1. Multiobjective optimization GAMS-Nimbus integration SUMMARY
Lecture 5
Multiobjective optimization
GAMS-Nimbus integration
SUMMARY
Timo Laukkanen

2. What is multiobjective optimization?
Optimization is the task of finding one or more solutions which correspond to minimizing (or maximizing) one or more objectives and which satifies all constraints
In a single-objective optimization problem there is one objective function (f.ex. Hot Utility Consumption) and a single solution, the optimal solution
In multiobjective optimization the task is to consider simultaneously several conflicting objectives (HU and Hex area). Typically there is no single solution, but a set of alternative mathematically equally good solutions (Pareto optimal solutions or non-dominated solutions)
Timo Laukkanen

3. What is multiobjective optimization?
Although multiple Pareto optimal solutions exist, the Decision Maker (DM) has to choose only one of these solutions as a final solution
In multiobjective optimization there are three equally important tasks:
Make an optimization MODEL that is solvable but still approximizes the reality closely enough
Find = OPTIMIZE all needed Pareto optimal solutions
Choose the single most preferred solution from all Pareto optimal solution i.e. MAKE A DECISION
Timo Laukkanen

4. What is multiobjective optimization?
Timo Laukkanen

5. What is multiobjective optimization?
Minimize {f1(x), f2(x),...,fk(x)}
Subject to x ε S
Involving k (≥ 2) conflicting objective functions fi: Rn R that are minimized simultaneously.
The decision variables x =(x1,x2,...,xn) belong to the nonempty feasible region S. This feasible set is defined by contraint functions.
The image of the feasible region in the objective space is called a feasible objective region Z=f(S)
In multiobjective optimization, objective vectors are regarded as optimal if none of their components can be improved without detoriation of at least one of the other components PARETO OPTIMALITY
Timo Laukkanen

6. What is multiobjective optimization?
When all objectives are minimized (min z= - max z), lower bounds of the Pareto optimal set are available in the ideal objective vector z*. This is obtained by minimizing each objective separatedly.
Upper bounds of the Pareto optimal set are available in the nadir objective vector znad.
Timo Laukkanen

7. Multiobjective optimization methods
BASIC METHODS
Weighting method
ε-constraint method
No-Preference methods
A Posteriori methods
A Priori methods
Interactive methods
Nimbus method
Timo Laukkanen

8. Multiobjective optimization methods
Weighting method
The different objectives are given weights, and the sum of these weighted objectives is minimized
Compare to the basic SYNHEAT-model
Timo Laukkanen

9. Multiobjective optimization methods
Challenge in OPTIMIZATION: Finding all Pareto optimal solutions
In engineering science (also in HENS) the different objectives are typically optimized using the so-called weighting method
The problem is that then in nonconvex problems (like the HENS) all Pareto- optimal solutions can not be found even if the weights are changed

10. Multiobjective optimization methods

11. Multiobjective optimization methods
ε-constraint method
Only one objective is minimized and the other objectives are contraints with varying upper bounds
Timo Laukkanen

12. Multiobjective optimization methods
No-Preference Methods
The preference of the DM is not taken into consideration
The solutions are compromize solutions and are ”in the middle” of the Pareto optimal set
Method of Global Criterion
The distance between some desirable reference point and the feasible objective space is minimized
Neutral Compromize Solution use
Timo Laukkanen

13. Multiobjective optimization methods
A Posteriori Methods
Methods for generating Pareto optimal solutions
All Pareto optimal solutions or a representation of these are generated
So the DM chooses from all Pareto optimal solutions
The computational burden to generate all Pareto optimal solutions can be expensive (ε-constraint method)
Timo Laukkanen

14. Multiobjective optimization methods
A Priori Methods
The DM specifies her/his preference information (for example as opinions to specified questions) before the solution process
Making the final decision can be easier (solutions in the same ”area”)
The DM might not know beforehand what is possible
Timo Laukkanen

15. Multiobjective optimization methods
Interactive methods
A decision maker plays an important role and the idea is to support the DM in searching for the most preferred solution
Steps of an iterative solution algorithm are repeated and the DM provides preference information so that the most preferred solution is found
Learning is important, the DM finds out what is possible
Types
Methods based on trade-off information
Reference point
Classification of objectives
NIMBUS
Timo Laukkanen

16. NIMBUS
In Nimbus (developed by Miettinen and Mäkelä at University of Jyväskylä) the DM classifies objectives into 5 groups
Timo Laukkanen

17. GAMS-NIMBUS integration
GAMS has world-class optimization solvers
GAMS does not have the ability to solve ”truly” multiobjective optimization problems
With the NIMBUS scalarization functions the multiobjective problem can be transfered into a single- objective problem that can find all the Pareto optimal solutions
Timo Laukkanen

