Optimal pricing and sourcing strategies in the presence of supply uncertainty and competition

Abstract

The implementation of cost leadership strategy can enable enterprises to obtain a lasting competitive advantage. In this paper, we construct a supply chain with two competing suppliers and two competing manufacturers. The suppliers act as the Stackelberg leaders, selling components to the follower manufacturers. The manufacturers use different sourcing strategies, one of which only uses the reliable supplier, while another adopts contingent dual sourcing. We derive the analytical form of the equilibrium solution of order quantities of manufacturers and wholesale prices of suppliers in different scenarios and compare the decision differences when reliable supplier stays in different game positions. We further investigate the impact of different cooperation contract between the manufacturer who adopts dual sourcing and unreliable supplier on the procurement and pricing strategies by numerical experiments. The results illustrate that reliable supplier acts as the Stackelberg leader are more beneficial to suppliers. Manufacturer with dual sourcing can work with the unreliable supplier by revenue sharing contract to achieve a win–win situation.

Appendix

Proof of Lemma 1

According to Eq. (6), it can be proved that $\varPi_{S1}^{WP}$ is concave functions for $w_{2}$. Let $w_{2WP}$ denote the solution of $\frac{{\partial \varPi_{S2}^{WP} }}{{\partial w_{2} }} = 0$, we have $w_{2WP} = \frac{{\mu w_{1} + c_{2} }}{2\mu }$. When $q_{{_{3} WP}}^{*} > 0$, then $w_{2WP} < w_{1}$.

Proof of Lemma 2

If $w_{1} > \frac{{c_{2} }}{\mu }$, based on Eq. (7) $\frac{{\partial \varPi_{S1}^{D} }}{{\partial w_{1}^{2} }} = \frac{{ - 4\upsigma^{ 2} - \left( {\uptheta + 2} \right)\upmu^{ 2} }}{{\left( {\uptheta + 2} \right)\upsigma^{ 2} }} < 0$. Let $\frac{{\partial \varPi_{S1}^{WP} }}{{\partial w_{1} }} = 0$, then $w_{{_{1} A}}^{C} = {{\left( {\left( {c_{1} + w_{2} } \right)\left( {\theta + 2} \right)\mu^{2} + 4\sigma^{2} \left( {a + c_{1} } \right)} \right)} \mathord{\left/ {\vphantom {{\left( {\left( {c_{1} + w_{2} } \right)\left( {\theta + 2} \right)\mu^{2} + 4\sigma^{2} \left( {a + c_{1} } \right)} \right)} {\left( {\left( {2\theta + 4} \right)\mu^{2} + 8\sigma^{2} } \right)}}} \right. \kern-0pt} {\left( {\left( {2\theta + 4} \right)\mu^{2} + 8\sigma^{2} } \right)}}$. If $w_{{_{1} A}}^{C} > {{c_{2} } \mathord{\left/ {\vphantom {{c_{2} } \mu }} \right. \kern-0pt} \mu }$, it is easy to prove that $\varPi_{S1}^{WP} \left( {w_{1A}^{C} } \right) > \varPi_{S1}^{WP} \left( {{{c_{2} } \mathord{\left/ {\vphantom {{c_{2} } \mu }} \right. \kern-0pt} \mu }} \right)$ and the optimal wholesale price of the reliable supplier is $w_{1WP}^{C} = w_{1A}^{C}$; if $w_{1A}^{C} \le {{c_{2} } \mathord{\left/ {\vphantom {{c_{2} } \mu }} \right. \kern-0pt} \mu }$, assume ${{c_{2} } \mathord{\left/ {\vphantom {{c_{2} } \mu }} \right. \kern-0pt} \mu } \le {{\left( {a + c_{1} } \right)} \mathord{\left/ {\vphantom {{\left( {a + c_{1} } \right)} 2}} \right. \kern-0pt} 2}$, it is more advantageous for the reliable supplier to resist the unreliable supplier by adopting low price policy and forming price barrier. So, the optimal wholesale price is $w_{1WP}^{C} = {{c_{2} } \mathord{\left/ {\vphantom {{c_{2} } \mu }} \right. \kern-0pt} \mu }$.

