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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 Solution
7. Unbalanced Transportation Problems
8. Degeneracy in Transportation Problems
9. More Than One Optimal Solution
10. 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 Problem
Des 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 Cleveland
Des 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 column
2. 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 Project
1 2 3Adams 11 14 6
Brown 8 10 11
Cooper 9 12 7
Initial TableInitial Table
Hungarian MethodHungarian Method
Person Project
1 2 3Adams 5 8 0
Brown 0 2 3
Cooper 2 5 0
Row ReductionRow Reduction
Hungarian MethodHungarian Method
Person Project
1 2 3Adams 5 6 0
Brown 0 0 3
Cooper 2 3 0
Column ReductionColumn Reduction
Hungarian MethodHungarian Method
Person Project
1 2 3Adams 5 6 0
Brown 0 0 3
Cooper 2 3 0
TestingTesting Covering Line 2
Covering Line 1
Hungarian MethodHungarian Method
Person Project
1 2 3Adams 3 4 0
Brown 0 0 5
Cooper 0 1 0
Revised Opportunity Cost TableRevised Opportunity Cost Table
Hungarian MethodHungarian Method
Person Project
1 2 3Adams 3 4 0
Brown 0 0 5
Cooper 0 1 0
TestingTestingCovering
Line 1
Covering Line 2
Covering Line 3
Hungarian MethodHungarian Method
Person Project
1 2 3Adams 6
Brown 10
Cooper 9
AssignmentsAssignments