An inventory model for non-instantaneous deteriorating items with partial backlogging, permissible delay in payments, inflation- and selling price-dependent demand and customer returns

Abstract

This paper develops an economic ordering policy model for non-instantaneous deteriorating items with selling price- and inflation-induced demand under the effect of inflation, permissible delay in payments and customer returns. Shortages are allowed and partially backlogged. The customer returns are assumed to increase with both the quantity sold and the product price. The main objective is to determine the optimal selling price, the optimal length of time in which there is no inventory shortage, and the optimal replenishment cycle simultaneously, to minimize the present value of the total profit. An efficient algorithm is presented to find the optimal solution of the developed model. Finally, a numerical example is extracted to solve the presented inventory model using the proposed algorithm.

Appendix

For a given value of $N$, the necessary optimality conditions for finding the optimal values $p^{*}$ and $t_1^*$ are given as follows and were implemented by us:

$$\begin{aligned}&\frac{\partial }{\partial p}f_1 \left( {p,t_1 ,T}\right) =((bkr^{2}c_3 e^{-rt_1 }(r+\theta )(k-1)(kr+\theta )(kr+\delta )e^{\left( {\delta +\left( {k-1} \right) r} \right) t_1 -\delta T} \\&\quad +\,bkr^{2}c_1 e^{-rt_d }\left( {k-1} \right) \left( {kr+\delta } \right) \left( {-r+kr+\delta } \right) e^{t_1 \left( {kr+\theta } \right) -t_d \left( {r+\theta } \right) } \\&\quad -\,\left( {kr+\theta } \right) \left( {-r+kr+\delta } \right) \left( {k-1} \right) \left( b( {( {e^{-rt_1 }}))^{2}c_2 -c_2 e^{-rt_1 }be^{-rT}+r\left( {\left( {c-2p} \right) b+a} \right) e^{-rT}} \right) \\&\qquad r\left( {r+\theta } \right) ke^{\left( {kr+\delta } \right) t_1 -\delta T}- bkrc_{2} e^{-rt_1 }\left( {r+\theta } \right) \left( {k-1} \right) \left( {kr+\theta } \right) \left( {-e^{-rt_1 }+e^{-rT}} \right) \\&\qquad \left( {-r+kr+\delta } \right) e^{\left( {-T+t} \right) \delta +krt} + bkre^{-t_d \left( {r+\theta } \right) }c_1 \left( {k-1} \right) \left( {kr+\delta } \right) \left( {-r+kr+\delta } \right) \left( {r+\theta } \right) \\&\qquad e^{t_1 \left( {kr+\theta } \right) +rt_d }+ b\theta e^{-t_d \left( {r+\theta } \right) }c_1 (k-1)\left( {kr+\delta } \right) \left( {-r+kr+\delta } \right) \left( {r+\theta } \right) e^{\left( {\left( {1+k} \right) r+\theta } \right) t_d} \\&\qquad bkr\left( {k-1} \right) \left( {kr+\delta } \right) \left( {-r+kr+\delta } \right) \left( {r+\theta } \right) \left( {e^{-\theta t_d }cr-e^{-t_d \left( {r+\theta } \right) }c_1 } \right) e^{t_1 \left( {kr+\theta } \right) } \\&\quad -\,\left( {c_1 \left( {r+\theta } \right) e^{-t_d \left( {r+\theta } \right) }+ ce^{-\theta t_d }r^{2}(k-1}) \right) b\left( {-r+kr+\delta } \right) \left( {r+\theta } \right) k\left( {kr+\delta } \right) \\&\qquad e^{t_d \left( {kr+\theta } \right) } + \left( {kr+\theta } \right) \left( {-r+kr+\delta } \right) rk(-bc_3 \left( {r+\theta } \right) e^{-rt_1 }-e^{-rt_d }bc_1 \\&\quad +\,\left( {r+\theta } \right) \left( {-2bp+a} \right) \left( {kr+\delta } \right) e^{rt_1 \left( {k-1} \right) }+ (bkrc_1 e^{-rt_d }\left( {kr+\delta } \right) \left( {-r+kr+\delta } \right) e^{rt_d \left( {k-11} \right) } \end{aligned}$$

$$\begin{aligned}&\quad +\,kr + \theta )(bc_1 e^{t_d \left( {r+\theta } \right) }\left( {kr+\delta } \right) \left( {-r+kr+\delta } \right) e^{-t_d \left( {r+\theta } \right) } + r(bkc_3 e^{-rt_1 }\delta \left( {kr+\delta } \right) e^{rT\left( {k-1} \right) } \\&\quad +\,(-r+kr + \delta )(\left( {kr+\delta } \right) k\left( -\alpha \left( {-2bp+a} \right) e^{krt_1 }+\left( {-2\beta pk-2\alpha bp+2\beta p+\alpha a} \right) e^{rt_1 }\right. \\&\quad +\,2p\beta ( {k-1} ) \Big )e^{-rt_1 } - bc\alpha \left( {k-1} \right) \left( {kr+\delta } \right) e^{krt_1 } + r\left( {\left( {c-2p} \right) b+a} \right) ke^{krT}\left( {k-1} \right) e^{-rT} \\&\quad -\,\left( {-bc\left( {k-1} \right) e^{krt_d }-\beta t_1 crk^{2}+\left( {\beta t_1 cr+\left( {-2p+c-c\alpha } \right) b+a} \right) k+bc\left( {-1+\alpha } \right) } \right) \\&\qquad \left( {kr+\delta } \right) ))))(r+\theta ))(-1+e^{-rH})/\left( (kr+\theta )(k-1)r^{2}(kr+\delta )k(r+\theta )(-r+kr+\delta )\right. \\&\qquad \left. (-1+e^{-\frac{rH}{N}}))+ \frac{SVe^{-rH}\left( {\alpha b-\alpha be^{krt_1 }+\beta t_1 kr} \right) }{kr}+\frac{c(-\beta t_1 kr+\alpha be^{krt_1 }-\alpha b}{kr}\right) \\&\quad +\,\frac{1}{r\left( {r+\theta } \right) \left( {-1+e^{-\frac{rH}{N}}} \right) }\left( I_p c\left( {a-bp} \right) \left( {-1+e^{-rH}} \right) \left( e^{-rM}\left( {e^{\left( {kr+\theta } \right) t_1 +rM}-e^{\left( {kr+\theta } \right) t_1 +rt_d }} \right) \right. \right. \\&\qquad \left( {r+\theta }\right) e^{\left( {-r-\theta } \right) t_d -rM} \left. \left. +re^{-rt_d }\left( {-e^{\left( {kr+\theta } \right) t_1 -\left( {r+\theta } \right) t_d }+e^{rt_1 \left( {k-1} \right) }} \right) \right) \right) =0 \end{aligned}$$

