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# Case 2: Optimizing network design to minimize costs
After successfully finishing your first engagement with Beanie Limited, Elisa
has praised you enthusiastically in conversations with Charles Murray, Beanie
Limited's COO (Chief Operations Officer). Charles is responsible for the entire
manufacturing and supply chain organization of Beanie Limited in Europe and
reports directly to Beanie Limited's CEO.
Given Charles role, he gets involved in larger, strategical and long-term
decisions when we compare with Elisa's domain. One of the hot topics that is
currently in Charles mind is the next plan for coffee beans reception in
European docks. Charles' office has reached out to you to prepare a report to
assist in this highly important decision by using optimisation techniques.
As you are probably aware of, coffee beans are not grown in Europe. World
production is divided across South/Central America, Africa and South East Asia.
Beanie Limited purchases all of its coffee beans from different providers in
America, mainly in Brasil and Colombia (these two countries account for almost
40% of the world production). Beanie Limited procurement teams execute
purchases with local providers at those locations and then the coffee beans are
shipped in container cargo ships to European docks.
Docks are a natural bottleneck in supply chains. Dock capacity in europe is
limited, and both storage and handling (the activities related with loading and
unloading goods on ships) in docks are expensive services. Charles thinks that
Beanie Limited has not paid enough attention to this part of the supply chain
since covid-19 radically changed the landscape (total costs for trans-oceanic
freight of standard containers have seen five-fold increases during the
pandemic) and wants to evaluate if the current way of working can be improved.
Currently, Beanie Limited takes a very simple approach: all the beans purchased
in America reach Europe on the Rotterdam docks (the largest in Europe). The
Amsterdam docks are used sparingly whenever there are high activity periods or
the Rotterdam docks are under a lot of stress. Charles believes that there is
an opportunity to reduce costs and make their supply chain more resilient if
the company distributes their reception points across more docks in Europe.
Charles' team has initiated conversations with 8 different docks throughout
Europe:
- Rotterdam
- Antwerp
- Hamburg
- Amsterdam
- Marseille
- Algeciras
- Valencia
- Genoa
In the upcoming months, Charles' team must sign contracts with some of these
docks to ensure incoming capacity for the coffee beans coming from America.
This means they will agree with the docks on the expected handled volumes and
thus they must decide with which docks they want to work and how much volume
they will assign to each. The team has collected the details provided by the
docks' companies for you to study. This information includes the capacity the
docks could commit for Beanie Limited, as well as the prices for handling
Beanie Limited's 40 feet containers through the dock.
Charles wants you to use your optimization skills to assess the different
options and recommend the best decisions for the company regarding how much
volume to put through each dock. He hopes you can come up with a plan where
Beanie Limited ensures it will have enough capacity, while at the same time
costs are as low as possible.
## Detailed task definition
- Below you will find four levels of questions. Levels 1 to 3 are compulsory.
Level 4 is optional.
- You need to write a report document where you answer the questions of the
different levels. This report should be directed towards Charles, should give
him clear recommendations and should justify these recommendations. It's
important for you to reflect your methodology to back your proposals.
- Each level is worth 2 points out of a total of 10. The 2 missing points will
grade the clarity and structure of your report and code.
- You need to use a Python notebook to solve all levels. A helper notebook is
provided. Please attach a notebook that shows your
solution/proposal/analysis.
- For the areas of the report where you have built an optimization model,
please provide a formal definition of the problem in mathematical terms. This
sould include the target function, decision variables and constraints.
- Include your team number, names and student IDs in all your deliverables.
## Data
A few facts:
- Charles' team estimates that Beanie Limited will need to receive 1,500,000
metric tons of coffee beans in European docks during next year.
- The category being negotiated is 40 feet containers. 40 feet containers have
a capacity of 66 cubic meters.
- One cubic meter of coffee beans weights approximately 450 kgrs.
Charles' team has shared with you a table (prices_and_capacities.csv) with the
next year's prices proposed by the different docks for 40ft containers. You can
also find the maximum capacity that the dock is willing to provide to Beanie
Limited (the capacity is described as a count of 40ft containers).
## Notebook
A notebook with some helping code has been provided. The code contains examples
on how to use `pulp`, a python package for modeling and solving optimization
problems. You can use it to get familiar with how to solve linear, integer and
mixed programming problems.
## Levels
### Level 1
Examine the data provided below and current proposals data. Do you think there
is a chance to reduce costs? Why? And will next year's total cost lower?
