Revised Simplex Algorithm Applied to Problems with Restrictions

10 Pages·0.577 MB·English

by  Yankov K.

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ARTTE Applied Researches in Technics, Technologies and Education
ISSN 1314-8788 (print), ISSN 1314-8796 (online)
The Scientific Journal of the Faculty of Technics and Technologies of Yambol,
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Editor-in-Chief

Krasimira Georgieva, PhD, Faculty of Technics and Technologies, Trakia University, Bulgaria

Co-Editors-in-Chief

Georgi Tassev, DSc, Faculty of Technics and Technologies, Trakia University, Bulgaria
Zlatina Kazlacheva, PhD, Faculty of Technics and Technologies, Trakia University, Bulgaria

Editorial Board

Nikolai Ganev, PhD, Czech Technical University of Prague, Czech Republic
Iliya Ginkov, PhD, Faculty of Technics and Technologies, Trakia University, Bulgaria
Savvas Vassiliadis, PhD, Technological Educational Institute of Piraeus, Greece
Mariana Ursache, PhD, “Gheorghe Asachi” Technical University of Iasi, Romania
Plamen Daskalov, PhD, University of Ruse "Angel Kanchev", Bulgaria
Marin Vlada, PhD, University of Bucharest, Romania
Penka Martincova, PhD, University of Zilina, Slovakia
Veselina Nedeva, PhD, Faculty of Technics and Technologies, Trakia University, Bulgaria
Tania Pehlivanova, PhD, Faculty of Technics and Technologies, Trakia University, Bulgaria
Nely Georgieva, PhD, Faculty of Technics and Technologies, Trakia University, Bulgaria
Albena Stoyanova, DSc, University of Food Technologies of Plovdiv, Bulgaria
Krasimira Dobreva, PhD, Faculty of Technics and Technologies, Trakia University, Bulgaria
Bianka Tornyova, PhD, Medical University of Plovdiv, Bulgaria
Margarita Pehlivanova, PhD, Faculty of Technics and Technologies, Trakia University,
Bulgaria

English Editor

Emilia Dimova, PhD, Faculty of Technics and Technologies, Trakia University, Bulgaria

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ICTTE 2013 Edition
International Conference on Technics, Technologies and Education
ICTTE 2013, Yambol, Bulgaria, October 30 and 31 2013
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CONTENTS

ICTTE 2013 PLENARY SESSION
GREEN SUPPLY CHAIN DESIGN AND MANAGEMENT FOR BIOFUEL:
OPPORTUNITIES AND CHALLENGES FOR BULGARIAN SCALE,
Boyan Ivanov, Dragomir Dobrudzaliev 87
ICTTE 2013 SESSION OF TECHNICS AND TECHNOLOGIES IN MECHANICAL,
AGRICULTURE AND TRANSPORT ENGINEERING
USE OF XRD AND HOLE-DRILLING METHOD FOR DETERMINATION OF
RESIDUAL STRESSES GRADIENT IN C45 STEEL AFTER LASER HARDENING,
Kamil Kolařík, Nikolaj Ganev, Jiří Čapek & Zdenek Pala 103
X-RAY DIFFRACTION RESIDUAL STRESS ANALYSIS IN MECHANICAL
ENGINEERING AND MATERIALS SCIENCE,
Nikolaj Ganev, Kamil Kolařík & Zdenek Pala 109
THE ISSUE OF GAUGE BLOCKS CALIBRATION BY INTERFEROMETRY
METHOD – WRINGING GAUGE BLOCKS,
Štěpánka Dvořáčková, Jiří Sobotka, Pavel Brdlík & František Dvořáček 117
NEW OPPORTUNITIES FOR CHEMICAI-HEAT TREATMENT OF TITANIUM AND
TITANIUM ALLOYS, Nevena Ivanova, Hristo Skulev 123
ICTTE 2013 SESSION OF ELECTRICAL ENGINEERING, ELECTRONICS,
AUTOMATICS, AND COMPUTER TECHNOLOGY
INFORMATION SYSTEM FOR SECURITY EVALUATION – DESIGN AND
IMPLEMENTATION, Penka Martincova, Karol Grondzak & Monika Vaclavkova 131
SOFTWARE PLATFORM FOR EXECUTION OF GRAPH DATABASE
APPLICATIONS, Georgi Pashev, Dilyana Budakova 137
RESTRICTIONS REVISED SIMPLEX ALGORITHM APPLIED TO PROBLEMS
WITH RESTRICTIONS, Kaloyan Yankov 143
ICTTE 2013 SESSION OF FOOD TECHNOLOGY
UTILIZATION OF VOLUME POROSITY IN SIEVES FABRICS FOR IMPROVING
SENSORY, RHAIOLOGY PROPERTIES AND ENERGY SAVING IN BREAD
MAKING, ElSayed A. Elnashar, Lamiaa M. Lotfy 151
ICTTE 2013 SESSION OF LIFELONG LEARNING
SERVITISATION: A STRATEGIC SERVICE BUSINESS MODEL FOR GROWTH,
Stela Baltova 166

