ᕷ༊㒆ศࢆ⪃៖ࡋࡓ㑅ᣲ༊⏬ᐃၥ㢟ࡢ᭱㐺ࣔࢹࣝ
ࠉ ࠉᇼ⏣ࠉᩗ
ࠉ ࠉA mathematical model with adjusting the population of
the divided city for the redistricting problem
ࠉ ࠉ
Keisuke Hotta
ࠉ ࠉAbstract
Let me consider the political redistricting problem for the single seat constituency system in the House of Representatives. In Japan, city is the smallest unit as one component of the electoral district. So basically it is prohibited to split the city for redictricting. However, in order to reduce the vote-value disparity it is approved as a special case. In fact, over 20 cities were divided for redrawing the electoral districts. Nemoto and Hotta[3, 4] treated this exception rule by proposing the pre-allocation method, which apportions the average population via a single seat of the prefecture as one district. In this research, I propose the mathematical model with adjusting the population of the divided city, and compare with the pre-allocation method.
1
ࡣࡌࡵ
⌧ᅾ㸪⾗㆟㝔㆟ဨࡣ㸪ᐃᩘ 480 ேࡢ࠺ࡕ 300 ேࢆᑠ㑅ᣲ༊ไࡢ㑅ᣲ࡛㑅ฟࡋ࡚࠸ࡿ㸬ࡇࡢ㑅 ᣲࢆᐇࡍࡿࡓࡵᚲせ࡞ᑠ㑅ᣲ༊ࡢ༊⏬ᐃࡣ㸪⾗㆟㝔㆟ဨ㑅ᣲ༊⏬ᐃᑂ㆟ࡼࡾ㸪ࡲࡎ 47 㒔㐨ᗓ┴㆟ဨᐃᩘ 300 ேࢆᐃᩘ㓄ศࡋ㸪ḟ㒔㐨ᗓ┴ẖ㑅ᣲ༊ࢆỴᐃࡋ࡚࠸ࡿ㸬ᅜໃㄪ ᰝేࡏ࡚ 10 ᖺ୍ᗘ㑅ᣲ༊ࡢぢ┤ࡋࡀ⾜ࢃࢀࡿࡀ㸪ࡇࡢᮇ࡞ࡿ࣓ࢹ➼ࡼࡾ㸪ࡑ ࡢⅬࡢ㑅ᣲ༊ྛ㑅ᣲ༊ேཱྀࡶ࡙࠸࡚ィ⟬ࡉࢀࡓࠕ୍⚊ࡢ㔜ࡳࡢ᱁ᕪࠖࡀ 2 ಸࢆ㉸࠼ ࡚࠸ࡿ≧ἣࡀᨺ⨨ࡉࢀ࡚࠸ࡿⅬࢆၥ㢟どࡉࢀࡿ㸬᱁ᕪࡀࡁࡃ࡞ࡿせᅉࢆㄪࡿࡓࡵ㸪ᐈほ ᛶࢆࡶࡗࡓᣦᶆࡋ࡚᭱㐺༊ࢆᑟฟࡋ㝈⏺᱁ᕪࢆ▱ࡿࡁ࡛࠶ࡿ᪨ࡢᥦゝࡀ࡞ࡉࢀ [1]㸪ⴭ⪅ ࡽࡢ㐣ཤࡢ◊✲ࡼࡾᐇ㝿ࡑࡢᑟฟࡀྍ⬟࡞ࡾ [3, 4]㸪ࡑࡢ⤖ᯝ㸪᱁ᕪ㛵ࡍࡿᵝࠎ࡞ศᯒ ࢆࡍࡿࡇࡀฟ᮶ࡓ [5, 6, 7, 8]㸬 ᑠ㑅ᣲ༊ࡢ༊ࢆ⾜࠺࠶ࡓࡗ࡚ࡣ㸪⾗㆟㝔㆟ဨ㑅ᣲ༊⏬ᐃᑂ㆟ࡀ༊సᡂ᪉㔪㸦cf.[2]㸧ࢆ ᥦ♧ࡋ࡚࠾ࡾ㸪ࡑࡢసᡂ᪉㔪ࡢ㦵Ꮚࡣ㸪 (1) ୍⚊ࡢ㔜ࡳࡢ᱁ᕪࡣ 2 ಸᮍ‶ࡀᇶᮏ㸪(2) ᕷ༊㒆ࡣཎ๎ศࡋ࡞࠸㸪 (3) 㑅ᣲ༊ෆ࡛㣕ࡧᆅࢆసࡽ࡞࠸㸪 (4) ᆅᇦࡢࡘ࡞ࡀࡾࢆ⪃៖ࡍࡿ㸪 ࡞ࡗ࡚࠸ࡿ㸬୍⚊ࡢ᱁ᕪṇࢆ⮳ୖ㢟ࡋ࡚࠸ࡿ᪥ᮏࡢሙྜ㸪(2)㹼(4) ࢆไ⣙ࡋ࡚ (1) ࢆ ᱁ᕪ᭱ᑠࡢ┠ⓗ㛵ᩘࡋࡓ᭱㐺ၥ㢟ᤊ࠼ࡿࡢࡀ⮬↛࡛࠶ࡿ㸬ࡇࢀࡀ㸪⾗㆟㝔ᑠ㑅ᣲ༊ไ࠾ ࡅࡿ༊⏬ᐃၥ㢟࡛࠶ࡿ㸬ࡇࡢၥ㢟ࡢཝᐦゎࢆồࡵ࡚᭱㐺༊ࢆᑟฟࡍࡿࡇࡀ㸪୍⚊ࡢ᱁ᕪ ࡢ㝈⏺ࢆ▱ࡿࡓࡵᚲせ࡞ᡭ㡰࡞ࡿ㸬ࡇ࠺ࡋ࡚ᑟฟࡋࡓ᪥ᮏᅜࡢ 300 ᑠ㑅ᣲ༊ࡢ࠺ࡕ㸪ே ཱྀࡀ᭱࡞ࡿ㑅ᣲ༊ࡢேཱྀ᭱ᑠ࡞ࡿ㑅ᣲ༊ࡢேཱྀࡢẚࢆ୍⚊ࡢ㝈⏺᱁ᕪࡼࡪ㸬 ᐇ㝿㸪ࡇࡢၥ㢟ࢆ㞟ྜศᆺ㸪ཬࡧࢢࣛࣇศᆺࣔࢹࣝࡋ㸪ཝᐦゎᑟฟࡢࡓࡵᗄࡘ ࡢᕤኵࢆࡍࡇ࡛ྛ㒔㐨ᗓ┴ࡢ᭱㐺༊ࢆᑟฟࡋ㸪㝈⏺᱁ᕪࢆồࡵ࡚࠸ࡿ [3, 4]㸬࠼ࡤ㸪㞟 ྜศᆺࡣ௨ୗࡢࡼ࠺ᐃᘧࡉࢀࡿ㸬 min . u/l s.t. l (1 xj) + qjxj (j; B) (1.1) qjxj u (j ; B) (1.2) P j;Bbijxj= 1 (i; N) (1.3) P j;Bxj = m (1.4) xj; {0, 1} (j; B) u, l 0. ࡇࡇ࡛㸪N :ᕷ༊㒆㞟ྜ㸪m:㑅ᣲ༊ᩘ㸪B:㑅ᣲ༊㞟ྜࡑࡢ⾲⌧⾜ิ [bij]㸪qj:ྛ㑅ᣲ༊ேཱྀࢆ ࡑࢀࡒࢀពࡋ㸪 :༑ศࡁ࡞ᩘ࡛࠶ࡿ㸬ࡲࡓ㸪u, l ࡣ㸦⤖ᯝⓗ㸧᭱ேཱྀ㑅ᣲ༊ࡢேཱྀ᭱ ᑠேཱྀ㑅ᣲ༊ࡢேཱྀࢆ♧ࡍࡇ࡞ࡿᐇᩘ್ኚᩘ㸪xjࡣ㑅ᣲ༊ೃ⿵ j ࢆ༊ࡋ࡚⏝࠸ࡿ ࠺ࢆ࠶ࡽࢃࡍ {0, 1}-ኚᩘ࡛࠶ࡿ㸬 ࡉ࡚㸪༊సᡂࡢ᪉㔪 (2) 㛵ࡋ࡚ࡣḟࡢࡼ࠺࡞እつᐃࡀᏑᅾࡍࡿࡓࡵ㸪ࡑࢀࢆ⪃៖ࡍࡿᚲ せࡀ࠶ࡿ㸬 (2a) 㐣ேཱྀᕷ༊2ࡀ࠶ࡿሙྜ㸪 (2b) 㐣ᑠேཱྀ㑅ᣲ༊3ࡢタ⨨ࢆ㑊ࡅࡿሙྜ ࡣᕷ༊㒆ࢆศࡍࡿࡇࡀチࡉࢀࡿ㸬༊⏬ᐃၥ㢟ࢆᢅ࠺ሙྜࡣ㸪እつᐃ࡛࠶ࡿࡇࡢᕷ༊㒆 ศࢆࡢࡼ࠺ᢅ࠺࠸࠺ၥ㢟ࡀᖖࡘ࠸࡚ࡲࢃࡿ㸬 ᰿ᮏ࣭ᇼ⏣ࡼࡿ᭱㐺༊ࢆᑟฟࡋࡓ◊✲࠾࠸࡚ࡣ㸪㐣࡞ࡿᕷ༊ࡸ㑅ᣲ༊ࡽ㸪ᙜヱ㒔 㐨ᗓ┴ࡢᖹᆒேཱྀ┦ᙜࡍࡿᆅᇦࢆྲྀࡾฟࡋ୍࡚㑅ᣲ༊ࡋ㸪ṧࡾࡢேཱྀ㞄᥋ࢢࣛࣇ㸪࠾ࡼ ࡧ 1 ࡘᑡ࡞࠸㑅ᣲ༊ᩘࡢၥ㢟ࢆ⏝࠸࡚ᨵࡵ࡚᭱㐺ࡍࡿ࠸࠺᪉ἲࢆࡗࡓ [3, 4]㸦௨ᚋ㸪ࡇࡢ ᪉ἲࢆ୍㑅ᣲ༊๓ᙜἲࡼࡪ㸧㸬ࡇࡢᡭἲࡢⅬࡣ㸪ேཱྀ࣭㑅ᣲ༊ᩘࡀᑠࡉࡃ࡞ࡿࡇࡼ ࡾィ⟬ࡀࡼࡾᐜ᫆࡞ࡿ࠸࠺Ⅼ㸪㐺⏝ࡢ⡆౽ࡉࡀ࠶ࡆࡽࢀࡿ㸬ࡋࡋ࡞ࡀࡽ㸪ࠕᙜヱ㒔㐨ᗓ 2ᅜࡲࡓࡣ㒔㐨ᗓ┴ࡢ 1 㑅ᣲ༊࠶ࡓࡾᖹᆒேཱྀࡢ 4/3 ಸࢆ㉸࠼ࡿேཱྀࢆᣢࡘᕷ༊ 3ᅜࡲࡓࡣ㒔㐨ᗓ┴ࡢ 1 㑅ᣲ༊࠶ࡓࡾᖹᆒேཱྀࡢ 2/3 ಸᮍ‶ࡢேཱྀࢆᣢࡘ㑅ᣲ༊ 42005 ᖺᗘᅜໃㄪᰝ㏿ሗ್ேཱྀᖹᡂࡢྜేࡀ୍ẁⴠࡋࡓẁ㝵 (2006 ᖺ 3 ᭶) ࡢᕷ༊㒆㞄᥋ࢢࣛࣇ
