```python
import csv
from pulp import *
```

\section{Fantasy Football}

In fantasy football every participant can assemble a team, that consists of

\begin{itemize}
\item 1 $\times$ Quarterback
\item 1 $\times$ Tight end
\item 2 $\times$ Running backs
\item 3 $\times$ Wide receivers
\item 1 $\times$ Defense & Special teams
\item 1 $\times$ Flex
\end{itemize}

Every Position is awarded points through a pre-defined rating system;
e.g. Rushing yards, Touchdowns etc.

Every draft has a salary/cost.
When picking a team, the salary/cost must not exceed the salary cap.

\section{The Data}

In this project, we consider fantasy football facilitated by https://www.draftkings.co.uk/.
The salary cap here is $50.000$, and we can download the data (like salary and position) of a player from there

```python
with open('DKSalaries.csv', 'r') as f:
            reader = list(csv.reader(f))
reader
```

    [['QB', 'RB', 'RB', 'WR', 'WR', 'WR', 'TE', 'FLEX', 'DST', '', 'Instructions'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '1. Locate the player you want to select in the list below '],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '2. Copy the ID of your player (you can use the Name + ID column or the ID column) '],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '3. Paste the ID into the roster position desired '],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      "4. You must include an ID for each player; you cannot use just the player's name "],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '5. You can create up to 500 lineups per file '],
     [' '],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'Position',
      'Name + ID',
      'Name',
      'ID',
      'Roster Position',
      'Salary',
      'Game Info',
      'TeamAbbrev',
      'AvgPointsPerGame'],
     ['',
      '',
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      '',
      '',
      '',
      '',
      '',
      'RB',
      'Christian McCaffrey (16006156)',
      'Christian McCaffrey',
      '16006156',
      'RB/FLEX',
      '9200',
      'CAR@WAS 12/27/2020 04:05PM ET',
      'CAR',
      '30.13'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'WR',
      'Tyreek Hill (16006352)',
      'Tyreek Hill',
      '16006352',
      'WR/FLEX',
      '9000',
      'ATL@KC 12/27/2020 01:00PM ET',
      'KC',
      '23.39'],
     ['',
      '',
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      '',
      '',
      '',
      'QB',
      'Patrick Mahomes (16006097)',
      'Patrick Mahomes',
      '16006097',
      'QB',
      '8500',
      'ATL@KC 12/27/2020 01:00PM ET',
      'KC',
      '27.80'],
     ['',
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      '',
      '',
      '',
      'WR',
      'Calvin Ridley (16006354)',
      'Calvin Ridley',
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      'WR/FLEX',
      '8500',
      'ATL@KC 12/27/2020 01:00PM ET',
      'ATL',
      '21.12'],
     ['',
      '',
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      '',
      '',
      '',
      '',
      'TE',
      'Travis Kelce (16006636)',
      'Travis Kelce',
      '16006636',
      'TE/FLEX',
      '8500',
      'ATL@KC 12/27/2020 01:00PM ET',
      'KC',
      '22.07'],
     ['',
      '',
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      '',
      '',
      '',
      'QB',
      'Lamar Jackson (16006098)',
      'Lamar Jackson',
      '16006098',
      'QB',
      '8000',
      'NYG@BAL 12/27/2020 01:00PM ET',
      'BAL',
      '23.40'],
     ['',
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      '',
      'RB',
      'Nick Chubb (16006158)',
      'Nick Chubb',
      '16006158',
      'RB/FLEX',
      '7800',
      'CLE@NYJ 12/27/2020 01:00PM ET',
      'CLE',
      '18.93'],
     ['',
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      '',
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      '',
      'WR',
      'DK Metcalf (16006356)',
      'DK Metcalf',
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      'WR/FLEX',
      '7800',
      'LAR@SEA 12/27/2020 04:25PM ET',
      'SEA',
      '19.31'],
     ['',
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      '',
      'RB',
      'David Montgomery (16006160)',
      'David Montgomery',
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      'RB/FLEX',
      '7700',
      'CHI@JAX 12/27/2020 01:00PM ET',
      'CHI',
      '17.42'],
     ['',
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      '',
      'WR',
      'Allen Robinson II (16006358)',
      'Allen Robinson II',
      '16006358',
      'WR/FLEX',
      '7700',
      'CHI@JAX 12/27/2020 01:00PM ET',
      'CHI',
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     ['',
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      '',
      'RB',
      'Austin Ekeler (16006162)',
      'Austin Ekeler',
      '16006162',
      'RB/FLEX',
      '7600',
      'DEN@LAC 12/27/2020 04:05PM ET',
      'LAC',
      '16.55'],
     ['',
      '',
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      '',
      '',
      '',
      'QB',
      'Deshaun Watson (16006099)',
      'Deshaun Watson',
      '16006099',
      'QB',
      '7600',
      'CIN@HOU 12/27/2020 01:00PM ET',
