mcgraw-hill/irwin © the mcgraw-hill companies, inc., 2003 6.1 transportation and assignment models

© The McGraw-Hill Companies, Inc., 20036.1McGraw-Hill/Irwin

Transportation and

Assignment Models


Outline

• Network models in general– Transportation, Assignment and Transshipment models belong to a

special class of linear programming problems called network flow problems

• Characteristics of Transportation models

• Characteristics of Assignment models

• Variations on a theme


Network Optimization Problems

Many optimization problems can be represented by a graphical network representation.

Some examples:– Distribution problems– Routing problems– Maximum flow problems– Designing computer / phone / road networks– Equipment replacement

Arcs

Nodes


Transportation problem• Frequently arises in planning for distribution of goods and

services from several supply locations to several demand locations.– Examples?

• Characteristics (typical)– Quantity of goods available at each supply location is limited.– Quantity of goods needed at each of several demand locations is known.– Usual objective of a transportation problem is to minimize the cost of shipping goods from the origins

to the destinations.

• Variations on the Transportation Problem theme– Total supply not equal to total demand.– Maximization objective function.– Route capacities or route minimums.– Unacceptable routes.


Example: Supply Chains – the generic model

Raw materials supplier

Manufacturing plant

Distribution center

Customers/

Retailers

upstream

downstream


Example: Forest industry supply chain

Wagner, H.M. (1975). Principles of Operations Research 2nd ed. Englewood Cliffs NJ: Prentice-Hall


Assumptions of Transportation Problems

• The Requirements Assumption– Each source has a fixed supply of units, where this entire supply must be distributed

to the destinations.– Each destination has a fixed demand for units, where this entire demand must be

received from the sources.

• The Feasible Solutions Property– A transportation problem will have feasible solutions if and only if the sum of its

supplies equals the sum of its demands.

• The Cost Assumption– The cost of distributing units from any particular source to any particular destination is

directly proportional to the number of units distributed.– This cost is just the unit cost of distribution multiplied by the number of units

distributed.


The network for a transportation problem

Variable costs - cij

Quantities - xij

Dem

an

ds - D

j

Cap

aci

ties

- K

i

Region 1

Region 2

Region 4

Region 3

Plant 1

Plant 2

Plant 3

The Decision Variable


A basic transportation model – Foster Generators

2

3

1

1

2

3

4

32

76

4

5

32

2

7

5

5

Transportation cost per unit

Plants (origin nodes)

Distribution Centers

Chicago 4000

St. Louis 2000

Bedford6000

Boston 6000

Distribution Routes (arcs)

Lexington 1500

York2500

Cleveland5000

Supplies /Plant Capacities (units)

Demands (units)


Min

n

i

m

jijij xc

1 1

st

n

ijij Dx

1 for j = 1,.., m

m

jiij Kx

1 for i = 1,.., n

0ijx for i = 1,.., n and j = 1,.., m

Parameters: cij variable costs per unit transported from plant i to region j Ki capacity for plant i Dj demand for region j m number of regions n number of plants

Decision variables: xij quantity transported from plant i to region j

Parameterscij variable costs per unit transported from Plant i to Region jKi capacity for Plant iDj demand for Region jm number of regionsn number of plants

Decision variablesxij quantity transported from Plant i to Region j

All demands are satisfied

No capacities are exceeded

Transportation Problem: Demand Allocation Model


A simple Transportation/Demand Allocation problem solved


The Capacitated Plant Location Model (CPLM)

• Open/Closed

• Fixed costs

• Capacities

• Variable costs

• Quantities

• Demands

Ware-house 1

Ware-house 2

Ware-house 3

Plant 1

Plant 2

Plant 3


The Capacitated Plant Location Model (CPLM)

Min

n

i

n

i

m

jijijii xcyf

1 1 1

st

m

jijii xyK

1

0 for i = 1,.., n

n

iijj xD

1

0 for j = 1,.., m

1/0iy for i = 1,.., n 0ijx

for i = 1,.., n and j = 1,.., m

Decision variablesyi binary variable indicating whether Plant i should be open (1) or closed (0)xij quantity transported from Plant i to Region j

ParametersFi fixed costs for Plant icij variable costs per unit transported from Plant i to Region jKi capacity for Plant iDj demand for Region jm number of regionsn number of potential plants


The CPLM with single sourcing

Ware-house 1

Ware-house 2

Ware-house 3

Plant 1

Plant 2

Plant 3

• Variable costs

• Open/Closed

• Fixed costs

• Capacities

• Assigning plants to warehouses

• Demands


The CPLM with single sourcing

Min

n

i

n

i

m

jijijjii xcDyf

1 1 1

st

n

iijx

1

1 for j = 1,.., m

ii

n

jijj yKxD

1 for i = 1,.., n

1/0iy for i = 1,.., n 1/0ijx

for i = 1,.., n and j = 1,.., m

ParametersFi fixed costs for Plant icij variable costs per unit transported from Plant i to Region jKi capacity for Plant iDj demand for Region jm number of regionsn number of potential plants

Decision variablesyi binary variable indicating whether Plant i should be open(1) or

closed (0)xij binary variable indicating whether Plant i should supply market in Region j


Transshipment model with sourcing

Market1

Market1

Market2

Market2

Market3

Market3

Plant 1Plant 1Supplier

1Supplier

1

Supplier2

Supplier2

Plant 2Plant 2

Plant 3Plant 3

Ware-house 1Ware-

house 1

Ware-house 2Ware-

house 2

Variable costs

Open/ ClosedQuantities

Capacities

Fixed costs

Demands

Open/ Closed

Quantities Quantities

Variable costs

Variable costs

Fixed costs

Capacities

Capacities

(Combines plant location, warehouse location and sourcing)



t

e

m

j

ejej

n

i

t

e

ieie

l

h

n

i

hihi

t

e

ee

n

i

ii xcxcxcyfyF1 11 11 111

min

subject to:

