ba 452 lesson b.1 transportation 1 1review we will spend up to 30 minutes reviewing exam 1 know how...
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BA 452 Lesson B.1 Transportation 11
Review
We will spend up to 30 minutes reviewing Exam 1• Know how your answers were graded.• Know how to correct your mistakes. Your final exam is
cumulative, and may contain similar questions.
Review
BA 452 Lesson B.1 Transportation 22
Readings
Readings
Chapter 6Distribution and Network Models
BA 452 Lesson B.1 Transportation 33
Overview
Overview
BA 452 Lesson B.1 Transportation 44
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 55
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 =):
Write destination constraints (with < or =):
1
1,2, , Demandm
ij ji
x d j n
Overview
1
1,2, , Supplyn
ij ij
x s i m
BA 452 Lesson B.1 Transportation 66
Tool Summary Identify implicit assumptions needed to complete a
formulation, such as all agents having an equal value of time.
Overview
BA 452 Lesson B.1 Transportation 77
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 99
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 general-purpose linear programming codes.
Network Models
BA 452 Lesson B.1 Transportation 1010
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.
BA 452 Lesson B.1 Transportation 1212
22
c11
c12
c13
c21
c22c23
d1
d2
d3
s1
s2
Sources Destinations
33
22
11
11
Transportation
Here is the network representation for a transportation problem with two sources and three destinations.
BA 452 Lesson B.1 Transportation 1313
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
Notation:
xij > 0 for all i and j
1
1,2, , Supplyn
ij ij
x s i m
1 1
Min m n
ij iji j
c x
1
1,2, , Demandm
ij ji
x d j n
=
Linear programming formulation (supply inequality, demand equality).
Transportation
BA 452 Lesson B.1 Transportation 1414
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.
Transportation
BA 452 Lesson B.1 Transportation 1515
Northwood Westwood Eastwood
Plant 1 24 30 40
Plant 2 30 40 42
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:
Formulate then solve the linear program that determines how shipments should be made to fill the orders above.
Transportation
BA 452 Lesson B.1 Transportation 1616
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
Transportation
BA 452 Lesson B.1 Transportation 1717
Transportation
BA 452 Lesson B.1 Transportation 1818
Define sources: Source 1 = Plant 1, Source 2 = Plant 2. Define destinations: 1 = Northwood, 2 = Westwood, 3 = Eastwood. Define costs:
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.
2+3=5 2x3 = 6
Supply s1 = 50
Demand d2 = 45
Cost c13 = 40
c11 = 24 c12 = 30 c13 = 40
c21 = 30 c22 = 40 c23 = 42
Transportation
BA 452 Lesson B.1 Transportation 1919
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
Variable names:
Xij = number of units shipped from Plant i to Destination j.
Destination 1 = Northwood; 2 = Westwood; 3 = Eastwood
Transportation
BA 452 Lesson B.1 Transportation 2020
Transportation
BA 452 Lesson B.1 Transportation 2121
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
Variable names:
Origin i = Plant i
Destination 1 = Northwood
Destination 2 = Westwood
Destination 3 = Eastwood Cost from Plant 1 to Northwood
Transportation
BA 452 Lesson B.1 Transportation 2222
Transportation with Modes of Transport
Transportation with Modes of Transport
BA 452 Lesson B.1 Transportation 2323
Overview
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. 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.
Transportation with Modes of Transport
BA 452 Lesson B.1 Transportation 2424
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:
Formulate then solve the linear program that determines shipping arrangements (mode, destination, and quantity) that minimize the total shipping cost.
Destination
Mode San Diego Norfolk Pensacola
Truck $12 $ 6 $ 5
Railroad $20 $11 $ 9
Airplane $30 $26 $28
Transportation with Modes of Transport
BA 452 Lesson B.1 Transportation 2525
Transportation with Modes of Transport
BA 452 Lesson B.1 Transportation 2626
Define the variables. We want to determine the pounds of material, xij , to be shipped by mode i to destination j.
Variable names:
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
San Diego Norfolk Pensacola
Truck x11 x12 x13
Railroad x21 x22 x23
Airplane x31 x32 x33
Transportation with Modes of Transport
BA 452 Lesson B.1 Transportation 2727
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
Transportation with Modes of Transport
BA 452 Lesson B.1 Transportation 2828
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
Transportation with Modes of Transport
BA 452 Lesson B.1 Transportation 2929
San Diego Norfolk Pensacola
Truck X11 X12 X13
Railroad X21 X22 X23
Airplane X31 X32 X33
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.
