Evolution model of collaborative production task responding to unplanned perturbations

3.1 Perturbations cases

Perturbations to CP are objective reality and cannot be avoided. Some perturbations such as technical improvement can be foreseen over duration of CP, which are called planned perturbations. Braha and Bar-Yam proved planned changes could decrease the value of sensitivity rates and increase the value of internal completion rates and consequently tend to increase the performance of the product development process [7]. However, the more awkward question is another case, which is called unplanned perturbations occurring in uncertain form at uncertain time such as customer requirements changes. Thus, the evolution proposed in the paper is applied to unplanned perturbations to resolve above questions.

Two categories of perturbations could conceivably result in modifications of tasks. Actually, fault-tolerant capability of CP is generally pre-set to cope with unexpected interference. Then some tiny disturbances would be counteracted to keep the whole system stable. In this case, adjustment is mainly manifested in the interior the enterprise of CP, which does not affect the efficiency and effectiveness of other enterprises. Thus, the scale and cooperative mechanism remain unchanged. On the other hand, once that the extent of alteration outstrips fault-tolerant capability, CP tasks may be exposed to redesigned and reallocated. Meanwhile, new cooperative partner maybe entry in and some impotent firms may be washed out. Anyway, perturbations to tasks can be generalized into the following two cases no matter how to change.

Case 1. Modifications of task load, a local adjustment without change networks scale, is applied to some tiny disturbances because the level of modifications generally will not exceed the capacity, which is a less exotic and better understood.

3.2 Modifications policies

Just like Matthew Effect [37], it is evidenced that customers tend to pay more attention to those functions consistent with their preference. Meanwhile, enterprises always spend more effort on some key loopholes. Then the tasks correspond to those functions and loopholes are often defined as critical parts. Naturally, modifications to those critical parts is the most likely to occur. Combined with case 1 and case 2, we define two modification policies of DWCPTN.

Out-information policy: First, modify the node with the highest out-strength, and continue to select and modify nodes in decreasing order of their out-strength connectivity. A high value of out-strength describes that node supplies more constraints to other nodes and with a high importance.

4.1 CA structure

A cellular automaton (CA) is a discrete model studied in computability theory, mathematics, physics, complexity science, theoretical biology and microstructure modelling [38]. A CA consists of a regular grid of cells, each in one of a finite number of states, such as on and off (in contrast to a coupled map lattice). The grid can be in any finite number of dimensions. For each cell, a set of cells called its neighbourhood is defined relative to the specified cell. An initial state (time t = 0) is selected by assigning a state for each cell [39]. A new generation is created (advancing t by 1), according to some fixed rule (generally, a mathematical function) that determines the new state of each cell in terms of the current state of the cell and the states of the cells in its neighbourhood. Typically, the rule for updating the state of cells is the same for each cell and does not change over time, and is applied to the whole grid simultaneously, though exceptions are known, such as the stochastic cellular automaton and asynchronous cellular automaton. Without loss of generality, A CA can be defined a quad:

CA = ( L , Q , N , f )(11)

Here, L, Q, N and f denote lattice space, finite state set, neighbour and state transition function respectively.

Anyway, CA is a discrete dynamic system emerging complex self-organizing behaviour but with straightforward constructs. Most of all, CA is able to describe the evolution of each node, which effectively overcome the defects of differential Fig. 6 based on mean field and stochastic model establish on Markov chain [39]. Motivated the aforementioned, in this paper, complexity and dynamic behaviours of DWCPTN is discussed based on CA theory.

In DWCPTN, each node is considered as a cellular, then the n-nodes network can be regarded as a cellular automata containing n cellulars, which constitutes a one-dimensional cellular space L. Define a network where each node (a task in the CP) could be in one of two states, unresolved or resolved, at any given moment. Then introduce a binary variable q _i  (t) indicating the state of node at time t, such that q _i  (t) =0 means that node i is unresolved and q _i  (t) =1 indicates that node i is resolved. Then Q = (0, 1) represents a state set of CA. Here, this paper considers a dynamic process occurring at discrete times, namely t = 0, 1, 2, ⋯.

