Problem:
Professional traders can find investment opportunities from day-to-day financial markets. Very often an investment strategy is required that determines how the investment should be planned across different days so that it can eventually lead to some profits. After months of study on the daily stock market, the traders of the international firm QuantTradeX have identified which stocks they should invest into so that they can receive almost-predictable returns. In particular, by investing in these stocks they have the following proprietary trading strategy. At the start of every day of the week (Monday through Friday), they can decide how much to invest on that day, say, X dollars. If in the next day they are able to match that initial investment (the X dollars) with another investment of half of that amount, i.e. X/2 dollars, then they can expect to receive a total return on the third day that doubles the initial amount of the investment, i.e. a total return of 2X dollars. For example, if the trader invests X dollars on Monday and another X/2 dollars on Tuesday, which in total is 1.5X dollars, they can expect to receive 2X dollars on Wednesday, which gives them a gain of 0.5X dollars in two days.
However, any initial investment that is not matched on the subsequent day will lose its whole value. Thus, considering any investment without a subsequent fund to match might not be meaningful.
Besides the stock market, Foreign Exchange (FX) market is another popular financial market that daily traders actively participate. Although the firm QuantTradeX used to invest mostly in the stock market, they cannot let go any good investment opportunity even if it is in the FX market. The firm has recently noticed that whenever the number of travelers increases during the weekend, the FX rate tends to increase also in the following few days and go back to normal by the end of the week. They believe that in this case if they are able to invest Y dollars in the FX market on the Monday that follows right after the weekend, they can receive its doubled amount, i.e. 2Y dollars next day (Tuesday). If the investment is made on Tuesday, they can receive 1.5Y dollars next day (Wednesday). However, if the investment is made on Wednesday, they only receive 1.2Y dollars in two days, i.e. Friday. No investment in FX markets will be considered for the days after Wednesday.
On any given day of a week, the traders of the firm can re-invest any return they receive from the previous investments made in the same week. If there is any amount in the firm's account that is not invested in any of the financial markets, that amount can accrue 1% of daily interest through money markets.
Every Sunday night, the traders of the firm need to determine an investment plan for the following weekdays, i.e. how money in their account should be invested in each day.
The plan is considered successful if it maximizes the amount of money they can have in their account by the end of the coming Saturday. It is assumed that on Saturday they can still receive returns from earlier investments in the same week.
Today is another Sunday and the firm needs to make its weekly investment plan. It has been just confirmed that the number of travelers has increased significantly this weekend. The firm currently has 1 million in its account.
You have been hired fairly recently and assigned to help the traders determine a successful investment plan for the coming week.
Report how you would solve this problem by formulating the algebraic formulation of the Linear Programming (LP) model. You should provide all the details, which includes but are not limited to the LP model, i.e. decision variables, constraints, and objective function, with detailed explanations, the Excel sheets including your LP model, and answer Report. Include "managerial statements" that communicate the results of the analyses (i.e. describe verbally the results).