Abstract
We address a variant of the single item lot sizing problem affected by proportional storage (or inventory) losses and uncertainty in the product demand. The problem has applications in, among others, the energy sector, where storage losses (or storage deteriorations) are often unavoidable and, due to the need for planning ahead, the demands can be largely uncertain. We first propose a twostage robust optimization approach with secondstage storage variables, showing how the arising robust problem can be solved as an instance of the deterministic one. We then consider a twostage approach where not only the storage but also the production variables are determined in the second stage. After showing that, in the general case, solutions to this problem can suffer from acausality (or anticipativity), we introduce a flexible affine rule approach which, albeit restricting the solution set, allows for causal production plans. A hybrid robuststochastic approach where the objective function is optimized in expectation, as opposed to in the worstcase, while retaining robust optimization guarantees of feasibility in the worstcase, is also discussed. We conclude with an application to heat production, in the context of which we compare the different approaches via computational experiments on realworld data.
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1 Introduction
Lot Sizing (LS) is a fundamental problem in a large part of modern production planning systems. In its basic version, given a demand for a single good over a finite time horizon, the problem calls for a production plan which minimizes storage, production, and setup costs, while keeping production and storage levels within the prescribed lower and upper bounds.
In the paper, we focus on a variant of LS where the storage suffers from proportional losses and the product demands are subject to uncertainty.^{Footnote 1} This variant suits the case of many applications in the energy sector where a small portion of the energy that is stored is lost over time and the demands (of, e.g., heat, as in the application that we will consider) are often not known in advance. This is especially relevant when the decision maker has to commit to a production plan ahead of when it becomes operational.
To cope with uncertainty, we develop a robust optimization approach. After observing that a fully firststage approach is not viable, we adopt an approach with firststage setup and production variables and secondstage storage variables. For it, we establish a reduction showing that this problem can be solved as an instance of its deterministic counterpart with suitably (re)defined demands and storage upper bounds. Then, we consider the case where the secondstage decisions comprise not just the storage but also the production variables, while still retaining firststage setup variables. After showing that solutions to this problem may be acausal (or anticipative), i.e., that they may contain production variables whose value, at time t, can only be determined by knowing the realization of the demand at time \(t^\prime > t\), we introduce an affine rule approach. This technique, although restricting the solution set, allows for causal production plans. We then illustrate a set of computational experiments in the context of an application to heat production.
The paper, which is an extended version of Coniglio et al. (2016), is organized as follows. In Sect. 2, we summarize some key results on the deterministic version of LS and present some relevant previous work on robust optimization. The version of LS with storage losses that we tackle in this paper is formally introduced in Sect. 3, together with its uncertainty aspect. In Sect. 4, we introduce the robust counterpart of the problem with secondstage inventory variables and present our reduction. The approach with secondstage production and storage variables is proposed in Sect. 5, while, in Sect. 6, we present a hybrid stochasticrobust extension. Computational results are reported and illustrated in Sect. 7. Concluding remarks are drawn in Sect. 8.
2 Previous work
We give a brief account of the most relevant works on different versions of LS, some also encompassing the uncertain case.
2.1 Deterministic LS
In the deterministic setting, LS has been extensively studied in many variants. Among the different versions, we mention that with backlogging, in which shortages in the inventory are allowed (or, said differently, unmet demand can be postponed to the future, at a cost), the uncapacitated one, in which production has no predefined bounds, and that with production capacities, either static (constant over time) or timedependent, where lower and upper bounds on the production are present. The storage is usually assumed to have either a zero or a static lower bound—in the latter case, the problem can be often brought to the case of a zero lower bound with a reformulation. Storage upper bounds are often assumed to be static, although many algorithms can be easily adapted to the timedependent case. For an extensive account on LS and many of its variants, we refer the reader to the monograph (Pochet and Wolsey 2006).
LS is known to be in \({\mathcal {P}}\) for the case with linear costs, complete conservation (i.e., no losses in the storage), zero or a static storage lower bounds, possibly nonzero and timedependent storage upper bounds, and no production bounds, as shown in Atamtürk and Küçükyavuz (2008). A similar result holds for nonnegative and nondecreasing concave cost functions, complete conservation, production bounds that are constant over time, and unrestricted storage, as shown in Hellion et al. (2012). For a polynomial time algorithm for the case with storage losses and nondecreasing concave costs, but no storage or production bounds, see Hsu (2000). As to \({\mathcal {NP}}\)hard cases, a number of examples are provided in Florian et al. (1980). These include the case of linear as well as fixed production costs, no inventory costs, no storage bounds, and no lower production bounds, but timedependent production upper bounds.
2.2 LS with uncertain demands
Classical approaches to handle uncertainties in LS are, historically, stochastic in nature, dating back as early as 1960 (Scarft 1960). The idea is to first assign a probability distribution to the uncertain demands and, then, to tackle the problem by looking for a solution of minimum expected cost, which is feasible with high probability.
Unfortunately, as pointed out in Liyanage and Shanthikumar (2005), even when the distribution is estimated within sufficient precision from historical data, such methods can yield solutions which, when implemented with the demand that realizes in practice, can be substantially more costly than those that were predicted with the stochastic approach (or infeasible, depending on how the problem is formulated). Moreover, and regardless of the accuracy of the estimation, such stochastic techniques are, in many cases, intrinsically doomed to suffer from the curse of dimensionality (Bertsimas and Thiele 2006). This is because they usually require a computing time which is, at least, linear in the size of the probability space (assumed to be discrete) to which the realized demand belongs—such space is, typically, exponential in the length of the time horizon. For an alternative way of modeling stochastic demands, via socalled servicelevel constraints, see Bookbinder and Tan (1988), Tempelmeier and Herpers (2011), and Tempelmeier (2013). Also see Calafiore (2008) for a stochastic programming approach where feasibility is enforced in expectation.
