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Adapting Thurstone’s Law of Comparative Judgment to fuse preference orderings in manufacturing applications

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Abstract

A rather common problem in the manufacturing field includes: (i) a collection of objects to be compared on the basis of the degree of some attribute, (ii) a set of judges that individually express their subjective judgments on these objects, and (iii) a single collective judgment, which is obtained by fusing the previous subjective judgments. The goal of this contribution is to develop a new technique that combines the Thurstone’s Law of Comparative Judgment with an ad hoc response mode based on preference orderings. Apart from being relatively practical and user-friendly, this technique allows to express the collective judgment of objects on a ratio scale and is applicable to a variety of practical contexts in the field of manufacturing. The description of the proposed technique is integrated with the application to a practical case study.

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Notes

  1. For simplicity, we consider single-ended psychological continua, in which the objects’ attribute progresses in one direction only, starting from the absolute-zero point; this is reasonable when the attribute has a positive connotation exclusively (e.g., the importance of a set of product requirements) (Torgerson 1958).

  2. This assumption is quite common for psychometric studies on subjective perceptions (Torgerson 1958; Lim 2011).

  3. According to the terminology introduced in Franceschini et al. (2007), the term “preference”—defined as subjective and non-empirical (i.e., which does not necessarily stem from a direct observation of reality) assignment of numbers/symbols to properties of objects—should be replaced with the term “evaluation”—defined as subjective and empirical (i.e., which stems from a direct observation of reality) assignment. Despite this, for the sake of simplicity the term “preference” will be hereafter used.

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Authors and Affiliations

  1. Department of Management and Production Engineering (DIGEP), Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129, Turin, Italy

    F. Franceschini & D. Maisano

Authors
  1. F. Franceschini
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  2. D. Maisano
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Correspondence to F. Franceschini.

Appendix

Appendix

A.1 Torgerson’s anchoring

This section exemplifies the anchoring technique by Torgerson (1958, p. 196), applying it to the LCJ scaling in Fig. 11. Focussing on this scaling process, it can be seen that the (input) paired-comparison relationships are identical to those in the example in Fig. 8, except that those with at least one of the dummy/anchor objects are not present. The resulting (non-anchored) scale is reported in Fig. 11d.

Fig. 11
figure 11

Example of application of the LCJ, considering the paired-comparison relationships by five judges (J1 to J5) on four objects (O1 to O4). These relationships are identical to those in the example in Fig. 9, except that those with at least one of the dummy/anchor objects are not present

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The rationale of the Torgerson’s anchoring is that results of the LCJ are (at least roughly) correlated with those resulting from the so-called Method of Single Stimuli, in which each judge directly assigns the objects’ scale values, with respect to two anchors: (1) a (presumed) absolute zero, corresponding to the absence of the attribute, and (2) the maximum-imaginable degree of the attribute, conventionally set to 5. While aware of the difficulty and potential roughness of these direct assignments, Torgerson (1958, p. 196) suggests their use just for the purpose of anchoring the LCJ scale.

Subsequently, judge assignments are aggregated—object by object—through a central tendency indicator, such as the mean or median value (g), and plot against the scale values (x) computed from the LCJ. Then, a straight line to the points is fitted and the intercept on the horizontal axis (g = 0) is taken as estimate of the position of the absolute-zero point (Z) and that on the horizontal line (g = 5) as estimate of the position of the point with maximum-imaginable degree (M) of the attribute.

Considering the example in Fig. 11, we hypothesize that the five judges directly assign the objects’ scale values on a rating scale from 0 to 5, with unitary resolution; the zero point corresponds to the absence while the maximum value (i.e., 5) corresponds to the maximum-imaginable degree of the attribute. Table 3 collects these assignments.

Table 3 Direct assignments of the scale values for six objects (i.e., O1 to O4, Z and M), by five judges (J1 to J5)
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Assignments are then aggregated using the arithmetic mean. The graph in Fig. 12 plots the resulting mean values (g) against the scale values (x) obtained through the LCJ (see Fig. 11). Then, a straight tendency line is fitted (through a linear least-squares regression) and the intersection of this line with the horizontal axis (g = 0) determines an estimates of the absolute-zero point (Z, i.e., first anchor), while that with the horizontal line g = 5 determines an estimate of the point (M, i.e., second anchor) of the maximum-imaginable degree of the attribute on the Thurstone’s scale. Next, the LCJ scale values are normalized in the conventional range [0, 100], through the linear transformation in Eq. 4. This scale can reasonably be considered as a ratio one (see Table 4).

Fig. 12
figure 12

Comparison of the scale values resulting from the Thurstone’s LCJ and those resulting from a direct scale-value assignment (Method of Single Stimuli) for four objects (O1 to O4)

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Table 4 Anchoring of the LCJ scale (in Fig. 11d), applying the technique by Torgerson
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We have verified that the new anchoring technique (presented in "Anchoring the Thurstone’s Scaling: the ZM-technique" section) provides results in line with those obtained from the Torgerson’s technique. e.g., Fig. 13 shows that these two anchoring techniques, when applied to the same scaling problem, are strongly correlated. Also, we have empirically observed that the correlation tends to increase for problems with a larger number of objects and/or judges.

Fig. 13
figure 13

Comparison between two anchoring techniques (i.e., that by Torgerson, exemplified in Table 4, and the proposed technique, exemplified in Fig. 9), with reference to the same LCJ-scaling problem

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A.2 Questionnaire

Figure 14 reports an example of questionnaire to guide the construction of preference orderings.

Fig. 14
figure 14

Example of questionnaire with guidance for the formulation of preference orderings

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Franceschini, F., Maisano, D. Adapting Thurstone’s Law of Comparative Judgment to fuse preference orderings in manufacturing applications. J Intell Manuf 31, 387–402 (2020). https://doi.org/10.1007/s10845-018-1452-5

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