Substance

Solving a Parallel Ordering Problem with Row 1 Having 6 Facilities and Row 2 Having 24 Facilities

Jsun Yui Wong

The lengths of the parallel row ordering problem (PROP, Amaral [5]) here come from Heragu and Kusiak [10, p. 9, Table 1], and the flow matrix comes from Nugent, Vollmann, and Ruml [13].

0 REM DEFDBL A-Z

1 DEFINT I, J, K, X

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), Y(33), C(33), CC(33)

3 DIM HS(49, 49)

4 DIM PE(49, 49)
5 DIM SD(49, 49)

17 HR(1) = 3: HR(2) = 9: HR(3) = 3: HR(4) = 7: HR(5) = 3: HR(6) = 7: HR(7) = 5: HR(8) = 9: HR(9) = 6

18 HR(10) = 5: HR(11) = 3: HR(12) = 9: HR(13) = 3: HR(14) = 7: HR(15) = 3: HR(16) = 7: HR(17) = 5: HR(18) = 9: HR(19) = 6: HR(20) = 5

29 HR(21) = 3: HR(22) = 9: HR(23) = 3

30 HR(24) = 7: HR(25) = 3: HR(26) = 7: HR(27) = 5: HR(28) = 9: HR(29) = 6: HR(30) = 5

31 FOR IL = 1 TO 30

32 FOR JL = 1 TO 30

33 READ HS(IL, JL)

34 NEXT JL

35 NEXT IL
36 REM

42 DATA 9999,3,2,0,0,2,10,5,0,5,2,5,0,0,2,0,5,6,3,0,1,10,0,10,2,1,1,1,0,1

43 DATA 3,9999,4,0,10,4,0,0,2,2,1,0,5,0,0,0,0,2,0,1,6,1,0,1,2,2,5,1,10,5

44 DATA 2,4,9999,3,4,0,5,5,5,1,4,1,0,4,0,4,0,6,3,2,5,5,2,1,0,0,3,1,0,2

45 DATA 0,0,3,9999,0,0,0,2,2,0,6,0,2,5,2,5,1,1,1,1,2,2,4,0,2,0,2,2,5,5

46 DATA 0,10,4,0,9999,5,2,0,0,0,0,2,0,0,0,0,2,1,0,0,2,0,5,1,0,2,1,0,2,1

47 DATA 2,4,0,0,5,9999,1,2,2,1,4,10,10,2,5,5,0,5,0,0,0,10,0,0,0,4,0,10,1,1

48 DATA 10,0,5,0,2,1,9999,10,10,5,10,10,6,0,0,10,2,1,10,1,5,5,2,3,5,0,2,0,1,3

49 DATA 5,0,5,2,0,2,10,9999,1,3,5,0,0,0,2,4,5,2,10,6,0,5,5,2,5,0,5,5,0,2

50 DATA 0,2,5,2,0,2,10,1,9999,10,2,1,5,2,0,3,0,2,0,0,4,0,5,2,0,5,2,2,5,2

51 DATA 5,2,1,0,0,1,5,3,10,9999,5,5,6,0,1,5,5,0,5,2,3,5,0,5,2,10,10,1,5,2

52 DATA 2,1,4,6,0,4,10,5,2,5,9999,0,0,1,2,1,0,2,0,0,0,6,6,0,4,5,3,2,2,10

53 DATA 5,0,1,0,2,10,10,0,1,5,0,9999,5,5,2,0,0,0,0,2,0,4,5,10,1,0,0,0,0,1

54 DATA 0,5,0,2,0,10,6,0,5,6,0,5,9999,2,0,4,2,2,1,0,6,2,1,5,5,0,0,1,5,5

55 DATA 0,0,4,5,0,2,0,0,2,0,1,5,2,9999,2,1,0,5,3,10,0,0,4,2,0,0,4,2,5,5

56 DATA 2,0,0,2,0,5,0,2,0,1,2,2,0,2,9999,4,5,1,0,1,0,5,0,2,0,0,5,1,1,0

57 DATA 0,0,4,5,0,5,10,4,3,5,1,0,4,1,4,9999,0,3,0,2,2,0,2,0,5,0,5,2,5,10

58 DATA 5,0,0,1,2,0,2,5,0,5,0,0,2,0,5,0,9999,2,2,0,0,0,6,5,3,5,0,0,5,1

59 DATA 6,2,6,1,1,5,1,2,2,0,2,0,2,5,1,3,2,9999,5,1,2,10,10,4,0,0,5,0,0,0

60 DATA 3,0,3,1,0,0,10,10,0,5,0,0,1,3,0,0,2,5,9999,0,5,5,1,0,5,2,1,2,10,10

61 DATA 0,1,2,1,0,0,1,6,0,2,0,2,0,10,1,2,0,1,0,9999,5,2,1,3,1,5,6,5,5,3

62 DATA 1,6,5,2,2,0,5,0,4,3,0,0,6,0,0,2,0,2,5,5,9999,4,0,1,0,0,0,5,0,0

63 DATA 10,1,5,2,0,10,5,5,0,5,6,4,2,0,5,0,0,10,5,2,4,9999,5,0,4,4,5,0,2,5

64 DATA 0,0,2,4,5,0,2,5,5,0,6,5,1,4,0,2,6,10,1,1,0,5,9999,0,4,4,1,0,2,2

65 DATA 10,1,1,0,1,0,3,2,2,5,0,10,5,2,2,0,5,4,0,3,1,0,0,9999,5,5,0,1,0,0

66 DATA 2,2,0,2,0,0,5,5,0,2,4,1,5,0,0,5,3,0,5,1,0,4,4,5,9999,1,0,10,1,0

67 DATA 1,2,0,0,2,4,0,0,5,10,5,0,0,0,0,0,5,0,2,5,0,4,4,5,1,9999,0,0,0,0

68 DATA 1,5,3,2,1,0,2,5,2,10,3,0,0,4,5,5,0,5,1,6,0,5,1,0,0,0,9999,0,0,10

69 DATA 1,1,1,2,0,10,0,5,2,1,2,0,1,2,1,2,0,0,2,5,5,0,0,1,10,0,0,9999,2,2

70 DATA 0,10,0,5,2,1,1,0,5,5,2,0,5,5,1,5,5,0,10,5,0,2,2,0,1,0,0,2,9999,2

71 DATA 1,5,2,5,1,1,3,2,2,2,10,1,5,5,0,10,1,0,10,3,0,5,2,0,0,0,10,2,2,9999

87 REM

88 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ

90 M = -1D+37

91 FOR J44 = 1 TO 30

93 A(J44) = J44

94 NEXT J44

111 IF RND < 1 / 4 THEN IMAX = 6 ELSE IF RND < 1 / 3 THEN IMAX = 6 ELSE IF RND < 1 / 2 THEN IMAX = 6 ELSE IMAX = 6

128 FOR I = 1 TO 10000

129 FOR KKQQ = 1 TO 30

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

133 III = 1 + FIX(RND * 30)

134 JJJ = 1 + FIX(RND * 30)

137 X(III) = A(JJJ)

139 X(JJJ) = A(III)

231 FOR J44 = 1 TO 30

233 FOR J45 = 1 TO 30

234 IF X(J44) = J45 THEN HHR(J44) = HR(J44) ELSE GOTO 238

237 Y(J45) = J44

238 NEXT J45

253 FOR ISE20 = 1 TO IMAX

254 C(ISE20) = .5 * HHR(Y(ISE20))

258 FOR ISE2000 = ISE20 + 1 TO IMAX

259 C(ISE20) = C(ISE20) + HHR(Y(ISE2000))

263 NEXT ISE2000

269 NEXT ISE20

386 FOR ISE20N = IMAX + 1 TO 30

387 C(ISE20N) = .5 * HHR(Y(ISE20N))

388 FOR ISE2000N = ISE20N + 1 TO 30

389 C(ISE20N) = C(ISE20N) + HHR(Y(ISE2000N))

390 NEXT ISE2000N

391 NEXT ISE20N

395 NEXT J44

401 FOR J77 = 1 TO 6

405 IF X(J77) > 6 THEN 1670

409 NEXT J77

411 PROD = 0

412 FOR J44 = 1 TO 30

413 FOR J45 = J44 + 1 TO 30

414 PROD = PROD - HS(Y(J44), Y(J45)) * ABS(C(J44) - C(J45))

415 NEXT J45

416 NEXT J44

422 P = PROD

1111 IF P <= M THEN 1670

1452 M = P

1453 FOR KLX = 1 TO 30

1454 CC(KLX) = C(KLX)

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1559 IIMAX = IMAX

1657 GOTO 128

1670 NEXT I

1889 IF M < -40700 THEN 1999

1891 PRINT A(1), A(2), A(3), A(4), A(5)

1892 PRINT A(6), A(7), A(8), A(9), A(10)

1893 PRINT A(11), A(12), A(13), A(14), A(15)

1894 PRINT A(16), A(17), A(18), A(19), A(20)

1896 PRINT A(21), A(22), A(23), A(24), A(25)

1897 PRINT A(26), A(27), A(28), A(29), A(30)

1929 PRINT M, JJJJ, A(1), A(2), A(3)

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [14]. The complete output through JJJJ=-31974 is shown below:

1       6       2       3       5
4       20       14       19       18
24       8       21       10       28
26       13       30       16       11
22       29       23       9       15
12       27       7       17       25
-40600       -31983       1       6       2

1       6       2       3       5
4       20       14       19       18
24       8       21       10       28
26       13       30       16       11
22       29       23       9       15
12       27       7       17       25
-40600       -31974       1       6       2

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [14], the wall-clock time for obtaining the output through JJJJ=-31974 was two minutes and a half.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.

[2] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.

[3] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.

[4] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.

[5] Andre R. S. Amaral (2013), A Parallel Ordering Problem in Facilities Layout. Computers and Operations Research, Vol. 40, Issue 12, December 2013, pp. 2930-2939.

[6] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.

[7] Miguel F. Anjos (2012), FLPLIB--Facility Layout Database. Retrieved on September 25 2012 from www.gerad.ca/files/Sites/Anjos/indexFR.html

[8] Miguel F. Anjos, FLPLIB--Facility Layout Database. www.miguelanjos.com.

[9] M. Anjos, G. Yen (2008). Provably Near-Optimal Solutions for Very Large Single-Row Facility Problems.
https://pdfs.semanticscholar.org/bef2/7ef4f8031887c78472f39bff82a60487bded.pdf

[10] S. S. Heragu, A. Kusiak. Efficient Models for the Facility Layout Problem. European Journal of Operational Research 53 (1), 1991, pp.1-13.

[11] Philipp Hungerlaender, Miguel F. Anjos (January 2012), A Semidefinite Optimization Approach to Free-Space Multi-Row Facility Layout. Les Cahiers du GERAD. Retrieved from www.gerad.ca/fichiers/cahiers/G-2012-03.pdf

[12] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[13] C. E. Nugent, T. E Vollmann, J. Ruml (1968), An Experimental Comparison of Techniques for the Assignment of Facilities to Locations. Operations Research, Vol. 16, pp. 150-173.

[14] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[15] Jsun Yui Wong (2012, September 17). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to the Corridor Allocation Problem with Eleven Facilities, Third Edition. https://myblogsubstance.typepad.com/substance/2012/09/index.html/

 

Solving a Parallel Ordering Problem of 30-Facility Layout, Second Edition

Jsun Yui Wong

In contrast to the facility lengths of the 2017 April 19 edition, the lengths here come from Heragu and Kusiak [10, p. 9, Table 1]. So line 17 through line 30 here are different from those lines of the earlier edition.

0 REM DEFDBL A-Z

1 DEFINT I, J, K, X

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), Y(33), C(33), CC(33)

3 DIM HS(49, 49)

4 DIM PE(49, 49)

5 DIM SD(49, 49)

17 HR(1) = 3: HR(2) = 9: HR(3) = 3: HR(4) = 7: HR(5) = 3: HR(6) = 7: HR(7) = 5: HR(8) = 9: HR(9) = 6

18 HR(10) = 5: HR(11) = 3: HR(12) = 9: HR(13) = 3: HR(14) = 7: HR(15) = 3: HR(16) = 7: HR(17) = 5: HR(18) = 9: HR(19) = 6: HR(20) = 5

