How are they coming up with part C, please explain? Johnson Research is a market

Question

How are they coming up with part C, please explain?

Johnson Research is a marketing research firm that handles consumer surveys for several clients. One client is a beverage producer that is planning to market a fruit juice product in a new market. The client wants to test the market by doing a survey before launching a TV commercial. Targeted costumers are individual households and are segmented by the gender and age of the household head. To draw statistically valid conclusions, the client and Johnson Research have determined the following requirements for the survey Survey at least 1,000 households in total Ensure that no more than 25% of the households surveyed have heads who are 51 or older Ensure that at least 20% of the households surveyed have heads who are 35 or younger Ensure that at least 45% of the households surveyed have female heads Ensure that no more than 60% of the households surveyed with heads 51 or older have male heads 4 Johnson Research decided that all surveys should be conducted in person. The estimated costs of reaching people in each age and gender category are as follows Age Gender unger Between 36 and 50 $28 51 or older S22 35 or you Female Male S25 23 26 You are required to formulate a linear programming model to determine the number of households in each age and gender category to minimize the total cost of survey while meeting the above requirements a. Define the decision variables. (5) X11-number of households with female heads 35 or younger x number of households with female heads between 36 and 50 13 number of households with female heads 51 or older x - number of households with male heads 35 or younger x22-number of households with male heads between 36 and 50 r23 - number of households with male heads 51 or older b. Formulate the objective function. (5) min 28xu + 25% 2 + 22x13 + 26x21 + 23x22 + 1 8x 23 c. Formulate the constraints. (10) +x +x23 2 1000 -0.25x -0.25xj +0.75x130.25x 0.25x2 +0.75x23 s 0.800.20xj2 -0.20x3 0.80x-0.20x220.20x23 2 0.55x1 1 +0.55x12 +0.55x13-0.45x21-0.45x22-0.45x23 -0.60x13 +0.40x

Explanation / Answer

It is formulation of all the constraints namely the conditions which act as restrictions.

The requirements for the survey to be represented as constraints:

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