A different option, often more affordable from a computational standpoint, is resorting to a robust optimization approach. The idea is of looking for a solution which is feasible for every realization of the uncertain demand belonging to a given uncertainty set and which, among all such feasible solutions, is of minimum cost in the worst case. Twostage robust optimization is an extension of this approach in which the variables are partitioned in two sets. The firststage (or hereandnow) variables are required to take a value independent of the realization of the uncertain parameters, while the secondstage (or waitandsee) variables are allowed to vary as a function of the specific realization that takes or has taken place, depending on whether the problem has a dynamical aspect or not.
Two seminal papers applying a robust optimization approach to LS are Bertsimas and Thiele (2004) and Bertsimas and Thiele (2006). The authors address the uncapacitated version of LS with backlogging, static storage, and production costs, with demands subject to a socalled \(\Gamma \)robustness model of uncertainty (Bertsimas and Sim 2003, 2004). When applied to LS, the idea of \(\Gamma \)robustness is of assuming that the uncertain parameters, i.e., the product demand for LS, (i) belong to symmetric intervals and (ii) given an integer \(\Gamma \), the total number of time steps in which the uncertain demand deviates from its nominal value to one of the extremes of the interval is bounded by \(\Gamma \), in any constraint of the problem.
Among other results, the authors of Bertsimas and Thiele (2004, 2006) show that the \(\Gamma \)robust counterpart of the variant of LS they consider (with backlogging) can be solved as an instance of the deterministic problem with modified demands, also for the case where production bounds are in place. Notice that, for this result to hold, bounds on the storage cannot be enforced. Unfortunately, such bounds are necessary in many applications, such as, e.g., those in the energy sector, where backlogging is not tolerable as the demand of energy (e.g., heat) must be typically satisfied when issued.
3 The problem: LS with storage losses and uncertain product demands
Consider a single product and a time horizon \(T = \{1, \ldots , n\}\). For each time step \(t \in T\), let \(d_t \ge 0\) be the product demand, \(q_t \ge 0\) the production variable, and \(u_t \ge 0\) the storage variable corresponding to the value of the storage at the end of time step t. Let \(z_t \in \{0,1\}\) be an indicator variable equal to 1 if production is active at time \(t \in T\), that is, if \(q_t > 0\), and let the constant \(u_0 \ge 0\) represent the initial storage value. Let also \(c_t^i\), \(c_t^v\), and \(c_t^c\) be, respectively, the inventory, production, and fixed setup cost (the latter is incurred when production is active) at time \(t \in T\). We consider timedependent bounds \(\underline{U}_t \le u_t \le \overline{U}_t\) on the storage, as well as timedependent bounds \(\underline{Q}_t \le q_t \le \overline{Q}_t\) on production, when active.
3.1 The deterministic case
The (DETerministic) variant of LS with storage losses tackled in this paper can be cast as the following MixedInteger Linear Programming (MILP) problem:
For notational consistency with the remainder of the paper, the problem is written in epigraph form, with the objective function moved into Constraint (1b). Constraints (1c) are balance constraints, which only differ from those in the standard LS problem for the (possibly timedependent) conservation factor \(\alpha _t \in (0,1]\). They account for the fact that, at the beginning of each time period \(t \in T\), a fraction \((1\alpha _t)\) of the previously stored product is lost, whereas, at the end of time period t, the remaining product is conserved. Constraints (1d) and (1e) impose lower and upper bounds on storage and production. Constraints (1f)–(1h) specify the nature of the decision variables.
3.2 The uncertain case
Let us assume an uncertain product demand d, belonging to an uncertainty set D —as better illustrated in Sect. 7, we will construct this set from a mixture of historical data and forecasts.
In the combinatorial optimization literature, two types of uncertainty sets are predominantly employed: finite uncertainty sets and polyhedral uncertainty sets. In the former case, D is assumed to be a finite set, albeit possibly extremely large.^{Footnote 2} In the latter, D is assumed to be a bounded polyhedron. The \(\Gamma \)robustness approach used in Bertsimas and Sim (2004) falls into this category. Its uncertainty set, the \(\Gamma \) uncertainty set, corresponds to assuming, for each time period \(t \in T\), that the uncertain demand \(d_t\) belongs to a symmetric interval \([\bar{d}_t  \hat{d}_t, \bar{d}_t + \hat{d}_t]\) centered around a nominal value \(\bar{d}_t\) and that, in each constraint of the problem, the demand could deviate from \(\bar{d}_t\) by \(\pm \hat{d}_t\) for, at most, \(\Gamma \) time steps.
4 Robust LS with firststage production and secondstage storage
As a first approach to the problem, we assume, as natural in a number of applications, that the storage variables u could be determined in a waitandsee fashion as a function of the realization of the uncertain demand d, while all the other variables are required to a take a hereandnow value independently of the realization. This corresponds to adopting a twostage setting with secondstage storage variables, as better explained in the following.^{Footnote 3} In terms of the production and setup variables, this approach can be regarded as an openloop approach (see Calafiore 2008), in which the value of the production variables at time t does not depend on the realization of d up to time t. Although we will consider more flexible approaches in the next section, this one is of interest since, as shown in this section, it allows for solving the corresponding robust problem very efficiently, as an instance of the deterministic problem, obtained after modifying a few of its givens.
