數(shù)學(xué)建模中的汽車租賃調(diào)度

\摘要Fg汽車租賃產(chǎn)業(yè)近年來(lái)快速開(kāi)展，其調(diào)度問(wèn)題的解決有著極強(qiáng)的實(shí)際意義。本文對(duì)汽車租賃業(yè)調(diào)度問(wèn)題進(jìn)展分析，利用層次分析法找出模型的關(guān)鍵因素，通過(guò)對(duì)上一年的調(diào)度情況進(jìn)展分析，找出了原有模型的優(yōu)劣，結(jié)合運(yùn)籌學(xué)中庫(kù)存論和規(guī)劃論的相關(guān)知識(shí)使用線性規(guī)劃制定出合理模型。在第一問(wèn)中根據(jù)最小二乘法的原理，制定出盡量滿足需求的調(diào)度模型并使用lingo軟件在盡量降低調(diào)度費(fèi)用的條件下調(diào)整出調(diào)度方案。二三問(wèn)中，增加了公司獲利、轉(zhuǎn)運(yùn)費(fèi)用以及短缺損失等因素的約束，利用matlab輔助，實(shí)現(xiàn)多目標(biāo)線性規(guī)劃，最終確定了調(diào)度方案。第四問(wèn)中綜合考慮到維修費(fèi)用，使用費(fèi)用，價(jià)格因素的影響，求解出汽車購(gòu)置模型。關(guān)鍵詞：汽車租賃調(diào)度、運(yùn)籌學(xué)、多目標(biāo)線性規(guī)劃、lingo、matlab軟件目錄一、問(wèn)題重述…………〔4〕二、問(wèn)題分析…………〔4〕三、模型的假設(shè)………〔5〕四、定義與符號(hào)說(shuō)明…………………〔5〕五、模型的建立與求解…………〔6-8〕六、模型的檢驗(yàn)………〔8〕六、模型評(píng)價(jià)與推廣…………………〔8〕七、參考文獻(xiàn)…………〔8〕八、附錄…………〔9-19〕-一、問(wèn)題重述國(guó)汽車租賃市場(chǎng)興起于1990年亞運(yùn)會(huì)，隨后在、、及等國(guó)際化程度較高的城市率先開(kāi)展，直至2000年左右，汽車租賃市場(chǎng)開(kāi)場(chǎng)在其他城市開(kāi)展。*城市有一家汽車租賃公司，此公司年初在全市圍有379輛可供租賃的汽車，分布于20個(gè)代理點(diǎn)中。每個(gè)代理點(diǎn)的位置都以地理坐標(biāo)*和Y的形式給出，單位為千米。假定兩個(gè)代理點(diǎn)之間的距離約為他們之間歐氏距離〔即直線距離〕的1.2倍。根據(jù)已有數(shù)據(jù)，我們要解決如下問(wèn)題：1．給出未來(lái)四周每天的汽車調(diào)度方案，在盡量滿足需求的前提下，使總的轉(zhuǎn)運(yùn)費(fèi)用最低；2．考慮到由于汽車數(shù)量缺乏而帶來(lái)的經(jīng)濟(jì)損失，給出使未來(lái)四周總的轉(zhuǎn)運(yùn)費(fèi)用及短缺損失最低的汽車調(diào)度方案；3．綜合考慮公司獲利、轉(zhuǎn)運(yùn)費(fèi)用以及短缺損失等因素，確定未來(lái)四周的汽車調(diào)度方案；4．為了使年度總獲利最大，從長(zhǎng)期考慮是否需要購(gòu)置新車？如果購(gòu)置的話，確定購(gòu)置方案〔考慮到購(gòu)置數(shù)量與價(jià)格優(yōu)惠幅度之間的關(guān)系，在此假設(shè)如果購(gòu)置新車，只購(gòu)置一款車型〕。二、問(wèn)題分析根據(jù)對(duì)問(wèn)題分析及文獻(xiàn)【1】，我們了解到運(yùn)籌學(xué)是以整體最優(yōu)為目標(biāo)，從系統(tǒng)的觀點(diǎn)出發(fā)，力圖以整個(gè)系統(tǒng)最正確的方式來(lái)解決該系統(tǒng)各部門之間的利害沖突。對(duì)所研究的問(wèn)題求出最優(yōu)解，尋求最正確的行動(dòng)方案，故我們結(jié)合運(yùn)籌學(xué)中規(guī)劃論和庫(kù)存論的知識(shí)對(duì)本問(wèn)題進(jìn)展了分析。問(wèn)題1：通過(guò)對(duì)【1】代理點(diǎn)的位置及年初擁有車輛數(shù)，【3】未來(lái)四周每個(gè)代理點(diǎn)每天的汽車需求量，【6】不同代理點(diǎn)之間的轉(zhuǎn)運(yùn)本錢的分析，為了獲取最低的費(fèi)用，我們采取線性規(guī)劃來(lái)求得最優(yōu)解，從而得到汽車代理點(diǎn)的實(shí)際供給矩陣。問(wèn)題2：該模型是關(guān)于多目標(biāo)線性規(guī)劃模型，由第一問(wèn)的汽車代理點(diǎn)的實(shí)際供給矩陣增加短缺損失這一約束條件通過(guò)matlab軟件計(jì)算出使未來(lái)四周總的轉(zhuǎn)運(yùn)費(fèi)用及短缺損失最低的汽車調(diào)度方案。問(wèn)題3：在該模型中，類比問(wèn)題1、問(wèn)題2，我們?cè)黾恿俗尮精@利最大的約束條件，由模型可得，轉(zhuǎn)移的汽車數(shù)量即可得到汽車調(diào)度方案。