Abstract
In this paper, we consider inventory control problems for deteriorating items with maximum serviceable lifetime under mixed sales situation, both of the demand and the deterioration rate are depending on time. A model is presented to formulate the process of mixed sales that deteriorated items are sold to consumers together with serviceable items, where penalty cost for the sales of deteriorated products is included. From the literature search, this study is one of the first researches on the joint inspection and inventory control policies under the mixed sales situation with timedependent demand and deterioration rate. An additional ordering contract is designed to improve the inventory holder’s profit. The optimal ordering time and ordering quantities are characterized for the additional ordering contract. We show that it would be more beneficial for the inventory holder to employ an additional order. Furthermore, two different inspection policies are considered in this study: (i) one inspection during the cycle; (ii) continuous monitoring in the cycle. The numerical results show that the net profit would increase if one inspection or continuous monitoring is conducted. These results provide useful insights to guide decisionmaking in inventory control problems of deteriorating products.
This is a preview of subscription content, access via your institution.
References
Ahmed, M. A., AlKhamis, T. A., & Benkherouf, L. (2013). Inventory models with ramp type demand rate, partial backlogging and general deterioration rate. Applied Mathematics and Computation, 219(9), 4288–4307.
Bakker, M., Riezebos, J., & Teunter, R. H. (2012). Review of inventory systems with deterioration since 2001. European Journal of Operational Research, 221(2), 275–284.
CáirdenasBarrón, L. E., Shaikh, A. A., Tiwari, S., & TreviñoGarza, G. (2020). An EOQ inventory model with nonlinear stock dependent holding cost, nonlinear stock dependent demand and trade credit. Computers and Industrial Engineering. https://doi.org/10.1016/j.cie.2018.12.004.
Chen, S. C., & Teng, J. T. (2015). Inventory and credit decisions for timevarying deteriorating items with upstream and downstream trade credit financing by discounted cash flow analysis. European Journal of Operational Research, 243(2), 566–575.
Chung, C. J., & Wee, H. M. (2011). Short lifecycle deteriorating product remanufacturing in a green supply chain inventory control system. International Journal of Production Economics, 129(1), 195–203.
Duan, Y., Cao, Y., & Huo, J. (2018). Optimal pricing, production, and inventory for deteriorating items under demand uncertainty: The finite horizon case. Applied Mathematical Modelling, 58, 331–348.
Dye, C. Y. (2013). The effect of preservation technology investment on a noninstantaneous deteriorating inventory model. Omega, 41(5), 872–880.
Ghare, P. M., & Schrader, G. F. (1963). A model for exponentially decaying inventory. Journal of Industrial Engineering, 14(5), 238–243.
Ghiami, Y., & Williams, T. (2015). A twoechelon productioninventory model for deteriorating items with multiple buyers. International Journal of Production Economics, 159, 233–240.
Gilding, B. H. (2014). Inflation and the optimal inventory replenishment schedule within a finite planning horizon. European Journal of Operational Research, 234(3), 683–693.
Hung, K. C. (2011). An inventory model with generalized type demand, deterioration and backorder rates. European Journal of Operational Research, 208(3), 239–242.
Jaggi, C. K., Gupta, M., Kausar, A., & Tiwari, S. (2019). Inventory and credit decisions for deteriorating items with displayed stock dependent demand in twoechelon supply chain using Stackelberg and Nash equilibrium solution. Annals of Operations Research, 274(1–2), 309–329.
Janssen, L., Claus, T., & Sauer, J. (2016). Literature review of deteriorating inventory models by key topics from 2012 to 2015. International Journal of Production Economics, 182, 86–112.
Khan, M. A., Shaikh, A. A., Panda, G. C., Bhunia, A. K., & Konstantaras, I. (2020). Noninstantaneous deterioration effect in ordering decisions for a twowarehouse inventory system under advance payment and backlogging. Annals of Operations Research, 289(2), 243–275.
Khanra, S., Ghosh, S. K., & Chaudhuri, K. S. (2011). An EOQ model for a deteriorating item with time dependent quadratic demand under permissible delay in payment. Applied Mathematics and Computation, 218(1), 1–9.
Lee, Y. P., & Dye, C. Y. (2012). An inventory model for deteriorating items under stockdependent demand and controllable deterioration rate. Computers and Industrial Engineering, 63(2), 474–482.
Luo, J., & Chen, X. (2015). Risk hedging via option contracts in a random yield supply chain. Annals of Operations Research, 257(1–2), 697–719.
Maihami, R., & Kamalabadi, I. N. (2012). Joint pricing and inventory control for noninstantaneous deteriorating items with partial backlogging and time and price dependent demand. International Journal of Production Economics, 136(1), 116–122.
Mishra, U., TijerinaAguilera, J., Tiwari, S., & CóirdenasBarrón, L. E. (2018). Retailer’s joint ordering, pricing, and preservation technology investment policies for a deteriorating item under permissible delay in payments. Mathematical Problems in Engineering. https://doi.org/10.1155/2018/6962417.
Pervin, M., Roy, S. K., & Weber, G. W. (2018). Analysis of inventory control model with shortage under timedependent demand and timevarying holding cost including stochastic deterioration. Annals of Operations Research, 260(1–2), 437–460.
Pervin, M., Roy, S. K., & Weber, G. W. (2019). Multiitem deteriorating twoechelon inventory model with price and stockdependent demand: A tradecredit policy. Journal of Industrial and Management Optimization, 15(3), 1345–1373.
