Customer requirement prioritization on QFD: a new proposal based on the generalized Yager’s algorithm

Abstract

The focus of this paper is on the prioritization of customer requirements (CRs) in the quality function deployment (QFD) process. There are numerous techniques for this task, which are based on different types of judgements (e.g., in the form of preference orderings, relative importance ratings, paired comparisons), generally collected by questionnaires/interviews. Unfortunately, many of these techniques violate some properties of the scales on which judgements are defined and/or are not flexible and user-friendly for respondents. This paper introduces a prioritization technique based on a recent algorithm—denominated as generalized Yager’s algorithm—aimed at fusing the preference orderings by multiple respondents into a single ordering. This technique can be applied even when (1) some CRs are tied or omitted in the individual preference orderings, and (2) respondents have a hierarchical importance ranking. Also, the procedure is automatable and relatively robust and provides a solution which is consistent with the preference orderings. The description is supported by a realistic application example, concerning the prioritization of QFD’s CRs in the design of an aircraft seat.

Appendix

1.1 Description of the GYA

This section briefly describes the GYA. For a more detailed explanation, we refer the reader to (Franceschini et al. 2015).

In general, the GYA is aimed at fusing the preference orderings of n alternatives (a, b, c, etc.), by multiple decision-making agents² (D ₁, D ₂, etc.), into a consensus fused orderings. In the problem of the prioritization of QFD’s CRs, decision-making agents would be the questionnaire/interview respondents and alternatives would be the CRs. For simplicity, in the remaining explanation, we will refer to this specific problem.

The GYA will be explained, assuming that the input preference orderings are non-strict linear (i.e., with no incomparabilities between CRs); however, this algorithm can be applied even in the more general case in which orderings are non-strict partial, i.e., when two or more CRs are incomparable (Franceschini et al. 2015).

The GYA description can be structured in three phases, described in the remaining subsections: (1) construction and reorganization of preference vectors, (2) definition of the reading sequence and (3) construction of the fused ordering.

1.1.1 Construction and reorganization of preference vectors

The goal of this phase is constructing preference vectors related to the input preference orderings. For the purpose of example, let us consider the same example presented in Fig. 2, about five fictitious respondents (D ₁ ~ D ₅) with five (non-strict linear) orderings of six CRs (a, b, c, d, e and f). It can be noticed that the only ordering including all the six CRs is that by D ₄. In the other ones, one or more CRs are systematically omitted. It is assumed that the non-strict linear ordering depicting the importance hierarchy between respondents is: D ₅ > (D ₃ ~ D ₄) > (D ₁ ~ D ₂).

The preference orderings should be turned into preference vectors. In this construction, we place the CRs as they appear in one ordering, with the most preferred one(s) in the top positions. If at any point t > 1 CRs are tied (i.e., indifferent), we place them in the same element and then place the null set (“Null”) in the next t − 1 lower positions. For example, when considering three CRs (a, b and c) with the ordering (a ~ b) > c, the resulting vector will conventionally be [{a ~ b}, Null, {c}]^T. By adopting this convention, the number of elements of a vector will coincide with the number of CRs.

Table 11 exemplifies the construction of the preference vectors related to the five preference orderings in Fig. 2. For simplicity, they will be denominated as the corresponding respondents (i.e., D _i ). Although there are six total CRs, some of them may be omitted in a certain vector; therefore, the number of elements (n _i ) can change from an ith vector to one other. It can be seen that each vector element is associated with a relative-position indicator, given by the cumulative relative frequency F _i,j —i.e., the ratio between the position (j) of an element—starting from the bottom—and n _i .

Table 11 Construction of preference vectors related to the orderings in Fig. 2

Next, preference vectors are transformed into so-called reorganized vectors, conventionally denominated as $D_{i}^{*}$. The vector reorganization stage is based on two steps: (1) D _i vectors are sorted decreasingly with respect to their importance, and (2) those with indifferent importance are aggregated. For example, D ₁ and D ₂ vectors have indifferent importance, so they are aggregated; the same applies to D ₃ and D ₄.

The aggregation is performed by merging the elements of the vectors to be aggregated and sorting them in descending order with respect to their F _i,j values. When the F _i,j values of two (or more) elements coincide, the CRs that they contain are considered as indifferent. After this reorganization, the resulting vectors are conventionally denominated as $D_{i}^{*}$, while the F _i,j values that they include are denominated as $F_{i,j}^{*}$ (see Table 12). It can be noted that $D_{2}^{*}$ and $D_{3}^{*}$—given by the aggregation of two pairs of vectors with indifferent importance (respectively, D ₃, D ₄ and D ₁, D ₂)—may contain two occurrences of some CRs.

