Mathew is the business owner of a laundry shop located at City Plaza. He has operated the business since June 2018 and after operating it for 6 months, he has realised that in certain months, the sales revenue is sufficient to cover the operating expenditures, while for certain months the sales revenue is not enough to cover the operating expenditures and he has to rely on his personal savings to tide through.

It is now the last week of December 2018 and he realises that moving forward, it is better for his business to have access to loan facility from the bank to ease out his operation. However, he is unsure of which loan package to sign up and has approached you, a close friend, to help him as you are trained in financial planning. To perform the analysis, you have requested Mathew to give a projection of the sales revenue and operating expenditures for the next twelve months.

Month	Sales Revenue ($)	Bills ($)
Jan	4,000	6,000
Feb	3,000	5,000
Mar	3,000	4,000
Apr	3,000	3,000
May	5,000	4,000
Jun	9,000	1,000
Jul	3,000	6,000
Aug	2,000	6,000
Sep	1,000	4,000
Oct	2,000	2,000
Nov	6,000	1,000
Dec	10,000	1,000

Based on the whole year projection, Mathew will make $8,000 net profit at the end of the year. However, since all expenditures must be paid in full by the end of every month, Mathew may be short on cash in some months until he sees the big sales in certain months, e.g. June and December. Mathew has two sources of loan:

Annual loan at 12% of interest per year, e.g. he borrows $100 at the beginning of January 2019 and pays back $112 at the end of December 2019. Early-pay-back is not allowed and Mathew can get an annual loan in January only.

Monthly loan at 2.5% of interest, e.g. he borrows $100 at the end of March and pays back $102.5 at the end of April. Early-pay-back is not allowed and Mathew cannot get a monthly loan in December.

He needs your help to determine whether he should just take up the annual loan with effect from January, or a mixture of both types of loan facilities. Assume that Mathew has zero cash balance at the beginning of 2019.

(a) Mathew aims to find the optimal loan plan that minimises the total amount of loan interest. Develop the problem as a linear programme. Use any computer software to solve the problem and describe the optimal loan plan. State assumptions you have made in your formulation. (30 marks)

(b) Use the software to generate a sensitivity report (5 marks). Answer the following questions by interpreting the report. Each question is worth 5 marks. (i) Suppose the interest rate for annual loan reduces to 10.5%. What will be the impact on the optimal loan plan? (ii) Suppose the bill to pay in October increases to $3,500. What will be the impact on the total amount of loan interest, if any? (iii) Suppose the sales in September are going to be better than it was expected. The revised forecast for September sales is now $2,500. Will your loan plan change? (20 marks)