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cachon+terwiesch.jl
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cachon+terwiesch.jl
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### A Pluto.jl notebook ###
# v0.19.11
using Markdown
using InteractiveUtils
# ╔═╡ 29d61cca-ad0f-4753-8789-f9da6e557b87
# Load the packages needed
using NewsvendorModel, Distributions
# ╔═╡ 1e8dae5b-fa24-46d0-b591-4c6a719e0a6c
md"""
This notebook illustrates the usage of [NewsvendorModel.jl](https://github.com/frankhuettner/NewsvendorModel.jl) with examples from the **excellent textbook by [Cachon & Terwiesch (3rd](http://cachon-terwiesch.net/3e/index.php)** or [4th](https://www.mheducation.com/highered/product/matching-supply-demand-introduction-operations-management-cachon-terwiesch/M9780078096655.html) edition).
These notes are not self-contained but should be considered as a companion to the above text book.
"""
# ╔═╡ e48eb4c3-a372-4335-8d04-05f76ccc06a7
md"You can load this notebook in [Pluto](https://github.com/fonsp/Pluto.jl) from here: [https://github.com/frankhuettner/NewsvendorModel.jl/blob/main/docs/src/textbook/cachon%2Bterwiesch.jl](https://github.com/frankhuettner/NewsvendorModel.jl/blob/main/docs/src/textbook/cachon%2Bterwiesch.jl)
"
# ╔═╡ 65eb4c20-8998-11ec-3abf-bb0dda80c6a9
md"""
## ONeill Hammer 3/2
The leading case of the book is about the **O'Neill Hammer 3/2**, with the following information (for a detailed treatment, it is referred to the sources below).
Unit values:
- cost per unit ordered = 110
- selling price per unit = 190
- salvage value = 190
Demand distribution:
- expected demand = 3192
- standard deviation of demand = 1181
"""
# ╔═╡ 11be7ca9-d279-4ff5-8b4d-4d1a5e144e43
# Define the model
oneill = NVModel(cost = 110,
price = 190,
salvage = 90,
demand = Normal(3192, 1181)
)
# ╔═╡ fa3ce275-8a61-4204-af49-01ea8592b288
# Solve the model
solve(oneill)
# ╔═╡ e20d3e01-39c4-46f2-8d8a-414bfd3048eb
md"""
Note that the result is slightly different from the textbook, which suggests the order quantity 4,184 (instead of 4,186). This is due to rounding errors in the textbook.
### 2-stage calculation
The authors also consider the scenario in which it is possible to make a second order later during the selling period (what follows is very brief and I recommend looking up the book). It is assumed that
* the unit cost for the second order are 20% higher ⇒ 20% * 110 = 22. Yet,
* demand for the rest of the season is assumed to be certain at that stage;
* the reorder stage is early enough to ensure that we have enough units until we receive our second delivery
Now there are two decision points:
1. How many units to order prior to the season starts.
2. How many units to order at the second stage.
To find the optimal order quantity for (1), we think of the product as follows:
- Having a unit leftover is effectively loosing Cₒ = 20
- Having a unit too little is creating not such a heavy damage anymore; instead, we can save the day be ordering remaining demand at the second stage, albeit at a worse price ⇒ we miss out Cᵤ = 22
This is as if we had a product traded for zero cost and zero price, but having a unit left over costs 20 ⇒ salvage = -20 ⇒ Cₒ = 20. Being short a unit costs a backorder penalty of 22 ⇒ Cᵤ = 22.
