# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

1 Page

### integer_programming_1

Course: IOE 310, Spring 2008
School: Michigan
Rating:

Word Count: 413

#### Document Preview

PROGRAMMING I Pure INTEGER Integer Program. max z = 3x1 + 2x2 x1 + x2 6 x1 int x2 int x1 0 x2 0 Mixed Integer Program. max z = 3x1 + 2x2 x1 + x2 6 x1 int x2 real x1 0 x2 0 Saigal (U of M) IOE 310 1/1 INTEGER PROGRAMMING 0-1 IP. max z = II 3x1 + 2x2 x1 + 2x2 2 2x1 - x2 1 x1 = 0 or 1 x2 = 0 or 1 LP Relaxation. max z = 3x1 + 2x2 x1 + x2 6 x1 0 x2 0 Saigal (U of M) IOE 310 2/1 FORMULATION Fixed...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Michigan >> Michigan >> IOE 310

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
PROGRAMMING I Pure INTEGER Integer Program. max z = 3x1 + 2x2 x1 + x2 6 x1 int x2 int x1 0 x2 0 Mixed Integer Program. max z = 3x1 + 2x2 x1 + x2 6 x1 int x2 real x1 0 x2 0 Saigal (U of M) IOE 310 1/1 INTEGER PROGRAMMING 0-1 IP. max z = II 3x1 + 2x2 x1 + 2x2 2 2x1 - x2 1 x1 = 0 or 1 x2 = 0 or 1 LP Relaxation. max z = 3x1 + 2x2 x1 + x2 6 x1 0 x2 0 Saigal (U of M) IOE 310 2/1 FORMULATION Fixed Charge Case I A company produces shirts, shorts and pants. It must rent machines for the production. The data follows: Item M/c Rental Fee Labor/unit Cloth/unit Sales Price Variable cost Shirt \$200 3 hrs 4 yrds \$ 12 \$6 Short \$150 2 hrs 3 yrds \$8 \$4 Pant \$100 6 hrs 4 yrds \$ 15 \$8 Max Avail 150 hrs 160 yrds Formulate an IP to maximize profit. Saigal (U of M) IOE 310 3/1 FORMULATION Continued I Decision Variables: x1 = number of shirts produced x2 = number of shorts produced x3 = number of pants produced y1 = 1 0 if any shirts produced if no shirts produced , y2 = 1 0 if any shorts produced if no shorts produced , y3 Saigal (U of M) IOE 310 4/1 FORMULATION Continued Rental Cost: I 200y1 + 150y2 + 100y3 . Profit: (12 - 6)x1 + (8 - 4)x2 + (15 - 8)x3 . THE MODEL: max 6x1 + 4x2 + 7x3 - 200y1 - 150y2 - 100y3 labor: 3x1 + 2x2 + 6x3 cloth: 150 4x1 + 3x2 + 4x3 160 In addition we need: x1 M1 y1 x2 M2 y2 x3 M3 y3 Saigal (U of M) IOE 310 5/1 FORMULATION Continued II Mi are large numbers. OPTIMUM SOLUTION: z = 75, x3 = 25, y3 = 1. Saigal (U of M) IOE 310 6/1 EITHER OR CONSTRAINTS I Given two constraints expressed by functions f and g : f (x1 , x2 , , xn ) 0 g (x1 , x2 , , xn ) 0 how do we express that at least one of these constraints is satisfied? Answer: Define y= 0 1 if constraint 1 must be satisfied if constraint 2 must be satisfied and add the constraints below to the formulation: f (x1 , x2 , , xn ) My g (x1 , x2 , , xn ) M(1 - y ) Saigal (U of M) IOE 310 7/1 EITHER OR CONSTRAINTS II where M is a very large number that renders the constraints f (x1 , x2 , , xn ) M g (x1 , x2 , , xn ) M redundant. Saigal (U of M) IOE 310 8/1 IF THEN CONSTRAINTS I In many cases if some constraint f (x1 , x2 , , xn ) > 0 is satisfied, then the constraint g (x1 , x2 , , xn ) 0 must be satisfied, i.e., f (x1 , x2 , , xn ) > 0 implies g (x1 , x2 , , xn ) 0 This is satisfied by including the following in the formulation: -g (x1 , x2 , , xn ) My f (x1 , x2 , , xn ) M(1 - y ) y = 0 or 1 Saigal (U of M) IOE 310 9/1
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

Michigan - IOE - 310
Introduction to Optimization - IOE 310Romesh Saigal Office Hours: E Mail Address: GSI 1 GSI 2: MIDTERM FINAL Wed 1:30-3:00pm and by appointment, Room IOE 2883 rsaigal@umich.edu Ankit Agarwal Yu Zhou Tuesday, October 23, 2007 Thursday, December 13,
