Impact of uncertainty ascribed to defective products on supply chains

ABSTRACT

Traditional inventory management has considered when and how much to order against the uncertainty of demand. Although many studies on inventory management have presumed that quality of products received from manufacturers is perfect, defective products would be, in actual, contained in the procurement to retailers. The inclusion of defective products causes the uncertainty in supply. The retailer does not necessarily require being responsible for all the loss by overstock and shortage of products due to the uncertainty ascribed to the inclusion of defective products. The purpose of this paper is to investigate the impact of the uncertainty ascribed to the inclusion of defective products on a supply chain. This paper first has formulated a simple supply chain model with defective products composed of the manufacturer and retailer. The model has made it clear how the uncertainty ascribed to the inclusion of defective products has an impact on the order quantity and respective profits. Then, this paper has discussed a method of assessing a latent loss as a monetary value. Combining contract techniques and game theory, this paper has proposed a contracting method that the manufacturer compensates the retailer for the latent loss to maintain a supply chain partnership.

KEYWORDS:

• Compensation contract
• coordination
• game theory
• incentive compatible condition
• newsvendor model
• supply chain management

1. Introduction

In common, traditional inventory management has considered when and how much to order against the uncertainty of demand. The inventory management methodology has been developed for overstock and shortage ascribed to the uncertainty of demand. The uncertainty of demand has caused a more crucial problem in supply chains. It is well known that fluctuations in demand at wholesaler and manufacturer levels can become larger than fluctuations in demand at a retailer level in supply chains. This phenomenon, especially, is called the bullwhip effect (H. L. Lee et al., 1997). The expansion of uncertainty in demand by the bullwhip effect is caused by various factors such as inaccurate forecasts and price variations.

In addition to the uncertainty of demand, the uncertainty of supply also must be considered in inventory management. Quality of products is considered as one of the factors causing the uncertainty in supply. Although the major part of research on inventory management has presumed that quality of products received from manufacturers is perfect, several defective products would, in actuality, have been contained in the procurement to retailers. The inclusion of defective products causes the uncertainty in supply, such as whether a defective product is contained or not, or how many defective products are contained. While the uncertainty of demand occurs at a retailer level and is propagated to wholesaler and manufacturer levels in supply chains, the uncertainty of supply occurs at a manufacturer level and is propagated to wholesaler and retailer levels in supply chains.

The increased sophistication and complexity of technologies and products, and also tough competition with other companies have increased the significance for the quality of products. Several studies on inventory management with defective products has been considered until now. Gao et al. (2016) have discussed quality improvement efforts coordination in supply chains with partial compensation cost allocation between a supplier and a manufacturer based on the information derived from failure root cause analysis under the situation that defective products are delivered to consumers. Yu and Chen (2018) have called a useful portion of defective products acceptable defective products, and investigated an integrated inventory model with acceptable defective products. Xie et al. (2021) have considered contract design considering recall efforts in a supply chain under the situation that some products are returned because of safety problems or product defects. The previous studies including mentioned above have treated several problems directly caused by defectives themselves, such as the cost allocation of compensation for the delivery of defective products to consumers and incentives for quality improvement efforts in supply chain members.

On one hand, the uncertainty ascribed to the inclusion of defective products is also a problem to be considered. Even though no defective product is delivered to a consumer, the inclusion of defective products causes a latent loss that should be avoided to supply chain members. For examples, shortage in stock is caused for thinking light of the inclusion of defective products. Production is halt for shortage of materials in the position of manufacturers and opportunity loss is increased for shortage of products in the position of retailers. Also, any extra in stock is held because of paying special attention to the inclusion of defective products.

The uncertainty of demand causes overstock and shortage of products and then the extra cost for overstock and shortage of products is imposed on the retailer. To minimize the loss by overstock and shortage of products, the retailer has an optimal plan for ordering quantity. In this case, the retailer is responsible for all the loss by overstock and shortage of products due to the uncertainty of demand. The loss by overstock and shortage of products is naturally not imposed on consumers. On one hand, the uncertainty ascribed to the inclusion of defective products also causes overstock and shortage of products and then the extra cost for overstock and shortage of products is imposed on the retailer. However, the retailer does not necessarily require being responsible for all the loss by overstock and shortage of products due to the uncertainty ascribed to the inclusion of defective products. That is, the compensation by the manufacturer to the retailer for the uncertainty ascribed to the inclusion of defective products should be considered, but compensation for the uncertainty ascribed to the inclusion of defective products remains poorly defined.

This paper focuses on the uncertainty ascribed to the inclusion of defective products. Then, the purpose of this paper is to investigate the impact of the uncertainty ascribed to the inclusion of defective products on a supply chain. As mentioned already, an explicit loss caused by defectives themselves can be estimated for compensation as a monetary value. The uncertainty ascribed to the inclusion of defective products can be evaluated as variation, but monetary compensation for the uncertainty ascribed to the inclusion of defective products remains unexplored. This paper discusses a method for assessing a latent loss causing the uncertainty ascribed to the inclusion of defective products as a monetary value. Through the discussion, we have proposed a contracting method that the manufacturer compensates the retailer for the latent loss ascribed to the uncertainty of defective products to maintain a supply chain partnership.

The rest of this paper is as follows: Several studies on inventory management related to defective products are introduced as a literature review at first. Then, we formulate and investigate a supply chain model with defective products. The subsequent section assesses the latent loss caused by the uncertainty ascribed to the inclusion of defective products as a monetary value and then discusses compensation to maintain a supply chain partnership combining contract techniques and game theory. Finally, we conclude this paper.

2. Literature review

We review several previous studies on production and inventory management associated with defective products in this section.

A lot of traditional studies has treated the situation that a portion of defective products is produced from a production process or is contained in a procurement from a supplier. Porteus (1986), which is one of the early studies on defective products in production and inventory management, has studied the impact of defective products on a basic lot-sizing problem under the assumption that a process goes out of control with a fixed probability. Rosenblatt and Lee (1986) also have investigated the impact of imperfect production on production cycles. Moinzadeh and Lee (1987), on one hand, have considered a continuous review inventory model with Poisson demand and defective products. Paknejad et al. (1995), in addition, have studied an optimal policy in a continuous review inventory model with defective products for any demand distribution. Several papers on production and inventory management associated with defective products has been published, see Salameh and Jaber (2000), Wu and Ouyang (2001), Eroglu and Ozdemir (2007), Khan et al. (2011), and Hsu and Hsu (2013, 2014). In recent years, Park (2021) has determined the optimal manufacturing quantity for an economic production lot-sizing model with a defective rate and preventive maintenance for a manufacturing machine. Zhang et al. (2023) have considered the integrated optimization of the economic manufacturing quantity and the condition-based maintenance policy in the situation that a proportion of defective products is fabricated under a specified state. The studies mentioned above have investigated an optimal policy of production or ordering quantity.

During the recent 20 years, many researchers regarding production and inventory management have considered supply chain models. The purpose of studies based on supply chain models consisting of a manufacturer and a retailer is to coordinate operations between a manufacturer and a retailer and then establish cooperation between a manufacturer and a retailer. So, for this purpose, some methods for the joint determination of a manufacturer and a retailer have been proposed under the situation that a proportion of defective products is produced or contained. Huang (2002, 2004) has extended the economical ordering quantity (EOQ) model in Salameh and Jaber (2000) to establish an integrated inventory policy for supply chain members where the manufacturer incurs a warranty cost for defective products. Khan and Jaber (2009) have proposed a three-level supply chain model by extending the EOQ model in Salameh and Jaber (2000) where a penalty cost arises in the case that the portion of defective products goes beyond a prescribed value. In recent years, Almaraj and Trafalis (2019, 2022) have considered a closed-loop supply chain model with an assumption of imperfect quality production.

Further, many researchers have considered several problems in supply chain models, such as compensation cost allocation in the case that a defective product is delivered to a consumer, and a mechanism of facilitating quality improvement efforts. Chao et al. (2009) first introduced a partial allocation contract between a manufacturer and a retailer for allocating a penalty against selling defective products to consumer. Gao et al. (2016) have shown that the partial cost allocation contract coordinates the quality improvement efforts of a manufacturer and a retailer. Yu and Chen (2018) have investigated an integrated inventory model with acceptable defective products under warranty and quality improvement. Under the situation that a retailer determines a retail price of a product to consumers while a manufacturer is responsible for the production and product quality, Yoo and Cheong (2018) have developed analytical models incorporating reward schemes as a mechanism of how the retailer can facilitate the quality improvement efforts by the manufacturer. Hu et al. (2019) have considered a manufacturer’s quality inspection avoidance behaviour model and investigated the impact of product quality effort by a manufacturer on profit and others of a retailer. Xie et al. (2021) have considered contract design considering recall efforts in a supply chain under the situation that some products are returned to a manufacturer because of safety problems or product defects. Gu et al. (2021) have considered a fresh-product supply chain model with quality-improvement effort and fresh-keeping effort provided that the products deteriorate.

