Appendix 1
For ARMA (1,1), the demand for period t is:
$$ {D_t}=\delta +\phi {D_t}_{-1 }+{\varepsilon_t}-\theta {\varepsilon_t}_{-1 } $$
(i)
The demand for period (t + 1) is,
$$ \begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {{D_t}_{+1 }=\delta +\phi {D_t}+{\varepsilon_t}_{+1 }-\theta {\varepsilon_t}} \\ {=\delta +\phi \delta +{\phi^2}{D_{t-1 }}+\phi {\varepsilon_t}-\phi \theta {\varepsilon_{t-1 }}+{\varepsilon_{t+1 }}-\theta {\varepsilon_t}} \\ {=\left( {1+\phi } \right)\delta +{\phi^2}Dt-1+{\varepsilon_{t+1 }}+\left( {\phi -\theta } \right){\varepsilon_t}-\phi \theta {\varepsilon_{t-1 }}} \\ \end{array}} & {\left[ {\mathrm{Substituting}\;\left( \mathrm{i} \right)} \right]} \\ \end{array}} \\ \end{array} $$
Similarly, the demand for period (t + 2) is:
$$ {D_t}_{+2 }=\left( {1+\phi +{\phi^2}} \right)\delta +{\phi^3}{D_t}_{+1 }+{\varepsilon_{t+2 }}-\left( {\phi -\theta } \right){\varepsilon_{t+1 }}-\phi \left( {\phi -\theta } \right){\varepsilon_t}-{\phi^2}\theta {\varepsilon_{t-1 }} $$
(ii)
Similarly, the demand for period (t + i) is:
$$ \begin{array}{*{20}c} {{D_t}_{+1 }=\left( {1+\phi +{\phi^2}+\ldots +{\phi^i}} \right)\delta +{\phi^{i+1 }}{D_{t-1 }}} \\ {\quad \quad \quad \quad \quad \quad + {\varepsilon_{t+i }}-\left( {\phi -\theta } \right){\varepsilon_{t+i-1 }}-\phi \left( {\phi -\theta } \right){\varepsilon_{t+i-2 }}\ldots -{\phi^i}\theta {\varepsilon_t}_{-1 }} \\ \end{array} $$
(iii)
Thus, the mean demand is given by,
$$ {{\hat{D}}_{{t + 1}}} = \left( {1 + \phi + {{\phi }^{2}} + \ldots + {{\phi }^{i}}} \right)\delta + {{\phi }^{{i + 1}}}{{D}_{{t - 1}}} $$
(iv)
(Since the mean of error variable terms are zero)
Appendix 2
Proceeding in the same way as in Appendix 1, i.e. substituting the expression for demand D
t
in the expression for demand D
t + 1, then substituting the expression for demand D
t + 1 in the expression for demand D
t + 2 and so on we get the following expressions.
Demand at period t is:
$$ {D_t}=\delta +{\phi_1}{D_t}_{-1 }+{\phi_2}{D_t}_{-2 }+{\varepsilon_t}-\theta {\varepsilon_t}_{-1 } $$
(i)
Substituting (i) in the expression for D
t + 1 we get,
$$ {D_{t+1 }}=\left( {1+{\phi_1}} \right)\delta +{\phi_1}{\phi_2}{D_{t-2 }}+{\phi_2}{D_{t-1 }}+\phi_1^2{D_{t-1 }}+{\varepsilon_{t+1 }}-\left( {\phi -\theta } \right){\varepsilon_t}-{\phi_1}\theta {\varepsilon_{t+1 }} $$
(ii)
Substituting (ii) in the expression for D
t + 2 we get,
$$ \begin{array}{*{20}c} {{D_{t+2 }}=\left( {1+{\phi_1}+\phi_1^2} \right)\delta +\phi_1^2{\phi_2}{D_{t-2 }}+{\phi_1}{\phi_2}{D_{t-1 }}+{\phi_2}{D_t}+\phi_1^3{D_{t-1 }}} \\ {\quad \quad \quad \quad \quad \quad \quad \quad +{\varepsilon_{t+2 }}-\left( {{\phi_1}-\theta } \right){\varepsilon_{t+1 }}-{\phi_1}\left( {{\phi_1}-\theta } \right){\varepsilon_t}-\phi_1^2\theta {\varepsilon_{t-1 }}} \\ \end{array} $$
