MA424 Modeling in Operations Research
Final Project 2019-2020
Background
The Amazoff Company is a new e-commerce company that is trying to base on the UK. They heard
about the LSE MA424 course, and they reach out to you looking for advice in order to setup part of
the logistics plan of the company. The purpose of this project is to find a plan for the warehousing,
routing and e-commerce for a period of a year.
1 Warehousing
The company is unsure about where to open the warehouses around the UK. They have information
about the potential clients and their demand, but the cost of opening a facility depends heavily on the
city and other factors. Your first task is to help the Amazoff Company to take a decision on where to
open facilities and how to assign them to the clients. For that, generate randomly 1 15 points in the
two dimensional square [−1, 1] × [−1, 1] representing the potential locations F to open a new facility,
and generate other 60 points in the same region representing the clients C. For each pair of points
(c, f) ∈ C × F, the cost of assigning a client to a given facility is given by the `1 distance between c
and f. The opening cost of a facility f ∈ F is given by 100 · 3
−kfk1 units. Find a model to decide on
the optimal way of opening and assigning facilities to clients.
Capacitated Model. Now suppose that each facility f ∈ F has a capacity of 100 · 2
kfk1
, and suppose
that each client has a demand that is uniformly distributed between one and 25 units. Modify the
above model to find the optimal way of opening and assigning facilities to clients, and at the same
time respecting the facilities capacities and the clients demands.
Balanced Loads. The company would like to maintain under control the difference between the facility
facing the highest demand and the one facing the lowest demand, where the demand faced by a facility
is the total demand of the clients assigned to that facility. Propose a variant of the capacitated model
in order to handle this. State clearly the idea behind your constraints, as well as the formulation and
its correctness.
2 Vehicle Routing
Your second task is to design the routing plan for the Amazoff Company. Suppose that the company
has a depot at the point (1, 1) from where the vehicles start the dispatching operations. In principle
suppose the company has access to a single vehicle for the dispatching operation and consider only
the 25 clients with highest demand, that we call high demand clients in what follows. Design a model
that decides on how to perform the routing operation for the single vehicle, starting on the depot
and finishing there. Recall that the vehicle should deliver at every high demand client as many units
as the demand. Suppose that the cost of going from client c to client ˆc is given by the `2 distance
between them.
1You can generate an initial set of locations and keep them for the rest of the task, is not necessary to generate a
new set every time you solve it. For example, you can generate the client and facility locations en Excel or R.
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Multiple Vehicles. Suppose that now the company has access to three different vehicles to perform the
dispatching operation. Design a new model to decide the routing plan for the three vehicles. Compare
the total cost obtained with the one faced by having a single vehicle. As before, the three vehicles
should start and finish at the depot.
Multiple Vehicles and Warehousing. Suppose that now the company has access to one vehicle per
facility open according to the facility location plan designed before. Taking the solution of the facility
location problem as given, design a model that finds the routing of each vehicle. Recall that a vehicle
of a given open facility should visit every client that was assigned to the facility in the plan found
before.
3 Routing Special Orders
The company deals sometimes with special orders that are to be routed with priority. Consider two
new clients drawn randomly from the square [−1, 1] × [−1, 1] and suppose that the company has a
vehicle that departs from the depot. Your task is to find a way of visiting each of the two new clients
in the cheapest way, but it should be done in a way that the vehicle visits on the way some other
clients. Suppose that the cost faced by the vehicle between two clients is given by their `1 distance,
and same for the distance between the depot and any client.
4 Internships
The company is looking for interns to be part of the analytics group, and they are implementing a
new system to manage the applications. The company is offering ten internships, and every applicant
should declare their preferences at the moment of applying to the internships. Namely, each applicant
should provide a ranking over the positions, where to top is the most preferred one and the last is
the less preferred. On the company side, they also have preferences over the applicants, depending o
their skills and experience, among others. The goal is to find a matching between the internships and
the applicants, covering each internship position. They heard that the MA424 students are familiar
with stable matchings, and they would like to implement the assignment in this way. Your task is to
help them to find such type of allocation. For that, suppose that you have 30 applicants, and each of
them has a full ranking list over the internships. On the other side, suppose that the company has
a full ranking over the applicants for each internship. Design a linear program to find an allocation
fulfilling the above requirements. To construct the instances suppose that the ranking of each agent
involved is constructed uniformly at random.
