authors/mikhail/Advertisement/Advertising_campaign_colab.ipynb

{
  "cells": [
    {
      "attachments": {},
      "cell_type": "markdown",
      "metadata": {
        "id": "7RjB1ipC7KbS"
      },
      "source": [
        "# Optimization of an advertising campaign for launching a new product on the market\n",
        "[![Advertising_campaign_colab.ipynb](https://img.shields.io/badge/github-%23121011.svg?logo=github)](https://github.com/ampl/colab.ampl.com/blob/master/authors/mikhail/Advertisement/Advertising_campaign_colab.ipynb) [![Open In Colab](https://colab.research.google.com/assets/colab-badge.svg)](https://colab.research.google.com/github/ampl/colab.ampl.com/blob/master/authors/mikhail/Advertisement/Advertising_campaign_colab.ipynb) [![Kaggle](https://kaggle.com/static/images/open-in-kaggle.svg)](https://kaggle.com/kernels/welcome?src=https://github.com/ampl/colab.ampl.com/blob/master/authors/mikhail/Advertisement/Advertising_campaign_colab.ipynb) [![Gradient](https://assets.paperspace.io/img/gradient-badge.svg)](https://console.paperspace.com/github/ampl/colab.ampl.com/blob/master/authors/mikhail/Advertisement/Advertising_campaign_colab.ipynb) [![Open In SageMaker Studio Lab](https://studiolab.sagemaker.aws/studiolab.svg)](https://studiolab.sagemaker.aws/import/github/ampl/colab.ampl.com/blob/master/authors/mikhail/Advertisement/Advertising_campaign_colab.ipynb) [![Hits](https://h.ampl.com/https://github.com/ampl/colab.ampl.com/blob/master/authors/mikhail/Advertisement/Advertising_campaign_colab.ipynb)](https://colab.ampl.com)\n",
        "\n",
        "Description:  The modern world is unthinkable without advertising. Advertising is the engine of progress.\n",
        "When developing and conducting advertising campaigns, companies strive to maximize the effectiveness of their advertising dollars. In this case, the effectiveness of advertising is measured by the degree to which the campaign goals are achieved. The main goals of an advertising campaign can be:\n",
        "-  Increase in sales;\n",
        "-  Attraction of new clients;\n",
        "- Launch of a new product on the market;\n",
        "- Increasing/maintaining market share or profit level.\n",
        "\n",
        "When developing and planning an advertising campaigns, it is necessary to take into account many factors that influence the final effectiveness of advertising, namely:\n",
        "- Characteristics of advertising channels (availability, cost, etc.);\n",
        "- Structure and composition of the target audience of each advertising channel;\n",
        "- Conversion rates for various advertising channels;\n",
        "- Limitation of the budget allocated for advertising;\n",
        "- Cost of 1 impression/click;\n",
        "- Localization of the target audience;\n",
        "- The intensity of advertising displays during the day, week;\n",
        "- Marketing activity of competitors;\n",
        "\n",
        "Mathematical optimization allows you to synchronize all the specified factors and limitations of advertising channels with each other, and determine, among many options, the most effective (from the point of view of campaign goals) plan for investing in advertising.\n",
        "\n",
        "Tags: Marketing, Advertisement, Deterministic model, Piecewise-linear, mip, ampl-only, cbc\n",
        "\n",
        "Notebook author: Mikhail Riabtsev <<mail@solverytic.com>>"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## 1. Problem statement"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Let's start by downloading the necessary extensions and libraries"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "id": "rQ50dBwL7KbX"
      },
      "outputs": [],
      "source": [
        "# Install dependencies\n",
        "%pip install -q amplpy pandas \n",
        "import pandas as pd                 # Loading panda to work with pandas.DataFrame objects (https://pandas.pydata.org/)\n",
        "import numpy as np                  # Loading numpy to perform multidimensional calculations numpy.matrix (https://numpy.org/)\n",
        "import matplotlib.pyplot as plt     # Loading library for creating static, animated, and interactive visualizations (https://matplotlib.org/)"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
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            "a0a3a52262a340debb5a29ce65170164"
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        },
        "id": "s7DEaRp87KbY",
        "outputId": "7ccc087a-1af8-445e-c56d-31f3371b0c38"
      },
      "outputs": [],
      "source": [
        "# Google Colab & Kaggle integration\n",
        "from amplpy import AMPL, ampl_notebook\n",
        "\n",
        "ampl = ampl_notebook(\n",
        "    modules=[\"cbc\"],  # modules to install\n",
        "    license_uuid=\"default\",  # license to use\n",
        ")  # instantiate AMPL object and register magics"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "The company plans to launch a new product on the market of 2 large cities (City1, City2). To inform customers about the new offer, the company is considering the possibility of placing the following types of advertisements:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 7,
      "metadata": {},
      "outputs": [
        {
          "data": {
            "text/html": [
              "<div>\n",
              "<style scoped>\n",
              "    .dataframe tbody tr th:only-of-type {\n",
              "        vertical-align: middle;\n",