0,,},1,0{,

0

0

1

1

11

1

1 1

1

hiieejei

j

t

e

ej

ee

m

j

ej

m

j

ej

n

i

ie

ii

t

e

ie

l

h

t

e

iehi

n

i

hhi

xxxyy

Dx

yWx

xx

yKx

xx

Sx

for i = 1, 2, 3, …, n

for i = 1, 2, 3, …, n

for e = 1, 2, 3, …, t

for e = 1, 2, 3, …, t

for j = 1, 2, 3, …, m

for h = 1, 2, 3, …, l

Warehouse capacity constraint

Warehouse flow balance

Market satisfaction

Plant flow balance

Plant capacity constraint

Source capacity constraint

Warehouse fixed

costs

Supplier-Plant

variable costs

Plant-Warehouse

variable costs

Warehouse-Market variable

costs

Plant fixed costs



Market 1D1

Market 1D1

Market 2D2

Market 2D2

Market mDm

Market mDm

Plant 1K1, F1, yi

Plant 1K1, F1, yi

Supplier 1S1

Supplier 1S1

Supplier lSl

Supplier lSl

Plant 2K2, F2 , yi

Plant 2K2, F2 , yi

Plant nKn, Fn , yi

Plant nKn, Fn , yi

Ware-house 1W1, f1 , ye

Ware-house 1W1, f1 , ye

Ware-house tWt, ft , ye

Ware-house tWt, ft , ye

X12, c12

Xtm, ctm

X21, c21

Xt2, ct2

X11, c11X11, c11

Xl2, cl2

Xln, cln

X11, c11

X21, c21

X2t, c2t

Xnt, cnt

xie, cie xej, cejxhi, chi



Parametersm number of markets or demand pointsn number of potential plant or factory

locationsl number of supplierst number of potential warehouse locations

Dj annual demand from customer j

Ki potential capacity of plant at site i

Sh supply capacity at supplier h

We potential warehouse capacity at site e

Fi fixed cost of locating a plant at site i

fe fixed cost of locating a warehouse at site e

chi cost of shipping one unit from supply source h to plant i

cie cost of producing and shipping one unit from plant i to warehouse e

cej cost of shipping one unit from warehouse e to customer j

Decision variables

yi 1 if plant is located at site i, 0 otherwise

ye 1 if warehouse is located at site e, 0 otherwise

xei quantity shipped from warehouse e to market j

xie quantity shipped from plant at site i to warehouse e

xhi quantity shipped from supplier h to plant at site i


The Assignment Problem• Arises in a variety of decision making situations:

– Jobs to machines– Agents to tasks– Sales personnel to sales territories– Contracts to bidders– Spacecraft to planetary missions– Nuclear warheads to targets

• Distinguishing feature of the basic assignment problem – One agent is assigned to one and only one task

• We seek a set of assignments that optimizes a stated objective– Minimize costs– Minimize time– Maximize profit– Maximize observation time– Maximize damage– …etc.


An Assignment Problem – Fowle Marketing

2

3

1

1

2

3

1015

9

3

18

5

6

9

14

Completion time in days

Project leaders (origin nodes)

Clients (destination

nodes)

Client 2 - 1

Client 3 - 1

Kari - 1

Client 1 - 1

Possible assignments

(arcs)

Gudmund - 1

Terry - 1

Supplies

Demands


The Assignment Problem

Parameterscij cost of assigning Agent i to Task jm number of Agentsn number of Tasks

Decision variablesxij assignment of Agent i to Task j, 0 if not assigned, 1 if assigned


A simple Assignment Problem solved


Table of ContentsChapter 6 (Transportation and Assignment

Problems)The P&T Company Distribution Problem (Section 6.1) 6.2–6.5Characteristics of Transportation Problems (Section 6.2) 6.6–6.14Variants of Transportation Problems: Better Products (Section 6.3) 6.15–6.17Variants of Transportation Problems: Nifty (Section 6.3) 6.18–6.20Applications of Transportation Problems: Metro Water (Section 6.4)

6.21–6.22Applications of Transportation Problems: Northern Airplane (Section 6.4)

6.23–6.25Applications of Transportation Problems: Middletown (Section 6.4) 6.26–6.28Applications of Transportation Problems: Energetic (Section 6.4) 6.29–6.31A Case Study: Texago Corp. Site Selection Problem (Section 6.5) 6.32–6.46Characteristics of Assignment Problems: Sellmore (Section 6.6) 6.47–6.51Variants of Assignment Problems: Job Shop (Section 6.7) 6.52-6.54Variants of Assignment Problems: Better Products (Section 6.7) 6.55Variants of Assignment Problems: Revised Middletown (Section 6.7)

6.56

Transportation & Assignment Problems (UW Lecture) 6.57–6.75These slides are based upon a lecture to second-year MBA students at the University of Washington that discusses transportation and assignment problems (as taught by one of the authors).


P&T Company Distribution Problem

CANNERY 1 Bellingham

CANNERY 2 Eugene

WAREHOUSE 1 Sacramento

WAREHOUSE 2 Salt Lake City

WAREHOUSE 3 Rapid City

WAREHOUSE 4 Albuquerque

CANNERY 3 Albert Lea


Shipping Data

Cannery OutputWarehous

eAllocation

Bellingham75

truckloadsSacramento

80 truckloads

Eugene125

truckloadsSalt Lake City

65 truckloads

Albert Lea100

truckloadsRapid City

70 truckloads

Total300

truckloadsAlbuquerque

85 truckloads

Total300

truckloads


Current Shipping Plan

Warehouse

From \ To SacramentoSalt Lake

CityRapid City

Albuquerque

Cannery

Bellingham 75 0 0 0

Eugene 5 65 55 0

Albert Lea 0 0 15 85


Shipping Cost per Truckload

Warehouse

From \ To Sacramento

Salt Lake City

Rapid CityAlbuquerq

ue

Cannery

Bellingham $464 $513 $654 $867

Eugene 352 416 690 791

Albert Lea 995 682 388 685

Total shipping cost = 75($464) + 5($352) + 65($416) + 55($690) + 15($388) + 85($685)

= $165,595


Terminology for a Transportation Problem

P&T Company Problem

Truckloads of canned peas

Canneries

Warehouses

Output from a cannery

Allocation to a warehouse

Shipping cost per truckload from a cannery to a warehouse

General Model

Units of a commodity

Sources

Destinations

Supply from a source

Demand at a destination

Cost per unit distributed from a source to a destination


Characteristics of Transportation Problems

• The Requirements Assumption– Each source has a fixed supply of units, where this entire supply must be

distributed to the destinations.– Each destination has a fixed demand for units, where this entire demand must be

received from the sources.

