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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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Transportation ProblemTransportation Problem
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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Transportation ProblemTransportation Problem
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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Transportation ProblemTransportation Problem
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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Example: BBC-1Example: BBC-1
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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Transportation ProblemTransportation Problem
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
SOURCESSOURCES DESTINATIONSDESTINATIONS
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Example: BBC-1Example: BBC-1
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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Example: BBC-1Example: BBC-1
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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Assignment ProblemAssignment Problem
Network RepresentationNetwork Representation
2222
3333
1111
2222
3333
1111cc1111
cc1212
cc1313
cc2121cc2222
cc2323
cc3131
cc3232
cc3333
WORKERSWORKERS JOBSJOBS
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Assignment ProblemAssignment Problem
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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Example: AssignmentExample: Assignment
Network RepresentationNetwork Representation
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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Example: AssignmentExample: Assignment
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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Example: AssignmentExample: Assignment
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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Example: AssignmentExample: Assignment
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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Example: AssignmentExample: Assignment
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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Transshipment ProblemTransshipment Problem
Network RepresentationNetwork Representation
2222
3333
4444
5555
6666
7777
1111
cc1313
cc1414
cc2323
cc2424
cc2525
cc1515
ss11
cc3636
cc3737
cc4646
cc4747
cc5656
cc5757
dd11
dd22
INTERMEDIATEINTERMEDIATE NODESNODES
SOURCESSOURCES DESTINATIONSDESTINATIONS
ss22
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Transshipment ProblemTransshipment Problem
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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Example: TransshippingExample: Transshipping
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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Example: TransshippingExample: Transshipping
Network RepresentationNetwork Representation
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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Example: TransshippingExample: Transshipping
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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Example: TransshippingExample: Transshipping
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
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