LP slides, meant for eLearning module.

1. Linear Programming Jelel EZZINE Jelel EZZINE Linear Programming

2. 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

3. 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 Next Next Linear programming: the objective function and the contraints have linear expressions

4. 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 Fomulation Given 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

5. Jelel EZZINE Linear Programming 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 30 20 10 Wind mill (wind energy) 40 35 20 Solar panel (solar energy) M.Nb E.Sv S/Unit Item

6. Jelel EZZINE Linear Programming Linear Programming Problem Energy related example: Linear Programming (STEP 2) Decision variables Number of wind mill to be installed Number of solar panel to be installed Objective function Minimize Maximize : maximize the energy saved for a given budget Maximization problem Energy saving device problem Or Decision variables

7. Jelel EZZINE Linear Programming Objective function Maximize Linear Programming Problem Energy related example: Linear Programming (STEP 2) Application : Mathematical formulation of the objective function E.Sv : Energy saved in Giga Joules per Unit per Year. Constraints ?? 20 Wind mill 35 Solar panel E.Sv Item

8. Jelel EZZINE Linear Programming Linear Programming Problem Energy related example: Linear Programming (STEP 2) Decision variables Number of smokleless wood stoves to be installed Number of improved kerosene stoves to be installed Constraints Constraint 1 Constraint m Where :

9. Jelel EZZINE Linear Programming 40 20 Solar panel 30 10 Wind mill M.Nb S/Unit Item 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. Constraint 2 Constraint 3 S/Unit : Cost per Unit ($). Maximum number of wind mill that can be installed per year < 30 Maximum number of Solar panel that can be installed per year < 40 The total cost of the items must not exceed the budget amount !! The total amount of the budget is equal to 200 $.

10. Jelel EZZINE Linear Programming Linear Programming Problem Energy related example: Linear Programming (STEP 2) The mathematical problem formulation : Subject to : Search for the optimal solution that maximize the objective function under the given constraints !! Geometrical approach

11. Jelel EZZINE Linear Programming Linear Programming Problem Energy related example: Geometrical Approach (STEP 3), Geometrical aspect of the constraints

12. Jelel EZZINE Linear Programming Linear Programming Problem Energy related example: Linear Programming (STEP 2) Constraints

13. Jelel EZZINE Linear Programming Linear Programming Problem Energy related example: Linear Programming (STEP 2) Geometrical aspect of the objective function

14. Jelel EZZINE Linear Programming Linear Programming Problem Energy related example: Linear Programming (STEP 2)