SOLUTION: MATPMD4 – Stochastic Processes and Optimisation

MATPMD4 – Stochastic Processes and OptimisationProject 2: OptimisationDeadline of 23:59 Friday 16 April 2021. Submit your project in via Canvas. The projectcan be typed using any word processing software (e.g. Microsoft Word or Latex); figurescan be drawn using any software you wish.PART 1: MAXIMISING A FUNCTION (15%)Objective: Find the maximum value of f(x; y; z), wheref(x; y; z) = esin(10z) + esin(40y) + sin(20ex) + sin[70 sin(x)] – sin[30 cos(z)]+ sin[sin(30y)] – sin[10(x + y)] + x2 + y2 + z26Constraints: The solution must be subject to the (hard) constraints:-2 ≤ x; y; z ≤ 2 and x; y; z 2 RYou should explain the approach taken, attaching any programming code that is used –provide comments to the code where appropriate.PART 2: MULTI-OBJECTIVE OPTIMISATION (25%)An investment firm has £50000 to invest and has 100 possible options to choose from.The option number i (column 1), expected return vi (column 2), and investment requiredci (column 3) for each option are given in the file Part2.txt (on Canvas).(i) Single Objective: Assuming the investment firm has a single objective, to maximisetheir total expected return, f1, defined asf1 = Xivixisolve this optimisation problem – here xi is a binary value denoting the investment in option i or not. Give the list of options they should invest in and their total expected return,ensuring that Pi cixi ≤ 50000.(ii) Multi-objective: Now assume the decision maker has two objectives: to maximisethe total expected return, f1, but also to minimise the total number of investments they make, f2, defined asf2 = XixiInvestigate the possible choices the firm could make in this situation.In both parts, you should explain the approach taken, attaching any programming codethat is used – provide comments to the code where appropriate.1MATPMD4 – Stochastic Processes and OptimisationPART 3: DISTRIBUTION NETWORK (30%)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 dailydelivery from their two warehouses to all 23 stores, with the vehicles returning to the warehouses at the end of the day. The geographical locations of the sites are shown below, withexact distances over the page:0204060801000 20 40 60 80 1001t2t3t4t5tt67t8tt9t10t11t12t13t14t15t16t 17t18t19t20t21t22t23~W1~W2(An Excel version of this data will be available on Canvas.)There are two types of vehicle that the supermarket can use: Cost per mileMaximum stores it can supplyVanLorry£1£3514 Given the aim is to minimise the total daily costs, find the best strategy you can such thatevery store receives its delivery and the warehouses have the correct number of vehicles atthe end of the day to carry out the deliveries the following day.Questions: Which stores should each warehouse supply? How many vans or lorries doeseach warehouse require? What routes should each vehicle take? What is the total cost?2MATPMD4 – Stochastic Processes and OptimisationThe distances from site i to site j, in miles, are given in the table below (table is symmetrical): 1 2 3 4 56 7 8 9 1011 12 13 14 1516 17 18 19 2021 22 23W1 W212345084 029 93 012 81 19 064 62 90 71 067891062 31 64 56 6720 65 42 23 4840 43 54 39 4678 47 71 69 9254 91 83 65 34046 026 22 026 67 50 086 49 61 111 0111213141538 64 64 45 2633 61 32 25 6856 29 70 55 4320 68 28 14 6368 56 93 74 856 23 31 81 3032 26 24 45 7424 37 16 50 6643 16 26 58 6464 51 46 89 42047 037 39 039 13 42 031 69 40 65 0161718192085 22 101 86 4654 65 41 43 8963 33 64 57 7181 52 104 86 2166 44 88 70 1747 66 47 67 7934 49 43 31 984 49 29 21 9067 63 54 92 5553 47 38 79 4957 70 31 73 3970 23 53 36 8960 33 29 44 6844 79 44 76 1431 62 29 60 11080 050 32 032 96 72 029 80 57 16 021222381 15 95 80 4838 52 60 42 3232 83 61 43 4039 61 41 60 7942 19 17 67 4472 29 46 95 2255 63 25 67 4115 37 23 32 3319 55 55 43 478 73 42 37 3148 58 46 44 2876 78 75 61 49045 074 32 0W1W272 42 67 64 8571 37 91 74 2518 60 42 8 10350 51 39 74 5773 39 42 52 8238 63 27 63 1861 29 14 85 7121 79 54 18 853 59 8723 33 57067 0 3MATPMD4 – Stochastic Processes and OptimisationPART 4: YOUR OWN REAL-LIFE EXAMPLE (30%) – Max 4 sidesGive 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 approximatingbehaviour with simulated data and your own model. It can incorporate problems fromother 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 neededto understand the problem (including references);(ii) Aim: Specify what is to be optimised – What is the main objective? What constraints will there be?(iii) Model: Convert your problem into a mathematical or statistical problem – whatis the form of your solutions? What is your objective function? What will the mathematical or statistical model be to get from your solutions to your objective? What constraintsexist on your possible solutions?(iv) Optimisation Method: Explain how you will solve the problem – mathematicalor computation approach, what algorithm(s) will you use etc. Submit your code/programso your results can be verified. Comment or explain how your code works – this can bedone a separate file, and is not included in the 4 pages.(v) Results: Give the results to your problem. Is there just one optima, or multipleoptima? Explain how you know you have got the optimal solution, or at least a solutionclose to the optimal, and that you are not at a local optima?(vi) Conclusion: Relate your results back in terms of the original problem. Critiqueyour results – what are the strengths and weaknesses of your work? (Weaknesses in yourmodel, 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!)4