Newsvendor Bounds and Heuristics for Series Systems

Transcription

1 Newsvendor Bounds and Heuristics for Series Systems August 6, 2002 Abstract We propose a heuristic for a multi-stage serial system based on solving a single newsvendor problem per stage. The heuristic is based on aggregating lead time demands over stages and using a modified holding cost rate that accounts for the time an item spends on each of the aggregated stages. Our numerical study indicates that the heuristic is very e ective in identifying near-optimal basestock levels. In addition, we provide distribution-free approximate bounds that accurately predict the sensitivity of the optimal average cost to the system parameters. 1

2 1 Introduction The study of multi-stage in-series inventory systems is central to the study of supply chain management both as a benchmark and as a building block for more complex supply networks. Unfortunately, policy evaluation and optimization are di cult to understand and communicate. Shang and Song (2001) have correctly identified the need to develop simple spreadsheet-based heuristics that are more accessible to practitioners and Operations Management students at business and engineering schools. The hope is that these heuristics would increase the understanding of multi-echelon base-stock systems among people who manage these systems in practice. Shang and Song developed a heuristic based on solving 2J newsvendor problems with lower and upper bounds on the holding cost. In this note, we develop amorerefined heuristic, based on solving only J newsvendor problems. Our heuristic is based on aggregating stages and using a modified holding cost rate that takes into account the proportion of time an item spends on each of the aggregated stages. Our numerical results indicate that our heuristic is near optimal, with an average penalty cost that is lower than the Shang and Song heuristic. In addition, we provide approximate distribution-free bounds that accurately reflect the sensitivity of the optimal average cost to changes in system parameters. 2 Series Systems Consider a series system consists of J stages as illustrated in the figure. Stage < Jprocures from sage + 1 and stage J replenishes from an outside supplier with ample stock. Customer demands occurs only at stage 1 and follows a (compound) Poisson process, {D(t),t 0}. IttakesL, =1,...,J units of time for a shipment to arrive to Stage from its predecessor. Ample Supply L J-1 L 1 J J-1 1 Demand 2

3 Unsatisfied demand is backordered at each stage but only Stage 1 incurs a linear backorder penalty cost b, per unit per unit time. We assume without loss of generality that each stage adds value as the item moves through the supply chain, so echelon holding costs h are positive. The system is operated under continuous review, so management orders every time a demand occurs. As pointed out by Zipkin (2000), this is ustified for expensive and/or slow moving items. For convenience, we will write h[1,]= k=1 h k, The following random variables describe the state of Stage in equilibrium: D = the leadtime demand, I = local on-hand inventory, IT = inventory in transit to stage, IO = inventory on order to stage, I = I + i + Ii), i<(it echelon inventory, IN = I B, echelon net inventory, IOP = IN + IT echelon inventory-order position, where B denotes the number of backorders at stage 1. Notice that IO IT since stage can be replenished only when stage + 1 has inventory. It is known that an echelon base stock policy s =(s J,...,s 1 ) is optimal for this series system (Zipkin (2000), Federgruen and Zipkin (1984) and the original work by Clark and Scarf (1960)). Under this policy, the manager continuously monitors the echelon inventory-order position at each stage and places an order from stage +1tobringitupto s whenever it is below this level. The optimal echelon base stock levels can be found through solving the following recursive optimization for = 1, 2,..., J: C (y) = E{h (y D )+C 1 (min[y D,s 1])} (1) s = min{y : C (y) C (x) for all x = y}, (2) where C 0 (y) =(b + h 1 )[y], see Gallego and Zipkin (1999). The optimal system wide average cost is given by C J (s J ). It is possible to evaluate any base stock policy using the above recursion by skipping the optimization step. 3 Newsvendor Bounds and Heuristics The recursive optimization (1) and (2) is di cult to describe to a practitioner. Shang and Song (2001) proposed a simple, and yet e ective, heuristic based on solving 2J newsvendor problems. 3

