E-companion to A risk- and ambiguity-averse extension of the max-min newsvendor order formula

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1 e-comanion to Han Du and Zuluaga: Etension of Scarf s ma-min order formula ec E-comanion to A risk- and ambiguity-averse etension of the ma-min newsvendor order formula Qiaoming Han School of Mathematics & Statistics Zhejiang University of Finance & Economics P R China qmhan@zufeeducn Donglei Du Faculty of Business Administration University of New Brunswick Fredericton NB Canada E3B 5A3 ddu@unbca Luis F Zuluaga Deartment of Industrial and Systems Engineering Lehigh University Bethlehem PA USA 5 luiszuluaga@lehighedu We highlight the main ideas behind the roof of Theorem The roofs resented here rely heavily on the duality (or geometric) aroach that has been widely used in the related literature from classical results (see eg Kemerman 96 Karlin and Studden 966 Isii 96) to very novel ones (see eg Chen et al Bertsimas and Poescu 5) Therein it is useful to let =( ):=( ) where i (i = ) reresents the i-th order non-central moments of the roduct s demand for given arameters > > That is the roduct s demand mean and variance are resectively given by and ( ) The roof of Theorem for Problem () can be divided into three stes Ste First of all in Aendi EC we consider the roblem (EC) of maimizing the variance of the rofit subject to an etra constraint E (( S ) + )=b v(; b) ma g ()=Var (min{ S} c) st E (S i )= i for i = E (( S ) + )=b a distribution on IR + (EC) where for any > >< = b : >: > + ale b ale <>: if q + ( ) + if ale 9 >= >; (EC) Here is the feasible region of b (See the e-comanion Aendi EC for details)

2 ec e-comanion to Han Du and Zuluaga: Etension of Scarf s ma-min order formula Theorem EC For any given b the otimal objective value of the above roblem (EC) is given by >< v(; b) = >: + ( + b) b( + b) if <ale and ( ) + ale b ale q ( ) + b( ) 4 b + if and Ste Second note that for any ale b ale ( )+ ( )( ) +( ) f () g ()= b c v(; b) imlying that h():=min f () g ()=min b b c v(; b) In Aendi EC we solve for otimal b Denote the following quantities: (EC3) b () = h () = ( b () = ) c + + h () = ( ( + ) + c) s b 3 () ( )( + ) +( ) A + h 3 () = c + ( ) +( ) +( ) Theorem EC For a given ifb () b () b 3 () then the otimal solution b and the corresonding value of the objective function h() for Problem (EC3) are given as follows for : (i) If ale then (b (b ()h h()) = ()) if > ( ) for + (b 3 ()h 3 ()) otherwise (ii) Otherwise >< (b 3 ()h 3 ()) if ale ( ) for (b + h()) = (b ()h ()) if >: ( ) < < + (b ()h ()) otherwise Ste 3 Finally Problem () is reduced to ma h() whose resolution resented in Aendi EC3 leads to Theorem ;

3 e-comanion to Han Du and Zuluaga: Etension of Scarf s ma-min order formula ec3 EC Proof of Theorem EC First simle algebra and the fact that E ((S )(S ) + )=E (((S ) + ) ) show that the objective function of Problems (EC) can be written as g () =Var (min{ S} c)=( ) Var Therefore Problem (EC) is reduced to S min Var + S i st E = i for i = i E ( S + )=b a distribution on IR + ; S + b ( )b By abusing the notations (S := S/ i := i / i ) the above roblem is simlified as q( b)= min Var (S ) + st E (S i ) = i for i = E ((S ) + ) = b a distribution on IR + (EC4) Consequently the rest of this section will focus on solving Problems (EC4) We first address the feasibility issue in Aendi EC and then resent closed-form solutions to Problem (EC4) in Aendi EC leading to the roof of Theorem EC EC Feasibility We recall two eisting results for bounding the eectation of the any random variable (S ) + subject to moments constraints: and ( ) = ma E ((S ) + ) st E (S i )= i for i = a distribution on IR + ( )= min E ((S ) + ) st E (S i )= i for i = a distribution on IR + The following result follows from the well-know formula by Scarf (cf Scarf 95) and from Zuluaga et al (9 Thm 3) Lemma EC If and then >< if > ( )= >: ( ) + ( ) +( ) if ale and ( )=( ) +

