数学建模中的汽车租赁调度

\摘要

Fg 汽车租赁产业近年来快速发展，其调度问题的解决有着极强的实际意义。本文对汽车租赁业调度问题进行分析，利用层次分析法找出模型的关键因素，通过对上一年的调度情况进行分析，找出了原有模型的优劣，结合运筹学中库存论和规划论的相关知识使用线性规划制定出合理模型。在第一问中根据最小二乘法的原理，制定出尽量满足需求的调度模型并使用lingo软件在尽量降低调度费用的条件下调整出调度方案。二三问中，增加了公司获利、转运费用以及短缺损失等因素的约束，利用matlab辅助，实现多目标线性规划，最终确定了调度方案。第四问中综合考虑到维修费用，使用费用，价格因素的影响，求解出汽车购买模型。

关键词：汽车租赁调度、运筹学、多目标线性规划、lingo、matlab软件

目录

一、问题重述 (4)

二、问题分析 (4)

三、模型的假设 (5)

四、定义与符号说明 (5)

五、模型的建立与求解…………………………………………（6-8 ）

六、模型的检验 (8)

六、模型评价与推广 (8)

七、参考文献 (8)

八、附录…………………………………………………………（9-19）-

一、问题重述

国汽车租赁市场兴起于1990年亚运会，随后在、、及等国际化程度较高的城市率先发展，直至2000年左右，汽车租赁市场开始在其他城市发展。某城市有一家汽车租赁公司，此公司年初在全市围有379辆可供租赁的汽车，分布于20个代理点中。每个代理点的位置都以地理坐标X和Y的形式给出，单位为千米。假定两个代理点之间的距离约为他们之间欧氏距离（即直线距离）的1.2倍。

根据已有数据，我们要解决如下问题：

1．给出未来四周每天的汽车调度方案，在尽量满足需求的前提下，使总的转运费用最低；

2．考虑到由于汽车数量不足而带来的经济损失，给出使未来四周总的转运费用及短缺损失最低的汽车调度方案；

3．综合考虑公司获利、转运费用以及短缺损失等因素，确定未来四周的汽车调度方案；

4．为了使年度总获利最大，从长期考虑是否需要购买新车？如果购买的话，确定购买计划（考虑到购买数量与价格优惠幅度之间的关系，在此假设如果购买新车，只购买一款车型）。

二、问题分析

根据对问题分析及文献【1】，我们了解到运筹学是以整体最优为目标，从系统的观点出发，力图以整个系统最佳的方式来解决该系统各部门之间的利害冲突。对所研究的问题求出最优解，寻求最佳的行动方案，故我们结合运筹学中规划论和库存论的知识对本问题进行了分析。

问题1：

通过对【附件1】代理点的位置及年初拥有车辆数，【附件3】未来四周每个代理点每天的汽车需求量，【附件6】不同代理点之间的转运成本的分析，为了获取最低的费用，我们采取线性规划来求得最优解，从而得到汽车代理点的实际供应矩阵。

