Chapter 10
Quantitative Facilities Planning
Models
Answers to Problems at the End of Chapter 10 10-1
SECTION 10.2
10.1 Using the half-sum method: half of the sum of the weights equals 35
(x*,y*) = (5,10), which is the location of existing facility 2
x-coord
wiSwiy-coord wiSwi
015 15 530 30
525 40>35 10 35 65>35
10 30 70 15 570
10.2 Using the half-sum method: half of the sum of the weights equals 15
10.3a Using the half-sum method; half of the sum of the weights equals 1725
x-coord
wiSwiy-coord wiSwi
-5 8 8 -3 3 3
-3 311 -1 811
1 4+3=7 18>15 0 4 15=15
2 7 25 5 7 22
3 5 30 6 3 25
8 5 30
Hospital Coordinate Weight
i
aiwi
3 8 300 300
110 450 750 < 1725
214 1200 1950 > 1725
432 1500 3450
x* = 14
Hospital Coordinate Weight
i
biwi
3 4 300 300 < 1725
4 6 1500 1800 > 1725
212 1200 3000
120 450 3450
y* = 6
10.4 Using Excel’s SOLVER tool,
Since SOLVER converged on an existing facility location, we check the necessary and
sufficient conditions for the optimum solution to the Weber problem to be an existing
facility location, say (ar,br):
The necessary and sufficient condition is satisfied, therefore (x*,y*) = (5,10).
10.5 Solving mathematically,
x* = [15(0)+25(5)+30(10)]/(15+20+5+30) = 6.07143
y* = [15(10)+20(10+5(15)+30(5)]/70 = 8.21429
Solving with Excel’s SOLVER tool,
Machine Weight
i
aibi(x – a i)2(y – b i)2wi
1 0 10 36.86 3.19 15
2 5 10 1.15 3.19 20
3 5 15 1.15 46.05 5
410 5 15.43 10.33 30
Obj Fn
x y f(X*)
5.00000 10.00000 312 SOLVER solution
Coordinates
Squared Difference
Decision Variables
Answers to Problems at the End of Chapter 10 10-4
10.6 Using the half-sum method: half of the sum of the weights equals 3.
10.7a Using the half-sum method: half the sum is 5.
10.7b Using the half-sum method: half of the sum is 10.
x-coord
wiSwiy-coord wiSwi
0 1 1 0 1 1
20 1 2 20 1 2
30 1 3=3 40 1 3=3
60 1 4 70 1 4
70 1 5 80 1 5
90 1 6 90 1 6
Station(s) Coordinate Weight
i
aiwi
1 & 2 4 2 2
3 & 4 6 2 4 < 5
5 & 9 8 2 6 > 5
7 & 10 10 2 8
6 & 8 12 210
x* = 8
Station(s) Coordinate Weight
i
aiwi
1 & 2 4 2 2
3 & 4 6 2 4 < 5
5 & 9 8 2 6 > 5
7 & 10 10 2 8
6 & 8 12 210
x* = 10
10.7c
10.7d Using the majority theorem, locate at x* = 10, y* = 10. (Note: the vertical line from
10.8 It is desired to determine the unweighted minimax location, i.e., it is important that the
farthest house be as close as possible, regardless of the monetary value of the house.
Work Addends
Station (i)aibigiai+b i-g iai+b i+g i-a i+b i-g i-a i+b i+g i
1 4 10 6 8 20 012
2 4 10 212 16 4 8
3 6 10 610 22 010
4 6 10 214 18 2 6
5 8 10 216 20 2 4
612 10 220 24 0 0
710 10 416 24 -6 4
812 10 418 26 -4 2
9 8 10 414 22 -6 6
10 10 10 614 26 -6 6
Min: 8 -6
Max: 26 12
Coordinates
a b a + b –a + b
20 15 35 -5
25 25 50 0
13 32 45 19
25 14 39 –11
421 25 17
18 826 –10
10.9a Since the weights are uniformly distributed over the indicated regions, the optimum x–
coordinate location will be the point at which no more than half the weight is to the left of
The optimum location (x*,y*) = (9, 6.33) is located in A4.
