Question 1. It is January 1, 2011. Beth is the finance manager at the Glop Foundation whose sole asset is $75,000 in cash in its checking account. The Foundation must make the following cash payments for the next three years:
2011: $25,000 2012: $25,500 2013: $26,000

Cash required for a year must be withdrawn from the bank and set aside at the beginning of that year. The Foundation is scheduled to shut down on January 1, 2014. On that date, all cash left over in its checking account will be withdrawn and paid to the Foundation's beneficiaries.
In addition to the checking account, the Foundation's bank offers two savings vehicles: a 2-year Certificate of Deposit (or CD) paying 3% interest per year and a 3-year CD paying 5% interest per year. Each CD can be purchased in any amount. Neither CD permits early withdrawal. Any cash that is not invested in the CD's on January 1, 2011, is left in the checking account that pays 2% interest per year. Interest (from the checking account and from the CDs) is deposited in the Foundation's checking account on January 1 of each year and is available for the Foundation's cash requirements for that year. For simplicity assume the checking account interest is paid at the end of the year balance.

Beth would like to develop a financing plan that maximizes the amount that can be paid to the Foundation's beneficiaries on January 1, 2014, while meeting the Foundation's payment requirements for the next three years.

Required:

• Develop a linear optimization model to accomplish this objective.
• Specify the decision variables.
• Specify the objective (as a function of the decision variables and/or intermediate variables).
• Specify the constraints (as a function of the decision variables and/or intermediate variables).
• Specify all intermediate variables (if any).
• Implement your model in Excel. What is the optimal investment strategy and how much can the Foundation's beneficiaries expect to get paid on January 1st 2014. Include Exhibit 1-1 of your implemented model that is clearly labeled (with row and column headings) and shows the formulas.

Question 2. Beta Mining owns a copper mine. The firm is going through a difficult period as the price of copper has tumbled in recent years. The annual revenue from the mine depends on the price of copper at the end of the year as shown in the table below.

It costs $800 million per year to operate the mine. Alternatively, the mine can be abandoned - wherein concrete is poured into its shafts and the mouth is boarded up at a cost of $50 million. Another alternative is to "moth-ball" the mine-wherein its operation is indefinitely suspended-at a cost of $100 million per year. Unlike an abandoned mine, a moth-balled mine can be restarted in subsequent years at no additional cost. The decision to operate, moth-ball or abandon the mine is made at the start of each year.

It is January 1, 2011. The price of copper today is $7,000/ton. Over the course of each of 2011 and 2012, it is estimated that there is a 60% chance that the price will increase by $1,000 and a 40% chance that it will decrease by $1,000. All costs and revenues are discounted to today's dollars.

Required:

Implement a decision tree to illustrate Beta Mining's decision-making about operating its mine over the next two years. Be careful in distinguishing between decision and chance nodes, and in naming the various nodes and branches. Include Exhibit 2-1 which is a clearly readable image of your implemented tree. Describe the optimal policy for Beta Mining over the next two years. The price evolution was modelled with a single parameter. How sensitive is the optimal course of action to this parameter, that is, to the probability that the price will increase? Briefly discuss the results and label any supporting documents as Exhibit 2-2 (max one page).

Question 3

The Blue Lagoon has long been producing a line of natural skin care products based on the mineral rich water in the blue lagoon. Their main sales outlet is at the Lagoon itself, with over 90% of overall sales. With the recent spike in number of tourists in Iceland, they have seen their inventory and production costs sky-rocket. A task force is looking at the production and inventory policies for their Algae & Mineral shower gel as a case study, and the task force is limiting its scope to the sales outlet at the Lagoon.

The weekly cost of holding one unit of shower gel in inventory is $3 (one unit is 100 cases of the shower gel). The marketing department estimates that weekly demand during the high season averages 12 units, with a standard deviation of 2 units and that it is reasonably well modeled by a normal distribution. If demand exceeds the amount of shower gel on hand, those sales are lost, as tourist visit the Lagoon once (at most) and do not return. The profit margin per case is $100.

The production department can produce at one of three levels: 11, 12 or 13 units per week. The cost of scaling up or scaling down the production level from one week to the next is $50. The task force is planning on evaluating the following production policy: If the current inventory is less than units, they will produce 13 units in the next week. If the current inventory is greater than units they will produce 11 units in the next week. Otherwise The Blue lagoon will continue the previous week's production level.

It is now early May 2015, and the high season of 16 weeks will start next week! The shower gel production is set to produce 11 units this week, and the team expects the Blue lagoon will start their high season with 6 units in stock.

Required:

Create a static Excel model for 16 weeks of operations at the Blue Lagoon Skin Care production facility. Make the simplifying assumption that production in a particular week is available for sale that same week. Further assume that inventory costs apply to the end of the week inventory status.

Assume the demand is 12 units/week for the whole 16 week season. What are the combined costs over the period? Add the randomness to the model, i.e. relax the assumption from a) that the demand is fixed at 12 units per week. Investigate whether the upper bound of units is the right inventory policy. In particular investigate values of U ranging from 3 to 10 in increments of 1 unit. Keep throughout your analysis.
Report the sample mean and the standard deviation of the 16 week costs under each policy in a well labeled table marked as Exhibit 3-4. What is the best value of U when L is kept at 3?