{"178111":{"#nid":"178111","#data":{"type":"news","title":"The Power of Mathematics","body":[{"value":"\u003Cp\u003E\u003Cstrong\u003EAs industrial engineers, we probably all \u003C\/strong\u003Eremember taking various prerequisitemathematics courses like calculusand linear algebra, then moving on tosome of the mathematically orientedcourses within the Stewart School ofIndustrial \u0026amp; Systems Engineering(ISyE) itself. Every once in a while,it\u2019s a good idea to take a step backand think about why all of that mathwas necessary as part of a strong ISyEexperience. This article offers somereminders about the uses and powerof mathematics in our discipline\u2015atleast for those who want to work oncutting-edge applications or emergingresearch areas.\u003C\/p\u003E\u003Cp\u003EThe short story in industrial engineering and operations research is that, for most practical purposes, all of the easy problems and results are gone, having been discovered and thoroughly studied long ago. That doesn\u2019t mean we can\u2019t go out into the world and solve real-life problems with appropriate existing technology; it just means we may have to roll up our mathematical sleeves a bit as we delve into applications that are becoming more and more challenging. For instance, it\u2019s quite easy to \u201csolve\u201d a steady-state single-server queuing system with some simple equations if the customer inter-arrival times and service times are independent and identically distributed exponential random variables. But what if you have an entire network of queues (like, say, a call center or a popular fast-food restaurant) experiencing transient arrival processes that vary throughout the day, or different server schedules and abilities, or equipment breakdowns? These types of problems obviously take a little more effort; a trivial equation isn\u2019t enough.\u003C\/p\u003E\u003Cp\u003EThis article will address some of the mathematics techniques that can be brought to bear on interesting ISyE applications and research problems. You would undoubtedly have been exposed to some of these methods in your travels as a student and in the real world (perhaps, at least, elementary versions), but some may be completely new to you. In any case, the idea is to provide a glimpse of the terrific power of mathematics that\u2019s available for use in problems important to industrial engineers and operations researchers.\u003C\/p\u003E\u003Cp\u003E\u003Cstrong\u003EGoing for a Walk\u003C\/strong\u003E\u003C\/p\u003E\u003Cp\u003ELet\u2019s begin with a discussion concerning a beautiful application of probability theory and stochastic processes. Of course, the most basic experiment in any probability course is that of flipping a coin. We\u2019ll show how this concept can be turned into something that\u2019s quite sophisticated from a mathematical point of view. Suppose every time I toss tails (T), I earn a dollar, and every time I toss heads (H), I lose a dollar. An interesting question involves that of determining how much money I will have after a certain number of tosses. Where do my total winnings stand after ten tosses? After 100? After one million? As an example, the ten-toss sequence TTTHHTTHTT would have given me a well-deserved net gain of $4.\u003C\/p\u003E\u003Cp\u003ESuch an experiment is called a random walk. Think of me taking a step to the left or a step to the right with equal probability (just like my earnings with the coin flips). In terms of my experiment, I\u2019d like to know where I stand after I\u2019ve been meandering around a while. What\u2019s the probability that I\u2019ll have at least $4 by the tenth toss? Will I earn $4 before I lose $4? But the random walk gives us so much more than a description of the probabilistic behavior of a finite number of coin flips. The magic happens as we increase the number of steps in the random walk, because the process then converges to what is known as Brownian motion.\u003C\/p\u003E\u003Cp\u003EHere is an example of what an exponential version of Brownian motion looks like \u003Cem\u003E(see photo with Associate Professor Shijie Deng)\u003C\/em\u003E. Notice that it bears a striking resemblance to a time series plot of stock prices. In fact, many financial engineers use Brownian motion to model stocks, options, and other financial instruments. Brownian motion is so important and mathematically deep that scientists have won at least two Nobel Prizes explaining it and using it in all sorts of applications. In ISyE, researchers use Brownian motion to:\u003C\/p\u003E\u003Cul\u003E\u003Cli\u003Eanalyze what goes on in busy queuing systems (like call centers);\u003C\/li\u003E\u003Cli\u003Estudy the movement of ants;\u003C\/li\u003E\u003Cli\u003Emodel how computer compilers process data lists;\u003C\/li\u003E\u003Cli\u003Efit complicated probability distributions;\u003C\/li\u003E\u003Cli\u003Edevelop efficient quality control charts;\u003C\/li\u003E\u003Cli\u003Eanalyze difficult data sets coming out of simulations; and, of course,\u003C\/li\u003E\u003Cli\u003Emodel stock and option process.