29 HR(21) = 3: HR(22) = 9: HR(23) = 3

30 HR(24) = 7: HR(25) = 3: HR(26) = 7: HR(27) = 5: HR(28) = 9: HR(29) = 6: HR(30) = 5

31 FOR IL = 1 TO 30

32 FOR JL = 1 TO 30

33 READ HS(IL, JL)

34 NEXT JL

35 NEXT IL

36 REM

42 DATA 9999,3,2,0,0,2,10,5,0,5,2,5,0,0,2,0,5,6,3,0,1,10,0,10,2,1,1,1,0,1

43 DATA 3,9999,4,0,10,4,0,0,2,2,1,0,5,0,0,0,0,2,0,1,6,1,0,1,2,2,5,1,10,5

44 DATA 2,4,9999,3,4,0,5,5,5,1,4,1,0,4,0,4,0,6,3,2,5,5,2,1,0,0,3,1,0,2

45 DATA 0,0,3,9999,0,0,0,2,2,0,6,0,2,5,2,5,1,1,1,1,2,2,4,0,2,0,2,2,5,5

46 DATA 0,10,4,0,9999,5,2,0,0,0,0,2,0,0,0,0,2,1,0,0,2,0,5,1,0,2,1,0,2,1

47 DATA 2,4,0,0,5,9999,1,2,2,1,4,10,10,2,5,5,0,5,0,0,0,10,0,0,0,4,0,10,1,1

48 DATA 10,0,5,0,2,1,9999,10,10,5,10,10,6,0,0,10,2,1,10,1,5,5,2,3,5,0,2,0,1,3

49 DATA 5,0,5,2,0,2,10,9999,1,3,5,0,0,0,2,4,5,2,10,6,0,5,5,2,5,0,5,5,0,2

50 DATA 0,2,5,2,0,2,10,1,9999,10,2,1,5,2,0,3,0,2,0,0,4,0,5,2,0,5,2,2,5,2

51 DATA 5,2,1,0,0,1,5,3,10,9999,5,5,6,0,1,5,5,0,5,2,3,5,0,5,2,10,10,1,5,2

52 DATA 2,1,4,6,0,4,10,5,2,5,9999,0,0,1,2,1,0,2,0,0,0,6,6,0,4,5,3,2,2,10

53 DATA 5,0,1,0,2,10,10,0,1,5,0,9999,5,5,2,0,0,0,0,2,0,4,5,10,1,0,0,0,0,1

54 DATA 0,5,0,2,0,10,6,0,5,6,0,5,9999,2,0,4,2,2,1,0,6,2,1,5,5,0,0,1,5,5

55 DATA 0,0,4,5,0,2,0,0,2,0,1,5,2,9999,2,1,0,5,3,10,0,0,4,2,0,0,4,2,5,5

56 DATA 2,0,0,2,0,5,0,2,0,1,2,2,0,2,9999,4,5,1,0,1,0,5,0,2,0,0,5,1,1,0

57 DATA 0,0,4,5,0,5,10,4,3,5,1,0,4,1,4,9999,0,3,0,2,2,0,2,0,5,0,5,2,5,10

58 DATA 5,0,0,1,2,0,2,5,0,5,0,0,2,0,5,0,9999,2,2,0,0,0,6,5,3,5,0,0,5,1

59 DATA 6,2,6,1,1,5,1,2,2,0,2,0,2,5,1,3,2,9999,5,1,2,10,10,4,0,0,5,0,0,0

60 DATA 3,0,3,1,0,0,10,10,0,5,0,0,1,3,0,0,2,5,9999,0,5,5,1,0,5,2,1,2,10,10

61 DATA 0,1,2,1,0,0,1,6,0,2,0,2,0,10,1,2,0,1,0,9999,5,2,1,3,1,5,6,5,5,3

62 DATA 1,6,5,2,2,0,5,0,4,3,0,0,6,0,0,2,0,2,5,5,9999,4,0,1,0,0,0,5,0,0

63 DATA 10,1,5,2,0,10,5,5,0,5,6,4,2,0,5,0,0,10,5,2,4,9999,5,0,4,4,5,0,2,5

64 DATA 0,0,2,4,5,0,2,5,5,0,6,5,1,4,0,2,6,10,1,1,0,5,9999,0,4,4,1,0,2,2

65 DATA 10,1,1,0,1,0,3,2,2,5,0,10,5,2,2,0,5,4,0,3,1,0,0,9999,5,5,0,1,0,0

66 DATA 2,2,0,2,0,0,5,5,0,2,4,1,5,0,0,5,3,0,5,1,0,4,4,5,9999,1,0,10,1,0

67 DATA 1,2,0,0,2,4,0,0,5,10,5,0,0,0,0,0,5,0,2,5,0,4,4,5,1,9999,0,0,0,0

68 DATA 1,5,3,2,1,0,2,5,2,10,3,0,0,4,5,5,0,5,1,6,0,5,1,0,0,0,9999,0,0,10

69 DATA 1,1,1,2,0,10,0,5,2,1,2,0,1,2,1,2,0,0,2,5,5,0,0,1,10,0,0,9999,2,2

70 DATA 0,10,0,5,2,1,1,0,5,5,2,0,5,5,1,5,5,0,10,5,0,2,2,0,1,0,0,2,9999,2

71 DATA 1,5,2,5,1,1,3,2,2,2,10,1,5,5,0,10,1,0,10,3,0,5,2,0,0,0,10,2,2,9999

87 REM

88 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ

90 M = -1D+37

91 FOR J44 = 1 TO 30

93 A(J44) = J44

94 NEXT J44

111 IF RND < 1 / 4 THEN IMAX = 15 ELSE IF RND < 1 / 3 THEN IMAX = 15 ELSE IF RND < 1 / 2 THEN IMAX = 15 ELSE IMAX = 15

128 FOR I = 1 TO 10000

129 FOR KKQQ = 1 TO 30

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

133 III = 1 + FIX(RND * 30)

134 JJJ = 1 + FIX(RND * 30)

137 X(III) = A(JJJ)

139 X(JJJ) = A(III)

231 FOR J44 = 1 TO 30

233 FOR J45 = 1 TO 30

234 IF X(J44) = J45 THEN HHR(J44) = HR(J44) ELSE GOTO 238

237 Y(J45) = J44

238 NEXT J45

253 FOR ISE20 = 1 TO IMAX

254 C(ISE20) = .5 * HHR(Y(ISE20))

258 FOR ISE2000 = ISE20 + 1 TO IMAX

259 C(ISE20) = C(ISE20) + HHR(Y(ISE2000))

263 NEXT ISE2000

269 NEXT ISE20

386 FOR ISE20N = IMAX + 1 TO 30

387 C(ISE20N) = .5 * HHR(Y(ISE20N))

388 FOR ISE2000N = ISE20N + 1 TO 30

389 C(ISE20N) = C(ISE20N) + HHR(Y(ISE2000N))

390 NEXT ISE2000N

391 NEXT ISE20N

395 NEXT J44

401 FOR J77 = 1 TO 15

405 IF X(J77) > 15 THEN 1670

409 NEXT J77

411 PROD = 0

412 FOR J44 = 1 TO 30

413 FOR J45 = J44 + 1 TO 30

414 PROD = PROD - HS(Y(J44), Y(J45)) * ABS(C(J44) - C(J45))

415 NEXT J45

416 NEXT J44

422 P = PROD

1111 IF P <= M THEN 1670

1452 M = P

1453 FOR KLX = 1 TO 30

1454 CC(KLX) = C(KLX)

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1559 IIMAX = IMAX

1657 GOTO 128

1670 NEXT I

1889 IF M < -22900 THEN 1999

1891 PRINT A(1), A(2), A(3), A(4), A(5)

1892 PRINT A(6), A(7), A(8), A(9), A(10)

1893 PRINT A(11), A(12), A(13), A(14), A(15)

1894 PRINT A(16), A(17), A(18), A(19), A(20)

1896 PRINT A(21), A(22), A(23), A(24), A(25)

1897 PRINT A(26), A(27), A(28), A(29), A(30)

1929 PRINT M, JJJJ, A(1), A(2), A(3)

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [14]. The complete output through JJJJ=-31982 is shown below:

4       13       10       15       12
3       7       5       11       8
6       1       9       14       2
24       18       19       23       29
25       20       21       17       22
16       27       30       28       26
-22888       -31995       4       13       10

4       13       11       15       12
3       7       5       10       8
6       1       9       14       2
26       18       19       23       29
27       20       21       17       22
16       24       30       28       25
-22836       -31986       4       13       11

4       15       11       14       12
2       7       5       10       8
6       1       9       13       3
26       20       18       23       28
27       19         21       17       22
16       24       30       29       25
-22889       -31983         4       15       11

4       13       11       15       12
3       7       5       10       8
6       1       9       14       2
26       18       19       23       29
27       20       21       17       22
16       24       30       28       25
-22836       -31982       4       13       11

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [14], the wall-clock time for obtaining the output through JJJJ=-31982 was 90 seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.

[2] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.

[3] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.

[4] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.

[5] Andre R. S. Amaral (2013), A Parallel Ordering Problem in Facilities Layout. Computers and Operations Research, Vol. 40, Issue 12, December 2013, pp. 2930-2939.

[6] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.

[7] Miguel F. Anjos (2012), FLPLIB--Facility Layout Database. Retrieved on September 25 2012 from www.gerad.ca/files/Sites/Anjos/indexFR.html

[8] Miguel F. Anjos, FLPLIB--Facility Layout Database. www.miguelanjos.com.

[9] M. Anjos, G. Yen (2008). Provably Near-Optimal Solutions for Very Large Single-Row Facility Problems.
https://pdfs.semanticscholar.org/bef2/7ef4f8031887c78472f39bff82a60487bded.pdf

[10] S. S. Heragu, A. Kusiak. Efficient Models for the Facility Layout Problem. European Journal of Operational Research, 53 (1), 1991, pp.1-13.

[11] Philipp Hungerlaender, Miguel F. Anjos (January 2012), A Semidefinite Optimization Approach to Free-Space Multi-Row Facility Layout. Les Cahiers du GERAD. Retrieved from www.gerad.ca/fichiers/cahiers/G-2012-03.pdf

[12] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[13] C. E. Nugent, T. E Vollmann, J. Ruml (1968), An Experimental Comparison of Techniques for the Assignment of Facilities to Locations. Operations Research, Vol. 16, pp. 150-173.

[14] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[15] Jsun Yui Wong (2012, September 17). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to the Corridor Allocation Problem with Eleven Facilities, Third Edition. https://myblogsubstance.typepad.com/substance/2012/09/index.html/

 

Solving Yet Another Parallel Ordering Problem of 30-Facility Layout

Jsun Yui Wong

Adapting to the computer program in Wong [13], the following computer program seeks to solve a parallel row ordering problem (PROP) of 30 facilities with the first row of 6 --due to L n/5 ] of Amaral [5]--facilities and the second row of 24 facilities. The 30x30 matrix below is based on a matrix in Nugent, Vollmann, and Ruml [11]. The 30 lengths of line 17 through line 30 are the first 30 lengths of the 40 lengths in Anjos [8, srflp.40.4].

0 REM DEFDBL A-Z

1 DEFINT I, J, K, X

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), Y(33), C(33), CC(33)

3 DIM HS(49, 49)

4 DIM PE(49, 49)

5 DIM SD(49, 49)

17 HR(1) = 10: HR(2) = 10: HR(3) = 9: HR(4) = 5: HR(5) = 6: HR(6) = 3: HR(7) = 9: HR(8) = 6: HR(9) = 6

18 HR(10) = 4: HR(11) = 9: HR(12) = 9: HR(13) = 7: HR(14) = 3: HR(15) = 1: HR(16) = 3: HR(17) = 10: HR(18) = 5: HR(19) = 1: HR(20) = 2

