Evolution model of collaborative production task responding to unplanned perturbations

Abstract

Inevitable perturbations have a significant impact on collaborative production (CP) cause that the complexity are greatly increased. In order to assess the impact accurately enterprises to make precise decisions, it is prerequisite to master the dynamic behaviour of CP when changes occur. Accordingly, in this paper, an evolution model for simulating CP task state to perturbations is proposed. Regard CP tasks as a directed weighted complex network (DWCPTN) firstly, and statistical properties of DWCPTN are analysed. Two perturbation cases and two modifications policies are defined followed. Based on above mentioned, evolution model based on cellular automaton (CA) and SIS (Susceptible-Infected-Susceptible) is presented to reveal the varying process by three indices (friction of resolved nodes, convergence time and economic behaviour). Finally, analytic results from a case of a chemical product CP network (CPNCP) are used to illustrate the model and method proposed in this paper. Simulation results shows that unplanned perturbations promote a negative role to CP but the improvements of self-healing can decrease the negative effects efficiently. With the analysis, it can provide insight into ways of improving the strategic and operational decision making for enterprises.

Keywords

Collaborative production task network evolution unplanned perturbations cellular automaton SIS

1 Introduction

Nowadays, an increasing number of enterprises or organizations combine to form a production alliance, namely collaborative production (CP), which can not only shorten the production cycle, rapidly respond to customer needs, but also reduce the risk of a single enterprise in market competition [1, 2]. Generally, CP encapsulates and integrates information, technology, materials and human resources, etc. scattered in different locations of enterprises or organizations, and eliminates the heterogeneity and distribution of these resources based on the network platform to allocate task to distributed agents [3]. That is to say, CP is a huge complex hierarchical system running in a wide range of distribution under the environment of constraint and multi-objective. Each enterprise or organization can be regarded as a subsystem, which assumes corresponding subtasks and may also consist of much many sub-subsystems corresponding sub-subtasks. Material flow and information flow between subtasks or sub-subtasks (subsystems or sub-subsystems) support the operation of CP [4].

Despite many advantages of CP, internal potentially explosive is worth vigilant. Normally, perturbations such as customer requirements changes and technical innovation are always inevitable [6]. These perturbations are highly possible cause the modifications of tasks several times [7]. Actually, a change rarely occurs alone due to the connections within complex systems [8]. A change to a task may trigger chain changes to other tasks, which is called changes propagation. It greatly increases the complexity of the CP process. If handled improperly, it may lead to retard the convergence or have a destabilizing effect on the system’s behaviour, and even delay scheme for CP [7]. Until now, it is difficult to systematically assess the potential change impacts due to knowledge or experience limitations [9]. In this situation, it more likely even leads to a wide range risk of product performance, reliability, cost and schedule etc.

Therefore, it is essential to have an all-round understanding about the complexity of CP and its dynamic behaviours response to perturbations. Thus, the evolution of CP would be discussed in this paper. Nof, S. Y. clarified the CP complexity in view of the associations between its components and believed that analysis of collaborative control theory is prerequisite for collaborative production management (CPM) [10 –12]. Eckert et al. regarded complexity as the structural complexity of components and connections, and the dynamic complexity of behaviour [8]. In any case, the internal relation between the various tasks makes CP fundamentally iterative [13], which is embodied in the iterations of tasks during the process. This kind of simulation copes with the work process of the enterprises, where the simulation model is concentrated on the enterprises’ task network [14]. Due to task-oriented simulation comprise complex manufacturing systems engineering and complicated process modelling, research findings on task-oriented simulation in CP are seldom reported. Approaches worth mentioning are mainly concentrated in the following: Harmonosky et al. presented a simulation method to select re-routing based on real-time. Simulation was used iteratively as a tool to find out the best policy from a set of alternative policies in real-time [15]. Laughery proposed a compendious task network modelling tool oriented to human performance to address the importance of some design issues such as task time, task allocation and workload [16 –19]. Zülch et al. presented an advanced approach for simulating work processes in manufacturing of a socio-technical system, and an object-oriented simulation tool, OSim, is developed [20]. Schlick et al. presented some effective simulation approaches on human work process, human error and abilities in a five-axis milling production cell [21]. This research places more emphasis on the human factors in production. To extend the capabilities of discrete event simulations, Baines and Ladbrook carried out an empirical study on worker performance and working efficiency for manufacturing simulation [22]. Liu et al. proposed a queuing network-model human processor (QN-MHP) to clarify performance of multi-task in human–machine systems [23]. Crist Kristy et al. used a scaled-down simulation model of a wafer fabrication facility to explore the impacts of several different policies for allocating resources to production and engineering work on the shop floor [24]. M. Gansterera developed a discrete-event simulation which mimics the production system based on a mathematical optimization model to identify good settings for three planning parameters, namely planned lead-times, safety stock, and lot sizes [25].

