Transportation and Transportation and Assignment ModelsAssignment Models

Chapter OutlineChapter Outline1. Introduction

2. Setting Up a Transportation Problem

3. Developing an Initial Solution:Northwest Corner Rule

4. Stepping-Stone Method: Finding a Least-Cost Solution

5. MODI Method

Chapter Outline - continuedChapter Outline - continued6. Vogel’s Approximation Method: Another

Way to Find an Initial Solution7. Unbalanced Transportation Problems8. Degeneracy in Transportation Problems9. More Than One Optimal Solution10. Facility Location Analysis

Learning ObjectivesLearning ObjectivesStudents will be able to

Structure special linear programming problems using the transportation and assignment models.

Use the northwest corner method and Vogel’s approximation method to find initial solutions to transportation problems.

Apply the stepping-stone and MODI methods to find optimal solutions to transportation problems.

Learning Objectives - continuedLearning Objectives - continuedSolve the facility location problem and other

application problems with the transportation model.

Solve assignment problems with the Hungarian (matrix reduction) method.

Specialized ProblemsSpecialized Problems Transportation Problem

Distribution of items from several sources to several destinations. Supply capacities and destination requirements known.

Assignment ProblemOne to one assignment of people to jobs, etc.

Specialized algorithms save time!Specialized algorithms save time!

Transportation ProblemTransportation ProblemDes Moines(100 units)capacity

Cleveland(200 units)required

Boston(200 units)required

Evansville(300 units)capacity

Ft. Lauderdale(300 units)capacity

Albuquerque(300 units)required

Transportation CostsTransportation Costs

From(Sources)

To(Destinations)

Albuquerque Boston ClevelandDes Moines

Evansville

Fort Lauderdale

$5

$8

$9

$4

$4

$7

$3

$3

$5

Unit Shipping Cost:1Unit, Unit Shipping Cost:1Unit, Factory to WarehouseFactory to Warehouse

Des Moines(D)

Evansville(E)

Ft Lauderdale(F)

WarehouseReq.

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity

5 4 3

3

57

48

9

Total Demand and Total SupplyTotal Demand and Total Supply

Des Moines(D)

Evansville(E)

Ft Lauderdale(F)

WarehouseReq.

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity

300 200 200 700

300

300

100

Transportation Table For Transportation Table For Executive Furniture Corp.Executive Furniture Corp.

Des Moines(D)

Evansville(E)

Ft Lauderdale(F)

WarehouseReq.

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity

300 200 200 700

300

300

1005 4 3

3

57

48

9

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

Start in the upper left-hand cell and allocate units to shipping routes as follows:Exhaust the supply (factory capacity) of each row

before moving down to the next row.Exhaust the demand (warehouse) requirements of

each column before moving to the next column to the right.

Check that all supply and demand requirements are met.

Initial SolutionInitial SolutionNorth West Corner RuleNorth West Corner Rule

Des Moines(D)

Evansville(E)

Ft Lauderdale(F)

WarehouseReq.

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity

300 200 200 700

300

300

1005 4 3

3

57

48

9

100

200 100

100 200

The Stepping-Stone MethodThe Stepping-Stone Method1. Select any unused square to evaluate.2. Begin 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).

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

4. 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 -.

5. Repeat steps 1 - 4 for each unused square.

Stepping-Stone Method - The Stepping-Stone Method - The Des Moines-to-Cleveland RouteDes Moines-to-Cleveland Route

Des Moines(D)

Evansville(E)

Ft Lauderdale(F)

WarehouseReq.

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity

300 200 200 700

300

300

1005 4 3

3

57

48

9

200

100

100

100 200

-- ++

--

++

++

--

Start

Stepping-Stone MethodStepping-Stone MethodAn Improved SolutionAn Improved Solution

Des Moines(D)

Evansville(E)

Ft Lauderdale(F)

WarehouseReq.

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity

300 200 200 700

300

300

1005 4 3

3

57

48

9

100

100

200

200100

Third and Final SolutionThird and Final Solution

Des Moines(D)

Evansville(E)

Ft Lauderdale(F)

WarehouseReq.

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity

300 200 200 700

300

300

1005 4 3

3

57

48

9

100

200

100200

100

MODI Method: 5 StepsMODI Method: 5 Steps1. Compute the values for each row and column: set Ri + Kj = Cij

for those squares currently used or occupied.2. After writing all equations, set R1 = 0.3. Solve the system of equations for Ri and Kj values.4. Compute the improvement index for each unused square by

the formula improvement index: Cij - Ri - Kj

5. Select the largest negative index and proceed to solve the problem as you did using the stepping-stone method.

Vogel’s ApproximationVogel’s Approximation For each row/column of table, find difference

between two lowest costs. (Opportunity cost) Find greatest opportunity cost. Assign as many units as possible to lowest cost

square in row/column with greatest opportunity cost.

