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On the impact of suboptimal decisions in the newsvendor model
Operations Research Letters 45 (2017) 84–89
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Operations Research Letters journal homepage: www.elsevier.com/locate/orl
On the impact of suboptimal decisions in the newsvendor model Avijit Khanra Indian Institute of Technology, Kanpur 208016, India
article
abstract
info
Article history: Received 19 May 2016 Received in revised form 18 September 2016 Accepted 11 December 2016 Available online 24 December 2016
We study the impact of suboptimal decisions in the newsvendor model, one of the popular inventory models. We establish a lower bound for the deviation of inventory cost from its minimum, when the order quantity is suboptimal. Demonstration of the bound shows the model to be sensitive to suboptimal decisions. © 2016 Elsevier B.V. All rights reserved.
Keywords: Inventory Newsvendor model Robustness and sensitivity analysis
1. Introduction The newsvendor problem is about stocking decision of a product with uncertain demand, where mismatch between demand and supply attracts penalty. This classic inventory problem was first addressed by Arrow et al. [1]. Owing to its wide applicability, the newsvendor problem attracted attentions of many scholars over the past six decades. Review of their works can be found in Khouja [8], Qin et al. [11], and Choi [3]. Simplest case of the newsvendor problem, known as the classical newsvendor problem, arises when we make certain simplifying assumptions about the demand and supply processes. Key assumptions are: (i) exogenous demand with known distribution, (ii) single procurement of any amount, and (iii) linearity of cost components. See Chapter 10 of Silver et al. [12] for the details. Let X denote the stochastic demand with distribution function F . Let co and cu denote the unit over-stocking and under-stocking costs. Let Q denote the order quantity. Then demand–supply mismatch cost, C (Q , X ) and its expected value are given by C (Q , X ) = co max{0, Q − X } + cu max{0, X − Q } E [C (Q )] =
Q
co (Q − x)dF (x) +
−∞
∞
cu (x − Q )dF (x) Q
= (co + cu ) ξ (µ − Q ) +
Q
F (x)dx
(1)
−∞
where ξ = cu /(co + cu ) is referred to as the critical fractile and µ is the mean demand. E [C (Q )] is convex in Q and the optimal solution is given by: F (Q ∗ ) = ξ .
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.orl.2016.12.007 0167-6377/© 2016 Elsevier B.V. All rights reserved.
Decision making in the classical newsvendor model requires the knowledge of demand distribution and cost parameters. When estimates of these parameters deviate from their true values, the actual decision (derived using the estimates) deviates from the optimum (calculated using the true values). Due to convex nature of the objective function, any deviation from the optimal decision increases cost. In this context, sensitivity analysis is performed to understand the impacts of (i) suboptimal decisions on expected cost and (ii) parameter estimation error on stocking decision. It shall be noted that the second question is of practical relevance when the impact of suboptimal decisions on expected cost is significant. Some of the popular inventory models, e.g., economic order quantity (EOQ), stochastic (r , Q ), and stochastic (s, S ) models have been found to be insensitive to suboptimal decisions [10,14,2]. In this sense, we should investigate question-(i) first. In the newsvendor literature, there are some papers (e.g., [5,4, 9,13]) that, to some extent, address question-(ii), i.e., the impact of parameter estimation error on stocking decision. However, question-(i), which is our focus, remains largely unanswered with the exception of one paper. In a recent article, Khanra et al. [7] established a lower bound for cost deviation, i.e., the deviation of expected cost from its minimum value. Demonstration of the lower bound showed the newsvendor model to be sensitive to suboptimal decisions. In a number of scenarios, cost deviation exceeded order quantity deviation, i.e., the deviation of order quantity from its optimal value. This behaviour of the newsvendor model is opposite to that of the EOQ, (r , Q ), and (s, S ) models. Khanra et al. [7] assumed the demand to follow symmetric unimodal distribution. In this paper, we study robustness of the newsvendor model to suboptimal decisions when the demand distribution is not necessarily symmetric. In particular, we establish a new lower bound for cost deviation when the demand follows general unimodal distribution. Demonstration of the
A. Khanra / Operations Research Letters 45 (2017) 84–89
(a) θ ̸= m.
85
(b) θ = m. Fig. 1. G ∈ UD a,b,c ,θ .
lower bound establishes sensitivity of the newsvendor model to suboptimal decisions. 2. New lower bound for cost deviation We need to decide measures for cost and order quantity deviations. We choose relative measure, i.e., cost deviation is measured by δC = (E [C (Q )] − E [C (Q ∗ )])/E [C (Q ∗ )], and order quantity deviation is measured by δQ = (Q − Q ∗ )/Q ∗ . These measures are unit-less fractions, hence, easy to compare. Using (1), we can express δC as a function of δQ as follows.
Q ∗ (1+δQ ) δC (δQ ) =
Q∗
ξ (µ −
Q ∗)
{F (x) − ξ }dx . Q∗ + −∞ F (x)dx
(2)
Let us adjust the unit of cost so that co + cu = 1. Then the numerator of δC (δQ ) in (2) is the absolute deviation of cost, denoted by ∆C (δQ ), and the denominator is the minimum mismatch cost, E [C (Q ∗ )]. We establish the lower bound for δC (δQ ) by combining a lower bound of ∆C (δQ ) and an upper bound of E [C (Q ∗ )]. To obtain these bounds for unimodal demand, first we need to characterize such distributions. 2.1. Unimodal demand distributions We call a distribution F to be unimodal if there exists c ∈ R such that F is convex in (−∞, c ] and concave in [c , ∞) [6]. Since we are dealing with demand distributions, we can safely assume a bounded support for the distribution. Let us denote the family of unimodal distributions with support [a, b], mode c, and F (c ) = θ by UD a,b,c ,θ . Let r = a/b denote ratio of the demand limits and m = (c − a)/(b − a) denote location of the mode. Non-negativity of demand ensures a ≥ 0. We assume a < c < b and strict monotony of F in [a, b]. Then r ∈ [0, 1), m ∈ (0, 1), and θ ∈ (0, 1). Note that every unimodal demand distribution can be ‘‘covered’’ by varying r, m, and θ in their respective ranges. Similarly, every cost structure (i.e., co and cu values) can be covered, if we vary ξ in (0, 1). In order to derive a lower bound for δC (δQ ), first we need to bound F itself. Given a, b, c, and θ , let us define G ∈ UD a,b,c ,θ as follows.
x − a θ c−a G(x) = 1 − b − x (1 − θ ) b−c
if x ∈ [a, c ) (3) if x ∈ [c , b].