Available research findings mentioned above mostly aim to resolve a key problem from a specific point such as task planning, task scheduling. In this article, a model from a global perspective will be established to explore the dynamic behaviour of CP, and research results will contribute to raise accuracy and reliability of the decision result.

The last few years have witnessed substantial and dramatic new advances in understanding the large-scale structural properties of many real world complex networks [26 –29]. The rise of the complex system theory provides a new way to scientific analysis interrelated problem [30]. As the abstract representation of complex system, complex networks, originated from the random graph theory in mathematics, reflects the complicated relationship between different subjects of the real world such as transportation network, power network, which all display a large heterogeneity in the capacity. Generally, complex networks are symbolized by the characteristics of scale-free and small world [28].

A CP task network is also heterogeneous and has the same statistical characterizations of weighted networks to identify the topology structure. Dan Braha examined, for the first time, the statistical properties of strategically important organizational networks of people engaged in distributed product development [7]. Then, they presented a detailed model and analysis of PD dynamics on complex networks based on the assumption that each task is equally influenced by its neighbouring tasks, meanwhile, they applied a mean-field approximation to the stochastic model [31]. This 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 CoPS. Fan and Qi et al. employed complex networks to build the product family structure model. They introduced the components relation networks model for product family structure [32]. Yang et al. applied the complex networks theory to solve the module partition problem based on the Girvan–Newman (GN) algorithm [33, 34].

These preliminary attempts have witnessed the advantages of complex networks in addressing the complex problems of product design. Motivated by these researches, in this paper, in order to examine an exquisite simulation of CP dynamic behaviour with respect to perturbations, a novel networks-based model is proposed firstly. Then evolution model is presented after perturbations and modification policies are defined. Simulation results are analysed finally to reveal the variations of CP. With the analysis, it can provide insight into ways of improving the strategic and operational decision making for enterprises.

This paper is organized as follows. The product networks and its statistics are presented in Section.2. Two changes cases and two modification policies are defined in Section 3. Section 4 presents evolution model. Results are illustrated in Section 5. Conclusions are presented in the final Section 6.

2 CP task networks

Camarinha M L and Afsarmanesh H defined cooperative behaviours of firms as collaborative network, which means a network system composed of different agents to achieve a common purpose through internet technology [3]. CP task, as previously stated, is regarded as a fundamental carrier over the duration of the process [4]. In this paper, an assumption that a firm is assigned only one task is defined. Figure 1 illustrate a cooperative relation network, where the nodes represent a task and the lines with arrows indicate the cooperative relation such as material flow, information flow, cash flow and energy flow et al.