As mentioned above, a variety of contract techniques have been proposed and practiced for solving any problems among supply chain members until now (Cachon, 2003). Arizono and Takemoto (2012) and Takemoto and Arizono (2013) have considered a buyback contract on publishing supply chains in Japan. Also, for solving a moral hazard problem in supply chains, Takemoto and Arizono (2020) have considered a supply chain contract with capacity reservation realizing collaborative coordination between supply chain members. C. H. Lee et al. (2013) have proposed the quality compensation contract in which a manufacturer compensates a retailer for defective products inadvertently sold to consumers. Their study has offered, for the first time as an incentive scheme in a supply chain, the quality compensation contract that has been long employed for quality control in numerous industries. Other related literature could be found in the review paper by Cogollo-Flórez and Correa-Espinal (2019).

The previous studies mentioned above have basically considered the direct impact of defective products themselves. On one hand, this paper focuses on the uncertainty ascribed to the inclusion of defective products. The retailer does not necessarily require being responsible for all the loss by overstock and shortage of products due to the uncertainty ascribed to the inclusion of defective products. The compensation for the uncertainty ascribed to the inclusion of defective products should be considered. Then, we discuss a method for assessing a latent loss caused by the uncertainty ascribed to the inclusion of defective products as a monetary value. we have proposed a contracting method that the manufacturer compensates the retailer for the latent loss ascribed to the uncertainty of defective products to maintain a supply chain partnership.

3. Model formulation and investigation

In this section, we formulate a mathematical model to analyse the uncertainty ascribed to the inclusion of defective products. We, especially, employ a single period newsvendor model as simple and clear because the traditional supply chain models have been frequently analysed using a newsvendor model. The following notations are defined in this paper:

$\mathit{r}$ : retail price at retailer.

$\mathit{w}$: wholesale price at manufacturer.

$\mathit{c}$: original cost at manufacturer.

$\mathit{s}$: salvage value for unsold products at retailer.

$\mathit{q}$: order quantity of retailer.

$\mathit{d}$: demand to retailer.

$\mathit{p}$: probability of being a defective product.

$\mathit{n}$: number of defective products contained in procurement.

Then, following assumptions are also considered:

• The demand $\mathit{d}$ is an integer number and a discrete random variable. In association with it, the order quantity $\mathit{q}$ is also an integer number and then a decision variable.

• The number of defective products $\mathit{n}$ follows a binomial distribution with $\mathit{q}$ and $\mathit{p}$.

• The salvage value for unsold products at the retailer $\mathit{s}$ is less than the wholesale price at the manufacturer $\mathit{w}$.

• All defective products contained in the procurement are assumed to be discovered and removed at the retailer level. In several previous studies, a portion of defective products has been provided to consumers. In such a case, it is difficult to estimate the impact of the uncertainty ascribed to the inclusion of defective products because the timing that the defective products are recognized as being defective is unknown. Hence, the assumption has been set.

• In this section, all defective products discovered by the retailer are returned to the manufacturer while the wholesale price $\mathit{w}$ is refunded to the retailer for every defective product, namely, full price refund policy. The defective products discovered in the procurement are returned to the manufacturer every period. Also, the return cost for defective products is imposed on the manufacturer as a necessary expense. The return cost is ignored in the respective profit functions because the return cost is not relevant to the uncertainty ascribed to the inclusion of defective products but relevant to defective products themselves.

• Inspection costs for products is ignored as a necessary expense. Also, since this paper has formulated a mathematical model as a single period newsvendor problem, the time progress in periods is ignored, that is, inspection time for products is ignored.

At first, the supply chain model consisting of a manufacturer and a retailer is formulated in the situation that defective products are not contained at all in the procurement, that is, $\mathit{p}=0$. The expected profit function of the retailer is defined as the following equation:

(1) $\begin{array}{c}{\mathrm{\Pi }}_0^R\left(\mathit{q}\right)=\sum _{\mathit{d}=0}^q\left\{\mathit{rd}-\mathit{wq}+\mathit{s}\left(\mathit{q}-\mathit{d}\right)\right\}Pr\left\{\mathit{D}=\mathit{d}\right\}+\sum _{\mathit{d}=\mathit{q}+1}^{\mathrm{\infty }}\left(\mathit{r}-\mathit{w}\right)\mathit{q}Pr\left\{\mathit{D}=\mathit{d}\right\}\\ =\left(\mathit{r}-\mathit{w}\right)\mathit{q}-\left(\mathit{r}-\mathit{s}\right)\mathit{E}\left[{\left(\mathit{q}-\mathit{d}\right)}^{+}\right],\end{array}$(1)

where $z^{+}=max\{0,\mathit{z}\}$ and $\mathit{E}[\mathit{z}]$ means the expectation of random variable $\mathit{z}$. Also, $\mathit{D}$ means a random variable expressing demand. Then, the expected profit function of the manufacturer in the situation that no defective product is contained in the procurement is given as follows:

(2) ${\mathrm{\Pi }}_0^M(\mathit{q})=(\mathit{w}-\mathit{c})\mathit{q}.$(2)

Next, the supply chain model consisting of a manufacturer and a retailer is formulated in the situation that some defective products are contained in the procurement. In this situation, we assume that all discovered defective products are returned to the manufacturer while the wholesale price $\mathit{w}$ per defective product is refunded to the retailer. The expected profit function of the retailer is defined as the following equation:

(3) $\begin{array}{c}{\Pi }_p^R\left(\mathit{q}\right)=\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=0}^{\mathit{q}-\mathit{n}}\left\{\mathit{rd}-\mathit{w}\left(\mathit{q}-\mathit{n}\right)+\mathit{s}\left(\mathit{q}-\mathit{n}-\mathit{d}\right)\right\}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ +\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=\mathit{q}-\mathit{n}+1}^{\mathrm{\infty }}\left(\mathit{r}-\mathit{w}\right)\left(\mathit{q}-\mathit{n}\right)\mathit{q}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ =\left(\mathit{r}-\mathit{w}\right)\left(1-\mathit{p}\right)\mathit{q}-\left(\mathit{r}-\mathit{s}\right)\mathit{E}\left[{\left(\mathit{q}-\mathit{n}-\mathit{d}\right)}^{+}\right],\end{array}$(3)

where $\mathit{N}$ means a random variable expressing the number of defective products in the order quantity $\mathit{q}$. Then, the expected profit function of the manufacturer in the situation that it is possible that some defective products are contained in the procurement with the probability $\mathit{p}$ is given as follows:

(4) ${\Pi }_p^M(\mathit{q})=\mathit{w}(1-\mathit{p})\mathit{q}-\mathit{cq}.$(4)

Note that Equations (3(3) $\begin{array}{c}{\Pi }_p^R\left(\mathit{q}\right)=\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=0}^{\mathit{q}-\mathit{n}}\left\{\mathit{rd}-\mathit{w}\left(\mathit{q}-\mathit{n}\right)+\mathit{s}\left(\mathit{q}-\mathit{n}-\mathit{d}\right)\right\}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ +\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=\mathit{q}-\mathit{n}+1}^{\mathrm{\infty }}\left(\mathit{r}-\mathit{w}\right)\left(\mathit{q}-\mathit{n}\right)\mathit{q}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ =\left(\mathit{r}-\mathit{w}\right)\left(1-\mathit{p}\right)\mathit{q}-\left(\mathit{r}-\mathit{s}\right)\mathit{E}\left[{\left(\mathit{q}-\mathit{n}-\mathit{d}\right)}^{+}\right],\end{array}$(3) ) and (4(4) ${\Pi }_p^M(\mathit{q})=\mathit{w}(1-\mathit{p})\mathit{q}-\mathit{cq}.$(4) ) are respectively reduced to Equations (1(1) $\begin{array}{c}{\mathrm{\Pi }}_0^R\left(\mathit{q}\right)=\sum _{\mathit{d}=0}^q\left\{\mathit{rd}-\mathit{wq}+\mathit{s}\left(\mathit{q}-\mathit{d}\right)\right\}Pr\left\{\mathit{D}=\mathit{d}\right\}+\sum _{\mathit{d}=\mathit{q}+1}^{\mathrm{\infty }}\left(\mathit{r}-\mathit{w}\right)\mathit{q}Pr\left\{\mathit{D}=\mathit{d}\right\}\\ =\left(\mathit{r}-\mathit{w}\right)\mathit{q}-\left(\mathit{r}-\mathit{s}\right)\mathit{E}\left[{\left(\mathit{q}-\mathit{d}\right)}^{+}\right],\end{array}$(1) ) and (2(2) ${\mathrm{\Pi }}_0^M(\mathit{q})=(\mathit{w}-\mathit{c})\mathit{q}.$(2) ) when $\mathit{p}=0$.

