The classical newsvendor model under normal demand with large coefficients of variation

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1 MPRA Munic Personal RePEc Arcive Te classical newsvendor model under normal demand wit large coefficients of variation George Halkos and Ilias Kevork University of Tessaly, Department of Economics August 0 Online at ttps://mpra.ub.uni-muencen.de/4044/ MPRA Paper No. 4044, posted. August 0 8:3 UTC

2 Te classical newsvendor model under normal demand wit large coefficients of variation By George E. Halkos and Ilias S. Kevork Department of Economics, University of Tessaly ABSTRACT In te classical newsvendor model, wen demand is represented by te normal distribution singly truncated at point ero, te standard optimality condition does not old. Particularly, we sow tat te probability not to ave stock-out during te period is always greater tan te critical fractile wic depends upon te overage and te underage costs. For tis probability we derive te range of its values. Writing te safety stock coefficient as a quantile function of bot te critical fractile and te coefficient of variation we obtain appropriate formulae for te optimal order quantity and te maximum expected profit. Tese formulae enable us to study te canges of te two target inventory measures wen te coefficient of variation increases. For te optimal order quantity, te canges are studied for different values of te critical fractile. For te maximum expected profit, its canges are examined for different combinations of te critical fractile and te loss of goodwill. Te range of values for te loss of goodwill ensures tat maximum expected profits are positive. Te sies of te relative approximation error wic result in by using te normal distribution to compute te optimal order quantity and te maximum expected profit are also investigated. Tis investigation is extended to different values of te critical fractile and te loss of goodwill. Te results indicate tat it is naïve to suggest for te coefficient of variation a maximum flat value under wic te normal distribution approximates well te target inventory measures. Keywords: Classical newsvendor model; truncated normal distribution; optimality condition; critical fractile; loss of goodwill; relative approximation error. JEL Codes: C4; C44; M; M.

3 . Introduction For te classical newsvendor model, te sufficient optimality condition to determine te order quantity is given by te well-known standard critical fractile formula (Kouja, 999). Tis formula states tat te probability not to observe stock-out during te demand life-cycle (oterwise called as te period) is equal to a critical fractile wic depends upon overage and underage costs. Te overage cost equals to unit purcase cost minus salvage value, and te underage cost is te difference between profit margin and loss of goodwill. Setting a-priori te probability not to ave stock-out during te period, te optimal order quantity is computed from te inverse cumulative distribution function of demand evaluated at te critical fractile. Wen demand is normally distributed, te standard critical fractile formula olds only wen te coefficient of variation is sufficiently small. In tis case, te probability to take negative demand is negligible. Taking te inverse of te cumulative distribution function evaluated at te critical fractile, te optimal order quantity is equal to te average demand plus te safety stock coefficient times te standard deviation of demand. In tis optimal order quantity equation, te safety stock coefficient is a quantile function of te critical fractile. So, te computed order quantity ensures tat te requested probability of not aving stock-out during te period is eventually attained. Te Normal distribution as been widely used in inventory management to model demand. A first reason is tat te teoretical properties of normal distribution enable us to derive exact expressions for target inventory measures suc as te optimal order quantity and te maximum expected profit. Te second reason is tat we can take good approximations for tese measures wen te coefficient of variation is low. Lau (997) offered a simple formula to compute te expected cost of te classical newsvendor model wen demand is normal wit a coefficient of variation less tan 0.3.

4 Perakis and Roels (008) derived order quantities tat maximie te newsvendor s maximum regret and tey stated tat a normal distribution wit small coefficient of variation is robust and is also entropy maximiing wen only mean and variance are known. To investigate purcase decision of newsvendor products wen demand distribution is unknown, Benion et al. (008) conducted an experiment during wic demand data were generated from a normal distribution wit coefficient of variation equal to one tird. Te normal distribution wit te same coefficient of variation was also used in a similar experiment conducted by Benion et al. (00) were alf of te participants knew te demand distribution and te oter alf did not. Wen te demand coefficient of variation is large, using again te normal distribution, te probability to take negative demand on a given period is not any more negligible. In suc case, and if data for demand are available, Gallego et al. (007) recommend te fit of te empirical distribution to one of te known non-negative random variables suc as te Gamma, or te Negative Binomial, or te Lognormal. For te tree distributions, te autors sowed tat te optimal order quantity first increase and ten decrease wen te demand standard deviation increases. To cope wit negative values for demand, Strijbosc and Moors (006) suggested two alternatives. Te first alternative is to interpret negative demand as purcases being sent back to stores. However, in markets of newsvendor products (magaines, cloting, perisable food etc.), tis explanation could not stand as it is very unlikely customers to ave te possibility to return back to stores purcases of suc products (unless te product is faulty). Te second alternative is to regret negative values. In suc case, te demand of te period sould be modeled by te normal distribution singly truncated at point ero. In te current paper, we follow te second alternative and illustrate tat wen demand follows te singly truncated normal distribution at point ero, te standard critical fractile 3