問(wèn)題4：根據(jù)上述模型，可以進(jìn)一步確定該公司為了使年度獲利最大，結(jié)合【5】，運(yùn)用層次分析法計(jì)算各個(gè)指標(biāo)權(quán)值，確定最優(yōu)購(gòu)置方案。三、模型的假設(shè)1、假設(shè)所有租賃車輛當(dāng)日租賃當(dāng)日還，不存在拖延現(xiàn)象；2、租賃汽車完好且在租賃過(guò)程中不損壞，無(wú)車輛維修費(fèi)用；3、假定兩個(gè)代理點(diǎn)之間的距離約為他們之間歐氏距離〔即直線距離〕的1.2倍；4、假設(shè)汽車使用年限、維修費(fèi)用和預(yù)期一樣5、假設(shè)物價(jià)不變動(dòng)，不考慮通貨膨脹和CPI的影響6、我們假設(shè)所有題目均在盡量滿足需求的前提下四、定義與符號(hào)說(shuō)明*:代理點(diǎn)地理位置橫坐標(biāo)Y:代理點(diǎn)地理位置縱坐標(biāo)W：費(fèi)用i,j:代理點(diǎn)編碼序列WIJ:第i個(gè)代理點(diǎn)調(diào)度到第j個(gè)代理點(diǎn)的轉(zhuǎn)運(yùn)費(fèi)用k:日期編碼序列Lkj:第k天第i個(gè)代理點(diǎn)的需求量L′kj:第k天第i個(gè)代理點(diǎn)的供給量*kij:第k天從第i個(gè)代理點(diǎn)轉(zhuǎn)運(yùn)到第i個(gè)代理點(diǎn)的汽車數(shù)目z:未來(lái)四周的總轉(zhuǎn)移費(fèi)用P:總短缺損失費(fèi)用T：未來(lái)四周總轉(zhuǎn)移費(fèi)用和短缺損失費(fèi)用五、模型的建立與求解問(wèn)題一：通過(guò)對(duì)【1】代理點(diǎn)的位置及年初擁有車輛數(shù)，【3】未來(lái)四周每個(gè)代理點(diǎn)每天的汽車需求量，【6】不同代理點(diǎn)之間的轉(zhuǎn)運(yùn)本錢的分析，為了獲取最低的費(fèi)用，我們采取線性規(guī)劃來(lái)求得最優(yōu)解。我們通過(guò)e*cel函數(shù)計(jì)算出各代理點(diǎn)之間的運(yùn)費(fèi)。為盡量滿足需求量，我們采用最小二乘法減小誤差明確每一天每個(gè)代理點(diǎn)的供給量，從而得到汽車代理點(diǎn)的實(shí)際供給矩陣。再通過(guò)lingo軟件獲得汽車調(diào)度轉(zhuǎn)運(yùn)費(fèi)用的最優(yōu)解以及調(diào)度方案。汽車每天的需求量通過(guò)【3】可以得知，在未來(lái)四周當(dāng)中，各代理點(diǎn)的每日總需求量有一局部多于其可供租賃車輛，另一局部少于可供租賃車輛。其數(shù)據(jù)可通過(guò)e*cel表格做出四周個(gè)帶搜點(diǎn)每日總需求量折線圖〔如下列圖〕。由題目可知，此公司年初在全市圍有379輛可供租賃的汽車。要使在盡量滿足需求的前提下，使總的轉(zhuǎn)運(yùn)費(fèi)用z最低。首先在lingo軟件中利用最小二乘法獲得其實(shí)際供給矩陣L’kj(如附錄1所示)，程序如附錄2所示，然后通過(guò)題目中所給【1】、【6】，利用e*cel電子表格函數(shù)計(jì)算公式，我們得出個(gè)代理點(diǎn)之間的轉(zhuǎn)運(yùn)費(fèi)用具體值WIJ計(jì)算過(guò)程如下：S．t：最終通過(guò)lingo軟件得到汽車每日調(diào)度車輛數(shù)*kij，進(jìn)而得到所求方案〔程序編程如附錄3所示〕。問(wèn)題2：我們所進(jìn)展的一切考慮都基于在盡量滿足需求量的條件下，在該問(wèn)題中模型中，我們需要增加短缺損失的約束進(jìn)展多目標(biāo)線性規(guī)劃。通過(guò)對(duì)【1】代理點(diǎn)的位置及年初擁有車輛數(shù)，【3】未來(lái)四周每個(gè)代理點(diǎn)每天的汽車需求量，【5】不同代理點(diǎn)的短缺損失費(fèi)及租賃收入【6】不同代理點(diǎn)之間的轉(zhuǎn)運(yùn)本錢的分析，我們可知，該模型是關(guān)于多目標(biāo)線性規(guī)劃模型，由第一問(wèn)的汽車代理點(diǎn)的實(shí)際供給矩陣通過(guò)matlab軟件計(jì)算出使未來(lái)四周總的轉(zhuǎn)運(yùn)費(fèi)用及短缺損失最低的汽車調(diào)度方案。T=min〔P+W〕符號(hào)定義：J；實(shí)際供給量為需求量誤差的平方P：短缺損失費(fèi)用W：已有車輛數(shù):各個(gè)代理點(diǎn)短缺損失費(fèi)用問(wèn)題3：汽車公司的目的是讓公司獲得最大的利益，讓使轉(zhuǎn)運(yùn)費(fèi)用及短缺損失最低，才可最終獲得最大利益。因此，在該模型中，類比問(wèn)題1、問(wèn)題2，結(jié)合【5】我們?cè)黾恿俗尮精@利最大的約束條件，由模型可得，轉(zhuǎn)移的汽車數(shù)量即可得到汽車調(diào)度方案。問(wèn)題4：根據(jù)上述模型，可以進(jìn)一步確定該公司為了使年度獲利最大，結(jié)合一、二、三問(wèn)的探究，我們確定需要購(gòu)置新車減少短缺損失以求獲得更大的效益。