运筹学案例分析机械产品生产计划问题

1/28运筹学案例分析报告机械产品生产计划问题班级1516122组号6姓名、学号（组长、分工）吴锴楠151612219、建立数学模型（组员、分工）张灿龙151612220、编写LINGO程序（组员、分工）游泽锋151612222、编写报告2/28一、案例描述机械加工厂生产7种产品产品1到产品7。该厂有以下设备四台磨床、两台立式钻床、三台水平钻床、一台镗床和一台刨床。每种产品的利润单位元/件,在这里,利润定义为销售价格与原料成本之差以及生产单位产品需要的各种设备的工时小时/件如表1所示,其中短划线表示这种产品不需要相应的设备加工。表1产品的利润和需要的设备工时产品1234567单位产品利润1000600300400100900300磨床050070030020050立钻01020003006水平钻020600080060镗床005003007010008刨床001005005从一月份至六月份,每个月中需要检修设备见表2所示在检修月份,被检修设备全月不能用于生产。每个月各种产品的市场销售量上限如表3所示。每种产品的最大库存量为100件,库存费用为每件每月05元,在一月初,所有产品都没有库存；而要求在六月底,每种产品都有50件库存。工厂每天开两班,每班8小时,为简单起见,假定每月都工作24天。表2设备检修计划月份计划检修设备及台数月份计划检修设备及台数一月一台磨床四月一台立式钻床二月二台立式钻床五月一台磨床和一台立式钻床三月一台镗床六月一台刨床和一台水平钻床表3产品的市场销售量上限件/月产品1234567一月5001000300300800200100二月6005002000400300150三月30060000500400100四月2003004005002000100五月010050010010003000六月500500100300110050060生产过程中,各种工序没有先后次序的要求。3/281制定六个月的生产、库存、销售计划,使六个月的总利润最大。2在不改变以上计划的前提下,哪几个月中哪些产品的售价可以提高以达到增加利润的目的。价格提高的幅度是多大3哪些设备的能力应该增加请列出购置新设备的优先顺序。4是否可以通过调整现有的设备检修计划来提高利润提出一个新的设备检修计划,使原来计划检修的设备在这半年中都得到检修而使利润尽可能的增加。5最优设备检修计划问题构造一个最优设备检修计划模型,使在这半年中各设备的检修台数满足案例中的要求而使利润为最大。二、问题分析及求解1、问题（1）的求解11、问题分析制定六个月的生产、库存、销售计划,使六个月的总利润最大。因为产品和设配都是完整的，所以该问题是一个整数规划问题，可以通过建立整数规划模型利用LINGO求解解决该问题。在该题中，有已知条件可得总利润总销售所获利润总库存所需金额。以此建立目标函数，通过分析，我们把决策变量定位每月各种产品的生产量、销售量以及库存量。考虑各产品的利润、设备使用时间、设备检修要求、库存量要求以及各产品每月销售上限的约束，建立整数规划模型，求解得出总利润最大的生产、库存、销售计划。12、模型构建121、决策变量设置用表示第个月第种产品的产量，；XIJJ7,216,21JI用表示第个月底第种产品的库存量,；Y用表示第个月第种产品的销售量,；S其中，当时，；1JYXS1当时，；6,32I所以，第种产品的总销售量为J；116IIJS用表示第种产品的单位利润，各产品的单位利润如下表所示A第种产品J1234567单位产品利润1000600300400100900300122、目标函数的确定；716617150JIIJYJYXAZMAX123、约束条件的确定4/28各个月设备使用时间的约束一月份设备使用时间2481X70516X5014X3012X1087353862237二月份设备使用时间2481X27056X2504X2301X2108173538637三月份设备使用时间2481X37056X3504X3012X31081758627四月份设备使用时间2481X47056X450X43012X41081735386227五月份设备使用时间2481X5706X504X53012X51081733862237六月份设备使用时间2480X6750X650X63120X61181473582237各个月各产品的销量限制5/28一月销量限制10Y7X26853410YX二月销量限制150Y27X3645435三月销量限制10Y372X4655431四月销量限制10Y473X62541五月销量限制0Y574X361521六月销量限制6/2860Y75X6134350Y62X1在六月底,每种产品都有50件库存,12J每种产品的最大库存量为100件76I0Y每个变量必须为整数；7/2813、模型求解结果通过LINGO软件的运行求解，最大利润为41464元，得到最高利润生产、库存、销售计划如下,（LINGO程序代码及结果见附录）表一1月2月3月4月5月6月生产量XI600002000550库