1 Slide

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Chapter 6: Transportation, Assignment, and Transshipment Problems

A network model is one which can be represented by a set of nodes, a set of arcs, and functions (e.g. costs, supplies, demands, etc.) associated with the arcs and/or nodes.

Examples include transportation, assignment, transshipment as well as shortest-route, maximal flow problems, minimal spanning tree and PERT/CPM problems.

All network problems can be formulated as linear programs. However, there are many computer packages that contain separate computer codes for these problems which take advantage of their network structure.

If the right-hand side of the linear programming formulations are all integers, then optimal solution of the decision variables will also be integers.

2 Slide

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Transportation Problem

The transportation problem seeks to minimize the total shipping costs of transporting goods from m origins (each with a supply si) to n destinations (each with a demand dj), when the unit shipping cost from an origin, i, to a destination, j, is cij.

The network representation for a transportation problem with two sources and three destinations is given on the next slide.

3 Slide

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Transportation ProblemNetwork Representation

2

c1

1 c12

c13c21

c22c23

d1

d2

d3

s1

s2

m Sources n Destinations

3

2

1

1

4 Slide

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Transportation Problem

Linear Programming Formulation Using the notation: xij = number of units shipped from origin i to destination j cij = cost per unit of shipping from origin i to destination j si = supply or capacity in units at origin i dj = demand in units at destination j

continued

5 Slide

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Transportation Problem

Linear Programming Formulation (continued)

1 1Min

m n

ij iji j

c x

1 1,2, , Supply

n

ij ij

x s i m

1 1,2, , Demand

m

ij ji

x d j n

xij > 0 for all i and j

6 Slide

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Example: Transportation Problem

The Navy has depots in Albany, BenSalem, and Winchester. Each of these three depots has 3,000 pounds of materials which the Navy wishes to ship to three installations, namely, San Diego, Norfolk, and Pensacola. These installations require 4,000, 2,500, and 2,500 pounds, respectively. The shipping costs per pound for are shown on the next slide. Formulate and solve a linear program to determine the shipping arrangements that will minimize the total shipping cost.

7 Slide

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DestinationSource San Diego Norfolk

PensacolaAlbany $12 $ 6

$ 5BenSalem 20 11

9Winchester 30 26

28

Example: Transportation Problem (Continued)

8 Slide

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Transportation Problem: Network Representation

2

3

1

2

3

1 c11 c12

c13

c21 c22

c23

c31 c32

c33

Source

Destination

Albany3000

BenSalem3000

Winchester3000

San Diego4000

Norfolk2500

Pensacola2500

9 Slide

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Define the Decision VariablesWe want to determine the pounds of material, xij , to be shipped by mode i to destination j. The following table summarizes the decision variables:

San Diego Norfolk Pensacola

Albany x11 x12 x13

BenSalem x21 x22 x23

Winchester x31 x32 x33

Example: Transportation Problem (Continued)

10 Slide

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Define the Objective Function Minimize the total shipping cost. Min: (shipping cost per pound for each

mode per destination pairing) x (number of pounds shipped by mode per destination pairing).

Min: 12x11 + 6x12 + 5x13 + 20x21 + 11x22 + 9x23

+ 30x31 + 26x32 + 28x33

Example: Transportation Problem (Continued)

11 Slide

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Define the Constraints Source availability: (1) x11 + x12 + x13 = 3000 (2) x21 + x22 + x23 = 3000 (3) x31 + x32 + x33 = 3000 Destination material requirements: (4) x11 + x21 + x31 = 4000 (5) x12 + x22 + x32 = 2500 (6) x13 + x23 + x33 = 2500 Non-negativity of variables: xij > 0, i = 1, 2, 3 and j = 1, 2,

3

Transportation Problem: Example #2

12 Slide

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Computer Output

OBJECTIVE FUNCTION VALUE = 142000.000 Variable Value Reduced

Cost x11 1000.000

0.000 x12 2000.000

0.000 x13 0.000

1.000 x21 0.000

3.000 x22 500.000

0.000 x23 2500.000

0.000 x31 3000.000

0.000 x32 0.000

2.000 x33 0.000

6.000

Example: Transportation Problem (Continued)

13 Slide

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Solution Summary• San Diego will receive 1000 lbs. from

Albanyand 3000 lbs. from Winchester.

• Norfolk will receive 2000 lbs. from Albany and 500 lbs. from BenSalem.• Pensacola will receive 2500 lbs. from

BenSalem. • The total shipping cost will be $142,000.

Transportation Problem: Example #2

14 Slide

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LP Formulation Special Cases• Total supply exceeds total demand:

• Total demand exceeds total supply: Add a dummy origin with supply equal to the shortage amount. Assign a zero shipping cost per unit. The amount “shipped” from the dummy origin (in the solution) will not actually be shipped.

