8_Transportation and Assignment__ Problems
第四章 运输问题(Transportation Problem)
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第三步：按σij=cij-(ui+vj), i,j∈N计算所有空格的检验数。如 σ11=c11-(u1+v1)=3-(0+2)=1， σ12=c12-(u1+v2)=11-(0+9)=2 这些计算可直接在上表中进行。 为了方便，特设计计算表，如下表所示
销地
B1
B2
B3
加工厂
B4
1.闭回路法； 2.位势法
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1 闭回路法
在给出调运方案的计算表上，如上例表，从每一空格出发找一条闭回路。
它是以某空格为起点，用水平或垂直线向前划，当碰到一数字格时可以
转90°后，继续前进，直到回到起始空格为止。闭回路如图(a),(b),(c)
等所示。
销 地 B1 B2 B3 B4 产
沃格尔法的步骤是：
第一步：在原表中分别计算出各行和各列的最 小运费和次小运费的差额，并填入该表的最右列 和最下行，见表4-1
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表4-1
销 地 B1 B2 B3 B4 行差额 产地
A1
3 11 3 10
0
A2
19 2 8
1
A3
7 4 10 5
1
列差额
25 1 3
第二步：从行或列差额中选出最大者，选择它所 在行或列中的最小元素。在表4-1中B2列是最大 差额所在列。B2列中最小元素为4，可确定A3 的产品先供应B2的需要。得表4-2：
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运输问题数学模型的一般形式
若用xij表示从Ai到Bj的运量，那么在产销平衡的条件下，要求得总
运费最小的调运方案，其数学模型为
Chapter06-运输问题和指派问题
Characteristics of Transportation Problems
• The Cost Assumption (成本假设)
The cost of distributing units from any particular source to any particular destination is directly proportional to the number of units distributed. (从任何 一个出发地到任何一个目的地的货物配送 成本和所配送的数量成线性比例关系)
Each destination has a fixed demand for units, where this entire demand must be received from the sources. (每一个目的地都 有一个固定的需求量，所有的需求量都必须由 出发地满足)
Characteristics of Transportation Problems
Chapter06-运输问题和 指派问题
2020年5月28日星期四
Table of Contents (主要内容)
• The P&T Company Distribution Problem (Section 6.1)(P&T公司的配送问题)
• Characteristics of Transportation Problems (Section 6.2)(运输问题的特征)
• The Feasible Solutions Property ( 可行解特性)
A transportation problem will have feasible solutions if and only if the sum of its supplies equals the sum of its demands. (当且仅当供应量的总 和等于需求量的总和时，运输问题才有 可行解)
lpSolve-5.6.20 用户指南 说明书
Package‘lpSolve’December10,2023Version5.6.20Title Interface to'Lp_solve'v.5.5to Solve Linear/Integer ProgramsAuthor Michel Berkelaar and othersMaintainer Gábor Csárdi<**********************>Description Lp_solve is freely available(under LGPL2)software forsolving linear,integer and mixed integer programs.In thisimplementation we supply a``wrapper''function in C and some Rfunctions that solve general linear/integer problems,assignment problems,and transportation problems.This versioncalls lp_solve version5.5.License LGPL-2URL https:///gaborcsardi/lpSolveBugReports https:///gaborcsardi/lpSolve/issuesEncoding UTF-8NeedsCompilation yesRepository CRANDate/Publication2023-12-1013:40:02UTCR topics documented:lp (2)lp.assign (4)lp.object (6)lp.transport (7)make.q8 (8)print.lp (9)Index1112lp lp Linear and Integer ProgrammingDescriptionInterface to lp_solve linear/integer programming systemUsagelp(direction="min",objective.in,const.mat,const.dir,const.rhs,transpose.constraints=TRUE,int.vec,presolve=0,compute.sens=0,binary.vec,all.int=FALSE,all.bin=FALSE,scale=196,dense.const,num.bin.solns=1,use.rw=FALSE,timeout=0L)Argumentsdirection Character string giving direction of optimization:"min"(default)or"max."objective.in Numeric vector of coefficients of objective functionconst.mat Matrix of numeric constraint coefficients,one row per constraint,one column per variable(unless transpose.constraints=FALSE;see below).const.dir Vector of character strings giving the direction of the constraint:each value should be one of"<,""<=,""=,""==,"">,"or">=".