Jsun Yui Wong
The following computer program seeks to solve Instance N-17 of Hungerlaender [22]; the data are from Anjos [6].
Line 100 through line 109 produce different starting solutions, good or bad. Because of line 111, the first row can only have 17 facilities.
Reference 42 presents another comnputational procedure.
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)
27 HR(1)=1:HR(2)=1:HR(3)=1:HR(4)=1:HR(5)=1:HR(6)=1:HR(7)=1:HR(8)=1: HR(9)=1:HR(10)=1
28 HR(11)=1:HR(12)=1:HR(13)=1:HR(14)=1:HR(15)=1:HR(16)=1
29 HR(17)=1
31 FOR IL=1 TO 17
32 FOR JL=1 TO 17
33 READ HS(IL,JL)
34 NEXT JL
35 NEXT IL
56 DATA 9999,0,10,0,10,4,20,6,2,10,10,10,0,0,10,8,8
57 DATA 0,9999,6,20,10,2,10,2,4,8,4,10,0,20,20,6,0
58 DATA 10,6,9999,4,0,10,4,8,8,10,0,0,0,10,2,0,0
59 DATA 0,20,4,9999,2,0,10,4,2,0,20,4,4,0,4,2,10
60 DATA 10,10,0, 2,9999,10,12,10,4,10,4,0,10,2,2,2,10
61 DATA 4,2,10, 0,10,9999,10,4,2,12,0,0,20,0,4,0,2
62 DATA 20,10,4, 10,12, 10,9999,0,0,0,10,20,4,4,10,2,4
63 DATA 6,2,8, 4,10, 4,0,9999,2,2,20,20,4,0,20,4,10
64 DATA 2,4,8, 2,4, 2,0, 2,9999,4,0,6,10,10,0,10,0
65 DATA 10,8,10, 0,10, 12,0, 2,4,9999,10,10,0,10,2,0,0
66 DATA 10,4,0, 20,4, 0,10, 20,0, 10,9999,10,4,10,2,20,0
67 DATA 10,10,0, 4,0, 0,20, 20,6, 10,10,9999,4,20,10,0,2
68 DATA 0,0,0, 4,10, 20,4, 4,10, 0,4,4,9999,4,4,2,0
69 DATA 0,20,10, 0,2, 0,4, 0,10, 10,10,20,4,9999,10,10,2
70 DATA 10,20,2, 4,2, 4,10, 20,0, 2,2,10,4,10,9999,6,0
71 DATA 8,6,0, 2,2, 0,2, 4,10, 0,20,0,2,10,6,9999,0
72 DATA 8,0,0, 10,10, 2,4, 10,0, 0,0,2,0,2,0,0,9999
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1D+37
91 FOR J44=1 TO 17
93 A(J44)=J44
94 NEXT J44
100 FOR JOC=1 TO 50
101 FFF=1+FIX(RND*17)
102 GGG=1+FIX(RND*17)
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=17 ELSE IF RND<1/3 THEN IMAX=17 ELSE IF RND<1/2 THEN IMAX=17 ELSE IMAX=17
126 REM IMAR=10+FIX(RND*1000)
128 FOR I=1 TO 200
129 FOR KKQQ=1 TO 17
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 III=1+FIX(RND*17)
134 JJJ=1+FIX(RND*17)
137 X(III)=A(JJJ)
139 X(JJJ)=A(III)
231 FOR J44=1 TO 17
233 FOR J45=1 TO 17
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
411 PROD=0
412 FOR J44=1 TO 17
413 FOR J45=J44+1 TO 17
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 17
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<-3390 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)
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)
1919 PRINT M,JJJJ,IIMAX,A(1),A(2)
1999 NEXT JJJJ
This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter. The complete output through JJJJ=-31942 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.
