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A simple approach for assessing the cost of system nervousness
Int. J. Production Economics 141 (2013) 619–625
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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
A simple approach for assessing the cost of system nervousness Huseyin Tunc a, Onur A. Kilic b, S. Armagan Tarim b, Burak Eksioglu a,n a b
Department of Industrial and Systems Engineering, Mississippi State University, P.O. Box 9542, Mississippi State, MS 39762, USA Department of Management, Hacettepe University, 06800 Beytepe, Ankara, Turkey
a r t i c l e i n f o
abstract
Article history: Received 23 January 2012 Accepted 14 September 2012 Available online 9 October 2012
A well-known problem in coordinating supply chain inventories is that replenishment decisions are revised due to stochastic demands. This issue is often referred to as system nervousness. The literature distinguishes between two types of nervousness: setup-oriented and quantity-oriented. It is widely accepted that cost of nervousness is difficult to measure. We argue that this cost can be evaluated by means of three well-established inventory control strategies: static uncertainty, dynamic uncertainty, and static-dynamic uncertainty. These strategies reflect extreme cases with regard to the setup- and the quantity-oriented nervousness, and provide a simple yet an objective measure to assess the cost of system nervousness. Our results are of practical importance. We highlight that the setup-oriented nervousness, which is considered to be the most critical in practice, can be eliminated with a minor cost penalty. This is, however, not the case for the quantity-oriented nervousness. & 2012 Elsevier B.V. All rights reserved.
Keywords: Setup-oriented nervousness Quantity-oriented nervousness Static uncertainty Dynamic uncertainty Static–dynamic uncertainty Non-stationary stochastic demand
1. Introduction Inventory control is far more challenging when demand is stochastic. The difficulty mainly springs from the fact that, under uncertainty, effective inventory control policies are needed to steer the inventory systems and efficient algorithms to compute policy parameters. A further complication with stochastic demands, particularly in supply chain networks and MRP environments, is that downstream players continually change the timing as well as the size of their replenishments. This issue is often referred to as the system nervousness, and regarded as an important performance measure for inventory control policies (De Kok and Inderfurth, 1997). The system nervousness arise due to revisions in the original plan which in turn result in different replenishment decisions in successive planning cycles. For example, a previously planned order may be revised or canceled following demand realizations in a particular period. The system nervousness leads to a lack of coordination in supply chains and MRP environments because nervousness on the top level propagates throughout the system (see e.g. De Kok and Inderfurth, 1997; Chatfield et al., 2004; Kaipia et al., 2006). The system nervousness is critical especially in systems characterized by low degrees of flexibility. In such environments, the cost of implementing revisions in
n
Corresponding author. Tel.: þ1 662 325 7625; fax: þ 1 662 325 7618. E-mail addresses: [email protected] (H. Tunc), [email protected] (O.A. Kilic), [email protected] (S.A. Tarim), [email protected] (B. Eksioglu). 0925-5273/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijpe.2012.09.022
replenishment schedules/quantities may overcome the advantage of employing a cost efficient control policy (Heisig, 2001). The literature distinguish between two types of system nervousness: setup-oriented and quantity-oriented. The former refers to revisions in replenishment schedules, whereas the latter refers to revisions in replenishment quantities. As mentioned by many authors (see e.g. Inderfurth, 1994; Heisig, 2001; Tarim and Kingsman, 2004) the setup-oriented nervousness is considered to be more critical as compared to the quantity-oriented nervousness. The nervousness syndrome has received only limited scientific attention in the literature. Most of the earlier studies on the subject employed simulation approaches in order to investigate the impact of different parameters on system nervousness (see e.g. Blackburn et al., 1986, 1987; Sridharan et al., 1988; Kadipasaoglu and Sridharan, 1997). More recent studies provided guidelines to assess the extent of system nervousness of different inventory policies (see e.g. Jensen, 1996; De Kok and Inderfurth, 1997; Heisig, 1998, 2001; Kilic and Tarim, 2011). These papers consider nervousness as an independent attribute of an inventory system and propose measures to assess the instability of associated inventory control systems. There have also been a few studies where system nervousness is incorporated to the total cost function by a given cost parameter (see e.g. Kropp et al., 1983; Kropp and Carlson, 1984). However, most of these works mentioned above are directed towards stationary systems and their results do not carry over to non-stationary demand environments. It is important to underline that non-stationary demands are very common in practice due to fast technological progress, changes in consumer
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preferences, seasonality, and trends (Kurawarwala and Matsuo, 1996). It is also known that it is very expensive to ignore the dynamic nature of demands in inventory control (see Tunc et al., 2010). Therefore, it is of significant practical importance to investigate system nervousness in non-stationary demand environments. In this paper, we aim to provide a pragmatic yet an objective measure to evaluate the cost of system nervousness in nonstationary stochastic demand environments. In order to capture both setup- and quantity-oriented nervousness, we convey our analysis building on the stochastic lot sizing problem where timing and quantity of replenishments is of concern. The proposed approach makes use of static, static-dynamic, and dynamic uncertainty strategies (Bookbinder and Tan, 1988). The static uncertainty strategy requires that the replenishment schedule and replenishment quantities are determined once and for all at the beginning of the planning horizon. The dynamic uncertainty strategy, on the other hand, offers an immense flexibility by allowing the revision of the replenishment schedule and replenishment quantities in response to realized demands throughout the planning horizon. The static-dynamic uncertainty strategy is a hybrid of the dynamic and the static uncertainty strategies. It is characterized by a fixed replenishment schedule where replenishment quantities are determined at replenishment epochs. These strategies pertain to different proposals with regard to coordinating supply chain inventories. We herein argue that these strategies establish a natural frame of reference by reflecting extreme cases for the setup- and the quantity-oriented system nervousness. In this context, when these strategies are exercised optimally the expected cost figures associated with them provide a yardstick for assessing the cost of system nervousness. It is important to note that the aforementioned assessment is mainly relates to the party that gives the order rather than the whole system. In this regard, we do not gauge the cost of overall system’s nervousness. Nevertheless, we do gauge the cost of determining nervousness-free ordering plans for upstream players in the supply chain. The approach provided in this paper is, indeed, one of the very first in the literature for measuring the cost of system nervousness. We conduct an extensive numerical study in order to explore how the cost of system nervousness as measured by means of the proposed framework is affected by inventory costs and demand parameters. The results of our study are of managerial significance. We show, for the majority of the cases, that the cost performance of the static uncertainty strategy is very poor as compared to the static-dynamic uncertainty strategy, whereas the cost performance of the static-dynamic uncertainty strategy is comparable to the dynamic uncertainty strategy. This reveals that the setup-oriented system nervousness, which is known to be critical in practice, can be avoided with a minor cost penalty, whereas, on the contrary, the quantity-oriented nervousness could only be eliminated at a significant expense. The rest of the paper is organized as follows. In Section 2, we present the preliminaries of our analysis and introduce the methods used to work out alternative control strategies. In Section 3, we conduct a numerical study and discuss its results. Finally, in Section 4 we conclude.