• The Feasible Solutions Property– A transportation problem will have feasible solutions if and only if the sum of its

supplies equals the sum of its demands.

• The Cost Assumption– The cost of distributing units from any particular source to any particular

destination is directly proportional to the number of units distributed.– This cost is just the unit cost of distribution times the number of units distributed.


The Transportation Model

• Any problem (whether involving transportation or not) fits the model for a transportation problem if:1. It can be described completely in terms of a table like Table 6.5 that

identifies all the sources, destinations, supplies, demands, and unit costs, and…

2. Satisfies both the requirements assumption and the cost assumption.

• The objective is to minimize the total cost of distributing the units.


The P&T Co. Transportation Problem data

Unit Cost

Destination(Warehous

e):Sacrame

ntoSalt Lake

CityRapid City

Albuquerque

Supply

Source (Cannery)

Bellingham $464 $513 $654 $867 75

Eugene 352 416 690 791 125

Albert Lea 995 682 388 685 100

Demand 80 65 70 85


Network Representation

S1

S2

S3

D4

D2

D1

D3

75

125

100

80

65

70

85

Supplies Demands

SourcesDestinations

(Bellingham)

(Eugene)

(Alber t Lea)

(Sacramento)

(Salt Lake City)

(Rapid City)

(Albuquerque)

464513

654867

352 416690

791

995 682

685

388


The Transportation Problem is an LP

Let xij = the number of truckloads to ship from cannery i to warehouse j(i = 1, 2, 3; j = 1, 2, 3, 4)

Minimize Cost = $464x11 + $513x12 + $654x13 + $867x14 + $352x21 + $416x22 + $690x23 + $791x24 + $995x31 + $682x32 + $388x33 + $685x34

subject to:Cannery 1: x11 + x12 + x13 + x14 = 75Cannery 2: x21 + x22 + x23 + x24 = 125Cannery 3: x31 + x32 + x33 + x34 = 100Warehouse 1: x11 + x21 + x31 = 80Warehouse 2: x12 + x22 + x32 = 65Warehouse 3: x13 + x23 + x33 = 70Warehouse 4: x14 + x24 + x34 = 85

andxij ≥ 0 (i = 1, 2, 3; j = 1, 2, 3, 4)

From Cannery 2 to all destinationsFrom Cannery 3 to all destinations

From Cannery 1 to all destinations

All canneries can supply all warehouses


Spreadsheet Formulation

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1011121314151617

B C D E F G H I JUnit Cost Destination (Warehouse)

Sacramento Salt Lake City Rapid City AlbuquerqueSource Bellingham $464 $513 $654 $867

(Cannery) Eugene $352 $416 $690 $791Albert Lea $995 $682 $388 $685

Shipment Quantity Destination (Warehouse)(Truckloads) Sacramento Salt Lake City Rapid City Albuquerque Total Shipped Supply

Source Bellingham 0 20 0 55 75 = 75(Cannery) Eugene 80 45 0 0 125 = 125

Albert Lea 0 0 70 30 100 = 100Total Received 80 65 70 85

= = = = Total CostDemand 80 65 70 85 $152,535


Integer Solutions Property

As long as all its supplies and demands have integer

values, any transportation problem with feasible

solutions is guaranteed to have an optimal solution with

integer values for all its decision variables. Therefore, it

is not necessary to add constraints to the model that

restrict these variables to only have integer values.


Distribution System at Proctor and Gamble

• Proctor and Gamble needed to consolidate and re-design their North American distribution system in the early 1990’s.– 50 product categories– 60 plants– 15 distribution centers– 1000 customer zones

• Solved many transportation problems (one for each product category).

• Goal: find best distribution plan, which plants to keep open, etc.

• Closed many plants and distribution centers, and optimized their product sourcing and distribution location.

• Implemented in 1996. Saved $200 million per year.For more details, see 1997 Jan-Feb Interfaces article, “Blending OR/MS,

Judgement, and GIS: Restructuring P&G’s Supply Chain”, downloadable at www.mhhe.com/hillier2e/articles


Better Products (Assigning Plants to Products)

The Better Products Company has decided to initiate the product of four new products, using three plants that currently have excess capacity.

Unit Cost

Product: 1 2 3 4

Capacity

Available

Plant

1 $41 $27 $28 $24 75

2 40 29 — 23 75

3 37 30 27 21 45

Required production

20 30 30 40

Question: Which plants should produce which products?


Transportation Problem Formulation

Unit Cost

Destination (Product): 1 2 3 4 Supply

Source(Plant)

1 $41 $27 $28 $24 75

2 40 29 — 23 75

3 37 30 27 21 45

Demand 20 30 30 40



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10111213141516

B C D E F G H IUnit Cost Product 1 Product 2 Product 3 Product 4

Plant 1 $41 $27 $28 $24Plant 2 $40 $29 - $23Plant 3 $37 $30 $27 $21

ProducedDaily Production Product 1 Product 2 Product 3 Product 4 At Plant Capacity

Plant 1 0 30 30 0 60 <= 75Plant 2 0 0 0 15 15 <= 75Plant 3 20 0 0 25 45 <= 45

Products Produced 20 30 30 40= = = = Total Cost

Required Production 20 30 30 40 $3,260


Nifty Co. (Choosing Customers)

• The Nifty Company specializes in the production of a single product, which it produces in three plants.

• Four customers would like to make major purchases. There will be enough to meet their minimum purchase requirements, but not all of their requested purchases.

• Due largely to variations in shipping cost, the net profit per unit sold varies depending on which plant supplies which customer.

Question: How many units should Nifty sell to each customer and how many units should they

ship from each plant to each customer?