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REVISED SIMPLEX ALGORITHM
APPLIED TO PROBLEMS WITH RESTRICTIONS

Kaloyan Yankov
Trakia University of Stara Zagora; Faculty of Technics and Technologies
Graf Ignatiev 38, 8600 Yambol, Bulgaria
e-mail: [email protected]

Abstract: The simplex method is one of the most commonly used for solving linear optimization
problems. In this work the problem is modeled using a dynamic data structure, which guarantees
unlimited number of parameters for optimization. Due to the applied heuristic for choosing between
equal elements for inclusion in the base set, it is possible to obtain a cyclic repetition of solutions. The
later results in termination of the optimization process, because of reaching either a time limit or
maximal number of iterations, but not an optimal solution in the chosen sense. The paper offers a
bijective indexing of the basic solutions. The stored indexes form the history of the optimization
process. The history allows for return at the beginning of the cycle formation and selection of an
alternative basis. The algorithm is applied in some software packages for optimizing animal feed and
feed rations of animals and humans. Future applications will be associated with processing and
analysis of images.
Keywords: simplex method, linear programming, iterative algorithms, optimal control, parallel
optimization, image analysis.

1. INTRODUCTION

The subject of mathematical optimization is the study of the methods and algorithms for
finding an optimal value (minimum or maximum) of an objective function z(x ,x ,…,x ). The
1 2 n
optimal value is sought after complying with a chosen criterion for acceptable values of the
argument X, which values belong to a well-defined subset of the space Rn. A typical
description of the feasible set is by a finite system of constraints f(x)≤ b:
i i

X ={x∈Rn : f (x) ≤b ,i =1,2,...,m} (1)
i i

Thus, a generic optimization problem is one of the form

extr{z(x): f (x)≤b ,i =1,2,...,m}
(2)
i i
x
where extr is one between a minimum or a maximum of the objective function z(x).
Linear Optimization (LO) is а part of optimization theory, those where the objective z(x) and
the constraints f(x) are linear functions of x.
i
The problem of linear optimization in its cannonic form is expressed in the following way:

minZ(x)=CT.X

A.X = B (3)

x ≥ 0, j =1,2,....,n
j

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Here: C∈Rn is the coeffcient vector of the objective function;
X∈Rn is the n-vector of decision variables x;
j
B∈Rm is the m- vector with non-negative values;
A∈Rm x n, n ≥ m - matrix of constraints

One of the most commonly used method for solving LO problems is the simplex method
(SM). Тoo many publications аrе devoted on it since 1947 when Dantzig published the SM
and it is one of the 10 algorithms with the greatest influence on the development and practice
of science and engineering in the 20-th century [13].

1.1. The Nature of the Simplex Method

The SM is easy to use and fast to converge, but requires to define an initializing strategy:
starting points, initial step and it also does not guarantee finding a global optimum [6].
As mentioned above, the simplex method is the subject of a lot of works in the field. In the
literature there are descriptions of different algorithms for its realization. In what follows we
look in detail at the parts of the algorithm related to our work.
The unknowns are divided into m basic variables. If they are not equal to zero, they form
basis vector X . The other (n-m) non-basic variables form the vector X . A basic solution X is
b N
one in which (n−m) variables are set equal to zero.

