1. INTRODUCTION TO STLPPFI
Linear Programming
Problems LPPs have been important subjects in solving optimization life
problems, and the simplex algorithm method has improved to solve them. This
improvement has led to extending Deterministic Transportation Linear
Programming Problems DTLPP from LPP as a special case of it, and it is one of
the most useful mathematical models. Additionally, DTLPP has three popular
algorithms: the North-West Corner Method, Matrix Minima Cost Method (MMCM) and
Vogel Approximation Method (VAM) for solving these kinds of problems and
finding Basic Feasible Solution (BFS) for DTLPP. However, these methods are not
sufficient to stop, so both algorithm methods, the Stepping Stone Method (SSM) and
Modify Distribution Method (MODI) are used to find optimal solution for DTLPP. Sometimes
the three methods mentioned above that are used to find BFS do not yield the optimal
solution. The mean goals of DTLPP are to minimize the total transport costs,
increasing the amount of transporting goods/objects as possible, increasing
availability/production sources and decreasing non-useful demand/requirement
endpoints as possible by removing unnecessary points and reducing waste points
which does not appear if and only if availability/production sources and demand/requirement
endpoints are balanced since the problem cannot be solved without a balance
condition (Reeb, James Edmund;Leavengood, Scott A, 2002; Winston, Wayne
L;Goldberg, Jeffrey B, 2004; Sharma, 1974; Sengamalaselvi, 2017).
Now, DTLPP steps to
complexity and challenges will be added to it via discovering real-life
problems as fuzziness and randomness of problem which are motivating to search
for more efficient algorithms and program outlines to solve Stochastic
Transportation Linear Programming Problems with Fuzzy Uncertainty Information
on Probability Distribution Space STLPPFI. Where problem formulation has
randomness in objective cost coefficients and fuzziness in linear inequalities
polyhedral sets of information probability distribution space, we focus on
transporting electricity power sector problems which has non-deterministic
values of transporting cost coefficients and non-deterministic values of
creation probability distribution parameters intervals with weight rank in
importance of probability distribution. The (LPP) mathematical formulation
contain a (max/min) linear objective function subject to set of linear
constraint satisfies equations/inequalities, with non-negative unrestricted
variable set. A DTLPP usually contains minimizing linear objective function or
minimizing total transporting cost of shipping objects, subject to both
availability and requirement linear constraints set where total availabilities
satisfy total requirements, parameters set are non-negative (Reeb,
James Edmund;Leavengood, Scott A, 2002; Sengamalaselvi, 2017; Sharma, 1974;
Winston, Wayne L;Goldberg, Jeffrey B, 2004).
An Stochastic
Transportation Linear Programming Problems (STLPP) is a DTLPP when parameters
are random and represented by probability distributions (Abdelaziz,
F Ben;Masri, Hatem, 2005; Abdelaziz, Fouad Ben;Masri, Hatem, 2010; Abdelaziz,
Fouad Ben;Masri, Hatem, 2009; Ameen, 2015; Guo, Haiying;Wang, Xiaosheng;Zhou,
Shaoling, 2015; Hamadameen, Abdulqader Othman;Hassan, Nasruddin, 2018). The
probability distribution space 
of an STLPP in
many cases is unknown, undetermined, and un-specified since it has fuzzy
information, unknown distribution, then should be determined/specified as first
step in solution procedures. Further, an STLPP’s under fuzzy information on
probability and described by fuzzy linear inequalities polyhedral set are
called Stochastic Transportation Linear Programming Problems with Fuzzy
Uncertainty Unknown Information on Probability Distribution Space STLPPFI (A.
Edward Samuel;M. Venkatachalapathy, 2011; Abdelaziz, F Ben;Masri, Hatem, 2005;
Abdelaziz, Fouad Ben;Masri, Hatem, 2009; Abdelaziz, Fouad Ben;Masri, Hatem,
2010; Ameen, 2015; Appati, Justice Kwame;Gogovi, Gideon Kwadwo;Fosu, Gabriel
Obed, 2015; Dharani, K;Selvi, D, 2018; Guo, Haiying;Wang, Xiaosheng;Zhou,
Shaoling, 2015; Mahdavi-Amiri, N;Nasseri, SH, 2006;
Mahdavi-Amiri;NezamNasseri;Seyed Hadi, 2007) and (Sengamalaselvi,
2017; Sakawa, Interactive multiobjective linear programming with fuzzy
parameters, 1993; Sakawa, Fundamentals of fuzzy set theory, 1993).
2. Preliminaries of Fuzzy
Concepts and Polyhedral Set Types
This section reviews some
necessary definitions of fuzzy concepts along with stating two kinds of
polyhedral sets that are related to certainty information on probability
distribution as follows:
2.1 Basic Definitions
2.1.1 Definition of Fuzzy Set: Let 
be a universal
set, 
. 
is called a
fuzzy/non-exact set that contains ordered pairs, 
where 
is membership
function of 
(i.e., a
characteristic/indicator function for that shows to
what degree ), if the
height of fuzzy set is one, then fuzzy set is normal, where the height of a
fuzzy set is the largest membership value attained by any point in the set (Ameen, 2015; Dharani, K;Selvi, D, 2018; Mahdavi-Amiri, N;Nasseri, SH, 2006;
Mahdavi-Amiri;NezamNasseri;Seyed Hadi, 2007; Sakawa, Fundamentals of fuzzy set
theory, 1993; Sakawa, Interactive multiobjective linear programming with fuzzy
parameters, 1993). Formulation (4.2-3) is one of the fuzzy set kinds.
2.1.2 Definition of Alpha-Level Set: The alpha−level
set of fuzzy set is a set 
. The lower and
upper bounds of alpha−level set of fuzzy set are finite
numbers represented by 
respectively (Ameen, 2015; Dharani, K;Selvi, D, 2018).When Formulation (4.2-1) convert to Formulation
(4.2-2) needs alpha-level set to finding efficient points via applying
conditions of fuzzy set, and also used in analyzing cases for second condition
of alpha-cut technique Formulation (4.1-3).
2.1.3
Definition of Convexity of Fuzzy Number: Fuzzy number is a
convex fuzzy set on 
if and only if
its membership function is piecewise continuous, and there exist have three
intervals 
and 
such that is increasing
on 
, equal to 1 on
, decreasing on
, and equal to
0 elsewhere, 
(Ameen, 2015; Dharani, K;Selvi, D, 2018; Mahdavi-Amiri, N;Nasseri, SH, 2006;
Mahdavi-Amiri;NezamNasseri;Seyed Hadi, 2007; Sakawa, Fundamentals of fuzzy set
theory, 1993; Sakawa, Interactive multiobjective linear programming with fuzzy
parameters, 1993). In Figure (4.2-1) and Formulation (4.2-4) shows seven
difference convex fuzzy numbers.
2.1.4 Definition of The Trapezoidal Fuzzy
Number 
: A trapezoidal fuzzy
number 
is 
, where 
is the modal
set of , and 
is the support
part set of (Ameen, 2015; Dharani, K;Selvi, D, 2018; Mahdavi-Amiri, N;Nasseri, SH, 2006;
Mahdavi-Amiri;NezamNasseri;Seyed Hadi, 2007; Sakawa, Fundamentals of fuzzy set
theory, 1993; Sakawa, Interactive multiobjective linear programming with fuzzy
parameters, 1993). The could be
illustrates in following figure:
Figure: 2.1-1:
Trapezoidal Fuzzy Number
Where the linear fuzzy membership
function LFMF for trapezoidal fuzzy number is as
following:
Formulation 2.1-1: The Linear Fuzzy Membership Function
LFMF For
2.1.5 Definition of The Triangle Fuzzy
Number 
: A trapezoidal fuzzy
number is reduced to
the triangular fuzzy number 
and denoted by
, where 
(Ameen, 2015; Dharani, K;Selvi, D, 2018), thus 
. The could be
illustrates as following figure:
Figure : 2.1-2:
Triangular Fuzzy Number
Where the linear fuzzy membership
function LFMF for triangular fuzzy number is as
following:
Formulation 2.1-2: The Linear Fuzzy Membership Function
LFMF For
2.1.6
Definition
of Ranking Function 
: A ranking function 
is a mapping
that transforms each fuzzy number into its corresponding real value in real
line, where a natural order exists (Ameen, 2015; Dharani, K;Selvi, D, 2018).
Formulation (4.2-5) and Formulation (4.2-6) are two kinds of ranking functions.
2.2 The Relating Polyhedral
Sets with Building Uncertainty Unknown Information on Probability Space
Two kinds of information on
probability distribution which are fuzzy and stochastic polyhedral sets are preferred
in this subsection.
2.2.1
The Fuzzy
Polyhedral Set 
: The fuzzy polyhedral
set contains fuzzy uncertainty unknown information on probability distribution
space 
and generated
by fuzzy/inexact inequalities on π which are for each probability 
of a given
event 
, and formed by:
Formulation 2.2-1: The Fuzzy Polyhedral
Set
Where 
and
are
and
dimensions
fixed fuzzy random matrices respectively, and 
was
a fuzzy/inexact inequality and crisp of 
which
meant that 
was
almost equal or less than (Abdelaziz,
F Ben;Masri, Hatem, 2005; Abdelaziz, Fouad Ben;Masri, Hatem, 2010; Abdelaziz,
Fouad Ben;Masri, Hatem, 2009; Ameen, 2015; Dharani, K;Selvi, D, 2018; Guo,
Haiying;Wang, Xiaosheng;Zhou, Shaoling, 2015; Sakawa, Interactive
multiobjective linear programming with fuzzy parameters, 1993; Sakawa,
Fundamentals of fuzzy set theory, 1993; Mahdavi-Amiri;NezamNasseri;Seyed Hadi,
2007; Mahdavi-Amiri, N;Nasseri, SH, 2006) and (Hamadameen, Abdulqader
Othman;Hassan, Nasruddin, 2018; Hamadameen, Abdulqader Othman;Zainuddin, Zaitul
Marlizawati, 2015).
2.2.2 The Stochastic Polyhedral Set 
: Where the information
on probability distribution space was stochastic
on 
and generated
by stochastic inequalities on π and crisp of which are for
each probability of a given
events 
, or converted from
fuzzy to stochastic, then it is called stochastic/default/known polyhedral set
and formed by:
Formulation 2.2-2:
The Stochastic Polyhedral Set
Where and
are
and
dimensions
fixed random matrices respectively (Abdelaziz,
F Ben;Masri, Hatem, 2005; Abdelaziz, Fouad Ben;Masri, Hatem, 2010; Abdelaziz,
Fouad Ben;Masri, Hatem, 2009; Ameen, 2015; Dharani, K;Selvi, D, 2018; Guo,
Haiying;Wang, Xiaosheng;Zhou, Shaoling, 2015; Sakawa, Interactive
multiobjective linear programming with fuzzy parameters, 1993; Sakawa,
Fundamentals of fuzzy set theory, 1993; Mahdavi-Amiri;NezamNasseri;Seyed Hadi,
2007; Mahdavi-Amiri, N;Nasseri, SH, 2006) and (Hamadameen, Abdulqader
Othman;Hassan, Nasruddin, 2018; Hamadameen, Abdulqader Othman;Zainuddin, Zaitul
Marlizawati, 2015).
Therefore, an LPPs with
Formulation (2.2-1) is then called linear programming problem with fuzzy
uncertainty unknown information on probability distribution space LPPFI.
Although, immediately every fuzzy polyhedral set 
Formulation
(2.2-1) should be converted to stochastic polyhedral set Formulation
(2.2-2) via alpha-cut technique approach, then after converting called linear programming
problem with certainty known information on probability distribution space LPP.
