PRELIM 4
Closed book exam. Justify all work. 1.
ORIE 3310/5310
April 28, 2009
a. (25 points) You work as an OR analyst for a manufacturing corporation which has production sites i = 1, . . . , m and distribution (sales) centers j = 1, . . . , n . The (int

PRELIM 1
ORIE 3310
July 16, 2012
Closed book exam. Justify all work.
1. Consider the following LP, to which the decomposition procedure is applied using two subproblems
(SUB1, SUB2). The rst two constraints are the linking constraints; the next two constr

PRELIM 1
ORIE 3310/5310
February 19, 2009
Closed book exam. Justify all work. 1. Consider again the the following problem from the homework assignment. The standard decomposition procedure is applied using two subproblems (SUB1, SUB2): ORIGINAL PROBLEM ma

ORIE 3310/5310 Practice Prelim 2 (Spr2009) Solutions
Spring 2010
1. (a) See class notes on Bellman's algorithm (for finding a shortest (1, n)-path in a directed acyclic graph). (b) Now suppose we wish to find a shortest (1, n)-path which contains a specif

ORIE 3310/5310
Optimization II
Spring 2010
Homework # 3
Solutions posted Thursday 3/11; homework quiz in recitation Friday 3/12 Wednesday 3/17. BHM refers to the text "Applied Mathematical Programming" by Bradley, Hax, and Magnanti (available on the cours

Spring 2013
Optimization II (ORIE 3310/5310/5311)
Homework 2 Solutions
Due at noon on Friday, February 8.
1. (25 points) An undirected bipartite graph G = (N, E ) is a graph with the property that we
can divide the node set N into two disjoint subsets N1

ORIE 3310/5310: Prelim 4 Solutions and grading scheme (problems 2, 3) 2(a) Give the cut developed by Gomory for the standard form integer programming problem with A, b, c integer-valued: maxcfw_cs | Ax = b, x 0, x integral Solution. Consider an optimal ba

HW 3 Solutions
Spring 2012
1. (a) The size of the basis is 5 + 5 1 = 9. Since only ve of the basic variables can take value
1, then the remaining four must be zero. So, each basic solution has four degenerate
variables. More generally, for an n n assignme

Spring 2013
Optimization II (ORIE 3310/5310/5311)
Homework 4: Prelim Review
Due at 3pm on Thursday, February 21.
1. (10 points, easy) Consider an assignment problem with 5 workers and 5 tasks with the
cost table given below. As usual, the table entry in r

ORIE 3310/5310
Optimization II
Spring 2014
Homework # 3
1. Given below is the utility data for a 5 5 assignment model (max total utility).
4
2
4
4
3
8
2
2
3
6
6
3
4
3
2
3
1
2
2
4
3
0
1
1
2
(a) What is the size of a basis for this problem? How many degener

Spring 2013
Optimization II (ORIE 3310/5310/5311)
Recitation 1 Solution
In this recitation, we will review how to download AMPL and to use AMPL to solve simple
linear and integer programs.
You can work in pairs or individually. Answer the questions in the

Spring 2013
Optimization II (ORIE 3310/5310/5311)
Homework 4: Prelim Review
Due at 3pm on Thursday, February 21.
1. (10 points, easy) Consider an assignment problem with 5 workers and 5 tasks with the
cost table given below. As usual, the table entry in r

PRELIM 2
Closed book exam. Justify all work. 1.
ORIE 3310/5310
March 12, 2009
a. (10) Explain Bellman's basic procedure for determining a shortest path in a directed acyclic graph; i.e., give (and explain) the optimal value function and recursive equation

PRELIM 3
Closed book exam. Justify all work.
ORIE 3310/5310
April 14, 2009
1. A supply-demand network is a digraph G = (V, E) , for which V = S D I , with the disjoint subsets S, D, I denoting, respectively, supply, demand, and intermediate nodes. For eac

PRELIM 1
ORIE 3310/5310/5311
February 27, 2014
Closed book exam. Justify all work.
1. For the following problem from Homework 1, the standard decomposition procedure is applied using
two subproblems (SUB1, SUB2):
ORIGINAL PROBLEM
max 6x1
s.t. x1
2x1
x1
+

PRELIM 2
ORIE 3310/5310
March 20, 2014
Closed book exam. Justify all work.
1. For the knapsack model (with positive integers B and pj , cj j )
Max p1 x1 + + pn xn
s.t. c1 x1 + + cn xn B
0 xj integer
we studied the recursion:
j = 1, . . . , n
vk (y) = max

Spring 2013
Optimization II (ORIE 3310/5310/5311)
Homework 4 Solution
Due at 3pm on Thursday, February 21.
1. (10 points, easy) Consider an assignment problem with 5 workers and 5 tasks with the
cost table given below. As usual, the table entry in row i a

ORIE 3310/5310
Optimization II
Spring 2014
Homework # 2
BHM refers to the text Applied Mathematical Programming by Bradley, Hax, and Magnanti (available
on the course Blackboard site).
Solutions posted Monday, March 17.
1. Problem #6, BHM Chapter 11.
2. S

Spring 2013
Optimization II (ORIE 3310/5310/5311)
Homework 2
Due at noon on Friday, February 8.
1. (25 points) An undirected bipartite graph G = (N, E ) is a graph with the property that we
can divide the node set N into two disjoint subsets N1 and N2 , s

Spring 2013
Optimization II (ORIE 3310/5310/5311)
Homework 5 Solutions
1. (25 points) In Lectures 9 and 10, we used dynamic programming to solve for the shortest
s t path in a layered graph G = (N, E ) (where N contains the origin s and the destination
t)

Outline ORIE 3310/5310/5311 Spring 2014
I.
II.
III.
IV.
V.
LP review: lectures 1/23 - 1/30
references: lecture notes on LP and historical notes on LP
Large-scale programming: lectures 2/4 2/20
references: large-scale programming notes; BHM text chapter 12

This problem is a case of the knapsack problem where we are trying to minimize the expected ordering costs for when a
part is needed and is not available in the sparepart kit.
Stages: Here, the stages correspond to the six spare parts, plus the terminal

Spring 2013
Optimization II (ORIE 3310/5310/5311)
Recitation 1
In this recitation, we will review how to download AMPL and to use AMPL to solve simple
linear and integer programs.
You can work in pairs or individually. Answer the questions in the blank sp

Spring 2013
Optimization II (ORIE 3310/5310/5311)
Homework 6 Solution
Due at noon on Friday, March 8.
1. (30 points) A government space agency is conducting research project on an engineering
problem that must be solved before people can safely y to Mars.

Spring 2013
Optimization II (ORIE 3310/5310)
Homework 8
Due at noon on Friday, April 19.
1. (20 points; From Hillier and Lieberman) Consider the integer program:
max
s.t.
5x1 + x2
x1 + 2x2 4,
x1 x2 1,
4x1 + x2 11,
x1 , x2 0, integers.
(a) Solve this probl