operations research modeling toolset

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Transportation-1 Operations Research Modeling Toolset Linear Programming Network Programming PERT/ CPM Dynamic Programming Integer Programming Nonlinear Programming Game Theor y Decision Analysis Markov Chains Queueing Theory Invento ry Theory Forecasting Markov Decision Processes Simulation Stochastic Programmin g

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Operations Research Modeling Toolset. Queueing Theory. Markov Chains. PERT/ CPM. Network Programming. Dynamic Programming. Simulation. Markov Decision Processes. Inventory Theory. Linear Programming. Stochastic Programming. Forecasting. Integer Programming. Decision Analysis. - PowerPoint PPT Presentation

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Page 1: Operations Research  Modeling Toolset

Transportation-1

Operations Research Modeling Toolset

Linear Programming

Network Programming

PERT/ CPM

Dynamic Programming

Integer Programming

Nonlinear ProgrammingGame

Theory

Decision Analysis

Markov Chains

Queueing Theory

Inventory Theory

Forecasting

Markov Decision

Processes

Simulation

Stochastic Programming

Page 2: Operations Research  Modeling Toolset

Transportation-2

Network Problems

• Linear programming has a wide variety of applications• Network problems

– Special types of linear programs– Particular structure involving networks

• Ultimately, a network problem can be represented as a linear programming model

• However the resulting A matrix is very sparse, and involves only zeroes and ones

• This structure of the A matrix led to the development of specialized algorithms to solve network problems

Page 3: Operations Research  Modeling Toolset

Transportation-3

Types of Network Problems

• Shortest Path Special case: Project Management with PERT/CPM

• Minimum Spanning Tree• Maximum Flow/Minimum Cut• Minimum Cost Flow

Special case: Transportation and Assignment Problems• Set Covering/Partitioning• Traveling Salesperson• Facility Locationand many more

Page 4: Operations Research  Modeling Toolset

Transportation-4

The Transportation Problem

Page 5: Operations Research  Modeling Toolset

Transportation-5

The Transportation Problem

• The problem of finding the minimum-cost distribution of a given commodity from a group of supply centers (sources) i=1,…,mto a group of receiving centers (destinations) j=1,…,n

• Each source has a certain supply (si)• Each destination has a certain demand (dj)• The cost of shipping from a source to a destination is

directly proportional to the number of units shipped

Page 6: Operations Research  Modeling Toolset

Transportation-6

Simple Network Representation

1

2

m

1

2

n

Sources Destinations

… …

Supply s1

Supply s2

Supply sm

Demand d1

Demand d2

Demand dn

xij

Costs cij

Page 7: Operations Research  Modeling Toolset

Transportation-7

Example: P&T Co.

• Produces canned peas at three canneriesBellingham, WA, Eugene, OR, and Albert Lea, MN

• Ships by truck to four warehousesSacramento, CA, Salt Lake City, UT, Rapid City, SD, and

Albuquerque, NM• Estimates of shipping costs, production capacities and

demands for the upcoming season is given• The management needs to make a plan on the least

costly shipments to meet demand

Page 8: Operations Research  Modeling Toolset

Transportation-8

Example: P&T Co. Map

1

23

1 2

3

4

Page 9: Operations Research  Modeling Toolset

Transportation-9

Example: P&T Co. Data

Warehouse

Cannery 1 2 3 4 Supply(Truckloads)

1 $ 464 $ 513 $ 654 $ 867 75

2 $ 352 $ 216 $ 690 $ 791 125

3 $ 995 $ 682 $ 388 $ 685 100

Demand (Truckloads) 80 65 70 85

Shipping cost per truckload

Page 10: Operations Research  Modeling Toolset

Transportation-10

Example: P&T Co.

• Network representation

Page 11: Operations Research  Modeling Toolset

Transportation-11

Example: P&T Co.

