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Multi-criteria design of mechanical system by using visual interactive analysis tool

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Nowadays, to design any mechanical system the engineer needs to analyse and evaluate among many criteria such as production cost, geometry, manufacturing technology, etc., which forms the terminology of multi-criteria design. In this paper, the design based on the product lifecycle management concept was proposed. A Visual Interactive Analysis Tool (VIAT) was used to deal with a model based multi-objective optimization including many parameters-variables, constraints and objective functions or criteria in the design process. In virtue of VIAT, both of engineer and related experts in the concerned system are able to perform analysis and evaluation, if there are inconsistencies, they can discuss openly before making a decision on the design option for the object. VIAT is a practical and user-friendly tool for the multi-criteria design of the mechanical system.
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Journal of Engineering Science and Technology
Vol. 14, No. 3 (2019) 1187 - 1199
© School of Engineering, Taylor’s University
1187
MULTI-CRITERIA DESIGN OF MECHANICAL SYSTEM
BY USING VISUAL INTERACTIVE ANALYSIS TOOL
DANG HOANG MINH1, PHUNG VAN BINH2,
BUI VAN PHUONG3, DUC VN4,5,*
1Industrial University of Ho Chi Minh City, Ho Chi Minh City, Vietnam
2Faculty of Aerospace Engineering, Le Quy Don Technical University
3Bauman Moscow State Technical University, Moscow, Russian Federation
4Division of Construction Computation, Institute for Computational Science,
Ton Duc Thang University, Ho Chi Minh City, Vietnam
5Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam
*Corresponding Author: nguyenvietduc@tdtu.edu.vn
Abstract
Nowadays, to design any mechanical system the engineer needs to analyse and
evaluate among many criteria such as production cost, geometry, manufacturing
technology, etc., which forms the terminology of multi-criteria design. In this paper,
the design based on the product lifecycle management concept was proposed. A
Visual Interactive Analysis Tool (VIAT) was used to deal with a model based multi-
objective optimization including many parameters-variables, constraints and
objective functions or criteria in the design process. In virtue of VIAT, both of
engineer and related experts in the concerned system are able to perform analysis and
evaluation, if there are inconsistencies, they can discuss openly before making a
decision on the design option for the object. VIAT is a practical and user-friendly tool
for the multi-criteria design of the mechanical system.
Keywords: Mechanical system, Multi-criteria design, Multi-objective optimization,
Product lifecycle management, Visual interactive analysis tool.
1188 D. H. Minh et al.
Journal of Engineering Science and Technology June 2019, Vol. 14(3)
1. Introduction
Up to the present moment, all kind of mechanical systems (equipment, tools, etc.)
have played a crucial role in our life. In fact, it has still been a big challenge for
every engineer and expert in the field to design them. Currently, the design might
be carried out by one of the following manners [1-3]:
Conventional method: first it needs to select technical parameters based on
mainly experience, and then check them whether they comply with the
requirements or not. In practice, this manner is the most commonly used due
to the limitation of budget and manpower for design at many institutions and
small companies, however, if the design based on experience is not suitable in
the manufacturing process, manufacturers will be very confused in looking for
the alternatives.
Design based on single-objective optimization: it focuses mostly on optimizing
a particular demand of the client such as production cost, or geometry, or
productivity, etc. Nevertheless, if the client requires those criteria
simultaneously, this manner will not meet the requirements.
Design based on multi-objective optimization or multi-criteria design: it
focuses on optimizing multi-criteria, for instance simultaneously production
cost, geometry, and productivity; thus, this manner seems to be the best option
for design engineers, though it has not been common-used yet in practice rather
than at research institutions and very large companies. The reason is generally
due to the fact that currently there is not a practical and user-friendly tool used
for multi-criteria design.
On the other hand, the design organization, i.e., the way how to unite numerous
experts in the mechanical system, is also a concerning topic. Frankly speaking, it is
truly useful for the creation of any product if there are contributions of knowledge
and experience from different professionals or experts. However, it is actually hard
to settle them work together because every expert owns different conception, bias,
and view on the significance of technical factor. Statnikov et al. [4] explained that
even sometimes their opinions on the same fact might be contradictory. According to
Saaksvuori and Immonen [5], the idea of united experts for a good reason can be
carried out by using the product lifecycle management concept. Doing so, if there are
inconsistencies among experts, they will be described by a mathematical model based
on multi-objective optimization and the resultant mutually-agreed solution will
relieve these issues. As a result of that, the objective of this paper is to propose the
idea of product lifecycle management with the aim to deal with the multi-criteria
design of the mechanical system. Besides, a visual interactive analysis method is used
for the solution search and decision-making process occurred smoothly.
2. Multi-Criteria Design Solutions for Mechanical System
Traditional approach
When the design is carried out by the traditional approach, a chief engineer needs
to acquire concept, knowledge, experience and useful data from other members in
the design team especially experts, as shown in Fig. 1. Based on this, he develops
a mathematical model based multi-objective optimization including parameters-
variables, constraints and criteria or objective functions. To deal with this problem,
he first has to determine weight coefficients of objective functions depending on
Multi-Criteria Design of Mechanical System by using Visual Interactive . . . . 1189
Journal of Engineering Science and Technology June 2019, Vol. 14(3)
criteria significance. Then, all of the objective functions are converted into an
equivalent single-objective function by mean of numerous techniques such as
weighted minimax (maximin), compromise programming, weighted Sum,
modified Tchebychev, weighted product, exponentially weighted sum, etc. The
terminology “convolutionpresents the above-mentioned conversion technique in
a general form, for instance, in case of weighted minimax (maximin) method
convolution =
 