The problem with secondstage u, which we report in full for notational convenience, reads:
It only differs from LSDET in that, here, we have a vector u(d) for each realization \(d \in D\).
As for the classical version of LS without storage losses, see, e.g., Bertsimas and Thiele (2006), Gaussian elimination shows that, in any feasible solution, the storage variables u are uniquely defined as a function of d and q:
with:
A compact formulation for this problem is thus obtained by substituting, in the previous formulation, the righthand side of Eq. (3) for u(d).
With the next theorem, we show how LS2RO\(_u\) can be reduced to an instance of LSDET with identical givens except for a different demand \(d^\prime \) and different storage upper bounds \(\overline{U}^\prime \), defined as follows:
The idea is that, with this definition, the newly introduced demand \(d'_t\) specifies a lower bound on the product which has to be available at time \(t \in T\) to ensure that every demand \(d \in D\) can be met. The value \(\Delta _t\) reduces the storage upper bounds of the transformed problem to prevent that, if a large production is realized but a large deficit in demand occurs (an event which would result in an overflow of storage), the storage upper bound \(\overline{U}_t\) is not exceeded. Formally:
Theorem 1
Assuming \(d'_t\) and \(\Delta _t\) are well defined for each \(t \in T\), LS2RO\(_u\) can be reduced, with respect to the firststage variables \((\eta ,z,q)\), to an instance of LSDET with \(d=d'\) and \(\overline{U} = \overline{U}'\), as defined in Eq. (4).
Proof
First note that, due to the assumptions on D, the values for \(d'_t\) and \(\overline{U}'_t\) can be computed iteratively from \(t=1\) to \(t=T\). Let \(u_t(d)\) be defined as in Eq. (3) and let:
For a given (z, q), adopting \(d = d'\) and \(\overline{U} = \overline{U}'\), we now show that \((\eta ,z,q,u(d))\), with u(d) and \(\eta \) defined as above, is feasible for LS2RO\(_u\) if and only if it is feasible for LSDET with demand \(d'\) and storage upper bounds \(\overline{U}'\). Following the derivations reported in “Appendix A”, we deduce:
What remains to be shown is that Constraints (2d) are satisfied by u(d) if and only if Constraints (1d) are satisfied by u. This is done by observing that, for all \(t \in T\), the following holds true:
Since the objective functions moved to Constraints (2b) and (1b) are equal up to a constant additive term, as highlighted in Eq. (6d), we deduce that \((\eta ,z,q)\) is optimal for LS2RO\(_u\) if and only if it is optimal for LSDET. \(\square \)
We remark that the reduction in Theorem 1 allows us to cast LS2RO\(_u\) as an instance of LSDET with static lower bounds on u. This can be quite relevant as, to the best of our knowledge, most of the algorithms for LSDET, such as the one by Atamtürk and Küçükyavuz (2008), require a storage with either zero or static lower bounds.
From a computational complexity perspective, Theorem 1 allows us to establish a direct relationship between LS2RO\(_u\) and LSDET:
Corollary 1
Given an uncertainty set D over which a linear function can be optimized in polynomial time, the reduction in Theorem 1 is a polynomial reduction. As such, LSROB2u is in \({\mathcal {P}}\) (respectively, \({\mathcal {NP}}\)hard) if and only if the corresponding version of LSDET is in \({\mathcal {P}}\) (respectively, \({\mathcal {NP}}\)hard).
As a consequence of Corollary 1, LSROB2u is in \({\mathcal {P}}\) for all the polynomially solvable cases of LSDET that we reported in Sect. 2, provided that their algorithm allows for the introduction of timedependent upper bounds \(\overline{U}\) on u. This is, for instance, the case of the problem studied in Hellion et al. (2012). Note that the result still holds if we introduce additional constraints on z and q or assumptions on the givens (except for d and \(\overline{U}\)). It is also valid for \(\overline{U}_t = \infty \) and for not necessarily nonnegative demands d.
The assumptions on the uncertainty set in Corollary 1 subsume the cases of many robustness models, including finite uncertainty sets, polyhedral uncertainty sets (such as the \(\Gamma \)robustness one), and ellipsoidal uncertainty sets, such as those employed in BenTal et al. (2009). The advantages in terms of computing times of this robust optimization approach are better illustrated in Sect. 7.
5 Robust LS with secondstage production and storage variables
In a number of applications, it is reasonable to assume that not only the storage variables but also the production variables could be adjusted (possibly, in real time) in a secondstage fashion, after the uncertain demand reveals.
Still retaining firststage setup variables, we can cast the problem with explicit secondstage variables u(d) and q(d), for all \(d \in D\), as this (infinite, for a continuous D) MILP:
LS2RO\(_{q,u}\) suffers from an important drawback: acausality (or anticipativity). Indeed:
Proposition 1
In the general case, optimal solutions to LS2RO\(_{q,u}\) can be acausal, with \(q_t(d)\) depending, for some \(t \in T\), from \(d_{t'}\) for \(t' > t\).
Proof
Let us consider a finite uncertainty set case with \(D = \{d^1, d^2\}\), where \(d^1 = (1,3,1)\) and \(d^2=(1,1,3)\), with \(u_0=0\), \(c^i=(1,1,1)\), \(\underline{U}=\underline{Q}=(0,0,0)\), \(\overline{U}=(2,2,2)\), \(\overline{Q}=(2,2,2)\), \(c^v=(1, 1, 1)\), and \(c^c=(0,0,0)\). Since the setup costs are identically zero, we can assume w.l.o.g. \(z_t=1\) for all \(t \in T\). It follows that LS2RO\(_{q,u}\) can be solved by solving two independent instances of LSDET, one for \(d=d^1\) and one for \(d=d^2\):

for \(d^1=(1,3,1)\), the unique optimal production vector is \(q^1=(2,2,1)\);

for \(d^2=(1,1,3)\), the unique optimal production vector is \(q^2=(1,2,2)\).
