Appropriate timing for SMEs to introduce an Internet-based online channel under uncertain operating costs: a real options analysis

Abstract

This study analyzes the strategic decisions of small- and medium-sized enterprises (SMEs) to determine their optimal timing for introducing an Internet-based online channel when faced with uncertain operating costs. Introductory threshold, option value of waiting, and expected waiting time are derived by constructing value functions for SMEs who consider introducing an Internet-based online channel in addition to a traditional brick-and-mortar channel when faced with uncertain operating costs through traditional net present value method and real options approach. In order to facilitate the use of the proposed models in practice, we propose a parameter estimation method for the uncertain operating costs according to historical data of the actual operating costs, and explain the procedures involved in the whole process of how our models can be used in practical applications. Furthermore, we examine the effect of various parameters, such as awareness of consumer consumption on the Internet, irreversible sunk investment cost, cost efficiency coefficient, risk-free interest rate, mean drift rate and uncertainty rate of the uncertain operating costs, on the optimal timing for the introduction of an Internet-based online channel. In this study, the strategic decision-making of optimal timing for introducing an Internet-based online channel under uncertain operating costs is investigated from a new perspective by seeing the introduction of an Internet-based online channel under uncertain operating costs as a real option. This paper is part of a growing literature in the field of distribution channel management at the intersection of operations and finance. This study enriches the research stream of dual-channel sales mode and expands the scope of application of the real options approach. The models and findings presented in this study can provide theoretical support and practical guidance for SMEs to decide whether and when to introduce an Internet-based online channel when faced with uncertain operating costs in practice.

Appendices

Appendix A: Derivation of the optimal profits

For the sake of simplicity, we will omit the specific time parameter in this section. When the company operating a single traditional brick-and-mortar channel, the profit function is described as

$$\prod_{tr} (p_{tr} ) = D_{tr} \, (p_{tr} - C_{tr} ).$$

(19)

Substituting Eq. (4) into Eq. (19) and maximizing Eq. (19), such that, the optimal price of the product in the traditional brick-and-mortar channel while the company operating a single traditional brick-and-mortar channel is given by

$$p_{tr}^{*} = \frac{1}{2}(1 + C_{tr} ).$$

(20)

Substituting Eq. (20) into Eq. (19), we can obtain the optimal profit while the company operating a single traditional brick-and-mortar channel as

$$\prod_{tr}^{*} = \frac{1}{4}(1 - C_{tr} )^{2} .$$

(21)

The profit function when the company operating a single Internet-based online channel is described as

$$\prod_{int} (p_{int} ) = D_{int} \, (p_{int} - C_{int} ).$$

(22)

Substituting Eq. (5) into Eq. (22) and maximizing Eq. (22), such that, the optimal price of the product on the Internet-based online channel while the company operating a single Internet-based online channel is given by

$$p_{int}^{*} = \frac{1}{2}(\eta + \lambda C_{tr} ).$$

(23)

Substituting Eq. (23) into Eq. (22), we can obtain the optimal profit while the company operating a single Internet-based online channel as

$$\prod_{int}^{*} = \frac{1}{4}(\eta - \lambda C_{tr} )^{2} .$$

(24)

When the company operating an integrated dual-channel sales mode, the profit function is described as

$$\prod_{du} (p_{td} ,p_{id} ) = D_{td} \, (p_{td} - C_{tr} ) + D_{id} \, (p_{id} - C_{int} ).$$

(25)

Substituting Eqs. (2) and (3) into Eq. (25), we have

$$\prod_{du} (p_{td} ,p_{id} ) = \frac{{(1 - \theta \eta - p_{td} + \theta p_{id} ) \, (p_{td} - C_{tr} )}}{{1 - \theta^{2} }} \, + \frac{{(\eta - \theta - p_{id} + \theta p_{td} ) \, (p_{id} - C_{int} )}}{{1 - \theta^{2} }}.$$

(26)

