OPSEARCH

, Volume 50, Issue 1, pp 60–74

# An effective heuristic for the P-median problem with application to ambulance location

## Authors

• Department of MathematicsThe University of Newcastle
• Janet Dzator
• School of BusinessThe University of Newcastle
Application Article

DOI: 10.1007/s12597-012-0098-x

Dzator, M. & Dzator, J. OPSEARCH (2013) 50: 60. doi:10.1007/s12597-012-0098-x

## Abstract

We consider the p-median problem which is to find the location of p-facilities so as to minimize the average weighted distance or time between demand points and service centers. Many heuristic algorithms have been proposed for this problem. In this paper we present a simple new heuristic which is effective for moderately size problem. The heuristic uses a reduction and an exchange procedure. Our methodology is tested on 400 randomly generated problems with 10 to 50 customer locations as well as 6 well known literature test problems. We also compare our method with the Branch and Bound method in terms of quality and computational time using a larger problem size of 150 customer locations. For the random problems the generated solutions were on average within 0.61 % of the optimum. A similar result was achieved for the literature test problems. A comparative analysis with literature heuristics supports the superiority of our method. The computational time of our heuristic is 0.75 % of the Branch and Bound Method. We also apply our heuristic to a case study involving the location of emergency vehicles (ambulances) in Perth City (Australia).

### Keywords

HeuristicsFacilitiesLocationP-median problem

## 1 Introduction

The fundamental objectives of locating facilities can be summarized into three categories. The first category refers to those designed to cover demand within a specified time or distance. This objective gives rise to location problems which are known as the Location Set Covering Problem (LSCP) and the Maximal Covering Location Problem (MCLP). The LSCP seeks to locate the minimum number of facilities required to ‘cover’ all demand or population in an area. The MCLP is to locate a predetermined number of facilities to maximize the demand or population that is covered. The second category refers to those designed to minimize maximum distance. This results in a location problem known as the p-center problem which addresses the difficulty of minimizing the maximum distance that a demand or population is from its closet facility given that p facilities are to be located. The third category refers to those designed to minimize the average weighted distance or time. This objective leads to a location problem known as the p-median problem. The p-median problem finds the location of p facilities to minimize the demand weighted average or total distance between demand or population and their closest facility.

A criterion for finding a good location for emergency facilities is the improvement of response times to emergency calls. The response time for these emergency facilities depend primarily on the distance between the emergency facilities and the emergency sites. An important aim is to locate these facilities such that the average (total) distance traveled by those who visit or use these facilities is minimized. This measures the effectiveness and efficiency of the emergency facilities. Thus, the utility derived from using these facilities increases as the distance between them decreases. In other words, as travel distances decrease, facility accessibility increases and the effectiveness of the located facilities increases giving rise to a decrease in response time.

In facility location problem an important application is the location of emergency facilities in city (Savas [1], Fitzsimmons [2], Swoveland et al. [3], Gendreau et al. [4], Repede and Bernardo [5], McAleer and Naqvi [6], Goldberg et al. [7], Fujiwara et al. [8]) A number of authors have used the p-median model to locate emergency facilities. These include Calvo and Marks [9]; Berlin et al. [10]; Mirchandani [11]; Carson and Batta [12]; Serra and Marianov [13]; Paluzzi [14] Caccetta and Dzator [15, 16] and Dzator [17]. Our focus in this paper is on the p-median problem. We focus on the p-median problem because unlike the covering problem, the coverage distance for the p-median problem is unrestricted and the number of facilities to be located is known. These properties of the p-median problem measure the effectiveness of the facility location by evaluating the average distance between the customers and facilities. Moreover, as reported by Uyeno and Seeberg [18] in their study, covering procedures tend to favor less congested rural areas over urban areas. In addition, the p-median problem can be formulated so as to minimize the average response time.

In this paper we focus on the simple heuristics which are based on Vertex Substitution (Exchange). We compare our method on a test set of 6 literature problems as well as a set of 400 randomly generated problems with n ranging from 10 to 50 with Myopic heuristic (Greedy), the neighborhood search heuristic of Maranzana [19] and the Exchange heuristic of Teitz and Bart [20]. We also compare our method with the Branch and Bound (BB) using the problem size of 150 for 20 different problems. The paper is organized as follows. In Section 2, the p-median problem is discussed. Section 3 briefly describes the literature heuristics that we use in our comparative analysis. Section 4 presents our new reduction based heuristic. Computational results are presented and discussed in Section 5. We apply our methodology to a case study concerning the location of ambulances in a city in Section 6.