結(jié)合【5】，綜合考慮汽車本錢、。維修費(fèi)用和使用年限的影響，進(jìn)展該三個(gè)條件的約束，利用lingo軟件制定了最優(yōu)購(gòu)車模型。六、模型評(píng)價(jià)與推廣本文給出的解決方案比擬合理，但是判定指標(biāo)有限，對(duì)多個(gè)指標(biāo)的權(quán)值缺乏論證，而是采取了平均權(quán)值的理想化處理。規(guī)劃論對(duì)解決汽車租賃調(diào)度問(wèn)題準(zhǔn)確而合理，不僅有效解決多個(gè)代理店協(xié)調(diào)問(wèn)題，還充分利用最優(yōu)理論給出合理的汽車購(gòu)置方案。但應(yīng)用這個(gè)模型時(shí)，缺乏對(duì)上一年數(shù)據(jù)的有效參考，僅有一年的數(shù)據(jù)也具有一定的局限性。對(duì)規(guī)劃論的相關(guān)知識(shí)結(jié)合得比擬簡(jiǎn)單。七、參考文獻(xiàn)【文獻(xiàn)1】(美)希利爾，"運(yùn)籌學(xué)導(dǎo)論"，清華大學(xué)，2007年8月【文獻(xiàn)2】(美)希利爾，"數(shù)學(xué)規(guī)劃導(dǎo)論"，清華大學(xué)，1995年【文獻(xiàn)3】盧開(kāi)明，"線性規(guī)劃"，清華大學(xué)，2009年八、附錄附錄1：27 26 23 16 11 27 15 25 19 19 30 22 24 30 29 13 22 15 29 1715 19 30 17 15 14 16 12 17 15 13 27 30 12 27 20 14 16 16 2420 15 20 11 11 27 26 12 14 30 28 15 13 16 22 15 26 14 24 1615 30 17 16 25 16 29 16 19 28 25 20 17 29 19 21 30 28 20 1312 28 19 14 12 29 16 20 24 25 24 15 24 16 12 11 20 24 30 1219 12 21 11 28 11 14 13 27 11 26 13 16 12 13 28 20 24 30 28附錄1model:Sets:Wh/w1..w29/:ak;Vd/v1..v20/:dj;links(wh,vd):c,*;endsetsData:ak=1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29dj=379c=22,18,19,18,24,16,19,17,22,15,18,23,14,18,18,17,21,23,18,1915,22,22,27,15,20,15,12,19,16,27,24,30,13,17,24,16,13,12,2816,28,25,15,28,24,25,19,18,13,17,18,26,19,15,30,28,12,28,1324,17,21,20,12,18,22,14,17,18,11,29,23,27,15,28,18,15,28,3014,23,30,20,17,19,21,18,25,13,23,13,20,15,20,29,30,25,30,1422,19,11,20,15,13,12,26,25,23,16,17,11,16,14,11,11,14,14,2619,12,27,20,12,15,22,17,18,19,16,14,20,12,15,30,30,11,12,1115,11,25,12,28,29,16,24,20,30,30,12,15,15,13,17,12,18,21,3022,29,11,15,18,14,24,23,27,14,27,16,29,11,14,18,15,22,22,2828,20,25,11,13,17,22,13,14,29,27,22,19,12,13,23,14,15,16,2629,16,14,12,21,20,14,28,11,11,12,23,30,18,19,16,25,29,25,2116,11,18,16,20,28,25,12,12,16,20,13,28,20,11,13,26,16,13,2522,13,13,18,11,19,18,16,30,11,13,22,17,17,28,19,17,14,19,2427,19,25,11,13,17,22,13,14,29,27,22,19,12,13,23,14,15,15,2527,20,14,20,19,15,30,28,25,11,29,11,15,17,16,30,12,20,28,1911,15,22,27,12,29,17,30,13,17,22,13,25,30,13,28,25,24,26,1323,20,19,26,26,30,13,16,20,23,22,15,18,27,13,18,18,15,24,1915,22,22,27,15,20,15,13,20,17,28,25,30,13,17,24,16,13,12,2815,22,22,27,20,20,15,12,19,16,27,24,30,13,17,24,16