MA3212 Optimisation and Operations Research Assignment 1
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MA3212 Optimisation and Operations Research
Assignment 1: Due on Friday Week 6, August 25, before 5 pm.
Submission: Pdf file online through JCU Learn. See end of assignment for details
Number of questions: 4
Question 1. For the following linear programming problems provide a graphical solution, if one exists, include the graph, and the final solution for the optimal values of the objective z and the decision variables x1 and x2 .
Graphs may be produced by hand or using software, your choice.
Question 2. (a) Find the optimal solution (if it exists) for the following LP problem
max z = 5x1 + 4x2 + 4x3
s.t. x1 + x2 + 2x3 ≤ 50
2x1 + x2 + x3 ≤ 120
x1 + x2 + x3 ≤ 60
x1 , x2 , x3 ≥ 0
(b) This problem comes from a factory that makes children’s toys. Each toy, A,B, C, uses 1, 1, and 2 metres of wood, respectively, and the factory only has 50 metres delivered each day week. The three toys, A, B, C, take 2 hours, 1 hour and 1 hour, respectively, to make, and the factory has only 120 hours of worker’s time available each week. Each toy is the same size but the storage room is only big enough for 60 toys. Each toy made produces $5, $4 and $4 profit, respectively.
(b) The management wants to use 18 hours per week for training for selected workers. Can they do this without reducing the factory’s profit? Explain why.
(c) Check your solution using the MATLAB function linprog(). Include a copy of your code.
Question 3. A printing company makes two types of paper using high quality or ordinary stock. They are required to supply at least 6000 cases of paper. They only have enough resources to produce 5000 cases of high quality paper, but more than enough stock for ordinary paper. Both types of paper are made with the same machinary and they have a limit of 1200 hours of machine time. High quality paper takes 20 minutes per case to produce and ordinary paper takes 10 minutes per case to produce. The profit on each case is $5/case and $3/case for high quality and ordinary paper , respectively.
a. Write the constraints for the problem.
b. Write the equation for the profit.
c. Graph the constraints and find the corner points
Question 4. Apply the standard simplex algorithm to the following LP problem and explain why a
max z = x1 + x2 + 2x3
s.t. − x1 − 2x2 − 3x3 ≤ 4
−3x1 − 2x2 − x3 ≤ 6
x1 , x2 , x3 ≥ 0
Submission of assignment:
• Submission is through JCU Learn. See link under Assessment, Assignment 1.
• Please include your name as part of the file name of the submitted file.
• Assignments can be typed or handwritten, your choice.
• Word files should be converted to pdf format.
• If handwritten, it can be scanned or photographed with a phone. Images should be converted to pdf, then software (such as pdfSam) can be used to combine each image into a single file.
• Multiple submissions can be made. Only your last submission before the due date will be marked. After the due date, only one submission will be allowed. Any late submissions will incur a 5% penalty as per JCU policy. It is recommended that you practice submitting early versions of your assignment if you are not familiar with submitting assignments on JCU.
• If you require an extension for medical reasons, a medical certificate will be required, as per JCU policy.
• pdfSam software: https:\\pdfsam.org
available for Windows or Mac. Other alternative software exists that merges pdf files into one file.
2023-08-23