Introduction to Operations Research
Summary
EBB829A05
Semester I B
Wouter Voskuilen
S4916344
Slides by dr. I. Bakir
The lectures contains some information that the book does not, and the book contains
some information that the lecture slides do not.
1
,Wouter Voskuilen Introduction to Operations Research
Lecture 1
Types of Input Data
− Type-A Data: available
− Type-B Data: not available but can be collected
− Type-C Data: not available and cannot be collected
Type-C Data need to be estimated
Modelling terminology
− Parameters: what imput data are known and needed to make the decisions.
− Decision variables: description of the set of decisions to be made. Indicate valid range
of all variables.
− Objective: measure to rank alternative solutions, maximize or minimize.
− Constraints: Limitations on the values of the decision variables. Develop mathematical
relationships to describe constraints.
”The optimal solution” versus ”an optimal solution”.
Always say ”an optimal solution” unless you can prove it is unique.
A hyperplane H ∈ Rn is an (n − 1)-dimensional collecction, described by
H = {x ∈ Rn |hT x = δ}, for h ∈ R and δ ∈ R (where h is called the normal vector).
A closed halfspace H ∈ Rn is a n-dimensional collection given by
H = {x ∈ R|hT x ≤ δ}, for h ∈ Rn , and δ ∈ R.
A polyhedron is the intersection of a number of halfspaces.
An LP model is built from m + n halfspace, so the feasible area of an LP-problem is a poly-
hedron.
The intersection of n linearly independent (n − 1)-dimensional hyperplanes defines a point.
Corner point feasible solutions (CPF solution):
− Feasible point: A point that satisfies all constraints.
− A feasible point that satisfies one or more constraints with equality (aTi x = bi ) is on
the boundary of the polyhedron.
2
, Wouter Voskuilen Introduction to Operations Research
− A corner point feasible solution is: A feasible point that is not on any segment con-
necting two other feasible solutions (no convex combinations possible).
− Each CPF is at the intersection of n hyperplanes.
− Two CPF solutions are adjacent if they share n − 1 hyperplanes and differ by only one.
Simplex Method
Property (1A)
If there is exactly one optimal solution, then it must be a CPF solution.
Property (1B)
If there are multiple optimal solutions then at least two must be adjacent CPF solutions
(provided we have a bounded feasible region).
From 1A and 1B we know: If one or more optimal solutions exist, then there is an opti-
mal solution that is a CPF solution.
Property (2)
m+n
The number of CPF solutions is bounded from above by n .
Property (3)
If a CPF has no adjacent CPF solutions that are better, then there are no better CPF solu-
tions anywhere.
This implies that we only need to check all adjacent solutions to verify whether a particular
solution is or is not optimal.
In the equation form:
− Original constraint: aTi x ≤ bi
− Equation form: aTi x + xn+i = bi
− The constraint in the original formulation is holding with equality if the slack variable
xn+i = 0 in the equation form.
− For this reason xn+i is called an indicating variable in this context.
− A basic solution is one with m basic variables and n nonbasic variables, with the
nonbasic variables all set to zero.
− A basic feasible solution is a basic solution with the additional requirements that all m
basic variables are nonnegative for a solution to the system of m equations.
3
Summary
EBB829A05
Semester I B
Wouter Voskuilen
S4916344
Slides by dr. I. Bakir
The lectures contains some information that the book does not, and the book contains
some information that the lecture slides do not.
1
,Wouter Voskuilen Introduction to Operations Research
Lecture 1
Types of Input Data
− Type-A Data: available
− Type-B Data: not available but can be collected
− Type-C Data: not available and cannot be collected
Type-C Data need to be estimated
Modelling terminology
− Parameters: what imput data are known and needed to make the decisions.
− Decision variables: description of the set of decisions to be made. Indicate valid range
of all variables.
− Objective: measure to rank alternative solutions, maximize or minimize.
− Constraints: Limitations on the values of the decision variables. Develop mathematical
relationships to describe constraints.
”The optimal solution” versus ”an optimal solution”.
Always say ”an optimal solution” unless you can prove it is unique.
A hyperplane H ∈ Rn is an (n − 1)-dimensional collecction, described by
H = {x ∈ Rn |hT x = δ}, for h ∈ R and δ ∈ R (where h is called the normal vector).
A closed halfspace H ∈ Rn is a n-dimensional collection given by
H = {x ∈ R|hT x ≤ δ}, for h ∈ Rn , and δ ∈ R.
A polyhedron is the intersection of a number of halfspaces.
An LP model is built from m + n halfspace, so the feasible area of an LP-problem is a poly-
hedron.
The intersection of n linearly independent (n − 1)-dimensional hyperplanes defines a point.
Corner point feasible solutions (CPF solution):
− Feasible point: A point that satisfies all constraints.
− A feasible point that satisfies one or more constraints with equality (aTi x = bi ) is on
the boundary of the polyhedron.
2
, Wouter Voskuilen Introduction to Operations Research
− A corner point feasible solution is: A feasible point that is not on any segment con-
necting two other feasible solutions (no convex combinations possible).
− Each CPF is at the intersection of n hyperplanes.
− Two CPF solutions are adjacent if they share n − 1 hyperplanes and differ by only one.
Simplex Method
Property (1A)
If there is exactly one optimal solution, then it must be a CPF solution.
Property (1B)
If there are multiple optimal solutions then at least two must be adjacent CPF solutions
(provided we have a bounded feasible region).
From 1A and 1B we know: If one or more optimal solutions exist, then there is an opti-
mal solution that is a CPF solution.
Property (2)
m+n
The number of CPF solutions is bounded from above by n .
Property (3)
If a CPF has no adjacent CPF solutions that are better, then there are no better CPF solu-
tions anywhere.
This implies that we only need to check all adjacent solutions to verify whether a particular
solution is or is not optimal.
In the equation form:
− Original constraint: aTi x ≤ bi
− Equation form: aTi x + xn+i = bi
− The constraint in the original formulation is holding with equality if the slack variable
xn+i = 0 in the equation form.
− For this reason xn+i is called an indicating variable in this context.
− A basic solution is one with m basic variables and n nonbasic variables, with the
nonbasic variables all set to zero.
− A basic feasible solution is a basic solution with the additional requirements that all m
basic variables are nonnegative for a solution to the system of m equations.
3