问题2：该模型是关于多目标线性规划模型，由第一问的汽车代理点的实际供应

矩阵增加短缺损失这一约束条件通过matlab软件计算出使未来四周总的转运费用及短缺损失最低的汽车调度方案。

问题3：在该模型中，类比问题1、问题2，我们增加了让公司获利最大的约束条件，由模型可得，转移的汽车数量即可得到汽车调度方案。

问题4：根据上述模型，可以进一步确定该公司为了使年度获利最大，结合【附件5】，运用层次分析法计算各个指标权值，确定最优购置方案。

三、模型的假设

1、假设所有租赁车辆当日租赁当日还，不存在拖延现象；

2、租赁汽车完好且在租赁过程中不损坏，无车辆维修费用；

3、假定两个代理点之间的距离约为他们之间欧氏距离（即直线距离）的1.2倍；

4、假设汽车使用年限、维修费用和预期相同

5、假设物价不变动，不考虑通货膨胀和CPI的影响

6、我们假设所有题目均在尽量满足需求的前提下

四、定义与符号说明

X:代理点地理位置横坐标

Y:代理点地理位置纵坐标

W：费用

i,j:代理点编码序列

W IJ:第i个代理点调度到第j个代理点的转运费用

k:日期编码序列

L kj:第k天第i个代理点的需求量

L′kj:第k天第i个代理点的供应量

x k ij:第k天从第i个代理点转运到第i个代理点的汽车数目

z:未来四周的总转移费用

P:总短缺损失费用

T：未来四周总转移费用和短缺损失费用

五、模型的建立与求解

问题一：

通过对【附件1】代理点的位置及年初拥有车辆数，【附件3】未来四周每个代理点每天的汽车需求量，【附件6】不同代理点之间的转运成本的分析，为了获取最低的费用，我们采取线性规划来求得最优解。我们通过excel函数计算出各代理点之间的运费。为尽量满足需求量，我们采用最小二乘法减小误差明确每一天每个代理点的供应量，从而得到汽车代理点的实际供应矩阵。再通过lingo软件获得汽车调度转运费用的最优解以及调度方案。

汽车每天的需求量通过【附件3】可以得知，在未来四周当中，各代理点的每日总需求量有一部分多于其可供租赁车辆，另一部分少于可供租赁车辆。其数据可通过excel表格做出四周个带搜点每日总需求量折线图（如下图）。

由题目可知，此公司年初在全市围有379辆可供租赁的汽车。要使在尽量满足需求的前提下，使总的转运费用z最低。首先在lingo软件中利用最小二乘法获得

其实际供应矩阵L ’kj (如附录1所示)，程序如附录2所示，然后通过题目中所给

【附件1】、【附件6】，利用excel 电子表格函数计算公式，我们得出个代理点之间的转运费用具体值W IJ

计算过程如下：W x 20j i 1x j

i,201j 291k ij Z ∑??? ??∑∑≠====

S ．t ：kj ij j i i ij j i i L x x ' L'1)j -(k 201201≥≠=≠=+-∑∑

最终通过lingo 软件得到汽车每日调度车辆数x k

ij ，进而得到所求方案（程序编程

如附录3所示）。

问题2：我们所进行的一切考虑都基于在尽量满足需求量的条件下，在该问题中模型中，我们需要增加短缺损失的约束进行多目标线性规划。

通过对【附件1】代理点的位置及年初拥有车辆数，【附件3】未来四周每个代理点每天的汽车需求量，【附件5】不同代理点的短缺损失费及租赁收入【附件6】不同代理点之间的转运成本的分析，我们可知，该模型是关于多目标线性规划模型，由第一问的汽车代理点的实际供应矩阵通过matlab 软件计算出使未来四周总的转运费用及短缺损失最低的汽车调度方案。

∑∑∑

=≠===291k 20j i 120

1j min W j L L ij

j ij -min J 20

1291k '∑∑=== T=min （P+W ）

符号定义：J ；实际供应量为需求量误差的平方

P ：短缺损失费用

W ：已有车辆数

t ij :各个代理点短缺损失费用

j k,201291k t 'P ?=∑∑==L L

ij j ij —

问题3：汽车公司的目的是让公司获得最大的利益，让使转运费用及短缺损失最低，才可最终获得最大利益。因此，在该模型中，类比问题1、问题2，结合【附件5】我们增加了让公司获利最大的约束条件，由模型可得，转移的汽车数量即可得到汽车调度方案。

问题4：根据上述模型，可以进一步确定该公司为了使年度获利最大，结合一、二、三问的探究，我们确定需要购买新车减少短缺损失以求获得更大的效益。结合【附件5】，综合考虑汽车成本、。维修费用和使用年限的影响，进行该三个条件的约束，利用lingo软件制定了最优购车模型。