3. STLPPFI PROBLEM FORMULATION and ITS MATLAB
CODE Program with Algorithms
This section discusses the
organization of STLPPFI model problem mathematically and data information of
STLPPFI model problem and MATLAB Code Program with its Algorithm as follows:
3.1 The Mathematical Formulation Problem
of STLPPFI
The mathematical
formulation of STLPPFI is shown as follows:
The Stochastic Unique-Objective
Function
Subject to:
With satisfying deterministic
balance condition:
With satisfying domain condition:
Formulation 3.1-1:
The Mathematical Formulation of
STLPPFI
Where stochastic
unique-objective function is optimality condition of LPP, and it is subject to
both of deterministic availability constraints and deterministic requirement
constraints respectively, as well as satisfying deterministic balance condition
and domain condition, where constraints are feasible solution region condition
of LPP, and deterministic balance condition mean that total availability
constraints satisfy total requirements constraints, and domain condition mean
that non-negativity of unknown variable set and belongings of stochastic
parameters to probability distribution space. Where 
are 
known vectors
production values of electricity in Kw/h and 
known vectors
requirement values of using electricity in Kw/h respectively, both 
are crisp and
does not appear scholastically, and 
is probably
estimated cost values of transporting electricity in IQD/Kw and it is not crisp
and appears scholastically should be determined it via both transformations,
and is 
random matrix
as shows in Table (3.1-1) as well as probably estimated cost values for each shows in Table
(3.1-2) respectively, and 
is unknown matrix
should be found it via suitable methods and it is amount of transporting
electricity in Kw/h. So, the Formulation (3.1-1) has fuzzy uncertainty unknown
information in probability distribution space . Depends on (Abdelaziz, Fouad Ben;Aouni, Belaid;El Fayedh, Rimeh, 2007; Abdelaziz, F Ben;Masri, Hatem,
2005; Abdelaziz, Fouad Ben;Masri, Hatem, 2009; Abdelaziz, Fouad Ben;Masri,
Hatem, 2010; Ameen, 2015). Formulation (3.1-1) can be defined in terms of some
probability distribution space , where 
is a discrete
set of events or a finite set of possible states of nature, 
is power set
of 
, and is fuzzy
uncertainty unknown probability distribution space that assigns to each 
probability of
occurrence 
(i.e., is 
matrix of
probabilities 
).
Although, the set is a known
polyhedral set of feasible solutions that includes deterministic constraints of
STLPPFI problem, solving STLPPFI Formulation (3.1-1) first needs to be converted
into DTLPP Formulation (3.2-1). Secondly, finding the set of non-negatives 
that minimize the
objective function, satisfy constraint conditions, balanced condition and
domain condition. Where the data is illustrated in Table (3.1-1) of STLPPFI Formulation (3.1-1), and note that the data in
Table (3.1-1) may be vague or containing
inaccurate values since there might be fuzzy information on probability
distribution or there is not any previous information on probability
distribution, so information and data will be distributed as follows:
Suppose that 
electricity
power production stations named 
and 
with 
cities need to
be supplied with electricity names 
and 
as following
balanced STLPPFI table:
Table: 3.1-1: The Data Distribution Table of STLPPFI
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Power Plants
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Cites
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Supply
Million Kw/h
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Demand
Million Kw/h
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Total =
(v) M
Kw/h
Balanced
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Where prices of transporting costs
are stochastically, and probably estimated cost values of random matrix of will be as
follows:
Table: 3.1-2: The Probably Estimated Cost Values of
Probability Distribution Space
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 IQD
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 IQD
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 IQD
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 IQD
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 IQD
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 IQD
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 IQD
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 IQD
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 IQD
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 IQD
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 IQD
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 IQD
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 IQD
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 IQD
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 IQD
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 IQD
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 IQD
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 IQD
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 IQD
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 IQD
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 IQD
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 IQD
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 IQD
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 IQD
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 IQD
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 IQD
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 IQD
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Where information of response
cities on electricity power plants are fuzzy distributed i.e., the information
probability distribution shown as fuzzy polyhedral set form
(Formulation (2.2-1)). The STLPPFI Formulation (3.1-1) has stochastically
uncertainty unknown expression in its objective function coefficients, and then
it has fuzzily uncertainty unknown expression in its information probability
distribution space . The
uncertainty has randomness for parameters and fuzziness for probability
distributing.
STLPPFI mathematical
formulation was introduced in subsection (3.1). Now standard Deterministic
Transportation Linear Programming Problems DTLPP need to be introduced. Since after
defuzzifying fuzziness of information on probability distribution space of
STLPPFI then STLPPFI convert to STLPP, then after derandomizing
stochastic/randomness of problem formulation of STLPP then immediately STLPP convert
to DTLPP. Now, DTLPP can be formulated and shown as follows:
The Deterministic Unique-Objective
Function
Subject to:
With satisfying deterministic
balance condition:
With satisfying domain condition:
Formulation 3.2-1:
The Mathematical Formulation of
DTLPP
Where the deterministic
unique-objective function is the optimality condition of LPP, and it is subject
to both deterministic availability constraints and deterministic requirement
constraints respectively, as well as satisfying the deterministic balance
condition and the domain condition, where constraints are the feasible solution
region condition of LPP, and the deterministic balance condition mean that the
total availability constraints satisfy the total requirements constraints, and
the domain condition means the non-negativity of unknown variable set. Where 
and 
are known matrices
of deterministic cost values of transporting electricity in IQD/Kw, known vector
production values of electricity in Kw/h, known vector
requirement values of using electricity in Kw/h respectively, and all of them
are deterministic and do not appear scholastically, and is unknown matrix
should be found it using a suitable method such as Dual-Simplex and VAM, and it
represents the amount of transporting electricity in Kw/h.
Although the set is a
polyhedral set of feasible solutions that includes deterministic constraints of
the DTLPP problem, to solve Formulation (3.2-1), we need to find a set of
non-negatives that minimize the
objective function, satisfy constraint conditions, the balanced condition and the
domain condition. The data distributed in Table (3.2-1) of DTLPP Formulation (3.2-1), and note that data in Table
(3.2-1) are most approximately equivalent
values of fuzzy random estimate values before and most approximately equivalent
values to exact values, since we convert fuzzy information on probability
distribution space to known information or we have approximately trust
information on probability distribution space now, so the information
distributed as follows:
Suppose that electricity
power product stations named and with cities need to
be supplied with electricity names and as following
balanced standard DTLPP table:
Table: 3.2-1: The Data Distribution Table of DTLPP
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Power Plants
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Cites
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Supply
Million Kw/h
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Demand
Million Kw/h
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Total =
(v) M
Kw/h
Balanced
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3.3
STLPPFI Algorithm
with its MATLAB Code Program
The STLPPFI algorithm with its
MATLAB code program was introduced in detail and STLPPFI MATLAB program will be
applied to solve illustrate example in section 5.
3.3.1
The
Algorithm Program Outline of STLPPFI
The program outline of
STLPPFI will be as follows:
Input: input fuzzy information on
probability distribution via three vectors 
and 
as credibility
degree of the DM about information on probability distribution space, vagueness
level vector and alpha-cut level vector respectively, then input an acceptance
vector via vector 
from 
matrix in each
row select one value where length of vector is 3/4/5, then
input 
matrix 
dimensions’
matrix as estimates cost values where matrix is 
estimates
distribution matrix with 
deterministic
acceptance vector face, then input 
value and 
value
respectively as length of row and column of deterministic cost values of matrix
will be in
final where should 
i.e., 
divide over 
if it is not
divided then the problem does not have solution, then finally input
availability constraints vector 
and input
requirement constraints vector 
where should
be total availability satisfy total requirement as applying balance condition
on it.
Step 1: Form the problem as Formulation (3.1-1).
Step 2: To transform Formulation (2.2-1) into Formulation (2.2-2)
use truth degrees’ algorithm which contains three algorithms respectively as
follows:
Step 2a: Use Formulation (4.1-3) i.e., (use alpha-cut algorithm) to
obtain alpha-cut technique probability interval from three vectors and as credibility
degree of the DM about information on probability distribution space, vagueness
level vector and alpha-cut level vector respectively to cover and determine
fuzzy uncertainty information on probability distribution.
Step 2b: Build the truth degrees set on probability distribution
via uses dividing each obtain alpha-cut technique probability interval to ten
continuous subintervals or eleven equal distance points in each row of degrees
of truth of fuzzy logical value i.e., (use linspace MATLAB function).
Step 2c: Build fuzzy truth degrees polyhedral set via using linear
fuzzy membership functions LFMF Formulation (4.1-1) and Formulation (4.1-2)
i.e., shown fuzzy truth degrees polyhedral set via both 
and 
in each row as
fuzzy truth degrees regions 
, where 
(1st 3
elements), 
(2nd 4
elements), 
(3rd 3
elements), 
(4th 4 elements),
(5th 3
elements), 
(6th 4
elements), 
(7th 3
elements) of FN matrix, and where fuzzy vector forms contain as 
, and for as 
. Then convert
fuzzy truth degrees polyhedral set to stochastic truth degrees polyhedral set
or deterministic values vector or nine important efficient points in each fuzzy
truth degrees regions after applying linear fuzzy ranking functions LFRF
Formulation (4.2-5) and Formulation (4.2-6) (i.e., use LFRF algorithm) to
deffuzzifier all seven fuzzy numbers column values for all 
in each row
and showing via obtain deterministic matrix values .
Step 3: Test second condition of alpha-cut technique probability
interval Formulation (4.1-3) (i.e., use acceptance P algorithm) which is from
deterministic vector in each row,
we take 
values to
obtain acceptance vectors to check that STLPP Formulation (3.1-1) with
stochastic polyhedral set Formulation (2.2-2) are acceptable for the entire
cases to go to next step or not (where cases are
passes via test 
cases via 
). Then create
logical comparative matrix to find and select each case equal one after , then select
locations of acceptance cases via create acceptance location vector, then
collect 
as
acceptances’ vectors of .
Step 4: Use Formulation (4.3-1) for deterministic acceptance
vectors that passes Step 3 (i.e., use EWS algorithm) to convert STLPP
Formulation (3.1-1) with stochastic polyhedral set Formulation (2.2-2) into
DTLPP Formulation (3.2-1). This step starts by inputting vector as an
acceptance vector that was obtained previously from matrix,
inputting 
dimensions’ estimates
distribution matrix as estimates cost values Table (3.1-2) with deterministic
acceptance vector face, inputting and scalar values
as length of row and column of final obtain deterministic cost value matrix in Table
(3.2-1). Then 
matrix
converts to 
matrix via
applying EWS algorithm process on matrix with dimension
acceptance vector to convert matrix to deterministic
cost value matrix in Table (3.2-1).
Step 5: Solve DTLLP Formulation (3.2-1) times with all
acceptance vectors that passes Step 3 via each step 5a & 5b separately to obtain
initial feasible solution IFS, where Formulation (3.2-1) contains the obtained deterministic
cost matrix in Table (3.2-1) with both 
and 
deterministic
vectors by inputting availability constraints vector and
requirement constraints vector in Table
(3.2-1) where the total availability should satisfy the total requirement.
Step 5a: Solve DTLPP via the Simplex Algorithm Method.
Step 5b: Solve DTLPP via the Vogel Approximation Algorithm Method.
Step 6: Find optimal solution among cases of
acceptance vectors that passes Step 3 for each two methods via the Modify
Distribution Method (MODI).
Step 7: Select the post optimal solution via deciding from
decision maker DM recommendation for a certain case.
Output: Optimal Solution that contain best basic feasible solution
BFS with minimum total cost transporting electricity objective function, with
selection post optimal solution value 
.