Proof of Corollary 1

If $\phi_{1} \le 0$, the reliable supplier will still monopolize the market. In this case, the profit of manufacturers will be $C\varPi_{M1}^{WP*} {\text{ = C}}\varPi_{M2}^{WP*} { = }\frac{{\left( {{\text{a}}\upmu - {\text{c}}_{ 2} } \right)^{ 2} }}{{\upmu^{ 2} \left( {\uptheta + 2} \right)^{ 2} }} > \frac{{\left( {{\text{a}} - {\text{c}}_{ 1} } \right)^{ 2} }}{{ 4\left( {\uptheta + 2} \right)^{ 2} }}$. If $\phi_{1} > 0$, the reliable supplier will coexist with the unreliable supplier. According to Eq. (8), $w_{{_{1} WP}}^{C*} < \frac{{a + c_{1} }}{2} < \frac{{3a_{1} - c_{1} }}{2}$ can be easily proved $C\varPi_{M1}^{WP*} - \varPi_{M1}^{S*} = \frac{{\left( {a - 2w_{1WP}^{C*} + c_{1} } \right)\left( {3a - 2w_{1WP}^{C*} - c_{1} } \right)}}{{4\left( {\theta + 2} \right)^{2} }}$; and $C\varPi_{M2}^{WP*} - C\varPi_{M1}^{WP*} = \frac{{\left( {w_{1WP}^{C*} - w_{2WP}^{C*} } \right)^{2} \mu^{2} }}{{4\sigma^{2} }}$. Corollary 1 can be proved.

Proof of Corollary 2

$\phi_{1} - \phi_{2} = \left( {c_{2} - c_{1} \mu } \right)\left( {\theta + 2} \right)\mu^{2}$. (1) If $c_{2} > c_{1} \mu$,$\phi_{1} > \phi_{2}$. (1) if $\phi_{2} \le 0$ and $\phi_{1} \le 0$, the reliable supplier will monopolize the market, and the wholesale price $w_{1WP}^{C*} = w_{1WP}^{S*} = \frac{{c_{2} }}{\mu }$; (2) if $\phi_{2} \le 0$ and $\phi_{1} > 0$, $w_{1WP}^{C*} > w_{1WP}^{S*} = \frac{{c_{2} }}{\mu }$. The unreliable supplier will coexist with the reliable supplier only when they are in an equal competitive position; (3) if $\phi_{2} > 0$ and $\phi_{1} > 0$, the unreliable supplier can always get an order. And

$$ w_{1WP}^{C*} - w_{1WP}^{S*} = - \frac{{\left( {\left( {c_{2} - c_{1} \mu } \right)\left( {\theta + 2} \right)\mu + 8\sigma^{2} \left( {a - c_{1} } \right)} \right)\left( {\theta + 2} \right)\mu^{2} }}{{2\left( {3\left( {\theta + 2} \right)\mu^{2} + 16\sigma^{2} } \right)\left( {\left( {\theta + 2} \right)\mu^{2} + 8\sigma^{2} } \right)}} < 0 $$

(A.1)

(2) If $c_{2} \le c_{1} \mu$, $\phi_{2} \ge \phi_{1} > 0$. Similarly, the unreliable supplier has access to the supply market. And $w_{1WP}^{C*} < w_{1WP}^{S*}$, if $\left( {c_{1} \mu - c_{2} } \right)\mu < \frac{{8\sigma^{2} \left( {a - c_{1} } \right)}}{{\left( {\theta + 2} \right)}}$; otherwise $w_{1WP}^{C*} \ge w_{1WP}^{S*}$.

Proof of Corollary 3

If $\phi_{i} \le 0,i = 1,2$, $w_{1WP}^{*} { = }w_{2WP}^{*} { = }\frac{{c_{2} }}{\mu }$. From proposition1, the production quantity, sales price, and profit are equal. Otherwise, $w_{1WP}^{*} \ge w_{2WP}^{*} { > }\frac{{c_{2} }}{\mu }$. $E\left( {q_{M2}^{WP} } \right) = q_{2WP}^{*} + q_{3WP}^{*} \mu = \frac{{a - w_{1WP}^{*} }}{\theta + 2}$, $\varPi_{M2}^{WP} { = }\frac{{\left( {w_{1WP}^{*} - w_{2WP}^{*} } \right)^{2} \mu^{2} }}{{4\sigma^{2} }} + \frac{{\left( {a - w_{1WP}^{*} } \right)^{2} }}{{\left( {\theta + 2} \right)^{2} }}$.