From Equation (3)(3) $\begin{array}{c}{\Pi }_p^R\left(\mathit{q}\right)=\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=0}^{\mathit{q}-\mathit{n}}\left\{\mathit{rd}-\mathit{w}\left(\mathit{q}-\mathit{n}\right)+\mathit{s}\left(\mathit{q}-\mathit{n}-\mathit{d}\right)\right\}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ +\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=\mathit{q}-\mathit{n}+1}^{\mathrm{\infty }}\left(\mathit{r}-\mathit{w}\right)\left(\mathit{q}-\mathit{n}\right)\mathit{q}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ =\left(\mathit{r}-\mathit{w}\right)\left(1-\mathit{p}\right)\mathit{q}-\left(\mathit{r}-\mathit{s}\right)\mathit{E}\left[{\left(\mathit{q}-\mathit{n}-\mathit{d}\right)}^{+}\right],\end{array}$(3) , the following proposition is obtained:

Proposition 1:

The expected profit function of the retailer ${\Pi }_p^R(\mathit{q})$ in Equation (3)(3) $\begin{array}{c}{\Pi }_p^R\left(\mathit{q}\right)=\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=0}^{\mathit{q}-\mathit{n}}\left\{\mathit{rd}-\mathit{w}\left(\mathit{q}-\mathit{n}\right)+\mathit{s}\left(\mathit{q}-\mathit{n}-\mathit{d}\right)\right\}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ +\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=\mathit{q}-\mathit{n}+1}^{\mathrm{\infty }}\left(\mathit{r}-\mathit{w}\right)\left(\mathit{q}-\mathit{n}\right)\mathit{q}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ =\left(\mathit{r}-\mathit{w}\right)\left(1-\mathit{p}\right)\mathit{q}-\left(\mathit{r}-\mathit{s}\right)\mathit{E}\left[{\left(\mathit{q}-\mathit{n}-\mathit{d}\right)}^{+}\right],\end{array}$(3) is concave in $\mathit{q}$ for any $\mathit{p}$. Therefore, an order quantity as maximizes ${\Pi }_p^R(\mathit{q})$ is existent uniquely. Such an order quantity as maximizes ${\Pi }_p^R(\mathit{q})$ is denoted by $q_p^{\ast }$. Then, the relationship of $q_0^{\ast }\le q_p^{\ast }$ is obtained for any $\mathit{p}$, where $q_0^{\ast }$ means the order quantity maximizing ${\Pi }_p^R(\mathit{q})$ under $\mathit{p}=0$, that is, Equation (1)(1) $\begin{array}{c}{\mathrm{\Pi }}_0^R\left(\mathit{q}\right)=\sum _{\mathit{d}=0}^q\left\{\mathit{rd}-\mathit{wq}+\mathit{s}\left(\mathit{q}-\mathit{d}\right)\right\}Pr\left\{\mathit{D}=\mathit{d}\right\}+\sum _{\mathit{d}=\mathit{q}+1}^{\mathrm{\infty }}\left(\mathit{r}-\mathit{w}\right)\mathit{q}Pr\left\{\mathit{D}=\mathit{d}\right\}\\ =\left(\mathit{r}-\mathit{w}\right)\mathit{q}-\left(\mathit{r}-\mathit{s}\right)\mathit{E}\left[{\left(\mathit{q}-\mathit{d}\right)}^{+}\right],\end{array}$(1) .

The proof of Proposition 1 is shown in Appendix A. When the inclusion of defective products in the procurement is anticipated, the optimal order quantity becomes larger than that of the situation where no defective product is contained at all in the procurement. This characteristic is corresponding to an intuitive standpoint.

Further, we have the following proposition:

Proposition 2:

The relationship of ${\mathrm{\Pi }}_0^R(q_0^{\ast })\ge {\Pi }_p^R(q_p^{\ast })\ge {\Pi }_p^R(q_0^{\ast })$ is satisfied for any $\mathit{p}$.

The proof of proposition 2 is shown in Appendix B. All defective products discovered by the retailer are returned to the manufacturer according to the full price refund policy. Hence, the retailer is compensated by the manufacturer for the direct damage from defective products themselves with the full price refunds. However, the relationship of ${\mathrm{\Pi }}_0^R(q_0^{\ast })\ge {\Pi }_p^R(q_p^{\ast })$ has been derived as shown in proposition 2. Remarkably, this reduction ${\mathrm{\Pi }}_0^R(q_0^{\ast })-{\Pi }_p^R(q_p^{\ast })\ge 0$ of the expected profit in the retailer is caused by the uncertainty ascribed to defective products. From this fact, we have found that although the retailer should choose $q_p^{\ast }(\ge q_0^{\ast })$ as the order quantity from the reason that it is possible that some defective products are contained in the procurement, the retailer would have some extra stocks if the number of defective products were less contained or not at all. That is, it means that the retailer has to bear the cost for holding or disposing some extra stocks. Also, the retailer would miss sale opportunity if defective products were contained more than expected. The uncertainty ascribed to defective products brings some losses not compensated by the full price refund policy to the retailer. The responsibility for the quality of products must be usually taken by the manufacturer. Also, the optimal policy of the retailer cannot recover those losses completely. Therefore, the wholesale price refunded to the retailer by the manufacturer is not full compensation for the uncertainty ascribed to defective products. Hence, in the case of investigating the optimal policy in the supply chain consisting of a manufacturer and a retailer, the impact of uncertainty ascribed to the defective products must be considered. In this paper, we consider compensation for a monetary loss due to the uncertainty ascribed defective products. We, especially, discuss a method for assessing a latent loss caused by the uncertainty ascribed to defective products as a monetary value and proposed a method of compensating for the latent loss ascribed to the uncertainty of defective products to maintain a supply chain partnership.

4. Loss assessment and compensation

In Proposition 1, the impact of the uncertainty ascribed to defective products in the procurement on the order quantity has been shown. As consequence, it has been found that the optimal order quantity of the situation that the inclusion of defective products in the procurement is anticipated becomes larger than that of the situation that no defective product is contained in the procurement. Then, the impact of the uncertainty ascribed to defective products in the procurement on the expected profit of the retailer has been shown in Proposition 2. Concretely, it has been seen that the reduction of the expected profit in the retailer is caused by the uncertainty ascribed to defective products.

In the previous section, it has been assumed that defective products are returned to the manufacturer while the wholesale price $\mathit{w}$ is refunded to the retailer, that is, full price refund policy. However, it has been shown that the full price refund policy does not recover all the loss in the retailer due to the impact of the uncertainty ascribed to defective products. On evaluating the impact of the uncertainty ascribed to defective products as a monetary loss, we introduce the compensation fee $\mathit{u}(>\mathit{w})$ per defective product paid to the retailer. That is, we consider a compensation contract with contract parameter $\mathit{u}$ for the uncertainty ascribed to defective products.

