MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01C6E7D9.BE8B0860" このドキュメントは単一ファイル Web ページ (Web アーカイブ ファイル) です。お使いのブラウザ、またはエディタは Web アーカイブ ファイルをサポートしていません。Microsoft Internet Explorer など、Web アーカイブをサポートするブラウザをダウンロードしてください。 ------=_NextPart_01C6E7D9.BE8B0860 Content-Location: file:///C:/72785C13/06dmt_3.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" 意思決定科学 3．LP復習

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マスタ タイトルの書式設定
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------=_NextPart_01C6E7D9.BE8B0860 Content-Location: file:///C:/72785C13/06dmt_3.files/pres.xml Content-Transfer-Encoding: quoted-printable Content-Type: text/xml; charset="utf-8" ------=_NextPart_01C6E7D9.BE8B0860 Content-Location: file:///C:/72785C13/06dmt_3.files/slide0001.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" 意思決定科学 3．LP復習
意思決定科学

線形計画法概観（復習= ）
情報学部
堀田敬介
2006/10/3, Tue.
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Content-Type: text/html; charset="shift_jis" 意思決定科学 3．LP復習
目次
•線形計画法
–数理モデル
–図的解法，単体法による解法
–双対問題，双対定理
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数理モデル
<= span style=3D'mso-fareast-hint:yes'>•例題：効率的なアルバイト=
•時給1200円の清掃作業，時給900円のウェイター2つ． <= /span>
•各仕事を行うとス= gレスがたまるが，各々5，3である．
•
•
•
•
<= span style=3D'font-size:83%'>•週末に5時間，アルバイトをする時間を取ることができる． <= /span>
<= span style=3D'font-size:83%'>•健康のため，ストレス許容量は21である． = ;
<= span style=3D'font-size:86%'>–さて，この条件のもとで，最大のアルバイト料を= 得るにはどちらのアルバイトをどれだけすればいいか?
時給1200円 = ≧ 時給900円
だから，5時間全てを清掃作業で！
= でも…，
ストレス：5×<= span lang=3DEN-US style=3D'mso-fareast-hint:yes;mso-fareast-language:JA'>5＝25　＞　21= （許容量） <= /span>
\1,200/h
5 stress
\900/h
3 stress
=
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数理モデル
•解説：効率的なア= ルバイト
&= #8226;時給1200円の清掃作業= C時給900円のウェイタ= [2つ．
&= #8226;各仕事を行うとストレスがたまるが，各々5，3である．
&= #8226;週末に5時間，アルバ= Cトをする時間を取ることができる．
&= #8226;健康のため，ストレス許容量は21である．
= max. 1200x1 + 900x2
 s. t.        = ;  x1  +      x2  ≦ 5
        = ;       5x1 +     3x2 ≦ 21
        = ;         x1,         x2 ≧ 0
アルバイト= 代最大化
アルバイト= 時間制約 =
許容ストレ= ス制約 =
アルバイト= 時間は非負&#= 13;
数理モデル
線= 形計画法，LP; Linear Program
数理モデル
•解説：どう解くか？　−その１−
max.1200x1+900x2
     s. t.=     x1  +  x2  ≦ 5
        = ;     5x1 + 3x2 ≦ 21
        = ;       x1,     x2 ≧ 0
図= 的解法 =
x1
x2
<= ![endif]> = <= ![endif]>
0
(1200, 900)
最適解： (x1, x<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";font-size:89%;position:relative; top:.37em;mso-text-raise:-25%;mso-fareast-language:JA'>2) =3D (3, 2)
ウェイターを3時間
清掃作業を2時間
最適値： \5,400
　<= /span>図的解法が使えるのは2次元（頑張って3次元）まで． = 　それ以上の高次元ではど= うするの？
3
2
注：(4, 3)方向
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数理モデル
•解説：どう解くか？　−その２−
max. 4x1 + 3x2
 s. t.    x1  +  x2  ≦  5
         5x1 + 3x2 ≦ 21
        = ;   x1,     x2  ≧ 0
単= 体法 (simplex method= )
max. z
 s. t.  z -4x1 - 3x2       =3D  0
        = ;       x1 +  x2 + s1         =3D  5
        = ;     5x1 + 3x2        + s2 =3D 21
        = ;       x1,     x2 ,  s1 ,   s2 ≧ 0
ratio = test
pivot<= /i>
reduced cost
max. z
 s. t.  z         +3/2s1+10/7s2=3D18= 3;
        = ;        x2 +5/2s1   -1/2s2=3D 2
        = ;    x1     &nb= sp; -3/2s1 &= nbsp; +1/2s2=3D 3
        = ;    x1, x2 ,      s1 ,        = ;   s2 ≧0
simplex tableau
= basic&= #13;
variab= le
nonbas= ic variable
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数理モデル
•単体法の考え方
max. 2x1 + 3x2 +   x3 + 2x4
 s. t.  2x1 + 7x2 + 3x3 + 3x4 =3D 6
        = ;  x1 ,     x2 ,    x3 ,    x4 ≧0
x1=3D6 -7/2x2 -3/2x3 -3/2x4
max. z =3D12 -   4x2 -  2x3  -     x4=
 s. t.  x1=3D 6 -7/2x2 -3/2x3 -3/2x4
        x1 ,        = ;    x2 ,      x3 ,     &nb= sp; x4 ≧0
被約費用（reduced c= ost）
最適解（an optimal solution）
x*=3D(6,0,0,0)
最適値（the optim= al value）
12
基底変数（basic variable）<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:Verdana;mso-f= areast-font-family: "ＭＳ Ｐゴシック";mso-special-format:lastCR;display:none'>
非基底変数（non-basic variable）<= /div>
辞書（dictionary）
最適辞書
（an optimal dictionary）
基底解（an basic solution）
x=3D(6,0,0,0)
実行可能基底解（an feasible basic solution）
x≧0 を満たす基底解
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Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhRAArAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAYAAQA9 ACMAgAAAAICAAAJkhI+py+0Ywpu0smizbnh7331iFY6mU54qJK3ukb5nLJt0/d34pu9Z77MAg5Mh EdU6gpJKHrMpfEIpxinLqqliAVps1/oFS7eXsXcR9j3TaiZ7F2m921wzGRa/P+J8fZmf54dQAAA7 ------=_NextPart_01C6E7D9.BE8B0860 Content-Location: file:///C:/72785C13/06dmt_3.files/slide0023_image115.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhPAArAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAYAAQA1 ACMAgAAAAICAAAJchI+py60RnJwUxYqzvbpT7oUMJJYLaJpkyqJsuL6lK2tx7dF4de+Z7pP0gh+i DWg8JTHD5cj5QUIN0mlzasE6rlpAFco1ur6+WzgZIwchkfOSrSbC4+u5uz7vIgoAOw== ------=_NextPart_01C6E7D9.BE8B0860 Content-Location: file:///C:/72785C13/06dmt_3.files/slide0024.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" 意思決定科学 3．LP復習
数理モデル
•単体法の幾何学的意味
max.   x1 +<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>   x2 +<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>   x3
 s. t.  2x1 +<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'> 6x2 +<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'> 3x3 ≦ <= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>24 = ;
         2x1        = ;      +   x3 ≦ 6
        = ;            &n= bsp;        x3 ≦ 2
        = ;   x1 ,  =   x2 ,  =   x3 ≧  0<= /span>
x<= i>6≧0 [x3≦2]
x<= i>5≧0
