Linear Programming :Applications Pertemuan 6 Matakuliah: K0442-Metode Kuantitatif Tahun: 2009.
Transportation, Assignment, and Transshipment Problems Pertemuan 7 Matakuliah: K0442-Metode...
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Transcript of Transportation, Assignment, and Transshipment Problems Pertemuan 7 Matakuliah: K0442-Metode...
Transportation, Assignment,and Transshipment Problems
Pertemuan 7
Matakuliah : K0442-Metode KuantitatifTahun : 2009
Bina Nusantara University 3
Material Outline
• Transportation ProblemNetwork RepresentationGeneral LP Formulation
• Assignment ProblemNetwork RepresentationGeneral LP Formulation
• Transshipment ProblemNetwork RepresentationGeneral LP Formulation
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.
• Transportation, assignment, and transshipment problems of this chapter as well as the PERT/CPM problems (in another chapter) are all examples of network problems.
Transportation, Assignment, and Transshipment Problems
• Each of the three models of this chapter can be formulated as linear programs and solved by general purpose linear programming codes.
• For each of the three models, if the right-hand side of the linear programming formulations are all integers, the optimal solution will be in terms of integer values for the decision variables.
• However, there are many computer packages (including The Management Scientist) that contain separate computer codes for these models which take advantage of their network structure.
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.
Transportation Problem• Network Representation
22
cc1111
cc1212
cc1313
cc2121
cc2222
cc2323
dd11
dd22
dd33
ss11
s2
SourcesSources DestinationsDestinations
33
22
11
11
Transportation Problem
• LP FormulationThe LP formulation in terms of the
amounts shipped from the origins to the destinations, xij , can be written as:
Min cijxij i j
s.t. xij < si for each origin i j
xij = dj for each destination j
i
xij > 0 for all i and j
• LP Formulation Special CasesThe following special-case modifications to
the linear programming formulation can be made:– 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
Example: Acme Block Co.
Acme Block Company has orders for 80 tons of
concrete blocks at three suburban locations
as follows: Northwood -- 25 tons,
Westwood -- 45 tons, and
Eastwood -- 10 tons. Acme
has two plants, each of which
can produce 50 tons per week.
Delivery cost per ton from each plant
to each suburban location is shown on the next slide.
How should end of week shipments be made to fill
the above orders?
AcmAcm
eeAcmAcm
ee
Delivery Cost Per Ton
Northwood Westwood Eastwood
Plant 1 24 30 40
Plant 2 30 40 42
Example: Acme Block Co.
Optimal Solution
From To Amount Cost
Plant 1 Northwood 5 120
Plant 1 Westwood 45 1,350
Plant 2 Northwood 20 600
Plant 2 Eastwood 10 420
Total Cost = $2,490
Example: Acme Block Co.
Partial Sensitivity Report (first half)
Adjustable CellsFinal Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease$C$12 X11 5 0 24 4 4$D$12 X12 45 0 30 4 1E+30$E$12 X13 0 4 40 1E+30 4$F$12 X21 20 0 30 4 4$G$12 X22 0 4 40 1E+30 4$H$12 X23 10,000 0,000 42 4 1E+30
Adjustable CellsFinal Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease$C$12 X11 5 0 24 4 4$D$12 X12 45 0 30 4 1E+30$E$12 X13 0 4 40 1E+30 4$F$12 X21 20 0 30 4 4$G$12 X22 0 4 40 1E+30 4$H$12 X23 10,000 0,000 42 4 1E+30
Example: Acme Block Co.
Partial Sensitivity Report (second half)
ConstraintsFinal Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease$E$17 P2.Cap 30.0 0.0 50 1E+30 20$E$18 N.Dem 25.0 30.0 25 20 20$E$19 W.Dem 45.0 36.0 45 5 20$E$20 E.Dem 10.0 42.0 10 20 10$E$16 P1.Cap 50.0 -6.0 50 20 5
ConstraintsFinal Shadow Constraint Allowable Allowable
Cell Name Value Price R.H. Side Increase Decrease$E$17 P2.Cap 30.0 0.0 50 1E+30 20$E$18 N.Dem 25.0 30.0 25 20 20$E$19 W.Dem 45.0 36.0 45 5 20$E$20 E.Dem 10.0 42.0 10 20 10$E$16 P1.Cap 50.0 -6.0 50 20 5
Example: Acme Block Co.
