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运筹学2
《运筹学》
姓名:
李思
班级:
应数1001
学号:
1304100809
2012年5月9日
案例C.9市场分配
应数1001李思学号:
1304100809
(一)问题的提出
某大型公司有D1和D2两个分公司,该公司向零售商供应油和酒:
公司打算将每位零售商分配给分公司D1或分公司D2,由分公司向零售商供应货物。
这种分配应尽可能使分公司D1控制40%销售量,D2控制其余60%销售量。
零售商的编号见下面给出的M1至M23。
每位零售商有个预计的油和酒的销售量。
编号M1至M8的零售商在1区,编号M9至M18在2区,编号M19至M23的在3区。
被认为有发展前途的一些零售商分在A组,其余的分在B组,每位零售商有如表所示给出的一些供应点,希望将如下各项按40/60的
比例分给D1和D2
1.有销售量的控制2.零售商的控制3.供应点的总数
4.酒销售量的控制5.1区油销售量的控制6.2区油销售量的控制
7.3区油销售量的控制8.A组零售商总数9.B组零售商总数这里有某些灵活性,任何分配可以变化+5%(包括零售商人数)。
简单的说,每种分配可在
35/65至45/55之间变动。
而目标是保持百分偏差之和为最小。
试建立模型,并确定此问题有无可行解。
如有则求出最优解。
数据如下表所示,其中销售量单位是(106)公升表
区位
零售商
油销量
供应点
酒销量
类型
1区
M1
9
11
34
A
M2
13
47
411
A
M3
14
44
82
A
M4
17
25
157
B
M5
18
10
5
A
M6
19
26
183
A
M7
23
26
14
B
M8
21
54
215
B
2区
M9
9
18
102
B
M10
11
51
21
A
M11
17
20
54
B
M12
18
105
0
B
M13
18
7
6
B
M14
17
16
96
B
M15
22
34
118
A
M16
24
100
112
B
M17
36
50
535
B
M18
43
21
8
B
3区
M19
6
11
53
B
M20
15
19
28
A
M21
15
14
69
B
M22
25
10
65
B
M23
39
11
27
B
(二)模型分析
因为目标是保持百分偏差之和为最小,所以在设计模型的时候,考虑到尽可能使分
公司D1控制40%销售量,D2控制其余60%销售量。
可将其列成整数规划模型,在模型中,23个零售商中每一位均用一个01变量xi来代表。
若xi=1,则把零售
商i分配到分公司D1,否则就把他分配到分公司D2。
可在约束条件上引入n1,n2,n3,n4,n5,n6,n7,n8,n9负偏差,p1,p2,p3,p4,p5,p6,p7,p8,p9正偏差,以提供所需目标,其中n1,n2,n3,n4,n5,n6,n7,n8,n9相当于松弛变量,p1,p2,p3,p4,p5,p6,p7,p8,p9相当于剩余变量。
n1,n2,n3,n4,n5,n6,n7,n8,n9,p1,p2,p3,p4,p5,p6,p7,p8,p9的上界为总量与5%的乘积目标就是使期望按比值40/60使“目标”所造成的绝对偏差之和为最小。
(三)模型建立
目标规划如下
min=n1+n2+n3+n4+n5+n6+n7+n8+n9+p1+p2+p3+p4+p5+p6+p7+p8+p9
s.t.
