A Systematic Approach in Shoe Last Design for Human Feet

A Systematic Approach in Shoe Last Design for Human Feet C. S. ang 1,. R. Chang 2, M. C. Lin 3 1 Department of Industrial Design, ung-hai University, aichung, aiwan 2 Department of Industrial Engineering and Management, Nan-Kai University of echnology, Nan-ou, aiwan 3 Department of Industrial Engineering and Enterprise Management, ung-hai University, aichung, aiwan E-mail: cswang@thu.edu.tw Abstract - he main purpose of this research is to build a process to find the most suitable shoe last for making shoes for human feet. A fitness function is defined to determine the optimum shoe last. Based on reverse engineering (RE) technology, this research scanned the surface of human foot and shoe last in SL (Stereo Lithography) format. e proposed the SL feature-based slicing algorithm to automatically construct the three most important girth characteristics for human feet and shoe last. Fuzzy theory was used to analyze and build the membership functions of these three important characteristics between the shoe last and human feet. he analytical hierarchy process (AHP) was applied to compare the important index and decide the weighting functions for each girth characteristic to determine the fitness function in all shoe last databases for the feet. Keywords - Reverse Engineering; Anthropometry; Shoe Last Design; SL Feature-based; Slicing Process; Fuzzy heory; Analytical Hierarchy Process I. INRODUCION he shoe last is designed according to the features and characteristics of human feet and is concerned with it s the shape and the functions of feet. he most important features of feet that the shoe last must take into account are length, width, ball girth, instep girth, and waist girth. Length and width are related to the size of the shoe last, and the three girths are related to comfort and fitness when people are wearing the shoes. he main purpose of this research is to design a process to find the most suitable shoe last in the database of shoe lasts for the human feet based on the length, width, and three main girths of shoe last. An SL (Stereo Lithography), featurebased, slicing algorithm was used to automatically construct the three most important feature-girth characteristics of human feet and shoe last. A fitness function was defined to determine the optimum shoe last for human feet. Fuzzy theory was used to analyze and build the membership functions for the three girths in the important features of shoe last and feet. he analytical hierarchy process (AHP) was applied to analyze the important indices from shoe experts to find the weighting factors for each girth to determine the fitness function between the shoe last and the feet. hree case studies were implemented to find the optimum shoe last in the database of 10 shoe lasts to prove the optimal solution in this research. II. MEHODOLOY he purpose of this research is to obtain the point clouds data of human feet and shoe lasts in the reverse engineering method by CMM and to search the coordinate data of the features of girths on the surface. Subsequently, fuzzy theory and AHP will be applied to find the optimal shoe last by analyzing the data to describe the characteristics of human feet and shoe lasts. he primary methods and the approach are described below (Fig. 1): 1) Scan human feet and shoe lasts to obtain surface and point cloud data [4, 5, 6]. 2) Compute and figure out the three important girths using a triangular, SL-slicing algorithm. 3) Determine the membership functions in shoe last girth features by fuzzy theory. 4) Find the weights for each girth by AHP analyzing process. 5) Case study and rank to find the most suitable shoe last for the human feet. Last and Human Feet Samples etting 3D appearance data by CMM Characteristics by Slicing Process Constructing Characteristics Data Last and Human Feet Outline Data Fuzzy Analyzing and Ranking through the irth Characteristics and the eights he Optimal Shoe Last for the Human Feet Pairwise Comparison Matrix of the irth by Questionnaires Find the Eigen Vector for the Matrix he eights of the irth Characteristics Fig. 1. Research Process A. Data of SL files Point cloud data for feet and shoe lasts can be obtained by CMM. Subsequently, the data can be processed to reduce noise, merged, assessed for uniform points procedures, and, finally, transformed and saved in SL format. his work can be accomplished by using reverse engineering software, specifically eomagic Studio. An SL-processed triangular format of a human foot is shown in Fig. 2. For the programming limitation, the triangular meshes of all objects are controlled within 50,000 meshes. Analytic Hierarchy Process