4 3.1 The Shang and Song Heuristic Let b = b + h[ +1,J], where sums over empty sets are defined to be zero. Let D[1,]= i=1 D i.the heuristic is based on solving the following two newsvendor problems for Stage. G u (y) = E{h[1,](y D[1,]) + + b (y D[1,]) }, s l = min{y : G u (y) G u (x) for all x = y}, G l (y) = E{h (y D[1,]) + + b (y D[1,]) }, s u = min{y : G l (y) G l (x) for all x = y}. Theorem 1 Shang and Song (2001): For any given and y: 1. G l (y) C (y) G u (y), 2. s l s su, where = 1 i=1 h[i +1,]ED i 1 is the average in-transit holding cost. 3.2 The Authors Heuristic The Shang and Song heuristic is based on using a lower and an upper bound on the holding cost for the subsystem {1,...,}, solving two newsvendor problems and either truncating or rounding the average of the two newsvendor problem solutions. In contrast, our approach consists of solving a single newsvendor problem based on approximate holding cost rate h GO (h,h[1,]). The idea is based on the approximate time an item spends at each stage of the subsystem. To obtain this approximation, let i = L i /L[1,], where L[1,]=L L,andset h GO = k h[k, ]. k=1 Using h GO as the approximate holding cost rate for a system that aggregates the demand over stages {1,...,} results in the newsvendor problem: G GO (y) = E{h GO (y D[1,]) + + b (y D[1,]) }, s GO = min{y : G GO (y) G GO (x) for all x = y}. 1 Shang and Song (2001) define = 1 hi+1edi. We think this is a typographical error. i=1 4

5 Theorem 2 For any given and y we have: 1. G l 2. s l (y) GGO (y) G u (y), sgo s u, 3. G GO (y) b h GO L[1,]E[X 2 ] where X is the random demand size of the compound Poisson process. Proof Notice that we have h h GO i=1 h i. Part 1 follows immediately from this inequality. Since the newsvendor cost functions are convex we also have G l (y) GGO (y) G u (y) where f(x) =f(x+1) f(x). This implies the second part of the Theorem. The last part is the distribution free bound as in Gallego and Moon (1993). The two Theorems imply that if the bounds suggested by Shang and Song (2001) are tight then s GO would be very close to the optimal base stock level, s. In the following section we illustrate how accurate this approximation is. If our approximation is close-to-optimal, the cost of managing the series system can also be bounded by a distribution free bound, that is C J (s J) bh GO J L[1,J]E[X 2 ]+ J. (3) This simple form enables sensitivity analysis. In particular, (1) the system cost is proportional to b, (2) downstream leadtimes have a larger impact on system performance than upstream leadtimes, (3) upstream echelon holding cost rates have a larger impact on the system performance than downstream echelon holding cost rates, (4) the system cost is proportional to and proportional to E[X 2 ]. As pointed out by Shang and Song, this type of parametric analysis enables a near characterization of system performance. Some system design issues may require investments in new processing plans or quicker but more expensive shipment methods. Marketing strategies could influence the demand as well as altering the cost of backlogging a customer. The closed form expression (3) facilitates gauging the benefit of any action on the inventory management costs, at least as a first cut. Our analysis suggests, for example, that management should focus on reducing the lead time at the upstream stages while reducing the holding cost at the down stream stages. If process resequencing is an option, the lowest value added processes with the longest processing times should be carried out sooner than later. 5

