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Int. J. Production Economics

Int. J. Production Economics 117 (2009) 136–141

0925-52

doi:10.1

� Cor

E-m

journal homepage: www.elsevier.com/locate/ijpe

A binary solution method for the multi-product newsboy problemwith budget constraint

Bin Zhang a,b, Xiaoyan Xu a, Zhongsheng Hua a,�

a School of Management, University of Science and Technology of China, Hefei, Anhui 230026, PR Chinab Lingnan College, Sun Yat-sen University, Guangzhou 510275, PR China

a r t i c l e i n f o

Article history:

Received 6 July 2006

Accepted 8 October 2008Available online 19 October 2008

Keywords:

Newsboy problem

Budget constraint

Optimization

73/$ - see front matter & 2008 Elsevier B.V. A

016/j.ijpe.2008.10.003

responding author. Tel.: +86 5513607792; fax

ail address: zshua@ustc.edu.cn (Z. Hua).

a b s t r a c t

Multi-product newsboy problem (MPNP) with budget constraint is a classical inventory

control/management problem. However, solution methods for MPNP under general

demand distributions are limited in the current literature. In this paper, by analyzing

properties of the optimal solution to the MPNP with a budget constraint, we develop a

solution algorithm for the constrained MPNP. The proposed algorithm is binary in

nature, and is applicable to general types of demand distribution functions, discrete as

well as continuous. For continuous demand distribution function, our approach can

obtain the optimal or near optimal solution to the constrained MPNP with polynomial

computation complexity of the o(n) order. On the other hand, for discrete demand

distribution functions, it can effectively provide good approximate solution. Numerical

experiments are presented to show the performance of our method.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

Multi-product newsboy problem (MPNP) with budgetconstraint, introduced firstly by Hadley and Whitin (1963),is a classical inventory control/management problem.Hadley and Whitin (1963) presents a solution method tothe constrained MPNP, which encounters difficulties,particularly when the number of products is rather large.After Hadley and Whitin’s early work, many researchershave developed different solution methods for MPNPswith different application background. Khouja (1999)presents a good literature review on these researches.

Erlebacher (2000) develops heuristic solutions for theMPNP with one capacity constraint. He begins by provingthe optimality of the order quantities for two specialcases, then he proceeds by developing heuristics for a fewspecific probability distribution functions. Vairaktarakis(2000) develops several minimax regret formulations forthe MPNP with a budget constraint. His approach can

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: +86 5513600025.

obtain the optimal solution only when the values ofdemand are intervallic or discrete. For MPNP undergeneral demand distributions, these approaches provideheuristic solutions rather than the optimal solutions.

Abdel-Malek et al. (2004) develops Lagrangian-basedmethods, which yield the optimal solution to the problemwhen the demand is uniformly distributed, and nearoptimal solution when the demand is other continuousdistributions. As it is pointed out by Abdel-Malek andMontanari (2005a), most of the existing Lagrangian-basedmethods do not pay much attention to the lower boundsof the order quantities (non-negativity constraints), e.g.,Abdel-Malek et al. (2004), Ben-Daya and Raouf (1993),Erlebacher (2000), Gallego and Moon (1993), Khouja(1999), Moon and Silver (2000), and Vairaktarakis(2000). This negligence, as observed by Lau and Lau(1995, 1996) could lead to infeasible order quantities(negative) for some of the considered products.

To address non-negativity constraints of the orderquantities, Abdel-Malek and Montanari (2005a) extendthe research of Abdel-Malek et al. (2004), and propose amodified Lagrangian-based method by analyzing thesolution space. Their approach, however, can be applied

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B. Zhang et al. / Int. J. Production Economics 117 (2009) 136–141 137

to obtain the optimal solutions only for special demanddistributions, e.g., uniform and normal demand.

In summary, the existing methods for the capacitatednewsboy problems have the following limitations: (1)many existing Lagrangian-based models could lead toinfeasible order quantities (negative) because of relaxingthe lower bounds of the demand (Lau and Lau, 1995,1996); (2) current solution methods can only solve theoptimal solution for the special cases (e.g., Erlebacher,2000; Abdel-Malek et al., 2004); (3) for general demanddistributions, the existing methods can only provideapproximate or heuristic solutions (e.g., Erlebacher,2000; Abdel-Malek and Montanari, 2005a).

In this paper, by analyzing properties of the optimalsolution to the MPNP with a budget constraint, wedevelop a solution algorithm for the constrained MPNP.The proposed algorithm can overcome some of theaforementioned limitations of the current methods.Additionally, it is applicable to both types of demanddistribution functions, discrete as well as continuous.

