Comparing Built-in Power Banks for a Smart Backpack Design Using an Auto-Weighting Fuzzy-Weighted-Intersection FAHP Approach

: Smart backpacks are a prevalent application of smart technologies, with functions such as motion recording, navigation, and energy harvesting and provision. Selecting a suitable built-in power bank is a critical task for a smart backpack design, which has rarely been investigated in the past. To fulﬁll this task, an auto-weighting fuzzy-weighted-intersection fuzzy analytic hierarchy process (FAHP) approach is proposed in this study. When decision makers lack an overall consensus, the auto-weighting fuzzy-weighted-intersection FAHP approach speciﬁes decision makers’ authority levels according to the consistency ratios of their judgments. In this way, the consensus among all decision makers can be sought. The auto-weighting fuzzy-weighted-intersection FAHP approach has been applied to compare six mobile power banks for a smart backpack design


Introduction
Some research results have shown that smart technologies can improve the quality of life of users [1][2][3][4].Among the existing smart technology applications, smart backpacks have considerable potential, because the existing market for backpacks is huge [5].However, the development of smart backpacks is not yet mature.In addition, the existing smart backpack designs, such as the applied smart technology, embedding location, and cost effectiveness, have rarely been optimized [6].In order to fill this gap, the problem of selecting the most suitable built-in power bank for a smart backpack design is investigated in this study, in which the suitability of the built-in power bank is optimized.To this end, an auto-weighting fuzzy-weighted-intersection fuzzy analytic hierarchy process (FAHP) approach is proposed in this study.The motives are explained as follows.
Selecting the most suitable built-in power bank for a smart backpack design is obviously a multiple-criteria decision-making problem [7] to which FAHP is a widely applied method [8,9].The incorporation of fuzzy logic is to consider the uncertainty inherent in the selection process [10,11], since the development of smart backpacks is still in its infancy.In addition, multiple decision makers are involved in the selection process to avoid personal bias and consider various viewpoints [12][13][14][15].As a result, the proposed methodology is a group-based FAHP method [16,17].
In the existing group-based FAHP methods, the consensus among decision makers is assumed to exist, and can be derived by calculating the fuzzy arithmetic average [18] or fuzzy geometric mean (FGM) [19] of decision makers' pairwise comparison results [20].However, the application of fuzzy Sankhe and Rodrigues [35] designed a smart backpack with three smart functions: the detection of obstacles in front (using ultrasonic sensors), automatic travelling (by driving the wheels so that the backpack moved forward by itself), and LED light warning.

FAHP Applications to Smart Backpacks
Subsequently, some applications of FAHP to smart backpacks are reviewed.In the view of Wu et al. [36], the five key factors affecting a smart backpack design were fashionable design, cheap price, many smart functions, high practicality, and light weight.Wu et al. proposed an FAHP method to compare the relative priorities of these critical factors.
Lin and Chen [7] proposed a multibelief analytic hierarchy process and nonlinear programming approach that was able to decompose a less consistent judgment matrix into several more consistent sub-judgment matrixes.The relative priority sets generated from these sub-judgment matrixes were very different from each other.Based on them, diversified smart backpack designs could be made [37,38].
In this study, the FWI operator proposed by Chen et al. [22] is incorporated into FAHP to select the most suitable built-in power bank for a smart backpack design.The FWI operator was originally designed for aggregating decision makers' judgments when they had unequal authority levels.However, an interesting property of FWI is that the aggregation result may not be an empty set even if decision makers lack an overall consensus, as illustrated in Figure 1.Therefore, by applying FWI instead of PCFI, it is no longer necessary to reduce the number of decision makers that reach a consensus.However, a prerequisite for applying FWI is that decision makers have unequal authority levels.If decision makers do not discriminate their authority levels, then a reasonable treatment is to assign the authority levels (or weights) of decision makers automatically based on the consistency ratios (CRs) of their pairwise comparison results.From this point of view, an auto-weighting fuzzy-weighted-intersection FAHP approach is proposed in this study.The simultaneous application of multiple positioning functions was to improve the accuracy of positioning.Sankhe and Rodrigues [35] designed a smart backpack with three smart functions: the detection of obstacles in front (using ultrasonic sensors), automatic travelling (by driving the wheels so that the backpack moved forward by itself), and LED light warning.