6 4 Numerical Study Here we report the performance of our heuristic and of the distribution free bound. We compare the exact solution based on equations (1) and (2) and report the percentage error i %= C J (s i J ) C J (s J ) C J (s J ) for i = {SS,GO}. By considering a larger set of experiments, we complete the numerical study in Shang and Song (2002). In particular, our numerical study includes unequal leadtimes. s GO s SS To manage the series system, we use an echelon base stock policy with echelon base stock levels for all. The approximate cost is given by G GO J (sgo )+ J. Similarly Shang and Song (2002) use = sl +su 2 and truncate this average when b 39 and round it otherwise. They approximate the optimal cost C J (s )byg u J (sl J )+ J instead of the average since the lower bounds become looser as the number of stages in the system increases. We study two sets of experiments: constant leadtime set and the randomized parameters set. The first set of experiments is similar to that of Gallego and Zipkin (1999) and Shang and Song (2002). The holding cost and the lead times are normalized so h[1,j]=1andl[1,j] = 1. We consider J {2, 4, 8, 16, 32, 64}; {16, 64}; andb {9, 39} (corresponding to fill rates of 90%, 97.5%). Within this group we consider linear holding-cost form (h =1/J); a ne holding cost form (h[1,]= +(1 )/J with =0.25 and 0.75); kink holding cost form (h =(1 )/J for J/2+1andh =(1+ )/J for <J/2+1with =0.25 and 0.75) and ump holding cost form (h = +(1 )/J for = N/2 and h =(1 )/J for = J/2 with =0.25 and 0.75). Notice that Shang and Song (2002) considers only the case for =64andb = 39. We report the results in Tables 1, 2 and 3. Out of 108 problem instances, in 24 cases the s GO and in 20 cases the s SS heuristic resulted in the exact solution, that is, the heuristics reached the same answer as the recursive optimization. The s GO (resp., s SS ) heuristic out performs in 48 (resp., 44) cases and they tie in 17 cases. The average error for s GO (resp., s SS )heuristicis0.195% (resp., 0.385%), while the maximum error is 3.68% and 1.24% for the GO and the SS heuristics respectively. The quality of the heuristics seems to deteriorate as the number of stages in the system exceeds 32. The SS heuristic seems to perform better for the ump holding cost case, while the GO heuristic tends to dominate in the other cases. The second set of experiments allow for unequal leadtimes. It is here that we expect the GO heuristic to perform better. To cover a wider range of problem instances we generate the leadtimes and holding costs from uniform distributions. In particular, we use the following set of parameters: h {Unif(0, 1), Unif(0, 5), Unif(1, 10)}, L {Unif(1, 2), Unif(1, 10), Unif(1, 40)}, 6

7 J {2, 4, 8, 16, 32} b {1, 9, 39, 49} {1, 3, 6}. We consider 25 combinations, taken at random, from the above parameters. For each subgroup we generate 40 problem instances and report the worst case ( i max max{ i 1,..., i 40 })aswellasthe 40 average performances ( i =1 avg = i 40 ). We report the summary statistics in Table 4. As an example, we present some of the problem instances from Case No. 17 in Table 5. Out of 1000 problem instances, in 188 cases the s GO and in 133 cases the s SS heuristic resulted in the exact solution. In 849 cases the error term for s GO heuristic is smaller or equal to that of s SS heuristic. The average error for the s GO (resp., s SS ) heuristic is 0.23% (resp., 0.83%). We observe that as the variance of the leadtimes across stages increases the average error term for s GO decreases (the average error for L Unif(1, 10) is 0.14% whereas it is 0.39% for L Unif(1, 2)). Similarly the s GO heuristic performs even better as the variance of echelon costs across stages in a series system increases. In Table 6 we present the ten worst cases that we encountered. The maximum worst case we observed so far was 4.36% for the s SS heuristic and 3.62% for the s GO heuristic. Figure 5 delineates the optimal cost function as well as the newsvendor costs for all stages of a four stage series system. Since all of the newsvendor cost functions use the same penalty cost the curves converge on the left side of the bowl shape curve and that they all are equal for the first stage. In most of our examples we observed that the C GO closely follows C rising when C does. This suggests that the second di erences for these cost functions are approximately the same. Hence the minimizers of the curve coincides with that of the C, which is the case in this particular example. In light of our numerical observations we suggest the s GO heuristics for a series system with small number of stages C GO and with high variance in leadtimes and echelon holding costs across stages. In Figure 1 (a) we plot the optimal cost of a four stage series system with h = 0.25, = 16 as a function of penalty costs and di erent leadtime configurations. The total system lead time and the echelon holding cost for the system is 1 and the three penalty cost b =1, 9, 99. The first system s leadtimes are (0.1, 0.1, 0.1, 0.7) whereas the fourth system s leadtimes are (0.7, 0.1, 0.1, 0.1). The optimal costs are for the first system and 5 for the fourth system. A cost reduction of 60.84%. In Figure 1 (b) we plot the optimal cost of a four stage series system with L =0.25, = 16 as a function of penalty cost and di erent echelon holding cost configuration. The first system s echelon holding cost is (0.1,0.1,0.1,0.7) whereas the fourth systems echelon holding costs are (0.7, 0.1, 0.1, 0.1). A crucial observation is that postponing the shortest and the most expensive processes to a later stage would significantly reduce the inventory related costs. Finally we compare the actual cost with the distribution free bounds In Table 7 and 8. We carry out a number of regressions of the bound to the actual cost for di erent parameter values and report the 7