We assume that all discovered defective products are returned to the manufacturer while the compensation fee $\mathit{u}(>\mathit{w})$ per defective product is refunded to the retailer. In this case, the expected profit function of the retailer is defined as the following equation:

(5) $\begin{array}{c}{\Pi }_p^R\left(\mathit{q},\mathit{u}\right)=\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=0}^{\mathit{q}-\mathit{n}}\left\{\mathit{rd}-\mathit{wq}+\mathit{s}\left(\mathit{q}-\mathit{n}-\mathit{d}\right)+\mathit{un}\right\}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ +\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=\mathit{q}-\mathit{n}+1}^{\mathrm{\infty }}\left\{\mathit{r}\left(\mathit{q}-\mathit{n}\right)-\mathit{wq}+\mathit{un}\right\}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ =\left(\mathit{r}-\mathit{w}\right)\left(1-\mathit{p}\right)\mathit{q}-\left(\mathit{r}-\mathit{s}\right)\mathit{E}\left[{\left(\mathit{q}-\mathit{n}-\mathit{d}\right)}^{+}\right]+\left(\mathit{u}-\mathit{w}\right)\mathit{pq}.\end{array}$(5)

Then, the expected profit function of the manufacturer is given as follows:

(6) ${\Pi }_p^M(\mathit{q},\mathit{u})=\mathit{w}(1-\mathit{p})\mathit{q}-\mathit{cq}-(\mathit{u}-\mathit{w})\mathit{pq}.$(6)

Note that Equations (5(5) $\begin{array}{c}{\Pi }_p^R\left(\mathit{q},\mathit{u}\right)=\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=0}^{\mathit{q}-\mathit{n}}\left\{\mathit{rd}-\mathit{wq}+\mathit{s}\left(\mathit{q}-\mathit{n}-\mathit{d}\right)+\mathit{un}\right\}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ +\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=\mathit{q}-\mathit{n}+1}^{\mathrm{\infty }}\left\{\mathit{r}\left(\mathit{q}-\mathit{n}\right)-\mathit{wq}+\mathit{un}\right\}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ =\left(\mathit{r}-\mathit{w}\right)\left(1-\mathit{p}\right)\mathit{q}-\left(\mathit{r}-\mathit{s}\right)\mathit{E}\left[{\left(\mathit{q}-\mathit{n}-\mathit{d}\right)}^{+}\right]+\left(\mathit{u}-\mathit{w}\right)\mathit{pq}.\end{array}$(5) ) and (6(6) ${\Pi }_p^M(\mathit{q},\mathit{u})=\mathit{w}(1-\mathit{p})\mathit{q}-\mathit{cq}-(\mathit{u}-\mathit{w})\mathit{pq}.$(6) ) are respectively reduced to Equations (3(3) $\begin{array}{c}{\Pi }_p^R\left(\mathit{q}\right)=\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=0}^{\mathit{q}-\mathit{n}}\left\{\mathit{rd}-\mathit{w}\left(\mathit{q}-\mathit{n}\right)+\mathit{s}\left(\mathit{q}-\mathit{n}-\mathit{d}\right)\right\}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ +\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=\mathit{q}-\mathit{n}+1}^{\mathrm{\infty }}\left(\mathit{r}-\mathit{w}\right)\left(\mathit{q}-\mathit{n}\right)\mathit{q}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ =\left(\mathit{r}-\mathit{w}\right)\left(1-\mathit{p}\right)\mathit{q}-\left(\mathit{r}-\mathit{s}\right)\mathit{E}\left[{\left(\mathit{q}-\mathit{n}-\mathit{d}\right)}^{+}\right],\end{array}$(3) ) and (4(4) ${\Pi }_p^M(\mathit{q})=\mathit{w}(1-\mathit{p})\mathit{q}-\mathit{cq}.$(4) ) when $\mathit{u}=\mathit{w}$.

From Equation (5)(5) $\begin{array}{c}{\Pi }_p^R\left(\mathit{q},\mathit{u}\right)=\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=0}^{\mathit{q}-\mathit{n}}\left\{\mathit{rd}-\mathit{wq}+\mathit{s}\left(\mathit{q}-\mathit{n}-\mathit{d}\right)+\mathit{un}\right\}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ +\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=\mathit{q}-\mathit{n}+1}^{\mathrm{\infty }}\left\{\mathit{r}\left(\mathit{q}-\mathit{n}\right)-\mathit{wq}+\mathit{un}\right\}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ =\left(\mathit{r}-\mathit{w}\right)\left(1-\mathit{p}\right)\mathit{q}-\left(\mathit{r}-\mathit{s}\right)\mathit{E}\left[{\left(\mathit{q}-\mathit{n}-\mathit{d}\right)}^{+}\right]+\left(\mathit{u}-\mathit{w}\right)\mathit{pq}.\end{array}$(5) , the following proposition is obtained:

Proposition 3:

The expected profit function of the retailer ${\Pi }_p^R(\mathit{q},\mathit{u})$ is concave in $\mathit{q}$ for any $\mathit{p}$. Therefore, an order quantity as maximizes ${\Pi }_p^R(\mathit{q},\mathit{u})$ is existent uniquely. Such an order quantity as maximizes ${\Pi }_p^R(\mathit{q},\mathit{u})$ is defined as $q_p^{†}$. In this case, the relationship of $q_p^{\ast }\le q_p^{†}$ is given, where $q_p^{\ast }$ is the order quantity maximizing ${\Pi }_p^R(\mathit{q})$ in Equation (3)(3) $\begin{array}{c}{\Pi }_p^R\left(\mathit{q}\right)=\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=0}^{\mathit{q}-\mathit{n}}\left\{\mathit{rd}-\mathit{w}\left(\mathit{q}-\mathit{n}\right)+\mathit{s}\left(\mathit{q}-\mathit{n}-\mathit{d}\right)\right\}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ +\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=\mathit{q}-\mathit{n}+1}^{\mathrm{\infty }}\left(\mathit{r}-\mathit{w}\right)\left(\mathit{q}-\mathit{n}\right)\mathit{q}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ =\left(\mathit{r}-\mathit{w}\right)\left(1-\mathit{p}\right)\mathit{q}-\left(\mathit{r}-\mathit{s}\right)\mathit{E}\left[{\left(\mathit{q}-\mathit{n}-\mathit{d}\right)}^{+}\right],\end{array}$(3) as mentioned before.

The proof of Proposition 3 is omitted because the proof is similar to that of Proposition 1. From Proposition 3, it is found that the risk avoidance by the compensation fee to the retailer makes the order quantity larger. In addition, we see that the fair compensation for the uncertainty ascribed to defective products by the manufacturer can be an incentive to the order of the retailer.

Then, we consider the assessment of the compensation fee for the monetary loss ascribed to the uncertainty of defective products. At first, the compensation fee $\mathit{u}$ has to secure the profit of the retailer at least. As mentioned above, the compensation fee can be an incentive to the order of the retailer. Hence, the following relationship is considered in the assessment of the compensation fee:

(7) ${\Pi }_p^R(\mathit{q},\mathit{u})-{\mathrm{\Pi }}_0^R(q_0^{\ast })\ge 0.$(7)

The compensation fee $\mathit{u}$ is decided so as to secure that the profit of the retailer, ${\Pi }_p^R(\mathit{q},\mathit{u})$, is equal to or greater than the profit of the retailer, ${\mathrm{\Pi }}_0^R(q_0^{\ast })$, in the case that there is no defective product in the procurement. Using Equations (3)(3) $\begin{array}{c}{\Pi }_p^R\left(\mathit{q}\right)=\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=0}^{\mathit{q}-\mathit{n}}\left\{\mathit{rd}-\mathit{w}\left(\mathit{q}-\mathit{n}\right)+\mathit{s}\left(\mathit{q}-\mathit{n}-\mathit{d}\right)\right\}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ +\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=\mathit{q}-\mathit{n}+1}^{\mathrm{\infty }}\left(\mathit{r}-\mathit{w}\right)\left(\mathit{q}-\mathit{n}\right)\mathit{q}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ =\left(\mathit{r}-\mathit{w}\right)\left(1-\mathit{p}\right)\mathit{q}-\left(\mathit{r}-\mathit{s}\right)\mathit{E}\left[{\left(\mathit{q}-\mathit{n}-\mathit{d}\right)}^{+}\right],\end{array}$(3) and (eq 5(5) $\begin{array}{c}{\Pi }_p^R\left(\mathit{q},\mathit{u}\right)=\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=0}^{\mathit{q}-\mathit{n}}\left\{\mathit{rd}-\mathit{wq}+\mathit{s}\left(\mathit{q}-\mathit{n}-\mathit{d}\right)+\mathit{un}\right\}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ +\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=\mathit{q}-\mathit{n}+1}^{\mathrm{\infty }}\left\{\mathit{r}\left(\mathit{q}-\mathit{n}\right)-\mathit{wq}+\mathit{un}\right\}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ =\left(\mathit{r}-\mathit{w}\right)\left(1-\mathit{p}\right)\mathit{q}-\left(\mathit{r}-\mathit{s}\right)\mathit{E}\left[{\left(\mathit{q}-\mathit{n}-\mathit{d}\right)}^{+}\right]+\left(\mathit{u}-\mathit{w}\right)\mathit{pq}.\end{array}$(5) ), Equation (7)(7) ${\Pi }_p^R(\mathit{q},\mathit{u})-{\mathrm{\Pi }}_0^R(q_0^{\ast })\ge 0.$(7) is transformed into

(8) $\mathit{u}\ge \mathit{w}+\frac{{\mathrm{\Pi }}_0^R(q_0^{\ast })-{\Pi }_p^R(\mathit{q})}{\mathit{pq}}.$(8)

Equation (8)(8) $\mathit{u}\ge \mathit{w}+\frac{{\mathrm{\Pi }}_0^R(q_0^{\ast })-{\Pi }_p^R(\mathit{q})}{\mathit{pq}}.$(8) indicates the condition of the compensation fee $\mathit{u}$ such that the expected profit of the retailer becomes equal to or greater than that of the retailer in the case that there is no defective product in the procurement for a given $\mathit{q}$. That is, the lower bound of the compensation fee for the monetary loss ascribed to the uncertainty of defective products has been shown.

