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1、单纯型表格算法单纯型表格算法9595Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpage page 2 27/30/20247/30/2024第七、八讲1 与基础解对应的单纯型表格（1） 关于单纯形法概念，以前已讲过，本章讲具体计算步骤。该计算方法在计算机上计算。令A为mn矩阵，mn,行线性独立，即秩为m。非退化线性规划的标准形式为：AX=b，X0，CTX=min(1) 如果b是A的不少于
2、m列的线性组合，则称非退化形式。设从 阶段2开始计算，即设已获得初始可行解X（这需阶段1，而 寻找第1个初始可行解亦需用到阶段2算法）。Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpage page 3 37/30/20247/30/2024第七、八讲Operations Research Operations Research Prof. Wang Prof. Wang School o
3、f Economics & ManagementSchool of Economics & Managementpage page 4 47/30/20247/30/2024第七、八讲Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpage page 5 57/30/20247/30/2024第七、八讲Operations Research Operations Research Prof.
4、Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpage page 6 67/30/20247/30/2024第七、八讲Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpage page 7 77/30/20247/30/2024第七、八讲Operations Research Op
5、erations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpage page 8 87/30/20247/30/2024第七、八讲Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpage page 9 97/30/20247/30/2024第七、
6、八讲1 与基础解对应的单纯型表格（8） 表格中，第一部分为A阵所有列关于当前基础列之表达式。第二部分为当前基础解的正分量。第三部分为当前基础阵之逆阵。Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpage page 10107/30/20247/30/2024第七、八讲2 新基础解表格与原表格的关系（1） 设在上述基础解的基础上，将非基矢量as引进基础解集，即asB集；而令原基础解集B中的V
7、r处原基矢量离开。这两种情况表达形式示于表1-3和表1-4表1-3原表格（用当前基础解表达矢量）a1asanbe1emV1t11t1st1nt10u11u1mVrtr1trstrntr0ur1urmVmtm1tmstmntm0um1ummOperations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpage page 11117/30/20247/30/2024第七、八讲2 新基础解表格与原表格的关系（2）
8、 表1-4新表格（用新基础解表达矢量）a1asanbe1em(V1)t11t1st1nt10u11u1mas(Vr)tr1trstrntr0ur1urm(Vm)tm1tmstmntm0um1ummOperations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpage page 12127/30/20247/30/2024第七、八讲2 新基础解表格与原表格的关系（3） 其中原表格是已经给出的与当前基础矢量相对
9、应的单纯形表格，新基础解是由原非基矢量as代入Vr所致。其新解所对应的表格形式如表1-4所示，那么新表中各个元素如何根据原表1-3求出呢？现在就对该问题进行推导。因为当前基础解和新基础解都假定是可行的非退化解，因此两表中：ti00，ti00(i=1,2,m)(16)Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpage page 13137/30/20247/30/2024第七、八讲2 新基
10、础解表格与原表格的关系（4） 则，当前基础可行解满足：AX=(17)因此，ti0就是X的正分量根据当前基础解来表达as：as=t1sV1+trsVr+tmsVm (18)由于基础矢量V1,Vr,Vm线性无关，因此trs0；Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpage page 14147/30/20247/30/2024第七、八讲2 新基础解表格与原表格的关系（5） 否则，从式(1
