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Chapter 08 - Location Planning and Analysis
8-21
Education.
Example Problem—Calculation of the Rectilinear Distance:
When answering this question, refer to textbook, Chapter 8, Problem 13.
a. Determine the Rectilinear distance between location A and B.
b. Determine the Rectilinear distance between location A and D.
c. Determine the Rectilinear distance between location B and D.
Solution to Example Problem—Measurement of Rectilinear Distance:
a. The Rectilinear distance between location A and B is:
The main advantage of this method is the ease of calculations, while the disadvantage is that
it generally overestimates the distance. Because of its propensity to overestimate distance,
Rectilinear distance is sometimes referred to as the pessimistic distance measurement
method. Rectilinear measure of distance is more commonly used in solving facility layout
problems.
3. Euclidean Method
This mathematical method is based on the Pythagorean Theorem. Euclidean distance
measures distance “as the crow flies” and is most applicable when a straight-line route is
possible.
The following formula for the hypotenuse of a right triangle provides us with the Euclidean
distance:
C2 = A2 + B2 or C =
22 BA
C is the side of a right triangle opposite the right angle, and A and B are the right degree sides
Chapter 08 - Location Planning and Analysis
Education.
22
GFGFe YYXXD
Where:
De = Euclidean distance measure between location F and location G;
XF = X coordinate of location F;
Example Problem—Calculation of the Euclidean Distance:
Refer to textbook, Chapter 8, Problem 13.
a. Determine the Euclidean distance between location A and B.
b. Determine the Euclidean distance between location A and D.
c. Determine the Euclidean distance between location B and D.
Solution to Example Problem—Calculation of the Euclidean Distance:
a.
The Euclidean distance is the direct or the shortest distance between two given sites.
Therefore, for any given pair of sites, the Euclidean distance provides the most direct or the
9765 22
22
e
BABAe
D
YYXXD
831.534
e
D
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4. Weighted Average Method
As we discussed in the previous sections, the Rectilinear method is a conservative,
pessimistic measure of distance, and the Euclidean method is an optimistic measure of
distance. Both Euclidean and Rectilinear measures of distance are frequently the only
standard distance options used as a part of many layout and location software programs.
However, some managers prefer the weighted average method that provides a compromise
distance between the optimistic Euclidean and the pessimistic Rectilinear methods.
If A and B are the locations in question, then the weighted average distance between A and B
is given by the following formula:
)()()()( rreewa DwDwD
Where:
Dr = Rectilinear distance measure between location A and location B;
De = Euclidean distance measure between location A and location B;
wr = weight associated with the Rectilinear distance;
we = weight associated with the Euclidean distance.
0
1
i
ii
w
w
Example Problem—Calculation of the Weighted Average Distance:
Refer to textbook, Chapter 8, Problem 13 and the two examples given earlier in this section.
Assume that the pessimistic (Rectilinear) weight is 0.4 and the optimistic (Euclidean) weight
is 0.6.
a. Determine the weighted average distance between location A and B.
b. Determine the weighted average distance between location A and D.
c. Determine the weighted average distance between location B and D.
Solution to Example Problem—Weighted Average
a. The weighted average distance between location A and B is:
Chapter 08 - Location Planning and Analysis
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Education.
Exercise Problems
1. Refer to textbook Problem 15 in Chapter 8, and measure the Rectilinear distance between
destination 1 and destination 5.
2. Refer to textbook Problem 15 in Chapter 8, and measure the Euclidean distance between
destination 1 and destination 5.
3. Refer to textbook Problem 15 in Chapter 8, and measure the Weighted Average distance
between destination 1 and destination 5 (wr = 0.3 and we = 0.7).
4. Refer to textbook Problem 15 in Chapter 8 and measure the Rectilinear distance between
destination 1 and destination 3.
5. Refer to textbook Problem 15 in Chapter 8, and measure the Euclidean distance between
destination 1 and destination 3.
6. Refer to textbook Problem 15 in Chapter 8, and measure the Weighted Average distance
between destination 1 and destination 3 (wr = 0.5 and we = 0.5).
Solutions to Exercise Problems
3862.4)1231.4)(7.0()5)(3.0( .3
1231.417 .2
53251 .1
WA
e
r
D
D
D
Chapter 08 - Location Planning and Analysis
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Education.
WD = weighted distance value for a given site
wi = weight associated with existing location i
di = the distance between existing site i and the proposed location
For a typical facility location problem, wi (weight) represents the amount of goods or units
shipped between the proposed location and the existing location i.
For a facility layout problem, wi (weight) represents the number of trips between the existing
location i and the proposed location.
Distance is calculated using either Rectilinear, Euclidean or Weighted Average methods. The site
with the lowest weighted-distance would be selected, because the lowest weighted-distance
usually results in the lowest transportation costs.