2. Methodology We consider the stochastic lot sizing problem which is a periodic-review finite-horizon inventory control problem with replenishment setup costs. The problem can be defined as follows. There is a planning horizon comprised N periods indexed by n A f1, . . . ,Ng. The demand, dn, in period n is an independent
random variable with a known probability distribution function, g n ðdn Þ, and occurs instantaneously at the beginning of the period. The demand is non-stationary, i.e. the demand rate may vary from period to period. A fixed ordering cost K is incurred each time an order is placed. Additionally, a holding cost h is incurred for each unit carried in inventory from one period to the next, and a shortage cost p is incurred for each unit of demand backordered. For the sake of brevity, we assume that there is no replenishment lead time and there is no direct item cost. The problem is to determine the replenishment schedule and replenishment quantities so as to minimize the sum of replenishment, holding, and penalty costs. A general mathematical formulation for the problem can be written as follows (see e.g. Sox, 1997; Tarim and Kingsman, 2006): " !þ ! # N n n n n X X X X X min E Kzn þ h xk dk þp xk dk n¼1
s:t:
zn ¼
zn A f0,1g
k¼1
1
if xn 4 0
0
otherwise and
xn A R þ
k¼1
k¼1
k¼1
8n A f1, . . . ,Ng 8n A f1, . . . ,Ng
where zn and xn are the respective decision variables indicating the replenishment action and the replenishment quantity for period n, and ðxÞ þ ¼ maxð0,xÞ and ðxÞ ¼ maxð0,xÞ. It is also assumed that the initial inventory is zero. Bookbinder and Tan (1988) proposed alternative strategies towards the stochastic lot sizing problem which differ from each other with respect to the time that replenishment decisions are made. We summarize these strategies in the following. The static uncertainty is a rigid control strategy which requires that all replenishment decisions are made at the very beginning of the planning horizon. Therefore, under the static uncertainty strategy zn and xn are fixed for all periods before any demand has been realized. The dynamic uncertainty is a flexible control strategy that postpones replenishment decisions until the latest time they need to be done. Here, zn and xn are dynamically determined at period n where the demand information regarding all periods up to period n has already been revealed. The static-dynamic uncertainty is a combination of the static and the dynamic uncertainty strategies which is characterized by a rigid replenishment schedule but flexible replenishment quantities. Thus, zn are determined for all periods at the beginning of the planning horizon, however, xn are dynamically determined at period n. In terms of cost-effectiveness and nervousness these strategies yield different performances. The cost-effectiveness is a positive function of the amount of demand information available at the time of making replenishment decisions. Thus, in terms of costeffectiveness, the dynamic uncertainty is the best strategy, and it is followed by the static-dynamic and the static uncertainty strategies, respectively. The nervousness performance, on the other hand, depends on the extent of replenishment decisions fixed in advance. Hence, in terms of nervousness performance, the static uncertainty is the best strategy, and it is followed by the static-dynamic and the dynamic uncertainty strategies respectively. This, in fact, reflects the fundamental trade-off between the cost-effectiveness and the nervousness. It is important to remark that these strategies bear very specific positions against each other with respect to the system nervousness. The static uncertainty strategy guarantees a nervousness-free inventory policy by imposing a rigid coordination over replenishments. The dynamic uncertainty strategy, on the other hand, leads to an inventory policy characterized by both the setup- and the quantityoriented nervousness because it makes replenishment decisions in a just-in-time fashion. As a hybrid of these two, the static-dynamic uncertainty suggests an inventory policy which yields no setup-
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oriented nervousness, but some extent of quantity-oriented nervousness. Therefore, when these strategies are exercised optimally, their expected cost figures stand for extreme case costs in terms of system nervousness, and the differences between these extreme case costs can be regarded as reflections of the cost of nervousness. More specifically, the difference between the expected costs of the static and the static-dynamic uncertainty strategies reflects the cost of the quantity-oriented nervousness, whereas the difference between the expected costs of the static-dynamic and the dynamic uncertainty strategies reflects the cost of the setup-oriented nervousness. The aforementioned strategies can be implemented in various ways in different settings, for example in a rolling horizon framework with forecast updates. However, for the purposes of this study, we are rather interested in their optimal applications. In what follows, we discuss the methods which are known to be optimal or near-optimal for the underlying strategies. The dynamic uncertainty strategy can be worked out optimally by using the non-stationary (s,S) policy. In his seminal paper Scarf (1960) proved the optimality of the (s,S) policy for the stochastic lot sizing problem. The policy manages inventories by placing a replenishment order whenever the inventory position drops below a critical reorder level so as to replenish up to a target level. Scarf’s proof is based on very mild assumptions and immediately extends to non-stationary demands. The policy becomes timedependent when demand is non-stationary since the critical levels vary from period to period. Finding the optimal parameters of the (s,S) policy is very demanding because it necessitates the recursive computation of a continuous cost function for each and every period. A possible approach to overcome the continuity issue is to assume discrete demands. This assumption is frequently employed in the literature since it leads to a discrete state space dynamic program which can be handled by standard dynamic programming approaches (see e.g. Bollapragada and Morton, 1999; Lulli and Sen, 2004; Guan et al., 2006; Huang and Kucukyavuz, 2008). Here, we take up the approach used