fi is given zero for Part IFi

Problem 2: Intergalactic Facility Location (35 pts) It is the year 160488 AD, an intergalactic company wants to open a set of factories on super- planets to supply the four dark galaxies with enriched sunlight. Three super-planets have the astrophysical characteristics and local resources to produce the required enriched sunlight. We assume that the cost to stream enriched sunlight between a super-planet and a galaxy is $1,500 per light-year. The distance (in light-years) between each galaxy and super-planet is given in Table 1. Galaxy 1 Galaxy 2 Galaxy 3 Galaxy 4 Galaxy 5 30 20 35 25 30 Super-planet 1 40 50 45 10 20 Super-planet 2 15 25 25 30 30 Super-planet 3 Table 1 Distance matrix (in light-years) Assume that there is no restriction on the amount of enriched sunlight that can be supplied by a super-planet. Let S be the set of super-planets and let G be the set of galaxies. We consider the following formulation for the Uncapacitated Facility Location model: min XcijXij +X fiyi (1) IES JEG IES Subject to: X xij = 1 VjEG (2) ies Xij Syi Xij binary Yi binary Vi ES, VEG Vi ES,VL EG Vies (3) (4) (5) Part I: Assume that there is no cost to open a factory on a super-planet, i.e. fi = 0, Vi E S. a. On which super-planet(s) should the company open a factory? (5 Points) b. Which super-planet will supply which galaxy? (6 Points) c. What is the total operations cost? (4 Points) Part II: Assume now that the cost of opening a factory is the same on every super-planet and is equal to $25,000. a. What is the new cost of the solution obtained in Part I.c? (5 Points) b. Are there cheaper solutions? Yes/No? If yes, give one cheaper solution and its total operations cost. (10 Points) Part III: After a series of business negotiations, the company has been able to obtain a tremendous discount of the opening cost of super-planet 2: the discounted opening cost is now $10,000 for this super-planet only. Can the company use this discount to further reduce its operations cost? (5 Points)