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Coursework Brief

Please answer ALL questions provided below. This coursework is worth 30% of your overall mark on the module. The total to obtain is 100 marks.

The coursework is due by 4pm on Monday, December 16th 2024. Please ensure your work is submitted before this deadline.

Submission is through KEATS only. Your submission must include a .zip file containing all of the items below:

1. A .pdf file with your workings and solutions (either typed or scanned). Any handwritten material including hand-drawn plots, please make sure they are clear and readable on a screen, otherwise your work will not be marked

2. Please provide clear and concice answers with clear indication of the part of each question (subques-tion) you are answering. Please include your code and its outputs in your .pdf document with your asnwers, under the corresponding subquestion heading, making sure it clearly corresponds to that answer heading.

3. .lp files of your LPSolve programs must also be included in your submission pack

4. .txt files with the outputs/solution of LPSolve for each of your programs. Again this is in addition to including these in the main .pdf document

1 Questions

Question 1

Consider the following linear program:

1. Plot the feasible region and the contour lines of the objective function to solve the program graphically.

2. Find all the extreme points of the feasible region and evaluate the objective function at each extreme point.                  [25 marks]

Question 2

Consider the following LP problem:

1. Solve with LPSolve and write down the optimal solution (include .lp file).

2. Convert the problem into canonical form.

3. Solve using the simplex method (by hand). Write down all simplex tableaus.

Show that you obtain the same result as in part 1.                                                            [25 marks]

Question 3

Consider the following ILP program:

Solve using the branch and bound method.

Use the branch and bound method, solving graphically, by hand the LP relaxations at each step.

Show your work: write down the LP relaxations, plot the graphs and write down the solutions and the bounds on the objective value at each node of the branch and bound tree; indicate the branching variables in each node and their respective values on each branch of the split.                                                           [25 marks]

Question 4

In the next 5 days you need to fly a total of 100 large boxes from England to Iceland.

Based on deals in place and the capacity of available aircrafts, the price and the maximum number of boxes you can send over on each day differs as shown in the table below:

The company in Iceland will pick up the boxes at the end of Friday. Any boxes sent over earlier would need to be kept in storage at a cost of £5 per box per night.

Your goal is to find a schedule to transfer all 100 boxes to Iceland at a minimum total cost, including transportation and storage.

Answer the following questions:

1. Formulate the problem above as an integer linear program: define decision variables and men-tion what each of them represents, formulate and explain the objective function and all neces-sary constraints.

2. Use LPSolve to solve the problem and write down the solution.                                                [25 marks]