Spring 2010, Professor Eckstein

Assignment 5

**Q1.** (35 points) You operate a farm market that sells fruit
grown by three organic farms in the surrounding region during the upcoming
8-week harvest season. Each farm operates on a "share" basis, as
follows

Total | Share | |||||||||

Week | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Pounds | Price |

Farm 1 | 0 | 10 | 20 | 30 | 20 | 10 | 0 | 0 | 90 | $ 95.00 |

Farm 2 | 0 | 0 | 15 | 25 | 30 | 20 | 10 | 5 | 105 | $ 120.00 |

Farm 3 | 10 | 15 | 20 | 25 | 20 | 5 | 0 | 0 | 95 | $ 105.00 |

For example, if you purchase a share of Farm 1, which costs $95, you would receive 90 pounds of fruit, 10 pounds delivered in week 2, 20 pounds delivered in week 3, 30 pounds in week 4, 20 pounds in week 5, and 10 pounds in week 6. You may buy as many shares as you like from each farm, and you may assume that fractional shares of any kind are permitted. You estimate the selling price and potential market for fruit over the 8-week season as follows:

Week | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |

Sales Price | $ 2.29 | $ 1.99 | $ 1.89 | $ 1.69 | $ 1.69 | $ 1.89 | $ 1.99 | $ 2.19 |

Max Sales | 400 | 420 | 450 | 500 | 500 | 450 | 400 | 400 |

For example, on week 1 your selling price will be $2.29 per pound, and you expect to be able to sell up to 400 pounds at that price. You have a special controlled-atmosphere refrigerator in which the fruit may be kept fresh indefinitely, and which can hold up to 600 pounds of fruit at a cost of $0.28 per pound per week (based on the average of the beginning and ending inventory). Each week, you may offer any amount of fruit for sale up to the total amount available from inventory and deliveries, and also not exceeding to the "max sales" limit shown above. You may offer less than the "max sales" amount, for example, in the hope of getting a higher price the following week. Assume that you have no inventory at the beginning of the season, and that the fruit has no value after the end of the 8-week season.

In order to maintain
good relations with your suppliers and not to become overly dependent on any one
of them, you have decided that at least 10% and at most 50% of your fruit,
measured in total pounds over the whole season, may come from any single
farm. How many shares should you buy from each farm, and how much fruit
should you sell each week, in order to maximize your profits? Hand in an
algebra formulation, and also solve using Excel and Solver, handing in the standard printouts.
[Note: the setup for this problem is quite like that for in investment problem
we covered in class 8 (February 11), except that you will also have variables
for how much you decide to sell in each period. To reduce data-entry time,
you may download the raw data from
http://eckstein.rutgers.edu/om/homework/farm-data.xls.]

**Q2.**
(30 points) Problem 79 on page 81 of the course pack (waste disposal).
Hand in an algebra formulation, and also solve using Excel and Solver, handing
in the standard printouts.

**Q3. **
(35 points) Refer to Problem 20(a) on page 104 of the course pack
(building a house). Instead of the instructions in in the course pack, do
the following:

- Draw the project diagram in the style I used in class
- You would like to complete the house as quickly as possible. Hand in an algebra formulation, with clear definitions of all decision variables.
- Solve the problem of part (b) using Excel and Solver, handing in the standard printouts.
- Using your results from (c), identify the critical path(s) on your project diagram.
- Using the data in table 4.28 of the book (just below the problem), assume that you have $100 to spend to speed up construction of the house. What is the fastest you can complete the house with this extra budget? Hand in an algebra formulation of this modified problems, with clear definitions of all decision variables. [Note: this problem is a bit different from the "crash" problem we studied in class, which was to find the cheapest way to meet a deadline; here, we want to find the most effective way to use a "crashing" budget.]
- Solve the problem of part (e) using Excel and Solver, handing in the standard printouts.