IOE 310 In-Class Assignment
Monday, April 18, 2011
A 4-ton vessel can be loaded with one or more of three items. The following table gives the unit
of weight, wi , in tons and the unit revenue in thousands of dollars, ri , for item i. How should the
vesse
IOE 310 In-Class Examples
Wednesday, April 13, 2011
1. The World Health Council is devoted to improving health care in the underdeveloped countries
of the world. It now has ve medical teams available to allocate among three such countries to
improve their
IOE 310 In-Class Assignment
Wednesday, April 6, 2011
Consider the following linear programming problem:
min 3x1 + 2x2
subject to:
4x1 + 5x2 20
x1 + 6x2 12
2x1 + x2 6
x1 6
x2 6
x1 , x2 0
1. Use the graph on the following page to draw the feasible region. I
IOE 310
Integer Programming Example Problems
3/28/2011
Example 1
The Research and Development Division of the Good Products Company has developed three
possible new products. The production cost per unit of each product would be essentially the same
in th
IOE 310 In-Class Assignment
Monday, March 21, 2011
Consider the following linear programming problem:
max Z = 2x1 + 7x2 + 4x3
x1 + 2x2 + x3 10
subject to:
3x1 + 3x2 + 2x3 10
x1 , x2 , x3 0
1. Find the optimal solution to the above LP.
The initial simplex
IOE 310 In-Class Assignment Solution
Wednesday, March 9, 2011
Consider the following network:
1. The shortest path = 17 and is O-C-F-G-T.
Solved nodes
connected
to unsolved
nodes
O
O
B
O
A
B
A
B
C
C
B
C
D
B
C
D
F
D
E
F
E
F
G
E
G
H
G
H
I
Closest
connected
IOE 310 In-Class Assignment Solution
Monday, February 21, 2011
Find the optimal solution for the LP below by completing the steps of the revised simplex method.
max x1 + 4x2
subject to:
x1 + 2 x2 6
2x1 + x2 8
x1 , x2 0
1. Write out the c vector, the A mat
IOE 310 In-Class Example Solution
Wednesday, February 16, 2011 and Monday, February 21, 2011
Find the optimal solution for the LP below by completing the steps of the revised simplex method.
min 3x1 + 2x2 + x3 + 3x4 + 4x5 + 10x6
subject to:
x1 + x4 + x6 =
IOE 310 In-Class Assignment
Wednesday, February 9, 2011
Consider the following problem:
max Z = x1 + x2 + x3 + x4
subject to:
x1 + x2 3
x3 + x4 2
and
xj 0
j = 1, 2, 3, 4
Work through the simplex method step by step to nd ALL the optimal basic feasible sol
IOE 310 In-Class Assignment Solution
Wednesday, January 26, 2011
Find the optimal solution for the LP below by completing the steps of the simplex method using
the simplex tableau method.
max 2x1 x2 + x3
subject to:
3x1 + x2 + x3 6
x1 x2 + 2 x3 1
x1 + x2
IOE 310 In-Class Assignment Solution
Wednesday, January 19, 2011
A paint company currently manufactures four types of paint and two types of thinners. Although
the paint company sells each individually, they are looking to market a blend of the paints and
IOE 310 In-Class Example
Wednesday, January 19, 2011
Every unit of product i that I produce uses dij units of resource j . I can sell a unit of product i
for $si . I have to purchase resource j at price $pj . There is an upper bound of uj on how much of
r
IOE 310 In-Class Assignment Solution
Monday, January 10, 2011
A university computing center receives a large number of jobs from students and faculty to be
executed on the computing facilities. Each student job requires 8 units of space on disk, and 7 uni
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Dynamic Programming
Introduction to optimization
Dynamic programming
Dynamic programming is a useful mathematical technique for making
a sequence of interrelated decisions.
It provides a systema
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Solving Nonlinear
Programming Problems
Introduction to optimization
Calculus Review
A function, f, defined on an interval (or on any convex set C of
some vector space) is called concave if, for
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Nonlinear Programming
Introduction to optimization
Nonlinear Programming
For practical purposes, we often assume that the
objective and constrain functions are linear. Frequently,
this assumptio
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Knapsack Problem
and
The Traveling Salesman
Problem
Introduction to optimization
The knapsack problem
Given a set of items, N, each with a weight, wi, and a
value, vi, determine the number of ea
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Solving Integer
Programming
Problems
Introduction to optimization
Solving Ip problems
Linear programming problems can be solved extremely
efficiently.
The only difference between LP problems and
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Integer Programming
and
Binary Integer Programming
Introduction to optimization
Integer programming
In many practical problems, the decision variables only make
sense if they are required to hav
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Methods for
Computing the Dual
Introduction to optimization
Example 1
Consider the following LP:
max Z = 6x1 + 8x2
s.t.:
5x1 + 2x2 20
x1 + 2x2 10
x1, x2 0
Example 1
Solving the primal with simpl
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Dual Simplex
and
Sensitivity Analysis
Introduction to optimization
Duality
Primal
Dual
Optimal Dual solution
The primal and dual solutions are so closely related that the
optimal solution of eit
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Duality
Introduction to optimization
Duality
Primal
Dual
Duality
Primal
Dual
x1
x2
x3
x4
x5
y1
y2
y3
c1
c2
c3
c4
c5
b1
b2
b3
a11
a12
a13
a14 a15
b1
a11
a21
a31
c1
a21 a22
a23 a24 a25
b2
a12 a22
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Sensitivity Analysis and
Duality
Introduction to optimization
Definition of a Linear program
Linear programs are mathematical models of optimization
problems in which:
The objective function is
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Network Optimization
Introduction to optimization
Network Terminology
A network consists of a set of nodes (aka vertices) which are
connected by arcs (aka links, edges, or branches).
If flow thr
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The Transportation Problem
and
The Assignment Problem
Introduction to optimization
The transportation problem
A transportation problem is concerned with distributing any
commodity from a group o
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Introduction to optimization
Revised Simplex Method
Revised Simplex Method
Revised Simplex Method
(4) Check for optimality.
If the reduced costs for nonbasic variables are strictly
positive in t
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Introduction to optimization
Simplex tableau
The simplex tableau stores all the variable
coefficients from the objective function and the
constraints.
For each iteration, we are only concerned w
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Introduction to optimization
Last Time: Big M Method
Sometimes we cannot start simplex by setting the
original variables = 0.
We construct an artificial problem with artificial
variables.
These
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Introduction to optimization
Simplex Method So Far
Simplex method
Graphically
Algebraically
Tableau Method
Steps of Simplex
1.
Identify an initial basic feasible solution (BFS).
2.
If the curren