The supply chain of interest is depicted in Figure 5.2.1. We have a setJ :={1,2, . . . , J} of locations. Each location consists of a production system with an attached inven-tory. The inventories are replenished by a supplier network which consists of a set M := {J+ 1, . . . , J +M} of workstations and manufactures raw material for all lo-cations, but distinguishes the replenishment orders from different locations. Each order for raw material is specified by a location j ∈ J and the resulting raw material is sent back exactly to the location which has placed the order.
Facilities in the supply chain. Each production system j ∈ J consists of a single server (machine) with infinite waiting room that serves customers on a make-to-order basis under FCFS regime. Customers arrive one by one at production systemj according to a Poisson process with rateλj >0and require service. To satisfy a customer’s demand the production system requires exactly one item of raw material, which is taken from the
Location 1
Location
Single server
Single server Waiting room Lost
sales
Lost sales
Replenishmen t order
Replenishmen t order Demand arrival
process
Demand arrival process
Inventory
Inventory
Replenishment
Replenishment Waiting room
Supplier with M stations
Figure 5.2.1.: Supply chain with base stock policy
associated local inventory. When a new customer arrives at a location while the previous customer’s processing is not finished, this customer will wait. If inventory is depleted at locationj, the customers who are already waiting in line will continue to wait, but newly arriving customers at this location will decide not to join the queue and are lost (“local lost sales”).
The service requests at the locations are exponentially-1 distributed. All service re-quests constitute an independent family of random variables which are independent of the arrival streams. The service at location j ∈ J is provided with local queue-length-dependent intensity. If there arenj >0 customers present at locationj either waiting or in service (if any) and if the inventory is not depleted, the service intensity isµj(nj)>0.
If the server is ready to serve a customer who is at the head of the line, and the inventory is not depleted, service immediately starts. Otherwise, the service starts at the instant when the next replenishment arrives at the local inventory.
The inventory at location j is controlled by prescribing a local base stock levelbj ≥1, which is the maximal size of the inventory there, we denoteb:= bj :j ∈J
.
Each workstation m ∈ M of the supplier network consists of a single server with in-finite waiting room under FCFS regime. The service requests at the workstations are exponentially-1 distributed. All service requests constitute an independent family of
5.2. Description of the model
random variables which are independent of the arrival streams. Service at workstation m∈M is provided with local queue-length-dependent intensity. If there are` >0orders present, the service intensity isνm(`)>0.
Routing in the supply chain. A served customer departs from the system (with the consumed material) immediately after service and at the same time an order for one item of raw material is placed at the supplier network (“base stock policy”).
To distinguish orders from different locations, each order is marked (tagged) by a “type”
which for simplicity is the index of the location, where the order is triggered. We found that Kelly’s deterministic routing scheme for “customers” in networks (cf. [Kel79, pp.
82ff.]) is a useful device to describe the interaction of inventories and supplier network.
It should be emphasized that the cycling “customer” represents an order in the supplier network and a item of raw material in the inventories.
An order triggered by location j follows a type-j-dependent route for eventual replen-ishment, denoted by r(j) =
r(j,1), . . . , r(j, S(j)−1), r(j, S(j))
. Here r(j, `) ∈M for
`= 2, . . . , S(j) is the identifier of the `-th workstation on the pathr(j), and S(j) is the number of stages of the route of type j. For completeness we fix r(j,1) := j ∈ J, and prescribe that a type-j order departing fromr(j, S(j))enters as an item of raw material immediately the inventory at locationj =r(j,1)to restart its cycle.
It is assumed that transmission times for orders are negligible and set to zero and that transportation times between the supplier network and the local inventories are negligible.
The usual independence assumptions are assumed to hold as well.
To obtain a Markovian process description of the integrated queueing-inventory system, we denote by Xj(t) the number of customers present at location j ∈J at time t≥0 either waiting or in service (queue length). ByYj(t) we denote the contents of the inventory at locationj∈J at time t≥0. ByWm(t)we denote the sequence of orders at workstation m∈M of the supplier network at timet≥0.
We denote by Km the set of possible states at node m ∈ M (local state space). The state km := [tm1, sm1;. . .;tm#km, sm#km] ∈Km indicates that there are #km orders at workstationm∈M, on positionp∈ {1, . . . ,#km}resides an order of typetmp∈J, which is on stage smp ∈ {1, . . . , S(tmp)} of its route r(tmp) =
r(tmp,1), . . . , r(tmp, S(tmp)) . Here(tm1, sm1)is the order at the head of the line, which is in service and(tm#km, sm#km) is the order at the tail of the line.
Notational convention. To make reading easier, we use a unified notation for the states of the inventories at the locations and the states of the workstations in the supplier network. In doing this we identify items of raw material arriving at the inventory j with the order sent out to the supplier network when an item is consumed by a departing customer. Therefore, adopting the state description of the workstations for that of the inventories, the state of the inventory at locationj∈J at timetis
Yj(t) =kj = [j,1;. . .;j,1]
| {z }
#kjitems
,
since the route of type j starts in the inventory at location j (i.e. tjp = j for the types
andsjp= 1 for allp∈ {1, . . . ,#kj}). A stage number sjp>1 indicates that the unit (as an order) is in the supplier network.
Summarizing, global states of the inventory-replenishment subsystem are
k=
inventories at locations
z }| { k1, . . . , kJ,
workstations at supplier network
z }| { kJ+1, . . . , kJ+M
∈K⊆
J+M
Y
j=1
Kj,
whereKj denotes the local state space atj ∈ J∪M and K denotes the feasible states composed of feasible local states.
For#kj = 0,j∈J, we read
tj1, sj1;. . .;tj#kj, sj#kj
=: [0], and for#km = 0,m∈M, we read
[tm1, sm1;. . .;tm#km, sm#km] =: [0].
We define the joint queueing-inventory process of this system by
Z = ((X1(t), . . . , XJ(t), Y1(t), . . . , YJ(t), WJ+1(t), . . . , WJ+M(t)) :t≥0).
Then, due to the usual independence and memoryless assumptions Z is a homogeneous Markov process, which we assume to be irreducible and regular. The state space ofZ is
E =
(n,k) :n∈NJ0,k∈K .
Discussion of the modelling assumptions. We have imposed simplifying assump-tions on the production-inventory system to obtain explicit and simple-to-calculate per-formance metrics of the system, which give insights into its long-time and stationary behaviour. This enables a parametric and sensitivity analysis that is easy to perform.
First, the assumption of exponentially distributed inter-arrival and service times are standard in the literature. The locally state-dependent service rates are also common and give quite a bit of flexibility. The lead time of an order is composed of the waiting times plus the service times in the supplier network. They are therefore more complex than exponential or even constant lead times.
Second, we assume that the local base stock levels are positive (i.e.bj ≥1 at location j). If bj = 0, all customers at location j would be lost, which is the same as excluding location j from the production-inventory system.
Third, the assumption of zero transportation times can be removed by inserting special (virtual)M/G/∞ workstations into the network.