首先说明，我的这篇短文仅仅是对当年大学生活的感慨，同徐水良没啥关系。
刚刚看到徐水良向大家推荐统计物理学教科书。
我无意继续同徐水良讨论这些问题。只是徐水良推荐的教科书不禁让我想起当年在大学时读过的几本《统计物理学》教科书，令我感慨万千。
相信在77年上大学的人都会有同样的经历。
我们的统计物理或称统计力学课用的是王竹溪编的一本薄薄的教科书，电动力学是曹昌琪的《电动力学》，还有一本很有名的教授的《电动力学》，电磁学是赵凯华（?）编辑的《电磁学》，褚圣麟的《原子物理学》 等等等等，相信当年各个大学都是通用的这几本教材，对这些教材，我们当年可都是能倒背如流了，现在全忘光了。如果曹昌琪不曾经给我当了三年的导师，我连他的名字也会忘得精光。
这些中文教科书都是让人兴味索然，我们就设法去买一些美国的通用教材。最著名的是伯克利的四大力学教材，有《波动学》（绿色封面）、《电动力学》、《统计物理》（蓝色封面）、《量子力学》(紫铜封面）等等。其中的《统计物理学》是我最喜欢读的，那简直就是一种享受，如同是看动画小人书或漫画一样。我至今还记得其中讲述不可逆过程的漫画：
一个酷似卓别林的小丑，提着拐杖，对着一个小木房丢了一颗原子弹。房子炸飞啦。小丑随后又对炸飞的房子废墟又丢了一个原子弹，结果是所有废墟碎片都按照被炸飞的轨迹逆向飞行，被炸飞的房子瞬间又被炸回来啦。
物理学和力学在当年好象区分不大。力学通常是指那些包含有微分方程式的物理学科就成为是力学。比如，电动力学，因为有麦克斯韦方程，就不叫做电动物理学了。电磁学的很多内容就是初级的电动力学。
电磁学的另一个升级版叫作电子线路，因为其中没有用到微分方程，就不敢自称为电磁力学了。
量子力学，因为有薛定谔方程，也就被称为是力学了。
统计物理学，在初级阶段叫做统计物理，高级的统计物理就称为是统计力学了。
光学，没什么微分方程，就一直没有升级为光力学。
热学，初级阶段叫做分子运动学或热物理，高级阶段就成为热力学了。
原子物理学一升级就成为量子力学了。
原子核物理学升级后就成为量子场论。
普通物理，一升级就成理论力学了。
还有，流体力学，弹性力学，材料力学，空气动力学，塑性力学, 等等等等，都要不叫作力学的初级课程。
但凡是没有用到微分方程的学科，都不敢称其为力学。
我这段话是经验之谈，开玩笑。
虽然是这样开玩笑，我们力学专业的人都认为力学没什么技术含量，认为力学系上古典学科，而物理系则是属于现代科学。更因为古典力学都让牛顿给糟蹋完了，剩下的任务就是完成牛顿还没来得及完成的修修补补工作，没什么机会得诺贝尔奖了，大家都争先恐后地转行学物理。
还记得有另一本英文版的著名的统计物理学，我忘记作者名字了，其中的题目特别好。我们科大图书馆只有一本。我们每个人都抢着去借那本书。我曾经借到过几次。就是现在，我还经常做梦我去图书馆去排队借那本书。
还有一本英文版统计物理学，好象是一位姓黄的美国华人编著的。也忘记名字了。后来到了单位工作后，我就将单位图书馆的那本黄某的统计物理学给包下来了。
前几天还记得有人在此提到铁摩欣科。我们在78年上《材料力学》是就是使用铁摩欣科的英文版《材料力学》，当时给我们上课的教授叫沈志荣，是美国的正宗博士，是当年同钱学森同一条轮船从美国回到祖国的，也是我毕业论文的导师。钱学森一直都是我们系的系主任。上沈教授的课，让我记住了一些相关的英文单词，我们学生都会牢记他经常使用的“homogeneous”一词。相信我们班上的学生都会不忘他传授的这个单词。我的毕业论文题目是“近海石油平台的震动模型分析”，其实就是帮助他查阅和归纳这方面的英文资料。沈志荣教授在那期间几乎每天下午都要去体育教研室，就是同宁柏下围棋。
当时科大的条件比较优越，我们所用的教科书基本上是免费发放，就连铁摩欣科的那本厚厚的英文硬皮书也是免费发放。如果不是免费发放的教科书，我们很多人就不买了。后来到了北大，才知道买书贵啊。
吉米多维奇的《高等数学解析题集》就更是我们那个年代尽人皆知几乎人手一本的参考书了。
最令人难忘的是上了方励之的《天体物理学和相对论》、阮图南的“量子场论”，我的物理学课都是到现代物理系或是物理系选修的。我有一个学期选了十几门课，既包括力学系的必修课，又同时去选修物理系和近代物理系的物理课程。我们那时是可以逃课的，参加期中和期末考试通过就给记学分。
凡是方励之授课，没上过的，我大多都会去听。因为那跟听评书一样令人开怀。方励之的那本科普读物《宇宙的創生》很有伯克利《统计物理学》的遗风，其中有大量的漫画。一方面在讲述宇宙的创生是始于和谐、美、平等。其中特别引用圆桌会议来比喻圆桌上平等和美的基本的要求，宇宙中就没有中心，处处是中心，圆形、椭圆形、球形是基本的存在形式，否则就会产生战争，产生灾难。而人类社会也是要求平等，最终的均衡是要逐渐地趋向于人人平等，没中心，没有国王，大家都是占有圆桌的平等的一个角，而不存在坐难坐北的席位，更不存在什么主席。
我们系的数学课要求比较高，各种数学课一直都是由徐澄波教授担任主教授，大概交了我们三年有余。徐澄波教授的授课令人难以忘怀，是北大数力系毕业，讲课的每一句话都配有十分有力度的动作。我们都不会忘记他讲授“Epsilon 非常小，小到不能再小”，那动作就如同是在反复地按死一个老共匪。徐澄波教授的夫人是浙大教授，后来，徐澄波教授也调到了浙大。我还同几位同学专程去他在浙大的家中拜访过他。
只有概率统计是同物理系一道上的大课。给我们上概率统计的老师也令人难以忘怀，最难忘记的是他几乎每堂课都会说几次“这个问题同学们并不百生”，引起一阵阵哄堂大笑，因为我们都知道他要说的是“陌生”，而他就一直都奇怪我们为何要笑。
刚刚想起那本蓝色硬皮的英文版《统计物理学》的作者叫黄克森，Kesen Huang.
徐水良 牛津大学研究生教材《统计物理学中的蒙特卡罗方法》等大学教材 2023-04-26 12:18:58 [点击:257]
赛昆、刘刚等等对统计学几乎一窍不通，不仅他们的统计计算很可笑，听不懂樊教授关于蒙特卡罗方法的介绍，赛昆还坚持说胡话疯话，非常可笑地一再一再说蒙特卡罗方法不是统计方法，不属于统计学，完全不知道自己是在一次又一次出丑。
参见：https://twishort.com/7Dsoc
https://facebook.com/xushuiliang/posts/pfbid02QebkdJ72nGX38cTZBfiuwsz776UEuTGdqT39qCBoyWS8hSgNKeS9qzdP79woj3u2l
我的统计学是40年前在监狱自学的，忘记用的是哪个大学的教科书。在与赛昆等家生共辩论蒙特卡罗法是不是统计方法等时候，发觉不仅赛昆、刘刚等对统计学几乎一窍不通，而且其他很多朋友似乎也不懂统计学。于是上网搜索相关大学教科书，看看有什么教科书，可供我们学习。这是一部分结果：
比较权威的有：《牛津大学研究生教材：统计物理学中的蒙特卡罗方法》等教科书，据说是经典的。纸本书籍，亚马逊网站和书店有售：
https://www.amazon.cn/dp/B00CE3TLB0
刘刚 浙大力学属于数力系能说明什么？ 2023-04-30 14:26:17 [点击:114]
