Operational Research (I)

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2 What is OR? Feb 4, 006 F. Che Page

3 Outlie for today Syllabus ad Course Structure Itroductio to OR Itroductio to Liear Programmig (first topic) Feb 4, 006 F. Che Page 3

4 About me Feg Che ( ) Office Hours : am 8:00 :30, pm :30-5:00 Office : Room 30, Mechaism Buildig, Xuhui Campus Tel : 6938(o), Feb 4, 006 F. Che /fche@sjtu.edu.c Page 4

5 Discussio ad Get Lectures Dowload lectures from ftp://public.sjtu.edu.c by fche (id) ad public (pw) 4 hours before class. Feb 4, 006 F. Che /fche@sjtu.edu.c Page 5

6 Text book Waye L. Wisto, Operatios Research: Applicatios ad Algorithms, third editio, 994. Feb 4, 006 F. Che Page 6

7 Course Objectives Lear some basic techiques ad methodologies of Operatio Research (OR). Lear to use some importat optimizatio software. Use these kowledge to solve some real problems. Feb 4, 006 F. Che Page 7

8 Class Structure Gradig ad Assigmets HW ad Participatio - 5% Presetatio & Programmig (Software) - 5% Middle Exam - 0% Fial Exam - 50% Attedace i class is required Abseces will be detrimetal to your stadig i the class. Feb 4, 006 F. Che /fche@sjtu.edu.c Page 8

10 Culture ad History Operatios Research started as a amed field i WWII(930s), thaks to physicists such as Philip M. Morse Empirical sciece: usig all relevat scietific methods to solve maagerial decisio problems Feb 4, 006 F. Che /fche@sjtu.edu.c Page 0

11 Quotes Quotes From Methods of Operatios Research, Morse ad Kimball Operatios Research is a scietific method of providig executive departmets with a quatitative basis for decisios regardig operatios uder their cotrol. Feb 4, 006 F. Che /fche@sjtu.edu.c Page

12 Quotes (co t) Operatios Research is a applied sciece utilizig all kow scietific techiques as tools i solvig a specific problem. Feb 4, 006 F. Che /fche@sjtu.edu.c Page

13 Quotes (co t) Operatios Research uses mathematics, but it is ot a brach of mathematics. Feb 4, 006 F. Che /fche@sjtu.edu.c Page 3

14 Quotes (co t) Operatios Research is ofte a experimetal sciece as well as a observatioal oe. Feb 4, 006 F. Che /fche@sjtu.edu.c Page 4

15 Quotes (co t) It ofte occurs that the major cotributio of the operatios research worker is to decide what is the real problem. Feb 4, 006 F. Che Page 5

16 Culture ad History Most Major Advaces i Operatios Research Have Come from Work o Real Problems A. K. Erlag, Daish telephoe egieer -- iveted queueig theory i work aimed to determie optimal capacity of ewly iveted cetral telephoe switchig ceters (95) Feb 4, 006 F. Che /fche@sjtu.edu.c Page 6

17 Chiese Postma Problem Mei-Ko Kwa( ),Graphic Programmig Usig Odd or Eve Poits, Chiese Mathematics, :73-77, 96. Whe the author was plottig a diagram for a mailma s route, he discovered the followig problem: A mailma has to cover his assiged segmet before returig to the post office. The problem is to fid the shortest walkig distace for the mailma. Feb 4, 006 F. Che /fche@sjtu.edu.c Page 7

the work of wheat harvestig i the Pekig suburbs was participated i by teachers ad studets The

18 A Facility Locatio Problem Hua Lo-Keg ad others, Applicatio of Mathematical Methods to Wheat Harvestig, Chiese Mathematics :77-9, 96. the work of wheat harvestig i the Pekig suburbs was participated i by teachers ad studets The objective was experimetal use of mathematical methods i the selectio of the threshig site most ecoomical of trasportatio. Feb 4, 006 F. Che /fche@sjtu.edu.c Page 8

19 Successful Applicatios. Police Patrol Officer Schedulig i Sa Fracisco (989) Save $ millio per year, improved respose times by 0%, other reveue $3,000,000 Reducig Fuel Costs i the Electric Power Idustry. Save $5 millio Feb 4, 006 F. Che /fche@sjtu.edu.c Page 9

20 Successful Applicatios(co t). Desigig a Igot Mold Strippig Facility at Bethlehem Steel. (989, $8 millio) Gasolie Bledig at Texaco(989,$30 mi) Schedulig Trucks at North America Va Lies. (988, $.5 millio) Ivetory Maagemet at Blue Bell. (985, 3% ivetory) Feb 4, 006 F. Che /fche@sjtu.edu.c Page 0

