0-1 Integer Linear Programming Model for Location Selection of Fire Station: A Case Study in Indonesia

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0-1 integer linear programming model for location selection of fire station: A case study in Indonesia

Susila Bahri

Citation: AIP Conference Proceedings 1723, 030004 (2016); doi: 10.1063/1.4945062 View online: http://dx.doi.org/10.1063/1.4945062

View Table of Contents: http://aip.scitation.org/toc/apc/1723/1 Published by the American Institute of Physics

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0-1 Integer Linear Programming Model for Location Selection of Fire Station: A Case Study in Indonesia

Susila Bahri

Department of Mathematics

Faculty of Mathematics and Natural Sciences Andalas University, Limau Manis Campus

Padang, Indonesia, 25163 [email protected]

Abstract. In this research, the minimization of the fire station model is constructed. The maximum time data required by the firefighter is used to construct the minimization model of the fire station in Padang. The model is used to determine the minimum number of the available fire station in Padang town. By using Matlab 2013a, the solution of the model can be found based on the Branch and Bound method. It denotes that the fire station must be built in Lubuk Begalung and Kuranji sub-districts.

INTRODUCTION

Nowadays, most fires which occur in Padang town announced by newspapers like Padang Ekspres and Singgalang have made the serenity and comfort of the Padang residents being disturbed. The fires are caused by various things such as a short circuit and a stove explosion so that every resident must be more careful and responsible for the fire incidents on his surrounding environment. However, we cannot deny that the tranquility in Padang town is the responsibility of Padang government.

The government of Padang through the fire extinguishing area disaster management agency (Badan Penanggulangan Bencana Daerah Pemadam Kebakaran/ BPBD PK) located on Rasuna Said street No.56, has done the best to serve the fire complaints from Padang residents. This can be proven by the availability of eleven firefighter buses together with their personnel. Nevertheless, to prevent or minimize the fire in eleven sub-districts of Padang town is not only determined by how many buses are available but more importantly by the accuracy and speed of the firefighter buses arrived at the occurrence location.

In fact, the government of Padang that has only two fire stations, often deals with problems how to prevent the fires because the distance between the occurrence location and the two stations is far away. The obstacles in terms of time which can fail or slow the efforts to extinguish the flames, can be overcome by knowing how many minimum fire stations that should be available. Moreover, we should build the fire station in the appropriate sub- district so that we can solve the problems or minimize the effects of fires.

This problem is a set covering problem that has been discussed by many authors [1] and [2]. In [1], Arogundade O.T et all studied about how to solve the fire facility location problem by using Balas Additive Algorithm whereas [2] told about set covering for determining the location problem of the fire stations.

INTEGER LINEAR PROGRAMMING

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¦

ⁿ

j j j

x c

1

Minimize

n j

m i

b x

a

_i

n j

j

ij

, 1 ,..., , 1 ,..., s

Constraint

1

¦ t

x

₁

,..., x

_n

t 0 and integer

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If all variables have the value 0 or 1, then the integer linear programming problem is called by the 0-1 integer linear programming problem, while if the value

b 1

, then the problem is so called by Set Covering Problem [3].

SET COVERING PROBLEMS

Suppose

S

₁

, ... , S

_n is a family subset of the set

S { 1 , 2 , ... , m }.

The covering set

S

is a subfamily

S

_jfor

1

j

such that

S jISj . We assume that every subset Sj has the cost cj ! 0 which is an integer and related to every subset. Then, we set the covering cost as the total amount of the subset cost of which is included in the cover. Next, a covering set determining problem with a minimum cost is an integer linear programming problem with a matrix A aij which has the size mu n where

¯ ®

j j

ij

if i S

S i a if

0 1

(2) In addition, we set

x

_j as a variable which has the value 0 or 1, the problem becomes [4]:

¦

ⁿ

j

c

j

x

j 1

Minimize

1 0

,..., 1 , ,..., 1 , 1 s

Constraint

1

or x

n j

m i

x a

j n

j ij j

t

¦

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BRANCH AND BOUND METHOD

The basic concept which is the steps of this method is dividing and conquering. For Dividing principle, the large initial problem is dividing the problem into small problems (new sub problems) through branching the tree search.

