Site Selection of New Facility Using Gravity Model and Mixed
Integer Linear Programming in Delivery and Logistic Company
Nafisha Herma Hanifha †
Ari Yanuar Ridwan
School of Industrial and System
Engineering
Telkom University
Bandung, Indonesia
nafisha.h3rma@gmail.com
School of Industrial and System
Engineering
Telkom University
Bandung, Indonesia
ari.yanuar.ridwan@gmail.com
ABSTRACT
Prafajar Suksessanno
Muttaqin
School of Industrial and System
Engineering
Telkom University
Bandung, Indonesia
prafajar37@gmail.com
ACM Reference format:
Nafisha Herma Hanifha, Ari Yanuar Ridwan, Prafajar Suksessanno
Muttaqin, 2020. In Proceedings of ACM APCORISE’20, June, 2020, Depok,
West Java, Indonesia. ACM, New York, NY, USA, 5 pages.
https://doi.org/10.1145/3400934.3400944
Delivery and logistic company is one of the businesses that
include in supply chain management business. There is one of
companies located in Indonesia that run in that field of business.
One of cities that has the highest demand in West Java is Bandung
with 50% of total demand at the province. Bandung itself already
has 13 delivery center in Bandung Raya, including Bandung City,
Bandung Regency, West Bandung Regency, and Cimahi City.
However, there are still unfulfilled demands that cause loss cost
for the company because the distance between delivery center to
customer is too far. Therefore, the company is willing to add one
more delivery center in Bandung Raya but outside Bandung City.
Gravity Model is used to determine the potential location of
facility in Soreang, Dayeuhkolot, and Ujung Berung. After the
potential locations are determined, Mixed Integer Linear
Programming is used for choose one of the potential locations to
be one location that has least cost between the three.
1 Introduction
In this new era, supply chain management is needed to support
businesses to develop rapidly [1]. Types of businesses that is
commonly found nowadays are delivery and logistic service
company. In Indonesia, there are a lot of company that provide
delivery and logistic service. There is one of companies that run
in that field of business in Indonesia.
Actually, in West Java, city that has the highest demand of the
company is Bandung, with average 570 thousand goods to send
per month, it is approximately 50% of total demand in West Java.
Bandung’s distribution center itself has 13 delivery centers in
Bandung Raya, including Bandung City, Bandung Regency, West
Bandung Regency, and Cimahi City. Common problem occurs in
the company is the number of unfulfilled demand, means that the
goods that are supposed to shipped to customers are delayed. If
goods are delayed, there are loss cost in the logistic cost.
Meanwhile, logistic cost is one of the important roles in a
company [2]. These are the percentage of unfulfilled demand and
the loss cost of each delivery center:
CCS CONCEPTS
• Industrial Engineering • Supply Chain Management • Facility
Problem
KEYWORDS
Facility Location, Delivery and Logistic, Gravity Model, Mixed
Integer Linear Programming
†Author
Footnote to be captured as Author Note
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Table 1: Unfulfilled Demand and Loss Cost Percentage
Delivery Center
Unfulfilled
Loss Cost in %
Demand in %
Asia Afrika
15
12.1
Cikutra
4.6
3.7
Cipedes
6.5
5.2
Situsaeur
15.6
12.5
Sekejati
9.4
7.5
Cimahi
6
7.2
Ujung Berung
8.1
9.8
APCORISE 2020, June 16–17, 2020, Depok, Indonesia
© 2020 Association for Computing Machinery.