5 formula does not old. Particularly, we sow tat te probability of not aving stock-out during te period is always greater tan te critical fractile. Writing te safety stock coefficient as a quantile function of bot te critical fractile and te coefficient of variation, we derive appropriate expressions for computing te optimal order quantity and te maximum expected profit. Tese expressions allow us to study analytically te canges of tese two target inventory measures wen te coefficient of variation is raising due to demand standard deviation increases. Particularly, tese canges are examined at different sies of te critical fractile. Te canges of maximum expected profits are also explored at different values of te loss of goodwill for wic te maximum expected profit is positive. Te use of te non-truncated normal distribution for approximating te exact values of te optimal order quantity and te maximum expected profit (wen te demand distribution is modeled by te truncated normal) is also investigated analytically. As criterion of investigation we use te approximation error as percentage of te exact values of te two target inventory measures. Te derived expressions of te two target inventory measures for te singly truncated normal allow us to evaluate tis criterion no matter wat values te average demand of te period, te selling price, te purcasing cost and te salvage value take on. So, for bot target inventory measures, tis relative approximation error is studied at different sies of te critical fractile. For te maximum expected profit te relative approximation error is also examined for different values of te loss of goodwill. Having establised tis experimental framework, to te extent of our knowledge, tis study using te singly truncated normal at point ero is performed for te first time. Our work comes closest to te paper of Hu and Munson (0). Assuming tat demand follows te truncated normal distribution at point ero, te autors gave an expression to compute te optimal order quantity, and using tis expression tey derived te function for te maximum expected profit. Keeping fixed te probability of not aving stock- 4

6 out during te period, tey examined te beavior of optimal order quantities and maximum expected profits wen te coefficient of variation was getting larger. However, teir study was based on Monte-Carlo simulations and on specific values for te average demand, te selling price, te purcase cost and te salvage value. Besides, teir simulation results were generated under te restrictive assumptions tat loss of goodwill is ero and te sum of overage and underage costs remains te same wen te probability to ave stock-out during te period is decreasing. Te last two remarks differentiate our work, wic is based on an analytic approac witout making specific assumptions about te values of te overage and te underage cost. Te paper of Strijbosc and Moors (006) also andles te case of modeling demand wit te normal distribution wit large coefficients of variation. For te (R,S) inventory control system, te autors used te censored and te truncated normal distribution to derive safety factors and order-up-to-levels. Halkos and Kevork (0) used tree alternative distributions to construct confidence intervals for te optimal order quantity and te maximum expected profit wen te demand follows te singly truncated normal at point ero. Tese distributions were te non-truncated normal, te lognormal and te exponential. Tey concluded tat only for very few combinations of te critical fractile and te sample sie te confidence intervals of te non-truncated normal and te lognormal distribution attained acceptable confidence levels. But tese intervals are caracteried by low precision and stability. Oter works wic dealt wit te prons and cons of using te normal distribution wit small or large coefficients of variation can be found in Janssen et al. (009). Te aforementioned arguments and discussion lead te rest of te paper to be structured as follows wen demand follows te singly truncated normal at point ero: In te next section, we derive te optimality condition and determine te range of values for te probability not to ave stock-out during te period. In section 3, we study te canges of te 5

7 optimal order quantity wen te coefficient of variation increases. In section 4, we derive te expression for te maximum expected profit, and determine te range of values for te loss of goodwill were maximum expected profit is positive. In te same section, we derive analytic forms for te relative approximation errors so tat to examine ow well te use of te nontruncated normal approximates te exact values of te two target inventory measures. Finally, te last section summaries te most important findings of te current work. In eac section, symbols wic are used in te analysis are explained wen required. Noneteless, for te reader s convenience, table provides te list of symbols, wic are used trougout tis paper, wit teir explanation.. Optimality Condition In te classical newsvendor model, te demand wic will occur during te period sould be a non-negative random variable X wit cumulative distribution function F ( x). Having specified at te start of te period te critical fractile R from te equation ( p c s) ( p v s) R, wen X is continuous te order quantity maximiing te expected profit (or te expected cost) of te period satisfies te sufficient optimality condition ( ) R F and tus is determined from te equation F - (. To satisfy te demand of te period, te newsvendor as available stock at te start of te period only te optimal order quantity,. Furter, receiving tis ordered quantity, e is not carged wit any fixed costs. Ordering, te critical fractile R expresses te probability te newsvendor not to experience a stock-out during te period. Following te rule of tumb suggested by Scweiter and Cacon (000), if R > 0. 5 (or alternatively R< 0. 5), te newsvendor product is classified as ig-profit (or low-profit respectively). Tis principle implies tat among different newsvendor products tat one wit te largest R as been purcased at te igest cost, it is sold at te igest price, and it yields te largest profit margin. 6

