《不确定情况下的决策问题》课程教学资料：A benchmark solution for the risk-averse newsvendor problem

Available online at www.sciencedirect.com ScienceDirect EUROPEAN JOURNAL OF OPERATIONAL RESEARCH ELSEVIER European Journal of Operational Research 174(2006)1643-1650 www.elsevier.com/locate/ejor Production,Manufacturing and Logistics A benchmark solution for the risk-averse newsvendor problem Baruch Keren a.",Joseph S.Pliskin b Sami Shamoon College of Engineering.The Department of Industrial Engineering and Management. Ben-Gurion University of the Negev.Bialik/Basel Streets,P.O.Box 45.Beer Sheva 84100,Israel Department of Industrial Engineering and Management.Ben-Gurion University of the Negev.Israel Received 19 January 2004:accepted 31 March 2005 Available online 19 July 2005 Abstract In this paper,we derive the first order conditions for optimality for the problem of a risk-averse expected-utility maximizer newsvendor.We use these conditions to solve a special case where the utility function is any increasing differentiable function,and the random demand is uniformly distributed.This special case has a simple closed form solution and therefore it provides an insightful and practical interpretation to the optimal point.We show some properties of the solution and also demonstrate how it can be used for assessing the newsvendor utility function parameters 2005 Elsevier B.V.All rights reserved. Keywords:Utility theory:Newsvendor problem:Risk-aversion:Uniform distribution 1.Introduction The interest in the newsvendor problem and its extensions has remained high since it was first intro- duced by Within (1955).A variety of extensions to the single period inventory problems,newsvendor problem and other newsvendor type problems have been presented and researched since then.Khouja's (1999)survey includes more than 10 types of extensions to the newsvendor problems.Kabak and Schiff (1978)dealt with the objective of maximizing the probability of achieving a target profit.Ismail and Loudberback (1979)examined the penalties for output not equaling demand for a firm facing an uncer- tain demand with a known probability function and under several alternatives for the objective function. Lau(1980)analyzes the newsvendor model under two different objective functions.In the first objective, Corresponding author.Tel.:+972 8 647 5641/42/46:fax:+972 8 647 5643. E-mail addresses:baruchke@sce.ac.il,baruchke@bgu.ac.il (B.Keren). 0377-2217/S-see front matter 2005 Elsevier B.V.All rights reserved doi:l0.1016/.ejor.2005.03.047

Production, Manufacturing and Logistics A benchmark solution for the risk-averse newsvendor problem Baruch Keren a,*, Joseph S. Pliskin b a Sami Shamoon College of Engineering, The Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Bialik/Basel Streets, P.O. Box 45, Beer Sheva 84100, Israel b Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Israel Received 19 January 2004; accepted 31 March 2005 Available online 19 July 2005 Abstract In this paper, we derive the first order conditions for optimality for the problem of a risk-averse expected-utility maximizer newsvendor. We use these conditions to solve a special case where the utility function is any increasing differentiable function, and the random demand is uniformly distributed. This special case has a simple closed form solution and therefore it provides an insightful and practical interpretation to the optimal point. We show some properties of the solution and also demonstrate how it can be used for assessing the newsvendor utility function parameters. 2005 Elsevier B.V. All rights reserved. Keywords: Utility theory; Newsvendor problem; Risk-aversion; Uniform distribution 1. Introduction The interest in the newsvendor problem and its extensions has remained high since it was first intro￾duced by Within (1955). A variety of extensions to the single period inventory problems, newsvendor problem and other newsvendor type problems have been presented and researched since then. Khoujas (1999) survey includes more than 10 types of extensions to the newsvendor problems. Kabak and Schiff (1978) dealt with the objective of maximizing the probability of achieving a target profit. Ismail and Loudberback (1979) examined the penalties for output not equaling demand for a firm facing an uncer￾tain demand with a known probability function and under several alternatives for the objective function. Lau (1980) analyzes the newsvendor model under two different objective functions. In the first objective, 0377-2217/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2005.03.047 * Corresponding author. Tel.: +972 8 647 5641/42/46; fax: +972 8 647 5643. E-mail addresses: baruchke@sce.ac.il, baruchke@bgu.ac.il (B. Keren). European Journal of Operational Research 174 (2006) 1643–1650 www.elsevier.com/locate/ejor

