Effects of Lot-Sizing Integration and Learning Effect on Managing Imperfect Items in a Manufacturer-Retailer Chain

Table 1

Summary of notations.

System
𝑘	The number of finished items
𝑖	The index of the finished items, 𝑖=1,2,…,𝑘
𝑗	The index of the decision policies, 𝑗∈{𝑁,𝐶}, where N denotes the Nash game policy and 𝐶 denotes the cooperation game policy
𝐷𝑖	The demand rate of finished item i in the marketplace
TC𝑗,𝐶	Total cost for the channel per unit time under policy 𝑗
Manufacturer
𝑡𝑝	Production time for each production cycle
𝑛𝑗,𝑖	The number of shipments of item i per batch production run under policy j
𝑚𝑗,𝑖	The number of shipments of raw material i per batch production run under policy 𝑗
𝑢𝑖	The usage rate of raw material for finished item 𝑖
𝐹	The major set up cost per production cycle
𝑓𝑖	The minor set up cost for adding finished item 𝑖 into the production schedule
𝑠𝑟𝑖	The ordering cost of the raw material for finished item 𝑖 per production cycle
ℎ𝑓𝑖	The inventory holding cost of finished item 𝑖 per unit time
ℎ𝑟𝑖	The inventory holding cost of the raw material for finished item i per unit time
𝑙	The learning rate in the production line, where 0.5<𝑙<1
𝑏	The learning coefficient, which decreases over the production time required per unit, 𝑏=log𝑙/log2
𝛽	The production rate of defective items per unit time
𝛾	The time required to produce the first unit of the production cycle
𝑡(𝑥)	The time to produce the xth unit item of the production cycle
TC𝑗,𝑀	The total cost per unit time under policy 𝑗
Retailer
𝑇𝑁,𝑖	The individual replenishment cycle of finished item𝑖 under the Nash game policy
𝑇𝐶	The common replenishment cycle of all finished items under the cooperation game policy
𝑅	The major set up cost per replenishment cycle
𝑟𝑖	The minor set up cost for adding finished item 𝑖 into the order
ℎ𝑖	The inventory holding cost of finished item 𝑖 per unit time
TC𝑗,𝑅	The total cost per unit time under policy 𝑗