农作物种植问题
问题摘
题合理安排农作物种植问题解决问题关键找种合理种植方案土面积灌溉水劳动时间定前提农场总收益数学规划问题
问题①设种植玉米x1英亩种植麦x2英亩种植燕麦x3英亩总收益z
问题②625英亩土5块120英亩1块25英亩设土分i123456玉米麦燕麦分j123 引入01变量xij选择第 i块土种植农作物 j记xij1否记xij0
二问题重述
定面积土现土种植农作物农作物分玉米麦燕麦等种农作物需灌溉水量劳动力收益定块土分配种植植物总收益
三模型假设
⑴种植作物需灌溉水超1000英亩周需劳动力超300时土面积超625英亩种作物水劳动力收益常数表格示
⑵种作物水量劳动力收益相互独立
⑶前年土种植作物年种植弄作物产量没影响
四符号定义
①问题设种植玉米x1英亩麦x2英亩燕麦x3英亩收益z元
②问题625英亩土5块120英亩1块25英亩设6块土分i123456玉米麦燕麦分记作j123引入01变量xij选择第 i块土种植农作物 j记xij1否记xij0收益Z∑∑CijXij
五模型建立
①问题设种植玉米x1英亩麦x2英亩燕麦x3英亩收益z美元
结合表格数作物总收益z MAX z400x1+300x2+250x3 ⑴
灌溉水超1000英亩 30x1+10x2+25x3<1000 ⑵
周劳动力超300时 08x1+02x2+03x3<300 ⑶
土总面积超625英亩 x1+x2+x3<625 ⑷
种植3种作物面积非负数 x1>0 x2>0 x3>0 ⑸
②问题625英亩土5块120英亩1块25英亩设6块土分i123456玉米麦燕麦分记作j123 ⑹
引入01变量xij选择第 i块土种植农作物 j记xij1否记xij0 收益Z∑∑CijXij ⑺
根题中约束条件块土种植种农作物：玉米麦燕麦
Xij应满足两约束条件：
①第i块土种植作物jCijXij表示土收益
Z∑∑CijXij j123 i123456
②块土必须种3种作物种种
∑xij1 j123
目标函数
Max Z400(120x11+120x21+120x31+120x41+120x51+25x61)
+300(120x12+120x22+120x32+120x42+120x52+25x62)
+250(120x13+120x23+120x33+120x43+120x53+25x63)
st ∑xij1 j123
xij{01}
六模型求解
问题①LINDO软件中输入文件：
Max 400x1+300x2+250x3
ST
30x1+10x2+25x3<1000
08x1+02x2+03x3<300
x1+x2+x3<625
x1>0
x2>0
x3>0
END
求解答案示：
LP OPTIMUM FOUND AT STEP 2
OBJECTIVE FUNCTION VALUE
1) 2062500
VARIABLE VALUE REDUCED COST
X1 187500000 0000000
X2 437500000 0000000
X3 0000000 125000000
ROW SLACK OR SURPLUS DUAL PRICES
2) 0000000 50000000
3) 62500000 0000000
4) 0000000 250000000
5) 187500000 0000000
6) 437500000 0000000
7) 0000000 0000000
NO ITERATIONS 2
RANGES IN WHICH THE BASIS IS UNCHANGED
OBJ COEFFICIENT RANGES
VARIABLE CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
X1 400000000 500000000 100000000
X2 300000000 100000000 166666672
X3 250000000 125000000 INFINITY
RIGHTHAND SIDE RANGES
ROW CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE DECREASE
2 1000000000 208333328 375000000
3 300000000 INFINITY 62500000
4 625000000 375000000 291666656
5 0000000 187500000 INFINITY
6 0000000 437500000 INFINITY
7 0000000 0000000 INFINITY
优解x11875 x24375 x30获总收益种植玉米1875英亩种植麦4375英亩
②问题LINDO软件中输入文件：
Max 48000x11+48000x21+48000x31+48000x41+48000x51+10000x61
+36000x12+36000x22+36000x32+36000x42+36000x52+7500x62
+30000x13+30000x23+30000x33+30000x43+30000x53+6250x63
ST x11+x12+x131
x21+x22+x231
x31+x32+x331
x41+x42+x431
x51+x52+x531
x61+x62+x631
360x11+360x21+360x31+360x41+360x51+75x61
+120x12+120x22+120x32+120x42+120x52+25x62
+300x13+300x23+300x33+300x43+300x53+625x63<1000
96x11+96x21+96x31+96x41+96x51+20x61
+ 24x12+24x22+24x32+24x42+24x52+5x62
+36x13+36x23+36x33+36x43+36x53+75x63<300
END
INT 18
LINDO软件中示：
OBJECTIVE FUNCTION VALUE
