Time-dependent and independent control rules for coordinated production and pricing under demand uncertainty and finite planning horizons

Abstract

We address the effect of uncertainty on a manufacturer’s dynamic production and pricing decisions over a finite planning horizon. The demand for products, which depends on their price, is characterized by two stochastic processes: potential demand and customer price sensitivity. An optimal policy for coordinating production and pricing is a time-dependent feedback rule with respect to the state of the manufacturer’s inventories. We show that when the volatility of customer sensitivity to the product price is negligible, the optimal policy can be obtained analytically. Moreover, our simulations demonstrate that the volatility of stochastic customer price sensitivity does not have a strong effect on the manufacturer’s expected profit. Therefore, the solution derived for the case of customer price sensitivity with zero volatility can serve as a good approximation heuristic for the optimal policy if the true volatility of customer price sensitivity is within 40 % of its mean and the volatility of potential demand is within 25 % of its mean. Moreover, under these conditions, a simplified, time-independent control rule deteriorates expected profits by only 1.5 %.

Appendices

Appendix 1

Here we detail the solution analysis of the dynamic coefficient $a_2 (t)$. Substituting solution (20) for $a_1 (t)$ in (21), we obtain a linear, first-order differential equation, with the general solution

$$\begin{aligned} a_2 (t)=e^{-\int {4\beta a_1 (t)} dt}\left[ {k_3 +\int {\left( {r_1 -2\alpha a_1 (t)} \right) e^{\int {4\beta a_1 (t)dt} }dt} } \right] , \end{aligned}$$

(33)

where $a_2 (T)=0$ is the terminal condition for determining constant $k_3$.

In (33), $\int {a_1 (t)dt}$ takes the following form

$$\begin{aligned} \int {a_1 (t)dt}&= \displaystyle \int \frac{\sqrt{\beta r_2 }\left( {e^{\left( {4\sqrt{\beta r_2 }} \right) t}-e^{4\sqrt{\beta r_2 }T}} \right) }{2\beta \left( {e^{\left( {4\sqrt{\beta r_2 }} \right) t}+e^{4\sqrt{\beta r_2 }T}} \right) }dt\\&= \int {\left[ {\frac{\sqrt{\beta r_2 }\left( {e^{4\sqrt{\beta r_2 }t}} \right) }{2\beta \left( {e^{4\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}} \right) }-\frac{\sqrt{\beta r_2 }\left( {e^{4\sqrt{\beta r_2 }T}} \right) }{2\beta \left( {e^{4\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}} \right) }} \right] dt} \\&= \frac{\sqrt{\beta r_2 }\ln \left[ {e^{4\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}} \right] }{2\beta \left( {4\sqrt{\beta r_2 }} \right) } \\&-\frac{\sqrt{\beta r_2 }\left( {e^{4\sqrt{\beta r_2 }T}} \right) }{2\beta }\int {\left( {\frac{1}{\left( {e^{\left( {4\sqrt{\beta r_2 }} \right) t}+e^{4\sqrt{\beta r_2 }T}} \right) }dt} \right) } \\&= \frac{\sqrt{\beta r_2 }\ln \left[ {e^{4\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}} \right] }{2\beta \left( {4\sqrt{\beta r_2 }} \right) }\\&-\frac{\sqrt{\beta r_2 }\left( {e^{4\sqrt{\beta r_2 }T}} \right) }{2\beta }\left[ {\frac{t}{e^{4\sqrt{\beta r_2 }T}}-\frac{\ln \left[ {e^{4\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}} \right] }{4\sqrt{\beta r_2 }e^{4\sqrt{\beta r_2 }T}}} \right] . \end{aligned}$$

That is,

$$\begin{aligned} \int {a_1 (t)dt} =\frac{\ln \left[ {e^{4\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}} \right] }{4\beta }-\frac{\sqrt{\beta r_2 }t}{2\beta }. \end{aligned}$$