in Bollapragada and Morton (1999). However, it is important to remark that even with the discrete demand assumption, the difficulty prevails because one needs to consider all possible inventory levels which, based on the precision of the discretization, could be arbitrarily large in number (Halman et al., 2006). The static-dynamic uncertainty strategy can be carried out optimally by means of the replenishment cycle policy (Ozen et al., 2012). This policy replenishes inventories up to a target level in cycles of a number of time periods. Thus, it involves the problem of finding the number and the length of replenishment cycles as well as the respective order-up-to quantities. The replenishment cycle policy has been addressed in many studies (see e.g. Askin, 1981; Bookbinder and Tan, 1988; Sox, 1997; Tarim and Kingsman, 2004, 2006). Recently, Rossi et al. (2008) and Ozen et al. (2012) introduced two different exact solution approaches based on constraint programming and dynamic programming techniques, respectively. Although these approaches provide the optimal solution, they are neither computationally efficient nor scalable (Tarim et al., 2011). Nevertheless, numerical studies in these papers reveal that the certainty equivalent mixed-integer programming formulation developed by Tarim and Kingsman (2006) provide near-optimal solutions for the replenishment cycle policy. Therefore, in this study we use the approach suggested in Tarim and Kingsman (2006). The static uncertainty strategy can be exercised optimally by using the stochastic version of the Wagner–Whitin dynamic lot size model (Vargas, 2009). This model provides an inventory policy which determines the length of re-order cycles and the corresponding order quantities at once at the beginning of the planning horizon, and it has been addressed by many authors under a variety of settings (see e.g. Lasserre et al., 1985; Vargas, 2009; Tempelmeier, 2011; Tempelmeier and Herpers, 2011). Vargas
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(2009) proved that the optimal replenishment quantities for a given replenishment cycle follow a critical ratio as in the case of the classical newsvendor problem. He also showed that although the Wagner-Whitin property does not extend to the stochastic case, the optimal replenishment schedule can be obtained by means of a shortest path problem on an acyclic network where arcs represent possible replenishment cycles. We employ this approach in the current study.
3. Numerical study In this section, we conduct a numerical study to explore the cost of nervousness by means of the proposed framework. We investigate how the relative importance of the setup- and the quantity-oriented nervousness are affected by inventory costs and demand parameters. In what follows, first we present the design of numerical experiments, and then we provide the results and discuss our findings. We use a test set involving a variety of demand and cost parameters. We assume that, for all problem instances, period demands are normally distributed. In order to control the extent of stochasticity in a straightforward manner, we characterize demand distributions by a fixed coefficient of variation over the planning horizon. Then, a problem instance can be defined by: the demand pattern p specifying the mean demands for all periods, planning horizon T, the coefficient of variation of demand r, setup cost K, penalty cost p, and holding cost h. We conduct a full factorial analysis by using the following sets of parameters: p A fSTAT,RAND, SIN1,SIN2,SEA1,SEA2,LCY1,LCY2g, T A f24,48g, r A f0:10,0:20,0:30g, K A f10,50,200,500,1000,3000g, p A f10,20,40g, and hA f1g. This setting leads to 864 test instances. Figs. 1 and 2 illustrate the demand patterns under consideration. Demand patterns used reflect common non-stationary demand structures. The STAT pattern reflects the case where demand is stationary over time. The RAND pattern, on the other hand, reflects the case where demand is erratic or random. Thus, the STAT and the RAND patterns represent the extreme cases of stationary/non-stationary dichotomy. The rest are different forms of sinusoidal, seasonal, and life-cycle structures. The sinusoidal patterns SIN1 and SIN2 are generated by using sinusoidal waves of different amplitudes. The seasonal demand patterns SEA1 and SEA2 exhibit few spikes throughout the planning horizon whereas a stationary pattern is followed during the rest of the time. The life-cycle patterns LCY1 and LCY2 are generated by imitating normal distribution with different standard deviations. We remark that for all demand patterns the average of mean demands over the planning horizon is fixed at 100 units. We solve each test instance by using the static, the dynamic, and the static-dynamic uncertainty strategies. We thus obtain the respective expected cost figures Csta, Cdyn, and C sta-dyn . Notice that the dynamic uncertainty strategy is cost-optimal. Thus, we measure the cost-effectiveness of the static and the staticdynamic uncertainty strategies against the dynamic uncertainty strategy. We define the following unified measures to express the costs of the setup- ðDs Þ and the quantity-oriented nervousness ðDq Þ as well as the sum of those two ðDs þ q Þ:
Ds ¼
C stadyn C dyn C
dyn
,
Dq ¼
C sta C stadyn C
dyn
,
Ds þ q ¼
C sta C dyn C dyn
We elaborate upon the effects of different parameters regarding demands and inventory costs. We report the average Ds , Dq , and Ds þ q for all problem instances characterized by the same pivot parameter. Table 1 presents the results of the numerical experiments. Below, we summarize our findings from the results of the numerical experiments referring Table 1. For all demand and cost
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Fig. 1. Demand patterns for 24-period test instances.
settings, the gap between the expected costs of the dynamic and the static-dynamic uncertainty strategies is fairly small, and Ds averages between 2.5 and 3.0% for both 24 and 48 period
instances. Notice that the length of the planning horizon does not significantly change the performance of static-dynamic uncertainty strategy. However, on the contrary, the static uncertainty
H. Tunc et al. / Int. J. Production Economics 141 (2013) 619–625
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Fig. 2. Demand patterns for 48-period test instances.