(iii)
Thus, the generalized expression for demand at period (t + i) is given by,
$$ \begin{array}{*{20}c} {{D_{{t+\mathrm{i}}}}=\left( {1+\phi +{\phi^2}+\ldots +{\phi^i}} \right)\delta +{\phi_2}\sum\limits_{j=2}^{i+2 } {\phi_1^{j-2 }} {D_{t+i}}_{-j }+{\phi^{i+1 }}{D_{t-1 }}} \\ {\quad \quad \quad \quad \quad +{\varepsilon_{t+i }}-\left( {\phi -\theta } \right){\varepsilon_{t+i-1 }}-\phi \left( {\phi -\theta } \right){\varepsilon_{t+i-2 }}\ldots -{\phi^i}\theta {\varepsilon_{t-1 }}} \\ \end{array} $$
(iv)
Thus, the expression for mean demand is,
$$ {{\widehat{D}}_{t+2 }}=\left( {1+{\phi_1}+\phi_1^2+\ldots +\phi_1^i} \right)\delta +{\phi_2}\sum\limits_{j=2}^{i+2 } {\phi_1^{j-2 }} {D_{{t+\mathrm{i}}}}_{-j }+{\phi^{i+1 }}{D_{t-1 }} $$
(v)
(Since the mean of error variable terms is zero and ε
t-1
is a constant term)
Appendix 3
Now we can derive the following expressions
$$ \begin{array}{*{20}c} {\sum\limits_{i=0}^{L-1 } {\left( {1-{\phi_1}^{i+1 }} \right)} } \\ {=\left( {1-{\phi_1}} \right)+\left( {1-{\phi_1}^2} \right)+\left( {1-{\phi_1}^3} \right)+\ldots +\left( {1-{\phi_1}^L} \right)} \\ {=(1+1+1+\ldots Lterms)-{\phi_1}\left( {1+{\phi_1}+{\phi_1}^2+\ldots +{\phi_1}{{{^L}}^{-1 }}} \right)} \\ {=\left( {L-{\phi_1}\frac{{\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}} \right)} \\ \end{array} $$
(i)
$$ \begin{array}{*{20}c} {\sum\limits_{i=0}^{L-1 } {{\phi_1}^{i+1 }{D_{t-1 }}} } \\ {={D_{t-1 }}\left[ {{\phi_1}+{\phi_1}^2+\ldots +{\phi_1}^L} \right]} \\ {={D_{t-1 }}{\phi_1}\left( {1+{\phi_1}+{\phi_1}^2+\ldots +{\phi_1}{{{^L}}^{-1 }}} \right)} \\ {=\frac{{{\phi_1}\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}{D_{t-1 }}} \\ \end{array} $$
(ii)
$$ \begin{array}{*{20}c} {\sum\limits_{i=0}^{L-1 } {\sum\limits_{k=2}^p {{\phi_k}\sum\limits_{j=k}^{i+k } {\phi_1^{j-k }} } } {D_{t+i}}_{-j }} \\ {=\sum\limits_{k=2}^p {{\phi_k}} \left[ {\sum\limits_{j=k}^k {\phi_1^{j-k }{D_t}_{-j }+} \sum\limits_{j=k}^{k+1 } {\phi_1^{j-k }{D_t}{{{_{-j}}}_{+1 }}+} \sum\limits_{j=k}^{k+2 } {\phi_1^{j-k }{D_t}{{{_{-j}}}_{+2 }}+} \ldots +\sum\limits_{j=k}^{k+L-1 } {\phi_1^{j-k }{D_t}{{{_{-j}}}_{+L-1 }}} } \right]} \\ {=\sum\limits_{k=2}^p {{\phi_k}} \left[ {{D_t}_{-k }+{D_t}_{-k+1 }+{\phi_1}{D_t}_{-k }+{D_t}_{-k+2 }+{\phi_1}{D_t}_{-k+1 }+\phi_1^2{D_t}_{-k }+\ldots +{D_t}{{{_{-k}}}_{+L-1 }}+{\phi_1}{D_t}{{{_{-k}}}_{+L-2 }}+\phi_1^2{D_t}{{{_{-k}}}_{+L-3 }}+\ldots +\phi_1^{L-1 }{D_t}_{-k }} \right]} \\ {=\sum\limits_{k=2}^p {{\phi_k}} \left[ {{D_t}_{-k}\left( {1+{\phi_1}+\phi_1^2+\ldots +\phi_1^{L-1 }} \right)+{D_t}{{{_{-k}}}_{+1 }}\left( {1+{\phi_1}+\phi_1^2+\ldots +\phi_1^{L-2 }} \right)+{D_t}{{{_{-k}}}_{+2 }}\left( {1+{\phi_1}+\phi_1^2+\ldots +\phi_1^{L-3 }} \right)+....