Each cellular represents a node in DWCPTN, and the neighbours are the nodes connected by edges. Due to the adjacency matrix W could reflect topology information of network, in this paper, use W to delineate the relationship between cellular and it’s each neighbour. Then, the neighbours of L _i , denoted by N _i could be represented by the set of nonzero elements in line i of W, namely N _i  ={ j|W _ij  ∈ W,  W _ij  ≠ 0 }.

All tasks will normally be resolved unless there is a deliberate termination or force majeure factors. Actually, error tolerance is planned to cope with expected or unexpected changes. Beforehand termination of CP, a cellular is always in one of two state, unresolved or resolved. At a moment, an unresolved cellular may be fully resolved with probability that depends on both its self-completion rate and on the number of unresolved neighbouring cellulars [39]. Meanwhile, a resolved cellular turns into unresolved depends on its incoming neighbouring cellulars. The larger the number of unresolved neighbours, the higher the probability of changing into unresolved. This repetition of tasks reflects input changes or iteration caused by other contiguous tasks. Let x _t and y _t represent the fraction of nodes unresolved and resolved at time t respectively, then x _t  + y _t  = 1.

Braha and Bar-Yam examined the dynamics of the product development (PD) process by analysing the sensitivity as well as robustness(error tolerance) of the PD network topology with respect to internal and external perturbations based on the assumption that each task is equally influenced by its neighbouring tasks and internal completion rates of all nodes are identical across tasks [7]. Then, a mean-field approximation was applied to the stochastic model. It effectively reveals the dynamic process of PD but largely simplifies the complexity of the problem, which causes that the simulation results are incapable of reflecting the tiny evolution possibly producing great costs or profits for complex systems. Therefore, research implemented by Braha and Bar-Yam may be not well suitable for CP. But fortunately is that explaining the systematic changes from the perspective of individual is precisely advantage of CA. Then, in this paper, a model based on CA simulating the state variation with respect to perturbations is proposed to reveal the dynamic behaviours of DWCPTN.

Introduce two indices firstly, load (Load _i denote the load of node i) and capacity (Cap _i denote the capacity of node i), to represent performance of the existing networks.Load i = ∑ k = j = 1 n Γ kj i Γ kj(12) Cap i = ( 1 + α ) Load i(13)

Here, Γ _kj  = |Dkj| represents the number of shortest distance between node k and node j, Γ kj i indicates the Γ kj i via i. α ≥ 0 is the tolerance parameter, which means the additional load of node i under the limitation Cap _i depends on the costs.

For CA, q _t  (i) represents the state of cellular c _i . Introduce a binary variable η, and evolution rules of complex product network by CA considering changes are defined as follows;

q i , t - u = q i , t - u - 1 + not ( q i , t - u - 1 ) η i 〈 Load i , Cap i 〉 0 ≤ u < t i , d(14) q i , t = not ( q i , t - u ) ( q i , t - 1 + not ( q i , t - u ) η i 〈 Load i , Cap i 〉 ) u ≥ t i , d(15) η i 〈 Load i , Cap i 〉 = { 0 Load i > Cap i 1 Load i ≤ Cap i(16)

Here, not () describes a negation operation. u is a time data. t_i,d indicates the self-completion time of cellular c _i , which depends on self-completion rate and on the number of unresolved neighbouring cellulars. Based on PARK J. [40], for most real world networks, nodes with higher degree (unweighted networks) or strength (weighted networks) are often get more attention. These nodes tend to lead themselves have stronger adaptability and self-healing ability. Thus, t_i,d and nodes degree are inverse ratio. In addition, let α denote cellular self-completion rate, which is up to production capacity of enterprise. Apparently, t_i,d is inversely proportional to α. Thus, for complex product networks,t i , d = 1 α s i − α(17)

Here, denotes the strength of node i. The meaning of Equations (14) and (15) could be illustrated by Figs. 2 and 3. As shown in Fig. 2, the modified nodes can be those removed or replaced. The node i state would maintain the original if the load staying in corresponding capacity. Actually, during CP process, in order to improve the flexibility of respond to emergencies, each parameter is set a certain margin. That is to say, in DWCPTN, each node and edge load is less than its capacity. When perturbations emerge, the excess portion of the load can absorb some impacts to improve the robustness and survivability. Conversely, once the load exceeds capacity, the state is turning into the reversal one. Moreover, it is remarkable that the state of neighbouring nodes may be changed because of the interconnections, which is up to comparison results of their load and capacity.