┴ࡢᖹᆒே୍ཱྀ࡛㑅ᣲ༊ࢆࡘࡃࡿࠖ࠸࠺࣮ࣝࣝࢆ᥇⏝ࡋࡓୖ࡛ࡢ᭱㐺༊ࢆồࡵ࡚࠸ࡿ࠸ ࠺Ⅼὀពࡀᚲせ࡛࠶ࡿ㸬ࡘࡲࡾ㸪ᖹᆒேཱྀࢆᣢࡘ୍ࡘࡢ㑅ᣲ༊ࢆồゎ๓సࡗ࡚ࡋࡲ࠺ࡓࡵ㸪 㑅ᣲ༊ࡢேཱྀࡋ࡚ᖹᆒࢆࡾᙜ࡚࡞࠸ሙྜ㸪᭱㐺ゎࡀᖜኚࢃࡾ㸪ࡼࡾᑠࡉ࠸᱁ᕪࢆᣢ ࡘ༊ࡀᏑᅾࡍࡿ࠸࠺ྍ⬟ᛶࢆྰᐃ࡛ࡁ࡞࠸㸬 ࡑࡇ࡛㸪ᮏ◊✲࡛ࡣࡑࡢⅬࢆ⪃៖ࡋ࡚㸪๓୍㑅ᣲ༊ࢆసࡿࡇ࡞ࡋ᭱㐺༊ࢆᑟฟࡍ ࡿ᪉ἲࢆᥦࡋ㸪2005 ᖺᗘேཱྀ 2006 ᖺᗘࡢ⾜ᨻ⏺ࢆ⏝࠸࡚᭱㐺༊ࢆᑟฟࡋ㸪୍㑅ᣲ༊๓ ᙜἲࡢ᱁ᕪࡢ㐪࠸ࡀチᐜ࡛ࡁ࡞࠸ࡁࡃ࡞ࡗ࡚ࡋࡲ࠺ࡢ࠺ࢆẚ㍑ࡋ㸪ᚑ᮶ἲࡢ ጇᙜᛶࢆ᳨ドࡍࡿ㸬
2
ᕷ༊㒆ศࢆ⪃៖ࡋࡓ᭱㐺ࣔࢹࣝ
2005 ᖺᗘࡢᅜໃㄪᰝ㏿ሗ್ேཱྀ 2006 ᖺᗘࡢ㞄᥋ࢢࣛࣇ4ࢆ⏝࠸࡚ᅜࡢ᭱㐺༊ࢆᑟฟࡍ ࡿ㸪24 ࡢᕷ༊㒆ࢆศࡍࡿࡇ࡞ࡿ [8]㸬ලయⓗࡣ㸪ᕷ༊㒆ศእつᐃ (2a) ヱᙜࡍ ࡿ㐣ேཱྀᕷ༊ࡀ 14㸦⾲ 2.1㸧㸪እつᐃ (2b) ࡼࡿࡶࡢࡀ 5 ᕷ㒆࡞ࡿ㸦⾲ 2.2㸧㸬 ⾲ 2.1 ࡢ㡯┠ྡࡢ࠺ࡕ㸪㆟ᖍᩘࡣ㸪2005 ᖺᗘேཱྀࡶ࡙࠸࡚ࠕ1 㸩᭱వἲࠖࢆ⏝࠸࡚㓄 ศࡋࡓ㆟ᖍᩘࢆពࡋ㑅ᣲ༊ᩘ➼ࡋ࠸㸬ୖ㝈㸪ୗ㝈್ࡣࡑࢀࡒࢀ b 㒔ᗓ┴ேཱྀ/㆟ᖍᩘ 4/3c, d 㒔ᗓ┴ேཱྀ/㆟ᖍᩘ 2/3e ࡛࠶ࡿ㸬 ⾲ 2.1: 㐣ேཱྀࡢࡓࡵศᑐ㇟࡞ࡿᕷ༊ࡑࡢேཱྀ 㒔ᗓ┴ 㒔ᗓ┴ேཱྀ ㆟ᖍᩘ ୖ㝈 ᖹᆒ ୗ㝈 㐣ேཱྀᕷ༊ ᕷ༊ேཱྀ ༓ⴥ 6,056,159 13 621,144 465,859 310,573 ⯪ᶫᕷ 569,829 ᮾி 12,570,904 26 644,661 483,496 322,331 ⏣༊ 665,370 ୡ⏣㇂༊ 841,399 ⦎㤿༊ 692,225 ㊊❧༊ 624,548 Ụᡞᕝ༊ 653,882 ⚄ዉᕝ 8,790,900 18 651,177 488,383 325,589 ┦ᶍཎᕷ 1 628,638 ᪂₲ 2,431,396 6 540,310 405,233 270,156 ᪂₲ᕷ 813,780 㟼ᒸ 3,792,457 9 561,845 421,384 280,923 ᯇᕷ 804,067 㜰 8,817,010 18 653,111 489,834 326,556 ሜᕷ 831,111 ᒸᒣ 1,957,056 5 521,881 391,411 260,941 ᒸᒣᕷ 674,605 ឡ 1,467,824 4 489,274 366,956 244,638 ᯇᒣᕷ 514,944 ⇃ᮏ 1,842,140 5 491,237 368,428 245,619 ⇃ᮏᕷ 669,541 㮵ඣᓥ 1,753,144 4 584,381 438,286 292,191 㮵ඣᓥᕷ 1 598,339 ᅜ 127,756,815 300 567,808 425,856 283,905⾲ 2.2: 㐣ᑠேཱྀ㑅ᣲ༊ࡢタ⨨ࢆ㑊ࡅࡿࡓࡵࡢศᑐ㇟ᕷ㒆ࡑࡢேཱྀ ᗓ┴ ᗓ┴ேཱྀ ㆟ᖍᩘ ୖ㝈 ᖹᆒ ୗ㝈 ᑐ㇟ᕷ㒆 ᕷ㒆ேཱྀ ⚟ 821,589 3 365,150 273,863 182,576 ⚟ᕷ 269,147 㜰 8,817,010 18 653,111 489,834 326,556 ᓊ⏣ᕷ 200,984 㧗▱ 796,211 3 353,871 265,404 176,936 㧗▱ᕷ 333,407 ࿃ᕝ㒆 49,916 Ἀ⦖ 1,360,830 4 453,610 340,208 226,805 㑣ぞᕷ 312,308 ศᑐ㇟࡞ࡿ㐣ேཱྀᕷ༊ࡢᇶ‽࡛࠶ࡿୖ㝈್ࡣ㸪ࠕᙜヱ㒔㐨ᗓ┴ࡢᖹᆒேཱྀࡢ 4/3 ಸࠖ ࠕᅜࡢᖹᆒேཱྀࡢ 4/3 ಸࠖࡢᑠࡉ࠸᪉ࡢ್࡛࠶ࡿࡇὀពࡉࢀࡓ࠸㸬࠼ࡤ㸪⯪ᶫᕷ㸦569, 829 ே㸧ࡣ༓ⴥ┴ࡢୖ㝈್㸦621, 144 ே㸧ࢆ㉸࠼࡚࠸࡞࠸ࡀ㸪ᅜࡢୖ㝈್㸦567, 808 ே㸧ࢆ㉸࠼࡚ ࠸ࡿࡓࡵศᑐ㇟࡞ࡿ㸦⾲ 2.1㸧㸬ᮾி㒔㊊❧༊㸦624, 548 ே㸧ࡶྠᵝ࡛࠶ࡿ㸬 㐣ᑠேཱྀ㑅ᣲ༊ࡢタ⨨ࢆ㑊ࡅࡿࡓࡵࡢᕷ㒆ศ࠾࠸࡚ࡣ㸪ᑐ㇟࡞ࡿᕷ㒆ࡣ⫤ࡁࡢ≧ែ㸪 ༶ࡕ㸪ศࡋࡓࡑࢀࡒࢀࡢ㒊ศࡀูࡢᕷ㒆㑅ᣲ༊ࢆᵓᡂࡏࡊࡿࢆᚓ࡞࠸ࡇὀពࡉࢀࡓ࠸㸬 ᐇ㝿㸪㧗▱ᕷ௨እࡢ 4 ࡘࡢᑐ㇟ᕷ㒆࡛ࡣࡑࡢࡼ࠺ศࡉࢀ࡚㑅ᣲ༊ࢆᵓᡂࡋ࡚࠸ࡿ [8]㸬 ṧࡾࡢ 5 ᕷࡣ 2/3 ಸࡽ 4/3 ಸࡢ⠊ᅖෆ࡛ࡣᙜヱ㒔㐨ᗓ┴ෆ࡛ᐇ⾜ྍ⬟࡞༊ࡀᏑᅾࡋ࡞࠸ ࡓࡵᚲせࡉࢀࡿศ࡛࠶ࡿ㸦⾲ 2.3㸧㸬ၥ㢟ࡢᛶ㉁ࡽ㸪ேཱྀࡢୖୗ㝈ไ⣙ࢆタࡅ࡞ࡅࢀࡤ ᐇ⾜ྍ⬟ᛶࡣ⮬᫂࡞ࡢ࡛㸪ᐇ㉁㸪᪉㔪 (2b) ヱᙜࡍࡿศ࡛࠶ࡿ⪃࠼࡚ࡶⰋ࠸㸬ᚲせᛂࡌ ࡚ (2c) 㐣ேཱྀ㑅ᣲ༊ࡢタ⨨ࢆ㑊ࡅࡿሙྜ ࡣศࢆチࡍ࠸࠺つᐃࢆຍ࠼࡚㸪እつᐃ (2c) ヱᙜࡍࡿศ࡛࠶ࡿࡍࡿࡇࡶ⪃࠼ࡽ ࢀࡿ㸬 ⾲ 2.3: ᙜヱ㒔㐨ᗓ┴ᐇ⾜ྍ⬟༊ࡀᏑᅾࡋ࡞࠸ࡓࡵศࡀᚲせ࡞ᕷࡑࡢேཱྀ 㒔㐨ᗓ┴ ேཱྀ ㆟ᖍᩘ ୖ㝈 ᖹᆒ ୗ㝈 ᑐ㇟ᕷ ᕷேཱྀ ༓ⴥ 6,056,159 13 621,144 465,859 310,573 ᕷᕝᕷ 466,408 ᯽ᕷ 381,016 ᐩᒣ 1,111,602 3 494,045 370,534 247,023 ᐩᒣᕷ 421,156 㤶ᕝ 1,012,261 3 449,893 337,420 224,947 㧗ᯇᕷ 417,986 బ㈡ 866,402 3 385,067 288,801 192,534 బ㈡ᕷ 206,973 ୍㑅ᣲ༊๓ᙜἲ࡛ࡣ㸪ᑐ㇟࡞ࡿ㐣ேཱྀᕷ༊ࡽᙜヱ㒔㐨ᗓ┴ࡢᖹᆒேཱྀศࢆྲྀࡾฟ ࡋ୍㑅ᣲ༊ࡋ࡚ᵓᡂࡋ㸪ṧࡾࡢேཱྀࢆࡑࡢᕷ༊ࡢேཱྀࡋ࡚᭱㐺ࡋࡓࡢ࡛࠶ࡿࡀ㸪ᮏ◊✲ ࡛ࡣ㸪๓ேཱྀࢆࡾࡗ࡚㑅ᣲ༊ࢆసࡿࡇࢆࡏࡎ㸪ḟࡢࡼ࠺ၥ㢟ࢆᩘ⌮ࣔࢹࣝࡍࡿ
