A computer start-up named Pear is considering entering the U.S. market with what they believe to be a smaller and faster computer than some of the existing products on the market. They want to get a feel for whether or not customers would be willing to switch from some of the existing bigger brands to consider their product. They want to collect a reasonable sample from across the United States representative of all age brackets. They have split the United States into 2 regions: East and West. They want at least 65% of their sample to cover the East and no fewer than 25% of the West. They also have divided the age groups into 3 categories: 18-35, 36-69, and 70 and up. They want at least 50% of their sample to be between 18 and 35 and at least 40% to be between 36 and 69. The costs per person surveyed is given in the table below:

Region	18-35	36-39	70 and up
East	$2.50	$2.00	$1.50
West	$3.50	$3.00	$2.00

Assume that at least 1,000 people are to be surveyed. The problem is for Pear Company to decide how many people to survey from each age bracket within each region in order to minimize costs while meeting requirements. Find the optimal solution and minimum cost.

 

FOR EAST REGION

AGE 18-35 = 375 people

AGE 36-69 = 300 people

AGE 70 and up = 75 people

FOR WEST REGION

AGE 18-35 = 125 people

AGE 36-69 = 100 people

AGE 70 and up = 25 people

minimum cost = $ 2,437.5

*please see explanation for the computation

Step-by-step explanation

since the requirements are at least 65% of the sample to cover east and no fewer than 25% to cover west, by looking at the cost per person table which is given in your problem, you can see that the cost in west is larger than east. so you better have more sample in the east to have minimum cost.

required: no fewer than 25% to west. use 25% for west

required: at least 65% to east. use 75%

total of 100%

total sample is 1000 people.

east sample = 1000 x 75% or .75 = 750 people

west sample = 1000 x 25% or .25 = 250 people

EAST SAMPLE (750 PEOPLE)

AGE 18-35 =50% OF SAMPLE

750 X 50% or .50= 375 PEOPLE

AGE 36-69= 40% OF SAMPLE

750 X 40% or .40 = 300 PEOPLE

AGE 70 AND UP = 10% OF SAMPLE

750 X 10% or .10= 75 PEOPLE

WEST SAMPLE (250 PEOPLE)

AGE 18-35 = 50% OF SAMPLE

250 X 50% or .50 = 125 PEOPLE

AGE 36-69 = 40% OF SAMPLE

250 X 40% or .40 = 100 PEOPLE

AGE 70 AND UP = 10% OF SAMPLE

250 X 10% or .10 = 25 PEOPLE

TO FIND MINIMUM COST

MULTIPLY THE NUMBER OF PEOPLE TO THE COST PER AGE BRACKET

EAST SAMPLE

375 X $ 2.5 = $ 937.5

300 X $ 2 = $ 600

75 X $ 1.5 = $ 112.5

TOTAL = 937.5 + 600 + 112.5 = $ 1,650

WEST SAMPLE

125 X $ 3.5 = $ 437.5

100 X $ 3 = $ 300

25 X $ 2 = $ 50

TOTAL = 437.5 + 300 + 50 = $ 787.5

TOTAL MINUMUM COST = EAST COST + WEST COST

1, 650 + 787.5 = $ 2,437.5