Transcript Transportation

Transportation
 Mode Selection
 Route Selection







Shortest Path
Minimum Spanning Tree
Transportation
Assignment
TSP
Route Sequencing
Tanker Scheduling
Mode Selection
More Variable
Ship
Rail
Truck
Plane...
Cheaper




Mode that minimizes Total
Cost
 Transportation Cost
 Inventory Costs
 Source
 Pipeline
 Destination - including safety stock
Costs
 Transportation Cost
 Cost per unit * Units
 Cost per unit
 $/CWT (based on origin, destination, freight,
weight)
 $/Time (leased, dedicated transportation)
Inventory
 At the plant
 1/2 “cycle quantity”
 At the warehouse
 1/2 “cycle quantity”
 Safety stock depends on lead time variability
 In the pipeline
 Annual Volume * Days in Transit
Days per year
Example (page 187)
Annual Volume
Unit Weight
Mode
Rail
Piggyback
Truck
Air
700,000
10 lbs
Rate
($/unit)
$
0.10
$
0.15
$
0.20
$
0.75
Time
21
14
5
2
Cost per Unit
$
Inventory Carrying
Std Dev. in LT
5
2
1
0.2
30.00
30%
Shipment
Size (units)
6,000
4,000
4,000
500
Transport
$
70,000
$ 105,000
$ 140,000
$ 525,000
Plant
Inventory
$
27,000
$
18,000
$
18,000
$
2,250
Warehouse
Inventory
$
27,000
$
18,000
$
18,000
$
2,250
$
$
$
$
Safety
Stock
172,603
69,041
34,521
6,904
$
$
$
$
Pipeline
362,466
241,644
86,301
34,521
$
$
$
$
Total
659,068
451,685
296,822
570,925
Multi-Modal Systems
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Ship from Japan to Long Beach
Rail from Long Beach to Terminals
Truck from Terminals to Dealerships
Where to change mode?
How many channels to operate?
Route Selection
 Getting From A to B
 Underlying Network
 Roads
 Airports
 Telecommunication links
 Costs of using each link
 Find the cheapest (shortest) path
Example (page 192)
B
A
90
138 66
C
348
84
I
84
132
120
90
F
156
D
E
60
132
48
G
126
H
126
48
J
150
Shortest Path Model
Distances
A
A
B
C
D
E
F
G
H
I
J
B
90
138
348
-
90
66
84
-
C
138
66
156
90
-
D
348
156
48
-
C
D
E
F
84
120
84
-
G
90
120
132
60
-
H
48
132
48
150
I
60
48
132
126
J
84
132
126
150
126
126
-
Route Chosen
A
A
B
C
D
E
F
G
H
I
J
Total To
Balance
Net
B
-
-
(1)
-
E
-
F
-
G
-
H
-
I
-
J
-
Total From
-
1
Costs Incurred
A
A
B
C
D
E
F
G
H
I
J
Total
B
-
C
-
D
-
E
-
F
-
G
-
H
-
I
-
J
-
Total
-
-
Limits
A
A
B
C
D
E
F
G
H
I
J
B
C
-
1
1
1
1
-
D
1
1
1
-
1
-
-
1
1
F
-
-
1
1
1
-
E
1
-
-
G
1
1
-
1
-
1
-
-
H
1
I
-
1
1
1
1
1
J
1
1
1
-
1
1
-
-
1
1
1
1
1
-
Applicability
 Single Origin
 Single Destination
 No requirement to visit intermediate
nodes
 No “negative cycles”
Tree of Shortest Paths
 Find shortest paths from Origin to each
node
 Send n-1 units from origin
 Get 1 unit to each destination
Shortest Path Problem
Distances
A
A
B
C
D
E
F
G
H
I
J
B
90
138
348
-
90
66
84
-
C
138
66
156
90
-
D
348
156
48
-
C
D
E
F
84
120
84
-
G
90
120
132
60
-
H
48
132
48
150
I
60
48
132
126
J
84
132
126
150
126
126
-
Route Chosen
A
A
B
C
D
E
F
G
H
I
J
Total To
Balance
Net
B
-
-
(1)
-
E
-
F
-
G
-
H
-
I
-
J
-
Total From
-
1
Costs Incurred
A
A
B
C
D
E
F
G
H
I
J
Total
B
-
C
-
D
-
E
-
F
-
G
-
H
-
I
-
J
-
Total
-
-
Limits
A
A
B
C
D
E
F
G
H
I
J
B
C
-
1
1
1
1
-
D
1
1
1
-
1
-
-
1
1
F
-
-
1
1
1
-
E
1
-
-
G
1
1
-
1
-
1
-
-
H
1
I
-
1
1
1
1
1
J
1
1
1
-
1
1
-
-
1
1
1
1
1
-
Minimum Spanning Tree
 Find the cheapest total cost of edges
required to tie all the nodes together
B
A
90
138 66
C
348
84
I
84
132
120
90
F
156
D
E
60
132
48
G
126
H
126
48
J
150
Greedy Algorithm
 Consider links from cheapest to most
expensive
 Add a link if it does not create a cycle with
already chosen links
 Reject the link if it creates a cycle.
