## Monday, January 16, 2012

### Dinkelbach’s Algorithm

The paper:

Dinkelbach’s Algorithm as an Efficient Method for Solving a Class of
MINLP Models for Large-Scale Cyclic Scheduling Problems  [link]

by Prof. Grossmann e.a. investigates some mixed-integer linear fractional programming problems. It compares some standard MINLP codes against an implementation of Dinkelbach’s algorithm and concludes that this algorithm performs very good. The algorithm consists of a solving a series of MIP problems with each time a different weighted objective function. This method is so simple it can be coded in GAMS in just a few lines.

If the problem is something like:

then we need to solve a series of MIP problems:

for different values of λ.

I wanted to see if I could use this on a (very) large mixed-integer linear fractional programming problem. Of course first try it on some very small examples. The first test is a continuous problem:

### Example 1: Linear fractional programming problem

 \$ontext    References:      Fengqi You, Pedro M. Castro, Ignacio E. Grossmann1      Dinkelbach’s Algorithm as an Efficient Method for Solving a Class of      MINLP Models for Large-Scale Cyclic Scheduling Problems      Said Tantawy      AN ITERATIVE METHOD FOR SOLVING LINEAR FRACTION      PROGRAMMING (LFP) PROBLEM WITH SENSITIVITY ANALYSIS \$offtext set    i 'products' /a1,a2/ ; parameters    uprofit(i) 'unit profit' /       a1 4       a2 2    /    ucost(i) 'unit cost' /       a1 1       a2 2    /    fixedcost /5/    fixedprofit /10/    matuse(i) 'raw material usage' /       a1 1       a2 3    /    rawavail 'raw material available' /30/ ; variables    x(i) 'production'    profit    cost    z    'objective variable' ; positive variables x,profit,cost; equations    eprofit 'calculate profit'    ecost   'calculate cost'    eraw    'raw material usage'    eprodcon  'production constraint'    eratio   'minlp objective' ; eprofit.. profit =e= fixedprofit + sum(i, uprofit(i)*x(i)); ecost..   cost =e= fixedcost + sum(i, ucost(i)*x(i)); eraw..    sum(i, matuse(i)*x(i)) =l= rawavail; eprodcon..  x('a1') + 5 =g= 2*x('a2'); eratio..   z =e= profit/cost; cost.lo = 1; *------------------------------------------------------------- * solve as nlp *------------------------------------------------------------- model m1 /all/; m1.solprint = 2; solve m1 maximizing z using nlp; display "------------------------------------",         "NLP Solver",         "------------------------------------",         z.l,x.l; *------------------------------------------------------------- * Dinkelbach's algorithm *------------------------------------------------------------- scalars   q  'optimal objective at end of algorithm' /0/   continue /1/   tol  /0.1/   iterations ; equation linobj; linobj .. z =e= profit - q*cost; model m2 /eprofit,ecost,eraw,eprodcon,linobj/; m2.solprint = 2; set iter /iter1*iter5/; loop(iter\$continue,    solve m2 maximizing z using lp;    if (z.l < tol,       continue = 0;       iterations = ord(iter);       display "------------------------------------",               "Dinkelbach algorithm",               "------------------------------------",               q,iterations,x.l;    else       q = profit.l/cost.l;    ); );

This continuous problem is very easy:

 ----     73 ------------------------------------             NLP Solver             ------------------------------------             VARIABLE z.L                   =        3.714  objective variable ----     73 VARIABLE x.L  production a1 30.000 ----    102 ------------------------------------             Dinkelbach algorithm             ------------------------------------             PARAMETER q                    =        3.714  optimal objective at end of algorithm             PARAMETER iterations           =            2  ----    102 VARIABLE x.L  production a1 30.000

The next small example is an integer problem:

### Example 2: an integer fractional programming problem

 \$ontext    References:      Fengqi You, Pedro M. Castro, Ignacio E. Grossmann1      Dinkelbach’s Algorithm as an Efficient Method for Solving a Class of      MINLP Models for Large-Scale Cyclic Scheduling Problems      Ildiko Zsigmond      Mixed Integer linear Fractional Programming By A Branch-and-Bound      Technique 1985 \$offtext variables    x1,x2    num  'numerator'    denom  'denominator'    z    'objective variable' ; integer variables x1,x2; x1.up = 5; x2.up = 4; equations    enum    'numerator'    edenom   'denominator'    e1,e2,e3    eratio ; enum.. num =e= 2*x1+x2-2; edenom..  denom =e= x1-x2+1; e1.. -5*x1 + 4*x2 =l= 0; e2.. -x1+x2 =l= 0.5; e3.. 2*x1+x2 =l= 11; eratio..   z =e= num/denom; denom.lo = .1; option optcr=0; model m1 /enum,edenom,e1,e2,e3,eratio/; m1.solprint = 2; solve m1 maximizing z using minlp; display "------------------------------------",         "MINLP Solver",         "------------------------------------",         z.l,x1.l,x2.l; *------------------------------------------------------------- * Dinkelbach's algorithm *------------------------------------------------------------- scalars   q  'optimal objective at end of algorithm' /0/   continue /1/   tol  /0.1/   iterations ; equation linobj; linobj .. z =e= num - q*denom; model m2 /enum,edenom,e1,e2,e3,linobj/; m2.solprint = 2; set iter /iter1*iter5/; display "------------------------------------"; loop(iter\$continue,    solve m2 maximizing z using mip;    if (z.l < tol,       continue = 0;       iterations = ord(iter);       display "------------------------------------",               "Dinkelbach algorithm",               "------------------------------------",               q,iterations,x1.l,x2.l;    else       q = num.l/denom.l;    ); );

The listing file shows:

 ----     51 ------------------------------------             MINLP Solver             ------------------------------------             VARIABLE z.L                   =        7.000  objective variable             VARIABLE x1.L                  =        3.000              VARIABLE x2.L                  =        3.000  ----     82 ------------------------------------             Dinkelbach algorithm             ------------------------------------             PARAMETER q                    =        7.000  optimal objective at end of algorithm             PARAMETER iterations           =        4.000              VARIABLE x1.L                  =        3.000              VARIABLE x2.L                  =        3.000

For our very large problem (approx. 1 million rows) this method was even more successful. All NLP solvers I have access to had troubles with the sheer size of the problem, even though the NLP relaxations are linearly constrained. But the MIP problems generated inside Dinkelbach’s algorithm turned out to be large but easy to solve. In addition the algorithm converged in about 5 major iterations.