0
max , 1..
i i i
iФ Ф i M



x
, for weighted sum method convolution
=
 
1
M
ii
i
Ф
x
, and for weighted product method convolution =
 
1
i
M
i
i
Ф


x
, etc.
Mechanical System
(MS)
Concept-
Design Analysis Material Technology Customer
Theoretical basis,
experience, information
Chief engineer
 
12
, , , n
x x x
and Conditions
 
12
, , , m
f f f
Variables
Functional
constraints
 
12
, , , M
Ф Ф Ф
Criteria
Weight
coefficients
 
, min max; 1..
eq i i
Ф convolution Ф i M
 
 
12
, , , M
Ф Ф Ф
 
Solution
Do they meet the
requirements?
YES
NO
NO
YES
END
MATHEMATICAL
MODEL
SINGLE-OBJECTIVE
OPTIMIZATION METHODS
Do they meet the
requirements?
and Conditions
and Conditions
Fig. 1. Scheme of multi-criteria design based on traditional approach.
1190 D. H. Minh et al.
Journal of Engineering Science and Technology June 2019, Vol. 14(3)
On the other hand, in order to find the extreme (min-max) of the equivalent
function, several methods can be used including direct search-descent methods,
genetic algorithms, particle swarm, etc., [1, 6]. By using these methods, the result
of the single-objective optimization can be obtained, and consequently, values of
initial objective functions
 
12
, , , M
Ф Ф Ф
 
can be pointed out. At this moment, the
engineer has to analyse whether these values comply with the requirements or not;
if not, he could modify the weight coefficients of objective functions and make a
calculation again up to the moment that there are desirable solutions; if yes, he
would send the results to experts in every field related to the concerned mechanical
system to analyse thoroughly. With feedback from the experts, he needs to correct
the model or conduct another optimal solution search.
An approach based on product lifecycle management concept
Since the traditional approach presents several drawbacks, a multi-criteria design
based on the product lifecycle management concept can be used, as the illustration
is given in Fig. 2. Looking into this figure, right at the first step every issue in the
lifecycle of the mechanical system is specified, arranged and given to an expert in
the field. Doing so, the system is studied by all of the related experts properly and
consistently [7].
Unlike the previous approach, the model based multi-objective optimization is
developed openly by the experts instead of the chief engineer on his own. Then, a
Visual Interactive Analysis Method (VIAM) is applied; indeed VIAM in the form
of software tool allows them to examine the parameter space that expressed in
determined ranges and existence tendency of valid solutions. At the same time, they
are also aware of constraints and objective functions or criteria of the concerned
mechanical system [8, 9].
It is important to note that by this approach to define parameter space,
constraints and objective functions is not a simple task at the beginning due to
varieties of experts’ views on the same fact. The virtue of VIAM is that when the
experts modify any determined range, the corresponding output results will yield
automatically and almost instantly.
If the outcome does not comply with the experts’ requirement, by using the
software tool they could keep adjusting the ranges up to the moment that there are
a valid result and the process finishes. Notably, all of these occur in real-time in
front of the experts, consequently, all of them together analyse the results in the
discussion before making a decision. In case there is no mutually agreed solution,
the tool would let them know, which criteria or determined range of parameter and
constraints are the reason for solution nonexistence.
Afterwards, in the dialogue, each of them needs to analyse thoroughly in
accordance with all of the data such as valid ranges in order to decide whether to
keep his opinion on criteria significance or change it. Besides, the tool is also able
to plot interactive graphs to illustrate the relation of criteria and parameter or others,
which efficiently support the experts in the analysis process. Definitely, the VIAM
tool provides a piece of useful information about design object that helps the
experts to make a decision.
Multi-Criteria Design of Mechanical System by using Visual Interactive . . . . 1191
Journal of Engineering Science and Technology June 2019, Vol. 14(3)
Concept-Design Compute/
Analysis
Material Technology
Customer
Chief engineer
Mechanical System
(MS)
VIAT VISUAL INTERACTIVE ANALYSIS TOOL
Variables Functional constraints Criteria
1
x
n
x
2
x
j
x
1
Ф
2
Ф
Ф
M
Ф
1
MINФ
2
MINФ
3
MINФ
MIN i
Ф
MIN M
Ф
1
MAXФ
2
MAXФ
3
MAXФ
MAX i
Ф
MAX M
Ф
Interactive Panel
1 1 1
22
ii
n n n
a x b
xb
xb
a x b


