paperresutls.pdf

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Fig.1. Gradual hub covering cases.

For further discussions, a simple example of gradual hub covering problem is used. In this example, three sets are shown in Figure .1; (A) not-covered, (B) partially covered and (C) fully covered. The ( a  , d  ) route is not covered at all via hubs ( b , c ) , the ( a  , d  ) route is partially covered via the mentioned hubs and ( a , d ) route is covered completely with these hubs. In set B, the gradually covering is depicted by the gradient between grey and white colour (i.e. Where the grey is more than white, the decay function value nears to 1 and where the white is more than grey, this function value near to 0.)

Results:

The performance is appraised rely on the relative gap (( Optimal Solution  OFV ) / OFV )  100 , where OFV is the obtained objective function value achieving with another method) which is the distance from optimal. In our experimental design, we take ⁿ ^  ¹⁰ ^, ¹⁵ ^, ²⁰ ^, ²⁵  ^,  ^  ⁰ ^. ⁶ ^, ⁰ ^. ⁷ ^, ⁰ ^. ⁸  and ^p ^   ¹ ^, ² ^, ³ for CAB dataset. We apply the AP dataset, which is derived from a real application to a postal delivery network. It includes up to 200 nodes, which portray postcode districts, the direction of their coordinates and flow volumes (mail flow); in addition, a feature of this data is that the flow matrix is not symmetrical (i.e., W

_ij

 W

_ji

and W

_ii

 0 ).

The results from the CPLEX solver cannot achieve to the optimum in 3600 second for n  40 , so the best-known result with this solver is shown. Moreover, CPLEX cannot obtain ever a feasible solution for n  200 in 7200 second.

The obtained gap ( (( TS  Best known ) / TS )  100 ) show the better performance of TS with resonable computational time.

In the Tables (1) to (3), we provide results from running the CPELX and TS on CAB dataset. The data in each cell of

CPLEX column consists of relative gap and TS column includes two results, relative gap (%)/ time (second). The gap

for CPLEX indicates the obtained solution in the same CPU time as TS (i.e. the CPLEX and TS search solution space

in the same computational time).

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Table 1.

Results (relative gap (%)/time (s)) for the TS algorithm and CPLEX (CAB data set,   0 . 6 ).

n CPLEX TS

10 15 20 25 10 15 20 25

p=1

β=1073,ω=810 0.027 0.029 0.025 0.041 0.027/0.1 0.029/0.2 0.025/0.3 0.041/0.4 β=973,ω=810 0.011 0.013 0.018 0.034 0.011/0.1 0.013/0.2 0.018/0.3 0.034/0.4 β=873,ω=810 0.046 0.033 0.000 0.000 0.046/0.1 0.033/0.2 0.000/0.3 0.000/0.4 β=1073,ω=710 0.067 0.028 0.041 0.058 0.067/0.1 0.028/0.2 0.041/0.3 0.058/0.4 β=973,ω=710 0.020 0.024 0.039 0.055 0.020/0.1 0.024/0.2 0.039/0.3 0.055/0.4 β=873,ω=710 0.064 0.057 0.051 0.058 0.064/0.1 0.057/0.2 0.051/0.3 0.058/0.4 β=1073,ω=610 0.048 0.020 0.049 0.061 0.048/0.1 0.020/0.2 0.049/0.3 0.061/0.4 β=973,ω=610 0.075 0.036 0.056 0.067 0.075/0.1 0.036/0.2 0.056/0.3 0.067/0.4 β=873,ω=610 0.071 0.066 0.084 0.078 0.071/0.1 0.078/0.2 0.084/0.3 0.074/0.4

p=2

p=3

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Table 2. Results (relative gap (%)/time (s)) for the TS algorithm and CPLEX (CAB data set,

  0 . 7

^).

n CPLEX TS

10 15 20 25 10 15 20 25

p=1

p=2

p=3

β=1073,ω=810 3.005 6.076 12.002 20.098 0.004/0.3 0.022/0.4 0.002/0.7 0.014/1.2 β=973,ω=810 3.005 6.016 12.006 20.048 0.005/0.3 0.014/0.4 0.008/0.7 0.034/1.2 β=873,ω=810 3.002 6.045 12.000 20.097 0.002/0.3 0.036/0.4 0.071/0.7 0.097/1.2 β=1073,ω=710 3.077 6.098 12.063 20.059 0.056/0.3 0.040/0.4 0.061/0.7 0.011/1.2 β=973,ω=710 3.004 6.092 12.042 20.037 0.004/0.3 0.006/0.4 0.022/0.7 0.035/1.2 β=873,ω=710 3.003 6.007 12.074 20.079 0.003/0.3 0.007/0.4 0.033/0.7 0.044/1.2 β=1073,ω=610 3.082 6.094 12.092 20.099 0.022/0.3 0.023/0.4 0.023/0.7 0.035/1.2 β=973,ω=610 3.000 6.05 12.041 20.098 0.004/0.3 0.026/0.4 0.041/0.7 0.058/1.2 β=873,ω=610 3.009 6.089 12.079 20.051 0.009/0.3 0.026/0.4 0.033/0.7 0.028/1.2 Table 3. Results (relative gap (%)/time (s)) for the TS algorithm and CPLEX (CAB data set,

  0 . 8

).

