solnpool.gms : Cplex Solution Pool for a Simple Facility Location Problem

**Description**

A simple version of a facility location problem is used to show how the solution pool and the tools associated with it work. This example is taken from the Cplex 11 User's Manual (ILOG, Cplex 11 User's Manual, 2007) A company is considering opening as many as four warehouses in order to serve nine different regions. The goal is to minimize the sum of fixed costs associated with opening warehouses as well as the various transportation costs incurred to ship goods from the warehouses to the regions. Whether or not to open a warehouse is represented by binary variable ow. Whether or not to ship goods from warehouse i to region j is represented by binary variable oa. Each region needs a specified amount of goods, and each warehouse can store only a limited quantity of goods. In addition, each region must be served by exactly one warehouse.

**Reference**

- ILOG, CPLEX 11 User's Manual, 2007.

**Small Model of Type :** MIP

**Category :** GAMS Model library

**Main file :** solnpool.gms

$Title Cplex Solution Pool for a Simple Facility Location Problem (SOLNPOOL,SEQ=326) $Ontext A simple version of a facility location problem is used to show how the solution pool and the tools associated with it work. This example is taken from the Cplex 11 User's Manual (ILOG, Cplex 11 User's Manual, 2007) A company is considering opening as many as four warehouses in order to serve nine different regions. The goal is to minimize the sum of fixed costs associated with opening warehouses as well as the various transportation costs incurred to ship goods from the warehouses to the regions. Whether or not to open a warehouse is represented by binary variable ow. Whether or not to ship goods from warehouse i to region j is represented by binary variable oa. Each region needs a specified amount of goods, and each warehouse can store only a limited quantity of goods. In addition, each region must be served by exactly one warehouse. The following GAMS program demonstrates a number of different approaches to collecting solution pools. GAMS will store the individual solutions in GDX containers/files which can then be further used by other programs or the same GAMS run. Cplex will name these GDX containers 'soln_loc_pNN.gdx', where NN will be the serial number of the solution found. To manage the different solutions, the names of the GDX containers created by Cplex will be stored in solnpool.gdx in the set 'index' using the set elements file*. Eight examples are solved in this gams run. $offtext Set i warehouses / w1*w4 / j regions / r1*r9 / Parameters f(i) fixed costs / w1 130, w2 150, w3 170, w4 180 / c(i) capacity / w1 90, w2 110, w3 130, w4 150 / d(j) demand / r1 10, r2 10, r3 12, r4 15, r5 15, r6 15, r7 20, r8 20, r9 30 /; Table t(j,i) transport costs w1 w2 w3 w4 r1 10 30 25 55 r2 10 25 25 45 r3 20 23 30 40 r4 25 10 26 40 r5 28 12 20 29 r6 36 19 16 22 r7 40 39 22 27 r8 75 65 55 35 r9 34 43 41 62; Variables totcost total cost fcost fixed cost tcost transportation cost ow(i) indicator for open warehouse oa(i,j) indicator for open shipment arc warehouse to region Binary variables ow, oa; Equations deftotcost definition total cost deffcost definition fixed cost deftcost definition transportation cost defwcap(i) limit utilization of warehouse by its capacity onew(j) only one warehouse per region defow(i,j) warehouse open if shipment from i to j; deftotcost.. totcost =e= fcost + tcost; deffcost.. fcost =e= sum(i, f(i)*ow(i)); deftcost.. tcost =e= sum((i,j), t(j,i)*oa(i,j)); defwcap(i).. sum(j, d(j)*oa(i,j)) =l= c(i); onew(j).. sum(i, oa(i,j)) =e= 1; defow(i,j).. ow(i) =g= oa(i,j); Model loc /all/ ; * --- Define sets, parameters and files to hold solutions Sets soln possible solutions in the