11
fc
22
fc
ii
fc
mm
fc
Visual graphs represent variables-variables,
criteria-criteria relations
 
12
, , , M
Ф Ф Ф
 
Mutually-agreed solution
PRODUCT LIFE-CYCLE
MANAGEMENT
Fig. 2. Scheme of multi-criteria design based
on product lifecycle management concept.
3. Toolkit for Multi-Criteria Analysis
Podinovskaya and Podinovski [10] and Podinovski [11] proposed to illustrate a multi-
criteria design based on the product lifecycle management concept, an algorithm
scheme on the basis of successive concession method and the parameters space
investigation method [4]. This approach is named as a visual interactive analysis
method or VIAM. The difference between VIAM and successive concession method is
an interactive panel. This panel shows data on a feasible range of objective functions
and mutual influence among them, thus, the expert could select the appropriate design
solution. Besides, VIAM is more effective than the parameter space investigation
method because the latter requires a lot of test-points (thousand, even million), which
in turn it is time-consuming.
1192 D. H. Minh et al.
Journal of Engineering Science and Technology June 2019, Vol. 14(3)
3.1. Details on the algorithm
The main idea is to use the single-objective optimization techniques as a device to
find valid solutions. The detailed steps in the solution search are described in Figure
3. Looking into this figure, the model {1} is the first step, hence, it is necessary to
optimize the vector
Φ
of M criteria with the parameter vector
x
, as well as
the constraints
f
(where, x = {x1, x2,…, xN}, f = f(x) = {f1(x), f2(x),…, fK(x)}, Ф
= Ф(x) = {Ф1(x), Ф2(x),…, ФM(x)}).
The condition of parameter vector
x
is as follows:
i i i
a x b
;
jj
ax
or
kk
xb
(i, j, k [1; N]); while, the condition of constraints
f
is as follows: fl(x)
0; fm(x) = 0; fn(x) < 0 or fo(x) 0 (l, m, n, o [1; K]). Next step {2} is to use
single-objective optimization algorithms [1] to determine the minimum and
maximum values or min Фi and max Фi, respectively, of every objective function,
and include these values into an interactive panel. To make it more user-friendly,
the optimal trends are provided with top-down order (from MAX to MIN). The
function, which needs to be maximized, is assigned with a negative sign. This panel
helps the experts to analyse and make a decision on the outcome. In case, the final
solution is a vector
 
12
, , , M
Ф Ф Ф
 
Φ
, i.e.,
 
MIN ;MAX ; 1..
i i i
Ф Ф Ф i M

.
Based on the panel in step {2}, the experts define a priority order for criteria or
objective functions, for instance in {3}:
 
12 M
Ф Ф Ф
. Accordingly, it
needs to optimize firstly the objective function Ф1 (step {4}) and the optimal value
MINФ1 is used for further steps. However, while setting a goal for Ф1 is such
superior, the obtained solution for other functions might be inferior and out of
requirement. In this case, the experts need to adjust the range of Ф1, and set its
value with a threshold
 
