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# MS5313 Homework 1: Optimization Models

## Instructions:

1. There are 4 questions. For each question, please present the mathematical formulation and the final solution in this word document; submit it together with your complete Excel file, containing the solver setup and solution.

## Question 1 (Revenue management):

NatBike is a bicycle manufacturer. Its customer demand curve is: d=15,000-10P, where P denotes the selling price and d denotes the total demand.  Assume that the cost of manufacturing each bicycle is \$200.

1.1. If NatBike charges a same price to all customers, what price should NatBike charge? What is the total profit?

1.2. If NatBike identifies two customer segments:

1. a) one that prefers a customized bicycle and is willing to pay a higher price, and
2. b) another that is willing to take a standardized bicycle.

Assume that the cost of manufacturing either bicycle is the same (\$200).

If NatBike may charge different prices to different customer segments, what price should NatBike charge each segment? What is the total profit?

1.3. Redo question 2, with additional condition that the total bicycle supply is no more than 8000 units.

## Question 2 (Advertising budget allocation):

A firm has a \$1,000,000 budget per year to be allocated among three mediums. The goal is to reach the largest high-potential audiences.The audience reached per advertising (AD), cost per AD, and maximum AD allowed per year is shown in the table below. Furthermore, the firm requires that at least 100days of advertising on social media platforms be placed per year.

2.1. How many units of advertising should the firm allocate to each medium and what is the maximum number of audiences reached?

2.2. After you solve the problem, identify which constraints are binding and non-binding, and what their corresponding shadow prices are.

 Medium Audience reached per AD Cost Per AD Max Ads per year TV (1min) 5000 8000 50 Daily newspaper (full page) 8500 9250 30 Social media platforms (1day) 6400 5000 150

## Question 3 (Farm management)

A farmer can plant up to 80 acres of land with wheat and corn. He can earn \$5,000 for every acre he plants with wheat and \$3,000 for every acre he plants with corn. His use of a necessary pesticide is limited by government regulations to 100 gallons for his entire 80 acres. Wheat requires 2 gallons of pesticide for every acre planted and corn requires just 1 gallon per acre. How should the farmer optimally allocate farm land to wheat and corn and what is the maximum profit he can make?

## Question 4 (Staffing problem)

A bus company launches a new line which will provide service 24 hours. There are 6 shifts, starting from 0am, 4am, 8am, 12pm, 4pm and 8pm, respectively. Each staff is supposed to continuouslywork exactly two shifts (total 8 hours). The minimum number of staffs needed for each shift is shown in the table below. The number of staffs to hire for each shift must be an integer number. What is the bus company’s optimal hiring plan in order to minimize the total number of staffs?

 Shift No. 1 2 3 4 5 6 Starting time 0am 4am 8am 12pm 4pm 8pm Minimum No. of StaffNeeded 60 70 50 60 20 30

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