The MATLAB code program of
STLPPFI problem solver technique is as follows:
% Alpha-cut technique part
format short; disp('input vector b
as a credibility degree of the DM about information on probability distribution
space'), b=input('b=');disp('input vector d as a vagueness
level'),d=input('d=');disp('input vector a as alpha-cut level'),a=input('a=');
b=b';d=d';a=a';n=length(a);p=zeros(n,2);
if length(b)==length(d) &&
length(b)==length(a) && length(d)==length(a)
disp('The problem has a
solution as follows:')
for k=1:1:n
p(k,[1,2])=[(b(k))-((d(k))*((1-a(k)))),(b(k))+((d(k))*((1-a(k))))];
end
else
disp('There is no solution since
all vectors you inputted are not in the same dimension')
end
if length(b)==length(d) &&
length(b)==length(a) && length(d)==length(a)
disp('The credibility degree
vector b is')
b=b';b
disp('The vagueness levels
vector d is')
d=d';d
disp('The alpha-cut levels
vector a is')
a=a';a
else
disp('please input all vectors
in same dimension')
end
% Truth Degrees technique part
b=size(p);n=b(1);l=zeros(n,11);FN=zeros(n,24);RN=zeros(n,9);TrFN1=0;TpFN2=0;TrFN3=0;TpFN4=0;TrFN5=0;TpFN6=0;TrFN7=0;RN1=0;RN2=0;RN3=0;RN4=0;RN5=0;RN6=0;RN7=0;
if b(2)==2
disp('The solution will be as
following')
for k=1:1:n
l(k,1:11)=linspace(p(k,1),p(k,2),11);
TrFN1(k,1:3)=l(k,[2,1,3]);TpFN2(k,1:4)=l(k,[3,5,2,6]);TrFN3(k,1:3)=l(k,[5,3,6]);TpFN4(k,1:4)=l(k,[5,7,4,8]);TrFN5(k,1:3)=l(k,[7,6,9]);TpFN6(k,1:4)=l(k,[7,9,6,10]);TrFN7(k,1:3)=l(k,[10,9,11]);FN(k,1:24)=[TrFN1(k,:),TpFN2(k,:),TrFN3(k,:),TpFN4(k,:),TrFN5(k,:),TpFN6(k,:),TrFN7(k,:)];RN1(k,1)=(FN(k,1))+((FN(k,3)-FN(k,2))/4);RN2(k,1)=((FN(k,4)+FN(k,5))/2)
+((FN(k,7)-FN(k,6))/4);RN3(k,1)=(FN(k,8))+ ((FN(k,10)-FN(k,9))/4);RN4(k,1)=((FN(k,11)
+FN(k,12))/2)+((FN(k,14)-FN(k,13))/4); RN5(k,1)=(FN(k,15))+((FN(k,17)-FN(k,16))/4);
RN6(k,1)=((FN(k,18)+FN(k,19))/2)+((FN(k,21)-FN(k,20))/4);RN7(k,1)=(FN(k,22))+((FN(k,24)-FN(k,23))/4);RN(k,1:9)=[p(k,1),RN1(k,:),RN2(k,:),RN3(k,:),RN4(k,:),RN5(k,:),RN6(k,:),RN7(k,:),p(k,2)];
end
else
disp('There is no solution since p
is not as a (n,2) dimension matrix')
end
if b(2)==2
disp('The Alpha-Cut Technique
probability intervals of each pi for all i in each row of (n,2) matrix p i.e.,
after applying Fuzzy Transformation of Probability Distribution Space via
Alpha-Cut Technique')
p
disp('The Truth Degrees
Process as follows applies: First, The Alpha-Cut Technique Probability
Intervals of each pi for all i divides to eleven equal distance points in each
row of following Degrees of Truth of fuzzy logical value')
l
disp('Second, Converting
Truth Degrees to both TrFN and TpFN in each row as following fuzzy Truth
Degrees region FN(k,1:24)=[TrFN1, TpFN2,TrFN3,TpFN4,TrFN5,TpFN6,TrFN7] Where
TrFN1(1st 3 elements),TpFN2(2nd 4 elements),TrFN3(3rd 3 elements), TpFN4(4th 4
elements),TrFN5(5th 3 elements),TpFN6(6th 4 elements),TrFN7(7th 3 elements) Where
fuzzy vector forms contain TrFN as (a, alpha, betta), and for TpFN as (a-lower,
a-upper, alpha, betta)')
FN
disp('The deterministic
vector values or nine important power points in each fuzzy Truth Degrees
regions after applying linear fuzzy ranking function LFRF to deffuzzifier fuzzy
for all column values for all i as each row of the following deterministic
values matrix RN')
RN
else
disp('please input p is a
(n,2) dimension matrix')
end
% Acceptance P technique part
kkk=size(RN);
if kkk(1)==3
disp('from deterministic
vector p in each row we take p1 p2 p3 ... pn vectors then we test 9^n cases via
sum(pi)=1 for all pi>0')
p1=RN(1,:);np1=length(p1);p2=RN(2,:);np2=length(p2);p3=RN(3,:);np3=length(p3);p123=zeros(np3,np1,np2);pp123=zeros(np3,np1,np2);
if
length(p1)==length(p2)&& length(p1)==length(p3)&&
length(p2)==length(p3)
for k=1:1:np3
for m=1:1:np1
for n=1:1:np2
p123(m,n,k)=p1(k)+p2(m)+p3(n);p123;
end
end
end
else
disp('There is not have
solution')
end
p123
disp('we find and select each
case equal one after sum(pi)=1 for all pi>0 then we collect pi as acceptance
vector'), pp123=logical(p123==1),[rowpp123,colpp123,volpp123]=find(pp123);rowpp123=rowpp123';colpp123=colpp123';volpp123=volpp123';rowpp123,colpp123,volpp123
else
disp('There is not have
solution since rows of RN more than 3')
end
% EWS technique part
disp('input vector p as an
acceptance vector from RN matrix in each row give one value where length of
vector p is 3/4/5'),p=input('p=');disp('input cs matrix
(mm*nn,3)/(mm*nn,4)/(mm*nn,5) 2-D dimensions matrix as a estimates cost values where
matrix cs is (m,n) distribution matrix with (1,k) stochastic vector face'),cs=input('cs=');
disp('please input mm and nn as what is length of row and column of
deterministic value matrix will be in final respectively Be attention should
(mm*nn)/(kk) i.e., (mm*nn) divide over (kk) if not divided not have solution')
mm=input('mm=');nn=input('nn=');kk=size(cs,2);
if rem(mm*nn,kk)==0
disp('we convert cs matrix (m*n,k)
2D matrix to css (m,k,n) 3D matrix')
if length(p)==3
css=cat(3,cs(1:mm,1:length(p)),cs((mm)+1:2*(mm),1:length(p)),cs((2*(mm))+1:3*(mm),1:length(p)));
elseif length(p)==4
css=cat(4,cs(1:mm,1:length(p)),cs((mm)+1:2*(mm),1:length(p)),cs((2*(mm))+1:3*(mm),1:length(p)),cs((3*(mm))+1:4*(mm),1:length(p)));
elseif length(p)==5
css=cat(5,cs(1:mm,1:length(p)),cs((mm)+1:2*(mm),1:length(p)),cs((2*(mm))+1:3*(mm),1:length(p)),cs((3*(mm))+1:4*(mm),1:length(p)),cs((4*(mm))+1:5*(mm),1:length(p)));
else
disp('There is not have
solution')
end
m=size(css,1);k=size(css,2);n=size(css,3);cd=zeros(m,k,n);css
if length(p)==k
disp('The solution will be as
following')
cd=pagetranspose(pagemtimes(css,p'));
else
disp('There is not have solution')
end
else
disp('There is not have solution')
end
if length(p)==k &&
length(p)==3
disp('The deterministic value
matrix cd is'),cd=[cd(:,:,1);cd(:,:,2);cd(:,:,3)];format long;cd
elseif length(p)==k &&
length(p)==4
disp('The deterministic value
matrix cd is'),cd=[cd(:,:,1);cd(:,:,2);cd(:,:,3);cd(:,:,4)];format long;cd
elseif length(p)==k &&
length(p)==5
disp('The deterministic value
matrix cd is'),cd=[cd(:,:,1);cd(:,:,2);cd(:,:,3);cd(:,:,4);cd(:,:,5)];format
long;cd
else
disp('please input (1,k) vector
p that have same 3rd dimension of cs (m,n,k) 3D matrix')
end
% Solving Deterministic TLPP part
disp('Now, we have deterministic
matrix cd, please input availability vector avb and input requirement vector
req where should be total availability satisfy total requirement')
avb=input('avb=');req=input('req=');
if sum(avb)==sum(req)
disp('The solution where uses dual-simplex
method'),[sol,zval,exitflag,
output]=DTLPPVogel(cd,avb,req);sol=reshape(sol,size(cd,1),size(cd,2));exitflag,
output, sol,format long;zval,disp('The solution where uses Vogel approximation
method'), [ibfs,objCost]=VogelBeModi2(cd,avb,req)
else
disp('There is not have
solution since total availability does not satisfy total requirement or
sum(avb) not equal to sum(req)')
end
Where three DTLPP problem solver MATLAB programs of previous works
are as follows:
function [sol,zval,exitflag,output]=
DTLPPVogel(cost,avb,req)
if sum(avb) ~= sum(req)
...exc=MException('tp:unbalancedProblem', ... 'Cannot solve unbalanced
problem.');
throw(exc);
end
x=optimvar('x',size(cost,1),size(cost,2),'LowerBound',0);z=sum(x.*cost,'all');numCons=size(cost,1)+size(cost,2);cons
= optimconstr(numCons, 1);count = 1;
for i=1:1:size(x,1)
cons(count)=sum(x(i,:))==avb(i);count=count+1;
end
for i = 1:1:size(x, 2)
cons(count)=sum(x(:,i))==req(i);count=count+1;
end
problem=optimproblem('Objective',z,'ObjectiveSense',
'min');problem.Constraints = cons;
show(problem),problem=prob2struct(problem);
[sol,zval,exitflag,output]=linprog(problem);
end
(Appati,
Justice Kwame;Gogovi, Gideon Kwadwo;Fosu, Gabriel Obed, 2015; Dharani, K;Selvi,
D, 2018; Sengamalaselvi, 2017)
function[ibfs,objCost]=VogelModi(data)
% ===DATA PREPARATION===
cost=data(1:end-1,1:end-1);demand=data(end,1:
end-1);supply=data(1:end-1,end)';ibfs=zeros(size(cost));
% ===VOGEL APPROXIMATION METHOD===
ctemp=cost; %temporal cost matrix
while
length(find(demand==0))<length(demand)||length(find(supply==0))<length(supply)
prow=sort(ctemp,1);prow=prow(2,:)-prow(1,:);
%row penalty
pcol=sort(ctemp,2);pcol=pcol(:,2)-pcol(:,1);
%column penalty
[rmax,rind]=max(prow);[cmax,cind]=max(pcol);
if rmax>cmax
[~,mind]=min(ctemp(:,rind));[amt,demand,supply,ctemp]=chkdemandsupply(demand,supply,rind,mind,ctemp);ibfs(mind,rind)=amt;
elseif cmax>= rmax
[~,mind]=min(ctemp(cind,:));[amt,demand,supply,ctemp]=chkdemandsupply(demand,supply,mind,cind,ctemp);ibfs(cind,mind)=amt;
end
end
objCost=sum(sum(ibfs.*cost));
%===MODIFIED DISTRIBUTION ===
val=-1;
while val<0
[prow,pcol]=find(ibfs>0);occupiedCells=[prow,pcol]';[prow,pcol]=find(ibfs==0);unoccupiedCells=[prow,pcol]';r=0;k
=[];
for i =
1:length(occupiedCells(1,:))
ri=occupiedCells(1,i);kj=occupiedCells(2,i);
[r,k]=occupiedSystemSolve(r,k,ri,kj,cost);
end
improvementIndex=zeros(length(unoccupiedCells(1,:)),3);
for i
=1:length(unoccupiedCells(1,:))
ri=unoccupiedCells(1,i);kj=unoccupiedCells(2,i);e=cost(ri,kj)-r(ri)-k(kj);improvementIndex(i
,:)=[ri,kj,e];
end
[val,ind]=min(improvementIndex(:,end));
if val< 0 %check whether
improvement is required
ri=improvementIndex(ind,1);kj=improvementIndex(ind,2);disp(['Create
a circuit around cell ('
num2str(ri) ',' num2str(kj) ')'
]); circuitImproved=[ri,kj,0];n=input('Enter number of element that forms the
circuit: ');
for i = 1:n
nCells = input(['Enter the index
of cell ' num2str(i) ' that forms the circuit: ']);
if mod(i,2) == 0
circuitImproved(i+1,:)=[nCells,ibfs(nCells(1),nCells(2))];
else
circuitImproved(i+1,:)=[nCells,-ibfs(nCells(1),nCells(2))];
end
end
ibfs=reallocateDemand(ibfs,circuitImproved);
disp(ibfs),objCost=sum(sum(ibfs.*cost));
end
end
% ===OTHER REQUIRED
FUNCTIONS===FUNCTION 1===
function
[r,k]=occupiedSystemSolve(r,k,ri,kj,cost)
if length(r)>=ri
k(kj)=cost(ri,kj)-r(ri);
else
r(ri)=cost(ri,kj)-k(kj);
end
% ===FUNCTION 2===
Function [y,demand,supply,ctemp]=chkdemandsupply(demand,supply,ded,sud,ctem)
tempd=demand;temps=supply;
if tempd(ded)>temps(sud)
temps(sud)=0;tempd(ded)=demand(ded)-supply(sud);y=supply(sud);ctem(sud,:)=inf;
elseif tempd(ded)<temps(sud)
tempd(ded)=0;temps(sud)=supply(sud)-demand(ded);y=demand(ded);ctem(:,ded)=inf;
elseif tempd(ded)==temps(sud)
tempd(ded)=0;temps(sud)=0;y=demand(ded);
ctem(:,ded)=inf;ctem(sud,:)=inf;
end
demand=tempd;supply=temps;ctemp =
ctem;
(Sengamalaselvi,
2017; Appati, Justice Kwame;Gogovi, Gideon Kwadwo;Fosu, Gabriel Obed, 2015)
4. Technique Approaches For Solving
STLPPFI
This section describes the
necessary technique approaches for solving STLPPFI. The algorithm program
outlines of those technique approaches and MATLAB code programs of those
technique approaches are stated in appendix section 7. Those algorithms and MATLAB
code programs are used for solving STLPPFI and then converting it to DTLPP followed
bt obtaining the g post optimal solution of STLPPFI with illustrations via
application example of section 5 for each step of the solution process.
