Location Models in Logistics Management
Vincent F. Yu
Department of Industrial Management
National Taiwan University of Science and Technology
Oct. 21, 2010
Vincent F. Yu (IM, NTUST)
Location Models
Oct. 21, 2010
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Outline
1
Location Models
2
Applications in Logistics Management: Location
Routing Problem and Its Variants
3
References
4
Q&A
Vincent F. Yu (IM, NTUST)
Location Models
Oct. 21, 2010
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Location Models
Location Models
Location! Location!! Location!!!
Location decisions arris in a variety of public and private sector
problems.
Public Sector: Hospitals, Fire Stations, Police Departments, Landfills
for Hazardous Waste, Public Schools, Bus/Train Stations, ..., etc.
Poor decisions lead to ....
Increased likelihood of property damage, lost of life, ...
Private Sector: Offices, Plants, Retail Stores, Distribution Centers,
Hubs, Call Centers, ..., etc.
Poor decisions lead to ....
Increased cost, decreased efficiency/competitiveness, ...
The success or failure of both public and private sector facilities
depends in part on the locations chosen for those facilities.
Vincent F. Yu (IM, NTUST)
Location Models
Oct. 21, 2010
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Location Models
Key Questions
1
How many facilities?
2
Where should each facility located?
3
How large should each facility be?
4
How should demand for the facilities’ services be allocated to the
facilities?
Vincent F. Yu (IM, NTUST)
Location Models
Oct. 21, 2010
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Location Models
Prototype Problems
1
Covering Problems.
2
Center Problems.
3
Median Problems.
4
Fixed Charge Facility Location Problems.
5
Extensions of Location Models:
Location Routing Problems.
Hub Location Problems.
Dispersion Models.
Location of Undesirable Facilities.
Hierarchical Location Models.
Multiobjective Problems, ...
Vincent F. Yu (IM, NTUST)
Location Models
Oct. 21, 2010
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Location Models
Application: Ambulance Location
Set Covering Model - minimize the number of ambulances needed so
that all demand nodes are within a given number of minutes.
Maximum Covering Model - maximize number of demands that can
be covered within a specified service standard using a given number
of vehicles.
P-center Problem - minimize the maximum response time using a
given number of vehicles.
P-medium Problem - minimize the average response time using a
given number of vehicles.
Other Considerations: stochastic demands, workload balance, service
type/characteristics, temporal variation (relocatable ambulances v.s.
fixed sites).
Vincent F. Yu (IM, NTUST)
Location Models
Oct. 21, 2010
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Location Models
Application: Siting Landfills for Hazardous
Wastes
P-medium Problem - minimize the average or total shipping distance.
Maxisum or Maximin Model - locate a given number of facilities to
maximize the (weighted) distance between population centers and the
nearest sites.
Fixed Charge Facility Location Problem - balance the initial capital
investment and ongoing operating costs.
Other Considerations: reduce the inequalities across communities
(spread the risk or disbenefit), routing, ...
Vincent F. Yu (IM, NTUST)
Location Models
Oct. 21, 2010
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Location Routing Problem and Its Variants
Location Routing Problem
Traveling Salesman Problem (TSP)
Vehicle Routing Problem (VRP)
Location Routing Problem (LRP): includes two key decision problems
in strategic logistics management
Determination of Facility Location
Planning of Vehicle Routes
These decisions are often interdependent. Therefore it is beneficial to
consider them simultaneously.
Vincent F. Yu (IM, NTUST)
Location Models
Oct. 21, 2010
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Location Routing Problem and Its Variants
Applications of LRP
Food and drink distribution
Blood bank location
Newspapers delivery
Waste collection/Reverse logistics
Bill delivery
Military application
Parcel delivery
Various consumer goods distribution
Vincent F. Yu (IM, NTUST)
Location Models
Oct. 21, 2010
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Location Routing Problem and Its Variants
Solution Strategy
The LRP belongs to the class of NP-hard problems.
As the problem size grows larger, heuristic approaches become the
only viable alternative.
Vincent F. Yu (IM, NTUST)
Location Models
Oct. 21, 2010
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Location Routing Problem and Its Variants
Problem Definition
G = (V , E , c) be a complete, weighted, and undirected network.
V : set of nodes comprised of a subset I of m potential depot sites
and a subset J = V \I of n customers.
cij : traveling cost (distance) between each pair of nodes i, j.