8 Wen X follows te normal distribution wit mean µ and variance σ, te application of te formula F ( ) R approximates well te optimal order quantity only wen te coefficient of variation ( CV σ µ ) is sufficiently small. Tis appens because te probability of taking negative demand is negligible and so, in te profit function of te newsvendor model (e.g. see Kouja, 999), it is legitimate to set te lower limit of X at minus infinity instead of ero. Te probability of negative demand equals to, were is te inverse of coefficient of variation, and is te cumulative distribution function of te standard normal evaluated at. Hence wen CV is sufficiently small, te optimal order quantity will be well approximated from te known formula (e.g. see Silver et al., 998) and te computed µ σ, () ap R is almost identical to R. ap will ensure tat te probability to observe a stock-out during te period On te contrary, wen CV is not sufficiently small, ow well formula () approximates te optimal order quantity is under question. Tis can be explained by considering a realiation of values from te normal random variable X for a sufficiently large number of consecutive periods. If CV is large, ten a significant number of negative values will appear in tis realiation. So regretting all te negative values, te remaining truncated part of te realiation will follow te singly truncated normal distribution at point ero, wic as probability density function ( x) [ σ ] ( π) σ x ( µ ) f e, and will be denoted as X ~ N ( µ, σ ). 7

9 Table : Notation and Terminology : p : c : v : Symbol Explanation Order quantity at te start of te period Selling price per unit Purcase cost per unit Salvage value s δ p c s : Loss of goodwill, were ( ) R : Requested critical fractile, were R ( p c s) ( p v s) X : Demand of te period F ( x) : Cumulative distribution function of demand X ~ N( µ, σ ) Demand follows te non-truncated normal distribution CV : Coefficient of variation, were CV σ µ R : Safety stock coefficient wen X ~ N(, σ ) ap : Optimal order quantity wen X ~ N( µ, σ ) ξ ap : Maximum expected profit wen X ~ N( µ, σ ) X ~ N ( µ, σ ) µ and sufficiently small CV Demand follows te normal distribution singly truncated at point ero : Inverse of coefficient of variation, were CV : Probability density function of te standard normal evaluated at ( µ ) σ, : Cumulative distribution function of te standard normal evaluated at ( µ ) σ, : Probability demand of te period to be positive ξ : Expected profit per period wen X ~ N ( µ, σ ) : Optimal order quantity maximiing ξ wen X ~ N ( µ, σ ) ξ : Maximum expected profit wen X ~ N ( µ, σ ) RAE Relative approximation error wen RAE ξ Relative approximation error wen ap is used to approximate ξ ap is used to approximate ξ Wen te demand wic will occur during te period follows function is derived in te Appendix and is given by X, te expected profit ξ ( p c) ( p v)( { µ σ ω) } ( p v s) ( µ ) σ. () Te explanations of symbols in () are given in Table. Maximiing ξ wit respect to, and using te derivatives, d dσ and d d σ, first and second order conditions are given respectively from te following expressions: 8

10 d ξ d ( p c s) ( p v s) 0, (3a) d ξ d ( p v s) σ < 0. (3b) From (3a) and (3b), te optimal order quantity satisfies te sufficient optimality condition µ ( X ) Pr Z Pr( Z ( ) ( ) Pr R σ From (4), we reac te first important conclusion. Since ( ( ) > 0 tat ( > R. (4), it follows and ence ( > R. So, if te probability of taking negative demand is not negligible, by ordering te optimal quantity,, te probability not to ave a stock-out during te period is always greater tan te critical fractile R. Te rate of cange of probability Pr( X ) wen CV is getting larger is studied in te next proposition. Proposition : Given te critical fractile R, te probability Pr( X ) Proof: See in te Appendix. is: (a) increasing in CV wit lower limit R and upper limit ( R ), (b) as inflection point at CV. An immediate implication of proposition is te following. Even if te critical fractile R is below 0.5, tere will be a range of values of te coefficient of variation (CV) were probability will exceed 0.5. In tis case, te rule of tumb using R to classify te product as low or ig-profit sould be used wit cautiousness. Wen demand follows distributions of non-negative random variables, like exponential, gamma, etc., tis rule of tumb works well since it olds Pr( X ) R. But wen demand follows te truncated normal X, tis equality is not true. So, wit R below 0.5, tere will be for certain a range of values for CV 9