              "    }\n",
              "\n",
              "    .dataframe tbody tr th {\n",
              "        vertical-align: top;\n",
              "    }\n",
              "\n",
              "    .dataframe thead th {\n",
              "        text-align: right;\n",
              "    }\n",
              "</style>\n",
              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>Type of advertising</th>\n",
              "      <th>Unit</th>\n",
              "      <th>Cost 1 unit</th>\n",
              "      <th>Audience, person</th>\n",
              "      <th>Conversion, impressions to clicks, %</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>TV: сhannel#1_time 0PM-5PM</td>\n",
              "      <td>minutes</td>\n",
              "      <td>1000</td>\n",
              "      <td>18000</td>\n",
              "      <td>3</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>TV: сhannel#1_time 5PM-10PM</td>\n",
              "      <td>minutes</td>\n",
              "      <td>1200</td>\n",
              "      <td>25000</td>\n",
              "      <td>3</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>Magazine A: cover</td>\n",
              "      <td>pages</td>\n",
              "      <td>200</td>\n",
              "      <td>7000</td>\n",
              "      <td>6</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>Magazine B: inner page</td>\n",
              "      <td>pages</td>\n",
              "      <td>220</td>\n",
              "      <td>8000</td>\n",
              "      <td>6</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>Radio A: time 5PM-10PM</td>\n",
              "      <td>minutes</td>\n",
              "      <td>200</td>\n",
              "      <td>11000</td>\n",
              "      <td>5</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>5</th>\n",
              "      <td>Radio B: time 5PM-10PM</td>\n",
              "      <td>minutes</td>\n",
              "      <td>250</td>\n",
              "      <td>15000</td>\n",
              "      <td>5</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>6</th>\n",
              "      <td>Subway: red line</td>\n",
              "      <td>page to week</td>\n",
              "      <td>250</td>\n",
              "      <td>15000</td>\n",
              "      <td>4</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>7</th>\n",
              "      <td>WebSite1</td>\n",
              "      <td>weeks</td>\n",
              "      <td>500</td>\n",
              "      <td>10000</td>\n",
              "      <td>15</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>8</th>\n",
              "      <td>WebSite2</td>\n",
              "      <td>weeks</td>\n",
              "      <td>400</td>\n",
              "      <td>7000</td>\n",
              "      <td>14</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>9</th>\n",
              "      <td>Billboard: 1place per week</td>\n",
              "      <td>place</td>\n",
              "      <td>400</td>\n",
              "      <td>3000</td>\n",
              "      <td>3</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>"
            ],
            "text/plain": [
              "           Type of advertising          Unit  Cost 1 unit  Audience, person  \\\n",
              "0   TV: сhannel#1_time 0PM-5PM       minutes         1000             18000   \n",
              "1  TV: сhannel#1_time 5PM-10PM       minutes         1200             25000   \n",
              "2            Magazine A: cover         pages          200              7000   \n",
              "3       Magazine B: inner page         pages          220              8000   \n",
              "4      Radio A: time 5PM-10PM        minutes          200             11000   \n",
              "5      Radio B: time 5PM-10PM        minutes          250             15000   \n",
              "6             Subway: red line  page to week          250             15000   \n",
              "7                     WebSite1         weeks          500             10000   \n",
              "8                     WebSite2         weeks          400              7000   \n",
              "9   Billboard: 1place per week         place          400              3000   \n",
              "\n",
              "   Conversion, impressions to clicks, %  \n",
              "0                                     3  \n",
              "1                                     3  \n",
              "2                                     6  \n",
              "3                                     6  \n",
              "4                                     5  \n",
              "5                                     5  \n",
              "6                                     4  \n",
              "7                                    15  \n",
              "8                                    14  \n",
              "9                                     3  "
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "broadcasters = pd.DataFrame([\n",
        "    ['TV: сhannel#1_time 0PM-5PM','minutes',    1000,   18000,  3   ], \n",
        "    ['TV: сhannel#1_time 5PM-10PM','minutes',   1200,   25000,  3   ], \n",
        "    ['Magazine A: cover','pages',               200,    7000,   6   ], \n",
        "    ['Magazine B: inner page','pages',          220,    8000,   6   ], \n",
        "    ['Radio A: time 5PM-10PM ','minutes',       200,    11000,  5   ], \n",