In above model, we assume that the resource utilization rate resource usage intensity and PCI (process capability index) required to accomplish the self-healing of the various tasks is uniform throughout the process. Meanwhile, product scale, budget, performance and other properties do not vary over the duration of the development.

Like other real-world networks, complex networks are characterized by dynamic process [30]. For last few years, exploring the dynamic problem of real-world network arouses great interest. One of the most plausible is the SIS (Susceptible-Infected- Susceptible) [39] proposed for epidemic spread. Especially, the SIS is modelled as differential equations:h t − h t − 1 ε = g t − τ − 1 − δ k h t g t g t − 1 − g t − 2 ε = − g t − 1 + δ k h t g t g t − 2 − g t − 3 ε = − g t − 2 + g t − 1 ⋯ ⋯ g t − τ − 1 − g t − τ − 2 ε = − g t − τ − 1 + g t − τ(18)where h _t and g _t respectively represent the fraction of infected nodes and susceptible nodes at time t, ɛ → 0, 0 < δ < 1 and τ represents the self-healing time of infected nodes turning into susceptible.

The SIS model reveals the evolution law of epidemic spread on homogeneous network. In DWCPTN, each node may be in a “resolved” or “unresolved” state. It may be affected by nodes that directly connect it, and could affect nodes that are directly reachable from it. The rule that a resolved node turns into unresolved is stochastically up to the number of unresolved connected nodes. The higher the number of unresolved connected nodes, the greater the probability of becoming unresolved. Similarly, an unresolved node may become resolved with a probability up to both its self-completion rate and on the number of unresolved neighbors. This progress is fundamentally consistent with SIS. However, it’s important to note that SIS is an unweighted network, which differ from DWCPTN. Motivated by the SIS model. Thus, is introduced to reflect the weight in this paper, a DWCPTN evolution model (DWCPTNEM) is constructed asfollows.y s , t - y s , t - 1 ɛ = x s , t - t d - 1 - λ sy s , t γ ij , λ x s , t - 1 - x s , t - 2 ɛ = x s , t - 1 - x s , t - 1 + λ sy s , t γ ij , λ x s , t - 2 - x s , t - 3 ɛ = - x s , t - 2 + x s , t - 1 … … … x s , t - t d - 1 - x s , t - t d - 2 ɛ = - x s , t - t d - 1 + x s , t - t d(19)

Here, x _s , t, y _s , t denote the relative fraction of unresolved cellulars and resolved cellulars with strength s at time t, and γ_ij,λ denotes the probability of cellular c _i linked to c _j , which describe the reachability of c _i , thus γ_ij,λ could be represented by nodal clustering coefficient CO _i .

Then, based on the Equations (14–19), the fraction of resolved cellulars could be calculated though iterations. y s , t = count { N s , i * | N s , i * ∈ N i , q s , i ( t ) = 1 , i = 1 , 2 , ⋯ , n } n(20)

Here, N s , i * denotes the set of cellulars with strength s in CA and count () is a counting function able to calculate the number required and q_s,i (t) represents the state of cellulars with strength s at time t. Then,y t = ∑ s = 0 s max y s , t(21)

Variations of y _t resulted from state transform of each cellular, which is equivalent to state changes of CP task. Namely, the dynamic characteristic of CP could be illustrated by the variations of y _t over the duration of product development.

Without loss of generality, at time t, perturbations emerge and suppose the modifications begin by node i and let σ _t denote the fraction of the modified node. A high value of σ _t describes a large degree of networks changes and a wide impact surface. Then, σ t = c o u n t { L i ' | L i ' ∈ L i , L o a d i > C a p i , i = 1 , 2 , ⋯ , n } n t(22)

As for the state of DWCPTN, , indicating the fraction of resolved celluars, could represented by Equation (23).y t ' = σ t y t(23)

Variations of caused by node modification reflect the state transform of each cellular, which is equivalent to state changes of DWCPTN. Accordingly, the dynamic characteristic of CP could be further illustrated by the variations of over the duration of the whole process.

