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A comparison of methods for lot sizing in a rolling horizon lotting paper

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					      A comparison of methods for lot-sizing in a rolling

                                horizon environment1

                      Wilco van den Heuvel2 • Albert P.M. Wagelmans

           Econometric Institute and Erasmus Research Institute of Management,

    Erasmus University Rotterdam, PO Box 1738, 3000 DR Rotterdam, The Netherlands

                                           September 2004

We argue that the superior performance of a recent method for lot-sizing in a rolling

horizon scheme is to a large extent due to the assumption that quite accurate future

demand estimates are available. We show that other methods, including a

straightforward one, can use this information just as effectively.



End Effects; Dynamic Lot-sizing; Rolling Horizon Scheme; Ending Inventory

Valuation




1. Introduction

Production planning decisions are usually made on a rolling horizon basis. That is, a

production planning is made for a fixed number of periods (T) for which the demand

is known. The first production decision is implemented and the horizon is rolled

forward to the period where the next production decision needs to be made. In this

1
    We would like to thank several participants of the International Workshop on Optimization in Supply

Chain Planning, held in Maastricht in June 2002, for their comments on this research. We also like to

thank Kamalini Ramdas for her useful comments on a previous version of the paper.



2
    Corresponding author. E-mail address: wvandenheuvel@few.eur.nl (W. van den Heuvel)


                                                   1
paper we consider such a situation with the following costs: a setup cost if production

takes place in a period, marginal production costs per unit produced and holding costs

for carrying inventory from one period to the next period. The goal is to construct a

production plan minimizing all relevant costs.



However, in practice, the ‘true’ horizon n beyond period T is typically not known. If

the firm makes an optimal decision for the T-period model horizon, this may not be

optimal for the n-period horizon. In particular, it is always optimal for the T-period

problem to leave zero inventory at the end of period T, but this may not be optimal for

the n-period problem. This is called an end-effect [5] or the truncated horizon effect

[3].



The question is now how to reduce these end-effects. In a recent paper, Fisher et al.

[4] present an ending inventory valuation (EIV) method that includes a valuation term

for end-of-horizon inventory in the objective function of the short-horizon model.

Their computational tests show that this method outperforms the Wagner-Whitin

algorithm and the Silver-Meal heuristic, under several demand patterns. In this paper,

we argue that the superior performance of the EIV method is to a large extent due to

the fact that the other two methods do not use any information about demand beyond

the planning horizon, whereas the EIV method assumes the availability of quite

accurate future demand estimates. To show this, we replicate their computational

tests, also including a method which incorporates this information about future

demand in a straightforward way. Furthermore, we include a method recently

proposed by [9]. Both methods turn out to have similar performance to the EIV




                                          2
method in comparison to the traditional methods. Moreover, the straightforward

method performs at least as well as all other methods for almost all demand patterns.



This paper is organized as follows. In section 2 we describe the methods that we are

going to compare. The computational tests are discussed in section 3. We end the

paper in section 4 with some concluding remarks and an interesting point for future

research.


2. Methods to mitigate end-effects

Before describing the different methods that try to reduce end-effects, we first show

how one can solve a T-period lot-sizing problem. This is based on the classical

solution method proposed by [11]. We introduce the following notation:

  dt = demand in period t

  Kt = setup cost in period t

  pt = marginal production cost in period t

  ht = holding cost in period t.

Wagner and Whitin [11] show that there exists an optimal solution that satisfies the

zero inventory property, i.e., production only takes place if the ending inventory drops

down to zero. This means that a production plan is completely determined by its

production periods. Let cs,t be the total cost for satisfying demand in periods s,…,t by

producing in period s (this is called a subplan) with 1 ≤ s ≤ t ≤ T. That is,

                                           t                i                  
                            c s,t = K s + ∑  p s d i +
                                                         ∑h         j −1   d i .
                                                                                
                                          i=s            j = s +1              

If f(t) denotes the minimal cost for the t-period production plan and we set f(0) = 0,

then the optimal production plan can be found recursively by




                                                3
                                      f (t ) = min { f (i − 1) + ci ,t }.
                                              i =1,...,t



It is not difficult to verify that this leads to an O(T2) algorithm.



In the remainder of the paper we will assume that production cost are zero and setup

and holding cost are constant, i.e., Kt = K, pt = 0 and ht = h for t = 1,…,T. We make

this assumption because the method of [4] can only handle this case. Note that taking

production cost equal to zero is reasonable, we may just consider the profit margin in

the objective function. In the next sections several methods proposed in the literature

are described.


