The Decision Rule Approach to Optimisation under Uncertainty: Methodology and Applications in Operations Management

Transcription

1 The Decision Rule Approach to Optimisation under Uncertainty: Methodology and Applications in Operations Management Angelos Georghiou, Wolfram Wiesemann, Daniel Kuhn Department of Computing, Imperial College London 180 Queen s Gate, London SW7 2AZ, United Kingdom. Abstract Decision-making under uncertainty has a long and distinguished history in operations research. However, most of the existing solution techniques suffer from the curse of dimensionality, which restricts their application to small and medium-sized problems, or they rely on simplifying modelling assumptions (e.g. absence of recourse actions). Recently, a new solution technique has been proposed, which we refer to as the decision rule approach. By approximating the feasible region of the decision problem, the decision rule approach aims to achieve tractability without changing the fundamental structure of the problem. In this paper, we survey the major theoretical results relating to this approach, and we investigate its potential in operations management. Keywords. Robust Optimisation, Decision Rules, Optimisation under Uncertainty. 1 Introduction Operations managers frequently take decisions whose consequences extend well into the future. Inevitably, the outcomes of such choices are affected by significant uncertainty: changes in customer taste, technological advances and unforeseen stakeholder actions all have a bearing on the suitability of these decisions. It is well documented in theory and practice that disregarding this uncertainty often results in severely suboptimal decisions, which can in turn lead to the underperformance or complete breakdown of production processes. Yet, 1

2 researchers and practitioners frequently neglect uncertainty and instead focus on the expected or most likely market developments. We argue in the following that this is caused by the inherent limitations of the mainstream approaches to decision-making under uncertainty. While our argument applies to other methods as well, we will restrict our discussion to stochastic programming and robust optimisation. The remainder of the paper advocates the decision rule approach as a promising candidate to remedy some of these shortcomings. Stochastic programming models the uncertain parameters of a decision problem as a random vector that follows a known distribution. The two major types of stochastic programs are recourse problems and chance-constrained problems. In the classical two-stage recourse problem, the decision maker selects a here-and-now decision before the realisations of the uncertain parameters are known. This decision is complemented by a wait-and-see action that is chosen after the parameter realisations are observed. The goal is to select here-andnow and wait-and-see decisions that optimise a risk functional, such as the expected value or the variance of a cost function. For example, a production planning problem may ask for a production schedule that minimises the expected inventory holding and backlogging costs when customer demands are uncertain. A two-stage recourse formulation of this problem could involve a here-and-now manufacturing decision that is complemented by a sales decision once the customer demands are observed. Two-stage recourse problems find their natural generalisation in multi-stage recourse problems, where a sequence of uncertain parameters is observed over time. In this setting, the decision maker chooses a recourse action whenever some of the parameter realisations are observed. For instance, a multi-stage formulation of our production planning problem could accommodate a weekly revision of the production schedule to avoid inventory depletions or overruns. Although two-stage and multi-stage recourse problems are very similar, their computational complexity differs considerably. While the exact solution of linear two-stage recourse problems is #P-hard [24], approximate solutions can be found quite efficiently via Monte Carlo sampling techniques [13,54]. Multi-stage recourse problems, on the other hand, are believed to be computationally intractable already when medium-accuracy solutions are sought [57]. For this reason, the majority of recourse problems studied in the operations management literature are based on two-stage formulations. Contrary to recourse formulations, chance-constrained problems traditionally only involve 2

3 here-and-now decisions. The goal is to optimise a deterministic objective, subject to the requirement that constraints involving uncertain parameters are satisfied with at least a prespecified probability. Going back to our production planning problem, a chance-constrained model could ask for a minimum cost production plan that satisfies the uncertain demands with a probability of at least 95%. Unfortunately, apart from some benign special cases, chance-constrained problems are notoriously difficult to solve [52]. Indeed, verifying the satisfaction of a chance constraint requires multidimensional integration, which becomes computationally demanding if there are more than a few uncertain problem parameters. As a result, one typically resorts to sampling approximations [14, 49] or reformulations using conditional value-at-risk constraints [53]. Chance-constrained problems involving wait-andsee decisions are studied in [26, 47]. A more detailed review of stochastic programming can be found in [13,54]. Classical robust optimisation can be viewed as a limiting case of chance-constrained problems where the constraints must be satisfied with probability one. Thus, a robust optimisation problem asks for a here-and-now decision that optimises a deterministic objective and at the same time satisfies a constraint set for every possibly realisation of the uncertain parameters. In our production planning example, a robust formulation may be appropriate if lost sales should be avoided at all costs, or if the decision maker is unable to assign a probability distribution to the uncertain customer demands. Robust optimisation has gained wide popularity in recent years due to its favourable computational complexity. Many classes of optimisation problems, including linear programs, conic-quadratic programs and mixedinteger linear programs, allow for robust formulations than can be solved with essentially the same effort as the corresponding deterministic problems [4,10]. Robust optimisation has been extended to accommodate chance constraints [16] and various measures of risk [46]. Similar to chance-constrained stochastic programming, however, robust optimisation traditionally only accounts for here-and-now decisions. The robust optimisation literature is surveyed in [4, 11]. The above-mentioned approaches to decision-making under uncertainty have in common that they either do not account for recourse actions or that they are computationally demanding. Neglecting recourse possibilities can lead to overly conservative decisions as operations managers typically have the option to revise plans once new information becomes available. Likewise, solution techniques must scale gracefully with the size of the problem to be useful 3

