6.4 Multi-Location Inventory Models with Lateral Transshipments
6.4.2 Features of the Simulation Model
The simulation model offers features that allow the mapping of very general sit- uations. The simulator is in principle suited for models with an arbitrary number of independent non-homogeneous locations, a single product, constant location- dependent delivery and transshipment lead times, and unlimited transportation re- sources. The most important extensions of existing models are the following ones.
With regard to the ordering mode, we assume a periodic review scheme with fixed lengthtP,iof the review period for orders at locationi. In principle arbitrary ordering policies can be realized within the simulation model and so far(si,Si)- and (si,nQi)- ordering policies have been implemented.
ri(t),yi(t)
t Si−ri(t)
yi(t) ri(t) Si
si
Fig. 6.3 (si,Si)-ordering policy. If the inventory positionriof locationidrops below the reorder pointsiat the end of an order period, an order is released to the order-up-to levelSi, i.e.,Si−ri product units are ordered. Analogously, but under continuous review, a transshipment order (TO) ofHi−fTO,i(t)product units is released, if the state functionfTO,i(t)falls belowhi
Clients arrive at the locations according to a compound renewaldemand process.
Such a process is described by two independent random variablesTiandBifor the inter-arrival time of clients at locationiand their demand,i=1. . .N, respectively.
Thus, exact holding and penalty cost can be calculated, which is not the case for models with discrete review, where the whole demand of a period is transferred to the end of a period. That disadvantage does not exist for models with continuous
review. However, in almost all such models a Poisson demand process is assumed – a strong restriction as well.
Concerning thedemand satisfaction mode, most models assume the back-order or the lost-sales cases. An arriving client is enqueued according to a specific ser- vice policy, such as First-In-First-Out (FIFO) and Last-In-First-Out (LIFO), sorting clients by their arrival time, Smallest-Amount-Next (SAN) and Biggest-Amount- Next (BAN), sorting clients by their unserved demand, and Earliest-Deadline-First (EDF). In addition, a random impatient time is realized for each client.
To balance excess and shortage, the simulation model permits all pooling modes from complete to time-dependent partial pooling. A symmetricN×NmatrixP= (pi j)defines pooling groups in such a way that two locationsiand jbelong to the same group if and only if pi j=1, pi j=0 otherwise. The following reflection is crucial. Transshipments allow the fast elimination of shortages, but near to the end of an order period transshipments may be less advantageous. Therefore, a parameter tpool,i∈[0,tP,i]is defined for each locationi. After thekthorder request, locationi can get transshipments from all other locations as long as for the actual timet≤ ktP,i+tpool,i holds. For all other times locationi can receive transshipments only from locations that are in the same pooling group. Thus, the transshipment policies become non-stationary in time.
Transshipments are in fact in the spotlight of this chapter. Regarding thetransship- ment mode, our simulation model allows transshipments at any time during an order cycle (continuous review) as well as multiple shipping points and partial deliveries to realize a transshipment decision (TD). To answer the question when to transship what amount between which locations, a great variety of rules can be defined. Broad applicability is achieved by three main ideas –priorities, introduction of astate func- tionand generalization of common transshipmentrules. Difficulties are caused by the problem to calculate the effects of a TD. Therefore, TDs should be based on ap- propriate forecasts for the dynamics of the model, especially the stock levels. The MLIMT simulator offers several possibilities. For each location transshipment orders (TO) and product offers (PO) are distinguished. Times for TOs or POs are the arrival times of clients or deliveries, respectively. Priorities are used to define the sequence of transshipments in one-to-many and many-to-one situations. Because of continuous time only such situations occur, and thus, all possible cases are considered. The three rules, Biggest-Amount-Next (BAN), Minimal-Transshipment-Cost per unit (MTC) and Minimal-Transshipment-Time (MTT) may be combined arbitrarily. State func- tions are used to decide when to release a TO or PO. The following variables for each locationiand timet≥0 are used in further statements:
yi(t) Inventory level
y±i (t) =max(±yi(t),0) On-hand stock(+)and shortage(−), respectively bord,i(t) Product units ordered but not yet delivered bord,ki Product units ordered in thek-th request btr,i(t) Transshipments on the way to locationi ri(t) =yi(t) +bord,i(t) +btr,i(t) Inventory position
tP,i Order period time
tA,i Delivery lead time of an order nord,i=⌊tA,i/tP,i⌋ Number of periods to deliver an order
To decide at time t in location i about a TO or PO, the state functions fTO,i(t) and fPO,i(t)are defined based on the available stock plus expected transshipments fTO,i(t) =yi(t) +btr,i(t)and the on-hand stock fPO,i(t) =y+i (t), respectively. Since fixed cost components for transshipments are feasible, a heuristic(hi,Hi)-rule for TOs is suggested in the following way, which is inspired by the(si,Si)-rule for or- der requests(hi≤Hi).
