䝽䘱䘀䗃ѝⲴ䖖䖶䐟 䰞仈⹄ウ㔬䘠㾱˖䝽䘱䘀䗃ѝⲴ䖖䖶䐟 䰞仈аⴤ 䘀ㆩ 亶 Ⲵ⹄ウ✝⛩ѻаDŽ Ҿ䖖䖶䐟 䰞仈Ⲵ ⭘㛼 ˈ 㓣 㔃Ҷ ㊫䰞仈 ≲䀓㇇⌅Ⲵ⹄ウ䘋 ˈ 䎻 ˈѪ⴨ 䢤DŽ䭞䇽˖⢙⍱䝽䘱˗䖖䖶䐟 䰞仈˗䙊⭘ ㇇⌅˗㔬䘠DŽThe Current Situation and Development Trends on Vehicle Routing Problems of distribution managementAbstract: Vehicle routing problem is one of the attractive research area in the circles of operations research. In this paper, on the basis of introducing briefly the application background, the research classified the vehicle routing problem, analyzed and summarized the progress of different type of problems and solution algorithms. Furthermore, the research progress of the problems is also discussed. It is expected to provide inference for relevant research work.Key words: distribution management; vehicle routing problem; heuristics; overview.䀶䲿⵰㓿⍾Ⲵ 、 Ⲵ䘋↕ˈ⢙⍱ӗъ䗵䙏 ˈ Ѫ ≁㓿⍾ Ⲵ 㜹 ⹰ӗъˈ ≤ 㺑䟿ањ ⧠ԓ 〻 㔬 Ⲵ䟽㾱 ḷDŽкц㓚80 ԓԕ ˈ ⢙⍱ӗъⲴ ˈ⢙⍱ъ Ѫањ⤜・Ⲵӗъ䗵䙏 䎧DŽ❦㘼ˈ ⢙⍱ъӽ Ҿ 㓗䱦⇥ˈо 䗮 ⴨∄䘈 а Ⲵ 䐍ˈ ѝ ケ Ⲵ䰞仈 ⢙⍱ 䖳儈DŽ 㔏䇑[1]ˈ ⢙⍱ 䍩⭘Ѫ7.1зӯ ˈ GDPⲴ∄䟽Ѫ17.8%ˈ㘼 㖾 ㅹ 䗮 ⢙⍱ 㓖 GDPⲴ10%ˈ丙 ㅹѝㅹ 䗮 Ⲵ∄䟽㓖Ѫ15%DŽ䗷儈Ⲵ⢙⍱ Ѫ 㓖 ≁㓿⍾ Ⲵ䟽㾱 ㍐DŽ ↔ˈ 儈⢙⍱ъⲴ、 ㇑⨶≤ ǃ䱽վ⢙⍱ ӏ䴰䀓 Ⲵ 䭞 䰞仈DŽ䘀䗃 ⢙⍱ Ⲵ䟽㾱㓴 䜘 DŽ 䘀䗃 ⢙⍱ 䍩⭘Ⲵ∄ 䖳儈ˈ㓖Ѫ50%[1]DŽ䱽վ䘀䗃 ˈ 儈䘀䗃 ⦷ǃ 䘋⢙⍱ъ 㔝 Ⲵ䟽㾱䙄 DŽѪ⢙⍱䝽䘱ѝⲴ 䭞а⧟ˈ䖖䖶䐟 䰞仈˄Vehicle Routing Problem, VRP˅ 䘀䗃㓴㓷Ո ⲴṨ 䰞仈ѻаDŽ㠚1959 Dantzig Ramser[2]俆⅑ ԕ ˈVRP Ѫ䘀ㆩ 㓴 Ո 亶 Ⲵ ⋯о⹄ウ✝⛩ѻаDŽ⧠ ⭏ӗѝˈ䛞 䘀䗃ǃ 䖖 ⊭䖖䈳 ㅹ䈨 䰞仈䜭 ԕ 䊑ѪVRPDŽ ↔ˈ VRPⲴ␡ ⹄ウˈ ⵰䟽㾱Ⲵ、 ѹ 〻 ⭘ԧ DŽ 㓣 㔃ҶVRP Ⲵ⹄ウ䘋 ˈ ҶVRPⲴ 䎻 ˈѪ⴨ 㓿傼 䐟DŽ1. 䖜䖼䐥 䰤从Ⲻ 䘦Ր㔏кˈ 䖖䖶䐟 䰞仈Ⲵа㡜 䘠Ѫ[3, 4]˖ а㌫ 㔉 Ⲵ ˄䘱䍗⛩ 䍗⛩˅ˈ⺞ 䘲 Ⲵ䝽䘱䖖䖶㹼傦䐟㓯ˈ Ӿ䝽䘱ѝ ˈ ⅑ 䇯䰞 њ ⛩ˈ 䘄 䝽䘱ѝ ˈ ┑䏣а Ⲵ㓖 Ԧл˄ 䖖䖶䖭䍗䟿ǃ 䴰≲䟿ǃ 䰤デ䲀 ㅹ˅ˈ 䘀䗃 ˄ ⭘䖖䖶 ǃ䖖䖶㹼傦䐟〻 䰤˅䗮 DŽ 1 ⽪ˈ ѝⲴ Ṷ㺘⽪䖖䖶 ⛩˄ 䖖 䝽䘱ѝ ˅ˈ ⛩㺘⽪䴰㾱䇯䰞Ⲵ ⛩ˈ㓯⇥㺘⽪є⛩ѻ䰤Ⲵ䘎 䐟⇥ˈ ѝ⇿ 㓯⇥ ⵰ањ䍩⭘˄ 䐍⿫ 㹼傦 䰤˅DŽ1 䖜䖼䐥 䰤从⽰Fig.1 Schematic figure of VRPⲴ䖖䖶䐟 䰞仈ѫ㾱 ԕлṨ 㾱㍐[5]˖䚃䐟㖁㔌˄Road Network˅ǃ ˄Customer˅ǃ䝽䘱ѝ ˄䖖 ˅˄Distribution Center, Depot˅ǃ䖖䖶˄Vehicle˅ǃ傮傦 ˄Driver˅ǃ䲀 Ԧǃ㹼傦䍩⭘ 㹼傦 䰤ˈԕ Ո ⴞḷ˄Objective˅DŽ њ㾱㍐Ⲵ⢩ 㿱㺘1DŽ㺞1 䖜䖼䐥 䰤从Ⲻ㓺 㾷㍖Tab. 1 Key elements of VRP㓴 㾱㍐⢩䚃䐟㖁㔌1. VRPⲴṨ 㾱㍐ѻа˗2. 䙊 ⭡ 㢲⛩ 㓴 Ⲵ䍻 㺘⽪DŽ 㺘⽪䐟⇥ˈ⛩㺘⽪䚃䐟Ӕ ⛩ǃ䝽䘱ѝ ˗3. ṩ 䐟㖁⴨㚄є⛩䰤Ⲵ䚃䐟⢩ ˈ⴨ Ⲵ Ѫ ˗ 䍻Ҹ⇿ 䶎䍏Ⲵ䍩⭘ 䟽ˈ є⛩䰤Ⲵ䘀㹼䐍⿫ˈ䘀㹼 䰤ㅹDŽ1. VRPѝ䖖䖶 Ⲵ 䊑ˈ Ҿ㖁㔌 ѝⲴ ⛩˗2. ⛩Ⲵ ㊫ ˖䘱䍗 ˄Delivery˅ǃ 䍗 ˄Pickup˅ є㘵˗3. ⛩Ⲵ 䴰≲䟿˖а⅑┑䏣ˈ ˄Split˅;4. ⛩Ⲵ 䰤˖䖖䖶 ӔԈ 䍗⢙ 㣡䍩Ⲵ 䰤˗5. ⛩Ⲵ 䰤デ˖ 㾱≲ Ⲵ 䰤 䲀ˈ Ѫ⺜ 䰤デ[6]䖟 䰤デ[7]˗6. ⛩ Ⲵ ⅑ ˖ Ո 㓗˗ ㊫ 亪ˈ 䘱䍗 䍗˗7. ⛩Ⲵ ㌫˖ ḀӋ лˈ ⛩Ⲵ䝽 ㌫ˈ Ӿањ 䍗❦ 䘱 оѻ䝽 Ⲵ⢩ 䘱䍗 ⛩DŽ䝽䘱ѝ 1. ⇿ 䖖䖶䐟㓯Ⲵ䎧⛩ 㓸⛩ˈ Ҿ㖁㔌 ѝⲴ ⛩˗2. 䖖 䟿˖ањ䖖 њ䖖 ˗3. 䖖䖶 䘄 䖖 ˖ VRP[5, 8]ˈ䰝 VRP˗4. 䖖 䰤デ˖ḀӋ лˈ䖖 䲀 ⢩↺㾱≲˄ Ⲵ 䰤 䰤˅˗5. 䖖 ѻ䰤Ⲵ ㌫˖ḀӋ лˈ䖖 䰤 䝽 ㌫ˈ Ӿањ䖖 䛓䟼 Ⲵ䖖䖶 享 оѻ䝽 Ⲵ Ⲵ䖖 DŽ䖖䖶1. 䖖䖶 㠚 䘈 』ˈ ԫ 䘄 ˗2. 䖖䖶㊫ ˖ VRPѝ 䇮䖖䖶Ѫ а㊫ ˈն 䱵䝽䘱㇑⨶ѝˈ䖖䱏 ⭡ н 㻵䖭㜭 ǃн ԕ Ⲵ 䖖䖶㓴 ˗3. 䖖䖶Ⲵ㻵䖭㜭 ˖䖖䖶 Ⲵ䖭䟽䟿 Ⲵ㻵䖭 䟿ㅹ˗4. 䖖䖶 ˖䖖䖶Ⲵ ˈ 䖖䖶䍝㖞䍩⭘ㅹ˗ ⭘ ս 䟼Ⲵ䍩⭘ ս 䰤Ⲵ䍩⭘ 㺑䟿˗5. 䖖䖶Ⲵ 㔝 ˄Duration˅˖䖖䖶 Ⲵ 䇨㹼傦䐍⿫ 䰤DŽ傮傦 1. 