Michigan - IOE - 310
A QUICK REVIEW OF MATERIAL YOU SHOULD KNOWIOE 310 Fall 2007INTRODUCTION TO LINEAR PROGRAMMINGWhat is linear programming. Graphical solution of a linear programming problem.IOE 310 Fall 2007AN EXAMPLEA manufacturer produces two toysSoldi
Michigan - IOE - 310
NETWORK MODELSIA Graph or equivalently a Network consists of a set V called vetrices, of nodes, and a set A called the set of arcs. Definition: An arc is an ordered set (called directed) or unordered set (called undirected) of vertices, represent
Michigan - IOE - 310
PERT AND CPMIPERT stands for Project Evaluation and Review Technique and CPM stands for the Critical Path Method. This is also called project networks, and these techniques have been very successfully used in many applications, including: Schedul
Michigan - IOE - 310
SENSITIVITY ANALYSIS: AN APPLIED APPROACHEffect of changes in objective function coefficients. Effect of changes in RHS coefficients. Shadow Prices. Importance of sensitivity analysis.Saigal (U of M)IOE 3101/6AN EXAMPLEmaximize z = 3x1 + 2x
Michigan - IOE - 310
OTHER FORMS AND CASES OF LPsObjective function may be a minimum. The linear Program may have multiple solutions. The linear program may be unbounded. The constraints may be equality or greater than inequality. Variables may be less that equal to 0 o
Michigan - IOE - 310
THE SIMPLEX ALGORITHMIStep 1 Convert LP into standard form. Step 2 Obtain a bfs (if possible) from standard form. Step 3 Determine if the bfs is optimal. Step 4 If non-optimal, determine: a nonbasic variable that should become basic. A basic vari
Michigan - IOE - 310
FORMULATING TRANSPORTATION PROBLEMSAn electric company has three plants that supply four cities. Supply demand data follow: To From City 1 City 2 City 3 City 4 Supply Plant 1 \$8 \$6 \$10 \$9 35 Plant 2 \$9 \$12 \$13 \$7 50 Plant 3 \$14 \$9 \$16 \$5 40 Demand
Michigan - IOE - 316
IOE 316 Introduction to Markov Processes Fall 2007 Midterm Exam 2Tuesday, December 11, 2007 10:40 am - 12:00 pmNAME UMIDSOLUTIONEngineering Honor Code (Make sure to sign it before turning in the exam): I have not received nor given any unautho
Michigan - IOE - 316
IOE 316 Fall 2007 Homework 1Due October 30, 20071. (20 points) Three white and three black balls are distributed in two urns in such a way that each contains three balls. We say that the system is in state i, i = 0, 1, 2, 3, if the first urn con
Michigan - IOE - 316
IOE 316 Homework 1 SolutionsDue October 30, 20071. (20 points) Three white and three black balls are distributed in two urns in such a way that each contains three balls. We say that the system is in state i, i = 0, 1, 2, 3, if the first urn con
Michigan - IOE - 316
IOE 316 Fall 2007 Homework 2Due November 6, 20071. (15 points) Consider the following Markov chain:(a) (5 points) Find the values of x and y. (b) (10 points) Find 2-step transition probability matrix P2 2. (25 points) A Markov chain {Xn , n 0
Michigan - IOE - 316
IOE 316 Fall 2007 Homework 2 SolutionsDue November 6, 20071. (15 points) Consider the following Markov chain:(a) (5 points) Find the values of x and y. (b) (10 points) Find 2-step transition probability matrix P2 SOLUTION: (a) (5 points) To fi
Michigan - IOE - 316