9x1+13x2+14x3+17x4+18x5+19x6+23x7+21x8+9x9+11x10+17x11+18x12+18x13
+17x14+22x15+24x16+36x17+43x18+6x19+15x20+15x21+25x22+39x23+n1p1=
179.6
x1+x2+x3+x4+x5+x6+x7+x8+x9+x10+x11+x12+x13+x14+x15+x16+x17+x18+x1
9+x20+x21+x22+x23+n2p2=9.2
11x1+47x2+44x3+25x4+10x5+26x6+26x7+54x8+18x9+51x10+20x11+105x12+7x
13+16x14+34x15+100x16+50x17+21x18+11x19+19x20+14x21+10x22+11x23+n3
p3=292
34x1+411x2+82x3+157x4+5x5+183x6+14x7+215x8+102x9+21x10+54x11+0x12+
6x13+96x14+118x15+112x16+535x17+8x18+53x19+28x20+69x21+65x22+27x23
+n4p4=958
9x1+13x2+14x3+17x4+18x5+19x6+23x7+21x8+n5p5=53.6
9x9+11x10+17x11+18x12+18x13+17x14+22x15+24x16+36x17+43x18+n6p6=86
6x19+15x20+15x21+25x22+39x23+n7p7=40x1+x2+x3+x5+x6+x10+x15+x20+n8p8=3.2x4+x7+x8+x9+x11+x12+x13+x14+x16+x17+x18+x19+x21+x22+x23+n9p9=6xi=0,1(i=1,2,…,23)
(四)模型求解
1.在软件上输入如下:
minobjs.t.
9x1+13x2+14x3+17x4+18x5+19x6+23x7+21x8+9x9+11x10+17x11+18x12+18x13+17x14+22x1
5+24x16+36x17+43x18+6x19+15x20+15x21+25x22+39x23+n1p1=179.6
x1+x2+x3+x4+x5+x6+x7+x8+x9+x10+x11+x12+x13+x14+x15+x16+x17+x18+x19+x20+x21+x
22+x23+n2p2=9.2
11x1+47x2+44x3+25x4+10x5+26x6+26x7+54x8+18x9+51x10+20x11+105x12+7x13+16x14+34
x15+100x16+50x17+21x18+11x19+19x20+14x21+10x22+11x23+n3p3=292
34x1+411x2+82x3+157x4+5x5+183x6+14x7+215x8+102x9+21x10+54x11+0x12+6x13+96x14+
118x15+112x16+535x17+8x18+53x19+28x20+69x21+65x22+27x23+n4p4=958
9x1+13x2+14x3+17x4+18x5+19x6+23x7+21x8+n5p5=53.6
9x9+11x10+17x11+18x12+18x13+17x14+22x15+24x16+36x17+43x18+n6p6=86
6x19+15x20+15x21+25x22+39x23+n7p7=40
x1+x2+x3+x5+x6+x10+x15+x20+n8p8=3.2
x4+x7+x8+x9+x11+x12+x13+x14+x16+x17+x18+x19+x21+x22+x23+n9p9=6
objn1p1n2p2n3p3n4p4n5p5n6p6n7p7n8p8n9p9=0
endintx1intx2intx3intx4intx5intx6intx7intx8intx9
intx10intx11intx12intx13intx14intx15intx16intx17intx18intx19intx20intx21intx22intx23
subn122.45subn21.15subn336.5subn4119.75subn56.7
subn610.75
subn75
subn80.4subn90.75subp122.45subp21.15subp336.5subp4119.75
subp56.7subp610.75subp75
subp80.4
subp90.75
2.在软件lindo上运行结果人如下:
LPOPTIMUMFOUNDATSTEP27
OBJECTIVEVALUE=0.888178420E15
DELETEX22ATLEVEL15
DELETEX11ATLEVEL14
DELETEX1ATLEVEL13
FLIPX15TO<=0AT12WITHBND=10.923302
SETX11TO>=1AT13,BND=10.92TWIN=0.1000E+3130951
SETX14TO>=1AT14,BND=10.92TWIN=0.1000E+3130951
SETX10TO>=1AT15,BND=11.73TWIN=40.0330973
SETX22TO>=1AT16,BND=11.73TWIN=0.1000E+3130973
