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Page 1: Transportation, Transshipment and Assignment Models and Assignment Models

TransportationTransportation, Transshipment, Transshipment

and and Assignment ModelsAssignment Models

Page 2: Transportation, Transshipment and Assignment Models and Assignment Models

Learning ObjectivesLearning Objectives

• Structure special LP network flow models.• Set up and solve transportation models • Extend basic transportation model to include

transshipment points.• Set up and solve facility location and other application

problems as transportation models.• Set up and solve assignment models

Page 3: Transportation, Transshipment and Assignment Models and Assignment Models

OverviewOverview

Part of a larger class of linear programming problems are known as network flow models.

They possess special mathematical features that enabled the development of very efficient, unique solution methods.

Page 4: Transportation, Transshipment and Assignment Models and Assignment Models

Transportation ModelTransportation Model

Transportation problem deals with the distribution of goods from several points of supply to a number of points of demand. They arise when a cost-effective pattern is needed to ship items from origins that have limited supply to destinations that have demand for the goods.

Resources to be optimally allocated usually involve a given capacity of goods at each source and a given requirement for the goods at each destination.

Most common objective of the transportation problem is to schedule shipments from sources to destinations so that total production and transportation costs are minimized

Page 5: Transportation, Transshipment and Assignment Models and Assignment Models

Transshipment ModelTransshipment Model

An extension of transportation problems is called transshipment problem in which a point can have shipments that both arrive as well as leave.

Example would be a warehouse where shipments arrive from factories and then leave for retail outlets

It may be possible for a firm to achieve cost savings (economies of scale) by consolidating shipments from several factories at a warehouse and then sending them together to retail outlets.

Page 6: Transportation, Transshipment and Assignment Models and Assignment Models

Assignment ModelAssignment Model

Assignment problem refers to a class of LP problems that involve determining most efficient assignment of:

People to projects,

Salespeople to territories,

Contracts to bidders,

Jobs to machines, and so on

Objective is to minimize total cost or total time of performing tasks at hand, although a maximization objective is also possible.

Page 7: Transportation, Transshipment and Assignment Models and Assignment Models

Transportation ModelTransportation Model

Page 8: Transportation, Transshipment and Assignment Models and Assignment Models

Problem definition

– There are m sources. Source i has a supply capacity of Si.

– There are n destinations.The demand at destination j is D j.

– Objective:

To minimize the total shipping cost of supplying the

destinations with the required demand from the available supplies at the sources.

Transportation ModelTransportation Model

Page 9: Transportation, Transshipment and Assignment Models and Assignment Models

The Transportation ModelThe Transportation ModelCharacteristicsCharacteristics

A product is to be transported from a number of sources to a number of destinations at the minimum possible cost.

Each source is able to supply a fixed number of units of the product, and each destination has a fixed demand for the product.

The linear programming model has constraints for supply at each source and demand at each destination.

All constraints are equalities in a balanced transportation model where supply equals demand.

Constraints contain inequalities in unbalanced models where supply is not equal to demand.

Page 10: Transportation, Transshipment and Assignment Models and Assignment Models

Transportation ModelTransportation Model- - Example 1Example 1

Executive Furniture Corporation

Page 11: Transportation, Transshipment and Assignment Models and Assignment Models

Transportation ModelTransportation Model Example 1Example 1

Executive Furniture Corporation

Transportation Costs Per Desk

Page 12: Transportation, Transshipment and Assignment Models and Assignment Models

Transportation ModelTransportation Model Example 1Example 1

Objective: minimize total shipping costs =

5 XDA + 4 XDB + 3 XDC + 3 XEA + 2 XEB +

1 XEC + 9 XFA + 7 XFB + 5 XFC

Executive Furniture Corporation:LP Transportation Model Formulation

Where: Xij = number of desks shipped from factory i to warehouse j

i = D (for Des Moines),

E (for Evansville), or

F (for Fort Lauderdale).

j = A (for Albuquerque),

B (for Boston), or

C (for Cleveland).

