In descriptive models, we describe optimisation

Question:

Background:

1. In descriptive models, we describe optimisation problems in words. These descriptions will help build optimisation models, whether in the form of mathematical formulations or models in optimisation software packages. Let’s look at a descriptive model: An investor wants to decide how to allocate her $10,000 investment. There are quite a few investment options available: fixed-term investment (expected 3% return), shares (expected 5% return), and options (expected 10% return). Each investment option takes investment of $1,000 as 1 unit (i.e., investment has to be in $1,000 increments). The investor wants to control her investment risk so she decides to diversify the portfolio. She sets a rule that the minimum amount of investment in each investment option is $2,000 and the maximum is $4,000. How should the investment portfolio be in order to maximise the expected return?

Based on this descriptive model, describe the decision variables, the objective function, and the constraints?

If we denote x1 as units of fixed-term investment, x2 as units of shares investment, and x3 as units of options investment (remember 1 unit = $1,000), how would we write the mathematical form for the objective function? (Note: as per the model description, fixed-term investment has $30 return per unit, shares investment has $50 return per unit, and options investment has $100 return per unit.)

2. You are looking for job opportunities right now and you received two job offers. Company A is in a traditional industry. You were offered a starting annual salary of $50,000. In the past, the company has average pay rises of 5% each year. However, given the current economic uncertainty, you believe the pay rise is only 20% likely to happen in the coming years. Company B is a high-tech start-up. You were offered a starting annual salary of $40,000 with a promise of 10% salary adjustment per year. Company B is in the process of IPO and it is said that 20% of chance it will succeed in IPO. If this is the case, each staff member will receive $40,000 as bonus.

Questions: What will be the expected monetary value for your INCOME during Year 1 (remember pay rise will only apply at the beginning of Year 2) if you choose Company A?

What will be the expected monetary value for your INCOME during Year 1 (remember pay rise will only apply at the beginning of Year 2) if you choose Company B?

What will be the expected monetary value for your SALARY at the beginning of Year 2 if you choose Company A? (1 mark) What will be the expected monetary value for your SALARY at the beginning of Year 2 if you choose Company B? (

Based on your calculations, if you focus on the income of Year 1, which company should you choose? What is the key information you take into consideration?

3. A winemaker in Barrossa Valley must decide how much of two types of wine she will produce from a particular variety of grapes. Each litre of table wine yields $8 profit, while each litre of dessert wine produces $5 profit. The labour (hour), bottling process time (hour), and grapes required (kg) used for each litre of table wine is 0.5, 0.2, and 1.2, respectively. The labour (hour), bottling process time (hour), and grapes required (kg) used for each litre of dessert wine is 0.6, 0.2, and 1.0, respectively. Resources available include 1000 labour hours, 800 hours of bottling process time, and 1800 kg of grapes.

Model the problem in Excel spreadsheet and solve it using Solver. What is the optimal production plan? How much profit would the winemaker be able to make?

Generate the sensitivity analysis report for the problem. Now, the winemaker has the opportunity to purchase additional labour and bottling hours at $2/hour, and the opportunity to purchase more grapes at $2/kg. If the winemaker has $1000 to spend, which resource should the winemaker purchase? Why? Please provide your answer based on the model that you developed.

4. Please provide the Excel spreadsheet model you developed for the Winemaker problem.