A comprehensive analysis of the Newsvendor model with unreliable supply

Abstract

A single-period, uncertain demand inventory model is analyzed under the assumption that the quantity ordered (produced) is a random variable. We first conduct a comprehensive analysis of the well known single period production/inventory model with random yield. Then, we extend some of the results existing in literature: our main contribution is to show that earlier results are only valid for a certain range of system parameters. Under the hypothesis that demand and the error in the quantity received from supplier are uniformly distributed, closed-form analytical solutions are obtained for all values of parameters. An analysis under normally distributed demand and error is also provided. The paper ends with an analysis of the benefit achieved by eliminating errors.

Appendices

Appendix 1. Technical details for configuration 1

Appendix 1.1

In configuration 1 we have three cost functions depending on the value of the received quantity compared with demand’s one. We have:

$$C^{1}_{1} {\left( {Q_{1} } \right)} = h{\int\limits_{Q_{A} = L_{{QA}} }^{U_{{QA}} } {{\int\limits_{x = L_{x} }^{L_{{QA}} } {{\left( {Q_{A} - x} \right)}} }} }f{\left( x \right)}g{\left( {Q_{A} } \right)}\operatorname{d} x\operatorname{d} Q_{A} $$

$$C^{2}_{1} {\left( {Q_{1} } \right)} = k.h{\int\limits_{Q_{A} = L_{{QA}} }^{U_{{QA}} } {{\int\limits_{x = U_{{QA}} }^{U_{x} } {{\left( {x - Q_{A} } \right)}} }} }f{\left( x \right)}g{\left( {Q_{A} } \right)}\operatorname{d} x\operatorname{d} Q_{A} $$

$$C^{3}_{1} {\left( {Q_{1} } \right)} = k.h{\int\limits_{Q_{A} = L_{{QA}} }^{U_{{QA}} } {{\int\limits_{x = Q_{A} }^{U_{{QA}} } {{\left( {x - Q_{A} } \right)}} }} }f{\left( x \right)}g{\left( {Q_{A} } \right)}\operatorname{d} x\operatorname{d} Q_{A} $$

$$\begin{array}{*{20}l} {{C^{2}_{1} {\left( {Q_{1} } \right)}} \hfill} & { = \hfill} & {{k.h{\int\limits_{Q_{A} = L_{{QA}} }^{U_{{QA}} } {{\int\limits_{x = U_{{QA}} }^{U_{x} } {{\left( {x - Q_{A} } \right)}} }} }f{\left( x \right)}g{\left( {Q_{A} } \right)}\operatorname{d} x\operatorname{d} Q_{A} } \hfill} \\ {{} \hfill} & { + \hfill} & {{h{\int\limits_{Q_{A} = L_{{QA}} }^{U_{{QA}} } {{\int\limits_{x = L_{{Q_{A} }} }^{Q_{A} } {{\left( {Q_{A} - x} \right)}} }} }f{\left( x \right)}g{\left( {Q_{A} } \right)}\operatorname{d} x\operatorname{d} Q_{A} } \hfill} \\ \end{array} $$

The total cost function is written as the following:

$$\begin{array}{*{20}l} {{C_{1} {\left( {Q_{1} } \right)}} \hfill} & { = \hfill} & {{C^{1}_{1} {\left( {Q_{1} } \right)} + C^{2}_{1} {\left( {Q_{1} } \right)} + C^{3}_{1} {\left( {Q_{1} } \right)}} \hfill} \\ {{} \hfill} & { = \hfill} & {{\frac{{h\left. {{\left( { - 6{\left( {k - 1} \right)}{\left( {Q_{1} - \mu _{x} } \right)}\sigma _{x} + {\sqrt 3 }{\left( {{\left( { - Q_{1} + \mu _{x} } \right)}^{2} + } \right)}} \right)}} \right)}}{{12\sigma ^{2}_{x} }}} \hfill} \\ \end{array} $$

The convexity of C ₁(Q ₁) is clear. Setting $\frac{{\partial C_{1} {\left( {Q_{1} } \right)}}}{{\partial Q_{1} }} = 0$ and solving, we get $Q^{*}_{1} = Q^{*}_{0} = \mu _{x} + \sigma _{x} {\sqrt 3 }\frac{{k - 1}}{{k + 1}}$ with an optimal cost function equal to: $C_{1} {\left( {Q^{*}_{1} } \right)} = \frac{{h{\left( {12k\sigma ^{2}_{x} + {\left( {k + 1} \right)}^{2} \sigma ^{2}_{\xi } } \right)}}}{{4{\sqrt 3 }{\left( {k + 1} \right)}\sigma _{x} }}$.

