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 1 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

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Page 1: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 2: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

BA 452 Lesson B.1 Transportation 22

Readings

Readings

Chapter 6Distribution and Network Models

Page 3: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

BA 452 Lesson B.1 Transportation 33

Overview

Overview

Page 4: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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.

Page 5: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 6: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 7: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

BA 452 Lesson B.1 Transportation 77

Network Models

Network Models

Page 8: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

BA 452 Lesson B.1 Transportation 88

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.

Page 9: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 10: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

BA 452 Lesson B.1 Transportation 1010

Transportation

Transportation

Page 11: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

BA 452 Lesson B.1 Transportation 1111

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.

Page 12: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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.

Page 13: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 14: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 15: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 16: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 17: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

BA 452 Lesson B.1 Transportation 1717

Transportation

Page 18: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 19: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 20: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

BA 452 Lesson B.1 Transportation 2020

Transportation

Page 21: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 22: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

BA 452 Lesson B.1 Transportation 2222

Transportation with Modes of Transport

Transportation with Modes of Transport

Page 23: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 24: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 25: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

BA 452 Lesson B.1 Transportation 2525

Transportation with Modes of Transport

Page 26: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 27: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 28: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 29: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 30: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 31: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

BA 452 Lesson B.1 Transportation 3131

Assignment

Assignment

Page 32: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 33: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 34: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 35: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 36: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 37: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 38: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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:

Page 39: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

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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

Page 41: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 42: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 43: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

BA 452 Lesson B.1 Transportation 4343

Assignment

Page 44: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 45: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

Page 46: BA 452 Lesson B.1 Transportation 1 1Review We will spend up to 30 minutes reviewing Exam 1 Know how your answers were graded.Know how your answers were

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

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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

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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

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BA 452 Lesson B.1 Transportation 4949

BA 452 Quantitative Analysis

End of Lesson B.1