Model and algorithm of two-stage distribution location routing with hard time window for city cold-chain logistics

applied sciences A icle Model and Algo i hm o Two-S age Dis ibu ion Loca ion Rou ing wi h Ha d Time Window o Ci y Cold-Chain Logis ics Liying Yan 1,2,3,4 , Manel G i oll 5and Pengjun Zheng 1,3,4,* 1Facul y o Ma i ime and T anspo a ion, Ningbo Uni e si y, Collabo a i e Inno a ion Cen e o Ningbo Po Logis ics Se ice Sys em, Ningbo 315211, Zhejiang, China; [email p o ec ed] 2 Depa men o Basic Cou ses, Ningbo Uni e si y o Finance & Economics, Ningbo 315211, Zhejiang, China 3Ningbo Uni e si y Sub-Cen e , Na ional T a ic Managemen Enginee ing& Technology Resea ch Cen e, Ningbo 315211, Zhejiang, China 4Collabo a i e Inno a ion Cen e o Mode n U ban T a ic Technologies, Nanjing 211189, Jiangsu, China 5Ba celona Inno a ion in T anspo (BIT), Ba celona School o Nau ical S udies, Uni e si a Poli ècnica de Ca alunya-Ba celonaTech, 08003 Ba celona, Spain; [email p o ec ed] *Co espondence: [email p o ec ed] Recei ed: 15 Ma ch 2020; Accep ed: 3 Ap il 2020; Published: 8 Ap il 2020   Abs ac : Taking cold-chain logis ics as he esea ch backg ound and combining wi h he o e all op imisa ion o logis ics dis ibu ion ne wo ks, we de elop wo-s age dis ibu ion loca ion- ou ing model wi h he minimum o al cos as he objec i e unc ion and a ying ehicle capaci y in di e en deli e y s ages. A hyb id gene ic algo i hm is designed based on coupling and collabo a ion o he wo-s age ou ing and ans e s a ions. The alidi y and easibili y o he model and algo i hm a e e i ied by conduc ing a andomly gene a ed es . The op imal solu ions o di e en objec i e unc ions o wo-s age dis ibu ion loca ion- ou ing a e compa ed and analysed. Resul s u n ou ha o di e en dis ibu ion objec i es, di e en dis ibu ion schemes should be employed. Finally, we compa e he wo-s age dis ibu ion loca ion- ou ing o single-s age ehicle ou ing p oblems. I is ound ha a wo-s age dis ibu ion loca ion- ou ing sys em is easible and e ec i e o he cold-chain logis ics ne wo k, and can dec ease dis ibu ion cos s o cold-chain logis ics en e p ises. Keywo ds: wo-s age dis ibu ion; loca ion- ou ing; hyb id gene ic algo i hm 1. In oduc ion Many ci ies ind i challenging o se up a high-e iciency ci y logis ics sys em o inc ease eigh e iciency and dec ease he impac s o ci y dis ibu ion on ci y li ing condi ions [ 1 ]. Macha is e al. [ 2 ] poin ed ou ha u ban goods dis ibu ion (UGD) has an impo an impac on he sus ainable de elopmen o ci ies. I is necessa y o ind new solu ions o he managemen o eigh dis ibu ion in o de o each a highe le el o e iciency. So a ci y logis ics model and some u ban eigh a ic policies ha e been s udied by many esea che s such as Taniguchi e al. [ 3 ], Allen e al. [ 4 ], Taniguchi e al. [ 5 ], Anand e al. [ 6 ], Danielis e al. [ 7 ]. Allen e al. [ 8 ] showed ha he use o u ban consolida ion cen e s (UCC) is p esumed o p o ide mo e e icien dis ibu ion in an u ban a ea, and i can dec ease ene gy use and en i onmen al impac . A UCC can be desc ibed as a logis ics acili y loca ed in ela i ely close p oximi y o he geog aphic a ea ha i se es, and wi h he ange o e ms used o e e o he UCC concep main including a public dis ibu ion depo , u ban ans-shipmen cen e , eigh pla o ms, coope a i e deli e y sys em, u ban dis ibu ion cen e , consolida ion cen e (some imes speci ic, e.g., e ail, cons uc ion), pick-up d op-o loca ion, o si e logis ics suppo concep and so on [ 9 ]. UCCs a e one o he mos equen ly implemen ed and s udied ci y logis ics ini ia i es, and acco ding o Lago io Appl. Sci. 2020,10, 2564; doi:10.3390/app10072564 www.mdpi.com/jou nal/applsci Appl. Sci. 2020,10, 2564 2 o 16 e al. [ 10 ] many case s udies ha e been p esen ed in li e a u e [ 11 , 12 ]. One o he mos e icien and ypical ways o implemen goods consolida ion is o adop mul i-s age dis ibu ion sys ems, especially wo-s age dis ibu ion sys em, whe e he deli e y om dis ibu ion cen e o cus ome s is managed by ou ing and consolida ing he eigh h ough in e media e depo s called ans e s a ions [ 13 ]. The e o e, selec ion o he loca ions o he ans e s a ions and planning o he wo-phase dis ibu ion