Assignment 1 (5 marks)

Due – 11:59pm, 9 August 2019

Try two of the four questions: one from Question 1 and Question 2 (LP problem), another from Question 3 and Question 4 (NLP problem).

Hint: First establish the mathematical model based on the lecture slides. Then get start with MATLAB. Get familiar with the two MATLAB program codes provided, L1_ex1_1.m and L1_ex1_3.m. Try to understand the functions in the programs by using MATLAB online help and reading the necessary document. Then obtain the graphical solutions to the problems you have chosen by making necessary changes to the codes. Please go to the tutorial classes to know more about MATLAB, the modelling, and the detailed requirements.

Question 1:

The Outdoor Furniture Corporation manufactures two products: benches and picnic tables for use in yards and parks. The firm has two main resources: its carpenters (labour) and a supply of redwood for use in the furniture. During the next production period, 1200 hours of manpower are available under a union agreement. The firm also has a stock of 6000 kilograms of quality redwood. Each bench that Outdoor Furniture produces requires 4 labour hours and 8 kilograms of redwood; each picnic table takes 6 labour hours and 35 kilograms of redwood. A completed bench yield a profit of $10 each, and table a profit of $30 each.

Decide the numbers of benches and tables they should produce to maximize the profit.

(1) Establish the model of the optimization problem (LP with 2 variables). State clearly in words what the decision variables are, what the objective function and the equation/inequality constraints represent.

(2) Obtain the graphical solution using MATLAB.

(3) Change one of the parameters to cause the optimal solution changes and solve the problem again.

Question 2:

The local bookstore must determine how many of each of the four new books on photonics it must order to satisfy the new interest generated in the discipline. Book 1 costs $75, will provide a profit of $13, and requires 3 inches of shelf space. Book 2 costs $85, will provide a profit of $10, and requires 4 inches of shelf space. Book 3 costs $65, will provide a profit of $8, and requires 2 inch of shelf space. Book 4 costs $95, will provide a profit of $15, and requires 5 inches of shelf space. Find the number of each type that must be ordered to maximize profit. Total shelf space is 100 inches. Total amount available for ordering is $4000. It has been decided to order at least a total of 10 Book 2 and Book 4.

(1) Establish the model of the optimization problem (LP with 4 variables). State clearly in words what the decision variables are, what the objective function and the equation/inequality constraints represent.

(2) Suppose the numbers of Book 2 and Book 4 are fixed as 10 and 3 respectively, establish the model of the new optimization problem (LP with 2 variables).

(3) For the model established in step (2), obtain the graphical solution using MATLAB.

(4) Change one of the parameters to cause the optimal solution changes and solve the problem again.

Question 3:

Determine the objective function for building a minimum-cost cylindrical refrigeration tank of volume 50m3, if the circular ends cost $10 per m2, the cylindrical wall costs $8 per m2, and it costs $100 per m2 for the protecting material over the whole surface.

(1) Establish the mathematical model of the optimization problem.

(2) Draw the graphs and find the optimal solution (graphical solution).

(3) Change one of the parameters to cause the optimal solution changes and solve the problem again.

Question 4:

A cylindrical coordinate robot is to be used for palletizing a rectangular area. Find the maximum rectangular area available within the annular foot-print of the robot workspace. Take r1=400mm and r2= 1000mm.

(1) Develop the mathematical model of the optimization problem.

(2) Draw the graphs and find the optimal solution (graphical solution).

(3) Change one of the parameters to cause the optimal solution changes and solve the problem again.

Assignment 1 Submission Guide

? Attempt only two of the four questions: one linear programming problem (questions 1 and 2) and one non–linear programming problem (questions 3 and 4)

? Write a short report in PDF or Word document format. Include the written code separately (MATLAB scripts) for each question you have attempted along with the report as zip archive. Make sure to name the files with the question number and name the zip archive with your student ID. Submit the zip file to TurnItIn using the link in the Assessment folder on UTSOnline. For example, the three files in the zip file could be:

Assignment_1_DOM_13263621.PDF, Qestion_2_DOM_13263621.m,

Qestion_4_DOM_13263621.m and the zip file name could be

Assignment_1_DOM_13263621.zip

? The assignment is worth 5 marks, each question is worth 2.5 marks ? For each question:

? Write the model of the optimisation problem (0.75 marks)

? Describe the decision variables, objective function, and equality/inequality constraints (0.75 marks)

? Find the optimal solution and write the values of the design variables, objective function, and equality/inequality constraints. Include an image of the graphical solution (0.5 marks)

? Change a parameter that causes the optimal solution to change. Find the optimal solution and write the values of the design variables, objective function, and equality/inequality constraints. Include an image of the graphical solution (0.5 marks)

? For question 2 you will write the model twice: once with four decision variables and once with two decision variables

? Include the code with proper comments in line to explain each code line that you have written to solve each question and attach it with the report.

? You will be asked to explain what you have written and how the code works. No mark will be given if you cannot explain the results and the code.

