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updated the formulations in the docs. #468

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updated the formulations in the docs. #468

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12 changes: 6 additions & 6 deletions notebooks/p-median.ipynb
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Original file line number Diff line number Diff line change
Expand Up @@ -17,14 +17,14 @@
"\n",
"**P-Median can be written as:**\n",
"\n",
"$\\begin{array} \\displaystyle \\textbf{Minimize} & \\displaystyle \\sum_{i \\in I}\\sum_{j \\in J}{a_i d_{ij} X_{ij}} &&& (1) \\\\\n",
">$\\begin{array} \\displaystyle \\textbf{Minimize} & \\displaystyle \\sum_{i \\in I}\\sum_{j \\in J}{a_i d_{ij} X_{ij}} &&& (1) \\\\\n",
"\\displaystyle \\textbf{Subject to:} & \\displaystyle \\sum_{j \\in J}{X_{ij} = 1} & \\forall i \\in I && (2) \\\\\n",
" & \\displaystyle \\sum_{j \\in J}{Y_{j} = p} &&& (3) \\\\\n",
" & X_{ij} \\leq Y_{j} & \\forall i \\in I & \\forall j \\in J & (4) \\\\\n",
" & X_{ij} \\in \\{0,1\\} & \\forall i \\in I & \\forall j \\in J & (5) \\\\\n",
" & Y_{j} \\in \\{0,1\\} & \\forall j \\in J && (6) \\\\ \\end{array}$\n",
"\n",
"$\\begin{array} \\displaystyle \\textbf{Where:}\\\\ & & \\displaystyle i & \\small = & \\textrm{index referencing nodes of the network as demand} \\\\\n",
">$\\begin{array} \\displaystyle \\textbf{Where:}\\\\ & & \\displaystyle i & \\small = & \\textrm{index referencing nodes of the network as demand} \\\\\n",
"& & j & \\small = & \\textrm{index referencing nodes of the network as potential facility sites} \\\\\n",
"& & d_{ij} & \\small = & \\textrm{shortest distance or travel time between nodes } i \\textrm{ and } j \\\\\n",
"& & p & \\small = & \\textrm{number of facilities to be located} \\\\\n",
Expand All @@ -39,7 +39,7 @@
" \\end{cases} \\\\ \n",
"\\end{array}$\n",
"\n",
"_The formulation above is adapted from Church and Murray (2018)_\n",
">_The formulation above is adapted from Church and Murray (2018)_\n",
"\n",
"This tutorial generates synthetic demand (clients) and facility sites near a 10x10 lattice representing a gridded urban core.\n",
"\n",
Expand Down Expand Up @@ -1676,15 +1676,15 @@
"\n",
"**Capacitated P-Median can be written as:**\n",
"\n",
"$\\begin{array} \\displaystyle \\textbf{Minimize} & \\displaystyle \\sum_{i \\in I}\\sum_{j \\in J}{a_i d_{ij} X_{ij}} &&& (1) \\\\\n",
">$\\begin{array} \\displaystyle \\textbf{Minimize} & \\displaystyle \\sum_{i \\in I}\\sum_{j \\in J}{a_i d_{ij} X_{ij}} &&& (1) \\\\\n",
"\\displaystyle \\textbf{Subject to:} & \\displaystyle \\sum_{j \\in J}{X_{ij} = 1} & \\forall i \\in I && (2) \\\\\n",
" & \\displaystyle \\sum_{j \\in J}{Y_{j} = p} &&& (3) \\\\\n",
" & \\displaystyle \\sum_{i \\in I}{a_i X_{ij} \\leq {c_j Y_{j}}}& \\forall j \\in J && (4) \\\\\n",
" & X_{ij} \\leq Y_{j} & \\forall i \\in I & \\forall j \\in J & (5) \\\\\n",
" & X_{ij} \\in \\{0,1\\} & \\forall i \\in I & \\forall j \\in J & (6) \\\\\n",
" & Y_{j} \\in \\{0,1\\} & \\forall j \\in J && (7) \\\\ \\end{array}$\n",
"\n",
"$\\begin{array} \\displaystyle \\textbf{Where:}\\\\ & & \\displaystyle i & \\small = & \\textrm{index referencing nodes of the network as demand} \\\\\n",
">$\\begin{array} \\displaystyle \\textbf{Where:}\\\\ & & \\displaystyle i & \\small = & \\textrm{index referencing nodes of the network as demand} \\\\\n",
"& & j & \\small = & \\textrm{index referencing nodes of the network as potential facility sites} \\\\\n",
"& & d_{ij} & \\small = & \\textrm{shortest distance or travel time between nodes } i \\textrm{ and } j \\\\\n",
"& & p & \\small = & \\textrm{number of facilities to be located} \\\\\n",
Expand All @@ -1700,7 +1700,7 @@
" \\end{cases} \\\\ \n",
"\\end{array}$\n",
"\n",
"_The formulation above is adapted from Church and Murray (2009)_"
">_The formulation above is adapted from Church and Murray (2009)_"
]
},
{
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