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See attached files. These are problems from an operations management course. I’m way behind and need help catching up. This assignment doesn’t have to be perfect, I just need something to turn it. I t was due a few days ago.
Submit complete solutions to the following problems to your instructor:
1. Cross Median Method: Text
Chapter 10 Exercise 6
2. Huff Method: Text
Chapter 10 Exercise 9
3. Set Covering Method: Text
Chapter 10 Exercise 12
Chapter 10 Exercise 6
10.6. You have been asked to help locate a catering service in the central business district of a city. The locations of potential customers on an xy coordinate grid areP1 = (4, 4), P2 = (12, 4), P3 = (2, 7), P4 = (11, 11), and P5 = (7, 14). The expected demand is weighted as w1 = 4, w2 = 3, w3 = 2, w4 = 4, and w5 = 1. Using the cross-median approach, recommend a location for the catering service that will minimize the total weighted distance traveled to serve the customers.
Chapter 10 Exercise 9
10.9. A community is currently being served by a single self-serve gas station with six pumps. A competitor is opening a new facility with 12 pumps across town. Table 10.12 shows the travel times in minutes from the four different areas in the community to the sites and the number of customers in each area.
a. Using the Huff retail location model and assuming that λ = 2, calculate the probability of a customer traveling from each area to each site.
b. Estimate the proportion of the existing market lost to the new competitor.
Table 10.2
Area 1 2 3 4
Old Station 5 1 9 15
New Competitor 20 8 12 6
Number of Customers 100 150 80 50
Chapter 10 Exercise 12
The Volunteer Fire Department serving the communities in Figure 10.8 has just purchased two used fire engines auctioned off by a nearby city.
a. Select all possible pairs of communities in which the fire engines could be located to ensure that all communities can be reached within 30 minutes or less.
b. What additional consideration could be used to make the final site selection from the community pairs found in part a?
Figure 10.8 Service Area Network
SEE ADDITIONAL ATTACHMENT FOR FIGURE 10.8 SERVICE AREA NETWORK
The physical location can be an important decision for many types of services such as
stores, restaurants and gas stations; a critical differentiator for defining success and failure.
Such services require the physical presence and participation of customers in the service
delivery process. The customers have to travel to these services to participate in the delivery
of the service. Locations that are closer to the customers or are in places that are frequented
by the customers tend to attract more customers.
Services that involve no or minimal face-to-face contact can be located just about anywhere.
Even services with high front office contact can have significant back office operations,
which can be located at distant places. Sometimes, employees working for the back office
can even work from home as they do not require significant face-to-face contact.
Location for most businesses is a strategic decision and involves a long term commitment.
Therefore, the location should be good and viable from a long term perspective and should
be able to withstand possible demographic and economic changes in that location.
Fitzsimmons and Fitzsimmons (2011) have identified the following strategic location
considerations:
Competitive clustering: Competitors, such as auto dealers, locate themselves in the
same area, since the customers of automobiles tend to do comparison shopping by
going to an area where multiple auto dealers are located.
Saturation marketing: Some firms, such as Dunkin Donuts, create a concentrated
cluster of many outlets in high traffic areas such as downtowns. This creates visibility
and awareness. The customer may not go to the first one they see but may be
tempted to enter the next one.
Marketing intermediaries: A service organization, such as Visa and Master card,
can reach millions more customers by using various banks and credit unions as
intermediaries.
Substitute communication for travel: Banks, such as ING, can reach customers
through internet and electronic transfers without the customers having to travel to a
physical location.
Separation of front from back office: In many services, such as dry cleaners, the
front and the back office operations need not be located in the same place. While the
front office location provides convenience for the customers, the centralized back
office location can provide economies of scale.
Impact of the internet on service location: The internet has revolutionized many
services. Many service organizations, such as Expedia, operate entirely through the
QSO 610 Module Seven 1
internet. Many brick and mortar organizations, such as AAA, now offer services
through the internet.
Before a site can be selected, information and data are collected on the important
characteristics of the available location alternatives in order to determine the best location.
Geographical Information Systems (GIS) are very useful in mapping information on different
geographical areas and sites. Site considerations, such as the following, are commonly
considered depending on the type of service facility to be located:
• Labor: Availability of labor with the right skills and the cost of labor.
• Competition: Competitors located in the vicinity.
• Traffic: Areas frequented by customers are preferred.
• Taxes: Property and other taxes can add to the cost of operating the site.
• Proximity to customers: Services, such as grocery stores, need to be located close to
where the customers live.
• Parking: Availability of adequate parking on premises or nearby.
• Access: Easy accessibility from highways and city roads, near public transportation, and
without too much traffic congestion.
• Expansion: The site should allow room for expansion in the future as the demand grows.
• Surroundings: Good and attractive surroundings tend to attract customers to the service
outlet.
• Visibility: The site should be clearly visible to the nearby traffic.
• Complementary Services: Locating restaurants near hotels and motels would attract
travelers.
• Special considerations: Particular services may have special and unique considerations,
such as rental car companies, need to be located at or near airports.
It is always good to start from a model that is suited to a particular location decision in order
to take the most important considerations into account and to narrow down the choices.
