Abstract
In this paper, we first give the definition of randomized time-varying knapsack problems (\(\textit{RTVKP}\)) and its mathematic model, and analyze the character about the various forms of \(\textit{RTVKP}\). Next, we propose three algorithms for \(\textit{RTVKP}\): (1) an exact algorithm with pseudo-polynomial time based on dynamic programming; (2) a 2-approximation algorithm for \(\textit{RTVKP}\) based on greedy algorithm; (3) a heuristic algorithm by using elitists model based on genetic algorithms. Finally, we advance an evaluation criterion for the algorithm which is used for solving dynamic combinational optimization problems, and analyze the virtue and shortage of three algorithms above by using the criterion. For the given three instances of \(\textit{RTVKP}\), the simulation computation results coincide with the theory analysis.
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Acknowledgments
Support in part by NSF of China (No. 11271257), the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20121303110005), the NSF of Hebei Province (No. A2013205021).
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Appendix: Three large-scale instances of \(fixRTVKP\)
Appendix: Three large-scale instances of \(fixRTVKP\)
For describing the instance of \(fixRTVKP\) succinctly, after sub-problem 0-1 \(KP(n, C_{i-1}, V_{i-1}, W_{i-1})\) changed to sub-problem 0-1 \(KP(n, C_i, V_i, W_i)\), let \((V_i\bigcup W_i)\setminus (V_{i-1}\bigcup W_{i-1})=\{V(l_k,w_k)| 1\le k\le N_1\}\bigcup \{W(l_j,w_j)|1\le j\le N_2\}\), where \(N_1+N_2 \le Threshold, V(l_k,v_k)\) represent that the profit of \(l_k\mathrm{th}\) item of sub-problem 0-1 \(KP_{i-1} (n, C_{i-1}, V_{i-1}, W_{i-1})\) has changed to \(v_k\) in sub-problem 0-1 \(KP_i (n, C_i, V_i, W_i). W(l_j,w_j)\) represent that the weight of \(l_j\mathrm{th}\) item of sub-problem 0-1 \(KP_{i-1} (n, C_{i-1}, V_{i-1}, W_{i-1})\) has changed to \(w_j\) in sub-problem 0-1 \(KP_i (n, C_i, V_i, W_i)\).
Instance 1 of \(fixRTVKP\).
Initial profit set of items is \(V_0[1\ldots 50]=\{\)220, 208, 198, 192, 180, 180, 165, 162, 160, 158, 155, 130, 125, 122, 120, 118, 115, 110, 105, 101, 100, 100, 98, 96, 95, 90, 88, 82, 80, 77, 75, 73, 72, 70, 69, 66, 65, 63, 60, 58, 56, 50, 30, 20, 15, 10, 8, 5, 3, 1\(\}\).
Initial weight set of items is \(W_0[1\ldots 50]=\{\)80, 82, 85, 70, 72, 70, 66, 50, 55, 25, 50, 55, 40, 48, 50, 32, 22, 60, 30, 32, 40, 38, 35, 32, 25, 28, 30, 22, 50, 30, 45, 30, 60, 50, 20, 65, 20, 25, 30, 10, 20, 25, 15, 10, 10, 10, 4, 4, 2, 1\(\}\).
Initial knapsack capacity is \(C_0=1000\).
The random variation period is 2 s, and \([A_v, B_v]=[1, 225], [A_w, B_w]=[1, 87], [A_c, B_c]=[930,1395]\).
The number of subproblems is \(m=10\), and \(Threshold\le 10\).
The random oscillation change of profit, weight of items and knapsack capacity are following:
\(C_1=1220, C_2=1000, C_3=1341, C_4=1285, C_5=1285, C_6=931, C_7=1119, C_8=947, C_9=1043\).
\((V_1\bigcup W_1)\setminus (V_0\bigcup W_0)=\{W\)(18,71), \(W\)(20,65), \(V\)(9,188), \(W\)(6,45), \(W\)(28,44), \(W\)(46,24), \(V\)(37,217), \(W\)(3,67), \(W\)(33,22), \(V\)(19,96)\(\}\).
