# Problem Description

## Collection Problem

$N_0$ (blood center)
$n$ (number of blood collection sites)
$N_i : i \in [1..n]$ (blood collection sites)
$Q_i$ (stock of bloods in collection site $N_i$)
$C_{ij} : i \ne j$ (time traveling between nodes $N_i$ and $N_j$)
$T_i$ (service time to collect blood from $N_i$)

$c'$ (maximum capacity of vehicle)
$c$ (minimum capacity of vehicle)
$t'$ (maximum time of vehicle for each collection routes)
$t$ (minimum time of vehicle for each collection routes)

$x_{i,j} = [1,0]$ (1 means vehicle directly travel from $N_i$ to $N_j$)
$X_{k} \in \{x_{i,j} \ : \ i,j \in [0..n] \}$ (collection of routes that starts from blood center and ends to blood center)

- Objective:
- Minimize overall time required to collect bloods from collection sites into blood center
- $\sum\limits_{i,j \in [0..n]} x_{ij} (C_{ij} + T_j)$
- Constraints:
- vehicle capacity constraint
- $\sum x_{ij} \ Q_j \ge c : x_{i,j} \in X_k \ \forall k$ (minimum)
- $\sum x_{ij} \ Q_j \le c' : x_{i,j} \in X_k \ \forall k$ (maximum)
- route time limit constraint
- $\sum x_{ij} (A_{ij} + T_j) \ge t : x_{i,j} \in X_k \ \forall k$ (minimum)
- $\sum x_{ij} (A_{ij} + T_j) \le t' : x_{i,j} \in X_k \ \forall k$ (maximum)
- each blood collection sites is visited once
- $\sum\limits_{i \in [0..n]} x_{ij} = 1 \ : \ \forall j \in [0..n]$
- $(i = j) \Rightarrow (x_{ij} = 0) \ : \ \forall i,j \in [0..n]$
- each vehicle starts from blood center and ends to blood center
- $\forall x_{i,j} \in X_k : \exists x_{0,j} \in X_k$
- $\forall x_{i,j} \in X_k : \exists x_{i,0} \in X_k$
- $\forall x_{i,j} \in X_k : \exists x_{j,h} \in X_k$

- Outputs:
- $k$ number of required routes ($X_k$) to collect all bloods from collection sites into blood center by minimizing overall traveling time. This value translates into decision of allocating the number of fleet to complete the task.
- suggested route $x \in X$
- Assumptions:
- There is no capacity limit of blood center. Thus all bloods from collection sites are transported to blood center
- Blood products are indifferentiable. Means that $Q_i$ and $c$ and $c'$ has single unit.

## Delivery Problem

*symmetric with collection?*

- Assumptions:
- Blood center has unlimited supply.
- Blood products are indifferentiable.