The document summarizes a thesis defense presentation on heuristic algorithms for truck scheduling at cross-docking terminals. It introduces the problem of sequencing inbound and outbound trucks at a single dock door to minimize makespan. Four heuristic algorithms are presented: local search, simulated annealing, large neighborhood search, and beam search. Computational experiments are reported to evaluate the heuristics on this truck scheduling problem.

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Thesis Defense

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Heuristics for Truck Scheduling at Cross
Docking Terminals
Supervisor: Dr. Ivan Contreras
Prepared by: Wenying Yan
April 2014

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we study a truck scheduling problem arising in cross-docking terminals.
It consists of sequencing a set of inbound and outbound trucks to a
single strip and stack door to minimize the makespan. We present four
different heuristic algorithms: a local search, a simulated annealing, a
large neighborhood search, and a beam search. Computational
experiments are reported.
Abstract

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 Introduction and Literature Review
 Problem Definition
 IP Formulations
 Heuristic Algorithms
 Computational Results
 Conclusions and Further Research
Outline

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The Importance of Logistics and Transportation
Logistics service providers GDP was predicted to increase by 40%
between 2007 and 2015, generating $56 billion. In 2011, truck
transportation shared the largest segment of logistics services and
accounted for 31% of the sector's share of GDP. (Transport Canada)

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Cross-docking
One innovative strategy in logistics and transportation that has increasingly
attracted industrial practitioners and researchers is cross-docking (CD).
The four main functions of a traditional distribution center: receiving, storage,
order picking, shipping.
CD however is an approach that eliminates the two most expensive handling
operations in a traditional distribution center: storage and order picking.

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The Definition of Cross-docking
Cross-docking is the process of consolidating freight with the same
destination (but coming from several origins), with minimal handling and
with little or no storage between unloading and loading of the goods .

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Literature Review Industry-Wise
Cross-docking was first used by the US trucking industry during the 1930s
(Arnaout et la., 2010)
From 1930s to now, there are many successful applications globally:
• In Hong Kong, All third-party logistics companies are applying CD systems
• In Germany, Deutsche Post World Net reduces the travel by 37-39%
• Many other companies have also reported the successful application of cross-docking
(e.g. UPS, Toyota , Walmart).

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Literature Review Research-Wise
Research-wise: ≥ 80% of total publications in CD have appeared
from 2004 until now.
0
10
20
30
40
50
60
70
NumberofPublications
Wael et al.,2013
Research Trend

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Literature Review Research-Wise
Three recently review papers:
• Van et al.(2012) provide an overview of CD concept.
• Agustina et al.(2010) provide a comprehensive review of mathematic models.
• Boysen et al.(2010) provide a detailed review to classify deterministic truck
scheduling in DC.
Classification of Problems:
• Strategic Decisions: Location of Cross-docks. (Sung et al.,2003)
• Strategic Decisions: Layout of Cross-docking Terminals. (Bartholdi et al.,2000)
• Tactical Decisions: Network Flow Optimization. (Lim et al.,2005)
• Operational Decisions: Vehicle Routing. (Laporte et al.,2009)
• Operational Decisions: Dock Door Assignment. (Yu et al.,2008)
• Operational Decisions: Truck Scheduling. (Boysen et al.,2012)

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Literature Review Research-Wise
Review Papers of Truck scheduling in CD
Single Inbound and Outbound Door:
• Yu et al.(2002): Operational Strategies for Cross Docking Systems
• Chen et al.(2009): Minimizing the makespan in a two-machine cross-docking flow shop
problem
• Boysen et al.(2010): Scheduling inbound and outbound trucks at cross docking
terminal
Scheduling of Inbound Trucks:
• Acar et al.(2004): Robust Dock Assignments at Less-than-Truckload Terminals
• Wang et al.(2008): Real-time trailer scheduling for cross-dock operations
• Rosales et al.(2009): Transfreight reduces costs and balances workload at
Georgetown cross-dock
Scheduling of Inbound and Outbound trucks:
• Lim et al.(2006): Truck dock assignment problem with time windows and capacity
constraint in transshipment network through cross-docks
• Boysen et al.(2010): Truck scheduling at zero-inventory cross docking terminals
• Kuo et al.(2013): Optimizing truck sequencing and truck dock assignment in a cross
docking system

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 Introduction and Literature Review
 Problem Definition
 IP Formulations
 Heuristic Algorithms
 Computational Results
 Conclusions and Further Research
Problem Definition

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Inbound
trucks
Outbound
trucks
Inbound
doors
Outbound
doors
A Cross-docking Terminal
Makespan: the time that the first inbound truck is
assigned to the time that the last outbound truck is
assigned.