11、8)中说明，as可由Vi(i=1,m,ir)线性组合，这说明新基础解不是线性无关的矢量构成，与假设矛盾。于是，Vr可根据式(18)表达为： Vr= (19)从假设知，新基础解基础矢量为：(V1)=V1,(Vr)=as,(Vm)=Vm(20)为获得新表格，系数tij应满足：Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpage page 15157/30/20247/30/2024第七、八讲2
12、 新基础解表格与原表格的关系（6） aj=t1j(V1)+trj(Vr)+tmj(Vm)这表明aj =trjas+(21)(式中，表示，下同)这唯一地定义了新表格中的系数。而根据当前基础解，存在： aj =trjVr+(22)Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpage page 16167/30/20247/30/2024第七、八讲2 新基础解表格与原表格的关系（7） 现在，把V
13、r用式(19)代入，则式(22)变为： aj =(23)将aj的两个表达式(21)和(23)的矢量系数加以比较，便可得出下述关系：当i=r时,trj=trj/trs (24)当ir时，tij=tij(tis/trs)trj (25)Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpage page 17177/30/20247/30/2024第七、八讲2 新基础解表格与原表格的关系（8） 对于
14、扩充表格右边系数uij的表达更没问题。可把单位矢量ej作为扩充矩阵A，I的列，就像阶段法1做的那样，可设：e1=an+1，e2=an+2，em=an+m(26)于是，我们可定义：uij=ti,，n+j(i,j=1,2,m)(27)在式(24)和(25)中，用n+j代替j可得当i=r时，urj=urj/trs (28)当ir时，uij=uij(tis/trs)urj (29)Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economic
15、s & Managementpage page 18187/30/20247/30/2024第七、八讲2 新基础解表格与原表格的关系（9） 上述公式的含义如下：如果用as代替Vr而形成新基础可行解，则称原表中的r行为支点行原表中的s列为支点列trs为支点元素支点行(r)：新表r行原表r行/trs非支点行(i):新表i行=原表i行i（原表r行）Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpa
16、ge page 19197/30/20247/30/2024第七、八讲2 新基础解表格与原表格的关系（10） 从上所述，如果知道原表格的支点行和支点列，那么，新的表格便可应孕而生。于是如何选择支点行和支点列便是单纯形表格的关键问题，下面就分别阐述这两个问题。Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpage page 20207/30/20247/30/2024第七、八讲3 如何选择支点
17、行 选择支点行即是假设支点列已选定（即已知道应引进的非基础矢量），应令哪一个基础矢量离开的问题。支点行的选择原则是新解可行，即新表格中的ti0（i=1,m）0(对于非退化)。即，若r行被选中，则必须满足：tr0/trs=minti0/tis：tis0，i=1,m这就是r行被中的充分必要条件，对于非退化情况，r行能被唯一确定。Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpage page 2
18、1217/30/20247/30/2024第七、八讲4 如何选择支点列 （1） 在选择支点行之前，必须首先选择支点列，其原则是，新矢量进入基础解使费用尽量小，即最优性选择。旧费用为： CTX=(32)表明基础矢量Vi的单位费用。如果决定选支点列s，那么新的解的费用是多少呢？根据： as =0 (33) 及 = AX = b (34)Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpage pa
19、ge 22227/30/20247/30/2024第七、八讲4 如何选择支点列 （2） 将(33)式乘以再加(34)式得：as+(35)于是，得出一组新解，其新解的费用为：cs +(36)而 =旧费用，令 (37)故 新费用=旧费用（zscs） (38)Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpage page 23237/30/20247/30/2024第七、八讲4 如何选择支点列
20、（3） 即，要使费用减少，只有zscs0才行，这和在第五节所得结论一致。即可得出下述结论：从式（35）看出，若所有tis0,则当时，X（）永远可行，若此时，zscs0,则最优目标值无界，即可无穷小。若zscs0且至少有一个tis0，则s列是被选中之一，若有若干列的zscs0,则通常选择最大的zscs作为支点列，然后根据前面讲的选择支点行标准选取支点元素并构成新的表格。Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics &
21、 Managementpage page 24247/30/20247/30/2024第七、八讲4 如何选择支点列 （4） 当选中s列作为支点列时，其新费用减少值为*（zscs）,其中，*是新基础解中as的系数。（as=(vr)）,我们有：*=tr0=tr0/trs如果，旧费用为z0,新费用为z0，则有 z0=z0tr0(39)对于原扩充表格：T；U=M1A，b；I，显然缺少一些信息，即检验数信息，现可把它放在表格最后一行：z1c1，,zncn，z0；y1，ym(40)Operations Research Operations Research Prof. Wang Prof. Wang S
22、chool of Economics & ManagementSchool of Economics & Managementpage page 25257/30/20247/30/2024第七、八讲4 如何选择支点列 （5） 这一行称为判断行或检验行。该行各元素含义如下：前n个元素之表达式为 zjcj= (j=1,n)(41)如果aj是当前基础解集，则：zjcj=0如果aj不是当前基础解集，则：元素z0是当前费用，(42)Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & Ma
23、nagementSchool of Economics & Managementpage page 26267/30/20247/30/2024第七、八讲4 如何选择支点列 （6） 最后m个元素定义为：(j=1,m) (43)其中uij=U是基础阵M之逆阵，则YT含义为：因此，YT是方程解，在最后阶段，当所有zjcj0时，YT将是下述对偶问题最优解：YTACT,YTb=max其中，YT将满足：YTaj=zjcj。Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & Manageme
24、ntSchool of Economics & Managementpage page 27277/30/20247/30/2024第七、八讲4 如何选择支点列 （7） 加进判断行后，表格变为：(44)判断行之计算：新判断行=旧判断行-0乘支点行 0=（zscs）/trs(45)证明从略。t11t1nt10u11u1mtm1tmntm0um1ummz1c1zncnz0y1ymOperations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economi
25、cs & Managementpage page 28287/30/20247/30/2024第七、八讲5 举例 （1） 至此，单纯形表格法的基本步骤已推导完毕，下面举例来说明单纯形表格的应用。例1-21已知线性规划为：AX=b，X0，CTX=min其中： A= ， b= ，CT=7，1，1(46) 例1-21应用阶段1，求出式(46)的初始基础可行解。解为了求初始基础可行解，要引进人工变量并构成新规划：AX=b，X0，CTX=min Operations Research Operations Research Prof. Wang Prof. Wang School of Economic
26、s & ManagementSchool of Economics & Managementpage page 29297/30/20247/30/2024第七、八讲5 举例 （2） a1a2a3e1e2be1123105e245601135790018其中，A= ，（X)T=(x1，x2，x3，s1，s2) b= (C)T=(0 0 0 1 1)现在变成求新规划最优解问题，显然，此规划的第1个基础可行解为：s1=5,s2=13,X=0,其相应表格为：(47)Operations Research Operations Research Prof. Wang Prof. Wang School
27、 of Economics & ManagementSchool of Economics & Managementpage page 30307/30/20247/30/2024第七、八讲5 举例 （3） 初始表格判断的计算很简单： ，例如z1c1=(1,4)c1=50=5从判断行看出，最大的判断元素为z3c3=9。故a3应进入基础解，支点列s=3（当然a1，a2进入也行），然后选支点行：故支点元素为t13=3,于是用公式可计算出第2个表格。Operations Research Operations Research Prof. Wang Prof. Wang School of Econ
28、omics & ManagementSchool of Economics & Managementpage page 31317/30/20247/30/2024第七、八讲5 举例（4）a1a2a3e1e2ba310e2210213210303(48)从表格（48）看出，max（zjcj）= z1c1=2,故令a1进入基础解集，然后选支点行r： Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managemen
29、tpage page 32327/30/20247/30/2024第七、八讲5 举例（5）支点元素为t21=2。经变换，得出下述表格(49)： a1a2a3e1e2ba301/212/3-1/67/6a111/20-11/23/2000-1-10(49)Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpage page 33337/30/20247/30/2024第七、八讲5 举例（6）判断行