Assign a zero shipping cost per unit

• Maximum route capacity from i to j: xij < Li

Remove the corresponding decision variable.

Transportation Problem

No modification of LP formulation is necessary.

15 Slide

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LP Formulation Special Cases (continued)• The objective is maximizing profit or

revenue:

• Minimum shipping guarantee from i to j: xij > Lij

• Maximum route capacity from i to j: xij < Lij

• Unacceptable route: Remove the corresponding

decision variable.

Transportation Problem

Solve as a maximization problem.

16 Slide

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Assignment Problem

An assignment problem seeks to minimize the total cost assignment of m workers to m jobs, given that the cost of worker i performing job j is cij. It assumes all workers are assigned and each job is performed. An assignment problem is a special case of a transportation problem in which all supplies and all demands are equal to 1; hence assignment problems may be solved as linear programs.The network representation of an assignment problem with three workers and three jobs is shown on the next slide.

17 Slide

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Assignment Problem

Network Representation

2

3

1

2

3

1 c11c12

c13

c21 c22

c23

c31 c32

c33

Agents Tasks

18 Slide

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Linear Programming Formulation Using the notation: xij = 1 if agent i is assigned

to task j 0 otherwise cij = cost of assigning agent i to

task j

Assignment Problem

continued

19 Slide

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Linear Programming Formulation (continued)

Assignment Problem

1 1Min

m n

ij iji j

c x

11 1,2, , Agents

n

ijj

x i m

11 1,2, , Tasks

m

iji

x j n

xij > 0 for all i and j

20 Slide

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An electrical contractor pays his subcontractors a fixed fee plus mileage for work performed. On a given day the contractor is faced with three electrical jobs associated with various projects. Given below are the distances between the subcontractors and the projects.

ProjectsSubcontractor A B C Westside 50 36 16

Federated 28 30 18 Goliath 35 32 20

Universal 25 25 14How should the contractors be assigned so that totalmileage is minimized?

Example: Assignment Problem

21 Slide

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Network Representation50

361628

301835 32

2025 25

14

West.

C

B

A

Univ.

Gol.

Fed.

ProjectsSubcontractors

Example: Assignment Problem

22 Slide

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Linear Programming Formulation

Min 50x11+36x12+16x13+28x21+30x22+18x23

+35x31+32x32+20x33+25x41+25x42+14x43 s.t. x11+x12+x13 < 1

x21+x22+x23 < 1 x31+x32+x33 < 1 x41+x42+x43 < 1 x11+x21+x31+x41 = 1 x12+x22+x32+x42 = 1 x13+x23+x33+x43 = 1 xij = 0 or 1 for all i and j

Agents

Tasks

Assignment Problem: Example

23 Slide

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The optimal assignment is:

Subcontractor Project Distance Westside C 16

Federated A 28Goliath (unassigned) Universal B 25

Total Distance = 69 miles

Assignment Problem: Example

24 Slide

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LP Formulation Special Cases• Number of agents exceeds the number of

tasks:

• Number of tasks exceeds the number of agents: Add enough dummy agents to equalize the number of agents and the number of tasks. The objective function coefficients for these new variable would be zero.

Assignment Problem

Extra agents simply remain unassigned.

25 Slide

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Assignment Problem

LP Formulation Special Cases (continued)• The assignment alternatives are evaluated in

terms of revenue or profit: Solve as a maximization problem.

• An assignment is unacceptable: Remove the corresponding decision

variable.

• An agent is permitted to work t tasks:

1 1,2, , Agents

n

ijj

x t i m

26 Slide

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Transshipment Problem

Transshipment problems are transportation problems in which a shipment may move through intermediate nodes (transshipment nodes)before reaching a particular destination node.Transshipment problems can be converted to larger transportation problems and solved by a special transportation program.Transshipment problems can also be solved by general purpose linear programming codes.The network representation for a transshipment problem with two sources, three intermediate nodes, and two destinations is shown on the next slide.

27 Slide

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Transshipment Problem

Network Representation

2

3

4

5

6

7

1c13

c14

c23

c24c25

c15

s1

c36

c37

c46c47

c56

c57

d1

d2

Intermediate NodesSources Destinationss2

DemandSupply

28 Slide

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Transshipment Problem

Linear Programming Formulation

Using the notation: xij = number of units shipped from node i to node j

cij = cost per unit of shipping from node i to node j

si = supply at origin node i dj = demand at destination node j

continued

29 Slide

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Transshipment Problem

all arcsMin ij ijc x

arcs out arcs ins.t. Origin nodes ij ij ix x s i

xij > 0 for all i and j

arcs out arcs in0 Transhipment nodesij ijx x

arcs in arcs out Destination nodes ij ij jx x d j

Linear Programming Formulation (continued)

continued

30 Slide

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Transshipment Problem

LP Formulation Special Cases• Total supply not equal to total demand• Maximization objective function• Route capacities or route minimums• Unacceptable routesThe LP model modifications required here areidentical to those required for the special

cases inthe transportation problem.