## 2 The P-median problem

The objective of the p-median problem is to find the locations of p facilities to minimize the demand weighted total distance (total cost) between each demand node and the nearest facility. For the p-median problem the cost of serving demands at node i is the product of the demand at node i and the distance between demand node i and the nearest facility to node i.
I

the set of demand nodes indexed by i

J

the set of candidate facility locations, indexed by j

p

the number of servers to be deployed or facilities to be located

ai

the population at the demand node i

dij

distance between demand node iI and candidate sites jJ

$$\begin{array}{*{20}{c}} {{{Y}_{{ij}}} = \left\{ {\begin{array}{*{20}{c}} {1,} \hfill & {{\text{if}}\:{\text{demands}}\:{\text{at}}\:{\text{node}}\:i \in I\:{\text{are}}\:{\text{assigned}}\:{\text{to}}\:{\text{a}}\:{\text{facility}}\:{\text{at}}\:{\text{candidate}}\:{\text{site}}\:j \in J} \hfill \\ {0,} \hfill & {{\text{otherwise}}} \hfill \\ \end{array} } \right.} \\ {{{X}_{j}} = \left\{ {\begin{array}{*{20}{c}} {1,} \hfill & {{\text{if}}\:{\text{we}}\:{\text{locate}}\:{\text{at}}\:{\text{candidate}}\:{\text{site}}\:j \in J} \hfill \\ {0,} \hfill & {{\text{otherwise}}} \hfill \\ \end{array} } \right.} \\ \end{array}$$
Minimize
$$\sum {_i\sum {_j{a_i}{d_{ij}}{Y_{ij}}} }$$
(1)
subject to
$$\matrix{{*{20}{c}} {\sum {_{j \in J}{Y_{ij}} = 1,} } \hfill & {\forall i \in I} \hfill \\ }$$
(2)
$$\sum {_{j \in J}{X_j} = p}$$
(3)
$$\matrix{{*{20}{c}} {{Y_{ij}} \leqslant {X_j},} \hfill & {\forall i \in I,j \in J} \hfill \\ }$$
(4)
$$\matrix{{*{20}{c}} {{Y_{ij}} \in \left\{ {0,1} \right\},} \hfill & {\forall i \in I,j \in J} \hfill \\ }$$
(5)
$$\matrix{{*{20}{c}} {{X_j} \in \left\{ {0,1} \right\},} \hfill & {\forall j \in J} \hfill \\ }$$
(6)

The objective (1) is to minimize the total distance from customers or clients to their nearest facility. Constraint (2) shows that the demand of each customer or client must be met. From constraint (3), the number of facilities to be located is p. Constraint (4) shows that customers must be supplied from open facility. Constraints (5) and (6) present the problem as a binary integer programming. The above formulation assumes that the potential facility sites are nodes on the network. Hakimi [21] showed that allowing facilities to be located on the arcs of the network instead of the nodes would not reduce total travel cost.

The p-median problem is computationally difficult to solve by exact methods because the problem is NP-hard on general networks as shown by Kariv and Hakimi [22]. Exact algorithms based on the methods of Branch and Bound (Daskin [23], Eaton et al. [24]) and Branch and Cut (Toregas et al. [25], Plane and Hendrick [26]) have been successfully used for small problems. However, solutions from the p-median model are considered efficient since they bring the facility locations into closer proximity of the users. The difficulty of solving the p-median problem by exact method has led researchers to consider sub optimal solutions generated by heuristic approaches. Heuristics for solving the p-median problem have been discussed in Daskin [23], Maranzana [19], Teitz and Bart [20] and Densham and Rushton [27], Ashayeri et al., [28], Simulated Annealing (Chiyoshi and Galvao; [29], Ringhini; [30]), Genetic Algorithm (Alp et al.; [31], Bozkaya et al.; [32], Chiou and Lan; [33], Dvorett; [34]) and Tabu Search (Salhi; [35], Rolland et al.; [36] and Voss; [37]).

