare, you should pay the holding cost for distributively left-over item, so co = h also makes sense. In sum, Theorem 1. 3 and Theorem 1. 4 are indeed the same result in di? erent forms. Example 1. 4. Suppose demand follows Poisson distribution with parameter 3. The cost parameters are cu = 10, cv = 5, co = 15. production plication that e? 3 ? 0. 0498. Answer The criterion value is cu ? cv 10 ? 5 = = 0. 2, cu + co 10 + 15 so we need to ? nd the smallest y such that makes FD (y) ? 0. 2. Compute the probability of possible demands. 30 ? 3 e = 0. 0498 0 31 PrD = 1 = e? 3 = 0. 1494 1 32 ? PrD = 2 = e = 0. 2241 2 PrD = 0 = 1. 3. INITIAL INVENTORY Interpret these values into FD (y). FD (0) =PrD = 0 = 0. 0498 0. 2 FD (1) =PrD = 0 + PrD = 1 = 0. 1992 0. 2 FD (2) =PrD = 0 + PrD = 1 + PrD = 2 = 0. 4233 ? 0. 2 Hence, the optimal production quantity here is 2. 15 1. 3 Initial Inventory Now let us extend our model a bit further. As opposed to the assumption that we had no gunstock at the beginning, suppose that we have m items when we decide how many we need to order. The solutions we have developed in previous sections assumed that we had no ancestry when placing an order.If we had m items, we should order y ? ? m items instead of y ? items. In other words, the optimal order or product ion quantity is in fact the optimal order-up-to or production-up-to quantity. We had another implicit assumption that we should order, so the ? xed cost did not matter in the previous model. However, if cf is very large, meaning that starting o? a production line or placing an order is very expensive, we may want to consider not to order. In such case, we have two scenarios to order or not to order. We will compare the expected cost for the two scenarios and choose the option with lower expected cost.Example 1. 5. Suppose understock cost is $10, overstock cost is $2, unit purchasing cost is $4 and ? xed ordering cost is $30. In other words, cu = 10, co = 2, cv = 4, cf = 30. take in that D ? Uniform(10, 20) and we already possess 10 items. Should we order or not? If we should, how many items should we order? Answer First, we need to compute the optimal amount of items we need to prepare for each day. Since cu ? cv 1 10 ? 4 = , = cu + co 10 + 2 2 the optimal order-up-to quantity y ? = 15 units. Hence, if we need to order, we should order 5 = y ? ? m = 15 ? 10 items. Let us examine whether we should actually order or not. . Scenario 1 non To Order If we decide not to order, we will not have to pay cf and cv since we order nothing actually. We just need to consider understock and overstock risks. We will ope straddle tomorrow with 10 items that we currently have if we decide not to order. ECost =cu E(D ? 10)+ + co E(10 ? D)+ =10(ED ? 10) + 2(0) = $50 16 CHAPTER 1. NEWSVENDOR PROBLEM Note that in this case E(10 ? D)+ = 0 because D is always great than 10. 2. Scenario 2 To Order If we decide to order, we will order 5 items. We should pay cf and cv accordingly. Understock and overstock risks also exist in this case.Since we will order 5 items to lift up the inventory level to 15, we will run tomorrow with 15 items instead of 10 items if we decide to order. ECost =cf + (15 ? 10)cv + cu E(D ? 15)+ + co E(15 ? D)+ =30 + 20 + 10(1. 25) + 2(1. 25) = $65 Since the expected cost of not ordering is lower than that of ordering, we should not order if we already have 10 items. It is obvious that if we have y ? items at hands safe now, we should order nothing since we already possess the optimal amount of items for tomorrows operation. It is also obvious that if we have nothing currently, we should order y ? items to prepare y ? tems for tomorrow. There should be a point between 0 and y ? where you are indi? erent between order and not ordering. Suppose you as a manager should give instruction to your assistant on when he/she should place an order and when should not. Instead of providing instructions for every possible current inventory level, it is easier to give your assistant just one number that sepa charge per units the decision. Let us call that number the vituperative level of current inventory m? . If we have more than m? items at hands, the expected cost of not ordering will be lower than the expected cost of ordering, so we should not order.Conversely, if we have less than m? items currently, we should order. Therefore, when we have take aimly m? items at hands right now, the expected cost of ordering should be equal to that of not ordering. We will use this intuition to obtain m? value. The decision butt against is summarized in the following ? gure. m* Critical level for placing an order y* Optimal order-up-to quantity Inventory If your current inventory lies here, you should order. Order up to y*. If your current inventory lies here, you should NOT order because your inventory is over m*. 1. 4. SIMULATION 17 Example 1. 6.Given the same settings with the previous example (cu = 10, cv = 4, co = 2, cf = 30), what is the critical level of current inventory m? that determines whether you should order or not? Answer From the answer of the previous example, we can infer that the critical value should be less than 10, i. e. 0 m? 10. Suppose we currently own m? items. Now, evaluate the expected costs of the two sc enarios ordering and not ordering. 