Advances in Economics, Business and Management Research, volume 128
International Scientific Conference "Far East Con" (ISCFEC 2020)
Solution of the Decentralized Task of Evaluating and
Improving Product Quality
G T Pipiay1, L V Chernenkaya1, V E Mager1
1
Highest School of Cyber-physic Systems and Control of the Institute for Computer
Sciences and Technologies of Peter the Great St.Petersburg Polytechnic University,
St.Petersburg 194021, Russia
E-mail: gogpipiy@ya.ru
Abstract. Management of products quality needs from a producer to decide following
tasks: technologies of production, economical issues, management of purchases. All
these tasks are aimed on the obtaining of a rational quality level of products. For this
various methods and models for monitoring a quality of product are developed. The
goal of this paper is to develop a method for assessing and improving product quality,
based on a multi-level optimization. The problem of quality evaluating is considered,
based on decentralization, quality objective functions, developed methodology for
assessing and improving product quality and proposed ways to improve the developed
methodology.
1. Introduction
One of the key factors for ensuring of competitiveness of instrumentation organizations is a rational
decision-making in regard to the quality of products. The rationality means that before deployment of
decision-making in regard to the quality we must analyze both qualitative and quantitative
information, and define best of possible alternatives.
The decision-making task becomes difficult, when a lot of factors should be taken into
consideration. The main factors include those, which effect on the effectiveness of the quality
management system, and the example of these is described in [1].
While solving a complex problem, which referred with uncertainty and difficulty of formalizing of
a decision-making system, methods based on the expert knowledge are used. The decision of a
problem, in this case, could be defined as a set-theoretic model:
Q, f1 ( A), f 2 ( A),.., f n ( A), R, K1 , K 2 ,..., K m , N ,
where f i ( A) is the preference relationship function of one alternative over others; R is a binary
relations on set; A , K j are comparisons criteria of alternatives; N is logics normalization of criteria;
Q is the objective function for finding the numerical value of the solution on multi- criteria on the
multicriteria set.
The task becomes more complicated, if several structural units are involved in the decision-making
process. If methods from classical qualimetry are used (or methods described above), the time for
decision-making increases, and objectivity of a product quality assessment is going down.
Copyright © 2020 The Authors. Published by Atlantis Press SARL.
This is an open access article distributed under the CC BY-NC 4.0 license -http://creativecommons.org/licenses/by-nc/4.0/.
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Among the most popular methods for solving decentralized decision-making tasks, methods from
the theory of multi-optimization are known. These methods were applied for managing of activities of
the railway transport hub [2], for optimal designing of load lifting mechanisms [3]. Other works, using
the theory of multi-optimization, are [4], [5].
2. Formulation of the problem of evaluating of the quality level of instrumentation products
Objective functions for the decentralization task, based on requirements to a quality monitoring model,
is defined in the following way: Q( x, y1, 2 ) is the performance monitoring model, F1 ( x, y1, 2 ) is a
function of quality costs, F2 ( x, y1, 2 ) is a function of purchasing management. The function of quality
is such, that Q : X Y1 Y2 R , where
y
is the quality criteria,
yi Yi R mi , where
Fi : X Yi R .
In the feasible region, for solving the above problem, we need to find a minimum point:
k
S {( x, y1, y2 ) X Y1 Y2 : Ax Bi yi b,Ai x Ci y bi , i 1,..., k} .
i 1
The structure of the task is shown in figure 1.
Figure 1. The structure of the task.
The task could be described by the following analytical model:
min Q ( x, y1 y, y 2 z , y3 w) cx d1 y d 2 z
xX
Ax B1 y1 B2 y 2 b1
min F ( x, y ) cx d1 y
yiY 1
Ax B1 y b
min F ( x, z ) cx d 2 z
ziZ 2
Ax B2 z b
(2)
The top level is intended for formulation of additional criteria for bottom levels. These criteria need
for managing of bottom levels. According with [6], “The leader goes first and chooses in an attempt to
optimize (maximize or minimize) his own objective function F ( x, y( x)) , subject to additional
constraints”.
The state of work for the top level is defined on following output properties:
the level of yield of products: x1 ;
the degree of effectiveness of the developed preventive activities: x2 ( y2 z 2 ) ;
the degree of effectiveness for new introduced technologies and techniques: x3 ( y3 z3 ) .
Variables x 2 , x3 will be defined the using of three variables of the bottom level y, z (more
detailed information about bottom variables is below).
The field of the definition of function Q(x) we’ll set through the polyhedron X 5 , with the
functional blocks of the device, shown in figure 2.
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Figure 2. The device.
The feasible region of Q(x) is defined by the following matrix:
a11x1 a12 x2 ( y 2 z 2 ) a13 x3 ( y3 z3 )
a21x1 a22 x2 ( y 2 z 2 ) a23 x3 ( y3 z3 )
a31x1 a32 x2 ( y 2 z 2 ) a33 x3 ( y3 z3 ) .
a41x1 a42 x2 ( y 2 z 2 ) a43 x3 ( y3 z3 )
a51x1 a52 x2 ( y 2 z 2 ) a53 x3 ( y3 z3 )
Criteria c1 x1 are calculated through an indicator of defect per unit in the following way:
d
a x 1 (1 ) ,
i1 1
m
(3)
where d is the number of founded defects, m is number of inspected units.