x* = 9
y* = 6.33
14
12
13
10
11
8
9
6
7
4
5
2
3
0
1
10
5
0
y-axis
30
25
20
15
50
45
40
35
70
65
60
55
90
85
80
75
x-axis
Cumulative Weight Consumed While Moving Along y-Axis
125
120
115
110
105
100
95
13
14
11
65
75
80
85
35
40
45
50
0
5
10
20
8
9
6
7
10
1
2
3
4
0
5
12
15
25
30
Cumulative Weight Consumed While Moving Along x-Axis
55
60
70
90
95
115
125
100
105
110
120
10.9b Since the facility cannot be located in A4, we search the border of A4. First, consider y =
10.9c Using Excel’s SOLVER tool,
10.9d Since the minimax location is to be determined with all points equally weighted, the
rectangular regions can be represented by their corners. Hence,
EF Rectilinear Weight
i
aibi|x 1 – a i||y 1 – b i|Distance wiwidi
A1 1.5 1.5 9.10 5.10 14.20 10 142.0
A2 412 6.60 5.40 12.00 15 180.0
A3 14 14 3.40 7.40 10.80 20 216.0
A4 12 4 1.40 2.60 4.00 30 120.0
P1 4 6 6.60 0.60 7.20 30 216.0
P2 10 10 0.60 3.40 4.00 15 60.0
P3 14 2 3.40 4.60 8.00 5 40.0
Obj Fn
x y f(X*)
10.60 6.60 216.0
Coordinates
Absolute Difference
Decision Variables
a b a + b –a + b a b a + b –a + b
2 2 4 0 8 4 12 -4
2 4 6 2 8 8 16 0
4 6 10 210 10 20 0
410 14 612 416 -8
412 16 812 820 -4
6 2 8 -4 12 12 24 0
6 4 10 -2 12 14 26 2
610 16 414 216 –12
612 18 614 12 26 -2
14 14 28 0
10.10a Since there are only 3 sites to consider, enumeration is the quickest means of solving the
problem.
10.10b With weights:
f(50,50) = max[600(|50-20|+|50-25|), 400(|50-36|+|50-18|), 500(|50-62|+|50-37|),
Without weights:
10.11a Using the half-sum method: half of the sum of the weights equals 1200.
10.11b Shown below is a partial contour line passing through the Building 2 location. Building 1
x-coord
wi∑wiy-coord wi∑wi
5200 200 5300 300
15 400 600 10 200 500
25 500 1100 15 400 900
30 600 1700 > 1200 20 400 1300 > 1200
35 300 200 25 500 1800
50 400 2400 30 600 2400
10.11c Enumerating the weighted maximum rectilinear distance for each of the three feasible
locations gives:
Building 2 is preferred. Note: if the print shop could be placed at any location, using
various starting solutions with SOLVER, we obtained the following solutions having a
locations gives:
Obj. Fcn
x y f(X*)
120 20 14,000
240 25 12,000
325 35 18,000
Building
Decision Variables
Obj. Fcn
x y f(X*)
120 20 35
240 25 50
325 35 45
Building
Decision Variables
10.11d Enumerating the weighted sum of squared Euclidean distances for each of the three
feasible locations gives:
Building 1 is preferred. Note: the unconstrained squared Euclidean optimum location is
10.11e Enumerating the weighted sum of Euclidean distances for each of the three feasible
locations gives:
Obj. Fcn
x y f(X*)
120 20 725,000
240 25 945,000
325 35 1,125,000
Building
Decision Variables
Obj. Fcn
x y f(X*)
