ipynb/mizuhiro_suzuki-kidney_exchange.ipynb

{
    "cells": [
        {
            "cell_type": "markdown",
            "id": "edcc75c5",
            "metadata": {},
            "source": [
                "# Recursive algorithm for kidney exchanges"
            ]
        },
        {
            "cell_type": "markdown",
            "id": "50a8f8da",
            "metadata": {},
            "source": [
                "In this notebook, I implement the recursive algorithm introduced in [Anderson et al (2015), \"Kidney Exchange and the Alliance for Paired Donation: Operations Research Changes the Way Kidneys Are Transplanted\"](https://dspace.mit.edu/handle/1721.1/111107)."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 1,
            "id": "38838b29",
            "metadata": {},
            "outputs": [],
            "source": [
                "using Pkg"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 2,
            "id": "06b2fcea",
            "metadata": {},
            "outputs": [],
            "source": [
                "using Graphs, GraphPlot\n",
                "using Random\n",
                "using StatsBase\n",
                "using Distributions\n",
                "using JuMP\n",
                "\n",
                "import HiGHS"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 15,
            "id": "2587cdc8",
            "metadata": {},
            "outputs": [],
            "source": [
                "Random.seed!(42);"
            ]
        },
        {
            "cell_type": "markdown",
            "id": "a02bef81",
            "metadata": {},
            "source": [
                "## Create a random graph"
            ]
        },
        {
            "cell_type": "markdown",
            "id": "c3556d4e",
            "metadata": {},
            "source": [
                "First I create a random graph of incompatible pairs.\n",
                "I consider two kinds of compatibility: tissue compatibility and blood-type compatibility.\n",
                "For the tissue compatibility, I assume that a donor is compatible to a patient with probability $p$, and for the blood-type compatibility, I randomly sample blood-types of patients and donors and consider compatibility between pairs.\n",
                "I consider two pairs are compatible if they are both tissue compatible and blood-type compatible."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 16,
            "id": "39c65817",
            "metadata": {},
            "outputs": [],
            "source": [
                "pairs = \"Pair_\" .* string.(collect('A':'Z'));\n",
                "numPairs = length(pairs);"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 17,
            "id": "665f8784",
            "metadata": {},
            "outputs": [],
            "source": [
                "bloodTypeVec = vec([join(x, \"_\") for x in Iterators.product([\"A\", \"B\", \"O\", \"AB\"], [\"A\", \"B\", \"O\", \"AB\"])]);"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 18,
            "id": "acf18086",
            "metadata": {},
            "outputs": [],
            "source": [
                "pairBloodTypes = sample(bloodTypeVec, numPairs, replace=true);"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 19,
            "id": "384f261a",
            "metadata": {},
            "outputs": [],
            "source": [
                "ABOCompatibility = Dict([(\"O\", [\"O\", \"A\", \"B\", \"AB\"]), (\"A\", [\"A\", \"AB\"]), (\"B\", [\"B\", \"AB\"]), (\"AB\", [\"AB\"])]);"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 20,
            "id": "32f5f10f",
            "metadata": {},
            "outputs": [],
            "source": [
                "pairABOCompatibilityMatrix = zeros(Int, numPairs, numPairs);\n",
                "\n",
                "for i = 1:numPairs, j = 1:numPairs\n",
                "    if i ≠ j\n",
                "        donorBloodType = split(pairBloodTypes[i], \"_\")[2]\n",
                "        patientBloodType = split(pairBloodTypes[j], \"_\")[1]\n",
                "        if patientBloodType in ABOCompatibility[donorBloodType]\n",
                "            pairABOCompatibilityMatrix[i, j] = 1\n",
                "        end\n",
                "    end\n",
                "end"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 21,
            "id": "ff000164",
            "metadata": {},
            "outputs": [],
            "source": [
                "p = 0.12;\n",
                "N = numPairs * numPairs;\n",
                "d = Binomial(1, p);\n",
                "\n",
                "pairTissueCompatibilityMatrix = reshape(rand(d, N), numPairs, numPairs);\n",