Based on Eq. (26), we can obtain the first- and second-order partial derivative of $\prod_{du} (p_{td} ,p_{id} )$ with respect to $p_{td}$ and $p_{id}$ as follows:

$$\begin{aligned} \frac{{\partial \prod_{du} }}{{\partial p_{td} }} & = \frac{{1 - \theta \eta - 2p_{td} + 2\theta p_{id} + C_{tr} - \theta \lambda C_{tr} }}{{1 - \theta^{2} }},\;\;\frac{{\partial \prod_{du} }}{{\partial p_{id} }} = \frac{{\eta - \theta - 2p_{id} + 2\theta p_{td} + \lambda C_{tr} - \theta C_{tr} }}{{1 - \theta^{2} }}, \\ \frac{{\partial^{2} \prod_{du} }}{{\partial p_{td}^{2} }} & = \frac{ - 2}{{1 - \theta^{2} }} < 0,\;\;\frac{{\partial^{2} \prod_{du} }}{{\partial p_{td} \partial p_{id} }} = \frac{{\partial^{2} \prod_{du} }}{{\partial p_{id} \partial p_{td} }} = \frac{2\theta }{{1 - \theta^{2} }},\;\;\frac{{\partial^{2} \prod_{du} }}{{\partial p_{id}^{2} }} = \frac{ - 2}{{1 - \theta^{2} }}. \\ \end{aligned}$$

Then, we can obtain that $\left( {\frac{{\partial^{2} \prod_{du} }}{{\partial p_{td} \partial p_{id} }}} \right)^{2} - \left( {\frac{{\partial^{2} \prod_{du} }}{{\partial p_{td}^{2} }}} \right) \, \left( {\frac{{\partial^{2} \prod_{du} }}{{\partial p_{id}^{2} }}} \right) = \frac{ - \,4}{{1 - \theta^{2} }} < 0$. Therefore, the profit function $\prod_{du} (p_{td} ,p_{id} )$ is jointly concave in $p_{td}$ and $p_{id}$. Consequently, there exists only one certain $p_{td}^{*}$ and $p_{id}^{*}$ that maximizes the profit function $\prod_{du} (p_{td} ,p_{id} )$ getting the maximum value $\prod_{du} (p_{td}^{*} ,p_{id}^{*} )$. Taking the first-order partial derivatives of $\prod_{du} (p_{td} ,p_{id} )$ with respect to $p_{td}$ and $p_{id}$, and letting the derivatives be zeros, we can obtain the maximum value of Eq. (25). Such that, the corresponding optimal prices of the product in the traditional brick-and-mortar channel and the Internet-based online channel while the company operating an integrated dual-channel sales mode are given by:

$$p_{td}^{*} = \frac{1}{2}(1 + C_{tr} ),$$

(27)

$$p_{id}^{*} = \frac{1}{2}(\eta + \lambda C_{tr} ).$$

(28)

Substituting Eqs. (27) and (28) into Eq. (26), we can obtain the optimal profit while the company operating an integrated dual-channel sales mode as

$$\prod_{du}^{*} = \frac{{(\lambda^{2} - 2\lambda \theta + 1)C_{tr}^{2} + 2(\eta \theta + \lambda \theta - \eta \lambda - 1)C_{tr} + \eta^{2} - 2\eta \theta + 1}}{{4(1 - \theta^{2} )}}.$$

(29)

Appendix B: Proofs of the propositions

Proof of Proposition 1

By comparing Eqs. (7) and (8), we can obtain that if $\Omega > 1$, then $\prod_{tr}^{*} > \prod_{int}^{*}$; and if $\Omega < 1$, then $\prod_{tr}^{*} < \prod_{int}^{*}$. Moreover,

$$\begin{aligned} \prod_{du}^{*} - \prod_{tr}^{*} = & \frac{{(\lambda^{2} - 2\lambda \theta + 1)C_{tr}^{2} + 2(\eta \theta + \lambda \theta - \eta \lambda - 1)C_{tr} + \eta^{2} - 2\eta \theta + 1}}{{4(1 - \theta^{2} )}} - \frac{1}{4}(1 - C_{tr} )^{2} \\ = & \frac{{(\lambda C_{tr} - \theta C_{tr} - \eta + \theta )^{2} }}{{4(1 - \theta^{2} )}} > 0, \\ \prod_{du}^{*} - \prod_{int}^{*} = & \frac{{(\lambda^{2} - 2\lambda \theta + 1)C_{tr}^{2} + 2(\eta \theta + \lambda \theta - \eta \lambda - 1)C_{tr} + \eta^{2} - 2\eta \theta + 1}}{{4(1 - \theta^{2} )}} - \frac{1}{4}(\eta - \lambda C_{tr} )^{2} \\ = & \frac{{(\lambda \theta C_{tr} - C_{tr} - \eta \theta + 1)^{2} }}{{4(1 - \theta^{2} )}} > 0. \\ \end{aligned}$$

