1-s2.0-s0377221713000416-main

European Journal of Operational Research 227 (2013) 483–493

Contents lists available at SciVerse ScienceDirect

European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Decision Support

Robust regret for uncertain linear programs with applicationto co-production models q

Tsan Sheng Ng ⇑Department of Industrial and Systems Engineering, National University of Singapore, Singapore 119260, Singapore

a r t i c l e i n f o

Article history:Received 23 May 2011Accepted 14 January 2013Available online 23 January 2013

Keywords:Uncertainty modellingLinear programmingMinimax regret

0377-2217/$ - see front matter � 2013 Elsevier B.V. Ahttp://dx.doi.org/10.1016/j.ejor.2013.01.014

q This work is sponsored by Grant NUS R-266-000-⇑ Tel.: +65 65162562.

E-mail address: [email protected]

a b s t r a c t

This paper considers the regret optimization criterion for linear programming problems with uncertaintyin the data inputs. The problems of study are more challenging than those considered in previous worksthat address only interval objective coefficients, and furthermore the uncertainties are allowed to arisefrom arbitrarily specified polyhedral sets. To this end a safe approximation of the regret function is devel-oped so that the maximum regret can be evaluated reasonably efficiently by leveraging on previousestablished results and solution algorithms. The proposed approach is then applied to a two-stage co-pro-duction newsvendor problem that contains uncertainties in both supplies and demands. Computationalexperiments demonstrate that the proposed regret approximation is reasonably accurate, and the corre-sponding regret optimization model performs competitively well against other optimization approachessuch as worst-case and sample average optimization across different performance measures.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

Linear programming is a widely used approach to model large-scale industrial optimization problems, such as inventory, produc-tion planning, and workforce planning problems. Given the re-quired data and parameter inputs, linear programs can be solvedvery efficiently using off-the-shelf platforms to generate planningsolutions. However, in reality, there are often uncertainties associ-ated with the data parameters required in the model. In particular,production problems involve parameters such as demands, yieldscoefficients, costs that can be uncertain at the time when decisionsneed to be generated and implemented. A solution obtained basedon a single realization of the input data (e.g. average case data) canbe of inferior quality. To mitigate the effects of planning uncertain-ties, stochastic optimization modelling is a well-known approachadopted among researchers and practitioners, where the uncertainparameters are modelled as random variables arising from someassumed distributions. On the other hand, in many practical situa-tions such as the case assumed in this paper, there may only bevery limited information regarding the uncertain parameters avail-able. For instance, only the bounds or support sets of the uncertainparameters may be known. In such cases, approaches such as ro-bust optimization, which is essentially a worst-case planning ap-proach, can be applied. However, when only support setinformation is used, robust optimization solutions may turn out

ll rights reserved.

043-133.

to be overly-conservative and performs poorly in the many situa-tions. As an alternative, in this work, rather than hedging againstworst-case realizations, another approach termed as regret optimi-zation, (or minimax regret, or the Savage Criterion (Savage, 1951))is considered for the uncertain linear programming problem.

1.1. Regret optimization in decision-making under uncertainty

The influence of regret on human decision-making behavior hasreceived extensive research interest in various fields, includingexperimental psychology (Kahneman and Tversky, 1982; Connollyand Zeelenberg, 2002), economics (Loomes and Sugden, 1987),consumer research (Simonson, 1992), organizational behavior (Ri-tov, 1996) and operations (Bell, 1982). Regret in decision-makingunder uncertainty is intimately related to the value of information,i.e. the discrepancy between the actual cost of a decision madeusing partial information, and the decision made with full knowl-edge of the future outcome. In many important situations, policiesand decisions are often subjected to post-audits (Gulliver, 1987;Neale and Holmes, 1990; Azzone and Maccarrone, 2001), andachieving low regret in decisions are desirable since managerswant to avoid being viewed as having exercised poor judgment.This motivates the use of prescriptive regret models as a decisionsupport tools for planning under uncertainties.

Minimax regret optimization has been applied to various prob-lems in operations research, e.g. production scheduling (Danielsand Kouvelis, 1995), portfolio selection (Inuiguchi and Tanino,2000), knapsack problems (Conde, 2004), econometrics and pricing(Manski, 2007; Bergemann and Schlag, 2008). Minimax regret

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484 T.S. Ng / European Journal of Operational Research 227 (2013) 483–493

ordering in newsvendor problems with only known demand inter-vals were studied by Kasugai and Kasegai (1961), and extensions tomulti-item newsvendor problems under budget constraints wereconsidered by Vairaktarakis (2000). More recently, Perakis andRoels (2008) employ the minimax regret approach to derive robustorder quantities for the newsvendor problem with partial informa-tion about the demand distribution (e.g., support, mean, variance,symmetry, unimodality), and apply entropy maximization crite-rion for selecting a probability distribution as an input to the news-vendor problem. Multiple market newsvendor problems using theminimax regret criterion is also recently studied by Lin and Ng(2011). These works motivate the relevance of the regret optimiza-tion models in operations research. On the other hand, most ofthese works are confined to very specific structures and simplifiedabstractions of planning problems. For instance, the newsvendorproblems only assume uncertainty in the demand information. Inthe application examples in this paper, the newsvendor problemsthat are considered can also have uncertainties in the supply andproduct yield information. In particular, the use of the minimax re-gret approach for generic linear programming models that mayhave uncertainties in any of the data coefficients is proposed in thiswork.

1.2. Regret optimization models for uncertain linear programs

The focus on this paper is the regret optimization criteria foruncertain linear programs. Inuiguchi and Sakawa (1995) first ad-dressed a linear programming problem with interval objectivefunction coefficients using the minimax regret approach. Theauthors propose a solution algorithm based on a relaxation proce-dure developed by Shimizu and Aiyoshi (1980). The algorithm re-quired solving a subproblem known as the candidate regretmaximizer problem repeatedly in order to obtain data realizationsthat yielded the maximum regret for a given planning solution. Thesubproblem is essentially a bilinear programming model in struc-ture, and the authors adopted a naive solution approach that re-quired the enumeration of all the vertices in the linearprogramming feasible space. This was clearly computationallyinfeasible even for problems with moderate size. For the sameproblem, Mausser and Laguna (1998a); Mausser and Laguna,1998b proposed a mixed integer formulation for the candidate re-gret sub-problem, and the computational study indicates that suchan mixed integer formulation significantly reduces solution timeby exploiting state-of-the-art integer solvers. The authors also de-velop a greedy heuristic procedure for the candidate regret maxi-mizer problem, and demonstrated its computational effectivenessfor large-scale problems. Averbakh and Lebedev (2005) showedthat the minimax regret linear programming problem with intervalobjective function coefficients is indeed strongly NP-hard.

This work extends the research of the above cited papers, and infact generalizes several of the past results. In particular, linear pro-grams with both uncertain objective coefficients and constraintcoefficients are considered. Rather than adopting the robust opti-mization approach that attempts to ensure that all constraintsare fully met, constraints are allowed to be violated at certain pen-alty costs. This is common in applications such as inventory andproduction planning problems where uncertainty in demandsand supplies are expected. However, as will be discussed in detail,this complicates the regret optimization problem significantly, sothat even the previous established results to evaluate the maxi-mum regret and the solution algorithms are rendered unsuitablefor all practical purposes. The main contribution in this work isthe development of a safe approximation of the regret functionfor this class of uncertain linear programs. While it is not the en-deavor of this work to resolve the computational complexity issuesof the regret optimization problem (since the problem considered

here are even harder than those considered by Averbakh and Lebe-dev (2005)), the proposed approximations retain the importantstructural properties of the problems in the previous works. Thisenables good leverage of the available results and solution algo-rithms to solve the problems efficiently.