The reminder of this paper is organized as follows. Wedescribe the constrained MPNP problem and the optimalsolution to the unconstrained MPNP in Section 2. InSection 3, by presenting the properties of the optimalsolution to the constrained MPNP, we develop a binarysolution method for the constrained MPNP under generaldemand distribution. Numerical examples are illustratedin Section 4. Section 5 briefly concludes the paper. Theproofs are presented in Appendix A.

2. The constrained MPNP

2.1. Mathematical model

In order to introduce a clear description of theconstrained MPNP, we first present the definition of thenotations in Table 1.

The model of the constrained MPNP can be expressed as

Min E ¼Xn

i¼1

EiðxiÞ

¼Xn

i¼1

½cixi þ hiEðxi � DiÞþþ viEðDi � xiÞ

þ�, (1)

Table 1Notations.

Notations Definitions

n total number of products

i product index

vi cost of revenue loss per unit of product i

hi cost incurred per product i for leftover at the end of the

specified period

ci cost per unit of product i

xi amount to be ordered of product i which is a decision variable

Di random demand of product i

fi (Di) probability density function of demand for product i

Fi (Di) cumulative distribution function of demand for product i

Ei expected cost function of product i

E total expected cost function

B budget function

BG available budget

subject to

B ¼Xn

i¼1

cixipBG, (2)

xiX0; i ¼ 1;2; . . . ;n (3)

The total expected cost in Eq. (1) is the sum of purchasecost, expected overage and underage costs of all products.Eq. (2) is a budget constraint, and Eq. (3) expresses thenon-negativity constraints on order quantities.

2.2. Optimal solution to the unconstrained problem

By relaxing the budget constraint, the constrainedMPNP becomes an unconstrained problem. The optimalsolution of the unconstrained problem can be solved bytaking the partial derivatives of E with respect to xi

(i ¼ 1,2,y,n) and setting them to zero (Hadley and Whitin,1963).

If the distribution of demand D is continuous, we caninterchange the derivative and the expectation operators,it follows that

qE=qxi ¼ ci þ hiEdðxi � DiÞ � viEdðDi � xiÞ; i ¼ 1;2; . . . ;n,

(4)

where d(x) ¼ 1 if xi40 and zero otherwise. SinceEd(xi�Di) ¼ Fi(xi) and Ed(Di�xi) ¼ 1�Fi(xi), Eq. (4) can berewritten as

qE=qxi ¼ ci � vi þ ðhi þ viÞFiðxiÞ; i ¼ 1;2; . . . ;n. (5)

Setting the derivative to zero reveals that

Fiðx�i Þ ¼ ðvi � ciÞ=ðhi þ viÞ ¼ yi; i ¼ 1;2; . . . ;n. (6)

Note that the expected cost is a convex function of xi,(i ¼ 1,2,y,n) since q2E=qx2

i ¼ f iðxiÞX0. If Fi(xi) is strictlyincreasing, Fi(xi) has an inverse function; otherwise wedefine the inverse function and yield the unique optimalsolution as follows:

x�i ¼ F�1i ðyiÞ ¼ inffxiX0jFiðxiÞXyig; i ¼ 1;2; . . . ;n. (7)

If the distribution of demand D is discrete, by writingEðxi � DiÞ

þ¼P1

j¼xiðxi � jÞPrðDi ¼ jÞ and working with the

forward difference, it is easy to see that the optimalsolution is given by x�i ¼ minfxi 2 FjFiðxiÞXyig, where F isthe support of the discrete demand D (you can refer tomathematics book).

3. The binary solution method

Before presenting the binary solution method, we firstpresent some properties of the optimal solution to theconstrained MPNP.

3.1. Properties of the optimal solution to the

constrained MPNP

To analyze properties of the optimal solution to theconstrained MPNP, in the spirit of Hua et al. (2006), we

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B. Zhang et al. / Int. J. Production Economics 117 (2009) 136–141138

define a marginal budget benefit function of product i as

riðxiÞ ¼ ðqE=qxiÞ=ðqB=qxiÞ

¼ ðci � vi þ ðhi þ viÞFiðxiÞÞ=ci; i ¼ 1;2; . . . ;n. (8)

The inverse function of ri(xi) can be expressed as

xiðriÞ ¼ F�1i ðððri � 1Þci þ viÞ=ðhi þ viÞÞ; i ¼ 1;2; . . . ;n (9)

By analyzing Eqs. (8) and (9), we have the followingLemma about the marginal budget benefit function:

Lemma 1. For i ¼ 1,2,y,n, we have

(a)

ri(xi) is a non-decreasing function of xi on xiX0; (b) 1�vi/cipri(xi)o0 for all xi 2 ½0; x

�i Þ.