What’s the difference
 Shortest Path Problem
 Rider’s version
 Consider the number of riders who will use it
 Spanning Tree Problem
 Builder’s version
 Consider only the cost of construction
Transportation Problem
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Sources with limited supply
Destinations with requirements
Cost proportional to volume
Multiple sourcing allowed
Example: Regal Company (Tools)
PROTRAC
Engine Distribution
Netherlands
500
Amsterdam
*
500
* Tilburg
800500
500
Antwer *
700
p
700
*
Belgium Liege
200
200
The Hague
*
800
*
900
900
Miles
0
50
Nancy
100
Germany
400
*
Leipzig
400
Transportation Costs
From Origin
Amsterdam
Antwerp
The Hague
Leipzig
120
61
102.5
To Destination
Nancy
Liege
130
41
40
100
90
122
Tilburg
62
110
42
Unit transportation costs from harbors to plants
Minimize
the transportation costs involved in moving
the engines from the harbors to the plants
A Transportation Model
PROTRAC Transportation Model
Unit Cost
From/To
Leipzig
Nancy
Liege
Tilburg
Amsterdam $
120.0 $
130.0 $
41.0 $
62.0
Antwerp
$
61.0 $
40.0 $
100.0 $
110.0
The Hague $
102.5 $
90.0 $
122.0 $
42.0
Shipments
From/To
Amsterdam
Antwerp
The Hague
Total
Required
Leipzig
400
Nancy
900
Liege
200
Tilburg Total
500
Total Cost
From/To
Amsterdam
Antwerp
The Hague
Total
Leipzig
-
Nancy
-
Liege
-
Tilburg
-
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
$
Total
$
$
$
$
-
-
Available
500
700
800
Crossdocking
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3 plants
2 distribution centers
2 customers
Minimize shipping costs
 Direct from plant to customer
 Via DC
A Network Model
Minimum Cost Network Flow Problem
Unit Shipping Costs
Transportation Costs ($ 000/Ton)
Plant to
DC
Plant 1 $
Plant 2 $
Plant 3 $
DC 1
5.0 $
1.0 $
1.0 $
DC 2
5.0
1.0
0.5
Plant to
Customer Customer 1
Plant 1 $
20.0
Plant 2 $
8.0
Plant 3 $
10.0
Customer 2
$
20.0
$
15.0
$
12.0
DC to
Customer Customer 1 Customer 2
DC 1 $
2.0 $
12.0
DC 2 $
2.0 $
12.0
Shipments
Plant to
DC
Plant 1
Plant 2
Plant 3
Total In
DC 1
200
200
Plant to
DC 2
Total Out
Customer Customer 1 Customer 2 Total
180
180
Plant 1
100
300
Plant 2
Plant 3
100
280
Total In
100
Out
100
DC to
Customer Customer 1 Customer 2 Total Out
DC 1
200
200
DC 2
200
80
280
Total In
400
80
Net Flows
Plant 1
Plant 2
Plant 3
Net Flow
Out
180
300
100
Supply
Net Flow In Demand
200 Customer 1
400
400 DC 1
300 Customer 2
180
180 DC 2
100
Net Flow
-
Arc Capacities
Transportation Capacities (Tons)