Prasad, K., & Mukherjee, B. (2014). Optimal inventory model under stock and time dependent demand for time varying deterioration rate with shortages. Annals of Operations Research, 243(1–2), 323–334.
Roy, S. K., Pervin, M., & Weber, G. W. (2020). Imperfection with inspection policy and variable demand under trade credit: A deteriorating inventory model. Numerical Algebra, Control and Optimization, 10(1), 45–74.
Sarkar, B. (2012). An EOQ model with delay in payments and time varying deterioration rate. Mathematical and Computer Modelling, 55(3), 367–377.
Sarkar, S., & Chakrabarti, T. (2013). An EPQ model having Weibull distribution deterioration with exponential demand and production with shortages under permissible delay in payments. Mathematical Theory and Modelling, 3(1), 1–7.
Sarkar, B., Saren, S., & CárdenasBarrón, L. E. (2015). An inventory model with tradecredit policy and variable deterioration for fixed lifetime products. Annals of Operations Research, 229(1), 677–702.
Tai, A. H., Xie, Y., & Ching, W. K. (2016). Inspection policy for inventory system with deteriorating products. International Journal of Production Economics, 173, 22–29.
Tai, A. H., Xie, Y., He, W., & Ching, W. K. (2019). Joint inspection and inventory control for deteriorating items with random maximum lifetime. International Journal of Production Economics, 207, 144–162.
Taleizadeh, A. A. (2014). An EOQ model with partial backordering and advance payments for an evaporating item. International Journal of Production Economics, 155, 185–193.
Taleizadeh, A. A., Tavassoli, S., & Bhattacharya, A. (2019). Inventory ordering policies for mixed sale of products under inspection policy, multiple prepayment, partial trade credit, payments linked to order quantity and full backordering. Annals of Operations Research. https://doi.org/10.1007/s1047901903369x.
Teng, J. T., CárdenasBarrón, L. E., Chang, H. J., Wu, J., & Hu, Y. (2016). Inventory lotsize policies for deteriorating items with expiration dates and advance payments. Applied Mathematical Modelling, 40(19–20), 8605–8616.
Tiwari, S., Jaggi, C. K., Bhunia, A. K., Shaikh, A. A., & Goh, M. (2017). Twowarehouse inventory model for noninstantaneous deteriorating items with stockdependent demand and inflation using particle swarm optimization. Annals of Operations Research, 254(1–2), 401–423.
Tiwari, S., Jaggi, C. K., Gupta, M., & CárdenasBarrón, L. E. (2018a). Optimal pricing and lotsizing policy for supply chain system with deteriorating items under limited storage capacity. International Journal of Production Economics, 200, 278–290.
Tiwari, S., CárdenasBarrón, L. E., Goh, M., & Shaikh, A. A. (2018b). Joint pricing and inventory model for deteriorating items with expiration dates and partial backlogging under twolevel partial trade credits in supply chain. International Journal of Production Economics, 200, 16–36.
Tiwari, S., CárdenasBarrón, L. E., Shaikh, A. A., & Goh, M. (2020). Retailer’s optimal ordering policy for deteriorating items under ordersize dependent trade credit and complete backlogging. Computers and Industrial Engineering, 139, 105559.
Tripathi, R. P., & Pandey, H. S. (2013). An EOQ model for deteriorating item with Weibull time dependent demand rate under trade credits. International Journal of Information and Management Sciences, 24, 329–347.
Wang, W. C., Teng, J. T., & Lou, K. R. (2014). Seller’s optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime. European Journal of Operational Research, 232(2), 315–321.
Wang, C., & Chen, X. (2013). Option contracts in fresh produce supply chain with circulation loss. Journal of Industrial Engineering and Management, 6(1), 104–112.
Wang, C., & Chen, X. (2016). Option pricing and coordination in the fresh produce supply chain with portfolio contracts. Annals of Operations Research, 248(1–2), 471–491.
Widyadana, G. A., & Wee, H. M. (2012). An economic production quantity model for deteriorating items with multiple production setups and rework. International Journal of Production Economics, 138(1), 62–67.
Wu, J., AlKhateeb, F. B., Teng, J. T., & CárdenasBarrón, L. E. (2016). Inventory models for deteriorating items with maximum lifetime under downstream partial trade credits to creditrisk customers by discounted cashflow analysis. International Journal of Production Economics, 171, 105–115.
Wu, J., & Chan, Y. L. (2014). Lotsizing policies for deteriorating items with expiration dates and partial trade credit to creditrisk customers. International Journal of Production Economics, 155, 292–301.
Wu, J., Teng, J. T., & Skouri, K. (2018). Optimal inventory policies for deteriorating items with trapezoidaltype demand patterns and maximum lifetimes under upstream and downstream trade credits. Annals of Operations Research, 264(1–2), 459–476.
Zhao, Y., Wang, S., Cheng, T. E., Yang, X., & Huang, Z. (2010). Coordination of supply chains by option contracts: A cooperative game theory approach. European Journal of Operational Research, 207(2), 668–675.
Author information
Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This research work was supported by Research Grants Council of Hong Kong under grant numbers 7301519, National Natural Science Foundation of China under grant number 11671158 , 72002201, 71701032 and 71603238, Zhejiang Provincial Natural Science Foundation of China under grant number LQ19G010002 and LY20G030023, and IMR and RAE Research Fund from Faculty of Science, the University of Hong Kong.
Appendix
Appendix
Proof of Proposition 1
Based on Eq. (10), it has
Set \(\displaystyle \frac{\partial {\varPi }_1}{\partial T_1}=0,\) it has \(T_1=\frac{(pc)(1+m)}{p+k+h(1+m)}\) or \(T_1=m.\)

