O’Kelly (1987) presented the first mathematical
explanation for the hub location problem in one of the
most important primary papers in the subject of HLPs. He
suggested a framework for quadratic integer
programming to solve the problem of reducing total
transportation costs for a set of hubs (p-hub median
problem) [1]. Campbell (1994) introduced p-hub center
and hub covering problems to the literature [2]. Then, first
formulation of multiple allocation is proposed by
Campbell (1992) for multiple p-median HLPs [3]. In
complete structures, every hub pair has a direct hub link
and these hub networks are usually considered in most of
the hub location studies. However, an efficiently
constructed incomplete hub network can occasionally
deliver almost the same service quality as a complete hub
network in terms of cost and/or service time. O’Kelly and
Miller (1994) introduced different hub networks, one being
the incomplete hub network design [4]. Calik et al. (2009)
presented a tabu-search based heuristic for the single
allocation hub covering problems over incomplete hub
networks [5]. Alumur et al. (2009) presented the first
modeling of an incomplete network [6].
HLP structure with 19 nodes and 4 hubs. The formulation presented by Alumur (2009) is used as
inspiration for the model [7]. The current network is as in Figure 4.
Linear mathematical formulation for Turkey Railway Freight Transportation network hub location
problem based on above descriptions. The parameters and variables are shown. The goal in this
paper's objective function is to minimize total cost. Total transportation costs are equal to the total
flow and total distance, which has a linear relationship between the two nodes. The objective function
and constraints are as follows.
Global Minimum
Hub Swap
Knowledge, Will, Courage
and Quickness!
Gap values are approximate to 0.
These values prove that good results are obtained according to the mixed
integer linear mathematical model.
Models with incomplete structures that GAMS cannot solve can be solved in the SA
algorithm for large data sets.
The incomplete model is analyzed in detail on GAMS, with α = 0.2, 0.6, 0.8, p =2,3,4,5 and q
values between 1 and 10 on the CAB25 data set.
The results of Transportation Cost and CPU Time obtained with the incomplete CAB25
data are compared with the results in the SA algorithm. When the results are examined,
the following points are obtained:
Figure 4 - HLP network structure with 19 nodes and 4 hubs
Network Planning Location Model For Turkey Railway Freight Transportation
Burak Alper Şahin, Işınsu Yıldız, Selen Dilara Kırklar, Abdullah Yıldızbaşı (Supervisor), Cihat Öztürk (Advisor)
Initialize parameters;
S = generate initial solution ( );
T = Tinitial ;
While ( T < Tfinal )
Until ( N ≤ I-Iter )
Generate Solution S' in the
neighborhood of S
if f(S'*) < f(S)
S <--- S'
else
Δ = f(S') - f(S)
r = random( );
if(r<exp(-Δ/k*T))
S <--- S'
T = T - 1
Return the best solution found;
Motivation
Department of Industrial Engineering
References
Conclusion
Results
Model
Hub Location Problems
Hub Location Problems (HLPs) are at the center of transportation and telecommunications network design planning. These
problems deal with determining the location of hub facilities and the allocation of the nodes to these located hub facilities.
When the authors' studies are examined in the literature, it is not seen any study about p-hub median incomplete HLP
structure with a developed meta-heuristic algorithm. For this purpose, single allocation incomplete p-hub median hub location
problem without hub capacity is proposed with Simulated Annealing algorithm for Turkey Railway Freight Transportation. The
model is evaluated with GAMS using the CAB, TR, and AP data sets to see the performance of the model. Complete and
incomplete HLP network designs can be shown in Figure 1 and Figure 2.
O’Kelly (1987) presented the first mathematical explanation for the hub location problem in one of the most
important primary papers in the subject of HLPs. He suggested a framework for quadratic integer programming to
solve the problem of reducing total transportation costs for a set of hubs (p-hub median problem) [1]. Campbell
(1994) introduced p-hub center and hub covering problems to the literature [2]. Then, first formulation of multiple
allocation is proposed by Campbell (1992) for multiple p-median HLPs [3]. In complete structures, every hub pair
has a direct hub link and these hub networks are usually considered in most of the hub location studies. However,
an efficiently constructed incomplete hub network can occasionally deliver almost the same service quality as a
complete hub network in terms of cost and/or service time. O’Kelly and Miller (1994) introduced different hub
networks, one being the incomplete hub network design [4]. Calik et al. (2009) presented a tabu-search based
heuristic for the single allocation hub covering problems over incomplete hub networks [5]. Alumur et al. (2009)
presented the first modeling of an incomplete network [6].
Model is tested on three data sets, CAB, AP and TR which are previously introduced in the literature. In our study, up to 25 data sets are studied in the incomplete structure
and up to 30 data sets in the complete structure. Incomplete and complete models are compared with these small datasets.