(c)

xi(ri) is a strictly increasing function of ri on 1�vi/cipri(xi)o0.

The proof of Lemma 1 can be found in Appendix A.1.

Let x��i (i ¼ 1,2,y,n) denote the optimal solution to theconstrained MPNP, and denote by O ¼ fijx��i a0; i ¼

1;2; . . . ;ng the set of product indexes of the optimal non-zero orders. Based on Lemma 1, we have the followingproposition:

Proposition 1. The optimal solution to the constrained

MPNP, x��i (i ¼ 1,2,y,n), has the following properties:

(a)

IfPn

i¼1cix�i pBG, then x��i ¼ x�i .P

(b)

If ni¼1cix

�i 4BG, then x��i ox�i for all (i ¼ 1,2,y,n), andPn

i¼1cix��i ¼ BG.

(c)

rjðx��j Þ ¼ rkðx

��k Þ for all j, kAO.

Fig. 1. Main steps of Algorithm 1.

Property (a) is intuitive since the budget constraint isnot active. The proofs of the last two properties can befound in Appendices A.2 and A.3.

Property (a) indicates that, if budget constraint isnot binding, then the optimal solution of the constrai-ned MPNP is the same as that of the unconstrainedproblem. Property (b) illustrates that, if the budgetconstraint is binding, i.e.,

Pni¼1cix

�i 4BG, the optimal

solution is smaller than that of the unconstrainedproblem. In this case, it is a necessary condition forthe optimal solution to the constrained MPNP that thebudget is fully utilized. Property (c) suggests that, forthe non-zero optimal orders, their marginal budgetbenefits should be equal, no matter whether the budgetconstraint is binding or not. This property is true sincethe total expected cost can be decreased by movingsome budget from the product with larger marginalbudget benefit to the product with smaller marginalbudget benefit.

3.2. Solution method

According to Lemma 1(c) and Proposition 1, theoptimal solution to the constrained MPNP is unique.When the budget constraint is binding, x��i (i ¼ 1,2,y,n) isthe optimal solution to the constrained MPNP under

continuous demand distribution if

Pni¼1

cix��i ¼ BG

x��i ¼ xiðr��Þ if ððr�� � 1Þci þ viÞ=ðhi þ viÞ4Fið0Þ;

x��i ¼ 0 otherwise

8>>>><>>>>:

(10)

where r�� ¼ riðx��i Þ is the marginal budget benefit at x��i

(iAO). Eq. (10) implies that, obtaining the exact value of r��

is the key in determining the optimal order quantity foreach of the competing products.

According Lemma 1(b), we have mini¼1;2;...;n

f1� vi=cigpr��o0. Because xi(ri) is a strictly increasingfunction of ri (Lemma 1(c)), we can determine r�� bymaximizing its value over the interval ½mini¼1;2;...;nf1�vi=cig;0� while satisfying Eq. (10). This can be done byapplying a binary search method over the interval.

Main steps of our approach for solving the optimalsolution to the constrained MPNP are summarized inAlgorithm 1 (as shown in Fig. 1).

In Algorithm 1, we first solve the unconstrained MPNP(Step 0) to obtain x�i . Then we judge whether x�i leads to abinding budget constraint or not (Step 1). If x�i leads to abinding budget constraint, then we apply the binarysearch procedure over interval [rL,rU] to determine x��i(i ¼ 1,2,y,n) (Steps 2–5). It is noteworthy that the optimalzero orders are identified in Step 3 through testing thecondition ((r�1)ci+vi)/(hi+vi)pFi(0). It is obvious thatAlgorithm 1 can be directly extended for the constrainedMPNP with non-zero lower bound constraints on orderquantities.

It is obvious that Algorithm 1 is a polynomial algorithmof the o(n) order. Since we do not assume any specificproperty on demand distribution, our approach can beapplied to the constrained MPNP under any theoreticaldemand distribution (e.g., normal, beta, uniform, bino-mial, etc.), even under an empirical demand distribution(e.g., Hua and Zhang, 2006; Hua et al., 2007).