Case 1
If \(p<c(1+m)+hm(1+m)+km,\) then \(\frac{(pc)(1+m)}{p+k+h(1+m)}<m.\) Then it has
$$\begin{aligned} \displaystyle \frac{\partial {\varPi }_1}{\partial T_1}\ge 0&\text {when}&T_1 \in \left( 0,\displaystyle \frac{(pc)(1+m)}{p+k+h(1+m)}\right] \\ \displaystyle \frac{\partial {\varPi }_1}{\partial T_1}< 0&\text {when}&T_1 \in \left( \displaystyle \frac{(pc)(1+m)}{p+k+h(1+m)},m\right] . \end{aligned}$$Therefore \({\varPi }_1\) reaches its maximum at \(T_1^*=\displaystyle \frac{(pc)(1+m)}{p+k+h(1+m)}.\)

Case 2
If \(p\ge c(1+m)+hm(1+m)+km,\) then \(\frac{(pc)(1+m)}{p+k+h(1+m)}\ge m.\) Then it has
$$\begin{aligned} \displaystyle \frac{\partial {\varPi }_1}{\partial T_1}\ge 0&\text {when}&T_1 \in \left( 0,m\right] . \end{aligned}$$Therefore \({\varPi }_1\) reaches its maximum at \(T_1^*=m.\)
The proof is completed.
Proof of Proposition 2
Suppose that \(c=b.\) Denote \(m_1=m+\frac{(pc)(1+m)}{p+k+h(1+m)}.\) Based on Eq. (14), it has
and

