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Linear Programming(An Optimization Technique)
General Optimization Problem I nvolves :
1. Decision Variables
2. Objective Criterion (Max/Min)
3. Constraints: (i) Equality (ii) Inequality
L inear Programming(LP) : It is one of the important Optimization
Techniques. Most Versatile, powerful & useful technique. Developed in 1947 by George B. Dantzig during
World War II to solve military problems, whileworking with US AirForce.
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(1)xcMin.)Zor(Max.Optimizen
1j
jj
The word Linear in LP indicates -relationship among variables both in objectivefunction and constraints.
while Programming means - SystematicMathematical technique.
General Linear Programming Problem (LPP)
with n variables and m constraints can be statedas follows:
)2(...2,1;),,(1
mibxa i
n
j
jij
)3(...2,1;0 njxj
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Types of Variables Decision Variables : xj are decision variables
Slack Variables
mibwxa in
j
ijij ...2,1;1
+
bxan
jijij
1
wi are slack variables.
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Artificial Variables :Required for solution methodology
Surplus Variables
bxa
n
jijij
1
mibsxa in
jijij ...2,1;
1
-
si are surplus variables
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Components of LPP :
The three important components of LPP
are as follows:
(a) Decision Variables
(b) Objective Function - Max. or Min.
(c) Constraints-limitations.
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1. Certainty:cj, aij, bi are known & constants.
2. Divisibility: Values of decision variable
can be integer or fractional.
3. Additivity: Total contribution = Sum of
contribution of all variables4. Linearity:Relationship among variables both
in objective function & constraints.
Assumptions of LP
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Formulation of LPP :
Steps to be followed.
1.To decide (define) decision variables.
2. To formulate objective function.
3. To formulate constraints.
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CASE - 1 Data for LP Formulation.
Factory: Two products P1 & P2
Profit/piece of P1 = Rs.3/-
Profit/piece of P2 = Rs.5/-
Time Constraint:
Time required/piece of P1 = 3 hrs.
Time required/piece of P2 = 2 hrs.
Max. time available = 18 hrs.
Material Constraint:
Max. Material available for P1 = 4 pieces.
Max. Material available for P2 = 6 pieces.
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CASE - 1 Formulation of LPP :
Let decision variable x1 = no of pieces of P1
x2 = no of pieces of P2
Max. Z = 3x1 + 5x2 Objective Equation
ConstraintTimeIxx )(182213 -+
ConstraintMaterialIIx )(41 -
ConditionsnegativityNonxx - 02,1
ConstraintMaterialIIIx )(62 -
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CASE - 2
If Time Constraint is modified to:
Time available is not less than 18 hours.
Max. Z = 3x1 + 5x2 Objective Equation
ConstraintTimeIxx )(182213 -+
ConstraintMaterialIIx )(41 -
ConditionsnegativityNonxx - 02,1
ConstraintMaterialIIIx )(62 -
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CASE - 3
If Time Constraint is modified to:
Time available is exactly 18 hours.
Max. Z = 3x1 + 5x2 Objective Equation
ConstraintTimeIxx )(182213 -+
ConstraintMaterialIIx )(41 -
ConditionsnegativityNonxx - 02,1
ConstraintMaterialIIIx )(62 -
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1. Feasible Solution: Solution with values of
decision variables(xj) which satisfy allConstrints & Non-negativity condition.
2. Basic Solution: For a set of m equations
in n variables (n>m), basic solution is a
solution obtained by setting (n-m)
variables equal to zero and solving for
remaining m equations in m variables.
Types of Solution
No. of possible Basic solutions = nCm
= n!/(m!.(n-m)!)
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(i) Basic Variables : Having values > 0.
(ii) Non-basic Variables :Having values = 0.
3. Basic Feasible Solution : It is a BasicSolution which satisfies (3).
4. Non-degenerate/degenerate B.F.S
5. Optimal Basic Feasible Solution : It is a
Basic Feasible Solution which optimizes
objective function.
6. Unbounded Solution :
7. Infeasible Solution :
8. Unique/Alternative Optimal Solutions :
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Methods to solve LPP :
1.Graphical Method - for only two variables.2. Simplex Method - Universal method.
3. Assignment Method - Special method.4. Transportation Method - Special method.
Note :Methods (2),(3) and (4) are iterative
methods.
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(1) Graphical Method
( for only two variables)
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3
6
9
2 4 6 x1
F.R.
Opt. Pt. (2,6)
Z=0.
Z=15.
Zopt = 36.
III
I
II
02,1
)(62)(41
)(182213/
2513
--
-+
+
xx
IIIxIIx
Ixxts
xxZMaxGraphical Method
Case 1.