Suppose that CP contains n tasks, V ={ v₁, v₂, ⋯ , v_n } and a task has m_i links denoted by E ={ e_i1, e_i2, ⋯ e_ij, ⋯ , e_imi } (i = 1, 2, ⋯ , n), where e_ij represents the connection from task i to task j, which means a type of connection flow transferred from i to j. Thus, e_ij is unequal to e_ji due to the flow direction. A set of values W ={ w_i1, w_i2, ⋯ , w_ij, ⋯ , w_imi } are real numbers associated with the links, where w_ij represents connection strength from i to j. Define tasks and their links as nodes and edges of the network respectively. Accordingly, a directed weighted CP task network (DWCPTN) can be constructed as follows. $G = (V, E, W)$ (1)

Such a complex network is represented as an adjacency matrix W with elements $w_{ij} = {\begin{matrix} w_{ij}, & if i and j are connected with the strength w_{ij} \\ 0, & otherwise \end{matrix}$ (2)

The important index expressing interdependent characteristics of tasks is W, which also implies the probability of interplay. A high value of W describe an intimate relationship of companies.

Suppose $R = {r_{κ} | κ = 1, 2, \dots, m_{i}}$ denote the collaborative relationship set, where r_κ represent κth collaborative relationship and $m_{i} = | | R | |$ indicates the number of collaborative relationship. Thus, Collaborative relationship matrix (CRM) is proposed to represent complicated dependence of DWCPTN. $CRM = {{crm}_{i, j, κ} | i, j = 1, 2, \dots, n; κ = 1, 2, \dots, m_{i}}$ (3)

Here, crm_i,j,κ denotes the number of relation r_κ from task i to task j. Generally, there are many parallel collaborative relationships between two companies. Let p_i,j,κ indicate the probability of interplay between task i and task j based on relation r_κ, and p_i,j,κ ∈ [0, 1]. Then, $W = {w_{i, j} = 1 - \prod_{k = 1}^{m_{i}} {(1 - p_{i, j, κ})}^{{crm}_{i, j, κ}} | 1 \leq i, j \leq n}$ (4)

Another key point is how to determine the value of p_i,j,κ. Based on expert scoring method, in this paper, triangular fuzzy number (TFN) [34] is introduced to map the clear value of the probability of collaborative relationship. A finite set of experts is defined as Ex ={ Ex_ℓ| ℓ =1, 2, …, Ω } and this article assumes that all experts have uniform weight. Meanwhile, the evaluation object set is just $R = {r_{κ} | κ = 1, 2, \dots, m_{i}}$ , of which evaluation index is p_i,j,κ. Linguistic variables set of experts, denoted by Θ ={ Θ_θ|θ = 0, 1, …, l - 1 }, is pre-defined by an odd number of elements. Suppose that the linguistic assessment information of r_κ between task i and task j from Ex_ℓ is φ_i,j,κℓ, then φ_i,j,κℓ ∈ Θ. And TFN of φ_i,j,κℓ can be expressed by Equation (5). $\begin{matrix} φ_{i, j, κ ℓ} = (φ_{i, j, κ ℓ}^{L}, φ_{i, j, κ ℓ}^{M}, φ_{i, j, κ ℓ}^{R}) \\ = (max (\frac{θ - 1}{l - 1}, 0), \frac{θ}{l - 1}, min (\frac{θ + 1}{l - 1}, 1)) \end{matrix}$ (5)

Let φ_i,j,κ indicate the linguistic assessment information of r_κ from all experts, then $φ_{i, j, κ} = (1 / Ω) \otimes (φ_{i, j, κ 1} \oplus φ_{i, j, κ 2} \oplus \dots \oplus φ_{i, j, κ Ω})$ (6)

Define $φ_{i, j, κ} = (φ_{i, j, κ}^{L}, φ_{i, j, κ}^{M}, φ_{i, j, κ}^{R})$ , thus $φ_{i, j, κ}^{L} = 1 / Ω \sum_{ℓ = 1}^{Ω} φ_{i, j, κ ℓ}^{L}$ (7) $φ_{i . j . κ}^{M} = 1 / Ω \sum_{ℓ = 1}^{Ω} φ_{i, j, κ ℓ}^{M}$ (8) $φ_{i, j, κ}^{R} = 1 / Ω \sum_{ℓ = 1}^{Ω} φ_{i, j, κ ℓ}^{R}$ (9)