Case 1
If \(p<c(1+m)+km+hm(1+m),\) then \(\frac{(pc)(1+m)}{p+k+h(1+m)}<m\) and \(m_1<2m.\)

(a)
If \(0<T<m_1,\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Set \(\displaystyle \frac{\partial {{\varPi }_2}}{\partial {t_1}}=0\), it yields \(t_1^*=\displaystyle \frac{T}{2}.\)

(b)
If \(T=m_1,\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}=0\) and \(\displaystyle \frac{\partial {{\varPi }_2}}{\partial {t_1}}=0.\) Since it requires that \(0<t_1\le m\) and \(0<Tt_1\le m,\) we have \(Tm\le t_1^* \le m.\)

(c)
If \(m_1<T<2m,\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}>0,\) and \({\varPi }_2\) is a convex function of \(t_1.\) Since \(Tm<\displaystyle \frac{T}{2}<m\) and \(m\displaystyle \frac{T}{2}=\displaystyle \frac{T}{2}(Tm),\) then \(t_1^*=m\,\,\text {or}\,\,Tm. \)

(a)

Case 2
If \(p\ge c(1+m)+km+hm(1+m),\) then \(\frac{(pc)(1+m)}{p+k+h(1+m)}\ge m\) and \(m_1\ge 2m.\) Hence \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0\) as \(T\in (0,2m),\) and \({\varPi }_2\) is a concave function of \(t_1.\) Set \(\displaystyle \frac{\partial {{\varPi }_2}}{\partial {t_1}}=0\), it yields \(t_1^*=\displaystyle \frac{T}{2}.\)
The proof is completed.
Proof of Proposition 3
Suppose that \(c>b.\) Denote
where
and
Based on Eq. (14), we have
and
Set \(\displaystyle \frac{\partial {\varPi }_2}{\partial t_1}=0\) and we have
Then it has

Case 1
If \(p<b(1+m)+km+h(1+m),\) it can be proved that
$$\begin{aligned} 0<m_{21}<m<m_{22}<m_{23}<m_{24}<m_{25}<2m. \end{aligned}$$
(a)
If \(0<T\le m_{21},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(T_{2c}\le 0,\) we have \(t_1^*=0.\)

(b)
If \(m_{21}<T\le m,\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(0<T_{2c}<T,\) we have \(t_1^*=T_{2c}.\)

(c)
If \(m<T\le m_{22},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(Tm<T_{2c}<m,\) we have \(t_1^*=T_{2c}.\)

(d)
If \(m_{22}<T\le m_{23},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(0<T_{2c}<Tm,\) we have \(t_1^*=Tm.\)

(e)
If \(m_{23}<T< m_{24},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(T_{2c}< Tm,\) we have \(t_1^*=Tm.\)

(f)
If \(T= m_{24},\) then \( \displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}=0,\) and \(\displaystyle \frac{\partial {{\varPi }_2}}{\partial {t_1}}<0.\) Since it requires that \(Tm\le t_{1}\le m,\) we have \(t_1^*=Tm.\)