21 Curret Hot Topics From Operatios Research (Very kow joural): Homelad Security Call ceters Iteret modelig Reveue maagemet Game-theoretic supply-chai maagemet Supply Chai ad Logistics Feb 4, 006 F. Che Page

22 Curret Hot Topics (co t) From Mathematics of Operatios Research: Iteret modelig (heavy tail distributios) Auctio theory Fiacial egieerig Price of aarchy Aims at aalyzig the differece betwee performace uder "selfish behavior" ad uder coordiated optimizatio. The methods here are game theoretic, ivolvig Nash equilibrium ad competitive equilibrium. Methods of both discrete optimizatio ad cotiuous optimizatio arise i the aalysis. Feb 4, 006 F. Che /fche@sjtu.edu.c Page

23 Curret Hot Topics (co t) From Maagemet Sciece: Social etworks Risk maagemet Data miig Strategic plaig Service operatios Feb 4, 006 F. Che Page 3

24 What We Will Lear Liear Programmig (Simplex Methods) Iteger Programmig (Brach ad Boud Methods) Dyamic Programmig (Forward ad Backward methods) Software (Lido & Excel) Feb 4, 006 F. Che Page 4

26 Mathematical Programmig (MP) What is MP? The format? objective style (MP) express s. t. g h j j ( x ( x f ( x,...,...,... ) ) 0, j =,..., l ) = 0, j = l +,..., m costrait Feb 4, 006 F. Che /fche@sjtu.edu.c Page 6

27 Geeral mathematical prog. What is MP? The format? style (MP) express miimize express s. t. g h j j ( x ( x f ( x,...,...,... ) ) 0, j =,..., l ) = 0, j = l +,..., m descriptio Feb 4, 006 F. Che /fche@sjtu.edu.c Page 7

28 Geeral mathematical prog. What is MP? The format? style (MP) express miimize express s. t. g h j j ( x ( x f ( x,...,...,... ) ) = ) 0, 0, j j =,..., l = l +,..., m descriptio Fid out a solutio which arrives the smallest objective Feb 4, 006 F. Che /fche@sjtu.edu.c Page 8

29 Geeral mathematical prog. What is MP? The format? style (MP) express miimize maximize express s. t. g h j j ( x ( x f ( x,...,...,... ) ) = ) 0, 0, j j =,..., l = l +,..., m descriptio Fid out a solutio which arrives the smallest objective Fid out a solutio which arrives the largest objective Feb 4, 006 F. Che /fche@sjtu.edu.c Page 9

30 Geeral mathematical prog. What is MP? The format? style (MP) express miimize maximize maxmi express s. t. g h j j ( x ( x f ( x,...,...,... ) ) 0, j =,..., l ) = 0, j = l +,..., m descriptio Fid out a solutio which arrives the smallest objective Fid out a solutio which arrives the largest objective Feb 4, 006 F. Che /fche@sjtu.edu.c Page 30

31 Geeral mathematical prog. What is MP? The format? objective (MP) express s. t. g h j j ( x ( x f ( x,...,...,... ) ) 0, j =,..., l ) = 0, j = l +,..., m f(x,x, x ) Liear descriptio x +x Feb 4, 006 F. Che /fche@sjtu.edu.c Page 3

32 Geeral mathematical prog. What is MP? The format? objective (MP) express s. t. g h j j ( x ( x f ( x,...,...,... ) ) 0, j =,..., l ) = 0, j = l +,..., m f(x,x, x ) descriptio Liear x +x Noliear x +x 3 Feb 4, 006 F. Che /fche@sjtu.edu.c Page 3

33 Geeral mathematical prog. What is MP? The format? (MP) express s. t. g h j j ( x ( x f ( x,...,...,... ) ) 0, j =,..., l ) = 0, j = l +,..., m g(x,x, x ) h(x,x, x ) descriptio costrait Liear x +x =5 Noliear x +x 3 <6 Feb 4, 006 F. Che /fche@sjtu.edu.c Page 33

34 Geeral mathematical prog. What is MP? The format? (MP) express s. t. g h j j ( x ( x f ( x,...,...,... ) ) 0, j =,..., l ) = 0, j = l +,..., m real umber, x R itegers oegative Feb 4, 006 F. Che /fche@sjtu.edu.c Page 34

35 Liear Programmig Defiitio A MP is called liear programmig, if f ad g j ad h j are all liear fuctio. (MP) express s. t. g h j j f ( x ( x ( x,...,...,... ) ) 0, j =,..., l ) = 0, j = l +,..., m express miimize maximize descriptio Fid out a solutio which arrives the smallest objective Fid out a solutio which arrives the largest objective Feb 4, 006 F. Che /fche@sjtu.edu.c Page 35