One the other hand, the monetary compensation for the uncertainty of defective products is considered from the viewpoint of the manufacturer. The profit of the manufacturer is decreased by the payment of the compensation fee. However, it might be expected that the compensation increases the profit of the manufacturer because the compensation fee can be an incentive to a larger order of the retailer. Hence, from the standpoint of the manufacturer, it is desirable to satisfy the following relationship on the assessment of the compensation fee:

(9) ${\Pi }_p^M(\mathit{q},\mathit{u})-{\mathrm{\Pi }}_0^M(q_0^{\ast })\ge 0.$(9)

The compensation fee can be decided so as to maintain that the profit of the manufacturer, ${\Pi }_p^M(\mathit{q},\mathit{u})$, is equal to or greater than the profit of the manufacturer, ${\mathrm{\Pi }}_0^M(q_0^{\ast })$, in the case that there is no defective product in the procurement. Using Equations (2(2) ${\mathrm{\Pi }}_0^M(\mathit{q})=(\mathit{w}-\mathit{c})\mathit{q}.$(2) ) and (6(6) ${\Pi }_p^M(\mathit{q},\mathit{u})=\mathit{w}(1-\mathit{p})\mathit{q}-\mathit{cq}-(\mathit{u}-\mathit{w})\mathit{pq}.$(6) ), Equation (9)(9) ${\Pi }_p^M(\mathit{q},\mathit{u})-{\mathrm{\Pi }}_0^M(q_0^{\ast })\ge 0.$(9) is transformed into

(10) $\mathit{u}\le \frac{\mathit{w}-\mathit{c}}{p}\left(1-\frac{q_0^{\ast }}{\mathit{q}}\right).$(10)

Equation (10)(10) $\mathit{u}\le \frac{\mathit{w}-\mathit{c}}{p}\left(1-\frac{q_0^{\ast }}{\mathit{q}}\right).$(10) indicates the condition of the compensation fee $\mathit{u}$ such that the expected profit of the manufacturer becomes equal to or greater than that of the manufacturer in the case that there is no defective product in the procurement for a given $\mathit{q}$. That is, the upper bound of the compensation fee desirable to the manufacturer can be considered.

As the results in Equations (8(8) $\mathit{u}\ge \mathit{w}+\frac{{\mathrm{\Pi }}_0^R(q_0^{\ast })-{\Pi }_p^R(\mathit{q})}{\mathit{pq}}.$(8) ) and (10(10) $\mathit{u}\le \frac{\mathit{w}-\mathit{c}}{p}\left(1-\frac{q_0^{\ast }}{\mathit{q}}\right).$(10) ), we have obtained the following relationship about the compensation fee and order quantity:

(11) $\mathit{w}+\frac{{\mathrm{\Pi }}_0^R(q_0^{\ast })-{\Pi }_p^R(\mathit{q})}{\mathit{pq}}\le \mathit{u}\le \frac{\mathit{w}-\mathit{c}}{p}\left(1-\frac{q_0^{\ast }}{\mathit{q}}\right).$(11)

The above condition is called the incentive compatible condition for compensation in this paper since both of the manufacturer and retailer have the incentive to conclude the compensation contract with contract parameters $\mathit{q}$ and $\mathit{u}$.

Then, some numerical examples about Equations (7(7) ${\Pi }_p^R(\mathit{q},\mathit{u})-{\mathrm{\Pi }}_0^R(q_0^{\ast })\ge 0.$(7) )-(11(11) $\mathit{w}+\frac{{\mathrm{\Pi }}_0^R(q_0^{\ast })-{\Pi }_p^R(\mathit{q})}{\mathit{pq}}\le \mathit{u}\le \frac{\mathit{w}-\mathit{c}}{p}\left(1-\frac{q_0^{\ast }}{\mathit{q}}\right).$(11) ) are illustrated. The parameters about revenue and expense are given as follows: $\mathit{r}=100.0,\mathit{w}=60.0,$ $\mathit{c}=35.0,\mathit{s}=\mathrm{10.0.}$Then, the demand distribution is assumed to be a discrete symmetrical triangle distribution with the range (400, 600), where the mean and standard deviation are respectively given as 500.0 and 40.8. In this case, the optimal order quantity and respective expected profits of the manufacturer and retailer in the situation that there is no defective product in the procurement are obtained as follows: $q_0^{\ast }=494,$${\mathrm{\Pi }}_0^R(q_0^{\ast })=18514.3,{\mathrm{\Pi }}_0^M(q_0^{\ast })=12350.0.$

Figures 1 and 2 show the result of Equations (8(8) $\mathit{u}\ge \mathit{w}+\frac{{\mathrm{\Pi }}_0^R(q_0^{\ast })-{\Pi }_p^R(\mathit{q})}{\mathit{pq}}.$(8) ) and (10) when $\mathit{p}=0.010$ and $\mathit{p}=0.025$, respectively. In Figure 1, the compensation fee $\mathit{u}$ is assessed higher as the order quantity of the retailer is larger. When the order quantity of the retailer is larger, it is expected that the impact of the uncertainty ascribed to defective products is significant. On one hand, the profit of the manufacturer increases when the order quantity of the retailer is large. Hence, the expensive compensation fee can be offered by the manufacturer. The gray area indicates the incentive compatible condition for compensation satisfying Equation (11)(11) $\mathit{w}+\frac{{\mathrm{\Pi }}_0^R(q_0^{\ast })-{\Pi }_p^R(\mathit{q})}{\mathit{pq}}\le \mathit{u}\le \frac{\mathit{w}-\mathit{c}}{p}\left(1-\frac{q_0^{\ast }}{\mathit{q}}\right).$(11) . In this case, the manufacturer can secure own profit in addition to compensating the retailer for the monetary loss due to the uncertainty of defective products. On the other hand, there is no area satisfying Equation (11)(11) $\mathit{w}+\frac{{\mathrm{\Pi }}_0^R(q_0^{\ast })-{\Pi }_p^R(\mathit{q})}{\mathit{pq}}\le \mathit{u}\le \frac{\mathit{w}-\mathit{c}}{p}\left(1-\frac{q_0^{\ast }}{\mathit{q}}\right).$(11) in Figure 2. When $\mathit{p}=0.025$, there are many defective products in the procurement. The manufacturer has a large payment due to the compensation for defective products. Then, the manufacturer cannot secure own profit when compensating the retailer for the monetary loss due to the uncertainty of defective products. From the viewpoint of the uncertainty of defective products, the importance of quality can be reconfirmed.

Figure 1. The relationship of Equations (8(8) $\mathit{u}\ge \mathit{w}+\frac{{\mathrm{\Pi }}_0^R(q_0^{\ast })-{\Pi }_p^R(\mathit{q})}{\mathit{pq}}.$(8) ) and (eq 10(10) $\mathit{u}\le \frac{\mathit{w}-\mathit{c}}{p}\left(1-\frac{q_0^{\ast }}{\mathit{q}}\right).$(10) ) when $\mathit{p}=\mathrm{0.010.}$

Figure 2. The relationship of Equations (8(8) $\mathit{u}\ge \mathit{w}+\frac{{\mathrm{\Pi }}_0^R(q_0^{\ast })-{\Pi }_p^R(\mathit{q})}{\mathit{pq}}.$(8) ) and (10) when $\mathit{p}=\mathrm{0.025.}$

As a comparative case of demand information, we also provide Figure 3, where the demand distribution is assumed to be a discrete symmetrical triangle distribution with the range (300, 700), and the mean and standard deviation are respectively given as 500.0 and 81.6. The situation in Figure 3 has large variability about demand in comparison with Figure 1. In this case, larger compensation fee is needed since larger uncertainty in demand makes order quantity increased.

Figure 3. The incentive compatible condition when $\mathit{p}=0.010$ and discrete symmetrical triangle distribution with the range (300, 700).

