spbenders5.gms : Stochastic Benders - Parallel MPI with GAMSModelInstance

**Description**

This example demonstrates a stochastic Benders implementation for the simple transport example. This is the fifth example of a sequence of stochastic Benders implementations using various methods to solve the master and subproblem. This fifth example implements the stochastic Benders algorithm using parallel implementation using MPI where the individual model (master or subproblem) in the GAMS jobs is implemented as a Python OO-API GamsModelInstance object. The advantage is that the models need to be generated only once and solved with varying data. Since the model rim of a GamsModelInstance cannot be changed, the master model includes all possible cuts with non-binding constraint at the beginning: sum(j, eps*received(j)) =l= bigM. During the cause of the algorithm the right hand side and the coefficients for received(j) are updated with the real cut data. In this example the model is started from the command line as follows: mpiexec -n 4 gams spbenders5 fileStemApFromEnv=PMI_RANK lo=2 On different platforms this command might look a little different. On HPC system, that often run Linux and use a specialized MPI, it might be more efficient (or even required) to use a Python version different from the one GAMS ships with. As documented in https://www.gams.com/latest/docs/API_PY_TUTORIAL.html#PY_GETTING_STARTED GAMS Python API packages must also be installed for this Python or an environment variable (e.g. PYTHONPATH) needs to point to the GAMS Python packages. Also the PMI_RANK variable might be called differently. In any case GAMS needs to be instructed to use an external Python installation by setting pySetup=0: mpirun -n 4 gams spbenders5 fileStemApFromEnv=OMPI_COMM_WORLD_RANK pySetup=0 lo=2 This command spawns card(s)+1 (=4 for the small example) copies of the spbenders4 model and the environment variable PMI_RANK decided which part the particular instance plays. The GAMS job with PMI_RANK=0 implements the master and the GAMS jobs PMI_RANK=1 to PMI_RANK=s implement the subproblems. The communication of the master variables received and the cut information happens via the Python package mpi4py and the GAMS embedded code facility. Keywords: linear programming, stochastic Benders algorithm, transportation problem, GAMS embedded code facility, Python, parallel computing, message passing interface

**Reference**

- Dantzig, G B, Chapter 3.3. In Linear Programming and Extensions. Princeton University Press, Princeton, New Jersey, 1963.