1. Scenario 1 Not Ordering ECost =cu E(D ? m? )+ + co E(m? ? D)+ =10(ED ? m? ) + 2(0) = 150 ? 10m? 2. Scenario 2 Ordering In this case, we will order.Given that we will order, we will order y ? ?m? = 15 ? m? items. Therefore, we will start tomorrow with 15 items. ECost =cf + (15 ? 10)cv + cu E(D ? 15)+ + co E(15 ? D)+ =30 + 4(15 ? m? ) + 10(1. 25) + 2(1. 25) = 105 ? 4m? At m? , (1. 14) and (1. 15) should be equal. 150 ? 10m? = 105 ? 4m? ? m? = 7. 5 units (1. 15) (1. 14) The critical value is 7. 5 units. If your current inventory is below 7. 5, you should order for tomorrow. If the current inventory is above 7. 5, you should not order. 1. 4 Simulation Generate 100 random demands from Uniform(10, 30). p = 10, cf = 30, cv = 4, h = 5, b = 3 1 p + b ? v 10 + 3 ? 4 = = p + b + h 10 + 3 + 5 2 The optimal order-up-to quantity from Theorem 1. 3 is 20. We will compare the executing between the policies of y = 15, 20, 25. Listing 1. 1 Continuous Uniform Demand Simulation Set up parameters p=10cf=30cv=4h=5b=3 How many random demands will be generated? n=100 Generate n random demands from the uniform distribution 18 Dmd=runif(n,min=10,max=30) CHAPTER 1. NEWSVENDOR PROBLEM Test the policy where we order 15 items for every period y=15 mean(p*pmin(Dmd,y)-cf-y*cv-h*pmax(y-Dmd,0)-b*pmax(Dmd-y,0)) 33. 4218 Test the policy where we order 20 items for every period y=20 mean(p*pmin(Dmd,y)-cf-y*cv-h*pmax(y-Dmd,0)-b*pmax(Dmd-y,0)) 44. 37095 Test the policy where we order 25 items for every period y=25 mean(p*pmin(Dmd,y)-cf-y*cv-h*pmax(y-Dmd,0)-b*pmax(Dmd-y,0)) 32. 62382 You can see the policy with y = 20 maximizes the 100-period average pro? t as promised by the theory. In fact, if n is relatively small, it is not guaranteed that we have maximized pro? t even if we run based on the optimal policy obtained from this section.The underlying assumption is that we should operate with this policy for a long epoch. Then, Theorem 1. 1 guarant ees that the average pro? t will be maximized when we use the optimal ordering policy. Discrete demand case can also be simulated. Suppose the demand has the following distribution. All other parameters remain same. d PrD = d 10 1 4 15 1 8 20 1 4 25 1 8 30 1 4 The theoretic optimal order-up-to quantity in this case is also 20. Let us test three policies y = 15, 20, 25. Listing 1. 2 Discrete Demand Simulation Set up parameters p=10cf=30cv=4h=5b=3 How many random demands will be generated? =100 Generate n random demands from the discrete demand distribution Dmd=sample(c(10,15,20,25,30),n,replace=TRUE,c(1/4,1/8,1/4,1/8,1/4)) Test the policy where we order 15 items for every period y=15 mean(p*pmin(Dmd,y)-cf-y*cv-h*pmax(y-Dmd,0)-b*pmax(Dmd-y,0)) 19. 35 Test the policy where we order 20 items for every period y=20 mean(p*pmin(Dmd,y)-cf-y*cv-h*pmax(y-Dmd,0)-b*pmax(Dmd-y,0)) 31. 05 Test the policy where we order 25 items for every period 1. 5. EXERCISE y=25 mean(p*pmin(Dmd,y)-cf-y* cv-h*pmax(y-Dmd,0)-b*pmax(Dmd-y,0)) 26. 55 19There are other distributions such as triangular, normal, Poisson or binomial distributions available in R. When you do your senior project, for example, you will observe the demand for a de fragmentisement or a factory. You ? rst approximate the demand using these theoretically established distributions. Then, you can simulate the performance of possible operation policies. 1. 5 Exercise 1. Show that (D ? y) + (y ? D)+ = y. 2. Let D be a discrete random variable with the following pmf. d PrD = d Find (a) Emin(D, 7) (b) E(7 ? D)+ where x+ = max(x, 0). 3. Let D be a Poisson random variable with parameter 3.Find (a) Emin(D, 2) (b) E(3 ? D)+ . Note that pmf of a Poisson random variable with parameter ? is PrD = k = ? k e . k 5 1 10 6 3 10 7 4 10 8 1 10 9 1 10 4. Let D be a continuous random variable and uniformly distributed between 5 and 10. Find (a) Emax(D, 8) (b) E(D ? 8)? where x? = min(x, 0). 5. Let D be an exponential random varia ble with parameter 7. Find (a) Emax(D, 3) 20 (b) E(D ? 4)? . CHAPTER 1. NEWSVENDOR PROBLEM Note that pdf of an exponential random variable with parameter ? is fD (x) = ? e x for x ? 0. 6. David buys fruits and vegetables wholesale and retails them at Davids Produce on La Vista Road.One of the more di? cult decisions is the amount of bananas to buy. Let us make some simplifying assumptions, and assume that David purchases bananas once a week at 10 cents per pound and retails them at 30 cents per pound during the week. Bananas that are more than a week old are too ripe and are sold for 5 cents per pound. (a) Suppose the demand for the good bananas follows the same distribution as D given in Problem 2. What is the expected pro? t of David in a week if he buys 7 pounds of banana? (b) Now assume that the demand for the good bananas is uniformly distributed between 5 and 10 like in Problem 4.What is the expected pro? t of David in a week if he buys 7 pounds of banana? (c) Find the expect ed pro? t if Davids demand for the good bananas follows an exponential distribution with mean 7 and if he buys 7 pounds of banana. 