OR Chapter 1. Introduction  Ex : Diet Problem Daily requirements : energy(2000kcal), protein(55g), calcium(800mg) Food Serving size Energy (kcal) Protein (g) Calcium (mg) Price per serving (cents) Max serving allowed Oatmeal28g Chicken100g Eggs2 large Wholemilk237cc Cherry pie170g Pork with beans 260g

OR  Formulation: Subject to

OR  Linear Programming Problem ( 선형계획법 문제 ) Subject to  objective function ( 목적함수 )  Constraints ( 제약식 )  nonnegativity constraints ( 비음제약식 ) (may not exist for some variables, then they are called unrestricted or free variables) right hand side ( 우변상수 )

OR Unusual formulations

OR ex) raws W=100 in., need 97 finals of width 45 in.610 finals of width 36 in. 395 finals of width 31 in.211 finals of width 14 in. Min x 1 + x 2 + x 3 + … + x 37 Note: number of patterns grows fast as problem becomes large (We don’t solve the problem with all columns in the model. We start with a few columns and solve the LP, and then identify and add a new column to the model and solve the LP again, … (column generation method). round down fractional optimal solution to LP to obtain integer solution, then use a few more raws to meet demands. extensions to 2-dimensional cutting stock (nesting problem), 3-D packing

OR

Linear Programming

OR  Minimization of piecewise linear convex function

OR

10

OR

OR

OR Terminology

OR Brief History of LP  Solving systems of linear inequalities : Fourier, 1826, not efficient (Chapter 16)  Simplex method : G. B. Dantzig, 1947  Ellipsoid method : L. G. Khachian, 1979 First polynomial time algorithm (theoretically efficient algorithm) for LP, practically not good.  Interior point method : L. Karmarkar, 1984 polynomial time algorithm, many variations, practically good performance. Recently, used for some nonlinear programming problems successfully (convex optimization)  Theory of LP provides important foundation for many other disciplines like integer programming, networks and graphs, nonlinear programming, etc.

OR Standard form

OR

OR

OR