Furthermore, this work expands on the uncertainty modelsassumed in the previous works (that only considered intervaldata). The expansion includes the consideration of the correla-tion among uncertain parameters through the use of a randomfactors model, and also the use of arbitrary support sets de-scribed with polyhedrons. Because these uncertainty modelsand the associated concepts are quite popular in the robust opti-mization parlance, the author term this work robust regretoptimization.

1.3. Outline

The outline of the rest of the paper is as follows. The next sec-tion introduces the uncertain linear programming problem ofinterest in this study, and the corresponding regret optimizationmodel. The issues related to the computational challenges of eval-uating and solving these regret models are discussed in detail. Anapproximate regret function is then developed along with asolution scheme of the proposed regret optimization model. Toshowcase the application of the proposed model, in Section 3, aco-production newsvendor model is introduced. Structuralproperties of the co-production newsvendor are then developedto enable the application of the regret optimization model. Sec-tion 4 present the results of some computational experimentsto evaluate and compare the performances of the proposed ap-proach with other commonly-used solution approaches such assample-average and worst-case optimization. Finally, Section 5concludes this work.

2. Robust regret for linear programming

2.1. Linear programming planning model

Let ci 2 RN , for all i = 0, . . . , I, c = [c0, . . . , cI], s 2 RI , andA 2 RM�N and a 2 RM be the input data to the following linear pro-gramming problem:

minx

c00x

s:t: c0ix� si P 0 8i ¼ 1; . . . ; I

Ax P a; x P 0

where x 2 RN are the decision variables, and si denote the ith ele-ment of s. In the situation of interest, the data inputs (c,s) can beuncertain at the point in time when decisions x need to bedetermined (we assume that A and a does not involve uncertain-ties). Hence, a solution obtained based on solving the above usinga given realization of the inputs may become infeasible. However,in most practical problems, when there are uncertainties to beexpected, constraints are allowed to be violated at a penalty cost.This is for instance in production-inventory models, where demandshortages are allowed at a shortage penalty cost, or excess inven-tory are stocked at penalty costs. The following problem is thusconsidered:

f ðc; sÞ ¼minx

c00xþXI

i¼1

gi si � c0ix� �þ

; s:t: Ax P a; x P 0 ð1Þ

where gi denotes the penalty cost per unit of the shortfall level [si -� cix]+. Denoting C ¼ ½c1; . . . ; cI � 2 RN�I , the linear programming for-mulation of (1) is:

T.S. Ng / European Journal of Operational Research 227 (2013) 483–493 485

f ðc; sÞ ¼ minx;y

c00xþXI

i¼1

giyi ð2Þ

s:t: yi þ c0ix P si 8i ¼ 1; . . . ; I ð3ÞAx P a; x P 0; y P 0 ð4Þ

Denoting the multipliers of (3) and (4) as k 2 RI and l 2 RM respec-tively, the dual formulation is then stated as:

f ðc; sÞ ¼ maxk;l

s0kþ a0l ð5Þ

s:t: Ckþ A0l 6 c0 ð6Þk 6 g; k P 0; l P 0 ð7Þ

Since any dual feasible solution (5)–(7) can be used to generate alower bound on (2)–(4), the dual formulation plays a pivotal rolein the development of a safe regret approximation in Section 2.4.Some basic characteristics of f as value functions of the inputs cand s are first stated in the following Proposition, the proof of whichfollows from basic results of linear programming duality (see for in-stance Bertsimas and Tsitsiklis, 1997) and is omitted here.

Proposition 1. For any given C and s, f is concave in c0. For any givenC and c0, f is convex in s.

2.2. Model for data uncertainties

To model the data uncertainties, the factor model approach com-monly used in robust optimization literature is followed; see forinstance, Ben-Tal and Nemirovski (1998); Chen et al. (2008); andGoh and Sim (2010). A factor model of uncertainty formulates eachuncertain parameter as an affine function of a set of random fac-tors. Let ~z ¼ ½~z1; . . . ;~zK � denote a K-vector of the random factors(also termed as primitive uncertainties), with the support set Z de-scribed by the polyhedron:

Z ¼ fz 2 RK jBz 6 b; z P 0g ð8Þ

Thus, all possible outcomes of z are contained in the assumedbounded set Z. Note that the definition of Z in (8) includes the spe-cial case of a hypercube set. The uncertain linear program parame-ters c are defined as affine functions of z using a set of factorcoefficients. It is assumed that the factor coefficients have been esti-mated (e.g. using past data studies) a priori. The factor model canthus be regarded as a first order approximation of the uncertainparameter. Specifically, for each i = 0, . . . , I, and k = 1, . . . , K, denotethe factor coefficients ck

i;n so that

ci;n ¼XK

k¼1

cki;nzk þ c0

i;n 8i ¼ 1; . . . ; I; n ¼ 0 . . . ;N

and similarly, the coefficients ski for k = 1, . . . , K, so that:

si ¼XK

k¼1

ski zk þ s0

i 8i ¼ 1; . . . ; I

Thus, the use of the factor coefficients cki;n and sk

i allows correlationamong the uncertain variables c to be modelled.

2.3. Uncertain linear programs and regret function

The regret r(x,z) associated with a feasible solution x for somegiven z is defined as that difference between the cost achievedby a solution x and the minimum cost achieved when z is fullyknown before solving the problem (1), i.e.

rðx; zÞ ¼ c00xþXI

i¼1

gi si � c0ix� �þ � f ðzÞ

The maximum regret associated with a decision x is then:

RðxÞ ¼maxz2Z

c00xþXI

i¼1

gi si � c0ix� �þ � f ðzÞ ð9Þ

Thus, R(x) represents the worst cost differential incurred by a decisionx due to the absence of complete exact planning data. The robustregret linear programming seeks x that minimizes the worst regret:

minx2X

RðxÞ ¼minx2X

maxz

c00xþXI

i¼1

gi si � c0ix� �þ � f ðzÞ ð10Þ

where X in the above denotes the feasible set of alternatives{x:Ax P a}. Basically, (10) is a minimax optimization problem,where the inner problem maximizes the regret associated with a gi-ven x, that is, equivalent to evaluating R(x) in (9). For this problem,Inuiguchi and Sakawa (1995) considered the case when there areonly uncertainties in the objective coefficients c0. In particular,the uncertainties arises from a hypercube set,fc0 2 RN jc0 6 c0 6 c0g. The authors then proposed the notions ofx-optimality and c-consistency to demonstrate that it suffices to con-sider the set of all extrema of the hypercube to locate the regretmaximizer. Note that this is the consequence of the fact that f(c0)is (piece-wise linear) concave in c0 given the rest of the inputs(Proposition 1), and hence r(�) conditioned on x, C, and s is convexin c0. The result then follows by noting that the maximum of a con-vex function over a convex set is always achieved at some extremaof the convex set. The key insight here is that the convexity of theregret function allows the reduction of the search space to the setof all extreme points of Z, which is finite. This is termed as the prop-erty of extrema sufficiency, and as a result the search for the regretmaximizer can be cast as a discrete optimization problem, wherethe solution of the discrete optimization model is an extreme pointof Z. In general the regret maximizer problem is known to be com-putationally NP-hard. Mausser and Laguna (1998a); Mausser andLaguna, 1998b proposed a linear mixed-integer programming(MIP) reformulation of the regret maximizer problem. While thisdoes not resolve the computational issues entirely, state-of-the-art large-scale MIP solvers are now easily available, and one can ex-ploit the power of such solvers conveniently.

