|
17 | 17 | "\n",
|
18 | 18 | "**P-Median can be written as:**\n",
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19 | 19 | "\n",
|
20 |
| - "$\\begin{array} \\displaystyle \\textbf{Minimize} & \\displaystyle \\sum_{i \\in I}\\sum_{j \\in J}{a_i d_{ij} X_{ij}} &&& (1) \\\\\n", |
| 20 | + ">$\\begin{array} \\displaystyle \\textbf{Minimize} & \\displaystyle \\sum_{i \\in I}\\sum_{j \\in J}{a_i d_{ij} X_{ij}} &&& (1) \\\\\n", |
21 | 21 | "\\displaystyle \\textbf{Subject to:} & \\displaystyle \\sum_{j \\in J}{X_{ij} = 1} & \\forall i \\in I && (2) \\\\\n",
|
22 | 22 | " & \\displaystyle \\sum_{j \\in J}{Y_{j} = p} &&& (3) \\\\\n",
|
23 | 23 | " & X_{ij} \\leq Y_{j} & \\forall i \\in I & \\forall j \\in J & (4) \\\\\n",
|
24 | 24 | " & X_{ij} \\in \\{0,1\\} & \\forall i \\in I & \\forall j \\in J & (5) \\\\\n",
|
25 | 25 | " & Y_{j} \\in \\{0,1\\} & \\forall j \\in J && (6) \\\\ \\end{array}$\n",
|
26 | 26 | "\n",
|
27 |
| - "$\\begin{array} \\displaystyle \\textbf{Where:}\\\\ & & \\displaystyle i & \\small = & \\textrm{index referencing nodes of the network as demand} \\\\\n", |
| 27 | + ">$\\begin{array} \\displaystyle \\textbf{Where:}\\\\ & & \\displaystyle i & \\small = & \\textrm{index referencing nodes of the network as demand} \\\\\n", |
28 | 28 | "& & j & \\small = & \\textrm{index referencing nodes of the network as potential facility sites} \\\\\n",
|
29 | 29 | "& & d_{ij} & \\small = & \\textrm{shortest distance or travel time between nodes } i \\textrm{ and } j \\\\\n",
|
30 | 30 | "& & p & \\small = & \\textrm{number of facilities to be located} \\\\\n",
|
|
39 | 39 | " \\end{cases} \\\\ \n",
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40 | 40 | "\\end{array}$\n",
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41 | 41 | "\n",
|
42 |
| - "_The formulation above is adapted from Church and Murray (2018)_\n", |
| 42 | + ">_The formulation above is adapted from Church and Murray (2018)_\n", |
43 | 43 | "\n",
|
44 | 44 | "This tutorial generates synthetic demand (clients) and facility sites near a 10x10 lattice representing a gridded urban core.\n",
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45 | 45 | "\n",
|
|
1676 | 1676 | "\n",
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1677 | 1677 | "**Capacitated P-Median can be written as:**\n",
|
1678 | 1678 | "\n",
|
1679 |
| - "$\\begin{array} \\displaystyle \\textbf{Minimize} & \\displaystyle \\sum_{i \\in I}\\sum_{j \\in J}{a_i d_{ij} X_{ij}} &&& (1) \\\\\n", |
| 1679 | + ">$\\begin{array} \\displaystyle \\textbf{Minimize} & \\displaystyle \\sum_{i \\in I}\\sum_{j \\in J}{a_i d_{ij} X_{ij}} &&& (1) \\\\\n", |
1680 | 1680 | "\\displaystyle \\textbf{Subject to:} & \\displaystyle \\sum_{j \\in J}{X_{ij} = 1} & \\forall i \\in I && (2) \\\\\n",
|
1681 | 1681 | " & \\displaystyle \\sum_{j \\in J}{Y_{j} = p} &&& (3) \\\\\n",
|
1682 | 1682 | " & \\displaystyle \\sum_{i \\in I}{a_i X_{ij} \\leq {c_j Y_{j}}}& \\forall j \\in J && (4) \\\\\n",
|
1683 | 1683 | " & X_{ij} \\leq Y_{j} & \\forall i \\in I & \\forall j \\in J & (5) \\\\\n",
|
1684 | 1684 | " & X_{ij} \\in \\{0,1\\} & \\forall i \\in I & \\forall j \\in J & (6) \\\\\n",
|
1685 | 1685 | " & Y_{j} \\in \\{0,1\\} & \\forall j \\in J && (7) \\\\ \\end{array}$\n",
|
1686 | 1686 | "\n",
|
1687 |
| - "$\\begin{array} \\displaystyle \\textbf{Where:}\\\\ & & \\displaystyle i & \\small = & \\textrm{index referencing nodes of the network as demand} \\\\\n", |
| 1687 | + ">$\\begin{array} \\displaystyle \\textbf{Where:}\\\\ & & \\displaystyle i & \\small = & \\textrm{index referencing nodes of the network as demand} \\\\\n", |
1688 | 1688 | "& & j & \\small = & \\textrm{index referencing nodes of the network as potential facility sites} \\\\\n",
|
1689 | 1689 | "& & d_{ij} & \\small = & \\textrm{shortest distance or travel time between nodes } i \\textrm{ and } j \\\\\n",
|
1690 | 1690 | "& & p & \\small = & \\textrm{number of facilities to be located} \\\\\n",
|
|
1700 | 1700 | " \\end{cases} \\\\ \n",
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1701 | 1701 | "\\end{array}$\n",
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1702 | 1702 | "\n",
|
1703 |
| - "_The formulation above is adapted from Church and Murray (2009)_" |
| 1703 | + ">_The formulation above is adapted from Church and Murray (2009)_" |
1704 | 1704 | ]
|
1705 | 1705 | },
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1706 | 1706 | {
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