Value-driven design for product families: a new approach for estimating value and a novel industry case study

Abstract

Advanced product platform and product family design methods are needed to define and optimize the value they bring to a company. Maximizing platform commonality and individual product performance often fails to realize the most valuable product family during optimization; however, few examples exist in the literature to explore these trade-offs. This paper introduces a novel industry case study to explore the differences between “traditional” multidisciplinary design optimization (MDO) and value-driven design (VDD) approaches to product family design. The case study involves a family of five commercially-available washing machines and integrates multidisciplinary analyses, simulations, mathematical models, and response surface models to obtain ratings for individual product attributes. These attributes are then aggregated into a value function for the product family using a novel approach to estimate sales volume and a demand sensitivity curve derived from publicly available data. We then formulate and solve a “traditional” MDO product family design problem using a multi-objective genetic algorithm to minimize performance deviation and a product family penalty function. A novel VDD formulation is then introduced and solved to maximize the net present value (NPV) for the firm producing the family of products. Visualization and comparison of the results illustrate that the “traditional” MDO formulation can find several promising solutions for the product family, but it fails to find solutions that maximize the value to the firm. The results also provide a benchmark for researchers to explore alternative value function formulations and solution approaches for product family design using the novel industry case study.

Appendix A. Engineering models for a front-loading washing machine

1.1 A.1 Dynamic model

It is well known that the vibration problems for front-loading washers usually occur during spinning cycles (Boyraz and Gündüz 2013; Lim et al. 2010; Nygårds and Berbyuk 2012). We perform vibration analysis of the spin mode by using a 2D dynamic model as seen in Fig. 10. The suspension system in front-loading washers consists of two springs and two dampers to minimize vibrations. In Fig. 10, θ_s is the angle of springs 1 and 2; and θ_d is the angle of dampers 1 and 2. The angle between the tub and spring, θ₁, the angle between tub and damper, θ₂, and the height of cabinet, h, are computed using

Fig. 10

2D dynamic model

$$ {\theta}_1={\cos}^{-1}\left(\frac{w_{\mathrm{s}}-2{l}_{\mathrm{s}}\cos \left({\theta}_{\mathrm{s}}\right)}{2{r}_{\mathrm{s}}}\right) $$

(A1)

$$ {\theta}_2={\cos}^{-1}\left(\frac{w_{\mathrm{d}}-2{l}_{\mathrm{d}}\cos \left({\theta}_{\mathrm{d}}\right)}{2{r}_{\mathrm{d}}}\right) $$

(A2)

$$ h={l}_{\mathrm{s}}\sin \left({\theta}_{\mathrm{s}}\right)+{l}_{\mathrm{d}}\sin \left({\theta}_{\mathrm{d}}\right)+{r}_{\mathrm{s}}\sin \left({\theta}_1\right)+{r}_{\mathrm{d}}\sin \left({\theta}_2\right) $$

(A3)

$$ where{w}_s=w\hbox{--} 2{w}_{ds}{w}_d=w\hbox{--} 2{w}_{dd} $$

(A4)

$$ {r}_s=r+{r}_{ds\kern0.75em }{r}_d=r+{r}_{dd} $$

(A5)

In (A1)–(A5), l_s is the length of springs 1 and 2; l_d is the length of dampers 1 and 2; w is the width of cabinet (i.e., the exterior of the washer); w_ds is the distance between the side cabinet and spring holder near the upper cabinet; w_dd is the distance between the side cabinet and the damper holder near the lower cabinet; r is the radius of tub; r_ds is the distance between tub and spring holder near the tub; and r_dd is the distance between the tub and the damper holder near the tub.

Similar types of 2D models have been used extensively to perform vibration analysis for front-loading washers in the literature (Boyraz and Gündüz 2013; Conrad and Soedel 1995; Nguyen et al. 2014a; Nguyen et al. 2014b; Yörükoğlu and Altuğ 2009). Boyraz and Gündüz (2013) define the equations of motion in the x and y axes as follows:

$$ {m}_{\mathrm{t}}\ddot{x}+{F}_{\mathrm{s}1,\mathrm{x}}-{F}_{\mathrm{s}2,\mathrm{x}}-{F}_{\mathrm{d}1,\mathrm{x}}+{F}_{\mathrm{d}2,\mathrm{x}}-{F}_{\operatorname{ext},\mathrm{x}}=0 $$