(g)
If \(m_{24}<T\le m_{25},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}>0,\) and \({\varPi }_2\) is a convex function of \(t_1.\) Since \(m<T_{2c},\) we have \(t_1^*=Tm.\)

(h)
If \(m_{25}<T< 2m,\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}>0,\) and \({\varPi }_2\) is a convex function of \(t_1.\) It shows that \(\frac{T}{2}<T_{2c}\) and \(Tm<T_{2c}<m.\) Since \(mT_{2c}T_{2c}(Tm))=T2T_{2c}<0,\) we have \(t_1^*=Tm.\)
Therefore, we have
$$\begin{aligned} t_1^*=\left\{ \begin{array}{ccc} 0, &{} T\in (0,m_{21}] \\ T_{2c},&{} T\in (m_{21},m_{22}] \\ Tm,&{}T\in (m_{22},2m) \end{array} \right. \end{aligned}$$ 
(a)

Case 2
If \(b(1+m)+km+hm(1+m)\le p<\frac{c+b}{2}(1+m)+km+hm(1+m),\) it can be proved that
$$\begin{aligned} 0<m_{21}<m<m_{22}<m_{23}<m_{24}<2m<m_{25}. \end{aligned}$$
(a)
If \(0<T\le m_{21},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(T_{2c}\le 0,\) we have \(t_1^*=0.\)

(b)
If \(m_{21}<T\le m,\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(0<T_{2c}<T,\) we have \(t_1^*=T_{2c}.\)

(c)
If \(m<T\le m_{22},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(Tm<T_{2c}<m,\) we have \(t_1^*=T_{2c}.\)

(d)
If \(m_{22}<T\le m_{23},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(0<T_{2c}<Tm,\) we have \(t_1^*=Tm.\)

(e)
If \(m_{23}<T< m_{24},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(T_{2c}< Tm,\) we have \(t_1^*=Tm.\)

(f)
If \(T= m_{24},\) then \( \displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}=0,\) and \(\displaystyle \frac{\partial {{\varPi }_2}}{\partial {t_1}}<0.\) Since it requires that \(Tm\le t_{1}\le m,\) we have \(t_1^*=Tm.\)

(g)
If \(m_{24}<T< 2m,\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}>0,\) and \({\varPi }_2\) is a convex function of \(t_1.\) Since \(T_{2c}>m,\) we have \(t_1^*=Tm.\)
Then we have
$$\begin{aligned} t_1^*=\left\{ \begin{array}{ccc} 0, &{} T\in (0,m_{21}] \\ T_{2c},&{} T\in (m_{21},m_{22}] \\ Tm,&{}T\in (m_{22},2m) \end{array} \right. \end{aligned}$$ 
(a)

Case 3
If \(\frac{c+b}{2}(1+m)+km+hm(1+m)\le p<c(1+m)+km+hm(1+m),\) it can be proved that
$$\begin{aligned} 0<m_{21}<m<m_{22}<2m<m_{24}<m_{23}<m_{25}. \end{aligned}$$
(a)
If \(0<T\le m_{21},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(T_{2c}\le 0\) and it requires that \(0\le t_1\le T\), we have \(t_1^*=0.\)

(b)
If \(m_{21}<T\le m,\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(0<T_{2c}<T,\) we have \(t_1^*=T_{2c}.\)

(c)
If \(m<T\le m_{22},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(Tm<T_{2c}<m,\) we have \(t_1^*=T_{2c}.\)

(d)
If \(m_{22}<T< 2m,\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(0<T_{2c}<Tm,\) we have \(t_1^*=Tm.\)
Then we have
$$\begin{aligned} t_1^*=\left\{ \begin{array}{ccc} 0, &{} T\in (0,m_{21}] \\ T_{2c},&{} T\in (m_{21},m_{22}] \\ Tm,&{}T\in (m_{22},2m) \end{array} \right. \end{aligned}$$ 
(a)

