Chapter 6: Transportation, Assignment, and Transshipment Problems
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Transcript of Chapter 6: Transportation, Assignment, and Transshipment Problems
1 Slide
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Chapter 6: 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.
Examples include transportation, assignment, transshipment as well as shortest-route, maximal flow problems, minimal spanning tree and PERT/CPM problems.
All network problems can be formulated as linear programs. However, there are many computer packages that contain separate computer codes for these problems which take advantage of their network structure.
If the right-hand side of the linear programming formulations are all integers, then optimal solution of the decision variables will also be integers.
2 Slide
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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.
3 Slide
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Transportation ProblemNetwork Representation
2
c1
1 c12
c13c21
c22c23
d1
d2
d3
s1
s2
m Sources n Destinations
3
2
1
1
4 Slide
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Transportation Problem
Linear Programming Formulation Using the notation: xij = number of units shipped from origin i to destination j cij = cost per unit of shipping from origin i to destination j si = supply or capacity in units at origin i dj = demand in units at destination j
continued
5 Slide
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Transportation Problem
Linear Programming Formulation (continued)
1 1Min
m n
ij iji j
c x
1 1,2, , Supply
n
ij ij
x s i m
1 1,2, , Demand
m
ij ji
x d j n
xij > 0 for all i and j
6 Slide
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Example: Transportation Problem
The Navy has depots in Albany, BenSalem, and Winchester. Each of these three depots has 3,000 pounds of materials which the Navy wishes to ship to three installations, namely, San Diego, Norfolk, and Pensacola. These installations require 4,000, 2,500, and 2,500 pounds, respectively. The shipping costs per pound for are shown on the next slide. Formulate and solve a linear program to determine the shipping arrangements that will minimize the total shipping cost.
7 Slide
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DestinationSource San Diego Norfolk
PensacolaAlbany $12 $ 6
$ 5BenSalem 20 11
9Winchester 30 26
28
Example: Transportation Problem (Continued)
8 Slide
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Transportation Problem: Network Representation
2
3
1
2
3
1 c11 c12
c13
c21 c22
c23
c31 c32
c33
Source
Destination
Albany3000
BenSalem3000
Winchester3000
San Diego4000
Norfolk2500
Pensacola2500
9 Slide
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Define the Decision VariablesWe want to determine the pounds of material, xij , to be shipped by mode i to destination j. The following table summarizes the decision variables:
San Diego Norfolk Pensacola
Albany x11 x12 x13
BenSalem x21 x22 x23
Winchester x31 x32 x33
Example: Transportation Problem (Continued)
10 Slide
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Define the Objective Function Minimize the total shipping cost. Min: (shipping cost per pound for each
mode per destination pairing) x (number of pounds shipped by mode per destination pairing).
Min: 12x11 + 6x12 + 5x13 + 20x21 + 11x22 + 9x23
+ 30x31 + 26x32 + 28x33
Example: Transportation Problem (Continued)
11 Slide
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Define the Constraints Source availability: (1) x11 + x12 + x13 = 3000 (2) x21 + x22 + x23 = 3000 (3) x31 + x32 + x33 = 3000 Destination material requirements: (4) x11 + x21 + x31 = 4000 (5) x12 + x22 + x32 = 2500 (6) x13 + x23 + x33 = 2500 Non-negativity of variables: xij > 0, i = 1, 2, 3 and j = 1, 2,
3
Transportation Problem: Example #2
12 Slide
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Computer Output
OBJECTIVE FUNCTION VALUE = 142000.000 Variable Value Reduced
Cost x11 1000.000
0.000 x12 2000.000
0.000 x13 0.000
1.000 x21 0.000
3.000 x22 500.000
0.000 x23 2500.000
0.000 x31 3000.000
0.000 x32 0.000
2.000 x33 0.000
6.000
Example: Transportation Problem (Continued)
13 Slide
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Solution Summary• San Diego will receive 1000 lbs. from
Albanyand 3000 lbs. from Winchester.
• Norfolk will receive 2000 lbs. from Albany and 500 lbs. from BenSalem.• Pensacola will receive 2500 lbs. from
BenSalem. • The total shipping cost will be $142,000.
