11 – 1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Location 11 For...

11 – 1Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall.

LocationLocation11

For For Operations Management, 9eOperations Management, 9e by by Krajewski/Ritzman/Malhotra Krajewski/Ritzman/Malhotra © 2010 Pearson Education© 2010 Pearson Education

PowerPoint Slides PowerPoint Slides by Jeff Heylby Jeff Heyl


Location DecisionsLocation Decisions

Location decisions affect processes and departments

Marketing Human resources Accounting and finance Operations International operations



Many factors Sensitive to location High impact on the company’s ability to meet its goals Divide location factors

Dominant factors in manufacturing Favorable labor climate Proximity to markets Quality of life Proximity to suppliers and resources Proximity to the parent company’s facilities Utilities, taxes, and real estate costs Other factors



Dominant factors in services

Impact of location on sales and customer satisfaction

Proximity to customers Transportation costs and proximity to markets Location of competitors Site-specific factors


Geographic Information SystemsGeographic Information Systems

GIS is a system of computer software, hardware, and data

Use to manipulate, analyze, and present information relevant to a location decision

Create a visual representation of a firm’s location choices

Useful decision-making tool

Using GIS to identify locations and demographic customer segments

Identifying locations that relate to target market Part of an array of decision-making tools


Locating a Single FacilityLocating a Single Facility

Expand onsite, build another facility, or relocate to another site Onsite expansion Building a new plant or moving to a new retail

or office space

Comparing several sites


Selecting a New FacilitySelecting a New Facility

Step 1: Identify the important location factors and categorize them as dominant or secondary

Step 2: Consider alternative regions; then narrow to alternative communities and finally specific sites

Step 3: Collect data on the alternatives

Step 4: Analyze the data collected, beginning with the quantitative factors

Step 5: Bring the qualitative factors pertaining to each site into the evaluation


EXAMPLE 11.1

A new medical facility, Health-Watch, is to be located in Erie, Pennsylvania. The following table shows the location factors, weights, and scores (1 = poor, 5 = excellent) for one potential site. The weights in this case add up to 100 percent. A weighted score (WS) will be calculated for each site. What is the WS for this site?

Calculating Weighted ScoresCalculating Weighted Scores

Location Factor Weight Score

Total patient miles per month 25 4

Facility utilization 20 3

Average time per emergency trip 20 3

Expressway accessibility 15 4

Land and construction costs 10 1

Employee preferences 10 5


SOLUTION

The WS for this particular site is calculated by multiplying each factor’s weight by its score and adding the results:

Calculating Weighted ScoresCalculating Weighted Scores

Location Factor Weight Score

Total patient miles per month 25 4

Facility utilization 20 3

Average time per emergency trip 20 3

Expressway accessibility 15 4

Land and construction costs 10 1

Employee preferences 10 5

WS =(25 4) + (20 3) + (20 3) + (15 4) + (10 1) + (10 5)

= 100 + 60 + 60 + 60 + 10 + 50

= 340

The total WS of 340 can be compared with the total weighted scores for other sites being evaluated.


Management is considering three potential locations for a new cookie factory. They have assigned scores shown below to the relevant factors on a 0 to 10 basis (10 is best). Using the preference matrix, which location would be preferred?

Application 11.1Application 11.1

LocationFactor Weight The

NeighborhoodSesameStreet

Ronald’sPlayhouse

Material Supply 0.1 5 9 8

Quality of Life 0.2 9 8 4

Mild Climate 0.3 10 6 8

Labor Skills 0.4 3 4 7


0.9

1.6

1.8

1.6

5.9

Management is considering three potential locations for a new cookie factory. They have assigned scores shown below to the relevant factors on a 0 to 10 basis (10 is best). Using the preference matrix, which location would be preferred?


0.5

1.8

3.0

1.2

6.5

0.8

0.8

2.4

2.8

6.8

LocationFactor Weight The

NeighborhoodSesameStreet

Ronald’sPlayhouse

Material Supply 0.1 5 9 8

Quality of Life 0.2 9 8 4

Mild Climate 0.3 10 6 8

Labor Skills 0.4 3 4 7


Load-Distance (Load-Distance (ldld) Method) Method

Identify and compare candidate locations Like weighted-distance method Select a location that minimizes the sum of

the loads multiplied by the distance the load travels

Time may be used instead of distance


Load-Distance (Load-Distance (ldld) Method) Method

Calculating a load-distance score Varies by industry Use the actual distance to calculate ld score Use rectangular or Euclidean distances Different measures for distance Find one acceptable facility location that

minimizes the ld score

ld = lidii

Formula for the ld score



What is the distance between (20, 10) and (80, 60)?

