Embed Size (px)
Citation preview
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-1 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Chapter 10Chapter 10
Transportation and Transportation and Assignment ModelsAssignment Models
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-2 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Learning ObjectivesLearning Objectives
Students will be able to
• Structure special LP problems using the transportation and assignment models.
• Use the N.W. corner, VAM, MODI, and stepping-stone method.
• Solve facility location and other application problems with transportation methods.
• Solve assignment problems with the Hungarian (matrix reduction) method
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-3 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Chapter OutlineChapter Outline
10.1 Introduction
10.2 Setting Up a Transportation Problem
10.2 Developing an Initial
Solution:Northwest Corner Rule
10.4 Stepping-Stone Method: Finding a
Least-Cost Solution
10.5 MODI Method
10.6 Vogel’s Approximation Method
10.7 Unbalanced Transportation Problems
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-4 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Chapter Outline - Chapter Outline - continuedcontinued
10.8 Degeneracy in Transportation Problems
10.9 More Than One Optimal Solution
10.10 Maximization Transportation Problems
10.11 Unacceptable or Prohibited Routes
10.12 Facility Location Analysis
10.13 Approach of the Assignment Model
10.14 Unbalanced Assignment Models
10.15 Maximization Assignment Problems
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-5 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Specialized ProblemsSpecialized Problems
• Transportation Problem• Distribution of items from several
sources to several destinations. Supply capacities and destination requirements known.
• Assignment Problem• One-to-one assignment of people
to jobs, etc.
Specialized algorithms save time!
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-6 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Importance of Special Importance of Special Purpose AlgorithmsPurpose Algorithms
• Fewer, less complicated,
computations than with simplex
• Less computer memory required
• Produce integer solutions
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-7 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Transportation Transportation ProblemProblem
Des Moines(100 units)
capacity
Cleveland(200 units)
required
Boston(200 units)
requiredEvansville(300 units)
capacity
Ft. Lauderdale(300 units)
capacity
Albuquerque(300 units)
required
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-8 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Transportation CostsTransportation Costs
From(Sources)
To(Destinations)
AlbuquerqueBoston
Cleveland
Des Moines
Evansville
Fort Lauderdale
$5
$8
$9
$4
$4
$7
$3
$3
$5
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-9 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Unit Shipping Cost:1Unit, Unit Shipping Cost:1Unit, Factory to WarehouseFactory to Warehouse
Des Moines(D)
Evansville(E)
Fort Lauderdale
(F)
WarehouseReq.
Albuquerque(A) Boston
(B)
Cleveland(C) Factory
Capacity
5 4 3
57
48
9
3
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-10 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Total Demand and Total Total Demand and Total SupplySupply
Des Moines(D)
Evansville(E)
Fort Lauderdale
(F)
WarehouseReq.
Albuquerque(A) Boston
(B)
Cleveland(C) Factory
Capacity
300 200 200 700
300
300
100
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-11 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Transportation Table For Transportation Table For Executive Furniture Corp.Executive Furniture Corp.
Des Moines(D)
Evansville(E)
Fort Lauderdale
(F)
WarehouseReq.
Albuquerque(A) Boston
(B)
Cleveland(C) Factory
Capacity
300 200 200 700
300
300
1005 4 3
3
57
48
9
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-12 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
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.
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-13 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Initial SolutionInitial SolutionNorth West Corner RuleNorth West Corner Rule
Des Moines(D)
Evansville(E)
Fort Lauderdale
(F)
WarehouseReq.
Albuquerque(A) Boston
(B)
Cleveland(C) Factory
Capacity
300 200 200 700
300
300
1005 4 3
3
57
48
9
100
200 100
100 200
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-14 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
The Stepping-Stone The Stepping-Stone MethodMethod
• 1. 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.
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-15 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Stepping-Stone Method - Stepping-Stone Method - The Des Moines-to-The Des Moines-to-
Cleveland RouteCleveland Route
Des Moines(D)
Evansville(E)
Fort Lauderdale
(F)
WarehouseReq.
Albuquerque(A) Boston
(B)
Cleveland(C) Factory
Capacity
300 200 200 700
300
300
1005 4 3
3
57
48
9
200
100
100
100 200
-- ++
--
++
++
--
Start
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-16 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Stepping-Stone MethodStepping-Stone MethodAn Improved SolutionAn Improved Solution
Des Moines(D)
Evansville(E)
Fort Lauderdale
(F)
WarehouseReq.