strategy performs rather poor as compared to both the staticdynamic and the dynamic uncertainty strategies. We observe that Ds þ q is on average around 63% and 103% for 24 and 48 period
instances respectively. The large difference between instances with short and long horizon lengths shows that the oversight of the static uncertainty strategy accumulates over time. These
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H. Tunc et al. / Int. J. Production Economics 141 (2013) 619–625
Table 1 The results of the numerical experiments. Average cost of nervousness (%) T¼24
Ds
T¼ 48
Dq
Ds þ q
Ds
Dq
Ds þ q
Demand pattern ðpÞ STAT 1.88 RAND 4.90 SIN1 2.20 SIN2 3.80 SEA1 2.03 SEA2 2.41 LCY1 1.92 LCY2 2.00
57.82 65.77 57.91 66.47 56.52 58.83 57.51 62.94
59.69 70.68 60.10 70.27 58.55 61.24 59.44 64.93
2.10 4.53 2.44 4.03 2.40 2.83 2.15 2.12
92.46 121.65 93.76 109.14 94.73 99.04 92.30 99.89
94.56 126.18 96.20 113.16 97.13 101.87 94.46 102.01
Coefficient of variation ðrÞ 0.1 0.84 0.2 2.47 0.3 4.62
45.36 62.30 73.75
46.20 64.77 78.36
0.82 2.62 5.04
74.44 103.33 123.34
75.26 105.95 128.38
Setup cost (K) 10 50 200 500 1000 3000
1.98 1.93 3.15 3.63 3.35 1.81
188.27 104.46 37.92 18.63 10.05 3.49
190.25 106.39 41.08 22.26 13.40 5.30
1.81 1.84 3.10 3.79 3.83 2.58
304.17 168.41 65.63 34.88 20.76 8.38
305.99 170.24 68.73 38.67 24.59 10.96
Penalty cost (p) 10 20 40
1.79 2.24 3.89
56.71 61.10 63.60
58.50 63.35 67.49
1.90 2.41 4.16
93.82 101.17 106.13
95.72 103.58 110.29
All instances
2.64
60.47
63.11
2.83
100.37
103.20
results, all together, suggest that the setup-oriented nervousness can be eliminated at a fairly small cost, whereas the quantityoriented nervousness is rather expensive to overcome. The cost-effectiveness of both the static and the staticdynamic strategies deteriorate as the erraticity of demand increases. This is particularly apparent when we make a pairwise comparison between the patterns RAND/STAT, LCY1/LCY2, SEA1/ SEA2, and SIN1/SIN2. Notice that these patterns reflect two different levels of erraticity of the same underlying demand pattern. We observe that both Ds and Ds þ q are higher for the more erratic counterpart. As a result of this, we can conclude that it is generally more costly to avoid both the setup- and the quantity-oriented nervousness in cases of highly erratic demands. The cost performance of the static and the static-dynamic strategies decline with increasing levels of demand uncertainty. For instance when T¼24, Ds increases from 0.84% to 4.62%, whereas Dq þ s increases from 46.20% to 78.36% as coefficient of variation increases from 0.1 to 0.3. The coefficient of variation defines the extent of the demand uncertainty. Thus, this result highlights the importance of using all available demand information – as is the case for the dynamic uncertainty strategy – when demand uncertainty is high. Also, it indicates that the cost of nervousness is much higher in highly stochastic environments. The magnitude of setup cost significantly affects the cost performance of control strategies. The static uncertainty strategy performs significantly better as setup cost increases. For example when T¼24, Dq þ s decreases from 190.25% to 5.30% as setup cost gradually increases from 10 to 3000. The static-dynamic uncertainty strategy, however, tends to perform better for relatively small and large values of setup cost. These results can be explained by looking at the two extreme cases of setup cost. When setup cost is zero, the inventory is replenished in every period. Thus, there is no setup-oriented nervousness, and the static-dynamic uncertainty is equivalent to the dynamic
uncertainty strategy. When setup cost is extremely large, the inventory is replenished only once probably at the very first period. Hence, there is neither setup- nor quantity-oriented nervousness, and all three strategies are equivalent. Therefore, the setup-oriented nervousness is more critical for the intermediate values of setup cost, whereas the quantity-oriented nervousness is more critical when setup cost is rather small. The cost-effectiveness of both the static and the staticdynamic uncertainty strategies consistently decreases with increasing penalty costs. For instance when T¼24, this decline is from 3.89% to 1.79% for Ds and 67.49% to 58.50% for Dq þ s . Thus, the cost of eliminating nervousness appears to be a positive function of penalty cost. This result underlines that the cost of using non-optimal safety stock levels – as is the case for the static and the static-dynamic strategies – is larger when penalty cost is large.
4. Conclusion In this paper, we analyzed three different control strategies for the stochastic lot sizing problem. These strategies reflect extreme cases with regard to the setup- and the quantity-oriented system nervousness. We argued that the expected cost figures associated with these strategies could provide an objective measure to assess the cost of system nervousness. We conducted a numerical study to compare these strategies in terms of cost-effectiveness under a variety of parameters regarding demands and inventory costs. This enabled us to quantify the costs of the setup- and the quantity-oriented nervousness. The results of our study are of practical importance. We observed that the static-dynamic uncertainty strategy performs nearly as good as the dynamic uncertainty strategy, and significantly outperforms the static uncertainty strategy. This points out that the setuporiented system nervousness, which is perceived as the most critical type of nervousness in practice, can be completely eliminated at a minor expense, whereas the quantity-oriented nervousness can only be avoided at a large cost penalty. The numerical study also indicated consistent links between the cost-effectiveness of control strategies and demand/cost parameters. We saw that the extent of the erraticity and the stochasticity of demand may significantly diminish the cost-effectiveness of the static and the static-dynamic uncertainty strategies. We observed that the static-dynamic uncertainty strategy performs relatively worse for intermediate values of setup cost. This is, however, the case for the static uncertainty strategy when setup cost is small. We also noticed that large penalty costs can diminish the cost performance of these strategies. These, all together, provide guidelines to assess how critical the setup- and the quantity-oriented nervousness are in different environments. The dynamic uncertainty is the cost-optimal strategy to mediate inventories. However, it is not necessarily recommended for general application in practice due to the nervousness syndrome (Bookbinder and Tan, 1988). The results presented in this study suggest that the static-dynamic uncertainty could be an effective strategy for coordinating supply chain inventories especially when setup-oriented system nervousness is of concern. Thus, we believe that it is practically important to develop models and solution approaches for the static-dynamic uncertainty strategy.