ou es a e key p oblems in ci y logis ics sys em op imiza ion. Wi h he de elopmen o he economy and he con inuous imp o emen o people’s li ing s anda ds in China, demand o cold-chain p oduc s has also inc eased o a la ge ex en , which p omo es apid de elopmen o he cold-chain logis ics. The cold chain has become an impo an pa o he u ban dis ibu ion sys em. The emainde o his pape is o ganised as ollows. Fi s ly, he loca ion- ou ing p oblem is desc ibed in a li e a u e e iew. Then a wo-s age loca ion- ou ing model is cons uc ed and a me aheu is ic algo i hm is implemen ed o sol e he model. The algo i hm is included as a hyb id gene ic algo i hm [ 14 ]. Finally, he easibili y and alidi y o he model and algo i hm a e demons a ed h ough a es example, and esul s o he s udy a e summa ised. 2. Li e a u e Re iew The concep o he loca ion- ou ing p oblem (LRP) can be aced back o 1961. Von Bo en e i s discussed he ela ionship be ween loca ion selec ion and anspo a ion cos in anspo a ion p oblems [ 15 ]. LRP and i s a ian s ha e been s udied ex ensi ely in he pas . Pe l and Daskin p oposed a mul i- ehicle and mul i- acili y wa ehouse loca ion- ou ing p oblem model wi h ehicle capaci y cons ain s [ 16 ]. Li e al. [ 17 ] s udied a h ee- ie dis ibu ion sys em wi h supplie s, wa ehouses, and mul iple geog aphically dispe sed e aile s, whe e e aile s could eplenish goods om wa ehouses and supplie s. A model was es ablished o minimize he long- e m a e age cos wi hin he sys em while mee ing he demand o each e aile . P odhon [ 18 ] pu o wa d a linea p og amming model o mul i-plan pe iodic loca ion pa hs by easonably de ining a iables and designed a hyb id e olu iona y algo i hm o sol e i . Xu [ 19 ] in oduced he damage cos o goods incu ed in ansi and es ablished a loca ion- ou e op imisa ion model conside ing wo-s age anspo a ion cos , damage cos , and penal y cos , and se ice le el. A Gene ic Algo i hm-Pa icle Swa m Op imiza ion algo i hm was designed o sol e he p oblem. Howe e , he ene gy consump ion cos was no conside ed in he model. Song e al. [ 20 ] s udied he LRP o a mul i-jou ney ehicle pa h, i.e., conside ing ha he ehicle uns mul iple ou es wi hin he a el cons ain ime, and a h ee-s age heu is ic algo i hm was designed o sol e he p oblem. Zhao e al. [ 21 ] build a he e ogeneous lee wo-echelon capaci a ed loca ion- ou ing model o join deli e y in ci y logis ics. Wang e al. [ 22 ] p oposed a bi-objec i e model, and designed an imp o ed algo i hm. The e ec i eness o he imp o ed algo i hm was demons a ed h ough a compa ison. Koç e al. [ 23 ] es ablished a loca ion- ou ing p oblem model conside ing a he e ogeneous lee and ime windows. A hyb id e olu iona y algo i hm combining mul iple heu is ics was designed o sol e he p oblem. Leng e al. [ 24 ] conside ed mul iple condi ions in he egional low-ca bon loca ion- ou ing p oblem, such as simul aneous pickup and deli e y, ime windows. Yu e al. [ 25 ] and Zhao e al. [ 26 ] s udied loca ion- ou ing p oblem wi h simul aneous pick-up and deli e y. To sol e he p oblem, a simula ed annealing (SA) heu is ic algo i hm was designed and a hype -heu is ic app oach based on i e a ed local sea ch was p oposed, espec i ely. Fo he cold-chain logis ics p oblems, Yang e al. [ 27 ] s udied a loca ion model o pe ishable p oduc s in wo- ie dis ibu ion cen e s, in oduced he cos o goods damage in o he objec i e unc ion, designed an imp o ed gene ic algo i hm o sol e he model, and demons a ed he e ec i eness o he model and algo i hm h ough an example. Howe e , he dis ance s udied was calcula ed using he adial shape o he loca ion and demand poin s. Zhao e al. [ 28 ] designed a sa is ac ion deg ee unc ion acco ding o se ice ime windows and in oduced a minimum en elope clus e ing analysis me hod and abu sea ch algo i hm o sol e he p oblem. Zheng e al. [ 29 ] cons uc ed loca ion in en o y ou ing p oblem unde a demand en i onmen in a cold-chain logis ics ne wo k, and non-domina ed Appl. Sci. 2020,10, 2564 3 o 16 so ing in oduced a mul i-objec i e gene ic algo i hm (GA). Wang e al. [ 30 ] s udied a cold-chain logis ics dis ibu ion ne wo k conside ing ca bon oo p in , he model o minimum o al cos including