In the world of global management, the division and span of control would be important. Since the Industrial Revolution and the division of labor, the problem and principle in the manufacturing are discovered in the beginning of 20 century, and developing at the scientific and economic theory [4] .

Generally, the production cost of lot-size (Q), T ( Q ) , is given by the sum of set-up, Z ( Q ) , and holding, L ( Q ) , costs as follows:

T ( Q ) = Z ( Q ) + L ( Q ) ,     0 < Q < ∞ . (2.1)

The optimization of the total costs (2.1) is easily obtained, and the solution is called the economic order quantity (EOQ) [8] .

Then, let us define the objective: T ( Q ) / 2 by U ( Q ) . From the Equation (2.1) and classical inequality, the following relation are then well-known:

u ( θ ) = { Z ( Q ) + L ( Q ) } / 2 ≥ Z ( Q ) L ( Q ) = W ( Q ) , (2.2)

by using Matsui’s equation: W = Z L , too.

If Z ( Q ) = L ( Q ) , the quantity Q * is become the EOQ formula under some demand (D), and the optimal cost is then given by 2U( Q * ) at equality: Z = L ( = W ) . The outline of the economic quantity, Q * , and the relations (2.2) are showed for the larger value (wealth) instead of costs, seen at Figure 1. For Figure 1, it is noted that the Z and L are regarded as the income (revenue) and number of peoples (here, fixed) in the system.

This paper aims the balancing issues at the maximization of total value (wealth) with respect to Q by the objective skewness as follows:

objective skewness:

E R ( income , Z ) × L T ( pepoles , L ) − { E R ( income , Z ) + L T ( peoples , L ) } / 2 → M a x ( Q ) , (2.3)

where ER and LT means the revenue (economics) and lead time (reliability), respectively.

Especially when ER is equal to LT, the system would attain the marginal value, and its value becomes maximal at profit. This principle could be guessed by the findings in pair-map method [9] .

For the class of multiple item, the ordered-entry (OE) type of flow-line system is here introduced, and is showed in Figure 2 [9] . The flow-line system of OE type consists of the customers (input) with income (Z) and arrival rate ( λ ), and the line with the processing stations of n ( > 1 ) and the respective processing times, m i ( 0 < m i < 1 ) , i = 1 , ⋯ , n .

Generally, the cycle time (Z) is given by the sum of the mean processing time

(m) and delay-time (D) [10] . If the system has the lost units (overflows), the cycle time, Z, is then become Z = d ( = 1 / λ ) in the queueing theory with no loss (d) [9] . That is, the cycle time, Z ( d ) , is written as follows:

Z ; d = m + D ,     d > 0 (2.4)

where d is called the mean inter-arrival (namely, inter-departure) time.

Also, the input of arrivals is assumed to have the income, ER, and the ER is given by the sum of mean operating cost (expense), EC, and benefit, EN. Then, the cycle time, Z, is as follows:

Z ; E R = E C + E N , (2.5)

similar to the Equation (2.4).

From Matsui’s queueing theory [10] , the probability of loss, B, is defined by B = m / Z , and, from the equation (2.5), another definition is given by

B = E C / E R ,     0 < B < 1 P = E N / E R ,     0 < P < 1 (2.6)

where P + B = 1 .

For the flowline system of OE type, the production rate (r) is defined by r = 1 / Z , and is then formulated as follows [7] :

r = λ { 1 − ∏ B i ( d , c ) } → max d min c (2.7)

where the vector c is the some variable of buffer effect as c = ( c 1 , c 2 , ⋯ , c i , ⋯ , c n ) , in which i means the time-range of look-ahead type (time buffer) on the flow line [11] .

Then, the optimality condition for the Equation (2.7) is obtained as the equilibrium solution:

( d * , c * ) such that B 1 = B 2 = ⋯ = B n , (2.8)

in the balancing of rates (See Appendix).

For the optimal vector, c, the monotonicity:

c 1 , c 2 , ⋯ , c i , ⋯ , c n ,     0 < c i < ∞ (2.9)

is also seen, and satisfies the following relation:

λ c i = λ i + 1 c i + 1 ,     i = 1 , 2 , ⋯ , n − 1. (2.10)

For the engine of higher growth, it is well-known that the mass production has results in the larger GDP since Ford system. This engine is the movement of larger lot-size with Q → ∞ , the modern economic growth is obtained in many developing countries, beginning at the advanced nations.

Now, let us consider the quantity, Q, as the input size. That is, the demand speed ( λ ) is regarded as λ = Q . For multiple items (classes), the lot-sizing issues could be classified as the three stages of economic growth by the larger ( Q → ∞ ), middle (EOQ), and smaller ( Q → 1 ).

Then, the following proposition is given:

Proposition 1. For 1 < Q < ∞ , the middle society with EOQ is called the Nash’s equilibrium.

line [ mass   production ] Q → ∞ ( Z > L ) ≤ lot [ middle   class ] EOQ ( Z = L ) ≤ ikko-nagashi [ gap-wider ] Q → 1 ( Z < L ) (3.1)

By the Proposition 1, the top of country rich could be seen at the EOQ age. Following the lot-size, Q, toward Q → 1 , the degree of demand-to-supply speed becomes promoted and visualized. At the smaller lot-size ( Q → 1 ), the squeal of flow line becomes larger, and grows the gap-wider society of wealth.