The solution process
contains several stages. First stage involves fuzzy transformation on
probability distribution space via alpha-cut technique approach applies, which
is defuzzification of fuzzy on probability distribution space of the original
STLPPFI problem. This process converts it to its corresponding equivalent to
Stochastic Transportation Linear Programming Problems with Certainty Known
Information on Probability Distribution Space (STLPP) by creating bounded
interval with unlimited possible known values from unknown probably/stochastic
values. Then the second stage uses the truth degrees technique approach for
creating fuzzy probability subintervals from the obtained interval, then Linear
Fuzzy Membership Functions (LFMF) will be found from fuzzy truth degrees set. This
will be followed by sketching them in one combinational figure for separating
various different fuzzy truth degrees regions. After that, the fuzzy truth
degrees regions will be defuzzified to stochastic truth degrees regions via
Linear Fuzzy Ranking Function (LFRF) to get finite discrete determined value
set. Then cases via testing second condition of alpha-cut technique polyhedral
set are analyzed to obtain acceptances’ vectors for the preparation of applying
stochastic transformation. The third stage is stochastic transformation of
formulation problem, which is derandomization of probably randomness value set
of random variables towards its corresponding equivalent deterministic values,
where stochastic transformation of objective function via EWS technique
approach applies to transforming STLPP to DTLPP. Then the fourth stage is solving
obtained DTLPP via Dual-Simplex Algorithm Method and VAM, then finding optimal
solution via the Modify Distribution Method (MODI), where the optimal solution of
an DTLPP model problems has the minimum objective function value (i.e., have
minimum objective total transportation costs). Finally, selecting post optimal
solution as a final result by deciding commands from decision makers (DM) among
the entire exit intervals for a certain case or analyzing results by answering
what is the perfect solution of STLPPFI entirely in a certain case.
4.1 Fuzzy Transformation on Probability
Distribution Space via Alpha-Cut Technique
The first transforming on
an STLPPFI Formulation (3.1-1) with Formulation (2.2-1) to STLPP Formulation
(3.1-1) with Formulation (2.2-2) involves transforming of the third component
of probability distribution space from fuzzy
uncertainty unknown information on probability distribution space that is
generated by fuzzy/inexact inequalities on and crisp of to stochastic
inequalities on and crisp of i.e., is in
Formulation (2.2-1) fuzzy polyhedral set 
. Then it should
be converted to Formulation (2.2-2) stochastic/default/known polyhedral set or default
probability distribution space which is the probabilities generated by
stochastic inequalities on via using
alpha-cut technique approach (Abdelaziz, 2012; Abdelaziz, F Ben;Masri, Hatem,
2005; Ameen, 2015). In general alpha-cut technique works on polyhedral sets to
convert them from probably estimated unknown values to bounded interval with
unlimited possible determined known values to obtain STLPP. Now, all fuzzy
inequalities 
of fuzzy
polyhedral set Formulation (2.2-1) could be shown as a Linear Fuzzy Membership
Function (LFMF) 
for and as following
two LFMF’s:
Formulation 4.1-1: The 
Linear
Fuzzy Membership Function (RLFMF)
Formulation 4.1-2:
The Linear
Fuzzy Membership Function (PLFMF)
Where 
is
a probability distribution space vector, 
is
a vagueness level vector and it is used with each value of exceeding
should
be neglected (Ameen, 2015), alpha-cut level vectors are 
and
the credibility degree of DM about information on probability distribution of 
are
around determinated values respectively i.e., 
then
we obtain 
.
The following alpha-cut technique will be applying for each fuzzy inequalities
in Formulation (2.2-1) as follows:
Formulation 4.1-3:
The Alpha-Cut Technique Formula
Therefore, immediately after applying fuzzy transformation
on probability distribution space by transferring polyhedral set from fuzzy
polyhedral set (2.2-1) to stochastic polyhedral set (2.2-2), we could apply the
truth degrees technique approach on probability distribution space, then
analyzing cases and finally stochastic transformation of objective function
problem formulation via EWS techniques approach (Ameen, 2015).
4.1.1
The
Alpha-Cut Technique Algorithm Program Outline
The program outline of Alpha-cut technique will be stated
in subsection (7.1.1).
4.1.2
The
Alpha-Cut Technique MATLAB Program
The MATLAB code program of Alpha-Cut technique will be
stated in subsection (7.1.2).
In this subsection as
subsection (4.1) also focuses is on probability distribution space. After
converting fuzzy uncertainty unknown information on probability distribution
space to certainty
known information on probability distribution space, the obtained probability
intervals could be defined as a fuzzy set of the truth degrees technique
approach on probability distribution space, where truth degrees technique improves
from fuzzy set of optimistic and pessimistic in probability distribution space
of researcher (Ameen, 2015). In addition, continuous interval of fuzzy
probability distribution polyhedral set is classified to ten equal subintervals
or eleven elementary equally distanced points, from which seven efficient
points are given via defuzzifying those eleven points by LFRF technique Formulation
(4.2-4) and Formulation (4.2-5) with two boundary points. Then we obtain nine
efficient points and mistake other unnecessary points (i.e., using LFRF
technique algorithm as a part of truth degrees technique algorithm). Note that
the obtained nine points from defuzzifying are different from first eleven
points since those eleven points in the first time were not necessarily to be
effective points but surely the obtained points are efficient ones.
The truth degrees
technique obtained from logic fuzziness of human normal languages via degrees
value of truth in numeric logical answering of a question. For example, when
anyone is asked to give opinions on the expected result of a random subject,
phenomenon, job, routine or health status, then the answers will be fuzzy logic
values, as this question (Are you fine?) the answer set is {1.0 Yes/Perfect,
0.9 Excellent, 0.8 Very Good, 0.7 Good, 0.6 Well, 0.5 Moderate, 0.4 Some, 0.3
Somewhat, 0.2 Little, 0.1 Very Little, 0 No/Bad}. In general, the truth degrees
technique on probability distribution space used for dividing continuous
intervals to deterministic discrete values set of efficient points. So, the
truth degree variable value 
contain terms 
and the series
of those terms are 
(Abdelaziz, F Ben;Masri, Hatem, 2005; Guo, Haiying;Wang, Xiaosheng;Zhou, Shaoling, 2015;
Ameen, 2015; Ben Abdelaziz, F;Masmoudi, Meryem, 2012; Dharani, K;Selvi, D,
2018; Hamadameen, Abdulqader Othman;Zainuddin, Zaitul Marlizawati, 2015;
Hamadameen, Abdulqader Othman;Hassan, Nasruddin, 2018). Then splitting each
continuous interval of the default probability distribution in (2.2-2)
into ten continuous subintervals as shown as following:
Where 
, with 
, and
Formulation 4.2-1:
The Splitting Original Interval of Probability Distribution Space
Such that those ten continuous
subintervals classified into seven stochastic truth degrees regions:
Formulation 4.2-2:
Stochastic Truth Degrees Regions Set
Where 
are Little,
Some, Moderate, Well, Good, Very Good and Excellent respectively, and also 
and 
are
pessimistic/fail region and optimistic/pass region probability distribution
respectively, and , as shown in
the following Figure (4.2-1).
When 
then we obtain
fuzzy truth degrees set of that could be
shown as:
Formulation 4.2-3:
Fuzzy Truth Degrees Regions Set
Note that fuzzy truth degrees set
Formulation (4.2-3) should be converted to stochastic truth degrees set
Formulation (4.2-2) via linear fuzzy ranking function LFRF technique
Formulation (4.2-5) and Formulation (4.2-6).
Figure: 4.2-1: Fuzzy Truth Degree Regions Set of Probability
Distribution Space 
The 
are linear
fuzzy membership functions LFMFs for entire both types trapezoidal fuzzy number
regions and
triangular fuzzy number regions of
fuzzy truth degrees set 
of each respectively
on information of probability distribution space will be as follows that
defined in and (Abdelaziz, F Ben;Masri, Hatem, 2005; Abdelaziz, Fouad Ben;Masri, Hatem, 2009; Ameen,
2015) were obtained from two Formulations (4.1-1) and (4.1-2):
Formulation 4.2-4:
The 
Linear
Fuzzy Membership Functions
Since LFRF is used as a technical
tool of fuzzy transformation of problem formulation, and also it is used as
particular step process of the truth degrees technique on probability
distribution to deffuzzifier fuzzy truth degrees intervals to stochastic truth
degrees intervals, then via LFRF technique finite deterministic bounded
discreate values set are obtained. In addition, to achieve fuzzy transformation,
LFRF to defuzzify fuzzy coefficients and parameters in Fuzzy Programing
Problems (FPP) is used. So, one of the most useful ranking functions proposed
by various researchers is (Mahdavi-Amiri, N;Nasseri, SH, 2006;
Mahdavi-Amiri;NezamNasseri;Seyed Hadi, 2007) since it could use for entire
types of because
immediately after transforming fuzzy numbers to real number, its value will
remain in the same interval and could be shown as average of fuzzy number
components (Ameen, 2015). Therefore, LFRF uses in converting intervals of truth
degrees to power useful important efficient points of its intervals.
For a 
, we could
formulate 
as following:
Formulation 4.2-5:
The Linear Fuzzy Ranking Function Technique Formula For 
Also, for a 
, we could
formulate as following:
Formulation 4.2-6:
The Linear Fuzzy Ranking Function Technique Formula For
Now, by using the above linear
fuzzy ranking function LFRF technique Formulation (4.2-4) and Formulation
(4.2-5) we defuzzifying those linear fuzzy
membership functions LFMF Formulation (4.2-3) for obtaining deterministic
discrete values for each .
4.2.1
The Ranking
Function Technique Program Outline
The program outline of linear fuzzy ranking function LFRF technique approach will be stated in subsection (7.1.3).
4.2.2
The Ranking
Function Technique MATLAB Program
The MATLAB Program of linear fuzzy ranking function LFRF technique approach will be stated in subsection (7.1.4).
4.2.3
The Truth
Degrees Technique Program Outline
The program
outline of truth degrees technique approach will be stated in subsection (7.1.5).
4.2.4
The Truth
Degrees Technique MATLAB Program:
The MATLAB Program of the
truth degrees technique approach will be
stated in subsection (7.1.6).
4.2.5
Analyzing
Cases: Analyzing cases starts after all of of probability
distribution space vector convert to crisp values, and then each caseq that
does not apply 
of Formulation
(4.1-3) should be neglected to get acceptance to stochastic transformation.
4.2.6
The
Analyzing Cases Test Program Outline
The program
outline of Analyzing Cases Test will be stated
in subsection (7.1.7).
