Waste Utilization Problem
This is a Multi-Objective Optimization Problem
- 2 cities $X$ and $Y$ produce garbage
- there are two incinerators $I_1$ and $I_2$ with come capacity
- each incinerator has some cost of utilization per unit of waste
- there are some transportation costs per unit from a city to an incinerator
How much garbage to send from $X$ and $Y$ to $I_1$ and $I_2$?
Example:
- $X$ produces 100 tons of garbage, $Y$ produces 150 tons
- transportation costs: $X \to I_1: 2, X \to I_2: 3; Y \to I_1: 3, Y \to I_2: 4$
- cost of utilization: $I_1: 2, I_2: 1$
- capacities of $I_1$ and $I_2$ are 150
Define the following variables:
- $XI_1, XI_2$ - garbage sent from $X$ to $I_1$ and $I_2$ respectively
- $YI_1, YI_2$ - garbage sent from $Y$ to $I_1$ and $I_2$ respectively
Constraints:
- amount of produced garbage = amount of incinerated garbage
- $XI_1 + XI_2 = 100$
- $YI_1 + YI_2 = 150$
- $I_1$ and $I_2$ have capacity:
- $XI_1 + YI_1 \leqslant 150$
- $XI_2 + YI_2 \leqslant 150$
- all must be positive
- $XI_1, XI_2, YI_1, YI_2 \geqslant 0$
So we have the following objectives:
- we want to minimize the total cost of incineration
- : $z_1 = 2 \cdot (XI_1 + YI_1) + 1 \cdot (XI_2 + YI_2)$
- and we want to minimize the transportation cost
- : $z_2 = 2 \cdot XI_1 + 3 \cdot YI_1 + 3 \cdot XI_2 + 4 \cdot YI_2$
We evaluate all feasible solutions against $z_1$ and $z_2$
- and get something similar to this
- the Pareto-optimal solutions dominate all other feasible solutions
How to select the best one?
- Weighed Sum
- will select only 4 solutions, the rest is ignored
- Ideal Point
- we find the closest point to the ideal
Links
- Routing Optimization for Waste Management http://www.ma.iup.edu/~jchrispe/ORArticles/WasteManagement.pdf
- Ant Colony optimization for Waste Utilization problem: http://thescipub.com/pdf/10.3844/jmssp.2009.199.205