Solution to Bonus Problem 3: Telephone Survey1
For a telephone survey, a marketing research group needs to contact at least 150 wives,
120 husbands, 100 single adult males, and 110 single adult females. It costs $2 to make a
daytime call and (because of higher labor costs) $5 to make an evening call. The table
below lists the results that can be expected. For example, 30% of all daytime calls are
answered by a wife, and 15% of all evening calls are answered by a single male. Because
of a limited staff, at most half of all phone calls can be evening calls.
Person Responding
Wife
Husband
Single male
Single female
None
Data for Phone Problem
Percentage of Daytime Calls
30
10
10
10
40
Percentage of Evening Calls
30
30
15
20
5
Based on 3-20 and 3-21 (p. 101) in Practical Management Science (2 nd ed., Winston and Albright, 2001
Duxbury Press). Solution by David Juran, 2001.
1
1. Determine how to minimize the cost of completing the survey.
Managerial Problem Definition
We want to minimize the total cost of completing the survey, subject to the various
probabilities of reaching certain types of people at certain times of the day, costs of
making calls, and minimum requirements for numbers of calls to certain demographic
groups.
Decision Variables
We need to decide how many evening calls and how many daytime calls to make.
Objective
Minimize the total cost.
Constraints
We need to contact 150 wives, 120 husbands, 100 single adult males, and 110 single
adult females. At most half of all phone calls can be evening calls.
Mathematical Formulation:
Decision Variables
X1 = Daytime Calls, X2 = Evening Calls
Objective
Minimize Z = 2X1 + 5X2
Constraints
0.30X1 + 0.30X2 ≥ 150
0.10X1 + 0.30X2 ≥ 120
0.10X1 + 0.15X2 ≥ 100
0.10X1 + 0.20X2 ≥ 110
1X1 ≥ 1X2
1X1, 1X2 ≥ 0
Decision Models
2
Prof. Juran
Here is a spreadsheet model for the problem:
A
B
1
Percentages
Daytime
2
Wife
30%
3
Husband
10%
4
Single male
10%
5
Single female
10%
6
None
40%
7
Sum
100%
8
9
Cost/call
$
2.00
10
11
Daytime
12
Calls made
1
13
14 Max evening calls
15
16
Contacts
Made
17
Wife
0.6
18
Husband
0.4
19
Single male
0.25
20
Single female
0.3
21
0
22
Total cost
$
7.00
23
24
25
Decision Models
C
Evening
30%
30%
15%
20%
5%
100%
$
D
E
F
G
H
5.00
Evening
1
<=
1
>=
>=
>=
>=
Total
2
=SUM(B12:C12)
=0.5*D12
Required
150
120
100
110
0
=SUMPRODUCT($B$12:$C$12,B5:C5)
=SUMPRODUCT($B$12:$C$12,B9:C9)
3
Prof. Juran
A
B
1
Percentages
Daytime
2
Wife
30%
3
Husband
10%
4
Single male
10%
5
Single female
10%
6
None
40%
7
Sum
100%
8
9
Cost/call
$
2.00
10
11
Daytime
12
Calls made
900
13
14 Max evening calls
15
16
Contacts
Made
17
Wife
300
18
Husband
120
19
Single male
105
20
Single female
110
21
0
22
Total cost
$ 2,300.00
C
Evening
30%
30%
15%
20%
5%
100%
$
D
5.00
Evening
100
<=
500
>=
>=
>=
>=
Total
1000
Required
150
120
100
110
0
The optimal solution is to make 900 Daytime calls and 100 Evening calls, for a total cost
of $2,300.
Decision Models
4
Prof. Juran
2. Starting with the optimal solution to the initial problem, use the SolverTable addin to investigate changes in the unit cost of either type of call. Specifically,
investigate changes in the cost of a daytime call, with the cost of an evening call
fixed, to see when (if ever) only daytime calls or only evening calls will be made.
Select Data — SolverTable, pick Oneway Table, and click OK.
The input cell is the value that we want to vary (in this case B9, the cost of a daytime
call). We specify a range of values for this cell (here, $0.00 to $20.00 in increments of
$1.00).
We also specify Output Cells (here, the numbers of each type of call — cells B12:C12,
and the total cost — cell B22).
Finally, we tell SolverTable to write its output starting in cell F1.
SolverTable does the rest, creating the following output:
Decision Models
5
Prof. Juran
F
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
G
Daytime
1200
1200
900
700
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
H
Evening
0
0
100
200
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
I
Total Cost
$
$ 1,200.00
$ 2,300.00
$ 3,100.00
$ 3,600.00
$ 4,000.00
$ 4,400.00
$ 4,800.00
$ 5,200.00
$ 5,600.00
$ 6,000.00
$ 6,400.00
$ 6,800.00
$ 7,200.00
$ 7,600.00
$ 8,000.00
$ 8,400.00
$ 8,800.00
$ 9,200.00
$ 9,600.00
$ 10,000.00
The SolverTable output can be used to create a chart:
Sensitivity Analysis
1400
$7,000
1200
$6,000
Daytime
Evening
$5,000
Total Cost
800
$4,000
600
$3,000
400
$2,000
200
$1,000
0
Total Cost
Calls Made
1000
$$-
$1.00
$2.00
$3.00
$4.00
$5.00
$6.00
$7.00
$8.00
$9.00
$10.00
Cost per Daytime Call
Conclusion: If daytime calls are very inexpensive, we can dispense with evening calls
altogether. However, we will always have to make at least 400 daytime calls, no matter
how expensive they are.
Decision Models
6
Prof. Juran
3. Repeat the analysis by changing the cost of an evening call and keeping the cost
of a daytime call fixed.
These results are consistent with those from Part 2. We will always make at least 400
daytime calls.
Sensitivity Analysis
1400
$3,000
1200
$2,500
1000
800
Daytime
$1,500
Evening
600
Total cost
Total Cost
Calls Made
$2,000
$1,000
400
$500
200
0
$$-
$1.00
$2.00
$3.00
$4.00
$5.00
$6.00
$7.00
$8.00
$9.00
$10.00
Cost per Evening Call
Decision Models
7
Prof. Juran