Before analyzing feet and shoe lasts, all SL data must be rotated and transformed to the correct position and the correct orientation, and then the correct characteristics data can be determined. As shown in Fig. 3, the Y-axis is set as the direction of the length, the X- axis is the direction of the width, and the XY plane is placed as the bottom of the foot and the shoe last. 2) Slicing Process he girths of feet and shoe lasts are obtained from the intersection of SL graphic triangular data with the characteristic plane. here is a need to determine which triangles intersect with the plane. If the intersection exists, there are five conditions in which a triangle can intersect with a plane [3], as shown in able 1. ABLE 1. AYS A RIANLE CAN INERSEC IH A PLANE (a) Intersects with two edges Fig. 2. A human foot Fig. 3. A shoe last B. Slicing for riangular SL Meshes he SL data file, which is constructed by triangles, is the de facto format for reverse engineering and rapid prototyping (RP). Each SL triangle is defined by three vertices and a normal vector. he point cloud data obtained by scanning human feet and shoe lasts were transformed and saved in an ASCII file of SL format. Slicing is the main process for reverse engineering [1]. he main task for slicing is to use the plane to intersect with the SL file to get the three important girth features needed for analysis. he section planes that intersect with the SL file and the slicing algorithm are two main components of this process. 1) Section planes of characteristics Both for feet and shoe lasts, the girth data are obtained by identifying or searching for the characteristic points on their appearance. herefore, the section planes of the girth can be established according to these points, and then the three girths are computed by the intersection of the section plane with the SL data. he section plane for the girth can be defined by three characteristic points on each girth. For the plane equation, ax + by + cz = d, a plane can be determined by abcd,,, using the coordinates of these points. For example, in the case of feet ball girths, inner and outer foot ball points must be identified as shown in Fig. 4, and these two points are defined individually as the maximum and the minimum coordinates of the X-axis or the width of the foot. So, if the coordinates of the two points are ( x 1, y 1, z 1 ) and ( x, y, z ), a plane that 2 2 2 passes through these two points and parallels the Z-axis is the ball girth section plane, and a= y y, 2 1 =, = 0 b x x 1 2 c, d x y x y 1 2 2 1 =. Fig. 4. Characteristic points of foot ball girth (b) Intersects with one vertex and one edge (c) Intersects with one edge within the slicing layer (d) Intersects with one vertex (e) Intersects with the triangle within the slicing layer As shown in able 1, in conditions (a), (b), and (c), there are two intersection points on each triangle. Because conditions (d) and (e) have the same intersection points as the triangles of conditions (b) and (c), they can be excluded as the special cases, although an intersection exists between them, as shown in Fig. 5. (i) Case (d) replaced by case (b) (ii) Case (e) replace by case (c) Fig. 5. Adjacent triangles intersect with a plane for case (d) and (e) herefore, only cases (a), (b), and (c) should be used to compute intersection points, and the points are estimated if the vertex is on the section plane and if the edge intersects with the section plane. And if there exists two intersection points between the SL triangles and the section plane, these two intersected points are the result as the girth points. he process for determining the intersection point of the section plane with the SL edge can be explained as follows. he two vertices of the SL triangular points are ( x, y, z ) and 1 1 1 ( x, y, z ). he section plane 2 2 2 ax + by + cz = d that passes through the line connecting these two points should satisfy the formula ( d ax by cz )( d ax by cz ) < 0, as 1 1 1 2 2 2 shown in Fig. 6. If an intersection exists between them, the coordinates ( x, yz, ) of the intersection point can be estimated by interpolation, and the result is x = x + t( x x ) y= y + t( y y ) (1) z = z + t( z z ), here t= ( d ax by cz )/[ a( x x) + b( y y ) + c( z z )] 1 1 2

Fig. 6. Intersection point C. Fuzzy Analysis of Shoe Lasts Different conditions, such as weather, time, and pressure can make the feet girths expand or shrink, but we do not consider these special conditions. Normally, there is a tolerance between the girth of the shoe last and the girth of the foot. Here, we can define the tolerance as the sense threshold ( S ), in length, for the difference in girth between the foot ( F ) and the shoe last ( L ). S = F L (2) For example, someone s foot length ( F ) is 260mm, L width ( F ) is 105mm and the ball girth ( F ) is 251 mm, so shoe lasts are produced that have various girths along with appropriate length and width according to the foot s length and width. After wear trials, the shoe with a girth of 245mm ( L ) among these lasts feels the most comfortable, so the sense threshold ( S ) can be defined as 6mm. In further trials, the lasts with the range of girth from 242 mm ( L ) to 249 mm ( L ) would be the R best fit. he former one (242 to 245 mm ) feel tight, and the latter one (245 to 249 mm ) feel loose. If it exceeds this range, it would be too tight or too loose and would feel uncomfortable. herefore, the range that should be chosen is LR L (249 242 = 7mm ), and 7 mm is the Fit Sense hreshold ( S ), where