Optimization of a periodically assessing model with manageable lead time under SLC with back order rebate for deteriorating items

Abstract

The study deals with periodic inventory model, wherein for deteriorating items back-order price rebate can be given to the end customer. Here under the service category constraint, the paper examines manageable lead time. Deterioration can take the form of decay, spoilage, damage, obsolescence and loss of original value of the product. The service level constraints have been defined as the expected demand shortages at the end of the cycle for a given safety factor divided by quantity available for satisfying the demand per cycle. The paper attempts to calculate the overall cost reduction in sourcing the products if the lead time in procurement is crashed component wise. This crashing cost is considered as a linear function and has been dealt with under two cases, viz, case (1) demand-distribution is known (approach of Gaussian-distribution); case (2) demand-distribution is un-known (approach of mini-max-distribution). An algorithm which jointly-optimises backorder-price-rebate, review-period and lead-time under the known service level for deteriorating-items has been developed. The basic assumption in the above calculations is that shortages are partially backlogged. The results show that considerable savings could be accomplished with manageable lead time, which can be passed on to the customers in the form of rebate.

Appendix

For a value of $L_{t} \in \left( {L_{ti} ,L_{t(i - 1)} } \right)$, Hessian Matrix H can be obtain as follows:

$$| H |= \left[ \begin{gathered} \frac{{\partial^{2} CAE^{\mathbb{R}} \left( {T,\pi_{m} ,L_{t} } \right)}}{{\partial T^{2} }}\frac{{\partial^{2} CAE^{\mathbb{R}} \left( {T,\pi_{m} ,L_{t} } \right)}}{{\partial T\partial \pi_{m} }} \hfill \\ \frac{{\partial^{2} CAE^{\mathbb{R}} \left( {T,\pi_{m} ,L_{t} } \right)}}{{\partial \pi_{m} \partial T}}\frac{{\partial^{2} CAE^{\mathbb{R}} \left( {T,\pi_{m} ,L_{t} } \right)}}{{\partial \pi_{m}^{2} }} \hfill \\ \end{gathered} \right]$$

Then, “calculating principal-minor H, first principal-minor of H” is

$$\begin{aligned} & \left| H \right| = \frac{{\partial^{2} CAE^{\mathbb{R}} \left( {T,\pi_{m} ,L_{t} } \right)}}{{\partial T^{2} }} \\ & CAE^{\mathbb{R}} (T,\pi_{m} ,L_{t} ) = \frac{{K + \sum\nolimits_{j = 1}^{i} {c_{j} } }}{T} + \frac{{\sigma \Psi (A)\left[ {\frac{{\beta_{0} \pi_{m}^{2} }}{{\pi_{n} }} + S + \pi_{n} - \frac{{\pi_{m} S\beta_{0} }}{{\pi_{n} }} - \pi_{m} \beta_{0} } \right]\left( {T + L_{t} } \right)^{\frac{1}{2}} }}{T} \\ & \quad + h_{c} \left[ {\frac{{D_{1} T}}{2} + \sigma A\left( {T + L_{t} } \right)^{\frac{1}{2}} + \sigma \left( {1 - \frac{{\beta_{0} \pi_{m} }}{{\pi_{n} }}} \right) \times \left( {T + L_{t} } \right)^{\frac{1}{2}} \times \Psi (A)} \right] \\ & \quad + \frac{1}{T}\left[ {\frac{{D_{1} + \theta s}}{\theta }\left( {e^{\theta T} - 1} \right) - D_{1} T} \right] \\ \end{aligned}$$

(19)

$$\begin{aligned} \frac{{CAE^{\mathbb{R}} (T,\pi_{m} ,L_{t} )}}{\partial T} & = - \frac{{\left( {K + \sum\nolimits_{j = 1}^{i} {c_{j} } } \right)}}{{T^{2} }} - \frac{{\sigma \times \psi (A) \times \left[ {\frac{{\beta_{0} \pi_{m}^{2} }}{{\pi_{n} }} + S + \pi_{n} - \frac{{S\beta_{0} \pi_{m} }}{{\pi_{n} }} - \beta_{0} \pi_{m} } \right] \times \left( {T + L_{t} } \right)^{\frac{1}{2}} }}{{T^{2} }} \\ & \quad + \frac{{\left[ {\frac{{\pi_{m}^{2} \beta_{0} }}{{\pi_{n} }} + \pi_{n} + S - \frac{{\pi_{m} \beta_{0} S}}{{\pi_{n} }} - \beta_{0} \pi_{m} } \right] \times \sigma \times \psi (A)}}{{2T\sqrt {T + L_{t} } }} \\ & \quad + h_{c} \left[ {\frac{{D_{1} }}{2} + \frac{\sigma A}{{2\sqrt {T + L_{t} } }} + \left( {1 - \frac{{\beta_{0} \pi_{m} }}{{\pi_{n} }}} \right) \times \frac{\sigma }{2} \times \psi (A) \times \frac{1}{{\sqrt {T + L_{t} } }}} \right] \\ & \quad + \frac{{\left( {D_{1} + \theta s} \right)}}{\theta } \times \frac{{\left( {\theta T - 1} \right)}}{{Te^{ - \theta T} }} + \frac{{\left( {D_{1} + \theta s} \right)}}{{T^{2} \theta }} \\ \end{aligned}$$

