**1. Introduction**

Production line-based assembly lines are still the most widely applied manufacturing systems [1]. Assembly-Line Balancing (ALB) [2] deals with the balanced assignment of tasks to the workstations, resulting in the optimisation of a given objective function without violating precedence constraints [3]. The efficiency of these optimisation tasks is mostly determined by the model of the manufacturing process represented [4].

The concept of Industry 4.0 has already had a significant influence on how production and assembly lines are designed [5] and managed [6]. The requirement of practical design at a high automation level ensures that sensors and equipment can be integrated in a fast, secure, and reliable way. In our research, we study how the interoperability capabilities of Industry 4.0 solutions can be improved and how the efficiency of solution development can be increased.

As Internet of Things-based products and processes are rapidly developing in the industry, there is a need for solutions that can support their fast and cost-effective implementation. There is a need for further standardization to achieve more flexible connectivity, interoperability, and fast application-oriented development; furthermore, advanced model-based control and optimisation functions require a better understanding of sensory and process data [7].

Usually, production systems include multiple subsystems and layers of connectivity. Thus, although research-based solutions for classical operations typically use a graph-based representation of problems and flow-based optimisation algorithms, conventional single-layer networks quickly become incapable of representing the complexity and connectivity of all the details of the production

line. With the overlapping data in Industry 4.0 solutions, it should be highlighted that multilayer networks are expected to be the most suitable options for representing modern production lines. The concept of a multilayer network was developed to represent multiple types of relationships [8], and these models have been proven to be applicable to the representation of complex connected systems [9]. Network-based models can also represent how products, resources and operators are connected [10], which is beneficial in terms of solving manufacturing cell formation problems [11]. This work demonstrates how the multilayer network representation of production lines can be utilised in line balancing.

In the proposed novel network model, the layers represent the skills of the operators, the tools required for the activities, and the precedence constraints of the activities. At the same time, a multi-objective optimisation algorithm designs the assignment of activities and operators to network layers. The proposed multilayer network approach supports the intuitive formulation of multi-objective line balancing optimisation tasks. Besides the utilisation of operators, the utilisation of the tools and the number of skills of an operator are also taken into account. The main advantage of the proposed network-based representation is that the latter two objectives are directly related to the structural properties of the optimised network.

Line balancing is a non-deterministic polynomial-time hard (NP-hard) optimisation problem, which means that the computational complexity of the optimisation problem increases exponentially as the dimensions of the problem increase. This challenge explains why numerous meta heuristic approaches such as simulated annealing (SA) [12,13], hybrid heuristic optimisation [13,14], chance-constrained integer programming [15], recursive and dynamic programming [16], as well as tabu search [17] have been utilised in the field of production management. Fuzzy set theory provides a transparent and interpretable framework to represent the uncertainty of information and solve the ALB problem [18,19]. Among the wide range of heuristic methods capable of achieving reasonable solutions [20], SA is the most widely used search algorithm [21], so it has already been applied to solve mixed and multi-model line-balancing problems [22].

To deal with the complexity of this problem, an SA algorithm was also developed. The proposed algorithm utilises a unique problem-oriented sequential representation of the assignment problem and applies a neighbourhood-search strategy that generates feasible task sequences for every iteration. Since the algorithm has to handle multiple aspects of line balancing, the analytic hierarchy process (AHP) technique is used to quantify the importance of the objectives, also known as Saaty's method [23]. AHP is a method for multi-criteria decision-making which is used to evaluate complex multiple criteria alternatives involving subjective judgments [24]. This method is a useful and practical approach to solving complex and unstructured decision-making problems by calculating the relative importance of the criteria based on the pairwise comparison of different alternatives [25]. The method has been widely applied thanks to its effectiveness and interpretability. Two papers were found in which it has already been applied to determine the cost function of multi-objective SA optimisation problems. In the first case study of supplier selection, AHP was applied to calculate the weight of every objective by applying the Taguchi method [26] (Adaptive Tabu Search Algorithm—ATSA [27]). In contrast, in the second report, this concept was applied to the maintenance of road infrastructure [28].

The novelties of the work are the following:


#### **2. Problem Formulation**

In this section, the problem formulation is presented. First, the representation of production line modelling with the multilayer network is introduced in Section 2.1. The details of the minimised function and its AHP-based aggregation are given in Section 2.2.

#### *2.1. Multilayer Network-Based Representation of Production Lines*

The proposed network model of the production line consists of a set of bipartite graphs that represent connections between operators, **o** = {*o*1,..., *oNo*}; skills of the operators needed to perform the given activity, **s** = {*s*1,...,*sNs*}; equipment, **e** = {*e*1,...,*eNe*}; activities (operations), **a** = {*a*1,..., *aNa*}; and the precedence constraints between activities, **a** - = - *a*1 - ,..., *a* - *Na* . The relationships between these sets are defined by bipartite graphs *Gi*,*<sup>j</sup>* = (*Oi*,*Oj*, *Ei*,*j*) represented by **A**[*Oi*,*Oj*] biadjacency matrices, where *Oi* and *Oj* denote a general representation of the sets of objects, such that *Oi*,*Oj* ∈ - **s**, **e**, **a** - , **a**, **o** .

The edges of these bipartite networks represent structural relationships; e.g., the biadjacency matrix **A**[**a**, **a**- ] represents the precedence constraints or **A**[**a**, **o**] represents the assignments of activities to operators. Moreover, the edge weights can be proportional to the number of shared components/resources or time/cost (see Table 1) [10].

**Table 1.** Definition of the biadjacency matrices of the bipartite networks used to illustrate how a multidimensional network can represent a production line.