Case 4
If \(c(1+m)+km+hm(1+m)\le p,\) it can be proved that
$$\begin{aligned} 0<m_{21}<m<2m<m_{22}<m_{24}<m_{23}<m_{25}. \end{aligned}$$
(a)
If \(0<T\le m_{21},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(T_{2c}\le 0\) and it requires that \(0\le t_1\le T\), we have \(t_1^*=0.\)

(b)
If \(m_{21}<T\le m,\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(0<T_{2c}<T/2,\) we have \(t_1^*=T_{2c}.\)

(c)
If \(m<T< 2m,\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(0<Tm<T_{2c}<m,\) we have \(t_1^*=T_{2c}.\)
Then we have
$$\begin{aligned} t_1^*=\left\{ \begin{array}{ccc} 0, &{} T\in (0,m_{21}] \\ T_{2c},&{} T\in (m_{21},2m) \end{array} \right. \end{aligned}$$ 
(a)
It shows that the results in Cases 1, 2 and 3 are the same, hence if \(p<c(1+m)+km+hm(1+m),\)
The proof is completed.
Proof of Proposition 4
Suppose that \(c<b.\) Denote
where
and
Based on Eq. (14), we have
and
Set \(\displaystyle \frac{\partial {\varPi }_2}{\partial t_1}=0\) and we have
Then it has

Case 1
If \(p<c(1+m)+km+h(1+m),\) it can be proved that
$$\begin{aligned} m_{21}<0<m_{31}<m<m_{25}<m_{24}<m_{23}<m_{22}<2m. \end{aligned}$$
(a)
If \(0<T\le m_{31},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(T<T_{2c},\) we have \(t_1^*=T.\)

(b)
If \(m_{31}<T\le m,\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(0<T_{2c}<T,\) we have \(t_1^*=T_{2c}.\)

(c)
If \(m<T\le m_{25},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(Tm<T_{2c}<m,\) we have \(t_1^*=T_{2c}.\)

(d)
If \(m_{25}<T< m_{24},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(m<T_{2c},\) we have \(t_1^*=m.\)

(e)
If \(T= m_{24},\) then \( \displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}=0,\) and \(\displaystyle \frac{\partial {{\varPi }_2}}{\partial {t_1}}>0.\) Since it requires that \(Tm\le t_{1}\le m,\) we have \(t_1^*=m.\)

(f)
If \(m_{24}<T\le m_{23},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}>0,\) and \({\varPi }_2\) is a convex function of \(t_1.\) Since \(T_{2c}<0\) and it requires that \(Tm\le t_{1}\le m,\) we have \(t_1^*=m.\)

(g)
If \(m_{23}<T\le m_{22},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}>0,\) and \({\varPi }_2\) is a convex function of \(t_1.\) Since \(T_{2c}<Tm,\) we have \(t_1^*=m.\)

(h)
If \(m_{22}<T< 2m,\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}>0,\) and \({\varPi }_2\) is a convex function of \(t_1.\) Since \(Tm<T_{2c}<m\) and \(T_{2c}<T/2,\) we have \(t_1^*=m.\)
Therefore, we have
$$\begin{aligned} t_1^*=\left\{ \begin{array}{ccc} T, &{} T\in (0,m_{31}] \\ T_{2c},&{} T\in (m_{31},m_{25}] \\ m,&{}T\in (m_{25},2m) \end{array} \right. \end{aligned}$$ 
(a)

Case 2
If \(c(1+m)+km+hm(1+m)\le p<\frac{c+b}{2}(1+m)+km+hm(1+m),\) it can be proved that
$$\begin{aligned} m_{21}<0<m_{31}<m<m_{25}<m_{24}<m_{23}<2m<m_{22}. \end{aligned}$$
(a)
If \(0<T\le m_{31},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(T<T_{2c}<m,\) we have \(t_1^*=T.\)

(b)
If \(m_{31}<T\le m,\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(0<T_{2c}<T,\) we have \(t_1^*=T_{2c}.\)

(c)
If \(m<T\le m_{25},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(Tm<T_{2c}<m,\) we have \(t_1^*=T_{2c}.\)

