有转换成本的生产计划

案例题目：有转换成本的生产计划七叶树制造厂（BuckeyeManufacturing）是为了卡车生产发动机头的制造企业。生产发动机头的生产线非常复杂，有900英尺长。这条生产线生产两种类型的发动机头：P型机头和H型机头。P头在重型卡车里使用，H头在小型卡车里使用。因为在一定时间只能生产一种类型的发动机头，生产线启动后只能生产P型机头或者生产H型机头，不能同时生产两种类型的机头。转换发生在每个周末，从P型机头转换变成H型机头的成本为500美元，反过来也是这样。当生产P型机头时，最大的产出率为每周100单位；当生产H型机头时，最大生产率为每周80单位。七叶树制造厂在使用生产线生产P型发动机头后关闭了一周。公司经理想安排未来8周的生产和生产转换。当前，七叶树的库存包括125单位的P型发动机头和143单位的H型发送机头。存货的年库存费用是存货价值的19.5%。P型发动机头的生产成本是225美元，H型发动机头的生产成本是310美元。安排生产计划的目标是使制造成本加上库存成本和转换成本的总和最小化。七叶树收到下面来自客户（机头组装厂）在未来9个星期的需求订单。安全库存的要求是周末库存水平必须至少达到下周需求水平的80%。产品需求周P型发动机头H型发动机头1553825538344304005454864548736588355793558

管理报告为七叶树的经理准备一份报告，阐述关于未来8周的生产和生产转换计划。一定要注意，按照这样的生产计划，总成本是多少，多少来源于库存成本，多少来源于生产转换成本。案例分析：1）建立模型：设定决策变量Xijk，i=第1,2,3,4,5,6,7,8周；j=P,H，代表生产机头类型；k=1or0,1代表转换生产，0代表不转换生产；年库存费用是存货价值的19.5%，那么周库存费用=19.5%/52=0.375%的存货价值；假设安全库存的要求是每周末库存水平必须至少达到下周需求水平的80%指的是P型或者H型车头均需满足。库存成本+制造成本+转换成本最小的目标函数如下：MinZ=225*100*(x1P1+x1P0+x2P1+x2P0+x3P1+x3P0+x4P1+x4P0+x5P1+x5P0+x6P1+x6P0+x7P1+x7P0+x8P1+x8P0)+310*80*(x1H1+x1H0+x2H1+x2H0+x3H1+x3H0+x4H1+x4H0+x5H1+x5H0+x6H1+x6H0+x7H1+x7H0+x8H1+x8H0)+500*(x1P1+x1H1+x2P1+x2H1+x3P1+x3H1+x4P1+x4H1+x5P1+x5H1+x6P1+x6H1+x7P1+x7H1+x8P1+x8H1)+225*125*0.375%*8+225*100*(x1P1+x1P0)*0.375%*8-225*55*0.375%*8+310*143*0.375%*8+310*80*(x1H1+x1H0)*0.375%*8-310*38*0.375%*8+225*100*(x2P1+x2P0)*0.375%*7-225*55*0.375%*7+310*80*(x2H1+x2H0)*0.375%*7-310*38*0.375%*7+225*100*(x3P1+x3P0)*0.375%*6-225*44*0.375%*6+310*80*(x3H1+x3H0)*0.375%*6-310*30*0.375%*6+225*100*(x4P1+x4P0)*0.375%*5+310*80*(x4H1+x4H0)*0.375%*5+225*100*(x5P1+x5P0)*0.375%*4-225*45*0.375%*4+310*80*(x5H1+x5H0)*0.375%*4-310*48*0.375%*4+225*100*(x6P1+x6P0)*0.375%*3-225*45*0.375%*3+310*80*(x6H1+x6H0)*0.375%*3-310*48*0.375%*3+225*100*(x7P1+x7P0)*0.375%*2-225*36*0.375%*2+310*80*(x7H1+x7H0)*0.375%*2-310*58*0.375%*2+225*100*(x8P1+x8P0)*0.375%-225*35*0.375%+310*80*(x8H1+x8H0)*0.375%-310*57*0.375%=23675x1P1+23175x1P0+26044x1H1+25544x1H0+23590.625x2P1+23090.625x2P0+25951x2H1+25451x2H0+23506.25x3P1+23006.25x3P0+25858x3