Thus, (33) is reduced to

$$\begin{aligned} a_2 (t)&= e^{-4\beta \left[ {\frac{\ln \left[ {e^{4\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}} \right] }{4\beta }-\frac{\sqrt{\beta r_2 }t}{2\beta }} \right] }\left[ {k_3 +\int {\left( {r_1 -2\alpha a_1 (t)} \right) e^{\int {4\beta a_1 (t)dt} }dt} } \right] \\&\!= e^{\left[ {2\sqrt{\beta r_2 }t\!-\!\ln \left( {e^{4\sqrt{\beta r_2 }t}\!+\!e^{4\sqrt{\beta r_2 }T}} \right) } \right] }\left[ {k_3 \!+\!\int {\left( {r_1 \!-\!2\alpha a_1 (t)} \right) e^{4\beta \left[ {\frac{\ln \left[ {e^{4\sqrt{\beta r_2 }t}\!+\!e^{4\sqrt{\beta r_2 }T}} \right] }{4\beta }\!-\!\frac{\sqrt{\beta r_2 }t}{2\beta }} \right] }dt} } \right] \\&= \frac{e^{2\sqrt{\beta r_2 }t}}{e^{4\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}}\left[ {k_3 +\int {\left( {r_1 -2\alpha a_1 (t)} \right) \frac{e^{4\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}}{e^{2\sqrt{\beta r_2 }t}}dt} } \right] . \end{aligned}$$

Therefore,

$$\begin{aligned} a_2 (t)&= \frac{k_3 e^{2\sqrt{\beta r_2 }t}}{e^{4\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}}+\frac{r_1 e^{2\sqrt{\beta r_2 }t}}{e^{4\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}}\int {\frac{e^{4\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}}{e^{2\sqrt{\beta r_2 }t}}dt} \\&-\frac{2\alpha e^{2\sqrt{\beta r_2 }t}}{e^{4\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}}\int {\left( {\frac{\sqrt{\beta r_2 }\left( {e^{\left( {4\sqrt{\beta r_2 }} \right) t}-e^{4\sqrt{\beta r_2 }T}} \right) }{2\beta \left( {e^{\left( {4\sqrt{\beta r_2 }} \right) t}+e^{4\sqrt{\beta r_2 }T}} \right) }} \right) \frac{e^{4\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}}{e^{2\sqrt{\beta r_2 }t}}dt} \\&= \frac{k_3 e^{2\sqrt{\beta r_2 }t}}{e^{4\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}}+\frac{r_1 e^{2\sqrt{\beta r_2 }t}}{e^{4\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}}\int {\left( {e^{2\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}e^{-2\sqrt{\beta r_2 }t}} \right) dt} \\&-\frac{\alpha \sqrt{\beta r_2 }}{\beta }\frac{e^{2\sqrt{\beta r_2 }t}}{e^{4\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}}\int {\left( {e^{2\sqrt{\beta r_2 }t}-e^{4\sqrt{\beta r_2 }T}e^{-2\sqrt{\beta r_2 }t}} \right) dt} \\&= \frac{k_3 e^{2\sqrt{\beta r_2 }t}}{e^{4\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}}+\frac{r_1 e^{2\sqrt{\beta r_2 }t}}{e^{4\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}}\left( {\frac{e^{2\sqrt{\beta r_2 }t}-e^{4\sqrt{\beta r_2 }T}e^{-2\sqrt{\beta r_2 }t}}{2\sqrt{\beta r_2 }}} \right) \\&-\frac{\alpha \sqrt{\beta r_2 }}{\beta }\frac{e^{2\sqrt{\beta r_2 }t}}{e^{4\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}}\left( {\frac{e^{2\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}e^{-2\sqrt{\beta r_2 }t}}{2\sqrt{\beta r_2 }}} \right) \end{aligned}$$