IOE 316 Fall 2007 Homework 3Due November 13, 20071. (25 points) Coin 1 comes up heads with probability 0.7 and coin 2 with probability 0.5. A coin is continually flipped until it comes up tails, at which time that coin is put aside and we start
Michigan - IOE - 316
IOE 316 Fall 2007 Homework 3 SolutionsDue November 13, 20071. (25 points) Coin 1 comes up heads with probability 0.7 and coin 2 with probability 0.5. A coin is continually flipped until it comes up tails, at which time that coin is put aside and
Michigan - IOE - 316
IOE 316 Fall 2007 Homework 4Due November 20, 20071. (10 points) As you arrive to a single-server fast-food restaurant you notice that there are three customers waiting in line and one additional customer is currently being serviced. You join the
Michigan - IOE - 316
IOE 316 Fall 2007 Homework 4 SolutionsDue November 20, 20071. (10 points) As you arrive to a single-server fast-food restaurant you notice that there are three customers waiting in line and one additional customer is currently being serviced. Yo
Michigan - IOE - 316
IOE 316 Fall 2007 Homework 5Due December 4, 20071. (25 points) Cars cross a certain point in the highway in accordance with a Poisson process with rate = 4 per minute. (a) (5 points) If Kate blindly runs across the highway, then what is the pro
Michigan - IOE - 316
IOE 316 Fall 2007 Homework 5 SolutionDue December 4, 20071. (25 points) Cars cross a certain point in the highway in accordance with a Poisson process with rate = 4 per minute. (a) (5 points) If Kate blindly runs across the highway, then what i
Michigan - IOE - 316
IOE 316 Fall 2007 Homework 6Due MONDAY December 10, 2007 NOON (12pm) Note: To turn in your homework, you can either slide it under instructor's office door (1783 IOE) by noon, or bring it to Arleigh's office hours that morning. No late homework wil
Michigan - IOE - 316
IOE 316 Fall 2007 Homework 6 SOLUTIONSDue MONDAY December 10, 2007 NOON (12pm) Note: To turn in your homework, you can either slide it under instructor's office door (1783 IOE) by noon, or bring it to Arleigh's office hours that morning. No late ho
Michigan - IOE - 316
Arizona - CHEM - 481
17 February, 2006 Name Social Security NumberCHEMISTRY 481 (Biophysical Chemistry) Midterm Exam I Instructions:(i) Take it easy and relax! If there are any difficulties or questions just ask the instructor. (ii) This exam consists of 07 printed pa
Arizona - CHEM - 481
10 March, 2006CHEMISTRY 481 (Biophysical Chemistry) Midterm Exam I1 Instructions:(i) Take it easy and relax! If there are any difficulties or questions just ask the instructor. (ii) This exam consists of 07 printed pages. Please make sure that yo
Arizona - CHEM - 481
24 April, 2006i . r : 21 n , Name Social Security Number Get-jd-hie/CHEMISTRY 481 (Biophysical Chemistry) Midterm Exam 111 Instructions:( i ) Take it easy and relax! If there are any difficulties or questions just ask the instructor.( This
Arizona - CHEM - 481
13 January, 2006 Michael F. Brown CHEMISTRY 481 (Biophysical Chemistry) Problem Set 01 To be turned in by:Monday, 20 January Worked examples for this course are dealt with primarily in the Discussion Section; whereas new concepts are introduced in th
Arizona - CHEM - 481
Arizona - CHEM - 481
Arizona - CHEM - 481
Arizona - CHEM - 481
