Sensitivity analysis of the newsvendor model Avijit Khanra (
[email protected]) Indian Institute of Management Ahmedabad, India Dr. Chetan A Soman Indian Institute of Management Ahmedabad, India Dr. Tathagata Bandyopadhyay Indian Institute of Management Ahmedabad, India
Abstract We perform sensitivity analysis of the classical newsvendor model. Conditions for symmetry/skewness of cost deviation (i.e., deviation of expected cost from its minimum) have been identified and its magnitude is studied by establishing a lower bound. We found the newsvendor model to be sensitive to sub-optimal ordering decisions. Keywords: Sensitivity analysis, Newsvendor model, Inventory management Introduction The newsvendor problem is one of the most well-researched and widely applicable inventory management problems (Choi, 2012). It is about deciding the order quantity of a product whose demand is unknown at the time of procurement decision and any mismatch between demand and supply at the end of the selling season attracts penalty. This problem was first addressed by Arrow et al. (1951) in their seminal paper “Optimal Inventory Policy”. Due to its wide applicability, different versions of the newsvendor model have been developed in the past six decades to fit into different situations. Review of these research works can be found in Silver et al. (1998), Khouja (1999), Qin et al. (2011) and Choi (2012). The simplest case of the newsvendor model is referred to as the classical newsvendor model (CNM). In CNM, i) objective is minimization of expected demand-supply mismatch cost, ii) demand is exogenous and stochastic, iii) only one procurement of any finite non-negative quantity is permitted, iv) the desired supply arrives before the selling season begins, v) left-over stock after the selling season (if any) is disposed off, vi) unmet demand (if any) is lost and vii) all cost components are linear with associated quantities. Other versions of the newsvendor model can be considered as extensions (by changing the assumptions) or generalizations (by relaxing the assumptions) of the CNM (Silver et al., 1998). In this paper, our focus is on the CNM. CNM, like any other inventory optimization model, requires knowledge of certain parameters for decision making. Optimum order quantity in the CNM depends on demand distribution function, over-stocking cost and under-stocking cost. All of these factors need not be known correctly in every situation. Newsvendor deals with stochastic demand whose realization takes place only after the procurement decision is made. Hence, theoretically speaking, correct knowledge of the demand
1
distribution function (i.e., the form and associated parameters) with certainty is impossible at the time of decision making. Over-stocking cost is the purchase cost less salvage value (of left-over stock). Purchase cost is generally known at the time of procurement decision. However, the same is not true for salvage value. Left-over stock is sold at a secondary market (if exists) and then the remaining stock (if any) is disposed off. This process begins after the selling season is over and it involves multiple agents, thereby making the assessment of salvage value at the time of procurement decision difficult. Under-stocking cost is the sum of unit profit and stock-out goodwill loss. Like purchase cost, unit profit is generally known, but stock-out goodwill loss is very difficult to estimate. Stock-out is reflected back as loss of future demand (of the concerned product as well as other products) through complex human interactions (Hadley and Whitin, 1963), which makes accurate assessment of the goodwill loss associated with stock-out very difficult. It is evident that parameter estimation error in CNM is unavoidable. Then the chances of the operational decision (derived using the parameter estimates) being the optimum (calculated using the true values) is very little. Given this unavoidable deviation of order quantity from its optimum (in short, order quantity deviation), it is important to study the nature of deviation of expected demand-supply mismatch cost from its minimum (in short, cost deviation). Surprisingly, very little is known about the behavior of cost deviation in the CNM, except for the fact that cost deviation is positive (i.e., expected demand-supply mismatch cost is greater than its minimum value) whenever order quantity deviation is non-zero. This is due to convexity of expected demand-supply mismatch cost in order quantity (Silver et al., 1998). In this work, to further our understanding of the nature of cost deviation, we want to i) identify conditions for symmetry/skewness of cost deviation and ii) study magnitude of cost deviation. If the decision maker is unsure about the optimum order quantity, answer to the first question tells which is better, ordering less or ordering more. Answer to the second question identifies the impact of sub-optimal ordering decisions on expected demand-supply mismatch cost. The remainder of this paper is organized as follows. We start with deriving an expression for cost deviation. Then we identify the necessary and sufficient conditions for symmetry/skewness of cost deviation. We study magnitude of cost deviation by establishing a lower bound for unimodal demand distributions. Then we briefly study order quantity deviation before concluding. Expression for cost deviation Let X be the random demand with support [a, b] ⊂ [0, ∞). Let F () and f () be the distribution and density functions of X. Let co and cu denote the unit over-stocking and under-stocking costs (positive). Let Q denote the order quantity (non-negative). The demand-supply mismatch cost, C is a random variable dependent on Q and X. It is given by C(Q, X) = Cost of over-stocking + Cost of under-stocking = co max{0, Q − X} + cu max{0, X − Q}. Expected demand-supply mismatch cost is given by Z Q Z b E[C(Q)] = co (Q − x)f (x)dx + cu (x − Q)f (x)dx a
Q
2
(1)
Z = (co + cu ) cf (µ − Q) +
Q
(Q − x)f (x)dx ,
(2)
a
where µ is the mean demand and cf = cu /(co + cu ) is known as the critical fractile. E[C(Q)] is convex in Q and the optimum order quantity, Q∗ is given by F (Q∗ ) = cf (Silver et al., 1998). Putting Q = Q∗ in (2), we get the minimum expected demand-supply mismatch cost as Z Q∗ ∗ E[C(Q )] = (co + cu ) cf µ − xf (x)dx . (3) a
We use ratio-based measure for deviation. Due to the unit-less nature of ratio-based measure, comparison among deviations is easy. Order quantity deviation, δQ = (Q − Q∗ )/Q∗ and cost deviation, δC = (E[C(Q)]−E[C(Q∗ )])/E[C(Q∗ )]. Note that δC depends on δQ as Q = Q∗ (1+δQ ). We want to understand the behavior of δC (δQ ) as δQ takes non-zero values. Using (2) for E[C(Q)] and (3) for E[C(Q∗ )], we get R Q∗ (1+δQ )
δC (δQ ) =
{F (x) − cf }dx . R Q∗ cf (µ − Q∗ ) + a F (x)dx Q∗
(4)
Note that δC (δQ ) depends on F and cf . Thus, the behavior of δC (δQ ) changes as the scenario (specified by F and cf ) changes. While answering the research questions, we try to accommodate the predominant scenarios if not all. Symmetry/skewness of cost deviation Let g be a real-valued function defined on an interval domain, I. Let x0 ∈ I and e0 > 0 such that x0 −e0 , x0 +e0 ∈ I. Then we say that g is i) symmetric in x0 ±e0 if g(x0 −e) = g(x0 +e) ∀e ∈ (0, e0 ], ii) left-skewed in x0 ± e0 if g(x0 − e) > g(x0 + e) ∀e ∈ (0, e0 ], iii) right-skewed in x0 ± e0 if g(x0 − e) < g(x0 + e) ∀e ∈ (0, e0 ] and iv) asymmetric in x0 ± e0 if none among the previous three holds. The following properties connect symmetry/skewness of δC (δQ ) in 0 ± e0 with that of f in Q∗ ± e0 Q∗ . Since δQ ≥ −1 (as Q ≥ 0), we