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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.
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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 - 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 Refinery Where 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) 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?
Design Dept Mold cutting workshop Injection Molding Machine Spray painting shop FACTORY BUILDINGS Raw materials warehouse Mold warehouse Product warehouse Plastics molding shop Product assembly shop Facilities Planning Problems: (c) Building Layout Problem - Determine the best size and shape of each department in a building 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
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 some Population centers.
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
d d b b a a c c Introduction and Terminology: Graphs Graph: G(V, E), V = a set of nodes and E = 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)} )
d d b b a a c c Graph terminology Path: a sequence of nodes, <n0, n1, …, nk+1> such that (i) each niV (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
d 7 6 5 b a 1 4 c 2 d c 3 3 1 b 2 e 1 3 2 f a Graph terminology.. Weighted graph: each edge a real weight Cycle: A closed path <n0, n1, …, nk, n0> 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 Degree of node: no. of edges incident on it Digraph: no. of incoming edges = indegree no. of incoming edges = outdegree
h h h f f f h h h f f f d d d c c c g g g g g g b b b e e e e e e e e e c c c c c c d d d d d d f f f b b b a a a b b b a a a a a a 2 2 2 d d d c c c 3 3 3 3 3 3 b b b e e e 2 2 2 1 1 1 f f f a a a 2 2 2 Graph terminology… Connected graph: a path between every pair of nodes unconnected connected Strongly connected digraph: each node reachable from every other node not strongly connected Strongly connected
2 2 2 d d d c c c d d d c c c 3 3 3 3 3 3 b b b b b b e e e e e e 2 2 2 1 1 1 f f f f f f a a a a a a 2 2 2 Graph terminology…. Acyclic graph: graph with no cycles A tree is an undirected, acyclic, connected graph
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 some Population centers.
Minimum spanning Trees: Example Redraw only the graph, with weights length of rail link.
Properties of optimum solution Property 1. The optimum set of connections is a sub-graph M( V’, E’) of G, such that V’ = V, and E’ E. Why? 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
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.
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 weightedge crossing the bag Light-edge
x x w w e e out out y y p Light - - edge heavy - edge e e c c in in b b a a 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?] => <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 lines connect buildings with networks connect town-areas with water/sewage channels … For real solutions: extra (redundant) links may be useful next topic: Transportation Planning: Shortest Paths
