Simplex Method Final

8/2/2019 Simplex Method Final

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CVNG 2010 (Civil Engineering Management)


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Simplex Method


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The simplex method is an algebraic methodfor solving linear programming problems.George Dantzig 'invented' the simplex methodin 1947 while looking for methods for solving

optimization problems. It is basically aprocess of taking various linear inequalitiesrelating to some situation, and finding the"best" value obtainable under thoseconditions. A typical example would be takingthe limitations of materials and labor, and thendetermining the "best" production levels formaximal profits under those conditions.

Brief History


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The simplex method operates on linear programs in standard form

Minimize, C.X

Subject to Ax= b, xi0

C represents the respective weights or cost of the variables xi theminimized statement is similarly called the cost of the solution. Thecoefficients of the system of equations are represented by A, and anyconstant values in the system of equations are combined on the righthand side of the inequality in the variables b. Combined, these statementsrepresent a linear program, to which we seek a solution .


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When given inequalities (constraints) in a simplex method problem

Example : x + 2y 14

3x- y 0

x y 2

they can be plotted on a graph to identify the corner points of thefeasible region. The feasible region is the portion of the graph that is

valid for all constraints.

Feasible

Region


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Simplex Method


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UseResourcesEffectively

ProjectObjectives


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OBJECTIVES

Within Budget

On Time

High Quality


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WHAT RESOURCES ARENEEDED?


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resources

Labour

Not Limitless Capital Costs

Materials

Equipment


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Working within Constraints !

Limited Resources = CONSTRAINTS

OBJECTIVES


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HOW IS THIS ACHIEVED ?

Linear Programming:

mathematical technique used to findthe maximum or minimum of a

linear functionfrom many variablessubject

to constraints.

Max : [ y = mx +c ]


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In other words

a way of defining and OPTIMIZING the

relationship among available resources in

order to generate a maximumoutput under

certain constraints.


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uy ng a ascar

Fuel efficient

Constraints:

Safe

Cheap


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Managerial Example

The Bess Block Company produces two types of

building blocks, concrete and clay. The concrete

blocks produce a gross profit of $10 per unit, while the

clay blocks produce a gross profit of $6 per unit.

For the week 180hrs of man-time are available to

produce these blocks, a concrete block requiring 6hrs

to produce, while a clay block requires

8hrs.Unfortunately, the warehouse can only

accommodate 250 s.m. of new production. What is

the production allocation required for each unit to

produce maximum profit, if the concrete blocks require

0.25 s.m. of space while the concrete requires

0.18s.m. of space.

$10 profit

$6 profit


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X = number concrete blocks needed

Y = number clay blocks needed

P = Profit

Bess Block Company

Max [P = 10 x + 6y]


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Bess Block Company

The Bess Block Company produces two types of

building blocks, concrete and clay. The concrete

blocks produce a gross profit of $10 per unit, while

the clay blocks produce a gross profit of $6 per unit.

For the week 180hrs of man-time are available to

produce these blocks, a concrete block requiring

6hrs to produce, while a clay block requires

8hrs.Unfortunately, the warehouse can onlyaccommodate 250 s.m. of new production. What is

the production allocation required for each unit to

produce maximum profit, if the concrete blocks

require 0.25 s.m. of space while the concrete

requires 0.18s.m. of space.

Time Constraint: Total Available Time: = 180hrs

6x + 8y 180

Concrete takes6hrs

Clay takes 8hrs


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Bess Block Company

The Bess Block Company produces two types of building

blocks, concrete and clay. The concrete blocks produce a

gross profit of $10 per unit, while the clay blocks produce a

gross profit of $6 per unit.

For the week 180hrs of man-time are available to produce

these blocks, a concrete block requiring 6hrs to produce,

while a clay block requires 8hrs.Unfortunately, the

warehouse can only accommodate 250 s.m. of newproduction. What is the production allocation required for

each unit to produce maximum profit, if the concrete blocks

require 0.25 s.m. of space while the concrete requires

0.18s.m. of space.

Space Constraint:

Concrete takes 0.25s.m.

Clay takes 0.18s.m.

Total Available Space = 250s.m.

0.25x + 0.18y


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Bess Block Company

The Bess Block Company produces two types of building

blocks, concrete and clay. The concrete blocks produce a

gross profit of $10 per unit, while the clay blocks produce a

gross profit of $6 per unit.

For the week 180hrs of man-time are available to produce

these blocks, a concrete block requiring 6hrsto produce,

while a clay block requires 8hrs.Unfortunately, the

warehouse can only accommodate 250 s.m. of newproduction. What is the production allocation required for

each unit to produce maximum profit, if the concrete blocks

require 0.25 s.m. of space while the concrete requires

0.18s.m. of space.