ࡇ࡛᭱㐺ゎࢆᑟฟࡍࡿ㸬 㞄᥋ࢢࣛࣇࢆ G = (V, E) ࡍࡿ㸬ࡇࡇ࡛㸪V ࡣᕷ༊㒆㞟ྜࢆ⾲ࡋ㸪E ࡣ㞄᥋ࡋ࡚࠸ࡿᕷ༊㒆 㛫ࡢᯞ㞟ྜ࡛࠶ࡿ㸬ศᑐ㇟࡞ࡿ㐣ᕷ༊ࢆ k(; V ) ࡋࡑࡢேཱྀࢆ pkࡍࡿ㸬㞄᥋ࢢࣛࣇ G ᪂ࡋ࠸ࣀ࣮ࢻࡋ࡚ࢲ࣑࣮ᕷ༊ dumkࢆຍ࠼㸪㐣ᕷ༊ k ࡢ㛫ࡢࡳᯞ ekࢆᙇࡿ㸦ᅗ 2.1㸧㸬ᣑ㞄᥋ࢢࣛࣇࡣ ˜G = (V {dumk}, E {ek}) ࡞ࡿ㸬ศᑐ㇟ᕷ༊㛫ࡢேཱྀㄪᩚࢆࡍ ᅗ 2.1: 㐣ᕷ༊ k ࡢศேཱྀㄪᩚ ࡿࡓࡵࡢ㠀㈇ᐇᩘኚᩘ k(; [0, pk 1]) ࢆᑟධࡋ㸪ࢲ࣑࣮ᕷ༊ dumkࡢேཱྀࢆ pk 1 k㸪ࡶ ࡢᕷ༊ k ࡢேཱྀࢆ 1 + kኚ᭦ࡍࡿ㸬2 ࡘࡢࡣࡶࡢேཱྀ pk୍⮴ࡍࡿࡇ㸪ཬࡧ㸪௬ k= 0 ࠸࠺ゎࡀᚓࡽࢀ࡚ࡶ㸪k ࡢேཱྀ 1 ேࡀಖドࡉࢀ㸪ศࡀ☜ᐇᐇ⾜ࡉࢀࡿࡇὀ ពࡉࢀࡓ࠸㸬ࡇࡢࡁ㸪᭱⤊ⓗᚓࡽࢀࡿ༊࡛ࡣ㸪ࢲ࣑࣮ᕷ༊ dumkࡀேཱྀ pk 1 kࢆ ᣢࡘ୍ࡘࡢ㑅ᣲ༊ࢆᵓᡂࡍࡿࡇ࡞ࡿ㸬 ḟ㸪㐣ᕷ༊ k ࢆྵࡴ㑅ᣲ༊㞟ྜࢆ B+㸪ྵࡲ࡞࠸㑅ᣲ༊㞟ྜࢆ Bࡍࡿ㸬ࡓࡔࡋ㸪dum k ࢆྵࡴ㑅ᣲ༊ࡣ㝖࠸࡚࠾ࡃ5㸬ࡲࡓ㸪ࡑࢀࡒࢀࡢ㞟ྜᑐᛂࡍࡿ⾲⌧⾜ิࢆ [b+ ij], [bij]㸪ேཱྀࢆ qj+, qj ࡋ㸪ࡑࡢ㑅ᣲ༊ࢆ༊ࡋ࡚᥇⏝ࡍࡿྰࢆỴᐃࡍࡿ {0, 1}-ኚᩘࢆ x+j, xj ࡍࡿ㸬 ௨ୖࡢ‽ഛࡢࡶ࡛㸪ศᑐ㇟ᕷ༊㒆ࡢேཱྀㄪᩚࢆຍࡍࡿ㞟ྜศᆺࣔࢹࣝࢆ௨ୗࡢࡼ࠺ ᐃᘧࡍࡿ㸬 5ࢲ࣑࣮ᕷ༊ dum kࡢேཱྀࡀୖ㝈 (b ┴ᖹᆒ × 4/3c) ࡼࡾࡁ࠸ሙྜࡣ dumkࢆྵࡴ㑅ᣲ༊ࡣ dumk༢⊂ᵓᡂࡢ㑅 ᣲ༊ࡢࡳ࡞ࡿࡀ㸪ࢲ࣑࣮ᕷ༊ dumkࡢேཱྀࡀୖ㝈ᮍ‶ࡢሙྜࡣ㸪ࣀ࣮ࢻ k ࠸ࡃࡘࡢࣀ࣮ࢻ࡛㑅ᣲ༊ࢆᵓᡂࡍࡿ ೃ⿵࡞ࡾ࠺ࡿࡢ࡛㸪ࡑࢀࢆ㞟ྜ B+, Bࡢ࠸ࡎࢀࡶྵࡵ࡞࠸ࡼ࠺㝖ࡋ࡚࠾ࡃ࠸࠺ࡇ
min . u/l s.t. l pk 1 k u (2.0) l (1 x+j) + qj+x+j + k (j; B+) (2.1a) q+jx+j + k u (j ; B+) (2.2a) l (1 xj ) + qjxj (j; B) (2.1b) qjxj u (j; B) (2.2b) X j;B+ b+ijx+j + X j;B bijxj = 1 (i; N) (2.3) X j;B+ x+j = 1, X j;B xj = m 2 (2.4) x+j ; {0, 1} (j ; B+), xj ; {0, 1} (j ; B+) u, l, k 0. ไ⣙ (2.0) ࡣ㸪ࢲ࣑࣮ᕷ༊ࡢࡳ୍࡛㑅ᣲ༊ࢆᵓᡂࡍࡿࡼ࠺㸪ேཱྀࢆ k࡛ୖୗ㝈ෆㄪᩚࡍࡿ ࡶࡢ㸬(2.1a),(2.1b) ࡣࡶ࡞ࡿ㞟ྜศᆺࡢไ⣙ (1.1) ᑐᛂࡋ࡚࠾ࡾ㸪ྠᵝ (2.2a),(2.2b) ࡣ (1.2) ᑐᛂࡍࡿ➼ᘧไ⣙࡛࠶ࡿ㸬ࡑࢀࡒࢀ㸪㑅ᣲ༊㞟ྜ B+, Bᇶ࡙࠸ࡓேཱྀୖୗ㝈ࢆ ⾲⌧ࡋ࡚࠸ࡿ㸬ไ⣙ (2.3) ࡣไ⣙ (1.3) ᑐᛂࡋ㸪௵ពࡢᕷ༊㒆 i ࡣ୍ࡘࡢ㑅ᣲ༊ࡢࡳྵࡲࢀ ࡿࡇࢆせㄳࡍࡿࡶࡢ㸬ไ⣙ (2.4) ࡣไ⣙ (1.4) ᑐᛂࡋ㸪ศࡉࢀࡓ㐣ᕷ༊ k ࢆྵࡴ㑅ᣲ༊ ࢆ୍ࡘ㸪ྵࡲ࡞࠸㑅ᣲ༊ࢆ m 2 ಶ㑅ࡪࡇࢆせồࡍࡿࡶࡢ࡛࠶ࡿ㸬(2.0)㹼(2.2) ࡛ m ಶࡢ㑅 ᣲ༊ࡢேཱྀୖୗ㝈ࢆつᐃࡋ㸪(2.3),(2.4) ࡛ศࢆᵓᡂࡍࡿ㸬 ࡇࡢ {0, 1}-ᩚᩘィ⏬ၥ㢟ࢆゎ࠸࡚ཝᐦゎࢆồࡵࡿࡇ࡛㸪ศࡋࡓᕷ༊ࡢேཱྀ㓄ศࢆ⪃៖ࡋ ࡓ᭱㐺༊ࢆồࡵࡿࡇࡀฟ᮶ࡿ㸬࡞࠾㸪ࡇࡢࣔࢹࣝࡣᑐ㇟ᕷ༊ࡀᙜヱ㒔㐨ᗓ┴ෆ 1 ࡘࡔࡅ Ꮡᅾࡍࡿሙྜࡢᐃᘧ࡛࠶ࡿࡀ㸪2 ࡘ௨ୖ࠶ࡿሙྜࡶ㸪↹㞧࡛ࡣ࠶ࡿࡀไ⣙ࢆྠᵝቑࡸࡍࡇ ࡛ゎࡃࡇࡀฟ᮶ࡿ6㸬ࡓࡔࡋ㸪࠼ࡤ㐣ேཱྀᕷ༊ࡀ k, l ࡢ 2 ᕷ༊࡛㸪ࡑࢀࡒࢀ k 1, k2, l1, l2 ศࡉࢀ㸪k1, l1࡛ྛࠎ୍㑅ᣲ༊ࢆᵓᡂࡍࡿࡇࢆᐃࡍࡿሙྜ㸪ṧࡾࡢ k2 l2ࡀ୍ࡘࡢ㑅 ᣲ༊ྵࡲࢀࡿࡇࢆチᐜࡍࡿ࠺࡛㸪ไ⣙➼ᘧࡢᐃᘧ㐪࠸ࡀฟࡿࡇὀពࡉࢀ ࡓ࠸㸬ࡲࡓ㸪ศࡍࡁᆅᇦࡀ」ᩘᕷ༊㒆࡛ᵓᡂࡉࢀ࡚࠸ࡿ㑅ᣲ༊ࢆྵࡴ㒔㐨ᗓ┴ࡢၥ㢟ࢆゎ ࡃሙྜ㸪ᙜヱ㑅ᣲ༊ࢆ୍ࡘࡢᕷࡳ࡞ࡋ࡚ࡇࡢࣔࢹࣝࢆ㐺⏝ࡍࡿࡇࡼࡾྠᵝࡢᐃᘧࢆ⏝ ࠸ࡿࡇࡀྍ⬟࡛࠶ࡿ㸬 ࡉ࡚㸪ࡇࡢᡭἲࢆ⏝࠸ࡿ㝿ὀពࡍࡁⅬࡀ୍ࡘ࠶ࡿ㸬୍㑅ᣲ༊๓ᙜἲ࡛㞟ྜศᆺࡢ ࣔࢹࣝࢆゎࡃሙྜࡣ㸪⪃៖ࡍࡁ㑅ᣲ༊ࢆิᣲࡋ࡚ࡽࢯࣝࣂ࣮ࢆ⏝ࡋ࡚ᩚᩘィ⏬ၥ㢟ࢆ ゎ࠸࡚࠸ࡿ [3, 4]㸬ࡑࡢࡓࡵ㸪࡚ࡢ㑅ᣲ༊ࢆิᣲࡋ࡚ࡋࡲ࠺ࡑࡢᩘࡣ⭾࡞ࡾ㸪⌧ᐇⓗ ࡣゎࡃࡇࡀฟ᮶࡞࠸㸬ᚑࡗ࡚㸪༊ࢆᵓᡂࡍࡿᐇ⾜ྍ⬟ゎࢆቯࡉ࡞࠸ࡼ࠺ิᣲࡍࡿ㑅ᣲ༊ 㞟ྜࢆไ㝈ࡍࡿࡇ࡛ࡑࡢᩘࢆῶࡽࡍᕤኵࢆࡋ࡚࠸ࡿ㸬ࡑࡇ࡛⏝࠸ࡓᕤኵࡣ㸪➨ 1 ጇᙜ㑅ᣲ༊ 62005 ᖺேཱྀ 2006 ᖺ㞄᥋ࢢࣛࣇࢆ⏝࠸㸪1+᭱వἲ࡛㆟ဨᐃᩘ㓄ศࡋࡓሙྜ㸪ศᑐ㇟ᕷ༊㒆ࢆ」ᩘᣢࡘ㒔 ᗓ┴ࡣ㸪ᮾி㒔㸦5 ༊㸧㸪༓ⴥ┴㸦3 ᕷ㸧㸪㜰ᗓ㸦2 ᕷ㸧㸪㧗▱┴㸦1 ᕷ 1 㒆㸧ࡢ 3 ᗓ┴㸦⾲ 2.1㹼2.3 ཧ↷㸧㸬
