Study of the Structural Significance of Supply Chain Elements with Variable Order Rate Alexander Pavlova ,b, Alexander Umarova and Yevgeny Aleshina a Mozhaisky Military Aerospace Academy, Zhdanovskaya street, St. Petersburg, 197198, Russia b Saint Petersburg Federal Research Center of the Russian Academy of Sciences (SPC RAS), 14th line of Vasilievsky island, 39, St. Petersburg, 199178, Russia Abstract One of the most significant stages of building a supply chain is the analysis of the criticality of the elements that make up it. When assessing the criticality of functional elements of adaptive supply chains, it is proposed to use the corresponding integral indicators. The introduced indicators, in addition, allow calculating the structural significance taking into account fluctuations in demand, based on the idea of a parametric genome of the structure of complex systems. This article presents the results of evaluating the significance of the functional elements of a certain supply chain with varying intensities of customer orders. Keywords Сriticality of failure of elements, integral indicator, parametric genome, complex multi-mode object, joint and separate receipt of customer orders 1. Introduction identification of goods, management of 1 acceptable risks in the transport and logistics The present and future of modernization system, which contributes to ensuring the of the economy, new relationships between integrated safety of the transportation process transport organizations and cargo owners in in supply chains [4-7]. Russia are developing in the direction of using Under these conditions, the question of innovative systems [1, 2]. Time, quality, studying the structural and functional safety, costs have become almost the most properties of supply chains (SC) from the critical factors in the management of transport standpoint of considering them as complex and logistics systems. According to the systems becomes relevant [8-10]. The constant authors, it is necessary to move from existing, increase in the complexity of the structural and predominantly functional, management functional features of the SC leads to the methods to process ones, which are based on spread of methods that take into account not risk management systems [3]. Solving this only the numerical values of the reliability complex problem requires not only process indicators of the functional elements (FE) of analysis, but also reliability management the supply chains (warehouses, manufacturing mechanisms. plants, suppliers, distributors, etc.), but also Ensuring security and control over more general assessments of the impact of transport and logistics processes in supply failures elements on the functioning of the chains is based on integrated risk, which is a entire SC under consideration, namely, new management tool for the transport sector. assessing the criticality of FE failures [11]. New approaches to improving the efficiency of Revealing the level (degree) of criticality of processes are based on modeling, labeling and failure for each element of the SC allows you to focus on improving the most important 1 Intelligent Transport Systems. Transport Security - 2021, May nodes in terms of SC functioning [12]. The 14, St. Petersburg, Russia criticality of FE failures must be considered as EMAIL: vka@mil.ru (A.N. Pavlov); vka@mil.ru (A.B. Umarov); vka@mil.ru (Ye.N. Aleshin). a complex property, which includes several ORCID: 0000-0002-6193-8882 (A.N. Pavlov); 0000-0003- particular indicators: the degree of redundancy 3308-8806 (A.B. Umarov); 0000-0003-1181-4416 (Ye.N. of the element; the likelihood of failure; Aleshin) ©� 2021 Copyright for this paper by its authors. Use permitted under Creative resistance of functional elements to external Commons License Attribution 4.0 International (CC BY 4.0). CEUR Workshop Proceedings (CEUR-WS.org) influences; structural significance, etc. [13]. Another important condition in solving introduced  earlier by the authors problems of evaluating, analyzing and χ (α1 , α 2 ,..., α m ) = ( χ 0 (α1 , α 2 ,..., α m ), synthesizing the appearance of a SC is the χ1 (α1 , α 2 ,..., α m ),..., χ n (α1 , α 2 ,..., α m ))T , need to take into account various options for –vector, the elements of which are the the receipt of customer orders, which coefficients of the probabilistic polynomial (2) significantly affect the survivability and of the successful functioning of the SC for the structural and functional reliability of both case of a homogeneous structure (equal individual elements and the entire SC as a probability of failure-free operation of the whole [11, 14-16]. FE P1= P2= ...= Pn= P ), The presented article presents the results of ℜ( P, α1 , α 2 ,..., α m ) = χ 0 (α1 , α 2 ,..., α m ) + a study of the structural significance of the (2) functional elements of supply chains, + χ1 (α1 , α 2 ,..., α m ) P + ... + χ n (α1 , α 2 ,..., α m ) P n , depending on the nature and intensity of the receipt of dynamic customer orders. we calculate the indicators of structural and functional reliability for a homogeneous and 2. Method for studying the inhomogeneous SC structure according to the structural significance of elements formulas (3) [18,19] 1   of the supply chain ∫0 ℜ(P,α1,...,α m )dP = Fhom og ( χ (α1 ,..., α m )) = 1 1 χ (α1 ,..., α m ) ⋅ (1, , ,..., 2 3 n +1 1 T ), To assess the structural significance of the (3) SC elements, it is advisable to use the   1 1 1 Fheterog ( χ (= α1 ,..., α m )) χ (α1 ,..., α m ) ⋅ (1, , 2 ,..., n )T , capabilities of the general logical-probabilistic 2 2 2  method and the program complex of logical- Fhom og poss ( χ (α1 ,..., α m= )) sup min{γ , G ({µ ℜ( µ , α1 ,..., α m ) ≥ γ })}. γ ∈[0,1] probabilistic modeling "Arbiter" [17], which is a universal graphical tool for the structural representation of the studied properties of To assess the structural significance of the complex objects. FE, we calculate the polynomial (4) taking into As a rule, the analysis of structurally account the dynamically changing intensities complex objects begins with the construction of customer orders and call it the significance of a functional integrity diagram (FID) of the polynomial. The resulting polynomial is object [3]. The structure of the constructed calculated by differentiating the probabilistic circuit includes functional elements (FE), polynomial of the successful functioning of the which are various technological operations, SC by the availability factor (probability of subsystems, blocks, nodes, connections of no-failure operation) of the i-th element [17]. various physical nature, elements, etc. ∂ℜ( P1 ,..., Pn , α1 ,..., α m ) ξi ( P1 ,..., Pn , α1 ,..., α m ) = = (4) Based on the FID, we will calculate the ∂Pi probabilistic polynomial of the successful = ℜ( P1 ,..., Pn , α1 ,..., α m ) P 1 = = − ℜ( P1 ,..., Pn , α1 ,..., α m ) P 0 . i i functioning of the SC [12,13,17], taking into Then each polynomial of significance account the conditions of joint and separate ξi ( P1 ,..., Pn , α1 ,..., α m ) (∀i =1,..., n) can be receipt of dynamic customer orders. Let the probabilistic polynomial of the associated with a parametric genome  successful