中国的各个系所包括的分支专业并不是什么科学问题。
中国文革前各个大学大多都有数力系。这仅仅是为了师资资源管理的方便，并不能据此来论证力学就是数学的一个分支。
将力学划分到哪个系，不是科学问题，而仅仅是管理问题。在那个年代，力学教师都需要很强的数学，而物理系规模通常是过于庞大，为了均衡，就将力学划分到数学系。中国高中里将力学划分到物理课中。力学究竟应该是划分到物理中还是划分到数学中，中国是根据师资和教学管理的需要来划分的。
中国在文革前很少有计算机系。计算机专业大多都划分到无线电系或者是数学系。这也是根据师资管理的需要。只是后来需要招募大量的计算机专业的学生后，才划分出计算机系。
中国将计算机或力学划分到哪个系不能作为证据来证明计算机或力学是属于哪种学科，这就如同将马克思主义、习近平思想划分到哲学系，并不能证明习近平思想就成了哲学。中国的数学系学生都要学习党史，这并不能证明中共党史就是数学的一个分支！
徐水良的辩论术真是风××。也许徐水良想说的就是他曾经是浙大非数力系的某个系的肄业生。
另外，Monte Carlo 方法也不是什么神秘的方法。可以简单地概括为：对随机事件进行大量随机抽样再对某个随机变量求平均值。
在计算机领域，通常将这种方法也称为Randomization Method。有很多计算机领域的专家会将Randomization Method列为自己的研究方向，但很少有人会将Monte Carlo方法作为自己的研究方向，就如同不会将“投硬币”、“掷骰子”列为自己的主攻方向一样。如果有人自称是Monte Carlo Simulation方面的专家，那是指他会使用Distributed computation 或Cloud Computation进行计算机计算，应该是指如何进行大规模计算，是计算机领域的专家，而不是统计学领域的专家。
Monte Carlo Simulation方法是属于一种“傻瓜方法”，是找不到更好算法时的无奈选择。这个方法本身不是什么难题，难的是如何实现这种算法中所需要的大量计算，那就是属于计算机领域的问题了。
徐水良在讨论Monte Carlo方法是属于数学领域还是统计学领域，这就如同是在讨论“掷骰子”是属于数学领域还是统计学领域一样地无知。
有很多人会在自己的Resume上将Monte Carlo Simuluation列为一个skill。当面试的人看到这个Skill时，这仅仅是表明他的计算机skill比较强，会使用Monte Carlo方法进行模拟计算，其中包括的内容会是Distributed Computation, Grids Computation, 而丝毫不包括什么统计学和其它的什么数学。其数学知识很可能仅仅是求平均值而已。
: An economic system that contains some sub systems. It can be as small as a family, a firm, and can be as large as a country or the whole world. 
is also called a macro system compared with its sub systems.
: An integer represents the total number of the sub systems in 
.
: An integer represents the total number of the products that are produced in 
.
: The ith unit or sub system in system 
. It could be an individual member, or a group of people, such as a firm or another economic system in 
. However, there are no overlaps between different Units. For example, if an individual has been included in one unit, the same individual can’t be included in another unit. 
is also called a micro system compared with its parent system.
: The kth product that can be produced in system 
. Actually, the product here means any labor activities that can be done by any unit in 
. It can be a real product, a project, a job, a task, and anything that need to be done or produced by any unit in 
.
: A time span that the above mentioned products are produced in 
.
: 
, which is the set of non-negative M-dimensional real vectors and with the original point 
excluded.
: The same as 
. However, we use it to particularly represent the Production Space with its kth Cartesian coordinate representing the amount of a parameter related to 
product.
: The demand amount of 
product that requested by 
.
: The maximum amount of 
product that can be produced by 
within time 
. Normally 
can be produced by 
when it uses all of its resources to work on product 
. All the 
make up an 
matrix, which is called the micro capability matrix of system 
.
: The amount of 
product that are actually produced by 
. All the 
make up a vector in 
, and is called micro supply vector.
: The amount of 
product produced by the macro system 
