DEPARTMENT OF MANAGEMENT SCIENCE MSCI 222: OPTIMISATION COURSEWORK EXERCISE Coursework should be submitted online via Moodle by 3pm on Monday 5 December of 2022. The exercise is to be done individually. Late work or evidence of collusion will be penalised in the manner specified in the department’s teaching code of practice. Your answers should clearly state your reasoning and any assumptions you make. You should define each variable you use in your models and summarise the reason for adding each constraint. In the following problem, you should replace X by the number between 0 and 99 formed by the 3rd and 4th digits of your library card number, Y by the number between 0 and 99 formed by the 5th and 6th digits of your library card number, and Z by the number between 0 and 99 formed by the 7th and 8th digits of your library card number. For example, if your library card number is 08123456, then X = 12, Y = 34 and Z = 56. Part A: The “SuperPaper Company” has two separate problems. First, they need to transport enough wood from forests to produce paper in their paper mills. They use 5 different forests (F1, F2, …, F5) to feed their 3 paper mills (P1, P2 and P3). Each of the first two paper mills (P1 and P2) needs to receive 1200 tonnes of wood every week to produce 300 tonnes of paper each (Please note that, every ton of wood received by any paper mill is converted into 0.25 tons of paper). The third paper mill produces the remaining required paper by receiving enough wood. Then the paper produced is sent to 5 five distribution centres (D1, D2, …, D5). The paper is distributed to seven markets (M1, M2, …, M7) through the distribution centres. The capacities of each distribution centre and forest, the demand of each market, and the unit transportation cost per ton of paper are given below. Market M1 M2 M3 M4 M5 M6 M7 Demand 30+X 40+Y 50+Z 120 140 130 150 Distribution Centres D1 D2 D3 D4 D5 Forests F1 F2 F3 F4 F5 Handling Capacity 200 250 220 210 300 Wood Quantity 600 700 1000 800 900 Table 1: Capacities and demand (tons of paper/week and tons of wood/week) P1 P2 P3 D1 D2 D3 D4 D5 M1 M2 M3 M4 M5 M6 M7 F1 54 78 70 P1 52 79 68 70 87 D1 93 78 59 57 72 67 65 F2 57 53 90 P2 71 76 68 65 80 D2 95 70 58 53 70 79 70 F3 92 90 65 P3 53 88 55 52 79 D3 79 71 55 59 86 81 54 F4 40 50 60 D4 91 79 82 71 80 64 61 F5 38 60 48 D5 87 97 94 76 51 70 62 Table 2: Transportation cost of wood and paper (￡/ton) The company wants to meet the demand at a minimum cost. Write the two mathematical models: 1. For the transport planning between the forests and paper mills 2. For the transport planning between paper mills and markets through distribution centres. Solve both of the problems using a software (Excel Solver, Classic Lindo or Lingo). Calculate the cost associated with your solution and present your solution in tabular form. (40 marks) Part B: After solving part A by formulating as two separate problems, you have learned that the paper mills have additional paper production capacities that can be used without investing any money on them. You think that considering the entire system with a single model instead of formulating it as two separate models can improve the cost associated with transporting wood and papers. Write a single mathematical model that minimises the cost of transporting papers and wood while satisfying the demand of the markets and the capacity constraints of distribution centres and forests. In addition, also include each paper mill’s production capacity that is given below, in your model. Use a software (Excel Solver, Classic Lindo or Lingo) to solve it. Calculate the cost associated with your solution and present your solution in tabular form. Calculate also how much the company could save by including the production decision in paper mills and transport decision from forests to paper mills together. (25 marks) Paper Mills P1 P2 P3 Production Capacity 400 450 500 Table 3: Production capacity (tons of paper/week) of each paper mill Part C: In the future, the company expects to have a 10% decrease in the demand of each market and therefore it plans to close one of the distribution centres. The objective of the company is to meet the demand at a minimum cost. Write a single mathematical model for this problem using the model built in Part B as your base model and use a software (Excel Solver, Classic Lindo or Lingo) to solve it. Calculate the cost associated with your solution and present your solution in a tabular form. The problem should be formulated as a single model and the answer should be acquired from this model. Approaches using multiple mathematical models or solver runs to solve the problem will not get any marks. (20 marks) Part D: Another option to address the 10% drop in demand is streamlining the transport operations between the distribution centres and the markets. The company said, if they could make sure that every market receives, at least 80% of their reduced demand from a single distribution centre, they may not need to close any distribution centres. Formulate a mathematical model considering all the constraints suggested in Part B plus every market should receive at least 80% of their demand from a single distribution centre. Do not include the distribution centre closing constraint in your mathematical model. Use a software (Excel Solver, Classic Lindo or Lingo) to solve it. Calculate the cost associated with your solution and present your solution in a tabular form. Compare your solution with the solution in the previous part and suggest the highest cost saving option. The problem should be formulated as a single model and the answer should be acquired from this model. Approaches using multiple mathematical models or solver runs to solve the problem will not get any marks. (15 marks)
