Combining Polyhedral Approaches and Stochastic Dual Dynamic Integer Programming for Solving the Uncapacitated Lot-Sizing Problem Under Uncertainty

We study the uncapacitated lot-sizing problem with uncertain demand and costs. The problem is modeled as a multistage stochastic mixed-integer linear program in which the evolution of the uncertain parameters is represented by a scenario tree. To solve this problem, we propose a new extension of the stochastic dual dynamic integer programming algorithm (SDDiP). This extension aims at being more computationally efficient in the management of the expected cost-to-go functions involved in the model, in particular by reducing their number and by exploiting the current knowledge on the polyhedral structure of the stochastic uncapacitated lot-sizing problem. The algorithm is based on a partial decomposition of the problem into a set of stochastic subproblems, each one involving a subset of nodes forming a subtree of the initial scenario tree. We then introduce a cutting plane–generation procedure that iteratively strengthens the linear relaxation of these subproblems and enables the generation of an additional strengthened Benders’ cut, which improves the convergence of the method. We carry out extensive computational experiments on randomly generated large-size instances. Our numerical results show that the proposed algorithm significantly outperforms the SDDiP algorithm at providing good-quality solutions within the computation time limit. Summary of Contribution: This paper investigates a combinatorial optimization problem called the uncapacitated lot-sizing problem. This problem has been widely studied in the operations research literature as it appears as a core subproblem in many industrial production planning problems. We consider a stochastic extension in which the input parameters are subject to uncertainty and model the resulting stochastic optimization problem as a multistage stochastic integer program. To solve this stochastic problem, we propose a novel extension of the recently published stochastic dual dynamic integer programming (SDDiP) algorithm. The proposed extension relies on two main ideas: the use of a partial decomposition of the scenario tree and the exploitation of existing knowledge on the polyhedral structure of the stochastic uncapacitated lot-sizing problem. We provide the results of extensive computational experiments carried out on large-size randomly generated instances. These results show that the proposed extended algorithm significantly outperforms the SDDiP at providing good-quality solutions for the stochastic uncapacitated lot-sizing problem. Although the paper focuses on a basic lot-sizing problem, the proposed algorithmic framework may be useful to solve more complex practical production planning problems.

Magnetic and Electrical Behaviors of the Homo- and Heterometallic 1D and 3D Coordination Polymers Based on the Partial Decomposition of the [Cr(C2O4)3]3− Building Block

One-dimensional (1D) oxalate-bridged homometallic {[Mn(bpy)(C2O4)]·1.5H2O}n (1) (bpy = 2,2’-bipyridine) and heterodimetallic {[CrCu3(bpy)3(CH3OH)(H2O)(C2O4)4][Cu(bpy)Cr(C2O4)3]·CH2Cl2·CH3OH·H2O}n (2) coordination polymers, as well as the three-dimensional (3D) heterotrimetallic {[CaCr2Cu2(phen)4(C2O4)6]·4CH3CN·2H2O}n (3) (1,10-phenanthroline) network, have been synthesized by a building block approach using a layering technique, and characterized by single-crystal X-ray diffraction, infrared (IR) and impedance spectroscopies and magnetization measurements. During the crystallization process partial decomposition of the tris(oxalato)chromate(III) happened and 1D polymers 1 and 2 were formed. The antiferromagnetic interactions between the manganese(II) ions were mediated by oxalate ligands in the chain [Mn(bpy)(C2O4)]n of 1, with intra-chain super-exchange interaction ? = (−3.134 ± 0.004) K; magnetic interaction between neighbouring chains is negligible making this system closer than other known Mn-chains to the ideal 1D Heisenberg antiferromagnet. Compound 2 comprises a 1D coordination anion [Cu(bpy)Cr(C2O4)3]nn− (Cr2–Cu4) with alternating [Cr(C2O4)3]3− and [Cu(bpy)]2+ units mutually bridged through the oxalate group. Another chain (Cr1–Cu3) is similar, but involves a homodinuclear unit [Cu(bpy)(H2O)(µ-C2O4)Cu(bpy)(CH3OH)]2+ (Cu1–Cu2) coordinated as a pendant group to a terminal oxalate oxygen. Magnetic measurements showed that the Cu1−Cu2 cationic unit is a strongly coupled antiferromagnetic dimer, independent from the other magnetic ions within ferromagnetic chains Cr1–Cu3 and Cr2–Cu4. A 3D polymer {[CaCr2Cu2(phen)4(C2O4)6]·4CH3CN·2H2O}n (3) comprising three different metal centers (Ca2+, Cr3+ and Cu2+) oxalate-bridged, contains Ca2+ atoms as nodes connected with four Cr3+ atoms through oxalate ligands. The network thus formed can be reduced to an underlying graph of diamondoid (dia) or (66) topology. Magnetization of 3 shows the ferromagnetic oxalate-bridged dimers [CuIICrIII], whose mutual interaction could possibly originate through the spin polarization of Ca2+ orbitals. Compounds 1 and 3 exhibit lower electrical conductivity at room temperature (RT) in comparison to compound 2.

Owning, Using, and Renting: Some Simple Economics of the “Sharing Economy”

New Internet-based “sharing-economy” markets enable consumer-owners to rent out their durable goods to nonowners. We model such markets and explore their equilibria both in the short run, in which ownership decisions are fixed, and in the long run, in which ownership decisions can be changed. We find that sharing-economy markets always expand consumption and increase surplus, but may increase or decrease ownership. Regardless, ownership is decoupled from individual preferences in the long run, as the rental rates and the purchase prices of goods become equal. If there are costs of bringing unused capacity to the market, they are partially passed through, creating a bias toward ownership. To test our theoretical work empirically, we conduct a survey of consumers, finding broad support for our modeling assumptions. The survey also allows us to offer a partial decomposition of the bring-to-market costs, based on attributes that make a good more or less amenable to being shared. This paper was accepted by Joshua Gans, business strategy.

In situ formation and solid-state oxidation of a triselenane NSeN-pincer MOF

Controlled partial decomposition of 2-selenonicotinic acid in the presence of Co2+ or Ni2+ resulted in the in situ formation of an unusual MOF based on triselenane ligands (RSeSeSeR) coordinated to M2+ centers as NSeN-pincers.