5 Uncertainty on the Demand
The company has an analytics group that is looking for certain advice on the following situation. The
demand for the upcoming months is clearly uncertain, but they have some data, which they would like
to exploit. In particular, they were thinking about placing orders for a set of different products taking
into account the available information. They heard that you are familiar with certain techniques on
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optimization under uncertainty. Can you provide some advice on how to handle this situation? In
this part you have to propose a way of dealing with this problem, only with the following information
at hand: For each product you have 100 samples of the demand, and you know the unit cost at
the moment of placing an order, the holding cost and the selling price. Feel free to make further
assumptions, in which case give a brief explanation on how they play a role and help to improve your
model. Think about providing keywords as well, or helpful references on this part.
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Deliverables and Report Contents
The Amazoff Company wants you to develop models for their problems. You can use Excel@Risk
and AMPL to implement and solve these models, but you are free to use other software as well if you
consider them helpful for the project, such as R, Octave, Matlab, etc. If you feel confident enough
in other programming language such as Python or Julia you are also free to use them. For that you
would need to get an academic license for the Gurobi/Cplex solvers. 2
It is expected that you will
then analyze and discuss your findings on the basis of the questions outlined in the previous sections.
You must deliver a Project Report containing your analysis and suggestions but also detailing the
modeling that you have undertaken. The Project report should consist of the following (explained
below in more detail):
(a) A brief Executive Summary, discussing your main findings. It should be completely free of
any mathematical terms and discussion of modeling technicalities. It should describe the main
characteristics of the solutions obtained and answer all the questions.
(b) A concise Management Report discussing all of your findings. The Management Report should
be independent of the Executive Summary (i.e., self-contained). Ideally, the Management Report
will avoid the use of unnecessary modeling technicalities.
(c) A number of Technical Appendices. The technical appendices will give a complete account of
the modeling development. In particular, an appendix detailing the development of the optimisation
models, where all entities of your models are defined and their meaning is explained in detail,
together with the models clearly stated.
2For information about how to get it, send an email to v.verdugo@lse.ac.uk
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Guidelines
• The deadline for submitting the final project is 17 March 2020 16:00. Two hard copies of the
report should be submitted anonymously (with your exam candidate number). The electronic
copy of the report, the model files and any other computer files you create should be contained
in one .zip file, to be uploaded on the moodle MA424 page by the same deadline.
• You are allowed to work in groups of size at most three on the models and associated files.
• All components of the report must be written individually. This includes: the executive
summary, the management report, any mathematics with associated explanation, any discussion
of the relationship between the mathematics and the software used, any computer code and any
description of the software features used. This is not an exhaustive list. You may not seek
advice from anyone else other than your fellow MA424 students (provided you report this) and
the MA424 lecturer.
• You must provide candidate exam number of all other students who you worked together with,
and a brief statement on the extent of collaboration and the individual contributions. (e.g.,
“we prepared the basic formulation together, and both of us had about the same contribution”,
“we prepared the basic formulation together, but most constraints were developed by me”, “we
worked mostly separately, but she explained to me the constraint about the maximum distance”).
Common elements of work that are not declared will be considered as cases of plagiarism.
• The report without the Technical Appendices should not exceed 12 pages with 11pts, single
spacing. Concise and clear reports are preferred.
• Any project submitted without all associated computer files will be given a mark of zero.
• The MA424 lecturer is permitted to give you help with the implementation of your models but
not the formulation.
• Recommendation: Try to ask the questions about the project during the lecturer’s office hours,
not by email.
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Marking scheme
Modeling 30
Implementation 20
Analysis 30
Organization/Presentation 20
Total 100
• Modeling. The basic ingredients of the model are appropriately and meaningfully defined. The
essential decision variables for the model are introduced with suitable types. Constraints are
formulated correctly.
• Implementation. All scenarios and the extension are implemented in a sound manner. Data
is provided in an appropriate form and separated from the model. Further, demonstrating
sufficient capacity to developing “good” models: avoiding unnecessary variables and constraints,
compactness of data representation as well as the use of more advanced features for defining
model entities.
• Analysis. The report should sufficiently demonstrate the ability to recognise and report all key
results of the models and relate these to the real-world problem as well as demonstrating sufficient
understanding of these results with a discussion of requisite insight. Discussing limitations of
the model, the possible sources of these and how they may impact on the problem decision.
• Organization/Presentation. A main report and technical appendices which are clear, concise,
well organised, well formatted and well presented with appropriate use of figures and tables, well
commented and easy to navigate files.
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