Eliminate row or column which has been completely satisfied.

Begin again, omitting eliminated rows/columns.

Vogel’s ApproximationVogel’s Approximation

Des Moines(D)

Evansville(E)

Ft Lauderdale(F)

WarehouseReq.

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity

300 200 200 700

300

300

1005 4 3

3

57

48

9

11 22

11

44

22

33

200

Vogel’s ApproximationVogel’s Approximation

Des Moines(D)

Evansville(E)

Ft Lauderdale(F)

WarehouseReq.

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity

300 200 200 700

300

300

1005 4 3

3

57

48

9

11

11

44

22

33

200100

Vogel’s ApproximationVogel’s Approximation

Des Moines(D)

Evansville(E)

Ft Lauderdale(F)

WarehouseReq.

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity

300 200 200 700

300

300

1005 4 3

3

57

48

9

44

11

22

33

200100

100

Vogel’s ApproximationVogel’s Approximation

Des Moines(D)

Evansville(E)

Ft Lauderdale(F)

WarehouseReq.

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity

300 200 200 700

300

300

1005 4 3

3

57

48

9

200100

100

100200

Special Problems in Special Problems in Transportation MethodTransportation Method

Unbalanced ProblemDemand Less than SupplyDemand Greater than Supply

Degeneracy More Than One Optimal Solution

Unbalanced ProblemUnbalanced ProblemDemand Less than SupplyDemand Less than Supply

Factory 1

Factory 2

Factory 3

Customer Requirements

Customer 1 Customer 2 Dummy FactoryCapacity

150 80 150 380

80

130

1708 5 16

7

109

1015

3

Unbalanced ProblemUnbalanced ProblemSupply Less than DemandSupply Less than Demand

Factory 1

Factory 2

Dummy

Customer Requirements

Customer 1 Customer 2 Customer 3 FactoryCapacity

150 80 150 380

80

130

1708 5 16

7

109

1015

3

DegeneracyDegeneracy

Factory 1

Factory 2

Factory 3

Customer Requirements

Customer 1 Customer 2 Customer 3 FactoryCapacity

100 100 100 300

80

120

1005 4 3

3

57

48

9

100

100

80

20

Degeneracy - Coming Up!Degeneracy - Coming Up!

Factory 1

Factory 2

Factory 3

Customer Requirements

Customer 1 Customer 2 Customer 3 FactoryCapacity

150 80 50 280

80

130

708 5 16

7

109

1015

3

70

80

50

50

30

Stepping-Stone Method - The Stepping-Stone Method - The Des Moines-to-Cleveland RouteDes Moines-to-Cleveland Route

Des Moines(D)

Evansville(E)

Ft Lauderdale(F)

WarehouseReq.

Albuquerque(A)

Boston(B)

Cleveland(C)

FactoryCapacity

300 200 200 700

300

300

1005 4 3

3

57

48

9

Start

200

100

100

100 200

-- ++

--

++

++

--

The Assignment MethodThe Assignment Method1. subtract the smallest number in each row from every

number in that row subtract the smallest number in each

column from every number in that column2. draw the minimum number of vertical and horizontal

straight lines necessary to cover zeros in the table if the number of lines equals the number of

rows or columns, then one can make an optimal assignment (step 4)

The Assignment Method The Assignment Method continuedcontinued

3. if the number of lines does not equal the number of rows or columnssubtract the smallest number not covered by a line

from every other uncovered numberadd the same number to any number lying at the

intersection of any two linesreturn to step 2

4. make optimal assignments at locations of zeros within the table

PG 10.13b

Hungarian MethodHungarian Method

Person Project1 2 3

Adams 11 14 6

Brown 8 10 11

Cooper 9 12 7

Initial TableInitial Table

Hungarian MethodHungarian Method

Person Project1 2 3

Adams 5 8 0

Brown 0 2 3

Cooper 2 5 0

Row ReductionRow Reduction

Hungarian MethodHungarian Method

Person Project1 2 3

Adams 5 6 0

Brown 0 0 3

Cooper 2 3 0

Column ReductionColumn Reduction

Hungarian MethodHungarian Method

Person Project1 2 3

Adams 5 6 0

Brown 0 0 3

Cooper 2 3 0

TestingTesting Covering Line 2

Covering Line 1

Hungarian MethodHungarian Method

Person Project1 2 3

Adams 3 4 0

Brown 0 0 5

Cooper 0 1 0

Revised Opportunity Cost TableRevised Opportunity Cost Table

Hungarian MethodHungarian Method

Person Project1 2 3

Adams 3 4 0

Brown 0 0 5

Cooper 0 1 0

TestingTestingCovering

Line 1

Covering Line 2

Covering Line 3

Hungarian MethodHungarian Method

Person Project1 2 3

Adams 6

Brown 10

Cooper 9

AssignmentsAssignments