7. Suppose we are selling lemonade during a football game. The lemonade sells for $18 per gallon but only costs $3 per gallon to make. If we run out of lemonade during the game, it will be unworkable to get more. On the other hand, leftover lemonade has a value of $1. live with that we believe the fans would buy 10 gallons with probability 0. 1, 11 gallons with probability 0. , 12 gallons with probability 0. 4, 13 gallons with probability 0. 2, and 14 gallons with probability 0. 1. (a) What is the mean demand? (b) If 11 gallons are prepared, what is the expected pro? t? (c) What is the best amount of lemonade to order forrader the game? (d) Instead, suppose that the demand was normally distributed with mean 1000 gallons and variance 200 gallons2 . How much lemonade should be ordered? 8. Suppose that a bakery specializes in chocolate cakes. strickle t he cakes retail at $20 per cake, but it takes $10 to prepare each cake. Cakes cannot be sold after one week, and they have a negligible salvage value.It is estimated that the weekly demand for cakes is 15 cakes in 5% of the weeks, 16 cakes in 20% of the weeks, 17 cakes in 30% of the weeks, 18 cakes in 25% of the weeks, 19 cakes in 10% of the weeks, and 20 cakes in 10% of the weeks. How many cakes should the bakery prepare each week? What is the bakerys expected optimal weekly pro? t? 1. 5. EXERCISE 21 9. A tv camera store specializes in a particular popular and fancy camera. Assume that these cameras become obsolete at the end of the month. They guarantee that if they are out of stock, they will special-order the camera and promise delivery the next day.In fact, what the store does is to purchase the camera from an out of state retailer and have it delivered through an express operate. Thus, when the store is out of stock, they actually lose the sales price of the camera and the sh ipping charge, but they maintain their good reputation. The retail price of the camera is $600, and the special delivery charge adds another $50 to the cost. At the end of each month, there is an inventory holding cost of $25 for each camera in stock (for doing inventory etc). Wholesale cost for the store to purchase the cameras is $480 each. (Assume that the order can only be made at the beginning of the month. (a) Assume that the demand has a discrete uniform distribution from 10 to 15 cameras a month (inclusive). If 12 cameras are ordered at the beginning of a month, what are the expected overstock cost and the expected understock or shortage cost? What is the expected follow cost? (b) What is optimal number of cameras to order to minimize the expected total cost? (c) Assume that the demand can be approximated by a normal distribution with mean 1000 and standard deviation 100 cameras a month. What is the optimal number of cameras to order to minimize the expected total cost? 10. Next months production at a manufacturing company will use a certain solvent for part of its production cultivate. Assume that there is an ordering cost of $1,000 incurred whenever an order for the solvent is placed and the solvent costs $40 per liter. Due to short product life cycle, unused solvent cannot be used in following months. There will be a $10 disposal charge for each liter of solvent left over at the end of the month. If there is a shortage of solvent, the production process is seriously disrupted at a cost of $100 per liter short. Assume that the initial inventory level is m, where m = 0, 100, 300, viosterol and 700 liters. a) What is the optimal ordering quantity for each case when the demand is discrete with PrD = 500 = PrD = 800 = 1/8, PrD = 600 = 1/2 and PrD = 700 = 1/4? (b) What is the optimal ordering policy for tyrannical initial inventory level m? (You need to specify the critical value m? in addition to the optimal order-up-to quantity y ? . When m ? m? , yo u make an order. Otherwise, do not order. ) (c) Assume optimal quantity will be ordered. What is the total expected cost when the initial inventory m = 0? What is the total expected cost when the initial inventory m = 700? 22 CHAPTER 1. NEWSVENDOR PROBLEM 11.Redo Problem 10 for the case where the demand is governed by the continuous uniform distribution varying between 400 and 800 liters. 12. An automotive company will make one last production run of separate for Part 947A and 947B, which are not interchangeable. These split are no longer used in new cars, but will be needed as replacements for warrantee work in existing cars. The demand during the warranty period for 947A is approximately normally distributed with mean 1,500,000 parts and standard deviation 500,000 parts, while the mean and standard deviation for 947B is 500,000 parts and 100,000 parts. (Assume that two demands are independent. Ignoring the cost of setting up for producing the part, each part costs only 10 cents to produce. However, if additional parts are needed beyond what has been produced, they will be purchased at 90 cents per part (the same price for which the automotive company sells its parts). Parts remaining at the end of the warranty period have a salvage value of 8 cents per part. There has been a proposal to produce Part 947C, which can be used to replace