The Simplex Method. Standard Linear Programming Problem Standard Maximization Problem 1. All...
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![Page 1: The Simplex Method. Standard Linear Programming Problem Standard Maximization Problem 1. All variables are nonnegative. 2. All the constraints (the conditions)](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649ec55503460f94bcfcc2/html5/thumbnails/1.jpg)
The Simplex Method
![Page 2: The Simplex Method. Standard Linear Programming Problem Standard Maximization Problem 1. All variables are nonnegative. 2. All the constraints (the conditions)](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649ec55503460f94bcfcc2/html5/thumbnails/2.jpg)
Standard Linear Programming Problem
Standard Maximization Problem
1. All variables are nonnegative.
2. All the constraints (the conditions) can be expressed as inequalities of the form:
ax + by ≤ c, where c is a positive constant
![Page 3: The Simplex Method. Standard Linear Programming Problem Standard Maximization Problem 1. All variables are nonnegative. 2. All the constraints (the conditions)](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649ec55503460f94bcfcc2/html5/thumbnails/3.jpg)
Illustrating Example (1)
Maximize the objective function:P(x,y) = 5x + 4ySubject to:x + y ≤ 202x + y ≤ 35-3x + y ≤ 12x ≥ 0y ≥ 0
![Page 4: The Simplex Method. Standard Linear Programming Problem Standard Maximization Problem 1. All variables are nonnegative. 2. All the constraints (the conditions)](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649ec55503460f94bcfcc2/html5/thumbnails/4.jpg)
Solution
![Page 5: The Simplex Method. Standard Linear Programming Problem Standard Maximization Problem 1. All variables are nonnegative. 2. All the constraints (the conditions)](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649ec55503460f94bcfcc2/html5/thumbnails/5.jpg)
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What about when all of the constraints (the inequalities) are of
the type “≤ positive constant”But we want to minimize the objective function instead of
maximizing.
![Page 12: The Simplex Method. Standard Linear Programming Problem Standard Maximization Problem 1. All variables are nonnegative. 2. All the constraints (the conditions)](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649ec55503460f94bcfcc2/html5/thumbnails/12.jpg)
Minimization with “≤” constraintsIllustrating Example (2)
Minimize the objective function:p(x,y) = -2x - 3ySubject to:5x + 4y ≤ 32x + 2y ≤ 10x ≥ 0y ≥ 0
![Page 13: The Simplex Method. Standard Linear Programming Problem Standard Maximization Problem 1. All variables are nonnegative. 2. All the constraints (the conditions)](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649ec55503460f94bcfcc2/html5/thumbnails/13.jpg)
SolutionLetq(x) = - p(x) = - ( -2x -3y) = 2x + 3yTo minimize p is to maximize q. Thus, we solve the
following standard maximization linear programming problem:
Maximize the objective function:q(x) = 2x + 3ySubject to:5x + 4y ≤ 32x + 2y ≤ 10x ≥ 0y ≥ 0
![Page 14: The Simplex Method. Standard Linear Programming Problem Standard Maximization Problem 1. All variables are nonnegative. 2. All the constraints (the conditions)](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649ec55503460f94bcfcc2/html5/thumbnails/14.jpg)
Rewriting the inequalities as equations, by introducing the “slack” variables u and v and the formula of the objective function as done in example (1).
5x + 4y ≤ 32 , x + 2y ≤ 10 and q = 2x +3y
Are transformed to:
5x + 4y + u = 32
x + 2y + v = 10
- 2x - 3y + q = 0
![Page 15: The Simplex Method. Standard Linear Programming Problem Standard Maximization Problem 1. All variables are nonnegative. 2. All the constraints (the conditions)](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649ec55503460f94bcfcc2/html5/thumbnails/15.jpg)
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![Page 16: The Simplex Method. Standard Linear Programming Problem Standard Maximization Problem 1. All variables are nonnegative. 2. All the constraints (the conditions)](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649ec55503460f94bcfcc2/html5/thumbnails/16.jpg)
