An imperfect quality economic order quantity with advanced receiving

Abstract

Differing from previous studies on economic order quantity model with imperfect quality, this paper assumes that a lot received contains either good, re-workable, or scrap items where the re-workable items are stored at a warehouse until a subsequent lot arrives and then these are sent back to the supplier to rework. We propose an imperfect quality inventory model with overlapped and advanced receiving in which the supplier provides a discount rate of procurement cost to compensate the buyer for the additional holding cost and maintain a cooperative relationship. As we expected, a lot sizing policy with overlapped and advanced receiving is developed to generalize the current literature models. It should be noted that the classical EOQ model and Maddah et al’s work (Comput Ind Eng 58:691–695, 2010) are two special cases of our model. A numerical example is given to illustrate the proposed model. In addition, a sensitivity analysis is made to investigate the effects of five important parameters on the optimal lot size and the expected total profit per cycle. Managerial insights are also presented.

Appendix 1

To obtain $\varphi _3 >1$, we should prove the following holds:

$$\begin{aligned} \left\{ {\frac{h_\mathrm{{g}} ( {{2D} / x+E\left[ {( {1-p})^2} \right] })}{h_\mathrm{{g}} ( {{2D} / x+E\left[ {( {1-p_\mathrm{{s}} })^2} \right] +( {D / x})( {{h_d } / {h_\mathrm{{g}} }-1})E\left[ {p_\mathrm{{s}} } \right] })-2cE\left[ {p_\mathrm{{r}} } \right] }} \right\} >1\qquad \end{aligned}$$

(20)

Considering Maddah et al. (2010) work, they assumed the re-workable items and scrap items are of defective items. This implies the percentage rate of defective items in Maddah et al’s work is $p =p_{\mathrm{{s}}}+p_{\mathrm{{r}}}$. Substituting $p$ as ($p_{\mathrm{{s}}}+p_{\mathrm{{r}}})$ in Eq. (20), we have

$$\begin{aligned}&\frac{h_\mathrm{{g}} ( {{2D} / x+E\left[ {( {1-p})^2} \right] })}{h_\mathrm{{g}} ( {{2D} / x+E\left[ {( {1-p_\mathrm{{s}} })^2} \right] +( {D / x})( {{h_d } / {h_\mathrm{{g}} }-1})E\left[ {p_\mathrm{{s}} } \right] })-2cE\left[ {p_\mathrm{{r}} } \right] }\nonumber \\&\quad =\frac{h_\mathrm{{g}} ( {{2D} / x+E\left[ {( {( {1-p_\mathrm{{s}} })-p_\mathrm{{r}} })^2} \right] })}{h_\mathrm{{g}} ( {{2D} / x+E\left[ {( {1-p_\mathrm{{s}} })^2} \right] +( {D / x})( {{h_d } / {h_\mathrm{{g}} }-1})E\left[ {p_\mathrm{{s}} } \right] })-2cE\left[ {p_\mathrm{{r}} } \right] }\nonumber \\&\quad =\frac{h_\mathrm{{g}} ( {{2D} / x+E\left[ {( {( {1-p_\mathrm{{s}} })^2-2( {1-p_\mathrm{{s}} })p_\mathrm{{r}} })+p_\mathrm{{r}}^2 } \right] })}{h_\mathrm{{g}} ( {{2D} / x+E\left[ {( {1-p_\mathrm{{s}} })^2} \right] +( {D / x})( {{h_d } / {h_\mathrm{{g}} }-1})E\left[ {p_\mathrm{{s}} } \right] })-2cE\left[ {p_\mathrm{{r}} } \right] }\nonumber \\&\quad =\frac{h_\mathrm{{g}} ( {{2D} / x+E\left[ {( {1-p_\mathrm{{s}} })^2} \right] })-h_\mathrm{{g}} ( {-E\left[ {p_\mathrm{{r}}^2 } \right] +2E\left[ {( {1-p_\mathrm{{s}} })} \right] E\left[ {p_\mathrm{{r}} } \right] })}{h_\mathrm{{g}} ( {{2D} / x+E\left[ {( {1-p_\mathrm{{s}} })^2} \right] +( {D / x})( {{h_d } / {h_\mathrm{{g}} }-1})E\left[ {p_\mathrm{{s}} } \right] })-2cE\left[ {p_\mathrm{{r}} } \right] }\qquad \end{aligned}$$

(21)

Observing Eq. (21), the first term in the numerator is greater than the first term in the denominator because $h_{\mathrm{{g}} }>h_{d}$. Furthermore, the second term in the numerator is less than the second term in the denominator because $c>>h_{\mathrm{{g}}}$. Therefore, the numerator in Eq. (21) is greater than the denominator in Eq. (21). This reveals $\varphi _3 >1$ and thus $Q^{*}> Q_{\mathrm{{Maddah}} (2010)}$. This completes the proof.