Posted: October 2nd, 2022

Business Decision Analytics under Uncertainty

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 discussion. You can use a graphing calculator or computer software to visualize the question
when applicable. Submit your work on Canvas as a single 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) commodity to each customer. The facility can be located anywhere within
the unit square 0  x1  1, 0  x2  1 (which can model e.g., a 10 km by 10 km rectangular region scaled to
the unit square). The customers are modeled as points pk = (pk1 , pk2) for k = 1, … , K located within the
unit square. Each customer’s yearly demand is assumed to be a known value, and we assume that all
demands are satisfied. However, customers are assumed to be of different relative “weight”, proportionate
to the size of their yearly demand for the commodity. This aspect is expressed by assigning weights wk for
k = 1, … , K to the customers. To illustrate the 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 for k = 1, … , K).
Assume that the distance between the facility location x and customer pk is expressed by the so-called
Manhattan (l1-norm) distance function
d(x, pk) = |x1 – pk1| + | x2 – pk2|.
Formulate a decision model that optimizes the location of the facility. The quality of a location is expressed
by the weighted sum of the Manhattan distances between the facility and the customers.
See page 2 for Questions 2 to 4.
Question 2 (20 points)
Determine the convexity / nonconvexity properties of your facility location problem. Based on the
discussion presented in the 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, x1
2 – x2 + 0.4 < 0, and x1
2 + 3 x2
2 < 0.5 must be
excluded from consideration for the possible location of the facility. Compared to the answer to 1.2, 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.