The problem considered here is essentially the 12-location problem in Hillier [1] and in Nugent, Vollmann, and Ruml [2] plus a change to more than 12 locations spread over three dimensions. The same interfacility flows [1, p. 33; 2, p. 168] are used here. So one knows the minimum objective function value to this newer problem is 289 [2, p. 159] or smaller.
Line 561 through line 626 of the computer program below extend to three dimensions the mathematical formulation appearing on page 6 of Heragu and Kusiak [3] and on page 140 of Heragu [4]. Line 1321 through line 1414 below use the interdepartmental flows presented in Hillier [1, p. 33] and in Nugent, Vollmann, and Ruml [2, p. 168].
The following computer program benefits from the computer programs of the present blog and the computer program on pages 229-232 of Conley [5].
2 DEFINT I,J,K
5 DIM B(99),N(99),A(99),H(99),L(99),U(99),X(1111),D(111),P(222)
12 FOR JJJJ=-32000 TO 32000
14 RANDOMIZE JJJJ
16 M=-1D+17
43 FOR IOC=1 TO 36
45 B(IOC)=0
47 NEXT IOC
51 FOR IOCTT=1 TO 36
53 N(IOCTT)=2
57 NEXT IOCTT
61 FOR KLQ=1 TO 36
62 A(KLQ)=B(KLQ)+RND*(N(KLQ)-B(KLQ))
63 NEXT KLQ
71 FOR KLR=1 TO 36
72 H(KLR)=3
73 NEXT KLR
88 FOR JJJ=1 TO 1000 STEP 10
90 FOR INEW=1 TO 1
94 FOR J=INEW*2 TO 0 STEP -1
102 FOR JJ=0 TO 36
128 FOR I=1 TO 3
129 FOR KKQQ=1 TO 36
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
132 FOR K=1 TO 36
133 IF RND<=.5 THEN 298 ELSE 230
134 REM
137 REM
230 IF A(K)-(N(K)-B(K))/H(K)^J
255 GOTO 265
260 L(K)=A(K)-(N(K)-B(K))/H(K)^J
265 IF A(K)+(N(K)-B(K))/H(K)^J>N(K) THEN 266 ELSE 268
266 U(K)=N(K)-L(K)
267 GOTO 272
268 U(K)=A(K)+(N(K)-B(K))/H(K)^J-L(K)
272 GOTO 300
298 X(K)=A(K)+2*RND*(2*RND-1)*(1/(1+RND*JJJ))*.05*A(K)
299 GOTO 302
300 X(K)=(L(K)+2*RND*RND*U(K))
302 NEXT K
304 X(JJ)=A(JJ)
306 KLPS=FIX(1+RND*108)
307 FOR KLA1=1 TO KLPS
308 KLA2=FIX(1+36*RND)
309 X(KLA2)=A(KLA2)
310 NEXT KLA1
343 IF RND>.1 GOTO 361
352 IF RND<.5 THEN 353 ELSE 355
353 X(JJ)=B(JJ)
354 GOTO 361
355 X(JJ)=N(JJ)
361 FOR I222=1 TO 36
364 X(I222)=FIX(X(I222))
368 NEXT I222
461 IF RND>.1 GOTO 531
465 IF RND<1/3 THEN 471 ELSE IF RND<1/2 THEN 491 ELSE 511
471 IOCT1=1+FIX(RND*12)
474 IOCT2=1+FIX(RND*12)
477 X(IOCT1)=A(IOCT2)
480 X(IOCT2)=A(IOCT1)
481 GOTO 531
491 IOCT6=13+FIX(RND*12)
494 IOCT7=13+FIX(RND*12)
497 X(IOCT6)=A(IOCT7)
500 X(IOCT7)=A(IOCT6)
502 GOTO 531
