Location

There are three plant location models. The first module is the standard qualitative/subjective weighting system. Several factors are identified that are considered to be important for the location decision. Weights are assigned to these factors and scores for these factors are determined for the various possible sites. The program computes the weighted sum of the scores (and identifies the site with the highest score).

The second and third methods are quantitative methods for location on a line (one dimensional) or a plane (two dimensional). In the one-dimensional case the coordinate or street number must be given, while in the two-dimensional case both a horizontal coordinate and a vertical coordinate must be given. In either case, the program will have a default weight of 1 trip per location, but this may be changed to reflect different numbers of trips or different weights of materials. The program will find the median location and the mean location and total weighted and unweighted distances from each location.

NOTE: Some of the location models called 1 and 2 dimensional location are also known as "center of gravity" models in some textbooks

The Qualitative (Weighting) Model

If the qualitative model is chosen, the general framework is given by the number of factors and the number of potential sites. In the screen following we show an example with seven factors and three potential sites.

Factor weights. Weights should be given for each factor. The weights can be given as whole numbers or fractions. Generally, weights sum to one or one hundred, but this is not a requirement.

Scores. The score of each city on each factor should be given.

Example 1 - Weighted location analysis

In the following screen we display a filled-in sample, along with the solution. Notice that the cities and the factors have been named.

The output is very straightforward and consists of the following.

Total weighted score. For each city, the weights are multiplied by the scores for each factor and summed. The total is printed at the bottom of each column. For example, the score for Philadelphia has been computed as

10*90 + 30*80 + 5*60 + 15*90 + 20*50 + 10*40 + 5*30 = 6500

which is listed at the bottom of the Philadelphia column.

The weighted average (total score/total weight) is also displayed for each location.

One Dimensional Siting

If one dimensional siting is chosen, the general framework consists of a column of weights or trips and a single coordinate or address column. The required information in order to get started is the number of sites to be included in the analysis.

The solution screen, which includes the data for a four site analysis, is given in the following screen.

The information to be filled in is:

Weight/trips. The weight or number of trips to and/or from each site. The default value is 1 for each location. This is what should be used when all customers/locations are considered to be equal. If more trips are made to one customer than another, this can be included in the weight/trip column. If the number of trips is the same but the weight of the materials differs, this should be included.

x coordinate. The coordinate of the locations must be given. This can be expressed in several different ways. These may be street addresses (on the same street since this is one dimensional), they may be floors in a building (it is possible for the dimension to go up instead of across), or they may be east-west or north-south coordinates where a negative number means west and a positive number means east or a negative number means south and a positive number means north.

Our sample problem with a solution appears above. The output again is very straightforward.

Total weight or number of trips. In order to find the mean or median location, it is necessary to determine the total number of trips or total weight. In the example, there are 13 total trips so the middle trip is the seventh.

The mean location. This is the location that minimizes the sum of the squares of the distances of the trips.

The median trip. The median trip is identified as trip number 7 and occurs from the location at 2800.

In general, an interesting question is whether a manager should minimize total distance or total distance squared. Notice in this example that one yields an answer at block 2800 and the other yields an answer at block 3100-3200.

Two Dimensional Siting

The information for two dimensional siting is analogous to the information required for one dimensional siting. Again, the only set-up information is the number of locations. The following screen contains the data and solution for a two-dimensional siting problem. The only difference between the data for one and two dimensional siting is the extra column that now appears for the second coordinate data. Data is to be entered in the same way as for one dimensional siting.

Example 3 - Two dimensional location

The solution screen for two dimensional siting has a large amount of information as exhibited in the preceding screen. Some of the information is the same as that of one dimensional siting and some is extra.

Weighted X-coordinate This is simply the coordinate multiplied by the number of trips. In the example, the number of trips is positive for each of the first five locations but 0 for the last two. The multiplications by the weights can be seen.

Weighted Y-Coordinate. This is identical to the previous column except that it is the y coordinate that is multiplied. These weighted columns demonstrate the computations that lead to the answers below the data. The average of these columns are the answers. Notice in this example that dividing by 7 yields the first ( unweighted ) average and dividing by the sum of the weights, which is 320, yields the second (weighted) average.

Median. The median trip is 160 (there is no 160.5) and the median x coordinate is

132 while the median y coordinate is 75.

Averages. The unweighted and weighted averages of the coordinates are displayed.

Table of distances

Tables of distances from point to point can be displayed.

Total Distance. The row named 'Total' contains the total distance from every site to this site. The means for computing the distance depends on the table (air distance vs. city block distance). The number 997.665 for raw material 1 means that the site of raw material 1 at coordinates 132, 123 has a total distance of 997.67 from each of the other six sites. That is, if you made one trip from each of the six sites to the site of raw material 1, it would cover a distance of 997.67. Another way to view this column is to say that potential site 1 is more central than potential site 2 because it has a distance of 921.71, which is the smaller than the 1018.99 of potential site 2.

Weighted total. The numbers in the distance row do not take into account that different numbers of trips are made between points or that different amounts of material are moved between points. This column multiplies the distance times the number of trips or amount of materials moved. Again, though, potential site 1 seems to have the advantage over potential site 2.

NOTE: It is possible, and maybe even useful, to solve the one dimensional problem by using the two dimensional model with one coordinate equal to 0 for all sites.