FAHP Applications to Smart Backpacks
Subsequently, some applications of FAHP to smart backpacks are reviewed.In the view of Wu et al. [36], the five key factors affecting a smart backpack design were fashionable design, cheap price, many smart functions, high practicality, and light weight.Wu et al. proposed an FAHP method to compare the relative priorities of these critical factors.
Lin and Chen [7] proposed a multibelief analytic hierarchy process and nonlinear programming approach that was able to decompose a less consistent judgment matrix into several more consistent sub-judgment matrixes.The relative priority sets generated from these sub-judgment matrixes were very different from each other.Based on them, diversified smart backpack designs could be made [37,38].
In this study, the FWI operator proposed by Chen et al. [22] is incorporated into FAHP to select the most suitable built-in power bank for a smart backpack design.The FWI operator was originally designed for aggregating decision makers' judgments when they had unequal authority levels.However, an interesting property of FWI is that the aggregation result may not be an empty set even if decision makers lack an overall consensus, as illustrated in Figure 1.Therefore, by applying FWI instead of PCFI, it is no longer necessary to reduce the number of decision makers that reach a consensus.However, a prerequisite for applying FWI is that decision makers have unequal authority levels.If decision makers do not discriminate their authority levels, then a reasonable treatment is to assign the authority levels (or weights) of decision makers automatically based on the consistency ratios (CRs) of their pairwise comparison results.From this point of view, an auto-weighting fuzzyweighted-intersection FAHP approach is proposed in this study.

Implementation Procedure
The auto-weighting fuzzy-weighted-intersection FAHP approach is proposed in this study for comparing built-in power banks for a smart backpack design.The operational procedure of the autoweighting fuzzy-weighted-intersection FAHP approach comprises the following steps:

Implementation Procedure
The auto-weighting fuzzy-weighted-intersection FAHP approach is proposed in this study for comparing built-in power banks for a smart backpack design.The operational procedure of the auto-weighting fuzzy-weighted-intersection FAHP approach comprises the following steps: Step 1.Each decision maker applies the FGM method [1,19,39,40] to evaluate the relative priorities of factors critical to a built-in power bank for a smart backpack design.Step 2. Evaluate the CR of the judgment matrix by each decision maker.
Step 3. Apply FI [23] to aggregate the relative priorities evaluated by decision makers.
Step 4. If all decision makers reach an overall consensus, go to Step 7; otherwise, go to Step 5.
Step 5. Calculate the authority level (or weight) of each decision maker.
Step 7. Applying the fuzzy technique for order preference by similarity to ideal solution (FTOPSIS) approach [18,[40][41][42] to assess the overall performance of a built-in power bank for a smart backpack design.
Step 8. Applying the center-of-gravity (COG) method [36,43,44] to defuzzify the assessment result, so as to generate an absolute ranking of built-in power banks for a smart backpack design.
A flowchart is provided in Figure 2 to illustrate the operational procedure.
Mathematics 2020, 8, x FOR PEER REVIEW 4 of 23 Step 1.Each decision maker applies the FGM method [1,19,39,40] to evaluate the relative priorities of factors critical to a built-in power bank for a smart backpack design.Step 2. Evaluate the CR of the judgment matrix by each decision maker.
Step 3. Apply FI [23] to aggregate the relative priorities evaluated by decision makers.
Step 4. If all decision makers reach an overall consensus, go to Step 7; otherwise, go to Step 5.
Step 5. Calculate the authority level (or weight) of each decision maker.
Step 7. Applying the fuzzy technique for order preference by similarity to ideal solution (FTOPSIS) approach [18,[40][41][42] to assess the overall performance of a built-in power bank for a smart backpack design.
Step 8. Applying the center-of-gravity (COG) method [36,43,44] to defuzzify the assessment result, so as to generate an absolute ranking of built-in power banks for a smart backpack design.
A flowchart is provided in Figure 2 to illustrate the operational procedure.