2.1 Fisher et al. [4]

Fisher et al. [4] propose the so-called ending inventory valuation (EIV) to deal with

end-effects. Because in an optimal solution the ending inventory I in period T may not

be zero, ending inventory I at the end of the short horizon is assigned the non-negative

value

                                                       h
                                    V(I) = K −           (EOQ − I)2 ,                       (1)
                                                      2D

where D reflects the demand rate (the specific choice depends on the demand pattern;

see section 3) and EOQ is the economic order quantity (i.e., EOQ = 2 KD/h ). The

periodic order quantity (POQ) can be calculated by POQ = EOQ/D. When production

in a certain period covers more than the demand up to period T, the ending inventory

in period T is assigned a value according to (1). One can derive from [4, p. 683]

(substitute equation (3.3) in equation (3.2)) that using the valuation function is the

same as changing the cost coefficients of the Wagner-Whitin recursion to:

             T − t +1    t + POQ −1
                                               T − t + 1 t + POQ −1
c    F
           =          K + ∑ h ( j − t ) d j − 1 −
                                                          ∑ h( j − t ) D for t > T − POQ + 1,
                                                   POQ  j =t
    t ,T
              POQ             j =t                       


                                                           4
where dj = D for j > T. The interpretation of the cost coefficients is as follows. The

first term corresponds to the fraction of setup cost that falls within the horizon T. The

second term consists of the holding cost for periods t,…,t + POQ -1. In the last term

we subtract the fraction of holding cost that fall beyond the horizon, where demand is

assumed to be equal to D for all periods. Again, a similar recursion as in the Wagner-

Whitin case can be applied and the first production decision will be implemented in

the production plan.


2.2 Stadtler [9]

Stadtler [9] uses the following approach to deal with end-effects. First for all periods

t = 1,…,T the time between order (TBO) τt is calculated, i.e., τt is the number of

periods for which demand is satisfied by producing in period t. Stadtler [9] uses

Groff’s heuristic [6] to calculate the ΤΒΟs in linear time. For the periods beyond

period T (these are the periods T + 1,…,Tmax) forecasted demands are used. (We will

come back to the choice of Tmax in section 3.) Next Stadtler [9] modifies the cost

coefficients for the subplans to

                              T − t +1    t +τ t −1
                                                            
                 c    S
                            =          K + ∑ h( j − t )d j  for t + τ t − 1 > T .
                                τt                         
                     t ,T
                                             j =t          

The idea is straightforward: costs for the subplan t,…,T are assigned a value

proportional to the number of periods that fall within the model horizon T. Now the

same recursion formula can be used as in the Wagner-Whitin algorithm.



If the TBOs and POQ (defined in section 2.1) are equal, then we have the following

relation between the cost coefficients of [9] and [4]:

                          T − t + 1 t + POQ −1
          ctFT = ctS,T + 1 −
                                     ∑ h( j − t )(d j − D) for t > T − POQ + 1
                              POQ  j =t
             ,
                                    


                                                    5
and if the demand forecasts and the demand rate are equal (i.e., dj = D for j > T):

                          T − t + 1 T
          ctFT = ctS,T + 1 −
                                   ∑ h( j − t )(d j − D) for t > T − POQ + 1.
                              POQ  j =t
             ,
                                   

So for small deviations of demand both methods are almost identical, and for constant

demand (dj = D) both methods coincide.


2.3 Extended Wagner Whitin

We propose a straightforward approach for extending the Wagner-Whitin algorithm.

As in [9] we assume that we have forecasted demands for the periods T + 1,…,Tmax.

Now we apply the Wagner-Whitin algorithm to the Tmax-period problem, so that we

have values f(t) for t = 1,…,Tmax. We call this method extended Wagner-Whitin

(EWW), because we extend the T-period model horizon. Russel and Urban [7] use a

similar idea of extending the horizon. If for some t ≥ T the last production period

occurs before or at period T, say in period s(t), then a value equal to the fraction of the

costs incurred in period s(t),…,t that falls beyond period T are subtracted from f(t). So

for periods t = T,…,Tmax, we have modified costs

                                      t −T                                        T − s (t ) + 1
             f ' (t ) = f (t ) −                  c s ( t ),t = f ( s (t ) − 1) +                c s ( t ),t .
                                   t − s (t ) + 1                                 t − s (t ) + 1

Now the first production decision corresponding to the plan with lowest cost f '(t) (t =

T,…,Tmax) is implemented in the rolling horizon scheme.



