# A Multi attribute Plant Location Model

A model that attempts to deal with the multidimensional location problem can be developed based on multi-attribute preference theory. One such model has been developed by Brown and Gibson (1972). We present here a modification of their model.

Classification of Criteria:

The model deals with any list of criteria set by management but classifies them as follows:

1) Critical: Criteria are critical if their nature may preclude the location of a plant at a particular site, regardless of other conditions that might exist. For example, a water oriented enterprise, such as a brewery, would not consider a site where a water shortage was a possibility. An energy oriented enterprise, such as an aluminum smelting plant, would not consider sites where low cost and plentiful electrical energy was not available. Critical factors have the effect of eliminating sites from consideration.
2) Objective: Criteria that can be evaluated in monetary terms, such as labor, raw material, utilizes, and taxes, are considered objective. A factor can be both objective and critical; for example, the adequacy of labor would be a critical factor, whereas labor cost would be an objective factor.
3) Subjective: Criteria characterized by a qualitative type of measurement are considered subjective. For example, the nature of community support may be evaluated, but its monetary equivalent cannot be established. Again, criteria can be classified as both critical and subjective.

Model Structure:

For each site, i, a location measure, LM, is defined that reflects the relative values for each criterion.

LMi = CFMi x [X x OFMi + ( 1 – x) x SFMi] —————eq 1

Where

CFMi = The critical factor measure for site i
( CFMi = 0 or 1)

OFMi = The objective factor measure for site i
( 0 ≤ OFMi ≤ 1)

SFMi = The subjective factor measure for site i
(0 ≤ SFMi ≤ 1)

X= The objective factor decision weight
( 0 ≤ X ≤ 1)
The critical factor measure CFMi is the product of the individual critical factor indexes for site i with respect to critical factor j. The critical index for each site is either 0 or 1, depending on whether the site has an adequacy of the factor or not. If any critical factor index is 0, then CFMi and the overall location measure LMi are also 0. Site i would therefore be eliminated from consideration.

The objective factor measure for site i, OFMi , in terms of the objective factor costs, OFCi is defined as follows:

OFMi = Maximum OFC – OFCi / Maximum OFC – Minimum OFC –eq 2

The effect of Equation 2 is that the site with the minimum cost will have OFMi =1, and the site with the maximum cost will have OFMi = 0.The sites with intermediate levels of cost are assigned OFMi values by the linear interpolation. For example, site 3 in Table will have

OFM3 = 5.5 – 4.1 / 5.5 — 3 = 5.5 – 4.1 / 5.5 – 3

= 1.4 / 2.5 = 0.56

The subjective factor measures, SFMi for each site is influenced by the relative weight of each subjective factor and the evaluation of site i relative to all other sites for each of the subjective factors. This results in the following statement:

SFMi = Σk ( SFWk x SWik) ————–eq4

Where

SFWk = The weight of subjective factor k relative to all subjective factors

SWik = The evaluation of site i relative to all potential sites for subjective factor k.

Note that Σ SFWk =1,0 ≤ SWik ≤ 1 and SWik for the site with the best score or performance on the subjective factor k is 1 and for the site with the worst score or performance on the subjective factor k is 0. a method to obtain SWik is simply to use a rating system by which the sites are rated on a 0 to 1 scale with respect to their relative desirability on the subjective factor k.

Finally, The objective factor decision weight, X, must be determined. This factor establishes the relative importance of the objective and subjective factors in the overall location problem. The decision is commonly based on managerial judgment that requires a careful analysis of the trade off between cost and the combined effect of the subjective factors.