+{D_t}{{{_{-k}}}_{+L-1 }}} \right]} \\ {=\sum\limits_{k=2}^p {{\phi_k}} \left[ {{D_t}_{-k}\frac{{\left( {1-\phi_1^L} \right)}}{{\left( {1-{\phi_1}} \right)}}+{D_t}_{-k+1}\frac{{\left( {1-\phi_1^{L-1 }} \right)}}{{\left( {1-{\phi_1}} \right)}}+{D_t}_{-k+2}\frac{{(1-\phi_1^{L-2 })}}{{(1-{\phi_1})}}+\ldots +{D_t}_{-k+L-1}\frac{{\left( {1-{\phi_1}} \right)}}{{\left( {1-{\phi_1}} \right)}}} \right]} \\ {=\frac{1}{{\left( {1-{\phi_1}} \right)}}\sum\limits_{k=2}^p {{\phi_k}} \left[ {{D_t}_{-k}\left( {1-\phi_1^L} \right)+{D_t}{{{_{-k}}}_{+1 }}\left( {1-\phi_1^{L-1 }} \right)+{D_t}{{{_{-k}}}_{+2 }}\left( {1-\phi_1^{L-2 }} \right)+\ldots +{D_t}{{{_{-k}}}_{+L}}_{-1}\left( {1-\phi_1^{{L-\left( {L-1} \right)}}} \right)} \right]} \\ {=\frac{1}{{\left( {1-{\phi_1}} \right)}}\sum\limits_{k=2}^p {{\phi_k}} \sum\limits_{j=0}^{L-1 } {{D_t}_{-k+j}\left( {1-\phi_1^{L-j }} \right)} } \\ \end{array} $$
(iii)
Thus, \( \widehat{D}_t^L \)
$$ \begin{array}{*{20}c} {=\sum\limits_{i=0}^{L-1 } {\left[ {\frac{{\left( {1-\phi_1^{i+1 }} \right)\left( {1-\sum\limits_{j=1}^p {{\phi_j}} } \right)}}{{\left( {1-{\phi_1}} \right)}}+\sum\limits_{k=2}^p {{\phi_k}} \sum\limits_{j=k}^{i+k } {\phi_1^{j-k }} {D_{t+i}}_{-j }+{\phi^{i+1 }}{D_{t-1 }}} \right]} } \\ {=\frac{{{\mu_d}\left( {1-\sum\limits_{j=1}^p {{\phi_j}} } \right)}}{{\left( {1-{\phi_1}} \right)}}\left( {L-{\phi_1}\frac{{\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}} \right)+\frac{{{\phi_1}\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}{D_{t-1 }}} \\ {+\frac{1}{{\left( {1-{\phi_1}} \right)}}\sum\limits_{k=2}^p {{\phi_k}} \sum\limits_{j=0}^{L-1 } {{D_{t-k}}_{+j}\left( {1-\phi_1^{L-j }} \right)} } \\ \end{array} $$
Appendix 4
4.1 Calculation of variance of lead time demand forecast error
By definition, the variance is given by,
$$ {{\left( {{{\widehat{\sigma}}_t}^L} \right)}^2}=\mathrm{VAR}\left( {{D_t}^L-{{\widehat{D}}_t}^L} \right) $$
(11)
Where,
$$ \begin{array}{*{20}{c}} {{{D}_{t}}^{L} - {{{\hat{D}}}_{t}}^{L} = \left( {{{D}_{t}} - {{{\hat{D}}}_{t}}} \right) + \left( {{{D}_{t}} + 1 - {{{\hat{D}}}_{{t + 1}}}} \right) + \ldots + \left( {{{D}_{{t + L - 1}}} - \widehat{{{{D}_{t}}_{{ + L - 1}}}}} \right)} \\ { = \left( {{{e}_{t}} + {{e}_{{t + 1}}} + \ldots + {{e}_{{t + L - 1}}}} \right) = \sum\limits_{{i = 0}}^{{L - 1}} {{{e}_{{t + i}}}} } \\\end{array} $$