[2x1+x<= i>3≦6]
x<= i>4≧0
[2x1+6x<= i>2+3x<= i>3= ≦24]
x3≧0
x1≧0
x2≧0
(<= /span>2,<= /span>0,<= /span>2)<= /span>
(<= /span>0,<= /span>0,<= /span>2)<= /span>
(<= /span>2,<= /span>7/<= /span>3,<= /span>2)<= /span>
(<= /span>0,<= /span>3,<= /span>2)<= /span>
(<= /span>3,<= /span>0,<= /span>0)<= /span>
(3,<= /span>3,<= /span>0)<= /span>
(0,<= /span>4,<= /span>0)<= /span>
= max.   x1 +<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>   x2 +<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>   x3
 s. t.  2x1 +<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'> 6x2 +<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'> 3x3 + = x<= /span>4        = ;          =3D 24
         2x1        = ;      +   x3        = ;  + x5         =3D 6
        = ;            &n= bsp;        x3        = ;          + x6 =3D 2 = ;
        = ;   x1 ,  =   x2 ,  =   x3 ,  x4 ,  x5 , x6 ≧0<= /span>
x<= i>1
x<= i>2
x<= i>3
(0,0,2,18,4,0)
(0,3,2,0,4,0)
(2,7/3,2,0,0,0)
(3,3,0,0,0,2)
基底変数 x4
↓入替↑
非基底変数 x2
基底変数 x6
↓入替↑
非基底変数 x3
※この例題= では　　　　　　　　全端点で非退化
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R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAJ AAoAgQAAAABmme360gECAwIWhBF3EcIrYkxNSmZvzTi7bVHcwjxAAQA7 ------=_NextPart_01C6E7D9.BE8B0860 Content-Location: file:///C:/72785C13/06dmt_3.files/slide0024_image145.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhCwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAK AAoAgQAAAABmme360gECAwIXhINhG+KCznsKtOkQpnerPn0eEy5HshQAOx== ------=_NextPart_01C6E7D9.BE8B0860 Content-Location: file:///C:/72785C13/06dmt_3.files/slide0017.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" 意思決定科学 3．LP復習
数理モデル
•線形計画問題を解く2つの解法
単= ﾌ法 (simplex method<= /i>)
内点法<= /span> (interior point meth= od)
参考：主双対内点法
Jacobi行列
Newton方向ベクトル
max. ctx&#= 13;
 s.t.   Ax =3D b&#= 13;
        = ;   x≧0
min. bty
 s.t. A= ty+s= =3D c
主・双対問題（行列表記）
x<= i>1
x<= i>2
x<= i>3
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演習1：数理モデルによる定式化
•最適生産量問題
–<= /span>ある工場では3つの製品A，B, Cを作っている．
•<= /span>A，B，Cを1単位作るのに，材料Pが其々6kg，2kg= ，3kg，材料Qが3kg, 2kg，5kg，材料Rが4l，3l2l，材料Sが5g，1g，9g必要．
•<= /span>この工場で使用できる材料P，Q，R，Sの量は，其々2500kg，3000kg，1800l，5000gである．
•<= /span>A，B，Cを1単位売って得られる利益が各々7万円，4万円，5万円．
•
–<= /span>利益最大となる，A，B，Cの生産単位はいくらか？ = ;
–　（ただし，小数以下= の生産単位も許す）
–
–数理モデルを作り，単体法により最適解と最適値を求めよ．
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双対問題
•主問題（Ｐ）
max. 4x1 + 3x2
 s. t.    x1  +  x2  ≦ 5
         5x1 + 3x2 ≦ 21
        = ;   x1,     x2 ≧ 0
•双対問題（Ｄ）
min. 5y1 + 21y2
 s. t.    y1 + 5y2 ≧ 4
        = ;   y1 + 3y2 ≧ 3
        = ;   y1,     y2 ≧ 0
Primal=
Dual
max. 4x1 + 3x2
 s. t.    x1  +  x2   =3D  5
         5x1 + 3x2  =3D 21
        = ;   x1,     x2 ≧ 0
min. 5y1 + 21y2
 s. t.    y1 + 5y2 ≧ 4
        = ;   y1 + 3y2 ≧ 3
対称型の主・双対問題
標準型の主・双対問題
=
一般的には…
<= ![endif]>
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双対問題
•双対問題の考え方
max. 15x1 + 13x2
 s. t.      x1  +  3x2  ≦  5  …@
        = ;   3x1 +     x2 ≦  7<= span style=3D'mso-spacerun:yes'>  …A
         11x1 +     x2 ≦ 17 …B
        = ;     x1,       x2 ≧ 0
@×<= /span>3 ＋A×<= /span>4 ＋B×<= /span>0
15x1 + 13x2 ≦ 43
目的関数値は= 43以下！&= #13;
@×<= /span>4 ＋A×<= /span>0 ＋B×<= /span>1
15x1 + 13x2 ≦ 37
目的関数値は= 37以下！&= #13;
@×<= /span>y1＋A×<= /span>y2＋B×<= /span>y3
(x1+3x2)y1+(3x1+x2 )y2+(11x1+x2 )y3≦ 5y1+7y2+17y3 
⇔ (y1+3y2+11y3)x1+(3y1+y2+y3)x2≦ 5y1+7y2+17y3 
15  x1        +       13  x2
3D"テキスト
≦
3D"テキスト
≦
minimize
= min. 5y1 +7y2 +17y3
 s. t.    y1 +3y2+11y3 ≧15
         3y1+  y2 +    y3 ≧13
        = ;   y1,   y2 ,     y3  ≧ 0
(P)=
(D)
※）y1,y<= i>2,y<= i>3≧0
※）x1,x<= i>2≧0
3D"テキスト
≦
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R0lGODlhAgBkAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAB AGIAgAAAAABmmQIIjI+py+0P4yoAOy== ------=_NextPart_01C6E7D9.BE8B0860 Content-Location: file:///C:/72785C13/06dmt_3.files/slide0025_image210.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgBWAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAQAB AFUAgAAAAABmmQIIjI+py+0PVwEAOw== ------=_NextPart_01C6E7D9.BE8B0860 Content-Location: file:///C:/72785C13/06dmt_3.files/slide0021.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" 意思決定科学 3．LP復習
演習2-1：主問題と双対問題
•双対問題を示せ
max.   5x1 + 2x2 + 4x3
 s. t.    2x1  +  x2  ｰ= 3x3  ≦ 5
         ｰ4x1 + 3x2  ｰ= 2x3  ≧ｰ2
        = ;   3x1 ｰ 2x2  +   x3  ≦ 4
        = ;     x1 + 4x2 + 5x3  ≦ 7
        = ;     x1,     x2 ,     x3 ≧ 0
(P)=
------=_NextPart_01C6E7D9.BE8B0860 Content-Location: file:///C:/72785C13/06dmt_3.files/slide0021_image211.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAgB/AHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAB AH0AgAAAAABmmQIKjI+py+0Po5ypAAA7 ------=_NextPart_01C6E7D9.BE8B0860 Content-Location: file:///C:/72785C13/06dmt_3.files/slide0030.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" 意思決定科学 3．LP復習
双対定理
•弱双対定理
R= 11;　任意の実行可能解 (x1 ,x2)<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; mso-fareast-hint:yes'>，(y1,<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>y<= /span>2) = について，
–4x1 + 3x<= /span>2 　≦　5<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>y<= /span>1 + 21y<= /span>2
R= 11;が成り立つ
max. 4x1 + 3x2
 s. t.    x1  +  x2  ≦ 5
         5x1 + 3x2 ≦ 21
        = ;   x1,     x2 ≧ 0
min. 5y1 + 21y2
 s. t.    y1 + 5y2 ≧ 4
        = ;   y1 + 3y2 ≧ 3
        = ;   y1,     y2 ≧ 0
(P)
(D)=
<= span style=3D'mso-fareast-hint:yes'>•証明
=
一般的には…