Based on the CFCS proposed by Opricovis S et al. [36], fuzzy value can turn into clear according to Equation (10). $\begin{matrix} p_{i, j, κ} = L + \frac{Δ [\begin{matrix} (φ_{i, j, κ}^{M} - L) {(Δ + φ_{i, j, κ}^{R} - φ_{i, j, κ}^{M})}^{2} (R - φ_{i, j, κ}^{L}) \\ + (φ_{i, j, κ}^{R} - L) {(Δ + φ_{i, j, κ}^{M} - φ_{i, j, κ}^{L})}^{2} \end{matrix}]}{\begin{matrix} (Δ + φ_{i, j, κ}^{M} - φ_{i, j, κ}^{L}) {(Δ + φ_{i, j, κ}^{R} - φ_{i, j, κ}^{M})}^{2} (R - φ_{i, j, κ}^{L}) \\ + (φ_{i, j, κ}^{R} - L) {(Δ + φ_{i, j, κ}^{M} - φ_{i, j, κ}^{L})}^{2} (Δ + φ_{i, j, κ}^{R} - φ_{i, j, κ}^{M}) \end{matrix}} \end{matrix}$ (10)

Here, $L = min {φ_{i, j, κ}^{L}}$ , $R = max {φ_{i, j, κ}^{R}}$ , Δ = R - L. Thus, according to Equations (3–10), W can be calculated.

3 Perturbations cases and modifications

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.

Case 2. Removing, adding or replacing tasks, usually caused by major revolution, are main methods to evolve a new CP network. For example, customers prefer to choose products with more features and capacities initially, but once actually worked with a product they will find the complex ones are too hard to use, and then some features may be abandoned. Consequently, the nodes expressing the corresponding feature would be removed and corresponding tasks could be modified.

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.

Additive policy: Same as in 1, but modify nodes according to the sum of their in-strength and out-strength when nodes out-strength is equal.

4 Evolution model based on CA

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 }.

4.2 Evolution rules

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. ${Load}_{i} = \sum_{k = j = 1}^{n} \frac{Γ_{kj}^{i}}{Γ_{kj}}$ (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;

$\begin{matrix} q_{i, t - u} & = q_{i, t - u - 1} + not (q_{i, t - u - 1}) η_{i} 〈 {Load}_{i}, {Cap}_{i} 〉 \\ 0 \leq u < t_{i, d} \end{matrix}$ (14) $\begin{matrix} q_{i, t} & = not (q_{i, t - u}) (q_{i, t - 1} + not (q_{i, t - u}) η_{i} 〈 {Load}_{i}, {Cap}_{i} 〉) \\ u \geq t_{i, d} \end{matrix}$ (15) $η_{i} 〈 {Load}_{i}, {Cap}_{i} 〉 = {\begin{matrix} 0 & {Load}_{i} > {Cap}_{i} \\ 1 & {Load}_{i} \leq {Cap}_{i} \end{matrix}$ (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, $t_{i, d} = \frac{1}{α} s {〈i〉}^{- α}$ (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.

4.3 Evolution model based on CA

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: $\begin{array}{l} \frac{h_{t} - h_{t - 1}}{ε} = g_{t - τ - 1} - δ 〈k〉 h_{t} g_{t} \\ \frac{g_{t - 1} - g_{t - 2}}{ε} = - g_{t - 1} + δ 〈k〉 h_{t} g_{t} \\ \frac{g_{t - 2} - g_{t - 3}}{ε} = - g_{t - 2} + g_{t - 1} \\ \dots \dots \\ \frac{g_{t - τ - 1} - g_{t - τ - 2}}{ε} = - g_{t - τ - 1} + g_{t - τ} \end{array}$ (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. $\begin{matrix} \frac{y_{s, t} - y_{s, t - 1}}{ɛ} = x_{s, t - t_{d} - 1} - λ {sy}_{s, t} γ_{ij, λ} \\ \frac{x_{s, t - 1} - x_{s, t - 2}}{ɛ} = x_{s, t - 1} - x_{s, t - 1} + λ {sy}_{s, t} γ_{ij, λ} \\ \frac{x_{s, t - 2} - x_{s, t - 3}}{ɛ} = - x_{s, t - 2} + x_{s, t - 1} \\ \dots \dots \dots \\ \frac{x_{s, t - t_{d} - 1} - x_{s, t - t_{d} - 2}}{ɛ} = - x_{s, t - t_{d} - 1} + x_{s, t - t_{d}} \end{matrix}$ (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. $y_{s, t} = \frac{count {N_{s, i}^{*} | N_{s, i}^{*} \in N_{i}, q_{s, i} (t) = 1, i = 1, 2, \dots, n}}{n}$ (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, $y_{t} = \sum_{s = 0}^{s_{max}} y_{s, t}$ (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, $σ_{t} = \frac{c o u n t {L_{i}^{'} | L_{i}^{'} \in L_{i}^{,} L o a d_{i} > C a p_{i}, i = 1, 2, \dots, n}}{n_{t}}$ (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)