Each sub problem has the bound value at the branching time. We solve it to get the solution for every sub problem.

Furthermore, we take the steps conquering principle. If the solution of the sub problem which has been approximated is the feasible solution, then branching cannot be continued or conquered and the bound value can be taken as an additional constraint for every new sub problem. Both steps is repeated until the optimal solution obtained. The model solution can be found if every variable value is an integer [5]. We apply the Branch and Bound method in linear programming with Matlab to solve the problem.

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GENERAL DESCRIPTION OF PADANG SUB-DISTRICTS

As we have mentioned before, the research data is collected by the fire extinguishing area disaster management agency of Padang. From the office, we know that the government of Padang has two fire stations namely in the East Padang and Kuranji sub-districts. Padang town consists of eleven sub-districts respectively as follows:

1. The sub-district of Bungus Teluk Kabung 2. The sub-district of Lubuk Kilangan 3. The sub-district of Lubuk Begalung 4. The sub-district of South Padang 5. The sub-district of East Padang (*) 6. The sub-district of West Padang 7. The sub-district of North Padang 8. The sub-district of Nanggalo 9. The sub-district of Kuranji (*) 10. The sub-district of Pauh

11. The sub-district of Koto Tangah

The sign (*) indicates that the location of the fire station is now located in East Padang and Kuranji sub-districts.

IMPLEMENTATION AND RESULT

From the map of Padang in FIGURE 1, we can obtain the adjacent neighbourhood; any neighbourhood with a non-zero border with its home neighbourhood for each sub-district.

FIGURE 1. The Map of Padang City

Based on the data obtained from the firefighter office, it was obtained that the best maximum travel time required by the firefighter bus to reach the fire location was 10 minutes. Moreover, in this research we use some assumptions as follows:

1. Damkar can move from any point in the particular sub-district to any point in other sub-districts provided that such a point that can pick the path traveled by the bus (Damkar).

2. The smooth access to go from one sub-district to other sub-districts without obstacles, congestion and 1

2 4 3

6 5

7 8

9

10 11

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4. The firefighter bus can be used to handle the fire occured in the sub-districts where the fire station takes place and its neighbourhood.

Furthermore, we present TABLE 1 to show the average maximum time that is need by the firefighter bus to reach the occurance location.

TABLE 1. The Average Maximum Time to Reach From One Subdistrct to Another in Minutes [6]

Subdis

tricts 1 2 3 4 5 6 7 8 9 10 11

1 0 10 10 15 15 15 15 20 20 20 25 2 10 0 10 15 15 15 15 20 20 10 20 3 10 10 0 5 10 15 15 15 15 15 20

4 15 15 5 0 5 5 15 15 15 15 20

5 15 15 10 5 0 5 10 10 10 10 15

6 15 15 15 5 5 0 5 10 10 15 15

7 15 15 15 15 10 5 0 5 10 15 15 8 20 20 15 15 10 10 5 0 10 15 10 9 20 20 15 15 10 10 10 10 0 10 10 10 20 10 15 15 10 15 15 15 10 0 15 11 25 20 20 20 15 15 15 10 10 15 0

The model of the fire minimization problem means that we minimize the number of fire stations in Padang as the set covering problem consisted of three linear programming components:

1. Objective function

Since the objective of modelling is to minimize the number of the fire stations, so the model of the objective function for this case is

11 10 9 8 7 6 5 4 3 2

Minimum Z x

1

x x x x x x x x x x

(4) which

x

_jdenotes the j –th sub-district with

j 1 ,..., 11

according to the sequence of the sub-districts

2. Constraints

From the TABLE 1, we can see that only the sub-district 1, 2 and 3 do not exceed the best maximum travel time 10 minutes. Hence, the contraint for the sub-district 1 is

x

₁

x

₂

x

₃

t 1

. The mathematical symbol

≥ means it is required at least one fire station to serve the sub-district 1 including its adjacent neighbourhood. Similarly, we can obtain the constrains for the sub-district 2 until 11, namely

