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?

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

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

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)} ) a b d c a b d c

Graph terminology Path:a sequence of nodes, such that (i) each n i  V (ii) (n i, n i+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 =, |P| = 3 a b d c a b d c

Graph terminology.. Directed graph, Digraph: each edge has a direction (tail, head) A directed edge isincident from the tail, incident to the head. Tail = = parent, Head = = child e c d f a b Degree of node: no. of edges incident on it Digraph:no. of incoming edges = indegree no. of incoming edges = outdegree Cycle: A closed path Weighted graph: each edge  a real weight a d b c

Graph terminology… Connected graph: a path between every pair of nodes e c d f a b e c d f a b e c d f a b e h f b d a g c e h f b d a g c e h f b d a g c e c d f a b e c d f a b e c d f a b e h f b d a g c e h f b d a g c e h f b d a g c Strongly connected digraph: each node reachable from every other node unconnected connected Strongly connected not strongly connected

Graph terminology…. e c d f a b e c d f a b e c d f a b e c d f a b e c d f a b e c d f a b A tree is an undirected, acyclic, connected graph Acyclic graph: graph with no cycles

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 2. The optimum solution is a tree. Proof (by contradiction): Assume existence of cycle. => ?? => 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.

Minimum spanning Trees: Example

Minimum spanning Trees: Example..

Minimum spanning Trees: Example…

Minimum spanning Trees: Example….

Minimum spanning Trees: Example…..

Minimum spanning Trees: Example……

Minimum spanning Trees: Example…….

Minimum spanning Trees: Example……..

Minimum spanning Trees: Example……...

Minimum spanning Trees: Example……….

Minimum spanning Trees: not unique

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 Light-edge

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?] heavy-edge e out w y x e w y x e in c b a e c b a 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