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When the demand distribution is discrete, because thefeasible values of the marginal budget benefit functionri(xi), i ¼ 1,2,y,n, are discontinuous, it is possible thatProposition 1(c) can be only approximately met. Inthis case, Algorithm 1 may stop when rL ¼ rU andPn

i¼1cix��i aBG, and the resulting solution is an approxi-

mate solution to the constrained MPNP. At the approx-imate solution, the budget may have gone beyond or mayhave leftover.

4. Numerical results

In this section, we present numerical examples toillustrate our approach. The first example is for the normaldemand distribution which comes from Abdel-Malek andMontanari (2005a). The second example is for betademand distribution, cited from Abdel-Malek et al.(2004). We also present numerical experiments to showthe applicability of our method to discrete demanddistribution.

4.1. The normal distribution example

In this example, demands of 17 products are allnormally distributed, and there is a budget constraintBG ¼ $2500. Table 2 presents the relevant information,where mi, si, i ¼ 1,2,y,n, are parameters of the mean andstandard deviation of the normal demand, x��i;GIM is thesolution cited from Abdel-Malek and Montanari (2005a),and x��i;Opt is the optimal solution obtained by applying ouralgorithm.

From Table 2, it can be observed that GIM and ourmethod provide similar results for normal demanddistribution.

4.2. The beta distribution example

When demands of six products are of beta distribu-tions and when there is a budget constraint BG ¼ $6500,

Table 2Parameters and solutions for normal demand distribution.

Product vi hi ci mi si x�i x��i;GIM x��i;Opt

1 7 1 4 102 51 85.75 0 0

2 12 2 8 73 18.3 62.64 0 0

3 30 4 19 123 30.8 108.90 0 0

4 30 4 17 95 23.8 87.88 0 0

5 40 2 23 62 15.5 58.26 0 0

6 45 5 15 129 43 139.89 106.86 106.85

7 16 1 10 69 34.5 55.98 0 0

8 21 2 10 83 41.5 80.74 14.02 14.01

9 42 3 40 120 30 68.96 0 0

10 34 5 20 89 22.3 80.95 0 0

11 20 3 10 115 38.3 108.71 15.58 15.65

12 15 5 7 91 30.3 83.32 42.20 42.25

13 10 3 4 52 17.3 50.33 34.56 34.60

14 20 3 12 76 38 61.14 0 0

15 47 2 33 66 16.5 56.66 0 0

16 35 4 21 147 36.8 133.71 0 0

17 22 1 11 104 34.7 102.11 15.23 15.13

E 30 887 39 821 39 821

we apply our approach to the constrained MPNP, andcompare our results with those of Abdel-Malek et al.(2004) in Table 3. In Table 3, xi,min, xi,max, ai and bi are theparameters of beta distribution, x��i;Opt is the optimalsolution obtained by applying our algorithm, and x��i;GIM isthe solution cited from Abdel-Malek et al. (2004).

According to the solution x��i , i ¼ 1,2,y,n, the expectedoverage and underage costs of product i can be, respec-tively, calculated by

hiEðxi � DiÞþ¼ hi

Z x��i

xi;min

f ðziðDÞ;ai;biÞðx��i � DÞdD,

i ¼ 1;2; . . . ;n, (11)

viEðDi � xiÞþ¼ vi

Z xi;max

x��i

f ðziðDÞ;ai;biÞðD� x��i ÞdD,

i ¼ 1;2; . . . ;n, (12)

where f(zi(D),ai,bi) is the probability density function ofbeta distribution with zi(D) ¼ (D�xi,min)/(xi,max�xi,min),i ¼ 1,2,y,n.

According to Eq. (1), the total expected cost can becalculated by summing up the purchase cost cix

��i , the

expected overage and underage costs calculated byEqs. (11) and (12) of all products. When calculating thecosts of product 6, because f ð0;a6;b6Þ ! 1 andf ð1;a6;b6Þ ! 1, we set the lower limit of integraloperator in Eq. (11) as (1+10�6)x6,min instead of x6,min,and set the upper limit of integral operator in Eq. (12) as(1�10�6)x6,max instead of x6,max.

As presented in Table 3, in comparison with x��i;Opt , x��i;GIM

is an approximate solution with a gap of 1.81%( ¼ (475140�466695)/466695) on the total expected cost.From Table 3, it can be observed that our method providesslightly better results than GIM does. In comparison withGIM suggested by Abdel-Malek and Montanari (2005a),our method is easy to follow and requires less computa-tional effort, since the GIM method needs additionalcomputation for dividing the solution space into threeregions.

