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Business Decision Analytics under Uncertainty - Fall 2022

Assignment 1

Please show your entire work with brief, but sufficiently detailed explanation. You can refer to the lectures to support your answer. You can use a graphing calculator or computer software to visualize parts of the question when applicable. Please submit your work on Canvas as a Word document; handwritten answers are not accepted. To help grading and adding comments, do not convert your work into a pdf file[s].

Question 1 (40 points)

You assist your client to select a location x = (x1 , x2) for a service facility that will serve K = 50 customers by providing a single (identical) service or commodity to each customer. The facility can be located anywhere within the unit square 0 < x1 < 1, 0 < x2 < 1 (the square can model e.g., a 10 km by 10 km rectangular region scaled to the unit square). The customers aremodeled as points pk = (pk1 , pk2) fork = 1, … , K located within the unit square. Each customer’s yearly demand for the service is assumed to be a known value: we also assume that all demands are satisfied. However, customers can have different relative “weights”, proportionate to the size of their yearly demand. This aspect is expressed by assigning weights wk fork = 1, … , K to the customers. To illustrate this problem, please see the figure below that shows the unit square (blue), a possible (but not optimized) location for the facility (black dot), and the locations of the “weighted” customers (red dots of radius wk  fork = 1, … , K).

Assume that the distance between the facility location x and customer k (point pk) is expressed by the so-

called Manhattan (l1-norm) distance function defined by

d(x, pk) = |x1  - pk1| + |x2  - pk2 |.

This notion corresponds to reaching the facility on a rectangular network of streets or roads. Formulate a decision model that optimizes the location of the facility. The quality of a location is expressed by the weighted sum of all Manhattan distances between the facility and the customers.

You can find a lot of Internet (and printed) literature on this important type of problem. Happy browsing!

See page 2 for Questions 2 to 4.

Question 2 (20 points)

Determine the convexity properties of your facility location model. Based on the discussion in the topical lectures, state whether this facility location problem is expected to be “easy” or “hard” to solve.

Question 3 (20 points)

Assume now that the regions |x1  – x2 | > 0.3, x1  + x2  > 1.5, x12  – x2  + 0.4 < 0, and x12 + 3 x22 < 0.5 must be excluded from consideration for the possible location of the facility. Compared to the answer to Question 2, state whether this facility location problem is expected to be “easier” or “harder” to solve.

Question 4 (20 points)

Suggest an initial solution that is likely to be a good “guess” of the solution to this facility location problem. Briefly explain your choice.