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.
Assignment Problem
• Network Representation
2222
3333
1111
2222
3333
1111cc1111
cc1212
cc1313
cc2121cc2222
cc2323
cc3131 cc3232
cc3333
AgentsAgents TasksTasks
• LP Formulation
Min cijxij i j
s.t. xij = 1 for each agent i j
xij = 1 for each task j i xij = 0 or 1 for all i and j
– Note: A modification to the right-hand side of the first constraint set can be made if a worker is permitted to work more than 1 job.
Assignment Problem
LP Formulation Special Cases
• Number of agents exceeds the number of tasks:
xij < 1 for each agent i j
• 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
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 a tasks:
xij < a for each agent i j
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 14
How should the contractors be assigned to minimize total mileage costs?
Example: Who Does What?
Example: Who Does What?
Network Representation
5050
3636
1616
2828
3030
1818
3535 3232
2020
25252525
1414
WestWest..WestWest..
CCCC
BBBB
AAAA
Univ.Univ.Univ.Univ.
Gol.Gol.Gol.Gol.
Fed.Fed. Fed.Fed.
ProjectsSubcontractors
Example: Who Does What? 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
Example: Who Does What?
• The optimal assignment is:
Subcontractor Project Distance Westside C 16
Federated A 28Goliath (unassigned) Universal B 25
Total Distance = 69 miles
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.
Transshipment Problem
• Network Representation
22 22
3333
4444
5555
6666
77 77
11 11cc1313
cc1414
cc2323
cc2424
cc2525
cc1515
ss11
cc3636
cc3737
cc4646
cc4747
cc5656
cc5757
dd11
dd22
Intermediate NodesIntermediate NodesSourcesSources DestinationsDestinations
ss22
DemandDemandSupplySupply
Transshipment Problem
• Linear Programming Formulation xij represents the shipment from node i to node j
Min cijxij i j
s.t. xij < si for each origin i j
xik - xkj = 0 for each intermediate i j node k
xij = dj for each destination j i xij > 0 for all i and j
Example: Zeron Shelving
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.
Example: Zeron Shelving
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
Example: Zeron Shelving
• Network Representation
ARNOLD
WASHBURN
ZROX
HEWES
7575
7575
5050
6060
4040
55
88
77
44
1155
88
33
4444
ArnoldArnold
SuperSuperShelfShelf
HewesHewes
ZroxZrox
ZeronZeronNN
ZeronZeronSS
Rock-Rock-RiteRite
Example: Zeron Shelving• 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
Example: Zeron Shelving
• Constraints DefinedAmount Out of Arnold: x13 + x14 < 75
Amount Out of Supershelf: x23 + x24 < 75
Amount Through Zeron N: x13 + x23 - x35 - x36 - x37 = 0Amount Through Zeron S: x14 + x24 - x45 - x46 - x47 = 0Amount Into Zrox: x35 + x45 = 50
Amount Into Hewes: x36 + x46 = 60
Amount Into Rockrite: x37 + x47 = 40
Non-negativity of Variables: xij > 0, for all i and j.
Example: Zeron Shelving Optimal Solution (from The Management Scientist )
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
Example: Zeron ShelvingExample: Zeron Shelving Optimal Solution
ARNOLD
WASHBURN
ZROX
HEWES
7575
7575
5050
6060
4040
55
88
77
44
1155
88
33 44
44
ArnoldArnold
SuperSuperShelfShelf
HewesHewes
ZroxZrox
ZeronZeronNN
ZeronZeronSS
Rock-Rock-RiteRite
7575
7575
5050
2525
3535
4040
Example: Zeron Shelving
Optimal Solution (continued)
Constraint Slack/Surplus Dual Prices
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
Example: Zeron Shelving Optimal Solution (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
Example: Zeron Shelving
Optimal Solution (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