4 ec4 e-comanion to Han Du and Zuluaga: Etension of Scarf s ma-min order formula E(S ) + Figure EC Illustration of Lemma EC Plots of ( )and( ) for =5 as a function of The bounds of Lemma EC are illustrated in Figure EC from which one can observe that E ((S ) + ) imlies that ale Therefore Problem (EC4) is feasible if and only if and b (see the definition of in (EC)) EC Closed-form solution for Problem (EC4) We solve Problems (EC4) by demonstrating a air of rimal and dual feasible solutions with equal objective values The dual roblem corresonding to Problem (EC4) can be written as: (Q d ) q d = ma y + y + y + by b b st y + y s + y s + by (s ) + ale ((s ) + ) s IR + Clearly weak duality holds between Problem (EC4) and (Q d ) that is q( b) q d Lemma EC If ( b) is feasible for Problem (EC4) and b< b = >< if s = s ( ) ( ) b = b (s)= = ( b ) if s = ( ) b >: 3 = + b if s = is feasible for Problem (EC4) then (EC5) Proof of Lemma EC First the lemma assumtion imlies that and b namely ( ) + ale b< (EC6)

5 e-comanion to Han Du and Zuluaga: Etension of Scarf s ma-min order formula ec5 The first inequality in (EC6) imlies that 3 = + b +( ) + The second inequality in (EC6) imlies that s > since >+ b and since ( ) b = b> b> (EC7) So (s) is a robability distribution Moreover s > from (EC7) which along with simle algebra shows that E () = E (S)= E (S )= and E ((S ) + )=b Lemma EC3 If ( b) is feasible for Problem (EC4) and b then (b+( )+r( b)) b ( ) r( b) >< = if s = s ( + ) = + b (s)= ( )+r( b) >: if s = s = ( +b) (EC) is feasible for Problem (EC4) where Proof of Lemma EC3 r( b)= ( ) (( ) +4b( ) 4b ) (EC9) We first show that ale s ale The lemma assumtion imlies that b and ale Therefore ( + b) ( ) = ( + b) ( + ) ( + ) ( ) =( )( ) Note that s if and only if (( + b) ( )) r ( b)=( ) (( ) +4b( ) 4b ) or equivalently (b + )( (b ) + ) after simle algebra which is evidently true On the other hand we will show that s < which is trivially true if ale In the case of > b ale ( ) + ( ) +( ) imlies that ( )( + b) ( ) < imlying further that ( )( + b) ( ) < r( b) or equivalently s < Then we will show that s > which is clearly true in case of > since ( ) r( b) from b Consider the case of ale It follows from b that ( ) ( ) b (( + )( ) +( ) ) Then simle algebra shows that ( ) ( ) b> r( b) or equivalently s > We may rewrite as = ( s ) ( s ) + imlying that ale ale So (s) is a robability distribution Finally s > and s < along with simle algebra show that E () = E (S)= E (S )= E ((S ) + )=b We now resent the closed-form solution for Problem (EC4) Theorem EC3 If ( b) is feasible for Problem (EC4) then ( ) b b >< if b< q( b)= >: b( ) + ( ) r( b) b if b where r( b) is defined in (EC9)

