Reference Model

In the following you find the complete model. Read closely through the constraints, as we will ask you what each constraint stands for in a quiz after this page.
 
Indices:
 

Set of biomass type: \( b \in B \)

Set of time periods: \(t \in T\) (Time period 1 models 2030 and the last on models 2050)

Set of regions, indices \(i,j,p \in I\) , we use \(i\) to model regions with biomass, \(p\) for regions we are building facilities and \(j\) for the demand.

Decision Variables:

\( \overline{x}_{bipt} \) Transport quantity of biomass \( b \) from region \( i \) to facility \( p \) in period \( t \)

\( x_{pjt} \) Transport quantity of biofuel from facility \( p \) to region \( j \) at time \( t \)

\( y_{pt} \) number of open production facilities \( p \)in period \( t \)

Parameters:

\( a_{bit} \)   maximum available amount of biomass \( b \) in region \( i \) in period \( t \)

\(f_{bit}^{pbm} \) price per unit of biomass \(b\) in region \(i\) at time \(t\)

\( HV^{bm} \) heating value of biomass \( b \)

\( HV^{bf} \) heating value of biofuel

\( c_{ij} \) distance between region \( i \) and \( j \)
\( t^{bf} \) transport cost for biofuel
 
\( t^{bm} \)  transport cost for biomass

\( Conv \)   conversion factor from units of biomass \( b \) to units of biofuel

\( d_{jt} \)  demand in region \( j \) at the time \( t \)

\( f^{fix} \) periodic fixed operating costs for a facility

\( f^{inv} \) construction costs for a facility

\( f^{run} \)  periodic variable operating costs for running a facility

\( s^{max} \) maximum capacity of a production facility
 

\( UT \) uptime per year of a facility

\( LT \) lifetime of a facility

 
Target Function:
 
Minimise costs resulting of purchase and transport of biomass, transport of biofuel as well as operation-independent, operation-dependent and construction costs of the production facilities
 
\[  min \sum_{t \in T}  (PC_t + TC_t + IC_t + OC_t) \]
 
with
 
\[ PC_t = \sum_{b \in B}\sum_{i \in I}\sum_{p \in I}f_{bit}^{pbm}{\overline{x}}_{bipt} \]
 
\[ TC_t = \sum_{b \in B}\sum_{i \in I}\sum_{p \in I} t^{bm} c_{ip}  \dfrac{{\overline{x}}_{bipt}}{HV_{b}} + \sum_{p \in I}\sum_{j \in I} t^{bf} c_{pj} \dfrac{x_{pjt}}{HV^{bf}} \]
 
\[ IC_t = \sum_{p \in I}^{}{(\frac{f^{Inv}}{LT}+f^{fix})y_{pt}} \]
 
\[ OC_t = \sum_{p \in I}\sum_{j \in I} f^{run} x_{pjt} \]
 
 
Constraints:

(1) \[\sum_{p \in I}{\overline{x}}_{bipt} \leq a_{bit}\ , \forall\ b \in B,\ i \in I,\ t \in T \]

(2) \[\sum_{b \in B}\sum_{i \in I}{\overline{x}}_{bipt} - Conv \sum_{j \in I}x_{pjt} \geq 0\ ,  \forall\ p \in I,\ t \in T\]

(3) \[\sum_{p \in I} x_{pjt} \geq d_{jt}\ , \forall\ j \in I,\ t \in T\]

(4) \[s^{max} * UT * y_{pt} \geq \sum_{j \in I} x_{pjt}\ , \forall\ p \in I,\ t \in T\]

(5) \[y_{p,t} \geq y_{p,t - 1}\ , \forall\ p \in I,\ t \in T\]

Non-negativity conditions:

\[\overline{x}_{bipt} \geq 0\ , \forall\ b \in B,\ i \in I,\ p \in I,\ t \in T \]

\[x_{pjt} \geq 0\ , \forall\ p \in I,j \in I,\ t \in T\]

\[y_{pt} \in \mathbb{N}_{0}\ , \forall\ p \in I,\ t \in T\]

 
 
 

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