## 3 The primary P-median heuristics

The p-median problem, as stated earlier, is a difficult problem to solve exactly so there are many heuristics which have been developed to solve it. The Myopic, Neighborhood and Exchange heuristics are examples of primary heuristics, applied to solve the p-median problem. These heuristics use all the values of the weighted distance matrix without deleting any extreme values before locating facilities. In considering column totals we may not select a location because of a very large cost entry. Since in practice one would never allocate a customer (demand) to very distant facility, it would make sense to eliminate large entries before taking column sums. Our new heuristic will implement this idea.
1. (i)

Myopic Algorithm (MA) for the P-Median Problem

The myopic heuristic is a greedy type which works in the following way. Firstly, a facility is located in such a way as to minimize the total cost for all customers. Facilities are then added one by one until p is reached. For this heuristic, the location that gives the minimum cost is selected. The main problem with this approach is that once a facility is selected it stays in all subsequent solutions. Consequently, the final solution attained may be far from optimal. Using other heuristics such as neighborhood search heuristic and other improvement heuristics (which are discussed later) will improve the solution obtained from the myopic algorithm. This heuristic is specifically known as Greedy-Add since facilities are added one-by-one to attain the required number of facilities. The reverse approach is known as Greedy-Drop which starts with facilities located at all potential facility sites and then eliminate (drop) the facility that has the least impact on the objective function. This method eliminates the facilities one-by-one until the required number of facilities p remains.

The outline of the Myopic algorithm as indicated by Daskin [23] is presented below as follows:-
1. Step 1:

Initialize k = 0 and Xk = {}, the empty set.

2. Step 2:

Increase k, the counter on the number of facilities located.

3. Step 3:

Compute $$Z_j^k = \sum {_i{h_i}d\left( {i,j \cup {X_{k - 1}}} \right)}$$ for each node j, which is not in the set Xk−1, where hi is the demand at node i.

4. Step 4:

Find the node j*(k) that minimizes $$Z_j^k$$. Add node j*(k) to the set Xk−1 to obtain the set Xk.

5. Step 5:

If k = P stop. Go to step 2 if k < P.

2. (ii)

Neighborhood Search Heuristic (NS)

Maranzana [19] proposed the Neighborhood search heuristic which is described as follows. We begin with any set of p facility nodes. The demand nodes are then divided into p subsets and for each subset, a demand node is allocated to the nearest facility node. The set of nodes assigned to a facility constitutes a “neighborhood” around that facility. Then within each neighborhood, the 1-median problem can be solved optimally by simply evaluating each potential site in the neighborhood and the best set of facilities is selected. The chosen facilities are then relocated to the optimal 1-median locations within each neighborhood. The node giving the optimal for each subset is found for each subset resulting in a new pattern of facility nodes. If any facility sites are relocated, new neighborhoods can be defined and the heuristic is repeated. This process is repeated until the facility nodes pattern remains the same as that in the previous step. That is the process is continued until there is no change in the facility sites or the neighborhoods.

The outline of this heuristic is presented as follows:
1. Step 1:

Select arbitrarily m distinct points $${p_{{x_1}}},{p_{{x_2}}}, \ldots, {p_{{x_m}}}$$ from set of m points P.

2. Step 2:

Determine a corresponding partition of P, $${P_{{x_1}}}, \ldots, {P_{{x_m}}},$$ which is associated with the array of m points, $${p_{{x_1}}}, \ldots, {p_{{x_m}}}$$ by putting $${P_{{x_i}}} = \left\{ {{p_k};{D_{k,{x_i}}} \leqslant {D_{k,{x_j}}}\quad {\text{for}}\;{\text{all}}\;j} \right\}$$, where $${D_{k,{x_i}}}\;{\text{and}}\;{D_{k,{x_j}}}$$ are the minimal path lengths from points pk to $${p_{{x_i}}}$$ and pk to $${p_{{x_j}}}$$, respectively.

3. Step 3:

Determine a center of gravity, $${c_{{x_i}}}$$ for each $${P_{{x_i}}}$$. (The center of gravity of the partitions QP for the point pj is defined as: $$\sum {_{{P_k} \in Q}{D_{j,k}}{w_k} \leqslant \sum {_{{P_k} \in Q}{D_{i,k}}{w_k}} }$$ for all i, where wk is the weight associated with the point pk.)

4. Step 4:

If $${c_{{x_i}}} = {p_{{x_i}}}$$ for all i, computation is stopped and the current values of $${p_{{x_i}}}\;{\text{and}}\;{P_{{x_i}}}$$ constitute the desired solution. Otherwise, set $${p_{{x_i}}} = {c_{{x_i}}}$$ and return to Step 2.