                "for i = 1:numPairs\n",
                "    pairTissueCompatibilityMatrix[i, i] = 0\n",
                "end"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 22,
            "id": "e88ccbe4",
            "metadata": {},
            "outputs": [],
            "source": [
                "pairCompatibilityMatrix = pairABOCompatibilityMatrix .* pairTissueCompatibilityMatrix;"
            ]
        },
        {
            "cell_type": "markdown",
            "id": "9f5d3be2",
            "metadata": {},
            "source": [
                "Also, later I introduce a non-directed donor.\n",
                "For a fair comparison, I assume that all donors are acceptable to the patient in Pair A."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 23,
            "id": "e579f2e6",
            "metadata": {},
            "outputs": [],
            "source": [
                "pairCompatibilityMatrix[:, 1] .= 1;\n",
                "pairCompatibilityMatrix[1, 1] = 0;"
            ]
        },
        {
            "cell_type": "markdown",
            "id": "2855b240",
            "metadata": {},
            "source": [
                "Here is the figure of a directed network."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 24,
            "id": "545f28cc",
            "metadata": {},
            "outputs": [
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                            "Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}, Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), Compose.UnitBox{Float64, Float64, Float64, Float64}(-1.2, -1.2, 2.4, 2.4, 0.0mm, 0.0mm, 0.0mm, 0.0mm), nothing, nothing, nothing, List([Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}, Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.LinePrimitive}(Compose.LinePrimitive[Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.14386263525616683cx, -0.06317801296454156cy), (-0.7196912585747259cx, -0.3383074487018238cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(0.019243768660375166cx, 0.012937792349256301cy), (0.8549074064164639cx, 0.39368601430495187cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.09142066622773591cx, 0.060901825878466265cy), (-0.3628817923275355cx, 0.8657083870063784cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.8353782751366288cx, 0.5873323798900133cy), (-0.13322130369557217cx, 0.03148401498927906cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.8287173841420824cx, 0.6894120210393615cy), (-0.46887282807892305cx, 0.9047698860407325cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.7469917054430284cx, -0.893419488385712cy), (-0.11833614599541768cx, -0.09508628850193224cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(0.12979285310362632cx, 0.02342683965335142cy), (0.024689673481829932cx, -0.002680908083479174cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(0.1985046976651485cx, -0.04292631188724169cy), (0.10746247068816556cx, -0.47312753329840695cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(0.27925852105795607cx, -0.02000011204941078cy), (0.5758735523334885cx, -0.3309569834947268cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.7196912585747259cx, -0.3383074487018238cy), (-0.14386263525616683cx, -0.06317801296454154cy)])  …  Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(0.21567519283805678cx, -0.8850275842599878cy), (-0.03496239673223028cx, -0.11001099117756558cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(0.15346053212440908cx, -0.9747253798412632cy), (-0.31572114134908075cx, -0.9959355459383118cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(0.29999519304764394cx, -0.900707789544042cy), (0.4396263018862407cx, -0.728215463913302cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(0.2115903332216374cx, -0.8864579872759127cy), (0.12060710465204688cx, -0.6453803649171617cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(0.2943213147513931cx, -0.8964731105033675cy), (0.5870410281604216cx, -0.4702684920481958cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.21521704905529176cx, -0.5189194990409008cy), (-0.08918628848586267cx, -0.11037845869176138cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.19166563509589685cx, -0.5301418731872374cy), (-0.049792605712767474cx, -0.3183075294219861cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.15198160013479567cx, -0.6114085045747885cy), (0.4064869614216994cx, -0.6517741311776643cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.2658263870589256cx, -0.5182012006984059cy), (-0.36698489118883626cx, -0.153884202944613cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.3628817923275355cx, 