Data for the Nifty Company

Unit Cost

Product: 1 2 3 4

Capacity

Available

Plant

1 $41 $27 $28 $24 75

2 40 29 — 23 75

3 37 30 27 21 45

Required production

20 30 30 40

Question: How many units should Nifty sell to each customer and how many units should

they ship from each plant to each customer?



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10111213141516171819

B C D E F G H IUnit Profit Customer 1 Customer 2 Customer 3 Customer 4

Plant 1 $55 $42 $46 $53Plant 2 $37 $18 $32 $48Plant 3 $29 $59 $51 $35

Total ProductionShipment Customer 1 Customer 2 Customer 3 Customer 4 Production Quantity

Plant 1 7,000 0 1,000 0 8,000 = 8,000Plant 2 0 0 0 5,000 5,000 = 5,000Plant 3 0 6,000 1,000 0 7,000 = 7,000

Min Purchase 7,000 3,000 2,000 0<= <= <= <= Total Profit

Total Shipped 7,000 6,000 2,000 5,000 $1,076,000<= <= <= <=

Max Purchase 7,000 9,000 6,000 8,000


Metro Water (Distributing Natural Resources)

Metro Water District is an agency that administers water distribution in a large geographic region. The region is arid, so water must be brought in from outside the region.

– Sources of imported water: Colombo, Sacron, and Calorie rivers.– Main customers: Cities of Berdoo, Los Devils, San Go, and Hollyglass.

Cost per Acre Foot

BerdooLos

DevilsSan Go

Hollyglass

Available

Colombo River $160 $130 $220 $170 5

Sacron River 140 130 190 150 6

Calorie River 190 200 230 — 5

Needed 2 5 4 1.5(million

acre feet)Question: How much water should Metro take from each river,

and how much should they send from each river to each city?



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1011121314151617

B C D E F G H IUnit Cost ($millions) Berdoo Los Devils San Go Hollyglass

Colombo River 160 130 220 170Sacron River 140 130 190 150Calorie River 190 200 230 -

Water Distribution Total(million acre-feet) Berdoo Los Devils San Go Hollyglass From River Available

Colombo River 0 5 0 0 5 <= 5Sacron River 2 0 2.5 1.5 6 <= 6Calorie River 0 0 1.5 0 1.5 <= 5Total To City 2 5 4 1.5

= = = = Total CostNeeded 2 5 4 1.5 ($million)

1,975


Northern Airplane (Production Scheduling)

Northern Airplane Company produces commercial airplanes. The last stage in production is to produce the jet engines and install them.

– The company must meet the delivery deadline indicated in column 2.– Production and storage costs vary from month to month.

Maximum Production

Unit Cost of Production ($million)

Unit Costof Storage($thousan

d)Month

ScheduledInstallatio

ns

Regular

TimeOvertim

e

Regular

TimeOvertim

e

1 10 20 10 1.08 1.10 15

2 15 30 15 1.11 1.12 15

3 25 25 10 1.10 1.11 15

4 20 5 10 1.13 1.15Question: How many engines should be produced in each of the four months so that the total of the production and storage costs will be

minimized?



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101112131415161718192021222324252627282930313233343536

B C D E F G H I JProduction Cost Regular Storage Cost($millions) Time Overtime ($millions per month)

Month 1 1.08 1.10 0.015Month 2 1.11 1.12Month 3 1.10 1.11Month 4 1.13 1.15

Unit Cost($millions) 1 2 3 4

1 (RT) 1.08 1.10 1.11 1.131 (OT) 1.10 1.12 1.13 1.152 (RT) - 1.11 1.13 1.14

Month 2 (OT) - 1.12 1.14 1.15Produced 3 (RT) - - 1.10 1.12

3 (OT) - - 1.11 1.134 (RT) - - - 1.134 (OT) - - - 1.15

MaximumUnits Produced 1 2 3 4 Produced Production

1 (RT) 10 5 0 5 20 <= 201 (OT) 0 0 0 0 0 <= 102 (RT) 0 10 0 0 10 <= 30

Month 2 (OT) 0 0 0 0 0 <= 15Produced 3 (RT) 0 0 25 0 25 <= 25

3 (OT) 0 0 0 10 10 <= 104 (RT) 0 0 0 5 5 <= 54 (OT) 0 0 0 0 0 <= 10

Installed 10 15 25 20= = = = Total Cost

Scheduled Installations 10 15 25 20 ($millions)77.4

Month Installed

Month Installed


Optimal Production at Northern Airplane

Month

1 (RT)

2 (RT)

3 (RT)

3 (OT)

4 (RT)

Production

20

10

25

10

5

Installations

10

15

25

0

20

Stored

10

5

5

10

0


Middletown School District

• Middletown School District is opening a third high school and thus needs to redraw the boundaries for the area of the city that will be assigned to the respective schools.

• The city has been divided into 9 tracts with approximately equal populations.

• Each school has a minimum and maximum number of students that should be assigned.

• The school district management has decided that the appropriate objective is to minimize the average distance that students must travel to school.

Question: How many students from each tract should be assigned to each school?


Data for the Middletown School District

Distance (Miles) to School

Tract 1 2 3Number of High School Students

1 2.2 1.9 2.5 500

2 1.4 1.3 1.7 400

3 0.5 1.8 1.1 450

4 1.2 0.3 2.0 400

5 0.9 0.7 1.0 500

6 1.1 1.6 0.6 450

7 2.7 0.7 1.5 450

8 1.8 1.2 0.8 400

9 1.5 1.7 0.7 500

Minimum enrollment

1,200 1,100 1,000

Maximum enrollment

1,800 1,700 1,500



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10111213141516171819202122232425262728293031

B C D E F G HDistance (Miles) School 1 School 2 School 3

Tract 1 2.2 1.9 2.5Tract 2 1.4 1.3 1.7Tract 3 0.5 1.8 1.1Tract 4 1.2 0.3 2Tract 5 0.9 0.7 1Tract 6 1.1 1.6 0.6Tract 7 2.7 0.7 1.5Tract 8 1.8 1.2 0.8Tract 9 1.5 1.7 0.7

Number of Total TotalStudents School 1 School 2 School 3 From Tract In Tract

Tract 1 0 500 0 500 = 500Tract 2 400 0 0 400 = 400Tract 3 450 0 0 450 = 450Tract 4 0 400 0 400 = 400Tract 5 350 150 0 500 = 500Tract 6 0 0 450 450 = 450Tract 7 0 450 0 450 = 450Tract 8 0 0 400 400 = 400Tract 9 0 0 500 500 = 500

Min Enrollment 1,200 1,500 1,350<= <= <= Total Distance

Total At School 1,200 1,500 1,350 (miles)<= <= <= 3,530

Max Enrollment 1,800 1,700 1,500


Energetic (Meeting Energy Needs)

• The Energetic Company needs to make plans for the energy systems for a new building.

