{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "1beb45a9-8301-48e0-8050-fe8f3e1d7191",
   "metadata": {},
   "source": [
    "The first part of the notebook is from Friday unchanged.  Scroll down about half way\n",
    "to where it says **Sensitivity Analysis** for what we did the following Monday."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6a6eebf6-87bb-4213-b779-06b973bcfa34",
   "metadata": {},
   "source": [
    "**Example 3.5.** A large construction firm is currently excavating at three sites.\n",
    "Meanwhile, they are also building at four additional sites, where they require fill\n",
    "dirt. The excavations at sites 1, 2, and 3 produce 150, 400, and 325 cubic yards\n",
    "of dirt per day. The building sites A, B, C, and D require 175, 125, 225, and\n",
    "450 cubic yards of dirt per day. Additional fill dirt can also be obtained from\n",
    "site 4 at a cost of 5 dollars per cubic yard. The cost of shipping fill dirt is about\n",
    "20 dollars per mile for one truckload, and a truck carries 10 cubic yards of dirt.\n",
    "Table 3.3 gives the distance between sites in miles. Find the transportation plan\n",
    "that minimizes the cost to the company."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5a04bc3c-1e09-4a64-bcd8-5dbf22a4e0fe",
   "metadata": {},
   "source": [
    "**Table 3.3**\n",
    "\n",
    "|site|A|B|C|D|\n",
    "|:--:|:--:|:--:|:--:|:--:|\n",
    "|**1**|5|2|6|10|\n",
    "|**2**|4|5|7|5|\n",
    "|**3**|7|6|4|4|\n",
    "|**4**|9|10|6|2|"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "a4988841-5024-443f-b786-6df9969f5761",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×4 Matrix{Int64}:\n",
       " 5   2  6  10\n",
       " 4   5  7   5\n",
       " 7   6  4   4\n",
       " 9  10  6   2"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D=[5 2 6 10; 4 5 7 5; 7 6 4 4; 9 10 6 2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "8c9087b7-5daf-4821-95f4-347606252de8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×4 Matrix{Int64}:\n",
       " 10   4  12  20\n",
       "  8  10  14  10\n",
       " 14  12   8   8\n",
       " 23  25  17   9"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "C=2*D\n",
    "C[4,:].+=5\n",
    "C"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "26113623-6e86-4261-bbc2-c6f6a0e3a357",
   "metadata": {},
   "outputs": [],
   "source": [
    "using JuMP, HiGHS"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "989eb0bc-de1d-4c17-94f2-6be027079715",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "A JuMP Model\n",
       "├ solver: HiGHS\n",
       "├ objective_sense: FEASIBILITY_SENSE\n",
       "├ num_variables: 0\n",
       "├ num_constraints: 0\n",
       "└ Names registered in the model: none"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "model=Model(HiGHS.Optimizer)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "29fe4479-cb12-4291-9976-50d0f01285cc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×4 Matrix{VariableRef}:\n",
       " x[1,1]  x[1,2]  x[1,3]  x[1,4]\n",
       " x[2,1]  x[2,2]  x[2,3]  x[2,4]\n",
       " x[3,1]  x[3,2]  x[3,3]  x[3,4]\n",
       " x[4,1]  x[4,2]  x[4,3]  x[4,4]"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@variable(model,x[1:4,1:4].>=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "07479977-8276-47c5-88f8-222286a820c6",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$ 10 x_{1,1} + 8 x_{2,1} + 14 x_{3,1} + 23 x_{4,1} + 4 x_{1,2} + 10 x_{2,2} + 12 x_{3,2} + 25 x_{4,2} + 12 x_{1,3} + 14 x_{2,3} + 8 x_{3,3} + 17 x_{4,3} + 20 x_{1,4} + 10 x_{2,4} + 8 x_{3,4} + 9 x_{4,4} $"
      ],
      "text/plain": [
       "10 x[1,1] + 8 x[2,1] + 14 x[3,1] + 23 x[4,1] + 4 x[1,2] + 10 x[2,2] + 12 x[3,2] + 25 x[4,2] + 12 x[1,3] + 14 x[2,3] + 8 x[3,3] + 17 x[4,3] + 20 x[1,4] + 10 x[2,4] + 8 x[3,4] + 9 x[4,4]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cost(x)=sum(C.*x)\n",
    "cost(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "0bbdfc56-dec4-421f-a04b-e91900798659",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10 x[1,1] + 8 x[2,1] + 14 x[3,1] + 23 x[4,1] + 4 x[1,2] + 10 x[2,2] + 12 x[3,2] + 25 x[4,2] + 12 x[1,3] + 14 x[2,3] + 8 x[3,3] + 17 x[4,3] + 20 x[1,4] + 10 x[2,4] + 8 x[3,4] + 9 x[4,4]\n"
     ]
    }
   ],
   "source": [
    "# Fix it so it doesn't scroll off the edge\n",
    "println(cost(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "ac7134d0-e23e-4b5c-8bf5-cf72b903ae8b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$ 10 x_{1,1} + 8 x_{2,1} + 14 x_{3,1} + 23 x_{4,1} + 4 x_{1,2} + 10 x_{2,2} + 12 x_{3,2} + 25 x_{4,2} + 12 x_{1,3} + 14 x_{2,3} + 8 x_{3,3} + 17 x_{4,3} + 20 x_{1,4} + 10 x_{2,4} + 8 x_{3,4} + 9 x_{4,4} $"
      ],
      "text/plain": [
       "10 x[1,1] + 8 x[2,1] + 14 x[3,1] + 23 x[4,1] + 4 x[1,2] + 10 x[2,2] + 12 x[3,2] + 25 x[4,2] + 12 x[1,3] + 14 x[2,3] + 8 x[3,3] + 17 x[4,3] + 20 x[1,4] + 10 x[2,4] + 8 x[3,4] + 9 x[4,4]"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "@objective(model,Min,cost(x))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "d4324889-bb0b-4eef-b2bf-81b26bac66ef",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{AffExpr}:\n",
       " x[1,1] + x[2,1] + x[3,1] + x[4,1]\n",
       " x[1,2] + x[2,2] + x[3,2] + x[4,2]\n",
       " x[1,3] + x[2,3] + x[3,3] + x[4,3]\n",
       " x[1,4] + x[2,4] + x[3,4] + x[4,4]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Amount of dirt received by site j\n",
    "r=[sum(x[:,j]) for j=1:4]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "99876ac8-aaa3-45b9-b053-27b835624d9b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.GreaterThan{Float64}}, ScalarShape}}:\n",
       " x[1,1] + x[2,1] + x[3,1] + x[4,1] ≥ 175\n",
       " x[1,2] + x[2,2] + x[3,2] + x[4,2] ≥ 125\n",
       " x[1,3] + x[2,3] + x[3,3] + x[4,3] ≥ 225\n",
       " x[1,4] + x[2,4] + x[3,4] + x[4,4] ≥ 450"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c1=@constraint(model,r.>=[175,125,225,450])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "c6c3aabf-fdfe-499c-8659-ed8f4f2cbdcb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{AffExpr}:\n",
       " x[1,1] + x[1,2] + x[1,3] + x[1,4]\n",
       " x[2,1] + x[2,2] + x[2,3] + x[2,4]\n",
       " x[3,1] + x[3,2] + x[3,3] + x[3,4]\n",
       " x[4,1] + x[4,2] + x[4,3] + x[4,4]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Amount of dirt produced by site i\n",
    "s=[sum(x[i,:]) for i=1:4]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "c4e1d7ef-be44-44d1-902e-1b96a4a5f3dd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{ConstraintRef{Model, MathOptInterface.ConstraintIndex{MathOptInterface.ScalarAffineFunction{Float64}, MathOptInterface.LessThan{Float64}}, ScalarShape}}:\n",
       " x[1,1] + x[1,2] + x[1,3] + x[1,4] ≤ 150\n",
       " x[2,1] + x[2,2] + x[2,3] + x[2,4] ≤ 400\n",
       " x[3,1] + x[3,2] + x[3,3] + x[3,4] ≤ 325"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c2=@constraint(model,s[1:3].<=[150,400,325])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "c07a34e8-9218-475a-b50f-87b65ed0bd80",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ \\begin{aligned}\n",
       "\\min\\quad & 10 x_{1,1} + 8 x_{2,1} + 14 x_{3,1} + 23 x_{4,1} + 4 x_{1,2} + 10 x_{2,2} + 12 x_{3,2} + 25 x_{4,2} + 12 x_{1,3} + 14 x_{2,3} + 8 x_{3,3} + 17 x_{4,3} + 20 x_{1,4} + 10 x_{2,4} + 8 x_{3,4} + 9 x_{4,4}\\\\\n",
       "\\text{Subject to} \\quad & x_{1,1} + x_{2,1} + x_{3,1} + x_{4,1} \\geq 175\\\\\n",
       " & x_{1,2} + x_{2,2} + x_{3,2} + x_{4,2} \\geq 125\\\\\n",
       " & x_{1,3} + x_{2,3} + x_{3,3} + x_{4,3} \\geq 225\\\\\n",
       " & x_{1,4} + x_{2,4} + x_{3,4} + x_{4,4} \\geq 450\\\\\n",
       " & x_{1,1} + x_{1,2} + x_{1,3} + x_{1,4} \\leq 150\\\\\n",
       " & x_{2,1} + x_{2,2} + x_{2,3} + x_{2,4} \\leq 400\\\\\n",
       " & x_{3,1} + x_{3,2} + x_{3,3} + x_{3,4} \\leq 325\\\\\n",
       " & x_{1,1} \\geq 0\\\\\n",