Thus,
$$ \begin{array}{*{20}c} {{e_{t+i }}={D_{t+i }}-{{\widehat{D}}_{t+i }}} \hfill \\ {={\varepsilon_{t+i }}+\left( {{\phi_1}-{\theta_1}} \right){\varepsilon_{t+i-1 }}+\left( {\phi_1^2-\phi 1{\theta_1}-{\theta_2}} \right){\varepsilon_{t+i-2 }}+\left( {\phi_1^3-\phi_1^2{\theta_1}-\phi 1{\theta_2}-{\theta_3}} \right){\varepsilon_{t+i-3 }}+\ldots } \hfill \\ \end{array} $$
Thus,
$$ \begin{array}{*{20}c} {{{{\left( {{{\widehat{\sigma}}_t}^L} \right)}}^2}=\mathrm{VAR}\left( {{D_t}^L-{{\widehat{D}}_t}^L} \right)} \hfill \\ {=\mathrm{VAR}\left( {{\varepsilon_{t+i }}+\left( {{\phi_1}-{\theta_1}} \right){\varepsilon_{t+i-1 }}+\left( {\phi_1^2-{\phi_1}{\theta_1}-{\theta_2}} \right){\varepsilon_{t+i-2 }}+\left( {\phi_1^3-\phi_1^2{\theta_1}-\phi 1{\theta_2}-{\theta_3}} \right){\varepsilon_{t+i-3 }}+\ldots } \right)} \hfill \\ {={\sigma^2}\left[ {1+\left( {{\phi_1}-{\theta_1}} \right)+\left( {\phi_1^2-{\phi_1}{\theta_1}-{\theta_2}} \right)+\left( {\phi_1^3-\phi_1^2{\theta_1}-{\phi_1}{\theta_2}-{\theta_3}} \right)+\ldots } \right]} \hfill \\ \end{array} $$
(12)
This is a constant, independent of time t.
Having found both \( {{\widehat{\sigma}}_t}^L \) and \( {{\widehat{D}}_t}^L \), we can now find the expression for order quantity various replenishment policies.
Appendix 5
$$ \begin{array}{*{20}c} {{q_t}={S_t}-{S_{t-1 }}+{D_{t-1 }}=\widehat{{{D_t}^L}}-\widehat{{{D_{t-1}}^L}}+{D_{t-1 }}} \\ {=\left[ {\frac{{{\mu_d}\left( {1-\sum\limits_{j=1}^p {{\phi_j}} } \right)}}{{\left( {1-{\phi_1}} \right)}}\left( {L-{\phi_1}\frac{{\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}} \right)+\frac{{{\phi_1}\left( {1-\phi {1^L}} \right)}}{{\left( {1-{\phi_1}} \right)}}{D_{t-1 }}+\frac{1}{{\left( {1-{\phi_1}} \right)}}\sum\limits_{k=2}^p {{\phi_k}} \sum\limits_{j=0}^{L-1 } {{D_{t-k}}_{+j}\left( {1-\phi_1^{L-j }} \right)} } \right]} \\ {-\left[ {\frac{{{\mu_d}\left( {1-\sum\limits_{j=1}^p {{\phi_j}} } \right)}}{{\left( {1-{\phi_1}} \right)}}\left( {L-{\phi_1}\frac{{\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}} \right)+\frac{{{\phi_1}\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}{D_{t-2 }}+\frac{1}{{\left( {1-{\phi_1}} \right)}}\sum\limits_{k=2}^p {{\phi_k}} \sum\limits_{j=0}^{L-1 } {{D_{t-k}}_{+j-1}\left( {1-\phi_1^{L-j }} \right)} } \right]} \\ {\quad \quad \quad \quad +{D_{t-1 }}} \\ {=\frac{{\left( {1-{\phi_1}^{L+1 }} \right)}}{{\left( {1-{\phi_1}} \right)}}{D_{t-1 }}-\frac{{{\phi_1}\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}{D_{t-2 }}+\frac{1}{{\left( {1-{\phi_1}} \right)}}\sum\limits_{k=2}^p {{\phi_k}} \sum\limits_{j=0}^{L-1 } {{D_{t-k}}_{+j}\left( {1-\phi_1^{L-j }} \right)} } \\ {-\frac{1}{{\left( {1-{\phi_1}} \right)}}\sum\limits_{k=2}^p {{\phi_k}} \sum\limits_{j=0}^{L-1 } {{D_{t-k}}_{+j-1}\left( {1-\phi_1^{L-j }} \right)} } \\ \end{array} $$
