LECTURE – 5
QUANTITATIVE MODELS FOR FACILITY
LOCATION:
BASED ON DIFFERENT OBJECTIVE FUNCTIONS OR
OPTIMIZATION CRITERIA
Learning Objective
1. To demonstrate the quantitative model to locate facility based on different
optimization criteria
6.13 Weighted Score Method
 Objective: Government of one nation wants to open up health check up clinic in a
village
 Decision: Where to locate health check up clinic out of three sites; site1- near bus
stop, site 2- center of the village, site 3– near mandi (vegetable and grocery market)
Factor to be considered:
1. Accessibility for all villagers
2. Annual lease cost
3. Delivery of medicines
STEPS OF WEIGHTED SCORE METHOD
List all the factors
Assign a weight to each factor, reflecting the relative importance of
factors given by service organization
Develop a scale for factors
Eg: 1 to 5 or 1 to 10
Assign score to each location or alternative for
each factor using a scale
Multiply the score and the weight for each
factor and total the score for each location
Arrange the total score in descending order.
The site with maximum score will be selected
SITE LOCATION FOR HEALTH CLINIC IN A VILLAGE
6.14 Single facility location problem: Minimize maximum distance
This set of problems usually occurs to meet the objective of some public sector or some
social initiative. Most of such facilities can be emergency hospital services, location of
ambulance stand and location of fire station. The main aim of such services is to locate
such a service facility at a location where the maximum distance from new emergency
facility to any existing user facilities is minimized. Here the criteria are that the farthest
customer has to walk the minimum distance to avail the facility. That is why such
problems are called Minimax location problem. We need the location coordinates of
existing facilities or user facilities. In such cases determining maximum delay is more
important to evaluate the effectiveness of service delivery than average or total delay in
providing service. Consider there are n existing facilities with location coordinates (x 1, y1),
(x2, y2), (x3, y3),… (xn, yn) in x – y plane. We want to locate new facility at (xR, yR) such
that the maximum distance from the new facility to any of the exciting facilities is
minimized.
So, we want to find the optimal location R with (xR, yR) coordinates.
The objective function for minimax location problem can be written as


f  xR , yR   min xR , yR max xi , yi  xR  xi  yR  yi 
where i = 1,…n
let’s take an example and find the solution of minimax location problem with the help of
example.
Example
State government wants to locate a fire station in a city region T. The region T has
residential area, some shops and few offices with the concentrated density located at six
locations with coordinates to be (10,13), (9,18), (10,8), (12,15), (15,15) and (18,6). The
objective is to locate the fire station in region T so that the farthest demand point can be
served with minimum time or be covered with minimum distance. This problem is a
minimax location problem with coordinates of existing locations presented in the table 6.6
below and figure 6.20.
TABLE 6.6: COORDINATES OF EXISTING FACILITY
Existing facility (T)
Coordinates of existing
facilities (xi,yi)
1
(x1,y1)  (10,13)
2
(x2,y2)  (9,18)
3
(x3,y3)  (10,8)
4
(x4,y4)  (12,15)
5
(x5,y5)  (15,15)
6
(x6,y6)  (18,6)
FIGURE 6.20: LOCATION OF EXISTING FACILITIES IN REGION T
Where to locate R (xR, yR)?
Intuitively, we want to plot a square which will cover all existing points so that we can find
a point or a line segment in a square which will be of minimum radius.
How to plot a diamond/square of smallest radius containing all the existing facilities
1. Cover all the existing points with the smallest rectangle so that all points are covered
within the rectangle. In an example we can plot a rectangle whose four edges pass
through the points (15, 15), (18, 6), (10, 8), (9, 18) as shown in figure 6.21.
FIGURE 6.21 PLOT OF A RECTANGLE TO COVER ALL POINTS
We want to draw a square, so while drawing rectangle ensure that each side of rectangle
is making an angle of  45 degree with an axis. It will help in extending the sides of
rectangle to get the shape of square/diamond.
2. How to know that which points will fall on the edges? We want that edges to make an
angle of  45 degrees that means the line passing through the point on edge will have
slope of either +1 or -1.That means we want the minimum intercept and maximum
intercept originating from the straight lines drawn from each existing point with both
+1 and -1 slope.
3. Determine the minimum and maximum intercept using following formulas to get the
edges of rectangle.
We want to find minimum of yi – (-1) xi and yi – (+1)xi
and maximum of yi – (-1) xi and yi – (+1)xi
For all i=1 …n which is given notation of C1 , C2 , C3 and C4 a as given below








