Optimizing setup cost in (R, T) inventory system model with imperfect production process, quality improvement, and partial backordering

Abstract

Managers always want to decrease the cost of the inventory system and increase the customers’ satisfaction. To access to the mentioned aim they should notice the different aspects of their inventory system to become successful in today competitive global marketing. In this research we model the periodic review inventory system by considering some important aspects of the system that are improving quality, reduction of setup cost, partial backordering and inspection process. The provided model is the first one that includes all of the mentioned aspects simultaneously. Finally, we provide the numerical result according to the certain algorithm and show how the sensitive parameters affect the total cost. We compare the total cost in two cases: at first, we consider setup cost and the probability of the production that results in producing defective items as parameters and in the second case we consider them as variables.

Appendix

The expected number of defective items is calculated as below according to the article that Porteus [23] has provided:

$$ \mathrm{Number}\ \mathrm{of}\ \mathrm{defective}\ \mathrm{items}= DT-\frac{\tilde{\phi}\left(1-{\phi}^{\tilde{D}T}\right)}{\phi } $$

(22)

As $\overset{\sim }{\phi }=1-\phi \cong 1$ we use the Taylor series expansion of $\overset{\sim }{\phi^{DT}}$ until 2nd order as ϕ is very small and we obtain:

$$ = DT-\frac{\tilde{\phi}\left(1-{\phi}^{\tilde{D}T}\right)}{\phi } $$

(23)

$$ = DT-\frac{1-1-\left(\ln \tilde{\phi}\right) DT-{\left[\frac{\left(\ln \tilde{\phi}\right) DT}{2}\right]}^2}{\phi } $$

(24)

$$ = DT-\frac{\frac{\phi }{\tilde{\phi}} DT-\frac{1}{2}{\left(\frac{\phi }{\tilde{\phi}} DT\right)}^2}{\phi } $$

(25)

$$ = DT-\frac{\phi DT-\frac{\phi^2{(DT)}^2}{2}}{\phi } $$

(26)

$$ =\frac{\phi {(DT)}^2}{2} $$

(27)

And the expected annual cost for defective items is computed as below:

$$ m\times \frac{\phi {(DT)}^2}{2}\times \frac{1}{T} $$

(28)