As can be seen in Figure 1, these bipartite networks are strongly connected. The proposed model can be considered as an interacting or interconnected network [8], where bipartite networks define the layers. Since different types of connections are defined, the model can also be handled as a multidimensional network. As illustrated in Figure 2, when relationships between the sets *Oi* and *Oj* are not directly defined, it is possible to evaluate the relationship between their elements *oi*,*<sup>k</sup>* and *oj*,*<sup>l</sup>* in terms of the number of possible paths or the length of the shortest path between these nodes [10].

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

**Figure 2.** Projection of a property connection.

In the case of connected unweighted multipartite graphs, the number of paths intersecting the set *O*<sup>0</sup> can be easily calculated based on the connected pairs of bipartite graphs as follows:

$$\mathbf{A}\_{\rm O\_0}[O\_i, O\_j] = \mathbf{A}[O\_{0\prime}, O\_i]^T \times \mathbf{A}[O\_{0\prime}, O\_j] \,. \tag{1}$$

In the proposed network model, the optimisation problem is defined by the allocation of tasks that require the allocation of different skills and tools to an operator that might necessitate extra training, labour and investment costs. The main benefit of the proposed network representation is that these costs can be directly evaluated based on the products of the biadjacency matrices **S** and **W**:

$$\mathbf{A}\left[\mathbf{s},\mathbf{o}\right] = \mathbf{A}\left[\mathbf{s},\mathbf{a}\right] \mathbf{A}\left[\mathbf{a},\mathbf{o}\right] = \mathbf{S}\mathbf{W}.\tag{2}$$

The resultant network **A**[**s**, **o**] represents how many times a given skill should be utilised by an operator, while its unweighted version **A***u*[**s**, **o**] models which skills the operators should have.

The design of the presented network model is based on the analysis of the semantically standardized models of production lines [29], and the experience gained in the development project connected to the proposed case study. The details of the multilayer network-based modelling of a wire-harness production process can be found in [10].

#### *2.2. The Objective Function*

A simple assembly line balancing problem (SALBP) assigns *Na* tasks/activities to *No* workstations/operators. Each activity is assigned to precisely one operator, and the sum of task times of workstation should be less or equal to the cycle time *Tc* [30]. Precedence relations between activities must not be violated [31]. There are two important variants of this problem [32]: SALBP-1 aims to minimise *No* for a given *Tc*, while the goal of SALBP-2 is to minimise *Tc* for a predefined *No* [1,33,34]. In this paper, the SALBP-2 problem was investigated and extended to include the following skill and equipment-related objective functions:

Station-time-related objective: The main objective of line balancing is to minimise the cycle time *Tc*, which is equal to the sum of the maximum of the station times *Tj*. The utilisation of the whole assembly line can be calculated as follows:

$$T\_c = \arg\max\_j T\_j = \sum\_{i=1}^{N\_x} w\_{i,j} t\_{i\prime} \tag{3}$$

where *ti* represents the elementary activity times of the *ai*-th activity.

As the theoretical minimum of *Tc* is

$$T\_c^\* = \frac{\sum\_{i=1}^{N\_d} t\_i}{N\_0} \,\tag{4}$$

the following ratio evaluates the efficiency of the balancing of the activity times:

$$Q\_T(\pi) = \frac{T\_c^\*}{T\_c} = \frac{\frac{\sum\_{i=1}^{N\_a} t\_i}{N\_o}}{\sum\_{i=1}^{N\_a} w\_{i,j} t\_i} \tag{5}$$

Skill-related (training) objective: The training cost is calculated with the node degree between skill-operator elements *s* − *o*. The number of skills *Ns* is divided by the sum of the node degrees *ki* between sub-networks *s* and *o* in the multilayer representation:

$$Q\_S(\pi) = \frac{N\_s}{\sum\_{i}^{s-o,o} k\_i!} \tag{6}$$

Equipment-related objective function: The equipment cost is calculated with the node degree between equipment-operator elements *e* − *o*. The number of pieces of equipment *Ne* is divided by the sum of the node degrees *ki* between sub-networks *e* and *o* in the multilayer representation:

$$Q\_E(\pi) = \frac{N\_\ell}{\sum\_{i}^{\iota - \sigma, \rho} k\_i!} \tag{7}$$

Since the importance of these objectives is difficult to quantify, a pairwise comparison is used to evaluate their relative importance, and the analytic hierarchy process (AHP) is used to determine the weights *λ* in the objective function:

$$Q(\pi) = \lambda\_1 Q\_T(\pi) + \lambda\_2 Q\_S(\pi) + \lambda\_3 Q\_E(\pi),\tag{8}$$

where *QT*(*π*) ∈ [0, 1] represents the balance of the production line, and *QS*(*π*) ∈ [0, 1] and *QE*(*π*) ∈ [0, 1] measure the efficiency of how the skills and tools are utilised, respectively.

The application of AHP-based weighting is beneficial to integrate the normalised values of the easy to evaluate station-time and equipment-related objectives, and the less specific training-related costs. Although the pairwise comparison of the importance of these objectives and cost-items is subjective, the consistency of the comparisons can be evaluated based on the numerical analysis of the resulted comparison matrices (which will be shown in the next section), which clarifies the reason for our choice of AHP as an ideal tool to extract expert knowledge for the formalisation of the cost function.

#### **3. Simulated Annealing-Based Line-Balancing Optimization**

This section presents the proposed optimisation algorithm. The representation of the SA problem is introduced in Section 3.1. Section 3.2 discusses how the precedence constraints of the activities are represented, while Section 3.3 presents how the assignment of activities to operators is formulated by a sequencing problem that can be efficiently solved by the proposed simulated annealing algorithm.