1644 B.Keren,J.S.Pliskin European Journal of Operational Research 174 (2006)1643-1650 the focus is on maximizing the decision maker's expected utility of the total profit.The second objective function is the maximization of the probability of achieving a certain level of profit.Atkinson(1979)mod- eled the issue of asymmetric demand information in a stochastic demand environment.His approach is consistent with the traditional principal-agent framework in which the objective is to allow an owner to provide proper incentives to a manager who has better demand information.He studied a contract in which a news vending manager receives a share of the additional profits that are generated as a result of using his order quantity instead of the one that would have been chosen by the owner.Anvari (1987) studied the newsboy problem using the capital asset pricing model(CAPM)while Chung(1990)provided an alternative derivation of Anvari's results.In this formulation,the newsboy borrows capital at time t=0 to invest in two independent projects,namely the newspaper operation and the financial market. Under the assumption that the expected return of this portfolio can be modeled using the one-period CAPM,Anvari (1987)computes the optimal inventory strategy in the case of a normally distributed newspaper demand. Eeckhoudt et al.(1995)assumed that the risk-averse and prudent newsvendor is allowed to obtain addi- tional newspapers if the demand exceeds his original order.They also assumed that the cost per unit of the additional newspapers,c,was higher than the original cost,c,but less than the selling price,S.They pre- sented as a natural assumption the relations 0 sv0 b salvage value for unsold unit,v-c(c+h)>0 D demand TP(O)objective function-total profit ux) a utility function.We assume that u'(x)>0 and for risk-averse u"(x)S,and the salvage value can be even less than zero.The holding cost is negative if v>0 and positive if the newsvendor must pay a disposal fee

the focus is on maximizing the decision makers expected utility of the total profit. The second objective function is the maximization of the probability of achieving a certain level of profit. Atkinson (1979) mod￾eled the issue of asymmetric demand information in a stochastic demand environment. His approach is consistent with the traditional principal-agent framework in which the objective is to allow an owner to provide proper incentives to a manager who has better demand information. He studied a contract in which a news vending manager receives a share of the additional profits that are generated as a result of using his order quantity instead of the one that would have been chosen by the owner. Anvari (1987) studied the newsboy problem using the capital asset pricing model (CAPM) while Chung (1990) provided an alternative derivation of Anvaris results. In this formulation, the newsboy borrows capital at time t = 0 to invest in two independent projects, namely the newspaper operation and the financial market. Under the assumption that the expected return of this portfolio can be modeled using the one-period CAPM, Anvari (1987) computes the optimal inventory strategy in the case of a normally distributed newspaper demand. Eeckhoudt et al. (1995) assumed that the risk-averse and prudent newsvendor is allowed to obtain addi￾tional newspapers if the demand exceeds his original order. They also assumed that the cost per unit of the additional newspapers, ^c, was higher than the original cost, c, but less than the selling price, S. They pre￾sented as a natural assumption the relations 0 6 v c ) (c + h)>0 D demand TP(Q) objective function—total profit u(x) a utility function. We assume that u0 (x) > 0 and for risk-averse u00(x) 0 and positive if the newsvendor must pay a disposal fee. 1644 B. Keren, J.S. Pliskin / European Journal of Operational Research 174 (2006) 1643–1650

B.Keren.J.S.Pliskin European Journal of Operational Research 174 (2006)1643-1650 1645 We formulate the objective function of the newsvendor model similarly to Arrow et al.(1958)and Had- ley and Whitin(1963)and others as follows:The risk-neutral newsvendor wishes to maximize the following profit function: Max TP(g)=S.Min(g,D)-cQ-h·[Max(2-D,0)]-元·[Max(D-2,0l. (1) Since the demand is a random variable and since we assume that the newsvendor is an expected utility max- imizer,the goal is Max Efu[TP()])= u[S.Min(g,D)-(c-元)g-(h+π)·Max(Q,D)+hDlf(D)dD Max Jo +D(D)dD. (2) The first order condition for an optimal solution is (S-c+)fu[(h+S)D-(c+h)olf(D)dD (c+)6WIS-c+π)0-π·Df(D)dD (3) For a risk-neutral newsvendor with linear utility (u(x)=a+bx and u'(x)=b),Eq.(3)gives the well- known critical ratio (S-c+n)bff(D)dD F(2) (c+)=b分fD)dD=1-F(g→F(0）=G+)+b: (4) 3.The risk-averse newsvendor faces a random uniform demand We assume a situation where the random demand for newspapers is uniformly distributed, fo(D)=a六0≤A≤D≤B.By substituting this into(4)we get S-c+为=2[h+S)D-(c+h)g]dD_+0-cwg+-ch☑ (h+S) c+h)o[(S-c+π0-π，D]dD S-c+0-R-S-c+0-☑ (5) 一 The critical ratio of the first order condition is h+SS-c+型_S-c)g-4S·A-cg)+h(4-1 (π(c+h) uS-c)g-uS-c)0+π(Q-B] (6) Given that the newsvendor orders a quantity O that satisfies Eq.(6),i.e.O=O*,then he may get a utility of ul(S-c)O*]as in the right hand side of(6)if the random demand will be exactly equal to the optimal quantity,i.e.D=O'.Hence we can conclude that u[(S-c)O]=u(TP(D=O")o=O).By similar con- siderations the critical ratio(6)can be presented as (h+S)(S-c+)u(TP(D=O)le=O)-u(TP(D=A)lO=O) (7) (π)(c+h) u(TP(D=g)I№=Q）-u(TP(D=B)=) given that the newsvendor sets his order quantity to the optimum O=O*.The optimum order quantity O* sets the difference between the utilities from the total profit at points D=O and D=4,divided by the difference between the utilities from the total profit at points D=O'and D=B to be a constant.It can

