Solution to Bonus Problem 3: Telephone Survey 1

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 Percentage of Daytime Calls Percentage of Evening Calls Wife 30 30 Husband 10 30 Single male 10 15 Single female 10 20 None 40 5 Data for Phone Problem

1 Based on 3-20 and 3-21 (p. 101) in Practical Management Science (2nd ed., Winston and Albright, 2001 Duxbury Press). Solution by David Juran, 2001. 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 C D E F G H 1 Percentages Daytime Evening 2 Wife 30% 30% 3 Husband 10% 30% 4 Single male 10% 15% 5 Single female 10% 20% 6 None 40% 5% 7 Sum 100% 100% 8 9 Cost/call $ 2.00 $ 5.00 10 11 Daytime Evening Total =SUM(B12:C12) 12 Calls made 1 1 2

13 <= =0.5*D12 14 Max evening calls 1 15 16 Contacts Made Required 17 Wife 0.6 >= 150 18 Husband 0.4 >= 120 19 Single male 0.25 >= 100 20 Single female 0.3 >= 110 21 0 0 22 Total cost $ 7.00 =SUMPRODUCT($B$12:$C$12,B5:C5) 23 =SUMPRODUCT($B$12:$C$12,B9:C9) 24 25

Decision Models 3 Prof. Juran A B C D 1 Percentages Daytime Evening 2 Wife 30% 30% 3 Husband 10% 30% 4 Single male 10% 15% 5 Single female 10% 20% 6 None 40% 5% 7 Sum 100% 100% 8 9 Cost/call $ 2.00 $ 5.00 10 11 Daytime Evening Total 12 Calls made 900 100 1000 13 <= 14 Max evening calls 500 15 16 Contacts Made Required 17 Wife 300 >= 150 18 Husband 120 >= 120 19 Single male 105 >= 100 20 Single female 110 >= 110 21 0 0 22 Total cost $ 2,300.00 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 add- in 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.

Decision Models 5 Prof. Juran SolverTable does the rest, creating the following output: F G H I 1 Daytime Evening Total Cost 2 0 1200 0 $ - 3 1 1200 0 $ 1,200.00 4 2 900 100 $ 2,300.00 5 3 700 200 $ 3,100.00 6 4 400 400 $ 3,600.00 7 5 400 400 $ 4,000.00 8 6 400 400 $ 4,400.00 9 7 400 400 $ 4,800.00 10 8 400 400 $ 5,200.00 11 9 400 400 $ 5,600.00 12 10 400 400 $ 6,000.00 13 11 400 400 $ 6,400.00 14 12 400 400 $ 6,800.00 15 13 400 400 $ 7,200.00 16 14 400 400 $ 7,600.00 17 15 400 400 $ 8,000.00 18 16 400 400 $ 8,400.00 19 17 400 400 $ 8,800.00 20 18 400 400 $ 9,200.00 21 19 400 400 $ 9,600.00 22 20 400 400 $ 10,000.00 The SolverTable output can be used to create a chart:

Sensitivity Analysis 1400 $7,000

1200 $6,000 Daytime Evening 1000 Total Cost $5,000 t e s d 800 $4,000 o a C M

l s a l t l o a 600 $3,000 T C

400 $2,000

200 $1,000

0 $- $- $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 $2,000 t e s d 800 o a C M

Daytime $1,500 l s a l t l o a 600 Evening T C Total cost $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