4.11. 配送経路
複数の車両を使用して、各顧客の品物を回収し、倉庫まで運びます。1 つの車両で複数の顧客から品物を回収することはできますが、収容できる容量には限りがあります。
基本例 (CVRP) のほかに、時間枠の設定が加わった例 (CVRPTW) もあります。
ハード制約:
- 車両の容量: 車両は、車載容量を超えて品物を運ぶことができない。
時間枠 (CVRPTW のみ):
- 移動時間: 別の場所に移動する場合には時間がかかる。
- 顧客対応の時間: 車両は顧客に対応している時間、顧客先にとどまる必要がある。
- 顧客の準備が整う時間: 顧客の準備が整う前に車両が到着する可能性があるが、準備ができるまで待機してから顧客に対応する必要がある。
- 顧客が設定した締め切り時間: 車両は、顧客が設定した締め切り時間までに到着する必要がある。
ソフト制約:
- 合計距離: 車両が移動する合計距離 (ガソリンの消費量) を最小限に抑える。
CVRP (Capacitated Vehicle Routing Problem) と CVRPTW (Capacitated Vehicle Routing Problem with Time Window) は、「VRP Web」で定義されています。
図4.8 価値提案
問題の規模
CVRP インスタンス (時間枠なし):
belgium-n50-k10 has 1 depots, 10 vehicles and 49 customers with a search space of 10^74. belgium-n100-k10 has 1 depots, 10 vehicles and 99 customers with a search space of 10^170. belgium-n500-k20 has 1 depots, 20 vehicles and 499 customers with a search space of 10^1168. belgium-n1000-k20 has 1 depots, 20 vehicles and 999 customers with a search space of 10^2607. belgium-n2750-k55 has 1 depots, 55 vehicles and 2749 customers with a search space of 10^8380. belgium-road-km-n50-k10 has 1 depots, 10 vehicles and 49 customers with a search space of 10^74. belgium-road-km-n100-k10 has 1 depots, 10 vehicles and 99 customers with a search space of 10^170. belgium-road-km-n500-k20 has 1 depots, 20 vehicles and 499 customers with a search space of 10^1168. belgium-road-km-n1000-k20 has 1 depots, 20 vehicles and 999 customers with a search space of 10^2607. belgium-road-km-n2750-k55 has 1 depots, 55 vehicles and 2749 customers with a search space of 10^8380. belgium-road-time-n50-k10 has 1 depots, 10 vehicles and 49 customers with a search space of 10^74. belgium-road-time-n100-k10 has 1 depots, 10 vehicles and 99 customers with a search space of 10^170. belgium-road-time-n500-k20 has 1 depots, 20 vehicles and 499 customers with a search space of 10^1168. belgium-road-time-n1000-k20 has 1 depots, 20 vehicles and 999 customers with a search space of 10^2607. belgium-road-time-n2750-k55 has 1 depots, 55 vehicles and 2749 customers with a search space of 10^8380. belgium-d2-n50-k10 has 2 depots, 10 vehicles and 48 customers with a search space of 10^74. belgium-d3-n100-k10 has 3 depots, 10 vehicles and 97 customers with a search space of 10^170. belgium-d5-n500-k20 has 5 depots, 20 vehicles and 495 customers with a search space of 10^1168. belgium-d8-n1000-k20 has 8 depots, 20 vehicles and 992 customers with a search space of 10^2607. belgium-d10-n2750-k55 has 10 depots, 55 vehicles and 2740 customers with a search space of 10^8380. A-n32-k5 has 1 depots, 5 vehicles and 31 customers with a search space of 10^40. A-n33-k5 has 1 depots, 5 vehicles and 32 customers with a search space of 10^41. A-n33-k6 has 1 depots, 6 vehicles and 32 customers with a search space of 10^42. A-n34-k5 has 1 depots, 5 vehicles and 33 customers with a search space of 10^43. A-n36-k5 has 1 depots, 5 vehicles and 35 customers with a search space of 10^46. A-n37-k5 has 1 depots, 5 vehicles and 36 customers with a search space of 10^48. A-n37-k6 has 1 depots, 6 vehicles and 36 customers with a search space of 10^49. A-n38-k5 has 1 depots, 5 vehicles and 37 customers with a search space of 10^49. A-n39-k5 has 1 depots, 5 vehicles and 38 customers with a search space of 10^51. A-n39-k6 has 1 depots, 6 vehicles and 38 customers with a search space of 10^52. A-n44-k7 has 1 depots, 7 vehicles and 43 customers with a search space of 10^61. A-n45-k6 has 1 depots, 6 vehicles and 44 customers with a search space of 10^62. A-n45-k7 has 1 depots, 7 vehicles and 44 customers with a search space of 10^63. A-n46-k7 has 1 depots, 7 vehicles and 45 customers with a search space of 10^65. A-n48-k7 