4.3. A binomial distribution experiment

In this experiment, demands of 5 products are allbinomially distributed, and there is a budget constraintBG ¼ $200. Table 4 gives the relevant information, whereNi (i ¼ 1,2,y,n) is the parameter of the binomial demand.

To investigate the applicability of our method todiscrete demand distribution, we randomly generate 100

Table 3Parameters and solutions for Beta demand distribution.

Product vi hi ci xi,min xi,max ai bi x�i x��i;GIM x��i;Opt

1 7 1 4 100 300 2 1 222.47 206.83 207.93

2 12 2 7 50 250 1 1.2 111.60 95.69 96.73

3 30 4 15 75 150 1 2 93.93 90.10 90.34

4 17 3 10 50 200 2 2 109.79 100.12 100.78

5 27 5 15 50 200 2 3 97.26 90.07 90.55

6 10 2 6 73 275 0.8 0.2 239.02 209.35 211.69

E 364 003 475140 466 695

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Table 4Parameters for binomial demand distribution.

Product vi hi ci Ni

1 7 1 4 11

2 12 2 8 11

3 30 4 14 12

4 30 4 22 11

5 40 2 25 10

Table 5Statistical results of our algorithm over 100 instances.

Solution

gap (%)

Absolute

budget bias (%)

Budget

excess (%)

Mean 2.23 4.06 4.62

Std. Dev. 2.13 3.00 2.76

95% Confidence interval

Lower 1.81 3.47 4.03

Upper 2.65 4.66 5.20

B. Zhang et al. / Int. J. Production Economics 117 (2009) 136–141140

instances by varying parameter pi (i ¼ 1,2,y,n) of thebinomial demand. In each instance, parameter pi

(i ¼ 1,2,y,n) is generated from the uniform distributionover the support of [0,1]. For the purpose of illustratinghow often the budget constraint is unsatisfied, we set thebudget constraint be active in all instances. We set asperformance criteria in our experiment: solution gap,absolute budget bias (including budget excess and left-over), and budget excess. Solution gap is computed by|E�OPT|/OPT�100%, where E is the objective value of ourapproach, and OPT is the optimal objective value solved bydynamic programming. In each instance, budget bias iscalculated by |B�BG|/BG�100%, where B is the spentbudget according to our approach. For the instances thatthe budget is exceeded at the obtained solution, budgetexcess is computed by (B�BG)/BG�100%. We solve the 100instances by applying our approach and report thestatistical results in Table 5.

In three of the 100 instances, our approach can obtainthe optimal solution. In the other 97 instances, ourapproach yields an approximate solution to the con-strained MPNP. In all 100 instances, our approach has theaverage gap of 2.23% on the total expected cost comparedwith the optimal solutions. When the budget constraintsare not satisfied (50 of the 100 instances), the budget isexceeded less than 5% on average. These results indicatethat our approach can provide good solutions to theconstrained MPNP under discrete demand distribution.

5. Conclusions

In this paper, we propose a binary solution algorithmfor multi-product newsboy problem (MPNP) with budgetconstraint by analyzing properties of its optimal solution.The proposed algorithm has three main advantages overcurrent approaches: (1) it can provide the optimal or nearoptimal solution to the constrained MPNP under general

continuous demand distribution; (2) it can effectivelyprovide good approximate solution to the constrainedMPNP under discrete demand distribution; (3) it is easyto follow, and can be extended for the constrainedMPNP with non-zero lower bound constraints on orderquantities.

There are many possible ways to extend this research.One is to find similar algorithm for the newsboyproblem with multiple constraints. Unlike the workspresented by Abdel-Malek and Montanari (2005b) andAbdel-Malek and Areeratchakul (2007), our methodcannot be directly extended for the constrained MPNPwith multiple constraints because defining a marginalbudget benefit with multiple constraints needs consider-able research works. Another way is to develop efficientalgorithm for solving some extension problems, e.g., thegardener problem studied by Abdel-Malek et al. (2008). Inaddition, the basic idea of this research can also be used tostudy solution methods for general convex non-linearproblems, which have been done in Zhang and Hua(2008).

Acknowledgments

The authors would like to thank the anonymousreferees for their constructive comments, which improvedthe original manuscript significantly. This research wassupported by the National Natural Science Foundation ofChina (Grants nos. 70725001, 70571073 and 70801065).

