m

Origins’ number

$\tilde{n}$

Destinations’ number

$I=\left\{1,2,\cdots ,m\right\}$

Origins’ set

$J=\left\{1,2,\cdots ,\tilde{n}\right\}$

Destinations’ set

i

Origin index, $i\in I$

j

Destination index, $j\in J$

k

Origin index or destination index, $k\in I$ or $k\in J$

$s_i^o$

Normal supply at origin $i\in I$

$d_j^o$

Normal demand at destination $j\in J$

$s_i$

Sudden increase supply at origin $i\in I$

$d_j$

Sudden increase demand at destination $j\in J$

$u_{ij}$

Shipping capacity of route from origin $i\in I$ to destination $j\in J$

${u^{\prime }}_{ij}$

Sudden increase shipping capacity of route from origin $i\in I$ to destination $j\in J$

T

Specified limited time

$t_{ij}$

Shipping time of route from origin $i\in I$ to destination $j\in J$

$I\times J=\left\{\left(i,j\right)|i\in I,j\in J\right\}$

Route set, i.e. the set for all routes from origin $i\in I$ to destination $j\in J$

$b_i^o$

Expansion cost per unit for the origin $i\in I$

$b_j^d$

Expansion cost per unit for the destination $j\in J$

$b_{ij}^e$

Expansion cost per unit for the route from origin $i\in I$ to destination $j\in J$

$v_i^s$

Operating speed of origin $i\in I$

$v_j^d$

Operating speed of destination $j\in J$

$v_{ij}^e$

Empty vehicle running speed of the route from origin $i\in I$ to destination $j\in J$

$d{t}_{ij}$

Distance of the route from origin $i\in I$ to destination $j\in J$

$x_{ij}^e$

Maximum transportation quantity of the route from origin $i\in I$ to destination $j\in J$ within the time-limited T

$x_{ij}$

Optimum shipping quantity of route from origin $i\in I$ to destination $j\in J$

$r_i^s$

Needed capacity expansion quantity for origin $i\in I$

$r_j^d$

Needed capacity expansion quantity for destination $j\in J$

$r_{ij}$

Needed capacity expansion quantity for the route from origin $i\in I$ to destination $j\in J$

z; z'

MCCETLTPD ’s optimum objective value, i.e. the minimum cost

$D^o={\left(d_j^o\right)}_{1\times \tilde{n}}$ ; $D={\left(d_j\right)}_{1\times \tilde{n}}$

Destination demand vector

$S^o={\left(s_i^o\right)}_{1\times m}$ ; $S={\left(s_i\right)}_{1\times m}$

Origin supply vector

$U={\left(u_{ij}\right)}_{m\times \tilde{n}}$

Shipping capacity matrix for route from origin $i\in I$ to destination $j\in J$

$Bs={\left(b_i^o\right)}_{1\times m}$

Expansion cost per unit vector for origin $i\in I$

$Bd={\left(b_j^d\right)}_{1\times \tilde{n}}$

Expansion cost per unit vector for destination $j\in J$

$Be={\left(b_{ij}^e\right)}_{m\times \tilde{n}}$

Expansion cost per unit matrix for route from origin $i\in I$ to destination $j\in J$

$Vs={\left(v_i^s\right)}_{1\times m}$

Operating speed vector of origin $i\in I$

$Vd={\left(v_j^d\right)}_{1\times \tilde{n}}$

Operating speed vector of destination $j\in J$

$Ve={\left(v_{ij}^e\right)}_{m\times \tilde{n}}$

Empty vehicle running speed matrix of route from origin $i\in I$ to destination $j\in J$

$Dt={\left(d{t}_{ij}\right)}_{m\times \tilde{n}}$

Shipping distance matrices of route from origin $i\in I$ to destination $j\in J$