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Inbound
trucks
Outbound
trucks
Inbound
door
Outbound
door
A Cross-dock with One Inbound and Outbound Door

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Product a b c d e
In Truck 1 2
2 2 1
3 2
4 1
Out Truck1 1
2 1 2
3 1
4 2 1
Total 2 2 1 2 1
The Information of Example Instance

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Time Slot 1 2 3 4 5 6 7
In Truck
Out Truck
Time slot 0
temporary
storage
outbound
door
inbound
door

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Time Slot 1 2 3 4 5 6 7
In Truck 1
Out Truck
Time slot 1
temporary
storage

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Time Slot 1 2 3 4 5 6 7
In Truck 1
Out Truck
Time slot 1
temporary
storage

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Time Slot 1 2 3 4 5 6 7
In Truck 1
Out Truck 1
Time slot 1
temporary
storage

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Time Slot 1 2 3 4 5 6 7
In Truck 1 2
Out Truck 1
Time slot 2
temporary
storage

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Time Slot 1 2 3 4 5 6 7
In Truck 1 2
Out Truck 1
Time slot 2
temporary
storage

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Time Slot 1 2 3 4 5 6 7
In Truck 1 2
Out Truck 1 2
Time slot 2
temporary
storage

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Time Slot 1 2 3 4 5 6 7
In Truck 1 2 3
Out Truck 1 2
Time slot 3
temporary
storage

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Time Slot 1 2 3 4 5 6 7
In Truck 1 2 3
Out Truck 1 2
Time slot 3
temporary
storage

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Time Slot 1 2 3 4 5 6 7
In Truck 1 2 3 4
Out Truck 1 2
Time slot 4
temporary
storage

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Time Slot 1 2 3 4 5 6 7
In Truck 1 2 3 4
Out Truck 1 2 3
Time slot 4
temporary
storage

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Time Slot 1 2 3 4 5 6 7
In Truck 1 2 3 4
Out Truck 1 2 3 4
Time slot 5
Total Makespan: 5
temporary
storage

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Time Slot 1 2 3 4 5 6 7
In Truck
Out Truck
Time slot 0

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Time Slot 1 2 3 4 5 6 7
In Truck 1
Out Truck
Time slot 1

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Time Slot 1 2 3 4 5 6 7
In Truck 1 2
Out Truck
Time slot 2

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Time Slot 1 2 3 4 5 6 7
In Truck 1 2 3
Out Truck
Time slot 3

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Time Slot 1 2 3 4 5 6 7
In Truck 1 2 3 4
Out Truck 1
Time slot 4

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Time Slot 1 2 3 4 5 6 7
In Truck 1 2 3 4
Out Truck 1 2
Time slot 5

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Time Slot 1 2 3 4 5 6 7
In Truck 1 2 3 4
Out Truck 1 2 3
Time slot 6

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Time Slot 1 2 3 4 5 6 7
In Truck 1 2 3 4
Out Truck 1 2 3 4
Time slot 7
Total Makespan: 7

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Product a b c d e
In Truck 1 2
2 2 1
3 2
4 1
Out Truck1 1
2 1 2
3 1
4 2 1
Total 2 2 1 2 1
Total numbers of combination A(5,5) = 14400
Small Example of Instance