1) 2020000
VARIABLE VALUE REDUCED COST
X11 1000000 48000000000
X21 0000000 48000000000
X31 0000000 48000000000
X41 0000000 48000000000
X51 0000000 48000000000
X61 1000000 10000000000
X12 0000000 36000000000
X22 1000000 36000000000
X32 1000000 36000000000
X42 1000000 36000000000
X52 1000000 36000000000
X62 0000000 7500000000
X13 0000000 30000000000
X23 0000000 30000000000
X33 0000000 30000000000
X43 0000000 30000000000
X53 0000000 30000000000
X63 0000000 6250000000
ROW SLACK OR SURPLUS DUAL PRICES
2) 0000000 0000000
3) 0000000 0000000
4) 0000000 0000000
5) 0000000 0000000
6) 0000000 0000000
7) 0000000 0000000
8) 85000000 0000000
9) 88000000 0000000
NO ITERATIONS 214
BRANCHES 12 DETERM 1000E 0
优解x111x611x221x321x421 x521种植玉米145英亩土种植麦480英亩土
③问题农作物灌溉水980英亩问题②中灌溉水1000英亩换980亩结果：
OBJECTIVE FUNCTION VALUE
1) 2020000
VARIABLE VALUE REDUCED COST
X11 0000000 48000000000
X21 0000000 48000000000
X31 0000000 48000000000
X41 0000000 48000000000
X51 1000000 48000000000
X61 1000000 10000000000
X12 1000000 36000000000
X22 1000000 36000000000
X32 1000000 36000000000
X42 1000000 36000000000
X52 0000000 36000000000
X62 0000000 7500000000
X13 0000000 30000000000
X23 0000000 30000000000
X33 0000000 30000000000
X43 0000000 30000000000
X53 0000000 30000000000
X63 0000000 6250000000
ROW SLACK OR SURPLUS DUAL PRICES
2) 0000000 0000000
3) 0000000 0000000
4) 0000000 0000000
5) 0000000 0000000
6) 0000000 0000000
7) 0000000 0000000
8) 65000000 0000000
9) 88000000 0000000
NO ITERATIONS 214
BRANCHES 10 DETERM 1000E 0
优解x511x611x121x221x321x421 种植玉米145英亩土种植麦480英亩土方案问题②改变方案
理灌溉水950英亩需1000英亩换950英亩结果：
OBJECTIVE FUNCTION VALUE
1) 2020000
VARIABLE VALUE REDUCED COST
X11 0000000 48000000000
X21 0000000 48000000000
X31 0000000 48000000000
X41 0000000 48000000000
X51 1000000 48000000000
X61 1000000 10000000000
X12 1000000 36000000000
X22 1000000 36000000000
X32 1000000 36000000000
X42 1000000 36000000000
X52 0000000 36000000000
X62 0000000 7500000000
X13 0000000 30000000000
X23 0000000 30000000000
X33 0000000 30000000000
X43 0000000 30000000000
X53 0000000 30000000000
X63 0000000 6250000000
ROW SLACK OR SURPLUS DUAL PRICES
2) 0000000 0000000
3) 0000000 0000000
4) 0000000 0000000
5) 0000000 0000000
6) 0000000 0000000
7) 0000000 0000000
8) 35000000 0000000
9) 88000000 0000000
NO ITERATIONS 158
BRANCHES 6 DETERM 1000E 0
优解x511x611x121x221x321x421 种植玉米145英亩土种植麦480英亩土方案问题②改变方案
七结果分析
针问题③知灌溉水改980英亩950英亩方案需改变
八模型优缺点
优点：01变量应方便问题求解
缺点：数复杂繁冗处理起较麻烦
九参考文献
[1]姜启源数学模型（第三版）北京高等教育出版社2003
应物理
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问题摘
