**Math and Comp Modeling in Systems Engineering**

**FINAL EXAMINATION**

**Spring 2021**

**INSTRUCTIONS – READ INSTRUCTIONS**

These instructions are not guidelines and must be followed. Failure to follow these instructions will result in a loss in marks:

· **Examination**: This examination contains six problems; you are to attempt **all** six problems.

· **Submission**: A ** single **Microsoft Word document should be submitted as the only examination solution document.

· This is the only document that will be marked.

· __Do not submit compressed files__.

· The file should be named according to the following format: “*your last name_final_exam.docx*”.

· **Supporting Material**: Do NOT submit supporting documents.

· **Submission Format:** Begin the solution of each problem on a new page. The heading of each page should contain your name and the problem number. Please explain carefully your solution procedures and include sufficient working to follow your solution processes. Generally, an attempt must be made to be awarded partial credit when your answers are supported by your work; however, answers alone cannot be awarded partial credit.

· **Submission Length:** Each question will take appropriate one page to answer. __If a question’s solutions longer than five-pages will be penalized with a loss of marks.__ Conciseness of writing is an important presentation skill and you should make an effort to make your solution clear.

· **Clarity: At least one mark from every question will be given for clarity.**

· **References and Assistance**: You are allowed to use written reference material as long as all such material is properly referenced and acknowledged. You are not allowed to receive assistance from other individuals; the work that you submit for this examination is to be exclusively your own.

· **Examination Problems**: If you believe a question statement is incomplete or ambiguous, you may make any reasonable assumptions required to complete or clarify the problem statement. Your assumptions must be stated clearly and, as long as they are not in conflict with the given problem statement or its logical context, will be considered as part of your problem solution.

**PROBLEM ONE (5 Points)**

You have the schedule the following jobs:

Job | A | B | C | D | E | F | G |

Processing time | 4 | 6 | 7 | 6 | 1 | 2 | 5 |

Deadline | 20 | 10 | 6 | 10 | 2 | 3 | 30 |

Schedule to:

1) Minimize the maximum tardiness

2) Minimize the summation of completion times

3) Minimize the number of late orders

Describe the approach you used and the meaning of its outcomes.

**PROBLEM TWO (5 Points)**

**Question 1: (3 point)**

a) Write a linear programming formulation for the following problem:

Billy quit university with a great idea to make widgets. There are two types of widget: A, and B. All widgets require glue, wood, and metal to make, shown in the table below. Billy has 20 glue, 10 wood, and 20 metal available. All widgets will sell for $5.

Widget | Glue | Wood | Metal |

A | 3 | 1 | 2 |

B | 2 | 2 | 2 |

b) What assumptions are you making?

**Question 2: (2 point)**

Enter problem into MS Excel and solve. Report your findings in a way you would to senior manager.

a) Why do you think one widget type is preferred over another?

**PROBLEM THREE (5 Points)**

Write out the mathematical equations relating to the modified water tank example:

Describe how the equations are related.

**PROBLEM FOUR (5 Points)**

Look at the “Sugarscape 2 Constant Growback” model from the Netlogo library. Write a report on it, this should include:

• A basic introduction to what phenomenon is being modelled

• What happens when you run the model including some screenshots.

• What happens when you vary the parameters.

Maximum of three pages allowed. Most marks for clarity.

Advice: Download Netlogo and make sure it works for you early.

**PROBLEM FIVE (5 Points)**

**Question 1: (2 point)**

Your model has three input variables that you use: (A, B, and C). Currently you use (2, 2, 2). You believe that actual values of these input variables follow a uniform distribution. The bounds for A are 0 and 10; B are 1 and 3; and C are 0 and 4.

You intend to do 10 runs for your sensitivity analysis. Create a Latin Hypercube sampling scheme for this sensitivity analysis. Describe the step you choose and provide an example input scheme.

**Question 2: (3 point)**

The mathematical model used in your experiment is as follows:

Run your sampling scheme on this model and report the correlation between the output variable and each input variable. What does this mean? What does the sensitivity analysis tell you about your original input guess? **PROBLEM SIX (5 Points)**

This question is about Game Theory and its solution methods.

1) Consider the normal-form game given below.

Left | Right | |

Up | 2, 2 | 0, 3 |

Down | 3, 0 | 1, 1 |

Solve this game, explaining your method in your own words, using **(2 points)**:

a) Maximin Criterion.

b) Dominance.

c) Nash Equilibrium Method.

2) Consider the normal-form game given below.

Left | Right | |

Up | 1, 3 | 1, 3 |

Down | 1, 3 | 1, 3 |

Try and solve this game using the following methods:

a) Maximin Criterion.

b) Dominance.

c) Nash Equilibrium Method.

Describe any problems with the solution methods. **(3 points)**

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