B.1 Transportation

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BA 452 Lesson B.1 Transportation
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Readings
Readings
Chapter 6
Distribution and Network Models
BA 452 Lesson B.1 Transportation
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Overview
Overview
BA 452 Lesson B.1 Transportation
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Overview
Network Models are nodes, arcs, and functions (costs, supplies, demands, etc.)
associated with the arcs and nodes, as in transportation, assignment,
transshipment, and shortest-route problems.
Transportation Problems are Resource Allocation Problems when outputs are
fixed, and when outputs and inputs occur at different locations, so goods must
be transported from origins to destinations.
Transportation Problems with Modes of Transport re-interpret some of the
different “origins” in a basic transportation problem to include not only location
but modes of transportation (truck, rail, …).
Assignment Problems are Transportation Problems when the “goods” are
workers that are transported to jobs, and each worker either does all of a job or
none of it, so the fraction completed is binary.
BA 452 Lesson B.1 Transportation
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Overview
Tool Summary
 Write the objective of maximizing a minimum as a linear
program.
• For example, maximize min {2x, 3y} as maximize M
subject to 2x > M and 3y > M.
 Define decision variable xij = units moving from origin i to
destination j.
 Write origin constraints (with < or =):
n
x
j 1

ij
 si
i 1, 2,
,m
Supply
Write destination constraints (with < or =):
m
x
i 1
ij
 dj
j 1, 2,
,n
Demand
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Overview
Tool Summary
 Identify implicit assumptions needed to complete a
formulation, such as all agents having an equal value of
time.
BA 452 Lesson B.1 Transportation
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Network Models
Network Models
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Network Models
Network Models are nodes, arcs, and functions (costs,
supplies, demands, etc.) associated with the arcs and
nodes. Transportation, assignment, transshipment, and
shortest-route problems are examples.
BA 452 Lesson B.1 Transportation
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Network Models



Each of the four network models (transportation,
assignment, transshipment, and shortest-route
problems) can be formulated as linear programs and
solved by general-purpose linear programming codes.
For each of the four models, if the right-hand side of the
linear programming formulations are all integers, the
optimal solution will be integer values for the decision
variables.
There are many computer packages (including The
Management Scientist) that contain convenient separate
computer codes for these models, which take advantage
of their network structure. But do not use such codes on
exams because they lack the flexibility of the generalpurpose linear programming codes.
BA 452 Lesson B.1 Transportation
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Transportation
Transportation
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Transportation
Overview
Transportation Problems are Resource Allocation Problems
when outputs are fixed, and when outputs and inputs occur
at different locations. Transportation Problems thus help
determine the transportation of goods from m origins (each
with a supply si) to n destinations (each with a demand dj)
to minimize cost.
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Transportation
Here is the network representation for a transportation
problem with two sources and three destinations.
1
d1
2
d2
3
d3
c11
s1
c12
1
c13
c21
s2
2
c23
c22
Sources
BA 452 Lesson B.1 Transportation
Destinations
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Transportation

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

Linear programming formulation (supply inequality, demand
equality).
Min
m
n
 c x
i 1 j 1
n
ij ij
x
ij
 si
i 1, 2,
,m
Supply
x
ij
 dj
=
j 1, 2,
,n
Demand
j 1
m
i 1
xij > 0 for all i and j
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Transportation
Possible variations:
• 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.
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Transportation
Question: Acme Block Company has orders for 80 tons of
concrete blocks at three suburban locations: Northwood -25 tons, Westwood -- 45 tons, and Eastwood -- 10 tons.
Acme has two plants, each of which can produce 50 tons
per week. Delivery costs per ton from each plant to each
suburban location are thus:
Plant 1
Plant 2
Northwood
24
30
Westwood
30
40
Eastwood
40
42
Formulate then solve the linear program that determines
how shipments should be made to fill the orders above.
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Transportation
Answer: Linear programming formulation (supply inequality,
demand equality).
 Variables: Xij = Tons shipped from Plant i to Destination j
 Objective:
Min 24 X11 + 30 X12 + 40 X13 + 30 X21 + 40 X22 + 42
X23
 Supply Constraints:
X11 + X12 + X13 < 50
X21 + X22 + X23 < 50
 Demand Constraints:
X11 + X21 = 25
X12 + X22 = 45
X13 + X23 = 10
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Transportation
BA 452 Lesson B.1 Transportation
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Transportation