      'HOU',
      '24.91'],
     ['',
      '',
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      '',
      '',
      '',
      '',
      '',
      'WR',
      'Keenan Allen (16006360)',
      'Keenan Allen',
      '16006360',
      'WR/FLEX',
      '7500',
      'DEN@LAC 12/27/2020 04:05PM ET',
      'LAC',
      '18.51'],
     ['',
      '',
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      '',
      '',
      '',
      '',
      'QB',
      'Justin Herbert (16006100)',
      'Justin Herbert',
      '16006100',
      'QB',
      '7400',
      'DEN@LAC 12/27/2020 04:05PM ET',
      'LAC',
      '24.24'],
     ['',
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      '',
      'RB',
      'Jonathan Taylor (16006164)',
      'Jonathan Taylor',
      '16006164',
      'RB/FLEX',
      '7300',
      'IND@PIT 12/27/2020 01:00PM ET',
      'IND',
      '15.54'],
     ['',
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      '',
      'QB',
      'Russell Wilson (16006101)',
      'Russell Wilson',
      '16006101',
      'QB',
      '7300',
      'LAR@SEA 12/27/2020 04:25PM ET',
      'SEA',
      '25.27'],
     ['',
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      '',
      '',
      'WR',
      'Robert Woods (16006362)',
      'Robert Woods',
      '16006362',
      'WR/FLEX',
      '7000',
      'LAR@SEA 12/27/2020 04:25PM ET',
      'LAR',
      '16.83'],
     ['',
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      '',
      '',
      '',
      '',
      'QB',
      'Jalen Hurts (16006102)',
      'Jalen Hurts',
      '16006102',
      'QB',
      '7000',
      'PHI@DAL 12/27/2020 04:25PM ET',
      'PHI',
      '6.88'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'RB',
      'Miles Sanders (16006166)',
      'Miles Sanders',
      '16006166',
      'RB/FLEX',
      '7000',
      'PHI@DAL 12/27/2020 04:25PM ET',
      'PHI',
      '14.54'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'WR',
      'Jarvis Landry (16006364)',
      'Jarvis Landry',
      '16006364',
      'WR/FLEX',
      '6900',
      'CLE@NYJ 12/27/2020 01:00PM ET',
      'CLE',
      '12.47'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'WR',
      'Julio Jones (16006366)',
      'Julio Jones',
      '16006366',
      'WR/FLEX',
      '6800',
      'ATL@KC 12/27/2020 01:00PM ET',
      'ATL',
      '17.23'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'RB',
      'James Robinson (16006168)',
      'James Robinson',
      '16006168',
      'RB/FLEX',
      '6800',
      'CHI@JAX 12/27/2020 01:00PM ET',
      'JAX',
      '18.81'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'WR',
      'Terry McLaurin (16006368)',
      'Terry McLaurin',
      '16006368',
      'WR/FLEX',
      '6700',
      'CAR@WAS 12/27/2020 04:05PM ET',
      'WAS',
      '15.49'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'RB',
      'Ezekiel Elliott (16006170)',
      'Ezekiel Elliott',
      '16006170',
      'RB/FLEX',
      '6700',
      'PHI@DAL 12/27/2020 04:25PM ET',
      'DAL',
      '15.28'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'RB',
      'Antonio Gibson (16006172)',
      'Antonio Gibson',
      '16006172',
      'RB/FLEX',
      '6600',
      'CAR@WAS 12/27/2020 04:05PM ET',
      'WAS',
      '15.93'],
     ['',
      '',
      '',
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      '',
      '',
      '',
      'WR',
      'Cooper Kupp (16006370)',
      'Cooper Kupp',
      '16006370',
      'WR/FLEX',
      '6600',
      'LAR@SEA 12/27/2020 04:25PM ET',
      'LAR',
      '14.58'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'RB',
      'Mike Davis (16006174)',
      'Mike Davis',
      '16006174',
      'RB/FLEX',
      '6500',
      'CAR@WAS 12/27/2020 04:05PM ET',
      'CAR',
      '14.19'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'WR',
      'Tyler Lockett (16006372)',
      'Tyler Lockett',
      '16006372',
      'WR/FLEX',
      '6500',
      'LAR@SEA 12/27/2020 04:25PM ET',
      'SEA',
      '16.5'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'RB',
      'Tony Pollard (16006176)',
      'Tony Pollard',
      '16006176',
      'RB/FLEX',
      '6500',
      'PHI@DAL 12/27/2020 04:25PM ET',
      'DAL',
      '7.96'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'RB',
      'J.D. McKissic (16006180)',
      'J.D. McKissic',
      '16006180',
      'RB/FLEX',
      '6400',
      'CAR@WAS 12/27/2020 04:05PM ET',
      'WAS',
      '11.51'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'QB',
      'Ben Roethlisberger (16006103)',
      'Ben Roethlisberger',
      '16006103',
      'QB',
      '6400',
      'IND@PIT 12/27/2020 01:00PM ET',
      'PIT',
      '18.91'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'RB',
      'Chris Carson (16006178)',
      'Chris Carson',
      '16006178',
      'RB/FLEX',
      '6400',
      'LAR@SEA 12/27/2020 04:25PM ET',
      'SEA',
      '16.66'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'WR',
      'Diontae Johnson (16006374)',
      'Diontae Johnson',
      '16006374',
      'WR/FLEX',
      '6300',
      'IND@PIT 12/27/2020 01:00PM ET',
      'PIT',
      '14.97'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'RB',