4.2.7
The
Analyzing Cases Test MATLAB Program
The MATLAB code
program of analysing cases test will be stated
in subsection (7.1.8).
4.3 Stochastic Transformation of Objective
Function Problem Formulation via The Expectation Weighted Summation EWS
Technique
The stochastic
transformation of objective functions of Formulation (3.1-1) via expected
weighted summation EWS technique approach on random objective coefficients
applies to convert STLPP to DTLPP, for a probability distribution space 
, then
stochastic transformation of objective function coefficients as following
applies:
Formulation 4.3-1:
The Expected Weighted Summation EWS Technique Procedure
4.3.1
The EWS
Technique Program Outline
The program outline of
expected weighted summation EWS technique will be stated in
subsection (7.1.9).
4.3.2
The EWS
Technique MATLAB Program
The MATLAB Program of the expected
weighted summation EWS technique will
be stated in subsection (7.1.10).
4.4
Finding Basic
Feasible Solution BFS for DTLPP via Dual-Simplex and Vogel Algorithm
When STLPPFI is converted
to DTLPP, its Basic Feasible Solution BFS can be found via various methods such
as North-West Corner Method, MMCM, Dual-Simplex Method and VAM. So, Dual-Simplex
with VAM are used, and both methods will be illustrated in the following
real-problem example (Reeb, James Edmund;Leavengood, Scott A, 2002; Winston,
Wayne L;Goldberg, Jeffrey B, 2004; Sharma, 1974; Sengamalaselvi, 2017). Where
algorithms of both methods are stated in a part of STLPPFI problem solver
MATLAB code program.
4.5
Finding Optimal
Solution for DTLPP via Modify Distribution Method MODI
When Basic Feasible
Solution BFS for DTLLP is found via Dual-Simplex and VAM, then Stepping Stone
Method SSM and Modify Distribution Method MODI are used to check if BFS is
optimal or not. Both methods are used to find the optimal solution when BFS is
not optimal. MODI is recommended for use in such cases, and it will be illustrated
in the following real-problem example as well (Reeb, James Edmund;Leavengood, Scott A, 2002; Winston, Wayne L;Goldberg, Jeffrey B, 2004;
Sharma, 1974; Sengamalaselvi, 2017). MODI algorithms are stated in a part of
STLPPFI problem solver MATLAB code program.
Selecting post optimal solution as
a final result via deciding commands from decision makers (DM) among the entire
exit intervals for a certain case or analyzing results by answering what
perfect solution is entirely in certain cases.
5. The Real-Life Application Problem
Example Using MATLAB
5.1
Illustrate Example
Suppose that three
electricity power production stations in Kurdistan region-Iraq namely: G1, G2
and G3 with four cities need to be supplied with electricity namely Hawler City,
Duhok City, Sulaymani City and Halabja City (Government, 2020; TV, 2021) as
following balanced STLPPFI problem table:
Table : 5.1-1: The Data Distribution Table of an
Electricity STLPPFI Real-Life Problem
|
Power Plants
|
Cites
|
Supply
Million Kw/h
|
|
HR
|
DK
|
SI
|
HA
|
|
|
|
|
|
|
36
|
|
|
|
|
|
|
51
|
|
|
|
|
|
|
42
|
|
Demand
Million Kw/h
|
46
|
22
|
31
|
30
|
Total =
129 M
Kw/h
Balanced
|
Where prices of transporting costs
are stochastically, and estimated cost values of random matrix of each will be (Government, 2020; TV, 2021) as follows:
Table: 5.1-2: The Probably Estimated Cost Values of
Probability Distribution Space
|
|
|
|
|
|
|
7.980 IQD
|
7.990 IQD
|
8.000 IQD
|
|
|
6.000 IQD
|
5.990 IQD
|
5.980 IQD
|
|
|
9.990 IQD
|
9.980 IQD
|
10.00 IQD
|
|
|
8.880 IQD
|
8.890 IQD
|
9.000 IQD
|
|
|
9.000 IQD
|
8.980 IQD
|
8.960 IQD
|
|
|
11.98 IQD
|
11.96 IQD
|
12.00 IQD
|
|
|
13.00 IQD
|
12.98 IQD
|
12.96 IQD
|
|
|
6.980 IQD
|
6.960 IQD
|
7.000 IQD
|
|
|
13.96 IQD
|
13.98 IQD
|
14.00 IQD
|
|
|
9.000 IQD
|
8.980 IQD
|
8.960 IQD
|
|
|
15.96 IQD
|
15.98 IQD
|
16.00 IQD
|
|
|
4.990 IQD
|
5.000 IQD
|
4.980 IQD
|
Where information regarding
the response of cities on electricity power plants is distributed as fuzzy
polyhedral set of uncertainty
unknown information on probability distribution (Government,
2020; TV, 2021) as follows:
Formulation 5.1-1: The
Fuzzy Polyhedral Information Set 
of Real-Life Problem
5.2 Solution Process
The problem could
formulate as follows: Objective Function:
Subject to both availability and
requirement constraints
Where balanced condition with
domain condition satisfies respectively
Formulation 5.2-1: The
STLPPFI Real-Life Application Problem
The problem has uncertainty
unknown expression in both randomness for objective function coefficients and
fuzziness for information probability distribution space . The STLPPFI
solution’s process starts from deffuzzifier fuzziness and derandomizing
randomness of it respectively via two main transformations followed by analyzing
cases, solving using Dual-Simplex and VAM, finding the optimal solution via
MODI and selecting post optimal solution. Where are crisp and
do not appear scholastically and they are 
and 
known vectors
as shown in Table (5.1-1) respectively, and 
is not crisp
and appears scholastically which needs to be determined via suitable
transformations, and is 
random matrix
as shown in Table (5.1-1) with estimated probably values for each shown in Table
(5.1-2)The matrix is unknown matrix
should be determined via Dual-Simplex and VAM, under fuzzy uncertainty unknown
information on probability distribution . The
Formulation (5.2-1) is defined in terms of some probability distribution space , where 
is a discrete
set of events or a finite set of possible states of nature, is power set
of , and is fuzzy
uncertainty unknown information on probability distribution space. Where that assigns to
each is the
probability of occurrence (i.e., is the 
matrix of
probabilities 
), with fuzzy
distribution information of response cities on electricity power plants are as
fuzzy polyhedral set Formulation
(5.1-1). Also, the set is a
polyhedral set of feasible solutions that includes determined constraints of
problem on probability distribution space . To solve
Formulation (5.2-1) we need to find a set of non-negatives that minimize
objective function, satisfies constraints, balances condition and domain
conditions.
Now, the solution process classified
over steps/stations as follows:
5.2.1
Station 1:
Suppositions of Stochastics: Suppose that
the credibility degree of DM about information on probability distribution of 
are around 
respectively,
where 
which is mean 
, since the
information on probability distribution are fuzzy. So, the suppositions have
been started, where the vagueness levels are 
and the
alpha-cut levels are 
. Note that the
vagueness levels and the alpha-cut levels control the length of interval in
each allowing for
their reuse 
-times as necessary
to obtain most close approximate values from exact values till we discover
which values have efficient on solution process results.
5.2.2
Station 2:
Fuzzy Transformation on Probability Distribution Space: Fuzzy
transformation on probability distribution will be applied via alpha-cut
technique Formulation (4.1-3) for each fuzzy inequalities polyhedral set
Formulation (5.1-1) as follows:
Then convert
Formulation (5.2-1) from STLPPFI to STLPP via apply above alpha-cut formula for
each 
, we
get:
Now, the fuzzy
transformation on the probability distribution space via alpha-cut technique is
applied, in which the fuzziness of probability distribution of STLPPFI is removed,
and STLPPFI transfers to STLPP by creating bounded interval with unlimited
possible known values from unknown probably/stochastic value i.e., 
converted
from stochastic values 
to
bounded interval with unlimited possible known values 
as 
,
. Then
fuzzy polyhedral set Formulation (5.1-1) of STLPPFI Formulation (5.2-1) convert
to stochastic polyhedral version set as follows:
5.2.3
Station
3: The Truth Degrees Technique on Probability Distribution Intervals: The current
step will be converting each bounded interval with unlimited possible known
values to bounded discrete finite possible known values set via truth degrees
technique approach, to choose the best effective values for each in each
interval we use truth degrees logical values. First, we divide each interval
into ten parts as follows:
5.2.4
Station 4:
Finding Modulus/Membership Function of Truth Degrees Set: Formulation
(4.2-3) with Figure (4.2-1) will be used to fuzzifier truth degrees set and to
find seven LFMF for each as follows:
Now, both kinds of fuzzy numbers are obtaining from above LFMFs’
of truth degrees set as follows:
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
Now, Linear Fuzzy Ranking Function LFRF Formulation (4.2-4) and
Formulation (4.2-5) will be used to defuzzify fuzzily above fuzzy numbers of
LFMFs’ of truth degrees set as follows:
Finally, nine efficient points are obtained for each in
each interval as follows:
5.2.5
Station 5:
Analyzing Cases: We have 9
cases for each , then totally obtain
cases to check.
So, for the current example 
cases are
existed. Now, to check all cases to state condition 
of alpha-cut technique as follows:
For case (1,1,1): let 
then
and
,
that is failed.
For case (7,8,8): let 
then
and
,
that is pass and so on for entire other cases. So, all of them failed except 3
cases which are 
,
and
are
passes as follows:
5.2.6
Station 6:
Stochastic Transformation of Objective Coefficients via Expected Weighted
Summation Technique Approach: We have 3 acceptance cases to apply stochastic
transformation of objective coefficients on it to convert from STLPP to DTLPP
by applying Formulation (4.3-1) on Table (5.1-2) with entire 3 accepted cases
of 
. The expectation weighted summation Formulation (4.3-1)
for each three above probability spaces 
, then
transformation of objective coefficients will be as follows:
5.2.7 Station 7: Form Problem as DTLPP
Standard Version: DTLPP is
obtained for 
as following
table:
Table 5.2-1: The Obtain DTLPP Data Distribution Table of an
Electricity STLPPFI Real-Life Problem for Case (7,8,8)
|
Power Plants
|
Cites
|
Supply
Million Kw/h
|
|
HR
|
DK
|
SI
|
HA
|
|
|
7.98625
|
5.99375
|
9.989
|
8.90375
|
36
|
|
|
8.9875
|
11.978
|
12.9875
|
6.978
|
51
|
|
|
13.9725
|
8.9875
|
15.9725
|
4.991
|
42
|
|
Demand
Million Kw/h
|
46
|
22
|
31
|
30
|
Total =
129 M
Kw/h
Balanced
|
5.2.8
Station 8:
Finding BFS for DTLPP via VAM Method: VAM will be applied to
find BFS for the above DTLPP via the difference of both least minimum cost in
each row and column of matrix as penalty
values, then enter maximum penalty in row or column, then select minimum cost
value cell for this maximum penalty value in row or column reduce the same
value for other costs that is not enter, repeat this till the end to get BFS (Reeb, James Edmund;Leavengood, Scott A, 2002; Winston, Wayne L;Goldberg, Jeffrey B, 2004;
Sharma, 1974; Sengamalaselvi, 2017; Karagul, Kenan;Sahin, Yusuf, 2020) as
follows:
Table 5.2-2: The DLPP Solution Steps via VAM to Get BFS for
Case (7,8,8)
|
7.98625
|
5.99375&10
|
9.989&26
|
8.90375
|
36//26//0
|
1.9925//2.00275
|
|
8.9875&46
|
11.978
|
12.9875&5
|
6.978
|
51//5//0
|
2.0095//2.9905//4
|
|
13.9725
|
8.9875&12
|
15.9725
|
4.991&30
|
42//12//0
|
3.9965//4.985//0
|
|
46//0
|
22//10//0
|
31//5//0
|
30//0
|
129
|
----
|
|
1.00125//0
|
2.99375//5.98425//0
|
2.9985
|
1.987//0
|
-----
|
Penalty values
|
Therefore, BFS is obtained based
on (Reeb, James Edmund;Leavengood, Scott A, 2002; Winston, Wayne L;Goldberg,
Jeffrey B, 2004; Sharma, 1974; Sengamalaselvi, 2017) as follows:
Table 5.2-3: The BFS Table of An Electricity STLPPFI Real-Life
Problem for Case (7,8,8)
|
0
|
10
|
26
|
0
|
|
46
|
0
|
5
|
0
|
|
0
|
12
|
0
|
30
|
5.2.9
Station 9:
Finding Optimal Solution for DTLPP via MODI Method:
The current solution is optimal, so the total
transporting cost is as follows:
Now, MODI is not necessary to be used
to find the optimal solution since immediately BFS is optimal solution based on
(Reeb, James Edmund;Leavengood, Scott A, 2002; Winston, Wayne L;Goldberg,
Jeffrey B, 2004; Sharma, 1974; Sengamalaselvi, 2017).
