Discrete Math B: Chapter 4, Linear Programming: The Simplex Method 1 |Chapter 4: Linear Programming! he Simplex MethodHHf U Day 1: (text pgl69-176) In chapter 3, we solved linear programming problems graphically. Since we can only easily graph with two variables (x and y), this approach is not practical for problems where there are more than two variables involved. To solve linear programming problems in three or more variables, we will use something called "The Simplex Method." 4.1 Slack Variables and the Pivot Getting Started: Variables: Use Xi, x2, x3(... instead of x, y, z,... Problems look like: Maximize z- 3x,+2x2+x3 A Standard Maximum Problem 4.1 Setup! 4.2 Solving! 2x, + x2 + x3 <150 Subject to 2x, + 2x2 + 8x3 < 200 2x, + 3x2 + x3 < 320 1. z is to be maximized 2. All variables, x1,x2,x3,...>0 3. All constraints are "less than or equal to" (i.e. < 4. 3/4.4 Look Different with x, > 0, x2 > 0, x3 > 0 To Use Simplex Method: Convert constraints (linear inequalities) into linear equations using SLACK VARIABLES. Example 1: Convert each inequality into an equation by adding a slack variable. Slack variables: si, s2, s3, etc. b) x, + 3x2 + 2.5X3 <100 a) 2X,+4.5X2<8 For example: If x,+x2<10 then x, +x2 +x, = 10 X, + JXI<-2.5)<3 * Sj-i ZX, tS, = g x, > 0 and "takes up any slack"
Examples for Operational Research - Simplex Method minimum finding
Generating Simplex Tables
Zadaci za Optimalno upravljanje korištenjem simplex metoda za dobijanje minimuma funkcije.
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Discrete Math B: Chapter 4, Linear Programming: The Simplex Method 1
|Chapter 4: Linear Programming!he Simplex MethodHHfU
Day 1:
(text pgl69-176)In chapter 3, we solved linear programming problems graphically. Since we can only easily graph with two
variables (x and y), this approach is not practical for problems where there are more than two variablesinvolved. To solve linear programming problems in three or more variables, we will use something called "TheSimplex Method."
4.1Slack Variables and the Pivot
Getting Started:
Variables: Use Xi, x2, x3(... instead of x, y, z,...
Problems look like:Maximize z- 3x,+2x2+x3
A Standard Maximum Problem4.1Setup!
4.2 Solving! 2x, + x2 + x3 <150
Subject to 2x, + 2x2 + 8x3 < 200
2x, +3x2 + x3 < 320
1. z is to be maximized
2. All variables, x1,x2,x3,...>03. All constraints are "less than or equal to" (i.e. <
4.3/4.4Look Different
with x, >0,x2 >0, x3 >0
To Use Simplex Method:
Convert constraints (linear inequalities) into linear equations using SLACK VARIABLES.
Example 1: Convert each inequality into an equation by adding a slack
variable.Slack variables:
si, s2, s3, etc.
b) x, +3x2 +2.5X3 <100a) 2X,+4.5X2<8For example:
If x,+x2<10
then x, +x2 +x, =10 X, + JXI<-2.5)<3 * Sj-iZX, tS,= g
x, >0 and "takes up any slack"
Discrete Math B: Chapter 4, Linear Programming: The Simplex Method 2
Example 2:a) Determine the number of slack variables needed
b) Name them
c) Use slack variables to convert each constraint into a linear equation
Maximize z — 3x]+2x2+x3
2x, +x2+x3 <150
Subject to 2x, +lx2 +8x3 < 200
2xt +3x2 + x3 < 320(one per constraint)oi) 3
VT) Si, S3
?)1
•2x1 2y 2. i- #x3 + S2. - 200
i- By 3. i- )c3 + S3 = 32o
with x, >0,x2 >0,x3 >0
n
rewr|TE the objective function so all the variables are on the left and the constants are on the right.
z-3x, +2x2 +X3 2 -3*,- -*3*0VPirvstlofi*-
ih. ctlpKo.order
|ÿ~*3y t -2-Xi 3 ?b — o
5230 WRITE the modified constraints (frSm step 1) and the objective function (from step 2) as an
augmented matrix. This is called the "simplex tableau."
There should be a row for each constraint.
The last row is the objective function.
EVERY variable used gets a column.
5, S*. S30 o
X Xz.
15o0I1% l
2. Z S 05
a 3
r*>CoKSfrafnb o o Zoo
0 o 1 o 32O
O O O \ Ocÿ>)ediÿ~p “2 “l
Discrete Math B: Chapter 4, Linear Programming: The Simplex Method 3
Example: Introduce slack variables as necessary, then write the initial simplex tableau for each linearprogramming problem.
lOv, -X-2. -X3 f Sr '3?
I3*i ¥-7x3 * S2- - 2-°5
I4-VC, +- Vi. '2*3 4- S3 - 3H:5
“lx, -3xz - *3 4- it - o
Ex 3) Find x, >0, x2 >0, andx3 >0 such that
10x, -x2 -x3 <138
13x, +6X2+7X3 < 205
14x, +x2 -2x3 <345
and z=7x, +3x2 +x3 is maximized.
X3 s, 62.*1 NL*.o I3So-I o10 I
20 5ot o(p 713 o
345OooH I -2
c;o 1o o-3 -1-7
Ex. 4) Find x, >0 and x2 >0 such that
2x, +12x2 <20
4x, +x2 <50
and z=8x, +5x2 is maximized.
2x 1 + — 10
4xt 4* y 2. t-Sa. - 5o
- Sx 2. it - o
X| S,I
2, 13 I 0 0 20
5ol o o
'6 0 0) l \ 0“8
Discrete Math B: Chapter 4, Linear Programming: The Simplex Method 4
Example 5: A businesswoman can travel to city A, city B, or city C. It is 122 miles to city A, 237 miles to city
B, and 307 miles to city C. She can travel up to 3000 miles. Dining and other expenses are $95 in city A, $130 in
city B, and $180 in city C. Her expense account allows her to spend $2000. A trip to city A will generate $800 in
sales, while a trip to city B will generate $1300 and a trip to city C will generate $1800. How many trips should
she make to each city to maximize sales? Write the initial simplex tableau.
Discrete Math B: Chapter 4, Linear Programming: The Simplex Method10
Day 4:
(Continued)4.2 Maximization Problems
Example 4: Solve using the Simplex Method
Kool T-Dogg is ready to hit the road and go on tour. He has a posse consisting of 150 dancers, 90 back-up
singers, and 150 different musicians and due to union regulations each performer can only appear once during
the tour. A small club tour requires1dancer,1back-up singer and 2 musicians for each show while a larger
arena tour requires 5 dancers, 2 back-up singer and1musician each night. If a club concert nets T-Dogg $175 anight while an arena show nets him $400 a night, how many of each show should he schedule so that his income
Discrete Math B: Chapter 4, Linear Programming: The Simplex Method
11
Example 5: Solve using the Simplex MethodThe Cut-Right Knife Company sells sets of kitchen knives. The Basic Set consists of 2 utility knives and1chefsknife. The Regular Set consists of 2 utility knives and1chef's knife and1bread knife. The Deluxe Set consists of
3 utility knives,1chefs knife, and1bread knife. Their profit is $30 on a Basic Set, $40 on a Regular Set, and $60on a Deluxe Set. The factory has on hand 800 utility knives, 400 chefs knives, and 200 bread knives. Assuming
all sets are sold, how many of set should be sold to maximize the profit. What is the maximum profit?
6) [>. X* *3 S, 9, a I nq,\tk>z 2 2 3 1 o o o |*» 1 * 1 ° 1