Proof of Proposition 5

According to Eq. (11), we can get the Hessian matrix of the profit function $\Pi _{\text{M2}}^{RS}$ as $H = \left[ {\begin{array}{*{20}c} { - 2} & {\left( { -\uplambda - 1} \right)\mu } \\ {\left( { -\uplambda - 1} \right)\mu } & { - 2\uplambda\left( {\upmu^{ 2} +\upsigma^{ 2} } \right)} \\ \end{array} } \right]$. When $4\uplambda \upsigma ^{ 2} > \left( {\uplambda - 1} \right)^{ 2}\upmu^{ 2}$, the matrix will be negative definite. Taking the derivative of $\Pi _{\text{M2}}^{RS}$ with respect to the variable $\left( {q_{2} ,q_{3} } \right)$, then we have:

$$ \left\{ \begin{aligned} & q_{2} = \frac{{\mu^{2} \left( {a - \theta q_{1} } \right)\lambda^{2} + \left( {\left( {\theta q_{1} - a + 2w_{1} - w_{2} } \right)\mu^{2} - 2\sigma^{2} \left( {a - \theta q_{1} - w_{1} } \right)} \right)\lambda - \mu^{2} w_{2} }}{{\left( {\lambda - 1} \right)^{2} \mu^{2} - 4\lambda \sigma^{2} }} \hfill \\ & q_{3} = \frac{{2\left( {w_{1} - w_{2} } \right)\mu - \left( {1 - \lambda } \right)\left( {a - \theta q_{1} + w_{1} } \right)\mu }}{{4\lambda \sigma^{2} - \left( {\lambda - 1} \right)^{2} \mu^{2} }} \hfill \\ \end{aligned} \right. $$

(A.2)

Taking the derivative of $\Pi _{\text{M1}}^{RS}$ with respect to the variable $q_{1}$, then we have:

$$ q_{1} = - \frac{1}{2}\mu \theta q_{3} - \frac{1}{2}\theta q_{2} + \frac{1}{2}a - \frac{1}{2}w_{1} $$

(A.3)

Simultaneous Eqs. (A.2) and (A.3), the quantity that the manufacturer ordered from each supplier when the reliable supplier coexists with the reliable supplier can get the order, as shown in Eq. (12).

Proof of Lemma 3

When $q_{3} > 0$, substitute $q_{3RS}^{*}$ into equal (13), then we can get

$$ \frac{{\partial \varPi_{S2}^{RS} }}{{\partial w_{2}^{2} }} = \frac{{ - 2\left( {\theta^{4} - 4} \right)^{2} \left( {\lambda + 1} \right)\sigma^{2} \mu^{2} }}{{\left( {\left( {\lambda - 1} \right)^{2} \left( {\theta^{2} - 2} \right)\mu^{2} + 2\sigma^{2} \left( {4 - \theta^{2} } \right)\lambda } \right)^{2} }} < 0 $$

(A.4)

Let $\frac{{\partial_{S2}^{RS} }}{{\partial w_{2}^{2} }} = 0$ and ${\text{w}}_{{ 2 {\text{B}}}}$ denote the optimal price, then

$$ {\text{w}}_{{ 2 {\text{B}}}} = \frac{{\left( {{\text{w}}_{ 1} \mu - {\text{c}}_{ 2} } \right)\left( {\uplambda - 1} \right)^{ 2} \left( {\uptheta^{ 2} - 2} \right)\upmu^{ 2} }}{{ 2\sigma^{ 2}\upmu\left( {\theta^{ 2} - 4} \right)\left( {\uplambda + 1} \right)}} + \frac{{\left( {\uptheta{\text{w}}_{ 1} + {\text{a}} + {\text{w}}_{ 1} } \right)\uplambda^{ 2} - {\text{a}} + {\text{w}}_{ 1} }}{{\left( {\theta { + 2}} \right)\left( {\uplambda + 1} \right)}}{ + }\frac{{\uplambda{\text{c}}_{ 2} }}{{\left( {\uplambda + 1} \right)\upmu}} $$

(A.5)

When $w_{2B} < \bar{w}_{2}$, $q_{3RS}^{*} > 0$ where $\bar{w}_{2} { = }\frac{{\left( {\theta w_{1} + a + w_{1} } \right)\lambda - a + w_{1} }}{\theta + 2}$.