either of the other two parts. The unit cost of 947C jumps from 10 to 14 cents, but all other costs remain the same. (a) presume 947C is not produced, how many 947A should be produced? b) Assuming 947C is not produced, how many 947B should be produced? (c) How many 947C should be produced in order to satisfy the same fraction of demand from parts produced in-house as in the ? rst two parts of this problem. (d) How much money would be saved or lost by producing 947C, but meeting the same fraction of demand in-house? (e) Is your answer to question (c), the optimal number of 947C to produce? If not, what would be the optimal num ber of 947C to produce? (f) Should the more expensive part 947C be produced instead of the two existing parts 947A and 947B. Why? Hint compare the expected total costs.Also, suppose that D ? Normal(, ? 2 ). q xv 0 (x? )2 1 e? 2? 2 dx = 2 q (x ? ) v 0 q (x? )2 1 e? 2? 2 dx 2 + = 2 v 0 (q? )2 (x? )2 1 e? 2? 2 dx 2 t 1 v e? 2? 2 dt + Pr0 ? D ? q 2 2 where, in the 2nd step, we changed variable by letting t = (x ? )2 . 1. 5. EXERCISE 23 13. A warranty department manages the after-sale dish up for a critical part of a product. The department has an obligation to replace any damaged parts in the next 6 months. The number of damaged parts X in the next 6 months is assumed to be a random variable that follows the following distribution x PrX = x 100 . 1 200 . 2 300 . 5 400 . 2The department currently has 200 parts in stock. The department needs to decide if it should make one last production run for the part to be used for the next 6 months. To start the production run, the ? xed cost is $2 000. The unit cost to produce a part is $50. During the warranty period of next 6 months, if a replacement request comes and the department does not have a part available in house, it has to buy a part from the spot-market at the cost of $100 per part. Any part left at the end of 6 month sells at $10. (There is no holding cost. ) Should the department make the production run? If so, how many items should it produce? 4. A store sells a particular brand of fresh juice. By the end of the day, any unsold juice is sold at a discounted price of $2 per gallon. The store gets the juice daily from a local producer at the cost of $5 per gallon, and it sells the juice at $10 per gallon. Assume that the daily demand for the juice is uniformly distributed between 50 gallons to 150 gallons. (a) What is the optimal number of gallons that the store should order from the distribution each day in order to maximize the expected pro? t each day? (b) If 100 gallons are ordered, what is the expected pro? t per day? 15. An auto company is to make one ? al purchase of a rare engine oil to ful? ll its warranty dishs for certain car models. The current price for the engine oil is $1 per gallon. If the company runs out the oil during the warranty period, it will purchase the oil from a supply at the market price of $4 per gallon. Any leftover engine oil after the warranty period is useless, and costs $1 per gallon to get rid of. Assume the engine oil demand during the warranty is uniformly distributed (continuous distribution) between 1 million gallons to 2 million gallons, and that the company currently has half million gallons of engine oil in stock (free of charge). a) What is the optimal amount of engine oil the company should purchase now in order to minimize the total expected cost? (b) If 1 million gallons are purchased now, what is the total expected cost? 24 CHAPTER 1. NEWSVENDOR PROBLEM 16. A company is obligated to provide warranty redevelopment for Product A to its guests next year. The warranty demand for the product follows the following distribution. d PrD = d 100 . 2 200 . 4 300 . 3 400 . 1 The company needs to make one production run to satisfy the warranty demand for entire next year. Each unit costs $100 to produce the penalty cost of a unit is $500.By the end of the year, the savage value of each unit is $50. (a) Suppose that the company has currently 0 units. What is the optimal quantity to produce in order to minimize the expected total cost? Find the optimal expected total cost. (b) Suppose that the company has currently 100 units at no cost and there is $20000 ? xed cost to start the production run. What is the optimal quantity to produce in order to minimize the expected total cost? Find the optimal expected total cost. 17. Suppose you are running a restaurant having only one menu, lettuce salad, in the Tech Square.You should order lettuce every day 10pm after closing. Then, your supplier delivers the ordered amount of lettuce 5am next morning. Store time of days is from 11am to 9pm every day. The demand for the lettuce salad for a day (11am-9pm) has the following distribution. d PrD = d 20 1/6 25 1/3 30 1/3 35 1/6 One lettuce salad requires two units of lettuce. The selling price of lettuce salad is $6, the buying price of one unit of lettuce is $1. Of course, leftover lettuce of a day cannot be used for future salad and you have to pay 50 cents per unit of lettuce for disposal. (a) What is the optimal order-up-to quantity of lettuce for a day? b) If you ordered 50 units of lettuce today, what is the expected pro? t of tomorrow? Include the purchasing cost of 50 units of lettuce in your calculation. Chapter 2 Queueing Theory Before getting into Discrete-time Markov Chains, we will learn about general issues in the queue uping theory. Queueing theory deals with a set of systems having hold space. It is a very powerful tool that can model a broad range of issues. Starting from analyzing a simple queue, a set of queues connected with each other will be cover as well in the end. This chapter will give you the background knowledge when you read the required book, The Goal.We will revisit the queueing theory once we have more advanced modeling techniques and knowledge. 