Such that (1) if $\Omega > 1$, then $\prod_{du}^{*} > \prod_{tr}^{*} > \prod_{int}^{*}$; and (2) if $\Omega < 1$, then $\prod_{du}^{*} > \prod_{int}^{*} > \prod_{tr}^{*}$. □

Proof of Proposition 4

Taking the first-order partial derivatives of $C_{ROP}^{*}$ with respect to $K$, $\lambda$, and $\eta$, we obtain that $\frac{{\partial C_{ROP}^{*} }}{\partial K} > 0$, $\frac{{\partial C_{ROP}^{*} }}{\partial \lambda } > 0$, $\frac{{\partial C_{ROP}^{*} }}{\partial \eta } < 0$. Therefore, $C_{ROP}^{*}$ is an increasing function of $K$ and $\lambda$, and a decreasing function of $\eta$. □

Proof of Proposition 5

As $C_{ROP}^{*} > C_{tr0}$, we have $\frac{{C_{ROP}^{*} }}{{C_{tr0} }} > 1$. Based on Eqs. (17) and (18), we can discuss the impacts of the risk-free interest rate, the mean drift rate and uncertainty rate of the uncertain operating cost on the introductory probability and the expected waiting time in three situations as follows:

• (1) When $\mu - \frac{1}{2}\sigma^{2} > 0$, we have $\frac{2}{{\sigma^{2} }}(\mu - \frac{1}{2}\sigma^{2} ) > 0$. Therefore, according to Eq. (17), we can obtain that the introductory probability $P(C_{tr0} \to C_{ROP}^{*} )$ will be certainly equal to 1. That is to say, the uncertain operating cost $C_{tr}$ evolves over time from the initial value $C_{tr0}$ to get to the introductory threshold $C_{ROP}^{*}$ will certainly occur. According to Eq. (18), we can obtain that the expected waiting time $E(T)$ is a certain value affected by the risk-free interest rate, mean drift rate, and uncertainty rate.

• (2) When $\mu - \frac{1}{2}\sigma^{2} = 0$, we have $\frac{2}{{\sigma^{2} }}(\mu - \frac{1}{2}\sigma^{2} ) = 0$. Similarly, according to Eq. (17), we can obtain that the introductory probability $P(C_{tr0} \to C_{ROP}^{*} )$ will be certainly equal to 1, which means that the uncertain operating cost $C_{tr}$ will certainly reach the introductory threshold $C_{ROP}^{*}$ in the dynamic evolves process from the initial value $C_{tr0}$. According to Eq. (18), we can obtain that the expected waiting time $E(T)$ will be infinite. That is to say, when the square of the uncertainty rate is equal to twice the mean drift rate of the uncertain operating cost, the expected waiting time tends to be infinite, although the uncertain operating cost will certainly reach the introductory threshold.

• (3) When $\mu - \frac{1}{2}\sigma^{2} < 0$, we have $\frac{2}{{\sigma^{2} }}\left( {\mu - \frac{1}{2}\sigma^{2} } \right) < 0$. Therefore, according to Eq. (17), we can obtain that the introductory probability $P(C_{tr0} \to C_{ROP}^{*} ) < 1$. Such that, the uncertain operating cost $C_{tr}$ evolves over time from the initial value $C_{tr0}$ to get to the introductory threshold $C_{ROP}^{*}$ with a certain probability less than 1. Therefore, in this case, the introductory probability is a certain value affected by the risk-free interest rate, mean drift rate, and uncertainty rate. Based on Eq. (18), we know that the expected waiting time $E(T)$ does not exist in this situation. □

Proof of Proposition 6

Basing on Eqs. (16) and (12), we have $\frac{{C_{ROP}^{*} }}{{C_{NPV}^{*} }} = \sqrt {\frac{{\beta_{1} }}{{\beta_{1} - 2}}} = \sqrt {1 + \frac{2}{{\beta_{1} - 2}}} > 1$. Therefore, the introductory threshold of the operating cost for introducing an Internet-based online channel in the real options approach is greater than the introductory threshold in the traditional NPV method.