Page 13: Transportation, Transshipment and Assignment Models and Assignment Models

Transportation ModelTransportation Model Example 1Example 1

Net flow at Des Moines = (Total flow in) - (Total flow out)

= (0) - (XDA + XDB + XDC)

Net flow at Des Moines =

-XDA - XDB - XDC = -100 (Des Moines capacity) and

-XEA - XEB - XEC = -300(Evansville capacity)

-XFA - XFB - XFC = -300 (Fort Lauderdale capacity)

Multiply each constraint by -1 and rewrite as:

XDA + XDB + XDC = 100 (Des Moines capacity)

XEA + XEB + XEC = 300 (Evansville capacity)

XFA + XFB + XFC = 300 (Fort Lauderdale capacity)

Executive Furniture Corporation: Supply Constraints

Page 14: Transportation, Transshipment and Assignment Models and Assignment Models

Transportation ModelTransportation Model Example 1Example 1

Executive Furniture Corporation: Demand Constraints Net flow at Albuquerque = (Total flow in) - (Total flow out)

= (XDA + XEA + XFA) - (0)

Net flow at Albuquerque =

XDA + XEA + XFA = 300 (Albuquerque demand) and

XDB + XEB + XFB = 200 (Boston demand)

XDC + XEC + XFC = 200 (Cleveland demand)

Page 15: Transportation, Transshipment and Assignment Models and Assignment Models

Transportation Model Transportation Model Example 1:Example 1:The Optimum SolutionThe Optimum Solution

SHIP:

100 desks from Des Moines to Albuquerque,

200 desks from Evansville to Albuquerque,

100 desks from Evansville to Boston,

100 desks from Fort Lauderdale to Boston,

and 200 desks from Fort Lauderdale to Cleveland.

Total shipping cost is $3,000.

Page 16: Transportation, Transshipment and Assignment Models and Assignment Models

Grain Elevator Supply Mill Demand

1. Kansas City 150 A. Chicago 200

2. Omaha 175 B. St. Louis 100

3. Des Moines 275 C. Cincinnati 300

Total 600 tons Total 600 tons

Transport Cost from Grain Elevator to Mill ($/ton)

Grain Elevator A. Chicago B. St. Louis C. Cincinnati 1. Kansas City 2. Omaha 3. Des Moines

$ 6 7 4

$ 8 11 5

$10 11 12

Transportation Model ExampleTransportation Model Example 2 2Problem Definition and DataProblem Definition and Data

Problem: How many tons of wheat to transport from each grain elevator to each mill on a monthly basis in order to minimize the total cost of transportation?

Data:

Page 17: Transportation, Transshipment and Assignment Models and Assignment Models

Transportation Model ExampleTransportation Model Example 2 2Model Formulation Model Formulation

Page 18: Transportation, Transshipment and Assignment Models and Assignment Models

Minimize Z = $6x1A + 8x1B + 10x1C + 7x2A + 11x2B + 11x2C + 4x3A + 5x3B + 12x3C

subject to:x1A + x1B + x1C = 150

x2A + x2B + x2C = 175 x3A + x3B + x3C = 275 x1A + x2A + x3A = 200 x1B + x2B + x3B = 100 x1C + x2C + x3C = 300 xij 0

xij = tons of wheat from each grain elevator, i, i = 1, 2, 3, to each mill j, j = A,B,C

Transportation Model ExampleTransportation Model Example 2 2Model Formulation Model Formulation

Page 19: Transportation, Transshipment and Assignment Models and Assignment Models

Transportation ModelTransportation Model- Example 3- Example 3

• Carlton Pharmaceuticals supplies drugs and other medical supplies.

• It has three plants in: Cleveland, Detroit, Greensboro.

• It has four distribution centers in: Boston, Richmond, Atlanta, St. Louis.

• Management at Carlton would like to ship cases of a certain vaccine as economically as possible.

