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Business Decision Analytics under Uncertainty

Assignment 1

Please show your entire work with brief, but sufficiently detailed explanation in a Word document. Start

your answer by typing your name, RUID and email address. You can refer to the course lectures to support

your answer. You can also use a graphing calculator or other computer software to visualize parts of the

questions 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 new service facility that will serve K = 50

customers by providing a single (identical type) service or commodity to each customer. As an example,

you can think of a centralized warehouse serving customers within a region.

The new facility can be located anywhere within the unit square : the square models a

100 km by 100 km rectangular region. The customers are modeled by points pk = (pk1 , pk2), k = 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 must be satisfied. Customers can have different relative weights,

proportionate to the size of their yearly demand. This aspect is expressed by assigning weights wk ≥ 0 to

each customer for k = 1, … , K. To illustrate the problem-type considered, 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 (displayed by red dots of radius wk for k = 1, … , K).

Assume that the distance between the facility location x and customer k (point pk) is expressed by the socalled Manhattan (l1-norm) distance function defined by

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

This definition corresponds to reaching the facility from a customer location pk by using 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 k = 1, … , K.

You can find a lot of Internet and printed literature on this important type of problem. You can do this

research at your discretion, it is not required to answer the assignment questions.

Question 2 (20 points)

Determine the convexity properties of your facility location model. Based on the discussion in the course

lectures, state whether this facility location problem-type 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 within the

unit square 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)

Propose an initial facility location that is likely to be a good “guess” of the solution to the problem. Briefly

explain your choice.

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