A lady named Susan Wong is budgeting for the coming year (Year X). During year X, she has to cover monthly expenses as well as irregular monthly financial obligations, and she plans to do so by investing the money not used to cover monthly expenses in either a 1-month, 3-month or 7-month investment scheme whose yields are 6%, 8% and 12% per year nominal respectively. When the investments mature, Susan will use the principals as part of her budget and invest all the interests in a long-term investment that is not considered in her budgeting process. Susan expects her expenses for the coming years to be as follows: Month Bills Month Bills
January $2,750 July $3,050
February $2,860 August $2,300
March $2,335 September $1,975
April $2,120 October $1,670
May $1,205 November $2,710
June $1,600 December $2,980
Table 1 – Expected monthly liabilities
Her net salary is $29,400 per year is to be received by her in 12 equal monthly paychecks deposited straight into her bank account. Additionally, she is currently having an extra $3,800 in her pocket to kick off her new year. Her objective is to gain as much interests as possible provided that all the monthly expenses are promptly and sufficiently covered. However, she is not sure whether she should use the extra $3,800 as part of her budget for the coming year (1) or invest some of it in a long-term investment (2). The case asks to help her complete the budgeting process in both (1) and (2) by forming a linear programming model.
A. Finding necessary data, assumptions and formulas:
In order to simplify and clarify the case, three assumptions are made 1.The concerned coming year is year X, meaning that currently Susan is in year X-1 and the year after year X is year X+1 2.The interests are compounded on a monthly basis.
3.The yield for the long-term investment in case (2) is higher than those of the three investment schemes Susan plans to put her money in. Regarding the compounding of interests, it is also important to find the salary Susan is to receive monthly and the monthly interest rate for two reasons. Firstly, the monthly receivable salaries are directly related to how much Susan can invest monthly, which in turn directly dictates the eventual interest receivables at the end of the year. Secondly, the nominal interest rate doesn’t indicate the true interest rate at which the interests are generated or compounded, because the nominal interest rate is not equal to the effective annual rate, which is the one at which the interests are usually compounded and generated yearly (1). -Monthly receivable salary = Yearly receivable salary/ 12 = $29,400/12 = $2,450 -Monthly interest rate = Yearly nominal interest rate/ 12. Hence, the monthly interest rate for: +1-month investment scheme: Yearly nominal interest rate/ 12 = 0.5% per month (a) +3-month investment scheme: Yearly nominal interest rate/ 12 = 0.067% per month (b) +7-month investment scheme: Yearly nominal interest rate/ 12 = 1% per month (c) Since the interests in this case are not compounded monthly and the investment is paid once only at the start of each investment scheme, the money gained at the end of each investment scheme is equal to the monthly interests time the number of months of each investment scheme, which is as follows: Total interests earned = Monthly interest rate * The principal deposited * Number of months of the investment scheme B. Forming a linear programming model
Applying linear programming to crack the case above as required, we have to progress through the following steps: Step 1: Define the necessary variables
Our group asserts that the principal amounts invested each month in each investment scheme, the surpluses available for investment purposes and the surpluses not invested each month are the variables in this case because they are the only factors that can be changed and at the same time...