. It is also called the macro supply of system 
. All the 
make up a vector in 
, and is called a macro supply vector of system 
.
; 
: The price of the 
product in system 
.
introduced above. An economic system is called a micro system if it is contained in a macro system, such as the sub-system 
described above.Definition: System Parameter: An economic parameter is called a system parameter if it is associated with an economic system. For example, the supply amount, demand amount, and price are all examples of system parameters.Definition: Micro Parameter and Macro Parameter: A system parameter is called a micro parameter if it is associated with a micro system, and is called a macro parameter if it is associated with a macro system. Some parameters may be associated with both micro system and macro system. As long as there is no confusion, we use the same alphabet character to represent a system parameter: if 
denotes a macro parameter associated with a macro system 
, then 
will be used to denote the same micro parameter associated with micro system 
.For example, 
and 
represent the macro and micro supply amounts of product 
, 
and 
denote the macro PPF of system 
and the micro PPF of the micro system 
correspondingly.Definition: Vector Parameter and Scalar Parameter: a system parameter may also be associated with each of the production, for example, supply amount normally means the amount of a product that can be provided by an economic system. In a system with 
products, a system parameter could be 
-dimensional with each component representing the amount associated with one product. These 
-dimensional parameters make up a vector in the production space 
, thus is called a Vector Parameter. A vector parameter can be represented by a vector in 
. Examples of vector parameters are: Supply, Demand, and Price.A system parameter is called a scalar parameter if it is one dimensional. Examples of scalar parameters are: income, GDP, revenue, etc.Additive Parameter: A system parameter is called additive parameter if the corresponding macro parameter can be derived by aggregating the same parameter over all of its micro systems. Or precisely, system parameter X is additive if
is a parameter reachable and meaningful to micro system 
, while 
is the same parameter reachable and meaningful to macro system 
.In our model, we tried to decompose the macro system 
into some micro systems 
, and divide the products into some products or tasks in such a way that some of the system parameters are additive, particularly, the supply amount, the demand amount, revenue, and income are all additive. However, the price parameter is not an additive parameter.Based on the above notations and definitions, we can say that the supply and demand are vector parameters and are additive, whereas the price is also a vector parameter, but not additive. The income, revenue, GDP, profit and loss are all additive scalar parameters. We are only interested in those parameters in this paper.
