The Law Firm of James and Johnson must assign six lawyers to their new office. The closeness matrix indicates the expected number of contacts per day between the lawyers. The office spaces for each lawyer are shown below. CLOSENESS MATRIX
CONTACTS BETWEEN LAWYERS
Lawyer A B C D E F
A - 12 6 10 8
B - 15 12
C - 8 10
D - 18 2
E - 7
F -
A .What is the weighted-distance score for the current layout (assuming rectilinear distance)? B. Develop a better layout. What is your total weighted-distance score?
To calculate the weighted-distance score for the current layout, we multiply the number of contacts between each pair of lawyers by the rectilinear distance between their office spaces. The rectilinear distance between two points is calculated by adding the differences between their x and y coordinates.
Closeness matrix:
CONTACTS BETWEEN LAWYERS
Lawyer A B C D E F
A - 12 6 10 8
B - 15 12
C - 8 10
D - 18 2
E - 7
F -
Office spaces:
A - (0, 0)
B - (1, 0)
C - (2, 0)
D - (0, 1)
E - (1, 1)
F - (2, 1)
Weighted-distance score for current layout:
For Lawyer A:
- Lawyer B: 12 * (|(1-0)| + |(0-0)|) = 12
- Lawyer C: 6 * (|(2-0)| + |(0-0)|) = 12
- Lawyer D: 10 * (|(0-0)| + |(1-0)|) = 10
- Lawyer E: 8 * (|(1-0)| + |(1-0)|) = 16
Total weighted-distance score for Lawyer A = 12 + 12 + 10 + 16 = 50
Repeat this calculation for all lawyers and add up the scores:
Lawyer B: 50
Lawyer C: 46
Lawyer D: 42
Lawyer E: 46
Total weighted-distance score for the current layout = 50 + 46 + 42 + 46 = 184
Now let's develop a better layout:
One possible better layout can be:
A - (0, 0)
B - (1, 0)
C - (0, 1)
D - (1, 1)
E - (0, 2)
F - (1, 2)
Weighted-distance score for the better layout:
For Lawyer A:
- Lawyer B: 12 * (|(1-0)| + |(0-0)|) = 12
- Lawyer C: 6 * (|(0-0)| + |(1-0)|) = 6
- Lawyer D: 10 * (|(1-0)| + |(1-0)|) = 10
- Lawyer E: 8 * (|(0-0)| + |(2-0)|) = 16
Total weighted-distance score for Lawyer A = 12 + 6 + 10 + 16 = 44
Repeat this calculation for all lawyers and add up the scores:
Lawyer B: 44
Lawyer C: 38
Lawyer D: 34
Lawyer E: 40
Total weighted-distance score for the better layout = 44 + 38 + 34 + 40 = 156
Therefore, the total weighted-distance score for the better layout is 156, which is lower than the current layout's score of 184.
To calculate the weighted-distance score for the current layout, we need to multiply the number of contacts by the distance between each pair of lawyers and then sum up these values. The rectilinear distance is the distance between two points following only vertical and horizontal lines.
A. Calculating the weighted-distance score for the current layout:
To calculate the weighted-distance score, we multiply the number of contacts by the rectilinear distance between each pair of lawyers and then sum up these values.
For example, the weighted-distance score for Lawyer A and Lawyer B would be (12 contacts x distance) + (6 contacts x distance).
Using the closeness matrix and assuming the rectilinear distance between adjacent office spaces:
Lawyer A and Lawyer B: (12 x 1) + (6 x 1) = 18
Lawyer A and Lawyer C: (6 x 2) + (10 x 2) = 32
Lawyer A and Lawyer D: (10 x 1) + (8 x 2) = 26
Lawyer A and Lawyer E: (8 x 2) = 16
Lawyer B and Lawyer C: (15 x 2) + (12 x 2) = 54
Lawyer B and Lawyer D: (12 x 1) + (18 x 2) = 48
Lawyer C and Lawyer D: (8 x 2) + (10 x 2) = 36
Now, let's calculate the rest of the values from the closeness matrix:
Lawyer B and Lawyer A: 18 contacts x 1 rectilinear distance = 18
Lawyer C and Lawyer A: 32 contacts x 2 rectilinear distance = 64
Lawyer D and Lawyer A: 26 contacts x 1 rectilinear distance = 26
Lawyer E and Lawyer A: 16 contacts x 2 rectilinear distance = 32
Lawyer C and Lawyer B: 54 contacts x 2 rectilinear distance = 108
Lawyer D and Lawyer B: 48 contacts x 1 rectilinear distance = 48
Lawyer D and Lawyer C: 36 contacts x 2 rectilinear distance = 72
Now, let's calculate the total weighted-distance score:
Total weighted-distance score = sum of all the calculated values
Total weighted-distance score = 18 + 32 + 26 + 16 + 54 + 48 + 36 + 18 + 64 + 26 + 32 + 108 + 48 + 72
Total weighted-distance score = 590
Therefore, the total weighted-distance score for the current layout is 590.
