5-5. The Reason of Setting the 1st and the 2nd Kinds of Manufacturing Overhead Applied
If standard burden rate is used in general cases, the managed-gross-profit chart by only use of manufacturing overhead applied AI is recommended, but there is a case in which it is better to set AII as well as AI in practical accounting. The overhead applied AII has the characteristics such as that it will not increase over a limited amount of AII if it reaches to the limited amount to be allocated. The property of AII is close to fixed costs at the year-end.
If we set such AII as mentioned above as the 1st kind, the slope of line 1 in Fig. 3 will change in monthly processes and the form of line 1 is unstable, though it passes　through the point A. This is my comment after I applied the managed-gross-profit chart to a profit management of my company.
5-6.  Another Type of Profit Chart
A profit chart, which is different from the managed-gross-profit chart, is presented in reference (1). This will be explained below.
Substituting Eq.(8) into Eq. (17) gives:

 P (ε) = QM (ε) + AXI (ρ) +∆X tan αXI (ρ) +∆ RXI (ρ, ε) + AXII (ε) - C I, II (ε) - η (ε) I, II - G (ε)
            =[QM (ε) - E (ρ, ε)] + ∆X tan αXI (ρ) (35)
where

E (ρ, ε) = (C I (ε) - AXI (ρ) - ∆ RXI (ρ, ε) ) + ( C II (ε) - AXII (ε) ) +G (ε)

 (36)
If we define notations, in Eq. (33), as follows:

P1 (ε) = QM (ε) - E (ρ, ε)

 (37)

                                         P2 (ε) = ∆X· tan αXI (ρ)

 (38)
we obtain P (ε) = P1 (ε) + P2 (ε).
If we put ε-data = ρ-data, The following equations by ρ-data are obtained:

P1 (ρ) = QM (ρ) - E (ρ)

 (39)
                                          P2 (ρ) = 0  (40)
where

E (ρ) = (C I (ρ) - AXI (ρ)) + ( C II (ρ) - AXII (ρ) ) +G (ρ)

 (41)
Suppose that there is a forecasted income statement, at the beginning of a year, which is assumed to be ρ-data. The second forecasted income statement at an interim or the final statement is assumed to be ε-data.
Two 45°line-coordinate systems (each is Cartesian plane with an added  line of slope 45°passing through the origin) are prepared as shown in Fig. 4(a) and Fig. 4(b).
I n Fig. 4(a), the horizontal axis is QM axis, and the vertical axis is cost axis. If we graduate, in the figure, the points of QM (ρ) and QM (ε) on the horizontal axis, the points of E (ρ) and E (ρ, ε) on the vertical axis respectively, and draw the line 1 and the line 2 through these points parallel to the horizontal axis, P1 (ε) can be expressed as the difference between the 45° line and the line 2 over QM (ε) as shown in that figure.
In Fig. (4b), the graduation of the vertical axis is the same of Fig. (4a). The horizontal axis is sales X with a proper scale and a proper interval of sales (the origin is not required to be placed). On the horizontal axis, the points of QM (ρ) and QM (ε) are placed. On the vertical axis the points of E (ρ, ε) and E (ρ) are placed, and the line 3 and the line 5 are drawn parallel to the horizontal axis. Beforehand the slope of tan αXI (ρ) is calculated by Eq. (9), and the line 4 is drawn through the point (X (ρ), E (ρ)). In addition, the line 6 through the point (X (ρ), E (ρ, ε)) is drawn parallel to the line 4. The profit P2 (ε) is expressed as the difference between the line 5 and the line 6 in the same figure.
If we superpose the two figures and look at it as one chart, we obtain one profit chart with two different scales of abscissas. This chart (superposed chart) was produced by my first idea, but it could not be used in practical profit management by me.