Define sources: Source 1 = Plant 1, Source 2 = Plant 2.
Define destinations: 1 = Northwood, 2 = Westwood, 3 = Eastwood.
Define costs: c11 = 24
c12 = 30
c13 = 40
c21 = 30



c22 = 40
c23 = 42
2+3=5
2x3 = 6
Define 2 supplies: s1 = 50, s2 = 50.
Define 3 demands: d1 = 25, d2 = 45, d3 = 10.
Define variables: Xij = number of units
shipped from Source i to Destination j.
Cost c13 = 40
Supply s1 = 50
Demand d2 = 45
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Transportation
Variable names:
Xij = number of units shipped from Plant i to Destination j.
Destination 1 = Northwood; 2 = Westwood; 3 = Eastwood
Optimal shipments:
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
BA 452 Lesson B.1 Transportation
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Transportation
BA 452 Lesson B.1 Transportation
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Transportation
Cost from Plant
1 to Northwood
Variable names:
Origin i = Plant i
Destination 1 = Northwood
Destination 2 = Westwood
Destination 3 = Eastwood
Optimal shipments:
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
BA 452 Lesson B.1 Transportation
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Transportation with Modes of Transport
Transportation with Modes of
Transport
BA 452 Lesson B.1 Transportation
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Transportation with Modes of Transport
Overview
Transportation Problems with Modes of Transport reinterpret some of the different “origins” in a basic
transportation problem to include not only location but
modes of transportation. For example, instead of “San
Diego” as an origin specifying location, we have “San
Diego by Truck” as an origin specifying both location and
mode of transportation.
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Transportation with Modes of Transport
Question: The Navy has 9,000 pounds of material in Albany, Georgia
that it wishes to ship to three installations: San Diego, Norfolk, and
Pensacola. They require 4,000, 2,500, and 2,500 pounds, respectively.
Government regulations require equal distribution of shipping among
the three carriers.
 The shipping costs per pound by truck, railroad, and airplane are:
Mode
Truck
Railroad
Airplane

Destination
San Diego Norfolk Pensacola
$12
$6
$5
$20
$11
$9
$30
$26
$28
Formulate then solve the linear program that determines shipping
arrangements (mode, destination, and quantity) that minimize the
total shipping cost.
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Transportation with Modes of Transport
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Transportation with Modes of Transport


Define the variables. We want to determine the pounds of material,
xij , to be shipped by mode i to destination j.
Variable names: San Diego Norfolk Pensacola
Truck
Railroad
Airplane

x11
x21
x31
x12
x22
x32
x13
x23
x33
Define the objective. Minimize the total shipping cost.
Min: (shipping cost per pound for each mode-destination pairing) x
(number of pounds shipped by mode-destination pairing).
Min: 12x11 + 6x12 + 5x13 + 20x21 + 11x22 + 9x23
+ 30x31 + 26x32 + 28x33
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Transportation with Modes of Transport


Define the constraints of equal use of transportation modes:
(1) x11 + x12 + x13 = 3000
(2) x21 + x22 + x23 = 3000
(3) x31 + x32 + x33 = 3000
Define the destination material constraints:
(4) x11 + x21 + x31 = 4000
(5) x12 + x22 + x32 = 2500
(6) x13 + x23 + x33 = 2500
BA 452 Lesson B.1 Transportation
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Transportation with Modes of Transport
Linear programming summary.
 Variables: Xij = Pounds shipped by Mode i to Destination j
 Objective:
Min 12 X11 + 6 X12 + 5 X13
+ 20 X21 + 11 X22 + 9 X23 + 30 X31 + 26 X32 + 28 X33
 Mode (Supply equality) Constraints:
X11 + X12 + X13 = 3000
X21 + X22 + X23 = 3000
X31 + X32 + X33 = 3000
 Destination Constraints:
X11 + X21 + X31 = 4000
X12 + X22 + X32 = 2500
X13 + X23 + X33 = 2500
BA 452 Lesson B.1 Transportation
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Transportation with Modes of Transport
Variable names:
Truck
Railroad
Airplane
San Diego Norfolk Pensacola
X11
X12
X13
X21
X22
X23
X31
X32
X33
Units to San Diego by truck
Solution Summary:
• San Diego receives 1000 lbs. by truck
and 3000 lbs. by airplane.
• Norfolk receives 2000 lbs. by truck
and 500 lbs. by railroad.
• Pensacola receives 2500 lbs. by railroad.
• The total shipping cost is $142,000.
BA 452 Lesson B.1 Transportation
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Transportation with Modes of Transport
The Management Science Transportation module is not
available. Remember, in that formulation, the supply
constraints are inequalities.
Min
m
n
 c x
i 1 j 1
n
x
j 1
m
x
i 1
ij ij
ij
 si
i 1, 2,
,m
Supply
ij
 dj
=
j 1, 2,
,n
Demand
xij > 0 for all i and j
But in Example 2, the “origins” are the modes of shipping,
and the supply constraint on each mode is an equality.
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Assignment
Assignment
BA 452 Lesson B.1 Transportation
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Assignment
Overview
Assignment Problems are Transportation Problems when
the “goods” are workers that are transported to jobs, and
each worker either does all of a job or none of it.
Assignment Problems thus minimize the total cost of
assigning of m workers (or agents) to m jobs (or tasks). The
simplest way to model all-or-nothing in any linear program
is to restrict the fraction of the job completed to be a binary
(0 or 1) decision variable.
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Assignment
An assignment problem is thus a special case of a
transportation problem in which all supplies and all
demands equal to 1; hence assignment problems may
be solved as linear programs. And although the only
sensible solution quantities are binary (0 or 1), the
special form of the problem and of The Management
Scientist guarantees all solutions are binary (0 or 1).
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Assignment
Here is the network representation of an assignment problem
with three workers (agents) and three jobs (tasks):
1
Agents
c11
1
c12
c13
c21
2
Tasks
c22
2
c23
c31
3
c33
c32
3
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Assignment
Notation:
xij =
1 if agent i is assigned to task j
0 otherwise
cij = cost of assigning agent i to task j
Min
m
n
 c x
i 1 j 1
n
s.t.
x
j 1
m
x
i 1
ij ij
ij
1
i 1, 2,
,m
Agents
ij
1
j 1, 2,
,n
Tasks
xij > 0 for all i and j
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Assignment
Possible variations:
• Number of agents exceeds the number of tasks:
Extra agents simply remain unassigned.
• An assignment is unacceptable:
Remove the corresponding decision variable.
• An agent is permitted to work t
n
x
j 1
ij
t
i 1, 2,
,m
tasks:
Agents
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Assignment
Question: Russell electrical contractors pay their
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.
Subcontractor
Westside
Federated
Goliath
Universal
Projects
A
B C
50 36 16
28 30 18
35 32 20
25 25 14
Assume each subcontractor can perform at most one
project. Formulate then solve the linear program that
assigns contractors to minimize total mileage costs.
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Assignment
Answer:
West.
Subcontractors
50
36
16
28
Fed.
18
35
Gol.
Univ.
A
20
25
Projects
30
B
32
C
25
14
BA 452 Lesson B.1 Transportation
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Assignment
Variable names:
Westside
Federated
Goliath
Universal
Project A Project B Project C
x11
x12
x13
x21
x22
x23
x31
x32
x33
x41
x42
x43


