On the capacitated vehicle routing problem

 We consider the Vehicle Routing Problem, in which a fixed fleet of delivery vehicles of uniform capacity must service known customer demands for a single commodity from a common depot at minimum transit cost. This difficult combinatorial problem contains both the Bin Packing Problem and the Traveling Salesman Problem (TSP) as special cases and conceptually lies at the intersection of these two well-studied problems. The capacity constraints of the integer programming formulation of this routing model provide the link between the underlying routing and packing structures. We describe a decomposition-based separation methodology for the capacity constraints that takes advantage of our ability to solve small instances of the TSP efficiently. Specifically, when standard procedures fail to separate a candidate point, we attempt to decompose it into a convex combination of TSP tours; if successful, the tours present in this decomposition are examined for violated capacity constraints; if not, the Farkas Theorem provides a hyperplane separating the point from the TSP polytope. We present some extensions of this basic concept and a general framework within which it can be applied to other combinatorial models. Computational results are given for an implementation within the parallel branch, cut, and price framework SYMPHONY. Received: October 30, 2000 / Accepted: December 19, 2001 Published online: September 5, 2002 Key words. vehicle routing problem – integer programming – decomposition algorithm – separation algorithm – branch and cut Mathematics Subject Classification (2000): 20E28, 20G40, 20C20     

Regressions and monotone chains II: The poset of integer intervals

A regressive function (also called a regression or contractive mapping) on a partial order P is a function mapping P to itself such that (x)x. A monotone k-chain for is a k-chain on which is order-preserving; i.e., a chain x ₁<...ksuch that (x ₁)...(x_k). Let P _nbe the poset of integer intervals {i, i+1, ..., m} contained in {1, 2, ..., n}, ordered by inclusion. Let f(k) be the least value of n such that every regression on P _nhas a monotone k+1-chain, let t(x,j) be defined by t(x, 0)=1 and t(x,j)=x ^t(x,j–1). Then f(k) exists for all k (originally proved by D. White), and t(2,k) < f(K) <t( + _k, k) , where _k 0 as k. Alternatively, the largest k such that every regression on P _nis guaranteed to have a monotone k-chain lies between lg^*(n) and lg^*(n)–2, inclusive, where lg^*(n) is the number of appliations of logarithm base 2 required to reduce n to a negative number. Analogous results hold for choice functions, which are regressions in which every element is mapped to a minimal element.     

The synthesis and the molecular structure of hexakis(carbonyl)hexafluoroantimonato(V)tungsten(II) undecafluorodiantimonate(V), [W(CO)6(FSbF5)][Sb2F11

The reaction of tungsten hexacarbonyl, W(CO)6, with antimony(V) fluoride, SbF5, in the conjugate Br?nsted-Lewis superacid HF-SbF5 at 40 degrees C produces quantitatively the salt [W(CO)6(FSbF5)][Sb2F11] as the main product. The observed 2e- oxidation without any loss of CO is unprecedented. The cation [W(CO)6(FSbF5)]+ is seven coordinated with a distorted C2v capped trigonal prismatic structure. [W(CO)6(FSbF5)][Sb2F11] crystallizes in the monoclinic space group P21 (No. 4). a = 8.2051(12) A, b = 16.511(3) A, c = 8.1432(2) A, beta = 111.5967(6) degrees, V = 1025.8(2) A3, Z = 2. Number of reflections measured = 9112, unique 4410. Residuals on F, I > 3 sigma (I): R (Rw) = 0.023 (0.023). In the [W(CO)6(FSbF5)]+ cation the FSbF5 group is very tightly coordinated to tungsten with the bridging fluorine nearly equidistant from W and Sb. The details of the molecular structure are compared to those to polymeric [[Mo(CO)4]2(cis-mu-F2SbF4)3]x[Sb2F11]x reported by us very recently.     

Enantio- and diastereodifferentiating cis,trans-photoisomerization of 2beta,3beta-diphenylcyclopropane-1alpha-carboxylic acid derivatives in organized media

[reaction: see text] Four methods of asymmetric induction in the cis, trans-photoisomerization of 2beta, 3beta-diphenylcyclopropane-1alpha-carboxylic acid derivatives were studied. Best results (ca. 80% de) were obtained by irradiation of chiral esters, amides, and salts in NaY and LiY zeolites and in the pure crystalline state.     

The chemistry of thujone. IX. Thujone as a chiral synthon for the synthesis of optically active steroid analogues. The vinylpicoline route

The thujone-derived enone 1 , upon base-catalyzed reaction with 2-methyl-6-vinylpyridine is converted to the pyridine analogue 5 (Scheme 1). Catalytic reduction of the latter to 6 generates two new centers of chirality which eventually become C(8) and C(14) in the ultimate synthetic steroid analogue 12 . An X-ray analysis of 6 establishes the structure and absolute configuration so as to determine its suitability in subsequent synthetic studies. The acetal derivative 7 , via Birch reduction, hydrolysis, and internal aldol cyclization, is converted into the cyclohexenone analogue 10 (Scheme 2). This ‘one-pot’ process affords an efficient conversion of the pyridine ring into a cyclohexenone system required for A-ring construction of the steroid skeleton. Finally, conversion of 10 , via the unsaturated diketone 11 , provides the chiral steroid analogue 12 .