㔉傮傦 䘱䍗ԫ ˈ 享ㅖ 䰤Ⲵ 㿴 DŽ⴨ 傮傦 䲀 Ԧа㡜䜭 ⴨ Ⲵ䖖䖶䲀 ԦѝDŽ䲀 Ԧ1. 䖖䖶Ⲵ 䍏䖭н㜭䎵䗷䖖䖶Ⲵ㻵䖭㜭 ˗2. 㾱≲䘱䍗ǃ 䍗ǃ 䘱䍗 ˗3. 㾱≲Ⲵ 䰤デ 傮傦 Ⲵ 䰤 ˗4. 䇯䰞 Ⲵ亪 㾱≲DŽ㹼傦䍩⭘㹼傦 䰤1. ⛩о ⛩ѻ䰤ǃ䝽䘱ѝ о ⛩ѻ䰤Ⲵ㹼傦䐍⿫ 㹼傦 䰤DŽՈ ⴞḷ1. 䘀䗃 ˈ Ҿ 䴰Ⲵ䖖䖶 ˄ 㓯䐟 ˅ǃ 㹼傦䐍⿫˄ 䰤˅˗2. о Ⲵн ㅹ Ⲵ 㖊 ˗3. 㺑 㓯䐟кⲴ㹼傦 䰤 䖖䖶䖭䟽䟿DŽ2. 䖜䖼䐥 䰤从Ⲻ ㊱䙊䗷 к䘠VRPṨ 㾱㍐䱴 н Ⲵ⢩ 䘲 Ⲵ ˈ 㹽⭏ ⿽н Ⲵ䖖䖶䐟 䰞仈㊫ DŽ ⴨ ⹄ウ[5, 9, 10]ˈ 㿱ⲴVRP ㊫ 㹽⭏㊫ DŽ VRP㊫ 㻵䖭㜭 ⲴVRPǃ 䐟〻䮯 ⲴVRPǃ 䰤デⲴVRPǃ 〻䘀䗃ⲴVRPˈԕ 䘱䍗ⲴVRP˗ ⁑ Ⲵ ⹰кˈ㔃 н Ⲵ㓖 Ԧˈ ҶVRPⲴ㹽⭏㊫ ˈ 䲿 VRPǃ⁑㋺VRPǃ VRPㅹDŽ њ㊫ Ⲵ⢩ 㿱㺘2DŽ㺞2 䖜䖼䐥 䰤从Ⲻ㊱Tab. 2 Types of VRP ㊫ ⢩㻵䖭㜭 ⲴVRP˄Capacitated VRP, CVRP˅1. VRPѝ Ⲵ ˗2. 䜭 Ҿ㾱䘱䍗Ⲵ 㾱 䍗Ⲵˈ 䴰≲䟿 ⸕ˈфн ˗3. 䖖䖶 ㊫ ф䜭 ањ䝽䘱ѝ ˗4. 䖖䖶 㻵䖭㜭 Ⲵ䲀 ˗5. Ո ⴞḷ Ⲵ 䍩⭘ DŽ䐟〻䮯 ⲴVRP˄Distance-Constrained andCapacitated VRP, DCVRP˅1. 䖖䖶㻵䖭㜭 䲀 ˈ 䐟〻䮯 䲀 DŽ䰤デⲴVRP˄VRP with time windows, VRPTW˅1. 䲔Ҷ䖖䖶㻵䖭㜭 Ⲵ㓖 ˈ⇿њ 䜭 ањоѻ⴨㚄㌫Ⲵ㾱≲ Ⲵ 䰤 䰤˗2. Ѫ⺜ 䰤デVRP 䖟 䰤デVRPDŽ⺜ 䰤デ ⇿亩ԫ 享 㾱≲Ⲵ 䰤 ˈ䖟 䰤デ Ḁ亩ԫ н㜭 㾱≲Ⲵ 䰤㤳 ˈ 㔉Ҹа Ⲵ 㖊[11]DŽ〻䘀䗃ⲴVRP˄VRP with backhauls, VRPB˅1. 䳶㻛 2њ 䳶˖㾱≲䘱 а 䟿䍗⢙Ⲵ 〻 ԕ 㾱≲ а 䟿䍗⢙䘀 䝽䘱ѝ Ⲵ 〻 ˗2. 〻 〻 Ⲵ䴰≲ ⸕ф ˗3. 〻 享 Ҿ 〻 DŽ䘱䍗ⲴVRP˄VRP with pickup and delivery˅1. 䖖䖶нӵ 䘱䍗 ˈҏ Ⲵ䍗⢙˗2. ⇿њ ⛩ˈ㿴 㻵DŽ㹽⭏㊫VRP[5]˄Open VRP, OVRP˅1.н㾱≲䖖䖶 ԫ 䘄 ⛩ˈ 㤕㾱≲䘄 ⛩ˈ ⋯ 〻䐟㓯䘄 DŽ䖖 VRP˄Multiple Depots VRP, MDVRP˅1. њ䝽䘱ѝ ˈ䖖䖶 ԕӾԫօањ䝽䘱ѝ ⍮ ˈ ԫ ˈ䖖䖶ҏ ԕ䘄 ԫօањ䝽䘱ѝ ˗2. 䰞仈 Ѫ DŽ䙊 䇮Ḁњ䖖 Ⲵ䖖䖶ӽ䴰䘄 䈕䖖 ˈ≲䀓 ˈ ➗Ḁ⿽㇇⌅ 䝽㔉Ḁњ䖖 ˈ❦ ➗ а䖖 VRP䘋㹼≲䀓[12]DŽ䖖 VRP[13]˄Heterogeneous Fleet VRP,HVRP˅1. 䖖䖶Ⲵ н ˈ䙊 䖖䖶Ⲵ䖭䟽䟿на㠤DŽ䴰≲ ⲴVRP[14]˄VRPwith Split Deliveries˅1. Ⲵ䴰≲ ԕ 㻛 њ䖖䖶 DŽ䲿 VRP˄Stochastic Vehicle Routing Problem, SVRP˅1. Ѫ䲿 VRPǃ䲿 䴰≲VRPǃ䲿 㹼傦 䰤VRP˗2. 䲿 VRP ⢙⍱亶 㓿 ⧠˗3. Ҿ䲿 䴰≲VRPˈ⺞ Ⲵ ⸕ˈն Ⲵ ⺞䴰≲䟿 ⸕ˈ ⟳⋩䝽䘱䰞仈˗4. Ҿ䲿 㹼傦 䰤VRPˈ⹄ウ䖳 ˈ㘼 䲿 㖁㔌 ⸝䐟 䰞仈Ⲵ⹄ウ䖳␡ [15, 16]DŽ⁑㋺VRP˄Fuzzy VRP, FVRP˅1. ḀӋ ˄ 䴰≲ǃ䐍⿫ǃ 䰤デ[3]˅ ⌅ ⺞ 䘠˗2. ⁑㋺ᾲ ⁑ ㇇⌅ 䀓 ↔㊫䰞仈DŽVRP˄Periodic VRP, PVRP˅1. VRPⲴ ˈVRP⹄ウⲴ 䖖䖶Ⲵ ˈ㘼PVRP 䖖䖶Ⲵањ Ⲵ ˈ ањ ˈ⇿њ ┑䏣䴰≲Ⲵ лˈ 㻛 а⅑DŽ䶎 〠㖁㔌VRP˄Asymmetric network VRP, AVRP˅1. ⧠ ѝˈ⭡Ҿ 㹼䚃 ⾱→ 䖜ㅹӔ䙊㇑ ˈ є 䘄Ⲵ䐍⿫ 䰤 н⴨ㅹ˗2. ⴞ Ⲵ≲䀓㇇⌅䜭 Ҿ䶎 〠TSP䰞仈Ⲵ㇇⌅[17]DŽVRP˄Dynamic VRP, DVRP˅1. 䖖䖶 ˈ䈳 н⺞ 䜘 н⺞ ˗ 䖖䖶 ˈ ⧠ Ⲵ䈳 ˗2. ѹкˈ䲿 VRPˈ⁑㋺VRPˈԕ 㖁㔌VRP䜭 ҾDVRPDŽ↔ ˈ䘈 ԕ 䰤デ㓖 о Ԇ㓖 Ԧ㔃 ˈ 䰤デ㓖 Ⲵ 䘱 ъVRPˈ 䰤デ㓖 Ⲵ 〻䘀䗃VRPㅹDŽ3. 䖜䖼䐥 䰤从Ⲻ≸䀙㇍⌋VRP ⭼ 䇔ⲴNP䳮䰞仈[18]DŽVRP㻛 ˈ ≲䀓㇇⌅Ⲵ 䙐аⴤ ⹄ウⲴ䟽⛩ 䳮⛩DŽⴞ ⿽≲䀓VRPⲴ㇇⌅ˈ 䍘к Ѫ㋮⺞㇇⌅ ㇇⌅є ㊫DŽ㋮⺞㇇⌅ Ҿ ѹ 䇱 ˈ ≲ Ո䀓Ⲵ㇇⌅DŽⴞ ⭘Ҿ≲䀓VRP ԓ㺘 Ⲵ㋮⺞㇇⌅ѫ㾱 ⭼䲀⌅[19]˄Branch and Bound Approach˅ǃ 䶒⌅[20]˄Cutting Planes Approach˅ǃ㖁㔌⍱㇇⌅˄Network Flow Approach˅[21] 㿴 ⌅˄Dynamic Programming Approach˅[22]DŽ⭡ҾVRP NP-䳮䰞仈ˈ ⴨ Ⲵ㋮⺞㇇⌅Ⲵ䇑㇇䟿а㡜䲿⵰䰞仈㿴⁑Ⲵ 䮯˗ф ⛩䎵䗷50 ˈ㋮⺞㇇⌅ на ≲ 䰞仈Ⲵ Ո䀓[23]DŽ⴨ ˈ ⭘䰞仈⢩ ѝㅹ䇑㇇ 䰤 㧧 VRPⲴ⅑Ո䀓 ┑ 䀓Ⲵ ㇇⌅ Ѫ 㘵Ԝ⹄ウⲴ䟽⛩DŽ ㇇⌅ Ѫ㓿 ㇇⌅˄Classical Heuristics˅ 䙊⭘ ㇇⌅˄Metaheuristics˅DŽ3.1 㔅 ㇍⌋˄1˅㢲㓖㇇⌅˄Saving Methods˅⭡Clark WrightҾ1964 俆⅑ [24]ˈ а⿽ Ҿ㢲㓖 Ⲵ䖖䖶䐟㓯䙀↕ 䙐㇇⌅DŽ Ѫ˖ ⇿њ䝽䘱⛩ Ѫа 㓯䐟ˈ 䜘㓯䐟 Ⲵ䳶 Ѫ 䀓DŽ ањ⛩о ањ⛩⴨䘎 а Ⲵ㓯䐟ˈ Ⲵ㓯䐟㜭┑䏣㓖 Ԧˈ 㹼Ⲵˈ Ⲵ㢲㓖 ˄ 䟼〻ǃ䰤ㅹ˅ ѹѪ䘎 䘉є 㓯䐟Ⲵ㢲㓖 DŽӾ 㓯䐟ѝ䘹 㢲㓖 