Appendix 1.2

The result presented above is valid till:

• 1. $U_{{Q_{A} }} \leqslant U_{x}$ for k≥1, so $Q_{1} + {\sqrt 3 }\sigma _{\xi } \leqslant \mu _{x} + {\sqrt 3 }\sigma _{x}$. Replacing Q ₁ by $Q^{*}_{1} = \mu _{x} + \sigma _{x} {\sqrt 3 }\frac{{k - 1}}{{k + 1}}$ and solving the last inequality, we get $\sigma _{\xi } \leqslant \frac{2}{{k + 1}}\sigma _{x}$. So, configuration 1 is defined for $\sigma _{\xi } \in {\left[ {0,\frac{{2k}}{{k + 1}}\sigma _{x} } \right]}$ if k≤1

• 2. $L_{x} \leqslant L_{{Q_{A} }}$ for k≤1, so $\mu _{x} - {\sqrt 3 }\sigma _{x} \leqslant Q_{1} - {\sqrt 3 }\sigma _{\xi }$. Again replacing Q ₁ by Q ₁* we get $\sigma _{\xi } \leqslant \frac{{2k}}{{k + 1}}\sigma _{x}$. So, configuration 1 is defined for $\sigma _{\xi } \in {\left[ {0,\frac{{2k}}{{k + 1}}\sigma _{x} } \right]}$ if k≤1

Appendix 2. Technical details for configuration 2

Appendix 2.1

For k≥1 we have:

$$C^{1}_{1} {\left( {Q_{1} } \right)} = h{\int\limits_{Q_{A} = U_{x} }^{U_{{Q_{A} }} } {{\int\limits_{x = L_{x} }^{U_{x} } {{\left( {Q_{A} - x} \right)}} }} }f{\left( x \right)}g{\left( {Q_{A} } \right)}\operatorname{d} x\operatorname{d} Q_{A} $$

$$\begin{array}{*{20}l} {{C^{2}_{1} {\left( {Q_{1} } \right)}} \hfill} & { = \hfill} & {{h{\int\limits_{Q_{A} = L_{{QA}} }^{U_{X} } {{\int\limits_{x = L_{X} }^{Q_{A} } {{\left( {Q_{A} - x} \right)}} }} }f{\left( x \right)}g{\left( {Q_{A} } \right)}\operatorname{d} x\operatorname{d} Q_{A} } \hfill} \\ {{} \hfill} & { + \hfill} & {{k.h{\int\limits_{Q_{A} = L_{{QA}} }^{U_{X} } {{\int\limits_{x = Q_{A} }^{U_{X} } {{\left( {x - Q_{A} } \right)}} }} }f{\left( x \right)}g{\left( {Q_{A} } \right)}\operatorname{d} x\operatorname{d} Q_{A} } \hfill} \\ \end{array} $$

The total cost function is written as the following:

$$C_{1} {\left( {Q_{1} } \right)} = C^{1}_{1} {\left( {Q_{1} } \right)} + C^{2}_{1} {\left( {Q_{1} } \right)} = \frac{1}{{72\sigma _{x} \sigma _{\xi } }}{\left( {h{\left( { - {\left( {k + 1} \right)}Q^{3}_{1} C^{3}_{\lambda } + {\left( {k + 1} \right)}\mu ^{2}_{x} + 3{\sqrt 3 }{\left( {k + 1} \right)}{\left( {\sigma _{x} \sigma _{\xi } } \right)}^{2} + 9\mu _{x} {\left( {{\left( {k + 1} \right)}\sigma ^{2}_{x} + 2{\left( {k - 3} \right)}\sigma _{x} \sigma _{\xi } + {\left( {k + 1} \right)}\sigma ^{2}_{\xi } } \right)} + 3{\left( {k + 1} \right)}Q^{2}_{1} {\left( {\mu _{x} + {\sqrt 3 }{\left( {\sigma _{x} + \sigma _{\xi } } \right)}} \right)} - 3Q_{1} {\left( {{\left( {k + 1} \right)}\mu ^{2}_{x} + 3{\left( {k + 1} \right)}\sigma ^{2}_{x} + 6{\left( {k - 3} \right)}\sigma _{x} \sigma _{\xi } + 3{\left( {k + 1} \right)}\sigma ^{2}_{\xi } + 2{\sqrt 3 }\mu _{x} {\left( {k + 1} \right)}{\left( {\sigma _{x} + \sigma _{\xi } } \right)}} \right)}} \right)}} \right)}$$

The second derivation of C ₁(Q ₁) is equal to $\frac{{h{\left( {k + 1} \right)}{\left[ {U_{x} - L_{{Q_{A} }} } \right]}}}{{12\sigma _{x} \sigma _{\xi } }}$ which is all the time positive since L _QA <U _x , so the convexity of C ₁(Q ₁).