In the case when C and s are uncertain, while the realized costfunction c0xþ

PIi¼1gi si � c0ix

� �þ is a convex function in the uncer-tain parameters, the optimal value function f(z) is generally notconcave, and consequently the regret function r(x,z) is non-convex.Even in the case where only the right-hand side vector s is uncer-tain, and f(s) convex (from Proposition 1), the regret function isstill neither concave nor convex in general. This can be verifiedeven in very simple problems, for instance in the co-productionproblem (Section 3) for a single and two product grade problem,as shown in Figs. 1 and 2 respectively. It can be noted that in bothcases the maximum regret does not occur at the extrema of theuncertain parameter space. In summary, without extrema suffi-ciency, evaluating the maximum regret for problems of even mod-est dimensions becomes practically impossible, and the regretmaximizers generally has to be located using global optimizationapproaches. However, global optimization methods typically onlylocate a lower bound on the maximum regret and cannot providea safe performance guarantee about the true maximum regret.Also, the regret maximization typically needs to be solved a largenumber of times iteratively in a solution algorithm.

In this work, C and s can uncertain in general, and the objectiveis to develop a safe approximation to the regret function that pre-serves extrema sufficiency, thus circumventing the abovemen-tioned issues.

2.4. Approximate regret function

In this section, an approximate regret function to the actualmaximum regret is proposed. The approximation is essentially

Fig. 1. Regret profile for the co-production newsvendor for a single product problem (J = 1).

Fig. 2. Regret profile for the co-production newsvendor for a two product problem (J = 2).


based on creating convex upper bounds on the actual regret usingLagrangian duality. This is presented formally in the followingproposition.

Proposition 2. Let

Rk;lðxÞ ¼ maxz2Z

c0xþXI

i¼1

gi si � c0ix� �þ � s0k� a0l ð11Þ

where k = [k1, . . . , kI], l = [l1, . . . , lN] and p = [p1, . . . , pN]. Define theset X to be the polyhedron defined as follows:

X ¼ fðk;l;pÞP 0jb0pn þ A0l 6 0;B0pn P Dnk� c0;n

8n ¼ 1; . . . ;Ng ð12Þ

where Dn denotes the K � I coefficient matrix so that, for eachi ¼ 1; . . . ; I;Dn

k;i ¼ cki;n, and c0;n ¼ c1

0;n; . . . ; cK0;n

h i. We then have that

Rk;lð�ÞP Rð�Þ for any (k,l,p) 2X.

Proof. For any k and l feasible in (6) and (7), i.e. the dual linearprogramming model, we have

rðx;zÞ ¼ c00xþXI

i¼1

gi si� c0ix� �þ � f ðzÞ6 c00xþ

XI

i¼1

gi si� c0ix� �þ �s0k�a0l

ð13Þ

where the inequality follows from the weak duality of linear pro-gramming. Since k and l must be feasible in (6), and since C is afunction of the random factors z, a given k and l feasible for some


realization of z might not be feasible in another. Consequently, inthe following we enforce the worst-case or robust constraint of (6)in considering the candidature of k and l. Let Cn denote the nthrow of C, i.e. Cn = [c1,n, . . . , cI,n]. The robust version of (6) is thenwritten as:

A0lþmaxz2ZfCnk� c0;ng 6 0 8n ¼ 1; . . . ;N ð14Þ

Clearly any l and k that are feasible in (14) is also feasible in (6) forany z 2 Z. For a given k, the maximization term on the left-hand-side of (14) can be evaluated as follows:

maxz

XK

k¼1

XI

i¼1

kicki;n � ck

0;n

!zk ð15Þ

s:t: Bz 6 b; z P 0 ð16Þ

This is a linear programming problem and the dual problem is:

min b0 pn

s:t: B0pn P Dnk� c0;n

pn P 0

where pn 2 RI are the dual variables, and Dn and c0,n are as definedin the statement of the proposition. Hence for a given k and l, ifthere exists p so that:

A0lþ b0pn 6 0 and B0pn P Dnk� c0;n; and p P 0 ð17Þ

we then have

0 P A0lþ bpn P A0lþmaxz2Z

Cnk� c0;n ð18Þ

where again the second inequality in (18) follows from weak dual-ity. In fact, since there exists p⁄ that achieves the dual optimal solu-tion, by the strong duality of linear programming the inequalitybecomes tight by choosing p⁄. Combining the above, k, l and p mustbe feasible in (17), which is the set X defined in (12). h

A direct corollary of Proposition 2 is that for a given l and k, theapproximate regret function

�rl;kðz; xÞ ¼ c0ðzÞ0xþXI

i¼1

gi½siðzÞ � ciðzÞ0x�þ � sðzÞ0k� a0l ð19Þ

is convex in z, since it is simply the sum convex functions in z. Con-sequently, the maximum approximate regret Rl,k(x) can be evalu-ated by solving a discrete optimization problem of investigating afinite set of extrema of Z. This will be elaborated in Section 2.5.The tightest maximum regret approximation for a given x is then gi-ven by:

RðxÞ ¼ minðk;l;pÞ2X

maxz2Z

�rk;l ð20Þ

Since the evaluation of maxz2Z�rl;kðz; xÞ is the point-wise maximumof a set of functions affine in l and k, the minimization over l and k

in the above is a (piece-wise linear) convex optimization problemover the polyhedron X and can be resolved easily using linear pro-gramming techniques. Finally, the minimax regret problem that wesolve is:

minx2X

RðxÞ ¼ minx2X;ðk;l;pÞ2X

maxz2Z

rk;lðz; xÞ ð21Þ

2.5. Candidate maximum regret problem

Consider now the resolution of the inner regret maximizationproblem, that is:

Rk;lðxÞ ¼maxz2Z

c0ðzÞ0xþXI

i¼1

gi ciðzÞ0x� siðzÞ� �þ � sðzÞ0k� a0l ð22Þ

For the case when only c0 is assumed to be uncertain, and that C is ahypercube, Mausser and Laguna (1998a); Mausser and Laguna,1998b proposed an integer programming formulation for the candi-date regret maximizer problem. While this does not resolve thecomputational issues entirely, state-of-the-art large-scale MIP solv-ers are now easily available, and one can exploit the power of suchsolvers conveniently. However, in many practical situations, thesupport sets Z may consist of several intersections of polyhedra thatdescribe known relationships of the parameters. For instance, inproduction processes, while product yields may be uncertain data,they have to add up to be one to be physically logical. In such casesit is generally not possible to directly identify and represent all theextrema explicitly, and hence the integer programming model ofMausser and Laguna (1998a); Mausser and Laguna, 1998b is not di-rectly applicable. Instead, in this work the following mixed integerprogramming reformulation to solve the candidate regret maxi-mizer problem is proposed.

Proposition 3. .Define y = [y1, . . . , yI] and s = [s1, . . . , sI] as a set ofreal-valued variables and binary-valued variables respectively. For agiven x a candidate regret maximizer can be obtained by solving thefollowing mixed integer programming problem:

Rk;lðxÞ ¼maxy;s;z

XI

i¼1

giyi þ c0ðzÞ0x� sðzÞ0k� a0l ð23Þ

s:t: yi 6 ciðzÞ0x� sðzÞ þMsi 8i ¼ 1; . . . ; I ð24Þyi 6 Mð1� siÞ 8i ¼ 1; . . . ; I ð25Þy 2 RI; s 2 f0;1g; z 2 Z ð26Þ

where M is some large enough number.