(i)
Thus, the required variance of order quantity:
$$ \begin{array}{*{20}c} {\mathrm{VAR}\left( {{q_t}} \right)} \\ {=\mathrm{VAR}\left( {\frac{{{\phi_1}\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}{D_{t-1 }}} \right)+\mathrm{VAR}\left( {\frac{{{\phi_1}\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}{D_{t-2 }}} \right)} \\ {+\mathrm{VAR}\left( {\frac{1}{{\left( {1-{\phi_1}} \right)}}\sum\limits_{k=2}^p {{\phi_k}} \sum\limits_{j=0}^{L-1 } {{D_{t-k}}_{+j}\left( {1-\phi_1^{L-j }} \right)} } \right)} \\ {+\mathrm{VAR}\left( {\frac{1}{{\left( {1-{\phi_1}} \right)}}\sum\limits_{k=2}^p {{\phi_k}} \sum\limits_{j=0}^{L-1 } {{D_{t-k}}_{+j-1 }(1-\phi_1^{L-j })} } \right)} \\ {-2\mathrm{COV}\left( {\frac{{\left( {1-{\phi_1}^{L+1 }} \right)}}{{\left( {1-{\phi_1}} \right)}}{D_{t-1 }},\frac{{{\phi_1}\left( {1-\phi {1^L}} \right)}}{{\left( {1-{\phi_1}} \right)}}{D_{t-2 }}} \right)} \\ {+2\mathrm{COV}\left( {\frac{{\left( {1-{\phi_1}^{L+1 }} \right)}}{{\left( {1-{\phi_1}} \right)}}{D_{t-1 }},\frac{1}{{\left( {1-{\phi_1}} \right)}}\sum\limits_{k=2}^p {{\phi_k}} \sum\limits_{j=0}^{L-1 } {{D_{t-k}}_{+j }(1-\phi_1^{L-j })} } \right)} \\ {-2\mathrm{COV}\left( {\frac{{\left( {1-{\phi_1}^{L+1 }} \right)}}{{\left( {1-{\phi_1}} \right)}}{D_{t-1 }},\frac{1}{{\left( {1-{\phi_1}} \right)}}\sum\limits_{k=2}^p {{\phi_k}} \sum\limits_{j=0}^{L-1 } {{D_{t-k}}_{+j-1}\left( {1-\phi_1^{L-j }} \right)} } \right)} \\ {-2\mathrm{COV}\left( {\frac{{{\phi_1}\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}{D_{t-2 }},\frac{1}{{\left( {1-{\phi_1}} \right)}}\sum\limits_{k=2}^p {{\phi_k}} \sum\limits_{j=0}^{L-1 } {{D_{t-k}}_{+j}\left( {1-\phi_1^{L-j }} \right)} } \right)} \\ {+2\mathrm{COV}\left( {\frac{{{\phi_1}\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}{D_{t-2 }},\frac{1}{{\left( {1-{\phi_1}} \right)}}\sum\limits_{k=2}^p {{\phi_k}} \sum\limits_{j=0}^{L-1 } {{D_{t-k}}_{+j-1}\left( {1-\phi_1^{L-j }} \right)} } \right)} \\ {-2\mathrm{COV}\left( {\frac{1}{{\left( {1-{\phi_1}} \right)}}\sum\limits_{k=2}^p {{\phi_k}} \sum\limits_{j=0}^{L-1 } {{D_{t-k}}_{+j}\left( {1-\phi_1^{L-j }} \right)}, \frac{1}{{\left( {1-{\phi_1}} \right)}}\sum\limits_{k=2}^p {{\phi_k}} \sum\limits_{j=0}^{L-1 } {{D_{t-k}}_{+j-1}\left( {1-\phi_1^{L-j }} \right)} } \right)} \\ \end{array} $$
(ii)
Treating individual terms of expression (ii), we get the following expressions.