and

$$\begin{aligned}&\frac{\partial }{\partial t_1 }f_1 ( {p,t_1,T} )=(-(k-1)( {r+\theta } )e^{-rt_1 }r( {a-bp} )( {-1+e^{-rH}} )( {( {-2+k} )r+\delta } )( {kr+\delta } )\\&\qquad c_3 e^{( {\delta +(k-1)r} )t_1 -\delta T}-(k-1)r(\delta +(k-1)r)(a-bp)c_1 (-1+e^{-rH})(kr+\delta )e^{-rt_d } \\&\qquad e^{t_1 ( {kr+\theta } )-t_d ( {r+\theta } )}+ (k-1)(( {( {-2+k} )r+\delta } )c_2 ( ( e^{-rt_1 } ))^{2}-e^{-rT}( {\delta +(k-1)r} )c_2 e^{-rt_1 } \\&\quad +\,e^{-rT}r( {kr+\delta } )( {c-p} ) ) ({r+\theta } )( {\delta +(k-1)r} )( {a-bp} )( {-1+e^{-rH}} )e^{( {kr+\delta } )t_1 -\delta T} \\&\quad -\,(k-1)( {r+\theta } )e^{-rt_1 }r( {\delta +k-1} )r )( {a-bp} )(-1+e^{-rH})c_2 ( {-2e^{-rt_1 }+e^{rT}} )e^{( {-T+t} )\delta +krt}\\&\quad +\,(-(k-1)( {r+\theta } )(\delta +(k-1)r( {a-bp} ) c_1 (-1+e^{-rH}) e^{-t_d ( {r+\theta } )}e^{t_1 ( {kr+\theta } )+rt_d } \\&\quad -\,(k-1)( {r+\theta } )r( {\delta +(k-1)r} )( {\alpha ( {a-bp} )e^{krt_1 }+\beta p} )( {c-SVe^{-rH}} )e^{-\frac{rH}{N}}\\&\quad -\,(k-1)(r+\theta )(\delta +(k-1)r)(e^{-\theta t_d }cr-e^{-t_d ( {r+\theta } )}c_1 )(a-bp)(-1+e^{-rH})e^{t_1 ( {kr+\theta } )} \\&\quad +\,r( ( c_3 ( {r+\theta } )( {-2+k} )e^{-rt_1 }+(k-1)( {e^{-rt_d }c_1 +p( {r+\theta } )} )( {\delta +(k-1)r} )( {a-bp} )\\&\qquad ({-1+e^{-rH}} ) e^{-rt_1 (k-1)}+( {r+\theta } )( \delta c_3 e^{-rt_1 }( {-1+e^{-rH}} )( {a-bp} )e^{rT(k-1)}+(k-1)\\&\qquad (({-SV+c-pe^{-rt_1 }} )e^{-rH} +pe^{-rt_1 } ) ({\delta +(k-1)r} )( {\alpha ( {a-bp} )e^{krt_1 }+\beta p} ) ) ) ) \\&\qquad (kr+\delta ))/((k-1)( {r+\theta } )r(\delta +(k-1)r)(-1+e^{-\frac{rH}{N}})(kr+\delta )) \\&\quad +\,\frac{1}{(k-1)^{2}( {kr+\theta } )kr^{2}( {r+\theta } )( {-1+e^{-\frac{rH}{N}}} )}(( {-1+e^{-rH}} ) \\&\qquad (-( {r+\theta } )(k-1)I_p be^{-rM}c(-(k-1)kre^{( {kr+\theta } )t_1 +rM}+( {r+\theta } )ke^{( {kr+\theta } )t_d +rM}\\&\quad +\,( {-kr-\theta } )e^{( {r+\theta } )t_d +krM}-e^{( {( {k+1} )r+\theta } )t_d }\theta (k-1) ) e^{( {-r-\theta } )t_d -rM}\\&\quad +\,k+k( r^{2}cI_p be^{-rt_d }(k-1)^{2}e^{( {kr+\theta } )t_1 -( {r+\theta } )t_d }\\&\quad +\,( {r+\theta } )I_e ( {kr+\theta } )( {-2bp+a} )( {krM-1-rM} )e^{rM(k-1)}+cbrI_p e^{-rt_d }( {r+\theta } )(k-1) \\&\qquad e^{rt_d (k-1)}+ ({-cbrI_p e^{-rt_d }(k-1)e^{rt_1 (k-1)}+( {r+\theta } )I_e ({-2bp+a})( {kr+\theta } )}) ))=0. \end{aligned}$$