/
selling-art-on-the-beach.jl
1471 lines (1196 loc) · 48.6 KB
/
selling-art-on-the-beach.jl
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### A Pluto.jl notebook ###
# v0.19.25
using Markdown
using InteractiveUtils
# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
quote
local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
local el = $(esc(element))
global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
el
end
end
# ╔═╡ b5566f57-1aac-42dc-89ed-79a0e948795a
using PrettyTables, JuMP, Cbc, PlutoUI, Plots, LinearAlgebra
# ╔═╡ bb186f30-ef53-11ed-3b49-b9cf8b30ab4c
md"""
# Selling your art on the beach
**A mathematical modeling story**
In Brazil, a common joke between undergrad students (usually during exams) is that "maybe I should give up and go sell my art on the beach".
In this series, let's follow a situation like that, by following the story of Javier, who just decided to leave his applied math program and sell beaded jewelry.
Javier spends his days on the beach, where he showcases his products, and while not selling, he can work on more of them.
It's a simple life.
However, while working on his art, Javier's mind wanders into his optimization class. He realizes that he could some use some of the basics of that class to improve his production.
"""
# ╔═╡ df1e6376-5c97-4ab7-a05e-50e6db8afe9a
md"""
Javier has 3 products:
- Necklaces
- Bracelets
- Earrings
To create these products, he uses 3 materials:
- Beads
- String
- Wire
Let's simplify, and assume all necklaces are equal, all bracelets are equal, etc.
For each product, you have a different use of materials, a different selling price, and a different time to assemble them, as described in the table below:
"""
# ╔═╡ b9bdde07-0cd6-4190-92a7-3fc03845b1ab
begin
materials = ["beads", "string", "wire"]
products = ["bracelet", "necklace", "earring"]
material_per_product = [
15 0.6 0.15;
80 2.5 0.2;
4 0.1 0.15;
]
selling_price = [25.0; 45.0; 10]
hours_to_assemble = [2.5; 4; 1.5]
units = " (" .* ["u", "m", "m", "€", "h"] .* ")"
pretty_table(
HTML,
[material_per_product selling_price hours_to_assemble],
header=[materials; "price"; "hours"] .* units,
row_labels=products,
)
end
# ╔═╡ 08252ffc-694d-4843-a659-c2e49fb15d63
md"""
Additionally, there are costs for the materials.
"""
# ╔═╡ 95872e39-a96d-43c6-8ee3-690f9d89bb2b
begin
material_cost = [0.25; 0.70; 1.5]
pretty_table(
HTML,
material_cost',
header=materials .* units[1:3],
row_labels=["cost (€)"],
)
end
# ╔═╡ 5fef189d-d525-41c2-85d4-6ca77707dc1a
md"""
Finally, Javier assumes that having variety is good for business, so he want at least some amount of each product.
"""
# ╔═╡ 48727e3f-febf-4f2c-a277-87039f2f6c50
begin
min_prod = [2, 2, 2]
pretty_table(
HTML,
min_prod',
header=products,
row_labels=["minimum production"],
)
end
# ╔═╡ 1914c98a-bc11-4a4f-82b5-8cdf5dc00dfb
minimum_required_time = sum(min_prod .* hours_to_assemble)
# ╔═╡ 4482fb99-d170-40ab-bf17-e0bdbe7e4ae5
md"""
## Modeling
The way that Javier learned to model was using the Sets, Parameters, and Variables.
