Welcome to the Course Page of

SC/MATH 4170 6.00 A

Operations Research II

2014-2015 Full Year


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

Important Links & Other Resources


This area will contain announcements and solutions to tests. 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.

April 17:

According to your requests, I am making available  solutions  to the selected


problems from Sections 11.9 and 11.10 of the textbook.

April 13:

Please note that the Final Examination for the course will take place on


Sunday, April 19, 2015 from 14:00 to 17:00 in ACE 012. The exam will


cover the entire material covered in class from Sections 11.1, 11.2, 11.3,


11.4, 11.5, 11.6, 11.7, 11.8, 11.9, 11.10, 11.11, 11.12 and 11.13 of the


textbook but with an emphasis on the more recent material covered after


the Midterm Test #2. If less common formulas are needed on the exam,


then those will be provided on the exam paper.


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


question will be indicated. You have to show all your work and should


explain your solutions where appropriate. Unofficial final grades for the


course will hopefully be posted to the course web page at the beginning


of May, 2015.

April 10:

Homework problems for Sections 11.11-11.13 and review problems:


Section 11.11: #1, 2, 4;


Section 11.12: #1, 2;


Section 11.13: #1, 2, 3, 4;


Chapter 11 Review Problems: #3(a), 6, 8, 9, 14, 15, 18, 21 (pages 705-706).

April 03:

As it was announced in the class yesterday, the student course evaluations are


available at the URL link:  http://courseevaluations.yorku.ca  until Friday,


April 17, 2015. It is the time for you to be the judge. Please go on-line and


complete the course evaluation. Thank you!

March 27:

Homework problems for Sections 11.9-11.10:


Section 11.9: #1, 3, 5, 6, 8, 10, 16;


Section 11.10: #1, 2, 5, 8.


Review Problems for Chapter 11: #8, 9, 14, 15, 18,21 (pages 705-706).

March 21:

As it was announced in the class on Thursday, we will write Midterm Test #2


during the lecture period from 19:00 to 20:20 on Thursday, March 26.


On the test you will be responsible for all the material covered in class


from Sections 11.1, 11.2, 11.3, 11.4, 11.5, 11.6, 11.7 and 11.8 of the textbook.


Please note that the classes will end on Friday, April 17. You may click here


to see a new course schedule.

March 16:

Please note that the classes will resume shortly. Click the link below for the details:



March 03:

Last night, in a very large membership meeting, all three units of CUPE 3903 voted


not to send the employer's final offer to a formal ratification vote. Overall, the vote was


70% against sending the offer to ratification vote. The union executive then decided


to call the strike, given that it had a strike mandate in hand from earlier in February.


That means that, as of Tuesday March 03 at 12:01am, CUPE 3903 is on strike.


The University’s Senate Executive decided on February 27, 2015 that all academic


activities, including classes are cancelled until the strike is settled. Unfortunately,


at this point there is no way to know how long the strike might last. In the meantime,


even though it is difficult, I strongly recommend that you take this opportunity to review


course material covered up to now, catch up with readings if you've fallen behind, and


even read ahead, and continue to work on your assignments. Although due dates are


frozen, course work will be due when we return to classes. Keeping up with work will


make the transition back to classes a little less shocking.


Please distribute this message to your class mates. Thank you.

February 28:

Please note that the Department of Mathematics and Statistics will be hosting


Science@50 anniversary event on Tuesday afternoon. The event will include


several short and informal lectures about cool Math in Simpsons cartoon,


followed by pizza and a career panel (a discussion with our alumni about


possible career paths in Math). The students who are interested to attend


this event should register at this website:




You can also find all information about the event at this website.

February 14:

As it was announced in the class on Thursday, we will write the


Medterm Test during the lecture period from 19:00 to 20:20 on


Thursday, March 05. On the test you will be responsible for all


the material covered in class from Sections 11.1, 11.2, 11.3, 11.4,


11.5, 11.6, 11.7 and 11.8 of the textbook.



February 08:

Homework problems for Sections 11.7-11.8:


Section 11.7: #2, 3, 4, 5;


Section 11.8: #2, 3, 4, 5, 6, 8.

January 31:

Homework problems for Sections 11.5-11.6:


Section 11.5: #1, 2, 4;


Section 11.6: #1, 3, 4, 6, 7, 8.

January 23:

Homework problems for Sections 11.3-11.4:


Section 11.3: #4, 6, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23;


Section 11.4: #1, 2, 3, 4, 6, 7, 13, 14, 16, 17;


Review Problems for Chapter 11: #2 (page 705).

