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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 [1] A. Aggarwal, J.K. Park, Improved algorithms for economic lot-size problems, Operations Research 14 (1993) 549–571. [2] A. Federgruen, M. Tzur, A simple forward algorithm to solve general dynamic lot sizing models with n periods in O(n log n) or O(n) time, Management Science 37 (1991) 909–925. [3] A. Federgruen, M. Tzur, Minimal forecast horizons and a new planning procedure for the general dynamic lot sizing model: Nervousness revisited, Operations Research 42 (1994) 456–468. [4] M. Fisher, K. Ramdas, Y. Zheng, Ending inventory valuation in multiperiod production scheduling, Management Science 47 (2001) 679–692. [5] R.C. Grinold, Model building techniques for the correction of end effects in multistage convex programs, Operations Research 31 (1983) 407–431. [6] G.K. Groff, A lot sizing rule for time-phased component demand, Production and Inventory Management 20 (1979) 47–53. [7] R.A. Russel, T. L. Urban, Horizon extensions for rolling production schedules: Length and accuracy requirements, International Journal of Production Economics 29 (1993) 111–122. 21 [8] E.A. Silver, H.C. Meal, A heuristic selecting lot size requirements for the case of a deterministic time-varying demand rate and discrete opportunities for replenishment, Production and Inventory Management 14 (1973) 64 –77. [9] H. Stadtler, Improved rolling schedules for the dynamic single-level lot-sizing problem, Management Science 46 (2000) 318–326. [10] A. Wagelmans, S. Van Hoesel, A. Kolen, Economic lot sizing: An O(n log n)- algorithm that runs in linear time in the Wagner-Whitin case, Operations Research 40 (1992) S145–S156. [11] H.M. Wagner, T.M. Whitin, Dynamic version of the economic lot size model, Management Science 5 (1958) 89–96. 22

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

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