0.8657083870063784cy), (-0.0914206662277359cx, 0.060901825878466265cy)])], Symbol(\"\"))]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(0.5883484054145521mm)]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.8274509803921568,0.8274509803921568,0.8274509803921568,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}, Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.LinePrimitive}(Compose.LinePrimitive[Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.6846696739998568cx, -0.3026214764737159cy), (-0.7196912585747259cx, -0.3383074487018238cy), (-0.6699246675340804cx, -0.33348184861549873cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(0.8192419101465962cx, 0.35864357752004705cy), (0.8549074064164639cx, 0.39368601430495187cy), (0.8050612070092904cx, 0.38976728942811995cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.33166105611410424cx, 0.8266537377275621cy), (-0.3628817923275355cx, 0.8657083870063784cy), (-0.3640691654982295cx, 0.8157224875450951cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.18067441028457312cx, 0.047238464362485175cy), (-0.13322130369557217cx, 0.03148401498927906cy), (-0.15944572745227814cx, 0.07405489234256736cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.5004069883450916cx, 0.8659678662955234cy), (-0.46887282807892305cx, 0.9047698860407325cy), (-0.5179708538422776cx, 0.8953155819340849cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.13396866025613685cx, -0.1425797038127773cy), (-0.11833614599541768cx, -0.09508628850193224cy), (-0.16083952558032796cx, -0.12141996891050848cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(0.06616594015991341cx, 0.0252425462595727cy), (0.024689673481829932cx, -0.002680908083479174cy), (0.07441118756325815cx, -0.007950731368024534cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(0.10045977753565224cx, -0.4236203389867542cy), (0.10746247068816556cx, -0.47312753329840695cy), (0.13392070760568972cx, -0.43070157780868157cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(0.5310694426320741cx, -0.30876257532817214cy), (0.5758735523334885cx, -0.3309569834947268cy), (0.555817903641614cx, -0.2851555538713175cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.17888421983103603cx, -0.09886398519264938cy), (-0.14386263525616683cx, -0.06317801296454154cy), (-0.19362922629681234cx, -0.0680036130508666cy)])  …  Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.004233663271791199cx, -0.1494539238768121cy), (-0.03496239673223028cx, -0.11001099117756558cy), (-0.036776249649552444cx, -0.15997807972186964cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.2695567386238587cx, -0.9767301320915989cy), (-0.31572114134908075cx, -0.9959355459383118cy), (-0.2680121551119484cx, -1.0108972514635894cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(0.423356241990989cx, -0.7754942662564761cy), (0.4396263018862407cx, -0.728215463913302cy), (0.39677248456725894cx, -0.7539749348160854cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(0.15319655244247238cx, -0.6833003845995818cy), (0.12060710465204688cx, -0.6453803649171617cy), (0.12119755122528902cx, -0.6953768785240537cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(0.5745376610977554cx, -0.518679914379049cy), (0.5870410281604216cx, -0.4702684920481958cy), (0.5463446308602146cx, -0.4993167836809873cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.08669539372503994cx, -0.1603163745802975cy), (-0.08918628848586267cx, -0.11037845869176138cy), (-0.11937762598265982cx, -0.15023423794840735cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.06172913757455695cx, -0.36686182075769075cy), (-0.049792605712767474cx, -0.3183075294219861cy), (-0.09014661648330538cx, -0.3478296197058957cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(0.3583917519317314cx, -0.6654434700101057cy), (0.4064869614216994cx, -0.6517741311776643cy), (0.3608574118171658cx, -0.6313304474859367cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.3379368331130551cx, -0.1945807669522297cy), (-0.36698489118883626cx, -0.153884202944613cy), (-0.3708920348173979cx, -0.20373131147325074cy)]), Compose.LinePrimitive{Tuple{Measures.Measure, Measures.Measure}}(Tuple{Measures.Measure, Measures.Measure}[(-0.12264140244116714cx, 0.09995647515728252cy), (-0.0914206662277359cx, 0.060901825878466265cy), (-0.09023329305704192cx, 0.1108877253397495cy)])], Symbol(\"\"))]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(0.5883484054145521mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.8274509803921568,0.8274509803921568,0.8274509803921568,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}, Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([]), List([Compose.Property{Compose.FontSizePrimitive}(Compose.FontSizePrimitive[Compose.FontSizePrimitive(4.