Jena Research Paers in Business and Economics A newsvendor model with service and loss constraints Werner Jammernegg und Peter Kischka 21/2008 Jenaer Schriften zur Wirtschaftswissenschaft Working and Discussion Paer Series School of Economics and Business Administration Friedrich-Schiller-University Jena ISSN 1864-3108 Publisher: Wirtschaftswissenschaftliche Fakultät Friedrich-Schiller-Universität Jena Carl-Zeiß-Str. 3, D-07743 Jena www.jbe.uni-jena.de Editor: Prof. Dr. Hans-Walter Lorenz h.w.lorenz@wiwi.uni-jena.de Prof. Dr. Armin Scholl armin.scholl@wiwi.uni-jena.de www.jbe.uni-jena.de
A newsvendor model with service and loss constraints Werner Jammernegg 1) Peter Kischka 2) 1) Vienna University of Economics and Business Institute of Production Management Nordbergstraße 15, 1090 Vienna, Austria E-Mail: werner.jammernegg@wu-wien.ac.at 2) Friedrich Schiller University Jena Deartment of Business Statistics Carl-Zeiß-Str. 3, 07743 Jena, Germany E-Mail:.kischka@wiwi.uni-jena.de Abstract Actual erformance measurement systems do not only consider financial measures like costs and rofits but also non-financial indicators with resect customer service, quality and flexibility. Using the newsvendor model we exlore the influence of ossibly conflicting erformance measures on imortant oerations decisions like the order quantity and the selling rice of a roduct. For rice-indeendent as well as rice-deendent demand distribution like in the classical newsvendor model the objective is to maximise the exected rofit. But the otimal decisions are comuted with resect to a service constraint a lower bound for the level of roduct availability and to a loss constraint an uer bound for the robability of resulting in loss. For the rice-indeendent model a condition for the existence of an otimal order quantity and its structure is resented. For the rice-setting newsvendor the admissible region of the order quantity and the selling rice is characterised for the additive and the multilicative model. Furthermore, it is shown that higher variability of demand leads to a smaller admissible region of the decision variables thereby easing the comutation of the otimal decisions. Keywords: Constrained Newsvendor Model, Price-Setting Newsvendor JEL Classification: C 44, M 11
2 1 Introduction The erformance of suly oerations should not be evaluated just by a single measure. Using multile erformance measures often results in tradeoffs like the one between inventory costs and level of roduct availability. Also erformance measurement systems try to reflect the interests of all stakeholders of a comany. E.g. the balanced scorecard system considers not only the financial ersective mainly of interest to the managers and owners of a comany but also the customer, sulier and emloyee ersectives. The like the to-level erformance measures of the suly chain oerations reference (SCOR) model are categorised into internal (costs, assets) and external (reliability, resonsiveness, flexibility) measures (see www. suly-chain.org). In this aer we use the newsvendor model to exlore the influence of ossibly conflicting erformance measures on imortant oerational measures like the order quantity and/or the selling rice of a roduct. The newsvendor model is one of the fundamental oerations models that is rich enough to gain imortant managerial insights (see Porteus (2008)). In the classical model for fixed rice the order quantity for a selling season is based on the maximisation of the exected rofit. It turns out that the otimal quantity solves the tradeoff resulting from the costs from ordering too many and the costs from ordering too few units of the roduct. The so-called critical ratio reresents the rofit value of the roduct and describes the cycle service level (in-stock robability) related to the otimal exected rofit. Therefore, the order quantity for high-rofit roducts is high and that of low-rofit roducts is low. But these findings may not be desirable for high-rofit and low-rofit roducts in general as is indicated by exerimental and emirical studies (see e.g. Schweitzer/Cachon (2000), Corbett/Fransoo (2007)). There exists a large body of literature concerning extensions of the classical newsvendor model (see e.g. Khouja (1999)). Instead of maximising the exected rofit there is a long history to maximise instead the robability of exceeding a secified minimum rofit (see e.g. Lau (1980)). Parlar/Weng (2003) modified this so-called satisficing objective to maximise the robability of exceeding a moving rofit target which they secify as the exected rofit. In Gan et al. (2004) exected rofit is maximised under a value at risk constraint. Another extension models the risk references of the newsvendor by using the exected utility framework (Eeckhoudt et al. (1995)), a loss-averse utility function (Wang/Webster (2009)) or the conditional value at risk (CVaR) (see e.g. Jammernegg/Kischka (2007)). With resect to customer-facing erformance measures the order quantity is determined for a given level of roduct availability (fill rate, cycle service level) (see e.g. Cachon/Terwiesch (2009), section 11.6). Another imortant stream of extensions allows the demand to be rice deendent (see e.g. Petruzzi/Dada (1999), Yao et al. (2006)). There are also rice-setting newsvendor models with other objectives than maximising exected rofit. In Lau/Lau (1988) the objective is to maximise the robability of exceeding a secified target rofit. For the additive and the multilicative demand models Chen et al. (2008) give sufficient conditions for the existence of an otimal order quantity and an otimal selling rice under the CVaR criterion.