㞟ྜ㸪➨ 2 ጇᙜ㑅ᣲ༊㞟ྜࢆᐃ⩏ࡋ㸪ࡑࢀࡽࢆ㡰ḟồࡵ㸪➨ 2 ጇᙜ㑅ᣲ༊ࢆิᣲࡍࡁ㑅ᣲ༊ ࡋ࡚ゎࢆồࡵࡿ࠸࠺ࡶࡢ࡛࠶ࡿ㸬 ➨ 1 ጇᙜ㑅ᣲ༊㞟ྜࡣ㸪ேཱྀࡢୖୗ㝈್ࡲࡿேཱྀࢆᣢࡘ㑅ᣲ༊ࡢ㞟ྜ࡛࠶ࡾ㸪➨ 2 ጇ ᙜ㑅ᣲ༊㞟ྜࡣ㸪ࡑࡢ㑅ᣲ༊ࢆ࠺ࡇࡼࡗ࡚ᙜヱ㒔㐨ᗓ┴ࡢ㞄᥋ࢢࣛࣇࡀᗄࡘࡢ㐃⤖ ᡂศศࡉࢀࡿሙྜ㸪ྛ㐃⤖ᡂศࡢேཱྀࡀୖୗ㝈್ࡲࡗ࡚࠸ࡿ࠺ࢆุᐃࡋ㸪ࡲࡗ ࡚࠸ࡿࡶࡢࡢࡳࢆ᭷ຠ࡞㑅ᣲ༊ࡋ࡚せ⣲ࡋࡓ㞟ྜ࡛࠶ࡿ㸬 ᐃ⩏ ேཱྀୖୗ㝈್ ¯u, ¯lᑐࡋ㸪 (1) 㑅ᣲ༊ j㸦ேཱྀ pj㸧ࡀ➨ 1 ጇᙜ㑅ᣲ༊ ¯l pj ¯u. (2) 㑅ᣲ༊ j㸦ேཱྀ pj㸧ࡀ➨ 2 ጇᙜ㑅ᣲ༊ j0, i ; {1, · · · , m 1}, ¯l pj0 i ¯u. ࡇࡇ࡛㸪j0ࡣ j ࢆ㝖ࡃࡇ࡛ㄏᑟࡉࢀࡿ㞄᥋ࢢࣛࣇ G ࡢ㐃⤖ᡂศ ᮏ◊✲ࡢࣔࢹࣝ࠸࡚ࡣ㸪ࡑࡢᐃᘧࡢ᪉ࡽ㸪⪃៖ࡍࡁ㑅ᣲ༊㞟ྜࡀ୍㑅ᣲ༊๓ ᙜἲࡼࡾࡶቑ࠼ࡿࡓࡵ㸪➨ 1࣭➨ 2 ጇᙜ㑅ᣲ༊㞟ྜࡢせ⣲ᩘࡶࡑࢀకࡗ࡚ቑຍࡍࡿࡇࡀண ࡉࢀࡿ㸬ࡑࡢ⤖ᯝ㸪⪃៖ࡍࡁ㑅ᣲ༊㞟ྜࡢせ⣲ᩘࡀ㸪ồゎᨭ㞀ࡀ⏕ࡌࡿ⛬ࡁࡃ࡞ࡿᜍࢀ ࡀ࠶ࡿ㸬ᮏ◊✲ࡢࣔࢹࣝ࠸࡚⪃៖ࡍࡁ➨ 2 ጇᙜ㑅ᣲ༊㞟ྜࡢせ⣲ᩘࡀࡢ⛬ᗘቑ࠼ࡿ ࢆ♧ࡋࡓࡢࡀḟ⾲ 2.4 ࡛࠶ࡿ㸬ྛ㒔ᗓ┴ࡢୖẁࡀ୍㑅ᣲ༊๓ᙜἲ࠾ࡅࡿྛ㞟ྜࡢせ⣲ᩘ㸪 ୗẁࡀᮏ◊✲ࡢࣔࢹࣝᑐࡍࡿྛ㞟ྜࡢせ⣲ᩘࢆ♧ࡍ㸬 ࡞࠾㸪୍㑅ᣲ༊๓ᙜἲ࡛ࡣ㸪๓సᡂࡋࡓᖹᆒேཱྀࢆࡶࡘ୍㑅ᣲ༊ࢆồゎࡢࣔࢹࣝ ࡽ㝖࠸࡚࠸ࡿࡓࡵ㸪ィ⟬ࡢ㝿⏝࠸ࡿ㆟ᖍᩘࡣ⾲ 2.4 ࡢᩘ್ࡼࡾ๓ࡾᙜ࡚ࡋࡓ㑅ᣲ༊ᩘศᑠ ࡉ࠸ࡇὀពࡉࢀࡓ࠸㸬ࡲࡓ㸪ᮏ◊✲ࡢ᪉ἲ࡛ࡣ㸪㑅ᣲ༊ิᣲࡢ㝿ࢲ࣑࣮ᕷ༊ dumkࢆ㝖࠸ ࡚ k ࡢேཱྀࢆ౽ᐅୖ 1 ࡋ࡚ィ⟬ࡋ࡚࠸ࡿࡓࡵ㸪㞟ྜࡢせ⣲ᩘࡀቑࡋ࡚࠸ࡿ㸬ྠࡌၥ㢟࡛ẚ ㍑ࡋࡓሙྜ㸪ୖグࡢⅬࡽ㸪୍㑅ᣲ༊๓ᙜἲࡀᮏ◊✲ࡢ᪉ἲࡼࡾࡶồゎࡀࡼࡾᐜ᫆࡞ࡿ㸬 ࡇࡢ⾲ࡼࡾ㸪➨ 2 ጇᙜ㑅ᣲ༊㞟ྜࡢせ⣲ᩘࡀ 3 ಸ⛬ᗘቑ࠼࡚࠸ࡿ㒔㐨ᗓ┴ࡶ࠶ࡿࡀ㸪ᒸᒣ┴ ࢆ㝖ࡁ༑ศồゎ࡛ࡁࡿ⠊ᅖ࡛࠶ࡿ7㸬 ࡲࡓ㸪➨ 2 ጇᙜ㑅ᣲ༊㞟ྜ࠸࠺ࢹࡀ㸪༊⏬ᐃၥ㢟ᑐࡋ࡚㠀ᖖ᭷ຠ࡛࠶ࡿࡇ ࡀᨵࡵ࡚☜ㄆࡉࢀࡓ㸬 7ᒸᒣ┴ࡣ㸪㸦⌧ᅾᚓࡽࢀ࡚࠸ࡿ༊ࡸィ⟬㏵ୖᚓࡽࢀࡿᬻᐃゎ࡞ࡢ㸧ᐇ⾜ྍ⬟ゎࢆ୍ࡘ⏝ࡍࡿࡇ࡛㸪᭱㐺ᛶ ࢆቯࡉ࡞࠸ࡼ࠺ୖୗ㝈್ࡢ⠊ᅖࢆ⊃ࡵࡿ࡞ࡢᕤኵࡼࡾ㸪ᚲせ࡞ิᣲᩘࢆᢚ࠼࡚ồゎྍ⬟࡛࠶ࡿ㸬
⾲ 2.4: ➨ 2 ጇᙜ㑅ᣲ༊㞟ྜࡢせ⣲ᩘࡢẚ㍑ ᗓ┴ ᗓ┴ேཱྀ ᕷ༊㒆ᩘ ㆟ᖍᩘ 㞟ྜ㸦2ᕷ༊㒆ᩘ 1㸧 ➨1ጇᙜ ➨2ጇᙜ ⚄ዉᕝ 8,790,900 51 18 2,251,799,813,685,250 2,651 1,021 4,503,599,627,370,490 4,946 1,580 ᪂₲ 2,431,396 31 6 2,147,483,647 37,228 4,759 4,294,967,295 230,515 18,136 㟼ᒸ 3,792,457 38 9 274,877,906,943 4,082 1,445 549,755,813,887 13,110 1,925 㜰 8,817,010 63 18 9,223,372,036,854,780,000 11,186 9,499 18,446,744,073,709,600,000 27,336 21,052 ᒸᒣ 1,957,056 27 5 134,217,727 474,702 15,293 268,435,455 3,560,000 62,801 ឡ 1,467,824 18 4 262,143 3,024 86 524,287 5,209 117 ⇃ᮏ 1,842,140 27 5 134,217,727 12,820 701 268,435,455 146,756 2,246 㮵ඣᓥ 1,753,144 34 4 17,179,869,183 4,262,860 165 34,359,738,367 51,320,176 174 ᐩᒣ 1,111,602 12 3 4,095 355 22 8,191 539 22 㤶ᕝ 1,012,261 15 3 32,767 150 25 32,767 3,447 50
3
ィ⟬⤖ᯝẚ㍑
ᑐ㇟࡞ࡿ㒔㐨ᗓ┴ࡢ࠺ࡕ㸪㐣ᕷ༊ࡀ 1 ࡘ㸪༶ࡕ 1 ㆟ᖍࡢࡳࢆ๓ᙜࡍࡿ 10 ᗓ┴ࡘ ࠸࡚㸪๓⠇ࡢࣔࢹࣝࢆ㐺⏝ࡋ࡚⤖ᯝࢆᚓࡓ㸬ィ⟬ࡣࢯࣝࣂ࣮ࡋ࡚ CPLEX 11.1 ࢆ⏝ࡋ㸪 CPU:Pentium4(2.8GHz)㸪Memory 1GB ࡢ PC ࢆ⏝࠸ࡓ㸬⾲ 3.1 㸪୍㑅ᣲ༊๓ᙜἲ࡛ᚓ ࡽࢀࡿ᭱㐺ゎᮏ◊✲࡛ᚓࡽࢀࡓ᭱㐺ゎࡢẚ㍑ࢆ♧ࡍ㸬⾲୰ࡢࠕ᭱ࠖࠕ᭱ᑠࠖࡣࡑࢀࡒࢀ᭱㐺 ༊࠾ࡅࡿேཱྀ᭱ࡢ㑅ᣲ༊ࡢேཱྀேཱྀ᭱ᑠࡢ㑅ᣲ༊ࡢேཱྀࢆ࠶ࡽࢃࡋ㸪ࠕ᱁ᕪࠖࡣࡑࡢẚ ࢆ♧ࡍ㸬ࡲࡓ㸪ྛ㒔㐨ᗓ┴ࡢୖẁࡀ୍㑅ᣲ༊๓ᙜἲࡼࡿ᭱㐺ゎ㸪ୗẁࡀᮏ◊✲࡛ồࡵࡓ᭱ 㐺ゎ࡛࠶ࡿ㸬ᗓ┴ẖࡢ㑅ᣲ༊ヲ⣽ࡢẚ㍑ࡣ⾲ 3.2㹼3.11 ♧ࡍ㸬 ⾲ 3.1 ࡽ㸪⚄ዉᕝ┴㸪᪂₲┴㸪㜰ᗓࡢ 3 ᗓ┴࡛ࡣ㸪᭱ேཱྀ᭱ᑠேཱྀኚࡣ࡞ࡃ㸪⤖ ᯝࡋ࡚ྠࡌ᱁ᕪࡀᚓࡽࢀࡓ㸬ࡓࡔࡋ㸪⚄ዉᕝ┴㜰ᗓࡣୖୗ㝈┦ᙜࡍࡿ㑅ᣲ༊ࡀࡓࡲࡓ ࡲྠࡌ࡛࠶ࡗࡓゝ࠺ࡔࡅ࡛㸪ேཱྀࡀࡑࡢ⠊ᅖෆྵࡲࢀࡿ㑅ᣲ༊ࡢᵓᡂࡣኚࢃࡗ࡚࠸ࡿ㸦⾲ 3.2, 3.5㸧㸬ࡲࡓ㸪ࡑࢀ௨እ࠾࠸࡚ࡣ㸪᭱ேཱྀ᭱ᑠேཱྀࡢ࠸ࡎࢀⱝᖸኚືࡀぢࡽࢀࡿ ࡢࡳ࡛㸪᱁ᕪࡢᕪࡣ࡛᭱ࡶ 0.02 ⛬ᗘ࡛࠶ࡿ㸬ᚑࡗ࡚㸪୍㑅ᣲ༊๓ᙜἲࡼࡗ࡚㝈ᐃࡉࢀ ࡚࠸ࡿ᭱㐺ゎࡣ㸪ᮏ᮶ࡢ᭱㐺ゎࡑࢀ㐪࠸ࡣ࡞ࡃ㸪୍⚊ࡢ㔜ࡳࡢ᱁ᕪࢆᖜࡺࡀࡵࡿࡼ ࠺࡞ࡇࡣࡋ࡚࠸࡞࠸ぢ࡚ྲྀࢀࡿ㸬ᐇ㝿㸪୍㑅ᣲ༊๓ᙜἲࡢ୍⚊ࡢ㔜ࡳࡢ᱁ᕪࡣ 2.153 ಸ 㸦᭱ேཱྀ㑅ᣲ༊㸸⚄ዉᕝ┴㸦566,460㸧㸪᭱ᑠேཱྀ㑅ᣲ༊㸸㧗▱┴㸦263,089㸧㸧࡛࠶ࡾ㸪ᮏ◊✲ ࡛ᚓࡽࢀࡿ୍⚊ࡢ㔜ࡳࡢ᱁ᕪࡶྠᵝ࡛࠶ࡿ㸬⾲ 3.1: ୍㑅ᣲ༊๓ᙜἲᮏ◊✲ࡢ᭱㐺ゎࡢẚ㍑ ᭱㐺ゎ [๓ᙜἲ] ᭱㐺ゎ [ᮏ◊✲] ᗓ┴ ᭱ ᭱ᑠ ᱁ᕪ ᭱ ᭱ᑠ ᱁ᕪ ⚄ዉᕝ 566,460 446,893 1.268 566,460 446,893 1.268 ᪂₲ 418,011 392,662 1.065 418,011 392,662 1.065 㟼ᒸ 443,679 401,285 1.106 438,110 401,285 1.092 㜰 582,723 446,643 1.305 582,723 446,643 1.305 ᒸᒣ 469,372 362,158 1.296 469,372 366,960 1.279 ឡ 374,777 360,741 1.039 374,777 363,796 1.030 ⇃ᮏ 389,720 318,446 1.224 382,013 318,446 1.200 㮵ඣᓥ 449,692 430,720 1.044 443,989 430,720 1.031 ᐩᒣ 376,827 364,241 1.035 376,827 367,387 1.026 㤶ᕝ 338,602 336,239 1.007 338,011 336,239 1.005 㒔㐨ᗓ┴ẖ㑅ᣲ༊ࢆヲ⣽ぢ࡚ࡳࡿ㸦⾲ 3.2㹼3.11㸧㸪ࡑࢀࡒࢀࡢ᭱㐺ゎ࠸࡚㑅ᣲ༊ ࡢᵓᡂࡣࡃኚࢃࡽࡎ㸪ศᑐ㇟࡞ࡗࡓ㐣ᕷ༊ࡢศ㛫ࡢேཱྀㄪᩚࡢࡳ࡛῭ࢇࡔࡢࡣ㸪᪂₲ ┴ (6)㸪ᐩᒣ┴ (3)㸪㤶ᕝ┴ (3)㸪㮵ඣᓥ┴ (4) ࡢ 4 ┴࡛࠶ࡿ㸦⾲ 3.3, 3.10, 3.11, 3.9㸧㸬ࡇࡇ࡛㸪 ᣓᘼෆࡢᩘᏐࡣ㆟ᖍᩘ㸦㑅ᣲ༊ᩘ㸧ࢆ࠶ࡽࢃࡍ㸬ྛ⾲ࡢᕥഃࡀ୍㑅ᣲ༊๓ᙜἲࡼࡿ᭱㐺㑅 ᣲ༊㸪ྑഃࡀᮏ◊✲ࡼࡾồࡵࡽࢀࡓ᭱㐺㑅ᣲ༊࡛࠶ࡿ㸬ᮏ◊✲ࡢ⤖ᯝḍ࡛ࡣ㸪ᕷ༊ᵓᡂ ேཱྀࡀኚࢃࡽ࡞࠸㑅ᣲ༊ࡣ┬␎ࡋ࡚࠶ࡿ㸬ྛ┴࡛㸪ศᑐ㇟࡞ࡿ᪂₲ᕷ㸪ᐩᒣᕷ㸪㧗ᯇᕷ㸪 㮵ඣᓥᕷ 1 ࡑࢀࡒࢀ࠾ࡅࡿศ㛫ࡢேཱྀㄪᩚࡢࡳࡀ␗࡞ࡿࡇࡀࢃࡿ㸬 ࡲࡓ㸪㑅ᣲ༊ࡢᵓᡂࡶኚࡋࡓ㒔㐨ᗓ┴ࡣ⚄ዉᕝ┴ (14)㸪㟼ᒸ┴ (9)㸪㜰ᗓ (18)㸪ᒸᒣ┴ (5)㸪ឡ┴ (4)㸪⇃ᮏ┴ (5) ࡢ 6 ᗓ┴࡛࠶ࡿ㸬⚄ዉᕝ┴ࡣ 18 㑅ᣲ༊ࡢ࠺ࡕ 9 㑅ᣲ༊㸦⾲ 3.2㸧㸪 㜰ᗓࡣ 18 㑅ᣲ༊ࡢ࠺ࡕ 6 㑅ᣲ༊㸦⾲ 3.5㸧ࡑࢀࡒࢀ㐣ேཱྀᕷ༊ࡢேཱྀㄪᩚࡑࢀక࠺ 㑅ᣲ༊ᵓᡂࡢኚ᭦ࡀ࠶ࡗࡓ㸬㟼ᒸ┴ࡣᯇᕷࡢศ㛫ேཱྀㄪᩚࡢ㸪2 ࡘࡢ㑅ᣲ༊࡛ᕷ༊㒆ᵓ ᡂࡢኚ᭦ࡀぢࡽࢀࡓ㸦⾲ 3.4㸧㸬ྠᵝ㸪ᒸᒣ┴ࡣᒸᒣᕷࡢศ㛫ேཱྀㄪᩚࡢ㸪2 ࡘࡢ㑅ᣲ ༊࡛ᕷ༊㒆ࡢධࢀ᭰࠼ࡀ㉳ࡇࡗ࡚࠾ࡾ㸦⾲ 3.6㸧㸪ឡ┴ࡣᯇᒣᕷࡢศ㛫ேཱྀㄪᩚࡢ㸪2 ࡘࡢ㑅ᣲ༊࡛ᕷ༊㒆ᵓᡂࡀኚࡋࡓ㸦⾲ 3.7㸧㸬⇃ᮏ┴ࡣ⇃ᮏᕷࡢศ㛫ேཱྀㄪᩚࡢ㸪2 ࡘ ࡢ㑅ᣲ༊࡛ᕷ༊㒆ࡢࡀ࠶ࡗࡓ㸦⾲ 3.8㸧㸬
4
ࡲࡵ
ᮏ◊✲࡛ࡣ㸪ᕷ༊㒆ศࢆᚲせࡍࡿ㒔㐨ᗓ┴ᑐࡋ㸪ࢲ࣑࣮ᕷ༊ dumkࢆ⏝࠸࡚ᣑ㞄᥋ࢢ ࣛࣇ ˜G ࢆᵓᡂࡋ㸪ேཱྀㄪᩚኚᩘ kࡼࡾศᕷ༊㒆ࡢேཱྀㄪᩚࢆຍࡋࡓࣔࢹࣝࢆᥦ㸪ࢯ ࣝࣂ࣮ࢆ⏝ࡋ࡚᭱㐺༊ࢆᑟฟࡋࡓ㸬ࡲࡓ㸪ᖹᆒேཱྀࢆ୍㑅ᣲ༊ࡋ࡚๓ࡾᙜ࡚࡚ゎ ࡃ୍㑅ᣲ༊๓ᙜἲẚ㍑ࡋࡓ⤖ᯝ㸪୍㑅ᣲ༊๓ᙜἲࡀ୍⚊ࡢ᱁ᕪࢆᴟ➃ࡺࡀࡵ࡚࠸ࡿࡇࡣ࡞࠸࠸࠺ࡇࢆ☜ㄆࡍࡿࡇࡀ࡛ࡁࡓ㸬 ཝᐦ᭱㐺ゎࢆồࡵࡿࡓࡵࡣ㸪ᮏ◊✲ࡢ᪉ἲࢆ⏝࠸ࡿ᪉ࡀࡼ࠸㸬ࡋࡋ࡞ࡀࡽ㸪ィ⟬ࡢᡭ 㛫ࡸồゎࡢ」㞧ࡉࡀࡸࡸቑࡍࡿࡓࡵ㸪⌧ᅾࡢ᪥ᮏࡢ⾜ᨻ⏺࣭ேཱྀ࡛⪃࠼ࡿ㝈ࡾ㸪㒔㐨ᗓ┴ෆ᱁ ᕪࡁ࡞ᙳ㡪ࢆ࠼࡞࠸୍㑅ᣲ༊๓ᙜἲࢆ⏝࠸ࡿ᪉ࡀ㸪ィ⟬ࢥࢫࢺࡸ࣮ࣝࣝࡢࢃࡾࡸ ࡍࡉ㸪⡆౽ࡉ࠸࠺Ⅼ࡛⏝ࡋࡸࡍ࠸ゝ࠼ࡿ㸬 ᮏ◊✲࠾ࡅࡿᐃᘧ࡛ࡣ㸪ศᑐ㇟࡞ࡿᕷ༊㒆ࡀ」ᩘᏑᅾࡍࡿሙྜ㸪ไ⣙ࡀ⤌ྜࡏⓗ ቑ࠼ࡿࡓࡵ㸪ồゎᅔ㞴ࡉࢆకࢃ࡞࠸ࡼ࠺㸪ࡼࡾ⡆౽࡞ᐃᘧࢆᶍ⣴ࡍࡿᚲせࡀ࠶ࡿ㸬 ࡲࡓ㸪➨ 2 ጇᙜ㑅ᣲ༊㞟ྜࡢせ⣲ᩘࡀቑຍࡍࡿࡓࡵ㸪ࡑࢀࢆඞ᭹ࡍࡿࢹࡋ࡚➨ 3 ጇᙜ 㑅ᣲ༊㞟ྜࢆᐃ⩏ࡋ࡚ᵓᡂࢆヨࡳࡓ㸬ࡇࡇ࡛㸪➨ 3 ጇᙜ㑅ᣲ༊㞟ྜࡣ㸪ࡑࡢ㑅ᣲ༊ࢆ࠺ࡇ ࡛㸪ᙜヱ㒔㐨ᗓ┴ࡢᵓᡂ㑅ᣲ༊ᩘ༊ೃ⿵ࡢ㑅ᣲ༊ᩘࡀࡁࡕࢇ୍⮴ࡍࡿ࠺ࢆุᐃ ࡋ㸪୍⮴ࡍࡿ㑅ᣲ༊ࡢࡳࢆせ⣲ࡍࡿ㞟ྜࡋ࡚ᐃ⩏ࡋࡓ㸬ࡋࡋ࡞ࡀࡽ㸪ヨ⟬࡛ࡣ➨ 2 ጇᙜ 㑅ᣲ༊㞟ྜࡽࡢⓗ࡞せ⣲ᩘࡢῶᑡࡣぢࡽࢀ࡞࠸ࡓࡵ㸪ᮏ◊✲࠾࠸࡚ࡣྲྀࡾᢅࢃ࡞ࡗࡓ㸬 ࡑࢀࡼࡾࡶ㸪ୖୗ㝈ேཱྀไ⣙ࢆ⊃ࡵࡿࡇ࡛㸪➨ 2 ጇᙜุᐃࡢ᮲௳ࢆཝࡋࡃࡋ㸪➨ 2 ጇᙜ㑅 ᣲ༊㞟ྜࡢせ⣲ᩘࢆῶࡽࡍࡇࡢ᪉ࡀຠᯝࡀ࠶ࡿᛮࢃࢀࡿ8㸬 8ึᮇタᐃ࡛ࡣ㸪༊సᡂ᪉㔪๎ࡾ㸪ୖୗ㝈ேཱྀไ⣙ࢆᙜヱ㒔㐨ᗓ┴ࡢᖹᆒேཱྀࡢ 4/3 ಸ 2/3 ಸࡋ࡚࠸ࡿ㸬ᚑࡗ ࡚㸪➨ 2 ጇᙜ㑅ᣲ༊ࡢᐃ⩏ (2) ࡼࡾ㸪ୖୗ㝈್ ¯u, ¯l ࡀࡇࡢ್ࡢࡣ㸪i = 1 ࡛ୗ㝈ไ⣙ࡢࡳㄪࢀࡤ༑ศ࡛࠶ࡿࡇ ὀព㸬¯u, ¯l ࡢ⠊ᅖࢆ⊃ࡵࢀࡤ㸪ୖୗ㝈ไ⣙ඹຠ࠸࡚ࡃࡿࡢ࡛㸪➨ 2 ጇᙜ㑅ᣲ༊㞟ྜࡢせ⣲ᩘࡀຍ㏿ᗘⓗῶᑡࡍࡿࡇ ࡀᮇᚅ࡛ࡁࡿ
ཧ⪃ᩥ⊩
