1 1 slide chapter 7 transportation, assignment, and transshipment problems n transportation problem...
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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.
3 3 Slide Slide
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
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.
5 5 Slide Slide
Transportation ProblemTransportation Problem
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
Transportation ProblemTransportation Problem
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
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 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
Example: BBCExample: BBC
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
Example: BBCExample: BBC
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
Example: BBCExample: BBC
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
Example: BBCExample: BBC
13 13 Slide Slide
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).
14 14 Slide Slide
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 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
Assignment ProblemAssignment Problem
Network RepresentationNetwork Representation
2222
3333
1111
2222
3333
1111cc1111
cc1212
cc1313
cc2121 cc2222
cc2323
cc3131 cc3232
cc3333
AGENTSAGENTS TASKSTASKS
16 16 Slide Slide
Assignment ProblemAssignment Problem
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.
Assignment ProblemAssignment Problem
18 18 Slide Slide
Assignment ProblemAssignment Problem
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
Example: Hungry OwnerExample: Hungry Owner
Network RepresentationNetwork Representation
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
Example: Hungry OwnerExample: Hungry Owner
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
Transshipment ProblemTransshipment Problem
Network RepresentationNetwork Representation
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
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
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
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 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
Example: TransshippingExample: Transshipping
Network RepresentationNetwork Representation
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
Example: TransshippingExample: Transshipping
Linear Programming FormulationLinear Programming Formulation
•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
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 = 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
Example: TransshippingExample: Transshipping
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
Example: TransshippingExample: Transshipping
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