Experience and judgment can be used last in making the final decision.
Techniques that involve distances between two locations can use either of the two measures
of distances:
Rectilinear distance: This is used when the movement is along perpendicular directions,
i.e., when a city is divided into rectangular blocks. With as the coordinates of
location i and as the coordinates of location
)y,x( ii
)y,x( jj j , the distance is computed as
follows, where the symbol | | denotes absolute value.
ijd
jijiij yyxxd −+−=
2 QSO 610 Module Seven
Euclidean distance: When the two locations are connected approximately by a straight line
(as the crow flies), the distance is determined on the basis of the Pythagoras theorem as
shown below:
22 )yy()xx(d jijiij −+−=
Example
If Erie is located at (50, 185) and State College is located at (175, 100), what is the
rectilinear distance between them? What is the Euclidean distance?
Solution
2108512510018517550 =+=−+−=ijdThe rectilinear distance
Euclidean distance
16151972572251562510018517550 22 .)()(dij ==+=−+−=
Regression Analysis
La Quinta Inn constructed a regression model based on the location of existing hotels
(Kimes and Fitzsimmons, 1990). It used operating margin, by adding depreciation and
interest expense to income and dividing by the total revenue, as the dependent variable. It
used 35 factors relating to the sites as independent variables. The regression yielded four
critical factors: price (room rate for the inn), income (average family income), college (college
enrollment) and state (state population per inn). Once the regression model was developed,
it was used to predict the operating margin at proposed new sites in order to determine the
best site alternative that would maximize the operating margin.
Cross Median Approach
This approach is based on rectilinear distances. According to this approach, the facility
should be located so that 50% of the concentration of total weight should be on either side of
the location. The weight is determined by the number of customers, demand, etc. Since the
weight may not be evenly divided on either side of the theoretical median, the sites that are
closest on either side of the median along both x and y axes are selected for inclusion in the
short list. Final selection is then made from this short list.
The following problem from Fitzsimmons and Fitzsimmons (2011) will be used to illustrate
the Cross Median method.
QSO 610 Module Seven 3
Problem
A pizza delivery service has decided to open a branch near off-campus student housing.
The project manager has identified five student apartment complexes in the northwest area
of the city, the locations of which, on an xy coordinate grid in miles, are C1 = (1, 2), C2 (2, 6),
C3 (3, 3), C4 = (4, 1), and C5 = (5, 4). The expected demand is weighted as w1 = 5, w2 = 4,
w3 = 3, w4 = 1, and w5 = 5. Using the cross-median approach, recommend a location for the
pizza branch that will minimize the total distance traveled. Show the recommended location
on a graph.
Solution
Step 1: Determine the total weight and the median.
w1 = 5, w2 = 4, w3 = 3, w4 = 1, w5 = 5
Median = (5 + 4 + 3 + 1 + 5)/2 = 9
Step 2: Plot the five apartment complexes on a grid.
C1 = (1, 2), C2 = (2, 6), C3 = (3, 3), C4 = (4, 1), C5 = (5, 4)
Figure 7-1
Step 3: Determine the x-median by moving from left to right until the cumulative weight
equals or exceeds the median weight. Repeat the same from right to left. If the two points
are different, connect the two points by a line.
4 QSO 610 Module Seven
xi wi Cumulated wi
(going right)
Cumulated wi
(going left)
C1 1 5 5 18
C2 2 4 9 ← 13
C3 3 3 12 9 ←
C4 4 1 13 6
C5 5 5 18 5
Table 7-1
Step 4: Determine the y-median by moving from bottom to top until the cumulative weight
equals or exceeds the median weight. Repeat the same from top to bottom. If the two points
are different, connect the two points by a line.
yi wi Cumulated wi
(going up)
Cumulated wi
(going down)
C4 1 1 1 18
C1 2 5 6 17
C3 3 3 9 ← 12
C5 4 5 14 9 ←
C2 6 4 18 4
Table 7-2
Step 5: Combine the x-median and y-median from previous steps to determine the cross-
median.
Using Cross-Median approach, the recommended location is the rectangular region [x, y]
between (2, 3) and (3, 4). See the shaded area in the graph.
Huff Model for Retail Outlet Location
Huff (1966) developed a model for locating a retail outlet that serves a number of customer
areas but competes with other similar retail outlets. His model assumes that the
attractiveness of a retail outlet is proportional to the size of the facility Sj but inversely
proportional to , where is the travel time from the customer area to facility
λ
ijT ijT i j . The
power λ called the propensity to travel depends on how the willingness of the customers to
travel to the service facility declines as the time to travel increases. Thus, the attractiveness
of the service facility is determined by the following formula:
The following problem from Fitzsimmons and Fitzsimmons (2011) will be used to explain this
method.
