A Stochastic Model for an Input Control Problem in a Two-Level Supply Chain for Production-Time-Dependent Products with Random Demands

Inspired by the poultry farming process, we studied an input control problem in a two-level supply chain for production-time-dependent products with random demands. Poultry farms deliver chicks in batches, and the raising process begins at timed intervals. Chicks become broilers after a predetermined raising time. The broilers in the process are shipped to manufacturing plants to satisfy the demand. The remaining chicks grow to the next product size to satisfy the demands for larger chickens. This procedure is repeated until the chicks are fully grown. After the chicks are grown to satisfy the demand for the largest size, the remaining chicks are discarded. In this paper, a stochastic model is presented to study an input control problem in the poultry farming process. Because of the production density, feed, and temperature control, one important issue in the operation of a poultry company is the determination of the raising interval and quantity of input (chicks). While existing mathematical models can provide effective information on the production-planning problem of systems, research has not been conducted on cases of random demand. Identifying a recursive structure and Markovian property for the number of raw materials (chicks) and the unfulfilled demand for each product type in the system, we demonstrate that embedded Markov chain models can be obtained. The equilibrium probabilities of the models can be calculated using matrix analytic methods or probability generating functions. Various numerical experiments are conducted to analyze how performance measures such as amount of disposal, unsatisfied demands, and total cost (considering disposal cost and opportunity cost) change with system parameters.