One of your friends knows that you are taking SC2x and that you learned how to calculate, for a given group of cities with their Cartesian coordinates, a location for a distribution center that would minimize the distance. "Maybe you can help me", he says. "I would like to calculate the coordinates for a distribution center, so that we minimize its distance to ten cities in Panama."
Your friend provides you with a list of ten major cities in Panama that serve as political capital of their respective provinces or regions. You
find a map of Panama online, and use an improvised grid to assign horizontal and vertical coordinates to the cities. These coordinates,
shown in the table below, use as origin a point in the top left corner of the map. You tried to make these units as close to kilometers as
possible. However, you prefer to think of them as relative coordinates on an arbitrary Euclidean grid.
"Do you have weights for these cities?", you ask your friend.
"No", he says, "but let's assume they all have equal weight, a weight of 1 for example".
Part 0:-
Giving each city equal weight, calculate the location for the distribution center that would minimize the distance to the ten cities
provided. Solve this as a continuous, distance-minimizing, location problem with equal weights for all cities.
Part 1:-
For the location whose coordinates you calculated above:
1. What is the exact horizontal coordinate?
2. What is the exact vertical coordinate?
Part 2:-
You ask your friend what he is working on, and he explains that he is volunteering for an organization that will soon distribute backpacks to
every school-aged child throughout Panama. You suggest that solving this problems with equal weights is not good enough, since some
cities surely have more schoolchildren than others. A quick online search provides you with approximate values for the populations of
the different provinces and regions where the cities are located. Using these population values, you propose to your friend using weight
values, as listed in the table below.
Using these weights, solve this as a continuous, demand-weighted distance-minimizing, location (or Weber) problem.
1. What is the horizontal coordinate?
2. What is the vertical coordinate?
Part 3:-
You decide to estimate the distance of distributing the backpacks from a DC in one city to the school children in another city. Given
that Panama is a small country and you only need a quick and dirty estimate, you assume an Euclidean space and a circuity factor of
1.25, which gives you the distances in the table below (where cities are identified using the first three letters in their name). Regardless
of which city the DC is located in, there will also be transportation from the DC to the school children within that same city. For
distances inside each city, you assume 5 distance units, except for Panama City, for which you use 10 distance units since it is a larger
city. The distances between cities are given in your arbitrary grid units, which may or may not be kilometers. You can download the
table in Excel and in LibreOffice.
After thinking about it for a while, your friend tells you that they may want to open two DCs. He wants these DCs to be located in two of
the ten cities from the list above, but they should not be located in the same city. The plan would be to distribute all the hundreds of
thousands of backpacks through these two DCs, using trucks. Let's assume that it is possible to open a DC in any of the ten given cities,
that the cost of opening and operating such a DC in any of these cities would be the about the same, and that the cost of outbound
transportation from the DC to the cities is about the same on a per unit per mile basis.
In which two cities would you recommend that they open these two DCs, in order to minimize the outbound transportation costs? Select the answer options that apply. Use the same demand weights as in Part 2. (Hint: If you are puzzled by the fact that we are asking for
this without providing any costs for DCs or trucks, you should start playing with different values and see how the model behaves.)
Part 4:-
You make some phone calls to carriers in Panama, and you calculate an average transportation cost of $1 per distance unit per 968 backpacks (e.g. you would pay $100 to move 968 backpacks 100 distance units). You also estimate that the fixed cost of operating each distribution center in one of these cities is $24,000 dollars per year. The yearly demands in the cities are approximated by the weights used in previous parts.
Assuming that these values are correct and applicable throughout the country, how many distribution centers would you recommend
opening?
Where would you recommend that they open these DCs, in order to minimize the outbound transportation costs?
Attachment:- SC2X.rar