We discuss the problem of computing optimal linearisation parameters for the first order loss function of a family of arbitrarily distributed random variable. We demonstrate that, in contrast to the problem in which parameters must be determined for the loss function of a single random variable, this problem is nonlinear and features several local optima and plateaus. We introduce a simple and yet effective heuristic for determining these parameters and we demonstrate its effectiveness via a numerical analysis carried out on a well known stochastic lot sizing problem

Hendrix, Eligius M.T.

Rossi, Roberto

English

Edinburgh Research Explorer

     Edinburgh Research Explorer                                      Piecewise linearisation of the first order loss function for familiesof arbitrarily distributed random variablesCitation for published version:Rossi, R & Hendrix, EMT 2014, 'Piecewise linearisation of the first order loss function for families ofarbitrarily distributed random variables' Paper presented at MAGO 2014 - XII Global Optimisation Workshop(GOW), Malaga, Spain, 1/09/14 - 4/09/14, .Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Preprint (usually an early version)Publisher Rights Statement:© Rossi, R., & Hendrix, E. M. T. (2014). Piecewise linearisation of the first order loss function for families ofarbitrarily distributed random variables. Paper presented at MAGO 2014 - XII Global Optimisation Workshop(GOW), Malaga, Spain.General rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 20. Feb. 2015Proceedings of MAGO 2014, pp. 1 – 4.Piecewise linearisation of the first order loss functionfor families of arbitrarily distributed random variablesRoberto Rossi1 and Eligius M. T. Hendrix21 Business School, University of Edinburgh, Edinburgh, United Kingdom, roberto.rossi@ed.ac.uk2 Department of Computer Architecture, Málaga University, Málaga, Spain eligius.hendrix@wur.nlAbstract We discuss the problem of computing optimal linearisation parameters for the ﬁrst order loss func-tion of a family of arbitrarily distributed random variable. We demonstrate that, in contrast to theproblem in which parameters must be determined for the loss function of a single random variable,this problem is nonlinear and features several local optima and plateaus. We introduce a simpleand yet eﬀective heuristic for determining these parameters and we demonstrate its eﬀectivenessvia a numerical analysis carried out on a well known stochastic lot sizing problem.Keywords: First order loss function, piecewise linear bounds, Jensen, Edmundson-Madanski, lot sizing1. IntroductionConsider a random variable ω with expected value µ and density function gω, and a scalarvariable x. The first order loss function is defined as Lω(x) = Eω[max(ω − x, 0)], where Edenotes the expected value. The complementary first order loss function is defined as L¯ω(x) =Eω[max(x− ω, 0)]. Note that Lω(x) = µ − x+ L¯ω(x). The first order loss function Lω(x) andits complementary function L¯ω(x) are extensively used in several application domains, such asinventory control [8] and finance (see e.g. [5]).In general, Lω(x) does not admit a closed form and cannot be evaluated without resorting tonumerical approximations (see e.g. [1]). Lω(x) and its numerical approximations are nonlinearin x and cannot directly be embedded in mixed integer linear programming (MILP) models.In [7] the authors discuss piecewise linear upper and lower bounds for the first order lossfunction, which can be immediately embedded in MILP models. These bounds are particularlyconvenient for a number of reasons: they rely on constant parameters that are independentof the mean and standard deviation of the normal distribution of interest; it is easy to obtainbounds for generic, i.e. non standard, normally distributed random variables via a simple lineartransformation. Optimal linearisation parameters are derived following an approach similar tothe one discussed in [2, 3], which minimise the maximum approximation error.In this work, we extend the approach in [7] to the case of generic, i.e. non normal, distributionsand we discuss how to embed into a MILP model piecewise linear upper and lower bounds ofthe first order loss function for a predefined family of generic random variables ω1,ω2, . . . ,ωN .These bounds are computed