4.4 Assessment indices

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]. $Cost (G) = \frac{\sum_{i \neq j \in V} w_{ij} ψ (D_{ij})}{\sum_{i \neq j \in V} ψ (D_{ij})}$ (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, $Cost (G) = \frac{\sum_{i \neq j \in V} w_{ij} ψ D_{ij}}{\sum_{i \neq j \in V} ψ D_{ij}}$ (26)

Based on Equation (26), sunk cost and extra cost of DWCPTN can be defined as follows: ${Cost}_{sunk} (t) = \frac{\sum_{i \neq j \in V_{res}} w_{ij, t} D_{ij, t}}{\sum_{i \neq j \in V_{res}} D_{ij, t}}$ (27) ${Cost}_{extra} (t) = = \frac{\sum_{i \neq j \in V_{\mod}} w_{ij, t} D_{ij, t}}{\sum_{i \neq j \in V_{\mod}} D_{ij, t}}$ (28)where V_res ={ V_i|q_i,t-1 = 1, i = 1, 2, ⋯ , n } and $V_{mod} = {V_{i} | q_{i, t} - q_{i, t}^{'} \neq 0, i = 1, 2, \dots, n}$ 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). $Cost (G) = \sum_{t = 1}^{T *} Cost (t)$ (30)

5 Simulation and analytical results: A case study

There is a CP network of chemical product (CPNCP), which contains multiple chemical production units, as shown in Fig. 4. In order to definitize the topological structure of CPNCP, eleven categories of collaborative relationship illustrated in Table 1 exist in these firms. Five experts are invited to score under the seven linguistic variables pre-set and listed in Table 2. Based on Equations (3–10), TFN of CPNCP can be found, as shown in Table 2. Then, according to Equations. (4), adjacency matrix W could be determined and the network structure consequently could be drawn in Fig. 5, which including 41 nodes and 55 edges.

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

5.2.1 Simulation without considering perturbations

One of the most practical aspects of a CP process is the total number of tasks problems remains slowdown as the project evolves over time, and eventually falls below an acceptable threshold within a specified time frame [30]. In order to validate the advantage of the model proposed aforementioned, we firstly assume that there are no perturbations. Based on Equations (14–19), fraction of resolved cellulars could be calculated. Figure 7 demonstrates variation trend of the relative fraction of resolved cellulars y_t in the context of td = 0, td = 5 and td = 10 without considering any perturbations. Moreover, corresponding results based on the model proposed by Braha and Yassine are also shown in Fig. 7.

As shown in Fig. 7, three curves all converge to 1 although at different times, which means all tasks are resolved finally under the assumption. It’s important to note that convergence time is strongly correlated to task self-completion ability. The tasks with td = 5 indicates that they have a more their self-completion ability than those with td = 10 and, accordingly, the corresponding tasks accomplished more quickly. Especially, td = 0 represents completing tasks instantly, which is an ideal. In addition, in the front of the process, the convergence speed is sharp, which may be due to the companies’ high passion and less interactive information. However, with the increase of mutual information over time, the convergence speed gradually slows down.

