Scheduling of the inbound and outbound trucks at a Cross-Docking platform considering mutualization of docks

$N$	Set of unloading docks
$M$	Set of loading docks
$I$	Set of inbound trucks
$O$	Set of outbound trucks
$R$	Set of requirements
$P$	Set of type of products
$s$	Time to change a truck at the dock
$t$	Time to transport a type of product unloaded from a truck
$B i g M$	Big number
$p_{l}$	Income for accepting requirement $l \in R$
$a_{l}$	The starting time needed by an inbound and outbound dock for the requirement $l \in R$
$b_{l}$	The ending time needed by an inbound and outbound dock for the requirement $l \in R$
$r e_{i}, r s_{j}$	Arrival time to the platform of unloading truck $i \in I$ and loading truck $j \in O$
$f_{i}, g_{j}$	Waiting cost at the platform per time unit of unloading truck $i \in I$ and loading truck $j \in O$
$q_{i k}$	Quantity of units that are unloaded at a platform of the product type $k \in P$ from truck $i \in I$
$d_{j k}$	Quantity of units that are loaded of the product type $k \in P$ from truck $j \in O$
$e$	The unloading time per unit
$o$	The loading time per unit
$w_{i}$	The starting time of the unloading from inbound truck $i \in I$ ’
$u_{i}$	The end time of the unloading from inbound truck $i \in I$ ’
$y_{j}$	The starting time of the loading of outbound truck $j \in O ´$ .
$v_{j}$	The end time of the loading of outbound truck $j \in O ´$
$α_{i}$	The waiting time of inbound truck $i \in I$
$β_{j}$	The waiting time of outbound truck $j \in O$
$x_{i j k}$	The number of units moved from inbound truck $i \in I$ to outbound truck $j \in O$ of the product type $k \in P$
$λ_{i j}$	1 if any type of product of the inbound truck $i \in I$ is moved to the outbound truck $j \in O$ , and 0 otherwise
$µ_{l}$	1 if the requirement $l \in R$ is accepted and 0 otherwise
$σ_{i j h}$	1 if the inbound truck $i \in I'$ precedes inbound truck $j \in I'$ at unloading dock $h \in N$ given that $i \neq j$ and 0 otherwise
$π_{i j h}$	1 if an outbound truck $i \in O'$ precedes outbound truck $j \in O'$ at loading dock $h \in M$ given that $i \neq j$ , and 0 otherwise
$ε_{l h}, θ_{l k}$	1 if the inbound $h \in N$ and outbound dock $k \in M$ docks are assigned to requirement $l \in R$
$δ_{l i}$	1 if the inbound truck $i \in I$ is served first at the dock and then the requirement $l \in R$ without overlapping
$τ_{l j}$	1 if the outbound truck $j \in O$ is served first at the dock and then the requirement $l \in R$ without overlapping
$ρ_{i h}$	1 if the truck $i \in I$ is assigned to the loading dock $h \in N$
$ω_{i h}$	1 if the truck $i \in O$ is assigned to the loading dock $h \in M$
$ψ_{l m h}$	Which only makes sense if the requirements $l$ and $m \in R$ are scheduled at the same unloading dock $h \in N$
$η_{l m h}$	Which only makes sense if the requirements $l$ and $m \in R$ are scheduled at the same loading dock $h \in M$

Parameter	Nomenclature	Value
Unloading docks	\|N\|	{2, 4}
Loading docks	\|M\|	{4, 6}
Inbound trucks	\|I\|	{4, 8}
Outbound trucks	\|O\|	{8, 16}
Requirements	\|R\|	{10, 20}
Product types	\|P\|	{20, 40}

Metric	Value
Number of optimal solutions	65
Average CPU time (sec.)	329.75
Maximum CPU time (sec.)	8103
Minimum CPU time (sec.)	0.86
Standard deviation (sec)	1,069

Metric	Value
Average optimality gap	8.95%
Maximum optimality gap	38.38%
Minimum optimality gap	0.06%
Standard deviation (%)	7.9%