restrict e0 ∈ (0, 1]. Property 1. δC (δQ ) is symmetric in 0 ± e0 if and only if the demand density function, f is symmetric almost everywhere in Q∗ ± e0 Q∗ . Density function of uniformly distributed demand is symmetric in Q∗ ± e0 Q∗ for any Q∗ (i.e., for any cf ) if e0 is not large (such that a ≤ Q∗ (1 − e0 ) and Q∗ (1 + e0 ) ≤ b). Then by the above property, δC (δQ ) is symmetric. The result regarding skewness is somewhat different. Property 2. δC (δQ ) is left (right) skewed in 0 ± e0 if the demand density function, f is left (right) skewed almost everywhere in Q∗ ± e0 Q∗ . Conversely, if δC (δQ ) is left (right) skewed in 0 ± e0 , f is left (right) skewed almost everywhere in Q∗ ± e00 Q∗ for some e00 ∈ (0, e0 ]. Property 1 states that symmetry of the demand density function, f almost everywhere in Q ± e0 Q∗ is both necessary and sufficient for the symmetry of cost deviation, δC (δQ ) in 0 ± e0 . These conditions are slightly different for skewness. By Property 2, left (right) skewness of f almost everywhere in Q∗ ± e00 Q∗ for some e00 ∈ (0, e0 ] is necessary, whereas the same in Q∗ ± e0 Q∗ is sufficient for the left (right) skewness of δC (δQ ) in 0 ± e0 . Thus, asymmetric demand density ∗
3
functions may lead to skewness of δC (δQ ), whereas only asymmetric demand density functions can lead to asymmetry of δC (δQ ). Property 1 and 2 have the following consequence for symmetric unimodal demand distributions. Property 3. For symmetric unimodal (in strong sense) demand distributions, δC (δQ ) is right-skewed in Q∗ ± e0 Q∗ if cf < 1/2, symmetric in Q∗ ± e0 Q∗ if cf = 1/2 and left-skewed in Q∗ ± e0 Q∗ if cf > 1/2 for every valid e0 > 0 (i.e., a ≤ Q∗ (1 − e0 ) < Q∗ (1 + e0 ) ≤ b). Commonly used demand distributions such as normal distribution (with equidistant truncation from the mode) and symmetric triangular distribution are symmetric and unimodal (in strong sense). In these cases, δC (−δQ ) < δC (δQ ) if cf < 1/2 and δC (−δQ ) > δC (δQ ) if cf > 1/2. Table 1 illustrates this observation. δC (δQ ) are calculated for different cf for normally distributed demand with cv = 0.25. It is evident that δC (δQ ) is right-skewed for cf = 0.25 and left-skewed for cf = 0.75. δC (δQ ) is symmetric for cf = 0.5. Table 1: Symmetry (skewness) of cost deviation Order quantity deviation (δQ ) cf
−5%
5%
−10%
10%
−15%
15%
−20%
20%
0.25
1.33
1.43
5.09
5.91
10.95
13.65
18.55
24.81
0.50
1.99
1.99
7.90
7.90
17.48
17.48
30.40
30.40
0.75
2.87
2.58
11.93
9.70
27.69
20.40
50.26
33.77
See our unpublished work (Khanra and Soman, 2013) for proofs of Property 1, 2 and 3. Magnitude of cost deviation δC (δQ ) depends on cf and F . Since cf ∈ (0, 1), we can capture dependence of δC (δQ ) on cf by considering different scenarios (i.e., different values of cf ). On the other hand, there is no straightforward way to capture dependence of δC (δQ ) on F by constructing scenarios. A possible solution to this problem lies in establishing F -independent bounds of δC (δQ ). A Lower bound of cost deviation We can rewrite (4) as δC (δQ ) = N (δQ )/D, where Z
Q∗ (1+δQ )
{F (x) − cf }dx,
N (δQ ) =
(5)
Q∗ ∗
Z
D = (µ − Q )cf +
Q∗
F (x)dx.
(6)
a
We establish a lower bound of δC (δQ ) for unimodal demand distributions by finding an upper bound of D and a lower bound of N (δQ ). Unlike the study of symmetry/skewness, our study of magnitude is restricted to unimodal demand distributions. Let us represent the family of unimodal distributions with support [a, b], mode c and F (c) = θ by UDa,b,c,θ . Let r = a/b and c = (1 − m)a + mb. By varying r ∈ [0, 1), m ∈ (0, 1) and θ ∈ (0, 1), every possible unimodal demand distribution can be covered. Let us define a special distribution in
4
UDa,b,c,θ . Given a, b, c and θ, let us define F0 as x−a θ if x ∈ [a, c), F0 (x) = c−a 1 − b−x (1 − θ) if x ∈ [c, b].
