The Taylor system has existed for over a century (Taylor, 1947), and the progress of GDP expansion has posed a challenge to the productivity limits of sequential conveyor flow. That is, the division and subdivision, of labor and small-lot Q to pursue Q → 1 since the time of Adam Smith has resulted in average minimization of delay (waste); however, this has simultaneously been hampered by the barrier of increased variability and the post-GDP problem. Therefore, a shift from lot Q to lot Q < 1 is required to break this barrier (Matsui, 2018, 2019).

This requires the division of labor in response to integration, and the harmonic balancing of sublation, which can only be achieved through the parallel ordered entry (OE) sequence equation (Matsui & Fukuta, 1977) shown in Figure 1, by following the loop conveyor type. This problem was first identified during a queueing cycle analysis of unloading delay (the CSPS model) (Matsui, 2005), which formed the inspiration for the author’s book on probability management (Matsui, 2009). Figure 1 depicts unloading and loading sides moving on a loop conveyor on the ground.

The loop conveyor, consisting of unloading and loading sides, usually rotates at a constant rate λ (>0). Loading delay is given by λD = 1 – ρ + η (η: mean number of follows) at Matsui’s “muda” law (Matsui, 2005). Here, production rate r is the reciprocal of cycle time Z, which is the sum of work time X and delay time D. Together with overflow rate v, this is given by:

r = 1 / Z and v = λ − r

The world on the ground is below the moving loop conveyor, and the operator, moving with acceleration, is stationary on the conveyor, a visualized world with a relativity bias when looking down on the ground, and at that point pursues subordinate balancing. A parallel-ordered entry-row division of labor in response to integration, and a harmonic balancing of sublation can produce a win-win & sharing world.

It is preferable to separate (bypass) going and returning to visualize the bidirectional movement between isolated islands. The loop-conveyor is a well-known systems theory in manufacturing, and based on this, we will proceed with a model example of “stationary in motion” (Mori, 1947). Loop-type flows are characterized by a parallel ordered-entry flow format rather than a linear format.

The definition of processing rate P and overflow rate B (Matsui et al., 1977; Matsui, 1982, 2011) is useful in the placement of the usables that the work targets when representing the flow. Figure 2 shows a closed-loop conveyor that repeats the unloading and loading on a conveyor flowing in one direction: if the balance between outflow (the former) and inflow (the latter) is maintained (e.g., synchronization), this forms a kind of circulating sustainable system.

Figure 2 shows the core of the pair-hierarchy (Matsui, 2020) where the artifact operators are located at the upper level (link) under profit maximization (revenue maximization, cost minimization) (Taylor, 1947). Figure 2 demonstrates that the loop conveyor on the ground is moving and its behavior depicts an unloading/loading waveform cycle.

The world on the ground is below the moving loop conveyor, and the operator, moving with acceleration, is stationary on the conveyor, a visualized world with a relativity bias when looking down on the ground, and at that point pursues subordinate balancing.

Each unloading r_i and loading r_j, usually consisting of one or more work stations of number n, are staggered, not equal, and are successively ordered entries, and the balancing mechanism of their outflow rate, r_i, versus inflow rate, r_j, is generally dynamic. The job at this point is not a sum of tasks, but a world of products, where the win-win & balancing (Matsui, 2018) solutions (equilibrium solution) are present. This issue can be discussed using two-level optimization in OR.

The flow time of the loop-conveyor in Figure 2 is similar to the job shop flow

shop type, and can be formulated using Matsui’s equation. Figure 2 is divided into unloading (a) and loading (b) sides, and modeled separately before formulation, as shown in Figure 3.

Let us consider Table 1 as an example flow shop problem (Job, n = 6) to present the formulation in the next section. Shop time F(W) in this case is obtained using the following equation of F W i , i = 1 , 2 , ⋯ , n (Conway et al., 1967):

F ( W ) = ∑ F W i .

Here, F W i is the job completion time, corresponding to W (=ZL) (Matsui, 2005) in Matsui’s formula.

*SPT: Shortest process time; **LPT: Latest process time.

Matsui’s equation is a determinant consisting of ki (Introduction), sho (Development), ten (transformation), ketsu (Conclusion), B (balance), and G (goal) (Matsui, 2014, 2019; Matsui et al., 2018). Loop-conveyors generally consist of two types of repetitions, with the unloading side corresponding to SPT rules, and the loading side corresponding to LPT rules. Case (a) and case (b) are presented in contrast using the difference between ten (T_a) and ten (T_b).

Figure 4 shows a sample formulation for Table 1, where the unloading side is given by Case (a) and the loading side by Case (b). The usables available for both cases are different, i.e., the usable in Case (a) and Case (b) are the “real” unit to be unloaded and the empty (hole) for loading, respectively.

The mathematical formulation of the loop conveyor is given in Figure 4 as a two-level optimization problem (Matsui, 2020; Conway et al., 1967) in OR terms using Matsui’s equation for the flow problem in the system. This unloading (U)-side formulation has been considered diagrammatically as a trickle-down type benefit-sharing problem in terms of the win-win & sharing problem in previous literature (Matsui, 2018; Matsui, 2020).

We aim for a harmonic (average) win-win and a harmonic shared balancing between layers, for the upper and lower problems, respectively. Here, the formulation for the unloading (U) and loading (L) sides are drawn using scheduling theory, where the two are considered formally separate and distinct. Additionally, they are drawn graphically and numerically, generally with a convex curve in Case (a) and a concave curve in Case (b).

There are limitations to the use of a solvable approach (Balakrishnan, 1972; Aiyoshi & Shimizu, 1981) for finding a solution to the two-level optimization example presented in the previous section; however, the numerical approach (Matsui, 2022) yields Figure 5 and Table 1. Figure 5 demonstrates that the solution is characterized by a type of Carnot-loop behavior, as the income distribution and growth problem can be thought of as a kind of GDP engine. This problem is analogous to the Carnot loop problem in thermodynamics, and is also a dissection of elliptic theory in pair maps.

Drawing on Figure 5, it can be surmised from Table 2 that win-win & sharing yields a harmonious world. Win-win is almost equally X = Y = Z in the Upper-, Middle-, and Lower-layer segments for both Push and Pull. However, for Sharing, the left (A) and right (B) vary in size on the Push side, but are equal on the pull side, with sharing maximized by balance.