Plant to
Plant to
Customer Customer 1 Customer 2
DC
Plant 1
200
200
Plant 1
Plant 2
200
200
Plant 2
Plant 3
200
200
Plant 3
DC 1
200
200
200
DC 2
200
200
200
DC to
Customer Customer 1 Customer 2
DC 1
200
200
DC 2
200
200
Incurred Costs
Plant to
Customer
Plant 1
Plant 2
Plant 3
Total In
Customer 1 Customer 2
$
$
$
$
$
$ 1,200
$
$ 1,200
Plant to
Total Out
DC
$
Plant 1
$
Plant 2
$
1,200
Plant 3
$
1,200 Total In
$
$
$
$
DC 1
200
200
$
$
$
$
DC 2
900
100
1,000
Total Out
$
900
$
300
$
$ 1,200
Good News
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Lots of applications
Simple Model
Optimal Solutions Quickly
Integral Data, Integral Answers
Bad News
 What’s Missing?
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Single Homogenous Product
Linear Costs
No conversions or losses
...
Homogenous Product
Linear Costs
 No Fixed Charges
 No Volume Discounts
 No Economies of Scale
Integer Models
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Can model nearly everything
Sometimes very difficult to solve
Foundation in Linear Models
Expand from there
The Rules
 Exactly like Linear Models except
 Some decision variables restricted to
 Binary - 0 or 1, Yes or No, True or False
 Integers
Steco Warehouse Location
Steco's Warehouse Location Model
Unit Costs
Warehouse
A
B
C
Monthly
Lease
($)
$ 7,750
$ 4,000
$ 5,500
Unit Cost per Truck to Sales District
1
2
3
4
$
170 $
40 $
70 $
160
$
150 $
195 $
100 $
10
$
100 $
240 $
140 $
60
Monthly Capacity
(Trucks)
200
250
300
Monthly Trucks From/To
Decisions Yes/No
Lease Warehouse A
0
Lease Warehouse B
0
Lease Warehouse C
0
Total TrucksTo
Monthly Demand (Trucks)
Warehouse A
Warehouse B
Warehouse C
Totals
Lease
Cost
$
$
$
$
-
1
0
0
0
0
100
$
$
$
$
To 1
-
2
0
0
0
0
90
$
$
$
$
To 2
-
3
0
0
0
0
110
$
$
$
$
To 3
-
4
0
0
0
0
60
$
$
$
$
To 4
-
Total
Trucks Effective
From Capacity
0
0
0
0
0
0
Total
Truck
Cost
$
$
$
$
-
Total
$
$
$
$
Cost
-
A Linear Model
 Ignore leasing for now -- all
warehouses are open
 Objective: Minimize Total Cost
 Decision Variables:
 Number of trucks from each
warehouse to each customer each
month
 Constraints
 Enough trucks to each customer
 Not too many trucks from each
Making Discrete Decisions
 New Decision Variables
 Do we lease warehouse or not -- binary
 New Constraints
 Effective Capacity depends on whether or
not warehouse is open.
 Warehouse A effective capacity is
 0 if we do not lease the warehouse
 200 if we do.
 This is Linear!
