Wasim Akram Mandal
University of Kalyani, India
E-mail: wasim0018@gmail.com
Sahidul Islam
University of kalyani, India
E-mail: drsahidul.math@gmail.com
Submission: 25/07/2018
Accept: 29/08/2018
ABSTRACT
A "signomial" is a mathematical function, contains one
or more independent variables. Richard J. Duffin
and Elmor L. Peterson (1967) introduced the term "signomial". Signomial geometric programming (SGP)
optimization technique often provides a much better mathematical result of
real-world nonlinear optimization problems. In this research paper, we have proposed
unconstrained and constrained signomial geometric programming (SGP) problem
with positive or negative integral degree of difficulty. Here a modified form
of signomial geometric programming (MSGP) has been developed and some theorems
have been derived. Finally, these are illustrated by proper examples and
applications.
Keywords: Polynomial; Constrained problem;
Signomial function; Modified Signomial Geometric Programming
1. INTRODUCTION
Geometric programming (GP) method is
a relatively new optimization method to solve a various types of non-linear
programming problem. The idea of geometric programming technique was fast
developed by Duffin, Peterson and Zener. Since 1960 ҆s geometric programming
(GP) has been known and used in various field like as operations management,
industrial engineering, structural problem etc..
In the late 1960s and early 1970s
the term “signomial” was introduced by Duffin and Peterson (1967) in their
seminal joint work on general algebraic optimization. A recent introductory
exposition is non-linear optimization problems. Although non-linear
optimization problems with constraints and/or objectives defined by signomials
are normally harder to solve than those defined by only posynimials (because
unlike posynomials, signomials are not guaranteed to be globally convex).
A signomial optimization geometric
programming technique often provides a much more accurate and better
mathematical representation of real-world non-linear optimization problems.
Passy and Wilde (1967) and Blau and
Wilde (1969) generalized some of the prototype concepts and theorems in order
to treat signomial geometric programming problems. In 1988 general type of
signomial geometric programming has been done by Charnes and Cooper, who
proposed methods for approximating signomial geometric programs with prototype
geometric programs. Recently Islam and Roy (2005) has been demonstrated
modified geometric programming problem and its application.
A “signomial” is a mathematical (algebraic) function of one or more
independent function. “signomial” is one of the most easily thought of as an
algebraic extension of multi-dimensional polynomials (posynomial) an extension
that permits exponents to be arbitrary real numbers (rather than just non
negative integers) while requiring the independent variables to be strictly
positive .
In this research paper we propose
unconstrained / constrained signomial geometric programming (SGP) and modified
form of signomial geometric programming (SGP) problem with positive or negative
integral of difficulty. Modified form of signomial geometric programming has
been demonstrated and some theorems have been presented here. Finally, these
are illustrated by proper real type of numerical examples and
applications.
2. SOME BASIC CONCEPTS
Monomial: The word ‘monomial’ comes from the
Latin word, mono meaning only one and mial meaning term. So a monomial is
an “expression in algebra that contains only one term”. The term monomial as used in the context of
geometric programming (GP) technique is similar to, but differs from the
standard definition of monomial used in algebra. In mathematics a monomial can
be a constant, a variable or the product of one or more constant and variables
but the exponent of variables cannot have a negative or fractional. Throughout
the exponent can be any real number, including fractional and negative.
So if
F(x)
where
c > 0 and
Polynomial: The word ‘polynomial’ comes from the
Latin word, polynomeaning many and mial meaning term. So a polynomial
is an “expression in algebra that contains many terms i.e., many monomial”.
So sum of one or more monomial function, i.e.,
any function of the form
F(x)
is
called polynomial function or in simply called polynomial.
Posynomial: If coefficients
So
sum of one or more monomials, i.e., any function of the form
F(x)
where
Note
1: The term posynomial function is mean to
suggest a combination of positive and polynomial function. So Positive
Signomial: A signomial function is differs from a
posynomial function in that the coefficient need not be positive. Let X is a
vector of real and positive numbers.
X
Then
the signomial function has the form as follows
Where
Signomial functions are closed under
addition, subtraction, multi-plication and scaling of basic mathematical
operation.