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双対定理
•双対定理=
R= 11;　主問題(P) に最適解(<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>x<= /span>1* , = x<= /span>2*) = が存在するならば，双対問題(D) にも最適解(y1*, = y<= /span>2*) = が存在し，最適値は等しい，即= ち，
–4x1* + 3x<= /span>2* 　＝　5y1* + 21y<= /span>2*
R= 11;　が成り立つ
•証明略
max. 4x1 + 3x2
 s. t.    x1  +  x2  ≦ 5
         5x1 + 3x2 ≦ 21
        = ;   x1,     x2 ≧ 0
min. 5y1 + 21y2
 s. t.    y1 + 5y2 ≧ 4
        = ;   y1 + 3y2 ≧ 3
        = ;   y1,     y2 ≧ 0
(P)
(D)=
=
一般的には…
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双対定理
•相補性定理
R= 11;　主問題(P) と双対問題= (D) の実行可能解 (x1 ,<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>x<= /span>2)<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; mso-fareast-hint:yes'>，(<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>y<= /span>1,<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>y<= /span>2) = が (P)，<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>(D)の最適解であるための必要十分条件は，
R= 11;
R= 11;　が成立することである．
•証明略
max. 4x1 + 3x2
 s. t.    x1  +  x2  ≦ 5
         5x1 + 3x2 ≦ 21
        = ;   x1,     x2 ≧ 0
min. 5y1 + 21y2
 s. t.    y1 + 5y2 ≧ 4
        = ;   y1 + 3y2 ≧ 3
        = ;   y1,     y2 ≧ 0
(P)
(D)=
=
一般的には…
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<= span style=3D'mso-fareast-hint:yes'>•（対称型の）主・双対線形計画問題を= 解くことは
•　　(i)&#= 13;
•
•　　(ii)&= #13;
•
•　　(iii)=
•
•　を満たす解 (x1 ,<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>x<= /span>2)<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; mso-fareast-hint:yes'>，(y1,<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>y<= /span>2) = を見つけること．
双対理論からの解法の考察
主実行可能条件
双対実行可能条件
相補性条件
（主）単体法 (<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>simplex method)
（主双対）内点法 (<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>primal= -dual IPM)
双対単体法 (<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>dual simplex method)
(i)，(iii)を満たしつつ，(ii)の成立で終了
(ii)，(iii)を満たしつつ，(i)の成立で終了
(i)，(ii)を満たしつつ，(iii)の成立で終了
(i)<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:Verdana;mso-f= areast-font-family: "ＭＳ Ｐゴシック";mso-fareast-hint:yes'>〜(iii)全てを満たす<= span style=3D'position:absolute;top:6.0%;left:66.29%;width:29.02%;height:4.0%'>= (<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>x<= /span>1* ,<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>x<= /span>2*)<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:"Times New Ro= man"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; mso-fareast-hint:yes'>，(y1*,<= span lang=3DEN-US style=3D'font-family:"Times New Roman";mso-ascii-font-family:= "Times New Roman"; mso-hansi-font-family:"Times New Roman";mso-fareast-language:JA'>y<= /span>2*)<= span lang=3DEN-US style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:= "Times New Roman"; mso-fareast-font-family:"ＭＳ Ｐゴシック";mso-hansi-font-family:"Times New= Roman"; font-size:50%;mso-fareast-language:JA;display:none'>
が（主・双= ﾎ）最適解
注：反復中(i)，(ii)<= /span>を満= たさないなど<= /span>バリエーショ<= /span>ンがある&#= 13;
  x1  +  x2  = 5
5x1 <= /span>+ 3x2 <= /span>= 21
y1 + 5<= /span>y2 ≧ 4
y1 + 3<= /span>y2 ≧ 3
x= 1, x2 <= /span>= 0
y= 1, y2 <= /span>= 0
=
一般的には…
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演習３：
•
–以下のLPについて，双= 対問題を作成し，弱双対定理が成り立っていることを確認せよ <= /div>
–また，単体法= により最適解を求め，双対定理，相補性定理が成り立っていることを確認せよ
max.     <= /span>x1 + 3x2 + 2x3
 s. t.    4x1  +2x2  ｰ  x3  ≦  8
        = ;   3x1 ｰ 2x2 + 4x3  ≦10
        = ;     x1,     x2 ,     x3 ≧ 0
(P)
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参考文献
<= span style=3D'font-family:"Times New Roman";font-size:75%;mso-fareast-hint:yes'= >•今野浩 「線形計画法」 日科技連(1987)
<= span style=3D'font-family:"Times New Roman";font-size:75%;mso-fareast-hint:yes'= >•反町洋一「線形計画法の実際」産業図書(1992) = ;
<= span style=3D'font-family:"Times New Roman";font-size:75%'>•H.P.Williams「数理計画モデルの作= 成法」産業図書(1995) = ;
<= span style=3D'font-family:"Times New Roman";font-size:75%;mso-fareast-hint:yes'= >•大山達雄「最適化モデル分析」日科技連(1993)
<= span style=3D'font-family:"Times New Roman";font-size:75%;mso-fareast-hint:yes'= >•福島雅夫 「数理計画入門」 朝倉書店(1996)
<= span style=3D'font-family:"Times New Roman";font-size:75%;mso-fareast-hint:yes'= >•田村明久・村松正和 共立出版(2002)
<= span style=3D'font-family:"Times New Roman";font-size:75%;mso-fareast-hint:yes'= >•藤田宏・今野浩・田邉國士 岩波書店(1994)
<= span style=3D'font-family:"Times New Roman";font-size:75%;mso-fareast-hint:yes'= >•小島正和・土谷隆・水野眞治・矢部博 「内点法」 朝倉書店(2001)
<= span style=3D'font-family:"Times New Roman";font-size:75%;mso-fareast-hint:yes'= >•森雅夫・松井知己 朝倉書店(2004)
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演習1解答：= 最適生産量問題
•max. 7x1+4x2+5x3
• s. t.  6x1+2x2+3x3 ≦2500
•         3x1+2x2+5x3 ≦3000
•         4x1+3x2+2x3 ≦1800
•         5x1+  x2+9x3 ≦5000
•        = ;   x1,   x2,      x3 ≧0
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双対問題：一般的な書式
•主問題（Ｐ）
max. c1x1 +…+ cnxn
 s. t. a11x1 +…+a1nxn  ≦ b1
        = ;            &n= bsp;  ：
　　  am1x1 +…+amnxn  ≦ bm
        = ;     x1,=   … ,     xn ≧ 0
•双対問題（Ｄ）
Primal=
Dual
min. b1y1 +…+ bmym
 s. t. a11y1 +…+am1ym ≧ c1
        = ;            &n= bsp;  ：
　　  a1ny1 +…+amnym ≧ cn
        = ;     y1,=   … ,     ym ≧ 0
m
n
m
n
目的関数：最大化=
　制約式 ： m 本=
　変　数  ： n  個
目的関数：最小化=
　制約式 ： n  本
　変　数  ： m 個
対称型の主・双対問題
=
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双対問題：行列表記
•主問題（Ｐ）
•双対問題（Ｄ）
Primal=
Dual
max. ctx
 s.t.   Ax ≦<= i> b
        = ;   x ≧ 0
min. bty
 s.t. Aty ≦ c
        = ;   y ≧ 0