ca bon emission cos was cons uc ed, and a hyb id algo i hm was designed o sol e he model. Mos o he exis ing LRP esea ch ocused on a gene al logis ics dis ibu ion ne wo k and single-s age loca ion- ou ing p oblems. When i comes o wo-s age LRP, mos o he models assume ha he anspo a ion pa h be ween he dis ibu ion cen e and he ans e s a ions is adial, i.e, he ehicle only p o ides se ices o one ans e s a ion a a ime and hen e u ns o he dis ibu ion cen e , and mos o he algo i hms a e in ol ed in single-s age op imisa ion o ans e s a ion selec ion and pa hs wi hou conside ing he coupling and collabo a i e op imisa ion o he wo s ages. In iew o his, we s udied a wo-s age cold-chain logis ics dis ibu ion ne wo k, including wo-s age op imisa ion o ans e s a ion loca ions and ou ing. A wo-s age loca ion- ou ing model wi h he minimum o al cos associa ed wi h he ha d ime window is cons uc ed, and di e en ypes o ehicles a e conside ed o dis ibu ion asks. An in eg a ed app oach is employed o design he algo i hm o ensu e he quali y o he solu ion. Finally, h ough a case s udy, we e i y he easibili y and e ec i eness o he model and algo i hm. 3. Model Fo mula ion 3.1. P oblem Desc ip ion The cold-chain logis ics p oblem conside ed in his s udy comp ises a cold-chain logis ics dis ibu ion cen e , mul iple po en ial cold-chain logis ics ans e s a ions and cus ome poin s. The dis ibu ion cen e need o deli e goods o he cus ome s h ough ans e s a ions wi hin a speci ied ime window. In he sys em, he dis ibu ion cen e , ans e s a ions, and ou ing be ween hem cons i u e he i s -s age ci y logis ics ne wo k. The ans e s a ions, cus ome s, and ou ing be ween hem cons i u e he second-s age ci y logis ics ne wo k. As shown in Figu e 1, we op imise loca ions o ans e s a ions and he wo-s age dis ibu ion ne wo k ehicle ou ing p oblem unde he condi ion o minimum o al cos . Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 o 17 Fo he cold-chain logis ics p oblems, Yang e al. [27] s udied a loca ion model o pe ishable 88 p oduc s in wo- ie dis ibu ion cen e s, in oduced he cos o goods damage in o he objec i e 89 unc ion, designed an imp o ed gene ic algo i hm o sol e he model, and demons a ed he 90 e ec i eness o he model and algo i hm h ough an example. Howe e , he dis ance s udied was 91 calcula ed using he adial shape o he loca ion and demand poin s. Zhao e al. [28] designed a 92 sa is ac ion deg ee unc ion acco ding o se ice ime windows and in oduced a minimum 93 en elope clus e ing analysis me hod and abu sea ch algo i hm o sol e he p oblem. Zheng e al. 94 [29] cons uc ed loca ion in en o y ou ing p oblem unde a demand en i onmen in a cold-chain 95 logis ics ne wo k, and non-domina ed so ing in oduced a mul i-objec i e gene ic algo i hm (GA). 96 Wang e al. [30] s udied a cold-chain logis ics dis ibu ion ne wo k conside ing ca bon oo p in , 97 he model o minimum o al cos including ca bon emission cos was cons uc ed, and a hyb id 98 algo i hm was designed o sol e he model. 99 Mos o he exis ing LRP esea ch ocused on a gene al logis ics dis ibu ion ne wo k and 100 single-s age loca ion- ou ing p oblems. When i comes o wo-s age LRP, mos o he models 101 assume ha he anspo a ion pa h be ween he dis ibu ion cen e and he ans e s a ions is 102 adial, i.e, he ehicle only p o ides se ices o one ans e s a ion a a ime and hen e u ns o he 103 dis ibu ion cen e , and mos o he algo i hms a e in ol ed in single-s age op imisa ion o ans e 104 s a ion selec ion and pa hs wi hou conside ing he coupling and collabo a i e op imisa ion o he 105 wo s ages. In iew o his, we s udied a wo-s age cold-chain logis ics dis ibu ion ne wo k, 106 including wo-s age op imisa ion o ans e s a ion loca ions and ou ing. A wo-s age 107 loca ion- ou ing model wi h he minimum o al cos associa ed wi h he ha d ime window is 108 cons uc ed, and di e en ypes o ehicles a e conside ed o dis ibu ion asks. An in eg a ed 109 app oach is employed o design he algo i hm o ensu e he quali y o he solu ion. Finally, h ough 110 a case s udy, we e i y he easibili y and e ec i eness o he model and algo i hm. 111 3. Model Fo mula ion 112 3.1. P oblem Desc ip ion 113 The cold-chain