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This paper analyzes a two-facility location problem under demand uncertainty. The maximum server for the ith facility is . It is assumed that primary service demand arrivals for the ith facility follow a Poisson process. Each customer chooses one of the facilities with a probability which depends on his or her distance to each facility. The service times are assumed to be exponential and there is no vacation or failure in the system. Both facilities are assumed to be substitutable which means that if a facility has no free server, the other facility is used to fulfill the demand. When there is no idle server in both facilities, each arriving primary demand goes into an orbit of unlimited size. The orbiting demands retry to get service following an exponential distribution. In this paper, the authors give a stability condition of the demand satisfying process, and then obtain the steady-state distribution by applying matrix geometric method in order to calculation of some key performance indexes. By considering the fixed cost of opening a facility and the steady state service costs, the best locations for two facilities are derived. The result is illustrated by a numerical example.

An order penetration point (OPP) is the boundary between make-to-order (MTO) policy and make-to-stock (MTS) policy. This study is dedicated to OPP strategic decision making which is concerned with the upstream and downstream decisions of OPP in supply chains. In this paper, we consider a two-echelon production supply chain in which in first production stage, supplier manufactures the semi-finished products on a MTS policy for a manufacturer in second production stage that will customize the products on a MTO policy. The manufacturer desires to find the optimal fraction of processing time fulfilled by its supplier and its optimal semi-finished products buffer storage capacity in OPP. Also, impatient customers are considered in the study. In order to calculate system performance indexes, we employ matrix geometric method and queuing steady state equations. Afterward, optimal solutions are obtained applying an optimization mathematical model with enumeration and direct search techniques.

This paper analyzes a two-facility location problem under demand uncertainty. The maximum server for the ith facility is It is assumed that primary service demand arrivals for the ith facility follow a Poisson process. Each customer chooses one of the facilities with a probability which depends on his or her distance to each facility. The service times are assumed to be exponential and there is no vacation or failure in the system. Both facilities are assumed to be substitutable which means that if a facility has no free server, the other facility is used to fulfill the demand. When there is no idle server in both facilities, each arriving primary demand goes into an orbit of unlimited size. The orbiting demands retry to get service following an exponential distribution. In this paper, the authors give a stability condition of the demand satisfying process, and then obtain the steady-state distribution by applying matrix geometric method in order to calculation of some key performance indexes. By considering the fixed cost of opening a facility and the steady state service costs, the best locations for two facilities are derived. The result is illustrated by a numerical example.