        "    ['Radio B: time 5PM-10PM ','minutes',       250,    15000,  5   ], \n",
        "    ['Subway: red line','page to week',         250,    15000,  4   ], \n",
        "    ['WebSite1','weeks',                        500,    10000,  15  ],\n",
        "    ['WebSite2','weeks',                        400,    7000,   14  ],\n",
        "    ['Billboard: 1place per week','place',      400,    3000,   3   ]], \n",
        "    columns = ['Type of advertising','Unit', 'Cost 1 unit', 'Audience, person','Conversion, impressions to clicks, %'])\n",
        "display(broadcasters)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        " *Table.1. Main characteristics of advertisements.*"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "The audience of advertising consists of the following user segments, % of the total audience:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 11,
      "metadata": {},
      "outputs": [
        {
          "data": {
            "text/html": [
              "<div>\n",
              "<style scoped>\n",
              "    .dataframe tbody tr th:only-of-type {\n",
              "        vertical-align: middle;\n",
              "    }\n",
              "\n",
              "    .dataframe tbody tr th {\n",
              "        vertical-align: top;\n",
              "    }\n",
              "\n",
              "    .dataframe thead th {\n",
              "        text-align: right;\n",
              "    }\n",
              "</style>\n",
              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>Type of advertising</th>\n",
              "      <th>14-17 years old</th>\n",
              "      <th>17-25 years old</th>\n",
              "      <th>25-40 years old</th>\n",
              "      <th>40-60 years old</th>\n",
              "      <th>over 60</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>TV: сhannel#1_time 0PM-5PM</td>\n",
              "      <td>7</td>\n",
              "      <td>25</td>\n",
              "      <td>25</td>\n",
              "      <td>27</td>\n",
              "      <td>16</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>TV: сhannel#1_time 5PM-10PM</td>\n",
              "      <td>5</td>\n",
              "      <td>18</td>\n",
              "      <td>32</td>\n",
              "      <td>30</td>\n",
              "      <td>15</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>Magazine A: cover</td>\n",
              "      <td>1</td>\n",
              "      <td>5</td>\n",
              "      <td>35</td>\n",
              "      <td>45</td>\n",
              "      <td>14</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>Magazine B: inner page</td>\n",
              "      <td>5</td>\n",
              "      <td>18</td>\n",
              "      <td>32</td>\n",
              "      <td>30</td>\n",
              "      <td>15</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>Radio A: time 5PM-10PM</td>\n",
              "      <td>15</td>\n",
              "      <td>32</td>\n",
              "      <td>18</td>\n",
              "      <td>30</td>\n",
              "      <td>5</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>5</th>\n",
              "      <td>Radio B: time 5PM-10PM</td>\n",
              "      <td>14</td>\n",
              "      <td>45</td>\n",
              "      <td>35</td>\n",
              "      <td>5</td>\n",
              "      <td>1</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>6</th>\n",
              "      <td>Subway: red line</td>\n",
              "      <td>16</td>\n",
              "      <td>27</td>\n",
              "      <td>25</td>\n",
              "      <td>25</td>\n",
              "      <td>7</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>7</th>\n",
              "      <td>WebSite1</td>\n",
              "      <td>5</td>\n",
              "      <td>18</td>\n",
              "      <td>32</td>\n",
              "      <td>30</td>\n",
              "      <td>15</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>8</th>\n",
              "      <td>WebSite2</td>\n",
              "      <td>15</td>\n",
              "      <td>32</td>\n",
              "      <td>18</td>\n",
              "      <td>30</td>\n",
              "      <td>5</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>9</th>\n",
              "      <td>Billboard: 1place per week</td>\n",
              "      <td>5</td>\n",
              "      <td>18</td>\n",
              "      <td>32</td>\n",
              "      <td>30</td>\n",
              "      <td>15</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>"
            ],
            "text/plain": [
              "           Type of advertising  14-17 years old  17-25 years old  \\\n",
              "0   TV: сhannel#1_time 0PM-5PM                7               25   \n",
              "1  TV: сhannel#1_time 5PM-10PM                5               18   \n",
              "2            Magazine A: cover                1                5   \n",
              "3       Magazine B: inner page                5               18   \n",
              "4      Radio A: time 5PM-10PM                15               32   \n",
              "5      Radio B: time 5PM-10PM                14               45   \n",
              "6             Subway: red line               16               27   \n",
              "7                     WebSite1                5               18   \n",