\((V_2\bigcup W_2)\setminus (V_1\bigcup W_1)=\{W\)(39,43), \(W\)(18,26), \(W\)(45,81), \(W\)(23,58), \(W\)(15,4), \(W\)(4,83), \(V\)(45,38), \(V\)(38,35), \(W\)(42,38), \(V\)(17,111)\(\}\).
\((V_3\bigcup W_3)\setminus (V_2\bigcup W_2)=\{W\)(39,5), \(W\)(43,38), \(V\)(47,181), \(W\)(30,11), \(W\)(7,43), \(W\)(49,55), \(W\)(35,32), \(V\)(41,17), \(V\)(32,209), \(W\)(40,6)\(\}\).
\((V_4\bigcup W_4)\setminus (V_3\bigcup W_3)=\{V\)(19,58), \(V\)(42,109), \(W\)(40,79), \(W\)(31,58), \(W\)(46,42), \(W\)(21,1), \(V\)(24,148), \(W\)(37,84)\(\}\).
\((V_5\bigcup W_5)\setminus (V_4\bigcup W_4)=\{W\)(32,47), \(W\)(1,64), \(W\)(17,38), \(V\)(42,8), \(V\)(8,138), \(W\)(34,69), \(V\)(10,84), \(W\)(39,72), \(V\)(7,206), \(W\)(19,31)\(\}\).
\((V_6\bigcup W_6)\setminus (V_5\bigcup W_5)=\{W\)(6,11), \(V\)(25,63), \(V\)(34,146), \(W\)(3,78), \(W\)(37,66), \(W\)(47,10), \(V\)(50,94), \(W\)(32,37), \(W\)(50,58), \(V\)(1,189)\(\}\).
\((V_7\bigcup W_7)\setminus (V_6\bigcup W_6)=\{V\)(44,124), \(W\)(8,73), \(W\)(18,20), \(W\)(10,48), \(W\)(2,69), \(V\)(20,57), \(V\)(4,150), \(V\)(45,210), \(V\)(3,46), \(W\)(44,76)\(\}\).
\((V_8\bigcup W_8)\setminus (V_7\bigcup W_7)=\{W\)(4,54), \(W\)(9,7), \(W\)(47,9), \(W\)(40,6), \(V\)(8,223), \(V\)(30,191), \(V\)(9,117), \(W\)(39,17), \(W\)(3,25)\(\}\).
\((V_9\bigcup W_9)\setminus (V_8\bigcup W_8)=\{V\)(47,38), \(V\)(26,84), \(V\)(43,220), \(V\)(32,49), \(W\)(2,17), \(V\)(8,127), \(W\)(40,30), \(W\)(1,75), \(W\)(20,27), \(V\)(2,115)\(\}\).
Instance 2 of \(fixRTVKP\).
Initial profit set of items is \(V_0[1\ldots 100]=\{\)117, 113, 113, 113, 112, 112, 112, 112, 112, 111, 110, 110, 109, 109, 108, 108, 108, 108, 108, 108, 108, 107, 106, 106, 105, 105, 105, 105, 104, 103, 102, 102, 102, 101, 101, 101, 101, 100, 100, 100, 100, 100, 100, 99, 99, 99, 99, 99, 99, 99, 99, 98, 98, 98, 98, 98, 98, 98, 98, 97, 97, 97, 97, 97, 97, 97, 97, 96, 96, 96, 96, 96, 96, 95, 95, 95, 95, 95, 94, 94, 94, 94, 94, 93, 93, 93, 92, 92, 92, 91, 91, 91, 90, 90, 89, 89, 88, 88, 87, 87\(\}\).
Initial weight set of items is \(W_0[1\ldots 100]=\{\)108, 98, 95, 107, 98, 100, 96, 105, 93, 112, 95, 105, 91, 96, 100, 103, 91, 96, 105, 90, 101, 110, 108, 95, 99, 96, 108, 101, 102, 100, 111, 88, 99, 112, 101, 105, 94, 113, 87, 101, 108, 96, 91, 89, 102, 99, 98, 93, 98, 99, 106, 112, 90, 100, 92, 94, 98, 97, 99, 95, 112, 108, 100, 98, 117, 98, 100, 98, 99, 113, 94, 111, 102, 99, 97, 87, 97, 103, 97, 89, 96, 94, 93, 104, 92, 109, 97, 109, 100, 88, 92, 108, 97, 106, 97, 97, 99, 94, 102, 95\(\}\).