Note that \(q_1^1 \ne q_1^2\), even though \(d^1_1 = d^2_1 = 1\). To be able to decide whether, at time \(t=1\), \(q_1 := q_1^1 = 2\) or \(q_1 := q_1^2 = 1\), we need to know \(d_2\) (so to be able to determine whether \(d= d^1\) or \(d=d^2\)). The (unique) optimal solution to the problem is, thus, acausal. \(\square \)
5.1 Affine decision rules
To prevent acausality, we resort to affine rules, a technique which has been successfully adopted in a number of optimization and control problems, see, e.g., BenTal et al. (2004), Calafiore (2008), and Bertsimas and Georghiou (2015).
While still allowing q(d) to be a function of d, the idea is to restrict the feasible region of the problem by requiring \(q_t(d)\) to be, for each \(t \in T\), an affine function of the uncertain demand vector d. We impose:
which stipulates that \(q_t(d)\) be composed of two parts: a firststage part \(\Xi _t\), which, as in LS2RO\(_u\), does not depend on the realization of d, and a secondstage part \(\sum _{j \in T: j \le t} \xi _{jt} d_j\), which does, but only for components of d up to time t.
Notice that, with this approach, the coefficients \(\xi _{it}\) and \(\Xi _t\) must be determined as firststage variables. As it is clear, this approach generalizes the case of LS2RO\(_u\) with a firststage q, as the latter is obtained when requiring \(\xi _{jt} = 0\) for all \(j,t \in T\).
Overall, we obtain the following (infinite, for a nonfinite D) MILP with Affine Rules (AR):
To arrive at a finite MILP, we can first derive a semiinfinite formulation by substituting the righthand sides of Eq. (3) and Constraints (9f) for variables u(d) and q(d). For finite scenario uncertainty sets, the problem thus obtained can be rewritten as a finite MILP by duplicating the constraints involving the demand vector d exactly D times. For the case of \(\Gamma \)robustness, we can obtain a reformulation of polynomial size via the dualization procedure introduced by Bertsimas and Sim (2004). We report the formulation thus obtained in “Appendix B”. Note that, since the feasible region of the problem belongs to a higherdimensional space than that of LS2RO\(_{u}\), we cannot hope for a result similar to that of Theorem 1.
5.2 A regularization approach
As mentioned in Caramanis (2006), affine rules add an affine regression subproblem to the robust counterpart of a deterministic problem.
Let \(q_t^*(d)\) be the (assumed unique, for simplicity) optimal production value at time t for LS2RO\(_{q,u}\) (in which the secondstage q is not restricted by affine rules and, thus, it can be anticipative), expressed as a function of the realization \(d \in D\). If LS2RO\(_{q,u}\)AR admits a solution with coefficients \(\xi _{jt}\) and \(\Xi _t\) satisfying:
\(q_t(d)\) can then take the same value that it takes in LS2RO\(_{q,u}\) and, thus, the affine rules pose no restriction to the problem. If Eq. (10) cannot be satisfied for all \(t \in T\) and \(d \in D\), LS2RO\(_{q,u}\)AR calls for coefficients which minimize the objective function loss that is incurred when adopting \(q_t(d) = \sum _{j \in T} \xi _{jt} d_j + \Xi _t\) instead of \(q_t^*(d)\).
As typical in many applications, D is usually an estimate of the “hidden” uncertainty set, let us call it \(\tilde{D}\), to which the realizations of the uncertain demand belong. In that case, one should look for coefficients which, rather than just minimizing the worstcase objective function value on demands \(d \in D\), are also likely to achieve a good objective function value on realizations \(d \in \tilde{D} \setminus D\), thus enjoying good generalization properties. In a number of machine learning works, see Bishop (2006), it is observed that, both in theory and in practice, a better generalization is obtained by introducing a regularization term which limits the magnitude of the coefficients of the function that is being learned—those in Eq. (8) in our case.
For these reasons, we also consider, in the computational experiments, an alternative approach encompassing a restriction on the norm of the vector of coefficients \(\xi _t = (\xi _{1t}, \dots , \xi _{jt}, \dots , \xi _{tt})\). More precisely, we consider, together with the original approach with unrestricted \(\xi _t\), a version where the coefficients \(\xi _{jt}\) are restricted to the interval \([1,1]\). This corresponds to introducing, for all \(t \in T\), the \(\infty \)norm constraint \(\xi _t_{\infty } \le 1\) on the vector of coefficients \(\xi _t\) adopted in each affine rule.
6 Hybrid robuststochastic approach
A robust approach is, arguably, bound not to be viable in a free market where the decision maker is facing a number of not riskaverse competitors. While interested in solutions which are feasible even in the worstcase (which is particularly relevant in the energy sector, as in the application considered in the next section), minimizing the objective function in expectation, as opposed to in the worstcase, is likely to offer solutions which, by being less riskaverse, can be more profitable in practice.
