CO 250 Assignment 4 – Fall 2013

Due: Friday October 11 by 10:00 am

Solutions should be deposited in the appropriate drop box slots (see table below, according to sectionand last name), outside MC 4066.

Section 1 (W.Cook, MWF 10:30–11:20am): Box 17 - Slot 6 (A-J)Box 17 - Slot 7 (K-S)Box 17 - Slot 8 (T-Z)

Section 2 (J.Cheriyan, TuTh 10:00–11:20am): Box 17 - Slot 9 (A-J)Box 17 - Slot 10 (K-S)Box 17 - Slot 11 (T-Z)

Assignment policy: While it is acceptable for students to discuss the course material and the assign-ments, you are expected to write solutions to assignments on your own. For example, copying or para-phrasing a solution from some fellow student or old solutions from previous offerings of related coursesqualifies as cheating and we will instruct the TA’s to actively look for suspicious similarities and evidenceof academic offenses when grading.All students found to be cheating will be given a mark of zero on the assignment (wherethe mark of zero will not be ignored as the lowest assignment). In addition, all academicoffenses are reported to the Associate Dean for Undergraduate Studies and are recordedin the student’s file (this may lead to further, more severe consequences). Furthermore,cheating students will receive a 5% penalty on their final mark; i.e., a mark of 74% wouldbe reduced to 69%.If you have any complaints about the marking of assignments, then you should first check your solutionsagainst the posted solutions. After that, if you see any marking error, then you should return yourassignment paper to the instructor of your section within one week and with written notes on all themarking errors; please write the notes on a new sheet and attach it to your assignment paper.

IMPORTANT: You MUST attach THE COVER SHEET that is available at the end of theassignment, or else you may make a copy of it on your own, and attach your copy. It isimperative that you indicate your full name and student ID (as we have many students withthe same last name).Explanations are required for all solutions; marks could be deducted for correct answers that have noexplanations.

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Problem 1: Simplex Algorithm (20 marks: (a), (b) 10 each)

Solve the described LP models using the dictionary form of the simplex algorithm. Show each dictionary.If the LP model has an optimal solution, then display an optimal solution and also a vector y that givesa certificate of optimality. If the LP model is unbounded, then display a feasible solution and a directionvector d that gives a certificate of unboundedness.

It is fine to check your arithmetic using Robert Vanderbei’s Simplex Pivot Tool

http://www.princeton.edu/~rvdb/JAVA7/pivot/simple.html

Be sure to toggle the menu (at the top right) to “Fraction” rather than “Decimal.”

(a)

max 3x1 + 2x2 + 4x3

subject to

x1 + x2 + 2x3 ≤ 4

2x1 + 3x3 ≤ 5

2x1 + x2 + 3x3 ≤ 7

x1 ≥ 0, x2 ≥ 0, x3 ≥ 0

(b)

max x1 + 3x2 − x3

subject to

2x1 + 2x2 − x3 ≤ 10

3x1 − 2x2 + x3 ≤ 10

x1 − 3x2 + x3 ≤ 10

x1 ≥ 0, x2 ≥ 0, x3 ≥ 0

Problem 2: Objective Equation in a Dictionary (10 marks)

Prove the following statement or give an example to show that the statement is false. For an LP problem(with a maximization objective), a feasible dictionary whose last row reads

z = z∗ + c̄1x1 + c̄2x2 + · · · + c̄nxn

describes an optimal solution if and only if c̄j ≤ 0 for all j. Here z∗ is a constant, giving the objectivevalue of the feasible LP solution described by the dictionary.

Problem 3: The Optimal Value (15 marks)

Let A be a matrix such that AT = −A and let x∗ be an optimal solution to the LP problem

minimize wTx subject to Ax ≥ −w, x ≥ 0.

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What is the value of wTx∗? Why?

Problem 4: Missing Data (15 marks: (a), (b), (c) 5 each)

The following is a dictionary for a maximization LP model in standard equality form.

x3 = 4 + x1 − µx2

x4 = 1 − αx1 + 4x2

x5 = β − γx1 − 3x2

z = 10 + δx1 − 2x2

The entries α, β, δ, γ, and µ are data. For each of the part below, find values for these data entries thatmake the statement true.

(a) The current solution is feasible but not optimal.

(b) The current solution is optimal.

(c) The LP model has unbounded objective value.

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Assignment 4 – COVER SHEET

Surname:

First Name:

Signature:

Id.#:

Section#:

“I have read the course policy on collaborations and academic offences,and have explicitly acknowledged all collaborations for this assignment.”

Signature:

Problem Value Marks Awarded

1 20

2 10

3 15

4 15

Total 60

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