Carlton Pharmateuticals

Page 20: Transportation, Transshipment and Assignment Models and Assignment Models

• Data– Unit shipping cost, supply, and demand

• Assumptions– Unit shipping cost is constant.– All the shipping occurs simultaneously.– The only transportation considered is between

sources and destinations.– Total supply equals total demand.

To From Boston Richmond Atlanta St. Louis Supply Cleveland $35 30 40 32 1200 Detroit 37 40 42 25 1000 Greensboro 40 15 20 28 800 Demand 1100 400 750 750

Page 21: Transportation, Transshipment and Assignment Models and Assignment Models

NETWORK

REPRESENTATION

Boston

Richmond

Atlanta

St.Louis

Destinations

Sources

Cleveland

Detroit

Greensboro

S1=1200

S2=1000

S3= 800

D1=1100

D2=400

D3=750

D4=750

37

40

42

32

35

40

30

25

4015

20

28

Page 22: Transportation, Transshipment and Assignment Models and Assignment Models

• The Mathematical Model

– The structure of the model is:

Minimize <Total Shipping Cost>

ST

[Amount shipped from a source] = [Supply at that source]

[Amount received at a destination] = [Demand at that destination]

– Decision variablesXij = amount shipped from source i to destination j.

where: i=1 (Cleveland), 2 (Detroit), 3 (Greensboro) j=1 (Boston), 2 (Richmond), 3 (Atlanta), 4(St.Louis)

Page 23: Transportation, Transshipment and Assignment Models and Assignment Models

Boston

Richmond

Atlanta

St.Louis

D1=1100

D2=400

D3=750

D4=750

The supply constraints

Cleveland S1=1200

X11

X12

X13

X14

Supply from Cleveland X11+X12+X13+X14 = 1200

DetroitS2=1000

X21

X22

X23

X24

Supply from Detroit X21+X22+X23+X24 = 1000

GreensboroS3= 800

X31

X32

X33

X34

Supply from Greensboro X31+X32+X33+X34 = 800

Page 24: Transportation, Transshipment and Assignment Models and Assignment Models

• The complete mathematical model

Minimize 35X11+30X12+40X13+ 32X14 +37X21+40X22+42X23+25X24+ 40X31+15X32+20X33+38X34

ST

Supply constrraints:X11+ X12+ X13+ X14 1200

X21+ X22+ X23+ X24 1000X31+ X32+ X33+ X34 800

Demand constraints: X11+ X21+ X31 1000

X12+ X22+ X32 400X13+ X23+ X33 750

X14+ X24+ X34 750

All Xij are nonnegative

===

====

Page 25: Transportation, Transshipment and Assignment Models and Assignment Models

Excel Optimal SolutionExcel Optimal Solution

CARLTON PHARMACEUTICALS

UNIT COSTSBOSTON RICHMOND ATLANTA ST.LOUIS SUPPLIES

CLEVELAND 35.00$ 30.00$ 40.00$ 32.00$ 1200DETROIT 37.00$ 40.00$ 42.00$ 25.00$ 1000GREENSBORO 40.00$ 15.00$ 20.00$ 28.00$ 800

DEMANDS 1100 400 750 750

SHIPMENTS (CASES)BOSTON RICHMOND ATLANTA ST.LOUIS TOTAL

CLEVELAND 850 350 0 0 1200DETROIT 250 0 0 750 1000GREENSBORO 0 50 750 0 800

TOTAL 1100 400 750 750 TOTAL COST = 84000

Page 26: Transportation, Transshipment and Assignment Models and Assignment Models

Range of optimality

WINQSB Sensitivity AnalysisWINQSB Sensitivity Analysis

If this path is used, the total cost will increase by $5 per unit shipped along it

Page 27: Transportation, Transshipment and Assignment Models and Assignment Models

Range of feasibility

Shadow prices for warehouses - the cost incurred from having 1 extra case of vaccine demanded at the warehouse

Shadow prices for plants - the cost savings realized for each extra case of vaccine available at the plant

Page 28: Transportation, Transshipment and Assignment Models and Assignment Models

Interpreting sensitivity analysis resultsInterpreting sensitivity analysis results

– Reduced costs

• The amount of transportation cost reduction per unit that

makes a given route economically attractive.