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Larger Example of Instance
a b c d e f g h i j k l m n
In Truck1 70 0 0 101 38 0 0 0 0 75 0 57 116 0
2 0 0 305 119 0 40 0 0 8 106 0 128 38 0
3 0 0 34 0 38 225 0 1000 43 29 504 0 114 34
4 0 0 0 71 0 0 0 0 157 21 0 120 29 0
5 0 0 0 72 0 231 0 0 0 43 0 0 0 80
6 0 485 0 40 38 132 0 0 125 112 0 96 135 0
7 0 0 158 14 0 0 0 0 0 34 0 47 147 0
8 0 0 0 10 0 133 0 0 0 109 0 0 78 471
9 0 0 0 113 0 0 0 0 161 81 0 45 140 408
10 0 0 140 0 0 0 0 0 199 84 0 132 53 0
11 0 0 98 22 0 140 0 0 0 91 496 123 22 0
12 930 515 0 103 86 34 0 0 0 98 0 94 0 0
13 0 0 0 73 277 65 1000 0 287 36 0 0 0 0
14 0 0 0 78 156 0 0 0 20 0 0 0 8 0
15 0 0 104 86 0 0 0 0 0 21 0 37 53 0
16 0 0 161 98 367 0 0 0 0 60 0 121 67 7
Out Truck1 2 0 0 110 107 53 173 102 0 0 0 0 0 0
2 116 0 0 104 0 0 0 98 195 0 0 0 34 0
3 0 0 285 7 24 58 0 63 0 0 168 799 0 0
4 48 336 139 16 111 66 74 107 0 0 0 0 185 107
5 0 0 77 108 76 0 0 0 39 1000 0 0 152 32
6 50 0 0 109 83 162 149 17 0 0 0 0 0 155
7 0 278 0 34 55 102 45 128 107 0 0 0 0 0
8 0 7 46 50 121 0 77 11 115 0 124 0 192 65
9 163 130 0 142 105 28 0 18 10 0 0 0 179 37
10 139 0 0 16 0 23 0 119 36 0 87 0 0 212
11 92 249 0 67 57 12 36 98 71 0 313 201 2 0
12 145 0 0 0 45 150 102 0 218 0 0 0 0 0
13 0 0 178 49 90 106 140 109 0 0 0 0 0 107
14 81 0 0 2 0 74 25 30 209 0 0 0 0 82
15 0 0 275 66 126 1 179 78 0 0 55 0 94 203
16 164 0 0 120 0 165 0 22 0 0 253 0 162 0

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4.3 × 1026
The total number of stars in the universe may be around 1.2×1023
(http://www.universetoday.com)
Total numbers of combination A(16, 16)
Complexity of the TRSP
The TRSP is NP-hard in the strong sense (Boysen 2010)

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Contributions of this Thesis
 Propose an integer programming formulation.
 Develop four heuristic algorithms:
• a local search (LS),
• a simulated annealing (SA),
• a large neighborhood search (LNS),
• a beam search (BS).
 Introduce two new sets of instances to assess the performance of
the proposed solution methods.

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 Introduction and Literature Review
 Problem Definition
 IP Formulations
 Heuristic Algorithms
 Computational Results
 Conclusions and Further Research
IP Formulations

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Minimise 𝐶 𝑋, 𝑌 = 𝐶 𝑚𝑎𝑥 (1)
Subject to:
𝐶 𝑚𝑎𝑥 ≥ 𝑦 𝑜𝑡 ∙ 𝑡 ∀ 𝑜 𝜖 𝑂; 𝑡 = 1, … , 𝑇 (2)
𝑡 = 1
𝑇
𝑥𝑖𝑡 = 1 ∀𝑖 𝜖 𝐼 (3)
𝑖 𝜖 𝐼 𝑥𝑖𝑡 ≤ 1 ∀ 𝑡 = 1, … , 𝑇 (4)
𝑡 = 1
𝑇
𝑦 𝑜𝑡 = 1 ∀ 𝑜 𝜖 𝑂 (5)
𝑜 𝜖 𝑂 𝑦 𝑜𝑡 ≤ 1 ∀ 𝑡 = 1, … , 𝑇 (6)
𝜏 = 1
𝑡
𝑖 𝜖 𝐼 𝑥𝑖𝜏 ∙ 𝑎𝑖𝑝 ≥ 𝜏 = 1
𝑡
𝑜 𝜖 𝑂 𝑦𝑜𝜏 ∙ 𝑏 𝑜𝑝 ∀ 𝑡 = 1, … , 𝑇; 𝑝 𝜖 𝑃 (7)
𝑥𝑖𝑡 𝜖 0, 1 ∀ 𝑖 𝜖 𝐼; 𝑡 = 1, … , 𝑇 (8)
𝑦 𝑜𝑡 𝜖 0, 1 ∀ 𝑜 𝜖 𝑂; 𝑡 = 1, … , 𝑇 (9)
Formulation Proposed by Boysen et al. (2012)

