Tasks Problem 1 (25 marks) A commercial museum in Australia is considering salvaging the wreck of RMS Titanic to be displayed in the museum at salvaging cost of $60,000. However, this project requires approval from relevant governmental agencies. The initial inquiry undertaken by a lobbyist, on behalf of the museum's management, has revealed that if the wreck is evaluated at a substantial value, the approval may be withheld, whereas if the value is low-moderate the approval may be granted. In order for an evaluation to be done in the first place, the museum's management has been advised to submit a bid to a nominated regulatory body. If the bid is too low, the regulatory body may not set up a mechanism to evaluate the wreck, because it will be considered not worth the hassle. However, in the event of losing the bid, the Museum could build a replica of the passenger liner at a cost of $135,000 which they consider an acceptable second-best alternative. The museum's management has estimated that they could get an evaluation done for $100,000, but this might lead the regulatory body to considering the wreck to be of substantial value and not approve their salvaging request. With a bid of less than $30,000 the agency is likely to not set up the evaluation mechanism in the first place. 1 Spring 2016 – ENGG953 Based on this scenario, develop a decision making model representing the options available for the museum. (Marks are not equally allocated for the following elements of your response): a) Articulation of the problem outlining the decision alternatives, chance events and their probabilities according to your knowledge and estimates, as well as possible outcomes; (7.5 marks) b) Construction of a decision tree showing the above information, including the calculation of expected values associated with each alternative; (7.5 marks) c) Interpretation of the results of your analysis, including acknowledgement of any other relevant factors that need to be considered in relation to this decision scenario; and (5.0 marks) d) Statement of all assumptions and estimates used, including necessary justifications. (5.0 marks) Problem 2 (25 marks) West Digital in Singapore has two manufacture lines to produce server hard disk drives (HDDs). The manufacturing process of the products through assembly, burn-in test, performance test, and packaging and distribution is shown Figure Q2. Assembly Burn-test Performance test Packaging & Distribution 6 3 1.2 1 1.3 7 10 2.4 4 1.0 2.1 1.5 8 2 5 1.1 11 2.0 1.7 9 2.3 0.95 Figure Q2 Production flow of hard disk drives The monthly production capacity at each assembly line (in 1000s of HDDs) is shown at nodes 1 and 2. The burn-test facilities at nodes 3, 4 and 5, performance test facilities at nodes 6, 7, 8 and 9, and Packaging & Distribution at nodes 10 and 11 have capacities of HDDs produced (1000s of HDDs) as shown in the figure. The various assembly, burn-test, and performance test costs per unit (one HDD) at each facility are shown in the following table: 2 Spring 2016 – ENGG953 Facility 1 2 3 4 5 6 7 8 9 10 11 Manufacture 8.0 10.0 1.6 1.8 1.4 1.7 0.8 1.0 1.2 0.6 0.5 unit cost ($) The monthly demand for the hard disk drives is 4,000. Tasks: 1) Formulate a linear programming model that indicates the number of hard disk drives must be produced at each facility to meet the monthly demand at the minimum cost. (10 marks) 2) Solve this model by using the computer to get solution of each of the model variables. (10 marks) 3) Interpret the results obtained in 2) to assist in determining number of HDDs that should be delivered from each facility node. (5 marks) Problem 3 (25 marks) A call centre has 70 staffs to be scheduled in three 8-hour shifts. However, the number of incoming calls varies significantly according to the time of day. The slowest period is between midnight and 4:00 am. It is estimated that the average number of calls served by a staff is 8 during that period. The average number of calls served by a staff in different time period of day is shown in the following table: Time period Duration Average number of calls served 1 0:00 am to 4:00 am 8 2 4:00 am to 8:00 am 15 3 8:00 am to 12:00 noon 30 4 12:00 noon to 4:00 pm 35 5 4:00 pm to 8:00 pm 32 6 8:00 pm to 12:00 midnight 20 To retain customers and acquire new ones, the call centre must maintain a high customer service level. To do so, it has determined the minimum number of staffs it needs to work during every 4-hour time segment is as follows: 8 from midnight to 4:00 am, 12 from 4:00 to 8:00 am, 26 from 8:00 am to noon, 30 from noon to 4:00 pm, 20 from 4:00 to 8:00 pm, and 14 from 8:00 pm to midnight. 