Wi , Oi : capacity and opening cost associated with each potential
depot site i.
Each customer j ∈ J has a demand dj which must be fulfilled be a
single vehicle.
A set K of identical vehicles with capacity Q is available.
Each vehicle, when used, incurs a fixed cost Fi and performs a single
route.
Vincent F. Yu (IM, NTUST)
Location Models
Oct. 21, 2010
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Location Routing Problem and Its Variants
Problem Definition (cont.)
Each route must start and terminate at the same depot.
Each route’s total load can not exceed vehicle capacity.
The total loads of the routes assigned to a depot can not exceed the
capacity of the depot.
The objective is to determine which depots should be opened and
which routes should be constructed to minimize the total traveling
cost.
Vincent F. Yu (IM, NTUST)
Location Models
Oct. 21, 2010
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Location Routing Problem and Its Variants
Mathematical Model
Define binary variables:
yi =1 iff depot i is opened; = 0 otherwise;
fij = 1 iff customer j is assigned to depot i; = 0 otherwise;
xjlk = 1 iff edge (j, l) is traversed from j to l in the route performed
by vehicle k.
Then the problem can be formulated as a binary integer program.
Vincent F. Yu (IM, NTUST)
Location Models
Oct. 21, 2010
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Location Routing Problem and Its Variants
Mathematical Model (cont.)
min z = ∑ Oi yi + ∑∑∑ cij xijk + ∑∑∑ Fi xijk
i∈I
i∈V j∈V k∈K
subject to
∑∑ x
= 1 ∀j ∈ J
k∈K i∈V
ijk
j∈J i∈V
j ijk
∑∑ d x
∑d
j∈J
(3)
f ≤ Wi yi ∀i ∈ I
(4)
j ij
ijk
∑∑ x
j∈V
jik
(1)
(2)
≤ Q ∀k ∈ K
∑ x −∑ x
j∈V
k∈K i∈I j∈J
= 0 ∀i ∈ V , k ∈ K
(5)
i∈I j∈J
ijk
≤ 1 ∀k ∈ K
(6)
i∈S j∈S
ijk
≤| S | −1 ∀S ⊆ J , k ∈ K
(7)
∑∑ x
∑x
u∈J
iuk
+
∑
u∈V \{ j }
xujk ≤ 1 + fij ∀i ∈ I , j ∈ J , k ∈ K
xijk ∈ {0,1} ∀i, j ∈ V , k ∈ K
(9)
yi ∈ {0,1} ∀i ∈ I
(10)
fij ∈ {0,1} ∀i ∈ I , j ∈ V
Vincent F. Yu (IM, NTUST)
(8)
(11)
Location Models
Oct. 21, 2010
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Location Routing Problem and Its Variants
Simulated Annealing Heuristic for LRP
Solution Representation
Initial Solution
Simulated Annealing Heuristic
Vincent F. Yu (IM, NTUST)
Location Models
Oct. 21, 2010
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Location Routing Problem and Its Variants
Solution Representation
24 0 0 25 10 9 17 2 23 14 15 16 19 0 8 11 6 22 0 0 4 1 12 18 20 13 5 7 3 21
Figure 1. An example of solution representation.
Figure 2. Visual illustration of the example solution given in Figure 1.
Vincent F. Yu (IM, NTUST)
Location Models
Oct. 21, 2010
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Location Routing Problem and Its Variants
Construction of Initial Solution
Step 1. Let U be the set of unused depots. For each depot i ∈ U, let
cc(i) be the number of unassigned customers whose closest
depot in U is depot i. Choose the depot in U with the
highest cc value. If there is a tie, select the depot with the
highest capacity.
Step 2. For all unassigned customers, assign them to the chosen
depot one by one in the increasing order of the distance
between the customer and the chosen depot. Stop when the
capacity of the depot is violated.
Step 3. Construct a TSP route which starts from and ends at the
depot using the Lin and Kernighan’s heuristic (1973).
Step 4. Split the TSP route into several routes so that the route
capacity constraint is not violated.
Step 5. If there are still unassigned customers, go to Step 1;
otherwise, terminate the procedure and encode the current
solution.