\u003C\/li\u003E\u003C\/ul\u003E\u003Cp\u003E\u003Cstrong\u003EGoing Nowhere Fast\u003C\/strong\u003E\u003C\/p\u003E\u003Cp\u003ESpeaking of queuing systems as in the last set of examples, how many of us have had to wait in lines a bit more than we would have liked at a store, on the phone, or at an amusement park? The science of queuing (line) theory allows us to analyze the flow of entities through all sorts of systems, where the terms \u201centities\u201d and \u201csystems\u201d can be quite general. For instance, we might be interested in a problem as simple as that of customer movement through a barber shop (perhaps encountering a tasty barber queue along the way), or more complicated systems such as airport baggage handling services, or a large call center handling millions of customer inquiries.\u003C\/p\u003E\u003Cp\u003EWhat are some of the issues involved in queuing theory and how can mathematics help us understand the performance of these types of systems? If you are a customer, you are certainly interested in moving through lines (queues) quickly and being served quickly. If you are the service provider, you may want to keep the lines short in order to save space and avoid customer dissatisfaction. On the other hand, if you are the post office, you\u2019ll likely want to keep the lines nice and long\u2014 to show your customers who\u2019s boss. In addition to the issue of line length, you\u2019d want to keep your servers relatively busy\u2015after all, an idle server is the devil\u2019s workshop (and is costing you money). A number of important questions arise from all of these considerations:\u003C\/p\u003E\u003Cul\u003E\u003Cli\u003EWhich one of several lines should I enter at the grocery store\u2019s checkout? Normally, you\u2019d pick the shortest one, but what happens if certain servers are quicker or more talented than others? What happens if you spot particularly slow customers in one of the lines? How about the self-service checkout machine?\u003C\/li\u003E\u003C\/ul\u003E\u003Cul\u003E\u003Cli\u003EHow many servers should I employ? Too many servers cost too much money; too few could cost customers.\u003C\/li\u003E\u003Cli\u003EShould we route different types of customers to different service stations in different orders?\u003C\/li\u003E\u003Cli\u003EWhat kind of cross-functionalities should our servers have in order to make the system more efficient?\u003C\/li\u003E\u003C\/ul\u003E\u003Cp\u003EISyE researchers study questions such as those described above using a combination of techniques arising from stochastic processes, differential equations, optimization, and computer simulation. The implications of such questions are tremendous and can generate considerations such as:\u003C\/p\u003E\u003Cul\u003E\u003Cli\u003EHow many medical personnel should we schedule in an emergency room?\u003C\/li\u003E\u003Cli\u003EWill we have enough voting machines and staff to carry out their proper functions during a national election? And, of course,\u003C\/li\u003E\u003Cli\u003EDoes The Varsity have enough space to accommodate customers before the next UGA game?\u003C\/li\u003E\u003C\/ul\u003E\u003Cp\u003E\u003Cstrong\u003E\u003Cbr \/\u003ETaking a Tour\u003C\/strong\u003E\u003C\/p\u003E\u003Cp\u003ESuppose you are a traveling salesman and you need to visit the following cities to show off your goods: Atlanta (A), Buffalo (B), Chicago (C), and Denver (D). Starting from and ending at Atlanta, what\u2019s the best way to do this? This is what ISyE researchers refer to as the Travelling Salesman Problem (TSP).\u003C\/p\u003E\u003Cp\u003EHere are the possible routes you could take: ABCDA, ABDCA, ACBDA, ACDBA, ADBCA, ADCBA\u003C\/p\u003E\u003Cp\u003ENotice that we have six potential routes (or \u201ctours\u201d), corresponding to the six permutations of the cities B, C, D. If we are interested in minimizing the distance travelled, then we really only have to look at the three tours ABCDA, ABDCA, ACBDA since, for example, the distance required for ABCDA is the same as that for ADCBA\u2014assuming we are comfortable walking backward. All we have to do is go on the web and look at the distances for the three routes to get our optimal answer. Pretty simple, right? But what happens if we have n cities on our agenda? Then it is very easy to show that we\u2019ll have to do the look-ups for (n-1)!