対称型の主・双対問題
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file:///C:/72785C13/06dmt_3.files/slide0028_image273.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhHwAfAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAe AB4AggAAAI6Wfr/JqfH72+360vX85AECAwECAwOEOLrc/qrASUcohejNu+/BlX1kSYSYqYLi6p7t qwUkOr4h7dlyHu6xlS/HSgmHxA1P6fs0izdYkvWDcobOKdM4e1J116C0+gVyx+AweRvFAtfds1sN h8nnXW08ikZ62V9+entvgmlhZ0yCJUs1g2Z8MpCSJo2UkAKZmpucnZwonqGimwEJADs= ------=_NextPart_01C6E7D9.BE8B0860 Content-Location: file:///C:/72785C13/06dmt_3.files/slide0031.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis" 意思決定科学 3．LP復習
双対定理
•弱双対定理
R= 11;　任意の実行可能解xi (i=3D1,…,n)，yj (j=3D1,…,m)　について，　　
(P)<= /div>
(D)<= /div>
•証明
=
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双対定理
•双対定理=
R= 11;　主問題（Ｐ）に最適解 xi* (i=3D1,…,n)，が存在するならば，双対問題（Ｄ）にも最適解 yj* (j=3D1,…,m)　が存在し，最適値は等しい= ，即ち，
R= 11;
R= 11;
R= 11;　が成り立つ．
(P)<= /div>
(D)<= /div>
•証明略
=
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双対定理
•相補性定理
R= 11;　主問題（Ｐ）と双対問題（Ｄ）の実行可能解 xi (i=3D1,…,n)，yj (j=3D1,…,m)　が，（Ｐ）（Ｄ）の最適解であるため= の必要十分条件は，
R= 11;
R= 11;
R= 11;
R= 11;　が成立することである．
(P)<= /div>
(D)<= /div>
•証明略
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<= span style=3D'mso-fareast-hint:yes'>•（対称型の）主・双対線形計画問題を= 解くことは
•　　(i)&#= 13;
•
•　　(ii)&= #13;
•
•　　(iii)=
•
•　を満たすxi (i=3D1,…,n)，yj (j=3D1,…,m)を見つけること．
双対理論からの解法の考察
主実行可能条件
双対実行可能条件
相補性条件
（主）単体法 (simplex method)
（主双対）内点法 (primal-dual IPM)<= /div>
双対単体法 (dual simplex method)
(i)，(iii)を満たしつつ，(ii)の成立で終了<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:Verdana;mso-f= areast-font-family: "ＭＳ Ｐゴシック";font-size:89%;mso-special-format:lastCR;display:none'>&#= 13;
(ii)，(iii)を満たしつつ，(i)の成立で終了<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:Verdana;mso-f= areast-font-family: "ＭＳ Ｐゴシック";font-size:89%;mso-special-format:lastCR;display:none'>&#= 13;
(i)，(ii)を満たしつつ，(iii)の成立で終了<= span style=3D'font-family:"ＭＳ Ｐゴシック";mso-ascii-font-family:Verdana;mso-f= areast-font-family: "ＭＳ Ｐゴシック";font-size:89%;mso-special-format:lastCR;display:none'>&#= 13;
(i)〜(iii)全てを満たす
x<= b>i (i=3D1,…,n)，y<= b>j (j=3D1,…,m)
が（主・双対）最適解=
注：反復中(i)，(ii)を満たさないなど<= /span>バリエーショ<= /span>ンがある
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<日付/時刻>
マスタ テキストの書式設定
第 2 レベル
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第 5 レベル
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InitHLinkArray() if( g_scaleInFrame||(IsWin("PPTSld") && parent.IsFullScrMode() ) ) document.body.scroll=3D"no" _RSW() if( IsWin("PPTSld") && parent.IsFullScrMode() ) FullScrInit(); =09 MakeSldVis(); ChkAutoAdv() if( runAnimations ) { if( document.all("NSPlay") ) document.all("NSPlay").autoStart =3D false; if( sld.filters && sld.filters.revealtrans ) setTimeout( "document.body.start()", sld.filters.revealtrans.duration * = 1000 ); else document.body.start(); } } function MakeSldVis()=20 { var fTrans=3Dg_showAnimation && SldHasTrans() if( fTrans )=09 { if( g_bgSound ) { idx=3Dg_bgSound.indexOf(","); pptSound.src=3Dg_bgSound.substr( 0, idx ); pptSound.loop=3D -(parseInt(g_bgSound.substr(idx+1))); } SlideObj.filters.revealtrans.Apply()=09 } SlideObj.style.visibility=3D"visible" if( fTrans ) SlideObj.filters.revealtrans.Play() } function MakeNotesVis()=20 { if( !IsNts() ) return false=20 SlideObj.style.display=3D"none" nObj =3D document.all.item("NotesObj") parent.SetHasNts(0) if( nObj ) {=20 nObj.style.display=3D"" parent.SetHasNts(1) } return 1 } function ChkAutoAdv() { if(SldHasTrans()) SlideObj.onfilterchange=3DAutoAdv else AutoAdv() } function AutoAdv() { if(!IsWin("PPTSld") || !gUseSldTimings )return var sld=3DGetCurSld() if( (sld.mAdvDelay>0) && !parent.IsFramesMode() ) setTimeout("parent.GoToNextSld()",sld.mAdvDelay) } function GetObj(id) { if(g_supportsPPTHTML) return document.all(id); else return document.getElementById(id); } function SldHasTrans() { return SlideObj.style.filter !=3D ""; } function GetSldId()=20 { var regExp =3D /file:\/\/\//i var pos =3D location.href.search(regExp) if (MHTMLPrefix !=3D "" && pos !=3D -1) sId =3D location.href.substring(pos) else { sId =3D RemoveFilePrefixFromHref(location.href); var regExp =3D /\// var fixedHref =3D sId var pos =3D -1 =09 pos =3D fixedHref.search(regExp) while (pos !=3D -1) { fixedHref =3D fixedHref.replace(regExp, "\\") pos =3D fixedHref.search(regExp) } =09 if (g_fBaseHyperlink =3D=3D true) sId =3D "file:///" + fixedHref; else sId =3D fixedHref.substring(fixedHref.lastIndexOf('\\') + 1) } =09 return sId } function HideMenu() { if( frames["PPTSld"] && PPTSld.document.all.item("ctx= tmenu") && PPTSld.ctxtmenu.style.display!=3D"none" ) { PPTSld.ctxtmenu.styl= e.display=3D'none'; return true } return false } function IsWin( name ) { return window.name =3D=3D name } function IsNts() { return IsWin("PPTNts") } function IsSldOrNts() { return( IsWin("PPTSld")||IsWin("PPTNts") ) } function SupportsPPTAnimation() { return( navigator.platform =3D=3D "Win32"= && navigator.appVersion.indexOf("Windows")>0 ) } function SupportsPPTHTML() { var appVer=3Dnavigator.appVersion, msie=3DappVer.indexOf("MSIE "), ver=3D0 if( msie >=3D 0 ) ver=3DparseFloat( appVer.substring( msie+5, appVer.indexOf(";",msie) ) ) else ver=3DparseInt(appVer) return( ver >=3D 4 && msie >=3D 0 ) } function _RSW() { if( !g_supportsPPTHTML || IsNts() || ( !g_scaleInFrame && (!IsWin("PPTSld") || !parent.IsFullScrMode()) ) ) return var padding=3D0; if( IsWin("PPTSld") && parent.IsFramesMode() ) padding=3D6 cltWidth=3Ddocument.body.clientWidth-padding cltHeight=3Ddocument.body.clientHeight-padding factor=3D(1.0*cltWidth)/g_origW if( cltHeight < g_origH*factor ) factor=3D(1.0*cltHeight)/g_origH newSize =3D g_origSz * factor if( newSize < 1 ) newSize=3D1 s=3DSlideObj.style s.fontSize=3DnewSize+"px" s.posWidth=3Dg_origW*factor s.posHeight=3Dg_origH*factor s.posLeft=3D(cltWidth-s.posWidth+padding)/2 s.posTop=3D(cltHeight-s.posHeight+padding)/2 if( g_scaleHyperlinks ) ScaleHyperlinks( factor ) } function _InitAnimations() { animRuntimeInstalled =3D ''+document.body.localTime !