1 1 Slide Chapter 7 Transportation, Assignment, and Transshipment Problems n Transportation Problem...

1 1 Slide Slide

Chapter 7Chapter 7Transportation, Assignment, and Transportation, Assignment, and

Transshipment ProblemsTransshipment Problems

Transportation ProblemTransportation Problem

Assignment ProblemAssignment Problem

The Transshipment ProblemThe Transshipment Problem

2 2 Slide Slide

Transportation, Assignment, and Transportation, Assignment, and Transshipment ProblemsTransshipment Problems

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

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Transportation, Assignment, and Transportation, Assignment, and Transshipment ProblemsTransshipment Problems

Each of the three models of this chapter Each of the three models of this chapter (transportation, assignment, and transshipment (transportation, assignment, and transshipment models) can be formulated as linear programs models) can be formulated as linear programs and solved by general purpose linear and solved by general purpose linear programming codes. programming codes.

For each of the three models, if the right-hand For each of the three models, if the right-hand side of the linear programming formulations are side of the linear programming formulations are all integers, the optimal solution will be in terms all integers, the optimal solution will be in terms of integer values for the decision variables.of integer values for the decision variables.

However, there are many computer packages However, there are many computer packages (including (including The Management ScientistThe Management Scientist) which ) which contain separate computer codes for these contain separate computer codes for these models which take advantage of their network models which take advantage of their network structure.structure.

4 4 Slide Slide


The The transportation problemtransportation problem seeks to minimize seeks to minimize the total shipping costs of transporting goods the total shipping costs of transporting goods from from mm origins (each with a supply origins (each with a supply ssii) to ) to nn destinations (each with a demand destinations (each with a demand ddjj), when ), when the unit shipping cost from an origin, the unit shipping cost from an origin, ii, to a , to a destination, destination, jj, is , is ccijij..

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

5 5 Slide Slide


Network RepresentationNetwork Representation

11

22

33

11

22

cc11

11cc1212

cc1313

cc2121 cc2222

cc2323

dd11

dd22

dd33

ss11

s2

SOURCESSOURCES DESTINATIONSDESTINATIONS

6 6 Slide Slide


LP FormulationLP Formulation

The LP formulation in terms of the The LP formulation in terms of the amounts shipped from the origins to the amounts shipped from the origins to the destinations, destinations, xxij ij , can be written as:, can be written as:

Min Min ccijijxxijij i ji j

s.t. s.t. xxijij << ssii for each origin for each origin ii jj

xxijij = = ddjj for each for each destination destination jj

ii

xxijij >> 0 0 for all for all ii and and jj

7 7 Slide Slide


LP Formulation Special CasesLP Formulation Special Cases

The following special-case modifications to The following special-case modifications to the linear programming formulation can be made:the linear programming formulation can be made:

•Minimum shipping guarantee from Minimum shipping guarantee from ii to to jj: :

xxijij >> LLijij

•Maximum route capacity from Maximum route capacity from ii to to jj::

xxijij << LLijij

•Unacceptable route: Unacceptable route:

Remove the corresponding decision variable.Remove the corresponding decision variable.

8 8 Slide Slide

Example: BBCExample: BBC

Building Brick Company (BBC) has orders Building Brick Company (BBC) has orders for 80 tons of bricks at three suburban locations for 80 tons of bricks at three suburban locations as follows: Northwood -- 25 tons, Westwood -- as follows: Northwood -- 25 tons, Westwood -- 45 tons, and Eastwood -- 10 tons. BBC has two 45 tons, and Eastwood -- 10 tons. BBC has two plants, each of which can produce 50 tons per plants, each of which can produce 50 tons per week. Delivery cost per ton from each plant to week. Delivery cost per ton from each plant to each suburban location is shown on the next each suburban location is shown on the next slide.slide.

How should end of week shipments be How should end of week shipments be made to fill the above orders?made to fill the above orders?