11 were R would classify te product as low-profit, wile te probability not to ave stock-out during te period would classify it as ig-profit. Te results of proposition are verified in Figure. Furter, table displays, for selected combinations of R and CV, te values of probability. Given CV, te error of approximating probability wit R depends upon te value of R. Setting te approximation error to be less tan a certain sie, R approximates well probability wen te coefficient of variation is below a critical value wic increases as R is getting larger. Setting, for example, te sie of te approximation error below 5 0, tis critical value ranges between 0. and 0.6 wen R is between 0.3 and Tis range of R is reasonable form te practice point of view. Figure : Grap of Pr( X ) wen CV is increasing Te data of table also verify te problem wic we raise for using R to classify te product as low or ig-profit wen R is below 0.5. As an example we mention te case of R0.4 and CV equal or greater tan.5. Using te rule of tumb wit R, te product is classified as low-profit. On te contrary, using te probability not to ave a stock-out during 0

12 te period, te product is ig-profit. Observe also tat wen CV is more tan 3, tis probability is greater tan 6%. Table : Values of probability Pr( X ) CV R 0.3 R 0.4 R 0.8 R Optimal ordering policy In te current section, we express te optimal order quantity,, as a function of te coefficient of variation (CV), and ten, given te critical fractile, R, we study ow canges wen CV is making larger. Using condition (4), te optimal order quantity is determined from te equation ( CV) µ σµ. (5) Given te average demand of period and increasing CV, te rate of cange of depends upon te properties of te function g ( CV) CV result (A9) of te Appendix, we obtain. Differentiating g given R, and using

13 dg d d ( CV. d Te derivative d d was obtained using formula () of Steinbrecer and Saw (008). Wen R exceeds 0.5, we sowed in te previous section tat > R > 0, and ence dg is positive for any CV> 0. But, wen R is set below 0.5, we sowed troug proposition tat tere is a range of values of CV for wic < 0. In tis case, dg made up of two terms of wic te first is negative and te second is positive. Ten to is determine te final sign of dg, we sall investigate its rate of cange wen CV is increasing. So we proceed to te next proposition: Proposition : Given tat R is below 0.5, Proof: See in te Appendix. R < 0 and upper limit ( ) 0 dg is increasing in CV wit lower limit R >. From proposition we deduce tat setting R below 0.5 and increasing CV, tere is only one value o CVo for wic te derivative dg becomes ero. Tis value o satisfies te equation ( o o o o. Summariing, we conclude tat for any 0 < R<, te optimal order quantity,, is a convex function of CV since dg > 0 from (A) in te appendix. Furter, te limiting values of are lim CV 0 µ ( lim( CV) ) µ ( R 0) µ CV 0, and lim CV ( lim ( CV) ) µ ( µ CV ( ).

14 Additionally, wen R > 0. 5, ten as global minimum at is increasing in CV for any 0 o CVo, and increases only wen CVo CV>, wile for R< 0. 5, CV>. To illustrate numerically te findings of tis section, we consider four ypotetical newsvendor products wose R as been set up at 0.3, 0.4, 0.8, and 0.95 respectively. From te principle of low/ig-profit products of Scweiter and Cacon (000), a product wit iger R as larger profit margin, wic follows from iger price and iger purcasing cost. However, wit larger R and iger price, p, te average demand of period, µ, sould be reduced suc tat a negative relationsip between p and µ to old. Te selected values of µ are displayed in table 3. In te same table we ave also included te values of te safety stock coefficient,, and te optimal order quantity,, computed from (5). For te selected values of µ and R te graps of µ g( CV) are presented in Figure 3. From te data of table 3 and te graps of figure 3, te remarks wic ave been made in tis section are verified. Observe also tat CV o, from wic is starting to increase, becomes smaller as R is approacing 0.5. So, for R0.3, wile for R0.4, CV o is approximately equal to CV o is approximately equal to 0.69, 4. Maximum Expected Profit Wen takes on its optimal value, using (3a) and R ( p c s) ( p v s) following relationsip is obtained, te c v R. (6) p v s Also from (5) and te definition of te standardied value, wic is given in Table, we take µ. (7) σ 3

15 Table 3: Exact values for te optimal order quantity wen demand follows te normal distribution singly truncated at ero CV R 0.3, µ 300 R 0.4, µ 00 R 0.8, µ 00 R 0.95, µ Figure : Grap of wen CV is increasing 4