=3D 'undefined'; isFullScreen =3D (window.name =3D=3D "PPTSld") && !parent.IsFramesMode(); g_animUseRuntime =3D g_showAnimation && animRuntimeInstalled && !(isFullSc= reen && parent.IsSldVisited()); if( g_animUseRuntime ) { collSeq =3D document.all.tags("seq"); if( collSeq !=3D null ) { for(ii=3D0;ii numSlds ) gSldJumpIdx =3D numSlds; if ( gSldJumpIdx >=3D 0 ) { if ( gSldJumpIdx =3D=3D 0 ) gSldJumpIdx =3D 1; var jumpTo =3D parseInt(gSldJumpIdx); gSldJump =3D 0; gSldJumpIdx =3D ""; win.GoToSld( parent.GetSldList().mList[jumpTo-1].mSldHref ) } } } function _KDH() { if( event.keyCode =3D=3D 8 ) { event.returnValue =3D 0; parent.GoToPrevSld(); } } var g_HLinkArray =3D new Array(); =20 function IMapAreaObj( areaObj, coords ) { this.areaObj =3D areaObj; this.coords =3D coords; } function InitHLinkArray() { var appVer =3D navigator.appVersion; var msie =3D appVer.indexOf ( "MSIE " ) var ver =3D 0; if ( msie >=3D 0 ) ver =3D parseInt ( appVer.substring( msie+5 ) ); linkNum =3D 0; for (i=3D0; i 4 ) ) g_HLinkArray[linkNum++] =3D new IMapAreaObj( areaObj, ParseCoords( ar= eaObj.coords ) ); } } function ScaleHyperlinks( factor ) { =20 for ( ii=3D0; ii< g_HLinkArray.length; ii++) { coordsStr=3D""; imaObj =3D g_HLinkArray[ii]; for ( jj=3D0; jj < imaObj.coords.length-1; jj++ ) coordsStr +=3D (imaObj.coords[jj]*factor) + ","; coordsStr +=3D (imaObj.coords[jj]*factor); imaObj.areaObj.coords =3D coordsStr; } } function ParseCoords( coordsStr ) { var numArray =3D new Array(); var i =3D curPos =3D commaPos =3D 0; while ( ( commaPos =3D coordsStr.indexOf(",", curPos) ) !=3D -1 ) {=20 numArray[i++] =3D parseInt( coordsStr.substr( curPos, commaPos ) ); curPos =3D commaPos + 1; } numArray[i] =3D parseInt( coordsStr.substr( curPos ) ); return numArray; } function DocumentOnClick() { if( IsNts() || parent.HideMenu() ) return; if( ( g_allowAdvOnClick && (window.name=3D=3D"PPTSld") && !parent.IsFrames= Mode() ) || (event && event.keyCode=3D=3D32) ) { =09 if( g_animUseRuntime && g_animMainSequence && g_animMainSequence.cangonex= t ) return; parent.GoToNextSld(); } } var g_supportsPPTHTML =3D SupportsPPTHTML(), g_scaleInFrame =3D 1, gId=3D""= , g_bgSound=3D"", g_scaleHyperlinks =3D false, g_allowAdvOnClick =3D 1, g_showInBrowser = =3D 0, gLoopCont =3D 0, gUseSldTimings =3D 1; var g_showAnimation =3D g_supportsPPTHTML && SupportsPPTAnimation() && ( (w= indow.name=3D=3D"PPTSld" && !parent.IsFramesMode()) || g_showInBrowser );va= r g_animManager =3D null; var g_animUseRuntime =3D false; var g_animItemsToHide, g_animInteractiveItems, g_animSlideTime; var g_animMainSequence =3D null; var ENDSHOW_MESG=3D"スライド ショーの最後です。 クリックすると終了します。"= , SCREEN_MODE=3D"Frames", gIsEndShow=3D0, NUM_VIS_SLDS=3D26, SCRIPT_HREF=3D= "script.js", FULLSCR_HREF=3D"fullscreen.htm"; var gCurSld =3D gPrevSld =3D 1, g_offset =3D 0, gNtsOpen =3D gHasNts =3D gO= tlTxtExp =3D 0, gHasNarration =3D 0, gOtlOpen =3D true window.gPPTHTML=3DSupportsPPTHTML() var g_fBaseHyperlink =3D false; var gMainDoc=3Dnew Array(new hrefList("slide0001.htm",1,-1,1),new hrefList(= "slide0029.htm",1,-1,1),new hrefList("slide0002.htm",1,-1,1),new hrefList("= slide0010.htm",1,-1,1),new hrefList("slide0011.htm",1,-1,1),new hrefList("s= lide0012.htm",1,-1,1),new hrefList("slide0023.htm",1,-1,1),new hrefList("sl= ide0024.htm",1,-1,1),new hrefList("slide0017.htm",1,-1,1),new hrefList("sli= de0003.htm",1,-1,1),new hrefList("slide0004.htm",1,-1,1),new hrefList("slid= e0025.htm",1,-1,1),new hrefList("slide0021.htm",1,-1,1),new hrefList("slide= 0030.htm",1,-1,1),new hrefList("slide0032.htm",1,-1,1),new hrefList("slide0= 034.htm",1,-1,1),new hrefList("slide0036.htm",1,-1,1),new hrefList("slide00= 20.htm",1,-1,1),new hrefList("slide0026.htm",1,-1,1),new hrefList("slide001= 4.htm",1,-1,1),new hrefList("slide0027.htm",1,-1,1),new hrefList("slide0028= .htm",1,-1,1),new hrefList("slide0031.htm",1,-1,1),new hrefList("slide0033.= htm",1,-1,1),new hrefList("slide0035.htm",1,-1,1),new hrefList("slide0037.h= tm",1,-1,1)); /********************************************* Frameset functions These functions control slide navigation and state of the frameset. **********************************************/ function RemoveFilePrefixFromHref(href) { var regExp =3D /^file:\/\/\//i; return href.replace(regExp, "") } function FullScrInit() { g_allowAdvOnClick =3D GetCurSld().mAdvOnClk document.body.style.backgroundColor=3D"black" document.oncontextmenu=3Dparent._CM; document.onkeydown =3D _KDH; document.ondragstart=3DCancel document.onselectstart=3DCancel self.focus() } function Redirect( frmId ) {=09 var str=3Ddocument.location.hash,idx=3Dstr.indexOf('#'), sId=3DGetSldId() if(idx>=3D0) str=3Dstr.substr(1); if( window.name !=3D frmId && ( sId !