3.4-10 a) This is a cost-benefit-tradeoff problem because it asks you to meet minimum
required benefit levels (number of consultants working each time period) at
minimum cost.
Let
f1 = number of full-time consultants working the morning shift (8 a.m.-4
p.m.),
f2 = number of full-time consultants working the afternoon shift (12
p.m.-8 p.m.),
f3 = number of full-time consultants working the evening shift (4 p.m.midnight),
p1 = number of part-time consultants working the first shift (8 a.m.-12
p.m.),
p2 = number of part-time consultants working the second shift (12 p.m.4 p.m.),
p3 = number of part-time consultants working the third shift (4 p.m.-8
p.m.),
p4 = number of part-time consultants working the fourth shift (8 p.m.midnight).
Minimize C = ($14 / hour)(8 hours)(f1 + f2 + f3) + ($5 / hour)(4 hours)(p1 + p2
+ p3 + p4)
subject to
f1 + p1 ≥ 6
f1 + f2 + p2 ≥ 8
f2 + f3 + p3 ≥ 12
f3 + p4 ≥ 6
f1 ≥ 2p1
f1 + f2 ≥ 2p2
f2 + f3 ≥ 2p3
f3 ≥ 2p4
and f1 ≥ 0, f2 ≥ 0, f3 ≥ 0, p1 ≥ 0, p2 ≥ 0, p3 ≥ 0, p4 ≥ 0.
b)
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
B
C
D
E
F
G
H
Full Time Full Time
Full Time
Part Time Part Time Part Time
Part Time
8am-4pm noon-8pm 4pm-midnight 8am-noon noon-4pm 4pm-8pm 8pm-midnight
Cost per Shift
$112
$112
$112
$48
$48
$48
$48
8am-noon
noon-4pm
4pm-8pm
8pm-midnight
1
1
Workers per Shift
4
Time of Day
8am-noon
noon-4pm
4pm-8pm
8pm-midnight
Total
Full Time
4
6
8
6
1
1
Shift Covers Time of Day? (1=yes, 0=no)
1
1
1
1
2
6
³
³
³
³
2
Times Total
Part Time
4
4
8
0
2
2
1
1
4
0
Total
Cost
$1,728
I
Total
Working
6
8
12
6
J
K
³
³
³
³
Total
Needed
6
8
12
6
3.4-14 a) Let xi = percentage of alloy i in the new alloy (i = 1, 2, 3, 4, 5).
(60%)x1 + (25%)x2 + (45%)x3 + (20%)x4 + (50%)x5 = 40%
(10%)x1 + (15%)x2 + (45%)x3 + (50%)x4 + (40%)x5 = 35%
(30%)x1 + (60%)x2 + (10%)x3 + (30%)x4 + (10%)x5 = 25%
x1 + x2 + x3 + x4+ x5 = 100%
b)
A
1
2
3
4
5
6
7
8
9
10
11
12
13
Cost ($/lb.)
Tin
Zinc
Lead
New Alloy Blend
B
Alloy 1
$22
60%
10%
30%
Alloy 1
4.3%
C
Alloy 2
$20
Alloy
25%
15%
60%
Alloy 2
28.3%
D
Alloy 3
$25
E
Alloy 4
$24
Composition
45%
20%
45%
50%
10%
30%
Alloy 3
67.4%
Alloy 4
0.0%
F
Alloy 5
$27
G
H
I
50%
40%
10%
New Alloy
Composition
40%
35%
25%
=
=
=
Desired
Composition
40%
35%
25%
Alloy 5
0.0%
Total Blend
100%
=
100%
Total Cost
$23.46
3.4-15 a)
Let xij = number of units produced at plant i of product j (i = 1, 2, 3; j = L, M, S).
Maximize Profit = $420(x1L + x2L + x3L) + $360(x1M + x2M + x3M) + $300(x1S +
x2S + x3S)
subject to
x1L + x1M + x1S ≤ 750
x2L + x2M + x2S ≤ 900
x3L + x3M + x3S ≤ 450
20x1L + 15x1M + 12x1S ≤ 13,000 square feet
20x2L + 15x2M + 12x2S ≤ 12,000 square feet
20x3L + 15x3M + 12x3S ≤ 5,000 square feet
x1L + x2L + x3L ≤ 900
x1M + x2M + x3M ≤ 1,200
x1S + x2S + x3S ≤ 750
(x1L + x1M + x1S) / 750 = (x2L + x2M + x2S) / 900
(x1L + x1M + x1S) / 750 = (x3L + x3M + x3S) / 450
and x1L ≥ 0, x1M ≥ 0, x1S ≥ 0, x2L ≥ 0, x2M ≥ 0, x2S ≥ 0, x3L ≥ 0, x3M ≥ 0, x3S ≥ 0.
b)
A
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
B
Large
$420
C
Medium
$360
D
Small
$300
Space Required
(sq.ft. per unit)
20
15
12
Production
Plant 1
Plant 2
Plant 3
Total Produced
Large
516.67
0
0
516.67
²
900
Medium
177.78
666.67
0
844.44
²
1200
Small
0
166.67
416.67
583.33
²
750
93%
93%
=
=
93%
93%
Unit Profit
Sales Forecast
Percentage of Plant 1 Capacity
Percentage of Plant 1 Capacity
E
Total
Produced
694.4
833.3
416.7
F
G
²
²
²
Capacity
750
900
450
Total Profit
$696,000
Percentage of Plant 2 Capacity
Percentage of Plant 3 Capacity
H
I
Space
Required
13,000
12,000
5,000
J
K
²
²
²
Space
Available
13,000
12,000
5,000