              "8                     WebSite2               15               32   \n",
              "9   Billboard: 1place per week                5               18   \n",
              "\n",
              "   25-40 years old  40-60 years old  over 60  \n",
              "0               25               27       16  \n",
              "1               32               30       15  \n",
              "2               35               45       14  \n",
              "3               32               30       15  \n",
              "4               18               30        5  \n",
              "5               35                5        1  \n",
              "6               25               25        7  \n",
              "7               32               30       15  \n",
              "8               18               30        5  \n",
              "9               32               30       15  "
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "segmentation_of_audience = pd.DataFrame([\n",
        "    ['TV: сhannel#1_time 0PM-5PM', 7,  25, 25, 27, 16  ], \n",
        "    ['TV: сhannel#1_time 5PM-10PM',5,  18, 32, 30, 15  ], \n",
        "    ['Magazine A: cover',          1,  5,  35, 45, 14  ],\n",
        "    ['Magazine B: inner page',     5,  18, 32, 30, 15  ], \n",
        "    ['Radio A: time 5PM-10PM ',    15, 32, 18, 30, 5   ], \n",
        "    ['Radio B: time 5PM-10PM ',    14, 45, 35, 5,  1   ], \n",
        "    ['Subway: red line',           16, 27, 25, 25, 7   ], \n",
        "    ['WebSite1',                   5,  18, 32, 30, 15  ],\n",
        "    ['WebSite2',                   15, 32, 18, 30, 5   ],\n",
        "    ['Billboard: 1place per week', 5,  18, 32, 30, 15  ]],\n",
        "    columns = ['Type of advertising','14-17 years old', '17-25 years old', '25-40 years old','40-60 years old', 'over 60'])\n",
        "display(segmentation_of_audience)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "*Table.2. Segment composition of advertising audiences, %.*"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "In addition, the audience of advertising media is distributed differently in cities (City1, City2), %:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 121,
      "metadata": {},
      "outputs": [
        {
          "data": {
            "text/html": [
              "<div>\n",
              "<style scoped>\n",
              "    .dataframe tbody tr th:only-of-type {\n",
              "        vertical-align: middle;\n",
              "    }\n",
              "\n",
              "    .dataframe tbody tr th {\n",
              "        vertical-align: top;\n",
              "    }\n",
              "\n",
              "    .dataframe thead th {\n",
              "        text-align: right;\n",
              "    }\n",
              "</style>\n",
              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>Type of advertising</th>\n",
              "      <th>City#1</th>\n",
              "      <th>City#2</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>TV: сhannel#1_time 0PM-5PM</td>\n",
              "      <td>60</td>\n",
              "      <td>40</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>TV: сhannel#1_time 5PM-10PM</td>\n",
              "      <td>70</td>\n",
              "      <td>30</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>Magazine A: cover</td>\n",
              "      <td>0</td>\n",
              "      <td>100</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>Magazine B: inner page</td>\n",
              "      <td>80</td>\n",
              "      <td>20</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>Radio A: time 5PM-10PM</td>\n",
              "      <td>80</td>\n",
              "      <td>20</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>5</th>\n",
              "      <td>Radio B: time 5PM-10PM</td>\n",
              "      <td>20</td>\n",
              "      <td>80</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>6</th>\n",
              "      <td>Subway: red line</td>\n",
              "      <td>0</td>\n",
              "      <td>100</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>7</th>\n",
              "      <td>WebSite1</td>\n",
              "      <td>75</td>\n",
              "      <td>25</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>8</th>\n",
              "      <td>WebSite2</td>\n",
              "      <td>80</td>\n",
              "      <td>20</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>9</th>\n",
              "      <td>Billboard: 1place per week</td>\n",
              "      <td>0</td>\n",
              "      <td>100</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>"
            ],
            "text/plain": [
              "           Type of advertising  City#1  City#2\n",
              "0   TV: сhannel#1_time 0PM-5PM      60      40\n",
              "1  TV: сhannel#1_time 5PM-10PM      70      30\n",
              "2            Magazine A: cover       0     100\n",
              "3       Magazine B: inner page      80      20\n",
              "4      Radio A: time 5PM-10PM       80      20\n",
              "5      Radio B: time 5PM-10PM       20      80\n",
              "6             Subway: red line       0     100\n",
              "7                     WebSite1      75      25\n",
              "8                     WebSite2      80      20\n",
              "9   Billboard: 1place per week       0     100"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "distrib_audience_by_city = pd.DataFrame([\n",