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}, Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}}(Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}[Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((-0.06265591258323344cx, -0.024377649657978506cy), 0.036w), Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((-0.9059436662489675cx, 0.6431940445372708cy), 0.036w), Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((-0.8026719388552126cx, -0.9641281272296657cy), 0.036w), Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((0.2171384391686897cx, 0.04512358122785076cy), 0.036w), Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((-0.8008979812476593cx, -0.37710781200838683cy), 0.036w), Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((-0.5581726164417133cx, -0.7316557550383422cy), 0.036w), Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((0.000289184149256716cx, -0.24352909453453975cy), 0.036w), Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((-0.13172413207517675cx, 1.0cy), 0.036w), Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((-1.0cx, 0.18807312917586816cy), 0.036w), Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((0.7829072979796712cx, 0.06027827922248141cy), 0.036w)  …  Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((-0.6129056229161798cx, 0.20932906569334198cy), 0.036w), Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((0.9368070876600725cx, 0.43100145631218667cy), 0.036w), Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((0.48099000853933327cx, 0.9932589075610554cy), 0.036w), Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((0.531082365580192cx, 0.19393522664267926cy), 0.036w), Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((-0.612173317617475cx, 0.6047343738986088cy), 0.036w), Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((0.21428233261081142cx, 0.5709420547869952cy), 0.036w), Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((-0.39106385328984083cx, -0.06716509556833528cy), 0.036w), Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((0.2433687086890599cx, -0.970660925779575cy), 0.036w), Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((-0.241747424957921cx, -0.6049203080746837cy), 0.036w), Compose.CirclePrimitive{Tuple{Measures.Measure, Measures.Measure}, Measures.Measure}((-0.39164654597203796cx, 0.9509878625428232cy), 0.036w)], Symbol(\"\"))]), List([Compose.Property{Compose.LineWidthPrimitive}(Compose.LineWidthPrimitive[Compose.LineWidthPrimitive(0.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.25098039215686274,0.8784313725490196,0.8156862745098039,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\")), Compose.Context(Measures.BoundingBox{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}, Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}((0.0w, 0.0h), (1.0w, 1.0h)), nothing, nothing, nothing, nothing, List([]), List([Compose.Form{Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}}(Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}[Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}((-0.06265591258323344cx, -0.024377649657978506cy), \"Pair_A\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}((-0.9059436662489675cx, 0.6431940445372708cy), \"Pair_B\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}((-0.8026719388552126cx, -0.9641281272296657cy), \"Pair_C\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}((0.2171384391686897cx, 0.04512358122785076cy), \"Pair_D\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}((-0.8008979812476593cx, -0.37710781200838683cy), \"Pair_E\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}((-0.5581726164417133cx, -0.7316557550383422cy), \"Pair_F\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}((0.000289184149256716cx, -0.24352909453453975cy), \"Pair_G\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}((-0.13172413207517675cx, 1.0cy), \"Pair_H\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}((-1.0cx, 0.18807312917586816cy), \"Pair_I\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}((0.7829072979796712cx, 0.06027827922248141cy), \"Pair_J\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm))  …  Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}((-0.6129056229161798cx, 0.20932906569334198cy), \"Pair_Q\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}((0.9368070876600725cx, 0.43100145631218667cy), \"Pair_R\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}((0.48099000853933327cx, 0.9932589075610554cy), \"Pair_S\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}((0.531082365580192cx, 0.19393522664267926cy), \"Pair_T\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}((-0.612173317617475cx, 