9 9 Slide Slide


Delivery Cost Per TonDelivery Cost Per Ton

NorthwoodNorthwood WestwoodWestwood EastwoodEastwood

Plant 1 24 Plant 1 24 30 30 40 40

Plant 2 Plant 2 30 40 30 40 42 42

10 10 Slide Slide


Partial Spreadsheet Showing Problem DataPartial Spreadsheet Showing Problem Data

A B C D E F G H

1

2 Constraint X11 X12 X13 X21 X22 X23 RHS

3 #1 1 1 1 50

4 #2 1 1 1 50

5 #3 1 1 25

6 #4 1 1 45

7 #5 1 1 10

8 Obj.Coefficients 24 30 40 30 40 42 30

LHS Coefficients

11 11 Slide Slide


Partial Spreadsheet Showing Optimal SolutionPartial Spreadsheet Showing Optimal Solution

A B C D E F G

10 X11 X12 X13 X21 X22 X23

11 Dec.Var.Values 5 45 0 20 0 10

12 Minimized Total Shipping Cost 2490

13

14 LHS RHS

15 50 <= 50

16 30 <= 50

17 25 = 25

18 45 = 45

19 10 = 10E.Dem.

W.Dem.

N.Dem.

Constraints

P1.Cap.

P2.Cap.

12 12 Slide Slide

Optimal SolutionOptimal Solution

FromFrom ToTo AmountAmount CostCost

Plant 1 Northwood 5 Plant 1 Northwood 5 120120

Plant 1 Westwood 45 Plant 1 Westwood 45 1,3501,350

Plant 2 Northwood 20 Plant 2 Northwood 20 600600

Plant 2 Eastwood 10 Plant 2 Eastwood 10 420420

Total Cost = $2,490Total Cost = $2,490


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Transportation Simplex MethodTransportation Simplex Method

The transportation simplex method requires that The transportation simplex method requires that the sum of the supplies at the origins equal the the sum of the supplies at the origins equal the sum of the demands at the destinations. sum of the demands at the destinations.

If the total supply is greater than the total If the total supply is greater than the total demand, a dummy destination is added with demand, a dummy destination is added with demand equal to the excess supply, and shipping demand equal to the excess supply, and shipping costs from all origins are zero. (If total supply is costs from all origins are zero. (If total supply is less than total demand, a dummy origin is added.)less than total demand, a dummy origin is added.)

When solving a transportation problem by its When solving a transportation problem by its special purpose algorithm, unacceptable shipping special purpose algorithm, unacceptable shipping routes are given a cost of +routes are given a cost of +MM (a large number). (a large number).

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An An assignment problemassignment problem seeks to minimize the seeks to minimize the total cost assignment of total cost assignment of mm workers to workers to mm jobs, jobs, given that the cost of worker given that the cost of worker ii performing job performing job jj is is ccijij. .

It assumes all workers are assigned and each job It assumes all workers are assigned and each job is performed. is performed.

An assignment problem is a special case of a An assignment problem is a special case of a transportation problemtransportation problem in which all supplies and in which all supplies and all demands are equal to 1; hence assignment all demands are equal to 1; hence assignment problems may be solved as linear programs.problems may be solved as linear programs.

The The network representationnetwork representation of an assignment of an assignment problem with three workers and three jobs is problem with three workers and three jobs is shown on the next slide.shown on the next slide.

15 15 Slide Slide



2222

3333

1111

2222

3333

1111cc1111

cc1212

cc1313

cc2121 cc2222

cc2323

cc3131 cc3232

cc3333

AGENTSAGENTS TASKSTASKS

16 16 Slide Slide


LP FormulationLP Formulation

Min Min ccijijxxijij i ji j

s.t. s.t. xxijij = 1 for each agent = 1 for each agent ii jj

xxijij = 1 for each task = 1 for each task jj ii

xxijij = 0 or 1 for all = 0 or 1 for all ii and and jj

•Note: A modification to the right-hand side of Note: A modification to the right-hand side of the first constraint set can be made if a the first constraint set can be made if a worker is permitted to work more than 1 job.worker is permitted to work more than 1 job.

17 17 Slide Slide

LP Formulation Special CasesLP Formulation Special Cases

•Number of agents exceeds the number of Number of agents exceeds the number of tasks:tasks:

xxijij << 1 for each agent 1 for each agent ii jj

•Number of tasks exceeds the number of Number of tasks exceeds the number of agents:agents:

Add enough dummy agents to Add enough dummy agents to equalize theequalize the

number of agents and the number of number of agents and the number of tasks.tasks.

The objective function coefficients for The objective function coefficients for thesethese

new variable would be zero.new variable would be zero.