Acknowledgments We would like to thank Roberto Rossi and the two anonymous reviewers for their valuable contributions. Two of the co-authors,
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Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe
A simple approach for assessing the cost of system nervousness Huseyin Tunc a, Onur A. Kilic b, S. Armagan Tarim b, Burak Eksioglu a,n a b
Department of Industrial and Systems Engineering, Mississippi State University, P.O. Box 9542, Mississippi State, MS 39762, USA Department of Management, Hacettepe University, 06800 Beytepe, Ankara, Turkey
a r t i c l e i n f o
abstract
Article history: Received 23 January 2012 Accepted 14 September 2012 Available online 9 October 2012
A well-known problem in coordinating supply chain inventories is that replenishment decisions are revised due to stochastic demands. This issue is often referred to as system nervousness. The literature distinguishes between two types of nervousness: setup-oriented and quantity-oriented. It is widely accepted that cost of nervousness is difficult to measure. We argue that this cost can be evaluated by means of three well-established inventory control strategies: static uncertainty, dynamic uncertainty, and static-dynamic uncertainty. These strategies reflect extreme cases with regard to the setup- and the quantity-oriented nervousness, and provide a simple yet an objective measure to assess the cost of system nervousness. Our results are of practical importance. We highlight that the setup-oriented nervousness, which is considered to be the most critical in practice, can be eliminated with a minor cost penalty. This is, however, not the case for the quantity-oriented nervousness. & 2012 Elsevier B.V. All rights reserved.
Keywords: Setup-oriented nervousness Quantity-oriented nervousness Static uncertainty Dynamic uncertainty Static–dynamic uncertainty Non-stationary stochastic demand
1. Introduction Inventory control is far more challenging when demand is stochastic. The difficulty mainly springs from the fact that, under uncertainty, effective inventory control policies are needed to steer the inventory systems and efficient algorithms to compute policy parameters. A further complication with stochastic demands, particularly in supply chain networks and MRP environments, is that downstream players continually change the timing as well as the size of their replenishments. This issue is often referred to as the system nervousness, and regarded as an important performance measure for inventory control policies (De Kok and Inderfurth, 1997). The system nervousness arise due to revisions in the original plan which in turn result in different replenishment decisions in successive planning cycles. For example, a previously planned order may be revised or canceled following demand realizations in a particular period. The system nervousness leads to a lack of coordination in supply chains and MRP environments because nervousness on the top level propagates throughout the system (see e.g. De Kok and Inderfurth, 1997; Chatfield et al., 2004; Kaipia et al., 2006). The system nervousness is critical especially in systems characterized by low degrees of flexibility. In such environments, the cost of implementing revisions in
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Corresponding author. Tel.: þ1 662 325 7625; fax: þ 1 662 325 7618. E-mail addresses: [email protected] (H. Tunc), [email protected] (O.A. Kilic), [email protected] (S.A. Tarim), [email protected] (B. Eksioglu). 0925-5273/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijpe.2012.09.022
replenishment schedules/quantities may overcome the advantage of employing a cost efficient control policy (Heisig, 2001). The literature distinguish between two types of system nervousness: setup-oriented and quantity-oriented. The former refers to revisions in replenishment schedules, whereas the latter refers to revisions in replenishment quantities. As mentioned by many authors (see e.g. Inderfurth, 1994; Heisig, 2001; Tarim and Kingsman, 2004) the setup-oriented nervousness is considered to be more critical as compared to the quantity-oriented nervousness. The nervousness syndrome has received only limited scientific attention in the literature. Most of the earlier studies on the subject employed simulation approaches in order to investigate the impact of different parameters on system nervousness (see e.g. Blackburn et al., 1986, 1987; Sridharan et al., 1988; Kadipasaoglu and Sridharan, 1997). More recent studies provided guidelines to assess the extent of system nervousness of different inventory policies (see e.g. Jensen, 1996; De Kok and Inderfurth, 1997; Heisig, 1998, 2001; Kilic and Tarim, 2011). These papers consider nervousness as an independent attribute of an inventory system and propose measures to assess the instability of associated inventory control systems. There have also been a few studies where system nervousness is incorporated to the total cost function by a given cost parameter (see e.g. Kropp et al., 1983; Kropp and Carlson, 1984). However, most of these works mentioned above are directed towards stationary systems and their results do not carry over to non-stationary demand environments. It is important to underline that non-stationary demands are very common in practice due to fast technological progress, changes in consumer
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preferences, seasonality, and trends (Kurawarwala and Matsuo, 1996). It is also known that it is very expensive to ignore the dynamic nature of demands in inventory control (see Tunc et al., 2010). Therefore, it is of significant practical importance to investigate system nervousness in non-stationary demand environments. In this paper, we aim to provide a pragmatic yet an objective measure to evaluate the cost of system nervousness in nonstationary stochastic demand environments. In order to capture both setup- and quantity-oriented nervousness, we convey our analysis building on the stochastic lot sizing problem where timing and quantity of replenishments is of concern. The proposed approach makes use of static, static-dynamic, and dynamic uncertainty strategies (Bookbinder and Tan, 1988). The static uncertainty strategy requires that the replenishment schedule and replenishment quantities are determined once and for all at the beginning of the planning horizon. The dynamic uncertainty strategy, on the other hand, offers an immense flexibility by allowing the revision of the replenishment schedule and replenishment quantities in response to realized demands throughout the planning horizon. The static-dynamic uncertainty strategy is a hybrid of the dynamic and the static uncertainty strategies. It is characterized by a fixed replenishment schedule where replenishment quantities are determined at replenishment epochs. These strategies pertain to different proposals with regard to coordinating supply chain inventories. We herein argue that these strategies establish a natural frame of reference by reflecting extreme cases for the setup- and the quantity-oriented system nervousness. In this context, when these strategies are exercised optimally the expected cost figures associated with them provide a yardstick for assessing the cost of system nervousness. It is important to note that the aforementioned assessment is mainly relates to the party that gives the order rather than the whole system. In this regard, we do not gauge the cost of overall system’s nervousness. Nevertheless, we do gauge the cost of determining nervousness-free ordering plans for upstream players in the supply chain. The approach provided in this paper is, indeed, one of the very first in the literature for measuring the cost of system nervousness. We conduct an extensive numerical study in order to explore how the cost of system nervousness as measured by means of the proposed framework is affected by inventory costs and demand parameters. The results of our study are of managerial significance. We show, for the majority of the cases, that the cost performance of the static uncertainty strategy is very poor as compared to the static-dynamic uncertainty strategy, whereas the cost performance of the static-dynamic uncertainty strategy is comparable to the dynamic uncertainty strategy. This reveals that the setup-oriented system nervousness, which is known to be critical in practice, can be avoided with a minor cost penalty, whereas, on the contrary, the quantity-oriented nervousness could only be eliminated at a significant expense. The rest of the paper is organized as follows. In Section 2, we present the preliminaries of our analysis and introduce the methods used to work out alternative control strategies. In Section 3, we conduct a numerical study and discuss its results. Finally, in Section 4 we conclude.