        "    ['TV: сhannel#1_time 0PM-5PM', 60,  40  ], \n",
        "    ['TV: сhannel#1_time 5PM-10PM',70,  30  ], \n",
        "    ['Magazine A: cover',          0,   100 ],\n",
        "    ['Magazine B: inner page',     80,  20  ], \n",
        "    ['Radio A: time 5PM-10PM ',    80,  20  ], \n",
        "    ['Radio B: time 5PM-10PM ',    20,  80  ], \n",
        "    ['Subway: red line',           0,   100 ], \n",
        "    ['WebSite1',                   75,  25  ],\n",
        "    ['WebSite2',                   80,  20  ],\n",
        "    ['Billboard: 1place per week', 0,   100 ]],\n",
        "    columns = ['Type of advertising','City#1', 'City#2'])\n",
        "display(distrib_audience_by_city)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "*Table.3. Distribution of the audience of broadcasters by city, %*"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "When developing an advertising campaign, it is necessary to take into account that the effectiveness of advertisements decreases as the number of views increases. So the first 10 views have the one indicated in Table 1. conversion rate, while the next 10 impressions are 10% less effective, and impressions >20 are 40% less effective.\n",
        "\n",
        "Data on the reduction rate of advertising effectiveness are given in the following table, %:"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 122,
      "metadata": {},
      "outputs": [
        {
          "data": {
            "text/html": [
              "<div>\n",
              "<style scoped>\n",
              "    .dataframe tbody tr th:only-of-type {\n",
              "        vertical-align: middle;\n",
              "    }\n",
              "\n",
              "    .dataframe tbody tr th {\n",
              "        vertical-align: top;\n",
              "    }\n",
              "\n",
              "    .dataframe thead th {\n",
              "        text-align: right;\n",
              "    }\n",
              "</style>\n",
              "<table border=\"1\" class=\"dataframe\">\n",
              "  <thead>\n",
              "    <tr style=\"text-align: right;\">\n",
              "      <th></th>\n",
              "      <th>Type of advertising</th>\n",
              "      <th>10-20 views</th>\n",
              "      <th>&gt;20 views</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>0</th>\n",
              "      <td>TV: сhannel#1_time 0PM-5PM</td>\n",
              "      <td>10</td>\n",
              "      <td>40</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>1</th>\n",
              "      <td>TV: сhannel#1_time 5PM-10PM</td>\n",
              "      <td>10</td>\n",
              "      <td>40</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>2</th>\n",
              "      <td>Magazine A: cover</td>\n",
              "      <td>5</td>\n",
              "      <td>20</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>3</th>\n",
              "      <td>Magazine B: inner page</td>\n",
              "      <td>5</td>\n",
              "      <td>20</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>4</th>\n",
              "      <td>Radio A: time 5PM-10PM</td>\n",
              "      <td>20</td>\n",
              "      <td>40</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>5</th>\n",
              "      <td>Radio B: time 5PM-10PM</td>\n",
              "      <td>20</td>\n",
              "      <td>40</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>6</th>\n",
              "      <td>Subway: red line</td>\n",
              "      <td>5</td>\n",
              "      <td>40</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>7</th>\n",
              "      <td>WebSite1</td>\n",
              "      <td>10</td>\n",
              "      <td>40</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>8</th>\n",
              "      <td>WebSite2</td>\n",
              "      <td>10</td>\n",
              "      <td>40</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>9</th>\n",
              "      <td>Billboard: 1place per week</td>\n",
              "      <td>10</td>\n",
              "      <td>20</td>\n",
              "    </tr>\n",
              "  </tbody>\n",
              "</table>\n",
              "</div>"
            ],
            "text/plain": [
              "           Type of advertising  10-20 views  >20 views\n",
              "0   TV: сhannel#1_time 0PM-5PM           10         40\n",
              "1  TV: сhannel#1_time 5PM-10PM           10         40\n",
              "2            Magazine A: cover            5         20\n",
              "3       Magazine B: inner page            5         20\n",
              "4      Radio A: time 5PM-10PM            20         40\n",
              "5      Radio B: time 5PM-10PM            20         40\n",
              "6             Subway: red line            5         40\n",
              "7                     WebSite1           10         40\n",
              "8                     WebSite2           10         40\n",
              "9   Billboard: 1place per week           10         20"
            ]
          },
          "metadata": {},
          "output_type": "display_data"
        }
      ],
      "source": [
        "reduce = pd.DataFrame([\n",
        "    ['TV: сhannel#1_time 0PM-5PM', 10,  40  ], \n",
        "    ['TV: сhannel#1_time 5PM-10PM',10,  40  ], \n",