X = X X T = X 0T
B N B

With the basis Xb is associated a bearer N+(x):

N (x)={ j∈n | X[j]>0 } (4)
+

If the vector X is a basis, the columns A[m, j] in matrix A[m,n] for j(cid:31)N (x) are linear
B +
B
independent. Basis X is nondegenerate if |N (x)| = |m| and the square basis matrix
B +
B
A[m,N (x)] has linear independent columns. The role of this condition is essential to prove
+
the finite convergence of the SM.
In the SM the goal is to find a basic solution X, which satisfy the restrictions AX=B.
A basic solution is obtained in the following way:
1. Set (n-m) variables equal to zero. These variables form vector X ∈R(n-m). The choice of a
N
set of variables to form an initial set of basic variables is not obvious. The number v of
possibilities to choose m basic variables out of a total of n variables is a combination of
n elements of m-th class:

⎛n⎞ n(n−1)(n−2)...(n−m+1)
v =Cnm =⎜⎜⎝m⎟⎟⎠= m(m−1)...2.1 (5)

This problem can be overcome by adding a set of m artifcial variables which form a set of
basic variables and which are penalized by a large coeffcients M in the objective function:

Z(x)=CTX +M.Y , M ∈R,Y∈Rm (6)

This penalty will cause the artifcial variables to become nonbasic as the algorithm proceeds.
2. Solve the system for the m remaining variables in the vector X .
b
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3. The vector of variables obtained is the basic solution X. It contains both basic X and
b
non-basic X variables:
N
Those initial setting steps are followed by iterative actions for improving the value of the goal
function. The algorithm boils down to applying rules to deduct the variable x from the basic
p
set and supplant it with the variable x from non-basic set, with this seeking the improvement
q
of the optimality of the goal function [7]. The operation of moving from a basic feasible
solution to an adjacent basic feasible solution is known as a pivot operation (Fig.1).

Figure 1. Pivot operation

4. The entering variable x corresponds to the column in matrix A which has the most
q
negative entry in the cost function row. If all cost function coeffcients are non-negative,
the cost cannot be lowered and an optimum is reached.
Due to the underdeterminded nature of the problem (n>m), it is possible to obtain several
basic solutions X The following algorithmic clauses are a premise for this.
.
• The choosing of the most negative coeffcient is only a heuristic. Besides there could be
several candidates with equal values to be included in the basis. The change of the
criterion for selection of entering variable is a prerequisite for finding optimality of the
algorithms accoding to an extra condition.
• The algorithm could enter in a cycle if a degenerate basis X is obtained. To avoid this
b
situation a choice of inpus and output variables from the basis is recommended to be
done amongst the variables with the smallest index. However following this approach
results in a loss of alternative solutions [10].
After the analysis above regarding the possibilities for multivariate solutions, the goal of the
present work is to suggest an approach allowing the realization of tracability of alternative
solutions, which could be obtained during the optimization procedure.

2. HISTORY OF THE OPTIMIZATION PROCESS

2.1. Indexing the basic solutions

As was mentioned, the simplex method can lead to a multitude of solutions, all of which
providing an extremum of the goal function. However the programming realization of the
algorithm gives a single solution. This is due to the aim for speeding the process, avoiding
cyclic repetition, and garanteeing convergence of the problem in acceptable for the user time
limits. The introduction of a suitabe system for identification and sorting of the basic
solutions, can provide a very comprehensive representation of the possible solutions.
Let for the decision varialbes set X={x ,x ,…x } exist a bijective mapping:
1 2 n

<:[1:n]→ X (7)

This mapping represents a linear ordering of the basic solutions Xj ∈ Rn, j=1,2,…v. Every Xj
obtained in the optimization process will be called a configuration of the opimization problem.
For j-th configuration X jcollate an index I j ∈Rn:
X

