Math 442/542 Theory of linear and nonlinear programming Winter 2011 Homework 1
(problems 4 and 5 are due Thursday, January 20)
1. (adapted from Bradley, Hax, Magnanti: Applied Math. Programming, p37) A strategic planner for an airline that flies to four d
Math 442/542 Applications of Linear Programming
1. Fair representation in electoral systems. Suppose that a state sends R persons to the U.S. House of Representatives. There are D counties in the state (D>R), and the state legislature wants to group these
Consultant hiring problem: Modeling on AMPL
1
Outline of modeling process
1. 2. Carefully read the problem to get a general idea Analyze each paragraph to extract all relevant data and information: a) Identify given data and define corresponding sets, par
Math 442/542
Winter 2011
Homework 1 Solution
1. There are many dierent ways of doing these formulations, and here we present just one of them.
Decision Variables:
Let xci , yci and zci represent the number of ights per day made to city c by the ith plane
Math 442/542
Winter 2011
Homework 2 Solution
1. The standard form of an LP looks like
max cT x
subject to Ax b
x0
(a) FALSE. An obvious counterexample is given in problem 2 where any solution on the line segment
connecting (5, 1)T and (0, 6)T is optimal,
Math 442/542
Winter 2011
Homework 3 Solution
1. For the Revised Simplex we rst need to x some representation of the problem (with equality constraints!) which we will use to get the data (A, b and c) in each step.
max 2x1
s.t. 5x1
3x1
x2
+2x2
+2x2
3x3
+7x
Math 442/542
Homework 4 Solution
1. Suppose the primal has a degenerate optimal basic solution. Then, an optimal tableau contains a row
whose r.h.s. column entry is 0. We can proceed with the dual simplex method by choosing the basic
variable in this row
Math 442/542
Homework 5 Solution
1. (H & L 9.6-2)
(a) The min-cost ow problem is represented by the following network.
[80]
[-60]
cF W = 7, uF W =
F
FACTORY 1
W
WAREHOUSE 1
cF D = 3, uF D = 50
[0]
cDW = 2, uDW = 50
D
Dist. Center
cGD = 4, uGD = 50
cDV =
Math 442/542
Handout 1 Solutions
Problem 1. Portfolio selection.
National Insurance Associates carries an investment portfolio of a variety of stocks,
bonds, and other investment alternatives. Currently $200,000 of funds have become
available and must be
Math 442/542
Handout 2 Solution
Problem 1. Linear regression.
The difficulty here lies in the fact that he optimization problem as it is stated in the
problem set is not linear: the absolute value or the maximum functions are not linear. So
we need to ref
Math 442/542
Handout 3 Solution
Problem 1. Facility Location problem.
Define the decision variables in the following way.
For i = NY, LA, Ch, At, let
1 if a warehouse is opened in city i
Wi =
0 otherwise
For i=NY,LA,Ch,At, and j=1,2,3, let
Xij = number
Math 442/542 Handout 3 Solution
Problem 1. Facility Location problem. Define the decision variables in the following way. For i = NY, LA, Ch, At, let 1 if a warehouse is opened in city i Wi = 0 otherwise For i=NY,LA,Ch,At, and j=1,2,3, let Xij = number of
Math 442/542 Applications of Linear Programming
1. Facility Location problem.
A company is considering opening warehouses in four cities: New York, Los Angeles, Chicago, and Atlanta. Each warehouse can ship 100 units per week. The weekly fixed cost of kee
Math 442/542 Handout 2 Solution
Problem 1. Linear regression. The difficulty here lies in the fact that he optimization problem as it is stated in the problem set is not linear: the absolute value or the maximum functions are not linear. So we need to ref
Math 442/542 Applications of Linear Programming Consultant Hiring Assume that you are responsible for hiring outside consultants for a large corporation. There are potential projects currently being considered by the corporation, each of which would requi
Math 442/542 Winter 2011
Homework 1 Solution
1. There are many different ways of doing these formulations, and here we present just one of them. Decision Variables: Let xci , yci and zci represent the number of flights per day made to city c by the i'th p
Math 442/542 Theory of linear and nonlinear programming Winter 2011
Homework 2
(Problems 2 and 6 are due Thursday, January 27)
1. Consider a Linear Programming Problem in Standard Form. Label each of the following statements as true or false and then just
Math 442/542 Winter 2011
Homework 2 Solution
1. The standard form of an LP looks like max cT x subject to Ax b x 0 (a) FALSE. An obvious counterexample is given in problem 2 where any solution on the line segment connecting (5, 1)T and (0, 6)T is optimal,
Math 442/542 Winter 2011
Homework 3 (Problems 2 and 3 are due Tuesday, February 15)
1. Recall the problem 4 from Homework 2. We are given the following linear program: min 2x1 + x2 + 3x3 s. t. 5x1 + 2x2 + 7x3 3x1 + 2x2 + 5x3 x1 , x2 , x3
= 420 280 0
After
Math 442/542 Winter 2011
Homework 3 Solution
1. For the Revised Simplex we first need to fix some representation of the problem (with equality constraints!) which we will use to get the data (A, b and c) in each step. max -2x1 s.t. 5x1 3x1 -x2 +2x2 +2x2 -
Math 442/542
Homework 4
(Problems 3(c,d,e) and 4 are due Thursday, Feb. 24 at class time)
1. Suppose that a primal problem has a degenerate BF solution (one or more basic variables equal to zero) as its optimal solution. What does this degeneracy imply ab
Math 442/542 Homework 5
(problems 1 and 2 are due Friday, March 11)
1. A company will be producing the same new product at two different factories, and then the product must be shipped to two warehouses. Factory 1 can send an unlimited amount by rail to w
Math 442/542 Examples of Modeling Real-life Situations as Linear Programs 1. Portfolio selection.
National Insurance Associates carries an investment portfolio of a variety of stocks, bonds, and other investment alternatives. Currently $200,000 of funds h
Math 442/542 Handout 1 Solutions Problem 1. Portfolio selection.
National Insurance Associates carries an investment portfolio of a variety of stocks, bonds, and other investment alternatives. Currently $200,000 of funds have become available and must be
# IP model of Consultant Hiring problem on AMPL set I; # set of projects param profitcfw_i in I; #profits from projects excluding consulting costs set J; # set of consultants param costcfw_j in J; # weekly cost of consultant j set ATTR; # set of K attribu