      'Cam Akers (16006182)',
      'Cam Akers',
      '16006182',
      'RB/FLEX',
      '6300',
      'LAR@SEA 12/27/2020 04:25PM ET',
      'LAR',
      '8.56'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'RB',
      'J.K. Dobbins (16006184)',
      'J.K. Dobbins',
      '16006184',
      'RB/FLEX',
      '6200',
      'NYG@BAL 12/27/2020 01:00PM ET',
      'BAL',
      '9.98'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'WR',
      'Brandin Cooks (16006376)',
      'Brandin Cooks',
      '16006376',
      'WR/FLEX',
      '6200',
      'CIN@HOU 12/27/2020 01:00PM ET',
      'HOU',
      '12.95'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'QB',
      'Baker Mayfield (16006104)',
      'Baker Mayfield',
      '16006104',
      'QB',
      '6100',
      'CLE@NYJ 12/27/2020 01:00PM ET',
      'CLE',
      '17.17'],
     ['',
      '',
      '',
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      '',
      '',
      '',
      '',
      'RB',
      'David Johnson (16006186)',
      'David Johnson',
      '16006186',
      'RB/FLEX',
      '6100',
      'CIN@HOU 12/27/2020 01:00PM ET',
      'HOU',
      '13.36'],
     ['',
      '',
      '',
      '',
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      '',
      '',
      '',
      '',
      '',
      'WR',
      'JuJu Smith-Schuster (16006378)',
      'JuJu Smith-Schuster',
      '16006378',
      'WR/FLEX',
      '6000',
      'IND@PIT 12/27/2020 01:00PM ET',
      'PIT',
      '13.71'],
     ['',
      '',
      '',
      '',
      '',
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      '',
      '',
      '',
      '',
      'RB',
      'Clyde Edwards-Helaire (16006188)',
      'Clyde Edwards-Helaire',
      '16006188',
      'RB/FLEX',
      '6000',
      'ATL@KC 12/27/2020 01:00PM ET',
      'KC',
      '14'],
     ['',
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      '',
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      '',
      '',
      '',
      '',
      '',
      'WR',
      'Chase Claypool (16006380)',
      'Chase Claypool',
      '16006380',
      'WR/FLEX',
      '5900',
      'IND@PIT 12/27/2020 01:00PM ET',
      'PIT',
      '13.46'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'RB',
      'Kareem Hunt (16006190)',
      'Kareem Hunt',
      '16006190',
      'RB/FLEX',
      '5900',
      'CLE@NYJ 12/27/2020 01:00PM ET',
      'CLE',
      '14.44'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'QB',
      'Jared Goff (16006105)',
      'Jared Goff',
      '16006105',
      'QB',
      '5900',
      'LAR@SEA 12/27/2020 04:25PM ET',
      'LAR',
      '18.88'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'WR',
      'DJ Moore (16006382)',
      'DJ Moore',
      '16006382',
      'WR/FLEX',
      '5800',
      'CAR@WAS 12/27/2020 04:05PM ET',
      'CAR',
      '15.13'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'RB',
      'James Conner (16006192)',
      'James Conner',
      '16006192',
      'RB/FLEX',
      '5800',
      'IND@PIT 12/27/2020 01:00PM ET',
      'PIT',
      '13.16'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'QB',
      'Matt Ryan (16006106)',
      'Matt Ryan',
      '16006106',
      'QB',
      '5800',
      'ATL@KC 12/27/2020 01:00PM ET',
      'ATL',
      '18.92'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'RB',
      "Le'Veon Bell (16006194)",
      "Le'Veon Bell",
      '16006194',
      'RB/FLEX',
      '5800',
      'ATL@KC 12/27/2020 01:00PM ET',
      'KC',
      '6.97'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'RB',
      'Wayne Gallman Jr. (16006196)',
      'Wayne Gallman Jr.',
      '16006196',
      'RB/FLEX',
      '5700',
      'NYG@BAL 12/27/2020 01:00PM ET',
      'NYG',
      '10.48'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'WR',
      'Marquise Brown (16006386)',
      'Marquise Brown',
      '16006386',
      'WR/FLEX',
      '5700',
      'NYG@BAL 12/27/2020 01:00PM ET',
      'BAL',
      '10.89'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Mark Andrews (16006638)',
      'Mark Andrews',
      '16006638',
      'TE/FLEX',
      '5700',
      'NYG@BAL 12/27/2020 01:00PM ET',
      'BAL',
      '12.48'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'QB',
      'Mitchell Trubisky (16006107)',
      'Mitchell Trubisky',
      '16006107',
      'QB',
      '5700',
      'CHI@JAX 12/27/2020 01:00PM ET',
      'CHI',
      '15.99'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'WR',
      'Amari Cooper (16006384)',
      'Amari Cooper',
      '16006384',
      'WR/FLEX',
      '5700',
      'PHI@DAL 12/27/2020 04:25PM ET',
      'DAL',
      '15.69'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'QB',
      'Philip Rivers (16006108)',
      'Philip Rivers',
      '16006108',
      'QB',
      '5600',
      'IND@PIT 12/27/2020 01:00PM ET',
      'IND',
      '16.91'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'RB',
      'Melvin Gordon III (16006198)',
      'Melvin Gordon III',
      '16006198',
      'RB/FLEX',
      '5600',
      'DEN@LAC 12/27/2020 04:05PM ET',
      'DEN',
      '13.81'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'WR',