• The energy needs fall into three categories:– electricity (20 units)– heating water (10 units)– heating space (30 units)

• The three possible sources of energy are– electricity– natural gas– solar heating unit (limited to 30 units because of roof size)

Question: How should Energetic meet the energy needs for the new building?


Cost Data for Energetic

Unit Cost

Energy Need: Electricity Water Heating Space Heating

Source of Energy

Electricity $400 $500 $600

Natural gas — 600 500

Solar heater — 300 400



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101112131415161718

B C D E F G H IEnergy Need

Unit Cost ($/day) Electricity Water Heating Space HeatingSource Electricity 400 500 600

of Natural Gas - 600 500Energy Solar Heater - 300 400

Energy Need TotalDaily Energy Use Electricity Water Heating Space Heating Used

Source Electricity 20 0 0 20of Natural Gas 0 0 10 10 Max Solar

Energy Solar Heater 0 10 20 30 <= 30Total Supplied 20 10 30

= = = Total CostDemand 20 10 30 ($/day)

24,000


Location of Texago’s Facilities

Type of Facility Locations

Oil fields 1. Several in Texas2. Several in California3. Several in Alaska

Refineries 1. Near New Orleans, Louisiana2. Near Charleston, South Carolina3. Near Seattle, Washington

Distribution Centers 1. Pittsburgh, Pennsylvania2. Atlanta, Georgia3. Kansas City, Missouri4. San Francisco, California


Potential Sites for Texago’s New Refinery

Potential Site Main Advantages

Near Los Angeles, California

1. Near California oil fields.2. Ready access from Alaska oil fields.3. Fairly near San Francisco distribution center.

Near Galveston, Texas 1. Near Texas oil fields.2. Ready access from Middle East imports.3. Near corporate headquarters.

Near St. Louis, Missouri 1. Low operating costs.2. Centrally located for distribution centers.3. Ready access to crude oil via the Mississippi River.


Production Data for Texago

Refinery

Crude OilNeeded Annually(Million Barrels) Oil Fields

Crude Oil Produced Annually(Million Barrels)

New Orleans

100 Texas 80

Charleston 60 California 60

Seattle 80 Alaska 100

New site 120 Total 240

Total 360 Needed imports = 360 – 240 = 120


Cost Data for Shipping to Refineries

Cost per Unit Shipped to Refinery or Potential Refinery

(Millions of Dollars per Million Barrels)

New Orlea

nsCharlest

onSeattl

e

Los Angel

esGalves

tonSt.

Louis

Source

Texas 2 4 5 3 1 1

California

5 5 3 1 3 4

Alaska 5 7 3 4 5 7

Middle East

2 3 5 4 3 4


Cost Data for Shipping to Distribution Centers

Cost per Unit Shipped to Distribution Center

(Millions of Dollars)

Pittsburgh Atlanta

Kansas City

San Francisco

Refinery

New Orleans 6.5 5.5 6 8

Charleston 7 5 4 7

Seattle 7 8 4 3

Potential Refinery

Los Angeles 8 6 3 2

Galveston 5 4 3 6

St. Louis 4 3 1 5

Number of units needed

100 80 80 100


Estimated Operating Costs for Refineries

Site Annual Operating Cost

(Millions of Dollars)

Los Angeles

Galveston

St. Louis

620

570

530


Basic Spreadsheet for Shipping to Refineries

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B C D E F G H I JRefineries

Unit Cost ($millions) New Orleans Charleston Seattle New SiteTexas 2 4 5

Oil California 5 5 3Fields Alaska 5 7 3

Middle East 2 3 5

Shipment Quantity Refineries(millions of barrels) New Orleans Charleston Seattle New Site Total Shipped Supply

Texas 0 0 0 0 0 = 80Oil California 0 0 0 0 0 = 60

Fields Alaska 0 0 0 0 0 = 100Middle East 0 0 0 0 0 = 120

Total Received 0 0 0 0= = = = Total Cost

Demand 100 60 80 120 ($millions)0


Shipping to Refineries, Including Los Angeles

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Unit Cost ($millions) New Orleans Charleston Seattle Los AngelesTexas 2 4 5 3

Oil California 5 5 3 1Fields Alaska 5 7 3 4

Middle East 2 3 5 4

Shipment Quantity Refineries(millions of barrels) New Orleans Charleston Seattle Los Angeles Total Shipped Supply






Shipping to Refineries, Including Galveston

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Unit Cost ($millions) New Orleans Charleston Seattle GalvestonTexas 2 4 5 1


Middle East 2 3 5 3

Shipment Quantity Refineries(millions of barrels) New Orleans Charleston Seattle Galveston Total Shipped Supply






Shipping to Refineries, Including St. Louis

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Unit Cost ($millions) New Orleans Charleston Seattle St. LouisTexas 2 4 5 1


Middle East 2 3 5 4

Shipment Quantity Refineries(millions of barrels) New Orleans Charleston Seattle St. Louis Total Shipped Supply






Basic Spreadsheet for Shipping to D.C.’s

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B C D E F G H I JDistribution Center

Unit Cost ($millions) Pittsburgh Atlanta Kansas City San FranciscoNew Orleans 6.5 5.5 6 8

Refineries Charleston 7 5 4 7Seattle 7 8 4 3

New Site

Shipment Quantity Distribution Center(millions of barrels) Pittsburgh Atlanta Kansas City San Francisco Total Shipped Supply

