*2.2. Characteristics of Methods for Optimizing the Arrangement of Workstations*

The classification of methods for optimizing the arrangement of workstations can be implemented, among others, according to the following criteria [24,25]:


When choosing a method to optimize the placement of stations for the planned research on the placement of 3D printing machines and devices, it was decided to take into account the most commonly used methods that simultaneously differed from each other in terms of the operation algorithm. The length of transport routes, the number of transport operations, transport costs, weight of transported materials, or the volume of transports can be used for optimization criteria. For the purposes of research, a total of five methods were selected for further analysis:


The CRAFT (computerized relative allocation of facilities technique) method was proposed in 1964 by E. Buff, G. Armour, and T. Vollmann. The CRAFT algorithm changes the positions of the stations in the initial layout to determine improved solutions based on the flow of materials; subsequent changes lead to the layout with the lowest total cost. CRAFT uses the material flow cost as a criterion to consider the best layout. The best design is the one that has a minimum total cost. CRAFT does not guarantee the least costly solution, as not all possible exchanges are considered. The quality of the final solution depends on the initial solution. Therefore, it is common practice to identify several different initial designs and try all the exchange combinations and then select the best solution generated [26–28].

In the CRAFT method, an existing layout or a completely new plan is considered as a block arrangement. The algorithm calculates the allocations of workstations and estimates the costs that will be incurred in the initial phase of the project. The impact on the cost measure is computed for two or more different position settings in the layout plan. The main goal of this method is to minimize the total cost of the TC function [16], which is defined by the formula:

$$TC = \sum\_{i=1}^{n} \sum\_{j=1}^{n} D\_{ij} \cdot \mathcal{W}\_{ij} \cdot \mathbb{C}\_{ij\prime} \tag{1}$$

where *Dij* is the distance between station *i* and station *j*; *Wij* is the volume of flow between station *i* and station *j*; *Cij* is the cost of transport between station *i* and station *j*; and *n* is the number of stations.

The key steps of the algorithm are the calculation of the total cost of the TC function for the distance matrix describing the examined layout and the part flow matrix built on the basis of the sequence of technological operations and the demand for parts at different time periods.

The Bloch-Schmigalla method was initiated in the 1950s by W. Bloch. It uses a mesh of equilateral triangles in order to find the correct distribution of positions. The method was further developed and modified by H. Schmigalla [29]. The Schmigalla method of triangles begins by determining the order in which the workstations will be located and their distribution at individual nodes of a mesh composed of equilateral triangles. One should start with setting up a pair of stations with the highest flow intensity. It was assumed that the optimal system will be obtained when the W value of the objective function is the smallest possible. The function W is expressed as follows:

$$\mathcal{W} = \sum\_{i=1}^{n} \sum\_{j=1}^{n} S\_{ij} \cdot L\_{ij} \to \text{minimum}, \tag{2}$$

where *Sij* is the amount of part flow between the stations *i* and *j*; and *Lij* is the distance between the stations, which is always equal to the side length of the equilateral triangle of the mesh.

The next steps of the algorithm provide for the construction of a matrix of the sequence of technological operations to select later, from the matrix of connections, the objects connected with the highest flow intensity; from these objects, we start arranging the positions on the mesh of equilateral triangles. A table of the intensity of connections between sites that have already been deployed and sites that have not yet been located helps in determining the location of subsequent objects.

The description of the ROC (rank order clustering) method was published in 1980 by J. King. ROC enables the arrangement of workstations by grouping them into manufacturing cells, taking into account families of parts classified according to technological similarity [30,31]. Its algorithm provides for the construction of a matrix of transport links and then, for each column of this matrix, its so-called binary weight is determined, according to the formula:

$$BW\_j = 2^{m-j} \, \_\prime \tag{3}$$

where *m* is the number of machines and *j* is the machine number. For the value of each row, the decimal binary equivalent is then calculated using the formula:

$$DE\_i = \sum\_{j=1}^{m} 2^{m-j} \cdot a\_{ij\prime} \tag{4}$$

where *aij* is the relationship between element *i* and position *j*, according to the matrix of transport connections. The effect of the ROC method is obtaining the arrangement of workstations with a job shop structure.

The modified spanning tree (MST) algorithm is based on the selection of adjacent pairs of workstations using a designated adjacency weight matrix. This method is similar to the spanning tree algorithm, which relies on a set of vertices and their connecting edges, where each edge with a separate weight connects two vertices. The vertices correspond to the workstations and the edges to the transport routes [32]. In the MST method, the adjacency weight matrix *f ij* is built based on Equation (5) [1]:

$$f\_{ij}' = \left(f\_{ij}\right)\left(d\_{ij} + 0, \mathbf{5}\cdot\left(l\_i + l\_j\right)\right),\tag{5}$$

where *f ij* is the adjacency weight matrix; *fij* is the flow matrix; *dij* is the recommended distances between *i* and *j* machines; *li* is the *i* machine dimension; and *lj* is the *j* machine dimension.

Equation (5) uses the product of the number of transport connections between workstations and the distance to be covered during the implementation of transport activities to determine the weight of the adjacency. It should be noted that the orientation of the machines is known in advance, or possibly presumed, which may be considered as a limitation of this method. As a consequence, it may also be necessary to prepare several variants of potential solutions that differ only in the orientation of machines and devices, and re-evaluate them by the MST algorithm.

From the adjacency weight matrix *f ij*, a pair of workstations connected by the highest weight are selected—they will be placed next to each other on the layout plan—a row and a column with a pair of devices already placed are plotted from the matrix. The next device is selected on the same principle, building a one-dimensional matrix of the order of arrangement of workstations.

The CORELAP (computerized relationship layout planning) method is one of the approximate methods. The use of this method is beneficial if there are various relationships between the objects being arranged that cannot be represented by one quantity [33].

The CORELAP method is based on determining the adjacency weight. The method consists of three stages: planning surfaces, their size and connections between them; calculating and designing CORELAP; and the final stage is drawing the layout plan. In order to define the relationship between the individual surfaces, each pair is assigned a symbol A, E, I, O, U, or X with a specific value, starting with "absolutely necessary" and ending with "undesirable". Relationships between positions recorded in the matrix are assessed by calculating the value of the proximity indicator (TCR). On this basis, the order of placing the positions is determined based on the six rules proposed in the CORELAP method. Depending on the adjacency method (fully contiguous, point contact, and non-contiguous, for which the adhesion coefficient is 1, 0.5, and 0, respectively), subsequent stations are arranged. The method was first used as an algorithm for arranging rooms or departments in industrial plants.

The presented characteristics of the methods of optimizing the arrangement of workstations show that their application for 3D printing machines is virtually problem-free with one exception. A significant limitation is related to the need to take into account auxiliary devices, often not connected to the main 3D printing device by means of transport. In this case, most of the methods based on material flow analysis simply omit these devices, and this may result in the lack of space for their placement on the layout plan. The CORELAP method does not have this disadvantage.
