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Kerimcan Ozcan MNGT 379 Operations Research 1
Transportation, Assignment,and Transshipment Problems
Chapter 7
MNGT 379 Operations Research 2Kerimcan Ozcan
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.
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.
MNGT 379 Operations Research 3Kerimcan Ozcan
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 below.
22
cc11
11cc1212
cc1313
cc2121
cc2222cc2323
dd11
dd22
dd33
ss11
s2
SourcesSources DestinationsDestinations
33
22
11
11
MNGT 379 Operations Research 4Kerimcan Ozcan
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 js.t. xij < si for each origin i
j
xij = dj for each destination j i
xij > 0 for all i and j
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.
MNGT 379 Operations Research 5Kerimcan Ozcan
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 below. How should end of week shipments be made to fill the above orders?
Northwood Westwood EastwoodPlant 1 24 30 40Plant 2 30 40 42
MIN 24x11+30x12+40x13+30x21+40x22+42x23
S.T.
1) 1x11+1x12+1x13<50 2) 1x21+1x22+1x23<50 3) 1x11+1x21=25 4) 1x12+1x22=45 5) 1x13+1x23=10
MNGT 379 Operations Research 6Kerimcan Ozcan
Example: Acme Block Co.Objective Function Value = 2490.000
Variable Value Reduced Costs -------------- --------------- ------------------ x11 5.000 0.000 x12 45.000 0.000 x13 0.000 4.000 x21 20.000 0.000 x22 0.000 4.000 x23 10.000 0.000
Constraint Slack/Surplus Dual Prices -------------- --------------- ------------------ 1 0.000 6.000 2 20.000 0.000 3 0.000 -30.000 4 0.000 -36.000 5 0.000 -42.000
OBJECTIVE COEFFICIENT RANGES
Variable Lower Limit Current Value Upper Limit ------------ --------------- --------------- --------------- x11 20.000 24.000 28.000 x12 No Lower Limit 30.000 34.000 x13 36.000 40.000 No Upper Limit x21 26.000 30.000 34.000 x22 36.000 40.000 No Upper Limit x23 No Lower Limit 42.000 46.000
RIGHT HAND SIDE RANGES
Constraint Lower Limit Current Value Upper Limit ------------ --------------- --------------- --------------- 1 45.000 50.000 70.000 2 30.000 50.000 No Upper Limit 3 5.000 25.000 45.000 4 25.000 45.000 50.000 5 0.000 10.000 30.000
MNGT 379 Operations Research 7Kerimcan Ozcan
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 below.
2222
3333
1111
2222
3333
1111 cc1111cc1212
cc1313
cc2121 cc2222
cc2323
cc3131 cc3232
cc3333
AgentsAgents
TasksTasks
MNGT 379 Operations Research 8Kerimcan Ozcan
Assignment Problem 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.
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.
MNGT 379 Operations Research 9Kerimcan Ozcan
Assignment Problem
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
MNGT 379 Operations Research 10Kerimcan Ozcan
Example: Who Does What?
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?
MNGT 379 Operations Research 11Kerimcan Ozcan
Example: Who Does What?
Network 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
MNGT 379 Operations Research 12Kerimcan Ozcan
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
The optimal assignment is: Subcontractor Project Distance Westside C 16 Federated A 28
Goliath (unassigned) Universal B 25 Total Distance = 69 miles
MNGT 379 Operations Research 13Kerimcan Ozcan
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 below.
22 22
3333
4444
5555
6666
77 77
11 11cc1313
cc1414
cc2323
cc2424cc2525
cc1515
ss11
cc3636
cc3737
cc4646
cc4747
cc5656
cc5757
dd11
dd22
SourcesSources DestinationsDestinationsss22
DemanDemandd
SupplSupplyy
Intermediate NodesIntermediate Nodes
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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
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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.
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
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Example: Zeron Shelving
Network Representation
7575
7575
5050
6060
4040
55
88
77
44
1155
88
33
4444
ArnoldArnold
SuperSuperShelfShelf
HewesHewes
ZroxZrox
ZeronZeronNN
ZeronZeronSS
Rock-Rock-RiteRite
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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 DefinedMinimize Overall Shipping Costs:
Min 5x13 + 8x14 + 7x23 + 4x24 + 1x35 + 5x36 + 8x37 + 3x45 + 4x46 + 4x47
Constraints DefinedAmount Out of Arnold: x13 + x14 < 75Amount Out of Supershelf: x23 + x24 < 75Amount Through Zeron N: x13 + x23 - x35 - x36 - x37 = 0Amount Through Zeron S: x14 + x24 - x45 - x46 - x47 = 0Amount Into Zrox: x35 + x45 = 50Amount Into Hewes: x36 + x46 = 60Amount Into Rockrite: x37 + x47 = 40
Non-negativity of Variables: xij > 0, for all i and j.
MNGT 379 Operations Research 18Kerimcan Ozcan
Example: Zeron Shelving
Objective Function Value = 1150.000
Variable Value Reduced Costs X13 75.000 0.000X14 0.000 2.000X23 0.000 4.000X24 75.000 0.000X35 50.000 0.000X36 25.000 0.000X37 0.000 3.000X45 0.000 3.000X46 35.000 0.000X47 40.000 0.000
Constraint Slack/Surplus Dual Prices
1 0.000 0.0002 0.000 2.0003 0.000 -5.0004 0.000 -6.0005 0.000 -6.0006 0.000 -10.0007 0.000 -10.000
MNGT 379 Operations Research 19Kerimcan Ozcan
Example: Zeron Shelving
Optimal SolutionOptimal Solution
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
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