2. 1 Introduction Think about a run system. All of you must have experienced wait in a expediency system. One example would be the Student Center or some restaurants. This is a human system. A bit more automated service system that has a queue would be a call center and automated answering machines. We can imagine a manufacturing system instead of a service system. These waiting systems can be generalized as a set of bu? ers and servers. There are key factors when you try to model such a system.What would you need to analyze your system? How frequently customers come to your system? Inter- reach Times How fast your servers can serve the customers? Service Times How many servers do you have? Number of Servers How large is your waiting space? Queue Size If you can collect data about these metrics, you can characterize your queueing system. In general, a queueing system can be denoted as follows. G/G/s/k 25 26 CHAPTER 2. QUEUEING THEORY The ? rst letter characterizes the distribution of inter-arrival times. The second letter characterizes the distribution of service times.The third number denotes the number of servers of your queueing system. The fourth number denotes the total capacity of your system. The fourth number can be omitted and in such case it means that your capacity is in? nite, i. e. your system can contain any number of people in it up to in? nity. The letter G represents a general distribution. Other candidate characters for this position is M and D and the meanings are as follows. G General Distribution M Exponential Distribution D Deterministic Distribution (or constant) The number of servers can vary from one to many to in? nity.The surface of bu? er can also be either ? nit e or in? nite. To simplify the model, assume that there is only a single server and we have in? nite bu? er. By in? nite bu? er, it means that space is so spacious that it is as if the limit does not exist. Now we set up the model for our queueing system. In terms of analysis, what are we interested in? What would be the performance measures of such systems that you as a manager should know? How long should your customer wait in line on average? How long is the waiting line on average? There are two concepts of average. One is average over time.This applies to the average number of customers in the system or in the queue. The other is average over people. This applies to the average waiting time per customer. You should be able to distinguish these two. Example 2. 1. Assume that the system is empty at t = 0. Assume that u1 = 1, u2 = 3, u3 = 2, u4 = 3, v1 = 4, v2 = 2, v3 = 1, v4 = 2. (ui is ith customers inter-arrival time and vi is ith customers service time. ) 1. What is the aver age number of customers in the system during the ? rst 10 proceeding? 2. What is the average queue size during the ? rst 10 minutes? 3.What is the average waiting time per customer for the ? rst 4 customers? Answer 1. If we draw the number of people in the system at time t with respect to t, it will be as follows. 2. 2. LINDLEY EQUATION 3 2 1 0 27 Z(t) 0 1 2 3 4 5 6 7 8 9 10 t EZ(t)t? 0,10 = 1 10 10 Z(t)dt = 0 1 (10) = 1 10 2. If we draw the number of people in the queue at time t with respect to t, it will be as follows. 3 2 1 0 Q(t) 0 1 2 3 4 5 6 7 8 9 10 t EQ(t)t? 0,10 = 1 10 10 Q(t)dt = 0 1 (2) = 0. 2 10 3. We ? rst need to compute waiting times for each of 4 customers. Since the ? rst customer does not wait, w1 = 0.Since the second customer arrives at time 4, while the ? rst customers service ends at time 5. So, the second customer has to wait 1 minute, w2 = 1. Using the similar logic, w3 = 1, w4 = 0. EW = 0+1+1+0 = 0. 5 min 4 2. 2 Lindley Equation From the previous example, w e now should be able to compute each customers waiting time given ui , vi . It requires too much e? ort if we have to draw graphs every time we need to compute wi . Let us generalize the logic behind calculating waiting times for each customer. Let us determine (i + 1)th customers waiting 28 CHAPTER 2. QUEUEING THEORY time.If (i + 1)th customer arrives after all the time ith customer waited and got served, (i + 1)th customer does not have to wait. Its waiting time is 0. Otherwise, it has to wait wi + vi ? ui+1 . Figure 2. 1, and Figure 2. 2 explain the two cases. ui+1 wi vi wi+1 Time i th arrival i th service start (i+1)th arrival i th service end Figure 2. 1 (i + 1)th arrival before ith service completion. (i + 1)th waiting time is wi + vi ? ui+1 . ui+1 wi vi Time i th arrival i th service start i th service end (i+1)th arrival Figure 2. 