511 IOCT8=25+FIX(RND*12)
514 IOCT9=25+FIX(RND*12)
517 X(IOCT8)=A(IOCT9)
520 X(IOCT9)=A(IOCT8)
522 GOTO 531
531 FOR IOCX=1 TO 36
533 IF X(IOCX)>2 THEN X(IOCX)=1
535 NEXT IOCX
541 FOR IOCY=1 TO 36
543 IF X(IOCY)<0 THEN X(IOCY)=0
545 NEXT IOCY
561 P(11)=ABS(X(1)-X(2))+ABS(X(13)-X(14))+ABS(X(25)-X(26))-1
562 P(12)=ABS(X(1)-X(3))+ABS(X(13)-X(15))+ABS(X(25)-X(27))-1
563 P(13)=ABS(X(1)-X(4))+ABS(X(13)-X(16))+ABS(X(25)-X(28))-1
564 P(14)=ABS(X(1)-X(5))+ABS(X(13)-X(17))+ABS(X(25)-X(29))-1
565 P(15)=ABS(X(1)-X(6))+ABS(X(13)-X(18))+ABS(X(25)-X(30))-1
566 P(16)=ABS(X(1)-X(7))+ABS(X(13)-X(19))+ABS(X(25)-X(31))-1
567 P(17)=ABS(X(1)-X(8))+ABS(X(13)-X(20))+ABS(X(25)-X(32))-1
568 P(18)=ABS(X(1)-X(9))+ABS(X(13)-X(21))+ABS(X(25)-X(33))-1
569 P(19)=ABS(X(1)-X(10))+ABS(X(13)-X(22))+ABS(X(25)-X(34))-1
570 P(20)=ABS(X(1)-X(11))+ABS(X(13)-X(23))+ABS(X(25)-X(35))-1
571 P(21)=ABS(X(1)-X(12))+ABS(X(13)-X(24))+ABS(X(25)-X(36))-1
572 P(22)=ABS(X(2)-X(3))+ABS(X(14)-X(15))+ABS(X(26)-X(27))-1
573 P(23)=ABS(X(2)-X(4))+ABS(X(14)-X(16))+ABS(X(26)-X(28))-1
574 P(24)=ABS(X(2)-X(5))+ABS(X(14)-X(17))+ABS(X(26)-X(29))-1
575 P(25)=ABS(X(2)-X(6))+ABS(X(14)-X(18))+ABS(X(26)-X(30))-1
576 P(26)=ABS(X(2)-X(7))+ABS(X(14)-X(19))+ABS(X(26)-X(31))-1
577 P(27)=ABS(X(2)-X(8))+ABS(X(14)-X(20))+ABS(X(26)-X(32))-1
578 P(28)=ABS(X(2)-X(9))+ABS(X(14)-X(21))+ABS(X(26)-X(33))-1
579 P(29)=ABS(X(2)-X(10))+ABS(X(14)-X(22))+ABS(X(26)-X(34))-1
580 P(30)=ABS(X(2)-X(11))+ABS(X(14)-X(23))+ABS(X(26)-X(35))-1
581 P(31)=ABS(X(2)-X(12))+ABS(X(14)-X(24))+ABS(X(26)-X(36))-1
582 P(32)=ABS(X(3)-X(4))+ABS(X(15)-X(16))+ABS(X(27)-X(28))-1
583 P(33)=ABS(X(3)-X(5))+ABS(X(15)-X(17))+ABS(X(27)-X(29))-1
584 P(34)=ABS(X(3)-X(6))+ABS(X(15)-X(18))+ABS(X(27)-X(30))-1
585 P(35)=ABS(X(3)-X(7))+ABS(X(15)-X(19))+ABS(X(27)-X(31))-1
586 P(36)=ABS(X(3)-X(8))+ABS(X(15)-X(20))+ABS(X(27)-X(32))-1
587 P(37)=ABS(X(3)-X(9))+ABS(X(15)-X(21))+ABS(X(27)-X(33))-1
588 P(38)=ABS(X(3)-X(10))+ABS(X(15)-X(22))+ABS(X(27)-X(34))-1
589 P(39)=ABS(X(3)-X(11))+ABS(X(15)-X(23))+ABS(X(27)-X(35))-1
590 P(40)=ABS(X(3)-X(12))+ABS(X(15)-X(24))+ABS(X(27)-X(36))-1
591 P(41)=ABS(X(4)-X(5))+ABS(X(16)-X(17))+ABS(X(28)-X(29))-1
592 P(42)=ABS(X(4)-X(6))+ABS(X(16)-X(18))+ABS(X(28)-X(30))-1
593 P(43)=ABS(X(4)-X(7))+ABS(X(16)-X(19))+ABS(X(28)-X(31))-1
594 P(44)=ABS(X(4)-X(8))+ABS(X(16)-X(20))+ABS(X(28)-X(32))-1