SOLUTION

Euclidean distance:

dAB = (xA – xB)2 + (yA – yB)2 = (20 – 80)2 + (10 – 60)2 = 78.1

Rectilinear distance:

dAB = |xA – xB| + |yA – yB| = |20 – 80| + |10 – 60| = 110



Management is investigating which location would be best to position its new plant relative to two suppliers (located in Cleveland and Toledo) and three market areas (represented by Cincinnati, Dayton, and Lima). Management has limited the search for this plant to those five locations. The following information has been collected. Which is best, assuming rectilinear distance?

Location x,y coordinates Trips/year

Cincinnati (11,6) 15

Dayton (6,10) 20

Cleveland (14,12) 30

Toledo (9,12) 25

Lima (13,8) 40



SOLUTION

Calculations:

Location x,y coordinates Trips/year

Cincinnati (11,6) 15

Dayton (6,10) 20

Cleveland (14,12) 30

Toledo (9,12) 25

Lima (13,8) 40

15(9) + 20(0) + 30(10) + 25(5) + 40(9) = 920

15(9) + 20(10) + 30(0) + 25(5) + 40(5) = 660

15(8) + 20(5) + 30(0) + 25(0) + 40(8) = 690

15(4) + 20(9) + 30(5) + 25(8) + 40(0) = 590

15(0) + 20(9) + 30(9) + 25(8) + 40(4) = 810Cincinnati =

Dayton =

Cleveland =

Toledo =

Lima =


Center of Gravity MethodCenter of Gravity Method

A good starting point Find x coordinate, x*, by multiplying each

point’s x coordinate by its load (lt), summing these products li xi, and dividing by li

The center of gravity’s y coordinate y* found the same way

Generally not the optimal location

x* =li xi

li

i

i

y* =li yi

li

i

i


Finding the Center of GravityFinding the Center of Gravity

EXAMPLE 11.2

A supplier to the electric utility industry produces power generators; the transportation costs are high. One market area includes the lower part of the Great Lakes region and the upper portion of the southeastern region. More than 600,000 tons are to be shipped to eight major customer locations as shown below:

Customer Location Tons Shipped x, y Coordinates

Three Rivers, MI 5,000 (7, 13)

Fort Wayne, IN 92,000 (8, 12)

Columbus, OH 70,000 (11, 10)

Ashland, KY 35,000 (11, 7)

Kingsport, TN 9,000 (12, 4)

Akron, OH 227,000 (13, 11)

Wheeling, WV 16,000 (14, 10)

Roanoke, VA 153,000 (15, 5)



What is the center of gravity for the electric utilities supplier? Using rectilinear distance, what is the resulting load–distance score for this location?



Fort Wayne, IN 92,000 (8, 12)

Columbus, OH 70,000 (11, 10)

Ashland, KY 35,000 (11, 7)


Akron, OH 227,000 (13, 11)

Wheeling, WV 16,000 (14, 10)

Roanoke, VA 153,000 (15, 5)SOLUTION

The center of gravity is calculated as shown below:

x* = =li xi

li

i

i

li = i

li xi =i

5 + 92 + 70 + 35 + 9 + 227 + 16 + 153 = 607

5(7) + 92(8) + 70(11) + 35(11) + 9(12) + 227(13) + 16(14) + 153(15) = 7,504

= 12.47,504607






Fort Wayne, IN 92,000 (8, 12)

Columbus, OH 70,000 (11, 10)

Ashland, KY 35,000 (11, 7)


Akron, OH 227,000 (13, 11)

Wheeling, WV 16,000 (14, 10)

Roanoke, VA 153,000 (15, 5)

x* = =li yi

li

i

i

li yi =i5(13) + 92(12) + 70(10) + 35(7) + 9(4) + 227(11) + 16(10) + 153(5) = 5,572

= 9.25,572607






Fort Wayne, IN 92,000 (8, 12)

Columbus, OH 70,000 (11, 10)

Ashland, KY 35,000 (11, 7)


Akron, OH 227,000 (13, 11)

Wheeling, WV 16,000 (14, 10)

Roanoke, VA 153,000 (15, 5)

The resulting load-distance score is

ld = lidi =i5(5.4 + 3.8) + 92(4.4 + 2.8) + 70(1.4 + 0.8) + 35(1.4 + 2.2) + 90(0.4 + 5.2) + 227(0.6 + 1.8) + 16(1.6 + 0.8) + 153(2.6 + 4.2)

= 2,662.4

wheredi = |xi – x*| + |yi – y*|



A firm wishes to find a central location for its service. Business forecasts indicate travel from the central location to New York City on 20 occasions per year. Similarly, there will be 15 trips to Boston, and 30 trips to New Orleans. The x, y-coordinates are (11.0, 8.5) for New York, (12.0, 9.5) for Boston, and (4.0, 1.5) for New Orleans. What is the center of gravity of the three demand points?