(A6)

$$ {m}_{\mathrm{t}}\ddot{y}+{F}_{\mathrm{s}1,\mathrm{y}}+{F}_{\mathrm{s}2,\mathrm{y}}+{F}_{\mathrm{d}1,\mathrm{y}}+{F}_{\mathrm{d}2,\mathrm{y}}-{F}_{\operatorname{ext},\mathrm{y}}=0 $$

(A7)

where m_t is the mass of the tub; F_s1 and F_s2 are the spring forces for springs 1 and 2, respectively; F_d1 and F_d2 are the damping forces for dampers 1 and 2, respectively; and F_ext is the external force. Boyraz and Gündüz (2013) validated the results of their 2D dynamic model against experimental measurements. In this paper, we briefly introduce a procedure to compute attributes related to vibration. The detailed computation procedure and validation result for the vibration and dynamic analysis is provided by Boyraz and Gündüz (2013).

Using (A6)–(A7), we can obtain the displacements of the tub in the x and y axes during spin mode. The maximum spin-speed of a drum is usually set between 1200 and 1300 rpm in front-loading washers (Kenmore 2015; LG 2015; Samsung 2015; Spelta et al. 2009). In this study, the maximum spin-speed is set to 1200 rpm. In (A8), the spin-speed of the drum is changed from 0 to 1200 rpm for 10 s. The angular velocity, β’ is given by

$$ {\beta}^{\prime }=N\left(1-{e}^{-\left(1/1.8\right)t}\right) $$

(A8)

where N is the maximum spin-speed converted to angular velocity and t is the time. When the spin-speed is increased from 0 to 1200 rpm, the normal and tangential forces, F_n and F_t, arise due to an unbalanced mass as seen in Fig. 10. The unbalanced mass, m_u, is usually set between 300 and 500 g (Lim et al. 2010; Spelta et al. 2009). In this work, m_u is set to 500 g.

$$ {F}_{\mathrm{n}}={m}_{\mathrm{u}}r{\left({\beta}^{\prime}\right)}^2 $$

(A9)

$$ {F}_{\mathrm{t}}={m}_{\mathrm{u}}r{\beta}^{{\prime\prime} } $$

(A10)

Then, the external forces for the x and y axes, F_ext,x and F_ext,y, are computed as follows:

$$ {F}_{\operatorname{ext},\mathrm{x}}={F}_{\mathrm{n}}\cos \left(\beta \right)-{F}_{\mathrm{t}}\cos \left(\beta \right) $$

(A11)

$$ {F}_{\operatorname{ext},\mathrm{y}}={F}_{\mathrm{n}}\sin \left(\beta \right)+{F}_{\mathrm{t}}\cos \left(\beta \right) $$

(A12)

Meanwhile, when the drum shifts during a spin cycle, the dynamic length of spring 1, l_{s1_dyn}, is defined by (A17), and then the spring forces for spring 1, F_s1,x and F_s1,y, in (A6) and (A7) are computed as follows (Boyraz and Gündüz 2013):>

$$ {x}_{\mathrm{s}}={l}_{\mathrm{s}}\cos \left({\theta}_{\mathrm{s}}\right) $$

(A13)

$$ {y}_{\mathrm{s}}={l}_{\mathrm{s}}\sin \left({\theta}_{\mathrm{s}}\right) $$

(A14)

$$ {x}_{\mathrm{s}1\_\mathrm{dyn}}(t)={x}_{\mathrm{s}}-x(t) $$

(A15)

$$ {y}_{\mathrm{s}1\_\mathrm{dyn}}(t)={y}_{\mathrm{s}}-y(t) $$

(A16)

$$ {l}_{\mathrm{s}1\_\mathrm{dyn}}(t)=\sqrt{{\left({x}_{\mathrm{s}1\_\mathrm{dyn}}(t)\right)}^2+{\left({y}_{\mathrm{s}1\_\mathrm{dyn}}(t)\right)}^2} $$

(A17)

$$ {F}_{\mathrm{s}1}(t)=k\left({l}_{\mathrm{s}}-{l}_{\mathrm{s}1\_\mathrm{dyn}}(t)\right) $$

(A18)

$$ {F}_{\mathrm{s}1,\mathrm{x}}(t)={F}_{\mathrm{s}1}(t)\frac{x_{\mathrm{s}1\_\mathrm{dyn}}(t)}{l_{\mathrm{s}1\_\mathrm{dyn}}(t)} $$