7 11 4 16 6
3 8 13 1 5
14 9 2 10 12
15 17
16.5 15.5 14.5 13.5 12.5
11.5 10.5 9.5 8.5 7.5
6.5 5.5 4.5 3.5 2.5
1.5 .5
-3388 -31981 17 7 11
7 11 4 16 6
2 8 13 3 5
14 9 1 10 12
15 17
16.5 15.5 14.5 13.5 12.5
11.5 10.5 9.5 8.5 7.5
6.5 5.5 4.5 3.5 2.5
1.5 .5
-3390 -31976 17 7 11
11 7 14 2 12
16 10 5 15 13
4 9 17 8 6
3 1
16.5 15.5 14.5 13.5 12.5
11.5 10.5 9.5 8.5 7.5
6.5 5.5 4.5 3.5 2.5
1.5 .5
-3390 -31955 17 11 7
11 7 14 2 12
16 10 5 15 13
4 9 17 8 6
3 1
16.5 15.5 14.5 13.5 12.5
11.5 10.5 9.5 8.5 7.5
6.5 5.5 4.5 3.5 2.5
1.5 .5
-3390 -31951 17 11 7
11 7 14 2 12
15 10 5 17 13
4 9 16 8 6
3 1
16.5 15.5 14.5 13.5 12.5
11.5 10.5 9.5 8.5 7.5
6.5 5.5 4.5 3.5 2.5
1.5 .5
-3388 -31942 17 11 7
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time from JJJJ=-32000 through JJJJ=-31942 was seven minutes.
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] Philipp Hungerlaender (April 2012), Single-Row Equidistant Facility Layout as a Special Case of Single-Row Facility Layout. Retrieved from www.optimization-online.org./DB_HTML/2012/04/3432.html
[23] 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.
[24] R. L. Karg, G. L. Thompson (1964), A Heuristic Approach to Solving Travelling Salesman Problems. Management Science, Vol. 10, Number 2, pp. 225-248.
[25] 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.
[26] A. H. Land, A. G. Doig, An Automatic Method of Solving Discrete Programming Problems. Econometrica, Vol. 28, No. 3 (Jul., 1960), pp. 497-520.
[27] 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.
[28] 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.
[29] 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.
[30] Katta G. Murty, Operations Research: Deterministic Optimization Models. Prentice Hall, Upper Saddle River, New Jersey 07458, 1994.
[31] 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.
[32] 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.
[33] Cliff T. Ragsdale, Spreadsheet Modeling and Decision Analysis, Fifth Edition. South-Westen, Cengage Learning, 2008.
[34] H. H. Rosenbrock (1960), An Automatic Method for Finding the Greatest or Least Value of a Function, Computer Journal Vol. 3, pp. 175-184.
[35] Donald M. Simmons (1969), One-Dimensional Space Allocation: An Ordering Algorithm. Operations Research, Vol. 17, No. 5 (Sep. - Oct., 1969), pp. 812-826.
[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, September 27). A General Nonlinear Integer/Discrete/Continuous Programming Solver Applied to a Linear Ordering Problem with 24 Facilities. Retrieved from http://myblogsubstance.typepad.com/substance/2012/09
[43] Ji-Hui Zhang, Xin-He Xu (1999), An Efficient Evolutionary Programming Algorithm. Computers and Operations Research 26, 645-663.
The following computer program seeks to solve Instance N-17 of Hungerlaender [22]; the data are from Anjos [6].
Line 100 through line 109 produce different starting solutions, good or bad. Because of line 111, the first row can only have 17 facilities.
Reference 42 presents another comnputational procedure.