The size of the option value of waiting can be reflected by the gap between the introductory threshold in the real options approach and the introductory threshold in the traditional NPV method. Therefore, a higher ratio of the two introductory thresholds means a larger option value of waiting, and vice versa.

As described in Appendix C, $\beta_{1} = \frac{1}{2} - \frac{\mu }{{\sigma^{2} }} + \sqrt {\left( {\frac{\mu }{{\sigma^{2} }} - \frac{1}{2}} \right)^{2} + \frac{2r}{{\sigma^{2} }}}$, we have $\frac{{\partial \beta_{1} }}{\partial \mu } = \frac{1}{{\sigma^{2} }}\left( {\frac{{\frac{\mu }{{\sigma^{2} }} - \frac{1}{2}}}{{\sqrt {\left( {\frac{\mu }{{\sigma^{2} }} - \frac{1}{2}} \right)^{2} + \frac{2r}{{\sigma^{2} }}} }} - 1} \right) < 0$, $\frac{{\partial \beta_{1} }}{\partial \sigma } = \frac{1}{{\sigma^{2} }}\left( {\frac{{\frac{\mu }{{\sigma^{2} }} - \frac{1}{2}}}{{\sqrt {\left( {\frac{\mu }{{\sigma^{2} }} - \frac{1}{2}} \right)^{2} + \frac{2r}{{\sigma^{2} }}} }} - 1} \right) < 0$, and $\frac{{\partial \beta_{1} }}{\partial r} = \frac{1}{{\sqrt {\left( {\frac{\mu }{{\sigma^{2} }} - \frac{1}{2}} \right)^{2} + \frac{2r}{{\sigma^{2} }}} \sigma^{2} }} > 0$. In addition, $\frac{{\partial \frac{{C_{ROP}^{*} }}{{C_{NPV}^{*} }}}}{{\partial \beta_{1} }} = - \,\frac{ 1}{{\sqrt {\frac{{\beta_{1} }}{{\beta_{1} - 2}}} \left( {\beta_{1} - 2} \right)^{ 2} }} < 0$. Thus, we have $\frac{{\partial \frac{{C_{ROP}^{*} }}{{C_{NPV}^{*} }}}}{\partial \mu } = \frac{{\partial \frac{{C_{ROP}^{*} }}{{C_{NPV}^{*} }}}}{{\partial \beta_{1} }} \cdot \frac{{\partial \beta_{1} }}{\partial \mu } > 0$, $\frac{{\partial \frac{{C_{ROP}^{*} }}{{C_{NPV}^{*} }}}}{\partial \sigma } = \frac{{\partial \frac{{C_{ROP}^{*} }}{{C_{NPV}^{*} }}}}{{\partial \beta_{1} }} \cdot \frac{{\partial \beta_{1} }}{\partial \sigma } > 0$, and $\frac{{\partial \frac{{C_{ROP}^{*} }}{{C_{NPV}^{*} }}}}{\partial r} = \frac{{\partial \frac{{C_{ROP}^{*} }}{{C_{NPV}^{*} }}}}{{\partial \beta_{1} }} \cdot \frac{{\partial \beta_{1} }}{\partial r} < 0$. □

Appendix C: Derivation of the value function

The process of solving for the value function $V(C_{tr} ,T)$ can analogy and reference to the solution of a perpetual American call option in the derivatives literature in the field of financial engineering. In order to obtain the optimal timing for the introduction of an Internet-based online channel, we need to solve a stochastic optimal stopping time problem. Based on the previous works of Dixit and Pindyck [15] and Grenadier and Weiss [17], the value function $V(C_{tr} ,T)$ meets the following Bellman Equation in continuous region,

$$\left[ {rV(C_{tr} ,T) - \prod_{tr}^{*} } \right]dt = E(dV(C_{tr} ,T)).$$

(30)

Be similar to Dixit and Pindyck [15], Grenadier and Weiss [17] and Schwartz and Zozaya-Gorostiza [35], we use Ito’s Lemma to manipulate $dV$, such that we have

$$dV = V^{\prime } dC_{tr} + Vdt + \frac{1}{2}\sigma^{2} C_{tr}^{2} V^{\prime\prime}dt,$$