Ⲵ䗩䘋㹼а⅑㓯䐟 ˈ н 㹼ˈ㇇⌅㔃 DŽ䈕㇇⌅ ԕ ┑ 䀓ˈնна 㜭 䇱 Ո䀓DŽ Ո⛩ ⨶ㆰ ф ⧠ˈⴞ Ѫ VRP ㇇⌅ѝӗ⭏ 䀓Ⲵ㇇⌅DŽ䈕㇇⌅ ԕ ˈ 㘵 䘋㇇⌅䘋㹼Ҷ⹄ウˈԕ ≲䀓Ⲵ䍘䟿ˈѫ㾱 ԕл2⿽䙄 ˖а 䙊䗷 Ⲵ 㔃 ㆆ⮕ ⨶㢲㓖 [25, 26]˗Ҽ Ҿ 䝽㇇⌅˄Matching Algorithm˅Ո 䈕㇇⌅ѝⲴ䐟㓯 䗷〻[27, 28]DŽ˄2˅ ㇇⌅˄Sweep Algorithm˅⭡Gillett MillerҾ1974 [29]ˈ ҾĀ 㓴 㓯䐟āⲴ㇇⌅DŽ 䉃Ā 㓴ā ⍮㔉⇿䖶䖖а㓴䝽䘱⛩DŽа⿽ㆰ Ⲵ 㓴 ⌅ ԕ䝽䘱ѝ Ѫ ⛩ˈ ḷ 䶒 Ѫ њ ˈ ↕Ѫ⇿њ Ⲵ⛩ ⍮а䖶䖖DŽ 䉃Ā㓯䐟ā ⇿њ ˈ 䟷⭘ ⌅䘹 䝽⛩ˈ❦ 䟷⭘ ㇇⌅ 㓯䐟DŽ 䘋㹼Ҷа⅑Ā 㓴о㓯䐟āⲴ䐟㓯 䙐 ˈ䘈 䝽⛩ˈ 㔗㔝䘋 Ā 㓴о䐟㓯ā〻 DŽ ↔ ˈⴤ Ⲵ⛩ 䝽 ∅Ѫ→DŽⲴ ㇇⌅а⅑ 㜭ӗ⭏а 䐟㓯ˈ Ѫ ㇇⌅Ⲵ ˈаӋ 㘵⹄ウҶа⅑⭏ 㤕 㹼䖖䖶䐟㓯Ⲵ ⌅ˈ❦ 䙊䗷≲䀓ањ䳶 䰞仈 ⺞ ՈⲴ㓴 ˈ∄䖳 Ⲵ 1-㣡⬓㇇⌅[30, 31] 2-㣡⬓㇇⌅[32]DŽ䴰㾱 Ⲵ ˈ ㇇⌅ 䘲 Ҿ 䶒㔃 ˄Plannar Structure˅ⲴVRPˈ Ҿ䛓Ӌ ḵṬ㺇䚃 ⲴVRPˈ䈕㇇⌅ н䘲⭘DŽ˄3˅䛫 ㇇⌅˄Nearest-Neighbor˅䈕㇇⌅ а⿽ 䙐 䐟㓯⌅ˈ Ӿа ањ䝽䘱⛩Ⲵ㓯䐟 (䙊 䐍⿫䝽䘱ѝ 䘁Ⲵ⛩)ˈӾ 䘹 ⛩ѝㆋ䘹 ⛩ˈ Ӿ ⛩ѝ䘹 ањ⛩ Ѫ 㓯䐟Ⲵ㓸⛩ˈ 㓯䐟Ⲵ DŽԕ↔ 㓯䐟н 䘋㹼 ˈⴤ 㓯䐟⋑ ⛩Ѫ→DŽ ⛩ 䘹 ˈ ㇇⌅㔃 ˗ ˈ ⭏ а Ⲵ 㓯䐟ˈ䟽 䶒Ⲵ㓯䐟 䗷〻ˈⴤ㠣 ⛩䜭 䘹 DŽ˄4˅ ㇇⌅䈕㇇⌅㔃 㢲㓖㇇⌅о䛫 ㇇⌅Ⲵ ˈ 䝽䘱⛩㓣 䐟 ѝԕ 䝽䘱㓯䐟DŽ 〻 о䛫 ㇇⌅⴨լˈ Ӿ 㓯䐟 ˈ 䙐 㓯䐟ˈ ⋑ 㹼 а 㓯䐟DŽ䈕㇇⌅Ⲵ 䭞 䘹 䘲Ⲵ 䝽⛩ 㓯䐟Ⲵ ս㖞䘋㹼 DŽ˄5˅є䱦⇥㇇⌅˄Two-phase Process˅VRPє䱦⇥ ㇇⌅⹄ウѝˈ ԓ㺘 ⲴѪChristofidesㅹӪ[33]ԕ Fisher Jaikumar[34] Ⲵ㇇⌅DŽChistofidesㅹ Ⲵє䱦⇥㇇⌅ѝˈ єњ䱦⇥˖ 䐟㓯 䙐䱦⇥ԕ 㹼䐟㓯 䙐䱦⇥ˈ 䍘 㜭≲䀓ḷ VRPDŽFisher Jaikumarㅹ俆 䙊䗷≲䀓ањ ѹ ⍮䰞仈˄Generalized Assignment ProblemˈGAP˅ ⺞ 㹼Ⲵ亮 ⛩ 㓴ˈ❦ ҾTSP㇇⌅ ⺞ ⇿њ 㓴 Ⲵ䖖䖶䐟㓯DŽGAP ањNP䳮䰞仈ˈ䙊 䙊䗷 Ṭ ㇇⌅ ≲䀓DŽFisher Jaikumar 䲿 Ⲵ⹄ウѝ 㔉 Ҷ亮 ⛩䘹 Ⲵ ⌅ Ṭ ⌅[35, 36]DŽ❦㘼ˈ䈕 ⌅ 䲀 ˖а 䶒ˈ㇇⌅н 㕆〻ˈ 䇑㇇䙏 ⿽ 亮 䘹 ԕ Ṭ 䗷〻 䖳 ˗ а 䶒ˈ㤕䟷⭘Fisher Jaikumar Ⲵ⿽ 亮 ㆋ䘹 ⌅ Ṭ ⌅ˈ ㇇⌅Ⲵ 䖳 ˈа㡜 䳮䗮 㘵 㔉 Ⲵ䇑㇇㔃 [32]DŽ Fisher Jaikumar⹄ウⲴ ⹰кˈBramel Simchi-Levi[37]䙊䗷≲䀓ањ㜭 㓖 䘹 䰞仈 ⺞ ⿽ 亮 ⛩ˈՈ ҶFisher Jaikumar ⿽ 亮 ⛩䘹 ⌅ˈն 㘵 㔉 Ҷ 㜭 㓖 VRPк䶒⍻䈅Ⲵ㔃 DŽ ԕⴻ ˈ 䜘 㓿 ㇇⌅ 1960 -1990 䰤 ⲴDŽ 䱵 ⭘ѝˈ ḷ 䀓 䙐 ⌅ 䀓 䘋 ⌅䜭 Ҿ㓿 ㇇⌅DŽа 䶒ˈ 㜭 䀓オ䰤ѝ䘋㹼 䲀 ㍒ˈ 䖳⸝Ⲵ 䰤 ≲ ┑ 䀓˗ а 䶒ˈ䘉㊫㇇⌅㜭 䖳Ѫ ˈԕ 䱵 ⭘ѝ ⻠ Ⲵ䈨 㓖 Ԧˈ ↔ ъ䖟Ԧ ѝ㻛 ⌋ ⭘DŽⴞ ˈ 㓿 ㇇⌅亶 ˈ Ѿ 䘋㹼䟽 䘋Ⲵ DŽ3.2 䙐⭞ ㇍⌋㠚20ц㓚90 ԓˈ䙊䗷⁑ 㠚❦⧠䊑 䗷〻ˈ䇨 㘵 ҶаӋ≲䀓VRPⲴ䙊⭘ ㇇⌅ˈѪ≲䀓 㿴⁑Ⲵ 䝽䘱䰞仈 Ҷ Ⲵ 䐟DŽ䈕㊫ ⌅ 亶 ㍒ ǃ䇠 㔃 ǃԕ 䀓Ⲵ䟽㓴㔃 䎧 ˈ 䈳 オ䰤䘋㹼 ㍒ˈ⢩ Ⲵ 䘋㹼㓥␡䈅 ˈ䘉ṧ 㜭 Ո䀓䳶ѝ ㍒ˈ 㜭 䐣 䜘 ㍒亶 ˈӾ㘼 䇱Ҷ⿽㗔Ⲵ ṧ ˈ䚯 䲧 䜘 Ոˈ 儈Ҷ Ո䀓Ⲵᾲ⦷DŽ䘁 ˈ䙊⭘ ㇇⌅ 䖳 ˈ ⁑ 䘰⚛㇇⌅ǃ⾱ ㍒㇇⌅ǃ䚇Ր㇇⌅ǃ㲱㗔㇇⌅ǃ㋂ 㗔㇇⌅ǃ⾎㓿㖁㔌ǃ␧ ㇇⌅ㅹ䜭 ⭘ ≲䀓VRPDŽ 㢲 ԓ㺘 Ⲵ䙊⭘ ㇇⌅䘋㹼㔬䘠DŽ˄1˅⁑ 䘰⚛㇇⌅˄Simulated Annealing, SA˅1982 ˈKirkpatrickfㅹ փ䘰⚛ 㓴 Ո 亶 ˈ Ҷа⿽≲䀓 㿴⁑VRPⲴ 䘁լ㇇⌅ˈ ⁑ 䘰⚛㇇⌅˄SA˅DŽSAⓀҾ փ䘰⚛䗷〻Ⲵ⁑ ˈ 䍘к а⿽䲿 Ⲵ ㍒ ⌅DŽ ѝˈSA䟷⭘Metropolis ˈ ⭘а㓴〠ѻѪ 䘋 㺘Ⲵ ㇇⌅䘋〻ˈ ㇇⌅ 亩 䰤䟼㔉 ањ䘁լ Ո䀓DŽо㓿 ㇇⌅⴨∄ˈSA 䘠ㆰǃ ⭘⚥⍫ǃ ⭘ ⌋ 䖳 Ԧ䲀 ㅹՈ⛩ˈⴞ 䇨 Ո 䰞仈ѝ ⭘DŽⴞ ҾSAⲴ⹄ウˈ 㔃 ԆՈ ㆆ⮕ ⌅DŽ Tian[38]ㅹӪ 䙐Ҷа⿽ Ҿ Ո ㆆ⮕ⲴSA㇇⌅ ≲䀓VRPˈ ⭘ ㇇⌅ 䙐 䀓ˈ SAо2-opt㔃 ˈ ㇇⌅ѝ ⭘⑙ 䟽 ˈ 䗮 㓸⑙ ˈṩ ⑙ Ո Ⲵ⑙ 䟽 ㊫ ㇇⌅DŽLi[39]ㅹӪ SAо⾱ ㍒㇇⌅⴨㔃 ˈ ⭘TS㇇⌅ѝⲴ〫ս Ӕ ㇇ 㓯䐟 㓯䐟䰤Ⲵ ˈ㘼 SA Ѫ 〻 ˈṩ Ⲵ Ո䀓 䇮 Ⲵ⑙ ˈ ㇇⌅䘋㹼 ㍒DŽTavakkoli-Moghaddam[40]ㅹӪ 䛫䘁㇇⌅оSA㔃 ˈԕ⭘ ≲䀓 䖖 VRP 䴰≲ Ⲵ 䖖 VRPDŽ㜑 Տ[41]ㅹ 䲿 䙐VRP 䀓Ⲵ ⹰кˈ㔃 ⁑ 䘰⚛㇇⌅ㆆ⮕ˈ䟷⭘䐟 䰤䈳 䐟 Ո ⌅ˈ 䙏 VRP䘋㹼Ҷ≲䀓DŽ⭡ҾSAⲴ 䙏 䖳 ˈфо Ԇ㇇⌅⴨∄ˈSA н㜭 Ⲵ䀓[42]ˈ ↔ⴞ SA VRPѝⲴ ⭘⋑ ⾱ ㍒㇇⌅ 䚇Ր㇇⌅ ⌋DŽ˄2˅⾱ ㍒㇇⌅˄Tabu Search, TS˅1986 Glover Ҷ⾱ ㍒㇇⌅˄TS˅[43]DŽ Ҿ Ӫ㊫ 〻Ⲵа⿽⁑ ˈTS 䜘亶 ㍒Ⲵа⿽ ˈ а⿽ 䙀↕ Ո㇇⌅ˈ ˖㔉 ањ 䀓˄ 䀓˅ 䘹䀓ӗ⭏ ˄亶 㔃 ˅ˈ 䀓Ⲵ亶 ѝ⺞ 㤕 䘹䀓˗㤕 䘹䀓 Ⲵⴞḷ ՈҾ ⴞ Ѫ→ ㍒ Ⲵ³ 䀓´˄Best-so-for˅ˈ 㿶 ⾱ ⢩ ˈ⭘ ԓ 䀓 ³ 䀓´˗㤕н к䘠 䘹䀓ˈ 䘹䀓䳶ѝ䘹 䶎⾱ Ⲵ 䘹䀓Ѫ Ⲵ 䀓ˈ㘼 㿶 о 䀓ⲴՈ ˗к䘠є⿽ л䜭 ⴨ Ⲵ 䊑 ⾱ 㺘ˈ ⾱ 㺘ѝ 䊑Ⲵԫ ˗ ↔䟽 к䘠䘝ԓ ㍒䗷〻ˈⴤ㠣┑䏣㓸→ DŽ ԕⴻ ˈ亶 㔃 ǃ 䘹䀓䳶ǃ⾱ 䊑ǃ㰀㿶 ǃ㓸→ ㅹ䜭 TS 㜭Ⲵ 䭞DŽˈWillard[44], Pureza Franca[45] TS ⭘ҾVRPˈն䜭 㜭≲ 䖳 Ⲵ㔃 DŽ Osman WassanⲴTS㇇⌅ѝ[46]ˈ ⭘є⿽ 䀓ӗ⭏ ⌅˖ањ 㢲㓖㇇⌅о ⌅㔃 ˈ ањ 㢲㓖㇇⌅о ⍮⌅㔃 ˗ 䛫 㔃 ањє 䐟 䰤Ⲵ њ 㘵єњ䘎㔝Ⲵ ⛩ⲴӔ ˗ ⾱ 䮯 ⭡ањ 〻 ㍒ 䰤䘋㹼 ѹDŽ⍻䈅㔃 ⽪ˈ䘉є⿽TS ⧠ 䜭 Ҷ䖳 Ⲵ㔃 DŽ1994 ˈGendreau[47] Ҷ㓿 ⲴTaburoute㇇⌅ˈ 亶 㔃 ањ亦⛩Ӿ Ⲵ㓯䐟ѝ ˈ а 㓯䐟ѝˈ㘼䘉 㓯䐟 ⭘ԆԜ Ⲵ≲䀓TSPⲴ ѹ ⌅˄GENI˅ ӗ⭏Ⲵpњ 䛫䘁䀓ѻа˗ Ҿ⾱ ˈ Ӿ 䰤[5, 10]ѝ䲿 ањ ˗ ⭘Ҷа⿽ ṧ ㆆ⮕ˈѪ 㘳㲁䛓Ӌ〫 䖳 Ⲵ亦⛩Ⲵ 㜭 ˈ 䛓Ӌ㓿 㻛〫 Ⲵ亦⛩䘋㹼 㖊˗ ⭘՚䎧⛩ˈ⭏ њ䀓 ⇿ањ䜭䘋㹼 䲀Ⲵ ㍒ˈ❦ 䘹 ѝ 㘵Ѫѫ ㍒䗷〻Ⲵ 䀓˗⍻䈅㔃 㺘 ˈ䈕㇇⌅㜭≲ 儈䍘䟿Ⲵ㔃 ˈф ≲ ⴞ Ѫ→Ⲵ 䀓DŽ1995 ˈRochat Taillard[48] Ҷ㠚䘲 䇠 ˄Adaptive Memory˅ᾲ ˈ 䘁 TS亶 ⧠Ⲵ Ⲵ䘋 ѻаDŽ ㍒䗷〻ѝ ⲴՈ䍘䀓 䎧 ˈ❦ 䙊䗷䘉Ӌ䀓Ⲵ㓴 ˈӗ⭏TSⲴ 䀓DŽањ㠚䘲 䇠 а㓴 њ ㍒䗷〻ѝн Ⲵ 䀓DŽ䘉Ӌ䀓ⲴḀӋ ㍐㻛 ˈ 䘋㹼н Ⲵ㓴 ԕӗ⭏ Ⲵ 䀓DŽ VRPѝˈӾ њ䀓ѝ 䘹 Ⲵ䖖䖶㓯䐟 㻛⭘ Ѫањ䎧⛩DŽ䘉њ ⌅ 㓿 ԆԜ 14њḷ VRP⍻䈅䰞仈ѝˈ≲ Ҷєњ ⴞ Ѫ→Ⲵ 䀓DŽ1998 ˈToth Vigo[49] Ҷа⿽ ⌋䘲⭘㤳 Ⲵㆋ㖁⾱ ㍒˄Granular Tabu Search, GTS˅DŽGTSⲴѫ㾱 ⓀҾ˖ањ ѝ䖳䮯Ⲵ䗩㻛 ањ Ո䀓ѝⲴ 㜭 DŽ ↔ˈ䙊䗷⎸ 䮯 䎵䗷ањ䟿 ˄Granularity˅Ⲵ Ⲵ䗩ˈ аӋ⋑ 䙄Ⲵ䀓 ㍒䗷〻ѝ н 㘳㲁DŽ㔃 㺘 ˈ GTS㜭 ⸝Ⲵ 䰤 ≲ Ⲵ䀓DŽ䘁 ˈBrandao[50] 䇮䇑ⲴTSѝˈ䛫 㔃 䘹䀓䳶 ⭡ ˈ ⭘Ҷй⿽〫 ˈ 䐟 䰤 ⛩ ǃ䐟 䰤 ⛩Ӕ ǃ䐟 ⛩Ӕ DŽ⾱ 䮯 ⭡[N/6ˈN/2]ѻ䰤Ⲵ䲿 ˈ ⭘ Ⲵ[N/3] Ѫ⾱ 䮯 䘋㹼 傼⭘ԕ ∄DŽBrandao Eglese[51]䇮䇑Ⲵ≲䀓 䐟 Ⲵ⾱ ㍒㇇⌅ѝˈԕ5⿽ ⌅㧧 䀓ˈ 䘹 ԧ䗩ǃ䘹 䍥䗩ǃ ǃ䘎 䗩 亦⛩(Connected Components)ǃ䐟 (Path Scanning)˗ 䛫 㔃 䗩 ǃ 䗩 є 䐟 кⲴ䗩Ӕ DŽ⭡ҾTS㇇⌅ 䀓 Ⲵ 䎆 ˈ 㘵ԕSolomon ⌅ Ѫ ㇇⌅ˈҏ 䟷⭘ ⌅ǃл ㇇⌅ǃK-tree㇇⌅ӗ⭏ 䀓DŽⴞ ˈTS 䖖VRPǃ 〻䖭䍗VRPǃ 䰤デ㓖 ⲴVRPǃԕ VRP䜭 䟿 ⭘DŽ˄3˅䚇Ր㇇⌅˄Genetic Algorithms, GA˅1975 ˈHolland[52] Ҷ䚇Ր㇇⌅˄GA˅DŽGA а⿽⁑ԯ⭏⢙䘋 䗷〻Ⲵ 䲿 ㍒ ⌅ˈ Ѫ˖ӾՈ 䰞仈Ⲵањ⿽㗔˄а㓴 㹼䀓˅ ˈ ➗䘲㘵⭏ Ո㜌 ⊠Ⲵ ⨶ˈ䙀ԓ˄Generation˅╄ ӗ⭏ 䎺 䎺 Ⲵањ⿽㗔˄а㓴 㹼䀓˅˗ ⇿аԓˈṩ њփ˄ 㹼䀓˅Ⲵ䘲 ˄ⴞḷ ˅ⲴՈ 䘹а䜘 Ո㢟њփ ˄㑱⇆˅ лаԓˈ 䘋㹼Ӕ ˈӗ⭏ ԓ㺘 Ⲵ䀓䳶 Ⲵ⿽㗔DŽ䘉њ䗷〻 㠤⿽㗔 㠚❦䘋 аṧⲴ ԓ⿽㗔∄⡦ԓ 䘲 Ҿ⧟ ˄ 㹼䀓∄ 㹼䀓 䘁䰞仈Ⲵ Ո䀓˅ˈ њ䘋 䗷〻ѝⲴ Ոњփ Ѫ䰞仈Ⲵ 㓸䀓DŽҾн ㊫ Ⲵ䰞仈ˈ օ䇮䇑а⿽ 㖾Ⲵ㕆⸱ Ṹаⴤ GAⲴ ⭘䳮⛩ѻаˈҏ GAⲴањ䟽㾱⹄ウ DŽҼ䘋 㕆⸱ ⌅ GAѝ ⭘Ⲵа⿽㕆⸱ ⌅ˈ❦㘼ˈ Ҿ≲䀓VRP䘉ṧⲴ 䰞仈 ˈҼ㓗 㕆⸱ а Ⲵ㕪䲧DŽ ↔ˈㅖ 㕆⸱ ⌅ ⌋䇔 DŽն ㅖ 㕆⸱ ⌅ ˈ 㔉GA Ҷ Ⲵ䰞仈ˈ㤕䟷 ㆰ Ӕ Պ ԓ㓯䐟ѝ Ӌ亦⛩䟽 ˈ аӋ亦⛩㻛䚇┿ˈ 㠤 㜭 䙽 亦⛩Ⲵ䶎⌅㓯䐟DŽ ↔ˈ 享 у䰘Ⲵ Ҿ亪 ⲴӔ ӗ⭏ Ⲵ ԓ DŽ䘁 㓿 Ҷ л ⌅[5]˖䜘 䝽Ӕ ⌅˄Partially Matched Crossover, PMX˅ǃ亪 Ӕ ⌅˄Order Crossover, OX˅ǃ ⧟Ӕ ⌅˄Cycle Crossover, CX˅ˈԕ 䗩䟽㓴Ӕ ⌅˄Edge Recombination Crossover, ERC˅DŽ Ҿ ˈҏ 㓿 ҶаӋу䰘Ⲵ ⌅ˈ ˄Remove and Reinsert˅ǃ ˄Swap˅ˈԕ 䘶䖜 ˄Inversion˅DŽⴞ ˈ≲䀓VRP OVRPⲴGA ⥞⴨ 䖳 ˈ㘼≲䀓 䰤デⲴVRP ㅹ ⲴGA ⥞⴨ 䖳 DŽ Ⲵ䲀 Ԧˈ⢩ 䰤デⲴ ˈаⴤ ⢥ ⵰ Ⲵ≲䀓 ⌅Ⲵ DŽ䘉㔉GA⭘ Ԉ 䲀 Ԧ 䶒Ⲵ ≲ ㄎҹ Ⲵ㔃 Ҷ ՊDŽ˄4˅㲱㗔㇇⌅˄Ant Colony Optimization, ACO˅㲱㗔㇇⌅ ⭡Dorigo, M., ㅹ[53]Ҿ1991 俆⅑ ⲴDŽACO 㠚❦⭼ѝⵏ 㲱㗔㿵伏㹼ѪⲴ 㘼 Ⲵа⿽⁑ 䘋 ㇇⌅DŽ 伏⢙ ˈ㲲㲱Պ 㓿䗷Ⲵ䐟 䙊䗷 а⿽ ◰㍐˄pheromoneˈ ㇇⌅ѝ〠Ѫ ㍐˅ ḷ䇠ˈ Ⲵ䟿 ṩ 䐟 䮯 伏⢙Ⲵㅹ㓗 DŽ䘉Ӌ ◰㍐Ѫ 㲲㲱 ˈ Ԝ 䘀伏⢙DŽⴞ ˈACO ⭘Ҿ ㊫VRPDŽBullnheimerㅹ[54]䪸 VRP䇮䇑Ҷу䰘Ⲵ㲱㗔䖜〫䘹 ㆆ⮕ˈ 