- January 4, 2021

MA424 Modelling in Operations Research Grammateia Kotsialou Final project (part 1) 2020 – 2021 OIL is a petrol company based in the UK. With this project, they reach out to you for advice on their future investment plans, e.g. on new products, whether to expand to more floors and on delivery options. 1 Production OIL produces three products of petrol that can be used for either cars, boats or planes. These products are made by combining two types of oil in different proportions. One tonne of petrol product 1 is made using between 50% and 70% of oil 1 and the remainder is oil 2. One tonne of the petrol product 2 is made using between 30% and 40% of oil 1 and remainder oil 2. The petrol product 3 is made using 10%-20% oil 1 and remainder oil 2. Furthermore, a tonne of the petrol product 1 requires 1 hour in the mixing machine, a tonne of product 2 requires 45 minutes and a tonne of product 3 requires 30 minutes (the company guarantees that any future product will require only 15 minutes in the mixing machine). The machine can be used for 600 hours per month. The cost of oil 1 is 200 per tonne, the cost of oil 2 is 350 per tonne. The supplier of oil 1 and oil 2 can supply unlimited amount of oil 2, but only 150.5 tonnes of oil 1. The selling price (per tonne) of the petrol product 1, 2 and 3 is £700, £800 and £900, respectively. The monthly demand for each product (made by OIL) is 100 tonnes. (i) OIL wants your advise about what amount of oil 1 and oil 2 to put into the three petrol products, in order to maximise its net profit. (ii) OIL decides to introduce two new additional products at the selling price of £1000 and £1050, respectively. Both new products use from 0% to 10% of oil 1 and the remainder is oil 2. At the same time, the company decides to produce exactly one among the three initial products but they are unsure which product that should be. Can you advise OIL on this? 2 Floor expansion To avoid the additional costs of using a warehouse, OIL decides to extend its own storage space (from the ground floor) into a three-floor extension as shown below. Both sides of each new floor (location points 1 – 6) are supported by a number of steel beams. The carrying capacity of each floor increases linearly with the number of beams that support it. Every floor has the same number of beams at each side. Half of the weight of each new floor is distributed to its beams on the left side and the other half to its beams on the right side. Each beam is 1m long except those in location point 6 which are 1 2m long. Each beam gives a weight capacity of 100 kg (independent of its length) for the floor that it supports. The amount of steel that OIL has budgeted for is enough to produce 240m of beams. Each floor has a maximum space capacity of 10 barrels and one barrel weighs 200 kg. The company wants to maximise the weight to be loaded on the extension. As this is only an initial drawing plan, OIL ignores the deeper physics of the construction, the weights of platforms and steel beams. (i) Should OIL proceed with building this three-floor extension? (ii) An alternative plan of OIL is to build a sixth-floor construction (three additional floors on top of the previous extension) such that the design pattern of each new floor i is as follows: one side rests on the i− 1 floor and the other side rests on the first floor. The left-hand side beams of each new floor are 1m tall and the right-hand side beams are as tall as they need to be (to connect to the first floor). Given your modelling of both extension plans, what would you advice OIL to build? 3 Petrol delivery OIL needs to deliver an order of 30 petrol barrels to a client and can do this using either one of two delivery companies (A and B) or both. The two companies offer the following deals: Company A charges £5 per barrel for the first 5 barrels. The next 5 barrels will cost £4 per barrel, and any additional barrel costs £3. Company B charges £4.5 per barrel if you send 10 or fewer barrels. However, company B offers a bulk discount such that if you send more than 10 barrels, you pay £4 per barrel. (i) What would you advice OIL to use for the delivery of this order? (ii) OIL wishes to use only one delivery company for the whole order. Can you suggest additional constraint(s) in your modelling to enforce this (independent of price changes)? Find an instance to demonstrate that your additional formulation works. 2 Deliverables and report content – final project part 1 OIL wants you to develop models for their problems. You can use AMPL and Excel to implement and solve these models. It is expected that you will analyse and discuss your findings on the basis of the questions outlined. You must deliver a project report with your analysis and suggestions but also the details of the modelling that you have undertaken. The project report should consist of the following: (a) A brief Executive Summary, discussing your main findings. It should be completely free of any mathematical terms and discussion of modelling technicalities. It should describe the main characteristics of the solutions obtained and answer all the questions. (b) A concise Management Report discussing all of your findings. The Management Report should be independent of the Executive Summary (i.e., self-contained). Ideally, the Management Report will avoid the use of unnecessary modelling technicalities. (c) A number of Technical Appendices. The technical appendices will give a complete account of the formulation/modelling development. In particular, an appendix detailing the development of the optimisation models, where all entities of the models you formulate are defined and their meaning is explained in detail, together with the models clearly stated. Do not add a printscreen of your code. Guidelines – final project part 1 (a) The deadline for submitting this project is 26/01/2021 midnight UK time. The electronic copy of the report (use a pdf file), the model files and any other computer files you create should be contained in one .zip file, to be uploaded on the moodle MA424 page anonymously – only with your exam candidate number – by the deadline. Name your zip. file with your exam candidate number only. (b) You are expected to work in the group of size at most three assigned to you by the course leaders. (c) All components of the report must be written individually. This includes: the executive summary, the management report, any mathematics with associated explanation, any discussion of the relationship between the mathematics and the software used, any computer code and any description of the software features used. This is not an exhaustive list. You may not seek advice from anyone else other than your fellow group members and the MA424 lecturers. (d) The computer code should also be written individually. You are expected to write brief explanation comments for each line of your code. (e) On a separate page at the beginning of your report, you must provide candidate exam number of all other students who you worked together with, and a brief statement on the extent of collaboration and the individual contributions. (e.g., “we prepared the basic formulation together, and both of us had about the same contribution”, “we prepared the basic formulation together, but most constraints were developed by me”, “we worked mostly separately, but she explained to me the constraint about the maximum distance”). Common elements of work that are not declared will be considered as cases of plagiarism. 3 (f) The report without the Technical Appendices should not exceed 8 pages (excluding the page described on point (e)) with 11pts, single spacing. Concise and clear reports are preferred. (g) Any project submitted without all associated computer files will be given a mark of zero. If an answer is given in the report that cannot be confirmed by your code, then this will be considered as plagiarism. (h) You are allowed to ask only clarification questions to the MA424 lecturers. (i) The clarification questions should be asked on the Moodle forum so that the answers are visible to all MA424 students. Marking scheme – final project part 1 Modelling 30 Implementation 20 Analysis 30 Organization/Presentation 20 Total 100 – Modelling. The basic ingredients of the model are appropriately and meaningfully defined. The es- sential decision variables for the model are introduced with suitable types. Constraints are formulated correctly. – Implementation. All scenarios and the extensions are implemented in a sound manner. Data is provided in an appropriate form and separated from the model. Further, demonstrating sufficient capacity to developing “good” models: avoiding unnecessary variables and constraints, compactness of data representation as well as the use of more advanced features for defining model entities. – Analysis. The report should sufficiently demonstrate the ability to recognise and report all key results of the models and relate these to the real-world problem as well as demonstrating sufficient understanding of these results with a discussion of requisite insight. Discussing limitations of the model, the possible sources of these and how they may impact on the problem decision. – Organisation/Presentation. A main report and technical appendices which are clear, concise, well organised, well formatted and well presented with appropriate use of figures and tables, well commented and easy to navigate files. 4 欢迎咨询51作业君