The advantage of the EWW method compared to the method proposed by [9] is that

we do not have to calculate the TBOs by applying a heuristic, because our method is

based on an exact approach. Moreover, the calculation of the TBOs takes O(Tmax)

time, while the calculation of f(Tmax) can also be performed in O(Tmax) time, because

of the stationarity of the cost coefficients (see [2], [10] and [1] for linear time



                                                            6
algorithms). This means that there is no loss in computation time, when applying our

method instead of Stadtler’s method. In fact, computation times are not an issue at all.

Namely, a 300-period lot-sizing problem can be solved by the classical Wagner-

Whitin algorithm in about 0.01 seconds on a Pentium IV 1.8 GHz with 256 MB

RAM.



An advantage of the EWW method with respect to the ending inventory valuation

method of [4] is that it can also be applied in the case when the cost coefficients are

non-stationary. This advantage also holds for Stadtler’s method in comparison to the

method of [4]. Furthermore, the EIV method computes the value of D in (1) in a

different way for different types of demand (see also section 3), whereas our method

and Stadtler’s method can be applied in the same way for each type of demand.


2.4 Wagner-Whitin and Silver-Meal

Two other methods often used to deal with end-effects are Wagner-Whitin (WW) and

Silver-Meal [8]. The Wagner-Whitin method solves the T-period problem and the first

production decision is implemented in the rolling horizon scheme. In the Silver-Meal

(SM) heuristic a production decision for periods 1 to t is made if the average cost of

producing for periods 1 to t + 1 exceeds the average production cost of producing for

periods 1 to t, or if t equals T.


3. Computational results

Computational tests of [4] show that the EIV method outperforms the Wagner-Whitin

algorithm and the Silver-Meal heuristic, under several demand patterns, within a

rolling horizon framework. To evaluate the significance of this result, we compare the

five methods described in section 2. Our comparison is based on the tests of [4].



                                           7
To eliminate the end-effects, the true horizon n is set to 300 periods and for every

period within this horizon non-negative demand is generated. Now for each of the

above methods a production plan is constructed using a rolling horizon scheme. The

short model horizon T ranges from 2 to 20 periods. For the 300 period lot-sizing

problem the optimal solution is determined (by applying the Wagner-Whitin

algorithm with horizon 300). For each setting of input parameters, we generate eight

problem instances and for each lot-sizing method the average percentage above

optimal cost is computed.



The EIV, ST (Stadtler’s method) and EWW production plans are constructed in such

a way that when arriving at or past period n – T during the rolling horizon scheme, the

remaining lot-sizing problem (consisting of less than T periods) is solved in an

optimal way so that the ending inventory always equals zero. In this way a fair

comparison is made, because the WW algorithm and SM heuristic also leave ending

inventory zero in period n.



In the following subsections the results of the replicated computational experiments

are discussed for different demand patterns.


3.1 Stationary demand

In this case demand is generated from a truncated normal distribution with mean µ =

100 and standard deviation σ = 0, 10, 22 and 43. The setup cost K is set to 800 and

holding cost h is normalized to 1. For a constant demand of 100, POQ

equals 2 K/Dh = 4 periods. For the EIV method it is assumed that the long run




                                          8
demand rate is known and so D is set to 100 and EOQ is set to 400 in (1). For ST and

EWW we make the same assumption and demand forecasts beyond the short model

horizon are set to 100 units of demand. Furthermore, we set Tmax = T + POQ – 1, so

that we do not use more than one cycle of forecasted demand. Table 1 shows the

results.




                                         9
Table 1: Percentage deviation from optimality for normally distributed demand (8

problem instances per scenario)

 T                  σ=0                               σ = 10                             σ = 22                              σ = 43

       WW      SM    EIV   EWW   ST     WW     SM       EIV EWW       ST     WW    SM      EIV EWW      ST     WW     SM        EIV EWW     ST

 2    28.57   28.57 0.00   0.00 0.00   29.33 29.33 15.47 15.47 15.47        31.41 31.41 18.84 18.84 18.84     36.84 36.84 21.49 21.49 21.49

 3     4.76    4.76 0.00   0.00 0.00    5.42   5.42 15.02 15.02 15.02        7.01 7.01 17.99 17.99 17.99      11.48 11.73 21.38 21.21 21.40