存量YI100000050产品1销售量SI50010002000500生产量XI12200107102109库存量YI0000250产品2销售量SI1220010710061生产量XI30020004006000库存量YI000010050产品3销售量SI300200040050050生产量XI30000500100350库存量YI0000050产品4销售量SI30000500100300生产量XI800500020011000库存量YI01000010050产品5销售量SI800400100200100050生产量XI30004500250550库存量YI10005050050产品6销售量SI2001004000300500生产量XI10025001001000库存量YI01000010050产品7销售量SI1001501001000502、问题（2）的求解在不改变以上计划的前提下,哪几个月中哪些产品的售价可以提高以达到增加利润的目的。价格提高的幅度是多大该问题是灵敏度分析的问题，通过灵敏度分析，得到各个月份各种产品销售价格可以提高的上限。在该问题中，决策变量及目标函数与第（1）题的相同，但由于LINGO无法解决整数规划灵敏度分析的问题，于是约束条件比问题（1）的约束条件少了各个决策变量的整数约束，形成新的模型，用LINGO编码时只要把问题（1）的源程序中整数约束的条件删掉即可，新模型的结果只有X261096667与原模型X2610900相差不大，可以近似认为求解结果相同。灵敏度分析见附录。在不改变问题（1）计划的前提下各个月各产品售价可提高以达到增加利润的目的的价格最大提高幅度如下表所示。8/28表二1月2月3月4月5月6月产品105610539产品216519404158315产品31305050211产品40629513091产品505190510产品651047513065产品711130283863、问题（3）的求解哪些设备的能力应该增加请列出购置新设备的优先顺序。该题可通过看影子价格的出结果，影子价格代表资源在最优解条件下每增加一单位效益的增加量，该题可以把设备视作资源，利润为效益。由于在（1）题模型中，各设备影子价格为0，故可在（2）题的模型寻找影子价格，各个设备各月份对应的影子价格如下表表三一月二月三月四月五月六月总和磨床0000000立钻01000316666701046667水平钻100003611112361111镗床00200000200刨床00000220220通过分析可得，增加一单位磨床可增加0利润，增加一单位立钻可增加1046667利润，增加一单位水平钻可增加2361111利润，增加一单位镗床可增加200利润，增加一单位刨床可增加220利润，所以，立钻，水平钻，镗床，刨床需要购置，购置的顺序依次为为刨床、镗床、立钻、水平钻。4、问题（4）的求解是否可以通过调整现有的设备检修计划来提高利润提出一个新的设备检修计划,使原来计划检修的设备在这半年中都得到检修而使利润尽可能的增加。该问题在问题（5）已经得到解决，具体操作见问题（5）。在问题（5）中，已经求出了使得利润最大的最优检修计划，为表五最优设备检修计划月份计划检修设备及台数月份计划检修设备及台数一月一台立式钻床四月一台立式钻床二月一台立式钻床五月两台磨床、一台立式钻床、一台水平钻床和一台镗床三月一台刨床六月无9/285、问题（5）的求解最优设备检修计划问题构造一个最优设备检修计划模型,使在这半年中各设备的检修台数满足案例中的要求而使利润为最大。41问题分析在该问题中，不同的设备检修计划会导致各设备在各个月的使用时间不同，故该问题可建立新的模型来求解，在新模型中，目标函数与问题（1）目标函数相同，约束条件中各设备每个月的使用时间约束不同，分别记磨床，立钻，水平钻，镗床，刨床依次为第1、2、3、4、5种设备，由此可引入新的决策变量6,54321K；5，4，3,21I，个月正常使用K台在第J种设备的第I，第1个月检修台在第种设备的第第，0ZIJK其中第一种设备（磨床）有4台，第二种设备（立式钻床）有2台，第三种设备（水平钻床）有3台，第四种设备（镗床）有1台，第五种设备（刨床）有1台,然后构建新的约束条件进行求解。42模型建立421、目标函数的确定；716617150JIIJYJYXAZMAX422、约束条件的确定各个月设备使用时间的约束一月份设备使用时间248Z51X7015X301248Z1X70425363862248Z17二月份设备使用时间248Z51X2705X2301248Z1X704536386248Z17三月份设备使用时间248Z513X7035X01248Z13X704256862248Z13710/28四月份设备使用时间248Z51X4705X4301248Z1X7025363862248Z17五月份设备使用时间248Z51X705X301248Z15X70425363862248Z157六月份设备使用时间248Z516X7065X301248Z16X704253382248