Example 1:
For
this signomial
Table 1: Compare between
monomial, polynomial and posynomial:
Monomial |
Polynomial |
Posynomial |
1) Contains
only one term. 2) Addition
of two or more monomials is not monomial. 3)
Subtraction of two or more monomials is not monomial. 4)
Multiplication of two or more monomials is a monomial. 5) Division
of a monomial by one or more other monomials is a monomial. 6) A monomial
is of the form, F(x) 7) Examples:
5, 2x, |
1) Contains
one or more term. 2) Addition
of two or more polynomials is polynomial. 3)
Subtraction of two or more polynomials is a polynomial. 4)
Multiplication of two or more polynomials is a polynomial. 5) Division
of a polynomial by one or more other monomials is a polynomial. 6) A
polynomial is of the form, F(x) 7) Examples:
5, 2x+y, |
1) Contains
one or more term. 2) Addition
of two or more posynomials is posynomial. 3)
Subtraction of two or more posynomials is not posynomial. 4)
Multiplication of two or more posynomials is a posynomial. 5) Division
of a posynomial by one or more other monomials is a posynomial. 6) A
posynomial is of the form, F(x) 7) Examples:
5, 2x+y, |
3. UNCONSTRAINED PROBLEM
3.1.
Signomial
geometric programming (SGP) problem:
3.1.1. Primal
program:
A
primal unconstrained signomial geometric programming (SGP) problem is of the
following form
Minimize
Subject to
Where
Here
3.1.2. Dual
program:
The dual geometric programming (DGP)
problem of the given primal geometric programming (GP) problem is
Maximize
Subject to
Case
I: If n>m+1,
(i.e. DD>0) then the dual program (DP) presents a system of linear equations
for the dual variables. Here the number of dual variables is more than the
number of linear equations. Here more optimal solutions of dual variable vector
exist. In order to find an optimal solution of the given dual program (DP), we
need to use some algorithmic techniques.
Case
II: If n<
m+1, (i.e. DD<0) then the dual program (DP) presents a system of linear
equations for the dual variables. Here the number of dual variables is less
than the number of linear equations. In this special case generally no solution
vector exists for the dual variables. However, using some special types of
techniques that is Least Square (LS) or Min-Max (MM) method one can get an
approximate solution for this type of problem.
Moreover
from the primal-dual relation,
Taking logarithms in (3.3), T0
log-linear simultaneous equations are transformed as
It
is a system ofn linear equations in tj (=log xj) for
j=1,2,...,m.
Note 2: A Weak Duality theorem say that
Corollary 1: When the values of
3.1.3.
Theorem: When
the values of
Proof
The expression for
Here the weights are
Now applying the AM
Or
Or
Or
=
=
i.e.,
This completes the proof.
Example
(Unconstrained SGP problem) 2:
Min Z(x, y)
Subject to x, y > 0.
Solution:
Assume
Corresponding
dual of the primal geometric problem is,
Maximize v(δ)
Subject to
From
(6)
So for primal optimal decision variables, the
following equations found
Taking
logarithm of each side yields equations which are linear in the logarithms of
decision the primal variables
In 10 + In x
In 5 + log x + log x
In 10 + 3log x
From
(3.7), optimal solutions are
x
y
And corresponding optimum value, Z(x, y)
3.2.
Modified
signomial GP problem:
3.2.1. Primal
program:
A
primal unconstrained modified signomial geometric programming (SGP) problem is
of the form
Minimize
Subject to
Where
Here
3.2.2. Dual
program:
Dual
geometric program (GP) problem of the given primal GP problem is
Maximize
Subject to
Case
I: If nk
Case
II: If
nk<nm+n, (i.e. DD<0) then the dual program (DP) presents a system of
linear equations for the dual variables. Here the number of dual variables is
less than the number of linear equations. In this case generally no solution
vector exists for the dual variables. However, using the Least Square (LS) or
the Min-Max (MM) technique one can get an approximate solution for this
non-linear system.
Furthermore
the primal-dual relation is
Note 3: A Weak Duality theorem would say that
Corollary 2: When the values of
3.2.3.
Theorem: When
Proof
The expression for
Here the weights are
Now applying AM-GM inequality, we get
Or
Or
Or
=
i.e.,
This completes the proof.
Example
(Unconstrained MSGP problem) 3:
Min Z(x)
Subject to
Solution:
Assume
Corresponding
dual of the given primal problem is
Maximize
Subject to
From
(3.12)
So for primal decision variables, the
following equations found
From (3.13), optimal solutions are
Corresponding optimal value is
Z(x)
4. CONSTRAINED PROBLEM
4.1.
