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Since Industrial Revolution and the division of labor, the wealth of nations is more and more increasing. Although the smaller division of labor is better in GDP, but mass production with larger lot (quantity) is better since Fordism and the division of quantity (lot-size Q, 0< Q < ∞ ) is well-known to be worse in manufacturing, for the sake of the larger set-up with penalty. This paper presents the progressive discipline for the contradiction on modern economic growth in the lot-sizing scheme. The theory would govern over from mass-production (larger Q), mid-lot (EOQ) and disparities (smaller Q), toward next to sharing equilibrium (0<Q<1). Especially, the Nash’s condition for the case of Q < 1 could be obtained by the duality of flow line vs. job shop. This theoretical review would give the further wealthy development to the gap-wider society of artifacts in the future, and point out that the shared society too could be balanced on the base of the harmonic mean under Industrial-financial capitalism.

Since 18-century, the modern economic growth in industry was seen and found first in the European countries by S. Kuznets [

By Industrial Revolution and the division of labor, the wealth of nations is more and more increasing [

However, it is remarked that the modern economic growth is now progressive under the smaller lot-size ( 0 < Q < ∞ ). Generally, the smaller lot-size accompanies with the negative effect of the larger set-up costs under EOQ (Economic order quantity) [

This paper presents a theory on modern economic growth from Mass production (larger Q), Middle class (EOQ) and Gap-wider stage (smaller Q), toward Sharing stage ( 0 < Q < 1 ). Because, the recent economic society promotes the smaller lot-size by the more speed of demand-to-supply, but the wealth of nations is more and more increasing [

Especially, it is noted that the shared equilibrium of Nash’s type becomes the stability solution with the duality of flowline vs. job shop in the gap-wider stage of wealth [

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 [

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) [

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

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 [

For the class of multiple item, the ordered-entry (OE) type of flow-line system is here introduced, and is showed in

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

(m) and delay-time (D) [

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 [

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 [

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 [

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 [

Now, the Pareto-like graph of income (

Then, the following proposition is given:

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

class (i) | processing time (x_{i}) |
---|---|

A | 7 |

B | 6 |

C | 4 |

D | 3 |

E | 2 |

F | 1 |

class (i) | processing time (i) | |
---|---|---|

line M 1 ( x 1 i ) | line M 2 ( x 2 i ) | |

1 | 3 | 2 |

2 | 1 | 6 |

3 | 8 | 7 |

4 | 4 | 6 |

5 | 11 | 4 |

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,

That is, the arrangement with LPT rule is better in

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

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

under the concave curve in

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).

The modern society faces the skewness of autonomous economic balancing by the more speed of demand-to-supply in global world. This problem is here regarded as that of smaller lot-size, Q → 0 , and is treated by Matsui’s equation and classical inequality at the lot-sizing class of multiple items in manufacturing.

This theory relates to the artifacts of economic body, and results in the advance of factory science in the multi-body with heterogeneity [

The furthermore problems would be the generalization of short proof ( n = 2 ) and its theory on economic ethics, although this former thing could be easily derived from the generality of classical inequality in mean. Also, the advanced theory on the closed OE system would be hoped in not only the coming recycle world, but also Adam-like invisible balancing at the shared society under a closed earth.

Matsui, M. (2018) A Theory of Modern Economic Growth toward Sharing Society. Theoretical Economics Letters, 8, 675-684. https://doi.org/10.4236/tel.2018.84045

This appendix gives the base on the scale change from the quantity to rate. For the purpose, Matsui’s equation: W = Z L and the probability of processing, P ( 0 < P ≤ 1 ) are here introduced.

Now, let us use the rate, x, by replace of P as follows: P = X / Z = L / Z = x ( 0 < x ≤ 1 ) , in which X = m . Then, the function, y, is defined by y = x in

Also, let us consider the functions of quantity, Q, as Z ( Q ) , L ( Q ) and W ( Q ) . When

Q → 0 ( Z → ∞ ) , x = P = L / Z → 0 and W → 0 . Also, when Q = Q ( Z = L ) , x − L / Z = 1 .

Probably, if the function, Z, is derivative in Q and monotone decreasing, then Z ″ < 0 . That is, W would be the concave function of Q in