distance from node i to node j is an important factor now, nor the travel time. The improvement method of Tsiligirides’s algorithm is not needed in our modified algorithm. The objective function of OTP increases score while a new node is inserted. Obviously, without increasing any score or decreasing the total score, a new node is not allowed to insert. In MBCPP applications, the objective is to seek a postman tour with maximized total benefits. In snowplow operations, a street somehow must be served more than once, because the benefit of link (i ,j) is positive and smaller than service cost, but still better than deadhead cost for objective function. The point is that Tsiligirides algorithm can’t handle negative score of node. We translate the negative score to a positive one and measure it by Di = Si
/
∑
= 41
k
S
k . A random numberR
∈U
(0,1) ranged between 0 and 1 is generated.The candidate node i is randomly selected with probability Pi as a new node inset to the current route. Nodes continuously are added to the current route until each edge (street) in E is served at least once. Then, apply the shortest path method from last node on the route back to the depot. The procedure is iterated until 400 routes are generated and a route with highest score is selected as the finally solution. The flow chart in Figure 9 provides an easy way to understand the sequence of logical operations of the modified Tsiligirides algorithm.
The bottleneck of the modified Tsiligirides algorithm is the shortest path segment which requires an order of O(n3) computations. Therefore, the complexity of modified Tsiligirides algorithm is O(n3).
6. Computational results
6.1 Modified Tsiligirides algorithm performance
To test and verify the proposed solution procedures, we generate six sets of test problems, which are randomly generated with the following characteristics described in Table 4.
Table 4. Characteristics of six sets of 210 problems.
Problem V E Density
A 10~40 30~780 0.7~1
B 50~90 860~4005
0.7~1
C 100~150 3465~11175 0.7~1
D 20~40 19~230 0.1~0.3
E 50~90 120~1334
0.1~0.3
F 100~150 559~3288 0.1~0.3
V = number of nodes; E = number of edges
The details of each set of test problems were described in Table 10 on appendix B. Costs and benefits are assumed to be proportional to the distance of each edge with the net-benefit-per-mile function introduced by Malandraki and Daskin (1993).
The problems A to C are dense, while D, E and F are sparse. We also compute average ratios to the upper bound of 210 problems due to their complexity. The modified Tsiligirides algorithm was coded in Visual Basic 6.0 and implemented on Pentium IV personal computer with CPU 2.4G and 512M memories.
Start
All edges in network are serviced at least once ?
Perform shortest path method, back to the Depot
Does all of candidate nodes associate with score s(i) < 0 ?
Compute "desirability" and randomly select a candidate node
inserted to the current route No No Yes
Translate the candidate score then compute "desirability"
and randomly select one node from four candidate nodes
Yes
Figure 9. Flow chart for the modified Tsiligirides algorithm
The modified Tsiligirides algorithm has five variations, which have different numbers of candidate nodes, noted by T1, T2, T3, T4 and T5. For instance, T1 has one candidate node, T2 has two candidate nodes, and so on. The comparisons of the performances of T1, T2, T3, T4 and T5 on the 210 test problems are summarized in Table 7, Table 8 and Table 9. The average ratio to the upper bound, the worst ratio to the upper bound, the number of best solutions obtained, and the number of worst solutions obtained are recorded. The networks of Table 7(a), Table 7(b) and Table 7(c) are dense. Since T5 achieved 42 best solutions in problem set A, B and C, it seems to be the best algorithm for dense networks. Out of the sparse networks shown in Table 7(d), Table 7(e) and Table (f), T2 achieved 30 best solutions, while T4 and T5 gained 24 best solutions and 23 best solutions, respectively. In the comparison of the average ratio to the upper bound between T2, T4 and T5 among problem set D, E and F, T2 reached 98.35% and T4 gained 98.38%, while T5 achieved up to 98.40%. Therefore, T5 is the best choice out of 5 variations of modified Tsiligirides algorithm no matter whether the network is dense or sparse. The modified Tsiligirides algorithm is based on inserting a randomly selected candidate node into the route. Therefore, if the number of run times increases, then the solution is near to the optimal one. In fact, the solution will not increase without limits. In Table 9(e), we varied the number of run times from 100 to 400 times. T5 is the best one of all algorithms when R = 100, 200, 300 and 400, since the average ratios to the upper bound for T5 are 99.008%, 99.079%, 99.1% and 99.122%, respectively. Obviously, when the number of run times increases, the average ratio to the upper bound goes up slightly. Out of 210 test problems, 65 solutions are the best for R = 100 as shown in Table 9(e). When R=
200 or 300, there are 49 best solutions and 96 worst ones. However, the numbers of the best and the worst solutions are approximately equal when the run times are 400. In fact, the 96 worst solutions were very close to improvement solutions out of 400 run times.
6.2 Benefit structure analysis
To investigate coefficient of benefit structure in MBCPP, considered the MBCPP network depicted in Figure 10 with symmetric cost functions produced from the Cartesian coordinates in Table 5, net-benefit-per-mile functions and three kinds of roads the same as example 1 of MBCPP.
4 7
9
1 3
2
8
10
6
5
Figure 10. The example that has ten arcs, ten nodes and three kinds of roads with depot is node 1
Table 5. The Cartesian coordinates for the MBCPP pathological example.