       " & x_{2,1} \\geq 0\\\\\n",
       " & x_{3,1} \\geq 0\\\\\n",
       " & x_{4,1} \\geq 0\\\\\n",
       " & x_{1,2} \\geq 0\\\\\n",
       " & x_{2,2} \\geq 0\\\\\n",
       " & x_{3,2} \\geq 0\\\\\n",
       " & x_{4,2} \\geq 0\\\\\n",
       " & x_{1,3} \\geq 0\\\\\n",
       " & x_{2,3} \\geq 0\\\\\n",
       " & x_{3,3} \\geq 0\\\\\n",
       " & x_{4,3} \\geq 0\\\\\n",
       " & x_{1,4} \\geq 0\\\\\n",
       " & x_{2,4} \\geq 0\\\\\n",
       " & x_{3,4} \\geq 0\\\\\n",
       " & x_{4,4} \\geq 0\\\\\n",
       "\\end{aligned} $$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "print(model)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "125af725-1e03-46d4-91eb-f29535e7de66",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Min 10 x[1,1] + 8 x[2,1] + 14 x[3,1] + 23 x[4,1] + 4 x[1,2] + 10 x[2,2] + 12 x[3,2] + 25 x[4,2] + 12 x[1,3] + 14 x[2,3] + 8 x[3,3] + 17 x[4,3] + 20 x[1,4] + 10 x[2,4] + 8 x[3,4] + 9 x[4,4]\n",
      "Subject to\n",
      " x[1,1] + x[2,1] + x[3,1] + x[4,1] ≥ 175\n",
      " x[1,2] + x[2,2] + x[3,2] + x[4,2] ≥ 125\n",
      " x[1,3] + x[2,3] + x[3,3] + x[4,3] ≥ 225\n",
      " x[1,4] + x[2,4] + x[3,4] + x[4,4] ≥ 450\n",
      " x[1,1] + x[1,2] + x[1,3] + x[1,4] ≤ 150\n",
      " x[2,1] + x[2,2] + x[2,3] + x[2,4] ≤ 400\n",
      " x[3,1] + x[3,2] + x[3,3] + x[3,4] ≤ 325\n",
      " x[1,1] ≥ 0\n",
      " x[2,1] ≥ 0\n",
      " x[3,1] ≥ 0\n",
      " x[4,1] ≥ 0\n",
      " x[1,2] ≥ 0\n",
      " x[2,2] ≥ 0\n",
      " x[3,2] ≥ 0\n",
      " x[4,2] ≥ 0\n",
      " x[1,3] ≥ 0\n",
      " x[2,3] ≥ 0\n",
      " x[3,3] ≥ 0\n",
      " x[4,3] ≥ 0\n",
      " x[1,4] ≥ 0\n",
      " x[2,4] ≥ 0\n",
      " x[3,4] ≥ 0\n",
      " x[4,4] ≥ 0\n"
     ]
    }
   ],
   "source": [
    "# Fix the scrolling left to right\n",
    "print(sprint(print,model))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "1c42a60e-9460-4752-8f9a-a870e6409ce4",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Running HiGHS 1.13.1 (git hash: 1d267d97c): Copyright (c) 2026 under Apache 2.0 license terms\n",
      "Using BLAS: blastrampoline \n",
      "LP has 7 rows; 16 cols; 28 nonzeros\n",
      "Coefficient ranges:\n",
      "  Matrix  [1e+00, 1e+00]\n",
      "  Cost    [4e+00, 2e+01]\n",
      "  Bound   [0e+00, 0e+00]\n",
      "  RHS     [1e+02, 4e+02]\n",
      "Presolving model\n",
      "7 rows, 16 cols, 28 nonzeros  0s\n",
      "7 rows, 14 cols, 24 nonzeros  0s\n",
      "Presolve reductions: rows 7(-0); columns 14(-2); nonzeros 24(-4) \n",
      "Solving the presolved LP\n",
      "Using dual simplex solver\n",
      "  Iteration        Objective     Infeasibilities num(sum)\n",
      "          0     0.0000000000e+00 Pr: 4(975) 0.0s\n",
      "          5     7.6500000000e+03 Pr: 0(0) 0.0s\n",
      "\n",
      "Performed postsolve\n",
      "Solving the original LP from the solution after postsolve\n",
      "\n",
      "Model status        : Optimal\n",
      "Simplex   iterations: 5\n",
      "Objective value     :  7.6500000000e+03\n",
      "P-D objective error :  0.0000000000e+00\n",
      "HiGHS run time      :          0.00\n"
     ]
    }
   ],
   "source": [
    "optimize!(model)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "c8a89858-855c-4e86-8d65-ecda18490416",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "7650.0"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "objective_value(model)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2717a9d1-78dc-4ee4-9796-2ae7428e0096",
   "metadata": {},
   "source": [
    "**Sensitivity Analysis**.  We check how sensitive the optimum is with respect\n",
    "to the constraints.\n",
    "\n",
    "The following has has been simplified from what we did interactively\n",
    "to make it easier to follow when reading the notebook."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "0da5c296-4652-4351-a830-62bb0204d6e5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{Float64}:\n",