595 P(45)=ABS(X(4)-X(9))+ABS(X(16)-X(21))+ABS(X(28)-X(33))-1
596 P(46)=ABS(X(4)-X(10))+ABS(X(16)-X(22))+ABS(X(28)-X(34))-1
597 P(47)=ABS(X(4)-X(11))+ABS(X(16)-X(23))+ABS(X(28)-X(35))-1
598 P(48)=ABS(X(4)-X(12))+ABS(X(16)-X(24))+ABS(X(28)-X(36))-1
599 P(49)=ABS(X(5)-X(6))+ABS(X(17)-X(18))+ABS(X(29)-X(30))-1
600 P(50)=ABS(X(5)-X(7))+ABS(X(17)-X(19))+ABS(X(29)-X(31))-1
601 P(51)=ABS(X(5)-X(8))+ABS(X(17)-X(20))+ABS(X(29)-X(32))-1
602 P(52)=ABS(X(5)-X(9))+ABS(X(17)-X(21))+ABS(X(29)-X(33))-1
603 P(53)=ABS(X(5)-X(10))+ABS(X(17)-X(22))+ABS(X(29)-X(34))-1
604 P(54)=ABS(X(5)-X(11))+ABS(X(17)-X(23))+ABS(X(29)-X(35))-1
605 P(55)=ABS(X(5)-X(12))+ABS(X(17)-X(24))+ABS(X(29)-X(36))-1
606 P(56)=ABS(X(6)-X(7))+ABS(X(18)-X(19))+ABS(X(30)-X(31))-1
607 P(57)=ABS(X(6)-X(8))+ABS(X(18)-X(20))+ABS(X(30)-X(32))-1
608 P(58)=ABS(X(6)-X(9))+ABS(X(18)-X(21))+ABS(X(30)-X(33))-1
609 P(59)=ABS(X(6)-X(10))+ABS(X(18)-X(22))+ABS(X(30)-X(34))-1
610 P(60)=ABS(X(6)-X(11))+ABS(X(18)-X(23))+ABS(X(30)-X(35))-1
611 P(61)=ABS(X(6)-X(12))+ABS(X(18)-X(24))+ABS(X(30)-X(36))-1
612 P(62)=ABS(X(7)-X(8))+ABS(X(19)-X(20))+ABS(X(31)-X(32))-1
613 P(63)=ABS(X(7)-X(9))+ABS(X(19)-X(21))+ABS(X(31)-X(33))-1
614 P(64)=ABS(X(7)-X(10))+ABS(X(19)-X(22))+ABS(X(31)-X(34))-1
615 P(65)=ABS(X(7)-X(11))+ABS(X(19)-X(23))+ABS(X(31)-X(35))-1
616 P(66)=ABS(X(7)-X(12))+ABS(X(19)-X(24))+ABS(X(31)-X(36))-1
617 P(67)=ABS(X(8)-X(9))+ABS(X(20)-X(21))+ABS(X(32)-X(33))-1
618 P(68)=ABS(X(8)-X(10))+ABS(X(20)-X(22))+ABS(X(32)-X(34))-1
619 P(69)=ABS(X(8)-X(11))+ABS(X(20)-X(23))+ABS(X(32)-X(35))-1
620 P(70)=ABS(X(8)-X(12))+ABS(X(20)-X(24))+ABS(X(32)-X(36))-1
621 P(71)=ABS(X(9)-X(10))+ABS(X(21)-X(22))+ABS(X(33)-X(34))-1
622 P(72)=ABS(X(9)-X(11))+ABS(X(21)-X(23))+ABS(X(33)-X(35))-1
623 P(73)=ABS(X(9)-X(12))+ABS(X(21)-X(24))+ABS(X(33)-X(36))-1
624 P(74)=ABS(X(10)-X(11))+ABS(X(22)-X(23))+ABS(X(34)-X(35))-1
625 P(75)=ABS(X(10)-X(12))+ABS(X(22)-X(24))+ABS(X(34)-X(36))-1
626 P(76)=ABS(X(11)-X(12))+ABS(X(23)-X(24))+ABS(X(35)-X(36))-1
788 FOR INSI=11 TO 76
791 IF P(INSI)<0 THEN P(INSI)=P(INSI) ELSE P(INSI)=0
795 NEXT INSI
1111 PSUM=0
1115 FOR IOCT=11 TO 76
1118 PSUM=PSUM+P(IOCT)
1121 NEXT IOCT
1131 PS1=99*555555!*PSUM
1321 P11B=5*ABS(X(1)-X(2))+2*ABS(X(1)-X(3))+4*ABS(X(1)-X(4))+1*ABS(X(1)-X(5))+0*ABS(X(1)-X(6))
1322 P12B=0*ABS(X(1)-X(7))+6*ABS(X(1)-X(8))+2*ABS(X(1)-X(9))+1*ABS(X(1)-X(10))+1*ABS(X(1)-X(11))+1*ABS(X(1)-X(12))
