The following article presents an approach to modelling, evaluating and analysing the structural and functional reliability and survivability of supply chains in conditions of fluctuating demand. The article proposes the concept of a parametric genome of the structure of complex multi-mode objects for calculating integral indicators of the structural and functional reliability of the supply chain with dynamic consumer orders.
1 Introduction1
In the process of implementing supply chain
management (SCM) in practice, managers have
faced with the problem of adapting to customers’
unplanned orders and individual technological
and economic requirements. How and with what
methods and technologies is it possible to assess
the reliability and stability of the supply chain in
the event of more or less serious deviations and
violations, fluctuations? So, finance losses
derived due to not received orders, fines and
penalties in certain supply chains reach 15% of
the annual turnover [
genome of the structure. Integral indicators of the structural and functional reliability of the supply chain
These indicators should be used as an
additional factors of economic efficiency to
characterize the target (functional) conflict
"efficiency or reliability" from an objective point
of view [
Figure 1 shows a diagram of the functional integrity of a certain adaptive supply chain, taking into account the structural and functional reserve. Apexes 1-10 reflect the performance of individual product suppliers, manufacturing plants, transport enterprises involved in meeting the needs of consumers, represented by vertices 11-14. Vertices from 15 to 33 are fictitious and are used to describe the logical relationships of the functional elements of the supply chain.
For example, apex 33 ensures the customer satisfaction that is a successful operation (achievement of the goal) of the supply chain. So, if there is no need in customer orders despite the refusals of suppliers and manufacturers, the goal of the supply chain will be achieved. It should be noted that orders from consumers in the supply chain differ in the nature and the intensity of their receipt. Firstly, certain orders can be main or auxiliary. In other words, the orders may be inconsistent (that is, they may be executed one at a time), and individual orders may come in at the same time as others. Secondly, orders may consist of different fractions of the arrival time at a given time interval or may have different values of the arrival probability at a given interval, that is, they can have a deterministic or random dynamic nature. Therefore, it is required to analyse and to evaluate indicators of the structural and functional reliability of the supply chain in the conditions of joint and separate receipt of dynamic customer orders.
Using the program complex of logical and
probabilistic modelling "Arbiter" [
Let us denote the intensity of receipt of orders from customers of the supply chain through i Pni , i 1,..., m . Further, based on the assumption that all functional elements of the supply chain are homogeneous in the probability of no-failure operation (P1 P2 ... Pn P) , the probabilistic polynomial of the successful functioning of the supply chain (1) can be transformed to the following form (2) (P,1,2 ,...,m ) 0 (1, 2 ,..., m ) 1(1, 2 ,..., m )P (2) 2 (1,2 ,..., m )P2 ... n (1, 2 ,..., m )Pn
By analogy with the concept of the structure
genome introduced in [
(1 , 2 ,..., m ) ( 0 (1 , 2 ,..., m ), 1 (1, 2 ,..., m ), 2 (1, 2 ,..., m ),..., n (1, 2 ,..., m ))Т the parametric genome of the structure.
Using the parametric genome of the supply chain structure, it is possible to calculate estimates of the structural and functional reliability of the supply chain, depending on the parameters 1 , 2 ,..., m of the intensity of customer orders.
So, in the case of a probabilistic description
of the failure-free operation of functional
elements for a homogeneous structure (the same
probability of failure-free operation of functional
elements), the function of the successful
functioning of the supply chain, represented by
the polynomial (P,1 , 2 ,..., m )
0 (1, 2 ,..., m ) 1(1, 2 ,..., m )P 2 (1, 2 ,..., m )P2
... n (1 , 2 ,..., m )Pn , changes its values in
the interval [
1 i 1 or
1 Fhomog ( (1, 2 ,..., m )) (P,1, 2 ,..., m )dP 0 (3) (1, 2 ,..., m ) (1, 1 , 1 ,..., 1 )T .
2 3 n 1
In the case of a heterogeneous structure (different probability of failure-free operation of functional elements), it can be used as an indicator of the structural and functional reliability of the supply chain
Fheterog ( (1 , 2 ,..., m ))
1 1 ... (P1 , P2 ,..., Pn ,1, 2 ,..., m )dP1dP2 ...dPn 0 0 or using the parametric genome structure formula (4) Fheterog ( (1 , 2 ,..., m )) (4) (1 , 2 ,..., m ) (1, 1 , 12 ,..., 1n )T .