Initial knapsack capacity is \(C_0=4995\).
The random variation period is 4 s, and \([A_v,B_v]=[71,121],[A_w,B_w]=[75,127],[A_c,B_c]=[4979,6971]\).
The number of subproblems is \(m=10\),and \(Threshold\le 20\).
The random oscillation change of profit,weight of items and knapsack capacity are following:
\(C_1=6021,C_2=5411,C_3=5900,C_4=6525,C_5=5102,C_6=5698,C_7=6058,C_8=4997,C_9=6414\).
\((V_1\bigcup W_1)\setminus (V_0\bigcup W_0)=\{W\)(68,102), \(W\)(70,111), \(V\)(59,105), \(W\)(6,77), \(W\)(28,125), \(W\)(96,92), \(V\)(37,77), \(W\)(3,122), \(W\)(83,112), \(V\)(19,76), \(V\)(27,103), \(V\)(70,93), \(V\)(100,72), \(W\)(4,89), \(W\)(34,99), \(W\)(42,101), \(W\)(69,76), \(W\)(63,78), \(V\)(60,73), \(W\)(30,111)\(\}\)
\((V_2\bigcup W_2)\setminus (V_1\bigcup W_1)=\{W\)(43,98), \(V\)(41,88), \(W\)(49,120), \(W\)(91,126), \(W\)(51,82), \(W\)(94,125), \(W\)(57,96), \(V\)(77,79), \(V\)(45,74), \(V\)(24,100), \(V\)(19,113), \(V\)(42,110), \(W\)(40,106), \(W\)(31,113), \(W\)(46,75), \(W\)(71,127), \(V\)(74,110), \(V\)(91,103)\(\}\)
\((V_3\bigcup W_3)\setminus (V_2\bigcup W_2)=\{V\)(56,90), \(W\)(53,77), \(W\)(42,122), \(W\)(8,123), \(V\)(88,80), \(W\)(46,80), \(V\)(59,118), \(V\)(23,78), \(V\)(31,113), \(V\)(1,73), \(W\)(56,101), \(V\)(25,106), \(V\)(84,75), \(W\)(3,98), \(W\)(37,121), \(W\)(97,87), \(V\)(100,104), \(W\)(82,92), \(W\)(100,99)\(\}\)
\((V_4\bigcup W_4)\setminus (V_3\bigcup W_3)=\{V\)(28,79), \(V\)(94,86), \(W\)(8,111), \(W\)(18,98), \(W\)(1,77), \(V\)(57,72), \(W\)(25,112), \(W\)(10,118), \(W\)(96,102), \(W\)(44,123), \(V\)(15,88), \(V\)(1,116), \(V\)(81,79), \(V\)(82,76), \(V\)(10,96), \(W\)(23,116), \(W\)(39,86), \(W\)(58,83), \(W\)(16,120)\(\}\) \((V_5\bigcup W_5)\setminus (V_4\bigcup W_4)=\{W\)(35,126), \(W\)(29,90), \(W\)(87,82), \(W\)(17,120), \(V\)(23,107), \(W\)(100,81), \(V\)(93,82), \(W\)(13,78), \(W\)(4,126), \(V\)(56,72), \(W\)(86,89), \(W\)(89,125), \(V\)(58,111), \(W\)(70,109), \(W\)(90,123), \(V\)(69,101), \(W\)(56,117), \(V\)(42,97)\(\}\)
\((V_6\bigcup W_6)\setminus (V_5\bigcup W_5)=\{V\)(54,94), \(W\)(80,101), \(V\)(30,78), \(V\)(67,117), \(W\)(96,86), \(V\)(87,111), \(V\)(83,82), \(W\)(15,116), \(V\)(72,94), \(W\)(14, 103), \(W\)(54,86), \(V\)(33,106), \(W\)(57,94), \(V\)(47,105), \(W\)(45,110), \(W\)(30,116), \(W\)(51,118), \(V\)(45,106), \(W\)(40,103), \(W\)(55,117)\(\}\)