Now, via using MATLAB code program
to solve the above example this process is explained in MATLAB as follows:
· Run STLPPFI problem solver MATLAB program
in Editor window of MATLAB Application.
· In command window: input fuzzy information
on probability distribution via three vectors and
as
credibility degree of the DM
about information on probability distribution space, vagueness level vector and
alpha-cut level vector respectively, then input an acceptance vector via vector
from
matrix
in each row select one value where length of vector is
3/4/5, then input matrix
dimensions’
matrix as estimates cost values where matrix is
estimates
distribution matrix with deterministic
acceptance vector face, then input value
and value
respectively as length of row and column of deterministic cost values of matrix
will
be in final where should i.e.,
divide
over if
it is not divided then the problem does not have solution, then finally input
availability constraints vector and
input requirement constraints vector where
should be total availability satisfy total requirement.
· In command window: The solution will be
shown optimal solution that contains the best BFS with minimum total cost
transporting electricity objective function, with selection post optimal
solution value.
Command Window
input vector b as a
credibility degree of the DM about information on probability distribution
space, input vector d as a vagueness level, input vector a as alpha-cut level
respectively
b=[1/2 1/5 1/10];d=[1/6
1/6 1/6];a=[1/2 1/2 1/2];
The problem has
solution as follows: The Alpha-Cut Technique probability intervals of each pi
for all i in each row of (n,2) matrix p after applying Fuzzy Transformation of
Probability Distribution Space via Alpha-Cut Technique
p=
0.4167 0.5833
0.1167 0.2833
0.0167 0.1833
The Truth Degrees process
is applied as follows: First, Alpha-Cut technique probability intervals of each
pi divides to eleven equals distance points in each row of following Degrees of
Truth of fuzzy logical value
l=
0.4167 0.4333 0.4500
0.4667 0.4833 0.5000 0.5167 0.5333 0.5500 0.5667 0.5833
0.1167 0.1333 0.1500
0.1667 0.1833 0.2000 0.2167 0.2333 0.2500 0.2667 0.2833
0.0167 0.0333 0.0500
0.0667 0.0833 0.1000 0.1167 0.1333 0.1500 0.1667 0.1833
Second, converting
Truth Degrees to both TrFN and TpFN in each row as follow: fuzzy Truth Degrees region
FN(k,1:24)=[TrFN1, TpFN2, TrFN3, TpFN4, TrFN5, TpFN6, TrFN7] Where TrFN1(1st 3
elements), TpFN2(2nd 4 elements), TrFN3(3rd 3 elements), TpFN4(4th 4 elements),
TrFN5(5th 3 elements), TpFN6(6th 4 elements), TrFN7(7th 3 elements). Where
fuzzy vector forms contain TrFN as (a, alpha, betta), and for TpFN as (a-lower,
a-upper, alpha, betta)
FN=
Columns 1 through 12
0.4333 0.4167 0.4500
0.4500 0.4833 0.4333 0.5000 0.4833 0.4500 0.5000 0.4833 0.5167
0.1333 0.1167 0.1500
0.1500 0.1833 0.1333 0.2000 0.1833 0.1500 0.2000 0.1833 0.2167
0.0333 0.0167 0.0500
0.0500 0.0833 0.0333 0.1000 0.0833 0.0500 0.1000 0.0833 0.1167
Columns 13 through 24
0.4667 0.5333 0.5167
0.5000 0.5500 0.5167 0.5500 0.5000 0.5667 0.5667 0.5500 0.5833
0.1667 0.2333 0.2167
0.2000 0.2500 0.2167 0.2500 0.2000 0.2667 0.2667 0.2500 0.2833
0.0667 0.1333 0.1167
0.1000 0.1500 0.1167 0.1500 0.1000 0.1667 0.1667 0.1500 0.1833
The deterministic
vector values or nine importance power points in each fuzzy truth degrees regions
after applying LFRF to deffuzzifier fuzzy for all column values for all i as
each row of following deterministic values RN matrix
RN=
0.4167 0.4417 0.4833
0.4958 0.5167 0.5292 0.5500 0.5750 0.5833
0.1167 0.1417 0.1833
0.1958 0.2167 0.2292 0.2500 0.2750 0.2833
0.0167 0.0417 0.0833
0.0958 0.1167 0.1292 0.1500 0.1750 0.1833
from deterministic
vector p in each row, we take p1 p2 p3 ... pn vectors then we test 9^n cases
via sum(pi)=1 for all pi>0
p123(:,:,1)=
0.5500 0.5750 0.6167
0.6292 0.6500 0.6625 0.6833 0.7083 0.7167
0.5750 0.6000 0.6417
0.6542 0.6750 0.6875 0.7083 0.7333 0.7417
0.6167 0.6417 0.6833
0.6958 0.7167 0.7292 0.7500 0.7750 0.7833
0.6292 0.6542 0.6958
0.7083 0.7292 0.7417 0.7625 0.7875 0.7958
0.6500 0.6750 0.7167
0.7292 0.7500 0.7625 0.7833 0.8083 0.8167
0.6625 0.6875 0.7292
0.7417 0.7625 0.7750 0.7958 0.8208 0.8292
0.6833 0.7083 0.7500
0.7625 0.7833 0.7958 0.8167 0.8417 0.8500
0.7083 0.7333 0.7750
0.7875 0.8083 0.8208 0.8417 0.8667 0.8750
0.7167 0.7417 0.7833
0.7958 0.8167 0.8292 0.8500 0.8750 0.8833
p123(:,:,2)=
0.5750 0.6000 0.6417
0.6542 0.6750 0.6875 0.7083 0.7333 0.7417
0.6000 0.6250 0.6667
0.6792 0.7000 0.7125 0.7333 0.7583 0.7667
0.6417 0.6667 0.7083
0.7208 0.7417 0.7542 0.7750 0.8000 0.8083
0.6542 0.6792 0.7208
0.7333 0.7542 0.7667 0.7875 0.8125 0.8208
0.6750 0.7000 0.7417
0.7542 0.7750 0.7875 0.8083 0.8333 0.8417
0.6875 0.7125 0.7542
0.7667 0.7875 0.8000 0.8208 0.8458 0.8542
0.7083 0.7333 0.7750
0.7875 0.8083 0.8208 0.8417 0.8667 0.8750
0.7333 0.7583 0.8000
0.8125 0.8333 0.8458 0.8667 0.8917 0.9000
0.7417 0.7667 0.8083
0.8208 0.8417 0.8542 0.8750 0.9000 0.9083
p123(:,:,3)=
0.6167 0.6417 0.6833
0.6958 0.7167 0.7292 0.7500 0.7750 0.7833
0.6417 0.6667 0.7083
0.7208 0.7417 0.7542 0.7750 0.8000 0.8083
0.6833 0.7083 0.7500
0.7625 0.7833 0.7958 0.8167 0.8417 0.8500
0.6958 0.7208 0.7625
0.7750 0.7958 0.8083 0.8292 0.8542 0.8625
0.7167 0.7417 0.7833
0.7958 0.8167 0.8292 0.8500 0.8750 0.8833
0.7292 0.7542 0.7958
0.8083 0.8292 0.8417 0.8625 0.8875 0.8958
0.7500 0.7750 0.8167
0.8292 0.8500 0.8625 0.8833 0.9083 0.9167
0.7750 0.8000 0.8417
0.8542 0.8750 0.8875 0.9083 0.9333 0.9417
0.7833 0.8083 0.8500
0.8625 0.8833 0.8958 0.9167 0.9417 0.9500
p123(:,:,4)=
0.6292 0.6542 0.6958
0.7083 0.7292 0.7417 0.7625 0.7875 0.7958
0.6542 0.6792 0.7208
0.7333 0.7542 0.7667 0.7875 0.8125 0.8208
0.6958 0.7208 0.7625
0.7750 0.7958 0.8083 0.8292 0.8542 0.8625
0.7083 0.7333 0.7750
0.7875 0.8083 0.8208 0.8417 0.8667 0.8750
0.7292 0.7542 0.7958
0.8083 0.8292 0.8417 0.8625 0.8875 0.8958
0.7417 0.7667 0.8083
0.8208 0.8417 0.8542 0.8750 0.9000 0.9083
0.7625 0.7875 0.8292
0.8417 0.8625 0.8750 0.8958 0.9208 0.9292
0.7875 0.8125 0.8542
0.8667 0.8875 0.9000 0.9208 0.9458 0.9542
0.7958 0.8208 0.8625
0.8750 0.8958 0.9083 0.9292 0.9542 0.9625
p123(:,:,5)=
0.6500 0.6750 0.7167
0.7292 0.7500 0.7625 0.7833 0.8083 0.8167
0.6750 0.7000 0.7417
0.7542 0.7750 0.7875 0.8083 0.8333 0.8417
0.7167 0.7417 0.7833
0.7958 0.8167 0.8292 0.8500 0.8750 0.8833
0.7292 0.7542 0.7958
0.8083 0.8292 0.8417 0.8625 0.8875 0.8958
0.7500 0.7750 0.8167
0.8292 0.8500 0.8625 0.8833 0.9083 0.9167
0.7625 0.7875 0.8292
0.8417 0.8625 0.8750 0.8958 0.9208 0.9292
0.7833 0.8083 0.8500
0.8625 0.8833 0.8958 0.9167 0.9417 0.9500
0.8083 0.8333 0.8750
0.8875 0.9083 0.9208 0.9417 0.9667 0.9750
0.8167 0.8417 0.8833
0.8958 0.9167 0.9292 0.9500 0.9750 0.9833
p123(:,:,6)=
0.6625 0.6875 0.7292
0.7417 0.7625 0.7750 0.7958 0.8208 0.8292
0.6875 0.7125 0.7542
0.7667 0.7875 0.8000 0.8208 0.8458 0.8542
0.7292 0.7542 0.7958
0.8083 0.8292 0.8417 0.8625 0.8875 0.8958
0.7417 0.7667 0.8083
0.8208 0.8417 0.8542 0.8750 0.9000 0.9083
0.7625 0.7875 0.8292
0.8417 0.8625 0.8750 0.8958 0.9208 0.9292
0.7750 0.8000 0.8417
0.8542 0.8750 0.8875 0.9083 0.9333 0.9417
0.7958 0.8208 0.8625
0.8750 0.8958 0.9083 0.9292 0.9542 0.9625
0.8208 0.8458 0.8875
0.9000 0.9208 0.9333 0.9542 0.9792 0.9875
0.8292 0.8542 0.8958
0.9083 0.9292 0.9417 0.9625 0.9875 0.9958
p123(:,:,7)=
0.6833 0.7083 0.7500
0.7625 0.7833 0.7958 0.8167 0.8417 0.8500
0.7083 0.7333 0.7750
0.7875 0.8083 0.8208 0.8417 0.8667 0.8750
0.7500 0.7750 0.8167
0.8292 0.8500 0.8625 0.8833 0.9083 0.9167
0.7625 0.7875 0.8292
0.8417 0.8625 0.8750 0.8958 0.9208 0.9292
0.7833 0.8083 0.8500
0.8625 0.8833 0.8958 0.9167 0.9417 0.9500
0.7958 0.8208 0.8625
0.8750 0.8958 0.9083 0.9292 0.9542 0.9625
0.8167 0.8417 0.8833
0.8958 0.9167 0.9292 0.9500 0.9750 0.9833
0.8417 0.8667 0.9083
0.9208 0.9417 0.9542 0.9750 1.0000 1.0083
0.8500 0.8750 0.9167
0.9292 0.9500 0.9625 0.9833 1.0083 1.0167
p123(:,:,8)=
0.7083 0.7333 0.7750
0.7875 0.8083 0.8208 0.8417 0.8667 0.8750
0.7333 0.7583 0.8000
0.8125 0.8333 0.8458 0.8667 0.8917 0.9000
0.7750 0.8000 0.8417
0.8542 0.8750 0.8875 0.9083 0.9333 0.9417
0.7875 0.8125 0.8542
0.8667 0.8875 0.9000 0.9208 0.9458 0.9542
0.8083 0.8333 0.8750
0.8875 0.9083 0.9208 0.9417 0.9667 0.9750
0.8208 0.8458 0.8875
0.9000 0.9208 0.9333 0.9542 0.9792 0.9875
0.8417 0.8667 0.9083
0.9208 0.9417 0.9542 0.9750 1.0000 1.0083
0.8667 0.8917 0.9333
0.9458 0.9667 0.9792 1.0000 1.0250 1.0333
0.8750 0.9000 0.9417
0.9542 0.9750 0.9875 1.0083 1.0333 1.0417
p123(:9) =
0.7167 0.7417 0.7833
0.7958 0.8167 0.8292 0.8500 0.8750 0.8833
0.7417 0.7667 0.8083
0.8208 0.8417 0.8542 0.8750 0.9000 0.9083
0.7833 0.8083 0.8500
0.8625 0.8833 0.8958 0.9167 0.9417 0.9500
0.7958 0.8208 0.8625 0.8750
0.8958 0.9083 0.9292 0.9542 0.9625
0.8167 0.8417 0.8833
0.8958 0.9167 0.9292 0.9500 0.9750 0.9833
0.8292 0.8542 0.8958
0.9083 0.9292 0.9417 0.9625 0.9875 0.9958
0.8500 0.8750 0.9167
0.9292 0.9500 0.9625 0.9833 1.0083 1.0167
0.8750 0.9000 0.9417
0.9542 0.9750 0.9875 1.0083 1.0333 1.0417
0.8833 0.9083 0.9500
0.9625 0.9833 0.9958 1.0167 1.0417 1.0500
Now, we need to find
and select each case equal one after sum(pi)=1 for all pi>0 then we collect
pi as acceptance vector 9×9×9 logical array
pp123(:,:,7)= pp123(:,:,8)=
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
rowpp123= colpp123=
volpp123=
8 8 7 62 70
71 1 1 1
input vector p as
deterministic acceptance vector from RN matrix in each row give one value where
length of vector p is 3/4/5
p=[0.55,0.275,0.175]
input cs matrix
(mm*nn,3)/(mm*nn,4)/(mm*nn,5) 2-D dimensions matrix as estimates cost values
where matrix cs is (m,n) distribution matrix with (1,k) deterministic
acceptance vector face.