(31)

where $V'$ and $V^{\prime\prime}$ are the first- and second-order derivative of the value function $V$ with respect to $C_{tr}$, respectively. Substituting equations $dC_{tr} (t) = \mu C_{tr} (t)dt + \sigma C_{tr} (t)dz$ and $E(dz) = 0$ into Eq. (31), and then combining with Eq. (30), we have

$$\mu C_{tr} V^{\prime} + \frac{1}{2}\sigma^{2} C_{tr}^{2} V^{\prime\prime} - rV + \prod_{tr}^{*} = 0.$$

(32)

In addition, the value function $V$ must satisfy the following conditions:

• (1) Boundary condition:

  $$V(0) = (NPV_{tr} \left| {C_{tr} } \right. = 0) = \left( {\frac{{r^{2} C_{tr}^{2} - r\mu C_{tr}^{2} + 2\mu^{2} + 4r\mu C_{tr} - 2r^{2} C_{tr} - 3r\mu + r^{2} }}{4(r - 2\mu )(r - \mu )r}\left| {C_{tr} } \right. = 0} \right) = \frac{1}{4r};$$

  (33)

• (2) Value matching condition:

  $$\begin{aligned} V(C_{ROP}^{*} ) & = EPV_{du} \left| {C_{tr0} = C_{ROP}^{*} } \right. - K \\ & = \frac{{\left\{ {\begin{array}{*{20}c} {\lambda^{2} r\mu C_{ROP}^{*} \,^{2} - 4r\mu C_{ROP}^{*} - r^{2} C_{ROP}^{*} \,^{2} + r\mu C_{ROP}^{*} \;^{2} + 3r\mu - r^{2} - 2\mu^{2} + 4\theta \eta r\mu C_{ROP}^{*} + 2\theta \eta r^{2} } \\ { - \eta^{2} r^{2} + 2\lambda \theta r^{2} C_{ROP}^{*} \,^{2} - 6\theta \eta r\mu - 2\lambda \theta r^{2} C_{ROP}^{*} + 4\theta \eta \mu^{2} + 4\lambda \theta r\mu C_{ROP}^{*} + 2\lambda \eta r^{2} C_{ROP}^{*} } \\ { - 2\theta \eta r^{2} C_{ROP}^{*} - 2\lambda \theta r\mu C_{ROP}^{*} \,^{2} + 3\eta^{2} r\mu - 4\lambda \eta r\mu C_{ROP}^{*} - 2\eta^{2} \mu^{2} - \lambda^{2} r^{2} C_{ROP}^{*} \,^{2} + 2r^{2} C_{ROP}^{*} } \\ \end{array} } \right\}}}{{4(r - 2\mu )(r - \mu )(\theta^{2} - 1)r}} - K; \\ \end{aligned}$$

  (34)

• (3) Smooth pasting condition:

  $$\frac{{\partial V(C_{ROP}^{*} )}}{{\partial C_{tr} }} = \frac{{\left\{ {\begin{array}{*{20}c} {\lambda^{2} \mu C_{ROP}^{*} + r - rC_{ROP}^{*} + \mu C_{ROP}^{*} + 2\theta \eta \mu + 2\lambda \theta rC_{ROP}^{*} - 2\lambda \theta \mu C_{ROP}^{*} } \\ { + 2\lambda \theta \mu + \lambda \eta r - \theta \eta r - \lambda \theta r - 2\lambda \eta \mu - \lambda^{2} rC_{ROP}^{*} - 2\mu } \\ \end{array} } \right\}}}{{2(r - 2\mu )(r - \mu )(\theta^{2} - 1)}}.$$

  (35)

According to Eq. (32), we can obtain that the general solution for the value function $V$ must have the form

$$V = AC_{tr}^{{\beta_{1} }} + BC_{tr}^{{\beta_{2} }} .$$

(36)

In the above equation, $\beta_{1} = \frac{1}{2} - \frac{\mu }{{\sigma^{2} }} + \sqrt {\left( {\frac{\mu }{{\sigma^{2} }} - \frac{1}{2}} \right)^{2} + \frac{2r}{{\sigma^{2} }}} > 2$ (since $r > 2\mu + \sigma^{2}$), $\beta_{2} = \frac{1}{2} - \frac{\mu }{{\sigma^{2} }} - \sqrt {\left( {\frac{\mu }{{\sigma^{2} }} - \frac{1}{2}} \right)^{2} + \frac{2r}{{\sigma^{2} }}} < 0$, and $A$ and $B$ are parameters to be determined.