logis ics p oblem conside ed in his s udy comp ises a cold-chain logis ics 114 dis ibu ion cen e , mul iple po en ial cold-chain logis ics ans e s a ions and cus ome poin s. The 115 dis ibu ion cen e need o deli e goods o he cus ome s h ough ans e s a ions wi hin a 116 speci ied ime window. In he sys em, he dis ibu ion cen e , ans e s a ions, and ou ing be ween 117 hem cons i u e he i s -s age ci y logis ics ne wo k. The ans e s a ions, cus ome s, and ou ing 118 be ween hem cons i u e he second-s age ci y logis ics ne wo k. As shown in Figu e 1, we op imise 119 loca ions o ans e s a ions and he wo-s age dis ibu ion ne wo k ehicle ou ing p oblem unde 120 he condi ion o minimum o al cos . 121 122 Figu e 1. Schema ic map o loca ion- ou ing in wo-s age dis ibu ion. 123 Figu e 1. Schema ic map o loca ion- ou ing in wo-s age dis ibu ion. 3.2. P oblem Assump ions To acili a e he s udy, he ollowing assump ions a e made: (1) he geog aphical loca ion o dis ibu ion cen e , po en ial ans e s a ions, ime window, demand o cus ome s a e known; (2) each cus ome can only be se ed by one deli e y ehicle; (3) he demand o a single cus ome is less han he ehicle capaci y. (4) a ic conges ion is no conside ed; (5) he ca go load o a ehicle should no exceed i s a ed load; Appl. Sci. 2020,10, 2564 4 o 16 (6) he uni ans e cos o each ans e s a ion is known and is a cons an ; empe a u e changes and he i s s age-ca go losses a e no conside ed; (7) ans e s a ions compe e wi h each o he , and en e p ises can choose di e en ans e s a ions o p o ide se ices acco ding o hei own cos minimiza ion; (8) he capaci y o e ige a ed anspo ehicles o dis ibu ion cen e ( i s -s age ou e) is known, and he demand o a ans e s a ion can be g ea e han he capaci y o a anspo ehicle, i.e. he demand o he i s -s age dis ibu ion pa h can be spli ; (9) he capaci y o e ige a ed anspo ehicles (second-s age ou e) o ans e s a ions is known, and can no be exceeded. Each ehicle will e u n o he s a ing poin a e comple ing asks; (10) ypes o ehicles a e di e en o deli e ies om dis ibu ion cen e and ans e s a ions, and capaci ies o same ype o ehicles a e he same, and he e a e enough anspo ehicles o he sys em. 3.3. Pa ame e and Va iables To build he model, he ollowing pa ame e s and a iables a e de ined: M: Se o candida e ans e s a ions and dis ibu ion cen e ; M1: Se o candida e ans e s a ions; Ng: Se o cus ome s assigned o ans e s a ion g, and ans e s a ion g; K: Se o e ige a ed ehicles in he dis ibu ion cen e ; Lg: Se o e ige a ed ehicles in he g ans e s a ion; dij: Dis ance be ween ipoin and jpoin ; 1: A e age speed o e ige a ed ehicles in he i s -s age ou e; 2: A e age speed o e ige a ed ehicles in he second-s age ou e; sl i: Se ice imes o ehicle l o he i- h cus ome (o ans e s a ion); c1: Use and consump ion cos o ehicles pe kilome e in he i s -s age ou e; c2: Use and consump ion cos o ehicles pe kilome e in he second-s age ou e; c0 1: Cos o he d i e in he i s -s age ou e; c0 2: Cos o he d i e in he second-s age ou e; uk j : Remaining ca go olume o he e ige a ed ehicle kwhen a i ing a he ans e s a ion o he cus ome poin j; bk j : Remaining ca go olume o he e ige a ed ehicle kwhen lea ing he ans e s a ion o cus ome poin j; P1: P ice o uni commodi y; θ1: Spoilage a e o p oduc in he anspo a ion p ocess; θ2: Spoilage a e o p oduc in unloading p ocess; W1: Fuel consumed in ope a ing he e ige a o du ing anspo a ion; W2: Fuel consumed in e ige a o ope a ion du ing unloading; θ3: Fuel p ice pe uni weigh ; l i: Time when he e ige a ed ehicle l eaches poin i; l ij: Time equi ed by a e ige a ed ehicle l o a el be ween wo poin s iand j; l 0g: Depa u e ime o e ige a ed ehicle l om he ans e s a ion g; λ1: T ansi cos pe uni o ime; Wg: Amoun o goods ans e ed a he g ans e s a ion; 3: T ans e poin unloading p ocessing speed; qi: Demand o ans e s a ion i; q0 i: Demand o cus ome i; Q1: Vehicle capaci y in he i s -s age ou e; Appl. Sci. 2020,10, 2564 5 o 16 Q2: Vehicle capaci y in he second-s age ou e; Xk ij is a 0–1 a iable: when Xk ij =1, he ehicle kpasses he oad be ween ans e s a ion(o dis ibu ion cen e ) iand ans e s a ion(o dis ibu ion cen e ) j; o he wise, Xk ij =0; xk jis a 0–1 a iable: when xk j=1, he ehicle kse ices o cus ome i; o he wise, xk j=0; xl ijg is a 0–1 a iable: when xl ijg =1, he ehicle lo ans e s a ion gpasses he oad be ween cus ome (o ans e s a ion ) iand cus ome (o ans e s a ion) j; o he wise, Xl ijg =0; Zgis a 0–1 a iable: when Zg=1, he ans e s a ion gis used; o he wise, Zg=0; yl ig is a 0–1 a iable: when yl ig =1, he ehicle lo ans e s a ion g p o ides se ice o cus ome i; o he wise, yl ig =0. 