Integrated inventory model with backorder and minimal repair in a supply chain

Abstract

This study explores the feasibility of applying an integrated inventory model with the consideration of incorporating production programs and maintenance to an imperfect process involving a deteriorating production system. The main target of this research is to build an integrated inventory model with the issues of backorder and repair. Additionally, our proposed model tries to provide an optimal number of shipments m and lower cost. Furthermore, this article offers the detailed discussion of two preventive maintenances of the production run period which are named as perfect preventive maintenance and imperfect preventive maintenance. Also, in this study, we develop various special cases that consider the failure rate such as Weibull, geometric and learning effect. Based on the results, the demand and production ratio influences the holding cost and purchase cost and has the highest impact on the integrated model.

Introduction

The inventory management plays an important role in the supply chain management which occupies at least 20%–30% of the total cost. In other words, the enterprises should control the inventory cost as well as the significant usefulness by reducing the total cost for whole supply chain system. Therefore, applying an integrated inventory approach might assist enterprises to find out the optimal order quantity and shipment policy in order to reach the total cost minimization. The findings of applying this integration not only ensure the benefits of whole supply chain members but also enhance customer profits in today’s increasingly turbulent market.

This article mainly focuses on the discussion of high degree of standardization production and the lot size is the decision variable. In addition, firms have the partner relationship with the suppliers and buyers as the key issue of current supply chain management. Therefore, the production processes, ordering and inventory control are remaining stable because of the information sharing by all chain members.

Literature review

In today’s highly competitive globalized environment, all chain members have to response the customers’ requirements quickly which help firms to gain competitive advantages. Based on this background, just-in-time (JIT) is a valuable technique to manage the supply chain management effectively. Liao applied JIT concept to the traditional economic order quantity (EOQ) model. They provide the effect of frequent shipments for a total cost and small lot size.^1,2 However, small lot delivery costs such as shipping and inspection costs were ignored. Ramasesh separated the total order cost of the EOQ model into the cost of placing a contact order with multiple small lots shipments. Martinich found that the development of long-term sole supplier relationship with a supplier will gain substantial benefits.^3,4

The past research mainly focused on traditional single EOQ model. We believe that how to represent the real-world situation should be more concerned. Based on this, we are trying to explore the integrated model to fill up the research gap between the past related researches.

In the JIT environment, the supplier and buyer coordinate closely as the stable long-term relationship which solved problems, negotiate and gain benefits together. ⁵ Banerjee ⁶ and Kim proposed a model which incorporated with JIT purchasing and JIT manufacturing. They also found that the benefit of a joint integrated inventory replenishment policy for the buyer and the supplier has been more significant than independently derived policies. In addition, they developed an integrated lot-splitting model of facilitating multiple shipments in small lots and compared with the existing approach in a simple JIT environment and showed that using the integrated approach can reduce the total cost for the vendor and the buyer over the existing approaches. ⁷ Gunasekaran ⁸ used long turn contract and small lots size to reduce lead times and inventory cost of supply chain members. Yang and Wee ⁹ considered deteriorated items with constant production and demand rate. They build an integrated mathematical model between a vendor and a buyer and show that the integrated approach results in an impressive cost-reduction compared with an independent decision by the buyer.

Goyal ¹⁰ illustrated the first integrated inventory model; he deduced that the optimal order time interval and production cycle time can be obtained by assuming that the supplier’s production cycle time is an integer multiple of the customer’s order time interval. Then, the research in Banerjee assumed that the vendor has a finite production rate and produces to order for the buyer on a lot-for-lot basis based on Goyal’s joint economic lot-size model. After that, Goyal extended Banerjee’s model by relaxing the lot-for-lot policy and he supposed that vendor’s economic production quantity (EPQ) must be an integer multiple of the buyer’s purchase quantity which provided a lower joint total relevant cost.^6,11 The integrated model can contribute significantly to improve the vendor–purchaser relationship.

Several studies have noted that the integrated model with stronger cooperation between the buyer and the supplier will bring success and performance. Gurnani’s ¹² research presented the models with quantity discount pricing policy and different ordering structures which containing a single supplier and various buyers in a chain system. Then, Woo et al. ¹³ proposed an integrated inventory model for a single vendor which purchases and processes raw materials in order to deliver final items to multiple buyers. After that, Khan and Sarker ¹⁴ proposed a framework for integrated inventory system with two-stage consideration and the integration of the JIT concept in the conventional joint batch-sizing problem. And then, the modified integrated model developed by Pan and Yang ¹⁵ assumed lead time as the controllable factor in a JIT environment. Afterward, Yang ¹⁶ applied the fuzzy theory to the single buyer and a single vendor integrated inventory model order policy to estimate the productivity and demand. Mohan et al. ¹⁷ investigated the optimal replenishment policy for multi-item with the permissible delay in payment and a budget constraint. Yang et al. ¹⁸ considered the time of the inventory model with single buyer and single vendor; the inventory cost will change following inventory cycle time. Yang and Lin ¹⁹ proposed a single-vendor and multiple buyers integrated inventory model with lead time demand following normal distribution.

In today’s highly competitive and uncertainty global market, the well-designed inventory and production policies for manufacturers are key factors to gain advantages. Barlow and Hunter ²⁰ proposed a paper of preventive maintenance (PM) to maintain the efficacy of the production system by regular maintenance. Groenevelt et al. ²¹ considered the production of equipment could be repaired immediately when production system failure occurs. Two production control policies are assumed for coping with these randomly interferences. The first policy assumes that production of the interrupted lots is not resumed after a breakdown. Instead, the on-hand inventory is depleted before a new cycle is initiated. In the second policy, production is immediately resumed after a breakdown, if the current on-hand inventory is below a certain threshold level. Sheu et al. ²² provided periodic PM policies, which maximize the availability of a repairable system with major repair at failure. In addition, there are three types of PMs: imperfect preventive maintenance (IPM), perfect preventive maintenance (PPM) and failed preventive maintenance (FPM).