Setting $\frac{{\partial C_{1} {\left( {Q_{1} } \right)}}}{{\partial Q_{1} }} = 0$ and solving we get $Q^{*}_{1} = Q^{*}_{0} + {\sqrt 3 }{\left( {{\sqrt {\sigma _{\xi } } } - {\sqrt {\frac{2}{{k + 1}}\sigma _{x} } }} \right)}^{2}$ where: $Q^{*}_{0} = \mu _{x} + \sigma _{x} {\sqrt 3 }.$

By doing the same for the case k≤1 we get:

$Q^{*}_{1} = Q^{*}_{0} - {\sqrt 3 }{\left( {{\sqrt {\sigma _{\xi } } } - {\sqrt {\frac{{2k}}{{k + 1}}\sigma _{x} } }} \right)}^{2}$ with an optimal cost function $C_{1} {\left( {Q^{*}_{1} } \right)} = {\sqrt 3 }hk{\left( {\sigma _{x} + \sigma _{\xi } } \right)} - 4\frac{{{\sqrt {2\sigma _{x} \sigma _{\xi } } }hk^{2} }}{{{\sqrt {3k{\left( {k + 1} \right)}} }}}.$

Appendix 2.2

The result obtained above is valid till:

• 1. If k≥1: L _QA ≥L _x and U _QA ≥U _x so $Q_{1} - {\sqrt 3 }\sigma _{\xi } \geqslant \mu _{x} - {\sqrt 3 }\sigma _{x}$ and $Q_{1} + {\sqrt 3 }\sigma _{\xi } \geqslant \mu _{x} + {\sqrt 3 }\sigma _{x}$. Again replacing Q ₁ by Q ₁* we get $\frac{2}{{k + 1}}\sigma _{x} \leqslant \sigma _{\xi } \leqslant \frac{{k + 1}}{2}\sigma _{x}$ and this inequality is verified since k≥1

• 2. If k≤1: by doing the same as the previous case we have $\frac{{2k}}{{k + 1}}\sigma _{x} \leqslant \sigma _{\xi } \leqslant \frac{{k + 1}}{{2k}}\sigma _{D}$. Again the inequality $\frac{{2k}}{{k + 1}}\sigma _{x} \leqslant \frac{{k + 1}}{{2k}}\sigma _{D}$ is well verified since k≤1

Appendix 3. Technical details for configuration 3

Appendix 3.1

We have:

$$C^{1}_{1} {\left( {Q_{1} } \right)} = k.h{\int\limits_{x - L_{x} }^{U_{x} } {{\int\limits_{Q_{A} = L_{{QA}} }^{L_{x} } {{\left( {x - Q_{A} } \right)}} }} }f{\left( x \right)}g{\left( {Q_{A} } \right)}\operatorname{d} Q_{A} \operatorname{d} x$$

$$C^{2}_{1} {\left( {Q_{1} } \right)} = h{\int\limits_{x = L_{x} }^{Ux} {{\int\limits_{Q_{A} = U_{x} }^{U_{{QA}} } {{\left( {Q_{A} - x} \right)}} }} }f{\left( x \right)}g{\left( {Q_{A} } \right)}\operatorname{d} Q_{A} \operatorname{d} x$$

$$\begin{array}{*{20}l} {{C^{3}_{1} {\left( {Q_{1} } \right)}} \hfill} & { = \hfill} & {{k.h{\int\limits_{x = L_{x} }^{U_{x} } {{\int\limits_{Q_{A} = L_{x} }^x {{\left( {x - Q_{A} } \right)}} }} }f{\left( x \right)}g{\left( {Q_{A} } \right)}\operatorname{d} Q_{A} \operatorname{d} x} \hfill} \\ {{} \hfill} & { + \hfill} & {{h{\int\limits_{x = L_{x} }^{U_{x} } {{\int\limits_{Q_{A} = x}^{U_{x} } {{\left( {Q_{A} - x} \right)}} }} }f{\left( x \right)}g{\left( {Q_{A} } \right)}\operatorname{d} Q_{A} \operatorname{d} x} \hfill} \\ \end{array} $$