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New Formulation Decision Variables
gitlp denotes the number of products of type p coming
from truck i moved from time slot t (receiving door) and
shipped by an outbound truck in time slot l (t <= l).

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Minimise 𝐶 𝑋, 𝑌 = 𝐶 𝑚𝑎𝑥 (1)
Subject to:
𝐶 𝑚𝑎𝑥 ≥ 𝑦 𝑜𝑡 ∙ 𝑡 ∀ 𝑜 𝜖 𝑂; 𝑡 = 1, … , 𝑇 (2)
𝑡 = 1
𝑇
𝑥𝑖𝑡 = 1 ∀𝑖 𝜖 𝐼 (3)
𝑖 𝜖 𝐼 𝑥𝑖𝑡 ≤ 1 ∀ 𝑡 = 1, … , 𝑇 (4)
𝑡 = 1
𝑇
𝑦 𝑜𝑡 = 1 ∀ 𝑜 𝜖 𝑂 (5)
𝑜 𝜖 𝑂 𝑦 𝑜𝑡 ≤ 1 ∀ 𝑡 = 1, … , 𝑇 (6)
𝜏 = 1
𝑡
𝑖 𝜖 𝐼 𝑥𝑖𝜏 ∙ 𝑎𝑖𝑝 ≥ 𝜏 = 1
𝑡
𝑜 𝜖 𝑂 𝑦𝑜𝜏 ∙ 𝑏 𝑜𝑝 ∀ 𝑡 = 1, … , 𝑇; 𝑝 𝜖 𝑃 (7)
𝑥𝑖𝑡 𝜖 0, 1 ∀ 𝑖 𝜖 𝐼; 𝑡 = 1, … , 𝑇 (8)
𝑦 𝑜𝑡 𝜖 0, 1 ∀ 𝑜 𝜖 𝑂; 𝑡 = 1, … , 𝑇 (9)
Formulation Proposed by Boysen et al. (2012)

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Minimize 𝐶 𝑋, 𝑌 = 𝐶 𝑚𝑎𝑥 10
Subject to:
𝐶 𝑚𝑎𝑥 ≥ 𝑦 𝑜𝑡 ∙ 𝑡 ∀ 𝑜 𝜖 𝑂; 𝑡 = 1, … , 𝑇 (11)
𝑡 = 1
𝑇
𝑥𝑖𝑡 = 1 ∀ 𝑖 𝜖 𝐼 (12)
𝑖 𝜖 𝐼 𝑥𝑖𝑡 ≤ 1 ∀ 𝑡 = 1, … , 𝑇 (13)
𝑡 = 1
𝑇
𝑦 𝑜𝑡 = 1 ∀ 𝑜 𝜖 𝑂 (14)
𝑜 𝜖 𝑂 𝑦 𝑜𝑡 ≤ 1 ∀ 𝑡 = 1, … , 𝑇 (15)
𝒍 = 𝒕
𝑻
𝒈𝒊𝒕𝒍𝒑 = 𝒂𝒊𝒑 𝒙𝒊𝒕 ∀ 𝒊 = 𝑰; 𝒑 𝝐 𝑷; 𝒕 𝝐 𝑻 (𝟏𝟔)
𝒊 𝝐 𝑰 𝒕 = 𝟏
𝒍
𝒈𝒊𝒕𝒍𝒑 ≥ 𝒃 𝒐𝒑 𝒚 𝒐𝒍 ∀ 𝒑 𝝐 𝑷; 𝒐 𝝐 𝑶; 𝒍 𝝐 𝑻 (𝟏𝟕)
𝑥𝑖𝑡 𝜖 0, 1 ∀ 𝑖 𝜖 𝐼; 𝑡 = 1, … , 𝑇 (18)
𝑦 𝑜𝑡 𝜖 0, 1 ∀ 𝑜 𝜖 𝑂; 𝑡 = 1, … , 𝑇 (19)
New Formulation

