Chapter 7: Transportation Models

Skip Ship Routing & Scheduling (pp. 212-214)

• Service Selection

• Shortest Path

• Transportation Problem

• Vehicle Routing & Scheduling

– One route: TSP

– Multiple routes: VRP

• Consolidation

Service Selection (Mode Selection)

• Most important factors:– Dependability (on-time delivery).

– Cost.

– Safety.

– Tracking.

• Different modes have different costs and characteristics.

• Lowest transportation cost is not always best.

Service Selection Tradeoff

• Transportation Cost vs. Inventory Cost.

• Shorter transit time: Higher transportation cost.

Fewer days held Lower inventory cost.

• Usually, Shorter transit time Smaller vehicles. More frequent trips Higher transportation cost.

Fewer units held Lower inventory cost .

Service Selection for Competing Suppliers

One buyer purchases 1000 cwt from each of two competing suppliers: A and B. Both use rail transport, but could use truck transport. Supplier profit = $20/cwt - transport cost.

Transport Cost Delivery TimeRail $2/cwt 6 daysTruck $5/cwt 3 days

Buyer offers to switch 100 cwt to supplier A from B for each day decrease in delivery time. For supplier A:

Sales ProfitRail (current) 1000 cwt 1000 cwt ($20/cwt - $2/cwt) = $18,000

Truck 1300 cwt 1300 cwt ($20/cwt - $5/cwt) = $19,500

Service Selection for Competing Suppliers

What if supplier B also switches to truck?

Buyer should give each equal business:

Sales ProfitSupplier A 1000 cwt 1000 cwt ($20/cwt - $5/cwt) = $15,000

Supplier B 1000 cwt 1000 cwt ($20/cwt - $5/cwt) = $15,000

So both suppliers are worse off than before! ($15,000 profit vs. $18,000 using rail)

Shortest Path Model

• Network includes:– Nodes: cities, customers, demand points

– Arcs or Links: Transportation links

– Number for each link to represent travel cost, time or distance.

A F

D

B

E

C

3

4

6

2

9

4

7

5

6

Shortest Path Problem

• Given:– A network with a specified origin and destination.– The distance (or travel time or cost) for each link.

• Determine the shortest path from the origin to the destination.

• Solution: Labeling algorithm (one of many)– Nodes are labeled as "solved" or "unsolved". – Solved = shortest path from the origin to that node is

known.

Shortest Path Labeling Algorithm

1. The origin is a solved node. All others are unsolved.

2. For each solved node, find the one unsolved node that is nearest and calculate the minimum total distance (origin to solved node + solved node to nearest unsolved node).

3. Make the unsolved node with the smallest total distance a solved node.

4. Repeat steps 2 and 3 until the destination is a solved node.

5. Trace the shortest path.

Shortest Path Example 1

A F

D

B

EC

30

20

22

6

17 10

518

27

1612

• Find the shortest path from A to F.

Shortest Path Example 1

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 20

A F

D

B

EC

30

20

22

6

17 10

518

27

1612*

Shortest Path Example 1

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 20 C 20 A-C

A F

D

B

EC

30

20

22

6

17 10

518

27

1612* *

Shortest Path Example 1

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 20 C 20 A-C A C

A F

D

B

EC

30

20

22

6

17 10

518

27

1612* *

Shortest Path Example 1

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 20 C 20 A-C A B 22 C

A F

D

B

EC

30

20

22

6

17 10

518

27

1612* *

Shortest Path Example 1

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 20 C 20 A-C A B 22 C D 26

A F

D

B

EC

30

20

22

6

17 10

518

27

1612* *

Shortest Path Example 1

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 20 C 20 A-C A B 22 B 22 A-B C D 26

A F

D

B

EC

30

20

22

6

17 10

518

27

1612*

*

*

Shortest Path Example 1

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A B C

A F

D

B

EC

30

20

22

6

17 10

518

27

1612*

*

*

Shortest Path Example 1

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A D 30 B C

A F

D

B

EC

30

20

22

6

17 10

518

27

1612*

*

*

Shortest Path Example 1

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A D 30 B E 32 C

A F

D

B

EC

30

20

22

6

17 10

518

27

1612*

*

*

Shortest Path Example 1

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A D 30 B E 32 C D 26

A F

D

B

EC

30

20

22

6

17 10

518

27

1612*

*

*

Shortest Path Example 1

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A D 30 B E 32 C D 26 D 26 C-D

A F

D

B

EC

30

20

22

6

17 10

518

27

1612*

*

*

*

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A D 30 B E 32 D 26 C-D C D 26 B C D