        "    ['Magazine A: cover',          5,   20  ],\n",
        "    ['Magazine B: inner page',     5,   20  ],\n",
        "    ['Radio A: time 5PM-10PM ',    20,  40  ], \n",
        "    ['Radio B: time 5PM-10PM ',    20,  40  ], \n",
        "    ['Subway: red line',           5,   40  ], \n",
        "    ['WebSite1',                   10,  40  ],\n",
        "    ['WebSite2',                   10,  40  ],\n",
        "    ['Billboard: 1place per week', 10,  20  ]],\n",
        "    columns = ['Type of advertising','10-20 views', '>20 views'])\n",
        "display(reduce)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "*Table.4. Indicators of decrease in the effectiveness of advertisements, %.*"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "One of the client’s requirements is the need to allocate at least 20% of the budget to TV advertising.\n",
        "\n",
        "**Optimization goals**\n",
        "\n",
        "It is necessary to distribute the advertising budget in such a way as to maximize the following indicators:\n",
        "1. Inform the maximum number of people about the release of a new product;\n",
        "2. Inform the maximum number of people from the target audience (persons 25-40 years old) about the release of a new product;\n",
        "3. Maximize the total number of conversions (purchases) of the target audience."
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## 2. Mathematical formulation of the problem"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "Let's start by describing **index sets**\n",
        "* $i$ -  list of available types of advertising, for $i$ = \"TV: сhannel#1_time 0PM-5PM\",..., I.\n",
        "* $i'$ -  list of TV types of advertising, $i' \\in$ I. $i'$ = 'TV: сhannel#1_time 0PM-5PM', 'TV: сhannel#1_time 5PM-10PM'.\n",
        "* $j$ -  segments of advertising audiences, for $j$ = \"14-17 years old\",..., J. \n",
        "* $j'$ - types of target segment. $j'$ = \"25-40\".\n",
        "* $k$ -  list of cities for broadcasting advertisements, for $k$ = \"City#1\",... K. \n",
        "\n",
        "$A_i$ - number of people in the advertising $i$ audience (Table 1).\n",
        "\n",
        "$B_i$ - the cost of displaying 1 unit of advertising $i$ (Table 1).\n",
        "\n",
        "$C_i$ - the conversion rate of advertising $i$ into targeted customer actions (Table 1).\n",
        "\n",
        "**Parameters**\n",
        "\n",
        "$D_{ij}$ - the share of various target groups ($j$) in the total number of advertising audiences ($i$) (Table 2).\n",
        "\n",
        "$F_{ij}$ - distribution of advertising audience ($i$) between cities ($j$) (Table 3).\n",
        "\n",
        "$M$ - available advertising budget, $.\n",
        "\n",
        "$R_i$ - The coefficient of decrease in advertising effectiveness $i$ with increasing number impressions (Table 4).\n",
        "\n",
        "To describe the conditions for changing the effectiveness of advertisements after n number of impressions, we use the concept of [Piecewise-Linear Programing](https://ampl.com/wp-content/uploads/Chapter-17-Piecewise-Linear-Programs-AMPL-Book.pdf). Thus, the objective function of the model is to maximize the number of targeted actions of the '25-40' audience as a result of advertising impressions.\n",
        "\n",
        "**The variables in this case are**\n",
        "* $x_i$: Number of advertisement broadcasts $i$, for $i$ = \"TV: сhannel#1_time 0PM-5PM\",..., N.\n",
        "\n",
        "**Objective function is**\n",
        "\n",
        "max  $ \\sum_i^I \\sum_k^K \\sum_{j'}^J X_i * A_i * D_{ij} * F_{ij} * R_i$ (goal 3)\n",
        "\n",
        " **The constraints are**\n",
        "\n",
        " s.t. $ \\sum_i^I X_i * B_i <= M $\n",
        " \n",
        "s.t. $ \\sum_{i'}^I X_{i'} * B_{i'} <= M * 0.2$"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## 3. AMPL formulation of the model"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        ">Use %%ampl_eval to evaluate AMPL commands and declarations"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 4,
      "metadata": {
        "id": "aEtyesqs7KbZ"
      },
      "outputs": [],
      "source": [
        "%%ampl_eval\n",
        "reset;\n",
        "### SETS \n",
        " set ADVERTISEMENT; \t\t# List of all available advertisements\n",
        " set AUDIENCE; \t\t\t\t# Broadcaster's audience segments\n",
        " set SET within AUDIENCE; \t# Types of target segments\n",
        " set CITY; \t\t\t\t\t# Target cities of the advertising campaign\n",
        " \n",
        "### PARAMETERS\n",
        " param Cost       {ADVERTISEMENT} >= 0; \t# Cost of 1 unit of advertising broadcasting for each type of advertising\n",
        " param Auditoria  {ADVERTISEMENT} >= 0; \t# Internal number of consumer groups for each advertisement\n",
        " param Conversion {ADVERTISEMENT} >= 0;\t\t# Conversion rate of advertisements into purchases\n",
        "\n",
        " param TargetAudience {ADVERTISEMENT, AUDIENCE} >= 0;\t# Segment composition of advertising audiences\n",
        " param City {ADVERTISEMENT, CITY} >= 0; \t\t\t\t# Distribution of the audience of broadcasters by city\n",
        " param Money; \t\t\t\t\t\t\t\t\t\t\t# Advertising campaign budget, USD\n",
        "\n",
        "### Indicators of reduction in advertising effectiveness (we use Piecewise-Linear Programing to describe this conception)\n",
        " # Number of efficiency ranges\n",
        " param npiece integer >= 1;\n",
