# A Detailed Examination of Sphicas (2014), Generalized EOQ Formula Using a New Parameter: Coefficient of Backorder Attractiveness

## Abstract

**:**

## 1. Introduction

## 2. Notation and Assumptions

- π = The backorder cost per unit (fixed backorder cost)
- h = The holding cost per unit, per unit of time
- p = The backorder cost per unit, per unit of time (linear backorder cost)
- r = An auxiliary expression, with $r=\left(h/p\right)$
- D = The demand rate per unit of time
- K = The ordering cost (setup cost) per order
- Q = The order quantity
- S = The backlogged amount
- Q − S = The initial inventory level, after backlogged quantity
- TC = The total cost per unit of time
- β = An auxiliary expression, with $\beta =\mathrm{max}\left\{0,1-\left({\pi}^{2}{D}^{2}/2DKh\right)\right\}$

- (1)
- There is one product in this inventory model.
- (2)
- The planning horizon is infinite, such that minimizing the average cost for the first planning horizon is the objective function.
- (3)
- Constant demand is assumed for the entire planning horizon.
- (4)
- Shortages are accepted and totally backordered.
- (5)
- There are two types of backlogged cost: (i) a fixed cost that is used for the maximum backlogged level that is not related to the waiting period, (ii) a linear backlogged cost that is used for accumulated backorders per unit of time.
- (6)
- (7)
- During our derivation, we assume two conditions: (C1) $h\ge p$ and (C2) $h<p$.
- (8)
- To study the intersection of ${Q}^{\ast}-{S}^{\ast}$ and ${S}^{\ast}$, we divide this problem into three cases: Case (a) $0<r<1$, Case (b) $r=1$, and Case (c) $r>1$.
- (9)
- When $p=0$, for $T{C}_{2}$, we divide our solution procedure into two cases: (i) $2DKh\ne {\pi}^{2}{D}^{2}$, and (ii) $2DKh={\pi}^{2}{D}^{2}$.
- (10)
- Under case (i), to compare the minimum between $\sqrt{2DKh}$ and $\pi D$, we further divide case (i) into two sub-cases: case (i-1) $2DKh<{\pi}^{2}{D}^{2}$, and case (i-2) $2DKh>{\pi}^{2}{D}^{2}$.

## 3. Review of Sphicas [5]

**Proposition**

**1 of Sphicas [5].**

**Proposition**

**2 of Sphicas [5].**

**Proposition**

**3 of Sphicas [5].**

**Proposition**

**4 of Sphicas [5].**

**Proposition**

**5 of Sphicas [5].**

## 4. A Detailed Examination of Sphicas [5]

- The first issue: the partition of the feasible domain in Sphicas [5] needs revision.
- The second issue: We provide a proof for the assertion $T{C}_{2}^{\ast}\ge T{C}_{1}^{\ast}$.
- The third issue: We derive a proof to show that $({Q}^{\ast}-{S}^{\ast})\left(\pi \right)$ increases.
- The fourth issue: the domain of $\beta $ in Proposition 1 needs revision, and only using $EO{Q}_{0}$, $r$ and $\pi $, Proposition 2 fails.
- The fifth issue: Proposition 3 is completely wrong.
- The sixth issue: the expression of ${S}^{\ast}/{Q}^{\ast}$ in Proposition 4 is tedious. In this paper, we provide a compact expression.
- The seventh issue: Proposition 5 contains a typo and the intersection of ${Q}^{\ast}-{S}^{\ast}$ and ${S}^{\ast}$ in the Figure 1 at $\beta =\left(3r-1\right)/r\left(r+1\right)$ is questionable.
- The eighth issue: Comment 1 in Section 4 with $\beta =1$ contains questionable results.
- The ninth issue: Comment 1 in Section 4 with ${Q}^{\ast}-{S}^{\ast}$ contains questionable results.
- The tenth issue: Comment 1 in Section 4 with $T{C}^{\ast}$ contains questionable results.
- The eleventh issue: Comment 2 in Section 4 to claim Equation (16) is superior to Equation (9) contains questionable results.
- The twelfth issue: the special cases in Section 5 for $p=0$ contain questionable results.

**Theorem**

**1.**

**Remark.**

**Proposition**

**1 of Sphicas [5].**

**Theorem**

**2.**

**Theorem**

**3.**

## 5. Direction for Future Research

## Funding

## Conflicts of Interest

## References

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**Figure 1.**${Q}^{\ast}=\sqrt{2KD/h}\sqrt{1+\beta r},{S}^{\ast}=\sqrt{2KD/h}r\left(\sqrt{1+\beta r}-\sqrt{1-\beta}\right)/\left(1+\mathrm{r}\right),{S}^{\ast}/{Q}^{\ast}=\left(\mathrm{r}/\left(1+\mathrm{r}\right)\right)(1-\sqrt{\left(1-\beta \right)/\left(1+\beta r\right)},{Q}^{\ast}-{S}^{\ast}=\sqrt{2KD/h}\left(\sqrt{1+\beta r}+r\sqrt{1-\beta}\right)/\left(1+\mathrm{r}\right)$ Reproduction of Figure 1 of Sphicas [5].

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**MDPI and ACS Style**

Luo, X.-R.
A Detailed Examination of Sphicas (2014), Generalized EOQ Formula Using a New Parameter: Coefficient of Backorder Attractiveness. *Symmetry* **2019**, *11*, 931.
https://doi.org/10.3390/sym11070931

**AMA Style**

Luo X-R.
A Detailed Examination of Sphicas (2014), Generalized EOQ Formula Using a New Parameter: Coefficient of Backorder Attractiveness. *Symmetry*. 2019; 11(7):931.
https://doi.org/10.3390/sym11070931

**Chicago/Turabian Style**

Luo, Xu-Ren.
2019. "A Detailed Examination of Sphicas (2014), Generalized EOQ Formula Using a New Parameter: Coefficient of Backorder Attractiveness" *Symmetry* 11, no. 7: 931.
https://doi.org/10.3390/sym11070931