六、模型评价与推广

本文给出的解决方案比较合理，但是判定指标有限，对多个指标的权值缺乏论证，而是采取了平均权值的理想化处理。规划论对解决汽车租赁调度问题准确而合理，不仅有效解决多个代理店协调问题，还充分利用最优理论给出合理的汽车购置方案。但应用这个模型时，缺乏对上一年数据的有效参考，仅有一年的数据也具有一定的局限性。对规划论的相关知识结合得比较简单。

七、参考文献

【文献1】(美)希利尔，《运筹学导论》，清华大学，2007年8月

【文献2】(美)希利尔，《数学规划导论》，清华大学，1995年

【文献3】卢开明，《线性规划》，清华大学，2009年

八、附录

附录1：

22 16 28 18 11 29 14 12 28 19 29 13 20 13 27 23 13 12 15 12

18 27 13 25 14 16 18 21 20 25 11 23 17 12 21 14 26 11 14 15

19 24 24 13 14 24 15 20 11 11 15 20 28 28 11 15 17 24 28 15 18 30 17 23 26 20 22 14 13 13 17 19 25 26 16 16 13 27 12 13 24 13 21 13 19 30 22 28 26 17 16 26 30 18 13 26 13 26 11 17 16 17 20 20 12 30 28 11 16 22 30 26 13 25 18 14 17 14 30 12 19 24 12 15 27 12 28 11 13 13 12 30 17 11 25 16 27 13 15 18 17 16 18 20 20 15 20 12 25 14 20 13 24 13 29 12 20 26 21 21 22 13 22 29 12 15 25 23 22 29 28 16 16 17 25 18 22 16 20 30 15 12 14 30 15 13 11 30 13 27 19 20 13 22 22 11 18 26 17 15 18 28 17 25 22 17 13 18 13 22 11 23 12 13 12 18 22 16 21 22 23 16 18 30 17 12 17 19 18 19 15 22 28 14 17 11 23 11 15 22 14 28 11 14 18 18 22 16 11 12 22 15 15 29 27 14 14 23 17 27 18 25 29 22 19 21 13 25 19 13 27 18 22 27 30 12 25 27 29 15 18 15 23 19 16 30 14 29 18 23 12 27 22 22 27 27 16 18 20 20 17 28 27 11 14 22 29 25 16 14 29 13 27 19 20 19 19 28 23 15

21 24 15 20 20 29 27 21 30 15 17 18 20 12 25 29 22 14 19 21 23 25 28 15 12 11 22 16 11 15 30 18 20 13 11 14 22 17 18 19

18 19 18 13 15 15 19 11 13 25 13 15 15 23 13 19 22 23 15 16

19 18 15 12 30 18 12 18 22 27 17 24 12 14 17 22 18 17 14 27 15 13 28 28 30 14 13 16 17 20 22 19 19 15 22 20 12 17 25 24 22 17 30 25 11 24 23 20 17 14 13 15 16 14 13 19 25 16 12 30 22 18 14 23 12 23 14 28 28 20 25 22 27 25 14 28 21 12 28 13 27 26 23 16 11 27 15 25 19 19 30 22 24 30 29 13 22 15 29 17 15 19 30 17 15 14 16 12 17 15 13 27 30 12 27 20 14 16 16 24 20 15 20 11 11 27 26 12 14 30 28 15 13 16 22 15 26 14 24 16 15 30 17 16 25 16 29 16 19 28 25 20 17 29 19 21 30 28 20 13 12 28 19 14 12 29 16 20 24 25 24 15 24 16 12 11 20 24 30 12 19 12 21 11 28 11 14 13 27 11 26 13 16 12 13 28 20 24 30 28

附录1

model:

Sets:

Wh/w1..w29/:ak;