120 20 36,156
240 25 43,864
325 35 45,150
Building
Decision Variables
Answers to Problems at the End of Chapter 10 10-11
10.12 The contour line passing through (1,5) is shown below.
10.13a The objective function for the optimum x-coordinate is
Center Euclidean Weight
i
aibi(x1 – a i)2(y1 – b i)2Distance wiwidi
A 5 10 401 224 25.012 200 5002
B50 15 623 100 26.888 400 10755
C25 25 0 0 0.039 500 20
D35 599 399 22.326 300 6698
E15 20 101 25 11.199 400 4480
F30 30 25 25 7.064 600 4239
Obj Fn
x y
f(X*)
25.032 24.977 31,193
Coordinates
Squared Difference
Decision Variables
Answers to Problems at the End of Chapter 10 10-12
10.13b The objective function for the optimum x and y coordinates is
f(X,Y) = 10[(x1 – 10)2 + (y1 – 25)2]½ + 6[(x1 – 10)2 + (y1 – 15)2]½ + 5[(x1 – 15)2
Machine Weight
iaibi|x1 – a i||y1 – b i||x 2 – a i||y 2 – b i| w 1iw2i
110 25 5 0 10 010 2
210 15 510 10 10 6 3
315 30 0 5 5 5 5 4
420 10 515 015 4 6
525 25 10 0 5 0 3 12
Weight
|x 1 – x 2|
|y 1 – y 2|
v12
5 0 4
Obj Fn
x1y1x2y2f(X*)
15 25 20 25 565
Coordinates
Absolute Difference
Absolute Difference
Decision Variables
Machine
i
aibi
(x1 – a i)2
(y1 – b i)2
(x2 – a i)2
(y2 – b i)2
d(X1,Pi)d(X2,Pi)w1iw2i
110 25 0 0 15 0 0.003 3.873 10 2
210 15 010 15 10 3.162 5.000 6 3
315 30 5 5 10 5 3.162 3.873 5 4
420 10 10 15 515 5.000 4.472 4 6
525 25 15 0 0 0 3.873 0.003 3 12
Distance Weight
(x1 – x 2)2
(y1 – y 2)2
d(X1,X2)v12
225 0 15.000 4
Obj Fn
x1y1x2y2f(X*)
10.0 25.0 25.0 25.0 191.541
Euclidean Distance
Weight
Squared Difference
Decision Variables
Coordinates
Squared Difference
10.14a
Answers to Problems at the End of Chapter 10 10-14
10.14c
Answers to Problems at the End of Chapter 10 10-15
10.15a Using the half-sum method with a total weight of 30,
Store x–coord weight Store y–coord weight
i
aiwiCum Sum i b iwiCum Sum
10 0 1 1 9 0 4 4
11 0 2 3 4 0 1 5
9 1 4 7 10 1 1 6
5 2 3 10 1 2 1 7
6 2 5 15 = 30/2 3 2 1 8
7 2 6 21 5 2 3 11
8 2 2 23 12 2 3 14 < 15
12 3 3 26 6 3 5 19 > 15
1 4 1 27 11 3 2 21
10.15b As shown below, we plot diamonds around each existing facility, with the radius of each
diamond equal to 1 mile. We also compute the slopes of contour lines in the grid squares
adjacent to the optimum location. In so doing, we find that (1.5, 3.5) and (2.5, 3.5) are
coordinate locations that are 1 mile from any store and have the smallest value of the
9
10
11
12
1
2
3
4
5
6
7
8
0
1
2
3
4
5
0
1
2
3
4
5
–4/3
4/3
8
–8
10.15c The optimum unweighted minimax location is obtained as follows:
Hence, the optimum minimax location is any point on the line segment connecting the
(2.5,2.5) and (3,2).
10.16a From the data given, it is a 4-story building, including a basement at z = -8. Three
departments (2, 3, and 4) are located on the first floor at z = 2, two departments are
Store
(i)aibiai + b i-a i+b i
1 4 2 6 -2
2 4 5 9 1
3 5 2 7 -3
4 5 1 6 -4
5 2 2 4 0
6 2 3 5 1
7 2 4 6 2
8 2 5 7 3
9 1 0 1 -1
10 0 1 1 1
11 0 3 3 3
12 3 2 5 -1
Min: 1 -4
Max: 9 3
Coordinates
Answers to Problems at the End of Chapter 10 10-18
10.16b Since the unconstrained optimum location coincides with the constrained values for x and
y, it will be the constrained location.