30、zjcj全部0，故达最优，且z0=0。说明原问题存在可行解，且第1个基础可行解即为新规划的最优解：x2=0,x3=7/6,x1=3/2。于是阶段1结束。例1-22求解（46）式的最优解。即进行阶段2。解将例1-21求得的新规划最优表格转换为原规划的初始单纯形表格。转换原则是：1）表的上面m行基本不变，新表右半部uij与原表人工变量矢量所对应的tij一致。2）表的判断行，需按照原规划的目标系数重算。Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool
31、 of Economics & Managementpage page 34347/30/20247/30/2024第七、八讲5 举例（7）计算的初始单纯形表格如表格(50)：比较(49)与(50)，看出：把e1，e2列移往右边a1a2a3be1e2a301a110103035/319/310/3(50)Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpage page 35357/30/20
32、247/30/2024第七、八讲5 举例（8）判断行不同，因阶段1时的新规划费用系数为CT=（00011），而原规划为CT=711。初始解的基础矢量为a3，a1，=(17)。例：z2c2=1+71=3z0=1+7=35/3y1=1+(-1)7=Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpage page 36367/30/20247/30/2024第七、八讲5 举例（9）从表格（50）看
33、出：z2c2=3为唯一正值，选s=2。选支点行：trs= V1 =a2(代替a3)最后表格：a1a2a3be1e2a20127/34/3-1/3a110-11/3-5/32/300-614/3-31/313/3Operations Research Operations Research Prof. Wang Prof. Wang School of Economics & ManagementSchool of Economics & Managementpage page 37377/30/20247/30/2024第七、八讲5 举例（10）则知，所有zjcj0(j =1，2，3)，已达最
34、优。最优基础可行解：x2=，x1=mincost=CTX =z0=最优对偶分量： y1=,y2=检查条件：AX=b,X0,YTACT,CTX=YTb。全部符合条件。eNbJ8G5D1A-w*t$qYnVkSgPdLaI7F3C0z)v&s#pXmUiRfOcK9H5E2B+x(u%rZoWkThQeMbJ8G4D1A-w*t!qYnVjSgPdLaI6F3C0y)v&s#pXlUiRfNcK9H5E2A+x(u$rZoWkThPeMbJ7G4D1z-w&t!qYmVjSgOdL9I6F3B0y)v%s#oXlUiQfNcK8H5E2A+x*u$rZnWkThPeMaJ7G4C1z-w&t!pY
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40、I7F3C0z)v&s!pXmUiRfOcK9H6E2B+x(u%rZoWlThQeMbJ8G4D1A-w*t!qYnVjSgPdLaI7F3C0y)v&s#pXmUiRfNcK9H5E2B+x(u$rZoWkThQeMbJ7G4D1z-w*t!qYmVjSgOdLaI6F3B0y)v%s#pXlUiQfNcK8H5E2A+x(u$rZnWkThPeMbJ7G4C1z-w&t!qYmVjRgOdL9I6F3B0y(v%s#oXlUiQfNbK8H5D2A+x*u$qZnWkShPeMaJ7F4C1z)w&t!pYmUjRgOcL9I6E3B0y(v%r#oXlTeMbJ7G4C1z-w&t!q
41、YmVjRgOdL9I6F3B0y(v%s#oXlUiQfNbK8H5D2A+x*u$qZnWkShPeMaJ7G4C1z)w&t!pYmVjRgOcL9I6E3B0y(v%r#oXlTiQfNbK8G5D2A-x*u$qZnVkShPdMaJ7F4C0z)w&s!pYmUjRfOcL9H6E3B+y(u%r#oWlTiQeNbK8G5D1A-x*t$qZnVkSgPdMaI7F4C0z)v&s!pXmUjRfOcK9H6E2B+y(u%rZoWlThQeNbJ8G4D1A-w*t$qYnVjSgPdLaI7F3C0z)v&s#pXmUiRfOcK9H5E2B+x(u%rZoWkThQeMbJ
42、8G4D1z-w*t!qYnVjSgOdLaI6F3C0y)v%s#pXlUiRfNcK8H5E2A+x(u$rZoWkThPeMbJ7G4D1z-w&t!qYmVjSgOdL9I6F3B0y)v%s#oXlUiQfNcK8H5D2A+x*u$rZnWkShPeMaJ7G4C1z)w&t!pYmVjRgOcL9I6E3B0y(v%s#oXlTiQfNbK8H5D2A-x*u$qZnWkShPdMaJ7F4C1z)w&s!pYmUjRgOcL9H6E3B+y(v%r#oWlTiQeNbK8G5D1A-x*t$qZnVkShPdMaI7F4C0z)w&s!pXmUjRfOcL9H6E2B+y(un