3 We roose newsvendor models with rice-indeendent as well as rice-deendent demand distribution where the objective is to maximise the exected rofit with resect to a lower bound for the level of roduct availability and an uer bound for the robability in resulting in loss. Instead of secifying shortage cost like in Wang/Webster (2009) in our aroach the newsvendor must decide on a minimal cycle service level for the roduct. In this way external and internal erformance measures are considered by adding two constraints to the classical newsvendor model. The aer is organised as follows. The classical newsvendor model with service and loss constraints is resented and analysed in section 2. A condition for the existence of an otimal order quantity is given and its structure is roven and discussed. In section 3 the rice-setting newsvendor model with service and loss constraints is investigated. The admissible region for the order quantity and the selling rice are characterized both for the multilicative and the additive demand model. Moreover it is shown that higher variability of demand leads to a smaller admissible region of the decision variables thereby easing the comutation of the otimal order quantity and the otimal selling rice. Finally, section 4 resents the conclusions from the main results of the aer. 2 Classical Newsvendor with Service and Loss Constraints 2.1 Model In this section we first introduce our notation for the classical newsvendor model and its erformance measures and then we resent our model. The random demand X is characterized by the distribution function F. The urchase rice er unit of the roduct is c. During the regular selling season, the roduct is sold to customers at a unit rice. Unsatisfied demand is lost, and leftover inventory of the roduct at the end of the selling season is sold in an other distribution channel at the salvage value er unit z. - c describes the cost of understocking by one unit, whereas c - z describes the cost of overstocking by one unit. It is assumed that > c > z holds. Let y denote the order quantity and g the rofit. g deends on y and the stochastic demand X and is given by g(y,x) = ( c) y ( z)(y X) + (1) + with (y X) = max (0,y X).
In the classical newsvendor model the otimal order quantity y* is derived by maximizing the exected rofit E(g(y,X)). The otimality condition is given by (see, e.g., Cachon and Terwiesch 2009, section 11.4): F(y*) = ( c) /( z). (2) Here and in the following, we assume that the distribution function F is strictly monotone increasing and continuous; then y* is defined uniquely by c y* = F z. (3) Now we can define the model with service and loss constraints: 4 s.t. Max E(g(y, X)) y 0 F(y) CSL, (4a) (4b) P(g(y,X) 0) PL. (4c) In (4b) F(y) describes the robability that there is no stock-out during the selling season. It is called cycle service level. By CSL we denote a given lower bound for the cycle service level. In (4c) the robability of loss is bounded from above by the secified value PL. The robability of loss is given by (c. Lau (1980), Jammernegg/Kischka (2007)): c z P(g(y,X) 0) = F y z. (5) Using (5) we can rewrite the model (4) as follows: Max E(g(y,X)) (6a) s.t. 1 1 z F (CSL) y F (PL) c. (6b) z For PL CSL an otimal order quantity exists in any case. For the reasonable case PL < CSL an otimal order quantity y exists rovided c F (PL) 1 z F (CSL). (7)
