Introduction to Linear Programming Consider the Diet problem: - n - - PowerPoint PPT Presentation

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Oct 19, 2023 327 likes •403 views

Introduction to Linear Programming Consider the Diet problem: - n food items, m nutrients - for every nutrient: the daily quota b j - for each item: cost per pound c i - for every item and nutrient: how much of the nutrient in a pound of item: a

SLIDE 1

Consider the Diet problem:

n food items, m nutrients
for every nutrient: the daily quota bj
for each item: cost per pound ci
for every item and nutrient: how much of the nutrient in a

pound of item: ai,j

Introduction to Linear Programming

SLIDE 2

Introduction to Linear Programming

SLIDE 3

A linear program looks like this:

Introduction to Linear Programming

Find x1, x2, …, xm which

maximize

c1x1 + c2x2 + … + cmxm

and satisfy these constraints:

a1,1x1 + a1,2x2 + … + a1,mxm · b1 a2,1x1 + a2,2x2 + … + a2,mxm · b2 … an,1x1 + an,2x2 + … + an,mxm · bn

SLIDE 4

A linear program in compressed form: Given a vector c in Rm, a vector b in Rn and a matrix A in Rn x m, find a vector x in Rm which satisfies xAT · b and maximizes cxT. Thm: Exists a polynomial-time algorithm solving linear programs. Caveat: Sometimes need integer programs (no algorithm for integer programs is likely to exist) !

Introduction to Linear Programming

SLIDE 5

Modeling via LP: Shortest paths

Introduction to Linear Programming

SLIDE 6

Consider the Diet problem:

pound of item: ai,j

Introduction to Linear Programming

Introduction to Linear Programming

A linear program looks like this:

Introduction to Linear Programming

Find x1, x2, …, xm which

c1x1 + c2x2 + … + cmxm

a1,1x1 + a1,2x2 + … + a1,mxm · b1 a2,1x1 + a2,2x2 + … + a2,mxm · b2 … an,1x1 + an,2x2 + … + an,mxm · bn

Introduction to Linear Programming

Modeling via LP: Shortest paths

Introduction to Linear Programming

Modeling via LP: Vertex Cover

Introduction to Linear Programming