Proof. For each c, we can re-write, for each i = 1, . . . , I,

gi c0ix� si� �þ ¼ max giyi

s:t: yi 6 c0ix� si� �

si; si 2 f0;1g

which can be linearized into the form of (24) and (25). The resultthen follows by combining each i = 1, . . . , I to formulate the objec-tive function, and noting that the regret maximizer is obtained bysearching over all z 2 Z. h

2.6. Master regret problem

Although the results in Sections 2.4 and 2.5 provide optimiza-tion formulations for evaluating the approximate maximum regretRk;lðxÞ, there is still no tractable formulation of (22) in the decisionvariables x for evaluating the minimax regret solution in (21). Thisissue is circumvented by using the notion of regret cuts, which issimilar to the approaches based on using sub-gradients in solvingstochastic linear programming models (Birge and Louveaux,1997). Since for a given z, the approximate regret (19) is convexin x, and the point-wise maximum of a set of convex functions re-tains convexity, lower bounds functions linear in x (termed as re-gret cuts here) can be used to approximate Rk;lðxÞ successively.The following is a procedure to generate valid regret cuts.

2.6.1. Regret cut generationProcedure GenCut

1. Solve (23)–(26) to generate a regret maximizer z 2 EðZÞ, whereEðZÞ is defined as the set of all extreme points of Z.

2. Fix cðzÞ in the problem:

miny

XI

i¼1

giyi

s:t: yi P ciðzÞ0x� siðzÞ 8i ¼ 1; . . . ; I; y P 0


The dual formulation of the above, with decision variables a, can bewritten as:

maxa

XI

i¼1

ðciðzÞ0x� siðzÞÞai ð27Þ

s:t: ai 6 gi 8i ¼ 1; . . . ; I ð28Þ

3. Solve for a and formulate the inequality:

r P c0ðzÞ0xþXI

i¼1

aiðciðzÞ � sðzÞÞ0x� sðzÞ0k� a0l ð29Þ

Note that while the dual formulation (27) and (28) is trivial tosolve, it is a more general approach for certain classes of two-stageproblems like the co-production problem in Section 3. Further-more, note that the set of all extrema in (27) and (28), denoted hereas C are fully characterized by the known parameters and does notdepend on the uncertain parameters z and the decisions x. Also,since the regret maximizers are located in the extreme point setEðZÞ of Z, in theory the minimax regret problem in (21) can bere-formulated through a complete enumeration of EðZÞ and C asfollows:

min r ð30Þ

s:t: rPc0ðzÞ0xþXI

i¼1

aiðciðzÞ�siðzÞÞ0x�sðzÞ0k�a0l 8z2E Zð Þ;8a2C ð31Þ

ðk;l;pÞ2X;x2X ; rP0 ð32Þ

The above problem is a linear program with the auxiliary variable rmodelling the approximate maximum regret. Unfortunately, evenfor Z defined as a hypercube by Inuiguchi and Sakawa (1995),jEðZÞj grows at 2K, and consequently solving (31) directly is highlyimpractical even with powerful computational resources. The situ-ation is further exacerbated with the I non-negative penalty termsgi c0ix� si� �þ, so that at worst the number of constraints in (31)

are jEðZÞj � jCj. The proposed approach is to solve the problemusing an iterative relaxation and cut generation approach, summa-rized as follows.

1. Initialization: Let V be a subset of the set of tuplesðz;aÞ 2 fEðZÞ � Cg.

2. Solve the following relaxed minimax regret problem:
f ¼min r ð33Þ
s:t: r P c0ðzÞ0xþXI

i¼1

aiðciðzÞ�siðzÞÞ0x�sðzÞ0k�a0l 8ðz; aÞ 2V

ð34Þðk;l;pÞ 2X; x2X ; r P 0

Let the solution to the above be x; k and l.3. Using x, k and l, apply Procedure GenCut. If Rk;lðxÞ 6 f , the cur-

rent solution x is the optimal solution, and the procedure termi-nates. Otherwise, append the generated cut (29)–(34), andupdate V :¼ V [ fðz; aÞg. Go to Step 1.

Note that in any iteration, the objective function value f in (33)is a valid lower bound to the minimax regret, while the evaluatedobjective function value Rk;lðxÞ in the candidate regret maximizerproblem (23)–(26) is an upper bound. Furthermore, the procedureis guaranteed to terminate since the sets EðZÞ and C are finite.

3. Application to a co-production newsvendor problem

3.1. Problem description

The objective of this section is to apply the robust regret ap-proach to a single period ordering problem under uncertainty. In

the problem, there are customer demands for a product that canhave different grades. Prior to learning the exact demands, an orderneed to be placed for the input material. This is then processed into aset of graded products, the proportions of which can also be uncer-tain. This is known as a co-production process (Bitran and Dasu,1992; Bitran and Leong, 1992; Bitran and Gilbert, 1994), motivatedby characteristics in semiconductor manufacturing. In semiconduc-tor production, processor chips are produced from the same stock ofwafer. Because not all chips cut from the wafers have identical oper-ating characteristics, they are separated into different gradesaccording to their speeds and other performance characteristics.

In co-production, it is often the case that some form of productsubstitution is allowed during the demand-fulfillment process. Inparticular, the practice of down-substitution is assumed in theworks of Bitran and Dasu (1992); Bitran and Leong (1992) and Bi-tran and Gilbert (1994). That is, products of a higher grade can beused to fill the demands of a lower grade (but not vice versa). Forinstance, in semiconductor manufacturing, excess supply of highclock-speed chips can be used to fill demands for lower end chipsif necessary. Generally, product down-substitution can improveflexibility in the demand fulfillment process.

The following notation pertains to the problem data. Othernotation will be introduced as and when necessary during themodel development.

3.1.1. Data and parameters

j
index for product grade, j = 1, . . . ,J J the number of product grades and demands c0 per unit production cost gj per unit revenue, or shortage penalty of grade j u vector of grade fraction realizations, u = [u1, . . . ,uJ] d vector of demand realizations, d = [d1, . . . ,dJ]
3.1.2. Decision variables

x
order quantity qj;j0 level of grade j product used to fulfill grade j0 demand yj shortage level of grade j
The co-production newsvendor problem with downward substi-tution considered is described in Hsu and Bassok (1999). The se-quence of events and actions of the co-production newsvendor isas follows. At the beginning of the planning, the co-productionnewsvendor needs to determine the order quantity x prior to learn-ing the actual demands d and grading fractions u of the products.The order arrives, and the levels of each graded product ujx and cus-tomer demands dj for all j = 1, . . . , J are observed. In the following itis assumed that a product indexed j is always of higher (better)grade than all products indexed j0 > j. The allocation qj;j0 from supplyof grade j to demand for grade j0 is then executed. The co-productionnewsvendor seeks to maximize his total profits, that is the total rev-enues less purchasing costs. It is assumed that the revenue per unitgj is non-increasing in j, i.e. gj P gj0 whenever j < j0. The problem fora given data realization z can be formulated as the following:

maxx;qP0

XJ

j¼1

gjdjðzÞ � gj djðzÞ �Xj

j0¼1

qj0j

24

35þ

� c0x ð35Þ

s:t:XJ

j0¼j

qjj0 6 ujðzÞx 8j ¼ 1; . . . J ð36Þ


In the objective function (35), the term djðzÞ � djðzÞ �Pj

j0¼1qj0 j

h iþcorresponds to the total sales volume of each product j, which isthe minimum of the demand realized and allocated quantityPj

