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:

1

TLCi/mi=21/2mi-2


z„ I               -Д ( Ia

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

(13)


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 :

1

δ( TL Ci / mi)  a [I(C + b di) + ʌl


δdi


1

21/2m1/2(Si+ a di)2


Ib (Si+ a di)2

21/2m1i/2[I(ci+bdi)+Si]21

(14)


and:

2

δ( TLCi/ mi )


δdi2


a2 22 [I(Ci + b di) + Si]2

3
. 1/2zo      ,5

4mi (Si+a di)2


1

Ia b 22

1/2                      1

4mi (Si+ adi)[I(ci+bdi)+


Ia b 2

1

4mi (Si+adi)2[I(ci+bdi)+Si]
21                     1

b2 I 22 (Si+a di)2


1-

Si]2


3

4mi [I(ci+bdi)+Si]2


(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 b2 (Si + a di)2               2Ia b

3                         3         >                                          1

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