1. An Adaptive Multiphase Approach for Large Unconditional and Conditional p-median Problems
Said Salhi
(With Drs Chandra Irawan & Maria Paola Scaparra)
Centre for Logistics & Heuristic Optimisation
(University of Kent, Canterbury, UK)
(COSI’2014- Bougie: 8-10 June 2014)

2. Centre for Logistics & Heuristic Optimisation (CLHO)
… we carry out research in various application areas of Combinatorial Optimisation, with a special focus on locational analysis and vehicle routing, using heuristic search techniques
Prof. Saïd Salhi
Dr. Gábor Nagy
Dr. Niaz Wassan
Dr. Paola Scaparra

3. Outline Introduction – Facility Location and Aggregation Method
The methodology - The overview of Adaptive Multiphase Approach (AA)
Computational Study
Existing Datasets
New datasets with guaranteed optimality
Adaptation of AA for conditional p-median problem
Conclusion and Future Research

4. A Logistic overview Location & Warehousing Transportation
Marketing
Location &
Warehousing
Transportation
& Distribution
Suppliers
Production
Sale strategies
Pricing
Promotion
Relationships
Inventory
Negotiation
Purchasing
Relationship
Training

5. Facility Problem

6. p-Median Problem [p=2] Solution Methods :
Exact Methods (n<400 and p<20)
Heuristics
Meta-heuristic

7. Introduction – Facility Location
Facility location is concerned with determining the location of facilities to fulfil objective functions and constrains.
Poor location decisions  serious losses
 Strategic problem
One possible classification of facility location :
Median Problems
Minimizing the average cost/time/distance
Centre Problems
Minimizing the maximum travel cost/time/distance
Covering Problems
Find value of p to cover all customers or maximizing # of covered customer when a critical coverage distance/cost/time is given

8. Possible Classes of Location-Allocation Problems

9. p-Median Problem Formulation
Originally formulated by ReVelle and Swain (1970).
Enhanced by Rosing et al. (1979)
(I,J) : set of customers and set of potential sites
wi : demand or weight of customer i;
di j : distance between customer i and potential site j;
p : number of facilities to locate;
Yi j = 1, if customer i is fully served by a facility at site j and = 0 otherwise;
X j = 1, if a facility is opened at potential site j and = 0 otherwise;
Fi : set of all sites except the p−1 furthest sites from demand point i.

10. p-Centre Problem Formulation (for infos only)*
(I,J) : set of customers and set of potential sites
r : the maximum distance between a customer and its closest facility;
d(i,j) : distance between customer i and potential site j;
p : number of facilities to locate;
Yi j = 1, if customer i is fully served by a facility at site j and = 0 otherwise;
X j = 1, if a facility is opened at potential site j and = 0 otherwise;

11. The Aggregation Problem in Location-Allocation Modelling Fotheringham et al. (1995)
Solution
Aggregated Zones and Centroids
Aggregated Spatial Units (ASU)
Location problems may consist of a large number of demand points.
It may be impossible and time consuming to solve optimally.
It is quite common to aggregate demand points.
The location problems are partitioned into smaller problems and hence they can be solved within a reasonable amount of computing time.
Individual Location /
Basic Spatial Units (BSU)

12. The effects of Aggregation
This aggregation may reduce the accuracy of the model.
It introduces error in the data used by location models and models output.
Types of errors:
ABC error
Cost error
Optimality error
Etc

13. ABC Error (Hillsman and Rhoda, 1978)
Potential facility site k j
dkj
Basic Spatial Unit (BSU)
i+1
Aggregated Spatial Unit (ASU)
Original Distance
i+2
Approximated Distance
Source A Error i
i+4 i k
k+1
i+2
i+1 d2 k d1
j+1
i+3
i+5
i+1 j j
i+2
Source C Error
Source B Error

14. Cost and Optimality Errors (Casillas, 1987)
Notation
C = (c1, c2,...,cn) : the list of BSUs
C’ = (c’1, c’2,...,c’m) : the list of ASUs
F : the optimal locations found with the original location model
F’ : the optimal locations found with the aggregated model
D(F,c) : the distance between a BSU and the closest element in F
f(F,C) : objective function evaluated using F and C
The cost error (ce)
ce can be defined by ce = f(F’,C) – f(F’,C’)
The difficulty to solve the original location problem is the main idea behind this error.
The optimality error (oe)
oe can be defined by oe = f(F’,C) – f(F,C)
The original location problem has to be solved first before the optimality error can be computed

15. Contribution of this research
A novel multiphase approach that incorporates aggregation, Variable Neighbourhood Search (VNS) and an exact method for solving large p- median problems,
New best solutions for some benchmark problems,
The construction of a new large dataset for p- median problems with guaranteed optimality,
An adaptation of the proposed approach for the conditional p-median problem.