• Linear programming formulationLet xij denote…

Minimize

subject to

Page 12: Operations Research  Modeling Toolset

Transportation-12

General LP Formulation for Transportation Problems

Page 13: Operations Research  Modeling Toolset

Transportation-13

Feasible Solutions

• A transportation problem will have feasible solutions if and only if

• How to deal with cases when the equation doesn’t hold?

n

jj

m

ii ds

11

Page 14: Operations Research  Modeling Toolset

Transportation-14

Integer Solutions Property: Unimodularity

• Unimodularity relates to the properties of the A matrix(determinants of the submatrices, beyond scope)

• Transportation problems are unimodular, so we get the integers solutions property:

For transportation problems, when every si and dj have an integer value, every BFS is integer

valued.

• Most network problems also have this property.

Page 15: Operations Research  Modeling Toolset

Transportation-15

Transportation Simplex Method

• Since any transportation problem can be formulated as an LP, we can use the simplex method to find an optimal solution

• Because of the special structure of a transportation LP, the iterations of the simplex method have a very special form

• The transportation simplex method is nothing but the original simplex method, but it streamlines the iterations given this special form

Page 16: Operations Research  Modeling Toolset

Transportation-16

Transportation Simplex Method

Initialization(Find initial CPF solution)

Is the current

CPF solution optimal?

Move to a better adjacent CPF solution

Stop

No

Yes

Page 17: Operations Research  Modeling Toolset

Transportation-17

The Transportation Simplex TableauDestination

Supply uiSource 1 2 … n

1c11 c12 …

c1n s1

2c21 c22 …

c2n s2

… … … … … …

mcm1 cm2 …

cmn sm

Demand d1 d2 … dnZ =

vj

Page 18: Operations Research  Modeling Toolset

Transportation-18

Prototype Problem

• Holiday shipments of iPods to distribution centers• Production at 3 facilities,

– A, supply 200k– B, supply 350k– C, supply 150k

• Distribute to 4 centers,– N, demand 100k– S, demand 140k– E, demand 300k– W, demand 250k

• Total demand vs. total supply

Page 19: Operations Research  Modeling Toolset

Transportation-19

Prototype ProblemDestination

Supply uiSource N S E W

A16 13 22 17

200

B14 13 19 15

350

C9 20 23 10

150

Dummy0 0 0 0

90

Demand 100 140 300 250Z =

vj

Page 20: Operations Research  Modeling Toolset

Transportation-20

Finding an Initial BFS

• The transportation simplex starts with an initial basic feasible solution (as does regular simplex)

• There are alternative ways to find an initial BFS, most common are– The Northwest corner rule– Vogel’s method– Russell’s method (beyond scope)

Page 21: Operations Research  Modeling Toolset

Transportation-21

The Northwest Corner Rule

• Begin by selecting x11, let x11 = min{ s1, d1 }• Thereafter, if xij was the last basic variable selected,

– Select xi(j+1) if source i has any supply left– Otherwise, select x(i+1)j

Page 22: Operations Research  Modeling Toolset

Transportation-22

The Northwest Corner RuleDestination

SupplySource N S E W

A16 13 22 17

200100 100

B14 13 19 15

35040 300 10

C9 20 23 10

150150

Dummy0 0 0 0

9090

Demand 100 140 300 250

Z = 10770

Page 23: Operations Research  Modeling Toolset

Transportation-23

Vogel’s Method

• For each row and column, calculate its difference:= (Second smallest cij in row/col) - (Smallest cij in row/col)

• For the row/col with the largest difference, select entry with minimum cij as basic