See Fig. 1 for a graphical depiction of G. When θ = m, G becomes the uniform distribution in [a, b]. We use QG∗ to denote the optimal order quantity when the demand distribution is G. Then EG [C (QG∗ )] denotes the minimum mismatch cost when G is the demand distribution. Using G, Lemma 1 offers a partial bound for every F ∈ UD a,b,c ,θ . Lemma 1. F (x) ≤ G(x) if x < c and F (x) ≥ G(x) if x ≥ c for every F ∈ UD a,b,c ,θ . Proof. We need to focus only on x ∈ (a, b) as G(x) = F (x) = 0 ∀x ≤ a and G(x) = F (x) = 1 ∀x ≥ b. Due to convexity of F in [a, c ], F (λa + (1 − λ)c ) ≤ λF (a) + (1 − λ)F (c ) = (1 − λ)θ ∀λ ∈ (0, 1). Replacing λa + (1 − λ)c by x, F (x) ≤ {(x − a)/(c − a)}θ = G(x) ∀x ∈ (a, c ). Similarly, due to concavity of F in [c , b], F (λc +(1 −λ)b) ≥ λF (c )+(1 −λ)F (b) = 1 −λ(1 −θ) ∀λ ∈ [0, 1). Replacing λc + (1 − λ)b by x, F (x) ≥ 1 − {(b − x)/(b − c )}(1 − θ ) = G(x) ∀x ∈ [c , b). Lemma 1 has the following consequence for Q ∗ , the optimal decision for F . Corollary 1. Q ∗ ≥ QG∗ if ξ < θ and Q ∗ ≤ QG∗ if ξ ≥ θ for every F ∈ UD a,b,c ,θ . Proof. If ξ < θ , Q ∗ < c. Then by Lemma 1, F (Q ∗ ) ≤ G(Q ∗ ). If, by contradiction, Q ∗ < QG∗ for some ξ < θ , ξ = F (Q ∗ ) ≤ G(Q ∗ ) < G(QG∗ ) = ξ , which is impossible. The strict inequality is due to strict monotony of G in [a, b]. Hence, Q ∗ ≥ QG∗ if ξ < θ . Similarly, if ξ ≥ θ , QG∗ ≥ c. Then by Lemma 1, F (QG∗ ) ≥ G(QG∗ ). Again, by contradiction, if Q ∗ > QG∗ for some ξ ≥ θ , ξ = G(QG∗ ) ≤ F (QG∗ ) < F (Q ∗ ) = ξ , which is impossible. The strict inequality is due to strict monotony of F in [a, b], which we assumed earlier. Hence, Q ∗ ≤ QG∗ if ξ ≥ θ . Lemma 1 and Corollary 1 are useful in establishing the lower bound for δC (δQ ). 2.2. Lower bound for the numerator Let us assume that the suboptimal decision, Q ∈ [a, b]. If a′ and b′ denote the observed lowest and highest demand, then the newsvendor is unlikely to order a quantity that is outside [a′ , b′ ]. Since [a′ , b′ ] ⊆ [a, b], we can assume that Q ∈ [a, b].
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A. Khanra / Operations Research Letters 45 (2017) 84–89
θ
θ
θ
ξ
ξ
ξ
0
a
Q
0
c
(a) δQ < 0.
a
Q
0
c
(b) 0 < δQ ≤ c /Q ∗ − 1.
a
c
Q
(c) δQ > c /Q ∗ − 1.
Fig. 2. ∆C (δQ ) when ξ < θ .
Proposition 1. For every F ∈ UD a,b,c ,θ , the absolute deviation of cost,
∗ ξQ∗ c 1−θ QG G 2 δ + (δ 2 − δ 2 ) 2 c − a ξ b − c Q c − QG∗ θ −ξ − (δQ − δ) if ξ < θ , +2 1−θ b−c ∆C (δQ ) ≥ ( 1 − ξ )c c θ QG∗ 2 δ + (δQ2 − δ 2 ) 2 b−c 1 −ξ c − a ∗ + 2 ξ − θ − QG − c (δ − δ ) if ξ ≥ θ , Q θ c−a where δ = min{δQ , c /QG∗ − 1} if ξ max{δQ , c /QG∗ − 1}.
< θ , otherwise δ =
Proof. If δQ = 0, δ = δQ in both the cases. Then the claim is that
Q ∆C (δQ ) ≥ 0, which is true as ∆C (δQ ) = Q ∗ {F (x) − ξ }dx = 0. We deal with the δQ ̸= 0 cases in two parts: (i) ξ < θ and (ii) ξ ≥ θ . Here, we present a proof for the first case. Proof for the second case is similar, and is provided in Appendix A of the online supplement. Hashed area in Fig. 2 exhibits ∆C (δQ ) when ξ < θ . By Corollary 1, a < QG∗ ≤ Q ∗ < c. The proof works differently when (a) δQ < 0, (b) 0 < δQ ≤ c /Q ∗ − 1, and (c) δQ > c /Q ∗ − 1. The first scenario is depicted in Fig. 2(a). Here, a ≤ Q < Q ∗ . Due to convexity of F in [a, Q ∗ ] ⊂ [a, c ], F (λa + (1 − λ)Q ∗ ) ≤ λF (a) + (1 − λ)F (Q ∗ ) = (1 − λ)ξ ∀λ ∈ [0, 1]. Replacing λa + (1 − λ)Q ∗ by x, ξ − F (x) ≥ λξ = ξ (Q ∗ − x)/(Q ∗ − a) ∀x ∈ [a, Q ∗ ] ⊇ [Q , Q ∗ ]. Then
∆C (δQ ) =
Q∗
{ξ − F (x)}dx ≥ Q
=
Q∗ Q ∗ (1+δQ )
ξ
Q∗ − x Q∗ − a
dx
δQ2 ξ (Q ∗ δQ )2 ξ QG∗ ξ QG∗ c 2 ≥ ≥ δQ . ∗ ∗ 2(Q − a) 2 1 − a/ Q 2 c−a
Here, δ = min{δQ , c /QG∗ − 1} = δQ . So the above inequality is same as the claim. The second scenario is depicted in Fig. 2(b). Here, Q ∗ < Q ≤ c. Due to unimodality of F , its density, f is increasing x in [a, c ]. Then ∀x ∈ [Q ∗ , Q ] ⊂ [a, c ], F (x) = F (Q ∗ ) + Q ∗ f (t )dt ≥
Q∗ ξ + f (Q ∗ )(x − Q ∗ ). For the same reason, F (Q ∗ ) = a f (t )dt ≤ f (Q ∗ )(Q ∗ − a) ⇒ f (Q ∗ ) ≥ ξ /(Q ∗ − a). Hence, ∀x ∈ [Q ∗ , Q ], F (x) − ξ ≥ ξ (x − Q ∗ )/(Q ∗ − a). Then Q Q ∗ (1+δQ ) x − Q∗ ∆C (δQ ) = {F (x) − ξ }dx ≥ ξ ∗ dx Q −a Q∗ Q∗ ξ QG∗ c 2 ≥ δQ , 2 c−a
just like the first scenario. Here too, δ = δQ . Hence the claim is valid. The third scenario is depicted in Fig. 2(c). Here, Q ∗ < c < Q ≤ b. Then
∆C (δQ ) =
c
Q∗
{F (x) − ξ }dx +
Q
{F (x) − ξ }dx.
