transportation, assignment, and transshipment

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Transportation, Transportation, Assignment, and Assignment, and TransshipmentTransshipment

Professor Ahmadi

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

Transshipment ProblemsTransshipment Problems

The Transportation Problem: The Network The Transportation Problem: The Network Model and a Linear Programming FormulationModel and a Linear Programming Formulation

The Assignment Problem: The Network Model The Assignment Problem: The Network Model and a Linear Programming Formulationand a Linear Programming Formulation

The Transshipment Problem: The Network The Transshipment Problem: The Network Model and a Linear Programming FormulationModel and a Linear Programming Formulation

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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.

Transportation, assignment, and Transportation, assignment, and transshipment problems are all examples of transshipment problems are all examples of network problems.network problems.

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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 (transportation, assignment, and transshipment models) can be formulated as transshipment models) can be formulated as linear programs. linear programs.

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

These three models can also be solved using a These three models can also be solved using a standard computer spreadsheet package.standard computer spreadsheet package.

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Transportation ProblemTransportation Problem

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.

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Network RepresentationNetwork Representation

11

22

33

11

22

cc1111

cc1212

cc1313

cc2121 cc2222

cc2323

dd11

dd22

dd33

ss11

s2

SOURCESSOURCES DESTINATIONSDESTINATIONS

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LP FormulationLP FormulationThe linear programming formulation in terms The linear programming formulation in terms

of the amounts shipped from the origins to the of the amounts shipped from the origins to the destinations, destinations, xxijij, 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 destination for each destination jj ii

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

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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 the linear programming formulation can be made:made:• Minimum shipping guarantees from Minimum shipping guarantees from ii to to jj::

xxijij >> LLijij

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

xxijij << LLijij

• Unacceptable routes:Unacceptable routes:

delete the variabledelete the variable

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Example: BBC-1Example: BBC-1

Building Brick Company (BBC) has orders for 80 tons Building Brick Company (BBC) has orders for 80 tons of bricks at three suburban locations as follows: of bricks at three suburban locations as follows: Northwood -- 25 tons, Westwood -- 45 tons, and Northwood -- 25 tons, Westwood -- 45 tons, and Eastwood -- 10 tons. Eastwood -- 10 tons. BBC has two plants. Plant 1 BBC has two plants. Plant 1 produces 50 and plant 2 produces 30 tons per week.produces 50 and plant 2 produces 30 tons per week.

How should end of week shipments be made to fill How should end of week shipments be made to fill the above orders given the following delivery cost the above orders given the following delivery cost per ton:per ton:

NorthwoodNorthwood WestwoodWestwood EastwoodEastwood

Plant 1 24 Plant 1 24 30 30 40 40

Plant 2 Plant 2 30 30 40 40 42 42

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LP FormulationLP Formulation• Decision Variables Defined Decision Variables Defined

xxijij = amount shipped from plant = amount shipped from plant ii to suburb to suburb jj

where where ii = 1 (Plant 1) and 2 (Plant 2) = 1 (Plant 1) and 2 (Plant 2)

jj = 1 (Northwood), 2 (Westwood), = 1 (Northwood), 2 (Westwood),

and 3 (Eastwood)and 3 (Eastwood)

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Network Representation of BBC-1Network Representation of BBC-1

Northwood1

Northwood1

Westwood2

Westwood2

Eastwood3

Eastwood3

Plant1

Plant1

Plant2

Plant2

2424

3030

4040

3030

4040

4242

2525

4545

1010

5050

30


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LP FormulationLP Formulation• Objective FunctionObjective Function

Minimize total shipping cost per week:Minimize total shipping cost per week:

Min 24Min 24xx1111 + 30 + 30xx1212 + 40 + 40xx1313 + 30 + 30xx2121 + 40 + 40xx2222 + + 4242xx2323

• ConstraintsConstraints

s.t. s.t. xx1111 + + xx1212 + + xx1313 << 50 (Plant 1 capacity) 50 (Plant 1 capacity)

xx2121 + + xx2222 + + xx23 23 << 30 (Plant 2 capacity) 30 (Plant 2 capacity)

xx1111 + + xx2121 = 25 (Northwood demand) = 25 (Northwood demand)

xx1212 + + xx2222 = 45 (Westwood demand) = 45 (Westwood demand)

xx1313 + + xx2323 = 10 (Eastwood demand) = 10 (Eastwood demand)

all all xxijij >> 0 (Non-negativity) 0 (Non-negativity)

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Optimal SolutionOptimal Solution

FromFrom ToTo AmountAmount CostCost

Plant 1 Northwood 5 120Plant 1 Northwood 5 120

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

Plant 2 Northwood 20 600Plant 2 Northwood 20 600

Plant 2 Eastwood 10 Plant 2 Eastwood 10 420420

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

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Assignment ProblemAssignment Problem

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 problem in which all supplies and transportation problem 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 network representation of an assignment The network representation 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.