January 12:

Homework problems for Sections 11.1-11.2:


Section 11.1: #1, 8, 9;


Section 11.2: #1, 2, 3, 5, 9, 10, 12.

January 04:

Please click  here  to see your December Exam grade.



December 09:

According to your requests, I am making available  solutions  to the selected


problems from Sections 13.1, 13.2, 13.4 and 13.5 of the textbook.

December 05:

As it was promised in the class yesterday, I am making available Lecture


Notes on Simulation.

December 04:

As it was announced earlier, we will write the Midterm Examination on


Wednesday, December 10 from 19:00 to 22:00 in TEL 1005. The exam


will be comprehensive, covering all the material covered in class according


to the Course Schedule but with some emphasis on the more resent material


that we have covered after the midterm test. Please note that there will be


several questions on the exam, some with few parts. Marks for each question


will be indicated. You have to show all your work to justify your answers


and should explain your solutions where appropriate.

November 30:

As it was asked in the class on Thursday, I am making available solutions


to the Homework Problems on Co-operative n-Person Games.

November 28:

Homework problems for Sections 13.4-13.5:


Section 13.4: #1, 2, 3, 9, 12, 14;


Section 13.5: #1, 2, 3, 4, 9.


Also, solve problem #2 and problem #5 from page 799 of the textbook.

November 19:

As it was promised in the last class, I am making available Additional


Notes on Decision Making.

November 15:

Extra homework problem for Section 14.7:


Prove that the vector of Shapley values vector is always an imputation.


Homework problems for Sections 13.1-13.2:


Section 13.1: #1, 4;


Section 13.2: #1, 2, 3, 7, 10, 13, 16.

November 05:

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


the class tomorrow.


I am making available  Notes  by Prof. Neal Madras on “Definitions for


Co-operative n-Person Game Theory”. Also, an interesting article about


game theory: The Arithmetics of Mutual Help by M.A. Nowak, R.M. May


and K. Sigmund (Scientific American, June 1995). (Not required for exam.)

October 21:

According to your requests, I am making available  solutions  to the 


selected problems from Sections 14.2 and 14.3 of the textbook.

October 18:

As it was announced in the class on Thursday, we will write Midterm Test #1


during the lecture period from 19:00 to 20:20 on Thursday, October 23.


On the test you will be responsible for all the material covered in class


from Sections 14.1, 14.2, 14.3 and 14.4 of the textbook.


Homework problems for Sections 14.5-14.7:


Section 14.7: #1, 2, 3, 7, 8, 10, 11;


Please note that in the answer for Problem #11 the vector of Shapley values


(in thousands of dollars) is: (305/3, 335/3, 266/3).


Review Problems for Chapter 14: #6.

October 14:

Please note that you may download free trial versions of LINDO and


LINGO packages at the following URLs, respectively:






Please be sure to download and use the Classic LINDO package.

October 06:

Please check the web site below for funding and awards:




Also, I am making available the URL for the newly created Facebook page of


the Department of Mathematics and Statistics:



October 04:

Homework problems for Section 14.3-14.4:


Section 14.3: #1(a), (b), (c), (d); 2; 3; 4; 5(a),(b);


Section 14.4: #1, 2, 3, 5.

September 23:

Please make note that we are going to have an information session on York's


Actuarial Science program on Thursday, September 25 at 4:00 PM in CLH D.


Also, right after the session (at 5:00 PM) a group of upper administration from


Sun Life Financial will speak about the profession, the hiring process in general


as well as about the set of skills Sun Life is seeking when recruiting. The students


will have a rare opportunity to address their questions directly to professional


actuaries that are involved in hiring. Beverages and snacks are to be provided.

September 21:

Homework problems for Section 14.2:


# 1, 3, 4, 5, 6; 9, 10, 11.

September 14:

Homework problems for Section 14.1:


# 1, 2, 3, 4.


As it was asked in the class, please also show that a Two-Person Zero-Sum


Game, with a payoff matrix of 3x3 cannot have exactly five saddle points.

