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# You arc a CFA (chartered financial analyst). Madonna has come to you because she needs help paying off her credit card bills. She owes the amounts on her credit cards shown in Table 43. Madonna is willing to allocate up to $5,000 per month to pay off these credit cards. All cards must be paid off within 36 months. Madonna’s goal is to minimize the total of all her payments. To solve this problem, you must understand how interest on a loan works. To illustrate, suppose Madonna pays$5,000 on Saks during month 1. Then her Saks balance at the beginning of month 2 is 20,000 - (5,000 - .005(20,000)) This follows because during month 1 Madonna incurs .005(20,000) in interest charges on her Saks card. Help Madonna solve her problems! Table 43: $$\begin{matrix} \text{Card} & \text{Balance }() & \text{Monthly Rate (\\%)}\\ \text{Saks Fifth Avenue} & \text{20,000} & \text{.5}\\ \text{Bloomingdale’s} & \text{50,000} & \text{1}\\ \text{Macys} & \text{40,000} & \text{1.5}\\ \end{matrix}$$

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We are helping to pay off a credit card bills. It is allocated up to $\5\,000$ per month. All cards must be paid off within $36$ months. The goal is to minimize the total of all payments. There are $3$ cards we are paying off: Saks, Bloomingdale's and Macys with balance of $20,50,40$ of thousands of dollars and monthly rate (in percentage) of $0.5,1,1.5$ respectively. Let us $s_t$ be the balance at the end of the month $t$ in Saks and similary $b_t$ and $m_t$ in Bloomingdale's and Macys respectively. Also let $x_t,y_t$ and $z_t$ be the payments in month $t$ for Saks, Bloomingdale's and Macys respectively. Let $z$ be the objective function. It follows:

$\min z=\sum_{t=1}^{36}x_t+y_t+z_t.$

Obivously the allocation of up to $\5\,000$ implies the following constraints:

$x_t+y_t+z_t\leq5000,t\in\{1,2,\dots,36\}.$

The fact that all cards will be paid off within $36$ months implies that $s_{36}=b_{36}=m_{36}=0.$ We can see that $s_0=20\,000,b_0=50\,000,m_0=40\,000,$ by the defintion. For the others we have:

$s_t=s_{t-1}-x_t+0.005\cdot s_{t-1}$

$b_t=b_{t-1}-y_t+0.01\cdot b_{t-1}$

$m_t=m_{t-1}-z_t+0.015\cdot m_{s-1}.$

Ofcourse, all variables are nonegative. Therefore, we are done.

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