# Analytic Hierarchy Process and Multilayer Network-Based Method for Assembly Line Balancing
## 2 Problem Formulation
### The Objective Function  
A simple assembly line balancing problem (SALBP) assigns \(N_{a}\) tasks/activities to \(N_{o}\) 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 \(T_{c}\)[30]. Precedence relations between activities must not be violated [31]. There are two important variants of this problem [32]: SALBP-1 aims to minimise \(N_{o}\) for a given \(T_{c}\), while the goal of SALBP-2 is to minimise \(T_{c}\) for a predefined \(N_{o}\)[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 \(T_{c}\), which is equal to the sum of the maximum of the station times \(T_{j}\). The utilisation of the whole assembly line can be calculated as follows:  
\[T_{c}=\arg\max_{j}T_{j}=\sum_{i=1}^{N_{a}}w_{i,j}t_{i}, \tag{3}\]  
where \(t_{i}\) represents the elementary activity times of the \(a_{i}\)-th activity.  
As the theoretical minimum of \(T_{c}\) is  
\[T_{c}^{*}=\frac{\sum_{i=1}^{N_{a}}t_{i}}{N_{o}}, \tag{4}\]  
the following ratio evaluates the efficiency of the balancing of the activity times:  
Figure 2: Projection of a property connection.  
\[Q_{T}(\pi)=\frac{T_{c}^{*}}{T_{c}}=\frac{\frac{\sum_{i=1}^{N_{a}}t_{i}}{N_{o}}}{ \sum_{i=1}^{N_{a}}w_{i,j}t_{i}}. \tag{5}\]  
Skill-related (training) objective: The training cost is calculated with the node degree between skill-operator elements \(s-o\). The number of skills \(N_{s}\) is divided by the sum of the node degrees \(k_{i}\) 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 \(N_{e}\) is divided by the sum of the node degrees \(k_{i}\) between sub-networks \(e\) and \(o\) in the multilayer representation:  
\[Q_{E}(\pi)=\frac{N_{e}}{\sum_{i}^{s-o,o}k_{i}}. \tag{7}\]  
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 \(\lambda\) in the objective function:  
\[Q(\pi)=\lambda_{1}Q_{T}(\pi)+\lambda_{2}Q_{S}(\pi)+\lambda_{3}Q_{E}(\pi), \tag{8}\]  
where \(Q_{T}(\pi)\in[0,1]\) represents the balance of the production line, and \(Q_{S}(\pi)\in[0,1]\) and \(Q_{E}(\pi)\in[0,1]\) 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.