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An optimisation approach to apparel sizing

OR seminar. An optimisation approach to apparel sizing. 937812 鍾孟容. Agenda. Introduction Background Main idea Result Conclusion. Introduction. 隨著生活水準的提升 ， 人們對於服裝的舒適度和合身度的要求越來越高 ， 雖然客制化的技術已趨而成熟，但其製造價格不菲且訂做時間長 ， 使得成衣仍為目前消費者選購服飾的主流。

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An optimisation approach to apparel sizing

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  1. OR seminar An optimisation approach to apparel sizing 937812 鍾孟容

  2. Agenda • Introduction • Background • Main idea • Result • Conclusion

  3. Introduction • 隨著生活水準的提升，人們對於服裝的舒適度和合身度的要求越來越高，雖然客制化的技術已趨而成熟，但其製造價格不菲且訂做時間長，使得成衣仍為目前消費者選購服飾的主流。 • Made4Me Corporation (2002)宣稱 60%的男性和女性找不到適合自己身體尺寸的服裝,且有50%的製造商在設計衣服時只依循公司內部的標準而非依據人體計測值。

  4. Introduction • 人類的體型會因不同的種族、 生活環境、年齡、性別和年代而相異，各個國家皆必須制定屬於自己的尺碼 • 由國家或紡織機構所訂定的尺碼標準涵蓋範圍廣大，尺碼個數往往過於繁多，造成廠商無所適從： • 要選擇哪一個尺碼﹖ • 要製作幾個尺碼﹖

  5. Topic Tryfos, P., 1986 , “An interger Programming Approach to the Apparel Sizing Problem”,The Journal of the Operational Research Society, Vol. 37, No. 10., pp. 1001-1006. 運用整數規劃法來決定尺碼個數以及選擇適當的尺碼 依據顧客需求發展出最佳的尺碼系統 McCulloch, C. E., Paal, B., Ashdown, S. P., 1998, “An Optimal Approach to Apparel Sizing”, Journal of the Operational Research Society, 49 (5), pp. 492-499.

  6. Background (Ashdown et al., 2002) • 何謂尺碼(Size)？ 用來標示服裝尺寸的大小分別，例如：在 市面上常用S、M、L等記號 • 控制維度(Control Dimensions)： 在製作服裝中所必須量測的重要的身體部 位，例如：上衣為胸圍和領圍。 • 允差(Tolerance)： 尺碼的適用範圍，假設允差=±1當尺碼為84 公分時，其適用範圍則為83-85公分。 • 合身性(Fit) ： 指身體和衣服在控制維度的距離越小越合 身。

  7. Main idea • Assumption • 服裝購買的機率與合身性有關 • 身體與服裝的尺寸差距越小購買率越高 Tryfos, P., 1986 , “An interger Programming Approach to the Apparel Sizing Problem”,The Journal of the Operational Research Society, Vol. 37, No. 10., pp. 1001-1006. • 尺碼太大、太小問題 • 假設某一鞋款尺碼長=11吋、寬=4吋 • 某甲的腳長=11.5吋、寬=4.5吋 • 某乙的腳長=10.5吋、寬=3.5吋 購買機率=0

  8. Formulation • and are the two control dimensions : a person with measurements : a size : the probability of this person buying c : a constant equal to the probability when

  9. Main idea • Assumption • 消費者會選擇 最大的尺碼 • 購買人數期望值： • N: the size of the target population • ri : the proportion of members of the target population in cell i

  10. 製造商的目標： 在於給定K個尺碼中，求其最大的銷售量 P-median problem

  11. P-median problem • P-median常用在解決都市公共設施區位問題，可同時決定設施最適數目、區位、規模和服務範圍等 • 將使用者視為均質狀態，其設施區位的選擇取決於該區人口數總需求量 • 其主要目的在於求解該地區使用者至設施區位點的最小總旅行距離。

  12. Formulation 目標式： 限制式： 每一個cell皆會被指派至一個尺碼 有k個尺碼必須被配置 cell只允許被指派至有尺碼的 位置 xij and yj are 0 or 1

  13. Optimal solutions

  14. Optimal sizes

  15. Comparison Eight sizes V.S. Optimal sizes

  16. as an “index of discomfort”or “loss” • as an “index of aggregate discomfort” • associated with a set of sizes. • increases as the distance between the measurements of class i and those of size j increase.

  17. Result • of eight sizes is 0.545, and these of optimal sizes determined is 0.651 • About 19% of aggregate discomfort is higher than the optimal. • Although these results depend on the assumed form of pij ,they are is indicative of the potential for improvement from an analytical approach to the sizing problem.

  18. Extension Tryfos, P., 1986 , “An interger Programming Approach to the Apparel Sizing Problem”,The Journal of the Operational Research Society, Vol. 37, No. 10., pp. 1001-1006. McCulloch, C. E., Paal, B., Ashdown, S. P., 1998, “An Optimal Approach to Apparel Sizing”, Journal of the Operational Research Society, 49 (5), pp. 492-499.

  19. Extension McCulloch, C. E., Paal, B., Ashdown, S. P., 1998, “An Optimal Approach to Apparel Sizing”, Journal of the Operational Research Society, 49 (5), pp. 492-499. • Control dimension • 2dimensions (stature and weight) • 5 dimensions (Neck girth, Chest girth, Shoulder girth, Steeve • outseam and Neck to buttock length ) • Objectives • Minimize an index of aggregate discomfort • Multiple objective • Increase accommodation of the population • Reduce the number of sizes in the system • Improve overall fit in accommodated individuals.