Based on aforementioned, changes impacts on DWCPTN could be observed though , which describe variation of structure and state respectively. But beyond that, the focus is as much on the convergence time and economic behaviour.

The convergence time, denoted by T^* means the time required to resolve all nodes. Then, whether design problems being solved or not at time t, it always falls below an acceptable threshold within a specified time frame and eventually approach to 1 at T^*.T * = ( t | y t ' = 1 )(24)

Another important index is the economic behaviour, which indicates the cost spent while rebuild a network caused by changes. Let Cost (t) denote the cost needed to complete the modifications. Generally, sunk cost is generated and unrecoverable once the task implemented. Cost _sunk  (t) is introduced to indicate the sunk cost. Meanwhile, extra cost, denoted by Cost _extra  (t), may be spent to modify corresponding relationship between tasks. V. Latora proposed a method to quantify the cost of a constant network [41].Cost ( G ) = ∑ i ≠ j ∈ V w ij ψ ( D ij ) ∑ i ≠ j ∈ V ψ ( D ij )(25)

Here, ψ (D _ij ) is the so-called cost evaluator function, which calculates the cost needed to build up a connection with a given length. Without loss of generality, ψ (D _ij ) is defined as the identity function ψ (D _ij ) = D _ij . Then,Cost ( G ) = ∑ i ≠ j ∈ V w ij ψ D ij ∑ i ≠ j ∈ V ψ D ij(26)

Based on Equation (26), sunk cost and extra cost of DWCPTN can be defined as follows:Cost sunk ( t ) = ∑ i ≠ j ∈ V res w ij , t D ij , t ∑ i ≠ j ∈ V res D ij , t(27) Cost extra ( t ) = = ∑ i ≠ j ∈ V mod w ij , t D ij , t ∑ i ≠ j ∈ V mod D ij , t(28)where V _res  ={ V _i |q_i,t-1 = 1, i = 1, 2, ⋯ , n } and V mod = { V i | q i , t - q i , t ′ ≠ 0 , i = 1 , 2 , ⋯ , n } mean the node set maintain resolved and modified with respect to perturbations.

Thus, at time t, the cost of DWCPTN evolves two parts: sunk cost and extra cost.Cost ( t ) = Cost sunk ( t ) + Cost extra ( t )(29)

Accordingly, the total cost Cost (G) spent to resolve all tasks can be expressed by Equation (30).Cost ( G ) = ∑ t = 1 T * Cost ( t )(30)

5.1 Topological properties

The evolution model of DWCPTN is proposed based on the characteristics of scale-free. Then, we begin analysis with the simulation of topology characteristics. The network topology characteristics depict its internal structure, whose distinction leads to the difference of function. Therefore, the description of topological character based on the statistical properties is a foundation of understanding complexity of DWCPTN. For weighted networks, the most important statistic is the nodes strength distribution and clustering coefficient. The former reflects the macro characteristic and the latter represents the small-world characteristic.

Based on the mentioned above, in-strength and out-strength are involved in case of the directed. According to the results of statistical analysis, the node in-strength and out-strength double logarithmic distribution function of the CPNCP are drawn in Fig. 6.

As shown in Fig. 6, the fronts of nodes in-strength and out-strength distribution curve are both approximately of a linear, which demonstrates a power-law regime. A distinction is still distinguished by centrality of points. It can be observed that points in Fig. 6(a) is more centralized than that in Fig. 6(b), which indicates that nodes in-strength play a more significant role than out-strength. This just proves that almost all the nodes in the network are inlinked by many other nodes while not all of the nodes provide information to others. The result clarifies that distinction between information generators and consumers indeed exist and a high generator may be a low consumer and vice versa.

Empirical studies show that most real networks are small-world, which is usually verified by coefficient transforms inequality proposed by Sporns [43]. By comparison with a random network with same scale and density, if CO/ CO _ran  > D/ D _ran , then the network is highly clustered. With regard for the CPNCP, results are shown in Table 3. After comparison, CPNCP is small-world. C and D denote clustering coefficient and average shortest path length. And the equations for C and D refer to [43].

5.2 Simulation of dynamic behaviours with respect to unplanned perturbations