At this stage, the present society might not be possible to catch up the global demand speed, and be being retarded to maintain the autonomous self-balancing by invisible hand. By queueing theory, it is well-known that, when the traffic intensity, ρ ( = m / d ) , is approaching to Q → 1 , the state of system could be become unstable.

For the case of twin item, Proposition 1 could be returned to the following problem of inequality:

Formulation 1:

( Z 1 L 1 + Z 2 L 2 2 ) Z > L ≤ { Z L } ∋ ( Z = L ) ≥ ( Z 1 L 1 + Z 2 L 2 2 ) Z < L . (3.2)

where the pair ( Z , L ) is an equilibrium solution.

Short proof: The proof is divided by the following two cases:

i) Z ≠ L   ( Z 1 L 1 ≠ Z 2 L 2 ) case

{ ( Z 1 L 1 + Z 2 L 2 ) / 2 } 2 − { ( Z 1 L 1 ) ( Z 2 L 2 ) } 2

⇒ ( Z 1 L 1 ) 2 + ( Z 2 L 2 ) 2 − 2 ( Z 1 L 1 ) ( Z 2 L 2 )         = { ( Z 1 L 1 ) 2 − ( Z 2 L 2 ) 2 } 2 ≥ 0. (3.3)

Also,

( Z L ) 2 − ( Z 1 L 1 ) ( Z 2 L 2 ) ≥ 0.

when

Z L > Z i L i ( Z i ≠ L i ) . (3.4)

ii) Z = L ( Z i < L i   or   Z i > L i ) case

Z 2 ( = L 2 ) − ( Z 1 L 1 + Z 2 L 2 ) / 2 = ( Z 2 + L 2 ) / 2 − ( Z 1 L 1 + Z 2 L 2 ) / 2 ≥ 1 2 { ( Z 2 + L 2 ) − 2 Z L } = ( Z − L ) 2 / 2 ≥ 0 , (3.5)

where Z L > Z i L i when Z = L .

Thus, the cases of Z ≠ L and Z = L complete the proof (3.2).

When the lot-size, Q , Q < 1 , the flow line of OE type is the useful scheme where the income could be regarded as the input to the system. Then, it is noted that this problem could be equivalent to the job shop scheduling problem of sequencing type with LPT (latest processing time) rule [6] , if the number of stations in flow line is corresponded to the number of jobs in job shop.

Now, the Pareto-like graph of income (Figure 3) could be considered under the sequencing problem of n / 1 / F ¯ (mean flow time) type (Table 1 [6] ). Similar to Table 1, the Pareto-like graph (Figure 4) is considered under the sequencing problem of n / 2 / F max (maximum flow time) type (Table 2 [12] ).

Then, the following proposition is given:

Proposition 2. The solution of shared-balancing under Q < 1 is a kind of

Nash’s equilibrium ( d , c ) and satisfies the condition: B 1 = B 2 = ⋯ = B n . In the objective (2.6), when λ → E R , the following formulation is obtained:

Formulation 2. The equilibrium problem of shared-balancing under Q < 1 is formulated as follows:

r = E R { 1 − ∏ B i ( d , c ) } → max E R ( d ) min c (4.1)

subjectto : W ( F Ⅰ ) = L ( F Ⅱ ) × Z ( E R ) . (4.2)

In Formulation 2, the constraints (4.2) means the condition of Matsui’s equation-type, and is related to the equivalence of sharing aspects (set of cells) under vertical vs. horizontal balancing. This is the condition of shared-balancing.

When λ 1 → E R 1 , the optimal (equilibrium) condition (4.1) is as follows:

E R 1 × c 1 = E R 2 × c 2 = ⋯ = E R n × c n , (4.3)

where there might be set to c 1 → E R 1 , c 2 → E R 1 + E R 2 , ⋯ , c n → E R 1 + E R 2 + ⋯ + E R n .

Therefore, from the Equation (4.3), the sequences { E R i } and { c i } could be obtained equivalently as follows:

E R i = ( c 1 / c i ) E R     or     c i = ( E R 1 / E R i ) c 1 ,     i = 1 , 2 , ⋯ , n . (4.4)

The formulation of sequencing problems, Table 1 and Table 2, gave the Pareto-like graph of the income, ER(Z), in Figure 3 and Figure 4, respectively. The former is called the problem of Johnson rule, and the latter is done the problem of Bowl phenomenon. However, the maximum in Pareto-like graph is not the SPT rule or Bowl phenomenon, but the LPT rule.

That is, the arrangement with LPT rule is better in Figure 3 and Figure 4, and the resulting concave curve shows the gap-wider skew of wealth in Pareto analysis. Furthermore, the Figure 3 and Figure 4 show the mechanism of sharing scheme as follows:

Vertical ( E R 1 ) → Horizontal ( E C 1 → E R 2 ) → Vertical ( E R 2 ) → ⋯ (4.5)

where the Equation (4.5) means the chain of sharing.

Thus, the constraints of Table 1 and Table 2 are outlined as follows:

( “ sharing ” vertiacl   sum W ( F I ) ) ↔ equivalent   to ( “ balancing ” horizontal   sum Z ( n ) × L ( F II ) ) (4.6)

under the concave curve in Figure 3 and Figure 4.

On the other hand, the objective (4.1) becomes then 1, because the processing time (x) is x > E R , the probability of processing, P, is P = 1 , and that of loss, B, B = 0 , in no overflow case. That is, this optimal condition shows the synchronization of flow time, and then, the production rat, r, r → E R in the objective (4.1).