(In each pair the two valuesare identical.)const.rhs Vector of numeric values for the right-hand sides of the constraints.transpose.constraintsBy default each constraint occupies a row of const.mat,and that matrix needs tobe transposed before being passed to the optimizing code.For very large con-straint matrices it may be wiser to construct the constraints in a matrix column-by-column.In that case set transpose.constraints to FALSE.int.vec Numeric vector giving the indices of variables that are required to be integer.The length of this vector will therefore be the number of integer variables.presolve Numeric:presolve?Default0(no);any non-zero value means"yes."Currently ignored.compute.sens Numeric:compute sensitivity?Default0(no);any non-zero value means"yes."binary.vec Numeric vector like int.vec giving the indices of variables that are required to be binary.all.int Logical:should all variables be integer?Default:FALSE.all.bin Logical:should all variables be binary?Default:FALSE.scale Integer:value for lpSolve scaling.Details can be found in the lpSolve docu-mentation.Set to0for no scaling.Default:196dense.const Three column dense constraint array.This is ignored if const.mat is supplied.Otherwise the columns are constraint number,column number,and value;thereshould be one row for each non-zero entry in the constraint matrix.lp3 num.bin.solns Integer:if all.bin=TRUE,the user can request up to num.bin.solns optimal so-lutions to be returned.use.rw Logical:if TRUE and num.bin.solns>1,write the lp out to afile and read it back in for each solution after thefirst.This is just to defeat a bug somewhere.Although the default is FALSE,we recommend you set this to TRUE if youneed num.bin.solns>1,until the bug is found.timeout Integer:timeout variable in seconds,defaults to0L which means no limit is set. DetailsThis function calls the lp_solve5.5solver.That system has many options not supported here.The current version is maintained at https:///5.5/Note that every variable is assumed to be>=0!ValueAn lp object.See lp.object for details.Author(s)Sam Buttrey,<***************>See Alsolp.assign,lp.transportExamples##Set up problem:maximize#x1+9x2+x3subject to#x1+2x2+3x3<=9#3x1+2x2+2x3<=15#f.obj<-c(1,9,1)f.con<-matrix(c(1,2,3,3,2,2),nrow=2,byrow=TRUE)f.dir<-c("<=","<=")f.rhs<-c(9,15)##Now run.#lp("max",f.obj,f.con,f.dir,f.rhs)##Not run:Success:the objective function is40.5lp("max",f.obj,f.con,f.dir,f.rhs)$solution##Not run:[1]0.04.50.0##The same problem using the dense constraint approach:#f.con.d<-matrix(c(rep(1:2,each=3),rep(1:3,2),t(f.con)),ncol=3)lp("max",f.obj,,f.dir,f.rhs,dense.const=f.con.d)##Not run:Success:the objective function is40.5##Get sensitivities#lp("max",f.obj,f.con,f.dir,f.rhs,compute.sens=TRUE)$sens.coef.from##Not run:[1]-1e+302e+00-1e+30lp("max",f.obj,f.con,f.dir,f.rhs,compute.sens=TRUE)$sens.coef.to##Not run:[1]4.50e+001.00e+301.35e+01##Right now the dual values for the constraints and the variables are#combined,constraints coming first.So in this example...#lp("max",f.obj,f.con,f.dir,f.rhs,compute.sens=TRUE)$duals##Not run:[1] 4.50.0-3.50.0-10.5##...the duals of the constraints are4.5and0,and of the variables,#-3.5,0.0,-10.5.Here are the lower and upper limits on these:#lp("max",f.obj,f.con,f.dir,f.rhs,compute.sens=TRUE)$duals.from##Not run:[1]0e+00-1e+30-1e+30-1e+30-6e+00lp("max",f.obj,f.con,f.dir,f.rhs,compute.sens=TRUE)$duals.to##Not run:[1]1.5e+011.0e+303.0e+001.0e+303.0e+00##Run again,this time requiring that all three variables be integer#lp("max",f.obj,f.con,f.dir,f.rhs,int.vec=1:3)##Not run:Success:the objective function is37lp("max",f.obj,f.con,f.dir,f.rhs,int.vec=1:3)$solution##Not run:[1]140##You can get sensitivities in the integer case,but they re harder to#interpret.#lp("max",f.obj,f.con,f.dir,f.rhs,int.vec=1:3,compute.sens=TRUE)$duals ##Not run:[1]10070##Here s an example in which we want more than one solution to a problem#in which all variables are binary:the8-queens problem,#with dense constraints.