(20)

This can be written as

$$\begin{aligned} & \frac{{\left( {K + \sum\nolimits_{j = 1}^{i} {c_{j} } } \right)}}{{T^{2} }} + \frac{{N(\pi_{x} ) \times \sigma \times \psi (A) \times \left( {T + L_{t} } \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} }}{{T^{2} }} = \frac{{N(\pi_{x} ) \times \sigma \times \psi (A)}}{{2T\sqrt {T + L_{t} } }} \\ & \quad + h_{c} \left[ {\frac{{D_{1} }}{2} + \frac{\sigma A}{{2\sqrt {T + L_{t} } }} + \left( {1 - \frac{{\beta_{0} \pi_{m} }}{{\pi_{n} }}} \right) \times \frac{1}{2} \times \sigma \times \psi (A) \times \frac{1}{{\sqrt {T + L_{t} } }}} \right] \\ & \quad + \frac{{\left( {D_{1} + \theta s} \right)}}{\theta } \times \frac{{\left( {\theta T - 1} \right)}}{{Te^{ - \theta T} }} + \frac{{\left( {D_{1} + \theta s} \right)}}{{\theta T^{2} }} \\ & {\text{where}}\quad N(\pi_{m} ) = \left[ {\frac{{\beta_{0} \pi_{m}^{2} }}{{\pi_{n} }} + S + \pi_{n} - \frac{{S\beta_{0} \pi_{m} }}{{\pi_{n} }} - \beta_{0} \pi_{m} } \right] \\ \end{aligned}$$

(21)

$$\begin{aligned} \frac{{\partial^{2} CAE^{\mathbb{R}} (T,\pi_{m} ,L_{t} )}}{{\partial T^{2} }} &= \frac{{2\left( {K + \sum\nolimits_{j = 1}^{i} {c_{j} } } \right)}}{{T^{3} }} + \frac{{\left( {3T + 4L} \right) \times N(\pi_{m} ) \times \sigma \times \psi (A)}}{{2T^{3} \sqrt {T + L_{t} } }} + \frac{{\left( {3T + 2L} \right) \times N(\pi_{m} ) \times \sigma \times \psi (A)}}{{4\sqrt {T + L_{t} } }} \\ & \quad - \frac{\sigma Ah}{{4(T + L_{t} )^{\frac{3}{2}} }} - \frac{\psi (A)\sigma h}{{4(T + L_{t} )^{\frac{3}{2}} }} + \frac{{\pi_{{m\beta_{0} }} }}{{4\pi_{n} }} \times \frac{\psi (A) \times \sigma }{{(T + L_{t} )^{\frac{3}{2}} }} + \frac{{\left( {D_{1} + \theta s} \right)}}{\theta } \times \left[ {\theta^{2} e^{\theta T} - \frac{{\left( {\theta T - 1} \right)}}{{e^{ - \theta T} T^{2} }} - \frac{2}{{T^{3} }}} \right] \\ & {\text{where}}\quad N(\pi_{m} ) = \left[ {\frac{{\beta_{0} \pi_{m}^{2} }}{{\pi_{n} }} + S + \pi_{n} - \frac{{S\beta_{0} \pi_{m} }}{{\pi_{n} }} - \beta_{0} \pi_{m} } \right] \\ \end{aligned}$$

