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If we are going to invest 40000$ to bank that yields 7% (expected return), bonds that yield 9% and stock that yields 14% with the following constrains:

  • Expected return on investment has to be at least 5000$

  • Investment on stock has to be at least 10000$

  • Amount invested into stock cannot exceed money invested into bank or bonds.

  • At least 5000$ but no more than 10000 must be invested into bank

  • Sum of all investments must be exactly 40000$

We can see that all these constraints cannot be satisfied at the same time so we relax on 3,4 so that we minimize the sum of deviations of them and form a new model and solve it.

So initially the model is something like this, where $C_1$,$C_2$,$C_3$ are monies invested into bank, bonds and stock: \begin{align*} \operatorname{maximize}\,f(C_1, C_2,C_3) &= 0.07 C_1 + 0.09 C_2 + 0.14 C_3\\ \text{Subject to}\;\; f(C_1, C_2,C_3) &\geq 5000\\ C_3 &\geq 10000\\ C_3 &< C_2 + C_1\\ 5000 &\leq C_1 \leq 10000\\ C_1 + C_2 + C_3 &= 40000 \end{align*} How does one minimize deviation of constrains (or goals I guess in this case). Do I form OLS type of equations for the constrains?

ELEC
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  • Your model looks right to me. But I do not understand your questions. To solve the problem you can apply the simplex method. If there is no solution, then the expected return on investment cannot be at least $$5000$. – callculus42 Nov 15 '15 at 19:06
  • Yeah but how does one "relax" on constraints (to goals) and minimize the sum of their deviations? – ELEC Nov 15 '15 at 19:13
  • You can "relax" the constraints one and two by introducing two variables: $$f(C_1,C_2,C_3)\geq 5000-y_1$$ $$C_3 \geq 10,000-y_2$$ The objective function becomes $$f(C_1,C_2,C_3)+y_1+y_2$$ – callculus42 Nov 15 '15 at 19:27
  • Ahhh. I'm supposed to relax on constrains three and four so would they would become: $C_3 < C_2 + C_1 + y_1$ and $5000 - y_2 \leq C_1 \leq 10000 - y_2$? – ELEC Nov 15 '15 at 19:36
  • @callculus could you explain your result a little further? – ELEC Dec 09 '15 at 15:26

1 Answers1

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For the constraint $C_1 \leq 10000$ you have to add $y_2$. If $y_2$ is positive then the RHS becomes larger and the value of $C_2$ doesn´t have to be smaller than $10,000$.

The other modifications are right. Additional you can weight the deviations. For this purpose you can add some coefficients at the objective function.

$\texttt{min} \ \ 0.07 C_1 + 0.09 C_2 + 0.14 C_3+w_1\cdot y_1+w_2 \cdot y_2$

If it is more important for you to minimize the deviation of the third constraint than the deviation of the fourth constraint, then $w_1$ has to be greater than $w_2$. In the opposite case $w_2$ has to be greater than $w_1$. With the choice of the value of the weights you can also determine the importance of the deviations in relation to $f(C_1,C_2,C_3)$.

callculus42
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  • Nice thanks! Any hints on how to solve the new objective function? Plot the constrains and see whats the feasible space etc? – ELEC Nov 15 '15 at 20:39
  • Since you have three variables and therefore a three dimensional space ($\mathbb R^3$) it is hard to plot the constraints and the objective function. Therefore you have to calculate the optimal solution by using for instance the simplex method. Maybe it is obvious but I should mention that $y_i\geq 0$. – callculus42 Nov 15 '15 at 21:06
  • You said: "For the constraint $C_1 \leq 10000$ you have to add $y_2$. If $y_2$ is positive then the RHS becomes larger and the value of $C_2$ doesn´t have to be smaller than $10,000$.". But $C_1$ yields the lowest expected return so why would we allow it to be larger? Shouldn't we allow less money to be invested into bank i.e $C_1 \leq 10000 - y_1$. And do you mean "value of $C_1$ instead of $C_2$ which you wrote. – ELEC Nov 16 '15 at 09:50
  • My understanding is that there 2 two ways to relax constrains 3,4 so that the first rule holds. Decrease the amount of money invested into bank (lowest yield) i.e $5000-y_2\leq C_1\leq 10000-y_2$. Or we could increase the sum of bank & bond investments such that we can put more money into stock (highest yield) i.e $C_3\leq C_2+C_1+y_1$. Or am I missing something? So if $y_2 = 4000$ and $y_1=10000$ we could have $C_1=5000-4000=1000$ and $C_2=14000$ and thus money allowed to stock is $25000$ because the third rule now holds if $y_1=10000$. $$C_3 \leq 1000+14000+10000\to C_3 = 25000$$ – ELEC Nov 16 '15 at 11:37
  • This configuration results into total return of $4830$ which is still not enough. So it seems that the deviation of $y_1$ dominates the expected returns. It has to be higher than $10000$. – ELEC Nov 16 '15 at 11:42