\((V_7\bigcup W_7)\setminus (V_6\bigcup W_6)=\{V\)(14,108), \(W\)(69,101), \(V\)(6,77), \(W\)(3,96), \(V\)(15,103), \(W\)(35,85), \(W\)(60,89), \(V\)(78,96), \(W\)(64,93), \(W\)(86,103), \(W\)(14,123), \(W\)(100,127), \(W\)(44,91), \(W\)(73,98), \(W\)(4,106), \(W\)(94,116), \(W\)(93,111), \(W\)(87,101), \(W\)(88,114)\(\}\)
\((V_8\bigcup W_8)\setminus (V_7\bigcup W_7)=\{W\)(71,98), \(W\)(12,77), \(V\)(68,114), \(W\)(41,95), \(W\)(25,110), \(W\)(77,92), \(V\)(3,102), \(V\)(79,71), \(W\)(85,86), \(W\)(20, 119), \(W\)(88,77), \(V\)(11,85), \(W\)(16,111), \(V\)(44,83), \(W\)(10,113), \(V\)(66,90), \(V\)(75,96), \(V\)(29,96), \(W\)(3,91), \(W\)(97,102)\(\}\)
\((V_9\bigcup W_9)\setminus (V_8\bigcup W_8)=\{W\)(26,120), \(W\)(3,82), \(W\)(27,95), \(W\)(12,76), \(W\)(21,79), \(W\)(89,108), \(W\)(52,81), \(V\)(1,87), \(V\)(79,72), \(V\)(78,98), \(V\)(40,79), \(W\)(58,120), \(V\)(9,121), \(V\)(2,73), \(W\)(29,83), \(W\)(6,96), \(W\)(5,84), \(V\)(73,75), \(W\)(57,77), \(V\)(58,80)\(\}\)
Instance 3 of \(fixRTVKP\).
Initial profit set of items is \(V_0[1\ldots 300]=\{\)383, 519, 420, 272, 166, 125, 354, 374, 44, 540, 9, 108, 13, 4, 403, 376, 599, 432, 184, 439, 114, 45, 333, 238, 95, 10, 195, 542, 231, 476, 129, 582, 223, 210, 442, 250, 116, 211, 342, 461, 300, 368, 327, 524, 460, 158, 171, 261, 24, 89, 174, 214, 455, 87, 222, 588, 25, 453, 256, 458, 375, 129, 104, 428, 344, 165, 556, 166, 359, 440, 373, 210, 576, 14, 548, 105, 396, 116, 243, 196, 583, 307, 141, 345, 544, 500, 250, 280, 449, 388, 107, 135, 182, 235, 521, 480, 48, 272, 17, 190, 122, 6, 380, 226, 243, 567, 513, 444, 469, 567, 86, 520, 573, 125, 494, 123, 30, 276, 288, 219, 191, 91, 531, 382, 508, 541, 574, 568, 111, 581, 452, 351, 74, 411, 239, 513, 39, 43, 213, 484, 189, 314, 240, 25, 253, 430, 239, 494, 71, 296, 568, 359, 460, 242, 307, 186, 366, 215, 347, 240, 386, 178, 510, 118, 487, 468, 116, 376, 136, 593, 500, 514, 294, 508, 514, 322, 164, 544, 20, 224, 408, 436, 418, 234, 102, 558, 452, 362, 527, 240, 288, 179, 544, 174, 498, 370, 325, 521, 543, 248, 341, 516, 49, 440, 319, 346, 551, 454, 587, 374, 29, 511, 424, 419, 127, 471, 596, 385, 578, 148, 28, 421, 542, 358, 108, 538, 143, 405, 59, 267, 300, 458, 140, 383, 364, 445, 424, 488, 42, 65, 179, 303, 435, 370, 304, 584, 277, 82, 33, 77, 382, 434, 438, 232, 169, 160, 390, 24, 340, 332, 541, 91, 574, 318, 317, 577, 356, 332, 237, 172, 415, 489, 444, 102, 46, 406, 122, 269, 18, 296, 516, 42, 490, 107, 109, 294, 391, 164, 162, 438, 518, 122, 290, 504, 448, 408, 205, 266, 390, 470\(\}, \)