ACM ISBN 978-1-4503-7600-6/20/06…$15.00
https://doi.org/10.1145/3400934.3400944
43
APCORISE’20, June, 2020, Depok, West Java, Indonesia
Delivery Center
Soreang
Dayeuh Kolot
Cikeruh
Padalarang
Lembang
Majalaya
Unfulfilled
Demand in %
13.4
12.3
3.9
2.5
1
1.7
N. Herma Hanifha et al
The process of calculating the distance between two locations in
this model is calculated as the geometric distance between two
locations using the following formula [9]:
Loss Cost in %
16.1
14.9
4.7
3
1.2
2
𝐷𝑛 = √(𝑥 − 𝑋𝑖 )2 + (𝑦 − 𝑌𝑖 )2
Where (𝑋𝑖 ; 𝑌𝑖 ) is the candidate coordinates for each facility in a
particular area and (𝑥;y) is the facility considered. The purpose of
this model is to get the location of the facility that minimizes the
total shipping costs that can be formulated as follows [9]:
𝑇𝐶 = ∑𝑘𝑛=1 𝑉𝑖 𝐷𝑛 𝐶𝑖
This company is willing to build a new delivery center to decrease
the number of unfulfilled demand. There are several reasons for
company to investing new facilities other than to fulfill demand.
Several of these reasons are to increase its production capacity, to
extend its product range, or to enter new market [3]. Based on the
table above, it can be seen that some of delivery centers have high
percentage of unfulfilled demand and loss cost. So, the new
delivery center must be in the near area of the problematic
delivery center in order to help that delivery center to fulfill the
demand. However, delivery centers that located in Bandung City
are excluded because the company prefer to build a new facility
outside the Bandung City. Delivery centers that have the higher
percentage, than average, in unfulfilled demand and loss cost are
Asia Afrika, Situsaeur, Ujung Berung, Soreang, and Dayeuh Kolot.
However, Asia Afrika and Situsaeur are excluded because these
two are located in Bandung City.
Notation:
𝑉𝑖 = Volume shipped from facility i
𝐷𝑛 = Distance between location of facility and location i, in
the nth iteration.
The result of the coordinates of the model will be located in one
of the districts in the area. All of villages that located in that
district will become the candidates of new facility locations. After
the potential location are determined, method that can be used is
mixed integer linear programming (MILP). The combination
between gravity method and MILP itself had been done before by
Brittany Cher Collins and Hao Wang in 2019. The difference is in
the objective of the gravity method.
MILP can be modelled when there is a set of candidate facility
locations problem [10]. The objective of MILP model is to identify
the optimal combination of location that leads to the minimum
total costs [11]. The input of MILP model are customer demand
and location, facility information and cost, and transportation
cost. This is the formula of MILP [12]
Build new facility locations are high cost and cannot be reverse,
also has long time impact [4]. On the other hand, determining the
location of facility has a positive impact on the distribution system
parameters including efficiency, cost, and time [5]. Alfred Weber
was the first one that started the location theory, he considered
how to locate a single warehouse in objective to minimize the total
distance between warehouse in 1990 [6].
Minimize:
To determine the potential location for the new facility in this
case, gravity model is used. This model has been a topic of
research instituted in the field of economics. Therefore, there is a
lot of research related to gravity models [7]. It was first applied in
Tinbergen in 1962 and continued in 1966 by Linnemann [8]. This
model is part of the supply chain management network
development strategy used to determine the location of a facility
(e.g. warehouse, factory, etc.). The input of gravity model are[9]:
1.
Volume shipped from delivery center
2.
Transportation cost
3.