3.4. Model De elopmen The wo-s age dis ibu ion loca ion- ou ing wi h ha d ime window o ci y cold-chain logis ics model cons uc ed in his pape akes he minimum o al cos as he objec i e unc ion. In consequence, he sub-cos should be analyzed i s ly. Then he o al cos o he wo-s age dis ibu ion loca ion- ou ing is ob ained by he a ious sub-cos s. 3.4.1. Objec i e Func ion Analysis o Model (1) T anspo a ion Cos The anspo a ion cos mainly includes ehicle use cos , uel used in he p ocess o anspo a ion, d i e ’s cos , and o he ac o s. Fo he con enience o esea ch, he anspo a ion cos is conside ed in wo pa s, e e ing o he use and consump ion cos o ehicles pe kilome e and d i e ’s cos . The anspo a ion cos o a e ige a ed ehicle in he i s -s age and second-s age ou es can be exp essed as ollows: C1=c1X k∈K X i,j∈M Xk ijdij+c0 1X k∈K X i,j∈M (Xk ij dij 1 +sk j)(1) C0 1=c2X g∈M1 X l∈Lg X i,j∈Ng Zgdijxl ijg +c0 2X g∈M1 X l∈Lg X i,j∈Ng Zg(dij 2 xl ijg +sl j)(2) (2) Damage Cos Commodi ies in he cold-chain dis ibu ion a e easily spoiled and he e o e need o be kep in an app op ia e low- empe a u e en i onmen . The quali y o pe ishable goods g adually declines o can lose alue wi h ime. When he quali y o he p oduc declines o a ce ain ex en , spoilage cos will be incu ed. The cos o damage is di ided in o wo pa s: he damage cos o goods accumula ed o e ime in he p ocess o anspo a ion and he damage cos incu ed when opening he doo in he p ocess o unloading. Because he i s -s age deli e y is usually ca ied ou in an enclosed en i onmen , spoilage cos will be minimal. We only conside ed he damage cos in he second-s age dis ibu ion. Hence, he o al damage cos can be exp essed as: C0 2=P1X g∈M1 X l∈Lg X j∈Ng Zgyl jg(1−e−θ1( l ij− l 0g))ul j+P1X g∈M1 X l∈Lg X j∈Ng Zgyl jg(1−e−θ2sl j)bl j(3) The i s pa ep esen s he cos o ca go damage in he anspo a ion p ocess, and he second pa ep esen s he cos o ca go damage in he unloading p ocess. (3) Re ige a ion Cos The cos o s o age associa ed wi h main aining he empe a u e and humidi y inside he ca iage is called he e ige a ion cos . The e ige a ion me hods employed in e ige a ed ehicles on he ma ke Appl. Sci. 2020,10, 2564 6 o 16 oday mainly include liquid ni ogen e ige a ion, mechanical e ige a ion, d y ice e ige a ion, and cold pla e e ige a ion. The e ige a ion me hod conside ed in his s udy is mechanical e ige a ion, and he cos gene a ed in he e ige a ion p ocess is ela ed o uel consump ion, ime, and weigh o goods. Li e a u e [31,32] p oposed a o mula o calcula ing he uel consump ion: W=ωεPe ξ∗10−3. The e ige a ion cos includes he cos associa ed wi h he ene gy consump ion o he ehicle in main aining a low- empe a u e en i onmen du ing deli e y and he cos o addi ional ene gy supplied o he e ige a ion sys em du ing he unloading p ocess. The e ige a ion cos in he i s -s age and second-s age ou e can be exp essed as ollows: C3=θ3W1X k∈K X i,j∈M dijxk ijuk j 1 +θ3W2X k∈K X j∈M xk jbk jsk j(4) C0 3=θ3W1X g∈M1 X l∈Lg X i,j∈Ng Zg dijxl ijgul j 2 +θ3W2X g∈M1 X l∈Lg X j∈Ng Zgsl jbl j(5) whe e, W is he uel consump ion (g/h); ω is he powe u iliza ion coe icien o he e ige a o ; ε is he uel consump ion a e (g/kW · h); Pe is he e ec i e powe o he e ige a o (kW); ξ is he speci ic g a i y o he uel; W1 is he uel oil consump ion du ing anspo a ion; and W2 is he uel consump ion du ing unloading. The i s pa ep esen s he e ige a ion cos o e ige a ed ehicles du ing anspo a ion, and he second pa ep esen s he e ige a ion cos o he e ige a ed ehicles du ing unloading. (4) Penal y Cos In an ac ual dis ibu ion p ocess, he dis ibu ion ehicles may no a i e on ime o a ious easons, his will incu a penal y cos . The concep o a ime window is in oduced. As he iming o i s -s age dis ibu