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How would you formulate this linear program in standard form? (ie objective function and constraints).

any help would be appreciated. I don't understand how to formulate this without having an equation for each month, but I'm not sure what that would equate to.

https://imgur.com/a/VzFWv

Could anyone give me any hints to begin?

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You are asked to find "The least expensive portfolio that meets or exceeds the scheduled tuition payments". So you have known payments $P_t$ in each future month. You can construct the known "cumulative payment curve" $CP_t=\sum_{i=1}^t P_i$ giving all payments due up to and including month t.

Any portfolio will produce cash outflows $O_t$ in each month, with cumulative curve $CO_t=\sum_{i=1}^t O_t$. You must minimize the cost of the portfolio today, suject to the inequalities $CO_t \ge CP_t$ for all t, which basically says that each step along the way the portfolio has generated enough cash for you to have been able to keep up with the required tuition payments so far.

Both the "cost of the portfolio today" and the "cumulative outflows" are linear functions of the decision variables, the amounts $w_j$ to be used to purchase the available securities $S_j$ for $j=1,\cdots,N$

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  • $\begingroup$ Thank you for the response. I somewhat understand what you are saying, but I am having trouble envisioning how to take what you have written and actually putting the constraints into a LP solver to find the solution. For example when they receive a payment that does not match the cash outflow, they will be paid a small amount of interest on this. I can only imagine modeling this so each period (month) get's it's own constraint with the annuity payment and tuition payment + cash earning interest. Is there a more efficient way to do this? $\endgroup$ – wu54656213 Oct 16 '17 at 2:21
  • $\begingroup$ You are right, my (simple) proposed solution assumes a zero interest rate on cash... I'll have to rethink it... $\endgroup$ – Alex C Oct 16 '17 at 2:38
  • $\begingroup$ Yes a "sources and uses of cash" equation in each period. Maybe something like this $h_{t-1}*(1+r)+O_t-P_t=h_t$. Where $h_t$ is the spare cash in period t, and r is the interest rate per period. Furthermore we have to constrain each $h_t \ge 0$. Hope this helps. $\endgroup$ – Alex C Oct 16 '17 at 3:16

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