3. (iii)

Teitz and Bart [20]-Exchange Heuristic (EH)

This is one of the early exchange heuristics developed by Teitz and Bart [20] for the p-median problem. The basic idea is to move a facility from the location it occupies in the current solution to an unused site. The heuristic starts by choosing an initial set of p number of nodes as the solution. Then a node which is not in the current solution is selected to substitute for each of the p nodes in turn. We find the objective value in each case and compare the changes in the objective function. The substitution leading to the biggest decrease in the objective function is selected and is exchanged for a node in the current solution. This exchange of nodes results in a new (improved) solution configuration and this process continues until there is no further improvement in the objective value. The solution thus obtained is a local optimum, not a global optimum.

The outline of the heuristic is presented as follows:
1. Step 1:

Select an initial set of any p potential facility sites among the n nodes and call this the current best facility set.

2. Step 2:

Let the candidate facility sites that are not in use in the current best facility sets be denoted by μ. If the set μ is empty then go to step 6. Otherwise, go to step 3.

3. Step 3:

Select a candidate facility site vμ, then remove facility site v from μ and go to step 4.

4. Step 4:

Calculate Δj for j = 1, 2, …p which denotes a change in the objective function with moving the jth facility site to candidate node v.

5. Step 5:

Define Δmin to be the minimum change for the objective value if any node in facility set is removed and replace by another one which is not in that in the set. If Δmin ≥ 0, go to step 6. Otherwise, create a new current facility set by replacing the location associated with the minimum Δj with facility site v and go to step 6.

6. Step 6:

If set μ is empty but currently updated which results in changes in facility sites in the current best solution since μ was last defined, then go to step 2. If set μ is empty and there is no change in the current best solution since μ was defined, stop, current solution is locally optimal. If μ is not the empty set, then go to step 3.

## 4 The new reduction heuristic (RH)

The primary heuristics described in the previous section use all values in the weighted distance matrix. Thus a location may not be selected because of a very large cost. In this section we will eliminate extreme values as would happen in real applications. To obtain the initial solution set for the heuristic we first of all eliminate the extreme values in each column. The number of extreme values to be eliminated depends on the size of the problem being considered. That is, the larger the size, the greater the number of extreme values to be eliminated. We define:
$$\alpha = \left\{ {\matrix{{*{20}{c}} {p,} \hfill & {if\;n \leqslant 29,} \hfill \\ {2p,} \hfill & {if\;30 \leqslant n \leqslant 39,} \hfill \\ {\left( {\left[ {\frac{n}{{10}}} \right] - 1} \right)p,} \hfill & {{\text{otherwise,}}} \hfill \\ } } \right.$$
(7)
where p is the number of facilities and n is the number of nodes.

### 4.1 Repeated reduction heuristic (RRH)

Input: n, p and α
1. Step 1:

Delete the highest α number of values from each column and let the resulting number of demand nodes be equal to n* (i.e. n* = nα).

2. Step 2:

Sum the values for each column after deleting the extreme values, then arrange the total values in ascending order of magnitude, and choose the first p nodes corresponding to the first p totals as the initial set.

3. Step 3:

Use the original weighted distance matrix and set the distance values (for both rows and columns) corresponding to the initial set of facilities to zero and sum the columns of the resulting distance matrix. For example, if the initial set is {1,2,3} then all values in rows and columns 1, 2 and 3 are changed to zero before the summation of each column.

4. Step 4:

Swap all the nodes which are not in the initial solution set with the nodes in the initial solution set. For example we select the nodes with the lowest number from the non-initial set and substitute for every node in the initial. We continue the process with the next lowest node number until all the nodes not in the initial set are used for swapping with the nodes of the initial set. This will lead to a number of possible solution set for Step 5.

5. Step 5:

Choose the set corresponding to the minimum value as the current solution.

6. Step 6:

With the current solution as the initial solution we return to Step 3. We continue this process until the objective value of the previous solution is the same as the current solution. We then consider the result as the final solution.

