Linear Programming - NPTELnptel.ac.in/courses/105108127/pdf/Module_3/M3L1slides.pdf · z Linear...
Transcript of Linear Programming - NPTELnptel.ac.in/courses/105108127/pdf/Module_3/M3L1slides.pdf · z Linear...
D Nagesh Kumar, IISc Optimization Methods: M3L11
Linear Programming
Preliminaries
D Nagesh Kumar, IISc Optimization Methods: M3L12
Objectives
To introduce linear programming problems (LPP)
To discuss the standard and canonical form of LPP
To discuss elementary operation for linear set of
equations
D Nagesh Kumar, IISc Optimization Methods: M3L13
Introduction and Definition
Linear Programming (LP) is the most useful optimization technique
Objective function and constraints are the ‘linear’ functions of ‘nonnegative’ decision variables
Thus, the conditions of LP problems are
1. Objective function must be a linear function of decision variables
2. Constraints should be linear function of decision variables
3. All the decision variables must be nonnegative
D Nagesh Kumar, IISc Optimization Methods: M3L14
Example
Conditionity Nonnegativ0,
Constraint3rd154
Constraint2nd113
Constraint1st532tosubject
FunctionObjective56Maximize
p
p
p
p
p
≥
≤+
≤+
≤−
+=
yx
yx
yx
yx
yxZ
This is in “general” form
D Nagesh Kumar, IISc Optimization Methods: M3L15
Standard form of LP problems
Standard form of LP problems must have following three characteristics:
1. Objective function should be of maximization type
2. All the constraints should be of equality type
3. All the decision variables should be nonnegative
D Nagesh Kumar, IISc Optimization Methods: M3L16
General form Vs Standard form
Violating points for standard form of LPP:
1. Objective function is of minimization type
2. Constraints are of inequality type
3. Decision variable, x2, is unrestricted, thus, may take negative values also.
General form
edunrestrict0
243
1532tosubject53Minimize
2
1
21
21
21
21
xx
xxxx
xxxxZ
≥≥+≤+≤−−−=
How to transform a general form of a LPP to the standard form ?
D Nagesh Kumar, IISc Optimization Methods: M3L17
General form Standard form
General form1. Objective function
2. First constraint
3. Second constraint
Standard form1. Objective function
2. First constraint
3. Second constraint
Transformation
21 53Minimize xxZ −−=
1532 21 ≤− xx 1532 321 =+− xxx
321 ≤+ xx 3421 =++ xxx
Variables and are known as slack variables3x 4x
21 53Maximize xxZZ +=−=′
D Nagesh Kumar, IISc Optimization Methods: M3L18
General form Standard form
General form4. Third constraint
Standard form4. Third constraint
Transformation
24 521 =−+ xxx24 21 ≥+ xx
5. Constraints for decision variables, x1 and x2
Variable is known as surplus variable5x5. Constraints for decision
variables, x1 and x2
edunrestrict2x01 ≥x 01 ≥x
222 xxx ′′−′=0,and 22 ≥′′′ xx
D Nagesh Kumar, IISc Optimization Methods: M3L19
Canonical form of LP Problems
The ‘objective function’ and all the ‘equality constraints’
(standard form of LP problems) can be expressed in canonical
form.
This is known as canonical form of LPP
Canonical form of LP problems is essential for simplex method
(will be discussed later)
Canonical form of a set of linear equations will be discussed
next.
D Nagesh Kumar, IISc Optimization Methods: M3L110
Canonical form of a set of linear equations
Let us consider the following example of a set of linear equations
(A0)
(B0)
(C0)
The system of equation will be transformed through ‘Elementary Operations’.
1023 =++ zyx
632 =+− zyx12 =−+ zyx
D Nagesh Kumar, IISc Optimization Methods: M3L111
Elementary Operations
The following operations are known as elementary operations:1. Any equation Er can be replaced by kEr, where k is a nonzero
constant.2. Any equation Er can be replaced by Er + kEs, where Es is
another equation of the system and k is as defined above.
Note: Transformed set of equations through elementary operations is equivalent to the original set of equations. Thus, solution of transformed set of equations is the solution of original set of equations too.
D Nagesh Kumar, IISc Optimization Methods: M3L112
Transformation to Canonical form: An Example
Set of equation (A0, B0 and C0) is transformed through elementary operations (shown inside bracket in the right side)
⎟⎠⎞
⎜⎝⎛ ==++ 01 A
31A
310
31
32 zyx
( )101 ABB38
38
380 −==+− zy
( )101 2ACC3
1735
310 −=−=−− zy
Note that variable x is eliminated from B0 and C0 equations to obtain B1 and C1. Equation A0 is known as pivotal equation.
D Nagesh Kumar, IISc Optimization Methods: M3L113
Transformation to Canonical form: Example contd.
