{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Mathematical Optimization Model for Labor Scheduling\n",
    "\n",
    "##  Indices and Sets\n",
    "- $E$: Set of employee types (e.g., permanent, contract, temporary)\n",
    "- $S$: Set of shift types (e.g., usual_time, evening_shift, overtime)\n",
    "\n",
    "*We have a target date. The date is a specific order deadline and on oneday we proceed the whole task of one day, and goes no further than 1 day order amount.\n",
    "\n",
    "## Parameters\n",
    "- $h_s$: Hours per shift type $s$\n",
    "- $m_e$: Productivity per hour for employee type $e$ \n",
    "    + Some employee type / work type have more/less than 1 hour productivity unit)\n",
    "- $c_{e,s}$: Cost per hour for employee type $e$ in shift type $s$\n",
    "- $L$: Total unit productivity required\n",
    "- $A_e$: Number of available employees of type $e$\n",
    "\n",
    "## Decision Variables\n",
    "- $x_{e,s} \\in \\mathbb{Z}_{\\geq 0}$: Number of employees of type $e$ assigned to shift type $s$\n",
    "    + $e$ : regular, temporary.. etc. \n",
    "    + $s$ : work hour type (evening shift, regular, late hour). \n",
    "    + Based on the employee type and work hour type, the productivity per hour and cost per hour differs\n",
    "\n",
    "---\n",
    "\n",
    "## Objective Function\n",
    "\n",
    "Minimize total labor cost:\n",
    "\n",
    "$$\n",
    "\\min \\sum_{e \\in E} \\sum_{s \\in S} x_{e,s} \\cdot h_s \\cdot c_{e,s}\n",
    "$$\n",
    "\n",
    "---\n",
    "\n",
    "## Constraints\n",
    "\n",
    "1. **Labor Demand Satisfaction**\n",
    "\n",
    "Ensure the total unit productivity covers the required total productivity:\n",
    "\n",
    "$$\n",
    "\\sum_{e \\in E} \\sum_{s \\in S} x_{e,s} \\cdot h_s \\cdot m_e \\geq L\n",
    "$$\n",
    "\n",
    "2. **Employee Availability**\n",
    "\n",
    "- Usual and evening shift is a discrete work type\n",
    "    + Those who come in the usual cannot work during the evening shift\n",
    "\n",
    "$$\n",
    "x_{e,\\text{usual}} + x_{e,\\text{evening}} \\leq A_e \\quad \\forall e \\in E\n",
    "$$\n",
    "\n",
    "3. **Overtime Limitation**\n",
    "\n",
    "Overtime must not exceed regular shift assignments. \n",
    "\n",
    "(Because one who works overtime must work during the regular shift)\n",
    "\n",
    "$$\n",
    "x_{e,\\text{overtime}} \\leq x_{e,\\text{usual}} \\quad \\forall e \\in E\n",
    "$$\n",
    "\n",
    "4. **Non-negativity and Integrality**\n",
    "\n",
    "All decision variables must be non-negative integers:\n",
    "\n",
    "$$\n",
    "x_{e,s} \\in \\mathbb{Z}_{\\geq 0} \\quad \\forall e \\in E, s \\in S\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": ".venv",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.12.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}