某发动机厂排气管车间生产计划的优化分析

【解】

（1）线性规划数学模型

车间基本生产工人工时计算如下：

S=27X（22X8-2-4）=4590人时/月

按产量1%的工废实际可用时间：4590/1.1=4544

每种设备月实际可利用工时如表C-2最后一列。

从表C-3中减去料废成本后每种排气管利润如下表所示。

产品

12345678

项目

毛坯价格981049411210697104102

料废成本1.962.081.882.242.121.942.082.04

辅料消耗22222222

动能消耗1010101010101010

工具等消耗10131214158911

管理费用1.4551.0991.211.441.1881.22651.3081.56

税收151614.81716.514.515.615.5

售价150160.1149172166145.6157.8155.8

利润（元）11.58511.92113.1113.3213.19210.933513.81211.7

根据给定的条件及要求，设

M，电…趣分别为第一种、第二种……第八种排气管的月产量，月产量单位为1000根/月。

将目标函数取为每月的总产品利润，得出关于月计划的线性规划模型为

maxZ=11.585石+11.92lx2+13.1+13.32兀+13.192x5+10.9335%+13.812^4-11.7^

41.9%+43.912+42.3XJ+47.8X4+46.5X5+39.7X6+49.2X7+45.7/工4544工时约束

4.0玉+4.5^2+4.8七+5.8为+5.25+4.0.%+4.6巧+5.6鼻-1378设备工时约束

3.9芭+4.5々+4.3刍+5.0X4+4.9/+4.44+5.1七+4.8/<1386

5.9%+5.8%+5.7七+6,3元&+6.5也+6.O4+6.6/+6.44<2061

3.5$+3.0X2+3.7/+4.0.V4+3.8巷+3.0x6+4.lx7+3.4虫<693

5.8%+6.2X2+5.7X3+6.4x4+6.3/+6.0x6+6.5x7+6.2/<2067

5.5内+5.79+4.7X3+6,0X4+5.9X5+5.24+6.2x7+5.6/<1382

5.9x,+3.8X2+4.OX3+4.Lq+3.7/+35%+4.Lv7+3.64<1362

4.1X1+4.0X2+4.0七+4.3.+4.2/+3.8%+4.3x7+4.34<1370

2.5再+2.9X2+2.7X3+3.0X4+3.0X5+2.54+3.1a+2.8/<693

2.8Xl+2.9X2+2.1刍+3.25+3.0/+2.5.%+3.2x7+3.04-693

x,>10

x2-x4=3

X3>5

/W60

Xy>10

可，工2，刍，“4，％5，%6，^，/NO

(2)用WinQSB软件求解:

Variable~>xiX2X3X4X5X6X7X8DirectionR.H.S.

Maximize11.58511.92113.1113.3213.19210.933513.812117

C141.943.942.347.846.539.749.2457<=4544

C244.5485.85.244.65.6<-1378

C33.94.54.354.94.45.14.B<=1386

C45.95.85.76.36.566.66.4<=2061

C53.533.743.834.13.4<■693

C65.86.25.76.46.366.56.2<=2067

C75.55.74.7G5.95.26.25.S<-1382

C83.93.84413.73.54.13.B1362

C94.1444.34.23.84.34.3<=1370

C102.52.92.7332.53.12.B<=693

C112.82.92.13.232.53.23<=693

C121000000D>=10

C13010•1000Ds3

C140010000D>=5

C150000010D<=60

C160000001D10

LowerBound0000000D

UpperBoundMMMMMMMM

VaciableTypeContinuoutContinuousContinuousContinuousContinuousContinuousContinuousContinuous

08:21:59TuesdayDecember142010

DecisionSolutionUnitCostorTotalReducedBasisAllowableAllowable

VariableValueProfitc(j)ContributionCostStatusMin.c(j)Max.c(i)