Further, supply chain coordination in this model is considered. The coordination approach is well known as one of solutions in supply chain contract problems (Cachon, 2003; Cachon & Lariviere, 2005). We apply the coordination approach to the model with the compensation for defective products in this paper. On applying the coordination approach, we define the total expected profit of the manufacturer and retailer, ${\mathrm{\Pi }}_p^{R+M}(\mathit{q},\mathit{u})$, as follows:

(12) $\begin{array}{c}{\mathrm{\Pi }}_p^{R+M}\left(\mathit{q},\mathit{u}\right)={\Pi }_p^R\left(\mathit{q},\mathit{u}\right)+{\Pi }_p^M\left(\mathit{q},\mathit{u}\right)\\ =\mathit{r}\left(1-\mathit{p}\right)\mathit{q}-\left(\mathit{r}-\mathit{s}\right)\mathit{E}\left[{\left(\mathit{q}-\mathit{n}-\mathit{d}\right)}^{+}\right]-\mathit{cq}\\ ={\Pi }_p^R\left(\mathit{q}\right)+{\Pi }_p^M\left(\mathit{q}\right).\end{array}$(12)

Note that the terms for the compensation fee $\mathit{u}$ are canceled in the total expected profit in Equation (12)(12) $\begin{array}{c}{\mathrm{\Pi }}_p^{R+M}\left(\mathit{q},\mathit{u}\right)={\Pi }_p^R\left(\mathit{q},\mathit{u}\right)+{\Pi }_p^M\left(\mathit{q},\mathit{u}\right)\\ =\mathit{r}\left(1-\mathit{p}\right)\mathit{q}-\left(\mathit{r}-\mathit{s}\right)\mathit{E}\left[{\left(\mathit{q}-\mathit{n}-\mathit{d}\right)}^{+}\right]-\mathit{cq}\\ ={\Pi }_p^R\left(\mathit{q}\right)+{\Pi }_p^M\left(\mathit{q}\right).\end{array}$(12) because the compensation is a simple transfer between the manufacturer and retailer. It is obvious that ${\mathrm{\Pi }}_p^{R+M}(\mathit{q},\mathit{u})$ is concave in $\mathit{q}$. Therefore, we obtain an order quantity as maximizes the total expected profit ${\mathrm{\Pi }}_p^{R+M}(\mathit{q},\mathit{u})$. Such an order quantity is noted by $q_p^C$, where $q_p^C$ is unique. Using $q_p^C$, Equation (11)(11) $\mathit{w}+\frac{{\mathrm{\Pi }}_0^R(q_0^{\ast })-{\Pi }_p^R(\mathit{q})}{\mathit{pq}}\le \mathit{u}\le \frac{\mathit{w}-\mathit{c}}{p}\left(1-\frac{q_0^{\ast }}{\mathit{q}}\right).$(11) is transformed into the following relationship:

(13) $\mathit{w}+\frac{{\mathrm{\Pi }}_0^R(q_0^{\ast })-{\Pi }_p^R(q_p^C)}{\mathit{p}{q}_p^C}\le \mathit{u}\le \frac{\mathit{w}-\mathit{c}}{p}\left(1-\frac{q_0^{\ast }}{q_p^C}\right).$(13)

Hence, the setting of the compensation fee in Equation (13)(13) $\mathit{w}+\frac{{\mathrm{\Pi }}_0^R(q_0^{\ast })-{\Pi }_p^R(q_p^C)}{\mathit{p}{q}_p^C}\le \mathit{u}\le \frac{\mathit{w}-\mathit{c}}{p}\left(1-\frac{q_0^{\ast }}{q_p^C}\right).$(13) satisfies the incentive compatible condition for compensation proposed in this paper. A numerical example about Equation (13)(13) $\mathit{w}+\frac{{\mathrm{\Pi }}_0^R(q_0^{\ast })-{\Pi }_p^R(q_p^C)}{\mathit{p}{q}_p^C}\le \mathit{u}\le \frac{\mathit{w}-\mathit{c}}{p}\left(1-\frac{q_0^{\ast }}{q_p^C}\right).$(13) is illustrated in Figure 4. The parameter settings are same as Figure 1. In this case, $q_p^C=530$ is obtained. Hence, the compensation fee $130\le \mathit{u}\le 165$ is required to achieve the supply chain coordination.

Figure 4. The expected profits and compensation fee when $q_p^C=530$.

The traditional coordination approach aims at maximizing the total expected profit ${\mathrm{\Pi }}_p^{R+M}(\mathit{q},\mathit{u})$ on the initiative of the manufacturer. The total expected profit ${\mathrm{\Pi }}_p^{R+M}(\mathit{q},\mathit{u})$ is maximized while the manufacturer determines such a compensation fee as maximizes own profit. In this case, the manufacturer should secure ${\mathrm{\Pi }}_0^R(q_0^{\ast })$ to the retailer at least. From proposition 2, the relation of ${\Pi }_p^R(q_p^C)\le {\mathrm{\Pi }}_0^R(q_0^{\ast })$ is satisfied. Therefore, the manufacturer offers the compensation fee $u^C$ that satisfies the following relation:

(14) ${\Pi }_p^R(q_p^C,u^C)-{\mathrm{\Pi }}_0^R(q_0^{\ast })=0.$(14)

Equation (14)(14) ${\Pi }_p^R(q_p^C,u^C)-{\mathrm{\Pi }}_0^R(q_0^{\ast })=0.$(14) shows the situation that the compensation fee $u^C$ secures the profit of the retailer in the case that there is no defective product in the procurement. In other words, the manufacturer offers the compensation fee $u^C$ while requests the order quantity $q_p^C$ to the retailer. From Equation (14)(14) ${\Pi }_p^R(q_p^C,u^C)-{\mathrm{\Pi }}_0^R(q_0^{\ast })=0.$(14) , we have the following equation:

(15) $u^C=\mathit{w}+\frac{{\mathrm{\Pi }}_0^R(q_0^{\ast })-{\Pi }_p^R(q_p^C)}{\mathit{p}{q}_p^C}.$(15)

Equation (15)(15) $u^C=\mathit{w}+\frac{{\mathrm{\Pi }}_0^R(q_0^{\ast })-{\Pi }_p^R(q_p^C)}{\mathit{p}{q}_p^C}.$(15) satisfies the incentive compatible condition for compensation in Equation (11)(11) $\mathit{w}+\frac{{\mathrm{\Pi }}_0^R(q_0^{\ast })-{\Pi }_p^R(\mathit{q})}{\mathit{pq}}\le \mathit{u}\le \frac{\mathit{w}-\mathit{c}}{p}\left(1-\frac{q_0^{\ast }}{\mathit{q}}\right).$(11) . Hence, the combination of $q_p^C$ and $u^C$ is given as a solution of the traditional coordination approach. However, the manufacturer monopolizes the increase in the total expected profit through the combination of $q_p^C$ and $u^C$. This result doesn’t always maintain the partnership between the manufacturer and retailer. Other theory and techniques of realizing a fair profit allocation for both manufacturer and retailer are needed to maintain partnership between the manufacturer and retailer in the supply chain.

By using the procedure of the Nash bargaining approach (Nash, 1950), we have the following Nash product:

(16) $\begin{array}{c}\mathit{G}(\mathit{q},\mathit{u})=\{{\Pi }_p^R(\mathit{q},\mathit{u})-{\mathrm{\Pi }}_0^R(q_0^{\ast })\}\{{\Pi }_p^M(\mathit{q},\mathit{u})-{\mathrm{\Pi }}_0^M(q_0^{\ast })\}\\ =\{{\Pi }_p^R(\mathit{q})-{\mathrm{\Pi }}_0^R(q_0^{\ast })+(\mathit{u}-\mathit{w})\mathit{pq}\}\{{\Pi }_p^M(\mathit{q})-{\mathrm{\Pi }}_0^M(q_0^{\ast })-(\mathit{u}-\mathit{w})\mathit{pq}\}.\end{array}$(16)