c
Following arguments in the second scenario, F (x) − ξ ≥ ξ (x − Q ∗ )/(Q ∗ − a) ∀x ∈ [Q ∗ , c ]. By Lemma 1, F (x) − ξ ≥ G(x) − ξ = (θ − ξ ) + (1 − θ )(x − c )/(b − c ) ∀x ∈ [c , b] ⊇ [c , Q ]. So
∆C (δQ ) ≥
c
Q∗
ξ
x − Q∗ Q∗ − a
Q
dx + c
x−c (θ − ξ ) + (1 − θ ) dx. b−c
We need to replace c suitably, otherwise the bound cannot be computed without knowing F . Let g1 denote the first integrand and g2 denote the second. Note that δ = min{δQ , c /QG∗ − 1} ≥ min{δQ , c /Q ∗ − 1} = c /Q ∗ − 1. Then Q ∗ < c ≤ Q ∗ (1 + δ) ≤ Q . Since g1 (x) ↑ x, for every x ∈ [c , Q ∗ (1 + δ)], g1 (x) ≤ g1 (Q ∗ (1 + δ)) ≤ g1 (cQ ∗ /QG∗ ) as δ ≤ c /QG∗ − 1. Now g1 (cQ ∗ /QG∗ ) = ξ (c − QG∗ )/(QG∗ − aQG∗ /Q ∗ ) ≤ ξ (c − QG∗ )/(QG∗ − a). From (3), QG∗ = a +(ξ /θ )(c − a). Hence, g1 (x) ≤ θ −ξ ≤ g2 (x) ∀x ∈ [c , Q ∗ (1 +δ)]. Then
∆C (δQ ) ≥
Q ∗ (1+δ)
Q∗
ξ
x − Q∗ Q∗ − a
Q ∗ (1+δQ )
+ Q ∗ (1+δ)
dx
x−c (θ − ξ ) + (1 − θ ) dx b−c
ξ (Q ∗ δ)2 = + Q ∗ (δQ − δ) 2(Q ∗ − a) 1 − θ Q∗ × (2 + δQ + δ) − c + (θ − ξ ) b−c 2 ξ QG∗ δ2 ≥ + QG∗ (δQ − δ)(1 − θ ) 2 1 − a/Q ∗ ∗ Q (δQ + δ) c − Q∗ θ −ξ × − + 2(b − c ) b−c 1−θ ∗ ξ QG c 2 ≥ δ + QG∗ (1 − θ ) 2 c−a QG∗ (δQ2 − δ 2 ) θ −ξ c − QG∗ × + − (δQ − δ) 2(b − c ) 1−θ b−c ∗ ξ QG∗ c 1−θ QG = δ2 + (δQ2 − δ 2 ) 2 c−a ξ b−c θ −ξ c − QG∗ +2 − (δQ − δ) . 1−θ b−c
A. Khanra / Operations Research Letters 45 (2017) 84–89
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Fig. 3. A and B for F ∈ UD a,b,c ,θ .
b
From the proof of Proposition 1, it appears that a lower bound for ∆C (δQ ) can be obtained even when Q ̸∈ [a, b]. Such a bound would have additional term(s), that would vanish when Q ∈ [a, b]. We do not derive that bound because Q ∈ [a, b] is a reasonable assumption.
c {F (x) − G(x)}dx = − a {G(x) − F (x)}dx + {F (x) − G(x)}dx = B − A. c ξ (B − A) if ξ < θ , ⇒ EG [C (QG∗ )] − E [C (Q ∗ )] ≥ −(1 − ξ )(B − A) if ξ ≥ θ .
2.3. Upper bound for the denominator
Fig. 3 depicts A and B. Clearly, min{A : F ∈ UD a,b,c ,θ } = AG = 0 and sup{A : F ∈ UD a,b,c ,θ } = (c − a)θ /2. Then 0 ≤ A < (c − a)θ /2. Note that maximum does not exist for {A : F ∈ UD a,b,c ,θ }. Similarly, 0 ≤ B < (b − c )(1 − θ )/2. Then, −(c − a)θ /2 < B − A < (b−c )(1−θ )/2. Thus, EG [C (QG∗ )]−E [C (Q ∗ )] > −ξ (c −a)θ /2 if ξ < θ , otherwise EG [C (QG∗ )] − E [C (Q ∗ )] > −(1 − ξ )(b − c )(1 − θ )/2.
Proposition 2. For every F ∈ UD a,b,c ,θ , the minimum mismatch cost, E [C (Q ∗ )] <
1 EG [C (QG∗ )] + ξ (c − a)θ
if ξ < θ ,
EG [C (Q ∗ )] + (1 − ξ )(b − c )(1 − θ ) G
if ξ ≥ θ .
2 1 2
Proof. Let us consider an arbitrary F ∈ UD a,b,c ,θ . Observe that
Q∗
E [C (Q ∗ )] = ξ (µ − Q ∗ ) + a F (x)dx is independent of δQ . Let µG denote the mean demand associated with G. Let ξ < θ . By Corollary 1, a < QG∗ ≤ Q ∗ < c. By Lemma 1, F (x) ≤ G(x) ∀x ∈ [a, c ). Since F is strictly increasing, F (x) < ξ ∀x < Q ∗ . Then
Then µG − µ =
b
1 EG [C (QG∗ )] + ξ (c − a)θ if ξ < θ , 2 ∗ ⇒ E [C (Q )] < 1 E [C (QG∗ )] + (1 − ξ )(b − c )(1 − θ ) G 2 if ξ ≥ θ . 2.4. The lower bound
EG [C (QG )] − E [C (Q )] = ξ (µG − µ) + ξ (Q − QG ) ∗
∗
QG∗
+
∗
{G(x) − F (x)}dx −
≥ ξ (µG − µ) +
∗
Q∗
F (x)dx
QG∗
a
QG∗
Combining Propositions 1 and 2, we get the lower bound for
δC (δQ ). After expressing QG∗ and EG [C (QG∗ )] in terms of a, b, c , θ , and ξ , we get the following inequality. Details are provided in Appendix B of the online supplement.
Q∗
{ξ − F (x)}dx ≥ ξ (µG − µ).
Let ξ ≥ θ . By Corollary 1, c ≤ Q ∗ ≤ QG∗ < b. By Lemma 1, F (x) ≥ G(x) ∀x ∈ [c , b]. Since F is strictly increasing, F (x) > ξ ∀x > Q ∗ . Then
δC (δQ ) ≥ k0 {k1 δ 2 + k2 (δQ2 − δ 2 ) + 2k3 |δQ − δ|},
Q∗
G
Q∗ G
{G(x) − F (x)}dx + F (x)dx Q∗ c = ξ (µG − µ) + {G(x) − F (x)}dx +
When ξ < θ k0
a
k1
a QG∗
QG∗
{F (x) − G(x)}dx + {F (x) − ξ }dx c Q∗ c {G(x) − F (x)}dx ≥ ξ (µG − µ) +
k2
−
a b
{F (x) − G(x)}dx = ξ (µG − µ) − (B − A),
− c
c
b where A = a {G(x)− F (x)}dx and B = c {F (x)− G(x)}dx. Note that b b b µ = a xf (x)dx = {bF (b) − aF (a)} − a F (x)dx = b − a F (x)dx.
(4)
where δ = min{δQ , q} if ξ < θ , otherwise δ = max{δQ , q}, and k0 , k1 , k2 , k3 , q are constants. We can express k0 , k1 , k2 , k3 , q in terms of r , m, θ , ξ as follows.
EG [C (QG∗ )] − E [C (Q ∗ )] = ξ (µG − µ) − ξ (QG∗ − Q ∗ )
a
k3 q
r /(1 − r ) + tm
(2 − t )m + (1 − θ )(1 − m) r /(1 − r ) + m m
(1 − θ ){r /(1 − r ) + tm} ξ (1 − m ) (θ − ξ )(θ − m) ξ θ (1 − m) (1 − t )m r /(1 − r ) + tm
When ξ ≥ θ r /(1 − r ) + m
(2 − t )(1 − m) + θ m r /(1 − r ) + m 1−m θ {1/(1 − r ) − t (1 − m)} (1 − ξ )m (ξ − θ )(m − θ) (1 − ξ )(1 − θ )m −(1 − t )(1 − m) 1/(1 − r ) − t (1 − m)
where t = ξ /θ if ξ < θ , otherwise t = (1 − ξ )/(1 − θ ). For symmetric unimodal distributions, m = θ = 1/2. However, if we put these values in (4), the bound that we get is not same as the bound in [7], who studied this special case. In fact, their bound
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A. Khanra / Operations Research Letters 45 (2017) 84–89
Fig. 4. Lower bound of δC (δQ ) for m = 0.5. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).