SETX16TO>=1AT17,BND=27.15TWIN=0.1000E+3130999
DELETEX16ATLEVEL17
DELETEX22ATLEVEL16
DELETEX10ATLEVEL15
DELETEX14ATLEVEL14
DELETEX11ATLEVEL13
DELETEX15ATLEVEL12
DELETEX13ATLEVEL11
DELETEX9ATLEVEL10
DELETEX3ATLEVEL9
FLIPX12TO>=1AT8WITHBND=8.3263254
SETX10TO<=0AT9,BND=8.326TWIN=0.1000E+3130999
SETX16TO<=0AT10,BND=8.326TWIN=0.1000E+3130999
SETX22TO>=1AT11,BND=8.326TWIN=0.1000E+3130999
SETX15TO>=1AT12,BND=12.38TWIN=27.6131075
SETX3TO<=0AT13,BND=12.38TWIN=0.1000E+3131075
SETX13TO>=1AT14,BND=12.38TWIN=0.1000E+3131075
SETX21TO>=1AT15,BND=13.37TWIN=16.1131128
SETX19TO<=0AT16,BND=13.37TWIN=0.1000E+3131128
SETX14TO<=0AT17,BND=16.04TWIN=0.1000E+3131149
SETX11TO>=1AT18,BND=16.04TWIN=0.1000E+3131149
SETX20TO<=0AT19,BND=16.04TWIN=0.1000E+3131149
DELETEX9ATLEVEL20
DELETEX20ATLEVEL19
DELETEX11ATLEVEL18
DELETEX14ATLEVEL17
DELETEX19ATLEVEL16
FLIPX21TO<=0AT15WITHBND=16.106396
SETX1TO<=0AT16,BND=16.11TWIN=0.1000E+3131172
SETX19TO<=0AT17,BND=16.11TWIN=0.1000E+3131172
SETX9TO<=0AT18,BND=36.30TWIN=21.1931200
DELETEX9ATLEVEL18
DELETEX19ATLEVEL17
DELETEX1ATLEVEL16
DELETEX21ATLEVEL15
DELETEX13ATLEVEL14
DELETEX3ATLEVEL13
DELETEX15ATLEVEL12
DELETEX22ATLEVEL11
DELETEX16ATLEVEL10
DELETEX10ATLEVEL9
DELETEX12ATLEVEL8
DELETEX6ATLEVEL7
DELETEX4ATLEVEL6
FLIPX23TO>=1AT5WITHBND=2.3200004
SETX6TO>=1AT6,BND=2.320TWIN=2.36931232
SETX15TO>=1AT7,BND=2.320TWIN=2.40031246
SETX16TO<=0AT8,BND=2.320TWIN=7.60031317
SETX9TO>=1AT9,BND=2.472TWIN=5.64031360
SETX14TO>=1AT10,BND=3.591TWIN=5.58831406
SETX5TO<=0AT13,BND=3.591TWIN=0.1000E+3131406
SETX7TO>=1AT13,BND=3.591TWIN=0.1000E+3131406
SETX10TO<=0AT13,BND=3.591TWIN=0.1000E+3131406
SETX3TO<=0AT14,BND=7.430TWIN=0.1000E+3131442
SETX4TO<=0AT15,BND=7.430TWIN=0.1000E+3131442
SETX12TO>=1AT16,BND=7.430TWIN=0.1000E+3131442
SETX13TO>=1AT17,BND=7.430TWIN=0.1000E+3131442
SETX19TO<=0AT18,BND=7.430TWIN=0.1000E+3131442
SETX21TO<=0AT19,BND=7.430TWIN=0.1000E+3131442
SETX22TO<=0AT20,BND=7.430TWIN=0.1000E+3131442
SETX11TO<=0AT21,BND=7.693TWIN=0.1000E+3131452
SETX1TO<=0AT22,BND=7.693TWIN=0.1000E+3131452
SETX20TO<=0AT23,BND=9.400TWIN=0.1000E+3131461
NEWINTEGERSOLUTIONOF9.39999962ATBRANCH832PIVOT
31461
BOUNDONOPTIMUM:
2.368894
DELETEX20ATLEVEL23
DELETEX1ATLEVEL22
DELETEX11ATLEVEL21
DELETEX22ATLEVEL20
DELETEX21ATLEVEL19
DELETEX19ATLEVEL18
DELETEX13ATLEVEL17
DELETEX12ATLEVEL16
DELETEX4ATLEVEL15