$$ \mathrm{VAR}\left( {\frac{{\left( {1-{\phi_1}^{L+1 }} \right)}}{{\left( {1-{\phi_1}} \right)}}{D_{t-1 }}} \right)={{\left( {\frac{{\left( {1-{\phi_1}^{L+1 }} \right)}}{{\left( {1-{\phi_1}} \right)}}} \right)}^{{^2}}}\sigma_d^2 $$
(iii)
$$ \mathrm{VAR}\left( {\frac{{{\phi_1}\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}{D_{t-2 }}} \right)={{\left( {\frac{{{\phi_1}\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}} \right)}^{{^2}}}\sigma_d^2 $$
(iv)
$$ \begin{array}{*{20}c} {\mathrm{VAR}\left( {\frac{1}{{\left( {1-{\phi_1}} \right)}}\sum\limits_{k=2}^p {{\phi_k}} \sum\limits_{j=0}^{L-1 } {{D_{t-k}}_{+j}\left( {1-\phi_1^{L-j }} \right)} } \right)} \hfill \\ {\quad =\frac{1}{{{{{\left( {1-{\phi_1}} \right)}}^2}}}\sum\limits_{k=2}^p {\phi_k^2} \left[ {L-2{\phi_1}\frac{{\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}+\frac{{\phi {1^2}\left( {1-{\phi_1}^{2L }} \right)}}{{\left( {1-\phi_1^2} \right)}}} \right]\mathrm{VAR}\left( {\sum\limits_{k=2}^p {\sum\limits_{j=0}^{L-1 } {{D_{t-k}}_{+j }} } } \right)} \hfill \\ \end{array} $$
[Since,
$$ \begin{array}{*{20}c} {\sum\limits_{j=0}^{L-1 } {\left( {1-\phi_1^{L-j }} \right)} =L-2{\phi_1}\frac{{\left( {1-\phi {1^L}} \right)}}{{\left( {1-{\phi_1}} \right)}}+\frac{{{\phi_1}^2\left( {1-{\phi_1}^{2L }} \right)}}{{\left( {1-\phi_1^2} \right)}}} \hfill \\ {=\frac{1}{{{{{\left( {1-{\phi_1}} \right)}}^2}}}\sum\limits_{k=2}^p {\phi_k^2} \left[ {L-2{\phi_1}\frac{{\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}+\frac{{{\phi_1}^2\left( {1-{\phi_1}^{2L }} \right)}}{{\left( {1-\phi_1^2} \right)}}} \right]\sigma_d^2\sum\limits_{k=2}^p L } \hfill \\ {=\frac{{L\left( {p-1} \right)\sigma_d^2}}{{{{{\left( {1-{\phi_1}} \right)}}^2}}}\sum\limits_{k=2}^p {\phi_k^2} \left[ {L-2{\phi_1}\frac{{\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}+\frac{{\phi {1^2}\left( {1-{\phi_1}^{2L }} \right)}}{{\left( {1-\phi_1^2} \right)}}} \right]} \hfill \\ \end{array} $$
(v)
Similarly,
$$ \begin{array}{*{20}c} {\mathrm{VAR}\left( {\frac{1}{{\left( {1-{\phi_1}} \right)}}\sum\limits_{k=2}^p {{\phi_k}} \sum\limits_{j=0}^{L-1 } {{D_{t-k}}_{+j-1}\left( {1-\phi_1^{L-j }} \right)} } \right)} \hfill \\ {\quad =\frac{{L\left( {p-1} \right)\sigma_d^2}}{{{{{\left( {1-{\phi_1}} \right)}}^2}}}\sum\limits_{k=2}^p {\phi_k^2} \left[ {L-2{\phi_1}\frac{{\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}+\frac{{{\phi_1}^2\left( {1-{\phi_1}^{2L }} \right)}}{{\left( {1-\phi_1^2} \right)}}} \right]} \hfill \\ \end{array} $$
(vi)
$$ \begin{array}{*{20}c} {\mathrm{COV}\left( {\frac{{\left( {1-{\phi_1}^{L+1 }} \right)}}{{\left( {1-{\phi_1}} \right)}}{D_{t-1 }},\frac{{{\phi_1}\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}{D_{t-2 }}} \right)} \hfill \\ {\quad =\frac{{\left( {1-{\phi_1}^{L+1 }} \right)}}{{\left( {1-{\phi_1}} \right)}}\frac{{{\phi_1}\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}{\phi_1}{\sigma^2}} \hfill \\ \end{array} $$
(vii)
[Since, COV(D
t−1, D
t−2) = ϕ
1
σ
2 (See Appendix 6)]
Since the second element in the fifth term in (ii) is not a single demand variable, but a sum of demands, thus the relation between a single demand (D
t−1) and the sum of demands is negligible since the demand at a particular time does not depend on the sum of demands of all periods. The same logic is applicable to other terms having a sum of terms instead of a single term. We know that the covariance of unrelated terms is zero. Thus, the covariances of the terms involving such unrelated elements are being neglected here.