Vd/v1..v20/:dj;

links(wh,vd):c,x;

endsets

Data:

ak=1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,2 6,27,28,29

dj=379

c=22,18,19,18,24,16,19,17,22,15,18,23,14,18,18,17,21,23,18,19

15,22,22,27,15,20,15,12,19,16,27,24,30,13,17,24,16,13,12,28

16,28,25,15,28,24,25,19,18,13,17,18,26,19,15,30,28,12,28,13

24,17,21,20,12,18,22,14,17,18,11,29,23,27,15,28,18,15,28,30

14,23,30,20,17,19,21,18,25,13,23,13,20,15,20,29,30,25,30,14

22,19,11,20,15,13,12,26,25,23,16,17,11,16,14,11,11,14,14,26

19,12,27,20,12,15,22,17,18,19,16,14,20,12,15,30,30,11,12,11

15,11,25,12,28,29,16,24,20,30,30,12,15,15,13,17,12,18,21,30

22,29,11,15,18,14,24,23,27,14,27,16,29,11,14,18,15,22,22,28

28,20,25,11,13,17,22,13,14,29,27,22,19,12,13,23,14,15,16,26

29,16,14,12,21,20,14,28,11,11,12,23,30,18,19,16,25,29,25,21

16,11,18,16,20,28,25,12,12,16,20,13,28,20,11,13,26,16,13,25

22,13,13,18,11,19,18,16,30,11,13,22,17,17,28,19,17,14,19,24

27,19,25,11,13,17,22,13,14,29,27,22,19,12,13,23,14,15,15,25

27,20,14,20,19,15,30,28,25,11,29,11,15,17,16,30,12,20,28,19

11,15,22,27,12,29,17,30,13,17,22,13,25,30,13,28,25,24,26,13

23,20,19,26,26,30,13,16,20,23,22,15,18,27,13,18,18,15,24,19

15,22,22,27,15,20,15,13,20,17,28,25,30,13,17,24,16,13,12,28

15,22,22,27,20,20,15,12,19,16,27,24,30,13,17,24,16,13,12,28 26,18,25,11,13,17,22,13,14,29,27,22,19,12,13,23,14,15,14,25 30,12,16,29,16,12,27,21,11,16,13,18,25,29,25,22,12,17,27,30 27,20,25,11,13,17,22,13,14,29,27,22,19,12,13,23,14,15,16,26 14,16,12,18,11,18,11,14,12,27,19,29,14,19,22,20,19,28,13,20 15,21,11,28,13,26,17,13,13,17,27,20,22,18,22,23,14,25,16,19 22,22,22,18,12,25,21,22,14,26,30,20,20,12,11,24,27,26,14,23 26,16,26,16,11,23,27,18,28,14,17,23,17,17,16,12,15,16,14,28 24,24,15,14,28,12,11,30,15,21,20,17,21,15,17,29,20,23,19,18 15,14,25,12,28,29,16,24,20,30,30,12,15,15,13,17,12,18,21,30 15,22,22,27,15,20,15,21,19,16,27,24,30,13,17,24,16,13,12,28 Enddata

Min=sum[links(k,j):(c(k,j)-x(k,j))^2];

for(wh(k):sum(links(k,j):c(k,j))<=dj(j))

end

附录2

model:

sets:

vd/a1..a20/:ai;

wh/b1..b29/:bj;

zx/c1..c20/:ck;

links(vd,wh,zx):w,x,m;

endsets

data:

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,

0.177609932

0.033941125, 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, 0.15668894

0.123328829, 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,

0.032467707

0.065612791, 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, 0.053005781 0.14456788, 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,