        " # Ad effectiveness value in each range\n",
        " param rate {ad in ADVERTISEMENT, p in 1..npiece} >= 0;\n",
        " param limit {p in 1..npiece-1} > if p=1 then 0 else limit [p-1];\n",
        "\n",
        "### VARIABLES\n",
        " var X {ADVERTISEMENT} >= 0;\t\t\t# Number/duration of each type of advertisement\n",
        " \n",
        "### OBJECTIVE\n",
        "maximize TotalAudienceInform:\t\t# 1. Inform the maximum number of people about the release of a new product\n",
        " sum {ad in ADVERTISEMENT, c in CITY, a in AUDIENCE} <<{p in 1..npiece-1} limit[p];\n",
        " {p in 1..npiece} rate[ad,p]>> X[ad] * Auditoria[ad] * City[ad,c];\n",
        "\n",
        " maximize TargetAudienceInform:\t\t# Inform the maximum number of people from the target audience (persons 25-40 years old) about the release of a new product\n",
        " sum {ad in ADVERTISEMENT, c in CITY, a in AUDIENCE: a in SET} <<{p in 1..npiece-1} limit[p];\n",
        " {p in 1..npiece} rate[ad,p]>> X[ad] * Auditoria[ad] * TargetAudience[ad,a] * City[ad,c];\n",
        "\n",
        " maximize TargetAudienceConversion:\t# 3. Maximize the total number of conversions (purchases) of the target audience\n",
        " sum {ad in ADVERTISEMENT, c in CITY, a in AUDIENCE: a in SET} <<{p in 1..npiece-1} limit[p];\n",
        " {p in 1..npiece} rate[ad,p]>> X[ad] * Auditoria[ad] * TargetAudience[ad,a] * Conversion[ad] * City[ad,c];\n",
        "\n",
        "### CONSTRAINTS\n",
        " subject to Budget:\t\t\t\t\t\t# Limit the cost of all advertisements to the allocated budget\n",
        "\tsum{ad in ADVERTISEMENT} X[ad] * Cost[ad]<= Money;\n",
        " subject to Budget_TV:\t\t\t\t\t# TV advertising costs must be at least 20% of the allocated budget\t\t\t\t\n",
        "\tsum{ad in ADVERTISEMENT: ad in {'TV: сhannel#1_time 0PM-5PM', 'TV: сhannel#1_time 5PM-10PM'}} X[ad] * Cost[ad] >= Money * 0.2;"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## 4. Load data"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {},
      "outputs": [],
      "source": [
        "%%writefile Advertisement.dat\n",
        " \n",
        " param Money = 50000;\n",
        "\t\n",
        " param: ADVERTISEMENT:\t\t\t\t\tCost\tAuditoria\tConversion:=\n",
        "\t'TV: сhannel#1_time 0PM-5PM'\t\t1000\t18000\t\t0.03\t\t\n",
        "\t'TV: сhannel#1_time 5PM-10PM'\t\t1200\t25000\t\t0.03\n",
        "\t'Magazine A: cover'\t\t\t\t\t200\t\t7000\t\t0.06\n",
        "\t'Magazine B: inner page'\t\t\t220\t\t8000\t\t0.06\n",
        "\t'Radio A: time 5PM-10PM' \t\t\t200\t\t11000\t\t0.05\n",
        "\t'Radio B: time 5PM-10PM' \t\t\t250\t\t15000\t\t0.05\n",
        "\t'Subway: red line'\t\t\t\t\t250\t\t15000\t\t0.04\n",
        "\t'WebSite1'\t\t\t\t\t\t\t500\t\t10000\t\t0.15\n",
        "\t'WebSite2'\t\t\t\t\t\t\t400\t\t7000\t\t0.14\n",
        "\t'Billboard: 1place per week'\t\t400\t\t3000\t\t0.03 ;\n",
        "\n",
        " set AUDIENCE = '14-17', '17-25', '25-40', '40-60', '>60';\n",
        "\t\n",
        " param TargetAudience: \t\t\t\t'14-17'\t'17-25'\t'25-40'\t'40-60'\t'>60':=\n",
        "\t'TV: сhannel#1_time 0PM-5PM'\t0.07\t0.25\t0.25\t0.27\t0.16\t\t\n",
        "\t'TV: сhannel#1_time 5PM-10PM'\t0.05\t0.18\t0.32\t0.30\t0.15\t\n",
        "\t'Magazine A: cover'\t\t\t\t0.01\t0.05\t0.35\t0.45\t0.14\t\n",
        "\t'Magazine B: inner page'\t\t0.05\t0.18\t0.32\t0.30\t0.15\t \n",
        "\t'Radio A: time 5PM-10PM'\t\t0.15\t0.32\t0.18\t0.30\t0.05\t\n",
        "\t'Radio B: time 5PM-10PM'\t\t0.14\t0.45\t0.35\t0.05\t0.01\t\n",
        "\t'Subway: red line'\t\t\t\t0.16\t0.27\t0.25\t0.25\t0.07\t\n",
        "\t'WebSite1'\t\t\t\t\t\t0.05\t0.18\t0.32\t0.30 \t0.15\t\n",
        "\t'WebSite2' \t\t\t\t\t\t0.15\t0.32\t0.18\t0.30\t0.05\t\n",
        "\t'Billboard: 1place per week'\t0.05\t0.18\t0.32\t0.30\t0.15\t;\n",
        "\t\n",
        " set SET='25-40';\n",
        "\n",
        " set CITY = City1, City2; \n",
        "\n",
        " param City:\t\t\t\t\t\tCity1\tCity2:=\n",
        "\t'TV: сhannel#1_time 0PM-5PM'\t0.6\t\t0.4\n",
        "\t'TV: сhannel#1_time 5PM-10PM'\t0.7\t\t0.3\n",
        "\t'Magazine A: cover'\t\t\t\t0\t\t1\n",
        "\t'Magazine B: inner page'\t\t0.8\t\t0.2\n",
        "\t'Radio A: time 5PM-10PM'\t\t0.8\t\t0.2\n",
        "\t'Radio B: time 5PM-10PM'\t\t0.2\t\t0.8\n",
        "\t'Subway: red line'\t\t\t\t0\t\t1\n",
        "\t'WebSite1'\t\t\t\t\t\t0.75\t0.25\n",
        "\t'WebSite2' \t\t\t\t\t\t0.8\t\t0.2\n",
        "\t'Billboard: 1place per week'\t0\t\t1\t;\n",
        "\t\n",
        " param npiece default 3;\n",
        "\t\n",
        " param rate:\t\t\t\t\t\t1\t\t2\t\t3:=\n",
        "\t'TV: сhannel#1_time 0PM-5PM'\t1\t\t0.9\t\t0.6\n",
        "\t'TV: сhannel#1_time 5PM-10PM'\t1\t\t0.9\t\t0.6\n",
        "\t'Magazine A: cover'\t\t\t\t1\t\t0.95 \t0.8\n",
        "\t'Magazine B: inner page'\t\t1\t\t0.95\t0.8\n",
        "\t'Radio A: time 5PM-10PM'\t\t1\t\t0.80\t0.6\n",
        "\t'Radio B: time 5PM-10PM'\t\t1\t\t0.80\t0.6\n",
        "\t'Subway: red line'\t\t\t\t1\t\t0.95\t0.6 \n",
        "\t'WebSite1'\t\t\t\t\t\t1\t\t0.9\t\t0.6\n",
        "\t'WebSite2'\t\t\t\t\t\t1\t\t0.9\t\t0.6\n",
        "\t'Billboard: 1place per week'\t1\t\t0.9\t\t0.8\t;\n",
        "\t\n",
        " param limit:=\t\t\t\t\t\t\n",
        " 1\t10\n",
        " 2\t20;"
      ]
    },
    {
      "attachments": {},
      "cell_type": "markdown",
      "metadata": {
        "id": "uftSUdTh9eZY"
      },
      "source": [