2. Methodology We consider the stochastic lot sizing problem which is a periodic-review finite-horizon inventory control problem with replenishment setup costs. The problem can be defined as follows. There is a planning horizon comprised N periods indexed by n A f1, . . . ,Ng. The demand, dn, in period n is an independent
random variable with a known probability distribution function, g n ðdn Þ, and occurs instantaneously at the beginning of the period. The demand is non-stationary, i.e. the demand rate may vary from period to period. A fixed ordering cost K is incurred each time an order is placed. Additionally, a holding cost h is incurred for each unit carried in inventory from one period to the next, and a shortage cost p is incurred for each unit of demand backordered. For the sake of brevity, we assume that there is no replenishment lead time and there is no direct item cost. The problem is to determine the replenishment schedule and replenishment quantities so as to minimize the sum of replenishment, holding, and penalty costs. A general mathematical formulation for the problem can be written as follows (see e.g. Sox, 1997; Tarim and Kingsman, 2006): " !þ ! # N n n n n X X X X X min E Kzn þ h xk dk þp xk dk n¼1
s:t:
zn ¼
zn A f0,1g
k¼1
1
if xn 4 0
0
otherwise and
xn A R þ
k¼1
k¼1
k¼1
8n A f1, . . . ,Ng 8n A f1, . . . ,Ng
where zn and xn are the respective decision variables indicating the replenishment action and the replenishment quantity for period n, and ðxÞ þ ¼ maxð0,xÞ and ðxÞ ¼ maxð0,xÞ. It is also assumed that the initial inventory is zero. Bookbinder and Tan (1988) proposed alternative strategies towards the stochastic lot sizing problem which differ from each other with respect to the time that replenishment decisions are made. We summarize these strategies in the following. The static uncertainty is a rigid control strategy which requires that all replenishment decisions are made at the very beginning of the planning horizon. Therefore, under the static uncertainty strategy zn and xn are fixed for all periods before any demand has been realized. The dynamic uncertainty is a flexible control strategy that postpones replenishment decisions until the latest time they need to be done. Here, zn and xn are dynamically determined at period n where the demand information regarding all periods up to period n has already been revealed. The static-dynamic uncertainty is a combination of the static and the dynamic uncertainty strategies which is characterized by a rigid replenishment schedule but flexible replenishment quantities. Thus, zn are determined for all periods at the beginning of the planning horizon, however, xn are dynamically determined at period n. In terms of cost-effectiveness and nervousness these strategies yield different performances. The cost-effectiveness is a positive function of the amount of demand information available at the time of making replenishment decisions. Thus, in terms of costeffectiveness, the dynamic uncertainty is the best strategy, and it is followed by the static-dynamic and the static uncertainty strategies, respectively. The nervousness performance, on the other hand, depends on the extent of replenishment decisions fixed in advance. Hence, in terms of nervousness performance, the static uncertainty is the best strategy, and it is followed by the static-dynamic and the dynamic uncertainty strategies respectively. This, in fact, reflects the fundamental trade-off between the cost-effectiveness and the nervousness. It is important to remark that these strategies bear very specific positions against each other with respect to the system nervousness. The static uncertainty strategy guarantees a nervousness-free inventory policy by imposing a rigid coordination over replenishments. The dynamic uncertainty strategy, on the other hand, leads to an inventory policy characterized by both the setup- and the quantityoriented nervousness because it makes replenishment decisions in a just-in-time fashion. As a hybrid of these two, the static-dynamic uncertainty suggests an inventory policy which yields no setup-
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oriented nervousness, but some extent of quantity-oriented nervousness. Therefore, when these strategies are exercised optimally, their expected cost figures stand for extreme case costs in terms of system nervousness, and the differences between these extreme case costs can be regarded as reflections of the cost of nervousness. More specifically, the difference between the expected costs of the static and the static-dynamic uncertainty strategies reflects the cost of the quantity-oriented nervousness, whereas the difference between the expected costs of the static-dynamic and the dynamic uncertainty strategies reflects the cost of the setup-oriented nervousness. The aforementioned strategies can be implemented in various ways in different settings, for example in a rolling horizon framework with forecast updates. However, for the purposes of this study, we are rather interested in their optimal applications. In what follows, we discuss the methods which are known to be optimal or near-optimal for the underlying strategies. The dynamic uncertainty strategy can be worked out optimally by using the non-stationary (s,S) policy. In his seminal paper Scarf (1960) proved the optimality of the (s,S) policy for the stochastic lot sizing problem. The policy manages inventories by placing a replenishment order whenever the inventory position drops below a critical reorder level so as to replenish up to a target level. Scarf’s proof is based on very mild assumptions and immediately extends to non-stationary demands. The policy becomes timedependent when demand is non-stationary since the critical levels vary from period to period. Finding the optimal parameters of the (s,S) policy is very demanding because it necessitates the recursive computation of a continuous cost function for each and every period. A possible approach to overcome the continuity issue is to assume discrete demands. This assumption is frequently employed in the literature since it leads to a discrete state space dynamic program which can be handled by standard dynamic programming approaches (see e.g. Bollapragada and Morton, 1999; Lulli and Sen, 2004; Guan et al., 2006; Huang and Kucukyavuz, 2008). Here, we