As customary in many results in robust optimization, such as, e.g., those in Bertsimas and Sim (2004) on \(\Gamma \)robustness, let us assume that \(d \in D\) is a symmetrically distributed random variable with \(E[d] = \bar{d}\). With this, the following holds:
Proposition 2
Let \(D \subseteq {\mathbb {R}}^n\) be a set centrally symmetric about \(\bar{d}\), \(d \in D\) a random variable with density function \(p: D \rightarrow [0,1]\) symmetric about \(\bar{d}\), and \(g:D \rightarrow {\mathbb {R}}\) an affine function. Then, \(E[g(d)] = \int _{x \in D} p(x)g(x) dx = g(\bar{d})\).
Proof
Let \(D' := \{d  \bar{d}: d \in D\}\). Since \(D'\) is centrally symmetric about 0, \(D' = D'\) holds. Let \(p'(x) := p (x + \bar{d})\) for \(x' \in D'\). Then, \(p'\) is symmetric about 0. Let f be a linear function. We have:
Thus, we conclude: \(\int _{D'} p(x')g(x')dx' = 0\). Since \(g(x)  g(0)\) is a linear function, we deduce:
which concludes the proof. \(\square \)
For LS2RO\(_{q,u}\) (the problem with secondstage u and q and affine rules), the objective function constraint of the hybrid stochasticrobust formulation reads:
In the next section, we will apply this approach for the case of \(\Gamma \)robustness.
For a finite D and assuming a uniform distribution with first moment \(\bar{d}\), the objective can be restated as:
In the next section, we adopt this approach for the case of finite uncertainty sets which, by construction (see further), are not symmetric in our application.
We remark that the hybrid approach is different from applying affine rules in a stochastic optimization setting, as, here, all constraints, except for the objective function one, are enforced in the worst case. We also remark that the hybrid stochasticrobust approach differs from the purely robust one only when affine rules are employed. This is because, if q is firststage, the objective function reads: \(\sum _{t \in T} (c^v_tq_t + c^c_tz_t + c_t^i (\alpha _{0t}u_0 + \sum _{i = 1}^t \alpha _{it} ( q_i  d_i) ))\). After rearranging the terms, it becomes: \(\sum _{t \in T} (c^v_tq_t + c^c_tz_t + c_t^i \alpha _{0t}u_0 + c_t^i\sum _{i = 1}^t \alpha _{it} q_i  c_t^i\sum _{i = 1}^t \alpha _{it} d_i)\). Since this function only depends on d in the additive term \(c_t^i\sum _{i = 1}^t \alpha _{it} d_i\) which involves no variables, the additive term can be dropped without affecting the optimality of the solutions that are found. The same holds in expectation over d, where, by linearity, we have: \(\sum _{t \in T} (c^v_tq_t + c^c_tz_t + c_t^i \alpha _{0t}u_0 + c_t^i\sum _{i = 1}^t \alpha _{it} q_i  c_t^i\sum _{i = 1}^t \alpha _{it} E[d_i])\), with d, here a random vector, only appearing in the additive term \(c_t^i\sum _{i = 1}^t \alpha _{it} E[d_i]\). Since the latter does not involve any variables, it can be dropped.
7 Computational results
We report on a set of computational results obtained with the models and methods that we have proposed and discussed. As an application, we consider a problem arising within the project Robuste Optimierung der Stromversorgungsysteme (Robust Optimization of Power Supply Systems), funded by the German Bundesministerium für Wirtschaft und Energie (Federal Ministry for Economic Affairs and Energy, BMWi). Originally, this is a Coproduction of Heat and Power (CHP) problem in which fuel is transformed into heat and power (at a fixed heattopower ratio) by a CHP device. When assuming that there is no internal power demand and that all the power which is produced can be sold to the market, the problem reduces to an instance of LS with heat as the unique good. Since heat can be stored (as hot water) for free, we set \(c_t^i = 0\) for all \(t \in T\). The cost of fuel is transferred to the static costs \(c^v\), together with the profit generated by the amount of power that is sold. More precisely, given a static fuel cost \(c^f\) per unit of heat, a heattofuel ratio \(\nu \), a heattopower ratio \(\rho \), and a static market profit \(c^m\) per unit of power, \(c^v\) is set to \(c^v := c^f \nu  c^m \rho \).
7.1 Instances and setup
We consider a dataset spanning a period of 2 years (with some missing months) with hourly time steps. The first half of the data set (338 days) is used as training set, the second half (232 days) as testing set. The heat demand is taken from historical data of a portion of Frankfurt (of around 50000 households). For all \(t \in T\), the storage bounds are set to \(\underline{U}_t = 0\) MWh and \(\overline{U}_t = 120\) MWh, while we set the production bounds to \(\underline{Q}_t = 37.5\) MWh and \(\overline{Q}_t = 125\) MWh. We also set \(u_0 = 36\) MWh. The market prices for the power market are taken from EPEX SPOT (the European Power Exchange).
We consider two cases: finite uncertainty sets and \(\Gamma \)robustness, with \(\Gamma \) uncertainty sets. Both sets are built based on the training set as well as on a heat demand forecast provided by our industrial partner ProCom GmbH. This forecast is generated in a twostage fashion, with an autoregressive component and a neural network one, relying on temperature and calendar events as main influence factors.
The finite uncertainty sets D are constructed as follows. From the original forecast \(\bar{d}\) for the current day, we single out the (up to 70) days from the set of historical time series (the training set) in which demand and forecast are closest in 1norm. After computing the forecast error between the two, we create a realization where such error is added to the forecast of the current day (for which the problem is being solved).