,13,12,2826,18,25,11,13,17,22,13,14,29,27,22,19,12,13,23,14,15,14,2530,12,16,29,16,12,27,21,11,16,13,18,25,29,25,22,12,17,27,3027,20,25,11,13,17,22,13,14,29,27,22,19,12,13,23,14,15,16,2614,16,12,18,11,18,11,14,12,27,19,29,14,19,22,20,19,28,13,2015,21,11,28,13,26,17,13,13,17,27,20,22,18,22,23,14,25,16,1922,22,22,18,12,25,21,22,14,26,30,20,20,12,11,24,27,26,14,2326,16,26,16,11,23,27,18,28,14,17,23,17,17,16,12,15,16,14,2824,24,15,14,28,12,11,30,15,21,20,17,21,15,17,29,20,23,19,1815,14,25,12,28,29,16,24,20,30,30,12,15,15,13,17,12,18,21,3015,22,22,27,15,20,15,21,19,16,27,24,30,13,17,24,16,13,12,28EnddataMin=sum[links(k,j):(c(k,j)-*(k,j))^2];for(wh(k):sum(links(k,j):c(k,j))<=dj(j))end附錄2model:sets:vd/a1..a20/:ai;wh/b1..b29/:bj;z*/c1..c20/:ck;links(vd,wh,z*):w,*,m;endsetsdata:ai=1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20;bj=1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29;ck=1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20;w=0 ,0.033941125, 0.123328829 ,0.065612791,0.14456788 ,0.10812586 ,0.254314203, 0.102247887,0.113913125, 0.205298818, 0.055887029, 0.0756, 0.061188234, 0.082222333, 0.07711631, 0.098920186, 0.094968003,0.096658119 ,0.220009782, 0.1776099320.033941125, 0 ,0.179929986 ,0.048371165,0.172816666, 0.124134765 ,0.092664867, 0.066269559 ,0.131072743, 0.137408803 ,0.105171289 ,0.061390879, 0.013682105, 0.042914054, 0.031486912,0.051919938, 0.078547861 ,0.083768681, 0.059427266, 0.156688940.123328829, 0.179929986 ,0 ,0.12612314 ,0.061983804, 0.15675459 ,0.053517847, 0.080517764 ,0.127647515, 0.037279657, 0.12084, 0.090859892, 0.111362328, 0.064935414, 0.064221417,0.170600671, 0.04032, 0.091746651 ,0.128336059, 0.0324677070.065612791, 0.048371165, 0.12612314 ,0 ,0.113954391, 0.085802098 ,0.043469161, 0.03954924 ,0.073946489, 0.05858778, 0.016970563, 0.072359104, 0.033641974, 0.02539496, 0.022710139, 0.039950449, 0.063801304,0.027216995, 0.054110287, 0.0530057810.14456788, 0.172816666, 0.061983804,0.113954391, 0 ,0.144828218,0.025183423, 0.124181216 ,0.053348065, 0.00648, 0.09672268, 0.175497692, 0.065453703, 0.099 ,0.069508906, 0.091296659, 0.033731101,0.137272657, 0.079664547, 0.0500904780.10812586, 0.124134765, 0.15675459, 0.085802098, 0.144828218 ,0 ,0.134410714, 0.045855076, 0.047001106,0.032315445, 