H1+25358x3H0+23421.875x4P1+22921.875x4P0+25765x4H1+25265x4H0+23337.5x5P1+22837.5x5P0+25672x5H1+25172x5H0+23253.125x6P1+22753.125x6P0+25579x6H1+25079x6H0+23168.75x7P1+22668.75x7P0+25486x7H1+24986x7H0+23084.375x8P1+22584.375x8P0+25393x8H1+24893x8H0-674.138约束条件见下：x1P1=0x1H0=0x1P1+x1P0+x1H1+x1H0≤1x2P1+x2P0+x2H1+x2H0≤1x3P1+x3P0+x3H1+x3H0≤1x4P1+x4P0+x4H1+x4H0≤1x5P1+x5P0+x5H1+x5H0≤1x6P1+x6P0+x6H1+x6H0≤1x7P1+x7P0+x7H1+x7H0≤1x8P1+x8P0+x8H1+x8H0≤1约束条件：安全库存的要求是周末库存水平必须至少达到下周需求水平的80%125+100*(x1P1+x1P0)-55≥55*80%143+80*(x1H1+x1H0)-38≥38*80%125+100*(x1P1+x1P0+x2P1+x2P0)-55-55≥44*80%143+80*(x1H1+x1H0+x2H1+x2H0)-38-38≥30*80%125+100*(x1P1+x1P0+x2P1+x2P0+x3P1+x3P0)-55-55-44≥0143+80*(x1H1+x1H0+x2H1+x2H0+x3H1+x3H0)-38-38-30≥0125+100*(x1P1+x1P0+x2P1+x2P0+x3P1+x3P0+x4P1+x4P0)-55-55-44≥45*80%143+80*(x1H1+x1H0+x2H1+x2H0+x3H1+x3H0+x4H1+x4H0)-38-38-30≥48*80%125+100*(x1P1+x1P0+x2P1+x2P0+x3P1+x3P0+x4P1+x4P0+x5P1+x5P0)-55-55-44-45≥45*80%143+80*(x1H1+x1H0+x2H1+x2H0+x3H1+x3H0+x4H1+x4H0+x5H1+x5H0)-38-38-30-48≥48*80%125+100*(x1P1+x1P0+x2P1+x2P0+x3P1+x3P0+x4P1+x4P0+x5P1+x5P0+x6P1+x6P0)-55-55-44-45-45≥36*80%143+80*(x1H1+x1H0+x2H1+x2H0+x3H1+x3H0+x4H1+x4H0+x5H1+x5H0+x6H1+x6H0)-38-38-30-48-48≥58*80%125+100*(x1P1+x1P0+x2P1+x2P0+x3P1+x3P0+x4P1+x4P0+x5P1+x5P0+x6P1+x6P0+x7P1+x7P0)-55-55-44-45-45-36≥35*80%143+80*(x1H1+x1H0+x2H1+x2H0+x3H1+x3H0+x4H1+x4H0+x5H1+x5H0+x6H1+x6H0+x7H1+x7H0)-38-38-30-48-48-58≥57*80%125+100*(x1P1+x1P0+x2P1+x2P0+x3P1+x3P0+x4P1+x4P0+x5P1+x5P0+x6P1+x6P0+x7P1+x7P0+x8P1+x8P0)-55-55-44-45-45-36-35≥35*80%143+80*(x1H1+x1H0+x2H1+x2H0+x3H1+x3H0+x4H1+x4H0+x5H1+x5H0+x6H1+x6H0+x7H1+x7H0+x8H1+x8H0)-38-38-30-48-48-58-57≥58*80%约束条件：如果换线，换线生产的产品必须与上一周生产的不同：x1P0-x2P0-x2H1≥0x1H1-x2P1-x2H0≥0x2P1+x2P0-x3P0-x3H1≥0x2H1+x2H0-x3P1-x3H0≥0x3P1+x3P0-x4P0-x4H1≥0x3H1+x3H0-x4P1-x4H0≥0x4P1+x4P0-x5P0-x5H1≥0x4H1+x4H0-x5P1-x5H0≥0x5P1+x5P0-x6P0-x6H1≥0x5H1+x5H0-x6P1-x6H0≥0x6P1+x6P0-x7P0-x7H1≥0x6H1+x6H0-x7P1-x7H0≥0x7P1+x7P0-x8P0-x8H1≥0x7H1+x7H0-x8P1-x8H0≥0Xijk=0,12）软件计算结果见下：3）验证结果是否与实际相符