ion is lexible, we only conside he ime window equi emen s o he cus ome s in he second-s age dis ibu ion. The ime window is di ided in o a ha d ime window and a so ime window. Conside ing he cha ac e is ics o cold-chain dis ibu ion, we calcula ed he penal y cos s associa ed wi h he ha d ime window. Assuming ha he ea lies se ice ime allowed by cus ome i is ET i , he la es se ice ime allowed by cus ome iis LT i , he se ice ime window equi ed by he cus ome iis [ETi,LTi]. The penal y unc ion equa ion can be exp essed as ollows [33]: C4=P( ) =          M <ET 0ET ≤ ≤LT M >LT whe e Mis in ini e. is he ime when he ehicle a i es a he cus ome . [ET,LT] is he se ice ime window equi ed by he cus ome . (5) T ans e Cos The cos a he ans e s a ion mainly consis s o he ime cos incu ed in he ans e o goods om la ge e ige a ed ehicles o small e ige a ed ehicles. The e o e, he ans e cos can be exp essed as ollows: C5=λ1X g∈M1 Zg Wg 3 (6) Appl. Sci. 2020,10, 2564 7 o 16 3.4.2. Model Se ing Based on he analysis o sub-cos in Sec ion 3.4.1, a wo-s age dis ibu ion loca ion- ou ing model wi h he ha d ime window cons ains o ci y cold-chain logis ics is es ablished as ollows: Minz1=C1+C0 1+C0 2+C3+C0 3+C4+C5(7) Subjec o: X i∈M1 Xk iqi≤Q1k∈K(8) X i∈Ng q0 iyl ig ≤Q2g∈M1,l∈Lg(9) X j∈M Xk ij =Xk ii∈M;k∈K(10) X i∈M Xk ij =Xk j,j∈M;k∈K(11) X i∈Ng X g∈M1 xl ijg =yl jg j∈Ng,l∈Lg(12) X j∈Ng X g∈M1 xl ijg =yl ig i∈Ng,l∈Lg(13) X g∈M1 X l∈Lg yl jg =1j∈Ng(14) X j∈Ng xl ijg =X j∈Ng xl jig ≤1i=g∈M1;l∈Lg(15) X j∈M1 Xk ij =X j∈M1 xk ji ≤1i=0; k∈K(16) X i∈Ng q0 iyig ≤Zgqgg∈M1(17) ETi≤ l i≤LTi(18) The objec i e unc ion o he model is shown in (7). Cons ain (8) shows ha a ehicle canno exceed i s maximum load in he i s le el dis ibu ion. Cons ain (9) shows ha ehicle can no exceed i s maximum load in he second-le el dis ibu ion. The i s s age o dis ibu ion low balance is shown in (10) and (11). The second-s age o dis ibu ion low balance is shown in (12) and (13). Cons ain (14) shows ha he e is only one e ige a ed ehicle p o iding a deli e y se ice o one cus ome . As pe cons ain s (15) and (16), he e ige a ed ehicles s a ing om a ans e s a ion mus e u n o he same a e se ing he cus ome , and he e ige a ed ehicles s a ing om he dis ibu ion cen e mus e u n o he dis ibu ion cen e a e se ing ans e s a ions. The o al cus ome equi emen s assigned o a ans e s a ion mus be lowe han o equal o he s o age capaci y o he ans e s a ion, which is imposed by (17). Cons ain (18) ep esen s he ime window o cus ome i. 4. Algo i hm Design Two-s age dis ibu ion loca ion- ou ing belong o he NP-Ha d p oblem, so we use heu is ic algo i hms o sol e his p oblem. A hyb id gene ic algo i hm is p oposed in he pape combining heu is ic ules and dis ance clus e ing. The low cha o he algo i hm is shown in Figu e 2. Appl. Sci. 2020,10, 2564 8 o 16 Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 o 17 Decoded ch omosome T ans e s a ion capaci y; ehicle capaci y; Whe he he e mina ion c i e ion is me ? End: S op i e a ion and ge he bes popula ion I he gene a ion o he new popula ion eaches he p e-se e olu iona y i d h ii id Fi ness e alua ion: Fi=1/Zi ;Roule e gambling Two-poin c osso e ; l Gene a ing easible ini ial popula ion a andom S a ing Ch omosome coding Each ch omosome consis s o h ee Mu a ion i C osso e i In e change mu a ion; In e sion mu a ion; Selec ion ope a ion 4. Algo i hm Design 261 Two-s age dis ibu ion loca ion- ou ing belong o he NP-Ha d p oblem, so we use heu is ic 262 algo i hms o sol e his p oblem. A hyb id gene ic algo i hm is p oposed in he pape combining 263 heu is ic ules and dis ance clus e ing. The low cha o he algo i hm is shown in Figu e 2. 