\/2 tours, and this number gets incredibly big very quickly.\u003C\/p\u003E\u003Cp\u003EIndeed, if we were to try to find the optimal tour by hand for just twenty cities, it would take a huge amount of effort, and it would be exceptionally tedious and time-consuming. Fifty cities by hand would be out of the question. Using mathematical tools from combinatorics, graph theory, and even topology (along with a liberal dose of computer science), ISyE researchers have optimally solved TSPs involving almost 100,000 cities\u2015and they can get nearly optimal solutions for much larger problems! This is not just a pie-in-the-sky mathematical exercise. You can use TSPs to:\u003C\/p\u003E\u003Cul\u003E\u003Cli\u003Efind the optimal route for a delivery truck;\u003C\/li\u003E\u003Cli\u003Edesign the optimal pattern for semiconductor chip etching;\u003C\/li\u003E\u003Cli\u003Edeliver meals on wheels to homebound infirmed patients; and\u003C\/li\u003E\u003Cli\u003Eschedule bus pickups for school children.\u003C\/li\u003E\u003C\/ul\u003E\u003Cp\u003E\u003Cbr \/\u003EFor more information on TSPs, visit\u003Cstrong\u003E http:\/\/www.tsp.gatech.edu\/\u003C\/strong\u003E.\u003C\/p\u003E\u003Cp\u003E\u003Cstrong\u003EGetting from Here to There\u003C\/strong\u003E\u003C\/p\u003E\u003Cp\u003ETravel is an aspect of all of our lives that ISyE touches in many ways. Think of an airline trip from Atlanta to New York. Typically, you start the process by going online to purchase your ticket at one of the major travel portals (or at the airline itself). The prices you see are dependent on a number of factors, such as time and date of your trip and class of service, and are actually determined by a combination of optimization, regression, and forecasting techniques. For instance, if your airline has determined via its data analytics that the Atlanta-New York route is popular on Labor Day weekend, it will likely try to take advantage of that forecast by keeping most of that route\u2019s prices higher than usual, reducing the number of low-price tickets, reducing the availability of free frequent flier tickets, and perhaps scheduling aircraft with greater capacities.\u003C\/p\u003E\u003Cp\u003EYour airline almost certainly makes multiple flights from Atlanta to New York every day on a variety of different planes with different capacities. How are the decisions made regarding which planes fly to which cities, at which times, carrying how many people? In particular:\u003C\/p\u003E\u003Cul\u003E\u003Cli\u003EHow does one assign the crew for a specific flight, especially in the presence of tight FAA safety restrictions regarding the amount of time that a crew can serve during a given time period?\u003C\/li\u003E\u003Cli\u003EHow does one determine flight schedules for a specific aircraft, while adhering to strict maintenance requirements?\u003C\/li\u003E\u003Cli\u003EShould the plane fly back and forth between two cities (e.g., Atlanta-New York), or is it more efficient to fly a larger circuit?\u003C\/li\u003E\u003Cli\u003EAre hub-and-spoke systems more efficient for your airline than direct point-to-point flights? Should your airline augment its route network with those of smaller commuter airlines?\u003C\/li\u003E\u003Cli\u003EShould overbookings be allowed, given proper statistical analysis with respect to no-shows?\u003C\/li\u003E\u003Cli\u003EHow should staff be assigned and how should the lines be configured at the airport\u2019s security checkpoints?\u003C\/li\u003E\u003C\/ul\u003E\u003Cp\u003EThese are all extremely difficult problems that require the use of optimization, statistical tools, and simulation (among others). ISyE is very lucky to have several researchers specializing in integer programming optimization techniques who are well-known for their work on many of the listed questions. The work\u2015though highly theoretical\u2015has financial consequences that result in millions and millions of dollars in savings.\u003C\/p\u003E\u003Cp\u003E\u003Cstrong\u003EFollow the Bouncing Ball\u003C\/strong\u003E\u003C\/p\u003E\u003Cp\u003EAs yet another example of mathematics used in our field, let\u2019s talk about something almost every sports fan can relate to: Which college basketball team is going to win the NCAA championship this year? This is a tough problem that involves a number of tricky aspects of probability, statistics, and optimization. The goal is to somehow use our analytical skills from these mathematical areas along with some intelligent data mining to make reasonable predictions (and to win our office pools). In terms of data mining, there\u2019s certainly a lot of information out there. For instance:\u003C\/p\u003E\u003Cul\u003E\u003Cli\u003EAre the bracket arrangements more helpful to some teams than others?