6 ec6 e-comanion to Han Du and Zuluaga: Etension of Scarf s ma-min order formula Proof of Theorem EC3 Case b< We consider two cases If( b) is feasible for Problem (EC4) then (s) in (EC5) is feasible for Problem (EC4) from Lemma EC and its corresonding objective value is equal to ( ) b b Thusq ale ( ) b b if ˆ< / Consider the dual solution for (Q d ) given by y = y = y = and by = This solution is feasible for (Q d ) and its corresonding objective value is equal to ( ) b b ale q d The weak duality imlies that q( b)=( ) b b when b< Case b If( b) is feasible for Problem (EC4) then (s) in (EC) is feasible for Problem (EC4) from Lemma EC3 and its corresonding objective value is equal to (b( ) + (( ) r( b))) b Thusq ale (b( ) + (( ) r( b))) b Consider the dual solution for (Q d ) given by y = (b ) + r( b) >< (( )b +( ) b( + ( ))) y = b r( b) (b ( ) + b( + ( 3))) y = r( b) >: (( ) +b( ) b ) by =( ) (( )( b)) r( b) Note that and that y = = 4 (( r( b) ) +b( ) b ) r( b) ( ) + ( ) r( b) + < y + y s + y s = y (s s ) ale y + y s + y s + by (s ) (s ) =(y )(s s ) ale (EC) where s and s are defined in (EC) That is the solution in (EC) is feasible for (Q d ) and its corresonding objective value is equal to (b( ) + (( ) r( b))) b ale q d The weak duality imlies that q( b)= (b( ) + (( ) r( b))) b when b Theorem EC3 after simle algebra yields the otimal solution for Problem (EC) given in Theorem EC EC Proof of Theorem EC For we need to solve the following otimization roblem (see (EC3)): min (b)= b c b v(; b) We consider two cases based on the range of b Case ( ) + ale b ale where<ale From Theorem EC we have that (b)= b c + ( + b) b( + b)

7 e-comanion to Han Du and Zuluaga: Etension of Scarf s ma-min order formula ec7 and the derivative of (b) is given by (b)= +! ( b) + ( + b) b( + b) For ale we have that (b) ale (hence (b) is non-increasing) since b ale that the minimizer of (b) is achieved at the right boundary b = first-order condition (b)= o ers the unique stationary oint b = minimizer of (b) can be therefore decided as follows: (i) If < then we have that + + > that is (b) < (hence (b) is decreasing) when ( minimizer of (b) is achieved at the right boundary ; ) + ale b ale ale imlying = b () For > the + + = b () The imlying that the b = = b () (ii) Otherwise we have minimizer of (b) is given by then( ) + ale + ale + and hence the b = + + = b () Case q ( ) where y = ale b ale ( )+ ( )( ) +( ) where (b)= b c (b)= y + From Theorem EC we have that + b( ) 4 b The derivative of (b) is given by + ( b) y The first-order condition (b)= o ers the unique stationary oint: s b 3 ( )( + ) +( ) A + The minimizer of (b) can be therefore decided as follows: (i) If > ( ) + then we have that b 3 () < when (ii) Otherwise we have ale b ale ( )+ ( )( ) +( ) imlying that b = + ale b 3 () ale at the stationary oint namely b = b 3 () ; that is (b) > (hence (b) is increasing) = b () imlying that the otimal solution is achieved

8 ec e-comanion to Han Du and Zuluaga: Etension of Scarf s ma-min order formula EC3 Proof of Theorem We rove two lemmas before stating the final roof Theorem Lemma EC4 For any if Proof of Lemma EC4 Case If ale and c then c we have + ( c) +4c( c) We consider three cases + ( c) +4c( c) Case If ale and < c then it follows that ale c < c imlying that 4 (( ) c) ( c) ale ale ( ) which is equivalent to the desired inequality Case 3 If > then it follows that 3 4 >c(6 ) c( + ) from c>c imlying that 4( c) ( ) ( c) which together with the assumtion ale c imlies that 4(( ) c) ( c) ( ) which is equivalent to the desired inequality Lemma EC5 Let If c and then Proof of Lemma EC5! = + ( c) +4c( c) ale ( ) + We consider two cases Case The case assumtion imlies that + > Moreover ale ale c imlies that ( ) ( ) +( ) c( c) or equivalently ( + ) ale ( )! + ( c) +4c( c) imlying the desired claim after simle algebra Case < We consider three subcases (i) First of all if < then the roof follows from the same argument as in Case since + >