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3
6
9
2 4 6 x1
F.R.
Opt. Pt.(4,6)
Z=15.
Zopt = 42
III
I
II02,1
)(62
)(41
)(182213/
2513
-
-
-+
+
xxIIIx
IIx
Ixxts
xxZMaxGraphical Method
Case 2.
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3
6
9
2 4 6 x1
F.L.
Opt. Pt.(2,6)
Z=15.
Zopt = 36
III
I
II
02,1
)(62)(41
)(182213/
2513
-
-
-+
+
xx
IIIxIIx
Ixxts
xxZMaxGraphical Method
Case 3.
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3
6
9
2 4 6 x1
F.R.
III
I
II
Graphical Method
Special Case.
Alternative OptimalSolutions
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3
6
9
2 4 6 x1
F.R.(Unbounded)
Opt. Pt.
(Min.)
III
II
I
GraphicalMethod
Special Case.
Z line
Opt. Sol. (Max)
(Unbounded)
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3
6
9
2 4 6 x1
F.R.Not
possible
InfeasibleSolution
GraphicalMethod
Special Case.
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4
5
4 5 x1
x2
20-2-3
2
-1
-1
-2
-3
-4
III
III
F.R.
Z=6
Opt. Pt. (0.5,
4.5)
Zopt = 12.5
02,1
)(521
)(3213
)(4212/
2312
-+
-+-
--
+-
xx
IIIxx
IIxx
Ixxts
xxZMax
GraphicalMethod
Special Case.
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Graphical Methodtodiscuss typical constraints
to identify regions of
constraints.
Typical Case ?62213 +l ttH
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4 5 x1
x2
0-2-3 -1
-1
-3
-2
1
2
3
21 3
Typical Case ?62213 -+ xxplottoHow
62213 -+ xx
),Point(xxWhen 022102 --
)Point(xxWhen 3,03201 --
Typical Case ?62213 l ttH
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2
3
4 5 x1
x2
10-2-3
1
-1
-1
-2
-3
2 3
Typical Case ?62213 - xxplottoHow
62213 - xx
),Point(xxWhen 022102
)Point(xxWhen 3,03201 --
Typical Case ?021 +l ttH
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4 5 x1
x2
0-2-3 -1
-1
-3
-2
1
2
3
21 3
Typical Case ?021 + xxplottoHow
021 +xx
21 xx - )Point(xxWhen 1,11211 --arrowFor
0102 xx
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Recapitulate
What is LPP ? Types of variables.
Components of LPP.
Assumptions of LP. Formulation of LPP.
Types of solutions.
Graphical Method.
Unique & Alternative Optimal Solution,
Unbounded FR, Infeasible solution.
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(2) Simplex Method
(Universal method)
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Simplex Method to solve
Max. Problem with All constraints.
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02,1
)(62
)(41
)(182213/
2513
-
-
-+
+
xx
IIIx
IIx
Ixxts
xxZMax
Standard Form:
Max Z = 3x1+5x2+0w1+0w2+0w3
3x1+2x2+w1+0w2+0w3 = 18
x1+0x2+0w1+w2+0w3 = 4
0x1+x2+0w1+0w2+w3 = 6
Simplex Method
Case 1.
To
prepare initial Tableau:
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ci xi bi x1 x2 w1 w2 w3
Ij 3 5 0 0 0
Ij = (cj -Ej) = Cj- (
aij.ci)
cJ 3 5 0 0 0
Toprepare initial Tableau:
Tableau - I
0 w1 18 3 2 1 0 0
0 w2 4 1 0 0 1 0
0 w3 6 0 1 0 0 1
Z = bi
.ci
= 0
Ej 0 0 0 0 0
Tableau I
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ci xi bi x1 x2 w1 w2 w3
Ij Z = 0 3 5 0 0 0
cj 3 5 0 0 0
Tableau - I
0 w1 18 3 2 1 0 0
0 w2 4 1 0 0 1 0
0 w3 6 0 1 0 0 1
Ratio18/2 = 9
4/0 =
6/1 = 6
Key Column Max +ve Ij Key Row Min positive ratio.
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How to get next tableau ?
Leaving variable : w3
Entering variable : x2
If the key element is 1, then key row remain same inthe new simplex tableau
If the key element is other than 1, then divide eachelement in the key row (including b value) by the keyelement to find new values of that row
Make the other elements in the key column equal to0 by performning elementary row operations withnew key row obtained as above.