3 Spring 2016 – ENGG953 Tasks: 1) Formulate and solve an integer programming model to help the call center schedule its staffs. (10 marks) 2) If the call centre has a maximum of only 13 staffs who will work the late shift from midnight to 8:00 am, reformulate the model to reflect this complication and solve it. (10 marks) 3) All staffs like to work the day shift from 8:00 am to 4:00 pm, so the call centre has decided to limit the number of staffs who work this 8-hour shift to 30. Reformulate the model in 2) to reflect this restriction and solve it. (5 marks) Problem 4 (25 marks) General Motor Company produced a lightweight, all-terrain vehicle named as Terrain-2000 for military use. The company is now planning to sell the vehicle to public. It has five plants that manufacture the vehicle and four regional distribution centres. The company is unsure of public demand for the Terrain-2000, so it is considering reducing its fixed operating costs by closing one or more plants, even though it would incur an increase in transportation costs. The relevant costs for the problem are provided in the following table (Table 1). The transportation costs are per thousand vehicles shipped; for example, the cost of shipping 1,000 vehicles from plant A to warehouse W_3 is $30,000. The annual demand from each warehouse of W_1, W_2, W_3 and W_4 is 15,000, 12,000, 9,000 and 11,000, respectively. The annual production capacity and annual fixed operating costs of each plant are given in the table; for example, the annual production capacity and annual fixed operating costs for Plant A is 12,000 and $ 2,000,000, respectively. Table 1 Data for Question 4 Transportation Costs ($1,000s) to Warehouse Annual Annual Fixed From Plant Production Operating W_1 W_2 W_3 W_4 Capacity Costs ($) A $52 $20 $30 $62 12,000 2,000,000 B $16 $45 $11 $32 15,000 900,000 C $10 $70 $35 $48 13,000 1,500,000 D $28 $22 $58 $25 10,000 1,000,000 E $42 $48 $24 $30 16,000 850,000 a) Formulate a linear programming model for this problem (10 marks) b) Solve this model by using the computer (10 marks) 4 Spring 2016 – ENGG953 c) Interpret the solution obtained in b) to assist the company in determining which plants should remain open and which should be closed and the number of vehicles that should be shipped from each plant to each warehouse to minimize the total cost. (5 marks) Learning Guides In order to complete the assignment tasks, you need to read Chapter 2 to Chapter 5, Chapter 9 and Chapter 12 of the prescribed textbook "Introduction to Management Science" by Bernard W. Taylor. While you are reading, you need to understand or answer the following questions: 1) Decision-making criteria and decision trees: How to draw a decision tree? 2) What is a sequential decision tree and how to draw it? 3) How to get a decision-making problem solution without and with probabilities? 4) Understand the linear programming model formulation steps: (1) Define the decision variables; (2) Define the objective function, and (3) Define the constraints. 5) Use graphical approach to get solution of a linear programming model. Know how to determine the optimal solution. 6) Learn the solution process of linear programming models using a software tool such as Excel Solver and QM for Windows. 7) Learn, practise and gain skills to reformulate a real problem into a linear programming model. 8) Practise and gain skills to get solution of a complicated problem using linear programming model by Excel. 9) Consider how to conduct a sensitivity analysis of a decision-making model. 10) Understand and use integer programming models: total integer model, 0–1 integer model, and mixed integer model. 11) Learn the approach to formulating multicriteria decision-making models and solution process using the computer, i.e. Excel Solver or QM for Windows. 5 Spring 2016 – ENGG953