Optimal Vehicle Size, Haulage Length, and the Structure of Transport Costs

transportation costs expression and then to divide by mito give a cost per ton measure,
thus:

_z„ I               -Д ( Ia

⁽Si + a di)2 [I(Ci + b di) + Si]2 + di — + b

To observe the relationship between total logistics-transportation costs per ton and
haulage distance we take the first and second derivatives of (13) with respect to
haulage distance di :

and:

(15)

Equations (14) and (15) are always positive and negative, respectively; i.e. equation
(13) is strictly concave in distance, if either a or b is zero. However, the general
conditions under which equation (13) is concave in distance is:

a² [I⁽ci + b di) + si J + I² b^{2 (}Si ⁺ a di⁾²               2Ia b

3                         3         >                                          1

^(Si+a d^{i)2 [}I⁽c^{i + b} dⁱ) ⁺ sⁱJ         ⁽Si ⁺ a di⁾²[I(C_i + b d_i) + S_i]²
thus:
a² [I(Ci+bdi)+Si]-2Iab(Si⁺ adi)[I(Ci+bdi)+Si]+I²b² (Si⁺ adi)² > 0

and:

{a [I(Ci+bdi)+Si]-Ib(Si+adi)}^{> 0}

This expression must always be true, except where:

a [I(Ci+bdi)+Si]=Ib(Si+adi)

i.e. when:

16

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