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For practical sized problems, the complexity is ON. PUBLICATION ABSTRACT A pharmaceutical company raised the question whether Links of London increased product portfolio could still be manufactured on the existing machinery. The proportional lotsizing and scheduling problem PLSP seemed to be most appropriate to answer this question. However, although there are papers dealing with a multilevel PLSP none allows a zero lead time offset which is a prerequisite for the case considered here. In this paper we will extend and modify an existing mixed integer linear programming MIP model formulation in two ways first, Links of London hot Pink Heart Charm will extend the singlelevel single machine PLSP to a multilevel single machine PLSP PLSPMLSM with a zero lead time offset. Second, we will describe a new and more compact model formulation incorporating period overlapping setup times and batch size constraints. Based on the realworld application several test instances have been generated to provide insights into those characteristics which make instances of the PLSPMLSM difficult to solve by a standard MIP solver. PUBLICATION ABSTRACT This paper considers a supplementary supplyorder system in a multiperiod situation. links of london sale each period, the buyer first places an initial order based on the demand prediction he has the opportunity to place a supplementary order with the supplier after the demand of that period is realized. The supplier maintains an inventory, and decides the quantity to be produced and the quantity to be provided for the supplementary order in each time period. We formulate the problem as a multiperiod inventory game, and derive the optimal production and order policies for the supplier and buyer, respectively. The existence and uniqueness of Nash equilibrium is proved in the generalized multiperiod setting, and the closedform Nash equilibrium solution is obtained when the parameters are stationary. Numerical study is performed to reveal more managerial insights. We find that the supplementary supplyorder mechanism, if designed properly, can effectively improve the multiperiod supply chain performance. PUBLICATION ABSTRACT Links of London Charms paper formulates an extension of the traveling purchaser problem where multiple types of commodities are sold at spatially distributed locations with stochastic prices each following a known probability distribution.