6524A Optimization Models
Optimization Models代写 For each question where you are requested a model,you must write it in a mathematical form.solve your model in Excel
1.WRITE your answers by hand or with a word processing
2. SUBMIT by March 11, 2019, 3:30 pm:
• your Excel files electronically using the Assignment tool underTake-home
3.For each question where you are requested a model, you must write it in a mathematical form. In addition, a linear model should be favored over a nonlinear
For each question that asks you to present the optimal solution, you must popularize this solution in the context of the problem (as if you were writing to a manager).Optimization Models代写
4.USE the following format when submitting electronically yourfiles:
• Firstname_Lastname_Problem-y.xlsx or
• Firstname_Lastname_Problem-y.zip
• where y refers to the problemnumber
Examples:
Aaron_Wang_Problem-1.xlsx
Aaron_Wang_Problem-2.zip
PROBLEM 1 (25 points) Optimization Models代写
The X-treme company manufactures all -terrain vehicles (ATVs) in its three plants located in Beijing, Shanghai and Suzhou. X-treme has four main clients that have ordered respectively 300, 500, 400, and 600 ATVs. To determine where ATVs will be manufactured, we must consider the manufacturing costs for each plant, the capacity of each plant, and the transportation costs from each plant to each client. The transportation is done by QuickExpress a third-party logistic firm. Transportation costs are computed according to the distance traveled (in km) and the number of ATVs transported.
The following table presents the number of kilometers between each plant and each client, the unit manufacturing costs for ATVs at each plant, and the capacity of each plant.Optimization Models代写
Plant | Client 1 | Client 2 | Client 3 | Client 4 | Unit cost | Capacity |
Beijing | 130 | 120 | 150 | 170 | 1,700 | 800 |
Shanghai | 160 | 140 | 180 | 130 | 2,000 | 600 |
Suzhou | 190 | 110 | 150 | 140 | 2,400 | 700 |
Demand | 300 | 500 | 400 | 600 |
QuickExpress charges $10 per vehicle transported plus $0.10 per vehicle and per kilometer. For example, transporting 300 vehicles from Beijing to Client 1 and 500 vehicles from Beijing to Client 2 will cost $17,900 = 300×(10 + 0.1×130) + 500×(10 + 0.1×120).
Question 1.1: (5 points)Optimization Models代写
Considering that QuickExpress will be transporting all the vehicles manufactured by X– treme, propose a mathematical model that will allow X-treme to determine its production plan for Beijing, Shanghai, and Suzhou that minimizes total production and transportation costs. Clearly define all the decision variables, the objective function and each of the constraints.
Question 1.2: (3 points)
Implement and solve your model in Excel. Clearly describe the optimal solution found.
During the latest board meeting at X-treme, the directors have realized that the transportation costs are high probably because of the lack of competition among logistic firms. X-treme has therefore asked different companies that specialize in transportation to provide them with an estimate for the transportation costs of its ATVs. Two (new) companies have submitted their offer, and X-treme has decided to go forward with only one of them. Before choosing which one, X-treme must examine the different possibilities.Optimization Models代写
The first company, BeijingExpress, has proposed the following transportation costs: $5,000 each time it must send a vehicle from a plant to a client (for each shipment from a plant to a client and independently of the number of ATVs transported) and an additional $0.05 per ATV and per km. For example, if BeijingExpress transports 300 vehicles from Beijing to Client 1 and 500 vehicles from Beijing to Client 2, the total transportation cost will be $14,950 = (5,000 + 300×0.05×130) + (5,000 + 500×0.05×120).
Question 1.3: (5 points)Optimization Models代写
Considering that both QuickExpress and BeijingExpress can be hired to transport the vehicles manufactured by X-treme, propose a new mathematical model that will allow X– treme to determine its production plan for Beijing, Shanghai, and Suzhou that minimizes total production and transportation costs. (We consider that each plant can have some of its production transported to a given client by QuickExpress, the remaining by BeijingExpress).
Question 1.4: (3 points)
Implement and solve your new model in Excel. Clearly describe the optimal solution found.Optimization Models代写
RapidTransans has also proposed to transport some ATVs. It proposes a transportation cost of $25 for the first 150 vehicles shipped from a plant to a client, and of $20 for all the other vehicles shipped from the same plant to the same client (independently of the distance). For example, if RapidTransans transports 300 vehicles from Beijing to Client 1 and 500 vehicles from Beijing to Client 2, the total transportation cost will be $17,500 = (150×25 + 150×20) + (150×25 + 350×20).
Question 1.5: (5 points)Optimization Models代写
Considering that both QuickExpress and RapidTransans can be hired to transport the vehicles manufactured by X-treme, propose a new mathematical model with that will allow X-treme to determine its production plan for Beijing, Shanghai, and Suzhou that minimizes total production and transportation costs. (We consider that each plant can have some of its production transported to a given client by QuickExpress, the remaining by RapidTransans).