Liao et al. ²³ presented an integrated EPQ model that incorporates EPQ and maintenance programs. This model included the considerations of imperfect repair, PM and rework on the damage of a deteriorating production system. Various special cases are considered, including the maintenance learning effect. Liao et al. ²³ extended earlier studies by relaxing the model undergoing a backorder owing to rejection of defective parts after a failure. This study found the optimal policy conditions, which demonstrate that this is more flexible than previously described policies. This study also presents the effects from other factors, such as the number of non-reworkable defective products and minimal repair cost.^23,24

In recent years, many researchers donated themselves to improving the integrated supply chain model with the issues of PM and backorder. Rahim and Fareeduddin ²⁵ created the mathematical model for determining an optimal production run length for a deteriorating production system with allowable shortages for products sold with a free minimal repair warranty period. Fitouhi and Nourelfath ²⁶ conducted the problem of integrating noncyclical PM and tactical production planning for a single machine. After that, Machani and Nourelfath ²⁷ explored a variable neighborhood search (VNS) to the integrated production and maintenance planning problem in multi-state systems. Then, Wang ²⁸ investigated the integrated problem of IPM and tactical production planning for a single machine system. Fitouhi and Nourelfath ²⁹ integrated noncyclical PM with tactical production planning in multi-state systems.

Model formulation

The purpose under this section is to establish a mathematical model. The notations and assumptions are defined as follows by Liao and colleagues.^30,31

The authors believe that the PM is the regular business activities; in other words, the repair costs such as repair engineers, materials and tools are kindly significant. Therefore, this research assumes that the repair cost should be a kind of fixed cost, scheduled and significant. Additionally, this research also assumes that repair times are implemented by the past experiences not sudden issues. Thus, the repair times in this study will be negligible (See all notations in Appendix 1).

Assumptions

1. In this study, the constants which are demand rate, setup cost, ordering cost and holding cost are known.

2. Backorder is permitted during the inventory depletion period.

3. The original system begins operating at time 0. The production process begins in an in-control state and produces perfect items.

4. Setup cost $C_s$ is incurred at the start of each inventory cycle. PM is performed following the production run period. The cost of each PM is $C_{\mathrm{pm}}$ .

5. A system has two types of PM at cumulative production run time j,T (j = 1, 2, 3…), based on outcome.

6. Type-II PM (perfect PM) makes the system as good as new, with probability $p_j={\overline{p}}_{j-1}-{\overline{p}}_j$ .

Following a perfect PM, the system returns to age 0.

If failure occurs before the scheduled PM, the system shifts into the “out-of-control” state, then minimal repair can be made immediately. Minimal repair merely restores the system to a functioning state following failure, so the production process returns to the in-control condition. The backorder occurs because of insufficient production following rejection of defective parts. The minimal repair cost at each failure is $C_m$ while the backorder cost per unit is $R_b$ . The number of defective products at each failure is $Q_d$ .

The repair times are negligible.

The PM cycle and inventory cycle are assumed to be same in this article.

Let $D_j$ denote the maintenance and backorder cost, including the backorder and minimal repair cost among the (j − 1)th PM and jth PM (production run period of inventory cycle j), and the PM cost of the jth PM (inventory depletion period of inventory cycle j). Let $Y_1,Y_2,\dots$ denote independent copies of Y.

Furthermore, the expected minimal repair and backorder cost is $\sum _{j=1}^{\infty }E[D_j]$ .

The expected failure number of periodic time T is

$$\sum _{j=1}^{\infty }\frac{{\overline{p}}_{j-1}}{\sum _{j=1}^{\infty }{\overline{p}}_{j-1}}\underset{j-1}{\overset{\mathrm{jT}}{\int }}r\left(t\right)\mathrm{dt}$$

(1)

Proof

$$\sum _{j=1}^{\infty }\frac{{\overline{p}}_{j-1}}{\sum _{j=1}^{\infty }{\overline{p}}_{j-1}}\underset{j-1}{\overset{\mathrm{jT}}{\int }}r\left(t\right)\mathrm{dt}$$

can be rewritten as

$$\frac{\sum _{j=1}^{\infty }{\overline{p}}_{j-1}\ln \left(\frac{\overline{F}\left((j-1)T\right)}{\overline{F}\left(\mathrm{jT}\right)}\right)}{\sum _{j=1}^{\infty }{\overline{p}}_{j-1}}\underset{j-1}{\overset{\mathrm{jT}}{\int }}r\left(t\right)\mathrm{dt}$$

Forming T is finite, $\overline{F}((j-1)T)/\overline{F}(\mathrm{jT})$ is finite and there exists a finite number V which satisfies $1<\overline{F}((j-1)T)/\overline{F}(\mathrm{jT})\leqq V$ , for j = 1, 2, …

Note that

$$\frac{\sum _{j=1}^{\infty }{\overline{p}}_{j-1}\ln (\frac{\overline{F}\left((j-1)T\right)}{\overline{F}\left(\mathrm{jT}\right)})}{\sum _{j=1}^{\infty }{\overline{p}}_{j-1}}<\frac{\sum _{j=1}^{\infty }{\overline{p}}_{j-1}\ln V}{\sum _{j=1}^{\infty }{\overline{p}}_{j-1}}=\ln V$$

therefore, this series

$$\sum _{j=1}^{\infty }\frac{{\overline{p}}_{j-1}}{\sum _{j=1}^{\infty }{\overline{p}}_{j-1}}\underset{j-1}{\overset{\mathrm{jT}}{\int }}r\left(t\right)\mathrm{dt}$$

Then, we have

$$\sum _{j=1}^{\infty }E[D_j,Y_1\ge \mathrm{jT}]=C_m$$

(2)

and

$$\sum _{j=1}^{\infty }E[D_j,(j-1)T<Y_1<\mathrm{jT}]=(C_m+Q_d{R}_b)\sum _{j=1}^{\infty }\frac{{\overline{p}}_{j-1}}{\sum _{j=1}^{\infty }{\overline{p}}_{j-1}}\underset{j-1}{\overset{\mathrm{jT}}{\int }}r\left(t\right)\mathrm{dt}$$

(3)

From equation (3), we get

$$\sum _{j=1}^{\infty }E[D_j,(j-1)T<Y_1<\mathrm{jT}]=(C_m+Q_d{R}_b)\sum _{j=1}^{\infty }\frac{{\overline{p}}_{j-1}-{\overline{p}}_j}{\sum _{j=1}^{\infty }{\overline{p}}_{j-1}}\underset{j-1}{\overset{\mathrm{jT}}{\int }}r\left(t\right)\mathrm{dt}$$

(4)

The vendor’s total expected cost

The inventory level of this model is illustrated in Figure 1. Once the vendor receives an order, the vendor will produce the items immediately until quantity reach to mQ. The item delivered from vendor to buyer by each Q unit, and there are m lots will deliver in an inventory cycle. The average inventory of vendors can be evaluated as follows

$$I_v=\frac{\left\{\left[mQ\left(\frac{Q}{P}+\left(m-1\right)\frac{Q}{D}\right)-\frac{m^2{Q}^2}{2p}\right]-\left[\frac{Q^2}{D}\left(1+2+\cdots +\left(m-1\right)\right)\right]\right\}}{\left(\frac{mQ}{D}\right)}=\frac{Q}{2}m\left(\left(1-\frac{D}{P}\right)-1+\frac{2D}{P}\right)$$

(5)

Figure 1.