C1  min xi  yi
i 1,..m
C2  max xi  yi
i 1,..m
C3  min - xi  yi
i 1,..m
C4  max - xi  yi
i 1,..m
In the figure 6.21 we can see that
C1 =min
C1 =18
(23, 27, 18, 27, 30, 24)
with coordinates (10, 8)
C2 =max
C2 =30
C3 =
C3
with coordinates (15, 15)
min (3, 9, -2, 3, 0, -12)
=-12 with coordinates (18, 6)
C4 =
C4
(23, 27, 18, 27, 30, 24)
max (3, 9, -2, 3, 0, -12)
=9 with coordinates (9, 18)
So, we can see the coordinates for value of C1, C2, C3, and C4 in the figure 6.22
FIGURE 6.22: COORDINATES OF C1, C2, C3 AND C4
The coordinates (10, 8), (15, 15), (18, 6), (9, 18) will lie on the edges of rectangle. We
can also see here that coordinates for C3 and C4 lie on the shorter edges of rectangle.
4. We need to extend the rectangle to a square. We know the shorter edges. We can
extend the shorter edges to obtain a square. Find the radius of a smallest square
(obtained by extending the rectangle) covering all the existing facility coordinates. It
can be determined by using following formula.
C5  max  C2  C1 , C4  C3 
i 1,..n
It will give the maximum length of rectangle which will become the length of square
after extending the shorter edges
C5  max  30  18,9  12 
i 1,..6
 max 12, 21
i 1,..6
C5  21
We want to extend the shorter edge of rectangle to make a square which is having
edge length of C5. That will also give the radius of square that is C5/2.
Radius of square is 21/2 = 10.5
5. We can see from the figure that we can extend the rectangle either upwards or
downwards. That means we can generate two squares with minimum radius of 10.5.
If we join the center points of two squares, we will get a line segment L joining the
following two points
 xR , y R 
1
 xR , y R 
2
1
C1  C3 , C1  C3  C5 
2
1
  C2  C4 , C2  C4  C5 
2

For our example, the two points are
 xR , y R 
1
 xR , y R 
2
1
18  12, 18  12  21
2
1
  30, 27 
2
 15, 13.5

1
 30  9, 30  9  21
2
1
  21, 18
2
 10.5, 9 

So, the line segment joining two points (15, 13.5) and (10.5, 9) will give the optimal
location for a fire station.
6.15 Single service facility location: Maximizing net profit
In most cases, the service organization consider maximizing net operating profits to be the
most important criteria while deciding the location of new facility. When we say net
operating profits that means sales or revenues, generated from some facility, operating cost
and size of facility are very important factors to be considered. The example of such a
facility is to locate a big retail outlet.
To make decision regarding facility location considering above mentioned factors. David
L. Huff (1966) proposed a useful solution for optimum retail location. The basic
assumptions of Huff model is that the expected sale at any potential site is dependent on
spatial location of that site and constrained by a maximum size limit. So, the model, under
above assumptions, determines the net operating profit for each potential location and
select the location which can generate maximum net operating profits.
The notations used for Huff model are presented below.
pj
Probability of a given alterative location j being chosen from among all
alternatives ranging from 1 to n (i.e. j=1,2, 3……n)
n
The number of alternative potential locations
uj
A positive pay off function or utility of jth location perceived by a customer
Sj
Size of a given facility j
Ci
The number of customers at a base region i where i= 1…. N
Tij
Distance travelled by ith customer base to location j in time units
Pij

Eij
Bik
Aijk
Probability of a customer from ith customer base traveling to a potential
facility location j
A parameter which is to be estimated empirically to reflect the effect of travel
time on various times of shopping trips
Expected number of customers from ithregion that are likely to travel to the
potential facility location j
The average annual amount budgeted B by customer at ith region for a given
product or product class k
The expected average annual expenditures A for a given product
k(k=1,2,3….p) by the customers at ith region at facility location j
6.15.1 Huff model to locate retail outlet to maximize net operating
profits
Step 1: The probability p of a given alternative facility, location j being chosen from
among all alternatives n is proportional to its utility u j perceived by the customer
pj 
uj
such that
n
u
j 1
j
n
p
j 1
j
 1 and 0  p j  1
The randomness involved in choosing the location is due to the fact the customer may not
be able to discriminate among choices of facilities perfectly specially when the perceived
difference among alternative choices are small. Customers may also be uncertain as to true
conditions associated with the fulfillment of shopping expectations at various alternative
facilities. To capture this kind of randomness and to consider all the facilities it is always
better to utilize the concept of relative utility of a location and probability of choosing that
facility.
STEP 2: Utility of a shopping facility j perceived by the customer base i.
The customer is unaware about the fact that to what extent any facility will meet his or her
requirements. But a customer does have a prior knowledge of the probability that various
shopping facilities might satisfy his or her shopping demands. Moreover, greater the
number and variety of products carried by a facility, the greater is the customer’s
expectation that his visit to the facility will be successful. We can term this concept as the
size of facility that the customer will be attracted more for large size of facility S j
U ij  S j
The utility of a facility is also influenced by the effort and expenses incurred by the
customer for traveling to alternative facility location in the form of transportation costs.
Hence, utility for any facility is inversely proportional to the time spent in traveling from
customer base to the facility (dependent on the distance). At the same time we should
incorporate the differences in terms of customers’ willingness to travel to various distances
for different types of products, which can be accounted for with the distance exponent λ.
So , we can say that
U ij 
1
Tij 
U ij 
Sj
We can write
Tij 
The probability a customer from customer base i traveling to a given facility j can be
written as
pij 
U ij
n
U
j 1
ij
Sj
Tij 
or  n
Sj