## 5 Computational results

We have implemented our new heuristic in C++ and tested on sets of 400 randomly generated sets of data for a [10, 100] matrix with n ranging from 10 to 50 in steps of ten and p ranging from 2 to 5 and 20 different problems of size 150. That is, for each problem size n and for locating 2, 3, 4 or 5 facilities, 80 uniformly distributed random problems are generated. We obtained the Branch and Bound (BB) method (BB) from the SITATION software. In addition, we apply our heuristic to 6 literature problems: the 55-node (Swain; [38]), a 42-node (Dantzig et al.; [39]), a 33-node (Karg and Thompson; [40]), a 30-node (Toregas et al.; [25]) a 12-node (Daskin; [23]) and a 9-node (Hribar and Daskin; [41]) problem. We compare the results from the heuristics with the optimal values obtained by complete enumeration. This will give an indication of whether the new heuristics can provide a good alternative to the exact solution techniques which are in many cases complex and expensive to apply. All computations were carried on a personal computer with an Intel Pentium 4 processor, 2.8GHZ and 448MB of RAM. The statistic used to measure the quality of the solution is given as $$\frac{{H - O}}{O} \times 100$$, where H is the optimal value resulting from the implementation of the heuristic and O is the true optimal value.

The 55-node data set represents 55 communities in the Washington D.C. (USA) area. Demands for each node were generated in pseudo-random manner with most large demands at the center of the region and most small demands at the outer region. The 42-node and 33-node problem represent 42 cities (Dantzig et al.; [39]) and 33 cities (Karg and Thompson; [40]) in USA respectively. The 30-node problem represents 30 communities in New York (Toregas et al.; [25]). We assume uniform demand for all of these problems. These test problems have been used by several authors including Khumawala [42]; Hillman and Rushton [43]; Church and Meadow [44]; Neebe [45] and Rahman and Smith [46]. The 12-node and 9-node problems represent network data from (Daskin; [23]) and a network data from (Hribar and Daskin; [41]), respectively.

The results of the literature heuristics were obtained from the SITATION software (Daskin, [23]). The solutions of the heuristics were compared with the optimal solutions, which were determined by the implementation of the Lagrangian relaxation in the SITATION software [23].

The results for the randomly generated text problem are given in Tables 1 and 2 whilst the results for literature test problems are given in Table 3 and Fig. 1. Note that for the literature problems we report the average ranging p from 1 to 10.
Table 1

Comparison performance of RRH and the existing heuristics using the 400 random data

Number of nodes (n)

Number of facilities (p)

MA

EH

NS

RRH

10

2

1.47

0.47

1.33

0.08

3

3.22

0.30

3.01

0.26

4

4.20

1.34

4.11

0.59

5

1.74

0

1.74

0.35

20

2

0.63

0.16

0.63

0

3

2.15

1.32

2.15

0.49

4

4.31

1.37

4.31

1.14

5

4.77

1.36

4.77

1.28

30

2

0.77

0.01

0.77

0

3

2.63

1.10

2.63

0.40

4

3.53

1.07

3.26

1.06

5

4.12

0.81

4.12

1.21

40

2

0.92

0.32

0.92

0.28

3

1.68

1.43

1.68

0.44

4

2.52

1.83

2.52

1.41

5

2.55

1.36

2.55

1.38

50

2

0.63

0.37

0.63

0.08

3

1.86

0.47

1.86

0.47

4

1.65

0.80

1.65

0.50

5

2.98

0.82

2.98

0.91

Average values for the heuristics

2.41

0.83

2.37

0.61

Table 2

Objective value and CPU time in seconds for different random data for n = 150 and p = 5 for 20 problems

Data

Objective value RRH

Objective value BB

$$\frac{{RRH - BB}}{{BB}} \times 100$$

CPU time (sec) RRH

CPU time (sec) BB

1

2846

2789

2.04

13.17

1,311.26

2

2815

2764

1.84

24.23

2,200.00

3

2908

2831

2.71

17.87

5,523.15

4

2763

2763

0

32.20

3,000.26

5

2731

2731

0

15.40

1,532.38

6

2864

2831

1.16

27.97

2,596.23

7

2799

2761

1.37

21.85

2,809.53

8

2898

2785

4.05

22.87

1,845.94

9

2720

2720

0

12.94

423.79

10

2874

2831

1.51

15.83

5,421.32

11

2776

2771

0.18

18.29

1,732.32

12

2839

2832

0.24

19.40

3,264.39

13

2824

2805

0.67

16.20

3,650.01

14

2836

2814

0.78

16.13

3,304.75

15

2837

2769

2.45

28.62

2,769.26

16

2727

2727

0

15.43

704.74

17

2769

2769

0

32.62

2,205.37

18

2809

2809

0

35.13

5,168.86

19

2842

2776

2.37

11.61

1,928.00

20

2850

2764

3.11

21.34

3,767.17

Average

1.22

21.05a

2,757.94

aThe computational time of our heuristic is 0.75 % of the Branch and Bound method