You can assume that last year, around 80% of Beanie Limited's containers
entered through Rotterdam and the remaining 20% through Amsterdam. Total volume
was around 24000 40ft containers. Although they don't have more granular
numbers at hand, Charles mentioned that they had a total expense of 13,700,00€
on docks fees.
### Level 2
With the price and capacity data provided for each dock, Charles would like you
to find out what is the optimal distribution across the different options for
next year. He is expecting you to provide the exact numbers, as well as the
expected costs that would result from your proposal.
### Level 3
As it tends to be in these projects, sometimes new information appears in the
middle of the project. Charles' team has reached out to let you know that they
have some additional details from the different ports.
The following docks have a sign-up fee. This means that, if Beanie Limited
wants to do any business with them at all, they need to pay this fee. This fee
is only paid once and is independent of the container volume that goes through
the dock.
- Algeciras: 800,000€
- Marseille: 500,000€
- Antwerp: 1,000,000€
Also, after presenting your initial results from level 2, Charles is very happy
with your work, although there is something bothering him. Even though your
proposal sounds reasonable, he is concerned that splitting the operations
between too many docks might make land logistics operations too complex and end
up causing higher costs in their truck transportation that cancel the potential
savings that could be obtained in docks operations.
Charles doesn't have numbers currently, so there is no way to tackle this
properly in an empirical way. To have the necessary information to think about
this in the future, he would like you to repeat your analysis with a new
condition in place: to only use a maximum of 3 docks in your proposal. Charles
wants to know what would be the optimal decision while restricting the used
docks to that number.
Charles wants you to reassess the situation with the new data and provide the
adequate solution with the new options. He also would like to understand how
these changes impact the solution you provided for level 2 and the overall
results for Beanie Limited.
### Level 4
Now that it seemed you finally had a proper proposal in place, Charles is
calling again with more concerns and asking you to go the extra mile with a
final assessment.
Apparently, some rumours have reached Charles ears. It seems that some new
regulations in Spain might affect taxes related to dock activities. Charles
believes this could have an impact on the prices in Valencia and Algeciras,
putting pressure on them to increase their prices because additional taxes
would increase their own costs. The issue is: Charles doesn't know for sure
what will happen. His expert opinion is that the following is possible:
- For Valencia, Charles thinks the price can end up somewhere between today's
price (310€/40ft) and 390€/40ft. He says the probability of any price is in
that range is equal.
- For Algeciras, Charles thinks the price can end up somewhere between today's
price (280€/40ft) and 330€/40ft. Again, he says the probability of any price
is in that range is equal.
Charles heard you did a great job by using simulation techniques with Elisa in
your previous project. Charles' can't request you to do any simulation work as
part of the current engagement, because that would be out of the scoped that
was agreed upon. But nevertheless, he would appreciate if you could explain to
him how would you use simulation to make the decision with these new
uncertainty in place. What would be your methodology? What kind of results
could you provide to him?
With that information, he might decide to spend some more money with Simiupf to
pursue that stream of work. And that would definetely make your bosses very
happy!

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{
"nbformat": 4,
"nbformat_minor": 0,
"metadata": {
"colab": {
"name": "case_2_student_notebook.ipynb",
"provenance": [],
"toc_visible": true,
"collapsed_sections": []
},
"kernelspec": {
"name": "python3",
"display_name": "Python 3"
},
"language_info": {
"name": "python"
}
},
"cells": [
{
"cell_type": "markdown",
"source": [
"# Introduction to PuLP\n",
"\n",
"For case 2, you will need to define and solve optimization problems. In this notebook, I'll help you understand how to use `pulp`, a Python package for modeling optimization problems. You might want to check the following links:\n",
"\n",
"- Documentation: https://coin-or.github.io/pulp/\n",
"- Homepage: https://github.com/coin-or/pulp\n",
"\n"
],
"metadata": {
"id": "eLvjUuJdzS7z"
}
},
{
"cell_type": "markdown",
"source": [
"# Installing and checking all is in place"
],
"metadata": {
"id": "HFavOEVS0dbY"
}
},
{
"cell_type": "markdown",
"source": [
"The first thing you need to do is to install `pulp`. `pulp` is not in the standard available packages in Colab, so you need to run the following cell once. "
],
"metadata": {
"id": "HgZwpjUG0PsK"
}
},
{
"cell_type": "code",
"source": [
"!pip install pulp"
],
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/"
},
"id": "ni6Q_YiO0nIm",
"outputId": "405d3f57-4502-4ed5-dcb9-d3204b585bb8"
},
"execution_count": null,
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"Collecting pulp\n",
" Downloading PuLP-2.6.0-py3-none-any.whl (14.2 MB)\n",
"\u001b[K |████████████████████████████████| 14.2 MB 8.9 MB/s \n",
"\u001b[?25hInstalling collected packages: pulp\n",
"Successfully installed pulp-2.6.0\n"
]
}
]
},
{
"cell_type": "markdown",
"source": [
"After doing that, you can import the library."