is stronger. In Appendix C of the online supplement, we derive a stronger bound than (4) for a subclass of UD a,b,c ,θ , where θ = m. When m = 1/2, this subclass becomes the family of symmetric unimodal distributions, and the new bound coincides with the bound in [7]. 3. Demonstration of the lower bound The lower bound of δC (δQ ) in (4) depends only on r , m, θ , and ξ . All of these parameters take values between zero and one. This allows us to construct different scenarios and study behaviour of the lower bound. Fig. 4 demonstrates the lower bound. Each diagram in Fig. 4 corresponds to a combination of r and ξ values. We considered low (0.25), medium (0.5), and high (0.75) values for both. We took m = 0.5 and varied θ around m. θ = m − 0.1 is indicated by red, θ = m is indicated by green, and θ = m + 0.1 is indicated by blue coloured curves. We do not consider scenarios with large difference between m and θ because
such distributions, being highly skewed, are generally unsuitable for demand modelling. For easy comparison, we included cost deviation for the EOQ model in Fig. 4 by the black curve. In the EOQ model, δC (δQ ) = δQ2 /{2(1 + δQ )} [10, pg. 208]. We considered another benchmark, the ±1 slope lines, which are indicated by dotted lines in Fig. 4. They separate the error ‘‘dampening’’ and ‘‘amplifying’’ zones. Since we assumed Q = Q ∗ (1 + δQ ) ∈ [a, b], δQ ∈ [a/Q ∗ − 1, b/Q ∗ − 1] ⊂ [a/b − 1, b/a − 1] = [r − 1, 1/r − 1]. This range shortens as r increases. For high r, i.e., r = 0.75, the range is [−0.25, 0.33]. Note that the actual range of δQ , for ξ = 0.75, is a subset of [−0.25, 0.33]. In Fig. 4, we consider δQ ∈ [−0.2, 0.3] for all values of ξ . We kept the range of δQ same for easy comparison among different diagrams in Fig. 4. If [−0.2, 0.3] ̸⊆ [a/Q ∗ − 1, b/Q ∗ − 1], i.e., the assumption of Q ∈ [a, b] is violated for some δQ ∈ [−0.2, 0.3], and yet we use (4) for computing the lower bound, we commit some error. However, such error is unlikely to be high enough to alter the following observations and the subsequent conclusions.
A. Khanra / Operations Research Letters 45 (2017) 84–89
Key observations (i) The lower bound lies above the EOQ curve in every scenario considered. (ii) For sufficiently large error, the lower bound lies above the ±1 slope lines. This threshold error decreases in r, and nears zero at r = 0.75. (iii) The lower bound increases as r increases, and its impact on the lower bound is much stronger than that of ξ and θ . Figures D.2 and D.3 in the online supplement demonstrate the lower bound corresponding to m = 0.35 and m = 0.65 respectively. The above observations are valid for those cases as well. A comparison of Fig. 4, D.2, and D.3 reveals that r dominates m in influencing the lower bound, just like it dominates ξ and θ . These observations are in sync with [7], who studied the special case of symmetric unimodal demand.
89
deviation increases in b, the upper limit of demand. Hence, we can say that the newsvendor model becomes more sensitive as r increases, unless b is small. Then the large value of δC (δQ ) at high r is influenced by small E [C (Q ∗ )]. A demonstration of the behaviour of ∆C (δQ ) is presented in Appendix E of the online supplement. So far, we studied the lower bound of cost deviation. The actual cost deviation, for a given unimodal demand distribution, can be much larger. Furthermore, our bound is not tight. Hence, we can conclude that the newsvendor model is sensitive to suboptimal decisions, when the demand distribution is unimodal. One can study robustness of the newsvendor model for nonunimodal distributions; however, such distributions are not suitable for demand modelling. Further research can be carried out to study how parameter estimation error translates into suboptimal decisions in the newsvendor model. Identifying the most important parameter(s) is another important research direction.
4. Conclusions Acknowledgements It is evident that the newsvendor model is more sensitive to suboptimal order quantities than the EOQ model. Stochastic (r , Q ) and (s, S ) models are even more robust than the EOQ model [14, 2]. This differing behaviour of the newsvendor model can be attributed to the absence of fixed cost. If we consider a fixed cost of K units per cycle in the newsvendor model, while co + cu = 1, then δC (δQ ) = ∆C (δQ )/{K + E [C (Q ∗ )]}. Generally K is large enough to significantly reduce δC (δQ ). Note that the newsvendor model with fixed cost is a case of stochastic (s, S ) model, which is quite robust. Among different factors, the ratio of lower and upper limits of demand (r) is most influential. Robustness of the newsvendor model deteriorates with increase in r. In fact, amplification of error occurs, i.e., δC (δQ ) > δQ , in scenarios with high values of r. This behaviour of the lower bound can be explained by the increase in absolute deviation. As r increases, the demand range, b − a = b(1 − r ) decreases, and the demand density function Q Q ‘‘rises’’. Then ∆C (δQ ) = Q ∗ {F (x) − ξ }dx = Q ∗ (Q − x)f (x)dx increases. However, a decreased demand range ‘‘narrows’’ the gap among a, Q ∗ , and b, which decreases the minimum mismatch cost,
Q∗
Q∗
E [C (Q ∗ )] = ξ (µ − Q ∗ ) + −∞ F (x)dx = (1 − ξ ) a F (x)dx + b ξ Q ∗ {1 − F (x)}dx. To understand what is increasing δC (δQ ) = ∆C (δQ )/E [C (Q ∗ )] as r increases, we can study the lower bound of absolute deviation, given by Proposition 1. Using the ratios, r and m, we can write Proposition 1 as follows. If Q ∈ [a, b], for every F ∈ UD a,b,c ,θ ,
ξ {r + (ξ /θ )m(1 − r )}b {k1 δ 2 + k2 (δQ2 − δ 2 ) 2 +2k (δ − δ)} if ξ < θ 3 Q ∆C (δQ ) ≥ ( 1 − ξ ){r + m(1 − r )}b {k1 δ 2 + k2 (δQ2 − δ 2 ) 2 +2k3 (δ − δQ )} if ξ ≥ θ ,
(5)
where δ = min{δQ , q} if ξ < θ , otherwise δ = max{δQ , q}, and k1 , k2 , k3 , q are constants as expressed in (4). Derivation of (5) is very similar to that of (4). It is evident from (5) that absolute
The author is grateful to the anonymous reviewer who gave insightful suggestions, and helped considerably in improving the manuscript. Appendix A. Supplementary material Supplementary material related to this article can be found online at http://dx.doi.org/10.1016/j.orl.2016.12.007. References [1] K.J. Arrow, T. Harris, J. Marschak, Optimal inventory policy, Econometrica 19 (3) (1951) 250–272. [2] F. Chen, Y.-S. Zheng, Sensitivity analysis of an (s, S ) inventory model, Oper. Res. Lett. 21 (1) (1997) 19–23. [3] T.-M. Choi (Ed.), Handbook of Newsvendor Problems, first ed., Springer, Hong Kong, 2012. [4] L. Eeckhoudt, C. Gollier, H. Schlesinger, The risk-averse (and prudent) newsboy, Manage. Sci. 41 (5) (1995) 786–794. [5] Y. Gerchak, D. Mossman, On the effect of demand randomness on inventories and costs, Oper. Res. 40 (4) (1992) 804–807. [6] B.V. Gnedenko, A.N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables (English translation K. L. Chung), Addison-Wesley Publishing Company, Massachusetts, 1968. [7] A. Khanra, C. Soman, T. Bandyopadhyay, Sensitivity analysis of the newsvendor model, European J. Oper. Res. 239 (2) (2014) 403–412. [8] M. Khouja, The single-period (news-vendor) problem: Literature review and suggestions for future research, Omega 27 (5) (1999) 537–553. [9] A.H.-L. Lau, H.-S. Lau, The effects of reducing demand uncertainty in a manufacturer-retailer channel for single-period products, Comput. Oper. Res. 29 (11) (2002) 1583–1602. [10] S. Nahmias, Production and Operations Analysis, sixth ed., McGraw-Hill, Boston, 2008. [11] Y. Qin, R. Wang, A.J. Vakharia, Y. Chen, M.M.H. Seref, The newsvendor problem: Review and directions for future research, European J. Oper. Res. 213 (2) (2011) 361–374. [12] E.A. Silver, D.F. Pyke, R. Peterson, Inventory Management and Production Planning and Scheduling, third ed., John Wiley & Sons, New York, 1998. [13] T. Wang, Sensitivity analysis for newsvendor model with general advertisement-sensitive demand, Procedia Eng. 15 (2011) 1209–1213. [14] Y.-S. Zheng, On properties of stochastic inventory systems, Manage. Sci. 38 (1) (1992) 87–103.