Table 3

Summary of comparison using the literature data

Literature data

Average value (%)

Myopic

Exchange

Neighborhood

RRH

55-node (Swain; 1971)

4.07

0.42

1.34

0.47

42-node (Dantzig et al.; 1964)

5.20

0.94

3.65

0.90

33-node (Karg and Thompson; 1964)

6.28

0.98

1.74

0.64

30-node (Toregas et al.; 1971)

1.46

0.04

1.17

0

13.46

2.38

6.70

0.24

Average values for 5 data sets

6.09

0.95

2.95

0.45

The average values for 10, 20 30, 40 and 50 nodes for RRH, EH, NS and MA in Table 1 are 0.61 %, 0.83 %, 2.37 % and 2.41 % respectively. Table 2 shows the comparison of RRH with the Branch and Bound (BB) method. RRH is within 1.22 % of BB but RRH is 0.75 % of BB computational time. The average values for each heuristic for the five literature data sets in ascending order in Table 3 are as: RRH-0.45 %, Exchange-0.95 %, Neighborhood-2.92 %, Myopic-6.09 %. RRH performs better as shown in Tables 1 and 3 which is confirmed by Fig. 1.

## 6 A case study: emergency facility location in Perth

In this section, we apply the new heuristic RRH discussed in Section 4 to locate ambulance stations in two sub-regions of the Perth Metropolitan area, namely the South East Metropolitan Region and the Central Metropolitan Region. For these two regions we compare the performance of the new heuristic locations with that of other existing heuristics. We also compare our locations with existing locations of the ambulance stations in the South East and Central Metropolitan regions and discuss the improvement achieved. The hospital emergency departments in Western Australia dealt with 837,504 attendances in 2009 [47]. At present, there are about 27 ambulance locations in the Perth Metropolitan area. The response time (time elapsed from the dispatch of an ambulance to its arrival at the emergency scene) which is one of the key indicators used to measure the performance of ambulance has been increasing in Perth metropolitan area for the period 1995 to 2010. Currently 87.6 % of emergency calls were responded within 15 min (Ahern [48]).

Perth, the metropolitan capital of Western Australia covers approximately 5,000 square kilometers, extends 70 km along the coast and had an estimated 2.30 million residents in 2010. The Perth metropolitan area is divided into five major statistical divisions, namely Central Metropolitan, East Metropolitan, North Metropolitan, South East Metropolitan and South West Metropolitan. We consider the South East Metropolitan area and the Central Metropolitan area.

All location-allocation heuristics need information about the distance or travel cost between the demand locations and the service or candidate locations. For this study the distance matrix from node to node for the study is determined by distance data developed by a company by the name travelmate [49]. This distance is the road network among the various suburbs in the Perth metropolitan area. Hence the distance values are the shortest road travel distances (equivalent to distance on a road map) between the origin and the destination. We weighted the distance by taking into consideration only the population of the origin suburb since we can also weigh by considering the nature of the road. The fact is that the better the road network the easier it is for a vehicle to move from one suburb to another. The weighted distance is thus the product of the weight assigned to the origin suburb and the distance between the origin suburb and the destination suburb. This weight is proportional to the population of each suburb which is shown in Tables 4 and 5. We note from the tables that the larger the population the larger the weight. This weighted distance is calculated for each of the 26 suburbs in South East Metropolitan region and the 25 suburbs in Central Metropolitan region. This calculation is done by noting the distance from a suburb to all other suburb and each value is multiplied by the weight of the origin suburb. This is repeated for every suburb and values are recorded as a 26 × 26 and 25 × 25 matrix representing South East Metropolitan and Central Metropolitan area respectively.
Table 4

Total demand weighted distance (Km) for heuristics and existing location for South East Metropolitan Region

Number of facilities

RRH

Myopic

Exchange (EH)

Neighborhood (NS)

Optimal (O)

Existing location (EL)

2

1369

1385

1385

1385

1369

1404

3

914

914

914

914

914

1101

4

696

696

696

696

696

818

5

557

607

557

557

557

688

6

473

521

473

502

473

650

7

418

431

418

418

416

608

8

361

380

361

361

361

487

9

306

325

307

307

306

418

Table 5

Total demand weighted distance (Km) for heuristics and existing location for Central Metropolitan Region

Number of facilities

RRH

Myopic

Exchange (EH)

Neighborhood (NS)

Optimal (O)

Existing location (EL)

2

401

463

401

401

401

485

3

303

329

303

303

303

426

4

246

281

246

250

246

304

5

210

238

210

210

202

209

6

166

196

183

183

166

174

7

138

169

139

139

138

147

8

117

150

117

117

117

125

9

103

131

103

103

103

105

We note from these calculations that the weighted distance of 26 suburbs of South East Metropolitan region ranges from about 4 to 1,009 km. In the case of Central Metropolitan region the minimum value is 1 km while the maximum is 171 km resulting in a range of 1 to 171 km.