Initial weight set of items is \(W_0[1\ldots 300]=\{\)653, 11, 543, 649, 278, 173, 879, 796, 710, 840, 238, 280, 844, 886, 522, 30, 982, 754, 182, 163, 155, 969, 766, 433, 710, 888, 802, 295, 386, 985, 8, 152, 483, 828, 488, 685, 373, 44, 117, 599, 369, 619, 543, 902, 177, 655, 842, 257, 945, 684, 238, 512, 570, 507, 516, 557, 27, 839, 566, 613, 612, 524, 456, 82, 485, 810, 492, 889, 729, 636, 263, 645, 191, 45, 109, 937, 688, 42, 634, 890, 431, 34, 291, 916, 478, 173, 258, 977, 443, 920, 643, 87, 91, 565, 822, 374, 438, 421, 759, 246, 791, 420, 714, 546, 134, 238, 173, 874, 904, 71, 624, 150, 778, 378, 607, 576, 686, 547, 249, 120, 483, 563, 733, 217, 108, 645, 898, 861, 646, 751, 422, 165, 528, 288, 590, 342, 683, 147, 495, 32, 676, 192, 464, 480, 853, 322, 978, 914, 126, 637, 673, 634, 194, 29, 659, 735, 477, 726, 996, 201, 336, 515, 533, 483, 434, 956, 139, 95, 448, 140, 362, 150, 777, 480, 731, 549, 49, 492, 324, 977, 252, 72, 837, 198, 746, 600, 770, 195, 736, 197, 956, 74, 464, 853, 273, 659, 926, 571, 527, 495, 563, 216, 784, 396, 510, 35, 926, 253, 877, 740, 85, 839, 447, 108, 575, 912, 639, 985, 738, 774, 948, 66, 544, 789, 905, 331, 347, 980, 951, 699, 653, 854, 488, 594, 99, 161, 698, 579, 476, 712, 782, 545, 29, 996, 818, 225, 44, 501, 93, 319, 565, 80, 101, 173, 846, 279, 264, 338, 784, 356, 976, 733, 536, 911, 607, 722, 167, 862, 93, 263, 334, 471, 727, 808, 648, 973, 396, 730, 927, 118, 455, 559, 771, 538, 306, 378, 478, 698, 469, 490, 140, 121, 396, 292, 722, 431, 830, 472, 174, 541\(\}\).
Initial knapsack capacity is \(C_0=84340\).
The random variation period is 8 s, and \([A_v,B_v]=[3,600],[A_w,B_w]=[3,998],[A_c,B_c]=[81750,117564]\).
The number of subproblems is \(m=10\), and \(Threshold\le 40\).
The random oscillation change of profit, weight of items and knapsack capacity are following:
\(C_1=95040,C_2=111407,C_3=103409,C_4=107377,C_5=113684,C_6=83289,C_7=112588,C_8=103113,C_9=91317\).
\((V_1\bigcup W_1)\setminus (V_0\bigcup W_0)=\{W\)(168,361), \(W\)(270,787), \(W\)(6,260), \(W\)(28,4), \(W\)(296,989), \(V\)(37,102), \(W\)(3,156), \(W\)(83,492), \(V\)(219,164), \(V\)(127,422), \(V\)(70,181), \(V\)(200,294), \(W\)(204,906), \(W\)(142,742), \(W\)(269,650), \(W\)(263,888), \(V\)(260,354), \(W\)(230,781), \(V\)(36,67), \(W\)(289,229), \(W\)(243,343), \(V\)(147,486), \(W\)(7,228), \(W\)(249,708), \(W\)(85,37), \(V\)(141,185), \(V\)(132,597), \(W\)(40,725), \(W\)(138,625), \(V\)(283,208), \(W\)(34,242), \(V\)(259,581), \(W\)(178,317), \(W\)(187,44), \(W\)(225,151), \(W\)(130,884), \(W\)(298,579)\(\}\).