1 1 1
MINФ Ф E  
(with the deviation
10E
). If so,
among the obtained solutions that Ф1
 
1
Ф
, there might be the most favourable
option for other objective functions. While searching the optimal value for other
objective functions
1
min i
Ф
(i=2…M), it needs to add the constraint
 
11
0ФФ
.
Notably, the upper index “1” implies that these are the minimum values of the rest
objective functions when the threshold
 
1
Ф
is concerned. Since there is a
constraint Ф1
 
1
Ф
, range values of objective functions are shrunken (Fig. 3), i.e.,
1
min MIN
ii
ФФ
{5}. Then, the value
1
min i
Ф
(I = 2…M) needs to be studied by
the experts, because there might be a situation that these ranges are invalid or they
do not comply with the expert requirements. This is an advantage of VIAM that it
checks the effect of
 
1
Ф
on all of the rest functions, while the successive
concession method only considers how the
 
1
Ф
influences on
2
Ф
. If so, the
threshold
 
1
Ф
needs to be modified again to resolve the step {5}. While, in the
case that these values
1
min i
Ф
comply with the requirement, similarly the experts
Multi-Criteria Design of Mechanical System by using Visual Interactive . . . . 1193
Journal of Engineering Science and Technology June 2019, Vol. 14(3)
set a threshold
 
1
2 2 2
minФ Ф E  
for the objective function Ф2 (with the
deviation
20E
) with the goal to optimize the succeeding functions (step {6}).
Fig. 3. Algorithm of visual interactive analysis method.
 
 
1
1 : , min min , , ?
M
constr ФФ 

xf Φ
MAXФ1MAXФ2
MAXФi
MAXФM
… …
MINФ1MINФ2MINФiMINФM
 
12 M
Ф Ф Ф
 
1 1 1
MIN Ф Ф E  
 
 
 
1
11
5 : , , 0 min ,( 2... )
i
constr Ф Ф Ф i M

 

xf
 
 
 
 
11 2
22
0
, , min ,( 3... )
0i
ФФ
constr Ф i M
ФФ



 



xf
 
 
1
1 1 1 1
Pareto Set , , , min M
M M M M
Ф Ф Ф Ф Ф Ф
 

  Φ
 
2:
 
YES
 
NO
 
NO
 
YES
 
 
11
1
11
0
, , min
0
M
M
PM
ФФ
const Ф
ФФ









xf
 
1
2 2 2
minФ Ф E  
 
2
3 3 3
minФ Ф E  
 
2
1 1 1
min M
M M M
Ф Ф E
 
 
Satisfy
Experts
 
YES
Experts c
o
Experts
Experts C
o\
Satisfy
Experts
 
NO
Experts
Experts Co
\
c
o
 
3:
 
4:
 
6:
 
8:
 
9:
Satisfy
Experts
 
7:
{10}:
 
11 :
OPTIMAL TREND
Interactive Panel
1194 D. H. Minh et al.
Journal of Engineering Science and Technology June 2019, Vol. 14(3)
While searching the optimal value for other objective functions
2
min i
Ф
(i=3…M), it needs to add two constraints
 
 
 
 
11
22
0
0
ФФ
ФФ


(step {7}). By the same
manner, the experts set a threshold
 
2
3 3 3
minФ Ф E  
(step {8}), and so on there
are a series of thresholds
 
1 2 1
, ,..., M
Ф Ф Ф
for the first (M-1) criteria (step {9}).
Based on (M-1) constraints of the corresponding threshold and necessity of
minimizing Mth criterion, eventually, the mutually-agreed solution with a desirable
priority order can be achieved (step {10}). The obtained result
 
12
, , , M
Ф Ф Ф
 
Φ
is the best mutually-agreed solution ({11}), which means
that there is no better solution for all of the functions simultaneously.
It is important to note that currently there are plenty of the single-objective
optimization methods available, hence, which one being effectively used in VIAM
depends on the mathematical model of particular objects.
3.2. Example of an interactive panel with four criteria
Based on the algorithm VIAM, it is possible to determine Visual interactive
analysis tool or VIAT, which is a computing application built in C++,
MATLAB, MAPLE etc., so that it supports the expert efficiently to have
multiple-criteria decision analysis and define the optimal design solution for
the mechanical system.
Assumed that there is a model with four criteria or objective functions need to
be optimized, and priority order of criteria (
 