A F

D

B

EC

30

20

22

6

17 10

518

27

1612*

*

*

*

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A D 30 B E 32 D 26 C-D C D 26 B E 32 C E 32 D E 31

A F

D

B

EC

30

20

22

6

17 10

518

27

1612*

*

*

*

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A D 30 B E 32 D 26 C-D C D 26 B E 32 C E 32 E 31 D-E D E 31

A F

D

B

EC

30

20

22

6

17 10

518

27

1612*

*

*

*

*

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A D 30 B E 32 D 26 C-D C D 26 B E 32 C E 32 E 31 D-E D E 31 B D E

A F

D

B

EC

30

20

22

6

17 10

518

27

1612*

*

*

*

*

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A D 30 B E 32 D 26 C-D C D 26 B E 32 C E 32 E 31 D-E D E 31 B F 49 D F 44 E F 47

A F

D

B

EC

30

20

22

6

17 10

518

27

1612*

*

*

*

*

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A D 30 B E 32 D 26 C-D C D 26 B E 32 C E 32 E 31 D-E D E 31 B F 49 D F 44 F 44 D-F E F 47

A F

D

B

EC

30

20

22

6

17 10

518

27

1612*

*

*

*

* *

Trace Shortest Path Backwards

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 20 C 20 A-C A B 22 B 22 A-B C D 26 A D 30 B E 32 D 26 C-D C D 26 B E 32 C E 32 E 31 D-E D E 31 B F 49 D F 44 F 44 D-F E F 47

A-C-D-F

Check Answer

A F

D

B

EC

30

20

22

6

17 10

518

27

1612

A-C-D-F Length = 20+6+18 = 44

Shortest Path Example 2

• Find the shortest path from A to K.

A K

H

E4

63

6

9

1

6

1

3

5

11

B

C

D G

JF

I2

3

3

5

4

4

A K

H

E4

63

6

9

1

6

1

3

5

11

B

C

D G

JF

I2

3

3

5

4

4

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 4 C 4 A-C

0

A K

H

E4

63

6

9

1

6

1

3

5

11

B

C

D G

JF

I2

3

3

5

4

4

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 4 C 4 A-C A C

0

4

A K

H

E4

63

6

9

1

6

1

3

5

11

B

C

D G

JF

I2

3

3

5

4

4

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 4 C 4 A-C A B 6 B 6 A-B C E 8

0

4

6

A K

H

E4

63

6

9

1

6

1

3

5

11

B

C

D G

JF

I2

3

3

5

4

4

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 4 C 4 A-C A B 6 B 6 A-B C E 8 B C

0

4

6

A K

H

E4

63

6

9

1

6

1

3

5

11

B

C

D G

JF

I2

3

3

5

4

4

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 4 C 4 A-C A B 6 B 6 A-B C E 8 B E 9 C E 8 E 8 C-E

0

4

6

8

A K

H

E4

63

6

9

1

6

1

3

5

11

B

C

D G

JF

I2

3

3

5

4

4

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 4 C 4 A-C A B 6 B 6 A-B C E 8 B E 9 C E 8 E 8 C-E B C E

0

4

6

8

A K

H

E4

63

6

9

1

6

1

3

5

11

B

C

D G

JF

I2

3

3

5

4

4

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 4 C 4 A-C A B 6 B 6 A-B C E 8 B E 9 C E 8 E 8 C-E B D 15 C F 10 E H 9 H 9 E-H

0

4

6

8

9

A K

H

E4

63

6

9

1

6

1

3

5

11

B

C

D G

JF

I2

3

3

5

4

4

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path B C E H

0

4

6

8

9

A K

H

E4

63

6

9

1

6

1

3

5

11

B

C

D G

JF

I2

3

3

5

4

4

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path B D 15 C F 10 F 10 C-F E I 14 H D 12

0

4

6

8

9

10

A K

H

E4

63

6

9

1

6

1

3

5

11

B

C

D G

JF

I2

3

3

5

4

4

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path B D 15 C F 10 F 10 C-F E I 14 H D 12 B C E H F

0

4

6

8

9

10

A K

H

E4

63

6

9

1

6

1

3

5

11

B

C

D G

JF

I2

3

3

5

4

4

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path B D 15 C F 10 F 10 C-F E I 14 H D 12 B D 15 C I 15 E I 14 H D 12 D 12 H-D F I 14

0

4

6

8

9

10

12

A K

H

E4

63

6

9

1

6

1

3

5

11

B

C

D G

JF

I2

3

3

5

4

4

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path C E H F D

0

4

6

8

9

10

12

A K

H

E4

63

6

9

1

6

1

3

5

11

B

C

D G

JF

I2

3

3

5

4

4

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path C I 15 E I 14 H K 14 K 14 H-K