, the production capability matrix for each micro system and for each product.
: The macro demand of 
product that needs to be accomplished by system 
.We assume the following parameters are unknowns and need to be resolved: Price, micro supply, macro supply, micro income, macro revenue.
, let us assume that it can arrange its resources to make any requested products. However, its production amount for product 
should be limited to or bounded by 
due to its limited resources and capabilities. Any of its feasible production state will be a point or a vector in the production space 
. All of its feasible production vectors should be a bounded range in the production space 
. We call this bounded range as its micro Production Possibility Range (PPR) and denoted as 
. The upper boundary of 
is called the micro PPF, denoted as 
. Normally, as long as the micro system 
is small enough compared with macro system 
, each of the micro PPF can be approximated by the linear plane curve that passing the 
axis at 
in production space 
.Given the micro capability matrix, the micro PPF 
can be formulated as:
can also be expressed in the following format:
is called the intrinsic price vector of the micro system.Actually, 
is a linear plane curve with vector 
as its normal vector. It passes the 
axis at the points:
is the unit vector on the 
axis. 
and is on the 
axis of the production space. We call these 
points as the micro vertex points.Given the price vector as 
and the micro supply vector 
of the micro system 
, the total income or revenue of the micro system 
can be expressed as:
is feasible to micro system 
, there should have some possible micro supply vectors that can maximize the income or revenue of the micro system 
. Such feasible micro supply vectors can be formulated as:
to be maximized is a linear function, according to Lagrange’s theorem, the max-min can only be found on the boundary of the region 
. Then the above equation is equivalent to:
vertex points listed in Equation (5). Then, the above equation can be further simplified as:
is an 
function defined as:
is the total number of vertex points that have 
maximized for 
. Number 
can be formulated as:
maximized. It gives at least one of the micro supply vectors 
that can have the micro income 
maximized. It is an explicit mathematical format for the micro supply as a function of the capability matrix and the price vector. However, it is a discrete function and thus is difficult to deal with. The key step and major contribution of our model is to formulate a continuous and smooth function to approximate the Equation (13). Here are the key steps. Equation (13) is equivalent to the following function:
is called the macro capability parameter defined as:
.In practical applications, we can simply drop the limited function by assigning 
with a large number, thus the Equation (14) can be simply expressed as:
for 
;Normally, it will be good enough if we assign 
as:
Equation (16) will be a good approximation for the micro supply function, and it gives the Micro PPF 
once the price vector goes through all possible directions in the production space 
. 
will be good enough for most of the practical applications. We use 
to get most of the results listed in this paper.
and the micro supply as a function of the price vector and the capability parameters, as shown in Equation (16). As each of the micro system 
has a maximum production frontier and a limited range as its PPR, the macro system should also have a macro PPR in the production space 
, and must have a macro PPF. As the supply parameter is an additive parameter, we can get the macro supply vector by aggregating all the micro supply vectors, that is:
, as shown in Figure 3. So, the above Equation is not only a supply function, but also a function that gives the macro PPF 
.Figure 4.shows the Supply-Price curves for various 
values. Once 
is big enough, the Supply-Price curves become step curves. Keeping in mind that each micro system is targeting at maximizing its income, so, when the price 
rises, a micro system 
may want to switch all of its resources to work on product 
, and thus to have the macro supply 
be suddenly increased by an amount of 
. At the same time, because 
switched from working on 
to work on some other products, such as 
(
), it will have the supply 
decreased suddenly by an amount of 
. This explains why the Supply-Price curve shows as a step curve, also explained the Law of Supply as discussed in later sections.The supply functions and the PPF given in Equation (17) are the key contributions of this paper. The rest of the paper discusses applications of this supply function, or compare our PPF with some other production possibility frontiers and see how much efficiency can be improved by using our proposed model.