(7)
b−c
In a 2D plot between x ∈ [a, b] and F (x), F0 is the straight lines connecting (a, 0), (c, θ) and (c, θ), (b, 1). F0 plays an important role in establishing bounds of D and N (δQ ). Property 4. Let D0 corresponds to F0 ∈ UDa,b,c,θ . For every F ∈ UDa,b,c,θ , D + 1 cf (c − a)θ if cf < θ, 0 2 D< D + 1 (1 − cf )(b − c)(1 − θ) if cf ≥ θ. 0 2 Property 5. Let Q∗0 corresponds to F0 ∈ UDa,b,c,θ . For every F ∈ UDa,b,c,θ , ∗ θ − cf c 2 1−θ Q0 2 c − Q∗0 Q∗0 cf 2 δ + (δ − δ ) + 2 − N (δQ ) ≥ (δQ − δ) 2 c−a cf b−c Q 1−θ b−c if cf < θ, where δ = min{δQ , c/Q∗0 − 1}, ∗ θ c(1 − cf ) c 2 Q0 2 cf − θ Q∗0 − c 2 δ + (δ − δ ) + 2 − N (δQ ) ≥ (δ − δQ ) 2 b−c 1 − cf c − a Q θ c−a if cf ≥ θ, where δ = max{δQ , c/Q∗0 − 1}. See our unpublished work (Khanra and Bandyopadhyay, 2013) for proofs of the above properties. To be able to use these results, we need to find expressions for Q∗0 and D0 . Putting F0 (Q∗0 ) = cf in (7), we get Q∗0 as a + cf (c − a) if cf < θ, θ (8) Q∗0 = b − 1−cf (b − c) if cf ≥ θ. 1−θ
Putting F0 (Equation 7) and Q∗0 (Equation 8) in (6), we get D0 as h i cf (1−cf )−(1−θ)2 (c − a) + (1 − θ)(b − c) if cf < θ, 2 θ h i D0 = 1−cf cf −θ2 θ(c − a) + 1−θ (b − c) if cf ≥ θ. 2
(9)
Using the bounds of N (δQ ) and D (and the expressions for D0 and Q∗0 ), we get the following lower bound of δC (δQ ) for unimodal demand distributions. 2 δC (δQ ) ≥ k0 {k1 δ 2 + k2 (δQ − δ 2 ) + 2k3 |δQ − δ|},
(10)
where δ = min{δQ , q} if cf < θ and δ = max{δQ , q} if cf ≥ θ. k0 , k1 , k2 , k3 , q are δQ -independent constants. Using r, m, θ, cf , we can represent these constants as following. See our unpublished work (Khanra and Bandyopadhyay, 2013) for the details. 5
k0 k2 q
when cf < θ r/(1 − r) + tm (2 − t)m + (1 − θ)(1 − m) (1 − θ){r + tm(1 − r)} cf (1 − m)(1 − r) m(1 − t) r/(1 − r) + tm
when cf ≥ θ r/(1 − r) + m , (2 − t)(1 − m) + θm θ{1 − t(1 − m)(1 − r)} , (1 − cf )m(1 − r) −(1 − m)(1 − t) , 1/(1 − r) − t(1 − m)
k1 k3
when cf < θ r + m(1 − r) m(1 − r) (θ − cf )(θ − m) cf θ(1 − m)
when cf ≥ θ r + m(1 − r) , (1 − m)(1 − r) (cf − θ)(m − θ) , (1 − cf )(1 − θ)m
where t = cf /θ if cf < θ and t = (1 − cf )/(1 − θ) if cf ≥ θ. Demonstration of the lower bound The lower bound of δC (δQ ) depends on r, m, θ and cf . This enables us to study its behavior in different scenarios by changing r ∈ [0, 1), m ∈ (0, 1), θ ∈ (0, 1) and cf ∈ (0, 1). We do not need to study scenarios with large |m − θ| values as such scenarios are impractical.