#chess.obj<-rep(1,64)q8<-make.q8()chess.dir<-rep(c("=","<"),c(16,26))chess.rhs<-rep(1,42)lp( max ,chess.obj,,chess.dir,chess.rhs,dense.const=q8, all.bin=TRUE,num.bin.solns=3)lp.assign Integer Programming for the Assignment ProblemDescriptionInterface to lp_solve linear/integer programming system specifically for solving assignment prob-lemsUsagelp.assign(cost.mat,direction="min",presolve=0,compute.sens=0)Argumentscost.mat Matrix of costs:the ij-th element is the cost of assigning source i to destination j.direction Character vector,length1,containing either"min"(the default)or"max"presolve Numeric:presolve?Default0(no);any non-zero value means"yes."Currently ignored.compute.sens Numeric:compute sensitivity?Default0(no);any non-zero value means"yes."In that case presolving is attempted.DetailsThis is a particular integer programming problem.All the decision variables are assumed to be integers;each row has the constraint that its entries must add up to1(so that there is one1and the remaining entries are0)and each column has the same constraint.This is assumed to be a minimization problem.ValueAn lp object.See documentation for details.The constraints are assumed(each row adds to1,each column adds to1,and no others)and are not returned.Author(s)Sam Buttrey,<***************>See Alsolp,lp.transportExamplesassign.costs<-matrix(c(2,7,7,2,7,7,3,2,7,2,8,10,1,9,8,2),4,4)##Not run:>assign.costs[,1][,2][,3][,4][1,]2771[2,]7729[3,]7388[4,]221026lp.object ##End(Not run)lp.assign(assign.costs)##Not run:Success:the objective function is8lp.assign(assign.costs)$solution##Not run:[,1][,2][,3][,4][1,]0001[2,]0010[3,]0100[4,]1000##End(Not run)lp.object LP(linear programming)objectDescriptionStructure of lp objectValueAn lp.object is a list containing the following elements:direction Optimization direction,as enteredx.count Number of variables in objective functionobjective Vector of objective function coefficients,as enteredconst.count Number of constraints enteredconstraints Constraint matrix,as entered(not returned by lp.assign or lp.transport) int.count Number of integer variablesint.vec Vector of integer variables’indices,as enteredobjval Value of objective function at optimumsolution Vector of optimal coefficientsnum.bin.solns Numeric indicator of number of solutions returnedstatus Numeric indicator:0=success,2=no feasible solutionAuthor(s)Sam Buttrey,<***************>See Alsolp,lp.assign,lp.transportlp.transport7 lp.transport Integer Programming for the Transportation ProblemDescriptionInterface to lp_solve linear/integer programming system specifically for solving transportation problemsUsagelp.transport(cost.mat,direction="min",row.signs,row.rhs,col.signs,col.rhs,presolve=0,compute.sens=0,integers=1:(nc*nr)) Argumentscost.mat Matrix of costs;ij-th element is the cost of transporting one item from source i to destination j.direction Character,length1:"min"or"max"row.signs Vector of character strings giving the direction of the row constraints:each value should be one of"<,""<=,""=,""==,"">,"or">=."