Criteria c2 x2 ( y 2 , z 2 ) are calculated as:
1c
a x (y ,z ) x (y ,z ) 2 ,
i2 2 2 2
2 2 2
(4)
Р
where x2 is the relation of the working corrective and preventive measures to the implemented
В
C ( Р)
ones (in the area of purchasing management and cost management), c2
is the relation of
C ( В)
expenses of working activities to the implemented cost (in the area of purchasing management and
cost management).
Criteria c3 x3 ( y3 , z3 ) are calculated in the same way, as the criteria c2 x2 ( y 2 , z 2 ) , where the
numerator is the ratio of working technologies, and the denominator is the number of embedded.
3. Formalization objective functions for bottom levels
The most popular methods for classifying of quality costs can be found in [7]. For evaluating costs of
quality F1 ( y ) on a manufacturing stage we introduce the following criteria:
costs of quality assessment - y1 ( x1 ) ;
costs of preventing of defects - y2 ( x1 ) ;
costs for eliminating of internal defects - y 3 .
In order to determine a feasible region for function F1 ( y ) . In articles [8], [9], a lot of causes are
listed for classification costs of quality. Considering the assessment of the quality level at the
manufacturing stage, we will determine the following causes:
costs for a control of technology stage - b1' ;
costs for inspection of products - b 2' ;
costs for analyzing of defects - b3' ;
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costs for eliminating of defects - b 4' ;
costs of contracting with the supplier - b5' ;
costs of providing quality of purchases - b6' .
The feasible region for function F1( y) will be introduced through the polyhedron Y 7 :
b11' y1
'
b22
y2
b32' y2
'
b43
y3
.
b53' y3
b61' y1 b61' y2
Searching for the optimum function values is done by the method proposed by A. Feigenbaum
[10]. A detailed description of this approach can be found in [11]. For calculating of the function
F1 ( y ) it is necessary to normalize its criteria.
y
y
Criteria d1 y1 , d 3 y3 and d 3 y3 will be calculated as d1 y1 ( x1 ) b / i1 ( x1 ) 1 ai1 ( x1 ) 1 ,
Pc
Pc
y
y
y
d 2 y2 ( x1 ) b / i 2 ( x1 ) 2 ai 2 ( x1 ) 2 , and d3 y3 b / i 3 3 , where d i/1 ( x1 ), d i/2 ( x1 ) are coefficients of
Pc
Pc
Pc
fixed costs for providing a level of a good final product, d i 3 1 is a cost depending on defects, Pc is
production cost.
Producers, who are guided by the requirements of ISO 9001-2015, need to determine a list and
suitability of suppliers through assessments of them by following criteria:
timely delivery - z1 ;
timeliness of defects elimination - z 2 ;
effectiveness of appeals to the supplier on emerging issues - z 3 .
/
For calculating of the partial criteria of the function F2 ( z ) we use an approach based on
comparative cost model. Analysis of this approach is presented in articles [12], [13]. Dimension of the
feasible region Bz for the function F2 ( z ) is equal to the number of suppliers involved in the
production process of flaw detectors.
Criterion d1 z1 could be found by the formula:
n
z (t )
1
bi1 z1 ( i 1
m
i
)
1
11d ,
z (t ) u
1
(5)
where z1 (t ) i is the time needed to eliminate defects or customer satisfaction, m is a number of
reference points, z1 (t )u is the time needed to eliminate defects or customer satisfaction under the
contract, d 1 is the ratio of the amount of the cost of components and the cost of applying them to their
destination to the cost of the flaw detector.
Criterion d 2 z 2 will be found by the formula:
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n
z (t )
bi2 z 2 ( i 1
2
n
i
)
1
11d ,
z 2 (t ) u
(6)
2
where z 2 (t ) i is time points for delivering components, n is the number of reference points, z 2 (t ) u is
the time to deliver components under the contract, d 2 is the ratio of the amount of the cost of
components and the cost of applying them to their destination to the cost of the flaw detector.
Criterion d 3 z3 will found by the formula:
bi3 z3 z3 ,
(7)
where z 3 is the relation of closed questions to the total number of questions.
4. Calculating the quality level of the product and defending ways for improving product
quality
The optimization of a double-level model with two followers could be done with using of KuhnTucker conditions and the simplex method. Let’s set dual variables u i R q , vi R m (i 1,..., k ) ,
i
i
which are related to functions Q( x, y, z ) , F1 ( y ) , F2 ( z ) . Constraints for systems of linear equations are
determined in the following way. If partial values of the i-th equation are differed from 1, we need to
put at the right-hand side the average value of these quantities with the sign ≥. If all of partial values of
the i-th equation are equal to 1, then at the right side we put number 1 with the sign ≤. If the equation
has one variable, then the constraint is put with the sign ≤.