0.6047343738986088cy), \"Pair_U\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}((0.21428233261081142cx, 0.5709420547869952cy), \"Pair_V\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}((-0.39106385328984083cx, -0.06716509556833528cy), \"Pair_W\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}((0.2433687086890599cx, -0.970660925779575cy), \"Pair_X\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}((-0.241747424957921cx, -0.6049203080746837cy), \"Pair_Y\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm)), Compose.TextPrimitive{Tuple{Measures.Length{:cx, Float64}, Measures.Length{:cy, Float64}}, Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}, Tuple{Measures.AbsoluteLength, Measures.AbsoluteLength}}((-0.39164654597203796cx, 0.9509878625428232cy), \"Pair_Z\", Compose.HCenter(), Compose.VCenter(), Compose.Rotation{Tuple{Measures.Length{:w, Float64}, Measures.Length{:h, Float64}}}(0.0, (0.5w, 0.5h)), (0.0mm, 0.0mm))], Symbol(\"\"))]), List([Compose.Property{Compose.FontSizePrimitive}(Compose.FontSizePrimitive[Compose.FontSizePrimitive(4.0mm)]), Compose.Property{Compose.StrokePrimitive}(Compose.StrokePrimitive[Compose.StrokePrimitive(RGBA{Float64}(0.0,0.0,0.0,0.0))]), Compose.Property{Compose.FillPrimitive}(Compose.FillPrimitive[Compose.FillPrimitive(RGBA{Float64}(0.0,0.0,0.0,1.0))])]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\"))]), List([]), List([]), 0, false, false, false, false, nothing, nothing, 0.0, Symbol(\"\"))"
                        ]
                    },
                    "execution_count": 24,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "g = SimpleDiGraph(pairCompatibilityMatrix);\n",
                "gplot(g, nodelabel=pairs, NODESIZE=0.09, arrowlengthfrac=0.05)"
            ]
        },
        {
            "cell_type": "markdown",
            "id": "4fd36f0e",
            "metadata": {},
            "source": [
                "There are 135 cycles in the graph, and the maximum length of cycles is 12."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 25,
            "id": "1cd679a1",
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "Number of cycles: 135\n",
                        "Maximum length of cycles: 12\n",
                        "[\"Pair_A\", \"Pair_E\", \"Pair_K\", \"Pair_X\", \"Pair_N\", \"Pair_G\", \"Pair_V\", \"Pair_H\", \"Pair_U\", \"Pair_I\", \"Pair_B\", \"Pair_Z\"]\n",
                        "[\"Pair_A\", \"Pair_E\", \"Pair_Q\", \"Pair_D\", \"Pair_N\", \"Pair_G\", \"Pair_V\", \"Pair_H\", \"Pair_U\", \"Pair_I\", \"Pair_B\", \"Pair_Z\"]\n"
                    ]
                }
            ],
            "source": [
                "println(\"Number of cycles: \", length(simplecycles(g)))\n",
                "maxCycleLength = maximum([length(x) for x in simplecycles(g)])\n",
                "println(\"Maximum length of cycles: \", maxCycleLength)\n",
                "\n",
                "for cycle in simplecycles(g)\n",
                "    if length(cycle) == maxCycleLength\n",
                "        println([pairs[x] for x in cycle])\n",
                "    end\n",
                "end"
            ]
        },
        {
            "cell_type": "markdown",
            "id": "2fdfb244",
            "metadata": {},
            "source": [
                "## Formulation without bounds for cycle lengths"
            ]
        },
        {
            "cell_type": "markdown",
            "id": "5dba4825",
            "metadata": {},
            "source": [
                "To obtain the cycles maximizing the number of exchanges, I solve the following maximization problem with $w_e = 1$ for all edges $e$:\n",
                "\n",
                "\\begin{align}\n",
                "    &\\max_{\\{y_e\\}} \\sum w_e y_e \\\\ \n",
                "    \\text{subject to} \n",
                "    & \\sum_{e \\in out(v)} y_e \\le \\sum_{e \\in in(v)} y_e \\quad \\text{for every pair $v$} \\\\\n",
                "    & \\sum_{e \\in in(v)} y_e \\le 1 \\quad \\text{for every pair $v$} \\\\\n",
                "    & y_e \\in \\{0, 1 \\}.\n",
                "\\end{align}\n",
                "\n",
                "Here $in(v)$ and $out(v)$ are the sets of edges that point to or are outgoing from a node $v$, respectively.\n",
                "The idea follows a [maximum flow problem](https://en.wikipedia.org/wiki/Maximum_flow_problem) to find a maximum possible flow from a feasible flow network.\n",
                "The first constraint corresponds to the flow conservation constraint, and the second constraint corresponds to the capacity constraint.\n",
                "\n",