• If the route is forced to be used under the current cost structure, for each item shipped along it, the total cost increases by an amount equal to the reduced cost.

– Shadow prices

• For the plants, shadow prices convey the cost savings realized for each extra case of vaccine available at plant.

• For the warehouses, shadow prices convey the cost incurred from having an extra case demanded at the warehouse.

Page 29: Transportation, Transshipment and Assignment Models and Assignment Models

Transportation ModelTransportation Model- Example 4- Example 4

Montpelier Ski Company: Using a Transportation model for production scheduling

– Montpelier is planning its production of skis for the months of July, August, and September.

– Production capacity and unit production cost will change from month to month.

– The company can use both regular time and overtime to produce skis.

– Production levels should meet both demand forecasts and end-of-quarter inventory requirement.

– Management would like to schedule production to minimize its costs for the quarter.

Page 30: Transportation, Transshipment and Assignment Models and Assignment Models

• Data:

– Initial inventory = 200 pairs

– Ending inventory required =1200 pairs

– Production capacity for the next quarter is shown on the table

– Holding cost rate is 3% per month per ski.

– Production capacity, and forecasted demand for this quarter (in pairs of skis), and production cost per unit (by months)

Forecasted Production Production Costs Month Demand Capacity Regular Time OvertimeJuly 400 1000 25 30August 600 800 26 32September 1000 400 29 37

Page 31: Transportation, Transshipment and Assignment Models and Assignment Models

• Analysis of demand:

– Net demand to satisfy in July = 400 - 200 = 200 pairs

– Net demand in August = 600

– Net demand in September = 1000 + 1200 = 2200 pairs

• Analysis of Supplies:

– Production capacities are thought of as supplies.

– There are two sets of “supplies”:

• Set 1- Regular time supply (production capacity)

• Set 2 - Overtime supply

Initial inventory

Forecasted demand In house inventory• Analysis of Unit costs

Unit cost = [Unit production cost] +

[Unit holding cost per month][the number of months stays in inventory]

Example: A unit produced in July in Regular time and sold in September costs 25+ (3%)(25)(2 months) = $26.50

Page 32: Transportation, Transshipment and Assignment Models and Assignment Models

Network representation

252525.7525.7526.5026.50 00 3030

30.9030.9031.8031.80

00+M+M

2626

26.7826.78

00

+M+M

3232

32.9632.96

00

+M+M

+M+M

2929

00

+M+M

+M+M

3737

00

ProductionMonth/period

Monthsold

JulyR/T

July O/T

Aug.R/T

Aug.O/T

Sept.R/T

Sept.O/T

July

Aug.

Sept.

Dummy

1000

500

800

400

400

200

200

600

300

2200

Demand

Prod

uctio

n Ca

pacit

y

July R/T

Page 33: Transportation, Transshipment and Assignment Models and Assignment Models

Source: July production in R/TDestination: July‘s demand.

Source: Aug. production in O/TDestination: Sept.’s demand

32+(.03)(32)=$32.96Unit cost= $25 (production)Unit cost =Production+one month holding cost

Page 34: Transportation, Transshipment and Assignment Models and Assignment Models

• Summary of the optimal solution

– In July produce at capacity (1000 pairs in R/T, and 500 pairs in

O/T). Store 1500-200 = 1300 at the end of July.

– In August, produce 800 pairs in R/T, and 300 in O/T. Store

additional 800 + 300 - 600 = 500 pairs.