5.2.2 Simulation considering perturbations

Based on the modification policies and evolution model proposed above, we further analyse the dynamic behaviour of DWCPTN with respect to unplanned perturbations. Similarity, the model assumes that designers, technology are changeless over the duration of process. Suppose that request to modify the task i at time t_e = 60, at which task i has been resolved. Based on Equations (14–19) simulation results can be calculated.

Figure 8 shows variation tendency of friction of the node resolved over the time on the context of different node strength. instantly fell sharply when perturbations occur because the original state is broken. Then, increasing momentum is similar to Fig. 7, where there are no interference factors. We further consider the dynamic behaviour if the modified nodes have different strength, and s (i) =0.3, s (i) =0.5, s (i) =0.8 are construed as an assumption. Actually, the nodes with higher strength supply more information to others, and changes of these nodes are quite possible to cause changes propagation and cascading failure due to the sophisticated correlation between tasks. Thus, it is plausible that the higher strength of the modified node, the greater impact generated to DWCPTN.

Figure 9 demonstrates the relationship of the convergence time to the time when change happens based on Equation. (24). As observed in Fig. 9, the later changes occur, the longer the convergence time. Actually, is unceasing increase along with the process going on. The later time of changes occurrence means that more tasks maybe reworked or modified. Furthermore, the convergence time is closely associated with the node strength, which is evidenced firstly in Fig. 7. eventually ascend to threshold 1 within a specified time, which is considered as the convergence time. Figure 9 provides a more comprehensible illustration about this. Consequently, the later time of changes occurrence and the high strength of modified node both result in the slower convergence.

Another important index to assess the impacts of perturbations is the economic behaviour, namely cost of DWCPTN. As shown in Fig. 10, the cost spent is proportional to the convergence time based on Equation (27–30). That is to say, the longer time the team takes to pay off reworks, the higher the costs for the business. Moreover, there is a much notable difference at t_e = 40 and later because the distinction of strength of modified nodes. Obviously, modified nodes with higher strength contributing to the convergence time more.

In conclusion, unplanned perturbations have resulted in some negative impacts on CPNCP. Task self-healing ability, task strength and the time of changes emerging all display substantial correlation with the performance of CPNCP, thus enhancing measures of production and prediction abilities maybe key emphasis in work, which proposes improvement recommendations for enterprises.

In order to validate the performance of the statistical analysis, some real datas about the chemical product were collected after the changes emerged. The comparison between the analytical results in this paper and the reality is shown in Fig. 11.

As shown in Fig. 11, a comparison (resolved node friction in (a), convergence time in (b) and cost in (c)) between analytical results and reality data shows the tendency of evolutionary processes compatibly, which verifies that the method and model proposed in this paper are available.

6 Conclusions

In order to have a better understanding of complexity and dynamic behaviour of CP to improve the ability of dealing with perturbations, in this paper, an evolution model of directed weighted product task network (DWCPTN) based on CA was proposed to reveal the evolution process of CP over the duration of the process considering interferences. The statistical properties such as scale-free and small world have analysed firstly. It was proved that DWCPTN presented properties similar to other real-world networks. Perturbations cases and modification policies were put forward followed. Based on the aforementioned, a DWCPTN evolution model (DWCPTNEM) based on CA and SIS was put forward to reveal varying pattern. The deterministic analysis including three indices (friction of resolved nodes, convergence time and economic behaviour) of the model demonstrated that the CP dynamic behaviour is strongly correlated to task self-completion ability, changes time and node strength. Moreover, CA introduced in DWCPTNEM shows advantages of more exquisite and more accurate than mean field method commonly used before. Actually, it is fairly difficult to predict and master customer requirements changes accurately due to uncertain man-made factors. Then, How to predict and determine the influence of the changes in a fuzzy environment should be further considered during modeling the product network.