                "Note that this does not impose any restriction on the length of cycles.\n",
                "The code below solves this problem."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 38,
            "id": "e53b60d7",
            "metadata": {},
            "outputs": [],
            "source": [
                "max_flow = Model(optimizer_with_attributes(HiGHS.Optimizer, \"output_flag\" => false))\n",
                "@variable(max_flow, 1 >= f[1:numPairs, 1:numPairs] >= 0, Int)\n",
                "# Capacity constraints\n",
                "@constraint(max_flow, [i = 1:numPairs, j = 1:numPairs], f[i, j] <= pairCompatibilityMatrix[i, j])\n",
                "@constraint(max_flow, [j = 1:numPairs], sum(f[:, j]) <= 1)\n",
                "# Flow conservation constraints\n",
                "@constraint(max_flow, [i = 1:numPairs], sum(f[i, :]) <= sum(f[:, i]))\n",
                "\n",
                "@objective(max_flow, Max, sum(f))\n",
                "optimize!(max_flow)\n"
            ]
        },
        {
            "cell_type": "markdown",
            "id": "4121f8e9",
            "metadata": {},
            "source": [
                "The solution is as follows.\n",
                "We can see that there is a very long cycle, which is operationally difficult since operations in a cycle need to be conducted simultaneously to prevent back-out of donors.\n",
                "In the next part, I introduce a recursive algorithm to impose such cycle length restrictions."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 40,
            "id": "7381e143",
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "Number of exchanges: 16\n",
                        "[\"Pair_A\", \"Pair_E\", \"Pair_Q\", \"Pair_D\", \"Pair_N\", \"Pair_G\", \"Pair_V\", \"Pair_H\", \"Pair_U\", \"Pair_I\", \"Pair_B\", \"Pair_Z\"]\n",
                        "[\"Pair_K\", \"Pair_X\"]\n",
                        "[\"Pair_O\", \"Pair_P\"]\n"
                    ]
                }
            ],
            "source": [
                "println(\"Number of exchanges: \", Int(objective_value(max_flow)))\n",
                "for cycle in simplecycles(SimpleDiGraph(value.(f)))\n",
                "    println([pairs[x] for x in cycle])\n",
                "end"
            ]
        },
        {
            "cell_type": "markdown",
            "id": "7d98093c",
            "metadata": {},
            "source": [
                "## Recursive formulation"
            ]
        },
        {
            "cell_type": "markdown",
            "id": "42584ac3",
            "metadata": {},
            "source": [
                "To avoid to have long cycles in a solution, after solving the maximization problem, I add the following restriction:\n",
                "$$\n",
                "    \\text{For each cycle $C$ whose length $> C_{max}$ }, \\sum_{e \\in C} y_e \\ge |C| - 1.\n",
                "$$\n",
                "This guarantees that the long cycle found in the previous optimization problem is removed from the following solution.\n",
                "I recursively add such constraint until I do not get any long cycles.\n",
                "The code below implements this idea, where the maximum length of cycle is set to be 3."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 41,
            "id": "7dee3df7",
            "metadata": {},
            "outputs": [],
            "source": [
                "cycleAdjacencyMatrixList = [];\n",
                "cycleLengthList = [];\n",
                "maxCycleLength = 3;\n",
                "optimalSolutionMatrix = zeros(Int, numPairs, numPairs);\n",
                "iter = 0;\n",
                "numExchanges = 0;"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 42,
            "id": "f3857d0a",
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        " 15.614115 seconds (9.84 M allocations: 583.603 MiB, 8.91% gc time, 2.21% compilation time)\n"
                    ]
                }
            ],
            "source": [
                "@time begin\n",
                "    while true\n",
                "\n",
                "        iter += 1\n",
                "        \n",
                "        max_flow = Model(optimizer_with_attributes(HiGHS.Optimizer, \"output_flag\" => false))\n",
                "        @variable(max_flow, 1 >= f[1:numPairs, 1:numPairs] >= 0, Int)\n",
                "        # Capacity constraints\n",
                "        @constraint(max_flow, [i = 1:numPairs, j = 1:numPairs], f[i, j] <= pairCompatibilityMatrix[i, j])\n",
                "        @constraint(max_flow, [i = 1:numPairs], sum(f[i, :]) <= sum(f[:, i]))\n",