has 1 depots, 7 vehicles and 47 customers with a search space of 10^68. A-n53-k7 has 1 depots, 7 vehicles and 52 customers with a search space of 10^77. A-n54-k7 has 1 depots, 7 vehicles and 53 customers with a search space of 10^79. A-n55-k9 has 1 depots, 9 vehicles and 54 customers with a search space of 10^82. A-n60-k9 has 1 depots, 9 vehicles and 59 customers with a search space of 10^91. A-n61-k9 has 1 depots, 9 vehicles and 60 customers with a search space of 10^93. A-n62-k8 has 1 depots, 8 vehicles and 61 customers with a search space of 10^94. A-n63-k9 has 1 depots, 9 vehicles and 62 customers with a search space of 10^97. A-n63-k10 has 1 depots, 10 vehicles and 62 customers with a search space of 10^98. A-n64-k9 has 1 depots, 9 vehicles and 63 customers with a search space of 10^99. A-n65-k9 has 1 depots, 9 vehicles and 64 customers with a search space of 10^101. A-n69-k9 has 1 depots, 9 vehicles and 68 customers with a search space of 10^108. A-n80-k10 has 1 depots, 10 vehicles and 79 customers with a search space of 10^130. F-n45-k4 has 1 depots, 4 vehicles and 44 customers with a search space of 10^60. F-n72-k4 has 1 depots, 4 vehicles and 71 customers with a search space of 10^108. F-n135-k7 has 1 depots, 7 vehicles and 134 customers with a search space of 10^240.
CVRPTW インスタンス (時間枠あり):
belgium-tw-d2-n50-k10 has 2 depots, 10 vehicles and 48 customers with a search space of 10^74. belgium-tw-d3-n100-k10 has 3 depots, 10 vehicles and 97 customers with a search space of 10^170. belgium-tw-d5-n500-k20 has 5 depots, 20 vehicles and 495 customers with a search space of 10^1168. belgium-tw-d8-n1000-k20 has 8 depots, 20 vehicles and 992 customers with a search space of 10^2607. belgium-tw-d10-n2750-k55 has 10 depots, 55 vehicles and 2740 customers with a search space of 10^8380. belgium-tw-n50-k10 has 1 depots, 10 vehicles and 49 customers with a search space of 10^74. belgium-tw-n100-k10 has 1 depots, 10 vehicles and 99 customers with a search space of 10^170. belgium-tw-n500-k20 has 1 depots, 20 vehicles and 499 customers with a search space of 10^1168. belgium-tw-n1000-k20 has 1 depots, 20 vehicles and 999 customers with a search space of 10^2607. belgium-tw-n2750-k55 has 1 depots, 55 vehicles and 2749 customers with a search space of 10^8380. Solomon_025_C101 has 1 depots, 25 vehicles and 25 customers with a search space of 10^40. Solomon_025_C201 has 1 depots, 25 vehicles and 25 customers with a search space of 10^40. Solomon_025_R101 has 1 depots, 25 vehicles and 25 customers with a search space of 10^40. Solomon_025_R201 has 1 depots, 25 vehicles and 25 customers with a search space of 10^40. Solomon_025_RC101 has 1 depots, 25 vehicles and 25 customers with a search space of 10^40. Solomon_025_RC201 has 1 depots, 25 vehicles and 25 customers with a search space of 10^40. Solomon_100_C101 has 1 depots, 25 vehicles and 100 customers with a search space of 10^185. Solomon_100_C201 has 1 depots, 25 vehicles and 100 customers with a search space of 10^185. Solomon_100_R101 has 1 depots, 25 vehicles and 100 customers with a search space of 10^185. Solomon_100_R201 has 1 depots, 25 vehicles and 100 customers with a search space of 10^185. Solomon_100_RC101 has 1 depots, 25 vehicles and 100 customers with a search space of 10^185. Solomon_100_RC201 has 1 depots, 25 vehicles and 100 customers with a search space of 10^185. Homberger_0200_C1_2_1 has 1 depots, 50 vehicles and 200 customers with a search space of 10^429. Homberger_0200_C2_2_1 has 1 depots, 50 vehicles and 200 customers with a search space of 10^429. Homberger_0200_R1_2_1 has 1 depots, 50 vehicles and 200 customers with a search space of 