A crossdock is being planned that will feed parts to three manufacturing facilities. The locations of the current plants with their coordinates and...

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The centroid method for such a case states that to find the optimal coordinates of a crossdock we have to calculate the weighed sums of the coordinates of the current plants. The weights mentioned are the volume requirements.

The formulas are  `x=(sum_(k=1)^n V_k*x_k)/(sum_(k=1)^n V_k)`  and  `y=(sum_(k=1)^n V_k*y_k)/(sum_(k=1)^n V_k),`

where `n=3,`  `x_k` and `y_k` are the coordinates given and `V_k` are the volume requirements.

Compute  `sum_(k=1)^n V_k` first, it is `5000+7000+1000=13000.`

Then  `sum_(k=1)^n V_k*x_k = 5000*55+7000*37+1000*80=614000`  

and  `sum_(k=1)^n V_k*y_k = 5000*20+7000*40+1000*90=470000.`

Therefore the coordinates of a new crossdock are

`x=614000/13000 approx 47`  and  `y=470000/13000 approx 36.`

That said, this method isn't correct even when its assumptions are held. As shown in the article (see the link), this method actually minimizes the sum of squares of shipping costs, not the sum of shipping costs itself.