凭栏袖手前些年公司有个项目没拿好，拿地价高了。因此下达项目计划时利润指标留了个负值，-300万元；今年尝试预结算时，发现决算值的利润大约为-500万。按制度规定，没完成利润指标要被罚，制度中约定的罚款计算公式大约如下：
罚款=（1-利润完成率）*考核系数*0.5*目标利润
那么问题来了，目标利润为负时还能按这个公式算吗？显然不行，百度后发现大家的意见意外地十分一致，都同意按下列公式计算负指标的完成率：
完成率=2-实际值/目标值 （公式一）
但是，没人能说清楚为什么要这么算，每当这个时候，我就异常兴奋起来，我觉得我有责任和义务把这个问题弄清楚。
首先，还是要建立数学模型。
既然要讨论问题，还是要适当一般化些为好。我们讨论多目标的情况：
在n元数组集合R∧n中，设
u=（u1，u2，…，un），v=（v1，v2，…，vn）
定义欧几里得内积为
（u，v）=∑ui*vi （i=1,2，…，n）
即R∧n为一个n维欧几里得空间。
我觉得在向量空间里讨论目标的实现情况的好处是向量不光可以度量，还有方向。那么究竟应该如何度量完成率呢？
上图我们用二维坐标表示n维向量空间，只要记住是向量空间就行了。
设目标值为 u=向量（A,B），实际值为v=向量（A,C），则 w=向量（C,B）=u-v
目标在B点，结果跑到C点了，其偏差为w，w的模（或距离）容易度量，但方向不好弄。
过C点作CD⊥AB与AB相交于D点，
我们有 a=向量（A,D），为实际值v在目标值u上的投影，设 ∠CAB=α＜π/2，由内积定义有
|u|=（u,u）∧(1/2)， |v|=（v,v）∧(1/2)， |a|=（a,a）∧(1/2)，
由于a与u方向相同，|a|可以看着是对u的贡献，
因此，我们可以定义完成率：
完成率（u,v）=|a|/|u| (公式二）
同样定义偏差率：
偏差率（u,v）=|DB|/|u|=(|u|-|a|)/|u|=1-|a|/|u|=1-完成率（u,v） (公式三）
另外，由 |a|=|v|*cosα
cosα=(u,v)/|u|*|v|
有 完成率（u,v）=|a|/|u|=|v|*cosα/|u|=|v|*{(u,v)/|u|*|v|}/|u|=(u,v)/|u|∧2 （公式四）
有一个问题，完成率是客观的吗？
我们认为，完成率跟主观立场有关。比方说我们的工薪收入，对自己家里来说是越多越好，但对公司来说，则是在完成一定工作量的前提下越少越好；那么到年末，我们家里和公司在测算薪金完成率时的立场和方法能一样吗？我们领导批评人说——你是在制造工作，离目标越来越远。真实世界的情况太复杂了，由于认知的局限性，我们很难找到正确的路径。以前，我们提倡计划生育，孩子越少越好，现在是越多越好；以前只看GDP，现在还要看绿色低碳环保，排放指标要花钱买。
因此，完成率是主观的，关键是看测算者自己的立场，上面公式二和公式四都是测算自己的立场与努力方向和目标向量一致时的完成率。下面，我们看看不一致时的情况。
假设，实际值为b=向量（A,E），b=-v，|b|=（b,b）∧(1/2)=|v|， ∠EAB=π-α＞π/2，
过E点作EG⊥AB与AB的反向延长线相交于G点，则知
c=向量（A,G）,为实际值b在目标向量上的投影，易知，c=-a，|c|=|a|，
现在的情况是c与u的方向相反，|c|可看着是对目标向量u做的负贡献；无法使用公式二和公式四。
我们需另辟蹊径，重新定义努力方向和目标向量不一致时的完成率。
关键是要将努力方向和目标向量设法重新变得一致。既然我们的努力方向是对的，那么目标的方向一定是错的，所以必须反过来，最直观的想法就是把目标设为-u，这样c与-u的方向一致，就可以用公式二计算了。
做一个坐标变换就可以达到目的了。把向量空间原点O（0,0）沿u移到H点，使|AH|=2*|AB|；
这样，在新坐标系里，目标值为“向量（H,B）”,显然，我们有：向量（H,B）=向量（A,B）-2*u=u-2*u=-u
实际值为“向量（H,G）”,我们有，向量（H,G）=向量（A,G）-2*u=c-2*u
因此，转换后实际值和目标值的方向一致，但由于是转化为模来计算，要解决保留方向信息的问题，先定义一个方向函数：
1 当向量u和v的夹角 0≤α＜π/2（等价简称向量u、v方向一致）
f(u,v)=｛
-1 当向量u和v的夹角π≥α＞π/2（等价简称向量u、v方向不一致）
套用公式二可得：
完成率（u,b）=|向量（H,G）|/|向量（H,B）|=(2*|u|-|c|*f(c,u))/|u|=2-|c|*f(c,u)/|u| (公式五）
同样，我们可以有
|c|=|b|*cos(α)
cos(α)=-cos(π-α)=-(u,b)/|u|*|b|
由公式五有
完成率（u,b）=2-|c|*f(c,u)/|u|=2-|b|*cos(α)*f(c,u)/|u|=2-|b|*【-(u,b)/|u|*|b|】*f(b,u)/|u|
=2+(u,b)*f(b,u)/|u|∧2 （公式六）
当n=1时，即单指标的情况下，由公式五易得公式一：完成率=2-实际值/目标值
下面开始总结：
1、负指标这个提法不合适。各人站在不同的立场上，看法肯定有所不同。我们认为，看努力方向和目标值是否一致这个说法较为客观；谁也没有绝对真理，指标本身无所谓正负（尤其是在向量空间中没有绝对的正向或反向之分），用什么公式只看使用者自己的主观立场。
2、如果你觉得你的努力方向和目标一致，你就用公式二和公式四计算完成率；反之，如果你认为你的努力方向和目标不一致，你就用公式五和公式六计算完成率。
3、大家从经验出发弄出的公式一还是靠谱的。
4、为什么指标系刚好要移动2*u？其一，如果移动u，则目标值变为零，不好比较计算完成率；其二，移动2*u之后，目标向量在新旧两个坐标系下，模相等，方向相反，体现了一种数学上的简洁及对称的美感，要知道在自然界，大多数情况下，简洁与美即意味着正确；其三，当我们发现努力方向和目标不一致时，第一反应就是——伙计，有没搞错？弄反了吧？因此，把目标值等量改变一下方向是最朴素的想法。
5、假设目标向量为c，如果你主观认为这个方向不符合你的要求，无论你的实际值是否与c一致，都可以通过移动坐标轴来把目标向量反过来，只不过现在要把新原点沿着c的方向移到两倍|c|的距离的地方；最后变换之后得到的结果和公式五及公式六完全一样。
6、你的立场及努力方向不一定要与实际值一致，因为可能有各种不可克服的因素导致你一时实现不了自己的主观理想。我们要再三强调的是努力方向是主观立场，不是你现在实际所处在的位子。比方说你的实际值可以与c同向，只要心里不认可，认为自己是被迫处在这个位子上的，属于身在曹营心在汉，这种情况下就要用公式五和公式六来测算完成率。
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