4 for practitioners. To date, this dichotomy between tractability and modelling accuracy has proved to be a major obstacle to the successful application of stochastic programming and robust optimisation to real-life operations management problems. In this paper, we make a case for the decision rule approach to decision-making under uncertainty. The decision rule approach allows us to approximately solve two-stage and multi-stage recourse problems, chance-constrained problems and robust optimisation problems in a computationally efficient way. The approach inherits the tractability of robust optimisation, but it permits a faithful modelling of the dynamic aspects of a decision problem. The purpose of this paper is twofold. Firstly, we survey the theoretical findings relating to the decision rule approach. To date, these results are scattered over several papers, and due to their technical terminology they attracted little attention outside the mathematical optimisation community. We aim to present the key findings of this field in a comprehensible and unified framework, and we extend the methodology to problems with integer here-andnow decisions. The second purpose of the paper is to critically examine the potential of the decision rule approach in operations management. To this end, we apply the approach to two stylised case studies, and we compare decision rules with alternative methods to decision-making under uncertainty. Stochastic programming techniques have a long history of applications in operations management. Recourse problems and chance-constrained problems have been developed for the design of supply chains [33,43,55], the planning of facility layouts [22,39,58], production planning and inventory control [3, 45], production scheduling [50, 62] and project management [21, 34]. There are much fewer applications of robust optimisation techniques to operations management problems. The paper [12] studies the optimal control of a multiechelon supply chain that is defined on a network. A robust inventory control problem with ambiguous demands is studied in [56]. Extension to simultaneous inventory control and dynamic pricing, as well as two-echelon supply chains with flexible commitments contracts, can be found in [2,5]. The papers [35,40] develop robust formulations of well known deterministic production scheduling problems, and robust project management problems are studied in [19, 63]. Besides stochastic programming and robust optimisation, decision problems with uncertain parameters are often formulated as stochastic dynamic programs. For textbook 4

5 introductions to stochastic dynamic programming, as well as its tractable approximation via neuro-dynamic programming, the reader is referred to [6, 51]. The remainder of the paper is structured as follows. In Section 2, we formulate the decision problems that we are interested in. We introduce linear decision rules in Section 3, and we generalise the approach to nonlinear decision rules in Section 4. Section 5 describes an extension to integer here-and-now decisions. We close with two operations management case studies in Section 6. Notation For a square matrix A R n n we denote by Tr(A) the trace of A, that is, the sum of its diagonal entries. For A,B R m n the inequalities A B and A B are understood to hold component-wise, and A denotes the transpose of A. For any real number c we define c + = max{c,0}. We define convx as the convex hull of a set X. 2 Decision-Making under Uncertainty 2.1 Problem Formulation We study dynamic decision problems under uncertainty of the following general structure. A decision maker first observes an uncertain parameter ξ 1 R k 1 and then takes a decision x 1 (ξ 1 ) R n 1. Subsequently, a second uncertain parameter ξ 2 R k 2 is revealed, in response to which the decision maker takes a second decision x 2 (ξ 1,ξ 2 ) R n 2. This sequence of alternating observations and decisions extends over T stages, where at any stage t = 1,...,T the decision maker observes an uncertain parameter ξ t and selects a decision x t (ξ 1,...,ξ t ). We emphasise that a decision taken at stage t depends on the whole history of past observations ξ 1,...,ξ t, but it may not depend on the future observations ξ t+1,...,ξ T. This feature reflects the non-anticipative nature of the dynamic decision problem and ensures its causality. To simplify notation, we define the history ofobservations upto time tas ξ t = (ξ 1,...,ξ t ) R kt, where k t = t s=1 k s. Moreover, we let ξ = (ξ 1,...,ξ T ) R k denote the vector concatenation of all uncertain parameters, where k = k T. 5

6 We assume that the decision taken at stage t incurs a linear cost c t (ξ t ) x t (ξ t ), where the vector of cost coefficients depends linearly on the observation history, that is, c t (ξ t ) = C t ξ t for some matrix C t R nt kt. We also assume that the decisions are required to satisfy a set of linear inequality constraints to be detailed below. The decision maker s objective is to select the functions or decision rules x 1 ( ),...,x T ( ), which map observation histories to decisions, such that the expected total cost is minimised while all inequality constraints are satisfied. Formally, this decision problem can be represented as follows. ) minimise E ξ ( T subject to E ξ ( T s=1 c t (ξ t ) x t (ξ t ) A ts x s (ξ s ) ξ t ) x t (ξ t ) 0 b t (ξ t ) ξ Ξ, t = 1,...,T Here, E ξ ( ) denotes expectation with respect to the random parameter ξ, while Ξ stands for the range of all possible values that ξ can adopt. Below, we will refer to Ξ as the uncertainty setandtoanyξ Ξasascenario. WewillhenceforthassumethatΞisaboundedpolyhedron of the form Ξ = {ξ R k : Wξ h} for some W R l k and h R l. Note that the stage-t decisions x t (ξ t ) in problem P are parameterised in ξ t, that is, every observation history ξ t corresponding to some scenario ξ Ξ gives rise to n t ordinary decision variables. Since the polyhedron Ξ typically contains infinitely many scenarios, problem P accommodates in fact infinitely many decision variables. (P) The inequality constraints in problem P are expressed in terms of deterministic constraint matrices A ts R mt ns and uncertainty-affected right-hand side vectors b t (ξ t ) R mt. We assume that b t (ξ t ) = B t ξ t for some matrices B t R mt kt. Note that the stage-t constraints are conditioned on the stage-t observation history ξ t, where E ξ ( ξ t ) denotes conditional expectation with respect to ξ given ξ t. Hence, E ξ ( ξ t ) treats ξ 1,...,ξ t as deterministic variables and takes the expectation only with respect to the future observations ξ t+1,...,ξ T. This implies that the stage-t constraints are parameterised in ξ t in a similar fashion like the stage-t decisions. Indeed, every ξ t corresponding to some scenario ξ Ξ gives rise to m t ordinary linear constraints. Since Ξ has typically infinite cardinality, problem P thus accommodates infinitely many constraints. Intuitively, problem P can therefore be viewed as an infinite-dimensional generalisation of the standard linear program. 6