For the value function $V$ to satisfy the boundary condition, it is certain that $B = 0$. Thus, the general solution for the value function $V$ must take the form $V = AC_{tr}^{{\beta_{1} }}$. Furthermore, the solution for the value function $V$ must satisfy the value matching and smooth pasting conditions. Therefore, basing on Eqs. (34) and (35), we can obtain the value function $V$ as follows:

$$V = \left\{ {\begin{array}{*{20}l} {AC_{tr}^{{\beta_{1} }} + \frac{{r^{2} C_{tr}^{2} - r\mu C_{tr}^{2} + 4r\mu C_{tr} - 2r^{2} C_{tr} + 2\mu^{2} - 3r\mu + r^{2} }}{4(r - 2\mu )(r - \mu )r} ,} \hfill & { \, C_{tr} { < }C_{ROP}^{*} } \hfill \\ {\frac{{\left\{ {\begin{array}{*{20}c} {2r^{2} C_{tr} - 4r\mu C_{tr} - r^{2} C_{tr}^{2} + r\mu C_{tr}^{2} + 3r\mu - r^{2} - 2\mu^{2} + 4\theta \eta r\mu C_{tr} + 2\theta \eta r^{2} } \\ { - \,\eta^{2} r^{2} + 2\lambda \theta r^{2} C_{tr}^{2} - 6\theta \eta r\mu - 2\lambda \theta r\mu C_{tr}^{2} + 4\theta \eta \mu^{2} + 4\lambda \theta r\mu C_{tr} + 2\lambda \eta r^{2} C_{tr} } \\ { - \,2\theta \eta r^{2} C_{tr} - 2\lambda \theta r^{2} C_{tr} + 3\eta^{2} r\mu - 4\lambda \eta r\mu C_{tr} - 2\eta^{2} \mu^{2} - \lambda^{2} r^{2} C_{tr}^{2} + \lambda^{2} r\mu C_{tr}^{2} } \\ \end{array} } \right\}}}{{4(r - 2\mu )(r - \mu )(\theta^{2} - 1)r}} - K,} \hfill & {C_{ROP}^{*} \le C_{tr} \le \bar{C}} \hfill \\ \end{array} } \right.,$$

(37)

where

$$C_{ROP}^{*} = \frac{{\sqrt {(\theta - \eta )^{2} \beta_{1} (r - 2\mu )^{2} r} - \sqrt {(r - 2\mu )r\beta_{1} (4Kr(1 - \theta^{2} )(\mu - r)^{2} - \mu^{2} \beta_{1} (\eta - \theta )^{2} )} }}{{\sqrt {(\theta - \lambda )^{2} (\beta_{1} - 2)(r - \mu )^{2} r^{2} } }}.$$

(38)

Appendix D: Parameter estimation of the uncertain operating cost

In this section, we propose a parameter estimation method for the uncertain operating cost. This method can estimate the parameters of the uncertain operating cost (including the mean drift rate $\mu$ and the uncertainty rate $\sigma$) according to historical data of the operating costs.

As described in Sect. 3.2, by assuming that the uncertain operating cost $C_{tr}$ evolves over time as GBM, such that $dC_{tr} (t) = \mu C_{tr} (t)dt + \sigma C_{tr} (t)dz$, $dz$ is a standard Wiener process with $E(dz) = 0$ and $Var(dz) = dt$, i.e., $dz \sim N(0,dt)$. The state variable, $C_{tr} (t)$ is lognormal in its distribution with Eqs. (39) and (40) following:

$$E[C_{tr} (t)] = C_{tr0} \cdot e^{\mu t}$$

(39)

$$Var[C_{tr} (t)] = C_{tr0}^{2} \cdot e^{2\mu t} (e^{{\sigma^{2} t}} - 1)$$

(40)

where $C_{tr0} = C_{tr} (0)$, and it represents the company’s initial operating cost.