denote the micro WPPF of micro system 
, and 
denote the macro WPPF of the macro system 
. 
is defined as the lower boundary under the conditions that all of its micro systems have reached their micro PPF 
. The WPPF 
is defined as the set of micro supply vectors that are on the 
and results a macro supply vector on the macro WPPF 
if aggregated for all micro systems.
and 
can be formulated using the same Equations as shown in Equation (16) and (17), as long as we set 
. We will not rewrite these Equations here.Let 
be the region bounded by 
and 
, and 
be the region bounded by 
and 
. 
is called the micro Maximum Production Possibility Range (MPPR), and 
is called the macro MPPR.Given a price vector 
, through Equations (16) and (17),we can get a macro supply vector 
on 
, a macro supply vector 
on 
, a micro supply vector 
on 
, and a micro supply vector 
on 
. It can be shown that the price vector 
is the normal vector of those curves at the corresponding points as indicated by these supply vectors [6].
denote the supply elasticity for the 
product. By applying partial differentiation to Equation (16) and (17), we can easily have:
given in Equation (19) is non-negative. Noting that 
and 
, thus all items in Equation (19) is non-negative, then the elasticity 
which is a sum of some non-negative numbers is also non-negative. Thus Equation (20) is proven.By applying partial differentiation to Equation (16), we have:
,
,for 
and 
Equation (21) follows the above two equations immediately, and thus the Law of Supply is proven.
and 
are functions of 
. Thus, the above general equilibrium equation gives 
equations with 
prices as unknowns. Generally, once we know the function format for 
and 
, the equilibrium price vector can be solved from Equation(22).We do not try to give a generic format for demand functions in this paper, instead, we just simply assume the demand is given as a vector with fixed direction, thus Equation (22) can be expressed as:
equations with 
for 
as the 
variables. Then the price victor 
can be solved by solving the above equation through many mature methods, such as Newton’s method [5, 9], Brent’s method [2, 13]. Thus all the micro supplies 
can be given through Equation (16), and all the macro supplies can be given through Equation (23) or Equation (17).
and 
as variables. It can be easily solved through the T-forward method [7, 8, 3]. Equation (24) and (25) should give the same solution.
. Similarly, we can construct a macro LPPF curve 
for the macro system 
, which can be formulated as:
and 
be the micro SPPF and macro SPPF for the Self Sufficient scenario. 
is the same as the plane curve 
as shown in Equation (26). However, the macro SPPF 
is different from the Macro LPPF 
.By definition, given the requested demand as 
, the micro SPPF 
can be expressed as:
can be formulated as:
is called the carpenter system, which contains 
units or members, and need to make 
products. The carpenter system needs to produce 4 table legs, 1 table top, and 6 chairs, which make up the 3 components of the demand vector. The micro capability matrix and demand vector are listed in Table 1. The requested amounts or demands are listed in row 3, and the production capabilities are listed in row 5 to 14.
, 
, 
, and 
are derived and are drawn in Figure 2. These four curves are all converged to the same plane curve.
, the micro WPPF 
, the micro LPPF 
, and the Micro SPPF 
are all shown as the same plane curve in the production space
and 
could be different from the plane curve for some small 
. However, the difference can be only shown up for 3 or more dimensional case. If we set all other prices to 0 except two of them, then the above four curves always converge to the plane curve.Figure 3. shows the macro PPF 
, along with some other macro curves, including WPPF 
, LPPF 
, and SPPF 
. Figure 4 draws the supply-price curves for various 
value. By comparing with Figure 1, our model gives the PPF and supply functions in more detailed formats and make the general supply-demand equilibrium solvable numerically.