We formulate the objective function of the newsvendor model similarly to Arrow et al. (1958) and Had￾ley and Whitin (1963) and others as follows: The risk-neutral newsvendor wishes to maximize the following profit function: Max Q TPðQÞ ¼ S MinðQ; DÞ cQ h ½MaxðQ D; 0Þ p ½MaxðD Q; 0Þ. ð1Þ Since the demand is a random variable and since we assume that the newsvendor is an expected utility max￾imizer, the goal is Max Q Efu½TPðQÞg ¼ Z 1 0 u½S MinðQ; DÞðc pÞQ ðh þ pÞ MaxðQ; DÞ þ hDf ðDÞdD ¼ Max Q Z Q 0 u½ðh þ SÞD ðc þ hÞQf ðDÞdD þ Z 1 Q u½ðS c þ pÞQ p Df ðDÞdD. ð2Þ The first order condition for an optimal solution is ðS c þ pÞ ðc þ hÞ ¼ R Q 0 u0 ½ðh þ SÞD ðc þ hÞQf ðDÞdD R 1 Q u0 ½ðS c þ pÞQ p Df ðDÞdD . ð3Þ For a risk-neutral newsvendor with linear utility (u(x) = a + bx and u0 (x) = b), Eq. (3) gives the well￾known critical ratio ðS c þ pÞ ðc þ hÞ ¼ b R Q 0 f ðDÞdD b R 1 Q f ðDÞdD ¼ F ðQ Þ 1 F ðQ Þ ) F DðQ Þ ¼ ðS þ pÞ c ðS þ pÞ þ h . ð4Þ 3. The risk-averse newsvendor faces a random uniform demand We assume a situation where the random demand for newspapers is uniformly distributed, fDðDeÞ ¼ 1 BA 0 6 A 6 De 6 B. By substituting this into (4) we get ðS c þ pÞ ðc þ hÞ ¼ R Q A u0 ½ðh þ SÞD ðc þ hÞQdD R B Q u0 ½ðS c þ pÞQ p DdD ¼ u½ðhþSÞQðcþhÞQu½ðhþSÞAðcþhÞQ ðhþSÞ u½ðScþpÞQpBu½ðScþpÞQpQ p . ð5Þ The critical ratio of the first order condition is ðh þ SÞðS c þ pÞ ðpÞðc þ hÞ ¼ u½ðS cÞQ u½ðS A cQÞ þ hðA QÞ u½ðS cÞQ u½ðS cÞQ þ pðQ BÞ . ð6Þ Given that the newsvendor orders a quantity Q that satisfies Eq. (6), i.e. Q = Q*, then he may get a utility of u[(S c)Q*] as in the right hand side of (6) if the random demand will be exactly equal to the optimal quantity, i.e. De ¼ Q . Hence we can conclude that u½ðS cÞQ  ¼ uðTPðDe ¼ Q ÞjQ ¼ Q Þ. By similar con￾siderations the critical ratio (6) can be presented as ðh þ SÞðS c þ pÞ ðpÞðc þ hÞ ¼ uðTPðDe ¼ Q ÞjQ ¼ Q Þ uðTPðDe ¼ AÞjQ ¼ Q Þ uðTPðDe ¼ Q ÞjQ ¼ Q Þ uðTPðDe ¼ BÞjQ ¼ Q Þ ; ð7Þ given that the newsvendor sets his order quantity to the optimum Q = Q*. The optimum order quantity Q* sets the difference between the utilities from the total profit at points De ¼ Q and De ¼ A, divided by the difference between the utilities from the total profit at points De ¼ Q and De ¼ B to be a constant. It can B. Keren, J.S. Pliskin / European Journal of Operational Research 174 (2006) 1643–1650 1645