j0¼1qj0 j. Note that the demand for grade j can only be fulfilled withgrades 1, . . . , j, and (36) imposes that product of grade j can only beused to fulfill demands for grades j0 P j. The above formulation canalso account for salvage values, by augmenting to the set of productgrades a dummy product grade J + 1 with uJ+1 = 0, and that dJ+1 andgJ+1 represent the salvage demand market size and salvage rate ofall unsold products. The salvage demand market size can be mod-elled as the uncertain parameter dJ+1(z), or can be set sufficientlyhigh (e.g. to be the sum of all maximum possible demand levels)if desired. (35) and (36) can be re-written in minimization format,and by introducing the demand shortfall variables yj, is equivalentto the following linear programming model:

f ðzÞ ¼ minx;y;qP0

c0xþXJ

j¼1

gjyj �XJ

j¼1

gjdjðzÞ ð37Þ

s:t: ð36Þ and

Xj

j0¼1

qj0 j þ yj ¼ djðzÞ 8j ¼ 1; . . . J ð38Þ

The following exposition focuses on the minimization format forconsistency with the development in Section 2, and gj is referredto as the shortage penalty rate of product j for convenience. Also,because the term

PJj¼1gjdj in the objective function (37) is indepen-

dent of the decision variables, it does not play an interesting role inthe optimization procedure and is hence omitted in the rest of thepaper of simplicity. For a given x, the second-stage allocation problemis defined as:

wðx; zÞ ¼ miny;qP0

XJ

j¼1

gjyj

s:t: ð36Þ and ð38Þ:ð39Þ

The problem (36)–(38) can then be re-written compactly as:

f ðzÞ ¼minxP0

cxþwðx; zÞ ð40Þ

The robust regret optimization problem for the co-productionnewsvendor with downward substitution is then:

minxP0

RðxÞ ¼ minxP0

maxz2Z

cxþwðx; zÞ � f ðzÞ ð41Þ

The following development will lead to demonstrate how the pro-posed approximate minimax regret approach in Section 2 can beapplied to co-production newsvendor problem with downwardsubstitution under uncertainty. Note that because the co-produc-tion with downward substitution is a two-stage decision-makingproblem with non-trivial recourse, the results in Section 2 mightnot be easily extended. To exploit these results, some additionalstructural properties of the allocation problem (39) will need tobe established first.

3.2. Characterizing the second-stage allocation problem

This section presents the results that allow the characterizationof the optimal solutions of the downward substitution problem di-rectly without the allocation variables qj;j0 . As mentioned, a productwith a higher grade is assumed to always have shortage penaltyrate no less than lower grade products. In such a situation, it isintuitively clear that an optimal allocation always follows by ful-filling a higher grade demand as much as possible first. Any out-standing supply is then down-substituted to feed the next lowergrade demand. The allocation procedure is then iterated for the

next lower grade, etc. The following proposition states this resultformally, the proof of which is provided in the Appendix section.

Proposition 4. An optimal solution y⁄ of the downward substitutionallocation problem is as follows:

y�1 ¼ ½d1 � u1x�þ ð42Þ

y�j ¼ dj �Xj�1

j0¼1

y�j0 þ uj0x� dj0

� �� ujx

24

35þ

8j ¼ 2; . . . ; J ð43Þ

Proof. See Appendix. h

The optimal shortages y⁄ in (42) and (43) can be evaluated bysolving a linear programming model for the down-substitutionproblem without the allocation variables qjj0 . This is presented inthe next Proposition.

Proposition 5. The optimal solution for the down-substitutionproblem in Proposition 4 is an optimal solution in the below linearprogram.

wðx;u;dÞ ¼minyP0

XJ

j¼1

gjyj ð44Þ

s:t:Xj

j0¼1

yj0 PXj

j0¼1

dj0 �Xj

j0¼1

uj0x 8j ¼ 1; . . . ; J ð45Þ


The next result shows how to formulate the down-substitutionproblem in Proposition 5 as a maximization problem instead, usingthe techniques of integer programming. The purpose of this is toobtain a single mixed integer programming formulation of the can-didate maximum regret problem similar to the approach presentedin Proposition 3.

Proposition 6. .Consider the following mixed-integer programmingproblem, where y are real variables, and s are defined as binaryvariables.

maxy;s

XJ

j¼1

XJ

j¼1

gjyj ð46Þ

s:t:Xj

j0¼1

yj0 6Xj

j0¼1

dj0 �Xj

j0¼1

uj0xþMsj 8j ¼ 1; . . . ; J ð47Þ

yj 6 Mð1� sjÞ 8j ¼ 1; . . . ; J ð48Þy P 0; s 2 f0;1g ð49Þ

The optimal objective function value of (46)–(49) is then equal to thatof the linear program (44) and (45). Furthermore, there always exist anoptimal solution to (46)–(49) with y⁄ taking the values of (42) and (43)in Proposition 4.


3.3. Robust regret for the co-production newsvendor

The co-production newsvendor desires to minimizes his maxi-mum regret in the decision-making:

minxP0;k;p

RkðxÞ;

s:t: ðk;pÞ 2 Xð50Þ

where the upper bound RkðxÞ on the maximum regret is stated as:


RkðxÞ ¼ maxyP0;z2Z

XJ

j¼1

gjyj þ c0x� k0d; s:t:ð47Þ � �ð49Þ

This can be obtained by following through the derivation in Propo-sition 2, and then applying the results obtained in Proposition 6. Asbefore, k and p refer to the dual multipliers of the linear program-ming models (36)–(38), (15) and (16) respectively, and X the corre-sponding feasible space of (k, p). The technical developmentessentially uses the same approach presented in Sections 2.3 and2.4, and will not be repeated in detail here.

3.3.1. Regret cut generationLet a = [a1, . . . , aJ] be the dual multipliers corresponding to the

demand shortage (45) of the second-stage allocation linear pro-gramming model (44) and (45) respectively. Thus using the gener-ated z obtained by solving the mixed integer model (46)–(49), thefollowing is a valid regret cut:

r P c0xþ kdðzÞ þXJ

j¼1

þXJ

j0¼1

aj

Xj

j0¼1

dj0 ðzÞ � uj0 ðzÞx ð51Þ

3.3.2. Master robust regret problemThe regret cut generation procedure in Section 2.4 is applied to

solve the robust regret optimization problem. In each pass of thesolution process, the following relaxed master problem is solved:

min r

s:t: r P cxþ k0dþXJ

j0¼1

aj

Xj

j0¼1

dj0 � uj0x� �

z 2 E0ðZÞ;a 2 C0

ðk;pÞ 2 X; x 2 X ; r P 0

where E0ðZÞ and C0 are subsets of EðZÞ and C respectively. These areupdated in each pass by applying Procedure CutGen in Section 2.4to obtain a candidate regret maximizer z and a, so thatE0ðZÞ :¼ E0ðZÞ [ z, C0 :¼ C0 [ a.

4. Computational study

In this section, computational studies of the co-productionnewsvendor problem with full downward substitution are per-formed. Section 4.1 first present some studies on small-size simu-lated problems, the objective of which is to evaluate theperformance of the robust regret model in approximating the trueregret function. Section 4.2 then expands the application of the ro-bust regret approach for a co-production system motivated by thecharacteristics of a semiconductor production problem. The objec-tive is to compare the performance of the robust regret approachwith other optimization approaches such as worst-case optimiza-tion and sample average approximation models. All the computa-tional experiments done are run in Intel Core Duo, 2.13 GHz CPUwith 2.99 GB of RAM desktops under Windows. The solution proce-

Table 1Regret evaluation RðxÞ=RðxÞ for test problems J = 3.

