Welcome to the Course Page of
MATH 3172 3.00 M

Combinatorial Optimization

S2 2015

Department of Mathematics and Statistics
Faculty of Science
York University
4700 Keele Street
Toronto, Ontario  M3J 1P3

Course Organization:


Course Information

Course Schedule


Exams Information

Links & Other Resources


This area will contain announcements and solutions to the tests and assignments. Announcements made in class will be posted here in reverse chronological order and will not be repeated in class. Please take a minute to read the announcements carefully, as they often get updated. You may need to Reload/Refresh this page.

August 20:

As it was announced in the class on yesterday, the Final Examination for


the course will take place on Tuesday, August 25 from 19:00 to 22:00 in


VH 2005. The exam will be comprehensive, covering the entire course


material according to the Course Schedule but with an emphasis on the more


recent material covered after the Midterm Test. If less common formulas are


needed on the exam, then those will be provided on the exam. There will be


several questions, some with a few parts. Marks for each question will be


indicated. You will have to show all your work, and explain your solutions


where appropriate.


Please note that I will be holding an additional office hour for last minute


questions on Tuesday, August 25 from 14:00 to 15:00.

August 17:

Homework problems for Sections 9.6, 9.7 and 9.8:


Section 9.6 #1, 2, 5 and 7;


Section 9.7 #1, 2 and 3;


Section 9.8 #1 and 3.

August 12:

Homework problems for Sections 9.2, 9.3 and 9.4:


Section 9.2 #1, 3, 10, 11, 12, 15, 20, 21, 22 and 23;


Section 9.3 #1, 2, 3, 7 and 9; 


Section 9.4 #1 and 3.

August 10:

Please note that the student course evaluations are available on-line


at the URL link  http://courseevaluations.yorku.ca  until August 21,


2015. It is the time for you to be the judge. Please go on-line and


complete the course evaluation.

August 03:

Please note that the graded Midterm Test papers will be returned in the


class on Wednesday, August 05.

July 28:

As it was announced in the class yesterday, we will write Midterm Test


during the lecture period on Wednesday, July 29. On the test you will be


responsible for all the material covered in class from Sections 6.3, 6.4,


6.5, 6.6, 6.7, 6.8, 6.9, 6.11, 7.1, 7.2, 7.3, 7.5 and 7.6 of the textbook.

July 24:

Homework problems for Chapter 8:


Section 8.2 #1, 2, 4, 6 and 9;


Section 8.3 #1, 4, 8, 9, 11 and 14;


Section 8.4 #2, 3, 4, 5, 14 and 16;


Section 8.6 #1 and 2.

July 14:

Homework problems for Chapter 7:


Section 7.1 # 1, 3, 5 and 7;


Section 7.2 #1 (for 7.1 Problem #1 only) and #3 (for 7.1 Problem #5 only);


Section 7.3  (for 7.1 Problems #1 and #5 only);


Section 7.5 #1, 2, 3, 4 and 6;


Section 7.6 #1, 2 and 3;


Chapter 7 Review Problem #5.

July 05:

Welcome back!

Please download the Trial Version of LINGO to your PC using the link:


Download free trial version of LINGO package


Please note that you may also use the package in the Gauss Lab at S110 Ross.

To view and/or print PDF files you need to download the Acrobat Reader  


Homework problems for Chapter 6:


Section 6.1 # 1, 2, 3, 5 (each Radio 2 requires 1 (not 2) Laborer-2 hour);


Section 6.2 # 1, 2;


Section 6.3 # 1, 2, 3, 4, 5.;


Section 6.5 # 1, 2, 3, 4, 5, 6;


Section 6.6 # 1, 2;


Section 6.7 # 1, 2, 3, 4;


Section 6.8 # 2, 3, 5, 6;


Section 6.9 # 1, 2, 3;


Section 6.10 # 1;


Section 6.11 # 1, 2, 3, 4.