 4     0.00    0.00 0.00   0.00 0.00    0.54   0.80     0.78    0.78 0.78    2.12 1.87     1.83   1.83 1.83    6.31   4.47     4.06   4.07 4.08

 5     2.86    0.00 0.00   0.00 0.00    3.44   0.74     0.50    0.50 0.50    5.21 1.07     0.89   0.88 0.88    7.50   1.86     1.68   1.89 1.89

 6     4.76    0.00 0.00   0.00 0.00    5.37   0.74     0.41    0.41 0.41    5.34 1.02     0.55   0.59 0.59    4.94   1.58     0.94   1.01 1.01

 7     4.76    0.00 0.00   0.00 0.00    1.58   0.74     0.42    0.42 0.42    1.79 1.02     0.54   0.49 0.49    2.31   1.47     0.55   0.65 0.65

 8     0.00    0.00 0.00   0.00 0.00    0.56   0.74     0.30    0.34 0.34    1.02 1.02     0.44   0.38 0.38    1.18   1.46     0.35   0.37 0.37

 9     0.00    0.00 0.00   0.00 0.00    1.22   0.74     0.27    0.25 0.25    1.31 1.02     0.34   0.32 0.32    1.02   1.46     0.29   0.26 0.26

10     2.86    0.00 0.00   0.00 0.00    1.75   0.74     0.24    0.25 0.25    1.28 1.02     0.23   0.29 0.29    0.62   1.46     0.23   0.17 0.17

11     4.67    0.00 0.00   0.00 0.00    0.86   0.74     0.24    0.25 0.25    0.70 1.02     0.21   0.21 0.21    0.63   1.46     0.18   0.17 0.17

12     0.00    0.00 0.00   0.00 0.00    0.47   0.74     0.22    0.23 0.23    0.34 1.02     0.18   0.16 0.16    0.41   1.46     0.08   0.12 0.12

13     0.00    0.00 0.00   0.00 0.00    0.60   0.74     0.18    0.16 0.16    0.47 1.02     0.14   0.13 0.13    0.26   1.46     0.10   0.10 0.10

14     0.00    0.00 0.00   0.00 0.00    0.82   0.74     0.14    0.13 0.13    0.41 1.02     0.11   0.11 0.11    0.19   1.46     0.09   0.06 0.06

15     4.57    0.00 0.00   0.00 0.00    0.51   0.74     0.12    0.12 0.12    0.29 1.02     0.10   0.08 0.08    0.11   1.46     0.02   0.03 0.03

16     0.00    0.00 0.00   0.00 0.00    0.23   0.74     0.11    0.13 0.13    0.24 1.02     0.07   0.09 0.09    0.09   1.46     0.02   0.01 0.01

17     0.00    0.00 0.00   0.00 0.00    0.38   0.74     0.11    0.11 0.11    0.14 1.02     0.10   0.10 0.10    0.07   1.46     0.03   0.03 0.03

18     0.00    0.00 0.00   0.00 0.00    0.48   0.74     0.08    0.08 0.08    0.16 1.02     0.05   0.07 0.07    0.05   1.46     0.02   0.02 0.02

19     4.57    0.00 0.00   0.00 0.00    0.28   0.74     0.09    0.08 0.08    0.11 1.02     0.04   0.03 0.03    0.05   1.46     0.01   0.01 0.01

20     0.00    0.00 0.00   0.00 0.00    0.18   0.74     0.06    0.07 0.07    0.08 1.02     0.02   0.03 0.03    0.02   1.46     0.01   0.01 0.01

AR     2.79    1.32 1.00   1.00 1.00    3.97   4.18     1.70    1.52 1.52    3.90 4.31     1.82   1.51 1.51    3.76   4.39     1.70   1.48 1.50

Note: µ = 100, K = 800, h = 1




                                                               10
It follows from table 1 that EIV, EWW and ST produce optimal quantities in the case

of a constant demand rate. The same holds for the SM heuristic if the model horizon

is sufficiently large. It can be shown that Groff’s heuristic leads to optimal TBOs for

constant demand, which implies that τt = POQ. In the previous section we showed for

this case that EIV and ST are identical. Furthermore, Fisher et al. [4] show that EIV

leads to optimal decisions for a constant demand rate. This proves that ST also leads

to optimal decisions for constant demand.