7 2.2 Expressiveness of Problem P The general decision problem under uncertainty described in Section 2.1 provides considerable modelling flexibility. Indeed, as we will show in this section, problem P encapsulates conventional deterministic and stochastic linear programs, robust optimisation problems and tight convex approximations of chance-constrained programs as special cases. Deterministic linear programs If the uncertainty set contains only one single scenario, that is, if Ξ = {ξ }, then problem P reduces to a deterministic linear program. In this case only the decisions and constraints corresponding to ξ = ξ are relevant, while all (conditional and unconditional) expectations become redundant and can thus be eliminated. Introducing the finite problem data A 11 A 1T b 1 (ξ 1 A =....., b = c 1 (ξ 1. and c =., A T1 A TT b T (ξ ) T c T (ξ ) T we can reformulate P as the standard linear program minimise c x subject to Ax b, x 0, (LP) whose finite-dimensional decision vector can be identified with (x 1 (ξ ),...,x 1 T (ξ )). T Remark 2.1 (Deterministic Decisions and Constraints). Deterministic decisions and constraints can conveniently be incorporated into the general problem P by requiring that ξ 1 is equal to 1 for all ξ Ξ. This can always be enforced by appending the equation ξ 1 = 1 to the definition of Ξ and has the effect that, in the first stage, only the decisions and constraints corresponding to the scenario ξ 1 = 1 have physical relevance. From now on, we will always assume that k 1 = 1 and that any ξ = (ξ 1,...,ξ T ) Ξ satisfies ξ 1 = 1. Stochastic programs Problem P can be specialised to a standard linear multistage stochastic program with recourse if we ensure that the stage-t constraints are not affected by the future decisions x t+1 (ξ t+1 ),...,x T (ξ T ). This is achieved by setting A ts = 0 for all t < s and has the effect that the term inside the conditional expectation of the stage-t constraint 7

8 becomes independent of ξ t+1,...,ξ T. Since E ξ ( ξ t ) treats ξ t as a constant, the conditional expectation thus becomes redundant and can be omitted. Therefore, problem P reduces to the following multistage stochastic program in standard form [13, 36]. ) ( T minimise E ξ c t (ξ t ) x t (ξ t ) t subject to A ts x s (ξ s ) b t (ξ t ) ξ Ξ, t = 1,...,T s=1 x t (ξ t ) 0 (SP) Robust optimisation problems If the distribution governing the uncertainty ξ is unknown or if the decision maker is very risk-averse, then it is not possible or unreasonable to minimise expected costs. In these situations a rational decision maker will minimise the worst-case costs, where the worst case (maximum) is evaluated with respect to all possible scenarios ξ Ξ; see e.g. [30] for a formal justification. As we have discussed in the introduction, such worst-case (robust) optimisation problems traditionally only involve here-and-now decisions [4, 11]. Problem P allows us to formulate a multi-stage generalisation of robust optimisation problems as follows. ( T ) minimise max c t (ξ t ) x t (ξ t ) ξ Ξ t subject to A ts x s (ξ s ) b t (ξ t ) ξ Ξ, t = 1,...,T s=1 x t (ξ t ) 0 (RO) In order to see that RO is a special case of P, we consider an epigraph reformulation of the worst-case objective, ( T ) c t (ξ t ) x t (ξ t ) max ξ Ξ = min τ R = min τ R { { τ : max ξ Ξ τ : ( T ) } c t (ξ t ) x t (ξ t ) τ } T c t (ξ t ) x t (ξ t ) τ ξ Ξ, (2.1) where τ R represents an auxiliary (deterministic) decision variable. Replacing the worstcase objective in RO with (2.1) transforms the robust optimisation problem RO into a variant of the stochastic programming problem SP with a particularly simple objective function (given by τ). As RO is a special case of SP and SP is a special case of P, we conclude that RO is indeed a special case of P. 8

9 Chance-constrained programs Let P ξ be the distribution of ξ. We can then formulate a multi-stage generalisation of chance-constrained programs as ( T ) minimise E ξ c t (ξ t ) x t (ξ t ) ( T ) (CC) subject to P ξ a it x t(ξ t ) b i (ξ) 1 ǫ i i = 1,...,I, x t (ξ t ) 0 ξ Ξ, t = 1,...,T, wherea it R nt, b i (ξ) Randǫ i (0,1]. Here, theithconstraint requiresthattheinequality T a it x t(ξ t ) b i (ξ) should be satisfied with probability at least 1 ǫ i. Chance constraints of this type are useful for modelling risk preferences and safety constraints in engineering applications. Note that a chance constraint with ǫ i = 1 reduces to a robust constraint that must hold for all ξ Ξ. Therefore, chance constraints with ǫ i < 1 can be viewed as soft versions of the corresponding robust constraints. We now demonstrate that CC has a tight conservative approximation of the form P. To this end, we introduce the loss functions L i (ξ) = b i (ξ) T a it x t(ξ t ). The ith chance constraint is therefore equivalent to the requirement that the smallest (1 ǫ i )-quantile of the loss distribution, which we denote by VaR ǫi (L i (ξ)), is nonpositive. To obtain a conservative approximation for the chance constraint, we introduce the conditional value-at-risk (CVaR) of L i (ξ) at level ǫ i, which is defined as CVaR ǫi (L i (ξ)) = min βi {β i + 1 ǫ i E ξ ([L i (ξ) β i ] + )}. Due to its favourable theoretical and computational properties, CVaR has become a popular risk measure in finance. Rockafellar and Uryasev [53] have shown that the optimal β i which solves the minimisation problem in the definition of CVaR coincides with VaR ǫi (L i (ξ)) and that the CVaR al level ǫ i coincides with the conditional expectation of the right tail of the lossdistribution above VaR ǫi (L i (ξ)). Thus, thefollowing implication holds; see alsofigure1. CVaR ǫi (L i (ξ)) 0 = VaR ǫi (L i (ξ)) 0 P ξ (L i (ξ) 0) 1 ǫ i As pointed out by Nemirovski and Shapiro [48], the CVaR constraint on the left-hand side represents the tightest convex approximation for the chance constraint on the right-hand side of the above expression. By linearising the term [L i (ξ) β i ] + in the definition of CVaR, the ith CVaR constraint can be re-expressed as the following system of linear inequalities β i + 1 ǫ i E ξ (z i (ξ)) 0, z i (ξ) b i (ξ) T a it x t(ξ t ) β i, z i (ξ) 0, (2.2) 9