[1] ᆏཱྀ⿱, ⏣῟୍㑻: 㑅ᣲ༊ࡾࡢ᭱㐺ࡘ࠸࡚, ୕⏣Ꮫ㞧ㄅ, 93-1 (2000) 109-137. [2] ⏣୰᐀Ꮥ: ᨻᨵ㠉 6 ᖺࡢ㐨⛬, ࡂࡻ࠺ࡏ࠸ (1997). [3] ᰿ᮏಇ⏨, ᇼ⏣ᩗ: ༊⏬ᐃၥ㢟ࡢࣔࢹࣝ᭱㐺༊ࡢᑟฟ, ࣮࢜࣌ࣞࢩࣙࣥࢬ࣭ࣜࢧ࣮ ࢳ, 48-3 (2003) 50-56. [4] ᰿ᮏಇ⏨, ᇼ⏣ᩗ: 㑅ᣲ༊᭱㐺༊ၥ㢟ࡢࣔࢹࣜࣥࢢཝᐦゎᑟฟ, ➨ 15 ᅇ RAMP ࢩࣥ ࣏ࢪ࣒࢘ㄽᩥ㞟 (2003) 104-117. [5] ᰿ᮏಇ⏨, ᇼ⏣ᩗ: ⾗㆟㝔ᑠ㑅ᣲ༊ไ࠾ࡅࡿ୍⚊ࡢ㔜ࡳࡢ᱁ᕪࡢ㝈⏺ࡑࡢ⪃ᐹ, 㑅ᣲ ◊✲, 20 (2005) 136-147. [6] ᰿ᮏಇ⏨, ᇼ⏣ᩗ: බᖹ࡞ᑠ㑅ᣲ༊ไࡢࡓࡵࡢᩘ⌮ࣔࢹࣝ, ࢩࢫࢸ࣒/ไᚚ/ሗ, 49-3 (2005) 2-7. [7] ᰿ᮏಇ⏨, ᇼ⏣ᩗ: ୍⚊ࡢ㔜ࡳࡢ᱁ᕪࡽほࡓᑠ㑅ᣲ༊ᩘ, 㑅ᣲ◊✲, 21 (2006) 169-181. [8] ᰿ᮏಇ⏨, ᇼ⏣ᩗ: ┴ቃ⦆ࡼࡿ୍⚊ࡢ㔜ࡳࡢ᱁ᕪࡢᙳ㡪ࡘ࠸࡚, ᪥ᮏ㑅ᣲᏛሗ ࿌ㄽᩥ (2009). [9] ᰿ᮏಇ⏨, ᇼ⏣ᩗ: ᖹᡂྜేࢆ⤒ࡓ⾗㆟㝔ᑠ㑅ᣲ༊ไ༊⎔ቃࡢኚ୍⚊ࡢ㔜ࡳࡢ᱁ ᕪ㸪TORSJ ᥖ㍕ணᐃ㸬 [10] ᇼ⏣ᩗ: 㑅ᣲ༊⏬ᐃၥ㢟ࡢ᭱㐺ࣔࢹࣝᑐࡍࡿᕷ༊㒆ศࡢ᪉⟇⪃ᐹ, ᪥ᮏ࣮࢜࣌ࣞ ࢩࣙࣥࢬ࣭ࣜࢧ࣮ࢳᏛ 2009 ᖺᗘ⛅Ꮨ◊✲Ⓨ⾲ ࣈࢫࢺࣛࢡࢺ㞟 (2009) 24-25.⾲ 3.2: ୍㑅ᣲ༊๓ᙜἲᮏ◊✲ࡢ᭱㐺ゎࡢẚ㍑㸸⚄ዉᕝ┴㸦18 㑅ᣲ༊㸧 ᭱㐺ゎ[๓ᙜἲ] ᭱㐺ゎ[ᮏ◊✲] 㑅ᣲ༊ 18 3 5 8 9 14 15 16 17 18 1 468,227 㭯ぢ༊ 264,450 ᕝᓮ༊ 203,777 2 446,893 ⚄ዉᕝ༊ 221,832 す༊ 84,960 ୰༊ 140,101 3 458,414 483,404 ༡༊ 196,787 196,787 ᡞሯ༊ 261,627 261,627 4 453,887 ಖᅵࢣ㇂༊ 204,179 ᪫༊ 249,708 5 509,091 484,101 ☾Ꮚ༊ 163,511 163,511 ༡༊ 221,777 221,777 ᰤ༊ 123,803 123,803 6 471,321 㔠ἑ༊ 210,642 㙊ᕷ 171,122 ㏸Ꮚᕷ 58,025 ୕ᾆ㒆 31,532 7 481,469 ༊ 311,654 ⥳༊ 169,815 8 476,793 500,945 ℩㇂༊ 127,393 127,393 ᕷ 221,218 221,218 ᗙ㛫ᕷ 128,182 128,182 9 548,331 446,893 Ἠ༊ 152,334 152,334 ⸨ἑᕷ 395,997 395,997 10 474,560 㟷ⴥ༊ 295,544 㒔⟃༊ 179,016 11 556,772 ᖾ༊ 144,513 ୰ཎ༊ 210,493 㧗ὠ༊ 201,766 12 566,460 ከᦶ༊ 205,407 ᐑ๓༊ 207,927 㯞⏕༊ 153,126 13 476,031 ᶓ㡲㈡ᕷ 426,162 ୕ᾆᕷ 49,869 14 487,447 550,308 ᖹሯᕷ 259,017 259,017 ⱴࢣᓮᕷ 228,430 228,430 15 496,858 456,479 ᑠ⏣ཎᕷ 198,722 198,722 ┦ᶍཎᕷ 2 39,045 39,045 ༡㊊ᕷ 44,130 44,130 ୰㒆 62,861 62,861 ㊊ୖ㒆 67,870 67,870 ㊊ୗ㒆 50,345 50,345 ὠஂ㒆 1 23,060 23,060 ὠஂ㒆 2 10,825 10,825 16 454,687 488,383 491,239 504,156 ┦ᶍཎᕷ 1 140,255 488,383 124,482 504,156 ⛙㔝ᕷ 168,317 168,317 ఀໃཎᕷ 100,573 100,573 ឡ⏥㒆 45,542 45,542 17 475,276 477,755 ཌᮌᕷ 222,349 222,349 ᾏ⪁ྡᕷ 123,708 123,708 ⥤℩ᕷ 81,758 81,758 㧗ᗙ㒆 47,461 47,461
⾲ 3.3: ୍㑅ᣲ༊๓ᙜἲᮏ◊✲ࡢ᭱㐺ゎࡢẚ㍑㸸᪂₲┴㸦6 㑅ᣲ༊㸧 ᭱㐺ゎ[๓ᙜἲ] ᭱㐺ゎ[ᮏ◊✲] 㑅ᣲ༊ 417,0921 418,0112 403,1923 395,2064 392,6625 405,2336 404,3141 418,0116 ᪂₲ᕷ 408,547 405,233 395,769 418,011 㛗ᒸᕷ 283,234 ୕᮲ᕷ 104,748 ᯽ᓮᕷ 94,644 ᪂Ⓨ⏣ᕷ 104,633 ᑠ༓㇂ᕷ 39,959 ຍⱱᕷ 31,481 ༑᪥⏫ᕷ 62,055 ぢ㝃ᕷ 42,668 ᮧୖᕷ 30,685 ⇩ᕷ 83,269 ⣒㨶ᕝᕷ 49,844 Ἠᕷ 56,965 ୖ㉺ᕷ 208,083 బΏᕷ 67,384 㜿㈡㔝ᕷ 47,043 㨶ᕷ 43,554 ༡㨶ᕷ 63,329 ጁ㧗ᕷ 37,829 ⫾ෆᕷ 32,813 ⵦཎ㒆 13,497 すⵦཎ㒆 8,545 8,545 ༡ⵦཎ㒆 13,363 ᮾⵦཎ㒆 14,704 ୕ᓥ㒆 5,338 㨶㒆 5,233 ༡㨶㒆 8,713 ୰㨶㒆 11,719 ส⩚㒆 4,806 ᒾ⯪㒆1 28,260 ᒾ⯪㒆2 19,218
⾲ 3.4: ୍㑅ᣲ༊ ๓ᙜἲ ᮏ◊✲ ࡢ᭱㐺ゎ ࡢẚ㍑ 㸸 㟼 ᒸ┴ 㸦 9 㑅 ᣲ༊㸧 ᭱㐺ゎ [๓ ᙜἲ ] ᭱㐺ゎ [ᮏ◊✲ ] 㑅ᣲ༊ 1 2 3 4 5 6 7 8 9 4 5 6 9 4 0 4 ,7 2 0 4 3 8 ,1 1 0 4 2 8 ,7 0 7 4 4 3 ,6 7 9 4 1 7 ,7 2 2 4 2 9 ,8 2 1 4 0 1 ,2 8 5 4 0 7 ,0 2 9 4 2 1 ,3 8 4 4 2 6 ,9 5 3 4 3 6 ,7 6 3 4 1 0 ,7 8 0 4 3 8 ,1 1 0 ⵇ༊ 2 6 2 ,7 6 9 㥴Ἑ༊ 2 0 8 ,0 4 3 ΎỈ༊ 1 2 3 0 ,0 6 7 ΎỈ༊ 2 1 2 ,8 3 7 ᯇᕷ 3 8 2 ,6 8 3 4 2 1 ,3 8 4 