Footnotes

Acknowledgments

This research is sponsored by National Natural Science Foundation of China (Project No. 71301176, 71401019). The authors also thank to Fundamental Research Funds for the Central Universities (Project No. CDJZR12110004) and Specialized Research Fund for the Doctoral Program of Higher Education (Project No. 20130191120001) for partial support of this work. We are grateful for the constructive suggestions provided by the reviewers, which would improve the paper.

References

Lin

Jiun-Shiung

, Ou-Yang

Chao

and Juan

Yeh-Chun

, Towards a standardised framework for a multiagent system approach for cooperation in an original design manufacturing company, International Journal of Computer Integrated Manufacturing 6 (2009), 494–514.

Levalle

Rodrigo Reyes

, Scavarda

Manuel

and Nof

Shimon Y.

, Collaborative production line control: Minimisation of throughput variability and WIP, International Journal of Production Research 23–24 (2013), 7289–7307.

Camarinha

M.L.

and Afsarmanesh

, Collaborative networks a new scientific discipline, Journal of Intelligent Manufacturing 5 (2005), 439–452.

Sang Ko

Hoo

and Nof

Shimon Y.

, Design and application of task administration protocols for collaborative production and service systems, International Journal of Production Economics 1 (2012), 177–189.

Camarinha

M.L.

, Afsarmanesh

, Galeano

, et al., Collaborative networked organizations Concepts and practice in manufacturing enterprises, Computers & Industrial Engineering 1 (2009), 46–60.

Eckert

C.M.

, Zanker

and Clarkson

P.J.

, Aspects of a better understanding of changes, In Proceedings of International Conference on Engineering Design, ICED 01 GLASGOW (AUGUS), 2001, pp. 21–23.

Dan

and Yaneer

, The statistical mechanics of complex product development: Empirical and analytical results, Management Science 7 (2007), 1127–1145.

Eckert

C.M.

, Clarkson

P.J.

and Zanker

, Change and customization in complex engineering domains, Research in Engineering Design 1 (2004), 1–21.

Keller

, Eckert

C.M.

and Clarkson

P.J.

, Multiple views to support engineering change management for complex products, Proceedings of the Third International Conference on Coordinated & Multiple Views in Exploratory Visualization, 2005, California.

10.

Nof

S.Y.

, Parallel and Distributed Models of Integration in Production systems, Proceedings of ICPR-13, Jerusalem, Israel 1995.

11.

Nof

S.Y.

, Collaborative control theory for E-Work, E-Production, and E-Service, Annual Reviews in Control 31 (2007), 281–292.

12.

Nof

S.Y.

and Velásquez

J.D.

, Collaborative E-Work, E-Business, and E-Service, In Handbook of Automation, London: Springer, 2009, pp. 1549–1576.

13.

Braha

and Maimon

, A Mathematical Theory of Design: Foundations, Algorithms, and Applications, KluwerAcademic Publishers, Boston, MA, 1998. pp. 211–213.,

14.

Zhang

, Schmidt

, Schlock

C.M.

, Reuth

and Luczak

, A human task-oriented simulation study in autonomous production cells, International Journal of Production Research 18 (2008), 5013–5041.

15.

Harmonosky

C.M.

, Farr

R.H.

and Ni

M.C.

Selective rerouting using simulated steady state system data, In Proceedings of the 1997 Winter Simulation Conference, ed. Andradottir

, Healy

K.J.

, Withers

D.H.

and Nelson

B.L.

, pp. 1293–1298.

16.

Laughery

, Computer simulation as a tool for studying human-centred systems, In Proceedings of the 1998 Winter Simulation Conference, Washington.

17.

Laughery

, Using discrete-event simulation to model human performance in complex systems, in Proceedings of the1999 Winter Simulation Conference, Phoenix, 1999.

18.

Laughery

K.R.

Jr , Archer

and Corker

, Modelling human performance in complex systems, In Handbook of Industrial Engineering, edited by Salvendy

, 3rd edn, Wiley: New York, NY, 2001, pp. 2409–2443.

19.

Laughery

K.R.