(1) If $w_{1} \le \frac{a - a\lambda }{\theta \lambda + \lambda + 1},\bar{w}_{2} \le 0$, The unreliable supplier needs to subsidize manufacturer 2 in order to receive an order. (2) If $w_{1} > \frac{a - a\lambda }{\theta \lambda + \lambda + 1},\;\bar{w}_{2} { > 0}$ and $w_{2B} - \bar{w}_{2} = \frac{{\left( {\left( {\lambda - 1} \right)^{2} \left( {\theta^{2} - 2} \right)\mu^{2} + 2\sigma^{2} \lambda \left( {4 - \theta^{2} } \right)} \right)\left( {\mu w_{1} - c_{2} } \right)}}{{2\left( {\theta^{2} - 4} \right)\mu \left( {\lambda + 1} \right)\sigma^{2} }}$. Note $4\uplambda \upsigma ^{ 2} > \left( {\uplambda - 1} \right)^{ 2}\upmu^{ 2}$, we have $\left( {\lambda - 1} \right)^{2} \left( {\theta^{2} - 2} \right)\mu^{2} + 2\lambda \sigma^{2} \left( {4 - \theta^{2} } \right) > 4\lambda \sigma^{2} - \left( {\lambda - 1} \right)^{2} \mu^{2} > 0$. Let $\psi \left( \theta \right){ = }\left( {\lambda - 1} \right)^{2} \left( {\theta^{2} - 2} \right)\mu^{2} + 2\lambda \sigma^{2} \left( {4 - \theta^{2} } \right) - \left( {4\lambda \sigma^{2} - \left( {\lambda - 1} \right)^{2} \mu^{2} } \right)$. Obviously, $\psi \left( \theta \right)$ is a monotone function when $\theta \in \left[ {0,1} \right]$, it is easy to verify that $\psi \left( \theta \right) > 0$. Therefore, if $\mu w_{1} > c_{2}$, ${\text{w}}_{{ 2 {\text{B}}}} < \overline{{w_{2} }}$ and $q_{3} > 0$; if $\mu w_{1} \le c_{2}$, ${\text{w}}_{{ 2 {\text{B}}}} \ge \overline{{w_{2} }}$ and $q_{3} { = }0$.

Proof of Lemma 4

If $w_{1} > \underline{{w_{1} }}$, $\frac{{\partial \varPi_{S1}^{RS} }}{{\partial w_{1}^{2} }} = \frac{{\left( {\theta - 2} \right)\left( {\theta \mu^{2} \lambda^{2} + \left( {4\mu^{2} + 4\sigma^{2} - 2\sigma^{2} \theta } \right)\lambda + \mu^{2} \theta } \right)}}{{\left( {\lambda - 1} \right)^{2} \left( {\theta^{2} - 2} \right) \cdot \mu^{2} + 2\sigma^{2} \left( {4 - \theta^{2} } \right)\lambda }} - 1$. Note $\left( {\lambda - 1} \right)^{2} \left( {\theta^{2} - 2} \right)\mu^{2} > 2\lambda \sigma^{2} \left( {4 - \theta^{2} } \right)$, then $\frac{{\partial \varPi_{S1}^{RS} }}{{\partial w_{1}^{2} }} < 0$ 。Let $\frac{{\partial \varPi_{S1}^{RS} }}{{\partial w_{1}^{2} }} = 0$, we have

$$ w_{1B}^{C} = \frac{{\left( {\theta - 2} \right)\left( {\left( {\theta + \lambda + 1} \right)w_{2} - a\left( {\lambda^{2} - \lambda } \right) + \left( {\theta + 2} \right)c_{1} \lambda } \right)\mu^{2} + \left( {a + c_{1} } \right)\left( {\lambda - 1} \right)^{2} \left( {1 - \theta } \right)\mu^{2} + 4\sigma^{2} \lambda \left( {\theta - 2} \right)\left( {a + c_{1} } \right)}}{{2\left( {1 - \theta } \right)\left( {\lambda - 1} \right)^{2} \mu^{2} + 2\lambda \left( {\theta - 2} \right)\left( {\left( {\theta + 2} \right)\mu^{2} + 4\sigma^{2} } \right)}} $$

(1) If $\underline{{w_{1} }} < \frac{{a + c_{1} }}{2}$ and $w_{1B}^{C} > \underline{{w_{1} }}$, supplier1 has two different pricing options: set price barriers for the unreliable supplier ($w_{1RS}^{C} { = }\underline{{w_{1} }}$) or coexist with the unreliable supplier ($w_{1RS}^{C} { = }w_{1B}^{C}$). It is easy to prove that $C\varPi_{S1}^{RS} \left( {w_{1} } \right)$ is a continuous function and increases with $w_{1}$ when $w_{1} \in \left( {\underline{{w_{1} }} ,w_{1B}^{C} } \right]$. It means $C\varPi_{S1}^{RS} \left( {w_{1B}^{S} } \right) > C\varPi_{S1}^{RS} \left( \underline{{w_{1} }} \right)$ and the optimal strategy for the reliable supplier coexists with the unreliable supplier.