– In September, produce 400 pairs (clearly in R/T). With 1000

pairs

retail demand, there will be

(1300 + 500) + 400 - 1000 = 1200 pairs available for shipment

Inventory + Production -

Demand

Page 35: Transportation, Transshipment and Assignment Models and Assignment Models

Unbalanced Transportation ProblemsUnbalanced Transportation Problems

• If supplies are not equal to demands, an unbalanced transportation model exists.

• In an unbalanced transportation model, supply or demand constraints need to be modified.

• There are two possible scenarios:

(1)Total supply exceeds total requirement.

(2)Total supply is less than total requirement.

Page 36: Transportation, Transshipment and Assignment Models and Assignment Models

Total Supply Exceeds Total RequirementTotal Supply Exceeds Total Requirement

Total flow out of Des Moines ( XDA + XDB + XDC) should be

permitted to be smaller than total supply (100).

The constraint should be written as

-XDA - XDB - XDC >= -100 (Des Moines capacity)

  -XEA - XEB - XEC >= -300 (Evansville capacity)

-XFA - XFB - XFC >= -300 (Fort Lauderdale capacity)

XDA + XDB + XDC <= 100

XEA + XEB + XEC <= 100

XFA + XFB + XFC <= 100

Page 37: Transportation, Transshipment and Assignment Models and Assignment Models

Total Supply Less Than Total RequirementTotal Supply Less Than Total Requirement

Total flow in to Albuquerque (that is, XDA + XEA + XFA) should

be permitted to be smaller than total demand (namely, 300).

This warehouse should be written as:

  XDA + XEA + XFA <= 300 (Albuquerque demand)

XDB + XEB + XFB <= 200 (Boston demand)

XDC + XEC + XFC <= 200 (Cleveland demand)

Page 38: Transportation, Transshipment and Assignment Models and Assignment Models

Develop the linear programming model and solve using Excel:

Construction site Plant A B C Supply (tons)

1 2 3

$ 8 15

3

$ 5 10 9

$ 6 12 10

120 80 80

Demand (tons) 150 70 100

Transportation Transportation Example Example 5: Formulation5: Formulation

Page 39: Transportation, Transshipment and Assignment Models and Assignment Models

Minimize Z = $8x1A + 5x1B + 6x1C + 15x2A + 10x2B + 12x2C + 3x3A + 9x3B + 10x3C

subject to:x1A + x1B + x1C = 120 x2A + x2B + x2C = 80

x3A + x3B + x3C = 80x1A + x2A + x3A 150 x1B + x2B + x3B 70

x1C + x2C + x3C 100 xij 0

Transportation Transportation Example Example 5: Formulation 5: Formulation

Page 40: Transportation, Transshipment and Assignment Models and Assignment Models

Transportation Model-Example 6Transportation Model-Example 6 Hardgrave Machine Company - New Factory Location

• Produces computer components at its plants in Cincinnati, Kansas City, and Pittsburgh.

• Plants not able to keep up with demand for orders at four warehouses in Detroit, Houston, New York, and Los Angeles.

• Firm has decided to build a new plant to expand its productive capacity.

• Two sites being considered:

– Seattle, Washington and

– Birmingham, Alabama.

• Both cities attractive in terms: labor supply, municipal services, and ease of factory financing.

Page 41: Transportation, Transshipment and Assignment Models and Assignment Models

Transportation Model-Example 6Transportation Model-Example 6Hardgrave Machine Company: Demand Supply Data and Production Costs

Page 42: Transportation, Transshipment and Assignment Models and Assignment Models

Transportation Model-Example 6Transportation Model-Example 6Hardgrave Machine Company: Shipping Costs

Page 43: Transportation, Transshipment and Assignment Models and Assignment Models

Transshipment ModelTransshipment Model

Page 44: Transportation, Transshipment and Assignment Models and Assignment Models

Transshipment ModelTransshipment Model

In a Transshipment Problem flows can occur both out of and into the

same node in three ways:

 1. If total flow into a node is less than total flow out from node,

node represents a net creator of goods (a supply point).

- Flow balance equation will have a negative right hand

side (RHS) value.