Inventory model for vendor.

Therefore, the vendor’s expected holding cost per year is

$$C_h{C}_v\left(\frac{Q}{2}(m(1-\frac{D}{P})-1+\frac{2D}{P})\right)$$

(6)

According to the assumptions and notations, the total expected annual cost for vendor is shown as below

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{TE}{C}_V=\mathrm{hoding\; cost}+\mathrm{setup\; cost}+\mathrm{PM\; cost}}\\ \multicolumn{1}{c}{+\mathrm{minimal\; repair\; cost}+\mathrm{backorder\; cost}}\\ \multicolumn{1}{c}{+\mathrm{rework\; cost}}\end{array}$$

And the costs of vendors are derived as follows:

• 1. Holding cost

$$C_h{C}_v\left[\frac{Q}{2}m((1-\frac{D}{P})-1+\frac{2D}{P})\right]-\frac{C_h{Q}_d}{2}\sum _{j=1}^{\infty }\frac{{\overline{p}}_{j-1}-{\overline{p}}_j}{\sum _{j=1}^{\infty }{\overline{p}}_{j-1}}\underset{0}{\overset{j\left(\frac{D}{P}\right)T}{\int }}r\left(t\right)\mathrm{dt}$$

(7)

We have $T=\mathrm{mQ}/D$

and

$$\frac{\mathrm{TD}}{2m}{C}_h{C}_v[m(1-\frac{D}{P})-1+\frac{2D}{P}]-\frac{C_h{Q}_d}{2}\sum _{j=1}^{\infty }\frac{{\overline{p}}_{j-1}-{\overline{p}}_j}{\sum _{j=1}^{\infty }{\overline{p}}_{j-1}}\underset{0}{\overset{j\left(\frac{D}{P}\right)T}{\int }}r\left(t\right)\mathrm{dt}$$

(8)

• 2. $\mathrm{Setup}\mathrm{cost}:C_s/T$

• 3. $\mathrm{PM}\mathrm{cost}:C_{\mathrm{pm}}/T$

• 4. $\mathrm{Minimal\; repair\; cost}:\frac{C_m}{T}{\displaystyle \sum _{j=1}^{\infty }\frac{{\overline{p}}_{j-1}-{\overline{p}}_j}{{\displaystyle \sum _{j=1}^{\infty }{\overline{p}}_{j-1}}}}{\displaystyle \underset{0}{\overset{j\left(\frac{D}{P}\right)T}{\int }}r}\left(t\right)dt\mathrm{during}T$

• 5. $\mathrm{Backorder\; cost}:\frac{Q_d{R}_b}{T}\sum _{j=1}^{\infty }\frac{{\overline{p}}_{j-1}-{\overline{p}}_j}{\sum _{j=1}^{\infty }{\overline{p}}_{j-1}}\underset{0}{\overset{j\left(\frac{D}{P}\right)T}{\int }}r\left(t\right)\mathrm{dt}$

Our production system of PM strategy is shown in Figure 2.

Figure 2.

Production system of PM strategy.

The vendor’s total expected cost can be obtained from the above equation

$$TE{C}_v=\frac{C_s}{T}+\frac{C_{pm}}{T}+\frac{TD}{2m}{C}_h{C}_v\left[\left(m\left(1-\frac{D}{P}\right)-1+\frac{2D}{P}\right)\right]+\left[\left(\frac{C_m}{T}+\frac{Q_d{R}_b}{T}+\frac{Q_{rw}{R}_{rw}}{T}-\frac{C_h{Q}_d}{2}\right){\displaystyle \sum _{j=1}^{\infty }\frac{{\overline{p}}_{j-1}-{\overline{p}}_j}{{\displaystyle \sum _{j=1}^{\infty }{\overline{p}}_{j-1}}}}{\displaystyle \underset{0}{\overset{j\left(\frac{D}{P}\right)T}{\int }}r}\left(t\right)dt\right]$$

(9)

The purchaser’s total expected cost

$$\mathrm{TE}{C}_p=\mathrm{ording\; cost}+\mathrm{holding\; cost}$$

And the costs of vendor are illustrated as follows:

1. Ordering cost of each cycle is: $C_{o}m/T$

2. Holding cost: $(C_h{C}_P)(Q/2)$

And we have

$$T=\frac{\mathrm{mQ}}{D}$$

And

$$C_h{C}_P\frac{\mathrm{TD}}{2m}$$

(10)

Therefore, the buyer’s expected cost can be obtained from the above equation as follows

$$\mathrm{TE}{C}_p=\frac{C_{0}m}{T}+C_h{C}_P\frac{\mathrm{TD}}{2m}$$

(11)

By adding the TEC_v and $\mathrm{TE}{C}_p$ , we can obtain the joint expected annual cost as follows

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{JTEC}(T,m)=\frac{C_s}{T}+\frac{C_{\mathrm{pm}}}{T}+\frac{\mathrm{TD}}{2m}{C}_h{C}_v}\\ \multicolumn{1}{c}{\left[(m(1-\frac{D}{P})-1+\frac{2D}{P})\right]}\\ \multicolumn{1}{c}{+\left[(\frac{C_m}{T}+\frac{Q_d{R}_b}{T}-\frac{C_h{Q}_d}{2})\sum _{j=1}^{\infty }\frac{{\overline{p}}_{j-1}-{\overline{p}}_j}{\sum _{j=1}^{\infty }{\overline{p}}_{j-1}}\underset{0}{\overset{j\left(\frac{D}{P}\right)T}{\int }}r\left(t\right)\mathrm{dt}\right]}\\ \multicolumn{1}{c}{+\frac{C_{0}m}{T}+C_h{C}_P\frac{\mathrm{TD}}{2m}}\end{array}$$

