Optimization and Simulation for Business
Second Midterm Exam 4/23/2024
Year 2023/24 Exam duration: 1 h 30 min
1. (2 points) Consider the following unconstrained nonlinear optimization problem
min f (x) ≡ 2x21 − 3x1 x2 + 3x32 − 6x1 x3 + 6x23 .
x
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(a) (0.5 points) Study if the point x∗ = 3 1 3/2 is a local minimizer for this problem by chacking
its necessary and sufficient optimality conditions.
(b) (1 point) Find all possible local minimizers for this problem, based on its necessary optimality
conditions.
(c) (0.5 points) Is this problem convex? Why?
2. (2.5 points) Given a nonlinear optimization problem defined as:
minx ax21 + 2x1 − 4x1 x2 + 2x22 + 4x2
s.t. 2x1 + x2 = 3
x1 − 2x2 ≤ 1.
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(a) (1 point) Determine if there exists a value for the parameter a such that the point x̂ = 1 1
satisfies the first-order necessary optimality conditions for the problem.
(b) (1.5 points) For a = 2, find all possible stationary points (points that satisfy the first-order necessary
conditions) for this problem.
3. (5.5 points) One of the tasks to be conducted in the management of an airport is the assignment of
boarding gates to flights. You are asked to help with the assignments of time slots at one gate to four
flights during a period of time of four hours. To carry out this assignment you must take into account the
following considerations:
• The values you need to determine are the moments from which the gate can be used by each one
of the flights; we will denote these times as ti , i = 1, 2, 3, 4. We will set the fixed value t1 = 0 as
our reference. The starting times for the following flights are chosen based on that time as their
reference. These times are measured in minutes.
• The minimum time between the starting times for successive flights is 30 minutes. That is, between
t2 and t1 there must be a difference of at least 30 minutes, and the same condition must hold between
t3 and t2 as well as between t4 and t3 .
• If the assigned times for successive flights is short, it is possible that any successive flights may be
affected by unplanned delays. It has been estimated that the average costs per passenger associated
to these delays for i = 1, 2, 3 (for the different flights, but note that these delays affect the following
assignments) are given by the function
c(s, t) = exp(−0.005(s − t)),
where s and t denote the times assigned to two consecutive flights and s > t. Applying this cost
function, the costs per passenger for each one of the flights will be given by c1 = exp(−0.005(t2 − t1 )),
c2 = exp(−0.005(t3 − t2 )), c3 = exp(−0.005(t4 − t3 )). Our model does not take into account the
delays suffered by the passengers in the first flight, as they would be caused by factors outside our
planning window.
The numbers of passengers estimated for each of the flights are:
Flight 2 3 4
Passengers 200 100 250
• If there is time left unassigned in the four hours available for these assignments, this time could be
assigned to other flights, or used for maintenance work, etc. Thus, this unused time has a value for
the airport; this value has been estimated to be v(t) = 0.8(270 − t), where t denotes the last assigned
time, t = t4 .
1
For example, the assignment t1 = 0, t2 = 35, t3 = 120, t4 = 210 is a valid assignment, as all the time
differences are larger than 30 minutes and the last one is smaller than 240 minutes. The costs per passenger
associated to each one of the flights will be given by c1 = exp(−0.005(35−0)), c2 = exp(−0.005(120−35)),
c3 = exp(−0.005(210 − 120)), and v = 0.8(270 − 210).
You are asked to determine the best possible values for the initial times assigned to each of the flights
for their access to the gate. Your goal would be to have the lowest possible total cost associated to the
possible delays, subtracting the value of the unassigned time at the end of the period.
(a) (2 points) Formulate an optimization problem that would find the best assignment of starting times
to each of the flights.
This question can be answered in writing. In this case you should upload a photo of your written
answers to Aula Global.
(b) (1.5 points) Introduce the preceding formulation in Excel and use the Solver to compute the optimal
time assignments. Interpret the solution that you obtain.
Upload the Excel file with your model and solution to Aula Global.
(c) (2 points) Verify that the first-order necessary optimality conditions are satisfied by the solution you
have obtained in the preceding question: i) Indicate the expressions for the optimality conditions for
the problem, ii) verify that these conditions are satisfied at the preceding solution, and iii) interpret
the value of the multipliers of the constraints at the solution of the problem.
You may obtain the optimal values of the multipliers from the sensitivity report generated by the
Solver.
This question can be answered in writing. In this case you should upload a photo of your written
answers to Aula Global.