1646 B.Keren,J.S.Pliskin European Journal of Operational Research 174 (2006)1643-1650 be verified that for a risk-neutral newsvendor with a linear utility (ux)=a+bx),Eq.(6)converts back to the critical ratio of(4). Now we use (6)and(7)to show that O*for a risk-averse newsvendor is less than O*for a risk-neutral one.For simplicity of the proof,we set all the utility function values at the right hand side of(6)and(7)to be positive.This can be done since a Von Neumann and Morgenstern(1944)utility function is defined up to a positive linear transformation.We use the theorem that states that the ranking of a group of alternatives will remain constant,using the utility functions u(x)or u*(x)=a+bu(x),independent of the values of x,as long as b>0.Proof for this is presented by Levy and Sarnat(1984,pp.121-122). The ratio on the left hand side of (7)is larger than 1,1.That can be verified since this con- (πj(c+) dition can be written as(S-c(S+h)+Sπ>π·c.Under our pre-assumptionsπ>O,S>cand (c+h)>0,therefore Sn>c and (S-c)(S+h)>0.An interesting conclusion here is that the maxi- mum expected profit determines o*,so that (TP(D=Blo=O))>(TP(D=Al=O)),i.e.given that the newsvendor orders o*,the total profit under this condition will be larger if D=B occurs,than if D=A occurs. Now,since u(TP(D =Blo=O))>u(TP(D=AlO=O)),the rank of the utilities from the basic state- of-natural events can be ordered u(TP(D==))>u(TP(D=BlO=))>u(TP(D=Al0=0))>0. (8) Our solution of Eq.(6)is a special case of Eeckhoudt et al.'s(1995)result.It presents a simple closed form solution and hence can serve as a benchmark. Claim.If the demand for newspapers is uniformly distributed over A,B],a risk-averse newsvendor with a uniformly more concave utility function sets his O*to a lower value than a less risk-averse newsvendor.We note here that a more concave utility function is more risk-averse according to Pratt's (1964)measure of risk, r(x)=-'(x)/u'(x) Proof.Let us assume that for a risk-averse newsvendor with utility function ui(x),the optimal order quan- tity Oi was found and all the values of the right hand side of(6)were set to be positive.Under this assump- tion,the first order condition (6)can be written as follows: (h+S)(S-c+)u1[(S-c)Oi]-u[(S+h)A-Oi(c+h)] (9) (π)(c+h) [(S-c)Q1-[(S-c)Q+π(-B)] We use the concave utility function fx)to generate a more concave utility function ua(x)=fu(x)).Now we replace u()by u()on the right hand side of(6).The result of this operation,as shown in Lemma 1(see Appendix A),is that the right hand side of(6)increases and therefore we get (h+S)(S-c+π)2[S-c)2]-2I(S+h)A-21(c+h)] (10) (π(c+h) 2[(S-c)21]-[(S-c)21+π(21-B)] According to Lemma 2(see Appendix A),the right hand side of(6)increases monotonically with O1.There- fore,in order to attain equality for (10),O must be decreased.The final conclusion is that >3. Corollary.A risk-averse newsvendor with a concave utility function sets O*less than a newsvendor who is risk- neutral. For the right hand side ratio of ()the ratiohas three important properties: (1)This ratio is 0 for o=4 and oo for O=B. (2)Given any value of O that satisfies the condition of Ratio >1,from this point and up to O=B,the ratio increases monotonically with O.The proof is given in Lemma 2