Standard Linear Programming Problem
Standard Minimization Problem
1. All variables are nonnegative.
2. All the constraints (the conditions) can be expressed as inequalities of the form:
ax + by ≥ c, where c is a positive constant
![Page 17: The Simplex Method. Standard Linear Programming Problem Standard Maximization Problem 1. All variables are nonnegative. 2. All the constraints (the conditions)](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649ec55503460f94bcfcc2/html5/thumbnails/17.jpg)
Solving
The Standard Minimization Problem
We use the fundamental theorem of Duality
![Page 18: The Simplex Method. Standard Linear Programming Problem Standard Maximization Problem 1. All variables are nonnegative. 2. All the constraints (the conditions)](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649ec55503460f94bcfcc2/html5/thumbnails/18.jpg)
Illustrating Example (3)
Minimize the objective function:p(x,y) = 6x + 8ySubject to:40x + 10y ≥ 240010x + 15y ≥ 21005x + 15y ≥ 1500x ≥ 0y ≥ 0
![Page 19: The Simplex Method. Standard Linear Programming Problem Standard Maximization Problem 1. All variables are nonnegative. 2. All the constraints (the conditions)](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649ec55503460f94bcfcc2/html5/thumbnails/19.jpg)
Minimize the objective function: p(x,y) = 6x + 8ySubject to:40x + 10y ≥ 2400, 10x + 15y ≥ 2100 , 5x + 15y ≥ 1500, x ≥ 0 and y ≥ 0We will refer to the above given problem by the primal (original) problem
First: We construct the following table, which we will refer to by the “primal” table:x y constant---------------------------------40 10 240010 15 21005 15 1500---------------------------------6 8
Second: We construct a dual (twin) table from interchanging the rows and columns in the primal table:
x' y' z' constant----------------------------------------------------------- 40 10 5 6 10 15 15 8---------------------------------------------------------2400 2100 1500
Third: We interpret the “dual table” as a standard maximization problem, which will refer to as the “dual problem” or “twin problem” of the “primal problem” or the “original problem”
Miaximoze the objective function: q( x ' , y ' , z ' ) = 2400x' + 2100y' + 1500z'Subject to:40x' + 10y' + 5z' ≤ 6, 10x' + 15y' + 15z' ≤ 8 , x' ≥ 0 and y' ≥ 0, z' ≥ 0
![Page 20: The Simplex Method. Standard Linear Programming Problem Standard Maximization Problem 1. All variables are nonnegative. 2. All the constraints (the conditions)](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649ec55503460f94bcfcc2/html5/thumbnails/20.jpg)
Fourth: We apply the simplex method explained in example (1) to solve this problem
Maximize the objective function: q(x,y,z) = 2400x' + 2100y' + 1500z'
Subject to:
40x' + 10y' + 5z' ≤ 6, 10x' + 15y' + 15z' ≤ 8 , x' ≥ 0 and y' ≥ 0, z' ≥ 0
4.a.Rewriting the inequalities and the formula of the objective function, with the slack variables being the same x and y (in that order) of the original (minimization) problem :
40x' + 10y' + 5z' + x = 6
10x' + 15y' + 15z' + y = 8
- 2400x' - 2100y' - 1500z‘ + q = 0
4.b. We construct the simplex table for this problem
![Page 21: The Simplex Method. Standard Linear Programming Problem Standard Maximization Problem 1. All variables are nonnegative. 2. All the constraints (the conditions)](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649ec55503460f94bcfcc2/html5/thumbnails/21.jpg)
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![Page 22: The Simplex Method. Standard Linear Programming Problem Standard Maximization Problem 1. All variables are nonnegative. 2. All the constraints (the conditions)](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649ec55503460f94bcfcc2/html5/thumbnails/22.jpg)
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![Page 23: The Simplex Method. Standard Linear Programming Problem Standard Maximization Problem 1. All variables are nonnegative. 2. All the constraints (the conditions)](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649ec55503460f94bcfcc2/html5/thumbnails/23.jpg)
Illustrating Example (4)
Minimize the objective function:p(x,y) = x + 2ySubject to:-2x + y ≥ 1- x + y ≥ 2x ≥ 0y ≥ 0
![Page 24: The Simplex Method. Standard Linear Programming Problem Standard Maximization Problem 1. All variables are nonnegative. 2. All the constraints (the conditions)](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649ec55503460f94bcfcc2/html5/thumbnails/24.jpg)
Minimize the objective function: p(x,y) = x + 2ySubject to:-2x + y ≥ 1, - x + y ≥ 2 We will refer to the above given problem by the primal (original) problem
First: We construct the following table, which we will refer to by the “primal” table:x y constant----------------------------------2 1 1-1 1 2---------------------------------1 2
Second: We construct a dual (twin) table from interchanging the rows and columns in the primal table:
x' y' constant------------------------------------------- -2 -1 1 1 1 2-----------------------------------------1 2
Third: We interpret the “dual table” as a standard maximization problem, which will refer to as the “dual problem” or “twin problem” of the “primal problem” or the “original problem”
Maximize the objective function: q( x ' , y ‘ ) = x' + 2y' Subject to:-2x' - y' ≤ 1, x' + y' ≤ 2 , x' ≥ 0 and y' ≥ 0
![Page 25: The Simplex Method. Standard Linear Programming Problem Standard Maximization Problem 1. All variables are nonnegative. 2. All the constraints (the conditions)](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649ec55503460f94bcfcc2/html5/thumbnails/25.jpg)
Fourth: We apply the simplex method explained in example (1) to solve this problem
Maximize the objective function: q( x ' , y ‘ ) = x' + 2y'
Subject to:
- 2x' - y' ≤ 1, x' + y' ≤ 2 , x' ≥ 0 and y' ≥ 0
4.a.Rewriting the inequalities and the formula of the objective function, with the slack variables being the same x and y (in that order) of the original (minimization) problem :
- 2x' - y' ' + x = 1
x' + y' + y = 2
- x' - 2y' + q = 0
4.b. We construct the simplex table for this problem
![Page 26: The Simplex Method. Standard Linear Programming Problem Standard Maximization Problem 1. All variables are nonnegative. 2. All the constraints (the conditions)](https://reader036.fdocuments.net/reader036/viewer/2022062314/56649ec55503460f94bcfcc2/html5/thumbnails/26.jpg)
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Homework
1. Using the simplex method, maximize: p = x + (6/5)y subject to:2x + y ≤ 180 , x + 3y ≤ 300 , x ≥ 0 , y ≥ 0Solution: p(48,84) = 148.8
2. Minimize: p(x,y) = - 5x - 4y Subject to: x + y ≤ 20 , 2x + y ≤ 35 , -3x + y ≤ 12 , x ≥ 0y ≥ 0Solution: p(15,5) = - 95
3. Using the dual theorem, minimize: p = 3x + 2y subject to:8x + y ≥ 80 , 8x + 5y ≥ 240 , x + 5y ≥ 100, x ≥ 0 , y ≥ 0Solution: p(20,16) = 92Maximize the objective function: