|
17 | 17 | "\n", |
18 | 18 | "**P-Median can be written as:**\n", |
19 | 19 | "\n", |
20 | | - "$\\begin{array} \\displaystyle \\textbf{Minimize} & \\displaystyle \\sum_{i \\in I}\\sum_{j \\in J}{a_i d_{ij} X_{ij}} &&& (1) \\\\\n", |
21 | | - "\\displaystyle \\textbf{Subject to:} & \\displaystyle \\sum_{j \\in J}{X_{ij} = 1} & \\forall i \\in I && (2) \\\\\n", |
22 | | - " & \\displaystyle \\sum_{j \\in J}{Y_{j} = p} &&& (3) \\\\\n", |
23 | | - " & X_{ij} \\leq Y_{j} & \\forall i \\in I & \\forall j \\in J & (4) \\\\\n", |
24 | | - " & X_{ij} \\in \\{0,1\\} & \\forall i \\in I & \\forall j \\in J & (5) \\\\\n", |
25 | | - " & Y_{j} \\in \\{0,1\\} & \\forall j \\in J && (6) \\\\ \\end{array}$\n", |
| 20 | + "$$\n", |
| 21 | + "\\begin{array}{lllll}\n", |
| 22 | + "\\textbf{Minimize} & \\displaystyle \\sum_{i \\in I} \\sum_{j \\in J} a_i d_{ij} X_{ij} &&& (1) \\\\\n", |
| 23 | + "\\textbf{Subject to:} & \\sum_{j \\in J} X_{ij} = 1 & \\forall i \\in I && (2) \\\\\n", |
| 24 | + "& \\sum_{j \\in J} Y_j = p &&& (3) \\\\\n", |
| 25 | + "& X_{ij} \\leq Y_j & \\forall i \\in I, j \\in J && (4) \\\\\n", |
| 26 | + "& X_{ij} \\in \\{0, 1\\} & \\forall i \\in I, j \\in J && (5) \\\\\n", |
| 27 | + "& Y_j \\in \\{0, 1\\} & \\forall j \\in J && (6) \\\\\n", |
| 28 | + "\\end{array}\n", |
| 29 | + "$$\n", |
| 30 | + "\n", |
| 31 | + "$$\n", |
| 32 | + "\\begin{array}{lllll}\n", |
| 33 | + "\\textbf{Where:} \\\\\n", |
| 34 | + "& i & = & \\text{index referencing demand nodes} \\\\\n", |
| 35 | + "& j & = & \\text{index referencing potential facility sites} \\\\\n", |
| 36 | + "& d_{ij} & = & \\text{distance or travel time between node } i \\text{ and } j \\\\\n", |
| 37 | + "& p & = & \\text{number of facilities to locate} \\\\\n", |
| 38 | + "& a_i & = & \\text{demand at node } i \\\\\n", |
| 39 | + "& X_{ij} & = &\n", |
| 40 | + " \\begin{cases}\n", |
| 41 | + " 1, & \\text{if demand } i \\text{ is assigned to facility } j \\\\\n", |
| 42 | + " 0, & \\text{otherwise}\n", |
| 43 | + " \\end{cases} \\\\\n", |
| 44 | + "& Y_j & = &\n", |
| 45 | + " \\begin{cases}\n", |
| 46 | + " 1, & \\text{if facility is located at node } j \\\\\n", |
| 47 | + " 0, & \\text{otherwise}\n", |
| 48 | + " \\end{cases}\n", |
| 49 | + "\\end{array}\n", |
| 50 | + "$$\n", |
26 | 51 | "\n", |
27 | | - "$\\begin{array} \\displaystyle \\textbf{Where:}\\\\ & & \\displaystyle i & \\small = & \\textrm{index referencing nodes of the network as demand} \\\\\n", |
28 | | - "& & j & \\small = & \\textrm{index referencing nodes of the network as potential facility sites} \\\\\n", |
29 | | - "& & d_{ij} & \\small = & \\textrm{shortest distance or travel time between nodes } i \\textrm{ and } j \\\\\n", |
30 | | - "& & p & \\small = & \\textrm{number of facilities to be located} \\\\\n", |
31 | | - "& & a_i & \\small = & \\textrm{service load or population demand at } i \\\\\n", |
32 | | - "& & X_{ij} & \\small = & \\begin{cases}\n", |
33 | | - " 1, \\textrm{if demand } i \\textrm{ is assigned to facility } j \\\\\n", |
34 | | - " 0, \\textrm{otherwise}\n", |
35 | | - " \\end{cases} \\\\\n", |
36 | | - "& & Y_{j} & \\small = & \\begin{cases}\n", |
37 | | - " 1, \\textrm{if node } j \\textrm{ has been selected for a facility} \\\\\n", |
38 | | - " 0, \\textrm{otherwise} \\\\\n", |
39 | | - " \\end{cases} \\\\ \n", |
40 | | - "\\end{array}$\n", |
41 | 52 | "\n", |
42 | 53 | "_The formulation above is adapted from Church and Murray (2018)_\n", |
43 | 54 | "\n", |
|
1696 | 1707 | "\n", |
1697 | 1708 | "**Capacitated P-Median can be written as:**\n", |
1698 | 1709 | "\n", |
1699 | | - "$\\begin{array} \\displaystyle \\textbf{Minimize} & \\displaystyle \\sum_{i \\in I}\\sum_{j \\in J}{a_i d_{ij} X_{ij}} &&& (1) \\\\\n", |
