Linear Programming
Jessica Faith Worrell
What is Linear Programming?
• A specialized mathematical decisionmaking aid used in industry and
government
• Helps interpret data and examine the
way things work or should work
• Goal is to find optimal solutions to
problems
History
• Early 1800s
– Fourier, a French mathematician,
formulated the linear programming
problem
• 1900s
– Kantorovich, a Russian mathematician, also
developed the problem
– Goal was to improve economic planning in
the USSR
History
• World War II provided both the urgency
and the funds for such research.
• Efficient resource allocation was
required for large scale military
planning such as fleets of cargo ships
and convoys
History
• 1947- George Dantzig along with
associates at the US Department of the
Air Force developed the simplex method
• 1951- T.C. Koopmans developed a
special linear programming solution
used to plan the optimal movement of
ships back and forth across the Atlantic
during the war
• 1975 Nobel Prize in Economic Science
Applications
• Petroleum refineries
• Maximize the value of oil inputs subject to
constraints on refinery equipment and gasoline
blend requirements
• Find best locations for pipelines
• Find best routes and schedules for tankers
• 5-10% of total computing time
Applications
• Armour Company
• Processed cheese spread specifications
• HJ Heinz Company
• Shipment schedules between factories and
warehouses
• Agriculture
• Minimize cost for cattle feed
Most Common Applications
• Minimize cost while meeting product
specifications
• Maximize profit with optimal production
processes or products
• Minimize cost in transportation routes
• Determine best schedules for
production and sales
Problem
• A company manufactures two types of
hand calculators, Model 1 and Model 2.
Model 1 takes one hour to manufacture
while Model 2 takes four hours. The
cost of manufacturing is $30 and $20
for each Model 1 and 2, respectively.
The company has 1,600 hours of labor
time available and $18,000 in running
costs. The profit on each Model 1 is
$10 and each Model 2 is $8.
Solution
• Involves analyzing systems of linear
equations
• Two methods
• Geometric
• Computational
Geometric Solution
• Four inequalities, or constraints, to
consider:
• Time constraint
x + 4y < 1,600
• Monetary constraint
30x + 20y < 18,000
• Nonnegativity constraint
x > 0 and y > 0
Feasible Set
• The set of all possible solutions to the
family of inequalities
• Size depends on the amount of
constraints
Geometric Solution
• Goal is to maximize the profit where
P = 10x + 8y
• Must be a point that lies in the feasible
set
Geometric Solution
• Maximum will occur at a vertex of the
feasible set.
P=
P=
P=
P=
0 at A (0, 0)
3200 at B (0, 400)
6400 at C (400, 300)
6000 at D (600, 0)
• Maximum profit is $6,400 when 400
Model 1 calculators are produced and
300 Model 2.
Computational Method
• Simplex method developed by Dantzig
• Involves using matrices to
systematically check each corner of the
feasible set
• Dantzig chose to go along an edge
guaranteed to maximize profit
• Must have linear equations
Simplex Method
• Change inequalities to equalities
x + 4y + u = 1600
30x + 20y + v = 18000
• Rewrite profit function
-10x - 8y + f = 0
Simplex Method
• Initial Simplex Tableau:
1
4
1
0
0 1600
30 20 0 1
0 18000
-10 -8
0 0
1
0
x
y u
v
f
• Corresponds to vertex A
• Basic variables
• Nonbasic variables
u
v
f
Simplex Method
• After using elementary row operations
to produce a one in place of the pivot,
and zeroes in the rest of the column,
we get a second matrix
0 10/3 1 -1/30 0 1000
u
1 2/3 0 1/30 0 600
x
0 -4/3 0 1/3 1 6000
f
• Corresponds to vertex D
Simplex Method
• Now, with 10/3 as the pivot, the
resulting matrix is as follows:
0 1
3/10 -1/100 0
300
y
1 0
-1/5 6/150 0
400
x
0 0
2/5 24/75 1
6400 f
• Corresponds to the point C
• Final Simplex Tableau
Simplex Method
• The final matrix gives the following
equations:
y + 3/10 u - 1/100 v = 300
x - 1/5 u + 6/150v = 400
2/5 u + 24/75 v + f = 6400
where u = 0, v = 0, y = 300, x = 400
and f = 6400.
• Thus, maximum profit is $6400.
Advantages of Simplex
Method
• Yields the same answer
• More practical for problems involving
more than two variables
• Readily programmable for a computer
Minimization
• Must find the maximum of -f
• Then the minimum is the negative value
of that maximum
Possible Outcomes
• Every linear programming problem falls
into one of three categories:
• The feasible set is empty.
If constraints are contradictory, such as
x + 2y > 4 and x + 2y < -2
Possible Outcomes
• The cost function is unbounded on the
set.
If two vertices of the feasible set satisfy the
maximum or minimum, then every point of
the line also does.
Leads to flexibility in production schedule.
Possible Outcomes
• The cost has a maximum or a minimum
on the feasible set
There is one point that is the optimal
solution
• The first two possibilities are
uncommon for real problems in
economics
Requirements
• Requires linearly proportional
relationships
Resources to be consumed by an activity
must be linearly proportional to the activity
• All activities must obey a materials
balance
Sum of the resource inputs = Sum of
product outputs
Conclusion
• Linear Programming offers industry a
way to inform the decision makers of all
the important information and the most
favorable decision.
• It is an asset to companies in today’s
growing economy.