      'Robby Anderson (16006388)',
      'Robby Anderson',
      '16006388',
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      'DEN@LAC 12/27/2020 04:05PM ET',
      'DEN',
      '0.62'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Austin Fort (16006774)',
      'Austin Fort',
      '16006774',
      'TE/FLEX',
      '2500',
      'DEN@LAC 12/27/2020 04:05PM ET',
      'DEN',
      '0'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Andrew Beck (16006776)',
      'Andrew Beck',
      '16006776',
      'TE/FLEX',
      '2500',
      'DEN@LAC 12/27/2020 04:05PM ET',
      'DEN',
      '0'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Chris Herndon (16006744)',
      'Chris Herndon',
      '16006744',
      'TE/FLEX',
      '2500',
      'CLE@NYJ 12/27/2020 01:00PM ET',
      'NYJ',
      '3.91'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Ryan Griffin (16006746)',
      'Ryan Griffin',
      '16006746',
      'TE/FLEX',
      '2500',
      'CLE@NYJ 12/27/2020 01:00PM ET',
      'NYJ',
      '1.96'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Trevon Wesco (16006748)',
      'Trevon Wesco',
      '16006748',
      'TE/FLEX',
      '2500',
      'CLE@NYJ 12/27/2020 01:00PM ET',
      'NYJ',
      '0.5'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Daniel Brown (16006750)',
      'Daniel Brown',
      '16006750',
      'TE/FLEX',
      '2500',
      'CLE@NYJ 12/27/2020 01:00PM ET',
      'NYJ',
      '0'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Ross Travis (16006752)',
      'Ross Travis',
      '16006752',
      'TE/FLEX',
      '2500',
      'CLE@NYJ 12/27/2020 01:00PM ET',
      'NYJ',
      '0'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Harrison Bryant (16006754)',
      'Harrison Bryant',
      '16006754',
      'TE/FLEX',
      '2500',
      'CLE@NYJ 12/27/2020 01:00PM ET',
      'CLE',
      '4.76'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'David Njoku (16006756)',
      'David Njoku',
      '16006756',
      'TE/FLEX',
      '2500',
      'CLE@NYJ 12/27/2020 01:00PM ET',
      'CLE',
      '4.71'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Stephen Carlson (16006758)',
      'Stephen Carlson',
      '16006758',
      'TE/FLEX',
      '2500',
      'CLE@NYJ 12/27/2020 01:00PM ET',
      'CLE',
      '0.68'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Eric Tomlinson (16006736)',
      'Eric Tomlinson',
      '16006736',
      'TE/FLEX',
      '2500',
      'NYG@BAL 12/27/2020 01:00PM ET',
      'BAL',
      '0'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Nick Boyle (16006738)',
      'Nick Boyle',
      '16006738',
      'TE/FLEX',
      '2500',
      'NYG@BAL 12/27/2020 01:00PM ET',
      'BAL',
      '4.14'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Kaden Smith (16006740)',
      'Kaden Smith',
      '16006740',
      'TE/FLEX',
      '2500',
      'NYG@BAL 12/27/2020 01:00PM ET',
      'NYG',
      '2.05'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Levine Toilolo (16006742)',
      'Levine Toilolo',
      '16006742',
      'TE/FLEX',
      '2500',
      'NYG@BAL 12/27/2020 01:00PM ET',
      'NYG',
      '1.09'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'DST',
      'Giants  (16006825)',
      'Giants ',
      '16006825',
      'DST',
      '2500',
      'NYG@BAL 12/27/2020 01:00PM ET',
      'NYG',
      '6.57'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Cethan Carter (16006722)',
      'Cethan Carter',
      '16006722',
      'TE/FLEX',
      '2500',
      'CIN@HOU 12/27/2020 01:00PM ET',
      'CIN',
      '0.66'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Mason Schreck (16006724)',
      'Mason Schreck',
      '16006724',
      'TE/FLEX',
      '2500',
      'CIN@HOU 12/27/2020 01:00PM ET',
      'CIN',
      '0'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'C.J. Uzomah (16006726)',
      'C.J. Uzomah',
      '16006726',
      'TE/FLEX',
      '2500',
      'CIN@HOU 12/27/2020 01:00PM ET',
      'CIN',
      '11.35'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Clark Harris (16006728)',
      'Clark Harris',
      '16006728',
      'TE/FLEX',
      '2500',
      'CIN@HOU 12/27/2020 01:00PM ET',
      'CIN',
      '0'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Pharaoh Brown (16006730)',
      'Pharaoh Brown',
      '16006730',
      'TE/FLEX',
      '2500',
      'CIN@HOU 12/27/2020 01:00PM ET',
      'HOU',
      '2.82'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Darren Fells (16006732)',
      'Darren Fells',
      '16006732',
      'TE/FLEX',
      '2500',
      'CIN@HOU 12/27/2020 01:00PM ET',
      'HOU',
      '6.09'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Kahale Warring (16006734)',
      'Kahale Warring',
      '16006734',
      'TE/FLEX',
      '2500',
      'CIN@HOU 12/27/2020 01:00PM ET',
      'HOU',
      '1.73'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      "James O'Shaughnessy (16006704)",
      "James O'Shaughnessy",
      '16006704',
      'TE/FLEX',
      '2500',
      'CHI@JAX 12/27/2020 01:00PM ET',
      'JAX',
      '3.72'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Eric Saubert (16006706)',
      'Eric Saubert',
      '16006706',