The lhs of (7) reresents the rofit value of the roduct. Therefore, for high-rofit roducts the existence of a solution is more likely than for low-rofit roducts. 5 For PL < CSL and z 0 we have the following sufficient condition for the existence of an otimal solution: cf (CSL) F (PL). This is an easy consequence from (7). For given demand distribution F and for given PL and CSL condition (7) can be used to describe the range of admissible rice arameters, c and z. Now we consider the influence of demand uncertainty on the existence of a solution. Let F and G be two distribution functions and assume that (7) is fulfilled for F. Obviously if G (PL) > F (PL) and G (CSL) < F (CSL) then (7) is fulfilled for distribution G, too. The revious condition holds in the following case: Let G (PL) < x0 < G (CSL) for some x 0 and G(x) F(x) for x x G(x) F(x) for x x. 0 0 (8) If a solution for distribution F exists, then also a solution for distribution G exists. Figure 1 illustrates condition (8). This condition is related to the stochastic order mean reserving sread (c. Müller/Stoyan (2002),. 28): Let G be uniformly distributed over [a, b] and F be uniformly distributed over [c, d] with c a b d and equal exected values; then F differs from G by a mean reserving sread. If PL < 0.5 < CSL then condition (8) is fulfilled.
6 F,G 1 G F x0 x Figure 1: Illustration of condition (8) for distribution functions F and G 2.2 Otimal Order Quantity Now we characterize the otimal order quantity. Proosition 1: If (7) is fulfilled the solution y* of model (6) is given by: c a) If F F ( CSL) z then y* = F (CSL), c z b) if F (CSL) F F (PL) z c z 1 c then y* = F, z c z c) if F F (PL) z c z c then y* = F (PL). z
7 Proof. We know from the classical unconstrained newsvendor model that E(g(y, X)) is a concave function of y. Note that its otimal solution is given by (3). With (7) it is guaranteed that an admissible solution (6b) exists. Part (a) describes the situation where the set of admissible solutions is right to the otimal unconstrained solution (3). Thus, the otimal solution y* is given by the lower bound of the admissible region, i. e. F (CSL). Part b) and c) can be roved similarly. If the demand distribution has bounded suort, in case c) the otimal order quantity must not exceed the maximal demand. In order to get further insight in the structure we consider uniformly distributed demand over [a,b], 0 a < b : F~Unif[a,b]. Then (7) is given by c (CSL PL)(b a). z CSL(b a) + a (9) If a = 0 the existence of an otimal order quantity is indeendent of the uer bound b. Contrary, for fixed a > 0 (9) is fulfilled for sufficiently small b. This means, the lower the demand variability the more likely an otimal order quantity exists. Examle: Let CSL = 0,8, PL = 0,1 and let F ~ Unif [a, b]. I) Low rofit value: e.g. = 8, c= 5, z= 2 From (9) we have: A solution exists if (b a)0,7 0,5 (b a)0,8 + a ; therefore, a solution exists for 0< a < b 8 a. 3
1 1 1 1 c Since F (CSL) F (0,8) F (0,5) F = > = z is ossible and we have for 0< a < b 8 a: 3 y* = F (CSL) = 0,8b + 0, 2a. II) High rofit value: e. g. = 8, c = 1, z = 0 only case a) of roosition 1 8 From (9) we have: A solution always exists. c Since F = F (0,875) > F (0,8) case a) from roosition 1 is not z ossible. Case b) is relevant, if F (0,875) F (0,1) 8 This is equivalent to b 7,93a and the solution is: y* = F (0,875) = 0,875b + 0,125a. Case c) is relevant, if F (0,1) 8 F (0,875) and therefore if b 7,93a. The solution is: y* F (0,1) 8 0,8b 0, 72a = = +. Note that in case c) the coefficient of variation is larger than in case b). For uniformly distributed demand over [0, b] the otimal solution deends on the function CSL(1 CSL). From Figure 2 it can be seen that for CSL in A loss aversion dominates and the otimal order quantity is given either by art b) or art c) of Proosition 1. Contrary, for CSL in B stockout aversion is dominating and the otimal solution in given either by art a) or by art b) of the Proosition 1.