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)
27 HR(1)=1:HR(2)=1:HR(3)=1:HR(4)=1:HR(5)=1:HR(6)=1:HR(7)=1:HR(8)=1: HR(9)=1:HR(10)=1
28 HR(11)=1:HR(12)=1:HR(13)=1:HR(14)=1:HR(15)=1:HR(16)=1
29 HR(17)=1
31 FOR IL=1 TO 17
32 FOR JL=1 TO 17
33 READ HS(IL,JL)
34 NEXT JL
35 NEXT IL
56 DATA 9999,0,10,0,10,4,20,6,2,10,10,10,0,0,10,8,8
57 DATA 0,9999,6,20,10,2,10,2,4,8,4,10,0,20,20,6,0
58 DATA 10,6,9999,4,0,10,4,8,8,10,0,0,0,10,2,0,0
59 DATA 0,20,4,9999,2,0,10,4,2,0,20,4,4,0,4,2,10
60 DATA 10,10,0, 2,9999,10,12,10,4,10,4,0,10,2,2,2,10
61 DATA 4,2,10, 0,10,9999,10,4,2,12,0,0,20,0,4,0,2
62 DATA 20,10,4, 10,12, 10,9999,0,0,0,10,20,4,4,10,2,4
63 DATA 6,2,8, 4,10, 4,0,9999,2,2,20,20,4,0,20,4,10
64 DATA 2,4,8, 2,4, 2,0, 2,9999,4,0,6,10,10,0,10,0
65 DATA 10,8,10, 0,10, 12,0, 2,4,9999,10,10,0,10,2,0,0
66 DATA 10,4,0, 20,4, 0,10, 20,0, 10,9999,10,4,10,2,20,0
67 DATA 10,10,0, 4,0, 0,20, 20,6, 10,10,9999,4,20,10,0,2
68 DATA 0,0,0, 4,10, 20,4, 4,10, 0,4,4,9999,4,4,2,0
69 DATA 0,20,10, 0,2, 0,4, 0,10, 10,10,20,4,9999,10,10,2
70 DATA 10,20,2, 4,2, 4,10, 20,0, 2,2,10,4,10,9999,6,0
71 DATA 8,6,0, 2,2, 0,2, 4,10, 0,20,0,2,10,6,9999,0
72 DATA 8,0,0, 10,10, 2,4, 10,0, 0,0,2,0,2,0,0,9999
88 FOR JJJJ=-32000 TO 32000
89 RANDOMIZE JJJJ
90 M=-1D+37
91 FOR J44=1 TO 17
93 A(J44)=J44
94 NEXT J44
100 FOR JOC=1 TO 50
101 FFF=1+FIX(RND*17)
102 GGG=1+FIX(RND*17)
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=17 ELSE IF RND<1/3 THEN IMAX=17 ELSE IF RND<1/2 THEN IMAX=17 ELSE IMAX=17
126 REM IMAR=10+FIX(RND*1000)
128 FOR I=1 TO 200
129 FOR KKQQ=1 TO 17
130 X(KKQQ)=A(KKQQ)
131 NEXT KKQQ
133 III=1+FIX(RND*17)
134 JJJ=1+FIX(RND*17)
137 X(III)=A(JJJ)
139 X(JJJ)=A(III)
231 FOR J44=1 TO 17
233 FOR J45=1 TO 17
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
411 PROD=0
412 FOR J44=1 TO 17
413 FOR J45=J44+1 TO 17
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 17
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<-3390 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)
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)
1919 PRINT M,JJJJ,IIMAX,A(1),A(2)
1999 NEXT JJJJ
This BASIC computer program was run via basica/D of Microsoft's GW-BASIC 3.11 interpreter. The complete output through JJJJ=-31942 is shown below. What follows is a hand copy from the computer-monitor screen; immediately below there is no rounding by hand.
7 11 4 16 6
3 8 13 1 5
14 9 2 10 12
15 17
16.5 15.5 14.5 13.5 12.5
11.5 10.5 9.5 8.5 7.5
6.5 5.5 4.5 3.5 2.5
1.5 .5
-3388 -31981 17 7 11
7 11 4 16 6
2 8 13 3 5
14 9 1 10 12
15 17
16.5 15.5 14.5 13.5 12.5
11.5 10.5 9.5 8.5 7.5
6.5 5.5 4.5 3.5 2.5
1.5 .5
-3390 -31976 17 7 11
11 7 14 2 12
16 10 5 15 13
4 9 17 8 6
3 1
16.5 15.5 14.5 13.5 12.5
11.5 10.5 9.5 8.5 7.5
6.5 5.5 4.5 3.5 2.5
1.5 .5
-3390 -31955 17 11 7
11 7 14 2 12
16 10 5 15 13
4 9 17 8 6
3 1
16.5 15.5 14.5 13.5 12.5
11.5 10.5 9.5 8.5 7.5
6.5 5.5 4.5 3.5 2.5
1.5 .5
-3390 -31951 17 11 7
11 7 14 2 12
15 10 5 17 13
4 9 16 8 6
3 1
16.5 15.5 14.5 13.5 12.5
11.5 10.5 9.5 8.5 7.5
6.5 5.5 4.5 3.5 2.5
1.5 .5
-3388 -31942 17 11 7
On a personal computer with a Pentium Dual-Core CPU E5200 @2.50GHz, 2.50 GHz, 960 MB of RAM, and the IBM basica/D interpreter, version GW BASIC 3.11, the wall-clock time from JJJJ=-32000 through JJJJ=-31942 was seven minutes.
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.
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