Solutions to Exam 2, STOR 415, Spring 2014
Multiple Choice Questions (5 points each).
1. D.
2. A
3. D
4. C
5. A
6. B
Free Response Questions.
7. (LP2) is feasible; a feasible solution is (0, 8, 2, 10)
8. (a) The dual:
min 3y1 + 5y2
y1 0
y2 0
y3 0
y1 + y2

STOR 415, Spring 2015
Solutions to Homework Assignment No. 4
1. (a)Choose x3 as the entering variable, and x1 as the leaving variable. Pivot on the
entry a33 .
(b)Unbounded.
(c)Stop with the current BFS being an optimal solution. (In fact, it is the uniqu

STOR 415, Spring 2014
Solutons to Homework Assignment No. 4
1. (a)Choose x3 as the entering variable, and x1 as the leaving variable. Pivot on the
entry a33 .
(b)Unbounded.
(c)Stop with the current BFS being an optimal solution. (In fact, it is the unique

Name:
STOR 415, Spring 2013
Final Exam
The exam starts at 4:00pm and ends at 7:00pm. It is closed book/notes. You can use calculators,
but not laptops. Write all answers in the blue book.
Multiple Choice Questions (make one choice for each question; 3.5 p

Handout Part I
STOR 415: Deterministic Models in Operations
Research
Shu Lu
Spring 2015
Contents
1 Introduction
1.1 Terminology in optimization
1.2 Linear programming . . . .
1.3 Solving an LP graphically .
1.4 Types of LP problems . . .
problems
. . . .

STOR 415, Spring 2015
Homework Assignment No. 4
1. Suppose that, after applying the simplex method to solve a maximization LP, you
obtain the tableau below, where a13 , a14 , a23 , a24 , a33 , a34 , b1 , b2 , b3 , c3 , c4 , d are real
numbers.
z x1
1 0
0

STOR 415, Spring 2014
Homework Assignment No. 3
1. Determine if each of the following linear programs is in standard form
or canonical form. For the linear programs that are in canonical form,
specify the isolated variables. For the linear programs that a

STOR 415, Spring 2014
Solutions to Homework Assignment No. 2
[
]
16 17
1. AB =
25 26
2. (a) The equality xT y = y x holds for any two n-dimensional column
vectors x and y.
Yes, because
x1
y1
n
x2
y2
[
]
[
]
xi yi = y1 , y2 , , yn . = y T x.
xT y

STOR 415, Practice Questions for Exam 2
Multiple Choice Questions.
1. Which of the following statements is true?
(a) If the primal LP is feasible, then the dual LP must be feasible.
(b) If the primal LP is infeasible, then the dual LP must be infeasible.

STOR 415, Spring 2014
Solutions to HW 7
1.
(a) Optimal sol: x = (0, 25, 25, 0, 0), optimal value: 300.
(b) In the new simplex tableau the reduced cost of x1 is 3 , and all other entries remain the
same. If 3 then this tableau continues to show an optimal

Name:
STOR 415, Spring 2012
Final Exam
The exam starts at 12:00pm and ends at 3:00pm. It contains 5 pages, with 10 multiple choice
questions and 4 free response questions. Each multiple choice question has only one correct answer.
The exam is closed book/

STOR 415, Spring 2013
Homework Assignment No. 3
1. Determine if each of the following LPs is in standard form or canonical form. For LPs in canonical form, specify the isolated variables.
For LPs not in standard form, convert them into standard form, and

STOR 415, Spring 2013
Homework Assignment No. 4
1. Suppose that, after applying the simplex method to solve a maximization LP, you
obtain the tableau below, where a13 , a14 , a23 , a24 , a33 , a34 , b1 , b2 , b3 , c3 , c4 , d are real
numbers.
z x1
10
00

STOR 415, Spring 2013
Homework Assignment No. 2
10
456
1. Let A =
, B = 0 1.
789
22
[
]
(a) Compute the product AB using the denition of matrix multiplication.
(b) Partition A into 2 blocks by a vertical line between its 2nd and 3rd
column, and partition