λ
ij
j
ij
T
S
A =
QSO 610 Module Seven 5
Problem
A locally owned department store samples two customers in each of five geographic areas
to estimate consumer spending in its home appliances department. It is estimated that these
customers are a good sample of the 10,000 customers the store serves. The number of
customers in each area is C1 = 1,500, C2 = 2,500, C3 = 1,000, C4 = 3,000, and C5 = 2,000. It
is found that the two consumers have the following budgets in dollars for home appliances
per year: B11 = 100, B12 = 150; B21 = 75, B22 = 100; B31 = 125, B32 = 125; B41 = 100, B42 =
120; and B51 = 120, B52 = 125.
a. Using the Huff retail location model, estimate annual home appliance sales for the
store.
b. Bull’s Eye, a chain department store, opens a branch in a shopping complex nearby.
The Bull’s-Eye branch is three times larger than the locally owned store. The travel
times in minutes from the five areas to the two stores (i = 1 for the locally owned
store, j = 2 for Bull’s Eye) are T11 = 20, T12 = 15; T21 = 35, = 20; T31 = 30, T32 = 25; T41
= 20, T42 = 25; and T51 = 25, T52 = 25. Use the Huff retail location model to estimate
the annual consumer expenditures in the home appliance section of each store
assuming that λ = 1.
Solution
a. The budget data for the customers in the five areas can be summarized as follows:
Customers Areas
1 2 3 4 5
1 $ 100 $ 75 $ 125 $ 100 $ 120
2 $ 150 $ 100 $ 125 $ 120 $ 125
Avg. budget (Bi) $ 125 $
87.50
$ 125 $ 110 $
122.50
No. customers (Ci) 1500 2500 1000 3000 2000
Sales (in
thousands)
187.5 218.75 125 330 145
Table 7-3
Annual home appliance sales can be computed as follows:
= 187.5+218.75+125+330+145
= $1106.25 (in thousands)
b. The attractiveness of customers from each of the areas to the locally owned store, to
be called “Local”, and the Bull’s Eyes Store can be calculated as follows:
6 QSO 610 Module Seven
Area 1 2 3 4 5
Local (S=1) 1/20 = .05 1 /35 = .0286 .0333 .05 .04
Bull’s Eye (S=3) 3/15 = .20 3/20 = .15 .12 .12 .12
Aij∑ 0.25 0.1786 0.1533 0.17 .16
Table 7-4
The probability a customer from an area would visit the local store or the Bull’s
Eye store is in proportion to the attractiveness of the two stores for the customers
from each area as computed below.
Store Customer Area
1 2 3 4 5
1 Local .05/(.05+.20) = .2 .0286/(.0286+.15) = .16 .217 .29 .25
2 Bull’s Eye .20/(.05+.20) = .8 .15/(.0286+.15)= .84 .783 .71 .75
Table 7-5
The sales from each area are then allocated according to the probabilities and the
total sales for each store are calculated:
Store Customer Area Sales (1000s)
1 2 3 4 5
1 Local $37.50 $35.00 $27.13 $95.70 $61.25 $256.58
2 Bull’s Eye $150.00 $183.75 $97.88 $234.30 $183.75 $849.68
*Sales (1000s) $187.5 $218.75 $125 $330 $245 $1106.25
Table 7-6
Set Covering Method for Locating Public Services
This method is used to locate public services where the objective is to be able to reach all
communities in the service area within a maximal time or distance. The following problem
from Fitzsimmons and Fitzsimmons (2011) will be used to explain this method.
Problem
A bank is planning to serve the rural communities shown in the figure below with automated
teller machines (ATMs). The travel time in minutes between communities in the service area
is shown on the network. The bank is interested in determining the number of location of
ATMs necessary to serve the communities so that a machine will be within 20 minutes’
travel time of any community.
QSO 610 Module Seven 7
Figure 7-2
Solution
Since the ATMs are to be located at selected communities, we make a list of what
communities can be served from each locations site as given below:
From Location Site Communities that can be reached within 20 minutes
1 1,2,3
2 1,2, 4,5
3 1, 3,4
4 2,3,4, 6,7
5 2, 5,6
6 4, 5, 6
7 4, 7
Table 7-7
Site 7 can be eliminated from the list since the sites served (4, 7) are a subset of the sites
served from 4 (2, 3, 4, 6, 7). In other words, site 4 is superior to site 7. The shortened list is
given below:
From Location Site Communities that can be reached within 20 minutes
1 1,2,3
2 1,2, 4,5
3 1, 3,4
4 2,3,4, 6,7
5 2, 5,6
6 4, 5, 6
Table 7-8
As we can see, site 4 must be selected since site 7 can only be served from there. Site 4
serves all sites except for 1 and 5. To serve 1 and 5, we must select site 2. Therefore, if
ATMs are located in sites 2 and 4, all communities will be able to reach the ATMs within 20
minutes of travel time.
8 QSO 610 Module Seven
QSO 610 Module Seven 9
References
Fitzsimmons, J. A., & Fitzsimmons, M. J. (2011). Service management: Operations, strategy,
information technology. (7th ed.). New York, NY: McGraw Hill.
Huff, D. L. (August, 1966), A programmed solution for approximating an optimum retail location. Land
Economics, pp. 293-303.
Kimes, S. E., & Fitzsimmons, J. A. (March, 1990). Selecting profitable hotel sites at La Quinta Motor
Inns. Interfaces, 20(2), 12-20.