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2222

3333

1111

2222

3333

1111cc1111

cc1212

cc1313

cc2121cc2222

cc2323

cc3131

cc3232

cc3333

WORKERSWORKERS JOBSJOBS

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Linear Programming FormulationLinear Programming Formulation

Min Min ccijijxxijij

i ji j

s.t. s.t. xxijij = 1 for each worker = 1 for each worker ii

jj

xxijij = 1 for each job = 1 for each job jj ii xxijij = 0 or 1 for all = 0 or 1 for all ii and and jj..

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

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Example: AssignmentExample: Assignment

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

ProjectProject AA BB CC

Westside 50 36 16Westside 50 36 16 Subcontractors Subcontractors 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 distance (and total cost)?minimize total distance (and total cost)?

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5050

3636

1616

28283030

1818

3535 3232

2020

2525 2525

1414

West.West.West.West.

CCCC

BBBB

AAAA

Univ.Univ.Univ.Univ.

Gol.Gol.Gol.Gol.

Fed.Fed.Fed.Fed.

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LP FormulationLP Formulation• Decision Variables DefinedDecision Variables Defined

xxij ij = 1 if subcontractor = 1 if subcontractor ii is assigned to is assigned to project project jj

= 0 otherwise= 0 otherwise

where: where: ii = 1 (Westside), 2 (Federated), = 1 (Westside), 2 (Federated),

3 (Goliath), and 4 3 (Goliath), and 4 (Universal)(Universal)

jj = 1 (A), 2 (B), and 3 (C) = 1 (A), 2 (B), and 3 (C)

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LP FormulationLP Formulation• Objective FunctionObjective Function

Minimize total distance:Minimize total distance:

Min 50Min 50xx1111 + 36 + 36xx1212 + 16 + 16xx1313 + 28 + 28xx2121 + 30 + 30xx2222 + + 1818xx2323

+ 35+ 35xx3131 + 32 + 32xx3232 + 20 + 20xx3333 + 25 + 25xx4141 + + 2525xx4242 + 14 + 14xx4343

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LP FormulationLP Formulation• ConstraintsConstraints

xx1111 + + xx1212 + + xx1313 << 1 (no more than one 1 (no more than one

xx2121 + + xx2222 + + xx2323 << 1 project assigned 1 project assigned

xx3131 + + xx3232 + + xx3333 << 1 to any one 1 to any one

xx4141 + + xx4242 + + xx4343 << 1 1 subcontractor) subcontractor)

xx1111 + + xx2121 + + xx3131 + + xx41 41 = 1 (each project must = 1 (each project must

xx1212 + + xx22 22 + + xx3232 + + xx4242 = 1 be assigned to just = 1 be assigned to just

xx1313 + + xx2323 + + xx3333 + + xx43 43 = 1 one subcontractor)= 1 one subcontractor)

all all xxijij >> 0 (non-negativity) 0 (non-negativity)

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Optimal AssignmentOptimal Assignment

SubcontractorSubcontractor ProjectProject DistanceDistance

Westside C 16Westside C 16

Federated A 28Federated A 28

Universal B 25Universal B 25

Goliath (unassigned) Goliath (unassigned)

Total Distance = 69 miles Total Distance = 69 miles

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Variations of Assignment ProblemVariations of Assignment Problem

Total number of agents not equal to total Total number of agents not equal to total number of tasksnumber of tasks

Maximization objective functionMaximization objective function Unacceptable assignmentsUnacceptable assignments

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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 as Transshipment problems can also be solved as linear programs.linear programs.

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.

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2222

3333

4444

5555

6666

7777

1111

cc1313

cc1414

cc2323

cc2424

cc2525

cc1515

ss11

cc3636

cc3737

cc4646

cc4747

cc5656

cc5757

dd11

dd22

INTERMEDIATEINTERMEDIATE NODESNODES


ss22

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Linear Programming FormulationLinear Programming Formulation

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

Min Min ccijijxxijij all arcsall arcs

s.t. s.t. xxijij - - xxijij << ssii for each origin for each origin node node ii

arcs outarcs out arcs inarcs in

xxijij - - xxijij = 0 for each = 0 for each intermediateintermediate

arcs outarcs out arcs inarcs in node node

xxijij - - xxijij = - = -ddjj for each for each destination destination arcs outarcs out arcs inarcs in node node j j (Note the (Note the order)order)

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

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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.

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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 cost to install the shelving at the The cost 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

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ARNOLD

WASHBURN

ZROX

HEWES

7575

7575

5050

6060

4040

55

88

77

44

1155

88

33

44

44

ArnoldArnold11

SupershelfSupershelf22

HewesHewes66

ZroxZrox55

ThomasThomas33

WashburnWashburn44

RockwrightRockwright77

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LP FormulationLP Formulation• Decision Variables Defined Decision 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

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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 = = 00

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 + + xx47 47 = 40 = 40

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

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Variations of Transshipment ProblemVariations of Transshipment Problem

Total supply not equal to total demandTotal supply not equal to total demand Maximization objective functionMaximization objective function Route capacities or route minimumsRoute capacities or route minimums Unacceptable routesUnacceptable routes

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The End of ChapterThe End of Chapter

transportation, assignment, and transshipment

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