F I 14 D J 15

0

4

6

8

9

10

12

14

Shortest Length = 14

Trace Shortest Path Backwards

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path B D 15 C F 10 F 10 C-F E I 14 H D 12 B D 15 C I 15 E I 14 H D 12 D 12 H-D F I 14 C I 15 E I 14 H K 14 K 14 H-K

F I 14 D J 15

Trace Shortest Path Backwards

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 4 C 4 A-C A B 6 B 6 A-B C E 8 B E 9 C E 8 E 8 C-E B D 15 C F 10 E H 9 H 9 E-H

A-C-E-H-K

Check Answer

A-C-E-H-K Length = 4+4+1+5 = 14

A K

H

E4

63

6

9

1

6

1

3

5

11

B

C

D G

JF

I2

3

3

5

4

4

Shortest Path Example 3

• Find the shortest path from A to K.

A K

H

E4

63

16

7

5

6

1

3

12

10

B

C

D G

JF

I12

16

8

5

4

4

A K

H

E4

63

16

7

5

6

1

3

12

10

B

C

D G

JF

I12

16

8

5

4

4

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 4 C 4 A-C A B 6 B 6 A-B C E 8 B E 9 C E 8 E 8 C-E

0

4

6

8

First 3 steps are same as Example 2!

A K

H

E4

63

16

7

5

6

1

3

12

10

B

C

D G

JF

I12

16

8

5

4

4

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 4 C 4 A-C A B 6 B 6 A-B C E 8 B E 9 C E 8 E 8 C-E B C E

0

4

6

8

A K

H

E4

63

16

7

5

6

1

3

12

10

B

C

D G

JF

I12

16

8

5

4

4

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 4 C 4 A-C A B 6 B 6 A-B C E 8 B E 9 C E 8 E 8 C-E B D 13 C I 14 E H 13

0

4

6

8

Tie for minimum distance

Select both!

A K

H

E4

63

16

7

5

6

1

3

12

10

B

C

D G

JF

I12

16

8

5

4

4

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path A C 4 C 4 A-C A B 6 B 6 A-B C E 8 B E 9 C E 8 E 8 C-E B D 13 D 13 B-D C I 14 E H 13 H 13 E-H

0

4

6

8

13

13

A K

H

E4

63

16

7

5

6

1

3

12

10

B

C

D G

JF

I12

16

8

5

4

4

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path C D E H

0

4

6

8

13

13

A K

H

E4

63

16

7

5

6

1

3

12

10

B

C

D G

JF

I12

16

8

5

4

4

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path C I 14 D G 29 E I 14 H K 25

0

4

6

8

13

13

Tie for minimum distance

Select both!

A K

H

E4

63

16

7

5

6

1

3

12

10

B

C

D G

JF

I12

16

8

5

4

4

Nearest Total MinimumSolved Unsolved Distance Nearest Distance Path C I 14 I 14 C-I D G 29 E I 14 I 14 E-I H K 25

0

4

6

8

13

13

There are two equal shortest paths from the origin to I!

Example 3 Answer

A-C-E-I-F-J-K

A-C-I-F-J-K

Length = 4+4+6+4+5+1 = 24

Length = 4+10+4+5+1 = 24

A K

H

E4

63

16

7

5

6

1

3

12

10

B

C

D G

JF

I12

16

8

5

4

4

Shortest Path Software in LogWare

• ROUTE module.

• For each node, enter:– Node number and name.

– X and Y coordinates if desired.

• For each link (arc), enter:– “From node” number.

– “To node” number.

– Cost.

– Save data.

• Click Solve to get shortest paths from node 1 to all other nodes.

LogWare

ROUTE Module in LogWare

Otherwise, click “Open file” and open Rfl01.dat.

If possible, click “Add row”; then enter data.

ROUTE Module: Edit as desired

Now, Delete and Add rows and edit data.

Save before solving.

ROUTE Module: Solution

Transportation Problem

• Given:– m origins (sources for product flows). – n destinations (sinks for product flows).– Supply at each origin.– Demand at each destination.– Shipping cost per unit of product from each origin to

each destination.

• Determine the minimum total cost shipping pattern to satisfy demand. – We will solve using TRANLP module of LogWare.

Transportation Problem Example

• 3 origins (sources) and 4 destinations (sinks)

Origin Supply Destination Demand

1 300 cwt. 1 400 cwt.

2 900 cwt. 2 300 cwt.