      'TE/FLEX',
      '2500',
      'CHI@JAX 12/27/2020 01:00PM ET',
      'JAX',
      '0.68'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Tyler Davis (16006708)',
      'Tyler Davis',
      '16006708',
      'TE/FLEX',
      '2500',
      'CHI@JAX 12/27/2020 01:00PM ET',
      'JAX',
      '0'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Ben Ellefson (16006710)',
      'Ben Ellefson',
      '16006710',
      'TE/FLEX',
      '2500',
      'CHI@JAX 12/27/2020 01:00PM ET',
      'JAX',
      '1.33'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Josh Oliver (16006712)',
      'Josh Oliver',
      '16006712',
      'TE/FLEX',
      '2500',
      'CHI@JAX 12/27/2020 01:00PM ET',
      'JAX',
      '0'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Demetrius Harris (16006714)',
      'Demetrius Harris',
      '16006714',
      'TE/FLEX',
      '2500',
      'CHI@JAX 12/27/2020 01:00PM ET',
      'CHI',
      '1.15'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'J.P. Holtz (16006716)',
      'J.P. Holtz',
      '16006716',
      'TE/FLEX',
      '2500',
      'CHI@JAX 12/27/2020 01:00PM ET',
      'CHI',
      '0'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Darion Clark (16006718)',
      'Darion Clark',
      '16006718',
      'TE/FLEX',
      '2500',
      'CHI@JAX 12/27/2020 01:00PM ET',
      'CHI',
      '0'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Patrick Scales (16006720)',
      'Patrick Scales',
      '16006720',
      'TE/FLEX',
      '2500',
      'CHI@JAX 12/27/2020 01:00PM ET',
      'CHI',
      '0'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Will Dissly (16006692)',
      'Will Dissly',
      '16006692',
      'TE/FLEX',
      '2500',
      'LAR@SEA 12/27/2020 04:25PM ET',
      'SEA',
      '4.11'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Colby Parkinson (16006694)',
      'Colby Parkinson',
      '16006694',
      'TE/FLEX',
      '2500',
      'LAR@SEA 12/27/2020 04:25PM ET',
      'SEA',
      '3.6'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Greg Olsen (16006696)',
      'Greg Olsen',
      '16006696',
      'TE/FLEX',
      '2500',
      'LAR@SEA 12/27/2020 04:25PM ET',
      'SEA',
      '5.14'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Tyler Ott (16006698)',
      'Tyler Ott',
      '16006698',
      'TE/FLEX',
      '2500',
      'LAR@SEA 12/27/2020 04:25PM ET',
      'SEA',
      '0'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Johnny Mundt (16006700)',
      'Johnny Mundt',
      '16006700',
      'TE/FLEX',
      '2500',
      'LAR@SEA 12/27/2020 04:25PM ET',
      'LAR',
      '3.1'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Brycen Hopkins (16006702)',
      'Brycen Hopkins',
      '16006702',
      'TE/FLEX',
      '2500',
      'LAR@SEA 12/27/2020 04:25PM ET',
      'LAR',
      '0'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Richard Rodgers (16006682)',
      'Richard Rodgers',
      '16006682',
      'TE/FLEX',
      '2500',
      'PHI@DAL 12/27/2020 04:25PM ET',
      'PHI',
      '8.81'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Joshua Perkins (16006684)',
      'Joshua Perkins',
      '16006684',
      'TE/FLEX',
      '2500',
      'PHI@DAL 12/27/2020 04:25PM ET',
      'PHI',
      '0'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Blake Bell (16006686)',
      'Blake Bell',
      '16006686',
      'TE/FLEX',
      '2500',
      'PHI@DAL 12/27/2020 04:25PM ET',
      'DAL',
      '2.05'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Sean McKeon (16006688)',
      'Sean McKeon',
      '16006688',
      'TE/FLEX',
      '2500',
      'PHI@DAL 12/27/2020 04:25PM ET',
      'DAL',
      '0'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'TE',
      'Blake Jarwin (16006690)',
      'Blake Jarwin',
      '16006690',
      'TE/FLEX',
      '2500',
      'PHI@DAL 12/27/2020 04:25PM ET',
      'DAL',
      '2.2'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'DST',
      'Cowboys  (16006826)',
      'Cowboys ',
      '16006826',
      'DST',
      '2400',
      'PHI@DAL 12/27/2020 04:25PM ET',
      'DAL',
      '4.43'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'DST',
      'Broncos  (16006827)',
      'Broncos ',
      '16006827',
      'DST',
      '2300',
      'DEN@LAC 12/27/2020 04:05PM ET',
      'DEN',
      '4.64'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'DST',
      'Bengals  (16006828)',
      'Bengals ',
      '16006828',
      'DST',
      '2300',
      'CIN@HOU 12/27/2020 01:00PM ET',
      'CIN',
      '3.21'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'DST',
      'Jaguars  (16006829)',
      'Jaguars ',
      '16006829',
      'DST',
      '2200',
      'CHI@JAX 12/27/2020 01:00PM ET',
      'JAX',
      '3.86'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'DST',
      'Jets  (16006830)',
      'Jets ',
      '16006830',
      'DST',
      '2000',
      'CLE@NYJ 12/27/2020 01:00PM ET',
      'NYJ',
      '4'],
     ['',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      '',
      'DST',
      'Falcons  (16006831)',
      'Falcons ',
      '16006831',
      'DST',
      '1900',
      'ATL@KC 12/27/2020 01:00PM ET',
      'ATL',
      '4.86']]