㲱㗔 ㍒ ˈ ⭘2-OptՈ 䐟 ˗Gambardellaㅹ[55] ACO⭘Ҿ≲䀓VRPTWˈ 㘵䟷⭘єњ⿽㗔 Ո 䰞仈ˈањՈ 䖖䖶 ⴞˈањՈ 㓯䐟䮯 ˗Donatiㅹ[56] ACO⭘Ҿ 㖁㔌VRPˈ ㇇⌅ѝ ㍐ ѹѪањ 䰤 䎆䟿ˈ а 〻 к䀓 Ҷ 䰤デⲴ 㖁㔌VRP˗Montemanniㅹ[57]⹄ウҶACO 䖖䖶䐟 䰞仈Ⲵ ⭘ˈ Ҷањ⭡һԦ㇑⨶ǃ㲱㗔㇇⌅ǃ ㍐ ㆆ⮕3њ ㍐㓴 Ⲵ㌫㔏Ṷ ˗Pellegrini ㅹ[58]⹄ウҶн 䖖䖶ǃ 䖖 ǃ 䰤デㅹ㓖 ⲴVRPˈ ⭘2⿽ ⿽ACOՈ ≲䀓˗Gajpal Abad[59]ԕ㋮㤡㲲㲱 䐟 ㆆ⮕Ⲵ㲱㗔㌫㔏≲䀓Ҷ 䘱 ъⲴVRP˗Ugur Aydin[60] ҾACO ҶањӪ ӔӂTSP ⁑ 䖟ԦDŽ˄5˅␧ ㇇⌅а䙊⭘ ㇇⌅⹄ウ␡ Ⲵ ˈ⹄ウӪ 㓿䖳 ⿽ аⲴ ㇇⌅㔃 䎧 ˈԕ 儈VRPⲴ≲䀓䙏 䀓Ⲵ䍘䟿DŽ1995 ˈGloverㅹ[61] 䈅 GA TS⴨㔃 ˗䛾㤲⾕ 㜑 㔗[62]䙊䗷 䜘 ㍒㜭 Ⲵ⡜ ㇇⌅о ㍒㜭 Ⲵ䚇Ր㇇⌅㔃 ˈ 䙐Ҷ≲䀓⢙⍱䝽䘱䐟 Ո 䰞仈Ⲵ␧ 䚇Ր㇇⌅ˈ 㢟 Ո㔃 ˗䛡 ẵ[63]ˈ[64] ⹄ウҶ㋂ 㗔㇇⌅о Ԇ㇇⌅Ⲵ㔃 VRP 䰞仈кⲴ ⭘˗б⿻䴧[65]ㅹ ⚮ ㇇ 㲱㗔㇇⌅ 䲧 䜘Ո Ⲵ㕪䲧˗Saez[66]ㅹ 䚇Ր㇇⌅о⁑㋺㚊㊫㔃 ˈ ↔ ⹰к Ҷа⿽ 㜭䘲 亴 ⌅≲䀓 䖖䖶 䘱 ъVRPDŽ㔬к 䘠ˈ ≲䀓VRPⲴ ⿽䙊⭘ ㇇⌅ ␧ ㇇⌅Ⲵ⹄ウ㺘 ˈ Ҿ Ⲯњ Ⲵ ˈ䘉Ӌ ⌅ѝ䜭 ԕ≲ 䶎 Ⲵǃ ⭊㠣 ՈⲴ䀓ˈ㲭❦䘀㇇ 䰤㾱䮯аӋDŽ փ 䈤ˈ ⴞ Ѫ→ˈ⾱ ㍒ ⧠ Ⲵ ⌅˗ Ҿ䚇Ր㇇⌅ ⾎㓿㖁㔌Ⲵ ⌅ 㜭Ո㢟˗㘼 Ҿ⁑ 䘰⚛ 㲱㗔㌫㔏Ⲵ ⌅ㄎҹ н DŽ❦㘼ˈ㘳㲁 ⿽ ⌅ 䙀⅑ ⧠ѝⲴ 㜭 䘋 ˈ ⿽␧ ㇇⌅ 䳶ѝ ㍒ ṧ ㍒ǃ ㇇⌅≲䀓䍘䟿 䇑㇇ 䰤ѻ䰤㜭㧧 䖳 Ⲵ 㺑ˈ␧ Ⲵ㲱㗔㌫㔏 䚇Ր㇇⌅ 㻛 ˈ о⾱ ㍒㇇⌅⴨ 㖾DŽ4. 㔉 䈣䖖䖶䐟 䰞仈 ㇑⨶ 䘀ㆩ 亶 䱄Ⲵ ⭘ԧ ⹄ウ DŽ䲿⵰⽮ՊⲴ ˈVRPҏ н Ⲵ ˈ 䲿 䴰≲VRPǃ䶎 〠㖁㔌VRPǃԃ -䝽䘱аփ 䐟 䰞仈ㅹ 䘋а↕⹄ウⲴ DŽ 㓣 㔃ҶVRP Ⲵ⹄ウ䘋 䎻 ˈѪ⴨ Ⲵ 䢤DŽ㘳 ⥞[1] 䶙 , 㔏䇑 , ѝ ⢙⍱о䟷䍝㚄 Պ. 2010 ⢙⍱䘀㹼 䙊 [M]. ѝ ⢙⍱о䟷䍝㚄 Պ, ѝ ⢙⍱ 䢤2011, Ӝ: ѝ ⢙䍴 ⡸⽮, 2011: 61-62[2] Dantzig, G.B., and Ramser, J.H., The truck dispatching problem [J].Management Science, 1959, 6: 80[3] ⇥仾 . 䖟 䰤デ㓖 Ⲵ 䖖䖶䐟 䰞仈⹄ウ ⭘[D]. 䮯⋉: ѝ , 2009[4] █・ . 䰤デ䖖䖶䐟 䰞仈 ㇇⌅⹄ウ[D]. 䮯⋉: ѝ , 2012[5] ㅖ . 䖖䖶䐟 䰞仈 ⭘⹄ウ[D]. 䮯⋉: ѝ , 2004[6] █・ , ㅖ . ≲䀓 ⺜ 䰤デ䖖䖶䐟 䰞仈Ⲵ ㇇⌅[J]. 䇑㇇ ⭘, 2012, 32(11): 3042-3043, 3070[7] 㛆䳱, ㅖ , 㛢 . 䖟 䰤デⲴ䖖䖶䐟 䰞仈 ⭘ 䇘[C]. 䮯⋉: ѝ 䘀ㆩ ՊㅜӔ⍱Պ, 2000: 634-638[8] ㅖ , 㙲䶆. 䖖䖶䐟 䰞仈 㤕 ⹄ウ䘋 [C]. ␡ : ѝ 䘀ㆩ Պㅜ Ӕ⍱Պ, 2006:395-400[9] Bodin, L.B., Golden, B.L., Assad, A.A., Ball, M.O., Routing and scheduling of vehicles and crews: the state ofthe art [J]. Computer Operation Research, 1983, 10: 163-211[10] Toth, P., Vigo, D. The Vehicle Routing Problem [M]. Society for Industrial and Applied Mathematics,Philadelphia, USA, 2002[11] 㛆 䖹. 䖖䖶䐟 Ո ⥞㔬䘠[J]. ь 㤳 䲒 ˄㠚❦、 ˅, 2010, 2: 31-37[12] Crevier, B., Cordeau, J.F., LaPorte, G., The multi-depot vehicle routing problem with inter-depot routes [J].European Journal of Operational Research, 2007, 176(2): 756-773[13] Golden, B., Assad, A., Levy, L., The fleet size and mix vehicle routing problem [J]. Computer and OperationsResearch, 11(1): 49-66[14] Drorand, M., Trudeau, P. Split delivery routing [J]. Naval Research Logistics, 1990, 37: 383-402[15] 䉒⿹⻺. 䲿 䖖䖶䐟 䰞仈⹄ウ[D]. 