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1 1 1 1 1 1 1 1 1 1

11 9

8

10 9 5

2

11 9

8 7 6 5

9 8 7 6 5

9 8 7 6 5 4

10 9 8 7 6 5 4 3

6 5 4 3

5 4 3 2 1

10 3

2 1

3 2 1

t

x x

x

x x x

x

x x

x x x x

x x x x x

x x x x x x

x x x x x x x x

x x x x

x x x x x

x x

x x x

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The first constraint is a constraint for the subdistrict 1. The second one is for the subdistrict 2 and so on.

3. Decision variables

The decision variable for this case is:

¯ ®

j x

_j

j

district -

sub in the built not is station the

if 0

district -

sub in the built is station the

if

1

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Combining three components of the 0-1 integer linear programming model, so the minimization model of the fire problem in Padang can be found.

Furthermore, the optimal value for the decision variable can be obtained by using the MATLAB R2013a software. Since the inequality on Matlab only hold for the sign “≤”, so both matrices A and b must be multiplied by - 1. Hence, the syntaxes for the objective function and constraints are

];

1 0 1 1 0 0 0 0 0 0 0

; 0 1 1 0 0 0 1 0 0 1 0

; 1 1 1 1 1 1 1 0 0 0 0

; 1 0 1 1 1 1 1 0 0 0 0

; 0 0 1 1 1 1 1 0 0 0 0

; 0 0 1 1 1 1 1 1 0 0 0

; 0 1 1 1 1 1 1 1 1 0 0

; 0 0 0 0 0 1 1 1 1 0 0

; 0 0 0 0 0 0 1 1 1 1 1

; 0 1 0 0 0 0 0 0 1 1 1

; 0 0 0 0 0 0 0 0 1 1 1 [

];

1

; 1

; 1 [

];

1

; 1

; 1 [

!!

A b

f

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0000 . 2 min

0 0 1 0 0 0 0 0 1 0 0

. min

] , [], [], , , , ( min]

, [

];

1

; 1

; 1 [

];

0

; 0

; 0 [

!!

f x

ated ter

on Optimizati

UB LB b

A f linprog f

x UB

LB

From the result, we have

f min 2

and

x

₃

x

₉

1

. It states respectively that the station should be built at least two stations and is located in the sub-district of Lubuk Begalung and Kuranji.

CONCLUSION

From this research, we conclude that is required at least two stations in Padang. The result leads the optimal solution for the location of fire station namely Lubuk Begalung and Kuranji subdistricts. Both subdistricts are the best fire station location because the time required by the bus to reach the fire location in adjacent neighbourhood is minimum. It is different from the location that ever existed before. For further research, we can construct the improving model with more variables.

ACKNOWLEDGMENTS

The author would like to acknowledge the research grant Hibah Bersaing from DIKTI and thank Faisal Asra for taking the data.

REFERENCES

1. Arogundade, O.T et all. A 0-1 Model for Fire and Emergency Service Facility Location Selection: A Case Study in Nigeria, Journal of Theoritical and Applied Information Technology, (2005).

2. G. Cornuejols, M. A. Trick, and M. J. Saltzman, A Tutorial on Integer Programming. (1995).

http://www.math.clemson.edu/~mjs/courses/mthsc.440/integer.

3. E. Neumann, Linear Programming with Matlab, Southern Illionis University, (2008)

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4. M. Borza, A. S. Rambely and M. Saraj, Mixed 0-1 Linear Programming for an Absolute Value Linear Fractional Programming with Interval Coefficients in the Objective Function, Applied Mathematical Sciences, 7 (2013), 3641 – 3653. http://dx.doi.org/10.12988/ams.2013.33196

5. E.L. Lawler, et all. The Traveling Salesman Problem, (John Wiley & Sons Ltd, New York, 1989), pp. 361–

401.

6. Padang Firefighter Station, (private communication, 2015).