An EPQ model with promotional demand in random planning horizon: population varying genetic algorithm approach

Abstract

One of the economic production quantity problems that have been of interest to researchers is the production with reworking of the imperfect items including waste most disposal form and vending the units. The available models in the literature assumed that the decay rate of the items is satisfied from three different points of view: (i) minimum demands of the customer’s requirement, (ii) demands to be enhanced for lower selling price and (iii) demands of the customers who are motivated by the advertisement. The model is developed over a finite random planning horizon, which is assumed to follow the exponential distribution with known parameters. The model has been illustrated with a numerical example, whose parametric inputs are estimated from market survey. Here the model is optimized by using a population varying genetic algorithm.

Appendix

A1

In the demand expression $D^1_1(S^1)=\frac{A^1(s^1_{max}-s^1)}{s^1-s^1_{max}}$ the input parameter $s^1_{min}$ is taken per unit total requirement input parameter cost. Other two parameters $A^1$, $s^1_{max}$ are computed by the solving following two regression lines.

$$\begin{aligned}&\frac{1}{n}\sum _{i=1}^{n}D^1(i)s^1(i)-\frac{s^1_{min}}{n}\sum _{i=1}^{n}D^1(i)\\&\quad =A^1\Big [s^1_{max}-\frac{1}{n}\sum _{i=1}^{n}s^1(i)\Big ]\\&\quad \text{ and }\,\, \frac{1}{n}\sum _{i=1}^{n}\{D^1(i)\}^2s^1(i)-\frac{s^1_{min}}{n}\sum _{i=1}^{n}\{D^1(i)\}^2\nonumber \\&\qquad =A^1\Big [s^1_{max}\frac{1}{n}\sum _{i=1}^{n}D^1(i)-\frac{1}{n}\sum _{i=1}^{n}D^1(i)s^1(i)\Big ],\,\,\text{ for } \text{ n=7 } \end{aligned}$$

A2

Similarly for the demand expression $D^1_2(\nu ^1)=\kappa ^1 (1-\frac{1}{\nu ^1+1})$ the input parameter $\kappa ^1$ is estimated by the solving following regression line.

$$\begin{aligned} k^1=\frac{1}{n}\sum _{i=1}^{n}D^1(i)-\frac{1}{n}\sum _{i=1}^{n}\frac{D^1(i)}{\nu ^1(i)},\,\,\text{ for } \text{ n=7 } \end{aligned}$$

B

The following formulae are used in ANOVA comparison:

For the groups data $X_1, X_2, \ldots , X_k$ of sizes $n_1, n_2, \ldots , n_k$ respectively,

$$\begin{aligned}&N=\sum _{i=1}^{k}n_i\,,\,\,\,\,\overline{X}=\frac{\sum _{i=1}^{k}X_i}{N},\\&SS_t=\sum _{i=1}^{k}\sum _{i=1}^{n}(X_i-\overline{X})^2\,\,\, \text{ with } \,\,df_t=N-1\\&\text{ and, }\,\, \overline{X_1}=\frac{\sum X_1}{n_1};\,\,\overline{X_2}=\frac{\sum X_2}{n_2};\ldots ;\\&\quad \overline{X_k}=\frac{\sum X_k}{n_k}\,\, SS_b=\sum _{i=1}^{k}[n_i(\overline{X_i}-\overline{X})^2]\, \text{ with }\,\, df_b=k-1\\&\text{ and, }\,\,SS_w=\sum _{i=1}^{k}\sum _{i=1}^{n}(X_i-\overline{X})^2\,\,\,\text{ with }\,\,df_w=N-k \end{aligned}$$

Finally, $s^2_t=\frac{SS_t}{df_t}=\frac{SS_t}{N-1};\,\,s^2_b=\frac{SS_b}{df_b}=\frac{SS_b}{k-1};\,\,s^2_w=\frac{SS_w}{df_w}=\frac{SS_w}{N-k}.$ and $F=\frac{\text{ greater } \text{ mean } \text{ square }}{\text{ lesser } \text{ mean } \text{ square }}=\frac{s^2_b}{s^2_w}.$

df of $F:k-1, N-k$.

