Assignment 1 (5 marks) Due -- 6pm on 22 March, 2017 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 tutorials to know more about MATLAB and the modelling. 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 5000 kilograms of quality redwood. Each bench that Outdoor Furniture produces requires 4 labour hours and 10 kilograms of redwood; each picnic table takes 7 labour hours and 35 kilograms of redwood. A completed bench yield a profit of $9 each, and table a profit of $20 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 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 2 inches of shelf space. Book 2 costs $85, will provide a profit of $10, and requires 3 inches of shelf space. Book 3 costs $65, will provide a profit of $8, and requires 1 inch of shelf space. Book 4 costs $100, will provide a profit of $15, and requires 4 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 $5000. 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 0 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 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 $6 per m2, and it costs $80 per m2 to refrigerate over the useful life. (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 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=300mm and r2= 600mm L w r1 r2 (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 and solve the problem again.