Wireless Network Design: Optimization Models and Solution ...

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8 The Design of Partially Survivable Networks 195 LP Heuristic 1 Heuristic 2 Heuristic 3 Col. Gen. Gap Time Gap Time Gap Time Gap Time (abs) (sec) (abs) (sec) (abs) (sec) (abs) (sec) 123397.81 2383.19 3.37 1018.19 17.60 1018.19 6.40 577.19 0.23 48404.98 1736.02 1.38 3868.02 8.02 2152.02 2.24 475.02 0.17 69759.00 441.00 3.45 441.00 3.37 441.00 3.49 111.00 0.17 113425.00 857.00 3.26 1146.00 52.61 1146.00 5.92 687.00 0.63 106138.00 671.00 2.67 7610.00 24.71 671.00 5.14 563.00 0.17 98469.34 1902.66 1.58 5106.66 22.98 1264.66 2.50 1200.66 0.63 182132.00 0.00 8.64 0.00 8.64 0.00 8.64 0.00 0.17 119684.00 3393.00 1.45 2189.00 2.11 2189.00 2.18 453.00 0.00 99243.35 2085.65 3.32 2981.65 52.03 2085.65 5.78 453.65 0.86 68149.79 766.21 5.05 1326.21 28.10 766.21 9.49 150.21 0.17 Table 8.2 Results from Data Set I LP Heuristic 1 Heuristic 2 Heuristic 3 Col. Gen. Gap Time Gap Time Gap Time Gap Time (abs) (sec) (abs) (sec) (abs) (sec) (abs) (sec) 236426.14 23.50 147.86 335.23 41.30 147.86 0.75 346227.56 1229.44 24.53 1229.44 866.85 709.44 47.48 189.44 1.21 151202.63 23.76 870.38 511.84 43.44 270.38 0.46 245311.00 0.00 23.29 0.00 23.29 0.00 23.29 0.00 0.23 414656.88 975.13 33.74 1623.13 153.13 921.13 56.62 273.13 3.51 225457.56 4435.44 28.13 1495.44 79.70 1936.44 54.21 256.44 2.53 206971.95 2656.05 20.18 4084.05 484.31 3489.05 36.50 361.05 1.09 352435.78 10590.22 27.31 11319.22 657.53 10590.22 48.75 573.22 7.26 204829.08 3214.92 21.91 4222.92 452.49 3739.92 42.18 295.92 11.00 288682.34 3591.66 17.92 5791.66 33.01 466.66 5.12 Table 8.3 Results from Data Set II ring while considering diversity requirements. It was reduced to a relatively small binary knapsack problem, and even the largest problem tried solved in a few milliseconds. The second problem extended the problem to a multi-period setting, and a column generation approach was used for solution. It solved very large problems efficiently, both in terms of time and solution quality. The third problem introduced capacity limitations at the hubs. A column generation procedure was proposed and the results from preliminary experiments look encouraging. Acknowledgements Portions of this chapter have appeared in print previously. I thank J. MacGregor Smith for providing the Mathematica code for data generation and heuristics used in Kubat et al. [9].
196 Syam Menon References 1. Balas, E., Zemel, E.: An Algorithm for Large Zero–One Knapsack Problems. Operations Research 28, 1130–1154 (1980) 2. Barnhart, C., Johnson, E., Nemhauser, G., Savelsbergh, M., Vance, P.: Branch and Price : Column Generation for Solving Huge Integer Programs. Operations Research 46, 316–329 (1998) 3. Cardwell, R., Monma, C., Wu, T.: Computer Aided Design Procedures for Survivable Fiber Optic Telephone Networks. IEEE Journal on Selective Areas in Communications 7(8), 1188– 1197 (1989) 4. Dutta, A., Kubat, P.: Design of Partially Survivable Networks for Cellular Telecommunications Systems. European Journal of Operations Research 118, 52–64 (1999) 5. Gilmore, P., Gomory, R.: A Linear Programming Approach to the Cutting Stock Problem. Operations Research 9, 849–859 (1961) 6. Gilmore, P., Gomory, R.: A Linear Programming Approach to the Cutting Stock Problem–Part II. Operations Research 11, 863–888 (1963) 7. Knuth, D.: The Art of Computer Programming, Volume 2: Seminumerical Algorithms. Third edn. Addison-Wesley Longman, Reading, MA (1997) 8. Kubat, P., Smith, J.: A Multi-period Network Design Problem for Cellular Telecommunication Systems. European Journal of Operations Research 134, 439–456 (2001) 9. Kubat, P., Smith, J., Yum, C.: Design of Cellular Networks with Diversity and Capacity Constraints. IEEE Transactions on Reliability 49(2), 165–175 (2000) 10. Land, A., Doig, A.: An Automatic Method for Solving Discrete Programming Problems. Econometrica 28(2), 497–520 (1960) 11. Lee, W.: Mobile Cellular Telecommunications: Analog and Digital Systems. Second edn. McGraw-Hill, New York (1995) 12. LINDO Systems, Inc.: LINDO Callable Library User’s Manual (2001) 13. Menon, S., Amiri, A.: Designing Partially Survivable Cellular Telecommunications Networks. Information Technology and Management 6(2/3), 293–303 (2005) 14. Menon, S., Amiri, A.: Multi-period Cellular Network Design via Price Influenced Simulated Annealing (PISA). IEEE Transactions on Systems, Man and Cybernetics – Part B 36(3), 600—610 (2006) 15. Pisinger, D.: A Minimal Algorithm for the 0-1 Knapsack Problem. Operations Research 45, 758–767 (1997) 16. Pisinger, D.: An Exact Algorithm for Large Multiple Knapsack Problems. European Journal of Operations Research 114, 528–541 (1999) 17. Sosonsky, J., Wu, T.: SONET Ring Applications for Survivable Fiber Loop Networks. IEEE Communications Magazine pp. 51–58 (1991) 18. Vance, P., Barnhart, C., Johnson, E., Nemhauser, G.: Airline Crew Scheduling : A New Formulation and Decomposition Algorithm. Operations Research 45, 188–200 (1997)
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International Series in Operations
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Editors Jeff Kennington Eli Olinick
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vi Preface assistance. The work of
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viii Contents 3.4.1 Experiments to
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x Contents 8.3.2 The Solution Proce