The Nash bargaining solution is obtained by maximizing the Nash product in Equation (16)(16) $\begin{array}{c}\mathit{G}(\mathit{q},\mathit{u})=\{{\Pi }_p^R(\mathit{q},\mathit{u})-{\mathrm{\Pi }}_0^R(q_0^{\ast })\}\{{\Pi }_p^M(\mathit{q},\mathit{u})-{\mathrm{\Pi }}_0^M(q_0^{\ast })\}\\ =\{{\Pi }_p^R(\mathit{q})-{\mathrm{\Pi }}_0^R(q_0^{\ast })+(\mathit{u}-\mathit{w})\mathit{pq}\}\{{\Pi }_p^M(\mathit{q})-{\mathrm{\Pi }}_0^M(q_0^{\ast })-(\mathit{u}-\mathit{w})\mathit{pq}\}.\end{array}$(16) . Note that the respective expected profits ${\mathrm{\Pi }}_0^R(q_0^{\ast })$ and ${\mathrm{\Pi }}_0^M(q_0^{\ast })$ of the manufacturer and retailer in the situation that there is no defective product are employed as a disagreement point or status-quo point in this paper. By differentiating Equation (16)(16) $\begin{array}{c}\mathit{G}(\mathit{q},\mathit{u})=\{{\Pi }_p^R(\mathit{q},\mathit{u})-{\mathrm{\Pi }}_0^R(q_0^{\ast })\}\{{\Pi }_p^M(\mathit{q},\mathit{u})-{\mathrm{\Pi }}_0^M(q_0^{\ast })\}\\ =\{{\Pi }_p^R(\mathit{q})-{\mathrm{\Pi }}_0^R(q_0^{\ast })+(\mathit{u}-\mathit{w})\mathit{pq}\}\{{\Pi }_p^M(\mathit{q})-{\mathrm{\Pi }}_0^M(q_0^{\ast })-(\mathit{u}-\mathit{w})\mathit{pq}\}.\end{array}$(16) by $\mathit{u}$ and letting it be equivalent to 0, we have

(17) $\mathit{u}=\mathit{w}+\frac{\{{\Pi }_p^M(\mathit{q})-{\mathrm{\Pi }}_0^M(q_0^{\ast })\}-\{{\Pi }_p^R(\mathit{q})-{\mathrm{\Pi }}_0^R(q_0^{\ast })\}}{2\mathit{pq}}.$(17)

By substituting $\mathit{u}$ in Equation (17)(17) $\mathit{u}=\mathit{w}+\frac{\{{\Pi }_p^M(\mathit{q})-{\mathrm{\Pi }}_0^M(q_0^{\ast })\}-\{{\Pi }_p^R(\mathit{q})-{\mathrm{\Pi }}_0^R(q_0^{\ast })\}}{2\mathit{pq}}.$(17) into $\mathit{G}(\mathit{q},\mathit{u})$ in Equation (16)(16) $\begin{array}{c}\mathit{G}(\mathit{q},\mathit{u})=\{{\Pi }_p^R(\mathit{q},\mathit{u})-{\mathrm{\Pi }}_0^R(q_0^{\ast })\}\{{\Pi }_p^M(\mathit{q},\mathit{u})-{\mathrm{\Pi }}_0^M(q_0^{\ast })\}\\ =\{{\Pi }_p^R(\mathit{q})-{\mathrm{\Pi }}_0^R(q_0^{\ast })+(\mathit{u}-\mathit{w})\mathit{pq}\}\{{\Pi }_p^M(\mathit{q})-{\mathrm{\Pi }}_0^M(q_0^{\ast })-(\mathit{u}-\mathit{w})\mathit{pq}\}.\end{array}$(16) , the following equation is obtained:

(18) $\mathit{G}(\mathit{q},\mathit{u})={\left(\frac{{\Pi }_p^M(\mathit{q})+{\Pi }_p^R(\mathit{q})-{\mathrm{\Pi }}_0^M(q_0^{\ast })-{\mathrm{\Pi }}_0^R(q_0^{\ast })}{2}\right)}^2$(18)

From the discussion for Equation (13)(13) $\mathit{w}+\frac{{\mathrm{\Pi }}_0^R(q_0^{\ast })-{\Pi }_p^R(q_p^C)}{\mathit{p}{q}_p^C}\le \mathit{u}\le \frac{\mathit{w}-\mathit{c}}{p}\left(1-\frac{q_0^{\ast }}{q_p^C}\right).$(13) , the order quantity $q_p^N$ maximizing $\mathit{G}(\mathit{q},\mathit{u})$ is given as $q_p^N=q_p^C$. Then, the compensation fee $u^N$ is given by Equation (17)(17) $\mathit{u}=\mathit{w}+\frac{\{{\Pi }_p^M(\mathit{q})-{\mathrm{\Pi }}_0^M(q_0^{\ast })\}-\{{\Pi }_p^R(\mathit{q})-{\mathrm{\Pi }}_0^R(q_0^{\ast })\}}{2\mathit{pq}}.$(17) and $q_p^N$. We have the combination $(q_p^N,u^N)$ as the solution in the compensation between the manufacturer and retailer using the Nash bargaining approach. Note that the increase in the total expected profit is equally divided into the manufacturer and retailer through the compensation fee when the compensation contract is operated using $(q_p^N,u^N)$. Therefore, it is more reasonable that the manufacturer and retailer agree with the compensation contract with $(q_p^N,u^N)$ in comparison to the compensation contract with $(q_p^C,u^C)$ by the coordination approach.

Figure 5 shows the solutions based on coordination, bargaining, and Stackelberg approaches, where the parameters are same as Figure 1. The detail of Stackelberg approaches is explained in Appendix C. Also, Table 1 shows the respective values of parameters and expected profits of the manufacturer and retailer in the situation of Figure 5. The parameters based on the Stackelberg approach don’t satisfy the incentive compatible condition for compensation, because the respective parameters are decided by the individuals under the Stackelberg approach. Then, the transaction based on the Stackelberg approach isn’t beneficial to both in this situation. The parameters based on the coordination approach satisfies the incentive compatible condition for compensation. However, the manufacturer monopolizes the increase in the total expected profit by the coordination approach. The parameters based on the coordination approach isn’t beneficial to the retailer in real sense, and then doesn’t always maintain the partnership between the manufacturer and retailer. On one hand, the parameters based on the bargaining approach satisfies the incentive compatible condition for compensation and the increase in the total expected profit is equally divided into the manufacturer and retailer by the compensation fee. Therefore, the bargaining approach that the manufacturer and retailer agree with the compensation fee and order quantity is more reasonable for both the manufacturer and retailer in comparison to the coordination approach. At least, the compensation technique is effective in solving the conflict and maintaining the partnership between the manufacturer and retailer.

Figure 5. The solutions based on coordination, bargaining, and Stackelberg approaches when $\mathit{p}=0.010$.

Table 1. The respective characteristics in case of $\mathit{p}=\mathrm{0.010.}$

Approach	$\mathit{q}$	$\mathit{u}$	${\Pi }_p^R(\mathit{q},\mathit{u})$	${\Pi }_p^M(\mathit{q},\mathit{u})$
No defective product	494	-	18514.3	12350.0
coordination	530	133.8	18514.3	12540.6
Nash bargaining	530	151.8	18609.6	12445.3
Stackelbelg	499	60.0	18512.2	12175.6

5. Concluding remarks

In this paper, we have considered the impact of the uncertainty ascribed to the inclusion of defective products in the procurement from a manufacturer in a supply chain. Although the major part of research on inventory management has presumed that quality of products received from manufacturers is perfect, several defective products would, in actuality, have been contained in the procurement to retailers. Then, several papers on inventory management with defective products has been published until now. The previous studies have concerned with defective products have treated several problems directly caused by defectives themselves in the situation a portion of defective products is delivered to consumers. Also, the uncertainty ascribed to the inclusion of defective products is also a problem to be considered. Even though no defective product is delivered to a consumer, the inclusion of defective products causes a latent loss that should be avoided to supply chain members. The retailer should not be responsible for the latent loss due to the uncertainty ascribed to the inclusion of defective products. The compensation for the uncertainty ascribed to the inclusion of defective products should be considered, but compensation for the uncertainty ascribed to the inclusion of defective products has remained poorly defined.

Then, this paper has focused on the uncertainty ascribed to the inclusion of defective products and investigated the impact of the uncertainty ascribed to the inclusion of defective products on a supply chain. As the result, we have derived the following managerial insights from the mathematical model with defective products:

• When the inclusion of defective products in the procurement is anticipated, the optimal order quantity becomes larger than that of the situation where no defective product is contained at all in the procurement. (Proposition 1)

• The uncertainty ascribed to the inclusion of defective products has brought some losses not compensated by the full price refund policy to the retailer, even though the retailer has practiced the optimal policy. (Proposition 2)

From the managerial insights above, we have discussed a method for assessing a latent loss causing the uncertainty ascribed to the inclusion of defective products as a monetary value. Then, we have investigated a method of compensating for the latent loss ascribed to the uncertainty of defective products to maintain a supply chain partnership.As the result, we have derived the following managerial insights from the mathematical model with defective products:

• The risk avoidance by the compensation fee to the retailer has made the order quantity larger. In addition, the fair compensation for the uncertainty ascribed to the inclusion of defective products by the manufacturer can be an incentive to the order of the retailer. (Proposition 3)

• The appropriate design of the compensation contract has brought the cooperative partnership between the manufacturer and retailer. Further, it has been shown that the compensation contract has a role of a commitment to show that the manufacturer offers good quality products, and gives the incentive for the improvement of quality to the manufacturer.