29 HR(21) = 9: HR(22) = 9: HR(23) = 10

30 HR(24) = 7: HR(25) = 10: HR(26) = 5: HR(27) = 7: HR(28) = 1: HR(29) = 10: HR(30) = 3

31 FOR IL = 1 TO 30

32 FOR JL = 1 TO 30

33 READ HS(IL, JL)

34 NEXT JL

35 NEXT IL
36 REM

42 DATA 9999,3,2,0,0,2,10,5,0,5,2,5,0,0,2,0,5,6,3,0,1,10,0,10,2,1,1,1,0,1

43 DATA 3,9999,4,0,10,4,0,0,2,2,1,0,5,0,0,0,0,2,0,1,6,1,0,1,2,2,5,1,10,5

44 DATA 2,4,9999,3,4,0,5,5,5,1,4,1,0,4,0,4,0,6,3,2,5,5,2,1,0,0,3,1,0,2

45 DATA 0,0,3,9999,0,0,0,2,2,0,6,0,2,5,2,5,1,1,1,1,2,2,4,0,2,0,2,2,5,5

46 DATA 0,10,4,0,9999,5,2,0,0,0,0,2,0,0,0,0,2,1,0,0,2,0,5,1,0,2,1,0,2,1

47 DATA 2,4,0,0,5,9999,1,2,2,1,4,10,10,2,5,5,0,5,0,0,0,10,0,0,0,4,0,10,1,1

48 DATA 10,0,5,0,2,1,9999,10,10,5,10,10,6,0,0,10,2,1,10,1,5,5,2,3,5,0,2,0,1,3

49 DATA 5,0,5,2,0,2,10,9999,1,3,5,0,0,0,2,4,5,2,10,6,0,5,5,2,5,0,5,5,0,2

50 DATA 0,2,5,2,0,2,10,1,9999,10,2,1,5,2,0,3,0,2,0,0,4,0,5,2,0,5,2,2,5,2

51 DATA 5,2,1,0,0,1,5,3,10,9999,5,5,6,0,1,5,5,0,5,2,3,5,0,5,2,10,10,1,5,2

52 DATA 2,1,4,6,0,4,10,5,2,5,9999,0,0,1,2,1,0,2,0,0,0,6,6,0,4,5,3,2,2,10

53 DATA 5,0,1,0,2,10,10,0,1,5,0,9999,5,5,2,0,0,0,0,2,0,4,5,10,1,0,0,0,0,1

54 DATA 0,5,0,2,0,10,6,0,5,6,0,5,9999,2,0,4,2,2,1,0,6,2,1,5,5,0,0,1,5,5

55 DATA 0,0,4,5,0,2,0,0,2,0,1,5,2,9999,2,1,0,5,3,10,0,0,4,2,0,0,4,2,5,5

56 DATA 2,0,0,2,0,5,0,2,0,1,2,2,0,2,9999,4,5,1,0,1,0,5,0,2,0,0,5,1,1,0

57 DATA 0,0,4,5,0,5,10,4,3,5,1,0,4,1,4,9999,0,3,0,2,2,0,2,0,5,0,5,2,5,10

58 DATA 5,0,0,1,2,0,2,5,0,5,0,0,2,0,5,0,9999,2,2,0,0,0,6,5,3,5,0,0,5,1

59 DATA 6,2,6,1,1,5,1,2,2,0,2,0,2,5,1,3,2,9999,5,1,2,10,10,4,0,0,5,0,0,0

60 DATA 3,0,3,1,0,0,10,10,0,5,0,0,1,3,0,0,2,5,9999,0,5,5,1,0,5,2,1,2,10,10

61 DATA 0,1,2,1,0,0,1,6,0,2,0,2,0,10,1,2,0,1,0,9999,5,2,1,3,1,5,6,5,5,3

62 DATA 1,6,5,2,2,0,5,0,4,3,0,0,6,0,0,2,0,2,5,5,9999,4,0,1,0,0,0,5,0,0

63 DATA 10,1,5,2,0,10,5,5,0,5,6,4,2,0,5,0,0,10,5,2,4,9999,5,0,4,4,5,0,2,5

64 DATA 0,0,2,4,5,0,2,5,5,0,6,5,1,4,0,2,6,10,1,1,0,5,9999,0,4,4,1,0,2,2

65 DATA 10,1,1,0,1,0,3,2,2,5,0,10,5,2,2,0,5,4,0,3,1,0,0,9999,5,5,0,1,0,0

66 DATA 2,2,0,2,0,0,5,5,0,2,4,1,5,0,0,5,3,0,5,1,0,4,4,5,9999,1,0,10,1,0

67 DATA 1,2,0,0,2,4,0,0,5,10,5,0,0,0,0,0,5,0,2,5,0,4,4,5,1,9999,0,0,0,0

68 DATA 1,5,3,2,1,0,2,5,2,10,3,0,0,4,5,5,0,5,1,6,0,5,1,0,0,0,9999,0,0,10

69 DATA 1,1,1,2,0,10,0,5,2,1,2,0,1,2,1,2,0,0,2,5,5,0,0,1,10,0,0,9999,2,2

70 DATA 0,10,0,5,2,1,1,0,5,5,2,0,5,5,1,5,5,0,10,5,0,2,2,0,1,0,0,2,9999,2

71 DATA 1,5,2,5,1,1,3,2,2,2,10,1,5,5,0,10,1,0,10,3,0,5,2,0,0,0,10,2,2,9999

87 REM

88 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ

90 M = -1D+37

91 FOR J44 = 1 TO 30

93 A(J44) = J44

94 NEXT J44

111 IF RND < 1 / 4 THEN IMAX = 6 ELSE IF RND < 1 / 3 THEN IMAX = 6 ELSE IF RND < 1 / 2 THEN IMAX = 6 ELSE IMAX = 6

128 FOR I = 1 TO 10000

129 FOR KKQQ = 1 TO 30

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

133 III = 1 + FIX(RND * 30)

134 JJJ = 1 + FIX(RND * 30)

137 X(III) = A(JJJ)

139 X(JJJ) = A(III)

231 FOR J44 = 1 TO 30

233 FOR J45 = 1 TO 30

234 IF X(J44) = J45 THEN HHR(J44) = HR(J44) ELSE GOTO 238

237 Y(J45) = J44

238 NEXT J45

253 FOR ISE20 = 1 TO IMAX

254 C(ISE20) = .5 * HHR(Y(ISE20))

258 FOR ISE2000 = ISE20 + 1 TO IMAX

259 C(ISE20) = C(ISE20) + HHR(Y(ISE2000))

263 NEXT ISE2000

269 NEXT ISE20

386 FOR ISE20N = IMAX + 1 TO 30

387 C(ISE20N) = .5 * HHR(Y(ISE20N))

388 FOR ISE2000N = ISE20N + 1 TO 30

389 C(ISE20N) = C(ISE20N) + HHR(Y(ISE2000N))

390 NEXT ISE2000N

391 NEXT ISE20N

395 NEXT J44

401 FOR J77 = 1 TO 6

405 IF X(J77) > 6 THEN 1670

409 NEXT J77

411 PROD = 0

412 FOR J44 = 1 TO 30

413 FOR J45 = J44 + 1 TO 30

414 PROD = PROD - HS(Y(J44), Y(J45)) * ABS(C(J44) - C(J45))

415 NEXT J45

416 NEXT J44

422 P = PROD

1111 IF P <= M THEN 1670

1452 M = P

1453 FOR KLX = 1 TO 30

1454 CC(KLX) = C(KLX)

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1559 IIMAX = IMAX

1657 GOTO 128

1670 NEXT I

1889 IF M < -38820 THEN 1999

1891 PRINT A(1), A(2), A(3), A(4), A(5)

1892 PRINT A(6), A(7), A(8), A(9), A(10)

1893 PRINT A(11), A(12), A(13), A(14), A(15)

1894 PRINT A(16), A(17), A(18), A(19), A(20)

1896 PRINT A(21), A(22), A(23), A(24), A(25)

1897 PRINT A(26), A(27), A(28), A(29), A(30)

1929 PRINT M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [12]. The complete output through JJJJ=-31977 is shown below:

3       5       4       2       6
1       16       18       14       15
17       10       13       24       25
19       7       26       21       23
30       27       9       11       8
12       28       22       29       20
-38818       -31982

3       5       4       2       6
1       16       18       14       15
17       10       13       24       25
19       7       27       21       23
30       28       9       11       8
12       26       22       29       20
-38801       -31978

3       5       4       2       6
1       16       18       14       15
17       10       13       24       25
19       7       27       21       23
30       28       9       11       8
12       26       22       29       20
-38801       -31977

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [11], the wall-clock time for obtaining the output through JJJJ=-31977 was two minutes and a half.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.

[2] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.

[3] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.

[4] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.

[5] Andre R. S. Amaral (2013), A Parallel Ordering Problem in Facilities Layout. Computers and Operations Research, Vol. 40, Issue 12, December 2013, pp. 2930-2939.

[6] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.

[7] Miguel F. Anjos (2012), FLPLIB--Facility Layout Database. Retrieved on September 25 2012 from www.gerad.ca/files/Sites/Anjos/indexFR.html

[8] Miguel F. Anjos, FLPLIB--Facility Layout Database. www.miguelanjos.com.

[9] Philipp Hungerlaender, Miguel F. Anjos (January 2012), A Semidefinite Optimization Approach to Free-Space Multi-Row Facility Layout. Les Cahiers du GERAD. Retrieved from www.gerad.ca/fichiers/cahiers/G-2012-03.pdf

[10] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[11] C. E. Nugent, T. E Vollmann, J. Ruml (1968), An Experimental Comparison of Techniques for the Assignment of Facilities to Locations. Operations Research, Vol. 16, pp. 150-173.

[12] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[13] Jsun Yui Wong (2012, September 17). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to the Corridor Allocation Problem with Eleven Facilities, Third Edition. https://myblogsubstance.typepad.com/substance/2012/09/index.html/

 

Solving Another Parallel Ordering Problem of 30-Facility Layout

Jsun Yui Wong

Adapting to the computer program in Wong [13], the following computer program seeks to solve a parallel row ordering problem (PROP) of 30 facilities with the first row of 7 --due to L n/4 ] of Amaral [5]--facilities and the second row of 23 facilities. The 30x30 matrix below is based on a matrix in Nugent, Vollmann, and Ruml [11]. The 30 lengths of line 17 through line 30 are the first 30 lengths of the 40 lengths in Anjos [8, srflp.40.4].

0 REM DEFDBL A-Z

1 DEFINT I, J, K, X

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), Y(33), C(33), CC(33)

3 DIM HS(49, 49)

4 DIM PE(49, 49)

5 DIM SD(49, 49)

17 HR(1) = 10: HR(2) = 10: HR(3) = 9: HR(4) = 5: HR(5) = 6: HR(6) = 3: HR(7) = 9: HR(8) = 6: HR(9) = 6

18 HR(10) = 4: HR(11) = 9: HR(12) = 9: HR(13) = 7: HR(14) = 3: HR(15) = 1: HR(16) = 3: HR(17) = 10: HR(18) = 5: HR(19) = 1: HR(20) = 2

29 HR(21) = 9: HR(22) = 9: HR(23) = 10

30 HR(24) = 7: HR(25) = 10: HR(26) = 5: HR(27) = 7: HR(28) = 1: HR(29) = 10: HR(30) = 3

31 FOR IL = 1 TO 30

32 FOR JL = 1 TO 30

33 READ HS(IL, JL)

34 NEXT JL

35 NEXT IL
36 REM

42 DATA 9999,3,2,0,0,2,10,5,0,5,2,5,0,0,2,0,5,6,3,0,1,10,0,10,2,1,1,1,0,1

43 DATA 3,9999,4,0,10,4,0,0,2,2,1,0,5,0,0,0,0,2,0,1,6,1,0,1,2,2,5,1,10,5

44 DATA 2,4,9999,3,4,0,5,5,5,1,4,1,0,4,0,4,0,6,3,2,5,5,2,1,0,0,3,1,0,2

45 DATA 0,0,3,9999,0,0,0,2,2,0,6,0,2,5,2,5,1,1,1,1,2,2,4,0,2,0,2,2,5,5

46 DATA 0,10,4,0,9999,5,2,0,0,0,0,2,0,0,0,0,2,1,0,0,2,0,5,1,0,2,1,0,2,1

47 DATA 2,4,0,0,5,9999,1,2,2,1,4,10,10,2,5,5,0,5,0,0,0,10,0,0,0,4,0,10,1,1

48 DATA 10,0,5,0,2,1,9999,10,10,5,10,10,6,0,0,10,2,1,10,1,5,5,2,3,5,0,2,0,1,3

49 DATA 5,0,5,2,0,2,10,9999,1,3,5,0,0,0,2,4,5,2,10,6,0,5,5,2,5,0,5,5,0,2

50 DATA 0,2,5,2,0,2,10,1,9999,10,2,1,5,2,0,3,0,2,0,0,4,0,5,2,0,5,2,2,5,2

51 DATA 5,2,1,0,0,1,5,3,10,9999,5,5,6,0,1,5,5,0,5,2,3,5,0,5,2,10,10,1,5,2

52 DATA 2,1,4,6,0,4,10,5,2,5,9999,0,0,1,2,1,0,2,0,0,0,6,6,0,4,5,3,2,2,10

53 DATA 5,0,1,0,2,10,10,0,1,5,0,9999,5,5,2,0,0,0,0,2,0,4,5,10,1,0,0,0,0,1

54 DATA 0,5,0,2,0,10,6,0,5,6,0,5,9999,2,0,4,2,2,1,0,6,2,1,5,5,0,0,1,5,5

55 DATA 0,0,4,5,0,2,0,0,2,0,1,5,2,9999,2,1,0,5,3,10,0,0,4,2,0,0,4,2,5,5

56 DATA 2,0,0,2,0,5,0,2,0,1,2,2,0,2,9999,4,5,1,0,1,0,5,0,2,0,0,5,1,1,0

57 DATA 0,0,4,5,0,5,10,4,3,5,1,0,4,1,4,9999,0,3,0,2,2,0,2,0,5,0,5,2,5,10

58 DATA 5,0,0,1,2,0,2,5,0,5,0,0,2,0,5,0,9999,2,2,0,0,0,6,5,3,5,0,0,5,1

59 DATA 6,2,6,1,1,5,1,2,2,0,2,0,2,5,1,3,2,9999,5,1,2,10,10,4,0,0,5,0,0,0

60 DATA 3,0,3,1,0,0,10,10,0,5,0,0,1,3,0,0,2,5,9999,0,5,5,1,0,5,2,1,2,10,10

61 DATA 0,1,2,1,0,0,1,6,0,2,0,2,0,10,1,2,0,1,0,9999,5,2,1,3,1,5,6,5,5,3

62 DATA 1,6,5,2,2,0,5,0,4,3,0,0,6,0,0,2,0,2,5,5,9999,4,0,1,0,0,0,5,0,0

63 DATA 10,1,5,2,0,10,5,5,0,5,6,4,2,0,5,0,0,10,5,2,4,9999,5,0,4,4,5,0,2,5

64 DATA 0,0,2,4,5,0,2,5,5,0,6,5,1,4,0,2,6,10,1,1,0,5,9999,0,4,4,1,0,2,2

65 DATA 10,1,1,0,1,0,3,2,2,5,0,10,5,2,2,0,5,4,0,3,1,0,0,9999,5,5,0,1,0,0

66 DATA 2,2,0,2,0,0,5,5,0,2,4,1,5,0,0,5,3,0,5,1,0,4,4,5,9999,1,0,10,1,0

67 DATA 1,2,0,0,2,4,0,0,5,10,5,0,0,0,0,0,5,0,2,5,0,4,4,5,1,9999,0,0,0,0

68 DATA 1,5,3,2,1,0,2,5,2,10,3,0,0,4,5,5,0,5,1,6,0,5,1,0,0,0,9999,0,0,10

69 DATA 1,1,1,2,0,10,0,5,2,1,2,0,1,2,1,2,0,0,2,5,5,0,0,1,10,0,0,9999,2,2

70 DATA 0,10,0,5,2,1,1,0,5,5,2,0,5,5,1,5,5,0,10,5,0,2,2,0,1,0,0,2,9999,2

71 DATA 1,5,2,5,1,1,3,2,2,2,10,1,5,5,0,10,1,0,10,3,0,5,2,0,0,0,10,2,2,9999

87 REM

88 FOR JJJJ = -32000 TO 32221

89 RANDOMIZE JJJJ

90 M = -1D+37

91 FOR J44 = 1 TO 30

93 A(J44) = J44

94 NEXT J44

111 IF RND < 1 / 4 THEN IMAX = 7 ELSE IF RND < 1 / 3 THEN IMAX = 7 ELSE IF RND < 1 / 2 THEN IMAX = 7 ELSE IMAX = 7

128 FOR I = 1 TO 10000

129 FOR KKQQ = 1 TO 30

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

133 III = 1 + FIX(RND * 30)

134 JJJ = 1 + FIX(RND * 30)

137 X(III) = A(JJJ)

139 X(JJJ) = A(III)

231 FOR J44 = 1 TO 30

233 FOR J45 = 1 TO 30

234 IF X(J44) = J45 THEN HHR(J44) = HR(J44) ELSE GOTO 238

237 Y(J45) = J44

238 NEXT J45

253 FOR ISE20 = 1 TO IMAX

254 C(ISE20) = .5 * HHR(Y(ISE20))

258 FOR ISE2000 = ISE20 + 1 TO IMAX

259 C(ISE20) = C(ISE20) + HHR(Y(ISE2000))

263 NEXT ISE2000

269 NEXT ISE20

386 FOR ISE20N = IMAX + 1 TO 30

387 C(ISE20N) = .5 * HHR(Y(ISE20N))

388 FOR ISE2000N = ISE20N + 1 TO 30

389 C(ISE20N) = C(ISE20N) + HHR(Y(ISE2000N))

390 NEXT ISE2000N

391 NEXT ISE20N

395 NEXT J44

401 FOR J77 = 1 TO 7

405 IF X(J77) > 7 THEN 1670

409 NEXT J77

411 PROD = 0

412 FOR J44 = 1 TO 30

413 FOR J45 = J44 + 1 TO 30

414 PROD = PROD - HS(Y(J44), Y(J45)) * ABS(C(J44) - C(J45))

415 NEXT J45

416 NEXT J44

422 P = PROD

1111 IF P <= M THEN 1670

1452 M = P

1453 FOR KLX = 1 TO 30

1454 CC(KLX) = C(KLX)

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1559 IIMAX = IMAX

1657 GOTO 128

1670 NEXT I

1889 IF M < -34915 THEN 1999

1891 PRINT A(1), A(2), A(3), A(4), A(5)

1892 PRINT A(6), A(7), A(8), A(9), A(10)

1893 PRINT A(11), A(12), A(13), A(14), A(15)

1894 PRINT A(16), A(17), A(18), A(19), A(20)

1896 PRINT A(21), A(22), A(23), A(24), A(25)

1897 PRINT A(26), A(27), A(28), A(29), A(30)

1929 PRINT M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [12]. The complete output through JJJJ=-31952 is shown below:

2       6       5       4       7
1       3       17       28       23
14       11       27       19       18
26       8       16       24       20
30       15       10       12       9
13       22       21       29       25
-34913       -31986

2 6 5 4 7
1 3 17 28 23
14 11 27 19 18
26 8 16 24 20
30 15 10 12 9
13 22 21 29 25
-34913 -31971

1       6       5       4       7
2       3       17       28       20
14       11       27       23       18
26       8       16       24       21
30       15       10       12       9
13       19       22       29       25
-34909       -31953

1 6 5 4 7
2 3 17 28 20
14 11 27 23 18
26 8 16 24 21
30 15 10 12 9
13 19 22 29 25
-34909 -31952

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [11], the wall-clock time for obtaining the output through JJJJ=-31952 was four minutes and a half.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.

[2] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.

[3] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.

[4] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.

[5] Andre R. S. Amaral (2013), A Parallel Ordering Problem in Facilities Layout. Computers and Operations Research, Vol. 40, Issue 12, December 2013, pp. 2930-2939.

[6] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.

[7] Miguel F. Anjos (2012), FLPLIB--Facility Layout Database. Retrieved on September 25 2012 from www.gerad.ca/files/Sites/Anjos/indexFR.html

[8] Miguel F. Anjos, FLPLIB--Facility Layout Database. www.miguelanjos.com.

[9] Philipp Hungerlaender, Miguel F. Anjos (January 2012), A Semidefinite Optimization Approach to Free-Space Multi-Row Facility Layout. Les Cahiers du GERAD. Retrieved from www.gerad.ca/fichiers/cahiers/G-2012-03.pdf

[10] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[11] C. E. Nugent, T. E Vollmann, J. Ruml (1968), An Experimental Comparison of Techniques for the Assignment of Facilities to Locations. Operations Research, Vol. 16, pp. 150-173.

[12] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[13] Jsun Yui Wong (2012, September 17). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to the Corridor Allocation Problem with Eleven Facilities, Third Edition. https://myblogsubstance.typepad.com/substance/2012/09/index.html/

 

Solving a Parallel Ordering Problem of 30-Facility Layout

Jsun Yui Wong

Adapting to the computer program in Wong [12], the following computer program seeks to solve the parallel row ordering problem (PROP)--Amaral [5]--of 30 facilities with the first row of 15 facilities and the second row of 15 facilities. The data can be found in Anjos [8].

0 REM DEFDBL A-Z

1 DEFINT I, J, K, X

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), Y(33), C(33), CC(33)

3 DIM HS(49, 49)

4 DIM PE(49, 49)

5 DIM SD(49, 49)

17 HR(1) = 10: HR(2) = 10: HR(3) = 9: HR(4) = 5: HR(5) = 6: HR(6) = 3: HR(7) = 9: HR(8) = 6: HR(9) = 6

18 HR(10) = 4: HR(11) = 9: HR(12) = 9: HR(13) = 7: HR(14) = 3: HR(15) = 1: HR(16) = 3: HR(17) = 10: HR(18) = 5: HR(19) = 1: HR(20) = 2

29 HR(21) = 9: HR(22) = 9: HR(23) = 10

30 HR(24) = 7: HR(25) = 10: HR(26) = 5: HR(27) = 7: HR(28) = 1: HR(29) = 10: HR(30) = 3

31 FOR IL = 1 TO 30

32 FOR JL = 1 TO 30

33 READ HS(IL, JL)

34 NEXT JL

35 NEXT IL
36 REM

42 DATA 9999,3,2,0,0,2,10,5,0,5,2,5,0,0,2,0,5,6,3,0,1,10,0,10,2,1,1,1,0,1

43 DATA 3,9999,4,0,10,4,0,0,2,2,1,0,5,0,0,0,0,2,0,1,6,1,0,1,2,2,5,1,10,5

44 DATA 2,4,9999,3,4,0,5,5,5,1,4,1,0,4,0,4,0,6,3,2,5,5,2,1,0,0,3,1,0,2

45 DATA 0,0,3,9999,0,0,0,2,2,0,6,0,2,5,2,5,1,1,1,1,2,2,4,0,2,0,2,2,5,5

46 DATA 0,10,4,0,9999,5,2,0,0,0,0,2,0,0,0,0,2,1,0,0,2,0,5,1,0,2,1,0,2,1

47 DATA 2,4,0,0,5,9999,1,2,2,1,4,10,10,2,5,5,0,5,0,0,0,10,0,0,0,4,0,10,1,1

48 DATA 10,0,5,0,2,1,9999,10,10,5,10,10,6,0,0,10,2,1,10,1,5,5,2,3,5,0,2,0,1,3

49 DATA 5,0,5,2,0,2,10,9999,1,3,5,0,0,0,2,4,5,2,10,6,0,5,5,2,5,0,5,5,0,2

50 DATA 0,2,5,2,0,2,10,1,9999,10,2,1,5,2,0,3,0,2,0,0,4,0,5,2,0,5,2,2,5,2

51 DATA 5,2,1,0,0,1,5,3,10,9999,5,5,6,0,1,5,5,0,5,2,3,5,0,5,2,10,10,1,5,2

52 DATA 2,1,4,6,0,4,10,5,2,5,9999,0,0,1,2,1,0,2,0,0,0,6,6,0,4,5,3,2,2,10

53 DATA 5,0,1,0,2,10,10,0,1,5,0,9999,5,5,2,0,0,0,0,2,0,4,5,10,1,0,0,0,0,1

54 DATA 0,5,0,2,0,10,6,0,5,6,0,5,9999,2,0,4,2,2,1,0,6,2,1,5,5,0,0,1,5,5

55 DATA 0,0,4,5,0,2,0,0,2,0,1,5,2,9999,2,1,0,5,3,10,0,0,4,2,0,0,4,2,5,5

56 DATA 2,0,0,2,0,5,0,2,0,1,2,2,0,2,9999,4,5,1,0,1,0,5,0,2,0,0,5,1,1,0

57 DATA 0,0,4,5,0,5,10,4,3,5,1,0,4,1,4,9999,0,3,0,2,2,0,2,0,5,0,5,2,5,10

58 DATA 5,0,0,1,2,0,2,5,0,5,0,0,2,0,5,0,9999,2,2,0,0,0,6,5,3,5,0,0,5,1

59 DATA 6,2,6,1,1,5,1,2,2,0,2,0,2,5,1,3,2,9999,5,1,2,10,10,4,0,0,5,0,0,0

60 DATA 3,0,3,1,0,0,10,10,0,5,0,0,1,3,0,0,2,5,9999,0,5,5,1,0,5,2,1,2,10,10

61 DATA 0,1,2,1,0,0,1,6,0,2,0,2,0,10,1,2,0,1,0,9999,5,2,1,3,1,5,6,5,5,3

62 DATA 1,6,5,2,2,0,5,0,4,3,0,0,6,0,0,2,0,2,5,5,9999,4,0,1,0,0,0,5,0,0

63 DATA 10,1,5,2,0,10,5,5,0,5,6,4,2,0,5,0,0,10,5,2,4,9999,5,0,4,4,5,0,2,5

64 DATA 0,0,2,4,5,0,2,5,5,0,6,5,1,4,0,2,6,10,1,1,0,5,9999,0,4,4,1,0,2,2

65 DATA 10,1,1,0,1,0,3,2,2,5,0,10,5,2,2,0,5,4,0,3,1,0,0,9999,5,5,0,1,0,0

66 DATA 2,2,0,2,0,0,5,5,0,2,4,1,5,0,0,5,3,0,5,1,0,4,4,5,9999,1,0,10,1,0

67 DATA 1,2,0,0,2,4,0,0,5,10,5,0,0,0,0,0,5,0,2,5,0,4,4,5,1,9999,0,0,0,0

68 DATA 1,5,3,2,1,0,2,5,2,10,3,0,0,4,5,5,0,5,1,6,0,5,1,0,0,0,9999,0,0,10

69 DATA 1,1,1,2,0,10,0,5,2,1,2,0,1,2,1,2,0,0,2,5,5,0,0,1,10,0,0,9999,2,2

70 DATA 0,10,0,5,2,1,1,0,5,5,2,0,5,5,1,5,5,0,10,5,0,2,2,0,1,0,0,2,9999,2

71 DATA 1,5,2,5,1,1,3,2,2,2,10,1,5,5,0,10,1,0,10,3,0,5,2,0,0,0,10,2,2,9999

87 REM

88 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ

90 M = -1D+37

91 FOR J44 = 1 TO 30

93 A(J44) = J44

94 NEXT J44

111 IF RND < 1 / 4 THEN IMAX = 15 ELSE IF RND < 1 / 3 THEN IMAX = 15 ELSE IF RND < 1 / 2 THEN IMAX = 15 ELSE IMAX = 15

128 FOR I = 1 TO 10000

129 FOR KKQQ = 1 TO 30

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

133 III = 1 + FIX(RND * 30)

134 JJJ = 1 + FIX(RND * 30)

137 X(III) = A(JJJ)

139 X(JJJ) = A(III)

231 FOR J44 = 1 TO 30

233 FOR J45 = 1 TO 30

234 IF X(J44) = J45 THEN HHR(J44) = HR(J44) ELSE GOTO 238

237 Y(J45) = J44

238 NEXT J45

253 FOR ISE20 = 1 TO IMAX

254 C(ISE20) = .5 * HHR(Y(ISE20))

258 FOR ISE2000 = ISE20 + 1 TO IMAX

259 C(ISE20) = C(ISE20) + HHR(Y(ISE2000))

263 NEXT ISE2000

269 NEXT ISE20

386 FOR ISE20N = IMAX + 1 TO 30

387 C(ISE20N) = .5 * HHR(Y(ISE20N))

388 FOR ISE2000N = ISE20N + 1 TO 30

389 C(ISE20N) = C(ISE20N) + HHR(Y(ISE2000N))

390 NEXT ISE2000N

391 NEXT ISE20N

395 NEXT J44

401 FOR J77 = 1 TO 15

405 IF X(J77) > 15 THEN 1670

409 NEXT J77

411 PROD = 0

412 FOR J44 = 1 TO 30

413 FOR J45 = J44 + 1 TO 30

414 PROD = PROD - HS(Y(J44), Y(J45)) * ABS(C(J44) - C(J45))

415 NEXT J45

416 NEXT J44

422 P = PROD

1111 IF P <= M THEN 1670

1452 M = P

1453 FOR KLX = 1 TO 30

1454 CC(KLX) = C(KLX)

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1559 IIMAX = IMAX

1657 GOTO 128

1670 NEXT I

1889 REM IF M < -999999 THEN 1999

1890 REM PRINT A(24), A(25), A(26), A(27), A(28),a(29),a(30)

1891 PRINT A(1), A(2), A(3), A(4), A(5)

1892 PRINT A(6), A(7), A(8), A(9), A(10)

1893 PRINT A(11), A(12), A(13), A(14), A(15)

1894 PRINT A(16), A(17), A(18), A(19), A(20), A(21), A(22), A(23)

1895 GOTO 1919

1896 PRINT A(26), A(27), A(28), A(29), A(30)

1905 PRINT CC(1), CC(2), CC(3), CC(4), CC(5)

1906 PRINT CC(6), CC(7), CC(8), CC(9)

1919 REM PRINT M, JJJJ, IIMAX, A(1), A(2)

1921 PRINT A(24), A(25), A(26), A(27), A(28), A(29), A(30)
1929 PRINT M, JJJJ

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [11]. The output through JJJJ=-31998 is summarized below:
.
.
.
-24826 -32000

.
.
.
-24481           -31999

15       2       3       5       1
10       12       8       4       11
7       14       13       6       9
25       30       19       23       21
16       26       18
29       28       27       20       24
17       22
-23714       -31998

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [11], the wall-clock time for obtaining the output through JJJJ=-31998 was 30 seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.

[2] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.

[3] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.

[4] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.