9 0,25 PL A B 0,5 1 Figure 2: Characterisation of the otimal solution for uniformly distributed demand over [0,b] 3 Price-Setting Newsvendor with Service and Loss Constraints 3.1 Model Now we consider a model where the order quantity y and also the selling rice are decision variables. We denote by X the random demand for rice with distribution function F. X is comosed of a deterministic art and a stochastic art (c. Petruzzi/Dada (1999)). The deterministic demand function d() is decreasing in ; we assume d() 0 for < and > 0 d( 0) = 0. The stochastic art is described by the random variable ε with distribution function H. The random rofit now deends also on and is given by (see (1)) + g(y,, X ) = ( c)y ( z)(y X ). (10)
10 Generalising the model (4) we consider now s.t. maxe(g(y,,x )) y, y 0,c< < F(y) CSL P(g(y,,X ) 0) PL 0 (11a) (11b) (11c) According to (5) we have c z P(g(y,,X ) 0) = F y z (12) We can rewrite model (11) as follows MaxE(g(y,,X )) (13a) s.t. y 0,c< < 0 1 1 z F (CSL) y F (PL) c z (13b) To gain further insight we consider two well known secial cases for random demand multilicative and the additive model (see e. g. Petruzzi/Dada (1999)). In the multilicative model X = d() ε we have X ; the x F (x) =Η for all x, < 0 d() In the additive model X = d() +ε we have (14) ( ) F (x) =Η x d() for all x, <. (15) 0 3.2 Admissible solutions for multilicative demand First we characterise the admissible regions of the decision variables and y. It is the set of all (, y) fulfilling
11 z Η (CSL)d() y Η (PL)d() c z c< < 0 (16) (16) follows immediately from (14) since F ( α ) =Η ( α )d(), 0<α< 1. Proosition 2. For model (13) with multilicative demand an admissible solution (, y) exists if Η (CSL) z + (c z) < 1 0,PL< CSL. (17) Η (PL) Proof. If PL < CSL then the lhs of (17) is larger than c. Η (CSL) Let = z + (c z) and define y= d() Η (CSL), the lower bound in (16). Inserting Η (PL) in the rhs of (16) shows that this order quantity y fulfills (16). Η (CSL) Moreover for = z + (c z) < 0 and y =Η (CSL)d( ) the constraints in (11b) Η (PL) and (11c) are fulfilled as equality. Like in the model with rice indeendent demand we investigate the influence of demand uncertainty on the region of admissible rices and order quantities. Proosition 3. Let Η and K be two distribution functions for random variable ε with Η (CSL) K (CSL), Η (PL) K (PL). (18) Then the set of admissible rices and order quantities (, y) corresonding for Η is a subset of the admissible set corresonding to K. This is obvious from (16). Therefore, if an otimal solution exists with resect to Η there is also an otimal solution with resect to K. We illustrate the structural roerties by means of some examles.
Let ε be uniformly distributed over [ 1 a,1+ a ],0< a 1. If Η ~ Unif(1 a,1+ a) and K~Unif(1 a',1+ a') then condition (18) is fulfilled for a a ' if PL<0.5<CSL. Consider the following examle with a linear deterministic demand function: d() = 10, c = 1, z = 0, CSL = 0.8, PL = 0.1, ε ~Unif(1 a,1+ a). 12 a 0.5 0.8 1.0 * 5.75 5.79 8 y* 6.38 6.44 3.2 CSL(*,y*) 1 0.83 0.8 PL(*,y*) 0 0.04 0.1 E(g(y*,*,X*)) 18.06 17.38 12.16 Table 1: Otimal decision variables and erformance measures for different values of arameter a In Table 1 * and y* denote the otimal selling rice and the otimal order quantity. The erformance measures in lines 4 to 6 corresond to these otimal values. As arameter a increases the rescribed bounds for CSL and PL become more relevant. For a = 1 the otimal solution is given by the secified and y in the roof of Proosition 2. For the examles resented in Table 1 the set of admissible rices and order quantities is shown in Figure 3. The admissible region is given by the resective straight line as lower bound and arabola as uer bound. Figure 3 illustrates the statement of Proosition 3.