The WW method, however, shows a striking pattern. One may expect a decrease in

deviation from optimality as T increases, because more information becomes

available. However, we see a pattern of returning non-optimal model horizons. This

can be explained as follows. In the optimal production plan production takes place

every 4 periods incurring cost 1400/4 = 350 per period, while for model horizon T = 5

it is optimal to produce for 5 periods incurring total cost of 1800/5 = 360 per period,

which is 2.87% above minimal cost. The same computations can be made for the

other model horizons. Note that in some cases multiple optimal solutions exist (e.g.,

for T = 7: produce for 3 and then for 4 periods or produce for 4 and then for 3

periods). How the Wagner-Whitin algorithm is implemented determines which

solution is selected. In our case the algorithm is implemented in such a way that the

first production decision is as short as possible.



Furthermore, table 1 shows that EIV, EWW and ST produce almost identical results.

In 47% of the problem instances (excluding the case σ = 0) the three methods produce

the same solution. The last row of the table contains the average rank (AR) with

respect to the solution of each method. First, we calculated the rank per problem



                                            11
instance for each method and then we averaged the rank over all model horizons. If

methods have the same solution for a problem instance, then they receive the same

rank. For example, if three methods perform best, they get all rank 1 and the next best

method gets rank 4.



It also follows from table 1 that EIV, EWW and ST outperform WW and SM. This

supports the conjecture that the superior performance of these methods is to a large

extent due to the fact that they make use of quite accurate knowledge about future

demand, whereas the two traditional methods do not use any information about

demand beyond the short model horizon. In our opinion, this fact does not get enough

attention in the discussion of the computational results by [4]. In fact, a comparison of

EIV to a straightforward method such as EWW or those discussed in [7] would have

been more informative.



For longer model horizons, we see in table 1 that as the standard deviation of demand

increases, WW, EIV and EWW perform better. Although Fisher et al. [4] also observe

this counterintuitive phenomenon, they do not give an explanation. We note, however,

that Federgruen and Tzur [3, p. 463-464] already observed that “the larger the inter-

period variability of the parameter values, the more likely it is that a specific choice f

for the first order period after period 1 has characteristics which are significantly

different from those pertaining to adjacent periods.” This means that the optimal first

order period after period 1 is easily identified.


3.2 Linearly increasing and decreasing demand

Now demand is generated by adding a normally distributed random variable to a

linearly increasing trend, i.e., demand in period t equals


                                            12
                       d t = µ + σε t + τ (t − 1) , t = 1,…, 300,                 (2)

where ε t is a normally distributed random variable and τ is the trend factor. A

linearly decreasing trend can be obtained by setting dt = d300-t+1 for t = 1,…,300. We

generate problem instances with µ = 100, σ = 10, K = 800, h = 1 and τ = 1, 10, 20

and 40. To determine the ending inventory in the EIV method in period t, Fisher et al.

[4] replace D by the expected demand in period t + T, which equals µ + τ(t + T – 1)

and EOQ is replaced by the corresponding economic order quantity. For EWW and

ST again we assume perfect demand knowledge, i.e., we assume that the trend is

known and the values of demand beyond the short model horizon are obtained by

substituting ε t = 0 in (2).




                                              13
Table 2: Percentage deviation from optimality for linearly increasing demand (8

problem instances per scenario)