	Parameters						Results
Instance ID	\|I\|:	\|O\|:	\|P\|:	\|N\|:	\|M\|:	\|R\|:	Gap	CPU time (s)	Objective function
1	4	8	20	2	4	10	0.00%	1.24	312,858.00
2	4	8	20	2	4	20	0.01%	70.95	255,583.00
3	4	8	20	2	6	10	0.00%	5.92	564,898.00
4	4	8	20	2	6	20	0.00%	15.42	345,280.00
5	4	8	20	4	4	10	0.00%	1.15	865,803.00
6	4	8	20	4	4	20	0.00%	5.34	138,327.00
7	4	8	20	4	6	10	0.00%	0.91	492,354.00
8	4	8	20	4	6	20	0.00%	6.52	328,345.00
9	4	8	40	2	4	10	0.00%	2.78	708,240.00
10	4	8	40	2	4	20	0.00%	5.4	629,168.00
11	4	8	40	2	6	10	0.00%	5.59	591,834.00
12	4	8	40	2	6	20	0.01%	6.35	603,579.00
13	4	8	40	4	4	10	0.00%	1.66	573,601.00
14	4	8	40	4	4	10	0.00%	0.86	406,284.00
15	4	8	40	4	6	10	0.00%	1.2	935,785.00
16	4	8	40	4	6	20	1.38%	10800	652,717.00
17	4	16	20	2	4	10	1.05%	10800	835,855.00
18	4	16	20	2	4	20	12.40%	10800	835,566.00
19	4	16	20	2	6	10	8.18%	10800	771,120.00
20	4	16	20	2	6	20	0.00%	249.25	380,905.00
21	4	16	20	4	4	10	6.91%	10800	719,676.00
22	4	16	20	4	4	20	19.18%	10800	951,634.00
23	4	16	20	4	6	10	10.38%	10800	187,914.00
24	4	16	20	4	4	20	27.28%	10800	690,746.00
25	4	16	40	2	4	10	9.04%	10800	1,428,986.00
26	4	16	40	2	4	20	4.24%	10800	1,783,752.00
27	4	16	40	2	6	10	7.52%	10800	912,581.00
28	4	16	40	2	6	20	4.86%	10800	1,095,480.00
29	4	16	40	4	4	10	10.81%	10800	819,178.00
30	4	16	40	4	4	20	3.23%	10800	1,194,469.00
31	4	16	40	4	6	10	5.61%	10800	1,437,353.00
32	4	16	40	4	6	10	2.75%	10800	1,202,794.00
33	8	8	20	2	4	10	0.01%	70.98	844,308.00
34	8	8	20	2	4	20	0.00%	23.96	546,287.00
35	8	8	20	2	6	10	0.00%	88.46	917,993.00
36	8	8	20	2	6	20	0.00%	41.25	673,694.00
37	8	8	20	4	4	10	0.01%	2054.39	497,865.00
38	8	8	20	4	4	20	0.00%	1404.75	350,849.00
39	8	8	20	4	6	10	0.06%	10800	294,047.00
40	8	8	20	4	6	20	0.00%	80.42	642,000.00
41	8	8	40	2	4	10	0.00%	368.2	1,058,835.00
42	8	8	40	2	4	10	0.01%	1161.51	886,919.00
43	8	8	40	2	6	10	0.00%	15.93	1,135,549.00
44	8	8	40	2	6	20	0.00%	22.66	1,142,830.00
45	8	8	40	4	4	10	0.00%	33.28	544,765.00
46	8	8	40	4	4	20	4.36%	10800	927,428.00
47	8	8	40	4	6	10	0.00%	39.96	464,805.00
48	8	8	40	4	6	20	0.00%	154.18	1,030,667.00
49	8	16	20	2	4	10	5.91%	10800	1,488,440.00
50	8	16	20	2	4	20	3.64%	10800	911,947.00
51	8	16	20	2	6	10	0.60%	10800	1,334,189.00
52	8	16	20	2	6	20	2.44%	10800	1,040,312.00
53	8	16	20	4	4	10	19.45%	10800	810,105.00
54	8	8	20	4	6	20	0.00%	110.78	1,073,301.00
55	8	16	20	4	6	10	1.62%	10800	533,236.00
56	8	16	20	4	4	20	19.72%	10800	542,777.00
57	8	16	40	2	4	10	27.40%	10800	2,292,650.00
58	8	16	40	2	4	20	9.87%	10800	1,787,240.00
59	8	16	40	2	6	10	13.60%	10800	2,066,403.00
60	8	16	40	2	6	20	0.41%	10800	1,414,612.00
61	8	16	40	4	4	10	16.10%	10800	1,763,647.00
62	8	16	40	4	4	20	11.98%	10800	1,337,360.00
63	8	16	40	4	6	10	4.81%	10800	1,218,799.00
64	8	16	40	4	6	20	0.62%	10800	868,508.00
65	4	8	20	2	4	10	0.00%	1.06	505,126.00
66	4	8	20	2	4	20	0.00%	8.65	646,574.00