3 6 5 ,9 5 7 4 3 8 ,1 1 0 ὠᕷ 2 0 8 ,0 0 1 2 0 8 ,0 0 1 ⇕ᾏᕷ 4 1 ,2 0 2 4 1 ,2 0 2 ୕ᓥᕷ 1 1 2 ,2 5 1 1 1 2 ,2 5 1 ᐩኈᐑᕷ 1 2 1 ,7 8 0 ఀᮾᕷ 7 2 ,4 3 5 7 2 ,4 3 5 ᓥ⏣ᕷ 9 6 ,0 7 1 ᐩኈᕷ 2 3 6 ,4 9 3 ☬⏣ᕷ 1 7 0 ,9 1 3 ↝ὠᕷ 1 2 0 ,1 1 1 ᕝᕷ 1 1 7 ,8 5 6 ⸨ᯞᕷ 1 2 9 ,2 5 6 ᚚẊሙᕷ 8 5 ,9 7 6 8 5 ,9 7 6 ⿄ᕷ 8 2 ,9 8 3 ୗ⏣ᕷ 2 6 ,5 5 7 2 6 ,5 5 7 〈㔝ᕷ 5 3 ,0 6 1 5 3 ,0 6 1 †すᕷ 4 4 ,0 5 8 4 4 ,0 5 8 ఀ㇋ᕷ 3 6 ,6 2 4 3 6 ,6 2 4 ᚚ๓ᓮᕷ 3 5 ,2 7 7 ⳥ᕝᕷ 4 7 ,5 2 4 ఀ ㇋ࡢᅜᕷ 5 0 ,0 0 5 5 0 ,0 0 5 ∾அཎᕷ 5 0 ,6 4 3 ㈡ⱱ㒆 5 1 ,9 4 4 5 1 ,9 4 4 ⏣᪉㒆 3 8 ,8 0 3 3 8 ,8 0 3 㥴ᮾ㒆 1 7 0 ,6 8 4 7 0 ,6 8 4 㥴ᮾ㒆 2 2 1 ,4 7 8 ᐩኈ㒆 9 ,6 9 5 ᗡཎ㒆 2 6 ,4 2 4 ᚿኴ㒆 1 1 2 ,6 9 5 ᚿኴ㒆 2 2 2 ,9 9 7 ᴵཎ㒆 1 2 8 ,6 4 8 ᴵཎ㒆 2 1 5 ,0 1 8 ࿘ᬛ㒆 2 0 ,2 7 3 ᘬబ㒆 1 6 ,9 3 8 1 6 ,9 3 8
⾲ 3.5: ୍㑅ᣲ༊๓ᙜἲᮏ◊✲ࡢ᭱㐺ゎࡢẚ㍑㸸㜰ᗓ㸦18 㑅ᣲ༊㸧 ᭱㐺ゎ [๓ᙜἲ] ᭱㐺ゎ [ᮏ◊✲] 㑅ᣲ༊ 18 1 2 3 4 9 18 1 478,801 464,228 㒔ᓥ༊ 99,839 99,839 ᮾᡂ༊ 79,485 79,485 ⏕㔝༊ 138,550 138,550 ᇛᮾ༊ 160,927 160,927 2 450,426 477,105 ⚟ᓥ༊ 60,929 60,929 す༊ 72,576 72,576 ᾉ㏿༊ 54,148 54,148 すᾷᕝ༊ 95,621 95,621 ༊ 100,392 100,392 ୰ኸ༊ 66,760 66,760 3 483,546 453,203 Ṉⰼ༊ 63,803 63,803 ༊ 83,156 83,156 ṇ༊ 73,204 73,204 すᡂ༊ 132,762 132,762 ఫஅỤ༊ 130,621 130,621 4 465,416 446,643 ኳ⋤ᑎ༊ 64,126 64,126 㜿ಸ㔝༊ 107,295 107,295 ఫྜྷ༊ 158,998 158,998 ᮾఫྜྷ༊ 134,997 134,997 5 532,210 ᮾᾷᕝ༊ 178,357 ྿⏣ᕷ 353,853 6 460,677 ᪫༊ 95,115 㭯ぢ༊ 107,410 ᮾᕷ 126,478 㛛┿ᕷ 131,674 7 555,848 ᾷᕝ༊ 169,215 ㇏୰ᕷ 386,633 8 473,964 ᖹ㔝༊ 200,490 ඵᑿᕷ 273,474 9 468,545 489,834 480,300 515,089 ሜᕷ 341,277 489,834 316,022 515,089 ᯇཎᕷ 127,268 127,268 10 582,723 ᓊ⏣ᕷ 200,984 ㈅ሯᕷ 90,312 Ἠబ㔝ᕷ 98,876 Ἠ༡ᕷ 64,686 㜰༡ᕷ 57,617 Ἠ༡㒆 1 44,505 Ἠ༡㒆 2 7,239 Ἠ༡㒆 3 18,504 11 533,569 ụ⏣ᕷ 101,643 Ⲉᮌᕷ 267,976 ⟪㠃ᕷ 127,132 ㇏⬟㒆 36,818 12 451,466 Ἠὠᕷ 77,674 Ἑෆ㛗㔝ᕷ 117,243 Ἠᕷ 177,837 㧗▼ᕷ 61,126 Ἠ㒆 17,586 13 465,854 㧗ᵳᕷ 351,803 ᦤὠᕷ 84,997 ୕ᓥ㒆 29,054 14 446,643 Ᏺཱྀᕷ 147,479 ᐷᒇᕝᕷ 241,825 ᅄᲄ␝ᕷ 57,339 15 481,647 ᯛ᪉ᕷ 404,004 㔝ᕷ 77,643 16 482,097 ᐩ⏣ᯘᕷ 123,800 ᯽ཎᕷ 77,065 ⩚᭥㔝ᕷ 118,686 ⸨ᑎᕷ 65,774 㜰⊃ᒣᕷ 58,207 ༡Ἑෆ㒆 38,565 17 513,744
⾲ 3.6: ୍㑅ᣲ༊๓ᙜἲᮏ◊✲ࡢ᭱㐺ゎࡢẚ㍑㸸ᒸᒣ┴㸦5 㑅ᣲ༊㸧 ᭱㐺ゎ[๓ᙜἲ] ᭱㐺ゎ[ᮏ◊✲] 㑅ᣲ༊ 368,6441 469,3722 362,1583 365,4714 391,4115 371,9211 367,1903 366,9604 381,6135 ᒸᒣᕷ 283,194 391,411 292,992 381,613 ᩜᕷ 469,372 ὠᒣᕷ 110,565 110,565 ⋢㔝ᕷ 67,009 67,009 ➟ᒸᕷ 57,266 57,266 ཎᕷ 45,107 45,107 ⥲♫ᕷ 66,589 66,589 㧗ᱱᕷ 38,796 38,796 ᪂ぢᕷ 36,062 36,062 ഛ๓ᕷ 40,221 40,221 ℩ᡞෆᕷ 39079 39,079 ㉥☬ᕷ 43,917 43,917 ┿ᗞᕷ 51,788 51,788 ⨾సᕷ 32,475 32,475 ὸཱྀᕷ 37,337 37,337 ᚚὠ㒆 6,521 6,521 ㉥☬㒆 14,900 14,900 Ẽ㒆 16,180 16,180 㒔❑㒆 11,920 11,920 ὸཱྀ㒆 10,822 10,822 ᑠ⏣㒆 15,715 15,715 ┿ᗞ㒆 1,020 1,020 Ɫ⏣㒆 14,059 14,059 ⏣㒆 17,735 17,735 ⱥ⏣㒆 1,684 1,684 ஂ⡿㒆 22,272 22,272 ຍ㈡㒆 14,040 14,040
