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This is how I would develop the formulation (conceptually).

• Investment costs, $Q$: $$Q = \sum_{i} m_{i} c_{i}$$

• Investment returns for $i$ at time $t$: we know that the return for $i$ is $m_{i} r_{i}$ if $t \geq t_{i}$, otherwise it is 0. To model this if logical condition: $$R = \sum_{i} \delta_{i} m_{i} r_{i}$$ $$\delta_{i} = 1, \text{ if } t \geq t_{i}$$ $$\delta_{i} = 0, \text{ if } t < t_{i}$$ where $R$ = overall returns. In this case, $t$ and $t_{i}$ are parameters, therefore $\delta_{i}$ are parameters too, not binary variables. They can be pre-calculated for a specified $t$.

In summary, your problem can be represented as follows:

$$\max_{m_{i}} (R - Q)$$ s.t. $$Q = \sum_{i} m_{i} c_{i}$$ $$R = \sum_{i} \delta_{i} m_{i} r_{i}$$ $$Q \leq C$$ $$m_{i} \geq 0\quad \forall i$$

where $C, c_{i}, r_{i}, \delta_{i}, t, t_{i}$ are parameters.

So, if $m_{i} \in \mathbb{R}$, this would be a linear program. If $m_{i} \in \mathbb{N}$, this becomes a mixed-integer linear program. (Note: for tractability in the integer case, you may need to specify a reasonably small upper-bound for $m_{i}$ or use partial integer variables)

For $t \rightarrow \infty$, infty$(i.e. all the investments have reached maturity), simply set all$\delta_{i} = 1$. 3 Omission - nonnegativity of m_i This is how I would develop the formulation (conceptually). • Investment costs,$Q$: $$Q = \sum_{i} m_{i} c_{i}$$ • Investment returns for$i$at time$t$: we know that the return for$i$is$m_{i} r_{i}$if$t \geq t_{i}$, otherwise it is 0. To model this if logical condition: $$R = \sum_{i} \delta_{i} m_{i} r_{i}$$ $$\delta_{i} = 1, \text{ if } t \geq t_{i}$$ $$\delta_{i} = 0, \text{ if } t < t_{i}$$ where$R$= overall returns. In this case,$t$and$t_{i}$are parameters, therefore$\delta_{i}$are parameters too, not binary variables. They can be pre-calculated for a specified$t$. In summary, your problem can be represented as follows: $$\max_{m_{i}} (R - Q)$$ s.t. $$Q = \sum_{i} m_{i} c_{i}$$ $$R = \sum_{i} \delta_{i} m_{i} r_{i}$$ $$Q \leq C$$ $$m_{i} \geq 0\quad \forall i$$ where$C, c_{i}, r_{i}, \delta_{i}, t, t_{i}$are parameters. So, if$m_{i} \in \mathbb{R}$, this would be linear program. If$m_{i} \in \mathbb{N}$, this becomes a mixed-integer linear program. For$t \rightarrow \infty$, simply set all$\delta_{i} = 1$. 2 Formatting This is how I would develop the formulation (conceptually). • Investment costs,$Q$: $$Q = \sum_{i} m_{i} c_{i}$$ • Investment returns for$i$at time$t$: we know that the return for $$m_{i} r_{i} \text{ i is m_{i} r_{i} if } t \geq t_{i}$$t_{i}$, otherwise it is 0. To model this if logical condition, we can write: $$R = \sum_{i} \delta_{i} m_{i} r_{i}$$ $$\delta_{i} = 1, \text{ if } t \geq t_{i}$$ $$\delta_{i} = 0, \text{ if } t < t_{i}$$ where $R$ = overall returns. In this case, $t$ and $t_{i}$ are parameters, therefore $\delta_{i}$ are parameters too, not binary variables. They can be pre-calculated for a specified $t$.

In summary, your problem can be represented as follows:

$$\max max_{m_{i}} (R - Q)$$ s.t. $$Q = \sum_{i} m_{i} c_{i}$$ $$R = \sum_{i} \delta_{i} m_{i} r_{i}$$ $$Q \leq C$$

where $C, c_{i}, r_{i}, \delta_{i}, t, t_{i}$ are parameters.

So, if $m_{i} \in \mathbb{R}$, this would be linear program. If $m_{i} \in \mathbb{N}$, this becomes a mixed-integer linear program. For $t \rightarrow \infty$, simply set all $\delta_{i} = 1$.

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