Supplement

5-10 Appendixes

Appendix-1: Derivation of Eq. (34)

Eq. (33) is given by :

f _C(ε) / X (φ) = P (ε) / (X (ε) - X (φ))

(1-1)

where, from Eq. (29),

f _C (ε) =η (ε) ^{I, II} + C^{I, II} (ε) + G (ε)

(1-2)

Eq. (1-1) is changed to be:

f _C (ε) = P (ε) / (X (ε) / X (φ) - 1)

(1-3)

X (ε) / X (φ) = 1+ P (ε) / f_C(ε)

= ( f _C(ε) + P (ε) ) / f _C(ε)

(1-4)

X (φ) / X (ε) = f _C(ε) / ( f _C(ε) + P (ε) )

(1-5)

Eq. (17) is given by:

P (ε) =Q^M(ε) +A^XI(ε) +A^XII(ε) - C^{I, II}(ε) - η (ε) ^{I, II} - G (ε)

(1-6)

From Eq. (1-2) and Eq. (1-6), we have

f _C(ε) + P (ε) = Q^M(ε) +A^XI(ε) +A^XII(ε)

(1-7)

Eq. (31) is given by:

Q^M(ε) = X (ε) - D^X(ε) - A^XI(ε) - A^XII(ε)

(1-8)

From Eq. (1-7) and Eq. (1-8), we obtain

f _C(ε) + P (ε)

= X (ε) - D^X(ε): denominator of Eq. (1-5)

(1-9)

Appendix-2: Derivation of break-even sales equation corresponding to Solomons’s problem when the 1st and the 2nd kinds of manufacturing overhead applied exist

The two equations, which give the break-even point, are the line-1 given by Eq. (2-1) and the line-2 given by Eq. (2-2) in Fig. 3.

Q^M / f (ε) + X / (f (ε) / tan α^XI(ε))=1

(2-1)

Q^M = - A^XII(ε) + tan β (ε) ·X

(2-2)

where

tan α^XI(ε) = (A^XI(ε) - G^V(ε)) / X (ε)

(2-3)

tan β (ε) = ( A^XII(ε) + Q^M(ε)) / X (ε)

(2-4)

f (ε) =f _C(ε) - A^XII(ε)

(2-6)

f _C(ε) =η (ε) ^{I, II} + C^{I, IIF} (ε) + G^F(ε)

(2-7)

Eq. (2-3) has been obtained referring to Eq. (9) in reference (3). In Eq. (2-3) and in Eq. (2-7), the superscripts V and F represent variable cost and fixed cost, respectively. Consequently we have the following equation:

G (ε) = G^F(ε) + G^V(ε)

(2-8)

Eq. (2-1) is changed to be

Q^M(ε) + tan α^XI(ε) X (ε)= f (ε)

(2-9)

At the break-even sales X (φ), Eq. (2-9) and Eq. (2-2) become:

Q^M(φ) + tan α^XI(ε) X (φ)= f (ε)

(2-10)

Q^M(φ) = - A^XII(ε) + tan β (ε) X (φ)

(2-11)

Substituting Eq. (2-11) into Eq. (2-10) gives

− A^XII(ε) + tan β (ε) X (φ) + tan α^XI(ε) X (φ) = f (ε)

(2-12)

Eq. (2-12) is changed to be

( tan α^XI(ε) + tan β (ε) )X (φ) = f (ε) + A^XII(ε)

X (φ) =( f (ε) + A^XII(ε) ) / (tan α^XI(ε) + tan β (ε) )

(2-13)

In Eq. (2-13) , the numerator and the denominator are:

f (ε) + A^XII(ε) = f _C(ε)

= η ^{I, II} (ε) + C^{I, II} (ε) + G^F(ε)

(2-14)

tan α^XI(φ) + tan β (ε)

= (A^XI(ε) - G^V(ε)) / X (ε) + ( A^XII(ε) + Q^M(ε)) / X (ε)

= (A^XI,II(ε) - G^V(ε) + Q^M (ε) ) / X (ε)

(2-15)

From Eq. (31), we have

Q^M(ε) = X (ε) - D^X(ε) - A^XI(ε) - A^XII(ε)

(2-16)

Substituting Eq. (2-16) into Eq. (2-15) gives

tan α^XI(ε) + tan β (ε)

=(X (ε) - D^X(ε) - G^V(ε) ) / X (ε)

(2-17)

Therefore, we have

X (φ) / X (ε) = (η ^{I, II} (ε) + C^{I, II} (ε) + G^F(ε) ) / (X (ε) - D^X(ε) - G^V(ε) )

(2-18)

When C ^{I, II} (ε)= C ^{I, IIF} (ε) + C ^{I, IIV} (ε), by similarity between ∆AHF and ∆DHC, fixed cost terms always go to numerator and variable cost terms always go to denominator in Eq. (2-18).

Appendix-3: Relationship between 45°- break-even chart and managed gross profit chart

When G^V(ε) =0 and α^XI(ε) is denoted as α, the relationship between the 45°- break-even chart and the managed gross profit chart is shown in Fig. 3-1.