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 Introduction and Literature Review
 Problem Definition
 IP Formulations
 Heuristic Algorithms
 Computational Results
 Conclusions and Further Research
Heuristic Algorithms

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Local Search
Inbound trucks neighborhood:
𝑁𝐼(S) = {S’ = (a’, b): ∃! (𝑖1, 𝑖2), a(𝑖1) = a’(𝑖2), a(𝑖2) = a’(𝑖1), 𝑖1 ≠ 𝑖2},
Outbound trucks neighborhood:
𝑁 𝑂(S) = {S’ = (a, b’): ∃! (𝑜1, 𝑜2), b(𝑜1) = b’(𝑜2), b(𝑜2) = b’(𝑜1), 𝑜1 ≠ 𝑜2}

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Local Search
Fix the outbound trucks sequences
Apply the best improvement strategy to inbound trucks
Fix the inbound trucks sequences
Apply the best improvement strategy to outbound trucks

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Pseudo Code of Local Search

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Pseudo Code of Simulated Annealing

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Simulated Annealing
1 2 3 4 5 6 7 8 9 10 11 12
1 2 3 10 5 6 7 8 9 4 11 12
S = F(S) = 20
S’ =
𝑖𝑓 𝐹 𝑆′ − 𝐹 𝑆 ≤ 0, 𝑠𝑎𝑦 F(S’) = 18
P{Accept 𝑆′
as next solution } = 1
𝑖𝑓 𝐹 𝑆′ − 𝐹 𝑆 > 0, 𝑠𝑎𝑦 F(S’) = 25
P{Accept 𝑆′
as next solution } = 𝑒𝑥𝑝 −
𝐹 𝑆′ −𝐹 𝑆
𝑇 𝑘
F(S’) = ?

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1 2 3 4 5 6 7 8 9 10 11 12 13
1 10 8 4 13 6 12 5 9 7 11 2 3
1 4 6 9 11 1
Lager Neighborhood Search
Destroy Strategy
Repair Strategy

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BS is an adaptation of the well-known branch and bound
(B&B) algorithm. But BS keeps only some promising
nodes and to permanently prune off other nodes.
Beam Search

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Beam Search
{i1, i2, i3, i5, i4}
{i3, i2, i4, i5, i1}

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Three different filtering approaches for inbound trucks:
 Makespan
 Lower bound (Boysen et al.)
 The difference between makespan and lower bound
The filtering approach for outbound trucks:
f(o) =
1
𝑝 𝜖 𝑃
𝑏 𝑜𝑝
𝜎 𝜖 𝑂 𝑏 𝜎𝑝
Beam Search

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 Introduction and Literature Review
 Problem Definition
 IP Formulations
 Heuristic Algorithms
 Computational Results
 Conclusions and Further Research
Computational Results

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First Set of Instances
Product a b c d e
Inbound trucks 1 50 0 0 0 0
2 0 30 0 0 0
3 0 0 40 0 0
4 0 0 0 20 0
5 0 0 0 0 60
Outbound trucks 1 17 13 0 9 18
2 0 0 18 0 14
3 18 0 13 7 13
4 0 17 0 2 15
5 15 0 9 2 0
Total units 50 30 40 20 60

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Product a1 a2 b1 b2 b3 c1 c2
Inbound trucks 1 50 10 0 0 0 0 0
2 20 30 0 0 0 0 0
3 0 0 10 30 10 0 0
4 0 0 20 30 20 0 0
5 0 0 40 10 30 0 0
6 0 0 0 0 0 50 10
7 0 0 0 0 0 60 40
Outbound trucks 1 30 0 24 13 18 13 15
2 15 13 0 16 0 26 8
3 0 11 21 17 13 17 0
4 25 0 25 0 15 22 16
5 0 14 0 24 14 32 11
Total units 70 40 70 70 60 110 50
Second Set of Instances