(2) If $\underline{{w_{1} }} < \frac{{a + c_{1} }}{2}$ and $w_{1B}^{C} \le \underline{{w_{1} }}$, the optimal wholesale price is $w_{1RS}^{C} { = }\underline{{w_{1} }}$ and the unreliable supplier will not enter the market.

(3) If $\underline{{w_{1} }} > \frac{{a + c_{1} }}{2}$, the appearance of the unreliable supplier will not affect the original monopoly position of The reliable supplier. Thus, $w_{1RS}^{C} { = }\frac{{a{ + }c_{1} }}{2}$.

Proof of Proposition 7

If $w_{2B} \ge 0$,$w_{1} \ge \widehat{{w_{1} }}$, where $\widehat{{w_{1} }} = \frac{{c_{2} \mu^{2} \left( {\lambda - 1} \right)^{2} \left( {2 - \theta^{2} } \right) + 2\sigma^{2} \left( {\theta - 2} \right)\left( {a\mu \left( {\lambda^{2} - 1} \right) + \lambda c_{2} \left( {\theta + 2} \right)} \right)}}{{\mu^{3} \left( {\lambda - 1} \right)^{2} \left( {2 - \theta^{2} } \right) + \left( {4 - 2\theta } \right)\mu \sigma^{2} \left( {\left( {\theta + 1} \right)\lambda^{2} + 1} \right)}}$.

Then,$\widehat{{w_{1} }} - \frac{{c_{2} }}{\mu } = \frac{{2\left( {\lambda + 1} \right)\left( {\theta - 2} \right)\left( {\left( {a\mu + \theta c_{2} + c_{2} } \right)\lambda - a\mu + c_{2} } \right)\sigma^{2} }}{{\mu \left( {\mu^{2} \left( {\lambda - 1} \right)^{2} \left( {2 - \theta^{2} } \right) + 2\left( {2 - \theta } \right)\sigma^{2} \left( {\left( {\theta + 1} \right)\lambda^{2} + 1} \right)} \right)}}$

$$ \widehat{{w_{1} }} - \frac{a - a\lambda }{\theta \lambda + \lambda + 1} = \frac{{\left( {\left( {\uplambda - 1} \right)^{ 2} \left( {\uptheta^{ 2} - 2} \right)\upmu^{ 2} + 2\uplambda \upsigma ^{ 2} \left( { 4-\uptheta^{ 2} } \right)} \right)\left( {\left( {{\text{a}}\upmu + {\text{c}}_{ 2}\uptheta + {\text{c}}_{ 2} } \right)\uplambda - {\text{a}}\upmu + {\text{c}}_{ 2} } \right)}}{{\mu \left( { 1+ \left( {\uptheta + 1} \right)\lambda } \right) \cdot \left( {\left( {\uplambda - 1} \right)^{ 2} \left( {\uptheta^{ 2} - 2} \right)\mu^{ 2} + 2\left( { 1+ \left( {\uptheta + 1} \right)\uplambda^{ 2} } \right)\left( {\uptheta - 2} \right)\upsigma^{ 2} } \right)}}. $$

Note $\frac{{c_{2} }}{\mu } < \frac{{a + c_{1} }}{2}$, if $w_{1} > \frac{a - a\lambda }{\theta \lambda + \lambda + 1}$ and $w_{1} > \frac{{c_{2} }}{\mu }$, substitute $w_{2RS} = w_{2B}$, $w_{2RS} = 0$ into the profit of the reliable supplier, respectively. We have

$$ {\text{w}}_{1B}^{S} = \frac{{\left( {{\text{c}}_{ 1}\upmu + {\text{c}}_{ 2} } \right)\left( {\uptheta +\uplambda + 1} \right)\upmu + 4\upsigma^{ 2} \left( {\uplambda + 1} \right)\left( {{\text{a}} + {\text{c}}_{ 1} } \right)}}{{ 2\left( {\uptheta +\uplambda + 1} \right)\upmu^{ 2} + 8\upsigma^{ 2} \left( {\uplambda + 1} \right)}},\quad w_{2RS} = w_{2B} $$