18 18 Slide Slide


LP Formulation Special Cases (continued)LP Formulation Special Cases (continued)

•The assignment alternatives are evaluated in The assignment alternatives are evaluated in terms of revenue or profit:terms of revenue or profit:

Solve as a maximization problem.Solve as a maximization problem.

•An assignment is unacceptable:An assignment is unacceptable:

Remove the corresponding decision Remove the corresponding decision variable.variable.

•An agent is permitted to work An agent is permitted to work aa tasks: tasks:

xxijij << aa for each agent for each agent ii jj

19 19 Slide Slide

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

ProjectsProjects SubcontractorSubcontractor AA BB CC Westside 50 36 16Westside 50 36 16

Federated 28 30 18 Federated 28 30 18 Goliath 35 32 20Goliath 35 32 20

Universal 25 25 14Universal 25 25 14

How should the contractors be assigned to How should the contractors be assigned to minimize total costs?minimize total costs?

Example: Hungry OwnerExample: Hungry Owner

20 20 Slide Slide



5050

3636

1616

28283030

1818

3535 3232

20202525 2525

1414

WestWest..WestWest..

CCCC

BBBB

AAAA

Univ.Univ.Univ.Univ.

Gol.Gol.Gol.Gol.

Fed.Fed. Fed.Fed.

ProjectsSubcontractors

21 21 Slide Slide


Linear Programming FormulationLinear Programming Formulation

Min 50Min 50xx1111+36+36xx1212+16+16xx1313+28+28xx2121+30+30xx2222+18+18xx2323

+35+35xx3131+32+32xx3232+20+20xx3333+25+25xx4141+25+25xx4242+14+14xx4343

s.t. s.t. xx1111++xx1212++xx13 13 << 1 1

xx2121++xx2222++xx23 23 << 1 1

xx3131++xx3232++xx33 33 << 1 1

xx4141++xx4242++xx43 43 << 1 1

xx1111++xx2121++xx3131++xx41 41 = 1= 1

xx1212++xx2222++xx3232++xx42 42 = 1= 1

xx1313++xx2323++xx3333++xx43 43 = 1= 1

xxijij = 0 or 1 for all = 0 or 1 for all ii and and jj

Agents

Tasks

22 22 Slide Slide

Transshipment ProblemTransshipment Problem

Transshipment problemsTransshipment problems are transportation are transportation problems in which a shipment may move through problems in which a shipment may move through intermediate nodes (transshipment nodes)before intermediate nodes (transshipment nodes)before reaching a particular destination node.reaching a particular destination node.

Transshipment problems can be converted to Transshipment problems can be converted to larger transportation problems and solved by a larger transportation problems and solved by a special transportation program.special transportation program.

Transshipment problems can also be solved by Transshipment problems can also be solved by general purpose linear programming codes.general purpose linear programming codes.

The network representation for a transshipment The network representation for a transshipment problem with two sources, three intermediate problem with two sources, three intermediate nodes, and two destinations is shown on the next nodes, and two destinations is shown on the next slide.slide.

23 23 Slide Slide



22 22

3333

4444

5555

6666

77 77

11 11

cc1313

cc1414

cc2323

cc2424

cc2525

cc1515

ss11

cc3636

cc3737

cc4646

cc4747

cc5656

cc5757

dd11

dd22

INTERMEDIATEINTERMEDIATE NODESNODES

SOURCESSOURCES DESTINATIONSDESTINATIONS

ss22

DemanDemandd

SupplySupply

24 24 Slide Slide



xxijij represents the shipment from node represents the shipment from node ii to node to node jj

Min Min ccijijxxijij

i ji j

s.t. s.t. xxijij << ssii for each origin for each origin ii j j

xxikik - - xxkjkj = 0 for each = 0 for each intermediateintermediate

ii jj node node kk

xxijij = = ddjj for each destination for each destination jj ii

xxijij >> 0 for all 0 for all ii and and jj

25 25 Slide Slide

Example: TransshippingExample: Transshipping

Thomas Industries and Washburn Thomas Industries and Washburn Corporation supply three firms (Zrox, Hewes, Corporation supply three firms (Zrox, Hewes, Rockwright) with customized shelving for its Rockwright) with customized shelving for its offices. They both order shelving from the same offices. They both order shelving from the same two manufacturers, Arnold Manufacturers and two manufacturers, Arnold Manufacturers and Supershelf, Inc. Supershelf, Inc.