For the \(\Gamma \)robustness approach, the hours of the training set days are partitioned into three groups: morning, midday, and evening. Similarly, the demand forecasts are partitioned in the groups low, medium, and high. This way, we can associate each pair \((t,d_t)\) with a category in \(\{\text {morning}, \text {midday}, \text {evening}\} \times \{\text {low}, \text {medium}, \text {high}\}\). We consider all the hours in a given category over all the training days and look at their deviation (i.e., the forecast error) between the historical demand and the forecast \(\bar{d}_t\). We then take the \(95\%\)quantiles w.r.t. positive and negative deviations and let the deviation \(\hat{d}_t\) at time t be equal to their maximum.
In our experiments, we do not introduce constraints on the endofhorizon value of \(u_{T}\). If required by the application though, it is possible to introduce a lower and/or upper bound on \(u_{T}\), solve the problem thus obtained and, in case of infeasibility, adjust those values, iterating until two values yielding a feasible solution are found.^{Footnote 4} See Calafiore (2008) for more details on a similar procedure.
The experiments are run on an Intel i73770 3.40 GHz machine with 32 GB RAM using CPLEX 12.6 and AMPL as modeling language. All the instances are solved to optimality within default precision.
7.2 Realized robustness
While we optimize over uncertainty sets built with a combination of training set and forecast for the current day, we evaluate the quality of the results on the testing set (i.e., out of sample) in terms of realized robustness, simulating a realworld situation in which a solution found via robust optimization is implemented with a realized demand \(\tilde{d}\) not necessarily belonging to the uncertainty set D. For a given \(\tilde{d}\), we first transfer any violation occurring in the lower and upper bound constraints imposed on u and q to the balance constraints. This is achieved by redefining \(q(\tilde{d})\) and \(u(\tilde{d})\), for all \(t \in T\), as follows:
where \(u_0' := u_0\). This way, the lower and upper bound constraints on storage and production are always satisfied. We then define the violation of the balance constraints for the newly introduced \(q'(\tilde{d})\) and \(u'(\tilde{d})\) as:
For the case of a firststage q, we simply let:
for all \(t \in T\), while adopting the same definition as in Eq. (11b) for \(u'(\tilde{d})\).
We remark that, if \(v'(\tilde{d}) \le 0\) and \(\tilde{d} + v'(\tilde{d}) \ge 0\), the solution with \(q'(\tilde{d})\) and \(u'(\tilde{d})\) corresponds to what a practitioner would arguably do when an infeasibility is detected: production and storage are adjusted so to stay within their prescribed bounds, while the infeasibility is transferred, if possible, to the customer, by satisfying only the reduced demand \(\tilde{d} + v'(\tilde{d})\) (remember that \(v'(\tilde{d}) \le 0\) is assumed to hold here), as opposed to the original \(\tilde{d}\).
For each instance of the problem (i.e., for each realization \(\tilde{d}\)), we evaluate the quality of a robust solution in terms of how it performs when implemented with the given realization \(\tilde{d}\) in terms of (total) realized violation:
and realized cost:
Note that the realized cost can, in general, be quite different from the objective function value achieved by an optimal robust solution. Indeed, the latter accounts for either the worstcase or the expected cost over D (depending on whether a worstcase or a stochastic objective function is employed), as opposed to the former, which amounts to the cost that realizes for a given \(\tilde{d}\).
The adoption of affine rules, although advantageous in terms of the extra flexibility it offers, introduces an element of socalled nervousness to the solutions as, for different realizations of \(\tilde{d}\), the realized production values \(q'_t(\tilde{d})\) are likely to be quite different. As mentioned in Tempelmeier and Herpers (2011), a large nervousness can have a negative impact on other managerial decisions which are taken after the lot sizing problem has been solved. To quantify this phenomenon, we also report a cumulated measure of nervousness, defined as:
where \(\bar{d}\) is the nominal value of d (the forecast).
7.3 Computational study
When affine rules are in place, all the experiments with either \(D = 0\) or \(\Gamma = 0\) (that is, those relying only on the forecast) are obtained by letting \(\xi _{jt} = 0\) for all \(j,t \in T\).
When presenting our results, we report the sum over all the instances in our data set of the realized violation, realized nervousness, and realized cost, for each value of D or \(\Gamma \). The charts that we report show the violationversuscost and nervousnessversuscost curves, thus allowing for, visually, assessing the tradeoff between these quantities that is achieved with the different approaches. The curves are obtained by joining each pair of points obtained with two consecutive values of D or \(\Gamma \). Computing times are reported in Tables 1, 2, and 3.
7.3.1 Firststage production and secondstage storage variables (LS2RO\(_u\))
The results for LS2RO\(_u\) (the approach with firststage production variables q and secondstage storage variables u) are obtained by solving the problem as an instance of LSDET, relying on Theorem 1. We let \(D = 0, 10, 20, 30, 50, 60, 70\) for the finite uncertainty set case and \(\Gamma = 0,1,2,3,4,5,6\) for \(\Gamma \)robustness. We do not go beyond \(\Gamma = 6\) as, for larger values, the problem becomes infeasible for at least a few instances. The results are reported in Table 1, with an illustration in terms of realized violations and costs provided in Fig. 1. Nervousness is not reported as, due to q being firststage, it is equal to zero for all instances.
Overall, we observe that the adoption of \(\Gamma \)robustness provides slightly better results, with a curve, as seen in Fig. 1, which dominates that for finite scenarios, providing smaller realized violations and costs, especially for small values of D and \(\Gamma \). Interestingly, as these two parameters increase, the two approaches become almost equivalent.
Although, as we will see in the remainder of the section, this approach is limited in terms of quality of the solutions, it comes at the advantage of being very fast from a computational standpoint and, by definition, of not suffering from nervousness.