take up the approach used in Bollapragada and Morton (1999). However, it is important to remark that even with the discrete demand assumption, the difficulty prevails because one needs to consider all possible inventory levels which, based on the precision of the discretization, could be arbitrarily large in number (Halman et al., 2006). The static-dynamic uncertainty strategy can be carried out optimally by means of the replenishment cycle policy (Ozen et al., 2012). This policy replenishes inventories up to a target level in cycles of a number of time periods. Thus, it involves the problem of finding the number and the length of replenishment cycles as well as the respective order-up-to quantities. The replenishment cycle policy has been addressed in many studies (see e.g. Askin, 1981; Bookbinder and Tan, 1988; Sox, 1997; Tarim and Kingsman, 2004, 2006). Recently, Rossi et al. (2008) and Ozen et al. (2012) introduced two different exact solution approaches based on constraint programming and dynamic programming techniques, respectively. Although these approaches provide the optimal solution, they are neither computationally efficient nor scalable (Tarim et al., 2011). Nevertheless, numerical studies in these papers reveal that the certainty equivalent mixed-integer programming formulation developed by Tarim and Kingsman (2006) provide near-optimal solutions for the replenishment cycle policy. Therefore, in this study we use the approach suggested in Tarim and Kingsman (2006). The static uncertainty strategy can be exercised optimally by using the stochastic version of the Wagner–Whitin dynamic lot size model (Vargas, 2009). This model provides an inventory policy which determines the length of re-order cycles and the corresponding order quantities at once at the beginning of the planning horizon, and it has been addressed by many authors under a variety of settings (see e.g. Lasserre et al., 1985; Vargas, 2009; Tempelmeier, 2011; Tempelmeier and Herpers, 2011). Vargas
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(2009) proved that the optimal replenishment quantities for a given replenishment cycle follow a critical ratio as in the case of the classical newsvendor problem. He also showed that although the Wagner-Whitin property does not extend to the stochastic case, the optimal replenishment schedule can be obtained by means of a shortest path problem on an acyclic network where arcs represent possible replenishment cycles. We employ this approach in the current study.
3. Numerical study In this section, we conduct a numerical study to explore the cost of nervousness by means of the proposed framework. We investigate how the relative importance of the setup- and the quantity-oriented nervousness are affected by inventory costs and demand parameters. In what follows, first we present the design of numerical experiments, and then we provide the results and discuss our findings. We use a test set involving a variety of demand and cost parameters. We assume that, for all problem instances, period demands are normally distributed. In order to control the extent of stochasticity in a straightforward manner, we characterize demand distributions by a fixed coefficient of variation over the planning horizon. Then, a problem instance can be defined by: the demand pattern p specifying the mean demands for all periods, planning horizon T, the coefficient of variation of demand r, setup cost K, penalty cost p, and holding cost h. We conduct a full factorial analysis by using the following sets of parameters: p A fSTAT,RAND, SIN1,SIN2,SEA1,SEA2,LCY1,LCY2g, T A f24,48g, r A f0:10,0:20,0:30g, K A f10,50,200,500,1000,3000g, p A f10,20,40g, and hA f1g. This setting leads to 864 test instances. Figs. 1 and 2 illustrate the demand patterns under consideration. Demand patterns used reflect common non-stationary demand structures. The STAT pattern reflects the case where demand is stationary over time. The RAND pattern, on the other hand, reflects the case where demand is erratic or random. Thus, the STAT and the RAND patterns represent the extreme cases of stationary/non-stationary dichotomy. The rest are different forms of sinusoidal, seasonal, and life-cycle structures. The sinusoidal patterns SIN1 and SIN2 are generated by using sinusoidal waves of different amplitudes. The seasonal demand patterns SEA1 and SEA2 exhibit few spikes throughout the planning horizon whereas a stationary pattern is followed during the rest of the time. The life-cycle patterns LCY1 and LCY2 are generated by imitating normal distribution with different standard deviations. We remark that for all demand patterns the average of mean demands over the planning horizon is fixed at 100 units. We solve each test instance by using the static, the dynamic, and the static-dynamic uncertainty strategies. We thus obtain the respective expected cost figures Csta, Cdyn, and C sta-dyn . Notice that the dynamic uncertainty strategy is cost-optimal. Thus, we measure the cost-effectiveness of the static and the staticdynamic uncertainty strategies against the dynamic uncertainty strategy. We define the following unified measures to express the costs of the setup- ðDs Þ and the quantity-oriented nervousness ðDq Þ as well as the sum of those two ðDs þ q Þ:
Ds ¼
C stadyn C dyn C
dyn
,
Dq ¼
C sta C stadyn C
dyn
,
Ds þ q ¼
C sta C dyn C dyn
We elaborate upon the effects of different parameters regarding demands and inventory costs. We report the average Ds , Dq , and Ds þ q for all problem instances characterized by the same pivot parameter. Table 1 presents the results of the numerical experiments. Below, we summarize our findings from the results of the numerical experiments referring Table 1. For all demand and cost
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Fig. 1. Demand patterns for 24-period test instances.
settings, the gap between the expected costs of the dynamic and the static-dynamic uncertainty strategies is fairly small, and Ds averages between 2.5 and 3.0% for both 24 and 48 period
instances. Notice that the length of the planning horizon does not significantly change the performance of static-dynamic uncertainty strategy. However, on the contrary, the static uncertainty
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Fig. 2. Demand patterns for 48-period test instances.