0.015832726, 0.098368971, 0.031926541, 0.036044972, 0.032019994,0.081258787, 0.043033847,0.020645581, 0.031151963, 0.0474205020.254314203, 0.092664867, 0.053517847,0.043469161, 0.025183423 ,0.134410714, 0, 0.108407764 ,0.071142595, 0.009790117, 0.138480208 ,0.199799964, 0.063645267, 0.102343891, 0.108849989,0.084552582, 0.070505796,0.071010196, 0.050235876, 0.1052516980.102247887, 0.066269559, 0.080517764,0.03954924 ,0.124181216 ,0.045855076,0.108407764, 0 ,0.044353273, 0.06644891, 0.089059809, 0.082687228, 0.09096285, 0.051929922, 0.03276, 0.08720855, 0.033041646, 0.025056736, 0.027808948, 0.0123693170.113913125, 0.131072743, 0.127647515,0.073946489 ,0.053348065, 0.047001106 ,0.071142595,0.044353273, 0 ,0.026563132, 0.03290014, 0.077813315, 0.096979082, 0.043878833, 0.07176, 0.04968884, 0.044794732, 0.037842833, 0.044009908, 0.0643987580.205298818, 0.137408803, 0.037279657,0.05858778 ,0.00648 ,0.032315445,0.009790117, 0.06644891 ,0.026563132, 0 ,0.13483323, 0.230005913, 0.120902129, 0.096631523, 0.152565359,0.14793703, 0.061846584,0.070194974, 0.04574519, 0.0804671360.055887029, 0.105171289, 0.12084, 0.016970563, 0.09672268,0.015832726 ,0.138480208, 0.089059809 ,0.03290014 ,0.13483323, 0 ,0.090966807, 0.082606382, 0.059899716, 0.04687471, 0.041460999, 0.05568, 0.042100679, 0.049601097, 0.071050540.0756, 0.061390879, 0.090859892,0.072359104 ,0.175497692, 0.098368971 ,0.199799964, 0.082687228 ,0.077813315, 0.230005913, 0.090966807, 0 ,0.065954712, 0.045235384, 0.081423692,0.050624216, 0.07626904, 0.103007262, 0.062971994, 0.1131683350.061188234, 0.013682105, 0.111362328,0.033641974 ,0.065453703, 0.031926541 ,0.063645267, 0.09096285 ,0.096979082, 0.120902129, 0.082606382, 0.065954712, 0 ,0.014934095, 0.390934879,0.012237647, 0.063690502,0.058139969, 0.025959969, 0.0324179950.082222333, 0.042914054, 0.064935414 ,0.02539496 ,0.099 ,0.036044972,0.102343891, 0.051929922 ,0.043878833, 0.096631523, 0.059899716, 0.045235384, 0.014934095 ,0, 0.017472264, 0.017132706,0.074924986, 0.05904439, 0.015479535, 0.047225756,0.077116317, 0.031486912, 0.064221417,0.022710139 ,0.069508906, 0.032019994 ,0.108849989, 0.03276 ,0.07176 ,0.152565359, 0.04687471, 0.081423692, 0.390934879 ,0.017472264, 0 ,0.037569775, 0.070292247,0.025996, 0.023233252, 