be verified that for a risk-neutral newsvendor with a linear utility (u(x) = a + bx), Eq. (6) converts back to the critical ratio of (4). Now we use (6) and (7) to show that Q* for a risk-averse newsvendor is less than Q* for a risk-neutral one. For simplicity of the proof, we set all the utility function values at the right hand side of (6) and (7) to be positive. This can be done since a Von Neumann and Morgenstern (1944) utility function is defined up to a positive linear transformation. We use the theorem that states that the ranking of a group of alternatives will remain constant, using the utility functions u(x) or u*(x) = a + bu(x), independent of the values of x, as long as b > 0. Proof for this is presented by Levy and Sarnat (1984, pp. 121–122). The ratio on the left hand side of (7) is larger than 1, ðhþSÞðScþpÞ ðpÞðcþhÞ > 1. That can be verified since this con￾dition can be written as (S c)(S + h) + Sp > p Æ c. Under our pre-assumptions p > 0, S > c and (c + h) > 0, therefore Sp > p Æ c and (S c)(S + h) > 0. An interesting conclusion here is that the maxi￾mum expected profit determines Q*, so that ðTPðDe ¼ BjQ ¼ Q ÞÞ > ðTPðDe ¼ AjQ ¼ Q ÞÞ, i.e. given that the newsvendor orders Q*, the total profit under this condition will be larger if De ¼ B occurs, than if De ¼ A occurs. Now, since uðTPðDe ¼ BjQ ¼ Q ÞÞ > uðTPðDe ¼ AjQ ¼ Q ÞÞ, the rank of the utilities from the basic state￾of-natural events can be ordered uðTPðDe ¼ Q jQ ¼ Q ÞÞ > uðTPðDe ¼ BjQ ¼ Q ÞÞ > uðTPðDe ¼ AjQ ¼ Q ÞÞ > 0. ð8Þ Our solution of Eq. (6) is a special case of Eeckhoudt et al.s (1995) result. It presents a simple closed form solution and hence can serve as a benchmark. Claim. If the demand for newspapers is uniformly distributed over [A, B], a risk-averse newsvendor with a uniformly more concave utility function sets his Q* to a lower value than a less risk-averse newsvendor. We note here that a more concave utility function is more risk-averse according to Pratts (1964) measure of risk, r(x) = u00(x)/u0 (x)). Proof. Let us assume that for a risk-averse newsvendor with utility function u1(x), the optimal order quan￾tity Q 1 was found and all the values of the right hand side of (6) were set to be positive. Under this assump￾tion, the first order condition (6) can be written as follows: ðh þ SÞðS c þ pÞ ðpÞðc þ hÞ ¼ u1½ðS cÞQ 1 u1½ðS þ hÞA Q 1ðc þ hÞ u1½ðS cÞQ 1 u1½ðS cÞQ 1 þ pðQ 1 BÞ . ð9Þ We use the concave utility function f(x) to generate a more concave utility function u2(x)  f(u1(x)). Now we replace u1(Æ) by u2(Æ) on the right hand side of (6). The result of this operation, as shown in Lemma 1 (see Appendix A), is that the right hand side of (6) increases and therefore we get ðh þ SÞðS c þ pÞ ðpÞðc þ hÞ Q 2. h Corollary. A risk-averse newsvendor with a concave utility function sets Q* less than a newsvendor who is risk￾neutral. For the right hand side ratio of (6), the ratio u½ðScÞQu½ðSþhÞAQðcþhÞ u½ðScÞQu½ðScÞQþpðQBÞ has three important properties: (1) This ratio is 0 for Q = A and 1 for Q = B. (2) Given any value of Q that satisfies the condition of Ratio > 1, from this point and up to Q = B, the ratio increases monotonically with Q. The proof is given in Lemma 2. 1646 B. Keren, J.S. Pliskin / European Journal of Operational Research 174 (2006) 1643–1650

B.Keren.J.S.Pliskin European Journal of Operational Research 174 (2006)1643-1650 1647 25 -u(x)=In(x) 20 -u(x)=sqrt(x) 15 晨 10 04 150 160 170 180 190 Q* Fig.1.The ratio as a function of the optimal quantity for two utility functions. (3)If 1and the utility function (is uniformly more concave than) then for all O that satisfies the condition >1,we obtain the following inequality: ul(s-c)e]-u[(S+h)4-e(c+h]u[(s-c)o]-ua[(S+h)4-(c+h (11) [(S-c)g-[(S-c)0+π(Q-B)】u2l(S-c)g-2[(S-c)2+π(9-B] This inequality is a direct result of Lemma 1. An illustration of this property is given in Fig.1.The utility function u(x)=In(x)is uniformly more con- cave than u(x)=vx.It can be verified since Pratt's (1964)measures of risk aversion -u"(x)/u'(x)are and respectively.The ratio for the more concave utility function u(x)=In(x)is higher than the less concave u(x)=vx. The parameters of the illustration are h=-5,S=50,c=18,n=20,A=100,B=200.We can see from Fig.I that for every given ratio,the more risk-averse newsvendor should set his O*to a lower value than the less risk averse one. 4.The effect of the variance on O* Another interesting question is how o*should be changed as a response to small changes in the variance of the uniform random demand.We assume that O*is set first and then the variance is increased,while at the same time the mean is preserved.The mean of the uniform distribution is E(D)=(A+B)/2 and the variance is VAR(D)=(B-A)/12.In order to increase the variance and to preserve the mean,we should broaden the distribution interval from [A,B]to [A-a,B+a].The parameter >0 is any small positive value. We substitute the new distribution interval in(6) (h+S(S-c+_u(S-cg]-S·A-c2)+h(4-)-·(S+h (12) (π(c+h) [(S-c)g-[(S-c)2+π(Q-B)-x·元 The left hand side of(12)is the same as(6)and does not include the parameter a.In order to preserve the equality in(12),O on the right hand side should be adjusted.The influence of a on O on the right hand side of (12)is ambiguous since both the numerator and the denominator increase.The conclusion is that