],
"metadata": {
"id": "k9YI0Kzw0qLT"
}
},
{
"cell_type": "code",
"source": [
"import pulp"
],
"metadata": {
"id": "hw6keX7x0tZ1"
},
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"source": [
"If all is good, running the following command will print a large log testing `pulp`. The last line should read \"OK\"."
],
"metadata": {
"id": "vD_rXehL1KXX"
}
},
{
"cell_type": "code",
"source": [
"pulp.pulpTestAll()"
],
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/"
},
"id": "Ney2a8mu1JqQ",
"outputId": "a6b32d96-b163-4fc5-b4fc-3d5673e5a40a"
},
"execution_count": null,
"outputs": [
{
"output_type": "stream",
"name": "stderr",
"text": [
"ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss.........."
]
},
{
"output_type": "stream",
"name": "stdout",
"text": [
"\t Test that logic put in place for deprecation handling of indexs works\n",
"\t Testing 'indexs' param continues to work for LpVariable.dicts\n",
"\t Testing 'indexs' param continues to work for LpVariable.matrix\n",
"\t Testing 'indices' argument works in LpVariable.dicts\n",
"\t Testing 'indices' param continues to work for LpVariable.matrix\n",
"\t Testing invalid status\n",
"\t Testing continuous LP solution - export dict\n",
"\t Testing export dict for LP\n",
"\t Testing export dict MIP\n",
"\t Testing maximize continuous LP solution\n",
"\t Testing continuous LP solution - export JSON\n",
"\t Testing continuous LP solution - export solver dict\n",
"\t Testing continuous LP solution - export solver JSON\n"
]
},
{
"output_type": "stream",
"name": "stderr",
"text": [
".........."
]
},
{
"output_type": "stream",
"name": "stdout",
"text": [
"\t Testing reading MPS files - binary variable, no constraint names\n",
"\t Testing reading MPS files - integer variable\n",
"\t Testing reading MPS files - maximize\n",
"\t Testing invalid var names\n",
"\t Testing logPath argument\n",
"\t Testing makeDict general behavior\n",
"\t Testing makeDict default value behavior\n",
"\t Testing measuring optimization time\n"
]
},
{
"output_type": "stream",
"name": "stderr",
"text": [
"............."
]
},
{
"output_type": "stream",
"name": "stdout",
"text": [
"\t Testing the availability of the function pulpTestAll\n",
"\t Testing zero subtraction\n",
"\t Testing inconsistent lp solution\n",
"\t Testing continuous LP solution\n",
"\t Testing maximize continuous LP solution\n",
"\t Testing unbounded continuous LP solution\n",
"\t Testing Long Names\n",
"\t Testing repeated Names\n",
"\t Testing zero constraint\n",
"\t Testing zero objective\n",
"\t Testing LpVariable (not LpAffineExpression) objective\n",
"\t Testing Long lines in LP\n",
"\t Testing LpAffineExpression divide\n"
]
},
{
"output_type": "stream",
"name": "stderr",
"text": [
"............."