16 So replacing wit in (), and using (6) and (7), te maximum expected profit is given by ξ ( p c)( µ σ) ( p v)( ωσ ) ( c v) σ ( p v s) σ ( p c) µ ( p v) σω ( p v s) σ ( c s) µ ( p v s) σω s( µσω) ( p v s) σ p, (8) were ω. Following Lapin (994), te loss of goodwill, s, is defined as te present value of future profits wic are expected to be lost from present unsatisfied customers wo will not come back to te store to purcase te same product in te future. So it is legitimate to set ( p c) s δ, were 0 δ. Hence, from R ( p c s) ( p v s) ( δ)( p c) we obtain p v s, (9) R and replacing (9) into (8), we take te final expression for te maximum expected profit ξ CV ω CV ω ( p c) µ δ ( R. (0) R Considering R, p c, and µ as given, te expression witin te brackets in (0) defines te sie of te maximum expected profit. To study te beavior of ξ wen CV is increasing, initially we sall determine te range of parameter δ, were maximum expected profits are non-negative. Te next proposition gives a prerequisite result to specify tis range. Proposition 3: For 0 R< Proof: See in te Appendix. u is decreasing in CV and as <, te function ( CV) ( upper limit ( ) ( 0 R > π. 5

17 Hence ξ is positive wen δ satisfies te inequality δ δ o R CV ω (. () For different values of R, figure 3 illustrates te grap of δ o wen CV is increasing. Given R, te range [, ] given, ten te range [, ] 0 δ is getting narrower as CV is getting larger. If we consider CV as o 0 δ becomes wider as R increases. o Figure 3: Grap of δ o wen CV is increasing Wen CV tends to infinity, ten for eac R, δ o tends to a corresponding limiting value π ( δ limδo. () CV π ( ( Tis limit is obtained since wen CV ten 0 and CV ω, (, and ( π). So, from () and () we deduce tat for any R, if δ is less tan or equal to 6

18 te corresponding limiting value δ, ten te maximum expected profit will be positive for any CV <. Tis is verified in Figure 4, were for different R s and wit δ to be equal to te corresponding δ, we display te grap of ξ [( p c) µ ] wen CV is increasing. For eac R, ξ decreases as CV is getting larger, and given CV, te expression witin te brackets in (0) takes on larger values wen R is increasing. Figure 4 :Grap of ξ [( p c) µ ] wen CV is increasing We are closing tis section by examining te approximation error in computing bot te optimal order quantity and te maximum expected profit if we used te non-truncated normal distribution. For te optimal order quantity, te relative approximation error is defined as ( ) ap, and using () and (5), tis is given by RAE ( ( CV) CV. (3) Wen demand is modeled by te non-truncated normal distribution, te maximum expected profit is computed from (e.g. see Kevork, 00), 7

19 ξ ( p c) µ ( δ) R CV R ap. So, te corresponding RAE is obtained using (0) from RAE ξ ξ ( δ) CV ω R ( δ) CV ω R R ξap ξ δ ω CV. (4) δ ω CV Table 4 displays te values of (3) and (4) wen CV is increasing. Te relative approximation error of bot and ξ depends upon te critical fractile. For ξ, its RAE ξ depends also on te loss of goodwill. For tis reason, te values of δ were restricted on te interval from ero up to δ. Tis range of δ values ensures tat at te corresponding R, maximum expected profits will be positive for any CV <. Given te value of CV, bot RAE and RAE ξ are decreasing as R is getting larger. Given R and CV, wit ero loss of goodwill RAE ξ is larger tan RAE. Furter, RAE ξ is making larger as te loss of goodwill is increasing. Te previous arguments indicate tat it seems naive to suggest for te coefficient of variation a maximum flat value under wic te optimal order quantity and te maximum expected profit to be well approximated by using te non-truncated normal distribution. Table 4, or alternatively formulae (3) and (4), offer te required information suc tat, according to te case and te desired sie of te approximation error, to be able to decide in favour of te normal distribution singly truncated at point ero. 8