The graphs above show how CPU Time in incomplete and complete
structure changes according to different α (discount factor), p and q values on AP, CAB and TR data sets.
The network structure for the CAB25 data set is shown on the maps. Selected
hubs, nodes assigned to hubs, node and hub connections can be seen on these
maps. These two maps show the position change of the hubs according to α
change from 0.2 to 0.6. Maps are shown in Figure 5 and Figure 6.
Comparison graphs of CPU Time for SA algorithm and GAMS are given in Figure 7 and
Figure 8. It has been observed that with the increase of the α value, the CPU Time also
increases in GAMS and SA. Also, resolution time of SA is shorter than GAMS. In Figure 8,
where the alpha and p values are constant and the q values are variable, it is observed
that as q increases, the solution time of GAMS shortens and SA increases. Finally, it
caused an increase in GAMS and SA Algorithm based on solution time when p values are
variable and other parameters are constant. The reason for the large time difference in
GAMS is due to the increase in the number of links to be tried.
It has been observed that there is no study about p-hub
median incomplete HLP structure with a developed
meta-heuristic algorithm. For this reason, the
performance of the algorithm is tested with the study in
the single allocation complete p-hub median structure
made by Kratica et al. (2007) [8]. The comparison is made
on the CAB and AP data sets together with the p, q and α
parameters. As a result of these comparisons, it is seen
that the results of SA algorithm are very close to the
Genetic Algorithm, even though it is worked in an
incomplete structure.
Obtained results for CAB and AP datasets are shown in
the Table 1.
[1] O'Kelly, M. E. (1987). “A quadratic integer programming for the location of interacting hub facilities”, European Journal of Operation Research, 393-404.
[2] Campbell, J. F. (1994). “A survey of network hub location”, Studies in Locational Analysis 6, 31-49.
[3] Campbell, J. F. (1992). “Location and allocation for distribution systems with transshipments and transportation economies of scale”, Annals of Operation Research, 77-99.
[4] O’Kelly, M.E., Miller, H.J. (1994). “The hub network design problem: a review and synthesis”, Journal of Transport Geography, 31–40.
[5] Calik, H., Alumur, S.A., Kara, B.Y., Karasan, O.E. (2009). “A tabu-search based heuristic for the hub covering problem over incomplete hub networks”, Computers and Operations Research, 3088-3096.
[6] Çağrı Özgün Kibiroğlu, Y. İ. (2019). Uncapacitated multiple allocation hub location problem under congestion.
[7] Sibel A. Alumur, B. Y. (2009). “The design of single allocation incomplete hub networks” , Transportation Research Part B: Methodological, 936-951
[8] Jozef Kratica, D. T. (2007). Two genetic algorithms for solving the uncapacitated single allocation p-hub median problem. European Journal of Operational Research, 15-28.
In general, there are different kinds of algorithms used for the complete hub location network structures. The
usage percentages of heuristic/metaheuristic and exact solution procedures used in the literature are given in the
Figure 3.
Single allocation incomplete p-hub median problem without hub capacity is presented as a mixed-integer linear formulation for Turkey Railway Freight Transportation.
Model is tested on GAMS using the CAB, TR, and AP data sets to see the performance of the linear mathematical model.
Simulated annealing is used to solve the developed model. This algorithm is compared with a study that uses Genetic Algorithm with a complete structure.
Results of SA and GAMS are compared on CAB25 dataset.
First study about p-hub median incomplete HLP structure with a developed meta-heuristic algorithm is presented to the literature.
Objective Function f(x)
variable x
Starting
Simulated Annealing Algorithm
Figure 3- Usage percentages of heuristic/metaheuristic and exact solution procedures
Node Swap
Hub Connection Swap
Parameters;
Initial Temperature (Ti)
Local Search Iteration (Nk)
Final Temperature (Tf)
Temperature Reduction Function (Tnew = Tcurrent-1)
3. Comparison of Simulated Annealing and Genetic Algorithms
1. CAB AP TR Dataset Results on GAMS
2. GAMS and Simulated Annealing Comparison
Proposed HLP Formulation and Network Structure
The graphs above show how transportation cost in incomplete and complete
structure changes according to different α (discount factor), p and q values on AP, CAB and TR data sets.
Figure 1 - Complete HLP network structure Figure 2 - Incomplete HLP network structure
Figure 7- CPU Time comparison on different α values
Figure 8- CPU Time comparison on different q values
α
Table 1
Performance comparison of Genetic and SA algorithms on CAB dataset
ααα
αααα
CPU Time CPU Time CPU Time
Cost Cost Cost Cost
CPU Time
α
CPU Time
q
Figure 5 - Network design structure with α = 0.2
Figure 6 - Network design structure with α = 0.6
