epscm.gms : eps-Constraint Method for Multiobjective Optimization

**Description**

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

**Reference**

- Mavrotas, G, Effective implementation of the ε-constraint method in Multi-Objective Mathematical Programming problems. Applied Mathematics and Computation 213, 2 (2009), 455-465.

**Small Model of Type :** LP

**Category :** GAMS Model library

**Main file :** epscm.gms

```
$title eps-Constraint Method for Multiobjective Optimization (EPSCM,SEQ=319)
$onText
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, Effective implementation of the eps-constraint method in
Multi-Objective Mathematical Programming problems.
Applied Mathematics and Computation 213, 2 (2009), 455-465.
Keywords: linear programming, eps-constraint method, multiobjective optimization
$offText
$inlineCom [ ]
$eolCom //
$stitle Example Model Definitions
Set
p 'power generation units' / Lignite, Oil, Gas, RES /
i 'load areas' / base, middle, peak /
pi(p,i) 'availability of unit for load types' / Lignite.(base,middle)
Oil.(middle,peak), Gas.set.i
RES.(base, peak) /
es(p) 'endogenous sources' / Lignite, RES /
k 'objective functions' / cost, CO2emission, endogenous /;
$set min -1
$set max +1
Parameter dir(k) 'direction of the objective functions'
/ cost %min%, CO2emission %min%, endogenous %max% /;
Set pheader / capacity, cost, CO2emission /;
Table pdata(pheader,p)
Lignite Oil Gas RES
capacity [GWh] 31000 15000 22000 10000
cost [$/MWh] 30 75 60 90
CO2emission [t/MWh] 1.44 0.72 0.45 0;
Parameter
ad 'annual demand in GWh' / 64000 /
df(i) 'demand fraction for load type' / base 0.6, middle 0.3, peak 0.1 /
demand(i) 'demand for load type in GWh';
demand(i) = ad*df(i);
Variable
z(k) 'objective function variables';
Positive Variable
x(p,i) 'production level of unit in load area in GWh';
Equation
objcost 'objective for minimizing cost in K$'
objco2 'objective for minimizing CO2 emissions in Kt'
objes 'objective for maximizing endogenous sources in GWh'
defcap(p) 'capacity constraint'
defdem(i) 'demand satisfaction';
* Objective functions
objcost.. sum(pi(p,i), pdata('cost',p)*x(pi)) =e= z('cost');
objco2.. sum(pi(p,i), pdata('CO2emission',p)*x(pi)) =e= z('CO2emission');
objes.. sum(pi(es,i), x(pi)) =e= z('endogenous');
defcap(p).. sum(pi(p,i), x(pi)) =l= pdata('capacity',p);
defdem(i).. sum(pi(p,i), x(pi)) =g= demand(i);
Model example / all /;
$STitle eps-Constraint Method
Set
k1(k) 'the first element of k'
km1(k) 'all but the first elements of k';
k1(k)$(ord(k) = 1) = yes;
km1(k) = yes;
km1(k1) = no;
Set kk(k) 'active objective function in constraint allobj';
Parameter
rhs(k) 'right hand side of the constrained obj functions in eps-constraint'
maxobj(k) 'maximum value from the payoff table'
minobj(k) 'minimum value from the payoff table';
Variable
a_objval 'auxiliary variable for the objective function'
obj 'auxiliary variable during the construction of the payoff table';
Positive Variable
sl(k) 'slack or surplus variables for the eps-constraints';
Equation
con_obj(k) 'constrained objective functions'
augm_obj 'augmented objective function to avoid weakly efficient solutions'
allobj 'all the objective functions in one expression';
con_obj(km1).. z(km1) - dir(km1)*sl(km1) =e= rhs(km1);
* We optimize the first objective function and put the others as constraints
* the second term is for avoiding weakly efficient points
augm_obj.. sum(k1,dir(k1)*z(k1)) + 1e-3*sum(km1,sl(km1)/(maxobj(km1) - minobj(km1))) =e= a_objval;
allobj.. sum(kk, dir(kk)*z(kk)) =e= obj;
Model
mod_payoff / example, allobj /
mod_epsmethod / example, con_obj, augm_obj /;
option limRow = 0, limCol = 0, solPrint = off, solveLink = %solveLink.CallModule%;
Parameter payoff(k,k) 'payoff tables entries';
Alias (k,kp);
* Generate payoff table applying lexicographic optimization
loop(kp,
kk(kp) = yes;
repeat
solve mod_payoff using lp maximizing obj;
payoff(kp,kk) = z.l(kk);
