A GENERALIZED DECOMPOSITION ALGORITHM FOR REAL-TIME TRUCK ROUTING PROBLEMS

(c) (d) Resource Constrains Resource Consumption a0 b0 a1 b1 Link hours miles Cost-Rev S 0 0 0 0 (S,D0) 2 0 50 D0 7 8 0 500 (S,D1) 1 0 50 D1 7 8 0 500 (D0,L0) 2 100 -110 L0 8 18 0 500 (D0,L2) 4 200 -100 L1 7 12 0 500 (D1,L1) 4 200 -110 L2 7 12 0 500 (D1,L3) 3 150 -120 L3 8 18 0 500 (L1,L0) 2 100 -110 T 12 19 0 500 (L2,L3) 3 150 -120 time window min and max miles (L0,T) 1 10 50 (L3,T) 1 10 50

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Table 2
Top 10 US Cities with the Largest Populations.

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Table 3
Forty-Five Non-directional City-Pairs.

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Table 4
Computational Results of 100 Tests.

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Table 5
Comparison Between Existing And New Methods.

(1)

(2)

(3)

(c)					(d)
Resource Constrains					Resource Consumption
	a0	b0	a1	b1	Link	hours	miles	Cost-Rev
S	0	0	0	0	(S,D0)	2	0	50
D0	7	8	0	500	(S,D1)	1	0	50
D1	7	8	0	500	(D0,L0)	2	100	-110
L0	8	18	0	500	(D0,L2)	4	200	-100
L1	7	12	0	500	(D1,L1)	4	200	-110
L2	7	12	0	500	(D1,L3)	3	150	-120
L3	8	18	0	500	(L1,L0)	2	100	-110
T	12	19	0	500	(L2,L3)	3	150	-120
	time window		min and max miles		(L0,T)	1	10	50
					(L3,T)	1	10	50

cityID	City	Population	Latitude	Longitude
1	New York	8,405,837	41	-74
2	Los Angeles	3,884,307	34	-118
3	Chicago	2,718,782	42	-88
4	Houston	2,195,914	30	-95
5	Philadelphia	1,553,165	40	-75
6	Phoenix	1,513,367	33	-112
7	San Antonio	1,409,019	29	-98
8	San Diego	1,355,896	33	-117
9	Dallas (depot)	1,257,676	33	-97
10	San Jose	998,537	37	-122

odID	Orig	Dest	Miles	odID	Orig	Dest	Miles
1	Chicago	Dallas	967	24	Houston	San Jose	1885
2	Chicago	Houston	1083	25	Los Angeles	New York	2778
3	Chicago	Los Angeles	2016	26	Los Angeles	Philadelphia	2710
4	Chicago	New York	791	27	Los Angeles	Phoenix	373
5	Chicago	Philadelphia	758	28	Los Angeles	San Antonio	1353
6	Chicago	Phoenix	1735	29	Los Angeles	San Diego	120
7	Chicago	San Antonio	1242	30	Los Angeles	San Jose	341
8	Chicago	San Diego	2139	31	New York	Philadelphia	97
9	Chicago	San Jose	2163	32	New York	Phoenix	2409
10	Dallas	Houston	967	33	New York	San Antonio	1821
11	Dallas	Los Angeles	1436	34	New York	San Diego	2799
12	Dallas	New York	1547	35	New York	San Jose	2944
13	Dallas	Philadelphia	1467	36	Philadelphia	Phoenix	2344
14	Dallas	Phoenix	1065	37	Philadelphia	San Antonio	1742
15	Dallas	San Antonio	274	38	Philadelphia	San Diego	2735
16	Dallas	San Diego	1358	39	Philadelphia	San Jose	2911
17	Dallas	San Jose	1687	40	Phoenix	San Antonio	981
18	Houston	Los Angeles	1548	41	Phoenix	San Diego	355
19	Houston	New York	1627	42	Phoenix	San Jose	711
20	Houston	Philadelphia	1548	43	San Antonio	San Diego	1276
21	Houston	Phoenix	1174	44	San Antonio	San Jose	1692
22	Houston	San Antonio	197	45	San Diego	San Jose	460
23	Houston	San Diego	1468