  20. The optimization problem • An effective and economical sizing system must satisfy multiple objectives. The most important of these are the following： • Accommodate as large a percentage of the population as possible with ready-made garments. • For accommodated individuals, provide as good a fit as possible. • Use as few sizes as possible.

  21. Main idea • Firstly, we frame the optimization problem as a continuous optimization of location of the sizes in the space of body dimensions. • Secondly, we allow some individuals to not be accommodated by the sizing system. • We define a system to be efficient if on the boundary where improvements on any objective come necessarily at the expense of deterioration of other objectives.

  22. Definition • The nth individual is represented by a vector of size I of their body measurements • Size groups s as a prototype body form with measurements • The sizing system we divide the population into S+1 groups. • S size groups • A group of individuals not accommodated • A：the set of those individuals accommodated by the size system • α：the proportion of disaccommodated individuals

  23. Definition • ：a dissimilarity measure exists which relates to • Each subject must be accommodated by only a single size, and the loss from imperfect fit to individual n is only dependent on ns distance to the closest prototype. • We define a loss function that puts an appropriate penalty on poor fit for an individual and aggregate these penalties into an overall measure of loss

  24. Formulation • For a given number of sizes , S, and a given disaccommodation rate , α, select y1, y2,…, ys so as to Subject to • The optimization problem is not amenable to standard optimization technology, so we reformulate it by introducing a modified loss function.

  25. Formulation • The disaccommodated individuals are clearly those with the largest values of • is the largest loss for accommodated individuals • is the smallest loss for disaccommodated individuals • Define by

  26. Formulation Constant in y1, y2,…, yn

  27. Specification of distance and loss functions • The more the individual’s measurements differ from the prototype, the worse the fit. • Fit is better predicted by proportional rather than absolute difference between individual and prototype measurements. • Mellian et al.(1990) give some empirical evidence which support this assumption. • One way to meet this requirement is to log transform the measurements.

  28. Specification of distance and loss functions • Perfect fit may occur in a range of values around the prototype. • A garment which is too small may not affect fit in the same way as one which is too large. • Discrepancies in certain dimensions are more critical to fit than others.

  29. Loss functions • The discrepancy still consistent with a perfect fit was based on generally accepted apparel design values selected by the authors for a person with the population average value of that variable. • The relative values of across measurements were chosen to reflect our judgment about the relative rate which increasing discrepancies in these measurements deteriorate fit. • A person being larger than the prototype was penalized three times more than being smaller. • ( , ) ( , )

  30. Distance function • We assume can be written as a sum of squared discrepancies over each of the I measurements. • The discrepancies in each measurement are given, in turn by

  31. Distance function

  32. Finding the efficient sizing system • This problem is high-dimensional, with 100 parameters (20 sizes by 5 variable ) to determine, and the distance function is not a differentiable function of the size. • We used the Nelder-Mead method for optimization.

  33. Nelder-Mead method • Nelder-Mead method 適用於無限制式下求最佳解的一種 • 直接搜尋法 • 由Nelder and Mead (1965) 提出，目的在尋找一個目標 • 函數 的最小值， ，即尋求滿足 • Nelder-Mead法是利用多面體來逐步逼近最佳點x*，在d • 維空間裡多面體有(d+1)個頂點，令 多面體 • 的頂點且滿足

  34. Nelder-Mead method • Nelder-Mead法試著將多面體中最差的頂點 （也就是函數值最大的點）以新的更佳的點替代掉，來更新多面體，使之逼近至最佳解。更新點的設定方式有四種： • 反射(reflection) • 擴展(expansion) • 外收縮(outside contraction) • 內收縮(inside contraction) • 如果上列四種更新方式皆不適用，就進行變小(shrink)的步驟。

  35. Nelder-Mead method

  36. Result • Individuals (2208) • One-quarter (552) of individuals for model assessment • Three-quarter (1656) to fit an optimal sizing system. • Our optimization converged to a sizing system with an aggregate loss across the 1656 individual of 1128.0.(S=20, cα=1.75 )

  37. Result

  38. Result

  39. Conclusions • The core of this approach to deriving a sizing system is to fix the number of sizes and disaccommodation rate and optimize the quality of the fit • It makes simultaneous the selection of disaccommodated individuals, the derivation of prototypes, and the assignment of individuals to size classes. • It creates a formal link between sizing goals and the methodology by which size are created.

  40. Reference • McCulloch, C. E., Paal, B., Ashdown, S. P., 1998, An Optimal Approach to Apparel Sizing, Journal of the Operational Research Society, 49 (5), pp. 492-499. • Tryfos, P., 1986 ,An interger Programming Approach to the Apparel Sizing Problem,The Journal of the Operational Research Society, Vol. 37, No. 10., pp. 1001-1006. • Tryfos, P., 1985 ,On the Qptimal Choice of Sizes,Operational Research, Vol. 33, No. 3., pp. 678-684. • Winks, J. M., 1997, Clothing Size International Standardization, Redwood Books, UK, pp.47-58. • Ashdown et al., 2002 ,Use of Body Scan Data to Design Sizing Systems Based on Target Markets, National Textile Center Project S01-CR01

  41. 敬請指教

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