DELETEX3ATLEVEL14
DELETEX10ATLEVEL13
DELETEX7ATLEVEL12
DELETEX5ATLEVEL11
FLIPX14TO<=0AT10WITHBND=5.5878572
SETX10TO<=0AT12,BND=5.588TWIN=0.1000E+3131461
SETX12TO>=1AT12,BND=5.588TWIN=0.1000E+3131461
SETX3TO<=0AT13,BND=6.556TWIN=0.1000E+3131527
SETX11TO>=1AT14,BND=6.556TWIN=0.1000E+3131527
SETX13TO>=1AT15,BND=6.556TWIN=0.1000E+3131527
SETX20TO<=0AT16,BND=6.556TWIN=0.1000E+3131527
SETX4TO<=0AT17,BND=13.27TWIN=113.031556
DELETEX4ATLEVEL17
DELETEX20ATLEVEL16
DELETEX13ATLEVEL15
DELETEX11ATLEVEL14
DELETEX3ATLEVEL13
DELETEX12ATLEVEL12
DELETEX10ATLEVEL11
DELETEX14ATLEVEL10
FLIPX9TO<=0AT9WITHBND=5.6395464
SETX1TO<=0AT10,BND=5.640TWIN=0.1000E+3131556
SETX3TO<=0AT11,BND=5.640TWIN=0.1000E+3131556
SETX5TO<=0AT12,BND=5.640TWIN=0.1000E+3131556
SETX19TO<=0AT13,BND=5.640TWIN=0.1000E+3131556
SETX20TO<=0AT14,BND=5.640TWIN=0.1000E+3131556
SETX21TO<=0AT15,BND=5.640TWIN=0.1000E+3131556
SETX22TO<=0AT16,BND=5.640TWIN=0.1000E+3131556
SETX4TO<=0AT17,BND=48.26TWIN=26.4731577
DELETEX4ATLEVEL17
DELETEX22ATLEVEL16
DELETEX21ATLEVEL15
DELETEX20ATLEVEL14
DELETEX19ATLEVEL13
DELETEX5ATLEVEL12
DELETEX3ATLEVEL11
DELETEX1ATLEVEL10
DELETEX9ATLEVEL9
FLIPX16TO>=1AT8WITHBND=7.5999956
SETX3TO<=0AT9,BND=7.600TWIN=0.1000E+3131577
SETX4TO<=0AT10,BND=7.600TWIN=0.1000E+3131577
SETX13TO>=1AT11,BND=7.600TWIN=0.1000E+3131577
SETX14TO<=0AT12,BND=7.600TWIN=0.1000E+3131577
SETX20TO<=0AT13,BND=7.600TWIN=0.1000E+3131577
SETX21TO<=0AT14,BND=7.600TWIN=0.1000E+3131577
SETX9TO<=0AT15,BND=9.275TWIN=38.0431603
SETX5TO<=0AT16,BND=9.275TWIN=0.1000E+3131603
SETX10TO<=0AT17,BND=9.275TWIN=0.1000E+3131603
SETX11TO>=1AT18,BND=9.275TWIN=0.1000E+3131603
SETX22TO<=0AT19,BND=9.275TWIN=0.1000E+3131603
SETX12TO<=0AT20,BND=25.76TWIN=0.1000E+3131625
DELETEX12ATLEVEL20
DELETEX22ATLEVEL19
DELETEX11ATLEVEL18
DELETEX10ATLEVEL17
DELETEX5ATLEVEL16
DELETEX9ATLEVEL15
DELETEX21ATLEVEL14
DELETEX20ATLEVEL13
DELETEX14ATLEVEL12
DELETEX13ATLEVEL11
DELETEX4ATLEVEL10
DELETEX3ATLEVEL9
DELETEX16ATLEVEL8
FLIPX15TO<=0AT7WITHBND=2.4003751
SETX4TO<=0AT8,BND=2.413TWIN=19.4231686
SETX9TO>=1AT9,BND=2.870TWIN=6.43831743
SETX16TO>=1AT10,BND=5.813TWIN=9.99431813
SETX1TO<=0AT11,BND=5.813TWIN=0.1000E+3131813
SETX3TO<=0AT12,BND=5.813TWIN=0.1000E+3131813
SETX5TO>=1AT13,BND=5.813TWIN=0.1000E+3131813
SETX11TO>=1AT14,BND=5.813TWIN=0.1000E+3131813
SETX13TO>