Appendix A

A.1. Proof of Lemma 1

Since Fi(xi) is a non-decreasing function, according tothe definition of ri(xi), it easily comes to Lemma 1(a).According to Lemma 1(a), we have

ci � vipci � vi þ ðhi þ viÞFiðxiÞpci � vi þ ðhi þ viÞFiðx�i Þ ¼ 0.

(A.1)

Dividing Eq. (A.1) by ci gives 1�vi/cipri(xi)p0. Accordingto the definition of xi(ri), we have ri(xi)o0 for 8xi 2 ½0; x

�i Þ.

Thus it gives Lemma 1(b). From the definitions of xi(ri)and the inverse function F�1

i , Lemma 1(c) can be easilyobtained.

A.2. Proof of Proposition 1(b)

This property can be proved by using the method ofreduction to absurdity. According to

Pni¼1cix

��i pBG andPn

i¼1cix�i 4BG, there exists at least one product k 2

f1;2; . . . ;ng satisfying x��k ox�k. According to Lemma 1(b),we have rkðx

��k Þo0.

(1) If x��j 4x�j for 8j 2 f1;2; . . . ;ng, then we set x���j ¼ x�j ,x���k ¼minfx��k þ ðx

��j � x�j Þcj=ck; x

�kg, and x���i ¼ x��i for all

iaj and iak, i ¼ 1,2,y,n. According to the convexity of theexpected cost function, we have Ejðx

���j ÞpEjðx

��j Þ and

Ekðx���k ÞoEkðx

��k Þ. It is in contradiction with the optimality

of x��i , i ¼ 1,2,y,n.

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B. Zhang et al. / Int. J. Production Economics 117 (2009) 136–141 141

(2) If x��j ¼ x�j for 8j 2 f1;2; . . . ;ng, then rjðx��j Þp0.

According to rkðx��k Þo0, there exists a ek40 satisfying

rkðx��k þ �kÞo0 and

rkðx��k þ �kÞorjðx

��j � �jÞ, (A.2)

where ej ¼ ekck/cj. Let x���k ¼ x��k þ �k, x���j ¼ x��j � �j, andx���i ¼ x��i for all iaj and iak, i ¼ 1,2,y,n. According to theconvexity of the expected cost function, we have

Ejðx���j Þ � Ejðx

��j Þo� �jcjrjðx

���j Þ (A.3)

and

Ekðx���k Þ � Ekðx

��k Þo�kckrkðx

���k Þ. (A.4)

Combining Eqs. (A.2)–(A.4) gives Ejðx���j Þ � Ejðx

��j Þþ

Ekðx���k Þ � Ekðx

��k Þo0. It is in contradiction with the optim-

ality of x��i , i ¼ 1,2,y,n.(3) If

Pni¼1cix

��i oBG, let x���k ¼minfx��k þ ðBG �

Pni¼1

cix��i Þ=ck; x

�kg and x���i ¼ x��i for all iak, i ¼ 1,2,y,n. Accord-

ing to the convexity of the expected cost function, we haveEkðx

���k ÞoEkðx

��k Þ. It is in contradiction with the optimality

of x��i , i ¼ 1,2,y,n.

A.3. Proof of Proposition 1(c)

IfPn

i¼1cix�i pBG, then riðx

��i Þ ¼ 0, i ¼ 1,2,y,n. When the

budget constraint is binding, this property can be provedby using the method of reduction to absurdity. If thereexists rjðx

��j Þarkðx

��k Þ for 8j, k 2 O, without loss the

generality, we assume that rjðx��j Þorkðx

��k Þ. According to

Lemma 1(b) and Proposition 1(b), we have rjðx��j Þo

rkðx��k Þo0. There exists a ek40 satisfying

rjðx��j þ �jÞorkðx

��k � �kÞ, (A.5)

where ej ¼ ekck/cj. Let x���j ¼ x��j þ �j, x���k ¼ x��k � �k, andx���i ¼ x��i for all iaj and iak, i ¼ 1,2,y,n. According to theconvexity of the expected cost function, we have

Ejðx���j Þ � Ejðx

��j Þp�jcjrjðx

���j Þ (A.6)

and

Ekðx���k Þ � Ekðx

��k Þp� �kckrkðx

���k Þ. (A.7)

Combining Eqs. (A.5)–(A.7) gives Ejðx���j Þ � Ejðx

��j Þþ

Ekðx���k Þ � Ekðx

��k Þo0. It is in contradiction with the optim-

ality of x��i , i ¼ 1,2,y,n.

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