N CPLEX TS

10 15 20 25 10 15 20 25

p=1

p=2

p=3

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In Table (4), we provide results from running TS on AP dataset with  ^  ⁰ ^. ⁶ ^, ⁰ ^. ⁸  ,    30 , 35  and

 20 , 25 

  for n  40 and    0 . 6 , 0 . 9  ,  ^  ¹⁰ ^, ¹⁵  and     9 , 5 for n  200 .

Table 4. Obtained objective function value using TS for AP dataset with larger nodes

p  β ω Best Known CPU TS CPU Gap %

40

2 0.6 30 20 1936.394 3600 2739.024 2.531 29.304 2 0.6 30 25 2279.256 3600 2979.404 2.421 23.500 2 0.6 35 20 2152.413 3600 2927.576 2.462 26.478 2 0.6 35 25 2371.631 3600 3167.299 2.442 25.121 2 0.8 30 20 1862.731 3600 2550.671 2.468 26.971 2 0.8 30 25 2083.942 3600 2858.456 2.398 27.096 2 0.8 35 20 2125.042 3600 2767.986 2.468 23.228 2 0.8 35 25 2284.892 3600 3042.297 2.659 24.896 4 0.6 30 20 2187.632 3600 3075.304 6.317 28.865 4 0.6 30 25 2342.727 3600 3332.956 6.476 29.710 4 0.6 35 20 2483.903 3600 3241.422 6.770 23.370 4 0.6 35 25 2573.730 3600 3462.368 6.353 25.666 4 0.8 30 20 2186.244 3600 2827.141 6.720 22.669 4 0.8 30 25 2374.865 3600 3214.966 6.707 26.131 4 0.8 35 20 2153.973 3600 3062.920 6.745 29.676 4 0.8 35 25 2526.863 3600 3229.480 6.594 21.756

200

2 0.6 10 5 - 7200 661.2087 203.23 -

2 0.6 10 9 - 7200 908.6192 205.27 -

2 0.6 15 5 - 7200 955.1068 202.63 -

2 0.6 15 9 - 7200 1151.3 204.34 -

4 0.6 10 5 - 7200 780.7048 498.54 -

4 0.6 10 9 - 7200 998.5768 488.4 -

4 0.6 15 5 - 7200 1083.345 485.09 -

4 0.6 15 9 - 7200 1340.523 496.13 -

2 0.9 10 5 - 7200 652.0087 198.43 -

2 0.9 10 9 - 7200 859.2817 201.64 -

2 0.9 15 5 - 7200 917.2129 199.12 -

2 0.9 15 9 - 7200 1112.878 202.53 -

4 0.9 10 5 - 7200 707.5947 480.55 -

4 0.9 10 9 - 7200 903.9127 489.9 -

4 0.9 15 5 - 7200 942.1618 487.87 -

4 0.9 15 9 - 7200 1192.2811 472.38 -

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Fig. 2. Objective function value plots

Fig. 2a shows the changes in objective function value when  changes for the case of p    1 , 2 , 3 . The service time

factor increases, the maximum value of flows increases. In Fig. 2b, it is seen that  increasing trend, imposes growing

mode for the objective function value of GHCP. This characteristic for  movement is alike to  . Moreover, Fig.2c

shows this attribute. Fig.4 shows the 95% confidence interval of gap for each test problem.

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paperresutls.pdf

Fig.1. Gradual hub covering cases.

Results:

 W

and W

 0 ).

The results from the CPLEX solver cannot achieve to the optimum in 3600 second for n  40 , so the best-known result with this solver is shown. Moreover, CPLEX cannot obtain ever a feasible solution for n  200 in 7200 second.

The obtained gap ( (( TS  Best known ) / TS )  100 ) show the better performance of TS with resonable computational time.

In the Tables (1) to (3), we provide results from running the CPELX and TS on CAB dataset. The data in each cell of

CPLEX column consists of relative gap and TS column includes two results, relative gap (%)/ time (second). The gap

for CPLEX indicates the obtained solution in the same CPU time as TS (i.e. the CPLEX and TS search solution space

in the same computational time).

Results (relative gap (%)/time (s)) for the TS algorithm and CPLEX (CAB data set,   0 . 6 ).

  0 . 7

  0 . 8

In Table (4), we provide results from running TS on AP dataset with    0 . 6 , 0 . 8  ,    30 , 35  and

 20 , 25 

  for n  40 and    0 . 6 , 0 . 9  ,    10 , 15  and     9 , 5 for n  200 .

Fig. 2. Objective function value plots

Fig. 2a shows the changes in objective function value when  changes for the case of p    1 , 2 , 3 . The service time

factor increases, the maximum value of flows increases. In Fig. 2b, it is seen that  increasing trend, imposes growing

mode for the objective function value of GHCP. This characteristic for  movement is alike to  . Moreover, Fig.2c

shows this attribute. Fig.4 shows the 95% confidence interval of gap for each test problem.

Fig. 4. Comparing results between average gap of the CPLEX and TS

In Table (4), we provide results from running TS on AP dataset with  ^  ⁰ ^. ⁶ ^, ⁰ ^. ⁸  ,    30 , 35  and

  for n  40 and    0 . 6 , 0 . 9  ,  ^  ¹⁰ ^, ¹⁵  and     9 , 5 for n  200 .