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Operations Research Letters journal homepage: www.elsevier.com/locate/orl
On the impact of suboptimal decisions in the newsvendor model Avijit Khanra Indian Institute of Technology, Kanpur 208016, India
article
abstract
info
Article history: Received 19 May 2016 Received in revised form 18 September 2016 Accepted 11 December 2016 Available online 24 December 2016
We study the impact of suboptimal decisions in the newsvendor model, one of the popular inventory models. We establish a lower bound for the deviation of inventory cost from its minimum, when the order quantity is suboptimal. Demonstration of the bound shows the model to be sensitive to suboptimal decisions. © 2016 Elsevier B.V. All rights reserved.
Keywords: Inventory Newsvendor model Robustness and sensitivity analysis
1. Introduction The newsvendor problem is about stocking decision of a product with uncertain demand, where mismatch between demand and supply attracts penalty. This classic inventory problem was first addressed by Arrow et al. [1]. Owing to its wide applicability, the newsvendor problem attracted attentions of many scholars over the past six decades. Review of their works can be found in Khouja [8], Qin et al. [11], and Choi [3]. Simplest case of the newsvendor problem, known as the classical newsvendor problem, arises when we make certain simplifying assumptions about the demand and supply processes. Key assumptions are: (i) exogenous demand with known distribution, (ii) single procurement of any amount, and (iii) linearity of cost components. See Chapter 10 of Silver et al. [12] for the details. Let X denote the stochastic demand with distribution function F . Let co and cu denote the unit over-stocking and under-stocking costs. Let Q denote the order quantity. Then demand–supply mismatch cost, C (Q , X ) and its expected value are given by C (Q , X ) = co max{0, Q − X } + cu max{0, X − Q } E [C (Q )] =
Q
co (Q − x)dF (x) +
−∞
∞
cu (x − Q )dF (x) Q
= (co + cu ) ξ (µ − Q ) +
Q
F (x)dx
(1)
−∞
where ξ = cu /(co + cu ) is referred to as the critical fractile and µ is the mean demand. E [C (Q )] is convex in Q and the optimal solution is given by: F (Q ∗ ) = ξ .
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.orl.2016.12.007 0167-6377/© 2016 Elsevier B.V. All rights reserved.
Decision making in the classical newsvendor model requires the knowledge of demand distribution and cost parameters. When estimates of these parameters deviate from their true values, the actual decision (derived using the estimates) deviates from the optimum (calculated using the true values). Due to convex nature of the objective function, any deviation from the optimal decision increases cost. In this context, sensitivity analysis is performed to understand the impacts of (i) suboptimal decisions on expected cost and (ii) parameter estimation error on stocking decision. It shall be noted that the second question is of practical relevance when the impact of suboptimal decisions on expected cost is significant. Some of the popular inventory models, e.g., economic order quantity (EOQ), stochastic (r , Q ), and stochastic (s, S ) models have been found to be insensitive to suboptimal decisions [10,14,2]. In this sense, we should investigate question-(i) first. In the newsvendor literature, there are some papers (e.g., [5,4, 9,13]) that, to some extent, address question-(ii), i.e., the impact of parameter estimation error on stocking decision. However, question-(i), which is our focus, remains largely unanswered with the exception of one paper. In a recent article, Khanra et al. [7] established a lower bound for cost deviation, i.e., the deviation of expected cost from its minimum value. Demonstration of the lower bound showed the newsvendor model to be sensitive to suboptimal decisions. In a number of scenarios, cost deviation exceeded order quantity deviation, i.e., the deviation of order quantity from its optimal value. This behaviour of the newsvendor model is opposite to that of the EOQ, (r , Q ), and (s, S ) models. Khanra et al. [7] assumed the demand to follow symmetric unimodal distribution. In this paper, we study robustness of the newsvendor model to suboptimal decisions when the demand distribution is not necessarily symmetric. In particular, we establish a new lower bound for cost deviation when the demand follows general unimodal distribution. Demonstration of the
A. Khanra / Operations Research Letters 45 (2017) 84–89
(a) θ ̸= m.
85
(b) θ = m. Fig. 1. G ∈ UD a,b,c ,θ .
lower bound establishes sensitivity of the newsvendor model to suboptimal decisions. 2. New lower bound for cost deviation We need to decide measures for cost and order quantity deviations. We choose relative measure, i.e., cost deviation is measured by δC = (E [C (Q )] − E [C (Q ∗ )])/E [C (Q ∗ )], and order quantity deviation is measured by δQ = (Q − Q ∗ )/Q ∗ . These measures are unit-less fractions, hence, easy to compare. Using (1), we can express δC as a function of δQ as follows.
Q ∗ (1+δQ ) δC (δQ ) =
Q∗
ξ (µ −
Q ∗)
{F (x) − ξ }dx . Q∗ + −∞ F (x)dx
(2)
Let us adjust the unit of cost so that co + cu = 1. Then the numerator of δC (δQ ) in (2) is the absolute deviation of cost, denoted by ∆C (δQ ), and the denominator is the minimum mismatch cost, E [C (Q ∗ )]. We establish the lower bound for δC (δQ ) by combining a lower bound of ∆C (δQ ) and an upper bound of E [C (Q ∗ )]. To obtain these bounds for unimodal demand, first we need to characterize such distributions. 2.1. Unimodal demand distributions We call a distribution F to be unimodal if there exists c ∈ R such that F is convex in (−∞, c ] and concave in [c , ∞) [6]. Since we are dealing with demand distributions, we can safely assume a bounded support for the distribution. Let us denote the family of unimodal distributions with support [a, b], mode c, and F (c ) = θ by UD a,b,c ,θ . Let r = a/b denote ratio of the demand limits and m = (c − a)/(b − a) denote location of the mode. Non-negativity of demand ensures a ≥ 0. We assume a < c < b and strict monotony of F in [a, b]. Then r ∈ [0, 1), m ∈ (0, 1), and θ ∈ (0, 1). Note that every unimodal demand distribution can be ‘‘covered’’ by varying r, m, and θ in their respective ranges. Similarly, every cost structure (i.e., co and cu values) can be covered, if we vary ξ in (0, 1). In order to derive a lower bound for δC (δQ ), first we need to bound F itself. Given a, b, c, and θ , let us define G ∈ UD a,b,c ,θ as follows.
x − a θ c−a G(x) = 1 − b − x (1 − θ ) b−c
if x ∈ [a, c ) (3) if x ∈ [c , b].