Instancenx 0 300 600 900

1 1.078 1.144 1.127 1.1752 1.125 1.16 1.082 1.1013 1.102 1.125 1.091 1.0724 1.063 1.085 1.104 1.1125 1.117 1.124 1.112 1.0306 1.104 1.077 1.041 1.0217 1.121 1.118 1.097 1.0458 1.085 1.081 1.078 1.082

dures are all coded in Java, and the linear and integer programmingmodels are solved by calling the CPLEX 11 libraries (www-01.ibm.-com/software/integration/optimization/cplex-optimizer).

4.1. Performance of regret approximation

The experimental setup for the computational study will be asfollows. Eight test problems of J = 3 are randomly generated. Foreach test problem, given a value of the order quantity x, the regretmaximizer problem to evaluate RðxÞ is first solved. The solution xthat minimizes R is then located by performing a line search of aset of values of x. Similarly, the actual maximum regret R(x) at eachof these values of x are also evaluated by an extensive grid searchover the space of Z. Clearly, this approach of evaluating the maxi-mum regret and then obtaining the minimax regret order quanti-ties is only computationally feasible for small test problems, andis performed here only for comparison purpose. Table 1 presentsthe results for the eight test instances for values of x 2 [0,1800],where the ratios of RðxÞ=RðxÞ are tabulated in the cells.

The results in Table 1 shows that the gap between the upperbound and the evaluated maximum regret is on average 7.3%across the instances. The largest gap is also no more than 17.5%.This verifies that the obtained upper bound approximation is rea-sonably tight. Next, Table 2 compares the results of the optimalsolutions of the actual minimax regret problem, x⁄, and the approx-imate minimax regret x. The actual maximum regret achieved by x,normalized by the true minimax regret values R(x⁄) are also tabu-lated. The evaluated maximum regret achieved by x was on aver-age 1.08 times of the minimax regret R(x⁄), and was no worsethan 1.13 times the minimax regret, across the eight test problems.The error in the optimal quantities ðx� x�Þ=x� was within �12.3%and 10% and across the test problems. In the results, the instanceswith order quantities x deviating further from the actual optimalsolution also had poorer regret performance (e.g. Instances 3 and5). On the other hand, the solutions in Instances 1, 2, and 8achieved more accurate approximations of the minimax regretsolution and correspondingly better regret performance. Theexperiment thus provides validation that the approximate mini-max regret model is effective in producing good quality solutionsthat are also close to the actual minimax regret solutionsconsistently.

4.2. Performance comparison

In this section, computational experiments are performed usinga larger scale problem of the co-production problem to comparethe performance of the robust regret model (50) with two otheroptimization modelling approaches described below. Comparisonsare made in terms of maximum realized regret, worst-case perfor-mance and computation times. For the computational experi-ments, data realizations were simulated based on estimates ofthe supports of the grading fractions and demands. Different prob-ability distributions were used to generate the data realizations.

1200 1500 1800 Average Maximum

1.120 1.140 1.136 1.13 1.1751.071 1.035 1.074 0.94 1.1251.153 1.144 1.120 1.12 1.1531.045 1.081 1.126 1.08 1.1261.048 1.045 1.086 1.08 1.1241.012 1.032 1.113 1.06 1.1131.083 1.084 1.112 1.09 1.1211.073 1.024 1.105 1.07 1.105

Table 2Comparison of minimax regret for test problems J = 3.

Instance x⁄ x x=x� RðxÞ=Rðx�Þ

1 1210.1 1200.3 0.99 1.0082 305.2 323.6 1.06 1.0233 1250.7 1097.2 0.87 1.1234 1640.3 1520.9 0.92 1.1155 890.1 979.2 1.10 1.1276 830.8 787.5 0.94 1.1067 720.2 787.8 1.09 1.0928 1090.4 1063.7 0.97 1.069

Table 4Performance comparison of WOP, SAA and AOP models to robust regret model.


4.2.1. Average-case-data optimization model (AOP)In this model, the uncertain parameters are assumed to be real-

ized at the mid-point values, and the solution �x based on that is ob-tained by solving the linear programming model (36)–(38).

4.2.2. Worst-case-data optimization model (WOP)In this case, the uncertain parameters are assumed to be real-

ized at their worst-case values, so that any costs achieved by asolution is as high as possible within the space of all possible real-izations. The optimization problem is then to obtain the solution x0

that minimizes the maximum cost of the problem.

minxP0

maxz2Z

gj½djðzÞ � ujðzÞx�þ þ c0x ð52Þ

The problem in (52) is also known as the robust optimization prob-lem and can be easily re-formulated as a linear programming prob-lem using strong duality results, see for instance (Bertsimas andSim, 2004).

4.2.3. Sample average approximation model (SAA)This approach assumes knowledge of the probability distribu-

tions of the uncertain parameters, which are used to generate aset of Q random sample realizations, i.e. zq, where q = 1, . . . , Q.The optimization problem is then to obtain the solution that min-imizes the sample average cost of the problem.

minxP0

c0xþ 1=QXQ

q¼1

wðx; zqÞ ð53Þ

The formulation (53) cab be easily re-formulated and solved as alinear programming model. In the computations the sample sizeis set to be R = 50.

4.2.4. Experiment set-upTable 3 summarizes the input parameters used in the computa-

tional study. There are 30 different input stocks whose levels needto be determined. After the testing stage, these stocks are sortedinto a total of 20 final product grades. The uncertainty ranges ofthe demand and grading fractions were estimated from a sampleset of past data of the semiconductor production. The cost rateparameters c were chosen from the interval [1,10] to model therelative per unit costs of the input stocks. Each uncertain variate

Table 3Computational study input parameters.

Parameter Values

Number of input stocks I 30Number of final products J 20Purchase cost rate c0 [1,10]Out-of-sample performance sample size K 300Estimated demand means [20,100]Estimated grading fractions means [0,1]Random factors ~z Beta (c1,c2)

~z 2 ½z; zþ Dz� with Dz P 0 is modeled as a zero-mean random var-iable as follows:

~z ¼ zþ Dz � Bðc1; c2Þ

where B is a beta-distributed random variable with shape parame-ters c1 and c2. The beta random variable is used to facilitate thecomputational experiments because of its flexibility in modelingdifferent bounded distributions. The mean of ~z is then

Eð~zÞ ¼ zþ Dz � c1

c1 þ c2

The shape parameters c1 and c2 are then chosen with the above isset to zero. In the problem case instances, many input stocks couldonly produce a subset of the final products. This can be accountedfor in our modelling framework by setting the appropriate Dz = 0above.

For performance comparison, the following performance mea-surements were evaluated, the sample maximum regret (SMR), sam-ple maximum costs (SMC) and sample average costs (SAC). Thesemeasures are aligned with the objective of each of the correspond-ing optimization models. The following procedure was used in theperformance evaluation of each model:

1. Solve planning model for optimal input stock levels x⁄.2. Generate K demand and grading fractions realizations according

to the assumed distribution.3. For each realization zk:� Solve the following second-stage allocation problem (39) to

evaluate the realized cost w(x⁄,zk).� Solve the co-production newsvendor (36)–(38), with full

information, i.e., zk for the optimal costs f(zk).� Evaluate the realized regret R(x⁄,zk) = c0x⁄ + w(x⁄,zk) � f(zk).

