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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
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