10 1 ǫ i VaR ǫi (L i (ξ)) CVaR ǫi (L i (ξ)) 0 L i (ξ) Figure 1: Relationship between VaR ǫi (L i (ξ)) and CVaR ǫi (L i (ξ)) for each constraint i, at level ǫ i. which involve the deterministic (first stage) variable β i R and a new stochastic (stage- T) variable z i (ξ) R. Replacing each chance constraint in CC with the corresponding system (2.2) of linear inequalities thus results in a problem of type P with expectation constraints. Therefore, chance-constrained problems of the type CC have tight conservative approximations within the class of problems P. We remark that the general decision problem P is flexible enough to cover also hybrid models which combine various aspects of deterministic, stochastic, robust and chanceconstrained programs in the same model. 3 The Decision Rule Approach In this section, we derive a tractable approximation to the decision problem P by restricting the space of the decision rules x t ( ), t = 1,...,T, to those that exhibit a linear dependence on the observed problem parameters ξ t. The second part of the section explains how we can efficiently measure the optimality gap that we incur through this simplification. 3.1 Determining the Best Linear Decision Rule Problem P generalises a number of difficult optimisation problems, including multi-stage stochastic programs. It is therefore clear that problem P is severely computationally in- 10

11 tractable itself. A simple but effective approach to improve the tractability of problem P is to restrict the space of the decision rules x t ( ), t = 1,...,T, to those that exhibit a linear dependence on the observation history ξ t. Remember that we stipulated in Remark 2.1 that k 1 = 1 and ξ 1 = 1 for all ξ Ξ. This implies that we actually optimise over all affine (i.e., linear plus a constant) decision functions of the non-degenerate uncertain parameters ξ 2,...,ξ T if we optimise over all linear functions of ξ = (ξ 1,...,ξ T ). IntherestofthepaperwewillassumethattheconditionalexpectationsE ξ (ξ ξ t )arelinear in the sense that there exist matrices M t R k kt such that E ξ (ξ ξ t ) = M t ξ t for all ξ Ξ. This assumption is non-restrictive. It is automatically satisfied, for instance, if the random parameters ξ t are mutually independent. In this case the conditional expectations reduce to simpler unconditional expectations, and thus we find E ξ (ξ ξ t ) = (ξ 1,...,ξ t,µ t+1,...,µ T ), where µ t denotes the unconditional mean value of ξ t. As ξ 1 = 1 for all ξ Ξ, we thus have E ξ (ξ ξ t ) = (ξ 1,...,ξ t,µ t+1 ξ 1,...,µ T ξ 1 ) ξ Ξ. The last expression is manifestly linear in ξ t. It is easy to verify that the conditional expectations remain linear when the process of the random parameters ξ t belongs to the large class of autoregressive moving-average models. For the further argumentation, we define the truncation operator P t : R k R kt through P t ξ = ξ t. Thus, P t maps any scenario ξ to the corresponding observation history ξ t up to stage t. If we model the decision rule x t (ξ t ) as a linear function of ξ t, it can thus be expressed as x t (ξ t ) = X t ξ t = X t P t ξ for some matrix X t R nt kt. Substituting these linear decision rules into P yields the following approximate problem. ) minimise E ξ ( T subject to E ξ ( T s=1 c t (ξ t ) X t P t ξ A ts X s P s ξ ξ t ) b t (ξ t ) X t P t ξ 0 ξ Ξ, t = 1,...,T (P u ) The objective function of P u can be simplified and re-expressed in terms of the second order moment matrix M = E(ξξ ) of the random parameters. Interchanging summation and 11

12 expectation and using the cyclicity property of the trace operator, we obtain ( T ) T E ξ c t (ξ t ) X t P t ξ = E ξ (ξ Pt Ct X t P t ξ) T = E ξ (Tr[P t ξξ Pt C t X t]) = T Tr(P t MP t C t X t). Similarly, we can reformulate the conditional expectation terms in the constraints of P u as ( T E ξ A ts X s P s ξ ) T ξ t = A ts X s P s M t P t ξ. s=1 s=1 Thus, the linear decision rule problem P u is equivalent to T minimise Tr(P t MPt C t X t) subject to ( T A ts X s P s M t P t B t P t )ξ 0 s=1 X t P t ξ 0 ξ Ξ, t = 1,...,T. (3.3) Although problem (3.3) has only finitely many decision variables, that is, the coefficients of the matrices X 1,...,X T encoding the linear decision rules, it is still not suitable for numerical solution as it involves infinitely many constraints parameterised by ξ Ξ. The following proposition, which captures the essence of robust optimisation, provides the tools for reformulating the ξ-dependent constraints in (3.3) in terms of a finite number of linear constraints [4, 11]. Proposition 3.1. For any p N and Z R p k, the following statements are equivalent. (i) Zξ 0 for all ξ Ξ, (ii) Λ R p l with Λ 0, ΛW = Z, Λh 0. Proof. We denote by Zπ the πth row of the matrix Z. Then, statement (i) is equivalent to Zξ 0 for all ξ subject to Wξ h { 0 min Z π ξ : Wξ h } π = 1,...,p ξ { 0 max h Λ π : W Λ π = Z π, Λ π 0 } (3.4) π = 1,...,p Λ π Λ π with W Λ π = Z π, h Λ π 0, Λ π 0 π = 1,...,p 12