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Third Set of Instances (Boysen et al.)
Product a b c d e f
In Truck1 70 19 0 101 38 0
2 0 0 305 119 0 0
3 0 110 34 0 38 225
4 80 0 0 71 0 0
5 0 0 162 72 0 131
6 66 485 0 40 38 132
Out Truck1 2 0 0 110 0 53
2 116 0 0 104 0 0
3 0 0 285 30 3 58
4 48 336 139 16 111 66
5 0 0 77 0 0 0
6 50 0 0 109 0 162
7 0 278 0 34 0 149
Total Units 216 614 501 403 114 488

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Size of Generated Instances

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Truck Information 1st set 2nd set 3rd set
InTruck OutTruck Product OTP Time(s) OTP Time(s) OTP Time(s)
14 14 14 23 90 16 28 15 26
14 16 14 25 60 19 11 17 96
14 18 14 26 195 22 14 18 34
16 14 16 25 7200 18 20 18 192
16 16 16 27 1500 20 25 18 776
16 18 16 29 1149 21 22 19 85
18 14 18 27 27420 19 61 20 95
18 16 18 29 510 20 30 19 402
18 18 18 31-30* 97230 23 34 19 482
Three Sets of Small Instances Solved with CPLEX
Boysen et al.(2010): Scheduling inbound and outbound trucks
at cross docking terminal
Dynamic programming cannot handle the large instance efficiently

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Three Sets of Large Instances Solved with CPLEX
Truck Information 1st set 3rd set
InTruck OutTruck Product UB LB Gap(%) Time(h) UB LB Gap(%) Time(h)
26 26 26 46 30 34% 24 36 22 39% 24
26 28 26 47 25 47% 24 40 20 50% 24
26 30 26 50 32 36% 24 45 22 51% 24
28 26 28 48 28 41% 24 40 26 35% 24
28 28 28 49 27 44% 24 49 27 44% 24
28 30 28 52 32 46% 24 39 19 51% 24
30 26 30 49 30 38% 24 49 20 59% 24
30 28 30 52 30 42% 24 49 25 48% 24
30 30 30 52 32 42% 24 35 14 14% 24

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Results of SA analysis
0
1
2
3
4
5
6
7
8
9
0.5 0.7 2.0 4.3 11.7 17.2 446.6
SA S1 SA S2 SA S3 SA S4 SA S5 SA S6 SA S7
AverageDeviation(%)
Time (s)

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Results of LNS analysis
0
0.5
1
1.5
2
2.5
3
3.5
4
96 133 381 824 2235 3296
LNS S1 LNS S2 LNS S3 LNS S4 LNS S5 LNS S6
AverageDeviation(%)
Time (s)

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Results of BS analysis
0
0.5
1
1.5
2
2.5
3
3.5
4
217 234 236 327 374 387 557 614 640
BS L5 9 BS G5 9 BS U5 9 BS L10 7 BS G10 7 BS U10 7 BS L10 12 BS G10 12BS U10 12
AverageDeviation(%)
Time (s)

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A Comparison of Heuristics with First Set of Large Instances
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
20.0
22.0
24.0
0.0 244.0 393.7 461.5 3600.0
LS BS U59 LNS S4 SA S7 CPLEX
AverageDeviation(%)
Time (s)

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A Comparison of Heuristics with Third Set of Large Instances
0.0
5.0
10.0
15.0
20.0
25.0
0.0 130.1 223.5 439.2 3600.0
LS LNS S4 BS L5 9 SA S7 CPLEX
AverageDeviation(%)
Time (s)

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Conclusions and Further Study
 Introduction and Literature Review
 Problem Definition
 IP Formulations
 Heuristic Algorithms
 Computational Results
 Conclusions and Further Research

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Conclusion
 “when |O|= 18 can be interpreted as an upper limit up to which the
BDP-approach can be reasonably applied.” ---- Boysen et al.
 Proposed algorithms are able to solve larger instances efficiently.
 Introduce two new sets of instances.
 Propose an integer programming formulation.

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Further Study
 Use decomposition technics to handle the proposed formulation.
 Develop an exact algorithm. (Branch and bound)
 Consider multiple inbound and outbound doors.
 Take into consideration a dynamic case.