Coordinate of existing and new facilities
𝑍 = ∑ ∑ 𝑐𝑖 𝑥𝑖𝑗 + ∑ 𝑓𝑖 𝑦𝑗
𝑖∈𝐼 𝑗∈𝐽
∑ 𝑥𝑖𝑗 ≥ 𝑑𝑖 , ∀𝑖 ∈ 𝐼
𝑖∈𝐼
𝑥𝑖𝑗 − 𝑀𝑦𝑗 ≤ 0, ∀𝑖 ∈ 𝐼, ∀𝑗 ∈ 𝐽
𝑥𝑖𝑗 ≥ 0
𝑦𝑗 = {0, 1}
∑ 𝑦𝑖 = 𝑛
Notation:
𝑥𝑖𝑗 = Unit volume shipped from facility j to customer node i
𝑑𝑖 = Volume of total demand for each customer node i
𝑦𝑗 = Indicates whether facility j is used or not
𝑓𝑗 = Fixed cost of facility
M = An arbitrary large number to link the volume with the
facility
Formula that used for determining the coordinate of new facilities
candidate is:
𝑥=
𝛴𝑖 𝑉𝑖 𝑋𝑖 𝐶𝑖
𝛴𝑖 𝑉𝑖 𝐶𝑖
;𝑦=
𝑗∈𝐽
Subject to:
𝛴𝑖 𝑉𝑖 𝑌𝑖 𝐶𝑖
𝛴𝑖 𝑉𝑖 𝐶𝑖
Notation:
𝑋𝑖 = X coordinate of facility i
𝑌𝑖 = Y coordinate of facility i
𝑉𝑖 = Volume shipped from facility i
𝐶𝑖 = Transportation rate from facility i
𝑥 = X coordinate of new candidate facility
𝑦 = Y coordinate of new candidate facility
n = Number of facilities need to be decided in the model
2 Methodology
These are steps to be taken in solving problem in this study. The
systematic problem solving is divided into five stages, namely (1)
44
Site Selection of New Facility Using Gravity Model and Mixed
Integer Linear Programming in Delivery and Logistic Company
APCORISE’20, June, 2020, Depok, West Java, Indonesia
Preliminary Stage that include Observation, Identification, and
Objective; (2) Data Collection Stage; (3) Data Processing Stage; (4)
Analysis; and (5) Conclusion Stage. This diagram below shows the
systematic of problem solving:
Figure 3: Customer Nodes Soreang
Figure 4: Customer Nodes Ujung Berung
3.2 Villages Coordinate
The data of customer nodes is classified into villages based on the
scope of each delivery center which are Dayeuh Kolot, Soreang,
and Ujung Berung. This table below shows the sample of villages
coordinate data in Dayeuh Kolot.
Table 2: Sample of Villages Coordinate Dayeuh Kolot
No
Village
X
Y
1
Ancolmekar
-7.092341
107.670363
2
Andir
-6.995162
107.616481
3
Arjasari
-7.064739
107.640131
4
Baleendah
-7.015652
107.63212
5
Banjaran Kulon
-7.053783
107.5805
6
Banjaran Wetan
-7.087402
107.609264
7
Baros
-7.060888
107.629329
8
Batukarut
-7.045685
107.597384
9
Bojongkunci
-7.018277
107.567217
10
Bojongmalaka
-6.993273
107.606791
..
….
…
…
53
Wargamekar
-7.02061
107.672832
Figure 1: Systematic of Problem Solving
3 Data Collection
3.1 Customer Nodes Classified by Villages
Figures below show the big picture of the scope of each delivery
centers. To make it simple, the customer nodes are classified into
villages.
3.3 Demand
Other than customer nodes and villages coordinate, data of
average demand per villages also needed. This table below depict
the sample of average demand per month of each villages and the
total average demand per month for delivery center that located
in Soreang.
Table 3: Sample of Average Demand of Soreang
No
Village
Demand
1
Bandasari
508
2
Banyusari
1015
Figure 2: Customer Nodes Dayeuh Kolot
45
APCORISE’20, June, 2020, Depok, West Java, Indonesia
No
3
4
5
6
7
8
9
10
..
46
Total
Village
Buninagara
Cangkuang
Cangkuang Kulon
Cibodas
Cigondewah Hilir
Cilame
Cilampeni
Ciluncat
….
