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Fix: properly render P-Median model formulation in docs (#466) #483

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Apr 25, 2025
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Fix: properly render P-Median model formulation in docs (#466) #483

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105 changes: 63 additions & 42 deletions notebooks/p-median.ipynb
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Original file line number Diff line number Diff line change
Expand Up @@ -17,27 +17,38 @@
"\n",
"**P-Median can be written as:**\n",
"\n",
"$\\begin{array} \\displaystyle \\textbf{Minimize} & \\displaystyle \\sum_{i \\in I}\\sum_{j \\in J}{a_i d_{ij} X_{ij}} &&& (1) \\\\\n",
"\\displaystyle \\textbf{Subject to:} & \\displaystyle \\sum_{j \\in J}{X_{ij} = 1} & \\forall i \\in I && (2) \\\\\n",
" & \\displaystyle \\sum_{j \\in J}{Y_{j} = p} &&& (3) \\\\\n",
" & X_{ij} \\leq Y_{j} & \\forall i \\in I & \\forall j \\in J & (4) \\\\\n",
" & X_{ij} \\in \\{0,1\\} & \\forall i \\in I & \\forall j \\in J & (5) \\\\\n",
" & Y_{j} \\in \\{0,1\\} & \\forall j \\in J && (6) \\\\ \\end{array}$\n",
"$$\n",
"\\begin{array}{lllll}\n",
"\\textbf{Minimize} & \\displaystyle \\sum_{i \\in I} \\sum_{j \\in J} a_i d_{ij} X_{ij} &&& (1) \\\\\n",
"\\textbf{Subject to:} & \\sum_{j \\in J} X_{ij} = 1 & \\forall i \\in I && (2) \\\\\n",
"& \\sum_{j \\in J} Y_j = p &&& (3) \\\\\n",
"& X_{ij} \\leq Y_j & \\forall i \\in I, j \\in J && (4) \\\\\n",
"& X_{ij} \\in \\{0, 1\\} & \\forall i \\in I, j \\in J && (5) \\\\\n",
"& Y_j \\in \\{0, 1\\} & \\forall j \\in J && (6) \\\\\n",
"\\end{array}\n",
"$$\n",
"\n",
"$$\n",
"\\begin{array}{lllll}\n",
"\\textbf{Where:} \\\\\n",
"& i & = & \\text{index referencing demand nodes} \\\\\n",
"& j & = & \\text{index referencing potential facility sites} \\\\\n",
"& d_{ij} & = & \\text{distance or travel time between node } i \\text{ and } j \\\\\n",
"& p & = & \\text{number of facilities to locate} \\\\\n",
"& a_i & = & \\text{demand at node } i \\\\\n",
"& X_{ij} & = &\n",
" \\begin{cases}\n",
" 1, & \\text{if demand } i \\text{ is assigned to facility } j \\\\\n",
" 0, & \\text{otherwise}\n",
" \\end{cases} \\\\\n",
"& Y_j & = &\n",
" \\begin{cases}\n",
" 1, & \\text{if facility is located at node } j \\\\\n",
" 0, & \\text{otherwise}\n",
" \\end{cases}\n",
"\\end{array}\n",
"$$\n",
"\n",
"$\\begin{array} \\displaystyle \\textbf{Where:}\\\\ & & \\displaystyle i & \\small = & \\textrm{index referencing nodes of the network as demand} \\\\\n",
"& & j & \\small = & \\textrm{index referencing nodes of the network as potential facility sites} \\\\\n",
"& & d_{ij} & \\small = & \\textrm{shortest distance or travel time between nodes } i \\textrm{ and } j \\\\\n",
"& & p & \\small = & \\textrm{number of facilities to be located} \\\\\n",
"& & a_i & \\small = & \\textrm{service load or population demand at } i \\\\\n",
"& & X_{ij} & \\small = & \\begin{cases}\n",
" 1, \\textrm{if demand } i \\textrm{ is assigned to facility } j \\\\\n",
" 0, \\textrm{otherwise}\n",
" \\end{cases} \\\\\n",
"& & Y_{j} & \\small = & \\begin{cases}\n",
" 1, \\textrm{if node } j \\textrm{ has been selected for a facility} \\\\\n",
" 0, \\textrm{otherwise} \\\\\n",
" \\end{cases} \\\\ \n",