That is,

$$\begin{aligned} a_2 (t)=\frac{k_3 e^{2\sqrt{\beta r_2 }t}}{e^{4\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}}+\frac{r_1 }{e^{4\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}}\left( {\frac{e^{4\sqrt{\beta r_2 }t}-e^{4\sqrt{\beta r_2 }T}}{2\sqrt{\beta r_2 }}} \right) -\frac{\alpha }{2\beta }. \end{aligned}$$

From the terminal condition, $a_2 (T)=0$, we have $k_3 =\frac{\alpha e^{2\sqrt{\beta r_2 }T}}{\beta }$. Therefore,

$$\begin{aligned} a_2 (t)=\frac{1}{\left( {e^{4\sqrt{\beta r_2 }t}+e^{4\sqrt{\beta r_2 }T}} \right) }\left( {\frac{\alpha e^{2\sqrt{\beta r_2 }T}e^{2\sqrt{\beta r_2 }t}}{\beta }+r_1 \left( {\frac{e^{4\sqrt{\beta r_2 }t}-e^{4\sqrt{\beta r_2 }T}}{2\sqrt{\beta r_2 }}} \right) } \right) -\frac{\alpha }{2\beta }\nonumber \\ \end{aligned}$$

(34)

Appendix 2

Table 5 presents the set of parameters for an additional experiment.

Table 5 List of parameters

We have computed the expected profits when varying both the volatility of potential demand, $\sigma _a$, and the volatility of customer price sensitivity, $\sigma _b$. As in the case of the original set of parameters (Fig. 2), the curve of the expected profit is concave. The observed maximum value is 7887.15 monetary units for $\sigma _a =0.5$ (around 5 % of the mean potential demand) and $\sigma _b =0.02$ (around 10 % of the mean customer price sensitivity).

Similarly to Fig. 4, three different volatility modes are examined. The first mode refers to $\sigma _a =0.5$ and $\sigma _b =0.1\mu _b$, which result in near-optimal performance. The second mode refers to raising the volatility, $\sigma _b$, up to 40 % of the mean, i.e., $\sigma _a =0.5,\sigma _b =0.4\mu _b$ reflecting very high volatility of customer sensitivity to price. In the third mode the volatility, $\sigma _a$, is set at 40 % of the mean, i.e., $\sigma _a =4$ and $\sigma _b =0.1\mu _b$, reflecting very high volatility of potential demand.

The difference between the profits obtained in modes 1 and 3 reflects the loss of profit due to the high volatility of potential demand. The difference between the profits obtained in modes 1 and 2 reflects the loss of profit due to the high volatility of customer price sensitivity. The higher the ratio of these differences, the lower the relative impact of the volatility of customer price sensitivity on profit as compared to the impact of the volatility of the potential demand on profit. This ratio, which is shown in Fig. 10, decreases from the factor of 4.24 for ${\mu _b }/{\mu _a }=0.005$, to 2.99 for ${\mu _b }/{\mu _a }=0.02$ (coincides with value of ${\mu _b }/{\mu _a }$ in Table 5) and then to 2.124 for ${\mu _b }/{\mu _a }=0.05$. Similarly to what was obtained for the original set, the conclusion is that the smaller the value of ${\mu _b }/{\mu _a }$, the lower the relative impact of volatility of customer sensitivity to price on the profit as compared to the impact of volatility of potential demand on the profit. Specifically, within the range of ${\mu _b }/{\mu _a }$ values [0, 0.0575], a range even larger than that obtained for the original set, the lower bound for this ratio is larger than 2, i.e., the impact of uncertainty in potential demand is two times greater than the impact of uncertainty in customer price sensitivity. Figure 10 shows that for the entire examined spectrum of ratios ${\mu _b }/{\mu _a }$, the expected profit for the cases of high volatility of potential demand is inferior to the expected profit for the cases of high volatility of customer price sensitivity.

Fig. 10

Ratio between the profit differences for different volatilities and ratios of ${\mu _b }/{\mu _a }$ (in units of $10^{-4})$ under the time-dependent control rule