[5] Andre R. S. Amaral (2013), A Parallel Ordering Problem in Facilities Layout. Computers and Operations Research, Vol. 40, Issue 12, December 2013, pp. 2930-2939.

[6] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.

[7] Miguel F. Anjos (2012), FLPLIB--Facility Layout Database. Retrieved on September 25 2012 from www.gerad.ca/files/Sites/Anjos/indexFR.html

[8] Miguel F. Anjos, FLPLIB--Facility Layout Database. www.miguelanjos.com.

[9] Philipp Hungerlaender, Miguel F. Anjos (January 2012), A Semidefinite Optimization Approach to Free-Space Multi-Row Facility Layout. Les Cahiers du GERAD. Retrieved from www.gerad.ca/fichiers/cahiers/G-2012-03.pdf

[10] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[11] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[12] Jsun Yui Wong (2012, September 17). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to the Corridor Allocation Problem with Eleven Facilities, Third Edition. https://myblogsubstance.typepad.com/substance/2012/09/index.html/

 

Solving Another Instance of a Parallel Ordering Problem of 23-Facility Layout

Jsun Yui Wong

Adapting to the computer program in Wong [12], the following computer program seeks to solve the parallel row ordering problem (PROP) of 23 facilities with the first row of 4 facilities and the second row of 19 facilities in Amaral [5, p. 2938, Table A4]. The data can be found in Amaral [5] and Anjos [8]. This is the instance {P23_e,
L n/5 ]}, Amaral [5, p. 2938].

0 REM DEFDBL A-Z

1 DEFINT I, J, K, X

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), Y(33), C(33), CC(33)

3 DIM HS(49, 49)

4 DIM PE(49, 49)

5 DIM SD(49, 49)

17 HR(1) = 13: HR(2) = 6: HR(3) = 8: HR(4) = 6: HR(5) = 8: HR(6) = 9: HR(7) = 14: HR(8) = 12: HR(9) = 11

18 HR(10) = 5: HR(11) = 6: HR(12) = 11: HR(13) = 12: HR(14) = 11: HR(15) = 14: HR(16) = 6: HR(17) = 12: HR(18) = 7: HR(19) = 8: HR(20) = 13

29 HR(21) = 5: HR(22) = 9: HR(23) = 6

31 FOR IL = 1 TO 23

32 FOR JL = 1 TO 23

33 READ HS(IL, JL)

34 NEXT JL

35 NEXT IL

42 DATA 9999,3,2,0,0,2,10,5,0,5,2,5,0,0,2,0,5,6,3,0,1,10,0

43 DATA 3,9999,4,0,10,4,0,0,2,2,1,0,5,0,0,0,0,2,0,1,6,1,0

44 DATA 2,4,9999,3,4,0,5,5,5,1,4,1,0,4,0,4,0,6,3,2,5,5,2

45 DATA 0,0,3,9999,0,0,0,2,2,0,6,0,2,5,2,5,1,1,1,1,2,2,4

46 DATA 0,10,4,0,9999,5,2,0,0,0,0,2,0,0,0,0,2,1,0,0,2,0,5

47 DATA 2,4,0,0,5,9999,1,2,2,1,4,10,10,2,5,5,0,5,0,0,0,10,0

48 DATA 10,0,5,0,2,1,9999,10,10,5,10,10,6,0,0,10,2,1,10,1,5,5,2

49 DATA 5,0,5,2,0,2,10,9999,1,3,5,0,0,0,2,4,5,2,10,6,0,5,5

50 DATA 0,2,5,2,0,2,10,1,9999,10,2,1,5,2,0,3,0,2,0,0,4,0,5

51 DATA 5,2,1,0,0,1,5,3,10,9999,5,5,6,0,1,5,5,0,5,2,3,5,0

52 DATA 2,1,4,6,0,4,10,5,2,5,9999,0,0,1,2,1,0,2,0,0,0,6,6

53 DATA 5,0,1,0,2,10,10,0,1,5,0,9999,5,5,2,0,0,0,0,2,0,4,5

54 DATA 0,5,0,2,0,10,6,0,5,6,0,5,9999,2,0,4,2,2,1,0,6,2,1

55 DATA 0,0,4,5,0,2,0,0,2,0,1,5,2,9999,2,1,0,5,3,10,0,0,4

56 DATA 2,0,0,2,0,5,0,2,0,1,2,2,0,2,9999,4,5,1,0,1,0,5,0

57 DATA 0,0,4,5,0,5,10,4,3,5,1,0,4,1,4,9999,0,3,0,2,2,0,2

58 DATA 5,0,0,1,2,0,2,5,0,5,0,0,2,0,5,0,9999,2,2,0,0,0,6

59 DATA 6,2,6,1,1,5,1,2,2,0,2,0,2,5,1,3,2,9999,5,1,2,10,10

60 DATA 3,0,3,1,0,0,10,10,0,5,0,0,1,3,0,0,2,5,9999,0,5,5,1

61 DATA 0,1,2,1,0,0,1,6,0,2,0,2,0,10,1,2,0,1,0,9999,5,2,1

62 DATA 1,6,5,2,2,0,5,0,4,3,0,0,6,0,0,2,0,2,5,5,9999,4,0

63 DATA 10,1,5,2,0,10,5,5,0,5,6,4,2,0,5,0,0,10,5,2,4,9999,5

64 DATA 0,0,2,4,5,0,2,5,5,0,6,5,1,4,0,2,6,10,1,1,0,5,9999

88 FOR JJJJ = -32000 TO 32133

89 RANDOMIZE JJJJ

90 M = -1D+37

91 FOR J44 = 1 TO 23

93 A(J44) = J44

94 NEXT J44

111 IF RND < 1 / 4 THEN IMAX = 4 ELSE IF RND < 1 / 3 THEN IMAX = 4 ELSE IF RND < 1 / 2 THEN IMAX = 4 ELSE IMAX = 4

128 FOR I = 1 TO 5000

129 FOR KKQQ = 1 TO 23

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

133 III = 1 + FIX(RND * 23)

134 JJJ = 1 + FIX(RND * 23)

137 X(III) = A(JJJ)

139 X(JJJ) = A(III)

231 FOR J44 = 1 TO 23

233 FOR J45 = 1 TO 23

234 IF X(J44) = J45 THEN HHR(J44) = HR(J44) ELSE GOTO 238

237 Y(J45) = J44

238 NEXT J45

253 FOR ISE20 = 1 TO IMAX

254 C(ISE20) = .5 * HHR(Y(ISE20))

258 FOR ISE2000 = ISE20 + 1 TO IMAX

259 C(ISE20) = C(ISE20) + HHR(Y(ISE2000))

263 NEXT ISE2000

269 NEXT ISE20

386 FOR ISE20N = IMAX + 1 TO 23

387 C(ISE20N) = .5 * HHR(Y(ISE20N))

388 FOR ISE2000N = ISE20N + 1 TO 23

389 C(ISE20N) = C(ISE20N) + HHR(Y(ISE2000N))

390 NEXT ISE2000N

391 NEXT ISE20N

395 NEXT J44

401 FOR J77 = 1 TO 4

405 IF X(J77) > 4 THEN 1670

409 NEXT J77

411 PROD = 0

412 FOR J44 = 1 TO 23

413 FOR J45 = J44 + 1 TO 23

414 PROD = PROD - HS(Y(J44), Y(J45)) * ABS(C(J44) - C(J45))

415 NEXT J45

416 NEXT J44

422 P = PROD

1111 IF P <= M THEN 1670

1452 M = P

1453 FOR KLX = 1 TO 23

1454 CC(KLX) = C(KLX)

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1559 IIMAX = IMAX

1657 GOTO 128

1670 NEXT I

1889 IF M < -29840 THEN 1999

1891 PRINT A(1), A(2), A(3), A(4), A(5)

1892 PRINT A(6), A(7), A(8), A(9), A(10)

1893 PRINT A(11), A(12), A(13), A(14), A(15)

1894 PRINT A(16), A(17), A(18), A(19), A(20), A(21), A(22), A(23)

1895 GOTO 1919

1896 PRINT A(26), A(27), A(28), A(29), A(30)

1905 PRINT CC(1), CC(2), CC(3), CC(4), CC(5)

1906 PRINT CC(6), CC(7), CC(8), CC(9)

1919 PRINT M, JJJJ, IIMAX, A(1), A(2)

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [11]. The complete output through JJJJ=-31898 shown below:

4       3       2       1       22
9       14       21       11       13
18       8       10       7       5
12       23       19       16       6
15       17       20
-29839.5       -31996       4       4       3

4       3       2       1       22
9       14       21       11       13
16       8       10       7       5
12       23       19       20       6
15       17       18
-29810.5       -31904       4       4       3

4       3       2       1       22
9       14       21       11       13
16       8       10       7       5
12       23       19       20       6
15       17       18
-29810.5       -31898       4       4       3

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [11], the wall-clock time for obtaining the output through JJJJ=-31898 was 150 seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.

[2] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.

[3] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.

[4] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.

[5] Andre R. S. Amaral (2013), A Parallel Ordering Problem in Facilities Layout. Computers and Operations Research, Vol. 40, Issue 12, December 2013, pp. 2930-2939.

[6] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.

[7] Miguel F. Anjos (2012), FLPLIB--Facility Layout Database. Retrieved on September 25 2012 from www.gerad.ca/files/Sites/Anjos/indexFR.html

[8] Miguel F. Anjos, FLPLIB--Facility Layout Database. www.miguelanjos.com.

[9] Philipp Hungerlaender, Miguel F. Anjos (January 2012), A Semidefinite Optimization Approach to Free-Space Multi-Row Facility Layout. Les Cahiers du GERAD. Retrieved from www.gerad.ca/fichiers/cahiers/G-2012-03.pdf

[10] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[11] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[12] Jsun Yui Wong (2012, September 17). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to the Corridor Allocation Problem with Eleven Facilities, Third Edition. https://myblogsubstance.typepad.com/substance/2012/09/index.html/

 

Solving Another Instance of a Parallel Ordering Problem of 23-Facility Layout

Jsun Yui Wong

Adapting to the computer program in Wong [12], the following computer program seeks to solve the parallel row ordering problem (PROP) of 23 facilities with the first row of 11 facilities and the second row of 12 facilities from Amaral [5, p. 2938, Table A4]. The data can be found in Amaral [5] and Anjos [8]. This is the instance {P23_b, L n/2 ]}, Amaral [5, p. 2938].

0 REM DEFDBL A-Z

1 DEFINT I, J, K, X

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), Y(33), C(33), CC(33)

3 DIM HS(49, 49)

4 DIM PE(49, 49)

5 DIM SD(49, 49)

17 HR(1) = 6: HR(2) = 7: HR(3) = 7: HR(4) = 11: HR(5) = 13: HR(6) = 14: HR(7) = 5: HR(8) = 5: HR(9) = 7

18 HR(10) = 12: HR(11) = 8: HR(12) = 6: HR(13) = 8: HR(14) = 14: HR(15) = 12: HR(16) = 13: HR(17) = 12: HR(18) = 5: HR(19) = 13: HR(20) = 14