The total cost function is written as the following:

$$\begin{array}{*{20}l} {{C_{1} {\left( {Q_{1} } \right)}} \hfill} & { = \hfill} & {{C^{1}_{1} {\left( {Q_{1} } \right)} + C^{2}_{1} {\left( {Q_{1} } \right)} + C^{3}_{1} {\left( {Q_{1} } \right)}} \hfill} \\ {{} \hfill} & { = \hfill} & {{\frac{{h{\left( { - 6{\left( {k - 1} \right)}{\left( {Q_{1} - \mu _{x} } \right)}\sigma _{\xi } + {\sqrt 3 }{\left( {k + 1} \right)}{\left( {{\left( { - Q_{1} + \mu _{x} } \right)}^{2} + \sigma ^{2}_{x} + 3\sigma ^{2}_{\xi } } \right)}} \right)}}}{{12\sigma _{\xi } }}} \hfill} \\ \end{array} $$

We can easily show that C ₁(Q ₁) is convex, and by setting $\frac{{\partial C_{1} {\left( {Q_{1} } \right)}}}{{\partial Q_{1} }} = 0$ and solving, we get $Q^{*}_{1} = {\left[ {\mu _{x} + \sigma _{\xi } {\sqrt 3 }\frac{{k - 1}}{{k + 1}}} \right]}$.

With an optimal cost function equal to: $C_{1} {\left( {Q^{*}_{1} } \right)} = \frac{{h{\left( {{\left( {k + 1} \right)}^{2} \sigma ^{2}_{x} + 12k\sigma ^{2}_{\xi } } \right)}}}{{4{\sqrt 3 }{\left( {k + 1} \right)}\sigma _{\xi } }}$.

Appendix 3.2

The result obtained above is valid till:

• 1. If k≥1 we have $Q_{1} + {\sqrt 3 }\sigma _{\xi } \geqslant \mu _{x} + {\sqrt 3 }\sigma _{D}$ and $Q_{1} - {\sqrt 3 }\sigma _{\xi } \geqslant 0$. So by replacing Q ₁ by Q ₁* we get $\frac{{k + 1}}{2}\sigma _{x} \leqslant \sigma _{\xi } \leqslant \frac{{k + 1}}{{{\sqrt {12} }}}\mu _{x}$

• 2. If k≤1 we have $Q_{1} - {\sqrt 3 }\sigma _{\xi } \leqslant \mu _{x} - {\sqrt 3 }\sigma _{x}$ and $Q_{1} - {\sqrt 3 }\sigma _{\xi } \geqslant 0$. Again, by replacing Q ₁ by Q ₁* we get $\frac{{k + 1}}{{2k}}\sigma _{x} \leqslant \sigma _{\xi } \leqslant \frac{{k + 1}}{{{\sqrt {12} }}}\mu _{x}$

For k≥1, it is easy to verify that $\frac{{k + 1}}{2}\sigma _{x} \leqslant \frac{{k + 1}}{{{\sqrt {12} }}}\mu _{x}$ since $L_{x} = \mu _{R} - {\sqrt 3 }\sigma _{x}$ is positive. $\frac{{k + 1}}{{2k}}\sigma _{x} \leqslant \frac{{k + 1}}{{{\sqrt {12} }}}\mu _{x}$ is verified for values of $CV_{x} = \frac{{\sigma _{x} }}{{\mu _{x} }}$ such that $CV_{x} \leqslant \frac{k}{{{\sqrt 3 }}}$, otherwise Configuration 4 does not exist and the maximum value that can take x _i is between $\frac{{2k}}{{k + 1}}$ and $\frac{{k + 1}}{{2k}}$ (Configuration 2) is positive.

Appendix 4. Extension to the case with initial inventory

In this appendix, we consider the case of multiplicative errors with an initial inventory I. By following the methodology developed in this paper, we extend our model and derive the optimal policy for each configuration.