Tables 4 and 5 show the comparison of total demand weighted distance for the new and existing heuristics corresponding to South East metropolitan area and Central metropolitan area respectively. Tables 6 and 7 shows the existing location and the location due to the new heuristic (RRH). The tables also shows the cost saved if ambulance stations are located using our new heuristic.
Table 6

Comparison of existing locations and RRH locations and minimum cost saving by RRH for South East Metropolitan Region

Number of facilities

Existing locations

RRH locations

Cost saved (%)

2

{13,22}

{14,25}

2.5

3

{1,13,22}

{1,14,24}

20.4

4

{1,11,13,22}

{1,8,14,24}

17.5

5

{1,7,11,13,22}

{1,8,14,23,26}

23.5

6

{1,7,11,13,16,22}

{1,8,13,14,23,26}

37.4

7a

{1,7,11,13,16,22,24}

{1,4,8,13,14,23,26}

46.1a

8

{1,7,11,13,14,16,22,24}

{1,2,8,9,13,14,23,26}

34.9

9

{1,7,8,11,13,14,16,22,24}

{1,2,4,8,9,13,14,23,25}

36.6

aThe number of ambulance stations at present is seven

Table 7

Comparison of existing locations and RRH locations and minimum cost saving by RRH for Central Metropolitan Region

Number of facilities

Existing locations

RRH locations

Cost saved (%)

2

{9,13}

{5,14}

20.9

3a

{9,13,23}

{3,5,20}

40.5a

4

{3,9,13,23}

{3,7,9,20}

23.5

5

{3,7,9,13,23}

{3,5,8,9,20}

−0.5

6

{3,5,7,9,13,23}

{3,5,8,9,13,19}

4.8

7

{1,3,5,7,9,13,23}

{2,5,8,9,13,14,18}

6.5

8

{1,3,5,7,9,13,14,23}

{1,3,5,8,9,13,14,18}

6.8

9

{1,3,5,7,9,13,14,18,23}

{1,2,3,5,8,9,13,14,18}

1.9

aThe number of ambulance stations at present is three

Currently, there are seven ambulance stations in the South East Metropolitan area while in the Central Metropolitan area there are three. In the case of the South East Metropolitan region the eighth and the ninth ambulance locations were chosen by considering the best (minimum) objective value if any location is added to the existing set of locations. We use a similar procedure to obtain the location for facilities that are less than seven facilities. For example, to obtain six facilities for existing locations, we drop a location one by one and choose the six locations that give the minimum value. The same procedure was used for the locations in the Central Metropolitan region. These locations are shown in Tables 6 and 7. There are 27 stations in the whole of the Perth Metropolitan area.

The new reduction heuristic (RRH) was applied to determine the optimal locations for ambulance stations. The solutions generated represent a significant improvement when compared to the existing location pattern. We note from this study that if RRH is used to locate the seven facilities in the South East Metropolitan area accessibility is increased by 45.5 %. This results in the improvement of the average response time by 6.82 min to 8.18 min when the new heuristic is used. In the case of the Central Metropolitan area accessibility is increased by 40.5 % giving a 6.07 min reduction in the average response time to 8.93 min.

## 7 Conclusions

This paper has focused on the facility location problem, which involves the determination of an optimal set of locations for the facilities. In particular, we consider the p-median problem as an effective tool for locating emergency facilities. This problem has been studied for the past 50 years and a number of heuristics have been proposed since the problem is NP-hard. The most common heuristics to solve the p-median are the ones based on vertex substitution. We have developed a new effective heuristic based on a reduction technique that eliminate outliers and exchange routine that improves the current solution. We tested our heuristic on a set of randomly generated test problems as well as a set of literature test problems. In addition we applied our methodology to locate ambulance stations in Perth Metropolitan. Our computational results demonstrate the effectiveness of our heuristic.