Figure 1: Lower bound of cost deviation 6
Figure 1 shows the lower bound of δC (δQ ) for F ∈ UDa,b,c,θ in different scenarios. We consider low, medium and high values for r and cf (0.25, 0.5, 0.75). Each diagram corresponds to a combination of r and cf values. We considered three values for θ too. θ = m − 0.1 is indicated by red, θ = m is indicated by green and θ = m + 0.1 is indicated by blue. We take m = 0.5. Low and medium values of m (0.35, 0.65) have been considered in Khanra and Bandyopadhyay (2013). Those cases are not significantly different from Figure 1. We also indicate δC (δQ ) for the EOQ model (black curve) for quick comparison. δC (δQ ) = 2 δQ /{2(1 + δQ )} for the EOQ model (Nahmias, 2001). Dotted lines with slope ±1 separate the error “dampening" and “amplifying" zones. If a curve (or part of it) lies below these lines, magnitude of the output error is less than that of the input error (i.e., dampening of error). Conversely, if a curve (or part of it) lies above these lines, the output error is more in magnitude than the input error (i.e., amplification of error). Greater steepness of the lower bound curves compared to the EOQ curve demonstrates that the newsvendor model is more sensitive to erroneous decisions than the EOQ model. Location of the lower bound curves with respect to ±1 slope lines implies that amplification of error occurs in many situations, particularly in high r scenarios. We also observe rise in magnitude of the lower bound as r increases. This behavior can be explained by flattening of the demand density function due to decreased r (i.e., increased demand range). Then N (δQ ) decreases, thereby decreasing δC (δQ ). The effect reverses when r increases. Other factors do not influence the lower bound strongly. Order quantity deviation: Some remarks We have showed that penalty for deviating from the optimum is quite high in the CNM. We took ±30% as the error range in the demonstration of the lower bound of cost deviation. Of course, the error range varies from situation to situation. Our conclusions hold if the error range is larger or somewhat smaller. However, if the error range is very small (say ±5%), some of our conclusions may not hold. Thus, we need to examine if such possibilities are common or not. Let Z = (X − µ)/σ be the standardized random demand where σ is the standard deviation. Let Fz be the distribution function associated with Z. Fz (z) = F (µ + zσ). The optimum order quantity in the CNM can be expressed as Q∗ = F −1 (cf ) = µ + σFz−1 (cf ). The operational order quantity c), where pb represents the estimate of p (representative). We assume b∗ = µ is given by Q b+σ bFz−1 (cf b∗ and Q∗ , that the form of demand distribution is correctly known. Using these expressions of Q ∗ ∗ ∗ b − Q )/Q can be expressed as order quantity deviation, δQ = (Q δQ =
δµ + cv {(1 + δσ )Fz−1 (cf (1 + δcf )) − Fz−1 (cf )} , 1 + cv Fz−1 (cf )
(11)
where cv is the coefficient of variation. δµ , δσ , δcf represent the deviations of µ, σ, cf from their c − cf )/cf is determined by δco and δcu (the deviations of co and cu respective true values. δcf = (cf from their respective true values) as δcf =
(1 − cf )(δcu − δco ) . 1 + cf δcu + (1 − cf )δco
(12)
We can rewrite (11) as δQ = (δµ +cv δσ×cf )/{1+cv Fz−1 (cf )} where δσ×cf = (1+δσ )Fz−1 (cf (1+ δcf ))−Fz−1 (cf ) is the combined effect of δσ and δcf . Impact of δµ and δσ×cf on δQ is straightforward, 7
δQ increases in both. If cv < 1 and magnitudes of δµ and δσ×cf are of same level, impact of δµ is stronger than that of δσ×cf . Thus, mean demand may be the most influential parameter in the CNM. We test our intuition with an example. Table 2 demonstrates δQ in different deviation scenarios for normal demand distribution with cv = 0.25. We take |δµ | = |δσ | = |δcu | = |δco | = 10%. ∆ = cv δσ×cf represents the joint impact of δσ and δcf on δQ . In Table 2, δQ varies form −17.9% to 17.3, while |δQ | varies from 4.4% to 17.9%. Clearly, order quantity deviation is not very small. It is comparable with the error in model parameters. Mean demand emerges as the most important parameter as |δµ | > |∆| in every scenario. Moreover, the signs of δµ and δQ are always same. Table 2: Order quantity deviation Deviation scenario
Low cf (0.25)
Medium cf (0.5)
High cf (0.75)
δµ
δσ
δ cu
δ co
∆
δQ
∆
δQ
∆
δQ
−
−
−
−
1.7
−10.0
0
−10.0
−1.7
−10.0
−