If an initial inventory is taken into account, we show that that the ordering quantity in all configurations, except configuration 1, is a non-linear function of the initial inventory. The following result summarizes the overall optimal policy:

Result 7

For a given vector (μ _x , σ _x , k, I), we distinguish two cases: Case A where $L_{{Q_{A} }} \leqslant L_{x}$ in the second configuration, i.e. $k \leqslant \frac{{L_{x} + 2U_{x} - 3I}}{{2L_{x} + U_{x} - 3I}}$ and Case B where $U_{{Q_{A} }} \geqslant U_{x}$ in the second configuration i.e. $k \geqslant \frac{{L_{x} + 2U_{x} - 3I}}{{2L_{x} + U_{x} - 3I}}$. Depending on system parameters, in both cases, 1,2 or 3 of the configurations presented throughout the paper may be observed. The expression of the optimal quantity to order for each configuration as well as the critical values of σ _ij can be determined by using the three steps approach described in Section 3.1:

Conf.	Interval of $\sigma _{\gamma }$	Q 1*
Conf. 1	[0,σ 12]	$\frac{{\mu _{\gamma } }}{{\mu ^{2}_{\gamma } + \sigma ^{2}_{\gamma } }}{\left[ {Q_{0} * - i} \right]}$
Conf. 2	[σ 12, σ 23]	Q 1* is obtained by solving $aQ^{{*3}}_{1} + bQ^{{*2}}_{1} + c = 0$
Conf. 3	${\left[ {\sigma _{{23}} ,\sigma _{{\gamma \;\max }} } \right]}$	$\frac{{{\sqrt {{\left( {k + 1} \right)}{\left( {{\left( {\mu _{x} - I} \right)}^{2} + \sigma ^{2}_{x} } \right)}} }}}{{{\sqrt { - 2{\sqrt 3 }{\left( {k - 1} \right)}\mu _{\gamma } \sigma _{\gamma } + {\left( {k + 1} \right)}{\left( {\mu ^{2}_{\gamma } + 3\sigma ^{2}_{\gamma } } \right)}} }}}$

$\begin{array}{*{20}l} {{{\text{Case}}\;A:k \leqslant \frac{{L_{x} + 2U_{x} - 3I}}{{2L_{x} + U_{x} - 3I}}} \hfill} \\ {{\sigma _{{12}} = \frac{{{\sqrt {{\left( {3{\left( {Q^{*}_{0} - I} \right)} - 2L_{x} } \right)}{\left( {2L_{x} + Q^{*}_{0} + I} \right)} - 16IQ^{*}_{0} } } - {\sqrt 3 }{\left( {Q^{*}_{0} - I} \right)}}}{{2{\left( {L_{x} - I} \right)}}}\mu _{\gamma } } \hfill} \\ {{\sigma _{{23}} = \frac{{{\left( {d - e} \right)} - {\sqrt {{\left( {d - e} \right)}^{2} - e^{2} } }}}{{{\sqrt 3 }e}}\mu _{\gamma } } \hfill} \\ {{\sigma _{{\gamma \;\max }} = \frac{{\mu _{\gamma } }}{{{\sqrt 3 }}}} \hfill} \\ \end{array}$

$\begin{array}{*{20}l} {{a = 2{\left( {k + 1} \right)}U^{3}_{\gamma } } \hfill} \\ {{b = 3kU_{x} {\left( {L^{2}_{\gamma } - U^{2}_{\gamma } } \right)} - 3L_{x} {\left( {U^{2}_{\gamma } + kL^{2}_{\gamma } } \right)} + 3{\left( {k + 1} \right)}IU^{2}_{\gamma } } \hfill} \\ {{c = {\left( {k + 1} \right)}{\left( {L_{x} - I} \right)}^{3} } \hfill} \\ {{d = k{\left( {U_{x} - I} \right)}^{2} } \hfill} \\ {{e = {\left( {{\sqrt 3 }{\left( {\mu _{x} - I} \right)} + \sigma _{x} } \right)}{\left( {k + 1} \right)}\sigma _{x} } \hfill} \\ \end{array}$

$\begin{array}{*{20}l} {{{\text{Case}}\;B:k \geqslant \frac{{L_{x} + 2U_{x} - 3I}}{{2L_{x} + U_{x} - 3I}}} \hfill} \\ {{\sigma _{{12}} = \frac{{{\sqrt 3 }{\left( {Q^{*}_{0} - I} \right)} - {\sqrt {{\left( {3{\left( {Q^{*}_{0} - I} \right)} - 2U_{x} } \right)}{\left( {2U_{x} + Q^{*}_{0} + I} \right)} - 16IQ^{*}_{0} } }}}{{2{\left( {U_{x} - I} \right)}}}\mu _{\gamma } } \hfill} \\ {{\sigma _{{23}} = \frac{{{\left( {d + e} \right)} - {\sqrt {{\left( {d + e} \right)}^{2} - e^{2} } }}}{{{\sqrt 3 }e}}\mu _{\gamma } } \hfill} \\ {{\sigma _{{\gamma \;\max }} = \frac{{\mu _{\gamma } }}{{{\sqrt 3 }}}} \hfill} \\ \end{array}$

