Lecture 5 Multiobjective optimization GAMS-Nimbus integration SUMMARY Timo Laukkanen

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

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

What is multiobjective optimization? Timo Laukkanen

What is multiobjective optimization? –Minimize {f 1 (x), f 2 (x),...,f k (x)} Subject to x ε S Involving k (≥ 2) conflicting objective functions f i : R n  R that are minimized simultaneously. The decision variables x =(x 1,x 2,...,x n ) 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

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 bounds of the Pareto optimal set are available in the nadir objective vector z nad. Timo Laukkanen

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

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

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

Multiobjective optimization methods

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

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

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

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

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

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

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

GAMS-NIMBUS integration Timo Laukkanen

GAMS-NIMBUS integration Nimbus user-interface Timo Laukkanen

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

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

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

Superstructure development

Heat recovery steam generator (HRSG)

High pressure boiler (fuel fired)

Medium pressure boiler (fuel fired)

Waste heat boiler (medium pressure)

Steam generation options

High pressure steam turbines

Complete superstructure

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...

Mass balances Indices, Sets and Variables

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

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

Momentum Balances Indices, Sets and Variables

Demands Indices, Sets and Variables Heat Electricity Mechanical Power

Lower and Upper Bounds non-power generating unit

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

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 CO 2 savings in the next 40 years CC (Composite Curves) –Different temperature cascades for hot and cold streams Timo Laukkanen

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

SUMMARY Grand Composite Curve (GCC) Pinch violations –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

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

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 Timo Laukkanen H1 Q11Q11 Q11Q11 C1 Q12Q12 Q12Q12 C2

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

SUMMARY What is multiobjective optimization GAMS-NIMBUS integration Timo Laukkanen