Carrier Selection

Mohawk Case

1

Variations on the shortest path problem

Arcs might represent roads, railways, bridges, or links in a communications network.Basic Problem: Find the shortest (fastest) path from node A to node H. Which arc would cause the greatest increase in the length of the shortest path, i.e. which would cause the maximum disruption in transportation/communications?

A

B

C

D

EG

H

F

4

32

1

3

3

4

2

6

9

2

A homeland security problem

Consider destroying one of the arcs in this network.

Protect the arc that causes the greatest increase in the length of the shortest path !

A

B

C

D

EG

H

F

4

32

1

3

3

4

2

6

9

2

The shortest path may change.

Finding the most reliable route

Associated with each arc is the probability that the arc is open/working.Find the most reliable route from A to HThe route with the highest probability that all of its arcs are open

(= product of probabilities of arcs along that route).

For example, the reliability of ACDH=0.8*0.8*0.9=0.648Often not the shortest route!

A

B

C

D

EG

H

F

.50

.80.80

1

.33

.20

.10

.80

.60

.90

.25

The assignment problem

An assignment problem can also be formulated as a network flow problemAn assignment problem example:A company has three workers: Ann, Bob, and Chris.On a particular day, there are three tasks to perform: 1, 2, and 3.Each worker can perform a task in a different length of time:

How to assign tasks to workers to minimize total time spent?

A B C

1 10 15 8

2 12 16 9

3 30 15 7

Graphical representation

A

B

C

1

2

3

(0,1) 10

1230

1615

15

8 9

7

0

0

0

0

0

0

(0,1)

(0,1)

(1,1)

(1,1)

(1,1)

XA1

XA2

XA3

Each worker is assigned to at most one task

Each task is assigned to one worker

Each arc has a binary decision variable: for example, XA3=1 if Ann is assigned to task 1 and 0 otherwise.

Graphical representation

A

B

C

1

2

3

(0,1) 10

1230

1615

15

8 9

7

0

0

0

0

0

0

(0,1)

(0,1)

(1,1)

(1,1)

(1,1)

What if you want to make sure Chris is not assigned to job 1?

Increase the cost of the C1 arc to a large number

Carrier Selection at Mohawk Paper Mills

Mill produces paper for customers in 12 citiesEach day the load planner determines the loads:# loads for each destination citydistances from mill to destination citiesinterim stops may be required for each destinationeach load is designed to fill one truck

Atlanta

Everett

Ephrata

Carrier Selection at Mohawk Paper Mills

Mill is served by six carriersCarriers have different per-mile rates for each destinationInterim stops cost extraThere is a minimum charge per truckloadCarriers have different numbers of available trucksSome carriers have contracts with minimum requirementsWhich loads should be assigned to which carriers?

Unassigned trucks

Graphical representation

1. ABCT

2. IRST

3. LAST

4. MRST

5. NEST

6. PSST

A. Atlanta

B. Everett

C. Ephrata

D. Riverview

E. Carson

F. Chamblee

G. Roseville

H. Hanover

I. Sparks

J. Parsippany

K. Effingham

L. Kearny

(1,4)

(7,8)

(6,7)

(0,7)

(0,3)

(4,4)

(4,4)

(1,1)

(3,3)

(5,5)

(1,1)

(1,1)

(1,1)

(1,1)

(2,2)

(1,1)

(5,5)

(7,7)

flow limits = (0, ∞)costs = transportation costs for each carrier-destination pair

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0Minimize overall flow cost

6 carrier nodes

12 destination nodes

Graphical representation

1. ABCT

2. IRST

3. LAST

4. MRST

5. NEST

6. PSST

A. Atlanta

B. Everett

C. Ephrata

D. Riverview

E. Carson

F. Chamblee

G. Roseville

H. Hanover

I. Sparks

J. Parsippany

K. Effingham

L. Kearny

(1,4)

(7,8)

(6,7)

(0,7)

(0,3)

(4,4)

(4,4)

(1,1)

(3,3)

(5,5)

(1,1)

(1,1)

(1,1)

(1,1)

(2,2)

(1,1)

(5,5)

(7,7)

flow limits = (0, ∞)costs = transportation costs for each carrier-destination pair

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

Minimum loadcommitments

Number ofavailable trucks

Number of loadsthat must be sentto Atlanta

flow = # of Atlanta loadsassigned to MRST

flow = total# loadsassigned toMRST

Decision variables

1. ABCT

2. IRST

3. LAST

4. MRST

5. NEST

6. PSST

A. Atlanta

B. Everett

C. Ephrata

D. Riverview

E. Carson

F. Chamblee

G. Roseville

H. Hanover

I. Sparks

J. Parsippany

K. Effingham

L. Kearny

X1

X2

X3

X4

X5

X6

XA

XB

XC

XD

XE

XF

XG

XH

XI

XJ

XK

XL

Xij where i{1,…,6} and j{A,…,L}

Input data

Decision variables in the worksheet

# of Atlanta loadsassigned to MRST

Total # loadsassigned to MRST

Total # loadsdelivered to Atlanta

Node constraints

1. ABCT

2. IRST

3. LAST

4. MRST

5. NEST

6. PSST

A. Atlanta

C. Ephrata

D. Riverview

E. Carson

F. Chamblee

G. Roseville

H. Hanover

I. Sparks

J. Parsippany

K. Effingham

L. Kearny

XA

XB

XC

XD

XE

XF

XG

XH

XI

XJ

XK

XL

Constraint for 1.ABCT: X1=X1A+X1B+X1C+X1D+X1E+X1F+X1G+X1H+X1I+X1J+X1K+X1L

Constraint for B.Everett: X1B+X2B+X3B+X4B+X5B+X6B=XB

One constraintfor each of the

18 nodes:inflow = outflow

X1

X2

X3

X4

X5

X6

B. Everett

Arc constraints

1. ABCT

2. IRST

3. LAST

4. MRST

5. NEST

6. PSST

A. Atlanta

B. Everett

C. Ephrata

D. Riverview

E. Carson

F. Chamblee

G. Roseville

H. Hanover

I. Sparks

J. Parsippany

K. Effingham

L. Kearny

XA

XB

XC

XD

XE

XF

XG

XH

XI

XJ

XK

XL

Two constraintsfor each of the

72 arcs:flow ≤ upper boundflow ≥ lower bound

Constraints for arc entering node 1.ABCT: X1≥1, X1≤4Constraints for arc connecting nodes 1.ABCT and C.Ephrata: X1C≥0, X1C≤∞

Constraints for arc leaving node C.Ephrata: XC≥3, XC≤3

X1

X2

X3

X4

X5

X6