New Orleans 0 0 0 0 0 = 100Refineries Charleston 0 0 0 0 0 = 60

Seattle 0 0 0 0 0 = 80New Site 0 0 0 0 0 = 120




Shipping to D.C.’s When Choose Los Angeles

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Los Angeles 8 6 3 2



Seattle 20 0 0 60 80 = 80Los Angeles 0 0 80 40 120 = 120


Demand 100 80 80 100 ($millions)1,570


Shipping to D.C.’s When Choose Galveston

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Galveston 5 4 3 6



Seattle 0 0 0 80 80 = 80Galveston 0 20 80 20 120 = 120


Demand 100 80 80 100 ($millions)1,630


Shipping to D.C.’s When Choose St. Louis

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St. Louis 4 3 1 5



Seattle 0 0 0 80 80 = 80St. Louis 0 20 80 20 120 = 120


Demand 100 80 80 100 ($millions)1,430


Annual Variable Costs

Site

Total Costof

ShippingCrude Oil

Total Costof Shipping

Finished Product

Operating Cost

for NewRefinery

TotalVariable

Cost

Los Angeles $880 million $1.57 billion $620 million$3.07 billion

Galveston 920 million 1.63 billion 570 million 3.12 billion

St. Louis 960 million 1.43 billion 530 million 2.92 billion


Sellmore Company Assignment Problem

• The marketing manager of Sellmore Company will be holding the company’s annual sales conference soon.

• He is hiring four temporary employees:– Ann– Ian– Joan– Sean

• Each will handle one of the following four tasks:– Word processing of written presentations– Computer graphics for both oral and written presentations– Preparation of conference packets, including copying and organizing materials– Handling of advance and on-site registration for the conference

Question: Which person should be assigned to which task?


Data for the Sellmore Problem

Required Time per Task (Hours)

TemporaryEmployee

WordProcessin

g Graphics PacketsRegistration

sHourlyWage

Ann 35 41 27 40 $14

Ian 47 45 32 51 12

Joan 39 56 36 43 13

Sean 32 51 25 46 15



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B C D E F G H I J

Required Time Word Hourly(Hours) Processing Graphics Packets Registrations Wage

Ann 35 41 27 40 $14Assignee Ian 47 45 32 51 $12

Joan 39 56 36 43 $13Sean 32 51 25 46 $15

WordCost Processing Graphics Packets Registrations

Ann $490 $574 $378 $560Assignee Ian $564 $540 $384 $612

Joan $507 $728 $468 $559Sean $480 $765 $375 $690

Word TotalAssignment Processing Graphics Packets Registrations Assignments Supply

Ann 0 0 1 0 1 = 1Assignee Ian 0 1 0 0 1 = 1

Joan 0 0 0 1 1 = 1Sean 1 0 0 0 1 = 1

Total Assigned 1 1 1 1= = = = Total Cost

Demand 1 1 1 1 $1,957

Task

Task

Task


The Model for Assignment Problems

Given a set of tasks to be performed and a set of assignees who are available to perform these tasks, the problem is to determine which assignee should be assigned to each task.

To fit the model for an assignment problem, the following assumptions need to be satisfied:

1. The number of assignees and the number of tasks are the same.

2. Each assignee is to be assigned to exactly one task.

3. Each task is to be performed by exactly one assignee.

4. There is a cost associated with each combination of an assignee performing a task.

5. The objective is to determine how all the assignments should be made to minimize the total cost.


The Network Representation

A2

A1

T4A4

T3A3

T2

T1

Assignees Tasks

490

540

468

690

(Ann)

(I an)

(Joan)

(Sean)

(Word processing)

(Graphics)

(Packets)

(Registrations)

574

378560

564

384612

507 728

559

480

765

375


Job Shop (Assigning Machines to Locations)

• The Job Shop Company has purchased three new machines of different types.

• There are five available locations where the machine could be installed.

• Some of these locations are more desirable for particular machines because of their proximity to work centers that will have a heavy work flow to these machines.

Question: How should the machines be assigned to locations?


Materials-Handling Cost Data

Cost per Hour

Location: 1 2 3 4 5

Machine

1 $13 $16 $12 $14 $15

2 15 — 13 20 16

3 4 7 10 6 7



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B C D E F G H I JCost ($/hour) Location 1 Location 2 Location 3 Location 4 Location 5

Machine 1 13 16 12 14 15Machine 2 15 - 13 20 16Machine 3 4 7 10 6 7

TotalAssignment Location 1 Location 2 Location 3 Location 4 Location 5 Assignments Supply

Machine 1 0 0 0 1 0 1 = 1Machine 2 0 0 1 0 0 1 = 1Machine 3 1 0 0 0 0 1 = 1

Total Assigned 1 0 1 1 0<= <= <= <= <= Total Cost

Demand 1 1 1 1 1 ($/hour)31


Better Products (No Product Splitting)

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B C D E F G H IUnit Cost Product 1 Product 2 Product 3 Product 4

Plant 1 $41 $27 $28 $24Plant 2 $40 $29 - $23Plant 3 $37 $30 $27 $21

Required Production 20 30 30 40

Cost ($/day) Product 1 Product 2 Product 3 Product 4Plant 1 $820 $810 $840 $960Plant 2 $800 $870 - $920Plant 3 $740 $900 $810 $840

TotalAssignment Product 1 Product 2 Product 3 Product 4 Assignments Supply

Plant 1 0 1 1 0 2 <= 2Plant 2 1 0 0 0 1 <= 2Plant 3 0 0 0 1 1 = 1

Total Assigned 1 1 1 1= = = = Total Cost

Demand 1 1 1 1 $3,290


Middletown School District (No Tract Splitting)

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B C D E F G H I J KDistance Number of Cost(Miles) School 1 School 2 School 3 Students (Miles) School 1 School 2 School 3