The projections are downloaded from the following website https://fantasyfootballanalytics.net/

```python
import pandas as pd
pd.read_csv('ffa_customrankings2020.csv').head()
```

<div>
<style scoped>
    .dataframe tbody tr th:only-of-type {
        vertical-align: middle;
    }

    .dataframe tbody tr th {
        vertical-align: top;
    }

    .dataframe thead th {
        text-align: right;
    }
</style>
<table border="1" class="dataframe">
  <thead>
    <tr style="text-align: right;">
      <th></th>
      <th>playerId</th>
      <th>player</th>
      <th>team</th>
      <th>position</th>
      <th>age</th>
      <th>exp</th>
      <th>bye</th>
      <th>points</th>
      <th>lower</th>
      <th>upper</th>
      <th>...</th>
      <th>dropoff</th>
      <th>tier</th>
      <th>ptSpread</th>
      <th>overallECR</th>
      <th>positionECR</th>
      <th>sdRank</th>
      <th>risk</th>
      <th>sleeper</th>
      <th>actualPoints</th>
      <th>salary</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <th>0</th>
      <td>2564832</td>
      <td>Jalen Hurts</td>
      <td>PHI</td>
      <td>QB</td>
      <td>50.0</td>
      <td>0.0</td>
      <td>NaN</td>
      <td>25.076629</td>
      <td>24.980000</td>
      <td>25.126470</td>
      <td>...</td>
      <td>1.756675</td>
      <td>1.0</td>
      <td>0.146470</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
    </tr>
    <tr>
      <th>1</th>
      <td>2558125</td>
      <td>Patrick Mahomes</td>
      <td>KC</td>
      <td>QB</td>
      <td>NaN</td>
      <td>3.0</td>
      <td>NaN</td>
      <td>25.011871</td>
      <td>23.510600</td>
      <td>25.909615</td>
      <td>...</td>
      <td>3.731826</td>
      <td>1.0</td>
      <td>2.399015</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
    </tr>
    <tr>
      <th>2</th>
      <td>2506363</td>
      <td>Aaron Rodgers</td>
      <td>GB</td>
      <td>QB</td>
      <td>37.0</td>
      <td>15.0</td>
      <td>NaN</td>
      <td>21.628036</td>
      <td>19.774000</td>
      <td>23.559190</td>
      <td>...</td>
      <td>0.850655</td>
      <td>2.0</td>
      <td>3.785190</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
    </tr>
    <tr>
      <th>3</th>
      <td>2560757</td>
      <td>Lamar Jackson</td>
      <td>BAL</td>
      <td>QB</td>
      <td>23.0</td>
      <td>2.0</td>
      <td>NaN</td>
      <td>20.932055</td>
      <td>18.058083</td>
      <td>22.349667</td>
      <td>...</td>
      <td>0.546536</td>
      <td>2.0</td>
      <td>4.291584</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
    </tr>
    <tr>
      <th>4</th>
      <td>2558063</td>
      <td>Deshaun Watson</td>
      <td>HOU</td>
      <td>QB</td>
      <td>25.0</td>
      <td>3.0</td>
      <td>NaN</td>
      <td>20.622707</td>
      <td>18.720000</td>
      <td>21.738160</td>
      <td>...</td>
      <td>0.703974</td>
      <td>2.0</td>
      <td>3.018160</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
      <td>NaN</td>
    </tr>
  </tbody>
</table>
<p>5 rows × 22 columns</p>
</div>

The data processing is handled through the custom class Data.

```python
from Data_processing import Data
data = Data()
data.Get_cost('DKSalaries.csv')
data.Get_proj('ffa_customrankings2020.csv',0.5)
data.match_data()
```

\section{Problem and Motivation}

Our goal is it now to find the team with the best possible projected score while staying below the salary cap.

\begin{equation}
\mathrm{max} \, \, \, c^\intercal x\\
\mathrm{s.t.}\,\,\, Ax \stackrel{\leq}{=} b.
\end{equation}

Let $N$ be the number of variables (determined by the number of players and teams in the NFL)

$c \in \mathbb{R}^N$ is the vector containing the projected points of each player

$x \in \{0,1\}^N$ represents the individual players,

$A$ enforces the necessary constraints (e.g. only one Quarterback)

Why do we need this? Could we not just brut force the solution?

