The document discusses network optimization problems and the shortest path problem. It provides terminology for network components like nodes and arcs. It also classifies network problems and gives examples. The shortest path problem aims to find the path between two nodes with minimum total distance. The document illustrates applying Dijkstra's algorithm to solve a shortest path problem on a sample network from node 0 to node T, showing the iterations and calculations to determine the shortest distance. Students are assigned to use the algorithm to find the shortest distance from node 0 to node 6 on a given network.

1. Minggu 3:
Masalah Jaringan
Part 1
Matakuliah
PENELITIAN OPERASIONAL 2
1

2. Kemampuan Akhir yang direncanakan (Sub-CPMK)
• Mahasiswa mampu memahami dan menemukan solusi Masalah
Optimasi Jaringan
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3. Bahan Kajian/Pokok Bahasan:
• Pengantar Permasalahan Optimasi Jaringan
• Struktur Permasalahan Optimasi Jaringan
• Jenis Permasalahan Optimasi Jaringan
• Contoh Permasalahan Jaringan
• Permasalahan jaringan Rute Terpendek
• Model Jaringan Rute Terpendek
• Penyelesaian dengan Algoritma Dijkstra
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4. Struktur
Jaringan
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5. TERMINOLOGY OF NETWORKS
• The points are called nodes
• The lines are called arcs
• If flow through an arc is allowed in
only one direction (e.g., a one-way
street), the arc is said to be a
directed arc.
• If flow through an arc is allowed in
either direction (e.g., a pipeline that
can be used to pump fluid in either
direction), the arc is said to be an
undirected arc or links.
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6. Klasifikasi
Masalah
Jaringan
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7. Klasifikasi Masalah Jaringan
• A network that has only directed arcs is called a directed network.
• if all its arcs are undirected, the network is said to be an undirected
network.
• A path between two nodes is a sequence of distinct arcs connecting
these nodes. For example, one of the paths connecting nodes O and T
is the sequence of arcs OB–BD–DT (O B D T), or vice versa.
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8. • A directed path from node i to node j is a sequence of connecting
arcs whose direction (if any) is toward node j, so that flow from node i
to node j along this path is feasible.
• An undirected path from node i to node j is a sequence of connecting
arcs whose direction (if any) can be either toward or away from node
j.
• A path that begins and ends at the same node is called a cycle.
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Connected Network with Cycle

9. • A spanning tree is a connected network for all n
nodes that contains no undirected cycles.
• The maximum amount of flow (possibly infinity) that
can be carried on a directed arc is referred to as the
arc capacity.
• A supply node (or source node or source) has the
property that the flow out of the node exceeds the
flow into the node.
• A demand node (or sink node or sink) is where the
flow into the node exceeds the flow out of the node.
• A transshipment node (or intermediate node)
satisfies conservation of flow, so flow in equals flow
out.
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Spanning Tree

10. Network Optimization Models
• Shortest-path problem
• Minimum spanning tree problem
• Maximum flow problem
• Travelling Salesman Problem
• Minimum cost flow problem
• Network simplex method
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11. Ilustrasi
masalah
Jaringan
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12. Contoh Masalah Jaringan
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13. Model dan Algoritma
Masalah Rute
Terpendek
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14. The Shortest-Path Problem
• Consider an undirected and connected network with two special
nodes called the origin and the destination.
• Associated with each of the links (undirected arcs) is a nonnegative
distance.
• The objective is to find the shortest path (the path with the minimum
total distance) from the origin to the destination.
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15. Algorithm for the Shortest-Path Problem
• Objective of nth iteration: Find the nth nearest node to the origin (to be repeated for n
1, 2, . . . until the nth nearest node is the destination.
• Input for nth iteration: n − 1 nearest nodes to the origin (solved for at the previous
iterations), including their shortest path and distance from the origin. (These nodes, plus
the origin, will be called solved nodes; the others are unsolved nodes.)
• Candidates for nth nearest node: Each solved node that is directly connected by a link to
one or more unsolved nodes provides one candidate—the unsolved node with the
shortest connecting link. (Ties provide additional candidates.)
• Calculation of nth nearest node: For each such solved node and its candidate, add the
distance between them and the distance of the shortest path from the origin to this
solved node. The candidate with the smallest such total distance is the nth nearest node
(ties provide additional solved nodes), and its shortest path is the one generating this
distance.
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16. Applying Dijkstra Algorithm to the Seervada Park
Shortest-Path Problem
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17. Iterasi 0
17
Distance:
O: 0
A: ∞
B: ∞
C: ∞
D: ∞
E: ∞
T: ∞
Unvisited Nodes: {O, A, B, C, D, E, T}

18. Iterasi 1
18
Distance:
O: 0
A: ∞, 2
B: ∞, 5
C: ∞, 4
D: ∞
E: ∞
T: ∞
Unvisited Nodes: {O, A, B, C, D, E, T}

19. Iterasi 2 & 3
19
Distance:
O: 0
A: ∞, 2
B: ∞, 5 vs 4
C: ∞, 4
D: ∞
E: ∞
T: ∞
Unvisited Nodes: {O, A, B, C, D, E, T}

20. Iterasi 4
20
Distance:
O: 0
A: ∞, 2
B: ∞, 4
C: ∞, 4
D: ∞, 8
E: ∞, 7
T: ∞
Unvisited Nodes: {O, A, B, C, D, E, T}

21. Iterasi 5
21
Distance:
O: 0
A: ∞, 2
B: ∞, 4
C: ∞, 4
D: ∞, 8 vs 8
E: ∞, 7
T: ∞,
Unvisited Nodes: {O, A, B, C, D, E, T}

22. Iterasi 6
22
Distance:
O: 0
A: ∞, 2
B: ∞, 4
C: ∞, 4
D: ∞, 8
E: ∞, 7
T: ∞, 13 vs 14
Unvisited Nodes: {O, A, B, C, D, E, T}

23. Seervada Park Shortest-Path Problem Solution
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24. Solution
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25. Solusi dengan Pemodelan OR
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26. Hasil
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27. Tugas
• Cari jarak terpendek dari 0 ke 6 untuk jaringan berikut:
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28. Reference
• Frederick S. Hillier, Gerald J. Lieberman - Introduction to operations
research-McGraw-Hill (2001)
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