Thus, the expression for variance of order quantity reduces to,
VAR(\( {q_t} \))
$$ \begin{array}{*{20}c} {={{{\left( {\frac{{\left( {1-{\phi_1}^{L+1 }} \right)}}{{\left( {1-{\phi_1}} \right)}}} \right)}}^{{^2}}}\sigma_d^2+{{{\left( {\frac{{{\phi_1}\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}} \right)}}^{{^2}}}\sigma_d^2} \hfill \\ {+\frac{{2L\left( {p-1} \right)\sigma_d^2}}{{{{{\left( {1-{\phi_1}} \right)}}^2}}}\sum\limits_{k=2}^p {\phi_k^2} \left[ {L-2{\phi_1}\frac{{\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}+\frac{{{\phi_1}^2\left( {1-{\phi_1}^{2L }} \right)}}{{\left( {1-\phi_1^2} \right)}}} \right]} \hfill \\ {-\frac{{2\left( {1-{\phi_1}^{L+1 }} \right)}}{{\left( {1-{\phi_1}} \right)}}\frac{{\phi 1\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}{\phi_1}{\sigma^2}} \hfill \\ {={{{\left( {\frac{{\left( {1-{\phi_1}^{L+1 }} \right)}}{{\left( {1-{\phi_1}} \right)}}} \right)}}^{{^2}}}\sigma_d^2-\frac{{\left( {1-{\phi_1}^{L+1 }} \right)}}{{\left( {1-{\phi_1}} \right)}}\frac{{{\phi_1}\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}{\phi_1}{\sigma^2}} \hfill \\ {+{{{\left( {\frac{{{\phi_1}\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}} \right)}}^{{^2}}}\sigma_d^2-\frac{{\left( {1-{\phi_1}^{L+1 }} \right)}}{{\left( {1-{\phi_1}} \right)}}\frac{{{\phi_1}\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}{\phi_1}{\sigma^2}} \hfill \\ {+\frac{{2L\left( {p-1} \right)\sigma_d^2}}{{{{{\left( {1-{\phi_1}} \right)}}^2}}}\sum\limits_{k=2}^p {\phi_k^2} \left[ {L-2{\phi_1}\frac{{\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}+\frac{{{\phi_1}^2\left( {1-{\phi_1}^{2L }} \right)}}{{\left( {1-\phi_1^2} \right)}}} \right]} \hfill \\ {=\left( {\frac{{\left( {1-{\phi_1}^{L+1 }} \right)}}{{\left( {1-{\phi_1}} \right)}}} \right)\sigma_d^2\left[ {\frac{{\left( {1-{\phi_1}^{L+1 }} \right)}}{{\left( {1-{\phi_1}} \right)}}-\frac{{\phi 1\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}{\phi_1}} \right]} \hfill \\ {+\frac{{\phi 1\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}\sigma_d^2\left[ {\frac{{{\phi_1}\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}-\frac{{\left( {1-{\phi_1}^{L+1 }} \right)}}{{\left( {1-{\phi_1}} \right)}}{\phi_1}} \right]} \hfill \\ {+\frac{{2L\left( {p-1} \right)\sigma_d^2}}{{{{{(1-{\phi_1})}}^2}}}\sum\limits_{k=2}^p {\phi_k^2} \left[ {L-2{\phi_1}\frac{{(1-{\phi_1}^L)}}{{\left( {1-{\phi_1}} \right)}}+\frac{{{\phi_1}^2\left( {1-\phi {1^{2L }}} \right)}}{{\left( {1-\phi_1^2} \right)}}} \right]} \hfill \\ {=\sigma_d^2\left[ {\frac{{(1-{\phi_1}^{L+1 })}}{{(1-{\phi_1})}}\left( {1+{\phi_1}-{\phi_1}^{L+1 }} \right)-\phi {1^{L+2 }}\frac{{\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}} \right]} \hfill \\ {+\frac{{2L\left( {p-1} \right)\sigma_d^2}}{{{{{\left( {1-{\phi_1}} \right)}}^2}}}\sum\limits_{k=2}^p {\phi_k^2} \left[ {L-2{\phi_1}\frac{{\left( {1-{\phi_1}^L} \right)}}{{\left( {1-{\phi_1}} \right)}}+\frac{{\phi {1^2}\left( {1-{\phi_1}^{2L }} \right)}}{{\left( {1-\phi_1^2} \right)}}} \right]} \hfill \\ \end{array} $$