1700 | | - "\\displaystyle \\textbf{Subject to:} & \\displaystyle \\sum_{j \\in J}{X_{ij} = 1} & \\forall i \\in I && (2) \\\\\n", |
1701 | | - " & \\displaystyle \\sum_{j \\in J}{Y_{j} = p} &&& (3) \\\\\n", |
1702 | | - " & \\displaystyle \\sum_{i \\in I}{a_i X_{ij} \\leq {c_j Y_{j}}}& \\forall j \\in J && (4) \\\\\n", |
1703 | | - " & X_{ij} \\leq Y_{j} & \\forall i \\in I & \\forall j \\in J & (5) \\\\\n", |
1704 | | - " & X_{ij} \\in \\{0,1\\} & \\forall i \\in I & \\forall j \\in J & (6) \\\\\n", |
1705 | | - " & Y_{j} \\in \\{0,1\\} & \\forall j \\in J && (7) \\\\ \\end{array}$\n", |
| 1710 | + "$$\n", |
| 1711 | + "\\begin{array}{lllll}\n", |
| 1712 | + "\\textbf{Minimize} & \\displaystyle \\sum_{i \\in I} \\sum_{j \\in J} a_i d_{ij} X_{ij} &&& (1) \\\\\n", |
| 1713 | + "\\textbf{Subject to:} & \\sum_{j \\in J} X_{ij} = 1 & \\forall i \\in I && (2) \\\\\n", |
| 1714 | + "& \\sum_{j \\in J} Y_j = p &&& (3) \\\\\n", |
| 1715 | + "& \\sum_{i \\in I} a_i X_{ij} \\leq c_j Y_j & \\forall j \\in J && (4) \\\\\n", |
| 1716 | + "& X_{ij} \\leq Y_j & \\forall i \\in I, j \\in J && (5) \\\\\n", |
| 1717 | + "& X_{ij} \\in \\{0, 1\\} & \\forall i \\in I, j \\in J && (6) \\\\\n", |
| 1718 | + "& Y_j \\in \\{0, 1\\} & \\forall j \\in J && (7) \\\\\n", |
| 1719 | + "\\end{array}\n", |
| 1720 | + "$$\n", |
1706 | 1721 | "\n", |
1707 | | - "$\\begin{array} \\displaystyle \\textbf{Where:}\\\\ & & \\displaystyle i & \\small = & \\textrm{index referencing nodes of the network as demand} \\\\\n", |
1708 | | - "& & j & \\small = & \\textrm{index referencing nodes of the network as potential facility sites} \\\\\n", |
1709 | | - "& & d_{ij} & \\small = & \\textrm{shortest distance or travel time between nodes } i \\textrm{ and } j \\\\\n", |
1710 | | - "& & p & \\small = & \\textrm{number of facilities to be located} \\\\\n", |
1711 | | - "& & a_i & \\small = & \\textrm{service load or population demand at } i \\\\\n", |
1712 | | - "& & c_j & \\small = & \\textrm{capacity of facility} j \\\\\n", |
1713 | | - "& & X_{ij} & \\small = & \\begin{cases}\n", |
1714 | | - " 1, \\textrm{if demand } i \\textrm{ is assigned to facility } j \\\\\n", |
1715 | | - " 0, \\textrm{otherwise}\n", |
1716 | | - " \\end{cases} \\\\\n", |
1717 | | - "& & Y_{j} & \\small = & \\begin{cases}\n", |
1718 | | - " 1, \\textrm{if node } j \\textrm{ has been selected for a facility} \\\\\n", |
1719 | | - " 0, \\textrm{otherwise} \\\\\n", |
1720 | | - " \\end{cases} \\\\ \n", |
1721 | | - "\\end{array}$\n", |
| 1722 | + "$$\n", |
| 1723 | + "\\begin{array}{lllll}\n", |
| 1724 | + "\\textbf{Where:} \\\\\n", |
| 1725 | + "& i & = & \\text{index referencing demand nodes} \\\\\n", |
| 1726 | + "& j & = & \\text{index referencing potential facility sites} \\\\\n", |
| 1727 | + "& d_{ij} & = & \\text{distance or travel time between nodes } i \\text{ and } j \\\\\n", |
| 1728 | + "& p & = & \\text{number of facilities to locate} \\\\\n", |
| 1729 | + "& a_i & = & \\text{demand at node } i \\\\\n", |
| 1730 | + "& c_j & = & \\text{capacity of facility at node } j \\\\\n", |
| 1731 | + "& X_{ij} & = &\n", |
| 1732 | + " \\begin{cases}\n", |
| 1733 | + " 1, & \\text{if demand } i \\text{ is assigned to facility } j \\\\\n", |
| 1734 | + " 0, & \\text{otherwise}\n", |
| 1735 | + " \\end{cases} \\\\\n", |
| 1736 | + "& Y_j & = &\n", |
| 1737 | + " \\begin{cases}\n", |
| 1738 | + " 1, & \\text{if facility is located at node } j \\\\\n", |
| 1739 | + " 0, & \\text{otherwise}\n", |
| 1740 | + " \\end{cases}\n", |
| 1741 | + "\\end{array}\n", |
| 1742 | + "$$\n", |
1722 | 1743 | "\n", |
1723 | 1744 | "_The formulation above is adapted from Church and Murray (2009)_" |
1724 | 1745 | ] |
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