functioning of the SC (customer χ i (α1 , α 2 ,..., α m ) , the substitution of which in service) have the form (1) formulas (3) allows calculating the ℜ( P1 ,..., Pn , Pn +1 ,..., Pn + m , Q1 ,..., Qn , Qn +1 ,..., Qn + m ) (1) significance of functional elements of где Pi (Qi = 1 − Pi ), i = 1,..., n - the probability of homogeneous and heterogeneous supply chains. failure-free operation (failure) of the FE SC, It is easy to see that the structural and Pn +i (Qn +i = 1 − Pn +i ), i = 1,..., m - will be significance of the FE of SC depends on the understood as the intensity of receipt (absence) nature and intensity of the receipt of dynamic of a customer order on the considered time customer orders [20,21]. We get that in interval, varying from 0 to 1, and denote them addition to separate or joint receipt of orders, = by α i P= n +i , i 1,..., m . the intensity of receipt can be equal Based on the concept of the parametric α1= α 2= ...= α m= α or unequal. For the genome of the structure of SC [18] four options for the receipt of dynamic customer orders considered in the example, (8). Let us denote the parametric genome of using the approach proposed in [11,17], to the significance polynomial of the element assess the structural significance of the FE of under consideration for the case of equivalent SC, we introduce the following integral (unequal) in intensity of incoming customer   indicators (5) - (8) orders by χ (α ) ( χ (α1 ,..., α m )) . In this case, 1/ m  J se= m ⋅ ∫ F* ( χ p (α )) dα , (5) for a homogeneous or inhomogeneous 0 structure, as in formulas (5) - (8), we use either 1  J je = ∫ F* ( χ c (α ))dα , (6)   1 1 1 T 0 F* ( χ (= α1 ,..., α m )) χ (α1 ,..., α m ) ⋅ (1, , ,..., ) ,  2 3 n +1 J su= m !⋅ ∫∫∫ F* ( χ p (α1 ,..., α m ))dα1dα 2 ...dα m , (7)   1 1 1 T α α 1 + ...+ m ≤1 0 ≤α i ≤1, i = 1,..., m F* ( χ (= α )) χ (α ) ⋅ (1, , ,..., ) , 1 1 1 2 3 n +1  J ju = ∫ ∫ ...∫ F* ( χ c (α1 ,..., α m ))dα1dα 2 ...dα m , (8) or 0 0 0   1 1 1 Integral indicators (5) and (6) are intended F* ( χ (= α1 ,..., α m )) χ (α1 ,..., α m ) ⋅ (1, , 2 ,..., n )T , to assess the structural significance of the FE 2 2 2 of homogeneous and heterogeneous SC with   1 1 1 T F* ( χ (= α )) χ (α ) ⋅ (1, , 2 ,..., n ) . respectively separate and joint receipt of 2 2 2 customers of equal intensity orders, and indicators (7) and (8) - with respectively The use of the polynomial for the separate and joint receipt of unequal in successful functioning of the supply chain (1) intensity orders. It is easy to understand that allows one to obtain parametric genomes of Fhom og , Fheterog or Fhom og poss can be used as both the entire SC and the values of its FE. It the integrand F* in formulas (5) - (8) for the should be noted that expression (1) has the form of a polynomial [17], the monomials of corresponding parametric genomes which include variables with degrees either 1     χ s (α ), χ j (α ), χ s (α1 ,..., α m ), χ j (α1 ,..., α m ) . or 0. In this case, the integrand can be Here represented in its most general form as follows  i    (9) χ s (α ) χ= = s (α ), χ j (α ) χ ij (α ), χ= s (α1 ,..., α m ) i  i  m m m χ=s (α1 ,..., α m ), χ j (α1 ,..., α m ) χ j (α1 ,..., α m ) β 0 + ∑ βiα i + ∑ ∑ βijα iα j + F* ( χ (α1 ,..., α m )) = i= 1 i = 1 j = i +1 (∀i =1,..., n) respectively, the parametric m m m (9) + ∑ ∑ ∑ βijk α iα jα l + ... + β12...mα1α 2 ...