# P.C. Gilmore's research while affiliated with IBM Research and other places

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## Publications (8)

In earlier papers on the cutting stock problem we indicated the desirability of developing fast methods for computing knapsack functions. A one-dimensional knapsack function is defined by: \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmar...

In earlier papers [Opns. Res. 9, 849–859 (1961), and 11, 863–888 (1963)] the one-dimensional cutting stock problem was discussed as a linear programming problem. There it was shown how the difficulty of the enormous number of columns occurring in the linear programming formulation could be overcome by solving a knapsack problem at every pivot step....

We consider a machine with a single real variable x that describes its state. Jobs J1, ', JN are to be sequenced on the machine. Each job requires a starting state A, and leaves a final state Bi. This means that Ji can be started only when x = Ai and, at the completion of the job, x = Bi. There is a cost, which may represent time or money, etc., fo...

In this paper, the methods for stock cutting outlined in an earlier paper in this Journal [Opns Res 9, 849--859 1961] are extended and adapted to the specific full-scale paper trim problem. The paper describes a new and faster knapsack method, experiments, and formulation changes. The experiments include ones used to evaluate speed-up devices and t...

## Citations

... The authors in [13,14] introduced the first model for this problem, [15] extended the first mathematical model including upper bounds on the number of leftovers, and [16] suggested the L.P. model for the one-dimensional bin packing problem. The first attempt to model two-dimensional bin packing/cutting stock problems was also made by [17], who proposed a column generation approach. The authors in [18] summarized the state-of-the-art of bin packing problem until 2006, and presented a new approach to bin packing problem which was different from previous approaches. ...

... Research on the irregular strip and bin packing problems can be traced back to the pioneering Linear Programming (LP) models for rectangular bin packing introduced by Gilmore and Gomory [41], and the pioneering heuristic methods for irregular strip packing proposed by Art [11], Adamowicz and Albano [2,1], and Albano and Sapuppo [5] in the late nineteen sixties and seventies. These early works introduce many of the basic ideas subsequently exploited by all heuristics methods reported in the literature, such as the notion of a feasible non-overlapping region between pieces based on the No-Fit Polygon (NFP) representation, and the sequential placement of pieces based on a bottom-left state-of-the-art in terms of performance among the family of exact continuous mathematical models for irregular strip packing. ...

... In 2005, this heuristic was modified by Moura and Oliveira [9]. The stack building or so-called tower building heuristic is proposed by Gilmore and Gomory [12] in 1965 and was used by Yap et al. [11] by towered up the boxes to fill the container. Nevertheless, cuboid or blocks arrangement were used to fill the container with cuboid arrangement of similar boxes. ...

... Gilmore and Gomory [16] used linear programming methods to build mathematical models. The postponed column generation method, improved in 1963 [17] and 1965 [18], is also applicable to the problem of twodimensional plate cutting problems and even multidimensional problems. ...

... For an overview of results for the offline version of Dial-a-Ride on the line, see [12]. Without release times, Gilmore and Gomory [14] and Atallah and Kosaraju [2] gave a polynomial time algorithm for closed, non-preemptive Dial-a-Ride on the line with capacity c = 1. Guan [15] showed that the closed, non-preemptive problem is hard for c = 2, and Bjelde et al. [9] extended this result for any finite capacity c ≥ 2 in both the open and the closed variant. ...

... Gilmore and Gomory [16] used linear programming methods to build mathematical models. The postponed column generation method, improved in 1963 [17] and 1965 [18], is also applicable to the problem of twodimensional plate cutting problems and even multidimensional problems. ...

... Problem ranca [4] predstavlja algoritam koji pomaže u rešavanju problema kombinatorike. Od svih varijacija ovog algoritma u ovom radu je, nakon niza testova, kao najpogodnija odabrana je primena 0-1 algoritma ranca i to koristeći pristup koji se oslanja na dinamičko programiranje. ...

... The enumeration of all tours for a given problem is not a possibly appropriate method. Corresponding to an n-city TSP, the possible permutations is equal to (n − 1)! Routes [18]. In addition, the exact methods are requested to constitute a search space of all possible permutations of cities, then to find the corresponding optimal solutions. ...