       " 0.0\n",
       " 0.0\n",
       " 0.0\n",
       " 0.0"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# All these zero mean the optimum lies on all constraints related to how\n",
    "# dirt is needed at sites A, B, C and D.  No more dirt than necessary was\n",
    "# supplied to any of the receiving sites.\n",
    "value(r)-[175,125,225,450]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "11e0c80d-178e-4b5a-9f72-34d6b7cfe4cc",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3-element Vector{Float64}:\n",
       "  25.0\n",
       " 225.0\n",
       "   0.0"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The first two are positive means that not all the dirt available at\n",
    "# sites 1 and 2 was used.  Thus, the amount of dirt available at the\n",
    "# first two sites did not affect the optimum.\n",
    "[150,400,325]-value(s[1:3])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fd03bbb4-6818-41f5-96c8-8fb139cb5fb5",
   "metadata": {},
   "source": [
    "As shown below, the\n",
    "shadow prices for sites 1 and 2 are zero, since there was excess dirt left at\n",
    "those sites.\n",
    "The shadow price for site 3 is −1 dollar, meaning that the total cost would\n",
    "increase by −1 dollar if this constraint were increased by 1 cubic yard of dirt."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "f4ad3435-1185-4003-b4e5-d1966e29c3a1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ x_{1,1} + x_{1,2} + x_{1,3} + x_{1,4} \\leq 150 $$"
      ],
      "text/plain": [
       "x[1,1] + x[1,2] + x[1,3] + x[1,4] ≤ 150"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# The dual of the constraint is called the shadow price.\n",
    "display(c2[1])\n",
    "dual(c2[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "23aac56c-703f-46f0-a992-4f66e25464b5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ x_{2,1} + x_{2,2} + x_{2,3} + x_{2,4} \\leq 400 $$"
      ],
      "text/plain": [
       "x[2,1] + x[2,2] + x[2,3] + x[2,4] ≤ 400"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "display(c2[2])\n",
    "dual(c2[2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "de60187c-7c89-4d13-ba4c-3b3c9b42e877",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ x_{3,1} + x_{3,2} + x_{3,3} + x_{3,4} \\leq 325 $$"
      ],
      "text/plain": [
       "x[3,1] + x[3,2] + x[3,3] + x[3,4] ≤ 325"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "-1.0"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Means that increasing the third element of the second\n",
    "# constraint group increases the price by -1.  That is,\n",
    "# decreases it by 1.\n",
    "display(c2[3])\n",
    "dual(c2[3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "007254d8-cd0c-44bf-a623-f6908bbfe85b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4-element Vector{Float64}:\n",
       " 8.0\n",
       " 4.0\n",
       " 9.0\n",
       " 9.0"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Simultaneously find the duals of all the c1 constraints\n",
    "dual.(c1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1c6927d9-c011-45fd-a3df-4b619d0ca701",
   "metadata": {},
   "source": [
    "As shown above, the duals of each of c1 constraints are non-zero, so\n",
    "the optimum lies on the boundary of those contraints.\n",
    "For example, if the contraint given by c1[3] is is\n",
    "increased by 1, then the cost increases by 9.\n",
    "\n",
    "Illustrate the shadow prices by changing the constraints\n",
    "and observing the change in price.  Note that if we change\n",
    "the constraint by too much, then other contraints may\n",
    "become important at which the shadow price might change."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "da85a392-62ef-4923-886e-3496cda8a149",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ x_{3,1} + x_{3,2} + x_{3,3} + x_{3,4} \\leq 326 $$"
      ],
      "text/plain": [