September 11:


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

Course Information


Term and Session:

Fall/Winter 2014-2015

MATH 4170





Thursday 19:00-21:50, BC 215



Dr. Iulduz Raguimov



S512 Ross



416-736-2100, Ext. 66092 or 416-736-5250


Office Hours:

MF 12:00-13:00 and T 16:00-17:00, also by appointment





Two midterm tests:

20% each


Two end-of-term exams:

30% each

Course Description: This course is integrated with the graduate courses GS/MATH 6900 3.00 F “Operations Research I” and GS/MATH 6901 3.00 W “Operations Research II”. The course deals with deterministic and probabilistic models based on optimization. The following topics will be discussed: 1) game theory (how to find the best strategies in a confrontation between two or more players with differing interests); 2) decision theory (how to act to minimize loss, subject to available data); 3) simulation (how to sample from a probability distribution and how to conduct statistical experiments by computer); 4) queueing theory (how to assess what may happen in a system where the customers arrive randomly, wait in line, and then get served); 5) nonlinear programming (how to optimize a nonlinear objective function subject to equality or inequality constraints). Time permitting, some additional topics (such as inventory theory or reliability) will also be discussed. 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 aid:

Student Solutions Manual for Operations Research, Fourth Edition


by Wayne L. Winston


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

Suggested supplementary

(a) Nonlinear Programming: Theory and Algorithms, Third Edition

reading for graduate students:

by Mokhtar S. Bazarra, Hanif D. Sherali, and C.M. Shetty


John Wiley & Sons, 2006;


(b) Probabilistic Models in Operations Research, by S. Guiasu


Nova Science Publishers, Inc., New York, 2009;


(c) Operations Research. An Introduction, Ninth Edition


by Hamdy A. Taha


Pearson Education, Inc., New Jersey, 2011;


(d) Introduction to Operations Research, by F.S. Hillier and G.J. Lieberman




(e) Non-antagonistic Games, by Yu. B. Germeier


D. Reidel Publishing Company, Boston, 1986.



Course Prerequisites:

Multivariable Calculus (SC/MATH 2010 3.00 or SC/MATH 2015 3.00


or MATH 2310 3.00), Elementary Probability (SC/MATH 2030 3.00) and


Operations Research I (SC/MATH 3170 6.00).

Course Credit

AS/MATH 4570 6.00.


Important Dates:

September 08: Classes commence.

September 22: The last date to enrol in the course without my permission.

October 14: The last date to enrol in the course with my permission.


October 29-November 02: Co-Curricular Days (no classes).

December 05: Fall classes end.


January 05: Classes resume.


February 06: The last date to drop the course without receiving a final grade.

February 14-20: Reading Week (no classes).


April 17: Classes end.

April 19-May 03: Examination period.

Course Schedule

(tentative, subject to any changes announced in class)




Sept 11

Introduction, 14.1

Classes begin on Thursday, September 11.

Sept 18


September 22 is the last day to enrol without my permission.

Sept 25



Oct 02



Oct 09



Oct 16


Oct 23

Test #1, 14.7

Test #1 will take place on Thursday, October 23. 

Oct 30

No classes

Co-Curricular Days.

Nov 06


Nov 13



Nov 20



Nov 27



Dec 04

21.4, 21.5, Review

Fall classes end on Friday, December 05.

Dec 05

Fall classes end

Last day to submit fall term work.

Dec 09-22

Fall Term Exam


Jan 08

11.1, 11.2

Winter classes begin on Monday, January 05.

Jan 15

11.2, 11.3


Jan 22

11.4, 11.5


Jan 29



Feb 05



Feb 12



Feb 19

No classes

Reading Week.

Feb 26


March 05, …



March 26

Test #2, 11.10

Test #2 will take place on Thursday, March 26. 

April 02

11.10, 11.11


April 09

11.12, 11.13


April 16

20.1, 20.2, 20.3, Review


April 17

Classes end

Apr 19-28

Winter Term Exam


Note: The course will not cover all the sections of each chapter from the Fourth Edition of the textbook.


Throughout the term, practice exercises will be posted to the Announcements section. 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.

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, calculator, 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 AUTOMATIC PREDETERMINED 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:
Two Tests (80 minutes each written tests held in lecture periods): 20% each (Tentative dates for the tests: Thursday, October 23, 2014 and Thursday, March 26, 2015);
Final Examination Fall Term
(three-hour exam scheduled by the Registar's Office): 30% of the overall grade, will take place in December 2014, during the examination period;
Final Examination Winter Term
(three-hour exam scheduled by the Registar's Office): 30% of the overall grade, will take place in April 2015, 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:
   1) 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;
   2) 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 December 18 is NOT a legitimate excuse for absence from a final exam on December 22.

Religious Observance and Accessibility:

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 weeks 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.

Important Links and Other Resources

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

© 2004-2015, All Rights Reserved, York University & Iouldouz S. Raguimov

Last updated April 17, 2015