   T                 τ=1                                τ = 10

         WW     SM      EIV EWW          ST    WW    SM EIV EWW        ST

   2     4.99 4.99 18.24 10.39 10.34           0.31 0.31 1.23    0.83 0.83

   3     3.30 0.43     2.86   1.45      1.41   0.38 0.03 0.15    0.06 0.15

   4     1.63 0.16     0.13   0.73      0.31   0.12 0.02 0.04    0.03 0.03

   5     2.89 0.18     0.16   0.30      0.28   0.26 0.02 0.01    0.02 0.02

   6     0.42 0.18     0.12   0.13      0.12   0.03 0.02 0.02    0.01 0.01

   7     2.07 0.18     0.10   0.35      0.20   0.22 0.02 0.01    0.02 0.02

   8     0.43 0.18     0.09   0.16      0.16   0.03 0.02 0.01    0.02 0.02

   9     1.15 0.18     0.13   0.14      0.13   0.17 0.02 0.00    0.00 0.00

  10     0.52 0.18     0.04   0.19      0.15   0.04 0.02 0.01    0.01 0.01

  11     0.89 0.18     0.08   0.09      0.09   0.14 0.02 0.00    0.01 0.00

  12     0.13 0.18     0.07   0.08      0.08   0.03 0.02 0.00    0.00 0.00

  13     0.88 0.18     0.07   0.16      0.11   0.15 0.02 0.00    0.01 0.01

  14     0.15 0.18     0.08   0.11      0.10   0.02 0.02 0.00    0.00 0.00

  15     0.58 0.18     0.09   0.10      0.10   0.13 0.02 0.01    0.01 0.01

  16     0.22 0.18     0.07   0.12      0.10   0.03 0.02 0.01    0.01 0.01

  17     0.54 0.18     0.08   0.09      0.08   0.13 0.02 0.01    0.01 0.01

  18     0.08 0.18     0.06   0.07      0.07   0.02 0.02 0.01    0.01 0.01

  19     0.54 0.18     0.05   0.09      0.08   0.08 0.02 0.01    0.01 0.01

  20     0.07 0.18     0.05   0.05      0.04   0.02 0.02 0.01    0.01 0.01

  AR     4.41 2.82     2.09   2.62      2.27   4.10 2.28 1.84    1.56 1.55

Note: µ = 100, σ = 10, K = 800, h = 1




                                                        14
The results for linearly increasing demand can be found in table 2. We only present

the results for trend factors 1 and 10. For trend factors 20 and 40 the tables mainly

consist of zeros, because when t in (2) is sufficiently large, demand is large relative to

the setup cost, so it is always optimal to produce each period and each method

produces a (near) optimal solution.



It follows from table 2 that EIV, EWW and ST outperform WW and SM.

Furthermore, we see that for τ = 1, EIV shows the best performance. We have the

following explanation for this.



                                                                        2K
In general it does not hold that the expected order cycle equals           , if demand
                                                                        µh

equals on average µ = E(D) (where D is stochastic and E(D) denotes the expected

demand). This follows from the fact that in general it holds that

                                    2K         2K
                                  Ε    
                                    Dh  ≠              .
                                              Ε( D ) h

In other words, the expected length of an optimal order cycle is not equal to the length

of an optimal order cycle when demand is equal to µ in all periods. For example, if we

have normally distributed demand, simulation experiments suggest that the average

order cycle will be somewhat larger, where the exact value depends on the variation

of demand. So, underestimating future demand may have a positive effect on the

results. We see that Fisher et al. [4] do underestimate future demand by assuming

constant demand after period t + T, while demand is in fact increasing. This

underestimation may lead to the better performance. It is interesting to note that

Fisher et al. [4] do not motivate why they underestimate demand. One would expect a



                                           15
choice for D that better reflects the trend. Possibly, this particular choice was made

because it simply gave the best results in the experiments.



The above argument is also supported by the fact that EIV does not have the best

performance in the case of linearly decreasing demand (see table 3), where Fisher et

al. [4] overestimate future demand. For this case both EWW and ST perform

somewhat better than EIV. It also follows from table 3 that EIV, EWW and ST

outperform the two classical methods. Furthermore, we see that EWW and EIV

produce almost similar results for all model horizons.




                                          16
Table 3: Percentage deviation from optimality for linearly decreasing demand (8

problem instances per scenario)