"""
# ╔═╡ af1c94ce-4900-4e4d-93b0-096917fd288d
oneill_1st_order_calculation = NVModel( demand = Normal(3192, 1181),
cost = 0,
price = 0,
salvage = -20,
backorder = 22,
)
# ╔═╡ 198bb582-9f88-48a7-85ea-d71559f0a34b
md"This yields the following critical fractile for the 1st stage:"
# ╔═╡ 60509577-f814-4770-ab01-41083bac00a5
critical_fractile(oneill_1st_order_calculation)
# ╔═╡ 84376bac-546a-4e44-bb97-7ecde967994b
md"This yields the following optimal order quantity for the 1st stage:"
# ╔═╡ ddb6f2c1-ca7a-4eb7-9f91-10323f21de44
q_opt(oneill_1st_order_calculation)
# ╔═╡ a372b600-4940-47da-861d-a903cdb77f8a
md"""
Ordering 3263 units results in the following expected lost sales, which is the expected order quantity for the second order:
"""
# ╔═╡ 44114735-088e-4223-b92a-b60da42f3fdc
lost_sales(oneill, 3263)
# ╔═╡ e1cd3166-7163-454a-8a6d-9fb8f6eabcb3
md"""
Ordering 3263 further yields the following expected profit (based on original unit values):
"""
# ╔═╡ 251daa01-9129-485c-8fd4-d660c5ebff6d
profit(oneill, 3263)
# ╔═╡ 981e3a79-a219-4870-9eae-2a01acd3e459
md"In total, this promises the following expected profit:"
# ╔═╡ 958d0d9a-81d9-40a7-a009-0ffbaeb26a0d
profit(oneill, 3263) + lost_sales(oneill, 3263) * (190 - 132)
# ╔═╡ 74e1fa73-67f4-473b-84a9-facd895aa666
md"""
## Selected Exercises
The book offers exercises with solutions. Here is a small selection that illustrates the convinience of Distributions.jl and NewsvendorModel.jl (everything is very brief and a detailed explanation can be found in the book).
### McClure Books
"""
# ╔═╡ 390a2fdd-0910-4941-ac1f-11bb3174c564
# Define demand
mcclure_demand = Normal(200, 80)
# ╔═╡ 1cc679f3-f753-487a-8cda-3e8369c5ffe1
# Define model
mcclure_nvm = NVModel(cost = 12, price = 20, salvage = 12 - 4,demand = mcclure_demand)
# ╔═╡ 98557010-627f-4de7-be6a-be583b48a121
md"##### a) Pr[demand > 400]"
# ╔═╡ db9a34bb-f82c-4360-8931-350b62de9816
1 - cdf(mcclure_demand, 400)
# ╔═╡ cffc5227-1acd-4f58-9c9d-5527c2755632
md"##### b) Pr[demand < 100]"
# ╔═╡ b59d4b3d-b8f8-440c-9640-9f0a78e8a6a9
cdf(mcclure_demand, 100)
# ╔═╡ 9a4131d2-0fb7-49b1-9fd6-f5b5f1cff5c5
md"##### c) Pr[160 < demand < 240]
"
# ╔═╡ a0ba77aa-d973-4f29-80da-c94f8cb177fb
cdf(mcclure_demand, 240) - cdf(mcclure_demand, 160)
# ╔═╡ 47ff9c6d-d508-457c-af75-2b9914d5ed53
md"##### d) Pr[160 < demand < 240]
"
# ╔═╡ ca0d15f8-0c0f-446e-9ebc-0d8b04a2e37e
q_opt(mcclure_nvm)
# ╔═╡ 8701aae7-c40b-4ef6-b347-2938d11c6ec8
md"##### e) Quantity such that Pr[demand ≤ q] = 95%
"
# ╔═╡ 33a8fdc2-3221-4540-b49c-d51c4fbd7ba0
quantile(mcclure_demand, 0.95)
# ╔═╡ ed7e61da-5905-40f0-8d57-0721093eef77
md"##### f)
"
# ╔═╡ 672bcd43-e09b-4215-bf40-101a7e1ae0ce
1 - 0.95
# ╔═╡ d792260e-41bc-4b77-8444-5ddb4f536376
md"##### g) Profit if q = 300
"
# ╔═╡ 36b39a81-a876-4a0e-9dd9-5224436e2ef1
profit(mcclure_nvm, 300)
# ╔═╡ 3daa3e0e-4aae-4b07-8ee8-e0fbef9abde2
md" ### EcoTable Tea "