7.3.2 Secondstage storage and production variables with affine rules (LS2RO\(_{q,u}\)AR)
We consider LS2RO\(_{q,u}\)AR (the two stage approach with secondstage u and q). We illustrate the results with finite uncertainty sets and \(\Gamma \)robustness, considering both the cases of a worstcase objective function and a stochastic one, with either unrestricted coefficients \(\xi _{tj}\) or coefficients restricted to \(\xi _{tj} \in [1,1]\) for all \(j,t \in T\).
The results for the case of finite uncertainty sets are reported in Table 2 and illustrated in Fig. 2.
Considering violations and costs, when adopting a worstcase objective, we register highly “unstable” solutions when no restrictions are imposed in \(\xi _{jt}\). The solutions that are obtained when imposing \(\xi _{tj} \in [1,1]\) for all \(j,t \in T\) are much more stable and largely dominate the previous ones. The introduction of a stochastic objective yields very large improvements w.r.t. both realized cost and violation, while also introducing a good deal of stability even for small values of D. The restriction of the coefficients provides an improvement also with a stochastic objective, but not as substantial as with a worstcase one.
Notice that, with \(D = 10,20\), the results are substantially worse than those obtained for larger values of D. This is a consequence of overfitting. Indeed, if \(D \ll T\), there are infinitely many (with \(DT\) degrees of freedom) affine functions satisfying Eq. (10) and, as such, we are likely to obtain solutions with a large fitting error on demand vectors \(\tilde{d} \notin D\), such as all those in the testing set.
We remark that, although \(D = 30 > 24 = T\), and, thus, the fitting subproblem (assuming the vectors \(d \in D\) are in general position) is wellposed, \(D=30\) seems to be still insufficient to suitably capture the structure of the “hidden” uncertainty set \(\tilde{D}\) as, for \(D\ge 40\), much better results are obtained. Overall, as expected, the quality of the results on the testing set increases with the size of the finite uncertainty set D.
As to nervousness, we observe larger values when unrestricted coefficients are used, as opposed to the restricted case, and when the objective function is minimized in the worstcase, as opposed to in expectation. Not surprisingly, higher values of nervousness are observed in presence of overfitting (\(D < T\)), which then start declining for \(D \ge 30\). Overall, the values of nervousness tend to decrease for larger values of D in all the configurations, although not completely monotonically. In all approaches but the worstcase unrestricted one, the nervousness levels seem to converge for larger D. Although it is unclear whether the nervousness values are converging for the worstcase unrestricted approach, they are clearly getting smaller for larger values of D.
Analogous experiments for the case of \(\Gamma \)uncertainty sets and \(\Gamma \)robustness are reported in Table 3 and illustrated in Fig. 3.
Considering violations and costs, the results seem quite stable, even with unrestricted coefficients. This is possibly due to the fact that \(\Gamma \) uncertainty sets are continuous and fulldimensional and, as a consequence, the set of points on top of which the affine rules are constructed always contains (except for \(\Gamma =0\)) sufficiently many data points in general position and, thus, overfitting cannot happen. What is more, we see that, in this case, restricting the coefficients hinders the quality of the solutions. As expected, we also observe that the results with a stochastic objective function dominate those with a worstcase one.
As to nervousness, we observe that, although its value increases substantially with small values of \(\Gamma \), it seems to converge to a constant for a larger \(\Gamma \). This value is comparable to that obtained with the three more “stable” approaches with finite uncertainty sets (worstcase restricted, stochastic unrestricted, and stochastic restricted).
7.3.3 A note on computing times
We remark that, for each setting, the hybrid stochasticrobust approach requires, for bigger uncertainty sets (larger D or larger \(\Gamma \)), much shorter computing times as opposed to the approach with a worstcase objective. For \(\Gamma \)robustness and with the largest values of \(\Gamma \), the difference in computing time can amount to a factor as large as eight.
We also observe that the \(\Gamma \)robustness approach is much faster than the finite uncertainty sets one. This is, most likely, due to the fact that, with a finite D, we have to replicate each constraint containing d exactly D times whereas, with \(\Gamma \)robustness, we can employ the compact reformulation reported in “Appendix B”.
As it is clear when comparing Tables 1, 2, and 3, the method with firststage production is faster than the other ones employing affine rules by almost two orders of magnitude.
7.3.4 Comparison of the best methods
We conclude with a final comparison of the best method, the one with secondstage production and affine rules employing a stochastic objective with unrestricted coefficients in the two cases of finite uncertainty sets and \(\Gamma \)robustness. For the sake of comparisons, we also consider LS2RO\(_{u}\) (that with firststage production) with \(\Gamma \)robustness (which performs better than the one with finite uncertainty sets).
The results are reported in Fig. 4. Here, LS2RO\(_{u}\) with \(\Gamma \)robustness is clearly dominated by the other two. While, with finite uncertainty sets, the affine rules fare rather well in terms of costs, the approach with \(\Gamma \)robustness yields solutions with substantially smaller violations, differently from the method with a finite uncertainty set, whose solutions converge, experimentally, to a total violation of 400, without further improvements for a larger D. Assuming the larger computational effort that is required to construct solutions with this method can be undertaken, it should be preferred due to offering higher quality solutions.
8 Concluding remarks
We have considered a variant of the lot sizing problem with storage losses subject to demand uncertainty. We have developed two twostage robust optimization approaches, both with firststage setup variables, one with secondstage storage variables but firststage production variables, and another one in which production and storage variables are both secondstage. For greater tractability, we have introduced affine rules to tackle the second problem, also encompassing a regularization aspect. We have then proposed a hybrid stochasticrobust approach with a stochastic objective function and robustness in all its constraints, arguably less conservative than the purely robust method with a worstcase objective.