8 R 2.Wefix all parameters except the parameter for which we investigate its e ect on the optimal cost and the bound. From these two tables we observe that the distribution free bound yields results that are close to the optimal cost. Note also that R 2 is close to 1 for all factors. This observation suggests that the bound can safely be used to investigate the impact of process and design changes on the cost of managing a series system. Notice that the bound only requires knowing h GO J, b, L[1,J], and E[X2 ]. Figure 1: (a) Optimal Cost vs. System Lead Time (b) Optimal Cost vs. Echelon Holding Costs C* b=1 b=9 b=99 C* b=1 b=9 b= (.1,.1,.1,.7) (.1,.1,.7,.1) (.1,.7,.1,.1) (.7,.1,.1,.1) (.1,.1,.1,.7) (.1,.1,.7,.1) (.1,.7,.1,.1) (.7,.1,.1,.1) Leadtime Echelon Holding Cost 5 Conclusion We presented a simple heuristic for multi-echelon serial systems based on solving one newsvendor problem for each stage of the system. The heuristic can be applied to assembly systems by applying Rosling (1989) s ideas. For distribution systems, the heuristic can be applied after using the decomposition principles in Gallego, Özer and Zipkin (1999). References [1] Clark, A. and H. Scarf Optimal Policies for a Multi Echelon Inventory Problem. Management Science [2] Federgruen, A. and P. Zipkin Computational Issues in an Infinite Horizon, Multiechelon Inventory Model. Operations Research. 32,

9 Table 1: Comparison of Optimal and Heuristic Policy:( =16,b= 39) N Form C(s SS J ) C(sGO J ) C(s J ) SS % GO % Form C(s SS J ) C(sGO J ) C(s J ) SS % GO % 64 Kink Kink = = A ne A ne = = Jump Linear = Out of 36 cases, heuristic s GO produces better results for 19 cases, heuristic s SS produces better results for 12 cases. [3] Gallego G. and I. Moon The Distribution Free Newsboy Problem: Review and Extensions. Journal of Oper. Res. Society. 44, [4] Gallego G, Ö. Özer, and P. Zipkin Bounds, Heuristics and Approximations for Distribution Systems. Working Paper. [5] Gallego G. and P. Zipkin Stock Positioning and Performance Estimation in Serial Production- Transportation Systems. Manuf. and Service Oper. Manag. 1, [6] Rosling, K Optimal Inventory Policies for Assembly Systems Under Random Demands. Operations Research 37, [7] Shang H. K. and J. Song Newsvendor Bounds and Heuristics for Optimal Policies in Serial Supply Chains. Working Paper. [8] Zipkin P Foundations of Inventory Management. The Irwin McGraw Hill Series. 9

