SA35 Linear Programming Spring 216 Asst. Prof. Nelson Uhan What is operations research? Lesson 1. Introduction The most influential academic discipline field you ve never heard of [Boston Globe, 24] perations Research (R) is the discipline of applying advanced mathematical methods to help make better decisions The Science of Better A liberal education in a technological world [INFRMS slogan] [Thomas Magnanti, former Dean of Engineering at MIT] Numerous applications, e.g. logistics manufacturing workforce scheduling finance marketing R and the military The military uses R to improve decision making in a variety of ways, e.g. force composition weapon selection search and detection flight operations scheduling training and personnel assignment Assessment Division (PNAV N81) at the Pentagon The Naval Postgraduate School has one of the oldest and most well-respected R departments in the US Naval Research Logistics is a prominent academic journal featuring research in R The traveling salesperson problem A salesperson located in Annapolis wants to visit clients in each of the 48 state capitals of the continental US and Washington DC What is shortest way of visiting all the capitals and then returning to Annapolis? Entire books have been written on the TSP 1
P A C 1 km N Honolulu 1 mi Salem lympia Sacramento REGN Carson City R U S S I A U.S. Department of the Interior U.S. Geological Survey Where We Are NEVADA Boise IDAH ARIZNA Phoenix UTAH A R C T I C C E A N 2 mi 2 km Helena Salt Lake City MNTANA Juneau WYMING Cheyenne Denver CLRAD Santa Fe NEW MEXIC NRTH DAKTA Bismarck SUTH DAKTA Pierre NEBRASKA KANSAS TEXAS Lincoln Topeka KLAHMA Austin MINNESTA klahoma City St Paul IWA Des Moines MISSURI Jefferson City ARKANSAS Little Rock WISCNSIN Madison LUISIANA Jackson L a k e S u p e r i o r ILLINIS Baton Rouge Springfield MISSISSIPPI 1 a n 1 2 2 Indianapolis Nashville L Lansing INDIANA Montgomery Columbus Frankfort KENTUCKY TENNESSEE ALABAMA 3 km 3 mi HI Atlanta H AWAII Charleston GERGIA A L ASK A PACIFIC CEA N Tallahassee L a k e WEST Columbia n t a r i o NEW YRK Harrisburg SUTH CARLINA FLRIDA C U B A Albany Annapolis Dover Richmond Raleigh Washington Trenton Concord Boston Hartford DELAWARE MARYLAND MAINE VERMNT Providence NEW HAMPSHIRE CNNECTICUT NEW JERSEY MASSACHUSETTS RHDE ISLAND The National Atlas of the United States of America R DC AT LA N T I C C E A N states_capitals2.pdf INTERIR-GELGICAL SURVEY, RESTN, -23 1962: contest by Proctor and Gamble - best TSP tour through 33 US cities 1998: The Florida Sun-Sentinel s Science page ponders Santa Claus s traveling problem The TSP has even been used to make art ne of the most popular problems in operations research Numerous applications in expected and unexpected places Circuit board manufacturing Genome sequencing Your turn! Try to find the shortest way of visiting all the capitals and then returning to Annapolis What about 13,59 cities in the US? Sophisticated mathematical techniques are our best bet The R approach Problem statement and data What is the shortest way of visiting all the capitals and then returning to Annapolis? Interpreting I F I C C E A CALIFRNIA nationalatlas.gov TM BERING SEA WASHINGTN GULF F Solution C A N A D A M E X I C C A N A D A M I C H I G A N L a k e M i c h i g G U L F F M E X I C a k e H u r o n STATES AND CAPITALS L a k e E r i e PENNSYLVANIA NRTH CARLINA Montpelier Augusta A T L A T H E B A H A M A S E A N N T I C C Modeling ptimization (this course), Stochastic processes Solving min {i, j} E c {i, j} x {i, j} s.t. {i,k} E x {i,k} = 2 i N {i, j} δ(s) x {i, j} 2 S N x {i, j} {, 1} {i, j} E Mathematical model Goals for this course Modeling Recognize opportunities for mathematical optimization Formulate optimization models linear programs that capture the essence of the problem Illustrate applications of real-world problems Solving Algorithms to solve these mathematical models Detailed topic list and schedule is on the syllabus 2
ptimization is everywhere Minimize time it takes to get from class to class Maximize the company s profits (Moneyball) Best lineup for the akland A s We are always trying to make decisions in a way that meets some objective subject to some constraints Some success stories of optimization helping solve complex real-world decision-making problems... Package delivery UPS has an air network consisting of 7 hubs, nearly 1 additional airports in the US, 16 aircraft of nine different types Constraints: UPS credits optimization-based planning tools with identifying operational changes that have saved over $87 million to date, reduced planning times, peak and non-peak costs, fleet requirements Sports scheduling ACC Basketball earns over $3 million in revenue annually, almost all from TV and radio TV networks need a steady stream of high quality games, NCAA rules, school preferences and traditions 3
Constraints: ptimization approaches yields reasonable schedules very quickly Radiation therapy High doses of radiation can kill cancer cells and/or prevent them from growing and dividing Can also kill healthy cells! Radiation can be delivered at different angles and intensities Constraints: Next... Many successes reported using different types of optimization models Formulating a small optimization model 4
nationalatlas.gov TM Where We Are STATES AND CAPITALS CA NADA ATLANTIC CEAN lympia WASHINGTN P A C I F I C C E A N Salem Sacramento REGN CALIFRNIA Carson City NEVADA Boise IDAH ARIZNA Phoenix UTAH Helena Salt Lake City MNTANA WYMING Cheyenne Denver CLRAD Santa Fe NEW MEXIC NRTH DAKTA Bismarck SUTH DAKTA Pierre NEBRASKA KANSAS TEXAS Lincoln Topeka KLAHMA MINNESTA klahoma City St Paul IWA Des Moines MISSURI Jefferson City ARKANSAS Little Rock WISCNSIN Madison Jackson L a k e S u p e r i o r ILLINIS Springfield MISSISSIPPI MICHIGAN a n L a k e M i c h i g Indianapolis Nashville L Lansing INDIANA Montgomery a k e H u r o n Columbus Frankfort KENTUCKY TENNESSEE ALABAMA Atlanta HI L a k e E r i e Charleston GERGIA PACIFIC CEAN L a k e Columbia n t a r i o NEW YRK PENNSYLVANIA WEST Harrisburg DC Albany Annapolis Dover Richmond Raleigh NRTH CARLINA SUTH CARLINA Washington Montpelier Augusta Trenton Concord Boston Hartford DELAWARE MARYLAND A T L A MAINE VERMNT NEW HAMPSHIRE Providence CNNECTICUT NEW JERSEY MASSACHUSETTS RHDE ISLAND E A N N T I C C Honolulu R U S S I A A R C T I C C E A N Austin LUISIANA Baton Rouge Tallahassee 1 km 1 mi ME G U L F F M E X I C FLRIDA BERING SEA GULF F 2 mi 2 km CANADA Juneau XIC 1 2 1 2 3 km 3 mi CU BA THE BAHAMAS U.S. Department of the Interior U.S. Geological Survey The National Atlas of the United States of America R states_capitals2.pdf INTERIR-GELGICAL SURVEY, RESTN, -23