B.1 Transportation

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


Readings

ReadingsChapter 6Distribution and Network Models


Overview

Overview


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.


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, , Demand

m

ij jix d j n

Overview

1 1,2, , Supply

n

ij ijx s i m


Tool Summary Identify implicit assumptions needed to complete a

formulation, such as all agents having an equal value of time.

Overview


Network Models

Network Models


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.


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


Transportation

Transportation


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.


2

c11c12

c13

c21

c22c23

d1

d2

d3

s1

s2

Sources Destinations

3

2

1

1

Transportation

Here is the network representation for a transportation problem with two sources and three destinations.


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, , Supply

n

ij ijx s i m

1 1Min

m n

ij iji j

c x

1 1,2, , Demand

m

ij jix d j n

=

Linear programming formulation (supply inequality, demand equality).

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.

Transportation


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


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 < 50X21 + X22 + X23 < 50 Demand Constraints:X11 + X21 = 25X12 + X22 = 45X13 + X23 = 10

Transportation


Transportation


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 = 40c21 = 30 c22 = 40 c23 =

42

Transportation


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


Transportation


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


Transportation with Modes of Transport



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.



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.

DestinationMode San Diego Norfolk PensacolaTruck $12 $ 6 $ 5Railroad $20 $11 $ 9Airplane $30 $26 $28





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



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



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 = 3000X21 + X22 + X23 = 3000X31 + X32 + X33 = 3000 Destination Constraints:X11 + X21 + X31 = 4000X12 + X22 + X32 = 2500X13 + X23 + X33 = 2500



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



The Management Science Transportation module is not available. Remember, in that formulation, the supply constraints are inequalities.


1 1Min

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, , Supply

n

ij ijx s i m

1 1,2, , Demand

m

ij jix d j n

=



Assignment

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.

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).

Assignment


2

3

1

2

3

1c11

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


Notation: xij = 1 if agent i is assigned to task j 0 otherwise

cij = cost of assigning agent i to task j


1 1Min

m n

ij iji j

c x

11 1,2, , Agents

n

ijjx i m

11 1,2, , Tasks

m

ijix j n

s.t.

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 tasks:

1 1,2, , Agents

n

ijjx t i m

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.

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


5036

16

2830

1835 32

2025 25

14

West.

C

B

A

Univ.

Gol.

Fed.

ProjectsSubcontractors

Assignment

Answer:


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


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



Goliath x31 x32 x33


Variable names:

Assignment




Goliath x31 x32 x33


Agent 1 capacity: x11+x12+x13 < 1

Task 3 done: x13+x23+x33+x43 = 1

Assignment


Project A Project B Project C

Westside x11 x12 x13


Goliath x31 x32 x33


Variable names:

Optimal assignment: Subcontractor Project Distance Westside C 16 Federated A 28

Goliath (unassigned) Universal B 25 Total distance = 69 miles

Assignment


Assignment


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


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


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


Goliath gets a task: GA+GB+GC = 1

Task A gets done: WA+FA+GA+UA=1

Assignment


Optimal assignment: Subcontractor Project Distance Westside C 16 Federated (unassigned)

Goliath B 32 Universal A 25 Total distance = 73 miles

Assignment


BA 452 Quantitative Analysis

End of Lesson B.1

Documents

B.1 Transportation