cs=[7.980,7.990,8.000;6.000,5.990,5.980;9.990,9.980,10.00;8.880,8.890,9.000;9.000,8.980,8.960;11.98,11.96,12.00;13.00,12.98,12.96;6.980,6.960,7.000;13.96,13.98,14.00;9.000,8.980,8.960;15.96,15.98,16.00;4.990,5.000,4.980]
please input mm and nn
as length of obtain row and column of deterministic value matrix must be in
final respectively. Be attention should (mm*nn)/(kk) since if (mm*nn) does not
divide over (kk) then we do not have solution.
mm=4, nn=3, we convert
cs (m*n,k) 2D matrix to css (m,k,n) 3D matrix
css(:,:,1)=
css(:,:,2)=
7.9800 7.9900 8.0000
9.0000 8.9800 8.9600
6.0000 5.9900 5.9800
11.980 11.960 12.000
9.9900 9.9800 10.000
13.000 12.980 12.960
8.8800 8.8900 9.0000
6.9800 6.9600 7.0000
css(:,:,3)=
13.960 13.980 14.000
9.0000 8.9800 8.9600
15.960 15.980 16.000
4.9900 5.0000 4.9800
The deterministic value
of cd matrix well be
cd=
7.98625 5.99375
9.98900 8.90375
8.98750 11.9780
12.9875 6.97800
13.9725 8.98750
15.9725 4.99100
Now, we have
deterministic cd matrix, please input availability vector avb and input
requirement vector req where should be total availability satisfy total
requirement.
avb=[36 51 42], req=[46
22 31 30]
The solution where uses
Vogel method
Optimization Problem: Solve
for: x
minimize:7.98625*x(1,1)+8.9875*x(2,1)+13.9725*x(3,1)+5.99375*x(1,2)+11.978*x(2,2)+8.9875*x(3,2)+9.989*x(1,3)+12.9875*x(2,3)+15.9725*x(3,3)+8.90375*x(1,4)+6.978*x(2,4)+4.991*x(3,4)
subject to:
x(1,1)+x(1,2)+x(1,3)+x(1,4)==36;x(2,1)+x(2,2)+x(2,3)+x(2,4)==51;x(3,1)+x(3,2)+x(3,3)+x(3,4)==42;x(1,1)+x(2,1)+x(3,1)==46;x(1,2)+x(2,2)+x(3,2)==22;x(1,3)+x(2,3)+x(3,3)==31;x(1,4)+x(2,4)+x(3,4)==30;
variable bounds:
0<=x(1,1);0<=x(2,1);0<=x(3,1);0<=x(1,2);0<=x(2,2);0<=x(3,2);0<=x(1,3);0<=x(2,3);0<=x(3,3);0<=x(1,4);0<=x(2,4);0<=x(3,4);
Optimal solution found:
BFS=
0 10 26 0
46 0 5 0
0 12 0 30
Zval ObjCost Optimal=
1055.594
5.2.10 Station 10: Selecting Post Optimal
Solution for DTLPP via Deciding from DM: By repeating steps from station 6 tell getting optimal
solution for 
, then optimal
solution is obtained, where the total transporting cost is as follows:
Also, by repeating steps from
station 6 tell getting optimal solution for 
, then optimal
solution is obtained, where the total transporting cost is as follows:
Finally, we have the optimal
solution among three cases which are 
, where the
total transporting cost is 
, where
analytic optimal solution for the above problem via crisp values version is 
.This solution
shows efficiency of STLPPFI algorithm via closing numerical solution from
analytic/exact solution.
Now, to selecting post optimal
solution for a certain amount among obtained solutions will be as follows:
1.
If
importance rank of probability problem is 
then
then
from
,
then DM should select as
a post optimal solution for certain problem.
2.
If
importance rank of probability problem is then
then
from
,
then DM should select 
as
a post optimal solution for certain problem with little additional penalty cost
which is 
.
3.
If
importance rank of probability problem is then
then
from
,
then DM should select 
as
a post optimal solution for certain problem with little additional penalty cost
which is 
.
Note that comparative comments on
both methods Dual-Simplex and VAM is that the Dual-Simplex Method immediately gives
the optimal solution without using MODI method after finding IBFS with more
elapsed time and more iterations and it is a more difficult way in great
problems, where VAM sometimes does not give optimal solution without using MODI
method with less elapsed time and less iterations and it is easy way in great
problems.
Conclusion
In this study, the MATLAB code program with its algorithm program of solving STLPPFI model
problems are proposed. The proposed method to solve the STLPPFI model problems
was based on several stages of solution process. The study considered STLPPFI
models. The method utilized concepts such
as LFMF, , , LFRF,
Alpha-Cut technique on probability distribution, Truth Degrees technique on
probability distribution, EWS technique and analyzing cases via second
condition test of alpha-cut technique. The study showed that STLPPFI model
problem solver is more efficient and has so close solution to analytic/exact
solution. The STLPPFI problem solver is used to convert STLPPFI into its
corresponding equivalent DTLPP via defuzzifying the probability distribution
and derandomization randomness of problem formulation respectively. The solution of numerical example in electricity field was
shown to find the optimal solution of it and selecting post optimality solution
for it. The result is evaluated and supports the entire stated technique
approaches, algorithms, and it supports MATLAB code program of STLPPFI problem
solver. The study showed that the theory was paralell with the algorithm and
its MATLAB program. The MATLAB code programs of both transformations with
followed technique approaches as Alpha-Cut technique, Truth Degrees technique, , , LFMF, LFRF,
EWS technique and analyzing/filtering cases test were simplified process of
finding the optimal solution to STLPPFI, and it issued the emerge of the entire
actual real-life problem situations. The
study illustrates that the proposed method implemented via MATLAB programming
is practical, easy and applicable to practical applications in the fields of
energy and industry. Moreover, the proposed method takes solution of the
problem at the lowest cost, least time running, maximum transporting amounts
and maximum profits.
References
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6. AppendEx
Technique approach algorithm
program outlines with its MATLAB code programs of them:
6.1.1
The
Alpha-Cut Technique Algorithm Program Outline
The program outline of Alpha-cut technique will be as follows:
Input: Fuzzy Probability Polyhedral Set 
Step 1: For 
,
Do Steps 2-6
Step 2: 
,
then define as
a 
Step 3: Fuzzifier in
via
Triangle or Trapezoid Linear Fuzzy Membership Function 
Step 4: Use Alpha-cut technique formula
Step 5: Defuzzifier which
is
stochastic now i.e., 
Step 6: 
Step 7: If 
,
then (Return to Step 1)
Step 8: If 
,
then (End of Process)
Output: Stochastic Probability Polyhedral Set .
6.1.2
The
Alpha-Cut Technique MATLAB Code Program
The MATLAB Program of the alpha-cut technique is as following:
format
short;disp('input vector b as credibility degree of DM about information on probability
distribution space'),b=input ('b=');disp('input vector d as vagueness
level'),d=input('d='); disp(' input vector a as alpha-cut level'), a=input('a=');b=b';d=d';a=a';n=length(a);p=zeros(n,2);
if length(b)==length(d)
&& length(b)==length(a) && length(d)==length(a)
disp('The solution
will be as following')
for k=1:1:n
p(k,[1,2])=[(b(k))-((d(k))*((1-a(k)))),(b(k))+((d(k))*((1-a(k))))];
end
else
disp('There is not have
solution since all vectors are you inputted are not in same dimension')
end
if length(b)==length(d)
&& length(b)==length(a) && length(d)==length(a)
disp('The credibility
degree vector b is'), b=b';b,disp('The vagueness levels vector d is'),
d=d';d,disp('The
alpha-cut levels vector a is'),
a=a';a,disp('The
probability intervals of each pi for all i as each row of following probability
interval matrix p i.e., applying Fuzzy Transformation of Probability
Distribution Space via Alpha-Cut Technique'),p
else
disp('please input
all vectors in same dimension')
end
6.1.3
The Ranking
Function Technique Program Outline
The program outline of linear
fuzzy ranking function LFRF technique approach will be as follows:
Input: Fuzzy Numbers 
Step 1: For ,
Do Steps 2-7
Step 2: Define Fuzzy
Step 3: Fuzzifier 
in via
Parametric Form then Ranking Function 
respectively
Step 4: Use Parametric
Form for Trapezoidal
Or Use Parametric Form
for Triangular
Step 5: Use LFRF
Technique for Trapezoidal
Or Use LFRF Technique
for Triangular
Step 6: Deffuzzifier 
which
is is
deterministic now i.e., 
Step 7:
Step 8: If 
,
then (Return to Step 1)
Step 9: If 
,
then (End of the Process)
Output: Real values of .