1X110.000011.5850115.85000basic-M12.9860

2X23.000011.921035.76300basic-M15.1005

3X382.773013.11001,085.15500basic12.0005M

4X4013.32000-3.1795atbound-M16.4995

5X5013.19200-1.2197atboundM14.4117

6X6010.9335o-1.3707atbound-M12.3042

7X710.000013.8120138.12000basic-M15.2485

8X8011.70000-2.4638atbound-M14.1638

ObjectiveFunction(Max.)=1,374.8880

LeftHandRightHandSlackShadowAllowableAllowable

ConstraintSideDirectionSideorSurplusPriceMin.RHSMax.RHS

1C14,544.0000<=4,544.000000.30991,254.20007,993.6190

2C2496.8106<=1,378.0000881.18930496.8107M

3C3459.4241<=1,386.0000926.57590459.4241M

4C4614.2064<=2J61.00001,446.79400614.2064M

5C5391.2603<=693.0000301.73970391.2603M

6C6613.4064<=2,067.00001,453.59400613.4064M

7C7523.1333<=1,382.0000858.86670523.1333M

8C8422.4922<=L362.0000939.50780422.4922M

9C9427.0922<=1J70.0000942.90780427.0922M

10C10288.1872<=693.0000404.81280288.1872M

11C11242.5234<=693.0000450.47660242.5234M

12Cl210.0000>=10.00000-1.4010088.5155

13Cl33.0000=3.00000-1.6849077.9385

14C1482.7730>=5.000077.77300-M82.7730

15C150<=60.000060.000000M

1601610.0000>=10.00000-1.4365076.8659

X=（10000,3000,82773,0.0,0,10,0）；Z=1374888（元）

（3）月生产应如下按排：第一〜第八种排气管分别按排10000,3000,82773,0,0,0,10000,0根。

月最大可获利润1374888元。

4.结果分析

（1）资源利用率如下表。

资源月可利用工时实际利用工时剩余资源实际利用率

I.平面铳床1378496.8106881.1936.05%

2.卧铳床1386459.4241926.5833.15%

3.组合钻2061614.20641446.7929.80%

4.单面铳床693391.2603301.7456.46%

5.攻丝床2067613.40641453.5929.68%

6.精铳床1382523.1333858.8737.85%

7.扩孔钻床1362422.4922939.5131.02%

8.摇臂钻床1370427.0922942.9131.17%

9.去毛刺机693288.1872404.8141.59%

1（）.清洗机693242.5234450.4835.00%

人工454445440.00100.00%

结果表明，各种设备负荷都远没有达到饱和，而人工已经满负荷运转，所以应提高人工的有

效工时。

(2)人工工时的影子价格等于0.3099,所有设备的影子价格全等于零，因此人工工时是制

约车间产量或利润增长的关键资源。

(3)将模型改为

maxZ=11.585x,+11.92l.r2+13.1Lv3+13.32^+13.192演+10.9335%+13.81+11.7公

4l.9.r,+43.9x2+42.3刍+478%+46.5/+39.74+49.2a+45.74<4544工时约束

4.0%+4.5/+4.8&+5.8%+5.2升+4.0%+4.6工7+5.6/<1378设备工时约束

3.9x(+4.5X2+4.3x,+5.0A4+4.9x$+4.4.%+5.1x7+4.8x8<1386

5.9%+5.8£+5.7,q+6.3A4+6.5.x5+6.0x6+6.6x7+6.4/<2061

3.5x)+3.0X2+3.7X3+4.0%+3.8.r5+3.0.r6+4.1x7+3.44《693

5.8x)+6.2X2+5.7.与+6.4%+6.3x5+6.0x6+6.5x7+6.2/<2067

5.5%+5.7X2+4.7,q+6.0.x4+5.9毛+5.2%+6.2x7+5.6/<1382

5.9%+3.8.G+4.0七+4.1Xj+3.7/+35%+4.1为+3.6/<1362

4.Lv1+4.0X2+4.0X3+4.3%+4.2/+3.8.%+4.3x7+4.3/<1370

2.5百+2,9x2+2.7/+3.0A4+3.0X5+25%+3.lx,+2.84<693

X

2.8%+2.9X2+2.1为+3.2.q+3.05+2.54+3.2x7+3.0/<693

%1>15,x2>5,x3>5.x4>3

x5>15,x(i>60,x-j>10,x8>60

用WinQSB软件求解，结果显示无可行解。见下表。

RightHandShadcwAddMoreThanAddUpTo

ConstraintDirectionSidePriceThisToRHSThisToRHS

1Cl<=4,544.000.2561.639.504,636.88

2C2<=1,378.000-896.4016M

3C3<=1,386.000-903.7013M

4C4<=2,061.000-1,400.60M

5C5<=6930-344.9759M

6C6<=2,067.000-1.410.93M

7C7<=1,382.000-800.4016M

8C8<=1,362.000-967.651M

9C9<=1,370.000-936.7637M

10CIO<-6930409.9508M

11Cll<=6930-403.6258M

12C12>=150.8579-M-3

13CI3>=50.6818-37.3462-37.3462

14C14>=52.2805-38.7589-38.7589

15C15>=30-M-3

16CI6>=150-M-15

17C17>=600.7696-M-35.8753

18C18>=100-M-10

191cl9|>=[60|0|-M-35.8753

令所有产量为等式约束，给定一个较大的人工工时数，如9000小时，求解后显示工时

的参数变化区间为(7516.4,M),见卜表所示。说明人工工时最少要加班7517—4544=2973

小时。设备时间有剩余。

DecisionSolutionUnitCostorTotalReducedBasisAllowableAllowable

VariableValueProfitc(j)ContributionCostStatusMin.c(j)Max.c(i)

T"XI15.000011.5850173.77500basic-MM

2X25.000011.921059.60500basicMM

3X35.000013.110065.55000basic-MM

4X43.000013.320039.96000basicMM

5X515.000013.1920197.88000basic-MM

6X660.000010.9335656.01000basic-MM

7X710.000013.8120138.12000basic-MM

8X860.000011.7000702.00000basic-MM

ObjectiveFunction(Max.)=2,032.9000

LeftHandRightHandSlackShadowAllowableAllowable

ConstraintSideDirectionSideorSurplusPriceMin.RHSMaxRHS

[C17,516.4000<=9,000.0000L483.600007,516.4000M

2C2823.9000<=1,378.000055410000823.9000M

3C3794.0000<=1,3