solution pool /file1*file1000/ solnpool(soln) actual solutions; Scalar cardsoln number of solutions in the pool; Alias (soln,s1,s2), (*,u); Parameters owX(soln,i) warehouse indicator by solution oaX(soln,i,j) arc indicator by solution xcostX(soln,*) cost structure by solution; files fsoln, fcpx / cplex.opt /; option limrow=0, limcol=0, optcr=0, mip=cplex; loc.optfile=1; loc.solprint=%solprint.Quiet%; loc.savepoint = 1; * The code to load different solution from gdx files will be used * several times in this program and we therefore copy it into an include file. $onecho > readsoln.gms execute_load 'solnpool.gdx', solnpool=index; cardsoln = card(solnpool); display cardsoln; oaX(soln,i,j) = 0; owX(soln,i) = 0; xcostX(soln,u) = 0; loop(solnpool(soln), put_utility fsoln 'gdxin' / solnpool.te(soln); execute_loadpoint; oaX(soln,i,j) = round(oa.l(i,j)); owX(soln,i) = round(ow.l(i)); xcostX(soln,'totcost') = totcost.l; xcostX(soln,'tcost') = tcost.l; xcostX(soln,'fcost') = fcost.l; xcostX(soln,'fcost^0.96') = fcost.l**0.96; ); * Restore the solution reported to GAMS execute_loadpoint 'loc_p.gdx'; $offecho * 1. Collect the incumbents found during the regular optimize procedure * The Cplex option 'solnpool' triggers the collection of solutions in * the GDX container solnpool. putclose fcpx 'solnpool solnpool.gdx'; solve loc min totcost using mip; $include readsoln display xcostX; * 2. Use the populate procedure instead of regular optimize procedure (option * 'solnpoolpop 2'). By default we will generate 20 solutions determined by * the default of option populatelim. This is a simple model which is quickly * solved with heuristics and cuts, so we need to let Cplex retain sufficient * exploration space to find alternative solutions. This is done with option * 'solnpoolintensity 4'. With solutions where the optimal solution can not * so quickly be found, the default for this option should be suffict. putclose fcpx 'solnpool solnpool.gdx' / 'solnpoolintensity 4' / 'solnpoolpop 2'; solve loc min totcost using mip; $include readsoln display xcostX; * 3. Now it gets more complicated. After the first call to populate we instruct * the GAMS/Cplex link to execute a GAMS program 'simple.gms' that decides if * the populate procedure should be called again. Moreover, we will instruct * Cplex to delete the solution 11 to 15 from the pool. After this two rounds * we will have 35 solutions in the pool. Note the use of %ncall% which expands * to an integer counting the number of previous execution of 'simple.gms'. * A nonzero return code of the gams run called by Cplex will signal to * Cplex that this is was the final call. putclose fcpx 'solnpool solnpool.gdx' / 'solnpoolintensity 4' / 'solnpoolpop 2' / 'solnpoolpoprepeat simple.gms' / 'solnpoolpopdel delsol.txt'; $onechoV > simple.gms $ife %ncalls%>0 abort 'Terminate search' file f /delsol.txt/; put f '11 12 13 14 15'; $offecho solve loc min totcost using mip; $include readsoln abort$(cardsoln <> 35) 'Expected to get 35 solutions'; display xcostX; * 4. Next we call the populate procedure, but we want solutions that are * within 3% of the optimum. If we find more than 20, we are willing to * call the populate procedure once more. It turns out that there are 11 * solutions of this quality, so no need to call populate again. putclose fcpx 'solnpool solnpool.gdx' / 'solnpoolintensity 4' / 'solnpoolpop 2' / 'solnpoolpoprepeat simple.gms' / 'solnpoolgap 0.03'; solve loc min totcost using mip; $include readsoln display xcostX; * 5. Lets look at the diversity of the solution by counting the differences * between the shipment indicator variables. Lets limit the number of * solutions in the pool by 10 and require solution within 10% of the * optimum. putclose fcpx 'solnpool solnpool.gdx' / 'solnpoolintensity 4' / 'solnpoolpop 2' / 'solnpoolcapacity 