(A.6)

$$ w_{1C}^{S} = \frac{{\left( {a + c_{1} } \right)\left( {\theta - 1} \right)\left( {4 \sigma^{2} \lambda - \left( {\lambda - 1} \right)^{2} \mu^{2} } \right) - 4 \sigma^{2} \left( {a + c_{1} } \right)\lambda + \left( {\theta^{2} - 4} \right)c_{1} \lambda \mu^{2} + a\left( {\theta - 2} \right)\left( {\lambda - \lambda^{2} } \right)\mu^{2} }}{{2 \cdot \left( {\mu^{2} \left( {\theta - 1} \right)\left( {\lambda - 1} \right)^{2} + \left( {2 - \theta } \right)\lambda \left( {\left( {2 + \theta } \right)\mu^{2} + 4 \sigma^{2} } \right)} \right)}},w_{2RS} = 0 $$

(A.7)

(1) If $\lambda \le \frac{{a\mu - c_{2} }}{{\left( {a\mu + \theta c_{2} + c_{2} } \right)}}$,$\widehat{{w_{1} }} > \frac{a - a\lambda }{\theta \lambda + \lambda + 1} > \frac{{c_{2} }}{\mu }$. The pricing of the reliable supplier affects the price and order of the unreliable supplier. The reliable supplier has three pricing options ${\text{w}}_{1RS}^{S}$.

• (1) if ${\text{w}}_{1RS}^{S} > \widehat{{w_{1} }}$, the reliable supplier coexists with the unreliable supplier and $w_{2RS} > 0$. The optimal price ${\text{w}}_{1RS}^{S} {\text{ = max}}\left\{ {{\text{w}}_{1B}^{S} ,\widehat{{w_{1} }}} \right\}$.

• (2) if $\frac{a - a\lambda }{\theta \lambda + \lambda + 1}{\text{ < w}}_{1RS}^{S} < \widehat{{w_{1} }}$, the reliable supplier coexists with the unreliable supplier and $w_{2RS} = 0$. The optimal price of supplier1 is ${\text{w}}_{1RS}^{S} {\text{ = max}}\left\{ {\frac{a - a\lambda }{\theta \lambda + \lambda + 1},{ \hbox{min} }\left\{ {{\text{w}}_{1C}^{S} ,\widehat{{w_{1} }}} \right\}} \right\}$

• (3) if ${\text{w}}_{1RS}^{S} < \frac{a - a\lambda }{\theta \lambda + \lambda + 1}$, the reliable supplier monopolizes the supply market and. $w_{2RS} = 0$${\text{w}}_{1RS}^{S} {\text{ = min}}\left\{ {\frac{a - a\lambda }{\theta \lambda + \lambda + 1},\frac{{a + c_{1} }}{2}} \right\}$.

(2) If $\lambda > \frac{{a\mu - c_{2} }}{{\left( {a\mu + \theta c_{2} + c_{2} } \right)}}$, $\widehat{{w_{1} }} < \frac{a - a\lambda }{\theta \lambda + \lambda + 1} < \frac{{c_{2} }}{\mu }$. The reliable supplier has two pricing options.

• (1) If ${\text{w}}_{1B}^{S} > \frac{{c_{2} }}{\mu }$, the reliable supplier coexists with the unreliable supplier and $w_{2RS} > 0$. ${\text{w}}_{1RS}^{S} {\text{ = w}}_{1B}^{S}$;

• (2) If ${\text{w}}_{1B}^{S} \le \frac{{c_{2} }}{\mu }$ the reliable supplier monopolizes the supply market, $w_{1RS}^{S*} = \frac{{c_{2} }}{\mu }$.

Let $w_{1D}^{S} = \mathop {\arg \hbox{max} }\limits_{{w_{1} \in Z}} S\varPi_{S1}^{RS} \left( {w_{1} } \right)$, $Z = \Big\{ { \hbox{max} }\left\{ {{\text{w}}_{1B}^{S} ,\hat{w}_{1} } \right\}, {\hbox{max} }\left\{ {\frac{a - a\lambda }{\theta \lambda + \lambda + 1} , {\text{min}}\left\{ {{\text{w}}_{1C}^{S} ,\hat{w}_{1} } \right\}} \right\},{ \hbox{min} }\left\{ {\frac{a - a\lambda }{\theta \lambda + \lambda + 1},\frac{{a + c_{1} }}{2}} \right\} \Big\}$.