Optimal Dynamic Production Planning for Supply Network with Random External and Internal Demands

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This paper focuses on joint production/inventory optimization in single and multiple horizons, respectively, within a complicated supply network (CSN) consisting of firm nodes with coupled demands and firm nodes with coupled demands. We first formulate the single-epoch joint optimal output model by allowing the production of extra quantity for stock underage, considering the fixed costs incurred by having inventory over demand and shortfalls. Then, the multi-temporal dynamic joint production model is further investigated to deal with stochastic demand fluctuations among CSN nodes by constructing a dynamic input–output model. The K-convexity defined in Rn space is proved to obtain the optimal control strategy. According to physical flow links, all demands associated to the nodes of CSN are categorized into the inter-node demand inside CSN (intermediate demand) and external demand outside CSN (final demand). We exploit the meliorated input–output matrix to describe demand relations, building dynamic input–output models where demand fluctuates randomly in single-cycle CSN and finite multi-cycle CSN. The novel monocyclic and multicyclic dynamic models have been developed to minimize system-wide operational costs. Unlike existent literature, we consider fixed costs incurred by overdemand and underdemand inventory into system operational cost functions and then demonstrate the convexity of objective functions. The cost function with two fixed penalty costs due to excess and shortage of inventory is developed in a multicycle model, and the K-convexity defined in Rn is proved to find out the optimal strategy for joint dynamic production of CSNs in the case of multi-products and multicycles.