(3) If 1 Q1, we obtain the following inequality: u1½ðS cÞQ u1½ðS þ hÞA Qðc þ hÞ u1½ðS cÞQ u1½ðS cÞQ þ pðQ BÞ 0 is any small positive value. We substitute the new distribution interval in (6) ðh þ SÞðS c þ pÞ ðpÞðc þ hÞ ¼ u½ðS cÞQ u½ðS A cQÞ þ hðA QÞ a ðS þ hÞ u½ðS cÞQ u½ðS cÞQ þ pðQ BÞ a p . ð12Þ The left hand side of (12) is the same as (6) and does not include the parameter a. In order to preserve the equality in (12), Q on the right hand side should be adjusted. The influence of a on Q on the right hand side of (12) is ambiguous since both the numerator and the denominator increase. The conclusion is that 0 5 10 15 20 25 150 160 170 180 190 Q* Ratio u(x)=ln(x) u(x)=sqrt(x) Fig. 1. The ratio as a function of the optimal quantity for two utility functions. B. Keren, J.S. Pliskin / European Journal of Operational Research 174 (2006) 1643–1650 1647

1648 B.Keren,J.S.Pliskin European Journal of Operational Research 174 (2006)1643-1650 Table 1 Optimal order quantity.as a function of variance [A,B].Variance Holding cost h=5 h=0 h=-5 h=-20 [100,2001,Var(D)=833 139.95 143.93 148.73 171.21 [95,205].Var(D)=1008 137.70 142.16 147.54 172.77 [90,210]Var(D)=1200 134.91 139.92 145.94 174.17 The fixed parameters for Table 1:S=50,c=30,=10,u(x)=v,E(D)=150.The contents of the table are values of o*. increasing a(i.e.increasing the variance)should increase or decrease O*.Table I includes examples for both cases which prove the claim that the variance has an ambiguous effect on O*.A higher variance should increase or decrease O*depending on the values of the problem's parameters. 5.An application-assessment of the newsvendor utility We can use Eq.(6)for assessment of the newsvendor utility parameters in a similar manner to the Schweitzer and Cachon (2000)experiments.A simple example is the following:A risk-averse newsvendor has an exponential utility function u(x)=1 -e#*,u>0.The problem's parameters are h=-5,S=50, c=18,n=20,A =100,B=200 and the newsvendor sets O*=190 as his optimal order quantity.Now we can compute the utility parameter u according to(13) 9=Ratio = 1-exp(u·[(S-c)g)-1-exp(u·I(S·A-c2)+h(A-]) (13) 1-exp(u·[S-c)g])-1-exp(u·[(S-c)2+π(Q-B)] This gives,by numerical computation,u=0.00051. A utility function with more than one parameter can be assessed by asking several hypothetical ques- tions.A quadratic utility function,for example,ux)=bx+cx2,needs at least two value points for assessment. Another method for assessment is to compare the newsvendor's risk-aversion to a benchmark risk-aver- sion and to find out if the newsvendor is more or less risk-averse than the benchmark.This can easily be done by comparing the optimal order quantity of the newsvendor to the O*of the benchmark newsvendor under the same parameters.We offer here a simple formula that can be used as a good benchmark.But,as Schweitzer and Cachon(2000)argued,part of their experimental findings(too low/too high patterns)can be explained by expected utility,prospect theory(risk seeking over losses and risk aversion over gains)and other models.However,it is unrealistic to expect that a human decision-maker would set his decision vari- ables to the optimal values in a complex environment. 6.Summary and conclusions We have solved a special case of the newsvendor problem.The solution provides an insightful and prac- tical interpretation to the optimal point.We can use the solution for the assessment of newsvendor utility parameters and even as a benchmark to more complex newsvendor problems.This method can be used to find the solutions of the newsvendor problem for other distributions.In future research,it can be applied to more complex versions of the newsvendor problem