(12)

Taking the partial derivatives of joint total expected cost (JTEC) (T,m) to find the optimal inventory runs time T and m, there exist a finite and unique optimal solution to minimize JTEC

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\partial \mathrm{JTEC}(T,m)}{\partial T}=-\frac{(C_{o}m+C_s+C_{\mathrm{pm}}+\left[(Q_d{R}_b+C_m)\sum _{j=1}^{\infty }\frac{{\overline{p}}_{j-1}-{\overline{p}}_j}{\sum _{j=1}^{\infty }{\overline{p}}_{j-1}}\underset{0}{\overset{j\left(\frac{D}{P}\right)T}{\int }}r\left(t\right)\mathrm{dt}\right])}{T^2}}\\ \multicolumn{1}{c}{+\left[(\frac{C_m}{T}+\frac{Q_d{R}_b}{T}+\frac{C_h{Q}_d}{2})\sum _{j=1}^{\infty }\frac{{\overline{p}}_{j-1}-{\overline{p}}_j}{\sum _{j=1}^{\infty }{\overline{p}}_{j-1}}j\frac{D}{P}r\left(\mathrm{jT}\frac{D}{P}\right)\right]+\frac{C_h{C}_{P}D}{2m}}\\ \multicolumn{1}{c}{+\frac{C_h{C}_{v}D}{2m}[m(1-\frac{D}{P})-1+\frac{2D}{P}]}\end{array}(13)$$

(13)

Let $\sum _{j=1}^{\infty }\frac{{\overline{p}}_{j-1}-{\overline{p}}_j}{\sum _{j=1}^{\infty }{\overline{p}}_{j-1}}\underset{0}{\overset{j\left(\frac{D}{P}\right)T}{\int }}r\left(t\right)\mathrm{dt}=Z$

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\partial \mathrm{JTEC}(T,m)}{\partial T}=-\frac{(C_{o}m+C_s+C_{\mathrm{pm}}+\left[(Q_d{R}_b+C_m+Q_{\mathrm{rw}}{R}_{\mathrm{rw}})Z\right])}{T^2}}\\ \multicolumn{1}{c}{+[(\frac{C_m}{T}+\frac{Q_d{R}_b}{T}\pm \frac{C_h{Q}_d}{2})Z\prime ]+\frac{C_h{C}_{P}D}{2m}}\\ \multicolumn{1}{c}{+\frac{C_h{C}_{v}D}{2m}[m(1-\frac{D}{P})-1+\frac{2D}{P}]}\end{array}(14)$$

(14)

And we do second-order partial derivatives, if equation (22) > 0, then JTEC will exist a local minimal solution

$$\begin{array}{c}\multicolumn{1}{c}{\frac{{\partial }^2\mathrm{JTEC}(T,m)}{{\partial }^{2}T}=\frac{2(C_{o}m+C_s+C_{\mathrm{pm}})}{T^3}}\\ \multicolumn{1}{c}{+\frac{2Z(Q_d{R}_b+C_m)}{T^3}-\frac{2Z\prime (Q_d{R}_b+C_m)}{T^2}}\\ \multicolumn{1}{c}{+\frac{Z\prime (Q_d{R}_b+C_m)}{T}-\frac{C_h{Q}_{d}Z″}{2}}\end{array}$$

(15)

Proof

See Appendix 2.

Probability model

Geometrical distribution

Here, a type II PM is performed as geometrical distribution.

This model is considered by Nakagawa. ${\overline{P}}_0=1;{\overline{P}}_1=q^j,0\le q<1,\overline{q}=1-q$ . The breakdown cost is as follows

$$C_m{\overline{q}}^{-2}\sum _{j=1}^{\infty }{q}^{j-1}\underset{0}{\overset{j\left(\frac{D}{P}\right)T}{\int }}r\left(t\right)\mathrm{dt}$$

So, JTEC will become

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{JTEC}=\frac{C_s}{T}+\frac{C_{\mathrm{pm}}}{T}+\frac{\mathrm{TD}}{2m}{C}_h{C}_v}\\ \multicolumn{1}{c}{\left[(m(1-\frac{D}{P})-1+\frac{2D}{P})\right]}\\ \multicolumn{1}{c}{+\left[(\frac{C_m}{T}+\frac{Q_d{R}_b}{T}-\frac{C_h{Q}_d}{2}){\overline{q}}^{-2}\sum _{j=1}^{\infty }{q}^{j-1}\underset{0}{\overset{j\left(\frac{D}{P}\right)T}{\int }}r\left(t\right)\mathrm{dt}\right]}\\ \multicolumn{1}{c}{+\frac{C_{0}m}{T}+C_h{C}_P\frac{\mathrm{TD}}{2m}}\end{array}$$

(16)

Let

$${\overline{q}}^{-2}\sum _{j=1}^{\infty }{q}^{j-1}\underset{0}{\overset{j\left(\frac{D}{P}\right)T}{\int }}r\left(t\right)\mathrm{dt}=Z$$

(17)

$$\frac{\partial JTEC\left(T,m\right)}{\partial T}=-\frac{\left(C_{o}m+C_s+C_{pm}+\left[\left(Q_d{R}_b+C_m\right)Z\right]\right)}{T^2}+\left[\left(\frac{C_m}{T}+\frac{Q_d{R}_b}{T}-\frac{C_h{Q}_d}{2}\right)Z^{\prime }\right]+\frac{C_h{C}_{P}D}{2m}+\frac{C_h{C}_{v}D}{2m}\left[m\left(1-\frac{D}{P}\right)-1+\frac{2D}{P}\right]$$

(18)

And we do second-order partial derivatives, if equation (22) > 0, then JTEC will exist a local minimal solution

$$\begin{array}{c}\multicolumn{1}{c}{\frac{{\partial }^2\mathrm{JTEC}(T,m)}{{\partial }^{2}T}=\frac{2(C_{o}m+C_s+C_{\mathrm{pm}})}{T^3}}\\ \multicolumn{1}{c}{+\frac{2Z(Q_d{R}_b+C_m)}{T^3}-\frac{2Z\prime (Q_d{R}_b+C_m)}{T^2}}\\ \multicolumn{1}{c}{+\frac{Z\prime (Q_d{R}_b+C_m)}{T}-\frac{C_h{Q}_{d}Z″}{2}}\end{array}$$

(19)

Proof

See Appendix 3.