(22)

$$\frac{{\partial^{2} CAE^{\mathbb{R}} (T,\pi_{m} ,L_{t} )}}{{\partial T^{2} }} = \psi (T) - \frac{{h_{c} \sigma }}{4}\left[ {\frac{A}{{(T + L_{t} )^{\frac{3}{2}} }} + \frac{\psi (A)}{{(T + L_{t} )^{\frac{3}{2}} }}} \right]$$

where

$$\begin{aligned} \psi (T) & = \frac{{2\left( {K + \sum\limits_{j = 1}^{i} {c_{j} } } \right)}}{{T^{3} }} + \frac{{\left( {3T + 4L_{t} } \right) \times N(\pi_{m} ) \times \sigma \times \psi (A)}}{{2T^{3} \sqrt {T + L_{t} } }} + \frac{{\left( {3T + 2L_{t} } \right) \times N(\pi_{m} ) \times \sigma \times \psi (A)}}{{4\sqrt {T + L_{t} } }} + \frac{{\pi_{m} \beta_{0} }}{{4\pi_{n} }} \times \frac{\psi (A)\sigma }{{(T + L_{t} )^{\frac{3}{2}} }} \\ & \quad + \frac{{\left( {D_{1} + \theta s} \right)}}{\theta } \times \left[ {\theta^{2} e^{\theta T} - \frac{{\left( {\theta T - 1} \right)}}{{e^{ - \theta T} T^{2} }} - \frac{2}{{T^{3} }}} \right] \\ \end{aligned}$$

(23)

Currently

$$\begin{aligned} & \left| {H_{11} } \right| = \frac{{\partial^{2} CAE^{\mathbb{R}} (T,\pi_{m} ,L_{t} )}}{{\partial T^{2} }} = \psi (T) - \frac{{h_{c} \sigma }}{4}\left[ {\frac{A}{{(T + L_{t} )^{\frac{3}{2}} }} + \frac{\psi (A)}{{(T + L_{t} )^{\frac{3}{2}} }}} \right] \\ & \delta (T) = \frac{{\left( {K + \sum\limits_{j = 1}^{i} {c_{j} } } \right)}}{{T^{2} }} + \frac{{N(\pi_{m} ) \times \sigma \times \psi (A) \times \left( {T + L_{t} } \right)^{\frac{1}{2}} }}{{T^{2} }} - \frac{{N(\pi_{m} ) \times \sigma \times \psi (A)}}{{2T\sqrt {T + L_{t} } }} + \frac{{\left( {D_{1} + \theta s} \right)}}{\theta } \times \frac{{\left( {\theta T - 1} \right)}}{{e^{ - \theta T} T}} + \frac{{\left( {D_{1} + \theta s} \right)}}{{\theta T^{2} }} \\ & {\text{Then}}\quad \delta (T) > \frac{{h_{c} \sigma }}{4}\left[ {\frac{A}{{(T + L_{t} )^{\frac{3}{2}} }} + \frac{\psi (A)}{{(T + L_{t} )^{\frac{3}{2}} }}} \right] \\ \end{aligned}$$

(24)

So from above, the following have

$$\left| {H_{11} } \right| > \psi (T) - \delta (T) > 0$$

This indicates $(S + \pi_{n} ) > \pi_{m} (\therefore s > 0 \,and \, \pi_{n} > \pi_{m} ).$

Therefore, $\left| {H_{11} } \right| > 0$

$$\begin{aligned} & \frac{{\partial^{2} CAE^{\mathbb{R}} (T,\pi_{m} ,L_{t} )}}{{\partial T\partial \pi_{m} }} = - \frac{{\left[ {\frac{{2\pi_{{m\beta_{0} }} }}{{\pi_{n} }} + S + \pi_{n} - \frac{{S\beta_{0} }}{{\pi_{n} }} - \beta_{0} } \right] \times \psi (A) \times \sigma \times \left( {T + L_{t} } \right)^{\frac{1}{2}} }}{{T^{2} }} \\ & \quad + \frac{{\left[ {\frac{{2\beta_{0} \pi_{m} }}{{\pi_{n} }} + S + \pi_{n} - \frac{{S\beta_{0} }}{{\pi_{n} }} - \beta_{0} } \right] \times \frac{\sigma }{2} \times \frac{1}{{\sqrt {T + L_{t} } }} \times \psi (A)}}{T} - \frac{{h_{c} \beta_{0} }}{{\pi_{n} }} \times \frac{\sigma }{2} \times \frac{1}{{\sqrt {T + L_{t} } }} \times \psi (A) \\ \end{aligned}$$

(25)

$$\frac{{\partial CAE^{\mathbb{R}} (T,\pi_{m} ,L_{t} )}}{{\partial \pi_{m} }} = \frac{{\left[ {\frac{{2\pi_{m} \beta_{0} }}{{\pi_{n} }} + \pi_{n} + S - \frac{{S\beta_{0} }}{{\pi_{n} }} - \beta_{0} } \right]}}{T} - \frac{{h_{c} \beta_{0} }}{{\pi_{n} }} \times \sigma \times \left( {T + L_{t} } \right)^{\frac{1}{2}} \times \psi (A)$$