20 5. Conclusions In te classical newsvendor model, te optimality condition for expected profit maximiation (or expected cost minimiation) states tat te probability not to ave stock-out during te period equals to a critical fractile wose value depends upon te overage and te underage cost. In te current paper we point out tat tis condition is true only wen demand of te period is a continuous non-negative random variable. If demand distribution is normal but wit coefficient of variation not sufficiently small, te probability te stocastic law of generating demand to give negative values is not negligible. In tis case, te demand sould be modeled by te normal distribution singly truncated at point ero. Wit demand to follow te truncated normal distribution, we prove tat for any value of te critical fractile, te probability not to experience a stock-out during te period is always greater tan te critical fractile. Particularly, we give for te first time, te range of values for tis probability at any critical fractile. Furtermore, if te critical fractile is less tan 0.5, we empasie tat te rule of tumb to classify te product as low or ig-profit sould be used wit cautiousness. We sow tat tere is a range of values for te coefficient of variation were te critical fractile would classify te product as low profit, wile te probability of not observing stock-out during te period would give a ig-profit product. Writing te safety stock coefficient as a quantile function of bot te coefficient of variation and te critical fractile, appropriate formulae to compute exactly te optimal order quantity and te maximum expected profit are developed wen demand follows te normal distribution singly truncated at point ero. Tese formulae allows to study te canges of te two target inventory measures wen te coefficient of variation is increasing, no matter wic values te selling price, te purcasing cost and te salvage value take on. So, te beavior of te optimal order quantity in canges of te coefficient of variation depends upon only te critical fractile. For te maximum expected profit, its beavior depends on te critical fractile 9

21 Table 4: Relative approximation error in computing te optimal order quantity and te maximum expected profit using te classical normal distribution instead of te truncated normal R 0.3 R 0.4 R 0.8 R 0.95 RAE RAE ξ RAE RAE ξ RAE RAE ξ RAE RAE ξ CV δ 0 δ 0.5 δ 0.3 δ 0 δ 0. δ 0.4 δ 0 δ δ δ 0 δ δ % 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% % 0.0% 0.0% 0.0% 0.00% 0.0% 0.0% 0.0% 0.00% 0.00% 0.0% 0.0% 0.00% 0.00% 0.00% 0.0% % 0.03% 0.03% 0.03% 0.00% 0.0% 0.0% 0.03% 0.00% 0.0% 0.0% 0.0% 0.00% 0.0% 0.0% 0.0% % 0.04% 0.05% 0.06% 0.00% 0.03% 0.04% 0.05% 0.00% 0.0% 0.0% 0.03% 0.00% 0.0% 0.0% 0.0% % 0.08% 0.09% 0.% 0.0% 0.06% 0.07% 0.08% 0.00% 0.03% 0.03% 0.04% 0.00% 0.0% 0.03% 0.04% % 0.% 0.5% 0.7% 0.0% 0.09% 0.% 0.3% 0.00% 0.04% 0.05% 0.07% 0.00% 0.03% 0.04% 0.06% % 0.9% 0.% 0.7% 0.0% 0.4% 0.7% 0.0% 0.0% 0.06% 0.08% 0.% 0.00% 0.05% 0.07% 0.09% % 0.8% 0.34% 0.40% 0.03% 0.% 0.5% 0.30% 0.0% 0.09% 0.% 0.6% 0.0% 0.07% 0.0% 0.4% % 0.40% 0.48% 0.58% 0.05% 0.9% 0.36% 0.43% 0.0% 0.3% 0.8% 0.4% 0.0% 0.0% 0.4% 0.0% % 0.56% 0.68% 0.8% 0.07% 0.4% 0.50% 0.6% 0.0% 0.8% 0.5% 0.33% 0.0% 0.4% 0.0% 0.8% % 0.76% 0.9%.% 0.09% 0.55% 0.68% 0.83% 0.03% 0.5% 0.33% 0.45% 0.0% 0.9% 0.7% 0.38% %.0%.3%.50% 0.3% 0.73% 0.90%.% 0.04% 0.3% 0.44% 0.60% 0.0% 0.6% 0.36% 0.50% % 3.8% 3.95% 4.97% 0.43%.8%.85% 3.6% 0.3% 0.98%.35%.9% 0.07% 0.76%.09%.60% % 3.4% 7.5% 3.6%.97% 9.4%.3% 6.5% 0.57% 3.84% 5.56% 8.40% 0.30%.96% 4.4% 6.99% % 3.6% 43.% 63.% 4.95%.9% 9.7% 4.6%.36% 8.59%.9% 0.8% 0.7% 6.53% 0.% 7.3% % 55.5% 79.% 5.0% 9.08% 38.% 53.6% 8.5%.4% 4.5%.5% 38.7%.3% 0.9% 7.5% 3.% 0.8.9% 8.5%.9% 07.8% 3.9% 56.3% 8.4% 3.% 3.60%.0% 33.4% 60.9%.8% 5.7% 5.8% 50.6% % 0.4% 68.3% 308.5% 9.% 75.%.% 88.7% 4.84% 7.6% 44.8% 86.0%.4% 0.5% 34.4% 7.6% 38.6% 38.% 6.0% 43.7% 4.4% 93.6% 4.5% 5.8% 6.09% 34.% 56.% 3.0% 3.0% 5.% 43.0% 94.4% % 56.% 435.8% 9.9% 48.% 7.6% 78.7% 59.8%.5% 6.0% 06.3% 54.9% 5.60% 44.3% 80.4% 8.0% 04.5% 339.% 603.% 94.4% 65.5% 8.% 38.% 90.% 5.5% 79.7% 4.9% 383.5% 7.45% 57.4% 07.6% 335.8% % 440.9% 80.5% 347.0% 87.6% 96.4% 5.8% 405.% 0.5% 0.5% 89.4% 58.3% 9.76% 73.4% 4.% 58.8% % 499.6% 95.0% % 00.5% 335.8% 59.% 753.3% 3.5% 5.6% 6.9% 79.%.% 8.5% 6.5% 67.4% 0

