Valid inequalities for the single-item capacitated lot sizing problem with step-wise costs

This paper presents a new class of valid inequalities for the single-item capacitated lotsizing problem with step-wise production costs (LS-SW). We first provide a survey of different optimization methods proposed to solve LS-SW. Then, flow cover and flow cover inequalities derived from the single node flow set are described in order to generate the new class of valid inequalities. The single node flow set can be seen as a generalization of some valid relaxations of LS-SW. A new class of valid inequalities we call mixed flow cover, is derived from the integer flow cover inequalities by a lifting procedure. The lifting coefficients are sequence independent when the batch sizes (V) and the production capacities (P) are constant and if V divides P. When the restriction of the divisibility is removed, the lifting coefficients are shown to be sequence independent. We identify some cases where the mixed flow cover inequalities are facet defining. A cutting plane algorithmis proposed for these three classes of valid inequalities. The exact separation algorithmproposed for the constant capacitated case runs in polynomial time. Finally, some computational results are given to compare the performance of the different optimization methods including the new class of valid inequalities.single-item capacitated lot sizing problem, flow cover inequalities, cutting plane algorithm

Exact methods for the single-item multi-plant capacitated lot sizing problem coupled with transportation

In this paper we study the integration of production, transportation and storage decisions in a multi plant-distribution center supply structure. Multiple plants produce one type of item, each of them with different production capacity and costs, and send finished goods to the distribution center (DC) using capacitated vehicles. Customer demand is known at DC level and has to be satisfied without backlogging. This problem contains classical capacitated lot sizing problem as a subpart, which, in the general case, is NP-hard. We propose an exact pseudo-polynominal dynamic programming algorithm and show that the problem is NP-hard in the ordinary sense. We then compare the computational time of this dynamic program to that of a mixed integer linear program (MILP) which is selected among 4 different MILP formulations based on its lower computational time.multi-plant production, transportation, single-item capacitated lot sizing problem, dynamic program

An inverse ellipsometric problem for thin film characterization: comparison of different optimization methods

International audienceIn this paper, an ill-posed inverse ellipsometric problem for thin film characterization is studied. The aim is to determine the thickness, the refractive index and the coefficient of extinction of homogeneous films deposited on a substrate without assuming any a priori knowledge of the dispersion law. Different methods are implemented for the benchmark. The first method considers the spectroscopic ellipsometer as an addition of single wavelength ellipsometers coupled only via the film thickness. The second is an improvement of the first one and uses Tikhonov regularization in order to smooth out the parameter curve. Cross-validation technique is used to determine the best regularization coefficient. The third method consists in a library searching. The aim is to choose the best combination of parameters inside a pre-computed library. In order to be more accurate, we also used multi-angle and multi-thickness measurements combined with the Tikhonov regularization method. This complementary approach is also part of the benchmark. The same polymer resist material is used as the thin film under test, with two different thicknesses and three angles of measurement. The paper discloses the results obtained with these different methods and provides elements for the choice of the most efficient strategy

Dynamic programming algorithms and MIP reformulations for production-transportation planning problems

International audienc

Exact methods for the single-item multi-plant capacitated lot sizing problem coupled with transportation

In this paper we study the integration of production, transportation and storage decisions in a multi plant-distribution center supply structure. Multiple plants produce one type of item, each of them with different production capacity and costs, and send finished goods to the distribution center (DC) using capacitated vehicles. Customer demand is known at DC level and has to be satisfied without backlogging. This problem contains classical capacitated lot sizing problem as a subpart, which, in the general case, is NP-hard. We propose an exact pseudo-polynominal dynamic programming algorithm and show that the problem is NP-hard in the ordinary sense. We then compare the computational time of this dynamic program to that of a mixed integer linear program (MILP) which is selected among 4 different MILP formulations based on its lower computational time

Valid inequalities for the single-item capacitated lot sizing problem with step-wise costs

This paper presents a new class of valid inequalities for the single-item capacitated lot sizing problem with step-wise production costs (LS-SW). We first provide a survey of different optimization methods proposed to solve LS-SW. Then, flow cover and flow cover inequalities derived from the single node flow set are described in order to generate the new class of valid inequalities. The single node flow set can be seen as a generalization of some valid relaxations of LS-SW. A new class of valid inequalities we call mixed flow cover, is derived from the integer flow cover inequalities by a lifting procedure. The lifting coefficients are sequence independent when the batch sizes (V) and the production capacities (P) are constant and if V divides P. When the restriction of the divisibility is removed, the lifting coefficients are shown to be sequence independent. We identify some cases where the mixed flow cover inequalities are facet defining. A cutting plane algorithm is proposed for these three classes of valid inequalities. The exact separation algorithm proposed for the constant capacitated case runs in polynomial time. Finally, some computational results are given to compare the performance of the different optimization methods including the new class of valid inequalities

Single-item lot sizing problem with carbon emissionunder the cap-and-trade policy

International audienc

Lot sizing problem with multi-mode replenishment and batch delivery

International audienc

Valid inequalities for the single-item capacitated lot sizing problem with step-wise costs.

International audienc