9 e-comanion to Han Du and Zuluaga: Etension of Scarf s ma-min order formula ec9 (ii) Second of all if and ale then imlies that + < for h any > Note that ( ) is an increasing function of in the range + decreasing in + Moreover ( ) lim!+ + = Therefore the desired result follows from the second case assumtion: ale < ( ) + and (iii) Now we consider the remaining case where < and > Note that > > ( ) c equivalently imlies that ( ) +( ) ( ) + c( c) or ( )! + ( c) ale +4c( c) imlying the desired claim after simle algebra Now we have all the ingredients to rove Theorem Proof of Theorem From Theorem EC for we consider three cases as follows: Case h() =h () = c under the condition > ( ) If + ale c thenh() being a linear function with a non-ositive sloe imlies that the maimizer of h() is ( ) = Otherwise > c namely h() has a ositive sloe then ( ) should assume the largest ossible value in its range Note that the right-hand-side of the case condition > ( ) as a function of is increasing imlying that the largest feasible + value of satisfies that = ( ) which shares the same assumtion as that for Case 3 + and hence the desired result is imlied by the argument therein Case h() =h () = + + c Thenh () being non-increasing imlies that the maimizer of h() is ( ) = Case 3 h()=h 3 () where h 3 ()= c + ( ) +( ) +( The first-order condition o ers the unique stationary oint of h() as! ( )= + ( c) c = + +4c( c) +4c( c) ) Lemmas EC4 and EC5 together imly that ( ) is the maimizer of h() and ale ( ( ) ) and hence + ( ) ( ) ( )

10 ec e-comanion to Han Du and Zuluaga: Etension of Scarf s ma-min order formula EC3 Worst-case distribution in Theorem Note that we let S := S/ in roblem (EC4) So in the worst-case distribution in equation (EC5) we should relace s with s Therefore oints s = and s = merge into one oint s = when tends to zero () when ˆ = = we have that =! () when ˆ = = ( ) + + ( ) ( ) ( + we have that =! once! + Now the worst-case distribution is given below in two cases:! and s ) = (! once ( + ) + ( ) ( + Case : c< : ( ) = The worst-case distribution is ( if s = s = (s)= if s = s = ) + )! and s = Case : c : s ( )= +( c)! = + +4c( c) c +4c( c) The worst-case distribution is >< = 4c if s = s 4c (s)= +( h( )= +4c( c) ) >: if s = s`( )= + ( +4c( c) ) c ( +4c( c)+ ) ( c) = + = +4c( c) c +4c( c)+ ( c) EC4 Risk-averse vs risk-neutral: close to worst-case distribution Both Scarf s and the risk and ambiguity-averse orders (cf Theorem ) are designed to immunize against otential worst-case scenarios Thus it is relevant to comare these orders when the out-ofsamle demand follows a distribution that belongs to the class of worst-case distributions associated with the solution of () and given in (4) (recall that the worst-case distribution associated with Scarf s order olicy corresonds to (4) when = ) In this section we resent results similar to the ones in Section 3 when the demand distribution is close to being a worst-case distribution (4) for a given value of By close we mean that the demand is equal to X + Y where X follows the worst-case distribution (4) and Y N ( (X)); that is Y follows a normal distribution with mean zero and a standard deviation that is a fraction ale < of the standard deviation of the worst-case distribution Throughout the eeriments in this section we set = Note that because the worst-case distributions in (4) are two-oint distributions adding the normal noise Y to them results in distributions that are more realistic

