Question

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	36-69	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.

How many people ages 18 - 15 should be interviewed? [a]

Answer 1

3) Let X1 = number of 18- to 35-year-olds interviewed in East

X2 = number of 36- to 69-year-olds interviewed in East

X3 = number of 70-year-olds and up interviewed in East

X4 = number of 18- to 35-year-olds interviewed in West

X5 = number of 36- to 69-year-olds interviewed in West

X6 = number of 70-year-olds and up interviewed in West

Minimize 2.5X1 + 2X2 + 1.5X3 + 3.5X4 + 3X5 + 2X6

Subject to: X1 + X2 + X3 ? 650

X4 + X5 + X6 ? 250

X1 + X4 ? 500

X2 + X5 ? 400

X1, X2, X3, X4, X5, X6 ? 0

X1 = 432,

X2 = 318,

X3 = 0,

X4 = 68,

X5 = 82,

X6 = 100              

Therefore Total no of people to be interviewed between 18-35 are (432+68)= 500.