Technical NoteA note on “Economic production quantity in batch manufacturing with imperfect quality, imperfect inspection, and destructive and non-destructive acceptance sampling in a two-tier market”
Graphical abstract
Introduction
In the context of Economic Order Quantity (EOQ) models, the literature on imperfect quality can be traced back to Porteus (1986), and then Rosenblatt and Lee (1986). Since the introduction of Salameh and Jaber (2000) on EOQ with imperfect production and inspection, this stream of literature has gained a lot of attention. Considering Economic Production Quantity (EPQ), Yoo, Kim, and Park (2009) introduce the concept of inspection error in this stream. Khan, Jaber, Guiffrida, and Zolfaghari (2011) provide a detailed review of the EPQ models for imperfect quality items. While most of the literature in this line assumes full inspection (Hsu and Hsu, 2013b, Pal and Mahapatra, 2017, Tai, 2013), there are studies that consider sampling inspection, mostly in the context of the EOQ model. Shih (1980) is one of the first papers which discusses EOQ with acceptance sampling. Pulak and Al-Sultan (1996) consider acceptance sampling for Economic Manufacturing Quantity (EMQ). Moussawi-Haidar, Salameh, and Nasr (2013) study EOQ with acceptance sampling considering an instantaneous replacement of defective items. Rezaei (2016) studies sampling inspection in the context of EOQ with imperfect quality and obtain higher profit compared to Salameh and Jaber (2000). While most of these studies are related to the EOQ model, Al-Salamah (2016) is possibly the only study that discusses an EPQ model combined with acceptance sampling.
Al-Salamah (2016) (henceforth AS) develops a novel economic production quantity (EPQ) model to find the optimal lot size for a manufacturer producing items in batches where the batches are subjected to inspection process before they can be sent out to the market. AS uses two different methods of inspection, namely, testing and screening. Testing method is applied only during sampling of the lot, and based on the sampling outcome, the entire lot is considered accepted or rejected. Two possible inspection errors can occur during the acceptance sampling process: Type 1 error (rejecting the lot when it should not be rejected) and Type 2 error (not rejecting the lot when it should be). If a lot is rejected, the non-sampled items from that lot go through a more expensive non-destructive screening stage to segregate items into non-defective, reworkable, and salvage, where non-defective items are sent to the secondary market. The reworkable items, after rework, are also sent to the secondary market. On the other hand, the accepted lots are not screened and directly sent to the primary market. A fraction of the returns from the primary market are reworkable; those are reworked upon and then sent to the secondary market, while the rest are salvaged. Depending on whether the testing is destructive or not, AS develops two different versions of the EPQ model. Additionally, AS considers inspection errors in the context of lot acceptance/rejection, while most of the literature assumes inspection errors at item level (for example, Duffuaa and El-Ga’aly, 2015, Khan et al., 2011, Yoo et al., 2009).
In this note, we identify two important amendments needed for the completeness of the study. Firstly, AS assumes that the accepted lot is sent to the primary market without screening, and a fraction of the lot, , returns from the primary market as defectives. This assumption by AS is common in the literature (see Hsu and Hsu, 2013a, Hsu and Hsu, 2013b, Khan, Jaber, Guiffrida, et al., 2011, Yoo et al., 2009). The inventory curve for primary market in AS, however, shows the number of items in the inventory after the removal of items due to destructive testing and non-tested defective items. Such a graph clearly violates the assumption in AS itself that the defective items are returned only after reaching the primary market. Not accounting for the non-tested defective items impacts the computations of inventory holding cost and cycle length. As a result, the calculation for the corresponding inventory position at the start of the dispatch cycle needs to be rectified.
Further, AS considers the proportion of defectives among the items dispatched to the primary market for the accepted lots, as well as the proportion of defectives among the items sent for screening for the rejected lots, to be equal to the proportion of true defectives in the produced lot, , in all scenarios. This equality holds in case of the destructive testing process, because the items sent to the market or screening, as the case may be, are independent of the tested items. However, in the case of the non-destructive acceptance sampling process, the tested items are also a part of the items sent to the primary market (for accepted lots) or screening (for rejected lots). AS has missed the fact that depending upon the results of the tests, the posterior probabilities of being defective will be different from for the tested items in the case of non-destructive sampling. We propose the necessary modifications to account for this as well.
In Section 2, we propose the modifications. In Section 3, we perform the numerical analysis with the modified expressions of the models and compare the same with the original paper. The notations listed in Table 1 are from Al-Salamah (2016), and are used throughout this article.
Section snippets
Destructive testing model: Revised inventory calculations
For destructive testing, AS considers that for the accepted lots, non-sampled items would be shipped to the primary market, while these items would be sent for extensive screening for the rejected lots, resulting in an expected number of items being sent to the primary market. Consequently, as each of these items is defective independently with probability , an expected number of items will be non-defectives, which will be sold; while the rest, , will be
Numerical analysis
For numerical analysis, we use the same values used by AS: and .
We first recalculate the optimal values of production quantity, profit as well as total cycle time for AS. Next we find the optimal solutions for destructive testing and non-destructive testing by revising and . Finally, for non-destructive testing we update the expression for in terms
CRediT authorship contribution statement
Apratim Guha: Conceptualization, Methodology, Software, Validation, Formal analysis, Writing - original draft, Writing - review & editing, Visualization. Dipankar Bose: Conceptualization, Methodology, Software, Validation, Formal analysis, Writing - original draft, Writing - review & editing, Visualization.
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