An Integer Model
Steco's Warehouse Location Model
Unit Costs
Warehouse
A
B
C
Monthly
Lease
($)
$ 7,750
$ 4,000
$ 5,500
Unit Cost per Truck to Sales District
1
2
3
4
$
170 $
40 $
70 $
160
$
150 $
195 $
100 $
10
$
100 $
240 $
140 $
60
Monthly Capacity
(Trucks)
200
250
300
Monthly Trucks From/To
Decisions Yes/No
Lease Warehouse A
0
Lease Warehouse B
0
Lease Warehouse C
0
Total TrucksTo
Monthly Demand (Trucks)
Warehouse A
Warehouse B
Warehouse C
Totals
Lease
Cost
$
$
$
$
-
1
0
0
0
0
100
$
$
$
$
To 1
-
2
0
0
0
0
90
$
$
$
$
To 2
-
3
0
0
0
0
110
$
$
$
$
To 3
-
4
0
0
0
0
60
$
$
$
$
To 4
-
Total
Trucks Effective
From Capacity
0
0
0
0
0
0
Total
Truck
Cost
$
$
$
$
-
Total
Cost
$
$
$
$
-
Special Case
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Assignment Problem
Supply at each source 1
Requirement at each destination 1
Match up suppliers with destinations
How’s this different from single sourcing
Assigning workers to tasks
“Application”
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Truckload shipping
Loads waiting at customers
Trucks sitting at locations
Which truck should handle which load?
No concern for what to do after that.
What are “sources”
What are “destinations”
What are costs?
Traveling Salesman
Problem
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Vehicle at depot
Customers to be served (visited)
Vehicle must visit all and return to depot
Minimize travel cost
Example
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Full Service Business
Driver at Service Center
Assigned vending machines to visit
What order should he visit to minimize the
time to complete the work and get back
to the depot?
Extensions
 If the “customers” involve transportation
 Customers = truck load shipments
 If more than one “salesman” involved
 Construct routes for the 7 drivers at the North
Metro Service center
 If the vehicle has capacity
 LTL deliveries
 If we intersperse pickups and deliveries
 If there are time windows on service
Basic TSP
 Data issues
 Estimate distance by location
 Calculate point to point distances
 Calculate point to point costs
Heuristics
 Cluster first Route Second
 Build delivery zones with approximately equal
work.
 Route a vehicle in each zone
 Clustering Approaches
 Assign most distant blocks first
 Sweep
 Space-filling curve
Space-Filling Curve
 Each point (X,Y) on the map
 Express X = string of 0’s and 1’s
 X = 16.5 = 10000.10
1*24+0*23+0*22+0*21+0*20+1*2-1 +0*2-2
 Express Y = string of 0’s and 1’s
 Y = 9.75 = 01001.11
0*24+1*23+0*22+0*21+1*20+1*2-1 +1*2-2
 Space Filling Number - interleave bits
 (X,Y) = 1001000001.1101
Properties
 Every pair (X,Y) has a unique point (X,Y)
 Every point on the line corresponds to a
single point (X,Y)
 If (X,Y) and (X’, Y’) are close together,
(X,Y) and (X’,Y’) tend to be close together.
Clustering
 Compute (X,Y) for each customer
 Sort the customers by their  values
 To build N routes
 Give first 1/Nth of customers to first route
 Give second 1/Nth of customers to second
route
 ...
Routing
 Each route visits the customers in order of
their  values.
  defines a route on the plane that visits
every point. We visit the customers in the
same order as that route
Route First
 Build a single large route
 Assign each vehicle a segment of the
route
Routing Heuristics
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Space-Filling Curve
Clarke-Wright Savings
Nearest Neighbor
Nearest Insertion
Farthest Insertion
2-interchange
2-Interchange
2-Interchange
2-Interchange
2-Interchange
Route Sequencing
Route
1
2
3
4
5
6
7
8
9
10
Departure
8:00
9:30
14:00
11:31
8:12
15:03
12:24
13:33
8:00
10:56
Return
10:25
11:45
16:53
15:21
9:52
17:13
14:22
16:43
10:34
14:25
Tanker Scheduling
Trip
1
2
3
4
From
Port2
Port1
Port1
Port2
Departure
0
8
32
49
To
Refinery3
Refinery1
Refinery2
Refinery3
Arrival
12
29
51
61
From\To Port1
Refinery1
Refinery2
Refinery3
Port2
21
19
13
16
15
12
Consolidation
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Cross docking
Multi-stop shipments
Larger Orders
Delay shipments
Consolidate Production