4. Evaluate the performance measures of:� Sample maximum regret SMR(x⁄) = maxk=1, . . .,KR(x⁄,zk).� Sample maximum costs SMC(x⁄) = maxk=1,. . ., Kc0x⁄ + w(x⁄,zk).� Sample average costs SACðx�Þ ¼ c0x� þ 1=K

PKk¼1wðx�; zkÞ.

4.2.5. Results and discussionTable 4 shows the results of the computational experiments for

the comparison of the robust regret optimization with the worst-case optimization (WOP), the sample-average approximation(SAA) and the average-case optimization (AOP) models. The tabu-lated values are the SMR, SMC and SAC of these models, normalizedwith respect to the corresponding performance measures of the ro-bust regret model. First, for the SMR, the average performance ofthe WOP, SAA, AOP across the eight test problems are 1.92, 1.24and 1.41 times that of the robust regret model respectively. Atworst, the solutions from these model solutions were 2.7, 1.6 and2.3 times that of the robust regret model solution respectively. Thisclearly demonstrates that while WOP and AOP are simple models toformulate and solve, their regret performance were very much

Instance SMR SMC SAC

WOP SAA AOP WOP SAA AOP WOP SAA AOP

1 1.47 1.07 1.08 1.25 1.10 1.13 1.33 0.96 0.972 2.39 1.12 1.09 1.23 1.21 1.17 1.28 1.01 1.233 2.31 1.16 1.21 1.06 1.13 1.14 1.18 0.95 1.174 1.11 1.08 1.2 1.07 1.15 1.16 1.10 0.87 0.985 2.69 1.32 1.65 1.15 1.21 1.23 1.26 1.05 1.126 2.77 1.65 2.31 1.14 1.08 1.11 1.41 0.98 1.247 1.6 1.48 1.51 1.21 1.17 1.17 1.23 0.97 1.318 1.06 1.17 1.26 0.95 1.06 1.14 1.09 1.10 0.98

Average 1.92 1.24 1.41 1.13 1.14 1.15 1.24 0.98 1.12


unstable and inferior compared to the robust regret. The SAA modelperformed relatively better than these two models, but there is stilla significant gap from the regret achieved by the robust regretmodel. This motivates the need for a model that explicitly accountsfor regret performance.

Next, for the SMC performance, we note that the WOP model at-tained a worst-case cost of on average 1.13 times that of the robustregret. This is surprising since WOP model explicitly hedges againstworst case performance. One possible reason is that the WOC isbased on optimizing around a rare event realization to hedgeagainst worst-case outcomes. For instance, in some realizationsthe WOP newsvendor is observed to take extreme positions suchas not to purchase and produce any products when variability ofthe yield and demand values are high. In fact, the performance ofthe WOP was only marginally better than the AOP (1.15 on average)and not significantly better than the SAA model (1.13 on average).The robust regret model on the other hand out-performed all threemodels in the SMC, and is hence relatively effective in hedgingagainst bad outcomes. This could be due to the nature of the regretoptimization, that tends to avoid solutions with large performancegaps across different data realizations.

For the performance of sample average costs, the WOC modelaveraged 1.23 times the sample average costs of the robust regretsolution. As before, this indicates the inferior performance of theWOC solution on average across many data realizations. The simpleaverage-data model AOP on the other hand performed relativelybetter, and in three test cases achieved an SAC lower than the ro-bust regret solution. However, it is also more exposed to severelybad outcomes, for instance in test Problems 2, 6 and 7. The SAAmodel outperformed all the other models and averaged 0.98 timesthe cost of the robust regret solution. This is not surprising sincethe SAA explicitly optimizes the expected costs of the problem.To put the results in perspective, however, the loss in performanceof the robust regret model in terms of the sample average costs ismarginal compared to the cost savings achieved in the measures ofSMR and SMC.

Finally, Table 5 shows the computational times of the eight testcases of the four models. The AOP and WOP averaged about 22.9and 34.1 seconds of solution time, while the SAA model averaged68.4 seconds, due to the larger problem formulation of the model.The robust regret model on average took 493.4 seconds to solve,and at worst about 960.3 seconds (�16 minutes) across the eighttest problems. Thus, it is clear that robust regret model requiressignificantly more computational time to solve compared to theother models. This is expected, since solving the robust regretmodel requires the resolution of the candidate maximum regretproblem repeatedly, which is a mixed integer programming prob-lem. The computational studies seem to suggest that the candidateregret problem indeed poses as a bottleneck in the solution pro-cess, since instances that require long solution times are also asso-ciated with those that require more regret cuts to be generated

Table 5Computational times of planning models (number of regret cuts in parenthesis).

Instance AOP model WOP model SAA model Robust regret model

1 15.6 30.8 72.8 215.5 (6)2 12.8 38.5 70.3 477.6 (9)3 29.1 44.3 65.6 960.3 (17)4 21.4 32.7 56.8 300.4 (6)5 28.7 29.4 72.5 819.8 (14)6 25.5 31.2 68.5 468.5 (10)7 31.6 37 63.8 274.6 (8)8 18.7 29.1 77.4 430.9 (10)

Average 22.9 34.1 68.4 493.4 (9.8)

(e.g. Instances 3 and 5). However, in view of the good performanceof the robust regret solutions, the expended computational timesmay still be considered acceptable if the planning cycles are per-formed on a daily or weekly basis, or if real-time re-generationof planning solutions are not required.

5. Conclusion

In this paper the regret optimization criterion for linear pro-gramming problems under uncertainty is proposed. This workbuilds on and extends the results of previous works by consider-ing linear programs with not only interval objective coefficientsbut also in the other data parameters. A safe approximation ofthe regret function is developed. This approximation preservesthe extrema sufficiency property so that the maximum regretcan be evaluated efficiently. The approach is further applied toa co-production newsvendor problem. Computational experi-ments first demonstrated that the regret approximation is reason-ably accurate. Next, comparison with other optimizationapproaches such as worst-case and sample average optimizationvalidate that the proposed model achieves a significantly superiorperformance in regret, and performs competitively across mea-sures such as average and worst case costs. In this work, severalspecial structural results were established and exploited in theco-production model for the application of the regret optimiza-tion approach. Future research includes extending the robust re-gret optimization approach to uncertain general two-stagelinear programs. Furthermore, recent developments in robustoptimization termed as distributionally robust optimization al-lows the incorporation of distributional information such as meanand variance in the specification of the uncertainties. Ongoing re-search includes incorporating these distributional information inthe robust regret modelling framework.