\u003C\/li\u003E\u003Cli\u003EDid certain teams already play each other and how did they do?\u003C\/li\u003E\u003Cli\u003EAre there any games with obvious home court advantages?\u003C\/li\u003E\u003Cli\u003EAre some of the teams currently on a hot streak?\u003C\/li\u003E\u003Cli\u003EDo any of the teams have injury issues?\u003C\/li\u003E\u003Cli\u003EHow have various seeds done in the past?\u003C\/li\u003E\u003Cli\u003ECan a team\u2019s margin of victory give us any clues about future performance?\u003C\/li\u003E\u003Cli\u003EWhat about a team\u2019s conference performance?\u003C\/li\u003E\u003C\/ul\u003E\u003Cp\u003ENCAA tournament prediction is clearly an active area, both from a seat-of-the-pants perspective as well as an analytical perspective. We are very lucky in ISyE to have a number of researchers who have developed an extremely successful prediction technology called the Logistic Regression \/ Markov Chain (LRMC) method. The interesting name reflects the statistical and probabilistic techniques the tool uses. What is nice about the LRMC ranking system is that it is designed to use only basic scoreboard data: which two teams played, whose court they played on, and what the margin of victory was\u2014though a new so-called Bayesian add-on has been developed recently that allows users to incorporate some gut feeling into the equation.\u003C\/p\u003E\u003Cp\u003EObviously, you have to go out and play the games, so you can\u2019t predict things correctly all of the time, but LRMC has done very well compared to just about any other prediction methodology, and ISyE has garnered a great deal of positive play from this terrific application of mathematics. If you would like more information about LRMC, visit \u003Cstrong\u003Ehttp:\/\/lrmc.isye.gatech.edu\/\u003C\/strong\u003E.\u003C\/p\u003E\u003Cp\u003E\u003Cstrong\u003EGetting Home Safe and Sound\u003C\/strong\u003E\u003C\/p\u003E\u003Cp\u003EAnother example involving mathematics and modeling in ISyE concerns the important problem of disease propagation. In 2009 and the early part of 2010, the northern hemisphere had to cope with the first waves of a new H1N1 influenza pandemic, also known as swine flu.\u003C\/p\u003E\u003Cp\u003EDespite high-profile vaccination campaigns in many countries, delays in the administration of vaccination programs were common, and high vaccination coverage levels were not achieved, so the disease was not effectively controlled.\u003C\/p\u003E\u003Cp\u003EWe were lucky this time. This particular strain of swine flu wasn\u2019t too awful in terms of mortality; in fact, it wasn\u2019t much worse than regular seasonal flu. Next time, things might not go our way. So what else could have been done to stem the march of a pandemic disease through the population? ISyE researchers have used a variety of mathematical tools to model the disease as well as certain mitigation strategies. These tools include everything from probability, statistics, differential equations, and optimization, which are then used in conjunction with computer simulations to come up with strategies to mitigate future pandemics. What kinds of strategies are out there?\u003C\/p\u003E\u003Cp\u003EHere are some possibilities:\u003C\/p\u003E\u003Cul\u003E\u003Cli\u003Eschool closure and social distancing\u003C\/li\u003E\u003Cli\u003Ebetter vaccination compliance\u003C\/li\u003E\u003Cli\u003Emore reliable vaccination supply chains\u003C\/li\u003E\u003Cli\u003Euse and procurement of more effective antiviral medicines\u003C\/li\u003E\u003Cli\u003Euse of face masks\u003C\/li\u003E\u003Cli\u003Eworking from home\u003C\/li\u003E\u003Cli\u003Eplacement of resources in locations that will allow healthcare officials to respond optimally to a pandemic\u003C\/li\u003E\u003C\/ul\u003E\u003Cp\u003EOf course, these strategies all cost a great deal of money and some work better than others. ISyE researchers are interested in optimizing health outcomes subject to budget constraints and are actively working in this area.