Z167各个月各产品的销量限制一月销量限制10Y7X26853410YX二月销量限制150Y27X3645435三月销量限制11/2810Y372X465540Y32X1四月销量限制10Y473X62541五月销量限制0Y574X361521六月销量限制60Y75X6134321在六月底,每种产品都有50件库存,12J每种产品的最大库存量为100件76I0Y每个变量必须为整数；各种设备需满足一定的检修数量要求12/2861K61K3J261K4J5Z47Z8Z所有的ZIJK为01变量43、模型求解结果通过LINGO软件的运行求解，得到最高利润为50500，此时Z221Z222Z513Z224Z115Z145Z225Z315Z4150其余的ZIJK1；即使在这半年中各设备的检修台数满足案例中的要求而使利润为最大的最优设备检修计划为表四最优设备检修计划月份计划检修设备及台数月份计划检修设备及台数一月一台立式钻床四月一台立式钻床二月一台立式钻床五月两台磨床、一台立式钻床、一台水平钻床和一台镗床三月一台刨床六月无三、结论1、决策效果（结果）的评价通过严格的案例分析以及我们组全体成员的研究讨论、反思修正以及LINGO软件的精密的计算，本案例得出最佳决策结果，（1）中所需要求的最大利润为41464元，具体生产、销售、库存数量方案见表一。（2），（3），（4）小题的计划或调整策略见求解结果中相应的表格。（5）小题，我们依照题意重新建立了一个新的模型，得出的在最优检修计划下利润最大为50500元，最优设备检修计划见表五。2、遇到的问题及解决方法（1）一开始我们的库存费只算到了第五个月底，我们刚开始认为第六个月底库存到七月才交库存费，但是，经过我们的讨论与思考，我们觉得有库存就得交库存费，所以要减去第六个月底的库存所需交的库存费，才能算作半年的13/28利润（2）题目中得出来结果中有多个数据要分析，如果不认真查看，很容易会把结果的数据混淆，而且题目要求的条件要比较多，输入数据时容易出现错误。解决方法反复检查，认真校对LINGO程序的结果与变量之间的对应关系。（3）理解题意错误导致计算结果与题目要求不一致。解决方法认真阅读题目，对题目进行多方面分析，反思纠错，探究讨论。14/28四、附录11、问题（1）LINGO代码及结果截图MAXX11X21X31X41X51X61Y6110X12X22X32X42X52X62Y626X13X23X33X43X53X63Y633X14X24X34X44X54X64Y644X15X25X35X45X55X65Y651X16X26X36X46X56X66Y669X17X27X37X47X57X67Y673Y11Y12Y13Y14Y15Y16Y17Y21Y22Y23Y24Y25Y26Y27Y31Y32Y33Y34Y35Y36Y37Y41Y42Y43Y44Y45Y46Y47Y51Y52Y53Y54Y55Y56Y57Y61Y62Y63Y64Y65Y66Y6705约束条件一月份工时050X11070X12000X13000X14030X15020X16050X1732824010X11200X12000X13030X14000X15060X16000X1722824020X11600X12080X13000X14000X15000X16060X1732824005X11003X12000X13007X14010X15000X16008X1712824000X11000X12001X13000X14005X15000X16005X1712824二月份工时15/28050X21070X22000X23000X24030X25020X26050X2742824010X21200X22000X23030X24000X25060X26000X2702824020X21600X22080X23000X24000X25000X26060X2732824005X21003X22000X23007X24010X25000X26008X2712824000X21000X22001X23000X24005X25000X26005X2712824三月份工时050X31070X32000X33000X34030X35020X36050X3742824010X31200X32000X33030X34000X35060X36000X3722824020X31600X32080X33000X34000X35000X36060X3732824005X31003X32000X33007X34010X35000X36008X3702824000X31000X32001X33000X34005X35000X36005X3712824四月份工时050X41070X42000X43000X44030X45020X46050X4742824010X41200X42000X43030X44000X45060X46000X4712824020X41600X42080X43000X44000X45000X46060