# ╔═╡ d64bf64a-92a7-4747-8099-d525308cb99d
# Define demand
ecotable_demand = Poisson(4.5)
# ╔═╡ 7127b4d0-87f5-42d7-a08c-0623bda740ae
# Define model
ecotable_nvm = NVModel(cost = 32, price = 55, salvage = 20, demand = ecotable_demand)
# ╔═╡ 8f7e2e1a-ac27-4a6f-9dda-e4364d1a69db
md"##### a) Pr[demand > 3]
"
# ╔═╡ 330bfc55-5197-4adb-a898-5adab6af407d
1 - cdf(ecotable_demand, 3)
# ╔═╡ 242c68ee-7147-4912-9652-960a459aca7e
md"##### b) Pr[demand < 7]
"
# ╔═╡ 5c05eb75-3e1f-4ef2-93d5-9d1a91286fea
cdf(ecotable_demand, 7)
# ╔═╡ 820727f5-21b3-4c4b-a632-822f4c41f69c
md"##### c) Optimal order quantity
"
# ╔═╡ cc2cfbf4-5e75-4de8-85dc-7cf24353164b
q_opt(ecotable_nvm)
# ╔═╡ 2914e282-8b19-4274-ba28-ae79ac03edb2
md"##### d) Expected sales at q = 4
"
# ╔═╡ b9873a29-a100-43d5-b40e-783b33f69613
sales(ecotable_nvm, 4)
# ╔═╡ f07108dd-7555-409a-85ca-5f7742a4a8dc
md"##### d) Expected leftover at q = 6
"
# ╔═╡ d89982d5-64de-46e7-8847-79b78ad71d56
leftover(ecotable_nvm, 6)
# ╔═╡ 82d30f07-364d-4d6f-a7ff-81f3ef3e4790
md"##### f) Smallest quantity q such that Pr[demand ≤ q] ≥ 90%
"
# ╔═╡ d76f07dc-1cb9-4d04-8fdf-e8be630de3e4
quantile(ecotable_demand, 0.90)
# ╔═╡ b1c38013-98b3-4edd-a87c-855ed4da4808
md"##### d) Expected profit at q = 8
"
# ╔═╡ a6b96124-c8e1-40f3-8f23-add5b6187704
profit(ecotable_nvm, 8)
# ╔═╡ ecff0cfe-8ea2-4eb3-a064-106ae6f2bfa8
md"### Pony Express"
# ╔═╡ fea33a55-38b8-44d7-b549-3174099c7b58
md"First we define the demand:"
# ╔═╡ 8aa6baf7-3128-47b8-8f45-35c6754ab29a
xs = vec([5_000 10_000 15_000 20_000 25_000 30_000 35_000 40_000 45_000 50_000 55_000 60_000 65_000 70_000 75_000])
# ╔═╡ 574c2a61-46fa-4f5c-8eac-7f08a507dc2a
ps = vec([0.0181 0.0733 0.1467 0.1954 0.1954 0.1563 0.1042 0.0595 0.0298 0.0132 0.0053 0.0019 0.0006 0.0002 0.0001])
# ╔═╡ a2daca19-41eb-4349-aa10-1a7518b0a0f9
# The distribution is nonparametric
elvis_demand = DiscreteNonParametric(xs,ps)
# ╔═╡ 47237478-72ba-4650-bd96-f8de2b5baf3d
# Define model
elvis_nvm = NVModel(cost = 6, price = 12, demand = elvis_demand, salvage = 2.5)
# ╔═╡ 814cbf4c-5b21-4374-b484-1380a03e449c
md"##### a)
"
# ╔═╡ f1e2056f-f376-4bf3-9f54-19a10491200c
cdf(elvis_demand, 30_000)
# ╔═╡ 3de90f41-a309-4e70-b48f-7f9bee92fb51
md"##### b)
"
# ╔═╡ 70463c90-3e9c-4b27-842f-a1f4e42f3b28
q_opt(elvis_nvm)
# ╔═╡ 5d4e2337-309e-49ba-a970-19b512ae58e1
md"##### c)
"
# ╔═╡ 9f81c748-52ce-43c5-8b60-d4ed62e3b7c9
quantile(elvis_demand, 0.9)
# ╔═╡ 85621539-1a57-4626-adde-251074f16803
md"##### d)
"
# ╔═╡ 69a85cc2-db01-4574-af5d-62bc9f9fd6ae
leftover(elvis_nvm, 50_000)
# ╔═╡ 56d4dcc9-562f-4915-9a78-ea9a49ee8d09
md"##### e)
"
# ╔═╡ f1b912de-b66f-4684-a29d-5b3279fb4b50
quantile(elvis_demand, 1.0)
# ╔═╡ 0a704099-7515-44fd-b329-34880166c4c3
profit(elvis_nvm, 75_000)
# ╔═╡ 0a2b7ffa-54ab-4aec-bd00-c8c6068f294f
md"The answer in the book appears to be confusing: IMHO lost sales = 0 if 75,000 wigs are ordered."
