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テレビCMのデータ分析における数理最適化の応用事例

motokawa tetsuya
February 19, 2024
470

 テレビCMのデータ分析における数理最適化の応用事例

数理最適化: Optimization Night #8 で発表した資料です
https://optimization.connpass.com/event/308608/

ソースコード
https://github.com/mtkwT/OptimizationNight_20240219

motokawa tetsuya

February 19, 2024
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Transcript

  1. ิ଍ɿάϩεϦʔνͱϢχʔΫϦʔν w άϩεϦʔνʢԆ΂ϦʔνɾΠϯϓϨογϣϯɾ(31ʣ w $.΁ͷԆ΂઀৮ਓ਺ʢׂ߹ʣ w ಉ͡ਓ͕ෳ਺ճ$.ʹ઀৮ͨ͠৔߹ɺԿ౓Ͱ΋Χ΢ϯτ͢Δ w ϢχʔΫϦʔνʢϦʔνɾϢχʔΫϢʔβׂ߹ʣ w

    $.ʹগͳ͘ͱ΋ճ઀৮ͨ͠ਓ਺ʢׂ߹ʣ w ಉ͡ਓ͕ෳ਺ճ$.ʹ઀৮ͯ͠΋ճ໨Ҏ߱͸Χ΢ϯτ͠ͳ͍ 7 ΑΓଟ͘ͷਓʹಧ͚Δͱ͍͏ҙຯͰ͸ɺ άϩεϦʔνͰ͸ͳ͘ϢχʔΫϦʔνΛ ޮ཰తʹ৳͹͢͜ͱ͕ॏཁ
  2. w ݸਓ ͕$.ʹҰ౓Ͱ΋઀৮ͨ͠Β ͠ͳ͚Ε͹ΛऔΔ֬཰ม਺Λ ͱ ͢ΔɻશମͷϢχʔΫϦʔνΛ ͱ͢Δͱɺ ͱͳΔ ΋֬཰ม਺ ɻ

    w ์ૹہ͸ ͷہͱ͢Δɻہ͝ͱͷάϩεϦʔν ग़ߘྔ Λ ͱ͠ɺہ͝ͱͷάϩεϦʔν͋ͨΓͷίετΛ ɺτʔλϧίετ εϙοτ$.શମͷ༧ࢉ Λ ͱ͢Δɻ w ͜ͷͱ͖ہ͝ͱͷ༧ࢉ഑෼Λௐ੔͠શମͰͷϢχʔΫϦʔνΛ࠷େԽ͢Δͱ͍͏ ໰୊͸ҎԼͷΑ͏ʹఆࣜԽͰ͖Δ i = 1,2,...,N Ri ¯ R ¯ R = 1 N N ∑ i=1 Ri ¯ R ℬ = {A, B, C, D, E} g = [gA , gB , gC , gD , gE ] c = [cA , cB , cC , cD , cE ] ¯ C max g E[ ¯ R|g] TU ∑ b∈ℬ cb gb = ¯ C, gb ≥ 0 (∀b ∈ ℬ) ࠷దԽ໰୊ͷఆࣜԽ 8
  3. ͷ෼ղ E[ ¯ R|g] w ͕ཅͳܗͰ෼͔Ε͹ղ͚Δɻͦ͜Ͱ໨తؔ਺Λ෼ղ͢Δɻ E[ ¯ R|g] 9

    max g E[ ¯ R|g] TU ∑ b∈ℬ cb gb = ¯ C, gb ≥ 0 (∀b ∈ ℬ) E[ ¯ R|g] = E[ 1 N N ∑ i Ri |g] = 1 N N ∑ i E[Ri |g] = 1 N N ∑ i Pr(Ri = 1|g) E[ ¯ R|g] ≈ 1 N ∑ i Pr(R = 1|g) = Pr(R = 1|g) ≈ 1 − ∏ b∈ℬ (1 − Pr(Rb = 1|gb ))  ͷ஋ΛऔΔ֬ ཰ม਺ͷظ଴஋͸Λ औΔ֬཰ʹ౳͍͠ ์ૹہ͝ͱͷ઀৮͕ಠཱͱԾఆ͢Δͱɺ์ૹہ߹ࢉ ͷ઀৮֬཰͸์ૹہ͝ͱͷඇ઀৮֬཰ͷੵΛ͔Β Ҿ͘͜ͱͰٻΊΒΕΔ ݸਓ ͝ͱʹάϩεϦʔνͰ৚݅෇͚ΒΕͨ ઀৮֬཰Λߟ͑Δͷ͸೉͍͠ͷͰɺ͜͜Ͱ ͸શݸਓͰҰఆͰ͋ΔͱԾఆ i
  4. ͷ෼ղ E[ ¯ R|g] w ์ૹہ߹ࢉͰͷ઀৮֬཰͸ɺ์ૹہ͝ͱͷ઀৮֬཰͕෼͔Ε͹ܭࢉՄೳ w ྫ͑͹ہͰߟ͑Δͱɺ  w

    ์ૹہ͝ͱͷग़ߘྔ ͔Βɺہ୯ମͰͷϢχʔΫϦʔν Λ༧ଌग़དྷΕ͹ྑ͍ R(gA , gB ) = 1 − (1 − RA (gA ))(1 − RB (gB )) = 1 − (1 − RA − RB + RA RB ) = RA + RB − RA RB gb Rb 10 R(g) ≈ 1 − ∏ b∈ℬ (1 − Rb (gb )) R(g) = Pr(R = 1|g), Rb (gb ) = Pr(Rb = 1|gb ) RA RB RA RB max g {1 − ∏ b∈ℬ (1 − Rb (gb ))} TU ∑ b∈ℬ cb gb = ¯ C, gb ≥ 0 (∀b ∈ ℬ) ͱ͢Δ RA ͱ͢Δ RB
  5. ਺஋࣮ݧ 12 ์ૹہ άϩεϦʔν(%) ϢχʔΫϦʔν(%) ౤Լίετ(ສԁ) A 38.7 31.0 483.8

    B 35.8 28.4 429.5 C 38.3 29.0 421.4 D 44.4 29.0 354.9 E 41.4 26.1 310.5 Total 198.6 81.6 2000.0 ์ૹہ άϩεϦʔν1% ͋ͨΓͷίετ A 12.5ສԁ B 12ສԁ C 11ສԁ D 8ສԁ E 7.5ສԁ Total Budget 2000ສԁ TDJQZPQUJNJ[FNJOJNJ[F Ͱ࠷దԽՄೳɻ໨తؔ਺ Ҿ਺໊GVO Ͱ͸ͦΕͧΕͷ์ૹہ͝ͱʹਪఆ͞ΕͨIJMMؔ਺Λ༻͍ͯɺ์ૹہ ͝ͱͷϦʔνΛܭࢉ͠ɺ Λܭࢉ͢Δɻ ੍໿৚݅ Ҿ਺໊DPOTUSBJOUT Ͱ͸ɺ์ૹہ͝ͱͷύʔίετ ͱάϩεϦʔνͷੵͷ࿨͕τʔλϧ༧ࢉͱҰக͢Δ͜ͱΛදࣜ͢ Λ༩͑Δʢ౳੍ࣜ໿ʣ ࠷దԽख๏ Ҿ਺໊NFUIPE Ͱ͸l4-421zΛࢦఆ͢Δ͜ͱͰɺ ੍໿෇͖࠷దԽͷ࣮ߦ͕Մೳɻ R(g) ≈ 1 − ∏ b∈ℬ (1 − Rb (gb )) ∑ b∈ℬ cb gb − ¯ C = 0 ࠷దԽ