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   "metadata": {},
   "source": [
    "This notebook contains the questions for the final review for MCS 472.  The solutions can be found on the lecture slides and in the lecture notes."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fa0b2937-3cbc-4b64-ba07-c7645c94dc1f",
   "metadata": {},
   "source": [
    "# Taguchi quality control"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6d9ed022-cebe-4c6b-b533-5bfd77e560ab",
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   "source": [
    "Suppose we measure the quality of watches as the deviation\n",
    "from the time at noon after one month without adjustments.\n",
    "\n",
    "Watches which show a deviation of more than 3 minutes have\n",
    "to be discarded at a cost of $\\$10$.\n",
    "\n",
    "We have to choose between two ways to produces watches:\n",
    "\n",
    "1. Watches produced by process $A$ give after one month an average\n",
    "   time of $12\\!:\\!01$ with a standard deviation of 1 minute.\n",
    "    \n",
    "2. Watches produced by process $B$ show $12\\!:\\!00$ on average,\n",
    "   but with a standard deviation of 2 minutes.\n",
    "\n",
    "Which process is best? $A$ or $B$?\n",
    "\n",
    "1. Set up the quality loss function.\n",
    "2. Use the expected loss to choose which process is best."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f575a8b9-f6fb-4fe1-889e-57ccf40a7dc9",
   "metadata": {},
   "source": [
    "# Linear Programming"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8349353d-d9cf-4023-a60e-79da0a44c108",
   "metadata": {},
   "source": [
    "A manufacturer produces two types of products, say $A$ and $B$.\n",
    "\n",
    "$A$ sells at $\\$5$ per unit, while $B$ sells at $\\$7$ per unit.\n",
    "\n",
    "We have to decide how many units of each product to produce,\n",
    "with the aim of maximizing the profit, subject to constraints:\n",
    "\n",
    "1. We must produce at least 20 units of $A$, and\n",
    "   offer at least $10$ units of $B$.\n",
    "    \n",
    "2. Our storage capacity is limited to 250 units.\n",
    "\n",
    "3. We have only 600 hours of labor available, \n",
    "\n",
    "   * one unit of $A$ requires two hours of labor,\n",
    "   * while one unit of $B$ takes three hours of labor.\n",
    "\n",
    "Do the following:\n",
    "\n",
    "1. Set up the linear programming model for this problem.\n",
    "2. Formulate the dual problem."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7bf011f6-4fbb-4f1c-a2a9-3154992d41f3",
   "metadata": {},
   "source": [
    "# Cost Benefit Analysis"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4c386a1c-3b12-421f-8b37-27d11bdc5177",
   "metadata": {},
   "source": [
    "An investment of $\\$3,000$ in information technology is expected\n",
    "to increase the monthly revenue by $\\$200$.\n",
    "\n",
    "Suppose the equipment has a life span of five years.\n",
    "\n",
    "1. Use a discount rate of 3% to determine the payback period.\n",
    "2. What is the return on investment?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f34af1e7-c71a-4bc2-8253-370e3479d62a",
   "metadata": {},
   "source": [
    "# Machine Learning"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2d3aee3a-bfa6-446b-ad14-7507b6d02a28",
   "metadata": {},
   "source": [
    "Consider the geographic prediction of the outcome of a referendum.\n",
    "\n",
    "Voters have three choices, labeled by\n",
    "\n",
    "  * $y = -1$ (if disapprove), or\n",
    "  * $y = +1$ (if approve), or \n",
    "  * $y = 0$ (if abstain).\n",
    "\n",
    "The pollsters collected 30 coordinates $(x_1, x_2)$,\n",
    "of 10 voters who disapproved, 10 who approved, and 10 who abstained.\n",
    "\n",
    "With a neural network we want to geographically predict\n",
    "the outcome of the referendum.\n",
    "\n",
    "1. Define the setup of the training data.\n",
    "2. Draw the two-layer neural network for this prediction,\n",
    "   using two neurons in the layer following the input layer.\n",
    "3. What are the dimensions of the neural network?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cbd936cd-4a39-4779-82dd-2cac02630376",
   "metadata": {},
   "source": [
    "# Ordinary Differential Equations"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "11cf9493-59c0-4522-a4ef-d4e72627a1fe",
   "metadata": {},
   "source": [
    "A crack in the hull of a ship at sea suddenly breaks open \n",
    "and the ship starts taking water in. \n",
    "\n",
    "As the crack is widening, the rate at which the volume of water \n",
    "inside the ship increases is one tenth of the volume already in the ship. \n",
    "\n",
    "The pump on board can pump out water at a rate of 0.5 ${\\rm m}^3$ per second.\n",
    "\n",
    "When the pump starts pumping water out,\n",
    "there is already 20 ${\\rm m}^3$ of water inside the ship.\n",
    "\n",
    "1. Set up the ODE to model the evolution of the volume of water\n",
    "   inside the ship as time advances.\n",
    "\n",
    "2. Only when the volume of water inside the ship becomes zero,\n",
    "   one can cover the crack in the hull and save the ship.\n",
    "\n",
    "   Solve the ODE.  Will the ship sink?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f22478fd-ba00-49d2-8f8a-6a3c8a80ca24",
   "metadata": {},
   "source": [
    "# Root Locus"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dd98086f-40c9-40e0-a231-77a4595f507e",
   "metadata": {},
   "source": [
    "Consider the plant with transfer function\n",
    "$$\n",
    "  P(s) = \\frac{5 s^2 - 88 s - 43}{-73 s^2 + 25 s + 4}.\n",
    "$$\n",
    "\n",
    "1. Show that the plant is unstable.\n",
    "2. Apply the root locus method to find a simple proportional\n",
    "   feedback constant $\\kappa$ to stabilize the plant."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0bfc9f8d-5c48-480a-9cf3-739336311b55",
   "metadata": {},
   "source": [
    "# Partial Differential Equations"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7d05835e-2dc5-4b26-9f1a-c0740864be19",
   "metadata": {},
   "source": [
    "Model the education process by a PDE.  \n",
    "\n",
    "* The function $u(x,t)$ models\n",
    "  the distribution of the scores at time $t$.  \n",
    "\n",
    "* For $s \\in [0,1]$, $\\displaystyle \\int_0^s u(x,t) dx$\n",
    "  gives the number of students with score $s$ or less.\n",
    "  The $s$ can be interpreted as a percentage.\n",
    "    \n",
    "* At $t=0$, $u(x,t)$ obviously corresponds to the Bell curve which models\n",
    "  the individual IQs.\n",
    "\n",
    "* As the semester progresses, the profile of the Bell curve changes.\n",
    "\n",
    "1. What type of PDE would you choose to model the education process?\n",
    "   Justify your choice.\n",
    "2. What are the critical parameters in your model?\n",
    "        \n",
    "   How does one characterize the best education process?"
   ]
  }
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