Tract 1 2.2 1.9 2.5 500 Tract 1 1100 950 1250Tract 2 1.4 1.3 1.7 400 Tract 2 560 520 680Tract 3 0.5 1.8 1.1 450 Tract 3 225 810 495Tract 4 1.2 0.3 2 400 Tract 4 480 120 800Tract 5 0.9 0.7 1 500 Tract 5 450 350 500Tract 6 1.1 1.6 0.6 450 Tract 6 495 720 270Tract 7 2.7 0.7 1.5 450 Tract 7 1215 315 675Tract 8 1.8 1.2 0.8 400 Tract 8 720 480 320Tract 9 1.5 1.7 0.7 500 Tract 9 750 850 350

TotalAssignment School 1 School 2 School 3 Assignments Supply

Tract 1 0 1 0 1 = 1Tract 2 1 0 0 1 = 1Tract 3 1 0 0 1 = 1Tract 4 0 1 0 1 = 1Tract 5 1 0 0 1 = 1Tract 6 0 0 1 1 = 1Tract 7 0 1 0 1 = 1Tract 8 0 0 1 1 = 1Tract 9 0 0 1 1 = 1

Total Assigned 3 3 3= = = Total Distance

Demand 3 3 3 (Miles)3560


The Transportation Problem

• A common problem in logistics is how to transport goods from a set of sources (e.g., plants, warehouses, etc.) to a set of destinations (e.g., warehouses, customers, etc.) at the minimum possible cost.

• Given– a set of sources, each with a given supply,– a set of destinations, each with a given demand,– a cost table (cost/unit to ship from each source to each destination)

• Goal– Choose shipping quantities from each source to each destination so as to

minimize total shipping cost.


The Network Representation

Sources Destinations

Supply1

Supply2

Supply3

Demand1

Demand2

Demand3

Demand4CostijShipment Quantityij


Transportation Problem ExampleA company has two plants (in Seattle and Atlanta) producing a certain product that is to be shipped to three distribution centers (in Sacramento, St. Louis, and Pittsburgh).

– The unit production costs are the same at the two plants, and the shipping costs per unit are shown in the table below.

– Shipments are made once per week.– During each week, each plant produces at most 60 units and each distribution center needs

at least 40 units.Unit Shipping Cost

Distribution Center

Sacramento

St. LouisPittsbur

gh

PlantSeattle $2 $6 $8

Atlanta $7 $5 $3Question: How many units should be shipped from

each plant to each distribution center?


Spreadsheet Solution

3

45678

9

101112131415

B C D E F G HSacramento St. Louis Pittsburgh

Cost Dist. Center Dist. Center Dist. CenterSeattle Plant $2 $6 $8Atlanta Plant $7 $5 $3

Shipment Sacramento St. Louis Pittsburgh

Quantities Dist. Center Dist. Center Dist. Center Shipped AvailableSeattle Plant 40 20 0 60 <= 60Atlanta Plant 0 20 40 60 <= 60

Shipped 40 40 40 Cost = $420>= >= >=

Needed 40 40 40


Shipping from D.C.’s to Customers

The same company ships one of its products from its three distribution centers to four different customers

– The shipping costs per unit are shown in the table below.– Shipments are made once per week.– During each week, each distribution center has received 40 units.– Customer demand is also shown in the table below.

Unit Shipping Cost

Customer

1 2 3 4

Distribution

Center

Sacramento

$8 $10 $7 $11

St. Louis $12 $11 $9 $6

Pittsburgh $10 $9 $15 $10

Customer Demand 40 30 25 25Question: How many units should be shipped from each distribution center to each customer?


Spreadsheet Solution

345678

9

10111213141516

B C D E F G H I

Cost Customer 1 Customer 2 Customer 3 Customer 4Sacramento DC $8 $10 $7 $11

St. Louis DC $12 $11 $9 $6Pittsburgh DC $10 $9 $15 $10

ShipmentQuantities Customer 1 Customer 2 Customer 3 Customer 4 Shipped Available

Sacramento DC 30 0 10 0 40 <= 40St. Louis DC 0 0 15 25 40 <= 40

Pittsburgh DC 10 30 0 0 40 <= 40Shipped 40 30 25 25 Cost = $965

>= >= >= >=Needed 40 30 25 25


Managing the Whole Supply Chain(Plant to D.C. to Customer)

3

45678

9

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1819

202122232425

26

272829303132333435

B C D E F G H ISacramento St. Louis Pittsburgh

Cost Dist. Center Dist. Center Dist. CenterSeattle Plant $2 $6 $8Atlanta Plant $7 $5 $3

Shipment Sacramento St. Louis Pittsburgh


Shipped 60 25 35 Cost = $350

Distribution from DC's to Customers


St. Louis DC $12 $11 $9 $6Pittsburgh DC $10 $9 $15 $10


Sacramento DC 35 0 25 0 60 <= 60St. Louis DC 0 0 0 25 25 <= 25

Pittsburgh DC 5 30 0 0 35 <= 35Shipped 40 30 25 25 Cost = $925

>= >= >= >=Needed 40 30 25 25

Total Cost = $1,275


Site Selection

• The lease is up on their distribution center in St. Louis. They now must decide whether to sign a new lease in St. Louis, or move the distribution center to a new location.

• One possible new location is Omaha, Nebraska, which is offering a better deal on the lease.

Question: Should they move their distribution center to Omaha?


Spreadsheet Solution to Site Selection

123

45678

9

1011121314151617

1819

202122232425

26

272829303132333435

A B C D E F G H I

Distribution from Plants to DC's

Sacramento Omaha PittsburghCost Dist. Center Dist. Center Dist. Center

Seattle Plant $2 $5 $8Atlanta Plant $7 $6 $3

Shipment Sacramento Omaha Pittsburgh


Shipped 60 0 60 Cost = $300

Distribution from DC's to Customers


Omaha DC $13 $10 $8 $8Pittsburgh DC $10 $9 $15 $10


Sacramento DC 35 0 25 0 60 <= 60Omaha DC 0 0 0 0 0 <= 0

Pittsburgh DC 5 30 0 25 60 <= 60Shipped 40 30 25 25 Cost = $1,025

>= >= >= >=Needed 40 30 25 25

Total Cost = $1,325


Distribution System at Proctor and Gamble

• Proctor and Gamble needed to consolidate and re-design their North American distribution system in the early 1990’s.