If fTO,i(t)<hi
release a TO forHi−fTO,i(t)product units.
However, in case of positive transshipment times it may be advantageous to take future demand into account. Thus, a TO is released on the basis of a forecast of the state function fTO,i(t′) for a time momentt′≥t, and the transshipment poli- cies become non-stationary in time. The MLIMT simulator offers three such time moments:t′=t, the current time (i.e., no forecast),t′=t1, the next order review moment, andt′=t2, the next potential moment of an order supply. For instance, the state functionfTO,i(t) =yi(t) +btr,i(t),t≥0 is considered. LetktP,i≤t<(k+1)tP,i, i.e., we assume that we are in the review period after thekthorder request. Thent1
is defined as follows.
t1= (k+1)tP,i. (6.1)
Fort2we introduce two eventsev(t)↔ {in the actual period there has not been an order supply untilt}andev(t)↔ {there has been an order supply untilt}.
t2= (k−nord,i)tP,i+tA,i+
0 ev(t)↔t<(k−nord,i)tP,i+tA,i
tP,i ev(t)↔t≥(k−nord,i)tP,i+tA,i. (6.2) Usingmi= Bi/ Tias long-run demand per time unit at locationi, the following forecasts are used, illustrated in Figures 6.4 and 6.5:
fˆTO,i(t) =fTO,i(t) =yi(t) +btr,i(t), (6.3) fˆTO,i(t1) =fTO,i(t)−mi(t1−t) +
bord,k′i,k′=k−nord,i ev(t)
0 ev(t) , (6.4)
fˆTO,i(t2) =fTO,i(t)−mi(t2−t). (6.5) Thus, replacing function fTO,i(t)by various forecast functions, a great variety of rules can be described to control the release of TOs. We remark that in case of lin- ear transshipment cost functions without set-up part the (hi,Hi)-rule degenerates to(Hi,Hi). A well-designed optimization algorithm will approximate that solution.
Therefore, we work generally with the(hi,Hi)-rule. To serve a TO, at least one loca- tion has to offer some product quantity. To decide when to offer what amount, an ad- ditional control parameter is introduced – the offering leveloi, corresponding to the
fˆi(t)
t Order
k
Order k+1 Supply
k−nord,i
Supply k+1−nord,i
fi(t) fˆi(t2)
fˆi(t1)
t t2
t1
Fig. 6.4 Forecast functions forev(t)↔t<(k−nord,i)tP,i+tA,i. In the actual period there has not been an order supply untilt. Thus, the time momentt2of the next order supplyk−nord,i is in the current period, and the supplied amount must be considered to forecast ˆfi(t1)
fˆi(t)
t Order
k
Order k+1 Supply
k−nord,i
Supply k+1−nord,i fi(t)
fˆi(t2) fˆi(t1)
t t2
t1
Fig. 6.5 Forecast functions forev(t)↔t≥(k−nord,i)tP,i+tA,i.In the actual period there has been an order supply untilt, and thus, the time momentt2 of the next order supply k+1−nord,iis in the next period, not affecting ˆfi(t1)
hold-back level introduced by Xu et al. [33]. Since only on-hand stock can be trans- shipped, the state function fPO,i(t) =y+i (t)is defined. The offered amounty+i (t)−oi
must not be smaller than a certain value∆omin,ito prevent undesirably small and frequent transshipments. Similar forecasts are applied to take future demand into account with forecast momentst,t1, andt2. For details we refer to Hochmuth [13].
Thus, the PO rule is as follows.
If ˆfPO,i(t)−oi≥∆omin,i
release a PO for ˆfPO,i(t)−oiproduct units.
Thus, the set of available transshipment policies is extended, including all com- monly used policies, and allowing multiple shipping points with partial deliveries.
In order to measure the system performance bycost and gain functions, order, holding, shortage (waiting) and transshipment cost functions may consist of fixed values, components linear in time, and components linear in time and units. Fixed cost arises from each non-served demand unit. All cost values are location-related.
The gain from a unit, sold by any location, is a constant. To track cost for infi- niteplanning horizons, appropriate approximations must be used. The only problem with respect to finite horizons is the increase in computing time to get a sufficiently accurate estimate, although the extent can be limited using parallelization.
Choosing cost function components in a specific way, cost criteriaas well as non-cost criteria can be used, e.g., the average ratio of customers experiencing a stock-out or the average queue time measured by out-of-stock cost, or the efficiency of logistics, indicated by order and transshipment cost.