43、WkShPdMaJ7F4C1z)w&s!pYmUjRgOcL9H6E3B+y(v%r#oWlTiQeNbK8G5D2A-x*t$qZnVkShPdMaI7F4C0z)w&s!pXmUjRfOcL9H6E2B+y(u%r#oWlThQeNbJ8G5D1A-w*t$qYnVkSgPdLaI7F3C0z)v&s!pXmUiRfOcK9H6E2B+x(u%rZoWlThQeMbJ8G4D1A-w*t!qYnVjSgPdLaI6F3C0y)v&s#pXlUiRfNcK9H5E2A+x(u$rZoWkThQeMbJ7G4D1z-w*t!qYmVjSgOdLaI6F3B0y)v%s#pXlUiQfNcK8H
44、5E2A+x*u$rZnWkThPeMaJ7G4C1z-w&t!pYmVjRgOdL9I6E3B0y(v%s#oXlUiQfNbK8H5D2A+x*u$qZnWkShPeMaJ7F4C1z)w&t!pYmUjRgOcL9I6E3B+y(v%r#oXlTiQeNbK8G5D2A-x*t$qZnVkShPdMaJ7F4C0z)w&s!pYmUjRfOcL9H6E3B+y(u%r#oWlTiQeNbJ8G5D1A-x*t$qYnVkSgPdMaI7F3y(v%r#oXlTiQeNbK8G5D2A-x*u$qZnVkShPdMaJ7F4C0z)w&s!pYmUjRfOcL9H6E3B+y(u%r#oW
45、lTiQeNbJ8G5D1A-x*t$qYnVkSgPdMaI7F3C0z)v&s!pXmUjRfOcK9H6E2B+y(u%rZoWlThQeNbJ8G4D1A-w*t$qYnVjSgPdLaI7F3C0y)v&s#pXmUiRfNcK9H5E2B+x(u$rZoWkThQeMbJ8G4D1z-w*t!qYnVjSgOdLaI6F3C0y)v%s#pXlUiRfNcK8H5E2A+x(u$rZnWkThPeMbJ7G4C1z-w&t!qYmVjRgOdL9I6F3B0y(v%s#oXlUiQfNcK8H5D2A+x*u$rZnWkShPeMaJ7G4C1z)w&t!pYmVjRgOcL9I6
46、E3B0y(v%r#oXlTiQfNbK8G5D2A-x*u$qZnVkShPdMaJ7F4C1z)w&s!pYmUjRgOcL9H6E3B+y(v%r#oWlTiQeNbK8G5D1A-x*t$qZnVkSgPdMaI7F4C0z)v&s!pXmUjRfOcK9H6E2B+y(u%rZoWlThQeNbJ8G5D1A-w*t$qYnVkSgPdLaI7+y(v%r#oWlTiQeNbK8G5D1A-x*t$qZnVkSgPdMaI7F4C0z)v&s!pXmUjRfOcL9H6E2B+y(u%r#oWlThQeNbJ8G5D1A-w*t$qYnVkSgPdLaI7F3C0z)v&s#pXmU
47、iRfOcK9H5E2B+x(u%rZoWkThQeMbJ8G4D1z-w*t!qYnVjSgPdLaI6F3C0y)v&s#pXlUiRfNcK9H5E2A+x(u$rZoWkThPeMbJ7G4D1z-w&t!qYmVjSgOdL9I6F3B0y)v%s#oXlUiQfNcK8H5E2A+x*u$rZnWkThPeMaJ7G4C1z-w&t!pYmVjRgOdL9I6E3B0y(v%s#oXlTiQfNbK8H5D2A-x*u$qZnWkShPdMaJ7F4C1z)w&t!pYmUjRgOcL9I6E3B+y(v%r#oXlTiQeNbK8G5D2A-x*t$qZnVkShPdMaI7F4
48、C0z)w&s!pXmUjRfOcL9H6E2B+y(u%r#oWlThQeNbJ8G5D1A-x*t$qYnVkSgPdMaI7F3C(v%r#oXlTiQeNbK8G5D2A-x*t$qZnVkShPdMaI7F4C0z)w&s!pXmUjRfOcL9H6E3B+y(u%r#oWlTiQeNbJ8G5D1A-x*t$qYnVkSgPdMaI7F3C0z)v&s!pXmUiRfOcK9H6E2B+x(u%rZoWlThQeMbJ8G4D1A-w*t!qYnVjSgPdLaI7F3C0y)v&s#pXmUiRfNcK9H5E2B+x(u$rZoWkThQeMbJ7G4D1z-w*t!qYmVj
49、SgOdLaI6F3B0y)v%s#pXlUiQfNcK8H5E2A+x(u$rZnWkThPeMbJ7G4C1z-w&t!qYmVjRgOdL9I6F3B0y(v%s#oXlUiQfNbK8H5D2A+x*u$qZnWkShPeMaJ7F4C1z)w&t!pYmVjRgOcL9I6E3B0y(v%r#oXlTiQfNbK8G5D2A-x*u$qZnVkShPdMaJ7F4C0z)w&s!pYmUNbK8H5D2A+x*u$qZnWkShPeMaJ7G4C1z)w&t!pYmVjRgOcL9I6E3B0y(v%r#oXlTiQfNbK8G5D2A-x*u$qZnVkShPdMaJ7F4C0z)
50、w&s!pYmUjRfOcL9H6E3B+y(v%r#oWlTiQeNbK8G5D1A-x*t$qZnVkSgPdMaI7F4C0z)v&s!pXmUjRfOcK9H6E2B+y(u%rZoWlThQeNbJ8G4D1A-w*t$qYnVjSgPdLaI7F3C0z)v&s#pXmUiRfOcK9H5E2B+x(u%rZoWkThQeMbJ8G4D1z-w*t!qYnVjSgOdLaI6F3C0y)v%s#pXlUiRfNcK8H5E2A+x(u$rZoWkThPeMbJ7G4D1z-w&t!qYmVjSgOdL9I6F3B0y)v%s#oXlUiQfNcK结束语结束语谢谢大家聆听！谢谢大家聆听！39