SOLUTION

x* = =li xi

li

i

i

y* = =li yi

li

i

i

[(20 11) + (15 12) + (30 4)]

(20 + 15 + 30)= 8.0

[(20 8.5) + (15 9.5) + (30 1.5)]

(20 + 15 + 30)= 5.5


Using Break-Even AnalysisUsing Break-Even Analysis

Compare location alternatives on the basis of quantitative factors expressed in total costs Determine the variable costs and fixed costs for

each site Plot total cost lines Identify the approximate ranges for which each

location has lowest cost Solve algebraically for break-even points over

the relevant ranges


Break-Even Analysis for LocationBreak-Even Analysis for Location

EXAMPLE 11.3

An operations manager narrowed the search for a new facility location to four communities. The annual fixed costs (land, property taxes, insurance, equipment, and buildings) and the variable costs (labor, materials, transportation, and variable overhead) are as follows:

Community Fixed Costs per Year Variable Costs per Unit

A $150,000 $62

B $300,000 $38

C $500,000 $24

D $600,000 $30



Step 1: Plot the total cost curves for all the communities on a single graph. Identify on the graph the approximate range over which each community provides the lowest cost.

Step 2: Using break-even analysis, calculate the break-even quantities over the relevant ranges. If the expected demand is 15,000 units per year, what is the best location?



SOLUTION

To plot a community’s total cost line, let us first compute the total cost for two output levels: Q = 0 and Q = 20,000 units per year. For the Q = 0 level, the total cost is simply the fixed costs. For the Q = 20,000 level, the total cost (fixed plus variable costs) is as follows:

Community Fixed CostsVariable Costs

(Cost per Unit)(No. of Units)Total Cost

(Fixed + Variable)

A $150,000

B $300,000

C $500,000

D $600,000



SOLUTION

To plot a community’s total cost line, let us first compute the total cost for two output levels: Q = 0 and Q = 20,000 units per year. For the Q = 0 level, the total cost is simply the fixed costs. For the Q = 20,000 level, the total cost (fixed plus variable costs) is as follows:

$62(20,000) = $1,240,000 $1,390,000

Community Fixed CostsVariable Costs

(Cost per Unit)(No. of Units)Total Cost

(Fixed + Variable)

A $150,000

B $300,000

C $500,000

D $600,000

$38(20,000) = $760,000 $1,060,000

$24(20,000) = $480,000 $980,000

$30(20,000) = $600,000 $1,200,000


A best B best C best


Figure 11.1 shows the graph of the total cost lines.

| | | | | | | | | | | |

0 2 4 6 8 10 12 14 16 18 20 22

1,600 –

1,400 –

1,200 –

1,000 –

800 –

600 –

400 –

200 –

–

An

nu

al

co

st

(th

ou

sa

nd

s o

f d

oll

ars

)

Q (thousands of units)

A

B

C

D

6.25 14.3

Break-even point

Break-even point

(20, 980)

(20, 1,390)

(20, 1,200)

(20, 1,060)

The line for community A goes from (0, 150) to (20, 1,390). The graph indicates that community A is best for low volumes, B for intermediate volumes, and C for high volumes. We should no longer consider community D, because both its fixed and its variable costs are higher than community C’s.

Figure 11.1 – Break-Even Analysis of Four Candidate Locations



Step 2: The break-even quantity between A and B lies at the end of the first range, where A is best, and the beginning of the second range, where B is best. We find it by setting both communities’ total cost equations equal to each other and solving:

(A) (B)

$150,000 + $62Q = $300,000 + $38Q

Q = 6,250 units

The break-even quantity between B and C lies at the end of the range over which B is best and the beginning of the final range where C is best. It is

(B) (C)

$300,000 + $38Q = $500,000 + $24Q

Q = 14,286 units



Step 2: The break-even quantity between A and B lies at the end of the first range, where A is best, and the beginning of the second range, where B is best. We find it by setting both communities’ total cost equations equal to each other and solving:

(A) (B)

$150,000 + $62Q = $300,000 + $38Q

Q = 6,250 units

The break-even quantity between B and C lies at the end of the range over which B is best and the beginning of the final range where C is best. It is

(B) (C)

$300,000 + $38Q = $500,000 + $24Q

Q = 14,286 units

No other break-even quantities are needed. The break-even point between A and C lies above the shaded area, which does not mark either the start or the end of one of the three relevant ranges.


