Facilities Planning

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

?

R

R

?

R

R

- 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

Ass

embl

y L

ine

Last Making

Legend:

Building

Road

Legend:

Building

Road

ToolWarehouse

Raw MaterialWarehouse

Finished goodsWarehouse

Machine Shop

Leather Stitching

Sole Making

Ass

embl

y L

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

IGA

CK

CP

KB

GK

RN

M

RG

SB

IGA

CK

CP

KB

GK

RN

M

RG

SB

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.

IGA

CK

CP

KB

GK

RN

M

RG

SB

IGA

CK

CP

KB

GK

RN

M

RG

SB

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)} )

a b

d

c

a b

d

c

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

d

c

a b

d

c

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

e

c d

f

a

b

2

3

1

31

2

3

2

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

1 4

6

7

5a

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

2

3

1

3

2

2

e

c d

f

a

b

2

3

1

3

2

2

e

c d

f

a

b

2

3

1

3

2

2

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

unconnectedconnected

Stronglyconnected

notstronglyconnected

Graph terminology….

e

c d

fa

b

2

3

1

3

2

2

e

c d

fa

b

2

3

1

3

2

2

e

c d

fa

b

2

3

1

3

2

2

e

c d

fa

be

c d

fa

be

c d

f

a

bA tree is an undirected, acyclic, connected graph

Acyclic graph: graph with no cycles

IGA

CK

CP

KB

GK

RN

M

RG

SB

IGA

CK

CP

KB

GK

RN

M

RG

SB

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

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

108

14

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

108

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

108

14

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.

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

108

14

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

108

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

108

14

Minimum spanning Trees: Example

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

10 8

14

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

10 8

14

Minimum spanning Trees: Example..

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

10 8

14

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

10 8

14

Minimum spanning Trees: Example…

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

108

14

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

108

14

Minimum spanning Trees: Example….

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

10 8

14

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

10 8

14

Minimum spanning Trees: Example…..

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

108

14

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

108

14

Minimum spanning Trees: Example……

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

10 8

14

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

10 8

14

Minimum spanning Trees: Example…….

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

108

14

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

108

14

Minimum spanning Trees: Example……..

Minimum spanning Trees: Example……...

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

108

14

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

108

14

Minimum spanning Trees: Example……….

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

108

14

MST length = 22+4+6+8+7+14+14+12=87 Km

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

108

14

MST length = 22+4+6+8+7+14+14+12=87 Km

Minimum spanning Trees: not unique

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

108

14

MST length = 22+4+6+8+7+14+14+12=87 Km

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

108

14

MST length = 22+4+6+8+7+14+14+12=87 Km

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

10 8

14

MST length = 22+4+6+8+7+14+14+12=87 Km

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

10 8

14

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

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

10 8

14

IGA

CK

CP

KB

GK

RN

M

RG

SB

20

16

619

4

10

7

12

14

15

12

14

17

22

24

10 8

14 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

eout

w

y

x

eout

w

y

x

ein cb

a

ein cb

a

p

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