1323 P13B=3*ABS(X(2)-X(3))+0*ABS(X(2)-X(4))+2*ABS(X(2)-X(5))+2*ABS(X(2)-X(6))+2*ABS(X(2)-X(7))
1324 P14B=0*ABS(X(2)-X(8))+4*ABS(X(2)-X(9))+5*ABS(X(2)-X(10))+0*ABS(X(2)-X(11))+0*ABS(X(2)-X(12))
1325 P15B=0*ABS(X(3)-X(4))+0*ABS(X(3)-X(5))+0*ABS(X(3)-X(6))+0*ABS(X(3)-X(7))+5*ABS(X(3)-X(8))
1326 P16B=5*ABS(X(3)-X(9))+2*ABS(X(3)-X(10))+2*ABS(X(3)-X(11))+2*ABS(X(3)-X(12))
1327 P17B=5*ABS(X(4)-X(5))+2*ABS(X(4)-X(6))+2*ABS(X(4)-X(7))+10*ABS(X(4)-X(8))+0*ABS(X(4)-X(9))+0*ABS(X(4)-X(10))+5*ABS(X(4)-X(11))+5*ABS(X(4)-X(12))
1328 P18B=10*ABS(X(5)-X(6))+0*ABS(X(5)-X(7))+0*ABS(X(5)-X(8))+0*ABS(X(5)-X(9))+5*ABS(X(5)-X(10))+1*ABS(X(5)-X(11))+1*ABS(X(5)-X(12))
1329 P19B=5*ABS(X(6)-X(7))+1*ABS(X(6)-X(8))+1*ABS(X(6)-X(9))+5*ABS(X(6)-X(10))+4*ABS(X(6)-X(11))+0*ABS(X(6)-X(12))
1330 P20B=10*ABS(X(7)-X(8))+5*ABS(X(7)-X(9))+2*ABS(X(7)-X(10))+3*ABS(X(7)-X(11))+3*ABS(X(7)-X(12))
1331 P21B=0*ABS(X(8)-X(9))+0*ABS(X(8)-X(10))+5*ABS(X(8)-X(11))+0*ABS(X(8)-X(12))
1332 P22B=0*ABS(X(9)-X(10))+10*ABS(X(9)-X(11))+10*ABS(X(9)-X(12))
1333 P23B=5*ABS(X(10)-X(11))+0*ABS(X(10)-X(12))
1334 P24B=2*ABS(X(11)-X(12))
1351 P25B=5*ABS(X(13)-X(14))+2*ABS(X(13)-X(15))+4*ABS(X(13)-X(16))+1*ABS(X(13)-X(17))+0*ABS(X(13)-X(18))
1352 P26B=0*ABS(X(13)-X(19))+6*ABS(X(13)-X(20))+2*ABS(X(13)-X(21))+1*ABS(X(13)-X(22))+1*ABS(X(13)-X(23))+1*ABS(X(13)-X(24))
1353 P27B=3*ABS(X(14)-X(15))+0*ABS(X(14)-X(16))+2*ABS(X(14)-X(17))+2*ABS(X(14)-X(18))+2*ABS(X(14)-X(19))
1354 P28B=0*ABS(X(14)-X(20))+4*ABS(X(14)-X(21))+5*ABS(X(14)-X(22))+0*ABS(X(14)-X(23))+0*ABS(X(14)-X(24))
1355 P29B=0*ABS(X(15)-X(16))+0*ABS(X(15)-X(17))+0*ABS(X(15)-X(18))+0*ABS(X(15)-X(19))+5*ABS(X(15)-X(20))
1356 P30B=5*ABS(X(15)-X(21))+2*ABS(X(15)-X(22))+2*ABS(X(15)-X(23))+2*ABS(X(15)-X(24))
1367 P31B=5*ABS(X(16)-X(17))+2*ABS(X(16)-X(18))+2*ABS(X(16)-X(19))+10*ABS(X(16)-X(20))+0*ABS(X(16)-X(21))+0*ABS(X(16)-X(22))+5*ABS(X(16)-X(23))+5*ABS(X(16)-X(24))
1368 P32B=10*ABS(X(17)-X(18))+0*ABS(X(17)-X(19))+0*ABS(X(17)-X(20))+0*ABS(X(17)-X(21))+5*ABS(X(17)-X(22))+1*ABS(X(17)-X(23))+1*ABS(X(17)-X(24))
1369 P33B=5*ABS(X(18)-X(19))+1*ABS(X(18)-X(20))+1*ABS(X(18)-X(21))+5*ABS(X(18)-X(22))+4*ABS(X(18)-X(23))+0*ABS(X(18)-X(24))
1370 P34B=10*ABS(X(19)-X(20))+5*ABS(X(19)-X(21))+2*ABS(X(19)-X(22))+3*ABS(X(19)-X(23))+3*ABS(X(19)-X(24))
1371 P35B=0*ABS(X(20)-X(21))+0*ABS(X(20)-X(22))+5*ABS(X(20)-X(23))+0*ABS(X(20)-X(24))