−
−
+
−0.9
−13.2
−2.8
−12.8
−4.4
−12.3
−
−
+
−
4.4
−6.8
2.8
−7.2
0.9
−7.7
−
−
+
+
1.7
−10.0
0
−10.0
−1.7
−10.0
−
+
−
−
−1.7
−14.1
0
−10.0
1.7
−7.1
−
+
−
+
−4.9
−17.9
−3.5
−13.5
−1.6
−9.9
−
+
+
−
1.6
−10.1
3.5
−6.5
4.9
−4.4
−
+
+
+
−1.7
−14.1
0
−10.0
1.7
−7.1
+
−
−
−
1.7
14.1
0
10.0
−1.7
7.1
+
−
−
+
−0.9
10.9
−2.8
7.2
−4.4
4.8
+
−
+
−
4.4
17.3
2.8
12.8
0.9
9.4
+
−
+
+
1.7
14.1
0
10.0
−1.7
7.1
+
+
−
−
−1.7
10.0
0
10.0
1.7
10.0
+
+
−
+
−4.9
6.1
−3.5
6.5
−1.6
7.2
+
+
+
−
1.6
14.0
3.5
13.5
4.9
12.8
+
+
+
+
−1.7
10.0
0
10.0
1.7
10.0
In this demonstration, we considered a “common" situation (i.e., a “common" demand distribution with “common" input error values) and observed that order quantity deviation is not very small. Hence, in general, our conclusions about robustness of the CNM hold. Conclusion Our work contributes to the literature in two distinct ways. First, we have identified conditions for symmetry and skewness of cost deviation. According to these conditions, for symmetric unimodal demand distributions (e.g., normal distribution), if one is uncertain about the optimum order quantity, it is better to under-estimate the order quantity if cf < 1/2 and it is better to over-estimate the order quantity if cf > 1/2. Based on experiments, Schweitzer and Cachon (2000) reported that managers order more than the optimum when cf < 1/2 and less than the optimum when cf > 1/2. Our finding suggests that cost minimizing managers are better off doing the opposite. 8
Second, we have established a lower bound of cost deviation for unimodal demand distributions. It’s demonstration revealed that the newsvendor model is sensitive to sub-optimal ordering decisions, much more sensitive than the EOQ model. This behavior of the newsvendor model is totally different from the robustness properties of other popular stochastic inventory models. Stochastic (r,Q) and (s,S) inventory systems have been found to be more robust than the EOQ model (Zheng, 1992; Chen and Zheng, 1997). We also have observed that cost deviation increases with the ratio of demand limits (r). However, in absence of a thorough investigation into the influence of r on order quantity deviation, we can not be conclusive. Our study focused solely on cost deviation. A thorough investigation into order quantity deviation is needed to understand the “end-to-end" effect (i.e., reflection of parameter estimation error into cost deviation). Still, we have established that the CNM is sensitive to sub-optimal ordering decisions. One way to improve the decision making is to improve accuracy of parameter estimation. Since the CNM is multi-parameter model, the question of parameter importance becomes important. Our brief study of order quantity deviation revealed that the mean demand is the most influential parameter in the CNM. However, this is inconclusive. Bibliography Arrow, K. J., Harris, T., and Marschak, J. (1951). Optimal inventory policy. Econometrica, 19(3):250–272. Chen, F. and Zheng, Y.-S. (1997). Sensitivity analysis of an (s,S) inventory model. Operations Research Letters, 21(1):19–23. Choi, T.-M., editor (2012). Handbook of Newsvendor Problems. Springer, Hong Kong, 1 edition. Hadley, G. and Whitin, T. M. (1963). Analysis of Inventory Systems. Prentice-Hall, Englewood Cliffs, N.J., 1 edition. Khanra, A. and Bandyopadhyay, T. (2013). Lower bound for cost deviation in the newsboy model. Khanra, A. and Soman, C. (2013). Sensitivity analysis of the newsboy model. Khouja, M. (1999). The single-period (news-vendor) problem: Literature review and suggestions for future research. Omega, 27(5):537–553. Nahmias, S. (2001). Production and Operations Analysis. McGraw-Hill, Boston, 4 edition. Qin, Y., Wang, R., Vakharia, A. J., Chen, Y., and Seref, M. M. H. (2011). The newsvendor problem: Review and directions for future research. European Journal of Operational Research, 213(2):361–374. Schweitzer, M. E. and Cachon, G. P. (2000). Decision bias in the newsvendor problem with a known demand distribution: Experimental evidence. Management Science, 46(3):404–420. Silver, E. A., Pyke, D. F., and Peterson, R. (1998). Inventory Management and Production Planning and Scheduling. John Wiley & Sons, New York, 3 edition. Zheng, Y.-S. (1992). On properties of stochastic inventory systems. Management Science, 38(1):87–103.
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