  T                  τ=1                          τ = 10

        WW     SM     EIV EWW       ST   WW    SM EIV EWW        ST

  2     5.21 5.21 18.38      9.71 9.47   0.43 0.43 1.47    0.96 0.94

  3     3.40 0.60    3.13    1.57 1.56   1.19 0.07 0.25    0.28 0.29

  4     1.80 0.24    0.29    0.29 0.29   0.17 0.12 0.10    0.06 0.06

  5     3.07 0.21    0.20    0.11 0.11   0.98 0.11 0.05    0.04 0.04

  6     0.40 0.21    0.12    0.11 0.11   0.03 0.11 0.04    0.02 0.02

  7     1.60 0.21    0.10    0.06 0.06   0.55 0.11 0.02    0.02 0.02

  8     0.57 0.21    0.07    0.08 0.08   0.10 0.11 0.03    0.02 0.02

  9     0.69 0.21    0.05    0.06 0.06   0.48 0.11 0.02    0.01 0.01

 10     0.38 0.21    0.04    0.04 0.04   0.03 0.11 0.02    0.00 0.00

 11     0.64 0.21    0.06    0.04 0.04   0.44 0.11 0.01    0.01 0.01

 12     0.13 0.21    0.03    0.02 0.02   0.03 0.11 0.02    0.02 0.02

 13     0.50 0.21    0.03    0.03 0.03   0.37 0.11 0.01    0.01 0.01

 14     0.15 0.21    0.03    0.03 0.03   0.03 0.11 0.01    0.01 0.01

 15     0.23 0.21    0.04    0.04 0.04   0.33 0.11 0.01    0.01 0.01

 16     0.11 0.21    0.03    0.04 0.04   0.01 0.11 0.01    0.01 0.01

 17     0.24 0.21    0.04    0.03 0.03   0.27 0.11 0.01    0.01 0.01

 18     0.07 0.21    0.02    0.03 0.03   0.01 0.11 0.01    0.00 0.00

 19     0.21 0.21    0.04    0.04 0.04   0.23 0.11 0.00    0.00 0.00

 20     0.06 0.21    0.05    0.05 0.05   0.01 0.11 0.00    0.00 0.00

 AR     4.34 3.77    2.25    1.56 1.57   3.80 4.00 2.19    1.46 1.48

Note: µ = 100, σ = 10, K = 800, h = 1




                                                    17
3.3 Seasonal demand

For this case demand is generated according to the following formula:

             d t = µ + σε t + a sin[2π/c(t + c/ 4 )] ,   t = 1,…, 300,              (3)

where a is the amplitude of the seasonal component and c is the length of the seasonal

cycle. To account for the seasonal demand pattern in period t for determining ending

inventory in the EIV algorithm, let the periodic order quantity POQ equal

  2 K /( µh) . Now Fisher et al. [4] replace D in (1) by the average demand in the first

POQ periods immediately following time t + T – 1 and EOQ is replaced by the

economic order quantity corresponding to this average demand. Again, we assume

perfect demand forecasts for EWW and ST (i.e., we assume ε t = 0 for t = T,…,Tmax).



To compare the different solution methods the same settings are used as in [4], i.e., µ

= 100, σ = 10, K = 800, h = 1, a = 20, 40, 60, 80 and c = 12. Again for each lot-sizing

method the average deviation from optimality is computed using eight iterations for

each parameter setting. The results of the tests are shown in table 4.




                                               18
Table 4: Percentage deviation from optimality for seasonal demand (8 problem

instances per scenario)

 T                  a = 20                            a = 40                              a = 60                               a = 80

      WW     SM        EIV EWW     ST     WW    SM       EIV EWW     ST     WW     SM        EIV EWW      ST     WW     SM        EIV EWW     ST

 2   29.67 29.67 16.65 15.42 15.58       31.14 31.14 16.54 16.17 16.58     35.09 35.09 16.29 20.24 17.35        40.74 40.74 14.23 20.56 19.90

 3    5.70   5.70 17.25 15.62 15.31       6.82 6.82 18.35 11.55 12.02      10.07 10.07 17.05       9.29 11.40   14.71 14.71 17.61       9.73 9.44

 4    0.80   1.01     3.27   3.91 0.97    1.95 1.76     5.21   4.51 3.98    5.02   4.32     4.83   4.11 4.18     9.37   8.03     5.03   4.56 5.03

 5    3.71   0.84     0.72   0.58 0.58    4.15 1.44     0.91   1.70 0.88    6.67   3.99     2.38   4.28 4.26     9.37   7.57     4.81   4.26 5.08

 6    4.41   0.79     0.46   0.48 0.46    2.36 1.33     1.00   0.87 0.86    1.76   2.58     1.38   2.94 1.57     3.07   3.11     2.62   4.20 5.17

 7    1.89   0.79     0.49   0.39 0.38    2.38 1.41     0.96   0.74 0.67    2.69   3.15     1.60   1.26 1.78     1.97   4.47     1.75   2.22 2.21

 8    0.61   0.79     0.46   0.48 0.45    1.19 1.44     0.67   0.62 0.76    2.43   3.36     0.47   0.53 0.74     2.92   3.82     0.68   0.68 2.61

 9    1.25   0.79     0.29   0.27 0.26    1.33 1.44     0.44   0.36 0.37    1.59   3.78     0.35   0.23 0.32     1.87   4.93     0.68   0.47 0.99

10    1.76   0.79     0.29   0.26 0.26    1.54 1.44     0.37   0.31 0.32    0.92   3.78     0.41   0.18 0.20     0.99   4.95     0.45   0.22 0.26

