Assume that it is the first of July and you are running a small shop. The sales revenue and the amount of bills you have to pay for the next six months are estimated as following:
In short, you will make $4000 profit at the end of the year. However since all bills must be paid in full every month, you may be short on cash in earlier months until you see the big sales in November and December. You have two sources of loan:
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Long term loan of six months at 10% of interest, e.g. you borrow $100$ now , and payback $110$ at the end of the year. Early payback is not allowed. You have to keep the money for six months at 10% of interest.
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Monthly loan at 4% of interest, e.g. you borrow $100$ at the beginning of any month and pay back $104$ at the end of the month. No monthly loan is available in December.
Assume you have zero cash at the moment.
Apply the linear programming to find the loan plan to minimize the interest payment.
For this question, I have selected 12 decision variables. $X_1-X_6$:
Sales Revenue $X_7- X_{12}$: Bills$X_7+ X_8+ X_9+ X_{10}+X_{11}+X_{12} \leq 20,000$ and $X_1+X_2+ X_3+X_4+X_5+X_6\leq 24,000$
I do not know if above constraints is actually correct, and I don't
know how to continue from there! Please help and maybe instead of
providing the answers, if any links or example of similar question, it
will also help!
Edit: Thanks for the explaination, however I still can't seem to find the answer.
During my lecture, the lecturer have told us that there are 12 decision variables.
Which from the answer you provided, it will be $L, S_1…S_5 and C_1 … C_6$.
My understanding on decision variables is that, it will have to be all included in the objective function.
Therefore my objective was
Min:
$1.1L + 1.04S – (C_1….C_6)$
Please kindly advise how to interpret the results for the loan plan base on the software.
Best Answer
When solving problems of this sort it helps to systematically follow something like the following procedure.
1) Write down the variables that you want to solve for.
In this case, you want to find the amount of each loan that you want to take out. Let us let $L$ be the number of long term loans that you want, and $S_1$ to $S_5$ be the amount of each short term loan, $S_1$ the loan amount for July, to $S_5$ the loan amount for November. Each of these loan amounts must be greater than or equal to 0.
2) Write down the constraints (adding internal variables if necessary).
Here there is only one type of constraint -- at the beginning of each month, after cash on hand along with last months income and loans have been applied to last months bills and expiring loans.
We see that cash on hand at the beginning of each month is a probably best described as a variable (otherwise we would need to repeat computations in the constraints). So to simplify the constraints, let us add variables $C_1$ to $C_6$ denoting cash on hand at the beginning of August to January. These variables are constrained to be greater than or equal to 0. This makes the 6 constraints:
$$L + S_1 + 3000 - 6000 = C_1$$ $$C_1 + S_2 + 2000 - 6000 - 1.04 S_1 = C_2$$ $$C_2 + S_3 + 1000 - 4000 - 1.04 S_2 = C_3$$ $$C_3 + S_4 + 2000 - 2000 - 1.04 S_3 = C_4$$ $$C_4 + S_5 + 6000 - 1000 - 1.04 S_4 = C_5$$ $$C_5 + 10000 - 1000 - 1.04 S_5 - 1.1 L = C_6$$
3) Write down the objective.
Here this is just the amount of money left over at the beginning of January, and we want as much of it as possible, so the Objective is to maximize $C_6$.
4) Solve.
In answer to your extended question: I am not familiar with the software you are using, so I cannot explain its results. It does seem that the answers that you are getting are absurd. For a problem like this that you can solve without resorting to a linear program (once you have chosen the amount of the long term loan, everything else is determined), solve it. Then compare your solution with the output of the software.