Transportation Problem: Example #2
14 Slide
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LP Formulation Special Cases• Total supply exceeds total demand:
• Total demand exceeds total supply: Add a dummy origin with supply equal to the shortage amount. Assign a zero shipping cost per unit. The amount “shipped” from the dummy origin (in the solution) will not actually be shipped.
Assign a zero shipping cost per unit
• Maximum route capacity from i to j: xij < Li
Remove the corresponding decision variable.
Transportation Problem
No modification of LP formulation is necessary.
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LP Formulation Special Cases (continued)• The objective is maximizing profit or
revenue:
• 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
Solve as a maximization problem.
16 Slide
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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.
17 Slide
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Assignment Problem
Network Representation
2
3
1
2
3
1 c11c12
c13
c21 c22
c23
c31 c32
c33
Agents Tasks
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Linear Programming Formulation Using the notation: xij = 1 if agent i is assigned
to task j 0 otherwise cij = cost of assigning agent i to
task j
Assignment Problem
continued
19 Slide
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Linear Programming Formulation (continued)
Assignment Problem
1 1Min
m n
ij iji j
c x
11 1,2, , Agents
n
ijj
x i m
11 1,2, , Tasks
m
iji
x j n
xij > 0 for all i and j
20 Slide
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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 14How should the contractors be assigned so that totalmileage is minimized?
Example: Assignment Problem
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Network Representation50
361628
301835 32
2025 25
14
West.
C
B
A
Univ.
Gol.
Fed.
ProjectsSubcontractors
Example: Assignment Problem
22 Slide
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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
Assignment Problem: Example
23 Slide
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The optimal assignment is:
Subcontractor Project Distance Westside C 16
Federated A 28Goliath (unassigned) Universal B 25
Total Distance = 69 miles
Assignment Problem: Example
24 Slide
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LP Formulation Special Cases• Number of agents exceeds the number of
tasks:
• 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
Extra agents simply remain unassigned.
25 Slide
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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 t tasks:
1 1,2, , Agents
n
ijj
x t i m
26 Slide
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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.
27 Slide
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Transshipment Problem
Network Representation
2
3
4
5
6
7
1c13
c14
c23
c24c25
c15
s1
c36
c37
c46c47
c56
c57
d1
d2
Intermediate NodesSources Destinationss2
DemandSupply
28 Slide
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Transshipment Problem
Linear Programming Formulation
Using the notation: xij = number of units shipped from node i to node j
cij = cost per unit of shipping from node i to node j
si = supply at origin node i dj = demand at destination node j
continued
29 Slide
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Transshipment Problem
all arcsMin ij ijc x
arcs out arcs ins.t. Origin nodes ij ij ix x s i
xij > 0 for all i and j
arcs out arcs in0 Transhipment nodesij ijx x
arcs in arcs out Destination nodes ij ij jx x d j
Linear Programming Formulation (continued)
continued
30 Slide
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Transshipment Problem
LP Formulation Special Cases• Total supply not equal to total demand• Maximization objective function• Route capacities or route minimums• Unacceptable routesThe LP model modifications required here areidentical to those required for the special
cases inthe transportation problem.
31 Slide
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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.
Transshipment Problem Example
32 Slide
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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
Transshipment Problem Example
33 Slide
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Network Representation
ARNOLD
WASHBURN
ZROX
HEWES
75
75
50
60
40
5
8
7
4
15
8
34
4
Arnold
SuperShelf
Hewes
Zrox
ZeronN
ZeronS
Rock-Rite
Transshipment Problem Example
34 Slide
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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
Transshipment Problem: Example
35 Slide
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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.
Transshipment Problem: Example
36 Slide
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Computer Output 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
Transshipment Problem: Example
37 Slide
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Solution
ARNOLD
WASHBURN
ZROX
HEWES
75
75
50
60
40
5
8
7
4
15
8
3 4
4
Arnold
SuperShelf
Hewes
Zrox
ZeronN
ZeronS
Rock-Rite
75
75
50
25
35
40
Transshipment Problem: Example
38 Slide
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Computer Output (continued) Constraint Slack/Surplus Dual
Values 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
Transshipment Problem: Example
39 Slide
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Computer Output (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
Transshipment Problem: Example
40 Slide
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Computer Output (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
Transshipment Problem: Example