                "        # Flow conservation constraints\n",
                "        @constraint(max_flow, [j = 1:numPairs], sum(f[:, j]) <= 1)\n",
                "\n",
                "        for (k, cycleAdjacencyMatrix) in enumerate(cycleAdjacencyMatrixList)\n",
                "            @constraint(\n",
                "                max_flow, \n",
                "                sum(f[i, j] * cycleAdjacencyMatrix[i, j] for i = 1:numPairs, j = 1:numPairs) <= cycleLengthList[k] - 1\n",
                "            )\n",
                "        end\n",
                "\n",
                "        @objective(max_flow, Max, sum(f))\n",
                "        optimize!(max_flow)\n",
                "\n",
                "        cycles = simplecycles(SimpleDiGraph(value.(f)));\n",
                "        if all([length(x) <= maxCycleLength for x in cycles]) == true\n",
                "            numExchanges = objective_value(max_flow)\n",
                "            optimalSolutionMatrix = value.(f)\n",
                "            break\n",
                "        else\n",
                "            for cycle in cycles\n",
                "                if length(cycle) > maxCycleLength\n",
                "                    tmpMat = SimpleDiGraph(zeros(numPairs, numPairs))\n",
                "                    for i = 1:(length(cycle) - 1)\n",
                "                        add_edge!(tmpMat, cycle[i], cycle[i + 1])\n",
                "                    end\n",
                "                    add_edge!(tmpMat, cycle[end], cycle[1])\n",
                "                    push!(cycleAdjacencyMatrixList, Matrix(adjacency_matrix(tmpMat)))\n",
                "                    push!(cycleLengthList, length(cycle))\n",
                "                end\n",
                "            end\n",
                "        end\n",
                "\n",
                "    end\n",
                "end"
            ]
        },
        {
            "cell_type": "markdown",
            "id": "4d7c1197",
            "metadata": {},
            "source": [
                "As a result of the optimization, I obtain the following cycles.\n",
                "The maximum length of the cycle is 3, but the number of exchanges decreased from 16 to 9.\n",
                "This is the cost of limited the length of cycles."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 43,
            "id": "006b1cb8",
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "Number of iterations: 85\n",
                        "[\"Pair_A\", \"Pair_E\", \"Pair_I\"]\n",
                        "[\"Pair_G\", \"Pair_V\"]\n",
                        "[\"Pair_K\", \"Pair_X\"]\n",
                        "[\"Pair_O\", \"Pair_P\"]\n",
                        "Number of exchanges: 9\n"
                    ]
                }
            ],
            "source": [
                "println(\"Number of iterations: \", iter)\n",
                "for cycle in simplecycles(SimpleDiGraph(optimalSolutionMatrix))\n",
                "    println([pairs[x] for x in cycle])\n",
                "end\n",
                "println(\"Number of exchanges: \", Int(numExchanges))"
            ]
        },
        {
            "cell_type": "markdown",
            "id": "e7e271d3",
            "metadata": {},
            "source": [
                "## Introduce an altruistic donor"
            ]
        },
        {
            "cell_type": "markdown",
            "id": "40c29301",
            "metadata": {},
            "source": [
                "Next, I introduce a non-directed donor (DNN), who does not need a kidney from someone else.\n",
                "This person can initiate a *chain*, which does not have to be a cycle.\n",
                "So that the solutions are cycles, I generate fake arrows from all the other donors to the DNN in the directed graph.\n",
                "In the example below, Pair A is the DNN.\n",
                "Also, I add the following restriction for the DNN so that he does not join to multiple cycles:\n",
                "$$\n",
                "    \\sum_{e \\in out(v)} y_e \\le 1.\n",
                "$$"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 44,
            "id": "9b8dcf39",
            "metadata": {},
            "outputs": [],
            "source": [
                "cycleAdjacencyMatrixList = [];\n",
                "cycleLengthList = [];\n",
                "maxCycleLength = 3;\n",
                "optimalSolutionMatrix = zeros(Int, numPairs, numPairs);\n",
                "iter = 0;\n",
                "numExchanges = 0;"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 45,