See Fig. 1 for a graphical depiction of G. When θ = m, G becomes the uniform distribution in [a, b]. We use QG∗ to denote the optimal order quantity when the demand distribution is G. Then EG [C (QG∗ )] denotes the minimum mismatch cost when G is the demand distribution. Using G, Lemma 1 offers a partial bound for every F ∈ UD a,b,c ,θ . Lemma 1. F (x) ≤ G(x) if x < c and F (x) ≥ G(x) if x ≥ c for every F ∈ UD a,b,c ,θ . Proof. We need to focus only on x ∈ (a, b) as G(x) = F (x) = 0 ∀x ≤ a and G(x) = F (x) = 1 ∀x ≥ b. Due to convexity of F in [a, c ], F (λa + (1 − λ)c ) ≤ λF (a) + (1 − λ)F (c ) = (1 − λ)θ ∀λ ∈ (0, 1). Replacing λa + (1 − λ)c by x, F (x) ≤ {(x − a)/(c − a)}θ = G(x) ∀x ∈ (a, c ). Similarly, due to concavity of F in [c , b], F (λc +(1 −λ)b) ≥ λF (c )+(1 −λ)F (b) = 1 −λ(1 −θ) ∀λ ∈ [0, 1). Replacing λc + (1 − λ)b by x, F (x) ≥ 1 − {(b − x)/(b − c )}(1 − θ ) = G(x) ∀x ∈ [c , b). Lemma 1 has the following consequence for Q ∗ , the optimal decision for F . Corollary 1. Q ∗ ≥ QG∗ if ξ < θ and Q ∗ ≤ QG∗ if ξ ≥ θ for every F ∈ UD a,b,c ,θ . Proof. If ξ < θ , Q ∗ < c. Then by Lemma 1, F (Q ∗ ) ≤ G(Q ∗ ). If, by contradiction, Q ∗ < QG∗ for some ξ < θ , ξ = F (Q ∗ ) ≤ G(Q ∗ ) < G(QG∗ ) = ξ , which is impossible. The strict inequality is due to strict monotony of G in [a, b]. Hence, Q ∗ ≥ QG∗ if ξ < θ . Similarly, if ξ ≥ θ , QG∗ ≥ c. Then by Lemma 1, F (QG∗ ) ≥ G(QG∗ ). Again, by contradiction, if Q ∗ > QG∗ for some ξ ≥ θ , ξ = G(QG∗ ) ≤ F (QG∗ ) < F (Q ∗ ) = ξ , which is impossible. The strict inequality is due to strict monotony of F in [a, b], which we assumed earlier. Hence, Q ∗ ≤ QG∗ if ξ ≥ θ . Lemma 1 and Corollary 1 are useful in establishing the lower bound for δC (δQ ). 2.2. Lower bound for the numerator Let us assume that the suboptimal decision, Q ∈ [a, b]. If a′ and b′ denote the observed lowest and highest demand, then the newsvendor is unlikely to order a quantity that is outside [a′ , b′ ]. Since [a′ , b′ ] ⊆ [a, b], we can assume that Q ∈ [a, b].
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A. Khanra / Operations Research Letters 45 (2017) 84–89
θ
θ
θ
ξ
ξ
ξ
0
a
Q
0
c
(a) δQ < 0.
a
Q
0
c
(b) 0 < δQ ≤ c /Q ∗ − 1.
a
c
Q
(c) δQ > c /Q ∗ − 1.
Fig. 2. ∆C (δQ ) when ξ < θ .
Proposition 1. For every F ∈ UD a,b,c ,θ , the absolute deviation of cost,
∗ ξQ∗ c 1−θ QG G 2 δ + (δ 2 − δ 2 ) 2 c − a ξ b − c Q c − QG∗ θ −ξ − (δQ − δ) if ξ < θ , +2 1−θ b−c ∆C (δQ ) ≥ ( 1 − ξ )c c θ QG∗ 2 δ + (δQ2 − δ 2 ) 2 b−c 1 −ξ c − a ∗ + 2 ξ − θ − QG − c (δ − δ ) if ξ ≥ θ , Q θ c−a where δ = min{δQ , c /QG∗ − 1} if ξ max{δQ , c /QG∗ − 1}.
< θ , otherwise δ =
Proof. If δQ = 0, δ = δQ in both the cases. Then the claim is that
Q ∆C (δQ ) ≥ 0, which is true as ∆C (δQ ) = Q ∗ {F (x) − ξ }dx = 0. We deal with the δQ ̸= 0 cases in two parts: (i) ξ < θ and (ii) ξ ≥ θ . Here, we present a proof for the first case. Proof for the second case is similar, and is provided in Appendix A of the online supplement. Hashed area in Fig. 2 exhibits ∆C (δQ ) when ξ < θ . By Corollary 1, a < QG∗ ≤ Q ∗ < c. The proof works differently when (a) δQ < 0, (b) 0 < δQ ≤ c /Q ∗ − 1, and (c) δQ > c /Q ∗ − 1. The first scenario is depicted in Fig. 2(a). Here, a ≤ Q < Q ∗ . Due to convexity of F in [a, Q ∗ ] ⊂ [a, c ], F (λa + (1 − λ)Q ∗ ) ≤ λF (a) + (1 − λ)F (Q ∗ ) = (1 − λ)ξ ∀λ ∈ [0, 1]. Replacing λa + (1 − λ)Q ∗ by x, ξ − F (x) ≥ λξ = ξ (Q ∗ − x)/(Q ∗ − a) ∀x ∈ [a, Q ∗ ] ⊇ [Q , Q ∗ ]. Then
∆C (δQ ) =
Q∗
{ξ − F (x)}dx ≥ Q
=
Q∗ Q ∗ (1+δQ )
ξ
Q∗ − x Q∗ − a
dx
δQ2 ξ (Q ∗ δQ )2 ξ QG∗ ξ QG∗ c 2 ≥ ≥ δQ . ∗ ∗ 2(Q − a) 2 1 − a/ Q 2 c−a
Here, δ = min{δQ , c /QG∗ − 1} = δQ . So the above inequality is same as the claim. The second scenario is depicted in Fig. 2(b). Here, Q ∗ < Q ≤ c. Due to unimodality of F , its density, f is increasing x in [a, c ]. Then ∀x ∈ [Q ∗ , Q ] ⊂ [a, c ], F (x) = F (Q ∗ ) + Q ∗ f (t )dt ≥
Q∗ ξ + f (Q ∗ )(x − Q ∗ ). For the same reason, F (Q ∗ ) = a f (t )dt ≤ f (Q ∗ )(Q ∗ − a) ⇒ f (Q ∗ ) ≥ ξ /(Q ∗ − a). Hence, ∀x ∈ [Q ∗ , Q ], F (x) − ξ ≥ ξ (x − Q ∗ )/(Q ∗ − a). Then Q Q ∗ (1+δQ ) x − Q∗ ∆C (δQ ) = {F (x) − ξ }dx ≥ ξ ∗ dx Q −a Q∗ Q∗ ξ QG∗ c 2 ≥ δQ , 2 c−a
just like the first scenario. Here too, δ = δQ . Hence the claim is valid. The third scenario is depicted in Fig. 2(c). Here, Q ∗ < c < Q ≤ b. Then
∆C (δQ ) =
c
Q∗
{F (x) − ξ }dx +
Q
{F (x) − ξ }dx.