C1

$$\begin{aligned}&\sum _{i=1}^{N^j}\Big [SR^j_i-PC^j_i-SC^j_i-RC^j_i-HC^j_i\Big ]\\&\quad =\Big [\frac{s^jD^j}{R}(1-e^{-RT^j})-\frac{1}{R}(P^j_0+P^j_1D^j)(1-e^{-Rt^j_1})\\&\qquad \{c^j_p + c^j_{sr}+r^j_c{\delta ^j(1-\beta ^j)}\}-\frac{h^j_c}{R^2}\Big \{\{\beta ^j+\delta ^j(1-\beta ^j)\}P^j_0\\&\quad +[\{\beta ^j+\delta ^j(1-\beta ^j)\}P^j_1-1]D^j\Big \}\{1-(1+Rt^j_1)e^{-Rt^j_1}\}\\&\quad -\frac{h^j_cD^j}{R^2}\Big \{e^{-RT^j}-\{R(T^j-t^j_1)+1\}e^{-Rt^j_1}\Big \}\Big ]\sum _{i=1}^{N^j}e^{-R(i-1)T^j}\\&\quad =\Big [\frac{s^jD^j}{R}(1-e^{-RT^j})-\frac{1}{R}(P^j_0+P^j_1D^j)(1-e^{-Rt^j_1})\\&\qquad \{c^j_p + c^j_{sr}+r^j_c{\delta ^j(1-\beta ^j)}\}\\&\quad -\frac{h^j_c}{R^2}\Big \{\{\beta ^j+\delta ^j(1-\beta ^j)\}P^j_0\\&\qquad +[\{\beta ^j+\delta ^j(1-\beta ^j)\}P^j_1-1]D^j\Big \}\{1-(1+Rt^j_1)e^{-Rt^j_1}\}\\&\quad -\frac{h^j_cD^j}{R^2}\Big \{e^{-RT^j}-\{R(T^j-t^j_1)+1\}e^{-Rt^j_1}\Big \}\Big ]\\&\qquad \frac{1-e^{-N^jRT^j}}{1-e^{-RT^j}} \end{aligned}$$

C2

$$\begin{aligned}&\sum _{N^j=0}^{\infty }\int _{N^jT^j}^{(N^j+1)T^j} TP(N^j,T^j)f(h)\,dh\\&\quad = \Big [\frac{s^jD^j}{R}(1-e^{-RT^j})-\frac{1}{R}(P^j_0+P^j_1D^j)(1-e^{-Rt^j_1})\\&\qquad \{c^j_p + c^j_{sr}+r^j_c{\delta ^j(1-\beta ^j)}\}\\&\quad -\frac{h^j_c}{R^2}\Big \{\{\beta ^j+\delta ^j(1-\beta ^j)\}P^j_0+[\{\beta ^j+\delta ^j(1-\beta ^j)\}P^j_1-1]D^j\Big \}\\&\qquad \{1-(1+Rt^j_1)e^{-Rt^j_1}\}\\&\quad -\frac{h^j_cD^j}{R^2}\Big \{e^{-RT^j}-\{R(T^j-t^j_1)+1\}e^{-Rt^j_1}\Big \}\Big ]\\&\qquad \sum _{N^j=0}^{\infty }\frac{1-e^{-N^jRT^j}}{1-e^{-RT^j}} \Big \{e^{-N^jT^j\lambda }-e^{-(N^j+1)T^j\lambda }\Big \}\\&\quad = \Big [\frac{s^jD^j}{R}(1-e^{-RT^j})-\frac{1}{R}(P^j_0+P^j_1D^j)(1-e^{-Rt^j_1})\\&\qquad \{c^j_p + c^j_{sr}+r^j_c{\delta ^j(1-\beta ^j)}\}\\&\quad -\frac{h^j_c}{R^2}\Big \{\{\beta ^j+\delta ^j(1-\beta ^j)\}P^j_0+[\{\beta ^j+\delta ^j(1-\beta ^j)\}P^j_1-1]D^j\Big \}\\&\qquad \{1-(1+Rt^j_1)e^{-Rt^j_1}\}\\&\quad -\frac{h^j_cD^j}{R^2}\Big \{e^{-RT^j}-\{R(T^j-t^j_1)+1\}e^{-Rt^j_1}\Big \}\Big ]\frac{e^{-\lambda T^j}}{1-e^{-(\lambda +R) T^j}} \end{aligned}$$

C3

$$\begin{aligned}&\int _{N^jT^j}^{h}{D^j}e^{-Rt}\,dt\\&\quad =\Big [\int _{N^jT^j}^{N^jT^j+t^j_1}{D^j}e^{-Rt}\,dt+ \int _{N^jT^j+t^j_1}^{h}D^je^{-Rt}\,dt\Big ]\\&\quad =\int _{N^jT^j}^{N^jT^j+t^j_1}{D^j}e^{-Rt}\,dt\\&\quad =\frac{D^j}{R}(e^{-RN^jT^j}-e^{-Rh}) \end{aligned}$$