Let's see how many combinations $C$ we need to try:

\begin{equation}
C = {\mathrm{\#WR} \brack 3} {\mathrm{\#RB} \brack 2} \times (\mathrm{\#FLX}-5)\times \mathrm{\#QB}\times \mathrm{\#TE}\times\mathrm{\#DST}
\end{equation}

```python
import scipy.special
(scipy.special.binom(len(data.Cost_RB.keys()),2)
 *
 scipy.special.binom(len(data.Cost_WR.keys()),3)
 *
 (len(data.Cost_FLX.keys())-5)*len(data.Cost_QB.keys())
 *
 len(data.Cost_DST.keys()))
```

    21599372793600.0

This is the number of possible teams (some of them will not be admissible, i.e. above the salary cap)

\section{Define the IP}
All variables and constraints are defined using pulp an open-source library for mixed-integer optimisation, that comes with a free solver.

We start by defining the problem variables; they will be saved in a dictionary. In this way, we can index Projections, Costs and our Variables with the same key (namely the name of the Player)

```python
QB = LpVariable.dicts('QB',{qb for qb in data.Cost_QB.keys()}, cat = LpBinary)
TE = LpVariable.dicts('TE',{te for te in data.Cost_TE.keys()}, cat = LpBinary)
RB = LpVariable.dicts('RB',{rb for rb in data.Cost_RB.keys()}, cat = LpBinary)
WR = LpVariable.dicts('WR',{wr for wr in data.Cost_WR.keys()}, cat = LpBinary)
DST = LpVariable.dicts('DST', {dst for dst in data.Cost_DST.keys()},
                       cat = LpBinary)
FLX = LpVariable.dicts('FLX', {flx for flx in data.Cost_FLX.keys()},
                       cat = LpBinary)
```

Initialize the problem

```python
prob = LpProblem('Fantasy Football',LpMaximize)
```

    /Users/jj3217/opt/anaconda3/lib/python3.7/site-packages/pulp/pulp.py:1198: UserWarning: Spaces are not permitted in the name. Converted to '_'
      warnings.warn("Spaces are not permitted in the name. Converted to '_'")

```python
print(DST)
type(DST['Broncos'])
```

    {'Chiefs': DST_Chiefs, 'Cowboys': DST_Cowboys, 'Seahawks': DST_Seahawks, 'Bears': DST_Bears, 'Falcons': DST_Falcons, 'Eagles': DST_Eagles, 'Giants': DST_Giants, 'Texans': DST_Texans, 'Chargers': DST_Chargers, 'Panthers': DST_Panthers, 'Steelers': DST_Steelers, 'Broncos': DST_Broncos, 'Jaguars': DST_Jaguars, 'Browns': DST_Browns, 'Rams': DST_Rams, 'Ravens': DST_Ravens, 'Jets': DST_Jets, 'Bengals': DST_Bengals, 'Colts': DST_Colts}

    pulp.pulp.LpVariable

\subsection{The cost function}
The cost function takes the following form:
\begin{equation}
\sum^N_{i=1} p_i x_i - r \sum^N_{i=1} r_i x_i,
\end{equation}
where $p_i$ denotes the projected score of player/special Team and $x_i$ dentoes the binary variable. We have also introduced the hyperparameter $r$, which penalises risky choicses. The risk of an player/sepcial team $r_i$ is essentially calcualted by looking at the standard deviation)

```python
r = 0.1
prob += (lpSum(QB[qb]*data.Proj_QB[qb] for qb in data.Cost_QB.keys())
         +
         lpSum(TE[te]*data.Proj_TE[te] for te in data.Cost_TE.keys())
         +
         lpSum(RB[rb]*data.Proj_RB[rb] for rb in data.Cost_RB.keys())
         +
         lpSum(WR[wr]*data.Proj_WR[wr] for wr in data.Cost_WR.keys())
         +
         lpSum([DST[dst]*data.Proj_DST[dst] for dst in data.Cost_DST.keys()])
         +
         lpSum([FLX[flx]*data.Proj_FLX[flx] for flx in data.Cost_FLX.keys()])
         -
         r*(lpSum(QB[qb]*data.Risk_QB[qb] for qb in data.Cost_QB.keys())
            +
            lpSum(TE[te]*data.Risk_TE[te] for te in data.Cost_TE.keys())
            +
            lpSum(RB[rb]*data.Risk_RB[rb] for rb in data.Cost_RB.keys())
            +
            lpSum(WR[wr]*data.Risk_WR[wr] for wr in data.Cost_WR.keys())
            +
            lpSum([DST[dst]*data.Risk_DST[dst] for dst in data.Cost_DST.keys()])
            +
            lpSum([FLX[flx]*data.Risk_FLX[flx] for flx
                   in data.Cost_FLX.keys()])))