(In each pair the two valuesare identical.)row.rhs Vector of numeric values for the right-hand sides of the row constraints.col.signs Vector of character strings giving the direction of the column constraints:each value should be one of"<,""<=,""=,""==,"">,"or">=."col.rhs Vector of numeric values for the right-hand sides of the column constraints.presolve Numeric:presolve?Default0(no);any non-zero value means"yes."Currently ignored.compute.sens Numeric:compute sensitivity?Default0(no);any non-zero value means"yes."integers Vector of integers whose ith element gives the index of the ith integer variable.Its length will be the number of integer variables.Default:all variables areinteger.Set to NULL to have no variables be integer.DetailsThis is a particular integer programming problem.All the decision variables are assumed to be integers,and there is one constraint per row and one per column(and no others).This is assumed to be a minimization problem.ValueAn lp object.Constraints are implicit and not returned.See documentation for details.Author(s)Sam Buttrey,<***************>8make.q8 ReferencesExample problem from Bronson(1981),Operations Research,Scahum’s Outline Series,McGraw-Hill.See Alsolp.assign,lp.transportExamples##Transportation problem,Bronson,problem9.1,p.86##Set up cost matrix#costs<-matrix(10000,8,5);costs[4,1]<-costs[-4,5]<-0costs[1,2]<-costs[2,3]<-costs[3,4]<-7;costs[1,3]<-costs[2,4]<-7.7costs[5,1]<-costs[7,3]<-8;costs[1,4]<-8.4;costs[6,2]<-9costs[8,4]<-10;costs[4,2:4]<-c(.7,1.4,2.1)##Set up constraint signs and right-hand sides.#row.signs<-rep("<",8)row.rhs<-c(200,300,350,200,100,50,100,150)col.signs<-rep(">",5)col.rhs<-c(250,100,400,500,200)##Run#lp.transport(costs,"min",row.signs,row.rhs,col.signs,col.rhs)##Not run:Success:the objective function is7790lp.transport(costs,"min",row.signs,row.rhs,col.signs,col.rhs)$solution##Not run:[,1][,2][,3][,4][,5][1,]010001000[2,]0030000[3,]0003500[4,]2000000[5,]5000050[6,]000050[7,]0010000[8,]00050100##End(Not run)make.q8Generate sparse constraint matrix for8-queens problemDescriptionGenerate sparse constraint matrix for8-queens problemUsagemake.q8()ArgumentsNone.DetailsSparse constraints come in a three-column matrix or data frame.Each row gives the row number, column number,and value of a particular non-zero entry in the constraint matrix.This function produces the sparse constraint matrix for the8-queens problem(in which the object is to place eight queens on a chessboard with no two sharing a row,column or diagonal).The resulting sparse representation is252x3,compared to42x64for the usual representation.ValueA252x3numeric matrix.See lp for the complete example.Author(s)Sam Buttrey,<***************>See Alsolpprint.lp Print an lp objectDescriptionPrint method for lp objectsUsage##S3method for class lpprint(x,...)Argumentsx List with items named objval and status.Normally this will have been called by lp,lp.assign,or lp.transport....Other arguments,all currently ignoredDetailsThis function prints the objective function value,together with the word"Success"if the operation is successful,or an indication of the error if not.If multiple solutions have been produced(because this was an all-binary problem and lp was called with num.bin.solns>1)the number of solutions is also displayed.ValueNoneAuthor(s)Sam Buttrey,<***************>See Alsolp,lp.assign,lp.transportIndex∗optimizelp,2lp.assign,4lp.object,6lp.transport,7make.q8,8print.lp,98-queens problem(make.q8),8lp,2,5–7,9,10lp.assign,3,4,6,8–10lp.object,3,6lp.transport,3,5,6,7,8–10make.q8,8print.lp,911。
Chapter 8 Transportation
2.2 Useful expressions
• Enquiry about freight rates and sailings ------ P143 • Replies to enquiries for freight rates ------ P143
• Urging shipment
1. We wish to call your attention to the fact that up to the present moment no news comes from you about the shipment under the captioned contract. 2. We are in urgent need of the goods in question and would have to request you to execute shipment within the time stipulated. 3. As the contract time of delivery is rapidly falling due (approaching), it is imperative that we receive your shipping advice without any further delay.