C4

$$\begin{aligned}&\sum _{N^j=0}^{\infty }\int _{N^jT^j}^{(N^j+1)T^j} PC^j_L f(h)\,dh\\&\quad =\sum _{N^j=0}^{\infty }\int _{N^jT^j}^{N^jT^j+t^j_1} PC^j_{L_1} f(h)\,dh\\&\qquad +\sum _{N^j=0}^{\infty }\int _{N^jT^j+t^j_1}^{(N^j+1)T^j} PC^j_{L_2} f(h)\,dh\\&\quad =\frac{c^j_p}{R}(P^j_0+P^j_1D^j)\Big [\frac{R}{R+\lambda }\{1-e^{-(R+\lambda ) t^j_1}\}\\&\qquad \quad -(1-e^{-R t^j_1})e^{-\lambda T^j})\Big ]\sum _{N^j=0}^{\infty }e^{-(\lambda +R)N^jT^j}\\&\quad =\frac{c^j_p(P^j_0+P^j_1D^j)}{R\{1-e^{-(R+\lambda )T^j}\}}\\&\qquad \Big [\frac{R}{R+\lambda }\{1-e^{-(R+\lambda ) t^j_1}\}-(1-e^{-R t^j_1})e^{-\lambda T^j})\Big ] \end{aligned}$$

C5

$$\begin{aligned}&\sum _{N^j=0}^{\infty }\int _{N^jT^j}^{(N^j+1)T^j} SC^j_L f(h)\,dh\\&\quad =\sum _{N^j=0}^{\infty }\int _{N^jT^j}^{N^jT^j+t^j_1} SC^j_{L_1} f(h)\,dh\\&\qquad +\sum _{N^j=0}^{\infty }\int _{N^jT^j+t^j_1}^{(N^j+1)T^j} SC^j_{L_2}f(h)\,dh\\&\quad =\frac{c^j_{sr}}{R}(P^j_0+P^j_1D^j)\Big [\frac{R}{R+\lambda }\{1-e^{-(R+\lambda ) t^j_1}\}\\&\qquad -(1-e^{-R t^j_1})e^{-\lambda T^j})\Big ]\sum _{N^j=0}^{\infty }e^{-(\lambda +R)N^jT^j}\\&\quad =\frac{c^j_{sr}(P^j_0+P^j_1D^j)}{R\{1-e^{-(R+\lambda )T^j}\}}\Big [\frac{R}{R+\lambda }\{1-e^{-(R+\lambda ) t^j_1}\}\\&\qquad -(1-e^{-R t^j_1})e^{-\lambda T^j})\Big ] \end{aligned}$$

C6

$$\begin{aligned}&\sum _{N^j=0}^{\infty }\int _{N^jT^j}^{(N^j+1)T^j} RC^j_L f(h)\,dh\\&\quad =\sum _{N^j=0}^{\infty }\int _{N^jT^j}^{N^jT^j+t^j_1} RC^j_{L_1} f(h)\,dh\\&\qquad +\sum _{N^j=0}^{\infty }\int _{N^jT^j+t^j_1}^{(N^j+1)T^j} RC^j_{L_2}f(h)\,dh\\&\quad =\frac{\delta ^j(1-\beta ^j)r^j_c}{R}(P^j_0+P^j_1D^j)\\&\qquad \Big [\frac{R}{R+\lambda }\{1-e^{-(R+\lambda ) t^j_1}\}\\&\qquad -(1-e^{-R t^j_1})e^{-\lambda T^j})\Big ]\sum _{N^j=0}^{\infty }e^{-(\lambda +R)N^jT^j}\\&\quad =\frac{\delta ^j(1-\beta ^j)r^j_c}{R\{1-e^{-(R+\lambda )T^j}\}}(P^j_0+P^j_1D^j)\\&\qquad \Big [\frac{R}{R+\lambda }\{1-e^{-(R+\lambda ) t^j_1}\}-(1-e^{-R t^j_1})e^{-\lambda T^j})\Big ] \end{aligned}$$

C7

$$\begin{aligned}&\sum _{N^j=0}^{\infty }\int _{N^jT^j}^{(N^j+1)T^j} SR^j_L f(h)\,dh\\&\quad =\sum _{N^j=0}^{\infty }\int _{N^jT^j}^{N^jT^j+t^j_1} SR^j_{L_1} f(h)\,dh\\&\qquad +\sum _{N^j=0}^{\infty }\int _{N^jT^j+t^j_1}^{(N^j+1)T^j} SR^j_{L_2} f(h)\,dh\\&\quad =\frac{D^js^j}{R}\left[ {(1-e^{-\lambda T^j})+\frac{\lambda }{R+\lambda }(e^{-(R+\lambda )T^j}-1)}\right] \\&\qquad \sum _{N^j=0}^{\infty }e^{-(\lambda +R)N^jT^j}\\&\quad =\frac{D^js^j}{R\{1-e^{-(R+\lambda )T^j}\}}\\&\qquad \left[ {(1-e^{-\lambda T^j})+\frac{\lambda }{R+\lambda }(e^{-(R+\lambda )T^j}-1)}\right] \end{aligned}$$