题合理安排农作物种植问题解决问题关键找种合理种植方案土面积灌溉水劳动时间定前提农场总收益数学规划问题
问题①设种植玉米x1英亩种植麦x2英亩种植燕麦x3英亩总收益z
问题②625英亩土5块120英亩1块25英亩设土分i123456玉米麦燕麦分j123 引入01变量xij选择第 i块土种植农作物 j记xij1否记xij0
二问题重述
定面积土现土种植农作物农作物分玉米麦燕麦等种农作物需灌溉水量劳动力收益定块土分配种植植物总收益
三模型假设
⑴种植作物需灌溉水超1000英亩周需劳动力超300时土面积超625英亩种作物水劳动力收益常数表格示
⑵种作物水量劳动力收益相互独立
⑶前年土种植作物年种植弄作物产量没影响
四符号定义
①问题设种植玉米x1英亩麦x2英亩燕麦x3英亩收益z元
②问题625英亩土5块120英亩1块25英亩设6块土分i123456玉米麦燕麦分记作j123引入01变量xij选择第 i块土种植农作物 j记xij1否记xij0收益Z∑∑CijXij
五模型建立
①问题设种植玉米x1英亩麦x2英亩燕麦x3英亩收益z美元
结合表格数作物总收益z MAX z400x1+300x2+250x3 ⑴
灌溉水超1000英亩 30x1+10x2+25x3<1000 ⑵
周劳动力超300时 08x1+02x2+03x3<300 ⑶
土总面积超625英亩 x1+x2+x3<625 ⑷
种植3种作物面积非负数 x1>0 x2>0 x3>0 ⑸
②问题625英亩土5块120英亩1块25英亩设6块土分i123456玉米麦燕麦分记作j123 ⑹
引入01变量xij选择第 i块土种植农作物 j记xij1否记xij0 收益Z∑∑CijXij ⑺
根题中约束条件块土种植种农作物：玉米麦燕麦
Xij应满足两约束条件：
①第i块土种植作物jCijXij表示土收益
Z∑∑CijXij j123 i123456
②块土必须种3种作物种种
∑xij1 j123
目标函数
Max Z400(120x11+120x21+120x31+120x41+120x51+25x61)
+300(120x12+120x22+120x32+120x42+120x52+25x62)
+250(120x13+120x23+120x33+120x43+120x53+25x63)
st ∑xij1 j123
xij{01}
六模型求解
问题①LINDO软件中输入文件：
Max 400x1+300x2+250x3
ST
30x1+10x2+25x3<1000
08x1+02x2+03x3<300
x1+x2+x3<625
x1>0
x2>0
x3>0
END
求解答案示：
LP OPTIMUM FOUND AT STEP 2
OBJECTIVE FUNCTION VALUE
1) 2062500
VARIABLE VALUE REDUCED COST
X1 187500000 0000000
X2 437500000 0000000
X3 0000000 125000000
ROW SLACK OR SURPLUS DUAL PRICES
2) 0000000 50000000
3) 62500000 0000000
4) 0000000 250000000
5) 187500000 0000000
6) 437500000 0000000
7) 0000000 0000000
NO ITERATIONS 2
RANGES IN WHICH THE BASIS IS UNCHANGED
OBJ COEFFICIENT RANGES
VARIABLE CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
X1 400000000 500000000 100000000
X2 300000000 100000000 166666672
X3 250000000 125000000 INFINITY
RIGHTHAND SIDE RANGES
ROW CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE DECREASE
2 1000000000 208333328 375000000
3 300000000 INFINITY 62500000
4 625000000 375000000 291666656
5 0000000 187500000 INFINITY
6 0000000 437500000 INFINITY
7 0000000 0000000 INFINITY
优解x11875 x24375 x30获总收益种植玉米1875英亩种植麦4375英亩
②问题LINDO软件中输入文件：
Max 48000x11+48000x21+48000x31+48000x41+48000x51+10000x61
+36000x12+36000x22+36000x32+36000x42+36000x52+7500x62
+30000x13+30000x23+30000x33+30000x43+30000x53+6250x63
ST x11+x12+x131
x21+x22+x231
x31+x32+x331
x41+x42+x431
x51+x52+x531
x61+x62+x631
360x11+360x21+360x31+360x41+360x51+75x61
+120x12+120x22+120x32+120x42+120x52+25x62
+300x13+300x23+300x33+300x43+300x53+625x63<1000
96x11+96x21+96x31+96x41+96x51+20x61
+ 24x12+24x22+24x32+24x42+24x52+5x62
+36x13+36x23+36x33+36x43+36x53+75x63<300
END
INT 18
LINDO软件中示：
OBJECTIVE FUNCTION VALUE
1) 2020000
