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1、. .第一章线性规划及单纯形法1用Xjj=1.25分别代表5中饲料的采购数，线性规划模型：2.解：设x表示在第i个时期初开场工作的护士人数，z表示所需的总人数，那么3解：设用i=1，2，3分别表示商品A，B，C，j=1，2，3分别代表前，中，后舱，Xij表示装于j舱的i种商品的数量，Z表示总运费收入那么：5.1 Z = 4 2解：如图：由图可得：即该问题具有唯一最优解3无可行解(4)如图：由图知，该问题具有无界解。61271系数矩阵A：B，b= y1=0，16/3，-7/6，0，0，0T同理y2=0，10，0，-7，0，0T y3=0， 3，0，0，7/2，0T y4=7/4，-4，0，0，0
2、，21/4T y5=0，0，-5/2，8，0，0Ty6=0，0，3/2，0，8，0T y7=1，0，-1/2，0，0，3T y8=0，0，0，3，5，0Ty9=5/4，0，0，-2，0，15/4Ty10=0， 3，-7/6，0，0，0T y11=0，0，-5/2，8，0，0Ty12=0，0，-5/2，3，5，0T y13=4/3，0，0，0，2，3/4Ty14=0，10，0，-7，0，0T y15=0， 3，0，0，7/3，0Ty16=0，0，3/2，0，8，0T基可行解：每个x值都大于0，y3，y6，y8，y12，y13，y15，y16最优解：y3，y6， y15，y16 Zmax=3p2
3、p3 p4，p2 p3 p5，p3 p4 p5，p2 p4 p5为奇异，只有16个基。解：2该线性问题最多有个根本解。根本解Z根本可行解最优解1X1X2X3X42-411/20032/5011/503-1/30011/6401/22050-1/202600118.基的定义X1 X2 X3所对应的列向量可以构成基 B 由 X1 X2 X3 列向量构成 = N 由 非基变量对应的向量构成 = B，b= B对应的基解：-13/5，37/5，0，0，3/59解：1由图知：单纯形法：化为标准形如下：C10500bCBXBX1X2X3XR0X3341090XR52018检验数1050000X3014/51
4、-3/521/510X112/50-1/58/5检验数010-2-165X2015/14-3/143/210X110-1/73/70检验数00-5/14-25/14-35/2所以：其中：9.2A点最大 Z= 8 化为标准形：C2-100bCBXBX1X2X3X40X33510150X4620124检验数2-1000X3041-1/23X111/301/64检验数0-10-1/3-8 0点0，0，15，24 A点4，0，3，0Zmax=810.解1要使A0，0成为最优解那么需C0且d0； 2要使B8/5，0成为最优解那么 C0且d=0或C0且d0； 3要使C1，3/2成为最优解那么 -5/2-C
5、/d-3/4且Cd0；即5/2C/d3/4且Cd0；4要使D0，9/4成为最优解那么C0或C=0，d011.(1)化为标准型：C2-11000bCBXBX1X2X3X4X5X60X4311100600X51-12010100X611-100120检验数2-1100000X404-51-30302X11-12010100X602-30-1110检验数01-30-20-200X40011-1-2102X1101/201/21/215-1X201-3/20-1/21/25检验数00-3/20-3/2-1/2-252C2350000bCBXBX1X2X3X4X5X6X70X42231000120X51
6、22010080X64060010160X7043000112检验数235000000X402010-1/2040X5-1/32001-1/303/85X32/301001/603/80X7-24000-1/214检验数-4/33000-5/60-40/30X410010-1/4-1/220X52/30001-1/12-1/22/35X32/301001/608/33X2-1/21000-1/81/41检验数1/60000-11/24-3/4-49/30X40001-2/3-1/81/412X110002/3-1/8-3/415X30010-11/41/223X201003/4-3/16-1/
7、83/2检验数0000-1/4-7/16-5/8-33/23标准型：C35000bCBXBX1X2X3X4X50X31010040X402010120X53200118检验数350000X3101004X20101/206X5300-116检验数300-5/20-30X30011/3-1/32X20101/206X1100-1/31/32检验数000-3/2-1-364标准型C-11-1-11-11-11-M-M-M0bCBXBX1X2X3X4X4“X5X5“X6X6“X7X8X9X10-MX71001-1001-110009-MX831-400002-201002-MX91.200-112-
8、2001060X10043000000000112检验数5M-1M+1-1-2MM-11-MM+11+M5M-11-5M000017M-MX70-1/34/31-1001/3-1/31-1/30028/3-1X111/3-4/300002/3-2/301/3002/3-MX90-1/310/300-114/3-4/30-1/31016/30X10043000000000112检验数04/3-2/3M14/5M-7/3M-11-MM-1M+15/3M-1/3-5/3M+1/30-5/3M+1/300-41/3M-2/3-MX70-1/501-12/5-2/5-1/51/51-1/5-2/5031