Variable names:
Units to San Diego by truck
Transportation with Modes of Transport
BA 452 Lesson B.1 Transportation 3030
The Management Science Transportation module is not available. Remember, in that formulation, the supply constraints are inequalities.
xij > 0 for all i and j
1 1
Min m n
ij iji j
c x
But in Example 2, the “origins” are the modes of shipping, and the supply constraint on each mode is an equality.
1
1,2, , Supplyn
ij ij
x s i m
1
1,2, , Demandm
ij ji
x d j n
=
Transportation with Modes of Transport
BA 452 Lesson B.1 Transportation 3131
Assignment
Assignment
BA 452 Lesson B.1 Transportation 3232
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.
Assignment
BA 452 Lesson B.1 Transportation 3333
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).
Assignment
BA 452 Lesson B.1 Transportation 3434
22
33
11
22
33
11c11
c12
c13
c21 c22
c23
c31 c32
c33
Agents Tasks
Here is the network representation of an assignment problem with three workers (agents) and three jobs (tasks):
Assignment
BA 452 Lesson B.1 Transportation 3535
Notation:
xij = 1 if agent i is assigned to task j
0 otherwise
cij = cost of assigning agent i to task j
xij > 0 for all i and j
1 1
Min m n
ij iji j
c x
1
1 1,2, , Agentsn
ijj
x i m
1
1 1,2, , Tasksm
iji
x j n
s.t.
Assignment
BA 452 Lesson B.1 Transportation 3636
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 tasks:
1
1,2, , Agentsn
ijj
x t i m
Assignment
BA 452 Lesson B.1 Transportation 3737
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.
ProjectsSubcontractor A B C Westside 50 36 16
Federated 28 30 18 Goliath 35 32 20
Universal 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.
Assignment
BA 452 Lesson B.1 Transportation 3838
50
36
16
2830
18
35 32
2025 25
14
West.West.
CC
BB
AA
Univ.Univ.
Gol.Gol.
Fed. Fed.
ProjectsSubcontractors
Assignment
Answer:
BA 452 Lesson B.1 Transportation 3939
Project A Project B Project C
Westside x11 x12 x13
Federated x21 x22 x23
Goliath x31 x32 x33
Universal x41 x42 x43
Variable names:
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.
Assignment
BA 452 Lesson B.1 Transportation 4040
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
Project A Project B Project C
Westside x11 x12 x13
Federated x21 x22 x23
Goliath x31 x32 x33
Universal x41 x42 x43
Variable names:
Assignment
BA 452 Lesson B.1 Transportation 4141
Project A Project B Project C
Westside x11 x12 x13
Federated x21 x22 x23
Goliath x31 x32 x33
Universal x41 x42 x43
Agent 1 capacity: x11+x12+x13 < 1
Task 3 done: x13+x23+x33+x43 = 1
Assignment
BA 452 Lesson B.1 Transportation 4242
Project A Project B Project C
Westside x11 x12 x13
Federated x21 x22 x23
Goliath x31 x32 x33
Universal x41 x42 x43
Variable names:
Optimal assignment:
Subcontractor Project Distance
Westside C 16
Federated A 28
Goliath (unassigned)
Universal B 25
Total distance = 69 miles
Assignment
BA 452 Lesson B.1 Transportation 4343
Assignment
BA 452 Lesson B.1 Transportation 4444
ProjectsSubcontractor A B C Westside 50 36
16Federated 28 30
18 Goliath 35 32
20Universal 25 25
14Optimal assignment:
Subcontractor Project Distance
Westside C 16
Federated A 28
Goliath (unassigned)
Universal B 25
Total distance = 69 miles
Assignment
BA 452 Lesson B.1 Transportation 4545
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?
Assignment
BA 452 Lesson B.1 Transportation 4646
Alternative notation:
WA = 0 if Westside does not get task A 1 if Westside does get task Aand so on.
Min 50WA+36WB+16WC+28FA+30FB+18FC
+35GA+32GB+20GC+25UA+25UB+14UC
s.t. WA+WB+WC < 1
FA+FB+FC < 1 GA+GB+GC = 1
UA+UB+UC < 1 WA+FA+GA+UA = 1
WB+FB+GB+UB = 1 WC+FC+GC+UC = 1
Agents
Tasks
Assignment
BA 452 Lesson B.1 Transportation 4747
Goliath gets a task: GA+GB+GC = 1
Task A gets done: WA+FA+GA+UA=1
Assignment
BA 452 Lesson B.1 Transportation 4848
Optimal assignment:
Subcontractor Project Distance
Westside C 16
Federated (unassigned)
Goliath B 32
Universal A 25
Total distance = 73 miles
Assignment
BA 452 Lesson B.1 Transportation 4949
BA 452 Quantitative Analysis
End of Lesson B.1