Due – 11:59pm, 9 August 2019

Try two of the four questions: one from Question 1 and Question 2 (LP problem), another from Question 3 and Question 4 (NLP problem).

Hint: First establish the mathematical model based on the lecture slides. Then get start with MATLAB. Get familiar with the two MATLAB program codes provided, L1_ex1_1.m and L1_ex1_3.m. Try to understand the functions in the programs by using MATLAB online help and reading the necessary document. Then obtain the graphical solutions to the problems you have chosen by making necessary changes to the codes. Please go to the tutorial classes to know more about MATLAB, the modelling, and the detailed requirements.

Question 1:

The Outdoor Furniture Corporation manufactures two products: benches and picnic tables for use in yards and parks. The firm has two main resources: its carpenters (labour) and a supply of redwood for use in the furniture. During the next production period, 1200 hours of manpower are available under a union agreement. The firm also has a stock of 6000 kilograms of quality redwood. Each bench that Outdoor Furniture produces requires 4 labour hours and 8 kilograms of redwood; each picnic table takes 6 labour hours and 35 kilograms of redwood. A completed bench yield a profit of $10 each, and table a profit of $30 each.

Decide the numbers of benches and tables they should produce to maximize the profit.

(1) Establish the model of the optimization problem (LP with 2 variables). State clearly in words what the decision variables are, what the objective function and the equation/inequality constraints represent.

(2) Obtain the graphical solution using MATLAB.

(3) Change one of the parameters to cause the optimal solution changes and solve the problem again.

Question 2:

The local bookstore must determine how many of each of the four new books on photonics it must order to satisfy the new interest generated in the discipline. Book 1 costs $75, will provide a profit of $13, and requires 3 inches of shelf space. Book 2 costs $85, will provide a profit of $10, and requires 4 inches of shelf space. Book 3 costs $65, will provide a profit of $8, and requires 2 inch of shelf space. Book 4 costs $95, will provide a profit of $15, and requires 5 inches of shelf space. Find the number of each type that must be ordered to maximize profit. Total shelf space is 100 inches. Total amount available for ordering is $4000. It has been decided to order at least a total of 10 Book 2 and Book 4.

(1) Establish the model of the optimization problem (LP with 4 variables). State clearly in words what the decision variables are, what the objective function and the equation/inequality constraints represent.

(2) Suppose the numbers of Book 2 and Book 4 are fixed as 10 and 3 respectively, establish the model of the new optimization problem (LP with 2 variables).

(3) For the model established in step (2), obtain the graphical solution using MATLAB.

(4) Change one of the parameters to cause the optimal solution changes and solve the problem again.

Question 3:

Determine the objective function for building a minimum-cost cylindrical refrigeration tank of volume 50m3, if the circular ends cost $10 per m2, the cylindrical wall costs $8 per m2, and it costs $100 per m2 for the protecting material over the whole surface.

(1) Establish the mathematical model of the optimization problem.

(2) Draw the graphs and find the optimal solution (graphical solution).

(3) Change one of the parameters to cause the optimal solution changes and solve the problem again.

Question 4:

A cylindrical coordinate robot is to be used for palletizing a rectangular area. Find the maximum rectangular area available within the annular foot-print of the robot workspace. Take r1=400mm and r2= 1000mm.

(1) Develop the mathematical model of the optimization problem.

(2) Draw the graphs and find the optimal solution (graphical solution).

(3) Change one of the parameters to cause the optimal solution changes and solve the problem again.

Assignment 1 Submission Guide

? Attempt only two of the four questions: one linear programming problem (questions 1 and 2) and one non–linear programming problem (questions 3 and 4)

? Write a short report in PDF or Word document format. Include the written code separately (MATLAB scripts) for each question you have attempted along with the report as zip archive. Make sure to name the files with the question number and name the zip archive with your student ID. Submit the zip file to TurnItIn using the link in the Assessment folder on UTSOnline. For example, the three files in the zip file could be:

Assignment_1_DOM_13263621.PDF, Qestion_2_DOM_13263621.m,

Qestion_4_DOM_13263621.m and the zip file name could be

Assignment_1_DOM_13263621.zip

? The assignment is worth 5 marks, each question is worth 2.5 marks ? For each question:

? Write the model of the optimisation problem (0.75 marks)

? Describe the decision variables, objective function, and equality/inequality constraints (0.75 marks)

? Find the optimal solution and write the values of the design variables, objective function, and equality/inequality constraints. Include an image of the graphical solution (0.5 marks)

? Change a parameter that causes the optimal solution to change. Find the optimal solution and write the values of the design variables, objective function, and equality/inequality constraints. Include an image of the graphical solution (0.5 marks)

? For question 2 you will write the model twice: once with four decision variables and once with two decision variables

? Include the code with proper comments in line to explain each code line that you have written to solve each question and attach it with the report.

? You will be asked to explain what you have written and how the code works. No mark will be given if you cannot explain the results and the code.

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