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| {{About|mathematics problem |military exercises to assess potential outcomes of an invasion of Iraq |"Desert Crossing" 1999}}The '''jeep problem''',<ref name=wolfram>{{MathWorld|urlname = JeepProblem|title = Jeep Problem}}</ref> '''desert crossing problem'''<ref>{{cite book
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| | last = Gardner
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| | first = Martin
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| | authorlink = Martin Gardner
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| | title = My Best Mathematical and Logic Puzzles
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| | publisher = Dover
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| | date = 1994
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| | pages = p53
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| | isbn = 0-486-28152-3}}</ref> or '''exploration problem'''<ref name=coxeter>"'''Exploration problems.''' Another common question is concerned with the maximum distance into a desert which could be reached from a frontier settlement by an explorer capable of carrying provisions that would last him for ''a'' days." [[W. W. Rouse Ball]] and [[H.S.M. Coxeter]] (1987). ''Mathematical Recreations and Essays'', Thirteenth Edition, Dover, p32. ISBN 0-486-25357-0.</ref> is a mathematics problem in which a [[jeep]] must maximise the distance it can travel into a desert with a given quantity of fuel. The jeep can only carry a fixed and limited amount of fuel, but it can leave fuel and collect fuel at fuel dumps anywhere in the desert.
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| The problem was solved by [[Nathan Fine|N. J. Fine]] in 1947.<ref name=wolfram/>
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| ==Problem==
| | It depends on the quality of the Wordpress theme but even if it's not a professional one you will be able to average 50-60$ EACH link. You can either install Word - Press yourself or use free services offered on the web today. This is a service where people write articles using a specific keyword you have given them. If you're using Wordpress and want to make your blog a "dofollow" blog, meaning that links from your blog pass on the benefits of Google pagerank, you can install one of the many dofollow plugins available. provided by Word - Press Automatic Upgrade, so whenever you need to update the new version does not, it automatically creates no webmaster. <br><br>If you have any thoughts pertaining to wherever and how to use [http://bbs.hst.net.tw/home/link.php?url=https://wordpress.org/plugins/ready-backup/ backup plugin], you can contact us at our own web site. Luckily, for Word - Press users, WP Touch plugin transforms your site into an IPhone style theme. The higher your blog ranks on search engines, the more likely people will find your online marketing site. It allows Word - Press users to easily use HTML5 the element enable native video playback within the browser. These four plugins will make this effort easier and the sites run effectively as well as make other widgets added to a site easier to configure. Once you've installed the program you can quickly begin by adding content and editing it with features such as bullet pointing, text alignment and effects without having to do all the coding yourself. <br><br>This gives a clearer picture that online shoppers are familiar with the WP ecommerce system. s cutthroat competition prevailing in the online space won. For a much deeper understanding of simple wordpress themes", check out Upon browsing such, you'll be able to know valuable facts. Thousands of plugins are available in Word - Press plugin's library which makes the task of selecting right set of plugins for your website a very tedious task. " Thus working with a Word - Press powered web application, making any changes in the website design or website content is really easy and self explanatory. <br><br>Digg Digg Social Sharing - This plugin that is accountable for the floating social icon located at the left aspect corner of just about every submit. This plugin allows a webmaster to create complex layouts without having to waste so much time with short codes. re creating a Word - Press design yourself, the good news is there are tons of Word - Press themes to choose from. If you are looking for Hire Wordpress Developer then just get in touch with him. It does take time to come up having a website that gives you the much needed results hence the web developer must be ready to help you along the route. <br><br>Under Settings —> Reading, determine if posts or a static page will be your home page, and if your home page is a static page, what page will contain blog posts. It can run as plugin and you can still get to that whole database just in circumstance your webhost does not have a c - Panel area. It can be concluded that white label SEO comprise of a third party who resells a contract involving IT expert or consultant, SEO professional and end user. Web developers and newbies alike will have the ability to extend your web site and fit other incredible functions with out having to spend more. Your topic is going to be the basis of your site's name. |
| There are ''n'' units of fuel stored at a fixed base. The jeep can carry at most 1 unit of fuel at any time, and can travel 1 unit of distance on 1 unit of fuel (the jeep's fuel consumption is assumed to be constant). At any point in a trip the jeep may leave any amount of fuel that it is carrying at a fuel dump, or may collect any amount of fuel that was left at a fuel dump on a previous trip, as long as its fuel load never exceeds 1 unit. There are two variants of the problem:
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| *'''Exploring the desert'''{{spaced ndash}} the jeep must return to the base at the end of every trip.
