MGSC 405v8 Problems Assignment
MGSC 405v8 Assignment 2
Instructions
Submit Assignment 2 after you have completed Lessons 6 through 9. This assignment is worth 20% of your final grade.
Read the requirements for each problem and plan your responses carefully. This assignment contains six problems. The relative weights of each problem differ and their respective values are noted beside each problem. Assignment 2 has a total of 100 marks.
Ensure that you answer all of the problem components as completely as possible. If supporting calculations are required, present them in good form.
Submit this assignment electronically as a single document in a word processing format (such as Microsoft Word) rather than in a spreadsheet format (such as Excel). It is easier for the evaluator to insert his or her comments into a Word document than into a spreadsheet file. If necessary to present results from Excel, copy and paste everything into the Word document. When preparing tabulated data, use the word processor’s “table” function or set and insert tabs (right-hand tabs for columns of figures) where required. Do not use default tabs.
Although these techniques may take a bit of time to master, you will find that they will significantly improve the appearance of many of the documents you must prepare for this course.
Your answers to all six problems should be submitted in one document.
Number your responses to match the assignment questions; do not retype the questions. Unless directed otherwise, answer in complete sentences and paragraphs.
Problem 1 (10 marks)
A network of streets connects the main points for entering and leaving a city area. Speed limits, road construction, and street sign restrictions lead to the flow diagram below, where the numbers represent how many vehicles can pass per hour between two nodes. Formulate (without solving) an LP to find the maximal flow in vehicles per hour from Node 1 to Node F.
Problem 2 (15 marks)
ABC minor league soccer team has one box office clerk. On average, tickets can be sold to 8 customers per minute. For normal games, customers arrive at the rate of 5 per minute. Assume arrivals follow the Poisson distribution and service times follow the exponential distribution.
a. What is the average number of customers waiting in line?
b. What is the average time a customer spends in the waiting line?
c. What is the average number of customers in the system?
d. What is a customer’s average time in the system?
e. What is the probability that someone will be buying tickets when an arrival occurs?
The team is playing in the league playoffs and anticipates more fans, estimating that the arrival rate will increase to 12 per minute. Output is supplied for a 2-cashier and a 3-cashier system.
| Number of Channels | 2 | 3 |
| Arrival Rate | 12 | 12 |
| Service Rate | 8 | 8 |
| Probability of No Units in System | .1429 | .2105 |
| Average Waiting Time | .1607 | .0197 |
| Average Time in System | .2857 | .1447 |
| Average Number Waiting | 1.9286 | .2368 |
| Average Number in System | 3.4286 | 1.7368 |
| Probability of Waiting | .6429 | .2368 |
| Probability of 7 in System | .0381 | .0074 |
a. The stadium has space for 6 customers to wait indoors to buy tickets. Which of the two systems shown above will be better, and why?
b. Which system do you advise them to use in the new situation, overall: 1-, 2-, or 3-cashier? Justify your answer.
Problem 3 (15 marks)
Solve the following problem graphically.
| Min | 6X + 11Y |
| s.t. | 9X + 3Y ≥ 27 |
| 7X + 6Y ≥ 42 | |
| 4X + 8Y ≥ 32 | |
| X, Y ≥ 0 and Y integer |
a. Graph the constraints for this problem. Indicate all feasible solutions.
b. Find the optimal solution to the LP Relaxation. Round up to find a feasible integer solution. Is this solution optimal?c.
Find the optimal solution, if different from solution above. Justify your answer.
Problem 4 (20 marks)
ABC Company has a contract to produce 10,000 garden hoses for a large discount chain. ABC has four different machines that can produce this kind of hose. Because these machines are from different manufacturers and use differing technologies, their specifications are not the same.
| Machine | Fixed Cost to Set Up Production Run |
Variable Cost per Hose |
Capacity |
| 1 | 750 | 1.25 | 6000 |
| 2 | 500 | 1.50 | 7500 |
| 3 | 1000 | 1.00 | 4000 |
| 4 | 300 | 2.00 | 5000 |
| a. | This problem requires two different kinds of decision variables. Clearly define each kind. |
| b. | The company wants to minimize total cost. Give the objective function and write the complete LP system of equations knowing that three additional constraints require that machine 1 is not being used if machine 4 is used, machine 3 must be used if machine 1 is used, and at least two machines must be used, overall. |
| c. | Solve the LP system above with computer (Excel Solver) and paste your solution into your answer document. |
Problem 5 (15 marks)
Jill has been employed at ABC Realty with a salary of $2,000 per month during the past year. Because Jill is considered to be a top salesperson, the manager of ABC is offering her one of three salary plans for the next year: (1) a 25% raise to $2,500 per month; (2) a base salary of $1,000 plus $600 per house sold; or, (3) a straight commission of $1,000 per house sold. Over the past year, Jill has sold up to 6 homes in a month.
| a. | Compute the monthly salary payoff table for Jill. | ||
| b. | For this payoff table, find Jill’s optimal decision using: (1) the conservative approach, (2) minimax regret approach. | ||
| c. | Suppose that the following is Jill’s distribution of home sales during the past year. If one assumes that this a typical distribution for Jill’s monthly sales, which salary plan should Jill select? | ||
| Home Sales | Number of Months | ||
| 0 | 1 | ||
| 1 | 2 | ||
| 2 | 1 | ||
| 3 | 2 | ||
| 4 | 1 | ||
| 5 | 3 | ||
| 6 | 2 | ||
Problem 6 (25 marks)
ABC Manufacturing Company has developed a unique new product and must now decide between two facility plans. The first alternative is to build a large new facility immediately. The second alternative is to build a small plant initially and to consider expanding it to a larger facility three years later if the market has proven favourable.
Marketing has provided the following probability estimates for a 10-year plan:
| First 3-Year Demand | Next 7-Year Demand | Probability |
| Unfavourable | Unfavourable | 0.2 |
| Unfavourable | Favorable | 0.0 |
| Favourable | Favorable | 0.7 |
| Favorable | Unfavourable | 0.1 |
If the small plant is expanded, the probability of demands over the remaining seven years is 7/8 for favourable and 1/8 for unfavourable. The accounting department has provided the payoff for each outcome.
| Demand | Facility Plan | Payoff |
| Favorable, favourable | 1 | $5,000,000 |
| Favorable, unfavourable | 1 | 2,500,000 |
| Unfavourable, unfavourable | 1 | 1,000,000 |
| Favorable, favourable | 2—expanded | 4,000,000 |
| Favorable, unfavourable | 2—expanded | 100,000 |
| Favorable, favourable | 2—not expanded | 1,500,000 |
| Favorable, unfavourable | 2—not expanded | 500,000 |
| Unfavourable, unfavourable | 2—not expanded | 300,000 |
Use these estimates to analyze ABC’s facility decision.
a. Perform a complete decision tree analysis.
b. Recommend a strategy to ABC based on the above analysis.
c. Determine what payoffs will result from your recommendation.