29 HR(21) = 6: HR(22) = 6: HR(23) = 12

31 FOR IL = 1 TO 23

32 FOR JL = 1 TO 23

33 READ HS(IL, JL)

34 NEXT JL

35 NEXT IL

42 DATA 9999,3,2,0,0,2,10,5,0,5,2,5,0,0,2,0,5,6,3,0,1,10,0

43 DATA 3,9999,4,0,10,4,0,0,2,2,1,0,5,0,0,0,0,2,0,1,6,1,0

44 DATA 2,4,9999,3,4,0,5,5,5,1,4,1,0,4,0,4,0,6,3,2,5,5,2

45 DATA 0,0,3,9999,0,0,0,2,2,0,6,0,2,5,2,5,1,1,1,1,2,2,4

46 DATA 0,10,4,0,9999,5,2,0,0,0,0,2,0,0,0,0,2,1,0,0,2,0,5

47 DATA 2,4,0,0,5,9999,1,2,2,1,4,10,10,2,5,5,0,5,0,0,0,10,0

48 DATA 10,0,5,0,2,1,9999,10,10,5,10,10,6,0,0,10,2,1,10,1,5,5,2

49 DATA 5,0,5,2,0,2,10,9999,1,3,5,0,0,0,2,4,5,2,10,6,0,5,5

50 DATA 0,2,5,2,0,2,10,1,9999,10,2,1,5,2,0,3,0,2,0,0,4,0,5

51 DATA 5,2,1,0,0,1,5,3,10,9999,5,5,6,0,1,5,5,0,5,2,3,5,0

52 DATA 2,1,4,6,0,4,10,5,2,5,9999,0,0,1,2,1,0,2,0,0,0,6,6

53 DATA 5,0,1,0,2,10,10,0,1,5,0,9999,5,5,2,0,0,0,0,2,0,4,5

54 DATA 0,5,0,2,0,10,6,0,5,6,0,5,9999,2,0,4,2,2,1,0,6,2,1

55 DATA 0,0,4,5,0,2,0,0,2,0,1,5,2,9999,2,1,0,5,3,10,0,0,4

56 DATA 2,0,0,2,0,5,0,2,0,1,2,2,0,2,9999,4,5,1,0,1,0,5,0

57 DATA 0,0,4,5,0,5,10,4,3,5,1,0,4,1,4,9999,0,3,0,2,2,0,2

58 DATA 5,0,0,1,2,0,2,5,0,5,0,0,2,0,5,0,9999,2,2,0,0,0,6

59 DATA 6,2,6,1,1,5,1,2,2,0,2,0,2,5,1,3,2,9999,5,1,2,10,10

60 DATA 3,0,3,1,0,0,10,10,0,5,0,0,1,3,0,0,2,5,9999,0,5,5,1

61 DATA 0,1,2,1,0,0,1,6,0,2,0,2,0,10,1,2,0,1,0,9999,5,2,1

62 DATA 1,6,5,2,2,0,5,0,4,3,0,0,6,0,0,2,0,2,5,5,9999,4,0

63 DATA 10,1,5,2,0,10,5,5,0,5,6,4,2,0,5,0,0,10,5,2,4,9999,5

64 DATA 0,0,2,4,5,0,2,5,5,0,6,5,1,4,0,2,6,10,1,1,0,5,9999

88 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ

90 M = -1D+37

91 FOR J44 = 1 TO 23

93 A(J44) = J44

94 NEXT J44

111 IF RND < 1 / 4 THEN IMAX = 4 ELSE IF RND < 1 / 3 THEN IMAX = 4 ELSE IF RND < 1 / 2 THEN IMAX = 4 ELSE IMAX = 4

128 FOR I = 1 TO 5000

129 FOR KKQQ = 1 TO 23

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

133 III = 1 + FIX(RND * 23)

134 JJJ = 1 + FIX(RND * 23)

137 X(III) = A(JJJ)

139 X(JJJ) = A(III)

231 FOR J44 = 1 TO 23

233 FOR J45 = 1 TO 23

234 IF X(J44) = J45 THEN HHR(J44) = HR(J44) ELSE GOTO 238

237 Y(J45) = J44

238 NEXT J45

253 FOR ISE20 = 1 TO IMAX

254 C(ISE20) = .5 * HHR(Y(ISE20))

258 FOR ISE2000 = ISE20 + 1 TO IMAX

259 C(ISE20) = C(ISE20) + HHR(Y(ISE2000))

263 NEXT ISE2000

269 NEXT ISE20

386 FOR ISE20N = IMAX + 1 TO 23

387 C(ISE20N) = .5 * HHR(Y(ISE20N))

388 FOR ISE2000N = ISE20N + 1 TO 23

389 C(ISE20N) = C(ISE20N) + HHR(Y(ISE2000N))

390 NEXT ISE2000N

391 NEXT ISE20N

395 NEXT J44

401 FOR J77 = 1 TO 4

405 IF X(J77) > 4 THEN 1670

409 NEXT J77

411 PROD = 0

412 FOR J44 = 1 TO 23

413 FOR J45 = J44 + 1 TO 23

414 PROD = PROD - HS(Y(J44), Y(J45)) * ABS(C(J44) - C(J45))

415 NEXT J45

416 NEXT J44

422 P = PROD

1111 IF P <= M THEN 1670

1452 M = P

1453 FOR KLX = 1 TO 23

1454 CC(KLX) = C(KLX)

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1559 IIMAX = IMAX

1657 GOTO 128

1670 NEXT I

1889 IF M < -29915 THEN 1999

1891 PRINT A(1), A(2), A(3), A(4), A(5)

1892 PRINT A(6), A(7), A(8), A(9), A(10)

1893 PRINT A(11), A(12), A(13), A(14), A(15)

1894 PRINT A(16), A(17), A(18), A(19), A(20), A(21), A(22), A(23)

1895 GOTO 1919

1896 PRINT A(26), A(27), A(28), A(29), A(30)

1905 PRINT CC(1), CC(2), CC(3), CC(4), CC(5)

1906 PRINT CC(6), CC(7), CC(8), CC(9)

1919 PRINT M, JJJJ, IIMAX, A(1), A(2)

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [11]. The complete output through JJJJ=-31926 shown below:

1       3       2       4       23
9       15       17       13       12
20       10       11       6       7
8       22       19       16       5
14       18       21
-29892.5       -31986       4       1       3

1       3       2       4       23
9       15       17       13       12
20       10       11       6       7
8       22       19       16       5
14       18       21
-29892.5       -31956       4       1       3

1       3       2       4       23
9       15       17       13       12
20       10       11       6       7
8       22       19       16       5
14       18       21
-29892.5       -31926       4       1       3

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [11], the wall-clock time for obtaining the output through JJJJ=-31926 was 110 seconds.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.

[2] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.

[3] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.

[4] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.

[5] Andre R. S. Amaral (2013), A Parallel Ordering Problem in Facilities Layout. Computers and Operations Research, Vol. 40, Issue 12, December 2013, pp. 2930-2939.

[6] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.

[7] Miguel F. Anjos (2012), FLPLIB--Facility Layout Database. Retrieved on September 25 2012 from www.gerad.ca/files/Sites/Anjos/indexFR.html

[8] Miguel F. Anjos, FLPLIB--Facility Layout Database. www.miguelanjos.com.

[9] Philipp Hungerlaender, Miguel F. Anjos (January 2012), A Semidefinite Optimization Approach to Free-Space Multi-Row Facility Layout. Les Cahiers du GERAD. Retrieved from www.gerad.ca/fichiers/cahiers/G-2012-03.pdf

[10] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[11] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[12] Jsun Yui Wong (2012, September 17). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to the Corridor Allocation Problem with Eleven Facilities, Third Edition. https://myblogsubstance.typepad.com/substance/2012/09/index.html/

 

Solving Another Instance of a Parallel Ordering Problem of 23-Facility Layout

Jsun Yui Wong

Adapting to the computer program in Wong [12], the following computer program seeks to solve the parallel row ordering problem (PROP) of 23 facilities with the first row of 11 facilities and the second row of 12 facilities from Amaral [5, p. 2935, Table A1]. The data can be found in Amaral [5] and Anjos [8]. This is the instance {P23_b,
L n/2 ]}, Amaral [5, p. 2935].

0 REM DEFDBL A-Z

1 DEFINT I, J, K, X

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), Y(33), C(33), CC(33)

3 DIM HS(49, 49)

4 DIM PE(49, 49)

5 DIM SD(49, 49)

17 HR(1) = 6: HR(2) = 7: HR(3) = 7: HR(4) = 11: HR(5) = 13: HR(6) = 14: HR(7) = 5: HR(8) = 5: HR(9) = 7

18 HR(10) = 12: HR(11) = 8: HR(12) = 6: HR(13) = 8: HR(14) = 14: HR(15) = 12: HR(16) = 13: HR(17) = 12: HR(18) = 5: HR(19) = 13: HR(20) = 14

29 HR(21) = 6: HR(22) = 6: HR(23) = 12

31 FOR IL = 1 TO 23

32 FOR JL = 1 TO 23

33 READ HS(IL, JL)

34 NEXT JL

35 NEXT IL

42 DATA 9999,3,2,0,0,2,10,5,0,5,2,5,0,0,2,0,5,6,3,0,1,10,0

43 DATA 3,9999,4,0,10,4,0,0,2,2,1,0,5,0,0,0,0,2,0,1,6,1,0

44 DATA 2,4,9999,3,4,0,5,5,5,1,4,1,0,4,0,4,0,6,3,2,5,5,2

45 DATA 0,0,3,9999,0,0,0,2,2,0,6,0,2,5,2,5,1,1,1,1,2,2,4

46 DATA 0,10,4,0,9999,5,2,0,0,0,0,2,0,0,0,0,2,1,0,0,2,0,5

47 DATA 2,4,0,0,5,9999,1,2,2,1,4,10,10,2,5,5,0,5,0,0,0,10,0

48 DATA 10,0,5,0,2,1,9999,10,10,5,10,10,6,0,0,10,2,1,10,1,5,5,2

49 DATA 5,0,5,2,0,2,10,9999,1,3,5,0,0,0,2,4,5,2,10,6,0,5,5

50 DATA 0,2,5,2,0,2,10,1,9999,10,2,1,5,2,0,3,0,2,0,0,4,0,5

51 DATA 5,2,1,0,0,1,5,3,10,9999,5,5,6,0,1,5,5,0,5,2,3,5,0

52 DATA 2,1,4,6,0,4,10,5,2,5,9999,0,0,1,2,1,0,2,0,0,0,6,6

53 DATA 5,0,1,0,2,10,10,0,1,5,0,9999,5,5,2,0,0,0,0,2,0,4,5

54 DATA 0,5,0,2,0,10,6,0,5,6,0,5,9999,2,0,4,2,2,1,0,6,2,1

55 DATA 0,0,4,5,0,2,0,0,2,0,1,5,2,9999,2,1,0,5,3,10,0,0,4

56 DATA 2,0,0,2,0,5,0,2,0,1,2,2,0,2,9999,4,5,1,0,1,0,5,0

57 DATA 0,0,4,5,0,5,10,4,3,5,1,0,4,1,4,9999,0,3,0,2,2,0,2

58 DATA 5,0,0,1,2,0,2,5,0,5,0,0,2,0,5,0,9999,2,2,0,0,0,6

59 DATA 6,2,6,1,1,5,1,2,2,0,2,0,2,5,1,3,2,9999,5,1,2,10,10

60 DATA 3,0,3,1,0,0,10,10,0,5,0,0,1,3,0,0,2,5,9999,0,5,5,1

61 DATA 0,1,2,1,0,0,1,6,0,2,0,2,0,10,1,2,0,1,0,9999,5,2,1

62 DATA 1,6,5,2,2,0,5,0,4,3,0,0,6,0,0,2,0,2,5,5,9999,4,0

63 DATA 10,1,5,2,0,10,5,5,0,5,6,4,2,0,5,0,0,10,5,2,4,9999,5

64 DATA 0,0,2,4,5,0,2,5,5,0,6,5,1,4,0,2,6,10,1,1,0,5,9999

88 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ

90 M = -1D+37

91 FOR J44 = 1 TO 23

93 A(J44) = J44

94 NEXT J44

111 IF RND < 1 / 4 THEN IMAX = 11 ELSE IF RND < 1 / 3 THEN IMAX = 11 ELSE IF RND < 1 / 2 THEN IMAX = 11 ELSE IMAX = 11

128 FOR I = 1 TO 5000

129 FOR KKQQ = 1 TO 23

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

133 III = 1 + FIX(RND * 23)

134 JJJ = 1 + FIX(RND * 23)

137 X(III) = A(JJJ)

139 X(JJJ) = A(III)

231 FOR J44 = 1 TO 23

233 FOR J45 = 1 TO 23

234 IF X(J44) = J45 THEN HHR(J44) = HR(J44) ELSE GOTO 238

237 Y(J45) = J44

238 NEXT J45

253 FOR ISE20 = 1 TO IMAX

254 C(ISE20) = .5 * HHR(Y(ISE20))

258 FOR ISE2000 = ISE20 + 1 TO IMAX

259 C(ISE20) = C(ISE20) + HHR(Y(ISE2000))

263 NEXT ISE2000

269 NEXT ISE20

386 FOR ISE20N = IMAX + 1 TO 23

387 C(ISE20N) = .5 * HHR(Y(ISE20N))

388 FOR ISE2000N = ISE20N + 1 TO 23

389 C(ISE20N) = C(ISE20N) + HHR(Y(ISE2000N))

390 NEXT ISE2000N

391 NEXT ISE20N

395 NEXT J44

401 FOR J77 = 1 TO 11

405 IF X(J77) > 11 THEN 1670

409 NEXT J77

411 PROD = 0

412 FOR J44 = 1 TO 23

413 FOR J45 = J44 + 1 TO 23

414 PROD = PROD - HS(Y(J44), Y(J45)) * ABS(C(J44) - C(J45))

415 NEXT J45

416 NEXT J44

422 P = PROD

1111 IF P <= M THEN 1670

1452 M = P

1453 FOR KLX = 1 TO 23

1454 CC(KLX) = C(KLX)

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1559 IIMAX = IMAX

1657 GOTO 128

1670 NEXT I

1889 IF M < -15900 THEN 1999

1891 PRINT A(1), A(2), A(3), A(4), A(5)

1892 PRINT A(6), A(7), A(8), A(9), A(10)

1893 PRINT A(11), A(12), A(13), A(14), A(15)

1894 PRINT A(16), A(17), A(18), A(19), A(20), A(21), A(22), A(23)

1895 GOTO 1919

1896 PRINT A(26), A(27), A(28), A(29), A(30)

1905 PRINT CC(1), CC(2), CC(3), CC(4), CC(5)

1906 PRINT CC(6), CC(7), CC(8), CC(9)

1919 PRINT M, JJJJ, IIMAX, A(1), A(2)

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [11]. The complete output through JJJJ=-31875 is shown below:

7       10       3       1       11
9       6       5       4       8
2       20       21       13       22
14       23       16       17       12
19       18       15
-15876.5       -31980       11       7       10

7 10 3 1 11
9 6 5 4 8
2 20 21 13 22
14 23 16 17 12
19 18 15
-15876.5 -31950 11 7 10

5       10       4       1       11
9       6       3       7       8
2       19       21       13       22
20       23       16       15       12
18       17       14
-15802.5       -31947       11       5       10

7 10 3 1 11
9 6 5 4 8
2 20 21 13 22
14 23 16 17 12
19 18 15
-15876.5 -31907 11 7 10

7 10 3 1 11
9 6 5 4 8
2 20 21 13 22
14 23 16 17 12
19 18 15
-15876.5 -31883 11 7 10

5 10 4 1 11
9 6 3 7 8
2 19 21 13 22
20 23 16 15 12
18 17 14
-15802.5       -31875       11       5       10

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [11], the wall-clock time for obtaining the output through JJJJ=-31875 was three minutes.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.

[2] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.

[3] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.

[4] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.

[5] Andre R. S. Amaral (2013), A Parallel Ordering Problem in Facilities Layout. Computers and Operations Research, Vol. 40, Issue 12, December 2013, pp. 2930-2939.

[6] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.

[7] Miguel F. Anjos (2012), FLPLIB--Facility Layout Database. Retrieved on September 25 2012 from www.gerad.ca/files/Sites/Anjos/indexFR.html

[8] Miguel F. Anjos, FLPLIB--Facility Layout Database. www.miguelanjos.com.