By chance, the Atlantic City Community Chest has to close temporarily for general repairs. They are considering four temporary office locations:


Property Address Move-in Costs Monthly Rent

Boardwalk $400 $50

Marvin Gardens $280 $24

St. Charles Place $350 $10

Baltic Avenue $60 $60

Use the graph on the next slide to determine for what length of lease each location would be favored? Hint: In this problem, lease length is analogous to volume.



| | | | | | | | |

0 1 2 3 4 5 6 7 8Months →

To

tal

Co

st →

500 –

–

400 –

–

300 –

–

200 –

–

100 –

–

–

Boardwalk

St Charles Place

Marvin Gardens

Baltic Avenue

Fs + csQ = FB + cBQ

Q =FB – Fs

cs – cB

= = 6 months– 300– 50

=$60 – $360$10 – $60

The short answer: Baltic Avenue if 6 months or less, St. Charles Place if longer

SOLUTION


Locating Within a NetworkLocating Within a Network

When a firm with a network of existing facilities plans a new facility, one of two conditions exists Facilities operate independently Facilities interact

The GIS method for locating multiple facilities The transportation method


A five step GIS frameworkStep 1: Map the data

Step 2: Split the area

Step 3: Assign a facility location

Step 4: Search for alternative sites

Step 5: Compute ld scores and check capacity

Other methods of location analysis Heuristics Simulation Optimization

Locating Within a NetworkLocating Within a Network


Locating Multiple FacilitiesLocating Multiple Facilities

EXAMPLE 11.4

Witherspoon Automotive remanufactures automotive components and subassemblies

Full truckloads of parts to and from customers

Two locations: Spartanburg, SC and Orlando, FL

Locations have a remanufacturing facility and a warehouse

Spartanburg covers a total of 362 customers

Orlando facility covers a total of 66 customers

Spartanburg DC shipped 17,219 and Orlando DC shipped 4,629 full truckloads last year

Operating regions and customer locations are shown in Figure 11.2.



Figure 11.2 – Operating Regions and Customer Location for Witherspoon Automotive



The senior management decided to close the Spartanburg facility and split the region into two each with its own manufacturing and distribution center

Five important location factors:

1. The new facilities should be located in a major metropolitan area

2. Total load–distance score should be minimized

3. Size of the two new facilities should not exceed a maximum of 9,500 truckloads of output per year

4. Customer truckloads allocated between the two facilities should be fairly balanced

5. New distribution network should be able to accommodate up to 1,000 full truckload shipments per year from the Alabama



Where should the two new facilities be opened, assuming that the Orlando DC will stay where it is, and that fixed cost differences in opening a new facility are comparable across most potential sites in the region?

SOLUTION

Using the data in their system and MapPoint, the managers at Witherspoon automotive overlaid the locations and the number of full truckload shipments delivered last year for each customer in the Spartanburg region onto a map. The Witherspoon Automotive video on myomlab shows how to perform this analysis using MapPoint. To achieve a greater degree of aggregation in customer base and to also give due consideration to the quality of life location factor, the map was changed from displaying data for each street address (customer) to an aggregate view that displays data for each Metropolitan Statistical Area (MSA).



Figure 11.3 – Truckload Concentration for Witherspoon



The darker the shade, the greater the number of truckloads in the MSA. It was visually clear that Atlanta and Charlotte were the major markets, with Columbia, South Carolina; Greenville, South Carolina; and Richmond, Virginia, also having a heavy concentration of customers. From this map in Figure 11.3, it was easy to see that the dark green shaded area in Atlanta has a heavy customer-trips concentration. It represents 4,475 full truckloads, which would easily support half a facility. It seems reasonable for the management to locate one of the two new facilities near Atlanta. This decision will also achieve two other management objectives, in that the facility is near a major metropolitan area and is also well placed to serve the proposed expansion of the northern Alabama market. Management stated that if it decides to locate a facility near the Atlanta area, it would be in Buford, Georgia.