Question 1.6: (3 points)
Implement and solve your new model in Excel. Clearly describe the optimal solution found.Optimization Models代写
Question 1.7: (1 point)
What is your final recommendation for X-treme: should the company contract BeijingExpress or RapidTransans?
PROBLEM 2 (30 points)Optimization Models代写
ABCGrid is planning a maintenance in five power plants (A, B, C, D, E). This maintenance occurs each four years and lasts three consecutive months at each plant. During that time, the capacity of the plant is reduced.
The two last columns of Table 2.1 indicate the months during which the maintenance can start at each plant. For example, maintenance at plant A can start at the earliest at the beginning of month 1, and at the latest at the beginning of month 3. The maintenance is specific to each plant. Consequently, the number of maintenance teams required per plant varies. The number of teams required for the maintenance of each plant is also presented in Table 2.1. For example, plant A needs 4 maintenance teams for the first month of the maintenance, 3 teams the following month, and 2 teams for the last month of the maintenance.Optimization Models代写
Plant | Number of maintenance teams required | Beginning of maintenance | |||
1st Month | 2nd Month | 3rd Month | Earliest | Latest | |
A | 4 | 3 | 2 | 1 | 3 |
B | 3 | 2 | 3 | 2 | 4 |
C | 4 | 3 | 3 | 3 | 4 |
D | 1 | 4 | 5 | 1 | 3 |
E | 6 | 5 | 3 | 2 | 4 |
Table 2.1: Maintenance information with respect to each power plant
ABCGrid wants to complete the maintenance of its five power plants in a delay of six months, that is, the maintenance must be completed at the end of month 6. The availability of the maintenance teams for the next six months is given in Table 2.2.
Month | 1 | 2 | 3 | 4 | 5 | 6 |
Availability | 7 | 17 | 18 | 12 | 7 | 10 |
Table 2.2: Number of teams available for maintenance in each month
The total cost for the maintenance of a plant varies depending on the month at which it is started. These costs are presented in Table 2.3. For example, for plant B, maintenance will cost $102,000 if it starts at the beginning of month 2, $100,000 at the beginning of month 3, and $99,000 at the beginning of month 4.Optimization Models代写
Plant | Month 1 | Month 2 | Month 3 | Month 4 |
A | 100 | 98 | 101 | |
B | 102 | 100 | 99 | |
C | 100 | 90 | ||
D | 110 | 105 | 100 | |
E | 96 | 98 | 100 |
Table 2.3: Cost (in thousands of dollars) when starting maintenance at a given month
Question 2.1: (10 points)Optimization Models代写
Propose a mathematical model that will help ABCGrid plan the maintenance of its five power plants at minimum cost. Clearly define all the decision variables, the objective function and each of the constraints.
Question 2.2: (5 points)
Implement and solve your model in Excel. Clearly describe the optimal solution found. When does the maintenance of all five power plants end?
Question 2.3: (10 points)
ABCGrid now wishes to complete the maintenance program of its five power plants as soon as possible (no matter the cost). propose a new mathematical model with that will allow to consider this new objective.Optimization Models代写
Question 2.4: (5 points)
Implement and solve your new model in Excel. Clearly describe the optimal solution found. Compare this new solution to the one you found in Question 2.2.
PROBLEM 3 (25 points)Optimization Models代写
The ABC investment firm has changed its investment strategies. It has sold all its assets that were in the aviation sector and wants to reinvest in the real estate sector. ABC has an
$8 million budget and monitors buildings for sale. So far, 15 projects are evaluated in the greater Beijing area. The price of each building, as well as the (discounted) total income expected over the next 10 years, are shown in the next table. Some of these projects are new buildings, part of building complexes. It is thus possible to buy more than one building. The last column gives the maximum number of buildings that can be purchased per project. Price and income (in millions of dollars) correspond to a single building.Optimization Models代写
Project | Municipality | Price | Income | Maximum |
1 | Beijing | 1.20 | 1.35 | 1 |
2 | Beijing | 1.14 | 1.23 | 1 |
3 | Beijing | 1.80 | 2.40 | 3 |
4 | Beijing | 2.16 | 2.30 | 4 |
5 | Beijing | 2.34 | 2.60 | 3 |
6 | Beijing | 1.75 | 2.35 | 4 |
7 | Hongzhou | 1.80 | 2.06 | 6 |
8 | Hongzhou | 2.97 | 3.27 | 3 |
9 | Hongzhou | 1.45 | 1.53 | 2 |
10 | Hongzhou | 1.23 | 1.35 | 1 |
11 | Hongzhou | 1.83 | 1.90 | 1 |
12 | Ningbo | 1.21 | 1.27 | 2 |
13 | Ningbo | 1.45 | 1.52 | 1 |
14 | Ningbo | 1.73 | 1.92 | 1 |
15 | Ningbo | 2.04 | 2.15 | 1 |
For each of the following questions, you are asked to formulate clearly a mathematical model and solve this model in Excel.