10 Table 2: Comparison of Optimal and Heuristic Policy:( =16,b=9) N Form C(s SS J ) C(sGO J ) C(s J ) SS % GO % Form C(s SS J ) C(sGO J ) C(s J ) SS % GO % 64 Kink Kink = = A ne A ne = = Jump Linear = Out of 36 cases, heuristic s GO produces better results for 18 cases, heuristic s SS produces better results for 9 cases. Table 3: Comparison of Optimal and Heuristic Policy:( =64,b= 39) N Form C(s SS J ) C(sGO J ) C(s J ) SS % GO % Form C(s SS J ) C(sGO J ) C(s J ) SS % GO % 64 Kink Kink alpha= alpha= A ne A ne alpha= alpha= Jump Linear alpha= Out of 36 cases, heuristic s GO produces better results for 11 cases, heuristic s SS produces better results for 22 cases. Note that this table reports the same experiments in Shang and Song (2002). 10

11 Table 4: Summary Statistics for 1000 Experiments No. N L h b SS max GO max SS avg GO avg 1 2 Unif(1,10) Unif(0,1) % 2.02% 0.15% 0.19% 2 4 Unif(1,10) Unif(0,1) % 1.36% 0.55% 0.20% 3 8 Unif(1,10) Unif(0,1) % 0.44% 0.95% 0.20% 4 16 Unif(1,10) Unif(0,1) % 0.21% 1.05% 0.12% 5 32 Unif(1,10) Unif(0,1) % 0.13% 0.95% 0.07% 6 2 Unif(1,2) Unif(0,1) % 2.28% 0.23% 0.09% 7 4 Unif(1,2) Unif(0,1) % 2.14% 0.65% 0.53% 8 8 Unif(1,2) Unif(0,1) % 1.23% 1.36% 0.41% 9 8 Unif(1,40) Unif(0,1) % 0.22% 0.55% 0.10% 10 4 Unif(1,40) Unif(0,1) % 0.88% 0.34% 0.14% 11 4 Unif(1,10) Unif(1,10) % 0.41% 0.31% 0.09% 12 8 Unif(1,10) Unif(0,5) % 0.43% 0.52% 0.10% 13 4 Unif(1,10) Unif(0,1) % 1.79% 0.21% 0.13% 14 8 Unif(1,10) Unif(0,5) % 0.65% 0.96% 0.20% 15 8 Unif(1,10) Unif(0,1) % 0.34% 0.41% 0.10% 16 8 Unif(1,10) Unif(0,5) % 0.41% 0.86% 0.08% 17 4 Unif(1,2) Unif(1,10) % 1.33% 1.59% 1.30% 18 8 Unif(1,2) Unif(1,10) % 0.77% 2.42% 0.25% Unif(1,2) Unif(1,10) % 0.51% 3.13% 0.26% 20 8 Unif(1,2) Unif(0,1) % 1.22% 0.64% 0.37% 21 4 Unif(1,10) Unif(0,1) % 1.94% 0.18% 0.25% 22 4 Unif(1,10) Unif(0,1) % 0.59% 0.37% 0.14% 23 4 Unif(1,10) Unif(0,1) % 0.42% 0.16% 0.08% 24 8 Unif(1,2) Unif(1,10) % 0.36% 1.33% 0.14% 25 8 Unif(1,2) Unif(0,1) % 0.32% 0.78% 0.15% 11