**Small Model of Type :** LP

**Category :** GAMS Model library

**Main file :** spbenders5.gms

```
$title Stochastic Benders - Parallel MPI with GAMSModelInstance (SPBENDERS5,SEQ=422)
$onText
This example demonstrates a stochastic Benders implementation for the
simple transport example.
This is the fifth example of a sequence of stochastic Benders
implementations using various methods to solve the master and
subproblem.
This fifth example implements the stochastic Benders algorithm using
parallel implementation using MPI where the individual model (master
or subproblem) in the GAMS jobs is implemented as a Python OO-API
GamsModelInstance object. The advantage is that the models need to be
generated only once and solved with varying data. Since the model rim
of a GamsModelInstance cannot be changed, the master model includes all
possible cuts with non-binding constraint at the beginning:
sum(j, eps*received(j)) =l= bigM. During the cause of the algorithm the
right hand side and the coefficients for received(j) are updated with
the real cut data.
In this example the model is started from the command line as follows:
mpiexec -n 4 gams spbenders5 fileStemApFromEnv=PMI_RANK lo=2
On different platforms this command might look a little different. On HPC
system, that often run Linux and use a specialized MPI, it might be more
efficient (or even required) to use a Python version different from the one
GAMS ships with. As documented in https://www.gams.com/latest/docs/API_PY_TUTORIAL.html#PY_GETTING_STARTED
GAMS Python API packages must also be installed for this Python or an
environment variable (e.g. PYTHONPATH) needs to point to the GAMS Python packages.
Also the PMI_RANK variable might be called differently. In any case GAMS needs
to be instructed to use an external Python installation by setting pySetup=0:
mpirun -n 4 gams spbenders5 fileStemApFromEnv=OMPI_COMM_WORLD_RANK pySetup=0 lo=2
This command spawns card(s)+1 (=4 for the small example) copies of the
spbenders4 model and the environment variable PMI_RANK decided which part
the particular instance plays. The GAMS job with PMI_RANK=0 implements the
master and the GAMS jobs PMI_RANK=1 to PMI_RANK=s implement the subproblems.
The communication of the master variables received and the cut information
happens via the Python package mpi4py and the GAMS embedded code facility.
Keywords: linear programming, stochastic Benders algorithm, transportation
problem, GAMS embedded code facility, Python, parallel computing,
message passing interface
$offText
$set MPI_RANK notSet
$if setenv PMI_RANK $set MPI_RANK %sysEnv.PMI_RANK%
$if setenv OMPI_COMM_WORLD_RANK $set MPI_RANK %sysEnv.OMPI_COMM_WORLD_RANK%
$if %MPI_RANK%==notSet $abort.noError This model needs to be started with mpiexec/mpirun. See preamble of the model source for details.
$set MPI_SIZE notSet
$if setenv PMI_SIZE $set MPI_SIZE %sysEnv.PMI_SIZE%
$if setenv OMPI_COMM_WORLD_RANK $set MPI_SIZE %sysEnv.OMPI_COMM_WORLD_SIZE%
$if %MPI_SIZE%==notSet $abort.noError This model needs to be started with mpiexec/mpirun. See preamble of the model source for details.
* On the major platforms, GMSPYTHONHOME gets set automatically, otherwise the
* user has to set it. This condition can also be removed, if one has set up its
* Python environment appropriately
$if not setEnv GMSPYTHONHOME
$if %gams.pySetup%==1 $abort.noError Embedded code Python not ready to be used
$if not setEnv GMSPYTHONHOME $log --- Using external Python
Set
i 'factories' / f1*f3 /
j 'distribution centers' / d1*d5 /;
Parameter
capacity(i) 'unit capacity at factories'
/ f1 500, f2 450, f3 650 /
demand(j) 'unit demand at distribution centers'
/ d1 160, d2 120, d3 270, d4 325, d5 700 /
prodcost 'unit production cost' / 14 /
price 'sales price' / 24 /
wastecost 'cost of removal of overstocked products' / 4 /;
Table transcost(i,j) 'unit transportation cost'
d1 d2 d3 d4 d5
f1 2.49 5.21 3.76 4.85 2.07
f2 1.46 2.54 1.83 1.86 4.76
f3 3.26 3.08 2.60 3.76 4.45;
$ifThen not set useBig
Set s 'scenarios' / lo, mid, hi /;
Table ScenarioData(s,*) 'possible outcomes for demand plus probabilities'
d1 d2 d3 d4 d5 prob
lo 150 100 250 300 600 0.25
mid 160 120 270 325 700 0.50
hi 170 135 300 350 800 0.25;
$else
$ if not set nrScen $set nrScen 10
Set s 'scenarios' / s1*s%nrScen% /;
Parameter ScenarioData(s,*) 'possible outcomes for demand plus probabilities';
option seed = 1234;
ScenarioData(s,'prob') = 1/card(s);
ScenarioData(s,j) = demand(j)*uniform(0.6,1.4);
$endIf
$eval cardS card(s)+1
$if not %cardS%==%MPI_SIZE% $abort MPI size needs to be %cardS%
$set solverlog
$if set useSolverLog $set solverlog output=sys.stdout
embeddedCode Python:
try:
from mpi4py import *
except:
raise Exception("Module mpi4py not found")
comm = MPI.COMM_WORLD