VARIABLE VALUE REDUCED COST
X11 1000000 48000000000
X21 0000000 48000000000
X31 0000000 48000000000
X41 0000000 48000000000
X51 0000000 48000000000
X61 1000000 10000000000
X12 0000000 36000000000
X22 1000000 36000000000
X32 1000000 36000000000
X42 1000000 36000000000
X52 1000000 36000000000
X62 0000000 7500000000
X13 0000000 30000000000
X23 0000000 30000000000
X33 0000000 30000000000
X43 0000000 30000000000
X53 0000000 30000000000
X63 0000000 6250000000
ROW SLACK OR SURPLUS DUAL PRICES
2) 0000000 0000000
3) 0000000 0000000
4) 0000000 0000000
5) 0000000 0000000
6) 0000000 0000000
7) 0000000 0000000
8) 85000000 0000000
9) 88000000 0000000
NO ITERATIONS 214
BRANCHES 12 DETERM 1000E 0
优解x111x611x221x321x421 x521种植玉米145英亩土种植麦480英亩土
③问题农作物灌溉水980英亩问题②中灌溉水1000英亩换980亩结果：
OBJECTIVE FUNCTION VALUE
1) 2020000
VARIABLE VALUE REDUCED COST
X11 0000000 48000000000
X21 0000000 48000000000
X31 0000000 48000000000
X41 0000000 48000000000
X51 1000000 48000000000
X61 1000000 10000000000
X12 1000000 36000000000
X22 1000000 36000000000
X32 1000000 36000000000
X42 1000000 36000000000
X52 0000000 36000000000
X62 0000000 7500000000
X13 0000000 30000000000
X23 0000000 30000000000
X33 0000000 30000000000
X43 0000000 30000000000
X53 0000000 30000000000
X63 0000000 6250000000
ROW SLACK OR SURPLUS DUAL PRICES
2) 0000000 0000000
3) 0000000 0000000
4) 0000000 0000000
5) 0000000 0000000
6) 0000000 0000000
7) 0000000 0000000
8) 65000000 0000000
9) 88000000 0000000
NO ITERATIONS 214
BRANCHES 10 DETERM 1000E 0
优解x511x611x121x221x321x421 种植玉米145英亩土种植麦480英亩土方案问题②改变方案
理灌溉水950英亩需1000英亩换950英亩结果：
OBJECTIVE FUNCTION VALUE
1) 2020000
VARIABLE VALUE REDUCED COST
X11 0000000 48000000000
X21 0000000 48000000000
X31 0000000 48000000000
X41 0000000 48000000000
X51 1000000 48000000000
X61 1000000 10000000000
X12 1000000 36000000000
X22 1000000 36000000000
X32 1000000 36000000000
X42 1000000 36000000000
X52 0000000 36000000000
X62 0000000 7500000000
X13 0000000 30000000000
X23 0000000 30000000000
X33 0000000 30000000000
X43 0000000 30000000000
X53 0000000 30000000000
X63 0000000 6250000000
ROW SLACK OR SURPLUS DUAL PRICES
2) 0000000 0000000
3) 0000000 0000000
4) 0000000 0000000
5) 0000000 0000000
6) 0000000 0000000
7) 0000000 0000000
8) 35000000 0000000
9) 88000000 0000000
NO ITERATIONS 158
BRANCHES 6 DETERM 1000E 0
优解x511x611x121x221x321x421 种植玉米145英亩土种植麦480英亩土方案问题②改变方案
七结果分析
针问题③知灌溉水改980英亩950英亩方案需改变
八模型优缺点
优点：01变量应方便问题求解
缺点：数复杂繁冗处理起较麻烦
九参考文献
[1]姜启源数学模型（第三版）北京高等教育出版社2003
应物理
文档香网(httpswwwxiangdangnet)户传
《香当网》用户分享的内容，不代表《香当网》观点或立场，请自行判断内容的真实性和可靠性！
该内容是文档的文本内容，更好的格式请下载文档
