LOCATION MODEL

SIMPLE MEDIAN MODEL BY

JOSEPH GEORGE KONNULLY

SIMPLE MEDIAN MODEL

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

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.

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 plant to existing plants, Li

755 900 450 500

TABLE 2.2

STEPS IN SMM MODELThe 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 load3. Find the y-coordinate of the new facility that sends or receives the median loadThe x,y co-ordinate found in steps 2 and 3 define the new plants location.

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.

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.

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