Weibull distribution

In the Weibull distribution, ${\overline{P}}_0=1;{\overline{P}}_j=q^{j\beta }(j=1,2,\dots )$ , $0\le q\le 1,0<\beta <1$

The breakdown cost is as follows

$$C_m\sum _{j=1}^{\infty }\frac{q^{{(j-1)}^{\beta }}-q^{j\beta }}{\sum _{j=1}^{\infty }{q}^{{(j-1)}^{\beta }}}\underset{0}{\overset{j\left(\frac{D}{P}\right)T}{\int }}r\left(t\right)\mathrm{dt}$$

So, the JTEC will become

$$\begin{array}{l}JTEC\left(T,m\right)=\frac{C_s}{T}+\frac{C_{pm}}{T}+\frac{TD}{2m}{C}_h\\ C_v\left[\left(m\left(1-\frac{D}{P}\right)-1+\frac{2D}{P}\right)\right]+\left[\left(\frac{C_m}{T}+\frac{Q_d{R}_b}{T}-\frac{C_h{Q}_d}{2}\right){\displaystyle \sum _{j=1}^{\infty }\frac{q^{{\left(j-1\right)}^{\beta }}-q^{j\beta }}{{\displaystyle \sum _{j=1}^{\infty }{q}^{{\left(j-1\right)}^{\beta }}}}}{\displaystyle \underset{0}{\overset{j\left(\frac{D}{P}\right)T}{\int }}r}\left(t\right)dt\right]\\ +\frac{C_{0}m}{T}+C_h{C}_P\frac{TD}{2m}\end{array}$$

(20)

Let

$$\sum _{j=1}^{\infty }\frac{q^{{(j-1)}^{\beta }}-q^{j\beta }}{\sum _{j=1}^{\infty }{q}^{{(j-1)}^{\beta }}}j\frac{D}{P}r\left(\mathrm{jT}\frac{D}{P}\right)=Z$$

(21)

$$\frac{\partial JTEC\left(T,m\right)}{\partial T}=-\frac{\left(C_{o}m+C_s+C_{pm}+\left[\left(Q_d{R}_b+C_m\right)Z\right]\right)}{T^2}+\left[\left(\frac{C_m}{T}+\frac{Q_d{R}_b}{T}-\frac{C_h{Q}_d}{2}\right)Z^{\prime }\right]+\frac{C_h{C}_{P}D}{2m}+\frac{C_h{C}_{v}D}{2m}\left[m\left(1-\frac{D}{P}\right)-1+\frac{2D}{P}\right]$$

(22)

And we do second-order partial derivatives, if equation (22) > 0, then JTEC will exist a local minimal solution

$$\begin{array}{c}\multicolumn{1}{c}{\frac{{\partial }^2\mathrm{JTEC}(T,m)}{{\partial }^{2}T}=\frac{2(C_{o}m+C_s+C_{\mathrm{pm}})}{T^3}}\\ \multicolumn{1}{c}{+\frac{2Z(Q_d{R}_b+C_m)}{T^3}-\frac{2Z\prime (Q_d{R}_b+C_m)}{T^2}}\\ \multicolumn{1}{c}{+\frac{Z\prime (Q_d{R}_b+C_m)}{T}-\frac{C_h{Q}_{d}Z″}{2}}\end{array}$$

(23)

Proof

See Appendix 4.

Learning effect

In this study, the learning effect is less noticeable while the number of PM times is becoming large. Therefore, the probability model is developed in the following discussion with the subsequent assumptions:

1. ${\overline{P}}_{j-1}>{\overline{P}}_j(j=1,2,\dots )$

2. ${\overline{P}}_0=1;{\overline{P}}_1\ne 0;{\overline{P}}_j={\overline{P}}_1\times j^b(j=1,2,\dots )$ , b is learning rate.

3. $b=\frac{\log r}{\log 2}$

The breakdown costs are shown as follows based on the research by Liao et al. ²³

$$C_m\left[\frac{1-{\overline{P}}_1}{1+{\overline{P}}_1{\displaystyle \sum _{j=2}^{\infty }{\left(j-1\right)}^b}}{\displaystyle \underset{0}{\overset{j\frac{D}{P}T}{\int }}r}\left(t\right)dt+{\overline{P}}_1{\displaystyle \sum _{j=2}^{\infty }\frac{{\left(j-1\right)}^b-j^b}{1+{\overline{P}}_1{\displaystyle \sum _{j=1}^{\infty }{\left(j-1\right)}^b}}}{\displaystyle \underset{0}{\overset{j\frac{D}{P}T}{\int }}r}\left(t\right)dt\right]$$

So, the JTEC will become

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{JTEC}(T,m)=\frac{C_s}{T}+\frac{C_{\mathrm{pm}}}{T}+\frac{\mathrm{TD}}{2m}{C}_h{C}_v\left[(m(1-\frac{D}{P})-1+\frac{2D}{P})\right]}\\ \multicolumn{1}{c}{+\left[(\frac{C_m}{T}+\frac{Q_d{R}_b}{T}-\frac{C_h{Q}_d}{2})Z\right]+\frac{C_{0}m}{T}+C_h{C}_P\frac{\mathrm{TD}}{2m}}\end{array}$$

(24)

Let

$$\left[\frac{1-{\overline{P}}_1}{1+{\overline{P}}_1{\displaystyle \sum _{j=2}^{\infty }{\left(j-1\right)}^b}}c{\displaystyle \underset{0}{\overset{j\frac{D}{P}T}{\int }}r}\left(t\right)dt+{\overline{P}}_1{\displaystyle \sum _{j=2}^{\infty }\frac{{\left(j-1\right)}^b-j^b}{1+{\overline{P}}_1{\displaystyle \sum _{j=1}^{\infty }{\left(j-1\right)}^b}}}{\displaystyle \underset{0}{\overset{j\frac{D}{P}T}{\int }}r}\left(t\right)dt\right]=Z$$

$$\begin{array}{l}\frac{\partial JTEC\left(T,m\right)}{\partial T}=-\frac{\left(C_{o}m+C_s+C_{pm}+\left[\left(Q_d{R}_b+C_m\right)Z\right]\right)}{T^2}\\ +\left[\left(\frac{C_m}{T}+\frac{Q_d{R}_b}{T}-\frac{C_h{Q}_d}{2}\right)Z^{\prime }\right]\\ +\frac{C_h{C}_{P}D}{2m}+\frac{C_h{C}_{v}D}{2m}\left[m\left(1-\frac{D}{P}\right)-1+\frac{2D}{P}\right]\end{array}$$

