Index sets

$f_1,\cdots ,F$ :

set of factories.

$p_1,\cdots ,P$ :

set of all different brands.

$d_1,\cdots ,D$ :

set of all distributors.

$w_1,\cdots ,W$ :

set of all internal warehouses.

$m_1,\cdots ,M$ :

set of production lines at various factories.

$v_1,\cdots ,V$ :

set of all vehicles in the transportation fleet.

$t_0,\cdots ,T$ :

time frame considered.

Parameters

$\delta$ :

amount of product p demanded by distributors d at time t.

${\eta }_{fcost}$ :

operational cost of production line $m$ at factory f.

${\eta }_{cap}$ :

production line m maximum production capacity of product p at factory f.

$S_{cap}$ :

warehouse w capacity.

$\gamma$ :

fuel price in litres.

${\rho }_v$ :

the number of kilometers to be travelled by vehicle v per litre of fuel.

${\beta }_v$ :

vehicle v maintenance cost per km.

${\nu }_{cap}$ :

capacities of vehicles.

${\zeta }_{fd}$ :

distance in km between factory f and distributors d.

${ϵ}_{fd}$ :

fixed driver cost working between factory f and distributors d.

${\varphi }_{fd}$ :

driver’s additional cost from factory f to distributors d.

Variables

${\nu }_{fd}$ :

vehicle v operational cost from factory f to distributors d.

${\sigma }_{act}$ :

production line m activated at factory f at time t.

$A_{prod}$ :

Number of products p produced by production line m in factory f at time t.

$A_{tfw}$ :

Number of products p at the warehouse w at time t.

$A_{twd}$ :

Number of products p transported from factory warehouse using vehicle v to distributors d at time t.

$A_{ware}$ :

total products p in warehouse w at time t.

VF:

number of each vehicles v used at factory at time t.

$N_{vfd}$ :

number of vehicles v used to transport products at time t.

$T_{vfd}$ :

total number of vehicle v used to transport products.

${\alpha }_{fd}$ :

driver total cost from factory f to distributors d.