| 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 |

${\eta}_{fcost}$ : | operational cost of production line $m$ at factory |

${\eta}_{cap}$ : | production line |

${S}_{cap}$ : | warehouse |

$\gamma $ : | fuel price in litres. |

${\rho}_{v}$ : | the number of kilometers to be travelled by vehicle |

${\beta}_{v}$ : | vehicle |

${\nu}_{cap}$ : | capacities of vehicles. |

${\zeta}_{fd}$ : | distance in km between factory |

${\u03f5}_{fd}$ : | fixed driver cost working between factory |

${\varphi}_{fd}$ : | driver’s additional cost from factory |

| Variables |

${\nu}_{fd}$ : | vehicle |

${\sigma}_{act}$ : | production line |

${A}_{prod}$ : | Number of products |

${A}_{tfw}$ : | Number of products |

${A}_{twd}$ : | Number of products |

${A}_{ware}$ : | total products |

VF: | number of each vehicles |

${N}_{vfd}$ : | number of vehicles |

${T}_{vfd}$ : | total number of vehicle |

${\alpha}_{fd}$ : | driver total cost from factory |