The next step was to partition the customers into 2 regions, each with a total demand of less than 9,500 truckloads. Because it seemed clear that the Atlanta area would have a facility, one region was circled around Atlanta as shown in Figure 11.4. Furthermore, if the northern Alabama market develops as hoped, it will handle an additional 1,000 truckloads. Because of this potential, the Atlanta region can only handle 8,500 truckloads for current customers. After a careful look at the map data, it was decided that Georgia, Tennessee, Alabama, and the parts of South Carolina that were within 2.5 hours of the Atlanta facility would be assigned to the Atlanta region. The Augusta/Aiken MSA that straddles the Georgia/South Carolina border was also added to this region to balance the two regions. This scenario results in the Atlanta region being assigned 8,397 truckloads and the second region having 8,822 trips, and achieving a 48.8 percent to 51.2 percent split while still allowing capacity in the Atlanta region for the proposed expansion of the northern Alabama market.



Figure 11.4 – Witherspoon’s Facilities Areas



In order to identify a good location for the second facility, the center of gravity for the second region was determined to find a good starting point. Looking at the map in Figure 11.4, it appears that the center of the second region is around Durham, North Carolina. However, the center of gravity for the second region is close to a National Forest in Randolph County, North Carolina—not too far from Charlotte but considerably south and west of Durham. Such an outcome is to be expected because the Charlotte and to a lesser extent Columbia, South Carolina, markets have such a large percentage of the truckload volume for this region. However, the center of gravity does not appear to be a promising site because it is only near one customer. Given this dilemma, the management of Witherspoon Automotive decided to pick a site next to the center of gravity as well as several sites in the general area of the center of gravity that are near Interstate I-85.



Load–distance scores were computed using driving mileage and driving time based on last year’s demand for each of the possible locations. The following results were obtained for load–distance calculations based on one-way trips:

Site CityLoad-Distance Using

One-Way MileageLoad-Distance Using

One-Way Travel Hours

1 Albemarle 1,331,608 22,194

2 Salisbury 1,075,839 18,541

3 Greensboro 1,222,675 20,378

4 Concord 1,037,424 17,938



As the Witherspoon Automotive managers reviewed the results, they noted that Concord will provide the minimum mileage and drive time, and have the additional advantage of being near Charlotte—fulfilling the managerial objective of being located near a major city. Another attractive feature of this solution is that the Greenville, South Carolina, and Augusta, Georgia, markets are almost as close to the Concord facility as they are to the Buford facility. Management can reassign customers in these markets to the Charlotte region at little additional cost if the northern Alabama market grows faster than expected.


An electronics manufacturer must expand by building a second facility. The search is narrowed to four locations, all of which are acceptable to management in terms of dominant factors. Assessment of these sites in terms of seven location factors is shown in Table 11.1. For example, location A has a factor score of 5 (excellent) for labor climate; the weight for this factor (20) is the highest of any. Calculate the weighted score for each location. Which location should be recommended?

Solved Problem 1Solved Problem 1



TABLE 11.1 | FACTOR INFORMATION FOR ELECTRONICS MANUFACTURER

Factor Score for Each Location

Location Factor Factor Weight A B C D

1. Labor climate 20 5 4 4 5

2. Quality of life 16 2 3 4 1

3. Transportation system 16 3 4 3 2

4. Proximity to markets 14 5 3 4 4

5. Proximity to materials 12 2 3 3 4

6. Taxes 12 2 5 5 4

7. Utilities 10 5 4 3 3


SOLUTION

Based on the weighted scores shown in Table 11.2, location C is the preferred site, although location B is a close second.


TABLE 11.2 | CALCULATING WEIGHTED SCORES FOR ELECTRONICS| MANUFACTURER

Weighted Score for each Location


1. Labor climate 20

2. Quality of life 16

3. Transportation system 16

4. Proximity to markets 14

5. Proximity to materials 12

6. Taxes 12

7. Utilities 10

Totals 100


TABLE 11.2 | CALCULATING WEIGHTED SCORES FOR ELECTRONICS| MANUFACTURER

Weighted Score for each Location


1. Labor climate 20

2. Quality of life 16

3. Transportation system 16

4. Proximity to markets 14

5. Proximity to materials 12

6. Taxes 12

7. Utilities 10

Totals 100

SOLUTION

Based on the weighted scores shown in Table 11.2, location C is the preferred site, although location B is a close second.


