## 2 Problem Description
### Basic Logistic Problem  
The model of a basic logistic problem (BLP) focuses on the minimisation of the cost function, which is related to the vehicles and the route they travel. The following sets are used in the problem:  
* vertices;
* arcs;
* streams;
* vehicle types;
* dimensions;
* logistic functions;
* logistics function performing activities at a node in the logistics network such as receiving, unloading, loading and dispatch of postal goods;
* logistics function performing activities at a node in the logistics network such as processing and sorting postal goods;
* logistics function that does not perform any activity in a logistics network node;  
The following objective cost function is optimised using an MIP algorithm subject to the constraints listed below.  
Objective cost function:  
\[\min\sum_{t\in\mathcal{T}}\sum_{a\in\mathcal{A}}y_{ta}\xi(t,a) \tag{1}\]  
Constraints:  
\[\sum_{a\in\delta^{+}(v)}x_{sadf}-\sum_{a\in\delta^{-}(v)}x_{sadf}=z_{svm(d)f}-z_ {s\text{s}df}+w_{svdfn(f)}-w_{s\text{s}df} \tag{2}\]  
\[\forall s\in\mathcal{S},\ \forall v\in\mathcal{V},\ \forall d\in\mathcal{D} \setminus\{DEL\}:d\geq d(s),\ \forall f\in\mathcal{F}\]  
\[\sum_{s\in\mathcal{S}}\sum_{d\in\mathcal{D}}\sum_{f\in\mathcal{F}}x_{sadf}l(s, m)\leq\sum_{t\in\mathcal{T}}y_{ta}c(t,m)\quad\forall a\in\mathcal{A},\ \forall m\in\mathcal{M} \tag{3}\]  
\[w_{s(s)d(s)NNEN}=1\quad\forall s\in\mathcal{S} \tag{4}\]  
\[z_{s\text{v}DELf}=0\quad\forall s\in\mathcal{S},\ \forall v\in\mathcal{V} \setminus\{b(s)\},\ \forall f\in\mathcal{F} \tag{5}\]  
\[z_{sb(s)DELf}=0\quad\forall s\in\mathcal{S},\ \forall f\in\mathcal{F}\setminus\{PSN\} \tag{6}\]  
\[y_{ta}\in\mathbb{Z}^{+}\quad\forall t\in\mathcal{T},\ \forall a\in\mathcal{A} \tag{7}\]  
\[x_{sadf}\in\mathbb{Z}^{+}\quad\forall s\in\mathcal{S},\ \forall a\in \mathcal{A},\ \forall d\in\mathcal{D},\ \forall f\in\mathcal{F} \tag{8}\]  
\[z_{s\text{s}df}\in\mathbb{Z}^{+}\quad\forall s\in\mathcal{S},\ \forall v\in \mathcal{V},\ \forall d\in\mathcal{D},\ \forall f\in\mathcal{F} \tag{9}\]  
\[w_{s\text{s}df}\in\mathbb{Z}^{+}\quad\forall s\in\mathcal{S},\ \forall v\in \mathcal{V},\ \forall d\in\mathcal{D},\ \forall f\in\mathcal{F}. \tag{10}\]  
Objective function (1) serves to minimise the total travel cost. Constraints (2) are the flow conservation constraints. On the left-hand side, we group the traffic entering and leaving a vertex, while on the right-hand side, we group the traffic that either becomes or ceases to be the considered traffic with given \(d\) and \(f\) by either being kept in a node (\(d\) is changing) or being subjected to a logistic function (\(f\) is changing). Constraints (3) assure that the adequate number of vehicles is provided on each arc of the network. Constraints (4) assure that each demand enters a network, while constraints (5) and (6) assure that each flow entering a network can leave it only with its destination vertex being completely sorted. In other words, (5) assures it cannot leave a network in a vertex which is not its destination vertex, while (6) assures that it cannot leave a network before being subjected to \(PSN\) logical function. Constraints (7)-(10) describe the admissible domain of the sets of variables used.