Quantitative Methods for Managerial Decision-Making ACN 309-5 Operations Management Transportation Models

Quantitative Methods for Managerial Decision-MakingACN 309-5

Operations Operations ManagementManagement

Transportation ModelsTransportation Models


OutlineOutline

TRANSPORTATION MODELING DEVELOPING AN INITIAL SOLUTION

The Northwest-Corner Rule The Intuitive Lowest-Cost Method

THE STEPPING-STONE METHOD SPECIAL ISSUES IN MODELING

Demand Not Equal to Supply Degeneracy


Learning ObjectivesLearning Objectives

After you read these notes, you should be able to

Identify or Define: Transportation modeling Facility location analysis

Explain or be able to use: Northwest-corner rule Stepping-stone method


Transportation ProblemTransportation Problem

DesMoines(100 unit capacity)

Fort Lauderdale(300 units capacity)

Cleveland(200 units required)

Evansville(300 units capacity)

Albuquerque(300 units required)

Boston(200 units required)


How much should be shipped from several sources to several destinations Sources: Factories, warehouses, etc. Destinations: Warehouses, stores, etc.

Transportation models Find lowest cost shipping arrangement Used primarily for existing distribution systems

Transportation ProblemTransportation Problem


A Transportation Model RequiresA Transportation Model Requires

The origin points, and the capacity or supply per period at each

The destination points and the demand per period at each

The cost of shipping one unit from each origin to each destination


Define problem Set up transportation table (matrix)

Summarizes all data Keeps track of computations

Develop initial solution Northwest corner rule

Find optimal solution Stepping stone method

Transportation ProblemTransportation ProblemSolution StepsSolution Steps


Transportation CostsTransportation Costs

FromTo

(Destination)(Sources) Albuquerque Boston Cleveland

Des Moines $5 $4 $3

Evansville $8 $4 $3

FortLauderdale

$9 $7 $5


Transportation TableTransportation Table

DestinationSource Supply

Demand

1

2

:

m

a1

a2

:

am

1 2 . . n

b 1 b 2 b n

Quantity demanded or required



DestinationSource 1 2 . . n Supply

1 x 11 c 11 x 12 c 12 . . x 1n c 1n a1

2 x 21 c 21 x 22 c 22 . . x 2n c 2n a2

: : : : : : : : : :

m x m1 c m1 x m2 c m2 . . x mn c mn am

Demand b 1 b 2 . . b n

Cost of supplying 1 unit from sources to destinations

Cost of supplying 1 unit from sources to destinations



DestinationSource Supply

Demand

1

2

:

m

a1

a2

:

am

1 2 . . n

b 1 b 2 b n

x11 x 12 . . x 1n

x 21 x 22 . . 2n

: : : : : : :

x m1 x m2 . . x mn

:

xQuantity supplied from sources to destinations



ToFrom

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity

Des Moines(D) 100

Evansville(E) 300

Fort Lauderdale(F) 300

WarehouseRequirements 300 200 200 700

5

8

9 7

4

4 3

3

5


Initial Solution Using the Initial Solution Using the Northwest Corner RuleNorthwest Corner Rule

ToFrom

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity

Des Moines(D) 100 100

Evansville(E) 200 100 300

Fort Lauderdale(F) 100 200 300


5

8

9 7

4

4 3

3

5


The Stepping Stone MethodThe Stepping Stone MethodSelect any unused square to evaluateBegin at this square. Trace a closed path back

to the original square via squares that are currently being used (only horizontal or vertical moves allowed)

Place + in unused square; alternate - and + on each corner square of the closed path

Calculate improvement index: add together the unit cost figures found in each square containing a +; subtract the unit cost figure in each square containing a -.

Repeat steps 1-4 for each unused square


Stepping-Stone Method: Tracing a Closed Path - Des Moines to Cleveland

ToFrom

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity



Fort Lauderdale(F) 100 200 300


5

8

9 7

4

4 3

3

5

Start

+

+

+ -

-

-


The Intuitive Lowest Cost MethodThe Intuitive Lowest Cost Method

Identify the cell with the lowest cost. Arbitrarily break any ties for the lowest cost.

Allocate as many units as possible to that cell without exceeding the supply or demand. Then cross out that row or column (or both) that is exhausted by this assignment.

Find the cell with the lowest cost from the remaining cells.

Repeat steps 2 & 3 until all units have been allocated.


ToFrom

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity

Des Moines(D)

100 100


Fort Lauderdale(F) 300 300

WarehouseRequirements

300 200 200 700

5

8

9 7

4

4 3

3

5

First, cross out top row

Second, cross out column C

Third, cross out row E

Initial Solution Using the Intuitive Lowest-Initial Solution Using the Intuitive Lowest-Cost MethodCost Method


Linear programming model is difficult to formulate & solve

Special purpose methods Are easier to formulate Are faster to compute Give integer solutions

Methods Stepping-stone MODI See your CD Tutorial © 1995 Corel Corp.

Specialized MethodsSpecialized Methods


Demand not equal to supply Called ‘unbalanced’ problem Add dummy source if demand > supply Add dummy destination if supply > demand

Degeneracy in Stepping Stone Method Too few shipping routes (cells) used

Number of occupied cells should be: m + n - 1 Create artificially occupied cell (0 value)

Represents fake shipment

Special Issues in theSpecial Issues in the Transportation Model Transportation Model


Transportation TableTransportation TableDemand Not Equal SupplyDemand Not Equal Supply

ToFrom

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity

Des Moines(D) 250

Evansville(E) 300

Fort Lauderdale(F) 300


5

8

9 7

4

4 3

3

5

0

0

0

Dummy

150

New Des Moines capacity


DegeneracyDegeneracy

ToFrom

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity





5

8

9 7

4

4 3

3

5


Degeneracy - ContinuedDegeneracy - Continued

ToFrom

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity




WarehouseRequirements

300 100 200 700

5

8

9 7

4

4 3

3

5

0

Documents

Quantitative Methods for Managerial Decision-Making ACN 309-5 Operations Management Transportation Models