X4732824005X41003X42000X43007X44010X45000X46008X4712824000X41000X42001X43000X44005X45000X46005X4712824五月份工时050X51070X52000X53000X54030X55020X56050X5732824010X51200X52000X53030X54000X55060X56000X5712824020X51600X52080X53000X54000X55000X56060X5732824005X51003X52000X53007X54010X55000X56008X5712824000X51000X52001X53000X54005X55000X56005X5712824六月份工时050X61070X62000X63000X64030X65020X66050X6742824010X61200X62000X63030X64000X65060X66000X6722824020X61600X62080X63000X64000X65000X66060X6722824005X61003X62000X63007X64010X65000X66008X6712824000X61000X62001X63000X64005X65000X66005X6702824各个月的销量限制一月销量限制X11Y11500X12Y121000X13Y13300X14Y14300X15Y15800X16Y16200X17Y17100二月销量限制X21Y11Y21600X22Y12Y22500X23Y13Y23200X24Y14Y240X25Y15Y2540016/28X26Y16Y26300X27Y17Y27150三月销量限制X31Y21Y31300X32Y23Y32600X33Y23Y330X34Y24Y340X35Y25Y35500X36Y26Y36400X37Y27Y37100四月销量限制X41Y31Y41200X42Y32Y42300X43Y33Y43400X44Y34Y44500X45Y35Y45200X46Y36Y460X47Y37Y47100五月销量限制X51Y41Y510X52Y42Y52100X53Y43Y53500X54Y44Y54100X55Y45Y551000X56Y46Y56300X57Y47Y570六月销量限制X61Y51Y61500X62Y52Y62500X63Y53Y63100X64Y54Y64300X65Y55Y651100X66Y56Y66500X67Y57Y6760在六月底,每种产品都有50件库存Y6150Y6250Y6350Y6450Y6550Y6650Y6750每种产品的最大库存量为100件Y11100Y12100Y13100Y14100Y15100Y16100Y17100Y21100Y22100Y23100Y24100Y25100Y26100Y27100Y31100Y32100Y33100Y34100Y35100Y36100Y37100Y41100Y42100Y43100Y44100Y45100Y46100Y47100Y51100Y52100Y53100Y54100Y55100Y56100Y57100Y61100Y62100Y63100Y64100Y65100Y66100Y6710017/28规定每个变量必须为整数GINX11GINX12GINX13GINX14GINX15GINX16GINX17GINX21GINX22GINX23GINX24GINX25GINX26GINX27GINX31GINX32GINX33GINX34GINX35GINX36GINX37GINX41GINX42GINX43GINX44GINX45GINX46GINX47GINX51GINX52GINX53GINX54GINX55GINX56GINX57GINX61GINX62GINX63GINX64GINX65GINX66GINX67GINY11GINY12GINY13GINY14GINY15GINY16GINY17GINY21GINY22GINY23GINY24GINY25GINY26GINY27GINY31GINY32GINY33GINY34GINY35GINY36GINY37GINY41GINY42GINY43GINY44GINY45GINY46GINY47GINY51GINY52GINY53GINY54GINY55GINY56GINY57GINY61GINY62GINY63GINY64GINY65GINY66GINY6712、问题（1）运行结果（部分）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、问题（2）灵敏度分析结果（部分）RANGESINWHICHTHEBASISISUNCHANGEDOBJECTIVE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