Comments on “ An economic order quantity ( EOQ ) for items with imperfect quality and inspection errors

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Introduction
A Khan et al. (2011) proposed an inventory model with imperfect processes and inspection errors.Later, Hsu (2012) found a contradiction in Khan' paper between the cycle length and the holding cost per cycle, then fixed this flaw and develop a corrected EOQ.However, there are still some queries to be discussed.This commenting paper points out three queries in Khan et al.'s (2011) article that need to be re-examined.Specifically, the revenue function derived in their article is unrealistic, and thus, this commenting paper further offers corrections to complement the shortcomings.The following notation is used throughout this comment (Please refer to Khan et al.'s (2011)

article).
D number of units demanded per year y order size c unit variable cost K fixed ordering cost A a parameter used for simplifying the holding cost in Eq.( 7) s unit selling price of a non-defective item v unit selling price of a defective item x screening rate d unit screening cost h unit holding cost T cycle length m 1 probability of Type I error(classifying a non-defective item as defective) m 2 probability of Type II error(classifying a defective item as non-defective) p probability that an item is defective t 1 inspection time in a cycle t 2 the remaining time in a cycle, after the defective items are screened out f (p) probability density function of p f (m 1 ) probability density function of m 1 f (m 2 ) probability density function of m 2 B 1 number of items that are classified as defective in one cycle B 2 number of defective items that are returned from the market in one cycle c a cost of accepting a defective item c r cost of rejecting a non-defective item In Khan et al. (2011), the authors established the total profit per cycle to be written as follows: Total profit per cycle TP(y) = total revenue per cycle -total cost per cycle = (the revenue from selling the good items + the revenue from selling the classified defective items)-(the procurement cost per cycle+ the screening cost per cycle + the holding cost per cycle).
Where the total cost per cycle is And the total revenue per cycle is

Revised model
Three queries are as follows: 1).On page 116 of Khan et al. (2011), "Figure 1 depicts the behavior of different types of inventory in the order cycle.The (red) triangle at the bottom represents the defective lot that is returned by the market and is accumulated into the salvaged lot."However, on page 114, the (red) triangle should start at 0, and end at T. The reason for this correction is that on page 115, the author states that "the screening and consumption of the inventory continues until time t 1 "; therefore, there would be some ) ) ) m e. ) From Eq.( 6), the total profit per cycle can now be written as

y Z t TP y sy p m vy p m vyp K cy dy c p ym c pym h Zt
where A=1-D/x-(m 1 +p)+p(m 1 +m 2 ).
From Eq.( 7), with , the expected total profit should be revised as Therefore, the first derivative of Eq. ( 8) is the same as Khan et al. (2011), and the optimal order size as well.

Numerical example
Using the same data from Khan et al. ( 2011

Conclusion
This commenting paper points out three queries in Khan et al.'s (2011) article; specifically, the revenue function derived in their article is unrealistic.This commenting paper further offers corrections to these shortcomings with a revised model.The numerical example is demonstrated for comparison.