2 (i + 1)th arrival after ith service completion. (i + 1)th waiting time is 0.Simply put, wi+1 = (wi + vi ? ui+1 )+ . This is called the Lindley E quation. Example 2. 2. Given the following inter-arrival times and service times of ? rst 10 customers, compute waiting times and system times (time spent in the system including waiting time and service time) for each customer. ui = 3, 2, 5, 1, 2, 4, 1, 5, 3, 2 vi = 4, 3, 2, 5, 2, 2, 1, 4, 2, 3 Answer Note that system time can be obtained by adding waiting time and service time. Denote the system time of ith customer by zi . ui vi wi zi 3 4 0 4 2 3 2 5 5 2 0 2 1 5 1 6 2 2 4 6 4 2 2 4 1 1 3 4 5 4 0 4 3 2 1 3 2 3 1 4 2. 3. TRAFFIC loudness 9 2. 3 Suppose Tra? c Intensity Eui =mean inter-arrival time = 2 min Evi =mean service time = 4 min. Is this queueing system stable? By stable, it means that the queue size should not go to the in? nity. Intuitively, this queueing system will not last because average service time is greater than average inter-arrival time so your system will soon explode. What was the logic behind this judgement? It was basically comparing the average inter-arr ival time and the average service time. To simplify the judgement, we come up with a new quantity called the tra? c intensity. De? nition 2. 1 (Tra? Intensity). Tra? c intensity ? is de? ned to be ? = 1/Eui ? = 1/Evi where ? is the arrival rate and is the service rate. Given a tra? c intensity, it will fall into one of the following three categories. If ? 1, the system is stable. If ? = 1, the system is unstable unless both inter-arrival times and service times are deterministic (constant). If ? 1, the system is unstable. Then, why dont we call ? workout instead of tra? c intensity? utilisation seems to be more intuitive and user-friendly name. In fact, utilization just happens to be same as ? if ? 1.However, the problem arises if ? 1 because utilization cannot go over 100%. workout is bounded above by 1 and that is why tra? c intensity is regarded more general notation to compare arrival and service rates. De? nition 2. 2 (Utilization). Utilization is de? ned as follo ws. Utilization = ? , 1, if ? 1 if ? ? 1 Utilization can also be interpreted as the long-run fraction of time the server is utilized. 2. 4 Kingman Approximation Formula Theorem 2. 1 (Kingmans High-tra? c Approximation Formula). Assume the tra? c intensity ? 1 and ? is close to 1. The long-run average waiting time in 0 a queue EW ? Evi CHAPTER 2. QUEUEING THEORY ? 1 c2 + c2 a s 2 where c2 , c2 are shape coe? cient of variation of inter-arrival times and service a s times de? ned as follows. c2 = a Varu1 (Eu1 ) 2, c2 = s Varv1 (Ev1 ) 2 Example 2. 3. 1. Suppose inter-arrival time follows an exponential distribution with mean time 3 minutes and service time follows an exponential distribution with mean time 2 minutes. What is the expected waiting time per customer? 2. Suppose inter-arrival time is constant 3 minutes and service time is also constant 2 minutes. What is the expected waiting time per customer?Answer 1. Tra? c intensity is ? = 1/Eui 1/3 2 ? = = = . 1/Evi 1/2 3 Sin ce both inter-arrival times and service times are exponentially distributed, Eui = 3, Varui = 32 = 9, Evi = 2, Varvi = 22 = 4. Therefore, c2 = Varui /(Eui )2 = 1, c2 = 1. Hence, s a EW =Evi =2 ? c2 + c2 s a 1 2 2/3 1+1 = 4 minutes. 1/3 2 2. Tra? c intensity remains same, 2/3. However, since both inter-arrival times and service times are constant, their variances are 0. Thus, c2 = a c2 = 0. s EW = 2 2/3 1/3 0+0 2 = 0 minutes It means that none of the customers will wait upon their arrival.As shown in the previous example, when the distributions for both interarrival times and service times are exponential, the squared coe? cient of variation term becomes 1 from the Kingmans approximation formula and the formula 2. 5. LITTLES LAW 31 becomes exact to compute the average waiting time per customer for M/M/1 queue. EW =Evi ? 1 Also note that if inter-arrival time or service time distribution is deterministic, c2 or c2 becomes 0. a s Example 2. 4. You are running a highway collect ing money at the entering toll gate. You lessen the utilization level of the highway from 90% to 80% by adopting car pool lane.How much does the average waiting time in front of the toll gate decrease? Answer 0. 8 0. 9 = 9, =4 1 ? 0. 9 1 ? 0. 8 The average waiting time in in front of the toll gate is reduced by more than a half. The Goal is about identifying bottlenecks in a plant. When you become a manager of a company and are running a expensive machine, you usually want to run it all the time with full utilization. However, the implication of Kingman formula tells you that as your utilization approaches to 100%, the waiting time will be skyrocketing. It means that if there is any uncertainty or random ? ctuation input to your system, your system will greatly su? er. In lower ? region, increasing ? is not that bad. If ? near 1, increasing utilization a little bit can lead to a disaster. Atlanta, 10 geezerhood ago, did not su? er that much of tra? c problem. As its tra? c infrast ructure capacity is getting closer to the demand, it is getting more and more fragile to uncertainty. A lot of strategies presented in The Goal is in fact to decrease ?. You can do various things to reduce ? of your system by outsourcing some process, etc. You can also strategically manage or balance the load on di? erent parts of your system.You may want to utilize customer service organization 95% of time, while utilization of sales people is 10%. 