c
Following arguments in the second scenario, F (x) − ξ ≥ ξ (x − Q ∗ )/(Q ∗ − a) ∀x ∈ [Q ∗ , c ]. By Lemma 1, F (x) − ξ ≥ G(x) − ξ = (θ − ξ ) + (1 − θ )(x − c )/(b − c ) ∀x ∈ [c , b] ⊇ [c , Q ]. So
∆C (δQ ) ≥
c
Q∗
ξ
x − Q∗ Q∗ − a
Q
dx + c
x−c (θ − ξ ) + (1 − θ ) dx. b−c
We need to replace c suitably, otherwise the bound cannot be computed without knowing F . Let g1 denote the first integrand and g2 denote the second. Note that δ = min{δQ , c /QG∗ − 1} ≥ min{δQ , c /Q ∗ − 1} = c /Q ∗ − 1. Then Q ∗ < c ≤ Q ∗ (1 + δ) ≤ Q . Since g1 (x) ↑ x, for every x ∈ [c , Q ∗ (1 + δ)], g1 (x) ≤ g1 (Q ∗ (1 + δ)) ≤ g1 (cQ ∗ /QG∗ ) as δ ≤ c /QG∗ − 1. Now g1 (cQ ∗ /QG∗ ) = ξ (c − QG∗ )/(QG∗ − aQG∗ /Q ∗ ) ≤ ξ (c − QG∗ )/(QG∗ − a). From (3), QG∗ = a +(ξ /θ )(c − a). Hence, g1 (x) ≤ θ −ξ ≤ g2 (x) ∀x ∈ [c , Q ∗ (1 +δ)]. Then
∆C (δQ ) ≥
Q ∗ (1+δ)
Q∗
ξ
x − Q∗ Q∗ − a
Q ∗ (1+δQ )
+ Q ∗ (1+δ)
dx
x−c (θ − ξ ) + (1 − θ ) dx b−c
ξ (Q ∗ δ)2 = + Q ∗ (δQ − δ) 2(Q ∗ − a) 1 − θ Q∗ × (2 + δQ + δ) − c + (θ − ξ ) b−c 2 ξ QG∗ δ2 ≥ + QG∗ (δQ − δ)(1 − θ ) 2 1 − a/Q ∗ ∗ Q (δQ + δ) c − Q∗ θ −ξ × − + 2(b − c ) b−c 1−θ ∗ ξ QG c 2 ≥ δ + QG∗ (1 − θ ) 2 c−a QG∗ (δQ2 − δ 2 ) θ −ξ c − QG∗ × + − (δQ − δ) 2(b − c ) 1−θ b−c ∗ ξ QG∗ c 1−θ QG = δ2 + (δQ2 − δ 2 ) 2 c−a ξ b−c θ −ξ c − QG∗ +2 − (δQ − δ) . 1−θ b−c
A. Khanra / Operations Research Letters 45 (2017) 84–89
87
Fig. 3. A and B for F ∈ UD a,b,c ,θ .
b
From the proof of Proposition 1, it appears that a lower bound for ∆C (δQ ) can be obtained even when Q ̸∈ [a, b]. Such a bound would have additional term(s), that would vanish when Q ∈ [a, b]. We do not derive that bound because Q ∈ [a, b] is a reasonable assumption.
c {F (x) − G(x)}dx = − a {G(x) − F (x)}dx + {F (x) − G(x)}dx = B − A. c ξ (B − A) if ξ < θ , ⇒ EG [C (QG∗ )] − E [C (Q ∗ )] ≥ −(1 − ξ )(B − A) if ξ ≥ θ .
2.3. Upper bound for the denominator
Fig. 3 depicts A and B. Clearly, min{A : F ∈ UD a,b,c ,θ } = AG = 0 and sup{A : F ∈ UD a,b,c ,θ } = (c − a)θ /2. Then 0 ≤ A < (c − a)θ /2. Note that maximum does not exist for {A : F ∈ UD a,b,c ,θ }. Similarly, 0 ≤ B < (b − c )(1 − θ )/2. Then, −(c − a)θ /2 < B − A < (b−c )(1−θ )/2. Thus, EG [C (QG∗ )]−E [C (Q ∗ )] > −ξ (c −a)θ /2 if ξ < θ , otherwise EG [C (QG∗ )] − E [C (Q ∗ )] > −(1 − ξ )(b − c )(1 − θ )/2.
Proposition 2. For every F ∈ UD a,b,c ,θ , the minimum mismatch cost, E [C (Q ∗ )] <
1 EG [C (QG∗ )] + ξ (c − a)θ
if ξ < θ ,
EG [C (Q ∗ )] + (1 − ξ )(b − c )(1 − θ ) G
if ξ ≥ θ .
2 1 2
Proof. Let us consider an arbitrary F ∈ UD a,b,c ,θ . Observe that
Q∗
E [C (Q ∗ )] = ξ (µ − Q ∗ ) + a F (x)dx is independent of δQ . Let µG denote the mean demand associated with G. Let ξ < θ . By Corollary 1, a < QG∗ ≤ Q ∗ < c. By Lemma 1, F (x) ≤ G(x) ∀x ∈ [a, c ). Since F is strictly increasing, F (x) < ξ ∀x < Q ∗ . Then
Then µG − µ =
b
1 EG [C (QG∗ )] + ξ (c − a)θ if ξ < θ , 2 ∗ ⇒ E [C (Q )] < 1 E [C (QG∗ )] + (1 − ξ )(b − c )(1 − θ ) G 2 if ξ ≥ θ . 2.4. The lower bound
EG [C (QG )] − E [C (Q )] = ξ (µG − µ) + ξ (Q − QG ) ∗
∗
QG∗
+
∗
{G(x) − F (x)}dx −
≥ ξ (µG − µ) +
∗
Q∗
F (x)dx
QG∗
a
QG∗
Combining Propositions 1 and 2, we get the lower bound for
δC (δQ ). After expressing QG∗ and EG [C (QG∗ )] in terms of a, b, c , θ , and ξ , we get the following inequality. Details are provided in Appendix B of the online supplement.
Q∗
{ξ − F (x)}dx ≥ ξ (µG − µ).
Let ξ ≥ θ . By Corollary 1, c ≤ Q ∗ ≤ QG∗ < b. By Lemma 1, F (x) ≥ G(x) ∀x ∈ [c , b]. Since F is strictly increasing, F (x) > ξ ∀x > Q ∗ . Then
δC (δQ ) ≥ k0 {k1 δ 2 + k2 (δQ2 − δ 2 ) + 2k3 |δQ − δ|},
Q∗
G
Q∗ G
{G(x) − F (x)}dx + F (x)dx Q∗ c = ξ (µG − µ) + {G(x) − F (x)}dx +
When ξ < θ k0
a
k1
a QG∗
QG∗
{F (x) − G(x)}dx + {F (x) − ξ }dx c Q∗ c {G(x) − F (x)}dx ≥ ξ (µG − µ) +
k2
−
a b
{F (x) − G(x)}dx = ξ (µG − µ) − (B − A),
− c
c
b where A = a {G(x)− F (x)}dx and B = c {F (x)− G(x)}dx. Note that b b b µ = a xf (x)dx = {bF (b) − aF (a)} − a F (x)dx = b − a F (x)dx.
(4)
where δ = min{δQ , q} if ξ < θ , otherwise δ = max{δQ , q}, and k0 , k1 , k2 , k3 , q are constants. We can express k0 , k1 , k2 , k3 , q in terms of r , m, θ , ξ as follows.