Tanjungsari
N. Herma Hanifha et al
Table 5: Soreang Iteration Result
Iteration
X
Y
Demand
503
513
1523
497
518
523
1000
528
…
559
40612
0
-6.995972
107.5497
Transportation
Cost / month
(Rp)
3,255,380.92
1
-6.99329
107.5524
3,219,455.85
2
-6.991439
107.5535
3,206,823.54
3
-6.990261
107.5539
3,202,497.59
4
-6.989638
107.5541
3,201,352.65
5
-6.989342
107.5542
3,201,080.90
6
-6.9892
107.5542
3,201,006.90
7
-6.989128
107.5543
3,200,982.40
8
-6.989088
107.5543
3,200,973.28
9
-6.989065
107.5544
3,200,969.73
10
-6.989051
107.5544
3,200,968.32
11
-6.989042
107.5544
3,200,967.75
12
-6.989037
107.5544
3,200,967.53
13
-6.989033
107.5544
3,200,967.44
14
-6.989031
107.5544
3,200,967.40
15
-6.98903
107.5544
3,200,967.39
16
-6.989029
107.5544
3,200,967.38
17
-6.989029
107.5544
3,200,967.38
The table above depicts the least transportation cost for the new
potential location is Rp3,200,967.38. With that least transportation
cost, the new location is coordinated with x -6.989029 and y
107.5544. This coordinate is located in Katapang District.
4 Result and Discussion
4.1 Gravity Model
Gravity model used to determine candidate locations for the new
facility by doing some iterations. The iteration stopped once the
transportation cost reach the minimum. These are the result of
gravity model of each area:
4.1.1 Dayeuh Kolot. For Dayeuh Kolot, 17 iterations are needed
to get the minimum total transportation cost.
Table 4: Dayeuh Kolot Iteration Result
Iteration
X
Y
Transportation
Cost / month
(Rp)
0
-7.040229
107.6077
2,646,939.43
1
-7.039595
107.6061
2,643,172.51
2
-7.039366
107.6056
2,642,813.81
3
-7.039173
107.6054
2,642,709.59
4
-7.039036
107.6053
2,642,672.89
5
-7.038946
107.6053
2,642,659.38
6
-7.038888
107.6052
2,642,654.33
7
-7.038851
107.6052
2,642,652.42
8
-7.038829
107.6052
2,642,651.70
9
-7.038814
107.6052
2,642,651.43
10
-7.038806
107.6052
2,642,651.32
11
-7.0388
107.6052
2,642,651.28
12
-7.038797
107.6052
2,642,651.27
13
-7.038795
107.6052
2,642,651.26
14
-7.038794
107.6052
2,642,651.26
15
-7.038793
107.6052
2,642,651.26
16
-7.038792
107.6052
2,642,651.26
17
-7.038792
107.6052
2,642,651.26
From the table above, it could be seen that the potential location
in Dayeuh Kolot area is located in coordinate x and y -7.038792
and 107.6052 respectively. To be exact, that coordinate is located
in Arjasari District with the least transportation cost is
approximately Rp2,642,651.
4.1.3 Ujung Berung. Different from Dayeuh Kolot and Soreang,
Ujung Berung only needs 15 iterations to determine the potential
location for the new facility.
Table 6: Ujung Berung Iteration Result
Iteration
X
Y
4.1.2 Soreang. Just like Dayeuh Kolot, the number of iteration
that needed for determine the new potential location in Soreang
is also 17.
0
-6.89536
107.658
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
-6.912717
-6.914564
-6.914564
-6.914257
-6.913993
-6.913831
-6.913742
-6.913696
-6.913673
-6.913661
-6.913655
-6.913652
-6.913651
-6.91365
-6.91365
107.6773
107.6818
107.6845
107.686
107.6867
107.6871
107.6873
107.6874
107.6874
107.6874
107.6875
107.6875
107.6875
107.6875
107.6875
Transportation
Cost / month
(Rp)
7,757,043.08
6,368,608.70
6,280,004.22
6,250,260.13
6,240,430.05
6,237,733.80
6,237,064.16
6,236,904.42
6,236,866.92
6,236,858.16
6,236,856.12
6,236,855.65
6,236,855.53
6,236,855.51
6,236,855.50
6,236,855.50
The table above shows that the potential location for the new
facility is located coordinate -6.91365 in x axis and 107.6875 in y
axis. This coordinate is located in Arcamanik District with the
least transportation cost Rp6,236,855.50.
