The document describes the Simple Median Model (SMM) for locating a new plant to minimize transportation costs between the plant and four existing facilities (F1-F4). The SMM considers loads transported in east-west and north-south directions only. It finds the optimal location in three steps: 1) identify the median load amount, 2) find the x-coordinate receiving this median load, 3) find the y-coordinate receiving this median load. Applying this to the four facilities, the model determines the new plant's optimal location is x=20, y=40, located between existing facilities F1 and F2.

1. LOCATION MODEL
SIMPLE MEDIAN MODEL
BY
JOSEPH GEORGE KONNULLY

2. SIMPLE MEDIAN MODEL
Suppose we want to
locate a new plant that
will annually receive
shipments of raw
materials from two
sources: F1 and F2. The
plant will create finished
goods that must be
shipped to two
distribution
warehouses, F3 and F4.
Given these four facilities
(Figure 2.1), where
should we locate the new
plant to
minimize annual
transportation costs for
this network of facilities?

3. THE SMM MODEL
The simple median model (SMM) can help answer this question.
• This model considers the volume of loads transported on rectangular paths.
• All movements are made in east-west or north-south directions;
• diagonal moves are not considered.
• SMM provides an optimal solution.
• this is discussed with the help of Figure 2.1 and the Table 2.2.
Let Li = Loads to be shipped annually between each existing facility Fi, a
Ci = Cost to move a load one distance unit to or from Fi.
Di = Distance units between facility Fi and the new plant.
Then, the total transit cost is the sum of the products CiLiDi for all i.

4. Existing Plant F1 F2 F3 F4
Location (X,Y) of Existing Plants (20,30) (10,40) (30,.50) (40,60)
Loads to be shifted from new 755 900 450 500
plant to existing plants, Li
TABLE 2.2

5. STEPS IN SMM MODEL
The SMM model consists in finding the X0,Y0
co-ordinates of the new plant that result in
minimum transportation costs. We follow
three steps:
1. Identify the median value of the Loads
moved
2.Find the the x-co-ordinate of the new
facility that sends or receives the median
load
3. Find the y-coordinate of the new facility
that sends or receives the median load
The x,y co-ordinate found in steps 2 and 3
define the new plants location.

6. IDENTIFY THE MEDIAN LOAD
• The total number of loads moved to and from the new plant
will be ∑ Li =755+900+450+500 = 2605.
• If we think of each load individually and number them
from 1 to 2605, then the median load number is the “
middle” number- that is , number for which the same
number of loads falls above and below.
For 2605 loads the median load number
is 1303 .
• If the total number of loads were even we consider both
‘middle ‘ numbers.

7. FIND THE X-COORDINATE OF MEDIAN LOAD
• First we consider movement of loads in the x-direction.
• Beginning at the origin of Figure 2.1 and moving to the right
along the x-axis, observe the number of loads moved to or from
existing facilities.
• Loads 1-900 are shipped by F2 from location x = 10. Loads
• 901-1,655 are shipped by F1 from x = 20.
• Since the median load falls in the interval 901-1,655, x = 20
is the desired x-coordinate location for the new plant.

8. FIND THE Y-CO-RDINATE OG MEDIAN LOAD
Now consider the y-direction of load movements.
• Begin at the origin of Figure 2.1 and move upward along the y-
axis.
• Movements in the y direction begin with
loads 1-755 being shipped by F1 from location y = 30.
• Loads 756-1,655 are shipped by F2 from location y = 40.
• Since the median load falls, in the interval 756-1,655, y = 40 is the
desired y- c oordinate for the new plant.
• The optimal plant location, x = 20 and y = 40, results in minimizing
annual transportation