11 e-comanion to Han Du and Zuluaga: Etension of Scarf s ma-min order formula ec but yet with a behavior very close to the original worst-case distribution The e ect of this noise in the moments of the demand s distribution is taken into account by setting the second (non-central) moment to ( + )+ to obtain the corresonding Scarf s and the risk- and ambiguity-averse order Figures EC EC3 below and Figure EC5 in Aendi EC5 comare the out-of-samle Share ratio for both Scarf s and the risk and ambiguity-averse order when the out-of-samle demand follows a distribution close to the worst-case distribution (4) for values = 5 5 To emhasize that the demand s distribution is not assumed to be fully known here we show the behaviour of the Share ratio for di erent choices of the risk-aversion arameter used to comute the risk- and ambiguity-averse order (ie not necessarily equal to the one s used to obtain the close to worst-case distributions) Analogous observations to those in Section 3 can be drawn for Figures EC and EC3; namely the risk- and ambiguity-averse aroach leads to either larger or comarable Share ratios than the risk-neutral and ambiguity-averse aroach Additional eeriments not resented here show that the this behavior in terms of out-of-samle Share ratio holds also when the demand distribution eactly follows a worst-case distribution (ie when no normal noise is added) 65 6 c>5 CV=6 = =5 = =5 6 5 c>5 CV=6 = =5 = = Share Ratio Share Ratio Figure EC Out-of-samle Share ratios vs risk-aversion for di erent coe cient of variation CV = / =65 c = 4/9 when the out-of-samle demand distribution follows the wort-case distribution (4) with values of risk-averse arameter = 5 5 The circled oints on the vertical ais identify the ma-min order formula of Scarf (95) Following the eeriments in Section 3 the results resented in Figure EC EC3 and EC5 are obtained by setting = 65 / =3 5 7 and c =3/4 4/9

12 ec e-comanion to Han Du and Zuluaga: Etension of Scarf s ma-min order formula c<5 CV=6 = =5 = = c<5 CV=6 = =5 = = Share Ratio Share Ratio Figure EC3 Out-of-samle Share ratios vs risk-aversion for di erent coe cient of variation CV = / =65 c = 3/4 when the out-of-samle demand distribution follows the wort-case distribution (4) with values of risk-averse arameter = 5 5 The circled oints on the vertical ais identify the ma-min order formula of Scarf (95) EC5 Additional Figures 4 c>5 CV=5 5 c<5 CV=5 3 Lognormal Gamma Beta Uniform 4 Lognormal Gamma Beta Uniform Share Ratio 9 7 Share Ratio Figure EC4 Out-of-samle Share ratios vs risk-aversion for di erent coe cient of variation CV = / =65 c = 3/4 (left) c = 4/9 (right) when the out-of-samle demand distribution follows common distributions The circled oints identify the ma-min order formula of Scarf (95)

13 e-comanion to Han Du and Zuluaga: Etension of Scarf s ma-min order formula ec3 3 c>5 CV=4 = =5 = =5 5 c<5 CV=4 = =5 = =5 6 Share Ratio 4 6 Share Ratio Figure EC5 Out-of-samle Share ratios vs risk-aversion for di erent coe cient of variation CV = / =65 c = 3/4 (left) c = 4/9 (right) when the out-of-samle demand distribution follows the wortcase distribution (4) with values of risk-averse arameter = 5 5 The circled oints identify the ma-min order formula of Scarf (95) References Bertsimas D and Poescu I (5) Otimal inequalities in robability theory: A conve otimization aroach SIAM J Otim 5(3):7 4 Chen L He S and Zhang S () Tight bounds for some risk measures with alications to robust ortfolio selection Oerations Research 59(4):47 65 Simchi-Levi D Chen X and Bramel J (4) The Logic of Logistics: Theory Algorithms and Alications for Logistics Management (Third Edition 4 Second Edition 5) Sringer-Verlag New York Karlin S and Studden W (966) Tchebyche Systems: with Alications in Analysis and Statistics Pure and Alied Mathematics Vol XV A Series of Tets and Monograhs Interscience Publishers John Wiley and Sons Kemerman J H B (96) The general moment roblem a geometric aroach The Annals of Mathematical Statistics 39():93 Isii K (96) The etrema of robability determined by generalized moments (i) bounded random variables Ann Inst Stat Math :9 33 Scarf H (95) A min-ma solution of an inventory roblem In Arrow K J Karlin S and Scarf H editors Studies in the Mathematical Theory of Inventory and Production ages 9 Stanford University Press Zuluaga L Peña J and Du D (9) Etensions of Lo s semiarametric bound for Euroean call otions Euroean Journal of Oerational Research 9:557 57