2 2 2
In the case, when performing functions by
functional elements included in the structure of
the supply chain, it is not possible to identify a
well-defined stochastic regularity of failure-free
operation, then it is proposed to use a
fuzzypossibility approach to describing the behavior of
functional elements, which is based on the
concept of space with a measure of possibility
[
So, as an indicator (5) of the structural and
functional reliability of the supply chain with a
fuzzy-possibility description of the behaviour of
its functional elements, one can use [
[
For this indicator, it is necessary to determine the measure of possibility G and, if possible, its distribution function g( ) .
For monotone homogeneous structures, the graph of the polynomial of the possibility of failure-free operation R( ,1, 2 ,..., m ) has the form shown in Figure 3. Here the polynomial R( ,1 , 2 ,..., m ) 0 (1 , 2 ,..., m ) 1 (1 , 2 ,..., m ) 2 (1 , 2 ,..., m ) 2 ... ... n (1 , 2 ,..., m ) n is obtained from the polynomial (P,1, 2 ,..., m ) by replacing the probability P of no-failure operation of functional elements with the possibility of no-failure operation of functional elements of the supply chain. As a measure of opportunity, we will use G({ R( ,1 , 2 ,..., m ) }) G(H ) sup A
AH
sup
R( ,1,2 ,...,m ) where
A is the Lebegue measure. Therefore, in the case of monotone homogeneous structures, the distribution function of the measure of possibility is g( ) 1 . {1 }, Fhomogposs 0.8 0.6 0.4 1.0 0.8 0.6 0.4 0.2 0.2 0
G H Figure 2. Graphic interpretation of finding the indicator of the possibility of failure-free operation of homogeneous structures
Then, for the considered case of a fuzzypossibility description of the failure-free operation of functional elements of the supply chain, the indicator of the possibility of failurefree operation of a monotonic homogeneous structure can be calculated by the formula (6) Fhomogposs ( (1,2 ,...,m )) 1 * , (6) where μ* (1 , 2 ,..., m ) (1, * , *2 ,..., *n )T 1 * .
is a solution to the equation
For non-monotone homogeneous structures, the uptime polynomial R( ,1 , 2 ,..., m ) either does not preserve "0" (R(0,1 , 2 ,..., m ) 1) , or does not preserve "1" (R(1,1 , 2 ,..., m ) 0) .
The graphs of the polynomials of the failurefree operation are shown in Figure 3.
When R(1,1 , 2 ,..., m ) 0 , the measure of possibility is G({ R( ,1 , 2 , ..., m ) }) G(H ) sup A
AH
sup R( ,1,2 ,..., m ) { max min }, where
max sup{ R(,1,2 ,...,m ) } and min inf{ R(,1,2 ,...,m ) } .
When R(0,1, 2,..., m ) 1, the measure of possibility G({ R( ,1 , 2 , ..., m ) }) G(H ) is sup A
AH
sup R( ,1, 2 ,..., m ) {1 ( max min )}, where
max sup{ R(,1,2 ,...,m ) } and min inf{ R(,1,2 ,...,m ) } .
The graphical interpretation of finding the indicator of the possibility of failure-free operation in this case is shown in Figure 3.
G 0.4 0.2 0.2 0
R(,1,...,m)
use the capabilities of the general
logicalprobabilistic method [
Fragment of the polynomial of the circuit functional integrity of the circuit shown in Figure 2 has the following form (P,1 , 2 ,..., m ) P4 (1 P)6 (1 2 )(1 4 ) P2 (1 P)3 (1 1 )(1 2 ) 3 (1 4 ) ... ... 4P5 (1 1 )(1 2 ) 3 (1 4 ).