=3D str) ) { obj =3D GetObj("Main-File") window.location.href=3Dobj.href+"#"+sId return 1 } return 0 } var MHTMLPrefix =3D CalculateMHTMLPrefix();=20 function CalculateMHTMLPrefix() { if ( document.location.protocol =3D=3D 'mhtml:') {=20 href=3Dnew String(document.location.href)=20 Start=3Dhref.indexOf('!')+1=20 End=3Dhref.lastIndexOf('/')+1=20 if (End < Start)=20 return href.substring(0, Start)=20 else=20 return href.substring(0, End)=20 } return ''; } function GetTags(base,tag) { if(g_supportsPPTHTML) return base.all.tags(tag); else return base.getElementsByTagName(tag); } function UpdNtsPane(){ if(frames["PPTNts"]) PPTNts.location.replace( MHTMLP= refix+GetHrefObj( gCurSld ).mNtsHref ) } function UpdNavPane( sldIndex ){ if(gNavLoaded) PPTNav.UpdNav() } function UpdOtNavPane(){ if(gOtlNavLoaded) PPTOtlNav.UpdOtlNav() } function UpdOtlPane(){ if(gOtlLoaded) PPTOtl.UpdOtl() } function SetHasNts( fVal ) { if( gHasNts !=3D fVal ) { gHasNts=3DfVal UpdNavPane() } } function ToggleOtlText() { gOtlTxtExp=3D!gOtlTxtExp UpdOtlPane() } function ClearMedia() { // Clear any sounds playing before launching another browser window. Other= wise, // in fullscreen mode, you'll continue to hear the sound in the frames mod= e. if (PPTSld.pptSound) PPTSld.pptSound.loop =3D 0; } function FullScreen() {=20 if ( PPTSld.g_animUseRuntime ) PPTSld.document.body.pause(); ClearMedia(); var href =3D ( document.location.protocol =3D=3D 'mhtml:') ? FULLSCR_HREF = : FULLSCR_HREF+"#"+GetHrefObj(gCurSld).mSldHref; if (MHTMLPrefix !=3D "") href =3D RemoveFilePrefixFromHref(href) if(PPTNav.event.ctrlKey) { var w =3D (window.screen.availWidth * 1.0) / 2.0 var h =3D w * (PPTSld.g_origH * 1.0) / PPTSld.g_origW win =3D window.open( MHTMLPrefix+href,null,"toolbar=3D0,resizable=3D1,top= =3D0,left=3D0," + "width=3D"+ w + ",height=3D" + h ); if( win.document.body && PPTSld.g_animUseRuntime ) win.document.body.PPTSldFrameset=3Dwindow; } else { win =3D window.open( MHTMLPrefix+href,null,"fullscreen=3Dyes" ); if( win.document.body && PPTSld.g_animUseRuntime ) win.document.body.PPTSldFrameset=3Dwindow; } } function ToggleVNarration() { rObj=3DPPTSld.document.all("NSPlay") if( rObj && !PPTSld.g_animUseRuntime ) { if( (rObj.playState =3D=3D 1)||(rObj.playState =3D=3D 0) ) rObj.Play() else if( rObj.playState =3D=3D 2 ) rObj.Pause() else return; } else if( PPTSld.g_animUseRuntime ) { narObj =3D PPTSld.document.all("narrationID") if( narObj ) narObj.togglePause() } } function GetCurSldNum() { =20 obj=3DGetHrefObj(gCurSld) if( obj.mOrigVis =3D=3D 1 ) return obj.mSldIdx else =20 return gCurSld } function GetNumSlds() { =20 if( GetHrefObj(gCurSld).mOrigVis =3D=3D 1 ) return GetSldList().mNumVisSlds; else return GetSldList().mList.length } function GetSldNum( href ) { for(ii=3D0; ii 1 ) PopSldList(); else if( !IsFramesMode() ) { if( gLoopCont ) GoToFirst() else EndShow() } } function GoToPrevSld() { ii=3DgCurSld-1 if( ii > 0 ) { obj=3DGetHrefObj(ii) while ( obj && ( obj.mVis =3D=3D 0 ) && ( ii>0 ) ) obj=3DGetHrefObj(--ii) if( ii =3D=3D 0 ) ii=3D1 GoToSldNum(ii) } } function GoToFirst(){ GoToSld( GetHrefObj(1).mSldHref ) } function GoToLast() { ii=3DGetSldList().mList.length if( ii !=3D gCurSld ) GoToSld( GetHrefObj(ii).mSldHref ) } function GoToSldNum( num ) { if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue obj =3D GetHrefObj( num ) obj.mVis=3D1 gPrevSld=3DgCurSld gCurSld =3D num; =09 if (MHTMLPrefix !=3D "") PPTSld.location.replace(MHTMLPrefix+RemoveFilePrefixFromHref(obj.mSldHref= )) else PPTSld.location.replace(obj.mSldHref) =09 if( IsFramesMode() ) { UpdNavPane(); UpdOtlPane(); UpdNtsPane() } } function GoToSld( href ) { if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue GetHrefObj( GetSldNum(href) ).mVis=3D1 if (MHTMLPrefix !=3D "") PPTSld.location.replace(MHTMLPrefix+RemoveFilePrefixFromHref(href)) else PPTSld.location.replace(href) } function SldUpdated( id ) { if( id =3D=3D GetHrefObj(gCurSld).mSldHref ) return gPrevSld=3DgCurSld gCurSld=3DGetSldNum(id) if( IsFramesMode() ) { UpdNavPane(); UpdOtlPane(); UpdNtsPane() } } function PrevSldViewed(){ GoToSld( GetHrefObj(gPrevSld).mSldHref ) } function HasPrevSld() { return ( gIsEndShow || ( gCurSld !=3D 1 && GetHrefO= bj( gCurSld-1 ).mVis =3D=3D 1 )||( GetCurSldNum() > 1 ) ) } function HasNextSld() { return (GetCurSldNum() !=3D GetNumSlds()) } function CloseWindow() { if( HideMenu() ) return; =09 var event =3D PPTSld.event; if( !IsFramesMode() && event && (event.keyCode=3D=3D27 || event.keyCode=3D= =3D32 || event.type=3D=3D"click" ) ) window.close( self ); CatchNumKeys( self, event ); } function Unload() { gIsEndShow=3D0; } function SetupEndShow() { gIsEndShow=3D1; PPTSld.document.body.scroll=3D"no"; PPTSld.document.onkeypress=3DCloseWindow; PPTSld.document.onclick=3DCloseWindow; PPTSld.document.oncontextmenu=3D_CM; } function EndShow() { if( IsFramesMode() ) return if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue doc=3DPPTSld.document var dir =3D doc.body.dir if( dir !=3D "rtl" ) dir =3D "ltr"; doc.open() doc.writeln('


' + ENDSHOW_MESG + '

