Mark E. Hartmann

ON INTEGER POINTS IN POLYHEDRA

We give an upper bound on the number of vertices ofPI, the integer hull of a polyhedronP, in terms of the dimensionn of the space, the numberm of inequalities required to describeP, and the size ϕ of these inequalities. For fixedn the bound isO(mnϕn−). We also describe an algorithm which determines the number of integer points in a polyhedron to within a multiplicative factor of 1+ε in time polynomial inm, ϕ and 1/ε when the dimensionn is fixed.

An improvement on paulson's procedure for selecting the poprlation with the largest mean from k normal populations with a common unknown variance

We obtain a tighter bound on the probability of making a correct selec tion in Paulsons procedure for selecting the normal population which has the largest population mean when the populations have a common unknown Tali ance. As a consequence, we are able to use a sharper value for the coristallt a∗ in Paulsons procedure. Simulation studies indicate that this leads to ari inlprovement in the expected number of stages to termination and expettetl total number of observations which is uniform in k, P∗s and σ/δ∗.

On the Chvátal rank of polytopes in the 0/1 cube

Abstract Given a polytope P⊆ R n , the Chvatal–Gomory procedure computes iteratively the integer hull PI of P. The Chvatal rank of P is the minimal number of iterations needed to obtain PI. It is always finite, but already the Chvatal rank of polytopes in R 2 can be arbitrarily large. In this paper, we study polytopes in the 0/1 cube, which are of particular interest in combinatorial optimization. We show that the Chvatal rank of any polytope P⊆[0,1]n is O (n 3 log n) and prove the linear upper and lower bound n for the case P∩ Z n =∅ .

A comparison of the performances of procedures for selecting the normal population having the largest mean when the populations have a common unknown variance

We study the performances of the two-stage non-eliminating procedure of Bechhofer, Dunnett and Sobel (1954), the two-stage eliminating procedure of Gupta and Kim (1984) and the multi-stage eliminating procedure of Paulson (1964) as modified by Hartmann (1990) for selecting the normal population which has the largest mean when the common variance is unknown. Constants to implement the procedures are provided. All of these procedures are open and guarantee the indifference-zone probability require-ment of Bechhofer (1954). The following performance characteristics are estimated by Monte Carlo sampling: the achieved probability of a correct selection, the expected number of vector-observations and the expected total number of observations to termination of experimentation. The critical role of the common initial sample size per population is discussed. Guidance is provided as to which procedure to use in different environments.

Facets of the p -cycle polytope

Abstract The purpose of this study is to provide a polyhedral analysis of the p -cycle polytope, which is the convex hull of the incidence vectors of all the p -cycles (simple directed cycles consisting of p arcs) of the complete directed graph K n . We first determine the dimension of the p -cycle, polytope, characterize the bases of its equality set, and prove two lifting results. We then describe several classes of valid inequalities for the case 2 p n , together with necessary and sufficient conditions for these inequalities to induce facets of the p -cycle polytope. We also briefly discuss the complexity of the associated separation problems. Finally, we investigate the relationship between the p -cycle polytope and related polytopes, including the p -circuit polytope. Since the undirected versions of symmetric inequalities which induce facets of the p -cycle polytope are facet-inducing for the p -circuit polytope, we obtain new classes of facet-inducing inequalities for the p -circuit polytope.

Outsourcing prioritized warranty repairs

Purpose – To consider the problem of outsourcing warranty repairs to outside vendors when items have priorities in service.Design/methodology/approach – The repair allocation problem is formulated as a convex minimum‐cost network flow problem and solve it by the successive shortest path algorithm. The computation issues involved with the problem are also discussed.Findings – Examples are provided to illustrate the cost benefits achieved due to the priority structure.Research limitations/implications – The research uses a closed static allocation model. Future research efforts can expand our model by considering a dynamic allocation method or an open population model.Practical implications – The paper can be a valuable resource to warranty managers who make decisions regarding the negotiations of warranty contracts and the allocation of items to outside repair vendors.Originality/value – A warranty manager can apply our results to receive insight on the value of giving priority in service to special custom...

Balancing problems in acyclic networks

Abstract A directed acyclic network with nonnegative integer arc lengths is called balanced if any two paths with common endpoints have equal lengths. In the buffer assignment problem such a network is given, and the goal is to balance it by increasing arc lengths by integer amounts (called buffers), so that the sum of the amounts added is minimal. This problem arises in VLSI design, and was recently shown to be polynomial for rooted networks. Here we give simple procedures which solve several generalizations of this problem in strongly polynomial time, using ideas from network flow theory. In particular, we solve a weighted version of the problem, extend the results to nonrooted networks, and allow upper bounds on buffers. We also give a strongly polynomial algorithm for solving the min-max buffer assignment problem, based on a strong proximity result between fractional and integer balanced solutions. Finally, we show that the problem of balancing a network while minimizing the number of arcs with positive buffers is NP-hard.

On the Chvátal Rank of Certain Inequalities

The Chvatal rank of an inequality ax ≤ b with integral components and valid for the integral hull of a polyhedron P, is the minimum number of rounds of Gomory-Chvatal cutting planes needed to obtain the given inequality. The Chvatal rank is at most one if b is the integral part of the optimum value z(a) of the linear program max{ax : x ∈ P}. We show that, contrary to what was stated or implied by other authors, the converse to the latter statement, namely, the Chvatal rank is at least two if b is less than the integral part of z(a), is not true in general. We establish simple conditions for which this implication is valid, and apply these conditions to several classes of facet-inducing inequalities for travelling salesman polytopes.

An analogue of Hoffman's circulation conditions for max-balanced flows

LetD=(V, A) be a directed graph. A real-valued vectorx defined on the arc setA is amax-balanced flow forD if for every cutW the maximum weight over arcs leavingW equals the maximum weight over arcs enteringW. For vectorsl⩽u defined onA, we describe an analogue of Hoffmans circulation conditions for the existence of a max-balanced flowx satisfyingl⩽x⩽u. We describe an algorithm for computing such a vector, but show that minimizing a linear function over the set of max-balanced flows satisfyingl⩽x⩽u is NP-hard. We show that the set of all max-balanced flows satisfyingl⩽x⩽u has a greatest element under the usual coordinate partial order, and we describe an algorithm for computing this element. This allows us to solve several related approximation problems. We also investigate the set of minimal elements under the coordinate partial order. We describe an algorithm for finding a minimal element and show that counting the number of minimal elements is #P-hard. Many of our algorithms exploit the relationship between max-balanced flows and bottleneck paths.

Integral bases and p -twisted digraphs

Abstract A well-known theorem in network flow theory states that for a strongly connected digraph D = ( V , A ) there exists a set of directed cycles the incidence vectors of which form a basis for the circulation space of D and integrally span the set of integral circulations; that is, every integral circulation can be written as an integral combination of these vectors. In this paper, we extend this result to general digraphs. Following a definition of Hershkowitz and Schneider, we call a digraph p -twisted if each pair of vertices is contained in a closed (undirected) walk with the property that as the walk is traversed there are no more than p changes in the orientations of the arcs. We show that for every p -twisted digraph there exists a set of p -twisted cycles the incidence vectors of which form a basis for the circulation space and integrally span the set of integral circulations. We show that such a set can be computed in O (|||) time.