22 and te sie of te loss of goodwill. To te extent of our knowledge, tis study is conducted for te first time in te context of te experimental framework wic we are establising in te current paper. Te beavior of te optimal order quantity in canges of te coefficient of variation is differentiated accordingly if te critical fractile is less or greater tan 0.5. Wen te critical fractile is below 0.5, ten as te coefficient of variation is getting larger, te optimal order quantity initially decreases, and ten tere is a turning point after wic it is starting to increase. Tis turning point corresponds to a value of te coefficient of variation wic becomes smaller as te critical fractile is approacing 0.5. If on te oter and te critical fractile is greater tan 0.5, te optimal order quantity is an increasing function of te coefficient of variation. For eac critical fractile we sowed tat tere is a range of values of te loss of goodwill were maximum expected profit is positive for any finite value of te coefficient of variation. For different values of te loss of goodwill witin tis range, we examined te canges of te maximum expected profit for different values of te critical fractile. We found out tat no matter if te critical fractile is less or greater tan 0.5, maximum expected profits are reducing wit increasing te coefficient of variation. On te oter and, given te value of te coefficient of variation, maximum expected profits become larger as te critical fractile is increasing. Finally, wen te sie of te coefficient of variation is not very small, we examined ow well te use of te non-truncated normal distribution can approximate te values of te optimal order quantity and te maximum expected profit. As criterion of evaluation we used te approximation error as percentage of te exact value of eiter te optimal order quantity or te maximum expected profit. For bot target inventory measures, tis relative

23 approximation error depends on te coefficient of variation and te critical fractile. For te maximum expected profit, it depends also upon te loss of goodwill. From te results wic were obtained, we concluded tat it is too simple to suggest a maximum flat value for te coefficient of variation under wic te use of te non-truncated normal distribution can give accurate approximations for te two target inventory measures. Te reason is tat te sie of te relative approximation error differs among te two target inventory measures. Wen te loss of goodwill is ero, maximum expected profit gives iger relative approximation errors tan te optimal order quantity. Wen te critical fractile is increasing, te relative approximation error for bot target inventory measures is reducing. And, as te loss of goodwill is rising, ten te relative approximation error for te maximum expected profit is making larger. Closing tis paper, for using te non-truncated normal distribution, we recommend researcers or practitioners first to specify te sie of te approximation error for eiter te optimal order quantity or te maximum expected profit. For te maximum expected profit, te knowledge of te loss of goodwill is necessary. And, unfortunately, tis is not easy to know in real life situations. Ten, using te formulae of te relative approximation errors, wic we offer, or alternatively from te data of table 4, to determine te maximum value of te coefficient of variation under wic te use of te non-truncated normal distribution is accepted.

24 Appendix Proof of () Using formula () of Koua (999), we give te following alternative form of te profit function, π ( p c) ( p v)( X ) ( p c) s( X ) Te expected value of (A) is if if X X. (A) > ξ 0 {( p c) ( p v)( x) } f ( x) dx {( p c) s( x) } f ( x) dx ( p c s) ( p v s) ( x) dx ( p v s) xf ( x) dx s ( x) dx. f xf (A) Te first integral in (A) is given in Hu and Manson (0) as f dx. (A3) ( x) To find te second integral, we take te intermediate result by setting v u, - u e π u du e π v dv ( ). (A4) Ten, using (A3) and (A4) 0 xf u ( x) dx ( µσu) e du ( x) π σ u f dx u e duµ. (A5) µ 0 - π σ Te last integral in (A) is te expected value of X, wic is obtained from formula (3.34) of Jonson et al. (994, p.56) as 3

25 xf ( x) dxµσ. (A6) 0 Replacing (A3), (A5), and (A6) into (A), ξ and finally ( p c) ( p v) ( p v s)( µ ) ( p v s) µ ( v s) p σ sµ sσ, ξ ( p c) ( p v)( { µ σ ω) } ( p v s) ( µ ) σ. Proof of proposition (i) lim ( lim R CV 0, (A7) R lim ( lim, (A8) CV 0 and d d d d d [ ( ] ( ( > 0. (A9) as d CV. d d d d d d d (ii) ( ( ( ( CV CV ( CV CV 3. 4

