线性规划在物流运输中数学模型及应用

目 录

线性规划在物流运输中数学模型及应用 ························································································ 1 摘 要 ··············································································································································· 1 关键词 ············································································································································· 1 引 言 ··············································································································································· 1 1、线性规划问题 ···························································································································· 1

1.1、线性规划问题的提出 ······································································································ 1 1.2、线性规划数学模型 ········································································································· 6 1.3、线性规划问题的标准形式 ······························································································ 7 1.4、线性规划问题解的概念 ·································································································· 8

1.4.1、可行解 ··················································································································· 9 1.4.2、基 ·························································································································· 9 1.4.3、基可行解 ············································································································· 10 1.4.4、可行基 ················································································································· 10

2、物流运输问题 ·························································································································· 10

2.1、物流运输 ······················································································································ 10 2.2、物流运输的规划设计 ···································································································· 11

2.2.1、运输成本 ············································································································ 11 2.2.2、运输速度 ············································································································ 11 2.2.3、运输的一致性 ···································································································· 11 2.2.4、与物流节点的匹配程度 ····················································································· 11 2.3、运输规划设计内容 ······································································································· 12

2.3.1、确定运输战略 ··································································································· 12 2.3.2、确定运输线路 ··································································································· 12 2.3.3、选择运输方式 ··································································································· 12 2.3.4、运输过程控制 ··································································································· 12 2.4、物流运输问题的提出 ··································································································· 12 2.5、物流运输问题的数学模型 ··························································································· 14 3、物流运输问题线性规划数学模型实例 ···················································································· 14

3.1、车辆调度问题 ··············································································································· 15 3.2、产销运输问题 ··············································································································· 17 3.3、物资调运问题： ··········································································································· 18 4、结束语······································································································································ 25 致谢 ··············································································································································· 25 参考文献 ······································································································································· 25 英文摘要 ······································································································································· 26 Linear Programming in logistics and ···················································································· 26 transportand application of mathematical models ····························································· 26

Abstract ································································································································ 26 Keywords ····························································································································· 26

线性规划在物流运输中数学模型及应用

线性规划在物流运输中数学模型及应用

摘 要：

本论文重要是对线性规划问题的提出、标准型、以及求解进行分析，然后建立

一些数学模型来解决一些实际问题。针对物流运输这个方面的实际应用建立一些特殊的数学模型用线性规划进行分析，让物流运输变的简单、快捷、节约成本。本文的关键是对物流运输中的问题建立的数学模型就行分析，利用线性规划来运算和求解，建立线性规划数学模型。

关键词：线性规划 物流运输 数学模型 车辆调用 物资调运 引 言：

物流是物品从供应地向接受地的实体流动过程。据数据统计，在机械产品的生

产过程中，加工时间仅占10％左右，而物流时间却占90％，很大一部分生产成本消耗在物流过程中。而运杂费接近总物流费用50％。因此，运输成了降低物流费用最有潜力的领域，它是物流活动的核心。在运输组织中，如何选择合理路线使运输费用最省，线性规划是实现运输管理最优化最成功的方法。线性规划创始人、美国G．Dantzig教授曾在一个学术会议上说，他除了发现单纯形法之外，还有两个功绩：一是总结人们的实践经验，认识到在管理科学中大多数的实际关系都可用线性公式来表示；二是明确提出应该使用目标函数作为最优方案的选择准则。为此，本文主要介绍在物流运输中如何建立它的线性规划数学模型。至于求解线性规划的单纯形法不在这里介绍，因为用单纯形法求解线性规划问题计算机应用软件包代替了人工计算，并能非常轻松地解决此问题。因此，现在物流业面临的新问题是针对具体的物资运输实物如何建立起数学模型，以及建立线性规划的条件。

1、线性规划问题

1.1、线性规划问题的提出

在生产管理和经营活动中经常提出一类问题，即如何合理地利用有限的人力、物力、财力等资源，以便得到最好的经济效益。

第 1 页 共 27 页

线性规划在物流运输中数学模型及应用

例2.1 饲料问题

饲料场的饲料由各种食物混合而成，要求各种营养素达到各自的一定限量。假定有n种食物f1,f2,?,fn可供选择，要求每天所供给的m种营养素v1,v2,?,vm量分别不少于b1,b2,?,bm单位，食物f1的单位重量的价格为ci,fi含vj的百分比aji，其中i?1,2,?m;i?1,2,?,n。

假定每天每份饲料含食物fi的重量为xi，其中i?1,2,?,n，则代价为

z?c1x1?c2x2???cnxn。要求在保证营养素vi不少于bi(i?1,2,?m)的条件下，使代价最小，则问题导致

min z?c1x1?c2x2???cnxn?a11x1?a12x2???a1nxn?b1??a21x1?a22x2???a2nxn?b2 ?s.t.???ax?ax???ax?bm22mnnn?m11??x1?0,x2?0,?,xn?0若考虑营养素vi不得少于bi，但不得超过bi,bi?bi，其中i?1,2,?,n，则问题导致

min z?c1x1?c2x2???cnxns.t.b1?a11x1?a12x2???a1nxn?b1b2?a21x1?a22x2???a2nxn?b2?bm?am1x1?am2x2???amnxn?bnx1?0,x2?0,?,xn?0如若进一步考虑饲料中食物fi的含量不得超过di单位，其中i?1,2,?n，则问题导致

第 2 页 共 27 页 2

线性规划在物流运输中数学模型及应用

min z?c1x1?c2x2???cnxns.t.b1?a11x1?a12x2???a1nxn?b1b2?a21x1?a22x2???a2nxn?b2?bm?am1x1?am2x2???amnxn?bn0?x1?d1,0?x2?d2,?,0?xn?dn

例2.2 生产计划问题

某工厂生产两种产品P1和P2。产品P1的单位售价29元，产品P2单位售价23元；产品P1每单位原材料费用为12元，而产品P2每单位原材料费用为11元；产品产品P2每单位需要机器 m1和机器P1每单位需要m1机器2小时和m2机器 1小时，

m2各1小时。产品P1每单位机器费用13元，产品P2每单位机器费用10元。该工厂机器 每天有100小时可供使用，机器每天有80小时可供使用。产品P1销售量不受限制，而产品P2最多只能卖出40个单位。问该厂应该如何安排使利润到达最大。

假定每日生产P1为x1单位，生产P2为x2单位。产品P1每单位的利润为29-12-13=4元，产品P2每单位的利润为28-11-10=2元。

总利润

z?4x1?2x2

约束条件

2x1?x2?100 x1?x2?80

x1?0,0?x2?40故生产计划问题导致下面的线性规划问题，即安排生产使总利润达到最大。

maxz?4x1?2x2

s.t.2x1?x2?100

x1?x2?80x1?0,0?x2?40第 3 页 共 27 页

3