Problem 1. (Dynamic Programming) The residents of the underground city of Zion defend themselves through a combination of kung fu, heavy artillery, and ecient algorithms. Recently they have become interested in automated methods that can help fend o attac

Homework 9 Solution April 14, 2009 Exercise 3, page 352 Suppose you have 7 full water bottles, 7 half-full, and 7 empty. You would like to divide the 21 bottles among three individuals so that each receives 7 bottles with the same quantity of water. Expre

Lecture 18 Dynamic Programming Problems
April 13, 2009
Dynamic Programming
The material is in Chapter 10
1
Dynamic Programming
Decompose a problem into several stages; Each stage comprise a much simpler problem, sometime called a subproblem; Require a r

Lecture 17 Solving Integer Programming Problems
April 2, 2009
Solving ILPs
ILPs can be used to formulate a lot of practical problems, but they are in general very hard to solve (NP-hard). There are three classes of algorithms for ILP. Exact algorithms. tr

Lecture 16 Integer and Mixed Integer Problems
March 30, 2009
Application: Set Covering
To promote on-campus safety, the U of A Security Department is in the process of installing emergency telephones at selected locations. The department wants to install

Lecture 15 Max-Flow - Min-Cut Maximal Flow Algorithm
March 18, 2009
Maximum Flow Problem: Example
Given a network G = (N, A) maximize
cfw_j :(s,j )A
xsj xij
cfw_j :(i,j )A cfw_j :(j,i)A
subject to
xji = 0
for all i = s
0 xij cij
for all (i, j ) A.
NOTE:

Lecture 14 Shortest Path Problems Dijkstras Algorithm, Max-Flow Problem
March 16, 2009
Shortest-route problem
Given a connected directed graph G(N, A) with a length (cost) ca associated with each a A, given a source node and a destination node, the shorte

Lecture 13 Network Models I
March 11, 2009
Graphs
A graph is a set of objects called nodes or vertices connected by links called edges. Usually we use G(N, E ) to denote a graph, where N is the set of all nodes is and E is the set of all edges. In a graph

Lecture 12 Transportation Algorithm
March 9, 2009
Last Lecture
The transportation model The transportation problem The LP formulation Variants of the transportation model The assignment model The transhipment model General minimum cost ow problems
2
The

Lecture 11 Transportation Model and Its Variants
March 4, 2009
Denition of the Transportation Model
Given m sources and n destinations, the supply at source i is ai and the demand at destination j is bj . The cost of shipping one unit of goods from source

Lecture 9: Duality and Post-Optimal Analysis
February 19, 2009
Lecture 9
Topic Outline
Dual of an LP Problem Primal-Dual Relationship Economic Interpretation of Duality Post-Optimal Analysis
Chapters 4.1, 4.2, 4.3, and 4.5.
GE330
1
Lecture 9
Today: LP Du

Lecture 8
Algebraic Sensitivity Analysis
February 17, 2009
Algebraical Sensitivity Analysis
TOYCO Model: TOYCO assembles three types of toystrains, trucks and carsusing three operations. The daily limits on the available times for the three operations are

Lecture 6
LP Sensitivity Analysis
February 12, 2009
Sensitivity Analysis
Study the impact of the changes in input data to the optimal value (solution). In general, there are three dierent types of changes in input data. Changes in the coecients in the ob

Lecture 6 Special Cases in LP
February 9, 2009
Degeneracy
A tie for the minimum ratio test can happen. Then in the next iteration, at lease one basic variable will be zero. The solution is called a degenerate solution. Degeneracy reveals that there is at

Lecture 5 Simplex Method II
February 4, 2009
Moving From One BFS to Another
Two adjacent vertices correspond to two BFSs whose basis have m 1 common basic variables (only one dierent basic variable). In Example 1: A and B are adjacent vertices, the basis

Lecture 4 Simplex Algorithm Path
February 2, 2009
Angelia Nedi c
Lecture 4
Path of the Simplex Method
Simplex method does not enumerate all the basic solutions It moves from a basic feasible to another (better) basic solution iteratively until it reaches

Lecture 3 Simplex Method I
January 27, 2009
Solving Linear Programs
The graphical method is only applicable for simple problems (e.g. problems with two variables). However, it provides some very important observations.
The feasible region has nite many v

Lecture 1 Introduction to Operations Research Methods for Prot and Value Engineering
January 16, 2009
Lecture 1
Example: Diet Problem
Design a one-week diet for a person getting all his/her food from a grocery store. The person must meet certain nutritio

Quiz Sample Solution
Due: February 24, 2009 Problem 1 [20 points] Top Brass Trophy Company makes large championship trophies for youth athletic leagues. They are planning production for fall sports:football and soccer. Each football trophy has a wood base

Quiz Sample
Due: February 24, 2009 Problem 1 [20 points] Top Brass Trophy Company makes large championship trophies for youth athletic leagues. They are planning production for fall sports:football and soccer. Each football trophy has a wood base, engrave

Homework 12 Solutions May 1, 2009 Exercise 1, page 669 Determine the stationary (extreme) points for the following functions: (a) f (x) = x3 + x. (b) f (x) = x4 + x2 . (c) f (x) = 4x4 x2 + 5. Solution: We nd the extreme (stationary) points as the solution

Homework 11 Solutions April 28, 2009 Exercise 5, page 555 The time between arrivals at the game room in the student union is exponential with mean of 10 minutes. (a) What is the arrival rate per hour? (b) What is the probability that no students will arri

Homework 11 Due April 28, 2009 Exercise 5, page 555 The time between arrivals at the game room in the student union is exponential with mean of 10 minutes. (a) What is the arrival rate per hour? (b) What is the probability that no students will arrive at

Homework 10 Solution Due April 21, 2009 Exercise 2, page 376, parts (a) and (d) Develop Branch and Bound (B&B) tree for each of the following problems. For convenience, use x1 as the rst branching variable at the starting node. (a) Maximize z = 3x1 + 2x2

Homework 10 Due April 21, 2009 Exercise 2, page 376, parts (a) and (d) Develop Branch and Bound (B&B) tree for each of the following problems. For convenience, use x1 as the rst branching variable at the starting node. (a) Maximize z = 3x1 + 2x2 , subject

Homework 9 Due April 14, 2009 Exercise 3, page 352 Suppose you have 7 full water bottles, 7 half-full, and 7 empty. You would like to divide the 21 bottles among three individuals so that each receives 7 bottles with the same quantity of water. Expressed

Homework 8 March 31, 2009 Solution to Exercise 2(a), page 241. The problem is the same as the Cable connection for Midwest TV Company with an additional link cfw_5, 6 of length 2. We can apply the minimum spanning tree algorithm starting with any node. It