Evaluating the Relative Priorities of Critical Factors
In the proposed methodology, at first, each decision maker evaluates the relative priorities of critical factors in pairs using the FGM method.The comparison results are expressed in linguistic terms such as "as equal as", "weakly more important than", "strongly more important than", "very

Evaluating the Relative Priorities of Critical Factors
In the proposed methodology, at first, each decision maker evaluates the relative priorities of critical factors in pairs using the FGM method.The comparison results are expressed in linguistic terms such as "as equal as", "weakly more important than", "strongly more important than", "very strongly more important than", "absolutely more important than", etc.These linguistic terms are usually mapped to triangular fuzzy numbers (TFNs) within [1,9,19].Chen [45] widened these TFNs to increase the possibility for decision makers to reach a consensus.To the contrary, Samanlioglu and Kaya [46] narrowed these TFNs to elevate the consistency of pairwise comparison results.

Based on pairwise comparison results, a fuzzy judgment matrix
The fuzzy eigenvalue and eigenvector of Ã, indicated with λ and x respectively, satisfy and where (−) and (×) denote fuzzy subtraction and multiplication, respectively.The FGM method [19,39,40] can be applied to estimate the relative priority of each critical factor ( w i ) as According to the arithmetic for TFNs, Equation ( 4) is equivalent to [47] In addition, the fuzzy maximal eigenvalue λ max can be estimated as which is decomposed into [48] The consistency of pairwise comparison results can be evaluated in terms of CR [49]: where RI is the random consistency index [49].CR should be less than 0.1 for a small FAHP problem, or less than 0.3 if the problem size is large or the problem is highly uncertain [50,51].

Auto-Weighting FWI for Aggregating the Relative Priorities
If the (overall) consensus among decision makers exists, FI can be applied to aggregate the relative priorities evaluated by them as follows [23].
Definition 1.The fuzzy intersection (FI) of the relative priorities evaluated by M decision makers for the i-th critical factor, indicated with w i (1)~ w i (M), is denoted by FI( w i (1), . . ., w i (M)) such that Otherwise, PCFI can be applied to aggregate the relative priorities evaluated by most decision makers [24,25] as follows.

Definition 2.
The H/ M partial consensus fuzzy intersection (PCFI) of the relative priorities evaluated by M decision makers for the i-th critical factor, indicated with w i (1)~ w i (M), is denoted by where g() The number of decision makers that reach a consensus is reduced by applying PCFI instead of FI, which is not a favorable property to decision makers.In contrast, FWI [22], as defined below, may find out a consensus without reducing the number of decision makers in this situation.

Definition 3. The fuzzy weighted intersection (FWI) of the relative priorities evaluated by M decision makers for the i-th critical factor, indicated with w
where ω m is the authority level of decision maker m; However, a prerequisite for applying FWI is that decision makers have unequal authority levels.If decision makers do not discriminate their authority levels, then the proposed methodology assigns different authority levels to decision makers automatically based on the CRs of their pairwise comparison results.Let the CR of the pairwise comparison results by expert m be denoted by CR(m).Obviously, Since 0.1 is a threshold for CR, which is inversely proportional to CR(m).A smaller value of CR(m) means higher consistency.Therefore, a reasonable choice of ω m is 0.1 (19) which satisfies the following properties: (1) 0 ≤ ω m ≤ 1; ( (3) ω m ∝ 1/ CR(m), i.e., the lower consistency ratio the higher the authority level.
To simplify the subsequent calculation, only the core of CR(m) is considered:

Assessing the Suitability of a Built-in Power Bank for a Smart Backpack Design
Subsequently, the prevalent FTOPSIS method [40][41][42] is applied to assess the suitability of a built-in power bank for a smart backpack design.First, the performance of a built-in power bank for a smart backpack design in optimizing each critical factor is normalized using the fuzzy distributive normalization [50]: where p qi is the performance of the q-th built-in power bank in optimizing the i-th critical factor; ρ qi is the normalized performance.Subsequently, the fuzzy weighted score is calculated based on the relative priorities derived using the auto-weighting FWI: However, FWI( w i (m) is a polygonal fuzzy number, while p qi is a TFN.Their combination is not easy to calculate.To tackle such complexity, FWI( w i (m) is approximated with a TFN as: as illustrated in Figure 3.In this way, the defuzzified value of the approximating TFN is equal to COG( FWI( w i (m) ) that is calculated as Mathematics 2020, 8, x FOR PEER REVIEW 8 of 23 where qi p  is the performance of the q-th built-in power bank in optimizing the i-th critical factor; qi ρ  is the normalized performance.Subsequently, the fuzzy weighted score is calculated based on the relative priorities derived using the auto-weighting FWI: However,  ({ ( )} i FWI w m  is a polygonal fuzzy number, while qi p  is a TFN.Their combination is not easy to calculate.To tackle such complexity,  ({ ( )} i FWI w m  is approximated with a TFN as: as illustrated in Figure 3.In this way, the defuzzified value of the approximating TFN is equal to Subsequently, the fuzzy ideal (zenith) point and the fuzzy anti-ideal (nadir) point are specified, respectively, as: The fuzzy distance from each built-in power bank to the two points are calculated, respectively, as: Subsequently, the fuzzy ideal (zenith) point and the fuzzy anti-ideal (nadir) point are specified, respectively, as: The fuzzy distance from each built-in power bank to the two points are calculated, respectively, as: Finally, the fuzzy closeness of each power bank is obtained as: A built-in power bank is more suitable if its fuzzy closeness is higher.To get an absolute ranking, the fuzzy closeness can be defuzzified using COG.

Application of the Proposed Methodology
A backpack company in Taipei City, Taiwan ran a project to investigate the potential opportunities of designing and manufacturing smart backpacks for the domestic market.To this end, the project team started from the selection of suitable built-in power banks for a smart backpack design.This topic is important because built-in power backs are the most common function of smart backpacks [29].The project team was composed of three members: an industrial engineering professor, a patent analyst, and a smart technology researcher.
In this era, mobile power banks are very prevalent.A user just needs to put a mobile power bank into his/her backpack before going out.Whether it is still necessary to build a power bank in a backpack is questionable.In our view, a built-in power bank is helpful for the following situations: (1) Marketing needs: if a smart backpack does not have a built-in power bank, it is no different from a normal backpack, because users only need to bring their own mobile power banks.(2) Protection: in normal backpacks, there is no dedicated space for placing a power bank, which is insufficient for the protection of the mobile power bank.(3) Convenience: most backpacks are not designed with a power interface, which will cause inconvenience when the user wants to charge.In addition, a smart backpack with a built-in power bank avoids the trouble of forgetting to carry a mobile power bank.
After reviewing the related literature and current practice, the following factors were considered critical to the selection of a built-in power bank for a smart backpack:

•
Weight: the lighter the better.According to the experimental results of Heuscher et al.
[52], for every 4 kg increase in the weight of a backpack, the user's chance of lower back pain will increase by 25%.If the weight of the backpack exceeds 10% of the user's weight, it may also cause long-term lower back pain.

