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Question:Santa Cruz Bottling is a manufacturer of organic soft drinks on the coast of central California. Its products are enjoying a growing reputation and increased demand throughout the American Southwest. Because of the high cost of transporting soft drinks, it is considering a new plant to serve the states of New Mexico and Arizona. A key concern in its search for a new location is the resultant transportation costs to serve its key markets. Following is a list of cities where its main wholesale customers are located, along with estimated annual demand in cases of product for each.

City	x-coordinate	y-coordinate	No. of cases
Phoenix	250	250	25,000
Tucson	350	125	20,000
Albuquerque	800	450	28,000
Santa Fe	850	520	17,000

a. Use the centroid method to recommend a location for the new bottling plant. Round your coordinates to one decimal place.

b. Do you have any concerns about the result? How would you deal with them?

Step by step solution

01

(a) Calculate the location with the help of the centroid method

$$\begin{gathered} C_x=\frac{\underset{}{∑}{d}_{ix}{V}_i}{\underset{}{∑}{V}_i} \\ {C}_y=\frac{\underset{}{∑}{d}_{iy}{V}_i}{\underset{}{∑}{V}_i} \\ \text{±Ê³Ü³Ù³Ù³ó±ð²õ±ð±¹²¹±ô³Ü±ð²õ,¾±²Ô³Ù³ó±ð²¹²ú´Ç±¹±ð´Ú´Ç°ù³¾³Ü±ô²¹â€“} \\ {C}_x=\frac{(25,000×250)+(20000×350)+(28000×800)+(17000×850)}{25000+20000+28000+17000} \\ =\frac{625000+7000000+22400000+14450000}{90000} \\ =\frac{50100000}{90000} \\ =556.7 \\ {C}_y=\frac{(25,000×250)+(20000×125)+(28000×450)+(17000×520)}{25000+20000+28000+17000} \\ =\frac{625000+2500000+12600000+8840000}{90000} \\ =\frac{30190000}{90000} \\ =335.4 \end{gathered}$$

02

Concern about the result.

From the above calculations, thex, and y coordinates for a new bottling do not lie near a single city. The company needs to establish a new bottling plant between City T and City A.

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