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0 if x ∈X , i=1,2,…,m
I j[i]= i N (8)
X 1 if x ∈X , j=1,2,…,v
i B

Thus to the configuration Xj corresponds a binary number and Xj obtained during the
optimization process can be ordered in an increasing order with respect to the magnitude of
their index.
Hamming distance r (Ip ,Iq ) between two indices Ip and Iq is defined as:
d X X x x

( ) n ( )
r Ip,Iq = ∑ Ip[i]⊕Iq[i] (9)
d X X X X
i=1
Where: 0 if Ip[i]= Iq[i], i=1,2,..,n
Ip[i]⊕Iq[i]= X X
X X 1 if Ip[i]≠ Iq[i], p,q=1.2...v, p≠q
X X

With each pivot operation the Hamming distance between the index Ip before and the
x
index Iq after the operation will ber (Ip,Iq )=2.
x d X X
Two configurations X p and Xq are equivalent if for their indices Ip and Iq is true that:
x x

( )
r Ip,Iq =0 (10)
d X X

2.2. A history of the basic solutions

Indices for checking optimality of basic solutions are memorized in a data structure called
HistorySet. For every newly generated basic solution Xj the index I j is created and a check
x
is made weather it is present in HistorySet. If present it means that a cycle has started and
another value should be chosen to enter the basis X . Otherwise the index is added to
b
HistorySet, after which the condition for optimality of the goal function is checked.
There are different options to organize HistorySet: linear list, binary tree, h-function, where
the index itself or part of it can be used as a key in the calculation of the address. In this work
a dynamic binary tree is used to contain the indices. An ordering criterion is the relation ‘<’,
and not the order of generation. Such ordering allows the coincidence search of basic
solutions to be performed with a bisection algorithm.

3. OPTIMIZATION PROCESS AND CONFIGURATION HISTORY

3.1. Optimization procedure in a linear algorithm

The algorithm in the case of a single-processor system is:
1. In the r-th consecutive pivot operation the index Ixq of the newly created basic solution X
is checked against the elements of the tree of the indixes.
2. If the index is present already in the tree, this indicates a cycle. Select the next entering
candidate.
3. If the index is not present in the tree, it is recorded in the HistorySet, after which a check
for optimality of the chosen basis is done.
The pseudo-code of the algorithm is on figure 2.

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procedure PivotOperation(A : DynamicArray);
var Xp,Xq : integer;
IndVal : TBits; // index value of configuration
begin // pivot operation
Xp:=DefineLeavingVar[A]; // choose leaving variable
repeat // search consecutive index
Xq:=DefineEnterVar(A); // choose entering variable
IndVal:= GetIndex(X, Xq); // create index Iq
until (not HistoryTree.HistorySearch(IndVal; Root); // Index is unique
HistoryTree.AddIndex(IndVal); // // adding an index to HistoryTree
end;

Figure 2. Formation of HistorySet

3.2. Parallel realization of the optimization procedure

Large-scale optimization problems can be solved efficiently using multiple processing units.
In order to make parallel computing possible, the algorithm has to be entirely or partially
composed of multiple independent computations [10]. Thus the problem in its different
versions can be solved simultaneously or concurrently, thereby increasing the overall
computational speed. Optimization problems can be solved in parallel mode using the
following scheme:
1. Let there be к candidates to enter the basis, and r be the number of possible parallel
processes. The number of necessary passes N to prove optimality is:
k

N = (k div r) + int(bool((k mod r)>0)) (11)
k

When N =[0,1] the number of candidates to enter the basis is no greater than the number of
k
possible parallel processes and the optimization can be performed without resorting to
HistorySet.
When N >1 a single pass parallel processing is impossible and necessitates the use of
k
HistorySet.
2. The configurations X1, X2,..,Xk , corresponding to all possible entering values are
generated.
3. If N >r, the first r configurations can be processed in parallel and the indices of the
k
remaining configurations Xr+1, Xr+2,..,Xk are written in THistoryTree for processing in the next
pass.
4. The values of the goal functions are calculated with a parallel operation.
5. The values coming out of the parallel processes are compared and following the decided
optimization criterion a solution is chosen.

