Chapter 7
Linear Programming
Models
Part One
 Basis
of Linear Programming
 Linear Program formulation
Linear Programming (LP)
Linear programming is a
optimization model with an
objective (in a linear function)
and a set of limitations (in linear
constraints).
A Linear Program
Max
S.T.
X1 + 2X2
3X1 + X2 <= 200
X2 <= 100
X1, X2 >= 0
LP Components
 Decision
variables - their values are
to be found in the solution.
 One objective function – tells our
goal.
 Constraints - reflect limitations.
 Only linear terms are allowed.
Linear Terms
A term is linear if it contains one variable
with exponent one, or if it is a constant.
 Examples of linear terms:

–
3.5X
2X 4

68.83
(3.78)6X1
7
X1
22
Examples of non-linear terms:
–
–
5X2
X2.5
X1X2
Log X
sin X
X3
5
8X
X
Format of a Linear Program
 Align
columns of inequality signs,
variable terms, and constants.
 Variable terms are at left, constant
terms are at right (called right-handside, RHS).
 Non-negative constraints must be
there.
LP Solution
A solution is a set of values each for a
variable.
 A feasible solution satisfies all
constraints.
 An infeasible solution violates at least one
constraint.
 The optimal solution is a feasible solution
that makes the objective function value
maximized (or minimized).

LP Solution Methods
Trial-and-Error
(brute force)
 Graphic Method
(Won’t work if more than 2 variables)
 Simplex Method (by George Dentzig)
(Elegant, but time-taking if by hand)
 Computerized simplex method
(We’ll use it!)

George Dantzig
1914-2005
Inventor of Simplex Method.
Professor of Operations
Research and
Computer Science
at Stanford University.
To Solve a Problem by Linear
Programming
 Formulate
the problem into a linear
program (LP).
 Enter the LP into QM.
 QM solves LP and provide the optimal
solution.
Formulate a Problem into LP
 To
formulate a decision making
problem into a linear program:
–
–
–
Understand the problem thoroughly;
Define decision variables in
unambiguous terms;
Describe the problem with one
objective function and a few
constraints, in terms of the variables.
Flair Furniture, Example, p.252
Products
Resources
Tables
Chairs
Carpentry
Painting
Profit per
unit
4 hrs/unit
2 hrs/unit
3 hrs/unit
1hr/unit
$70
$50
Resource
available
240 hrs
100 hrs
Find how many tables and chairs should be
produced to maximize the total profit.
Flair Furniture, Example, p.252

Definitions of variables:

LP formulation:

Solution from QM
Tips of Formulating LP
 What
–
Those amounts you want to decide.
 What
–
is the ‘objective’?
Profit (or cost) you do not know but you
want to maximize (or minimize).
 What
–
are variables?
are ‘constraints’?
Restrictions of reaching your ‘objective’.
Holiday Meal Turkey Ranch, p.270
Composition (oz/pound)
Brand 1
Brand 2
Min. req.
Ingredient
feed
feed
per turkey
A
5
10
90 oz
B
4
3
48 oz
C
0.5
0
1.5 oz
Cost per
2 cents
3 cents
pound
Find how many pounds of brand 1feed and
brand 2 feed should be purchased with lowest
cost, which meet the minimum requirements of
a turkey for each ingredient.
Holiday Meal Turkey Ranch, p.270

Definitions of variables:

LP formulation:

Solution from QM:
Formulating
To formulate a business problem into a linear
program is to re-describe the problem with a
‘language’ that a computer understands.
 The key concern of formulation is:

–

whether the LP tells the story exactly the same as
the original one.
Formulating is synonymous with ‘describing’
and ‘translating’. It is NOT ‘solving’.
“Team Work”
 The
process of solving a business
problem by using linear programming
is a team work between us and
computers:
–
–
We formulate the problem in LP so that
computers can understand;
Computers solve the LP, providing us
with the solution to the problem.
Irregular LP Problems
A regular LP has one optimal solution.
 An irregular LP has no or many optimal
solutions:

–
–
–

Infeasible problem
Unbounded problem
Multiple optimal solutions
Redundancy refers to having extra and
un-useful constraints.
Part Two
 Shadow
Price (Dual Value)
 Sensitivity Analysis
Dual Price
 Each
dual price is associated with a
constraint. It is the amount of
improvement in the objective function
value that is caused by a one-unit
increase in the RHS of the
constraint.
 It is also called Shadow Price.
In a product-mix problem

As in the Flair Furniture example, a dual
price is:
–
–
–
the contribution of an additional unit of a
resource to the objective function value
(total profit), i.e.,
the marginal value of a resource, i.e.,
The highest “price” the company would be
willing to pay for one additional unit of a
resource.
Primal and Dual in LP
Each linear program has another
associated with it. They are called a pair
of primal and dual.
 The dual LP is the “transposition” of the
primal LP.
 Primal and dual have equal optimal
objective function values.
 The solution of the dual is the dual prices
of the primal, and vice versa.

More on Dual Price:

A dual price can be negative, which
shows a negative ( or worse off)
contribution to the objective function
value by an additional unit of RHS
increase of the constraint.
Sensitivity Analysis (S.A.)
S.A. is the analysis of the effect of
parameter changes on the optimal
solution.
 S. A. is conducted after the optimal
solution is obtained.

S.A. on Objective Coefficients

Sensitivity range for an objective
coefficient is the range of values over
which the coefficient can change without
changing the current optimal solution.
S.A. on RHS
 Sensitivity
range for a RHS value is
the range of values over which the
RHS value can change without
changing the dual prices.
S.A. on other changes
 To
see sensitivities on following
changes, one must solve the changed
LP again:
–
–
–
Changing technological (constraint)
coefficients
Adding a new constraint
Adding a new variable
Why doing S.A.?
LP is used for decision making on
something in the future.
 Rarely does a manager know all of the
parameters exactly. Many parameters are
inaccurate “estimates” when a model is
formed and solved.
 We want to see to what extent the optimal
solution is stable to the inaccurate
parameters.

Sensitive or In-sensitive?
Do we want a model more sensitive or less
sensitive to the inaccuracies (changes) of
parameters in it ?
 Answer:

Less sensitive.

Why?
–
An optimal solution that is insensitive to
inaccuracies of parameters is more likely valid
in the real world situation.