36 Liear Programmig Defiitio (Gia) max z = s. t. x x x x 3x + x + x + x Feb 4, 006 F. Che /fche@sjtu.edu.c Page 36

37 The History of Liear Programmig. George Datzig Leoid Katorovich Feb 4, 006 F. Che Page 37

38 The History of Liear Programmig 936 W.W. Leotief published "Quatitative Iput ad Output Relatios i the Ecoomic Systems of the US" which was a liear model without objective fuctio. 939 Katoravich (Russia) actually formulated ad solved a LP problem 94 Hitchcock poses trasportatio problem (special LP) WWII Allied forces formulate ad solve several LP problems related to military A breakthrough occurred i Feb 4, 006 F. Che /fche@sjtu.edu.c Page 38

39 The History of Liear Programmig (C) US Air Force wated to ivestigate the feasibility of applyig mathematical techiques to military budgetig ad plaig. George Datzig had proposed that iterrelatios betwee activities of a large orgaizatio ca be viewed as a LP model ad that the optimal program (solutio) ca be obtaied by miimizig a (sigle) liear objective fuctio. Air Force iitiated project SCOOP (Scietific Computig of Optimum Programs) Feb 4, 006 F. Che /fche@sjtu.edu.c Page 39

40 Today s LP A large variety of Simplex-based algorithms exist to solve LP problems. Other (polyomial time) algorithms have bee developed for solvig LP problems: Khachia algorithm (979) Kamarkar algorithm (AT&T Bell Labs, mid 80s) oe of these algorithms have bee able to beat Simplex i actual practical applicatios. Simplex (i its various forms) is ad will most likely remai the most domiat LP algorithm for at least the ear future Feb 4, 006 F. Che /fche@sjtu.edu.c Page 40

41 Typical Applicatios of Liear Programmig. A maufacturer wats to develop a productio schedule ad ivetory policy that will satisfy sales demad i future periods ad same time miimize the total productio ad ivetory cost. Feb 4, 006 F. Che /fche@sjtu.edu.c Page 4

42 Typical Applicatios of Liear Programmig. A maufacturer wats to develop a productio schedule ad ivetory policy that will satisfy sales demad i future periods ad same time miimize the total productio ad ivetory cost.. A fiacial aalyst must select a ivestmet portfolio from a variety of stock ad bod ivestmet alteratives. He would like to establish the portfolio that maximizes the retur o ivestmet. Feb 4, 006 F. Che /fche@sjtu.edu.c Page 4

43 Typical Applicatios of Liear Programmig cotiued 3. A marketig maager wats to determie how best to allocate a fixed advertisig budget amog alterative advertisig media such as radio, TV, ewspaper, ad magazies. The goal is to maximize advertisig effectiveess. 4. A compay has warehouses i a umber of locatios throughout the coutry. For a set of customer demads for its products, the compay would like to determie how much each warehouse should ship to each customer so that the total trasportatio costs are miimized. Feb 4, 006 F. Che /fche@sjtu.edu.c Page 43

44 Typical Applicatios of Liear Programmig cotiued 3. A marketig maager wats to determie how best to allocate a fixed advertisig budget amog alterative advertisig media such as radio, TV, ewspaper, ad magazies. The goal is to maximize advertisig effectiveess. 4. A compay has warehouses i a umber of locatios throughout the coutry. For a set of customer demads for its products, the compay would like to determie how much each warehouse should ship to each customer so that the total trasportatio costs are miimized. Feb 4, 006 F. Che /fche@sjtu.edu.c Page 44

45 Example (Gia) mi z = s. t. x x x x 3x, + x + x x + x () () (3) (4) (5) Objective Costraits Sig costraits Feb 4, 006 F. Che /fche@sjtu.edu.c Page 45

46 Basic termiologies Objective fuctio coefficiet (Gia) max z = s. t. x x x x 3x + x + x + x Feb 4, 006 F. Che /fche@sjtu.edu.c Page 46

47 Basic termiologies Objective fuctio coefficiet Techological coefficiet (Gia) max z = s. t. x x x x 3x + x + x + x Feb 4, 006 F. Che /fche@sjtu.edu.c Page 47

48 Basic termiologies Objective fuctio coefficiet Techological coefficiet Right-had side ( rhs ) (Gia) max z = s. t. x x x x 3x + x + x + x Feb 4, 006 F. Che /fche@sjtu.edu.c Page 48

49 Homework Read page -5 by yourself. Feb 4, 006 F. Che Page 49

Product Mix Problem with Radom Return and Preference of Production Quantity. Osaka University Japan

Product Mix Problem with Radom Retur ad Preferece of Productio Quatity Hiroaki Ishii Osaka Uiversity Japa We call such fiace or idustrial assets allocatio problems portfolio selectio problems, ad various