Question 3.1: (10 points)
Assume that it is possible to invest in a portion of a building, that is to say, if ABC buys1.2 buildings in project 4, it will cost $2,592,000, and the expected revenue would be $2,760,000. Propose a mathematical model to help ABC allocate its budget in order to maximize its total income. Solve your model and present clearly your solution.Optimization Models代写
Question 3.2: (3 points)
Assume now that it is not possible to invest only in part of a building, i.e. ABC must purchase each building entirely (or not at all). How should the model for Questions 3.1 be modified? How should then ABC allocate its budget in order to maximize its total income? Is there a big difference between the answers of Questions 3.1 and 3.2?
Question 3.3: (3 points)
Assume again it is not possible to invest only in part of a building. A 16th project (with a single building) is under consideration. Its selling price is $1.25 million. What should be its minimum income to make it worthwhile to buy? Clearly explain the process used to arrive at your answer.Optimization Models代写
Question 3.4:
Assume again it is not possible to invest only in part of a building. Note that the first six projects are on the island of Beijing, projects 7 to 11 are on the north shore (Hongzhou) and the last four are on the south shore (Ningbo).
a) If ABC wants to have at least one building in each of the three regions, how should the model proposed for Question 3.2 be modified to take this requirement into account? How should ABC invest? (3points)Optimization Models代写
b) If ABC wants to have all its buildings in the same area (Beijing, Hongzhou or Ningbo), how should the model proposed for Question 3.2 be modified to take this requirement into account? How should ABC invest? (3points)
c) If ABC wants to have either all its buildings in Beijing or all buildings outside Beijing (Hongzhou or Ningbo), how should the model proposed for Question 3.2 be modified to take this requirement into account? How should ABC invest? (3points)
PROBLEM 4 (20 points)
A recent earthquake has caused major damages to the facilities of a nuclear plant.
Most importantly, the earthquake has damaged four nuclear reactors, which temperature have dramatically risen in the last hours. The current temperature of the four reactors is presented in the following table:
Reactor | #1 | #2 | #3 | #4 |
Temperature in °F | 453 | 122 | 517 | 212 |
To avoid a complete melt down of the reactors, by the end of the day their temperature should not exceed 800°F. Conversely, it would be unnecessary to cool a reactor to a temperature below 100°F.Optimization Models代写
The authorities have planned a series of measures that would cool down the reactors and must now decide which of them will be implemented on each reactor over the next 24 hours. In order to better control the staff’s exposure to radiation every measure should be performed during an integer number of hours. On the site, 1,200 people are currently present. Each of them should only work a maximum of one hour a day to limit his exposure to radiation.
The following table shows the various measures that can be applied.
For each measure, it also presents the personnel needed to apply any measure for an hour to a reactor and its impact on the temperature of this reactor.
Measure |
Personnel needed
(per hour) |
Impact on temperature
(in °F/h) |
1) Do nothing | 0 | +7 |
2) Inject sea water in the reactor | 17 | -15 |
3) Spray water on reactor’s building | 3 | -5 |
4) Drop water on reactor’s building from a helicopter |
3 |
-3 |
5) Use electric pumps | 10 | -10 |
6) Use the reactor’s cooling system | 20 | -25 |
For security reasons, it is impossible at any point of time to apply more than one of these measures on a given reactor.
The use of the reactor’s cooling system (Measure 6) and the use of electric pumps (Measure 5) are subject to one additional restriction. Prior to applying these measures, the reactor must first be connected to the electric network. None of the reactors is currently connected to this network. Such an operation would require the use of 3 people per hour for 4 hours for each reactor. Again, for safety reasons, it is impossible to link a reactor to the electric network while another operation is in progress on this reactor.Optimization Models代写
Question 4.1: (17 points)
Propose a mathematical model that could help determine, for each reactor, the amount of time in the next 24 hours that should be dedicated to each measure in order to minimize the average temperature of the four reactors at the end of the day. Note that your model does not need to determine the order in which each measure will be applied. Do not solve your model.
Question 4.2: (3 points)Optimization Models代写
Based on the model developed in the previous questions, you have determined that an average temperature of 378°F can be reached. However, implementing the resulting solution would cause Reactor #3 to reach the temperature of 790°F at the end of the day. The authorities consider this temperature to be too high and are asking if you could find a solution that reaches the same average temperature while ensuring that the temperature of the warmest reactor be as low as possible. Propose changes to the model of Question 4.1 to reflect this new information. Do not solve your new model.
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