A perfect information lower bound for robust lot-sizing problems

Abstract

Robust multi-stage linear optimization is hard computationally and only small problems can be solved exactly. Hence, robust multi-stage linear problems are typically addressed heuristically through decision rules, which provide upper bounds for the optimal solution costs of the problems. We investigate in this paper lower bounds inspired by the perfect information relaxation used in stochastic programming. Specifically, we study the uncapacitated robust lot-sizing problem, showing that different versions of the problem become tractable whenever the non-anticipativity constraints are relaxed. Hence, we can solve the resulting problem efficiently, obtaining a lower bound for the optimal solution cost of the original problem. We compare numerically the solution time and the quality of the new lower bound with the dual affine decision rules that have been proposed by Kuhn et al. (Math Program 130:177–209, 2011).

Appendices

Reformulation of affine decision rules

Plugging (14)–(16) into $\mathcal {P}$ and enforcing that y does not depend on $\xi$, we obtain the formulation below.

Although the number of variables in the problem described above is polynomial, we still have to deal with an infinite number of constraints. As said already, one easily sees that we can restrict ourselves to the extreme points of $\varXi$; yet, this typically leads to an exponential number of constraints. An alternative and more compact approach applies classical tools from robust optimization to reformulate each robust constraint as a polynomial number of deterministic constraints plus a polynomial number of additional real variables, see Ben-Tal and Nemirovski (2000). We skip the details of that approach and provide below the resulting MILP.

We can obtain similarly the formulation for the problem without setup by removing the variable y from the above formulation.

Reformulation of dual affine decision rules

In this appendix, we sketch how to obtain a MILP reformulation for the lower bound of the robust lot-sizing following the method presented in Kuhn et al. (2011). The method requires to introduce an artificial probability measure with our uncertainty set $\varXi$. To keep the presentation simple, we will exemplify the framework from Kuhn et al. (2011) with the discrete probability measure $\mathbb {P} (\xi )$ defined over the set of extreme points of $\varXi$. Hence, $\mathbb {P}$ is any vector in $\mathbb {R}_+^{\vert {{\mathrm{ext}}}(\varXi )\vert }$ that satisfies

$$\begin{aligned} \sum _{\xi \in {{\mathrm{ext}}}(\varXi )}\mathbb {P} (\xi )=1. \end{aligned}$$

The first step of the approach of Kuhn et al. (2011) relaxes the robust constraints to expectation constraints as follows.

where $\psi$ and $\theta$ are additional slack variables. One readily verifies that, for any probability distribution $\mathbb {P} (\xi )$, the above problem provides a lower bound for the optimal solution of problem $\mathcal {PI}_{1}$.

Then, using advanced dualization techniques inspired by the dualization used in classical robust optimization, the authors of Kuhn et al. (2011) are able to reformulate the above lower bound as the following mixed-integer linear program, where matrix $\mathcal M$ is defined as $\mathcal {M} = \mathbb {E} (\xi \xi ^{T})$, $\mathcal W$ is the matrix defined as $\mathcal {W} = (W - qe^{T}_{1})\mathcal {M}$, and $e_{1}$ is the vector with all entries equals to one. Specifically, the following mixed-integer linear program is obtained by applying reformulation (4.6) from Kuhn et al. (2011) to $\mathcal {PI}_{1}$.

We can obtain similarly the formulation for the problem without setup by removing the vriable y from the above formulation.

Proof of Lemma 2

Let us detail the inner maximization of (26) as

By definition, $\beta _{kl}\ge 0$ for each $k\in K,l\in L(k)$, so that we can relax constraints (34) to $\xi \ge 0$ without affecting the optimal solution. Thanks to the strong duality in linear programming, the optimal solution cost of the above problem is equal to the optimal solution cost of its dual, given by

$$\begin{aligned}&\min \quad \varGamma \theta + \sum _{k\in K} \varphi _k \\&\text{ s.t. }\quad \theta + \varphi _k \ge \sum _{l\in L(k)} \beta _{kl} z_{l},\quad k\in K \\&\quad \qquad \theta ,y\ge 0. \end{aligned}$$

Substituting $\varphi _k$ by $\max (0,\sum _{l\in L(k)} \beta _{kl} z_{l}-\theta )$ for each $k\in K$, we can further reformulate (26) as

$$\begin{aligned} \min _{x\in \mathcal {Z},\theta \ge 0} \varGamma \theta + \sum _{l\in L}\alpha _l z_{l}+\sum _{k\in K} \max \left( 0,\sum _{l\in L(k)} \beta _{kl} z_{l}-\theta \right) . \end{aligned}$$

(35)

The crucial step of our proof (which differs from Theorem 3 from Bertsimas and Sim (2003)) is that, because the constraint $\sum _{l\in L(k)}z_l=1$ holds for each $k\in K$, we can further reformulate (35) as

$$\begin{aligned} \min _{x\in \mathcal {Z},\theta \ge 0} \varGamma \theta + \sum _{l\in L}\alpha _l z_{l}+\sum _{k\in K}\sum _{l\in L(k)}z_{l} \max (0,\beta _{kl}-\theta ). \end{aligned}$$

The rest of the proof is identical to the proof of Theorem 3 from Bertsimas and Sim (2003).