2.1. Supply Network Model
The supply network (SN) is also known as the supply chain network. The SN can be simplified to a single supply chain when there is only one chain in it. In the supply chain, raw materials, intermediate materials, and finished goods are procured exclusively. A simple supply chain has been described previously by a set of differential equations [
31,
32,
33]. This model can express the complex behavior of the supply chain perfectly. However, it cannot express complex network structures, meaning it is only suitable for a single-chain structure.
With the increase of supply objects and purchase objects, the supply chain structure is gradually upgraded to a network structure. The primary goal of the SN is to satisfy customer demands at the lowest cost.
From the perspective of structure, SN can be viewed as a collection of nodes V = {vi} (retailers, distributors, and manufacturers) and arcs E = {eij} (the set of relations between the enterprise nodes). SN can be defined as a set of self-organizing agents that interact through a series of links. Therefore, SN can be expressed as G = (V, E).
Since there isn’t a central, authoritative organizing node, the SN is a self-organizing system. A dynamic SN is formed when each node is treated as a dynamic system. Each node has the behavior of ordering goods, production, and sales, whether it is a retailer, distributor, or manufacturer. In view of these studies [
31,
32,
33], the status of each node in the SN can be described as
The status of each enterprise can be expressed as Equation (2). The parameters are expressed in
Table 1.
The variation of demand quantity
relates to the dissatisfaction
m in the preceding period. The variation of the supply quantity
is influenced by the information distortion
r and the supply quantities
xi2 in the preceding period. It also needs to take into account both the demand quantities
xi1 and the inventory
xi3. The variation of inventory
not only relates to the demand quantities
xi1 and the supply quantities
xi2, but also depends on safety stock [
34].
The state of each enterprise can be seen as a process, including order-taking, product-making, and goods distribution; similarly, the status of each node is affected by its neighboring nodes [
35,
36,
37]. This is expressed by Equation (3).
In Equation (3), the first item represents the state of node vi itself, and the second item represents the impact on the state of node vi by the neighboring nodes with a supply relationship. The state of each node is mainly adjusted by its own demand quantities xi1, supply quantities xi2, and inventory xi3. In SN, there is not a one-to-one relationship, but a very complicated relationship.
There are interactions between supply quantities, demand quantities, and inventory. The supply quantity from vi to vj is not the demand quantity form vj to vi. Therefore, the state of the node is also adjusted by the state of the neighbor nodes with a supply relationship, using the second term.
The set of neighbors of node vi is denoted by Ni. The neighbor of node vi means that there is a supply relationship with the node vi. The number of nodes vj satisfying shows how many nodes supply node vi. The supply relationship may be one-to-one, or several-to-one, which is expressed by the adjacency matrix G = [gij]. When there is a business relationship between node vi and vj, gij = 1, otherwise gij = 0. G is symmetric and non-negative. The inner coupling matrix Γ = diag(c1, c2, c3) is a diagonal matrix to indicate coupling strength: where c is the coupling coefficient; c1 represents the influence of neighbors of node xi on the first component, demand quantities; c2 represents the influence of neighbors of node xi on the second component, supply quantities; c3 represents the influence of neighbors of node xi on the third component, inventory. The physical interpretation of the inner coupling matrix is the ability of mutual control between nodes.
For example, there is a supply relationship between a raw material node xj and a factory node xi. Since the raw material node xj does not only supply to the factory node xj, and the factory node xi does not only demand from the raw material node xj, the supply quantities of the raw material node xj cannot be coupled to the demand quantities of the factory node xi directly. In Equation (3), the demand quantities of the factory node xi (xi1) are related to the supply quantities of the raw material node xj (xj2) (i.e., gij = 1, xi1–xj2). Furthermore, in the first item of Equation (3), Fj(xj), the supply quantities of the raw material node xj (xj2) are related to the demand quantities of the raw material node xj (xj1) (i.e., xj2–xj1). So, the supply quantities of the raw material node are coupled to the supply quantities of the factory node xi (i.e., xj2–xj1), which expresses the connections between nodes in Equation (3).
If the nodes are only interconnected through the demand relationship, then the nodes are coupled through the first component. In this scenario, Γ = diag(c, 0, 0).
The configuration matrix
A = [
aij] is used, therein
aij = −
gij (
i ≠
j),
, then Equation (3) becomes Equation (4)
Therein,
α is the vector with scheduled parameters,
φi is the scheduled status,
f1i is the function without parameters, and
f2i is the function with parameters. The validity of the proposed SN model has been presented in our previous study [
30].
2.3. The Resilient Recovery Measurement Based on the Outer Synchronization Error
Resilience is the ability to accomplish the original plan in any case [
39]. As shown in
Figure 4, a quantifiable and time-dependent performance function is the basis for the measurement of the resilience [
6,
13,
40,
41]. The dotted curve
FT(
t) denotes the targeted performance function of SN if not affected by disruption. The disruption deteriorates the performance to the level
F(
td) at time
td. Then, the performance function of SN
F(
t) will be improved by recovery action and reach the targeted level
FT(
t) at a later time
tr.
Let
R1(
t) be the resilient recovery ratio of SN at time
t (
t >
td).
R1(
t) describes the cumulative functionality that has been restored at time
t, normalized by the expected cumulative functionality if the SN has not been affected by disruption.
R1(
t) is given as Equation (6), where
R1(
t) is quantified by the ratio of the area with diagonal stripes
S1 to the area of the shaded part
S2 [
6].
Outer synchronization error ei → 0 shows that the actual status restores itself to the scheduled status. That is to say, the smaller the outer synchronization error, the better the resilient recovery.
Total outer synchronization error of all nodes is expressed as Equation (7):
The outer synchronization error of all nodes E(t) = |F(t) − FT(t)| can be used to measure the resilience.
Taking advantage of the outer synchronization error, a function for the assessment of the resilience is illustrated in
Figure 5. It is a quantifiable and time-dependent performance function, also known as the resilient recovery ratio, which can be measured by
R(
t):
Equation (8) focuses on the resilient recovery in the range of [0, 1]. R(t) = 0 when E(t) = ET(t). This means that SN has not recovered from its disrupted state (there has been no “resilience” action); R(t) = 1 when E(t) = 0, which corresponds to the ideal case, where SN recovers to its scheduled state immediately after disruption.
This resilient recovery quantification is capable of measuring both the magnitude and rapidity of recovery. Moreover, this measurement of resilient recovery is not memoryless, since it considers the cumulative recovery of the functionality.