13 The equivalence in the third line follows from linear programming duality. Interpreting Λ π as the πth row of a new matrix Λ R p l shows that the last line in (3.4) is equivalent to assertion (ii). Thus, the claim follows. Using Proposition 3.1, one can reformulate the inequality constraints in (3.3) to obtain minimise subject to T Tr(P t MP t C t X t) T A ts X s P s M t P t B t P t = Λ t W,Λ t h 0,Λ t 0 s=1 X t P t = Γ t W,Γ t h 0,Γ t 0 t = 1,...,T. ( P u ) The decision variables in P u are the entries of the matrices X t R nt kt, Λ t R mt l and Γ t R nt l for t = 1,...,T. Note that the objective function as well as all constraints are linear in these decision variables. Thus, Pu constitutes a finite linear program, which can be solved efficiently with off-the-shelf solvers such as IBM ILOG CPLEX [1]. A major benefit of using linear decision rules is that the size of the approximating linear program P u grows only moderately with the number of time stages. Indeed, the number of variables and constraints is quadratic in k, l, m = T m t, and n = T n t. Note that these numbers usually scale linearly with T, and hence the size of P u typically grows only quadratically with the number of decision stages. We close this section with two remarks about alternative approximation methods to convert problem P to a finite linear program that is amenable to numerical solution. Remark 3.2 (Scenario tree approximation). Instead of approximating the functional form of the decision rules x t ( ), t = 1,...,T, we can can improve the tractability of problem P by replacing the underlying process ξ 1,...,ξ T of random parameters with a discrete stochastic process. The resulting process can be visualised as a scenario tree, which ramifies at all time points at which new problem data is observed (Figure 2). Scenario tree approaches to stochastic programming have been studied extensively over the past decades; see e.g. the survey paper [23] that accounts for the developments up to the year More recent contributions are listed in the official stochastic programming bibliography [60]. In contrast to the decision rule approach, scenario tree methods typically scale exponentially with the number of decision stages. Figure 3 compares the scenario tree and the decision rule approximations. 13

14 Figure 2: Scenario tree with samples ξ s 1,..., ξ s 9. Remark 3.3 (Sample-Based Optimisation). We derived a tractable approximation for problem P in two steps. First, we restricted the decision rules x t ( ), t = 1,...,T, to be linear functions of the observation histories ξ t. Afterwards, we used linear programming duality to obtain a finite problem. We can derive a different approximation for problem P if we enforce the semi-infinite constraints in P u only over a finite subset of samples { ξs 1,..., ξ K} s Ξ. It has been shown in [14,61] that a modest number K of samples suffices to satisfy the semi-infinite constraints in P u with high probability. The advantage of such sampling-based approaches is that they allow us to model more general dependencies between the problem data (A, b, c) and the random parameters. However, we are not aware of any methods to measure the optimality gap that we incur with sample-based methods. 3.2 Suboptimality of the Best Linear Decision Rule The price that we have to pay for the favourable scaling properties of the linear decision rule approximation is a potential loss of optimality. Indeed, the best linear decision rule can result in a substantially higher objective value than the best general decision rule (which is typically nonlinear). The difference u = min P u minp between the optimal values of the approximate and the original decision problem can be interpreted as the approximation error associated with the linear decision rule approximation. As P u is a restriction of the minimisation problem P, u is necessarily nonnegative. Modellers should estimate u in 14

15 x(ξ) x( ξ s ) x(ξ) ξ s Ξ Ξ Figure 3: Comparison of the scenario tree (left) and the decision rule approximation (right). Scenario trees replace the process ξ 1,...,ξ T of random parameters with a discrete stochastic process. The decision rule approach retains the original stochastic process, but it restricts the functional form of the decision rules x t ( ), t = 1,...,T. order to assess the appropriateness of the linear decision rule approximation for a particular problem instance: a small u indicates that implementing the solution of Pu will incur a negligible loss of optimality, while a large u may prompt us to be more cautious and to try to improve the approximation quality (e.g. by using more flexible piecewise linear decision rules; see Section 4). Generally speaking, there are two ways to measure the approximation error u. We can derive generic a priori bounds on the maximum value of u that can be incurred over a class of instances, or we can measure u a posteriori for a specific problem instance. A priori bounds on u have a long history. In particular, linear decision rules have been proven to optimally solve the linear quadratic regulator problem [6], while piecewise linear decision rules optimally solve two-stage stochastic programs [28]. More recently, linear decision rules have been shown to optimally solve a class of one-dimensional robust control problems [9] and two-stage robust vehicle routing problems [32]. On the other hand, the worst-case approximation ratio for linear decision rules applied to two-stage robust optimisation problems with m linear constraints has been shown to be of the order O( m), see [7]. Similar results have been derived for two-stage stochastic programs in [8]. Given their scarcity and their somewhat limited scope, it seems fair to say that a priori 15

16 bounds on u are at most indicative of the expressive power of linear and piecewise linear decision rules. It thus seems natural to consider a posteriori bounds on u that exploit the specific structure of individual instances of the problem P. Unfortunately, the direct computation of u for a specific instance of P would require the solution of P itself, which is intractable. In this section we demonstrate, however, that an upper bound on u can be obtained efficiently by studying a dual decision problem associated with P. It is well known that any primal linear program min x {c x : Ax b, x 0} has an associated dual linear program max y {b y : A y c, y 0}, which is based on the same problem data (A,b,c), such that the following hold: the minimum of the primal is never smaller than the maximum of the dual (weak duality), and if either the primal or the dual is feasible then the minimum of the primal coincides with the maximum of the dual (strong duality) [18]. There is a duality theory for decision problems of the type P which is strikingly reminiscent of the duality theory for ordinary linear programs. Following Eisner and Olsen [25], the dual problem corresponding to P can be defined as ) maximise E ξ ( T subject to E ξ ( T s=1 b t (ξ t ) y t (ξ t ) A st y s(ξ s ) ξ t ) y t (ξ t ) 0 c t (ξ t ) ξ Ξ, t = 1,...,T. Note that the dual maximisation problem D is stated in terms of the same problem data as the primal minimisation problem P. As for ordinary linear programs, dualisation transposes the constraint matrices and swaps the roles of the objective function and right-hand side coefficients. Dualisation also reverts the temporal coupling of the decision stages in the sense that the sums in the constraints of D now run over the first index of the constraint matrices. Thus, even if the primal stage-t constraint is independent of ξ t+1,...,ξ T, implying that theconditional expectation E ξ ( ξ t ) becomes redundant in P, thedual stage-tconstraint usually still depends on ξ t+1,...,ξ T, implying that the conditional expectation E ξ ( ξ t ) has a non-trivial effect in D. Therefore, in hindsight we realise that the inclusion of conditional expectation constraints in P was necessary to preserve the symmetry of the employed duality scheme. (D) As in the case of ordinary linear programming, there exist weak and strong duality results for problems P and D [25]. In particular, the minimum of P is never smaller than 16