Following similar procedure, y is eliminated from equation A1 and C1considering B1 as pivotal equation:
⎟⎠⎞
⎜⎝⎛ −==++ 212 B
32AA40 zx
⎟⎠⎞
⎜⎝⎛ −=−=−+ 12 B
83B10 zy
⎟⎠⎞
⎜⎝⎛ +=−=−+ 212 B
31CC6200 z
D Nagesh Kumar, IISc Optimization Methods: M3L114
Transformation to Canonical form: Example contd.
Finally, z is eliminated form equation A2 and B2 considering C2 as pivotal equation :
( )323 CAA100 −==++x
( )323 CBB200 +==++ y
⎟⎠⎞
⎜⎝⎛ −==++ 23 C
21C300 z
Note: Pivotal equation is transformed first and using the transformed pivotal equation other equations in the system are transformed.
The set of equations (A3, B3 and C3) is said to be in Canonical form which is equivalent to the original set of equations (A0, B0 and C0)
D Nagesh Kumar, IISc Optimization Methods: M3L115
Pivotal Operation
Operation at each step to eliminate one variable at a time, from all equations except one, is known as pivotal operation.
Number of pivotal operations are same as the number of variables in the set of equations.
Three pivotal operations were carried out to obtain the canonical form of set of equations in last example having three variables.
D Nagesh Kumar, IISc Optimization Methods: M3L116
Transformation to Canonical form: Generalized procedure
Consider the following system of n equations with n variables
)(
)(
)(
2211
222222121
111212111
nnnnnnn
nn
nn
Ebxaxaxa
Ebxaxaxa
Ebxaxaxa
=+++
=+++
=+++
LLL
MM
MM
LLL
LLL
D Nagesh Kumar, IISc Optimization Methods: M3L117
Transformation to Canonical form: Generalized procedure
Canonical form of above system of equations can be obtained by performing n pivotal operations
Variable is eliminated from all equationsexcept j th equation for which is nonzero.
General procedure for one pivotal operation consists of following two steps,1. Divide j th equation by . Let us designate it as , i.e.,
2. Subtract times of equation from
k th equation , i.e.,
( )nix i L1=
jia
jia )( jE ′
kia )( jE ′
( )njjk ,,1,1,2,1 LL +−=
ji
jj a
EE =′
jkik EaE ′−
D Nagesh Kumar, IISc Optimization Methods: M3L118
Transformation to Canonical form: Generalized procedure
After repeating above steps for all the variables in the system of equations, the canonical form will be obtained as follows:
)(100
)(010
)(001
21
2221
1121
cnnn
cn
cn
Ebxxx
Ebxxx
Ebxxx
′′=+++
′′=+++
′′=+++
LLL
MM
MM
LLL
LLL
It is obvious that solution of above set of equation such as is the solution of original set of equations also.
ii bx ′′=
D Nagesh Kumar, IISc Optimization Methods: M3L119
Transformation to Canonical form: More general case
Consider more general case for which the system of equations has m equation with n variables ( )
It is possible to transform the set of equations to an equivalent canonical form from which at least one solution can be easily deduced
mn ≥
)(
)(
)(
2211
222222121
111212111
mmnmnmm
nn
nn
Ebxaxaxa
Ebxaxaxa
Ebxaxaxa
=+++
=+++
=+++
LLL
MM
MM
LLL
LLL
D Nagesh Kumar, IISc Optimization Methods: M3L120
Transformation to Canonical form: More general case
By performing n pivotal operations for any m variables (say, , called pivotal variables) the system of equations reduced to canonical form is as follows
Variables, , of above set of equations is known as nonpivotal variables or independent variables.
mxxx L,, 21
)(100
)(010
)(001
11,21
22211,221
11111,121
cmmnmnmmmm
cnnmmm
cnnmmm
Ebxaxaxxx
Ebxaxaxxx
Ebxaxaxxx
′′=′′++′′++++
′′=′′++′′++++
′′=′′++′′++++
++
++
++
LLLLLL
MM
MM
LLLLLL
LLLLLL
nm xx ,,1 L+
D Nagesh Kumar, IISc Optimization Methods: M3L121
Basic variable, Nonbasic variable, Basic solution, Basic feasible solution
One solution that can be obtained from the above set of equations is
mxxx L,, 21
nm xx ,,1 L+
( ) nmixmibx
i
ii
,,1for0,,1forL
L
+===′′=
This solution is known as basic solution.
Pivotal variables, , are also known as basic variables.
Nonpivotal variables, , are known as nonbasic variables.
Basic solution is also known as basic feasible solution because it satisfies all the constraints as well as nonnegativity criterion for all the variables
D Nagesh Kumar, IISc Optimization Methods: M3L122
Thank You