Albuquerque(A) Boston
(B)
Cleveland(C) Factory
Capacity
300 200 200 700
300
300
1005 4 3
3
57
48
9
100
100
200
200100
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-17 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
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
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-18 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
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.
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-19 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Vogel’s ApproximationVogel’s Approximation1. For each row/column of table,
find difference between two lowest costs. (Opportunity cost)
2. Find greatest opportunity cost.
3. Assign as many units as possible to lowest cost square in row/column with greatest opportunity cost.
4. Eliminate row or column which has been completely satisfied.
4. Begin again, omitting eliminated rows/columns.
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-20 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Special Problems in Special Problems in Transportation MethodTransportation Method
• Unbalanced Problem• Demand Less than Supply• Demand Greater than Supply
• Degeneracy
• More Than One Optimal Solution
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-21 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Unbalanced ProblemUnbalanced ProblemDemand Less than SupplyDemand Less than Supply
Factory 1
Factory 2
Factory 3
Customer Requirements
Customer 1 Customer
2
DummyFactory
Capacity
150 80 150 380
80
130
1708 5 0
0
09
1015
3
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-22 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
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
00
1015
0
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-23 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
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
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-24 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
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
7
9
16
10
1015
3
70
80
50
50
30
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-25 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Stepping-Stone Method - Stepping-Stone Method - The Des Moines-to-The Des Moines-to-
Cleveland RouteCleveland Route
Des Moines(D)
Evansville(E)
Fort 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
-- ++
--
++
++
--
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-26 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
The Assignment ProblemThe Assignment Problem
Project
Person 1 2 3
Adams $11 $14 $6
Brown $8 $10 $11
Cooper $9 $12 $7
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-27 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
The Assignment The Assignment MethodMethod
1. 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)
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-28 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
The Assignment The Assignment Method - continuedMethod - continued
3. if the number of lines does not equal the number of rows or columns
• subtract the smallest number not covered by a line from every other uncovered number
• add the same number to any number lying at the intersection of any two lines
• return to step 2
4. make optimal assignments at locations of zeros within the table
PG 10.13b
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-29 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Hungarian MethodHungarian Method
Initial Table
Person Project
1 2 3
Adams 11 14 6
Brown 8 10 11
Cooper 9 12 7
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-30 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Hungarian MethodHungarian Method
Row Reduction
Person Project
1 2 3
Adams
Brown
Cooper
5 8 0
0 2 3
2 5 0
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-31 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Hungarian MethodHungarian Method
Column Reduction
Person Project
1 2 3
Adams 5 6 0
Brown 0 0 3
Cooper 2 3 0
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-32 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Hungarian MethodHungarian Method
Person Project
1 2 3
Adams
Brown
Cooper
5 6 0
0 0 3
2 3 0
TestingCovering
Line 2
Covering Line 1
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-33 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Hungarian MethodHungarian Method
Person Project
1 2 3
Adams 3 4 0
Brown 0 0 5
Cooper 0 1 0
Revised Opportunity Cost Table
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-34 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Hungarian MethodHungarian Method
Person Project
1 2 3
Adams3 4 0
Brown0 0 5
Cooper0 1 0
TestingCovering
Line 1
Covering Line 2
Covering Line 3
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-35 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Hungarian MethodHungarian MethodAssignments
Person Project
1 2 3
Adams 6
Brown 10
Cooper 9
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-36 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Maximization Assignment Maximization Assignment ProblemProblem
Project
1 2 3 Dummy
Adams $11 $14 $6 $0
Brown $8 $10 $11 $0
Cooper $9 $12 $7 $0
Davis $10 $13 $8 $0
To accompany Quantitative Analysis for Management, 8e by Render/Stair/Hanna
10-37 © 2003 by Prentice Hall, Inc. Upper Saddle River, NJ 07458
Maximization Assignment Maximization Assignment ProblemProblem
Project
1 2 3 Dummy
Adams $32 $0 $8 $14
Brown $6 $4 $3 $14
Cooper $5 $2 $77 $14
Davis $4 $1 $6 $14