264 Figu e 2. Flow cha o hyb id gene ic algo i hm. 265 4.1. Ch omosome Coding 266 Each ch omosome consis s o h ee sub-s ings: Sub-s ing 1 in ol es encoding leng h J. The 267 in ege o gene 1 − J is a non- epe i i e sequence, ep esen ing he p io i y selec ion o de and 268 deli e y o de o he ans e s a ion. Sub-s ing 2 is encoded as an in ege wi h leng h N and gene 1 269 − N, ep esen ing he dis ibu ion o de o he demand poin s. Sub-s ing 3 is encoded as an in ege 270 code o leng h 1, indica ing he numbe o ans e s a ion selec ed. Fo example, when J = 5 and N = 271 6, namely, he e a e 5 ans e s a ion and 6 cus ome poin , o he ch omosome shown in Figu e 3, 272 he p io i y o he ansi ion o be selec ed is 5, 4, 2, 1, 3. The dis ibu ion pa h o he demand poin s 273 is 2-3-4-1-5-6; 2 indica es ha he i s wo bi s o he i s laye o coding a e selec ed o se up a 274 ans e s a ion. Mo eo e , 5, 4 coded in he i s laye is he o de o dis ibu ion o he ans e 275 s a ion. 276 N Y Figu e 2. Flow cha o hyb id gene ic algo i hm. 4.1. Ch omosome Coding Each ch omosome consis s o h ee sub-s ings: Sub-s ing 1 in ol es encoding leng h J. The in ege o gene 1 − Jis a non- epe i i e sequence, ep esen ing he p io i y selec ion o de and deli e y o de o he ans e s a ion. Sub-s ing 2 is encoded as an in ege wi h leng h Nand gene 1 − N, ep esen ing he dis ibu ion o de o he demand poin s. Sub-s ing 3 is encoded as an in ege code o leng h 1, indica ing he numbe o ans e s a ion selec ed. Fo example, when J=5 and N=6, namely, he e a e 5 ans e s a ion and 6 cus ome poin , o he ch omosome shown in Figu e 3, he p io i y o he ansi ion o be selec ed is 5, 4, 2, 1, 3. The dis ibu ion pa h o he demand poin s is 2-3-4-1-5-6; 2 indica es ha he i s wo bi s o he i s laye o coding a e selec ed o se up a ans e s a ion. Mo eo e , 5, 4 coded in he i s laye is he o de o dis ibu ion o he ans e s a ion. Appl. Sci. 2019, 9, x FOR PEER REVIEW 10 o 17  3 21 2|6-5-1-4-3-2|3-1-2-4-5 subs ing subs ingsubs ing  277 Figu e 3. Ch omosome coding example. 278 4.2. Gene a e he Ini ial Popula ion a Random 279 Gene ic algo i hm s a s om he popula ion o he p oblem solu ions, so i is necessa y o 280 gene a e an ini ial popula ion as he s a ing poin o e olu ion. Acco ding o he encoding me hod, 281 N ch omosomes a e andomly gene a ed. 282 4.3. C osso e Ope a ion 283 To main ain he di e si y o he popula ion, we conduc ed c osso e ope a ions o each 284 sub-s ing in he ch omosome. Two-poin c osso e ope a ion is ca ied ou in sub-s ing 1. Cycle 285 c osso e ope a ion is chosen o pe o m on sub-s ing 2. Two-poin c osso e ope a ion is used in 286 sub-s ing 3. The de ailed desc ip ion o c osso e ope a ion is as ollows: 287 (1) C osso e ope a ion o sub-s ing 1 288 Two-poin c osso e : i s ly, wo ch omosomes a e andomly selec ed as he pa en , and 289 gene a ing wo andom na u al numbe s 1 and 2. Secondly, he gene agmen s be ween 1 and 2 290 o wo pa en ch omosomes a e exchanged, wo o sp ing ch omosomes a e ob ained. Finally, he 291 ch omosomes o he wo o sp ing a e modi ied so ha no con lic occu s. Fo example, pa en 292 ch omosome 1: 1 3 2 5 4, pa en ch omosome 2: 1 2 4 5 3, andom numbe s 1 = 2, 2 = 4. The 293 o sp ing ch omosomes a e c ossing a e o sp ing ch omosome 1: 1 2 4 5 4 and o sp ing 294 ch omosome 2: 1 3 2 5 3. Finally, wo new o sp ing ch omosome a e ob ained by epai ing, new 295 o sp ing ch omosome 1: 1 2 4 5 3, new o sp ing ch omosome 2: 1 3 2 5 4. 296 (2) C osso e ope a ion o sub-s ing 2 297 Cycle c osso e : i s ly, a cycle will be ound acco ding o he co esponding gene posi ion o 298 he pa en ch omosome. Secondly, he ci cula ing gene is eplica ed o he o sp ing. Thi dly, he 299 emaining genes a e iden i ied o he o sp ing, and he emaining genes ou sides he pa en 300 ch omosome cycle a e used o ill wi h he o iginal o sp ing. Finally, o ming new descendan s. Fo 301 example : Fi s ly, pa en ch omosome 1 : 2 7 5 6 4 8 9 1 3, pa en ch omosome 2 : 4 3 6 8 9 7 1 2 5, 302 ind cycle 1 : 2 4 9 1 2, cycle 2：4 2 1 9 4. Secondly, o sp ing ch omosome 1: 2 * * * 4 * 9 1 * , o sp ing 303 ch omosome 2 : 4 * * * 9 * 1 2 *. Thi dly, pa en 1 emaining ch omosome: * 3 6 8 * 7 * * 5, pa en 2 304 emaining ch omosome: * 7 5 6 * 8 * * 3. Finally, new o sp ing ch omosome 1: 2 3 6 8 4 7 9 1 5; new 305 o sp ing ch omosome 2: 4 7 5 6 9 8 1 2 3. 306 (3) C osso e ope a ion o sub-s ing 3 307 Two-poin c osso e : andom selec ion o wo ch omosomes as pa en s and wo o sp ing 308 ch omosomes a e ob ained by di ec exchange o wo pa en ch omosomes. Fo example, selec wo 309 pa en ch omosomes 1 and 2, he ch omosomes o he c ossed o sp ing a e 2 and 1. 