•
Battery capacity (mAh): the larger the better; • Price (cost): the cheaper the better; • Size: the smaller the better; • Brand awareness: the higher the better.
These critical factors are not easy to optimize simultaneously and need to be compensated.To evaluate the relative priorities of these critical factors, each decision maker (i.e., project team member) utilized linguistic terms to express his/her belief about the relative priority of a critical factor over another.Based on these beliefs, three fuzzy pairwise comparison matrixes were constructed, as shown in Table 1.
Each decision maker applied the FGM method to derive the fuzzy maximal eigenvalue and relative priorities from the corresponding fuzzy pairwise comparison matrix.As a result, the derived fuzzy maximal eigenvalues were showing certain levels of consistency.In addition, the relative priorities evaluated by the decision makers are summarized in Figure 4.
Each decision maker applied the FGM method to derive the fuzzy maximal eigenvalue and relative priorities from the corresponding fuzzy pairwise comparison matrix.As a result, the derived fuzzy maximal eigenvalues were  The overall consensus reached by all the decision makers, represented by the FI results of the relative priorities derived by them, are summarized in Figure 5. Obviously, all the decision makers reached an overall consensus regarding the values of 1 w  and 2 w  .However, an overall consensus regarding the values of other relative priorities was lacking, because the FI results were empty sets.As a result, existing fuzzy group decision-making methods assuming the existence of an overall consensus, such as Lin et al. [1], Chen [15], Chen and Lin [23], Samanlioglu and Kaya [46], and Gao et al. [53] were logically not applicable.To solve this problem, the auto-weighting FWI operator was applied to find out weighted consensus among all decision makers instead.First, the authority levels (or weights) of the decision makers were determined according to Equation ( 19) as 0.402, 0.295, and 0.302, respectively.The overall consensus reached by all the decision makers, represented by the FI results of the relative priorities derived by them, are summarized in Figure 5. Obviously, all the decision makers reached an overall consensus regarding the values of w 1 and w 2 .However, an overall consensus regarding the values of other relative priorities was lacking, because the FI results were empty sets.As a result, existing fuzzy group decision-making methods assuming the existence of an overall consensus, such as Lin et al. [1], Chen [15], Chen and Lin [23], Samanlioglu and Kaya [46], and Gao et al. [53] were logically not applicable.To solve this problem, the auto-weighting FWI operator was applied to find out weighted consensus among all decision makers instead.First, the authority levels (or weights) of the decision makers were determined according to Equation ( 19) as 0.402, 0.295, and 0.302, respectively.
Based on the assigned authority levels, FWI was applied to aggregate the relative priorities evaluated by the decision makers.The results are summarized in Figure 6.
To facilitate the subsequent calculation, the FWI results were approximated with TFNs according to Equation (26).The approximation results are shown in Figure 7. Obviously, "price" was the most critical factor, followed by "weight".Based on the assigned authority levels, FWI was applied to aggregate the relative priorities evaluated by the decision makers.The results are summarized in Figure 6.Based on the assigned authority levels, FWI was applied to aggregate the relative priorities evaluated by the decision makers.The results are summarized in Figure 6.To facilitate the subsequent calculation, the FWI results were approximated with TFNs according to Equation (26).The approximation results are shown in Figure 7. Obviously, "price" was the most critical factor, followed by "weight".Among the five critical factors, "battery capacity" and "brand awareness" were the-higher-thebetter performance, whereas the others were the-lower-the-better performances.The performances in optimizing these critical factors were evaluated according to the rules depicted in Table 2.The formulation of these rules referred to [36,37].To facilitate the subsequent calculation, the FWI results were approximated with TFNs according to Equation (26).The approximation results are shown in Figure 7. Obviously, "price" was the most critical factor, followed by "weight".Among the five critical factors, "battery capacity" and "brand awareness" were the-higher-thebetter performance, whereas the others were the-lower-the-better performances.The performances in optimizing these critical factors were evaluated according to the rules depicted in Table 2.The formulation of these rules referred to [36,37].Among the five critical factors, "battery capacity" and "brand awareness" were the-higher-the-better performance, whereas the others were the-lower-the-better performances.The performances in optimizing these critical factors were evaluated according to the rules depicted in Table 2.The formulation of these rules referred to [36,37].
Based on the derived relative priorities, six existing mobile power banks that met the following requirements were compared:

Critical Factor Rule
Weight where x q is weight.
Battery capacity where x q is battery capacity.
Price (cost) where x q is price.

Size
where x q is size in terms of thickness.
Brand awareness where x q is the number of products of the brand.
(4) The ranking results using the two methods are compared in Figure 8.There were considerable differences between the ranking results using the two methods.One possible reason for this was that the FGM-FGM-FWA approach assigned a heavier weight to battery capacity about which the overall consensus among decision makers was insufficient.
Asus ZenPower 10,000 (4) The ranking results using the two methods are compared in Figure 8.There were considerable differences between the ranking results using the two methods.One possible reason for this was that the FGM-FGM-FWA approach assigned a heavier weight to battery capacity about which the overall consensus among decision makers was insufficient.(5) In this experiment, decision makers lacked an overall consensus.The FGM-FGM-FWA method could not deal with this problem, but it directly aggregated decision makers' judgments.The result obtained in this way was unconvincing.In contrast, the proposed methodology reasonably adjusted the weights of decision makers to generate an overall consensus.The weight of a decision maker was proportional to the consistency of his/her judgment.In this way, the selection result would be more convincing.This is the advantage of the proposed methodology over the FGM-FGM-FWA method.