3 800 cwt. 3 700 cwt.

4 600 cwt.

Shipping

cost ($/cwt): D1 D2 D3 D4

O1 3 3 4 2

O2 2 4 3 7

O3 2 5 1 5

Transportation Problem Example

• 3 origins (sources) and 4 destinations (sinks)

300 O1

D4 600

D3 700

D2 300

D1 4003

342

24

37

5

152

800 O3

900 O2

Transportation Problem Example

• A feasible solution: flows are in blue.

300 O1

D4 600

D3 700

D2 300

D1 4003

342

24

37

5

152

800 O3

900 O2

300

100

300

500

200

600

Cost = 300x3+100x2+300x4+500x3+200x5+600x5 = 7500

Solving Transportation Problems

• Place data in Transportation Matrix.

From/To T1 T2 T3 T4 Supply

F1 3 3 4 2 300

F2 2 4 3 7 900

F3 2 5 1 5 800

Demand 400 300 700 600

Enter data into TRANLP and solve.

1.Open a file.

2. Change Problem label and specify number of rows and columns.

3. Enter data (use Backspace to erase entries).

4. Save data.

5. Click Solve.

TRANLP in LogWare

If possible, enter “No. of rows” and “No. of columns”.

If not, then click “Open file” and open TRAN01.dat.

File TRAN01.dat

Now, enter “No. of rows” and “No. of columns”.

Then, edit data.

Save before solving.

TRAN01.dat Solution

Solution.

Click “Report” for more...

TRANLP OutputProblem label: Example OPTIMUM SUPPLY SCHEDULE ----------- Cell ------------ Unit Cell Units Source name Sink name cost cost allocated F1 T1 3.00 .00 0 F1 T2 3.00 .00 0 F1 T3 4.00 .00 0 F1 T4 2.00 600.00 300 Totals 600.00 300 Source capacity = 300 Slack capacity = 0

F2 T1 2.00 800.00 400 F2 T2 4.00 1,200.00 300 F2 T3 3.00 .00 0 F2 T4 7.00 1,400.00 200 Totals 3,400.00 900 Source capacity = 900 Slack capacity = 0

F3 T1 2.00 .00 0 F3 T2 5.00 .00 0 F3 T3 1.00 700.00 700 F3 T4 5.00 500.00 100 Totals 1,200.00 800 Source capacity = 800 Slack capacity = 0

Total allocated = 2,000 Slack required = 2,000Total cost = 5,200.00

TRANLP OutputProblem label: Example OPTIMUM SUPPLY SCHEDULE ----------- Cell ------------ Unit Cell Units Source name Sink name cost cost allocated F1 T1 3.00 .00 0 F1 T2 3.00 .00 0 F1 T3 4.00 .00 0 F1 T4 2.00 600.00 300 Totals 600.00 300 Source capacity = 300 Slack capacity = 0

F2 T1 2.00 800.00 400 F2 T2 4.00 1,200.00 300 F2 T3 3.00 .00 0 F2 T4 7.00 1,400.00 200 Totals 3,400.00 900 Source capacity = 900 Slack capacity = 0

F3 T1 2.00 .00 0 F3 T2 5.00 .00 0 F3 T3 1.00 700.00 700 F3 T4 5.00 500.00 100 Totals 1,200.00 800 Source capacity = 800 Slack capacity = 0

Total allocated = 2,000 Slack required = 2,000Total cost = 5,200.00

Optimal Cost

Optimal flows

Optimal Solution

• Optimal solution: flows are in blue.

300 O1

D4 600

D3 700

D2 300

D1 4003

342

24

37

5

152

800 O3

900 O2

300

400

300

200700

100

Cost = 300x2+400x2+300x4+200x7+700x1+100x5 = 5200

TRANLP OutputProblem label: Example OPTIMUM SUPPLY SCHEDULE ----------- Cell ------------ Unit Cell Units Source name Sink name cost cost allocated F1 T1 3.00 .00 0 F1 T2 3.00 .00 0 F1 T3 4.00 .00 0 F1 T4 2.00 600.00 300 Totals 600.00 300 Source capacity = 300 Slack capacity = 0

F2 T1 2.00 800.00 400 F2 T2 4.00 1,200.00 300 F2 T3 3.00 .00 0 F2 T4 7.00 1,400.00 200 Totals 3,400.00 900 Source capacity = 900 Slack capacity = 0

F3 T1 2.00 .00 0 F3 T2 5.00 .00 0 F3 T3 1.00 700.00 700 F3 T4 5.00 500.00 100 Totals 1,200.00 800 Source capacity = 800 Slack capacity = 0

Total allocated = 2,000 Slack required = 2,000Total cost = 5,200.00

slack capacity=0 means all is sent from every source

Total allocated = Slack required means each destination receives what it needs.