T bl I
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ci xi bi x1 x2 w1 w2 w30 w1 18 3 2 1 0 0
0 w2 4 1 0 0 1 0
0 w3 6 0 1 0 0 1
Ij Z = 0 -3 -5 0 0 0
Ratio18/2 = 9
4/0 =
6/1 = 6
cj 3 5 0 0 0Tableau - I
18 18 - (6*2)/1 = 6
R1 2R3 gives
2 2-2(1) = 0 , 3 3- 2(0) = 3 and 1 12(0) = 1
T bl I I
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0 w1 6 3 0 1 0 -2
0 w2 4 1 0 0 1 0
Ij Z = 30 3 0 0 0 -5
Ratio6/3 = 2
4/1 = 4
6/0 =
cj 3 5 0 0 0
Key Column Max +ve Ij
Key Row Min positive ratio.
Tableau - I I
5 x2 6 0 1 0 0 1
ci xi bi x1 x2 w1 w2 w3
Tableau I I I
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3 x1 2 1 0 1/3 0 -2/3
Ij Z = 36 0 0 - 1 0 - 3
cj 3 5 0 0 0Tableau - I I I
5 x2 6 0 1 0 0 1
This is the final Tableau.
The Optimal Solution is x1 = 2, x2 = 6
giving Z = 36
ci xi bi x1 x2 w1 w2 w3
0 w2 2 0 0 -1/3 1 2/3
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Interpretation of
Simplex Method through
Graphical method.
I nterpretation of Simplex Method
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3
6
9
2 4 6 x1
F.R.
Opt. Pt. (2,6)
Z=15.
Zopt = 36.
III
I
II
02,1
)(62
)(41
)(182213/
2513
-
-
-+
+
xx
IIIx
IIx
Ixxts
xxZMax
I nterpretation of Simplex Method
Through Graphical Method
T1
T3
T2
GATE 2002
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GATE - 2002(1) A furniture manufacture produces Chairs & Tables.
The wood working department is capable of producing
200 chairs or 100 tables or any proportionatecombinations of these per week. The weekly demand
for chairs and tables is limited to 150 and 80 units
respectively.The profit from a chair is Rs. 100 and that
from a table is Rs. 300.
Set up the problem as a Linear Program.
Determine optimal product mix and optimal valueof objective function.
If a profit of each table drops to Rs. 200 per unit,
what is the product mix and profit ?
(a)
(b)
(c)
GATE 2002
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GATE - 2002
MaxZ = 100x1+ 300x2
S/t1
100
2
200
1+
xx
x1 150
x2 80
x1,x2 0
Formulation(a)
(b) Optimal Solution
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(40, 80) Z = 28000
Z = 15000
100
200
150
50
80
x2
x1
(b) Optimal Solution
(c) Multiple Optimal Solutions
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(40, 80) Z = 20000
Z = 15000
100
200
150
50
80
x2
x1
75 Opt. Line
(c) Multiple Optimal Solutions
GATE 2003
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GATE - 2003(1) A manufacture produces two types of products, 1 and
2, at production levels of x1 and x2 respectively. The
profit is given 2x1 + 5x2. The production constraints
are :
The maximum profit which can meet the constraint is
(A) (D)(C)(B)29 38 44 75
x1 + 3x2 40
3x1 + x2 24
x1 + x2 10
x1 0, x2 0
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40
x2
x1
10
10
20
Z = 50
Opt. Pt. (0,10) Z = 50
Ans. = (C) 44
25
GATE 2000
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GATE - 2000
Max Z = 4x1+ 6x2 + x3
S/t
x1, x2, x3 0
2x1+ x2 + 3x3 5
Solve :
If x2 2 is added then what will be the
solution ?
GATE 2000
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GATE - 2000
Solution :
x1 = 0, x2 = 5, x3 = 0 giving z = 30
Through Simplex Method, in
two iterations the solution ofbasic problem is :
If x2 2 is added then the solution through
Simplex Method, in three iterations will be :
x1
= 3/2, x2
= 2, x3
= 0 giving z = 18
GATE - 2008
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GATE 2008Max Z = 4x1
+ 6x2
x1, x2 0
3x1+ 2x2 6
2x1+ 3x2 6
Q.1 After introducing slack variables w1 and w2, the initial
feasible solution is represented by the tableau below.
ci xi bi x1 x2 w1 w2
Ij 0 -4 -6 0 0
cj 4 6 0 0
0 w1 6 3 2 1 0
0 w2 6 2 3 0 1
GATE - 2008
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GATE 2008
After some Simplex interactions, the following tableau is obtained.
ci xi bi x1 x2 w1 w2
Ij 12 0 0 0 2
cj 4 6 0 0
0 w1
2 5/3 0 1 -1/3
6 x2 2 2/3 1 0 1/3