12 Table 5: Some of the problem instances from Case No. 17 in Table 4 (L 4,L 3,L 2,L 1) (h 4,h 3,h 2,h 1) s SS C 4(s SS 4 ) sgo C 4(s GO 4 ) s C 4(s 4 ) SS % GO % (1.698,1.067,1.274,1.676) (9.928,2.889,4.290,1.521) (8,7,6,5) (7,7,6,5) (7,7,5,5) % 0.04% (1.241,1.200,1.442,1.939) (5.818,7.261,2.996,6.118) (8,7,6,4) (7,6,6,4) (7,6,6,4) % 0% (1.412,1.749,1.071,1.798) (9.804,5.981,4.367,1.712) (8,8,6,5) (8,7,5,5) (7,7,5,5) % 1.33% (1.545,1.662,1.503,1.700) (5.982,4.047,4.553,2.421) (9,8,6,4) (9,7,6,4) (8,7,6,4) % 1.10% (1.077,1.186,1.291,1.082) (6.992,7.452,6.848,6.558) (6,6,4,3) (6,5,4,3) (6,5,4,3) % 0% (1.840,1.019,1.772,1.663) (9.896,2.546,2.596,8.907) (8,7,6,3) (8,7,6,3) (8,7,6,3) % 0% (1.969,1.575,1.250,1.872) (2.681,8.524,2.728,5.110) (10,7,6,4) (9,7,6,4) (9,6,6,4) % 0.83% (1.434,1.382,1.818,1.403) (3.391,5.041,2.926,6.195) (9,7,6,3) (8,7,6,3) (8,7,6,3) % 0% (1.204,1.600,1.800,1.885) (5.879,4.950,8.401,9.215) (9,8,6,4) (8,7,5,4) (8,7,5,4) % 0% (1.032,1.664,1.813,1.687) (3.185,2.163,1.310,5.019) (9,9,7,4) (9,8,7,4) (9,8,7,4) % 0% Table 6: Some of the observed Worst Cases for Both Heuristics When N =4 (L 4,L 3,L 2,L 1) (h 4,h 3,h 2,h 1) s SS C 4(s SS 4 ) sgo C 4(s GO 4 ) s C 4(s 4 ) SS % GO % (1.222,1.765,1.938,1.732) (7.719,9.942,4.913,9.118) (9,8,6,3) (8,7,6,3) (8,7,6,3) % 0% (1.862,1.057,1.462,1.842) (7.542,1.643,3.786,5.147) (9,8,6,4) (8,7,6,4) (8,7,6,4) % 0% (1.561,1.254,1.644,1.979) (3.155,8.428,9.658,4.546) (9,7,6,4) (8,6,5,4) (8,6,5,4) % 0% (1.599,1.422,1.013,1.103) (6.410,3.175,2.005,1.374) (8,7,5,4) (7,6,5,4) (7,6,5,4) % 0% (1.159,1.583,1.929,1.404) (8.716,1.984,3.435,3.370) (9,8,6,4) (8,8,6,4) (8,8,6,4) % 0% (1.412,1.749,1.071,1.798) (9.804,5.981,4.367,1.712) (8,8,6,5) (8,7,5,5) (7,7,5,5) % 1.33% b =49, =1 (1.949,1.921,1.550,1.346) (0.472,0.375,0.847,0.317) (7,6,3,3) (6,5,3,3) (6,5,3,3) % 0% (1.009,1.919,1.276,1.273) (0.588,0.691,0.838,0.726) (5,4,3,2) (5,4,3,2) (4,4,3,2) % 2.133% (1.693,1.303,1.427,1.07) (0.967,0.683,0.153,0.877) (4,4,4,2) (5,4,4,2) (4,4,4,2) % 2.137% (1.545,1.448,1.409,1.299) (0.466,0.501,0.153,0.323) (6,5,4,3) (6,5,5,3) (5,5,5,3) % 1.845% b =1, =1 Table 7: Performance of the Distribution Free Bound for factors N,b, and. b =10 =16 R 2 = N =4 =16 R 2 = N =4 b =10 R 2 = h i =0.25 L i = % h i =0.25 L i = % h i =0.25 L i = % N DFB C* b DFB C* DFB C*

13 Table 8: Performance of Distribution Free Bound wrt factors h 4,h 1,L 4,andL 1 When N =4 b =10 =16 R 2 = b =10 =16 R 2 = b =10 =16 R 2 = b =10 =16 R 2 = h i =0.25 L i = % h i =0.25 L i = % h i =0.25 L i = % h i =0.25 L i = % h 4 DFB C* h 1 DFB C* L 4 DFB C* L 1 DFB C* Figure 2: Cost at Each Stage As a Function of Inventory Levels:N =4, =16,b=9,h i = L i =.25 C u C, C * GO C l * C