Optimal analysis of a multi-server queue with preprocessing time and replenishment inventory

ABSTRACTRecently, e-commerce has had the advantage of lower rental expenses. Convenient online payment has rapidly grown and provides a great business opportunity for enterprises. Inventory management plays a critical role in the competitiveness of most electronic business enterprises. Before the products hit the store shelves or are available online to customers, the quantity and quality of the products should be confirmed and inspected. Thus, in addition to the replenishment lead time, the preprocessing time has essential effects on inventory management. To reduce the inventory cost and determine a proper replenishment policy, we propose a multi-server inventory queue with a preprocessing time, which is seldom mentioned by the existing studies. The steady-state probabilities, system stability conditions, and expressions of several critical system characteristics are derived. Subsequently, the operating cost function is developed to determine the optimal reorder point and an appropriate number of servers with the minimum cost. The established numerical results and sensitivity analysis help managers identify critical variables and enhance operational efficiency.CO EDITOR-IN-CHIEF: Hsieh, Sun-Yuan, Pang, Ai-ChunASSOCIATE EDITOR: He, DebiaoKEYWORDS: Inventory queuematrix-geometric methodoptimal replenishment policypreprocessing time Nomenclature 0=the zero row vectorc=number of serversc∗,s∗,γ∗=the optimization solution at the minimum costCc=cost per serverCh=holding cost of each customer in the systemCI=holding cost per inventory in the systemCγ=cost for providing a specific processing ratee=the identity column vectorE[C]=the expected number of customers in the systemE[I]=the expected number of inventories in the systemF=the expected cost functionM=the replenishment quantityO=the zero matrixP=the steady-state vectorP[ID]=the probability of no customer or inventoryP[IDC]=the probability of an empty systemPi,jk=the steady-state probabilityQ=the transition matrixR=the rate matrixs=reorder point(S)=states for material preprocessing(W)=states for normal operatingx=the invariant probabilityα=the replenishment rateε=criteria for convergence determinationλ=the mean arrival rateμ=the mean service rateΠi=the steady-state sub-vectorγ=the preprocessing rateΩ=set of system statesΩS=set of system states for material preprocessingΩW=set of system states for normal operatingDisclosure statementNo potential conflict of interest was reported by the author(s).