F SF = LR L. (3) By the definition and considering the concept of sense threshold mentioned above, there exists a fuzzy relationship in design for the feet girth and the shoe last girth. herefore, the evaluation and estimation mode that searches and ranks are proposed initially based on fuzzy theory, and the optimal shoe last can be determined from the database of the shoe lasts according to this evaluation mode. Fuzzy sets theory was first proposed by Zadeh [8]. In this research, a proposed membership function μ ( x) in a triangular shape for the foot girth and the shoe last girth is shown in Fig. 7. L is the optimal shoe last girth designed from the foot girth and the sense threshold, and the difference between L and L is the fit sense R threshold ( S ). herefore, if the shoe last girth is smaller F than L, it would cause discomfort by being too tight, and if the shoe last girth is larger than L R, it would cause discomfort by being too loose. If the shoe last girth is between L and L R, a membership grade of [0, 1] would be estimated from the membership function μ ( x). 0, x< L.. (4) ( x L ) /( L L ), L x L... μ A μ( x) = ( L x) /( L L ), L x L... μ R R R B 0, x> LR Fig. 7. Membership Function of irth For the membership grade estimated at this stage, the number 1 means that the shoe last girth fits the foot the best, and the number 0 means it doesn t fit the foot. As for the membership grade between 0 and 1, the larger means the level at which the shoe last girth fits the foot girth. he more items in membership of the shoe last girths is not 0, the higher in ranking that the shoe last fits the feet. So, the initial ranking can be created after analyzing the membership grades of these three girths, and they can be described as: 1) all membership grades of the girth items are not 0; 2) two membership grades of the girth items are not 0; 3) only one membership grade of the girth items is not 0; and 4) all membership grades of the three girths are 0. D. Analytical Hierarchy Process Although the initial ranking can be established through analyzing the fuzzy membership grade of each girth, several shoe lasts could simultaneously occur that cannot be distinguished from each other for the membership grades with a value of 0 for the girth. herefore, for the subsequent work, each girth must be assigned a weight related to its relative importance. Each shoe last would be evaluated by the fitness function R, so the optimal shoe last can be determined through further, accurate ranking according to R. As to the weights of these three girths, the analytical hierarchy process (AHP) would be used to figure them out. R = wm B B + wm I I + wm,..(5) M is the Membership rade of Ball irth; w B B is the eight of Ball irth; M is Membership rade of Instep I irth; w I is eight of Instep irth, M is Membership rade of aist irth; and w is eight of aist irth. AHP has become preferred by decision-makers as a reliable tool since it ranks the evaluation factors according to their relative importance, then assesses the decision points for every factor, and, finally, uses a mathematical method to find the percentage distribution of decision points in terms of the weights that affect the decision [7]. hrough AHP, by gathering the comments and the estimation of shoe experts or decision makers, a pair-wise comparison matrix of factors is created according to a

nominal scale, and then the priority vector and maximum eigen value would be determined. Finally, the result would be estimated if there is consistency. If not, it would not be accepted. he eigen method is used to solve AHP, and the eigen-vector w obtained from the pair-wise comparison matrix is the weights related to all factors. After determining the eigen-vector, the result must be examined. Reference [2] proposed the Consistency Index ( CI ) to estimate this value: CI = ( λmax n)/( n 1).(6) here λ is the largest or principal eigen-value of the max comparison matrix, and n is the number of factors or the order of the comparison matrix. In general, w= λmaxw and if CI 0.1, the matrix is consistent and the eigen-vector w obtained from the matrix can represent the weights related to all factors. III. RESULS he user interface that implements the method mentioned above and obtains the measurements of the characteristic features is programmed by MALAB, as shown in Fig. 8. he girths can be sent to eomagic Studio or AutoCAD formats for further analysis and visualization, as shown in Fig. 9 and Fig. 10. and right foot are chosen, but deformed, pathological, or flat feet that are abnormal would not be appropriate for the study and should be excluded. For the shoe lasts, the style is a size 9 men s leather shoe. ABLE 2. SAMPLES FOR 10 SHOE LASS 1 (ab0017) 2 (ab0155) 3 (ab9998) 4 (ab8562) 5 (ab9647) 6 (tw698) 7 (tw736) 8 (tw9615) Fig. 8. Operating User Interface programmed by Matlab 9 (tw1127) Fig. 9. Output the girth to eomagic for further analysis 10 (tw1247) Fig. 10. Output the girth to Auto CAD here are three examinee s feet and 10 shoe lasts (able 2) for the verification of the research. For the adult male examinee, the shoe size No. 9 in the U.S. standard, normal he first step for AHP is to determine the importance of the factors in three girths. By sending questionnaires to the experts and to manufacturers in the shoe industry, we obtained the related criteria for these girths. All 10 questionnaires were received, and the pair-wise comparison matrix from the questionnaires was estimated by the Eigen method to get the weights. Due to the fact that there were only six questionnaires that passed consistency index ( CI ) test, the weights that were estimated from these six questionnaires were used. hen, the geometric mean method was used to obtain the synthesized weights from these six consistency