(d)
If \(m_{25}<T< m_{24},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(m<T_{2c},\) we have \(t_1^*=m.\)

(e)
If \(T= m_{24},\) then \( \displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}=0,\) and \(\displaystyle \frac{\partial {{\varPi }_2}}{\partial {t_1}}>0.\) Since it requires that \(Tm\le t_{1}\le m,\) we have \(t_1^*=m.\)

(f)
If \(m_{24}<T\le m_{23},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}>0,\) and \({\varPi }_2\) is a convex function of \(t_1.\) Since \(T_{2c}<0<Tm<m,\) we have \(t_1^*=m.\)

(g)
If \(m_{23}<T< 2m,\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}>0,\) and \({\varPi }_2\) is a convex function of \(t_1.\) Since \(0<T_{2c}<Tm\) and it requires that \(Tm\le t_{1}\le m,\) we have \(t_1^*=m.\)
Therefore, we have
$$\begin{aligned} t_1^*=\left\{ \begin{array}{ccc} T, &{} T\in (0,m_{31}] \\ T_{2c},&{} T\in (m_{31},m_{25}] \\ m,&{}T\in (m_{25},2m) \end{array} \right. \end{aligned}$$ 
(a)

Case 3
If \(\frac{c+b}{2}(1+m)+km+hm(1+m)\le p<b(1+m)+km+hm(1+m),\) it can be proved that
$$\begin{aligned} m_{21}<0<m_{31}<m<m_{25}<2m<m_{23}<m_{24}<m_{22}. \end{aligned}$$
(a)
If \(0<T\le m_{31},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(T<T_{2c},\) we have \(t_1^*=T.\)

(b)
If \(m_{31}<T\le m,\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(0<T_{2c}<T,\) we have \(t_1^*=T_{2c}.\)

(c)
If \(m<T\le m_{25},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(Tm<T_{2c}<m,\) we have \(t_1^*=T_{2c}.\)

(d)
If \(m_{25}<T< 2m,\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(m<T_{2c}\) and it requires that \(Tm\le t_{1}\le m,\) we have \(t_1^*=m.\)
Therefore, we have
$$\begin{aligned} t_1^*=\left\{ \begin{array}{ccc} T, &{} T\in (0,m_{31}] \\ T_{2c},&{} T\in (m_{31},m_{25}] \\ m,&{}T\in (m_{25},2m) \end{array} \right. \end{aligned}$$ 
(a)

Case 4
If \(b(1+m)+km+hm(1+m)\le p,\) it can be proved that
$$\begin{aligned} m_{21}<0<m_{31}<m<2m<m_{25}<m_{23}<m_{24}<m_{22}. \end{aligned}$$
(a)
If \(0<T\le m_{31},\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(T<T_{2c}\) and it requires that \(0\le t_1\le T,\) we have \(t_1^*=T.\)

(b)
If \(m_{31}<T\le m,\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(0<T_{2c}<T\) and it requires that \(0\le t_1\le T,\) we have \(t_1^*=T_{2c}.\)