\((V_2\bigcup W_2)\setminus (V_1\bigcup W_1)=\{W\)(37,446), \(V\)(256,29), \(W\)(53,461), \(W\)(142,811), \(W\)(8,384), \(V\)(288,248), \(W\)(246,944), \(V\)(159,304), \(V\)(123,431), \(V\)(131,270),,\(W\)(156,481), \(V\)(25,208), \(V\)(184,90), \(W\)(3,449), \(W\)(237,413), \(W\)(97,120), \(V\)(100,535), \(W\)(182,753), \(W\)(200,525), \(V\)(168,143), \(W\)(49,574), \(V\)(22,559), \(V\)(14,547), \(V\)(117,400), \(V\)(57,77), \(W\)(225,55), \(W\)(210,52), \(W\)(196,568), \(W\)(244,646), \(V\)(215,536), \(V\)(1,305), \(V\)(181,65), \(V\)(282,456), \(V\)(110,250), \(W\)(223,613), \(W\)(39,278)\(\}\).
\((V_3\bigcup W_3)\setminus (V_2\bigcup W_2)=\{V\)(192,84), \(V\)(257,152), \(W\)(235,371), \(W\)(229,737), \(W\)(170,225), \(W\)(217,968), \(V\)(123,116), \(W\)(300,572), \(V\)(293,472), \(W\)(213,611), \(W\)(4,976), \(W\)(289,456), \(V\)(256,280), \(W\)(86,281), \(W\)(89,553), \(V\)(58,267), \(W\)(270,165), \(W\)(90,59), \(V\)(269,589),,\(V\)(242,545), \(W\)(61,805), \(W\)(240,474), \(V\)(197,544), \(V\)(50,196), \(W\)(298,499), \(V\)(206,571), \(W\)(156,837), \(W\)(2,443), \(W\)(87,314), \(W\)(56,312), \(W\)(13,851), \(W\)(246,332), \(V\)(122,425), \(V\)(83,484), \(W\)(97,305), \(V\)(62,156), \(W\)(74,509)\(\}\).
\((V_4\bigcup W_4)\setminus (V_3\bigcup W_3)=\{W\)(40,324), \(W\)(55,629), \(W\)(250,241), \(W\)(275,137), \(V\)(219,252), \(W\)(259,422), \(W\)(26,584), \(W\)(15,927), \(W\)(75,471), \(V\)(134,565), \(V\)(98,345), \(V\)(74,579), \(W\)(269,263), \(W\)(103,515), \(W\)(128,821), \(W\)(125,70), \(W\)(162,240), \(W\)(133,576), \(W\)(226,86), \(W\)(143,293), \(V\)(165,129), \(V\)(261,290), \(W\)(171,289), \(W\)(234,790), \(W\)(297,686), \(W\)(251,143), \(W\)(296,711), \(V\)(26,267), \(W\)(159,409), \(W\)(267,697), \(W\)(152,587), \(W\)(101,162), \(W\)(127,49), \(V\)(71,584), \(V\)(228,86), \(V\)(265,328), \(W\)(287,684), \(V\)(178,125)\(\}\).
\((V_5\bigcup W_5)\setminus (V_4\bigcup W_4)=\{V\)(129,310), \(W\)(3,25), \(W\)(296,656), \(V\)(231,595), \(W\)(172,438), \(V\)(154,593), \(W\)(225,311), \(W\)(141,934), \(W\)(30,27), \(W\)(59,649), \(W\)(108,524), \(W\)(259,466), \(W\)(161,436), \(W\)(178,131), \(W\)(188,880), \(W\)(61,519), \(W\)(85,484), \(W\)(212,819), \(W\)(257,99), \(W\)(224,544), \(V\)(117,433), \(V\)(127,204), \(V\)(172,550), \(W\)(297,77), \(V\)(213,162), \(V\)(186,124), \(W\)(130,328), \(W\)(260,967), \(V\)(156,96), \(W\)(285,853), \(W\)(173,544), \(V\)(33,574), \(V\)(84,316), \(W\)(168,939), \(W\)(39,234), \(W\)(55,538), \(W\)(76, 546), \(W\)(222,941)\(\}\).