17 the maximum of D (weak duality), and if some technical regularity conditions hold, then the minimum of P coincides with the maximum of D (strong duality). The symmetry between P and D enables us to solve D with the linear decision rule approach that was originally designed for P. Indeed, if we model the dual decision rule y t (ξ t ) as a linear function of ξ t, then it can be expressed as y t (ξ t ) = Y t ξ t for some matrix Y t R mt kt. Substituting these dual linear decision rules into D yields an approximate problem D l, which can be shown to be equivalent to the following tractable linear program. maximise subject to T Tr(P t MP t B t Y t) T A st Y sp s M t P t C t P t = Φ t W,Φ t h 0,Φ t 0 s=1 Y t P t = Ψ t W,Ψ t h 0,Ψ t 0 t = 1,...,T. ( D l ) The decision variables in D l are the entries of the matrices Y t R mt kt, Φ t R nt l and Ψ t R mt l for t = 1,...,T. In analogy to the primal approximation error u, the dual approximation error can be defined as l = maxd max D l. As D l is a restriction of the maximisation problem D, l is necessarily nonnegative. It quantifies the loss of optimality of the best linear dual decision rule with respect to the best general dual decision rule. Unfortunately, l is usually unknown as its computation would require the solution of the original dual decision problem D. However, thejointprimalanddualapproximationerror = min P u max D l isefficiently computable; it merely requires the solution of two tractable finite linear programs. Note that constitutes indeed an upper bound on both u and l since = min P u max D l = min P u minp +minp maxd +maxd max D l = u +minp maxd + l u + l, where the last inequality follows from weak duality. We conclude that for any decision problem of the type P the best primal and dual linear decision rules can be computed efficiently by solving tractable finite linear programs. 17

18 The corresponding approximation errors are bounded by, which can also be computed efficiently. Moreover, the optimal values of the approximate problems P u and D l provide upper and lower bounds on the optimal value of the original problem P, respectively. 4 Nonlinear Decision Rules A large value of indicates that either the primal or the dual approximation (or both) are inadequate. If the loss of optimality of linear decision rules is unacceptably high, modellers will endeavour to find a less conservative (but typically more computationally demanding) approximation. Ideally, one would choose a richer class of decision rules over which to optimise. In this section we show that the techniques developed for linear decision rules can also be used to optimise efficiently over more flexible classes of nonlinear decision rules. The underlying theory has been developed in a series of recent publications [4,17,29,31]. We will motivate the general approach first through an example. Example 4.1. Assume that a two-dimensional random vector ξ = (ξ 1,ξ 2 ) is uniformly distributed on Ξ = {1} [ 1,1]. This choice of Ξ satisfies the standard assumption that ξ 1 = 1 for all ξ Ξ. Any scalar linear decision rule is thus representable as x(ξ) = X 1 ξ 1 + X 2 ξ 2, where X 1 denotes a constant offset (since ξ 1 is equal to 1 with certainty), while X 2 characterises the sensitivity of the decision with respect to ξ 2 ; see Figure 4 (left). To improve flexibility, one may introduce a breakpoint at ξ 2 = 0 and consider piecewise linear continuous decision rules that are linear in ξ 2 on the subintervals [ 1,0] and [0,1], respectively. These decision rules are representable as x(ξ) = X 1 ξ 1 +X 2 (min{ξ 2,0})+X 3 (max{ξ 2,0}), (4.5) where X 1 denotes again a constant offset, while X 2 and X 3 characterise the sensitivities of the decision with respect to ξ 2 on the subintervals [ 1,0] and [0,1], respectively; see Figure 4 (centre). We can now define a new set of random variables ξ 1 = ξ 1, ξ 2 = min{ξ 2,0} and ξ 3 = max{ξ 2,0}, which are completely determined by ξ. In particular, the support for ξ = (ξ 1,ξ 2,ξ 3 ) is given by Ξ = {ξ R 3 : ξ 1 = 1, ξ 2 [ 1,0], ξ 3 [0,1], ξ 2 ξ 3 = 0}. Note also that ξ is uniformly distributed on Ξ. We will henceforth refer to ξ as the lifted random 18

19 x(ξ) x(ξ) x (ξ ) X 3 X 3 1 X 2 X 1 X 2 X ξ X 1 ξ 2 ξ 3 X Ξ Ξ 1 Ξ ξ 2 Figure 4: Illustration of the linear and piecewise linear decision rules in the original and the lifted space. Note that the set Ξ = L(Ξ) is non-convex, represented by the thick line in the right diagram. Since the decision rule x (ξ ) is a linear function of the random parameters ξ, however, it is nonnegative over Ξ if and only if it is nonnegative over the convex hull of Ξ, which is given by the dark shaded region. vector as it ranges over a higher-dimensional lifted space. Moreover, the function L(ξ) = (L 1 (ξ),l 2 (ξ),l 3 (ξ)) = (ξ 1,min{ξ 2,0},max{ξ 2,0}), which maps ξ to ξ, will be referred to as a lifting. By construction, the piecewise linear decision rule (4.5) in the original space is equivalent to the linear decision rule x (ξ ) = X 1 ξ 1 +X 2 ξ 2 +X 3 ξ 3 in the lifted space; see Figure 4 (right). Moreover, due to the linearity of x (ξ ) in ξ, the decision rule x (ξ ) is nonnegative over the nonconvex set Ξ if and only if x (ξ ) is nonnegative over the convex hull of Ξ. We can therefore replace the nonconvex support Ξ in the lifted space with its (polyhedral) convex hull, which can be represented as an intersection of halfspaces as required in Section 2.1. Hence, all techniques developed for linear decision rules can also be used for piecewise linear decision rules of the form (4.5). To solve a general decision problem of the type P in nonlinear decision rules, we define a lifting operator L(ξ) = (L 1 (ξ 1 ),...,L T (ξ T )), whereeachl t (ξ t )representsacontinuousfunctionfromr kt tor k t forsomek t k t. Usingthe lifting operator, we can construct a lifted random vector ξ = (ξ 1,...,ξ T ), where ξ Rk t = L t (ξ t ), t = 1,...,T, and k = k 1+ +k T. The distribution of the lifted random vector ξ is completely determined by that of the primitive random vector ξ, andthe support of ξ can be defined as Ξ = L(Ξ). As for the primitive uncertainties it proves useful to define observation 19