(25)

And we do second-order partial derivatives, if equation (22) > 0, then JTEC will exist a local minimal solution

$$\begin{array}{c}\multicolumn{1}{c}{\frac{{\partial }^2\mathrm{JTEC}(T,m)}{{\partial }^{2}T}=\frac{2(C_{o}m+C_s+C_{\mathrm{pm}})}{T^3}}\\ \multicolumn{1}{c}{+\frac{2Z(Q_d{R}_b+C_m)}{T^3}-\frac{2Z\prime (Q_d{R}_b+C_m)}{T^2}}\\ \multicolumn{1}{c}{+\frac{Z\prime (Q_d{R}_b+C_m)}{T}-\frac{C_h{Q}_{d}Z″}{2}}\end{array}$$

(26)

Proof

See Appendix 5.

Numerical example

To illustrate the effectiveness of the models presented above, we examine a simplified numerical study. In this example, we are trying to find out an optimal value of T and m which is shown by the following four steps:

• Step 1. Let m be equal to the minimum feasible value 1.

• Step 2. Calculate T using relation.

• Step 3. Calculate JTEC by embedding the last calculates T and m. If $T{C}_m<T{C}_{m-1}$ , let m = m + 1 and go to Step 2; otherwise, go to Step 4.

• Step 4. Find the minimum JTEC and the corresponding value of decision variables T and m as the optimal solution.

In order to illustrate the above solution procedure, we consider an inventory problem has the following data which are based on our discussion (Table 1) in order to represent more real-world situations.

Table 1.

Parameter setting.

Parameter	Cost/number
D	600 unit/year
P	1000 unit/year
$C_h$	0.2 per unit
$C_v$	$20 per unit
$C_p$	$25 per unit
$C_o$	$20 each order
$C_s$	$30 per run
$C_{\mathrm{PM}}$	$20 per run
$C_m$	$10 each time
$Q_d$	1
$R_b$	$6
r(t)	Following a uniform distribution with a = 0.1, b = 0.4
g(t)	Following a Gamma distribution with $\alpha =0.02$ , $\beta =2$

Numerical results sensitive analysis

In this part, we use the uniform and gamma distribution into the shot down rate, and the results are illustrated as follows.

From Tables 2–9 and Figures 3–10, we can see when lot size m is 3; JTECs are decreasing. So we can find the corresponding value of the decision variables T and m as the optimal solution and the range of JTEC is 704–733.

Table 2.

Basic model with uniform distribution.

m	p = 0.01		p = 0.05		p = 0.1		p = 0.15
	JTEC	T	JTEC	T	JTEC	T	JTEC	T
1	819.3926	0.17773	818.141	0.177723	816.5766	0.177715	815.0122	0.177707
2	727.7938	0.258578	726.5551	0.258563	725.0075	0.258199	723.4586	0.258525
3	713.338	0.322742	712.1096	0.322718	710.5751	0.322139	709.0387	0.322657
4	719.9683	0.377749	718.7487	0.377716	717.2256	0.37693	715.6998	0.377633
5	734.8329	0.42665	733.6211	0.426608	732.1081	0.425628	730.5918	0.426505
6	753.419	0.471084	752.2143	0.471035	750.7106	0.469871	749.2028	0.470912
7	773.7792	0.512061	772.5811	0.512004	771.0858	0.510667	769.5859	0.511863
8	794.9451	0.550268	793.7531	0.550204	792.2657	0.548703	790.7732	0.550046
9	816.3519	0.586232	815.1657	0.586162	813.6857	0.584506	812.2001	0.585987
10	837.9306	0.620155	836.7498	0.620078	835.2769	0.618274	833.7978	0.619888

JTEC: joint total expected cost.

Table 3.

Geometrical model with uniform distribution.

m	p = 0.01		p = 0.05		p = 0.1		p = 0.15
	JTEC	T	JTEC	T	JTEC	T	JTEC	T
1	816.6609	0.177651	814.0137	0.177647	813.8364	0.177643	813.2417	0.177823
2	725.2204	0.25839	722.5813	0.258381	722.4121	0.258371	721.4551	0.258802
3	710.8903	0.322443	708.2543	0.32243	708.0947	0.322413	706.8503	0.323098
4	717.6283	0.377343	715.001	0.377322	714.8438	0.377302	713.3529	0.378231
5	732.5886	0.426143	729.9662	0.426117	729.8139	0.426092	728.1039	0.427252
6	751.2617	0.470483	748.6438	0.470452	748.4958	0.470422	746.5868	0.471799
7	771.7021	0.51137	769.0883	0.511335	768.9444	0.511299	766.8518	0.512883
8	792.9429	0.549492	790.3328	0.549452	790.1927	0.549413	787.9289	0.55119
9	814.4201	0.585376	811.8136	0.585332	811.6771	0.585289	809.2522	0.587251
10	836.0652	0.619222	833.4621	0.619174	833.329	0.619127	830.752	0.621264

JTEC: joint total expected cost.

Table 4.

Weibull model with uniform distribution.

m	p = 0.01		p = 0.05		p = 0.1		p = 0.15
	JTEC	T	JTEC	T	JTEC	T	JTEC	T
1	817.5988	0.169365	814.2299	0.177703	810.8122	0.177707	808.17	0.177739
2	726.2454	0.244228	722.6844	0.258515	719.2586	0.258525	716.5517	0.258601
3	711.921	0.30381	708.271	0.322642	704.8387	0.322657	702.0805	0.322779
4	718.6303	0.355175	714.9376	0.377612	711.4998	0.377633	708.6976	0.377798
5	733.5449	0.401099	729.8344	0.426479	726.3918	0.426505	723.5505	0.426712
6	752.1637	0.443045	748.4499	0.470882	745.0028	0.470912	742.1259	0.471158
7	771.1906	0.510667	768.8371	0.511828	765.3859	0.511863	762.4763	0.512146
8	793.7255	0.518281	790.0282	0.550006	786.5732	0.550046	783.633	0.550363
9	815.1416	0.552641	811.4587	0.585943	808.0001	0.585987	805.2065	0.574073
10	835.4152	0.62604	833.0598	0.61984	829.5978	0.619888	826.6017	0.620269

JTEC: joint total expected cost.