in such a way as to minimise the maximum approximation errorover the given family of random variables. We demonstrate the effectiveness of this technique toaddress a well known stochastic lot sizing problem. Because of the relation that exists betweenthe function Lω(x) and its complement L¯ω(x), the following discussion will be limited to thecomplementary first order loss function L¯ω(x).2 Roberto Rossi and Eligius M. T. Hendrix2. Piecewise linear upper and lower boundsThe complementary first order loss function function L¯ω(x) is convex in x regardless of thedistribution of ω. For this reason, both Jensen’s lower bound and Edmundson-Madanski upperbound are applicable [4]. Consider a partition of the support Ω of ω into W disjoint compactsubregions Ω1, . . . ,ΩW . We define, for all i = 1, . . . ,Wpi = Pr{ω ∈ Ωi} =∫Ωigω(t) dt and E[ω|Ωi] =1pi∫Ωitgω(t) dt. (1)Lemma 1. Let Ω1, . . . ,ΩW be a partition of the support Ω of random variable ω, pi and E[ω|Ωi]defined by (1) and lower bounding function Λiω(x) = x∑ik=1pk−∑ik=1pkE[ω|Ωk]. Then lowerbounding functionΛω(x) = max(maxiΛiω(x), 0)≤ L¯ω(x)is a piecewise linear function with W + 1 segments.This lower bound is a direct application of Jensen’s inequality. Let us then consider themaximum approximation error eW = maxx(L¯ω(x) − Λω(x)) for the lower bound in Lemma1 associated with a given partition. A piecewise linear upper bound, i.e. the Edmundson-Madanski bound, is Λω(x)+ eW , which is obtained by shifting up the lower bound in Lemma 1by a value eW . These lower and upper bounds for any random variable ω can directly be usedin an MILP model.Having established this results, the question is how to partition the support Ω in order toobtain good bounds. A number of works discussed how to obtain an optimal partitioning ofthe support under a framework that minimises the maximum approximation error [2, 3]. Inshort, these works demonstrate that, in order to minimise the maximum approximation error,one must find parameters ensuring approximation errors at piecewise function breakpoints areall equal. This result unfortunately does not hold when optimal linearisation parameters mustbe found for complementary first order loss functions of a family of generic random variables.20 40 60 80 100x0.0050.0100.0150.0200.0250.030PDFHΩLFigure 1a. Probability density function (PDF) ofω1.20 40 60 80 100 120 140x0.0050.0100.0150.0200.025PDFHΩLFigure 1b. Probability density function (PDF) ofω2.Consider the complementary first order loss functions for a family of generic random variablesω1, . . . ,ωn, . . . ,ωN . From (1) it is clear that E[ω|Ωi] is uniquely determined by the choice ofpi. From this choice of the coefficients pi follows for each ωn the function Λω(x). Within anMILP model, one can select the desired bounding function via a binary selector variable yn:Λ(x) =N∑n=1max(maxi(xi∑k=1pk − yni∑k=1pkE[ωn|Ωk]), 0)adding∑Nn=1yn = 1. These expressions generalise those discussed in [7], which only hold fornormally distributed random variables.Piecewise linearisation of the first order loss function for families of arbitrarily distributed random variables 3The challenge is, of course, to compute an optimal partition of random variable supportsinto W disjoint compact subregions with probability masses p1, . . . , pi, . . . , pW . We shall firstdemonstrate that this is a nonconvex optimisation problem in contrast to computing optimallinearisation parameters for a single loss function. To do so, we consider a simple instanceinvolving two random variables ω1 and ω2 with probability density function as shown in Fig.1a and 1b.We split the support of ω1 and ω2 into five regions with probability mass p1, . . . , p5, respec-tively. In Fig. 2 we plot the maximum approximation erroreW = max(maxx(L¯ω1(x)− Λω1(x)),maxx(L¯ω2(x)− Λω2(x))),when p1 and p4 are free to vary, p2 = 0.3, p3 = 0.1 and p5 = 1−p1−p2−p3−p4; note that thisis a standard 2-simplex in R3 projected in R2 and coloured to reflect the value of eW . It is clearthat this function has a number of local minima. In fact, the function is also constant in some0.1 0.2 0.3 0.4 0.50.50.60.70.80.91.02345xyFigure 2. Maximum approximation error of ourpiecewise linear lower bound when p1 + p4 + p5 ≤1−p2−p3; the x axis represents p1, i.e. the slope ofsegment 1, the y axis represents p1 + p2 + p3 + p4,i.e. the slope of segment 4; darker regions meanlower error.SegmentsOptimality gap %2 3 4 5 6 7 8 9 10 110.010.10110100Figure 3. Optimality gap trend as a function ofthe number of segments used in the linearisation.The optimality gap is presented as a percentageof the optimal solution obtained with the modelembedding a piecewise linear upper bound.regions, i.e. it has wide plateaus. It is intuitive to observe this, if one considers the fact thatthe slope of the i-th linear segment is given by∑ki=1pi and that by varying the slope of one ormore segments the maximum approximation error — attained at one or more breakpoints —may easily remain the same. Finding a global optimum of this function is a challenge.Therefore we developed a metaheuristic to find good, but not necessarily optimal parametervalues. The heuristic is a combination of simple random sampling (SRS) and coordinate descent(CD) from the best solution produced by the SRS. For the above example, this strategy producedthe following partitioning: p1 = 0.24, p2 = 0.18, p3 = 0.215, p4 = 0.175, p5 = 0.19, withassociated maximum approximation error of 0.639. This approximation error amounts to 1.5%of the expected value of ω1 and to 1.3% of the expected value of ω2. As we will demonstrate,this strategy produced fairly good outcomes in practical applications.4 Roberto Rossi and Eligius M. T. Hendrix3. An application to stochastic lot-sizingWe applied the metaheuristic to compute near optimal linearisation parameters for the stochas-tic lot sizing problem discussed in [6]. We tested the approach over a test bed discussed in[6] comprising 270 instances. In our experiments, in contrast to the original test bed in whichdemand is normally distributed, demand follows different distributions in different periods: nor-mal, Poisson, exponential and uniform. All instances took just a few seconds to be solved. Notethat each of these instances comprises N = 15 periods in which demand is observed. Con-sequently, there are N(N + 1)/2 loss functions in the family for which optimal linearisationparameters must be computed; these parameters must be computed for each instance sepa-rately, this is why a lightweight heuristic is desirable. The average optimality gap trend —the difference between the optimal solution of the model embedding our piecewise linear upperbound and that of the model embedding our piecewise linear lower bound — as a function ofthe number of segments used in the linearisation is shown in Fig. 3. As we see, the gap dropsinitially with the number of segments and it is well below 1% of the optimal cost even whenjust four segments are employed. For higher number of segments the gap fluctuates. This isdue to the fact that the simple heuristic here proposed gets stuck in local minima for highdimensional spaces. Future research should therefore investigate more effective approaches tocomput optimal parameters for linearisations involving a large number of segments.4. ConclusionsWe have shown that finding optimal linearisation parameters to approximate loss functionswhen several non-normal random variables are considered, is a challenging global optimisationproblem. We discuss how to handle this in a practical setting to generate reasonable resultsthat can be used in MILP models for inventory control.References[1] Steven K. De Schrijver, El-Houssaine Aghezzaf, and Hendrik Vanmaele. Double precision rational approxi-mation algorithms for the standard normal ﬁrst and second order loss functions. Applied Mathematics andComputation, 219(4):2320–2330, November 2012.[2] Momčilo M. Gavrilović. Optimal approximation of convex curves by functions which are piecewise linear.Journal of Mathematical Analysis and Applications, 52(2):260–282, November 1975.[3] Alyson Imamoto and Ben Tang. Optimal piecewise linear approximation of convex functions. In S. I.Ao, Craig Douglas, W. S. 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Piecewise linearisation of the first order loss function for families of arbitrarily distributed random variables

Rossi, R.

Hendrix, E.M.T.

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Wageningen Yield 

Piecewise linearisation of the first order loss function for families of arbitrarily distributed random variables

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