EG [C (QG∗ )] − E [C (Q ∗ )] = ξ (µG − µ) − ξ (QG∗ − Q ∗ )
a
k3 q
r /(1 − r ) + tm
(2 − t )m + (1 − θ )(1 − m) r /(1 − r ) + m m
(1 − θ ){r /(1 − r ) + tm} ξ (1 − m ) (θ − ξ )(θ − m) ξ θ (1 − m) (1 − t )m r /(1 − r ) + tm
When ξ ≥ θ r /(1 − r ) + m
(2 − t )(1 − m) + θ m r /(1 − r ) + m 1−m θ {1/(1 − r ) − t (1 − m)} (1 − ξ )m (ξ − θ )(m − θ) (1 − ξ )(1 − θ )m −(1 − t )(1 − m) 1/(1 − r ) − t (1 − m)
where t = ξ /θ if ξ < θ , otherwise t = (1 − ξ )/(1 − θ ). For symmetric unimodal distributions, m = θ = 1/2. However, if we put these values in (4), the bound that we get is not same as the bound in [7], who studied this special case. In fact, their bound
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A. Khanra / Operations Research Letters 45 (2017) 84–89
Fig. 4. Lower bound of δC (δQ ) for m = 0.5. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).
is stronger. In Appendix C of the online supplement, we derive a stronger bound than (4) for a subclass of UD a,b,c ,θ , where θ = m. When m = 1/2, this subclass becomes the family of symmetric unimodal distributions, and the new bound coincides with the bound in [7]. 3. Demonstration of the lower bound The lower bound of δC (δQ ) in (4) depends only on r , m, θ , and ξ . All of these parameters take values between zero and one. This allows us to construct different scenarios and study behaviour of the lower bound. Fig. 4 demonstrates the lower bound. Each diagram in Fig. 4 corresponds to a combination of r and ξ values. We considered low (0.25), medium (0.5), and high (0.75) values for both. We took m = 0.5 and varied θ around m. θ = m − 0.1 is indicated by red, θ = m is indicated by green, and θ = m + 0.1 is indicated by blue coloured curves. We do not consider scenarios with large difference between m and θ because
such distributions, being highly skewed, are generally unsuitable for demand modelling. For easy comparison, we included cost deviation for the EOQ model in Fig. 4 by the black curve. In the EOQ model, δC (δQ ) = δQ2 /{2(1 + δQ )} [10, pg. 208]. We considered another benchmark, the ±1 slope lines, which are indicated by dotted lines in Fig. 4. They separate the error ‘‘dampening’’ and ‘‘amplifying’’ zones. Since we assumed Q = Q ∗ (1 + δQ ) ∈ [a, b], δQ ∈ [a/Q ∗ − 1, b/Q ∗ − 1] ⊂ [a/b − 1, b/a − 1] = [r − 1, 1/r − 1]. This range shortens as r increases. For high r, i.e., r = 0.75, the range is [−0.25, 0.33]. Note that the actual range of δQ , for ξ = 0.75, is a subset of [−0.25, 0.33]. In Fig. 4, we consider δQ ∈ [−0.2, 0.3] for all values of ξ . We kept the range of δQ same for easy comparison among different diagrams in Fig. 4. If [−0.2, 0.3] ̸⊆ [a/Q ∗ − 1, b/Q ∗ − 1], i.e., the assumption of Q ∈ [a, b] is violated for some δQ ∈ [−0.2, 0.3], and yet we use (4) for computing the lower bound, we commit some error. However, such error is unlikely to be high enough to alter the following observations and the subsequent conclusions.
A. Khanra / Operations Research Letters 45 (2017) 84–89
Key observations (i) The lower bound lies above the EOQ curve in every scenario considered. (ii) For sufficiently large error, the lower bound lies above the ±1 slope lines. This threshold error decreases in r, and nears zero at r = 0.75. (iii) The lower bound increases as r increases, and its impact on the lower bound is much stronger than that of ξ and θ . Figures D.2 and D.3 in the online supplement demonstrate the lower bound corresponding to m = 0.35 and m = 0.65 respectively. The above observations are valid for those cases as well. A comparison of Fig. 4, D.2, and D.3 reveals that r dominates m in influencing the lower bound, just like it dominates ξ and θ . These observations are in sync with [7], who studied the special case of symmetric unimodal demand.
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deviation increases in b, the upper limit of demand. Hence, we can say that the newsvendor model becomes more sensitive as r increases, unless b is small. Then the large value of δC (δQ ) at high r is influenced by small E [C (Q ∗ )]. A demonstration of the behaviour of ∆C (δQ ) is presented in Appendix E of the online supplement. So far, we studied the lower bound of cost deviation. The actual cost deviation, for a given unimodal demand distribution, can be much larger. Furthermore, our bound is not tight. Hence, we can conclude that the newsvendor model is sensitive to suboptimal decisions, when the demand distribution is unimodal. One can study robustness of the newsvendor model for nonunimodal distributions; however, such distributions are not suitable for demand modelling. Further research can be carried out to study how parameter estimation error translates into suboptimal decisions in the newsvendor model. Identifying the most important parameter(s) is another important research direction.
4. Conclusions Acknowledgements It is evident that the newsvendor model is more sensitive to suboptimal order quantities than the EOQ model. Stochastic (r , Q ) and (s, S ) models are even more robust than the EOQ model [14, 2]. This differing behaviour of the newsvendor model can be attributed to the absence of fixed cost. If we consider a fixed cost of K units per cycle in the newsvendor model, while co + cu = 1, then δC (δQ ) = ∆C (δQ )/{K + E [C (Q ∗ )]}. Generally K is large enough to significantly reduce δC (δQ ). Note that the newsvendor model with fixed cost is a case of stochastic (s, S ) model, which is quite robust. Among different factors, the ratio of lower and upper limits of demand (r) is most influential. Robustness of the newsvendor model deteriorates with increase in r. In fact, amplification of error occurs, i.e., δC (δQ ) > δQ , in scenarios with high values of r. This behaviour of the lower bound can be explained by the increase in absolute deviation. As r increases, the demand range, b − a = b(1 − r ) decreases, and the demand density function Q Q ‘‘rises’’. Then ∆C (δQ ) = Q ∗ {F (x) − ξ }dx = Q ∗ (Q − x)f (x)dx increases. However, a decreased demand range ‘‘narrows’’ the gap among a, Q ∗ , and b, which decreases the minimum mismatch cost,
Q∗
Q∗
E [C (Q ∗ )] = ξ (µ − Q ∗ ) + −∞ F (x)dx = (1 − ξ ) a F (x)dx + b ξ Q ∗ {1 − F (x)}dx. To understand what is increasing δC (δQ ) = ∆C (δQ )/E [C (Q ∗ )] as r increases, we can study the lower bound of absolute deviation, given by Proposition 1. Using the ratios, r and m, we can write Proposition 1 as follows. If Q ∈ [a, b], for every F ∈ UD a,b,c ,θ ,
ξ {r + (ξ /θ )m(1 − r )}b {k1 δ 2 + k2 (δQ2 − δ 2 ) 2 +2k (δ − δ)} if ξ < θ 3 Q ∆C (δQ ) ≥ ( 1 − ξ ){r + m(1 − r )}b {k1 δ 2 + k2 (δQ2 − δ 2 ) 2 +2k3 (δ − δQ )} if ξ ≥ θ ,
(5)
where δ = min{δQ , q} if ξ < θ , otherwise δ = max{δQ , q}, and k1 , k2 , k3 , q are constants as expressed in (4). Derivation of (5) is very similar to that of (4). It is evident from (5) that absolute
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