       "x[3,1] + x[3,2] + x[3,3] + x[3,4] ≤ 326"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Change the constraint with shadow price -1\n",
    "# by deleting it and adding a constraint that is 1 greater.\n",
    "delete(model,c2[3])\n",
    "c2[3]=@constraint(model,s[3]<=326)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "63f1e845-fec6-416c-a7c7-c7ccfb1d7998",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "LP has 7 rows; 16 cols; 28 nonzeros\n",
      "Coefficient ranges:\n",
      "  Matrix  [1e+00, 1e+00]\n",
      "  Cost    [4e+00, 2e+01]\n",
      "  Bound   [0e+00, 0e+00]\n",
      "  RHS     [1e+02, 4e+02]\n",
      "Solving LP with useful basis so presolve not used\n",
      "Using dual simplex solver\n",
      "  Iteration        Objective     Infeasibilities num(sum)\n",
      "          0    -1.0000001711e+00 Ph1: 2(2); Du: 1(1) 0.0s\n",
      "          2     7.6490000000e+03 Pr: 0(0) 0.0s\n",
      "\n",
      "Model status        : Optimal\n",
      "Simplex   iterations: 2\n",
      "Objective value     :  7.6490000000e+03\n",
      "P-D objective error :  0.0000000000e+00\n",
      "HiGHS run time      :          0.00\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "7649.0"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimize!(model)\n",
    "objective_value(model)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4f03608e-8f26-4af0-bee9-991a3d91b4fe",
   "metadata": {},
   "source": [
    "Note that 7649 is one less that 7650.  This is consistent\n",
    "with the shadow price since increasing by -1 is\n",
    "the same as decreasing by 1."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "c39c5f29-618b-4372-9e5f-c527a1b977db",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$ x_{1,3} + x_{2,3} + x_{3,3} + x_{4,3} \\geq 226 $$"
      ],
      "text/plain": [
       "x[1,3] + x[2,3] + x[3,3] + x[4,3] ≥ 226"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Change the constraint with shadow price 9\n",
    "# by deleting it and adding a constraint that is 1 greater.\n",
    "delete(model,c1[3])\n",
    "c1[3]=@constraint(model,r[3]>=226)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "2521aab1-7f92-40f2-a7f5-cde70284dca0",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "LP has 7 rows; 16 cols; 28 nonzeros\n",
      "Coefficient ranges:\n",
      "  Matrix  [1e+00, 1e+00]\n",
      "  Cost    [4e+00, 2e+01]\n",
      "  Bound   [0e+00, 0e+00]\n",
      "  RHS     [1e+02, 4e+02]\n",
      "Solving LP with useful basis so presolve not used\n",
      "Using dual simplex solver\n",
      "  Iteration        Objective     Infeasibilities num(sum)\n",
      "          0    -1.9000008583e+01 Ph1: 4(8); Du: 3(19) 0.0s\n",
      "          2     7.6580000000e+03 Pr: 0(0) 0.0s\n",
      "\n",
      "Model status        : Optimal\n",
      "Simplex   iterations: 2\n",
      "Objective value     :  7.6580000000e+03\n",
      "P-D objective error :  0.0000000000e+00\n",
      "HiGHS run time      :          0.00\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "7658.0"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimize!(model)\n",
    "objective_value(model)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3f2ff989-4aff-4ac1-9ba9-7800f7aeb618",
   "metadata": {},
   "source": [
    "Note that 7658 is nine more than 7649.  This is consistent\n",
    "with the shadow price being 9."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "48b3f598-08d8-4410-96fc-baaeffd3b8ab",
   "metadata": {},
   "source": [
    "In both of the above cases changing the contraint by a\n",
    "small amount did not affect which constraints were active\n",
    "so the shadow price exactly predicted by how much the\n",
    "objective function changed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "f3ceccbf-8fa5-438c-b226-368515aab684",
   "metadata": {},
   "outputs": [],
   "source": []
  }
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