2. 5 Littles Law L = ? W The Littles Law is much more general than G/G/1 queue. It can be applied to any black box with de? nite boundary. The Georgia Tech campus can be one black box. ISyE building itself can be another. In G/G/1 queue, we can easily get average size of queue or service time or time in system as we di? erently draw box onto the queueing system. The following example shows that Littles faithfulness can be applied in broader context than the queueing theory. 32 CHAPTER 2. QUEUEING THEORY Example 2. 5 (Merge of I-75 and I -85).Atlanta is the place where two interstate highways, I-75 and I-85, merge and cross each other. As a tra? c manager of Atlanta, you would like to estimate the average time it takes to drive from the north con? uence point to the south con? uence point. On average, 100 cars per minute enter the merged area from I-75 and 200 cars per minute enter the same area from I-85. You also dispatched a chopper to take a aerial snapshot of the merged area and counted how many cars are in the area. It turned out that on average 3000 cars are within the merged area. What is the average time between entering and exiting the area per vehicle?Answer L =3000 cars ? =100 + 200 = 300 cars/min 3000 L = 10 minutes ? W = = ? 300 2. 6 Throughput Another focus of The Goal is set on the throughput of a system. Throughput is de? ned as follows. De? nition 2. 3 (Throughput). Throughput is the rate of output ? ow from a system. If ? ? 1, throughput= ?. If ? 1, throughput= . The bounding constraint of throug hput is either arrival rate or service rate depending on the tra? c intensity. Example 2. 6 (Tandem queue with two stations). Suppose your factory production line has two stations linked in series. Every raw material coming into your line should be graceful by Station A ? rst.Once it is processed by Station A, it goes to Station B for ? nishing. Suppose raw material is coming into your line at 15 units per minute. Station A can process 20 units per minute and Station B can process 25 units per minute. 1. What is the throughput of the entire system? 2. If we double the arrival rate of raw material from 15 to 30 units per minute, what is the throughput of the whole system? Answer 1. First, obtain the tra? c intensity for Station A. ?A = ? 15 = = 0. 75 A 20 Since ? A 1, the throughput of Station A is ? = 15 units per minute. Since Station A and Station B is linked in series, the throughput of Station . 7. SIMULATION A becomes the arrival rate for Station B. ?B = ? 15 = = 0. 6 B 25 33 Also, ? B 1, the throughput of Station B is ? = 15 units per minute. Since Station B is the ? nal stage of the entire system, the throughput of the entire system is also ? = 15 units per minute. 2. Repeat the same steps. ?A = 30 ? = = 1. 5 A 20 Since ? A 1, the throughput of Station A is A = 20 units per minute, which in turn becomes the arrival rate for Station B. ?B = A 20 = 0. 8 = B 25 ?B 1, so the throughput of Station B is A = 20 units per minute, which in turn is the throughput of the whole system. 2. 7 SimulationListing 2. 1 Simulation of a Simple Queue and Lindley Equation N = 100 Function for Lindley Equation lindley = function(u,v) for (i in 1length(u)) if(i==1) w = 0 else w = append(w, max(wi-1+vi-1-ui, 0)) return(w) u v CASE 1 Discrete Distribution Generate N inter-arrival times and service times = sample(c(2,3,4),N,replace=TRUE,c(1/3,1/3,1/3)) = sample(c(1,2,3),N,replace=TRUE,c(1/3,1/3,1/3)) Compute waiting time for each customer w = lindley(u,v) w CASE 2 Deterministic Distribution All inter-arrival times are 3 minutes and all service times are 2 minutes Observe that nobody waits in this case. 4 u = rep(3, 100) v = rep(2, 100) w = lindley(u,v) w CHAPTER 2. QUEUEING THEORY The Kingmans approximation formula is exact when inter-arrival times and service times follow iid exponential distribution. EW = 1 ? 1 We can con? rm this equation by simulating an M/M/1 queue. Listing 2. 2 Kingman Approximation lambda = arrival rate, mu = service rate N = 10000 lambda = 1/10 mu = 1/7 Generate N inter-arrival times and service times from exponential distribution u = rexp(N,rate=lambda) v = rexp(N,rate=mu) Compute the average waiting time of each customer w = lindley(u,v) mean(w) 16. 0720 Compare with Kingman approximation rho = lambda/mu (1/mu)*(rho/(1-rho)) 16. 33333 The Kingmans approximation formula becomes more and more accurate as N grows. 2. 8 Exercise 1. Let Y be a random variable with p. d. f. ce? 3s for s ? 0, where c is a constan t. (a) Determine c. (b) What is the mean, variance, and squared coe? cient of variation of Y where the squared coe? cient of variation of Y is de? ned to be VarY /(EY 2 )? 