$\begin{array}{*{20}l} {{a = 2{\left( {k + 1} \right)}L^{3}_{\gamma } } \hfill} \\ {{b = 3L_{x} {\left( {U^{2}_{\gamma } - L^{2}_{\gamma } } \right)} - 3U_{x} {\left( {U^{2}_{\gamma } + kL^{2}_{\gamma } } \right)} - 3L^{2}_{\gamma } {\left( {k + 1} \right)}I} \hfill} \\ {{c = {\left( {k + 1} \right)}{\left( {U_{x} - I} \right)}^{3} } \hfill} \\ {{d = {\left( {L_{x} - I} \right)}^{2} } \hfill} \\ {{e = {\left( {{\sqrt 3 }{\left( {\mu _{x} - I} \right)} - \sigma _{x} } \right)}{\left( {k + 1} \right)}\sigma _{x} } \hfill} \\ \end{array}$

For each critical value σ _ij expressed in the table above, a condition on I should be satisfied to assure σ _Ij ∈R ⁺ and as a consequence, Configuration j exists. This condition is expressed in the form of an interval of variation of I, as represented in the table below:

Case A:$k \leqslant \frac{{L_{x} + 2U_{x} - 3I}}{{2L_{x} + U_{x} - 3I}}$		Case B:$k \geqslant \frac{{L_{x} + 2U_{x} - 3I}}{{2L_{x} + U_{x} - 3I}}$
Interval of I	Possible conf.	Interval of I	Possible conf.
${\left[ {0,\mu _{x} - \sigma _{x} \frac{{{\sqrt 3 } + {\sqrt {{\left( {3 - k} \right)}{\left( {k + 1} \right)}} }}}{k}} \right]}$	1–2–3	[0, L x ]	1–2–3
${\left[ {0,\frac{1}{3}{\left( {2L_{x} + Q^{*}_{0} } \right)}} \right]}$	1–2	[0, 3Q 0*−2U x ]	1–2
[0,Q 0*]	1	[0, Q 0*]	1
[Q 0*,+∞]	Do not order	[Q 0*,+∞]	Do not order

Remarks

• By setting I=0 we find the results pertaining to the multiplicative errors case which are developed in this paper

• For Case A, an additional condition on model parameters should be made to ensure that the lower boundary of the received quantity reaches zero in Configuration 3 and as a consequence to ensure that Configuration 3 exists. This condition is as follows: $2 - \frac{{3L_{x} {\left( {L_{x} + U_{x} } \right)}}}{{L^{2}_{x} + L_{x} U_{x} + U^{2}_{x} }} \leqslant k \leqslant 3$

• For Case B, an additional condition should be made to ensure the existence of Configuration 2 and also to assure that 3Q ₀*−2U _x ≥0. This condition is as follows: $k \geqslant 2 - 3\frac{{L_{x} }}{{U_{x} }}$

• For the particular case considered in Inderfurth (2004) (L _x =0 and $L_{\gamma } = 0$), our result confirm the optimal policy provided by the author:

  $$Q^{*}_{1} = \left\{ {\begin{array}{*{20}l} {{\frac{1}{{U_{\gamma } }}{\sqrt {\frac{{{\left( {k + 1} \right)}{\left( {U_{x} - I} \right)}^{3} }}{{3U_{x} }}} }\;{\text{if}}\;I \in {\left[ {0,3Q^{*}_{0} - 2U_{x} } \right]}} \hfill} \\ {{\frac{{\mu _{\gamma } }}{{\mu ^{2}_{\gamma } + \sigma ^{2}_{\gamma } }}{\left[ {Q^{*}_{0} - I} \right]}\;{\text{if}}\;I \in {\left[ {3Q^{*}_{0} - 2U_{x} ,Q^{*}_{0} } \right]}} \hfill} \\ {{0\;{\text{if}}\;I \in {\left[ {Q^{*}_{0} , + \infty } \right]}} \hfill} \\ \end{array} } \right.$$