10^429. Homberger_0200_R2_2_1 has 1 depots, 50 vehicles and 200 customers with a search space of 10^429. Homberger_0200_RC1_2_1 has 1 depots, 50 vehicles and 200 customers with a search space of 10^429. Homberger_0200_RC2_2_1 has 1 depots, 50 vehicles and 200 customers with a search space of 10^429. Homberger_0400_C1_4_1 has 1 depots, 100 vehicles and 400 customers with a search space of 10^978. Homberger_0400_C2_4_1 has 1 depots, 100 vehicles and 400 customers with a search space of 10^978. Homberger_0400_R1_4_1 has 1 depots, 100 vehicles and 400 customers with a search space of 10^978. Homberger_0400_R2_4_1 has 1 depots, 100 vehicles and 400 customers with a search space of 10^978. Homberger_0400_RC1_4_1 has 1 depots, 100 vehicles and 400 customers with a search space of 10^978. Homberger_0400_RC2_4_1 has 1 depots, 100 vehicles and 400 customers with a search space of 10^978. Homberger_0600_C1_6_1 has 1 depots, 150 vehicles and 600 customers with a search space of 10^1571. Homberger_0600_C2_6_1 has 1 depots, 150 vehicles and 600 customers with a search space of 10^1571. Homberger_0600_R1_6_1 has 1 depots, 150 vehicles and 600 customers with a search space of 10^1571. Homberger_0600_R2_6_1 has 1 depots, 150 vehicles and 600 customers with a search space of 10^1571. Homberger_0600_RC1_6_1 has 1 depots, 150 vehicles and 600 customers with a search space of 10^1571. Homberger_0600_RC2_6_1 has 1 depots, 150 vehicles and 600 customers with a search space of 10^1571. Homberger_0800_C1_8_1 has 1 depots, 200 vehicles and 800 customers with a search space of 10^2195. Homberger_0800_C2_8_1 has 1 depots, 200 vehicles and 800 customers with a search space of 10^2195. Homberger_0800_R1_8_1 has 1 depots, 200 vehicles and 800 customers with a search space of 10^2195. Homberger_0800_R2_8_1 has 1 depots, 200 vehicles and 800 customers with a search space of 10^2195. Homberger_0800_RC1_8_1 has 1 depots, 200 vehicles and 800 customers with a search space of 10^2195. Homberger_0800_RC2_8_1 has 1 depots, 200 vehicles and 800 customers with a search space of 10^2195. Homberger_1000_C110_1 has 1 depots, 250 vehicles and 1000 customers with a search space of 10^2840. Homberger_1000_C210_1 has 1 depots, 250 vehicles and 1000 customers with a search space of 10^2840. Homberger_1000_R110_1 has 1 depots, 250 vehicles and 1000 customers with a search space of 10^2840. Homberger_1000_R210_1 has 1 depots, 250 vehicles and 1000 customers with a search space of 10^2840. Homberger_1000_RC110_1 has 1 depots, 250 vehicles and 1000 customers with a search space of 10^2840. Homberger_1000_RC210_1 has 1 depots, 250 vehicles and 1000 customers with a search space of 10^2840.
4.11.1. 配送経路のドメインモデル
時間枠ありの配送経路のドメインモデルでは、シャドウ変数の機能を多用します。こうすることで、arrivalTime や departureTime などのプロパティーがドメインモデルで直接利用できるため、制約をより自然に表現できます。
直線距離ではなく道路の距離
車は、直線距離を移動するのではなく、道路や高速道路を使用する必要があります。ビジネスの観点からすると、これは非常に重要です。
最適化アルゴリズムでは、2 点の距離を検索できている (できれば、事前に計算されている) 場合には、これは特に重要ではありません。移動のコストについては、距離の代わりに移動時間、ガソリン代、またはこれらの重み関数をベースにすることも可能です。GraphHopper (埋め込み可能なオフライン Java エンジン)、Open MapQuest (web サービス)、Google Maps Client API (web サービス) など、移動コストを事前に計算する技術があります。
また、Leaflet や Google Maps for developers など、移動コストをレンダリングする技術もあります。optaplanner-webexamples-*.war には、このようなレンダリングのデモ例が含まれています。
GraphHopper または Google Map Directions を使用して実際の経路をレンダリングすることも可能ですが、高速道路で経路が重なるため、停止する順番を確認するのが困難になります。
2 点間の移動コストは、Planner で使用したのと同じ最適化条件を使用する点に注意してください。たとえば、GraphHopper はデフォルトで、最短ではなく、最速の経路を返します。「最速」の GPS 経路の km (またはマイル) の距離を使用して、Planner で「最短」の移動を最適化しないようにしてください。以下のように、準最適な解が導き出される可能性があります。
一般的な考え方とは異なり、多くのユーザーは最短の経路ではなく、最速の経路を使用したいと考えます。通常の道路よりも高速道路の使用を好みます。舗装されていない道よりも舗装されている道路を好みます。実際には、最速の経路と、最短の経路が同じであることはほとんどありません。