To study the structural and functional reliability of the supply chain, we will use formulas (3), (4), (5). In this case, we will assume that all orders can be fulfilled both individually and jointly.
The calculation results for i {0, 0.2, 0.4, 0.6, 0.8,1}, i 1,..., 4, are shown in Figure 4.
Variant №1 corresponds to the absence of orders (1 0; 2 0; 3 0; 4 0) . Variants from №2 to №6 reflect a gradual increase from 0.2 to 1.0 in the intensity of order 1 (the apex of the functional integrity diagram №11) with a step of 0.2. Variants from No. 7 to No. 11, from No. 12 to No. 16, from No. 17 to No. 21 correspond to an increase in the intensity of orders 2, 3, 4 (apexes No. 12, No. 13, No. 14).
Further options from No. 22 to No. 26, from No. 27 to No. 31, from No. 32 to No. 36 reflect the structural and functional reliability of a gradual increase in the joint receipt of two orders 1 and 2, 1 and 3, 1 and 4, respectively. Then the options for the joint receipt of three orders are located (1, 2, 3; 1, 2, 4; 1, 3, 4; 2, 3, 4). And finally, options from 57 to 61 - a joint receipt of four orders. It should be noted that with the individual receipt of orders, the structural and functional reliability of the supply chain is higher than with the joint receipt of these orders. In addition, it can be seen from the graphs that the structural and functional reliability has a piecewise linear relationship with changes in the intensities of the receipt of individual orders. Moreover, for a homogeneous supply chain, it is impossible to clearly identify the best option for the joint receipt of orders, in contrast to a heterogeneous system.
In Figure 5 shows the changes in the values of indicators of the structural and functional reliability of the supply chain during the transition from joint to separate receipt of orders. Let us comment on the results obtained.
The first 20 options reflect a single receipt of orders with a gradual increase in intensity. It should be noted that in this case, the values of the indicators Fheterog ( (1,...,4 )) , Fhomog ( (1 ,...,4 )) , Fhomogposs( ( 1,..., 4 )) do not change for the indicated variants, since there is an individual receipt of orders.
Variants from No. 21 to No. 29, from No. 30 to No. 38, from No. 39 to No. 47 reflect changes in the structural and functional reliability of the supply chain with a gradual increase in the intensity of uniform separate and joint receipt of two orders 1 and 2, 1 and 3, 1 and 4 respectively. Variants from No. 48 to No. 56 - a gradual increase in the intensity of the uniform separate and joint receipt of three orders (2, 3, 4; 1, 2, 3; 1, 2, 4; 1, 3, 4). Then there are variants from No. 57 to No. 60 - a gradual increase in the intensity of uniform separate and joint receipt of four orders.
It should be noted that with the joint receipt of several orders (two, three, four), the feasible assessment of the reliability of a homogeneous structure increases only by 4 6% . The probabilistic assessment of the reliability of a homogeneous or heterogeneous structure can increase for two orders by a maximum of 12% , for three and four orders - by a maximum of 14.5 15.2% . Moreover, the maximum value is achieved with the uniform use of these orders.
Within the framework of this article, a methodology and technology for assessing the reliability and survivability of the supply chain in the event of more or less serious fluctuations in demand are proposed. The proposed approach to the study of the structural and functional reliability of the supply chain under the conditions of changing customer orders is based on the parametric genome of the structure. The analysis of the above results showed that when creating and designing a supply chain, it is necessary to take into account various options (joint-incompatible, equivalent-unequal, homogeneous-heterogeneous) of dynamic customer orders, which significantly affect the reliability and survivability of the supply chain.
Research carried out on this topic was carried out with partial financial support from RFBR grants (No. 17-29-07073, 18-07-01272, 18-0801505, 19–08–00989, 20-08-01046), under the budget theme 0073–2019–0004.