Course Information:



S2 2015





Lectures (day, time and venue):

M/W 18:00 – 21:00, ACW 205



Dr. Iouldouz Raguimov



S512 Ross






416-736-2100, Ext. 66092


Office Hours:

T 13:00-14:00, W 16:00 – 17:00, also by appointment


Midterm Test:

40% of the overall grade for the course


Final Examination:


Course Description: This course introduces students to combinatorial optimization (integer programming) including problem formulation, branch-and-bound method, cutting-plane method, implicit enumeration, and its business and industrial applications including transportation problem, network flow optimization etc. For a more detailed list of topics with references to the textbook, please see Course Schedule.


Operations Research: Applications and Algorithms, Fourth Edition

by Wayne L. Winston 


Books/Cole, Thomson, 2004. ISBN 0-534-42358-2

Optional Aids:

Student Solutions Manual for Operations Research, Fourth Edition


by Wayne L. Winston


Books/Cole, Thomson, 2004. ISBN 0-534-42360-4


Course Prerequisites:

SC/MATH 3171 3.00, SC/MATH 1021 3.00 or AP/SC/MATH 1025 3.00


or SC/MATH 2221 3.00


Students who do not have the prerequisites must see me within the first two


weeks of the classes.



Course Credit

AP/ECON 3120 3.00, AP/ADMS 3331 3.00, AP/ADMS 3351 3.00,


GL/MATH 3660 6.00, CS/MATH 3170 6.00.


Important Dates:

July 06: Classes commence

July 10: The last date to enrol in the course without my permission

July 20: The last date to enrol in the course with my permission

August 10: The last date to drop the course without receiving a final grade


August 21: Classes end

August 25: Final examination.

Course Schedule:
(subject to any changes announced in class)




July 06-10

6.4, 6.5, 6.6, 6.7 (omit pp 319-21), 6.8

Classes begin on Monday, July 06. 

July 13-17

6.9, 6.11, 7.1 (omit pp 369-370), 7.2


July 20-24

7.3, 7.5, 7.6

No classes on Wednesday, July 22.

July 27-31

8.1, 8.2, Midterm Test

Midterm Test will take place on Wednesday, July 29.

August 03-07

8.3, 8.6, 8.4

No classes on Monday, August 03 (Civic Holiday).

August 10-14

9.1, 9.2 (omit pp 499-502), 9.3, 9.4


August 17-21

9.6, 9.8, 18.1, Review


August 24-28

Final Examination

August 25, 2015.

Note: The course will not cover all the sections of the chapters of the textbook.


Throughout the term, homework problems will be posted to the Announcements Section of the course web page. In order to develop your understanding of the course material you must do all assigned problems. These will not be collected for grading and hence will not directly affect your grade. However, it is extremely important that you complete all assigned exercises. Do not memorize them. Rather, learn how to solve problems like them. Learning Mathematics is not a spectacular sport, it is a participatory event. There is nothing like a bit of computation to strengthen understanding in mathematics! Use the Solutions Manual only after you have tried the problem. Successful students must keep up with homework and seek help for points they do not understand as soon as possible. Do not fall behind! It is recommended to read the relevant sections of the textbook before every class. Next to the classes, working out the answers to the problems is the most important preparation for the tests and exam that will contain for the most part but not exclusively, questions very similar to those from the textbook. It is quite practical to work on your own or together in small groups. Each student should do at least three hours of independent study for every lecture hour. The amount you learn in this course and the grade you receive will be proportional to the amount of time you spend working on problems.
Note: Help is also available in the Math Lab in S525 Ross.

Exams Information:

Lecture/Exam Rules:

All internet connected and other distractive electronic devices (cell phones, pagers, head phones, digital dictionaries, CD players, laptops) need to be turned off during class, i.e. during lectures, tutorials, tests and exams. All such devices as well as all books, papers, knapsacks, and briefcases must be left at the front of a lecture hall during tests and exams. Anyone caught with electronic devices will be charged with Academic Dishonesty (see the next page). The only items you may have at your seat during exams and tests are pens, pencils, eraser, nonprogrammable, nongraphing calculators, student ID and purses. Notes need to be taken manually; use of cellphone cameras in class is not allowed. You are responsible for all material covered in lectures.
Note: Photo identification and signing-in are required at all tests and exams.