67	4	8	20	2	6	10	0.00%	2.22	527,179.00
68	4	8	20	2	6	20	0.00%	16.56	524,141.00
69	4	8	20	4	4	10	0.00%	5.06	202,053.00
70	4	8	20	4	4	20	0.00%	32.48	301,253.00
71	4	8	20	4	6	10	0.00%	2.28	219,005.00
72	4	8	20	4	6	20	0.00%	19.17	428,856.00
73	4	8	40	2	4	10	0.00%	77.35	127,437.00
74	4	8	40	2	4	20	0.01%	198.05	815,770.00
75	4	8	40	2	6	10	0.00%	3.24	701,028.00
76	4	8	40	2	6	20	0.00%	5.86	583,738.00
77	8	8	40	4	6	20	0.01%	1692.53	426,410.00
78	4	8	40	4	4	20	0.00%	4.3	500,186.00
79	4	8	40	4	6	10	0.00%	1.97	660,610.00
80	4	8	40	4	6	20	0.00%	3.28	211,708.00
81	4	16	20	2	4	10	5.87%	10800	1,796,690.00
82	4	16	20	2	4	20	16.68%	10800	585,802.00
83	4	16	20	2	6	10	0.01%	234.26	832,645.00
84	4	16	20	2	6	20	6.85%	10800	623,745.00
85	4	16	20	4	4	10	0.01%	8103.08	844,798.00
86	8	16	20	4	6	20	3.16%	10800	742,344.00
87	4	16	20	4	6	10	0.01%	958.22	541,751.00
88	8	16	20	4	6	20	0.95%	10800	788,795.00
89	4	16	40	2	4	10	6.33%	10800	490,816.00
90	4	16	40	2	4	20	5.83%	10800	892,754.00
91	4	16	40	2	6	10	10.89%	10800	1,044,698.00
92	4	16	40	2	6	20	2.31%	10800	751,663.00
93	4	16	40	4	4	10	10.93%	10800	1,550,992.00
94	4	16	40	4	4	20	38.38%	10800	344,941.00
95	4	16	20	4	6	20	7.53%	10800	736,943.00
96	4	16	40	4	6	20	11.85%	10800	551,485.00
97	8	8	20	2	4	10	0.00%	405.39	513,908.00
98	8	8	20	2	4	20	0.01%	258.47	606,950.00
99	8	8	20	2	6	10	0.00%	228.94	1,014,807.00
100	8	8	20	2	6	20	0.00%	48.66	608,272.00
101	8	8	20	4	4	10	0.00%	5.85	299,956.00
102	8	8	20	4	4	20	8.37%	10800	405,949.00
103	8	8	20	4	6	10	0.00%	14.93	466,094.00
104	4	16	20	4	6	20	7.78%	10800	1,293,690.00
105	4	16	40	4	6	20	17.27%	10800	649,728.00
106	8	8	40	2	4	20	0.01%	53.75	461,588.00
107	8	8	40	2	6	10	0.00%	43.23	855,876.00
108	8	8	40	2	6	20	0.01%	1232.06	813,503.00
109	8	8	40	4	4	10	0.00%	72.47	547,090.00
110	8	8	40	4	4	20	0.01%	945.76	457,167.00
111	8	8	40	4	6	10	0.00%	261.11	879,908.00
112	8	8	40	2	4	20	0.01%	354.82	1,101,907.00
113	8	16	20	2	4	10	4.99%	10800	1,129,864.00
114	8	16	20	2	4	20	2.47%	10800	1,525,942.00
115	8	16	20	2	6	10	1.59%	10800	906,274.00
116	8	16	20	2	6	20	0.00%	83.88	652,993.00
117	8	16	20	4	4	10	21.05%	10800	892,154.00
118	8	16	20	4	4	20	11.37%	10800	745,172.00
119	8	16	20	4	6	10	0.20%	10800	823,104.00
120	8	16	40	4	6	20	1.42%	10800	706,096.00
121	8	16	40	2	4	10	12.19%	10800	1,754,913.00
122	8	16	40	2	4	20	10.33%	10800	1,239,179.00
123	8	16	40	2	6	10	27.34%	10800	2,007,234.00
124	8	16	40	2	6	20	3.23%	10800	1,257,199.00
125	8	16	40	4	4	10	7.48%	10800	1,133,588.00
126	8	16	40	4	4	20	17.23%	10800	1,215,819.00
127	8	16	40	4	6	10	4.79%	10800	1,164,960.00
128	4	8	40	4	4	20	0.00%	3.68	716,780.00

[1] * jmonto36@eafit.edu.co

Brasil

Brasil

Scheduling of the inbound and outbound trucks at a Cross-Docking platform considering mutualization of docks

Abstract

Paper aims

Originality

Research method

Main findings

Implications for theory and practice