Mini-course 1 Decision Analysis (Dr. Mariya Sodenkamp) / Class 4 / Pareto Optimality Power Plant Solution
.pdf
Example: Building a power sta5on
We want to build a hydraulic power sta4on with maximal power output in shortest possible 4me.
The resul4ng power sta4on can have power output in the range between 0 MW and 20’000 MW.
The cost for the power plant can be calculated as C=P/(20’000 MW) * 10 Mio €.
The construc4on of the power plant will take 2 years. We can reduce this 4me by at most 10 months, but reduc4on of the 4me by a month will increase the overall cost by 1 Mio €.
The available budget for the project is 15 Mio €.
What are the Pareto op4mal configura4ons for this power plant?
IT in Business| © Sodenkamp
Building a power plant (Solu5on)
Power Output
0
Step 1. Iden4fy the criteria.
• “We want to build a hydraulic power sta4on with maximal power output in shortest possible 4me.”
• We have two criteria to be maximized: “Power output” and “Reduc4on in construc4on 4me”. Represent them with a graphic.
Step 2. Draw all feasible solu4ons
• All solu4ons with posi4ve power output and and posi4ve reduc4on in construc4on 4me are feasible so far
Reduc.on in construc.on .me
IT in Business| © Sodenkamp
Building a power plant (Solu5on)
Power Output (P)
20’000 MW
0
10 months
Step 3. Add the constraints on the power output
• “The resul4ng power sta4on can have power output (P) in the range between 0 MW and 20’000 MW.”
• 0< P < 20’000
Step 4. Add the constraints on the reduc4on in construc4on 4me (T)
• “We can reduce this 4me by at most 10 months”
• 0 < T < 10
Reduc.on in construc.on .me (T)
IT in Business| © Sodenkamp
Building a power plant (Solu5on)
Power Output |
|
Step 5. Add the constraints on the costs |
||
|
• |
“The available budget for the project is 15 Mio €.” |
||
|
|
|||
|
|
• The total costs can be calculated from the power |
||
15 Mio € |
|
|
output and reduc4on in construc4on 4me: “The cost |
|
|
|
for the power plant can be calculated as C=P/(20’000 |
||
|
|
|
||
|
|
|
MW) * 10 Mio €.” and “… reduc4on of the 4me (T) |
|
|
|
• |
by a month will increase the overall cost by 1 Mio €.” |
|
|
|
C= P/20’000*10’000’000+T*1’000’000 |
||
20’000 |
|
• |
We need to remove all points (T, P) with the costs |
|
MW |
|
|
over the budget. Using this dependency we can |
|
|
|
|
calculate all solu4ons with the cost of 15 Mio € and |
|
|
|
|
remove all points that lie above. This can be done |
|
|
|
|
using one of the following methods: |
|
|
|
|
• |
Calculate the equa4on |
|
|
|
|
P = (15 Mio - T * 1Mio)/10 Mio * 0.02 Mio |
|
|
|
• Find two solu4ons (points) and connect them |
|
0 |
|
|
|
with a line, e.g., points (10 Months, 10’000 |
10 |
Reduc.on in |
|
MW) and (5 Months, 20’000 MW). |
|
|
|
|||
|
months |
construc.on .me |
|
|
IT in Business| © Sodenkamp
Building a power plant (Solu5on)
Power Output
15 Mio €
20’000 MW
6. Find a set of Pareto op4mal solu4ons
• All points that are not dominated by any other point
Pareto Set
0 |
10 |
Reduc.on in |
|
|
|||
|
months |
construc.on .me |
|
IT in Business| © Sodenkamp