**Operations Research **

**Operations Research**

**Duration: 2h00/2h30**

**Duration: 2h00/2h30**

运筹学考试代考 1.Consider the following Linear Programming Model for a production planning problem of four types of juice: J1, J2, J3and J4.

**1.**

Consider the following Linear Programming Model for a production planning problem of four types of juice: **J**** _{1}**,

**J****,**

_{2}

**J****and**

_{3}

**J****. The decision variables,**

_{4}

**x****, represent the thousand liters daily produced of juice**

_{i}

**J****, i=1, 2, 3, 4. The first constraint is concerned with the daily availability of machine hours whereas the second one is related with the availability of a fruit concentrate. The third constraint guarantees the production rule required by the sales department and the last constraint represents a commitment with customers. The objective is to maximize the daily profit.**

_{i}

** a) **Describe the optimal production planning of juices. Mention the quantities to beproduced and the total profit and analyze each of the constraints.

**[E:1.5/T: 2.0]**** b) **Would the production plan change if the profit, per thousand liters, of the juice

**is set to 3 c.u.? And the daily profit? Justify.**

**J1**

**[E:1.0]**** c) **Due to a machine breakdown, the available daily machine hours have been reduced by 4 hours. Will there be changes in the types of juice to be produced? Will daily profit be affected? Justify.

**[E:1.0/T: 1.5]**** d) **Given the latest sales report, the department suggests that the difference between the joint production levels of

**and**

**J1****and the production level of**

**J3****becomes 3 thousand liters. Should this new production rule be applied? Justify.**

**J2**

**[E:1.0/T: 1.5]**** e) **Analyze the possible weaknesses and potentialities associated with the currentproduction plan of juices. Justify.

**[E:1.5]**** **

**2. 运筹学考试代考**

The factory ** Electre **makes three types of electrical components,

**,**

**C1****and**

**C2****, which will be used in the production of two new products,**

**C3****and**

**P1****. The components are processed on the same production line, with a weekly availability of 350 hours. In theproduction line, one unit of**

**P2****takes 1 hour, one unit of**

**C1****takes 45 minutes, and one unit of**

**C2****takes 30 minutes. After production,**

**C3****must be hand finished, which takes 15 minutes per unit. The weekly hand finishing time available is 45 hours. The units of components necessary to produce one unit of each product (**

**C3****and**

**P1****), as well as the sales price and the maximum estimated sale of each product, are given in the following table:**

**P2**

**a) **

**a)**

Write down the linear programming model that allows determining the weeklyproduction plan of ** Electre **that maximizes the total value of the sale. Mention the meaning of the decision variables and give a brief explanation about the meaning of the objective function and the constraints.

**[E:1.5/T: 1.75]****b) **

**b)**

Assume that the ** Electre **plant will have to produce at least 90 units of

**or at least 50 units of**

**P1****. Make the necessary changes to the model presented in**

**P2****in order tocontemplate the new condition.**

**a)**

**[E:1.0/T: 1.25]****c) 运筹学考试代考**

**c) 运筹学考试代考**

To increase the quality of products ** P1 **and

**, the**

**P2****plant has an extraproduction line for the**

**Electre****This line production requires the use of an additional machine, which has a weekly availability of 60 hours and a setup cost of 110 c.u. Each unit of component**

**C3****requires 50 minutes on this additional machine. The**

**C3****can produce products**

**Electre****and**

**P1****of regular quality (those that are made up of**

**P2****components that have not passed through the extra production line) and of**

**C3***quality (those that are made up of*

*premium***components that have passed through the extra production line). Assuming that each unit of premium quality product is sold by more 3 c.u. than each one that has a regular quality, reformulate the problem presented in**

**C3****in orderto accomplish the new situation of**

**a)****.**

**Electre**

**[E:1.5/T: 2.0]**** **

**3. 运筹学考试代考**

A maintenance company received, for tomorrow, four repairment requests for a type of equipment that require specialized technicians. The company only has three teams of technicians available with the necessary qualifications for this type of repairment. Each of these teams is located in one of the company’s repair centers around the country. Each repair takes a long time, making it impossible for the same team of technicians to carry out more than one repair on the same day. The round trip costs (c.u.), from the repair center where each team is located and the customers, are as follows:

Client 1 | Client 2 | Client 3 | Client 4 | |

Team 1 | 60 | 50 | 110 | 40 |

Team 2 | 30 | 90 | 75 | 50 |

Team 3 | 45 | 100 | 75 | 80 |

The company wants to know which repair requests to accept, such that the total traveling cost is as low as possible. Represent the problem in a network and identify the Network Model that allows to find the optimal solution. Justify. **[E: 2.75/T: 3.25] **

** **

**4. 运筹学考试代考**

Consider the following information concerned with a project:

Activity | A | B | C | D | E | F | G | H |

Predecessors | — | — | A | A,B | B | C,D | E | C,D,E |

Draw an AOA (Activities on Arcs) network and number the nodes. **[E: 1.25/T: 1.5] **

**5.**

Consider the project which data and AON network are the following:

Activity | Normal Duration (days) | Reduced Duration (days) | Cost in Normal Duration (c. u.) | Cost in Reduced Duration (c. u.) | Daily Resources Required |

A | 3 | 3 | 120 | 120 | 4 |

B | 4 | 3 | 350 | 550 | 3 |

C | 3 | 2 | 100 | 150 | 3 |

D | 4 | 2 | 200 | 350 | 4 |

E | 5 | 2 | 400 | 700 | 3 |

F | 3 | 3 | 210 | 210 | 5 |

** a) **Find the minimum project duration. Justify.

**[E: 0.75/T: 1.0]**** b) **Determine the critical activities and the critical path(s). Justify.

**[E: 1.25/T: 1.75]****For the Test: choose between c) and d). For the Exam: answer both questions. **

** c) **Suppose that there are 6 units of resource available per day. In order to obtain aprecedence-feasible and resourcefeasible schedule apply the SGS heuristic. Consider as priority rules the longest duration and, to break ties, the maximum resource required. Write the schedule and the respective project duration. Justify.

**[E: 2.0/T: 2.5]**** d) **Suppose that it is intended to reduce the project duration by 2 days with the least increase on direct cost. Which activities should be reduced? By how many days? What will be the new direct cost? Justify.

**[E: 2.0/T: 2.5]**

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