Steady-state Analysis And Optimal Production Strategy Of Production-service Inventory Model With Server’s Vacation

The enterprise’s reasonable inventory management can not only avoid oversupply and cause waste of resources,but also avoid oversupply and lead to the loss of customers.It is an inevitable choice to adapt the rapid development of modern economy to integrate traditional production enterprises with emerging service companies.Inventory management in an enterprise must not only reduce its own expenses,but also meet customer needs,and it also needs to increase employee motivation.Based on different vacation strategies,this paper studies the production-service inventory model with server’s vacation.The equilibrium condition of the system and the steady state probability of the system were obtained by comprehensively appling the following mathematical tools and analysis methods of stochastic process such as Markov theory,quasi-birth and death process and matrix-geometric solution method.The optimal inventory strategy was obtained by using genetic algorithm.The details are as follows:Firstly,the M/M/1 production-service inventory system with server’s vacation was studied,where the server takes vacation only when the server faces the empty inventory.The steady-state conditions of the system are obtained using the quasi-birth and death process theory.Then,the steady-state analysis of the system is performed to prove that the steadystate probability distribution of the system has a product form solution.On this basis,the calculation formulas of some performance measures and cost function related to this model are further obtained.The optimal(s,S)inventory was computed numerically.The impact of some system parameters on performance measures was investigated.Secondly,the M/M/1 production service inventory system with server’s vacation,where the server takes vacation only when the server faces the empty queue.The matrix geometric solution of steady-state probability is obtained by using the quasi-birth and death process theory.On this basis,some performance measures and cost function of the model are further obtained.Then the numerical analysis of the effect system parameters on some performance measures and the cost function of the system.We compare the two cases for the performance measures of the M/M/1 production-service inventory system with server’s vacation:one case is that the server takes vacations only when the server faces the empty inventory,and another case is that the server takes vacations only when the server faces the empty queue.Finally,the M/PH/1 production-service inventory system with server’s vacation was studied,where the server takes vacation only when the server faces the empty inventory.In this model,the PH distribution of the service time was considered.Using the quasi-birth and death process theory,the matrix geometric solution of steady state probability is obtained.Based on this,some performance measures and cost function of the model are further obtained.Then the effects of parameters on performance measures under three special cases of different PH distribution are compared and analyzed.The effect of the system parameters on some performance measures and cost function are analyzed numerical.