α m . genomes of the structural significance of the i= 1 j= i +1 k =+ j 1 functional elements of the adaptive supply Using the obtained equality (9), we obtain chain with an incompatible (separate) receipt simplified expressions for the integral of customers of equal intensity, with a joint indicators of the structural significance of the receipt of orders of equal intensity, with a FE of a homogeneous or inhomogeneous SC. separate receipt of orders of unequal intensity, Then, as can be seen from the reasoning, with a joint receipt of customers with unequal formulas (5), (6) and (8) take the following intensity of orders. form 1/m 3. Conversion of expressions in  m ⋅ ∫ F* ( χ (α ))dα = J se = integrated indicators for 0 1/m simplified estimation of the m m m = m ⋅ ∫ ( β 0 + ∑ βiα + ∑ ∑ βijα 2 + ... + β12...mα m )dα = structural significance of elements 0 i= 1 i = 1 j = i +1 of SC m m m βi ∑ ∑ βij To calculate the values of integral β0 ∑ = = + β = + i =1 + i 1 j 2i 1 + ... + m 12...m , indicators of the structural significance of m 0 ⋅1 m ⋅ 2 m ⋅3 m ⋅ (m + 1) functional elements, taking into account the nature and intensity of the receipt of dynamic customer orders, we will use formulas (5) - 1 1 1−α  m! 1 =J je ∫= F* ( χ (α ))dα ⋅ ∫ α1dα1 ∫ α 2 dα 2 ... 0 1⋅ 2 ⋅ ... ⋅ (m − k ) 0 0 1 m m m 1−α1 −α 2 −...−α k −1 = ∫ ( β 0 + ∑ βiα + ∑ ∑ βijα 2 + ... + β12...mα m )dα= 0 i= 1 i = 1 j = i +1 ... ∫ 0 (1 − α1 − α 2 − ... − α k ) m − k ⋅ α k dα k . m m m β0 ∑ βi = = + ∑ ∑ βij β12...m = + i =1 + i 1 j i 1 + ... + , It is easy to see that the last integral of this 1 2 3 (m + 1) expression is equal to 1 1 1 1−α1 −α 2 −...−α k −1  J ju ∫= ∫ ...∫ F* ( χ (α1 ,..., α m ))dα1dα 2 ...dα m 0 0 0 ∫ 0 (1 − α1 − α 2 − ... − α k ) m − k ⋅ α k dα k = 1 1 1 m m m (1 − α1 − α 2 − ... − α k −1 ) m − k + 2 = ∫ ∫ ...∫ (β0 + ∑ βiα i + ∑ ∑ βijα iα j + ... + 0 0 0 i= 1 i = 1 j = i +1 = (m − k + 1) ⋅ (m − k + 2) . + β12...mα1α 2 ...α m )dα1dα 2 ...dα m = As a result, we get m m m β0 ∑β ∑ ∑ β β12...m ∫∫∫ ij m !⋅ α i1 ⋅ α i2 ⋅ ... ⋅ α ik dα1dα 2 ...dα m = i = 1 j = i +1 = 0 + i =1 1 + i 2 + ... + m . 2 2 2 2 α1 +...+α m ≤1 0≤α i ≤1,i =1,..., m Simplifying expression (7), we obtain  m! J su = m !⋅ ∫∫∫ F* ( χ p (α1 ,..., = α m ))dα1dα 2 ...dα m = ⋅ α1 + ...+α m ≤1 0 ≤α i ≤1, i = 1,..., m m !⋅ (m + 1) ⋅ (m + 2) ⋅ ... ⋅ (m + k − 2) m m m 1 ( β 0 + ∑ β iα i + ∑ ∑ β ijα iα j + ... + 1 = m !⋅ ∫∫∫ ⋅∫ (1 − α1 )m+ k −2 ⋅ α1dα1 = . α1 + ...+α m ≤1 0 ≤α i ≤1, i = 1,..., m i= 1 i = 1 j = i +1 0 ( m + 1) ⋅ ( m + 2) ⋅ ... ⋅ ( m + k ) m β ∑β i + β12...mα1α 2 ...α m )dα1dα 2 ...dα m =0 + i =1 + 4. Results of calculating the 1 m +1 m m structural significance of elements ∑∑β i = 1 j = i +1 ij β12...m of SC with variable intensities of + + ... + . (m + 1) ⋅ (m + 2) (m + 1) ⋅ (m + 2) ⋅ ... ⋅ (m + m) customer orders Let's calculate the structural significance of In this m-fold integral, the value of the the elements of some adaptive supply chain in integral of any monomial consisting of k ≤ m the context of dynamically changing customer different variables and included in polynomial orders. In this article, we will use the results (9) is a constant value equal to obtained in [11], namely: as functional elements, as already mentioned above, in m !⋅ ∫∫∫ α1 +...+α m ≤1 α i ⋅ α i ⋅ ... ⋅ α i dα1dα 2 ...dα m = 1 2 k supply chains can be distributors, warehouses, 0 ≤α i ≤1,i = 1,..., m manufacturing plants, providers, suppliers, etc. 1−α1 1 Scheme of functional integrity of some. The = m!⋅ ∫ α1dα1 ∫ α 2 dα 2 ... 0 0 SC, taking into account various options for the 1−α1 −α 2 −...−α k −1 1−α1 −α 2 −...−α k 1−α1 −α 2 −...