18. GAMS-NIMBUS integration
Timo Laukkanen

19. GAMS-NIMBUS integration Nimbus user-interface
Timo Laukkanen

20. GAMS-NIMBUS integration used to solve the SYNHEAT problem
Example:
Stream data for Example 1 taken from Table 1 in Björk and Westerlund (2002a).
Stream Tin (◦C) Tout (◦C) Fcp (kW/K) h (kW/m2K) H H C C
Hot utility
Cold utility
Timo Laukkanen

21. HeVI
A computer software for automatically generating the stream grid from the results of the SYNHEAT-model
Timo Laukkanen

22. Optimal Synthesis and Operation of Utility Plants
Given a set of demands of electricity, mechanical power and steam at different pressure levels, design a utility plant at minimum cost by determining the equipment configuration and operating conditions

23. Superstructure development

24. Heat recovery steam generator (HRSG)

25. High pressure boiler (fuel fired)

26. Medium pressure boiler (fuel fired)

27. Waste heat boiler (medium pressure)

28. Steam generation options

29. High pressure steam turbines

30. Complete superstructure

31. Equations for the Feasible Region
Equations of Change
Mass Balance (continuity)
Momentum (motion)
Energy
Demands
Heating
Electricity
Mechanical Power
Logic
Selections
Conditional Constraints
Economic
Cost Functions
Physical Properties
Enthalpy, Entropy, Steam Quality...

32. Mass balances
Indices, Sets and Variables

33. Mass balances A B D C Indices, Sets and Variables
I(n,m) "input flowrates to units"
/ B .(A,C)
D .(D) /; C
Indices, Sets and Variables

34. Energy balances Indices, Sets and Variables
external power demand (net)
electricity
produced
steam produced
energy with the
flow into the unit
energy with the
flow out of the unit
waste heat duty
energy from
burner
Indices, Sets and Variables
efficiency of combustion

35. Momentum Balances
Indices, Sets and Variables

36. Demands
Heat
Electricity
Mechanical Power
Indices, Sets and Variables

37. Lower and Upper Bounds
non-power generating unit

38. Only one steam turbine for each demand
Turbine units of high pressure ext. turbine

39. COURSE SUMMARY
Process Integration/Heat Exchanger Network Synthesis (HENS) is an important step in process design
Energy saving is very often also economically feasible
Energy saving in industry is a major contributor in CO2 savings in the next 40 years
CC (Composite Curves)
Different temperature cascades for hot and cold streams
Timo Laukkanen

40. SUMMARY Problem Table Algorithm (PTA) Adjust (shift) the temperatures
Find the temperature intervals
Calculate the enthalpy balance for each interval
heat surplus (+) and deficit (-)
Cascade the enthalpy
Add largest deficit at the top
Make the heat cascade thermodynamically feasible
Timo Laukkanen

41. SUMMARY Grand Composite Curve (GCC) Pinch violations Stream grid
Don’t transfer heat across pinch
Don’t use hot utility below pinch
Don’t use cold utility above pinch
Stream grid
Maximum Energy Recovery (MER) Network
Targeting for minimum number of Units
Timo Laukkanen

42. SUMMARY
A LP (linear programming) transshipment model for minimizing utility consumption
Only the starting temperatures are used to develop the temperature intervals
Energy balance equations around each temperature interval
Heat residuals to cascade heat into a lower temperature interval
Minimize the utilities
A MILP (Mixed Integer Linear Programming) extended transhipment model for minimizing the number of units
With the utility consumptions, and pinch point known
Energy balance equations around each temperature interval for each stream
Heat residuals for each stream to cascade heat into a lower temperature interval
Big-M formulations to define the existance of heat exchange matches
Minimize the number units
Timo Laukkanen

43. SUMMARY
Cost optimization of HENS (area minimization) with an NLP superstructure
One superstructure for all streams
Embed ALL alternative network structures
Key elements:
Heat exchanger units
Mixers at inlets of each heat exchanger
Splitters at the outlets of each heat exchanger
Mixer at output of stream
Splitter at input of stream
Minimize the total cost of the network H1
Q11 C1
Q12 C2
Timo Laukkanen

44. SUMMARY
SYNHEAT, a stagewise superstructure for simultaneous synthesis of heat exchanger networks
min Total Cost =Area Cost +Units Fixed Cost +Utility Cost
stage k=1
stage k=2
H1-C1
H1-C1
tH1,1
tH1,2
tH1,3 H1 CW
H1-C2
H1-C2
tC1,1
tC1,2
tC2,1 S C1
tC1,3
tC2,2
tC2,3 S C2
H2-C1
H2-C1
tH2,1
tH2,2
tH2,3 H2 CW
H2-C2
H2-C2
temperature
location k=1
temperature
location k=2
temperature
location k=3
Timo Laukkanen

45. SUMMARY What is multiobjective optimization GAMS-NIMBUS integration
Timo Laukkanen