• Eliminate any row/col with no supply/demand left from further steps

• Repeat until BFS found

Page 24: Operations Research  Modeling Toolset

Transportation-24

Vogel’s Method (1): calculate differencesDestination

Supply diffSource N S E W

A16 13 22 17

200 3

B14 13 19 15

350 1

C9 20 23 10

150 1

Dummy0 0 0 0

90 0

Demand 100 140 300 250

diff 9 13 19 10

Page 25: Operations Research  Modeling Toolset

Transportation-25

Vogel’s Method (2): select xDummyE as basic variable

DestinationSupply diff

Source N S E W

A16 13 22 17

200 3

B14 13 19 15

350 1

C9 20 23 10

150 1

Dummy0 0 0 0

90 0

Demand 100 140 300 250

diff 9 13 19 10

90

Page 26: Operations Research  Modeling Toolset

Transportation-26

Vogel’s Method (3): update supply, demand and differences

DestinationSupply diff

Source N S E W

A16 13 22 17

200 3

B14 13 19 15

350 1

C9 20 23 10

150 1

Dummy0 0 0 0

--- ---

Demand 100 140 210 250

diff 5 0 3 5

90

Page 27: Operations Research  Modeling Toolset

Transportation-27

Vogel’s Method (4): select xCN as basic variable

DestinationSupply diff

Source N S E W

A16 13 22 17

200 3

B14 13 19 15

350 1

C9 20 23 10

150 1

Dummy0 0 0 0

--- ---

Demand 100 140 210 250

diff 5 0 3 5

90

100

Page 28: Operations Research  Modeling Toolset

Transportation-28

Vogel’s Method (5): update supply, demand and differences

DestinationSupply diff

Source N S E W

A16 13 22 17

200 4

B14 13 19 15

350 2

C9 20 23 10

50 10

Dummy0 0 0 0

--- ---

Demand --- 140 210 250

diff --- 0 3 5

90

100

Page 29: Operations Research  Modeling Toolset

Transportation-29

Vogel’s Method (6): select xCW as basic variable

DestinationSupply diff

Source N S E W

A16 13 22 17

200 4

B14 13 19 15

350 2

C9 20 23 10

50 10

Dummy0 0 0 0

--- ---

Demand --- 140 210 250

diff --- 0 3 5

90

100 50

Page 30: Operations Research  Modeling Toolset

Transportation-30

Vogel’s Method (7): update supply, demand and differences

DestinationSupply diff

Source N S E W

A16 13 22 17

200 4

B14 13 19 15

350 2

C9 20 23 10

--- ---

Dummy0 0 0 0

--- ---

Demand --- 140 210 200

diff --- 0 3 2

90

100 50

Page 31: Operations Research  Modeling Toolset

Transportation-31

Vogel’s Method (8): select xAS as basic variable

DestinationSupply diff

Source N S E W

A16 13 22 17

200 4

B14 13 19 15

350 2

C9 20 23 10

--- ---

Dummy0 0 0 0

--- ---

Demand --- 140 210 200

diff --- 0 3 2

90

100 50

140

Page 32: Operations Research  Modeling Toolset

Transportation-32

Vogel’s Method (9): update supply, demand and differences

DestinationSupply diff

Source N S E W

A16 13 22 17

60 5

B14 13 19 15

350 4

C9 20 23 10

--- ---

Dummy0 0 0 0

--- ---

Demand --- --- 210 200

diff --- --- 3 2

90

100 50

140

Page 33: Operations Research  Modeling Toolset

Transportation-33

Vogel’s Method (10): select xAW as basic variable

DestinationSupply diff

Source N S E W

A16 13 22 17

60 5

B14 13 19 15

350 4

C9 20 23 10

--- ---

Dummy0 0 0 0

--- ---

Demand --- --- 210 200

diff --- --- 3 2

90

100 50

140 60

Page 34: Operations Research  Modeling Toolset

Transportation-34

Vogel’s Method (11): update supply, demand and differences

DestinationSupply diff

Source N S E W

A16 13 22 17

--- ---

B14 13 19 15

350 4

C9 20 23 10

--- ---

Dummy0 0 0 0

--- ---

Demand --- --- 210 140

diff --- ---

90

100 50

140 60

Page 35: Operations Research  Modeling Toolset

Transportation-35

Vogel’s Method (12): select xBW and xBE as basic variables

DestinationSupply diff

Source N S E W

A16 13 22 17

--- ---

B14 13 19 15

---

C9 20 23 10

--- ---

Dummy0 0 0 0

--- ---

Demand --- --- --- ---

diff --- ---

90

100 50

140 60

140210

Z = 10330

Page 36: Operations Research  Modeling Toolset

Transportation-36

Optimality Test

• In the regular simplex method, we needed to check the row-0 coefficients of each nonbasic variable to check optimality and we have an optimal solution if all are 0