T
ij
j 1
Step 3: Expected number of customer from region I to visit facility j
We can find the expected number of customers from a customer base region i visiting to
facility j, Eij, by multiplying the number of customers at I, Ci, with the probability that
customers from I will select j for shopping pij
Hence, Eij  pij .Ci
Step 4: Expected average annual expenditures by customers
The total sales for a given facility will depend also on the annual expenditures to be spent
at a facility for particular product. So, the expected average annual expenditures A for a
given product k (k=1, 2….p) by the customers from region i at facility j is determined as
Aijk  Eij .Bik
Where Bik is the average annual amount budgeted B by customers at i for a given product k.
Summing Aijk for all m customer regions for each product k will give the total expected
annual sales for a given facility. Deducting operating costs from sales revenues will give
total net profit.
Example:
A retailer wants to locate a supermarket in a city with 5 customer zones with different
demand densities located with coordinates give below in table. The distance is in
Kilometers.
TABLE 6.7: DEMAND DENSITIES OF ALL CUSTOMER ZONES
Customer
zones (i)
Coordinates of
customers zones (xi, yi)
Demand
density at i
1
(2,3)
7000
2
(1,4)
1000
3
(3,1)
2000
4
(3,5)
3000
5
(4,1)
5000
There is one competitor already located at (2, 2). Retailer wish to explore two more sites at
(3, 2) and at (3, 3) with double the capacity of competitor to be at (3, 3) with double the
capacity for supermarket.
Assume λ =2 for convenience stores. Let’s consider that retailer wants to maximize the sale
of women’s dress material for which customers can have annual amount of budget of Rs
500. Which site will fetch more expected sales and maximum net profits? Consider the
operating costs are double for two potential sites than the competitor. Locate the
coordinates of customer zones 1,…,5 as shown in Figure 6.23. The competitors location is
represented by A, R1 and R2 are two potential sites out of which the retailer has to choose
one. Retailer also wants to know whether locating at either R1 or R2 will be profitable over
competitor A.
FIGURE 6.23: LOCATION OF CUSTOMER ZONES
We can assume metropolitan metric for calculating distance. In metropolitan metric the
customer can take turns only at right angles (90o) only.
We will first determine the travel distance from each customer region A, R1 and R2 as
determined in Table 6.8
TABLE 6.8: TRAVEL DISTANCE FROM CUSTOMER REGION TO SITE
Travel distance in km(Tij)
Site(i,j)
Customer region(i)
1
2
3
4
5
R1
2
3
1
3
1
R2
1
2
2
2
3
A
1
2
2
4
3
Now we will find the utility Uij perceived by i=1...5 for facility j= R1, R2 and A as given in
table 6.9. The size of A will be taken as 1 and for R1 and for R2, it can be taken as 2.
TABLE 6.9: UTILITY PERCEIVED BY CUSTOMERS
Uij
Customer region(i)
Site(j)
1
R1(size=2) 2/22 = 0.5
2
3
4
5
2/32=0.22
2/12=2
2/32=0.22
2/22=0.5
2/22=0.5
2/22=0.5
2/32=0.22
R2(size=2)
2/12 =2
2/22=0.5
A(size=1)
1/12 =1
1/22=0.25 1/22=0.25 1/42=0.06 1/32=0.11
Total
3.5
0.97
2.75
0.78
0.83
We will use the utility to find the probability of a customer i travelling to R1, R2 and A as
given in Table 6.10.
TABLE 6.10: PROBABILITY OF CUSTOMERS TRAVELLING TO SITES
Site(j)
pij
Customer region(i)
1
R1
2
0.5/3.5=0.143 0.22/0.97=0.227
3
4
5
2/2.75
0.22/0.78
0.5/0.83
R2
2/3.5=0.571
0.5/0.97
0.5/2.75
0.5/0.78
0.22/0.83
A
1/3.5=0.285
0.25/0.97
0.25/2.75
0.06/0.78
0.11/0.83
Expected number of customers Eijwill be determined below in Table 6.11.
TABLE 6.11: EXPECTED NUMBER OF CUSTOMERS VISITING SITES
Eij
Site
(j)
Customer region(i)
1
2
3
4
5
R1
1001
227
1454
846
3010
R2
3997
515
364
1923
795
A
1995
258
182
231
665
For k to be women’s dress material monthly total expected revenues generated at R1, R2
and A are given in Table 6.12.
TABLE 6.12: MONTHLY TOTAL EXPECTED REVENUES FOR DIFFERENT
ITEMS
Site
(j)
Aijk(in Rs)
Monthly
Total
Customer region(i)
1
2
3
4
5
R1
100100
22700
145400
84600
301000
653800
R2
399700
51500
36400
192300
79500
759400
A
199500
25800
18200
23100
66500
333100
So, we can see that the location R2 (3, 3) will generate more revenues, hence retailer can
locate the supermarket at (3, 3).