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

Additional data is shown on the next slide. Additional data is shown on the next slide.

26 26 Slide Slide


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

ThomasThomas WashburnWashburn Arnold 5 8Arnold 5 8 Supershelf 7 4Supershelf 7 4

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

ZroxZrox HewesHewes RockwrightRockwright Thomas 1 5 8Thomas 1 5 8

Washburn 3 4 4Washburn 3 4 4

27 27 Slide Slide



ARNOLD

WASHBURN

ZROX

HEWES

7575

7575

5050

6060

4040

55

88

77

44

1155

88

33

4444

ArnoldArnold

SuperSuperShelfShelf

HewesHewes

ZroxZrox

ThomasThomas

Wash-Wash-BurnBurn

Rock-Rock-WrightWright

28 28 Slide Slide



•Decision Variables DefinedDecision Variables Defined

xxijij = amount shipped from manufacturer = amount shipped from manufacturer ii to supplier to supplier jj

xxjkjk = amount shipped from supplier = amount shipped from supplier jj to customer to customer kk

where where ii = 1 (Arnold), 2 (Supershelf) = 1 (Arnold), 2 (Supershelf)

jj = 3 (Thomas), 4 (Washburn) = 3 (Thomas), 4 (Washburn)

kk = 5 (Zrox), 6 (Hewes), 7 (Rockwright) = 5 (Zrox), 6 (Hewes), 7 (Rockwright)

•Objective Function DefinedObjective Function Defined

Minimize Overall Shipping Costs: Minimize Overall Shipping Costs:

Min 5Min 5xx1313 + 8 + 8xx1414 + 7 + 7xx2323 + 4 + 4xx2424 + 1 + 1xx3535 + 5 + 5xx3636 + + 88xx3737

+ 3+ 3xx45 45 + 4+ 4xx4646 + 4 + 4xx4747

29 29 Slide Slide


Constraints DefinedConstraints Defined

Amount Out of Arnold: Amount Out of Arnold: xx1313 + + xx1414 << 75 75

Amount Out of Supershelf: Amount Out of Supershelf: xx2323 + + xx2424 << 75 75

Amount Through Thomas: Amount Through Thomas: xx1313 + + xx2323 - - xx3535 - - xx3636 - - xx3737 = 0= 0

Amount Through Washburn: Amount Through Washburn: xx1414 + + xx2424 - - xx4545 - - xx4646 - - xx4747 = 0= 0

Amount Into Zrox: Amount Into Zrox: xx3535 + + xx4545 = 50 = 50

Amount Into Hewes: Amount Into Hewes: xx3636 + + xx4646 = 60 = 60

Amount Into Rockwright: Amount Into Rockwright: xx3737 + + xx4747 = 40 = 40

Non-negativity of Variables: Non-negativity of Variables: xxijij >> 0, for all 0, for all ii and and jj..

30 30 Slide Slide


Optimal Solution (from Optimal Solution (from The Management The Management Scientist Scientist ))

Objective Function Value = Objective Function Value = 1150.0001150.000

VariableVariable ValueValue Reduced Reduced CostsCosts

X13 75.000 X13 75.000 0.0000.000

X14 0.000 X14 0.000 2.0002.000

X23 0.000 X23 0.000 4.0004.000

X24 75.000 X24 75.000 0.0000.000

X35 50.000 X35 50.000 0.0000.000

X36 25.000 X36 25.000 0.0000.000

X37 0.000 X37 0.000 3.0003.000

X45 0.000 X45 0.000 3.0003.000

X46 35.000 X46 35.000 0.0000.000

X47 40.000 X47 40.000 0.0000.000

31 31 Slide Slide


Optimal SolutionOptimal Solution

ARNOLD

WASHBURN

ZROX

HEWES

7575

7575

5050

6060

4040

55

88

77

44

1155

88

33 44

44

ArnoldArnold

SuperSuperShelfShelf

HewesHewes

ZroxZrox

ThomasThomas

Wash-Wash-BurnBurn

Rock-Rock-WrightWright

7575

7575

5050

2525

3535

4040

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