Multilayer Network-Based Production Flow Analysis

Essential information about the products to be assembled, parts to be manufactured, materials to be used, methods and techniques to convert the material to the required finished components, and manpower to operate the plant is usually available to a company, but rarely in an appropriate form for ease of digestion by the manager. In this section, we propose a network-based model to study the relationship between these elements.

As can be seen in Figure 1, the proposed network consists of a set of bipartite graphs representing connections between the sets of products $\mathbf{p}=\left\{p_{1}, \ldots, p_{N_{p}}\right\}$, machines/workstations $\quad \mathbf{w}=\left\{w_{1}, \ldots, w_{N_{w}}\right\}$, parts/components $\mathbf{c}=\left\{C_{1}, \ldots, C_{N_{C}}\right\}$, activities (operations) $\mathbf{a}=\left\{a_{1}, \ldots, a_{N_{a}}\right\}$, and their categorical properties (referred as activity types) $\mathbf{t}=\left\{t_{1}, \ldots, t_{N_{t}}\right\}$ and skills of the operators needed to perform the given activity $\mathbf{s}=\left\{s_{1}, \ldots, s_{N_{s}}\right\}$.

Figure 1 Illustrative network representation of a production system. The definitions of the symbols are given in Table 1.

The relationships among these sets are defined by bipartite graphs $G_{i, j}=\left(O_{i}, O_{j}, E_{i, j}\right)$ represented by $\mathbf{A}\left[O_{i}, O_{j}\right]$ biadjacency matrices, where $O_{i}$ and $O_{j}$ are used as a general representation of a sets of objects, as $O_{i}, O_{j}, \in\{\mathbf{p}, \mathbf{w}, \mathbf{c}, \mathbf{a}, \mathbf{t}, \mathbf{s}\}$.

The edges of these bipartite networks can represent material, energy or information flows, structural relationships, assignments, attributes, and preferences, and the edge weights can be proportional to the number of shared components/resources or time/cost needed to produce a given product (see Table 1).

Table 1 Definition of the biadjacency matrices of the bipartite networks used to illustrate how a production system can be represented by a multidimensional network.

The proposed model can be considered as an interacting or interconnected network, where the family of bipartite networks defines crossed layers. Since different types of connections between the nodes can be defined, the model can also be handled as a multidimensional network. Both of these models are the special cases of multilayer networks, which representation is beneficial, since the layers represent the direct connections defined by the bipartite graphs, while the interlayer connections help in term of the visualization of the complex system by arranging the corresponding nodes at the same place within the layers (as it is illustrated in Figure 2).

Figure 2 Visualization of the illustrative network as a multilayer/multiplex network highlights how the complex production system can be grouped into modules based on the “viewpoints” of the layers.

Table 3 Node types of the proposed network.

Table 4 Node edge matchings in the proposed network.

Table 5 The ADACOR predicates can be directly applied to define layers of the network (please note that we use the term activity to refer to operations).