26 d Hence 0 < > wen CV <. > Proof of proposition Using (A9), we take ( d d d and d d But dg d ( d d ( ) ( ) R R d. d d d d d d d d d d d ( )( ) ( )( ) ( ( 3. (A0) Hence d g 3 ( ( > 0 (A) Te positive sign of (A) is explained as follows. Wen 0. 5, ten 0 and ence d g is positive. On te oter and, wen < 0. 5, ten < 0 and tus te expression inside te brackets of (A) is made up of two terms wit te first one to be positive and te second one negative. Even in tis case, te net result of te sum of te two terms in brackets is positive, and tis is explained by using te following relationsips wic are deduced from te properties of te standard normal curve: 5

27 (a) As > R, it olds tat ( < < R. (b) Since R R < 0, we take ( > R, and R < (. (c) If ( < 0. 5, it follows tat Pr( Z ) < and ( <. From (a), (b) and (c), we reac te inequalities ( < < we obtain ( wen < ( and > from wic <. Tus te expression inside te brackets of (A) is also positive Te proof is completed by taking te following limits using (A7) and (A8): dg lim cv 0 lim cv 0 ( ( lim ) ( ( 0) < 0 lim CV 0 R R R dg 0 lim ( ( lim ) ( ( ) > 0 cv lim R cv 0 π lim ( R ) CV Proof of proposition 3 From (A0) and (A), du d d d ( ) 3 ( ( < 0. Using (A7) and (A8), 6

28 lim CV lim ( lim ( ) ( ) CV 0 R u R R, CV 0 lim 0 lim lim lim CV ( lim CV u ( π (. CV ( lim To prove tat 0 lim 0 limu is positive, we take te following limits CV { } π 0 lim π 0. 5 R 0 Furter, and ( ( { π } π π 0 lim ( (. R d dr { } d π d, R d R d R dr d R π ( ( R R π, d dr { } R π π π < 0 ( ( d R d R d dr 8 R. Hence π ( 0 ( >, since tis function is concave for any R, and its lower and upper limit is ero. 7

29 REFERENCES Benion, U., Coen, Y., Peled, R., Savit, T. (008). Decision making and te newsvendor problem: an experimental study. Journal of te Operational Researc Society 59: Benion, U., Coen, Y., Savit, T. (00). Te newsvendor problem wit unknown distribution. Journal of te Operational Researc Society 6: Gallego, G., Katircioglu, K., Ramacandran, B. (007). Inventory management under igly uncertain demand. Operations Researc Letters 35: Halkos, G.E., Kevork, I.S. (0). Non-negative demand in newsvendor models: Te case of singly truncated normal samples. MPRA Paper No. 384, Online at ttp://mpra.ub.unimuencen.de/384/. Hu, J., Munson, C.L. (0). Improved profit functions for newsvendor models wit normally distributed demand. International Journal of Procurement Management 4: Janssen, E., Strijbosc, L., Brekelmans, R. (009). Assessing te effects of using demand parameters estimates in inventory control and improving te performance using a correction function. International Journal of Production Economics 8: Jonson, N.L., S. Kot, N, Bakakrisnan Continuous Univariate Distributions. Vol., nd Edition. Wiley New York. Kevork, I.S. (00). Estimating te optimal order quantity and te maximum expected profit for single-period inventory decisions. Omega 38: 8 7. Kouja, M. (999). Te single-period (news-vendor) problem: literature review and suggestions for future researc. Omega 7: Lapin, L.L. (994). uantitative metods for business decision wit cases, 6 t ed. Duxbury Press, An International Tomson Publising Company. 8

30 Lau, H. (997). Simple Formulas for te Expected Costs in te Newsboy Problem: an educational note. European Journal of Operational Researc 00: Perakis, G., Roels, G. (008). Regret in te newsvendor model wit partial information. Operations Researc 56: Scweiter, M.E., Cacon, G.P. (000). Decision Bias in te Newsvendor Problem wit a Known Demand Distribution: Experimental Evidence. Management Science 46: Silver, E.A., Pyke, D.F., Peterson, R. (998). Inventory management and production planning and sceduling, 3 rd ed. New York, NY: Jon Wiley & Sons. Steinbrecer, G., Saw, W.T. (008). uantile mecanics. European Journal of Applied Matematics 9: 87-. Strijbosc, L.W.G., Moors, J.J.A. (006). Modified normal demand distributions in (R,S)- inventory control. European Journal of Operational Researc 7: 0-. 9