\u003C\/p\u003E\u003Cp\u003EOne advantage of this work is that it can be extended to other healthcare arenas, for instance:\u003C\/p\u003E\u003Cul\u003E\u003Cli\u003Emeasles outbreaks\u003C\/li\u003E\u003Cli\u003Emalaria\u003C\/li\u003E\u003Cli\u003Echolera\u003C\/li\u003E\u003C\/ul\u003E\u003Cp\u003E\u003Cstrong\u003E\u003Cbr \/\u003EConclusion\u003C\/strong\u003E\u003C\/p\u003E\u003Cp\u003EThis article has just touched the surface of how the mathematical tools used by ISyE folks can be adopted to solve a variety of theoretical and applied problems. Some of these mathematical technologies are available through courses in ISyE (or from a good math department), but there is no doubt that such cutting-edge methods are required reading for today\u2019s modern practitioners of industrial engineering and operations research.\u003C\/p\u003E\u003Cp\u003EThis article was written by Professor Dave Goldsman and first appeared in the 2012 edition of the \u003Ca href=\u0022http:\/\/issuu.com\/isyealumnimagazine\/docs\/2012\u0022\u003EISyE Alumni Magazine\u003C\/a\u003E.\u003C\/p\u003E","summary":null,"format":"limited_html"}],"field_subtitle":"","field_summary":[{"value":"\u003Cp\u003EThis article, written by Professor Dave Goldsman, discusses the uses and power of mathematics for those who want to work on cutting-edge applications or emerging research areas in the field of industrial engineering.\u003C\/p\u003E","format":"limited_html"}],"field_summary_sentence":"","uid":"27511","created_gmt":"2012-12-18 09:21:20","changed_gmt":"2016-10-08 03:13:22","author":"Ashley Daniel","boilerplate_text":"","field_publication":"","field_article_url":"","dateline":{"date":"2012-12-18T00:00:00-05:00","iso_date":"2012-12-18T00:00:00-05:00","tz":"America\/New_York"},"extras":[],"hg_media":{"178121":{"id":"178121","type":"image","title":"Associate Professor Shijie Deng illustrates a financial model incorporating Brownian motion.","body":null,"created":"1449179039","gmt_created":"2015-12-03 21:43:59","changed":"1475894825","gmt_changed":"2016-10-08 02:47:05","alt":"Associate Professor Shijie Deng illustrates a financial model incorporating Brownian motion.","file":{"fid":"195948","name":"graphdryerase_blueyellow.jpg","image_path":"\/sites\/default\/files\/images\/graphdryerase_blueyellow_0.jpg","image_full_path":"http:\/\/hg.gatech.edu\/\/sites\/default\/files\/images\/graphdryerase_blueyellow_0.jpg","mime":"image\/jpeg","size":1435931,"path_740":"http:\/\/hg.gatech.edu\/sites\/default\/files\/styles\/740xx_scale\/public\/images\/graphdryerase_blueyellow_0.jpg?itok=OqL-1J4e"}},"178131":{"id":"178131","type":"image","title":"Jose Sarmiento, ISyE undergraduate student; Prof. Anton Kleywegt; Kyungha Lim, ISyE undergraduate student; \u0026 Xinchang Wang, ISyE PhD student, plying the tools of the trade at Delta\u0027s Tech Ops Center.","body":null,"created":"1449179039","gmt_created":"2015-12-03 21:43:59","changed":"1475894825","gmt_changed":"2016-10-08 02:47:05","alt":"Jose Sarmiento, ISyE undergraduate student; Prof. Anton Kleywegt; Kyungha Lim, ISyE undergraduate student; \u0026 Xinchang Wang, ISyE PhD student, plying the tools of the trade at Delta\u0027s Tech Ops Center.","file":{"fid":"195949","name":"13c2306-p4-029.jpg","image_path":"\/sites\/default\/files\/images\/13c2306-p4-029_0.jpg","image_full_path":"http:\/\/hg.gatech.edu\/\/sites\/default\/files\/images\/13c2306-p4-029_0.jpg","mime":"image\/jpeg","size":2499806,"path_740":"http:\/\/hg.gatech.edu\/sites\/default\/files\/styles\/740xx_scale\/public\/images\/13c2306-p4-029_0.jpg?itok=Ks9VYDFm"}}},"media_ids":["178121","178131"],"groups":[{"id":"1242","name":"School of Industrial and Systems Engineering (ISYE)"}],"categories":[{"id":"145","name":"Engineering"}],"keywords":[{"id":"53351","name":"Applicable Mathematics"},{"id":"53391","name":"Brownian motion"},{"id":"53381","name":"Combinatronics"},{"id":"53361","name":"Dave Goldsman"},{"id":"1362","name":"efficiency"},{"id":"109","name":"Georgia Tech"},{"id":"2612","name":"Graph Theory"},{"id":"426","name":"isye"},{"id":"233","name":"Logistics"},{"id":"53341","name":"Mathematics in Real Life"},{"id":"1377","name":"optimization"},{"id":"169545","name":"Stewart School of Industrial \u0026 Systems Engineering"},{"id":"14817","name":"topology"},{"id":"53371","name":"TSP"}],"core_research_areas":[{"id":"39431","name":"Data Engineering and Science"},{"id":"39541","name":"Systems"}],"news_room_topics":[],"event_categories":[],"invited_audience":[],"affiliations":[],"classification":[],"areas_of_expertise":[],"news_and_recent_appearances":[],"phone":[],"contact":[{"value":"\u003Cp\u003E\u003Ca href=\u0022mailto:barbara.christopher@isye.gatech.edu\u0022\u003E\u003Cstrong\u003EBarbara Christopher\u003C\/strong\u003E\u003C\/a\u003E\u003Cbr \/\u003EIndustrial and Systems Engineering\u003Cbr \/\u003E\u003Cstrong\u003E404.385.3102\u003C\/strong\u003E\u003C\/p\u003E","format":"limited_html"}],"email":[],"slides":[],"orientation":[],"userdata":""}}}