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「一変量の分布」で実施できる「工程能力分析」のレポートを徹底理解(計算式を示したスクリプト付き)

JMPで工程能力分析をする際には、「工程のスクリーニング」や「工程能力」のプラットフォームがありますが、最もわかりやすいのは「一変量の分布」プラットフォームで [工程能力] のオプションを使用する方法でしょう。

 

JMPのサンプルデータ「Pickles.jmp」に対して、連続尺度の列「酸度」について「一変量の分布」で仕様限界(下側 = 8、上側 = 17)を設定し、工程能力分析を実行すると、次のレポートが表示されます。

 

Masukawa_Nao_0-1722933796046.png

 

レポートには、「ヒストグラム」、「工程の要約」、「群内シグマ 工程能力」、「全体シグマ 工程能力」、「不適合率」の項目があり、これらのグラフや統計量を理解しておくことは、工程能力分析を実施する上で重要です。

 

単にCpkやPpkといった工程能力指数を算出することが目的であり、レポートの該当の箇所を見れば良いという考えもあるかもしれませんが、これらの指標をどのようにして算出しているかを理解することも重要です。

 

そこで本ブログでは、上記で示した「一変量の分布」の工程能力分析のレポートの見方や、表示される統計量の計算方法などを徹底解説します。また、計算式を示したスクリプト(Calculate_Capability.jsl)をダウンロードすることができます。

 

注意

  • 提示する例は目標値を設定せずに、下側仕様限界(LSL)、上側仕様限界(USL)を指定したものです。
  • 工程能力指数に対する信頼区間の計算は複雑なので、ここでは解説しません。
  • サブグループの列を指定して工程能力分析を実施できますが、ここではその例を示していません。
  • 今回説明するレポートは、工程の分布として正規分布を仮定しています。JMPでは正規分布以外の分布に対する工程能力分析が行えますが、ここではその解説をしません。

 

工程の要約:群内シグマ、全体シグマとは

Masukawa_Nao_1-1722994632490.png

 

Yに指定した列「酸度」に対するヒストグラムが、指定した仕様限界線(赤色の実線)とともに表示されます。このグラフにより、視覚的にLSLを下回るものやUSLを上回るデータがどれぐらいあるかを確認できます。

 

群内シグマ:短期的に計算されたばらつきの推定値(管理図のσを使用)

全体シグマ:長期的に計算されたばらつきの推定値(データか全体から求められる標準偏差を使用)

 

工程能力分析は、管理図などを用い対象となる工程が安定しているかどうかを確認して用いることをお勧めします。工程が安定している場合は、群内シグマと全体シグマの値はほぼ等しい値になりますが、工程が安定していないとこれらの値は乖離します。

 

ヒストグラムには、標本平均と群内シグマをパラメータとした正規分布曲線(青色)、標本平均と全体シグマをパラメータとした正規分布曲線(黒の点線)が描かれています。2つの曲線を比較すると、群内の曲線の方が、この工程の分布としては適切であるように見えます。

 

工程が安定しているとみなせる場合は全体シグマの値を検討すると良いでしょう。工程が安定していない場合、全体シグマの値の信ぴょう性は失われます。その際は群内シグマの値を検討すると良いでしょう。

 

逆に、全体シグマの値と群内シグマと値を比較することによって、工程が安定しているかどうかを判断することができます。その指標が、次の式で定義される安定指数です。

 

安定指数:全体シグマ ÷ 群内シグマ

 

工程が安定している場合、前述の通り群内シグマと全体シグマの値がほぼ等しい値にあるため、安定指数は1に近い値をとります。値が大きいほど、工程が安定していないことを示します。この例での安定指数は1.446と大きいため、工程が安定していないと言えます。

 

*「工程のスクリーニング」プラットフォームの「工程性能グラフ」で多数の工程を比較できますが、このグラフでは、安定指数の閾値として1.25を用いています。1.25より大きいと工程が不安定とみなします。

 

計算方法

群内シグマは、移動範囲の平均に基づいて算出されます。移動範囲とは、現在の値から1つ前の値を引き算して絶対値をとったものです。R管理図に相当するシグマを算出することになり、|2行目の値 - 1行目の値|、|3行目の値 - 2行目の値| ・・・ |N番目の値 - (N-1)番目の値| を計算し、これらの和をとります。この和をd2(2) という定数で割り算します。d2(2)とはサブグループが2(移動範囲の場合は、現在の値から一つ前までをサブグループとみなすため)のときのd2の値です。管理図のシグマの計算でも用いられる定数で、この値は1.128となります。さらにこの値を、サブグループの数(この例のデータ数は24であり、サブグループの数はデータ数から1引いた23となる)で割り算します。