2. Consider a single server queue. Initially, there is no customer in the system.Suppose that the inter-arrival times of the ? rst 15 customers are 2, 5, 7, 3, 1, 4, 9, 3, 10, 8, 3, 2, 16, 1, 8 2. 8. EXERCISE 35 In other words, the ? rst customer will arrive at t = 2 minutes, and the second will arrive at t = 2 + 5 minutes, and so on. Also, suppose that the service time of the ? rst 15 customers are 1, 4, 2, 8, 3, 7, 5, 2, 6, 11, 9, 2, 1, 7, 6 (a) Compute the average waiting time (the time customer spend in bu? er) of the ? rst 10 departed customers. (b) Compute the average system time (waiting time plus service time) of the ? st 10 departed customers. (c) Compute the average queue size during the ? rst 20 minutes. (d) Compute the average server utilization during the ? rst 20 minutes. (e) Does the Littles law of hold for the average queue size in the ? rst 20 minutes? 3. We want to decide whether to employ a human operator or buy a machine to paint nerve beams with a rust inhibitor. Steel beams are produced at a constant rate of one every 14 minutes. A skilled human operator takes an average time of 700 seconds to paint a steel beam, with a standard deviation of 300 seconds.An automatic painter takes on average 40 seconds more than the human painter to paint a beam, but with a standard deviation of only 150 seconds. Estimate the expected waiting time in queue of a steel beam for each of the operators, as well as the expected number of steel beams waiting in queue in each of the two cases. Comment on the e? ect of variability in service time. 4. The arrival rate of customers to an ATM machine is 30 per hour with exponentially distirbuted in- terarrival times. The transaction times of two customers are independent and identically distributed.Each transaction time (in minutes) is distributed according to the following pdf f (s) = where ? = 2/3. (a) What is the average waiting for each customer? (b) What is the average number of customers waiting in line? (c) What is the average number of customers at the site? 5. A production line has two machines, Machine A and Machine B, that are arranged in series. Each job needs to processed by Machine A ? rst. Once it ? nishes the affect by Machine A, it moves to the next station, to be processed by Machine B. Once it ? nishes the processing by Machine B, it leaves the production line.Each machine can process one job at a time. An arriving job that ? nds the machine busy waits in a bu? er. 4? 2 se? 2? s , 0, if s ? 0 otherwise 36 CHAPTER 2. QUEUEING THEORY (The bu? er sizes are assumed to be in? nite. ) The processing times for Machine A are iid having exponential distribution with mean 4 minutes. The processing times for Machine B are iid with mean 2 minutes. Assume that the inter-arrival times of jobs arriving at the production line are iid, having exponential distribution with mean of 5 minutes. (a) What is the utilization of Machine A?What is the utilization of Machine B? (b) What is the throughput of the production system? (Throughput is de? ned to be the rate of ? nal output ? ow, i. e. how many items will exit the system in a unit time. ) (c) What is the average waiting time at Machine A, excluding the service time? (d) It is known the average time in the entire production line is 30 minutes per job. What is the long-run average number of jobs in the entire production line? (e) Suppose that the mean inter-arrival time is changed to 1 minute. What are the utilizations for Machine A and Machine B, respectively?What is the throughput of the production system? 6. An auto collision discover has roughly 10 cars arriving per week for repairs. A car waits exterior until it is brought inside for bumping. After bumping, the car is painted. On the average, there are 15 cars waiting outside in the yard to be re paired, 10 cars inside in the bump area, and 5 cars inside in the painting area. What is the average length of time a car is in the yard, in the bump area, and in the painting area? What is the average length of time from when a car arrives until it leaves? 7. A small vernacular is sta? d by a single server. It has been observed that, during a normal business day, the inter-arrival times of customers to the bank are iid having exponential distribution with mean 3 minutes. Also, the the processing times of customers are iid having the following distribution (in minutes) x PrX = x 1 1/4 2 1/2 3 1/4 An arrival ? nding the server busy joins the queue. The waiting space is in? nite. (a) What is the long-run fraction of time that the server is busy? (b) What the the long-run average waiting time of each customer in the queue, excluding the processing time? c) What is average number of customers in the bank, those in queue plus those in service? 2. 8. EXERCISE (d) What is the throughput o f the bank? 37 (e) If the inter-arrival times have mean 1 minute. What is the throughput of the bank? 8. You are the manager at the Student Center in charge of running the food court. The food court is composed of two parts cooking station and cashiers desk. Every person should go to the cooking station, place an order, wait there and pick up ? rst. Then, the person goes to the cashiers desk to check out. After checking out, the person leaves the food court.The coo