Final Grade:

The official grades at York University are LETTER grades. Qualitative descriptions of these grades can be found in the York Calendar at this link. Your test marks will be on a numerical scale because these have to be weighted and averaged in various ways, but THERE IS NO AUTOMATICPREDETERMINED CORRELATION BETWEEN NUMERICAL GRADES AND LETTER GRADES in this course. In particular, the "numerical" grades you receive are NOT to be interpreted as percentages; they are merely 'raw scores'. When the tests are returned in class, the distribution of'raw scores' will be announced along with an approximate conversion to letter grades. The median letter grade for the course will likely fall between C and C+. 
The composition of the final grade is as follows:
Midterm Test (two-hour written test held in lecture periods): 40% of the overall grade (Tentative date for the midterm test: Wednesday, July 29, 2015);
Final Examination 
(three-hour exam scheduled by the Registar's Office): 60% of the overall grade, will take place during the examination period.
 Students have five days from the date of the return of test papers to appeal their marks. Some test papers (randomly selected) are photocopied before they are handed back, so we can detect subsequent alterations.

Make-up Policy:

No permission will be given to a student to write tests in advance of their scheduled dates. No make-up will be done for the tests. A missed test will be counted as zero, except under extreme circumstances in which case the corresponding percentage of the overall grade will be "forwarded" to the final exam. If you miss a test for medical reasons, you must turn in within one week following the test date the Attending Physician's Statement form provided here after getting it filled in by your doctor. No other type of medical note will be accepted. However, missing tests is extremely dangerous and not recommended. As experience has shown, students who miss class tests because of some 'mysterious illnesses' will usually average 30% on the final exam.
Student who misses the final examination will be allowed to write a make-up exam only if both of the following conditions are met:
the student notifies me (raguimov@mathstat.yorku.ca) or the Department of Mathematics and Statistics (Undergraduate Office, N502/503 Ross Building, 416-736-2100, Ext. 55902 or 33969) in advance that the exam will be missed;
 the student submits within one week following the exam date the Attending Physician's Statement form provided here after getting it filled in by his/her doctor.
Students who miss the final examination and do not meet both conditions will receive a grade of F. It is student's responsibility to fill out and submit the Deferred Standing Agreement form.
Note: Do not make vacation/job plan until the final exam date is known: having a plane ticket for Banff or Hawaii on August 20 is NOT a legitimate excuse for absence from a final exam on August 27.

Religious Observance:

Special needs students, please identify yourselves to the course instructor as soon as possible. York University is committed to respecting the religious beliefs and practices of all members of the community and making accommodations for observances of special significance to adherents. If any of the dates specified in the course schedule for in-class tests pose such a conflict, students should contact me (raguimov@mathstat.yorku.ca) within the first week of classes. Please note that if the final exam date poses a conflict, students must complete the Final Exam Accommodation form, which can be obtained from the Registrar’s Office. Please note that York University welcomes feedback on accessibility issues. Your feedback may be provided by sending e-mail to: mathstat@yorku.ca

Academic Honesty:

Students are responsible for familiarizing themselves with University Policies and Regulations. Conduct that violates the ethical or legal standards of the university community or of one’s programme or specialization may result in serious consequences. Refer to the Senate Policy on Academic Honesty. Photo identification and signing-in are required at all tests and exams. Also, some test papers (randomly selected) will be photocopied before they are returned so that subsequent alterations can be detected.


Individual questions can be discussed by e-mail or in person after class, or during office hours. Please send all e-mail notes as plain text within the body of the message. Do not send attachments or HTML-formatted e-mail. No texting, please. Also, if the name of your account is an alias, I will not know who the mail is from unless you sign it; it also risks being accidentally discarded as junk mail. The course web page will be up-dated regularly to include important announcements made in class, such as the material to be covered on the tests. E-mail notes requesting such information contained on the web page will be answered the last.

Links and Other Resources:

1- York Undergraduate Mathematics Program
2- York University Library
Infinity Club
- Online Forum
- Download free trial version of LINDO package
- Download free trial version of LINGO package
7- Local time in Toronto

Questions and comments regarding this Web page please send to raguimov@mathstat.yorku.ca

© 2006-2015, All Rights Reserved, York University & Iouldouz S. Raguimov

Last updated on August 20, 2015