') doc.close() } function SetSldVisited(){ GetSldList().mList[gCurSld-1].mVisited=3Dtrue } function IsSldVisited(){ return GetSldList().mList[gCurSld-1].mVisited } function hrefList( sldHref, visible, advDelay, advClk ) { this.mSldHref=3D this.mNtsHref =3D sldHref this.mOrigVis=3D this.mVis =3D visible this.mVisited=3D false this.mAdvDelay=3D advDelay this.mAdvOnClk=3D advClk } function SldList(arr,curSld,fEnd) { this.mCurSld =3D curSld; this.mList =3D new Array(); var idx =3D 1; for(ii=3D0;ii 0) { PushSldList(sldList,fEnd); gCurSld =3D 1; } else if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue } function PushSldList(arr,fEnd) { var ii =3D gSldStack.length; gSldStack[ii] =3D new SldList(arr,gCurSld,fEnd); GoToSld( gSldStack[ii].mList[0].mSldHref ); } function PopSldList() { if (gSldStack[gSldStack.length-1].fEndShow) EndShow() else { gCurSld =3D gSldStack[gSldStack.length-1].mCurSld; gSldStack[gSldStack.length-1] =3D null; gSldStack.length--; var sldList =3D gSldStack[gSldStack.length-1]; GoToSld( sldList.mList[gCurSld - 1].mSldHref ); } } var custShowList=3Dnew Array(); /********************************************* Navigation button implementation There are 2 types of buttons: ImgBtn, TxtBtn implemented as function objects. They share a similiar interface so the event handlers can call SetActive, for example, on a button=20 object without needing to know exactly=20 what type of button it is. **********************************************/ //---------------------------------- function ImgBtn( oId,bId,w,action ) //---------------------------------- { var t=3Dthis t.Perform =3D _IBP t.SetActive =3D _IBSetA t.SetInactive=3D _IBSetI t.SetPressed =3D _IBSetP t.SetDisabled=3D _IBSetD t.Enabled =3D _IBSetE t.ChangeIcon =3D null t.UserAction =3D action t.ChgState =3D _IBUI t.mObjId =3D oId t.mBorderId=3D bId t.mWidth =3D w t.mIsOn =3D t.mCurState =3D 0 } function _IBSetA() { if( this.mIsOn ) { obj=3Dthis.ChgState( gHiliteClr,gShadowClr,2 ) obj.style.posTop=3D0 } } function _IBSetI() { if( this.mIsOn ) { obj=3Dthis.ChgState( gFaceClr,gFaceClr,1 ) obj.style.posTop=3D0=20 } } function _IBSetP() { if( this.mIsOn ) { obj=3Dthis.ChgState( gShadowClr,gHiliteClr,2 ) obj.style.posLeft+=3D1; obj.style.posTop+=3D1 } } function _IBSetD() { =20 obj=3Dthis.ChgState( gFaceClr,gFaceClr,0 ) obj.style.posTop=3D0=20 } function _IBSetE( state ) { var t=3Dthis GetObj( t.mBorderId ).style.visibility=3D"visible" if( state !=3D t.mIsOn ) { t.mIsOn=3Dstate if( state ) t.SetInactive() else t.SetDisabled() } } function _IBP() { var t=3Dthis if( t.mIsOn ) { if( t.UserAction !=3D null ) t.UserAction() if( t.ChangeIcon ) { obj=3DGetObj(t.mObjId) if( t.ChangeIcon() ) obj.style.posLeft=3Dobj.style.posLeft+(t.mCurState-4)*t.mWidth else obj.style.posLeft=3Dobj.style.posLeft+(t.mCurState-0)*t.mWidth } t.SetActive() } =20 } function _IBUI( clr1,clr2,nextState ) { var t=3Dthis SetBorder( GetObj( t.mBorderId ),clr1,clr2 ) obj=3DGetObj( t.mObjId ) obj.style.posLeft=3Dobj.style.posLeft+(t.mCurState-nextState)*t.mWidth-obj= .style.posTop t.mCurState=3DnextState return obj } //----------------------------------------- function TxtBtn( oId,oeId,action,chkState ) //----------------------------------------- { var t=3Dthis t.Perform =3D _TBP t.SetActive =3D _TBSetA t.SetInactive=3D _TBSetI t.SetPressed =3D _TBSetP t.SetDisabled=3D _TBSetD t.SetEnabled =3D _TBSetE t.GetState =3D chkState t.UserAction =3D action t.ChgState =3D _TBUI t.mObjId =3D oId t.m_elementsId=3D oeId t.mIsOn =3D 1 } function _TBSetA() { var t=3Dthis if( t.mIsOn && !t.GetState() ) t.ChgState( gHiliteClr,gShadowClr,0,0 ) } function _TBSetI() { var t=3Dthis if( t.mIsOn && !t.GetState() ) t.ChgState( gFaceClr,gFaceClr,0,0 ) } function _TBSetP() { if( this.mIsOn ) this.ChgState( gShadowClr,gHiliteClr,1,1 ) } function _TBSetD() { =20 this.ChgState( gFaceClr,gFaceClr,0,0 ) this.mIsOn =3D 0 } function _TBSetE() { var t=3Dthis if( !t.GetState() ) t.ChgState( gFaceClr,gFaceClr,0,0 ) else t.ChgState( gShadowClr,gHiliteClr,1,1 ) t.mIsOn =3D 1 } function _TBP() { var t=3Dthis if( t.mIsOn ) {=20 if( t.UserAction !=3D null ) t.UserAction() if( !t.GetState ) return if( t.GetState() ) t.SetPressed() else t.SetActive() } =20 } function _TBUI( clr1,clr2,lOffset,tOffset ) { SetBorder( GetObj( this.mObjId ),clr1,clr2 ) Offset( GetObj( this.m_elementsId ),lOffset,tOffset ) } function Offset( obj, top, left ){ obj.style.top=3Dtop; obj.style.left=3Dle= ft } function SetBorder( obj, upperLeft, lowerRight ) { s=3Dobj.style; s.borderStyle =3D "solid" s.borderWidth =3D 1=20 s.borderLeftColor =3D s.borderTopColor =3D upperLeft s.borderBottomColor=3D s.borderRightColor =3D lowerRight } function GetBtnObj(){ return gBtnArr[window.event.srcElement.id] } function BtnOnOver(){ b=3DGetBtnObj(); if( b !=3D null ) b.SetActive() } function BtnOnDown(){ b=3DGetBtnObj(); if( b !=3D null ) b.SetPressed() } function BtnOnOut(){ b=3DGetBtnObj(); if( b !=3D null ) b.SetInactive() } function BtnOnUp() { b=3DGetBtnObj() if( b !=3D null ) b.Perform() else Upd() } function GetNtsState(){ return parent.gNtsOpen } function GetOtlState(){ return parent.gOtlOpen } function GetOtlTxtState(){ return parent.gOtlTxtExp } function NtsBtnSetFlag( fVal ) { s=3Ddocument.all.item( this.m_flagId ).style s.display=3D"none" if( fVal ) s.display=3D"" else s.display=3D"none" } function _BSetA_Border(){ b =3D gBtnArr[this.mObjId]; if( b !=3D null ) b.S= etActive() } function _BSetI_Border(){ b =3D gBtnArr[this.mObjId]; if( b !=3D null ) b.S= etInactive() } function _BSetP_Border(){ b =3D gBtnArr[this.mObjId]; if( b !=3D null ) b.S= etPressed() } function _BSetA_BorderImg() {=20 b =3D gBtnArr[this.mBorderId]=20 if( b !