46
Site Selection of New Facility Using Gravity Model and Mixed
Integer Linear Programming in Delivery and Logistic Company
APCORISE’20, June, 2020, Depok, West Java, Indonesia
with only 16 million difference, approximately. The last is
Katapang District (Soreang) with the minimum cost
Rp685,689,988.
4.2 Mixed Integer Linear Programming
After the potential locations for the new facility are determined,
the mixed integer linear programming is performed by comparing
the total cost of each area of potential locations.
Table 7: MILP Results
District
Arjasari
(Dayeuhkolot)
Katapang
(Soreang)
Arcamanik
(Ujung Berung)
Village
Arjasari
Mekarjaya
Ancolmekar
Mangunjaya
Wargaluyu
Lebakwangi
Baros
Batukarut
Patrolsari
Pinggirsari
Rancakole
Total Cost
Cilampeni
Banyusari
Katapang
Pangauban
Sangkanhurip
Sekarwangi
Sukamukti
Gandasari
Total Cost
Arcamanik
Endah
Bina Harapan
Cisaranten
Endah
Cisaranten
Kulon
Sukamiskin
Total Cost
5 Conclusion
In conclusion, to find the location for the new facility in this case,
gravity model and mixed integer linear programming (MILP) are
needed. Gravity model used for determine potential location in
Dayeuh Kolot, Soreang, and Ujung Berung. There are three
districts in result of gravity model. These are Arjasari, Katapang,
and Arcamanik. All of the villages that located in those 3 districts
are become the candidate of the new facility location. After got
the candidate locations, MILP is done to choose one of the
locations that has the minimum total cost. As a result,
combination of Arjasari and Rancakole villages with a total cost
of nearly 500 million rupiah was chosen. This villages are located
in Arjasari District in Dayeuh Kolot. Therefore, the new delivery
center will be built in that area.
Open/Not Open
Open
Not Open
Not Open
Not Open
Not Open
Not Open
Not Open
Not Open
Not Open
Not Open
Open
Rp459,061,630
Not Open
Open
Open
Not Open
Not Open
Not Open
Not Open
Not Open
Rp685,689,988
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A. J. Arumugham, “Solving Supply Chain Network Gravity Location Model
Using LINGO Investigations on Design of Supply Chain Networks for
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(Institute of Industrial Engineers), vol. 38, no. 7. pp. 547–564, Jul-2006.
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S. Shahriar, L. Qian, S. Kea, and N. M. Abdullahi, “The Gravity Model of Trade:
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R. Guo, “Determinants of spatial (dis)integration,” in China’s Spatial
(Dis)integration, Elsevier, 2015, pp. 67–105.
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D. O. Effendi and N. Siswanto, “Determination of Provincial Level of
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Not Open
Open
Not Open
Not Open
Open
Rp475,272,617
From the table above, it could be seen that each district has vary
number of villages. Arjasari district with 11 villages which are
Arjasari, Mekarjaya, Ancolmekar, Mangunjaya, Wargaluyu,
Lebakwangi, Baros, Batukarut, Patrolsari, Pinggirsari, and
Rancakole. Katapang with 8 villages that are Banyusari,
Cilampeni, Katapang, Pangauban, Sangkanhurip, Sekarwangi,
Sukamukti, and Gandasari. Lastly, 5 villages of Arcamanik, these
are Arcamanik Endah, Bina Harapan, Cisaranten Endah,
Cisaranten Kulon, and Sukamiskin.
From vary number of villages, two villages was chosen from each
district with the minimum cost. The least total cost is performed
by the combination of Arjasari and Rancakole villages in Arjasari
district (Dayeuh Kolot) with Rp459,061,630. While the least total
cost of Arcamanik (Ujung Berung) is slightly higher than Arjasari
47