strategy performs rather poor as compared to both the staticdynamic and the dynamic uncertainty strategies. We observe that Ds þ q is on average around 63% and 103% for 24 and 48 period
instances respectively. The large difference between instances with short and long horizon lengths shows that the oversight of the static uncertainty strategy accumulates over time. These
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Table 1 The results of the numerical experiments. Average cost of nervousness (%) T¼24
Ds
T¼ 48
Dq
Ds þ q
Ds
Dq
Ds þ q
Demand pattern ðpÞ STAT 1.88 RAND 4.90 SIN1 2.20 SIN2 3.80 SEA1 2.03 SEA2 2.41 LCY1 1.92 LCY2 2.00
57.82 65.77 57.91 66.47 56.52 58.83 57.51 62.94
59.69 70.68 60.10 70.27 58.55 61.24 59.44 64.93
2.10 4.53 2.44 4.03 2.40 2.83 2.15 2.12
92.46 121.65 93.76 109.14 94.73 99.04 92.30 99.89
94.56 126.18 96.20 113.16 97.13 101.87 94.46 102.01
Coefficient of variation ðrÞ 0.1 0.84 0.2 2.47 0.3 4.62
45.36 62.30 73.75
46.20 64.77 78.36
0.82 2.62 5.04
74.44 103.33 123.34
75.26 105.95 128.38
Setup cost (K) 10 50 200 500 1000 3000
1.98 1.93 3.15 3.63 3.35 1.81
188.27 104.46 37.92 18.63 10.05 3.49
190.25 106.39 41.08 22.26 13.40 5.30
1.81 1.84 3.10 3.79 3.83 2.58
304.17 168.41 65.63 34.88 20.76 8.38
305.99 170.24 68.73 38.67 24.59 10.96
Penalty cost (p) 10 20 40
1.79 2.24 3.89
56.71 61.10 63.60
58.50 63.35 67.49
1.90 2.41 4.16
93.82 101.17 106.13
95.72 103.58 110.29
All instances
2.64
60.47
63.11
2.83
100.37
103.20
results, all together, suggest that the setup-oriented nervousness can be eliminated at a fairly small cost, whereas the quantityoriented nervousness is rather expensive to overcome. The cost-effectiveness of both the static and the staticdynamic strategies deteriorate as the erraticity of demand increases. This is particularly apparent when we make a pairwise comparison between the patterns RAND/STAT, LCY1/LCY2, SEA1/ SEA2, and SIN1/SIN2. Notice that these patterns reflect two different levels of erraticity of the same underlying demand pattern. We observe that both Ds and Ds þ q are higher for the more erratic counterpart. As a result of this, we can conclude that it is generally more costly to avoid both the setup- and the quantity-oriented nervousness in cases of highly erratic demands. The cost performance of the static and the static-dynamic strategies decline with increasing levels of demand uncertainty. For instance when T¼24, Ds increases from 0.84% to 4.62%, whereas Dq þ s increases from 46.20% to 78.36% as coefficient of variation increases from 0.1 to 0.3. The coefficient of variation defines the extent of the demand uncertainty. Thus, this result highlights the importance of using all available demand information – as is the case for the dynamic uncertainty strategy – when demand uncertainty is high. Also, it indicates that the cost of nervousness is much higher in highly stochastic environments. The magnitude of setup cost significantly affects the cost performance of control strategies. The static uncertainty strategy performs significantly better as setup cost increases. For example when T¼24, Dq þ s decreases from 190.25% to 5.30% as setup cost gradually increases from 10 to 3000. The static-dynamic uncertainty strategy, however, tends to perform better for relatively small and large values of setup cost. These results can be explained by looking at the two extreme cases of setup cost. When setup cost is zero, the inventory is replenished in every period. Thus, there is no setup-oriented nervousness, and the static-dynamic uncertainty is equivalent to the dynamic
uncertainty strategy. When setup cost is extremely large, the inventory is replenished only once probably at the very first period. Hence, there is neither setup- nor quantity-oriented nervousness, and all three strategies are equivalent. Therefore, the setup-oriented nervousness is more critical for the intermediate values of setup cost, whereas the quantity-oriented nervousness is more critical when setup cost is rather small. The cost-effectiveness of both the static and the staticdynamic uncertainty strategies consistently decreases with increasing penalty costs. For instance when T¼24, this decline is from 3.89% to 1.79% for Ds and 67.49% to 58.50% for Dq þ s . Thus, the cost of eliminating nervousness appears to be a positive function of penalty cost. This result underlines that the cost of using non-optimal safety stock levels – as is the case for the static and the static-dynamic strategies – is larger when penalty cost is large.
4. Conclusion In this paper, we analyzed three different control strategies for the stochastic lot sizing problem. These strategies reflect extreme cases with regard to the setup- and the quantity-oriented system nervousness. We argued that the expected cost figures associated with these strategies could provide an objective measure to assess the cost of system nervousness. We conducted a numerical study to compare these strategies in terms of cost-effectiveness under a variety of parameters regarding demands and inventory costs. This enabled us to quantify the costs of the setup- and the quantity-oriented nervousness. The results of our study are of practical importance. We observed that the static-dynamic uncertainty strategy performs nearly as good as the dynamic uncertainty strategy, and significantly outperforms the static uncertainty strategy. This points out that the setuporiented system nervousness, which is perceived as the most critical type of nervousness in practice, can be completely eliminated at a minor expense, whereas the quantity-oriented nervousness can only be avoided at a large cost penalty. The numerical study also indicated consistent links between the cost-effectiveness of control strategies and demand/cost parameters. We saw that the extent of the erraticity and the stochasticity of demand may significantly diminish the cost-effectiveness of the static and the static-dynamic uncertainty strategies. We observed that the static-dynamic uncertainty strategy performs relatively worse for intermediate values of setup cost. This is, however, the case for the static uncertainty strategy when setup cost is small. We also noticed that large penalty costs can diminish the cost performance of these strategies. These, all together, provide guidelines to assess how critical the setup- and the quantity-oriented nervousness are in different environments. The dynamic uncertainty is the cost-optimal strategy to mediate inventories. However, it is not necessarily recommended for general application in practice due to the nervousness syndrome (Bookbinder and Tan, 1988). The results presented in this study suggest that the static-dynamic uncertainty could be an effective strategy for coordinating supply chain inventories especially when setup-oriented system nervousness is of concern. Thus, we believe that it is practically important to develop models and solution approaches for the static-dynamic uncertainty strategy.
Acknowledgments We would like to thank Roberto Rossi and the two anonymous reviewers for their valuable contributions. Two of the co-authors,
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