(A19)

$$ {F}_{\mathrm{s}1,\mathrm{y}}(t)={F}_{\mathrm{s}1}(t)\frac{y_{\mathrm{s}1\_\mathrm{dyn}}(t)}{l_{\mathrm{s}1\_\mathrm{dyn}}(t)} $$

(A20)

where x_s and y_s are the initial lengths of the spring in the x and y axes, respectively; x_{s1_dyn} and y_{s1_dyn} are the dynamic lengths of spring 1 in the x and y axes, respectively; and k is the spring coefficient. Similar to spring 1, the spring forces for spring 2, F_s2,x and F_s2,y, can also be computed using the analyses in (A13), (A14), (A15), (A16), (A17), (A18), (A19), and (A20). In this case, we use (A21) instead of (A15) because spring 2 has a positive displacement in the x axis unlike spring 1 when the drum shifts to the right (Boyraz and Gündüz 2013).

$$ {x}_{\mathrm{s}2\_\mathrm{dyn}}(t)={x}_{\mathrm{s}}+x(t) $$

(A21)

Also, the dynamic length of damper 1, l_{d1_dyn}, and the damping forces for damper 1, F_d1,x and F_d1,y, in (A6) and (A7) are computed as follows (Boyraz and Gündüz 2013):

$$ {x}_{\mathrm{d}}={l}_{\mathrm{d}}\cos \left({\theta}_{\mathrm{d}}\right) $$

(A22)

$$ {y}_{\mathrm{d}}={l}_{\mathrm{d}}\sin \left({\theta}_{\mathrm{d}}\right) $$

(A23)

$$ {x}_{\mathrm{d}1\_\mathrm{dyn}}(t)={x}_{\mathrm{d}}-x(t) $$

(A24)

$$ {y}_{\mathrm{d}1\_\mathrm{dyn}}(t)={y}_{\mathrm{d}}+y(t) $$

(A25)

$$ {l}_{\mathrm{d}1\_\mathrm{dyn}}(t)=\sqrt{{\left({x}_{\mathrm{d}1\_\mathrm{dyn}}(t)\right)}^2+{\left({y}_{\mathrm{d}1\_\mathrm{dyn}}(t)\right)}^2} $$

(A26)

$$ {v}_1={l}_{\mathrm{d}1\_\mathrm{dyn}}^{\prime } $$

(A27)

$$ {F}_{\mathrm{d}1}(t)={v}_1c $$

(A28)

$$ {F}_{\mathrm{d}1,\mathrm{x}}(t)={F}_{\mathrm{d}1}(t)\frac{x_{\mathrm{d}1\_\mathrm{dyn}}(t)}{l_{\mathrm{d}1\_\mathrm{dyn}}(t)} $$

(A29)

$$ {F}_{\mathrm{d}1,\mathrm{y}}(t)={F}_{\mathrm{d}1}(t)\frac{y_{\mathrm{d}1\_\mathrm{dyn}}(t)}{l_{\mathrm{d}1\_\mathrm{dyn}}(t)} $$

(A30)

where x_d and y_d are the initial lengths of the damper in the x and y axes, respectively; x_{d1_dyn} and y_{d1_dyn} are the dynamic lengths of damper 1 in the x and y axes, respectively; and c is the damping coefficient. To compute the damping forces for damper 2, F_d2,x and F_d2,y, we can also utilize the same analyses in (A22), (A23), (A24), (A25), (A26), (A27), (A28), (A29), to (A30), but we use (A31) instead of (A24):

$$ {x}_{\mathrm{d}2\_\mathrm{dyn}}(t)={x}_{\mathrm{d}}+x(t) $$

(A31)

When optimizing the dynamic model, Boyraz and Gündüz (2013) suggest three kinds of attributes that should be minimized: (1) wobbling movements, (2) transient vibration, and (3) steady-state vibration. In Fig. 11, the wobbling movements are observed in the spin mode of front-loading washers. To reduce the vibration in a washer, it is essential to minimize the wobbling movements. This attribute can be defined as follows:

Fig. 11

Wobbling movements

$$ WO=\max \left(\sqrt{x^2+{y}^2}\right) $$

(A32)

Another important attribute that should be considered is the span of transient vibration. As seen in Fig. 12, transient vibrations in the x and y axes occur during spinning cycles. The transient vibration span is given by