䜭: 㾯 Ӕ䙊 , 2003[16] Bent, R.W., and Hentenryck, P.V., Scenario-Based Planning for Partially Dynamic Vehicle Routing withStochastic Customers [J]. Operations Research, 2004, 52(6): 977-987[17] Oppen, J., L okketangen, A., Arc routing in a node routing environment [J]. Computers and OperationsResearch, 2006, 33(4): 1033-1055[18] Lenstra, J.K., and Rinnooy, K., Complexity of vehicle routing and scheduling problem [J]. Network, 1981, 11:221-227[19] . Ҿ ⭼䲀⌅Ⲵ 㹼 ㍒㇇⌅⹄ウ[D]. 㾯 : 㾯 ⭥ 、 , 2011[20] Տ. а⿽ Ҿ䀓 ѝ 䶒Ⲵ ㊫㇇⌅[D]. 䘎: 䘎⨶ , 2009[21] . CVRP 䝽㖁㔌⍱㇇⌅[D]. ⍾ : ь、 , 2003[22] 㕚 㣜, 䛥 . 㿴 ㇇⌅Ⲵ ⨶ ⭘[J]. ѝ 、 , 2005(21): 42-42[23] Golden, B.L., Wasil, E.A., Kelly, J.P., and Chao, I.M., Meatheuristics in vehicle routing [C]. In Crainic, T.G.,and Laporte, G. editors, Fleet Management and Logistics, Kluwer Academic Publishers, London, 1998, 33-56 [24] Clark, G. and Wright, J.R., Scheduling of vehicle routing problem from a central depot to a number ofdelivery points [J]. Operations Research, 1964, 12: 568-581[25] Nelson, M.D., Nygard, K.E., Griffin, J.H., and Shreve, W.E., Implementation techniques for the vehiclerouting problem [J]. Computer and Operations Research, 1985, 12(3): 273-283[26] Paessens, H., The savings algorithm for the vehicle routing problem [J]. European Journal of OperationalResearch, 1988, 34(3): 336-344[27] Altinkemer K., Gavish B., Parallel savings based heuristic for the delivery Problem [J]. Operations Research,1991, 39: 456-469[28] Wark, P., and Holt, J., A repeated matching heuristic for the vehicle routing problem [J]. Journal ofOperational Research Society, 1994, 45: 1156-1167[29] Gillett, B.E., and Miller, L.R., A heuristic algorithm for the vehicle dispatch problem [J]. Operations Research,1974, 22: 340-349[30] Foster, B., and Ryan, D., An integer Programming approach to the vehicle scheduling problem [J]. OperationResearch, 1976, 27: 307-384[31] Ryan, D.M., Hjorring, C., and Glover, F. Extension of the Petal method for vehicle routing [J]. Journal ofOperational Research Society, 1993, 44: 289-296[32] Cordeau, J.F., Gendreau, M., Laporte, G., Potvin, J.Y., and Semet, F., A guide to vehicle routing heuristics [J].Journal of Operational Research Society, 2002, 53(5): 512-522[33] Christofides, N., Mingozzi, A., and Toth, P., The vehicle routing problem [C]. Combinatorial optimization,Chichester, UK: Wiley, 1979: 315-338[34] Fisher, M.L., Jaikumar, R., A generalized assignment heuristic for vehicle routing [J]. Network, 1981, 11:109-124[35] Fisher, M.L., Greenfield, A.J., Jaikumar, R., and Lester, J., A computerized vehicle routing application [J].Interfaces, 1982, 12(4): 42-52[36] Fisher, M.L., Jaikumar, R., and