• There is an efficient way to find these row-0 coefficients for a given BFS to a transportation problem:– Given the basic variables, calculate values of dual variables

• ui associated with each source • vj associated with each destination

using cij – ui – vj = 0 for xij basic, or ui + vj = cij

(let ui = 0 for row i with the largest number of basic variables)– Row-0 coefficients can be found from c’

ij=cij-ui-vj for xij nonbasic

Page 37: Operations Research  Modeling Toolset

Transportation-37

Optimality Test (1)Destination

Supply uiSource N S E W

A16 13 22 17

200

B14 13 19 15

350

C9 20 23 10

150

Dummy0 0 0 0

90

Demand 100 140 300 250

vj

90

140

100

60

140210

50

Page 38: Operations Research  Modeling Toolset

Transportation-38

Optimality Test (2)• Calculate ui, vj using cij – ui – vj = 0 for xij basic

(let ui = 0 for row i with the largest number of basic variables)Destination

Supply uiSource N S E W

A16 13 22 17

200 0

B14 13 19 15

350 -2

C9 20 23 10

150 -7

Dummy0 0 0 0

90 -21

Demand 100 140 300 250vj 16 13 21 17

90

140

100

60

140210

50

Page 39: Operations Research  Modeling Toolset

Transportation-39

Optimality Test (3)• Calculate c’

ij=cij-ui-vj for xij nonbasic

DestinationSupply uiSource N S E W

A16 13 22 17

200 00 1

B14 13 19 15

350 -20 2

C9 20 23 10

150 -714 9

Dummy0 0 0 0

90 -215 8 4

Demand 100 140 300 250

vj 16 13 21 17

90

140

100

60

140210

50

Page 40: Operations Research  Modeling Toolset

Transportation-40

Optimal Solution

A

B

C

N

S

W

Sources Destinations

Supply = 200

Supply = 350

Supply = 150

Demand = 100

Demand = 140

Demand = 250

E Demand = 300 (shortage of 90)

60

140

210

140

50

100

Cost Z = 10330

Page 41: Operations Research  Modeling Toolset

Transportation-41

An Iteration

• Find the entering basic variable– Select the variable with the largest negative c’

ij

• Find the leaving basic variable– Determine the chain reaction that would result from increasing

the value of the entering variable from zero– The leaving variable will be the first variable to reach zero

