\$title Three-dimensional Noughts and Crosses Multiple Solutions (CUBESOLN,SEQ=371) \$onText White and black balls are to be arranged in a cube one to a cell in such a way as to minimize the number of lines with balls of equal color. For this example the length of the cube is three. a total of 49 lines exist in a cube of 3x3x3. 27 lines by changing only one coordinate, 18 diagonals within a plane, and 4 diagonals going across planes. This variation of the model uses a simple "binary cut" to eliminate a solution so a subsequent solve produces an alternative solution. This is done in a loop to produce altenative solutions. It also uses Cplex' solution pool facility to generate multiple solutions. There are many symmetries in the cube model so both procedures find multiple optimal solutions. Williams, H P, Experiments in the formulation of Integer Programming Problems. Mathematical Programming Study 2 (1974). Keywords: mixed integer linear programming, mathematics, CPLEX solution pool, mathematical games, noughts and crosses, tic-tac-toe, cut generation \$offText Set s 'domain for line identification' / a, b, c, incr, decr / x(s) 'coordinate labels' / a, b, c / d(s) 'directions' / incr, decr / b 'bounds' / low, high /; Alias (x,y,z), (d,dp), (s,sp,spp); Set ld(s,sp,spp) 'line definition'; ld("incr",y,z) = yes; ld(x,"incr",z) = yes; ld(x,y,"incr") = yes; ld("incr",d,z) = yes; ld(x,"incr",d) = yes; ld(d,y,"incr") = yes; ld("incr",d,dp) = yes; display ld; Parameter ls(b) 'sign for line definitions' / low 1, high -1 / lr(b) 'rhs for line definitions' / low 2, high -1 / df(x,s) 'line definition function'; df(x,y) = ord(y) - ord(x); df(x,"decr") = 1 + card(x) - 2*ord(x); display df; Variable core(x,y,z) 'placement of balls (white 0 black 1)' line(s,sp,spp) 'line identification' num 'number of lines of equal color'; Binary Variable core; Positive Variable line; Equation nbb 'total number of balls definition' ldef(s,sp,spp,b) 'line definitions' ndef 'number of lines definition'; nbb.. sum((x,y,z), core(x,y,z)) =e= floor(card(x)**3/2); ldef(s,sp,spp,b)\$ld(s,sp,spp).. ls(b)*sum(x, core(x+df(x,s),x+df(x,sp),x+df(x,spp))) =l= line(s,sp,spp) + lr(b); ndef.. num =e= sum((s,sp,spp)\$ld(s,sp,spp), line(s,sp,spp)); Model cube / all / option optCr = 0, resLim = 10, limRow = 0, limCol = 0, solPrint = silent; solve cube minimizing num using mip; * Cuts to cut off binary solution Set cuts / c1*c5 / cc(cuts) 'dynamic cuts'; Parameter sol 'solutions to cut off'; Equation cut(cuts); cut(cc).. sum((x,y,z)\$sol(cc,x,y,z), core(x,y,z)) + sum((x,y,z)\$(sol(cc,x,y,z) = 0), 1 - core(x,y,z)) =l= card(x)*card(y)*card(z) - 1; Model cubeCut / all /; loop(cuts, sol(cuts,x,y,z) = round(core.l(x,y,z)); sol(cuts,'','','obj') = num.l; cc(cuts) = yes; solve cubeCut minimizing num using mip; abort.noerror\$(cubeCut.modelStat <> %modelStat.optimal% and cubeCut.modelStat <> %modelStat.integerSolution%) sol; ); display 'more solutions available', sol, core.l, num.l; option clear = sol; * Cplex solution pool facility \$onEcho > cplex.opt solnPool solnpool.gdx solnPoolPop 2 solnPoolIntensity 4 populateLim 41 solnPoolGap 0.03 \$offEcho cube.optFile = 1; option mip = cplex; solve cube minimizing num using mip; display core.l, num.l; Set soln 'possible solutions in the solution pool' / file1*file40 / solnpool(soln) 'actual solutions'; execute_load 'solnpool.gdx', solnpool = index; loop(solnpool(soln), put_utility 'gdxin' / solnpool.te(soln); execute_loadpoint; sol(soln,x,y,z) = round(core.l(x,y,z)); sol(soln,'','','obj') = num.l; ); display\$(card(solnpool) = card(soln)) 'more solutions available'; display sol;