Fig. 12

Transient and steady-state vibration span

$$ {x}_{\mathrm{span}}^{\mathrm{tr}}={x}_{\mathrm{max}}-{x}_{\mathrm{min}} $$

(A33)

$$ {y}_{\mathrm{span}}^{\mathrm{tr}}={y}_{\mathrm{max}}-{y}_{\mathrm{min}} $$

(A34)

$$ TR={x}_{\mathrm{span}}^{\mathrm{tr}}+{y}_{\mathrm{span}}^{\mathrm{tr}} $$

(A35)

The span of the steady-state vibration should be also minimized when designing front-loading washers. In Fig. 12, the steady-state vibrations occur after transient vibrations. The span of steady-state vibration is defined as

$$ {x}_{\mathrm{span}}^{\mathrm{st}}={x}_{\mathrm{max}}-{x}_{\mathrm{min}} $$

(A36)

$$ {y}_{\mathrm{span}}^{\mathrm{st}}={y}_{\mathrm{max}}-{y}_{\mathrm{min}} $$

(A37)

$$ SS={x}_{\mathrm{span}}^{\mathrm{st}}+{y}_{\mathrm{span}}^{\mathrm{st}} $$

(A38)

Meanwhile, if the horizontal force, F_hor, affecting the cabinet is greater than the friction force between the front-loading washer and the floor, then the washer may exhibit an oscillatory behavior that mimics walking (Boyraz and Gündüz 2013; Conrad and Soedel 1995; Nygårds and Berbyuk 2014). To prevent walking during spinning cycles, the following constraint should be satisfied (Boyraz and Gündüz 2013):

$$ \left({F}_{\mathrm{ver}}+ mg\right)\mu -{F}_{\mathrm{hor}}>0 $$

(A39)

where F_ver and F_hor are the vertical and horizontal forces acting on the cabinet, respectively; g is the acceleration of gravity; and μ is the coefficient of friction.

1.2 A.2 Strength analysis of the spider

The spider subassembly has three arms that connect the rear of the drum with the motor that rotates the drum (see Fig. 13). As seen in the left-hand side of Fig. 13, when an unbalanced load occurs during a spinning cycle, the centrifugal force due to the unbalanced load creates tension and bending in the spider subassembly (Gorguluarslan et al. 2014). Therefore, the total maximum stress including the bending and normal stresses should be minimized in the design process of front-loading washers. Gorguluarslan et al. (2014) introduced an analytical method to compute the total maximum stress on the spider subassembly. Since the load is imposed on the three arms of the spider equally (Gorguluarslan et al. 2014), the maximum moment affecting each spider arm, M_max, can be defined as

Fig. 13

2D model for the drum and spider

$$ {M}_{\mathrm{max}}=\frac{1}{3}{F}_{\mathrm{n}}d $$

(A40)

where d is the depth of the tub.

The maximum bending stress on the spider, s_b,max, and the normal stress due to tension, s_n,max, can now be analytically computed as follows:

$$ {s}_{\mathrm{b},\max }=\frac{M_{\mathrm{max}}q}{I} $$

(A41)

$$ {s}_{\mathrm{n},\max }=\frac{F_{\mathrm{n}}}{A} $$

(A42)

where q is the distance from the neutral axis of the cross-section; I is the area moment of inertia; and A is the cross-sectional area. The total maximum stress on the spider is

$$ {s}_{\mathrm{t},\max }={s}_{\mathrm{b},\max }+{s}_{\mathrm{n},\max } $$

(A43)

This analytical approach to compute the maximum stress on the spider has been shown to provide similar values compared to finite-element analysis (FEA) results (Gorguluarslan et al. 2014).

1.3 A.3 Load torque analysis for the direct drive motor

The direct drive motor used to rotate a tub in front-loading washing machines delivers power more efficiently than conventional washing machines using a belt and pulley (Kalkat 2014). Unlike the vibration and strength analyses introduced in the previous sections, the maximum load torque for the motor occurs during washing cycles (Kim and Jung 2012; Lee et al. 2007). In the washing mode, the laundry repeatedly falls within the drum, and the rotating laundry may result in intense load torque on the motor.