13 Figure 3: Admissible regions for the multilicative model with uniform distribution Dotted lines: Unif(0.5,1.5), dashed lines: Unif(0.2,1.8), solid lines: Unif(0,2) It has to be noticed that the otimal order quantity cannot exceed the maximal demand for a given selling rice. Therefore for distributions with bounded suort [α, β] condition (16) can be relaced by z Η (CSL)d() y Min H (PL)d(),d() β. (19) c z
In Figure 4 the relevant admissible region is shown for by the area bounded by the three lotted corners. 14 ε ~ Unif(0.5,1.5). It is characterized Figure 4: Relevant admissible region for the multilicative model with uniform distribution For less rofitable roducts no solution may exist. E. g. this is the case for d() = 10, c = 5, z = 2, CSL = 0.8, PL = 0.1, ε ~ Unif(0.3,1.7). Here condition (17) is violated. We resent a second set of examles based on the Weibull distribution. Let ε be Weibull distributed with location arameter 1 and scale arameter γ, i. e. γ v (v) 1 e. Η = If Η ~ Weib ( γ ) and K ~ Weib ( δ ) then condition (18) is fulfilled for 1 PL < 1 < CSL. e γ δ if In Figure 5 the admissible regions for different scale arameters of the Weibull distribution are shown.
15 Figure 5: Admissible regions for the multilicative model with Weibull distribution Dotted lines: Weib(3), dashed lines: Weib(2), solid lines: Weib(1) 3.3 Admissible solutions for additive demand For the additive model the admissible region is the set of all (, y) fulfilling z Η (CSL) + d() y ( Η (PL) + d()) c z c< < 0. (20)
(20) follows immediately from (15)since Condition (20) is fulfilled if F ( α ) =Η ( α ) + d(), 0<α< 1. 16 Η +Η +Η (CSL)(c z) (PL)z ( c)d() (PL) and c< < 0 holds. Like in the multilicative model the qualitative roerties resented in Proosition 3 hold. Examle Let ε be uniformly distributed over [ a,a ], a > 0. If Η ~ Unif( a, a) and K ~ Unif( a ',a ') then (18) is satisfied for a a' if PL < 0.5 < CSL. Note that this is analogous to the condition in the multilicative model. In Figure 6 this is illustrated for a linear deterministic demand function. For ε~ Unif( 5,5) no admissible solution exists. Figure 6: Admissible regions for the additive model with uniform distribution Dotted lines: Unif(-1,+1), dashed lines: Unif(-3,+3), solid lines: Unif(-5,+5)
4 Conclusions 17 In this aer we resent a newsvendor model with the objective to maximise the exected rofit with resect to a service constraint and a loss constraint. First we consider just the ordering decision. The condition for the existence of an otimal order quantity can be used to secify rice arameters of the model. The structure of the otimal decision shows that the otimal order quantity for low rofit roducts in general is higher than that of the classical newsvendor whereas the otimal ordering decision for high rofit roducts is limited by the secified robability of loss. In the second art of the otimal ordering and ricing decisions are investigated where the stochastic demand deends on the selling rice. For the multilicative and the additive model the admissible region of the decision variables is characterised. It turns out that for the multilicative demand model the region of admissible solutions does not deend on the deterministic demand function. For both models the influence of demand uncertainty on the region of the admissible decision variables is analysed. The roosed comarison of stochastic demand variables is related to the stochastic order mean reserving sread. For all models higher variability of demand leads to a smaller region of the admissible decision variables order quantity and selling rice. The managerial imlication of this finding is that the higher demand variability is the more the rescribed erformance measures cycle service level and robability of loss determine the otimal decisions. The comutational consequence is that for high variability the otimal solution is given by the model with equality constraints. This leads to a considerable reduction of the effort necessary for comuting the otimal decisions. Moreover, if the demand variability is too high it is ossible that no admissible order quantity and selling rice exist.
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