11    0.83   0.79     0.32   0.20 0.20    1.03 1.44     0.24   0.17 0.18    0.90   3.78     0.21   0.19 0.22     0.44   4.95     0.17   0.25 0.28

12    0.40   0.79     0.22   0.20 0.20    0.19 1.44     0.16   0.16 0.19    0.20   3.78     0.36   0.41 0.53     0.26   4.95     0.44   0.44 0.47

13    0.55   0.79     0.14   0.15 0.14    0.55 1.44     0.17   0.14 0.16    0.39   3.78     0.33   0.30 0.39     0.40   4.95     0.43   0.40 0.56

14    0.72   0.79     0.18   0.15 0.15    0.50 1.44     0.16   0.12 0.13    0.50   3.78     0.09   0.14 0.34     0.52   4.95     0.25   0.46 0.72

15    0.37   0.79     0.17   0.11 0.11    0.40 1.44     0.14   0.10 0.11    0.42   3.78     0.12   0.07 0.20     0.51   4.95     0.17   0.24 0.57

16    0.28   0.79     0.15   0.12 0.12    0.20 1.44     0.10   0.08 0.09    0.27   3.78     0.15   0.06 0.09     0.49   4.95     0.17   0.13 0.38

17    0.24   0.79     0.11   0.10 0.09    0.18 1.44     0.07   0.06 0.06    0.19   3.78     0.06   0.06 0.07     0.23   4.95     0.12   0.08 0.16

18    0.38   0.79     0.10   0.08 0.08    0.22 1.44     0.10   0.06 0.07    0.10   3.78     0.07   0.03 0.05     0.12   4.95     0.10   0.06 0.09

19    0.28   0.79     0.07   0.08 0.08    0.14 1.44     0.11   0.09 0.10    0.15   3.78     0.06   0.03 0.03     0.10   4.95     0.08   0.05 0.07

20    0.14   0.79     0.07   0.10 0.09    0.12 1.44     0.09   0.07 0.08    0.10   3.78     0.02   0.01 0.04     0.12   4.95     0.03   0.02 0.03

AR    3.88   4.21     2.52   1.89 1.69    3.81 4.28     2.55   1.84 2.01    3.61   4.46     2.16   1.84 2.47     3.38   4.59     2.04   1.87 2.73

Note: µ = 100, σ = 10, c = 12, K = 800, h = 1




                                                               19
It follows from table 4 that EIV, EWW and ST outperform WW and SM.

Furthermore, WW performs better than SM, especially for larger values of T. This is

also what we see in table 1, where SM performs poorly for large variation of demand.

Table 4 also shows that EWW outperforms EIV on average for all values of the

seasonal component. We explain this better performance by the fact that it

incorporates the seasonal component in a better way. EIV averages the first POQ

periods of expected demand to account for its cyclical behavior, but EWW

incorporates the actual expected cyclical behavior. The better performance of EWW

with respect to ST (for a ≥ 40) may be explained by the fact that Groff’s heuristic is

used to calculate the TBOs, whereas our method is based on an exact approach.


4. Concluding remarks

To apply the approaches discussed in this paper, the required information must be

available. In general, when reasonable (or even better) estimates of future data are

available, it is quite natural to make use of these. One could argue, however, that the

simulation experiments do not describe a very realistic situation, if only because the

parameters of the demand patterns are not known in practice but have to be estimated.

Nevertheless, the computational results do show that there is a significant cost

reduction possible, if additional demand information is known (and used in a proper

way). So, from the computational tests one could derive the value of additional

demand information.



As already mentioned in section 3.2, the expected length of an order cycle is not equal

to the length of an order in the case with demand equal to the expected demand for all

                   (           )
periods (because Ε 2 K /( Dh) ≠ 2 K /( E ( D)h) in general). This also means that




                                          20
average   cost   per   order   cycle   is    not   equal   to    2 KE ( D ) h   (because

 (        )
Ε 2 KDh ≠ 2 KΕ( D)h in general). It may be interesting to devise a method that

incorporates the ‘true’ expected order cycles and/or costs. Maybe this will lead to

better methods that can be applied in a rolling horizon scheme to mitigate end-effects.


References

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                                            21
[8]    E.A. Silver, H.C. Meal, A heuristic selecting lot size requirements for the case

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[9]    H. Stadtler, Improved rolling schedules for the dynamic single-level lot-sizing

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                                         22

				
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Description: A comparison of methods for lot sizing in a rolling horizon lotting paper