testID	Existing Method		New Method		Driver Reduced (%)	LF Improved (%)
testID	num drivers	LF	num drivers	LF
1	18	63	15	70	20	11.1
2	16	69	14	73	14.3	5.8
3	15	70	14	73	7.1	4.3
4	17	66	15	71	13.3	7.6
5	17	68	14	74	21.4	8.8
6	19	62	16	66	18.8	6.5
7	18	63	16	68	12.5	7.9
8	18	62	15	69	20	11.3
9	16	69	14	75	14.3	8.7
10	18	62	16	68	12.5	9.7
11	19	58	17	66	11.8	13.8
12	17	68	14	76	21.4	11.8
13	17	65	15	72	13.3	10.8
14	17	65	16	67	6.3	3.1
15	17	67	15	72	13.3	7.5
16	16	69	13	79	23.1	14.5
17	18	61	14	73	28.6	19.7
18	17	66	15	71	13.3	7.6
19	18	62	16	65	12.5	4.8
20	19	59	17	63	11.8	6.8
21	17	67	15	69	13.3	3
22	17	66	15	72	13.3	9.1
23	17	65	14	75	21.4	15.4
24	17	65	15	69	13.3	6.2
25	16	71	14	74	14.3	4.2
26	19	58	16	66	18.8	13.8
27	17	69	14	77	21.4	11.6
28	17	66	15	74	13.3	12.1
29	17	65	15	73	13.3	12.3
30	16	69	15	71	6.7	2.9
31	18	63	14	75	28.6	19
32	17	65	14	74	21.4	13.8
33	18	61	15	68	20	11.5
34	18	61	15	69	20	13.1
35	16	72	14	73	14.3	1.4
36	18	64	16	68	12.5	6.3
37	18	61	16	66	12.5	8.2
38	18	64	16	67	12.5	4.7
39	17	65	15	71	13.3	9.2
40	17	63	16	67	6.3	6.3
41	16	68	14	76	14.3	11.8
42	17	66	15	69	13.3	4.5
43	17	68	14	77	21.4	13.2
44	17	68	14	72	21.4	5.9
45	16	69	13	76	23.1	10.1
46	16	71	14	74	14.3	4.2
47	18	62	15	71	20	14.5
48	16	69	15	69	6.7	0
49	17	65	13	77	30.8	18.5
50	17	66	14	72	21.4	9.1
51	18	64	16	68	12.5	6.3
52	17	68	14	73	21.4	7.4
53	18	59	17	62	5.9	5.1
54	17	63	16	64	6.3	1.6
55	19	60	17	63	11.8	5
56	17	65	14	72	21.4	10.8
57	17	67	14	74	21.4	10.4
58	19	62	16	66	18.8	6.5
59	18	64	16	67	12.5	4.7
60	18	63	15	68	20	7.9
61	17	64	15	70	13.3	9.4
62	18	62	15	69	20	11.3
63	17	65	15	70	13.3	7.7
64	18	61	14	73	28.6	19.7
65	17	64	15	71	13.3	10.9
66	19	58	18	59	5.6	1.7
67	17	68	14	76	21.4	11.8
68	18	64	14	73	28.6	14.1
69	16	71	13	78	23.1	9.9
70	17	64	13	79	30.8	23.4
71	19	61	17	64	11.8	4.9
72	17	68	14	75	21.4	10.3
73	18	63	15	70	20	11.1
74	18	60	18	60	0	0
75	17	65	14	74	21.4	13.8
76	17	68	15	72	13.3	5.9
77	17	63	16	65	6.3	3.2
78	18	65	15	71	20	9.2
79	17	69	13	80	30.8	15.9
80	17	66	13	77	30.8	16.7
81	16	70	15	71	6.7	1.4
82	18	66	15	71	20	7.6
83	18	62	14	72	28.6	16.1
84	17	66	16	68	6.3	3
85	18	64	15	71	20	10.9
86	17	65	15	71	13.3	9.2
87	16	67	15	73	6.7	9
88	16	71	14	73	14.3	2.8
89	18	65	13	78	38.5	20
90	16	70	16	67	0	-4.3
91	17	63	14	74	21.4	17.5
92	17	68	15	70	13.3	2.9
93	17	64	15	71	13.3	10.9
94	17	66	14	76	21.4	15.2
95	17	65	14	73	21.4	12.3
96	19	60	17	64	11.8	6.7
97	17	66	14	72	21.4	9.1
98	19	60	17	64	11.8	6.7
99	18	62	15	70	20	12.9
100	17	67	14	73	21.4	9

Existing Method		New Method		Driver	LF
avg drivers	avg LF	avg drivers	avg LF	Reduced (%)	Improved (%)
17.3	65	14.9	70.9	16.3	9.1

R_{k} = E F F {\begin{matrix} (D_{k}^{*}, C_{k}) = [(\max (a_{k}^{l}, D_{i}^{l} + d_{i k}^{l}), l = 1,2,..., L), C_{i} + C_{i k}] \\ | \forall (i, k) \in A, (D_{i}^{*}, C_{i}) \in Q_{i}, and D_{k}^{*} belonging to urcentainty zone \end{matrix}}

R P_{k} = E F F {\begin{matrix} (D_{k}^{*}, C_{k}) = [(\max (a_{k}^{l}, D_{i}^{l} + d_{i k}^{l}), l = 1,2,..., L), C_{i} + C_{i k}] \\ | \forall (i, k) \in A, (D_{i}^{*}, C_{i}) \in P_{i}, and D_{k}^{*} belonging to uncertainty zone \end{matrix}}

\begin{array}{l} P_{S} = Q_{S} = (0,0,0), P_{D 0} = P_{D 1} = P_{L 0} = P_{L 1} = P_{L 2} = P_{L 3} = P_{T} = \emptyset, \\ Q_{D 0} = (7,0, - \infty), Q_{D 1} = (7,0, - \infty), Q_{L 0} = (8,0, - \infty), Q_{L 1} = (7,0, - \infty), Q_{L 2} = (7,0, - \infty), Q_{L 3} = (8,0, - \infty), Q_{T} = (12,0, - \infty), \\ d_{D 0} = (2,0,50), d_{D 1} = (1,0,50), d_{L 0} = (2,100, - 110), d_{L 1} = (4,200, - 110), d_{L 2} = (4,200, - 100), d_{L 3} = (3,150, - 120), d_{T} = (1,10,50), and \\ O = {{(7,0, - \infty)}_{D 0}, {(7,0, - \infty)}_{D 1}, {(7,0, - \infty)}_{L 1}, {(7,0, - \infty)}_{L 2}, {(8,0, - \infty)}_{L 0}, {(8,0, - \infty)}_{L 3}, {(12,0, - \infty)}_{T}} . \end{array}

[1] *Corresponding author.

(a)				(b)
Driver Information				Load Information
ID	Loc	Avail	Home	ID	Pkup	Due	Dest
D0	Phila	07:00	New York	L0	Phila	1800	New York
D1	Boston	07:00	New York	L1	Boston	1200	Phila
st. work hours <= 10, miles <= 500				L2	Phila	1200	Houston
Ending at home				L3	Boston	1800	New York
				st. Delivery time <= Due

Brasil

Brasil

A GENERALIZED DECOMPOSITION ALGORITHM FOR REAL-TIME TRUCK ROUTING PROBLEMS

ABSTRACT