Table 5.

Learning curve model with uniform distribution.

m	r = 0.8		r = 0.75		r = 0.7		r = 0.65
	JTEC	T	JTEC	T	JTEC	T	JTEC	T
1	797.803	0.177619	799.3675	0.177627	804.7712	0.177491	808.7543	0.177675
2	706.4271	0.258314	707.9754	0.258333	713.6532	0.258008	717.2654	0.258448
3	692.1482	0.322322	693.6837	0.322352	699.5788	0.321836	702.8968	0.322535
4	698.9301	0.377178	700.4547	0.377219	706.536	0.376519	709.6019	0.377467
5	713.9295	0.425937	715.4443	0.425989	721.6912	0.425115	724.5329	0.426298
6	732.638	0.470238	734.144	0.470299	740.5412	0.469262	743.1794	0.470667
7	753.1111	0.511089	754.6089	0.511159	761.1448	0.509968	763.5953	0.511581
8	774.3823	0.549176	775.8725	0.549255	782.5377	0.547918	784.8131	0.549729
9	795.8882	0.585027	797.3712	0.585114	804.1581	0.583639	806.2688	0.585638
10	817.5604	0.618842	819.0366	0.618937	825.9383	0.617331	827.8936	0.619507

JTEC: joint total expected cost.

Table 6.

Basic model with gamma distribution.

m	p = 0.01		p = 0.05		p = 0.1		p = 0.15
	JTEC	T	JTEC	T	JTEC	T	JTEC	T
1	803.6668	0.177649	803.1144	0.177646	802.3408	0.177642	801.5672	0.177638
2	712.2315	0.258384	711.6854	0.258377	710.9197	0.258367	710.154	0.258358
3	697.9055	0.322434	697.3644	0.322422	696.6049	0.322407	695.8455	0.322392
4	704.6471	0.37733	704.1102	0.377313	703.3562	0.377293	702.6021	0.377273
5	719.6106	0.426127	719.0775	0.426106	718.3282	0.426081	717.5789	0.426056
6	738.2865	0.470463	737.757	0.470439	737.012	0.470409	736.267	0.470379
7	758.7296	0.511347	758.2032	0.51132	757.4622	0.511285	756.7213	0.511251
8	779.9727	0.549466	779.4494	0.549435	778.7121	0.549397	777.9749	0.549358
9	801.4523	0.585348	800.9317	0.585313	800.198	0.585271	799.4642	0.585228
10	823.0996	0.619191	822.5816	0.619154	821.8512	0.619108	821.1208	0.619061

JTEC: joint total expected cost.

Table 7.

Geometrical model with gamma distribution.

m	p = 0.01		p = 0.05		p = 0.1		p = 0.15
	JTEC	T	JTEC	T	JTEC	T	JTEC	T
1	805.7789	0.177579	803.9034	0.17761	803.4099	0.17765	802.9116	0.177691
2	714.4824	0.258218	712.5454	0.258292	711.9714	0.258388	711.3904	0.258486
3	700.2665	0.322169	698.2807	0.322288	697.6429	0.32244	696.9963	0.322596
4	707.1024	0.376971	705.0749	0.377132	704.3824	0.377338	703.6795	0.37755
5	722.1498	0.425679	720.0851	0.42588	719.3439	0.426137	718.5911	0.426401
6	740.902	0.469931	738.8035	0.47017	738.0812	0.470475	737.2199	0.470789
7	761.4153	0.510736	759.2857	0.51101	758.4596	0.511361	757.6194	0.511722
8	782.724	0.54878	780.5653	0.549088	779.7013	0.549482	778.8224	0.549358
9	804.2652	0.58459	802.0792	0.58493	801.1794	0.585365	800.2634	0.585812
10	825.9706	0.618367	823.7589	0.618737	822.8254	0.61921	821.8747	0.619696

JTEC: joint total expected cost.

Table 8.

Weibull model with gamma distribution.

m	p = 0.01		p = 0.05		p = 0.1		p = 0.15
	JTEC	T	JTEC	T	JTEC	T	JTEC	T
1	801.8766	0.177639	801.1804	0.177636	800.7797	0.177637	799.1249	0.177653
2	710.4603	0.258362	709.7711	0.258353	709.3681	0.258356	707.6801	0.258395
3	696.1493	0.322398	695.4658	0.322384	695.0608	0.322389	693.3466	0.322451
4	702.9037	0.377281	702.2251	0.377263	701.8185	0.377269	700.0817	0.377354
5	717.8787	0.426066	717.2043	0.426043	716.7963	0.426051	715.0395	0.426157
6	736.565	0.470391	735.8946	0.470364	735.4853	0.470373	733.7102	0.470499
7	757.0177	0.511265	756.3508	0.511234	755.9403	0.511244	754.1485	0.511388
8	778.2698	0.549373	777.6062	0.549339	777.1947	0.54935	775.3872	0.549512
9	799.7577	0.585245	799.0974	0.585207	798.6847	0.58522	796.8625	0.585399
10	821.4129	0.61908	820.7555	0.619038	820.3419	0.619052	818.5058	0.619247

JTEC: joint total expected cost.

Table 9.

Learning curve model with gamma distribution.

m	r = 0.8		r = 0.75		r = 0.7		r = 0.65
	JTEC	T	JTEC	T	JTEC	T	JTEC	T
1	797.6969	0.177619	799.2436	0.177627	802.1608	0.17804	806.2831	0.178181
2	706.322	0.258312	707.8528	0.258331	709.9372	0.259322	713.7747	0.259662
3	692.0439	0.32232	692.5734	0.32235	694.9852	0.323927	698.5963	0.324469
4	698.8266	0.377175	699.3453	0.377216	701.19	0.379357	704.6067	0.380094
5	713.8266	0.425934	714.3357	0.425984	715.6762	0.428659	718.92	0.429581
6	732.5358	0.470234	733.0361	0.470294	733.9184	0.473472	737.0051	0.474568
7	753.0094	0.511084	753.5016	0.511153	753.9615	0.514806	756.9032	0.516067
8	774.2812	0.549171	774.7658	0.549249	774.8316	0.553351	777.6382	0.554768
9	795.7876	0.585021	796.2651	0.585107	795.96	0.589637	798.6393	0.591203
10	817.4602	0.618836	817.9311	0.61893	817.2761	0.623864	819.8354	0.62557

JTEC: joint total expected cost.

