Table 1 Some significant features of the optimal policies for different combinations of demand curves, amounts of the resource available, discount coefficients and planning horizons

\( Q \cdot e^{ - \lambda \cdot p} \)

\( R \)

\( 1\,\,\forall t \)

\( \begin{aligned} T \ge \hfill \\ R\cdot\frac{e}{Q} \hfill \\ \end{aligned} \)

\( R/T \)

\( - \frac{1}{\lambda }\cdot\ln \frac{1}{T} \)

\( - \frac{R}{T}\cdot\frac{1}{\lambda }\cdot\ln \frac{1}{T} \)

\( \left\lceil {R\cdot\frac{e}{Q}} \right\rceil \)

\( 150 \cdot e^{ - p} \)

180

\( 0.95^{t - 1} \)

10

\( q_{10}^{*} /q_{1}^{*} = 0.59 \)

\( p_{10}^{*} /p_{1}^{*} = 1.28 \)

\( I_{10}^{*} /I_{1}^{*} = 0.76 \)

4

\( q_{t} = 10,000 \cdot \left( {1 + \frac{t - 1}{4}} \right)\cdot\left( {1 - \frac{p}{{1000\cdot\left( {1 + \frac{t - 1}{4}} \right)}}} \right) \)

18,000

\( 0.875^{t - 1} \)

>11

\( \begin{aligned} > 0\,\,\forall t \le 11 \hfill \\ = 0\,\,\forall t > 11 \hfill \\ \end{aligned} \)

3

11

Max at \( t = 6 \) \( q_{11}^{*} /q_{1}^{*} = 0.68 \)

Min at t = 2 \( p_{11}^{*} /p_{1}^{*} = 2.91 \)

\( I_{11}^{*} /I_{1}^{*} = 1.98 \)

\( 0.9^{t - 1} \)

≤8

\( > 0\,\forall t \)

>9

\( \begin{aligned} > 0\,\,\forall t|2 \le t \le 13 \hfill \\ = 0\,\,\forall t > 13 \hfill \\ \end{aligned} \)

13

Max at \( t = 8 \) \( q_{13}^{*} /q_{2}^{*} = 3.75 \)

\( p_{13}^{*} /p_{2}^{*} = 3.19 \)

\( I_{13}^{*} /I_{2}^{*} = 11.96 \)

1000

\( 0.9^{t - 1} \)

13

\( >0\,\,t = 7, \ldots ,9 \)

\( q_{9}^{*} /q_{7}^{*} = 0.31 \)

\( p_{9}^{*} /p_{7}^{*} = 1.22 \)

\( I_{9}^{*} /I_{7}^{*} = 0.38 \)

1

\( q_{t} = 10,000\cdot\left( {1 + \frac{t - 1}{100}} \right)\cdot\left( {1 - \frac{p}{{1000\cdot\left( {1 + \frac{t - 1}{100}} \right)}}} \right) \)

18,000

\( 0.875^{t - 1} \)

9

\( q_{9}^{*} /q_{1}^{*} = 0.08 \)

\( p_{9}^{*} /p_{1}^{*} = 1.54 \)

\( I_{9}^{*} /I_{1}^{*} = 0.12 \)

4

\( 0.999^{t - 1} \)

11

\( q_{11}^{*} /q_{1}^{*} = 1.33 \)

\( p_{11}^{*} /p_{1}^{*} = 1.06 \)

\( I_{11}^{*} /I_{1}^{*} = 1.41 \)