=3D null && this.mIsOn && !b.GetState() ) { obj=3Dthis.ChgState( gHiliteClr,gShadowClr,2 ) obj.style.posTop=3D0 } } function _BSetI_BorderImg() {=20 b =3D gBtnArr[this.mBorderId] if( b !=3D null && this.mIsOn && !b.GetState() ) { obj=3Dthis.ChgState( gFaceClr,gFaceClr,1 ) obj.style.posTop=3D0 } } var gHiliteClr=3D"THREEDHIGHLIGHT",gShadowClr=3D"THREEDSHADOW",gFaceClr=3D"= THREEDFACE" var gBtnArr =3D new Array() gBtnArr["nb_otl"] =3D new TxtBtn( "nb_otl","nb_otlElem",parent.ToggleOtlPan= e,GetOtlState ) gBtnArr["nb_otlElem"] =3D new TxtBtn( "nb_otl","nb_otlElem",parent.ToggleOt= lPane,GetOtlState ) gBtnArr["nb_nts"] =3D new ImgBtn( "nb_nts","nb_ntsBorder",10,parent.ToggleN= tsPane ) gBtnArr["nb_nts"].SetActive =3D _BSetA_BorderImg; gBtnArr["nb_nts"].SetInactive =3D _BSetI_BorderImg; gBtnArr["nb_ntsBorder"] =3D new TxtBtn( "nb_ntsBorder","nb_ntsElem",parent.= ToggleNtsPane,GetNtsState ) gBtnArr["nb_ntsElem"] =3D new TxtBtn( "nb_ntsBorder","nb_ntsElem",parent.To= ggleNtsPane,GetNtsState ) gBtnArr["nb_prevBorder"] =3D gBtnArr["nb_prev"]=3D new ImgBtn( "nb_prev","n= b_prevBorder",30,parent.GoToPrevSld ) gBtnArr["nb_nextBorder"] =3D gBtnArr["nb_next"]=3D new ImgBtn( "nb_next","n= b_nextBorder",30,parent.GoToNextSld ) gBtnArr["nb_sldshw"]=3D new ImgBtn( "nb_sldshw","nb_sldshwBorder",18,parent= .FullScreen ) gBtnArr["nb_sldshwBorder"] =3D new TxtBtn( "nb_sldshw","nb_sldshwBorder",pa= rent.FullScreen,null ) gBtnArr["nb_sldshwBorder"].SetActive =3D _BSetA_Border; gBtnArr["nb_sldshwBorder"].SetInactive =3D _BSetI_Border; gBtnArr["nb_sldshwText"] =3D new TxtBtn( "nb_sldshw","nb_sldshwText",parent= .FullScreen,null ) gBtnArr["nb_sldshwText"].SetActive =3D _BSetA_Border; gBtnArr["nb_sldshwText"].SetInactive =3D _BSetI_Border; gBtnArr["nb_voice"] =3D gBtnArr["nb_voiceBorder"] =3D new ImgBtn( "nb_voice= ","nb_voiceBorder",18,parent.ToggleVNarration ) gBtnArr["nb_otlTxtBorder"] =3D gBtnArr["nb_otlTxt"]=3D new ImgBtn( "nb_otlT= xt","nb_otlTxtBorder",23,parent.ToggleOtlText ) gBtnArr["nb_ntsBorder"].m_flagId=3D "nb_nts" gBtnArr["nb_ntsBorder"].SetFlag =3D NtsBtnSetFlag gBtnArr["nb_otlTxt"].ChangeIcon=3D GetOtlTxtState /********************************************* Context menu implementation _CM() is the function that's hooked up to the oncontextmenu event. Once we're asked to show the menu, we first build it by creating DIVs on-the-fly. Then we position it=20 within the screen area so it doesn't get clipped. Creating the DIVs using createElement() means we don't have to write out any extra HTML into the slide HTML files. **********************************************/ var sNext=3D"次へ",sPrev=3D"前へ",sEnd=3D"スライド ショーの終了",sFont=3D"= lＳ 明朝",sArrow=3D"矢印",sFreeform=3D"フリーフォーム",sRect=3D"四角形",sOv= al=3D"楕円" function ShowMenu() { BuildMenu(); var doc=3DPPTSld.document.body,x=3DPPTSld.event.clientX+doc.scrollLeft,y= =3DPPTSld.event.clientY+doc.scrollTop m =3D PPTSld.document.all.item("ctxtmenu") m.style.pixelLeft=3Dx if( (x+m.scrollWidth > doc.clientWidth)&&(x-m.scrollWidth > 0) ) m.style.pixelLeft=3Dx-m.scrollWidth m.style.pixelTop=3Dy if( (y+m.scrollHeight > doc.clientHeight)&&(y-m.scrollHeight > 0) ) m.style.pixelTop=3Dy-m.scrollHeight m.style.display=3D"" } function _CM() { if( !parent.IsFullScrMode() ) return; if(!PPTSld.event.ctrlKey) { ShowMenu() return false } else HideMenu() } function BuildMenu() { if( PPTSld.document.all.item("ctxtmenu") ) return var mObj=3DCreateItem( PPTSld.document.body ) mObj.id=3D"ctxtmenu" mObj.style.visibility=3D"hidden" var s=3DmObj.style s.position=3D"absolute" s.cursor=3D"default" s.width=3D"120px" SetCMBorder(mObj,"menu","black") var iObj=3DCreateItem( mObj ) SetCMBorder( iObj, "threedhighlight","threedshadow" ) iObj.style.padding=3D2 CreateMenuItem( iObj,sNext,M_GoNextSld,M_True ) CreateMenuItem( iObj,sPrev,M_GoPrevSld,M_HasPrevSld ) =09 CreateSeparator( iObj ) CreateMenuItem( iObj,sEnd,M_End,M_True ) mObj.style.visibility=3D"visible" } function Cancel() { window.event.cancelBubble=3Dtrue; window.event.returnVa= lue=3Dfalse } function Highlight() { ChangeClr("activecaption","threedhighlight") } function Deselect() { ChangeClr("threedface","menutext") } function Perform() { e=3DPPTSld.event.srcElement if( e.type=3D=3D"menuitem" && e.IsActive() ) e.Action() else PPTSld.event.cancelBubble=3Dtrue } function ChangeClr( bg,clr ) { e=3DPPTSld.event.srcElement if( e.type=3D=3D"menuitem" && e.IsActive() ) { e.style.backgroundColor=3Dbg e.style.color=3Dclr } } function M_HasPrevSld() { return( parent.HasPrevSld() ) } function M_GoNextSld() { if( gIsEndShow ) M_End(); else GoToNextSld() } function M_GoPrevSld() { if( gIsEndShow ) { gIsEndShow=3D0; history.back();= PPTSld.event.cancelBubble=3Dtrue; } else GoToPrevSld() } function M_True() { return true } function M_End() { window.close( self ) } function CreateMenuItem( node,text,action,eval ) { var e=3DCreateItem( node ) e.type=3D"menuitem" e.Action=3Daction e.IsActive=3Deval e.innerHTML=3Dtext if( !e.IsActive() ) e.style.color=3D"threedshadow" e.onclick=3DPerform e.onmouseover=3DHighlight e.onmouseout=3DDeselect s=3De.style; 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charset="shift_jis" 意思決定科学 3．LP復習 ------=_NextPart_01C6E7D9.BE8B0860 Content-Location: file:///C:/72785C13/06dmt_3.files/outline.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="shift_jis"
 ノ= ート
スラ= Cド ショー
アウトライン
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