1 2 4
Ф Ф Ф
) is considered as
shown in Fig. 4. The valid range of objective functions and their optimal trends are
defined and illustrated in the interactive panel (Fig. 4).
Bearing in mind that values of objective functions have to be expressed with
the right magnitudes and units that allow experts to study and analyse determinedly.
To simplify the nomination, the expert or group of experts or a chief engineer
is named as a decision-maker (DM). According to an interactive panel, DM selects
a threshold
 
1
Ф
for the objective function Ф1.
This threshold turns into a constraint to define the optimal values of the rest of
functions, as it is illustrated in Fig. 5. An influence of Ф1 on the rest of functions
can be observed visually by adjusting the threshold. Thus, DM can easily see the
“price to pay” to improve the Ф1 with a magnitude of
1
E
. Yet, the panel also shows
a new valid range and an inaccessible domain at which, there would be never a
valid solution with the threshold
 
1
Ф
.
If the new range does not comply with the requirements of experts, they can
adjust the threshold up to the moment there is a mutually-agreed solution.
Multi-Criteria Design of Mechanical System by using Visual Interactive . . . . 1195
Journal of Engineering Science and Technology June 2019, Vol. 14(3)
MINФ1MINФ2
[Ф1]
OPTIMAL TREND
MINФ3
MAXФ1MAXФ2MAXФ3
Feasible value region of objective functions
Ф1Ф2Ф3MINФ4
Ф4
MAXФ4
[Фi]Threshold value of objective functions adjustable value
The priority descending order of objective functions
The best value
The worst value
Desired value
region
Decision makers (Group of experts)
DM
DM
Fig. 4. Interactive panel-step 1.
Based on a valid range for the function Ф2, DM continues selecting a threshold
 
2
Ф
for this function. The threshold
 
2
Ф
also turns into a constraint to define the
optimal value the functions Ф3 and Ф4.
At this moment, a valid range becomes narrower; while an inaccessible domain
is expanded, as it can be seen in Fig. 5. Carry on studying, DM picks a threshold
 
3
Ф
for the function Ф3. Here, the optimal value
4
minФ
can be defined on the basis
of the selected thresholds
 
1
Ф
,
 
2
Ф
,
 
3
Ф
, as it can be observed in Fig. 6. Indeed,
at this step, the obtained solution is the best mutually-agreed one among experts
with respect to the initial concerned priority order, as shown in Fig. 7.
It is evident that at every step there is an intervention of experts to select, adjust
and analyse the thresholds. Besides, the adjustment of thresholds points out a
mutual influence of one function to another and alteration of the valid range.
Thus, the design object is analysed comprehensively and entirely. Doing so, the
final design option is incontrovertibly the most optimal one.
1196 D. H. Minh et al.
Journal of Engineering Science and Technology June 2019, Vol. 14(3)
MINФ1MINФ2
[Ф1]
MINФ3
MAXФ1MAXФ2MAXФ3
[Ф2]
Ф1Ф2Ф3
MINФ4
Ф4
MAXФ4
The best value of objective functions when considering the
threshold [Ф1]
Region that can not
reach agreement
DM
The best value
The worst value
[Фi]Threshold value of objective functions adjustable value
Decision makers (Group of experts)
DM
Desired value
region
Desired value region of objective functions
The priority descending order of objective functions
OPTIMAL TREND
Feasible value region of objective functions
1
2
minФ
1
3
minФ
1
4
minФ
1
min i
Ф
Fig. 5. Interactive panel-step 2.
MINФ1MINФ2
[Ф1]
MINФ3
MAXФ1MAXФ2MAXФ3
Ф1Ф2Ф3
MINФ4
Ф4
MAXФ4
[Ф3]
DM
[Ф2]
The best value when considering thresholds [Ф1] and [Ф2]
Region that can not
reach agreement
The best value
The worst value
[Фi]Threshold value of objective functions adjustable value
Decision makers (Group of experts)
DM
Desired value
region
Desired value region of objective functions
The priority descending order of objective functions
OPTIMAL TREND
Feasible value region of objective functions
1
2
minФ
2
3
minФ
2
4
minФ
2
min i
Ф
Fig. 6. Interactive panel-step 3.
Multi-Criteria Design of Mechanical System by using Visual Interactive . . . . 1197
Journal of Engineering Science and Technology June 2019, Vol. 14(3)
MINФ1MINФ2
[Ф1]
MINФ3
MAXФ1MAXФ2MAXФ3
[Ф2]
Ф1Ф2Ф3
 