There will be 1 variable
for each agent-task
pair, so 12 variables all
together.
There will be 1
constraint for each
agent and for each
task, so 7 constraints
all together.
BA 452 Lesson B.1 Transportation
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Assignment
Variable names:
Westside
Federated
Goliath
Universal
Project A Project B Project C
x11
x12
x13
x21
x22
x23
x31
x32
x33
x41
x42
x43
Min
50x11+36x12+16x13+28x21+30x22+18x23
+35x31+32x32+20x33+25x41+25x42+14x43
s.t.
x11+x12+x13 < 1
x21+x22+x23 < 1
Agents
x31+x32+x33 < 1
x41+x42+x43 < 1
x11+x21+x31+x41 = 1
x12+x22+x32+x42 = 1 Tasks
x13+x23+x33+x43 = 1
xij = 0 or 1 for all i and j
BA 452 Lesson B.1 Transportation
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Assignment
Agent 1 capacity: x11+x12+x13 < 1
Westside
Federated
Goliath
Universal
Project A Project B Project C
x11
x12
x13
x21
x22
x23
x31
x32
x33
x41
x42
x43
Task 3 done: x13+x23+x33+x43 = 1
BA 452 Lesson B.1 Transportation
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Assignment
Variable names:
Westside
x13
Federated
Goliath
Universal
Project A Project B Project C
x11
x12
x21
x31
x41
x22
x32
x42
x23
x33
x43
Optimal assignment:
Subcontractor Project Distance
Westside
C
16
Federated
A
28
Goliath
(unassigned)
Universal
B
25
Total distance = 69 miles
BA 452 Lesson B.1 Transportation
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Assignment
BA 452 Lesson B.1 Transportation
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Assignment
Projects
A B C
50 36
Subcontractor
Westside
16
Federated
28 30
18
Goliath
35 32
20
Universal
25 25
14
Optimal assignment:
Subcontractor Project Distance
Westside
C
16
Federated
A
28
Goliath
(unassigned)
Universal
B
25
Total distance = 69 miles
BA 452 Lesson B.1 Transportation
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Assignment
Question: Now change Example 3 to take into account the
recent marriage of the Goliath subcontractor to your
youngest daughter. That is, you have to assign Goliath
one of the jobs.
How should the contractors now be assigned to minimize
total mileage costs?
BA 452 Lesson B.1 Transportation
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Assignment
Alternative notation:
WA =
0 if Westside does not get task A
1 if Westside does get task A
and so on.
Min
s.t.
50WA+36WB+16WC+28FA+30FB+18FC
+35GA+32GB+20GC+25UA+25UB+14UC
WA+WB+WC < 1
FA+FB+FC
<1
Agents
GA+GB+GC = 1
UA+UB+UC < 1
WA+FA+GA+UA = 1
WB+FB+GB+UB = 1 Tasks
WC+FC+GC+UC = 1
BA 452 Lesson B.1 Transportation
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Assignment
Goliath gets a task: GA+GB+GC = 1
Task A gets done: WA+FA+GA+UA=1
BA 452 Lesson B.1 Transportation
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Assignment
Optimal assignment:
Subcontractor Project Distance
Westside
C
16
Federated
(unassigned)
Goliath
B
32
Universal
A
25
Total distance = 73 miles
BA 452 Lesson B.1 Transportation
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BA 452
Quantitative Analysis
End of Lesson B.1
BA 452 Lesson B.1 Transportation
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