Figure 3.

Basic model with uniform distribution.

Figure 4.

Geometrical model with uniform distribution.

Figure 5.

Weibull model with uniform distribution.

Figure 6.

Learning curve model with uniform distribution.

Figure 7.

Basic model with gamma distribution.

Figure 8.

Geometrical model with gamma distribution.

Figure 9.

Weibull model with gamma distribution.

Figure 10.

Learning curve model with gamma distribution.

In the basic model, geometrical and Weibull with uniform and gamma distribution; when maintenance factor increases, the JTEC will decrease. In the learning curve model, the basic value of the maintenance factor differs from other models. If the maintenance factor decreases, the JTEC will increase.

Sensitive analysis

In this section, we apply the sensitive analysis to analyze our integrated inventory mode three key impact factors: D/P, $C_{\mathrm{pm}}$ and $Q_d$ . The abbreviations of eight cases are shown in Table 10.

Table 10.

The abbreviations of eight cases.

Case	Full name	Abbreviation
1	Basic model with uniform distribution	B-U
2	Geometrical model with uniform distribution	G-U
3	Weibull model with uniform distribution	W-U
4	Learning curve model with uniform distribution	L-U
5	Basic model with gamma distribution	B-G
6	Geometrical model with gamma distribution	G-G
7	Weibull model with gamma distribution	W-G
8	Learning curve model with gamma distribution	L-G

According to the results from Table 11, the joint total cost is increasing while the demand and production ratio D/P is increasing. That is because the D/P ratio influences the holding cost and purchase cost. Table 12 shows that when we increase the PM cost $C_{\mathrm{pm}}$ , the joint total cost will increase.

Table 11.

Sensitive analysis of D/P ratio for JTEC.

Case	Ratio
	0.4	0.5	0.6	0.7	0.8
1	616.5152	667.0782	713.338	755.2945	792.9478
2	603.615	654.0886	700.2665	742.1488	779.7355
3	620.7779	668.3748	711.921	751.4163	786.8609
4	627.8517	678.3488	724.5482	766.45	804.0542
5	601.1754	651.69	697.9055	739.8219	777.4391
6	587.8031	638.2762	684.4538	726.3357	763.9221
7	599.4299	649.9389	696.1493	738.061	775.6741
8	595.3481	645.8448	692.0439	733.9455	771.5494

Table 12.

Sensitive analysis of PM cost for JTEC.

Case	$C_{\mathrm{pm}}$
	15	20	25	30	35
1	697.8458	713.338	728.8302	744.3244	759.8147
2	684.7482	700.2665	715.7848	731.3031	746.8215
3	695.4633	711.921	728.3786	744.8362	761.2939
4	709.0357	724.5482	740.0606	755.5731	771.0855
5	682.3984	697.9055	713.4126	728.9196	744.4267
6	668.934	684.4538	699.9736	715.4934	731.0132
7	680.6405	696.1493	711.6581	727.1668	742.6756
8	676.5314	692.0439	707.5565	723.069	738.5816

Then, Table 13 shows that when we increase the number of non-reworkable defective products $Q_d$ , the joint total cost will increase. It is because that the backorder cost is higher than the reduction in holding cost of $Q_d$ ; therefore, the total joint total cost increases while $Q_d$ increases. And, Table 14 indicates the definition of X-scale from Figures 11 to 18 for clear presentations.

Table 13.

Sensitive analysis of $Q_d$ for JTEC.

Case	$Q_d$
	1	3	5	7	9
1	713.338	734.5419	755.7457	776.9496	798.1535
2	700.2665	713.1786	726.0908	739.0029	751.915
3	711.921	733.605	750.8971	770.3852	789.8732
4	690.601	698.5746	705.001	711.4275	717.8539
5	697.9055	709.5183	718.6348	728.9995	748.0737
6	684.4538	694.5384	701.405	708.5236	715.3902
7	696.1493	706.4712	714.5997	723.8249	733.0501
8	692.0439	699.1624	704.7512	711.1049	717.4586

Table 14.

The definition of X-scale (from Figures 11 to 18).

X-scale	1	2	3	4	5
D/P	0.4	0.5	0.6	0.7	0.8
$C_{\mathrm{pm}}$	15	20	25	30	35
$Q_d$	1	3	5	7	9

Figure 11.

Sensitive analysis of basic model with uniform distribution.

Figure 12.

Sensitive analysis of geo model with uniform distribution.

Figure 13.

Sensitive analysis of Weibull model with uniform distribution.

Figure 14.

Sensitive analysis of learning curve model with uniform distribution.

Figure 15.

Sensitive analysis of basic model with gamma distribution.

Figure 16.

Sensitive analysis of geo model with gamma distribution.

Figure 17.

Sensitive analysis of Weibull model with gamma distribution.

Figure 18.

Sensitive analysis of learning curve model with gamma distribution.

From Figures 11–18, we can see that among the three impact factors, the D/P ratio has the highest influence with integrated model.

Conclusion

Basically, in this article, we are considering the situation of both vendors and buyers which indicate that the buyer’s cycle is Q/D and the whole integrated cycle is DQ/m. In other words, our research extends past model based on the buyer’s ordering cycle which is different against Liao’s model. To sum up, this research focuses on the integrated inventory model development which has the difference of Liao’s EPQ model.

In this research, five of these findings are worth summarizing: (1) the analysis of PM strategy, production process’s fail and minimal repair; (2) the considerations of reworkable defective products at each failure and backorder with non-reworkable defective products at each failure during the production processes; (3) developing an integrated inventory model within the vendor’s inventory model and buyer’s inventory model; (4) the application of four PM probability and two shot down probability in our proposed model; and finally (5) providing an example to illustrate how the proposed model works. To sum up, we present the results of our integrated model with one vendor one buyer consideration and an effect of the modified production processes via the detailed numerical example.

Based on the results, the demand and production (D/P ratio) influences the holding cost and purchase cost and has the highest impact on the integrated model.

Future work will hopefully involve the real-world concern to our research which will be our main target. In addition, we do hope our research can be extended into more considerable areas such as multi-buyer or multi-vendor problems. Also we can apply more considerable variables such as PM time interval, permissible delay in payments and test error.

In spite of all the limitations of our research, we believe that the findings from our study are intriguing enough to encourage further research into the topic of an integrated inventory model.