Appendix A

A.1. Proof of Proposition 4

In the following, define Fj for all j = 1, . . . , J as the total supply ofgrades 1, . . . , j � 1 NOT used to fulfill demands for gradesj = 1, . . . , j � 1, i.e. the outstanding supply after fulfilling demandsfor grades j = 1, . . . , j � 1. For the case of j = 1 we define F1 = 0. Anoptimal shortage level for grade j, for all j = 1, . . . , J, can then bewritten as y�j ¼ ½dj � Fj � ujx�þ, since gj P gj0 for all j0 > j, and henceone always do no worse by satisfying as much of dj as possible withFj compared to satisfying demands of grades j0 > j. Extending thesame reasoning to the next grade j + 1,

Fjþ1 ¼ ½ujxþ Fj � dj�þ

That is, in the optimal solution, Fj+1 is that outstanding supply afterfulfilling d1, . . . , dj as much as possible. Consequently, to show theexpressions for y��j in (42) and (43), it suffices to show that

Fj ¼Xj�1

j0¼1


� �8j ¼ 2; . . . ; J

First, note that for product grade j = 1, (42) is clear, i.e.y�j ¼ ½d1 � u1x�þ, since it is always no worse to use u1x fill demandsfor j = 1 as much as possible. Now suppose by hypothesis that theabove is true for some j = 2, . . . , J. Indeed, based on y�1; F2 can bewritten as

F2 ¼ y�1 þ u1x� d1


where it is noted that F2 = u1x � d1 if y�1 ¼ 0, and F2 = 0 if y�1 > 0. Forthe general case, the following can be written:

Fjþ1 ¼ ½ujxþ Fj � dj�þ

¼ ujxþXj�1

j0¼1


� �� dj

24

35þ

¼ ujx� dj þXj�1

j0¼1


� �

þ dj � ujx�Xj�1

j0¼1

y�j þ uj0x� dj0

� �24

35þ

¼Xj

j0¼1


� �

Hence by induction the hypothesis holds true for all j = 2, . . . , J.Thus, the solution presented in (42) and (43) is at least as good asany optimal solution, and must also be optimal. h


In the following it is shown that there exists an optimal solutionto the linear program in (44) and (45) with

y�j ¼ dj �Xj�1

j0¼1


� �� ujx

24

35þ

for all j ¼ 1; . . . ; J ð54Þ

First consider (45) re-written as follows:

yj P dj �Xj�1

j0¼1

yj0 þ uj0x� dj0� �

� ujx 8j ¼ 1; . . . ; J

Clearly y�j P dj �Pj�1

j0¼1 y�j0 þ uj0x� dj0

� �� ujx

h iþby feasibility for any

optimal solution y⁄. Furthermore, there is always an optimal solu-tion with the above inequality in (54) tight. This is so since any

y�j > dj �Xj�1

j0¼1


� �� ujx

24

35þ

can always be decreased by some non-zero level, and y�j0 for some j0 > j

can be increased by the same amount, resulting in a nonnegative cost

decrease of ðgj � gj0 Þ y�j � dj �Pj�1

j0¼1 y�j0 þ uj0x� dj0

� �� ujx

h iþ� . Since

this is at least as good as the optimal solution, there must be an opti-mal solution with y�j as defined in (54). Finally, applying the result ofProposition 4, the optimal solution to the down-substitution problemis hence also an optimal solution to the linear program (44) and(45). h


It suffices to show that the optimal solution of (46)–(49),

yj ¼Pj

j0¼1dj0 �Pj

j0¼1uj0xh iþ

for all j. First in (47), if the right-hand

side termsPj

j0¼1dj0 �Pj

j0¼1uj0x < 0, then we must have sj = 1 by fea-

sibility. Otherwise, it is always optimal to set sj = 0, and set

yj ¼Pj

j0¼1dj0 �Pj

j0¼1uj0x, since because IP is maximizing in the

objective function. h

References

Averbakh, I., Lebedev, V., 2005. On the complexity of minimax regret linearprogramming. European Journal of Operational Research 160, 227–231.

Azzone, G., Maccarrone, P., 2001. The design of the investment post-audit process inlarge organisations: evidence from a survey. European Journal of InnovationManagement 4, 7387.

Bell, D.E., 1982. Regret in decision making under uncertainty. Operations Research30, 961–981.

Ben-Tal, A., Nemirovski, A., 1998. Robust convex optimization. Mathematics inOperations Research 23, 769–805.

Bergemann, D., Schlag, K.H., 2008. Pricing without priors. Journal of the EuropeanEconomic Association 6, 560–569.

Bertsimas, D., Tsitsiklis, J.N., 1997. Introduction to Linear Optimization. AthenaScientific.

Birge, J.R., Louveaux, F., 1997. Introduction to Stochastic Programming. Springer,New York.

Bitran, G.R., Dasu, S., 1992. Ordering policies in an environment of stochastic yieldsand substitutable demands. Operations Research 40, 999–1017.

Bitran, G.R., Leong, T.Y., 1992. Deterministic approximations to co-productionproblems with service constraints and random yields. Management Science 38,724–742.

Bitran, G.R., Gilbert, S.M., 1994. Co-production processes with random yields in thesemiconductor industry. Operations Research 42, 476–491.

Chen, X., Sim, M., Sun, P., Zhang, J., 2008. A linear decision-based approximationapproach to stochastic programming. Operations Research 56, 344–357.

Conde, E., 2004. An improved algorithm for selecting p items with uncertain returnsaccording to the minimax-regret criterion. Mathematical Programming 100,345–353.

Connolly, T., Zeelenberg, M., 2002. Regret in decision making. Current Directions inPsychological Science 11, 212–216.

Daniels, R.L., Kouvelis, P., 1995. Robust scheduling to hedge against processing timeuncertainty in single-stage production. Management Science 41, 363–376.

Goh, J., Sim, M., 2010. Distributionally robust optimization and its tractableapproximations. Operations Research 58, 902–917.

Gulliver, F.R., 1987. Post-project appraisals pay. Harvard Business Review March–April, 128–132.

Hsu, A., Bassok, Y., 1999. Random yield and random demand in a production systemwith downward substitution. Operations Research 47, 277–290.

Inuiguchi, M., Sakawa, M., 1995. Minimax regret solution to linear programmingproblems with an interval objective function. European Journal of OperationalResearch 86, 526–536.

Inuiguchi, M., Tanino, T., 2000. Portfolio selection under independent possibilisticinformation. Fuzzy Sets and Systems 115, 83–92.

Kahneman, D., Tversky, A., 1982. The Psychology of Preferences, Scientific American.Kasugai, H., Kasegai, T., 1961. Note on minimax regret ordering policy – static and

dynamic solutions and a comparison to maximin policy. Journal of theOperations Research Society of Japan 3, 155169.

Lin, J., Ng, T.S., 2011. Robust multi-market newsvendor models with intervaldemands data. European Journal of Operational Research 212, 361–373.

Loomes, G., Sugden, R., 1987. Testing for regret and disappointment in choice underuncertainty. The Economic Journal 97, 118–129.

Manski, C.F., 2007. Minimax-regret treatment choice with missing outcome data.Journal of Econometrics 139, 105–115.

Mausser, H.E., Laguna, M., 1998a. A new mixed integer formulation for themaximum regret problem. International Transactions in Operational Research5, 389–403.

Mausser, H.E., Laguna, M., 1998b. A heuristic to minimax absolute regret for linearprograms with interval objective function coefficients. European Journal ofOperational Research 117, 157–174.

Neale, C.W., Holmes, D.E.A., 1990. Post-auditing capital projects. Long RangePlanning 23, 88–96.

Perakis, G., Roels, G., 2008. Regret in the newsvendor model with partialinformation. Operations Research 56, 188–203.

Ritov, I., 1996. Probability of regret: anticipation of uncertainty resolution in choice.Organizational Behavior and Human Decision Processes 66, 228–236.

Savage, L.J., 1951. The theory of statistical decision. Journal of the AmericanStatistical Association 46, 55–67.

Shimizu, K., Aiyoshi, E., 1980. Necessary conditions for min–max problems andalgorithms by a relaxation procedure. IEEE Transactions on Automatic Control25, 62–66.

Simonson, I., 1992. The influence of anticipating regret and responsibility onpurchase decisions. The Journal of Consumer Research 19, 105–118.

Vairaktarakis, G.L., 2000. Robust multi-item newsboy models with a budgetconstraint. International Journal of Production Economics 66, 213–226.

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