Facilities Planning
Objectives and Agenda:
1. Different types of Facilities Planning Problems
2. Intro to Graphs as a tool for deterministic optimization
3. Finding the Minimum Spanning Tree (MST) in a graph
4. Optimum solution of a Facilities Planning Problem using MST
Facilities Planning Problems: (a) Site Location Problem
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- Where to locate a new/additional facility
Issues: Cost, labor availability, wage levels, govt. subsidies,transportation costs for materials, taxes, legal issues, …
Example: New China Oil co.7 oil wells 1 RefineryWhere to locate the refinery to minimize pipeline costs.
Facilities Planning Problems: (b) Site planning
- How many buildings are required at a site, their locations, sizes, and connections (materials, data)
Legend:
Building
Road
ToolWarehouse
Raw MaterialWarehouse
Finished goodsWarehouse
Machine Shop
Leather Stitching
Sole Making
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Last Making
Legend:
Building
Road
Legend:
Building
Road
ToolWarehouse
Raw MaterialWarehouse
Finished goodsWarehouse
Machine Shop
Leather Stitching
Sole Making
Ass
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ine
Last Making
Example: Athletic Shoe Co.(a) What are the issues used to determine building locations?(b) Optimum layout of underground data cables to connect all buildings?
Facilities Planning Problems: (c) Building Layout Problem
- Determine the best size and shape of each department in a building
Mold cutting workshop Injection Molding Machine Spray painting shop
Plastics molding shop
FACTORY BUILDINGS
Raw materials warehouse
Product assembly shop
Design Dept
Mold warehouse
Product warehouse
Example:Plastic Mold Co.
Facilities Planning Problems: (d) Department Layout Problem
- How to layout the machines, work stations, etc. in a department
Example: Old China Bicycle Co.How will you design the assembly line for assembling 100 bikes/day?
Facilities Planning Problems
Most Facility Planning Problems have many constraints
Mathematical models are very complex
[Why do we need to make mathematical model ?]
We will study one (simple) example of the Site planning Problem
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Example: Site Planning Problem
- Join N population centers of a city by Train System (MTR)- Direct connection lines can be built between some pairs- Cost of Train network total length of lines- Each pair of Stations must have some train route between them
Example:Map of Delhi and somePopulation centers.
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Example: Site Planning Problem
We will use ‘Graphs’ to solve the example
- Graph theory (in Mathematics) is useful to solve many problems- We will use one Graph method: Minimum Spanning Trees (MST)- MST can be used for many different problems
Introduction and Terminology: Graphs
Graph: G(V, E),V = a set of nodes andE = a set of edges.
Each edge links exactly two nodes, (node1, node2)
An edge is incident on each node on its ends.
Example:G(V, E) = ( { a, b, c, d}, { (a, b), (b, c), (b, d), (c, d), (a, d)} )
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Graph terminology
Path: a sequence of nodes, <n0, n1, …, nk+1> such that(i) each ni V(ii) (ni, ni+1) E, for each i = 0, .., k
Moving on a path: traversing the graph
The length of a path = number of edges in the path
Example:P = <a, b, c, d>, |P| = 3 a b
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Graph terminology..
Directed graph, Digraph: each edge has a direction (tail, head)
A directed edge is incident from the tail,incident to the head.
Tail = = parent, Head = = child
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Degree of node: no. of edges incident on itDigraph: no. of incoming edges = indegree
no. of incoming edges = outdegree
Cycle: A closed path <n0, n1, …, nk, n0>
Weighted graph: each edge a real weight
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Graph terminology…
Connected graph: a path between every pair of nodes
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Strongly connected digraph: each node reachable from every other node
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Graph terminology….
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bA tree is an undirected, acyclic, connected graph
Acyclic graph: graph with no cycles
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Example: (repeat)
- Join N population centers of a city by Train System (MTR)- Direct connection lines can be built between some pairs- Cost of Train network total length of lines- Each pair of Stations must have some train route between them
Example:Map of Delhi and somePopulation centers.
Minimum spanning Trees: Example
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Redraw only the graph, with weights length of rail link.
Properties of optimum solution
Property 2. The optimum solution is a tree.
Proof (by contradiction):
Assume existence of cycle <na, nb, …, nk, na>.
=> ??
=> Optimum set of railway links is a minimum spanning tree
Property 1. The optimum set of connections is a sub-graph M( V’, E’) of G, such that V’ = V, and E’ E.
Why?
Minimum spanning Trees: Prim’s method
Step 1. Put the entire graph (all nodes and edges) in a bag.
Step 2. Select any one node, pull it out of the bag;(edges incident on this node will cross the bag)
Step 3. Among all edges crossing the bag, pick the one with MIN weight.
Add this edge to the MST.
Step 4. Select the node inside the bag connected to edge selected in Step 3.
Step 5. Pull node selected in Step 5 out of bag.
Step 6. Repeat steps 3, 4, 5 until the bag is empty.
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Minimum spanning Trees: Example
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Minimum spanning Trees: Example..
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Minimum spanning Trees: Example…
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Minimum spanning Trees: Example….
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Minimum spanning Trees: Example…..
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Minimum spanning Trees: Example……
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Minimum spanning Trees: Example…….
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Minimum spanning Trees: Example……..
Minimum spanning Trees: Example……...
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Minimum spanning Trees: Example……….
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MST length = 22+4+6+8+7+14+14+12=87 Km
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MST length = 22+4+6+8+7+14+14+12=87 Km
Minimum spanning Trees: not unique
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MST length = 22+4+6+8+7+14+14+12=87 Km
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MST length = 22+4+6+8+7+14+14+12=87 Km
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MST length = 22+4+6+8+7+14+14+12=87 Km
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MST length = 22+4+6+8+7+14+14+12=87 Km
Proof of correctness, Prim’s algorithm
Proof by induction: At the i-th step:
we have a partial MST “outside the bag”
we select Least weight edge crossing the bag
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Proof of correctness, Prim’s algorithm..
Assume: Light-edge is not part of MST
=> Some other “bag-crossing-edge” must be part of MST [WHY?]
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Light-edge-
=> <p, light-edge>: cycle
=> cut heavy-edge, join light-edge reduce cost (contradiction!)
Concluding remarks
Minimum spanning Trees provide
good starting solutions
For problems of the type:
connect towns with roads,connect factories with supply linesconnect buildings with networksconnect town-areas with water/sewage channels…
For real solutions: extra (redundant) links may be useful
next topic: Transportation Planning: Shortest Paths
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