By the way, there are some bargaining solutions except the Nash bargaining solution shown in this paper. One of them is the generalized Nash bargaining solution (Nagaraja & Sosic, 2008; Roth, 1979). In recent years, a new approach is given as an expansion of the Nash bargaining approach by Kato et al. (2018). Their approaches mentioned above considers the power balance between a manufacturer and a retailer. The improvement of compensation contract based on various bargaining approach will be considered as a future issue.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Notes on contributors

Yasuhiko Takemoto

Yasuhiko Takemoto received his BE, ME and PhD from Osaka Prefecture University, all in the Department of Industrial Engineering, in 2000, 2002 and 2004, respectively. From April 2004 to September 2007, he was a research associate in the Department of Strategic Management, School of Business Administration, University of Hyogo. Since October 2007 to March 2017, he was an associate professor in the Faculty of Management and Information Systems, Prefectural University of Hiroshima. Since April 2017, he is an associate professor in the Faculty of Science and Engineering, Kindai University. His research interests centre in the statistical process control and supply chain management.

Ikuo Arizono

Ikuo Arizono received his BE, ME and PhD from Osaka Prefecture University, all in the Department of Industrial Engineering, in 1982, 1984 and 1988, respectively. From 1985 to 1990 he was a research associate, from 1990 to 1994, he was an assistant professor and since 1994, he was an associate professor in the Graduate School of Engineering, Osaka Prefecture University. Since 2011, he is a professor in the Graduate School of Natural Science and Technology, Okayama University. His research interests centre in the statistical quality control and operational research.

Appendix A:

Proof of Proposition 1

Assume that $f_i(\mathit{x}),\mathit{i}=0,1,2,\cdots ,$ is a concave function in $\mathit{x}$ and the top of $f_i(\mathit{x})$ is given at $x_i^{\ast }$, where $x_i^{\ast }\le x_{i+1}^{\ast }$. The function $\mathit{F}(\mathit{x})$ defined as $\mathit{F}(\mathit{x})=\sum _i{k}_i{f}_i(\mathit{x})$ is concave and $x^{\ast }$ giving the top of $\mathit{F}(\mathit{x})$ has the relation of $x_0^{\ast }\le x^{\ast }$, where $k_i>0$ for any $\mathit{i}$.

From Equation (3)(3) $\begin{array}{c}{\Pi }_p^R\left(\mathit{q}\right)=\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=0}^{\mathit{q}-\mathit{n}}\left\{\mathit{rd}-\mathit{w}\left(\mathit{q}-\mathit{n}\right)+\mathit{s}\left(\mathit{q}-\mathit{n}-\mathit{d}\right)\right\}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ +\sum _{\mathit{n}=0}^q\sum _{\mathit{d}=\mathit{q}-\mathit{n}+1}^{\mathrm{\infty }}\left(\mathit{r}-\mathit{w}\right)\left(\mathit{q}-\mathit{n}\right)\mathit{q}Pr\left\{\mathit{D}=\mathit{d}\right\}Pr\left\{\mathit{N}=\mathit{n}\right\}\\ =\left(\mathit{r}-\mathit{w}\right)\left(1-\mathit{p}\right)\mathit{q}-\left(\mathit{r}-\mathit{s}\right)\mathit{E}\left[{\left(\mathit{q}-\mathit{n}-\mathit{d}\right)}^{+}\right],\end{array}$(3) , we obtain

$\begin{array}{c}{\Pi }_p^R\left(\mathit{q}\right)=\left(\mathit{r}-\mathit{w}\right)\left(1-\mathit{p}\right)\mathit{q}-\left(\mathit{r}-\mathit{s}\right)\mathit{E}\left[{\left(\mathit{q}-\mathit{n}-\mathit{d}\right)}^{+}\right]\\ =\sum _{\mathit{n}=0}^q\left\{\left(\mathit{r}-\mathit{w}\right)\left(\mathit{q}-\mathit{n}\right)-\left(\mathit{r}-\mathit{s}\right)\right\}\mathit{E}[{\left(\mathit{q}-\mathit{n}-\mathit{d}\right)}^{+}|\mathit{N}=\mathit{n}]Pr\left\{\mathit{N}=\mathit{n}\right\}\\ \equiv \sum _{\mathit{n}=0}^q{g}_n\left(\mathit{q}\right)Pr\left\{\mathit{N}=\mathit{n}\right\}.\end{array}$

Note that ${\mathrm{\Pi }}_0^R(\mathit{q})=g_0(\mathit{q})$. Therefore, we have the relation of $q_0^{\ast }\le q_p^{\ast }$.

Appendix B:

Proof of Proposition 2

From Equation (1)(1) $\begin{array}{c}{\mathrm{\Pi }}_0^R\left(\mathit{q}\right)=\sum _{\mathit{d}=0}^q\left\{\mathit{rd}-\mathit{wq}+\mathit{s}\left(\mathit{q}-\mathit{d}\right)\right\}Pr\left\{\mathit{D}=\mathit{d}\right\}+\sum _{\mathit{d}=\mathit{q}+1}^{\mathrm{\infty }}\left(\mathit{r}-\mathit{w}\right)\mathit{q}Pr\left\{\mathit{D}=\mathit{d}\right\}\\ =\left(\mathit{r}-\mathit{w}\right)\mathit{q}-\left(\mathit{r}-\mathit{s}\right)\mathit{E}\left[{\left(\mathit{q}-\mathit{d}\right)}^{+}\right],\end{array}$(1) , we obtain

$\begin{array}{c}{\mathrm{\Pi }}_0^R(q_0^{\ast })=(\mathit{r}-\mathit{w})q_0^{\ast }-(\mathit{r}-\mathit{s})\mathit{E}[(q_0^{\ast }-\mathit{d}{)}^{+}]\\ =\sum _n\{(\mathit{r}-\mathit{w})q_0^{\ast }-(\mathit{r}-\mathit{s})\mathit{E}[{(q_0^{\ast }-\mathit{d})}^{+}]\}Pr\{\mathit{N}=\mathit{n}\}\\ \ge \sum _n\{(\mathit{r}-\mathit{w})(q_p^{\ast }-\mathit{n})-(\mathit{r}-\mathit{s})\mathit{E}[{(q_p^{\ast }-\mathit{n}-\mathit{d})}^{+}|\mathit{N}-\mathit{n}]\}Pr\{\mathit{N}=\mathit{n}\}\\ ={\Pi }_p^R(q_p^{\ast }).\end{array}$

Then, it is obvious that ${\Pi }_p^R(q_q^{\ast })\ge {\Pi }_p^R(q_0^{\ast })$. Therefore, we have Proposition 2.

Appendix C:

Stackelberg approach

In this paper, the manufacturer and retailer decide the parameters in the transaction based on the cooperation from both. However, it is possible that the respective parameters are decided by the individuals. In this case, the manufacturer decides the compensation fee $\mathit{u}$ while the retailer decides the order quantity $\mathit{q}$. On the transaction of sales, the retailer decides the order quantity$\mathit{q}$ for a given $\mathit{u}$. In other words, the manufacturer decides the compensation fee $\mathit{u}$ on the best response of the retailer for any $\mathit{u}$. The solution obtained by this procedure is called the Stackelberg solution. In this appendix, we show the decision procedure of parameters using the Stackelberg approach.

The retailer decides the order quantity $\mathit{q}$ for a given $\mathit{u}$ as follows:

$q_p^S(\mathit{u})=arg\underset{q}{max}{\Pi }_p^R(\mathit{q},\mathit{u})$

Note that the optimal order quantity $q_p^S(\mathit{u})$ is a function of $\mathit{u}$. Then, the manufacturer decides the compensation fee based on $q_p^S(\mathit{u})$ as follows.

$u_p^S=arg\underset{u}{max}{\Pi }_p^M(q_p^S(\mathit{u}),\mathit{u})$

The combination of the contract parameters $u_p^S$ and $q_p^S(u_p^S)$ is the Stackelberg solution.