increasing a (i.e. increasing the variance) should increase or decrease Q*. Table 1 includes examples for both cases which prove the claim that the variance has an ambiguous effect on Q*. A higher variance should increase or decrease Q* depending on the values of the problems parameters. 5. An application—assessment of the newsvendor utility We can use Eq. (6) for assessment of the newsvendor utility parameters in a similar manner to the Schweitzer and Cachon (2000) experiments. A simple example is the following: A risk-averse newsvendor has an exponential utility function u(x)=1 e l Æ x , l > 0. The problems parameters are h = 5, S = 50, c = 18, p = 20, A = 100, B = 200 and the newsvendor sets Q* = 190 as his optimal order quantity. Now we can compute the utility parameter l according to (13) 9 ¼ Ratio ¼ 1 expðl ½ðS cÞQÞ 1 expðl ½ðS A cQÞ þ hðA QÞÞ 1 expðl ½ðS cÞQÞ 1 expðl ½ðS cÞQ þ pðQ BÞ . ð13Þ This gives, by numerical computation, l = 0.00051. A utility function with more than one parameter can be assessed by asking several hypothetical ques￾tions. A quadratic utility function, for example, u(x) = bx + cx2 , needs at least two value points for assessment. Another method for assessment is to compare the newsvendors risk-aversion to a benchmark risk-aver￾sion and to find out if the newsvendor is more or less risk-averse than the benchmark. This can easily be done by comparing the optimal order quantity of the newsvendor to the Q* of the benchmark newsvendor under the same parameters. We offer here a simple formula that can be used as a good benchmark. But, as Schweitzer and Cachon (2000) argued, part of their experimental findings (too low/too high patterns) can be explained by expected utility, prospect theory (risk seeking over losses and risk aversion over gains) and other models. However, it is unrealistic to expect that a human decision-maker would set his decision vari￾ables to the optimal values in a complex environment. 6. Summary and conclusions We have solved a special case of the newsvendor problem. The solution provides an insightful and prac￾tical interpretation to the optimal point. We can use the solution for the assessment of newsvendor utility parameters and even as a benchmark to more complex newsvendor problems. This method can be used to find the solutions of the newsvendor problem for other distributions. In future research, it can be applied to more complex versions of the newsvendor problem. Table 1 Optimal order quantity, Q*, as a function of variance [A, B], Variance Holding cost h = 5 h = 0 h = 5 h = 20 [100, 200], VarðDeÞ ¼ 833 139.95 143.93 148.73 171.21 [95, 205], VarðDeÞ ¼ 1008 137.70 142.16 147.54 172.77 [90, 210], VarðDeÞ ¼ 1200 134.91 139.92 145.94 174.17 The fixed parameters for Table 1: S = 50, c = 30, p = 10, uðxÞ ¼ ffiffi x p ; EðDeÞ ¼ 150. The contents of the table are values of Q*. 1648 B. Keren, J.S. Pliskin / European Journal of Operational Research 174 (2006) 1643–1650

B.Keren.J.S.Pliskin European Journal of Operational Research 174 (2006)1643-1650 1649 Appendix A Lemma 1.Given three scalars C>B>A>0 and a concave function fx),f(x)>0,f"(x)tan(B)and (c1,u(TP(D=))>u(TP(D=B))>u(TP(D=4))>0 as was explained for(8),under our assumptions,u'()>0,(S-c)>0 and (c+h)>0.Hence the first two expressions of the numerator are positive.The third term is also positive since (TP(D=))<(TP(D=B)).The utility function uis fC) fB) f(C)fB) B 0 (C-B) C)A) fA) l (C-A) B C Fig.2.A concave function fx)

Appendix A Lemma 1. Given three scalars C > B > A > 0 and a concave function f(x), f0 (x) > 0, f00(x) tan(b) and f ðCÞf ðBÞ ðCBÞ 1. The claim is that the right hand side of Eq. (6), u½ðScÞQu½ðSAcQÞþhðAQÞ u½ðScÞQu½ðScÞQþpðQBÞ increases monotonically with Q in the range where the left hand side of (6) is larger than 1. Proof. Given that the newsvendor orders Q = Q*, the right hand side ratio of (7) is Ratio ¼ uðTPðDe ¼ Q ÞÞ uðTPðDe ¼ AÞÞ uðTPðDe ¼ Q ÞÞ uðTPðDe ¼ BÞÞ. Taking the derivative with respect to Q to generate the first order condition for optimality oðRatioÞ oQ ¼ ½uðTPðDe ¼ Q ÞÞ uðTPðDe ¼ BÞÞðS cÞu0 ðTPðDe ¼ Q ÞÞ þ½uðTPðDe ¼ Q ÞÞ uðTPðDe ¼ BÞÞðc þ hÞu0 ðTPðDe ¼ AÞÞ ½uðTPðDe ¼ Q ÞÞ uðTPðDe ¼ AÞÞ ½ðS cÞu0 ðTPðDe ¼ Q ÞÞ ðS c þ pÞu0 ðTPðDe ¼ BÞÞ * + ½uðTPðDe ¼ Q ÞÞ uðTPðDe ¼ BÞÞ2 . For Ratio > 1, uðTPðDe ¼ Q ÞÞ > uðTPðDe ¼ BÞÞ > uðTPðDe ¼ AÞÞ > 0 as was explained for (8), under our assumptions, u0 (Æ) > 0, (S c) > 0 and (c + h) > 0. Hence the first two expressions of the numerator are positive. The third term is also positive since u0 ðTPðDe ¼ Q ÞÞ < u0 ðTPðDe ¼ BÞÞ. The utility function u is AB C (C-A) O (C-B) f(C)-f(B) f(C) f(B) f(A) f(C)-f(A) α1 β α2 Fig. 2. A concave function f(x). B. Keren, J.S. Pliskin / European Journal of Operational Research 174 (2006) 1643–1650 1649