⾲ 3.7: ୍㑅ᣲ༊๓ᙜἲᮏ◊✲ࡢ᭱㐺ゎࡢẚ㍑㸸ឡ┴㸦4 㑅ᣲ༊㸧 ᭱㐺ゎ[๓ᙜἲ] ᭱㐺ゎ[ᮏ◊✲] 㑅ᣲ༊ 365,3501 360,7412 374,7773 366,9564 363,7961 365,4542 363,7974 ᯇᒣᕷ 147,988 366,956 151,147 363,797 ᕷ 173,985 173,985 Ᏹᓥᕷ 89,445 ඵᖭᕷ 41,265 ᪂ᒃᕷ 123,952 123,952 す᮲ᕷ 113,369 113,369 Ὢᕷ 50,785 ఀணᕷ 39,494 ᅄᅜ୰ኸᕷ 92,854 92,854 すணᕷ 44,949 ᮾ ᕷ 35,279 35,279 ㉺ᬛ㒆 8,098 8,098 ୖᾋ✰㒆 10,946 10,946 ఀண㒆 52,989 ႐ከ㒆 19,620 19,620 すᏱ㒆 12,095 Ᏹ㒆 17,122 ༡Ᏹ㒆 26,633
⾲ 3.8: ୍㑅ᣲ༊๓ᙜἲᮏ◊✲ࡢ᭱㐺ゎࡢẚ㍑㸸⇃ᮏ┴㸦5 㑅ᣲ༊㸧 ᭱㐺ゎ[๓ᙜἲ] ᭱㐺ゎ[ᮏ◊✲] 㑅ᣲ༊ 318,4461 382,0132 389,7203 383,5334 368,4285 379,8323 379,8364 382,0135 ⇃ᮏᕷ 301,113 368,428 287,528 382,013 ඵ௦ᕷ 136,885 ேྜྷᕷ 37,579 Ⲩᑿᕷ 55,968 55,968 Ỉಛᕷ 29,119 ⋢ྡᕷ 71,853 71,853 ᒣ㮵ᕷ 57,727 57,727 ⳥ụᕷ 51,864 51,864 Ᏹᅵᕷ 38,022 ୖኳⲡᕷ 32,504 Ᏹᇛᕷ 63,091 㜿⸽ᕷ 29,639 29,639 ྜᚿᕷ 51,647 51,647 ኳⲡᕷ 96,477 ୗ┈ᇛ㒆1 27,543 ୗ┈ᇛ㒆2 12,254 ⋢ྡ㒆1 5,626 5,626 ⋢ྡ㒆2 40,484 40,484 㮵ᮏ㒆 30,773 30,773 ⳥ụ㒆 61,535 61,535 㜿⸽㒆1 15,024 15,024 㜿⸽㒆2 25,688 ୖ┈ᇛ㒆 89,761 ඵ௦㒆 13,233 ⴺ㒆 26,266 ⌫☻㒆 63,110 ኳⲡ㒆 8,927
⾲ 3.9: ୍㑅ᣲ༊๓ᙜἲᮏ◊✲ࡢ᭱㐺ゎࡢẚ㍑㸸㮵ඣᓥ┴㸦4 㑅ᣲ༊㸧 ᭱㐺ゎ[๓ᙜἲ] ᭱㐺ゎ[ᮏ◊✲] 㑅ᣲ༊ 449,6921 434,4462 430,7203 438,2864 443,9891 443,9894 㮵ඣᓥᕷ1 160,053 438,286 154,350 443,989 㮵ඣᓥᕷ2 6,048 㮵ᒇᕷ 106,220 ᯖᓮᕷ 25,150 25,150 㜿ஂ᰿ᕷ 25,074 ฟỈᕷ 57,908 ཱྀᕷ 22,118 ᣦᐟᕷ 46,820 46,820 すஅ⾲ᕷ 18,198 18,198 ᆶỈᕷ 18,926 ⸃ᦶᕝෆᕷ1 96,164 ⸃ᦶᕝෆᕷ2 6,206 ᪥⨨ᕷ 52,414 ᭮ᕷ 42,287 ࠸ࡕࡁ୵ᮌ㔝ᕷ 32,990 㟝ᓥᕷ 127,289 ༡ࡉࡘࡲᕷ 41,676 ᚿᕸᚿᕷ 34,777 ⨾ᕷ1 42,808 42,808 ⨾ᕷ2 6,784 6,784 㮵ඣᓥ㒆1 462 462 㮵ඣᓥ㒆2 673 673 ᥬᐟ㒆 14,126 14,126 ᕝ㎶㒆 28,066 28,066 ⸃ᦶ㒆 25,690 ฟỈ㒆 11,959 ఀబ㒆 9,380 ጸⰋ㒆1 74,831 ጸⰋ㒆2 12,565 ᭮㒆 15,303 ⫢ᒓ㒆 45,341 ⇃ẟ㒆 29,705 29,705 ᓥ㒆1 62,265 62,265 ᓥ㒆2 14,582 14,582
⾲ 3.10: ୍㑅ᣲ༊๓ᙜἲᮏ◊✲ࡢ᭱㐺ゎࡢẚ㍑㸸ᐩᒣ┴㸦3 㑅ᣲ༊㸧 ᭱㐺ゎ[๓ᙜἲ] ᭱㐺ゎ[ᮏ◊✲] 㑅ᣲ༊ 364,2411 376,8272 370,5343 367,3871 367,3883 ᐩᒣᕷ 50,622 370,534 53,768 367,388 㧗ᒸᕷ 181,231 㨶ὠᕷ 46,330 46,330 ịぢᕷ 54,493 ᕝᕷ 34,007 34,007 㯮㒊ᕷ 42,695 42,695 ◝Ἴᕷ 49,429 ᑠ▮㒊ᕷ 33,537 ༡◝ᕷ 58,137 ᑕỈᕷ 94,160 94,160 ୰᪂ᕝ㒆 53,721 53,721 ୗ᪂ᕝ㒆 42,706 42,706 ⾲ 3.11: ୍㑅ᣲ༊๓ᙜἲᮏ◊✲ࡢ᭱㐺ゎࡢẚ㍑㸸㤶ᕝ┴㸦3 㑅ᣲ༊㸧 ᭱㐺ゎ[๓ᙜἲ] ᭱㐺ゎ[ᮏ◊✲] 㑅ᣲ༊ 338,6021 336,2392 337,4203 338,0111 338,0113 㧗ᯇᕷ 80,566 337,420 79,975 338,011 டᕷ 110,080 ᆏฟᕷ 57,268 57,268 ၿ㏻ᑎᕷ 35,495 ほ㡢ᑎᕷ 65,217 ࡉࡠࡁᕷ 55,753 55,753 ᮾࡀࢃᕷ 35,929 35,929 ୕㇏ᕷ 71,178 ᑠ㇋㒆 33,666 33,666 ᮌ⏣㒆 28,794 28,794 㤶ᕝ㒆 3,537 3,537 ⥤ḷ㒆1 25,629 25,629 ⥤ḷ㒆2 17,460 17,460 ௰ከᗘ㒆1 30,656 ௰ከᗘ㒆2 23,613