 

群内シグマ =  (移動範囲の和 /  d2(2) ) /  23 =  1.160

 

全体シグマは、データから計算される標準偏差で計算されます。「要約統計量」の標準偏差の値と一致していることが確認できます。

 

Masukawa_Nao_1-1722997685823.png

 

工程能力指数

Masukawa_Nao_1-1722997859960.png

 

JMPのマニュアルにも掲載されていますが、群内シグマの工程能力指数は次の式で算出します。

 

Masukawa_Nao_2-1722998028767.png

 

この式において、μは標本平均で推定し、σは短期シグマで推定します。前述した「工程の要約」のレポートに表示される値を用いて次のように工程能力指数の推定値が計算されます。

 

Cp  = (17 -   /  (6 *  1.1598)  = 1.293

Cpl  = (10.5667 -   /  (3 *  1.1598)  = 0.738

Cpu = (17 - 10.5667) /(3 *  1.1598) = 1.849

Cpk = min(0.738, 1.849) = 0.738

 

全体シグマの工程能力指数では、上記の式のσを全体シグマによって推定します。

 

Pp  = (17 - /  (6 *  1.6766)  = 0.895

Ppl  = (10.5667 -   /  (3 *  1.6766)  = 0.510

Ppu = (17 - 10.5667) /(3 *  1.6766) = 1.279

Ppk = min(0510, 1.279) = 0.510

 

この例ではCpkは0.738、Ppk は0.510と低い値になっています。

 

不適合率

Masukawa_Nao_3-1722998939811.png

 

観測%:仕様限界外となったデータの割合です。

この例では、LSL未満のデータは1件、USL超えのデータは無いので、これらの値をデータ数(24)で割り算します。

 

LSL未満  = (1 / 24 )* 100  =  4.1667

USL超え  = (0/ 24 )* 100  =  0

 

群内σ %や全体σ %は、ヒストグラムに示した正規分布曲線に対し、LSL未満となる確率、USL超えとなる確率を示しています。

 

Masukawa_Nao_0-1722999499986.png

 

群内%: 群内シグマを用いた正規分布密度曲線に対する仕様限界外の面積(確率)

 

正規分布の累積分布関数 Normal Distribution(q, mu, sigma)関数を使って次の式で計算できます。

 

LSL未満  = Normal Distribution(LSL,標本平均,群内シグマ) * 100 =  1.3448

USL超え  = (1- Normal Distribution(USL,標本平均,群内シグマ)) * 100 = 0.0000

 

全体%: 全体シグマを用いた正規分布密度曲線に対する仕様限界外の面積(確率)

 

LSL未満  = Normal Distribution(LSL,標本平均,全体シグマ) * 100 =  6.2901

USL超え  = (1- Normal Distribution(USL,標本平均,全体シグマ)) * 100 = 0.0062

 

 

工程能力を向上させるには

前述したように、この例の工程に対しCpkは0.738、Ppk は0.510なので、工程能力は低いと言えるでしょう。では、工程能力を向上させるには、工程をどのように改善すればよいでしょうか。JMPではこの問いに対して数値的なヒントを与えるオプションが用意されています。

 

レポート「工程能力分析」の左にある赤い三角ボタンから [対話的工程能力プロット] のオプションを選択します。すると、次のようなレポートが追加されます。グラフには、現在のデータに対する(標本)平均、全体シグマをパラメータとした正規分布、LSL、USLの線が描かれています。これが現在の工程能力を示しています。

 

Masukawa_Nao_0-1723006678592.png

 

では、仮に工程を改善できて、平均を今よりも大きい方にシフトし、ばらつきを今より小さくできたら工程能力はどれぐらい改善できるのでしょうか? それを「新しい工程能力」のスライダーを動かすことによって確認できます。例えば、平均を12.5 に、全体シグマ(ばらつき)を1.1にしたときのレポートを示します。

 

Masukawa_Nao_1-1723006699477.png

 

これにより、Ppk は1.364に向上し、よく用いられる目安の値 1.33を超えていることになります。

あくまでも仮想的なケースをシミュレーションしているだけですが、今後の工程改善で目安的な値を知っておくのは有用かと思います。

 

 

今回紹介した「一変量の分布」の工程能力分析も含む、「工程能力」、「工程のスクリーニング」プラットフォームを用いた工程能力分析について、以下の動画で紹介しています。

JMPによる工程能力分析(日本語)

 

by 増川 直裕(JMP Japan)

Naohiro Masukawa - JMP User Community

 

Last Modified: Aug 7, 2024 2:41 AM