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epscm.gms : eps-Constraint Method for Multiobjective Optimization

The eps-Constraint Method

This is a GAMS implementation of the augmented eps-constraint method
for generating the efficient (Pareto optimal, nondominated) solutions
in multiobjective problems. The eps-constraint method optimizes one of
the objective functions using the remaining objective functions as
constraints, varying their right hand side.

The generated optimal solutions proved to be efficient solutions of
the multiobjective problem under certain conditions.

The eps-constraint method consists of two phases:
1. Creation of the payoff table
2. Use the ranges from the payoff table in order to apply the method

The augmented eps-constraint uses lexicographic optimization in the
construction of the payoff table (in order to secure the Pareto
optimality of the individual optima) and a slightly modified objective
function in order to ensure the production of Pareto optimal (and not
weakly Pareto optimal) solutions. In addition, it performs early exit
from infeasible loops improving the performance of the algorithm in
multi-objective problems with several objective functions.

The algorithm can work also with MIP models. Actually the advantages
of the eps-constraint method over the weighting method are more
apparent for MIP problems where the non supported Pareto optimal
solutions can be produced.

A simplified power generation problem is used to illustrate the
method:

Four types of power generation units are considered in a region,
namely, lignite fired, oil fired, natural gas fired and units
exploiting renewable energy sources (RES) which are mostly small hydro
and wind.  The power generation characteristics of these units are
shown in table pdata.

The yearly demand is 64000 GWh and is characterized by a load duration
curve which can be divided into three type of loads: base load (60%),
medium load (30%) and peak load (10%). The lignite fired units can be
used only for base and middle load, the oil fired units for middle and
peak load, the RES units for base and peak load and the natural gas
fired units for all type of loads. The endogenous sources are lignite
and RES.

We consider three objective functions: the minimization of production
cost, the minimization of CO2 emissions and the minimization of
external dependence (i.e. oil and natural gas) by maximizing the
endogenous energy sources.  The task is to generate the payoff table
and the Pareto optimal (efficient, non-dominated) solutions of the
problem.

Additional information can be found at:

http://www.gams.com/modlib/adddocs/epscm.pdf

• Mavrotas, G, Generation of efficient solutions in Multiobjective Mathematical Programming problems using GAMS. Tech. rep., School of Chemical Engineering, National Technical University of Athens, 2006.