1372 P36B=0*ABS(X(21)-X(22))+10*ABS(X(21)-X(23))+10*ABS(X(21)-X(24))
1373 P37B=5*ABS(X(22)-X(23))+0*ABS(X(22)-X(24))
1374 P38B=2*ABS(X(23)-X(24))
1401 P39B=5*ABS(X(25)-X(26))+2*ABS(X(25)-X(27))+4*ABS(X(25)-X(28))+1*ABS(X(25)-X(29))+0*ABS(X(25)-X(30))
1402 P40B=0*ABS(X(25)-X(31))+6*ABS(X(25)-X(32))+2*ABS(X(25)-X(33))+1*ABS(X(25)-X(34))+1*ABS(X(25)-X(35))+1*ABS(X(25)-X(36))
1403 P41B=3*ABS(X(26)-X(27))+0*ABS(X(26)-X(28))+2*ABS(X(26)-X(29))+2*ABS(X(26)-X(30))+2*ABS(X(26)-X(31))
1404 P42B=0*ABS(X(26)-X(32))+4*ABS(X(26)-X(33))+5*ABS(X(26)-X(34))+0*ABS(X(26)-X(35))+0*ABS(X(26)-X(36))
1405 P43B=0*ABS(X(27)-X(28))+0*ABS(X(27)-X(29))+0*ABS(X(27)-X(30))+0*ABS(X(27)-X(31))+5*ABS(X(27)-X(32))
1406 P44B=5*ABS(X(27)-X(33))+2*ABS(X(27)-X(34))+2*ABS(X(27)-X(35))+2*ABS(X(27)-X(36))
1407 P45B=5*ABS(X(28)-X(29))+2*ABS(X(28)-X(30))+2*ABS(X(28)-X(31))+10*ABS(X(28)-X(32))+0*ABS(X(28)-X(33))+0*ABS(X(28)-X(34))+5*ABS(X(28)-X(35))+5*ABS(X(28)-X(36))
1408 P46B=10*ABS(X(29)-X(30))+0*ABS(X(29)-X(31))+0*ABS(X(29)-X(32))+0*ABS(X(29)-X(33))+5*ABS(X(29)-X(34))+1*ABS(X(29)-X(35))+1*ABS(X(29)-X(36))
1409 P47B=5*ABS(X(30)-X(31))+1*ABS(X(30)-X(32))+1*ABS(X(30)-X(33))+5*ABS(X(30)-X(34))+4*ABS(X(30)-X(35))+0*ABS(X(30)-X(36))
1410 P48B=10*ABS(X(31)-X(32))+5*ABS(X(31)-X(33))+2*ABS(X(31)-X(34))+3*ABS(X(31)-X(35))+3*ABS(X(31)-X(36))
1411 P49B=0*ABS(X(32)-X(33))+0*ABS(X(32)-X(34))+5*ABS(X(32)-X(35))+0*ABS(X(32)-X(36))
1412 P50B=0*ABS(X(33)-X(34))+10*ABS(X(33)-X(35))+10*ABS(X(33)-X(36))
1413 P51B=5*ABS(X(34)-X(35))+0*ABS(X(34)-X(36))
1414 P52B=2*ABS(X(35)-X(36))
1443 P1=P11B+P12B+P13B+P14B+P15B+P16B+P17B+P18B+P19B+P20B
1444 P2=P21B+P22B+P23B+P24B+P25B+P26B+P27B+P28B+P29B+P30B
1445 P3=P31B+P32B+P33B+P34B+P35B+P36B+P37B+P38B
1446 P4=P39B+P40B+P41B+P42B+P43B+P44B+P45B+P46B+P47B+P48B
1447 P5=P49B+P50B+P51B+P52B
1448 P6=P1+P2+P3+P4+P5
1450 P=-P6+PS1
1451 IF P<=M THEN 1670
1452 M=P
1453 PP1=P6
1454 FOR KLX=1 TO 36
1455 A(KLX)=X(KLX)
1456 NEXT KLX
1457 GOTO 128
1670 NEXT I
1702 NEXT JJ
1706 NEXT J
1777 NEXT INEW
1778 REM
1888 NEXT JJJ
1890 IF M>-280 THEN 1916 ELSE 1999
1916 PRINT JJJJ,M,PP1
1917 PRINT A(1),A(2),A(3),A(4),A(5)
1918 PRINT A(6),A(7),A(8),A(9),A(10)
1920 PRINT A(11),A(12)
1922 PRINT A(13),A(14),A(15),A(16),A(17)
1924 PRINT A(18),A(19),A(20),A(21),A(22)
1926 PRINT A(23),A(24)
1928 PRINT A(25),A(26),A(27),A(28),A(29)
