Homework1

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School

University of California, Berkeley *

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Course

240

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Industrial Engineering

Date

Oct 30, 2023

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pdf

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Uploaded by BaronSkunk2569

IEOR 240 - Fall 2023 Homework 1 Due Date: 09/22/2023 (Friday) at 23:59 (11:59 PM). Submission requirement For problems that require a mathematical formulation, please clearly state the definition of decision variables and the indexes of them. Write a sentence to explain each constraint and objective function. Also include an explanation of the optimal solution if it exists. Submit your homework (typed, scanned or photographed) and code (Feel free to use any software you like) in a single pdf file. You can choose to merge the pdf files together or include screenshots of your code and solutions in your homework pdf. 1. Consider a manufacturing company that produces two types of products, A and B. The profit per unit of product A is 30 and per unit of product B is 20. The company has a maximum of 120 hours of labor available and 100 kg of raw material. It takes 2 hours of labor and 5 kg of raw material to produce one unit of product A, and 1 hour of labor and 4 kg of raw material to produce one unit of product B. Formulate a linear programming model to maximize the company’s profit. Solve it graphically and by using software. Comment on whether the problem is infeasible, unbounded, has unique optimal solution, or has infinitely many optimal solutions. 2. A nutritionist wants to create a meal plan consisting of two foods, A and B. Each serving of food A contains 2 units of protein, 3 units of fat, and 1 unit of carbohy- drates. Each serving of food B contains 1 unit of protein, 2 units of fat, and 2 units of carbohydrates. The nutritionist wants to have at least 10 units of protein, 12 units of fat, and 12 units of carbohydrates in the meal plan. Food A costs 4 per serving and food B costs 3 per serving. Formulate the problem as a linear programming problem to minimize the cost. Solve it graphically and by using software. Comment on whether the problem is infeasible, unbounded, has unique optimal solution, or has infinitely many optimal solutions. 3. A company wants to invest in two projects, A and B. The company has 10 , 000 available for investment. Each dollar invested in project A will return 1 . 5 times the investment, and each dollar invested in project B will return 2 times the investment. The company wants to invest at least 6 , 000 in project A and at least 7 , 000 in project B. Formulate a linear programming model to maximize the company’s profit and solve it graphically and by using software. Comment on whether the problem is infeasible, unbounded, has unique optimal solution, or has infinitely many optimal solutions.

4. A company wants to invest in two projects, each dollar invested in project A will return 1 . 5 times the investment, and each dollar invested in project B will return 2 times the investment. The only restriction imposed by the company’s board is that the difference in the investment amounts for the two projects cannot exceed 1 dolllar. Formulate the problem as a linear programming problem to maximize the profit. Solve it graphically and by using software. Comment on whether the problem is infeasible, unbounded, has unique optimal solution, or has infinitely many optimal solutions.

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