```

\subsection{The Constraints}
First we habe to mae sure that the exact number of required players is chosen e.g.
\begin{equation}
\sum_{x_i \mathrm{is} \, \mathrm{WR}} x_i = 3
\end{equation}

```python
prob +=(lpSum(QB[qb] for qb in data.Cost_QB.keys()) == 1)
prob +=(lpSum(TE[te] for te in data.Cost_TE.keys()) == 1)
prob +=(lpSum(RB[rb] for rb in data.Cost_RB.keys()) == 2)
prob +=(lpSum(WR[wr] for wr in data.Cost_WR.keys()) == 3 )
prob +=(lpSum(DST[dst] for dst in data.Cost_DST.keys()) == 1 )
prob +=(lpSum(FLX[flx] for flx in data.Cost_FLX.keys()) == 1 )
```

The next constraint makes sure that we do not select the same player as a Wide Receiver/Running back and Flex. We see again why storing the variables in a dictionary is useful

```python
for wr in data.Cost_WR.keys():
    prob += (FLX[wr]+WR[wr] <= 1)
for rb in  data.Cost_RB.keys():
    prob += (RB[rb]+FLX[rb] <= 1)
```

We now make sure that the lineup stays below the salary cap
\begin{equation}
\sum^N_{i=1} \mathrm{salary}_i x_i \leq 50.000
\end{equation}

```python
salary_cap = 50000
prob += (lpSum(QB[qb]*data.Cost_QB[qb] for qb in data.Cost_QB.keys())
         +
         lpSum(TE[te]*data.Cost_TE[te] for te in data.Cost_TE.keys())
         +
         lpSum(RB[rb]*data.Cost_RB[rb] for rb in data.Cost_RB.keys())
         +
         lpSum(WR[wr]*data.Cost_WR[wr] for wr in data.Cost_WR.keys())
         +
         lpSum(DST[dst]*data.Cost_DST[dst] for dst in data.Cost_DST.keys())
         +
         lpSum(FLX[flx]*data.Cost_FLX[flx] for flx in data.Cost_FLX.keys())
         <= salary_cap)

```

This constraint is optional and again depends on the Hyperparameter Max_per_team.
It enforces that our lineup contains a maximum of Max_per_team players from one team.

```python
max_per_team =2
for t in data.Teams:
    prob += (lpSum(QB[qb] for qb in data.Cost_QB.keys()
                   if data.Player_Team[qb] == t)
             +
             lpSum(TE[te] for te in data.Cost_TE.keys()
                   if data.Player_Team[te] == t)
             +
             lpSum(RB[rb] for rb in data.Cost_RB.keys()
                   if data.Player_Team[rb] == t)
             +
             lpSum(WR[wr] for wr in data.Cost_WR.keys()
                     if data.Player_Team[wr] == t)
             +
             lpSum(DST[dst] for dst in data.Cost_DST.keys()
                   if data.Player_Team[dst] == t)
             +
             lpSum(FLX[flx] for flx in data.Cost_FLX.keys()
                   if data.Player_Team[flx] == t)
        <= max_per_team)
```

We now call the solver provided by pulp that solves the optimisation problem in no time.

```python
prob.solve()
```

    1

This is the optimal team for the given week:

```python
for v in prob.variables():
    if v.varValue > 0:
        print( v.name, "=", v.varValue)

```

    DST_Texans = 1.0
    FLX_David_Johnson = 1.0
    QB_Jalen_Hurts = 1.0
    RB_Austin_Ekeler = 1.0
    RB_Nick_Chubb = 1.0
    TE_Dallas_Goedert = 1.0
    WR_Amari_Cooper = 1.0
    WR_Jerry_Jeudy = 1.0
    WR_Marquise_Brown = 1.0

```python
prob += (lpSum(v for v in prob.variables() if v.varValue > 0)
    <= 0)
prob.solve()
```

    1

It is possible to add new constraints and solve the optimisation problem again.
Here we chose to add the constraint that no player of the previous line can be chosen again. In principle this can be limited to any number e.g. we can add a constraint that allows to reuse $n$ players.

```python
for v in prob.variables():
    if v.varValue > 0:
        print( v.name, "=", v.varValue)
```

    DST_Ravens = 1.0
    FLX_Nick_Chubb = 1.0
    QB_Patrick_Mahomes = 1.0
    RB_David_Johnson = 1.0
    RB_David_Montgomery = 1.0
    TE_Hayden_Hurst = 1.0
    WR_Brandin_Cooks = 1.0
    WR_Cam_Sims = 1.0
    WR_Marvin_Hall = 1.0

```python

```