100 80 80 100

32 48 64 16

48 64 48 32

70 42 56 56

24 36 36 48

24 60 60 48

50 40 30 30

348 370 374 330


The operations manager for Mile-High Lemonade narrowed the search for a new facility location to seven communities. Annual fixed costs (land, property taxes, insurance, equipment, and buildings) and variable costs (labor, materials, transportation, and variable overhead) are shown in Table 11.3.


a. Which of the communities can be eliminated from further consideration because they are dominated (both variable and fixed costs are higher) by another community?

b. Plot the total cost curves for all remaining communities on a single graph. Identify on the graph the approximate range over which each community provides the lowest cost.

c. Using break-even analysis, calculate the break-even quantities to determine the range over which each community provides the lowest cost.



TABLE 11.3 | FIXED AND VARIABLE COSTS FOR MILE-HIGH LEMONADE

Community Fixed Costs per Year Variable Costs per Barrel

Aurora $1,600,000 $17.00

Boulder $2,000,000 $12.00

Colorado Springs $1,500,000 $16.00

Denver $3,000,000 $10.00

Englewood $1,800,000 $15.00

Fort Collins $1,200,000 $15.00

Golden $1,700,000 $14.00



Lo

cati

on

co

sts

(in

mill

ion

s o

f d

olla

rs)

Barrels of lemonade per year (in hundred thousands)

10 –

8 –

6 –

4 –

2 –

–| | | | | | |

0 1 2 3 4 5 6

Fort Collins Boulder Denver

Golden

Break-even point

Break-even point

2.67

Figure 11.5 – Break-Even Analysis of Four Candidate Locations



SOLUTION

a. Aurora and Colorado Springs are dominated by Fort Collins, because both fixed and variable costs are higher for those communities than for Fort Collins. Englewood is dominated by Golden.

b. Figure 11.5 shows that Fort Collins is best for low volumes, Boulder for intermediate volumes, and Denver for high volumes. Although Golden is not dominated by any community, it is the second or third choice over the entire range. Golden does not become the lowest-cost choice at any volume.



c. The break-even point between Fort Collins and Boulder is

$1,200,000 + $15Q =$2,000,000 + $12Q

Q =266,667 barrels per year

The break-even point between Denver and Boulder is

$3,000,000 + $10Q =$2,000,000 + $12Q

Q =500,000 barrels per year



The new Health-Watch facility is targeted to serve seven census tracts in Erie, Pennsylvania, whose latitudes and longitudes are shown in Table 11.4. Customers will travel from the seven census-tract centers to the new facility when they need health care. What is the target area’s center of gravity for the Health-Watch medical facility?

TABLE 11.4 | LOCATION DATA AND CALCULATIONS FOR HEALTH WATCH

Census Tract Population Latitude Longitude Population Latitude

Population Longitude

15 2,711 42.134 –80.041 114,225.27 –216,991.15

16 4,161 42.129 –80.023 175,298.77 –332,975.70

17 2,988 42.122 –80.055 125,860.54 –239,204.34

25 2,512 42.112 –80.066 105,785.34 –201,125.79

26 4,342 42.117 –80.052 182,872.01 –347,585.78

27 6,687 42.116 –80.023 281,629.69 –535,113.80

28 6,789 42.107 –80.051 285,864.42 –543,466.24

Total 30,190 –1,271,536.04 –2,416.462.80



SOLUTION

We use MapPoint in this solution, with coordinates represented in the form of latitude and longitude rather than an (x, y) grid, to calculate the center of gravity. First the target area is displayed on the map of Erie, Pennsylvania, using MapPoint. In Figure 11.6 a pushpin is placed in the approximate geographical center of the census tracts. The location sensor is then turned on. By moving the cursor over the pushpin, the location sensor will register the longitude and latitude for the pushpin. The population of each census tract is added to the map using MapPoint’s built-in demographic data. Thus, we obtain the following table in which latitudes and longitudes for each of the seven census-tracts are given, along with their actual populations, in thousands.



Figure 11.6 – Center of Gravity for Health-Watch



Next we solve for the center of gravity x* and y*. Because the coordinates are given as longitude and latitude, x* is the longitude and y* is the latitude for the center of gravity.

x* = = 42.11781,271,536.05

30,190

y* = = – 80.0418– 2,416,462.81

30,190

The center of gravity is (42.12 North, 80.04 West), and is shown on the map to be fairly central to the target area.

Active Model 11.2 in myomlab confirms these calculations for the center of gravity, and allows us to explore other alternative locations as well.

11 – 1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Location 11 For...

Documents

Transcript of 11 – 1 Copyright © 2010 Pearson Education, Inc. Publishing as Prentice Hall. Location 11 For...