16. The overview of Adaptive Multiphase Approach (AA)
The method consists of four phases.
The method solves very large p-median problems with n customers.
Customer sites are potential facility sites.
Solving (n, n, p) p-median problems

17. The illustration of Adaptive Approach (AA)

18. The main steps of AA – Phase 1

19. The main steps of AA – Phase 2

20. The main steps of AA – Phases 3 & 4

21. Distance Calculation i
i+1
i+2 k j
dkj

22. Aggregation Method
n = 80 points
m = 8 points

23. The “Mini VNS” (Local search with shaking)
One shaking and one call to the local search only
Our local search
Enhancement of the fast interchange heuristic (Whitaker, 1983) with the use of an efficient data structure (Resende and Werneck 2007) .
swap a chosen facility with the demand point served by the chosen facility
when calculating the saving, we just include the demand points served by the neighbouring facilities
The shaking
Using the kth neighbourhood (Nk)
k  (1, kmax) and kmax = p

24. Written in C++ .Net 2010 and using CPLEX 12.5.
Computational Study
Dataset
BIRCH instances  Avella et al. (2012)
TSP instances
Newly generated data sets : Here the optimal solution is guaranteed geometrically.
Written in C++ .Net 2010 and using CPLEX
Parameters: m = 0.1n and T = 10.
The seed for the random generator = m

25. Experiment on BIRCH instances
n = 25,000 to 89,600 and p = 25 to 64
The results of our experiments are compared with the ones obtained by:
Irawan and Salhi (2013)  IS
Avella et al. (2012)  AV
Hansen et al. (2009)  VNSH

27. Experiment on TSP instances
n = 16,862 to 71,009 and p = 25 to 100 with an increment of 25, totalling 16 instances.
compared with the ones obtained by IS (2013)
In general, AA produces better results than IS and produces 13 new best solutions.
Average CPU time, the AA is faster than IS

28. Geometric Datasets
N = 500, p =4
N = 20000, p = 100

29. Experiment on Geometric instances
n = 20,000 to 60,000 with increment of 10,000
p = 0.5%n and p = 1%n
HA produces the optimal solutions for all instances
CPU time  when p  rather than n 
It seems that the circle dataset can be solved quite easily by HA.

30. HMH method on Geometric instances

31. Conditional p-median problem
The p-median problem becomes the conditional problem when some (say q) facilities already exist in the study area
The aim is to locate p new facilities given the existing q facilities.
This problem is also known as the (p, q) median problem.
A customer can be served by one of the existing or the new open facilities whichever is the closest to the customer.

32. The adaptation of AA for (p,q) median problems
The aggregation method
The q existing facility locations are always included in the promising facility locations
The “Local Search with Shaking”
The shaking
When finding the best facility to be removed (say facility j), facility j is not one of the existing facilities.
The local search
The implementation of the best improvement strategy does not include the existing facilities
The exact method (same formulation + the following constraints with Q being the set of existing facilities)

33. (p, q)-Median problem (AAq)
Computational Study for (p,q) median problems
The adapted method was assessed on the TSP dataset.
The results also reveal that our method performs quite well when compared against the results of the unconditional p- median problem which are used as lower bounds.
Description
p-Median problem (AA)
(p, q)-Median problem (AAq) p Z
Time (seconds) q
Deviation (%)
China Data (n=71,009) 50
78,526,961.86
3,404.75 25
82,437,517.47
4.98
911.16 75
63,786,587.67
3,339.16
66,448,542.12
4.17
2,068.54
66,367,405.04
4.05
792.73
100
54,865,606.72
2,958.15
57,296,333.65
4.43
1,353.44
56,649,744.56
3.25
705.66

34. Conclusion Future Research Our approach performs well and runs fast.
Three types of datasets were used to test our method.
Future Research
This study could be extended to investigate other related location problems such as the large p-center problem.
Also can be adapted for clustering of large datasets with higher dimension as part of data mining.

35. THANK YOU