Transportation Problem

• In last problem total supply = total demand.– Each origin sends all it has.– Each destination receives all it demands.

• Other possibilities:– Total Supply > Total Demand

• Some origins will keep some of the supply.

– Total Supply < Total Demand

• Some destinations will not receive all they demand.

TRANLP Output #2Problem label: Example OPTIMUM SUPPLY SCHEDULE ----------- Cell ------------ Unit Cell Units Source name Sink name cost cost allocated F1 T1 3.00 .00 0 F1 T2 3.00 .00 0 F1 T3 4.00 .00 0 F1 T4 2.00 600.00 300 Totals 600.00 300 Source capacity = 300 Slack capacity = 0

F2 T1 2.00 800.00 400 F2 T2 4.00 1,200.00 300 F2 T3 3.00 .00 0 F2 T4 7.00 .00 0 Totals 2,000.00 700 Source capacity = 900 Slack capacity = 200

F3 T1 2.00 .00 0 F3 T2 5.00 .00 0 F3 T3 1.00 700.00 700 F3 T4 5.00 500.00 100 Totals 1,200.00 800 Source capacity = 800 Slack capacity = 0

Total allocated = 1,800 Slack required = 1,800Total cost = 3,800.00

What is happening here?

TRANLP Output #2Problem label: Example OPTIMUM SUPPLY SCHEDULE ----------- Cell ------------ Unit Cell Units Source name Sink name cost cost allocated F1 T1 3.00 .00 0 F1 T2 3.00 .00 0 F1 T3 4.00 .00 0 F1 T4 2.00 600.00 300 Totals 600.00 300 Source capacity = 300 Slack capacity = 0

F2 T1 2.00 800.00 400 F2 T2 4.00 1,200.00 300 F2 T3 3.00 .00 0 F2 T4 7.00 .00 0 Totals 2,000.00 700 Source capacity = 900 Slack capacity = 200

F3 T1 2.00 .00 0 F3 T2 5.00 .00 0 F3 T3 1.00 700.00 700 F3 T4 5.00 500.00 100 Totals 1,200.00 800 Source capacity = 800 Slack capacity = 0

Total allocated = 1,800 Slack required = 1,800Total cost = 3,800.00

Total allocated = Slack required means each destination receives what it needs.

Supply > Demand

slack capacity=200 means 200 is left at origin 2

TRANLP Output #3Problem label: Example OPTIMUM SUPPLY SCHEDULE ----------- Cell ------------ Unit Cell Units Source name Sink name cost cost allocated F1 T1 3.00 .00 0 F1 T2 3.00 .00 0 F1 T3 4.00 .00 0 F1 T4 2.00 600.00 300 Totals 600.00 300 Source capacity = 300 Slack capacity = 0

F2 T1 2.00 800.00 400 F2 T2 4.00 1,200.00 300 F2 T3 3.00 .00 0 F2 T4 7.00 1,400.00 200 Totals 3,400.00 900 Source capacity = 900 Slack capacity = 0

F3 T1 2.00 .00 0 F3 T2 5.00 .00 0 F3 T3 1.00 700.00 700 F3 T4 5.00 500.00 100 Totals 1,200.00 800 Source capacity = 800 Slack capacity = 0

Total allocated = 2,000 Slack required = 2,200Total cost = 5,200.00

What is happening here?

TRANLP Output #3Problem label: Example OPTIMUM SUPPLY SCHEDULE ----------- Cell ------------ Unit Cell Units Source name Sink name cost cost allocated F1 T1 3.00 .00 0 F1 T2 3.00 .00 0 F1 T3 4.00 .00 0 F1 T4 2.00 600.00 300 Totals 600.00 300 Source capacity = 300 Slack capacity = 0

F2 T1 2.00 800.00 400 F2 T2 4.00 1,200.00 300 F2 T3 3.00 .00 0 F2 T4 7.00 1,400.00 200 Totals 3,400.00 900 Source capacity = 900 Slack capacity = 0

F3 T1 2.00 .00 0 F3 T2 5.00 .00 0 F3 T3 1.00 700.00 700 F3 T4 5.00 500.00 100 Totals 1,200.00 800 Source capacity = 800 Slack capacity = 0

Total allocated = 2,000 Slack required = 2,200Total cost = 5,200.00

Total allocated < Slack required means some destination(s) did not receive what they need. Can not tell which one(s) without input data.

Supply < Demand