LOCATION MODELLING AND ECONOMICS

1. SIMPLE MEDIAN MODELSuppose 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 ModelThe simple median model (SMM) can help answer this question. This model considers thevolume of loads transported on rectangular paths. All movements are made in east-west or north-southdirections; diagonal moves are not considered. The SMM provides an optimal solution. This is discussedwith the help of Figure 2.1 and the Table 2.2.Let Li = Loads to be shipped annually between each existing facility Fi, andCi = 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.

A new plant has to be located which will supply raw materials to a set of existing plants in a group of companies. There are four existing plants which require material movement with new plant. The material transported (in tons) per year from new plant to existing plants are as given below. Determine the optimum location for the new plant such that the distance move is minimum.

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

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.

(i) 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.

(ii) Find the x – coordinate of the 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 ofloads moved to or from existing facilities. Loads 1-900 are shipped by F2 from location x = 10. Loads901-1,655 are shipped by F1 from x = 20. Since the median load falls in the interval 901-1,655, x = 20is the desired x-coordinate location for the new plant.

(ii) Find the y – coordinate of the median load

Now consider the y-direction of load movements. Begin atthe origin of Figure 2.1 and move upward along the y-axis. Movements in the y direction begin withloads 1-755 being shipped by F1 from location y = 30. Loads 756-1,655 are shipped by F2 fromlocation y = 40. Since the median load falls, in the interval 756-1,655, y = 40 is the desired ycoordinatefor the new plant.• The optimal plant location, x = 20 and y = 40, results in minimizing annual transportation costs for thisnetwork of facilities. The calculation is shown in Table 2.4.Remarks:• First, we have considered the case in which only one new facility is to be added.• Second, we have assumed that any point in x-y coordinate system is an eligible point for locating thenew facility. The model does not consider road availability, physical terrain, population densities, orany other considerations.

2. Factor Rating MethodThe process of selecting a new facility location involves a series of following steps:1. Identify the important location factors.2. Rate each factor according to its relative importance, i.e., higher the ratings is indicativeof prominent factor.3. Assign each location according to the merits of the location for each factor.4. Calculate the rating for each location by multiplying factor assigned to each location withbasic factors considered.5. Find the sum of product calculated for each factor and select best location having highesttotal score.ILLUSTRATION 1: Let us assume that a new medical facility, Health-care, is to belocated in Delhi. The location factors, factor rating and scores for two potential sites areshown in the following table. Which is the best location based on factor rating method?

The total score for location 2 is higher than that of location 1. Hence location 2, is the bestchoice.

3..Break-even AnalysisBreak even analysis implies that at some point in the operations, total revenue equals total cost.Break even analysis is concerned with finding the point at which revenues and costs agreeexactly. It is called ‘Break-even Point’. The Fig. 4.3 portrays the Break Even Chart:Break even point is the volume of output at which neither a profit is made nor a loss is incurred.

Plotting the break even chart for each location can make economic comparisons of locations.This will be helpful in identifying the range of production volume over which location can beselected.

ILLUSTRATION 5: Potential locations X, Y and Z have the cost structures shown below.The ABC company has a demand of 1,30,000 units of a new product. Three potentiallocations X, Y and Z having following cost structures shown are available. Select whichlocation is to be selected and also identify the volume ranges where each location is suited?

LOCATION X LOCATION Y LOCATION Z

FIXED COST Rs. 150,000 Rs. 350,00 Rs. 950,000

VARIABLE COST Rs. 10 Rs. 8 Rs. 6

SOLUTION: Solve for the crossover between X and Y:10X + 150,000 = 8X + 350,0002X = 200,000X = 100,000 unitsSolve for the crossover between Y and Z:8X + 350,000 = 6X + 950,0002X = 600,000X = 300,000 unitsTherefore, at a volume of 1,30,000 units, Y is the appropriate strategy.

From the graph (Fig. 4.4) we can interpret that location X is suitable up to 100,000 units,location Y is suitable up to between 100,000 to 300,000 units and location Z is suitable if thedemand is more than 300,000 units.

LOCATIONAL ECONOMICS

An ideal location is one which results in lowest production cost and least distribution cost per unit.These costs are influenced by a number of factors as discussed earlier. The various costs whichdecide locational economy are those of land, building, equipment, labour, material, etc. Other factorslike community attitude, community facilities and housing facilities will also influence the selectionof best location. Economic analysis is carried out to decide as to which locate best location.The following illustration will clarify the method of evaluation of best layout selection.ILLUSTRATION 6: From the following data select the most advantageous location forsetting a plant for making transistor radios.