A Study of Compensation Problem for Uncertainty of Quality in Supply Chain Yasuhiko Takemoto, Ikuo Arizono Abstract—The uncertainty of demand occurs at a down- stream side and spreads over an upstream side in a supply chain. To the contrary, the uncertainty of supply occurs at an upstream side and spreads over an downstream side in a supply chain. The quality of items is particularly important as a factor in the uncertainty. Imperfect quality items included in the procurement from a supplier brings the uncertainty of supply, that is, whether imperfect quality items are included or not, or how many imperfect quality items are included. An intangible loss due to the uncertainty of imperfect quality items is unavoidable even if the downstream side (taht is, a manufacrurer or retailer) prevented all imperfect quality items from being distributed to customers and society by perfect inspection. Therefore, in this paper, we consider the impact of the uncertainty of imperfect quality items in the situation that the imperfect quality items included in the procurement from the supplier are perfectly retrieved in the downstream side. In particular, the impact of the uncertainty of imperfect quality items is evaluated as an object of a monetary compensation. Hence, we propose a condition of the monetary compensation to maintain supply chain partnarship. Through some numerical examples, the condition of the monetary compensation for the uncertanty of imperfect quality items is illustrated. Index Terms—compensation, coordination, incentive compat- ible condition, newsvendor model, I. I NTRODUCTION I N common production and inventory systems, the uncer- tainty about demand of customers exists. In particular, the spread of the uncertainty about the demand in a supply chain is well known as the bullwhip effect [1]. The uncertainty of demand occurs at a downstream side in a supply chain. Then, the uncertainty of demand spreads over an upstream side in a supply chain. To the contrary, the uncertainty of supply occurs at an upstream side and spreads over a down- stream side. The quality of items is particularly important as a factor in the uncertainty. In many researches regarding production and inventory systems, it has been assumed that the quality of items received from a supplier are perfect. However, imperfect quality items are sometimes included in the procurement to a manufacturer or retailer as some results of imperfect production and inspection by a supplier and damage in transport. The impact of imperfect quality items on customers and society is by no means trivial. It is also known that quality has a great impact on a reputation and customer confidence. Manuscript received Nov 28, 2017; revised Jan 12, 2018. This work was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 26350433: “The rationalization and optimization of distribu- tion systems based on mechanism design in supply chain environments” and Grant Number 17K01266: “The investigation of data visualization and its application to production and operation management”. Y. Takemoto is with the Fucalty of Science and Engineering, Kindai University, 577-8502 Osaka, JAPAN (e-mail:takemoto@mech.kindai.ac.jp). I. Arizono is with the Graduate School of Natural Science and Technology, Okayama University, 700-8530 Okayama, JAPAN (e- mail:arizono@okayama-u.ac.jp). Theerefore, the compensation for imperfect quality items to customers and society has to be made. Then, the cost allocation problems regarding the compensation between supply chain members have been considered until now, such as Gao et al.[2]. On one hand, imperfect quality items included in the procurement brings the uncertainty of supply. That is, it is unknown whether imperfect quality items are included or not, or how many imperfect quality items are included. As mentioned above, the uncertainty of supply occuring at an upstream side spreads over a downstream side. That is, a manufacturer or retailer suffers intangible loss due to the uncertainty of containing imperfect quality items in the procurement from a supplier in addition to physical and mental losses due to imperfect quality items themselves. Note that the mental loss due to imperfect quality items in this paper means damage due to a reputation and customer confidence. Even if the manufacturer or retailer prevented all imperfect quality items from being distributed to customers and society by perfect inspection, the intangible loss due to the uncertainty of imperfect quality items is unavoidable. In this paper, we consider the impact of the uncertainty of containing imperfect quality items in the procurement from a supplier in a supply chain composed of a supplier and a retailer. In particular, we investigate the case where the imperfect quality items included in the procurement from the supplier are perfectly retrieved in the retailer. In this case, we evaluate exactly the impact of the uncertainty of imperfect quality items as a monetary loss. Then, a monetary compensation is calculated in the viewpoint of the uncertanty of imperfect quality items. Further, we propose a condition of the monetary compensation to maintain supply chain partnarship. Through some numerical examples, the condition of the monetary compensation for the uncertanty of imperfect quality items is illustrated. II. LITERATURE REVIEW As early literature about imperfect quality items in the pro- duction and inventory management, Porteus [3] has studied an impact of imperfect qualitty items on a basic lot sizing problem under the assumption that a process goes out of control with a fixed probability. Rosenblatt and Lee [4] have studied an impact of imperfect production on production cycles. Moinzadeh and Lee [5] have dealt with a continuous review inventory model with Poisson demand and imperfect quality items. Paknejad et al. [6] have studied an optimal policy in a continuous review inventory model with imperfect quality items for any demand distribution. Salameh and Jaber [7] have developed an extended economical ordering quantity (EOQ) model where imperfect quality items are salvaged at a discount price. Further, many researchers have considered the model with imperfect quality items such as Wu and Ouyang Proceedings of the International MultiConference of Engineers and Computer Scientists 2018 Vol II IMECS 2018, March 14-16, 2018, Hong Kong ISBN: 978-988-14048-8-6 ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online) IMECS 2018