20 February, 2006 Michael F. Brown CHEMISTRY 487 (Biophysical Chemistry) Problem Set 05To be turned in by: Monday, 27 FebruaryWorked examples for this course are dealt with primarily in the Discussion Section; whereas new concepts are introduced i
Arizona - CHEM - 481
27 February, 2006 Michael F. Brown CHEMISTRY 481 (Biophysical Chemistry) Problem Set 06To be turned in by: Monday, 06 MarchWorked examples for this course are dealt with primarily in the Discussion Section; whereas new concepts are introduced in t
Arizona - CHEM - 481
06 March, 2006 Michael F. Brown CHEMISTRY 481 (Biophysical Chemistryl Problem Set 07To be turned in by: NOT TO BE GRADED!Worked examples for this course are dealt with primarily in the Discussion Section; whereas new concepts are introduced in the
Arizona - CHEM - 481
22 March, 2006 Michael F. BrownCHEMISTRY 481 (Biophysical Chemistry) Problem Set 08To be turned in by: Monday, 27 MarchlWorked examples for this course are dealt with primarily in the Discussion Section; whereas new concepts are introduced in the
Arizona - CHEM - 481
Arizona - CHEM - 481
Arizona - CHEM - 481
Arizona - CHEM - 481
Arizona - CHEM - 481
12 February, 2007 Michael F. Brown CHEMISTRY 481 (Biophysical Chemistry) Problem Set 05 (STUDY GUIDE - NOT TO BE GRADED)To be turned in by: NEVERWorked examples for this course are dealt with primarily in the Discussion Section; whereas new concep
Arizona - CHEM - 481
26 February, 2007 Michael F. Brown CHEMISTRY 481 (Biophysical Chemistry) Problem Set 07 (STUDY GUIDE - NOT TO BE GRADED)To be turned in by: NEVERWorked examples for this course are dealt with primarily in the Discussion Section; whereas new concep
Arizona - CHEM - 481
19 March, 2007 Michael F. Brown CHEMISTRY 481 (Biophysical Chemistry) Problem Set 08To be turned in by: Monday, 26 MarchWorked examples for this course are dealt with primarily in the Discussion Section; whereas new concepts are introduced in the
Arizona - CHEM - 481
02 April, 2007 Michael F. Brown CHEMISTRY 481 (Biophysical Chemistry) (STUDY GUIDE - NOT TO BE GRADED) Problem Set 09To be turned in by: NEVERWorked examples for this course are dealt with primarily in the Discussion Section; whereas new concepts
Arizona - CHEM - 481
30 April, 2007 Michael F. Brown CHEMISTRY 481 (Biophysical Chemistry) Problem Set 12 - STUDY GUIDETo be turned in by: NEVER Background reading: Chapter 9 Chapter 10 Chapter 11.111.3, 11.7 Chapter 13.113.3, 13.10 Chapter 15 Back of Chapter Problems r
Arizona - BIOC - 462b
Name: _ Section: _TA: _Problem Set 1 Biochemistry 462b Spring 2008 Due Jan 25, 2007 Question 1. (3 points) The parietal cells of the stomach lining contain membrane pumps that transport hydrogen ions from the cytosol of these cells (pH 7.0) into
Arizona - BIOC - 462b
11Name: _ TA: _ Problem Set 3 Biochemistry 462b Spring 2008 Due Feb 8, 2008Section: _The following questions are based on the papers that will be presented during the discussion session on Thursday, Feb 7. In the last problem set, you saw how ske
Arizona - BIOC - 462b
Name: _ Section: _TA: _Problem Set 2 Biochemistry 462b Spring 2008 Due Feb 1, 2008 Research in metabolsim and metabolic regulation, unlike many other branches of biochemistry, often requires you to look at the whole organism. The pioneers of meta
Arizona - BIOC - 466
Problem Set 3. PyMOL and DNA structureDiscussion 2/5/08 Due: 2/8/08Run the program PyMOL by double clicking on the PyMOL icon. Two windows will appear. In the top window (the PyMOL Tcl/Tk GUI), left click on File, choose Open, and select DNA1.pdb