4. DATA STRUCTURES

The main data structures are determined by the formulation of the opitmization problem (3).
The classic static structure „array” with its necessity of precisely defined size of rows and
columns is unacceptable. This is why a dynamic data structures are used, whose size is
defined runtime and is dependent on the order of the problem. For the vectors such as the
coeffcient vector C, decision vector X and B a dynamic list is used. For 2D matrices as the
ones of the limits A(m x n) an approach is used similar to graph presentation with dynamic
data structures [12]. The framework of the data structure HistorySet for storing and search of
the indexes of configurations is yet different. This structure has to realize a binary tree. Each
node of the tree should contain the value of the configurations index, the entering and
leaving values of variables and pointers to the left and right childs. It is defined as a
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successor of the base class TObject. The new class THistoryNode comprises all common
fields and methods of its descendant classes (Figure 3).

TNodePtr = ^ THistoryNode;
THistoryNode = class(TObject)
private
IndVal : TBits; // index value
Leaving, Entering : integer; // leaving and entering variables
LeftNode, RightNode : TNodePtr; // ponters to the left and right descendants
protected
constructor Create(IndVal);
destructor Destroy;
procedure Set_Node(IndVal, Leaving, Entering);
function Get_LeftNode; // access to the left descendant
function Get_RightNode; // access to the right descendant
function Add_LeftNode(IndVal) : TNodePtr;
function Add_RightNode(IndVal) : TNodePtr;
…………………………..
end;

Figure 3. Abstract class THistoryNode

The next structure in the hierarchy is THistoryTree. This is a structure that provides access to
and control of the encapsulated class THistoryNode. THistoryTree possesses the methods to
support the history of the configurations as wells the very important pivot operation (Fig.4).

THistoryTree = class(THistoryNode);
public
inherited methods;
function HistorySearch(const IndVal; var Root: TNodePtr) :boolean;
function AddIndex(const IndVal) : TNodePtr; // add new indev
procedure PivotOperation(A : DynamicArray); // fig.1 and fig.2
…………………………………………..
end;

Figure 4. Structure of THistoryTree

The hierarchy of the newly created objects with regard to the system-defined class TObject is
presented on figure 5.

Figure 5. Class hierarchy

On figure 6 is shown the recursive method for searching an index in THistoryThee and its
recording in cases where the corresponding configuration was not generated.

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function HistorySearch(const IndVal; var Root: TNodePtr): boolean;
begin
Result:=false;
if (Root=nil) // IndVal is not in HistorySet
then Root.Create(IndVal) // write IndVal in the HistorySet
             else  if (IndVal < Root^.Value)
then Self.HistorySearch(IndVal,Root^.left)
else if (IndVal > Root^.Value) then Self.HistorySearch(IndVal,Root^.right)
else result:=true // IndVal is in History set
end;

Figure 6. Method HistorySearch

5. REALIZATION OF THE SIMPLEX METHOD

The algorithm is applied in the following software packages [14].
• KOMFU 2000 – a software for optimization of raw material composition of compound
cattle fodder applying as criterion minimum price [5].
• RUMIX – a package for compilation and optimization of individual and group feeding
rations for ruminants and other animals.
• PREMIX – a software for compiling and optimizing recipes for premixes.
• Simplex-Diet – a software for calculating the individual nutritional norm for human diet to
meet minimum calorific value, while ensuring complete nutrition.
This packages are developed by ExBit, Stara Zagora and are on the market since the mid-
90s in more than a hundred companies in Bulgaria, Macedonia, Belgium and Kazakhstan.