is a convex curve, the macro WPPF 
and the macro SPPF 
are concave curves, whereas the macro LPPF 
is a linear plane curve in the production space. The maximum PPR 
is the region bounded by 
and 
. The extension line of the demand vector 
intersects with these 4 curves at the points 
, 
, 
, and 
respectively. These curves and points are used for improved efficiency analysis
has reached its micro PPF
, then the aggregated macro state of the macro system must be in the macro MPPR
. Also, suppose the macro system 
is requested to produce the demand vector
. The demand vector 
is as shown as the line 
in Figure 3. Figure 3. also shows the four points
, 
, 
, and
, which are the intersection points of the line 
with the curves
, 
, 
, and
.For all possible production states in
, the Line 
gives all possible states that are proportional to the requested demand
. Any other states in 
may have some products wasted or not needed compared with the requested demand
. So, for all possible states in
, only the states on line 
can best meet the requested demand
. We are only interested in the states that are on the line
.For all states on the line
, the point 
is the best scenario, because it gives the maximum possible amount for all supplies, 
is the worst scenario, whereas 
and 
are somewhere in the middle. Point 
can be treated as the average state for all points in the macro MPPR
.Let us assume that point 
is the average state that the macro system reached without using any optimal management method. Now, let 
denote the improved production efficiency by comparing the best scenario 
with the average scenario
, then 
can be formulated as:
rises and all other prices remain unchanged, the supply 
rises, whereas the supply for other products decreases, for example, Supply 
decreases. When 
is large, the supply curves 
and 
become step curves
and proportional to the requested production demand vector 
.The last column gives the Earnings or Incomes for each team member.The row with header “Supply/Demand” gives the number
defined in Equation (23). It is the ratio between the optimal solution and the requested demand amount.The row with header “Optimal Intrinsic Price” gives the intrinsic prices which can be used to calculate the earnings for each team member. The intrinsic price vector for these 3 products should be (0.51, 0.68, 0.52). Once this intrinsic price vector is known, each team member will reach his maximum income by making the products with the amounts as requested in the optimal solution. In other words, the whole team will realize the optimal solution automatically as long as each sub unit has maximized its income.The row with header “Self Sufficient Supply” lists the supply state 
on the SPPF 
.The row with header “Linear Macro Supply” lists the supply state 
on the LPPF 
.The row with header “Improved Efficiency” lists the improved efficiency ratio by the optimal solution 
compared with the Linear Macro Scenario
.The row with header “Loop Count” lists the number of loops for our numerical calculation to converge to the results with required precision.The optimal solution not only depends on the capability matrix, but also depends on the direction of the demand vector. Given a team member and his capability matrix, we cannot tell which product is the most specialized product for him. The optimal solution may request him to work on one product. However, once the direction of the demand vector changes, the same team member might be requested to work on some other products to have the whole team to reach the optimal solution.
can be explained as the delivery ratio (or amount) of the kth project 
by team member 
. The demand for each project is 1 within required delivery time.Let 
denote the required delivery time for the kth project 
, and 
denote the fastest delivery time of team member 
to deliver the kth project 
under the condition that 
works only on project 
. Then the equivalent demand vector can be expressed as:
, we can find the improved efficiency ratio by the optimal solution. The improved efficiency for the carpenter team is:
in this section. Let 
be the intersection point of the demand vector 
(or its extension) with the curve 
, and 
denote the length of the line 
. Then 
can be easily found as:
is a convex curve, we can use 
to approximate the PPF 
, with 
to be a value that can make the 
most close to 
. Normally, once we find one point on 
, we can calculate the value of 
which can make the 
most close to the 
. Further, we assume the average production state is roughly at the point 
on the curve 
(or denoted as
). If 
is used to approximate 
, the improved efficiency by reaching the macro PPF can be estimated as:
is a good approximation for 
when
; and even 
will be a good approximation for 
when 
and
.To make it more conservative, we use 
to approximate the macro PPF
. Then the production efficiency ratio that can be improved by the optimal solution can be estimated as:
products.Based on our test results and the above analysis, our optimal solution for team work management normally can improve the production efficiency by 40% for most of team work problems!
can be used to approximate the PPF 
and LPPF 
. Here all the curves are shown in 2-dimensional. You need to imagine it in M-dimensional
, and the micro income amount 
. Using these micro information, a team manager can tell each team member 
to work on the 
with the amount 
as indicated in the optimal solution. Note that most of the team members may work on only one product. The team manager can also tell each team member 
about how much he should be paid, which would be 
. So, with micro management, a team manager manages each of the team members in micro details, including what and how much need to be done, as well as how much should be paid as return. That is why we call this management method as Micro Management.
. Given this intrinsic price vector, each team member will reach his maximum income by working on the product with the amount as requested by the optimal solution. Suppose each team member is targeting at maximizing his income, the team manager needs only declare the intrinsic prices, then, every team member will automatically to work on the product with the amount such that the whole team will reach its optimal solution as indicated in the optimal solution.The team manager doesn’t manage the team members in micro details, but manage it in macro level, which simply tells how much will be paid for product or project to be delivered. That is why we call this management method as macro management.