questionnaires, and the weights related to these three girths were determined to be: ball weight w = 0.9772; waist weight w w = 0.1848; and instep weight B w I = 0.0816. As a result of the analysis, the important sequence of these three girths is Ball > aist > Instep, and the evaluation value R can be estimated from the weights using AHP analysis and the membership grades from fuzzy sets. hen, a further, more accurate ranking can be determined according to R. Fig. 11(a). Shoe last No. ab9998 Fig. 12(a). Shoe last tw736 Fig. 11(b). Shoe last No. ab9998 Fig. 12(b). Shoe last tw736 IV. DISCUSSION Although the ranking of the optimal shoe last can be obtained by the estimation obtained from the proposed method using fuzzy theory and AHP, the shoe last that fits each examinee the best can be determined according by the ranking. Once the database that has considerable data related to shoe last design is set up with the increase of the lasts, there will be a need for faster ways to obtain a more accurate ranking among all shoe lasts and to determine the optimal shoe last for each person. 1) Case 1: First Examinee here are no shoe lasts for which all the membership grades of these three girths are not 0 among these 10 lasts, so there was no optimal shoe last for first examinee among these 10 lasts. However, for Nos. ab9998, ab8652, and tw9615, these three shoe lasts all have only two girth items for which the membership grades are not 0. herefore, the optimal shoe last among these three lasts can be estimated through the accurate ranking of the formula, R= wbmb+ wimi + wm, and the result is shown below: Rab9998 = 0.9772 0.671+ 0.0816 0.885 + 0.1848 0 = 0.7279 Rab8652 = 0.9772 0.648 + 0.0816 0 + 0.1848 0.315 = 0.6914 R = 0.9772 0 + 0.0816 0.523 + 0.1848 0.307 = 0.0994 tw9615 R > R > R, which means As a result, ab9998 ab8652 tw9615 the No. ab9998 shoe last would be the optimal last for the first examinee among these 10 lasts, and it is shown in Fig. 11(a) for SL and in Fig. 11(b) for the sample. 2) Case 2 - Second Examinee he No. ab9998 last has all the membership grades that are not 0 for these three girth items. herefore, it would be the optimal last for the second examinee among these 10 lasts, and it is shown in Fig. 11(a) for SL and in Fig. 11(b) for the sample. 3) Case 3 - hird Examinee Last No. tw736 is the only one for which all the membership grades of these three girth items are not 0. herefore, it would be the optimal last for the third examinee among these 10 lasts, and it is shown in Fig. 12(a) and Fig. 12(b). V. CONCLUSION he concrete efforts in this paper that have been completed are listed below: 1. A programmed graphic user interface, which automatically estimates and obtains the data characteristics of the shoe lasts and human feet, has been created, allowing human errors to be decreased. 2. A concept of the sense threshold applying to shoe last and feet from fuzzy theory and the weights of girths by AHP is proposed. A fitness function in shoe last and feet is proposed, and the optimal shoe last for individual feet can be found. All the shoe lasts for one examinee can be ranked by using the fitness function. 3. All the characteristics and the graphic data of the shoe lasts and human feet estimated within this research can be stored to set up a database that can be used in subsequent work as a reference for the modification or design of shoe lasts. REFERENCES [1] Choi, S.H., and Kwok, K.., 2002, Hierarchical slice contours for layered-manufacturing, Computers in Industry, Vol. 48, pp. 219-239. [2] Douligeris, C., and Pereira, I., 1992, An analytical hierarchy process approach to the analysis of quality in telecommunication systems, IEEE Proceedings on SMC, pp. 1684-1688. [3] ibson, I., 2002, Software solutions for rapid prototyping, Ch 5, John iley. [4] Kouchi, M., 1995, Analysis of foot shape variation based on the medial axis of foot outline, Ergonomics, Vol.38, No.9, pp.1911-1920. [5] Kouchi, M., and sutsumi, E., 1996, Relation between the medial axis of the foot outline and 3-D foot shape, Ergonomics, Vol. 39, No. 6, pp.853-861. [6] Kouchi, M., and Mochimaru, M., 2001, Development of low cost foot-scanner for a custom shoe making system, Proceeding of 5 th Symp. of Footwear Biomechanics, pp. 58-59. [7] Saaty,.L., 1980, he Analytic Hierarchy Process, Mcraw-Hill, New York. [8] Zadeh, L.A., 1965, Fuzzy sets, Information and Control, Vol. 8, pp. 338-353.