0.050090478

0.10812586, 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,

0.047420502

0.254314203, 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,

0.105251698

0.102247887, 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,

0.012369317

0.113913125, 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,

0.064398758

0.205298818, 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, 0.080467136

0.055887029, 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, 0.07105054

0.0756, 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, 0.113168335

0.061188234, 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,

0.032417995

0.082222333, 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, 0.068530822

0.098920186, 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,

0.060193727

0.094968003, 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, 0.010733126

0.096658119, 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, 0.014126401

0.220009782, 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, 0.030926493

0.177609932, 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, 0

mi=22, 16, 28, 18, 11, 29, 14, 12, 28, 19, 29, 13, 20, 13, 27, 23, 13, 12, 15,

12

18, 27, 13, 25, 14, 16, 18, 21, 20, 25, 11, 23, 17, 12, 21, 14, 26, 11, 14, 15 19, 24, 24, 13, 14, 24, 15, 20, 11, 11, 15, 20, 28, 28, 11, 15, 17, 24, 28, 15 18, 30, 17, 23, 26, 20, 22, 14, 13, 13, 17, 19, 25, 26, 16, 16, 13, 27, 12, 13 24, 13, 21, 13, 19, 30, 22, 28, 26, 17, 16, 26, 30, 18, 13, 26, 13, 26, 11, 17 16, 17, 20, 20, 12, 30, 28, 11, 16, 22, 30, 26, 13, 25, 18, 14, 17, 14, 30, 12

17, 16, 18, 20, 20, 15, 20, 12, 25, 14, 20, 13, 24, 13, 29, 12, 20, 26, 21, 21 22, 13, 22, 29, 12, 15, 25, 23, 22, 29, 28, 16, 16, 17, 25, 18, 22, 16, 20,

30

15, 12, 14, 30, 15, 13, 11, 30, 13, 27, 19, 20, 13, 22, 22, 11, 18, 26, 17, 15 18, 28, 17, 25, 22, 17, 13, 18, 13, 22, 11, 23, 12, 13, 12, 18, 22, 16, 21, 22 23, 16, 18, 30, 17, 12, 17, 19, 18, 19, 15, 22, 28, 14, 17, 11, 23, 11, 15, 22 14, 28, 11, 14, 18, 18, 22, 16, 11, 12, 22, 15, 15, 29, 27, 14, 14, 23, 17, 27 18, 25, 29, 22, 19, 21, 13, 25, 19, 13, 27, 18, 22, 27, 30, 12, 25, 27, 29, 15 18, 15, 23, 19, 16, 30, 14, 29, 18, 23, 12, 27, 22, 22, 27, 27, 16, 18, 20, 20 17, 28, 27, 11, 14, 22, 29, 25, 16, 14, 29, 13, 27, 19, 20, 19, 19, 28, 23, 15 21, 24, 15, 20, 20, 29, 27, 21, 30, 15, 17, 18, 20, 12, 25, 29, 22, 14, 19, 21 23, 25, 28, 15, 12, 11, 22, 16, 11, 15, 30, 18, 20, 13, 11, 14, 22, 17, 18, 19 18, 19, 18, 13, 15, 15, 19, 11, 13, 25, 13, 15, 15, 23, 13, 19, 22, 23, 15, 16 19, 18, 15, 12, 30, 18, 12, 18, 22, 27, 17, 24, 12, 14, 17, 22, 18, 17, 14, 27 15, 13, 28, 28, 30, 14, 13, 16, 17, 20, 22, 19, 19, 15, 22, 20, 12, 17, 25, 24 22, 17, 30, 25, 11, 24, 23, 20, 17, 14, 13, 15, 16, 14, 13, 19, 25, 16, 12, 30 22, 18, 14, 23, 12, 23, 14, 28, 28, 20, 25, 22, 27, 25, 14, 28, 21, 12, 28, 13 27, 26, 23, 16, 11, 27, 15, 25, 19, 19, 30, 22, 24, 30, 29, 13, 22, 15, 29, 17 15, 19, 30, 17, 15, 14, 16, 12, 17, 15, 13, 27, 30, 12, 27, 20, 14, 16, 16, 24 20, 15, 20, 11, 11, 27, 26, 12, 14, 30, 28, 15, 13, 16, 22, 15, 26, 14, 24, 16 15, 30, 17, 16, 25, 16, 29, 16, 19, 28, 25, 20, 17, 29, 19, 21, 30, 28, 20, 13

19, 12, 21, 11, 28, 11, 14, 13, 27, 11, 26, 13, 16, 12, 13, 28, 20, 24, 30, 28 Enddata

min=sum(links:(i,j):sum(x(j,k))*w(j,k));

for(links:vd(i):zx(k):wh(j):sum(x(i,k)-sum(x(k,i)+l(j-1,i)>=l(j,i);

end