[9] Philipp Hungerlaender, Miguel F. Anjos (January 2012), A Semidefinite Optimization Approach to Free-Space Multi-Row Facility Layout. Les Cahiers du GERAD. Retrieved from www.gerad.ca/fichiers/cahiers/G-2012-03.pdf

[10] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[11] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[12] Jsun Yui Wong (2012, September 17). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to the Corridor Allocation Problem with Eleven Facilities, Third Edition. https://myblogsubstance.typepad.com/substance/2012/09/index.html/

 

Solving a Parallel Ordering Problem of 23-Facility Layout

Jsun Yui Wong

Adapting to the computer program in Wong [12], the following computer program seeks to solve the parallel row ordering problem (PROP) of 23 facilities with the first row of 11 facilities and the second row of 12 facilities in Amaral [5, p. 2935, Table A1]. The data can be found in Amaral [5] and Anjos [8]. This is the instance {P23_a, L n/2 ]}, Amaral [5, p. 2935].

0 REM DEFDBL A-Z

1 DEFINT I, J, K, X, A

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), Y(33), C(33), CC(33)

3 DIM HS(49, 49)

4 DIM PE(49, 49)

5 DIM SD(49, 49)

17 HR(1) = 3: HR(2) = 9: HR(3) = 3: HR(4) = 7: HR(5) = 3: HR(6) = 7: HR(7) = 5: HR(8) = 9: HR(9) = 6

18 HR(10) = 5: HR(11) = 3: HR(12) = 9: HR(13) = 3: HR(14) = 7: HR(15) = 3: HR(16) = 7: HR(17) = 5: HR(18) = 9: HR(19) = 6: HR(20) = 5

29 HR(21) = 3: HR(22) = 9: HR(23) = 3

31 FOR IL = 1 TO 23

32 FOR JL = 1 TO 23

33 READ HS(IL, JL)

34 NEXT JL

35 NEXT IL

42 DATA 9999,3,2,0,0,2,10,5,0,5,2,5,0,0,2,0,5,6,3,0,1,10,0

43 DATA 3,9999,4,0,10,4,0,0,2,2,1,0,5,0,0,0,0,2,0,1,6,1,0

44 DATA 2,4,9999,3,4,0,5,5,5,1,4,1,0,4,0,4,0,6,3,2,5,5,2

45 DATA 0,0,3,9999,0,0,0,2,2,0,6,0,2,5,2,5,1,1,1,1,2,2,4

46 DATA 0,10,4,0,9999,5,2,0,0,0,0,2,0,0,0,0,2,1,0,0,2,0,5

47 DATA 2,4,0,0,5,9999,1,2,2,1,4,10,10,2,5,5,0,5,0,0,0,10,0

48 DATA 10,0,5,0,2,1,9999,10,10,5,10,10,6,0,0,10,2,1,10,1,5,5,2

49 DATA 5,0,5,2,0,2,10,9999,1,3,5,0,0,0,2,4,5,2,10,6,0,5,5

50 DATA 0,2,5,2,0,2,10,1,9999,10,2,1,5,2,0,3,0,2,0,0,4,0,5

51 DATA 5,2,1,0,0,1,5,3,10,9999,5,5,6,0,1,5,5,0,5,2,3,5,0

52 DATA 2,1,4,6,0,4,10,5,2,5,9999,0,0,1,2,1,0,2,0,0,0,6,6

53 DATA 5,0,1,0,2,10,10,0,1,5,0,9999,5,5,2,0,0,0,0,2,0,4,5

54 DATA 0,5,0,2,0,10,6,0,5,6,0,5,9999,2,0,4,2,2,1,0,6,2,1

55 DATA 0,0,4,5,0,2,0,0,2,0,1,5,2,9999,2,1,0,5,3,10,0,0,4

56 DATA 2,0,0,2,0,5,0,2,0,1,2,2,0,2,9999,4,5,1,0,1,0,5,0

57 DATA 0,0,4,5,0,5,10,4,3,5,1,0,4,1,4,9999,0,3,0,2,2,0,2

58 DATA 5,0,0,1,2,0,2,5,0,5,0,0,2,0,5,0,9999,2,2,0,0,0,6

59 DATA 6,2,6,1,1,5,1,2,2,0,2,0,2,5,1,3,2,9999,5,1,2,10,10

60 DATA 3,0,3,1,0,0,10,10,0,5,0,0,1,3,0,0,2,5,9999,0,5,5,1

61 DATA 0,1,2,1,0,0,1,6,0,2,0,2,0,10,1,2,0,1,0,9999,5,2,1

62 DATA 1,6,5,2,2,0,5,0,4,3,0,0,6,0,0,2,0,2,5,5,9999,4,0

63 DATA 10,1,5,2,0,10,5,5,0,5,6,4,2,0,5,0,0,10,5,2,4,9999,5

64 DATA 0,0,2,4,5,0,2,5,5,0,6,5,1,4,0,2,6,10,1,1,0,5,9999

88 FOR JJJJ = -32000 TO 32000

89 RANDOMIZE JJJJ

90 M = -1D+37

91 FOR J44 = 1 TO 23

93 A(J44) = J44

94 NEXT J44

111 IF RND < 1 / 4 THEN IMAX = 11 ELSE IF RND < 1 / 3 THEN IMAX = 11 ELSE IF RND < 1 / 2 THEN IMAX = 11 ELSE IMAX = 11

128 FOR I = 1 TO 5000

129 FOR KKQQ = 1 TO 23

130 X(KKQQ) = A(KKQQ)

131 NEXT KKQQ

133 III = 1 + FIX(RND * 23)

134 JJJ = 1 + FIX(RND * 23)

137 X(III) = A(JJJ)

139 X(JJJ) = A(III)

231 FOR J44 = 1 TO 23

233 FOR J45 = 1 TO 23

234 IF X(J44) = J45 THEN HHR(J44) = HR(J44) ELSE GOTO 238

237 Y(J45) = J44

238 NEXT J45

253 FOR ISE20 = 1 TO IMAX

254 C(ISE20) = .5 * HHR(Y(ISE20))

258 FOR ISE2000 = ISE20 + 1 TO IMAX

259 C(ISE20) = C(ISE20) + HHR(Y(ISE2000))

263 NEXT ISE2000

269 NEXT ISE20

386 FOR ISE20N = IMAX + 1 TO 23

387 C(ISE20N) = .5 * HHR(Y(ISE20N))

388 FOR ISE2000N = ISE20N + 1 TO 23

389 C(ISE20N) = C(ISE20N) + HHR(Y(ISE2000N))

390 NEXT ISE2000N

391 NEXT ISE20N

395 NEXT J44

401 FOR J77 = 1 TO 11

405 IF X(J77) > 11 THEN 1670

409 NEXT J77

411 PROD = 0

412 FOR J44 = 1 TO 23

413 FOR J45 = J44 + 1 TO 23

414 PROD = PROD - HS(Y(J44), Y(J45)) * ABS(C(J44) - C(J45))

415 NEXT J45

416 NEXT J44

422 P = PROD

1111 IF P <= M THEN 1670

1452 M = P

1453 FOR KLX = 1 TO 23

1454 CC(KLX) = C(KLX)

1455 A(KLX) = X(KLX)

1456 NEXT KLX

1559 IIMAX = IMAX

1657 GOTO 128

1670 NEXT I

1889 IF M < -10255 THEN 1999

1891 PRINT A(1), A(2), A(3), A(4), A(5)

1892 PRINT A(6), A(7), A(8), A(9), A(10)

1893 PRINT A(11), A(12), A(13), A(14), A(15)

1894 PRINT A(16), A(17), A(18), A(19), A(20), A(21), A(22), A(23)

1895 GOTO 1919

1896 PRINT A(26), A(27), A(28), A(29), A(30)

1905 PRINT CC(1), CC(2), CC(3), CC(4), CC(5)

1906 PRINT CC(6), CC(7), CC(8), CC(9)

1919 PRINT M, JJJJ, IIMAX, A(1), A(2)

1999 NEXT JJJJ

This computer program was run with qb64v1000-win [11]. The complete output through JJJJ=-31764 is shown below:

8       11       3       1       10
9       5       2       6       7
4       23       19       12       21
14       22       15       17       13
18       20       16
-10251       -31937       11       8       11

7       11       3       1       10
9       5       2       6       8
4       23       20       12       21
14       22       15       17       13
18       19       16
-10242       -31884       11       7       11

7       11       3       1       10
9       5       2       6       8
4       23       20       12       21
14       22       15       17       13
18       19       16
-10242       -31764       11       7       11

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [11], the wall-clock time for obtaining the output through JJJJ=-31764 was five minutes.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.

[2] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.

[3] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.

[4] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.

[5] Andre R. S. Amaral (2013), A Parallel Ordering Problem in Facilities Layout. Computers and Operations Research, Vol. 40, Issue 12, December 2013, pp. 2930-2939.

[6] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.

[7] Miguel F. Anjos (2012), FLPLIB--Facility Layout Database. Retrieved on September 25 2012 from www.gerad.ca/files/Sites/Anjos/indexFR.html

[8] Miguel F. Anjos, FLPLIB--Facility Layout Database. www.miguelanjos.com.

[9] Philipp Hungerlaender, Miguel F. Anjos (January 2012), A Semidefinite Optimization Approach to Free-Space Multi-Row Facility Layout. Les Cahiers du GERAD. Retrieved from www.gerad.ca/fichiers/cahiers/G-2012-03.pdf

[10] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[11] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[12] Jsun Yui Wong (2012, September 17). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to the Corridor Allocation Problem with Eleven Facilities, Third Edition. https://myblogsubstance.typepad.com/substance/2012/09/index.html/

 

A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Double-Row Facility Layout Problem of 25 Facilities, Second Edition

Jsun Yui Wong

Based on the computer program in [42], the following computer program seeks to solve Instance N-25-05 on page 11 of Ghosh and Kothari [16]; the data are from Anjos [6]. This is a corridor allocation problem [4].

Line 91 through line 109 produce different starting solutions, good or bad.

To reach optimality with one computer run is the goal of line 88 and line 111, which are

88 FOR JJJJ = -32000 TO 32000 STEP .1
and 111 IF RND < 1 / 4 THEN IMAX = 10 ELSE IF RND < 1 / 3 THEN IMAX = 11 ELSE IF RND < 1 / 2 THEN IMAX = 12 ELSE IMAX = 13, respectively.

0 REM DEFDBL A-Z
1 DEFINT I, K, X

2 DIM B(99), N(99), A(2002), H(99), L(99), U(99), X(2002), D(111), P(111), PS(33), J44(2002), J(99), AA(99), HR(32), HHR(32), Y(33), C(33), CC(33)
3 DIM HS(49, 49)
4 DIM PE(49, 49)
5 DIM SD(49, 49)
26 HR(1) = 6: HR(2) = 2: HR(3) = 4: HR(4) = 3: HR(5) = 5
27 HR(6) = 5: HR(7) = 3: HR(8) = 1: HR(9) = 5: HR(10) = 3
28 HR(11) = 4: HR(12) = 5: HR(13) = 6: HR(14) = 5: HR(15) = 2
29 HR(16) = 3: HR(17) = 6: HR(18) = 6: HR(19) = 3: HR(20) = 5
30 HR(21) = 1: HR(22) = 3: HR(23) = 5: HR(24) = 1: HR(25) = 2
31 FOR IL = 1 TO 25
32 FOR JL = 1 TO 25
33 READ HS(IL, JL)
34 NEXT JL
35 NEXT IL
61 DATA 999,3,2,0,0,10,5,0,5,2,0,0,2,0,5,3,0,1,10,0,2,1,1,1,0
62 DATA 3,999,4,0,10,0,0,2,2,1,5,0,0,0,0,0,1,6,1,0,2,2,5,1,10
63 DATA 2,4,999,3,4,5,5,5,1,4,0,4,0,4,0,3,2,5,5,2,0,0,3,1,0
64 DATA 0,0,3,999,0,0,2,2,0,6,2,5,2,5,1,1,1,2,2,4,2,0,2,2,5
66 DATA 0,10,4,0,999,2,0,0,0,0,0,0,0,0,2,0,0,2,0,5,0,2,1,0,2
67 DATA 10,0,5,0,2,999,10,10,5,10,6,0,0,10,2,10,1,5,5,2,5,0,2,0,1
68 DATA 5,0,5,2,0,10,999,1,3,5,0,0,2,4,5,10,6,0,5,5,5,0,5,5,0
69 DATA 0,2,5,2,0,10,1,999,10,2,5,2,0,3,0,0,0,4,0,5,0,5,2,2,5
70 DATA 5,2,1,0,0,5,3,10,999,5,6,0,1,5,5,5,2,3,5,0,2,10,10,1,5
71 DATA 2,1,4,6,0,10,5,2,5,999,0,1,2,1,0,0,0,0,6,6,4,5,3,2,2
72 DATA 0,5,0,2,0,6,0,5,6,0,999,2,0,4,2,1,0,6,2,1,5,0,0,1,5
73 DATA 0,0,4,5,0,0,0,2,0,1,2,999,2,1,0,3,10,0,0,4,0,0,4,2,5
74 DATA 2,0,0,2,0,0,2,0,1,2,0,2,999,4,5,0,1,0,5,0,0,0,5,1,1
75 DATA 0,0,4,5,0,10,4,3,5,1,4,1,4,999,0,0,2,2,0,2,5,0,5,2,5
76 DATA 5,0,0,1,2,2,5,0,5,0,2,0,5,0,999,2,0,0,0,6,3,5,0,0,5
77 DATA 3,0,3,1,0,10,10,0,5,0,1,3,0,0,2,999,0,5,5,1,5,2,1,2,10
78 DATA 0,1,2,1,0,1,6,0,2,0,0,10,1,2,0,0,999,5,2,1,1,5,6,5,5
79 DATA 1,6,5,2,2,5,0,4,3,0,6,0,0,2,0,5,5,999,4,0,0,0,0,5,0
80 DATA 10,1,5,2,0,5,5,0,5,6,2,0,5,0,0,5,2,4,999,5,4,4,5,0,2
81 DATA 0,0,2,4,5,2,5,5,0,6,1,4,0,2,6,1,1,0,5,999,4,4,1,0,2
82 DATA 2,2,0,2,0,5,5,0,2,4,5,0,0,5,3,5,1,0,4,4,999,1,0,10,1
83 DATA 1,2,0,0,2,0,0,5,10,5,0,0,0,0,5,2,5,0,4,4,1,999,0,0,0
84 DATA 1,5,3,2,1,2,5,2,10,3,0,4,5,5,0,1,6,0,5,1,0,0,999,0,0
85 DATA 1,1,1,2,0,0,5,2,1,2,1,2,1,2,0,2,5,5,0,0,10,0,0,999,2
86 DATA 0,10,0,5,2,1,0,5,5,2,5,5,1,5,5,10,5,0,2,2,1,0,0,2,999
88 FOR JJJJ = -32000 TO 32000 STEP .1