First, the load torque due to the moments of the drum subassembly and laundry, T_drum and T_laundry, respectively, can be computed as

$$ {M}_{\mathrm{drum}}={M}_{\mathrm{fd}}+{M}_{\mathrm{cd}}+{M}_{\mathrm{rd}}+{M}_{\mathrm{as}}+{M}_{\mathrm{cs}}+{M}_{\mathrm{sh}} $$

(A44)

$$ {T}_{\mathrm{drum}}={M}_{\mathrm{drum}}\alpha $$

(A45)

$$ {T}_{\mathrm{laundry}}={M}_{\mathrm{laundry}}\alpha $$

(A46)

where M_drum, M_fd, M_cd, M_rd, M_as, M_cs, and M_sh are the moments of inertia of drum subassembly, front, center, and rear of drum, spider arms, center of spider, and shaft, respectively; and M_laundry is the moment of inertia of the laundry absorbing water.

In washing mode, there also exists a torque load due to the force of gravity on the laundry. The magnitude of the torque load changes according to the location of the laundry within the drum. For example, the load torque is maximum when the angle, θ, in Fig. 14 is 90°. Therefore, the maximum torque load due to gravity, T_g,max, is defined as

Fig. 14

Load torque due to the gravity

$$ {T}_{\mathrm{g}}={m}_{\mathrm{laundry}} gr\sin \left(\pi -\theta \right) $$

(A47)

$$ {T}_{\mathrm{g},\max }={m}_{\mathrm{laundry}} gr $$

(A48)

To estimate the total torque load more accurately, other types of torques (e.g., reverse load torque due to the viscosity between the laundry and water (Lee et al. 2007)) should also be considered. However, the influences of other loads are ignored in this work because the magnitudes of them are relatively small (Lee et al. 2007), and it is difficult to estimate accurate torque values (Kim and Jung 2012). Thus, the total load torque for the direct drive motor, T_total, is computed as

$$ {T}_{\mathrm{total}}={T}_{\mathrm{drum}}+{T}_{\mathrm{laundry}}+{T}_{\mathrm{g},\max } $$

(A49)

1.4 A.4 Response surface models based on ENERGY STAR and Consumer Reports data

In this section, we introduce a method to estimate attributes for (1) energy consumption, (2) water consumption, (3) washing performance, (4) gentleness, (5) noise, and (6) reliability. Unlike the performance analyses in the previous sections, it is difficult to estimate the values of these six attributes by using analytical methods or simulation models. In this work, we estimate the performance values based on data obtained from ENERGY STAR (2015) and Consumer Reports (2015). ENERGY STAR is an Environmental Protection Agency (EPA) voluntary program in the USA that protects the environment by promoting energy efficiency. ENERGY STAR provides data on the energy levels of different types of products such as appliances, food service equipment, and electronics. In particular, available data on front-loading washing machines includes the integrated modified energy factor (IMEF) and the integrated water factor (IWF), which are the levels of energy and water consumption, respectively (ENERGY STAR 2015). Washing machines must satisfy US federal standards for IMEF and IWF (i.e., IMEF ≥ 1.84, and IWF ≤ 4.7) (ENERGY STAR 2015). Meanwhile, Consumer Reports publishes reviews and performance ratings of various products based on customer surveys and test results in their laboratory. The five levels of ratings they provide are as follows: excellent, very good, good, fair, and poor (Consumer Reports 2015). In this paper, we mapped each level to the numbers 1 to 5 (i.e., excellent = 5, very good = 4, good = 3, fair = 2, and poor = 1).

Based on the data provided by ENERGY STAR and Consumer Reports, we create response surface models (RSMs) for each attribute. It is highly likely that different kinds of variables such as drum capacity, water temperature, program options, revolutions per minute (RPM) during each cycle, and cycle time may affect the performance of washing machines. In this work, the capacity of the drum and total cycle time are selected as the design (input) variables to create RSMs and estimate performance values, because ENERGY STAR and Consumer Reports provide only the data on capacity and cycle time of each washer (Consumer Reports 2015; ENERGY STAR 2015). Thus, the value of each attribute is estimated by using a surrogate model in terms of the capacity of drum, V, and total cycle time, CT. The capacity of the drum is approximated as a function of the radius of the drum, r, and the depth of the drum, d, as follows:

$$ V\approx \pi {r}^2d $$

(A50)

In Table 12, we summarize the R² values for the linear regression model of each brand’s attribute as a function of V and CT. The coefficient of determination, R² is computed as