 
 
11
22
33
3
44
Solution
,
,
,
min
ФФ
ФФ
ФФ
ФФ
Φ
MINФ4
Ф4
MAXФ4
[Ф3]
The mutually-agreed solution
DM
minФiThe best value when considering thresholds [Ф1], [Ф2]and [Ф3]
The region that can not
reach agreement
The best value
The worst value
[Фi]Threshold value of objective functions adjustable value
Decision makers (Group of experts)
DM
Desired value
region
Desired value region of objective functions
The priority descending order of objective functions
OPTIMAL TREND
Feasible value region of objective functions
1
2
minФ
2
3
minФ
3
4
min
Fig. 7. Interactive panel-step 4.
3.3. VIAT applicability
The benefits of VIAT are the following:
Provide information on the feasible range of objective functions and their
alterations in the feasible solution search;
Evaluate the “price to pay” to improve a particular function and its effect on
the rest of functions;
Support efficiently a decision-maker to select a suitable threshold of the
objective function in the valid range at which, there exists a valid solution.
Eventually, allow for an automatic determination of mutually-agreed solutions
among experts for a multi-criteria design option.
By virtue of VIAT, the authors have found the solution for automation and
management of design and production of composite pressure vessel by the winding
method [8], multi-criteria design of an innovative frame saw machine [12, 13] and
determination of optimal manufacturing modes in the metal cutting process [14].
4. Conclusion
In this paper, a multi-criteria design of the mechanical system was studied. The
design based on the product lifecycle management concept was concerned. A
Visual Interactive Analysis Method (VIAM) together with a tool named VIAT was
used. VIAT has provided a useful database that allows related experts in the
mechanical system to analyse conveniently a mutual effect of one objective
1198 D. H. Minh et al.
Journal of Engineering Science and Technology June 2019, Vol. 14(3)
function on another. Without any bias and overbearing opinion, they conducted a
straightforward discussion and evaluation accurately on the same fact with the
utmost aim to define the optimal mutually-agreed solution for the design object.
VIAT is actually a very user-friendly tool that makes the multi-criteria design more
practical especially for the mechanical system.
Acknowledgement
The authors would like to express their gratitude to Rector of Industrial University
of Ho Chi Minh City and Dean of Faculty of Mechanical Engineering for the
interest, help and invaluable contributions to the paper.
Nomenclatures
ai
Lower limits for ith parameter
bi
Upper limit for ith parameter
fi=fl(x)
ith functional constraints
K
Number of functional constraints
N
Number of parameters
M
Number of criteria
Greek Symbols
i
E
Deviation for ith functional constraints
x
Parameter vector
xi
ith parameter
Ф(x)
Criteria vector
Фi =Фi(x)
ith criterion
Φ
Criteria vector in final solution
i
Ф
ith criterion in final solution
 
i
Ф
Threshold for ith criterion
Min Фi
Minimum value of ith criterion
MaxФi
Maximum values of ith criterion
min k
i
Ф
Minimum values of ith criterion, when first k thresholds
 
1... k
ФФ
,
 
1
Ф
are concerned
Abbreviations
DM
Decision-Maker
MS
Mechanical System
VIAM
Visual Interactive Analysis Method
VIAT
Visual Interactive Analysis Tool
References
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Chapter
Introduction Characteristics of a Constrained Problem Random Search Methods Complex Method Sequential Linear Programming Basic Approach in the Methods of Feasible Directions Zoutendijk's Method of Feasible Directions Rosen's Gradient Projection Method Generalized Reduced Gradient Method Sequential Quadratic Programming Transformation Techniques Basic Approach of the Penalty Function Method Interior Penalty Function Method Convex Programming Problem Exterior Penalty Function Method Extrapolation Techniques in the Interior Penalty Function Method Extended Interior Penalty Function Methods Penalty Function Method for Problems with Mixed Equality and Inequality Constraints Penalty Function Method for Parametric Constraints Augmented Lagrange Multiplier Method Checking the Convergence of Constrained Optimization Problems Test Problems MATLAB Solution of Constrained Optimization Problems References and Bibliography Review Questions Problems