]
},
{
"output_type": "stream",
"name": "stdout",
"text": [
"\t Testing MIP solution\n",
"\t Testing MIP solution with floats in objective\n",
"\t Testing Initial value in MIP solution\n",
"\t Testing fixing value in MIP solution\n",
"\t Testing MIP relaxation\n",
"\t Testing feasibility problem (no objective)\n",
"\t Testing an infeasible problem\n",
"\t Testing an integer infeasible problem\n",
"\t Testing another integer infeasible problem\n",
"\t Testing column based modelling\n",
"\t Testing dual variables and slacks reporting\n"
]
},
{
"output_type": "stream",
"name": "stderr",
"text": [
"..........ssssssssssssssssssssssssssssssssssssssssssssssssssssss"
]
},
{
"output_type": "stream",
"name": "stdout",
"text": [
"\t Testing fractional constraints\n",
"\t Testing elastic constraints (no change)\n",
"\t Testing elastic constraints (freebound)\n",
"\t Testing elastic constraints (penalty unchanged)\n",
"\t Testing elastic constraints (penalty unbounded)\n",
"\t Testing timeLimit argument\n"
]
},
{
"output_type": "stream",
"name": "stderr",
"text": [
"ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss\n",
"----------------------------------------------------------------------\n",
"Ran 840 tests in 15.681s\n",
"\n",
"OK (skipped=784)\n"
]
}
]
},
{
"cell_type": "markdown",
"source": [
"# Defining and solving problems\n",
"\n",
"The following cells show you the absolute minimum to model and solve a problem with `pulp`. The steps are:\n",
"\n",
"1. Define decision variables\n",
"2. Define the target function\n",
"3. Define the constraints\n",
"4. Assemble the problem\n",
"5. Solve it\n",
"6. Examine results\n",
"\n",
"For more flexibility, options and interesting stuff, please check up the PuLP documentation."
],
"metadata": {
"id": "oiXz40NR1whf"
}
},
{
"cell_type": "markdown",
"source": [
"## Define decision variables"
],
"metadata": {
"id": "nq5bcQs03g0j"
}
},
{
"cell_type": "code",
"source": [
"x = pulp.LpVariable(\n",
" name=\"x\",\n",
" cat=pulp.LpContinuous \n",
" )\n",
"\n",
"y = pulp.LpVariable(\n",
" name=\"y\",\n",
" cat=pulp.LpInteger # This will make the variable integer only\n",
" )\n",
"\n",
"z = pulp.LpVariable(\n",
" name=\"z\",\n",
" cat=pulp.LpBinary # This will make the variable binary (only 0 or 1)\n",
")"
],
"metadata": {
"id": "0SPhww4L3buh"
},
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"source": [
"## Define the target function"
],
"metadata": {
"id": "uhlbq2oO35kp"
}
},
{
"cell_type": "code",
"source": [
"target_function = 10 * x - 5 * y + z"
],
"metadata": {
"id": "pu3Im9DH39CN"
},
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"source": [
"## Define constraints"
],
"metadata": {
"id": "lqD0dD474Izw"
}
},
{
"cell_type": "code",
"source": [
"constraint_1 = x >= 0\n",
"constraint_2 = y >= 0\n",
"constraint_3 = x >= 10\n",
"constraint_4 = y <= 50"
],
"metadata": {
"id": "5Cu51lYj4OUC"
},
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"source": [
"## Assemble the problem\n",
"\n",
"To put all the parts together, you need to declare a problem and specify if you want to minimize or maximize the target function.\n",
"\n",
"Once you have that:\n",
"- First, you \"add\" the target function.\n",
"- After, you \"add\" all the constraints you want to include."
],
"metadata": {
"id": "d5nq94IM4kSU"
}
},
{
"cell_type": "code",
"source": [
"problem = pulp.LpProblem(\"my_silly_problem\", pulp.LpMinimize)\n",
"\n",
"problem += target_function\n",
"\n",
"for constraint in (\n",
" constraint_1,\n",
" constraint_2,\n",
" constraint_3,\n",
" constraint_4\n",
" ):\n",
" problem += constraint"
],
"metadata": {
"id": "yI-Oiwh64mRc"
},
"execution_count": null,
"outputs": []
},
{
"cell_type": "markdown",
"source": [
"## Solve it\n",
"\n",
"The problem object is now unsolved. You can call the `solve` method on it to find a solution."
],
"metadata": {
"id": "RJTWfR8-5fBd"
}
},
{
"cell_type": "code",
"source": [
"f\"Status: {pulp.LpStatus[problem.status]}\"\n",
"problem.solve()"
],
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/"
},
"id": "4Fbltpbp5mRi",
"outputId": "07f9c959-e9b0-4fe7-e7ea-c698703111ff"
},
"execution_count": null,
"outputs": [
{
"output_type": "execute_result",
"data": {
"text/plain": [
"1"
]
},
"metadata": {},
"execution_count": 9
}
]
},
{
"cell_type": "markdown",
"source": [
"## Examine results\n",
"\n",
"After calling `solve` on a problem, you can access:\n",
"- The status of the problem. It can be solved, but also it might show to be not feasible.\n",
"- The values assigned to each decision variable.\n",
"- The final value for the target function.\n",
"\n"
],
"metadata": {
"id": "0pc9RmrO7FKo"
}
},
{
"cell_type": "code",
"source": [
"print(f\"Status: {pulp.LpStatus[problem.status]}\")\n",
"for v in problem.variables():\n",
" print(v.name, \"=\", v.varValue)\n",
" \n",
"print(pulp.value(problem.objective))"
],
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/"
},
"id": "8U4xVvUg9W07",
"outputId": "32a330f1-65ab-4903-f29b-368f2bacaf94"
},
"execution_count": null,
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"Status: Optimal\n",
"x = 10.0\n",
"y = 50.0\n",
"z = 0.0\n",
"-150.0\n"
]
}
]
},
{
"cell_type": "markdown",
"source": [
"# Peanut Butter Example\n",
"\n",
"As an additional example, you can find below the model and solver for the Peanut Butter Sandwich example we discussed on lecture 6."