Wassenbove, L.N., Multiplier adjustment method for the generalizedassignment problem [J]. Management Science, 1986, 32: 1095-1103[37] Bramel, J.B., and Simchi-Levi, D., A location based heuristic for general routing problems [J]. OperationsResearch, 1995, 43(4): 649-660[38] Tian, P., Ma, J., and Zhang, D.M., Application of the simulated annealing algorithm to the combinatorialoptimization problem with permutation property: an investigation of generation mechanism [J].European Journal of Operational Research, 1999, 118(1): 81-94[39] Li, H., Lim, A., Huang, J., Local search with annealing-like restarts to solve VRPTW [J]. European Journal ofOperational Research, 2003, 150(1): 115-127[40] Tavakkoli-Moghaddam , R. Safaei, N., Gholipour, Y., A hybrid simulated annealing for capacitated vehiclerouting problems with the independent route length [J]. Applied Mathematics and Computation, 2006, 176(2): 445-454[41] 㜑 Տ, , 㜑 . 䖖䖶䐟 䰞仈Ⲵ⁑ 䘰⚛㇇⌅[J]. ѝ 䐟 , 2006, 19(4): 123-126[42] Breedam, A.V., Improvement heuristics for the vehicle routing problem based on simulated annealing [J].European Journal of Operation Research, 1995, 86(3): 480-490[43] Glover, F. Future paths for integer programming and links to artificial Intelligence [J]. Computers andOperations Research, 1986, 13:533-549[44] Willard, J.A.G., Vehicle routing using r-optimal tabu search [M]. London: The Imperial College, 1989[45] Pureza, V.M., and Franca, P.M., Vehicle routing problems via tabu search metaheuristic [R].Technical ReportCRT-347, Centre for Research on Transportation, Montreal, Canada, 1991[46] OsmanˈI., and Wassan, N., A reactive tabu search metaheuristic for the vehicle routing problem with back-hauls [J]. Journal of Scheduling, 2002, 5(4): 263-285[47] Gendreau, M., Hertz, A., L aporte, G., A tabu search heuristic for the vehicle routing problem [J].Management Science, 1994, 40: 1276-1290[48] Rochat, Y., and Taillard, E.D., Probabilistic diversification and intensification in local search for vehiclerouting [J].Journal of Heuristics, 1995, 1: 147-167[49] Toth, P., and Vigo, D., The granular tabu search and its application to the vehicle routing problem[R].Technical Report OR19819, DEIS, Italy, 1998[50] Brandao, J., A new tabu search algorithm for the vehicle routing problem with backhauls [J]. EuropeanJournal of Operation Research, 2006, 173(2): 540-555[51] Brandao, J., and Eglese, R., A deterministic tabu search algorithm for the capacitated arc routing problem [J].Computers and Operations Research, 2008, 35(4): 1112-1126[52] Holland, J., Adaptation in natural and artificial systems [D]. University of Michigan Wesley, 1975[53] Colorni, A., Dorigo, M., Maniezzo, V., Distributed optimization by ant colonies [J]. In: Proc of the FirstEuropean Conference of Artificial Life. Paris, France Elsevier Publishing, 1991: 134-142[54] Bullnheimer, B., Hartl, R.F., and Strauss, C., Applying ant system algorithm to the vehicle routing problem[A]. Advances and Trends in Local Search Paradigms for Optimizaiton [C], Kluwer Acedenics, 1998: 109-120[55] Gambardella, L.M., Taillard, E., Agazzi, G., MACS-VROTW: a multiple ant colony system for vehiclerouting problems with time window [A]. New ides in optimization [C], London, U.K.: McGraw-Hill, 1999: 63-76[56] Donati, A.V., Montemanni, R., Casagrande, N., Rizzoll, A.E., and Gambardella, L.M., Time dependentvehicle routing problem with a multi ant colony system [R]. Technical Report TR-17-03, IDSIA, Galleria2, Manno, 6928, Switzerland, 2003[57] Montemanni, R., Gambardella, L.M., Rizzoll, A.E., and Donati, A.V., A new algorithm for a dynamic vehiclerouting problem based on ant colony system [R]. Technical Report TR-23-02, IDSIA, Galleria2, Manno, 6928, Switzerland, 2004[58] Pellegrini, P., Favaretto, D., Moretti, E., Multiple ant colony optimization for a rich vehicle routing problem:a case study [J]. Lecture Notes in Computer Science, 2007, 4639: 627-634[59] Gajpal, Y., and Abad, P.L., Multi-ant colony system (MACS) for a vehicle routing problem with backhauls[J]. European Journal of Operation Research, 2009, 196: 102-117[60] Ugur, A., and Aydin, D., An interactive simulation and analysis software for solving TSP using ant colonyoptimization algorithms [J]. Advances in Engineering Software, 2004, 40: 341-349[61] Glover, F., Kelly, J.P., and L aguna, M., Genetic algorithms and tabu search: hybrids for optimization [J].Computers Operation and Research, 1995, 22(1): 111-134[62] 䛾㤲⾕, 㜑 㔗. ⭘␧ 䚇Ր㇇⌅≲䀓⢙⍱䝽䘱䐟 Ո 䰞仈Ⲵ⹄ウ[J]. ѝ ㇑⨶、 , 2002, 10(5): 51-56[63] 䛡 , ẵ. Ҿ 䘋㋂ 㗔㇇⌅Ⲵ ս-䘀䗃䐟㓯䰞仈⹄ウ[J].ѝ Ỡ 〻, 2006, 17(22):2359-2361[64] . 䖖䖶䐟 䰞仈Ⲵ㋂ 㗔㇇⌅⹄ウо ⭘[D]. ⎉⊏: ⎉⊏ ъ , 2007[65] б⿻䴧, 㜑⾕ , ≨ . ≲䀓 䰤デⲴ䖖䖶䐟 䰞仈Ⲵ␧ 㲱㗔㇇⌅[J]. ㌫㔏 〻⨶䇪о 䐥, 2007,10: 98-104[66] Saez, D., Cortes, C.E., and Nunez, A., Hybrid adaptive predictive control for the multi-vehicle dynamic pick-up and delivery problem based on genetic algorithms and fuzzy clustering. Computers and Operations Research, 2008, 35: 3412-3438。