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xii Contents References . . . . . .
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List of Contributors Khaldoun Al Ag
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List of Contributors xvii Nitin Sal
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Chapter 1 Introduction to Optimizat
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1 Introduction to Optimization in W
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1 Introduction to Optimization in W
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Part I Background
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10 Dinesh Rajan The goal of this ch
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12 Dinesh Rajan The source encoder
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14 Dinesh Rajan ping (FH) CDMA, and
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16 Dinesh Rajan of SNR is given in
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18 Dinesh Rajan 2.3 Information The
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20 Dinesh Rajan Perror ≤ e −nE
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22 Dinesh Rajan For instance, for t
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24 Dinesh Rajan 2.4.1 Block Codes I
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26 Dinesh Rajan B(z) = 1 � (1 + z
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28 Dinesh Rajan layers. Two of the
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30 Dinesh Rajan nel can support a p
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32 Dinesh Rajan 2.5.2 Physical laye
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34 Dinesh Rajan matched filter for
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36 Dinesh Rajan codes) leads to bet
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38 Dinesh Rajan a length N Inverse
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40 Dinesh Rajan the transmission of
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42 Dinesh Rajan setting, a node may
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44 Dinesh Rajan weighting of the si
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46 Dinesh Rajan 29. Oestges, C., Cl
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48 K. V. S. Hari terrain of operati
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50 K. V. S. Hari fading is calculat
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54 K. V. S. Hari Fig. 3.1 Normalize
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56 K. V. S. Hari Table 3.3 Impulse
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58 K. V. S. Hari where m antennas a
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60 K. V. S. Hari 3.5.2.4 Dominant R
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62 K. V. S. Hari March, 1984-13 Sep
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64 K. V. S. Hari 47. Van Rees, J.:
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Part II Optimization Problems for N
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102 Eli Olinick carrying capacity.
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104 Eli Olinick 93 85 160 16 38 21
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106 Eli Olinick Amaldi et al. [7] p
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108 Eli Olinick levels (possibly le
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110 Eli Olinick gmℓ ∑ ∑ m∈M
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112 Eli Olinick Using bk to denote
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114 Eli Olinick 5.3.5 Additional De
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116 Eli Olinick capacity, provide n
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118 Eli Olinick ficult optimization
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Chapter 6 Optimization Based WLAN M
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6 Optimization Based WLAN Modeling
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10 Integer Programming Models for P
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11 Improving Network Connectivity i
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12 Simulation-Based Methods for Net
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296 Hang Jin and Li Guo formance in
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298 Hang Jin and Li Guo (ISI) as lo
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300 Hang Jin and Li Guo ARQ (HARQ)
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302 Hang Jin and Li Guo 13.3 Radio
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304 Hang Jin and Li Guo Table 13.2
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306 Hang Jin and Li Guo is transmit
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308 Hang Jin and Li Guo uplink powe
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312 Hang Jin and Li Guo where ρ mi
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314 Hang Jin and Li Guo 13.4.4.3 Ex
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318 Hang Jin and Li Guo 1. Guarante
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14 Cross Layer Scheduling in Wirele
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15 Major Trends in Cellular Network
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