0.0685308220.098920186, 0.051919938, 0.170600671,0.039950449 ,0.091296659, 0.081258787 ,0.084552582, 0.08720855 ,0.04968884 ,0.14793703, 0.041460999, 0.050624216, 0.012237647 ,0.017132706, 0.037569775, 0 ,0.099045444,0.018440998, 0.031587593, 0.0601937270.094968003, 0.078547861, 0.04032, 0.063801304 ,0.033731101, 0.043033847 ,0.070505796, 0.033041646 ,0.044794732, 0.061846584, 0.05568, 0.07626904, 0.063690502, 0.074924986, 0.070292247,0.099045444 ,0, 0.040010239, 0.027690634, 0.0107331260.096658119, 0.083768681, 0.091746651,0.027216995 ,0.137272657, 0.020645581 ,0.071010196, 0.025056736 ,0.037842833, 0.070194974, 0.042100679, 0.103007262, 0.058139969 ,0.05904439, 0.025996 ,0.018440998, 0.040010239, 0, 0.014599123, 0.0141264010.220009782, 0.059427266, 0.128336059,0.054110287 ,0.079664547, 0.031151963 ,0.050235876, 0.027808948 ,0.044009908, 0.04574519, 0.049601097, 0.062971994, 0.025959969 ,0.015479535, 0.023233252, 0.031587593 ,0.027690634, 0.014599123, 0, 0.0309264930.177609932, 0.15668894, 0.032467707,0.053005781,0.050090478, 0.047420502 ,0.105251698, 0.012369317 ,0.064398758, 0.080467136, 0.07105054, 0.113168335, 0.032417995 ,0.047225756, 0.068530822 ,0.060193727, 0.010733126 ,0.014126401, 0.030926493, 0mi=22, 16, 28, 18, 11, 29, 14, 12, 28, 19, 29, 13, 20, 13, 27, 23, 13, 12, 15, 1218, 27, 13, 25, 14, 16, 18, 21, 20, 25, 11, 23, 17, 12, 21, 14, 26, 11, 14, 1519, 24, 24, 13, 14, 24, 15, 20, 11, 11, 15, 20, 28, 28, 11, 15, 17, 24, 28, 1518, 30, 17, 23, 26, 20, 22, 14, 13, 13, 17, 19, 25, 26, 16, 16, 13, 27, 12, 1324, 13, 21, 13, 19, 30, 22, 28, 26, 17, 16, 26, 30, 18, 13, 26, 13, 26, 11, 1716, 17, 20, 20, 12, 30, 28, 11, 16, 22, 30, 26, 13, 25, 18, 14, 17, 14, 30, 1219, 24, 12, 15, 27, 12, 28, 11, 13, 13, 12, 30, 17, 11, 25, 16, 27, 13, 15, 1817, 16, 18, 20, 20, 15, 20, 12, 25, 14, 20, 13, 24, 13, 29, 12, 20, 26, 21, 2122, 13, 22, 29, 12, 15, 25, 23, 22, 29, 28, 16, 16, 17, 25, 18, 22, 16, 20,3015, 12, 14, 30, 15, 13, 11, 30, 13, 27, 19, 20, 13, 22, 22, 11, 18, 26, 17, 1518, 28, 17, 25, 22, 17, 13, 18, 13, 22, 11, 23, 12, 13, 12, 18, 22, 16, 21, 2223, 16, 18, 30, 17, 12, 17, 19, 18, 19, 15, 22, 28, 14, 17, 11, 23, 11, 15, 2214, 28, 11, 14, 18, 18, 22, 16, 11, 12, 22, 15, 15, 29, 27, 14, 14, 23, 17, 2718, 25, 29, 22, 19, 21, 1