10' / 'solnpoolgap 0.1'; solve loc min totcost using mip; $include readsoln Scalar aggdiff aggregated differences /0/; loop((s1,s2)$(not sameas(s1,s2) and solnpool(s1) and solnpool(s2)), aggdiff = aggdiff + sum((i,j), oaX(s1,i,j) xor oaX(s2,i,j)); ); display aggdiff; * 6. We repeat the experiment by now setting the solution pool replacement * strategy to 'diversity' and let the populate procedure find many more * solutions, we should see an increase in the aggregated difference putclose fcpx 'solnpool solnpool.gdx' / 'solnpoolintensity 4' / 'solnpoolpop 2' / 'solnpoolcapacity 10' / 'solnpoolgap 0.1' / 'populatelim 10000' / 'solnpoolreplace 2'; solve loc min totcost using mip; $include readsoln Scalar aggdiffX aggregated differences /0/; loop((s1,s2)$(not sameas(s1,s2) and solnpool(s1) and solnpool(s2)), aggdiffX = aggdiffX + sum((i,j), oaX(s1,i,j) xor oaX(s2,i,j)); ); display aggdiffX; abort$(aggdiffX < aggdiff) 'We expected *more* diversity'; * 7. We can fine tune diversity by using a diversity filter. Suppose that * facilities w1 and w2 are open. Let a solution keeping those two facilities * open be the reference. We use a diversity filter to stipulate that any * solution added to the solution pool must differ from the reference by * decisions to open or close at least two other facilities. The following * filter enforces this diversity by specifying a minimum diversity of 2. * Note that the reference solution becomes the incumbent and is reported as * the first solution in the pool. put fcpx 'solnpool solnpool.gdx' / 'solnpoolintensity 4' / 'solnpoolpop 2' / 'divfltlo 2' / 'writeflt ow2.flt'; parameter owrefsol(i) / w1 1, w2 1, w3 0, w4 0 /; loop(i, put / 'ow.divflt("' i.tl:0 '") ' owrefsol(i):0:0); putclose fcpx; solve loc min totcost using mip; $include readsoln display owX; loop(solnpool(soln)$(ord(soln)>1), abort$(sum(i,abs(owX(soln,i) - owrefsol(i)))<2) 'solution differs to little'); * 8. We can also implement more complicated filters using the incumbent filter. * For example, we want to enforce that transportation cost is less than * fixedcost**0.96. The incumbent filter is implemented as part of the BCH * facility (http://www.gams.com/docs/bch.htm). Whenever a new incumbent is * found by Cplex, Cplex will run the GAMS program 'incbflt'. The incumbent * is provided in a GDX container called 'bchout_i'. The GAMS program can * inspect and decide if the incumbent should be accepted or rejected by * Cplex. A nonzero return code of the gams run called by Cplex will signal * to Cplex that the incumbent is accecpted. putclose fcpx 'solnpool solnpool.gdx' / 'solnpoolintensity 4' / 'solnpoolpop 2' / 'userincbcall incbflt.gms'; $onecho > incbflt.gms variables tcost, fcost; $gdxin bchout_i $load tcost fcost abort$(tcost.l < fcost.l**0.96) 'Accept'; $offecho solve loc min totcost using mip; $include readsoln display xcostX; loop(solnpool(soln), abort$(xcostX(soln,'tcost')>=xcostX(soln,'fcost')) 'tcost too big'); * 9. We might sometimes want to order the solutions, for example, best * to worst. For this we can use the gdxrank utility. Note that gdxrank * returns the sorted indices. putclose fcpx 'solnpool solnpool.gdx' / 'solnpoolintensity 4' / 'solnpoolpop 2'; solve loc min totcost using mip; $include readsoln Parameter uval(soln) unsorted objective values idx(soln) sorted index position; uval(soln)=xcostX(soln,'totcost'); * sort solutions via gdxrank $libinclude rank uval solnpool idx file fsorted / 'sorted.txt' /; put fsorted 'Sorted solutions:'; * Get the best five in a file loop(solnpool(soln), if (idx(soln) <= 5, put #(idx(soln)+1) @1 soln.tl:0 ':' @10 uval(soln) ); ); putclose; * Get all solutions sorted in a new parameter parameter sval(soln) sorted objective values; sval(soln + (idx(soln)- Ord(soln)))$solnpool(soln) = uval(soln); display sval;