6. FUTURE DEVELOPMENT

The approach that developed in this work for finding multivariate solutions has shown its
reliability for many years of usage.
Another sphere of application of the simplex method besides economics is its active use in
image analysis and image processing [11].
In [9] the simplex optimization determined the parameter values of selected image quality
operators. The processing algorithm modified the image with selected quality operators
whose parameter values were “guessed” by the Simplex algorithm. The objective function
evaluation algorithm assessed the performance of the image processing algorithm. When the
optimization criterion was reached, the optimization stopped. Once the image was optimized,
the image classification component was used to create a classification model.
Mehfuza and Thakar [4] use the simplex algorithm for optimization a sequence of similarity
transformations from the sensed image to the goal – the reference image.
In [3] the author measures the similarity between two medical images using the mutual
information (MI) criterion. Minimization of the MI function is done by the simplex method. The
optimization method searches the parameter space to find the coordinates that give a global
minimum value for the function to minimize.
The referenced results as well as other results that are not mentioned here are a prerequisite
to seek a similar solution in the evaluation and classification of images during the quality
control in the food industry.
The simplex algorithm will be tested in a project regarding formulation and classification of
certain quality characteristics of pork, depending on the parameters of the image obtained
from the computer vision system [1, 2, 8].

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7. CONCLUSIONS

1. This works formulates an indexing scheme for the basic solutions when applying the
simplex method.
2. The indexes obtained during the optimization process are written on a dynamic binary
tree, which represents the history of the optimization process.
3. Using the index tree gives the means to trace all possible solution configurations and to
find optimality according to additional conditions.
4. The suggested indexing scheme for the solutions allows parallel processing.

8. ACKNOWLEDGEMENTS

This study has been supported by Grant 4ФТТ/13.05.2013 from the Faculty of Technics and
Technologies, Trakia University - Yambol.

9. REFERENCES

[1] Bochev V., Zl. Zlatev, Kr. Dobreva. (2012a). Application of computer vision systems for
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9.2 Biotechnologies and food technologies. pp. 116-119.
[2] Bochev V., Zl. Zlatev, Kr. Dobreva. (2012b). Development of a computer system to
assess the quality of meat and meat products by color features. University of Ruse
“Angel Kanchev” Vol.51, book 9.2 Biotechnologies and food technologies. pp. 125-129.
[3] Hamre, Bjorn. (1998) Three-dimensional image registration of magnetic resonance (MRI)
head volumes. Ph.D.Thesis, University of Bergen, Norway.
[4] Holia M., V.K.Thakar. (2009) Image registration for recovering affine transformation
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[5] Kazandjieva R., Todorov N.(1998), KOMFU - Software System for Establishing and
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the Development of Fundamental and Applied Sciences".June, 5-6, 1998, Stara Zagora,
Bulgaria, vol.2. ISBN 954-8180-65-0, pp. 503-507.
[6] Martin V., M. Thonnat. (2007). A Cognitive Vision Approach to Image Segmentation.
INRIA Sophia Antipolis Méditerranée, project-team PULSAR, France. 19th IEEE Int.
Conf. on Tools with Artificial Intelligence - Vol.1 29-31 Oct. 2007, Patras, pp.265- 294.
[7] Nemirovski A. (2013). Optimization I: Introduction to Linear Optimization. Georgia
Institute of Technology, School of Industrial and Systems Engineering.
[8] Nedeva V., Zl. Zlatev (2012) Contactless Methods for Quality Evaluation of Pork Meat.
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[9] Shuo Yan (2008). Adaptive Image Quality Improvement with Bayesian Classification for
in-Line Monitoring. Ph.D. Thesis, University of Toronto.
[10] Singiresu S. Rao. (2009). Engineering Optimization: Theory and Practice, Fourth Edition
Copyright John Wiley & Sons, Inc..p.813.
[11] Velho L., P. Carvalho, et.al. (2008) Mathematical Optimization in Computer Graphics
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[12] Yankov K., N. Ivanova. (1992). Graphs Presentation with Dynamic Structures. Third
Scientific Conf. "Modern Tendencies in the Development of Fundamental and Applied
Sciences" June 4-5, 1992, Stara Zagora.
[13] http://www.computer.org/csdl/mags/cs/2000/01/c1022.html
[14] http://exbit.bg/index.html

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