because of this chain reaction

Page 42: Operations Research  Modeling Toolset

Transportation-42

DestinationSupply uiSource N S E W

A16 13 22 17

200 03 2

B14 13 19 15

350 0- 2

C9 20 23 10

150 -5- 2 12 9

Dummy0 0 0 0

90 -15- 1 2 - 4

Demand 100 140 300 250

vj 16 13 19 15

Initial Solution Obtained by the Northwest Corner Rule

100

150

1030040

100

90

Page 43: Operations Research  Modeling Toolset

Transportation-43

DestinationSupply uiSource N S E W

A16 13 22 17

200 0

B14 13 19 15

350 0

C9 20 23 10

150 -5

Dummy0 0 0 0

90 -15

Demand 100 140 300 250

vj 16 13 19 15

Iteration 1

100

150

1030040

100

90?+

- +

-

Page 44: Operations Research  Modeling Toolset

Transportation-44

End of Iteration 1

DestinationSupply uiSource N S E W

A16 13 22 17

200

B14 13 19 15

350

C9 20 23 10

150

Dummy0 0 0 0

90

Demand 100 140 300 250

vj

100

150

10021040

100

90

Page 45: Operations Research  Modeling Toolset

Transportation-45

Optimality Test

DestinationSupply uiSource N S E W

A16 13 22 17

200 03 2

B14 13 19 15

350 0- 2

C9 20 23 10

150 -5- 2 12 9

Dummy0 0 0 0

90 -193 6 4

Demand 100 140 300 250

vj 16 13 19 15

100

150

10021040

100

90

Page 46: Operations Research  Modeling Toolset

Transportation-46

Iteration 2

DestinationSupply uiSource N S E W

A16 13 22 17

200 0

B14 13 19 15

350 0

C9 20 23 10

150 -5

Dummy0 0 0 0

90 -19

Demand 100 140 300 250

vj 16 13 19 15

100

150

10021040

100

90

?+

- +

-

Page 47: Operations Research  Modeling Toolset

Transportation-47

End of Iteration 2

DestinationSupply uiSource N S E W

A16 13 22 17

200

B14 13 19 15

350

C9 20 23 10

150

Dummy0 0 0 0

90

Demand 100 140 300 250

vj

60

150

100210

140

90

40

Page 48: Operations Research  Modeling Toolset

Transportation-48

Optimality Test

DestinationSupply uiSource N S E W

A16 13 22 17

200 21 0

B14 13 19 15

350 02

C9 20 23 10

150 -50 14 9

Dummy0 0 0 0

90 -195 8 4

Demand 100 140 300 250

vj 14 11 19 15

60

150

100210

140

90

40

Z = 10330

Page 49: Operations Research  Modeling Toolset

Transportation-49

Optimal Solution

A

B

C

N

S

W

Sources Destinations

Supply = 200

Supply = 350

Supply = 150

E

60

140

40

100

150

210

Demand = 100

Demand = 140

Demand = 250

Demand = 300 (shortage of 90)

Cost Z = 10330

Page 50: Operations Research  Modeling Toolset

Transportation-50

The Assignment Problem

• The problem of finding the minimum-costly assignment of a set of tasks (i=1,…,m) to a set of agents (j=1,…,n)

• Each task should be performed by one agent• Each agent should perform one task• A cost cij associated with each assignment

• We should have m=n (if not…?)

• A special type of linear programming problem, and• A special type of transportation problem,

with si=dj= ?

Page 51: Operations Research  Modeling Toolset

Transportation-51

Prototype Problem

• Assign students to mentors• Each assignment has a ‘mismatch’ index• Minimize mismatches

MentorSupply

Student Snape McGonagall Lupin

Harry5 2 3

1

Draco1 4 5

1

Goyle2 4 4

1

Demand 1 1 1

Page 52: Operations Research  Modeling Toolset

Transportation-52

Prototype Problem

• Linear programming formulationLet xij denote…

Minimize

subject to

Page 53: Operations Research  Modeling Toolset

Transportation-53

General LP Formulation for Assignment Problems

Page 54: Operations Research  Modeling Toolset

Transportation-54

Solving the Assignment Problem

• It is a linear programming problem, so we could use regular simplex method

• It is a transportation problem, so we could use transportation simplex method

• However, it has a very special structure, such that it can be solved in polynomial time

• Many such algorithms exist, but the best known (and one of the oldest) is the Hungarian Method

Page 55: Operations Research  Modeling Toolset

Transportation-55

The Hungarian Method

1. Subtract row minimums from each element in the row2. Subtract column minimums from each element in the column3. Cover the zeroes with as few lines as possible4. If the number of lines = n, then optimal solution is hidden in zeroes5. Otherwise, find the minimum cost that is not covered by any lines

1. Subtract it from all uncovered elements2. Add it to all elements at intersections (covered by two lines)

6. Back to step 3

Page 56: Operations Research  Modeling Toolset

Transportation-56

The Hungarian Method – Optimal Solution

How to identify the optimal solution:• Make the assignments one at a time in positions that

have zero elements. • Begin with rows or columns that have only one zero.

Cross out both the row and the column involved after each assignment is made.

• Move on to the rows and columns that are not yet crossed out to select the next assignment, with preference given to any such row or column that has only one zero that is not crossed out.

• Continue until every row and every column has exactly one assignment and so has been crossed out.

Page 57: Operations Research  Modeling Toolset

Transportation-57

Hungarian MethodMentor

Student Snape McG Lupin

Harry 5 2 3Draco 1 4 5Goyle 2 4 4

Mentor

Student Snape McG Lupin

Harry

Draco

Goyle

Mentor

Student Snape McG Lupin

Harry

Draco

Goyle

Mentor

Student Snape McG Lupin

Harry

Draco

Goyle