"\\end{array}$\n",
"\n",
"_The formulation above is adapted from Church and Murray (2018)_\n",
"\n",
Expand Down Expand Up @@ -1696,29 +1707,39 @@
"\n",
"**Capacitated P-Median can be written as:**\n",
"\n",
"$\\begin{array} \\displaystyle \\textbf{Minimize} & \\displaystyle \\sum_{i \\in I}\\sum_{j \\in J}{a_i d_{ij} X_{ij}} &&& (1) \\\\\n",
"\\displaystyle \\textbf{Subject to:} & \\displaystyle \\sum_{j \\in J}{X_{ij} = 1} & \\forall i \\in I && (2) \\\\\n",
" & \\displaystyle \\sum_{j \\in J}{Y_{j} = p} &&& (3) \\\\\n",
" & \\displaystyle \\sum_{i \\in I}{a_i X_{ij} \\leq {c_j Y_{j}}}& \\forall j \\in J && (4) \\\\\n",
" & X_{ij} \\leq Y_{j} & \\forall i \\in I & \\forall j \\in J & (5) \\\\\n",
" & X_{ij} \\in \\{0,1\\} & \\forall i \\in I & \\forall j \\in J & (6) \\\\\n",
" & Y_{j} \\in \\{0,1\\} & \\forall j \\in J && (7) \\\\ \\end{array}$\n",
"$$\n",
"\\begin{array}{lllll}\n",
"\\textbf{Minimize} & \\displaystyle \\sum_{i \\in I} \\sum_{j \\in J} a_i d_{ij} X_{ij} &&& (1) \\\\\n",
"\\textbf{Subject to:} & \\sum_{j \\in J} X_{ij} = 1 & \\forall i \\in I && (2) \\\\\n",
"& \\sum_{j \\in J} Y_j = p &&& (3) \\\\\n",
"& \\sum_{i \\in I} a_i X_{ij} \\leq c_j Y_j & \\forall j \\in J && (4) \\\\\n",
"& X_{ij} \\leq Y_j & \\forall i \\in I, j \\in J && (5) \\\\\n",
"& X_{ij} \\in \\{0, 1\\} & \\forall i \\in I, j \\in J && (6) \\\\\n",
"& Y_j \\in \\{0, 1\\} & \\forall j \\in J && (7) \\\\\n",
"\\end{array}\n",
"$$\n",
"\n",
"$\\begin{array} \\displaystyle \\textbf{Where:}\\\\ & & \\displaystyle i & \\small = & \\textrm{index referencing nodes of the network as demand} \\\\\n",
"& & j & \\small = & \\textrm{index referencing nodes of the network as potential facility sites} \\\\\n",
"& & d_{ij} & \\small = & \\textrm{shortest distance or travel time between nodes } i \\textrm{ and } j \\\\\n",
"& & p & \\small = & \\textrm{number of facilities to be located} \\\\\n",
"& & a_i & \\small = & \\textrm{service load or population demand at } i \\\\\n",
"& & c_j & \\small = & \\textrm{capacity of facility} j \\\\\n",
"& & X_{ij} & \\small = & \\begin{cases}\n",
" 1, \\textrm{if demand } i \\textrm{ is assigned to facility } j \\\\\n",
" 0, \\textrm{otherwise}\n",
" \\end{cases} \\\\\n",
"& & Y_{j} & \\small = & \\begin{cases}\n",
" 1, \\textrm{if node } j \\textrm{ has been selected for a facility} \\\\\n",
" 0, \\textrm{otherwise} \\\\\n",
" \\end{cases} \\\\ \n",
"\\end{array}$\n",
"$$\n",
"\\begin{array}{lllll}\n",
"\\textbf{Where:} \\\\\n",
"& i & = & \\text{index referencing demand nodes} \\\\\n",
"& j & = & \\text{index referencing potential facility sites} \\\\\n",
"& d_{ij} & = & \\text{distance or travel time between nodes } i \\text{ and } j \\\\\n",
"& p & = & \\text{number of facilities to locate} \\\\\n",
"& a_i & = & \\text{demand at node } i \\\\\n",
"& c_j & = & \\text{capacity of facility at node } j \\\\\n",
"& X_{ij} & = &\n",
" \\begin{cases}\n",
" 1, & \\text{if demand } i \\text{ is assigned to facility } j \\\\\n",
" 0, & \\text{otherwise}\n",
" \\end{cases} \\\\\n",
"& Y_j & = &\n",
" \\begin{cases}\n",
" 1, & \\text{if facility is located at node } j \\\\\n",
" 0, & \\text{otherwise}\n",
" \\end{cases}\n",
"\\end{array}\n",
"$$\n",
"\n",
"_The formulation above is adapted from Church and Murray (2009)_"
]
Expand Down