2. If total flow into a node exceeds total flow out from node,

node represents a net consumer of goods, (a demand point).

- Flow balance equation will have a positive RHS value.

3. If total flow into a node is equal to total flow out from node,

node represents a pure transshipment point.

- Flow balance equation will have a zero RHS value.

Page 45: Transportation, Transshipment and Assignment Models and Assignment Models

It is an extension of the transportation model.

Intermediate transshipment points are added between the sources and destinations.

Items may be transported from:

Sources through transshipment points to destinations

One source to another

One transshipment point to another

One destination to another

Directly from sources to to destinations

Some combination of these

The Transshipment ModelThe Transshipment ModelCharacteristicsCharacteristics

Page 46: Transportation, Transshipment and Assignment Models and Assignment Models

Executive Furniture Corporation – RevisitedExecutive Furniture Corporation – Revisited

Assume it is possible for Executive Furniture to ship desks from Evansville factory to its three warehouses at very low unit shipping costs.

Consider shipping all desks produced at other two factories (Des Moines and Fort Lauderdale) to Evansville.

Consider using a new shipping company to move desks from Evansville to all its warehouses.

Page 47: Transportation, Transshipment and Assignment Models and Assignment Models

Executive Furniture Corporation - RevisitedExecutive Furniture Corporation - Revisited

• Revised unit shipping costs are shown here.

• Note Evansville factory shows up in both the “From” and “To” entries.

Page 48: Transportation, Transshipment and Assignment Models and Assignment Models

LP Model for theTransshipment Problem Two new additional decision variables for new shipping

routes are to be added.

XDE= number of desks shipped from Des Moines to Evansville

XFE = number of desks shipped from Fort Lauderdale to Evansville

 Objective Function: minimize total shipping costs =

5XDA + 4XDB + 3XDC + 2XDE + 3XEA + 2XEB +

+1XEC + 9XFA + 7XFB + 5XFC + 3XFE

 

Executive Furniture CoExecutive Furniture Corporation Revisitedrporation Revisited

Page 49: Transportation, Transshipment and Assignment Models and Assignment Models

Executive Furniture CoExecutive Furniture Corporation Revisitedrporation Revisited

• Relevant flow balance equations written as:

 (0) - (XDA + XDB + XDC + XDE) = -100 (Des Moines capacity)

(0) - (XFA + XFB + XFC + XFE) = -300 (Fort Lauderdale capacity)

• Supplies have been expressed as negative numbers in the RHS.

 Net flow at Evansville = (Total flow in) - (Total flow out)

= (XDE + XFE) - (XEA + XEB + XEC)

• Net flow equals total number of desks produced (the supply) at Evansville.

Net flow at Evansville = (XDE + XFE) - (XEA + XEB + XEC) = -300

• No change in demand constraints for warehouse requirements: 

XDA + XEA + XFA = 300 (Albuquerque demand)

XDB + XEB + XFB = 200 (Boston demand)

XDC + XEC + XFC = 200 (Cleveland demand)

LP Model for theTransshipment Problem

Page 50: Transportation, Transshipment and Assignment Models and Assignment Models

Extension of the transportation model in which intermediate transshipment points are added between sources and destinations. An example of a transshipment point is a distribution center or warehouse located between plants and stores

Data:

Stores Warehouses 6. Chicago 7. St. Louis 8. Cincinnati 3. Kansas 4. Omaha 5. Des Moines

$6 7

4

8 11 5

10 11 12

Transshipment Model ExampleTransshipment Model Example 2 2Problem Definition and Data Problem Definition and Data

Grain Elevator Farm 3. Kansas City 4. Omaha 5. Des Moines 1. Nebrasca 2. Colorado

$16 15

10 14

12 17

Page 51: Transportation, Transshipment and Assignment Models and Assignment Models

Transshipment Model ExampleTransshipment Model Example 2 2Problem Definition and Data Problem Definition and Data