Table 12 R² of attributes for each brand

$$ {R}^2=1-\frac{\sum {\left({y}_i-{\hat{y}}_i\right)}^2}{\sum {\left({y}_i-\overline{y}\right)}^2} $$

(A51)

where y_i is the ith original data value, ${\hat{y}}_i$ is the ith predicted value, and $\overline{y}$ is the mean of original data values. In this study, higher-order models are not considered to minimize the nonlinearity in the analysis. Some values of R² are not computed as seen in Table 12; e.g., the value of IWF for LG does not exist because all values of IWF for LG washers are the same. On the other hand, Samsung and Kenmore provide data to construct RSMs for all kinds of attributes unlike the other brands. The question posed here is the following: Which brands’ RSMs should be selected to estimate attribute values? For example, some surrogate models for Whirlpool, Electrolux, GE, and Maytag have higher R² values than the R² values of RSMs for LG, Samsung, and Kenmore. However, the RSMs for Whirlpool, Electrolux, GE, and Maytag are not considered in this study because the range of the capacity, V, used to create RSMs is relatively small; so, the RSMs may estimate inaccurate performance values (e.g., the front-loading washers for Electrolux only have two kinds of capacities: (1) 4.2 and (2) 4.3 cu. ft.). In this work, IMEF and IWF are estimated by using the RSMs for Samsung as seen in Fig. 15a and b. On the other hand, WP (Washing Performance), GE (Gentleness), and NS (Noise) are computed employing the RSMs for Kenmore in Fig. 15c, d, and e. Figure 15 shows the linear regression models and their mathematical expressions.

Fig. 15

RSMs to estimate performances

As shown in Table 12, the values of R² of the selected surrogate models (i.e., R² values highlighted in italics in Table 12) are higher than those of the other brands’ models except for Whirlpool, Electrolux, GE, and Maytag. Thus, we assume that the front-loading washers are designed based on the characteristics of the washers for Samsung and Kenmore for this initial formulation. In industry, it is highly likely that each washer company has data on numerous design variables and performance data, which is not available to the public; so, it is possible in practice to add more variables to create more accurate RSMs based on water temperature, cycle times during washing/rinsing/spinning modes, RPMs in each cycle, etc. Thus, companies can create more accurate RSMs based on their own data for existing washing machines should they wish to apply our integrated approach.

Meanwhile, Consumer Reports also provides information on brand reliability by estimating failure rates for 4-year-old washers based on a survey of washing machines purchased from 2007 to 2014 (Consumer Reports 2015). For example, the values of failure rates for LG and Samsung are 12% and 14%, respectively, which are lower than those for other brands. On the other hand, GE and Frigidaire have relatively high failure rates compared to other brands (i.e., GE 24% and Frigidaire 25%). The lower failure rate for a brand means that fewer repairs and problems have occurred. We approximate the failure rate, FR, based on the current value of reliability for a company as

$$ FR={FR}_0\left(\left(1-\sum \limits_{i=1}^{n_{\mathrm{at}}}{w}_i\right)+\sum \limits_{i=1}^{n_{\mathrm{at}}}{w}_i{Q}_i\right) $$

(A52)

$$ \mathrm{where}\kern0.6em {Q}_i=\frac{f_i-{f}_{\min, i}}{f_{\max, i}-{f}_{\min, i}}\kern0.6em \mathrm{or} $$

(A53)

$$ {Q}_i=\frac{f_{\max, i}-{f}_i}{f_{\max, i}-{f}_{\min, i}} $$

(A54)

In (A52), FR₀ is the current value of failure rate (e.g., Samsung: FR₀ = 14%); n_at is the number of attributes affecting reliability; w_i is the weight for R_i; and f_i, f_min,i, and f_max,i are the current, minimum, and maximum values of the ith attribute, respectively. In this work, we assume that the failure rate changes linearly according to the levels of attributes. The four kinds of attributes are selected to compute FR: (1) vibration, (2) noise, (3) maximum stress on the spider, and (4) load torque for the motor, which can be obtained from the other analysis disciplines. Q_i for each attribute is computed using (A53) for smaller-is-better (SIB) attributes or (A54) for larger-is-better (LIB) attributes. In (A52), the value of Q_i should be close to 0 in order to obtain a better FR value (i.e., minimize FR) compared to FR₀. On the other hand, if Q_i becomes 1, then FR is equal to FR₀.