# ╔═╡ 67ce094a-9cad-42f3-bd17-ac0ed3ecfc91
md"""### Flextrola
We just investigate part **(j)**: Variation with log normal demand
"""
# ╔═╡ b13b3d84-d43c-464f-ba1b-30e1fc7dc8fa
# Parameter σ² of log normal distribution from mean= 1000 and standard deviation = 600
σ² = log(1 + 600^2 / 1000^2)
# ╔═╡ ecbcc004-012a-4ac3-99ba-cc6194e22700
# Parameter μ of log normal distribution from mean = 1000 and standard deviation = 600
μ = log(1000^2 / √(600^2 + 1000^2))
# ╔═╡ be3145d3-1e72-4b27-818d-6d29b925edf1
flextrola_demand = LogNormal(μ, √σ²)
# ╔═╡ 27342a58-0bd2-43b4-9b44-3a04e9115086
# Let us plot this as in the textbook
begin
using StatsPlots
plot(xlims=(0, 2500), xlabel = "Demand", ylabel = "Probability")
plot!(Normal(1000, 600), marker = :dot, label="Normal(Normal(1000, 600))")
plot!(flextrola_demand, marker = :hex, label="LogNormal{Float64}(μ=6.86, σ=0.307)")
end
# ╔═╡ 0dedd60f-7a88-4570-bd74-080eb47b80db
# Indeed, the parameters yield the desired mean
mean(flextrola_demand)
# ╔═╡ b5e7e67e-df34-498c-b97c-d34f6edde014
# Indeed, the parameters yield the standard deviation
std(flextrola_demand)
# ╔═╡ 74ee066c-bc06-4a35-8945-148b3f546f91
# Define the model
flextrola_nvm = NVModel(cost = 72, price = 121, salvage = 50, demand=flextrola_demand)
# ╔═╡ eb040544-c8b3-4156-a1fd-04fcc0a6d431
solve(flextrola_nvm)
# ╔═╡ 8e239e87-682e-4244-8b2e-8c73c7a4737d
md"""
## References
- [NewsvendorModel.jl](https://github.com/frankhuettner/NewsvendorModel.jl)
- Gerard Cachon, Christian Terwiesch, [Matching Supply with Demand: An Introduction to Operations Management. McGraw-Hill (2012)](http://cachon-terwiesch.net/3e/index.php), Chapters 12 and 13.
- Gerard Cachon, Christian Terwiesch, Matching Supply with Demand: An Introduction to Operations Management. McGraw-Hill ([2018](https://www.mheducation.com/highered/product/)), Chapters 14 and 15.
"""
# ╔═╡ 00000000-0000-0000-0000-000000000001
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StatsPlots = "f3b207a7-027a-5e70-b257-86293d7955fd"
[compat]
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"""
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