Page 52: Transportation, Transshipment and Assignment Models and Assignment Models

Minimize Z = $16x13 + 10x14 + 12x15 + 15x23 + 14x24 + 17x25 + 6x36 + 8x37 + 10x38 + 7x46 + 11x47 +

11x48 + 4x56 + 5x57 + x58

subject to: x13 + x14 + x15 = 300x23+ x24 + x25 = 300x36 + x46 + x56 = 200x37+ x47 + x57 = 100x38 + x48 + x58 = 300x13 + x23 - x36 - x37 - x38 = 0x14 + x24 - x46 - x47 - x48 = 0x15 + x25 - x56 - x57 - x58 = 0xij 0

Transshipment Model ExampleTransshipment Model Example 2 2Model FormulationModel Formulation

Page 53: Transportation, Transshipment and Assignment Models and Assignment Models

Assignment ModelAssignment Model

Page 54: Transportation, Transshipment and Assignment Models and Assignment Models

The Assignment ModelThe Assignment Model

Problem definition– m workers are to be assigned to m jobs

– A unit cost (or profit) Cij is associated with worker i performing job j.

– Minimize the total cost (or maximize the total profit) of assigning workers to jobs so that each worker is assigned a job, and each job is performed.

Page 55: Transportation, Transshipment and Assignment Models and Assignment Models

It is a special form of linear programming models similar to the transportation model.

Supply at each source and demand at each destination is limited to one unit.

In a balanced model supply equals demand.

In an unbalanced model supply is not equal to demand.

The Assignment ModelThe Assignment ModelCharacteristicsCharacteristics

Page 56: Transportation, Transshipment and Assignment Models and Assignment Models

– The number of workers is equal to the number of jobs.

– Given a balanced problem, each worker is assigned exactly once, and each job is performed by exactly one worker.

– For an unbalanced problem “dummy” workers (in case there are more jobs than workers), or “dummy” jobs (in case there are more workers than jobs) are added to balance the problem.

The Assignment ModelThe Assignment Model Assumptions Assumptions

Page 57: Transportation, Transshipment and Assignment Models and Assignment Models

Fix-It Shop Example

Received three new rush projects to repair: (1) a radio, (2) a toaster oven, and (3) a broken coffee table. Three workers (each has different talents and abilities).Estimated costs to assign each worker to each of the three projects.

Assignment ModelAssignment Model Example 1 Example 1

Page 58: Transportation, Transshipment and Assignment Models and Assignment Models

Assignment Model Assignment Model Example 1 Example 1 Fix-It Shop

• Rows denote people or objects to be assigned, and columns denote tasks or jobs assigned.

• Numbers in table are costs associated with each particular assignment.

Page 59: Transportation, Transshipment and Assignment Models and Assignment Models

Assignment Assignment Model Example 1Model Example 1

• Owner's objective is to assign three projects to workers in a way that result is lowest total cost.

Fix-It Shop: Assignment Alternatives and Costs

Page 60: Transportation, Transshipment and Assignment Models and Assignment Models

Assignment Assignment Model Example 1 Model Example 1

• Owner's objective is to assign three projects to workers in a way that results in lowest total cost.

Fix-It Shop

Page 61: Transportation, Transshipment and Assignment Models and Assignment Models

Assignment ModelAssignment Model Example 1 Example 1

Formulate LP model -

Xij = “Flow” on arc from node denoting worker i to node denoting

project j.