(viii)
Appendix 6
$$ \mathrm{COV}\left( {{D_{t-1 }},{D_{t-2 }}} \right)=E\left[ {{D_{t-1 }}{D_{t-2 }}} \right]-E\left[ {{D_{t-1 }}} \right]E\left[ {{D_{t-2 }}} \right] $$
Now we can write,
$$ \begin{array}{*{20}c} {{D_{t-1 }}{D_{t-2 }}} \hfill \\ {=\left( {\delta +{\phi_1}{D_{t-2 }}+{\phi_2}{D_{t-3 }}+\ldots +{\phi_p}{D_{t-p }}+{\varepsilon_{t-1 }}-{\theta_1}{\varepsilon_{t-2 }}-{\theta_2}{\varepsilon_{t-3 }}-\ldots -{\theta_q}{\varepsilon_{t-q }}} \right){D_{t-2 }}} \hfill \\ {=\left( {1-\sum\limits_{i=1}^p {{\phi_i}} } \right){\mu_d}{D_{t-2 }}+{\phi_1}D_{t-2}^2+{\phi_2}{D_{t-3 }}{D_{t-2 }}+\ldots +{\phi_p}{D_{t-p }}{D_{t-2 }}} \hfill \\ {\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad +{\varepsilon_{t-1 }}{D_{t-2 }}-{\theta_1}{\varepsilon_{t-2 }}{D_{t-2 }}-{\theta_2}{\varepsilon_{t-3 }}{D_{t-2 }}-\ldots -{\theta_q}{\varepsilon_{t-q }}{D_{t-2 }}} \hfill \\ {E\left[ {{D_{t-1 }}{D_{t-2 }}} \right]} \hfill \\ {=\left( {1-\sum\limits_{i=1}^p {{\phi_i}} } \right){{{\left( {E\left[ {{D_{t-2 }}} \right]} \right)}}^2}+{\phi_1}E\left[ {D_{t-2}^2} \right]+{\phi_2}{{{\left( {E\left[ {{D_{t-2 }}} \right]} \right)}}^2}+\ldots +{\phi_p}{{{\left( {E\left[ {{D_{t-2 }}} \right]} \right)}}^2}} \hfill \\ {+E\left[ {{\varepsilon_{t-1 }}{D_{t-2 }}-{\theta_1}{\varepsilon_{t-2 }}{D_{t-2 }}-{\theta_2}{\varepsilon_{t-3 }}{D_{t-2 }}-\ldots -{\theta_q}{\varepsilon_{t-q }}{D_{t-2 }}} \right]} \hfill \\ \end{array} $$
Since error terms are independent and mean of error terms are zero, thus all the terms involving error terms will be zero. Thus,
$$ \begin{array}{*{20}c} {\mathrm{COV}\left( {{D_{t-1 }},{D_{t-2 }}} \right)=E\left[ {{D_{t-1 }}{D_{t-2 }}} \right]-E\left[ {{D_{t-1 }}} \right]E\left[ {{D_{t-2 }}} \right]} \hfill \\ {={{{\left( {E\left[ {{D_{t-2 }}} \right]} \right)}}^2}+{\phi_1}E\left[ {D_{t-2}^2} \right]-{\phi_1}{{{\left( {E\left[ {{D_{t-2 }}} \right]} \right)}}^2}-{\phi_2}{{{\left( {E\left[ {{D_{t-2 }}} \right]} \right)}}^2}+{\phi_2}{{{\left( {E\left[ {{D_{t-2 }}} \right]} \right)}}^2}-\ldots -{\phi_p}{{{\left( {E\left[ {{D_{t-2 }}} \right]} \right)}}^2}+{\phi_p}{{{\left( {E\left[ {{D_{t-2 }}} \right]} \right)}}^2}-{{{\left( {E\left[ {{D_{t-2 }}} \right]} \right)}}^2}} \hfill \\ {={\phi_1}E\left[ {D_{t-2}^2} \right]-{\phi_1}{{{\left( {E\left[ {{D_{t-2 }}} \right]} \right)}}^2}} \hfill \\ {={\phi_1}\sigma_d^2} \hfill \\ \end{array} $$
Thus, \( \mathrm{COV}\left( {{D_{t-1 }},{D_{t-2 }}} \right)={\phi_1}\sigma_d^2 \)