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| *'''Crossing the desert'''{{spaced ndash}} the jeep must return to the base at the end of every trip except for the final trip, when the jeep travels as far as it can before running out of fuel.
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| In either case the objective is to maximise the distance travelled by the jeep on its final trip. Alternatively, the objective may be to find the least amount of fuel required to produce a final trip of a given distance.
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| In the classic problem the fuel in the jeep and at fuel dumps is treated as a [[continuous function|continuous]] quantity. More complex variations on the problem have been proposed in which the fuel can only be left or collected in discrete amounts.<ref>[http://page.mi.fu-berlin.de/rote/Papers/pdf/Optimal+logistics+for+expeditions+-+the+jeep+problem+with+complete+refilling.pdf Optimal Logistics for Expeditions: the Jeep Problem with Complete Refilling], Gunter Rote and Guochuan Zhang, June 1996</ref>
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| ==Solution==
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| [[File:Jeep problem 1.png|thumb|250px|Solution to "exploring the desert" variant for ''n'' = 3, showing fuel contents of jeep and fuel dumps at start of each trip and at turnround point on each trip.]]
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| A strategy that maximises the distance travelled on the final trip for the "exploring the desert" variant is as follows:
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| *The jeep makes ''n'' trips. On each trip it starts from base with 1 unit of fuel.
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| *On the first trip the jeep travels a distance of 1/(2''n'') units and leaves (''n'' − 1)/''n'' units of fuel at a fuel dump. The jeep still has 1/(2''n'') units of fuel – just enough to return to base.
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| *On each of the subsequent ''n'' − 1 trips the jeep collects 1/(2''n'') units of fuel from this first fuel dump on the way out, so that it leaves the fuel dump carrying 1 unit of fuel. It also collects 1/(2''n'') units of fuel from this first fuel dump on the way back, which is just enough fuel to return to base.
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| *On the second trip the jeep travels to the first fuel dump and refuels. It then travels a distance of 1/(2''n'' − 2) units and leaves (''n'' − 2)/(''n'' − 1) units of fuel at a second fuel dump. The jeep still has 1/(2''n'' − 2) units of fuel, which is just enough to return to the first fuel dump. Here it collects 1/(2''n'') units of fuel, which is just enough fuel to return to base.
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| *On each of the subsequent ''n'' − 2 trips the jeep collects 1/(2''n'' − 2) units of fuel from this second fuel dump on the way out, so that it leaves the fuel dump carrying 1 unit of fuel. It also collects 1/(2''n'' − 2) units of fuel from the second fuel dump on the way back, which is just enough fuel to return to the first fuel dump.
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| *The jeep continues in this way, so that on trip ''k'' it establishes a new ''k''th fuel dump at a distance of 1/(2''n'' − 2''k'' + 2) units from the previous fuel dump and leaves (''n'' − ''k'')/(''n'' − ''k'' + 1) units of fuel there. On each of the subsequent ''n'' − ''k'' trips it collects 1/(2''n'' − 2''k'' + 2) units of fuel from the ''k''th dump on its way out and another 1/(2''n'' − 2''k'' + 2) units of fuel on its way back.