            "id": "464702b0",
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "  1.069255 seconds (213.03 k allocations: 15.769 MiB, 37.29% gc time, 85.78% compilation time)\n"
                    ]
                }
            ],
            "source": [
                "@time begin\n",
                "    while true\n",
                "    \n",
                "        iter += 1\n",
                "\n",
                "        max_flow = Model(optimizer_with_attributes(HiGHS.Optimizer, \"output_flag\" => false))\n",
                "        @variable(max_flow, 1 >= f[1:numPairs, 1:numPairs] >= 0, Int)\n",
                "        # Capacity constraints\n",
                "        @constraint(max_flow, sum(f[1, :]) <= 1) # For a DDN\n",
                "        @constraint(max_flow, [j = 1:numPairs], sum(f[:, j]) <= 1)\n",
                "        @constraint(max_flow, [i = 1:numPairs, j = 1:numPairs], f[i, j] <= pairCompatibilityMatrix[i, j])\n",
                "        # Flow conservation constraints\n",
                "        @constraint(max_flow, [i = 1:numPairs], sum(f[i, :]) <= sum(f[:, i]))\n",
                "\n",
                "        for (k, cycleAdjacencyMatrix) in enumerate(cycleAdjacencyMatrixList)\n",
                "            @constraint(\n",
                "                max_flow, \n",
                "                sum(f[i, j] * cycleAdjacencyMatrix[i, j] for i = 1:numPairs, j = 1:numPairs) <= cycleLengthList[k] - 1\n",
                "            )\n",
                "        end\n",
                "\n",
                "        @objective(max_flow, Max, sum(f))\n",
                "        optimize!(max_flow)\n",
                "\n",
                "        cycles = simplecycles(SimpleDiGraph(value.(f)));\n",
                "        if all([(length(x) <= maxCycleLength) | (1 in x) for x in cycles]) == true\n",
                "            numExchanges = objective_value(max_flow)\n",
                "            optimalSolutionMatrix = value.(f)\n",
                "            break\n",
                "        else\n",
                "            for cycle in cycles\n",
                "                if (length(cycle) > maxCycleLength) & !(1 in cycle)\n",
                "                    tmpMat = SimpleDiGraph(zeros(numPairs, numPairs))\n",
                "                    for i = 1:(length(cycle) - 1)\n",
                "                        add_edge!(tmpMat, cycle[i], cycle[i + 1])\n",
                "                    end\n",
                "                    add_edge!(tmpMat, cycle[end], cycle[1])\n",
                "                    push!(cycleAdjacencyMatrixList, Matrix(adjacency_matrix(tmpMat)))\n",
                "                    push!(cycleLengthList, length(cycle))\n",
                "                end\n",
                "            end\n",
                "        end\n",
                "\n",
                "    end\n",
                "end"
            ]
        },
        {
            "cell_type": "markdown",
            "id": "60661bad",
            "metadata": {},
            "source": [
                "The solution has a long chain (I know that this is a chain because the DNN is included in the cycle) and two short cycles.\n",
                "This results in 16 exchanges, larger than the number of exchanges without a DNN.\n",
                "This example shows the value of having a DNN in a platform for more kidney exchanges."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 33,
            "id": "cb485070",
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": [
                        "Number of iterations: 1\n",
                        "[\"Pair_A\", \"Pair_E\", \"Pair_Q\", \"Pair_D\", \"Pair_N\", \"Pair_G\", \"Pair_V\", \"Pair_H\", \"Pair_U\", \"Pair_I\", \"Pair_B\", \"Pair_Z\"]\n",
                        "[\"Pair_K\", \"Pair_X\"]\n",
                        "[\"Pair_O\", \"Pair_P\"]\n",
                        "Number of exchanges: 16\n"
                    ]
                }
            ],
            "source": [
                "println(\"Number of iterations: \", iter)\n",
                "for cycle in simplecycles(SimpleDiGraph(optimalSolutionMatrix))\n",
                "    println([pairs[x] for x in cycle])\n",
                "end\n",
                "println(\"Number of exchanges: \", Int(numExchanges))"
            ]
        }
    ],
    "metadata": {
        "kernelspec": {
            "display_name": "Julia 1.9.3",
            "language": "julia",
            "name": "julia-1.9"
        },
        "language_info": {
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