89 RANDOMIZE JJJJ
90 M = -1D+37
91 FOR J44 = 1 TO 25
93 A(J44) = J44
94 NEXT J44
100 FOR JOC = 1 TO 50
101 FFF = 1 + FIX(RND * 25)
102 GGG = 1 + FIX(RND * 25)
103 A(FFFF) = A(FFF)
104 A(FFF) = A(GGG)
107 A(GGG) = A(FFFF)
109 NEXT JOC
111 IF RND < 1 / 4 THEN IMAX = 10 ELSE IF RND < 1 / 3 THEN IMAX = 11 ELSE IF RND < 1 / 2 THEN IMAX = 12 ELSE IMAX = 13

126 REM IMAR=10+FIX(RND*1000)
128 FOR I = 1 TO 500

129 FOR KKQQ = 1 TO 25
130 X(KKQQ) = A(KKQQ)
131 NEXT KKQQ
133 III = 1 + FIX(RND * 25)
134 JJJ = 1 + FIX(RND * 25)
221 X(III) = A(JJJ)
223 X(JJJ) = A(III)
231 FOR J44 = 1 TO 25
233 FOR J45 = 1 TO 25
234 IF X(J44) = J45 THEN HHR(J44) = HR(J44) ELSE GOTO 238
237 Y(J45) = J44
238 NEXT J45
253 FOR ISE20 = 1 TO IMAX
254 C(ISE20) = .5 * HHR(Y(ISE20))
258 FOR ISE2000 = ISE20 + 1 TO IMAX
259 C(ISE20) = C(ISE20) + HHR(Y(ISE2000))
263 NEXT ISE2000
269 NEXT ISE20
386 FOR ISE20N = IMAX + 1 TO 25
387 C(ISE20N) = .5 * HHR(Y(ISE20N))
388 FOR ISE2000N = ISE20N + 1 TO 25
389 C(ISE20N) = C(ISE20N) + HHR(Y(ISE2000N))
390 NEXT ISE2000N
391 NEXT ISE20N
395 NEXT J44
411 PROD = 0
412 FOR J44 = 1 TO 25
413 FOR J45 = J44 + 1 TO 25
414 PROD = PROD - HS(Y(J44), Y(J45)) * ABS(C(J44) - C(J45))
415 NEXT J45
416 NEXT J44
422 P = PROD
1111 IF P <= M THEN 1670
1452 M = P
1453 FOR KLX = 1 TO 25
1454 CC(KLX) = C(KLX)
1455 A(KLX) = X(KLX)
1456 NEXT KLX
1559 IIMAX = IMAX
1657 GOTO 128
1670 NEXT I
1889 IF M < -7819 THEN 1999

1890 REM GOTO 1919
1891 PRINT A(1), A(2), A(3), A(4), A(5)
1892 PRINT A(6), A(7), A(8), A(9), A(10)
1893 PRINT A(11), A(12), A(13), A(14), A(15)
1894 PRINT A(16), A(17), A(18), A(19), A(20)
1895 PRINT A(21), A(22), A(23), A(24), A(25)

1905 PRINT CC(1), CC(2), CC(3), CC(4), CC(5)

1906 PRINT CC(6), CC(7), CC(8), CC(9), CC(10)
1907 PRINT CC(11), CC(12), CC(13), CC(14), CC(15)
1908 PRINT CC(16), CC(17), CC(18), CC(19), CC(20)
1909 PRINT CC(21), CC(22), CC(23), CC(24), CC(25)
1919 PRINT M, JJJJ, IIMAX
1999 NEXT JJJJ

This computer program was run with qb64v1000-win [35]. The complete output through JJJJ=-31569.79 is shown below:

11      24      4      3      12
8      17      20      10      5
23      1      13      15      21
19      14      25      18      16
7      22      2      6      9
42.5 37.5 33.5 30 26 5
24.5 23.5 20.5 17 13.5
8 2.5 46 .40 34.5
29.5 25.5 22.5 19.5 17.5
16 13.5 10 7 3
-7819 -31884.15 12

10 24 15 14 11
19 5 20 9 16
23 12 1 3 21
7 2 25 6 4
18 22 13 17 8
46 40 34.5 29.5 25.5
22.5 19.5 17 13.5 8
2.5 42.5 37.5 33.5 30
26.5 24.5 23.5 20.5 17.5
16 13.5 10 7 3
-7819 -31569.79 11

Above there is no rounding by hand; it is just straight copying by hand from the monitor screen.

On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM and qb64v1000-win [35], the wall-clock time for obtaining the output through JJJJ=-31569.79 was 35 minutes.

Acknowledgment

I would like to acknowledge the encouragement of Roberta Clark and Tom Clark.

References

[1] Andre R. S. Amaral (2006), On the Exact Solution of a Facility Layout Problem. European Journal of Operational Research 173 (2006), pp. 508-518.

[2] Andre R. S. Amaral (2008), An Exact Approach to the One-Dimensional Facility Layout Problem. Operations Research, Vol. 56, No. 4 (July-August, 2008), pp. 1026-1033.

[3] Andre R. S. Amaral (2011), Optimal Solutions for the Double Row Layout Problem. Optimization Letters, DOI 10.1007/s11590-011-0426-8, published on line 30 November 2011, Springer-Verlag 2011.

[4] Andre R. S. Amaral (2012), The Corridor Allocation Problem. Computers and Operations Research 39 (2012), pp. 3325-3330.

[5] Miguel F. Anjos, Anthony Vannelli, Computing Globally Optimal Solutions for Single-Row Layout Problems Using Semidefinite Programming and Cutting Planes. INFORMS Journal on Computing, Vol. 20, No. 4, Fall 2008, pp. 611-617.

[6] Miguel F. Anjos (2012), FLPLIB--Facility Layout Database. Retrieved on September 25 2012 from www.gerad.ca/files/Sites/Anjos/indexFR.html

[7] David L. Applegate, Robert E. Bixby, Vasek Chvatal, William J. Cook, The Traveling Salesman Problem: A Computational Study. Princeton and Oxford: Princeton University Press, 2006.

[8] L. L. Barachet (1957), Graphic Solution of the Travelling Salesman Problem. Operations Research, Vol. 5, pp. 841-845.

[9] T. Y. Chen, H. C. Chen (2009), Mixed-Discrete Structural Optimization Using a Rank-Niche Evolution Strategy, Engineering Optimization, 41:1, 39-58.

[10] Leon Cooper, U. N. Bhat, L. J. LeBlanc, Introduction to Operations Research Models. W. B. Saunders Company, 1977.

[11] George B. Dantzig, Discrete-Variable Extremum Problems. Operations Research, Vol. 5, No. 2 (Apr., 1957), pp. 266-277.

[12] G. Dantzig, R. Fulkerson, S. Johnson, Solution of a Large-Scale Traveling-Salesman Problem.
Operations Research, Vol. 2, No. 4 (Nov., 1954), pp. 393-410.

[13] Zvi Drezner (1980), DISCON: A New Method for the Layout Problem. Operations Research, Vol. 28, No. 6 (Nov. - Dec., 1980), pp. 1375-1384.

[14] W. J. Fabrycky, P. M. Ghare, P. E. Torgersen, Applied Operations Research and Management Science. Englewood Cliffs, N. J. 07362: Prentice-Hall, Inc., 1984.

[15] Billy E. Gillett, Introduction to Operations Research: A Computer-Oriented Algorithmic Approach. McGraw-Hill, Inc., 1976.

[16] Diptesh Ghosh, Ravi Kothari, Population Heuristics for the Corridor Allocation Problem, W.P. No. 2012-09-02, September 2012. Retrieved on September 14 2012 from Google search.

[17] Martin Grotschel, Olaf Holland (1991), Solution of Large-Scale Symmetric Travelling Salesman Problems, Mathematical Programming 51, pp. 141-202.

[18] Sunderesh S. Heragu, Andrew Kusiak (1988), Machine Layout Problem in Flexible Manufacturing Systems. Operations Research, Vol. 36, No. 2, Operations Research in Manufacturing (Mar. - Apr., 1988), pp. 258-268.

[19] S. S. Heragu, A. Kusiak (1991), Efficient Models for the Facility Layout Problem. European Journal of Operational Research, Vol. 53 (1991), pp. 1-13.

[20] S. S. Heragu, Facilities Design, Third Edition. Boca Raton, Florida: CRC Press, 2008.

[21] Philipp Hungerlaender, Miguel F. Anjos (January 2012), A Semidefinite Optimization Approach to Free-Space Multi-Row Facility Layout. Les Cahiers du GERAD. Retrieved from www.gerad.ca/fichiers/cahiers/G-2012-03.pdf

[22] Michael Junger, Thomas M. Liebling, Dennis Naddef, George L. Nemhauser, William R. Pulleybank, Gerhart Reinelt, Giovanni Rinaldi, Lawrence A. Wolsey--Editors, 50 Years of Integer Programming 1958-2008. Berlin: Springer, 2010.

[23] R. L. Karg, G. L. Thompson (1964), A Heuristic Approach to Solving Travelling Salesman Problems. Management Science, Vol. 10, Number 2, pp. 225-248.

[24] K. Ravi Kumar, George C. Hadjinicola, Ting-li Lin (1995), A Heuristic Procedure for the Single-Row Facility Layout Problem. European Journal of Operational Research 87 (1995) 65-73.

[25] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.

[26] E. L. Lawler, M. D. Bell, A Method for Solving Discrete Optimization Problems. Operations Research, Vol. 14, No. 6 (Nov.-Dec., 1966), pp. 1098-1112.

[27] Robert F. Love, Jsun Y. Wong (1976), On Solving a One-Dimensional Space Allocation Problem with Integer Programming. INFOR, Vol. 14, No. 2, June 1976, pp. 139-143.

[28] Microsoft Corp., BASIC, Second Edition (May 1982), Version 1.10. Boca Raton, Florida: IBM Corp., Personal Computer, P. O. Box 1328-C, Boca Raton, Florida 33432, 1981.

[29] Katta G. Murty, Operations Research: Deterministic Optimization Models. Prentice Hall, Upper Saddle River, New Jersey 07458, 1994.

[30] C. E. Nugent, T. E. Vollmann, J. Ruml (1968), An Experimental Comparison of Techniques for the Assignment of Facilities to Locations, Operations Research 16 (1968), pp. 150-173.

[31] Christopher J. Picone, Wilbert E. Wilhelm (1986), "A Perturbation Scheme To Improve Hillier's Solution to the Facilities Layout Problem": a Clarification. Management Science Vol. 32, No. 2, February 1986, pp. 253-254.

[32] Cliff T. Ragsdale, Spreadsheet Modeling and Decision Analysis, Fifth Edition. South-Westen, Cengage Learning, 2008.

[33] H. H. Rosenbrock (1960), An Automatic Method for Finding the Greatest or Least Value of a Function, Computer Journal Vol. 3, pp. 175-184.

[34] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. - Oct., 1969), pp. 812-826.

[35] Wikipedia, QB64, https://en.wikipedia.org/wiki/QB64

[36] Wayne L. Winston, Munirpallam Venkataramanan, Introduction to Mathematical Programming: Operations Research, Volume One, Fourth Edition. Pacific Grove, CA 93950: Brooks/Cole--Thomson Learning, 2003.

[37] Wayne L. Winston, S. Christian Albright, Practical Management Science, Third Edition. Thomson/South-Western, 2007.

[38] Jsun Yui Wong (2009, July 18). An Integer Programming Computer Program Applied to One-Dimensional Space Allocation. Retrieved from http://wongsllllblog.blogspot.com/2009/07/

[39] Jsun Yui Wong (2009, November 2). A Computer Program for Relative Location of Circular Facilities. Retrieved on September 3, 2012, from http://wongsllllblog.blogspot.ca/2009/11/

[40] Jsun Yui Wong (2009, December 18). A Heuristic Nonlinear Integer Solver Applied to a Problem of Assignment of Facilities to Locations. Retrieved from http://wongsnewnewblog.blogspot.ca/2009/12/

[41] Jsun Yui Wong (2011, October 27). A Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a New Formulation of a Twelve-City Traveling Salesman Problem, Sixth Edition. Retrieved from http://computationalresultsfromcomputerprograms.wordpress.com/2011/10/27/

[42] Jsun Yui Wong (2012, October 02). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Double-Row Facility Layout Problem of 25 Facilities. Retrieved from https://myblogsubstance.typepad.com/substance/2012/10/index.html

[43] Ji-Hui Zhang, Xin-He Xu (1999), An Efficient Evolutionary Programming Algorithm. Computers and Operations Research 26, 645-663.