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| When the jeep starts its final trip, there are ''n'' − 1 fuel dumps. The farthest contains 1/2 of a unit of fuel, the next farthest contain 1/3 of a unit of fuel, and so on, and the nearest fuel dump has just 1/''n'' units of fuel left. Together with 1 unit of fuel with which it starts from base, this means that the jeep can travel a total round trip distance of
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| :<math>1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} = \sum_{k=1}^n \frac{1}{k}</math>
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| units on its final trip (the maximum distance travelled into the desert is half of this).<ref name=coxeter/> It collects half of the remaining fuel at each dump on the way out, which fills its tank. After leaving the farthest fuel dump it travels 1/2 a unit further into the desert and then returns to the farthest fuel dump. It collects the remaining fuel from each fuel dump on the way back, which is just enough to reach the next fuel dump or, in the final step, to return to base.
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| [[File:Jeep problem 2.png|thumb|250px|Solution to "crossing the desert" variant for ''n'' = 3, showing fuel contents of jeep and fuel dumps at start of each trip, at turnround point on first two trips, and at end of final trip.]]
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| The distance travelled on the last trip is the ''n''th [[harmonic number]], ''H''<sub>''n''</sub>. As the harmonic numbers are unbounded, it is possible to exceed any given distance on the final trip, as along as sufficient fuel is available at the base. However, the amount of fuel required and the number of fuel dumps both increase exponentially with the distance to be travelled.
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| The "crossing the desert" variant can be solved with a similar strategy, except that there is now no requirement to collect fuel on the way back on the final trip. So on trip ''k'' the jeep establishes a new ''k''th fuel dump at a distance of 1/(2''n'' − 2''k'' + 1) units from the previous fuel dump and leaves (2''n'' − 2''k'' − 1)/(2''n'' − 2''k'' + 1) units of fuel there. On each of the next ''n'' − ''k'' − 1 trips it collects 1/(2''n'' − 2''k'' + 1) units of fuel from the ''k''th dump on its way out and another 1/(2''n'' − 2''k'' + 1) units of fuel on its way back.
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| Now when the jeep starts its final trip, there are ''n'' − 1 fuel dumps. The farthest contains 1/3 of a unit of fuel, the next farthest contain 1/5 of a unit of fuel, and so on, and the nearest fuel dump has just 1/(2''n'' − 1) units of fuel left. Together with 1 unit of fuel with which it starts from base, this means that the jeep can travel a total distance of
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| :<math>1 + \frac{1}{3} + \frac{1}{5} + \cdots + \frac{1}{2n-1} = \sum_{k=1}^n \frac{1}{2k-1}=H_{2n-1}-\frac{1}{2}H_{n-1}</math>
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| units on its final trip.<ref name=wolfram/><ref name=coxeter/> It collects all of the remaining fuel at each dump on the way out, which fills its tank. After leaving the farthest fuel dump it travels a further distance of 1 unit.
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| Note that
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| :<math>\sum_{k=1}^n \frac{1}{2k-1} > \sum_{k=1}^n \frac{1}{2k} = \frac{1}{2}H_{n}</math>
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| so it is possible in theory to cross a desert of any size given enough fuel at the base. As before, the amount of fuel required and the number of fuel dumps both increase exponentially with the distance to be travelled.
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| ==Practical applications==
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| The problem can have a practical application in wartime situations, especially with respect to fuel efficiency. In the context of the bombing of Japan in [[World War II]] by [[B-29]]s, [[Robert McNamara]] says in the film ''[[The Fog of War]]'' that understanding the fuel efficiency issue caused by having to transport the fuel to forward bases was the main reason why the strategy of launching bombing raids from mainland China was abandoned in favor of the [[island hopping]] strategy:
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| {{quote|"We had to fly those planes from the bases in Kansas to India. Then we had to fly fuel over the hump into China. [...] We were supposed to take these [[B-29]]s—there were no [[tanker aircraft]] there. We were to fill them with fuel, fly from [[India]] to [[Chengdu|Chengtu]]; offload the fuel; fly back to India; make enough missions to build up fuel in Chengtu; fly to [[Yawata]], [[Japan]]; bomb the [[steel mills]]; and go back to India.