20 histories ξ t = (ξ 1,...,ξ t ), k t = k Rk t 1 + +k t, and truncation operators P t : R Rk k t which map ξ to ξ t, respectively. Our goal is to solve problem P in nonlinear decision rules of the form x t (ξ t ) = X t P tl(ξ), which constitute linear combinations of the component functions of the lifting operator. The matrices X t contain the coefficients of these Rnt k t linear combinations, while the truncation operators P t eliminate those components of L(ξ) that depend on the future uncertainties ξ t+1,...,ξ T, thereby ensuring non-anticipativity. By construction, the nonlinear decision rules x t (ξ t ) = X t P tl(ξ) depending on the primitive uncertainties are equivalent to linear decision rules x t (ξ t ) = X t ξ t depending on the lifted uncertainties. Note that ordinary linear decision rules in the primitive uncertainties can be recovered by choosing a trivial lifting operator L(ξ) = ξ. For the further argumentation we require that the lifting preserves the degeneracy of the first random parameter, that is, ξ 1 = 1 for all ξ Ξ. Moreover, we assume that there is a linear retraction operator R(ξ ) = (R 1 (ξ 1 ),...,R T(ξ T )), which allows us to express the primitive random vector ξ as a linear function of the lifted random vector ξ. To this end, we assume that each R t represents a linear function from R k t to R k t. The best nonlinear decision rule can then be computed by solving the following optimisation problem. ( T ) minimise E ξ c t (P t R(ξ )) X tp tξ ( T subject to E ξ A ts X ) s P s ξ ξ t b t (P t R(ξ )) s=1 X t P t ξ 0 ξ Ξ, t = 1,...,T (P u ) Note that the observation histories in the objective function and in the right-hand side coefficients have been expressed as ξ t = P t ξ = P t R(ξ ). The optimisation variables in P u are the entries of the matrices X t Rnt k t. The key observation is that the approximate problems P u and P u have exactly the same structure. The only difference is that Ξ = L(Ξ) is typically not a polyhedron because of the nonlinearity of the lifting operator. In the following, we assume that an exact representation (or outer approximation) of the convex hull of Ξ is available in the form of l inequality constraints: ˆΞ := {ξ R k : W ξ h }, 20

21 where convξ = ˆΞ (exact representation) or convξ ˆΞ (outer approximation). Such representations can be determined efficiently for polyhedral supports, see [29]. The nonlinear decision rule problem P u can then be transformed into a tractable linear program in the same way as the linear decision rule problem P u was converted to P u, see Section 3. minimise subject to T T s=1 Tr(P t M R P t C t X t ) A ts X sp sm tp t B t P t R= Λ t W,Λ t h 0,Λ t 0 X t P t = Γ t W,Γ t h 0,Γ t 0 t = 1,...,T ( P u ) Here, the matrix R R k k is defined through Rξ = R(ξ ) for all ξ Ξ, M = E ξ (ξ ξ ) R k k denotes the second order moment matrix associated with ξ, and the conditional expectations are assumed to satisfy E ξ (ξ ξ t ) = M tξ t for some matrices M t R k k t and all ξ Ξ. The decision variables in P u are the entries of the matrices X t R nt k t, Λ t R mt l and Γ t R nt l for t = 1,...,T. Similarly, we can measure the suboptimality of nonlinear decision rules by solving the following dual approximate problem (see Section 3.2). maximise subject to T T s=1 Tr(P t M RP t B t Y t ) A st Y s P s M t P t C tp t R= Φ t W,Φ t h 0,Φ t 0 Y t P t = Ψ t W,Ψ t h 0,Ψ t 0 t = 1,...,T ( D l ) ThedecisionvariablesofthisproblemaretheentriesofthematricesY t Rmt k t, Φ t R nt l and Ψ t R mt l for t = 1,...,T. One can show that the finite-dimensional primal approximation P u is equivalent to the semi-infinite primal problem P u if ˆΞ coincides with convξ, and P u provides an upper bound on the optimal value of P u if ˆΞ is an outer approximation of convξ. Likewise, the finite-dimensional dual approximation D l is equivalent to the semi-infinite dual problem D l (not shown here) if ˆΞ coincides with convξ, and D l provides a lower bound on the optimal value of D l if ˆΞ is an outer approximation of convξ. In particular, the finite-dimensional approximate problems P u and D l still bracket the optimal value of the problem P in nonlinear decision rules if we employ an outer approximation ˆΞ of the convex hull of Ξ. The situation is illustrated in Figure 5. 21

22 Figure 5: Relationship between the primal bounds P i u and the dual bounds D i l for an exact representation of convξ (i = 1) and an outer approximation of convξ (i = 2). 5 Incorporating Integer Decisions Optimisation problems often involve decisions that are modelled through integer variables. These problems are still amenable to the decision rule techniques described in Sections 3 and 4 if the integer variables do not depend on the uncertain problem parameters ξ. In order to substantiate this claim, we consider a variant of problem P in which the right-hand side vectors of the constraints may depend on a vector of integer variables z Z d. ( T ) minimise E ξ c t (ξ t ) x t (ξ t ) subject to z Z E ξ ( T s=1 A ts x s (ξ s ) ξ t ) x t (ξ t ) 0 b t (z,ξ t ) ξ Ξ, t = 1,...,T (P MI ) As usual, we assume that b t (z,ξ t ) depends linearly on ξ t, that is, b t (z,ξ t ) = B t (z)ξ t for some matrix B t (z) R mt kt. We further assume that B t (z) depends linearly on z and that Z R d results from the intersection of Z d with a convex compact polytope. In order to apply the decision rule techniques from Sections 3 and 4 to P MI, we study the parametric program P(z), which is obtained from P MI by fixing the integer variables z. By construction, P(z) is a decision problem of the type P, which is bounded above and below by the linear programs P u (z) and D l (z) associated with the primal and dual linear decision rule approximations, respectively. Thus, an upper bound on P MI is obtained by minimising the optimal value of Pu (z) over all z Z. The resulting optimisation problem, which we 22