\((V_6\bigcup W_6)\setminus (V_5\bigcup W_5)=\{W\)(284,945), \(V\)(80,578), \(W\)(135,644), \(W\)(157,4), \(V\)(206,387), \(V\)(282,303), \(W\)(142,934), \(W\)(243,509), \(W\)(278,507), \(V\)(253,506), \(V\)(74,475), \(W\)(276,155), \(V\)(11,422), \(W\)(213,818), \(V\)(132,309), \(V\)(186,555), \(V\)(190,279), \(W\)(154,393), \(W\)(41,135), \(V\)(36,390), \(W\)(106,367), \(W\)(4,844), \(W\)(209,411), \(V\)(150,257), \(W\)(128,494), \(W\)(30,367), \(W\)(75,684), \(V\)(116,314), \(W\)(293,822), \(W\)(29,142), \(W\)(176,861), \(V\)(71,310), \(V\)(205,506), \(W\)(164,729), \(W\)(11,262), \(V\)(241,207), \(W\)(120,996), \(W\)(105,945), \(W\)(151,308)\(\}\).
\((V_7\bigcup W_7)\setminus (V_6\bigcup W_6)=\{W\)(23,185), \(V\)(85,495), \(W\)(165,8), \(W\)(246,827), \(V\)(79,576), \(W\)(120,44), \(V\)(45,476), \(W\)(146,611), \(V\)(271,36), \(W\)(133,571), \(W\)(188,685), \(W\)(202,174), \(W\)(228,647), \(V\)(216,186), \(V\)(244,82), \(V\)(64,268), \(W\)(89,389), \(V\)(261,294), \(V\)(255,559), \(V\)(91,178), \(W\)(131,723), \(V\)(68,353), \(W\)(36,357), \(W\)(139,892), \(V\)(125,203), \(W\)(60,388), \(V\)(145,8), \(V\)(27,298), \(W\)(256,128), \(V\)(197,49), \(W\)(265,437), \(W\)(214,209), \(V\)(49,557), \(V\)(207,462), \(V\)(5,130), \(V\)(113,242), \(W\)(66,103), \(V\)(42,319), \(W\)(157,679)\(\}\).
\((V_8\bigcup W_8)\setminus (V_7\bigcup W_7)=\{W\)(231,912), \(W\)(212,768), \(V\)(191,457), \(W\)(35,936), \(W\)(117,507), \(V\)(63,110), \(W\)(104,701), \(W\)(157,151), \(W\)(18,232), \(V\)(29,499), \(W\)(19,925), \(W\)(156,488), \(W\)(204,691), \(W\)(210,961), \(W\)(290,51), \(W\)(154,797), \(W\)(3,950), \(W\)(170,893), \(V\)(109,496), \(W\)(4,36), \(W\)(105,443), \(W\)(114,725), \(V\)(123,559), \(V\)(111,368), \(W\)(209,4), \(V\)(194,322), \(V\)(38,168), \(V\)(1,220), \(W\)(118,282), \(W\)(16,405), \(V\)(213,587), \(W\)(155,152), \(W\)(185,873), \(W\)(116,686), \(W\)(99,168), \(V\)(278,195), \(V\)(190,402), \(V\)(42,245)\(\}\).
\((V_9\bigcup W_9)\setminus (V_8\bigcup W_8)=\{V\)(285,263), \(W\)(272,892), \(W\)(172,380), \(V\)(54,544), \(V\)(126,111), \(V\)(151,209), \(V\)(194,503), \(W\)(217,300), \(V\)(29,370), \(W\)(22,440), \(V\)(217,343), \(V\)(167,386), \(W\)(165,347), \(V\)(22,303), \(W\)(65,428), \(W\)(203,992), \(W\)(242,876), \(W\)(224,642), \(V\)(269,351), \(W\)(108,406), \(W\)(113,223), \(W\)(157,341), \(W\)(297,271), \(W\)(146,767), \(V\)(292,559), \(W\)(115,506), \(V\)(123,405), \(W\)(153,372), \(V\)(239,114), \(V\)(67,320), \(V\)(287,269), \(W\)(234,209), \(W\)(247,467), \(V\)(26,484), \(V\)(3,312), \(V\)(231,591), \(W\)(79,882), \(W\)(49,937), \(W\)(116,594)\(\}\).
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He, Y., Zhang, X., Li, W. et al. Algorithms for randomized time-varying knapsack problems. J Comb Optim 31, 95–117 (2016). https://doi.org/10.1007/s10878-014-9717-1
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DOI: https://doi.org/10.1007/s10878-014-9717-1