def symbols2List(*symbols):
return list(map(lambda x: list(gams.get(x)), symbols))
def syncData(db_from, db_to, *args):
for sym in args:
db_from[sym].copy_symbol(db_to[sym])
def solveMI(mi, symIn=[], symOut=[]):
for sym in symIn:
gams.db[sym].copy_symbol(mi.sync_db[sym])
mi.solve(%solverlog%)
for sym in symOut:
mi.sync_db[sym].copy_symbol(gams.db[sym])
pauseEmbeddedCode
abort$execerror 'Problems setting up mpy4py or some other Python error. Check the log';
Parameter r(j) / #j 0/, osub(s), cconst(s), ccoeff(s,j);
$ifThen.MPI 0==%MPI_RANK%
* Benders master problem
$if not set maxiter $set maxiter 25
Set iter 'max Benders iterations' / 1*%maxiter% /;
Alias (iter,it);
Parameter
cutconst(iter) 'constants in optimality cuts' / #iter 0 /
cutcoeff(iter,j) 'coefficients in optimality cuts' / #iter.#j 0 /;
Variable
ship(i,j) 'shipments'
product(i) 'production'
received(j) 'quantity sent to market'
zmaster 'objective variable of master problem'
theta 'future profit';
Positive Variable ship;
Equation
masterobj 'master objective function'
production(i) 'calculate production in each factory'
receive(j) 'calculate quantity to be send to markets'
optcut(iter) 'Benders optimality cuts';
masterobj.. zmaster =e= theta - sum((i,j), transcost(i,j)*ship(i,j))
- sum(i,prodcost*product(i));
receive(j).. received(j) =e= sum(i, ship(i,j));
production(i).. product(i) =e= sum(j, ship(i,j));
optcut(iter)..
theta =l= cutconst(iter) + sum(j, cutcoeff(iter,j)*received(j));
product.up(i) = capacity(i);
Model masterproblem / all /;
Scalar
rgap 'relative gap' / 0 /
lowerBound 'global lower bound' / -inf /
upperBound 'global upper bound' / +inf /
objMaster / 0 /
objSub / 0 /;
* Initialize cut to be non-binding
cutconst(iter) = 1e15;
cutcoeff(iter,j) = eps;
$libInclude pyEmbMI miMaster 'masterproblem max zmaster using lp' -all_model_types=cplexd cutconst.Accumulate cutcoeff.Accumulate
option limRow = 0, limCol = 0, solPrint = silent, solver = cplexd, solveLink = %solveLink.loadLibrary%;
r(j) = 0;
objMaster = 0;
loop(it,
continueEmbeddedCode:
comm.bcast([[0], list(gams.get('r'))], root=0)
cut = comm.gather(None, root=0)[1:]
gams.set('osub', [c[0][0] for c in cut])
gams.set('cconst', [c[1][0] for c in cut])
gams.set('ccoeff', [rec for s in cut for rec in s[2]])
pauseEmbeddedCode osub, cconst, ccoeff
* The clear of the cut data below goes together with the Accumulate updateType
* of the update symbols. It also work without the clear and updateType BaseCase
* but requires much more data exchange because we communicate in every iteration
* the data of all cuts generated so far
option clear = cutconst, clear = cutcoeff;
objSub = sum(s, osub(s));
cutconst(it) = eps + sum(s, cconst(s));
cutcoeff(it,j) = eps + sum(s, ccoeff(s,j));
if(lowerBound < objMaster + objSub, lowerBound = objMaster + objSub);
rgap = (upperBound - lowerBound)/(1 + abs(upperBound));
break$(rgap < 0.001);
continueEmbeddedCode:
solveMI(miMaster,['cutconst','cutcoeff'],['received','zmaster','theta'])
pauseEmbeddedCode received, zmaster, theta
upperBound = zmaster.l;
objMaster = zmaster.l - theta.l;
r(j) = received.l(j);
);
* Terminate sub jobs
continueEmbeddedCode:
comm.bcast([[1],[]], root=0)
endEmbeddedCode
abort$(rgap >= 0.001) 'need more iterations', lowerbound, upperbound;
display 'optimal solution', lowerbound, upperbound;
$else.MPI
* Benders' subproblem
Variable
sales(j) 'sales (actually sold)'
waste(j) 'overstocked products'
zsub 'objective variable of sub problem';
Positive Variable sales, waste;
Equation
subobj 'subproblem objective function'
selling(j) 'part of received is sold'
market(j) 'upperbound on sales';
subobj.. zsub =e= sum(j, price*sales(j)) - sum(j, wastecost*waste(j));
selling(j).. sales(j) + waste(j) =e= r(j);
market(j).. sales(j) =l= demand(j);
Model subproblem / subobj, selling, market /;
* Infinite loop
Singleton set ss(s);
ss(s) = ord(s)=%MPI_RANK%;
demand(j) = scenarioData(ss,j);
Scalar done;
$libInclude pyEmbMI miSub 'subproblem max zsub using lp' -all_model_types=cplexd r.Zero
option limRow = 0, limCol = 0, solPrint = silent, solver = cplexd, solveLink = %solveLink.loadLibrary%;
while(1,
continueEmbeddedCode:
rx = comm.bcast(None, root=0)
gams.set('done', rx[0])
miSub.sync_db['r'].clear()
if not rx[0][0]:
for rec in rx[1]:
miSub.sync_db['r'].add_record(rec[0]).value = rec[1]
miSub.solve(%solverlog%)
syncData(miSub.sync_db,gams.db,'market','selling','zsub')
pauseEmbeddedCode done, market, selling, zsub
abort.noerror$done 'terminating subprocess';
osub(ss) = eps + ScenarioData(ss,'prob')*zsub.l;
cconst(ss) = eps + ScenarioData(ss,'prob')*sum(j, market.m(j)*demand(j));
ccoeff(ss,j) = eps + ScenarioData(ss,'prob')*selling.m(j);
continueEmbeddedCode:
comm.gather(symbols2List('osub', 'cconst', 'ccoeff'), root=0 )
pauseEmbeddedCode
);
$endIf.MPI
```