23 denote by P MI u, represents a mixed-integer linear program (MILP). minimise T Tr(P t MP t C t X t) subject to z Z T A ts X s P s M t P t B t (z)p t = Λ t W,Λ t h 0,Λ t 0 s=1 X t P t = Γ t W,Γ t h 0,Γ t 0 t = 1,...,T ( P u MI ) Similarly, a lower bound on P MI is obtained by minimising the optimal value of D l (z) over all z Z. The resulting min-max problem has a bilinear objective function that is linear in the integer variables z and in the coefficients of the dual decision rules Y t. In order to convert this problem to an MILP, we follow the exposition in [38] and dualise D l (z). One can show that strong duality holds whenever the original problem P MI is feasible. By construction, we thus obtain a lower bound on P MI minimising the dual of D l (z) over all z Z. The resulting optimisation problem, which we denote by P MI l, again represents an MILP. minimise T Tr(P t MP t C t X t) subject to z Z T A ts X s P s M t P t +S t P t = B t (z)p t s=1 ( ) W he 1 MP t Xt 0 ( ) W he 1 MP t St 0 t = 1,...,T ( P l MI ) The optimisation variables in P MI l are the matrices X t R nt kt and S t R mt kt, as well as the binary variables z Z. Problem P MI is also amenable to the refined approximation methods based on piecewise linear decision rules as discussed in Section 4. Further details can be found in [29]. Remark 5.1. We assumed that the integer decisions z in problem P MI do not impact the coefficients c t ( ) and A of the objective function and the constraints, respectively. It is straightforward to apply the techniques presented in this section to a generalised problem where the objective function and constraint coefficients depend linearly on z. Using a Big-M reformulation, the resulting primal and dual approximate problems can again be cast as MILPs. 23

24 6 Case Studies In the following, we apply the decision rule approach to two well known operations management problems, and we compare the method with alternative approaches to account for data uncertainty. All of our numerical results are obtained using the IBM ILOG CPLEX 12 optimisation package on a dual-core 2.4GHz machine with 4GB RAM [1]. 6.1 Production Planning Our first case study concerns a medium-term production planning problem for a multiproduct plant with uncertain customer demands and backlogging. We assume that the plant consists of a single processing unit that is capable of manufacturing different products in a continuous single-stage process. We first elaborate a formulation that disregards changeovers, and we afterwards extend the model to sequence-dependent changeover times and costs. We wish to determine a production plan that maximises the expected profit for a set of products I and a weekly planning horizon T = {1,...,T}. To this end, we denote by p ti the amount of product i I that is produced during week t T \{T}. The processing unit can manufacture r i units of product i per hour, and it has an uptime of R hours per week. At the beginning of each week t T, we observe the demand ξ ti that arises for product i during week t. We assume that the demands ξ 1i in the first week are deterministic, while the other demands ξ ti, t > 1, are stochastic. Having observed the demands ξ ti, we then decide on the quantity s ti of product i that we sell during week t at a unit price P ti. We also determine the orders b ti for product i that we backlog during week t at a unit cost CB ti. We assume that the sales s ti in week t must be served from the stock produced in week t 1 or before. Once the sales decisions s ti for week t have been made, the inventory level I ti for product i during week t is known. Each unit of product i held during period t leads to inventory holding costs CI ti, and we require that the inventory levels I ti satisfy the lower and upper inventory bounds I ti and I ti, respectively. Deterministic versions of this problem have been studied in [15, 41]. For literature surveys on related production planning problems, we refer to [27,62]. The temporal structure of the problem is illustrated in Figure 6. 24

25 week t 1 week t week t+1 p t 1,i p t,i p t+1,i I t 1,i I t,i I t+1,i s t 1,i s t,i s t+1,i b t 1,i b t,i b t+1,i ξ t,i ξ t+1,i supply demand Figure 6: Temporal structure of the production planning model. The decisions with subscript t may depend on all demands realised in weeks 1,...,t. We can formulate the production planning problem as follows. [ ] maximise E P ti s ti (ξ t ) CB ti b ti (ξ t ) CI ti I ti (ξ t ) t T i I subject to p ti (ξ t )/r i R i I p ti (ξ t ) 0 i I b ti (ξ t ) = b t 1,i (ξ t 1 )+ξ ti s ti (ξ t ) I ti (ξ t ) = I t 1,i (ξ t 1 )+p t 1,i (ξ t 1 ) s ti (ξ t ) b 1i (ξ t ) = b 0i +ξ 1i s 1i (ξ 1 ), I 1i (ξ t ) = I 0i s 1i (ξ 1 ) b ti (ξ t ) 0, s ti (ξ t ) 0, I ti I ti (ξ t ) I ti t T \{T} } t T \{1}, i I t T, i I We require the constraints to be satisfied for all realisations ξ Ξ of the uncertain customer demands. The parameters b 0i and I 0i specify the initial backlog and inventory, respectively. One can easily show that the production planning problem is an instance of the problem P studied in Section 2.1. The same applies to variations of the problem where the prices and/or costs are uncertain, as well as variations where the product demand is only known at the end of each week. For the sake of brevity, we disregard these variants here. So far, our production planning problem does not account for changeovers between consecutively manufactured products. Frequent changeovers are undesirable as the involved clean-up, set-up and start-up activities result in both delays and costs. To incorporate changeovers, we follow the approach presented in [41] and introduce binary variables c tij, t T \{T} and i,j I, that indicate whether a changeover from product i to product j occurs in week t. Likewise, we introduce binary variables c tij, t T \{T} and i,j I, that 25