10.16c Using SOLVER to obtain the weighted minimax location gives:
10.16d With the constrained minimax location, fixed distances are traveled by departments that
aiwiSum b iwiSum c iwiSum
8 2 2 10 1 1 -8 2 2
12 1 3 20 2 3 2 1 3
15 1 4 20 2 5 2 1 4
25 1 5 25 1 6 2 2 6
25 2 7 35 1 7 12 2 8
25 2 9 35 2 9 12 3 11
30 3 12 35 2 11 22 1 12
40 2 14 40 3 14 22 2 14
x* = 25 y* = 35 z* = 12
EF Rectilinear Weight
i
aibici|x – a i||y – b i||z – ci|Distance wiwidi
1820 -8 16.73 0.73 19.73 37.20 274.4
212 10 212.73 10.73 9.73 33.20 133.2
325 35 20.27 14.27 9.73 24.27 124.3
440 20 215.27 0.73 9.73 25.73 251.5
525 35 12 0.27 14.27 0.27 14.80 229.6
630 40 12 5.27 19.27 0.27 24.80 374.4
715 25 22 9.73 4.27 10.27 24.27 124.3
825 35 22 0.27 14.27 10.27 24.80 249.6
Obj Fn
x y z f(X*)
24.73 20.73 11.73 74.4
Coordinates
Absolute Difference
Decision Variables
Answers to Problems at the End of Chapter 10 10-19
Now, assume the mail room is located on the first floor.
The maximum weighted distance is reduced to 84. What if the mail room is located in the
basement? As shown, using SOLVER, f* = 84 with x* = 26, y* = 36, and z* = -8.
aibici|x – a i||y – bi| |z – c i|
1
25 35 12 252 5.00 6.50 0.00 11.50 75.0
2
25 35 12 148 5.00 35.00 12.00 40.00 88.0
3
25 35 12 110 5.00 35.00 12.00 40.00 50.0
4
25 35 12 240 5.00 35.00 12.00 40.00 120.0
5
25 35 12 205.00 35.00 12.00 40.00 80.0
6
30 40 12 300.00 40.00 12.00 40.00 120.0
7
25 35 12 130 5.00 35.00 12.00 40.00 70.0
8
25 35 12 210 5.00 35.00 12.00 40.00 90.0
Obj Fn
x y z f(X*)
30 29 12 120.0
gi+w idi
Rectilinear
Distance
Absolute Difference
Decision Variables
coordinates
wi
gi
Dept
aibici|x – a i| |y – b i| |z – c i|
125 35 2 2 84 0.00 0.00 0.00 0.00 84.0
212 10 2 1 0 13.00 25.00 0.00 38.00 38.0
325 35 2 1 0 0.00 0.00 0.00 0.00 0.0
440 20 2 2 0 15.00 15.00 0.00 30.00 60.0
525 35 2 2 20 0.00 0.00 0.00 0.00 20.0
630 40 2 3 30 5.00 5.00 0.00 10.00 60.0
725 35 2 1 40 0.00 0.00 0.00 0.00 40.0
825 35 2 2 40 0.00 0.00 0.00 0.00 40.0
Obj Fn
x y z f(X*)
25 35 2 84.0
Rectilinear
Distance
gi+w idi
Decision Variables
Dept
coordinates
wi
gi
Absolute Difference
aibici|x – a i| |y – b i| |z – c i|
125 35 2 2 84 0.00 0.00 0.00 0.00 84.0
212 10 2 1 0 13.00 25.00 0.00 38.00 38.0
325 35 2 1 0 0.00 0.00 0.00 0.00 0.0
440 20 2 2 0 15.00 15.00 0.00 30.00 60.0
525 35 2 2 20 0.00 0.00 0.00 0.00 20.0
630 40 2 3 30 5.00 5.00 0.00 10.00 60.0
725 35 2 1 40 0.00 0.00 0.00 0.00 40.0
825 35 2 2 40 0.00 0.00 0.00 0.00 40.0
Obj Fn
x y z f(X*)
25 35 2 84.0
Rectilinear
Distance
gi+w idi
Decision Variables
Dept
coordinates
wi
gi
Absolute Difference