4.2 Single-Item-Models
4.2.1 Deterministic Models
Normally the demand is not known. In case of standard models therefore it is necessary to substitute the real existing process of demand by a row of forecasts d(t+τ), τ = 0,1,2. . .. Hered(t+τ) means the forecast of demand in periodt+τ; t is the current period. Those forecasts will be repeated as often as necessary, in order to minimise errors. Thus standard models are used in a continuous way:
after every forecast the order policies are calculated anew. But not every forecast error can be averted; therefore the safety stock SBt is necessary.
For static time series the methods of moving average and exponential smooth-ing of 1. order are available. If there is a trend, then exponential smoothsmooth-ing 2.
order or linear regression analysis can be used. For seasonal fluctuation, one can avail the forecast method of Winters.
Figure 4.2: Scheme of single-item-models
Classical Lot Size Model
Instead of the classical lot size model one speaks of the Andler-, Harris- or Wilson-model. Precondition is the assumption that demand is constant and continuous.
Moreover there is no delivery time and no shortage. The problem is to determine the best order size q and the best length of the order interval T in such a way that the sum of order and storage costs is minimal. The relevant order costsB(q) are fix costs that arise by ordering.
B(q) =
½ cB ∀ q >0
0 ∀ q= 0 (4.3)
Storage costs L(q) of the order cycle T are L(q) =Tq
2cL (4.4)
with 2q asaverage inventoryandcLas costs per unit and time intervall. In order to determine the optimum order policy, following procedure is applied: the average total costs C per time unit are
C = 1
T(cB+Tq 2cL)
= cB
T + q 2cL
= d
qcB+qcL
2 (4.5)
with T = dq and d as demand rate. The average order costs cTB = dqcB and the average storage costs L(q)T = 2qcL are illustrated in Figure 4.3:
Figure 4.3: Classical lot size model; Left: inventory process. Right: different costs.
While the average storage costs rise linearly with the order size, the order costs diminish in a hyperbolic way. There is a minimum of total costs for a certain order quantity. The minimum can be calculated by differentiating 4.5 after q:
∂C
∂q =−d
q2cB+cL
2 = 0
=⇒q∗ =
r2dcB
cL
(4.6) q∗ is the classical lot size. For the optimal length of cycle T∗ and the optimal average costs following holds:
T∗ = q∗ r =
r2cB
dcL
(4.7) C∗ =p
2dcBcL (4.8)
The classical lot size model can easily consider many restrictions, for example delivery dates, continuous inflow, shortages and similiar things. In this way the lot size model is a special case of many complex models.
Wagner-Whitin-Model
The Wagner-Whitin-model is marked by a row of specialisations of the general deterministic system. It is defined by the deterministic dynamic decision problem of Table 4.1
(1) yt: inventory at the beginning of periodt; t = 0,1, . . . , N Yt=yt:yt ≥0 state domain
y0: initial inventory
inventories can take every positive value;
that means there is neither a restriction on storage capacity nor shortfalls.
(2) qt: order in period t;t = 0,1, . . . , N −1 orders can take every positive value.
size restrictions and quantisations do not exist;
but the orders have to avert shortfalls.
(3) dt: Demand in the intervall of inspection t, t+ 1;
t= 0,1, . . . , N −1
(4) yt+1 =yt+qt−dt: equation of stock balance (5) Cost Criterion: min C =PN−1
t=0 (B(qt) +L(yt+1)) Cost of Ordering: B(qt) = cB(qt) =cB for qt6= 0
B(qt) = cB(qt) = 0 else Cost of Storage: L(yt) =cL(yt−d2t)
Table 4.1: Deterministic decision problem
In this model delivery time was omitted. But that is no constraint; it only serves the simplification of notation. There are only fixed costs for orders. Quan-tity based costs are not considered, because they don’t influence the moment of ordering or the order size. Quantity dependent, non-proportional costs are not included in the cost criterion above. Storage costs evaluate an average inventory with a storage cost rate of cL. The special model structure implies two essential simplifications for the determination of the optimal order policy:
1. If there is an empty stock or the inventory is fallen to a minimum level, an order will be placed; otherwise there would be unnecessary storage costs.
2. The consolidated demand of future periods will be ordered; otherwise there would be unnecessary storage costs as well.
These evident conditions to an optimal policy lead to a basic restriction of differ-ent policies. The saturation of conditions 1 and 2 leads to the so called Wagner-Whitin-algorithm.
At first the periods 0 and 1, then 0,1,2, then 0,1,2,3, and so on, are opti-mised; thereby the results of the last optimisation (with one period less) are used.
Because of that just a fraction of different policies has to be considered.
Heuristic Methods
Although the Wagner-Whitin-model is a very efficient algorithm, several methods have been developed as approximation to this model. Contrary to Wagner-Whitin those heuristic algorithms don’t consider the whole planning period. Therefore the computation time and the solution quality are lower. The following methods check, whether the demand of a period can be satisfied by the last order or whether a new order has to be dismissed. The first order is determined by the first period with a demand. The next order quantities are satisfied by one order as long as a special criterion is fulfilled; otherwise a new order is dismissed.
Least-Unit-Cost: The order quantity in periodtis increased with future ma-terial requirements as long as the average costs per quantity unit can be reduced.
If there is an order in period τ and the demand is covered up to period j with j > τ , the average costs are defined by:
cunitτ j = cB+cL Thus those order quantity of period τ has to be determined that leads to a min-imum of equation 4.9. The decision problem of period τ then can be formulated
as:
max{j|cunitτ j < cunitτ j−1} (4.10) Thus the highest j has to be found, which fulfils conditioncunitτ j < cunitτ j−1. In other words: that period has to be found, whose demand can be satisfied by the order in period τ without an increase in average costs.
Part-Period-Balancing: The key-note of this heuristic is that an order reaches for as many periods as the storage costs are equal to the order fix costs:
cL j∗
X
t=τ+1
(t−τ)dt≤cB (4.11)
where j∗ is the period up to which the order reaches.
Silver-Meal: In the style of the classical lot size model the Silver-Meal method tries to minimise costs per time unit or period. If there is an order in period τ which covers the material requirement up to periodj, j > τ, following costs have to be considered:
cperiodτ j = cB+cLPj
t=τ+1(t−τ)dt
j−τ + 1 (4.12)
In periodτ those j is sought, which fulfils following condition:
max{j|cperiodτ j < cperiodτ,j−1} (4.13) This means that the costs per period shall be minimised.
Other Algorithms: Further onGroffandSavingsare two methods for deter-mining order time and quantity. They work similiar to the previous algorithms;
therefore they shall not be presented here. For further reading [Te03] is recom-mended.