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Linear Programming
Jelel EZZINE
Jelel EZZINE Linear Programming
Jelel EZZINE Linear Programming
L. V. Kantorovich
1939
« A technique for distributing raw materials to maximize output »
Year
Preliminary work
G. B. Dantzig
Algorithm for solving real planning problems (SIMPLEX method)
Military application: organize and expediate supplies to troops
T. J. Koopmans (1910-1985)
Economic application of linear programming models
1975
The theory of distribution of ressources and its correlation
to linear programming
Wide range of applications:agriculture, natural science, social
science, transportation, energy, etc.
1945
Brief History
L. Khachiyan
1979
Introduction of ellipsoid method for solving linear programming problems
1984
N. Karmarkar
New interior point projective
method for linear programming
Jelel EZZINE Linear Programming
Linear Programming Problem Overview
Step 1
Step 2
Given a problem with a minimization or maximization objective
Mathematical Model
construction
Solution of the mathematical
model
LP Geometry
SIMPLEX
Objective function
Decision variables
Constraints
Nex
tN
ext
Linear programming: the objective function and the contraints have linear expressions
Jelel EZZINE Linear Programming
Linear Programming Problem Energy related example: Problem Formulation
Growing energy crisis
Usage of energy saving devices.
Major problem:
Counter measure
Renewable energy
Problem FomulationGiven various sources of renewable energy or energy saving devices, Maximize the energy saved for a given budget amount
Minimize the amount of budget to achieve a certain target of saved energy
Jelel EZZINE Linear Programming
302010Wind mill (wind energy)
403520Solar panel (solar energy)
M.NbE.Sv S/UnitItem
Linear Programming Problem Energy related example: Problem Formulation (STEP 1)
Nomenclature: 1 Item : Renewable energy source. 2 S/Unit : Cost per Unit ($). 3 E.Sv : Energy saved in Giga Joules per Unit per Year. 4 M.Nb : maximum number of item that can be installed per year.
Let the total amount of the budget be equal to 200 $.
How much items (wind mill and solar panels) should be installed in order to maximize the energy saved for the given budget ??
Problem Formulation
Jelel EZZINE Linear Programming
Linear Programming Problem Energy related example: Linear Programming (STEP 2)
Decision variables
xy
Number of wind mill to be installed
Number of solar panel to be installed
Objective function
Minimize
Maximize1coef 1var 2coef 2var ncoef varn:
maximize the energy saved for a given budget
Maximization problem
Energy saving device problem
Or
Decision variables
Jelel EZZINE Linear Programming
Item E.Sv
Wind mill 20
Solar panel 35
Objective function
Maximize 1coef 1var 2coef 2var20
Linear Programming Problem Energy related example: Linear Programming (STEP 2)
Application: Mathematical formulation of the objective function
35x y
E.Sv : Energy saved in Giga Joules per Unit per Year.
Constraints ??
Jelel EZZINE Linear Programming
Linear Programming Problem Energy related example: Linear Programming (STEP 2)
Decision variables
xy
Number of smokleless wood stoves to be installedNumber of improved kerosene stoves to be installed
Constraints
Constraint 1
Constraint m
11coef 1var 12coef 2var 1ncoef varn Op 1Rhs
1mcoef 1var 2mcoef 2var mncoef varn Op MRhs
OpWhere :
Jelel EZZINE Linear Programming
4020Solar panel
3010Wind millM.NbS/UnitItem
Linear Programming Problem Energy related example: Linear Programming (STEP 2)
Constraints
Constraint 1
2
M.Nb : maximum number of item that can be installed per year.
11coef 1var Op 1Rhsx
Constraint 2
y
Constraint 3
11coef 1var 12coef 2var Op 1Rhs10 x 20 y 200
S/Unit : Cost per Unit ($).
Maximum number of wind mill that can be installed per year < 3030
Maximum number of Solar panel that can be installed per year < 4040
The total cost of the items must not exceed the budget amount !!
The total amount of the budget is equal to 200 $.
Jelel EZZINE Linear Programming
Linear Programming Problem Energy related example: Linear Programming (STEP 2)
The mathematical problem formulation :
max 20 35x y
Subject to :30; 40x y
10 20 200x y
Search for the optimal solution that maximize the objective function under the given constraints !!
Geometrical approach
Jelel EZZINE Linear Programming
Linear Programming Problem Energy related example: Geometrical Approach (STEP 3),
Geometrical aspect of the constraints
Jelel EZZINE Linear Programming
Linear Programming Problem Energy related example: Linear Programming (STEP 2)
30; 40x y
10 20 200x y
Constraints
Jelel EZZINE Linear Programming
Linear Programming Problem Energy related example: Linear Programming (STEP 2)
Geometrical aspect of the objective function
Jelel EZZINE Linear Programming
Linear Programming Problem Energy related example: Linear Programming (STEP 2)