31 Slide

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The Northside and Southside facilities of Zeron Industries supply three firms (Zrox, Hewes, Rockrite) with customized shelving for its offices. They both order shelving from the same two manufacturers, Arnold Manufacturers and Supershelf, Inc.

Currently weekly demands by the users are 50 for Zrox, 60 for Hewes, and 40 for Rockrite. Both Arnold and Supershelf can supply at most 75 units to its customers.

Additional data is shown on the next slide.

Transshipment Problem Example

32 Slide

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Because of long standing contracts based on past orders, unit costs from the manufacturers to the suppliers are:

Zeron N Zeron S Arnold 5 8 Supershelf 7 4

The costs to install the shelving at the various locations are:

Zrox Hewes Rockrite Thomas 1 5 8

Washburn 3 4 4

Transshipment Problem Example

33 Slide

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Network Representation

ARNOLD

WASHBURN

ZROX

HEWES

75

75

50

60

40

5

8

7

4

15

8

34

4

Arnold

SuperShelf

Hewes

Zrox

ZeronN

ZeronS

Rock-Rite

Transshipment Problem Example

34 Slide

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Linear Programming Formulation• Decision Variables Defined

xij = amount shipped from manufacturer i to supplier j

xjk = amount shipped from supplier j to customer k

where i = 1 (Arnold), 2 (Supershelf) j = 3 (Zeron N), 4 (Zeron S) k = 5 (Zrox), 6 (Hewes), 7

(Rockrite)• Objective Function Defined

Minimize Overall Shipping Costs: Min 5x13 + 8x14 + 7x23 + 4x24 + 1x35 +

5x36 + 8x37 + 3x45 + 4x46 + 4x47

Transshipment Problem: Example

35 Slide

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Constraints DefinedAmount Out of Arnold: x13 + x14 < 75Amount Out of Supershelf: x23 + x24 < 75Amount Through Zeron N: x13 + x23 - x35 - x36 - x37 = 0Amount Through Zeron S: x14 + x24 - x45 - x46 - x47 = 0Amount Into Zrox: x35 + x45 = 50Amount Into Hewes: x36 + x46 = 60Amount Into Rockrite: x37 + x47 = 40

Non-negativity of Variables: xij > 0, for all i and j.

Transshipment Problem: Example

36 Slide

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Computer Output Objective Function Value =

1150.000 Variable Value Reduced

Costs X13 75.000

0.000 X14 0.000

2.000 X23 0.000

4.000 X24 75.000

0.000 X35 50.000

0.000 X36 25.000

0.000 X37 0.000

3.000 X45 0.000

3.000 X46 35.000

0.000 X47 40.000

0.000

Transshipment Problem: Example

37 Slide

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Solution

ARNOLD

WASHBURN

ZROX

HEWES

75

75

50

60

40

5

8

7

4

15

8

3 4

4

Arnold

SuperShelf

Hewes

Zrox

ZeronN

ZeronS

Rock-Rite

75

75

50

25

35

40

Transshipment Problem: Example

38 Slide

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Computer Output (continued) Constraint Slack/Surplus Dual

Values 1 0.000

0.000 2 0.000

2.000 3 0.000

-5.000 4 0.000

-6.000 5 0.000

-6.000 6 0.000 -

10.000 7 0.000 -

10.000

Transshipment Problem: Example

39 Slide

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Computer Output (continued)

OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value Upper

Limit X13 3.000 5.000 7.000 X14 6.000 8.000

No Limit X23 3.000 7.000

No Limit X24 No Limit 4.000

6.000 X35 No Limit 1.000

4.000 X36 3.000 5.000

7.000 X37 5.000 8.000

No Limit X45 0.000 3.000

No Limit X46 2.000 4.000

6.000 X47 No Limit 4.000

7.000

Transshipment Problem: Example

40 Slide

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Computer Output (continued)

RIGHT HAND SIDE RANGES Constraint Lower Limit Current Value Upper

Limit 1 75.000 75.000

No Limit 2 75.000 75.000

100.000 3 -75.000 0.000

0.000 4 -25.000 0.000

0.000 5 0.000 50.000

50.000 6 35.000 60.000

60.000 7 15.000 40.000

40.000

Transshipment Problem: Example