Conclusions
Smart technology applications are penetrating into our daily lives.Smart backpacks with builtin power backs are one of the most prevalent smart technology applications.However, selecting a suitable mobile power back for a smart backpack design is a challenging task, because the designer has to make a trade-off among several critical factors.To address this challenge, an auto-weighting fuzzy-weighted-intersection FAHP approach is proposed in this study.The auto-weighting fuzzyweighted-intersection FAHP approach aggregates multiple decision makers' judgments in a reasonable manner when an overall consensus among these decision makers is lacking.Compared to the existing methods based on PCFI that seek the partial consensus of only some experts, the autoweighting fuzzy-weighted-intersection FAHP approach applies the FWI operator by specifying decision makers' authority levels according to the consistency ratios of their judgments, so that the number of decision makers that reach a consensus is not reduced.In this way, the group decisionmaking result will be more acceptable to all decision makers.
The proposed methodology has been applied to compare six existing mobile power banks for a smart backpack design to illustrate its applicability.After analyzing the experimental results, the following conclusions were drawn: (5) In this experiment, decision makers lacked an overall consensus.The FGM-FGM-FWA method could not deal with this problem, but it directly aggregated decision makers' judgments.The result obtained in this way was unconvincing.In contrast, the proposed methodology reasonably adjusted the weights of decision makers to generate an overall consensus.The weight of a decision maker was proportional to the consistency of his/her judgment.In this way, the selection result would be more convincing.This is the advantage of the proposed methodology over the FGM-FGM-FWA method.

Conclusions
Smart technology applications are penetrating into our daily lives.Smart backpacks with built-in power backs are one of the most prevalent smart technology applications.However, selecting a suitable mobile power back for a smart backpack design is a challenging task, because the designer has to make a trade-off among several critical factors.To address this challenge, an auto-weighting fuzzy-weighted-intersection FAHP approach is proposed in this study.The auto-weighting fuzzy-weighted-intersection FAHP approach aggregates multiple decision makers' judgments in a reasonable manner when an overall consensus among these decision makers is lacking.Compared to the existing methods based on PCFI that seek the partial consensus of only some experts, the auto-weighting fuzzy-weighted-intersection FAHP approach applies the FWI operator by specifying decision makers' authority levels according to the consistency ratios of their judgments, so that the number of decision makers that reach a consensus is not reduced.In this way, the group decision-making result will be more acceptable to all decision makers.
The proposed methodology has been applied to compare six existing mobile power banks for a smart backpack design to illustrate its applicability.After analyzing the experimental results, the following conclusions were drawn: (1) Among the six compared mobile power banks for a smart backpack design, iNeno M12, a mobile power back with high battery capacity and low weight, was evaluated as the most suitable built-in power bank.In contrast, tsoe SPB-S10 was considered the least suitable owing to the low awareness of the brand.(2) The ranking result using the proposed methodology was slightly different from that using an existing method.The reason was that whether an overall consensus existed among decision makers was not emphasized in the existing method.

Figure 1 .
Figure 1.The fuzzy weighted intersection (FWI) result when decision makers lack an overall consensus.

Figure 1 .
Figure 1.The fuzzy weighted intersection (FWI) result when decision makers lack an overall consensus.

Figure 4 .
Figure 4.The relative priorities evaluated by the decision makers.

Figure 4 .
Figure 4.The relative priorities evaluated by the decision makers.

Figure 5 .
Figure 5.The fuzzy intersection(FI) results of the relative priorities.

Figure 6 .
Figure 6.FWI results of the relative priorities based on the assigned authority levels.

Figure 6 .
Figure 6.FWI results of the relative priorities based on the assigned authority levels.

Figure 6 .
Figure 6.FWI results of the relative priorities based on the assigned authority levels.

Figure 8 .
Figure 8.Comparison of the ranking results.

Figure 8 .
Figure 8.Comparison of the ranking results.

Table 1 .
Fuzzy pairwise comparison matrixes constructed by three decision makers.

Table 1 .
Fuzzy pairwise comparison matrixes constructed by three decision makers.

Table 2 .
Rules for evaluating the performances in optimizing critical factors.