STOR 415, Spring 2013
Homework Assignment No. 1
1. Furnco manufactures desks and chairs. Each desk uses 4 units of wood, and each
chair uses 3. A desk contributes $40 to prot, and a chair contributes $25. Marketing restrictions require that the number of

STOR 415, Spring 2014
Solutions to Homework Assignment No. 3
1. (a)
min
s.t
z = 3x1 + x2
x1
x1 + x2
2x1 x2
x1 , x2
3,
4,
= 3,
0
Not in standard form or canonical form. Convert it into standard
form to obtain the following LP:
min
s.t
z = 3x1 + x2
x1 s1

* Cuppa Coffee
*Robust Joe must consist of 60\%-75\% Sumatran beans and at least 10\% Columbian beans.
*Light Joe: must consist of 50\%-60\% Brazilian beans and no more than 20\% Sumatran beans.
set I /Columbian, Brazilian, Sumatran/,
J /Robust, Light/;

* CSL work scheduling (lp)
set
t months /1*5/;
* scalars and parameters
scalars iniWorkers "skilled technicians at the beginning of month 1" /50/,
workHours "Number of hours a skilled technician can work per month" /160/,
trainHours "Number of hours to

* Data Corporal problem (modeled as a MCNFP)
* With the restrition that flow on each arc is bounded by 200
set N/B,R,C,A,L/;
scalar uplimit /200/;
parameter s(N) /B 400, R 300, C 0, A -300, L -400/;
* node B:boston R:raleigh C: chicargo A:austin L: los an

* cake (lp)
set
i types of cakes /cc, bf/,
t months /1*3/;
scalar
upbdd upper bdd for number of cakes produced each month /65/;
parameters
h(i) holding cost in dollars
/cc 0.5
bf 0.4/;
table d(i,t) demand of each type in each month
1 2 3
cc 40 30 20
bf

* A transportation problem. A company supplies goods to three customers
* from three warehouses.
set
i warehouses /w1, w2, w3, dummy/
j custumers /c1*c3/;
* Because total demand exceeds total supply we add a dummy supply point with
* supply equal to total

STOR 415 Handout
Basic feasible solutions and extreme points
Consider a polyhedron in standard form
P = cfw_x Rn | Ax = b, x 0.
(1)
Assume that A is a mn matrix whose rows are linearly independent. Recall
that a vector x Rn is a basic solution for P if an

STOR 415, Spring 2015
Homework Assignment No. 1
Regarding homework grading:
There are 3 questions in this homework assignment. The grader will pick 2 questions to grade. The grades of your homework will be based on grades you receive
on the two graded qu

STOR 415, Spring 2015
Solutions to Homework Assignment No. 2
1. (a) Graphically show the feasible set. Label the vertices (i.e., corners)
of the feasible set with their coordinates.
The vertices are (4, 0) and (3, 1).
(b) Find the optimal solution(s) and

STOR 415, Spring 2015
Solutions to Homework Assignment No. 1
1. (a) Formulate an LP to maximize Furncos prot.
maximize
subject to
z = 40x1 + 25x2
4x1 + 3x2
2x1 x2
x1
x2
(b)
(c)
(d)
(e)
20,
0,
0,
0.
Is (0, 0) a feasible solution? Yes
Is (3, 2) a feasib

STOR 415, Spring 2015
Homework Assignment No. 6
1. Determine the feasibility of each LP below by solving the Phase-1 LP.
(If an LP is not in standard form, rst convert it into standard form.)
For each feasible LP determined as feasible, write down its fea

STOR 415, Spring 2015
Solutions to Homework Assignment No. 3
1. (a)
min
s.t
z = 3x1 + x2
x1
x1 + x2
2x1 x2
x1 , x2
3,
4,
= 3,
0
Not in standard form or canonical form. Convert it into standard
form to obtain the following LP:
min
s.t
z = 3x1 + x2
x1 s1

STOR 415, Spring 2015
Homework Assignment No. 5
1. Suppose the basic feasible solution shown in a simplex tableau is nondegenerate.
Which of the following statements is always true?
(a) The reduced costs of nonbasic variables in this tableau are all stric