Modern society is being formed by the worldwide division of work as we move towards globalization. Since 1776, there is the problem of invisible hand by Smith [5] , and this problem become more important at global economics. At the classics, the price is central (Smith), and the production quantity is at next considered as the newer actor (Keynes [6] ). Recently, it is pointed out in Matsui [7] that the demand speed would be near to the invisible hand, because it would bring each maximal-profit (re-) balancing in the changeable market economy.

Generally, the body-centered network (SCM/GDP) might be able to be invisibly balanced and cooperated by the demand speed (God hand) and cloud computing [8] . However, there is the profit-balancing at series’ types, otherwise may the cost-relative balancing at parallel types [9] . Thus, this God-like hand would be alive and the win-win would be attainable at not only SCM networks but also Smith’s world only under equally partnership.

As an example, let us consider the two- or three-center model consisting of sales, assembly and fabrication centers [10] . The two main types of configuration are series and parallel systems. The SCM is a series type, and ERP (enterprise resource planning) is a parallel type. For a series class, the profit maximization is attainable under the demand speed given (shared), even if each heterogeneous agent (enterprises) pursues the self-goal in non-cooperation under indivisible environment.

Thus, each unit-optimization in profit gives the total optimization in sum under non-cooperation, and the point (balancing) occurs at near middle lead-time (reliability). This class is called the integral optimization, and might be governed by the Ellipse map and strategy in Matsui [11] .

Our profit is corresponded to the marginal profit/value in Accounting/GDP, and it is similar to the medium criterion in [12] . In the classics, A. Smith presents the first concept on the invisible hand in 1759 [13] , prior to 1776. This concept might be near to that of medium criterion in our balancing issues.

The stochastic balancing problem is a class of the Conveyor-Serviced Production Station (CSPS) and its networks [14] . This study is called the station-centered approach to the power-conveyor system, and this station is here corresponded to the individual bodies (human, house, enterprise, etc.) in the society or country.

In the body balancing system, the definition of stochastic balancing is here as follows: A stabling phenomenon of transient, bottlenecked vs. balanced, state of the object system. The balanced solution by this stochastic balancing becomes quasi-optimal.

Recently, the medium approach to the body-balancing system is outlined to SCM type in Figure 1 [12] . In Figure 1, is the demand speed (cycle time), and also, and are the medium criterion, and moving-standard inventory in the individual body, , respectively.

The medium criterion, , is controlled by, in which the parameters, and

are the cost coefficients (penalties) of inventory holdings, excess inventory and shortage inventory, respectively.

Now, the Newsvendor’s condition [1] is the followings:

where is the distribution function of the inventory in the in the individual body,.

Then, the balancing goal is given by the following objective function:

where is the weight factor,.

An optimal condition (balancing) is assumed from the classical inequality and Matsui’s equation [12] as follows:

In (3), and correspond to and, respectively, and means a balancing value at the equilibrium.

Two principles on the medium balancing are here presented and considered. At first, the -balancing problem (2) is seen on the upper level of 2-level scheme in Figure 1. This main problem is easily decomposed of the dual problem:

in the respective body of entity. Matsui’s point, , is based on the so-called Chameleon’s criteria [12] .

Now, the following condition is considered under the demand speed (cycle time), , and the exponential service with the mean, (supply speed). That is,

and the demand speed, , is as follows:

On d-balancing, the following relation is obtained from (6):

Especially, for Poisson service, the optimal condition is

where is Poissonian type distribution.

These relations are outlined in Figure 2. From Figure 2 and Matsui’s equation [12] , the balance equation is

and the balancing principle is

from (9) and the classic inequality.

On the lower level, the entity is state of the body and period, and the network flow is seen in Figure 3. This network flow would be behaviored under demand speed, , in the body-centered balancing.

Now, the following notations are introduced in each and from [14] :

: Order scheduled for body and period;

: Expected demand in body and period;

: Next inventory at the end of body and period

.

where is the backorder position (quantity).

For each, the balance relation on demand and supply is from Figure 3 and [15] as follows:

From (11), the second balancing principle is obtained as follows:

where the second and third terms of right hand are corresponded to the coordinating of the moving marginal inventory, , and end inventory, , at, respectively.

The equation (11) is similar to that of progressive control in [12] . Thus, our network flow is governed under -balancing by the order quantity, , in (11).

The usual conveyor systems are the two types of the stations with or without stopper. These systems are treated in [11] by a stochastic approach. This stochastic treatment aids at the minimization of the irregular interruption in delay and idleness at the series type.

A cost approach to the delay and idleness is seen in [11] , and this balancing problem is considered by the 2- stage method. An example of stopper type is seen in Figure 4, the cycle time is 1.5 minutes, and the number of stations is.

The Newsboy method to this case [16] gives the penalty cost function as follows:

where the right hand consists of the objective of -th station:

in which is the p.d.f. of work time at the -th station.

Then, the optimal condition is obtained by a differenciation method in the followings:

The cycle time, , is determined by the Equation (14), and this equation is here available in replace of (2).

For example, let us, and. From (15), the optimal cycle time, , is given by for the Erlangian with phase in the work time of stations.

Now, let us apply the simulator of SALPS soft [12] to this case. The simulation result is seen in Table 1, and the optimal cycle time, , is with a slight difference. In addition, the total cost would be negligible.

Another verification is considered by Figure 5. From Figure 5, the bottleneck of the system is seen in the stations, 2 and 5, in Table 1. The balancing hypothesis in (3) would be valid.

Next, let us consider the two-stage supply chain (SCM) of a franchise type in Figure 5. This case consists of a series of factory, , and warehouse, , and is seen in Figure 5 [16] .

In the case, the outflow is uncontrollable, and the system is only controllable by inflow.

For this case, the balancing problem is the minimization of the total inventory between suppliers. That is,

The objective (16) is here available in replace of (2).

For this case, the balancing table is presented and considered on the network flow balancing. The total inventory in one year is 5176, and is improved much more than the practice (8074).

From Table 2, the medium balancing holds except July, September and April as follows:

It is them noted that any bottleneck phenomena would be doubtful in July, September and April.

This result would show the effectiveness of medium balancing method, and the hypothesis (3) is ascertained.