Reference no: EM132489826

Optimisation

PART 1: MAXIMISING A FUNCTION

Objective: Find the maximum value of f (x, y, z) you can, where

f (x, y, z) = e^sin(40z) + e^sin(50x) + sin(60e^y) + sin[70 sin(x)] + sin[70 cos(z)]

+ sin[sin(80y)] - sin[10(x + y)] + (x² + y² + z²)/4

Constraints: The solution must be subject to the (hard) constraints:
-1 ≤ x, y, z ≤ 1 and x, y, z ∈ R

You should explain the approach taken, attaching any programming code that is used -provide comments to the code where appropriate

PART 2: DISTRIBUTION NETWORK

A major supermarket is updating its delivery network. They have 2 main warehouses (W1 and W2) and 23 stores at locations (1-23). Each day they must carry out a daily delivery from their two warehouses to all 23 stores, with the vehicles returning to the ware- houses at the end of the delivery. The geographical locations of the sites are shown below, with exact distances over the page:

(An Excel version of this data will be available on Canvas.) There are two types of vehicle that the supermarket can use:

Given the aim is to minimise the total daily costs, find the best strategy you can such that every store receives its delivery and the warehouses have the correct number of vehicles at the end to carry out the deliveries the following day.

Questions: Which stores should each warehouse supply? How many vans or lorries should they use? What routes should each vehicle take? What is the total cost?

PART 3: YOUR OWN REAL-LIFE EXAMPLE

Give an example of a real-life optimisation problem. This can be any example from business, government, leisure or sport. It may involve using existing data, or simply approximating behaviour with simulated data and your own model. It can incorporate problems from other modules, but must not repeat work.

The key points you must include in your report are:

(i) Background: Introduce the situation, including any relevant information that is needed to understand the problem (including references if required);

(ii) Aim: Specify what is to be optimised - What is the main objective? What con- straints will there be?

(iii) Model: Convert your problem into a mathematical or statistical problem - what is the form of your solutions? What is your objective function? What will the mathemati- cal or statistical model be to get from your solutions to your objective? What constraints exist on your possible solutions?

(iv) Optimisation Method: Explain how you will solve the problem - mathematical or computation approach, what algorithm(s) will you use etc. (Submit your code/program so your results can be verified. Comment or explain how your code works - this can be done as an Appendix, in addition to the 4 pages.)

(v) Results: Give the results to your problem. Is there just one optima, or multiple optima? How do you know you have got the optimal solution, or at least a solution close to the optimal, and that you are not at a local optima?

(vi) Conclusion: Put your results back in terms of the original problem. Critique your results - what are the strengths and weaknesses of your work? (Weaknesses in your model, for example due to the assumptions you make to simplify it, are not a bad thing, as long as you are aware of them. Remember, no model is perfect!)

Attachment:- Project-Optimisation.rar