(c)
If \(m<T< 2m,\) then \(\displaystyle \frac{\partial ^2{{\varPi }_2}}{\partial {t_1}^2}<0,\) and \({\varPi }_2\) is a concave function of \(t_1.\) Since \(Tm<T_{2c}<m\) and it requires that \(Tm \le t_1\le T,\) we have \(t_1^*=T_{2c}.\)
Therefore, we have
$$\begin{aligned} t_1^*=\left\{ \begin{array}{ccc} T, &{} T\in (0,m_{31}] \\ T_{2c},&{} T\in (m_{31},2m) \end{array} \right. \end{aligned}$$ 
(a)
It shows that the results in Cases 1, 2 and 3 are the same, hence if \( p<b(1+m)+km+hm(1+m),\)
The proof is completed.
Proof of Lemma 1
Denote \(T_s=T\tau .\)The total number of items sold during the time period \([\tau , T]\) is
Since \(0\le T_s\le m\tau \) and \(\displaystyle \frac{\partial Q_{\tau T}}{T_s}>0,\) it implies that \(Q_{\tau T}\) increases as \(T_s\) increases. Hence
Then if \(Q_{I\tau }\ge M_Q,\) \(T_s^*=m\tau \) and \(T^*=m;\)
If \(Q_{I\tau }< M_Q,\) we have
Then \(T_s^*=m\tau \sqrt{(m\tau )^22\frac{m}{D}Q_{I\tau }},\) and \(T^*=m\sqrt{(m\tau )^22\frac{m}{D}Q_{I\tau }}.\)
Proof of Lemma 2
Based on lemma 1, if \(Q_{I\tau }> M_Q,\) the inventory holder can increase the profit by decrease the quantity of Q. Therefore it has
that is
Then
where
where \(v_1=\displaystyle \frac{3(1+m)\sqrt{(1+m)^2+8(1+m)}}{4}\) and \(v_1=\displaystyle \frac{3(1+m)+\sqrt{(1+m)^2+8(1+m)}}{4}.\)
V is concave in \([0,1+m(1+m)^{1/3})\) and convex in \([1+m(1+m)^{1/3}, m).\) Since
V reaches its maximum at \(\tau =v_1.\) Hence \(Q\le V(\tau =v_1).\) The proof is completed.
Proof of Lemma 4
If the serviceable items are sold out before time m, it has
Denote \(D_t= \displaystyle \frac{1+m}{m}(Dt+D\ln (1+mt)D\ln (1+m)),\) then
Hencex
and
The proof is completed.
Proof of Proposition 5
Based on Eq. (27), we have
Then the average cost of one product is
Therefore, we suppose that the sales price is greater than \(c+d+g/D.\)
We have
where \(y=1+mT.\)
Denote
then \( \displaystyle \frac{\partial ^2 {\varPi } _4}{\partial T^2}<0\) when \(T\in (0,t_d),\) \( \displaystyle \frac{\partial ^2 {\varPi } _4}{\partial T^2}=0\) when \(T=t_d,\) \( \displaystyle \frac{\partial ^2 {\varPi } _4}{\partial T^2}>0\) when \(T\in (t_d,m),\) which implies that \({\varPi } _4\) is concave in \((0,t_d)\) and convex in \((t_d,m).\)
Denote
Set \(\displaystyle \frac{\partial {\varPi } _4}{\partial T}=0,\) then \(y_1=\displaystyle \frac{G+\sqrt{G^2H}}{2pD}\) and \(y_2=\displaystyle \frac{G\sqrt{G^2H}}{2pD}\). \(T_{y1}=1+my_1\) and \(T_{y2}=1+my_2.\) It can be proved that \(1<y_1<m+1\) when \(c+d+g/D<p.\) It also has \(y_2<1.\)

(a)
If \(c+d+g/D<p<(c+d)(1+m),\) then \(0<t_d<m\) and it can be proved that \(0<T_{y1}<t_d<m<T_{y2}.\) Then the optimal replenishment cycle is \(T^*=1+my_1.\)

(b)
If \((c+d)(1+m)\le p,\) then \(m\le t_d.\) It can be proved that \(0<T_{y1}<m<t_d\) and \(m<T_{y2}.\) Then the optimal replenishment cycle is \(T^*=1+my_1.\)
Therefore the optimal replenishment cycle is \(T^*=1+my_1\) as \(c+d+g/D<p\), and the optimal quantity is
The proof is completed.
Rights and permissions
About this article
Cite this article
Xie, Y., Tai, A.H., Ching, WK. et al. Joint inspection and inventory control for deteriorating items with timedependent demand and deteriorating rate. Ann Oper Res (2021). https://doi.org/10.1007/s10479021039432
Accepted:
Published:
Keywords
 Deteriorating items
 Mixed sales
 Additional order
 Inspection policy
 Timedependent demand and deterioration rate
 Inventory system