Problem 1
Based on the destination locations and quantities given in textbook problem 15, the company is
considering two locations for a new plant. The coordinates of the first plant location are: (x = 2, y
= 3) and the coordinates of the second location are: (x = 4, y = 3).
a. Determine the Euclidean distance from the first proposed plant location to all of the
destination locations.
b. Determine the Rectilinear distance from the first proposed plant location to all of the
destination locations.
c. Determine the Euclidean distance from the second proposed plant location to all of the
destination locations.
d. Determine the Rectilinear distance from the second proposed plant location to all of the
destination locations.
e. Determine the Weighted Distance value for the first and second proposed plant locations
based on Euclidean distance and decide where the new plant should be located.
f. Determine the Weighted Distance value for the first and second proposed plant locations
based on Rectilinear distance and decide where the new plant should be located.
Solution to Problem 1
Destination
X coordinate
Y coordinate
Q
D1
1
2
900
D2
2
4
300
D3
3
1
700
D4
4
2
600
D5
5
3
1200
Chapter 08 - Location Planning and Analysis
Education.
a. Plant Site 1—Euclidean Distance
From Plant Site 1 (P1) to Destination j (Dj)
11)43()22()( DistanceEuclidean
414.12)23()12()( DistanceEuclidean
22
22
21
22
11
DP
DP
Chapter 08 - Location Planning and Analysis
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d. Plant Site 2—Rectilinear Distance
From Plant Site 2 (P2) to Destination j (Dj)
13354)( Distancer Rectilinea
12344)( Distancer Rectilinea
34324)( Distancer Rectilinea
42314)( Distancer Rectilinea
52
42
32
22
12
DP
DP
DP
DP
e. Weighted Distance Values Based on Euclidean Distances
P1 = Plant 1
P2 = Plant 2
k = # of existing locations
k
i
iidw
1
WDP1 = 900(1.414) + 300(1) + 700(2.236) + 600(2.236) + 1,200(3) = 8,079.4
WDP2 = 900(3.162) + 300(2.236) + 700(2.236) + 600(1) + 1,200(1) = 6,881.8
Because 6,881.8 < 8,079.4, choose proposed Plant Location 2.
f. Weighted distance values based on Rectilinear distances
WDP1 = 900(2) + 300(1) + 700(3) + 600(3) + 1,200(3) = 9,600
C. Factor Scoring Model
This simplistic selection procedure has many areas of application. The two most common areas of
application are: 1) Facility Location; 2) Product Selection.
Factor Scoring is a very flexible method that considers both tangible and intangible factors. It has
the capability to consider multiple decision criteria simultaneously.
Steps of the Factor Scoring Model:
1. Develop a list of (factors) criteria to be considered. The decision maker should consider these
factors important in evaluating each decision alternative.
2. Assign a weight to each factor that describes the factor’s relative importance.
Chapter 08 - Location Planning and Analysis
Let wi = the weight for factor i, where i = 1, 2, ….., F
F = the number of factors considered.
The higher the weight, the more important the criterion.
We will use a five-point scale to establish the relative importance of the factors considered.
The following table provides an interpretation of the weight scale:
Factor Weight Interpretation Table
Importance
Weight
Very important
5
Somewhat important
4
Average importance
3
Somewhat unimportant
2
Very unimportant
1
For example, if a factor has a weight of 4, it is somewhat more important than the average
factor.
If a factor has a weight of 1, relative to the other factors being considered, the factor in
question is very unimportant.
3. Determine a list of decision alternatives. Let dj = decision alternative j and assign a rating for
each factor/decision alternative combination.
Let rij = the rating for factor i and decision alternative j.
Where j = 1, 2,……, D and
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For the purposes of rating each decision alternative/factor combination, we use the following
9-point scale.
Level of Satisfaction
Rating
Extremely High
9
Very high
8
High
7
Slightly High
6
Average
5
Slightly Low
4
Low
3
Very Low
2
Extremely Low
1
For example, a score of 7 for a given decision alternative would indicate that the decision
maker (manager) rates this decision alternative (location) high with respect to a given factor.
4. Compute the factor score for each decision alternative. Let Sj = factor score
F
i
ijij rwS
5. Sequence the decision alternatives from the highest score to the lowest score. The decision
alternative with the highest factor score is the recommended decision alternative. The
decision alternative with the second highest factor score is the second choice decision
1. Microwave Ovens
2. Refrigerators
3. Stoves
Management thinks that the following decision criteria should be used in selecting the product:
1. Manufacturing capability/cost
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Education.
2. Market demand
3. Unit profit margin
4. Long term profitability/growth
5. Transportation costs
6. Useful life
The company has determined the following weights for the decision criteria (factors):
FACTOR
WEIGHT
Manufacturing capability/cost
4
Market demand
5
Unit profit margin
3
Long term profitability/growth
5
Transportation costs
2
Useful life
1
The decision factor ratings for each criteria are given in the following table:
DECISION FACTOR RATINGS
FACTOR
MICROWAVE
REFRIGERATOR
STOVE
1
4
3
8
2
8
4
2
3
6
9
5
4
3
6
7
5
9
2
4
6
1
5
6
Based on the information provided, determine the factor scores for all three products. What is the
best choice for the appliance manufacturing company? What is the second best choice?