For Maximum Profit:

0.25x + 0.18y

250

Pmax = 10 x + 6y

Subject to:

6x + 8y 180


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Solution ?

SimplexMethod


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Summary

Simplex

Methodmarketing mix determination

financial decision making

production scheduling

workforce assignment, and resource

blending


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Worked Example

Courtesy

math.uww.edu

22.06.2007


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Example

Maximize

Z=X1 + 2X2 X3Subject to

2X1 + 2X2 + X3 144X1 + 2X2 + 3X3 282X

1+ 5X

2+ 5X

3 30

X1 0, X2 0, X3 0


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Slack Variables

2X1 + 2X2 + X3 + S1 + 0 + 0 = 144X

1

+ 2X2

+ 3X3

+ 0 + S2 + 0 = 28

2X1 + 5X2 + 5X3 + 0 + 0 + S3 = 30


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Simplex Tableau

X1 X2 X3 S1 S2 S3

2 1 1 1 0 0 14

4 2 3 0 1 0 28

2 5 5 0 0 1 30

-1 -2 1 0 0 0 0

Ratios 14/1 28/2 30/5

(1/5)r3=R3

r1 r3 = R1

r2 2r3 = R2

r4 + 2r3 = R4

X1 X2 X3 S1 S2 S3

2 1 1 1 0 0 144 2 3 0 1 0 28

2/5 1 1 0 0 1/5 6

-1/5 0 3 0 0 2/5 12


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X1 X2 X3 S1 S2 S38/5 0 0 1 0 -1/5 8

16/5 0 1 0 1 -2/5 16

2/5 1 1 0 0 1/5 6

-1/5 0 3 0 0 2/5 12

Ratios 8/ (8/5)

16/ (16/5)

6/ (2/5)

(5/16)r2 = R2

r1 8/5r2 = R1

r3 2/5r2 = R3

r4 + 1/5r2 = R4

X1 X2 X3 S1 S2 S3

8/5 0 0 1 0 -1/5 8

1 0 5/16 0 5/16 -1/8 5

2/5 1 1 0 0 1/5 6

-1/5 0 3 0 0 2/5 12


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X1 X2 X3 S1 S2 S3

0 0 -1/2 1 -1/2 0 0

1 0 5/16 0 5/16 -1/8 5

0 1 7/8 0 -1/8 1/4 4

0 0 49/16 0 1/16 3/8 13

All indicators (0, 0, 49/16, 0, 1/16 and 3/8) are now zero or bigger (13

is NOT an indicator.

X3=S2=S3=0

1/2 X3 + 1S1 + 1/2S2 = 0X1 + 5/16X3 + 5/16S2 1/8S3 = 5X2 + 7/8X3 1/8S2 + 1/4S3 = 4

Z = 13


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Simplex Method


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Pros

Simplex method is essentially an algorithm used in linearprogramming

Advantages

Easily programmed on a computer

Its an algorithm that can be easily programmed on acomputer.

Any Function used in the method can be quickly andeasily adapted in a software program as only theevaluation of the function needs to be altered.

Its ability to be used on computers in softwarespeeds up the problem solving process, as opposedto doing it manually.


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Pros (contd)

Easy to use

Generally the method is very simple to us, oncethe language is familiar it is fairly easy toimplement

when compared to the graphical method, thismethod allows a problem to be addressed withmore than 2 decision variables.

When compared to the least-squares method,in that it does not require a derivative functionand the orthogonality condition is not relevant.


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Cons

When using this method it can often becomedifficult to notice mistakes

This method is time consuming when donemanually i.e. by hand

Limited applications in terms of solvingprogramming problems, its use is limited fore.g. In business situations it only applieswhere a decimal quantity is appropriate. It isalso only appropriate when a few variables are

at play. In These situations, the method isvery efficient. Many problems with a real lifepractical interest have many variables


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Cons (contd)

Difficult requirements

Only problems that can be expressed in astandard from with 3 conditions can be

solved with the this algorithm. The constraints of the problem must also

use non negative constraints for allvariables. And it must be expressed in the

form where the number on the right side ispositive.


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Conclusion

Generally the method is fairlystraightforward and easy to learn can beeasily programmed into computers.

Making it easier to carry out problemsolving. Its major downfall being that itcan only work for specific problems instandard form . however for these typeof problems it is quite efficient .


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Simplex Method

Simplex Method Final

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