Table 12 Construction of reorganized vectors related to the preference vectors in Table 11

We remark that the vector aggregation is performed by using the information on the relative position of the elements (i.e., F _i,j ). The underlying assumption is that the degree of preference of the CRs in different preference vectors depends on their relative position. For a certain aggregated vector, the relevant $F_{i,j}^{*}$ values which contain the information on the relative position of the elements in the original orderings.

1.1.2 Definition of the reading sequence

The object of this phase is determining a sequence for reading the elements of the reorganized vectors. We remark that (1) these elements can be read according to a bottom-up or top-down sequence and that (2) the importance of $D_{i}^{*}$ vectors should be taken into account in this phase.

As regards the CRs’ prioritization problem, the top-down approach is probably more appropriate since it is focused on the upper positions of the preference orderings. For this reason, the remaining description of the GYA will refer to this approach. The flowchart in Fig. 5a illustrates the algorithm for constructing the fused ordering.

Fig. 5

Flowcharts illustrating the second and third phase of the top-down variant of the GYA: (a) definition of the sequence for reading the elements of the reorganized vectors and (b) procedure for constructing the fused ordering

Let us now focus on the criterion for switching from one element to one other. The first element to be read is that with highest position, in the most important vector ($D_{1}^{*}$). Having read a certain vector element, the next potentially readable $D_{i}^{*}$ vectors are those for which the not-yet-read element with highest position has $F_{i,j}^{*}$ higher than or equal to that of the last element read in the preceding vector (i.e., $D_{i - 1}^{*}$).

Reversing the perspective, a $D_{i}^{*}$ vector is temporarily “locked” (i.e., it cannot be read) if the $F_{i,j}^{*}$ value of the not-yet-read element with highest position is overcome by that of the last element read in the preceding vector. The set A includes the subscripts of the potentially readable (or “unlocked”) vectors. In formal terms:

$$ A = \left\{ {i \in \left\{ {2, \ldots ,m} \right\}} \right\}:F_{{i,\hbox{max} \left( {j : {\text{not - yet - read}}} \right)}}^{*} \ge F_{{i - 1,\hbox{min} \left( {j : {\text{read}}} \right)}}^{*} . $$

(1)

Among the vectors indicated in A, the one to be read is that with subscript:

$$ i = \hbox{min} \left( A \right). $$

(2)

Equation 2 entails that, among the “unlocked” vectors, priority is given to the one of highest importance. Having determined the vector to be read, the next element is the one not-yet-read with highest position. If there is no unlocked vector (i.e., A = “Null”), the next element is that (not-yet-read) with highest position in the most important not-yet-completely-read vector.

For further information about this sequencing logic and the rationale behind it, we refer the reader to (Franceschini et al. 2015).

1.1.3 Construction of the fused ordering

The flowchart in Fig. 5b illustrates the procedure for determining the fused ordering. A kth CR is included into the fused ordering when the gradual number of occurrences (O _k ) in the reading sequence reaches a certain threshold, i.e.,:

$$ T_{k,x} = x \cdot O_{k}^{\text{TOT}} , $$

(3)

being x a conventional percentage of the total number of occurrences ($O_{k}^{\text{TOT}}$) in the $D_{i}^{*}$ vectors’ elements. Table 13 shows the T _k,x values related to the CRs; x was conventionally set to 50 %.

Table 13 Thresholds for the selection of the CRs, in the case exemplified in Table 11; x is conventionally set to 50 %

It is worth noting that for a CR to be in the upper positions of the fused ordering, a predetermined portion of the occurrences larger than or equal to (x) should be in the upper positions in any of the individual preference orderings.

Applying the algorithm to the reorganized vectors in Table 12 and using the thresholds in Table 13, the fused preference ordering is: a > c > b > d > f > e. Table 14 shows the step-by-step construction. The last columns contain the gradual ordering.

Table 14 Step-by-step application of the GYA, in the case exemplified in Table 11

The robustness of the solution can be evaluated through a sensitivity analysis aimed at evaluating the stability of the fused ordering with respect to small variations in the T _k,x values.