310 4.4. Mu a ion Ope a ion 311 To main ain he di e si y o he popula ion, we conduc ed mu a ion ope a ion o each 312 sub-s ing in he ch omosome. In e change mu a ion ope a ion is ca ied ou in sub-s ing 1. 313 In e sion mu a ion ope a ion is chosen o pe o m on sub-s ing 2. Single-poin mu a ion ope a ion 314 is used in sub-s ing 3. The de ailed desc ip ion o mu a ion ope a ion is as ollows: 315 (1) Mu a ion ope a ion o sub-s ing 1 316 Figu e 3. Ch omosome coding example. Appl. Sci. 2020,10, 2564 9 o 16 4.2. Gene a e he Ini ial Popula ion a Random Gene ic algo i hm s a s om he popula ion o he p oblem solu ions, so i is necessa y o gene a e an ini ial popula ion as he s a ing poin o e olu ion. Acco ding o he encoding me hod, N ch omosomes a e andomly gene a ed. 4.3. C osso e Ope a ion To main ain he di e si y o he popula ion, we conduc ed c osso e ope a ions o each sub-s ing in he ch omosome. Two-poin c osso e ope a ion is ca ied ou in sub-s ing 1. Cycle c osso e ope a ion is chosen o pe o m on sub-s ing 2. Two-poin c osso e ope a ion is used in sub-s ing 3. The de ailed desc ip ion o c osso e ope a ion is as ollows: (1) C osso e ope a ion o sub-s ing 1 Two-poin c osso e : i s ly, wo ch omosomes a e andomly selec ed as he pa en , and gene a ing wo andom na u al numbe s 1 and 2. Secondly, he gene agmen s be ween 1 and 2 o wo pa en ch omosomes a e exchanged, wo o sp ing ch omosomes a e ob ained. Finally, he ch omosomes o he wo o sp ing a e modi ied so ha no con lic occu s. Fo example, pa en ch omosome 1: 1 3 2 5 4, pa en ch omosome 2: 1 2 4 5 3, andom numbe s 1 =2, 2 =4. The o sp ing ch omosomes a e c ossing a e o sp ing ch omosome 1: 1 2 4 5 4 and o sp ing ch omosome 2: 1 3 2 5 3. Finally, wo new o sp ing ch omosome a e ob ained by epai ing, new o sp ing ch omosome 1: 1 2 4 5 3, new o sp ing ch omosome 2: 1 3 2 5 4. (2) C osso e ope a ion o sub-s ing 2 Cycle c osso e : i s ly, a cycle will be ound acco ding o he co esponding gene posi ion o he pa en ch omosome. Secondly, he ci cula ing gene is eplica ed o he o sp ing. Thi dly, he emaining genes a e iden i ied o he o sp ing, and he emaining genes ou sides he pa en ch omosome cycle a e used o ill wi h he o iginal o sp ing. Finally, o ming new descendan s. Fo example: Fi s ly, pa en ch omosome 1: 2 7 5 6 4 8 9 1 3, pa en ch omosome 2: 4 3 6 8 9 7 1 2 5, ind cycle 1: 2 4 9 1 2, cycle 2: 4 2 1 9 4. Secondly, o sp ing ch omosome 1: 2 * * * 4 * 9 1 *, o sp ing ch omosome 2: 4 * * * 9 * 1 2 *. Thi dly, pa en 1 emaining ch omosome: * 3 6 8 * 7 * * 5, pa en 2 emaining ch omosome: * 7 5 6 * 8 * * 3. Finally, new o sp ing ch omosome 1: 2 3 6 8 4 7 9 1 5; new o sp ing ch omosome 2: 4 7 5 6 9 8 1 2 3. (3) C osso e ope a ion o sub-s ing 3 Two-poin c osso e : andom selec ion o wo ch omosomes as pa en s and wo o sp ing ch omosomes a e ob ained by di ec exchange o wo pa en ch omosomes. Fo example, selec wo pa en ch omosomes 1 and 2, he ch omosomes o he c ossed o sp ing a e 2 and 1. 4.4. Mu a ion Ope a ion To main ain he di e si y o he popula ion, we conduc ed mu a ion ope a ion o each sub-s ing in he ch omosome. In e change mu a ion ope a ion is ca ied ou in sub-s ing 1. In e sion mu a ion ope a ion is chosen o pe o m on sub-s ing 2. Single-poin mu a ion ope a ion is used in sub-s ing 3. The de ailed desc ip ion o mu a ion ope a ion is as ollows: (1) Mu a ion ope a ion o sub-s ing 1 In e change mu a ion: Two andom poin s a e selec ed in he encoding s ing, and hei posi ions a e swapped. Fo example, selec exchange poin s 3 and 7 om ch omosomes 1 4 3| 926 7| 5 8, he ch omosome ob ained a e exchange mu a ion is 1 4 7|9263|5 8. (2) Mu a ion ope a ion o sub-s ing 2 Appl. Sci. 2020,10, 2564 16 o 16 32. Wang, G. Op imiza ion Resea ch o Common Dis ibu ion Rou ing P oblem o Cold Chain Logis ics Based on Join Dis ibu ion Mode. Mas e ’s Thesis, Beijing Jiao ong Uni e si y, Beijing, China, 2018. 33. Shi, C.C.; Wang, X.; Ge, X.L. Resea ch on ehicle scheduling p oblem o mul i-dis ibu ion cen e s wi h ime window. Compu . Eng. Appl. 2009,45, 21–24. © 2020 by he au ho s. Licensee MDPI, Basel, Swi ze land. This a icle is an open access a icle dis ibu ed unde he e ms and condi ions o he C ea i e Commons A ibu ion (CC BY) license (h p://c ea i ecommons.o g/licenses/by/4.0/).