Vincent F. Yu (IM, NTUST)
Location Models
Oct. 21, 2010
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Location Routing Problem and Its Variants
SA Procedure
SALRP(T0, TF, α, K, P, Nnon-improving, Iiter)
⎡
Step 1: Let Ndummy = ⎢
⎢
∑Q
i
di ⎤
⎥.
k ⎥
Generate the initial solution X by the greedy heuristic.
Step 2: Let T=T0; I=0; N=0; Fbest=obj(X, P); Xbest=X;
Step 3: I=I+1;
Step 4: (Generate a solution Y based on X)
Step 4.1: Generate r = random (0, 1);
Step 4.2: Case r≤1/3: Generate a new solution Y from X by random swap operation;
Case 1/3<r≤2/3: Generate a new solution Y from X by random insertion operation;
Case 2/3<r≤1: Generate a new solution Y from X by random 2-opt operation;
Step 5: If Δ=obj(Y, P) - obj(X, P) ≤0 {Let X=Y;}
Else {
Generate r = random (0,1);
If r<exp(-Δ/KT) {Let X=Y;}
Vincent F. Yu (IM, NTUST)
Location Models
Oct. 21, 2010
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Location Routing Problem and Its Variants
≤
SA Procedure (cont.)
≤
Δ
≤
Generate r = random (0,1);
If r<exp(-Δ/KT) {Let X=Y;}
}
Step 6: If (obj(X, P)<Fbest and X is feasible){Xbest=X; Fbest=obj(X, P);N=0;}
Step 7: If I=Iiter {
T=αT; I=0;N=N+1;
Perform Local search based on swap operation on Xbest;
Perform Local search based on insertion operation on Xbest;
}
Else {Go to Step 3;}
Step 8: If T<TF or N=Nnon-improving {Terminate the SA heuristic;}
Else {Go to Step 3;}
Vincent F. Yu (IM, NTUST)
Location Models
Oct. 21, 2010
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Location Routing Problem and Its Variants
Computational Study
Benchmark instances
Barreto (2004): 19 instances.
Prins et al. (2004): 30 instances.
Tuzun and Burke (1999): 36 instances.
Parameter setting
Computational Results
Barreto (2004): 18/19 best solutions (best parameters; new: 8; gap:
0.40%); 19/19 best solutions (parameter analysis; new: 9)
Prins et al. (2004): 16/30 best solutions (best parameters; new: 7;
gap: 0.38%); 30/30 best solutions (parameter analysis; new: 18)
Tuzun and Burke (1999): : 7/36 best solutions (best parameters; new:
5; gap: 1.1%); 29/36 best solutions (parameter analysis; new: 25)
Vincent F. Yu (IM, NTUST)
Location Models
Oct. 21, 2010
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Location Routing Problem and Its Variants
Conclusions and Future Directions
Special solution encoding method
Better results
41/85 best solutions using best parameters; 20 of them are new.
78/85 best solutions found during parameter analysis; 52 of them are
new.
Longer computational time
Extensions
Vincent F. Yu (IM, NTUST)
Location Models
Oct. 21, 2010
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Location Routing Problem and Its Variants
LRP Variants
LRP variants with applications in reverse logistics and integrated logistics:
Multiple echelon LRP
LRP with simultaneous pickup and delivery
...
Vincent F. Yu (IM, NTUST)
Location Models
Oct. 21, 2010
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References
Reference
Mark S. Daskin (1995), Network and Discrete Location - Models,
Algorithms, and Applications, John Wiley & Sons.
Zvi Drezner and Horst W. Hamacher (2001), Facility Location Applications and Theory, Springer.
A simulated annealing heuristic for the capacitated location routing
problem, Computers & Industrial Engineering, 58 (2010), pp.
288-299. ( Co-authors: Prof. Shih-Wei Lin (CGU), Prof. Wenyih Lee
(CGU), and Prof. Ching-Jung Ting (YZU))
Vincent F. Yu (IM, NTUST)
Location Models
Oct. 21, 2010
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Q&A
Q&A
Vincent F. Yu
Department of Industrial Management
National Taiwan University of Science and Technology
43, Sec. 4, Keelung Rd., Taipei 106, Taiwan
Tel: 02-2737-6333
E-mail: vincent@mail.ntust.edu.tw
Web: web.ntust.edu/∼vincent
Vincent F. Yu (IM, NTUST)
Location Models
Oct. 21, 2010
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