The analytical model of optimization task (2) looks in following way:
min Q( x, y, z ) Cx d1 y d 2 z
Ax By Bz b, Ax Ci yi b j
ut B st et
ut (bt Ax By ) st y 0
vt B st et
vt (bt Ax Bz ) st z 0
( x, y, z ) 0, ut 0, vt 0.
(8)
The example of task is listed in the table 1.
Table 1. The example.
Q
E1
E2
E3
E4
E5
F1
E1
E2
E3
E4
E5
E6
F2
E1
E2
x1 x2(y2) z2 y3 z3
b
0,8 0.3 0,1 0,8 0,8 ≥0,68
0,3 0,1 0,6 0,9 1 ≥0,63
0.7 0.9
1 0,7 1 ≥0,84
0,1 0.9
1
1 0,5 ≥0,68
1
1
1
1
1 ≤1,00
x1 x2(y2) z2 y3 z3
b/
0,1
≤0,1
0,5
≤0,5
1
≤1,0
1
≤1,0
- 0,9 ≤0,9
0,9 0,7
≥0,8
x1 x2(y2) z2 y3 z3
b//
0.3
0,5 - 0,8 ≥0,53
1
1
1
≤1
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Taking into account the example table 1, the task (8) will look like:
min Q ( x, y , z ) 1 x1 y 2 z 2 y3 z3
0,8 x1 0,3 y 2 0,1z 2 0,8 y3 0,8 z3 0,68
0,3 x1 0,1 y 2 0,6 z 2 0,9 y3 z3 0,63
0,7 x1 0,9 y 2 z 2 0,7 y3 z3 0,84
0,1x1 0,9 y 2 z 2 y3 0,5 z3 0,68
x1 y 2 z 2 y3 z3 1,0,1x1 0,1
0,5 x2 0,5, x2 1, y3 1
0,9 y3 0,9,0,9 x1 0,7 x2 0,8
0,3 x1 0,5 z 2 0,8 z3 0,53
x1 z 2 z3 1, u1 0,9u 2 u3 1
u1 (1 y3 ) 0, u 2 (0,9 0,9 y3 ) 0, u3 ( y3 ) 0
0,5v1 v2 v3 1,0,8v1 v2 v4 1
v1 (0,3 z1 0,5 z 2 0,8 z3 0,53) 0
v2 (1 z1 z 2 0 z3 ) 0
v3 ( z1 ) 0, v4 ( z 2 ) 0, ( x, y , z ) 0, ui 0, vi 0.
Optimization model with using of Kuhn-Tucker theorem will be as following:
min Q ( x, y, z ) 1 x1 y2 z 2 y3 z3
0,8 x1 0,3 y2 0,1z 2 0,8 y3 0,8 z3 0,22
0,3 x1 0,1 y2 0,6 z 2 0,9 y3 z3 0,08
0,7 x1 0,9 y 2 z 2 0,7 y3 z3 0,32
0,1x1 0,9 y 2 z 2 y3 0,5 z3 0,02
x1 y2 z 2 y3 z3 1
0,1x1 0,1,0,5 x2 0,5, x2 1, y3 0
0,9 x1 0,7 x2 0,8
0,3 x1 0,5 z 2 0,8 z3 0,1
x1 z 2 z3 1, z 2 0, z3 0
u1 0,9u 2 u3 1
u1 (1 y3 ) 0,u 2 (0,9 0,9 y3 ) 0,u3 ( y3 ) 0
0,5v1 v2 v3 1,0,8v1 v2 v4 1
v1 (0,3 z1 0,5 z 2 0,8 z3 0,53) 0
v2 (1 z1 z 2 0 z3 ) 0
v3 ( z1 ) 0, v4 ( z 2 ) 0, ( x, y , z ) 0, ui 0, vi 0.
More detail information about bi-level programming one can see in [14]. Based on the results of
implementation of simplex method, the quality level of flaw detector at the production stage was
determined as Q( x, y, z) 0,67 . Values of variables are introduced in the Table 2.
Table 2. Values of variables.
N
1
2
3
x
y
z
1
2
0,3333333 0,7142857
0
0,6666667
0,00
0,00
3
0
0
0,00
Thus, for enhancement of the product quality it is needed:
To replace suppliers, or to update procedures that regulate relations with suppliers;
To reduce expenses for preventive measures (or update them), and update procedures for
assessment of product quality in operational inspection.
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Advances in Economics, Business and Management Research, volume 128
5. Conclusions
Presented approaches for assessing of the level of quality possess advantages in comparing with
standard methods of qualimetry. These advantages are achieved by including of a lot of factors into
the model, which affect on the quality of products. The fact that the model has many factors makes the
method more complicated for calculating the quality of product. For this it is recommended to use at
intermediate stage with multidimensional cluster analysis methods or fuzzy logic methods.
Recommended to change the linear type of model optimization for nonlinear, in order to increase
the feasible region. To assess the accuracy of decision-making, it is necessary to involve a probabilitytheoretic approach in the proposed methodology.
The developed methodology based on multi-level optimization allows to take into account many
factors, and flexibly manage the quality of the product. Quality management is carried out by making
changes to certain indicators of product quality, both from the side of economic losses and from the
side of production technology.
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