9、/5-1X111/5000-2/52/56/5-6/501/52/5014/5-1X30-1/10100-3/103/102/5-2/50-1/103/1008/50X10043/100009/10-9/10-6/56/503/10-9/10136/5检验数0-1/5M+11/100M-11-M2/5M-17/10-2/5M+17/103/5-1/5M1/5M-3/50-6/5M-7/5M0-31M/5-22/55解：标准化：C62108000bCBXBX1X2X3X4X5X6X70X556-4-4100200X63-328010250X74-21300110检验数6210800000X521
10、-208114600X6-510200-2510X34-21300110检验数-34220-2200-10-1000X5110012120702X2-510201-2510X3-601702-320检验数7600-660-2234-210由表可得，因此问题的解无界。6化为:标准形：Z=-ZIC-x-1-1000bCBXBX1X2X3X4X5X6-XX1100-40-25-1X20102-313-1X30012-565检验数0004-4x-87-2x5x+8 如图：1. 1. X7/2 时，检验数0 ，最优解：5，3，5，0，0T2. 2. 1X7/2时，4-4X0由I得：C-x-1-1000b
11、CBXBX1X2X3X4X5X6-XX1101/3-10/3-5/3020/3-1X201-1/65/3-13/6013/60X6001/61/3-5/615/6检验数00X/3-7/6-10X/3+5/3-5X/3-13/6020X/3+13/63. X -2X+70C-x-1-1000bCBXBX1X2X3X4X5X6-XX11200-6011-1X201/201-3/21/23/2-1X30-110-252检验数02x-200-6X-2511X+2检验数0，列系数0，所以解无界。4.-3/2X4-4X0C-x-1-1000bCBXBX1X2X3X4X5X6-XX1100-10/3-5/3
12、020/3-1X20105/3-13/6013/6-1X60011/3-5/615/6检验数X/3-7/6-10X/3+5/3-5X/3-13/620X/3+13/6判断检验数的符号：1)1/2X 1 ，所有检验数 01/2X2-1.3X1/2时表AC-x-1-1000bCBXBX1X2X3X4X5X6-XX11200-600110X403/5-1/101-13/100013/100X60-1/51/50-2/5112/5检验数02X-1-10-6X011X对-6X讨论，令-6X=0 X=01 0X1/2时，检验数00X1/22-1.3X0 又 X5列的系数 0 ，所以解无界3 -1.5X0
13、，又 X5的列的系数0，所以解无界7解：化为标准形：C164000000-M-M-MbCBXBX1X2X3X4X5X6X7X8X9X10X11X120X4-122100000000130X54-41010000000200X612100110000017-MX101000000001001-MX110100000-100102-MX1200100000-10013检验数1+M6+M4+M000-M-M-M0000X4-1021000200-2090X54010100-40040260X61010010200-2013-MX10100000-10010016X20100000-100102-MX
14、1200100000-10013检验数1+M04+M000-M6-M0-6-M00X4-1001000220-2-230X54000100-4104-1250X61000010210-2-110-MX10100000-10010016X20100000-1001024X300100000-10013检验数1+M00000-M6410-6-M-4-M0X4000100-1221-2-240X500001004-41-44-1210X6000001121-1-2-191X1100000-10010016X20100000-1001024X300100000-10013检验数0000001640X8
15、0001/200-1/2111/2-1-120X5000210205-20-5290X6000-10120-1-20151X1100000-10010016X20101/200-1/2011/21/2-144X300100000-10013检验数000-30040-20X80001/401/4013/40-1-3/413/40X500031-100600-6240X7000-1/201/210-1/2-101/25/21X1100-1/201/200-1/2001/27/26X20101/401/4003/400-3/421/44X300100000-10013检验数000-10-2000-4
16、7即：8解：化为标准形：C1-11-31-1-3-M-M-M-MbCBXBX1X2X3X4X5X6X7X8X9X10X11-MX8003011010006-MX9012-1000010010-MX10-100001000100-MX11001001100016检验数1-MM-16M+1-M-31+M-1+3M3+M00001X30011/31/301/30002-MX9010-1-2/3-2/30-2/31006-MX10-100001000100-MX110000-1/32/31-1/30014检验数1-MM-10-M-32/3-MM-1/3M-30001X300101/31/301/300