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Thanks for your attention
Questions?

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S
F(S)
T0
T1
Tf
Always accept S’ as current solution
Accept S’ with P = 𝑒𝑥𝑝 −
𝐹 𝑆′ −𝐹 𝑆
𝑇 𝑘
S0

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0
0.2
0.4
0.6
0.8
1
1.2
Simulated Annealing
𝑒𝑥𝑝 −
𝐹 𝑆′
− 𝐹 𝑆
𝑇𝑘

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Assumptions
• There are only one receiving and shipping door in the cross-dock and there are
located at different places of the terminal (segregated mode of service).
• The time of processing (i.e. unloading or loading processes) for each truck is the
same and within a certain time slot (e.g. few hours).
• All inbound and outbound trucks are available at the beginning of the time horizon.
There are no predefined restrictions on truck assignments to slots (e.g. release or
due dates)
• The input data is known in advance and deterministic.
• The time for delivering products from the receiving door to the shipping door is
constant and therefore can be ignored when model the problem.
• The numbers of the product in the inbound trucks are equal to the numbers of
products required by the outbound trucks.
• The size of the temporary stock is unlimited.

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Notation for the Formulation
I: Set of inbound trucks (index i)
O: Set of outbound trucks (index o)
T: (Maximum) number of time slots available for (un-)loading trucks
(index t)
P: Set of products (index p)
aip: Quantity of product type p carried by inbound truck i
bop: Quantity of product type p required by outbound truck o

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Formulation Proposed by Boysen et la. (2012)
xit =
1, 𝑖𝑓 𝑖𝑛𝑏𝑜𝑢𝑛𝑑 𝑡𝑟𝑢𝑐𝑘 𝑖 𝑖𝑠 𝑎𝑠𝑠𝑖𝑔𝑛𝑒𝑑 𝑡𝑜 𝑠𝑙𝑜𝑡 𝑡
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
yot =
1, 𝑖𝑓 𝑜𝑢𝑡𝑏𝑜𝑢𝑛𝑑 𝑡𝑟𝑢𝑐𝑘 𝑜 𝑖𝑠 𝑎𝑠𝑠𝑖𝑔𝑛𝑒𝑑 𝑡𝑜 𝑠𝑙𝑜𝑡 𝑡
0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

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SA cooling parameters
S1 S2 S3 S4 S5 S6 S7
InTo 1000 1000 1000 1000 1000 1000 1000
InTf 1 1 1 1 1 1 1
InK 37 111 37 111 37 111 37
InL 5 2 10 5 10 10 150
OutTo 1000 1000 1000 1000 1000 1000 1000
OutTf 1 1 1 1 1 1 1
OutK 37 111 37 111 111 111 37
OutL 5 2 10 5 20 10 150

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Lower Bound
𝑠 𝑜𝑝 = min 𝑡 = 𝑠 𝑜−1 𝑝 + 1 , … , 𝑇 𝜏=1
min{ 𝐼 ;𝑡}
𝑎 𝜋 𝜏
𝑝
𝑝 ≥ 𝜏=1
𝑜
𝑏 𝜇 𝜏
𝑝
𝑝 ∀ 𝑜 𝜖 𝑂; 𝑝 = 𝑃
C2 = 𝑚𝑎𝑥 𝑝 𝜖 𝑃{𝑠|𝑂|𝑝}

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Lower Bound
0
10
20
30
40
50
60
0 16 3600
Proposed LB1 Proposed LB2 CPLEX LB
Time (s)
AverageDeviation(%)

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Dynamic programming

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Dispatching Rule
f(o) =
1
𝑝 𝜖 𝑃
𝑏 𝑜𝑝
𝜎 𝜖 𝑂 𝑏 𝜎𝑝
(the fraction of total product volume).
f(1) =
1
2
2+2+6
+
3
3 +3+3
+
4
4+5+5
The first truck carries 2 units of a, 3 units of b and 4 units of c; the
second truck carries 2 units of a, 3 units of b and 5 units of c; the third
truck carries 6 units of a, 3 units of b and 5 units of c.