],
"metadata": {
"id": "I2lNaFm2XVK1"
}
},
{
"cell_type": "code",
"source": [
"pb = pulp.LpVariable(\n",
" name=\"Peanut Butter grams\",\n",
" cat=pulp.LpContinuous \n",
" )\n",
"\n",
"b = pulp.LpVariable(\n",
" name=\"Bread grams\",\n",
" cat=pulp.LpContinuous \n",
" )"
],
"metadata": {
"id": "HI4E2dNoXVK4"
},
"execution_count": null,
"outputs": []
},
{
"cell_type": "code",
"source": [
"target_function = 5.88 * pb + 2.87 * b"
],
"metadata": {
"id": "PfTxq8R0XVLB"
},
"execution_count": null,
"outputs": []
},
{
"cell_type": "code",
"source": [
"no_negative_pb = pb >= 0\n",
"no_negative_b = b >= 0\n",
"max_pb_we_have = pb <= 200\n",
"max_b_we_have = b <= 300\n",
"doctors_dietary_restriction = pb <= 0.13 * b"
],
"metadata": {
"id": "2X1AzQM8XVLD"
},
"execution_count": null,
"outputs": []
},
{
"cell_type": "code",
"source": [
"problem = pulp.LpProblem(\"sandwich_problem\", pulp.LpMaximize)\n",
"\n",
"problem += target_function\n",
"\n",
"for constraint in (\n",
" no_negative_pb,\n",
" no_negative_b,\n",
" max_pb_we_have,\n",
" max_b_we_have,\n",
" doctors_dietary_restriction\n",
" ):\n",
" problem += constraint"
],
"metadata": {
"id": "3oEoQXebXVLE"
},
"execution_count": null,
"outputs": []
},
{
"cell_type": "code",
"source": [
"f\"Status: {pulp.LpStatus[problem.status]}\"\n",
"problem.solve()\n",
"print(f\"Status: {pulp.LpStatus[problem.status]}\")\n",
"for v in problem.variables():\n",
" print(v.name, \"=\", v.varValue)\n",
" \n",
"print(f\"Final calories: {pulp.value(problem.objective)}\")"
],
"metadata": {
"colab": {
"base_uri": "https://localhost:8080/"
},
"outputId": "d873d58d-6d9e-459e-d66f-127f436b1aab",
"id": "u1vI73kiXVLF"
},
"execution_count": null,
"outputs": [
{
"output_type": "stream",
"name": "stdout",
"text": [
"Status: Optimal\n",
"Bread_grams = 300.0\n",
"Peanut_Butter_grams = 39.0\n",
"1090.32\n"
]
}
]
},
{
"cell_type": "markdown",
"source": [
"# Case 2\n",
"\n",
"You can use the rest of the notebook to work on the different parts of case 2."
],
"metadata": {
"id": "6kWgbTjU-LaN"
}
},
{
"cell_type": "code",
"source": [
"# Good luck!"
],
"metadata": {
"id": "aYzseTWh-Sal"
},
"execution_count": null,
"outputs": []
}
]
}

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dock,40_ft_container_price_eur,max_capacity
Rotterdam,470,33000
Antwerp,470,25000
Hamburg,480,44000
Amsterdam,610,11000
Marseille,380,9000
Algeciras,280,20000
Valencia,310,11000
Genoa,340,7500
1 dock 40_ft_container_price_eur max_capacity
2 Rotterdam 470 33000
3 Antwerp 470 25000
4 Hamburg 480 44000
5 Amsterdam 610 11000
6 Marseille 380 9000
7 Algeciras 280 20000
8 Valencia 310 11000
9 Genoa 340 7500