1650 B.Keren,J.S.Pliskin European Journal of Operational Research 174(2006)1643-1650 concave and therefore for u(TP(D=))>u(TP(D=B))we must have (u(TP(D="))- u'(TP(D=B)))0 and (Ratio)>0.口 a0 References Anvari,M.,1987.Optimality criteria and risk in inventory models:The case of the newsboy problem.Journal of Operational Research Society38.625-632. Arrow,J.K.,Karlin,S.,Scarf,H.,1958.Studies in the Mathematical Theory of Inventory and Production.Stanford University Press, Stanford,CA. Atkinson,A.,1979.Incentives,uncertainty,and risk in the newsboy problem.Decision Sciences 10.341-353. Chung,K.,1990.Risk in inventory models:The case of the newsboy problem,optimality conditions.Journal of Operational Research Society41.173-176. Eeckhoudt,L.,Gollier,C.,Schlesinger,H.,1995.The risk-averse (and prudent)newsboy.Management Science 41 (5),786-794. Hadley,G..Whitin,T.M.,1963.Analysis of Inventory Systems.Prentice-Hall,Englewood,Cliffs,NJ. Ismail,B.E..Loudberback,J.G.,1979.Optimizing and satisfying in stochastic cost-volume-analysis.Decision Sciences 10.205-217. Kabak.I.W.,Schiff,A.I.1978.Inventory models and management objectives.Sloan Management Review 19(2).53-59. Khouja,M.,1999.The single-period(newsvendor)problem:Literature review and suggestions for further research.Omega 27,537- 553. Lau,H.S.,1980.The newsboy problem under alternative optimization objectives.Journal of the Operations Research Society 31 (1), 525-535. Levy,H.,Sarnat,M.,1984.Portfolio and Investment Selection:Theory and Practice.Prentice-Hall International.Inc. Pratt,W.J.,1964.Risk aversion in the small and in the large.Econometrica 32,122-136. Schweitzer,M.E..Cachon,G.P.,2000.Decision bias in the newsvendor problem with a known demand distribution:Experimental evidence.Management Science 46(3),404-421. Von Neumann,J.,Morgenstern,O..1944.Theory of Games and Economics Behavior.Princeton University Press,Princeton,NJ. Within.T.M.,1955.Inventory control and price theory.Management Science 2,61-80

concave and therefore for u0 ðTPðDe ¼ Q ÞÞ > u0 ðTPðDe ¼ BÞÞ we must have ðu0 ðTPðDe ¼ Q ÞÞ u0 ðTPðDe ¼ BÞÞÞ 0 and oðRatioÞ oQ > 0. References Anvari, M., 1987. Optimality criteria and risk in inventory models: The case of the newsboy problem. Journal of Operational Research Society 38, 625–632. Arrow, J.K., Karlin, S., Scarf, H., 1958. Studies in the Mathematical Theory of Inventory and Production. Stanford University Press, Stanford, CA. Atkinson, A., 1979. Incentives, uncertainty, and risk in the newsboy problem. Decision Sciences 10, 341–353. Chung, K., 1990. Risk in inventory models: The case of the newsboy problem, optimality conditions. Journal of Operational Research Society 41, 173–176. Eeckhoudt, L., Gollier, C., Schlesinger, H., 1995. The risk-averse (and prudent) newsboy. Management Science 41 (5), 786–794. Hadley, G., Whitin, T.M., 1963. Analysis of Inventory Systems. Prentice-Hall, Englewood, Cliffs, NJ. Ismail, B.E., Loudberback, J.G., 1979. Optimizing and satisfying in stochastic cost-volume-analysis. Decision Sciences 10, 205–217. Kabak, I.W., Schiff, A.I., 1978. Inventory models and management objectives. Sloan Management Review 19 (2), 53–59. Khouja, M., 1999. The single-period (newsvendor) problem: Literature review and suggestions for further research. Omega 27, 537– 553. Lau, H.S., 1980. The newsboy problem under alternative optimization objectives. Journal of the Operations Research Society 31 (1), 525–535. Levy, H., Sarnat, M., 1984. Portfolio and Investment Selection: Theory and Practice. Prentice-Hall International, Inc. Pratt, W.J., 1964. Risk aversion in the small and in the large. Econometrica 32, 122–136. Schweitzer, M.E., Cachon, G.P., 2000. Decision bias in the newsvendor problem with a known demand distribution: Experimental evidence. Management Science 46 (3), 404–421. Von Neumann, J., Morgenstern, O., 1944. Theory of Games and Economics Behavior. Princeton University Press, Princeton, NJ. Within, T.M., 1955. Inventory control and price theory. Management Science 2, 61–80. 1650 B. Keren, J.S. Pliskin / European Journal of Operational Research 174 (2006) 1643–1650