**Sets**:
- Products: $p \in \mathcal{P}$
- Materials: $m \in \mathcal{M}$
**Parameters**:
| Name | Unit | Identifier | Set |
|:--|:--|:--|:--|
| Time availability | h | $\text{time\_available}$ | |
| Product $p$ selling price | € / unit | $\text{selling\_price}_p$ | $p \in \mathcal{P}$ |
| Material $m$ cost | € / [$m$] | $\text{material\_cost}_m$ | $m \in \mathcal{M}$ |
| Hours to assemble product $p$ | h / unit | $\text{assemble\_time}_p$ | $p \in \mathcal{P}$ |
| Minimum production of product $p$ | unit | $\text{min\_prod}_p$ | $p \in \mathcal{P}$ |
| Amount of $m$ per unit of $p$ | [$m$] / unit | $\text{mat\_per\_prod}_{p,m}$ | $p \in \mathcal{P}, m \in \mathcal{M}$ |
**Variables**:
- Amount of product $p$ assembled and sold (unit): $\text{prod}_p, \quad p \in \mathcal{P}$
**Expressions**:
- Revenue:
$\displaystyle \text{revenue} = \sum_{p \in \mathcal{P}} \text{selling\_price}_p \times \text{prod}_p$
- Total Used Material:
$\displaystyle \text{total\_used\_material}_m = \sum_{p \in \mathcal{P}} \text{mat\_per\_prod}_{p,m} \times \text{prod}_p, \quad m \in \mathcal{M}$
- Total Material Cost:
$\displaystyle \text{total\_material\_cost} = \sum_{m \in \mathcal{M}} \text{total\_used\_mat}_m \times \text{material\_cost}_m, \quad m \in \mathcal{M}$
- Time expenditure:
$\text{time\_spent} = \displaystyle \sum_{p \in \mathcal{P}} \text{assemble\_time}_p \times \text{prod}_p$
**Objective**:
- Maximize profit:
$\text{maximize} \quad \text{revenue} - \text{total\_material\_cost}$
**Constraints**:
- Time Availability:
$\text{spent\_time} \leq \text{time\_available}$
- Minimum production:
$\text{prod}_p \geq \text{min\_prod}_p, \quad p \in \mathcal{P}$
- Integer production:
$\text{prod}_p \in \mathbb{Z}$
"""
# ╔═╡ 02f10634-74f9-436b-8a32-6d63bda26c7e
products
# ╔═╡ d058c09f-b434-4ded-ac11-92ae312d4d84
begin
# using JuMP, Cbc
function optimize_production(time_availability)
model = Model(Cbc.Optimizer)
set_attribute(model, "logLevel", 0)
products_index = Dict(p => i for (i, p) in enumerate(products))
materials_index = Dict(m => i for (i, m) in enumerate(materials))
@variable(model,
prod[p=products] ≥ min_prod[products_index[p]],
Int
)
@expression(model,
total_used_material[m=materials],
sum(
material_per_product[products_index[p], materials_index[m]] * prod[p]
for p in products
)
)
# total_mat_cost = sum(
# total_used_material[m] * material_cost[m]
# for m, _ in enumerate(materials)
# )
total_mat_cost = sum(total_used_material .* material_cost)
# total_mat_cost = dot(total_used_material, material_cost)
revenue = sum(prod .* selling_price)
profit = revenue - total_mat_cost
time_expenditure = sum(hours_to_assemble .* prod)
@objective(model, Max, profit)
@constraint(model, time_expenditure ≤ time_availability)
optimize!(model)
return [value(prod[p]) for p in products], objective_value(model), model
end
end
# ╔═╡ c6349108-bc5e-4745-9d11-197482dd21a1
md"""
Time Availability = $(@bind slider_time_availability Slider(ceil(Int, minimum_required_time):40, show_value=true))
"""
# ╔═╡ c0ba129e-51a6-462c-8ed8-6e0c62e8b9da
let
production, _ = optimize_production(slider_time_availability)
pretty_table(
HTML,
production',
header=products,
row_labels=["production"],
)
end
# ╔═╡ 848b3a22-0ff0-4539-b6bb-d9f629c292d0
let
time_availabilities = ceil(Int, minimum_required_time):40
productions = getfield.(optimize_production.(time_availabilities), 1)
revenues = dot.(productions, Ref(selling_price))
costs = dot.(productions, Ref(material_per_product), Ref(material_cost))
profits = revenues - costs
slacks = dot.(productions, Ref(hours_to_assemble))
plt = plot(layout=grid(1, 2), size=(800, 400))
bar!(
plt[1],
time_availabilities,
time_availabilities - slacks,
lab="Unnalocated time (h)",
)
plot!(
plt[2],
time_availabilities,
profits,
lab="profits",
m=(3, :lightblue),
)
xlabel!("Time availability")
end
# ╔═╡ 2e4475fd-4386-409e-929b-20d5092a83d6
let
production, profit, model = optimize_production(slider_time_availability)
println(model)
end
# ╔═╡ 00000000-0000-0000-0000-000000000001
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PrettyTables = "08abe8d2-0d0c-5749-adfa-8a2ac140af0d"
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Cbc = "~1.1.1"
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# ╔═╡ 00000000-0000-0000-0000-000000000002
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