Question 4 of 6: Facility Location by Weighted Factor Rating

Question 4: Facility Location by Weighted Factor Rating (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Three candidate locations whose quantitative costs are already known to be roughly equal, rated by the site selection committee against five qualitative factors on a 5-point scale, with the factor weights shown.

Find. The composite weighted score of each location, the recommended site, and how robust that recommendation is to the weights the committee chose.

Approach. Form the weighted sum of the factor scores for each location, compare, and then test the margin by asking how far the weights would have to move before the ranking changes — because the weights are the softest input in the whole calculation.

1. Confirm the weights are a valid weighting. $$\sum_k w_k = 0.20 + 0.30 + 0.25 + 0.15 + 0.10 = 1.00$$ so the composite scores will land on the same 1-to-5 scale as the individual ratings and can be compared directly against the verbal anchors of that scale.
2. Compute the composite score of each location. The composite is $$V_j = \sum_{k} w_k\,s_{jk}$$ with $w_k$ the weight of factor $k$ and $s_{jk}$ the committee's score for location $j$ on that factor. For location A, $$V_A = 0.20(5) + 0.30(3) + 0.25(5) + 0.15(4) + 0.10(5) = 1.00 + 0.90 + 1.25 + 0.60 + 0.50$$ and evaluating this and the two companion sums, $$\boxed{V_A = 4.25, \qquad V_B = 3.75, \qquad V_C = 3.95}$$
3. Select the site. Location A has the highest composite score and is the recommendation. Reading down the stacked bars in the figure shows where the margin comes from: A wins on labor supply, supporting services and community attitude, and loses only on labor relations, where it scores 3 against C's 5. Since the transportation and processing costs were stated to be roughly equal, this qualitative composite is the deciding criterion and no further cost trade-off is needed.
4. Test how safe the recommendation is. The margin over the runner-up is $$V_A - V_C = 4.25 - 3.95 = 0.30$$ on a five-point scale, which is 7.5 per cent of the range — a real but not overwhelming lead. The obvious challenge is that labor relations, where A is weakest, already carries the largest weight; suppose the committee decided it should carry more, taking the increment from supporting services, where A is strongest. Shifting a weight $\delta$ from supporting services to labor relations changes the two composites by $$\Delta V_A = (3-5)\delta = -2\delta, \qquad \Delta V_C = (5-3)\delta = +2\delta$$ so the 0.30 gap closes at a rate of $4\delta$ and the two tie at $$\boxed{\delta = \frac{0.30}{4} = 0.075 \quad\Rightarrow\quad w_{\text{labor relations}} = 0.375,\ w_{\text{supporting services}} = 0.175,\ V_A = V_C = 4.10}$$
5. State the recommendation with its qualification. Location A is selected, but the ranking survives only a 0.075 shift of weight from supporting services to labor relations — a re-weighting the committee could plausibly make. The professional answer therefore recommends A and asks the committee to confirm that labor relations really is worth no more than 0.30, since that single judgment, not the arithmetic, decides the outcome. Locations B and C should not be released until it is confirmed.