z.fx(kk) = z.l(kk); // freeze the value of the last objective optimized
kk(k++1) = kk(k); // cycle through the objective functions
until kk(kp);
kk(kp) = no;
* release the fixed values of the objective functions for the new iteration
z.up(k) = inf;
z.lo(k) = -inf;
);
if(mod_payoff.modelStat <> %modelStat.Optimal% and
mod_payoff.modelStat <> %modelStat.FeasibleSolution%,
abort 'no feasible solution for mod_payoff');
display payoff;
minobj(k) = smin(kp,payoff(kp,k));
maxobj(k) = smax(kp,payoff(kp,k));
$set fname p.%gams.scrext%
File fx 'solution points from eps-method' / "%gams.scrdir%%fname%" /;
$if not set gridpoints $set gridpoints 10
Set
g 'grid points' / g0*g%gridpoints% /
grid(k,g) 'grid';
Parameter
gridrhs(k,g) 'rhs of eps-constraint at grid point'
maxg(k) 'maximum point in grid for objective'
posg(k) 'grid position of objective'
firstOffMax 'counter'
lastZero 'counter'
numk(k) 'ordinal value of k starting with 1'
numg(g) 'ordinal value of g starting with 0';
lastZero = 1;
loop(km1,
numk(km1) = lastZero;
lastZero = lastZero + 1;
);
numg(g) = ord(g) - 1;
grid(km1,g) = yes; // Here we could define different grid intervals for different objectives
maxg(km1) = smax(grid(km1,g), numg(g));
gridrhs(grid(km1,g))$(%min% = dir(km1)) = maxobj(km1) - numg(g)/maxg(km1)*(maxobj(km1) - minobj(km1));
gridrhs(grid(km1,g))$(%max% = dir(km1)) = minobj(km1) + numg(g)/maxg(km1)*(maxobj(km1) - minobj(km1));
display gridrhs;
* Walk the grid points and take shortcuts if the model becomes infeasible
posg(km1) = 0;
repeat
rhs(km1) = sum(grid(km1,g)$(numg(g) = posg(km1)), gridrhs(km1,g));
solve mod_epsmethod maximizing a_objval using lp;
if(mod_epsmethod.modelStat <> %modelStat.Optimal%, // not optimal is in this case infeasible
lastZero = 0;
loop(km1$(posg(km1) > 0 and lastZero = 0), lastZero=numk(km1));
posg(km1)$(numk(km1) <= lastZero) = maxg(km1); // skip all solves for more demanding values of rhs(km1)
else
loop(k, put fx z.l(k):12:2); put /;
);
* Proceed forward in the grid
firstOffMax = 0;
loop(km1$(posg(km1) < maxg(km1) and firstOffMax = 0),
posg(km1) = posg(km1) + 1;
firstOffMax = numk(km1);
);
posg(km1)$(numk(km1) < firstOffMax) = 0;
until sum(km1$(posg(km1) = maxg(km1)),1) = card(km1) and firstOffMax = 0;
putClose fx; // close the point file
* Get unique solutions from the point file using some Posix Tools (awk, (g)sort, uniq) that come with GAMS
$set awkscript awk.%gams.scrext%
File fa / "%gams.scrdir%%awkscript%" /;
put fa 'BEGIN { printf("Table solutions(*,*)\n$ondelim\nsol';
loop(k, put ',' k.tl:0);
putClose '\n"); }' / '{ print NR,$0 }' / 'END { print ";" }';
$if %system.filesys% == UNIX execute 'cd "%gams.scrdir%" && sort %fname% | uniq | awk -f %awkscript% > g.%gams.scrext% && gams g.%gams.scrext% o=gx.%gams.scrext% lo=0 gdx=soleps';
$if not %system.filesys% == UNIX execute 'cd "%gams.scrdir%" && gsort %fname% | uniq | awk -f %awkscript% > g.%gams.scrext% && gams g.%gams.scrext% o=gx.%gams.scrext% lo=0 gdx=soleps';
execute 'mv -f "%gams.scrdir%soleps.gdx" .';
Set s 'solutions' / 1*50 /;
Parameter solutions(s,k) 'unique solutions';
execute_load 'soleps', solutions;
display solutions;
$exit
$onText
The display should produce a table with 18 unique solutions:
---- 203 PARAMETER solutions Unique solutions
cost CO2emissi~ endogenous
1 3075000.000 62460.000 31000.000
2 3078000.000 62316.000 31200.000
3 3099000.000 61308.000 32600.000
4 3111000.000 60732.000 33400.000
5 3120000.000 60300.000 34000.000
6 3141000.000 59292.000 35400.000
7 3147000.000 59004.000 35800.000
8 3162000.000 58284.000 36800.000
9 3183000.000 57276.000 38200.000
10 3204000.000 56268.000 39600.000
11 3219000.000 55548.000 40600.000
12 3225000.000 55260.000 41000.000
13 3315000.000 53820.000 39000.000
14 3423000.000 52092.000 36600.000
15 3531000.000 50364.000 34200.000
16 3639000.000 48636.000 31800.000
17 3747000.000 46908.000 29400.000
18 3855000.000 45180.000 27000.000
$offText
```