6.1.4
The Ranking
Function Technique MATLAB Program
The MATLAB Program of Linear Fuzzy Ranking Function
technique is as following:
format short;disp('input
n * 3 or n * 4 matrix (FN) as a vectors of TpFN or TrFN in each row where each
row of n * 3 matrix (FN) contains (a, alpha, betta) as a vector of TrFN or each
row of n * 4 matrix (FN) contains (aL, aU, alpha, betta) as a vector of
TpFN'),FN=input ('FN=');b=size(FN);n=b(1);RN=zeros(n,1);
if b(2)==3
disp('it is TrFN')
for k=1:1:n
RN(k,1)=(FN(k,1))+((FN(k,3)-FN(k,2))/4);
end
elseif b(2)==4
disp('it is TpFN')
for k=1:1:n
RN(k,1)=((FN(k,1)+FN(k,2))/2)+((FN(k,4)-FN(k,3))/4);
end
else
disp('There is not
have solution since FN is not TpFN nor TrFN')
end
if b(2)==3 || b(2)==4
disp('The fuzzy
number values before converting will be as follows vector/matrix'),FN,disp('The
deterministic real values after converting from fuzzy numbers will be as
follows vector'),RN
else
disp('please input
RN vector/matrix as a TpFN or TrFN, or correct your input matrix into n * 3 or
n * 4 matrix (FN) as a vector of TpFN or TrFN')
end
6.1.5
The Truth
Degrees Technique Program Outline: The program outline of truth degrees technique
will be as follows:
Input: Stochastic
Probability Polyhedral Set
Step 1: For ,
Do Steps 2-8
Step 2: Define each as
a continuous interval 
Step 3: Divide of
each as
ten continuous subintervals 
Step 4: Replace at
each ten subintervals as fuzzy truth degrees regions membership functions 
Step 5: is fuzzy number i.e., 
Step 6: Deffuzzifier via
linear fuzzy ranking function technique formula to get stochastic truth degrees
regions from fuzzy truth degrees regions i.e., 
Step 7: Deffuzzifier i.e.,
or is
deterministic discrete value now
Step 8:
Step 9: If ,
then (Return to Step 1)
Step 10: If ,
then (End of Process)
Output: Real values of .
6.1.6
The Truth
Degrees Technique MATLAB Program
The MATLAB code
program of the truth degrees technique is as follows:
format
short;disp('input (n,2) matrix p as Alpha-Cut Technique probability intervals
of each pi for all i in each row of matrix p after applying Fuzzy
Transformation of Probability Distribution Space via Alpha-Cut
Technique'),p=input('p=');b=size(p);n=b(1);l=zeros(n,11);FN=zeros(n,24);RN=zeros(n,9);TrFN1=0;TpFN2=0;TrFN3=0;TpFN4=0;TrFN5=0;TpFN6=0;TrFN7=0;RN1=0;RN2=0;RN3=0;RN4=0;RN5=0;RN6=0;RN7=0;
if b(2)==2
disp('The solution
will be as following')
for k=1:1:n
l(k,1:11)=linspace(p(k,1),p(k,2),11);TrFN1(k,1:3)=l(k,[2,1,3]);TpFN2(k,1:4)=l(k,[3,5,2,6]);TrFN3(k,1:3)=l(k,[5,3,6]);TpFN4(k,1:4)=l(k,[5,7,4,8]);TrFN5(k,1:3)=l(k,[7,6,9]);TpFN6(k,1:4)=l(k,[7,9,6,10]);TrFN7(k,1:3)=l(k,[10,9,11]);FN(k,1:24)=[TrFN1(k,:),TpFN2(k,:),TrFN3(k,:),TpFN4(k,:),TrFN5(k,:),TpFN6(k,:),TrFN7(k,:)];RN1(k,1)=(FN(k,1))+((FN(k,3)-FN(k,2))/4);RN2(k,1)=
((FN(k,4)+FN(k,5))/2)+((FN(k,7)-FN(k,6))/4);
RN3(k,1)=(FN(k,8))+((FN(k,10)-FN(k,9))/4);
RN4(k,1)=((FN(k,11)+FN(k,12))/2)+((FN(k,14)-FN(k,13))/4);RN5(k,1)=(FN(k,15))+((FN(k,17)-FN(k,16))/4);RN6(k,1)=((FN(k,18)+FN(k,19))/2)+((FN(k,21)-FN(k,20))/4);RN7(k,1)=(FN(k,22))
+((FN(k,24)-FN(k,23))/4);RN(k,1:9)=[p(k,1),
RN1(k,:),RN2(k,:),RN3(k,:),RN4(k,:),RN5(k,:),RN6(k,:),RN7(k,:),p(k,2)];
end
else
disp('There is not have
solution since p is not as a (n,2) dimension matrix')
end
if b(2)==2
disp('The Alpha-Cut
Technique probability intervals of each pi for all i in each row of matrix
p'),p,disp('The Truth Degrees Process as follows applies: First, The Alpha-Cut
Technique Probability Intervals of each pi for all i divides to 11 points in
each row of following Degrees of Truth of fuzzy logical value'),l,
disp('Second, Converting Truth Degrees to both TrFN and TpFN in each row as
following fuzzy Truth Degrees region FN(k,1:24)=[TrFN1,TpFN2,
TrFN3,TpFN4,TrFN5,TpFN6,TrFN7] Where TrFN1(1st 3 elements),TpFN2(2nd 4
elements), TrFN3(3rd 3 elements), TpFN4(4th 4 elements), TrFN5(5th 3 elements),
TpFN6(6th 4 elements), TrFN7(7th 3 elements). Where fuzzy vector forms for TrFN
is (a, alpha, betta),for TpFN is (a-lower, a-upper, alpha,
betta)'),FN,disp('The deterministic vector values or importance power points in
each fuzzy Truth Degrees regions after applying linear fuzzy ranking function
LFRF to deffuzzifier fuzzy for all column values for all i as each row of the
following deterministic values matrix rf'),RN
else
disp('please input p is
a (n,2) dimension matrix')
end
6.1.7
The
Analyzing Cases Test Program Outline
The program
outline of Analyzing Cases Test will be as follows:
Input: Deterministic
matrix RN values or importance efficient points that divides as each row of
matrix RN are deterministic vector values Probability Polyhedral Set .
Step 1: Define 
, and each 
.
Step 2: For , for 
, for 
,
Do Steps 3-8.
Step 3: Do 
, 
of
entire location 
of matrix
.
Step 4: Replace 
at
place of matrix
.
Step 5: Create
comparative matrix
as
logical answer of question is 
or
not, if yes take logical value 1 and if no take logical value 0.
Step 6: Find entire
locations of logical matrix that
have value 1, to getting and finding location of of matrix
,
then separate location to 
and
.
Step 7: Create entire
acceptance vectors 
,
where each 
.
Step 8: if
loop applies,
or 
if
loop applies,
or 
if
loop applies.
Step 9: If and
and
,
then (Return to Step 2).
Step 10: If and
and
,
then (End of Process).
Output: Real values of
acceptance vectors .
6.1.8
The
Analyzing Cases Test MATLAB Program The MATLAB code program of analyzing cases test is as
follows:
format short;disp('input
(3,n) matrix RN as 3 obtained deterministic vectors p in each row take p1 p2 p3
... pn as n-dimensional vectors'),RN=input('RN=');kkk=size(RN);
if kkk(1)==3
disp('from
deterministic vector p in each row take p1 p2 p3 ... pn vectors then we test
9^n cases via sum(pi)=1 for all pi>0'),p1=
RN(1,:);np1=length(p1);p2=RN(2,:);np2=length(p2);p3=RN(3,:);np3=length(p3);p123=zeros(np3,np1,np2);pp123=zeros(np3,np1,np2);
if
length(p1)==length(p2)&&length(p1)
==length(p3)&&length(p2)==length(p3)
for k=1:1:np3
for
m=1:1:np1
for
n=1:1:np2
p123(m,n,k)=p1(k)+p2(m)+p3(n);p123;
end
end
end
else
disp('There is
not have solution')
end
p123,disp('we find and
select each case equal one after sum(pi)=1 for all pi>0 then we collect pi
as acceptance vector'),pp123=logical
(p123==1),[rowpp123,colpp123,volpp123]=find(pp123);rowpp123=rowpp123';colpp123=colpp123';volpp123=volpp123';rowpp123,colpp123,volpp123
else
disp('There is not
have solution since rows of RN more than 3 or less than 3')
end
6.1.9
The EWS
Technique Program Outline
The program
outline of expected weighted summation EWS technique will be as follows:
Input: Stochastic objective
function coefficients equation
Step 1: For ,
Do Steps 2-5
Step 2: Define
stochastic objective coefficients
Step 3: Fuzzifier
stochastic objective coefficients via the EWS approach 
Step 4: Deffuzzifier
stochastic objective coefficients to deterministic
Step 5:
Step 6: If ,
then (Return to Step 1)
Step 7: If ,
then (End of Process)
Output: Deterministic objective
function coefficients equation .
6.1.10
The EWS
Technique MATLAB Program
The MATLAB
Program of the expected weighted summation EWS technique is as following:
format
short;disp('input (1,3)/(1,4)/(1,5) vector p as acceptance probability
distribution vector'),p=input('p=');disp(' input cs matrix
(mm*nn,3)/(mm*nn,4)/(mm*nn,5) 2D matrix as estimate cost values where cs matrix
contains (m,n) distribution matrix with (1,k) acceptance weighted vector
face'),cs=input('cs='); disp('please input mm and nn as what is length of row
and column of deterministic value matrix will be in final respectively Be
attention should (mm*nn)/(kk) i.e., (mm*nn) divided over (kk) if not divided
not have solution'),mm=input ('mm=');nn=input('nn=');kk=size(cs,2);
if rem(mm*nn,kk)==0
disp('we convert cs
matrix (m*n,k) 2D matrix to css (m,k,n) 3D matrix')
if length(p)==3
css=cat(3,cs(1:mm,1:length(p)),cs((mm)+1:2*(mm),1:length(p)),cs((2*(mm))+1:3*(mm),1:length(p)));
elseif length(p)==4
css=cat(4,cs(1:mm,1:length(p)),cs((mm)+1:2*(mm),1:length(p)),cs((2*(mm))+1:3*(mm),1:length(p)),cs((3*(mm))+1:4*(mm),1:length(p)));
elseif length(p)==5
css=cat(5,cs(1:mm,1:length(p)),cs((mm)+1:2*(mm),1:length(p)),cs((2*(mm))+1:3*(mm),1:length(p)),cs((3*(mm))+1:4*(mm),1:length(p)),cs((4*(mm))+1:5*(mm),1:length(p)));
else
disp('There is not
have solution')
end
m=size(css,1);k=size(css,2);n=size(css,3);cd=zeros(m,k,n);css
if length(p)==k
disp('The solution
will be as following')
cd=pagetranspose(pagemtimes(css,p'));
else
disp('There is not have
solution')
end
else
disp('There is not have
solution')
end
if length(p)==k
&& length(p)==3
disp('The deterministic
value matrix cd is'), cd=[cd(:,:,1);cd(:,:,2);cd(:,:,3)];cd
elseif length(p)==k
&& length(p)==4
disp('The deterministic
value matrix cd is'),
cd=[cd(:,:,1);cd(:,:,2);cd(:,:,3);cd(:,:,4)];cd
elseif length(p)==k
&& length(p)==5
disp('The deterministic
value matrix cd is'),
cd=[cd(:,:,1);cd(:,:,2);cd(:,:,3);cd(:,:,4);cd(:,:,5)];cd
else
disp('please input
(1,k) vector p that have same 3rd dimension of cs (m,n,k) 3D matrix')
end
, Triangular
Fuzzy Number 
, Linear Fuzzy
Ranking Function (LFRF), Expectation Weighted Summation technique (EWS) and
analyzing cases via second condition test of alpha-cut technique. The STLPPFI
problem solver is utlized to convert STLPPFI into its corresponding equivalent
Deterministic Transportation Linear Programming Problem (DTLPP) via
defuzzifying from fuzziness on probability distribution space and
derandomization randomness of problem formulation respectively. In addition,
Dual-Simplex algorithm method with Vogel Approximation Algorithm Method (VAM)
are used to obtain optimal solution from DTLPP. All MATLAB code programs with their
proposed algorithm outlines are new except Dual-Simplex and VAM. The MATLAB
code program of STLPPFI problem solver are more efficient along with a
numerical example on electricity field illustrating practicability of this
proposed MATLAB code program with its algorithm. Finally, the solution
procedure illustrates the MATLAB code program of the proposed method is
practical and applicable in the fields of energy and industry as it facilitates
the method of transforming the energy at the lowest cost, least time running
and is commercially applicable. Comparative comments are provided between
Dual-Simplex and VAM in solution process.