Computational experiments on a heat production problem have shown that the twostage approach with secondstage production and storage variables outperforms that with firststage production variables. The solutions that we have obtained in this setting with \(\Gamma \)robustness are better in terms of cost and violations than those obtained with a finite uncertainty set, and are also much faster to be computed from a computational perspective. We have found that the hybrid stochasticrobust approach yields the overall best results, resulting in smaller costs and smaller violations when compared to the one with a worstcase objective, and also being, with \(\Gamma \)robustness, much faster than the former.
Notes
The variant admits the classical version of LS as a special case.
Some authors adopt the expression discrete, as opposed to finite. See, e.g., Büsing et al. (2011).
Note that a completely firststage approach, calling for a solution \((\eta ,z,q,u)\) which is feasible for all realizations of the uncertain demand \(d \in D\), which is obtainable by solving the formulation for LSDET after imposing Constraints (1c) for all realizations \(d \in D\), yields an infeasible problem as soon as \(D > 1\), even with a single time period.
Empirically, we observe that the solutions that we obtain tend to reduce the value of the storage at time \(t = T\) to a small nonzero quantity, produced in previous periods \(t < T\) with low production costs, which is then used to cope with potential positive deviations in the demand due to uncertainty—if no such deviations realize, the remaining unutilized product is stored in \(u_{T}\).
References
Atamtürk A, Küçükyavuz S (2008) An algorithm for lot sizing with inventory bounds and fixed costs. Oper Res Lett 36(3):297–299
BenTal A, Goryashko A, Guslitzer E, Nemirovski A (2004) Adjustable robust solutions of uncertain linear programs. Math Program 99(2):351–376
BenTal A, El Ghaoui L, Nemirovski A (2009) Robust optimization. Princeton University Press, Princeton
Bertsimas D, Georghiou A (2015) Design of near optimal decision rules in multistage adaptive mixedinteger optimization. Oper Res 63(3):610–627
Bertsimas D, Sim M (2003) Robust discrete optimization and network flows. Math Program 98(1–3):49–71
Bertsimas D, Sim M (2004) The price of robustness. Oper Res 52(1):35–53
Bertsimas B, Thiele A (2004) A robust optimization approach to supply chain management. IPCO 2004:86–100
Bertsimas D, Thiele A (2006) A robust optimization approach to inventory theory. Oper Res 54(1):150–168
Bishop CM (2006) Pattern recognition. Mach Learn 128
Bookbinder JH, Tan JY (1988) Strategies for the probabilistic lotsizing problem with servicelevel constraints. Manag Sci 34(9):1096–1108
Büsing C, Koster AMCA, Kutschka M (2011) Recoverable robust knapsacks: the discrete scenario case. Optim Lett 5(3):379–392
Calafiore GC (2008) Multiperiod portfolio optimization with linear control policies. Automatica 44(10):2463–2473
Caramanis C (2006) Adaptable optimization: theory and algorithms. PhD thesis, Massachusetts Institute of Technology
Coniglio S, Koster AMCA, Spiekermann N (2016) On robust lot sizing problems with storage deterioration, with applications to heat and power cogeneration. Lecture notes in computer science. pp 1–12
Florian M, Lenstra JK, Rinnooy Kan AHG (1980) Deterministic production planning: algorithms and complexity. Manag Sci 26(7):669–679
Hellion B, Mangione F, Penz B (2012) A polynomial time algorithm to solve the singleitem capacitated lot sizing problem with minimum order quantities and concave costs. Eur J Oper Res 222(1):10–16
Hsu VN (2000) Dynamic economic lot size model with perishable inventory. Manag Sci 46(8):1159–1169
Liyanage LH, Shanthikumar JG (2005) A practical inventory control policy using operational statistics. Oper Res Lett 33(4):341–348
Pochet Y, Wolsey LA (2006) Production planning by mixed integer programming. Springer, New York
Scarft H (1960) The optimality of \((s, S)\) policies in the dynamic inventory problem. In: Arrow KJ, Karlin S, Suppes P (eds) Mathemtical methods in the social sciences, vol 1. Stanford University Press, p 196
Tempelmeier H (2013) Stochastic lot sizing problems. In: Handbook of stochastic models and analysis of manufacturing system operations. Springer, New York, pp 313–344
Tempelmeier H, Herpers S (2011) Dynamic uncapacitated lot sizing with random demand under a fillrate constraint. Eur J Oper Res 212(3):497–507
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This work is partially supported by the German Federal Ministry for Economic Affairs and Energy, BMWi, Grant 03ET7528B.
Appendices
Appendix A: Derivations in the proof of Theorem 1
For \(t>1\), define \(\check{d} \in D\) such that the following holds:
As a consequence, we deduce:
With this, we can then show:
Similarly, we deduce:
Lastly, we compute \(\eta \):
Appendix B: MILP formulation for LS2RO\(_{q,u}\)AR and \(\Gamma \)robustness
The MILP formulation with uncertainty intervals \([\bar{d}_t  \hat{d}_t, \bar{d}_t + \hat{d}_t]\) reads:
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Coniglio, S., Koster, A.M.C.A. & Spiekermann, N. Lot sizing with storage losses under demand uncertainty. J Comb Optim 36, 763–788 (2018). https://doi.org/10.1007/s1087801701478
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DOI: https://doi.org/10.1007/s1087801701478
Keywords
 Lot sizing
 Storage losses/deterioration
 Uncertain demand
 Twostage robust optimization
 Affine rules
 Stochastic programming