Jr , Lebiere

and Archer

, Modelling human performance in complex systems, In Handbook of Human Factors and Ergonomics, edited by Salvendy

, 3rd edn, Wiley: New York, NY, 2006, pp. 967–996

20.

Zülch

, Fischer

and Jonsson

, An integrated object model for activity network based simulation, In Proceedings of the 2000 Winter Simulation Conference, Orlando, 2000.

21.

Schlick

, Reuth

and Luczak

, Comparative simulation study of work processes in autonomous production cells, Human Factors and Ergonomics in Manufacturing 1 (2002), 31–54.

22.

Baines

and Ladbrook

, Using empirical evidence of variations in worker performance to extend the capabilities of discrete event simulations, Proceedings of the 2003 Winter Simulation Conference, New Orleans, Atlanta.

23.

Liu

Y.L.

, Feyen

and Tsimhoni

, Queuing network-model human processor (QN-MHP): A computational architecture for multitask performance in human–machine systems,Human Interaction, ACM Trans Computer–Human Interaction 1 (2006), 37–70.

24.

Kristy

Crist

and Uzsoy

Reha

, Prioritising production and engineering lots in wafer fabrication facilities: A simulation study, International Journal of Production Research 11 (2011), 3105–3125.

25.

Gansterera

Margaretha

, Almederb

Christian

and Richard

, Hartla, Simulation-based optimization methods for setting production planning parameters, International Journal of Production Economics 5 (2014), 206–213.

26.

Watts

D.J.

and Strogatz

S.H.

, Collective dynamics of ‘small-world’ networks, Nature 6684 (1998), 440–442.

27.

Barabasi

A.L.

and Albert

, Emergence of scaling in random networks, Science 5439 (1999), 509–512.

28.

Strogatz

S.H.

, Exploring complex networks, Nature 3 (2001), 268–276.

29.

Suksmono

A.B.

and Hirose

, Intelligent beamforming by using a complex-valued neural network, J Intell Fuzzy Syst 3-4 (2004), 139–147.

30.

Dan

and Yaneer

, The topology of large-scale engineering problem-solving networks, Physical Review E 69 (2004), 1–7.

31.

Marro

and Dickman

, None equilibrium Phase Transitions in Lattice Models, Cambridge University Press, Cambridge, UK, 1999.

32.

Fan

and Qi

G.N.

, Modelling of product family stricture and module analysis method based on complex network, Chinese Journal of Mechanical Engineering 3 (2007), 187–192.

33.

Yang

, Module partition method based on complex network theory, Journal of Graphics 6 (2012), 69–75.

34.

Newman

M.E.J.

, Analysis of weighted networks, Physical Review E 5 (2004), 1311–1319.

35.

Yen

K.K.

, Ghoshray

and Roig

, A linear regression model using triangular fuzzy number coefficients, Fuzzy Sets and Systems 2 (1999), 167–177.

36.

Opricovis

and Tzeng

G.H.

, Defuzzification within a multicriteria decision model, Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 5 (2003), 635–652.

37.

Merton

Robert K.

, The matthew effect in science, Science 3810 (1968), 56–63.

38.

Precharattana

Monamorn

and Triampo

Wannapong

, Physica A: Statistical Mechanics and its Applications 407 (2014), 303–311.

39.

Wang

Y.Q.

and Jiang

G.P.

, Epidemic spreading in complex networks with spreading delay based on cellular automata, Acta Physica Sinica 8 (2011), 080510

40.

Park

and Sahni

, Maximum lifetime broadcasting in wireless networks, IEEE Transactions on Computers 9 (2005), 1081–1090.

41.

Romualdo

P.S.

and Alessandro

, Epidemic spreading in scale-free networks, Physical Review Letters 3200 (2001).

42.

Latora

and Marchiori

, Economic small-world behaviour in weighted networks, European Journal of Physics B 32 (2003), 249–263.

43.

Sporns

, Honey

C.J.

and Kotter

, Identification and classification of hubs in brain networks, PLoS One 10 (2007), 1–14.