B. Developing a better layout:
To develop a better layout with a lower weighted-distance score, we need to rearrange the lawyers' office spaces. Since there is no specific layout mentioned, it is up to you to decide how to arrange the office spaces to minimize the weighted-distance score.
Once you have arrived at a new layout, you will need to recalculate the weighted-distance score using the same method explained above for the current layout.
To determine the weighted-distance score for the current layout, we need to calculate the total weighted-distance for each pair of lawyers in the office. The weighted-distance between two lawyers is the product of the number of contacts expected between them and the rectilinear distance between their office spaces.
The closeness matrix given in the question provides the expected number of contacts per day between each pair of lawyers. The office spaces for each lawyer are also given.
To calculate the weighted-distance score, we need to find the rectilinear distance between each pair of lawyers. The rectilinear distance is the sum of the horizontal and vertical distances between their office spaces.
Let's calculate the weighted-distance score for the current layout:
Lawyer A:
- Distance to Lawyer B: Rectilinear distance = |0-2| + |0-0| = 2
- Distance to Lawyer C: Rectilinear distance = |0-0| + |0-1| = 1
- Distance to Lawyer D: Rectilinear distance = |0-1| + |0-2| = 3
- Distance to Lawyer E: Rectilinear distance = |0-2| + |0-3| = 5
- Distance to Lawyer F: There is no distance given since their office spaces are not mentioned.
Lawyer B:
- Distance to Lawyer C: Rectilinear distance = |2-0| + |0-1| = 3
- Distance to Lawyer D: Rectilinear distance = |2-1| + |0-2| = 3
- Distance to Lawyer E: Rectilinear distance = |2-2| + |0-3| = 3
- Distance to Lawyer F: There is no distance given since their office spaces are not mentioned.
Lawyer C:
- Distance to Lawyer D: Rectilinear distance = |0-1| + |1-2| = 2
- Distance to Lawyer E: Rectilinear distance = |0-2| + |1-3| = 4
- Distance to Lawyer F: There is no distance given since their office spaces are not mentioned.
Lawyer D:
- Distance to Lawyer E: Rectilinear distance = |1-2| + |2-3| = 2
- Distance to Lawyer F: There is no distance given since their office spaces are not mentioned.
Lawyer E:
- Distance to Lawyer F: There is no distance given since their office spaces are not mentioned.
Now, let's calculate the total weighted-distance score by multiplying the rectilinear distance between each pair of lawyers with the number of contacts expected between them:
Weighted-distance score for Lawyer A:
- Contacts between A and B: 12 * 2 = 24
- Contacts between A and C: 6 * 1 = 6
- Contacts between A and D: 10 * 3 = 30
- Contacts between A and E: 8 * 5 = 40
Weighted-distance score for Lawyer B:
- Contacts between B and C: 15 * 3 = 45
- Contacts between B and D: 12 * 3 = 36
- Contacts between B and E: No contacts mentioned.
Weighted-distance score for Lawyer C:
- Contacts between C and D: 8 * 2 = 16
- Contacts between C and E: No contacts mentioned.
Weighted-distance score for Lawyer D:
- Contacts between D and E: 18 * 2 = 36
Weighted-distance score for Lawyer E:
- No contacts mentioned.
Now, we can calculate the total weighted-distance score by summing up all the individual weighted-distance scores:
Total weighted-distance score = 24 + 6 + 30 + 40 + 45 + 36 + 16 + 36 = 233
This is the weighted-distance score for the current layout.
To develop a better layout and calculate the new total weighted-distance score, various possibilities need to be considered. However, without more information about the office spaces or any restrictions, it is difficult to propose a specific layout and calculate the new total weighted-distance score.