– 50 product categories– 60 plants– 15 distribution centers– 1000 customer zones

• Solved many transportation problems (one for each product category).

• Goal: find best distribution plan, which plants to keep open, etc.

• Closed many plants and distribution centers, and optimized their product sourcing and distribution location.

• Implemented in 1996. Saved $200 million per year.

For more details, see 1997 Jan-Feb Interfaces article, “Blending OR/MS, Judgement, and GIS: Restructuring P&G’s Supply Chain”, downloadable at

www.mhhe.com/hillier2e/articles


The Assignment Problem

• The job of assigning people (or machines or whatever) to a set of tasks is called an assignment problem.

• Given– a set of assignees– a set of tasks– a cost table (cost associated with each assignee performing each task)

• Goal– Match assignees to tasks so as to perform all of the tasks at the minimum

possible cost.


Network Representation

Assignees Tasks

Costij


Assignment Problem Example

The coach of a swim team needs to assign swimmers to a 200-yard medley relay team (four swimmers, each swims 50 yards of one of the four strokes). Since most of the best swimmers are very fast in more than one stroke, it is not clear which swimmer should be assigned to each of the four strokes. The five fastest swimmers and their best times (in seconds) they have achieved in each of the strokes (for 50 yards) are shown below.

Backstroke

Breaststroke

Butterfly Freestyle

Carl 37.7 43.4 33.3 29.2

Chris 32.9 33.1 28.5 26.4

David 33.8 42.2 38.9 29.6

Tony 37.0 34.7 30.4 28.5

Ken 35.4 41.8 33.6 31.1

Question: How should the swimmers be assigned to make the fastest relay team?


Algebraic Formulation

Let xij = 1 if swimmer i swims stroke j; 0 otherwisetij = best time of swimmer i in stroke j

Minimize Time = ∑ i ∑ j tij xij

subject to:

each stroke swum: ∑ i xij = 1 for each stroke j

each swimmer swims 1: ∑ j xij ≤ 1 for each swimmer i

andxij ≥ 0 for all i and j.



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10

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B C D E F G H I

Best Times Backstroke Breastroke Butterfly FreestyleCarl 37.7 43.4 33.3 29.2Chris 32.9 33.1 28.5 26.4David 33.8 42.2 38.9 29.6Tony 37.0 34.7 30.4 28.5Ken 35.4 41.8 33.6 31.1

Assignment Backstroke Breastroke Butterfly FreestyleCarl 0 0 0 1 1 <= 1Chris 0 0 1 0 1 <= 1David 1 0 0 0 1 <= 1Tony 0 1 0 0 1 <= 1Ken 0 0 0 0 0 <= 1

1 1 1 1 Time = 126.2= = = =1 1 1 1


Bidding for Classes

• In the MBA program at a prestigious university in the Pacific Northwest, students bid for electives in the second year of their program.

• Each of the 10 students has 100 points to bid (total) and must take two electives.

• There are four electives available:– Quantitative Methods– Finance– Operations Management– Accounting

• Each class is limited to 5 students.

Question: How should students be assigned to the classes?


Points Bid for Electives

Electives

Student

Quantitative

Methods Finance

OperationsManageme

nt Accounting

George 60 10 10 20

Fred 20 20 40 20

Ann 45 45 5 5

Eric 50 20 5 25

Susan 30 30 30 10

Liz 50 50 0 0

Ed 70 20 10 0

David 25 25 35 15

Tony 35 15 35 15

Jennifer 60 10 10 20


Spreadsheet Solution(Maximizing Total Points)

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15

1617181920212223242526272829

B C D E F G H I J K

Points QMETH Finance Op Mgt. AccountingGeorge 60 10 10 20

Fred 20 20 40 20Ann 45 45 5 5Eric 50 20 5 25

Susan 30 30 30 10Liz 50 50 0 0Ed 70 20 10 0

David 25 25 35 15Tony 35 15 35 15


Total Classes StudentAssignment QMETH Finance Op Mgt. Accounting Classes to Take Points

George 1 0 0 1 2 = 2 80Fred 0 0 1 1 2 = 2 60Ann 1 1 0 0 2 = 2 90Eric 0 1 0 1 2 = 2 45

Susan 0 1 1 0 2 = 2 60Liz 1 1 0 0 2 = 2 100Ed 1 0 1 0 2 = 2 80

David 0 1 1 0 2 = 2 60Tony 0 0 1 1 2 = 2 50

Jennifer 1 0 0 1 2 = 2 805 5 5 5

<= <= <= <= Total Points = 705Capacity 5 5 5 5


Spreadsheet Solution(Maximizing the Minimum Student Point

Total)

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15

161718192021222324252627282930

B C D E F G H I J K L M

Points QMETH Finance Op Mgt. AccountingGeorge 60 10 10 20

Fred 20 20 40 20Ann 45 45 5 5Eric 50 20 5 25

Susan 30 30 30 10Liz 50 50 0 0Ed 70 20 10 0

David 25 25 35 15Tony 35 15 35 15


Total Classes MinAssignment QMETH Finance Op Mgt. Accounting Classes to Take Points Points

George 1 0 0 1 2 = 2 80 >= 50Fred 0 1 1 0 2 = 2 60 >= 50Ann 0 1 0 1 2 = 2 50 >= 50Eric 1 0 1 0 2 = 2 55 >= 50

Susan 0 1 1 0 2 = 2 60 >= 50Liz 0 1 0 1 2 = 2 50 >= 50Ed 1 0 0 1 2 = 2 70 >= 50

David 0 1 1 0 2 = 2 60 >= 50Tony 1 0 1 0 2 = 2 70 >= 50

Jennifer 1 0 0 1 2 = 2 80 >= 505 5 5 5

<= <= <= <= Total Points = 635Capacity 5 5 5 5

Min Points = 50

mcgraw-hill/irwin © the mcgraw-hill companies, inc., 2003 6.1 transportation and assignment models

Documents

mcgrawhill companies

transportation cost

general transportation

theme slide

network optimization

distribution of goods

supply locations

downstream slide