1930 PRINT A(30),A(31),A(32),A(33),A(34)
1932 PRINT A(35),A(36),JJJJ,M,PP1
1999 NEXT JJJJ
This BASIC computer program was run with the IBM basica/D interpreter, and the best candidate solutions produced from JJJJ=-32000 through JJJJ=-31658 (in compressed form and to be interpreted in accordance with line 1916 through line 1932; copied manually from the computer monitor) are presented below.
-31989 -262 262
2 0 1 2 0
0 1 2 1 0
1 2
0 0 0 1 1
1 0 0 1 0
1 1
1 1 1 0 1
0 0 0 1 0
0 1 -31989 -262 262
-31716 -262 262
0 2 1 0 2
2 1 0 1 2
1 0
1 1 1 0 0
0 1 1 0 1
0 0
0 0 0 1 0
1 1 1 0 1
1 0 -31716 -262 262
-31658 -263 263
1 1 0 1 1
0 0 0 1 0
0 1
2 0 2 2 0
0 1 2 1 0
1 1
1 1 1 0 0
0 0 0 1 1
1 0 -31658 -263 263
The best candidate solutions shown above are at JJJJ=-31989 and at JJJJ=-31716 with an objective function value of 262. 289, the minimum objective function value for the older problem [2, p. 159], to 262 is 1.103 to 1.
The output above was produced in 7 hours on a personal computer with an Intel 2.66 GHz. chip and the IBM interpreter, which is slower than the corresponding compiler.
If one makes changes to the computer program above, such as a change of changing line 128 above to 128 FOR I=1 TO 4, one may or may not obtain 262, which may or may not be optimal. That leads to a general remedy that in general, in order to increase the chance of getting an optimal solution, one preferably should run several computers simultaneously, each with a different line 128, for example; instead of line 128, another line can be chosen. This multicomputer approach can mitigate the possible danger of missed optimality and can produce a usable solution faster than just running one computer.
References
[1] F.S. Hillier, Quantitative tools for plant layout analysis, J. Indust. Eng. 14 (1963) 33-40.
[2] C.E. Nugent, T.E. Vollmann, J. Ruml, An experimental comparison of techniques for the assignment of facilities to locations, Operations Research 16 (1968) 150-173.
[3] S.S. Heragu, A. Kusiak, Efficient models for the facility layout problem, European Journal of Operational Research 53 (1991) 1-13.
[4] S.S. Heragu, Recent models and techniques for solving the layout problem, European Journal of Operational Research 57 (1992) 136-144.
[5] W.C. Conley, Optimization: A Simplified Approach, Petrocelli, Princeton, New Jersey, 1981.