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| We had so little training on this problem of maximizing [fuel] efficiency, we actually found to get some of the B-29s back instead of offloading fuel, they had to take it on. To make a long story short, it wasn't worth a damn. And it was [[Curtis LeMay|LeMay]] who really came to that conclusion, and led the [[Combined Chiefs of Staff|Chiefs]] to move the whole thing to the [[Marianas]], which devastated Japan."<ref>[http://www.errolmorris.com/film/fow_transcript.html Fog of War transcript], www.errolmorris.com</ref>}}
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| (The [[Atomic_bombings_of_Hiroshima_and_Nagasaki|atomic bombing missions]] at the end of World War II were flown using B-29 [[Superfortresses]] from the [[Pacific Ocean|Pacific]] island of [[Tinian]] in the [[Northern Marianas Islands]].)
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| ==See also==
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| *[[Optimization (mathematics)]]
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| *[[Dynamic programming]]
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| == References ==
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| {{reflist}}
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| ==External links==
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| *[http://video.google.com/videosearch?q=the+fog+of+war&ie=UTF-8&oe=utf-8&rls=org.mozilla:en-US:official&client=firefox-a&um=1&sa=X&oi=video_result_group&resnum=4&ct=title# The Fog of War]
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| [[Category:Mathematical optimization]]
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| [[Category:Operations research]]
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| [[Category:Recreational mathematics]]
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It depends on the quality of the Wordpress theme but even if it's not a professional one you will be able to average 50-60$ EACH link. You can either install Word - Press yourself or use free services offered on the web today. This is a service where people write articles using a specific keyword you have given them. If you're using Wordpress and want to make your blog a "dofollow" blog, meaning that links from your blog pass on the benefits of Google pagerank, you can install one of the many dofollow plugins available. provided by Word - Press Automatic Upgrade, so whenever you need to update the new version does not, it automatically creates no webmaster.
If you have any thoughts pertaining to wherever and how to use backup plugin, you can contact us at our own web site. Luckily, for Word - Press users, WP Touch plugin transforms your site into an IPhone style theme. The higher your blog ranks on search engines, the more likely people will find your online marketing site. It allows Word - Press users to easily use HTML5 the element enable native video playback within the browser. These four plugins will make this effort easier and the sites run effectively as well as make other widgets added to a site easier to configure. Once you've installed the program you can quickly begin by adding content and editing it with features such as bullet pointing, text alignment and effects without having to do all the coding yourself.
This gives a clearer picture that online shoppers are familiar with the WP ecommerce system. s cutthroat competition prevailing in the online space won. For a much deeper understanding of simple wordpress themes", check out Upon browsing such, you'll be able to know valuable facts. Thousands of plugins are available in Word - Press plugin's library which makes the task of selecting right set of plugins for your website a very tedious task. " Thus working with a Word - Press powered web application, making any changes in the website design or website content is really easy and self explanatory.
Digg Digg Social Sharing - This plugin that is accountable for the floating social icon located at the left aspect corner of just about every submit. This plugin allows a webmaster to create complex layouts without having to waste so much time with short codes. re creating a Word - Press design yourself, the good news is there are tons of Word - Press themes to choose from. If you are looking for Hire Wordpress Developer then just get in touch with him. It does take time to come up having a website that gives you the much needed results hence the web developer must be ready to help you along the route.
Under Settings —> Reading, determine if posts or a static page will be your home page, and if your home page is a static page, what page will contain blog posts. It can run as plugin and you can still get to that whole database just in circumstance your webhost does not have a c - Panel area. It can be concluded that white label SEO comprise of a third party who resells a contract involving IT expert or consultant, SEO professional and end user. Web developers and newbies alike will have the ability to extend your web site and fit other incredible functions with out having to spend more. Your topic is going to be the basis of your site's name.