Signomial
geometric programming (SGP) problem:
4.1.1. Primal
program:
A
constrained primal signomial geometric programming (SGP) programming problem is
of the form
Minimize
Subject to
Where
4.1.2. Dual
program:
Dual
geometric program (GP) problem of the given primal signomial program (SGP)
problem is
Maximize
Subject to
Case
I: If n
Case
II: If n<
m+1, (i.e. DD <0), In this case generally no solution vector exists for the
dual signomial variables. But using Least the Square (LS) or the Min-Max (MM)
method one can get an approximate solution for this system.
Furthermore the primal-dual
relation is
And
Example
(Constrained SGP problem) 4:
Min
Subject to
x, y, z > 0.
Solution:
Assume
Corresponding
dual of the given primal is
Maximize v(δ)
Subject to
From
(4.5),
i.e.,
Corresponding dual of primal is
Maximize v(δ)
Subject to
From
(4.7),
i.e.,
From the primal-dual relation
Taking
logarithm of each side yields equations which are linear in the logarithms of
the primal variables
In 2 + 2In y + 4Inz
In 5 + 2log x
In 2 + 2log x
From (4.8), optimal solutions are
4.2. Modified Signomial geometric
programming (MSGP) programming:
4.1.3. Primal
program:
A
primal modified signomial GP programming problem is of the form
Minimize
Subject to
Where
4.1.4. Dual
program:
Dual
GP problem of the given primal MSGP problem is
Maximize
Subject to
Case
I: If nk
Case
II: If
nk<nm+n, (i.e. DD <0), In this case generally no solution vector exists
for the dual signomial variables. But using the Least Square (LS) or the
Min-Max (MM) method one can get an approximate solution for this system.
Furthermore from the primal-dual
relation
and
Example
(Constrained MSGP problem) 5:
Min Z(x)
Subject to
Sol.
Taking
Here the primal problem is
Min Z(x)
Subject to
And corresponding dual problem
is
Subject to
Approximate solutions of this
system of linear equations are
And
Then the objective function is
And
from primal-dual relation following system of equations gives optimal primal
variables
10
Solving the system of equations,
optimal solutions are
and the value of objective
function is Z(x)
5. APPLICATION
5.1.
Problem
(Unconstrained SGP):
The demand (d) of an item is uniform
at the rate of 10 units per month. The set-up cost (
As stated, this problem can be
formulated as the unconstrained SGP problem
It is unconstrained signomial
geometric programming problem. The optimal solution is
5.2.
Problem
(Unconstrained MSGP):
In a multi-item (n items) inventory
model, demand of an item is uniform at the rate of
As stated, this problem can be
formulated as the unconstrained MSGP problem
In
particular here we assume n
i |
|
|
|
|
1 |
10 |
20 |
50 |
50 |
2 |
15 |
10 |
125 |
25 |
And the out-put value is
i |
|
|
|
1 |
4.00 |
2.00 |
177.45 |
2 |
3.87 |
1.94 |
5.3.
Problem
(Constrained SGP):
A constrained signomial geometric
programming (SGP) problem is of the following form
It
is constrained signomial geometric programming problem. The optimal solution is
5.4.
Problem
(Constrained SMGP):
A
constrained modified signomial geometric programming (MSGP) problem is of the
following form
It
is constrained MSGP problem. In particular here we assume n
i |
|
|
|
|
1 |
-2 |
1 |
5 |
-2 |
2 |
-1 |
1 |
3 |
And the out-put value is
i |
|
|
|
|
|
1 |
0.408 |
0.004 |
18.57 |
16.00 |
69.09 |
2 |
0.716 |
0.006 |
7.663 |
16.00 |
6. CONCLUSION:
In this paper we have developed
signomial geometric programming (SGP) problem with positive or negative
integral degree of difficulty. Modified form of signomial geometric programming
technique has been demonstrated and some theorems have been derived here. In
this research paper we have developed the technique only in a crisp environment
but in future research of uncertainty in fuzzy non-linear programming (NLP)
model, by using different type of fuzzy numbers (fuzzy coefficients, random
fuzzy number, generalized fuzzy number) such as random fuzzy number,
institutional fuzzy number or adaptive fuzzy demand rate analytically should be
more challenging and interesting.
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