Solution value will equal 1 if worker i is assigned to project j :

i = A (for Adams), B (for Brown), or C (for Cooper)

j = 1 (for project 1), 2 (for project 2), or 3 (for project 3)

Objective Function: minimize total assignment cost =

11XA1 + 14XA2 + 6XA3 + 8XB1 + 10XB2 + 11XB3 +

+ 9XC1 + 12XC2 + 7XC3

Fix-It Shop

Page 62: Transportation, Transshipment and Assignment Models and Assignment Models

Assignment Model Example 1 Assignment Model Example 1 Fix-It Shop

Constraints expressed using standard flow balance equations are as

follows:

-XA1 - XA2 - XA3 = -1 (Adams availability)

-XB1 - XB2 - XB3 = -1 (Brown availability)

-XC1 - XC2 - XC3 = -1 (Cooper availability)

XA1 + XB1 + XC1 = 1 (Project 1 requirement)

XA2 + XB2 + XC2 = 1 (Project 2 requirement)

XA3 + XB3 + XC3 = 1 (Project 3 requirement)

Page 63: Transportation, Transshipment and Assignment Models and Assignment Models

Assignment Model- Example 2 Assignment Model- Example 2

Ballston Electronics• Five different electrical devices produced on five

production lines, are needed to be inspected.• The travel time of finished goods to inspection areas

depends on both the production line and the inspection area.

• Management wishes to designate a separate inspection area to inspect the products such that the total travel time is minimized.

Page 64: Transportation, Transshipment and Assignment Models and Assignment Models

• Data: Travel time in minutes from assembly lines to inspection areas.

Inspection AreaA B C D E

1 10 4 6 10 12Assembly 2 11 7 7 9 14 Lines 3 13 8 12 14 15

4 14 16 13 17 175 19 17 11 20 19

Inspection AreaA B C D E

1 10 4 6 10 12Assembly 2 11 7 7 9 14 Lines 3 13 8 12 14 15

4 14 16 13 17 175 19 17 11 20 19

Assignment Model- Example 2Assignment Model- Example 2

Page 65: Transportation, Transshipment and Assignment Models and Assignment Models

Assignment Model Example 2: Network RepresentationAssignment Model Example 2: Network Representation(3 of 3)(3 of 3)

1

2

3

4

5

Assembly Line Inspection AreasA

B

C

D

E

S1=1

S2=1

S3=1

S4=1

S5=1

D1=1

D2=1

D3=1

D4=1

D5=1

Page 66: Transportation, Transshipment and Assignment Models and Assignment Models

• Computer solutions– A complete enumeration is not efficient even

for moderately large problems (with m=8, m! > 40,000 is the number of assignments to enumerate).

– The Hungarian method provides an efficient solution procedure.

• Special cases– A worker is unable to perform a particular job.– A worker can be assigned to more than one job.– A maximization assignment problem.

Page 67: Transportation, Transshipment and Assignment Models and Assignment Models

Assignment Model ExampleAssignment Model Example 3 3 Problem Definition and DataProblem Definition and Data

Problem: Assign four teams of officials to four games in a way that will minimize total distance traveled by the officials. Supply is always one team of officials, demand is for only one team of officials at each game.

Data:

Page 68: Transportation, Transshipment and Assignment Models and Assignment Models

Minimize Z = 210xAR + 90xAA + 180xAD + 160xAC + 100xBR + 70xBA + 130xBD + 200xBC + 175xCR + 105xCA +

140xCD + 170xCC + 80xDR + 65xDA + 105xDD + 120xDC

subject to: xAR + xAA + xAD + xAC = 1 xij 0xBR + xBA + xBD + xBC = 1xCR + xCA + xCD + xCC = 1xDR + xDA + xDD + xDC = 1xAR + xBR + xCR + xDR = 1xAA + xBA + xCA + xDA = 1xAD + xBD + xCD + xDD = 1xAC + xBC + xCC + xDC = 1

Assignment Model ExampleAssignment Model Example 3 3 Model FormulationModel Formulation

Page 69: Transportation, Transshipment and Assignment Models and Assignment Models

SummarySummary

Three Three network flow modelsnetwork flow models have been have been presented:presented:

1.1. Transportation modelTransportation model deals with distribution of goods from several supplier to a number of demand points.

2.2. Transshipment modelTransshipment model includes points that permit goods to flow both in and out of them.

3.3. Assignment modelAssignment model deals with determining the most efficient assignment of issues such as people to projects.