−α m−1 receipt of dynamic customer orders, is shown ... ∫ 0 α k dα k ∫ 0 dα k +1 ... ∫ 0 dα m . in Figure 1. It should be noted that in the presented If we integrate this monomial over the FID, vertices 1-10 reflect the operability (probability of no-failure operation) of the variables α k +1 , α k + 2 ,..., α m , we get the elements of the SC under consideration, following expression vertices 11-14 reflect the intensity of incoming dynamic customer orders (or can be interpreted as probabilities of incoming customer orders), and vertices 15-33 are fictitious and describe the logical relationships corresponding parametric genomes   between the elements of the supply chain. χ ij (α1 , α 2 , α 3 , α 4 ), χ si (α1 , α 2 , α 3 , α 4 ) . Using the program complex "Arbiter" [17], for the FID of the SC we obtain two Then, for example, for an element polynomials reflecting the probability of its represented on the FID by vertex 1, the successful functioning (10) required polynomials (9) will have the ℜ j ( P1 ,..., P10 , P11 ,..., P14 , Q1 ,..., Q10 , Q11 ,..., Q14 ), following form  (10) Fhom og ( χ 1j (α1 ,...,α 4 ) = 0,25α1 + 0,066667 α 2 + 0,433333α 3 + ℜs ( P1 ,..., P10 , P11 ,..., P14 , Q1 ,..., Q10 , Q11 ,..., Q14 ), +0,433333α 4 −0,11667α1α 2 −0,48333α1α 3 −0,48333α1 α 4 − where ℜ s ( P1 ,..., P10 , P11 ,..., P14 , Q1 ,..., Q10 , Q11 ,..., Q14 ) −0,46429α 2α 3 −0,13095α 2 α 4 − 0,55357 α 3α 4 + - is a function of the probability of satisfying +0,482143α1α 2α 3 +0,14881α1α 2α 4 + customer orders, which is a group of +0,577381α1α 3α 4 + 0,386508α 2α 3α 4 − 0,40079α1α 2α 3α 4 , incompatible events (GIE); and  Fheterog ( χ 1j (α1 ,...,α 4 ) = 0,375α1 + 0,09375α 2 + 0,65625α 3 + ℜ j ( P1 ,..., P10 , P11 ,..., P14 , Q1 ,..., Q10 , Q11 ,..., Q14 ) +0,65625α 4 −0,1875α1α 2 −0,75α1α 3 −0,75α1 α 4 − 0,5α 2α 3 − - is a function of the probability of satisfying orders from non-GIE customers; −0,4375α 2 α 4 − 0,82031α 3α 4 + 0,523438α1α 2α 3 + Pi (Qi ), i = 1,10 +0,460938α1α 2α 4 +0,851563α1α 3α 4 + 0,5625α 2α 3α 4 − - probability of uptime (failure) of supply chain elements, and −0, 57617α1α 2α 3α 4 , P10+i (Q10+i ), i = 1, 4 - is the rate of receipt of  Fhom og ( χ s1 (α1 ,..., α 4 ) = 0, 25α1 + 0,066667 α 2 + dynamic customer orders. 33 +0, 433333α 3 +0, 433333α 4 ,  Fheterog ( χ 1j (α1 ,..., α 4 ) = 0, 375α1 + +0, 09375α 2 + 0, 65625α 3 +0, 65625α 4 . 13 14 11 12 The results of the study of the structural significance of elements of the SC with 31 32 29 30 separate (taking into account GIE) and joint (excluding GIE) receipt of equal or unequal in 25 26 27 28 23 24 intensity orders from customers of a 21 22 19 20 homogeneous or inhomogeneous structure of SC are given in Table 1 and Figure 2. 15 16 17 18 Table 1 7 8 9 10 1 2 3 4 5 6 Values of structural significance of SC elements Num Joint receipt of orders Separate receipt of orders ber of Unequal Equal Unequal Equal Figure 1: Example figure eleme Hom Hete Hom Hete Hom Hete Hom Hete nts og rog og rog og rog og rog Let us calculate the polynomials (9) for the 1 0,20 794 0,293 09 0,16 617 0,226 56 0,23 667 0,356 25 0,14 792 0,222 66 heterogeneous   2 0,13 0,172 0,11 0,141 0,09 0,131 0,05 0,082 ( Fheterog ( χ ij (α1 ,..., α 4 )), Fheterog ( χ si (α1 ,..., α 4 ))) 3 008 0,12 24 0,171 119 0,11 60 0,140 000 0,09 25 0,131 625 0,05 03 0,082 860 26 000 82 000 25 625 03 and homogeneous   4 0,16 0,231 0,13 0,184 0,16 0,243 0,10 0,152 ( Fhom og ( χ ij (α1 ,..., α 4 )), Fhom og ( χ si (α1 ,..., α 4 ))) 895 57 927 38 333 75 208 34 5 0,04 0,071 0,04 0,063 0,03 0,056 0,02 0,035 structure of the SC, having determined for 878 66 474 22 667 25 292 16 6 0,04 0,071 0,04 0,063 0,03 0,056 0,02 0,035 each i-th functional element the significance 878 66 474 22 667 25 292 16 polynomial according to formula (4) for the 7 0,04 0,075 0,04 0,065 0,03 0,056 0,02 0,035 998 56 508 17 667 25 292 16 cases of the presence and absence of GNS 8 0,05 0,076 0,04 0,065 0,03 0,056 0,02 0,035 among the incoming customer orders and the 146 54 627 95 667 25 292 16 9 0,04 0,075 0,04 0,065 0,03 0,056 0,02 0,035 998 56 508 17 667 25 292 16 10 0,05 0,076 0,04 0,065 0,03 0,056 0,02 0,035 146 54 627 95 667 25 292 16 Figure 2: Structural Significance of SC elements in conditions of demand fluctuations Analysis of the results of calculating the significance of the first four functional structural significance of the SC elements elements becomes more homogeneous, while makes it possible to draw the following for others it doubles in contrast to the option of conclusions. receiving orders clients representing GIE. Regardless of the nature and intensity of orders, the first four elements have the greatest 5. Conclusion structural significance, and the maximum To calculate the structural significance of value of the indicators of their significance is the functional elements of a certain supply achieved with a separate arrival of orders of chain, the significance polynomial was unequal intensity, when the supply chain under determined for each individual element as a consideration consists of elements that are result of differentiating the probabilistic heterogeneous in terms of the probability of polynomial of the successful functioning of the failure-free operation. The rest of the SC SC with respect to the variable characterizing elements have approximately equal the probability of failure-free operation of this significance, significantly different from the element. Based on the concept of the significance of elements 1-4. parametric genome and using the derived In addition, taking the value of the formulas for calculating the structural and indicator of the structural significance of FE1 functional reliability of the SC, expressions as 1, with the joint receipt of customer orders, were obtained for the indicators of the regardless of the equivalence of the receipt of structural significance of elements of the SC, orders and the homogeneity of the SC which reflect the contributions to the structural structure, the structural significance of the and functional reliability of the SC when remaining FE will have the following shares of transferring elements from an inoperative state this value: for FE4 0.79-0.84, for FE2 and FE3 to an operable one, taking into account - 0.59-0.67, for others it will be approximately fluctuations in demand. Finally, an example of 0.23-0.29. And in the case of receipt of orders a study of the structural significance of from customers that are GIE: for FE4 it is functional elements of an adaptive supply 0.68-0.69, for FE2 and FE3 - 0.37-0.38, for chain in the context of dynamic customer others - about 0.15. This suggests that with the orders is considered. joint receipt of dynamic customer orders while The analysis of the results obtained in this achieving the goal of the supply chain - article allows us to conclude that the change in satisfying customer orders, the structural the intensity and nature of the receipt of customer orders has a significant effect on the demand uncertainties: A real-life case. values of integral indicators of the structural European Journal of Operational significance of the SC elements, which Research 227 (1), 199–215. (2013) predetermines the need to take into account [9] Acar, Y., Kadipasaoglu, S., Schipperijn, such changes in further studies of the P. 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