        "## 5. Solve problem"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        ">Use %%ampl_eval to evaluate AMPL commands and declarations"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": 11,
      "metadata": {},
      "outputs": [
        {
          "name": "stdout",
          "output_type": "stream",
          "text": [
            "\n",
            "Substitution eliminates 10 variables.\n",
            "10 piecewise-linear terms replaced by 30 variables and 10 constraints.\n",
            "Adjusted problem:\n",
            "30 variables, all linear\n",
            "2 constraints, all linear; 36 nonzeros\n",
            "\t2 inequality constraints\n",
            "1 linear objective; 30 nonzeros.\n",
            "\n",
            "cbc 2.10.10:cbc 2.10.10: optimal solution; objective 32164.7\n",
            "0 simplex iterations\n"
          ]
        }
      ],
      "source": [
        "%%ampl_eval\n",
        "\n",
        "reset data;                 # Clearing the data\n",
        "data Advertisement.dat;     # Loading data from a file\n",
        "\n",
        "option solver cbc;          # Choosing a solver\n",
        "\n",
        "# Defining Output Settings \n",
        "option show_stats 1;        # (1) Show statistical information about the size of the problem. Default 0 (statistics are not displayed)\n",
        "option display_1col 0;      # Data Display Settings\n",
        "option omit_zero_rows 1;    # Hide rows with 0 values. Default (0)\n",
        "\n",
        "# Selecting an Objective Function (choose one of)\n",
        "#objective TotalAudienceInform;     # goal 1\n",
        "#objective TargetAudienceInform;    # goal 2\n",
        "objective TargetAudienceConversion; # goal 3\n",
        "\n",
        "solve;                      # Solve the model (Objective function TotalAudienceInform)"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        "## 6. Display the solution"
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {},
      "source": [
        ">Use %%ampl_eval"
      ]
    },
    {
      "cell_type": "code",
      "execution_count": null,
      "metadata": {},
      "outputs": [],
      "source": [
        "%%ampl_eval\n",
        "\n",
        "display X,                                  # Show the value of the variables for the resulting solution\n",
        "{ad in ADVERTISEMENT} X[ad] * Cost[ad],     # Show the budget spent on each type of advertising\n",
        "\n",
        "{c in CITY, ad in ADVERTISEMENT, a in AUDIENCE} X[ad] * Auditoria[ad] * TargetAudience[ad,a] * City[ad,c],                      # Number of views\n",
        "sum {c in CITY, ad in ADVERTISEMENT, a in AUDIENCE} X[ad] * Auditoria[ad] * TargetAudience[ad,a] * City[ad,c],                  # Total number of views\n",
        "\n",
        "{c in CITY, ad in ADVERTISEMENT, a in AUDIENCE} X[ad] * Auditoria[ad] * TargetAudience[ad,a] * Conversion[ad] * City[ad,c],     # Number of conversions\n",
        "sum {c in CITY, ad in ADVERTISEMENT, a in AUDIENCE} X[ad] * Auditoria[ad] * TargetAudience[ad,a] * Conversion[ad] * City[ad,c]; # Total number of conversions"
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      "source": [
        "> ## 7. Retrieve solution as a pandas dataframe"
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              "      <th>X.val</th>\n",
              "    </tr>\n",
              "  </thead>\n",
              "  <tbody>\n",
              "    <tr>\n",
              "      <th>Billboard: 1place per week</th>\n",
              "      <td>0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>Magazine A: cover</th>\n",
              "      <td>0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>Magazine B: inner page</th>\n",
              "      <td>0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>Radio A: time 5PM-10PM</th>\n",
              "      <td>20.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>Radio B: time 5PM-10PM</th>\n",
              "      <td>0.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>Subway: red line</th>\n",
              "      <td>0.000000</td>\n",
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              "    <tr>\n",
              "      <th>TV: сhannel#1_time 0PM-5PM</th>\n",
              "      <td>0.000000</td>\n",
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              "    <tr>\n",
              "      <th>TV: сhannel#1_time 5PM-10PM</th>\n",
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              "    </tr>\n",
              "    <tr>\n",
              "      <th>WebSite1</th>\n",
              "      <td>64.000000</td>\n",
              "    </tr>\n",
              "    <tr>\n",
              "      <th>WebSite2</th>\n",
              "      <td>10.000000</td>\n",
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              "                                 X.val\n",
              "Billboard: 1place per week    0.000000\n",
              "Magazine A: cover             0.000000\n",
              "Magazine B: inner page        0.000000\n",
              "Radio A: time 5PM-10PM       20.000000\n",
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              "Subway: red line              0.000000\n",
              "TV: сhannel#1_time 0PM-5PM    0.000000\n",
              "TV: сhannel#1_time 5PM-10PM   8.333333\n",
              "WebSite1                     64.000000\n",
              "WebSite2                     10.000000"
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