By using Riemann summation method, we can obtain a stochastic differential equation of $C_{tr} (t)$ as follows:

$$C_{tr} (t) = C_{tr0} \cdot e^{{\left( {\mu - \frac{1}{2}\sigma^{2} } \right)t + \sigma zt}}$$

(41)

Let $F = \ln C_{tr}$, then we have

$$^{{dF = \left( {\mu - \frac{1}{2}\sigma^{2} } \right)dt + \sigma dz}}$$

(42)

If we take equidistant samples by dividing the time series of historical data of the company’s operating cost in the finite time interval $[0,T]$, and let the sampling interval be $\Delta t$, then we could obtain $N + 1$ samples, i.e., $\{ C_{tr} (t_{i} )\}_{i = 0}^{N}$, where $N = \frac{T}{\Delta t}$, $t_{i} = i \cdot \Delta t$, and $i = 0,\;1,\;2, \ldots ,N$. Based on Eq. (42), if we let $Y_{i} = dF = \ln \frac{{C_{tr} (t_{i} )}}{{C_{tr} (t_{i - 1} )}}$, then $Y_{i}$ follows a normal distribution with a mean of $\left( {\mu - \frac{1}{2}\sigma^{2} } \right)\Delta t$ and a variance of $\sigma^{2} \Delta t$ in the finite time interval $[0,T]$, such that the probability density function (PDF) of $Y$ is as follows:

$$f\left( {y/\mu ,\sigma } \right) = \frac{1}{{\sigma \sqrt {2\pi \Delta t} }}\exp \left[ {\frac{{\left( {y - \mu \Delta t + \frac{1}{2}\sigma^{2} \Delta t} \right)^{2} }}{{2\sigma^{2} \Delta t}}} \right]$$

(43)

The likelihood function is the product of all of the $N + 1$ sample observations of the PDF, such that

$$L\left( {y/\mu ,\sigma } \right) = \prod\limits_{i = 1}^{n} {f\left( {y/\mu ,\sigma } \right)} = \frac{1}{{(\sigma \sqrt {2\pi \Delta t} )^{n} }}\exp \left[ { - \,\sum\limits_{i = 1}^{n} {\frac{{\left( {y_{i} - \mu \Delta t + \frac{1}{2}\sigma^{2} \Delta t} \right)^{2} }}{{2\sigma^{2} \Delta t}}} } \right]$$

(44)

Then, we can obtain the log-likelihood function as follows:

$$\ln L\left( {y/\mu ,\sigma } \right) = - \,\sum\limits_{i = 1}^{n} {\frac{{\left( {y_{i} - \mu \Delta t + \frac{1}{2}\sigma^{2} \Delta t} \right)^{2} }}{{2\sigma^{2} \Delta t}}} - n\ln \left( {\sigma \sqrt {2\pi \Delta t} } \right)$$

(45)

First, let $\Delta t = 1$, and then taking the first-order partial derivatives of Eq. (45) with respect to $\mu$ and $\sigma$ and letting the derivatives be zeros, we can obtain the following:

$$\frac{\partial L}{\partial \mu } = \sum\limits_{i = 1}^{n} {\frac{{2y_{i} - 2\mu + \sigma^{2} }}{{2\sigma^{2} }}} = 0$$

(46)

$$\frac{\partial L}{\partial \sigma } = \sum\limits_{i = 1}^{n} {\frac{{(y_{i} - \mu )^{2} }}{{2\sigma^{2} }}} - \,\frac{1}{4}n\sigma - \frac{n}{\sigma } = 0$$

(47)

Basing on Eqs. (46) and (47), we can obtain the maximum likelihood estimators for the mean drift rate $\mu$ and the uncertainty rate $\sigma$ of the uncertain operating cost as follows:

$$\mu = \frac{1}{2}(\overline{{y^{2} }} - (\overline{y} )^{2} + 2\overline{y} )$$

(48)

$$\sigma = \sqrt {\overline{{y^{2} }} - (\overline{y} )^{2} }$$

(49)

where $\overline{y}$ and $\overline{{y^{2} }}$ represent the mean and the second-order origin moment of the $N + 1$ samples (i.e., $\{ C_{tr} (t_{i} )\}_{i = 0}^{N}$), respectively. Therefore, basing on Eqs. (48) and (49), we can obtain the maximum likelihood estimators for the mean drift rate $\mu$ and the uncertainty rate $\sigma$ of the uncertain operating cost according to the time series of historical data of the company’s operating cost.