Arizona - BIOC - 462b
Name:_Answer Key_ TA_ Section:_ PROBLEM SET 6 Biochemistry 462b Spring 2008 Total 20 points - Answer BRIEFLY!] 1. (4 pts) Standard reduction potentials of flavins vary from protein to protein, depending upon the protein environment of the flavin. The
Arizona - CHEM - 481
27 March, 2006 Michael F. BrownCHEMISTRY 481 (Biophysical Chemistry) Problem Set 09To be turned in by: Monday, 03 AprilWorked examples for this course are dealt with primarily in the Discussion Section; whereas new concepts are introduced in the
Arizona - CHEM - 481
03 April, 2006 Michael F, BrownCHEMISTRY 481 (Biophysical Chemist* Problem Set 10To be turned in by: Monday, 10 AprilWorked examples for this course are dealt with primarily in the Discussion Section; whereas new concepts are introduced in the Le
Arizona - CHEM - 481
10 April, 2006 Michael F. BrownCHEMISTRY 487 (Biophysical Chemistry) Problem Set 77To be turned in by: Monday, 17 AprilWorked examples for this course are dealt with primarily in the Discussion Section; whereas new concepts are introduced in the
Arizona - CHEM - 481
17 April, 2006 Michael F. Brown CHEMISTRY 481 (Biophysical Chemistry) Problem Set 12To be turned in by: NOT TO BE GRADED!Worked examples for this course are dealt with primarily in the Discussion Section; whereas new concepts are introduced in the
Arizona - CHEM - 481
Arizona - CHEM - 481
Arizona - CHEM - 481
Arizona - CHEM - 481
19 February, 2007 Michael F. Brown CHEMISTRY 481 (Biophysical Chemistry) Problem Set 06To be turned in by: Monday, 26 FebruaryWorked examples for this course are dealt with primarily in the Discussion Section; whereas new concepts are introduced i
Arizona - CHEM - 481
16 April, 2007 Michael F. Brown CHEMISTRY 481 (Biophysical Chemistry) Problem Set 10To be turned in by: Monday, 23 AprilWorked examples for this course are dealt with primarily in the Discussion Section; whereas new concepts are introduced in the
Arizona - CHEM - 481
22 April, 2007 Michael F. Brown CHEMISTRY 481 (Biophysical Chemistry) Problem Set 11To be turned in by: Monday, 30 AprilWorked examples for this course are dealt with primarily in the Discussion Section; whereas new concepts are introduced in the
Arizona - BIOC - 462b
Name: _ Net ID: _ Section: _EXAM IBIOCHEMISTRY 462B - SPRING 2008Write your name and UA Net ID on ALL pages! This exam has 10 questions on 6 pages (including this face page) and a total of 100 points. Read all questions carefully Show your work o
Arizona - BIOC - 462b
Name: _Net ID: _Section: _EXAM II Key March 12, 2008BIOCHEMISTRY 462B - SPRING 2007Write your name and UA Net ID on ALL pages! There are 9 questions and 100 POINTS total in this exam. This Exam has 7 pages (including this face page). R
Arizona - BIOC - 462b
Name _ Section _Net ID _Exam IIIApril 9, 2007 Biochemistry 462bWrite your name and UA NetID on all pages1. _ 2. _ 3. _ 4. _ 5. _ 6. _ Total _1Name _ Section _Net ID _1. (8 pts total) The structural diversity of the compounds that are i
Arizona - BIOC - 462b
Name _NetID_Biochemistry 462b Exam IV Wednesday May 9, 2007 _ Page 2 _Page 3 _Page 4 _Page 5 _Page 6 _Page 7 _Page 8 _Page 9 _ Page 10 _ Total1Name _NetID_1. (8 pts) G-protein coupled receptor (GPCR) systems have evolved complex mechanism