(26)

$$\frac{{\partial^{2} CAE^{\mathbb{R}} (T,\pi_{m} ,L_{t} )}}{{\partial \pi_{n}^{2} }} = \frac{{2\beta_{0} }}{{\pi_{n} }}$$

(27)

Similarly

The “second principal minor” H is

$$\begin{aligned} \left| {H_{22} } \right| & = \left[ \begin{gathered} \frac{{\partial^{2} CAE^{\mathbb{R}} (T,\pi_{m} ,L_{t} )}}{{\partial T^{2} }}\frac{{\partial^{2} CAE^{\mathbb{R}} (T,\pi_{m} ,L_{t} )}}{{\partial T\partial \pi_{x} }} \hfill \\ \frac{{\partial^{2} CAE^{\mathbb{R}} (T,\pi_{m} ,L_{t} )}}{{\partial \pi_{m} \partial T}}\frac{{\partial^{2} CAE^{\mathbb{R}} (T,\pi_{m} ,L_{t} )}}{{\partial \pi_{m}^{2} }} \hfill \\ \end{gathered} \right] \\ \left| {H_{22} } \right| & = \left[ {\frac{{\partial^{2} CAE^{\mathbb{R}} (T,\pi_{m} ,L_{t} )}}{{\partial T^{2} }} \times \frac{{\partial^{2} CAE^{\mathbb{R}} (T,\pi_{m} ,L_{t} )}}{{\partial \pi_{m}^{2} }} - \left( {\frac{{\partial^{2} CAE^{\mathbb{R}} (T,\pi_{m} ,L_{t} )}}{{\partial T\partial \pi_{m} }}} \right)^{2} } \right] \\ & \quad \therefore = \frac{{2\beta_{0} }}{{\pi_{n} }} \times \left\{ \begin{gathered} \frac{{2\left( {K + \sum\limits_{j = 1}^{i} {c_{j} } } \right)}}{{T^{3} }} + \frac{{\left( {3T + 4L_{t} } \right) \times N(\pi_{m} ) \times \sigma \times \psi (A)}}{{2T^{3} \left( {T + L_{t} } \right)^{\frac{1}{2}} }} + \frac{{\left( {3T + 4L_{t} } \right) \times N(\pi_{m} ) \times \sigma \times \psi (A)}}{{2T^{3} \left( {T + L_{t} } \right)^{\frac{1}{2}} }} \hfill \\ - \frac{{h_{c} A\sigma }}{{4(T + L_{t} )^{\frac{3}{2}} }} - \frac{{\psi (A)\sigma h_{c} }}{{4(T + L_{t} )^{\frac{3}{2}} }} + \frac{{\pi_{m} \beta_{0} }}{{4\pi_{n} }} \times \sigma \times \frac{\psi (A)}{{(T + L_{t} )^{\frac{3}{2}} }} \hfill \\ + \left[ {\frac{{\theta^{2} }}{{e^{ - \theta T} }} - \frac{2}{{T^{3} }} - \frac{{\left( {\theta T - 1} \right)}}{{e^{ - \theta T} T^{2} }}} \right] \times \frac{{\left( {D_{1} + \theta s} \right)}}{\theta } \hfill \\ \end{gathered} \right\} \\ & \quad - \left\{ \begin{gathered} - \frac{{\left[ {\frac{{2\beta_{0} \pi_{m} }}{{\pi_{n} }} + S + \pi_{n} - \frac{{S\beta_{0} }}{{\pi_{n} }} - \beta_{0} } \right] \times \sigma \times \psi (A) \times \sqrt {T + L_{t} } }}{{T^{2} }} + \frac{{\left[ {\frac{{2\beta_{0} \pi_{m} }}{{\pi_{n} }} + S + \pi_{n} - \frac{{S\beta_{0} }}{\pi n} - \beta_{0} } \right] \times \sigma \times \psi (A)}}{{2T\sqrt {T + L_{t} } }} \hfill \\ - \frac{{h_{c} \beta_{0} }}{{\pi_{n} }} \times \frac{\sigma }{2} \times \left( {T + L_{t} } \right)^{{\frac{ - 1}{2}}} \times \psi (A) \hfill \\ \end{gathered} \right\}^{2} > 0 \\ \end{aligned}$$

(28)

for $(S + \pi_{n} ) > \pi_{m} .$