17、02-1X2010-1-2/3-2/3001006-MX10-1000010-2/30100-MX110000-1/32/31-1/30014检验数1-M00-45/3M-2M-3001X300101/20-1/21/200-1/20-1X2010-1-101-110110-MX10-10001/20-3/21/201-1/2-61X60000-1/213/2-1/2003/26检验数1-M00-41/2M-1001X3111000100-110-1X2-200-100-2012-2-21X5-200010-3102-3-12-1X6-10000100100检验数-100-400-3121解：
18、标准形：1C2-12000-M-M-MbCBXBX1X2X3X4X5X6X7X8X9-MX7111-1001006-MX8-1010-100102-MX902-100-10010检验数2-1+3M2+M-M-M-M000-MX7103/2-101/210-1/26-MX8-1010-100102-1X201-1/200-1/2001/20检验数205/2M+3/2-M-M1/2M-1/200-MX75/200-13/21/21-3/2-1/232X3-1010-100102-1X2-1100-1/2-1/201/21/21检验数5/2M+7/200-M3/2M1/2M02X1100-2/53/
19、51/52/5-3/5-1/56/52X3001-2/5-2/51/52/52/5-1/516/5-1X2010-1/5-1/5-2/51/51/52/58/5检验数0007/5-3/5-6/5由表可知此题解无界。2得一辅助问题：C000000-1-1-1bCBXBX1X2X3X4X5X6X7X8X9-1X7111-1001006-1X8-1010-100102-1X902-100-10010检验数031-1-1-1000-1X7103/2-101/210-1/26-1X8-1010-1001020X201-1/200-1/2001/20检验数005/2-1-11/200-3/2-1X75/2
20、00-13/21/21-3/2-1/230X3-1010-1001020X2-1100-1/2-1/201/21/21检验数5/200-13/200X1100-2/53/51/52/5-3/5-1/56/50X3001-2/5-2/51/52/52/5-1/516/50X2010-1/5-1/5-2/51/51/52/58/5检验数000000-1-1-10C2-12000bCBXBX1X2X3X4X5X62X1100-2/53/51/5-1X3001-2/5-2/51/52X2010-1/5-1/5-2/5检验数0-334/5由表知此题属于解无界2大M法，先化为标准形：Z=-ZC-4-100
21、-M-MbCBXBX1X2X3X4X5X60X41201004-MX51100103-MX643-10016检验数5M-44M-1-M0009M0X405/41/4105/2-MX501/41/4013/2-4X113/4-1/4003/2检验数0M/4+2M/4-1003M/2+6-1X2011/54/502-MX5001/5-1/511-4X110-2/5-3/500检验数00M/5-7/5-M/5-8/50M+2-1X2010110X3001-15-4X1100-12000-39原问题唯一最优解二阶段法：引入人工变量 X5 X6得原问题的一个辅助问题：C0000-1-1bCBXBX1X2
22、X3X4X5X60X41201004-MX51100103-MX643-10016检验数54-100090X405/41/4105/2-MX501/41/4013/2-4X113/4-1/4003/2检验数01/41/4003/2-1X2011/54/502-MX5001/5-1/511-4X110-2/5-3/500检验数001/5-1/501-1X2010110X3001-15-4X1100-12检验数000-10C-4-100bCBXBX1X2X3X4-1X2001-150X301011-4X1100-12检验数000-393标准形：C-23-1000-M-MbCBXBX1X2X3X4X
23、5X6X7X8-MX7142-100108-MX83200-10016检验数-2-4M6M+32M-1-M-M0003X21/411/2-1/4001/402-MX85/20-11/2-10-1/212检验数5/2M-11/40-M-5/21/2M+3/4-M003X2013/5-3/101/1003/10-1/109/5-2X110-2/51/5-2/50-1/52/54/5检验数00-18/513/1003X25/2100-1/2001/230X450-21-20-124检验数0-103/20-M-M由表知此题解无界；6两阶段法：C000000-1-1bCBXBX1X2X3X4X5X6X7X8-1X7142-100108-1X83200-10016检验数462-1-10000X21/411/2-1/4001/402-1X85/20-11/2-10-1/212检验数5/2M-11/40-M-5/21/2M+3/4-M000X2013/5-3/101/1003/10-1/109/50X110-2/51/5-2/50-1/52/54/5检验数000000-1