Node X(i) Y(i)
1 73 36 2 24 2 3 52 47 4 0 85 5 87 83 6 7 20 7 90 81 8 46 57 9 47 13 10 86 9
The network density of the pathological example is 0.2, which belongs to sparse network. It has an upper bound 241.746. The branch and bound method was solved the pathological example to obtain an optimal solution 120.64, and the optimal ratio to the upper bound near to 50%. The optimal route of the MBCPP is 1-2-3-10-3-5-7-6-5-7-5-3-4-8-4-9-4-3-2-1. Also, we applied the T5 of modified Tsiligirides algorithm to find the optimal solution 120.64 within five run times for this example. In addition, the route was obtained 1-2-3-10-3-5-7-6-5-7-5- 3-4-8-4-9-4-3-2-1 for the T5 of modified Tsiligirides algorithm. Therefore, the modified Tsiligirides algorithm can be applying in the MBCPP which was after network transformation. To investigate the impact of the benefit structure on the solutions of T5 algorithm, we decreased the net-benefit-per-mile functions by 10%
each time, and verified 10 test problems from problem set D, E, and F (sparse
networks), including D1, D11, D21, E1, E11, E21, E31, E41, F1 and F11. The
results showed that the performance of net-benefit-per-mile functions decreased
while coefficient of benefit structure was reduced, as depicted in Figure 11 and recorded in Table 6. Obviously, T5 algorithm performs well for dense or large size networks. We point out that the net-benefit-per-mile function is an important factor for our modified Tsiligirides algorithm. If sparse and small networks have low coefficient (0.3 ~ 0.1) of benefit structure, then the solution procedure could produce solutions of poor quality.
Relevance of benefit
Figure 11. Performance of net-benefit-per-mile functions as decreasing 10% each time
Table 6. The results of sparse network with reduce 10% coefficient of benefit each times.
D1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
Best sol. 407.045 345.954 284.862 224.188 163.734 103.280 44.457 -11.101 -68.691 UB 485.573 431.621 377.668 323.715 269.763 215.810 161.858 107.905 53.953 Ratio 83.83% 80.15% 75.43% 69.25% 60.70% 47.86% 27.47% -10.29% -127.32%
D11
Best sol. 1134.797 970.827 785.400 678.211 477.343 381.803 215.338 69.878 -100.009 UB 1273.084 1131.630 990.171 848.723 707.265 565.815 424.361 282.906 141.454 Ratio 89.14% 85.79% 79.32% 79.91% 67.49% 67.48% 50.74% 24.70% -70.70%
D21
Best sol. 2269.629 1988.850 1703.564 1434.424 1170.003 875.208 609.173 336.962 -17.677 UB 2390.735 2125.098 1859.459 1593.824 1328.185 1062.548 796.912 531.274 265.637 Ratio 94.93% 93.59% 91.62% 90.00% 88.09% 82.37% 76.44% 63.43% -6.65%
E1
Best sol. 2889.330 2538.229 2198.138 1897.316 1507.595 1189.161 797.915 453.843 95.033 UB 3054.513 2715.123 2375.730 2036.342 1696.950 1357.560 1018.171 678.780 339.390 Ratio 94.59% 93.48% 92.52% 93.17% 88.84% 87.60% 78.37% 66.86% 28.00%
E11
Best sol. 6127.019 5433.122 4693.601 3975.813 3248.827 2471.900 1772.732 956.046 393.466 UB 6452.262 5735.344 5018.419 4301.508 3584.585 2867.668 2150.754 1433.834 716.918 Ratio 94.96% 94.73% 93.53% 92.43% 90.63% 86.20% 82.42% 66.68% 54.88%
E21
Best sol. 7242.989 6382.623 5473.091 4690.783 3801.397 2923.253 2042.630 1179.815 403.954 UB 7528.877 6692.335 5855.787 5019.251 4182.705 3346.164 2509.626 1673.082 836.542
Ratio 96.20% 95.37% 93.46% 93.46% 90.88% 87.36% 81.39% 70.52% 48.29%
E31
Best sol. 10547.249 9308.129 8065.303 6868.281 5744.037 4311.264 3184.760 2043.656 643.973 UB 10834.319 9630.506 8426.691 7222.879 6019.065 4815.252 3611.440 2407.626 1203.813 Ratio 97.35% 96.65% 95.71% 95.09% 95.43% 89.53% 88.19% 84.88% 53.49%
E41
Best sol. 12426.788 10949.028 9530.659 8094.778 6722.719 5191.226 3760.000 2452.508 999.352 UB 12717.436 11304.388 9891.336 8478.291 7065.240 5652.192 4239.145 2826.096 1413.048 Ratio 97.71% 96.86% 96.35% 95.48% 95.15% 91.84% 88.70% 86.78% 70.72%
F1
Best sol. 17321.913 15306.686 13341.553 11361.702 9309.869 7393.545 5385.887 3373.817 1448.918 UB 17677.764 15713.568 13749.372 11785.176 9820.975 7856.780 5892.588 3928.390 1964.196 Ratio 97.99% 97.41% 97.03% 96.41% 94.80% 94.10% 91.40% 85.88% 73.77%
F11
Best sol. 37777.108 33500.228 29180.155 24931.678 20677.240 16335.981 12119.757 7889.587 3524.952 UB 38163.471 33923.085 29682.695 25442.314 21201.925 16961.540 12721.157 8480.770 4240.386 Ratio 98.99% 98.75% 98.31% 97.99% 97.53% 96.31% 95.27% 93.03% 83.13%