4.1. Problem Modelling, Pair-Wise Comparison, Priority Derivation, and Consistency Evaluation
The aim of this work is to select a joint having the best combination of interlocking, head height, and strength. To achieve this goal, six criteria (
), including rivet’s head height (
), bottom thickness (
), minimum bottom thickness (
), deformed rivet diameter (
), shear strength (
), and peel strength (
), were taken into account. Therefore, the decision hierarchy structure for selecting a joint with the declared criteria among
(
) number of the joints is the one shown in
Figure 7. For all criteria, except the head height, a higher amount means better quality. In the case of head height, it is important that the head has good contact with the top sheet, as moisture can gather and cause corrosion issues, which means the ideal head height is zero. Also, it is possible to have a head height with an amount of less than zero, which is evident when the rivet is beneath the sheet’s surface. Therefore, Equation (2) cannot be applied for evaluation. To overcome this problem, a reference head height,
, is introduced as the goal for the criteria. Hence, the target for a joint is to have a relative height,
, close to 1. The absolute value is used to fit the AHP model requirement, although h
i can be negative in practice. In this case, using Equation (2), the entries of the pair-wise comparison matrix for the head height (
) are:
From Equations (3) and (10), the pair-wise comparison matrices for each of the criteria are as below:
and
The next step is to construct the priority vector for the matrices of Equations (10) to (14). Using Equation (5), priority vectors are defined as:
and
At this step, the pair-wise comparison matrix (
) for the whole hierarchy can be derived by assembling all the priority vectors (Equations (17–22)). For this purpose, each of the priority vectors forms one of the columns in the comparison matrix (
):
Before further progress, the consistency of the pair-wise comparison matrix () should be checked using Equation (9).
Until now, the pair-wise comparison matrix was formed without any prioritization among the criteria. For this reason, a decision maker should prioritize each of the criteria by giving them a weight (
) using Saaty’s scale (
Table 1). Applying the same procedure as that used for the criteria, the weight vector of the whole hierarchy is defined as:
By multiplying to (), the rank of the alternatives is defined by the resultant vector.
From all of the above procedures, one can simply select the best condition of riveting following the steps of the flowchart shown in
Figure 8.
4.2. Ranking the Alternatives
Before starting the analysis for the current empirical case, it is important to consider the possibility of insertion of all of the experimental situations regardless of their quality and whether they can pass the predefined limits or not. However, to reduce the size of analysis and to decrease the number of calculations, it is better to decrease the number of alternatives by eliminating unacceptable conditions, considering the point that the low quality conditions spontaneously will be placed at the end of the ranking by AHP. Therefore, eliminating the poor conditions does not have any effect on the final assessment. Inserting data in
Table 4 into Matrix (
) (Equation (23)), the comparison matrix is defined as:
Using Equation (8), with inconsistency issues in consideration, GCI is calculated as zero (), which means that the comparisons are completely consistent. This is reasonable, since all of the presented data in the current work are quantitative.
Furthermore, in this step, it is necessary to rank the criteria based on
Table 1, which means the pair-wise comparison of the criteria. It is the decision maker’s responsibility to verbally judge between the criteria based on their knowledge, scientific or industrial requirements and standards, and the customer’s demand. The selected Saaty’s scale is summarized in
Table 5. The scales in
Table 1 are used when
has an equal or higher importance in comparison to
. This means that if the weight of
over
is
according to Saaty’s scale, then the weight of
over
is simply
, which could not be found in
Table 1. The most important parameter in performance assessment of a joint is its strength. Therefore, the highest relative importance numbers are associated with shear and peel strength over others. However, the relative importance of shear and peel strengths were assumed to be the same, although in real applications they may have different importance. The second important parameter is the head height and minimum bottom thickness. There are strict rules and standards on head height in different applications, especially in the automobile industry. Hence, giving a high priority to head height is meaningful. Also, the minimum bottom thickness is crucial because of its role in the failing of a joint. Although the importance of the bottom thickness is lower than that of head height/minimum bottom thickness, it is more important than deformed rivet diameter. As long as the rivet is stable (do not lose its strength or fail), a higher deformed rivet diameter can provide a stronger interlock.
Using scales in
Table 5 and Equation (24), the weight vector of the whole hierarchy is calculated as:
By multiplying Equation (25) with Equation (26), final priorities of the alternatives is obtained. The final priorities and rank of the joints are presented in
Table 6.
is ranked as first. Considering the properties of this joint in
Table 4, it is evident that the highest shear strength and the lowest head height correspond to this joint. Its peel strength is the same as that of
, which is placed third in
Table 6. The second rank in
Table 6 is occupied by
, which has the highest peel strength. Also, it has a higher deformed rivet diameter and lower bottom and minimum bottom thicknesses in comparison to
. Paying attention to the condition of the joints in
Table 3, it is interesting to see that for the rivets with a hardness of
, the best joint was achieved by applying a velocity of
. On the other hand, for the rivets with a lower hardness of
, the performance of the joint is the best at a higher velocity of 360 mm/s. This contradicts a pre-assumption that using the lower hardness rivets and keeping all other parameters unchanged, a joint with the same performance can be achieved with a lower force. This is a simple example to declare the importance of AHP in multi-criteria problems in materials science, such as SPR joints.
Figure 9 shows the microstructure changes in different areas of the best quality joint (
). As seen in
Figure 9a, the microstructure of the as-received sheet consists of almost equi-axed grains, with an average grain size of 45 µm. In the area near the rivet’s tip grain refinement happens (see
Figure 9b) as a result of the deformation that happens in this area and the compressive stress applied by the insertion of the rivet. Also, the grains are elongated as a result of the material flow. The average grain size in this area is approximately 40 µm. By approaching the rivet’s tail, grains become more elongated and the grain refinement increases. The approximate grain size of the grains in this region (
Figure 9c) is 25 µm. This means the decrease in the grain size by 38% is noteworthy. As seen in
Figure 9d, the microstructure around the joint is more equi-axed in comparison to the other regions and the microstructure is more similar to the as-received one, with an average grain size of 36 µm.
Another stimulating result of the analysis is . It is ranked third, however this joint is standing higher than , , and , despite its lower shear strength. This confirms that it is not correct to select the condition of the best joint based on only one parameter, even though that parameter is the most important one (shear strength) among the others. Also, this shows that by lowering the die depth, longer rivets with lower hardness are required for an acceptable joint.
Finally, it is possible to insert other criteria into this analysis, depending on decision makers’ requirements. Also, it is possible to use qualitative criteria, such as the appearance of surface cracks and deflection of the sheet, in the mentioned model. The point is that in this case, a consistency check will be an important step and can be used either for the pair-wise comparison matrixes of each criterion, or the total pair-wise comparison matrix.