• in sb’ s favor \ in favor of sb: regarding somebody as the beneficiary e.g. 我方已以你方为受益人开出了信用证。 我方已以你方为受益人开出了信用证。 we have opened a letter of credit in your favor. • invite / draw/ call sb’s attention to …: ask sb to pay attention to sth e.g. We would like to invite your attention to our Order No.123. • recommend to: turn over to 交付 e.g. We are in urgent need of the goods to recommend to our clients.
八年级英语下册 Unit 8《Transportation》Part 2同步练习1 北师大版
Unit 8 TransportationPart 2I. 用must，have to 和mustn’t的正确形式填空。
1. To buy a ticket, you _____ line up at this window.2. To get to the train, you _______ put your ticket in this machine here.3. Don’t stand close to the tracks. You _______ stand behind thi s yellow line.4. You _______ take animals on the subway. It’s against the rules.5. You ______ give rides to other people. It’s dangerous.II. 完成句子。
1. 过马路时要走斑马线。
When you _____ ____ ______, you ____ ______ cross it at a zebra crossing. 2. 他妈妈病了，他得在家照顾她。
His mother is ill. He ____ _____ look after her at home.3. 每个司机都要有驾驶执照。
Every driver ____ ______ a driver’s license.4. 不准在街上踢球，很危险。
You _____ ______ ______ in the street . It is dangerous.5. 在路上开车时，一定要等交通灯。
When you drive along the road, you ____ _____ for the traffic lights to change.6. 红灯亮时，一定要停车。
When the traffic lights ___ ___, you ____ ______ your car.7. 他感冒了，他必须休息。
Unit_8_transport_and_insurance
Export packing
Export packing
Export packing
Export packing
Export packing
Export packing
Export packing
Arrange adequate packing, including shipping marks
I. Transport
Receive the bill of lading from the shipping company
Pay the freight bill
Arrange shipment
Parcel Post Insurance
II. Insurance
3. PICC、CIC、ICC 中国人民保险公司(The People’s Insurance Company of China; PICC)制定的中国保险 条款（China Insurance Clauses; CIC） 伦敦保险协会（Insurance Institute of London）制定的协会货物条款（Institute Cargo Clauses; ICC）
II. Insurance
4、海洋运输货物保险的三个基本险别中FPA主要包 括全部损失（Total Loss）和共同海损 （General Average）；WPA则再加上单独海 损（Particular Average）；AR除WPA的范围外， 还包括11种一般附加险（Extraneous Risks）。 所以FPA<WPA<AR。 5、一般附加险全部包括在一切险中。因此如投保 了一切险，则无需再投保一般附加险。如投保 了平安险或水渍险后，可以根据需要加保一般 附加险中的一种或几种险别。 6、特别附加险和特殊附加险需由投保人向保险公 司提出申请，经保险公司特别同意，且投保人 投保了基本险后方可以加保。
Chapter06运输问题和指派问题
The Transportation Problem is an LP
•subject to (约束)
Cannery 1: x11 + x12 + x13 + x14 = 75 Cannery 2: x21 + x22 + x23 + x24 = 125 Cannery 3: x31 + x32 + x33 + x34 = 100 Warehouse 1: x11 + x21 + x31 = 80 Warehouse 2: x12 + x22 + x32 = 65 Warehouse 3: x13 + x23 + x33 = 70 Warehouse 4: x14 + x24 + x34 = 85 and xij ≥ 0 (i = 1, 2, 3; j = 1, 2, 3, 4)
PPT文档演模板
Chapter06运输问题和指派问题
P&T Company Distribution Problem
•试建立该网络 配送问题的数 学模型？
PPT文档演模板
Chapter06运输问题和指派问题
运输问题
•运输问题关心的是以最 低的总配送成本把出发 地的任何产品运送到每 一个目的地
PPT文档演模板
Network Representation
PPT文档演模板
Chapter06运输问题和指派问题
运输问题的网络表述
• 忽略出发地和目的地在地理上的 布局
• 左边一列为出发地(S)，旁边的数 字代表供应量
• 右边一列为目的地(D)，旁边的 数字代表需求量
• 箭头表示可能的运输途径，其上 面的数字代表单位运输成本