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Linear Programming Jelel EZZINE Jelel EZZINE Linear Programming

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Page 1: LP eLearning module

Linear Programming

Jelel EZZINE

Jelel EZZINE Linear Programming

Page 2: LP eLearning module

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

Page 3: LP eLearning module

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

Page 4: LP eLearning module

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

Page 5: LP eLearning module

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

Page 6: LP eLearning module

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

Page 7: LP eLearning module

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 ??

Page 8: LP eLearning module

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 :

Page 9: LP eLearning module

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 $.

Page 10: LP eLearning module

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

Page 11: LP eLearning module

Jelel EZZINE Linear Programming

Linear Programming Problem Energy related example: Geometrical Approach (STEP 3),

Geometrical aspect of the constraints

Page 12: LP eLearning module

Jelel EZZINE Linear Programming

Linear Programming Problem Energy related example: Linear Programming (STEP 2)

30; 40x y

10 20 200x y

Constraints

Page 13: LP eLearning module

Jelel EZZINE Linear Programming

Linear Programming Problem Energy related example: Linear Programming (STEP 2)

Geometrical aspect of the objective function

Page 14: LP eLearning module

Jelel EZZINE Linear Programming

Linear Programming Problem Energy related example: Linear Programming (STEP 2)